Particle Physics and Dark Energy: Beyond Classical Dynamics

Physik-Department Max-Planck-Institutfuumlr Kernphysik

Particle Physics andDark Energy

Beyond Classical Dynamics

Doctoral Thesis in Physics

submitted by

Mathias Garny

2008

Technische Universitaumlt Muumlnchen

TECHNISCHE UNIVERSITAumlT MUumlNCHEN

Max-Planck-Institut fuumlr Kernphysik



Mathias Garny

Vollstaumlndiger Abdruck der von der Fakultaumlt fuumlr Physik der Technischen Universitaumlt Muumlnchen zurErlangung des akademischen Grades eines

Doktors der Naturwissenschaften (Dr rer nat)

genehmigten Dissertation

Vorsitzender Univ-Prof Dr Lothar Oberauer

Pruumlfer der Dissertation1 Prof Dr Manfred Lindner

Ruprecht-Karls-Universitaumlt Heidelberg

2 Univ-Prof Dr Alejandro Ibarra

Die Dissertation wurde am 24092008 bei der Technischen Universitaumlt Muumlncheneingereicht und durch die Fakultaumlt fuumlr Physik am 24102008 angenommen

Particle Physics and Dark Energy Beyond Classical Dynamics

Abstract

In this work quantum corrections to classical equations of motion are investigated for dynamicalmodels of dark energy featuring a time-evolving quintessence scalar field Employing effective quan-tum field theory the robustness of tracker quintessence potentials against quantum corrections aswell as their impact on cosmological observables are discussed Furthermore it is demonstrated thata rolling quintessence field can also play an important role for baryogenesis in the early universe Themacroscopic time-evolution of scalar quantum fields can be described from first principles withinnonequilibrium quantum field theory based on Kadanoff-Baym equations derived from the 2PI ef-fective action A framework for the nonperturbative renormalization of Kadanoff-Baym equations isprovided Renormalized Kadanoff-Baym equations are proposed and their finiteness is shown for aspecial case

Zusammenfassung

In dieser Arbeit werden Quantenkorrekturen klassischer Bewegungsgleichungen in dynamischen Mo-dellen der Dunklen Energie untersucht welche ein zeitabhaumlngiges Quintessenz-Skalarfeld beinhaltenIm Rahmen effektiver Quantenfeldtheorie wird die Stabilitaumlt von Quintessenz-Potentialen bezuumlglichQuantenkorrekturen sowie deren Einfluszlig auf kosmologische Parameter diskutiert Daruumlber hinauswird gezeigt daszlig ein zeitabhaumlngiges Quintessenzfeld auch fuumlr die Baryogenese im fruumlhen Univer-sum eine wichtige Rolle spielen kann Die makroskopische Zeitentwicklung von skalaren Quanten-feldern kann basierend auf Grundprinzipien der Nichtgleichgewichtsquantenfeldtheorie mittels Ka-danoff-Baym Gleichungen beschrieben werden Es wird ein Formalismus fuumlr die nichtperturbativeRenormierung von Kadanoff-Baym Gleichungen entwickelt renormierte Kadanoff-Baym Gleichun-gen vorgeschlagen und deren Endlichkeit fuumlr einen Spezialfall nachgewiesen

Contents

1 Introduction 1

2 Dynamical Dark Energy 521 Quintessence Cosmology 622 Tracking Solutions 923 Interacting Quintessence 12

3 Quantum Effective Action 1531 1PI Effective Action 1632 2PI Effective Action 1933 nPI Effective Action 21

4 Quantum Corrections in Quintessence Models 2341 Self-Interactions 2442 Matter Couplings 4443 Gravitational Coupling 5144 Summary 60

5 Leptonic Dark Energy and Baryogenesis 6151 Quintessence and Baryogenesis 6152 Creation of a BminusL-Asymmetry 6253 Stability 66

6 Quantum Nonequilibrium Dynamics and 2PI Renormalization 6761 Kadanoff-Baym Equations from the 2PI Effective Action 6862 Nonperturbative 2PI Renormalization at finite Temperature 73

7 Renormalization Techniques for Schwinger-Keldysh Correlation Functions 7971 Non-Gaussian Initial States 7972 Nonperturbative Thermal Initial Correlations 8673 Renormalized Kadanoff-Baym Equation for the Thermal Initial State 103

8 Renormalization of Kadanoff-Baym Equations 10581 Kadanoff-Baym Equations and 2PI Counterterms 10582 Renormalizable Kadanoff-Baym Equations from the 4PI Effective Action 10683 Impact of 2PI Renormalization on Solutions of Kadanoff-Baym Equations 11284 Summary 129

9 Conclusions 131

viii CONTENTS

A Conventions 135

B Effective Action Techniques 137B1 Low-Energy Effective Action 137B2 Effective Action in Curved Background 138B3 Renormalization Group Equations 141

C Resummation Techniques and Perturbation Theory 145C1 Relation between 2PI and 1PI 145C2 Resummed Perturbation Theory 146

D Quantum Fields in and out of Equilibrium 151D1 Thermal Quantum Field Theory 151D2 Nonequilibrium Quantum Field Theory 157

E Nonperturbative Renormalization Techniques 163E1 Renormalization of the 2PI Effective Action 163E2 Renormalization of 2PI Kernels 165E3 Two Loop Approximation 167E4 Three Loop Approximation 168

F Integrals on the Closed Real-Time Path 171

Acknowledgements 173

Bibliography 175

Chapter 1

Introduction

According to the standard model of cosmology the evolution of our universe experienced a rapidlyinflating and highly correlated phase at its beginning This phase ended in an explosive entropyproduction (reheating) during which all kinds of sufficiently light particles were produced and ther-malized most of them highly relativistic Reheating was followed by a controlled expansion duringwhich the temperature decreased and more and more massive species became non-relativistic (radi-ation domination) Subsequently pressure-less baryonic and cold dark matter became the dominantcontribution to the total energy density and underwent gravitational clustering (matter domination)However in recent cosmic history the expansion of the universe started to accelerate This may beattributed to the so-called dark energy which became more and more important and makes up overtwo third of the energy density of the universe todayAll that is known about dark energy is based on its gravitational interaction While the total energydensity can be measured by observations of the anisotropy of the cosmic microwave background(CMB) the forms of energy which cluster gravitationally can be inferred from large-scale structuresurveys together with appropriate models of structure formation However the clustered energy ismuch less than the total energy density such that an additional homogeneously distributed com-ponent is required On top of that such a dark energy component can precisely account for theaccelerated expansion observed by measurements of the luminosity of distant supernovae [133] Thisconcordance of different observations makes the need for dark energy convincing and the questionabout its nature one of the most outstanding questions in astro-particle physicsThe inclusion of a cosmological constant in Einsteinrsquos equations of General Relativity provides aparameterization of dark energy which is compatible with cosmological observations [89] The cos-mological constant can be viewed as a covariantly conserved contribution to the energy-momentumtensor which is invariant under general coordinate transformations For any quantum field theory forwhich coordinate invariance is unbroken this is precisely the property of the vacuum expectationvalue of the energy-momentum tensor Therefore the cosmological constant may be interpreted asthe vacuum energy within quantum field theory [188] However since quantum field theory togetherwith classical gravity determines the vacuum energy only up to a constant it is impossible to predictthe value of the cosmological constant Furthermore the naiumlve summation of zero-point energies ofall momentum modes of a free quantum field leads to a divergent result Once a cutoff between theTeV and the Planck scale is imposed this amounts to a value which is between 60 and 120 ordersof magnitude too large This fact is known as the cosmological constant problem [178] If the valueinferred from cosmological observations is taken at face value an enormous hierarchy between thevacuum energy density and the energy density of radiation and matter must have existed in the earlyuniverse (smallness problem) Subsequently radiation and matter get diluted due to the cosmic ex-

2 1 Introduction

pansion and the cosmological constant becomes of comparable order of magnitude precisely in thepresent cosmological epoch (coincidence problem)These unsatisfactory features of the cosmological constant have motivated an extensive search foralternative explanations of dark energy Apart from attempts to explain cosmic acceleration by modi-fications of the equations of General Relativity [74 151] models of dynamical dark energy [65 162]explore the possibility that the dark energy density might evolve with time and become diluted duringcosmic expansion similar to the radiation and matter components In this way its smallness todaycould be attributed to a dynamical mechanism and the huge age of the universeSimilar dynamical mechanisms are well-known in cosmology For example cosmic inflation pro-vides a dynamical mechanism leading to a spatially flat universe in which the total energy densityis naturally very close to the critical energy density [108] as observed by CMB measurements [89]Another example is provided by baryogenesis Here the observed baryon density (as well as theabsence of antibaryons) is attributed to a dynamically produced asymmetry If the three Sakharovconditions [163] are fulfilled in the early universe namely violation of baryon number conservationviolation of charge-conjugation and its combination with parity and departure from thermal equilib-rium a baryon asymmetry can develop For specific realizations the final observable value of theasymmetry is even insensitive to a primordial asymmetry [48 71] Both examples show that a dy-namical mechanism can help to explain a measurable quantity which would otherwise have requiredan enormous amount of fine-tuning of the ldquoinitialrdquo state after the Big BangDynamical models for dark energy typically require the introduction of new degrees of freedom Forexample cosmic acceleration could be powered by a slowly rolling scalar field [157 182] calledquintessence field similar to the inflaton field in the early universe A special class of quintessencemodels featuring so-called tracking solutions [169] exhibits a dynamical self-adjusting mechanism ofthe dark energy density This means that the evolution of the dark energy density today is insensitive tothe amount of primordial dark energy in the early universe Therefore the energy densities of matterand dark energy can be comparable not only in the present epoch but also in the early universe Forspecific models both energy densities are even of comparable magnitude during the entire history ofthe universe [85157] These features represent advantages of tracker quintessence models comparedto the cosmological constantHowever quintessence models cannot address the fundamental cosmological constant problem ofquantum field theory Additionally introducing scalar fields brings up even more theoretical ques-tions on the quantum level Above all this includes the hierarchy problem It states that a scalarfield is unprotected against large quantum corrections to its mass originating in quadratically diver-gent loop corrections (where ldquolargerdquo refers to an ultraviolet embedding scale) Nevertheless particlephysicists and cosmologists commonly resort to scalar fields The most prominent examples are theHiggs field in the Standard Model and the inflaton field in cosmology However up to now no directexperimental evidence for the existence of an elementary scalar field exists

In the context of quintessence models it is therefore an urgent question what role quantum correctionsplay for the dynamics of the quintessence scalar field In particular the quintessence field is charac-terized by two striking properties which deserve special attention These are (i) the quintessencetracker potential and (ii) the macroscopic time-evolution of the field value over cosmic time-scales

Quintessence tracker potentials have a form which is not well-known within particle physics in-volving exponentials and inverse powers of the field Therefore it is important to investigate therobustness of such exceptional potentials with respect to quantum correctionsTypically tracker quintessence fields feature non-renormalizable self-interactions suppressed by in-verse powers of the Planck scale This indicates that tracker potentials may result from integrating

1 Introduction 3

out some unknown degrees of freedom at the Planck scale Below this scale effective quantum fieldtheory can be employed The ignorance about the superior theory is encapsulated into a few effectiveparameters (like the potential energy at a certain field value eg today) and the ultraviolet embeddingscaleIn order to assess the self-consistency of quintessence tracker models it is crucial to investigate theirrobustness with respect to quantum corrections originating from self-interactions In particular itis necessary to investigate whether the asymptotic flatness of the potential is stable under radiativecorrectionsPhenomenological signatures which could reveal the existence of a rolling quintessence field in-clude time-varying fundamental lsquoconstantsrsquo as well as apparent violations of the equivalence prin-ciple [157] Both effects result from couplings between quintessence and Standard Model particlesHowever once quantum corrections are taken into account such couplings destroy the desired prop-erties of the quintessence field if they are too large Therefore it is important to investigate theirquantum backreaction and to obtain quantitative upper boundsAdditionally it is necessary to check whether radiatively induced non-minimal gravitational cou-plings are in conflict with experimental tests of General Relativity For example non-minimal cou-plings of the quintessence field can lead to a time-variation of the effective Newton constant overcosmological time-scales [181]

The second characteristic property of the quintessence field mentioned above is its macroscopic time-evolution over cosmological time-scales Therefore the question arises how to calculate radiativecorrections for a time-evolving scalar field If the kinetic energy of the field is much smaller than thepotential energy and if its environment can be approximated by a vacuum or a thermal background itis possible to use a derivative expansion of the effective action in vacuum or at finite temperature re-spectively At leading order this amounts to replacing the classical potential by the effective potentialin the equations of motionQuantum corrections within quintessence models as described in this work employ the derivativeexpansion of the effective action The latter is applicable since the quintessence field is slowly rollingtoday However this might not have been the case in the early universe Therefore it is necessaryto develop methods that can describe the quantum dynamics of scalar fields beyond the limitations ofthe derivative expansion This falls into the realm of nonequilibrium quantum field theoryNote that similar questions arise for other nonequilibrium phenomena within astro-particle and high-energy physics like inflation and reheating as well as baryogenesis or heavy ion collisions Tradi-tionally these processes are modeled by semi-classical approximations These include Boltzmannequations hydrodynamic transport equations or effective equations of motion for a coherent scalarfield expectation value for example based on mean-field approximations [18 63 130]Since it is of great importance to assess the reliablity of these approximations a comparison witha completely quantum field theoretical treatment is desirable In recent years it has been demon-strated that scalar (and fermionic) quantum fields far from equilibrium can be described based on firstprinciples by Kadanoff-Baym equations [1 2 25 32 142] These are evolution equations for the fullone- and two-point correlation functions obtained from the stationarity conditions of the 2PI effectiveaction [66] The advantages of this treatment are twofold First its conceptual simplicity is veryattractive The only assumption entering the derivation of Kadanoff-Baym equations is the truncationof the so-called 2PI functional which amounts to a controlled approximation in the coupling constantor the inverse number of field degrees of freedom for specific quantum field theories [25] Other-wise no further assumptions are required In particular no assumptions that would only hold close tothermal equilibrium or in the classical limit are required Furthermore for any time-reversal invari-ant quantum field theory the Kadanoff-Baym equations are also time-reversal invariant in contrast

4 1 Introduction

to Boltzmann equations Second Kadanoff-Baym equations inherently incorporate typical quantum(eg off-shell) effects as well as ldquoclassicalrdquo (eg on-shell) effects in a unified manner Thereforethey are very versatile and can be employed both to assess the validity of conventional semi-classicalapproximations (eg for baryogenesis and leptogenesis) and in situations where a single effectivedescription does not exist (eg for (p)reheating by inflaton decay and subsequent thermalization)In addition Kadanoff-Baym equations can describe the quantum dynamics of a time-evolving scalarfield beyond the lsquoslow-rollrsquo approximation (eg for inflation and quintessence)It has been shown that numerical solutions of Kadanoff-Baym equations not only provide a descrip-tion of the quantum thermalization process of relativistic quantum fields for closed systems [30 3233] but also feature a separation of time-scales between kinetic and chemical equilibration (prether-malization) [31] Furthermore they have been compared to semi-classical transport equations forbosonic and fermionic systems [1 123 142 143] Moreover Kadanoff-Baym equations can describethe decay of a coherent oscillating scalar field expectation value under conditions where parametricresonance occurs [33] and have also been investigated in curved space-time [115 170]These successes of nonequilibrium quantum field theory make it worthwhile and in view of realisticapplications necessary to answer remaining conceptual questions like renormalization There areseveral reasons why a proper renormalization of Kadanoff-Baym equations is essential First it isrequired for a quantitative comparison with semi-classical Boltzmann equations which are finiteby construction Second renormalization has an important quantitative impact on the solutions ofKadanoff-Baym equations and therefore affects thermalization time-scales Third it is crucial foridentifying physical initial states meaning all nonequilibrium initial states that can occur as realstates of the physical ensemble The fact that this class excludes for example an initial state featuringbare particle excitations shows that this is of significance Finally a proper renormalization leadsto a stabilization of the computational algorithm used for the numerical solution of Kadanoff-Baymequations such that its range of applicability is extended and its robustness is improved

In chapter 2 quintessence models with tracking solutions are briefly reviewed and in chapter 3 anoverview over perturbative as well as nonperturbative calculation techniques of the quantum effectiveaction is given In chapter 4 the robustness of tracker quintessence models with respect to quantumcorrections is studied Quantum corrections induced by the self-interactions of the quintessence fieldby couplings to Standard Model particles and by the gravitational interaction are investigated andconsequences for cosmology as well as for observational signatures of a rolling quintessence field arediscussed Next in chapter 5 it is demonstrated that the quintessence field can also play an importantrole in the early universe This is done by presenting a model where baryogenesis and late-timecosmic acceleration are both driven by a time-evolving complex quintessence fieldThe derivation of Kadanoff-Baym equations starting from the 2PI effective action is briefly reviewedin chapter 6 as well as the nonperturbative renormalization procedure of the 2PI effective action inthermal equilibrium which has recently been formulated [28 29 37 173ndash175]The remaining part of this thesis is dedicated to the renormalization of Kadanoff-Baym equationsThis requires two steps First in chapter 7 the nonperturbative renormalization procedure for the 2PIeffective action in vacuum and in thermal equilibrium is adapted to the closed Schwinger-Keldyshreal-time contour which is the starting point for nonequilibrium quantum field theory Second inchapter 8 extended Kadanoff-Baym equations that can be used to describe systems featuring non-Gaussian initial correlations are derived from the 4PI effective action An ansatz for renormal-ized Kadanoff-Baym equations within λΦ4-theory is proposed and verified analytically for a specialcase Furthermore properties expected from solutions of renormalized Kadanoff-Baym equationsare checked and the importance of renormalization for nonequilibrium quantum dynamics is demon-strated

Chapter 2

Dynamical Dark Energy

In the following the main theoretical motivations for dynamical dark energy models are reviewedand it is briefly discussed in how far dynamical dark energy and specifically quintessence models withtracking solutions can address the problems connected to the cosmological constant Furthermorepossible observational signatures of a quintessence field are reviewed For a detailed discussion ofthe observational evidence for accelerated expansion and dark energy it is referred to Refs [89 100133 160]In order to be able to distinguish clearly between the different cosmological questions it is useful tomake a detailed definition

QFT smallness problem Why is there no huge cosmological constant contributing a vacuum en-ergy density of order M4

pl M4GUT M4

SUSY or M4elweak

Cosmological smallness problem How can one explain a small nonzero cosmological constant ordark energy density

Coincidence of scales The present dark energy and matter densities are1

ρde asymp 13 middot10minus123 M4pl and ρM asymp 05 middot10minus123 M4

pl

Coincidence of epochs In our present cosmological epoch the expansion of the universe changesfrom decelerated to accelerated [160]

The last two items are observational statements The question is whether there is a natural explanationfor these coincidences or whether they are just an ldquoaccidentrdquoIt appears likely that these questions cannot be answered by a single approach On the one handa mechanism (or a symmetry) is needed that explains why the huge field theoretical contributionsincluding contributions from potential shifts do not exist at all or at least why they do not act as asource of gravity On the other hand the observed acceleration of the universe has to be explainedThe cosmological standard model with a cosmological constant and a cold dark matter component(ΛCDM) is in accordance with all present observations inside the errorbars [89] However it doesnot answer any of the four cosmological questions above The value of the cosmological constant hasto be fine-tuned to fulfill the two ldquocoincidencesrdquo At the Planck epoch there is a hierarchy of order10minus123 between the energy density of the cosmological constant and the relativistic matter content inthis model

1 The values are based on the ldquoconcordance modelrdquo ΩDE = 07 ΩM = 03 and use H0 = 70kmsMpc

6 2 Dynamical Dark Energy

Figure 21 Schematic illustration of the evolution of the radiation matter and dark energy densitiesfor the cosmological constant (left) and a tracking quintessence model (right)

Starting point for dynamical dark energy models is the ldquocosmological smallness problemrdquo The aim isto explain the smallness of dark energy by the huge age of the universe Therefore a ldquotime-dependentcosmological constantrdquo can be introduced that decays (similar to matter or radiation density) duringcosmic evolution thus providing a natural explanation for its smallness today (see figure 21) At thePlanck scale the dark energy content of the universe does not have to be fine-tuned to an extraordi-narily small numberGeneral covariance of the equations of motion dictates that the dark energy cannot only depend ontime but is given by a space-time dependent field2which has to be added to the Lagrangian of thetheory as a new dynamical degree of freedom This opens up a whole field of possibilities mani-festing themselves in a huge variety of scalar-field-based models like Chaplygin Gas (a cosmic fluidderived from a Born-Infeld Lagrangian with equation of state p sim minus1ρ) phantom energy (derivedfrom a scalar-field Lagrangian with kinetic term with a ldquowrong signrdquo and with pressure p lt minusρ) ork-essence (with nonlinear kinetic term) and of course most straightforward and probably most elab-orated quintessence with a standard kinetic term and a self-interaction described by the quintessencepotential to name only a few (see [65 162] for reviews [16])The details of the decaying field are important when addressing the ldquocoincidence of scalesrdquo Gener-ally it will therefore depend on the specific model in how far a natural explanation for this remarkablecoincidence is found Quintessence provides a special class of so-called tracking solutions that ac-counts for this coincidence which will be discussed in the followingThe ldquocoincidence of epochsrdquo is not generically addressed by dynamic dark energy models In somemodels the two coincidences are linked (like for a cosmological constant) while in other models theyhave to be discussed separately

21 Quintessence Cosmology

The framework of cosmology is the general theory of relativity and cosmological models with dy-namical dark energy can be formulated within this setting However one should keep in mind thatcosmology is based on some fundamental assumptions like isotropy and large-scale homogeneity ofall components of our universe Their validity is assumed in the following Scalar-fields in cosmol-ogy are actually not unusual Already shortly after the big bang the universe may have undergone

2Just replacing the cosmological constant by a function Λ(t) is not possible because the Einstein equations can only besolved for covariant conserved energy-momentum tensors Tmicroν ρ = 0 However (Λgmicroν )ρ = 0 only if Λequiv const

21 Quintessence Cosmology 7

an accelerated phase the cosmic inflation which is often described by a slowly rolling scalar-fieldcalled inflaton [108139140] In this section the quintessence scalar-field will be introduced into thegeneral theory of relativity in close analogy to the inflaton scalar-field3 Starting point is the gravi-tational action with a standard kinetic term and a potential for the quintessence scalar-field φ givenby [157 182]

S =int

d4xradicminusg(x)

(minus R

16πG+

12

gmicroνpartmicroφpartνφ minusV (φ)+LB

) (21)

where G is Newtonrsquos constant and LB is the Lagrangian describing all other forms of energy like darkmatter baryonic matter radiation and neutrinos which will be called ldquobackgroundrdquo Furthermoreg(x) is the determinant of the metric gmicroν(x) and R is the curvature scalar as defined in appendix A Thecoupling of the quintessence field to gravity is called minimal in this case since there are no explicitcoupling terms like φ 2R It is only mediated through the integration measure and the contractionof the space-time derivatives in the kinetic term dictated by general coordinate invariance Possibleconstant contributions in the action (ie the cosmological constant) are assumed to be absorbed intothe potential V (φ) Variation of the action with respect to the metric yields the Einstein equations

Rmicroν minusR2

gmicroν = 8πG(T Bmicroν +T Q

microν) (22)

with the Ricci-tensor Rmicroν the energy-momentum tensor for the background T Bmicroν = 2radic

minusgδ (radicminusgLB)

δgmicroν and

T Qmicroν = partmicroφpartνφ minusgmicroν

(12(partφ)2minusV

)equiv (ρφ + pφ )umicrouν minusgmicroν pφ (23)

The energy-momentum tensor can be expressed in analogy to a perfect fluid with unit 4-velocityvector umicro = partmicroφ

radic(partφ)2 and energy density and pressure given by

ρφ =12(partφ)2 +V (φ) and pφ =

12(partφ)2minusV(φ) (24)

Variation of the action with respect to φ leads to the equation of motion for the quintessence field4

2φ +dV (φ)

dφ= 0 (25)

with the covariant DrsquoAlembertian for a scalar-field

2 = DmicroDmicro =1radicminusg

partmicro

radicminusgpart

micro

Under the assumptions of isotropy homogeneity and a spatially flat universe the Robertson-Walker-Metric for comoving coordinates xmicro = (tx) with a dimensionless scalefactor a(t) can be used

ds2 = gmicroνdxmicrodxν = dt2minusa(t)2dx2

After specializing the energy-momentum tensors to contain only space-independent densities ρB(t)and ρφ (t) and pressures5 pB(t) and pφ (t) the Einstein equations reduce to the Friedmann equations

3M2plH

2 = ρφ +ρB (26)

3M2pl

aa

= minus12(ρφ +3pφ +ρB +3pB)

3It is also possible to construct models where the quintessence and the inflaton fields are identical [154]4 If the background Lagrangian LB contains φ (eg quintessence-dependent couplings) the right hand side of the

equation of motion has to be replaced by δLBδφ For the basic discussion of quintessence it will be assumed that thisterm has a negligible influence on the dynamics of the φ field

5The energy momentum tensors for the background and the φ field are assumed to be of the form of an ideal fluidT i

microν = (ρi + pi)umicro uν minusgmicroν pi with umicro = (10)


Figure 22 Schematic illustration of the equation of motion of the quintessence field

with the Hubble parameter H = aa and the Planck-Mass Mpl = 1radic

8πG The critical density isdefined as ρc equiv 3M2

plH2 The first Friedmann equation is often written in terms of Ωi equiv ρiρc

1 = Ωφ +ΩB

In the case of a spatially homogeneous scalar field φ(t) the covariant DrsquoAlembertian is

2 = aminus3partt a3

partt = part2t +3Hpartt

yielding an equation of motion from (25) for the homogeneous quintessence field

φ +3Hφ +dV (φ)

dφ= 0 (27)

Illustratively the derivative of the potential acts like a force which accelerates the scalar field valuetowards smaller potential energies thereby being ldquodampedrdquo by the 3Hφ -term However the dampingdepends on the contents of the universe including quintessence itself which means there is a back-reaction (see figure 22) The latter is responsible for the existence of non-trivial ldquotrackingrdquo solutions

The equation of motion is equivalent to the ldquofirst law of thermodynamicsrdquo

d(a3ρφ )dt =minuspφ da3dt (28)

which can also be obtained from the requirement of covariant conservation of the energy-momentumtensor T Q ν

microν = 0 Actually this law is also valid for each independent6 species i in the background

d(a3ρi)dt =minuspi da3dt (29)

Furthermore it can be shown that the corresponding equation for the total energy density ρtotal equivρφ +sumi ρi and the (analogically defined) total pressure ptotal can be derived from the Friedmann equa-tions Thus assuming N species in the background there are 4 + N independent equations (secondorder differential equations are counted twice) from (29 27 26) with 4+2N independent variables

6An independent species should have negligible interaction with other species

22 Tracking Solutions 9

a aφ φ ρi pi This means the system can only be solved by specifying N additional equationsconventionally taken to be the equations of state for the N background species

pi = pi(ρi)equiv ωiρi (210)

A constant ldquoequation of state parameterrdquo ωi together with the first law of thermodynamics (29)yields the scaling behavior of the most important background components7

ωM = 0 ρM prop aminus3 nonrelativistic matter ωR = 13 ρM prop aminus4 relativistic matter ωΛ = minus1 ρΛ prop a0 cosmological constant

It is useful to define the equation of state parameter ωφ analogously to the background for the quint-essence field

ωφ =pφ

ρφ

=φ 22minusVφ 22+V

(211)

However the crucial difference is that this parameter will in general not be a constant Therefore thescaling behavior of quintessence cannot be integrated as easily as for matter and radiation Like ininflationary scenarios it is used that ωφ can be close to minus1 if the scalar-field is slowly rolling (ieφ 22 V ) down its potential It can be seen from the second Friedmann equation (26) that it is anecessary condition for an accelerated expansion of the universe that ωφ ltminus13 If the quintessencefield is static (φ = 0) it acts like a cosmological constant V with ωφ =minus1 On the other hand a freelyrolling field (φ 22V ) has ωφ = +1 and scales like aminus6 In any intermediate case one has

minus1le ωφ le+1

if the potential is positive Models with ωφ lt minus1 can be obtained by flipping the sign of the kineticterm in the Lagrangian (tachyonic or phantom dark energy) or by introducing new terms in the actionleading to cosmologies with a Big Rip in the future Such models allow superluminal velocities andare unstable on the quantum level since the energy density is not bounded from below [162] Thesemodels are not considered here Instead the focus lies on those models which are able to address theldquocosmological smallness problemrdquo most efficiently

22 Quintessence with Tracking Solutions

Within quintessence cosmology specific models are obtained from specific choices of the potentialA priori the potential may be an arbitrary function of the field value From the point of view of par-ticle physics a polynomial which contains quadratic and quartic terms similar to the standard Higgspotential would be the most straightforward choice since it is renormalizable and well-understoodFurthermore such a potential furnishes the simplest model of cosmic inflation in the early universewhich is compatible with all observational constraints [89] However for dynamical dark energy arenormalizable potential suffers from several shortcomings First it would be necessary to fine-tunethe mass and the coupling constant to extraordinarily small values8 in order to prevent the field from

7The cosmological constant is only given for completeness It does not appear in the background since it is absorbedinto the potential V

8For a quadratic potential the typical relaxation time-scale is given by the mass Requiring that this time-scale is of theorder of the age of the universe means that the mass has to be of the order of the Hubble constant H0 sim 10minus33 eV Whena quartic term is present it is additionally required that the quartic coupling constant is extremely tiny λ H2

0 M2pl A

similar constraint is well-known for chaotic inflation λ H2inf M2

pl 10minus10


reaching the stable potential minimum already long before the present epoch and thereby disqualifyas dynamical dark energy Second even if the fine-tuning of the mass and the coupling constant ispermitted it would additionally be necessary to fine-tune the initial conditions of the field in the earlyuniverse in order to achieve precisely the observed dark energy density todayOn the other hand it is possible to specify desired properties of dynamical dark energy and then tryto construct potentials which yield solutions featuring these properties This philosophy has beenfollowed in Ref [157] and generalized in Ref [169] leading to the notion of tracker quintessencemodels which are characterized by the following properties First the dynamics of the quintessencefield today should be insensitive to the initial value in the early universe Second it should be possibleto explain the smallness of the quintessence energy density today due to its dilution caused by thecosmic expansion similar to the dark matter density Thereby it is desired that the ratio of darkenergy and dark matter densities stays ideally of order unity during the complete cosmic history suchthat their similarity is not a special ldquocoincidencerdquo at all Third a necessary property is the cross-over from matter domination to dark energy domination The last property is the only one shared bythe cosmological constant which however is absolutely sensitive to the ldquoinitialrdquo value since it is aconstant and requires a huge hierarchy between the dark matter and dark energy densities in the earlyuniverseAs has been shown in Ref [169] the upper properties are realized for quintessence potentials whichfulfill the so-called tracker condition It states that the dimensionless function

Γ(φ)equiv V (φ)V primeprime(φ)V prime(φ)2

has to be larger or equal to unity and (approximately) constant for all field values for which V (φ) isbetween the critical energy density today and after inflation The latter requirement can be shown toguarantee the existence of attractors in phase space which wipe out the dependence on initial con-ditions for all solutions which approach the attractor solution [169] Thus the first desired propertyis fulfilled For the attractor solution the quintessence field dilutes with cosmic expansion with anapproximately constant equation of state [169]

ωlowastφ = ωBminus

Γminus1Γminus 1

2

(1+ωB) (212)

where ωB = 13 during radiation domination and ωB = 0 during matter domination The equation ofstate parameter determines the evolution of the quintessence energy density in the expanding universeFor a quintessence potential where Γ sim 1 the quintessence equation of state ωlowast

φis close to ωB such

that the quintessence energy density evolves with time approximately proportional to the dominantbackground density Thus for a quintessence potential where Γsim 1 the dark energy density ldquotracksrdquofirst the radiation density and then the matter density and thereby meets the second desired propertyFor Γ = 1 the ratio of the dark energy and dark matter densities would even be exactly constant duringmatter domination and exhibit perfect tracking behaviour For Γ gt 1 however one has that ωlowast

φlt

ωB This means that the ratio of the quintessence energy density and the background energy densityincreases with time Therefore a cross-over from matter domination to dark-energy domination hasto occur at some point which was the third desired propertyThe prototype tracker potentials are those for which Γ(φ) is precisely constant They are given by

V (φ) =

M4

pl exp(minusλ

φ

Mpl

)for Γ = 1

c middotφminusα for Γ gt 1 with Γ = 1+ 1α


Both the exponential and the inverse power law potentials decrease monotonously with φ and ap-proach their minimal value (zero) asymptotically for infinitely large field values For the tracker solu-tion the field slowly rolls down the potential with φ lowast(t)|trarrinfinrarrinfin Their properties have been studiedextensively in the literature [910 34 157169 182] and will therefore only be briefly sketched hereFurthermore many alternative potentials for which Γ is only approximately constant are typicallybuilt up from combinations of the prototype potentials like the inverse exponential potential [169] orthe so-called SUGRA potential [42 43] and share many of their basic properties

Exponential potential For the exponential potential the quintessence energy density is preciselyproportional to the radiation density during radiation domination (with Ωφ = 4λ 2) and to the matterdensity during matter domination (with Ωφ = 3λ 2) Therefore the exponential potential motivatesthe search for early dark energy which clearly discriminates it from the cosmological constant Con-straints on early dark energy arise from its impact on BBN structure formation and the CMB [8586]A typical upper bound for the dark energy fraction at redshifts z amp 2 is Ωφ lt 005 which impliesthat λ gt 775 For a single exponential potential Ωφ would always remain constant and no cross-over towards accelerated expansion would occur which disqualifies it as a viable dark energy modelHowever the tracking attractor just exists if λ gt

radic3(1+ωB) ie if the potential is steep enough

Otherwise the exponential potential features an attractor for which the quintessence energy densitydominates over the radiation and matter densities with equation of state ωlowast

φ= minus1 + λ 23 such that

accelerated expansion occurs when λ is small enough Therefore viable models can be constructedfor which the cross-over is triggered by an effective change in the slope of the exponential potentialThis can be accomplished by a potential which is given by the sum of two exponentials with differentslope [21] or by a ldquoleaping kinetic termrdquo [111] For the cross-over to occur now it is necessary toadjust the relative size of the exponentials which may be considered as an unavoidable tuning of thepotential In Ref [111] it is argued however that the tuning is much less severe as required for thecosmological constant (over two instead of 120 orders of magnitude)

Inverse power law potential The inverse power law potential alone already leads to a viable dy-namical dark energy model for which the dark energy density dilutes during cosmic expansion ac-cording to the tracking solution but the fraction Ωφ grows At some point the quintessence densitybecomes comparable to the dark matter density and then leads to the onset of a dark energy domi-nated epoch of accelerated expansion This cross-over occurs when the field value is of the order ofthe Planck scale Therefore it happens in the present epoch if V (Mpl)simM2

plH20 Thus the pre-factor

cequiv Λ4+α of the inverse power law potential has to have the order of magnitude

Λ =O

((H0

Mpl

) 24+α

Mpl

)=O

(10minus

1224+α Mpl

)

For example Λsim 10keV for α = 1 The smaller the inverse power law index α the more shallow isthe potential Since the field rolls more slowly in shallow potentials its equation of state today is themore negative the smaller the inverse power law index A conservative upper bound ωφ lt minus07 onthe dark energy equation inferred from SN1a and CMB measurements leads to an upper bound α 2for the inverse power law index [84]

Self-adjusting mass

For tracking solutions not only the potential energy of the quintessence field decreases with time butalso the effective time-dependent mass m2

φ(t) equiv V primeprime(φ(t)) of the quintessence field which is given


by the second derivative of the potential approaches zero for t rarr infin For the tracking solution it isexplicitly given by [169]

m2φ (t) = V primeprime(φ lowast(t)) =

92

Γ

(1minusω

lowastφ

2)

H(t)2 (213)

Thus for tracker quintessence potentials the classical dynamics drive the mass of the quintessencefield towards a value which is of the order of the Hubble parameter It is emphasized that on theclassical level this is a self-adjusting mechanism for the mass since even if one starts with a differentvalue the mass converges towards the value given above since the tracking solution is an attractorsolution A mass of the order of the Hubble scale which corresponds to the inverse size of thehorizon is also desirable for stability reasons since it inhibits the growth of inhomogeneities in thequintessence field [157]

Possible origins of tracker potentials

Exponential and inverse power law potentials are very unusual from the point of view of high energyphysics Nevertheless some attempts have been made to obtain such potentials from a superior theoryIn Ref [34] it was proposed that the quintessence field can be interpreted as a fermion condensate in astrongly interacting supersymmetric gauge theory whose dynamics may under certain assumptionsbe describable by an inverse power law potential An extension of the upper scenario to supergravitydiscussed in Ref [42] leads to the so-called SUGRA-potential Exponential potentials may occur inthe low-energy limit of extradimensional theories [165] or could result from the anomalous breakingof dilatation symmetry [182] In any case the quintessence field is an effective degree of freedomdescribed by an effective theory which is valid below an ultraviolet embedding scale The aim ofthe present work is to investigate the robustness of tracker potentials under quantum corrections ina model-independent way which includes a wide range of possibilities for the unknown underlyingUV completion

23 Interacting Quintessence

Interactions between the rolling quintessence field and Standard Model fields lead to striking phe-nomenological consequences [157 172 184] which can be tested experimentally in many ways Ingeneral interactions of the quintessence field are expected if it is embedded in an effective field theo-ry framework [51] For a neutral scalar field there are plenty of possibilities for couplings betweenquintessence and Standard Model fields [11 15 36 44 46 56 64 83 87 95 137 145 183 184 186]For tracker quintessence potentials it is plausible that also the couplings may have a non-trivial de-pendence on the quintessence field The effects described below are generic for quintessence modelsand are treated as model-independent as possibleIn principle one can discriminate between direct effects of the quintessence coupling on the proper-ties of the Standard Model particles and indirect backreaction effects of the Standard Model fieldson the quintessence dynamics [96] The quantum vacuum contribution of the latter is discussed insection 42 Here the most prominent direct effects are briefly mentioned

Apparent violations of the equivalence principle Yukawa-type couplings between the quint-essence field and fermion fields ψi may be parameterized as [157]

LYuk =minussumi

Fi(φ) ψiψi (214)

23 Interacting Quintessence 13

Each function Fi(φ) gives a φ -dependent contribution to the mass (mi) of each fermion species Sincethe field value φ(t) changes during cosmic evolution the fermion masses are also time-varying oncosmological time-scales Actually this is a very typical feature of quintessence models Of coursethe time-variation of the fermion mass is supposed to be tiny in comparison to the total massThe fermions ψi do not need to be fundamental fermions but should be understood as effective fieldseg describing neutrons or protons with effective Yukawa couplings Fi(φ) In this case the φ -dependence of the nucleon masses could also be mediated by a φ -dependence of the QCD scale thatcould for example result from a φ -dependent unified gauge coupling in some GUT theory [184]The Yukawa couplings (214) mediate a long-range interaction by coherent scalar-boson exchangebetween the fermions [157] This interaction can be described by a Yukawa potential between twofermions of type i and j of spatial distance r

UYukawa(r) =minusyi y jeminusmφ r

r (215)

with couplings yi equiv dFidφ and the dynamical quintessence mass m2φ

= V primeprime(φ) As mφ is typicallyof the order H inside the horizon (mφ r 1) this interaction is a long-range interaction like gravityTherefore it can be seen as a correction to the Newtonian potential

U(r) =minusGmi m j1r

(1+8πM2

plyi

mi

y j

m j

) (216)

where the first term in the brackets represents the Newtonian contribution and the second term thequintessence contribution for an interaction of species i with j One consequence of the speciesdependence is a violation of the equivalence principle This turns out to put the most stringent boundon the couplings yi The acceleration of different materials towards the sun has been shown to bethe same up to one part in 1010 [157] from which a bound for the Yukawa couplings of neutrons andprotons can be derived9 [157]

yn yp 10minus24 (217)

This means a coupling of quintessence to baryonic matter has to be highly suppressed In other wordsthe strength of the interaction for baryonic matter is of the order y2

nm2n sim y2

pm2p sim (1024GeV)minus2 and

thus 10 orders of magnitude weaker than the gravitational coupling Gsim (1019GeV)minus2

Time-variation of masses and couplings Not only the fermion masses but basically all ldquocon-stantsrdquo in the Standard model (and beyond) could depend on the quintessence field10 A time-variationof fundamental gauge couplings can be induced by the term

LGauge =12

Z(φ)Tr(FmicroνFmicroν)

where Fmicroν is the field strength tensor of some gauge symmetry [184] The time-dependent normaliza-tion can also be expressed by replacing the gauge coupling g according to g2rarr g2Z(φ) which leadsto a time-dependent effective coupling For the photon field this leads to a time-varying fine-structureldquoconstantrdquo αem Actually a detection of such a variation could be considered as a possible signal forquintessence [82] Furthermore a variation in the strong coupling (and thereby the QCD scale) could

9Numerically this bound corresponds to M2ply

2m2 lt 10minus10 where m is the nucleon mass10The presence of the non-constant field φ will also alter the classical conservation laws since it is possible that eg

energy and momentum is exchanged with the quintessence field However the total energy and momentum are still con-served


lead to varying masses of baryons If the Standard Model is embedded in a GUT theory it is evenpossible to relate the variation of the various gauge couplings yielding interrelations between thevariation of nucleon masses and the fine-structure constant [184] Thus quintessence could predict arelation between the violation of the equivalence principle and the change of αemThe effect of changing fundamental constants can show up in many different ways giving the pos-sibility to extract experimental bounds (see [184]) Besides geonuclear bounds (Oklo |∆αem(z asymp013)|αem lt 10minus7) and astronuclear bounds (decay rates in meteorites |∆αem(z asymp 045)|αem lt3 middot 10minus7) there are measurements from the observation of absorption lines in Quasars (typically∆αem(z asymp 2)αem sim minus7 middot 10minus6 with errors of the same order [168 176]) Furthermore Big BangNucleosynthesis (BBN) constrains |∆ΛQCD(z asymp 1010)|ΛQCD lt 10minus2 and |∆αem(z asymp 1010)|αem lt10minus2(10minus4) where the latter bound applies if a GUT-motivated relation between αem and ΛQCD isused [50 75 118 172] Possible time variations of the electron to proton mass ratio are investigatedin Refs [119 158] The experimental bounds imply that the functions Z(φ) and Fi(φ) may only varyslightly while φ changes of the order Mpl or more during a Hubble time

Time-variation of the effective Newton constant Non-minimal gravitational couplings of thequintessence field lead to modifications of Einstein gravity [52 55 73 94 155 171] A non-minimalcoupling which is linear in the curvature scalar can be understood as an additional contribution to theNewton constant in the Einstein-Hilbert actionint

d4xradicminusg(minus R

16πGminus f (φ)R+

)equivint


16πGeff+

)

where1

16πGeff=

116πG

+ f (φ)

Hereby Geff is an effective Newton constant which appears in the gravitational force law for systemswhich are small compared to the time- and space-scales on which φ(x) varies analogically to Brans-Dicke scalar-tensor theories [41] For a scalar field with time-dependent field value a non-minimalcoupling which is linear in R thus leads to a time-variation of the effective Newton constant overcosmological time-scales Of course a variation in the strength of gravity is highly restricted byexperiments [155181] Laboratory and solar system experiments testing a time variation of G restricttodayrsquos value to

∣∣GeffGeff∣∣today le 10minus11yrminus1 and an independent constraint from effects induced on

photon trajectories gives∣∣ f prime2( f minus116πG)

∣∣todayle 1500 The requirement that the expansion time-

scale Hminus1 during BBN may not deviate by more than 10 from the standard value means that thevalue of the gravitational constant during BBN may not have differed by more than 20 from todayrsquosvalue [181] This can be rewritten in the form∣∣∣∣(Geff)BBNminus (Geff)today

(Geff)today

∣∣∣∣le 02

Chapter 3

Quantum Effective Action

The effective action contains the complete information about a quantum theory In this chapterapproximation techniques for the effective action of a scalar quantum field in Minkowski space-timeare reviewed which is described by the classical action

S[φ ] =int

d4x(

12(partφ)2minusVcl(φ)

) (31)

The extension to curved space-time and the calculation of the contribution to the effective action fromcouplings between the scalar field and heavier degrees of freedom is discussed in appendix BThe quantum field operator Φ(x) and its conjugate partx0Φ(x) obey equal-time commutation relations(units where h = 1 are used hereafter)

[Φ(x0x)Φ(x0y)]minus = 0

[Φ(x0x)partx0Φ(x0y)]minus = ihδ(3)(xminusy) (32)

[partx0Φ(x0x)partx0Φ(x0y)]minus = 0

A statistical ensemble of physical states in the Hilbert space belonging to the real scalar quantum fieldtheory can be described by a density matrix ρ In any orthonormal basis |n〉 of the Hilbert spacethe density matrix

ρ = sumn

pn|n〉〈n| (33)

describes a statistical ensemble in which the state |n〉 can be found with probability pn The expecta-tion value of an observable described by the operator O is given by

〈O〉= Tr(ρO) (34)

Total conservation of probability implies that Trρ = 1 Since 0le pn le 1 it follows that Trρ2 le 1 IfTrρ2 = 1 the ensemble can be described by a pure state1 |ψ〉 with density matrix ρ = |ψ〉〈ψ| Anexample for the latter case is an ensemble in the vacuum state |0〉

ρ = |0〉〈0| (35)

The vacuum state is defined as the eigenstate of the Hamiltonian

H(x0) =int

d3x(

12(Φ(x))2 +

12(nablaΦ(x))2 +Vcl(Φ(x))

)(36)

1This can easily be seen by choosing a basis of the Hilbert space which contains the state |ψ〉

16 3 Quantum Effective Action

with lowest energy For any external classical source J(x) coupled to the quantum field Φ(x) the state|0〉J is defined as the eigenstate of the Hamiltonian

HJ(x0) =int

d3x(

12(Φ(x))2 +

12(nablaΦ(x))2 +Vcl(Φ(x))minus J(x)Φ(x)

)(37)

with lowest energy The density matrix of a canonical ensemble in thermal equilibrium2 at tempera-ture T is known explicitly

ρ =1Z

exp(minusβH) (38)

where3 β = 1(kT ) and Zminus1 = Tr exp(minusβH) The vacuum ensemble is obtained from the thermalensemble in the limit T rarr 0 Any density matrix which can not be written in the form of eq (35)or eq (38) characterizes a nonequilibrium ensemble The computation of the effective action for en-sembles which are characterized by a Gaussian density matrix at some initial time tinit = 0 is treated inappendix D and the generalization to arbitrary density matrices with initial non-Gaussian correlationscan be found in section 71

31 1PI Effective Action

In this section the effective action for ensembles described by the density matrix

ρ = |0〉J J〈0| (39)

including the vacuum state for vanishing external source J(x) = 0 is treated The expectation valueof the field operator Φ(x) in the presence of the external classical source J(x)

φ(x)equiv Tr(ρ Φ(x)) =δW [J]δJ(x)

(310)

can be obtained from the derivative of the generating functional W [J] for connected correlation func-tions which is given by the path integral [180]

exp(

iW [J])

=intDϕ exp

(iS[ϕ]+ i

intd4xJ(x)ϕ(x)

) (311)

The effective action Γ[φ ] is the Legendre transform of W [J]

Γ[φ ] = W [J]minusint

d4xJ(x)φ(x) (312)

where the dependence on J is expressed by a dependence on φ using relation (310) By constructionthe equation of motion determining the field expectation value φ(x) including all quantum correctionsfor vanishing external source is obtained from the stationary point of the effective action

δΓ[φ ]δφ(x)

= 0 (313)

2When considering a quantum field theory with conserved global charges there is an additional contribution fromthe corresponding chemical potentials in the equilibrium density matrix For the real scalar quantum field there are nosymmetries which could lead to conserved charges and thus the chemical potential vanishes in thermal equilibrium

3In the following units where k = 1 are used

31 1PI Effective Action 17

The effective action can be calculated using its expansion in terms of ldquoone-particle-irreduciblerdquo (1PI)Feynman diagrams [122]

Γ[φ ] = S[φ ]+i2

Tr lnGminus10 +Γ1[φ ] (314)

iΓ1[φ ] = + +

=18

intd4x [minusiV (4)

cl (φ(x))]G0(xx)2 +112

intd4xint

d4y [minusiV primeprimeprimecl (φ(x))]G0(xy)3[minusiV primeprimeprimecl (φ(y))]

+

The functional iΓ1[φ ] is equal to the sum of all 1PI Feynman diagrams [122] without external linesA Feynman diagram is ldquoone-particle-reduciblerdquo (1PR) if it can be separated into two disconnectedparts by cutting one of its internal lines Conversely a Feynman diagram is 1PI if it is not 1PR Thelines of the 1PI Feynman diagrams represent the classical field-dependent propagator

Gminus10 (xy) =

minusiδ 2S[φ ]δφ(x)δφ(y)

= i(2x +V primeprimecl(φ(x)))δ 4(xminus y) (315)

and the field-dependent interaction vertices are given by the third and higher derivatives of the classi-cal action

iδ 3S[φ ]δφ(x1) δφ(x3)

= minusiV primeprimeprimecl (φ(x1))δ 4(x1minus x2)δ 4(x2minus x3)


= minusiV (4)cl (φ(x1))δ 4(x1minus x2)δ 4(x2minus x3)δ 4(x3minus x4)

(316)

and so onEach 1PI Feynman diagram contributing to the loop expansion of the effective action formulatedin terms of the field-dependent classical propagator G0(xy) and the field-dependent classical ver-tices (316) resums an infinite set of Feynman diagrams which are being composed of the free field-independent propagator

Gminus10 (xy) = i(2x +V primeprimecl(0))δ 4(xminus y)

and the field-independent vertices which are given by the derivatives iδ kS[φ ]δφ k|φ=0 (k ge 3) of theclassical action evaluated at φ = 0 and an arbitrary number of external lines given by the field valueφ(x) This infinite resummation can be recovered from each 1PI Feynman diagram by replacing theclassical propagator G0(xy) by its Schwinger-Dyson expansion around the free propagator G0(xy)

G0(xy) = G0(xy)+int

d4vG0(xv)[minusiV primeprimecl(φ(v))minus iV primeprimecl(0)]G0(vy)

= G0(xy)+int

d4vG0(xv)[minusiV primeprimecl(φ(v))minus iV primeprimecl(0)]G0(vy) +

+int

d4vint

d4uG0(xv)[minusiV primeprimecl(φ(v))minus iV primeprimecl(0)]G0(vu)[minusiV primeprimecl(φ(u))minus iV primeprimecl(0)]G0(uy)

+

and performing a Taylor expansion with respect to the field value φ around φ = 0

V primeprimecl(φ(x)) = V primeprimecl(0)+V primeprimeprimecl (0)φ(x)+12

V (4)cl (0)φ(x)2 + (317)


as well as inserting a similar Taylor expansion of the higher derivatives of the classical potentialinto the classical field-dependent vertices (316) In general the effective action can equivalently beexpanded in terms of Feynman diagrams involving the classical propagator and in terms of Feynmandiagrams involving the free propagator The former possibility has the advantage that only a finitenumber of Feynman diagrams contributes to the effective action at each loop order since no infiniteresummation of external lines is required as in the latter case [122] Furthermore the 1PI resummedloop expansion in terms of the classical propagator has a larger range of applicability In the caseof spontaneous symmetry breaking for example the free propagator is formally ill-defined sinceV primeprimecl(0) lt 0 This is due to an unsuitable choice of the expansion point (here φ = 0) in the field Incontrast to that the 1PI resummed loop expansion does not require a Taylor expansion in the fieldand is therefore manifestly independent of the expansion point It is well-defined for all field values φ

where V primeprimecl(φ) gt 0 and is therefore applicable to theories with spontaneous symmetry breaking [122]Alternatively to the expansion in 1PI Feynman diagrams the effective action can be expanded inpowers of space-time derivatives of the field φ(x)

Γ[φ ] =int

d4x(minusVeff (φ)+

Z(φ)2

(partφ)2 +

) (318)

The lowest order of the derivative expansion is called effective potential The next Lorentz-invariantorder contains two derivatives Both expansions may be combined to obtain an expansion of theeffective potential in terms of 1PI Feynman diagrams

Veff (φ) = Vcl(φ)+12

int d4k(2π)4 ln

(k2 +V primeprimecl(φ)

k2

)+V1(φ)

minusV1(φ) = + +

=18

[minusV (4)

cl (φ)][int d4k

(2π)41

k2 +V primeprimecl(φ)

]2

+1

12[minusV primeprimeprimecl (φ)

]2 int d4k(2π)4

int d4q(2π)4

1(k2 +V primeprimecl)(q2 +V primeprimecl)((q+ k)2 +V primeprimecl)

+

formulated in Euclidean momentum space using the Euclidean classical propagator

Gminus10 (k) = k2 +V primeprimecl(φ)

The momentum integral over ln((k2 +V primeprimecl(φ))k2) in the first line is obtained from the one-loopcontribution i2Tr lnGminus1

0 to the effective action see eq (314) up to a field-independent constantThe Feynman diagrams are obtained from the Feynman rules given above transferred to Euclideanmomentum space ie with lines representing the field-dependent classical propagator G0(k) andfield-dependent classical vertices given by minusV (k)

cl (φ) (k ge 3)The integrals over the loop momenta contain ultraviolet (UV) divergences Therefore it is eithernecessary to remove these divergences by a suitable renormalization of the parameters appearing inthe classical action which is for a given fixed UV regulator possible for the renormalizable classicalpotential

Vcl(φ) = V0 + micro3φ +

12

m2φ

2 +13

gφ3 +

14

λφ4 (319)


or to embed the quantum theory at a physical UV scale and treat it as an effective field theory In thelatter case the loop momenta are confined to be below the UV scale since the theory is only validup to this scale such that there are no UV divergences Instead the result explicitly depends on theenergy scale of the UV embedding


The 2PI effective action is a straightforward generalization of the expansion of the effective action interms of 1PI Feynman diagrams It can be derived from the generating functional W [JK] includinglocal and bilocal external classical sources J(x) and K(xy)

exp(

iW [JK])

=intDϕ exp

(iS[ϕ]+ iJϕ +

i2

ϕKϕ

) (320)

with the short-hand notation

Jϕ =int

d4xJ(x)ϕ(x) ϕKϕ =int

d4xint

d4yϕ(x)K(xy)ϕ(y) (321)

The field expectation value and the connected two-point correlation function (ldquofull propagatorrdquo) inthe presence of the sources J(x) and K(xy) can be obtained from the derivatives of the generatingfunctional W [JK]

φ(x) equiv Tr(ρ Φ(x)) =δW [JK]

δJ(x) (322)

G(xy) equiv Tr(ρ (Φ(x)minusφ(x)(Φ(y)minusφ(y)) =2δW [JK]δK(yx)

minusφ(x)φ(y)

The 2PI effective action is defined as the double Legendre transform of the generating functional

Γ[φ G] = W [JK]minusint

d4xJ(x)φ(x)minus 12

intd4xint

d4yK(yx)(G(xy)+φ(x)φ(y)) (323)

The equations of motion of the field expectation value φ(x) and the full propagator G(xy) are

δΓ[φ G]δφ(x)

=minusJ(x)minusint

d4yK(xy)φ(y) δΓ[φ G]δG(xy)

=minus12

K(xy) (324)

For vanishing external sources the equations of motion including all quantum corrections are byconstruction given by the stationarity conditions of the 2PI effective action

δΓ[φ G]δφ(x)

= 0δΓ[φ G]δG(xy)

= 0 (325)

The 2PI effective action can be calculated using its expansion in terms of ldquotwo-particle-irreduciblerdquo(2PI) Feynman diagrams [66]

Γ[φ G] = S[φ ]+i2

Tr lnGminus1 +i2

Tr(Gminus1

0 G)+Γ2[φ G] (326)

iΓ2[φ G] = + + (327)

=18

intd4x [minusiV (4)

cl (φ(x))]G(xx)2 +112

intd4xint

d4x [minusiV primeprimeprimecl (φ(x))]G(xy)3[minusiV primeprimeprimecl (φ(y))]

+


The functional iΓ2[φ G] is equal to the sum of all 2PI Feynman diagrams [66] without external linesA Feynman diagram is ldquotwo-particle-reduciblerdquo (2PR) if it can be separated into two disconnectedparts by cutting two of its internal lines A Feynman diagram is 2PI if it is not 2PR The field-dependent interaction vertices of the 2PI Feynman diagrams are given by the third and higher deriva-tives of the classical action as before see eq (316) However in contrast to the 1PI effective actionthe lines of the 2PI Feynman diagrams contributing to the 2PI effective action represent the full prop-agator G(xy)Using the upper parameterization of the 2PI effective action the equation of motion for the fullpropagator G(xy) is

δΓ[φ G]δG(yx)

= 0 hArr Gminus1(xy) = Gminus10 (xy)minus 2iδΓ2[φ G]

δG(yx) (328)

This equation of motion can be written in the form of a self-consistent Schwinger-Dyson equation

Gminus1(xy) = Gminus10 (xy)minusΠ(xy) (329)

where the self-energy Π(xy) is obtained from opening one line of each 2PI Feynman diagram con-tributing to the 2PI functional Γ2[φ G]

Π(xy)equiv 2iδΓ2[φ G]δG(yx)

(330)

In contrast to the perturbative Schwinger-Dyson equation the self-energy contains Feynman dia-grams with lines given by the full propagator G(xy) which appears also on the left hand side of theself-consistent Schwinger-Dyson equation Therefore the self-consistent Schwinger-Dyson is an im-plicit ie nonperturbative and in general non-linear equation for the propagator G(xy) In spite ofthese complications the self-consistency of the 2PI formalism has some advantages which are indis-pensable when studying the time-evolution of quantum fields For example approximations basedon a loop truncation of the 2PI effective action lead to evolution equations for the two-point func-tion which are free of the secularity-problem (see appendix D) in contrast to approximations basedon a loop truncation of the 1PI effective action which break down at late times even for arbitrarilysmall values of the coupling constant Thus approximations based on a loop truncation of the 2PIeffective action have a larger range of applicability than those based on a loop truncation of the 1PIeffective action This is similar to the difference between free perturbation theory and 1PI resummedperturbation theory discussed in the previous sectionFor the exact theory the 2PI effective action evaluated with the field-dependent solution G[φ ] of theself-consistent Schwinger-Dyson equation agrees with the 1PI effective action [66]

Γ[φ G[φ ]] = Γ[φ ] (331)

Truncations of the 2PI effective action for example up to a certain loop order correspond to aninfinite resummation of 1PI Feynman diagrams of all loop orders but with certain restrictions ontheir topology [3766] Assume the 2PI functional is truncated such that it contains just some finite orinfinite subset of all 2PI diagrams denoted by iΓtrunc

2 [φ G] Then the propagator in this approximationis determined by solving the equation of motion

Gminus1(xy) = Gminus10 (xy)minusΠ

trunc(xyG) (332)

where the self-energy Πtrunc(xyG) is derived from iΓtrunc2 [φ G] but still contains the propagator

G(xy) ie the equation of motion is still a self-consistent equation [120] The solution of this equa-tion for a given φ denoted by G[φ ] is therefore called the ldquofullrdquo propagator [120] (even though it is

33 nPI Effective Action 21

not the exact propagator due to the truncation of iΓ2[φ G]) An approximation to the exact effectiveaction is obtained by inserting G[φ ] into the truncated 2PI effective action Γappr[φ ] = Γtrunc[φ G[φ ]]In principle the same approximation can also be obtained via the perturbative expansion of the effec-tive action in terms of 1PI Feynman diagrams containing the classical propagator However even ifjust one single Feynman diagram was kept in the 2PI functional iΓtrunc

2 [φ G] it yields an approxima-tion Γappr[φ ] to the effective action which corresponds to a selective infinite series of perturbative 1PIFeynman diagrams [120] (see also appendix C1) In the following the superscripts are omitted andtruncations of the 2PI functional are also denoted by iΓ2[φ G]

33 nPI Effective Action

The nPI effective action is derived from the generating functional W [J1 Jn] including externalclassical sources Jk(x1 xk) for 1le k le n

exp(

iW [J1 Jn])

=intDϕ exp

(iS[ϕ]+ i

n

sumk=1

1k

J12middotmiddotmiddotk ϕ1ϕ2 middot middot middotϕk

) (333)


J12middotmiddotmiddotk ϕ1ϕ2 middot middot middotϕk =int

d4x1 middot middot middotint

d4xn J(x1 xk)ϕ(x1) middot middot middotϕ(xk) (334)

The nPI effective action is obtained by the multiple Legendre transform

Γ[φ GV3 Vn] = W [J1 Jn]minusn

sumk=1

J12middotmiddotmiddotkδW

δJ12middotmiddotmiddotk (335)

The equations of motion of the field expectation value φ(x) the full propagator G(xy) and the fullconnected vertex functions Vk(x1 xk) including all quantum corrections for vanishing externalsources are by construction given by the stationarity conditions of the nPI effective action

δΓ

δφ(x)= 0

δΓ

δG(xy)= 0

δΓ

δV12middotmiddotmiddotk= 0 (336)

For the exact theory all nPI effective actions with propagator and vertices evaluated at the stationarypoint agree with the 1PI effective action in the absence of sources

Γ[φ ] = Γ[φ G] = Γ[φ GV3] = = Γ[φ GV3 Vn]

Loop approximations still obey an equivalence hierarchy for vanishing sources [26]

Γ[φ ]1minusloop = Γ[φ G]1minusloop = Γ[φ GV3]1minusloop = Γ[φ GV3V4]1minusloop = Γ[φ G]2minusloop = Γ[φ GV3]2minusloop = Γ[φ GV3V4]2minusloop =

Γ[φ GV3]3minusloop = Γ[φ GV3V4]3minusloop =

4PI Effective Action

As an example the 4PI effective action Γ[GV4] = Γ[0G0V4] for a theory with Z2-symmetry φ rarrminusφ is considered In this case the connected two- and four-point functions are given by

G(x1x2) = G12 =2δW [KL]

δK12 V4(x1x2x3x4) =

4δW [KL]δL1234

minusG12G34minusG13G24minusG14G23


in terms of the generating functional W [KL] = W [0K0L] For λΦ44-theory the three-loopapproximation of the 4PI effective action reads [27]

Γ[GV4] =i2

Tr lnGminus1 +i2

Tr(Gminus1

0 G)+Γ2[GV4] (337)

iΓ2[GV4] =

=18

intd4x [minusiλ ]G(xx)2 (338)

+1

24

intd4x1234

intd4y [iA4(x1x2x3x4)]G(x1y)G(x2y)G(x3y)G(x4y)[minusiλ ]

minus 148

intd4x1234

intd4y1234 [iA4(x1x2x3x4)]G(x1y1)G(x2y2)times

timesG(x3y3)G(x4y4)[iA4(y1y2y3y4)]

where a compact notation d4x1234 = d4x1 middot middot middotd4x4 is used and the kernel A4 is defined via

V4(x1x2x3x4) =int

d4y1234 G(x1y1)G(x2y2)G(x3y3)G(x4y4)[iA4(y1y2y3y4)]

The equation of motion for V4 in the absence of sources is obtained from the stationarity condition

δΓ[GV4]δV4

= 0 hArr iA4(x1x2x3x4) =minusiλδ4(x1minus x2)δ 4(x1minus x3)δ 4(x1minus x4)

Thus the full 4-point function V4(x1x2x3x4) is in this approximation given by the classical vertexwith four full propagators attached to it Inserting the 4-point kernel into the 4PI effective actionyields the corresponding approximation of the 2PI effective action

iΓ2[G] = iΓ2[GV4] =18

intd4x [minusiλ ]G(xx)2 +

148

intd4xint

d4y [minusiλ ]G(xy)4[minusiλ ]

This is precisely the three-loop approximation of the 2PI effective action Γ[G] = Γ[φ = 0G] ie

Γ[G]3minusloop = Γ[GV4]3minusloop

for vanishing sources According to the equivalence hierarchy one would expect that only the nPIeffective actions for n ge 3 coincide at three-loop level However due to the Z2-symmetry all corre-lation functions involving an odd number of fields vanish such that 2PI and 3PI also coincide andtherefore also 2PI and 4PI

Chapter 4

Quantum Corrections in QuintessenceModels

Quintessence models admitting tracking solutions [169] feature attractors in phase-space which wipeout the dependence on the initial conditions of the field in the early universe as discussed in chapter 2Furthermore tracking solutions exhibit a dynamical self-adjusting mechanism yielding an extremelysmall time-evolving classical mass mφ (t)sim H(t) of the quintessence field of the order of the Hubbleparameter The smallness of mφ (t) inhibits the growth of inhomogeneities of the scalar field [157] andmakes quintessence a viable dark energy candidate In this context it is an important question whetherthe self-adjusting mechanism for the classical mass and its smallness are robust under quantum cor-rections [224383102132152159171] The long-standing ldquocosmological constant problemrdquo canbe reformulated as the problem to determine the overall normalization of the effective quintessencepotential Apart from that quantum corrections can influence the dynamics by distorting the shapeor the flatness (ie the derivatives) of the scalar potential Vcl(φ)rarr Veff (φ) Additionally quantumcorrections can induce non-minimal gravitational couplings between the field φ and the curvaturescalar R or a non-standard kinetic termNote that the fundamental ldquocosmological constant problemrdquo of quantum field theory is not addressedin this work Since quantum field theory together with classical gravity determines the effectivepotential only up to a constant it will always be assumed here that the freedom to shift the potentialby an arbitrary constant Veff (φ)rarr Veff (φ)+ const is used in such a way that it yields the observedvalue for dark energy in the present cosmological epoch However as mentioned above even withthis assumption there remain quantum corrections to the dynamics of the quintessence field whichcan be addressed by quantum field theory In this chapter these impacts of quantum fluctuations onthe dynamics of a light quintessence field from three different sources are investigated These sourcesare self-couplings couplings to Standard Model particles and couplings to gravityIn section 41 quantum corrections to the shape of the scalar potential originating from the quint-essence self-couplings are investigated in the framework of effective field theory In this frameworkit is assumed that the quintessence field arises from a high-energy theory which is governed by a UV-scale of the order of the GUT or Planck scales This is possible since the self-couplings of the darkenergy field although typically non-renormalizable are Planck-suppressed in tracking quintessencemodels [9 10 34 157 169 182] Suitable approximations of the effective action are discussed andprevious studies [4383] are extended by identifying and resumming the relevant contributions whichexplicitly depend on the UV-scale For two exemplary classes of models the resulting effective po-tential is used to study their robustnessIn section 42 quantum corrections induced by couplings between the quintessence field and Stan-

24 4 Quantum Corrections in Quintessence Models

dard Model particles are investigated The low-energy effective action is studied which contains thequintessence-field-dependent contributions of the Standard Model fields to the vacuum energy [2081] Even under relatively conservative assumptions these contributions dominate the effective po-tential unless the couplings are tiny [2081] Upper bounds on the couplings of a tracker quintessencefield are quantified and translated into upper bounds for time-variations of Standard Model particlemasses on cosmological time-scales caused by these couplings as well as into upper bounds on thecoupling strength to a long-range fifth force mediated by the quintessence field These are linkedto potentially observable effects like a variation of the electron to proton mass ratio [119 158] overcosmological time-scales or tiny apparent violations of the equivalence principle [172 184]In section 43 it is investigated which kinds of non-minimal gravitational couplings are induced byquantum fluctuations of the dark energy scalar field Gravitational couplings of the quintessence fieldare a crucial property of dark energy The minimal gravitational coupling contained in the covariantderivative in the kinetic term of the quintessence action and the covariant integration measure are re-quired due to general coordinate invariance Non-minimal gravitational couplings between the rollingscalar field and the curvature scalar lead to a time-variation of the effective Newton constant over cos-mological time-scales This is constrained observationally by solar system tests of gravity and by BigBang Nucleosynthesis [39 52 53 55 73 94 101 155] The non-minimal couplings which are gener-ated radiatively for a tracker quintessence field in one-loop approximation are derived and comparedto the observational bounds Corrections to the kinetic term are also discussed in section 43

41 Quantum Corrections from Self-Interactions

If the light scalar field responsible for dark energy has itself fluctuations described by quantum fieldtheory quantum corrections induced by its self-interactions do contribute to the quantum effectiveaction In this section this contribution is investigated Typical potentials used in the context ofquintessence contain non-renormalizable self-couplings involving eg exponentials of the fieldVcl(φ) = V0 exp

(minusλφMpl

)[9 10 34 157 169 182] These enter the effective action via the field-

dependent vertices (see eq (316))

minus iV (k)cl (φ) =minusiVcl(φ)Mk M = Mplλ simMpl

radicΩde3 (41)

which are suppressed by a scale M between the GUT and the Planck scale Such couplings couldarise from an effective theory by integrating out some unknown high-energy degrees of freedom atan ultraviolet scale ΛsimO(M) The effective field theory is only valid up to this physical embeddingscale Λ and the quantum effective action explicitly depends on the value of Λ Ultraviolet divergentcontributions to the effective action lead to marginal dependence prop lnΛ (for logarithmic divergences)or relevant dependence prop Λn (eg n = 2 for quadratic divergences) on the embedding scale Λ In thesimplest case Λ can be imagined as a cutoff for the momentum cycling in the loops of the FeynmandiagramsIt turns out that it is useful to keep track of the dependence on the suppression scale M of the verticesand the embedding scale Λ separately although they are closely related in a way depending on theunknown underlying high-energy theory Since the suppression scale M is of the order of the GUT orthe Planck scale the same is possibly true for Λ Because unknown quantum gravity effects dominateabove the Planck scale an upper bound Λ Mpl is assumed In order to establish a meaningfulapproximation it is desirable to resum all relevant contributions proportional to powers of

Λ2M2 simO(1)

41 Self-Interactions 25

whereas the tiny mass m2φsim V primeprime(φ) of the quintessence field which is typically of the order of the

Hubble scale admits a perturbative expansion in powers of

V primeprime(φ)M2 simV (φ)M4 ≪ 1

In section 411 power counting rules for tracker potentials within effective field theory are derivedand used to identify the dependence of Feynman diagrams on V (φ) M and Λ within this schemeIn section 412 an approximation to the effective action which resums the field-dependent relevantcontributions at leading order in V (φ)M4 is discussed In section 413 the same approximationis applied to a quantum field theory in 1+1 space-time dimensions where the effective potential isknown independently due to the symmetry properties of the theory and it is demonstrated that theresummation introduced in section 412 yields concordant results In section 414 the robustnessof the prototype tracker potentials namely the exponential and the inverse power-law potential isstudied

411 Effective Field Theory for Tracker Potentials

An effective theory describes the dynamics of a system by reducing it to effective degrees of freedomwith effective interactions which are not fundamental but only exist up to a certain energy scale ΛAbove this ultraviolet scale Λ of the effective theory it has to be replaced by another (effective orfundamental) theoryAn example for an effective field theory is the Fermi model of β -decay [97] based on an effectivepoint-like 4-fermion interaction between the electron the neutrino the neutron (down quark) and theproton (up quark) The interaction strength is given by the Fermi constant GF = 1166 middot10minus5GeVminus2The non-renormalizable effective interaction has to be replaced by the electroweak W -boson exchangeat the UV scale of the order Λsim 1

radicGF

An example for a loop calculation within an effective field theory is provided by the NambundashJona-Lasinio model [149] which features a 4-fermion self-interaction which is invariant under the chiraltransformation ψ rarr eiαγ5ψ

L= ψiγmicropartmicroψ +

G4[(ψψ)2minus (ψγ5ψ)2]

Similar to the Fermi model it is an effective field theory with UV scale Λsim 1radic

G If the interactionstrength is stronger than a critical value the chiral symmetry is broken dynamically such that the vac-uum expectation value 〈ψψ〉 equiv minus2MG is non-zero The scale M of the dynamical chiral symmetrybreaking is determined by a self-consistent Schwinger-Dyson equation (gap equation) which involvesa one-loop ldquotadpolerdquo Feynman integral If the UV scale of the theory is implemented by a Lorentzinvariant cutoff for the Euclidean loop momentum the gap equation reads [149]

M = = 2GMint

k2ltΛ2

d4k(2π)4

1k2 +M2 = 2GM

Λ2

16π2 f1(M2Λ2) (42)

with f1(M2Λ2) = 1+ M2

Λ2 ln(

M2

Λ2+M2

) f1(0) = 1 It has a non-zero solution M if G gt Gcrit = 8π2Λ2

Loop integrals in effective field theory

In order to resum the relevant contributions to the quantum effective action for the scalar field de-scribed by the action (31) with a tracker potential Vcl(φ) it is important to identify the dependence


on the embedding scale Λ In analogy to the NambundashJona-Lasinio model the embedding scale isassumed to cut off the ultraviolet divergences in the loop integrals However the form of this cut-off depends on the unknown degrees of freedom at the embedding scale In general this lack ofknowledge can be captured by a form factor FΛ(k) which parameterizes the cutoff-function For ourpurpose it is not required to know this form factor in detail but it is sufficient to know its asymptoticbehaviour

FΛ(k) =

1 for |kmicro | Λ 0 for |kmicro | Λ

(43)

The form factor modifies the high-momentum contribution of the loop integrals accomplished bymodifying the integration measure1

d4krarr d4k FΛ(k)equiv d4Λk

A hard momentum cutoff in Euclidean momentum space corresponds to a form factor FΛ(k) = θ(k2minusΛ2) As an illustrative example the two-loop contributions to the effective action (see eq (314)) areconsidered The same parameterization of the quadratically divergent Feynman integral (ldquotadpolerdquo)is used as in eq (42) int d4

Λk

(2π)41

k2 +m2 =Λ2

16π2 f1(m2Λ2) (44)

where the shape of the dimensionless function f1(x) depends on the form factor but as above isof order one for m2 Λ2 ie f1(x) sim O(1) for 0 le x 1 Similarly the following quadraticallydivergent two-loop Feynman integral (ldquosetting sunrdquo) is parameterized asint d4

Λk

(2π)4

int d4Λq

(2π)41

(k2 +m2)(q2 +m2)((q+ k)2 +m2)=

Λ2

(16π2)2 f2(m2Λ2)

where the dimensionless function f2(x) has been defined such that f2(x)simO(1) for 0le x 1 Withthese definitions the two-loop contributions to the effective action in the limit m2

φ= V primeprimecl(φ) ≪ Λ2

can be evaluated

=18

V (4)cl (φ)

[int d4Λk

(2π)41


]2

(45)

=18

V (4)cl (φ)

[Λ2

16π2 f1(V primeprimeclΛ2)]2

asymp 18

V (4)cl (φ)

[Λ2

16π2 f1(0)]2

= Vcl(φ) middot

λ 4

8M4pl

[Λ2

16π2 f1(0)]2

for Vcl(φ) = V0 exp(minusλφMpl)

=1

12[V primeprimeprimecl (φ)

]2 intint d4Λkd4

Λq

(2π)81

(k2+V primeprimecl)(q2+V primeprimecl)((q+k)2+V primeprimecl)(46)

=112[V primeprimeprimecl (φ)

]2 Λ2

(16π2)2 f2(V primeprimeclΛ2) asymp 1


]2 Λ2

(16π2)2 f2(0)

= Vcl(φ) middot Vcl(φ)M4

pl︸︷︷︸10minus120

middot

λ 6

12M2pl

Λ2

(16π2)2 f2(0)


1The most general form factor FΛ(k1 kn) for overlapping loop integrals can depend on all loop momenta k1 knHere it is assumed for simplicity that FΛ(k1 kn) = FΛ(k1)FΛ(k2) middot middot middotFΛ(kn) This choice is sufficient to identify therelevant contributions The results below do not depend on this assumption


As an example the two diagrams are also evaluated for an exponential potential First it can be ob-served that both are proportional to the classical potential Vcl(φ) in this case Second it is emphasizedthat the second diagram is suppressed with respect to the first one by a relative factor

Vcl(φ)M4pl asymp ρφM4

pl asymp 10minus120

The value 10minus120 applies for the present epoch Even if the quintessence energy density was muchlarger in cosmic history the ratio ρφ (t)M4

pl ≪ 1 is a very small number2 It turns out that thesuppression of the non-local diagram with two vertices with respect to the local diagram with onevertex is a result which can be generalized for tracker potentials

Power counting rules for tracker potentials

In order to identify proper approximations for quintessence tracker potentials it is necessary to esti-mate the orders of magnitude of the contributions to the effective action Since these involve deriva-tives of the (classical) quintessence potential it is desirable to set up a power counting rule giving anestimate of their order of magnitudeFor tracker quintessence potentials it turns out that the scale height M yields such an estimate

V (k)cl (φ)simVcl(φ)Mk (47)

It is an exact relation for exponential potentials see eq (41) where Vcl(φ) is of the order of thecritical energy density simM2

plH2 and M is between the GUT and the Planck scales For inverse power

law potentials the scale height depends on the field value Msim φ However during the present epochthe field value is also of the order of the Planck scaleBy dimensional analysis a 2PI Feynman diagram with V vertices and L loops can within effectivefield theory be estimated with the upper power counting rule For example an extension of the upperanalysis leads to

Diagrams with V = 1 sim Vcl(φ) middot

Λ2M2L

Diagrams with V = 2 sim Vcl(φ) middot Vcl(φ)M4 middot

Λ

2M2Lminus1

In general only the maximally divergent L-loop diagrams yield relevant contributions which are notsuppressed by powers of V (φ)M4

pl ≪ 1 compared to the classical potential These diagrams areprecisely those which only involve ldquotadpolerdquo integrals ie those with one vertex Apart from theldquodouble bubblerdquo diagram discussed above all higher-dimensional operators suppressed by powers ofM yield a ldquomulti bubblerdquo diagram with one vertexMotivated by the above estimate it will be shown in the next section that for tracker potentialsthe leading quantum correction to the classical potential can be obtained in terms of 2PI Feynmandiagrams with V = 1 but with arbitrarily high number of loops The resummation of all diagramswith V = 1 is accomplished by a generalized Hartree-Fock approximation of the 2PI effective action

412 Hartree-Fock Approximation

Within the framework of the 2PI effective action the Hartree-Fock approximation consists of a trun-cation of the 2PI functional iΓ2[φ G] containing all local 2PI Feynman diagrams [66] In the context

2An upper bound ρφ lt ρmax for the energy density of the quintessence field is assumed where ρmax is the maximalenergy density at the end of the inflation ρmax simM2

plH2inf sim 10minus8M4

pl(Hinf 1014GeV)2


of λΦ4-theory there is only a single local 2PI Feynman diagram the ldquodouble bubblerdquo diagram whichis the first contribution in eq (328) In general a 2PI Feynman diagram F contained in iΓ2[φ G]is ldquolocalrdquo if its contribution to the 2PI self-energy Π(xy) see eq (330) can be written in the form2δFδG(xy) = minusiΠloc(x)δ 4(xminus y) ie it is only supported at coincident space-time points Fora general scalar potential Vcl(φ) of interest here there are infinitely many local 2PI Feynman dia-grams which are precisely given by all diagrams with one vertex (ldquomulti-bubblerdquo diagrams) The 2PIeffective action in Hartree-Fock approximation is thus given by

iΓ2[φ G] =infin

sumL=2

12LL

intd4x(minusiV (2L)

cl (φ(x)))G(xx)L (48)

where the L = 2 contribution is the ldquodouble-bubblerdquo The factor 12LL takes into account the correctsymmetry factor for the ldquomulti-bubblerdquo contributions which contain a 2L-vertexThe self-consistent Schwinger-Dyson equation for the full propagator G(xy) in Hartree-Fock ap-proximation follows from the stationarity condition of the 2PI action see eqs (329330)

Gminus1(xy) = Gminus10 (xy)minus 2iδΓ2[φ G]

δG(yx)(49)

= i(2x +V primeprimecl(φ(x))δ 4(xminus y) minusinfin

sumL=2

L2LL

(minusiV (2L)cl (φ(x)))G(xx)Lminus1

δ4(xminus y)

Due to the locality of the self-energy it is possible to make the Hartree-Fock ansatz

Gminus1(xy) = i(2x +M2eff (x))δ

4(xminus y) (410)

for which the full propagator in Hartree-Fock approximation is parameterized by a local effectivemass Meff (x) The upper self-consistent Schwinger-Dyson equation is indeed solved by a propagatorof this form3 which reduces to a self-consistent ldquogap equationrdquo for the effective mass M2

eff (x)

M2eff (x) = V primeprimecl(φ(x))+

infin

sumL=2

L2LL

V (2L)cl (φ(x))G(xx)Lminus1

This equation can equivalently be written in a compact form with an exponential derivative operator

M2eff (x) = exp

[12

G(xx)d2

dφ 2

]V primeprimecl(φ(x)) (411)

The gap-equation is still a self-consistent equation for M2eff (x) since the effective mass enters also in

the propagator G(xx) on the right-hand side The effective potential is obtained from the effectivemass in the limit of a space-time independent field value (see below) In this limit the effective massis also space-time independent and the self-consistency of the gap equation can explicitly be seen byswitching to Euclidean momentum space

M2eff = exp

[12

(int d4Λk

(2π)41

k2 +M2eff

)d2

dφ 2

]V primeprimecl(φ)

3Note that this is due to the structure of the Hartree-Fock approximation For truncations containing non-local dia-grams one indeed has to solve the equation in the complete xminus y plane if the self-consistency should not be sacrificedThis is important for nonequilibrium quantum fields discussed in chapter 6 and also for the renormalizability of generalapproximations based on the 2PI formalism see appendix E


In order to obtain the effective potential Veff (φ) at some range of field values φ the gap equation hasto be solved for these values of φ Since the solution depends on φ it is denoted by Meff (φ) anddetermined by the requirement

M2eff (φ) = exp

[12

(int d4Λk

(2π)41

k2 +m2

)d2

dφ 2

]V primeprimecl(φ)

∣∣∣∣∣m2=M2

eff (φ)

(412)

More generally for a space-time dependent field φ(x) the solution of the gap equation (411) is afunction of the space-time point x and a functional of the field φ(middot) which is denoted by Meff (xφ) Itis determined by the requirement

M2eff (xφ) = exp

[12

G(xx)d2

dφ 2

]V primeprimecl(φ(x))

∣∣∣∣∣G(xx)=G(xx M2

eff (middot φ)) (413)

where for any function M2(x) G(xy M2(middot)

)is the solution of the equation(

2x +M2(x))

G(xy M2(middot)

)=minusiδ 4(xminus y)

The Hartree-Fock approximation to the effective action Γhf [φ ] follows from inserting the field-depen-

dent full propagator G[φ ](xy)equivG(

xy M2eff (middot φ)

)determined by the solution M2

eff (xφ) of the gapequation into the 2PI effective action (see section 32 [66]) Up to a field-independent constant theeffective action is obtained from eqs (48 326 410)

Γhf [φ ] = Γ[φ G[φ ]]

=int

d4x(

12(partφ)2minusVhf (φ)

)+

i2

Tr[

ln(2x +M2

eff (xφ))minus iM2

eff (xφ)G[φ ]]

where

Vhf (φ(x))equiv exp[

12

G(xxφ)d2

dφ 2

]Vcl(φ(x)) (414)

The effective potential in Hartree-Fock approximation is the lowest order contribution to the derivativeexpansion of Γhf [φ ]

V hfeff (φ) = Vhf (φ)+

12

int d4Λk

(2π)4

[ln

(k2 +M2

eff (φ)

k2

)minus

M2eff (φ)

k2 +M2eff (φ)

] (415)

where Vhf (φ) can be written as

Vhf (φ) = exp[

12

(int d4Λk

(2π)41

k2 +m2

)d2

dφ 2

]Vcl(φ)

∣∣∣∣∣m2=M2

eff (φ)

= V (φ m2)∣∣m2=M2

eff (φ)

In order to simplify the notation an auxiliary potential has been introduced

V (φ m2)equiv exp[

12

(int d4Λk

(2π)41

k2 +m2

)d2

dφ 2

]Vcl(φ) (416)

which is obtained from applying the exponential derivative operator containing a propagator with anauxiliary mass m2 to the classical potential Vcl(φ) The gap equation for M2

eff (φ) can also be expressedvia the auxiliary potential

M2eff (φ) =

part 2V (φ m2)partφ 2

∣∣∣∣m2=M2

eff (φ) (417)


Resummed perturbation theory

In order to check the validity of the Hartree-Fock approximation it is necessary to have a formalismavailable which allows to estimate the corrections Since the Hartree-Fock approximation is basedon the intrinsically nonperturbative self-consistent gap equation derived from the 2PI effective actionthe calculation of corrections to this approximation is not straightforward as in perturbation theoryInstead the exact propagator has to be expanded around the self-consistently determined Hartree-Fock propagator similar to the expansion of the full propagator around the classical propagator (seeappendix C1) in order to obtain an expansion of the exact effective action around the Hartree-Fockresult In appendix C2 it is shown that this yields an expansion of the exact effective action in termsof tadpole-free 1PI Feynman diagrams with dressed propagators and dressed vertices Applying theresult from eq (C9) to the lowest order of the derivative expansion of the effective action yields acorresponding expansion of the exact effective potential V exact

eff (φ) in terms of 1PI Feynman diagramswithout tadpoles

V exacteff (φ) = V hf

eff (φ)+V notadeff (φ) (418)

minusV notadeff (φ) = +

=112

[minusV (3)(φ)

]2 int d4Λk

(2π)4

int d4Λq

(2π)41

(k2 +V (2))(q2 +V (2))((q+ k)2 +V (2))+

where V hfeff (φ) is the effective potential in Hartree-Fock approximation as given in eq (415) and

minusV notadeff (φ) is the sum of all 1PI Feynman diagrams without tadpoles with lines representing the

field-dependent dressed propagator in Euclidean momentum space

Gminus1hf (k) = k2 +M2

eff (φ) = k2 +V (2)(φ)

determined self-consistently by the solution of the gap equation (412) and field-dependent dressedvertices given by the derivatives of the auxiliary potential (416) evaluated with auxiliary mass m2 =M2

eff (φ)

minus iV (k)(φ)equiv minusipart kV (φ m2)partφ k

∣∣∣∣m2=M2

eff (φ) (419)

for k ge 3 The gap equation (412) can be rewritten as M2eff (φ) = V (2)(φ) (see also eq 417) which

was already used above A Feynman diagram contains a ldquotadpolerdquo if it contains at least one linewhich begins and ends at the same vertex The effective potential expanded in terms of the dressedpropagator and vertices defined above contains only Feynman diagrams which have no ldquotadpolesrdquo

Hartree-Fock approximation for tracker potentials

The gap equation and the effective potential in Hartree-Fock approximation are now evaluated withineffective field theory for a tracker potential characterized by the power-counting rules discussed insection 411 The dependence of the effective mass on the UV embedding scale Λ is obtained byinserting eq (44) into the gap equation (412)

M2eff (φ) = exp

[12

(Λ2

16π2 f1(m2Λ2))

d2

dφ 2

]V primeprimecl(φ)

∣∣∣∣∣m2=M2

eff (φ)


In the limit M2eff (φ) Λ2 the gap equation has an approximate solution

M2eff (φ) exp

[Λ2

32π2 f1(0)d2

dφ 2

]V primeprimecl(φ) middot

1+O

(M2

eff

Λ2

)

This solution can be trusted for all values of φ where the approximate solution fulfills the assumptionM2

eff (φ) Λ2 Within the range of field values where this is the case the approximate solution of thegap equation can be used in order to obtain a corresponding approximation of the effective potentialusing eq (415) The momentum integral in the second term in eq (415) is only logarithmicallydivergent since the quadratic divergences of the two contributions to the integrand cancel (this canbe verified using ln(1+x)asymp x) Therefore it has a marginal dependence on the UV scale Λ and maybe parameterized in the formint d4

Λk

(2π)4

[ln(

k2 +m2

k2

)minus m2

k2 +m2

]=

m4

16π2 f0(m2Λ2) (420)

where f0(x) sim O(1) (for 0 le x 1) contains a logarithmic dependence on Λ Thus all relevantcontributions are captured by the first term in eq (415) Using that M2

eff (φ) Λ one finally obtainsthe effective potential in leading order in M2

eff Λ2 and Veff M4 from eq (415)

Veff (φ) exp[

Λ2

32π2 f1(0)d2

dφ 2

]Vcl(φ) middot

1+O

(M2

eff

Λ2

)+O

(Veff

M4

) (421)

where for simplicity the effective potential is denoted by Veff (φ) equiv V hfeff (φ) unless otherwise stated

Here the suppression scale M is defined as the scale height of the effective potential Veff (φ)

V (k)eff (φ)simVeff (φ)Mk (422)

analogously to the scale height M of the classical potential Vcl(φ) In section 414 it will be shownthat the effective potentials obtained for classical tracker potentials indeed fulfill a relation of this typeThe corrections of the order M2

eff Λ2 are inherited from the corrections to the approximate solutionof the gap equation and the corrections of order Veff M4 originate from the marginal contributions tothe effective potential which have been neglected The latter can be seen in the following way Themarginal contributions can be written in the form

δV marginaleff (φ) =

12

M4eff (φ)

16π2 f (M2eff (φ)Λ

2)

where f (x)equiv f0(x)+δ f (x)simO(1) (for 0le x 1) contains a logarithmic Λ-dependence Here f0(x)is the marginal contribution to the effective potential in Hartree-Fock approximation (see eqs (415420)) and δ f (x) stands for marginal corrections to the Hartree-Fock approximation (see also below)The power counting rule (422) for the effective tracker potential directly yields that V primeprimeeff sim Veff M2ie the order of magnitude of the effective mass can be estimated as M2

eff sim Veff M2 at leading

order in M2eff Λ2 Thus the marginal corrections δV marginal

eff simM4eff middot lnΛ sim [Veff middot (Veff M4) middot lnΛ] are

suppressed by a factor of the order Veff M4 compared to the leading contribution to the effectivepotentialUsing the resummed perturbation theory the order of magnitude of corrections to the Hartree-Fockeffective potential can also be estimated The first correction comes from the non-local tadpole-free


1PI Feynman diagrams with two vertices connected by l + 1 lines (l ge 2) Within effective fieldtheory their contribution is of the order (see eq (46))

δV nonloceff (φ) =

infin

suml=2

12(l +1)

[V (l+1)(φ)

]2 Λ2(lminus1)

(16π2)l fl(M2eff (φ)Λ

2)

where again fl(x) sim O(1) (for 0 le x 1) Using the upper power counting rule (422) the dressedvertices (419) for the effective potential (421) can be estimated as V (l+1) sim Veff Ml+1 such that[V (l+1)]2Λ2(lminus1) sim Veff middotVeff M4 middot (ΛM)2(lminus1) Thus δV nonloc

eff sim Veff middotVeff M4 middotF(ΛM) is also sup-pressed by the tiny factor of order Veff M4 ≪ 1 where F(ΛM) contains a resummation of thesubleading relevant contributions sim (ΛM)2(lminus1) fl(0)[2(l +1)(16π2)l] O(1) (for Λ M)In summary the approximation to the effective potential from eq (421) can be used in the range offield values φ where the conditions

M2eff (φ) Λ

2 M2 and Veff (φ) M4

are fulfilled For a quintessence tracker potential both conditions are in fact identical if the UVembedding scale and the suppression scale of the non-renormalizable interactions are of the sameorder (as expected for an effective field theory) Λ sim M since M2

eff sim Veff M2 at leading order inM2

eff Λ2 Furthermore for exponential tracker potentials the suppression scale M simM Mpl turnsout to be close to the Planck scale (see section 414) such that the corrections to the leading effectivepotential in eq (421) within the effective field theory framework are indeed of the order4 Veff M4

pl sim10minus120 during the present cosmological epoch Clearly the corrections are negligible even if some ofthe upper assumptions are relaxed for example if a UV embedding scale ΛMpl is allowed as willbe discussed in detail in section 414For simplicity it may be assumed that the function f1(x) appearing in the parameterization of theldquotadpolerdquo Feynman integral in eq (44) is normalized to f1(0) = plusmn1 This can be achieved withoutloss of generality by rescaling the precise value of Λ by a factor of order one For generality thepossibility that f1(0) can be positive or negative has been included for the following reason Sincethe Feynman integral (44) has a relevant dependence on Λ the value of the integral is dominated bycontributions close to the UV embedding scale at which the unknown underlying theory becomesimportant Thus although the integral (44) is of the order of magnitude sim Λ2 the precise numericalvalue will strongly depend on the form factor FΛ(k) Therefore due to the unknown shape of the formfactor it cannot be decided a priori whether f1(x) is positive or negative even though the integrandwithout the form factor is positive definite There are also similar examples like the Casimir effectwhere the sign of the renormalized 0-0-component of the energy-momentum tensor can be positiveor negative depending eg on boundary conditions and geometry even though the unrenormalizedcontribution is positive definite [35]Finally an approximation of the effective potential is obtained which resums all relevant contribu-tions for quintessence tracker potentials (which are characterized by the power-counting rule (47))and which explicitly depends on the UV embedding scale Λ

Veff (φ) exp[plusmn Λ2

32π2d2

dφ 2

]Vcl(φ) (423)

The corrections have been estimated to be of the order M2eff (φ)Λ2 and Veff (φ)M4 This result can

be compared to the one-loop analysis of Refs [43 83] The one-loop result can be recovered by

4 As mentioned in the beginning it is assumed here that the freedom to shift the effective potential by a constant is usedto match the present quintessence energy density with the observed value today


inserting the Taylor-expansion exp[c middotpart 2] = suminfinL=0 cLpart 2LL of the exponential derivative operator up

to first order

V1minusloop(φ)[

1plusmn Λ2

32π2d2

dφ 2

]Vcl(φ)

For tracker potentials obeying the power counting rule (47) the higher-order contributions which areresummed by the Taylor-series of the exponential derivative operator are proportional to

Λ2LM2L L = 234

These relevant corrections are unsuppressed for an effective theory where the UV embedding scaleΛ is of the order of the suppression scale M of non-renormalizable interactions and therefore itis important to take them into account As discussed above this is accomplished by the effectivepotential (423) in Hartree-Fock approximation which is valid as long as the effective quintessencemass and potential energy are much smaller than ΛsimM MplIt should be mentioned that the upper results are valid under the assumption that the embedding scaleΛ itself does not depend (strongly) on the value of the scalar field φ This is a reasonable assumptionif the UV completion is generically connected to quantum gravity effects in which case Λ sim Mplcan be expected [58 65] which is also compatible with M sim Mpl On the other hand in principlethe UV embedding scale Λ might depend on the field value φ in a way which is specific for the UVcompletion If for example the quintessence field influences the size R(φ) of a compactified extra-dimension and if the embedding scale Λ prop Rminus1(φ) corresponds to the compactification scale of thisextra-dimension it might depend on φ The parametric dependence of Λ on φ thus has to be studiedcase-by-case for any possible UV completion and will depend on the details of the embedding Inorder to be able to investigate the robustness of tracker potentials in a model-independent way theanalysis is restricted to those classes of UV completions where the field-dependence of the embeddingscale is negligible compared to the field-dependence of the classical tracker potential in the Hartree-Fock approximated effective potential (423) An analogous restriction has also been made in theone-loop analysis of Refs [43 83]

413 Manifestly finite Effective Potential in 1+1 Dimensions

Before studying the robustness of quintessence potentials using the generalized Hartree-Fock approx-imation it will be applied to quantum field theory in 1+1 space-time dimensions in order to checkwhether the approximation introduced above yields correct results in a case where the exact effec-tive potential is known independently due to the symmetry properties of the theory Furthermoreit turns out that the generalized Hartree-Fock approximation can be used efficiently to compute therenormalized effective potential for a scalar quantum field in 1+1 dimensions with non-derivativeself-interactionsThe Hartree-Fock approximation discussed in section 412 can be extended in a straightforward wayto d-dimensional quantum field theory described by the classical action

S[φ ] =int

ddx(


) (424)

Since the action is dimensionless (h = 1 in natural units) the field has mass-dimension [φ ] = (dminus2)2 The expansion of the effective action in terms of 1PI or 2PI diagrams described in sections 31and 32 respectively can be transferred to d dimensions by replacing all 4-dimensional integrals inposition and momentum space by d-dimensional integrals d4xrarr ddx d4k(2π)4rarr ddk(2π)d aswell as δ 4(xminus y)rarr δ d(xminus y)


For d = 1 + 1 ie for two-dimensional Minkowski space the field value φ is dimensionless andtherefore all non-derivative k-point self-interactions with classical vertices given by the derivativesminusiV (k)

cl (φ) of the potential (k ge 3) are renormalizable It will now be shown that it is even possibleto perform the renormalization explicitly for the self-consistent Hartree-Fock approximation and fora general potential Vcl(φ) in d = 1+1The effective action in d dimensions in Hartree-Fock approximation is given by eq (414) with d4xrarrddx and with a full propagator G(xy) parameterized as in eq (410) by an effective mass M2

eff whichis determined self-consistently by the field-dependent gap equation (413) For simplicity the lowestorder of the derivative expansion of the effective action ie the effective potential is treated hereThe effective potential in d dimensions in Hartree-Fock approximation is up to a field-independentconstant given by (see eq 415)


12

int ddk(2π)4

[ln

(k2 +M2

eff (φ)

k2

)minus

M2eff (φ)

k2 +M2eff (φ)

] (425)

As above (see eqs (412 414 417)) M2eff (φ) and Vhf (φ) can be rewritten as

M2eff (φ) =


∣∣∣∣m2=M2

eff (φ) Vhf (φ) = V (φ m2)

∣∣m2=M2

eff (φ) (426)

using the d-dimensional auxiliary potential

V (φ m2)equiv exp[

12

(int ddk(2π)4

1k2 +m2

)d2

dφ 2

]Vcl(φ) (427)

Renormalization in 1+1 dimensions

For d = 1 + 1 the momentum integral in the second term in eq (425) is convergent and can beexplicitly calculated such that the effective potential in Hartree-Fock approximation is (the effectivepotential has mass-dimension two in d = 1+1)


18π

M2eff (φ) =

(V (φ m2)+

18π


)m2=M2

eff (φ) (428)

In the second expression on the right-hand side the effective potential is rewritten in terms of theauxiliary potential V (φ m2) Obviously the effective mass and the effective potential are finite ifV (φ m2) is finite In order to completely renormalize all divergences in Hartree-Fock approximationit is thus sufficient (in d = 1 + 1) to introduce counterterms which remove the divergences of theldquotadpolerdquo Feynman integral appearing in eq (427) Note that this integral is only logarithmically di-vergent in d = 1+1 such that dimensional regularization [61] may be used without loss of generalityWith ε equiv 1minusd2 for d near 2 the dimensionally regulated ldquotadpolerdquo integral is given by

int ddk(2π)d

1k2 +m2 =

Γ(ε)(4π)d2 mdminus2 =

microminus2ε

4π

(1ε

+ ln4πeminusγ micro2

m2 +O(ε))

(429)

where the renormalization scale micro has been introduced in the last equality and γ asymp 05772 is Eulerrsquosconstant To keep the field value a dimensionless quantity as in d = 2 the replacement φ rarr microminusεφ

is made In order to remove the term which diverges when ε rarr 0 all coupling constants appearing


in the (bare) classical potential V Bcl (φ) are split into a renormalized part and a counterterm5 and all

renormalized terms are collected in V Rcl (φ) and all counterterms in δVcl(φ) to get

V Bcl (φ) = V R

cl (φ)+δVcl(φ)

Here the bare classical potential V Bcl (φ) can be identified with the potential appearing in the (bare)

classical action (424) such that the dimensionally regulated auxiliary potential is

V (φ m2) = exp[

18π

(1ε


m2 +O(ε))

d2

dφ 2

]V B

cl (φ)

The auxiliary potential can be renormalized according to the minimal subtraction scheme if the coun-terterms are chosen according to

δVcl(φ)equiv(

exp[minus 1

8π

1ε

d2

dφ 2

]minus1)

V Rcl (φ) (430)

Note that the counterterms do not depend on m2 which is crucial for the self-consistency of the gapequation (426) With this the auxiliary potential can be written in terms of the renormalized classicalpotential (for ε rarr 0)

V (φ m2) = exp[

18π

lnmicro2

m2d2

dφ 2

]V R

cl (φ) (431)

where micro2 equiv 4πeminusγ micro2 The auxiliary potential is thus manifestly finite for an arbitrary finite renormal-ized classical potential V R

cl (φ) and arbitrary auxiliary mass m2 and depends on the renormalizationscale micro Consequently it can be seen from eqs (426) and (428) that the effective mass M2

eff (φ) andthe effective potential Veff (φ) in Hartree-Fock approximation are also manifestly finite in d = 1 + 1In particular the self-consistent gap equation which determines the field-dependent effective masscan be rewritten in terms of the renormalized classical potential

M2eff (φ) = exp

[1

8πln

micro2

m2d2

dφ 2

]V R

clprimeprime(φ)

∣∣∣∣∣m2=M2

eff (φ)

(432)

and is also manifestly finite in d = 1+1

Renormalized resummed perturbation theory

Before calculating the renormalized effective potential for a specific example it should be notedthat the counterterms contained in δVcl(φ) as defined in eq (430) are actually already the exactcounterterms ie the exact effective potential is rendered finite by this choice of δVcl(φ) This canbe seen using the resummed perturbation theory discussed above (see also appendix C2) where anexpansion of the exact effective action in terms of 1PI Feynman diagrams without tadpoles but withdressed propagators and vertices has been derivedThe corresponding expansion (418) of the exact effective potential can easily be transferred to anarbitrary dimension d In d = 1 + 1 dimensions it was shown above that the auxiliary potentialV (φ m2) is rendered finite by the counterterms (430) for arbitrary auxiliary masses m2 Thereforethe dressed propagator Ghf (k) and the dressed vertices (419) minusiV (k)(φ) are themselves finite in d =1 + 1 and can be calculated explicitly from the manifestly finite expression (431) for V (φ m2)

5 A field rescaling Z is not introduced here since this in not necessary in d = 1+1


Furthermore there is only one type of Feynman integral which is divergent in d = 1 + 1 given bythe logarithmically divergent ldquotadpolerdquo integral6 (429) Since the expansion (418) of the effectivepotential is characterized by the property that it just contains Feynman diagrams without tadpolesand precisely these diagrams do not contain any divergent loop integrals the effective potential ind = 1+1 is completely renormalized by the counterterms (430)This result can be interpreted in the following way All divergences have been resummed into thedressed propagator and the dressed vertices (419) introduced above which are renormalized by thecounterterms (430) The Feynman diagrams without tadpoles contributing to V notad

eff (φ) according tothe expansion (418) are convergent in d = 1 +1 and thus no further counterterms are required Forexample the two loop contribution to V notad

eff (φ) is convergent and equal to

=1

(8π)2

ψ prime(16)+ψ prime(1

3)minusψ prime(23)minusψ prime(5

6)54

(V (3)(φ)

)2

V (2)(φ) (433)

where ψ prime(x) = dψ(x)dx is the first derivative of the digamma function ψ(x) = Γprime(x)Γ(x) Notethat due to the self-consistently determined dressed propagator and dressed vertices this diagramcorresponds to an infinite resummation of perturbative diagrams (see section 32 and appendix C)Since all contributions to V notad

eff (φ) are convergent it is possible to calculate an arbitrary Feynmandiagram up to its numerical prefactor by dimensional analysis Let F be a diagram contributing toV notad

eff (φ) with Vk vertices with k legs (k ge 3) Then it has V = sumkVk vertices P = sumk kVk2 internallines and L = PminusV+ 1 loops [179] Since all vertices have mass-dimension two in d = 1 + 1 theirproduct contributes a factor with dimension 2V Since F has also mass-dimension two and the onlyfurther scale which appears in the convergent loop integrals is the effective mass M2

eff =V (2) containedin the dressed propagator Ghf (k) the diagram can be written as

F =1

(8π)L g(F)prod

kge3

(V (k)(φ)

)Vk

(V (2)(φ)

)Vminus1 (434)

with a constant numerical prefactor denoted by g(F) For example for the two loop diagram (433)it is g(F) = (ψ prime(1

6)+ψ prime(13)minusψ prime(2

3)minusψ prime(56))54asymp 0781

Altogether it was shown that the exact and completely renormalized effective potential (418) for ascalar quantum field in 1+1 dimensions with non-derivative self-interactions can be written as


eff (φ) + V notadeff (φ) (435)

=(

V (φ)+1

8πV (2)(φ)

)+ sum

F

g(F)(8π)L

prodkge3

(V (k)(φ)

)Vk

(V (2)(φ)

)Vminus1

where V hfeff (φ) is the effective potential in Hartree-Fock approximation (428) which was rewritten

using V (φ)equivV (0)(φ) =Vhf (φ) and V (2)(φ) = M2eff (φ) The sum runs over all 1PI Feynman diagrams

6 A Feynman diagram for a scalar quantum field is convergent if the superficial degree of divergence of the diagram andall its subdiagrams is negative [38 61 113 177 191] For a diagram with V momentum-independent vertices P internalscalar lines and an arbitrary number of external lines the superficial degree of divergence is D = dLminus 2P [179] whereL = PminusV + 1 is the number of loops In d = 1 + 1 D = 2Lminus2P = minus2(V minus1) ie only (sub-)diagrams with one vertexcan contain divergences The internal lines of loop diagrams with one vertex have to begin and end at this vertex ie theyare ldquotadpolesrdquo attached to this vertex


F without tadpoles for which the dimensionless numerical constants g(F) are defined via eq (434)and with dressed vertices (419)

V (k)(φ)equiv part kV (φ m2)partφ k

∣∣∣∣m2=M2

eff (φ)

derived from the ldquotadpole-resummedrdquo auxiliary potential V (φ m2) (431) evaluated with the effectivemass m2 = M2

eff (φ) determined by the renormalized gap equation (432)

Exponential potential mdash Liouville theory

In this section the Hartree-Fock approximation is applied to a quantum field with an exponentialpotential

Vcl(φ) = V0 exp(minusλφ) (436)

with a dimensionless parameter λ known as Liouville Theory [76 148] In 1+1 dimensions thisis a renormalizable potential In the following it will be show that the effective potential can berenormalized and computed explicitly with the techniques introduced above and yields a result whichagrees with an independent method based on the conformal symmetry of Liouville Theory [67 76156] (which exists for the exponential potential in 1+1 dimensions only)The Hartree-Fock approximation is ideally suited for the exponential potential It is possible to findan exact solution of the gap equation (432) since the derivative d2dφ 2 appearing in the exponentialderivative operator can be just replaced by λ 2

M2eff = exp

[λ 2

8πln

micro2

M2eff

]V R

clprimeprime(φ) = λ

2V0 exp

[λ 2

8πln

(micro2

M2eff

)minusλφ

] (437)

Inserting eq (436) for V Rcl (φ) the gap equation can be easily solved algebraically for each value of

φ by dividing the equation by the renormalization scale micro2 and taking the logarithm on both sides

ln

(M2

eff

micro2

)= ln

(λ 2V0

micro2

)+

λ 2

8πln

(micro2

M2eff

)minusλφ

rArr ln

(M2

eff (φ)

micro2

)=

11+λ 2(8π)

[ln(

λ 2V0

micro2

)minusλφ

]

The solution of the gap equation thus reads

ln

(M2

eff (φ)

micro2

)= ln

(M2

r

micro2

)minus φ

λminus1 +λ(8π)

where ln(M2r micro2)equiv [ln(λ 2V0micro2)](1+λ 2(8π)) Furthermore using eqs (426 431) yields

Vhf (φ) = exp

[λ 2

8πln

(micro2

M2eff (φ)

)]V R

cl (φ) = Vr exp[minus φ

λminus1 +λ(8π)

]

where ln(VrV0)equivminus[ln(M2r micro2)]λ 2(8π) Together with the solution of the gap equation the effec-

tive potential in Hartree-Fock approximation is obtained from eq (428)


18π

M2eff (φ) =

(Vr +

18π

M2r

)exp[minusλ φ

]


The effective potential in Hartree-Fock approximation is also an exponential of the field φ with arenormalized pre-factor Vr +M2

r (8π) and with slope given by

λminus1 = λ

minus1 +λ(8π)

The upper relation can also be obtained completely independently from the transformation propertiesof the energy-momentum tensor which is highly constrained by the conformal symmetry of LiouvilleTheory in 1+1 dimensions [67 76]Using the expansion of the exact effective potential in terms of 1PI tadpole-free Feynman diagramswith dressed propagator and dressed vertices it is additionally possible to show that the effectivepotential in Hartree-Fock approximation captures basically already all quantum corrections to thepotential The dressed vertices and propagator for the exponential potential (436) are given by

V (k)(φ) =part kV (φ m2)

partφ k

∣∣∣∣m2=M2

eff (φ)= (minusλ )k Vhf (φ) = (minusλ )k Vr exp

[minusλ φ

]

Using this it can be seen from eq (434) that the contribution from a tadpole-free diagram F with Vkvertices with k legs (kge 3) ie with V = sumkVk vertices P = sumk kVk2 internal lines and L = PminusV+1loops has the form

F =1

(8π)L g(F)prod

kge3

((minusλ )kVhf (φ)

)Vk

(λ 2Vhf (φ)

)Vminus1 =1

(8π)L g(F)(minusλ )2P

λ 2(Vminus1)Vhf (φ) = g(F)(

λ 2

8π

)L

Vhf (φ)

Thus all contributions to the effective potential are proportional to Vhf (φ) Consequently using eq(435) the exact effective potential is obtained

V exacteff (φ) = VR exp

[minusλ φ

] (438)

where all contributions have been resummed into the constant prefactor

VR = Vr

(1+

λ 2

8π+sumF

g(F)(

λ 2

8π

)L)

= Vr

(1+

λ 2

8π+0781

(λ 2

8π

)2

+

)

The sum runs over all 1PI Feynman diagrams F without tadpoles L ge 2 is the number of loopsof F and g(F) is the dimensionless numerical prefactor defined in eq (434) This diagrammaticcalculation of the effective potential also agrees with the result given in Ref [76] without derivation

414 Robustness of Quintessence Potentials

For tracker potentials which obey the power-counting rule (47) non-renormalizable interactions aresuppressed by a high-energy scale M Mpl Within effective field theory embedded at a UV scaleΛ sim M the effective potential (423) obtained from the Hartree-Fock approximation is the leadingcontribution to the effective potential for classical tracker potentials Therefore eq (423) yields auseful prescription to estimate the stability of tracker quintessence potentials Vcl(φ) under quantumcorrections induced by its self-interactions This prescription consists of applying the exponentialderivative operator

exp[plusmn Λ2

32π2d2

dφ 2

](439)

to the classical potential Vcl(φ) In the following the effect of this operator on the prototype trackerquintessence potentials is investigated Furthermore the dependence on the embedding scale Λ isdiscussed as well as the validity conditions of the Hartree-Fock approximation The impact on cos-mological tracking solutions is studied for some examples


Exponential potential

One prototype class of tracker potentials are (combinations of [21 150]) exponential potentials [10157 182] Remarkably an exponential of the field φ is form-invariant under the action of the opera-tor (439) Consider eg the following finite or infinite sum of exponentials

Vcl(φ) = sumj

Vj exp(minusλ j

φ

Mpl

) (440)

The only effect of applying the operator (439) is a simple rescaling of the prefactors Vj according to

Vj rarr Vj exp

[plusmn

λ 2j Λ2

32π2M2pl

] (441)

This extends the result of Ref [83] for the one-loop case which would correspond to the first termin a Taylor expansion of (439) Note that if ΛsimMpl the correction can be of an important size andcan influence the relative strength of the exponentials in (440) The necessary conditions of validityV primeprimeeff (φ) Λ2 and Veff (φ) M4 simM4

pl for the Hartree-Fock approximation are both fulfilled when

Veff (φ) Λ2M2

pl M4pl

which implies that it is applicable if Λ Hmax where Hmax is the maximum value of the Hubbleparameter where the field φ plays a role For example Hmax could be the inflationary scale Hinf Forchaotic inflation with quadratic potential it is typically of the order Hinf sim (δTCMBTCMB) middotMpl sim10minus5Mpl sim 1013GeV [140 141] Furthermore note that the effective potential indeed fulfills thepower-counting rule (422) for tracker potentials with scale-height of the order M sim M Mpl forλ j ampO(1)Altogether it is found that exponential potentials are stable under radiative corrections from self-interactions in the domain of validity of the Hartree-Fock approximation within effective field theoryIn particular ultraviolet embedding scales up to the Planck scale Λ Mpl are possible The subleadingcorrections which would lead to a distortion of the exponential shape are suppressed by a factor ofthe order of Veff (φ)M4

pl This is an extremely tiny number of the order H2M2pl in the context of

quintessence models

Inverse power law potential

The second prototype class of tracker potentials are (combinations) of inverse powers of the quint-essence field φ [43 83 157 169]

Vcl(φ) = sumα

cαφminusα (442)

The action of the operator (439) yields

Veff (φ) = sumα

cαφminusα

Γ(α)

infin

sumL=0

Γ(α +2L)L

(plusmnΛ2

32π2φ 2

)L

= sumα

cαφminusα

Γ(α)

intinfin

0dt tαminus1 exp

(minustplusmn Λ2

32π2φ 2 t2)

(443)

where the Γ-function inside the sum over L has been replaced by an integration over the positivereal axis in the second line by using its definition This integral gives a finite result if the negativesign in the exponent is used which will therefore be assumed from now on First two limiting cases


1

101

102

103

104

105

106

107

108

10-4 10-3 10-2 10-1 1

Vef

f(φ

) V

(φ0)

φ φ0

φ0 = φ(t0) = O(Mpl)Vcl(φ)

Veff(φ)

Vone-loop(φ)

Vtwo-loop(φ)

Figure 41 Comparison of the Hartree-Fock approximation of the effective potential Veff (φ) (red)with the leading one- and two-loop contributions as given by the Taylor expansion of the derivativeoperator (439) up to first and second order respectively (blue) as well as the classical potentialVcl(φ) prop φminusα (black) for α = 2 The loop expansion breaks down at small field values φ ΛThe non-perturbative ldquomulti-bubblerdquo resummation accomplished by the Hartree-Fock approximationallows to extend the range of validity to the complete admissible range of field values φ gt 0

will be discussed where the integral can be solved analytically For large field values φ Λ whichcorresponds to small potential energy and -curvature the second term in the exponent appearing inthe second line of eq (443) can be neglected which implies that asymptotically

Veff (φ)rarrVcl(φ)equivsumα

cαφminusα φ rarr infin (444)

This means the low energy regime where the potential and its derivatives go to zero is not changed byquantum corrections For the opposite limit where φ Λ the integral in the last line of (443) can becalculated by neglecting the first term in the argument of the exponential

Veff (φ) rarr sumα

cαφminusα

Γ(α)12

Γ(α

2)(

Λ2

32π2φ 2

)minus α

2

= sumα

Γ(α

2 )2Γ(α)

cα

(Λ

4πradic

2

)minusα

= const (445)

Thus the effective potential approaches a constant finite value for φ Λ(4πradic

2) of the order Vcl(Λ)in the small-field limit φ Λ (see figures 41 and 42) Furthermore it is easy to see that also thesecond derivative of the effective potential approaches a constant value

V primeprimeeff (φ)rarrsumα

Γ(α+22 )

2Γ(α)cα

(Λ

4πradic

2

)minus(α+2)

(446)

Similarly all higher derivatives approach constant values for φ Λ Therefore the effective poten-tial Veff (φ) fulfills the power-counting rule (422) with scale height given by

M sim

Λ for φ Λ

φ for φ Λ (447)


1

103

106

109

1012

110-110-210-310-410-510-6

0

10

102

103

104

105

106

Vef

f(φ

) V

(φ0)

z (f

or tr

acke

r so

lutio

n)

φ φ0

today

CMB

Λ = φ0 asymp Mpl

V(φ) prop φ- α

Λ = φ01024 asymp 10-3Mpl

Figure 42 Dependence of the effective potential Veff (φ) on the UV embedding scale Λ for an inversepower law potential Vcl(φ) prop φminusα with α = 2 The potential is normalized to the value of the potentialVcl(φ0) at redshift z = 0 From top to bottom Λ is enlarged by a factor 2 for each red line Theblack line is the classical potential Vcl(φ) which is a straight line due to the double logarithmicscale For φ Λ the effective potential Veff (φ) approaches a constant value whereas Vcl(φ) growsunboundedly The redshift-scale on the right-hand side applies for the classical tracking solution onlyand illustrates when the deviations of the effective potential Veff (φ) from the classical potential Vcl(φ)become relevant in cosmic history going backward from φφ0 = 1 (today)

The scale-height M of the effective potential approaches a constant value for small field values φ in contrast to the scale height M sim φ of the classical potential Vcl(φ) Thus the singularity of theclassical potential Vcl(φ) see eq (442) for φ rarr 0 is not present for the effective potential Veff (φ)where a constant value of the order Vcl(Λ) is approached insteadThe Hartree-Fock approximation requires that V primeprimeeff (φ)Λ2 M2 and Veff (φ) M4 From eq (447)it can be seen that the requirement Λ2 M2 is fulfilled in the whole range of possible field valuesφ gt 0 In order to check the other conditions of validity the case where the potential consists of onlyone inverse power-law term Vcl(φ) = cαφminusα will be treated first for simplicity In the range φ Λthe limits of the effective potential (444) and the effective mass (445) can be used

V primeprimeeff (φ)sim cαΛminus(α+2) Λ2 hArr Λ c1(α+4)α

Veff (φ)sim cαΛminusα M4 sim Λ4 hArr Λ c1(α+4)α

Thus both conditions of validity yield the same lower bound on the embedding scale Λ The condi-tions of validity in the range φ Λ can be evaluated using that Veff (φ)Vcl(φ) in this range

V primeprimeeff (φ)sim cαφminus(α+2) Λ2 hArr Λ c1(α+4)α (Λφ)

α+2α+4

Veff (φ)sim cαφminusα M4 sim φ 4 hArr Λ c1(α+4)α (Λφ)

Since Λφ 1 by assumption the bounds obtained in the large-field range are weaker than thebounds obtained in the small-field range All conditions of validity are thus fulfilled if the embedding


-1

-09

-08

-07

-06

-05

0 02 04 06 08 1

ωφ

Ωφ

toda

y (z

=0)ΛMpl =

0103

05

07

09

11 1315

29

Figure 43 Evolution in the (Ωφ ωφ )-plane for the effective potential Veff (φ) of an inverse power-law potential Vcl(φ) prop φminusα with α = 1 for various values of Λ keeping H0 = 73kmsMpc and Ωde equivΩφ (z = 0) = 076 fixed The UV embedding scale Λ is enlarged by 02Mpl for each red line startingfrom Λ = 01Mpl The black line is the tracking solution in the classical potential Vcl(φ) from whichthe solutions deviate considerably for embedding scales Λ close to the Planck scale The four arrowson each trajectory mark the points with redshifts z = 210501 from left to right

scale fulfills the lower bound Λ c1(α+4)α For the classical potential (442) which contains a sum

of inverse power-laws the generalized bound is

Λmaxα

c1(α+4)α

For a single inverse power-law the order of magnitude of the constant cα required to reproduce thecorrect abundance of dark energy is [169]

c1(α+4)α sim

(H2

0 Mα+2pl

)1(α+4)sim((100MeV)6Mαminus2

pl

)1(α+4)

Thus the lower bound on the embedding scale is a relatively mild restriction Λ 100MeV forobservationally allowed [169] values of the inverse power-law index α 2 For extremely steeppotentials αrarrinfin the lower bound asymptotically approaches the Planck scale It is emphasized thatloop approximations to the effective potential break down in the limit φ rarr 0 whereas the Hartree-Fock approximation is applicable (see figure 41) The dependence of the effective potential on theUV embedding scale Λ is shown in figure 42 for the case Vcl(φ) prop φminus2Finally the question in how far typical tracking quintessence models are changed by considering theeffective potential from eq (443) is investigated Since the field value today is typically of the orderof the Planck scale [169] the large-field limit eq (444) where the effective potential approaches theclassical potential and the corrections are negligible is only applicable when Λ ≪ Mpl For valuesup to Λ Mpl10 the field φ can have a tracking solution The redshift zquant in cosmic history wherethe effective potential starts to deviate from the classical tracking potential see figure 42 gives arough estimate at which redshift the tracking sets in For a potential dominated by a single inverse


α

ΛM

pl

02 04 06 08 1 12 14 16 18 2 0

05

1

15

2

25

3

-09

8-0

95

-09

-08 -07

-06

Figure 44 Contour plot of the equation of state ωde today (z = 0) using the effective potentialVeff (φ) obtained from the classical potential Vcl(φ) prop φminusα depending on the embedding scale Λ andthe inverse power-law index α The limit Λ = 0 corresponds to the classical limit Veff equiv Vcl AgainH0 and Ωde = 076 are chosen as in figure 43

power-law Vcl(φ) prop φminusα

zquant sim

(Mpl10

Λ(4πradic

2α(α +1))

) α+23(1+ωB)

is obtained by requiring a deviation of the effective potential of less than 1 and using the trackingsolution during matter and radiation domination with equation of state ωφ = α

α+2(1+ωB)minus1 [169]with ωB = 013 respectively For example assuming ΛsimMpl100 (where Mpl equiv 1

radicG) the track-

ing sets in at redshift zquant sim 300 for α = 2 and zquant sim 130 for α = 1 Similar bounds also holdfor other types of potentials eg like the SUGRA-potential [43] which are dominated by an inversepower-law behaviour at redshifts z 05 For values Λ amp Mpl10 there are large deviations fromthe tracking solution even at low redshifts and today as is shown in figure 43 for an exemplary casewith Vcl(φ) prop φminusα If the UV embedding scale Λ is of the order of the Planck scale there is a directtransition from the slow roll regime with φ Λ equation of state ωφ simminus1 and dark energy fractionΩφ ≪ 1 in the flattened effective potential Veff (φ) to the dark energy dominated accelerating solutionfor φ amp Mpl with Ωφ rarr 1 and ωφ rarrminus1 Thus the solution never performs tracking with ωφ =minus 2

α+2as for the classical potential Vcl(φ) In the case α = 1 the equation of state today ωde equiv ωφ (z = 0)is enhanced for 01 ΛMpl 13 compared to the tracking value and gets smaller for even larger7

Λ see figures 43 and 44 Moreover the sign of dωφdz can change depending on the value of theembedding scale Λ

7Note that even when Λ amp Mpl the pre-factor of the tadpole integral (44) is still sub-Planckian due to the loop factor116π2


42 Quantum Corrections from Matter Couplings

If the quintessence dynamics is governed by a low-energy effective theory which is determined byintegrating out some unknown high energy degrees of freedom involving eg quantum gravity stringtheory or supergravity [58 65] the low-energy theory should generically contain couplings and self-couplings of the quintessence field suppressed by some large scale eg the Planck scale In thissection radiative corrections induced by couplings between the quintessence field and ldquolow-energyrdquoparticle species will be investigated In this context ldquolow-energyrdquo stands for degrees of freedomwhich exist well below the UV embedding scale of the quintessence field including the well-knownStandard Model particlesOn the one hand such couplings can influence the properties of the Standard Model particles Therolling quintessence field can for example drive a time-variation of particle masses and couplingsover cosmological time-scales Quintessence models leading to time-varying Standard Model massesand couplings as well as mass-varying neutrinos (MaVaNs) have been frequently investigated seeeg [11 15 36 44 46 56 64 83 87 95 137 145 183 184 186] In some cases such couplings can bedirectly constrained observationally like for a coupling to Standard Model gauge fields [51] Forthe photon quintessence couplings can lead to tiny time-variations of the fine-structure constantαem [54176] and a coupling to the gluons could manifest itself by a tiny time-variation of the protonmass [119 158] over cosmic history Such time-variations can be tested observationally for exampleby comparing the frequency of spectral lines which depends on first and second powers of αem re-spectively from spectra emitted by quasars at various redshifts [54176] Other constraints arise fromthe impact of time-varying couplings and masses on Big Bang Nucleosynthesis [50 75 172] predic-tions Additionally the coupling to a light quintessence field mediates a gravity-like long range forceleading to tiny apparent violations of the equivalence principle [172 184] which is constrained byhigh-precision test of General Relativity [155 181] A significant interaction with dark matter is lessconstrained [13] and is considered in many contexts eg [14 96 117 189 190] often accompaniedby a varying dark matter mass (varying mass particles VAMP) [62 99 114 161]On the other hand the interactions of matter with the quintessence field can also influence the dy-namics of the quintessence field itself via the backreaction effect ie due to the contributions tothe equation of motion of the scalar field originating from its matter interaction [96] Illustrativelyclassical backreaction occurs due to a background matter density which the quintessence field feelsdue to the matter interaction As a consequence only the sum of the energy-momentum tensor ofthe quintessence field and of the interacting particles are conserved Such a backreaction effect mighttrigger the cross-over from matter domination to quintessence domination For example a couplingto neutrinos which leads to growing neutrino masses slows down the rolling quintessence field dueto the presence of the cosmic neutrino background If the increase of the neutrino masses becomesstrong enough the rolling quintessence field gets stopped and yields a cosmic expansion similar tothe cosmological constant which can be linked to the neutrino mass scale in specific models [11]Due to the presence of vacuum quantum fluctuations the interactions of the quintessence field leadto a backreaction effect even in the limit of vanishing background matter density For cosmologicalmatter densities it turns out that this ldquoquantum vacuumrdquo backreaction generically overwhelms theclassical backreaction for particle species much heavier than the dark energy scale around simmeV aswill be investigated in the following using the low-energy effective actionNote that the low-energy effective action as defined in appendix B1 captures quantum fluctuationsof (renormalizable) Standard Model degrees of freedom ie the quintessence field is treated as aclassical background field here Thus the opposite limit as in the previous section is taken wherethe impact of quantum fluctuations of the quintessence field itself has been investigated but mattercouplings have been assumed to be absent As discussed in appendix B1 the full quantum effective

42 Matter Couplings 45

action for a coupled quintessence field can be obtained in two steps by first calculating the low-energyeffective action by a path integral over the matter fields and then calculating the effective action bya path integral over the quintessence field This means if the low-energy effective action discussedhere is considered as the input for the ldquoclassicalrdquo action in the previous section one could recover inprinciple the full effective action for a coupled quintessence field8At lowest order in a derivative expansion of the low-energy effective action the quantum vacuumbackreaction is determined by the response of the quantum vacuum energy to variations of the quint-essence field value This response in turn is given by the quintessence-field-dependence of the low-energy effective potential obtained from integrating out all matter fields heavier than the quintessencefield

421 Quantum Backreaction

Generically the light classical mass m2φ(φ) = V primeprimecl(φ) of the quintessence field is unprotected against

huge corrections induced by quantum fluctuations of heavier degrees of freedom coupled to the quint-essence field (ldquohierarchy problemrdquo) Furthermore this is not only the case for the classical mass butalso for all higher derivatives V (k)

cl (φ) and the slope V primecl(φ) of the classical potential as well as thetotal potential energy Vcl(φ) The latter is the ldquoold cosmological constant problemrdquo which is not ad-dressed here As before the freedom to shift the effective potential by an arbitrary field-independentamount will be used instead such that the total effective potential energy today has the value requiredfor dark energy Furthermore if a huge amount of fine-tuning is accepted also the quintessence massand slope can be chosen to have the required values today by a suitable renormalization of the quan-tum fluctuations of (renormalizable) heavier degrees of freedom coupled to the quintessence field likethe Standard Model particles However even in this case there may still be huge corrections to theclassical potential and its derivatives evaluated at a quintessence field value which is slightly displacedfrom todays value Since the scalar field is rolling such corrections would affect the behaviour of thequintessence field in the past and could destroy some of the desired features (like tracking behaviour)of dynamical dark energy if they are too largeThe effective quintessence potential slope and mass are given by the first and second field derivativesof the low-energy effective quintessence potential respectively Their values today may be fixed byimposing renormalization conditions on the low-energy effective quintessence potential Even if theseare chosen such that the corrections to the quintessence potential are minimized today the quantumvacuum still leads to a remaining ldquominimal responserdquo on the dynamics of the quintessence field Inthe following the minimal response of one-loop quantum fluctuations of Standard Model particleson the quintessence field will be calculated It will be shown that the low-energy effective potentialcan be renormalized by imposing three independent renormalization conditions (linked to the quarticquadratic and logarithmic divergences) in this case The minimal response is obtained by choosingthe three renormalization conditions such that the quantum contributions to the low-energy effectivepotential Veff (φ) and its first and second derivative vanish today

Veff (φ = φ0) = Vcl(φ = φ0) V primeeff (φ = φ0) = V primecl(φ = φ0) (448)

V primeprimeeff (φ = φ0) = V primeprimecl(φ = φ0)

8This would require however to know details about the UV completion of the quintessence field combined with theStandard Model which imposes constraints on the combination of the field-dependence of the self-interactions and thefield-dependence of the couplings At the level of approximation represented by the low-energy effective action radiativecorrections induced by quintessence couplings can be investigated in a model-independent way ie no information aboutthe details of the unknown UV completion is required


where φ0 equiv φ(t0) is the quintessence field value today (t = t0) Here Vcl(φ) represents the (renor-malized) classical quintessence potential in terms of which the low-energy effective potential can beexpanded as

Veff (φ) = Vcl(φ)+V1L(φ)+

where V (φ)1L denotes the (renormalized) one-loop contribution Since the quintessence field generi-cally changes only slowly on cosmological time-scales one expects that the leading effect of quantumfluctuations is suppressed by a factor of the order

V primeprimeprimecl (φ = φ0)1L(φ(t0)∆t)3 (449)

with ∆t of the order of a Hubble time compared to the classical potential Vcl(φ)The coupling between quintessence and any massive particle species j is modeled by assuming ageneral dependence of the mass on the quintessence field This general form includes many interestingand potentially observable possibilities like a time-varying (electron- or proton-) mass m j(φ(t))a Yukawa coupling dm jdφ to fermions (eg protons and neutrons) mediating a new long-rangegravity-like force or a coupling between dark energy and dark matter (dm) of the form (see eg [13])

ρdm +3Hρdm = ρdmd lnmdm(φ)

dφφ (450)

In terms of particle physics a dependence of the mass on the dark energy field φ could be producedin many ways which are just briefly mentioned here One possibility would be a direct φ -dependenceof the Higgs Yukawa couplings or of the Higgs VEV For Majorana neutrinos the Majorana massof the right-handed neutrinos could depend on φ leading to varying neutrino masses via the seesawmechanism [107 186] The mass of the proton and neutron could also vary through a variationof the QCD scale for example induced by a φ -dependence of the GUT scale [185] Additionallya variation of the weak and electromagnetic gauge couplings could directly lead to a variation ofthe radiative corrections to the masses [81] Possible parameterizations of the φ -dependence arem(φ) = m0(1+β f (φMpl)) with a dimensionless coupling parameter β and a function f (x) of orderunity or m(φ) = m0 exp(βφMpl) [83]

One-loop low-energy effective potential

The one-loop contribution to the low-energy effective potential for the quintessence field can be cal-culated from the functional determinants of the propagators with mass m(φ) (see section B1)

V1L(φ) =12

int d4k(2π)4

(sumB

gB ln(k2 +mB(φ)2)minussumF

gF ln(k2 +mF(φ)2)

) (451)

where B and F run over all bosons and fermions with internal degrees of freedom gB and gF respec-tively The momentum has been Wick-rotated to Euclidean space To implement the renormalizationconditions (448) the following integrals are considered

I0(m2) equivint d4k

(2π)4 ln(k2 +m2) (452)

Il(m2) equivint d4k

(2π)41

(k2 +m2)l =(minus1)lminus1

(lminus1)dl

(dm2)l I0(m2)


which are finite for l ge 3 Following the procedure described in Ref [179] the divergences in I0 I1and I2 are isolated by integrating I3 with respect to m2 yielding

I0(m2) = 2int m2

dm23

int m23dm2

2

int m22dm2

1 I3(m21) + D0 +D1m2 +D2m4 (453)

with infinite integration constants D0 D1 and D2 Thus one is led to introduce three countertermsproportional to m0 m2 and m4 to cancel the divergences which can be easily reabsorbed by a shiftof the scalar potential Vcl(φ) This leaves a finite part Ifinite

0 of the same form as (453) but withthe three infinite constants replaced by three finite parameters that have to be fixed by the threerenormalization conditions (448) The appropriate choice can be expressed by choosing the lowerlimits in the integration over the mass m2 to be equal to its todays value m2

0

Ifinite0 (m2m2

0) = 2int m2

m20

dm23

int m23

m20

dm22

int m22

m20

dm21 I3(m2

1)

=1

32π2

(m4(

lnm2

m20minus 3

2

)+2m2m2

0minus12

m40

) (454)

where I3(m2) = 1(32π2m2) has been usedThus the renormalized one-loop contribution to the low-energy effective potential which fulfills therenormalization conditions (448) is uniquely determined to be

V1L(φ) =12

(sumB

gBIfinite0 (mB(φ)2mB(φ0)2)minussum

FgF Ifinite

0 (mF(φ)2mF(φ0)2)

) (455)

The higher loop corrections involve interaction vertices of the (Standard Model) matter particles Theone-loop result is exact in the limit of vanishing interaction strength Thus the best approximationto the full low-energy effective potential is obtained by applying the one-loop approximation to theeffective low-energy degrees of freedom of the Standard Model ie to nucleons instead of quarksThe low-energy effective potential renormalized in this way can be regarded as the result of a fine-tuning of the contributions from the quantum fluctuations of heavy degrees of freedom to the quint-essence potential energy slope and mass at its todays values ie evaluated for φ = φ0 Howeverwhen the quintessence field had different values in the cosmic history the cancellation does not occurany more and one expects the huge corrections of order m4 to show up again unless the coupling isextremely weak Indeed this argument yields extremely strong bounds for the variation of the masseswith the rolling field φ [20 81] To obtain a quantitative limit it is required that the one-loop contri-bution to the potential should be subdominant during the relevant phases of cosmic history up to nowwhich is taken to be of the order of a Hubble time in order to ensure that the quintessence dynamicseg tracking behaviour are not affected For the corresponding φ -values this means that

V1L(φ)Vcl(φ) (456)

is required If the one-loop effective potential (455) is Taylor-expanded around todays value φ0 thefirst non-vanishing contribution is by construction of third order

V1L(φ) asymp 13

V primeprimeprime1L(φ0)(φ minusφ0)3 asymp 13

132π2 sum

j

(minus1)2s j g j

m j(φ0)2

(dm2

j

dφ(φ minusφ0)

)3

asymp 196π2 sum

j(minus1)2s j g jm j(φ0)4

(d lnm2

j

d lnV primeprimeclln

V primeprimecl(φ)V primeprimecl(φ0)

)3

(457)


Here the index j runs over bosons B and Fermions F (with spin s j) and eq (454) has been used Inthe last line the dependence on the quintessence field φ has been rewritten as a dependence on itsmass m2

φequivV primeprimecl(φ) Today the mass is of the order of the Hubble constant H0 sim 10minus33eV For tracking

quintessence models [169] the quintessence mass also scales proportional to the Hubble parameterH during cosmic evolution Therefore it is assumed that

lnV primeprimecl(φ)V primeprimecl(φ0)sim lnH2H20 3ln(1+ z) (458)

In order to investigate under which conditions the inequality (456) is fulfilled up to a redshift zmax themost conservative assumption is to replace the logarithm in the last line in (457) by its maximal valueof order 3 ln(1+zmax) and the right hand side of (456) by the minimal value Vcl(φ0) Furthermore theinequality (456) is certainly fulfilled if each individual contribution to the one-loop potential (455)respects it Altogether under these assumptions the requirement (456) that the quintessence dynam-ics are unaltered up to a redshift zmax yields a bound for the variation of the mass m j of a species j(with g j internal degrees of freedom) with the quintessence mass scale V primeprimecl sim H2

∣∣∣∣∣d lnm2j

d lnV primeprimecl

∣∣∣∣∣ 13ln(1+ zmax)

(96π2Vcl(φ0)g jm j(φ0)4

) 13

(459)

This bound is the main result of this section It scales with mass like mminus43 ie the bound gets tighterfor heavier particles Inserting zmax sim zeq sim 103 and expressing the potential energy

Vcl(φ0) =1minusωde

2Ωde

3H20

8πG

in terms of the dark energy fraction Ωde and equation of state ωde with H0 sim 70kmsMpc yields∣∣∣∣∣d lnm2j

d lnV primeprimecl

∣∣∣∣∣(

1minusωde

2Ωde

07

) 13 1

3radicg j

(13meVm j(φ0)

) 43

(460)

Finally it should be remarked that there remains the possibility that several masses m j(φ) change insuch a way that the total contribution to the low-energy effective potential stays small [81] Generi-cally this would require an additional dynamical mechanism or symmetry which leads to such fine-tuned correlated changes at the required level The total low-energy effective action would thendepend on the details of such an unknown explicit mechanism presumably closely related to the UVembedding An example for such a mechanism could be based on supersymmetry where the massesof fermions and their superpartners would have to change in the same way if SUSY was unbrokenso that their contributions in eq (451) would always cancel However this is not the case below theSUSY breaking scale The bound (459) which applies for mass-variations with arbitrary relativesize for all species is independent of the details of the unknown UV completion

422 Bounds on Quintessence Couplings

The upper bound (459) can be directly related to upper bounds for the coupling strength to the long-range force mediated by the light scalar field and for cosmic mass variation The relative change ofthe mass m j since redshift z can be related to the derivative d lnm2

jd lnV primeprimecl using eq (458)

∆m j

m jasymp

d lnm2j



3ln(1+ z)d lnm2

j

d lnV primeprimecl (461)


ν

emicro

p b Zt

bound from radiative corrections

100

10-5

10-10

10-15

10-20

101210910610310010-3

∆mm

meV

typical range forbounds from obseg ∆αα

Figure 45 Bounds for cosmic mass variation since redshift zsim 2 from the radiative correction to thequintessence potential in dependence of the mass m The red (vertical) lines mark the masses of someStandard Model particles The limits inferred from observations eg of ∆αemαem strongly dependon the considered particle type and further assumptions but typically lie around 10minus4 to 10minus5 [172]

which means the bound (460) directly gives an upper limit for the relative mass variation of speciesj since redshift z For example for the variation of the electron mass since zsim 2 the upper bound

∆me

me 07 middot10minus11

(1minusωde

2Ωde

07

) 13

(462)

is obtained which is at least six orders of magnitude below direct observational constraints for achange in the electron-proton mass ratio [172] For heavier particles the bounds are even strongerby a factor (mem)43 see figure 45 eg of the order ∆mpmp 10minus15 for the proton It shouldbe emphasized that these upper bounds are valid under the assumption that the mass-variation isdriven by a rolling scalar field with tracker properties and in the absence of cancellations among thecontributions from different particle species In this case however the upper bound is a conservativeupper bound due to the renormalization conditions which correspond to the ldquominimal responserdquo Thismeans that for any other choice of renormalization conditions the upper bounds will be even stronger

The only known particles which could have a sizeable mass variation due to the bound (460) areneutrinos Thus models considering mass-varying neutrinos or a connection between dark energyand neutrinos (see eg [11 45 95]) are not disfavored when considering quantum fluctuations Ifthe bound (460) is saturated quantum backreaction effects are of the same order of magnitude asclassical backreaction effects and can have an impact on the quintessence dynamics in the recentpast where the turnover to a dark energy dominated cosmos occursFermions with quintessence-field-dependent masses are subject to a Yukawa-like interaction medi-ated by the quintessence field (ldquofifth forcerdquo) with typical range given by the inverse mass of thequintessence field mminus1

φsim Hminus1

0 and Yukawa coupling strength given by the derivative of the fermionmass [157]

y j =dm j(φ)

dφ


which can be described by a Yukawa potential (see section 23) Since this interaction leads to anapparent violation of the equivalence principle an upper bound on the effective quintessence Yukawacouplings for nucleons can be inferred [157] On the other hand for a rolling quintessence field thecoupling strength is constrained by the bound (459) via the relation

y j =dm j

dφ=

12

m jV primeprimeprimeclV primeprimecl

d lnm2j

d lnV primeprimeclequiv

m j

2M

d lnm2j

d lnV primeprimecl

where the scale height M equiv (d lnV primeprimecldφ)minus1 of the quintessence mass was introduced which is typi-cally of the order of the Planck scale today [169] For the proton and neutron an upper limit

ypn 04 middot10minus35(

Mpl

M

)(1GeVmpn

) 13(

1minusωde

2Ωde

07

) 13

(463)

is obtained which is far below the limit from the tests of the equivalence principle [157] see eq (217)These limits can be compared to the corresponding gravitational coupling given by m jMpl eg ofthe order 10minus19 for the nucleons Thus the bound in eq (460) also directly gives a bound for therelative suppression

β j equivy j

m jMpl=

d lnm j

d(φMpl)

of the coupling strength to the fifth force mediated by the quintessence field compared to the gravita-tional coupling giving (for M simMpl ωde +1 1 Ωde sim 07)

β j ∆m j

m j 4

(meVm j

)43

sim 10minus11(

me

m j

)43

(464)

Note that the bound from eq (463) also holds for other species (with mass-scaling sim mminus13) whosequintessence couplings are in general not constrained by the tests of the equivalence principle [157]This is also true for dark matter if it consists of a new heavy species like eg a weakly interactingmassive particle (WIMP) which severely constrains any coupling via a φ -dependent mass

ydm = dmdmdφ 10minus36 (TeVmdm)13

corresponding to a limit of the order

∆mdmmdm 10minus19 (TeVmdm)43

for a mass variation between zsim 2 and now from eq (464)

43 Gravitational Coupling 51

43 Quantum Corrections from Gravitational Coupling

Since any dynamical dark energy scenario is necessarily situated in a curved space-time setting forexample described by a Robertson-Walker metric it is important to study the quantum correctionson such a background In φ 4-theory one-loop radiative corrections induce a non-minimal coupling(NMC)

ξ Rφ22

between the curvature scalar R and the scalar field φ with a dimensionless coupling ξ [35] Even ifthe renormalization condition

ξ (micro0) = 0

is chosen at some renormalization point characterized by a scale micro = micro0 the corresponding renormalization-group improved effective action which is applicable at very different scales micro 6= micro0 contains a non-zero non-minimal coupling as described by the renormalization group running of ξ (micro) [92 116]For a scalar field with non-zero field expectation value φ the non-minimal coupling ξ Rφ 22 leads toa rescaling of the Newton constant G = Mminus2

pl (see section 23)

116πGeff (φ)

=1

16πG+

12

ξ φ2

where the effective Newton constant Geff (φ) appears in the gravitational force law for systems whichare small compared to the time- and space-scales on which φ = φ(x) varies A rolling quintessencefield with a non-minimal coupling which is linear in R thus leads to a time-variation of the (effective)Newton ldquoconstantrdquo on cosmic time-scales

∆Geff

Geffequiv

Geff (φ(t))minusGeff (φ(t0))Geff (φ(t0))

=minusξ

2(φ

2(t)minusφ2(t0)

)16πGeff (φ(t))

which is constrained by precision tests of General Relativity and Big Bang Nucleosynthesis [55155181]For tracking quintessence models the scalar field value today is of the order of the Planck scaleφ(t0)2 sim M2

pl = 1G Thus a non-minimal coupling of the form Rφ 2 potentially yields a largecontribution to the effective Newton constant unless the coupling ξ is small enough For inverse-power-law potentials constraints on the time-variation of the Newton constant lead to an upper limit|ξ | 3 middot10minus2 [55 155]Radiative corrections which lead to a non-minimal coupling of the form Rφ 2 as for the φ 4-theorycould thus lead to a conflict with experimental constraints on a time-varying Newton constant How-ever dynamical dark energy scenarios making use of a scalar field involve non-renormalizable inter-actions suppressed by some high-energy scale up to the Planck scale described by a tracker potentialVcl(φ) with properties which are very different compared to a φ 4-potential Therefore it is importantto include the non-renormalizable interactions in the investigation of radiatively induced non-minimalcouplings between the dark energy scalar field and gravityIn the following this analysis will be performed based on the semi-classical9 one-loop effective ac-tion on a curved background discussed in appendix B2 which is obtained using Heat Kernel Expan-sion [35] and zeta-function regularization [91 110]

9The metric gmicroν (x) is treated as a classical background field in this approach


431 Radiatively induced Non-minimal Coupling for φ 4-Theory

The action of a scalar field in curved space-time with standard kinetic term

S[φ gmicroν ] =int

d4xradicminusg(

12

gmicroνpartmicroφpartνφ minusVcl(φ)

) (465)

contains minimal couplings to the metric via the integration measure and the contraction of the space-time derivatives in the kinetic term required by general coordinate invariance In quantum field theoryradiative corrections to the classical action furthermore lead to additional non-minimal couplings togravityBefore investigating non-minimal couplings for a quintessence theory the calculation of radiativecorrections in curved space-time will be reviewed for a theory described by the φ 4-potential

Vcl(φ) = Λ+m2φ

22+λφ44

in order to compare the generalized formalism discussed in appendix B which is also suitable forthe quintessence case with known results The minimal scalar action in curved space-time which isstable under one-loop quantum corrections is [35 92 116]


d4xradicminusg L(φ(x)gmicroν(x)) (466)

=int

d4xradicminusg(

12

gmicroνpartmicroφpartνφ minusV (φ R)+ ε1C + ε2G+2B(φ R)

)

where

V (φ R) = Vcl(φ)+12

ξ Rφ2 +

R16πG

+ ε0R2 (467)

B(φ R) = ε3φ2 + ε4R

C = Rmicroνρσ Rmicroνρσ minus2RmicroνRmicroν +R23

G = Rmicroνρσ Rmicroνρσ minus4RmicroνRmicroν +R2

with dimensionless constants εi and including the Einstein-Hilbert term linear in R10 The necessityto include all the upper terms can be seen from the renormalization group improved effective actionwhich arises in the following way Assume that some given approximation to the effective actioncontains parameters which can describe the dynamics around a typical energy scale micro0 At anotherenergy scale micro 6= micro0 radiative corrections may change the effective values of these parameters asdescribed by the renormalization group Then the renormalization group improved effective action isan improved approximation to the effective action where the running of the parameters is incorporatedsuch that it is applicable also at scales micro 6= micro0 (see appendix B)The renormalization-group improvement of the one-loop effective action (ldquoleading logarithm approx-imationrdquo) is accomplished by starting with the classical action at the reference scale micro0 and takingthe running into account as described by the renormalization group equations obtained from the one-loop approximation As shown in appendix B the renormalization-group improved effective actionin leading logarithm approximation for a scalar in curved space and for the renormalization schemediscussed in section B2 is

ΓLL[φ gmicroν micro] =int

d4xradicminusg(

12

gmicroνpartmicroφpartνφ minusVLL(φ R micro) (468)

+ ε1(micro)C + ε2(micro)G+2BLL(φ R micro))

10The latter two terms are total derivatives and thus not relevant for the dynamics but they are needed for the cancellationof divergences and do appear in the dynamics if their running is considered [92 116]


where for φ 4-theory it is possible to make the ansatz

VLL(φ R micro) = Λ(micro)+m2(micro)

2φ

2 +λ (micro)

4φ

4 +12

ξ (micro)Rφ2 +

R16πG(micro)

+ ε0(micro)R2

BLL(φ R micro) = ε3(micro)φ 2 + ε4(micro)R

Inserting the ansatz into the partial differential equations (B19) for VLL and BLL yields

partVLL

part t=

dΛ

dt+

12

dm2

dtφ

2 +14

dλ

dtφ

4 +12

dξ

dtRφ

2minus R16πG2

dGdt

+dε0

dtR2

=1

64π2

(part 2VLL

partφ 2 minusR6

)2

=1

64π2

(m(micro)2 +

λ (micro)2

φ2 +(

ξ (micro)minus 16

)R)2

part2BLL

part t=

dε3

dt2φ

2 +dε4

dt2R

=1

192π2

(part 22VLL

partφ 2 minus 2R5

)=

1192π2

(λ (micro)

22φ

2 +(

ξ (micro)minus 15

)2R)

where t = ln(micro2micro20 ) By comparing the coefficients of the terms proportional to φ 2 φ 4 Rφ 2 R

R2 and φ 0R0 = const in the two upper expressions for partVLLpart t and the coefficients of the termsproportional to 2φ 2 and 2R in the two upper expressions for part2BLLpart t the one-loop renormalizationgroup equations for φ 4-theory in curved space [92 116] within the renormalization scheme fromsection B2 are obtained

dλ

dt=

3λ 2

32π2 dm2

dt=

λm2

32π2

dGdt

= minus8πG2m2(ξ minus 1

6)32π2

dΛ

dt=

m4

64π2

dξ

dt=

λ (ξ minus 16)

64π2 dε0

dt=

(ξ minus 16)2

64π2

dε1

dt= minus 1

120 middot32π2 dε2

dt= minus 1

360 middot32π2

dε3

dt=

λ

12 middot32π2 dε4

dt=

ξ minus 15

6 middot32π2

(469)

where the β -functions from eq (B17) for the parameters ε1 and ε2 were also included The firstline which describes the running of the quartic coupling and the mass is identical to the MS resultin flat space The second line describes the running of the Newton- and the cosmological constantsThe running of the non-minimal coupling ξ is given in the third line along with the running ofthe coefficients of higher curvature scalars whose presence in the action leads to modifications ofstandard General Relativity For non-zero quartic coupling λ the renormalization group equation forthe non-minimal coupling ξ has no fixed point at ξ = 0 Thus even if the renormalization conditionξ (micro0) = 0 is imposed at the reference scale micro = micro0 a radiatively induced non-minimal coupling isgenerated in the renormalization-group improved effective action applicable at other scales micro 6= micro0For generic values λ 6= 0 m2 6= 0 and ξ 6= 16 the same is true for all the running parameters forwhich reason the action (466) is indeed the minimal scalar action in curved space which is stableunder one-loop renormalization group running Note that the fixed point ξ = 16 of the non-minimalcoupling corresponds to the value of ξ for which the classical action is conformal invariant in thelimit mΛGminus1rarr 0 [35]


432 Radiatively induced Non-minimal Coupling for Quintessence

In order to study radiatively induced non-minimal couplings for a quintessence field it is desirable togeneralize the renormalization group equations to general scalar potentials Vcl(φ) for which effectivefield theory is applicable Within effective field theory ultraviolet divergences are absent since thetheory is only valid up to the UV embedding scale Λ Nevertheless for a given approximation to theeffective action within effective field theory which can describe the dynamics around an energy scalemicro0 Λ radiative corrections can lead to a rescaling of the effective parameters at different scalesmicro 6= micro0 micro Λ Similarly as before this scale-dependence can be incorporated in a renormalizationgroup improved effective action which yields generalized renormalization group equations for aneffective field theory below the embedding scaleFor a quintessence field the UV embedding scale is typically of the order of the Planck or the GUTscale whereas the dynamical scale is of the order of the Hubble scale micro simH(t) ≪ Λ If it is assumedfor example that non-minimal gravitational couplings of the quintessence field are absent for somereference scale micro0 sim H(t0) ≪ Λ non-minimal couplings can be generated radiatively at differentscales micro sim H(t) Since the dynamical scale H(t) changes (slowly) in cosmic history radiativelygenerated non-minimal couplings could manifest themselves as described above by a time-variationof the effective Newton constant In general non-minimal couplings between the field φ and thecurvature scalar R which are linear in R ie of the form f1(φ)R with some (scale-dependent) functionf1(φ) lead to an effective Newton ldquoconstantrdquo

116πGeff (φ)

=1

16πG+ f1(φ)

which varies over cosmic time-scales due to the rolling quintessence field φ(t) Such a time-variationis constrained observationally between Big Bang Nucleosynthesis (BBN) H(tBBN) sim T 2

BBNMpl sim10minus15eV and today H0 sim 10minus33eV to be less than sim 20 [181] Therefore it is important that radia-tively induced non-minimal couplings from renormalization group running between these scales donot violate this bound Since both scales are far below the UV scale Λ and far below any other thresh-olds of known particle masses one may focus on the logarithmic scale dependence sim ln(micro2micro2

0 )as described by the renormalization group derived from the one-loop β -functions obtained via zeta-function regularization [110] in curved space (see appendix B)In the following it will be shown that the minimal scalar action in curved space-time with generalscalar potential Vcl(φ) which is stable under one-loop quantum corrections has the same form asfor φ 4-theory see eq (466) however with a ldquogeneralized potentialrdquo V (φ R) and a function B(φ R)with a more general dependence on φ and R In order to capture radiatively induced non-minimalcouplings involving higher powers of φ and R the ansatz

V (φ R) = sumnm

cnmφnRm (470)

B(φ R) = sumnm

cnmφnRm

is made with coefficients cnm and cnm respectively This ansatz is possible for all functions which canbe written as a Taylor series around φ = 0 and R = 0 Equivalently it is possible to expand around anyother values φ = φ0 and R = R0 if necessary Since the final result does not depend on the choice ofthe expansion point it is set to zero for simplicity It should be emphasized however that the result isapplicable to all theories where V (φ R) and B(φ R) including especially the potential Vcl(φ) possessTaylor expansions around at least one arbitrary expansion point which does not necessarily have tobe at φ = R = 0 The generalized potential V (φ R) and the function B(φ R) from eq (467) for


φ 4-theory correspond to the choice

c00 = Λ c20 =m2

2 c40 =

λ

4 c21 =

ξ

2 c01 =

116πG

c02 = ε0 c20 = ε3 c01 = ε4

The one-loop effective action for the action given in eq (466) with V (φ R) and B(φ R) parameterizedas in the ansatz (470) has been derived in appendix B2 Inserting the first three terms of the HeatKernel Expansion (B15) into eq (B14) yields

Γ[φ gmicroν ]1L =int d4x

32π2

radicminusg[minus (XminusR6)2

2

(ln

XminusR6micro2 minus 3

2

)(471)

minus(

1120

Cminus 1360

Gminus 130

2R+162X)

lnXminusR6

micro2 +infin

sumj=3

g j(xx)( jminus3)(XminusR6) jminus2

]equiv Γ1L[φ gmicroν micro]+Γ1LHD[φ gmicroν ]

whereX = X(φ R) = part

2V (φ R)partφ2

and micro is the renormalization scale In the last line of eq (471) the contribution Γ1LHD[φ gmicroν ] isdefined which contains the sum over the higher terms of the Heat Kernel Expansion ( j ge 3) Theseinvolve curvature scalars built from higher powers of the curvature tensor and higher derivative termswhich are independent of the renormalization scale (see appendix B2 and Ref [121]) In contrast tothis the first two terms (which correspond to j = 02 see eq(B15)) denoted by Γ1L[φ gmicroν micro] dodepend on micro In appendix B the renormalization group improved effective action for the one-loop effective ac-tion (471) was derived It has a similar form as for φ 4-theory see eq (468) However it containsa renormalization group improved ldquogeneralized potentialrdquo VLL(φ R micro) and a function BLL(φ R micro)with a more general dependence on φ and R compared to φ 4-theory The scale-dependence of VLL

and BLL is determined by the partial differential equations (see eq (B19) t = ln(micro2micro20 ))

part

part tVLL(φ R micro) =

164π2

(part 2VLL(φ R micro)

partφ 2 minus R6

)2

VLL(φ R micro0) = V (φ R)

part

part t2BLL(φ R micro) =

1192π2


partφ 2 minus 2R5

) 2BLL(φ R micro0) = 0

This result is indeed independent of the choice of the expansion point in eq (470) The running ofthe parameters ε1(micro) and ε2(micro) in the action (468) is identical to that of φ 4-theory (see eqs (B17)and (469))In order to investigate the radiatively induced non-minimal couplings the ldquogeneralized potentialrdquoVLL(φ R micro) is expanded in powers of R

VLL(φ R micro) = f0(φ micro)+ f1(φ micro)R+ f2(φ micro)R2 + middot middot middot

As discussed above the non-minimal coupling of the form f1(φ micro)R which is linear in R resultsin a time-variation of the effective Newton constant The partial differential equation determiningVLL(φ R micro) yields a hierarchy of partial differential equations for fk(φ micro) |0le kle N The lowesttwo are

part

part tf0(φ micro) =

164π2

(part 2 f0(φ micro)

partφ 2

)2

f0(φ micro0) = Vcl(φ) (472)

part


132π2

part 2 f0(φ micro)partφ 2


partφ 2 minus 16

) f1(φ micro0) = f1(φ)


The renormalization group equation in the first line describes the running of the quintessence po-tential and the second line yields the running of the non-minimal coupling which is linear in R(ldquoNMCrdquo) The renormalization group equations for φ 4-theory are recovered by inserting f0(φ micro) =Λ(micro) + m2(micro)φ 22 + λ (micro)φ 44 and f1(φ micro)R = R(16πG(micro)) + ξ (micro)Rφ 22 It is emphasizedthat in general the functional dependence of f0(φ micro) and f1(φ micro) on the field is only subject to therestriction that it can be written as a Taylor series around some field value φ = φ0 which need notnecessarily be φ0 = 0 The partial differential equation for BLL(φ R micro) can be decomposed similarlyby an expansion in RHere it is demanded that the potential is given by a (tracker) quintessence potential Vcl(φ) at thereference scale micro0 Furthermore a renormalization condition f1(φ micro0) = f1(φ) is imposed on thenon-minimal coupling parameterized by the function f1(φ) If

part f1(φ micro0)partφ = part f1(φ)partφ equiv 0 (mNMC) (473)

is set ie f1(φ micro) equiv const then the quintessence field is minimally coupled at the reference scalemicro0 (eg micro0 sim H(tBBN)sim 10minus15eV) Note that the partial differential equation describing the runningof f1(φ micro) does not have a fixed point at f1(φ micro)equiv const Therefore the renormalization group im-proved effective action contains a non-vanishing NMC at all scales micro 6= micro0 (eg micro sim H0 sim 10minus33eV)even though part f1(φ micro0)partφ equiv 0 which is purely generated by radiative corrections Since this non-minimal gravitational coupling is unavoidably present in the theory it is denoted by mNMC (ldquomini-mal NMCrdquo)Note that the scale-dependence of the functions f0(φ micro) and f1(φ micro) already includes the runningof the ldquocosmological constantrdquo Λ(micro) equiv f0(φ micro)|φ=0 and the ldquoNewton constantrdquo 1(16πG(micro)) equivf1(φ micro)|φ=0 respectively In fact the non-minimal coupling11 f1(φ(t) micro(φ(t))) for a rolling fieldφ(t) evaluated with a renormalization scale of the order of the dynamical scale of the quintessencefield micro2(φ(t)) sim m2

φ(φ(t)) encodes the time-variation of the effective Newton ldquoconstantrdquo (which is

relevant for astrophysical and laboratory measurements since it appears in the gravitational force law)

116πGeff (φ(t))

=1

16πG+ f1(φ(t) micro(φ(t)))

caused by both the renormalization group running and the rolling quintessence field in a unifiedmanner12 It is emphasized that the choice of the renormalization scale micro is not free here but is fixedby the matching of the renormalization group improved effective potential with the one-loop effectivepotential (see appendix B and Ref [60])

micro2(φ) equiv V primeprimecl(φ)+

(ξ0minus

16

)R (474)

=[

92

Γ

(1minusω

lowastφ

2)

+9(

ξ0minus16

)(ωBminus

13

)]H2

prop H2

where the renormalization condition f1(φ micro0) = ξ0φ 22 + const has been inserted as an exampleas well as the dynamical mass (213) of a tracker quintessence potential Vcl(φ) and the curvaturescalar R of a FRW solution with ωB = 013 during matterradiation domination The mNMC (473)corresponds to the choice ξ0 = 0

11For the rolling quintessence field φ(t) t denotes the time12Similarly the time-variation of the effective energy density ρφ = 1

2 φ 2 + f0(φ(t) micro(φ(t))) encodes the time-variationof dark energy caused by both the rolling quintessence field and the renormalization group running of the cosmologicalconstant due to quantum fluctuations of the quintessence field in a unified manner However the latter is negligible here(see below)


Finally note that the renormalization group equation (472) for the non-minimal coupling f1(φ micro)Rhas fixed-points of the form

f1(φ micro) = f lowast1 (φ)equiv 116πG

+bφ +12

ξlowastφ

2

for the ldquoconformal couplingrdquo ξ lowast = 16 and arbitrary constant values G and b

433 Robustness of Quintessence Actions

The impact of radiative corrections which are not encoded in the effective potential ie non-minimalgravitational couplings and corrections to the kinetic term on tracker quintessence fields will now beinvestigated Therefore the results of the previous section are applied to a quintessence field withclassical action containing a tracker potential Vcl(φ) characterized by the power-counting rules (47)

Linear non-minimal gravitational coupling

The renormalization group improved effective action contains the scale-dependent ldquogeneralized po-tentialrdquo VLL(φ R micro) = sum

infink=0 fk(φ micro)Rk which simultaneously encodes the renormalization group

running of the potential f0(φ micro) and all non-minimal couplings between the field φ and the curva-ture scalar R in leading logarithm approximation It is determined by the partial differential equa-tion (B19) which can be decomposed into a hierarchy of partial differential equations for the contri-butions fk(φ micro) see eq (472)For scales where |t|= | ln(micro2micro2

0 )| 32π2 the solution of the renormalization group equations (472)for fk(φ micro) (k = 01) in linear approximation is

f0(φ micro) = Vcl(φ)+t

64π2

(V primeprimecl(φ)

)2+O

( t32π2

)2 (475)

f1(φ micro) = f1(φ)+t

32π2V primeprimecl(φ)(

f primeprime1 (φ)minus 16

)2

+O( t

32π2

)2

For example for the running between the Big Bang Nucleosynthesis era micro0 sim H(tBBN) sim 10minus15eVand today micro sim H0 sim 10minus33eV |t|(32π2) asymp 026 According to the power counting rules (47) therunning of the quintessence potential is completely negligible since the scale-dependent part propor-tional to V primeprimecl(φ)2 sim Vcl(φ)(Vcl(φ)M4) is suppressed by the tiny factor Vcl(φ)M4 ≪ 1 which is ofthe order 10minus120 today compared to the classical potential This is in agreement with the suppressionof logarithmic corrections with respect to the UV scale found in section 41Assuming for example that the non-minimal coupling at the reference scale is quadratic in the fieldf1(φ micro0) = f1(φ) = ξ0φ 22+ const the radiative correction to the non-minimal coupling is

f1(φ micro) = f1(φ micro0)+t


ξ0minus16

)2

+O( t

32π2

)2 (476)

The combined effect of the rolling quintessence field and the running non-minimal coupling thusleads to a time-variation of the effective Newton constant given by

∆Geff

Geff=


= minus(

f1(φ(t) micro)minus f1(φ(t0) micro0))

16πGeff (φ(t))

= minus ξ0

2

(φ

2(t)minusφ2(t0)

)16πGeff (φ(t))

minus 132π2 ln

(micro2(φ(t))

micro20

)V primeprimecl(φ(t))

(ξ0minus

16

)2

16πGeff (φ(t))


where the renormalization scale is given by eq (474) The first contribution is the classical contri-bution and the second is the one induced by radiative corrections Even if the non-minimal couplingat the reference scale micro0 vanishes ie ξ0 = 0 radiative corrections induce a non-minimal coupling(ldquomNMCrdquo) which leads to a time-variation of the effective Newton constantFor tracker quintessence fields the time variation of the effective Newton constant between BBN andtoday is (Geff equiv Gobs = 1M2

pl ∆φ 2 equiv φ 2(t)minusφ 2(t0))

∆Geff

Geffasymp minus8πξ0

∆φ 2

M2plminus 1

32π2 ln(

H20

H2BBN


H20

(ξ0minus

16

)2 16πH20

M2pl

The first term on the right-hand side is the classical contribution It vanishes if the quintessence fieldis minimally coupled at the reference (BBN) scale ie in the limit ξ0 rarr 0 The second term onthe right-hand is the quantum contribution It denotes the non-minimal coupling which is generatedradiatively between the reference scale and todayThe agreement between the abundances of light elements and predictions from BBN lead to the upperbound ∆Geff Geff 20 [181] Since the rolling quintessence field is of the order of the Planck scaletoday ∆φ 2M2

pl can be of order one Therefore the BBN bound yields restrictive upper bounds on|ξ0| 005 [55155] However the radiatively induced contribution to the non-minimal coupling (themNMC) is suppressed by the tiny factor H2

0 M2pl Therefore if the non-minimal coupling |ξ0| is small

enough at the BBN scale tracker quintessence models are robust against radiative corrections to thenon-minimal coupling between the BBN scale and todayNote that the linear approximation in t to the solutions (475) of the renormalization group equationshas to be extended if the scope of the running is enlarged for example to be between the GUTscale and today Using the power-counting rules (47) for tracker potentials it is found that thecoefficients of the contributions proportional to higher powers of t(32π2) are highly suppressed bypowers of Vcl(φ)M4 sim Vcl(φ)M4

pl However it is also possible to show that for specific exampleseg Vcl(φ) prop exp(minusλφMpl) the expansion in powers of t is an asymptotic expansion in which casea non-perturbative treatment is obligatory for |t|(32π2)rarr infin

Nonlinear non-minimal gravitational coupling

Apart from the non-minimal coupling which is linear in the curvature scalar R the scale-dependentldquogeneralized potentialrdquo VLL(φ R micro) = sum

infink=0 fk(φ micro)Rk also encodes the running of non-minimal cou-

plings fk(φ micro) between the scalar field and higher powers of R for k ge 2The presence of nonlinear terms in the curvature scalar leads to modifications of General Relativitywhich are suppressed if their contribution to the action is suppressed with respect to the Einstein-Hilbert term [12] This is the case if fk(φ micro) M2

plR1minusk for all relevant values of the curvaturescalar R For cosmology the curvature scalar is of the order of the Hubble scale Rsim H2The running of the non-minimal coupling f2(φ micro)R2 as obtained from eq (B19) is given by thepartial differential equation

part


164π2

[2


part 2 f2(φ micro)partφ 2 +


partφ 2 minus 16

)2]

f2(φ micro0) = f2(φ) (477)

For φ 4-theory f2(φ micro)equiv ε0(micro) does not explicitly depend on φ The running of the coupling ε0(micro)in φ 4-theory is recovered by inserting part 2 f2(φ micro)partφ 2 = 0 and part 2 f1(φ micro)partφ 2 = ξ (micro)


In order to estimate the radiatively induced non-minimal coupling prop R2 the initial conditions

f2(φ micro0) = f2(φ)equiv ε0 equiv const and f1(φ micro0) = f1(φ)equiv ξ0φ22+ const

are assumed With this choice the field is minimally coupled at the reference scale micro = micro0 for ξ0 = 0The approximate solution of the renormalization group equation is

f2(φ micro) = ε0 +t

64π2

(ξ0minus

16

)2

+12

( t32π2

)2V (4)

cl (φ)(

ξ0minus16

)3

+O( t

32π2

)3

Up to linear order in t = ln(micro2micro20 ) f2(φ micro) does not explicitly depend on φ similar to φ 4-theory

A non-minimal coupling prop V (4)cl (φ)R2 arises at order t2 which is extremely suppressed by the factor

V (4)cl (φ)simVcl(φ)M4 for a tracker potential

For a potential Vcl(φ) involving higher-dimensional operators radiative corrections also induce non-minimal couplings between the field and higher powers Rk k ge 3 of the curvature scalar For ex-ample for a potential which contains a dimension six (or higher) operator a radiatively inducednon-minimal coupling prop V (6)

cl (φ)R3 arises at order t3

f3(φ micro) =13

( t32π2

)3V (6)

cl (φ)(

ξ0minus16

)4

+O( t

32π2

)4

where f3(φ micro0) = 0 was assumed For a tracker potential this is extremely suppressed compared tothe linear term prop R(16πG)sim RM2

pl since

V (6)cl (φ)R3(RM2

pl)sim (M2M2pl) middot (Vcl(φ)M4) middot (R2M4)

where Rsim H2 and M simMpl

Kinetic term

The one-loop effective action (471) contains apart from one-loop non-minimal gravitational cou-plings also the one-loop higher-derivative contributions to the effective action The first contributionto the derivative expansion (318) has the form of a modification of the kinetic term Z(φ)(partφ)22 Inthe flat space-time limit the one-loop contribution obtained from the Heat Kernel Expansion (471)is

Γ[φ ηmicroν ]1L =int d4x

32π2

[minusV1L(φ)minus

(162X)

lnXmicro2

+(minus 1

12partmicroXpart

microXminus 160

22X)

1X

+infin

sumj=4

g j(xx)( jminus3)X jminus2

]

=int d4x

32π2

[minusV1L(φ)+

112X

partmicroXpartmicroX +O

(part

4)]=

int d4x32π2

[minusV1L(φ)+

12

Z1L(φ)(partφ)2 +O(part

4)]

where the third coefficient of the Heat Kernel Expansion g3(xx) (see Ref [121]) was inserted in theMinkowski limit in the second line The one-loop correction to the kinetic term is thus given by

Z(φ) = 1+Z1L(φ) Z1L(φ) =[V primeprimeprimecl (φ)

]2V primeprimecl(φ)

It is independent of the renormalization scale micro in accordance with the vanishing anomalous dimen-sion see eq (B17) For a tracker potential the one-loop correction to the kinetic term is suppressedby the factor Z1L simV primeprimeprimecl (φ)2V primeprimecl(φ)simVcl(φ)M4 ≪ 1 compared to the classical value Z = 1


44 Summary

In this chapter quantum corrections to quintessence models have been investigated These provide aform of dynamical dark energy for which an extremely light rolling scalar field is responsible for thepresent cosmic acceleration similar to the inflaton in the early universe

First an approximation scheme suitable to investigate the impact of quintessence self-couplings onthe shape of the effective potential has been introduced An additive constant has been fine-tunedto be zero thus bypassing the unresolved ldquocosmological constant problemrdquo It has been shown thatthe quantum corrections to the scalar potential can be self-consistently calculated in leading order inV primeprime(φ)Λ2 Hereby Λ denotes the embedding scale characteristic for an underlying theory and V primeprime(φ)denotes the square of the quintessence mass which is of the order of the Hubble parameter for track-ing solutions While potentials involving exponentials just get rescaled inverse power law potentialsare flattened at small field values The effective potential approaches a finite maximum value thustruncating the singular behaviour of the inverse power law This distortion of the potential directlyplays a role cosmologically if Λ is large roughly Λ amp Mpl10 and affects general properties liketracking behaviour

Second couplings between the quintessence field and heavier degrees of freedom like the StandardModel fermions or dark matter have been discussed The discussion has been constrained to cou-plings that can effectively be written in the form of quintessence-field-dependent mass terms Thequantum corrections induced by these couplings have been described by the low-energy effectiveaction obtained from integrating out the Standard Model degrees of freedom An upper bound forthe couplings was quantified under the assumption that fine-tuning in the form of renormalizationconditions for the low-energy effective potential is admitted This fine-tuning was used to minimizethe quantum corrections in the present cosmological epoch The remaining corrections constitute theminimal quantum vacuum backreaction of the Standard Model fields on the dynamics of the quint-essence fieldNext the upper bounds on the couplings have been translated into upper bounds for potentially ob-servable effects like tiny time-variations of particle masses between redshift z sim 2 and now or tinyapparent violations of the equivalence principle Note that it has been assumed that the mass varia-tions are uncorrelated In this case they are constrained to be far below observational bounds for allStandard Model particles The latter are of the order |∆mm| 10minus5 [119158] However it has beenfound that massive neutrinos can have large relative mass variations of order one The bound can beavoided for correlated mass variations of different species which are finely tuned in such a way thattheir quintessence-field-dependent contributions to the vacuum energy cancel

Third non-minimal gravitational couplings induced by quantum corrections have been investigatedFor φ 4-theory a non-minimal coupling of the form φ 2R is induced by radiative corrections in theeffective action where R denotes the curvature scalar For a tracker potential however all couplingsof the form φ nRm with integers n and m have to be included at one loop level and will be inducedby quantum corrections unless the field is exactly conformally coupled Potentially non-minimalcouplings of the quintessence field can lead to conflicts with tests of General Relativity However fortracker potentials it has been shown that the radiatively induced non-minimal couplings as obtainedfrom the one-loop renormalization group analysis are suppressed by powers of H2M2

pl ≪ 1 andtherefore do not lead to sizeable deviations from General Relativity

Chapter 5

Leptonic Dark Energy and Baryogenesis

Scalar fields with time-dependent expectation value are not only considered in quintessence modelsbut are commonly invoked in cosmology above all to describe the inflationary phase [108] of theearly universe Furthermore rolling fields are the basis of a number of baryogenesis models [8 78]and also play an important role in the context of a possible time-variation of fundamental constantsover cosmological time-scales [172] Due to the similarity of the underlying concepts it is an inter-esting question whether some of these issues could be related This has been studied for example forthe early- and late time acceleration called quintessential inflation [154] or for the combination ofspontaneous lepto- and baryogenesis with quintessence [138 187] and quintessential inflation [72]Here a toy model is discussed where baryogenesis and cosmic acceleration are driven by a leptonicquintessence field coupled indirectly to the Standard Model sector via a massive mediating scalarfield It does not require the introduction of new interactions which violate baryon (B) or lepton (L)number below the inflationary scale Instead a BminusL-asymmetry is stored in the quintessence fieldwhich compensates for the corresponding observed baryon asymmetry

51 Quintessence and Baryogenesis

Complex scalar fields have been discussed as candidates for dynamical dark energy [40 106] whichoffer the possibility that the field carries a U(1)-charge [8 78] and thus could itself store a baryon orlepton density [23] This approach can very well be accommodated within the so-called ldquobaryosym-metric baryogenesisrdquo [79 80] scenario where one attempts to explain the overabundance of matterover antimatter without postulating new baryon- or lepton number violating interactions neverthelessstarting with no initial asymmetry This requires the introduction of an invisible sector in which anasymmetry is hidden that exactly compensates the one observed in the baryon (and lepton) sectorthereby circumventing one of the Sakharov conditions [163] Here a possible realization is discussedwhere the anomaly-free combination BminusL is conserved below the inflationary scale and the invisiblesector which compensates for the BminusL-asymmetry of the Standard Model (SM) baryons and leptonsis leptonic dark energy [23 103] For other realizations involving dark matter or neutrinos see egRefs [77 79]

Toy Model

In this section the question is addressed of how BminusL-asymmetries in the dark energy sector real-ized by a complex quintessence field charged under BminusL and in the SM sector can be created by adynamical evolution within an underlying BminusL-symmetric theory For this it is necessary to con-

62 5 Leptonic Dark Energy and Baryogenesis

sider a suitable interaction between both sectors Direct couplings between the quintessence field andSM fields are commonly investigated for example in the context of time-varying coupling constantsandor -masses [172] or violations of the equivalence principle [157] which leads to strong constraintsin the case of a coupling eg to photons or nucleons [51 102 157] (see also section 42) Here a toymodel is discussed where it is assumed that direct interactions between the quintessence field φ andthe SM are sufficiently suppressed allowing however an indirect interaction mediated by a ldquomediat-ing fieldrdquo χ which couples to φ and the SM In the late universe the χ-interactions should freeze outThis means that the massive scalar χ is hidden today and also that the transfer of asymmetry betweenthe quintessence and the SM sector freezes out Thus once an asymmetry has been created in eachsector in the early universe it will not be washed out later on In the specific setup considered herethe quintessence field is taken to carry lepton number minus2 so that it carries a BminusL-density given by

nφ =minus2|φ |2θφ (with φ equiv |φ |eiθφ ) (51)

and analogously for the mediating field χ which carries the same lepton number The effective toy-model Lagrangian for φ and χ is

L =12(partmicroφ)lowast(part micro

φ)minusV (|φ |)+12(partmicro χ)lowast(part micro

χ)minus 12

micro2χ |χ|2

minus12

λ1|φ |2|χ|2minus14

λ2(φ 2χlowast2 +hc)+LSM(χ )

with dimensionless coupling constants λ1 gt 0 and λ2 lt λ1 responsible for the coupling betweenthe quintessence and the mediating field This Lagrangian has a global U(1)-symmetry under acommon phase rotation of φ and χ which corresponds to a BminusL-symmetric theory The coupling ofthe mediating field to the SM encoded in the last contribution should also respect this symmetry Thisis compatible eg with a Yukawa-like coupling of the form LSM 3 minusgχνc

RνR + hc to right-handedneutrinos see Ref [23] For the quintessence potential an exponential potential of the form [21 98157 182]

V (|φ |) = V0

(eminusξ1|φ |Mpl + keminusξ2|φ |Mpl

)(52)

is assumed which leads to tracking of the dominant background component and a crossover towardsan accelerating attractor at the present epoch for ξ1

radic3 ξ2 and a suitable choice of k [21] For the

dynamics in the early universe one can safely neglect the second term Since the vacuum expectationvalue (VEV) of φ increases and typically |φ |amp Mpl today the effective mass m2

χ asymp micro2χ +λ1|φ |2 of the

mediating field gets huge and the field indeed decouples the quintessence and the SM sectors in thelate universe However before the electroweak phase transition the dynamics of φ and χ can lead toa creation of the baryon asymmetry

52 Creation of a BminusL-Asymmetry

To study the evolution of the scalar fields φ and χ in the early universe it is described by a flatFRW metric after the end of inflation with a Hubble parameter H = Hinf and with VEVs φ = φ0 andχ = χ0eminusiα0 inside our Hubble patch which are displaced by a relative angle α0 in the complex planeThese initial conditions correspond to dynamical CP violation if sin(2α0) 6= 0 which is necessary forthe formation of an asymmetry [1980] Under these conditions the fields start rotating in the complexplane and thus develop a BminusL-density see eq (51) This asymmetry is then partially transferred tothe SM by the BminusL-conserving decay of the χ-field into SM particles leading to a decay term for the

52 Creation of a BminusL-Asymmetry 63

10-2

10-1

100

101

102

103

|φ(t

)|H

Inf

ΓasympHφ0HInf =

10010101001

-3sdot10-3-2sdot10-3-1sdot10-3

01sdot10-3

0 02 04 06 08 1

θ φ =

arg

(φ)

t HInf

Figure 51 Numerical solution for the absolute value of the quintessence VEV |φ | (upper) and itscomplex phase (lower) for various initial conditions φ0 and the choice λ1 = 1λ2 = 01V0ρ0 =10minus5ξ1 = 7χ0 = Hinf = 1012GeVα0 = π4g = 1 of parameters

χ-field in the equations of motion [23]

φ +3Hφ = minus2partVpartφ lowastminusλ1|χ|2φ minusλ2φ

lowastχ

2

χ +3H χ +3ΓχrarrSMχ = minusmicro2χ χminusλ1|φ |2χminusλ2χ

lowastφ

2

where ΓχrarrSM = g2

8πmχ is the decay rate and g2 stands for the squared sum of the Yukawa couplings

corresponding to the relevant decay channels Provided that the quintessence behaviour is dominatedby the exponential and not by the mixing terms (which is roughly the case if |V prime(φ0)| χ2

0 φ0χ30 ) it

will roll to larger field values with only small changes in the radial direction (see figure 51) whereasthe χ-field oscillates and decays once ΓχrarrSM amp H (see figure 52)Due to the BminusL-symmetry the total BminusL-density is conserved and thus the asymmetries stored in thedifferent components always add up to the initial value which was assumed to be zero after inflationie

nφ +nχ +nSM equiv 0 (53)

After the decay of the χ-field the comoving asymmetry freezes (see figure 53) since there is no moreexchange between the quintessence and the SM sectors1 [23]

nSMa3rarrminusnφ a3rarr const =int

infin

0dt a3

ΓχrarrSM middotnχ equiv Ainfin (54)

and thus the BminusL-asymmetry in the SM is exactly compensated by the BminusL-asymmetry stored in thequintessence field The final yield of the BminusL-asymmetry

nSMs = D middotκ equiv D middot minusAinfin

32ρ340

prop Ainfin (55)

1Here t equiv 0 and aequiv 1 at the end of inflation


0

02

04

06

08

1

0 02 04 06 08 1 12 14 16

|χ(t

)| a

32 H

Inf

t HInf

exact

WKB

Figure 52 Numerical and approximate WKB solution for the absolute value of the mediating fieldVEV |χ| for the same parameter values as in figure 51 despite φ0 = Hinf

(where ρ0 equiv 3H2inf M

2pl) can actually be calculated either numerically or for a restricted parameter

range analytically via the integral in eq (54) using an approximate WKB solution for χ(t) [23] (seefigure 52 and figure 53)

κ asympminusN2

sin(2α0)(

χ0

Hinf

)2

middot

36 middot10minus10 φ0

1013GeV

(Hinf

1012GeV

)12

if φ30 χ

20 φ0 |V prime(φ0)|

17 middot10minus8(

ξ1

7V0

ρ0

)13(

Hinf

1012GeV

)76

if |V prime(φ0)| φ30 χ

30

(56)where N equiv N (λ1λ2g) contains the the dependence on the coupling constants with N sim 1 forg2(8π) sim λ2λ1 λ1 sim 1 [23] The analytic estimate agrees well with the numerical results (seefigure 53) inside the respective domains of validity In the notation of eq (55) κ prop Ainfin is thecontribution which depends on the dynamics of the quintessence and the mediating field and D is afactor of proportionality which depends on the expansion history of the universe after inflation andcan vary in the range 1 amp D amp 10minus6 for various models of inflation and repreheating [23] Thusarriving at the observed value2 nSMs sim 10minus10 is possible if the asymmetry parameter κ is in therange

10minus10 κ 10minus4 (57)

which is indeed the case for a broad range of values for the initial energy density and VEV of thequintessence field (see figure 54)

2Note that the BminusL-asymmetry and the baryon asymmetry differ by an additional sphaleron factor of order one seeRef [109]


-4sdot10-10

-2sdot10-10

0

2sdot10-10

4sdot10-10

0 02 04 06 08 1 12 14 16 18

n φχ a

3 [3

2 ρ 0

34 ]

t HInf

g=05

Ainfin

nφsdota3

nχsdota3

nχsdota3 WKB

Figure 53 Time-evolution of the comoving asymmetry of the quintessence (red) and the mediating(blue) fields for the same parameters as in figure 51 despite g = 05 After an initial phase of os-cillations the χ-field decays and the asymmetry stored in the quintessence field goes to a constantasymptotic value Ainfin which is of equal amount but opposite sign as the asymmetry created in the SMThe analytic WKB approximation for nχ is also shown (dashed)

φ0 HInf

V0

ρ0

10-3 10-2 10-1 100 101 102 103 104 105 106

10-810-710-610-510-410-310-210-1

10-10

10-9

10-8 10-7 10-6 10-5

2sdot10-10

5sdot10-10

Figure 54 Contour plot of the created asymmetry κ prop Ainfin V0ρ0 corresponds to the fraction of quint-essence energy density after inflation and φ0 is the initial quintessence VEV The other parameters arechosen as in figure 51 The dashed lines divide the regions where the analytic approximations fromeq (56) are valid


53 Stability

An important issue in the context of complex quintessence models is to study the stability against theformation of inhomogeneities which could otherwise lead to the formation of so-called Q-balls [59]and destroy the dark energy properties Once the comoving asymmetry is frozen one can estimatefrom eq (51) the phase velocity θφ which is necessary to yield an asymmetry nφssim 10minus10

|θφ |H

=|nφ |

2H|φ |2sim 10minus10 2π2

45glowastS(T )

T 3

2H|φ |2 10minus8 (HMPl)32

2H|φ |2 10minus8 (58)

where it was assumed that glowastS(T ) sim 100 and |φ | amp Mpl Thus the field is moving extremely slowlyin the radial direction compared to the expansion rate of the universe which is exactly the oppositelimit as that which was studied for example in the spintessence models [40] Quantitatively onecan show [134] that there exist no growing modes for linear perturbations in |φ | and θφ for anywavenumber k provided that

θ2φ lt

3H +2ϕϕ

3H +6ϕϕV primeprime (59)

(with ϕ equiv |φ | V primeprime equiv d2Vdϕ2) Since the mass V primeprime sim H2 of the quintessence field tracks the Hubblescale [169] and since ϕϕ gt 0 this inequality is safely fulfilled once the tracking attractor is joinedand thus there are no hints for instabilities in this regime Details of the analysis including also theearly moments of evolution as well as additional particle processes can be found in Ref [23]Finally it is mentioned that since the underlying Lagrangian is BminusL-symmetric it offers a possibil-ity to combine Dirac-neutrinos with baryogenesis aside from the Dirac-leptogenesis mechanism [77]Note that the lepton-asymmetry in the SM is of opposite sign compared to Dirac-leptogenesis Fur-thermore there is no specific lower bound on the reheating temperature like in thermal leptogene-sis [70]In conclusion the coupled leptonic quintessence model discussed here can account for the observedbaryon asymmetry of the universe without introducing new BminusL-violating interactions below theinflationary scale by storing a lepton asymmetry in the dark energy sector

Chapter 6

Quantum Nonequilibrium Dynamicsand 2PI Renormalization

The standard big bang paradigm implies that cosmology is nonequilibrium physics As has been seenin the previous chapters nonequilibrium phenomena do not only occur in the early universe (likebaryogenesis) A rolling quintessence field for which the expectation value evolves with time duringall cosmological epochs also provides an example for a nonequilibrium situationThe description of nonequilibrium phenomena within quantum field theory has traditionally beenlimited to semi-classical approximations These can either describe highly correlated systems likea system with time-varying field expectation value or systems where correlations are quickly lostbut which are nevertheless sufficiently dilute such that quantum nonequilibrium effects like off-shelleffects are sufficiently suppressed and Boltzmann equations may be used However in situationswhere neither of the two limits described above can be applied a full quantum field theoretical de-scription is required An example is a system where a time-evolving field expectation value and anon-thermal distribution of particle-like excitations have to be treated simultaneously as it occurs forthe inflaton field during reheating and could also occur for a quintessence fieldA self-consistent quantum field theoretical description of quantum fields far from equilibrium is avail-able in the form of Kadanoff-Baym Equations derived from the 2PI effective action and many inter-esting nonequilibrium questions have been addressed within this framework in the recent years Theirderivation is briefly reviewed in section 61Due to the inherently nonperturbative structure of Kadanoff-Baym equations their renormalizationis still an unresolved question which is tackled in chapters 7 and 8 of this work There are variousreasons why a proper renormalization of Kadanoff-Baym equations is desirable as mentioned in theintroduction In particular it is required for quantitative comparisons with semi-classical approachesRenormalization is indispensable in order to obtain reliable predictions from realistic applications ofKadanoff-Baym equationsThe renormalization techniques for Kadanoff-Baym equations developed in this work are based onthe nonperturbative renormalization procedure of the 2PI effective action which has been recentlyformulated at finite temperature and which is reviewed in section 62For concreteness the nonequilibrium formalism is discussed for a real scalar λΦ44 quantum fieldtheory although the underlying concepts are more general and can be adapted to more realistic quan-tum field theories The fundamental action in Minkowski space is given by

S[φ ] =int

d4x(

12(partφ)2minus 1

2m2

φ2minus λ

4φ

4)

(61)

68 6 Nonequilibrium Dynamics and 2PI Renormalization

61 Kadanoff-Baym Equations from the 2PI Effective Action

The closed real-time path

Within quantum nonequilibrium dynamics one is interested in the time-evolution of correlation func-tions for a system which can be described by a density matrix ρ at a given initial time tinit equiv 0 Ingeneral the correlation functions are defined as expectation values of products of field operators andtheir conjugates with respect to the statistical ensemble Such expectation values can be calculatedusing the so-called in-in or closed-time-path (CTP) formalism [68 126 166] In contrast to the usualin-out formalism the calculation of expectation values requires the evaluation of matrix elementswhere the left state and the right state are both specified at the initial time For a Heisenberg operatorOH(t) which may be an arbitrary product of field operators and their conjugates all evaluated at acommon time argument t the expectation value is given by [68]

〈OH(t)〉= Tr(

ρ U(tinit t)OI(t)U(t tinit))

(62)

= Tr(

ρ T[

exp(

+iint t

tinit

dt primeHI(t prime))]OI(t) T

[exp(minusiint t

tinit

dt primeHI(t prime))])

where OI(t) = exp(itH0)OH(0)exp(minusitH0) denotes the interaction picture operator The interactionpicture time-evolution operator is given by [68]

U(t t prime) = exp(itH0)exp(minusi(tminus t prime)H

)exp(minusit primeH0

)(63)

=

T[exp(minusiint t

t primedt primeprimeHI(t primeprime))]

for t gt t prime

T[exp(+iint t prime

t dt primeprimeHI(t primeprime))]

for t lt t prime

where H0 is the quadratic part of the Hamiltonian and the interactions are contained in HI(t) =exp(itH0)(HminusH0)exp(minusitH0) T and T denote the chronological and the antichronological time-ordering operator respectively The product of operators appearing in the trace (62) contains achronologically ordered part and an antichronologically ordered part Therefore the contour C shownin figure 61 is defined which is running along the real axis from tinit to tmax = t and back to tinit aswell as a time-ordering operator TC on the contour The time arguments of the operators may alsobe assigned to the contour C The operator TC becomes the chronological time-ordering operator onthe branch running forward in time and the antichronological time-ordering operator on the branchrunning backward in time All operators belonging to the antichronological branch Cminus are placedleft of the operators belonging to the chronological branch C+ In this way the expectation value ineq (62) can be written as

〈OH(t)〉= Tr(

ρ TC

[exp(minusiint

CdtHI(t)

)OI(t)

]) (64)

where the time integral is performed along the contour C = C+ +Cminus Note that it is possible to extendthe contour to a maximal time tmax gt t by inserting the unity operator 1 = U(t tmax)U(tmax t) left orright of the operator OI(t) in eq (62)

The Schwinger-Keldysh propagator

The Schwinger-Keldysh propagator is defined as the connected two-point correlation function on theclosed real-time contour C

G(xy) = 〈TC Φ(x)Φ(y)〉minus〈Φ(x)〉〈Φ(y)〉 (65)

61 Kadanoff-Baym Equations from the 2PI Effective Action 69

Figure 61 Closed real-time contour [68 126 166]

The Schwinger-Keldysh propagator can be obtained by functional differentiation from the generatingfunctional for correlation functions formulated on the closed real-time path The generating functionalin the presence of a local external source J(x) and a bilocal external source K(xy) written down usinga complete basis of common eigenstates of the field operator Φ(x) at the initial time tinit equiv 0

Φ(0x)|ϕ0〉= ϕ(x)|ϕ0〉 (66)

is given by

Zρ [JK] = Tr(

ρ TC

[exp(

iint

Cd4xJ(x)Φ(x)+

i2

intCd4xint

Cd4yΦ(x)K(xy)Φ(y)

)])=

intDϕ+

intDϕminus 〈ϕ+0 |ρ|ϕminus0〉times

langϕminus0

∣∣∣∣TC

[exp(

i JΦ+i2

ΦKΦ

)]∣∣∣∣ϕ+0rang

where the short hand notation (321) applies (withintrarrintC) The second matrix element can be

expressed by a path integral over all field configurations ϕ(x) with time argument attached to thecontour C fulfilling the boundary conditions ϕ(0plusmnx) = ϕplusmn(x) [49]

Zρ [JK] =intDϕ+

intDϕminus 〈ϕ+0 |ρ|ϕminus0〉

ϕ(0minusx)=ϕminus(x)intϕ(0+x)=ϕ+(x)

Dϕ exp(

iS[ϕ]+ i Jϕ +i2

ϕ Kϕ

)

equivintDϕ 〈ϕ+0 |ρ|ϕminus0〉 exp

(iS[ϕ]+ i Jϕ +

i2

ϕ Kϕ

) (67)

The information about the initial state enters via the matrix element of the density matrix The stan-dard case which has been used for numerical studies so far is a Gaussian initial state

2PI effective action for a Gaussian initial state

A Gaussian initial state is an initial state for which all connected n-point correlation functions vanishfor nge 3 The density matrix element for a Gaussian initial state can be parameterized as

〈ϕ+0 |ρ|ϕminus0〉= exp(

iα0 + iα1ϕ +i2

ϕα2ϕ

) (68)


Figure 62 Diagrams contributing to the three-loop truncation of the 2PI effective action in thesymmetric phase (setting-sun approximation) for a Gaussian initial state

Therefore in the Gaussian case the contribution of the density matrix to the generating functional (67)can be absorbed into the external sources J + α1rarr J and K + α2rarr K (the constant α0 can be ab-sorbed into the normalization of the path integral measure)The 2PI effective action is the double Legendre transform of the generating functional (67) withrespect to the external sources The latter has the same structure as the generating functional (320) invacuum except that all time-integrations are performed over the closed real-time path Consequentlythe 2PI effective action for a Gaussian initial state is obtained from the parameterization given ineq (326) by replacing the time-integrations

intrarrintC For example the three-loop truncation of the

2PI effective action Γ[G]equiv Γ[φ = 0G] in the Z2-symmetric phase (〈Φ(x)〉= 0) which is referred toas setting-sun approximation is given by (see figure 62)

Γ[G] =i2

Tr lnGminus1 +i2

Tr(Gminus1

0 G)+Γ2[G] (69)

iΓ2[G] =minusiλ

8

intCd4xG(xx)2 +

(minusiλ )2

48

intCd4xint

Cd4yG(xy)4

Here Gminus10 (xy) = i(2x +m2)δ 4

C(xminus y) is the free inverse Schwinger-Keldysh propagator which con-tains the (bare) mass and the Dirac distribution on the time path C

Kadanoff-Baym equations for a Gaussian initial state

The equation of motion for the full Schwinger-Keldysh propagator is obtained from evaluating thefunctional derivative of the 2PI effective action with respect to the two-point function (which yields astationarity condition for vanishing external source K(xy) by construction)

δ

δG(xy)Γ[G] =minus1

2K(xy) (610)

Here the external sources are formally not zero for the physical situation but J(x) = α1(x) andK(xy) = α2(xy) due to the density matrix element However their contribution to the equationof motion will be omitted below because it vanishes in the Kadanoff-Baym equations Instead theinformation about the initial state only enters via the initial conditions for the two-point function fora Gaussian initial state (see appendix D)In setting-sun approximation the equation of motion for the propagator is given by (see figure 63)


Π(xy) equiv 2iδΓ2[G]δG(yx)

=minusiλ

2G(xx)δ 4

C(xminus y)+(minusiλ )2

6G(xy)3 (612)

where Π(xy) is the full self-energy The Kadanoff-Baym equations are an equivalent formulation of


Figure 63 Diagrams contributing to the self-energy Π(xy) in setting-sun approximation for a Gaus-sian initial state

the equation of motion They are obtained by convolving eq (611) with the full propagator(2x +m2 +

λ

2G(xx)

)G(xy) = minus iδ 4

C(xminus y)minus iint

Cd4zΠ(xz)G(zy) (613)

and inserting the decomposition of the full two-point function into the statistical propagator GF(xy)and the spectral function Gρ(xy)

G(xy) = GF(xy)minus i2

sgnC(x0minus y0)Gρ(xy) (614)

The Kadanoff-Baym equations read(2x +M2(x)

)GF(xy) =

int y0

0d4zΠF(xz)Gρ(zy)

minusint x0

0d4zΠρ(xz)GF(zy) (615)

(2x +M2(x)

)Gρ(xy) =

int y0

x0

d4zΠρ(xz)Gρ(zy)

The effective mass M2(x) = m2 + λ

2 G(xx) contains the bare mass and the local part of the self-energy (612) The non-local part of the self-energy can be decomposed into statistical and spectralcomponents similarly as the propagator In setting-sun approximation one has

Πnonminuslocal(xy) = ΠF(xy)minus i2

sgnC(x0minus y0)Πρ(xy) =(minusiλ )2

6G(xy)3 (616)

A more detailed derivation can be found in appendix DFor a Gaussian initial state the complete information about the initial state enters via the initialvalues of the connected one- and two-point functions and their time derivatives (up to one derivativeof each time argument see section D22) For the spectral function these initial conditions arefixed by the equal-time commutation relations (see eq (D49)) For the statistical propagator itis convenient to parameterize the initial conditions in terms of an effective kinetic energy densityω(t = 0k) and effective particle number density n(t = 0k) at the initial time t = 0 for each spatialmomentum mode k (see eq (D51)) The definitions obtained from the free-field ansatz [25] (whereG(x0y0k) =

intd3xeminusik(xminusy) G(xy))

ω2(tk) =

(partx0party0GF(x0y0k)

GF(x0y0k)

)∣∣∣∣∣x0=y0=t

n(tk) = ω(tk)GF(t tk)minus 12

(617)

have proven to yield meaningful results although there is no unique definition


Quantum dynamics far from equilibrium

With the formalism presented above it is possible to answer the question of how a quantum fieldevolves out of equilibrium for a wide class of circumstances In particular the quantum thermalizationprocess can be studied from first principles for a closed system [32] It is interesting to note that thederivation of Kadanoff-Baym equations within quantum field theory does not require any furtherapproximations or assumptions The Kadanoff-Baym equation (615) is an exact evolution equationfor the full two-point correlation function (the approximation enters on the level of a truncation ofthe self-energy Π(xy) like in eq (612)) In particular no assumptions are required which wouldonly hold for systems close to equilibrium [32] Kadanoff-Baym equations are suitable to studyquantum fields arbitrarily far from equilibrium as long as the underlying quantum field theory is validFurthermore Kadanoff-Baym equations do not violate time-reversal invariance [32] in contrast egto Boltzmann-equations [164] Due to the unitary time-evolution thermal equilibrium can neverbe reached completely Nevertheless observables like the two-point correlation function have beenshown to converge towards a thermal value at late times for closed systems involving scalar quantumfields on a lattice in 1+1 [32] 1+2 [123] and 1+3 [33142] space-time dimensions (see also [69] for thenonrelativistic case) as well as for fermionic quantum fields in 1+3 space-time dimensions [30 143]Furthermore in contrast to semi-classical descriptions given eg by Boltzmann equations [164]Kadanoff-Baym equations include memory effects since they are non-local in time and are capableof describing scattering processes which involve exchange of virtual (quasi-)particles (ldquooff-shellrdquo)as well as on-shell particles in a unified quantum-field theoretical manner Therefore in situationswhere the upper effects become important the application of standard Boltzmann equations includingeg the lowest order 2-to-2 scattering process might lead to quantitatively or even qualitativelyincorrect results [142 143 147] Since standard Boltzmann equations are widely used in all areasof physics it seems worth to investigate under which circumstances they are reliable and in howfar various extensions of Boltzmann-equations [147] can capture the off-shell and memory effectsincluded in the quantum-field theoretical Kadanoff-Baym treatment For such a comparison to workquantitatively it is desirable to have a proper renormalization procedure available which allows tocompare the evolution of semi-classical Boltzmann-ensembles with physical renormalized excitedstates rather than bare excited statesThere are also situations where semi-classical descriptions are not available eg for highly correlatedsystems which may undergo an instability A typical situation of this type is the decay of a scalarcondensate A coherent scalar condensate which periodically oscillates in its potential starts to de-cay due to its couplings into (quasi-)particle excitations This decay may additionally be resonantlyenhanced if parametric resonance conditions are fulfilled [127 128] creating a highly non-thermalpopulation of field quanta which are then expected to thermalize on a much longer time-scale How-ever this subsequent thermalization process cannot be described in the conventional 1PI frameworkWithin a quantum field theoretical treatment based on Kadanoff-Baym equations the evolution ofthe system can be followed at all stages starting from the coherent condensate to the thermalizedplasma [33] If the oscillating field is the inflaton the upper scenario is known as reheating (or pre-heating if parametric resonance occurs) [5 128 129 167] Using Kadanoff-Baym equations it is thuspossible to explore the period between the end of inflation and the beginning of the radiation dom-inated regime [3 4] This is relevant eg for the production of primordial gravitational waves [88]which will be tested by future precision measurements of the polarization of the cosmic microwavebackground [125] and for the reheating temperature This is the maximal temperature of the plasmain the early universe which is relevant eg for leptogenesis [70] and the production of long-livedthermal relics (ldquogravitinosrdquo) [93]In principle Kadanoff-Baym equations can even be applied in regimes where a priori no well-defined

62 Nonperturbative 2PI Renormalization at finite Temperature 73

notion of (quasi-)particle excitations exists as might occur in strongly coupled theories under extremenonequilibrium conditions [27] Such a situation may be encountered in high-energy Heavy IonCollisions performed at RHIC and planned at the LHC [6 7]Finally it is mentioned that it is possible to analyze kinetic and chemical equilibration using Kada-noff-Baym equations Kinetic equilibration requires energy-momentum exchange between differentmomentum modes eg via quantum scattering processes while chemical equilibration occurs dueto energy-momentum transfer between different species eg via decay and recombination processesDue to these different underlying microscopic processes one expects that kinetic and chemical equi-libration occur on different time-scales Such a separation of time-scales has indeed been found forthe quantum equilibration process described by Kadanoff-Baym equations [143] Microscopic kineticequilibration already occurs long before macroscopic observables have reached their final equilibriumvalues [31] An important requirement for the applicability of effective eg hydrodynamic descrip-tions of nonequilibrium processes is the validity of local thermal equilibrium [112] The ldquoprethermal-izationrdquo [31] featured by solutions of Kadanoff-Baym equations is a justification from first principlesregarding the domain of applicability of hydrodynamic equations used eg for the interpretation ofdata from high-energy Heavy Ion Collisions [131]

62 Nonperturbative Renormalization of the 2PI Effective Action at fi-nite Temperature

The 2PI effective action provides the appropriate framework for the investigation of quantum nonequi-librium dynamics However due to its nonperturbative nature renormalization is more complicatedcompared to the conventional perturbative approachIn general a perturbative approximation (for example a loop approximation of the 1PI effective ac-tion) is compatible with the renormalizability of the underlying quantum field theory if the followingcondition holds LetM denote the set of perturbative Feynman diagrams belonging to the approxi-mation of interest Then for any diagram inM it is necessary that all diagrams which are requiredto cancel its UV divergences and subdivergences (as determined by the BPHZ renormalization pro-cedure [38 113 191]) do also belong toMSince the solution of the self-consistent equation of motion for the full 2PI propagator correspondsto a selective infinite resummation of perturbative Feynman diagrams it is non-trivial whether anapproximation based on a truncation of the 2PI effective action is compatible with renormalizabilityRecently it has been shown [28 29 37 173ndash175] that systematic (eg loop 1N) truncations ofthe 2PI effective action lead to approximations which are compatible with renormalizability and acompletely nonperturbative renormalization procedure for the 2PI effective action in vacuum andat finite temperature has been formulated The 2PI vacuum counterterms which render all n-pointfunctions finite have to be determined self-consistentlyThe derivation of the nonperturbative renormalization procedure at finite temperature is briefly re-viewed in this section for the setting-sun approximation (69) of the 2PI effective action

The thermal time path

The density matrix ρ = Zminus1 exp(minusβH) in thermal equilibrium at temperature T = 1β is explicitlyknown in terms of the full Hamiltonian The exponential appearing in the thermal density matrixcan be interpreted as the full time-evolution operator exp(minusitH) evaluated for the imaginary timet = minusiβ Accordingly the matrix element of the thermal density matrix can be written as a pathintegral over field configurations ϕ(x) with time argument on a time contour I running along the


Figure 64 Thermal time contour C+I [136]

imaginary axis from t = 0 to t =minusiβ [49] (see section D11)

〈ϕ+0 |ρ|ϕminus0〉 =

ϕ(minusiβ x)=ϕ+(x)intϕ(0minusx)=ϕminus(x)

Dϕ exp(

iint

Id4xL(x)

) (618)

The upper path integral representation of the thermal density matrix element yields a generating func-tional for the thermal state by concatenating the time contours C and I (the derivation is analogousto that of eq (67))

Zβ [JK] = Tr(

ρ TC+I

[exp(

iint

C+Id4xJ(x)Φ(x)+

i2

intC+I

d4xint

C+Id4yΦ(x)K(xy)Φ(y)

)])=

intDϕ exp

(iint

C+Id4x L(x)+ J(x)ϕ(x)+ i

2

intC+I

d4xint

C+Id4yϕ(x)K(xy)ϕ(y)

) (619)

The path integral is performed over all field configurations ϕ(x) with time argument attached to thethermal time path C+I (see figure 64) which fulfill the periodicity relation ϕ(0+x) = ϕ(minusiβ x)The time arguments of the external sources are also attached to the thermal time path C+I

The thermal propagator

The thermal propagator is defined as the connected two-point correlation function on the thermal timecontour C+I

Gth(xy) = 〈TC+I Φ(x)Φ(y)〉minus〈Φ(x)〉〈Φ(y)〉 (620)

The thermal propagator can be obtained from the generating functional (619) for correlation functionsformulated on the thermal time path by functional differentiationFor calculations in thermal equilibrium it is sometimes convenient to use a pure imaginary time for-malism by setting tmax = 0 such that only the path I contributes Since thermal correlation functionsconsidered here are space-time translation invariant it is convenient to Fourier transform the thermaltwo-point function with respect to the relative imaginary times and spatial coordinates

Gth(xy) =int

qeiq(xminusy)Gth(q) for x0y0 isin I (621)


The meaning ofint

q depends on the context For zero-temperature calculationsint

q equivint d4q(2π)4 denotes

the integral over Euclidean momentum space For finite-temperature calculations howeverint

q equivint Tq equiv T sumn

int d3q(2π)3 where q0 = iωn and the sum runs over the Matsubara frequencies ωn = 2πnT (see

section D12)

2PI effective action and Schwinger-Dyson equation

The 2PI effective action in thermal equilibrium is the double Legendre transform of the generatingfunctional (619) with respect to the external sources The latter has the same structure as the gen-erating functional (320) in vacuum except that all time-integrations are performed over the thermaltime path Consequently the 2PI effective action in thermal equilibrium is obtained from the parame-terization given in eq (326) by replacing the time-integrations

intrarrintC+I Especially the setting-sun

approximation of the 2PI effective action is obtained from eq (69) by replacingintCrarr

intC+I

The equation of motion for the full thermal propagator is obtained from the stationarity condi-tion (328) of the 2PI effective action with respect to variations of the two-point function In setting-sun approximation it is given by

Gminus1th (xy) = Gminus1

0th(xy)minusΠth(xy) (622)

Πth(xy) equiv2iδΓ2[Gth]δGth(yx)

=minusiλ

2Gth(xx)δ 4

C+I(xminus y)+(minusiλ )2

6Gth(xy)3 (623)

where Πth(xy) is the full thermal self-energy and Gminus10th(xy) = i(2x + m2)δ 4

C+I(xminus y) is the freeinverse thermal propagator Note that x0y0 isin C+I take real as well as imaginary values

Nonperturbative renormalization procedure mdash derivation

Starting from the bare classical action

S[φ ] =int

d4x(

12(partφ)2minus 1

2m2

Bφ2minus λB

4φ

4)

(624)

the field is rescaled and the bare mass mB and the bare coupling λB are split into renormalized partsand counterterms

φR = Zminus12φ Zm2

B = m2R +δm2 Z2

λB = λR +δλ Z = 1+δZ (625)

where Z is the rescaling factor of the field value The equation of motion for the renormalized 2PIpropagator GthR = Zminus1Gth in setting-sun approximation and at finite temperature is obtained by usingeq (622) on the imaginary time path I and switching to 4-momentum space

Gminus1thR(k) = k2 +m2

RminusΠthR(k) (626)

ΠthR(k) = minusδZ0k2minusδm20minus

λR +δλ0

2

intqGthR(q)+

λ 2R

6

intpq

GthR(p)GthR(q)GthR(kminusqminus p)

Here δZ0 δm20 and δλ0 denote the 2PI counterterms in setting-sun approximation which have to

be chosen such that the divergences in the tadpole- and setting-sun contributions to the renormalizedself-energy ΠthR(k) as well as the divergences hidden in the full propagator are removed indepen-dent of the temperature As will be shown in the following this is accomplished by imposing tworenormalization conditions

Gminus1T0

(k = k) = k2 +m2R

ddk2 Gminus1

T0(k = k) = +1 (627)


Figure 65 Bethe-Salpeter equation

for the propagator supplemented by a third renormalization condition for an appropriate 4-point func-tion Here GT0(k) is the solution of eq (626) for an (arbitrary) reference temperature T0 and k is an(arbitrary) reference scale (eg T0 = 0 k = 0)The aim is to find a set of counterterms which also renormalizes the propagator GT (k) for all T 6= T0the equation for which can be written as

Gminus1T (k) = Gminus1

T0(k)minus∆Π(k)

∆Π(k) = minusλR +δλ0

2

[int T

qGT (q)minus

int T0

qGT0(q)

](628)

+λ 2

R

6

[int T

pqGT (p)GT (q)GT (kminusqminus p)minus

int T0

pqGT0(p)GT0(q)GT0(kminusqminus p)

]

Inverting the first line yields an expansion

GT (k) = GT0(k)+∆G(k) = GT0(k)+GT0(k)∆Π(k)GT0(k)+∆2G(k) (629)

At large momenta k2 T 2T 20 both propagators agree asymptotically such that ∆Π(k) sim c1 lnk +

c2(lnk)2 + (with coefficients ci) just grows logarithmically Thus ∆G(k) and ∆2G(k) fall off likekminus4 and kminus6 times powers of logarithms respectively Furthermore

int Tq equiv

int T0q +

int∆Tq where the latter

is exponentially suppressed for q2 T 2T 20 Altogether using Weinbergs theorem [177] one finds

that

∆Π(k) =12

[int T0

q∆G(q)+

int∆T

qGT (q)

]ΛT0(qk)+F(k) (630)

where F(k) contains all finite contributions (and falls off like kminus2 times powers of logarithms) andwhere ΛT0(qk) is equal to

ΛR(qk) =minusλRminusδλ0 +λ2R

intpGthR(p)GthR(kminusqminus p) (631)

evaluated at temperature T0 Using the second part of eq (629) in eq (630) one can writeint T0

q∆Π(q)

[δ (qminus k)minus 1

2G2

T0(q)ΛT0(qk)

]=

12

[int T0

q∆

2G(q)+int

∆T

qGT (q)

]ΛT0(qk)+F(k)

Multiplying by δ (kminus p)+G2T0

(k)VT0(k p)2 (with VT0 arbitrary) and integrating over k yields

int T0

q∆Π(q)

δ (qminus p)minus 1

2G2

T0(q)[VT0(q p)minusΛT0(q p)minus 1

2

int T0

kΛT0(qk)G2

T0(k)VT0(k p)

]=

=12

int T0

k

[int T0

q∆

2G(q)+int

∆T

qGT (q)

]ΛT0(qk)+F(k)

(δ (kminus p)+

12

G2T0

(k)VT0(k p))

If one demands that VT0(q p) fulfills the ldquoBethe-Salpeter equationrdquo (see figure 65) at temperature T0


VR(q p) = ΛR(q p)+12

intkΛR(qk)G2

thR(k)VR(k p) (632)

it is possible to encapsulate all divergences of the upper equation into VT0(q p)

∆Π(p) =12

int T0

q

[∆

2G(q)+F(q)G2T0

(q)]VT0(q p)+

12

int∆T

qGT (q)VT0(q p)

The momentum integrals are finite provided that the 2PI 4-point function VT0(q p) is finite and growsat most logarithmically when one of its arguments tends towards infinity while the other is fixed Itturns out [28] that this is achieved by requiring VT0(q p) to be finite at the renormalization point

VT0(q = k p = k) =minusλR (633)

Finally since ∆Π(p) is finite eq (628) implies that the renormalized 2PI propagator GT (k) is finitefor all temperatures T In summary the renormalization conditions eq (627) for the propagator GthR(k) and eq (633) forthe 4-point function VR(q p) (evaluated at some arbitrary reference temperature T0) together with thenonperturbative Schwinger-Dyson equation (626) and Bethe-Salpeter equation (632) form a closedset of equations for the determination of the 2PI counterterms

Chapter 7

Renormalization Techniques forSchwinger-Keldysh CorrelationFunctions

In this chapter a framework appropriate for the nonperturbative renormalization of Kadanoff-Baymequations is developed and applied to the three-loop truncation of the 2PI effective action

The nonperturbative 2PI renormalization procedure is transferred to the 2PI effective action formu-lated on the closed Schwinger-Keldysh real-time contour Therefore a Kadanoff-Baym equation forthe full thermal propagator formulated on the closed real-time contour is derived This requires the in-corporation of initial states characterized by non-Gaussian n-point correlation functions (for arbitraryn) into the Kadanoff-Baym equations

In section 71 Kadanoff-Baym equations for non-Gaussian initial states are derived In section 72it is shown how to calculate the thermal values of the non-Gaussian n-point correlation functions fora given truncation of the 2PI effective action and a Kadanoff-Baym equation for the thermal initialstate is derived This equation can then be renormalized explicitly by transferring the renormalizationprocedure of the 2PI effective action at finite temperature to the closed real-time contour which isdone in section 73

These renormalized Kadanoff-Baym equations for thermal equilibrium then provide the basis for thetransition to renormalized nonequilibrium dynamics

71 Kadanoff-Baym Equations for Non-Gaussian Initial States

A statistical ensemble of physical states in the Hilbert space belonging to the real scalar λΦ44quantum field theory is considered which is described by a density matrix ρ at the time tinit equiv 0The generating functional Zρ [JK] for nonequilibrium correlation functions in the presence of a localexternal source J(x) and a bilocal external source K(xy) can be conveniently formulated on theclosed real-time path C (see figure 61) and has a path integral representation given in eq (67) Theinformation about the initial state of the system enters via the matrix element of the density matrixwith respect to two arbitrary eigenstates Φ(0x)|ϕplusmn0〉= ϕplusmn(x)|ϕplusmn0〉 of the quantum field operatorevaluated at the initial time

80 7 Renormalization Techniques for Schwinger-Keldysh Correlation Functions

Non-Gaussian Initial State

The matrix element of the density matrix ρ is a functional of the field configurations ϕ+(x) andϕminus(x) which can be written as [49]

〈ϕ+0 |ρ|ϕminus0〉= exp(iF [ϕ]) (71)

For a Gaussian initial state F [ϕ] is a quadratic functional of the field (see eq (68)) For a generalinitial state the functional F [ϕ] may be Taylor expanded in the form

F [ϕ] = α0 +int

Cd4xα1(x)ϕ(x)+

12

intCd4xd4yα2(xy)ϕ(x)ϕ(y)

+13

intCd4xd4yd4zα3(xyz)ϕ(x)ϕ(y)ϕ(z)+ (72)

where the integrals have been written in four dimensions Since F [ϕ] only depends by definition onthe field configuration ϕplusmn(x) = ϕ(0plusmnx) evaluated at the boundaries of the time contour the kernelsαn(x1 xn) for nge 1 are only nonzero if all their time arguments lie on the boundaries of the timecontour With the notation δ+(t) = δC(t minus 0+) and δminus(t) = δC(t minus 0minus) they can be written in theform

αn(x1 xn) = αε1εnn (x1 xn)δε1(x

01) middot middot middotδεn(x

0n) (73)

where δC denotes the Dirac distribution on C and summation over ε j = +minus is implied In this waythe explicit dependence of the functional F [ϕ] on the field configurations ϕ+(x) and ϕminus(x) may berecovered

F [ϕ] = α0 +int

d3xαε1 (x)ϕε(x)+

12

intd3xint

d3yαε1ε22 (xy)ϕε1(x)ϕε2(x)+ (74)

Thus the kernels αn contribute only at the initial time Furthermore the complete set of kernels αn

for n ge 0 encodes the complete information about the density matrix characterizing the initial stateNot all the kernels are independent The Hermiticity of the density matrix ρ = ρdagger implies that

iαε1εnn (x1 xn) =

(iα(minusε1)(minusεn)

n (x1 xn))lowast

(75)

If the initial state is invariant under some symmetries there are further constraints For example foran initial state which is invariant under the Z2-symmetry Φrarr minusΦ all kernels αn(x1 xn) withodd n vanish If the initial state is homogeneous in space the initial correlations αn(x1 xn) areinvariant under space-translations xirarr xi +a of all arguments for any real three-vector a and canbe conveniently expressed in spatial momentum space


int d3k1

(2π)3 middot middot middotint d3kn

(2π)3 ei(k1x1++knxn)

(2π)3δ

3(k1 + +kn) iαε1εnn (k1 kn) (76)

Altogether the generating functional for a statistical ensemble is given by

Zρ [JK] =intDϕ exp

(i

S[ϕ]+ Jϕ +12

ϕKϕ +F3[ϕα3α4 ])

where the kernels α0 α1 and α2 have been absorbed into the measure Dϕ and into the sources J andK respectively The functional F3[ϕα3α4 ] contains the contributions of third fourth and higherorders of the Taylor expansion (72) It vanishes for a Gaussian initial state

71 Non-Gaussian Initial States 81

711 2PI Effective Action for Non-Gaussian Initial States

The 2PI effective action in the presence of non-Gaussian correlations is obtained from the standardparameterization [66] of the 2PI effective action applied to a theory described by the modified classicalaction S[φ α3α4 ]equiv S[φ ]+F3[φ α3α4 ]

Γ[φ G] = S[φ ]+i2

Tr lnGminus1 +i2

Tr(Gminus1

0 Gminus1)+ Γ2[φ G] (77)

where iGminus10 equiv δ 2S[φ ]

δφ(x)δφ(y) This parameterization may be rewritten by splitting it into a part whichcontains the contributions from non-Gaussian initial correlations and one which resembles the pa-rameterization obtained in the Gaussian case (D31)

Γ[φ G] = S[φ ]+i2

Tr lnGminus1 +i2

Tr(Gminus1

0 Gminus1)+Γ2[φ G]+ΓnG[φ Gα3α4 ] (78)

where iGminus10 equiv

δ 2S[φ ]δφ(x)δφ(y) is the classical inverse propagator (D32) and the non-Gaussian contribution

is obtained by comparing eq (78) and eq (77)

ΓnG[φ Gα3α4 ] = F3[φ α3α4 ]+12

Tr(

δ 2F3

δφδφG)

+Γ2nG[φ Gα3α4 ] (79)

The 2PI functional

iΓ2[φ Gα3α4 ]equiv iΓ2[φ G]+ iΓ2nG[φ Gα3α4 ] (710)

is equal to the sum of all 2PI Feynman diagrams with lines given by the full propagator G(xy) andwith vertices given by the derivatives of the modified classical action S[φ α3α4 ] Apart from theclassical three- and four-point vertices given by eq (D33) for a general non-Gaussian initial statethe initial n-point correlations (with nge 3) lead to additional effective non-local vertices connecting nlines (see figure 71) They result from the contribution of the corresponding sources αm(x1 xm)mge n contained in the contribution F3[φ α3α4 ] to the generating functional (77) and are givenby

iδ nF3[φ α3α4 ]δφ(x1) δφ(xn)

= iαn(x1 xn)+int

Cd4xn+1iαn+1(x1 xn+1)φ(xn+1)

+12

intCd4xn+1d4xn+2iαn+2(x1 xn+2)φ(xn+1)φ(xn+2)+

equiv iαn(x1 xn) (711)

Note that since the sources αm(x1 xm) are only supported at the initial time all the upper inte-grals along the time contour C just depend on powers of the initial value of the field expectation valueφ(x)|x0=0 Therefore the effective non-local n-point vertex iαn(x1 xn) indeed encodes informa-tion about the initial state and is in particular independent of the subsequent time-evolution of φ(x)Analogously

iα2(xy)equiv iα2(xy)+ iδ 2F3[φ α3α4 ]

δφ(x)δφ(y) (712)

is defined For a Z2-symmetric initial state the field expectation value vanishes φ(x)|x0=0 = 0 suchthat αn(x1 xn) = αn(x1 xn) From eq (73) it can be seen that the effective non-local verticesare supported at the initial time similarly to the sources iαn(x1 xn)

iαn(x1 xn) = iαε1εnn (x1 xn)δε1(x


0n) (713)


Figure 71 Non-local effective vertices iαn(x1 xn) connecting n lines for n = 3456 encodingthe non-Gaussian three- four- five- six- -point correlations of the initial state

Thus the contribution of these effective non-local vertices will be most important in the first momentsof the nonequilibrium evolution In particular eg the four-point source α4(x1 x4) can lead toa non-vanishing value of the connected four-point correlation function at the initial time which isimpossible for a Gaussian initial stateThe 2PI functional (710) is thus equal to the sum of all 2PI Feynman diagrams with lines given bythe full propagator G(xy) and with n-point vertices (n ge 3) given by eq (711) as well as classicalthree- and four-point vertices given by eq (D33) Note that those 2PI diagrams which contain ex-clusively the classical vertices given in eq (D33) by definition contribute to the functional iΓ2[φ G]Therefore the diagrams contributing to the non-Gaussian part iΓ2nG[φ Gα3α4 ] contain at leastone effective vertex from eq (711) involving a source αn(x1 xn) (nge 3) Thus the non-Gaussiancontribution to the 2PI effective action defined in eq (78) indeed vanishes for Gaussian initial con-ditions (D24)

ΓnG[φ Gα3 = 0α4 = 0 ] = 0 (714)

As an example an initial 4-point correlation is considered for an initial state which is Z2-symmetricsuch that φ(x) = 0 Then the 2PI functional Γ2[Gα4]equiv Γ2[φ = 0Gα3 = 0α4α5 = α6 = = 0]in ldquonaiumlverdquo 1 three loop approximation reads (see figure 72)

iΓ2[Gα4] =18

intC

d4x1234

[minus iλδ12δ23δ34 + iα4(x1 x4)

]G(x1x2)G(x3x4)

+148

intC

d4x1234d4x5678


]G(x1x5)times (715)

timesG(x2x6)G(x3x7)G(x4x8)[minus iλδ56δ57δ58 + iα4(x5 x8)

]

where a compact notation δ12 = δC(x1minus x2) and d4x1234 = d4x1 middot middot middotd4x4 has been used Note that thecontribution to the mixed ldquobasketballrdquo diagram in the second and third line with one classical and oneeffective vertex appears twice which accounts for the symmetry factor 124 This truncation of the2PI functional is also referred to as setting-sun approximation in the following

712 Self-Energy for Non-Gaussian Initial States

The equation of motion for the full propagator is obtained from the stationarity condition of the 2PIeffective action in the presence of the source α2(xy) δΓ[G]δG(yx) = minusα2(xy)2 Using theparameterization (77) and eq (712)

Gminus1(xy) = Gminus10 (xy)minusΠ(xy)minus iα2(xy) (716)

1This means no difference is made between diagrams with or without non-local effective vertices when counting loops


Figure 72 Diagrams contributing to the three-loop truncation of the 2PI effective action in the sym-metric phase (setting-sun approximation) in the presence of an effective non-local four-point vertex

is obtained where the self-energy Π(xy) is given by

Π(xy) =2iδ Γ2[φ Gα4α6 ]

δG(yx)=

2iδΓ2[φ G]δG(yx)

+2iδΓ2nG[φ Gα4α6 ]

δG(yx) (717)

For the non-Gaussian case the self-energy can be decomposed as

Π(xy) = ΠG(xy)+Π

nG(xy) (718)

ΠG(xy) = minusiΠloc(x)δC(xminus y)+Π

Gnonminusloc(xy)

ΠnG(xy) = Π

nGnonminusloc(xy)+ iΠnG

surface(xy)

where ΠG = 2iδΓ2δG contains the contributions to the self-energy which are also present for aGaussian initial state and ΠnG = 2iδΓ2nGδG contains all contributions which contain at least onenon-Gaussian initial correlation The latter can be further decomposed into a non-Gaussian non-localpart ΠnG

nonminusloc(xy) which contains diagrams where both external lines are attached to a local standardvertex and a part iΠnG

surface(xy) which contains all non-Gaussian contributions which are supportedonly at the initial time surface where x0 = 0 or y0 = 0 In general such contributions can arise in thefollowing ways

1 From diagrams where both external lines are connected to an effective non-local vertex as givenin eq (711) They are supported at x0 = y0 = 0

2 From diagrams where one of the two external lines is connected to an effective non-local vertexwhile the other one is connected to a classical local vertex as given in eq (D33) They aresupported at x0 = 0y0 ge 0 or vice-versa

3 Via the contribution iα2(xy) of the initial two-point source which is supported at x0 = y0 = 0This is the only Gaussian surface-contribution

Accordingly the contributions to the self-energy which are supported at the initial time surface canbe further decomposed as

Πsurface(xy) = ΠnGsurface(xy)+ α2(xy) = Παα(xy)+Πλα(xy)+Παλ (xy) (719)

where

Παα(xy) = δε1(x0)Πε1ε2αα (xy)δε2(y0)

Πλα(xy) = Πε

λα(x0xy)δε(y0) (720)

Παλ (xy) = δε(x0)Πε

αλ(xy0y) = Πλα(yx)

Παα contains all contributions of type (1) and (3) Diagrams of type (2) contribute to Πλα or Παλ

depending which external line is attached to the effective non-local vertex and which to the classical


Figure 73 Contribution Πλα(xy) to the self-energy Π(xy) where the left line is connected to aclassical vertex and the right line to an effective non-local vertex

local vertex Thus for all diagrams contributing to Πλα the left line is connected to the classical four-or three-point vertex which means that it can always be written in the form (see figure 73)

iΠλα(xy) =minusiλ

6

intCd4x123G(xx1)G(xx2)G(xx3)iAnG

4 (x1x2x3y)

+minusiλφ(x)

2

intCd4x12G(xx1)G(xx2)iAnG

3 (x1x2y) (721)

The non-local part contains all diagrams where both external lines are attached to a classical localvertex as given in eq (D33) It can be split into statistical and spectral components similarly to theGaussian case

Πnonminusloc(xy) = ΠGnonminusloc(xy)+Π

nGnonminusloc(xy) = ΠF(xy)minus i

2sgnC(x0minus y0)Πρ(xy) (722)

The local part does not receive any changes in the non-Gaussian case and is included in an effectivetime-dependent mass term

M(x)2 = m2 +λ

2φ

2(x)+Πloc(x) = m2B +

λ

2φ

2(x)+λ

2G(xx) (723)

For the setting-sun approximation from eq (715) the self-energy is given by (see figure 74)

ΠGnonminusloc(xy) =

(minusiλ )2

6G(xy)3 M(x)2 = m2 +

λ

2G(xx)

ΠnGnonminusloc(xy) = 0


6

intd4x123 G(xx1)G(xx2)G(xx3) iα4(x1x2x3y) (724)

iΠαα(xy) = iα2(xy)+12

intd4x34 iα4(xyx3x4)G(x3x4)

+16

intd4x234567 iα4(xx2x3x4)G(x2x5)G(x3x6)G(x4x7) iα4(x5x6x7y)

A comparison with eq (721) yields that

iAnG4 (x1x2x3x4) = iα4(x1x2x3x4) iAnG

3 (x1x2x3) = 0

for the upper approximation


Multiplying eq (716) with the propagator and integrating yields(2x +M2(x)

)G(xy) = minusiδC(xminus y)minus i

intCd4z [Π(xz)+ iα2(xy)]G(zy) (725)

= minusiδC(xminus y)minus iint

Cd4z[Π

Gnonminusloc(xz)+Π

nGnonminusloc(xz)+ iΠλα(xz)

]G(zy)


Figure 74 Diagrams contributing to the self-energy Π(xy) in setting-sun approximation in thepresence of an effective non-local four-point vertex From left to right the diagrams contribute toΠloc ΠG

nonminusloc Πλα Παλ and the last two both contribute to Παα

The second line follows from using the parameterization (718) of the self-energy and assumingx0 gt 0 and y0 gt 0 Using eqs (720722) yields the Kadanoff-Baym equations for GF(x0y0k) andGρ(x0y0k) for an (arbitrary) non-Gaussian initial state

(part

2x0 +k2 +M2(x0)

)GF(x0y0k) =

int y0

0dz0

ΠF(x0z0k)Gρ(z0y0k)

minusint x0

0dz0

Πρ(x0z0k)GF(z0y0k) (726)

+ΠλαF(x0k)GF(0y0k)

+14

Πλαρ(x0k)Gρ(0y0k) (part

2x0 +k2 +M2(x0)

)Gρ(xy) =

int y0

x0

dz0Πρ(x0z0k)Gρ(z0y0k)

where

ΠλαF(x0k) = Π+λα

(x0k)+Πminusλα

(x0k)

Πλαρ(x0k) = 2i(Π

+λα

(x0k)minusΠminusλα

(x0k))

(727)

Using eq (721) yields an equivalent formulation

(2x +M2(x)

)GF(xy) =

int y0

0d4zΠF(xz)Gρ(zy)

minusint x0


minus λ

6V nG

4 (xxxy)minus λφ(x)2

V nG3 (xxy) (

2x +M2(x))

Gρ(xy) =int y0

x0

d4zΠρ(xz)Gρ(zy)

where

V nG4 (x1x2x3x4) equiv

intC

d4y1234 G(x1y1)G(x2y2)G(x3y3)G(x4y4) iAnG4 (y1y2y3y4)

V nG3 (x1x2x3) equiv

intC

d4y123 G(x1y1)G(x2y2)G(x3y3) iAnG3 (y1y2y3) (729)

denote the four- and three-point functions constructed from the kernels AnG4 and AnG

3 appearing inthe initial-time-surface contribution Πλα(xy) to the self-energy respectively (see eq (721) and fig-ure 73) Note that these new contributions on the right hand side of the Kadanoff-Baym equations


do not have to vanish in the limit x0y0rarr 0 unlike the memory integrals This is due to the fact thatthe higher non-Gaussian correlations of the initial state can lead to a non-vanishing value of the con-nected four- and three-point correlation functions at the initial time In contrast to this for a Gaussianinitial state all higher correlations vanish at the initial time by definition

72 Kadanoff-Baym Equations with Nonperturbative Thermal InitialCorrelations

The Kadanoff-Baym equations discussed in section 71 are in principle capable to describe the time-evolution of the full two-point correlation function for a statistical ensemble which is described by anarbitrary state at some initial time tinit = 0 Since the nonperturbative renormalization is establishedat finite temperature it is an important step to show that the full equilibrium propagator is indeed asolution of the nonperturbatively renormalized Kadanoff-Baym equations for a thermal initial stateThis requires the incorporation of appropriate thermal initial correlations into the Kadanoff-Baymequations However since the underlying approximation based on the truncation of the 2PI effectiveaction is highly non-perturbative the choice of appropriate thermal initial correlations is not straight-forward For example for the three-loop truncation of the 2PI effective action the thermal n-pointcorrelation functions for all n = 246 are non-zero although only two diagrams have been kept inthe 2PI effective action Therefore one has to expect that non-Gaussian initial n-point correlations forall n = 246 are required to describe thermal equilibrium with Kadanoff-Baym equations In thefollowing it is shown how to construct the thermal initial correlations required for a given truncationof the 2PI effective action explicitly This is accomplished by matching the nonperturbative equationof motion for the propagator formulated on the thermal time path with the Kadanoff-Baym equationfor a non-Gaussian initial state formulated on the closed real-time pathThus it is necessary to relate the following two equivalent descriptions of thermal equilibrium

1 Via the thermal time contour (ldquoC+I rdquo)

2 Via the closed real-time contour C and a thermal initial state characterized by thermal initialcorrelations α th

n (x1 xn) (ldquoC+α rdquo)

The first formulation exploits the explicit structure of the thermal density matrix whereas the secondone can easily be generalized to a nonequilibrium ensemble

The thermal value of any (nonperturbative) Feynman diagram can directly be computed via the ther-mal time contour C+I if the thermal (nonperturbative) propagator for real and imaginary timesis available For the computation of the corresponding (nonperturbative) Feynman diagram via theclosed real-time contour C only real times appear However it requires the knowledge of the thermalinitial correlations α th

n (x1 xn) which are appropriate for the considered approximation

Since nonequilibrium Kadanoff-Baym equations are formulated on the closed real-time path C itis required to use the second approach In the following it is shown how to construct the thermalcorrelations α th

n (x1 xn) explicitly for a given truncation of the 2PI effective action Before turningto the nonperturbative case the relation between the two descriptions of thermal equilibrium will bediscussed within perturbation theory

72 Nonperturbative Thermal Initial Correlations 87

721 Thermal Initial Correlations mdash Perturbation Theory

Thermal time contour C+I

The free thermal propagator defined on C+I is (see also eq (D10))

iGminus10th(xy) =

(minus2xminusm2)

δC+I(xminus y) for x0y0 isin C+I (730)

which may be decomposed into the free thermal statistical propagator and the free thermal spectralfunction

G0th(xy) = G0F(xy)minus i2

sgnC+I(x0minus y0)G0ρ(xy)

The explicit solution of the equation of motion is

G0F(x0y0k) =nBE(ωk)+ 1

2ωk

cos(ωk(x0minus y0)

) (731)

G0ρ(x0y0k) =1

ωksin(ωk(x0minus y0)

)for x0y0 isin C+I

where nBE(ωk) is the Bose-Einstein distribution function

nBE(ωk) =1

eβωk minus1 ωk =

radicm2 +k2

Each of the two time arguments of the propagator can either be real or imaginary which yields fourcombinations GCC

0th GCI0th GIC

0th GII0th These appear in perturbative Feynman diagrams which are

constructed with the free propagator G0th and the classical vertices In position space each internalvertex of a Feynman diagram is integrated over the thermal time contour C+I In order to disentanglethe contributions from the real and the imaginary branch of the time contour the following Feynmanrules are defined

GCC0th(xy) = GCI

0th(xy) =

GII0th(xy) = GIC

0th(xy) =

minusiλint

Cd4x = minusiλ

intId4x = minusiλ

intC+I

d4x =

(732)

The filled circles denote a real time and the empty circles denote an imaginary time As an examplethe perturbative setting-sun diagram is considered with propagators attached to both external linesand evaluated for real external times x0y0 isin C Both internal vertices are integrated over the twobranches C and I respectively Using the upper Feynman rules the resulting four contributions canbe depicted as

S0(xy)equiv =(minusiλ )2

6

intC+I

d4uint

C+Id4vG0th(xu)G0th(uv)3G0th(vy)

=


Closed real-time contour with thermal initial correlations C+α

In this paragraph it will be shown how to explicitly construct the perturbative setting-sun S0(xy)(or any other perturbative diagram) from corresponding perturbative Feynman diagrams which justinvolve real times which lie on the closed real-time contour C as well as the appropriate thermalinitial correlations α th

n (x1 xn) As discussed in section 71 initial correlations appear in Feynmandiagrams as additional effective non-local vertices which are supported only at the initial time tinit equiv0 at which the closed contour C starts (t = 0+) and ends (t = 0minus)Starting from the diagram on the thermal time contour C+I one would like to obtain the functionS0(xy) without reference to imaginary times The parts with imaginary and those with real times areconnected by the free propagator evaluated with one imaginary and one real time Using the explicitsolution (731) together with elementary trigonometric addition theorems it can be written as

GIC0th(minusiτy0k) =

GII0th(minusiτ0k)G0th(00k)

GCC0F(0y0k)+ ipartτGII

0th(minusiτ0k)GCC0ρ(0y0k)

Next the unequal-time statistical propagator and the spectral function are rewritten as

GCC0F(0y0k) =

intCdz0

δs(z0)GCC0th(z

0y0k) (733)

GCC0ρ(0y0k) = minus2i

intCdz0

δa(z0)GCC0th(z

0y0k) (734)

where

δs(z0) =12(δC(z0minus0+)+δC(z0minus0minus)

)

δa(z0) =12(δC(z0minus0+)minusδC(z0minus0minus)

) (735)

Combining the upper equations a helpful expression for the free propagator evaluated with one imag-inary and one real time is obtained


intCdz0

∆0(minusiτz0k)GCC0th(z

0y0k)

=(736)

where the free ldquoconnectionrdquo defined in eq (D7) was inserted In terms of the symmetric and anti-symmetric Dirac-distributions δsa(z0) the free connection reads

∆0(minusiτz0k) = ∆s0(minusiτk)δs(z0)+∆

a0(minusiτk)δa(z0) (737)

=

(GII

0th(minusiτ0k)G0th(00k)

)δs(z0)+

2partτGII0th(minusiτ0k)

δa(z0)

= (738)

Analogously the free propagator evaluated with one real and one imaginary time can be written as

GCI0th(y

0minusiτk) =int

Cdz0 GCC

0th(x0z0k)∆T

0 (z0minusiτk)

= (739)


where ∆T0 (z0minusiτk) = ∆0(minusiτz0k) =

The connections ∆0 and ∆T0 are attached to an imaginary and a real vertex on the left and right sides

respectively Their Fourier transform into position space is

∆0(vz) =int d3k

(2π)3 e+ik(vminusz)∆0(v0z0k) for v0 isin Iz0 isin C

as well as ∆T0 (zv) = ∆0(vz) Conversely the Fourier transform of the connection with respect to the

imaginary time is a function of one Matsubara frequency ωn = 2πβn and one real time z0 isin C

∆0(ωnz0k) =

(GII

0th(ωnk)G0th(00k)

)δs(z0)+

2iωnGII0th(ωnk)

δa(z0) (740)

and ∆T0 (z0ωnk) = ∆0(ωnz0k) Eq (736) for the free propagator with one imaginary and one real

time then becomes

GIC0th(ωny0k) =

intCdz0

∆0(ωnz0k)GCC0th(z

0y0k) (741)

By replacing all free propagators which connect an imaginary and a real time inside a perturbativeFeynman diagram via the convolution of the connection and the real-real propagator it is possible toencapsulate the parts of the diagram which involve ldquoimaginaryrdquo vertices represented by empty circlesFor example the setting-sun diagram with one real and one imaginary vertex can be rewritten as

= equiv equiv th

0L

According to the symbolic notation employed here the subdiagram containing the imaginary vertexmarked by the box can be encapsulated into an effective non-local 4-point vertex Its structure isdetermined by the connections ∆0 and ∆T

0 as can be seen by rewriting the above diagrams in terms ofthe corresponding formal expressions

(minusiλ )2

6

intCd4uint

Id4vG0th(xu)G0th(uv)3G0th(vy)

=(minusiλ )2

6

intCd4uint

Cd4z1

intCd4z2

intCd4z3

intCd4z4 G0th(xu)G0th(uz1)G0th(uz2)G0th(uz3)[int

Id4v∆

T0 (z1v)∆T

0 (z2v)∆T0 (z3v)∆0(vz4)

]G0th(z4y)

equiv minusiλ6

intCd4uint

Cd4z1

intCd4z2

intCd4z3

intCd4z4 G0th(xu)G0th(uz1)G0th(uz2)G0th(uz3)[

αth40L(z1z2z3z4)

]G0th(z4y)

In the last line the thermal effective 4-point vertex has been introduced

αth40L(z1z2z3z4) = minusiλ

intId4v∆0(vz1)∆0(vz2)∆0(vz3)∆0(vz4)

th

0L= equiv


Since the connection ∆0(vzi) is only supported at the initial time z0i = 0plusmn the effective 4-point vertex

vanishes as soon as one of the four real times z01 z

40 lies beyond the initial time Thus the effective

4-point vertex has precisely the same structure as the non-local effective vertices describing the initialcorrelations for arbitrary initial states (see section 71) Within the perturbative framework treatedhere the above 4-point vertex constitutes the leading order contribution to the loop expansion of thethermal initial 4-point correlation function (see section D11)

αth4 (z1z2z3z4) = α

th40L(z1z2z3z4) + α

th41L(z1z2z3z4) +

th=

th

0L+

th

1L+

In general for any thermal diagram on C+I with V vertices there are 2V possibilities to combinethe integration over C or I at each vertex For each of these 2V contributions all lines connectinga real and an imaginary vertex are replaced using relation (736) Thereby the parts containing I-integrations are encapsulated into non-local effective vertices Thus any thermal diagram on C+Ican be equivalently represented by 2V diagrams on C which contain the classical vertex along withappropriate non-local effective verticesThese non-local effective vertices indeed match the thermal initial correlations α th

n discussed in sec-tion D11 This has been demonstrated above for the setting-sun diagram which contains a singleimaginary vertex For diagrams which contain internal lines which connect two imaginary verticesrepresenting the propagator GII

0th(minusiτminusiτ primek) the following relation can be employed

GII0th(minusiτminusiτ primek) = D0(minusiτminusiτ primek)

+int

Cdw0

intCdz0

∆0(minusiτw0k)GCC0th(w

0z0k)∆T0 (z0minusiτ primek)

= D0(minusiτminusiτ primek)+∆s0(minusiτk)G0th(00k)∆s

0(minusiτ primek)

= +

(742)

which can be verified by explicit calculation from eqs (D6 731 737) Hereby the propagatorD0(minusiτminusiτ primek) which is defined in eq (D6) is represented by the dotted line which connectstwo imaginary times It furnishes the perturbative expansion of the thermal initial correlations (seesection D11) By applying the upper relation to the setting-sun diagram with two imaginary verticesit can be rewritten as

= equiv =

= + + +

= + + +

In the first step the propagators connecting real and imaginary vertices were replaced by the convo-lution of the connection and the real-real propagator This already yields an effective non-local two-vertex as indicated in the third diagram in the first line In order to check that this effective non-local


two-vertex is indeed composed from the thermal initial correlations the three propagators connectingthe two imaginary vertices are replaced using relation (742) such that it falls apart into eight termswhich combine to the four inequivalent contributions shown in the second line2 Finally the partswhich contain imaginary vertices and dotted lines can be identified with the corresponding contri-butions to the perturbative expansion of the thermal initial correlations discussed in section D11which is represented graphically by encapsulating the subdiagrams inside the boxes In the third linethe first diagram thus contains a thermal effective two-point vertex which itself appears at two-looporder in the perturbative expansion of the thermal initial correlations Similarly the thermal effectivefour- and six-point vertices contained in the second and third diagram respectively appear at one-and zero-loop order in the perturbative expansion of the thermal initial correlations The two effectivefour-point vertices contained in the fourth diagram are identical to those already encountered aboveThus using the representation (736) of the free propagator connecting a real and an imaginary timeany perturbative thermal Feynman diagram formulated on the thermal time contour C+I can berelated with a set of perturbative Feynman diagrams formulated on the closed real-time contour Cand the required approximation to the full thermal initial correlations α th

n can be explicitly constructedwith the help of the formalism introduced here For example for the perturbative setting sun diagramthe equivalence between C+I and C+α can in summary be written as

S0(xy) = =

+ + +

+ + +

Within perturbation theory the dotted and dashed propagators as well as the connection are knownexplicitly They are given in terms of elementary functions such that the upper equivalence can becross-checked by an explicit calculation of both types of diagrams After this reassuring exercise onecan proceed to the nonperturbative case

722 Thermal Initial Correlations mdash 2PI


The full thermal propagator defined on C+I fulfills the nonperturbative Schwinger-Dyson equationderived from the 2PI effective action (see also eq (622))

Gminus1th (xy) = i(2x +m2)δC+I(xminus y)minusΠth(xy) for x0y0 isin C+I (743)

2Note that the symmetry factors are taken into account properly For example the symmetry factor of the seconddiagram in the second line is one third times the symmetry factor of the original diagram in the first line Since there arethree possibilities to obtain this diagram from the first one it is obtained with the correct prefactor


It furnished the expansion of the 2PI effective action in terms of 2PI Feynman diagrams Similar tothe perturbative case the following Feynman rules are defined

GCCth (xy) = GCI

th (xy) =

GIIth (xy) = GIC

th (xy) =

minusiλint

Cd4x = minusiλ

intId4x = minusiλ

intC+I

d4x =

(744)

in order to disentangle the contributions from the real and the imaginary branch of the thermal timecontour In order to derive a nonperturbative generalization of eq (736) it is helpful to define aldquomixed propagatorrdquo which coincides with the full propagator on the imaginary branch I of the ther-mal time contour and obeys the free equation of motion on the real branch C

Mixed thermal propagator

It is helpful to define projections on the parts C and I of the thermal time contour

1I(x0) =

0 if x0 isin C1 if x0 isin I 1C(x0) =

1 if x0 isin C0 if x0 isin I (745)

which fulfill the relation

1I(x0)+1C(x0) = 1 for all x0 isin C+I (746)

The mixed thermal propagator is defined by the following equation of motion

Gminus1mth(xy) = i(2x +m2

B)δC+I(xminus y)minus1I(x0)1I(y0)Πth(xy) for x0y0 isin C+I (747)

where Πth(xy) is the full thermal self-energy It can be decomposed into statistical and spectralcomponents

Gmth(xy) = GmF(xy)minus i2

sgnC+I(x0minus y0)Gmρ(xy)

The equation of motion for the mixed propagator can equivalently be written as(2x +m2)Gmth(xy) =minusiδC+I(xminus y)minus i1I(x0)

intId4zΠmth(xz)Gmth(zy) (748)


mth GCImth GIC

mth GIImth The mixed propagator evaluated with two imaginary time

arguments is identical to the full thermal 2PI propagator

GIImth(xy) = GII

th (xy) for x0y0 isin I (749)

whereas the mixed propagators evaluated with two real time arguments GCCmth(xy) as well as GCI

mth(xy)(where x0 isin Cy0 isin I) fulfill the equation of motion of the free propagator(

2x +m2B)

GCImth(xy) =

(2x +m2

B)

GCCmF(xy) =

(2x +m2

B)

GCCmρ(xy) = 0 (750)

At the initial time x0 = y0 = 0 the propagators on all branches of the thermal time path agree

GCCmth(xy)|x0=y0=0 = GCI

mth(xy)|x0=y0=0 = GICmth(xy)|x0=y0=0 = GII

mth(xy)|x0=y0=0 = Gth(xy)|x0=y0=0


Thus the initial value of the mixed propagator at x0 = y0 = 0 is given by the full thermal propagatorFor the mixed propagator with one imaginary and one real time GIC

mth(xy) (x0 isin Iy0 isin C) theequation of motion transformed to spatial momentum space reads(

minuspart2τ +k2 +M2

th)

GICmth(minusiτy0k) =minus

intβ

0dτprimeΠ

IIth (minusiτminusiτ primek)GIC

mth(minusiτ primey0k)

Compared to the corresponding equation (D17) for the full thermal propagator the memory integralalong the real axis is absent Next a Fourier transformation with respect to the imaginary time isperformed using in particularint

β

0dτ eminusiωnτ

part2τ GIC

mth(minusiτy0k) =minusω2n GIC

mth(ωny0k)+disc(iωn GICmth +partτGIC

mth)(y0k)

where a possible contribution from boundary terms has to be taken into account

disc(iωn GICmth +partτGIC

mth)(y0k) =

[(iωn GIC

mth +partτGICmth)(minusiτy0k)

]τ=β

τ=0

The Fourier transformed equation for the mixed propagator reads(ω

2n +k2 +M2

th)

GICmth(ωny0k) = (751)

=minusΠIIth (ωnk)GIC


mth)(y0k)

The boundary terms have to fulfill the equation of motion(part

2y0 +k2 +m2

B

)disc(GIC

mth)(y0k) =

(part

2y0 +k2 +m2

B

)disc(partτGIC

mth)(y0k) = 0

which follows from using GICmth(ωny0k) = GCI

mth(y0ωnk) and the equation of motion (750) for

GCImth Furthermore the initial conditions at y0 = 0 are fixed by the periodicity relation of the thermal

propagator as well as the equal-time commutation relations

disc(GICmth)(0k) = Gth(00k)minusGth(minusiβ 0k) = 0

party0 disc(GICmth)(0k) = party0Gth(00k)minusparty0Gth(minusiβ 0k)

= party0Gth(00k)minuspartx0Gth(00k) = i

disc(partτGICmth)(0k) = partτGth(00k)minuspartτGth(minusiβ 0k) = 1

party0 disc(partτGICmth)(0k) = party0partτGth(00k)minusparty0partτGth(minusiβ 0k) = 0

The statistical and spectral components GCCmF(0y0k) and GCC

mρ(0y0k) of the mixed propagatorare two linearly independent solutions of the free equation of motion Since it is a second orderdifferential equation any solution can be expressed as a linear combination especially

disc(GICmth)(y

0k) = GICmth(minusiτy0k)

∣∣τ=β

τ=0 = minusiGCCmρ(0y0k) (752)

disc(partτGICmth)(y

0k) = partτGICmth(minusiτy0k)

∣∣τ=β

τ=0 =GCC

mF(0y0k)Gth(00k)

Inserting this result together with the self-consistent Schwinger-Dyson equation (D20) for the fullthermal propagator into eq (751) finally yields

GICmth(ωny0k) =

(GII

th (ωnk)Gth(00k)

)GCC

mF(0y0k)minus(

iωnGIIth (ωnk)

)GCC

mρ(0y0k)

=int

Cdz0

∆m(ωnz0k)GCCmth(z

0y0k) (753)


where in the second line an integration over the closed real-time path C was inserted as well as theldquomixed connectionrdquo

∆m(ωnz0k) =(

GIIth (ωnk)

Gth(00k)

)δs(z0)+

(2iωnGII

th (ωnk))

δa(z0)

= ∆sm(ωnk)δs(z0)+∆

am(ωnk)δa(z0) (754)

=

which is only supported at the initial time z0 = 0plusmn Furthermore the transposed connection is definedas ∆T

m(z0ωnk) = ∆m(ωnz0k) Eq (753) for the mixed propagator is the extension of eq (736)for the free propagator Thus the mixed propagator evaluated with one real and one imaginary timeis decomposed into the convolution of the mixed connection which involves the full 2PI propagatorand the real-real mixed propagator which obeys the free equation of motion

Full thermal propagator

Using the equation of motion (747) of the mixed propagator the self-consistent equation of mo-tion (743) of the full propagator can be rewritten as


mth(xy)minus[1minus1I(x0)1I(y0)

]Πth(xy) for x0y0 isin C+I

By convolving this equation with Gth from the left and with Gmth from the right the integratedSchwinger-Dyson equation is obtained

Gth(xy) = Gmth(xy)+int

C+Id4uint

C+Id4vGth(xu)

[1minus1I(u0)1I(v0)

]Πth(uv)Gmth(vy) (755)

Evaluating it for x0 isin C and y0 isin I and performing a Fourier transformation with respect to therelative spatial coordinate xminusy as well as the imaginary time y0 gives

GCIth (x0ωnk) = GCI

mth(x0ωnk)+

intC+I

du0int

Cdv0(

Gth(x0u0k)Πth(u0v0k)GCImth(v

0ωnk))

minus iint

Cdu0 GCC

th (x0u0k)Πth(u0ωnk)GIImth(ωnk)

Next GCImth(x

0ωnk) and GCImth(v

0ωnk) are replaced using eq (753) with interchanged arguments

GCIth (x0ωnk)

=int

Cdz0[

GCCmth(x

0z0k)+int

C+Idu0

intCdv0(

Gth(x0u0k)Πth(u0v0k) GCCmth(v

0z0k))]

∆Tm(z0ωnk)

minus iint

Cdu0 GCC

th (x0u0k)Πth(u0ωnk)GIIth (ωnk)

=int

Cdz0[

GCCth (x0z0k)minus

intCdu0int

Idv0(

GCCth (x0u0k)Πth(u0v0k) GIC

mth(v0z0k)

)]∆

Tm(z0ωnk)

minus iint

Cdu0 GCC


=int

Cdz0 GCC

th (x0z0k)

∆

Tm(z0ωnk)minus iΠth(z0ωnk)GII

th (ωnk)

minusint

Cdu0int

Idv0

Πth(z0v0k)GICmth(v

0u0k)∆Tm(u0ωnk)


where GIImth(ωnk) = GII

th (ωnk) has been used (see eq (749)) In the second step the Schwinger-Dyson equation (755) evaluated for x0z0 isin C was used again In the third step the full real-realpropagator was factored out by interchanging the integration variables u0harr z0 in the second and thirdterm The last line can be simplified by Fourier transforming the imaginary time v0 and performingthe integral over C using eq (754)int

Cdu0int

Idv0

Πth(z0v0k)GICmth(v

0u0k)∆Tm(u0ωnk) =

= minusiT suml

intCdu0

Πth(z0ωlk)GICmth(ωlu0k)∆T

m(u0ωnk)

= minusiT suml

Πth(z0ωlk)GICmth(ωl0k)∆s

m(ωnk)

= minusiT suml

Πth(z0ωlk)GIIth (ωlk)

GIIth (ωnk)

Gth(00k)

Finally a decomposition of the full thermal 2PI propagator evaluated with one real time and oneMatsubara frequency is obtained

GCIth (x0ωnk) =

intCdz0 GCC

th (x0z0k)

∆


th (ωnk)

minus iT summ

Πth(z0ωmk)GIIth (ωmk)

GIIth (ωnk)

Gth(00k)

=

intCdz0 GCC

th (x0z0k)∆T (z0ωnk) (756)

In the last line the ldquofull connectionrdquo was introduced

∆T (z0ωnk) = ∆

Tm(z0ωnk)minus iT sum

mΠth(z0ωmk)

[δnm

TGII

th (ωnk)minusGII

th (ωmk)GIIth (ωnk)

Gth(00k)

]equiv ∆


mΠth(z0ωmk)D(ωmωnk) (757)

with ∆(ωnz0k) = ∆T (z0ωnk) Compared to the mixed connection the full connection contains anadditional term which is the convolution of the thermal self-energy evaluated with one real time andone Matsubara frequency with the propagator D(ωmωnk) defined in the last line This propagatorcan be rewritten as

D(ωnωmk) =δnm

TGII

th (ωnk)minusGII

th (ωnk)GIIth (ωmk)

Gth(00k)(758)

=δnm

TGII

th (ωnk)minus∆sm(ωnk)Gth(00k)∆s

m(ωmk)

=δnm

TGII

th (ωnk)minusint

Cdw0

intCdz0

∆m(ωnw0k)Gth(w0z0k)∆Tm(z0ωmk)

=δnm

TGII

th (ωnk)minusint

Cdw0

intCdz0

∆(ωnw0k)Gth(w0z0k)∆T (z0ωmk)

In the last lineintCdw0int

Cdz0 X(ωnw0k)Gth(w0z0k)Πth(z0ωmk)= 0 was used where X isin∆ΠthThe propagator D has the properties

D(ωnωmk) = D(ωmωnk) T summ

D(ωnωmk) = 0 (759)


From the last property it can be inferred that only the non-local part of the thermal self-energyΠth(z0ωmk) = Πloc

th + Πnlth(z

0ωmk) contributes in eq (757) since the local part is independentof the Matsubara frequency (see eq D13)

T summ

Πth(z0ωmk)D(ωmωnk) = T summ

Πnlth(z

0ωmk)D(ωmωnk)

By applying the inverse Fourier transformation with respect to imaginary time using in particular

D(minusiτminusiτ primek) = T 2sumnm

eiωnτminusiωmτ primeD(ωnωmk)

the full thermal 2PI propagator with one imaginary and one real time can be decomposed as

GCIth (x0minusiτk) =

intCdz0 GCC

th (x0z0k)∆T (z0minusiτk)

=

GICth (minusiτy0k) =

intCdz0

∆(minusiτz0k)GCCth (z0y0k) (760)

=

where the full connection is given by

∆(minusiτz0k) = ∆m(minusiτz0k) +int

Idv0 D(minusiτv0k)Πnl

th(v0z0k) (761)

= ∆s(minusiτk)δs(z0)+∆

a(minusiτk)δa(z0) +int


th(v0z0k)

= = +

∆T (z0minusiτk) = ∆(minusiτz0k) = (762)

The coefficients ∆sa(minusiτk) are derived from eq (753) They are given in terms of the full thermal2PI propagator evaluated on the imaginary contour I

∆s(minusiτk) = ∆

sm(minusiτk) =

GIIth (minusiτ0k)Gth(00k)

∆a(minusiτk) = ∆

am(minusiτk) = 2partτGII

th (minusiτ0k) (763)

Eqs (760761763) constitute the nonperturbative generalizations of eqs (736737) The nonper-turbative generalization of eq (742) is obtained from eq (758) using eq (D19)

GIIth (minusiτminusiτ primek) = D(minusiτminusiτ primek)+

intCdw0

intCdz0

∆(minusiτw0k)Gth(w0z0k)∆T (z0minusiτ primek)

= D(minusiτminusiτ primek)+∆sm(minusiτk)Gth(00k)∆s

m(minusiτ primek) (764)

= +

= +

Note that only the parts of the connections which are proportional to δsa(w0) and δsa(z0) contributeto the integrals in the first line The parts involving Πnl

th do not contribute since the integrals over theclosed real-time path in the first line vanish for them This is due to the fact that GII

th and D purelydepend on imaginary time arguments



Similar to the free propagator the full propagator connecting imaginary and real times can be de-composed into a convolution of the full ldquoconnectionrdquo ∆(minusiτz0k) and the full real-real propagatorHowever equation (761) for the full connection is an implicit equation due to the extra contributionof the non-local part of the full thermal self-energy For example for the 2PI three loop approxima-tion in the Z2-symmetric phase the thermal self-energy is given by the tadpole- and the setting-sundiagrams which itself contain the full propagator Only the latter contributes to the non-local partsuch that eq (761) takes the form

= +

The full connection within a given 2PI truncation is the exact solution of equation (761) Formally itcan be expanded in an infinite series obtained from iteratively inserting the mixed connection for thefull connection

∆(0)(minusiτz0k) = ∆m(minusiτz0k)

∆(k+1)(minusiτz0k) = ∆m(minusiτz0k)+

intIdv0 D(minusiτv0k) Π

nlth(v

0z0k)∣∣GIC

th rarr∆(k)lowastGCCth

(765)

For example for the 2PI three loop approximation in the Z2-symmetric phase the first steps of thisiteration can be depicted as

= + +

+

+

where the first line represents the zeroth step and the first step and the second line shows all diagramscontributing at the second step The diagrams in the third line appear at the third step The diagramsare generated with the correct symmetry factors (see footnote 2 on p 91) Obviously the expansioncan be re-organized as an expansion in the number of mixed connections contained in each diagram


Similar to the perturbative case the formalism established above can be used to relate any Feynmandiagram formulated on the thermal time path (ldquoC+Irdquo) the lines of which are given by the fullpropagator with a set of Feynman diagrams formulated on the closed real-time path containing non-local effective vertices representing the thermal initial correlations (ldquoC+ αrdquo) This is accomplishedby three steps

1 First the contour integrations over the thermal time path C+I associated with interaction ver-tices are split into two integrations over C and I A diagram with V vertices is thus decomposedinto 2V contributions

2 Second all internal propagator lines connecting a real and an imaginary time are replaced bythe convolution of the full connection with the real-real propagator according to eq (761) Ad-ditionally the internal propagator lines connecting two imaginary times are replaced accordingto eq (764) The parts containing imaginary times are encapsulated which can be visualizedby joining the full ldquoconnectionsrdquo to boxes surrounding the imaginary vertices

3 Third the series expansion of the full connection in terms of the mixed connection is insertedEach resulting contribution can be identified as a diagram formulated on the closed real-timepath C containing non-local effective vertices αn The latter are constructed explicitly asappropriate for the underlying 2PI approximation

The first two steps are analogous to the perturbative case with full propagators and connectionsinstead of free ones The third step is special for the nonperturbative case It results in contributionswhich contain non-local effective vertices α th

n of arbitrarily high order n These take into accountthermal initial n-point correlations which are present for all n due to the underlying nonperturbativeapproximation For example for the full setting sun diagram step one and two can be written as

S(xy) = =(minusiλ )2

6

intC+I

d4uint

C+Id4vGth(xu)Gth(uv)3Gth(vy)

=

= + + +

+ + +

(766)The symmetry factors of all diagrams are taken into account properly (see footnote 2 on page 91)


For the setting-sun diagram with one real and one imaginary vertex the third step can be written as

=

+

+

The first diagram in the second line is obtained by inserting the zeroth iteration for the four fullconnections ∆rarr ∆(0) = ∆m The other diagrams are obtained by inserting the first iteration ∆rarr ∆(1)The ellipsis stand for the contributions obtained by inserting the second and higher iterations of thefull connection All diagrams shown above are generated with the correct symmetry factorEach of the boxes with thin lines represents a non-local effective vertex encoding the correlationsof the initial state Accordingly a thin box which is attached to n propagator lines represents acontribution to the initial correlation αnThe thermal initial correlations are determined by the matrix element of the thermal density matrixAs has been shown in section D11 the thermal initial correlations can be expanded in a seriesof connected Feynman diagrams with propagator D0(minusiτminusiτ primek) (see eq (D6)) and ldquoimaginaryrdquovertices within perturbation theory Moreover in section 721 it has been shown that these appear assub-diagrams inside the perturbative non-local effective vertices denoted by the thin boxesWithin the 2PI framework the thermal effective non-local vertices are also given by subdiagramsinside the thin boxes however with lines representing the propagator D(minusiτminusiτ primek) which is deter-mined by the full thermal propagator (see eq (764)) These subdiagrams represent the approximationof the full thermal initial correlations which are appropriate in the nonperturbative case Within theformalism established above these can be constructed explicitly For example the lowest order non-perturbative thermal 4-point and 6-point initial correlations are given by

αth40L2PI(z1z2z3z4) = minusiλ

intId4v∆m(vz1)∆m(vz2)∆m(vz3)∆m(vz4)

th

0L 2PI= equiv (767)

αth60L2PI(z1z2 z6) = (minusiλ )2

intId4vint

Id4w∆

Tm(z1v)∆T

m(z2v)∆Tm(z3v)D(vw)

∆m(wz4)∆m(wz5)∆m(wz6)

th

0L 2PI= equiv (768)


723 Kadanoff-Baym Equation for the Thermal Initial State

On the one hand the equation of motion for the full thermal propagator defined on the closed real-time contour C is given by the Kadanoff-Baym equation for a thermal initial state represented bythermal initial correlations α th

n (ldquoC+αrdquo) The latter is a special case of the Kadanoff-Baym equationfor a non-Gaussian initial state (see eq (725)) which has the form(

part2x0 +k2 +M2

th)

Gth(x0y0k) =minusiδC(x0minus y0) (769)

minus iint

Cdz0 [

ΠGthnl(x

0z0k)+ΠnGthnl(x

0z0k)+ iΠthλα(x0z0k)]

Gth(z0y0k)

where ΠGthnl(x

0z0k) and ΠnGthnl(x

0z0k) denote the Gaussian- and non-Gaussian parts of the non-local self-energy respectively and

Πthλα(x0z0k) = ΠthλαF(x0k)δs(z0)minus i2

Πthλαρ(x0k)δa(z0)

denotes the contribution from the non-Gaussian initial correlations which is only supported at theinitial time surface z0 = 0 (see section 71)On the other hand the equation of motion of the full thermal propagator based on the thermal timecontour (ldquoC+Irdquo) evaluated for x0y0 isin C (see eq D14) is(

part2x0 +k2 +M2

th)

Gth(x0y0k) =minusiδC(x0minus y0)minus iint

C+Idz0

Πnlth(x

0z0k)Gth(z0y0k)

For example for the three-loop truncation of the 2PI effective action in the Z2-symmetric phase(setting-sun approximation) the convolution of the thermal non-local self energy and the full thermalpropagator is

intC+I

dz0Π

nlth(x

0z0k)Gth(z0y0k) =

Using the full connection (761) the integral over the imaginary contour I can be rewritten asintIdz0

Πnlth(x

0z0k)Gth(z0y0k) =int

Idv0

Πnlth(x

0v0k)int

Cdz0

∆(v0z0k)Gth(z0y0k)

=int

Cdz0[int

Idv0

Πnlth(x

0v0k)(

∆m(v0z0k)+int

Idw0 D(v0w0k)Πnl

th(w0z0k)

)]Gth(z0y0k)

Inserting this into the upper equation of motion it takes precisely the form of the Kadanoff-Baymequation for a non-Gaussian initial state By comparison the non-Gaussian contributions to the self-energy for the thermal initial state can be inferred

ΠGthnl(x

0z0k) = Πnlth(x

0z0k)∣∣x0z0isinC (770)

ΠnGthnl(x

0z0k) =int

Idv0int

Idw0

Πnlth(x

0v0k)D(v0w0k)Πnlth(w

0z0k)∣∣∣∣x0z0isinC

iΠthλα(x0z0k) =int

Idv0

Πnlth(x

0v0k)∆m(v0z0k)∣∣∣∣x0z0isinC

For the setting-sun approximation the steps listed above leading from the formulation of the Kada-noff-Baym equation on the thermal time path (ldquoC+Irdquo) to the formulation on the closed real-time


path with thermal initial correlations (ldquoC+αrdquo) are

Thus the Gaussian and non-Gaussian contributions to the self-energy in setting-sun approximationfor a thermal initial state are

ΠGthnl(x

0z0k) =

ΠnGthnl(x

0z0k) = (771)

iΠthλα(x0z0k) =

In order to explicitly obtain the thermal initial correlations which are appropriate for a specific 2PIapproximation the iterative expansion (765) of the full connection in terms of the mixed connectionhas to be inserted This yields a series expansion of the non-Gaussian self-energies

Πthλα =infin

sumk=0

Π(k)thλα

ΠnGthnl =

infin

sumk=0

Π(k)nGthnl (772)

where

Π(0)thλα

(x0z0k) = Πthλα(x0z0k)∣∣∣∣GIC

th rarr∆(0)lowastGCCth

Π(k)thλα



minus Π(kminus1)thλα

(x0z0k)

and analogously for Π(k)nGthnl For example in setting-sun approximation the thermal initial correla-

tions obtained from inserting the zeroth first and second iteration of the full connection are

iΠ(0)thλα

(x0z0k) = =th

0L 2PI(773)


iΠ(1)thλα

(x0z0k) =

iΠ(2)thλα

(x0z0k) =

The zeroth contribution contains the thermal non-local effective 4-point vertex (767) The first con-tribution contains three diagrams with thermal effective 6- 8- and 10-point vertices and the seconditeration yields six contributions with thermal effective 8- 10- 12- (two diagrams) 14- and 16-pointvertices the smallest and largest of which are depicted in the last line of (773) The expansion ofΠnG

thnl contains thermal non-local effective vertices of order six and higher

Π(0)nGthnl (x0z0k) = =

th

0L2PI

Π(1)nGthnl (x0z0k) =

The zeroth contribution contains the thermal non-local effective 6-point vertex (768) The first con-tribution contains 15 diagrams with thermal effective vertices of order 8 to 18The order of the thermal initial correlations appearing up to the fifth contribution in setting-sun ap-proximation are shown in table 71 Only a single term contains an initial 4-point correlation whichis given in the first line of eq (773)

iΠ(0)thλα

(xz) =minusiλ

6

intCd4x123Gth(xx1)Gth(xx2)Gth(xx3) iα th

40L2PI(x1x2x3z) (774)

Furthermore the upper term yields the only contribution to the Kadanoff-Baym equation (769) forthe thermal initial state which does not contain an internal ldquorealrdquo vertex Thus all other contribu-tions contain at least one contour integral over the closed real-time path C associated to internal realvertices These integrals have to vanish when all external times approach the initial time since theintegrations over the two branches of the closed real-time contour yield identical contributions withopposite sign Therefore in the limit x0 y0rarr 0 only the diagram containing the initial 4-point cor-relation given in the first line of eq (773) contributes to the right hand side of the Kadanoff-Baymequation (769) for the thermal initial state in setting-sun approximation(

part2x0 +k2 +M2

th)

GthF(x0y0k)∣∣x0=y0=0 = Π

(0)thλαF(x0k)GthF(0y0k)

∣∣∣x0=y0=0

(part

2x0 +k2 +M2

th)

Gthρ(x0y0k)∣∣x0=y0=0 = 0 (775)

73 Renormalized Kadanoff-Baym Equation for the Thermal Initial State 103

Πthλα(x0z0k)

4 6 8 10 12 14 16 middot middot middot 22 middot middot middot 28 middot middot middot 34 middot middot middot0 times1 times times times2 times times times times times3 times times times times middotmiddot middot times4 times times times middotmiddot middot times middot middot middot times5 times times middotmiddot middot times middot middot middot times middot middot middot times

ΠnGthnl(x

0z0k)

4 6 8 10 12 14 16 18 middot middot middot 30 middot middot middot 42 middot middot middot 54 middot middot middot 66 middot middot middot0 times1 times times times times times times2 times times times times times middotmiddot middot times3 times times times times middotmiddot middot times middot middot middot times4 times times times middotmiddot middot times middot middot middot times middot middot middot times5 times times middotmiddot middot times middot middot middot times middot middot middot times middot middot middot times

Table 71 Thermal initial correlations in 2PI setting-sun approximation The column number is theorder n = 46 of the thermal initial n-point correlation The row number k = 01 shows whichinitial correlations contribute to Π

(k)thλα

(upper table) and Π(k)nGthnl (lower table) respectively Due to

the Z2-symmetry only even correlations are non-zero

In summary the formulation of the equation of motion for the thermal propagator derived from the2PI effective action on the closed real-time path can now serve as the link required to combine thenonperturbative 2PI renormalization with Kadanoff-Baym equations

73 Renormalized Kadanoff-Baym Equation for the Thermal InitialState

On the one hand the nonperturbative renormalization procedure of the 2PI effective action describedin section 62 renders the thermal propagator defined on the thermal time path finite On the otherhand the Schwinger-Keldysh propagator which is the solution of the Kadanoff-Baym equations forthe thermal initial state coincides with the thermal propagator on the real time axis Therefore thenonperturbative renormalization procedure of the 2PI effective action also renders the Kadanoff-Baym equations for the thermal initial state finite The corresponding renormalized thermal initialcorrelations

αnR(x1 xn) = Zn2αn(x1 xn) (776)

are obtained by transferring the renormalized Schwinger-Dyson (622) equation formulated on thethermal time path C+I to the formulation on the closed real-time path with initial correlations (C+α)as described above


The renormalized Kadanoff-Baym equation for the thermal initial state thus reads(2x +δZ02x +m2

R +δm20 +

λR +δλ0

2GthR(xx)

)GthR(xy) =minusiδ 4

C(xminus y) (777)

minus iint

Cd4z[Π

GthnlR(xz)+Π

nGthnlR(xz)+ iΠthλαR(xz)

]GthR(zy)

where δZ0 δm20 and δλ0 are the 2PI vacuum counterterms as determined by the nonperturba-

tive renormalization procedure of the 2PI effective action at finite temperature and GthR(xy) =Zminus1Gth(xy) The renormalized self-energies for the thermal initial state are obtained from eq (770)

ΠGthnlR(xz) = Z Π

Gthnl(xz) = Π

nlthR(xz)

∣∣x0z0isinC (778)

ΠnGthnlR(xz) = Z Π

nGthnl(xz) =

intId4vint

Id4wΠ

nlthR(xv)DR(vw)Πnl

thR(wz)∣∣∣∣x0z0isinC

iΠthλαR(xz) = Z iΠthλα(xz) =int

Id4vΠ

nlthR(xv)∆m(vz)

∣∣∣∣x0z0isinC

where DR(xy) = Zminus1D(xy) is the renormalized propagator from which the thermal initial correla-tions are constructed via the iterative expansion (772)In the three-loop approximation of the 2PI effective action the non-local part of the renormalizedthermal self-energy which is given by the setting-sun diagram

ΠnlthR(xy) =

(minusiλR)2

6GthR(xy)3

contains the renormalized coupling Therefore all thermal initial correlations which are generated viathe iterative expansion (772) also contain the renormalized coupling For example the contributionof the zeroth iteration (which is the only one containing an initial 4-point correlation) is given by

iΠ(0)thλαR(xz) =

minusiλR

6

intCd4x123GthR(xx1)GthR(xx2)GthR(xx3) iα th

40L2PIR(x1x2x3z) (779)

where the renormalized thermal initial 4-point correlation is given by

iα th40L2PIR(z1z2z3z4) =minusiλR

intId4v∆m(vz1)∆m(vz2)∆m(vz3)∆m(vz4) (780)

Altogether it has been possible to explicitly construct a class of renormalized solutions of Kada-noff-Baym equations (namely those for thermal initial states) which can serve as the basis to deriverenormalized Kadanoff-Baym equations for nonequilibrium initial states

Chapter 8

Renormalization of Kadanoff-BaymEquations

In recent years it turned out that the 2PI effective action [66] defined on the closed real-time path [68126166] is an excellent starting point to study quantum fields out of thermal equilibrium [122532]So far however in this highly nonperturbative context the issue of renormalization has not beenaddressed properlyAs mentioned in the introduction there are several reasons why a proper renormalization of Kadanoff-Baym equations derived from the 2PI effective action is desirable Most important it is required for aquantitative comparison with semi-classical approximations like Boltzmann equations Furthermorerenormalization can have an important quantitative impact on solutions of Kadanoff-Baym equationsis crucial in order to identify physical initial states and enhances the robustness of the computationalalgorithm [147]In this chapter nonperturbatively renormalized Kadanoff-Baym equations are proposed and theirfiniteness is verified analytically for a special case The relevance of renormalization for Kadanoff-Baym equations is illustrated by means of numerical solutionsIn section 81 it is shown that it is necessary to extend the Kadanoff-Baym equations (615) (whichhave been the basis for numerical investigations so far) in order to be compatible with renormalizationThen the tools derived in chapter 7 are used in order to tackle the nonperturbative renormalization ofKadanoff-Baym equations which is done in section 82 by including an initial 4-point correlation Animportant reference value for the latter is the thermal value for which the connection to the nonper-turbative renormalization procedure of the 2PI effective action is demonstrated explicitly Finally therelevance of nonperturbative counterterms as well as non-Gaussian initial correlations for numericalsolutions of Kadanoff-Baym equations is demonstrated in section 83

81 Kadanoff-Baym Equations and 2PI Counterterms

On the one hand it has been shown [28] that nonperturbative 2PI vacuum counterterms render alln-point functions derived from the 2PI effective action finite in thermal equilibrium In particularthis means that these 2PI counterterms can be chosen independent of the temperatureOn the other hand it has been shown [32] that Kadanoff-Baym equations respect late-time univer-sality meaning that the late-time behavior depends only on conserved quantities like average energydensity and global charges but not on the details of the initial conditions and that the solutionsasymptotically approach a stationary state for which the effective particle number distribution con-verges towards a thermal Bose-Einstein distribution

106 8 Renormalization of Kadanoff-Baym Equations

Altogether this suggests that the 2PI vacuum counterterms are adequate to renormalize the solutionsof Kadanoff-Baym equations for late times for any appropriate initial condition However as will beshown below inserting the 2PI counterterms into the Kadanoff-Baym equations (615) is not sufficientfor their renormalization Instead it is additionally required to remove the restriction to a Gaussianinitial stateBy splitting the bare mass- and coupling appearing in the bare classical action (624) into renormal-ized parts and counterterms (see eq 625) and rescaling the field value the self-consistent Schwinger-Dyson equation (611) derived from the 2PI effective action (69) for a Gaussian initial state formu-lated on the closed real-time path can be written as

Gminus1R (xy) = i

(2x +m2

R)

δ4C(xminus y)minusΠR(xy) (81)

ΠR(xy) = minusi(

δZ02x +δm20 +

λR +δλ0

2GR(xx)

)δ

4C(xminus y)+

(minusiλR)2

6GR(xy)3

It is equivalent to the Kadanoff-Baym equations (615) The full connected Schwinger-Keldysh prop-agator GR(xy) equiv Zminus1G(xy) also appears in the self-energy ΠR(xy) which is given in ldquosetting-sunapproximationrdquo (see section 61) here It contains counterterms parameterized analogously to thecorresponding Schwinger-Dyson equation (626) in thermal equilibriumOne peculiarity of the Kadanoff-Baym equations (615) is that at the initial time only the local part ofthe self-energy (which is proportional to δ 4

C(xminusy)) contributes while the non-local part is suppresseddue to the memory integrals which vanish at the initial time Since both parts of the self-energycontain divergences it is thus impossible to choose the counterterms such that the Kadanoff-Baymequations for a Gaussian initial state are finite at t = 0 and t gt 0 simultaneouslySo far an approximate perturbative renormalization prescription has been used by default [17] Thisprescription is designed such that it is appropriate at the initial time t = 0 while it misses divergencesoccurring at t gt 0 In contrast to this the nonperturbative renormalization procedure (see section 62)which can as explained above be expected to be correct for trarr infin fails at t = 0 for a Gaussian ini-tial state since the divergence contained in the setting-sun diagram which is to be canceled by thecoupling counterterm vanishes at the initial time The reason for this are the missing higher correla-tions at the initial time Therefore it is necessary to extend the Kadanoff-Baym equations (615) tonon-Gaussian initial states

82 Renormalizable Kadanoff-Baym Equations from the 4PI EffectiveAction

In thermal equilibrium the full thermal 4-point correlation function carries logarithmic divergenceswhich are accounted for by the 2PI renormalization prescription However for a Gaussian initial statethe connected 4-point correlation function vanishes at the initial time by construction In order totransfer the 2PI renormalization prescription to Kadanoff-Baym equations it is therefore importantto take a 4-point correlation into account from the beginning onThe 4PI effective action provides an efficient framework to derive Kadanoff-Baym equations forinitial states featuring a non-Gaussian 4-point correlation for which reason its three-loop truncationis employed below1

1Note however that it is also possible to derive these equations without reference to the 4PI effective action This has theadvantage that completely general initial states (featuring also initial n-point correlations for n gt 4) as well as truncationsof the 2PI effective action which cannot be obtained via the 4PI effective action [26] can also be incorporated on the samefooting The general formalism can be found in section 71

82 Renormalizable Kadanoff-Baym Equations from the 4PI Effective Action 107

821 4PI Effective Action with Initial 4-Point Correlation

The generating functional for nonequilibrium correlation functions describing an ensemble charac-terized by the density matrix ρ at an initial time tinit equiv 0 in the presence of classical external 2- and4-point sources can be represented by the path integral (see section 61)

Zρ [KL] =intDϕ 〈ϕ+0 |ρ|ϕminus0〉 exp

(iS[ϕ]+

i2

ϕ Kϕ +i

4L1234ϕ1ϕ2ϕ3ϕ4

) (82)

The density matrix element for an initial state featuring a non-Gaussian 4-point correlation can beparameterized as


iα0 +i2

ϕα2ϕ +i

4(α4)1234ϕ1ϕ2ϕ3ϕ4

) (83)

where the short-hand notations (321 334) apply (withintrarrintC) Here only the Z2-symmetric case

where all odd correlation functions vanish at all times is covered for simplicity The generalizationcan be found in section 71 The kernels characterizing the initial correlations are supported at theinitial time only (ie for t = 0+ and t = 0minus on C)

αn(x1 xn) = sumε1=plusmnmiddot middot middotsum

εn=plusmnα

ε1εnn (x1 xn)δC(x0

1minus0ε1) middot middot middotδC(x0nminus0εn) (84)

In this case the contribution of the density matrix to the generating functional can be absorbed into theexternal sources K +α2rarrK and L+α4rarr L (the constant α0 can be absorbed into the normalizationof the path integral measure)The 4PI effective action Γ[GV4] is the double Legendre transform of the generating functional (82)with respect to the external sources The latter has the same structure as the corresponding generatingfunctional (333) with 2- and 4-point sources in vacuum except that all time-integrations are per-formed over the closed real-time path Consequently the 4PI effective action for the initial state (83)is obtained from the parameterization given in eq (337) by replacing the time-integrations

intrarrintC

822 Kadanoff-Baym Equation with Initial 4-Point Correlation

The equation of motion for the connected 4-point function derived from the 4PI effective action is

δ

δV4(xyzw)Γ[GV4] =minus

14

L(xyzw) (85)

and the equation of motion for the Schwinger-Keldysh propagator reads

δ

δG(xy)Γ

L[G] =minus12

K(xy) (86)

Here the external sources are formally not zero for the physical situation but K(xy) = α2(xy) andL(xyzw) = α4(xyzw) due to the density matrix element (83) Furthermore ΓL[G] denotes the2PI effective action obtained from inserting the solution V4 of eq (85) into the 4PI effective actionand performing the inverse Legendre transform with respect to the 4-point source (where d4x1234 =d4x1 middot middot middotd4x4 and G12 = G(x1x2))

ΓL[G] = Γ[GV4]+

14

intCd4x1234 L(x1x2x3x4) [V4(x1x2x3x4)+G12G34 +G13G24 +G14G23]


In the following the three-loop approximation (setting-sun approximation) of the 4PI effective action(see section 33) is considered for concreteness Although the three-loop 2PI and three-loop 4PIapproximations agree in the absence of sources this is not the case here due to the initial 4-pointcorrelation L = α4 6= 0 Instead the solution of eq (85) obtained from eq (337) is

δΓ

δV4=minusα4

4hArr iA4(x1x2x3x4) =minusiλδ

4C(x1minus x2)δ 4

C(x1minus x3)δ 4C(x1minus x4)+ iα4(x1x2x3x4)

Thus the kernel iA4 equiv iAG4 + iAnG

4 is given by the sum of the classical vertex which is also presentin the Gaussian case and the non-Gaussian initial 4-point correlation AnG

4 equiv α4 Accordingly the4-point function has two contributions given by

V4(x1x2x3x4) =int

Cd4y1234 G(x1y1)G(x2y2)G(x3y3)G(x4y4)[(iAG

4 + iAnG4 )(y1y2y3y4)]

equiv V G4 (x1x2x3x4)+V nG

4 (x1x2x3x4) (87)

The corresponding 2PI effective action ΓL[G] is obtained by inserting V4 into the 4PI effective actionand setting L = α4 The result coincides with the 2PI effective action (715) considered in section 71Therefore the Kadanoff-Baym equations are

(2x +M2(x)

)GF(xy) =

int y0

0d4zΠF(xz)Gρ(zy)minus

int x0

0d4zΠρ(xz)GF(zy)

minus λ

6V nG

4 (xxxy) (88)(2x +M2(x)

)Gρ(xy) =

int y0

x0

d4zΠρ(xz)Gρ(zy)

They constitute an extension of the Kadanoff-Baym equations (615) incorporating a non-Gaussianinitial 4-point correlation which leads to the additional contribution in the second line It has to beemphasized that in contrast to the memory integrals this contribution does not have to vanish whenx0y0rarr 0 The effective mass M2(x) and the non-local self-energies ΠFρ(xy) are identical to thosein the Gaussian case (see eq (616))

823 Renormalization

Motivated by the parameterization (626) of the renormalized 2PI effective action at finite temper-ature as well as the renormalized Kadanoff-Baym equation (777) for the thermal initial state thefollowing ansatz for the Kadanoff-Baym equation determining the renormalized Schwinger-Keldyshpropagator GR(xy) = Zminus1G(xy) is proposed


(2x +δZ02x +m2

R +δm20 +

λR +δλ0

2GR(xx)

)GFR(xy)

=int y0

0d4zΠFR(xz)GρR(zy) minus

int x0

0d4zΠρR(xz)GFR(zy)

minus λR

6V nG

4R(xxxy) (89a)

(2x +δZ02x +m2

R +δm20 +

λR +δλ0

2GR(xx)

)GρR(xy)

=int y0

x0

d4zΠρR(xz)GρR(zy) (89b)

Here δZ0 δm20 and δλ0 denote the 2PI vacuum counterterms determined by the nonperturbative

renormalization procedure The non-local part of the renormalized self-energy ΠR(xy) = Z Π(xy)is given by the setting-sun diagram with renormalized couplings

ΠnonminuslocalR(xy) = ΠFR(xy)minus i2

sgnC(x0minus y0)ΠρR(xy) =(minusiλR)2

6GR(xy)3

and V nG4R = Zminus2V nG

4 is given by the renormalized initial 4-point correlation α4R = Z2α4

V nG4R(xxxy) =

intCd4y1234 GR(xy1)GR(xy2)GR(xy3)[iα4R(y1y2y3y4)]GR(y4y)

Although the initial 4-point correlation α4R is only supported at the initial time it does lead to a non-zero contribution to the Kadanoff-Baym equations for non-zero times x0y0 ge 0 This can be seen byinserting the parameterization (84) into the upper equation

V nG4R(xxxy) =

intd3y1234 GR(xyε1)GR(xyε2)GR(xyε3)[iα

ε1ε2ε3ε44R (y1y2y3y4)]GR(yε4 y)

The four time integrations over the closed contour are annihilated by the four Dirac distributions ofthe initial correlation Above summation over εi =plusmn is implied and

GR(xyε) = GR(x0x0ε y) = GFR(x0x0y)minus iε2

GρR(x0x0y) for ε isin +minus

The non-Gaussian contribution to the Kadanoff-Baym equations (89) may also be understood as acontribution to the self-energy which is only supported at the initial time surface y0 = 0plusmn

minus λR

6V nG

4R(xxxy)equivint

Cd4y4 ΠλαR(xy4)GR(y4y) (810)

where

iΠλαR(xy) =16

intCd4y123 [minusiλR]GR(xy1)GR(xy2)GR(xy3)[iα4R(y1y2y3y)]

equiv iΠλαFR(x0xy)δs(y0)minus i2

iΠλαρR(x0xy)δa(y0) (811)

with δsa(y0) equiv[δC(y0minus0+)plusmnδC(y0minus0minus)

]2 Due to the structure of the initial correlation the

three propagators appearing in the non-Gaussian contribution ΠλαR(xy) to the self-energy are evalu-ated at the times t = x0 and tinit = 0 For sufficiently dense and strongly coupled systems the unequal-time propagators GFρR(x00k) are damped exponentially for each momentum mode k (see left part


0001

001

01

1

0 5 10 15 20 25 30 35 40 45 50

G(t

0k

=0)

t mR

GF(t0k)

Gρ(t0k)

-06-04-02

0 02 04 06 08

1 12 14 16

0 05 1 15 2 25 3 35 4

Πλα

(tk

=0)

t mR

ΠλαF(tk)

Πλαρ(tk)

Figure 81 Left The unequal-time propagator is damped exponentially The damping rate increaseswith the density and the coupling strength of the system Right The non-Gaussian contribution tothe self-energy is strongly damped Thus the contribution of the initial 4-point correlation is mostrelevant close to the initial time t = 0

of figure 81) Therefore also ΠλαFρR(x0k) =intd3xeminusik(xminusy) ΠλαFρR(x0xy) is damped expo-

nentially with respect to x0 (see right part of figure 81) Hence the contribution of the initial 4-pointcorrelation to the Kadanoff-Baym equation is suppressed for times much larger than the characteristicdamping time-scale This means in particular that all properties of solutions of Kadanoff-Baym equa-tions at late times including universality and thermalization are not changed Instead the influenceof the initial 4-point correlation is maximal near the initial time Additionally the memory integralsvanish for x0y0rarr tinit = 0 such that the non-Gaussian contribution minusλRV nG

4R(xxxy)6 makes upthe only non-zero term on the right-hand side of the Kadanoff-Baym equations in this limitIn section 81 it was observed that the 2PI vacuum counterterms renormalizing the 2PI effective actionin equilibrium which can be expected to be correct at late times fail for x0y0 rarr 0 for a Gaussianinitial state The reason was that the divergence contained in the memory integral which is to becanceled by the coupling counterterm vanishes at the initial time Now however it is possible toinvestigate whether the non-Gaussian initial 4-point correlation can be chosen such as to remedy thisshortcoming of the Gaussian initial state

824 Finiteness for Renormalized Initial States

In order to verify the ansatz (89) for renormalized Kadanoff-Baym equations it will be shown inthe following (as a first step) that the 2PI vacuum counterterms determined via the nonperturbativerenormalization procedure indeed render the Kadanoff-Baym equations finite in the limit x0y0rarr 0for the special case where both the initial 2- and 4-point correlations take their thermal valuesNote that nevertheless this initial state corresponds to a nonequilibrium situation since all highercorrelations are omitted However it represents the choice for which the deviation from thermal equi-librium is minimal within the class of initial states characterized by a density matrix of the form (83)In setting-sun approximation the renormalized thermal initial 4-point correlation is given by (seeeq (780))

iα th4R(z1z2z3z4) =minusiλR


where ∆m(vz) =int d3k

(2π)3 eik(vminusz) T sumn eiωnτ∆m(ωnz0k) for v = (minusiτv) denotes the Fourier trans-formed ldquomixed connectionrdquo defined in eq (754) For thermal initial 2- and 4-point correlations the


2- and 4-point functions in the limit x0y0rarr 0 are thus given by (see chapter 7)

GFR(xy)|x0y0=0 = GthR(xy)|x0y0=0 (813)

V nG4R(x1x2x3x4)|x0

i =0 = minusiλR

intId4vGthR(x1v)GthR(x2v)GthR(x3v)GthR(x4v)|x0

i =0

where GthR is the solution of the renormalized Schwinger-Dyson equation (626) obtained from thethree-loop truncation of the 2PI effective action at finite temperature Inserting this into the Kada-noff-Baym equation (89) for the statistical propagator evaluated at x0 = y0 = 0 yields (after dividingby Z = 1+δZ0)

part2x0GFR(xy)|x0y0=0 = minus

[minusnabla2 +Zminus1

(δm2

0 +m2R +

λR +δλ0

2GthR(xx)

)]GthR(xy)|x0y0=0

minus Zminus1 λR

6(minusiλR)

intId4vGthR(xv)3GthR(vy)|x0y0=0

After Fourier transforming with respect to (xminusy) as well as inserting the Fourier transformation ofthe thermal propagator with respect to the 4-momentum k = (ωnk) the upper equation becomes

part2x0GFR(x0y0k)|x0y0=0 = minusT sum

neiωnτ

[k2 +Zminus1

(δm2

0 +m2R +

λR +δλ0

2

intqGthR(q)

minus λ 2R

6

intpq

GthR(p)GthR(q)GthR(kminusqminus p))]

GthR(ωnk)|τrarr0

= minusT sumn

eiωnτ[k2 +Zminus1 (m2

R +ΠthR(k)minusδZ0k2)]GthR(ωnk)|τrarr0

The combination of the thermal tadpole- and setting-sun contributions in the inner brackets of thefirst line is precisely the same as for the renormalized thermal self-energy (626) which has beeninserted in the second line The nonperturbative renormalization procedure is designed such thatΠthR(k) is finite Therefore the thermal setting-sun contribution which stems from the contributionof the initial 4-point correlation is crucial for renormalization Next it is used that the thermal 2PIpropagator fulfills the self-consistent Schwinger-Dyson equation (626)


neiωnτ

[k2 +Zminus1

(Gminus1

thR(ωnk)minusZk2)]

GthR(ωnk)|τrarr0

= minusT sumn

eiωnτ[Zminus1minusω

2n GthR(ωnk)

]τrarr0

= minuspart2τ GthR(minusiτ0k)|τrarr0

where k2 = ω2n +k2 and T sumn eiωnτ = 0 for τ 6= 0 has been used The last expression is manifestly

finite since the full renormalized thermal propagator GthR(minusiτ0k) is finite for 0 le τ le β TheKadanoff-Baym equation for the spectral function does not involve any divergences for x0y0rarr 0

Outlook

It has been shown that the Kadanoff-Baym equations (89) supplied with 2PI vacuum countertermsderived from the three-loop truncation of the 2PI effective action with thermal initial 2- and 4-pointcorrelation are rendered finite in the limit x0y0 rarr 0 As discussed above in the opposite limitx0y0rarr infin where thermal equilibrium is approached the nonperturbative renormalization procedureof the 2PI effective action at finite temperature can also be expected to be appropriate In order to


show that the Kadanoff-Baym equations with thermal initial 2- and 4-point correlation are also ren-dered finite at intermediate times it is required to show that the truncation of the higher thermaln-point correlations for n ge 6 does not introduce any divergences Furthermore if the initial 2-pointcorrelation deviates from its thermal value it can be expected that the initial 4-point correlation alsohas to be modified such that the Kadanoff-Baym equations stay finite In order to investigate thisquestion it is necessary to expand the Kadanoff-Baym equations with nonequilibrium initial condi-tions around the renormalized Kadanoff-Baym equations for thermal equilibrium which have beenderived in chapter 7 Therefore it is required to formulate the Bethe-Salpeter equation encounteredin section 62 on the closed real-time path In this way it should be possible to derive criteria whichthe nonequilibrium initial state of the ensemble has to fulfill in order to be compatible with renormal-ization Only these ldquorenormalized initial statesrdquo may occur as real physical states of the ensembleAbove already one class of renormalized initial states could be identified namely those with thermalinitial 2- and 4-point correlation functions

83 Impact of 2PI Renormalization on Solutions of Kadanoff-BaymEquations

The Kadanoff-Baym equations (89) for the renormalized Schwinger-Keldysh propagator containcounterterms determined according to the nonperturbative renormalization procedure of the 2PI effec-tive action and take into account a non-Gaussian initial state featuring an initial 4-point correlation Inthis section the relevance of nonperturbative 2PI counterterms as well as the initial 4-point correlationis investigated by means of numerical solutions of Kadanoff-Baym equationsIn order to compare the nonperturbatively renormalized Kadanoff-Baym equations to the conven-tionally used Kadanoff-Baym equations which contain approximate perturbative counterterms andGaussian initial correlations both sets of equations are given in section 831 Next the numericalcomputation of the 2PI counterterms is discussed in section 832The impact of the non-Gaussian initial 4-point correlation is investigated in section 833 There-fore solutions of Kadanoff-Baym equations with Gaussian and non-Gaussian initial states but withidentical (2PI) counterterms are comparedThe impact of the renormalization prescription is investigated in section 834 by comparing solutionsof Kadanoff-Baym equations with approximate perturbative counterterms and with nonperturbative2PI counterterms but with identical (Gaussian) initial stateFinally in section 835 it is shown that the nonequilibrium time-evolution of the renormalizedSchwinger-Keldysh propagator is compatible with time-independent countertermsThe nonperturbative 2PI counterterms were determined with the renormalize program which wasdeveloped following the lines of Ref [29] Furthermore the numerical solutions of the Kadanoff-Baym equations are based on an extended version of the kadanoffBaymmm program [146 147]

831 Kadanoff-Baym Equations with Nonperturbative 2PI Counterterms and InitialFour-Point Correlation Function

The general form of the evolution equation for the full connected Schwinger-Keldysh two-point func-tion (Kadanoff-Baym equation) for a space-translation invariant system without further approxima-tions is

(part

2x0 +k2 +M2(x0)

)GF(x0y0k) =

int y0

0dz0


83 Impact of 2PI Renormalization on Solutions of Kadanoff-Baym Equations 113

minusint x0

0dz0

Πρ(x0z0k)GF(z0y0k)+ΠλαF(x0k)GF(0y0k)+14

Πλαρ(x0k)Gρ(0y0k)

complemented by a similar equation for the spectral function (see eq (726)) The information aboutthe underlying 2PI (loop) approximation and renormalization prescription is encoded in the expres-sions for the self-energies which will be given below for the cases of interest In particular non-Gaussian initial correlations enter via the contributions ΠλαFρ which vanish for a Gaussian initialstateThe Kadanoff-Baym equations (89) can be brought into the upper form by Fourier transforming withrespect to the relative spatial coordinate and parameterizing it in terms of ldquobarerdquo propagators G = ZGR

and self-energies Π = Zminus1ΠR (where Z = 1 + δZ0) Furthermore the parameterization of the initial4-point correlation described in eqs (810 811) is used Before presenting the resulting expressionsfor the self-energies corresponding to the full nonperturbative renormalization procedure those forthe approximate perturbative renormalization prescription are given for comparison

Approximate perturbative renormalization

So far when solving Kadanoff-Baym equations an approximate perturbative renormalization pre-scription has been used by default [17] Here only the mass is renormalized at one-loop order ofstandard perturbation theory while the coupling remains unchanged The bare mass is then given by

m2B = m2

Rminusλ

2

int d3p(2π)3

1

2radic

m2R +p2

reg

where the momentum integral is calculated employing a regulator (which is provided by the latticediscretization in the case of numerical calculations) As the coupling constant is unchanged the effec-tive mass and the nonlocal self-energies are given by

M2(x0) = m2B +

λ

2

int d3p(2π)3 GF(x0x0p)

ΠF(x0y0k) = minusλ 2

6

([GF lowastGF lowastGF ](x0y0k)minus 3

4[Gρ lowastGρ lowastGF ](x0y0k)

)

Πρ(x0y0k) = minusλ 2

6

(3[GF lowastGF lowastGρ ](x0y0k)minus 1

4[Gρ lowastGρ lowastGρ ](x0y0k)

)

The non-local parts contain the double convolutions

[GF lowastGF lowastGF ](x0y0k) =int d3p

(2π)3d3q

(2π)3 GF(x0y0p)GF(x0y0q)GF(x0y0kminuspminusq)

with similar expressions involving Gρ The approximate perturbative renormalization prescription isdesigned for a Gaussian initial state for which

ΠλαF(x0k) = Πλαρ(x0k) = 0

It is important to note that this perturbative renormalization prescription suffers from several short-comings First it neglects the renormalization of the coupling Second it does not take into accountcontributions from higher loop orders And third it ignores the nonperturbative nature of the under-lying 2PI formalism


Full nonperturbative renormalization

The Kadanoff-Baym equations for the renormalized Schwinger-Keldysh propagator which have beenproposed in eq (89) contain mass and coupling counterterms determined according to the full non-perturbative renormalization procedure of the 2PI effective action as well as an initial 4-point corre-lation function

Nonperturbative counterterms The Kadanoff-Baym equations (89) contain the full 2PI coun-terterms Their determination requires the solution of a self-consistent Schwinger-Dyson equationfor the full thermal propagator together with a Bethe-Salpeter equation for the appropriate 4-pointkernel (see section 62) Evaluated for the 3-loop truncation of the 2PI effective action both equationsread

Gminus1(k) = k2 +m2B +

λB

2

intq

G(q)minus λ 2R

6Z4

intpq

G(p)G(q)G(kminus pminusq)

V (k) = λBminusλB

2

intq

G2(q)V (q) (814)

minus λ 2R

Z4

intq

G(q)G(kminusq)+λ 2

R

2Z4

intpq

G(p)G(kminusqminus p)G2(q)V (q)

where G(k) equiv ZGthR(k) V (k) equiv ZVR(kq = 0) Z = 1 + δZ0 m2B = (m2

R + δm20)Z and λB = (λR +

δλ0)Z2 For given bare mass m2B and bare coupling λB the renormalized mass m2

R the renormalizedcoupling λR and the field renormalization Z are determined by the renormalization conditions

Zd

dk2 Gminus1vac(k = 0) = +1

Z Gminus1vac(k = 0) = m2

R (815)

Z2Vvac(k = 0) = λR

where Gvac (k) and Vvac (k) denote the solutions of eqs (814) obtained at zero temperature Desiredvalues for the renormalized mass and coupling can be achieved by an appropriate choice of the baremass and coupling (see section 832)

Initial 4-point correlation It is convenient to expand the initial 4-point correlation in terms of thesymmetric and antisymmetric Dirac distributions δsa(t) defined below eq (811)

α4(xyzw) = sumi jklisinsa

αi jkl4 (xyzw)δi(x0)δ j(y0)δk(z0)δl(w0)

which is equivalent to the expansion (84) The possible combinations of the upper indices togetherwith the Hermiticity condition (75) imply that it is parameterized by 16 real functions of four spatialpoints However only five of them are independent namely αssss

4 αaaaa4 αssaa

4 αsssa4 and αsaaa

4 while the other components are obtained by permutation of the four arguments If in addition thecontribution of the 4-point correlation to the density matrix (83) is real (which turns out to be true forall cases considered below) the latter two vanish such that only three independent functions remain

Self-energy The nonperturbatively renormalized effective mass and non-local self-energies aregiven by

M2 (x0) = m2B +

λB

2

int d3p(2π)3 GF(x0x0p) (816)


ΠF(x0y0k) = minus λ 2R

6Z4



)

Πρ(x0y0k) = minus λ 2R

6Z4



)

In addition a real initial 4-point correlation can be incorporated in the non-Gaussian self-energiesgiven by

ΠλαF(x0k) = minus λR

6Z2

([GF middotGF middotGF middot iαssss

4 ] (x00k)minus 34[Gρ middotGρ middotGF middot iαaass

4](x00k)

)

Πλαρ(x0k) = minus λR

6Z2

(3[GF middotGF middotGρ middot iαssaa

4](x00k)minus 1

4[Gρ middotGρ middotGρ middot iαaaaa

4](x00k)

)

Here the spatial Fourier transform of the initial 4-point correlation enters according to

[GF middotGF middotGF middot iαssss4 ] (x00k) =

=int d3p

(2π)3d3q

(2π)3 GF(x00p)GF(x00q)GF(x00kminuspminusq) iαssss4 (pqkminuspminusqminusk)

with similar expressions involving Gρ

832 Numerical Computation of Nonperturbative Counterterms

In order to be able to solve Kadanoff-Baym equations containing 2PI counterterms it is necessaryto compute the latter according to the nonperturbative renormalization procedure of the 2PI effectiveaction [2829] This has to be done numerically for two reasons First it is required to compute thesecounterterms with the identical regulator as for the Kadanoff-Baym equations which is providedby the lattice discretization Second the Schwinger-Dyson and Bethe-Salpeter equations cannot besolved analytically Accordingly these equations are solved numerically on a lattice with the samesize N3

s and lattice spacing as for the spatial coordinates as is used for the solution of the Kadanoff-Baym equations (typical values are Ns = 32 and asmR = 05) in order to obtain the 2PI countertermsfor the same regulatorThe discretization of the temporal direction determines the temperature according to T = 1(Ntat)The temporal lattice spacing at is chosen small enough such that the continuum limit is approached2If appropriate at may be chosen to coincide with the time-step used for the solution of the Kadanoff-Baym equations The lattice cutoff is then determined by the spatial spacing Λsim πasThe 2PI counterterms are determined by solving eqs (814 815) at a reference temperature T0 mR which is sufficiently close to the zero-temperature (infinite volume) limit by choosing Nt 10(mRat) Using the counterterms determined at the reference temperature the thermal propaga-tor at some temperature T 6= T0 is determined by solving eqs (814) on a lattice where Nt = 1(Tat)while at as and Ns remain fixedIn the course of this work the numerical computation of 2PI counterterms has been achieved follow-ing the lines of Ref [29] Starting from some initial values of the bare parameters the Schwinger-Dyson and Bethe-Salpeter equations are solved iteratively (see figure 82) simultaneously for all mo-mentum modes and the renormalized quantities are then extracted from the renormalization condi-tions Then the values of the bare parameters are adjusted and the upper iteration is repeated until

2 The discretization required to solve Kadanoff-Baym equations apparently breaks Lorentz invariance as does thenonequilibrium ensemble itself This singles out a preferred frame where the expectation value of the total momentum ofthe ensemble vanishes (center of mass frame) The field renormalization can be obtained by evaluating the 4-momentumderivative in eq (815) via spatial (Zs) or temporal (Zt ) lattice points It has been checked that both possibilities lead tonegligible differences in the results


1

12

14

16

18

2

22

24

0 10 20 30 40 50 60 70 80 90 100

mR

min

it

iteration

14

15

16

17

18

19

20

21

1 10

V(k

)

kminit

0th iteration

10th iteration

20th iteration

30th iteration

90th iterationV(k=0) = λRZ2

Figure 82 Left Renormalized mass extracted from the iterative solution of the Schwinger-Dysonequation for the propagator G(k) according to the renormalization condition (815) Right Iterativesolution of the Bethe-Salpeter equation for the kernel V (k)

the result yields the desired values of the renormalized mass and coupling The renormalized vacuummass mR is used to set the scale for all simulations The dependence of the 2PI counterterms on thecoupling λR is shown in figure 83For the subsequent calculation of the thermal propagator at some temperature T T0 it is onlynecessary to perform the iteration once since the bare parameters are fixed to those determined atthe reference temperature The thermal mass can then be extracted via the zero-mode of the thermalpropagator

m2th = Z Gminus1

th (k = 0) (817)

833 Gaussian versus Non-Gaussian Initial State

In order to verify the full nonperturbative renormalization procedure of Kadanoff-Baym equations itis instructive to investigate solutions which minimally deviate from thermal equilibrium for severalreasons First it permits a detailed comparison with renormalized equilibrium quantities The lattercan independently be computed within thermal quantum field theory for which the renormalizationof the 2PI effective action is known Second it provides the possibility to show the importance ofthe non-Gaussian 4-point correlation of the initial state for renormalization Furthermore the ther-mal limit is valuable in order to investigate the dependence on the cutoff provided by the (lattice)regulator the elimination of which is the ultimate goal of renormalization Finally a reasonable de-scription of the thermal limit within Kadanoff-Baym equations is the basis for a controlled transitionto nonequilibriumThe reason for the existence of a minimal deviation of solutions of Kadanoff-Baym equations fromthermal equilibrium is the following Describing thermal equilibrium requires to incorporate thermalinitial n-point correlation functions for all nisinN into Kadanoff-Baym equations as has been shown inchapter 7 Therefore for Kadanoff-Baym equations incorporating initial n-point correlations for finiten the thermal propagator is no ldquofixed-pointrdquo solution3 Since numerical investigations are confined tofinite n (actually already the inclusion of n = 4 requires a sophisticated algorithm) it is a non-trivialquestion how large the unavoidable deviations from thermal equilibrium are for a given truncation ofthe thermal initial correlations

3 In contrast to this standard (classical) Boltzmann equations do possess a ldquofixed-pointrdquo solution for thermal one-particledistribution functions


001

01

1

10

100

1000

0 05 1 15 2

δλ

λR24

Nonpert 3-loop renasmR = 05

asmR = 025

asmR = 0125 01

1

10

100

1000

0 05 1 15 2

|δm

2 mR

2 |

λR24

Nonpert 3-loop

Pert 1-loop

Figure 83 Left Dependence of the nonperturbative 2PI coupling counterterm on the renormalizedcoupling for three different lattice spacings as For a given regulator the coupling counterterm di-verges at some maximal value of the renormalized coupling This maximal value becomes smallerwhen decreasing as ie when increasing the cutoff (triviality) Right Comparison of the nonpertur-bative 2PI mass counterterm and the approximate perturbative mass counterterm for asmR = 05

In the case of Kadanoff-Baym equations for Gaussian initial states only the initial 2-point correlationis retained Since the 4-point function carries logarithmic divergences this means that Gaussian initialstates feature an unavoidable cutoff-dependent offset from thermal equilibriumIn contrast to this Kadanoff-Baym equations incorporating a thermal initial 2- and 4-point correlationcoincide with those for thermal equilibrium in the limit x0y0rarr 0 (see section 72) In particular thismeans that the initial values of thermal masses or energy densities coincide with those in thermalequilibrium which are renormalized by the 2PI countertermsThe thermal n-point correlations for nge 6 are suppressed due to two reasons First since the effectivenon-local n-point vertices describing the initial n-point correlations are supported only at the initialtime they would enter the Kadanoff-Baym equations accompanied by n propagators GR(t0k) eval-uated at t = x0y0 which are damped exponentially for t mminus1

R (see figure 81) Thus the memoryto n-point correlations of the initial state is lost the more rapidly the higher n Second for Φ4-theorythe contribution of initial correlations higher than 4 is also suppressed when approaching the initialtime as has been shown in section 723In the following a detailed comparison between the Kadanoff-Baym equations with and withoutthermal initial 4-point correlation is presented In both cases the full nonperturbative renormaliza-tion procedure is employed For the first set of solutions however a Gaussian initial state is usedFor the second set of solutions the non-Gaussian thermal initial 4-point correlation is added The2PI counterterms and the initial conditions for the thermal 2-point correlation are identical for bothsets The solutions with initial 4-point correlation are used to show the relevance of non-Gaussiancorrelations for renormalization Finally the cutoff dependence is investigated

Renormalized thermal initial 2- and 4-point correlation

2-point correlation The thermal initial 2-point correlation is encoded in the initial conditions forthe statistical propagator For the thermal case they are given by

GF(x0y0k)∣∣x0=y0=0 = Gth(k)

partx0GF(x0y0k)∣∣x0=y0=0 = 0 (818)


partx0party0GF(x0y0k)∣∣x0=y0=0 = ωth(k)Gth(k)

where

Gth(k) = Gth(minusiτ0k)|τ=0 = T sum

nGth(ωnk)

ωth(k)2 =(

part 2τ Gth(minusiτ0k)Gth(minusiτ0k)

)∣∣∣∣τrarr0

=T sumn

(1minusω2

n Gth(ωnk))

Gth(k) (819)

and Gth(ωnk) is a solution of the thermal self-consistent Schwinger-Dyson equation (814) at tem-perature T = 1β

4-point correlation The full thermal initial 4-point correlation appearing in the in setting-sun ap-proximation is derived in chapter 7 It is given by (see eqs 774 767)

iα i jkl4 th(k1k2k3k4) = minusλR

Z2

βint0

dτ ∆i(minusiτk1)∆ j(minusiτk2)∆k(minusiτk3)∆l(minusiτk4)

where i jkl isin sa and


s(minusi(β minus τ)k) =Gth(minusiτ0k)

Gth(k)=

T sumn eiωnτGth(ωnk)Gth(k)

∆a(minusiτk) = minus∆

a(minusi(β minus τ)k) = 2partτGth(minusiτ0k) = T sumn

eiωnτ 2iωnGth(ωnk)

Using the (anti-)symmetry relations which follow from the periodicity of the thermal propagatorone can rewrite the upper integral according to

int β

0 rarr 2int β2

0 Furthermore the anti-symmetry of∆a(minusiτk) implies that the correlations αsssa

4 th and αsaaa4 th indeed vanish

Comparison of solutions with and without thermal initial 4-point correlation function

The comparison is based on two sets of numerical solutions [146] of Kadanoff-Baym equationsone with and one without thermal initial 4-point correlation on a lattice with 323times 20002 latticesites and lattice spacings of asmR = 05 and atmR isin 0010025 (the latter was used for solutionscovering a total time range t middotmR gt 103 in order to reduce computational costs) For both sets the2PI counterterms and the thermal propagator which is required for the computation of the thermalinitial correlations were obtained by independently solving the Schwinger-Dyson and Bethe-Salpeterequations (814) on a lattice of the same spatial size and with identical spatial lattice spacing For thetemporal lattice spacing atmR = 001 was used throughout in order to minimize numerical errors Forthe computation of the 2PI counterterms a number Nt = 1024 of sites along the time direction wasused while Nt = 1(Tat) for the thermal propagator at temperature T

Energy conservation One of the most attractive properties of approximations derived from nPIeffective actions is their compatibility with conserved charges of the underlying theory [24] In thecase of real scalar Φ4-theory in Minkowski space-time this means that total energy and momentumare conserved by solutions of Kadanoff-Baym equations Extending the derivation in Ref [147] of


03

04

05

06

07

08

001 01 1 10 100

GF(t

tk)

t mR

k = 0

k = mR

k = 2mR

500 1000 1500 2000t mR

KB Gauss

KB Non-Gauss

ThQFT

Figure 84 Time evolution of the equal-time propagator GF(t tk) obtained from Kadanoff-Baymequations with thermal initial 2-point correlation function (red lines) as well as thermal initial 2-and 4-point correlation functions (green lines) for three momentum modes respectively The bluehorizontal lines show the renormalized thermal propagator Gth(k) which serves as initial conditionat t = 0

the energy-momentum tensor from the 2PI effective action defined on the closed real-time contour tonon-Gaussian initial states yields for the total renormalized energy density

Etotal(t) =int d3k

(2π)3

[12

(partx0party0 +k2 +m2

B +λB

4

int d3q(2π)3 GF(t tq)

)GF(x0y0k)|x0=y0=t

minus 14

int t

0dz0 (

ΠF(tz0k)Gρ(z0 tk)minusΠρ(tz0k)GF(z0 tk))

minus 14

(ΠλαF(tk)GF(0 tk)+

14

Πλαρ(tk)Gρ(0 tk))]

+ const (820)

It has been checked that the total energy density is conserved by the numerical solutions used below toan accuracy of lt 10minus3 for Gaussian initial conditions and lt 10minus4 for non-Gaussian initial conditionsFurthermore similar to the Kadanoff-Baym equations it is possible to show that the total energydensity is formally finite in the limit trarr 0 and for thermal 2- and 4-point initial correlation functions(up to a time- and temperature-independent constant) provided the self-energies are chosen accordingto the full nonperturbative renormalization procedure

Minimal offset from thermal equilibrium In thermal equilibrium the propagator Gth(x0y0k)depends only on the difference x0minus y0 of its two time arguments Therefore the thermal equal-time propagator Gth(t tk) = Gth(k) is given by a time-independent constant for all momentummodes The Schwinger-Keldysh propagator G(x0y0k) obtained from solving Kadanoff-Baym equa-tions with nonequilibrium initial conditions approaches thermal equilibrium at late times such thatG(t tk) = GF(t tk) evolves with time but converges towards a constant value for t rarr infin How-ever even in the case where the initial conditions of the Schwinger-Keldysh propagator are chosen


to coincide with the thermal propagator G(t tk) does depend on time since all higher thermal cor-relations have been truncated at the initial time due to the restriction to Gaussian initial conditionsThus this unavoidable time-dependence of the equal-time propagator is a measure of the impact ofthe truncation of higher correlations It reveals the minimal deviation of solutions of Kadanoff-Baymequations from thermal equilibrium For the extended Kadanoff-Baym equations which take intoaccount an initial 4-point correlation function both the propagator and the non-Gaussian 4-point cor-relation function can be chosen to coincide with their respective values in thermal equilibrium at theinitial time Therefore one expects that the time-dependence of the equal-time propagator and there-fore the minimal deviation from thermal equilibrium is smaller compared to the case without initial4-point correlation functionIn figure 84 the time-evolution of the equal-time propagator is shown for two solutions which rep-resent the minimal deviation from thermal equilibrium for Gaussian Kadanoff-Baym equations aswell as non-Gaussian Kadanoff-Baym equations including a thermal initial 4-point correlation func-tion For both solutions the initial values of the propagator are chosen to coincide with the thermalpropagator at temperature T = 2mR For the Gaussian case the equal-time propagator immediatelystarts to oscillate for times t middotmR amp 1 and then drifts towards a stationary value which is slightly dis-placed from the initial value For the non-Gaussian case the time-dependence is indeed considerablyreduced and the Schwinger-Keldysh propagator always remains close to the renormalized thermalpropagator The residual time-dependence can be attributed to the truncation of the higher thermaln-point correlation functions for n gt 4 as well as to numerical errors (the latter can be reduced bychoosing a smaller time-step at) Qualitatively a similar behaviour is found when varying the initialtemperature and the lattice cutoff Λ prop aminus1

s

Offset between initial and final Temperature Due to the truncation of higher correlations theKadanoff-Baym equations for Gaussian initial states as well as those incorporating an initial 4-pointcorrelation function cannot describe thermal equilibrium exactly However the minimal offset fromthermal equilibrium is considerably reduced when taking a thermal initial 4-point correlation intoaccountApart from that a qualitative difference between both types of equations exists which has the follow-ing reason As has been shown in section 723 the 4-point correlation of the initial state contributesto the Kadanoff-Baym equations in the limit x0y0rarr 0 whereas the contributions from even higherthermal correlations are suppressed since these enter Kadanoff-Baym equations exclusively via mem-ory integrals within Φ4-theory The same is true for the total energy density (820) Therefore thetotal energy density Einit equiv Etotal(t = 0) computed at the initial time using thermal initial 2- and 4-point correlation functions corresponding to a temperature Tinit coincides with the total energy Eeq(T )of an ensemble in complete thermal equilibrium at the same temperature ie Einit = Eeq(Tinit) Fort rarr infin solutions of Kadanoff-Baym equations asymptotically approach thermal equilibrium Due touniversality [32] the final temperature Tfinal is uniquely characterized by the value of the total energydensity ie Efinal = Eeq(Tfinal) Furthermore the initial and final total energy agree since the totalenergy is conserved Therefore also the initial and final temperatures have to agree ie Tinit = TfinalIn contrast to this if only a Gaussian thermal 2-point correlation at temperature Tinit is used theresulting total energy does not coincide with the corresponding value in thermal equilibrium ieEinit 6= Eeq(Tinit) due to the missing contribution from the thermal 4-point correlation function Never-theless for trarr infin complete thermal equilibrium is approached asymptotically ie Efinal = Eeq(Tfinal)for some final value of the temperature Tfinal Since the total energy is also conserved the initial andfinal temperatures can not agree ie one expects that Tinit 6= Tfinal for a Gaussian initial stateFor solutions of Kadanoff-Baym equations which minimally deviate from thermal equilibrium an


-03-02-01

0 01 02 03 04

001 01 1 10 100

microm

R

t mR

2

21

22

23T

mR

500 1000 1500 2000t mR

KB Gauss

KB Non-Gauss

Thermal Eq

0

1

0 8

f BE(n

)

ωmR0

1

0 8

f BE(n

)

ωmR

Figure 85 Time evolution of the temperature and chemical potential obtained from a fit of the ef-fective particle number density n(tk) to a Bose-Einstein distribution for Kadanoff-Baym equationswith thermal initial 2-point correlation function (red lines) as well as thermal initial 2- and 4-pointcorrelation functions (green lines) The shaded areas illustrate qualitatively the deviation of the mo-mentum dependence of n(tk) from the Bose-Einstein distribution function They are obtained fromthe asymptotic standard error of the fit (via least-square method) magnified by a factor 10 for bettervisibility Nevertheless the errors become invisibly small at times t middotmR 10 The insets show afunction fBE(n) of the effective particle number density n(tk) plotted over the effective energy den-sity ω(tk) The function is chosen such that a Bose-Einstein distribution corresponds to a straightline the slope and y-axis intercept of which determine the temperature and the chemical potential(here fBE(n) = ln(1n + 1)minusωTre f was used with Tre f = 275mR) At the initial time (left inset)the particle number densities of both solutions agree with a Bose-Einstein distribution with the sametemperature and therefore lie on top of each other At the largest time (right inset) the slope of thered line is smaller which corresponds to an increase in temperature Inside the insets the underlyinggrey lines show the best-fit Bose-Einstein distribution function

effective time-dependent temperature T (t) and chemical potential micro(t) may be obtained by fitting theeffective particle number density n(tk) (see eq (617)) to a Bose-Einstein distribution function foreach time t

nfit(tk) =[

exp(

ω(tk)+ micro(t)T (t)

)minus1]minus1

The time evolution of the effective temperature and chemical potential obtained from numerical solu-tions of Gaussian Kadanoff-Baym equations with thermal initial 2-point correlation function as wellas non-Gaussian Kadanoff-Baym equations with thermal initial 2- and 4-point correlation functionsare shown in figure 85 Due to the thermal initial 2-point correlation function the effective particlenumber densities agree with a Bose-Einstein distribution at the initial time with identical initial tem-perature for both types of equations For trarrinfin the effective particle number densities also agree witha Bose-Einstein distribution very well as expected However for the solution without initial 4-point


correlation the final temperature has increased compared to the initial value In contrast to this theinitial and final values of the temperature agree up to 05 for the solution with thermal initial 4-pointcorrelation functionThe offset between the initial and final temperature is a quantitative measure of the unavoidableoffset from thermal equilibrium which occurs for a Gaussian initial state Equivalently it may beparameterized by the energy difference

∆E = Eeq(Tfinal)minusEeq(Tinit) = EfinalminusEeq(Tinit) = EinitminusEeq(Tinit)

=14

int d3k(2π)3 ΠλαF(tk)GF(0 tk)

∣∣t=0 = minus λR

24Z2 V nG4 (xxxx)

∣∣x0=0

=λ 2

R

24

intkpq

GthR(p)GthR(q)GthR(kminus pminusq)GthR(minusk)

which is equal to the contribution of the initial thermal 4-point correlation function to the total energyThis contribution contains a (quadratic and quartic) UV divergence and therefore the Kadanoff-Baymequations for a Gaussian initial state feature a divergent offset from thermal equilibrium Since the 2PIcounterterms renormalize the divergences in thermal equilibrium they cannot do so for a Gaussianinitial state as well On the other hand if a thermal 4-point correlation of the initial state is taken intoaccount then ∆E = 0 and no (divergent) offset occursThe temperature-offset implies that also all quantities derived from the Schwinger-Keldysh propaga-tor like the total number density N(t) equiv

intd3k(2π)3 n(tk) or the effective thermal mass mth(t) equiv

ω(tk = 0) feature an offset between their initial values and their late-time asymptotic values for aGaussian initial state (see figure 86)

Thermalization For a real scalar quantum field the chemical potential vanishes in thermal equilib-rium due to the absence of global conserved charges apart from energy and momentum In figure 85it can be seen that the effective chemical potential micro(t) is indeed very close to zero at the initial timewhich shows that the initial thermal propagator indeed yields a thermal effective number density dis-tribution Furthermore micro(t) also approaches zero in the late-time limit which means that thermalequilibrium has effectively been reached for times t middotmR gt 2000 for both types of equation For thesolution with thermal initial 4-point correlation function the effective particle number density re-mains very close to a Bose-Einstein distribution also at intermediate times and the time-variation ofthe corresponding effective temperature and chemical potential is significantly smaller compared tothe solution without initial 4-point correlation function Furthermore for the latter also the deviationfrom the Bose-Einstein distribution is larger at intermediate times which is illustrated by the shadedareas in figure 85 It is interesting to note that for a Gaussian initial state the solution which mini-mally deviates from thermal equilibrium resembles a typical non-equilibrium solution The quantumthermalization process is characterized by a phase of kinetic equilibration after which the effectiveparticle number is already close to a Bose-Einstein distribution however with non-zero chemicalpotential (prethermalization [31]) In figure 85 this corresponds to the phase when the shaded areabecomes invisibly small Subsequently the chemical potential approaches its equilibrium value (zero)on a much longer time-scale as can be seen on the right part of figure 85 Altogether it is concludedthat a controlled transition from equilibrium to nonequilibrium cannot be achieved for a Gaussianinitial state

Matching of Kadanoff-Baym equations with thermal quantum field theory In order to quan-titatively compare solutions of Kadanoff-Baym equations which are formulated on the closed real-time path with numerical solutions of the Schwinger-Dyson equation at finite temperature which are


15

155

16

165

17

175

18

185

19

001 01 1 10 100

mth

(t)

mR

t mR

KB Gauss (A)KB Gauss (B)KB Non-Gauss (A)KB Non-Gauss (B)ThQFT (A)ThQFT (B) 17

18

10 100

Figure 86 Time evolution of the effective thermal mass mth(t) = ω(tk = 0) obtained from Kada-noff-Baym equations with (green) and without (red) a thermal initial 4-point correlation function Thehorizontal line (blue) shows the value obtained from thermal quantum field theory within 2PI 3-loopapproximation according to definition (A) in table 81 In the inset also the thermal mass accordingto definition (B) is shown The red and green circles give the values of the thermal mass obtained fromevaluating definition (B) for Gaussian and non-Gaussian Kadanoff-Baym equations respectively

Kadanoff-Baym Thermal QFT

mAth ω(tk = 0) =

radicpartx0 party0 GFR(x0y0k)

GFR(x0y0k)

∣∣∣x0=y0=t

ωth(k = 0) =radic

part 2τ GthR(minusiτ0k)GthR(minusiτ0k)

∣∣∣τ=0

mBth

(limsrarrinfin

int s0 dsprimeGρR(t + sprime

2 tminus sprime2 k)

)minus12 (GthR(ωn = k = 0)

)minus12

Table 81 The two rows show two definitions of the effective thermal mass as observed at differentenergy scales Both definitions can be evaluated on the real time path (left column) or the imaginarytime path (right column) and coincide in thermal equilibrium The expressions in the left column canalso be evaluated in a nonequilibrium situation

solved on the imaginary time path it is necessary to identify quantities which can be computed in bothcases One such quantity is the two-point function evaluated for coincident time arguments as hasbeen discussed above (see figure 84) The effective thermal mass mth(t) = ω(tk= 0) obtained fromthe zero-mode of the effective energy density for Kadanoff-Baym equations corresponds within ther-mal quantum field theory to the zero-mode of the thermal effective energy density ωth(k= 0) definedin eq (819) However the thermal mass mth = GthR(ωn = k = 0)minus12 defined in eq (817) consti-tutes an inequivalent definition for non-zero coupling The latter corresponds to the infrared-limit ofthe two-point correlation function while the former is related to its oscillation frequency and thereforetheir difference is a manifestation of the scale-dependence of physical observables The thermal massaccording to both definitions can be computed for solutions of Kadanoff-Baym equations as well as


13

14

15

16

1 15 2 25 3 35

mth

mR

(as mR)-1

T = 15 mR

14

15

16

17

mth

mR

T = 17 mR

KB GaussKB Non-GaussThQFT

16

17

18

19m

thm

RT = 20 mR

001 01 1 10 100t mR

Figure 87 Cutoff dependence of the effective thermal mass mth(t) = ω(tk = 0) obtained from Ka-danoff-Baym equations with (green) and without (red) a thermal initial 4-point correlation functionfor three different initial temperatures The areas shaded light and dark grey (as well as the errorbarsin the Gaussian case) show the maximal and the minimal values of the thermal mass in the interval0le t middotmRle 100 and the circles show the value of the thermal mass which is approached at the largesttime For one exemplary case the determination of these values is shown in the inset in the upperright corner In the non-Gaussian case the time-variation of the thermal mass is very small suchthat it remains close to the thermal mass computed within thermal quantum field theory (blue) at alltimes

for the full thermal propagator parameterized by imaginary times (see table 81)In figure 86 the time-evolution of the effective mass according to definition (A) for Kadanoff-Baymequations is shown For the solution with thermal initial 2- and 4-point correlation function thethermal mass is nearly constant and therefore agrees with the initial equilibrium value very well Thesecond definition (B) of the thermal mass provides an independent consistency check Its computationfor Kadanoff-Baym equations amounts to the limiting value of the integral of the spectral functionover the relative time which is obtained by extrapolating the value of the integral with definite upperboundary for the available times Since the spectral function is damped exponentially with respect tothe relative time good convergence is achieved provided the maximal relative time is large comparedto the inverse damping rate As shown in the inset of figure 86 the thermal masses computed ac-cording to definition (B) also agree very well with the equilibrium value for the non-Gaussian caseBelow definition (A) is used throughout unless otherwise stated

Cutoff dependence Figure 87 displays the thermal masses obtained from solutions of Kadanoff-Baym equations solved on five different lattice configurations corresponding to five values of the UVcutoff (with constant IR cutoff) as well as three different values of the initial temperature respec-tively Additionally the renormalized thermal mass computed within thermal quantum field theory


employing the nonperturbative renormalization of the 2PI effective action is shown which indeedbecomes independent of the cutoff when aminus1

s T For lower values of the cutoff a residual cut-off dependence occurs which can be attributed to the Boltzmann-tail of the thermal particle numberdistribution Since the Boltzmann-tail is exponentially suppressed for smaller temperatures also theresidual cutoff dependence decreases for smaller temperatures as can bee seen in figure 87For the solutions of the Gaussian Kadanoff-Baym equations with thermal initial 2-point correlationfunction the errorbars in figure 87 represent the maximal the minimal and the final value4 of the ef-fective thermal mass mth(t) in the time interval 0le t middotmR le 100 while the initial value is given by therenormalized thermal mass computed within thermal quantum field theory at the initial temperatureFor the solutions of the non-Gaussian Kadanoff-Baym equations with thermal initial 2- and 4-pointcorrelation functions the effective thermal mass always remains very close to the renormalized ther-mal mass for all values of the cutoff and of the initial temperatureFor the Gaussian case an offset between the initial and the final value of the thermal mass occursThis offset is equivalent to the temperature-offset discussed above It is a measure for the influenceof the higher correlations which have been neglected in the Gaussian case Since the non-Gaussian 4-point correlation function contains divergences one expects that the offset increases with the cutoffIn figure 87 the offset corresponds to the difference between the dashed and the dotted lines Itindeed increases with the cutoff for the considered parameters

834 Approximate Perturbative versus Nonperturbative Counterterms

In this section the impact of the renormalization prescription on nonequilibrium solutions of Ka-danoff-Baym equations is investigated Therefore two distinct prescriptions are used in order todetermine the mass and coupling counterterms appearing in the Kadanoff-Baym equations First themass is renormalized using the approximate perturbative prescription at one-loop order while leavingthe coupling unchanged [17] Second the full nonperturbative 2PI renormalization procedure [2829]is employed to determine the mass and coupling counterterms in vacuum Then solutions of the Ka-danoff-Baym equations for both renormalization procedures are compared For this comparison aGaussian initial state is used in both cases in order to guarantee an identical initial stateIt is emphasized that even for a Gaussian initial state the approximately renormalized and the non-perturbatively renormalized Kadanoff-Baym equations are genuinely inequivalent for two reasonsFirst for the approximate perturbative renormalization prescription the coupling constants which ap-pear in front of the tadpole and setting-sun contributions in the self-energy are identical whereas thebare coupling appears in front of the tadpole and the renormalized coupling appears in front of thesetting-sun diagram of the nonperturbatively renormalized Kadanoff-Baym equations Second theratio of the bare and the renormalized masses are different and in particular also the ratio of the bareand the renormalized coupling are specific for the renormalization prescriptionThe Kadanoff-Baym equations were solved [146] for both renormalization procedures and two val-ues of the (renormalized) coupling respectively on a lattice with 323times10002 lattice sites and latticespacings of asmR = 05 and atmR = 005 For the approximate perturbative renormalization prescrip-tion the corresponding value of the bare mass is given in the left column of table 82 The baremass and coupling obtained by the full nonperturbative renormalization procedure are given in theright column of table 82 The initial conditions for the propagator are determined in accordance withRef [25 142] and correspond to an initial effective particle number distribution which is peakedaround the momentum |k|= 3mR In Figure 88 the time evolution of the statistical equal-time prop-agator for the four parameter sets introduced above and identical initial conditions is shown For

4It has been checked that the effective thermal mass has indeed reached its final value already for times t middotmR 100 incontrast to the effective temperature and chemical potential


0

05

1

15

2

25

3

35

001 01 1 10 102 103 104

GF(t

tk=

0)

t mR

A24 A18

E18

E24

Figure 88 Statistical equal-time propagator over time for the four different parameter sets shown intable 82

parameter set A24 the Kadanoff-Baym numerics is very unstable and breaks down already for veryearly times Decreasing the coupling the numerics can be stabilized as can be seen for parameter setA18 The curve for parameter set E24 shows two features First the numerics is stable although boththe bare and the renormalized coupling are greater or equal to the value used for parameter set A24Second although both couplings are strictly greater than the value chosen for parameter set A18 thethermalization time is dramatically larger Thus the exact nonperturbative renormalization procedureindeed has a stabilizing virtue for the computational algorithm and also has a significant quantitativeimpact on the numerical solutions of Kadanoff-Baym equations Furthermore it is important to notethat qualitative features of Kadanoff-Baym equations like late-time universality and prethermalizationare independent of the renormalization procedure

835 Renormalized Nonequilibrium Dynamics

Above it has been shown that extended Kadanoff-Baym equations which take into account an initialstate featuring a 4-point correlation function possess solutions which come very close to the renor-malized thermal state as obtained from the three-loop truncation of the 2PI effective action at finitetemperature This provides the possibility for a controlled transition to a nonequilibrium situation bydistorting the thermal initial 2- and 4-point correlation functions However these distortions cannotbe chosen arbitrarily if one demands that the nonequilibrium state should also be renormalized bythe identical 2PI counterterms One of these restrictions is that the nonequilibrium initial correla-

A18 λ = 18 m2B =minus687 m2

R E18 λR = 18 λB = 3718 m2B =minus1439 m2

R

A24 λ = 24 m2B =minus949 m2

R E24 λR = 24 λB = 6343 m2B =minus2514 m2

R

Table 82 Counterterms for the two sets of couplings and the approximate perturbative renormaliza-tion prescription (left column) as well as the exact nonperturbative renormalization procedure (rightcolumn)


095

1

105

0 1 2 3 4 5 6 7GF(t

tk)

GF(0

0k

)

kmR

t mR = 2000

095

1

105

GF(t

tk)

GF(0

0k

)

t mR = 10

095

1

105

GF(t

tk)

GF(0

0k

)

t mR = 05

095

1

105

GF(t

tk)

GF(0

0k

)

t mR = 00 KB Gauss

095

1

105

GF(t

tk)

Gth

(k)

t mR = 20

0 1 2 3 4 5 6 7

095

1

105

kmR

t mR = 2000

095

1

105t mR = 10

095

1

105t mR = 05

095

1

105t mR = 00 KB Non-Gauss

095

1

105t mR = 20

Figure 89 Momentum dependence of the equal-time propagator for five different times t middotmR =000520102000 obtained from Kadanoff-Baym equations with (green lines right side) and with-out (red lines left side) thermal initial 4-point correlation function respectively The shaded areasshow the maximum and minimum values of GF(t tk)Gth(k) for all times

tion functions coincide with the thermal values asymptotically for large spatial momenta since thisasymptotic behaviour determines the divergences which are to be canceled by the counterterms Fur-thermore one may expect that the distortions of the 2- and the 4-point correlations cannot be chosencompletely independently but have to be related in such a way that the Kadanoff-Baym equationsremain finiteAdditionally since the counterterms are given by fixed time-independent numbers a necessary con-dition for the finiteness of Kadanoff-Baym equations at all times is that the divergences are alsotime-independent Since the divergences are related to the asymptotic behaviour of the full propa-gator at large momenta this can only be the case if this asymptotic behaviour does not change withtimeIn figures 89 and 810 the ratio of the equal-time propagator over the thermal propagator is plot-ted over the absolute spatial momentum for five different times The largest spatial momentum isdetermined by the size of the spatial lattice spacing providing the UV cutoffFor the solutions shown in figure 89 a thermal initial 2-point correlation function has been usedTherefore at the initial time the ratio of the equal-time propagator and the thermal propagator isunity However for the solution without initial 4-point correlation function all momentum modesof the propagator are excited as soon as t middotmR amp 1 (see left part of figure 89) This indicates thatrenormalization with time-independent counterterms is impossible in this case In contrast to this thesolution with thermal initial 4-point correlation function always remains close to the renormalizedthermal propagator for all momentum modes (see right part of figure 89) It has been checked that


09 1

11 12

0 1 2 3 4 5 6 7

GF(t

tk)

Gth

(k)

kmR

t mR = 2000

09 1

11 12

GF(t

tk)

Gth

(k)

t mR = 10

09 1

11 12

GF(t

tk)

Gth

(k)

t mR = 05

09 1

11 12

GF(t

tk)

Gth

(k)

t mR = 00 KB Gauss

09 1

11 12

GF(t

tk)

Gth

(k)

t mR = 20

0 1 2 3 4 5 6 7

09 1 11 12

kmR

t mR = 2000

09 1 11 12

t mR = 10

09 1 11 12

t mR = 05

09 1 11 12

t mR = 00 KB Non-Gauss

09 1 11 12

t mR = 20

Figure 810 Momentum dependence of the ratio of the equal-time propagator and the thermal prop-agator for five different times t middotmR = 000520102000 obtained from Kadanoff-Baym equationswith (green lines right side) and without (red lines left side) thermal initial 4-point correlation func-tion as well as identical nonequilibrium initial conditions for the 2-point function respectively Theshaded areas show the maximum and minimum values of GF(t tk)Gth(k) for all times

this behaviour stays the same when the cutoff is variedFor the solutions shown in figure 810 the initial 2-point correlation function has been distorted suchthat it corresponds to a nonequilibrium initial condition At large values of the momentum it coin-cides with the thermal propagator as required for renormalizability Furthermore the nonequilibriuminitial condition has been chosen such that the energy density is identical to the case with thermalinitial correlation For the solution without initial 4-point correlation function it is found again thatall momentum modes of the propagator are excited as soon as t middotmR amp 1 up to the highest momentum(see left part of figure 810) In contrast to this when employing a thermal initial 4-point correlationfunction the high momentum modes of the propagator are not excited considerably Instead thenonequilibrium correlation relaxes by exciting the low momentum modes of the two-point function(see right part of figure 810) This is precisely the property required for renormalization with time-independent counterterms It is an indication that the renormalization of Kadanoff-Baym equations isindeed possible within the framework presented hereWhen going to initial conditions which deviate more strongly from equilibrium it may be expectedthat also the initial 4-point correlation function has to be modified accordingly in order to preservethe renormalization However this is beyond of the scope of the present workAltogether it is concluded that the Kadanoff-Baym equations (89) are a good candidate to describerenormalized nonequilibrium dynamics Furthermore they provide the possibility for a controlledtransition from renormalized thermal equilibrium to nonequilibrium quantum dynamics

84 Summary 129

84 Summary

In this and the previous chapter a framework appropriate for the nonperturbative renormalization ofKadanoff-Baym equations has been developed and an ansatz for renormalized Kadanoff-Baym equa-tions has been proposed For the three-loop truncation of the 2PI effective action it has been shownanalytically that these Kadanoff-Baym equations are indeed finite for one special class of renormal-ized initial conditions and close to the initial time Additionally it has been demonstrated that theirnumerical solutions possess properties which are expected from renormalized Kadanoff-Baym equa-tions

The renormalization of Kadanoff-Baym equations is based on the nonperturbative renormalizationprocedure of the 2PI effective action which has been formulated recently at finite temperature [2829 37 173ndash175]

In chapter 7 the nonperturbative renormalization procedure of the 2PI effective action at finite tem-perature has been transferred to the closed real-time path In order to do so it is necessary to explicitlyspecify all thermal correlation functions characterizing the thermal state which plays the role of theldquoinitialrdquo state on the closed real-time path It has been shown that thermal n-point correlation func-tions have to be taken into account for all n ge 0 within the nonperturbative 2PI formalism Further-more an iterative computation prescription for the nonperturbative thermal initial correlations whichare appropriate for a given truncation of the 2PI effective action has been developed and applied tothe three-loop truncation Finally renormalized Kadanoff-Baym equations which describe thermalequilibrium on the closed real-time path have been derived

In this chapter an ansatz for renormalized Kadanoff-Baym equations describing nonequilibrium en-sembles has been proposed These contain mass and coupling counterterms determined accordingto the nonperturbative renormalization prescription of the 2PI effective action [28 29] and take intoaccount a non-Gaussian 4-point correlation function of the initial state [32 49 57] They can be con-veniently derived from the 4PI effective action For the three-loop truncation it has been verifiedanalytically that these Kadanoff-Baym equations are rendered finite close to the initial time and forinitial conditions which correspond to the minimal deviation from thermal equilibrium In contrastto this Kadanoff-Baym equations for a Gaussian initial state feature a divergent offset from renor-malized thermal equilibrium which means that they cannot be renormalized with time-independentcounterterms This qualitative difference could also be demonstrated by means of numerical solu-tions It has been found that the Kadanoff-Baym equations containing nonperturbative 2PI countert-erms and a non-Gaussian initial 4-point correlation possess particular solutions which remain closeto the renormalized thermal propagator for all times For Gaussian Kadanoff-Baym equations it wasfound that the offset from thermal equilibrium which was mentioned above indeed increases whenthe cutoff is increasedSo far approximate perturbative counterterms have been used when solving Kadanoff-Baym equa-tions [17] It has been demonstrated that numerical instabilities which occur when the coupling isincreased can be alleviated if nonperturbative 2PI counterterms are used instead

A necessary requirement for the renormalizability of Kadanoff-Baym equations with time-independentcounterterms is that the divergences are also time-independent Therefore it is required that theasymptotic behaviour of the Schwinger-Keldysh propagator at large momenta is universal ie time-independent It was demonstrated that this is indeed the case for nonequilibrium solutions of Kada-noff-Baym equations containing nonperturbative 2PI counterterms and a non-Gaussian initial 4-pointcorrelation function In contrast to this all momentum modes are excited when Gaussian Kadanoff-Baym equations are employed

Chapter 9

Conclusions

In this work the quantum dynamics of time-evolving scalar fields has been studied in a cosmologicalcontext In particular the robustness of quintessence tracker potentials with respect to quantum cor-rections has been investigated and nonequilibrium renormalization techniques for Kadanoff-Baymequations have been developedThe classical dynamics of the quintessence field is described by its self-interaction potential Quint-essence potentials featuring tracking solutions avoid some of the problems connected to the cosmo-logical constant Therefore it is important to investigate quantum corrections for these exceptionalpotentialsQuantum field theory together with classical gravity determines the effective quintessence potentialonly up to a constant Therefore it was assumed here that the freedom to shift the potential by anarbitrary constant is used in such a way that the quintessence energy density matches the observedvalue for dark energy in the present cosmological epoch However even with this assumption thereremain quantum corrections to the dynamics of the quintessence field which can be addressed byquantum field theory These quantum corrections arise from the self-interactions of the scalar fieldcouplings to Standard Model particles and couplings to gravity

Quantum corrections induced from self-interactions have been investigated for two classes of pro-totype tracker potentials namely exponential and inverse power law potentials In particular therobustness of the shape of the potential was analyzed within the framework of effective field theoryTherefore a suitable Hartree-Fock approximation scheme has been developed which resums all rel-evant contributions Its validity has been verified by applying it to Liouville quantum field theoryFurthermore it has been shown that corrections to the Hartree-Fock approximation are suppressed bypowers of the ratio of the Hubble parameter and the Planck scale for typical tracker potentialsRemarkably for a classical exponential potential the Hartree-Fock approximation yields an effectivepotential which also features an exponential dependence on the field value This extends the one-loopresult of Ref [83] For the inverse power law potential the one-loop approximation breaks down nearthe singularity of the classical potential [83] In contrast to this it could be shown that the Hartree-Fock effective potential does not have a singularity but instead approaches a finite maximum valueand thus is applicable in the whole range of admissible field values Furthermore it was shown thatthe effective potential leads to a modification of the tracking solution compared to the classical caseIf the ultraviolet embedding scale of the effective theory is close to the Planck scale the predictionfor the dark energy equation of state differs significantly from the classical value

Quantum corrections induced from couplings of the quintessence field to Standard Model particleshave been investigated employing the low-energy effective action obtained from integrating out the

132 9 Conclusions

Standard Model degrees of freedom If the couplings are too large these quantum corrections woulddestroy the desired properties of the tracker potential An upper bound for the couplings was obtainedunder the assumption of minimal quantum vacuum backreaction These indirect bounds were com-pared to direct observational bounds The latter result for example from tests of a time-variation ofthe electron-proton mass ratio and of the equivalence principle

Quantum corrections induced by the gravitational coupling of the quintessence field have been in-vestigated using the one-loop renormalization group improved effective action in curved space-timeThey have been found to be negligibly small for tracker potentials

Quintessence fields can also be important in the early universe in contrast to the cosmological con-stant In this work this has been demonstrated by presenting an explicit model where baryogenesisand the present-day acceleration are both driven by a complex quintessence field which carries lep-ton number The introduction of new interactions which violate baryon or lepton number is notnecessary Instead a lepton asymmetry is stored in the quintessence field It has been shown that theobserved baryon asymmetry can be explained quantitatively by the semi-classical dynamics resultingfrom the considered model

The nonequilibrium processes that occur in the early universe until now eg baryogenesis (p)re-heating or a rolling quintessence field are typically described by semi-classical approximations likeBoltzmann equations or by effective equations of motion for a coherent scalar field expectation valueIn order to assess the validity of these approximations a quantitative comparison with the evolutionequations for the full quantum dynamics is necessary The latter is provided by Kadanoff-Baymequations For this purpose a proper renormalization of Kadanoff-Baym equations is an indispens-able preconditionIn this thesis a framework for the nonperturbative renormalization of Kadanoff-Baym equations hasbeen developed In particular the nonperturbative renormalization procedure of the 2PI effectiveaction at finite temperature has been transferred to the closed real-time path which is the startingpoint for nonequilibrium quantum field theoryFurthermore an ansatz for renormalized Kadanoff-Baym equations has been proposed within λΦ4-theory These equations contain mass and coupling counterterms determined according to the nonper-turbative renormalization procedure of the 2PI effective action in vacuum Additionally it has beenshown that renormalization requires the extension of Kadanoff-Baym equations to non-Gaussian ini-tial states Such an extension has been derived from the 4PI effective action It features a non-Gaussian initial 4-point correlation function The ansatz for renormalized Kadanoff-Baym equationscould be verified analytically for the three-loop (setting-sun) approximation for a special class ofrenormalized initial conditions and close to the initial timeFinally it has been demonstrated that the Kadanoff-Baym equations containing nonperturbative 2PIcounterterms and a non-Gaussian initial 4-point correlation function possess solutions with propertieswhich are expected from renormalized Kadanoff-Baym equations

Thus it could be shown that the methods used for describing the nonequilibrium quantum dynamicsof scalar fields are indeed considerably improved by the renormalization techniques developed in thiswork Applying these techniques is essential for a quantitative description of quantum fields far fromthermal equilibriumTherefore the renormalization of Kadanoff-Baym equations is an important step towards realisticapplications within astro-particle and high-energy physics In particular renormalized Kadanoff-Baym equations provide the basis for describing time-evolving scalar fields beyond the limitationsof the derivative expansion of the effective action The derivative expansion is used for example to

9 Conclusions 133

describe cosmic inflation and has also been used for the quintessence field above Within inflationarymodels predictions like the spectral index are directly tested by CMB measurements Since thesepredictions rely on the underlying derivative expansion it is important to assess its validityFurthermore renormalized Kadanoff-Baym equations can also be applied to study the quantum dy-namics of other nonequilibrium processes like for example for preheating baryogenesis or heavyion collisions In view of these applications it is important to note that the renormalization of Kada-noff-Baym equations presented above can be transferred to quantum field theories including fermionsand gauge fields In particular renormalized Kadanoff-Baym equations provide a quantum field the-oretical generalization of semi-classical Boltzmann equations The latter are used for example todescribe the formation of a lepton asymmetry within the leptogenesis framework However for spe-cific realizations of leptogenesis quantum corrections may play an important role In this contextthe renormalization techniques developed above are required in order to describe leptogenesis withinnonequilibrium quantum field theory

Appendix A

Conventions

The Minkowski metric sign convention (+1minus1minus1minus1) is used In General Relativity the signconvention according to the classification of Misner-Thorne-Wheeler [144] is (minus++) In this con-vention the curvature tensor is

Rαmicroνλ = +

(partνΓ

α

microλminuspartλ Γ

αmicroν +Γ

η

microλΓ

αην minusΓ

η

microνΓα

ηλ

)

with the Christoffel symbols

Γαmicroν =

12

gαβ(partmicrogβν +partνgmicroβ minuspartβ gmicroν

)

and the Ricci tensor is given byRmicroλ = +Rα

microαλ

The curvature scalar isR = gmicroλ Rmicroλ = Rmicro

micro

Throughout energy momentum frequency time length and temperature are all measured in naturalunits for which h = c = k = 1

Appendix B

Effective Action Techniques

B1 Low-Energy Effective Action

The contribution to the effective action for a scalar field from quantum fluctuations of degrees offreedom much heavier than the scalar field is discussed in this section This is the typical situation foran extremely light quintessence field φ coupled to Standard Model fields1 ψ j described by the action

S[φ ψ j] =int

d4x(

12(partφ)2minusVcl(φ)+L(φ ψ j)

) (B1)

where L(φ ψ j) contains the Standard Model Lagrangian as well as couplings between operatorsOSMk

composed from the fields ψ j and the scalar field φ

L(φ ψ j) = LSM(ψ j)+sumk

fk(φ)OSMk (B2)

As before the effective action Γ[φ ] is the Legendre transform of the generating functional

exp(

iW [J])

=intDϕ

int (prod

jDψ j

)exp(

iS[ϕψ j]+ iint

d4xJ(x)ϕ(x))

(B3)

In order to obtain the impact of the fluctuations of the fields ψ j on the evolution of the field φ it isconvenient to perform the path integrals in two steps In the first step the path integral over the heavyfields ψ j yields the semi-classical low-energy effective action Seff [φ ]

exp(

iSeff [φ ])equivint (

prodjDψ j

)exp(iS[ϕψ j]) (B4)

where the fields ψ j are ldquointegrated outrdquo and the scalar field is treated as a classical background fieldThe complete effective action is obtained in the second step from the path integral over ϕ

exp(

iΓ[φ ])

= exp(

iW [J]minus iint

d4xJ(x)φ(x))

=intDϕ exp

(iSeff [ϕ]+ i

intd4x J(x)(ϕ(x)minusφ(x))

)

1A coupling of the field φ to particles beyond the Standard Model like dark matter can easily be included here

138 B Effective Action Techniques

which can be recognized as the effective action for an uncoupled scalar field φ described by thelow-energy effective action Seff [φ ] Thus Seff [φ ] is the leading contribution to the effective actionfrom quantum fluctuations of degrees of freedom much heavier than the scalar field As for theeffective action the low-energy effective potential Veff (φ) can be defined as the lowest-order con-tribution to the derivative expansion of Seff [φ ] defined analogously to eq (318) For non-derivativecouplings between φ and ψ j the low-energy effective potential in one-loop approximation is given byeq (451) [60 105]Note that the one-loop low-energy effective action is analogous to the Heisenberg-Euler effectiveaction [90] which describes the impact of quantum (vacuum) fluctuations of the Standard Modelfermions predominantly the electron being the lightest charged particle on a classical electromag-netic background fieldIn the context of a rolling quintessence field quantum (vacuum) fluctuations of the Standard Modelfields lead to quantum corrections to the equation of motion of the scalar field In other wordsstandard-model couplings of the quintessence field lead to a quantum backreaction on its dynamics(see [96] for a discussion of the classical backreaction of Standard Model particles and dark matter)It should be emphasized that the quantum corrections to the equation of motion of the scalar fieldφ captured by the low-energy effective action Seff [φ ] have their origin in the quantum fluctuationsof the degrees of freedom ψ j For a quintessence field φ coupled to standard-model particles theseldquoheavyrdquo degrees of freedom are well-known In fact for typical quintessence masses of the order ofthe Hubble parameter mφ sim H even masses at the neutrino energy scale simmeV are ldquoheavyrdquo

B2 Effective Action in Curved Background

In this section the calculation of the one-loop effective action in a non-trivial background geometrygiven by the metric gmicroν using Heat Kernel Expansion [35] and zeta-function regularization [91 110]is briefly reviewed Similarly to dimensional regularization zeta-function regularization exploits theanalyticity properties of Feynman integrals but is more convenient in curved space-time [110] Theone-loop higher derivative contributions to the effective action see eq (318) can be obtained by thesame formalism A generalization of the classical action (31) to curved space-time is consideredusing the covariant integration measure d4x

radicminusg


d4xradicminusg(

12(partφ)2minusV (φ R)+ ε1C + ε2G+2B(φ R)

) (B5)

V (φ R) is a generalized potential which depends on φ and the curvature scalar R and terms pro-portional to the square of the Weyl tensor C = Rmicroνρσ Rmicroνρσ minus 2RmicroνRmicroν + 1

3 R2 and proportional tothe Gauss-Bonnet invariant G = Rmicroνρσ Rmicroνρσ minus 4RmicroνRmicroν + R2 have been added Furthermore anadditional term 2B(φ R) is included where B(φ R) is a (so far arbitrary) function of φ and R and 2

is the covariant DrsquoAlembert operator The form of the action is chosen in anticipation of the resultthat it includes all terms needed for the cancellation of divergences [35] The latter two terms aretotal derivatives and thus not relevant for the dynamics but are also required for the cancellation ofdivergences [35] and do appear in the dynamics if their running is considered [92] Note that theEinstein-Hilbert term minusR(16πG) as well as a possible cosmological constant have been absorbedinto the generalized potential V (φ R) Minimal coupling between R and φ in the classical action isrealized for the choice V (φ R) = Vcl(φ)+ f (R) Standard General Relativity is then recovered forf (R) =minusR(16πG) and ε1 = 0The effective action can be calculated analogously to flat space by an expansion in 1PI Feynman

B2 Effective Action in Curved Background 139

diagrams with the classical propagator2

Gminus10 (xy) =

minusiδ 2S[φ gmicroν ]δφ(x)δφ(y)

= i

(2x +

δ 2V (φ R)δφ 2

∣∣∣∣φ(x)R(x)

)δ

4(xminus y) (B6)

and interaction vertices given by the third and higher derivatives iδ kS[φ ]δφ(x1) middot middot middotδφ(xk) (kge 3) ofthe classical action In one-loop approximation the effective action is (see eq (314))

Γ[φ gmicroν ] = S[φ gmicroν ]+i2

Tr lnGminus10 (B7)

Rewriting the trace of a logarithm as the logarithm of the determinant the one-loop contribution tothe effective action for the action (B5) is

Γ[φ gmicroν ]1L =i2

lndet A (B8)

with the operator

Aequiv2x +X(x) X(x) =δ 2V (φ R)

δφ 2

∣∣∣∣φ(x)R(x)

(B9)

The generalized zeta-function for A is ζA(ν) equiv summ λminusνm where λm are the eigenvalues of A Using

zeta-function regularization (see eg [35 110]) the determinant can be written as

Γ[φ gmicroν ]1L =i2 sum

mln

λm

micro2 =12i

(ζ primeA(0)+ζA(0) ln micro2) (B10)

where ζ primeA = dζAdν and an arbitrary renormalization scale micro was introduced in order to obtain dimen-sionless quantities in the logarithm by shifting the effective action by a field-independent constantThe zeta-function can also be expressed via the heat kernel K(xys) fulfilling the heat equation

ipart

part sK(xys) = A(x)K(xys)

with boundary condition K(xy0) = δ 4(xminus y) The name of the ldquoheat equationrdquo originates fromthe Helmholtz equation with a ldquoproper timerdquo ldquo imiddots rdquo and the Laplace operator A =4 In terms of acomplete set of normalized eigenfunctions Aφm(x) = λmφm(x) the solution of the heat equation is3

K(xys) = summ

eminusλm isφm(x)φ lowastm(y)

such that the zeta-function has the representation

ζA(ν) = summ

iΓ(ν)

infinint0

ds(is)νminus1 eminusλm is =i

Γ(ν)

infinint0

ds(is)νminus1int

d4xK(xxs) (B11)

where the integral representation Γ(ν) = iλ νint

infin

0 ds(is)νminus1eminusλ is of the Γ-function and the normaliza-tion of the eigenfunctions has been used The ansatz for the solution K(xys) of the heat equation ofRefs [121 153] is

K(xys) =i∆12

V M(xy)(4πis)2 G(xys)exp

(minusσ(xy)

2isminus is

(X(y)minus R(y)

6

)) (B12)

2The Dirac δ -distribution in curved space-time is defined through the requirement thatintd4xradicminusg(x)δ 4(xminus y) f (x) =

f (y) for test functions f (x) [121]3 The boundary condition K(xy0) = δ 4(xminus y) follows directly from the completeness relation of the eigenfunctions


where σ(xy) is the proper arclength along the geodesic from x to y and ∆V M the van Vleck-Morettedeterminant [35]

∆VM(xy) =minus 1radicg(x)g(y)

det[minuspart 2σ(xy)

partxmicropartyν

] (B13)

fulfilling ∆V M(xx) = minusg(x) After inserting this ansatz together with the expansion G(xys) =sum

infinj=0(is)

jg j(xy) of the Heat Kernel into eq (B11) the integration over s can be performed usingagain the integral representation of the Γ-function

ζA(ν) =i

Γ(ν)

int d4x16π2

radicminusg

infin

sumj=0

g j(xx)Γ(ν + jminus2)

(XminusR6)ν+ jminus2

= iint d4x

16π2

radicminusg

g0(xx)(XminusR6)2minusν

(νminus1)(νminus2)+ g1(xx)

(XminusR6)1minusν

νminus1

+ g2(xx)(XminusR6)minusν +infin

sumj=3


Γ(ν)(XminusR6)ν+ jminus2

)

where Γ(α + 1) = αΓ(α) was used to rewrite the first three terms of the Heat Kernel Expansionexplicitly From the previous relation it can be inferred that Γ(ν + jminus2)Γ(ν) = (ν + jminus3)(ν + jminus4) middot middot middotν for j ge 3 Therefore the limit ν rarr 0 for ζA(ν) and ζ primeA(ν) can be performed and eq (B10)finally yields for the one-loop contribution to the effective action


32π2

radicminusg[minusg0(xx)

X2

2

(ln

Xmicro2 minus

32

)+ g1(xx)X

(ln

Xmicro2 minus1

)minusg2(xx) ln

Xmicro2 +

infin

sumj=3

g j(xx)( jminus3)

X jminus2

]

(B14)

where X equiv X minusR6 The coincidence limits yrarr x of the coefficients g j(xy) of the Heat KernelExpansion can be calculated recursively The results for the lowest orders from Ref [121] are

g0(xx) = 1 (B15)

g1(xx) = 0

g2(xx) =1

180(Rmicroνρσ RmicroνρσminusRmicroνRmicroν)minus 1

302R+

162X

=1

120Cminus 1

360Gminus 1

302R+

162X

where C and G are the Weyl- and Gauss-Bonnet terms as given above The coefficients g j(xx) withj ge 3 contain higher-order curvature scalars built from the curvature- and Ricci tensors and space-time derivatives of R and X They correspond to finite contributions to the one-loop effective action(B14) whereas the j = 012-contributions come along with divergences proportional to g0X2 g1Xand g2 Using eq (B15) one can see that it is necessary to introduce counterterms proportionalto X2 = (part 2Vpartφ 2minusR6)2 2(X minusR5) = 2(part 2Vpartφ 2minusR5) C and G in order to cancel thesedivergences which is already done implicitly in the result (B14) for the effective action through thezeta-function regularization [110] Nevertheless all operators contained in the counterterms shouldbe already present in the tree level action [90]

B3 Renormalization Group Equations 141

B3 Renormalization Group Equations

Callan-Symanzik Equation

Within the renormalization scheme provided eg by the zeta-function regularization [110] the renor-malized one-loop effective action explicitly depends on a renormalization scale micro In contrast to thisthe exact effective action is by construction independent of the renormalization scale It can be equiv-alently written either entirely in terms of bare parameters which are manifestly scale-independentor in terms of scale-dependent renormalized parameters In the latter case the vanishing total micro-derivative of the effective action yields the Callan-Symanzik equation for the effective action

0 =d

d ln micro2 Γ[φi] =

(part

part ln micro2 +sumN

βNpart

partcN+sum

i

intd4xγiφi(x)

δ

δφi(x)

)Γ[φi]

where all parameters of the theory are denoted collectively by cN and all fields by φi(x) For ascalar field in curved space φi(x) = φ(x)gmicroν(x) Furthermore for example for φ 4-theorycN sup Λm2λ ξ Gε0 ε4 The coefficients βN (β -functions) and γi (anomalous dimensions)are functions of these parameters The Callan-Symanzik equation is a partial differential equationwhich possesses characteristic solutions given by trajectories in parameter space cN(micro) and fieldspace φi(x micro) parameterized by the renormalization scale micro along which the effective action isconstant These trajectories are determined by definition by the renormalization-group equations

dd ln micro2 cN(micro) = βN(cN(micro)) and

dd ln micro2 φi(x micro) = γi(cN(micro))φi(x micro)

Renormalization Group Improved Effective Action

If the exact β -functions and anomalous dimensions were known as well as the exact effective actionfor one set of parameters cN(micro0) and one field configuration φi(x micro0) the renormalization groupequations yield the effective action along the complete trajectory for all scales micro The effective actionat micro = micro0 then yields the initial conditions for the renormalization group equations In practice onlyapproximations to the effective action are known Using the one-loop β -functions and anomalousdimensions as well as the classical (zero-loop) action S[φi] for one set of parameters cN(micro0) andone field configuration φi(x micro0) as initial condition at the scale micro = micro0 the renormalization groupequations yield an improved approximation (ldquoleading logarithm approximationrdquo) ΓLL[φi micro] to theeffective action for all scales micro This renormalization-group improved effective action is determinedby the partial differential equation

part

part tΓLL[φi micro] = minus

(sumN

βN(micro)part

partcN+sum

i

intd4xγi(micro)φi(x micro)

δ

δφi(x)

)ΓLL[φi micro]

ΓLL[φi micro0] = S[φi] (B16)

where t = ln(micro2micro20 ) The solutions of the one-loop renormalization group equations have to be

inserted for βN(micro)equiv βN1L(cN(micro)) and γi(micro)equiv γi1L(cN(micro))

One-Loop Renormalization Group Equations

The one-loop β -functions and the one-loop anomalous dimensions are obtained by matching thepartial differential equation (B16) at micro = micro0 with the one-loop effective action (471) The one-loopβ -functions will now be derived in this way for the action (466) of a scalar field in curved space with


generalized potential V (φ R) and B(φ R) from eq (470) On the one hand the classical action (466)can be inserted into the partial differential eq (B16) evaluated at micro = micro0

part

part tΓLL[φ gmicroν micro]

∣∣∣∣micro=micro0

= minus

(sumN

βNpart

partcN+sum

i

intd4xγiφi(x)

δ

δφi(x)

)S[φi]

= minusint

d4xradicminusg[sumnm

βnmφnRm +βε1C +βε2G+sum

nmβ nm2(φ nRm)

+ γφ φ

(2φ minus partV (φ R)

partφ

)+ γgmicroν

gmicroν

(12

gmicroνL(φ gmicroν)+δL(φ gmicroν)

δgmicroν

)]

where the β -functions βnm and β nm control the running of the coefficients cnm and cnm respec-tively On the other hand it can be used that the first derivative with respect to t = ln(micro2micro2

0 ) ofthe renormalization-group improved effective action and of the one-loop effective action coincide atthe reference scale micro = micro0 [60] For the one-loop effective action (471) the following is obtained

part



= minus part

part ln micro20

Γ1L[φ gmicroν micro0]

=int d4x

32π2


2minus 1

120Cminus 1

360Gminus 1

302R+

162X]

= minusint d4x

32π2

radicminusg

[12 sum

nm

(n

sumk=0

m

suml=0

dkldnminuskmminusl

)φ

nRmminus 1120

Cminus 1360

G

+16 sum

nm

((n+2)(n+1)cn+2mminus

15

δn0δm1

)2(φ nRm)

]

where the parameterizations (470) were inserted for X = partV (φ R)partφ 2 with dnm equiv (n + 2)(n +1)cn+2mminus δn0δm16 where δnm = 1 if n = m and zero otherwise The one-loop β -functions areobtained by comparing the coefficients of both upper expressions

βnm =1

64π2

n

sumk=0

m

suml=0

dkldnminuskmminusl

β nm =1

192π2


15

δn0δm1

) (B17)

βε1 = minus 132π2

1120


1360

γφ = 0 γgmicroν= 0

It is convenient to define a renormalization-group improved generalized potential VLL(φ R micro) and arenormalization-group improved function BLL(φ R micro)

VLL(φ R micro) = sumnm

cnm(micro)φ nRm BLL(φ R micro) = sumnm

cnm(micro)φ nRm (B18)

where the coefficients are solutions of the one-loop renormalization group equations dcnmdt = βnm

and dcnmdt = β nm Using the one-loop β -functions (B17) gives

part


164π2


partφ 2 minus R6

)2

part

part tBLL(φ R micro) =

1192π2


partφ 2 minus R5

) (B19)


Thus the solution of the partial differential equation (B16) for the renormalization-group improvedeffective action can be rewritten as


d4xradicminusg(

12

gmicroνpartmicroφpartνφ minusVLL(φ R micro)ε1(micro)C + ε2(micro)G+2BLL(φ R micro)

)

The initial condition at micro = micro0 in eq (B16) yields the initial conditions

VLL(φ R micro0) = V (φ R) BLL(φ R micro0) = 0

In the second equation it was used that the initial condition for BLL(φ R micro) can be chosen arbitrarilysince it appears as a total derivative in the action4

Sliding Renormalization Scale

The renormalization-group improved effective action ΓLL[φ gmicroν micro] yields an approximation to theeffective action which is applicable around the scale micro It is desirable to have one approximationΓCW [φ gmicroν ] available which simultaneously describes the dynamics for a certain range of scalesFor a single scalar field this is accomplished by exploiting the fact that the choice of the scale micro

in ΓLL[φ gmicroν micro] is free In fact so far no assumptions have been made which would restrict micro toa constant (see footnote 4) Evaluating the renormalization-group improved effective action with afield-dependent scale parameter t = ln(micro2micro2

0 ) yields [60]

ΓCW [φ gmicroν ] = ΓLL[φ gmicroν micro]

∣∣∣∣∣t=ln

(part2Vpartφ2minusR6

micro20

)+Γ1LHD[φ gmicroν ]

where the second term denotes the scale-independent part of the one-loop effective action (471)The choice for the field-dependent scale is obtained from requiring that ΓCW [φ gmicroν ]rarr S[φ gmicroν ] +Γ1L[φ gmicroν micro0] for trarr 0 [60]

4In factradicminusg middot2BLL(φ R micro) is not a total derivative if a field-dependent scale micro = micro(φ(x) ) is chosen and therefore

it contributes to the effective action in this case However since the reference scale micro0 is a constantradicminusg middot2BLL(φ R micro0)

is a total derivative (recall thatradicminusg middot2 = part micro

radicminusg middotpartmicro when applied to a Lorentz scalar)

Appendix C

Resummation Techniques andPerturbation Theory

C1 Relation between 2PI and 1PI

The equation of motion for the full connected two-point correlation function G(xy) derived from the2PI effective action has the form of a self-consistent Schwinger-Dyson equation [66]

Gminus1(xy) = Gminus10 (xy)minusΠ[φ G](xy) where Π[φ G](xy) =


(C1)

It is an inherently nonperturbative equation since the self-energy Π[φ G] is given by an expressionwhich also involves the full propagator G(xy) As explained in section 32 approximations withinthe 2PI formalism are achieved by truncating the 2PI functional Γ2[φ G] which is equal to the sum ofall 2PI diagrams with lines representing the full propagator and without external lines The full prop-agator is the solution of the self-consistent Schwinger-Dyson equation (C1) where the expressionfor the self-energy is obtained from the functional derivative of the truncated 2PI functional Γ2[φ G]Equivalently the full propagator G(xy) can also be expressed in terms of perturbative Feynmandiagrams involving the classical propagator G0(xy) In section 32 it has been mentioned that evenif only a very limited number of 2PI diagrams is retained in the truncated 2PI functional Γ2[φ G]the resulting full propagator corresponds to an infinite set of perturbative Feynman diagrams In thissection the construction of this infinite set is reviewed following Refs [37 147] By convolvingeq (C1) with the classical propagator from the left and with the full propagator from the right theintegrated form of the Schwinger-Dyson equation is obtained

G(xy) = G0(xy)+int

d4uint

d4v G0(xu)Π[φ G](uv)G(vy) (C2)

This equation permits an iterative solution starting from the classical propagator

G(0)(xy) = G0(xy)

G(k+1)(xy) = G0(xy)+int

d4uint

d4v G0(xu)Π(k)(uv)G(k+1)(vy)

= G0(xy)+int

d4uint

d4v G0(xu)Π(k)(uv)G0(xu)(vy)+

+int

d4uint

d4vint

d4zint

d4w G0(xu)Π(k)(uv)G0(vz)Π(k)(zw)G0(wy)+

146 C Resummation Techniques and Perturbation Theory

The self-energy appearing in the kth step is obtained by inserting the propagator G(k)(xy) into theexpression Π[φ G] for the self-energy derived from the (truncated) 2PI functional

Π(k)(uv) = Π[φ G(k)](uv)

The propagator G(k)(xy) is itself given by the Schwinger-Dyson series involving the self-energyΠ(kminus1)(uv) Employing a compact notation by suppressing the space-time integrations yields

Π(0) = Π[φ G0]

Π(k) = Π[φ G(k)] = Π[φ G0

infin

sumn=0

(Π(kminus1)G0)n]

Thus Π(k) is obtained from attaching self-energy insertions given by Π(kminus1) to the internal lines of theldquoskeletonrdquo diagrams contained in Π[φ G] Therefore for krarr infin this leads to an infinite hierarchy ofFeynman diagrams each of which is composed from nested skeleton diagrams with lines representingthe classical propagator G0 Since

Π[φ G[φ ]] = limkrarrinfin

Π(k)

where G[φ ] is the solution of the self-consistent Schwinger-Dyson equation (C1) the full propagatorobtained from the 2PI effective action indeed corresponds to an infinite summation of perturbativediagramsIf the self-energy Π[φ G] is derived from the exact 2PI functional Γ2[φ G] the self-energy Π[φ G[φ ]]equals the sum of all perturbative 1PI self-energy diagrams Furthermore these are obtained from theiterative procedure described above with the correct symmetry factors [37] If the self-energy Π[φ G]is derived from a truncation of the 2PI functional Γ2[φ G] then Π[φ G[φ ]] corresponds to an infinitesubset of all perturbative 1PI self-energy diagrams This subset is characterized by restrictions onthe topology of the perturbative diagrams since only a restricted set of skeletons is used for theirconstruction Even if only a single 2PI diagram is retained in the 2PI functional the correspondinginfinite subset contains perturbative diagrams of arbitrarily high loop orderAn approximation of the effective action can be obtained by inserting the full propagator G[φ ] intothe truncated 2PI effective action (see section 32)

C2 Resummed Perturbation Theory

Effective action from the 2PI Hartree-Fock approximation

For the extended Hartree-Fock approximation of the 2PI effective action derived in section 412 thesolution of the self-consistent Schwinger-Dyson equation can be written in the form

Gminus1hf (xy) = i(2x +M2

eff (xφ))δ d(xminus y) (C3)

where Ghf equiv G[φ ] is the full propagator in Hartree-Fock approximation The effective mass is deter-mined by the Hartree-Fock gap equation

M2eff (xφ) = exp

[12

G(xx)d2

dφ 2


∣∣∣∣∣G(xx)=G(xx M2

eff (middot φ)) (C4)



2x +M2(x))

G(xy M2(middot)

)=minusiδ d(xminus y)

C2 Resummed Perturbation Theory 147

An approximation of the effective action is obtained by inserting the full propagator in Hartree-Fockapproximation into the 2PI effective action

Γhf [φ ] = Γ[φ G[φ ]] (C5)

=int

ddx(


)+

i2

Tr[

ln(2x +M2

eff (xφ))minus iM2

eff (xφ)G[φ ]]

where


12

G(xxφ)d2

dφ 2

]Vcl(φ(x)) (C6)

has been defined Furthermore it is convenient to define an auxiliary potential

V (φ(x) M2(middot))equiv exp[

12

G(xx M2(middot)

) d2

dφ 2

]Vcl(φ(x)) (C7)

in terms of which the effective mass and Vhf (φ(x)) can be written as

M2eff (xφ) =

part 2V (φ M2(middot))partφ 2

∣∣∣∣φ=φ(x)M2(middot)=M2

eff (middot φ)

Vhf (φ(x)) = V (φ M2(middot))∣∣φ=φ(x)M2(middot)=M2

eff (middot φ) (C8)

Expansion of the exact effective action in terms of 1PI Feynman diagrams without tadpoles

It is possible to expand the exact effective action around the Hartree-Fock approximation (C5)

Γexact[φ ] = Γhf [φ ]+Γnotad[φ ] (C9)

iΓnotad[φ ] = +

=112

intddxint

ddy [minusiV (3)(φ(x))]Ghf (xy)3[minusiV (3)(φ(y))]+

where iΓnotad[φ ] is equal to the sum of all 1PI Feynman diagrams without tadpoles with lines repre-senting the field-dependent dressed propagator

Gminus1hf (xy) = i(2x +V (2)(φ(x)))δ d(xminus y)

determined self-consistently by the solution of the gap equation (C4) and field-dependent dressedvertices given by the derivatives of the auxiliary potential

minusiV (k)(φ(x))equiv minusipart kV (φ M2(middot))partφ k


eff (middot φ)

for kge 3 The gap equation (C4) can be rewritten as M2eff (xφ) = V (2)(φ(x)) which has already been

used above A Feynman diagram contains a ldquotadpolerdquo if it contains at least one line which begins andends at the same vertex The effective action expanded in terms of the dressed propagator and verticesdefined above only contains Feynman diagrams which have no ldquotadpolesrdquo


Derivation

The upper expansion of the effective action can be derived in two steps In the first step an ex-pansion of the exact propagator around the full Hartree-Fock propagator is performed Subtractingthe equation of motion of the exact propagator from the equation of motion of the full Hartree-Fockpropagator yields

Gminus1(xy)minusGminus1hf (xy) =minusΠ[φ G](xy)+Πhf [φ Ghf ](xy)equivminusΠnotad(xy)

where Πhf denotes the expression for the self-energy derived from the Hartree-Fock truncation (48)of the 2PI effective action An expansion of the self-energy Πnotad defined above in terms of 1PIdiagrams with lines representing the Hartree-Fock propagator and vertices given by the derivativesof the classical potential can be obtained by an iterative expansion similar to the one discussed insection C1

Π(0)notad = Π[φ Ghf ]minusΠhf [φ Ghf ]

Π(k)notad = Π[φ Ghf

infin

sumn=0

(Π(kminus1)notad Ghf )n]minusΠhf [φ Ghf ] (C10)

According to Ref [37] any 1PI Feynman diagram with two external lines (ldquoself-energy diagramrdquo)can be decomposed into a unique skeleton diagram (obtained from opening one line of a 2PI diagramwithout external lines) and a set of self-energy sub-diagrams which are attached to the internal lines ofthe skeleton as insertions The Hartree-Fock self-energy Πhf [φ Ghf ] consists of the sum of all tadpoleself-energy diagrams which are called tadpole-skeletons or tadpole-insertions in the followingThe 0th iteration Π

(0)notad consists of all possible skeleton diagrams evaluated with the propagator

Ghf except those contained in Πhf [φ Ghf ] ie except tadpole-skeletons Furthermore the skele-ton diagrams themselves do by definition not contain any insertions and therefore especially notadpole-insertions Thus Π

(k)notad for k = 0 does not contain any tadpole-skeletons or diagrams carry-

ing tadpole-insertions It can be proven by induction that this is also true for all k ge 0 and thereforefor Πnotad itself The tadpole-skeletons are explicitly subtracted at each step of the iteration (C10)Furthermore the diagrams contained in Π

(kminus1)notad are the insertions of the diagrams contributing to

Π(k)notad Since the former contain no tadpole-skeletons the latter contain no tadpole-insertions

The fact that Πnotad does neither contain tadpole-skeletons nor diagrams carrying tadpole-insertionscan also be formulated in the following way When all tadpoles appearing in any self-energy diagramcontributing to Πnotad are removed the remaining diagram is still 1PI All contributions to Πnotadthat do contain tadpoles can be generated from such diagrams by adding tadpoles at the verticesSumming over the number of tadpoles attached to each vertex is equivalent to replacing the verticesaccording to

minus iV (k)(φ(x))rarrminusiexp(

12

Ghf (xx)d2

dφ 2

)V (k)(φ(x)) =minusiV (k)(φ(x)) (C11)

which can be seen from a Taylor expansion of the exponential The term of order L corresponds to Ltadpoles It remains to be shown that the diagrams are generated with the correct symmetry factorsLet F be a Feynman diagram contributing to Πnotad and let Fγ be the unique diagram obtained byremoving all tadpoles from F with γ = γ1 γl l ge 2 the unique set of tadpoles contained atthe vertices 1 l of F Then Fγ isin ΠNoTad and Fγ has the same number of vertices as F sinceF does not contain any tadpole-insertions Due to the exponential in eq (C11) the tadpoles γi aregenerated with correct symmetry factors N(γi) FurthermoreFγ isinΠNoTad has the correct symmetry


factor N(Fγ) However there can be several possibilities how to attach the tadpoles in γ to Fγ

leading to the same diagram F Let K(F) be the number of these possibilities Then it is to be shownthat

K(F) middot 1N(Fγ) prod

li=1 N(γi)

F =1

N(F)F (C12)

where F denotes the diagram F without symmetry factor and N(middot) equiv |S(middot)| denotes the symmetryfactor equal to the order of the symmetry group S(middot) of a given diagram Thus eq (C12) is equivalentto

K(F) =

∣∣∣∣∣S(Fγ)otimesl

prodi=1

S(γi)

∣∣∣∣∣ |S(F)| (C13)

Since S(F) is a subgroup of S(Fγ)otimesprodli=1 S(γi) the expression on the right-hand side of eq (C13)

is an integer and equal to the order of the set of co-sets S(Fγ)otimesprodli=1 S(γi)S(F) Each co-set

corresponds to one of the possible attachments counted by K(F) [61]Altogether it is found that Πnotad(xy) is equal to the sum of all 1PI Feynman diagrams with twoexternal lines internal lines representing the Hartree-Fock propagator Ghf (xy) dressed vertices

minusiV (k)(φ(x)) obtained from the derivatives of the auxiliary potential and without any tadpoles

In the second step it is shown that Γnotad[φ ] equiv Γexact[φ ]minusΓtad [φ ] can analogously be expressed interms of 1PI Feynman diagrams with propagator Ghf (xy) dressed vertices minusiV (k)(φ(x)) withoutexternal lines and without any tadpoles Therefore it will first be shown that the Feynman diagramscontributing to Γnotad[φ ] are neither ldquomulti-bubblerdquo diagrams (see section 412) nor carry tadpole-insertions when formulated in terms of the propagator Ghf (xy) and classical vertices minusiV (k)(φ(x))Second the remaining tadpoles are resummed by replacing the classical vertices by the dressed ver-tices according to the rule (C11)Using the parameterization (326) of the exact 2PI effective action and eq (C9) for Γtad [φ ] one finds

Γnotad[φ ] =i2

Tr ln(1minusΠnotadGhf )+i2

TrΠnotadG+ Γ2[φ Ghf ] (C14)

where

Γ2[φ Ghf ] = Γ2[φ G]minusΓ2hf [φ Ghf ]minusTrδΓ2hf [φ Ghf ]

δGhf

(GminusGhf

) (C15)

Here Γ2[φ G] denotes the exact 2PI functional evaluated with the exact propagator and Γ2hf [φ Ghf ]denotes the Hartree-Fock truncation (48) of the 2PI functional which resums the multi-bubble dia-grams evaluated with the Hartree-Fock propagatorAn expansion of Γnotad[φ ] in terms of 1PI Feynman diagrams with propagator Ghf (xy) and classicalvertices minusiV (k)(φ(x)) is obtained from eq (C14) by Taylor expanding the logarithm in the first termon the right-hand side in powers of ΠnotadGhf and by inserting the Schwinger-Dyson sum

G = Ghf

infin

sumn=0

(ΠnotadGhf )n equiv Ghf +∆G (C16)

for the exact propagator Then multi-bubble diagrams or diagrams carrying tadpole-insertions couldarise in eq (C14) from the following terms

(i) The linear term in the expansion of i2 Tr ln(1minusΠnotadGhf ) in powers of ΠnotadGhf

(ii) The linear term in the expansion of i2 TrΠnotadG in powers of ΠnotadGhf


(iii) Diagrams contributing to Πnotad which carry tadpole-insertions

(iv) Diagrams contributing to Γ2[φ Ghf ]

The contributions from (i) and (ii) cancel and (iii) cannot occur as was shown in the first step ofthe derivation In order to investigate (iv) the 2PI functional Γ2[φ G] equiv Γ2hf [φ G]+ Γ2notad[φ G]is split into a Hartree-Fock part containing (local) multi-bubble diagrams evaluated with the exactpropagator and a non-local part Inserting eq (C16) into the former yields

Γ2hf [φ G] = Γ2hf [φ Ghf ] + TrδΓ2hf [φ Ghf ]

δGhf∆G + O (∆G)2

Multi-bubble diagrams arise from the first term on the right-hand side and diagrams carrying tadpoleinsertions from the second However precisely those are cancelled in the expression for Γ2[φ Ghf ]which can be seen from eq (C15) Thus Γnotad[φ ] does not contain multi-bubble diagrams or dia-grams carrying tadpole-insertions when formulated in terms of the propagator Ghf (xy) and classicalvertices minusiV (k)(φ(x))Similar to self-energy diagrams any 1PI Feynman diagram without external lines can be decomposedinto a 2PI skeleton diagram without external lines and a set of self-energy sub-diagrams which areattached to the internal lines of the skeleton as insertions However in contrast to the self-energy di-agrams this decomposition is not unique Therefore it is important to check that every 1PI Feynmandiagram without tadpole-insertions contributes exactly once to Γnotad[φ ] ie that no over-countingoccurs The argument is analogous to the expansion of the 2PI effective action in terms of perturba-tive Feynman diagrams [37] The three contributions on the right-hand side of eq (C14) count everydiagram with a multiplicity factor nc minusnl and ns respectively where nc is the number of circles nlthe number of lines in circles and ns the number of skeletons of a given 1PI diagram without externallines as defined in Ref [37] Due to the relation ncminusnl +ns = 1 [37] every diagram is counted onceAny diagram F contributing to Γnotad[φ ] can be composed from a unique 1PI diagram without anytadpoles Fγ by attaching tadpoles γ = γ1 γl l ge 2 at the l vertices of Fγ Thus it followsanalogously to the first step of the derivation that F can be generated with correct symmetry factorfrom the diagram Fγ formulated with dressed vertices minusV (k)(φ) by expanding the exponential ineq (C11) Due to the uniqueness of Fγ for any F isin Γnotad[φ ] no over-counting can occur hereSince also Fγ isin Γnotad[φ ] all 1PI diagrams without any tadpoles are included in Γnotad[φ ]Finally it is found that iΓnotad[φ ] is equal to the sum of all 1PI Feynman diagrams with internallines representing the Hartree-Fock propagator Ghf (xy) with no external lines with dressed vertices

minusV (k)(φ) derived from the auxiliary potential (C7) and without any tadpoles

Appendix D

Quantum Fields in and out ofEquilibrium

D1 Thermal Quantum Field Theory

Thermal quantum field theory describes quantum fields in thermal equilibrium In section D11two alternative representations of the density matrix element of the thermal density matrix withinperturbation theory are reviewed Furthermore in section D12 an equation of motion for the fullthermal propagator is derived from the 2PI effective action formulated on the thermal time path

D11 Thermal State

A statistical ensemble in a thermal state at temperature T = 1β is described by the density matrix

ρ =1Z

exp(minusβH)

where the partition function Z is chosen such that Trρ = 1 [124 135 136] The interaction termscontained in the full Hamiltonian H lead to the presence of higher correlations and make the thermalstate a highly non-Gaussian state In contrast to any generic nonequilibrium density matrix thethermal density matrix has the property to lead to correlation functions which are invariant under timetranslations [104136] This means that the thermal state indeed describes an ensemble in equilibriumThe exponential appearing in the thermal density matrix can be interpreted as the full time-evolutionoperator exp(minusitH) evaluated for the imaginary time t = minusiβ Accordingly the matrix element ofthe thermal density matrix (see eq (67)) can be written as a path integral over field configurationsϕ(x) with time argument on a time contour I running along the imaginary axis from t = 0 to t =minusiβ [49] Alternatively the matrix element can be represented by a Taylor expansion in terms ofthermal correlation functions α th

n (x1 xn) as in eqs (7172)


ϕ(minusiβ x)=ϕ+(x)intϕ(0x)=ϕminus(x)

Dϕ exp(

iint

Id4xL(x)

)C+I

exp

(i

infin

sumn=0

αth12middotmiddotmiddotnϕ1ϕ2 middot middot middotϕn

)C+α

(D1)

where the short-hand notation from eq (334) applies Thus for the thermal state there exist twopossibilities how to calculate thermal correlation functions Either by extending the closed real-time

152 D Quantum Fields in and out of Equilibrium

path C in the generating functional (67) by the imaginary path I (ldquoC+I rdquo) or by keeping the closedreal-time contour C in the generating functional (67) and inserting the thermal initial correlationsα th

n (x1 xn) (ldquoC+ α rdquo) Within perturbation theory the latter can be obtained from a perturbativeexpansion of the thermal density matrix element Since extensive use of both formulations is madeboth are reviewed here


By using the path integral representation of the thermal density matrix a path integral representationof the generating functional for the thermal state can be obtained by concatenating the time contoursC and I (the derivation is analogous to the steps leading from eq (67) to eq (619))

Zβ [J] = Tr(

ρ TC+I

[exp(

iint

C+Id4xJ(x)Φ(x)

)])=intDϕ exp

(iint

C+Id4x L(x)+ J(x)ϕ(x)

)

The part of the time path along the imaginary axis prepares the system in a thermal state at the initialtime tinit = 0 and is characteristic for thermal equilibrium whereas the part of the time path alongthe real axis yields the time-evolution of the system For calculations in thermal equilibrium it issometimes convenient to use a pure imaginary time formalism by setting tmax = 0 such that only thepath I contributes However here the real-time evolution of correlation functions (with a finite initialtime) is of interest in which case the full thermal time path is requiredThe time arguments of the thermal propagator can also be attached to the thermal time path andusing the time-ordering operator TC+I it reads

Gth(xy) = 〈TC+I Φ(x)Φ(y)〉minus〈Φ(x)〉〈Φ(y)〉 = minusδ 2 lnZβ [J]δJ(x)δJ(y)

∣∣∣∣∣J=0

(D2)

The thermal propagator evaluated with imaginary time arguments fulfills the relation

Gth(minusiτx0y) = Gth(0xminusi(β minus τ)y) for 0le τ le β

ie it is periodic with period β which can be seen using cyclic invariance of the trace

Tr(

eminusβHΦ(minusiτx)Φ(0y)

)= Tr

(eminusβHeτH

Φ(0x)eminusτHΦ(0y)

)= Tr

(eminusτH

Φ(0y)eminus(βminusτ)HΦ(0x)

)= Tr

(eminusβH

Φ(minusi(β minus τ)y)Φ(0x))

Due to time-translation invariance the thermal field expectation value is constant in time Thermal n-point correlation functions can be calculated by taking the nth derivative of the generating functionalZβ [J] with respect to the external source J(x)


Alternatively one can describe the generating functional for thermal correlation functions withoutreference to imaginary times by inserting the representation of the thermal density matrix element inthe second line of eq (D1) into the generating functional (67) For this approach it is required tocalculate the thermal correlation functions α th

n (x1 xn) explicitly This requires to match the twoformulations of the thermal density matrix element in eq (D1)For an interacting theory the thermal density matrix element cannot be calculated exactly However itcan be expanded perturbatively [49] starting from the density matrix ρ0 = 1

Z0exp(minusβH0) containing

D1 Thermal Quantum Field Theory 153

the free Hamiltonian H0 which is quadratic in the field such that the path integral in eq (D1) can beperformed

〈ϕ+0 |ρ0|ϕminus0〉=N0 exp[

iS0[φ0]]

=N0 exp[

iint

Id4x(

12(partφ0)2minus 1

2m2

φ20

)]

Here N0 is a normalization factor which is independent of ϕplusmn and φ0(x) is the solution of the freeequation of motion δS0δφ = (minus2minusm2)φ0 = 0 on I subject to the boundary conditions

φ0(0x) = φ0(0minusx) = ϕminus(x) and φ0(minusiβ x) = φ0(0+x) = ϕ+(x)

The solution is uniquely determined and in spatial momentum space given by

φ0(minusiτk) =sinh(ωkτ)sinh(ωkβ )

ϕ+(k)+sinh(ωk(β minus τ))

sinh(ωkβ )ϕminus(k) (D3)

where ω2k = m2 +k2 The exponent of the free thermal density matrix element is quadratic in φ0

Therefore it describes a Gaussian initial state Using the explicit form of φ0(minusiτk) it can be writtenas in eq (D25) with

ξ2k =

nBE(ωk)+ 12

ωk ηk = 0

σ2k

4ξ 4k

= ω2k where nBE(ωk) =

1eβωk minus1

(D4)

The full thermal initial correlations can be obtained by perturbing the full Hamiltonian H around H0

〈ϕ+0 |ρ|ϕminus0〉= exp[iF [φ0]

] iF [φ0] = lnN + iS0[φ0]+ iFint [φ0]

where N is a normalization factor iS0[φ0] is the free contribution and iFint [φ0] is the sum of allconnected Feynman diagrams with vertices

iδ 4Sintδφ4 =minusiλδI(x1minus x2)δI(x1minus x3)δI(x1minus x4) =

which are integrated over the imaginary contour I denoted by the empty circle The boundaryconditions of the path integral (D1) are formally taken into account by the field ldquoexpectationrdquo value

φ0(minusiτk) = (D5)

along the imaginary contour I as well as the propagator

D0(minusiτminusiτ primek) = (D6)

=sinh(ωkτ)sinh(ωk(β minus τ prime))Θ(τ primeminus τ)+ sinh(ωkτ prime)sinh(ωk(β minus τ))Θ(τminus τ prime)

ωk sinh(ωkβ )

which is the Greens function for solutions of the free equation of motion that vanish at the boundariesτ = 0β denoted by the dotted line To first order in λ iFint [φ0] is given by

iFint [φ0] = + O(λ 2)

=minusiλ4

intI

d4x

3D0(xx)2 +6φ0(x)2D0(xx)+φ0(x)4 + O(λ 2)


The field-independent diagrams like the first one above can be absorbed into the normalization N The perturbative expansions of the thermal initial correlations α th

n are obtained by the n-th functionalderivative with respect to the field

iα thn (x1 xn) =

(δ iF [φ0]

δϕε1(x1) middot middot middotδϕεn(xn)

∣∣∣∣φ0=0

)δC(x0

1minus0ε1) middot middot middotδC(x0nminus0εn)

to which all diagrams with n insertions of φ0 contribute Here it can explicitly be seen that the initialcorrelations are supported only at the initial time as required Formally the functional derivativecorresponds to replacing the field insertions by (distinguishable) external lines in the diagrammaticexpansion of iFint [φ0] according to

φ0(minusiτk) 7rarr ∆0(minusiτx0k) equiv sinh(ωkτ)sinh(ωkβ )

δC(x0minus0+)+sinh(ωk(β minus τ))

sinh(ωkβ )δC(x0minus0minus)

equiv ∆+0 (minusiτk)δC(x0minus0+)+∆

minus0 (minusiτk)δC(x0minus0minus)

7rarr (D7)

For example the thermal four-point initial correlation function obtained from the fourth derivative ofiFint [φ0] is

iα th4 (x1x2x3x4) = minusiλ

intId4v∆0(vx1)∆0(vx2)∆0(vx3)∆0(vx4)+O(λ 2)

= + O(λ 2) (D8)

where ∆0(vx) =int d3k(2π)3 e+ik(vminusx) ∆0(v0x0k) for v0 isin Iz0 isin C Switching again to momentum

space an explicit expression for the leading contribution to the perturbative thermal initial four-pointcorrelation function is obtained

iα thε1ε2ε3ε44 (k1k2k3k4) =

= minusλ

intβ

0dτ ∆

ε10 (minusiτk1)∆ε2

0 (minusiτk2)∆ε30 (minusiτk3)∆ε4

0 (minusiτk4) + O(λ 2)

For example for ε1 = ε2 = ε3 = ε4 = + or minus

iα th++++4 (k1k2k3k4) = iα thminusminusminusminus

4 (k1k2k3k4) =

= minusλ

intβ

0dτ

sinh(ωk1τ)sinh(ωk1β )




+ O(λ 2)

rarr minusλ

ωk1 +ωk2 +ωk3 +ωk4

+ O(λ 2) for β rarr infin (D9)

The last line represents the zero-temperature limit The correlations with mixed upper indices vanishin the zero-temperature limit as required for a pure initial state Altogether a diagrammatic expansionof the matrix element of the thermal density matrix in terms of perturbative Feynman diagrams hasbeen developed as suggested in Ref [49] This allows to explicitly calculate thermal correlationfunctions order by order in the quartic coupling constant The lowest-order perturbative result (D8)may be compared to the nonperturbative 2PI result (767)


D12 Nonperturbative Thermal 2PI Propagator on the Thermal Time Path

In this section an equation of motion for the full thermal propagator is derived from the stationaritycondition of the 2PI effective action formulated on the thermal time path C+I This self-consistentequation of motion is the analogon of the Kadanoff-Baym equation on the closed real-time path CThe classical thermal propagator defined on C+I is (φ(x)equiv φ = const in equilibrium)

iGminus10th(xy) =

(minus2xminusm2minus λ

2φ

2)

δC+I(xminus y) for x0y0 isin C+I (D10)

The full thermal propagator is determined by the equation of motion derived from the 2PI effectiveaction defined on the thermal time contour C+I which is given by the self-consistent Schwinger-Dyson equation


0th(xy)minusΠth(xy) for x0y0 isin C+I (D11)

The thermal propagator can be decomposed into the statistical propagator and the spectral function

Gth(xy) = GF(xy)minus i2

sgnC+I(x0minus y0)Gρ(xy) for x0y0 isin C+I (D12)

where sgnC+I(x0minus y0) is the signum function defined on the path C+I It is equal to +1 if x0

corresponds to a ldquolaterrdquo time than y0 along the time path where ldquolaterrdquo refers to the time-orderingoperator TC+I In particular all times on the imaginary branch I are ldquolaterrdquo than all times on theantichronological branch Cminus and these are ldquolaterrdquo than all times on the chronological branch C+The thermal self-energy can be decomposed similarly as in eqs (D42 D44)

Πth(xy) = minusiΠlocth (x)δC+I(xminus y)+Π

nlth(xy)

Πnlth(xy) = ΠF(xy)minus i

2sgnC+I(x0minus y0)Πρ(xy)

=(minusiλ )2

2φ

2Gth(xy)2 +(minusiλ )2

6Gth(xy)3 (D13)

M2th = m2 +

λ

2φ

2 +Πlocth (x) = m2 +

λ

2φ

2 +λ

2Gth(xx)

where in the third line as an example the 2PI-O(λ 2) approximation is given (see section D2)This approximation coincides with the setting-sun approximation for vanishing field expectationvalue The thermal effective mass M2

th is time-independent in equilibrium Convolving the ther-mal Schwinger-Dyson equation with Gminus1

th yields an equation of motion for the thermal propagator onthe thermal time path C+I(

2x +M2th)

Gth(xy) =minusiδC+I(xminus y)minus iint

C+Id4zΠ

nlth(xz)Gth(zy) (D14)


th GCIth GIC

th GIIth The equation of motion evaluated for two real arguments yields

an equation for GCCth etc The four equations of motion for GCC

th GCIth GIC

th and GIIth are coupled due

to the contour integral on the right hand side For example the equation for GCCth is

(2x +M2

th)

GCCth (xy) = minusiδC(xminus y)minus i

intCd4zΠ

CCth (xz)GCC

th (zy)

minusiint

Id4zΠ

CIth (xz)GIC

th (zy) (D15)


Similar to the Kadanoff-Baym equations on the closed real-time contour the upper equation canbe decomposed into an equation for the thermal statistical propagator GCC

F and the thermal spectralfunction GCC

ρ (2x +M2

th)

GCCF (xy) =

int y0

0d4zΠ

CCF (xz)GCC

ρ (zy)

minusint x0

0d4zΠ

CCρ (xz)GCC

F (zy) (D16)

minusint

β

0dτ

intd3zΠ

CIth (x(minusiτz))GIC

th ((minusiτz)y)

(2x +M2

th)

GCCρ (xy) =

int y0

x0

d4zΠCCρ (xz)GCC

ρ (zy)

For the propagators GICth GCI

th and GIIth one finds analogously(

2x +M2th)

GICth (xy) =

int y0

0d4zΠ

ICth (xz)GCC

ρ (zy)

minusint

β

0dτ

intd3zΠ

IIth (x(minusiτz))GIC

th ((minusiτz)y)

(2x +M2

th)

GCIth (xy) = minus

int x0

0d4zΠ

CCρ (xz)GCI

th (zy) (D17)

minusint

β

0dτ

intd3zΠ

CIth (x(minusiτz))GII

th ((minusiτz)y) (2x +M2

th)

GIIth (xy) = minus iδI(xminus y)

minusint

β

0dτ

intd3zΠ

IIth (x(minusiτz))GII

th ((minusiτz)y)

The equation of motion for the purely imaginary-time propagator is independent of the other equa-tions which is an reflection of causality Since thermal correlations are invariant under space andtime translations it is convenient to switch to momentum space In addition to the spatial Fouriertransform (D48) a temporal Fourier transformation can be performed for all times which lie on theimaginary part I of the thermal time contour

Gth(x0y0k) =int

d3xeminusik(xminusy) Gth(x0xy0y)

GIIth (k0k) =

intβ

0dτ eminusik0(τminusτ prime) GII

th (minusiτminusiτ primek) (D18)

GICth (k0y0k) =

intβ

0dτ eminusik0τ GII

th (minusiτy0k)

and analogously for GCIth Since the thermal propagator is periodic on the finite interval I it is

sufficient to know its Fourier transform for the Matsubara frequencies

k0 = ωn =2π

Tn = 2πβn n = 0plusmn1plusmn2

The inverse Fourier transformation with respect to the imaginary time is thus given by the discreteFourier sum

GIIth (minusiτminusiτ primek) = T sum

neiωn(τminusτ prime) GII

th (ωnk)

GICth (minusiτy0k) = T sum

neiωnτ GIC

th (ωny0k) (D19)

D2 Nonequilibrium Quantum Field Theory 157

By applying the Fourier transformation to the last equation in (D17) the nonperturbative Schwinger-Dyson equation for the full thermal Matsubara propagator is obtained(

ω2n +k2 +M2

th)

GIIth (ωnk) = 1minusΠ

IIth (ωnk)GII

th (ωnk) (D20)

whereint β

0 dτ(minusiδI(minusiτminus iτ prime)) = 1 was used

D2 Nonequilibrium Quantum Field Theory

Within nonequilibrium quantum field theory nonperturbative approximations of the full effective ac-tion based on the 2PI formalism [66] can be used to describe the quantum equilibration process [27]In contrast to this perturbative approximations based on the usual (1PI) effective action cannot de-scribe thermalization even for arbitrarily small couplings λ due to secular behaviour [27] This meansthat the perturbative approximation fails for late times λ t amp 1 The derivation of the 2PI effective ac-tion for ensembles out of equilibrium and the resulting Kadanoff-Baym equations which describe thetime-evolution of the full connected two-point correlation function is reviewed below for Gaussianinitial states For an introduction to nonequilibrium quantum field theory it is referred to Ref [27]As was shown in section 61 the information about the initial state enters via the matrix elementof the density matrix describing the statistical ensemble at some initial time tinit equiv 0 which canbe characterized by an infinite set of initial n-point correlation functions αn(x1 xn) according toeqs (7172) In the following the form of these initial correlations is discussed for two specialclasses of initial states

D21 Pure Initial States

If the complete statistical ensemble is in a definite state |ψ〉 in Hilbert space (pure initial state) thedensity matrix has the form ρ = |ψ〉〈ψ| In this case the density matrix element (71) is of the form

〈ϕ+0 |ρ|ϕminus0〉= 〈ϕ+0 |ψ〉〈ψ|ϕminus0〉 equiv exp(iFψ [ϕ+]

)exp(minusiFlowastψ [ϕminus]

) (D21)

where exp(iFψ [ϕ]

)equiv 〈ϕ0|ψ〉 Thus for a pure initial state the functional defined in eq (71) splits

up into two separate contributions where the first one depends only on ϕ+(x) = ϕ(0+x) and thesecond one depends only on ϕminus(x) = ϕ(0minusx)

F [ϕ] = Fψ [ϕ+]minus iFlowastψ [ϕminus] (D22)

The coefficients of the Taylor expansion (72) thus cannot contain any mixed terms with respect tothe upper indices for a pure initial state

αn(x1 xn) = α++ middotmiddotmiddot+n (x1 xn)δ+(x0

1) middot middot middotδ+(x0n)

+αminusminusmiddotmiddotmiddotminusn (x1 xn)δminus(x0

1) middot middot middotδminus(x0n) (D23)

D22 Gaussian Initial States

A Gaussian initial state is characterized by the absence of higher correlations

αn(x1 xn) = 0 for nge 3 (Gaussian initial state) (D24)

The most general Gaussian initial state can thus be parameterized as


i

α0 +int


12

intd3xd3yϕε1(x)αε1ε2

2 (xy)ϕε2(y))


For an initial state which is invariant under spatial translations it is convenient to switch to spatialmomentum space and use αε

1 (x) = αε1 = const and α

ε1ε22 (xy) =

int d3k(2π)3 eik(xminusy) α

ε1ε22 (k)


i

α0 +αε0 ϕε(0)+

12

int d3k(2π)3 ϕε1(k)αε1ε2

2 (k)ϕε2(minusk))

(D25)

Due to the Hermiticity of the density matrix the initial correlations have to fulfill the relations α+1 =

minusαminus1lowast α

++2 =minusα

minusminus2lowast and α

+minus2 =minusα

minus+2lowast Within real scalar theory the initial correlations α

ε1εnn

may additionally be chosen to be totally symmetric in the upper indices For a Gaussian initial statethis is equivalent to α

+minus2 = α

minus+2 Thus αε

1 (x) and αε1ε22 (xy) can be described by two and three

real-valued functions respectively1 One may completely parameterize these independent degreesof freedom of the Gaussian state by the initial expectation values of the field operator and of itsconjugate [27]

φ(x)|x0=0 = Tr(

ρ Φ(x))∣∣∣∣

x0=0 φ(x)|x0=0 = Tr

(ρ partx0Φ(x)

)∣∣∣∣x0=0

(D26)

together with the initial values of the three real correlation functions

G(xy)|x0=y0=0 =

Tr(

ρ Φ(x)Φ(y))minusφ(x)φ(y)

∣∣∣∣x0=y0=0

(partx0 +party0)G(xy)∣∣x0=y0=0 =

Tr(

ρ

[Φ(x)party0Φ(y)+partx0Φ(x)Φ(y)

])(D27)

minus(φ(x)φ(y)+ φ(x)φ(y)

)∣∣x0=y0=0

partx0party0G(xy)∣∣x0=y0=0 =

Tr(ρ partx0Φ(x)party0Φ(y)

)minus φ(x)φ(y)

∣∣x0=y0=0

The relations between the upper initial conditions for the one- and two-point function and the densitymatrix (D25) are obtained by evaluating the Gaussian integrals [27] For an initial state which isinvariant under spatial translations one obtains

φ(x)|x0=0 =intDϕ ϕ(x)〈ϕ0 |ρ|ϕ0〉= ξ

2k=0 sum

ε=plusmniαε

1 (D28)

φ(x)|x0=0 =intDϕ

minusipartpartϕ(x)

langϕ0 |ρ|ϕ prime0

rang∣∣∣∣ϕ prime=ϕ

=12i

(sum

ε=plusmnε iαε

1 +2iηk=0 ξk=0 sumε=plusmn

iαε1

)

Setting G(xy) =int d3k

(2π)3 eik(xminusy) G(x0y0k) one obtains similarly

G(x0y0k)∣∣x0=y0=0 = ξ

2k

(partx0 +party0)G(x0y0k)∣∣x0=y0=0 = 2ηkξk (D29)

partx0party0G(x0y0k)∣∣x0=y0=0 = η

2k +

σ2k

4ξ 2k

with

1ξ2k = minus sum

ε j=plusmniαε1ε2

2 (k)

1The constant α0 is determined by the normalization condition Trρ = 1 of the density matrix


Figure D1 Diagrams contributing to iΓ2[φ G] at two- and three-loop order with less than threevertices (2PI-O(λ 2)-approximation)

2iηkξk = sumε j=plusmn

ε1iαε1ε22 (k) = sum

ε j=plusmnε2iαε1ε2

2 (k)

σ2kξ

2k = minus sum

ε j=plusmnε1ε2iαε1ε2

2 (k)

(D30)

From eq (D23) it can be seen that the Gaussian density matrix (D25) describes a pure initial state ifηk = 0 and σ2

k = 1

D23 2PI Effective Action for Gaussian Initial States

As has been discussed in section 61 the 2PI effective action formulated on the closed real-time pathC can be parameterized in the standard form [66]

Γ[φ G] = S[φ ]+i2

Tr lnGminus1 +i2

Tr(Gminus1

0 G)+Γ2[φ G] (D31)

for a nonequilibrium ensemble which is characterized by a Gaussian initial state While the derivationof Kadanoff-Baym equations discussed in section 61 has been restricted to the setting-sun approxi-mation the general derivation is reviewed here The general form of the Kadanoff-Baym equationsincludes also a non-vanishing field expectation value φ(x)Within λΦ44-theory the inverse classical propagator is given by

iGminus10 (xy)equiv δ 2S[φ ]

δφ(x)δφ(y)=(minus2xminusm2minus λ

2φ(x)2

)δC(xminus y) (D32)

The functional iΓ2[φ G] is the sum of all two particle irreducible (2PI) Feynman diagrams with linesgiven by the full propagator G(xy) and without external lines [66] The vertices of the graphs con-tained in iΓ2[φ G] are given by the third and fourth derivatives of the classical action S[φ ]

=iδ 4S[φ ]

δφ(x1) δφ(x4)= minusiλδC(x1minus x2)δC(x2minus x3)δC(x3minus x4)

=iδ 3S[φ ]

δφ(x1) δφ(x3)= minusiλφ(x1)δC(x1minus x2)δC(x2minus x3) (D33)

The initial one- and two-point correlation functions parameterizing the Gaussian initial density ma-trix (D25) do not appear explicitly in the 2PI effective action which is a peculiarity of the Gaus-sian initial state Instead the initial state enters via the initial conditions for the one- and two-pointfunctions φ(x) partx0φ(x) G(xy) (partx0 + party0)G(xy) and partx0party0G(xy) at x0 = y0 = 0 (see eqs (D28)and (D29))The two- and three-loop contributions to iΓ2[φ G] with less than three vertices are (see figure D1)

iΓ2[φ G] =minusiλ

8

intCd4xG(xx)2 +

(minusiλ )2

12

intCd4xd4yφ(x)G(xy)3

φ(y)

+(minusiλ )2

48

intCd4xd4yG(xy)4 +O(λ 3) (D34)


Figure D2 Diagrams contributing to the self-energy Π(xy) at two- and three-loop order with lessthan three vertices (2PI-O(λ 2)-approximation)

The 2PI-O(λ 2)-approximation of iΓ2[φ G] coincides with the setting-sun approximation for vanish-ing field expectation value

Equation of motion for the full propagator

The equation of motion for the full propagator is obtained from evaluating the functional derivativeδΓ[φ G]δG(xy) = minusK(xy)2 of the 2PI effective action (see eq (324)) using the parameteriza-tion (D31)

Gminus1(xy) = Gminus10 (xy)minusΠ(xy)minus iK(xy) (D35)

where for generality the bilocal source K(xy) was included and the self-energy Π(xy) was intro-duced which is defined as


(D36)

In 2PI-O(λ 2)-approximation the self energy can be calculated using eq (D34)

Π(xy) =minusiλ

2G(xx)δC(xminus y)+

(minusiλ )2

2φ(x)G(xy)2

φ(y)+(minusiλ )2

6G(xy)3 +O(λ 3) (D37)

Since the diagrams contributing to the self-energy Π(xy) contain the full propagator G(xy) theldquogap equationrdquo (D35) is an intrinsically non-perturbative equation for the two-point function Itcan be compared to the usual perturbative Schwinger-Dyson equation which has a similar form aseq (D35) However in the perturbative case the self-energy is evaluated using the perturbativepropagator G0(xy) In contrast to the perturbative case the gap equation (D35) which determinesthe full propagator may be viewed as a self-consistent Schwinger-Dyson equation It is preciselythis self-consistency of the 2PI formalism which leads to well-behaved nonequilibrium evolutionequations for the two-point function in contrast to perturbative approaches which suffer from thesecularity problem [27] The bilocal source K(xy) may be split into two parts

K(xy) = α2(xy)+Kext(xy) (D38)

where the first contribution stems from the initial two-point correlations encoded in the source α2(xy)and the second contribution is an additional external bilocal source term In a physical situation thebilocal external source vanishes Kext(xy) = 0 such that K(xy) is only supported at initial timesx0 = y0 = 0 This source term fixes the initial condition for the propagator at x0 = y0 = 0

D24 Kadanoff-Baym Equations for Gaussian Initial States

The Kadanoff-Baym equations for the two-point function are obtained by multiplying the equation ofmotion (D35) Gminus1(xz) = Gminus1

0 (xz)minusΠ(xz)minus iα2(xz) with G(zy) and integrating over z(2x +m2 +

λ

2φ(x)2

)G(xy) =minusiδC(xminus y)minus i

intCd4z(Π(xz)+ iα2(xz))G(zy) (D39)


where the inverse classical propagator Gminus10 from eq (D32) was inserted It is useful to decompose the

two-point function into the statistical propagator GF(xy) and the spectral function Gρ(xy) whichare defined via the anticommutator and commutator of the field operator respectively

GF(xy) =12〈 [Φ(x)Φ(y)]+ 〉minus〈Φ(x)〉〈Φ(y)〉

Gρ(xy) = i〈 [Φ(x)Φ(y)]minus 〉 (D40)

such that the Schwinger-Keldysh propagator can be written in the form


sgnC(x0minus y0)Gρ(xy) (D41)

Furthermore the self-energy contains local and non-local parts

Π(xy) =minusiΠloc(x)δC(xminus y)+Πnonminusloc(xy) (D42)

The local part can be included in an effective time-dependent mass term

M(x)2 = m2 +λ

2φ

2(x)+Πloc(x) = m2 +λ

2φ

2(x)+λ

2G(xx) (D43)

and the non-local part can be split into statistical and spectral components similar to the propagator

Πnonminusloc(xy) = ΠF(xy)minus i2

sgnC(x0minus y0)Πρ(xy) (D44)

In 2PI-O(λ 2)-approximation the non-local self-energies are given by

ΠF(xy) =(minusiλ )2

2φ(x)

(GF(xy)2minus 1

4Gρ(xy)2

)φ(y)

+(minusiλ )2

6

(GF(xy)3minus 3

4GF(xy)Gρ(xy)2

)+O(λ 3)

Πρ(xy) =(minusiλ )2

2φ(x)

(2GF(xy)Gρ(xy)

)φ(y) (D45)

+(minusiλ )2

6

(3GF(xy)2Gρ(xy)minus 1

4Gρ(xy)3

)+O(λ 3)

Using the equal-time commutation relations (32) of the quantum field gives

Gρ(xy)∣∣x0=y0 = 0 partx0Gρ(xy)

∣∣x0=y0 = δ

(3)(xminusy) (D46)

With the help of these relations it is found that

part2x0G(xy) = part

2x0GF(xy)minus i

2sgnC(x0minus y0)part

2x0Gρ(xy)

minus iδC(x0minus y0)partx0Gρ(xy)minus ipartx0

[δC(x0minus y0)Gρ(xy)

]= part

2x0GF(xy)minus i


2x0Gρ(xy)minus iδC(x0minus y0)δ (3)(xminusy)

Using this relation along with the integration rules on the closed real-time path (see appendix F)the real and causal Kadanoff-Baym equations are finally obtained from inserting the decomposi-tions (D41 D42 D44) of the propagator and the self-energy into the equation of motion (D39)(

2x +M2(x))

GF(xy) =int y0


int x0

0d4zΠρ(xz)GF(zy)

(2x +M2(x)

)Gρ(xy) =

int y0

x0

d4zΠρ(xz)Gρ(zy) (D47)


The Kadanoff-Baym equations split into two coupled integro-differential equations for GF(xy) andGρ(xy) For a system with spatial translation invariance it is convenient to perform a Fourier trans-formation with respect to the relative spatial coordinate (xminusy)

G(x0y0k) =int

d3xeminusik(xminusy) G(xy) (D48)

and similarly for Π(xy) For isotropic systems the propagator G(x0y0k) depends only on theabsolute value |k| of the spatial momentum k The Kadanoff-Baym equations in the upper formhave been used successfully as a basis to study quantum fields far from equilibrium during the lastdecade [2 25 32 123 142] (see also section 61) In section 71 a generalization of these equationsfor general initial states which may contain non-Gaussian initial correlations is discussedNote that the two-point source α2(xy) has been dropped since it vanishes for x0 gt 0 Howeverit fixes the initial conditions for the statistical propagator GF(xy) at x0 = y0 = 0 see eq (D29)The initial conditions for the spectral function Gρ(xy) are fixed by eq (D46) obtained from theequal-time commutation relations (32)

GF(x0y0k)∣∣x0=y0=0 = ξ

2k

partx0GF(x0y0k)∣∣x0=y0=0 = ηkξk (D49)

partx0party0GF(x0y0k)∣∣x0=y0=0 = η

2k +

σ2k

4ξ 2k

Gρ(x0y0k)∣∣x0=y0=0 = 0

partx0Gρ(x0y0k)∣∣x0=y0=0 = 1 (D50)

partx0party0Gρ(x0y0k)∣∣x0=y0=0 = 0

The first derivatives with respect to y0 are related to the first derivatives with respect to x0 in thesecond and fifth line due to the symmetry property GF(xy) = GF(yx) and the antisymmetry propertyGρ(xy) = minusGρ(yx) which follow directly from the definition (D40) A physical interpretation ofthe initial conditions for the statistical propagator GF(xy) can be obtained by parameterizing it interms of the initial effective particle- and energy number densities (617)

ξ2k =

n(t = 0k)+ 12

ω(t = 0k) ηk = 0

σ2k

4ξ 4k

= ω2(t = 0k) (D51)

The ldquomemory integralsrdquo on the right hand side of the Kadanoff-Baym equations imply that the time-evolution of G(xy) near the point (x0y0) in the x0-y0-plane depends on the value of the propagatorG(uv) during the entire history 0 lt u0 lt x0 0 lt v0 lt y0 from the initial time tinit = 0 on Theldquomemory integralsrdquo turn out to be crucial for the successful description of the quantum thermalizationprocess [32]

Appendix E

Nonperturbative RenormalizationTechniques

Truncations of the 2PI effective action yield self-consistent and nonperturbative approximations tothe equations of motion for the two-point correlation function These equations contain ultravioletdivergences which commonly occurs in relativistic quantum field theory However due to theirself-consistent structure the isolation and removal of divergences requires much more sophisticatedtechniques for these equations compared to perturbative calculations The proper renormalizationrequires nonperturbative techniques which have been formulated recently [28 29 37 174 175] forsystems in thermal equilibrium and at zero temperature It has been found that approximations basedon systematic (eg loop) truncations of the 2PI functional are indeed renormalizable and that thevacuum counterterms are sufficient to remove all divergences at finite temperature The determinationof the vacuum counterterms by solving self-consistent equations for the two- and four-point functionswill be discussed in the following based on Refs [28 29]

E1 Renormalization of the 2PI Effective Action

It is convenient to split the action into a free and an interaction part

S0[φ ] =int

d4x(

12(partφ)2minus 1

2m2

Bφ2)

Sint [φ ] =minusint

d4xλB

4φ(x)4 (E1)

such that the 2PI Effective Action can be written as

Γ[φ G] = S0[φ ]+i2

Tr lnGminus1 +i2

TrGminus10 G+Γint [φ G] (E2)

where iGminus10 (xy) = (minus2xminusm2

B)δ (xminus y) is the free perturbative propagator and

Γint [φ G] = Sint [φ ]+12

Trpart 2Sint

partφpartφG+Γ2[φ G] (E3)

Here iΓ2[φ G] is the sum of all 2PI vacuum diagrams with lines representing the full propagatorG(xy) The equations of motion for the field expectation value and the full propagator are obtainedfrom the stationarity conditions (325) of the 2PI effective action For the full propagator G(xy) theequation of motion takes the form of a self-consistent Schwinger-Dyson equation1

Gminus1(xy) = Gminus10 (xy)minusΠ(xy) (E4)

1 The Schwinger-Dyson equation can equivalently be written in the two forms Gminus1 = Gminus10 minus 2iδΓintδG = Gminus1

0 minus2iδΓ2δG The latter corresponds to eq (329) Here the first form is more convenient

164 E Nonperturbative Renormalization Techniques

where the self-energy is given by Π(xy) = 2i δΓint [φ G]δG(yx)

Definition of counterterms

For the purpose of renormalization the action is rewritten by rescaling the field φ and splitting thebare mass mB and coupling λB into a renormalized part and a counterterm respectively


B = m2R +δm2 Z2

λB = λR +δλ (E5)

The action expressed in terms of renormalized quantities can be written as

SR[φR] = S[φ ] = S0R[φR]+Sint [φR]λBrarrλR+δλ +12

intxy

φR iδGminus10 φR (E6)

with the renormalized free action

S0R[φR] =int

d4x(

12(partφR)2minus 1

2m2

Rφ2R

) (E7)

and a contribution containing the counterterms δZ = Z minus 1 and δm2 of the form iδGminus10 (xy) =

(minusδZ2xminusδm2)δ (xminusy) Similarly the 2PI effective action can be expressed in terms of the rescaledfield expectation value φR = Zminus12φ and the rescaled full propagator GR = Zminus1G

ΓR[φRGR] = Γ[φ G] = S0R[φR]+i2

Tr lnGminus1R +

i2

TrGminus10RGR +Γ

Rint [φRGR] (E8)

where iGminus10R(xy) = (minus2xminusm2

R)δ (xminus y) is the renormalized free perturbative propagator and

ΓRint [φRGR] =

12

intxy

φR iδGminus10 φR +

i2

TrδGminus10 GR +Γint [φRGR]λBrarrλR+δλ (E9)

To derive the last relation Γint [φ G] = Γint [φRGR]λrarrλR+δλ was used For each 2PI vacuum diagramcontributing to Γint [φ G] this follows from the relation 4V = 2P+E between the number of verticesV the number of propagators P and the number of field expectation values E

E11 Divergences and Counterterms in 2PI Kernels

Due to the self-consistent nature of the 2PI formalism the structure of the Schwinger-Dyson equa-tions determining the complete propagator is inherently nonperturbative and corresponds to theresummation of an infinite set of perturbative diagrams [37] As a consequence the renormaliza-tion of approximations based on truncations of the 2PI functional is highly nontrivial It has beenshown [28 37 174 175] recently that systematic truncations indeed lead to renormalizable approx-imations Besides the divergences which can be identified and subtracted via the BPHZ construc-tion [38 113 191] the divergent contributions hidden in the nonperturbative propagator have to beaccounted for in a way compatible with the self-consistent structure of the Schwinger-Dyson equa-tions (see section 62)

E12 Parameterization of the Renormalized 2PI Effective Action

In order to renormalize the 2PI effective action completely counterterms which cancel all types ofdivergences indicated above have to be included For a given truncation of the 2PI functional it canbe necessary to keep only some parts of the full counterterms which are appropriate for the considered

E2 Renormalization of 2PI Kernels 165

δZ2 δm22 δZ0 δm2

0

Figure E1 Diagrams containing mass and field counterterms

approximation Thus the counterterms which appear in different places may be different parts of thefull counterterms Here a parameterization of the renormalized 2PI effective action is used followingRef [28]

ΓRint [φRGR] =

12

intxy

φR iδGminus102φR +

i2

TrδGminus100GR +Γint [φRGR]λBrarrλR+δλi (E10)

where the mass- and wavefunction renormalization counterterms are given by (see figure E1)

iδGminus102(xy) = (minusδZ22xminusδm2

2)δ (xminus y)


0)δ (xminus y) (E11)

The coupling counterterms δλi are chosen in the following way (see figure E2)

Γint [φRGR]λBrarrλR+δλi = minusλR +δλ4

4

intxφ

4R(x)minus λR +δλ2

4

intxφ

2R(x)GR(xx)

minus λR +δλ0

8

intxG2

R(xx)+ γR[φRGR] (E12)

where γR[φRGR] stands for the contributions from nonlocal diagrams which just contain the BPHZcounterterms to the appropriate order

δλ4 δλ2 δλ0

Figure E2 Local diagrams containing coupling counterterms

E2 Renormalization of 2PI Kernels

The counterterms are determined by imposing renormalization conditions for the two- and four-pointfunctions Therefore the two-point kernels

ΠR(xy) =2iδΓR

int

δGR(yx) ΠR(xy) =

iδ 2ΓRint

δφR(x)δφR(y) (E13)

are defined in terms of which the renormalized Schwinger-Dyson equation for the full propagatorGR(xy) can be expressed as

Gminus1R (xy) = Gminus1

0R(xy)minusΠR(xy) (E14)


Furthermore the four-point kernels

Λ(xyuv) =4δ 2Γint

δG(xy)δG(uv) Λ(xyuv) =

2δ 3Γint

δφ(x)δφ(y)δG(uv) (E15)

are defined Due to the self-consistent structure of the 2PI formalism the four-point kernels Λ and Λ

do only contribute to the complete n-point functions via the resummed kernels V and V which aresolutions of the Bethe-Salpeter equations [28]

V (xyuv) = Λ(xyuv)+i2

intabcd

Λ(xyab)G(ac)G(db)V (cduv)

(E16)


intabcd

V (xyab)G(ac)G(db)Λ(cduv)

The solutions of the Bethe-Salpeter equations can formally be obtained by an iteration which yields aresummation of ladder diagrams where the ladder steps are given by the kernel Λ and the connectionsof the steps are given by the complete propagator G Note that the nonperturbative renormalization ofthe four-point kernels can formally be understood as being built up of two steps First the divergencescontained in the diagrammatic contributions to the kernels Λ and Λ are subtracted via an appropriatechoice of BPHZ counterterms δλ BPHZ

0 and δλ BPHZ2 respectively Second the additional divergences

appearing in the renormalized solutions V R equiv Z2V and VR equiv Z2V of the Bethe-Salpeter equation areremoved by additional counterterms ∆λ0 and ∆λ2 such that the complete counterterms are given bythe sum δλ0 = δλ BPHZ

0 +∆λ0 and δλ2 = δλ BPHZ2 +∆λ2 In practice the full counterterms δλ0 and

δλ2 can be determined in one step by imposing a renormalization condition on the kernels V R and VR

Renormalization conditions

For the vacuum theory it is most convenient to work in Euclidean momentum space qmicro = (iq0q) byperforming a Fourier transformation and a Wick rotation along the q0-axis The Euclidean propagatoris given by

G(xy) =int

qeminusiq(xminusy)G(q)

and the four-point kernel in momentum space is given by

(2π)4δ

(4)(p1 + p2 + p3 + p4)Λ(p1 p2 p3 p4) =int

x1x2x3x4

eisumn pnxnΛ(x1x2x3x4)

An analogous transformation holds for the other four-point functionsThe renormalization conditions can be imposed at an arbitrary subtraction point q in momentumspace However it is important that the same point is used consistently for all 2PI kernels

ΠR(q = q) = ΠR(q = q) = 0

ddq2 ΠR(q = q) =

ddq2 ΠR(q = q) = 0 (E17)

V R(pi = q) = VR(pi = q) = Γ(4)R (pi = q) = minusλR

Especially the renormalization conditions for the kernels V R and VR coincide with the one for thefour-point function Γ

(4)R equiv Z2Γ(4)

Γ(4)(xyuv) =

d4Γ[φ G[φ ]]dφ(x)dφ(y)dφ(u)dφ(v)

(E18)

E3 Two Loop Approximation 167

where G[φ ] denotes the solution of the Schwinger-Dyson equation (E4) for a given field configurationφ(x) The renormalization conditions for ΠR are equivalent to the conditions

Gminus1R (q = q) = q2 +m2

R d

dq2 Gminus1R (q = q) = +1 (E19)

for the complete propagator The seven conditions (E17) determine the counterterms δm20 δZ0

δm22 δZ2 δλ0 δλ2 and δλ4 A simplification occurs for approximations where all contributions

to ΠR(xy) and ΠR(xy) are identical In this case also the corresponding counterterms agreeδm2

0 = δm22 δZ0 = δZ2 and δλ0 = δλ2 In the following the subtraction point will be chosen at

zero momentum q = 0 Another interesting choice is q2 = minusm2R which corresponds to the on-shell

renormalization scheme

E3 Two Loop Approximation

The 2PI two-loop approximation corresponds to a Hartree-Fock approximation and can be used tocheck the nonperturbative renormalization procedure explicitly It corresponds to a truncation ofthe 2PI functional where only the local two-loop O(λ ) contributions are retained in which caseeqs (E8) (E10) and (E12) with γR(φRGR) equiv 0 define the renormalized 2PI effective action com-pletely Furthermore the symmetric phase with vanishing field expectation value φ = 0 is consideredIn this case the 2PI two-point kernels ΠR(xy) and ΠR(xy) are given by

ΠR(xy) = minusi(

δZ02x +δm20 +

λR +δλ0

2GR(xx)

)δ (xminus y)

ΠR(xy) = minusi(

δZ22x +δm22 +

λR +δλ2

2GR(xx)

)δ (xminus y) (E20)

and the 2PI four-point kernels are given by

Z2Λ(xyuv) = minus(λR +δλ0)δ (xminus y)δ (xminusu)δ (xminus v)

Z2Λ(xyuv) = minus(λR +δλ2)δ (xminus y)δ (xminusu)δ (xminus v) (E21)

Z2Γ

(4)(xyuv) = minus(λR +δλ4minus3δλ0)δ (xminus y)δ (xminusu)δ (xminus v)

Since the kernels ΠR(xy) and ΠR(xy) have an identical structure the renormalization conditions(E17) can be satisfied by identical counterterms ie δm2

0 = δm22 δZ0 = δZ2 and δλ0 = δλ2

From eq (E20) the renormalized Schwinger-Dyson equation (E14) in two-loop approximation inEuclidean momentum space is obtained

Gminus1R (k) = k2 +m2

R +δZ0k2 +δm20 +

λR +δλ0

2

intq

GR(q)

Using the renormalization conditions for the propagator (E19) immediately yields the mass- and fieldcounterterms

δZ0 = 0 δm20 =minusλR +δλ0

2

intq

GR(q) (E22)

and the complete propagator in two-loop approximation is simply given by

Gminus1R (k) = Gminus1(k) = k2 +m2

R (E23)


In order to determine the coupling counterterm the Schwinger-Dyson equation has to be supple-mented by the Bethe-Salpeter equation (E16) in two-loop approximation

V R(p1 p2 p3 p4) =minus(λR +δλ0)minusλR +δλ0

2

intq

GR(q+ p1 + p2)GR(q)V R(q+ p1 + p2minusq p3 p4)

which is obtained by inserting the two-loop 2PI kernel from eq (E21) into eq (E16) and performinga Fourier transformation For the determination of the counterterm it suffices to solve this equationfor VR(k)equivminusV R(kminusk00)

VR(k) = λR +δλ0minusλR +δλ0

2

intq

G2R(q)VR(q) (E24)

Obviously this equation has a constant solution VR(k) = VR(0) = λR where the last equality followsfrom the renormalization condition for V R in eq (E17) Thus the Bethe-Salpeter equation in two-loop approximation reduces to an algebraic equation for the counterterm δλ0 It is most convenientto rewrite the Bethe-Salpeter equation and eq (E22) in terms of Z = 1+δZ0 and the bare quantitiesλB = Zminus2(λR +δλ0) and m2

B = Zminus1(m2R +δm2

0)

Z = 1

m2B = m2

RminusλB

2

intq

G(q) (E25)

λminus1B = λ

minus1R minus

intq

G2(q)

These equations together with eq (E23) form a closed set of equations for the determination ofthe nonperturbative 2PI counterterms δm2

0 = δm22 δZ0 = δZ2 and δλ0 = δλ2 in two-loop approxi-

mation It is understood that the momentum integrals are suitably regularized eg by dimensionalor lattice regularization Additionally the counterterm δλ4 has to be determined by imposing therenormalization condition (E17) on the four-point function Γ(4) from eq (E21) yielding

δλ4 = 3δλ0 (E26)

E4 Three Loop Approximation

The 2PI three-loop approximation includes non-local contributions and therefore yields non-localequations of motion for nonequilibrium initial conditions This approximation has frequently beenused to study quantum dynamics far from equilibrium [1 2 25 32 142] and therefore the nonper-turbative renormalization within this approximation is of interest Truncating all diagrams whichcontribute to the 2PI functional to more thanO(λ 2) the renormalized 2PI effective action is given byeqs (E8) (E10) and (E12) where the non-local contributions are given by

iγR(φRGR) =(minusiλR)2

12

intxy

φR(x)GR(xy)3φR(y) +

(minusiλR)2

48

intxy

GR(xy)4 (E27)

Thus γR(φRGR) contains diagrams up to three-loop order which are shown in figure E3 Evaluatingthe 2PI two-point kernels ΠR(xy) and ΠR(xy) using the definitions in eq (E13) for the symmetricphase φ = 0 yields

ΠR(xy) = minusi(

δZ02x +δm20 +

λR +δλ0

2GR(xx)

)δ (xminus y)minus λ 2

R

6G3

R(xy)

(E28)

ΠR(xy) = minusi(

δZ22x +δm22 +

λR +δλ2

2GR(xx)


R

6G3

R(xy)

E4 Three Loop Approximation 169

Figure E3 Nonlocal diagrams contributing up to three-loop 2PI-O(λ 2) order

The 2PI four-point kernels defined in eqs (E15) and the four-point function given by eq (E18) inthree-loop approximation read

Z2Λ(xyuv) = minus(λR +δλ0)δ (xminus y)δ (xminusu)δ (xminus v)+ iλ 2

RG2R(xy)δ (xminus z)δ (yminusw)



Z2Γ

(4)(xyuv) = minus(λR +δλ4)δ (xminus y)δ (xminusu)δ (xminus v)+ (E29)

+(V RminusZ2Λ)(xyuv)+(V RminusZ2

Λ)(xuyv)+(V RminusZ2Λ)(xvuy)

As for the two-loop approximation the two-point kernels ΠR(xy) and ΠR(xy) have an identicalstructure which implies that the renormalization conditions (E17) can be satisfied by identical coun-terterms ie δm2

0 = δm22 δZ0 = δZ2 and δλ0 = δλ2 and that the four-point kernels Λ and Λ as

well as V and V coincide From eq (E20) the renormalized Schwinger-Dyson equation (E14) inthree-loop approximation in Euclidean momentum space is obtained


R +δZ0k2 +δm20 +

λR +δλ0

2

intq

GR(q)

minus λ 2R

6

intpq

GR(p)GR(q)GR(kminusqminus p) (E30)

The Bethe-Salpeter equation in three-loop approximation is obtained analogously to the two-loopcase by inserting the three-loop 2PI kernel from eq (E29) into eq (E16) After performing a Fouriertransformation the Bethe-Salpeter equation for the kernel VR(k)equivminusV R(kminusk00) reads2

VR(k) = λR +δλ0minusλ2R

intq

GR(q)GR(kminusq)minus λR +δλ0

2

intq

G2R(q)VR(q)

+λ 2

R

2

intpq

GR(p)GR(kminusqminus p)G2R(q)VR(q) (E31)

For a numerical solution it is convenient to rewrite the Bethe-Salpeter equation and the Schwinger-Dyson equation in terms of Z = 1 + δZ0 and the bare quantities λB = Zminus2(λR + δλ0) and m2

B =Zminus1(m2

R +δm20)


λB

2

intq

G(q)

minus Zminus4λ 2R

6

intpq

G(p)G(q)G(kminusqminus p) (E32)

V (k) = λBminusZminus4λ

2R

intq

G(q)G(kminusq)minus λB

2

intq

G2(q)V (q)

+Zminus4λ 2

R

2

intpq

G(p)G(kminusqminus p)G2(q)V (q) (E33)

2 The kernel VR(q p) defined in section 62 is related to the 4-point kernel via VR(q p) = V R(qminusqminusp p)


The renormalization conditions (E17) written in terms of G(k) and V (k) read

ZGminus1(k = 0) = m2R Z

ddq2 Gminus1(k = 0) = +1 Z2V (k = 0) = λR (E34)

The Bethe-Salpeter equation (E33) and the Schwinger-Dyson equation (E32) together with the upperrenormalization conditions form a closed set of equations for the determination of the nonperturbative2PI counterterms δm2

0 = δm22 δZ0 = δZ2 and δλ0 = δλ2 in three-loop approximation Finally the

counterterm δλ4 is determined by imposing the renormalization condition (E17) on the four-pointfunction Γ(4) from eq (E29) yielding

δλ4 = 3δλ0minus3λ2R

intq

G2R(q) (E35)

Appendix F

Integrals on the Closed Real-Time Path

Nonequilibrium as well as thermal correlation functions can conveniently be calculated by attachingthe time arguments to the closed real-time contour C (see figure 61) and the imaginary time contourI (see figure 64) respectively In general any time contour P is a complex valued curve which canbe parameterized by a mapping tp [ab]rarrC u 7rarr tp(u) from a real interval into the complex planeThe integral of a function f Crarr C along the time contour P is given by the curve integralint

Pdtp f (tp) =

binta

dudtp(u)

duf (tp(u))

Furthermore for space-time points xmicrop = (x0

px) with zero-component on the time contour P intP

d4x =intP

dx0p

intd3x

is defined The signum function on a time contour P is defined as

sgnP(tp(u1)minus tp(u2)) = sgn(u1minusu2) =

+1 if u1 gt u2 0 if u1 = u2 minus1 if u1 lt u2

for u1u2 isin [ab]Let f Rrarr C be a continous function with time argument attached to the real axis Then its integralover the closed real-time path C vanishes since the contributions from the chronological and theantichronological parts cancel int

Cdtc f (tc) = 0

For the derivation of the Kadanoff-Baym equations (615) the following relations which involve thesignum function on the closed real-time path are requiredint

Cdtc sgnC(t1minus tc) f (tc) = 2

t1intt0

dt f (t)

intCdtc sgnC(t1minus tc)sgnC(tcminus t3) f (tc) = 2sgnC(t1minus t3)

t1intt3

dt f (t)

Note that the upper relations are true irrespective of whether the times t1 and t3 belong to the chrono-logical or the antichronological part of the closed real-time path Therefore the upper compact nota-tion is unambiguous

Danksagung

An dieser Stelle moumlchte ich mich bei allen bedanken die zum Gelingen dieser Arbeit bei-getragen haben Insbesondere danke ich

bull meinem Betreuer Herrn Prof Dr Manfred Lindner Er hat mir diese Arbeitan einem sehr interessanten und vielseitigen Thema ermoumlglicht Auszligerdemhat er fuumlr exzellente Arbeitsbedingungen gesorgt und hat die Teilnahme anmehreren Sommerschulen und Konferenzen gefoumlrdert

bull Florian Bauer Marc-Thomas Eisele und Markus Michael Muumlller (ldquoMMMrdquo)fuumlr die gute Zusammenarbeit

bull Markus Michael Muumlller fuumlr die Erstellung von numerischen Loumlsungen derKadanoff-Baym Gleichungen und das Probelesen der Arbeit

bull allen Mitgliedern des ehemaligen Lehrstuhls T30d sowie der Abteilung fuumlrTeilchen- und Astroteilchenphysik fuumlr die anregende Arbeitsatmosphaumlre undinteressante Diskussionen uumlber physikalische und unphysikalische Themen

bull den Sekretaumlrinnen Karin Ramm und Anja Berneiser fuumlr die freundliche Un-terstuumltzung bei buumlrokratischen Angelegenheiten

bull den Systemadministratoren sowie Herrn Koumlck fuumlr die Bereitstellung vonRechnerressourcen

bull A Anisimov E Babichev J Berges S Borsanyi H Gies U ReinosaA Vikman und C Wetterich fuumlr hilfreiche Kommentare und Diskussionen

bull dem Perimeter Institute fuumlr die Finanzierung einer Sommerschule

bull meinen Zimmerkollegen Florian Bauer Michael Schmidt und Viviana Nirofuumlr die gute Gemeinschaft und die Auflockerungen zwischendurch

bull der Deutschen Bahn AG dafuumlr daszlig ich das Leben zwischen Heidelberg undMuumlnchen in vollen Zuumlgen genieszligen durfte

bull und dem birthday-script fuumlr die Versuumlszligung vieler Nachmittage

Ganz besonders danke ich meiner Lebensgefaumlhrtin Sylvia die mir jederzeit tatkraumlftig undliebevoll zur Seite gestanden ist sowie meinen Eltern Cornelia und Michael und meinerSchwester Hella die mich immerzu verstaumlndnisvoll unterstuumltzt haben Einen groszligen Dankhaben Gisela und Wilfried verdient insbesondere fuumlr die unkomplizierte Hilfe bei praktischenAspekten der doppelten Haushaltsfuumlhrung und Angelika fuumlr vielerlei hilfreiche Ratschlaumlge

Bibliography

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[2] G Aarts and J M Martiacutenez Resco Transport coefficients from the 2PI effective action PhysRev D68 085009 (2003) hep-ph0303216

[3] G Aarts and A Tranberg Particle creation and warm inflation Phys Lett B650 65ndash71(2007) hep-ph0701205

[4] G Aarts and A Tranberg Thermal effects on inflaton dynamics (2007) arXiv0712

1120[hep-ph]

[5] L F Abbott E Farhi and M B Wise Particle Production in the New Inflationary CosmologyPhys Lett B117 29 (1982)

[6] J Adams et al Experimental and theoretical challenges in the search for the quark gluonplasma The STAR collaborationrsquos critical assessment of the evidence from RHIC collisionsNucl Phys A757 102ndash183 (2005) nucl-ex0501009

[7] K Adcox et al Formation of dense partonic matter in relativistic nucleus nucleus collisionsat RHIC Experimental evaluation by the PHENIX collaboration Nucl Phys A757 184ndash283(2005) nucl-ex0410003

[8] I Affleck and M Dine A new mechanism for baryogenesis Nucl Phys B249 361 (1985)

[9] A Albrecht and C Skordis Phenomenology of a realistic accelerating universe using onlyPlanck-scale physics Phys Rev Lett 84 2076ndash2079 (2000) astro-ph9908085

[10] L Amendola Coupled quintessence Phys Rev D62 043511 (2000) astro-ph9908023

[11] L Amendola M Baldi and C Wetterich Growing matter (2007) arXiv07063064

[12] L Amendola R Gannouji D Polarski and S Tsujikawa Conditions for the cosmologicalviability of f(R) dark energy models Phys Rev D75 083504 (2007) gr-qc0612180

[13] L Amendola C Quercellini D Tocchini-Valentini and A Pasqui Constraints on the inter-action and self-interaction of dark energy from cosmic microwave background Astrophys J583 L53 (2003) astro-ph0205097

[14] L Amendola and D Tocchini-Valentini Stationary dark energy the present universe as aglobal attractor Phys Rev D64 043509 (2001) astro-ph0011243

176 BIBLIOGRAPHY

[15] L Anchordoqui and H Goldberg Time variation of the fine structure constant driven by quint-essence Phys Rev D68 083513 (2003) hep-ph0306084

[16] C Armendariz-Picon V F Mukhanov and P J Steinhardt Essentials of k-essence Phys RevD63 103510 (2001) astro-ph0006373

[17] A Arrizabalaga J Smit and A Tranberg Equilibration in ϕ4 theory in 3+1 dimensions PhysRev D72 025014 (2005) hep-ph0503287

[18] J Baacke K Heitmann and C Patzold Nonequilibrium dynamics of fermions in a spatiallyhomogeneous scalar background field Phys Rev D58 125013 (1998) hep-ph9806205

[19] K R S Balaji T Biswas R H Brandenberger and D London Dynamical CP violation inthe early universe Phys Lett B595 22ndash27 (2004) hep-ph0403014

[20] T Banks M Dine and M R Douglas Time-varying alpha and particle physics Phys RevLett 88 131301 (2002) hep-ph0112059

[21] T Barreiro E J Copeland and N J Nunes Quintessence arising from exponential potentialsPhys Rev D61 127301 (2000) astro-ph9910214

[22] N Bartolo and M Pietroni Scalar tensor gravity and quintessence Phys Rev D61 023518(2000) hep-ph9908521

[23] F Bauer M-T Eisele and M Garny Leptonic dark energy and baryogenesis Phys Rev D74023509 (2006) hep-ph0510340

[24] G Baym Selfconsistent approximation in many body systems Phys Rev 127 1391ndash1401(1962)

[25] J Berges Controlled nonperturbative dynamics of quantum fields out of equilibrium NuclPhys A699 847ndash886 (2002) hep-ph0105311

[26] J Berges n-PI effective action techniques for gauge theories Phys Rev D70 105010 (2004)hep-ph0401172

[27] J Berges Introduction to nonequilibrium quantum field theory AIP Conf Proc 739 3ndash62(2005) hep-ph0409233

[28] J Berges S Borsanyi U Reinosa and J Serreau Nonperturbative renormalization for 2PIeffective action techniques Annals Phys 320 344ndash398 (2005) hep-ph0503240

[29] J Berges S Borsanyi U Reinosa and J Serreau Renormalized thermodynamics from the 2PIeffective action Phys Rev D71 105004 (2005) hep-ph0409123

[30] J Berges S Borsanyi and J Serreau Thermalization of fermionic quantum fields Nucl PhysB660 51ndash80 (2003) hep-ph0212404

[31] J Berges S Borsanyi and C Wetterich Prethermalization Phys Rev Lett 93 142002 (2004)hep-ph0403234

[32] J Berges and J Cox Thermalization of quantum fields from time-reversal invariant evolutionequations Phys Lett B517 369ndash374 (2001) hep-ph0006160

BIBLIOGRAPHY 177

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[34] P Binetruy Models of dynamical supersymmetry breaking and quintessence Phys Rev D60063502 (1999) hep-ph9810553

[35] N D Birrell and P C W Davies Quantum fields in curved space (Cambridge UniversityPress Cambridge UK 1982)

[36] O E Bjaelde et al Neutrino Dark Energy ndash Revisiting the Stability Issue (2007) arXiv07052018

[37] J-P Blaizot E Iancu and U Reinosa Renormalization of phi-derivable approximations inscalar field theories Nucl Phys A736 149ndash200 (2004) hep-ph0312085

[38] N N Bogoliubov and O S Parasiuk On the Multiplication of the causal function in thequantum theory of fields Acta Math 97 227ndash266 (1957)

[39] B Boisseau G Esposito-Farese D Polarski and A A Starobinsky Reconstruction of ascalar-tensor theory of gravity in an accelerating universe Phys Rev Lett 85 2236 (2000)gr-qc0001066

[40] L A Boyle R R Caldwell and M Kamionkowski Spintessence New models for dark matterand dark energy Phys Lett B545 17ndash22 (2002) astro-ph0105318

[41] C Brans and R H Dicke Machrsquos principle and a relativistic theory of gravitation Phys Rev124 925ndash935 (1961)

[42] P Brax and J Martin Quintessence and supergravity Phys Lett B468 40ndash45 (1999)astro-ph9905040

[43] P Brax and J Martin The robustness of quintessence Phys Rev D61 103502 (2000)astro-ph9912046

[44] P Brax and J Martin Dark energy and the MSSM Phys Rev D75 083507 (2007) hep-th0605228

[45] A W Brookfield C van de Bruck D F Mota and D Tocchini-Valentini Cosmology withmassive neutrinos coupled to dark energy Phys Rev Lett 96 061301 (2006) astro-ph0503349

[46] N Brouzakis N Tetradis and C Wetterich Neutrino Lumps in Quintessence Cosmology(2007) arXiv07112226

[47] I L Buchbinder S D Odintsov and I L Shapiro Effective action in quantum gravity (IOPPublishing Bristol UK 1992)

[48] W Buchmuumlller P Di Bari and M Pluumlmacher Leptogenesis for pedestrians Ann Phys 315305ndash351 (2005) hep-ph0401240

[49] E Calzetta and B L Hu Nonequilibrium quantum fields closed time path effective actionWigner function and Boltzmann equation Phys Rev D37 2878 (1988)

178 BIBLIOGRAPHY

[50] B A Campbell and K A Olive Nucleosynthesis and the time dependence of fundamentalcouplings Phys Lett B345 429ndash434 (1995) hep-ph9411272

[51] S M Carroll Quintessence and the rest of the world Phys Rev Lett 81 3067ndash3070 (1998)astro-ph9806099

[52] F C Carvalho and A Saa Non-minimal coupling exponential potentials and the w lt -1 regimeof dark energy Phys Rev D70 087302 (2004) astro-ph0408013

[53] R Catena N Fornengo A Masiero M Pietroni et al Dark matter relic abundance andscalar-tensor dark energy Phys Rev D70 063519 (2004) astro-ph0403614

[54] H Chand R Srianand P Petitjean and B Aracil Probing the cosmological variation of thefine-structure constant Results based on VLT-UVES sample Astron Astrophys 417 853(2004) astro-ph0401094

[55] T Chiba Quintessence the gravitational constant and gravity Phys Rev D60 083508(1999) gr-qc9903094

[56] T Chiba and K Kohri Quintessence cosmology and varying alpha Prog Theor Phys 107631ndash636 (2002) hep-ph0111086

[57] K Chou Z Su B Hao and L Yu Equilibrium and Nonequilibrium Formalisms Made UnifiedPhys Rept 118 1 (1985)

[58] D J H Chung L L Everett and A Riotto Quintessence and the underlying particle physicstheory Phys Lett B556 61ndash70 (2003) hep-ph0210427

[59] S R Coleman Q balls Nucl Phys B262 263 (1985)

[60] S R Coleman and E Weinberg Radiative Corrections as the Origin of Spontaneous SymmetryBreaking Phys Rev D7 1888ndash1910 (1973)

[61] J C Collins Renormalization (Cambridge University Press Cambridge UK 1984)

[62] D Comelli M Pietroni and A Riotto Dark energy and dark matter Phys Lett B571 115ndash120 (2003) hep-ph0302080

[63] F Cooper et al Nonequilibrium quantum fields in the large N expansion Phys Rev D502848ndash2869 (1994) hep-ph9405352

[64] E J Copeland N J Nunes and M Pospelov Models of quintessence coupled to the elec-tromagnetic field and the cosmological evolution of alpha Phys Rev D69 023501 (2004)hep-ph0307299

[65] E J Copeland M Sami and S Tsujikawa Dynamics of dark energy Int J Mod Phys D151753ndash1936 (2006) hep-th0603057

[66] J M Cornwall R Jackiw and E Tomboulis Effective Action for Composite Operators PhysRev D10 2428ndash2445 (1974)

[67] T L Curtright and C B Thorn Conformally Invariant Quantization of the Liouville TheoryPhys Rev Lett 48 1309 (1982)

BIBLIOGRAPHY 179

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[69] P Danielewicz Quantum theory of nonequilibrium processes II Application to nuclear colli-sions Annals Phys 152 305ndash326 (1984)

[70] S Davidson and A Ibarra A lower bound on the right-handed neutrino mass from leptogenesisPhys Lett B535 25ndash32 (2002) hep-ph0202239

[71] S Davidson E Nardi and Y Nir Leptogenesis (2008) 08022962

[72] A De Felice S Nasri and M Trodden Quintessential baryogenesis Phys Rev D67 043509(2003) hep-ph0207211

[73] R de Ritis A A Marino C Rubano and P Scudellaro Tracker fields from nonminimallycoupled theory Phys Rev D62 043506 (2000) hep-th9907198

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[76] E DrsquoHoker and R Jackiw Liouville field theory Phys Rev D26 3517 (1982)

[77] K Dick M Lindner M Ratz and D Wright Leptogenesis with Dirac neutrinos Phys RevLett 84 4039ndash4042 (2000) hep-ph9907562

[78] M Dine and A Kusenko The origin of the matter-antimatter asymmetry Rev Mod Phys 761 (2004) hep-ph0303065

[79] S Dodelson and L M Widrow Baryon symmetric baryogenesis Phys Rev Lett 64 340ndash343(1990)

[80] A D Dolgov Non GUT baryogenesis Phys Rept 222 309ndash386 (1992)

[81] J F Donoghue Spatial gradients in the cosmological constant JHEP 03 052 (2003) hep-ph0101130

[82] M Doran Can we test dark energy with running fundamental constants JCAP 0504 016(2005) astro-ph0411606

[83] M Doran and J Jaeckel Loop corrections to scalar quintessence potentials Phys Rev D66043519 (2002) astro-ph0203018

[84] M Doran M Lilley and C Wetterich Constraining quintessence with the new CMB dataPhys Lett B528 175ndash180 (2002) astro-ph0105457

[85] M Doran and G Robbers Early dark energy cosmologies JCAP 0606 026 (2006)astro-ph0601544

[86] M Doran G Robbers and C Wetterich Impact of three years of data from the WilkinsonMicrowave Anisotropy Probe on cosmological models with dynamical dark energy Phys RevD75 023003 (2007) astro-ph0609814

180 BIBLIOGRAPHY

[87] M Doran and C Wetterich Quintessence and the cosmological constant Nucl Phys ProcSuppl 124 57ndash62 (2003) astro-ph0205267

[88] J F Dufaux A Bergman G N Felder L Kofman et al Theory and Numerics of Grav-itational Waves from Preheating after Inflation Phys Rev D76 123517 (2007) arXiv

07070875

[89] J Dunkley et al Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) ObservationsLikelihoods and Parameters from the WMAP data (2008) arXiv08030586

[90] G V Dunne Heisenberg-Euler effective Lagrangians Basics and extensions (2004) hep-th0406216

[91] E Elizalde Ten physical applications of spectral zeta functions Lect Notes Phys M35 1ndash224(1995)

[92] E Elizalde K Kirsten and S D Odintsov Effective Lagrangian and the back reaction prob-lem in a selfinteracting O(N) scalar theory in curved space-time Phys Rev D50 5137ndash5147(1994) hep-th9404084

[93] J R Ellis J E Kim and D V Nanopoulos Cosmological Gravitino Regeneration and DecayPhys Lett B145 181 (1984)

[94] V Faraoni Inflation and quintessence with nonminimal coupling Phys Rev D62 023504(2000) gr-qc0002091

[95] R Fardon A E Nelson and N Weiner Dark energy from mass varying neutrinos JCAP 0410005 (2004) astro-ph0309800

[96] G R Farrar and P J E Peebles Interacting dark matter and dark energy Astrophys J 6041ndash11 (2004) astro-ph0307316

[97] E Fermi An attempt of a theory of beta radiation 1 Z Phys 88 161ndash177 (1934)

[98] P G Ferreira and M Joyce Structure formation with a self-tuning scalar field Phys Rev Lett79 4740ndash4743 (1997) astro-ph9707286

[99] U Franca and R Rosenfeld Age constraints and fine tuning in variable-mass particle modelsPhys Rev D69 063517 (2004) astro-ph0308149

[100] J Frieman M Turner and D Huterer Dark Energy and the Accelerating Universe (2008)arXiv08030982

[101] R Gannouji D Polarski A Ranquet and A A Starobinsky Scalar-tensor models of normaland phantom dark energy JCAP 0609 016 (2006) astro-ph0606287

[102] M Garny Quantum corrections in quintessence models Phys Rev D74 043009 (2006)hep-ph0606120

[103] M Garny B-L-symmetric baryogenesis with leptonic quintessence J Phys A40 7005ndash7010(2007) hep-ph0612145

[104] F Gelis The Effect of the vertical part of the path on the real time Feynman rules in finitetemperature field theory Z Phys C70 321ndash331 (1996) hep-ph9412347

BIBLIOGRAPHY 181

[105] E Gildener and S Weinberg Symmetry Breaking and Scalar Bosons Phys Rev D13 3333(1976)

[106] J-A Gu and W-Y P Hwang Can the quintessence be a complex scalar field Phys LettB517 1ndash6 (2001) astro-ph0105099

[107] P Gu X Wang and X Zhang Dark energy and neutrino mass limits from baryogenesis PhysRev D68 087301 (2003) hep-ph0307148

[108] A H Guth The inflationary universe a possible solution to the horizon and flatness problemsPhys Rev D23 347ndash356 (1981)

[109] J A Harvey and M S Turner Cosmological baryon and lepton number in the presence ofelectroweak fermion number violation Phys Rev D42 3344ndash3349 (1990)

[110] S W Hawking Zeta function regularization of path intergrals in curved space CommunMath Phys 55 133 (1977)

[111] A Hebecker and C Wetterich Natural quintessence Phys Lett B497 281ndash288 (2001)hep-ph0008205

[112] U W Heinz and P F Kolb Early thermalization at RHIC Nucl Phys A702 269ndash280 (2002)hep-ph0111075

[113] K Hepp Proof of the Bogolyubov-Parasiuk theorem on renormalization Commun MathPhys 2 301ndash326 (1966)

[114] M B Hoffman Cosmological constraints on a dark matter Dark energy interaction (2003)astro-ph0307350

[115] A Hohenegger A Kartavtsev and M Lindner Deriving Boltzmann Equations from Kadanoff-Baym Equations in Curved Space-Time (2008) arXiv08074551

[116] B L Hu and D J OrsquoConnor Effective lagrangian for lambda phi4 theory in curved space-time with varying background fields quasilocal approximation Phys Rev D30 743 (1984)

[117] G Huey and B D Wandelt Interacting quintessence the coincidence problem and cosmicacceleration Phys Rev D74 023519 (2006) astro-ph0407196

[118] K Ichikawa and M Kawasaki Constraining the variation of the coupling constants with bigbang nucleosynthesis Phys Rev D65 123511 (2002) hep-ph0203006

[119] A Ivanchik et al A new constraint on the time dependence of the proton-to- electron massratio Analysis of the Q 0347-383 and Q 0405- 443 spectra Astron Astrophys 440 45ndash52(2005) astro-ph0507174

[120] Y B Ivanov J Knoll and D N Voskresensky Self-consistent approximations to non-equilibrium many- body theory Nucl Phys A657 413ndash445 (1999) hep-ph9807351

[121] I Jack and L Parker Proof of summed form of proper time expansion for propagator in curvedspace-time Phys Rev D31 2439 (1985)

[122] R Jackiw Functional evaluation of the effective potential Phys Rev D9 1686 (1974)

182 BIBLIOGRAPHY

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[124] J I Kapusta and C Gale Finite-temperature field theory Principles and applications (Cam-bridge University Press Cambridge UK 2006)

[125] B G Keating A G Polnarev N J Miller and D Baskaran The Polarization of the CosmicMicrowave Background Due to Primordial Gravitational Waves Int J Mod Phys A21 2459ndash2479 (2006) astro-ph0607208

[126] L V Keldysh Diagram technique for nonequilibrium processes Sov Phys JETP 20 1018(1965)

[127] S Y Khlebnikov and I I Tkachev Classical decay of inflaton Phys Rev Lett 77 219ndash222(1996) hep-ph9603378

[128] L Kofman A D Linde and A A Starobinsky Reheating after inflation Phys Rev Lett 733195ndash3198 (1994) hep-th9405187

[129] L Kofman A D Linde and A A Starobinsky Non-Thermal Phase Transitions After InflationPhys Rev Lett 76 1011ndash1014 (1996) hep-th9510119

[130] L Kofman A D Linde and A A Starobinsky Towards the theory of reheating after inflationPhys Rev D56 3258ndash3295 (1997) hep-ph9704452

[131] P F Kolb and U W Heinz Hydrodynamic description of ultrarelativistic heavy-ion collisions(2003) nucl-th0305084 nucl-th0305084

[132] C F Kolda and D H Lyth Quintessential difficulties Phys Lett B458 197ndash201 (1999)hep-ph9811375

[133] M Kowalski et al Improved Cosmological Constraints from New Old and Combined Super-nova Datasets (2008) arXiv08044142

[134] A Kusenko and M E Shaposhnikov Supersymmetric Q-balls as dark matter Phys LettB418 46ndash54 (1998) hep-ph9709492

[135] N P Landsman and C G van Weert Real and Imaginary Time Field Theory at Finite Temper-ature and Density Phys Rept 145 141 (1987)

[136] M Le Bellac Thermal Field Theory (Cambridge University Press Cambridge UK 1996)

[137] S Lee K A Olive and M Pospelov Quintessence models and the cosmological evolution ofalpha Phys Rev D70 083503 (2004) astro-ph0406039

[138] M-z Li X-l Wang B Feng and X-m Zhang Quintessence and spontaneous leptogenesisPhys Rev D65 103511 (2002) hep-ph0112069

[139] A D Linde A New Inflationary Universe Scenario A Possible Solution of the Horizon Flat-ness Homogeneity Isotropy and Primordial Monopole Problems Phys Lett B108 389ndash393(1982)

[140] A D Linde Chaotic Inflation Phys Lett B129 177ndash181 (1983)

BIBLIOGRAPHY 183

[141] A D Linde Particle Physics and Inflationary Cosmology (Harwood Chur Switzerland1990) hep-th0503203

[142] M Lindner and M M Muumlller Comparison of Boltzmann equations with quantum dynamicsfor scalar fields Phys Rev D73 125002 (2006) hep-ph0512147

[143] M Lindner and M M Muumlller Comparison of Boltzmann Kinetics with Quantum Dynamicsfor a Chiral Yukawa Model Far From Equilibrium Phys Rev D77 025027 (2008) arXiv07102917

[144] C Misner K Thorne and J Wheeler Gravitation (Freeman New York 1973)

[145] D F Mota V Pettorino G Robbers and C Wetterich Neutrino clustering in growing neutrinoquintessence Phys Lett B663 160ndash164 (2008) arXiv08021515

[146] M M Muumlller private communications

[147] M M Muumlller Comparison of Boltzmann Kinetics with Quantum Dynamics for RelativisticQuantum fields PhD thesis Munich Tech U (2006)

[148] Y Nakayama Liouville field theory A decade after the revolution Int J Mod Phys A192771ndash2930 (2004) hep-th0402009

[149] Y Nambu and G Jona-Lasinio Dynamical model of elementary particles based on an analogywith superconductivity I Phys Rev 122 345ndash358 (1961)

[150] I P Neupane Accelerating cosmologies from exponential potentials Class Quant Grav 214383ndash4397 (2004) hep-th0311071

[151] S Nojiri and S D Odintsov Introduction to modified gravity and gravitational alternative fordark energy ECONF C0602061 06 (2006) hep-th0601213

[152] V K Onemli and R P Woodard Super-acceleration from massless minimally coupled phi4Class Quant Grav 19 4607 (2002) gr-qc0204065

[153] L Parker and D J Toms New form for the coincidence limit of the feynman propagator orheat kernel in curved space-time Phys Rev D31 953 (1985)

[154] P J E Peebles and A Vilenkin Quintessential inflation Phys Rev D59 063505 (1999)astro-ph9810509

[155] F Perrotta C Baccigalupi and S Matarrese Extended quintessence Phys Rev D61 023507(2000) astro-ph9906066

[156] A M Polyakov Quantum geometry of bosonic strings Phys Lett B103 207ndash210 (1981)

[157] B Ratra and P J E Peebles Cosmological consequences of a rolling homogeneous scalarfield Phys Rev D37 3406 (1988)

[158] E Reinhold et al Indication of a Cosmological Variation of the Proton - Electron Mass Ra-tio Based on Laboratory Measurement and Reanalysis of H(2) Spectra Phys Rev Lett 96151101 (2006)

[159] A Riazuelo and J-P Uzan Cosmological observations in scalar-tensor quintessence PhysRev D66 023525 (2002) astro-ph0107386

184 BIBLIOGRAPHY

[160] A G Riess et al Type Ia Supernova Discoveries at zgt1 From the Hubble Space TelescopeEvidence for Past Deceleration and Constraints on Dark Energy Evolution Astrophys J 607665ndash687 (2004) astro-ph0402512

[161] R Rosenfeld Relic abundance of mass-varying cold dark matter particles Phys Lett B624158ndash161 (2005) astro-ph0504121

[162] V Sahni Dark matter and dark energy Lect Notes Phys 653 141ndash180 (2004) astro-ph0403324

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[164] F Schwabl Statistische Mechanik (Springer Berlin Germany 2000)

[165] J-M Schwindt and C Wetterich Dark energy cosmologies for codimension-two branes NuclPhys B726 75ndash92 (2005) hep-th0501049

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[167] Y Shtanov J H Traschen and R H Brandenberger Universe reheating after inflation PhysRev D51 5438ndash5455 (1995) hep-ph9407247

[168] R Srianand H Chand P Petitjean and B Aracil Limits on the time variation of the electro-magnetic fine- structure constant in the low energy limit from absorption lines in the spectra ofdistant quasars Phys Rev Lett 92 121302 (2004) astro-ph0402177

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[175] H van Hees and J Knoll Renormalization of self-consistent approximation schemes II Ap-plications to the sunset diagram Phys Rev D65 105005 (2002) hep-ph0111193

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BIBLIOGRAPHY 185

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[181] C Wetterich Cosmologies with variable Newtonrsquos ldquoconstantrdquo Nucl Phys B302 645 (1988)

[182] C Wetterich Cosmology and the fate of dilatation symmetry Nucl Phys B302 668 (1988)

[183] C Wetterich Conformal fixed point cosmological constant and quintessence Phys Rev Lett90 231302 (2003) hep-th0210156

[184] C Wetterich Crossover quintessence and cosmological history of fundamental rsquoconstantsrsquoPhys Lett B561 10ndash16 (2003) hep-ph0301261

[185] C Wetterich Probing quintessence with time variation of couplings JCAP 0310 002 (2003)hep-ph0203266

[186] C Wetterich Growing neutrinos and cosmological selection Phys Lett B655 201ndash208(2007) 07064427

[187] M Yamaguchi Generation of cosmological large lepton asymmetry from a rolling scalar fieldPhys Rev D68 063507 (2003) hep-ph0211163

[188] Y B Zelrsquodovich The Cosmological constant and the theory of elementary particles Sov PhysUsp 11 381ndash393 (1968)

[189] X Zhang Coupled quintessence in a power-law case and the cosmic coincidence problemMod Phys Lett A20 2575 (2005) astro-ph0503072

[190] W Zimdahl and D Pavon Interacting quintessence Phys Lett B521 133ndash138 (2001)astro-ph0105479

[191] W Zimmermann Convergence of Bogolyubovrsquos method of renormalization in momentumspace Commun Math Phys 15 208ndash234 (1969)

Introduction


Quintessence Cosmology

Tracking Solutions

Interacting Quintessence




nPI Effective Action

Quantum Corrections in Quintessence Models

Self-Interactions

Matter Couplings

Gravitational Coupling

Summary


Quintessence and Baryogenesis

Creation of a B-L-Asymmetry

Stability

Quantum Nonequilibrium Dynamics and 2PI Renormalization

Kadanoff-Baym Equations from the 2PI Effective Action

Nonperturbative 2PI Renormalization at finite Temperature

Renormalization Techniques for Schwinger-Keldysh Correlation Functions

Non-Gaussian Initial States

Nonperturbative Thermal Initial Correlations

Renormalized Kadanoff-Baym Equation for the Thermal Initial State

Renormalization of Kadanoff-Baym Equations

Kadanoff-Baym Equations and 2PI Counterterms

Renormalizable Kadanoff-Baym Equations from the 4PI Effective Action

Impact of 2PI Renormalization on Solutions of Kadanoff-Baym Equations

Summary

Conclusions

Conventions


Low-Energy Effective Action

Effective Action in Curved Background

Renormalization Group Equations

Resummation Techniques and Perturbation Theory

Relation between 2PI and 1PI

Resummed Perturbation Theory

Quantum Fields in and out of Equilibrium

Thermal Quantum Field Theory

Nonequilibrium Quantum Field Theory

Nonperturbative Renormalization Techniques

Renormalization of the 2PI Effective Action

Renormalization of 2PI Kernels

Two Loop Approximation

Three Loop Approximation


Acknowledgements

Bibliography

TECHNISCHE UNIVERSITAumlT MUumlNCHEN

Max-Planck-Institut fuumlr Kernphysik



Mathias Garny

Vollstaumlndiger Abdruck der von der Fakultaumlt fuumlr Physik der Technischen Universitaumlt Muumlnchen zurErlangung des akademischen Grades eines

Doktors der Naturwissenschaften (Dr rer nat)

genehmigten Dissertation

Vorsitzender Univ-Prof Dr Lothar Oberauer

Pruumlfer der Dissertation1 Prof Dr Manfred Lindner

Ruprecht-Karls-Universitaumlt Heidelberg

2 Univ-Prof Dr Alejandro Ibarra

Die Dissertation wurde am 24092008 bei der Technischen Universitaumlt Muumlncheneingereicht und durch die Fakultaumlt fuumlr Physik am 24102008 angenommen


Abstract


Zusammenfassung


Contents

1 Introduction 1








9 Conclusions 131

viii CONTENTS

A Conventions 135







Bibliography 175

Chapter 1

Introduction


2 1 Introduction




1 Introduction 3



4 1 Introduction



Chapter 2




pl M4GUT M4

SUSY or M4elweak




pl












S =int

d4xradicminusg(x)

(minus R

16πG+

12


) (21)


Rmicroν minusR2


microν) (22)



δgmicroν and


(12(partφ)2minusV







2φ +dV (φ)

dφ= 0 (25)



partmicro

radicminusgpart

micro




3M2plH

2 = ρφ +ρB (26)

3M2pl

aa











1 = Ωφ +ΩB


2 = aminus3partt a3



φ +3Hφ +dV (φ)

dφ= 0 (27)















ωφ =pφ

ρφ

=φ 22minusVφ 22+V

(211)


minus1le ωφ le+1






0 M2pl A


pl 10minus10






Γminus1Γminus 1

2

(1+ωB) (212)




φlt


V (φ) =

M4

pl exp(minusλ

φ

Mpl

)for Γ = 1












Λ =O

((H0

Mpl

) 24+α

Mpl

)=O

(10minus

1224+α Mpl

)


Self-adjusting mass






92

Γ

(1minusω

lowastφ

2)

H(t)2 (213)







LYuk =minussumi

Fi(φ) ψiψi (214)




r (215)




(1+8πM2

plyi

mi

y j

m j

) (216)




nm2n sim y2




LGauge =12










16πGminus f (φ)R+

)equivint


16πGeff+

)

where1

16πGeff=

116πG

+ f (φ)






(Geff)today

∣∣∣∣le 02

Chapter 3



S[φ ] =int

d4x(


) (31)






ρ = sumn

pn|n〉〈n| (33)


〈O〉= Tr(ρO) (34)


ρ = |0〉〈0| (35)


H(x0) =int

d3x(

12(Φ(x))2 +


)(36)




HJ(x0) =int

d3x(

12(Φ(x))2 +


)(37)


ρ =1Z

exp(minusβH) (38)




ρ = |0〉J J〈0| (39)



(310)


exp(

iW [J])

=intDϕ exp

(iS[ϕ]+ i

intd4xJ(x)ϕ(x)

) (311)



d4xJ(x)φ(x) (312)


δΓ[φ ]δφ(x)

= 0 (313)





Γ[φ ] = S[φ ]+i2


iΓ1[φ ] = + +

=18

intd4x [minusiV (4)

cl (φ(x))]G0(xx)2 +112

intd4xint


+


Gminus10 (xy) =








(316)




G0(xy) = G0(xy)+int


= G0(xy)+int


+int

d4vint


+



V (4)cl (0)φ(x)2 + (317)




Γ[φ ] =int

d4x(minusVeff (φ)+

Z(φ)2

(partφ)2 +

) (318)



int d4k(2π)4 ln


k2

)+V1(φ)

minusV1(φ) = + +

=18

[minusV (4)

cl (φ)][int d4k

(2π)41


]2

+1


]2 int d4k(2π)4

int d4q(2π)4


+







12

m2φ

2 +13

gφ3 +

14

λφ4 (319)





exp(

iW [JK])

=intDϕ exp

(iS[ϕ]+ iJϕ +

i2

ϕKϕ

) (320)


Jϕ =int


d4xint




δJ(x) (322)


minusφ(x)φ(y)




intd4xint



δΓ[φ G]δφ(x)

=minusJ(x)minusint


=minus12

K(xy) (324)


δΓ[φ G]δφ(x)


= 0 (325)


Γ[φ G] = S[φ ]+i2

Tr lnGminus1 +i2

Tr(Gminus1

0 G)+Γ2[φ G] (326)

iΓ2[φ G] = + + (327)

=18

intd4x [minusiV (4)

cl (φ(x))]G(xx)2 +112

intd4xint


+



δΓ[φ G]δG(yx)


δG(yx) (328)





(330)


Γ[φ G[φ ]] = Γ[φ ] (331)




trunc(xyG) (332)








exp(

iW [J1 Jn])

=intDϕ exp

(iS[ϕ]+ i

n

sumk=1

1k


) (333)







sumk=1




δΓ

δφ(x)= 0

δΓ

δG(xy)= 0

δΓ









G(x1x2) = G12 =2δW [KL]

δK12 V4(x1x2x3x4) =

4δW [KL]δL1234




Γ[GV4] =i2

Tr lnGminus1 +i2

Tr(Gminus1

0 G)+Γ2[GV4] (337)

iΓ2[GV4] =

=18


+1

24

intd4x1234


minus 148

intd4x1234




V4(x1x2x3x4) =int



δΓ[GV4]δV4



iΓ2[G] = iΓ2[GV4] =18


148

intd4xint





Chapter 4







(minusλφMpl




radicΩde3 (41)


Λ2M2 simO(1)








radicGF






M = = 2GMint

k2ltΛ2

d4k(2π)4

1k2 +M2 = 2GM

Λ2

16π2 f1(M2Λ2) (42)

with f1(M2Λ2) = 1+ M2

Λ2 ln(

M2

Λ2+M2






FΛ(k) =


(43)




Λk

(2π)41

k2 +m2 =Λ2

16π2 f1(m2Λ2) (44)


Λk

(2π)4

int d4Λq

(2π)41

(k2 +m2)(q2 +m2)((q+ k)2 +m2)=

Λ2

(16π2)2 f2(m2Λ2)



can be evaluated

=18

V (4)cl (φ)

[int d4Λk

(2π)41


]2

(45)

=18

V (4)cl (φ)

[Λ2


asymp 18

V (4)cl (φ)

[Λ2

16π2 f1(0)]2

= Vcl(φ) middot

λ 4

8M4pl

[Λ2

16π2 f1(0)]2


=1


]2 intint d4Λkd4

Λq

(2π)81



]2 Λ2



]2 Λ2

(16π2)2 f2(0)



middot

λ 6

12M2pl

Λ2

(16π2)2 f2(0)






pl asymp 10minus120










Λ2M2L


Λ

2M2Lminus1







pl(Hinf 1014GeV)2



iΓ2[φ G] =infin

sumL=2

12LL

intd4x(minusiV (2L)

cl (φ(x)))G(xx)L (48)



δG(yx)(49)


sumL=2

L2LL


δ4(xminus y)



4(xminus y) (410)


eff (x)


infin

sumL=2

L2LL



M2eff (x) = exp

[12

G(xx)d2

dφ 2




M2eff = exp

[12

(int d4Λk

(2π)41

k2 +M2eff

)d2

dφ 2

]V primeprimecl(φ)




M2eff (φ) = exp

[12

(int d4Λk

(2π)41

k2 +m2

)d2

dφ 2

]V primeprimecl(φ)

∣∣∣∣∣m2=M2

eff (φ)

(412)


M2eff (xφ) = exp

[12

G(xx)d2

dφ 2


∣∣∣∣∣G(xx)=G(xx M2




2x +M2(x))

G(xy M2(middot)







Γhf [φ ] = Γ[φ G[φ ]]

=int

d4x(


)+

i2

Tr[

ln(2x +M2

eff (xφ))minus iM2

eff (xφ)G[φ ]]

where


12

G(xxφ)d2

dφ 2

]Vcl(φ(x)) (414)



12

int d4Λk

(2π)4

[ln

(k2 +M2

eff (φ)

k2

)minus

M2eff (φ)

k2 +M2eff (φ)

] (415)


Vhf (φ) = exp[

12

(int d4Λk

(2π)41

k2 +m2

)d2

dφ 2

]Vcl(φ)

∣∣∣∣∣m2=M2

eff (φ)

= V (φ m2)∣∣m2=M2

eff (φ)


V (φ m2)equiv exp[

12

(int d4Λk

(2π)41

k2 +m2

)d2

dφ 2

]Vcl(φ) (416)



M2eff (φ) =


∣∣∣∣m2=M2

eff (φ) (417)








=112

[minusV (3)(φ)

]2 int d4Λk

(2π)4

int d4Λq

(2π)41

(k2 +V (2))(q2 +V (2))((q+ k)2 +V (2))+





eff (φ) = k2 +V (2)(φ)


eff (φ)


∣∣∣∣m2=M2

eff (φ) (419)





M2eff (φ) = exp

[12

(Λ2

16π2 f1(m2Λ2))

d2

dφ 2

]V primeprimecl(φ)

∣∣∣∣∣m2=M2

eff (φ)



M2eff (φ) exp

[Λ2

32π2 f1(0)d2

dφ 2


1+O

(M2

eff

Λ2

)



Λk

(2π)4

[ln(

k2 +m2

k2

)minus m2

k2 +m2

]=

m4

16π2 f0(m2Λ2) (420)




Veff (φ) exp[

Λ2

32π2 f1(0)d2

dφ 2

]Vcl(φ) middot

1+O

(M2

eff

Λ2

)+O

(Veff

M4

) (421)







12

M4eff (φ)


2)









infin

suml=2

12(l +1)

[V (l+1)(φ)

]2 Λ2(lminus1)


2)



M2eff (φ) Λ







32π2d2

dφ 2

]Vcl(φ) (423)






to first order

V1minusloop(φ)[

1plusmn Λ2

32π2d2

dφ 2

]Vcl(φ)


Λ2LM2L L = 234




S[φ ] =int

ddx(


) (424)







12

int ddk(2π)4

[ln

(k2 +M2

eff (φ)

k2

)minus

M2eff (φ)

k2 +M2eff (φ)

] (425)


M2eff (φ) =


∣∣∣∣m2=M2


∣∣m2=M2

eff (φ) (426)


V (φ m2)equiv exp[

12

(int ddk(2π)4

1k2 +m2

)d2

dφ 2

]Vcl(φ) (427)




18π

M2eff (φ) =

(V (φ m2)+

18π


)m2=M2

eff (φ) (428)


int ddk(2π)d

1k2 +m2 =


microminus2ε

4π

(1ε


m2 +O(ε))

(429)






V Bcl (φ) = V R

cl (φ)+δVcl(φ)



V (φ m2) = exp[

18π

(1ε


m2 +O(ε))

d2

dφ 2

]V B

cl (φ)


δVcl(φ)equiv(

exp[minus 1

8π

1ε

d2

dφ 2

]minus1)

V Rcl (φ) (430)


V (φ m2) = exp[

18π

lnmicro2

m2d2

dφ 2

]V R

cl (φ) (431)




M2eff (φ) = exp

[1

8πln

micro2

m2d2

dφ 2

]V R

clprimeprime(φ)

∣∣∣∣∣m2=M2

eff (φ)

(432)









=1

(8π)2



6)54

(V (3)(φ)

)2

V (2)(φ) (433)





F =1

(8π)L g(F)prod

kge3

(V (k)(φ)

)Vk

(V (2)(φ)

)Vminus1 (434)







=(

V (φ)+1

8πV (2)(φ)

)+ sum

F

g(F)(8π)L

prodkge3

(V (k)(φ)

)Vk

(V (2)(φ)

)Vminus1







∣∣∣∣m2=M2

eff (φ)







M2eff = exp

[λ 2

8πln

micro2

M2eff

]V R


2V0 exp

[λ 2

8πln

(micro2

M2eff

)minusλφ

] (437)



ln

(M2

eff

micro2

)= ln

(λ 2V0

micro2

)+

λ 2

8πln

(micro2

M2eff

)minusλφ

rArr ln

(M2

eff (φ)

micro2

)=

11+λ 2(8π)

[ln(

λ 2V0

micro2

)minusλφ

]


ln

(M2

eff (φ)

micro2

)= ln

(M2

r

micro2

)minus φ

λminus1 +λ(8π)


Vhf (φ) = exp

[λ 2

8πln

(micro2

M2eff (φ)

)]V R


λminus1 +λ(8π)

]




18π

M2eff (φ) =

(Vr +

18π

M2r

)exp[minusλ φ

]




λminus1 = λ

minus1 +λ(8π)



partφ k

∣∣∣∣m2=M2


[minusλ φ

]


F =1

(8π)L g(F)prod

kge3


)Vk

(λ 2Vhf (φ)

)Vminus1 =1



λ 2

8π

)L

Vhf (φ)



[minusλ φ

] (438)


VR = Vr

(1+

λ 2

8π+sumF

g(F)(

λ 2

8π

)L)

= Vr

(1+

λ 2

8π+0781

(λ 2

8π

)2

+

)




exp[plusmn Λ2

32π2d2

dφ 2

](439)





Vcl(φ) = sumj

Vj exp(minusλ j

φ

Mpl

) (440)


Vj rarr Vj exp

[plusmn

λ 2j Λ2

32π2M2pl

] (441)



Veff (φ) Λ2M2

pl M4pl



quintessence models



Vcl(φ) = sumα

cαφminusα (442)


Veff (φ) = sumα

cαφminusα

Γ(α)

infin

sumL=0

Γ(α +2L)L

(plusmnΛ2

32π2φ 2

)L

= sumα

cαφminusα

Γ(α)

intinfin

0dt tαminus1 exp

(minustplusmn Λ2

32π2φ 2 t2)

(443)



1

101

102

103

104

105

106

107

108

10-4 10-3 10-2 10-1 1

Vef

f(φ

) V

(φ0)

φ φ0


Veff(φ)

Vone-loop(φ)

Vtwo-loop(φ)







cαφminusα

Γ(α)12

Γ(α

2)(

Λ2

32π2φ 2

)minus α

2

= sumα

Γ(α

2 )2Γ(α)

cα

(Λ

4πradic

2

)minusα

= const (445)




Γ(α+22 )

2Γ(α)cα

(Λ

4πradic

2

)minus(α+2)

(446)


M sim

Λ for φ Λ

φ for φ Λ (447)


1

103

106

109

1012

110-110-210-310-410-510-6

0

10

102

103

104

105

106

Vef

f(φ

) V

(φ0)

z (f

or tr

acke

r so

lutio

n)

φ φ0

today

CMB

Λ = φ0 asymp Mpl

V(φ) prop φ- α








α+2α+4




-1

-09

-08

-07

-06

-05

0 02 04 06 08 1

ωφ

Ωφ

toda

y (z

=0)ΛMpl =

0103

05

07

09

11 1315

29




Λmaxα

c1(α+4)α


c1(α+4)α sim

(H2

0 Mα+2pl


pl

)1(α+4)



α

ΛM

pl

02 04 06 08 1 12 14 16 18 2 0

05

1

15

2

25

3

-09

8-0

95

-09

-08 -07

-06



zquant sim

(Mpl10

Λ(4πradic

2α(α +1))

) α+23(1+ωB)



radicG) the track-
























dφφ (450)




V1L(φ) =12

int d4k(2π)4

(sumB


gF ln(k2 +mF(φ)2)

) (451)


I0(m2) equivint d4k

(2π)4 ln(k2 +m2) (452)

Il(m2) equivint d4k

(2π)41


(lminus1)dl

(dm2)l I0(m2)



I0(m2) = 2int m2

dm23

int m23dm2

2

int m22dm2

1 I3(m21) + D0 +D1m2 +D2m4 (453)



0

Ifinite0 (m2m2

0) = 2int m2

m20

dm23

int m23

m20

dm22

int m22

m20

dm21 I3(m2

1)

=1

32π2

(m4(

lnm2

m20minus 3

2

)+2m2m2

0minus12

m40

) (454)


V1L(φ) =12

(sumB


FgF Ifinite

0 (mF(φ)2mF(φ0)2)

) (455)


V1L(φ)Vcl(φ) (456)


V1L(φ) asymp 13


132π2 sum

j

(minus1)2s j g j

m j(φ0)2

(dm2

j

dφ(φ minusφ0)

)3

asymp 196π2 sum


(d lnm2

j



)3

(457)








d lnV primeprimecl

∣∣∣∣∣ 13ln(1+ zmax)


) 13

(459)



2Ωde

3H20

8πG


d lnV primeprimecl

∣∣∣∣∣(

1minusωde

2Ωde

07

) 13 1

3radicg j

(13meVm j(φ0)

) 43

(460)





∆m j

m jasymp

d lnm2j



3ln(1+ z)d lnm2

j



ν

emicro

p b Zt


100

10-5

10-10

10-15

10-20

101210910610310010-3

∆mm

meV




∆me


(1minusωde

2Ωde

07

) 13

(462)



φsim Hminus1


y j =dm j(φ)

dφ



y j =dm j

dφ=

12


d lnm2j


m j

2M

d lnm2j

d lnV primeprimecl



Mpl

M

)(1GeVmpn

) 13(

1minusωde

2Ωde

07

) 13

(463)


β j equivy j

m jMpl=

d lnm j

d(φMpl)


β j ∆m j

m j 4

(meVm j

)43

sim 10minus11(

me

m j

)43

(464)









ξ Rφ22


ξ (micro0) = 0


pl (see section 23)

116πGeff (φ)

=1

16πG+

12

ξ φ2


∆Geff

Geffequiv


=minusξ

2(φ

2(t)minusφ2(t0)

)16πGeff (φ(t))








d4xradicminusg(

12


) (465)


Vcl(φ) = Λ+m2φ

22+λφ44




=int

d4xradicminusg(

12


)

where

V (φ R) = Vcl(φ)+12

ξ Rφ2 +

R16πG

+ ε0R2 (467)

B(φ R) = ε3φ2 + ε4R





d4xradicminusg(

12







2φ

2 +λ (micro)

4φ

4 +12

ξ (micro)Rφ2 +

R16πG(micro)

+ ε0(micro)R2



partVLL

part t=

dΛ

dt+

12

dm2

dtφ

2 +14

dλ

dtφ

4 +12

dξ

dtRφ

2minus R16πG2

dGdt

+dε0

dtR2

=1

64π2

(part 2VLL

partφ 2 minusR6

)2

=1

64π2

(m(micro)2 +

λ (micro)2

φ2 +(

ξ (micro)minus 16

)R)2

part2BLL

part t=

dε3

dt2φ

2 +dε4

dt2R

=1

192π2

(part 22VLL

partφ 2 minus 2R5

)=

1192π2

(λ (micro)

22φ

2 +(

ξ (micro)minus 15

)2R)



dλ

dt=

3λ 2

32π2 dm2

dt=

λm2

32π2

dGdt


6)32π2

dΛ

dt=

m4

64π2

dξ

dt=

λ (ξ minus 16)

64π2 dε0

dt=

(ξ minus 16)2

64π2

dε1

dt= minus 1


dt= minus 1

360 middot32π2

dε3

dt=

λ

12 middot32π2 dε4

dt=

ξ minus 15

6 middot32π2

(469)





116πGeff (φ)

=1

16πG+ f1(φ)




V (φ R) = sumnm

cnmφnRm (470)

B(φ R) = sumnm

cnmφnRm




c00 = Λ c20 =m2

2 c40 =

λ

4 c21 =

ξ

2 c01 =

116πG

c02 = ε0 c20 = ε3 c01 = ε4



32π2


2

(ln


2

)(471)

minus(

1120

Cminus 1360

Gminus 130

2R+162X)

lnXminusR6

micro2 +infin

sumj=3




2V (φ R)partφ2



part


164π2


partφ 2 minus R6

)2


part


1192π2


partφ 2 minus 2R5





part


164π2


partφ 2

)2


part


132π2



partφ 2 minus 16








116πGeff (φ(t))

=1




(ξ0minus

16

)R (474)

=[

92

Γ

(1minusω

lowastφ

2)

+9(

ξ0minus16

)(ωBminus

13

)]H2

prop H2







+bφ +12

ξlowastφ

2










64π2

(V primeprimecl(φ)

)2+O

( t32π2

)2 (475)




)2

+O( t

32π2

)2




ξ0minus16

)2

+O( t

32π2

)2 (476)


∆Geff

Geff=


= minus(


16πGeff (φ(t))

= minus ξ0

2

(φ

2(t)minusφ2(t0)

)16πGeff (φ(t))

minus 132π2 ln

(micro2(φ(t))

micro20


(ξ0minus

16

)2

16πGeff (φ(t))




∆Geff


∆φ 2

M2plminus 1

32π2 ln(

H20

H2BBN


H20

(ξ0minus

16

)2 16πH20

M2pl











part


164π2

[2




partφ 2 minus 16

)2]

f2(φ micro0) = f2(φ) (477)







64π2

(ξ0minus

16

)2

+12

( t32π2

)2V (4)

cl (φ)(

ξ0minus16

)3

+O( t

32π2

)3






f3(φ micro) =13

( t32π2

)3V (6)

cl (φ)(

ξ0minus16

)4

+O( t

32π2

)4


pl since

V (6)cl (φ)R3(RM2



Kinetic term



32π2

[minusV1L(φ)minus

(162X)

lnXmicro2

+(minus 1

12partmicroXpart

microXminus 160

22X)

1X

+infin

sumj=4


]

=int d4x

32π2

[minusV1L(φ)+

112X


(part

4)]=

int d4x32π2

[minusV1L(φ)+

12


4)]






44 Summary






Chapter 5





Toy Model








χ)minus 12

micro2χ |χ|2

minus12





V (|φ |) = V0


)(52)









10-2

10-1

100

101

102

103

|φ(t

)|H

Inf

ΓasympHφ0HInf =

10010101001


01sdot10-3

0 02 04 06 08 1

θ φ =

arg

(φ)

t HInf




lowastχ

2


lowastφ

2




0 φ0χ30 ) it





infin

0dt a3




32ρ340

prop Ainfin (55)



0

02

04

06

08

1

0 02 04 06 08 1 12 14 16

|χ(t

)| a

32 H

Inf

t HInf

exact

WKB





κ asympminusN2

sin(2α0)(

χ0

Hinf

)2

middot


1013GeV

(Hinf

1012GeV

)12

if φ30 χ


17 middot10minus8(

ξ1

7V0

ρ0

)13(

Hinf

1012GeV

)76


30






-4sdot10-10

-2sdot10-10

0

2sdot10-10

4sdot10-10

0 02 04 06 08 1 12 14 16 18

n φχ a

3 [3

2 ρ 0

34 ]

t HInf

g=05

Ainfin

nφsdota3

nχsdota3

nχsdota3 WKB


φ0 HInf

V0

ρ0

10-3 10-2 10-1 100 101 102 103 104 105 106

10-810-710-610-510-410-310-210-1

10-10

10-9

10-8 10-7 10-6 10-5

2sdot10-10

5sdot10-10



53 Stability


|θφ |H

=|nφ |


45glowastS(T )

T 3


2H|φ |2 10minus8 (58)


θ2φ lt

3H +2ϕϕ



Chapter 6



S[φ ] =int

d4x(

12(partφ)2minus 1

2m2

φ2minus λ

4φ

4)

(61)





〈OH(t)〉= Tr(


(62)

= Tr(

ρ T[

exp(

+iint t

tinit


[exp(minusiint t

tinit





)(63)

=

T[exp(minusiint t


for t gt t prime

T[exp(+iint t prime


for t lt t prime


〈OH(t)〉= Tr(

ρ TC

[exp(minusiint

CdtHI(t)

)OI(t)

]) (64)








Φ(0x)|ϕ0〉= ϕ(x)|ϕ0〉 (66)

is given by

Zρ [JK] = Tr(

ρ TC

[exp(

iint

Cd4xJ(x)Φ(x)+

i2

intCd4xint

Cd4yΦ(x)K(xy)Φ(y)

)])=

intDϕ+


langϕminus0

∣∣∣∣TC

[exp(

i JΦ+i2

ΦKΦ

)]∣∣∣∣ϕ+0rang



Zρ [JK] =intDϕ+



Dϕ exp(

iS[ϕ]+ i Jϕ +i2

ϕ Kϕ

)


(iS[ϕ]+ i Jϕ +

i2

ϕ Kϕ

) (67)





iα0 + iα1ϕ +i2

ϕα2ϕ

) (68)






Γ[G] =i2

Tr lnGminus1 +i2

Tr(Gminus1

0 G)+Γ2[G] (69)

iΓ2[G] =minusiλ

8

intCd4xG(xx)2 +

(minusiλ )2

48

intCd4xint

Cd4yG(xy)4





δ


2K(xy) (610)




=minusiλ

2G(xx)δ 4


6G(xy)3 (612)





λ

2G(xx)








)GF(xy) =

int y0

0d4zΠF(xz)Gρ(zy)

minusint x0


(2x +M2(x)

)Gρ(xy) =

int y0

x0

d4zΠρ(xz)Gρ(zy)





6G(xy)3 (616)



ω2(tk) =


GF(x0y0k)

)∣∣∣∣∣x0=y0=t


(617)
















Dϕ exp(

iint

Id4xL(x)

) (618)


Zβ [JK] = Tr(

ρ TC+I

[exp(

iint

C+Id4xJ(x)Φ(x)+

i2

intC+I

d4xint


)])=

intDϕ exp

(iint


2

intC+I

d4xint


) (619)






Gth(xy) =int



The meaning ofint






section D12)





intC+I





=minusiλ

2Gth(xx)δ 4


6Gth(xy)3 (623)





S[φ ] =int

d4x(

12(partφ)2minus 1

2m2

Bφ2minus λB

4φ

4)

(624)



B = m2R +δm2 Z2

λB = λR +δλ Z = 1+δZ (625)





λR +δλ0

2

intqGthR(q)+

λ 2R

6

intpq




Gminus1T0

(k = k) = k2 +m2R

ddk2 Gminus1

T0(k = k) = +1 (627)





T0(k)minus∆Π(k)


2

[int T

qGT (q)minus

int T0

qGT0(q)

](628)

+λ 2

R

6

[int T


int T0


]





int Tq equiv

int T0q +



that

∆Π(k) =12

[int T0

q∆G(q)+

int∆T

qGT (q)

]ΛT0(qk)+F(k) (630)





q∆Π(q)


2G2

T0(q)ΛT0(qk)

]=

12

[int T0

q∆

2G(q)+int

∆T

qGT (q)

]ΛT0(qk)+F(k)



int T0

q∆Π(q)


2G2


2

int T0

kΛT0(qk)G2

T0(k)VT0(k p)

]=

=12

int T0

k

[int T0

q∆

2G(q)+int

∆T

qGT (q)

]ΛT0(qk)+F(k)

(δ (kminus p)+

12

G2T0

(k)VT0(k p))




intkΛR(qk)G2

thR(k)VR(k p) (632)


∆Π(p) =12

int T0

q

[∆

2G(q)+F(q)G2T0

(q)]VT0(q p)+

12

int∆T

qGT (q)VT0(q p)




Chapter 7













F [ϕ] = α0 +int

Cd4xα1(x)ϕ(x)+

12


+13





0n) (73)


F [ϕ] = α0 +int


12

intd3xint






n (x1 xn))lowast

(75)



int d3k1



(2π)3δ

3(k1 + +kn) iαε1εnn (k1 kn) (76)



(i

S[ϕ]+ Jϕ +12






Γ[φ G] = S[φ ]+i2

Tr lnGminus1 +i2

Tr(Gminus1

0 Gminus1)+ Γ2[φ G] (77)



Γ[φ G] = S[φ ]+i2

Tr lnGminus1 +i2

Tr(Gminus1





ΓnG[φ Gα3α4 ] = F3[φ α3α4 ]+12

Tr(

δ 2F3

δφδφG)

+Γ2nG[φ Gα3α4 ] (79)

The 2PI functional




= iαn(x1 xn)+int


+12





δφ(x)δφ(y) (712)




0n) (713)




ΓnG[φ Gα3 = 0α4 = 0 ] = 0 (714)


iΓ2[Gα4] =18

intC

d4x1234


]G(x1x2)G(x3x4)

+148

intC

d4x1234d4x5678


]G(x1x5)times (715)


]










δG(yx)=



δG(yx) (717)


Π(xy) = ΠG(xy)+Π

nG(xy) (718)


Gnonminusloc(xy)

ΠnG(xy) = Π


surface(xy)









where


Πλα(xy) = Πε










6


4 (x1x2x3y)

+minusiλφ(x)

2


3 (x1x2y) (721)






M(x)2 = m2 +λ

2φ


λ

2φ

2(x)+λ

2G(xx) (723)



(minusiλ )2

6G(xy)3 M(x)2 = m2 +

λ

2G(xx)



6




+16




3 (x1x2x3) = 0







Cd4z[Π

Gnonminusloc(xz)+Π


]G(zy)





(part

2x0 +k2 +M2(x0)

)GF(x0y0k) =

int y0

0dz0


minusint x0

0dz0



+14


2x0 +k2 +M2(x0)

)Gρ(xy) =

int y0

x0


where


(x0k)+Πminusλα

(x0k)


+λα


(x0k))

(727)


(2x +M2(x)

)GF(xy) =

int y0

0d4zΠF(xz)Gρ(zy)

minusint x0


minus λ

6V nG


V nG3 (xxy) (

2x +M2(x))

Gρ(xy) =int y0

x0

d4zΠρ(xz)Gρ(zy)

where


intC



intC




















iGminus10th(xy) =

(minus2xminusm2)







2ωk

cos(ωk(x0minus y0)

) (731)

G0ρ(x0y0k) =1


)for x0y0 isin C+I


nBE(ωk) =1

eβωk minus1 ωk =

radicm2 +k2


0th GCI0th GIC



GCC0th(xy) = GCI

0th(xy) =

GII0th(xy) = GIC

0th(xy) =

minusiλint

Cd4x = minusiλ

intId4x = minusiλ

intC+I

d4x =

(732)



6

intC+I

d4uint


=










GCC0F(0y0k) =

intCdz0

δs(z0)GCC0th(z

0y0k) (733)


intCdz0

δa(z0)GCC0th(z

0y0k) (734)

where


)


) (735)



intCdz0


0y0k)

=(736)




=

(GII


)δs(z0)+


δa(z0)

= (738)


GCI0th(y

0minusiτk) =int

Cdz0 GCC

0th(x0z0k)∆T

0 (z0minusiτk)

= (739)





∆0(vz) =int d3k




∆0(ωnz0k) =

(GII

0th(ωnk)G0th(00k)

)δs(z0)+

2iωnGII0th(ωnk)

δa(z0) (740)


time then becomes

GIC0th(ωny0k) =

intCdz0


0y0k) (741)


= equiv equiv th

0L



(minusiλ )2

6

intCd4uint


=(minusiλ )2

6

intCd4uint

Cd4z1

intCd4z2

intCd4z3


Id4v∆

T0 (z1v)∆T

0 (z2v)∆T0 (z3v)∆0(vz4)

]G0th(z4y)

equiv minusiλ6

intCd4uint

Cd4z1

intCd4z2

intCd4z3


αth40L(z1z2z3z4)

]G0th(z4y)




th

0L= equiv








th41L(z1z2z3z4) +

th=

th

0L+

th

1L+





+int

Cdw0

intCdz0




0(minusiτ primek)

= +

(742)


= equiv =

= + + +

= + + +





S0(xy) = =

+ + +

+ + +









GCCth (xy) = GCI

th (xy) =

GIIth (xy) = GIC

th (xy) =

minusiλint

Cd4x = minusiλ

intId4x = minusiλ

intC+I

d4x =

(744)




1I(x0) =














mth GCImth GIC



GIImth(xy) = GII




2x +m2B)

GCImth(xy) =

(2x +m2

B)

GCCmF(xy) =

(2x +m2

B)

GCCmρ(xy) = 0 (750)









th)


intβ

0dτprimeΠ




β

0dτ eminusiωnτ

part2τ GIC



mth)(y0k)



mth)(y0k) =

[(iωn GIC


]τ=β

τ=0


2n +k2 +M2

th)




mth)(y0k)


2y0 +k2 +m2

B

)disc(GIC

mth)(y0k) =

(part

2y0 +k2 +m2

B

)disc(partτGIC

mth)(y0k) = 0












disc(GICmth)(y


∣∣τ=β




∣∣τ=β

τ=0 =GCC

mF(0y0k)Gth(00k)


GICmth(ωny0k) =

(GII

th (ωnk)Gth(00k)

)GCC

mF(0y0k)minus(

iωnGIIth (ωnk)

)GCC

mρ(0y0k)

=int

Cdz0


0y0k) (753)



∆m(ωnz0k) =(

GIIth (ωnk)

Gth(00k)

)δs(z0)+

(2iωnGII

th (ωnk))

δa(z0)



=










C+Id4uint

C+Id4vGth(xu)

[1minus1I(u0)1I(v0)




mth(x0ωnk)+

intC+I

du0int

Cdv0(


0ωnk))

minus iint

Cdu0 GCC


Next GCImth(x

0ωnk) and GCImth(v


GCIth (x0ωnk)

=int

Cdz0[

GCCmth(x

0z0k)+int

C+Idu0

intCdv0(


0z0k))]

∆Tm(z0ωnk)

minus iint

Cdu0 GCC


=int

Cdz0[

GCCth (x0z0k)minus

intCdu0int

Idv0(


mth(v0z0k)

)]∆

Tm(z0ωnk)

minus iint

Cdu0 GCC


=int

Cdz0 GCC

th (x0z0k)

∆


th (ωnk)

minusint

Cdu0int

Idv0

Πth(z0v0k)GICmth(v

0u0k)∆Tm(u0ωnk)




Cdu0int

Idv0

Πth(z0v0k)GICmth(v


= minusiT suml

intCdu0


m(u0ωnk)

= minusiT suml


m(ωnk)

= minusiT suml


GIIth (ωnk)

Gth(00k)


GCIth (x0ωnk) =

intCdz0 GCC

th (x0z0k)

∆


th (ωnk)

minus iT summ


GIIth (ωnk)

Gth(00k)

=

intCdz0 GCC

th (x0z0k)∆T (z0ωnk) (756)


∆T (z0ωnk) = ∆


mΠth(z0ωmk)

[δnm

TGII

th (ωnk)minusGII


Gth(00k)

]equiv ∆




D(ωnωmk) =δnm

TGII

th (ωnk)minusGII


Gth(00k)(758)

=δnm

TGII


m(ωmk)

=δnm

TGII

th (ωnk)minusint

Cdw0

intCdz0


=δnm

TGII

th (ωnk)minusint

Cdw0

intCdz0





D(ωnωmk) = 0 (759)



th + Πnlth(z


T summ


Πnlth(z

0ωmk)D(ωmωnk)






intCdz0 GCC


=


intCdz0


=




th(v0z0k) (761)




th(v0z0k)

= = +




sm(minusiτk) =







intCdw0

intCdz0




= +

= +







= +





nlth(v

0z0k)∣∣GIC


(765)


= + +

+

+










6

intC+I

d4uint


=

= + + +

+ + +




=

+

+




th

0L 2PI= equiv (767)


intId4vint

Id4w∆

Tm(z1v)∆T



th

0L 2PI= equiv (768)





part2x0 +k2 +M2

th)


minus iint

Cdz0 [

ΠGthnl(x

0z0k)+ΠnGthnl(x


Gth(z0y0k)

where ΠGthnl(x






part2x0 +k2 +M2

th)


C+Idz0

Πnlth(x

0z0k)Gth(z0y0k)


intC+I

dz0Π

nlth(x

0z0k)Gth(z0y0k) =


Πnlth(x


Idv0

Πnlth(x

0v0k)int

Cdz0


=int

Cdz0[int

Idv0

Πnlth(x

0v0k)(

∆m(v0z0k)+int

Idw0 D(v0w0k)Πnl

th(w0z0k)

)]Gth(z0y0k)


ΠGthnl(x

0z0k) = Πnlth(x


ΠnGthnl(x

0z0k) =int

Idv0int

Idw0

Πnlth(x




Idv0

Πnlth(x






ΠGthnl(x

0z0k) =

ΠnGthnl(x

0z0k) = (771)

iΠthλα(x0z0k) =


Πthλα =infin

sumk=0

Π(k)thλα

ΠnGthnl =

infin

sumk=0

Π(k)nGthnl (772)

where

Π(0)thλα



Π(k)thλα




(x0z0k)



iΠ(0)thλα

(x0z0k) = =th

0L 2PI(773)


iΠ(1)thλα

(x0z0k) =

iΠ(2)thλα

(x0z0k) =




th

0L2PI



iΠ(0)thλα

(xz) =minusiλ

6


40L2PI(x1x2x3z) (774)


part2x0 +k2 +M2

th)



∣∣∣x0=y0=0

(part

2x0 +k2 +M2

th)

Gthρ(x0y0k)∣∣x0=y0=0 = 0 (775)


Πthλα(x0z0k)


ΠnGthnl(x

0z0k)



(k)thλα










R +δm20 +

λR +δλ0

2GthR(xx)


C(xminus y) (777)

minus iint

Cd4z[Π

GthnlR(xz)+Π


]GthR(zy)



ΠGthnlR(xz) = Z Π

Gthnl(xz) = Π

nlthR(xz)



nGthnl(xz) =

intId4vint

Id4wΠ

nlthR(xv)DR(vw)Πnl



Id4vΠ

nlthR(xv)∆m(vz)



ΠnlthR(xy) =

(minusiλR)2

6GthR(xy)3


iΠ(0)thλαR(xz) =

minusiλR

6


40L2PIR(x1x2x3z) (779)





Chapter 8







Gminus1R (xy) = i

(2x +m2

R)


ΠR(xy) = minusi(

δZ02x +δm20 +

λR +δλ0

2GR(xx)

)δ

4C(xminus y)+

(minusiλR)2

6GR(xy)3










(iS[ϕ]+

i2

ϕ Kϕ +i

4L1234ϕ1ϕ2ϕ3ϕ4

) (82)



iα0 +i2

ϕα2ϕ +i

4(α4)1234ϕ1ϕ2ϕ3ϕ4

) (83)




εn=plusmnα




intrarrintC



δ


14

L(xyzw) (85)


δ

δG(xy)Γ

L[G] =minus12

K(xy) (86)


ΓL[G] = Γ[GV4]+

14




δΓ

δV4=minusα4


4C(x1minus x2)δ 4





V4(x1x2x3x4) =int


4 + iAnG4 )(y1y2y3y4)]


4 (x1x2x3x4) (87)


(2x +M2(x)

)GF(xy) =

int y0


int x0

0d4zΠρ(xz)GF(zy)

minus λ

6V nG

4 (xxxy) (88)(2x +M2(x)

)Gρ(xy) =

int y0

x0

d4zΠρ(xz)Gρ(zy)


823 Renormalization



(2x +δZ02x +m2

R +δm20 +

λR +δλ0

2GR(xx)

)GFR(xy)

=int y0


int x0


minus λR

6V nG

4R(xxxy) (89a)

(2x +δZ02x +m2

R +δm20 +

λR +δλ0

2GR(xx)

)GρR(xy)

=int y0

x0






6GR(xy)3



V nG4R(xxxy) =



V nG4R(xxxy) =







minus λR

6V nG

4R(xxxy)equivint


where

iΠλαR(xy) =16








0001

001

01

1

0 5 10 15 20 25 30 35 40 45 50

G(t

0k

=0)

t mR

GF(t0k)

Gρ(t0k)

-06-04-02

0 02 04 06 08

1 12 14 16

0 05 1 15 2 25 3 35 4

Πλα

(tk

=0)

t mR

ΠλαF(tk)

Πλαρ(tk)














V nG4R(x1x2x3x4)|x0

i =0 = minusiλR


i =0




(δm2

0 +m2R +

λR +δλ0

2GthR(xx)

)]GthR(xy)|x0y0=0

minus Zminus1 λR

6(minusiλR)




neiωnτ

[k2 +Zminus1

(δm2

0 +m2R +

λR +δλ0

2

intqGthR(q)

minus λ 2R

6

intpq


GthR(ωnk)|τrarr0

= minusT sumn





neiωnτ

[k2 +Zminus1

(Gminus1

thR(ωnk)minusZk2)]

GthR(ωnk)|τrarr0

= minusT sumn


2n GthR(ωnk)

]τrarr0




Outlook








(part

2x0 +k2 +M2(x0)

)GF(x0y0k) =

int y0

0dz0



minusint x0

0dz0







m2B = m2

Rminusλ

2

int d3p(2π)3

1

2radic

m2R +p2

reg


M2(x0) = m2B +

λ

2



6



)


6



)



(2π)3d3q










λB

2

intq

G(q)minus λ 2R

6Z4

intpq


V (k) = λBminusλB

2

intq

G2(q)V (q) (814)

minus λ 2R

Z4

intq

G(q)G(kminusq)+λ 2

R

2Z4

intpq






Zd



R (815)

Z2Vvac(k = 0) = λR






4 αaaaa4 αssaa




M2 (x0) = m2B +

λB

2

int d3p(2π)3 GF(x0x0p) (816)



6Z4



)


6Z4



)



6Z2



4](x00k)

)


6Z2


4](x00k)minus 1


4](x00k)

)



=int d3p

(2π)3d3q








1

12

14

16

18

2

22

24

0 10 20 30 40 50 60 70 80 90 100

mR

min

it

iteration

14

15

16

17

18

19

20

21

1 10

V(k

)

kminit

0th iteration

10th iteration

20th iteration

30th iteration




m2th = Z Gminus1

th (k = 0) (817)





001

01

1

10

100

1000

0 05 1 15 2

δλ

λR24


asmR = 025

asmR = 0125 01

1

10

100

1000

0 05 1 15 2

|δm

2 mR

2 |

λR24

Nonpert 3-loop

Pert 1-loop







partx0GF(x0y0k)∣∣x0=y0=0 = 0 (818)



where


nGth(ωnk)

ωth(k)2 =(



=T sumn

(1minusω2

n Gth(ωnk))

Gth(k) (819)




Z2

βint0





Gth(k)=






int β

0 rarr 2int β2







03

04

05

06

07

08

001 01 1 10 100

GF(t

tk)

t mR

k = 0

k = mR

k = 2mR

500 1000 1500 2000t mR

KB Gauss

KB Non-Gauss

ThQFT



Etotal(t) =int d3k

(2π)3

[12


B +λB

4


)GF(x0y0k)|x0=y0=t

minus 14

int t

0dz0 (


minus 14


14


+ const (820)





s



-03-02-01

0 01 02 03 04

001 01 1 10 100

microm

R

t mR

2

21

22

23T

mR

500 1000 1500 2000t mR

KB Gauss

KB Non-Gauss

Thermal Eq

0

1

0 8

f BE(n

)

ωmR0

1

0 8

f BE(n

)

ωmR



nfit(tk) =[

exp(


)minus1]minus1





=14



24Z2 V nG4 (xxxx)

∣∣x0=0

=λ 2

R

24

intkpq








15

155

16

165

17

175

18

185

19

001 01 1 10 100

mth

(t)

mR

t mR


18

10 100



mAth ω(tk = 0) =


GFR(x0y0k)

∣∣∣x0=y0=t

ωth(k = 0) =radic


∣∣∣τ=0

mBth

(limsrarrinfin


2 tminus sprime2 k)


)minus12




13

14

15

16

1 15 2 25 3 35

mth

mR

(as mR)-1

T = 15 mR

14

15

16

17

mth

mR

T = 17 mR


16

17

18

19m

thm

RT = 20 mR

001 01 1 10 100t mR











0

05

1

15

2

25

3

35

001 01 1 10 102 103 104

GF(t

tk=

0)

t mR

A24 A18

E18

E24





A18 λ = 18 m2B =minus687 m2

R E18 λR = 18 λB = 3718 m2B =minus1439 m2

R

A24 λ = 24 m2B =minus949 m2

R E24 λR = 24 λB = 6343 m2B =minus2514 m2

R



095

1

105

0 1 2 3 4 5 6 7GF(t

tk)

GF(0

0k

)

kmR

t mR = 2000

095

1

105

GF(t

tk)

GF(0

0k

)

t mR = 10

095

1

105

GF(t

tk)

GF(0

0k

)

t mR = 05

095

1

105

GF(t

tk)

GF(0

0k

)

t mR = 00 KB Gauss

095

1

105

GF(t

tk)

Gth

(k)

t mR = 20

0 1 2 3 4 5 6 7

095

1

105

kmR

t mR = 2000

095

1

105t mR = 10

095

1

105t mR = 05

095

1


095

1

105t mR = 20




09 1

11 12

0 1 2 3 4 5 6 7

GF(t

tk)

Gth

(k)

kmR

t mR = 2000

09 1

11 12

GF(t

tk)

Gth

(k)

t mR = 10

09 1

11 12

GF(t

tk)

Gth

(k)

t mR = 05

09 1

11 12

GF(t

tk)

Gth

(k)

t mR = 00 KB Gauss

09 1

11 12

GF(t

tk)

Gth

(k)

t mR = 20

0 1 2 3 4 5 6 7

09 1 11 12

kmR

t mR = 2000

09 1 11 12

t mR = 10

09 1 11 12

t mR = 05

09 1 11 12


09 1 11 12

t mR = 20



84 Summary 129

84 Summary






Chapter 9

Conclusions




132 9 Conclusions






9 Conclusions 133


Appendix A

Conventions


Rαmicroνλ = +

(partνΓ

α


αmicroν +Γ

η

microλΓ

αην minusΓ

η

microνΓα

ηλ

)


Γαmicroν =

12


)


microαλ


micro


Appendix B




S[φ ψ j] =int

d4x(


) (B1)




fk(φ)OSMk (B2)


exp(

iW [J])

=intDϕ

int (prod

jDψ j

)exp(

iS[ϕψ j]+ iint

d4xJ(x)ϕ(x))

(B3)


exp(


prodjDψ j



exp(

iΓ[φ ])

= exp(

iW [J]minus iint

d4xJ(x)φ(x))

=intDϕ exp

(iSeff [ϕ]+ i


)






radicminusg


d4xradicminusg(


) (B5)






Gminus10 (xy) =


= i

(2x +

δ 2V (φ R)δφ 2

∣∣∣∣φ(x)R(x)

)δ

4(xminus y) (B6)



Tr lnGminus10 (B7)



lndet A (B8)

with the operator


δφ 2

∣∣∣∣φ(x)R(x)

(B9)




mln

λm

micro2 =12i



ipart



K(xys) = summ



ζA(ν) = summ

iΓ(ν)

infinint0


Γ(ν)

infinint0

ds(is)νminus1int

d4xK(xxs) (B11)


infin


K(xys) =i∆12


(minusσ(xy)

2isminus is

(X(y)minus R(y)

6

)) (B12)







partxmicropartyν

] (B13)


infinj=0(is)


ζA(ν) =i

Γ(ν)

int d4x16π2

radicminusg

infin

sumj=0



= iint d4x

16π2

radicminusg



(XminusR6)1minusν

νminus1


sumj=3



)



32π2


X2

2

(ln

Xmicro2 minus

32

)+ g1(xx)X

(ln

Xmicro2 minus1

)minusg2(xx) ln

Xmicro2 +

infin

sumj=3

g j(xx)( jminus3)

X jminus2

]

(B14)


g0(xx) = 1 (B15)

g1(xx) = 0

g2(xx) =1


302R+

162X

=1

120Cminus 1

360Gminus 1

302R+

162X






0 =d


(part


βNpart

partcN+sum

i

intd4xγiφi(x)

δ

δφi(x)

)Γ[φi]






part


(sumN

βN(micro)part

partcN+sum

i


δ

δφi(x)

)ΓLL[φi micro]








part



= minus

(sumN

βNpart

partcN+sum

i

intd4xγiφi(x)

δ

δφi(x)

)S[φi]

= minusint



nmβ nm2(φ nRm)

+ γφ φ


partφ

)+ γgmicroν

gmicroν

(12


δgmicroν

)]



part



= minus part

part ln micro20


=int d4x

32π2


2minus 1

120Cminus 1

360Gminus 1

302R+

162X]

= minusint d4x

32π2

radicminusg

[12 sum

nm

(n

sumk=0

m

suml=0

dkldnminuskmminusl

)φ

nRmminus 1120

Cminus 1360

G

+16 sum

nm


15

δn0δm1

)2(φ nRm)

]


βnm =1

64π2

n

sumk=0

m

suml=0

dkldnminuskmminusl

β nm =1

192π2


15

δn0δm1

) (B17)


1120


1360








part


164π2


partφ 2 minus R6

)2

part


1192π2


partφ 2 minus R5

) (B19)




d4xradicminusg(

12


)







0 ) yields [60]


∣∣∣∣∣t=ln


micro20







Appendix C






(C1)


G(xy) = G0(xy)+int

d4uint



G(0)(xy) = G0(xy)


d4uint


= G0(xy)+int

d4uint


+int

d4uint

d4vint

d4zint






Π(0) = Π[φ G0]

Π(k) = Π[φ G(k)] = Π[φ G0

infin

sumn=0

(Π(kminus1)G0)n]



Π(k)








M2eff (xφ) = exp

[12

G(xx)d2

dφ 2


∣∣∣∣∣G(xx)=G(xx M2




2x +M2(x))

G(xy M2(middot)




Γhf [φ ] = Γ[φ G[φ ]] (C5)

=int

ddx(


)+

i2

Tr[

ln(2x +M2

eff (xφ))minus iM2

eff (xφ)G[φ ]]

where


12

G(xxφ)d2

dφ 2

]Vcl(φ(x)) (C6)



12

G(xx M2(middot)

) d2

dφ 2

]Vcl(φ(x)) (C7)


M2eff (xφ) =



eff (middot φ)






iΓnotad[φ ] = +

=112

intddxint







eff (middot φ)




Derivation






infin

sumn=0











12

Ghf (xx)d2

dφ 2







li=1 N(γi)

F =1

N(F)F (C12)


K(F) =


prodi=1

S(γi)

∣∣∣∣∣ |S(F)| (C13)






Γnotad[φ ] =i2



where


δGhf

(GminusGhf

) (C15)


G = Ghf

infin

sumn=0













Appendix D




D11 Thermal State


ρ =1Z

exp(minusβH)





Dϕ exp(

iint

Id4xL(x)

)C+I

exp

(i

infin

sumn=0


)C+α

(D1)







Zβ [J] = Tr(

ρ TC+I

[exp(

iint

C+Id4xJ(x)Φ(x)

)])=intDϕ exp

(iint


)



∣∣∣∣∣J=0

(D2)




Tr(


)= Tr

(eminusβHeτH


)= Tr

(eminusτH


)= Tr

(eminusβH










iS0[φ0]]

=N0 exp[

iint

Id4x(

12(partφ0)2minus 1

2m2

φ20

)]









ξ2k =

nBE(ωk)+ 12

ωk ηk = 0

σ2k

4ξ 4k


1eβωk minus1

(D4)











ωk sinh(ωkβ )


iFint [φ0] = + O(λ 2)

=minusiλ4

intI

d4x

3D0(xx)2 +6φ0(x)2D0(xx)+φ0(x)4 + O(λ 2)




iα thn (x1 xn) =

(δ iF [φ0]


∣∣∣∣φ0=0

)δC(x0








7rarr (D7)




= + O(λ 2) (D8)




= minusλ

intβ

0dτ ∆






4 (k1k2k3k4) =

= minusλ

intβ

0dτ





+ O(λ 2)

rarr minusλ







iGminus10th(xy) =


2φ

2)











nlth(xy)



=(minusiλ )2

2φ


6Gth(xy)3 (D13)

M2th = m2 +

λ

2φ


λ

2φ

2 +λ

2Gth(xx)




2x +M2th)


C+Id4zΠ



th GCIth GIC



th GCIth GIC



(2x +M2

th)


intCd4zΠ

CCth (xz)GCC

th (zy)

minusiint

Id4zΠ

CIth (xz)GIC

th (zy) (D15)




ρ (2x +M2

th)

GCCF (xy) =

int y0

0d4zΠ

CCF (xz)GCC

ρ (zy)

minusint x0

0d4zΠ

CCρ (xz)GCC

F (zy) (D16)

minusint

β

0dτ

intd3zΠ


th ((minusiτz)y)

(2x +M2

th)

GCCρ (xy) =

int y0

x0

d4zΠCCρ (xz)GCC

ρ (zy)



2x +M2th)

GICth (xy) =

int y0

0d4zΠ

ICth (xz)GCC

ρ (zy)

minusint

β

0dτ

intd3zΠ


th ((minusiτz)y)

(2x +M2

th)

GCIth (xy) = minus

int x0

0d4zΠ

CCρ (xz)GCI

th (zy) (D17)

minusint

β

0dτ

intd3zΠ



th)


minusint

β

0dτ

intd3zΠ


th ((minusiτz)y)


Gth(x0y0k) =int


GIIth (k0k) =

intβ



GICth (k0y0k) =

intβ


th (minusiτy0k)



k0 = ωn =2π





th (ωnk)


neiωnτ GIC

th (ωny0k) (D19)



ω2n +k2 +M2

th)


IIth (ωnk)GII

th (ωnk) (D20)

whereint β








) (D21)

where exp(iFψ [ϕ]














i

α0 +int


12


2 (xy)ϕε2(y))




ε1ε22 (xy) =


ε1ε22 (k)


i

α0 +αε0 ϕε(0)+

12


2 (k)ϕε2(minusk))

(D25)



++2 =minusα


+minus2 =minusα


ε1εnn


+minus2 = α

minus+2 Thus αε



φ(x)|x0=0 = Tr(

ρ Φ(x))∣∣∣∣

x0=0 φ(x)|x0=0 = Tr

(ρ partx0Φ(x)

)∣∣∣∣x0=0

(D26)


G(xy)|x0=y0=0 =

Tr(


∣∣∣∣x0=y0=0


Tr(

ρ


])(D27)


)∣∣x0=y0=0



)minus φ(x)φ(y)

∣∣x0=y0=0



2k=0 sum

ε=plusmniαε

1 (D28)

φ(x)|x0=0 =intDϕ

minusipartpartϕ(x)



=12i

(sum

ε=plusmnε iαε


iαε1

)



G(x0y0k)∣∣x0=y0=0 = ξ

2k



2k +

σ2k

4ξ 2k

with

1ξ2k = minus sum


2 (k)







2 (k)

σ2kξ

2k = minus sum


2 (k)

(D30)


k = 1



Γ[φ G] = S[φ ]+i2

Tr lnGminus1 +i2

Tr(Gminus1

0 G)+Γ2[φ G] (D31)




2φ(x)2



=iδ 4S[φ ]


=iδ 3S[φ ]




8

intCd4xG(xx)2 +

(minusiλ )2

12


φ(y)

+(minusiλ )2

48










(D36)


Π(xy) =minusiλ


(minusiλ )2

2φ(x)G(xy)2

φ(y)+(minusiλ )2

6G(xy)3 +O(λ 3) (D37)







λ

2φ(x)2














M(x)2 = m2 +λ

2φ


2φ

2(x)+λ

2G(xx) (D43)






2φ(x)

(GF(xy)2minus 1

4Gρ(xy)2

)φ(y)

+(minusiλ )2

6

(GF(xy)3minus 3

4GF(xy)Gρ(xy)2

)+O(λ 3)


2φ(x)

(2GF(xy)Gρ(xy)

)φ(y) (D45)

+(minusiλ )2

6


4Gρ(xy)3

)+O(λ 3)



∣∣x0=y0 = δ

(3)(xminusy) (D46)


part2x0G(xy) = part

2x0GF(xy)minus i


2x0Gρ(xy)



]= part

2x0GF(xy)minus i




2x +M2(x))

GF(xy) =int y0


int x0

0d4zΠρ(xz)GF(zy)

(2x +M2(x)

)Gρ(xy) =

int y0

x0




G(x0y0k) =int



GF(x0y0k)∣∣x0=y0=0 = ξ

2k



2k +

σ2k

4ξ 2k

Gρ(x0y0k)∣∣x0=y0=0 = 0

partx0Gρ(x0y0k)∣∣x0=y0=0 = 1 (D50)



ξ2k =

n(t = 0k)+ 12

ω(t = 0k) ηk = 0

σ2k

4ξ 4k

= ω2(t = 0k) (D51)


Appendix E





S0[φ ] =int

d4x(

12(partφ)2minus 1

2m2

Bφ2)


d4xλB

4φ(x)4 (E1)


Γ[φ G] = S0[φ ]+i2

Tr lnGminus1 +i2





Trpart 2Sint











B = m2R +δm2 Z2




intxy



S0R[φR] =int

d4x(

12(partφR)2minus 1

2m2

Rφ2R

) (E7)




Tr lnGminus1R +

i2

TrGminus10RGR +Γ

Rint [φRGR] (E8)



ΓRint [φRGR] =

12

intxy


i2









0



ΓRint [φRGR] =

12

intxy


i2




2)δ (xminus y)





4

intxφ


4

intxφ

2R(x)GR(xx)

minus λR +δλ0

8

intxG2



δλ4 δλ2 δλ0




ΠR(xy) =2iδΓR

int

δGR(yx) ΠR(xy) =

iδ 2ΓRint









2δ 3Γint





intabcd


(E16)


intabcd









G(xy) =int



(2π)4δ

(4)(p1 + p2 + p3 + p4)Λ(p1 p2 p3 p4) =int

x1x2x3x4



ΠR(q = q) = ΠR(q = q) = 0

ddq2 ΠR(q = q) =

ddq2 ΠR(q = q) = 0 (E17)



(4)R equiv Z2Γ(4)

Γ(4)(xyuv) =


(E18)




R d

dq2 Gminus1R (q = q) = +1 (E19)









ΠR(xy) = minusi(

δZ02x +δm20 +

λR +δλ0

2GR(xx)

)δ (xminus y)

ΠR(xy) = minusi(

δZ22x +δm22 +

λR +δλ2

2GR(xx)





Z2Γ






R +δZ0k2 +δm20 +

λR +δλ0

2

intq

GR(q)



2

intq

GR(q) (E22)



R (E23)




2

intq




2

intq

G2R(q)VR(q) (E24)



0)

Z = 1

m2B = m2

RminusλB

2

intq

G(q) (E25)

λminus1B = λ

minus1R minus

intq

G2(q)




δλ4 = 3δλ0 (E26)




12

intxy


(minusiλR)2

48

intxy

GR(xy)4 (E27)


ΠR(xy) = minusi(

δZ02x +δm20 +

λR +δλ0

2GR(xx)


R

6G3

R(xy)

(E28)

ΠR(xy) = minusi(

δZ22x +δm22 +

λR +δλ2

2GR(xx)


R

6G3

R(xy)








Z2Γ








R +δZ0k2 +δm20 +

λR +δλ0

2

intq

GR(q)

minus λ 2R

6

intpq




intq


2

intq

G2R(q)VR(q)

+λ 2

R

2

intpq



B =Zminus1(m2

R +δm20)


λB

2

intq

G(q)

minus Zminus4λ 2R

6

intpq



2R

intq


2

intq

G2(q)V (q)

+Zminus4λ 2

R

2

intpq











intq

G2R(q) (E35)

Appendix F



Pdtp f (tp) =

binta

dudtp(u)

duf (tp(u))



d4x =intP

dx0p

intd3x





Cdtc f (tc) = 0



t1intt0

dt f (t)


t1intt3

dt f (t)


Danksagung














Bibliography





1120[hep-ph]











176 BIBLIOGRAPHY



















BIBLIOGRAPHY 177


















178 BIBLIOGRAPHY



















BIBLIOGRAPHY 179




















180 BIBLIOGRAPHY



07070875

















BIBLIOGRAPHY 181



















182 BIBLIOGRAPHY



















BIBLIOGRAPHY 183




















184 BIBLIOGRAPHY




















BIBLIOGRAPHY 185














Introduction



Tracking Solutions







Self-Interactions

Matter Couplings


Summary




Stability












Summary

Conclusions

Conventions

















Acknowledgements

Bibliography


Abstract


Zusammenfassung


Contents

1 Introduction 1








9 Conclusions 131

viii CONTENTS

A Conventions 135







Bibliography 175

Chapter 1

Introduction


2 1 Introduction




1 Introduction 3



4 1 Introduction



Chapter 2




pl M4GUT M4

SUSY or M4elweak




pl












S =int

d4xradicminusg(x)

(minus R

16πG+

12


) (21)


Rmicroν minusR2


microν) (22)



δgmicroν and


(12(partφ)2minusV







2φ +dV (φ)

dφ= 0 (25)



partmicro

radicminusgpart

micro




3M2plH

2 = ρφ +ρB (26)

3M2pl

aa











1 = Ωφ +ΩB


2 = aminus3partt a3



φ +3Hφ +dV (φ)

dφ= 0 (27)















ωφ =pφ

ρφ

=φ 22minusVφ 22+V

(211)


minus1le ωφ le+1






0 M2pl A


pl 10minus10






Γminus1Γminus 1

2

(1+ωB) (212)




φlt


V (φ) =

M4

pl exp(minusλ

φ

Mpl

)for Γ = 1












Λ =O

((H0

Mpl

) 24+α

Mpl

)=O

(10minus

1224+α Mpl

)


Self-adjusting mass






92

Γ

(1minusω

lowastφ

2)

H(t)2 (213)







LYuk =minussumi

Fi(φ) ψiψi (214)




r (215)




(1+8πM2

plyi

mi

y j

m j

) (216)




nm2n sim y2




LGauge =12










16πGminus f (φ)R+

)equivint


16πGeff+

)

where1

16πGeff=

116πG

+ f (φ)






(Geff)today

∣∣∣∣le 02

Chapter 3



S[φ ] =int

d4x(


) (31)






ρ = sumn

pn|n〉〈n| (33)


〈O〉= Tr(ρO) (34)


ρ = |0〉〈0| (35)


H(x0) =int

d3x(

12(Φ(x))2 +


)(36)




HJ(x0) =int

d3x(

12(Φ(x))2 +


)(37)


ρ =1Z

exp(minusβH) (38)




ρ = |0〉J J〈0| (39)



(310)


exp(

iW [J])

=intDϕ exp

(iS[ϕ]+ i

intd4xJ(x)ϕ(x)

) (311)



d4xJ(x)φ(x) (312)


δΓ[φ ]δφ(x)

= 0 (313)





Γ[φ ] = S[φ ]+i2


iΓ1[φ ] = + +

=18

intd4x [minusiV (4)

cl (φ(x))]G0(xx)2 +112

intd4xint


+


Gminus10 (xy) =








(316)




G0(xy) = G0(xy)+int


= G0(xy)+int


+int

d4vint


+



V (4)cl (0)φ(x)2 + (317)




Γ[φ ] =int

d4x(minusVeff (φ)+

Z(φ)2

(partφ)2 +

) (318)



int d4k(2π)4 ln


k2

)+V1(φ)

minusV1(φ) = + +

=18

[minusV (4)

cl (φ)][int d4k

(2π)41


]2

+1


]2 int d4k(2π)4

int d4q(2π)4


+







12

m2φ

2 +13

gφ3 +

14

λφ4 (319)





exp(

iW [JK])

=intDϕ exp

(iS[ϕ]+ iJϕ +

i2

ϕKϕ

) (320)


Jϕ =int


d4xint




δJ(x) (322)


minusφ(x)φ(y)




intd4xint



δΓ[φ G]δφ(x)

=minusJ(x)minusint


=minus12

K(xy) (324)


δΓ[φ G]δφ(x)


= 0 (325)


Γ[φ G] = S[φ ]+i2

Tr lnGminus1 +i2

Tr(Gminus1

0 G)+Γ2[φ G] (326)

iΓ2[φ G] = + + (327)

=18

intd4x [minusiV (4)

cl (φ(x))]G(xx)2 +112

intd4xint


+



δΓ[φ G]δG(yx)


δG(yx) (328)





(330)


Γ[φ G[φ ]] = Γ[φ ] (331)




trunc(xyG) (332)








exp(

iW [J1 Jn])

=intDϕ exp

(iS[ϕ]+ i

n

sumk=1

1k


) (333)







sumk=1




δΓ

δφ(x)= 0

δΓ

δG(xy)= 0

δΓ









G(x1x2) = G12 =2δW [KL]

δK12 V4(x1x2x3x4) =

4δW [KL]δL1234




Γ[GV4] =i2

Tr lnGminus1 +i2

Tr(Gminus1

0 G)+Γ2[GV4] (337)

iΓ2[GV4] =

=18


+1

24

intd4x1234


minus 148

intd4x1234




V4(x1x2x3x4) =int



δΓ[GV4]δV4



iΓ2[G] = iΓ2[GV4] =18


148

intd4xint





Chapter 4







(minusλφMpl




radicΩde3 (41)


Λ2M2 simO(1)








radicGF






M = = 2GMint

k2ltΛ2

d4k(2π)4

1k2 +M2 = 2GM

Λ2

16π2 f1(M2Λ2) (42)

with f1(M2Λ2) = 1+ M2

Λ2 ln(

M2

Λ2+M2






FΛ(k) =


(43)




Λk

(2π)41

k2 +m2 =Λ2

16π2 f1(m2Λ2) (44)


Λk

(2π)4

int d4Λq

(2π)41

(k2 +m2)(q2 +m2)((q+ k)2 +m2)=

Λ2

(16π2)2 f2(m2Λ2)



can be evaluated

=18

V (4)cl (φ)

[int d4Λk

(2π)41


]2

(45)

=18

V (4)cl (φ)

[Λ2


asymp 18

V (4)cl (φ)

[Λ2

16π2 f1(0)]2

= Vcl(φ) middot

λ 4

8M4pl

[Λ2

16π2 f1(0)]2


=1


]2 intint d4Λkd4

Λq

(2π)81



]2 Λ2



]2 Λ2

(16π2)2 f2(0)



middot

λ 6

12M2pl

Λ2

(16π2)2 f2(0)






pl asymp 10minus120










Λ2M2L


Λ

2M2Lminus1







pl(Hinf 1014GeV)2



iΓ2[φ G] =infin

sumL=2

12LL

intd4x(minusiV (2L)

cl (φ(x)))G(xx)L (48)



δG(yx)(49)


sumL=2

L2LL


δ4(xminus y)



4(xminus y) (410)


eff (x)


infin

sumL=2

L2LL



M2eff (x) = exp

[12

G(xx)d2

dφ 2




M2eff = exp

[12

(int d4Λk

(2π)41

k2 +M2eff

)d2

dφ 2

]V primeprimecl(φ)




M2eff (φ) = exp

[12

(int d4Λk

(2π)41

k2 +m2

)d2

dφ 2

]V primeprimecl(φ)

∣∣∣∣∣m2=M2

eff (φ)

(412)


M2eff (xφ) = exp

[12

G(xx)d2

dφ 2


∣∣∣∣∣G(xx)=G(xx M2




2x +M2(x))

G(xy M2(middot)







Γhf [φ ] = Γ[φ G[φ ]]

=int

d4x(


)+

i2

Tr[

ln(2x +M2

eff (xφ))minus iM2

eff (xφ)G[φ ]]

where


12

G(xxφ)d2

dφ 2

]Vcl(φ(x)) (414)



12

int d4Λk

(2π)4

[ln

(k2 +M2

eff (φ)

k2

)minus

M2eff (φ)

k2 +M2eff (φ)

] (415)


Vhf (φ) = exp[

12

(int d4Λk

(2π)41

k2 +m2

)d2

dφ 2

]Vcl(φ)

∣∣∣∣∣m2=M2

eff (φ)

= V (φ m2)∣∣m2=M2

eff (φ)


V (φ m2)equiv exp[

12

(int d4Λk

(2π)41

k2 +m2

)d2

dφ 2

]Vcl(φ) (416)



M2eff (φ) =


∣∣∣∣m2=M2

eff (φ) (417)








=112

[minusV (3)(φ)

]2 int d4Λk

(2π)4

int d4Λq

(2π)41

(k2 +V (2))(q2 +V (2))((q+ k)2 +V (2))+





eff (φ) = k2 +V (2)(φ)


eff (φ)


∣∣∣∣m2=M2

eff (φ) (419)





M2eff (φ) = exp

[12

(Λ2

16π2 f1(m2Λ2))

d2

dφ 2

]V primeprimecl(φ)

∣∣∣∣∣m2=M2

eff (φ)



M2eff (φ) exp

[Λ2

32π2 f1(0)d2

dφ 2


1+O

(M2

eff

Λ2

)



Λk

(2π)4

[ln(

k2 +m2

k2

)minus m2

k2 +m2

]=

m4

16π2 f0(m2Λ2) (420)




Veff (φ) exp[

Λ2

32π2 f1(0)d2

dφ 2

]Vcl(φ) middot

1+O

(M2

eff

Λ2

)+O

(Veff

M4

) (421)







12

M4eff (φ)


2)









infin

suml=2

12(l +1)

[V (l+1)(φ)

]2 Λ2(lminus1)


2)



M2eff (φ) Λ







32π2d2

dφ 2

]Vcl(φ) (423)






to first order

V1minusloop(φ)[

1plusmn Λ2

32π2d2

dφ 2

]Vcl(φ)


Λ2LM2L L = 234




S[φ ] =int

ddx(


) (424)







12

int ddk(2π)4

[ln

(k2 +M2

eff (φ)

k2

)minus

M2eff (φ)

k2 +M2eff (φ)

] (425)


M2eff (φ) =


∣∣∣∣m2=M2


∣∣m2=M2

eff (φ) (426)


V (φ m2)equiv exp[

12

(int ddk(2π)4

1k2 +m2

)d2

dφ 2

]Vcl(φ) (427)




18π

M2eff (φ) =

(V (φ m2)+

18π


)m2=M2

eff (φ) (428)


int ddk(2π)d

1k2 +m2 =


microminus2ε

4π

(1ε


m2 +O(ε))

(429)






V Bcl (φ) = V R

cl (φ)+δVcl(φ)



V (φ m2) = exp[

18π

(1ε


m2 +O(ε))

d2

dφ 2

]V B

cl (φ)


δVcl(φ)equiv(

exp[minus 1

8π

1ε

d2

dφ 2

]minus1)

V Rcl (φ) (430)


V (φ m2) = exp[

18π

lnmicro2

m2d2

dφ 2

]V R

cl (φ) (431)




M2eff (φ) = exp

[1

8πln

micro2

m2d2

dφ 2

]V R

clprimeprime(φ)

∣∣∣∣∣m2=M2

eff (φ)

(432)









=1

(8π)2



6)54

(V (3)(φ)

)2

V (2)(φ) (433)





F =1

(8π)L g(F)prod

kge3

(V (k)(φ)

)Vk

(V (2)(φ)

)Vminus1 (434)







=(

V (φ)+1

8πV (2)(φ)

)+ sum

F

g(F)(8π)L

prodkge3

(V (k)(φ)

)Vk

(V (2)(φ)

)Vminus1







∣∣∣∣m2=M2

eff (φ)







M2eff = exp

[λ 2

8πln

micro2

M2eff

]V R


2V0 exp

[λ 2

8πln

(micro2

M2eff

)minusλφ

] (437)



ln

(M2

eff

micro2

)= ln

(λ 2V0

micro2

)+

λ 2

8πln

(micro2

M2eff

)minusλφ

rArr ln

(M2

eff (φ)

micro2

)=

11+λ 2(8π)

[ln(

λ 2V0

micro2

)minusλφ

]


ln

(M2

eff (φ)

micro2

)= ln

(M2

r

micro2

)minus φ

λminus1 +λ(8π)


Vhf (φ) = exp

[λ 2

8πln

(micro2

M2eff (φ)

)]V R


λminus1 +λ(8π)

]




18π

M2eff (φ) =

(Vr +

18π

M2r

)exp[minusλ φ

]




λminus1 = λ

minus1 +λ(8π)



partφ k

∣∣∣∣m2=M2


[minusλ φ

]


F =1

(8π)L g(F)prod

kge3


)Vk

(λ 2Vhf (φ)

)Vminus1 =1



λ 2

8π

)L

Vhf (φ)



[minusλ φ

] (438)


VR = Vr

(1+

λ 2

8π+sumF

g(F)(

λ 2

8π

)L)

= Vr

(1+

λ 2

8π+0781

(λ 2

8π

)2

+

)




exp[plusmn Λ2

32π2d2

dφ 2

](439)





Vcl(φ) = sumj

Vj exp(minusλ j

φ

Mpl

) (440)


Vj rarr Vj exp

[plusmn

λ 2j Λ2

32π2M2pl

] (441)



Veff (φ) Λ2M2

pl M4pl



quintessence models



Vcl(φ) = sumα

cαφminusα (442)


Veff (φ) = sumα

cαφminusα

Γ(α)

infin

sumL=0

Γ(α +2L)L

(plusmnΛ2

32π2φ 2

)L

= sumα

cαφminusα

Γ(α)

intinfin

0dt tαminus1 exp

(minustplusmn Λ2

32π2φ 2 t2)

(443)



1

101

102

103

104

105

106

107

108

10-4 10-3 10-2 10-1 1

Vef

f(φ

) V

(φ0)

φ φ0


Veff(φ)

Vone-loop(φ)

Vtwo-loop(φ)







cαφminusα

Γ(α)12

Γ(α

2)(

Λ2

32π2φ 2

)minus α

2

= sumα

Γ(α

2 )2Γ(α)

cα

(Λ

4πradic

2

)minusα

= const (445)




Γ(α+22 )

2Γ(α)cα

(Λ

4πradic

2

)minus(α+2)

(446)


M sim

Λ for φ Λ

φ for φ Λ (447)


1

103

106

109

1012

110-110-210-310-410-510-6

0

10

102

103

104

105

106

Vef

f(φ

) V

(φ0)

z (f

or tr

acke

r so

lutio

n)

φ φ0

today

CMB

Λ = φ0 asymp Mpl

V(φ) prop φ- α








α+2α+4




-1

-09

-08

-07

-06

-05

0 02 04 06 08 1

ωφ

Ωφ

toda

y (z

=0)ΛMpl =

0103

05

07

09

11 1315

29




Λmaxα

c1(α+4)α


c1(α+4)α sim

(H2

0 Mα+2pl


pl

)1(α+4)



α

ΛM

pl

02 04 06 08 1 12 14 16 18 2 0

05

1

15

2

25

3

-09

8-0

95

-09

-08 -07

-06



zquant sim

(Mpl10

Λ(4πradic

2α(α +1))

) α+23(1+ωB)



radicG) the track-
























dφφ (450)




V1L(φ) =12

int d4k(2π)4

(sumB


gF ln(k2 +mF(φ)2)

) (451)


I0(m2) equivint d4k

(2π)4 ln(k2 +m2) (452)

Il(m2) equivint d4k

(2π)41


(lminus1)dl

(dm2)l I0(m2)



I0(m2) = 2int m2

dm23

int m23dm2

2

int m22dm2

1 I3(m21) + D0 +D1m2 +D2m4 (453)



0

Ifinite0 (m2m2

0) = 2int m2

m20

dm23

int m23

m20

dm22

int m22

m20

dm21 I3(m2

1)

=1

32π2

(m4(

lnm2

m20minus 3

2

)+2m2m2

0minus12

m40

) (454)


V1L(φ) =12

(sumB


FgF Ifinite

0 (mF(φ)2mF(φ0)2)

) (455)


V1L(φ)Vcl(φ) (456)


V1L(φ) asymp 13


132π2 sum

j

(minus1)2s j g j

m j(φ0)2

(dm2

j

dφ(φ minusφ0)

)3

asymp 196π2 sum


(d lnm2

j



)3

(457)








d lnV primeprimecl

∣∣∣∣∣ 13ln(1+ zmax)


) 13

(459)



2Ωde

3H20

8πG


d lnV primeprimecl

∣∣∣∣∣(

1minusωde

2Ωde

07

) 13 1

3radicg j

(13meVm j(φ0)

) 43

(460)





∆m j

m jasymp

d lnm2j



3ln(1+ z)d lnm2

j



ν

emicro

p b Zt


100

10-5

10-10

10-15

10-20

101210910610310010-3

∆mm

meV




∆me


(1minusωde

2Ωde

07

) 13

(462)



φsim Hminus1


y j =dm j(φ)

dφ



y j =dm j

dφ=

12


d lnm2j


m j

2M

d lnm2j

d lnV primeprimecl



Mpl

M

)(1GeVmpn

) 13(

1minusωde

2Ωde

07

) 13

(463)


β j equivy j

m jMpl=

d lnm j

d(φMpl)


β j ∆m j

m j 4

(meVm j

)43

sim 10minus11(

me

m j

)43

(464)









ξ Rφ22


ξ (micro0) = 0


pl (see section 23)

116πGeff (φ)

=1

16πG+

12

ξ φ2


∆Geff

Geffequiv


=minusξ

2(φ

2(t)minusφ2(t0)

)16πGeff (φ(t))








d4xradicminusg(

12


) (465)


Vcl(φ) = Λ+m2φ

22+λφ44




=int

d4xradicminusg(

12


)

where

V (φ R) = Vcl(φ)+12

ξ Rφ2 +

R16πG

+ ε0R2 (467)

B(φ R) = ε3φ2 + ε4R





d4xradicminusg(

12







2φ

2 +λ (micro)

4φ

4 +12

ξ (micro)Rφ2 +

R16πG(micro)

+ ε0(micro)R2



partVLL

part t=

dΛ

dt+

12

dm2

dtφ

2 +14

dλ

dtφ

4 +12

dξ

dtRφ

2minus R16πG2

dGdt

+dε0

dtR2

=1

64π2

(part 2VLL

partφ 2 minusR6

)2

=1

64π2

(m(micro)2 +

λ (micro)2

φ2 +(

ξ (micro)minus 16

)R)2

part2BLL

part t=

dε3

dt2φ

2 +dε4

dt2R

=1

192π2

(part 22VLL

partφ 2 minus 2R5

)=

1192π2

(λ (micro)

22φ

2 +(

ξ (micro)minus 15

)2R)



dλ

dt=

3λ 2

32π2 dm2

dt=

λm2

32π2

dGdt


6)32π2

dΛ

dt=

m4

64π2

dξ

dt=

λ (ξ minus 16)

64π2 dε0

dt=

(ξ minus 16)2

64π2

dε1

dt= minus 1


dt= minus 1

360 middot32π2

dε3

dt=

λ

12 middot32π2 dε4

dt=

ξ minus 15

6 middot32π2

(469)





116πGeff (φ)

=1

16πG+ f1(φ)




V (φ R) = sumnm

cnmφnRm (470)

B(φ R) = sumnm

cnmφnRm




c00 = Λ c20 =m2

2 c40 =

λ

4 c21 =

ξ

2 c01 =

116πG

c02 = ε0 c20 = ε3 c01 = ε4



32π2


2

(ln


2

)(471)

minus(

1120

Cminus 1360

Gminus 130

2R+162X)

lnXminusR6

micro2 +infin

sumj=3




2V (φ R)partφ2



part


164π2


partφ 2 minus R6

)2


part


1192π2


partφ 2 minus 2R5





part


164π2


partφ 2

)2


part


132π2



partφ 2 minus 16








116πGeff (φ(t))

=1




(ξ0minus

16

)R (474)

=[

92

Γ

(1minusω

lowastφ

2)

+9(

ξ0minus16

)(ωBminus

13

)]H2

prop H2







+bφ +12

ξlowastφ

2










64π2

(V primeprimecl(φ)

)2+O

( t32π2

)2 (475)




)2

+O( t

32π2

)2




ξ0minus16

)2

+O( t

32π2

)2 (476)


∆Geff

Geff=


= minus(


16πGeff (φ(t))

= minus ξ0

2

(φ

2(t)minusφ2(t0)

)16πGeff (φ(t))

minus 132π2 ln

(micro2(φ(t))

micro20


(ξ0minus

16

)2

16πGeff (φ(t))




∆Geff


∆φ 2

M2plminus 1

32π2 ln(

H20

H2BBN


H20

(ξ0minus

16

)2 16πH20

M2pl











part


164π2

[2




partφ 2 minus 16

)2]

f2(φ micro0) = f2(φ) (477)







64π2

(ξ0minus

16

)2

+12

( t32π2

)2V (4)

cl (φ)(

ξ0minus16

)3

+O( t

32π2

)3






f3(φ micro) =13

( t32π2

)3V (6)

cl (φ)(

ξ0minus16

)4

+O( t

32π2

)4


pl since

V (6)cl (φ)R3(RM2



Kinetic term



32π2

[minusV1L(φ)minus

(162X)

lnXmicro2

+(minus 1

12partmicroXpart

microXminus 160

22X)

1X

+infin

sumj=4


]

=int d4x

32π2

[minusV1L(φ)+

112X


(part

4)]=

int d4x32π2

[minusV1L(φ)+

12


4)]






44 Summary






Chapter 5





Toy Model








χ)minus 12

micro2χ |χ|2

minus12





V (|φ |) = V0


)(52)









10-2

10-1

100

101

102

103

|φ(t

)|H

Inf

ΓasympHφ0HInf =

10010101001


01sdot10-3

0 02 04 06 08 1

θ φ =

arg

(φ)

t HInf




lowastχ

2


lowastφ

2




0 φ0χ30 ) it





infin

0dt a3




32ρ340

prop Ainfin (55)



0

02

04

06

08

1

0 02 04 06 08 1 12 14 16

|χ(t

)| a

32 H

Inf

t HInf

exact

WKB





κ asympminusN2

sin(2α0)(

χ0

Hinf

)2

middot


1013GeV

(Hinf

1012GeV

)12

if φ30 χ


17 middot10minus8(

ξ1

7V0

ρ0

)13(

Hinf

1012GeV

)76


30






-4sdot10-10

-2sdot10-10

0

2sdot10-10

4sdot10-10

0 02 04 06 08 1 12 14 16 18

n φχ a

3 [3

2 ρ 0

34 ]

t HInf

g=05

Ainfin

nφsdota3

nχsdota3

nχsdota3 WKB


φ0 HInf

V0

ρ0

10-3 10-2 10-1 100 101 102 103 104 105 106

10-810-710-610-510-410-310-210-1

10-10

10-9

10-8 10-7 10-6 10-5

2sdot10-10

5sdot10-10



53 Stability


|θφ |H

=|nφ |


45glowastS(T )

T 3


2H|φ |2 10minus8 (58)


θ2φ lt

3H +2ϕϕ



Chapter 6



S[φ ] =int

d4x(

12(partφ)2minus 1

2m2

φ2minus λ

4φ

4)

(61)





〈OH(t)〉= Tr(


(62)

= Tr(

ρ T[

exp(

+iint t

tinit


[exp(minusiint t

tinit





)(63)

=

T[exp(minusiint t


for t gt t prime

T[exp(+iint t prime


for t lt t prime


〈OH(t)〉= Tr(

ρ TC

[exp(minusiint

CdtHI(t)

)OI(t)

]) (64)








Φ(0x)|ϕ0〉= ϕ(x)|ϕ0〉 (66)

is given by

Zρ [JK] = Tr(

ρ TC

[exp(

iint

Cd4xJ(x)Φ(x)+

i2

intCd4xint

Cd4yΦ(x)K(xy)Φ(y)

)])=

intDϕ+


langϕminus0

∣∣∣∣TC

[exp(

i JΦ+i2

ΦKΦ

)]∣∣∣∣ϕ+0rang



Zρ [JK] =intDϕ+



Dϕ exp(

iS[ϕ]+ i Jϕ +i2

ϕ Kϕ

)


(iS[ϕ]+ i Jϕ +

i2

ϕ Kϕ

) (67)





iα0 + iα1ϕ +i2

ϕα2ϕ

) (68)






Γ[G] =i2

Tr lnGminus1 +i2

Tr(Gminus1

0 G)+Γ2[G] (69)

iΓ2[G] =minusiλ

8

intCd4xG(xx)2 +

(minusiλ )2

48

intCd4xint

Cd4yG(xy)4





δ


2K(xy) (610)




=minusiλ

2G(xx)δ 4


6G(xy)3 (612)





λ

2G(xx)








)GF(xy) =

int y0

0d4zΠF(xz)Gρ(zy)

minusint x0


(2x +M2(x)

)Gρ(xy) =

int y0

x0

d4zΠρ(xz)Gρ(zy)





6G(xy)3 (616)



ω2(tk) =


GF(x0y0k)

)∣∣∣∣∣x0=y0=t


(617)
















Dϕ exp(

iint

Id4xL(x)

) (618)


Zβ [JK] = Tr(

ρ TC+I

[exp(

iint

C+Id4xJ(x)Φ(x)+

i2

intC+I

d4xint


)])=

intDϕ exp

(iint


2

intC+I

d4xint


) (619)






Gth(xy) =int



The meaning ofint






section D12)





intC+I





=minusiλ

2Gth(xx)δ 4


6Gth(xy)3 (623)





S[φ ] =int

d4x(

12(partφ)2minus 1

2m2

Bφ2minus λB

4φ

4)

(624)



B = m2R +δm2 Z2

λB = λR +δλ Z = 1+δZ (625)





λR +δλ0

2

intqGthR(q)+

λ 2R

6

intpq




Gminus1T0

(k = k) = k2 +m2R

ddk2 Gminus1

T0(k = k) = +1 (627)





T0(k)minus∆Π(k)


2

[int T

qGT (q)minus

int T0

qGT0(q)

](628)

+λ 2

R

6

[int T


int T0


]





int Tq equiv

int T0q +



that

∆Π(k) =12

[int T0

q∆G(q)+

int∆T

qGT (q)

]ΛT0(qk)+F(k) (630)





q∆Π(q)


2G2

T0(q)ΛT0(qk)

]=

12

[int T0

q∆

2G(q)+int

∆T

qGT (q)

]ΛT0(qk)+F(k)



int T0

q∆Π(q)


2G2


2

int T0

kΛT0(qk)G2

T0(k)VT0(k p)

]=

=12

int T0

k

[int T0

q∆

2G(q)+int

∆T

qGT (q)

]ΛT0(qk)+F(k)

(δ (kminus p)+

12

G2T0

(k)VT0(k p))




intkΛR(qk)G2

thR(k)VR(k p) (632)


∆Π(p) =12

int T0

q

[∆

2G(q)+F(q)G2T0

(q)]VT0(q p)+

12

int∆T

qGT (q)VT0(q p)




Chapter 7













F [ϕ] = α0 +int

Cd4xα1(x)ϕ(x)+

12


+13





0n) (73)


F [ϕ] = α0 +int


12

intd3xint






n (x1 xn))lowast

(75)



int d3k1



(2π)3δ

3(k1 + +kn) iαε1εnn (k1 kn) (76)



(i

S[ϕ]+ Jϕ +12






Γ[φ G] = S[φ ]+i2

Tr lnGminus1 +i2

Tr(Gminus1

0 Gminus1)+ Γ2[φ G] (77)



Γ[φ G] = S[φ ]+i2

Tr lnGminus1 +i2

Tr(Gminus1





ΓnG[φ Gα3α4 ] = F3[φ α3α4 ]+12

Tr(

δ 2F3

δφδφG)

+Γ2nG[φ Gα3α4 ] (79)

The 2PI functional




= iαn(x1 xn)+int


+12





δφ(x)δφ(y) (712)




0n) (713)




ΓnG[φ Gα3 = 0α4 = 0 ] = 0 (714)


iΓ2[Gα4] =18

intC

d4x1234


]G(x1x2)G(x3x4)

+148

intC

d4x1234d4x5678


]G(x1x5)times (715)


]










δG(yx)=



δG(yx) (717)


Π(xy) = ΠG(xy)+Π

nG(xy) (718)


Gnonminusloc(xy)

ΠnG(xy) = Π


surface(xy)









where


Πλα(xy) = Πε










6


4 (x1x2x3y)

+minusiλφ(x)

2


3 (x1x2y) (721)






M(x)2 = m2 +λ

2φ


λ

2φ

2(x)+λ

2G(xx) (723)



(minusiλ )2

6G(xy)3 M(x)2 = m2 +

λ

2G(xx)



6




+16




3 (x1x2x3) = 0







Cd4z[Π

Gnonminusloc(xz)+Π


]G(zy)





(part

2x0 +k2 +M2(x0)

)GF(x0y0k) =

int y0

0dz0


minusint x0

0dz0



+14


2x0 +k2 +M2(x0)

)Gρ(xy) =

int y0

x0


where


(x0k)+Πminusλα

(x0k)


+λα


(x0k))

(727)


(2x +M2(x)

)GF(xy) =

int y0

0d4zΠF(xz)Gρ(zy)

minusint x0


minus λ

6V nG


V nG3 (xxy) (

2x +M2(x))

Gρ(xy) =int y0

x0

d4zΠρ(xz)Gρ(zy)

where


intC



intC




















iGminus10th(xy) =

(minus2xminusm2)







2ωk

cos(ωk(x0minus y0)

) (731)

G0ρ(x0y0k) =1


)for x0y0 isin C+I


nBE(ωk) =1

eβωk minus1 ωk =

radicm2 +k2


0th GCI0th GIC



GCC0th(xy) = GCI

0th(xy) =

GII0th(xy) = GIC

0th(xy) =

minusiλint

Cd4x = minusiλ

intId4x = minusiλ

intC+I

d4x =

(732)



6

intC+I

d4uint


=










GCC0F(0y0k) =

intCdz0

δs(z0)GCC0th(z

0y0k) (733)


intCdz0

δa(z0)GCC0th(z

0y0k) (734)

where


)


) (735)



intCdz0


0y0k)

=(736)




=

(GII


)δs(z0)+


δa(z0)

= (738)


GCI0th(y

0minusiτk) =int

Cdz0 GCC

0th(x0z0k)∆T

0 (z0minusiτk)

= (739)





∆0(vz) =int d3k




∆0(ωnz0k) =

(GII

0th(ωnk)G0th(00k)

)δs(z0)+

2iωnGII0th(ωnk)

δa(z0) (740)


time then becomes

GIC0th(ωny0k) =

intCdz0


0y0k) (741)


= equiv equiv th

0L



(minusiλ )2

6

intCd4uint


=(minusiλ )2

6

intCd4uint

Cd4z1

intCd4z2

intCd4z3


Id4v∆

T0 (z1v)∆T

0 (z2v)∆T0 (z3v)∆0(vz4)

]G0th(z4y)

equiv minusiλ6

intCd4uint

Cd4z1

intCd4z2

intCd4z3


αth40L(z1z2z3z4)

]G0th(z4y)




th

0L= equiv








th41L(z1z2z3z4) +

th=

th

0L+

th

1L+





+int

Cdw0

intCdz0




0(minusiτ primek)

= +

(742)


= equiv =

= + + +

= + + +





S0(xy) = =

+ + +

+ + +









GCCth (xy) = GCI

th (xy) =

GIIth (xy) = GIC

th (xy) =

minusiλint

Cd4x = minusiλ

intId4x = minusiλ

intC+I

d4x =

(744)




1I(x0) =














mth GCImth GIC



GIImth(xy) = GII




2x +m2B)

GCImth(xy) =

(2x +m2

B)

GCCmF(xy) =

(2x +m2

B)

GCCmρ(xy) = 0 (750)









th)


intβ

0dτprimeΠ




β

0dτ eminusiωnτ

part2τ GIC



mth)(y0k)



mth)(y0k) =

[(iωn GIC


]τ=β

τ=0


2n +k2 +M2

th)




mth)(y0k)


2y0 +k2 +m2

B

)disc(GIC

mth)(y0k) =

(part

2y0 +k2 +m2

B

)disc(partτGIC

mth)(y0k) = 0












disc(GICmth)(y


∣∣τ=β




∣∣τ=β

τ=0 =GCC

mF(0y0k)Gth(00k)


GICmth(ωny0k) =

(GII

th (ωnk)Gth(00k)

)GCC

mF(0y0k)minus(

iωnGIIth (ωnk)

)GCC

mρ(0y0k)

=int

Cdz0


0y0k) (753)



∆m(ωnz0k) =(

GIIth (ωnk)

Gth(00k)

)δs(z0)+

(2iωnGII

th (ωnk))

δa(z0)



=










C+Id4uint

C+Id4vGth(xu)

[1minus1I(u0)1I(v0)




mth(x0ωnk)+

intC+I

du0int

Cdv0(


0ωnk))

minus iint

Cdu0 GCC


Next GCImth(x

0ωnk) and GCImth(v


GCIth (x0ωnk)

=int

Cdz0[

GCCmth(x

0z0k)+int

C+Idu0

intCdv0(


0z0k))]

∆Tm(z0ωnk)

minus iint

Cdu0 GCC


=int

Cdz0[

GCCth (x0z0k)minus

intCdu0int

Idv0(


mth(v0z0k)

)]∆

Tm(z0ωnk)

minus iint

Cdu0 GCC


=int

Cdz0 GCC

th (x0z0k)

∆


th (ωnk)

minusint

Cdu0int

Idv0

Πth(z0v0k)GICmth(v

0u0k)∆Tm(u0ωnk)




Cdu0int

Idv0

Πth(z0v0k)GICmth(v


= minusiT suml

intCdu0


m(u0ωnk)

= minusiT suml


m(ωnk)

= minusiT suml


GIIth (ωnk)

Gth(00k)


GCIth (x0ωnk) =

intCdz0 GCC

th (x0z0k)

∆


th (ωnk)

minus iT summ


GIIth (ωnk)

Gth(00k)

=

intCdz0 GCC

th (x0z0k)∆T (z0ωnk) (756)


∆T (z0ωnk) = ∆


mΠth(z0ωmk)

[δnm

TGII

th (ωnk)minusGII


Gth(00k)

]equiv ∆




D(ωnωmk) =δnm

TGII

th (ωnk)minusGII


Gth(00k)(758)

=δnm

TGII


m(ωmk)

=δnm

TGII

th (ωnk)minusint

Cdw0

intCdz0


=δnm

TGII

th (ωnk)minusint

Cdw0

intCdz0





D(ωnωmk) = 0 (759)



th + Πnlth(z


T summ


Πnlth(z

0ωmk)D(ωmωnk)






intCdz0 GCC


=


intCdz0


=




th(v0z0k) (761)




th(v0z0k)

= = +




sm(minusiτk) =







intCdw0

intCdz0




= +

= +







= +





nlth(v

0z0k)∣∣GIC


(765)


= + +

+

+










6

intC+I

d4uint


=

= + + +

+ + +




=

+

+




th

0L 2PI= equiv (767)


intId4vint

Id4w∆

Tm(z1v)∆T



th

0L 2PI= equiv (768)





part2x0 +k2 +M2

th)


minus iint

Cdz0 [

ΠGthnl(x

0z0k)+ΠnGthnl(x


Gth(z0y0k)

where ΠGthnl(x






part2x0 +k2 +M2

th)


C+Idz0

Πnlth(x

0z0k)Gth(z0y0k)


intC+I

dz0Π

nlth(x

0z0k)Gth(z0y0k) =


Πnlth(x


Idv0

Πnlth(x

0v0k)int

Cdz0


=int

Cdz0[int

Idv0

Πnlth(x

0v0k)(

∆m(v0z0k)+int

Idw0 D(v0w0k)Πnl

th(w0z0k)

)]Gth(z0y0k)


ΠGthnl(x

0z0k) = Πnlth(x


ΠnGthnl(x

0z0k) =int

Idv0int

Idw0

Πnlth(x




Idv0

Πnlth(x






ΠGthnl(x

0z0k) =

ΠnGthnl(x

0z0k) = (771)

iΠthλα(x0z0k) =


Πthλα =infin

sumk=0

Π(k)thλα

ΠnGthnl =

infin

sumk=0

Π(k)nGthnl (772)

where

Π(0)thλα



Π(k)thλα




(x0z0k)



iΠ(0)thλα

(x0z0k) = =th

0L 2PI(773)


iΠ(1)thλα

(x0z0k) =

iΠ(2)thλα

(x0z0k) =




th

0L2PI



iΠ(0)thλα

(xz) =minusiλ

6


40L2PI(x1x2x3z) (774)


part2x0 +k2 +M2

th)



∣∣∣x0=y0=0

(part

2x0 +k2 +M2

th)

Gthρ(x0y0k)∣∣x0=y0=0 = 0 (775)


Πthλα(x0z0k)


ΠnGthnl(x

0z0k)



(k)thλα










R +δm20 +

λR +δλ0

2GthR(xx)


C(xminus y) (777)

minus iint

Cd4z[Π

GthnlR(xz)+Π


]GthR(zy)



ΠGthnlR(xz) = Z Π

Gthnl(xz) = Π

nlthR(xz)



nGthnl(xz) =

intId4vint

Id4wΠ

nlthR(xv)DR(vw)Πnl



Id4vΠ

nlthR(xv)∆m(vz)



ΠnlthR(xy) =

(minusiλR)2

6GthR(xy)3


iΠ(0)thλαR(xz) =

minusiλR

6


40L2PIR(x1x2x3z) (779)





Chapter 8







Gminus1R (xy) = i

(2x +m2

R)


ΠR(xy) = minusi(

δZ02x +δm20 +

λR +δλ0

2GR(xx)

)δ

4C(xminus y)+

(minusiλR)2

6GR(xy)3










(iS[ϕ]+

i2

ϕ Kϕ +i

4L1234ϕ1ϕ2ϕ3ϕ4

) (82)



iα0 +i2

ϕα2ϕ +i

4(α4)1234ϕ1ϕ2ϕ3ϕ4

) (83)




εn=plusmnα




intrarrintC



δ


14

L(xyzw) (85)


δ

δG(xy)Γ

L[G] =minus12

K(xy) (86)


ΓL[G] = Γ[GV4]+

14




δΓ

δV4=minusα4


4C(x1minus x2)δ 4





V4(x1x2x3x4) =int


4 + iAnG4 )(y1y2y3y4)]


4 (x1x2x3x4) (87)


(2x +M2(x)

)GF(xy) =

int y0


int x0

0d4zΠρ(xz)GF(zy)

minus λ

6V nG

4 (xxxy) (88)(2x +M2(x)

)Gρ(xy) =

int y0

x0

d4zΠρ(xz)Gρ(zy)


823 Renormalization



(2x +δZ02x +m2

R +δm20 +

λR +δλ0

2GR(xx)

)GFR(xy)

=int y0


int x0


minus λR

6V nG

4R(xxxy) (89a)

(2x +δZ02x +m2

R +δm20 +

λR +δλ0

2GR(xx)

)GρR(xy)

=int y0

x0






6GR(xy)3



V nG4R(xxxy) =



V nG4R(xxxy) =







minus λR

6V nG

4R(xxxy)equivint


where

iΠλαR(xy) =16








0001

001

01

1

0 5 10 15 20 25 30 35 40 45 50

G(t

0k

=0)

t mR

GF(t0k)

Gρ(t0k)

-06-04-02

0 02 04 06 08

1 12 14 16

0 05 1 15 2 25 3 35 4

Πλα

(tk

=0)

t mR

ΠλαF(tk)

Πλαρ(tk)














V nG4R(x1x2x3x4)|x0

i =0 = minusiλR


i =0




(δm2

0 +m2R +

λR +δλ0

2GthR(xx)

)]GthR(xy)|x0y0=0

minus Zminus1 λR

6(minusiλR)




neiωnτ

[k2 +Zminus1

(δm2

0 +m2R +

λR +δλ0

2

intqGthR(q)

minus λ 2R

6

intpq


GthR(ωnk)|τrarr0

= minusT sumn





neiωnτ

[k2 +Zminus1

(Gminus1

thR(ωnk)minusZk2)]

GthR(ωnk)|τrarr0

= minusT sumn


2n GthR(ωnk)

]τrarr0




Outlook








(part

2x0 +k2 +M2(x0)

)GF(x0y0k) =

int y0

0dz0



minusint x0

0dz0







m2B = m2

Rminusλ

2

int d3p(2π)3

1

2radic

m2R +p2

reg


M2(x0) = m2B +

λ

2



6



)


6



)



(2π)3d3q










λB

2

intq

G(q)minus λ 2R

6Z4

intpq


V (k) = λBminusλB

2

intq

G2(q)V (q) (814)

minus λ 2R

Z4

intq

G(q)G(kminusq)+λ 2

R

2Z4

intpq






Zd



R (815)

Z2Vvac(k = 0) = λR






4 αaaaa4 αssaa




M2 (x0) = m2B +

λB

2

int d3p(2π)3 GF(x0x0p) (816)



6Z4



)


6Z4



)



6Z2



4](x00k)

)


6Z2


4](x00k)minus 1


4](x00k)

)



=int d3p

(2π)3d3q








1

12

14

16

18

2

22

24

0 10 20 30 40 50 60 70 80 90 100

mR

min

it

iteration

14

15

16

17

18

19

20

21

1 10

V(k

)

kminit

0th iteration

10th iteration

20th iteration

30th iteration




m2th = Z Gminus1

th (k = 0) (817)





001

01

1

10

100

1000

0 05 1 15 2

δλ

λR24


asmR = 025

asmR = 0125 01

1

10

100

1000

0 05 1 15 2

|δm

2 mR

2 |

λR24

Nonpert 3-loop

Pert 1-loop







partx0GF(x0y0k)∣∣x0=y0=0 = 0 (818)



where


nGth(ωnk)

ωth(k)2 =(



=T sumn

(1minusω2

n Gth(ωnk))

Gth(k) (819)




Z2

βint0





Gth(k)=






int β

0 rarr 2int β2







03

04

05

06

07

08

001 01 1 10 100

GF(t

tk)

t mR

k = 0

k = mR

k = 2mR

500 1000 1500 2000t mR

KB Gauss

KB Non-Gauss

ThQFT



Etotal(t) =int d3k

(2π)3

[12


B +λB

4


)GF(x0y0k)|x0=y0=t

minus 14

int t

0dz0 (


minus 14


14


+ const (820)





s



-03-02-01

0 01 02 03 04

001 01 1 10 100

microm

R

t mR

2

21

22

23T

mR

500 1000 1500 2000t mR

KB Gauss

KB Non-Gauss

Thermal Eq

0

1

0 8

f BE(n

)

ωmR0

1

0 8

f BE(n

)

ωmR



nfit(tk) =[

exp(


)minus1]minus1





=14



24Z2 V nG4 (xxxx)

∣∣x0=0

=λ 2

R

24

intkpq








15

155

16

165

17

175

18

185

19

001 01 1 10 100

mth

(t)

mR

t mR


18

10 100



mAth ω(tk = 0) =


GFR(x0y0k)

∣∣∣x0=y0=t

ωth(k = 0) =radic


∣∣∣τ=0

mBth

(limsrarrinfin


2 tminus sprime2 k)


)minus12




13

14

15

16

1 15 2 25 3 35

mth

mR

(as mR)-1

T = 15 mR

14

15

16

17

mth

mR

T = 17 mR


16

17

18

19m

thm

RT = 20 mR

001 01 1 10 100t mR











0

05

1

15

2

25

3

35

001 01 1 10 102 103 104

GF(t

tk=

0)

t mR

A24 A18

E18

E24





A18 λ = 18 m2B =minus687 m2

R E18 λR = 18 λB = 3718 m2B =minus1439 m2

R

A24 λ = 24 m2B =minus949 m2

R E24 λR = 24 λB = 6343 m2B =minus2514 m2

R



095

1

105

0 1 2 3 4 5 6 7GF(t

tk)

GF(0

0k

)

kmR

t mR = 2000

095

1

105

GF(t

tk)

GF(0

0k

)

t mR = 10

095

1

105

GF(t

tk)

GF(0

0k

)

t mR = 05

095

1

105

GF(t

tk)

GF(0

0k

)

t mR = 00 KB Gauss

095

1

105

GF(t

tk)

Gth

(k)

t mR = 20

0 1 2 3 4 5 6 7

095

1

105

kmR

t mR = 2000

095

1

105t mR = 10

095

1

105t mR = 05

095

1


095

1

105t mR = 20




09 1

11 12

0 1 2 3 4 5 6 7

GF(t

tk)

Gth

(k)

kmR

t mR = 2000

09 1

11 12

GF(t

tk)

Gth

(k)

t mR = 10

09 1

11 12

GF(t

tk)

Gth

(k)

t mR = 05

09 1

11 12

GF(t

tk)

Gth

(k)

t mR = 00 KB Gauss

09 1

11 12

GF(t

tk)

Gth

(k)

t mR = 20

0 1 2 3 4 5 6 7

09 1 11 12

kmR

t mR = 2000

09 1 11 12

t mR = 10

09 1 11 12

t mR = 05

09 1 11 12


09 1 11 12

t mR = 20



84 Summary 129

84 Summary






Chapter 9

Conclusions




132 9 Conclusions






9 Conclusions 133


Appendix A

Conventions


Rαmicroνλ = +

(partνΓ

α


αmicroν +Γ

η

microλΓ

αην minusΓ

η

microνΓα

ηλ

)


Γαmicroν =

12


)


microαλ


micro


Appendix B




S[φ ψ j] =int

d4x(


) (B1)




fk(φ)OSMk (B2)


exp(

iW [J])

=intDϕ

int (prod

jDψ j

)exp(

iS[ϕψ j]+ iint

d4xJ(x)ϕ(x))

(B3)


exp(


prodjDψ j



exp(

iΓ[φ ])

= exp(

iW [J]minus iint

d4xJ(x)φ(x))

=intDϕ exp

(iSeff [ϕ]+ i


)






radicminusg


d4xradicminusg(


) (B5)






Gminus10 (xy) =


= i

(2x +

δ 2V (φ R)δφ 2

∣∣∣∣φ(x)R(x)

)δ

4(xminus y) (B6)



Tr lnGminus10 (B7)



lndet A (B8)

with the operator


δφ 2

∣∣∣∣φ(x)R(x)

(B9)




mln

λm

micro2 =12i



ipart



K(xys) = summ



ζA(ν) = summ

iΓ(ν)

infinint0


Γ(ν)

infinint0

ds(is)νminus1int

d4xK(xxs) (B11)


infin


K(xys) =i∆12


(minusσ(xy)

2isminus is

(X(y)minus R(y)

6

)) (B12)







partxmicropartyν

] (B13)


infinj=0(is)


ζA(ν) =i

Γ(ν)

int d4x16π2

radicminusg

infin

sumj=0



= iint d4x

16π2

radicminusg



(XminusR6)1minusν

νminus1


sumj=3



)



32π2


X2

2

(ln

Xmicro2 minus

32

)+ g1(xx)X

(ln

Xmicro2 minus1

)minusg2(xx) ln

Xmicro2 +

infin

sumj=3

g j(xx)( jminus3)

X jminus2

]

(B14)


g0(xx) = 1 (B15)

g1(xx) = 0

g2(xx) =1


302R+

162X

=1

120Cminus 1

360Gminus 1

302R+

162X






0 =d


(part


βNpart

partcN+sum

i

intd4xγiφi(x)

δ

δφi(x)

)Γ[φi]






part


(sumN

βN(micro)part

partcN+sum

i


δ

δφi(x)

)ΓLL[φi micro]








part



= minus

(sumN

βNpart

partcN+sum

i

intd4xγiφi(x)

δ

δφi(x)

)S[φi]

= minusint



nmβ nm2(φ nRm)

+ γφ φ


partφ

)+ γgmicroν

gmicroν

(12


δgmicroν

)]



part



= minus part

part ln micro20


=int d4x

32π2


2minus 1

120Cminus 1

360Gminus 1

302R+

162X]

= minusint d4x

32π2

radicminusg

[12 sum

nm

(n

sumk=0

m

suml=0

dkldnminuskmminusl

)φ

nRmminus 1120

Cminus 1360

G

+16 sum

nm


15

δn0δm1

)2(φ nRm)

]


βnm =1

64π2

n

sumk=0

m

suml=0

dkldnminuskmminusl

β nm =1

192π2


15

δn0δm1

) (B17)


1120


1360








part


164π2


partφ 2 minus R6

)2

part


1192π2


partφ 2 minus R5

) (B19)




d4xradicminusg(

12


)







0 ) yields [60]


∣∣∣∣∣t=ln


micro20







Appendix C






(C1)


G(xy) = G0(xy)+int

d4uint



G(0)(xy) = G0(xy)


d4uint


= G0(xy)+int

d4uint


+int

d4uint

d4vint

d4zint






Π(0) = Π[φ G0]

Π(k) = Π[φ G(k)] = Π[φ G0

infin

sumn=0

(Π(kminus1)G0)n]



Π(k)








M2eff (xφ) = exp

[12

G(xx)d2

dφ 2


∣∣∣∣∣G(xx)=G(xx M2




2x +M2(x))

G(xy M2(middot)




Γhf [φ ] = Γ[φ G[φ ]] (C5)

=int

ddx(


)+

i2

Tr[

ln(2x +M2

eff (xφ))minus iM2

eff (xφ)G[φ ]]

where


12

G(xxφ)d2

dφ 2

]Vcl(φ(x)) (C6)



12

G(xx M2(middot)

) d2

dφ 2

]Vcl(φ(x)) (C7)


M2eff (xφ) =



eff (middot φ)






iΓnotad[φ ] = +

=112

intddxint







eff (middot φ)




Derivation






infin

sumn=0











12

Ghf (xx)d2

dφ 2







li=1 N(γi)

F =1

N(F)F (C12)


K(F) =


prodi=1

S(γi)

∣∣∣∣∣ |S(F)| (C13)






Γnotad[φ ] =i2



where


δGhf

(GminusGhf

) (C15)


G = Ghf

infin

sumn=0













Appendix D




D11 Thermal State


ρ =1Z

exp(minusβH)





Dϕ exp(

iint

Id4xL(x)

)C+I

exp

(i

infin

sumn=0


)C+α

(D1)







Zβ [J] = Tr(

ρ TC+I

[exp(

iint

C+Id4xJ(x)Φ(x)

)])=intDϕ exp

(iint


)



∣∣∣∣∣J=0

(D2)




Tr(


)= Tr

(eminusβHeτH


)= Tr

(eminusτH


)= Tr

(eminusβH










iS0[φ0]]

=N0 exp[

iint

Id4x(

12(partφ0)2minus 1

2m2

φ20

)]









ξ2k =

nBE(ωk)+ 12

ωk ηk = 0

σ2k

4ξ 4k


1eβωk minus1

(D4)











ωk sinh(ωkβ )


iFint [φ0] = + O(λ 2)

=minusiλ4

intI

d4x

3D0(xx)2 +6φ0(x)2D0(xx)+φ0(x)4 + O(λ 2)




iα thn (x1 xn) =

(δ iF [φ0]


∣∣∣∣φ0=0

)δC(x0








7rarr (D7)




= + O(λ 2) (D8)




= minusλ

intβ

0dτ ∆






4 (k1k2k3k4) =

= minusλ

intβ

0dτ





+ O(λ 2)

rarr minusλ







iGminus10th(xy) =


2φ

2)











nlth(xy)



=(minusiλ )2

2φ


6Gth(xy)3 (D13)

M2th = m2 +

λ

2φ


λ

2φ

2 +λ

2Gth(xx)




2x +M2th)


C+Id4zΠ



th GCIth GIC



th GCIth GIC



(2x +M2

th)


intCd4zΠ

CCth (xz)GCC

th (zy)

minusiint

Id4zΠ

CIth (xz)GIC

th (zy) (D15)




ρ (2x +M2

th)

GCCF (xy) =

int y0

0d4zΠ

CCF (xz)GCC

ρ (zy)

minusint x0

0d4zΠ

CCρ (xz)GCC

F (zy) (D16)

minusint

β

0dτ

intd3zΠ


th ((minusiτz)y)

(2x +M2

th)

GCCρ (xy) =

int y0

x0

d4zΠCCρ (xz)GCC

ρ (zy)



2x +M2th)

GICth (xy) =

int y0

0d4zΠ

ICth (xz)GCC

ρ (zy)

minusint

β

0dτ

intd3zΠ


th ((minusiτz)y)

(2x +M2

th)

GCIth (xy) = minus

int x0

0d4zΠ

CCρ (xz)GCI

th (zy) (D17)

minusint

β

0dτ

intd3zΠ



th)


minusint

β

0dτ

intd3zΠ


th ((minusiτz)y)


Gth(x0y0k) =int


GIIth (k0k) =

intβ



GICth (k0y0k) =

intβ


th (minusiτy0k)



k0 = ωn =2π





th (ωnk)


neiωnτ GIC

th (ωny0k) (D19)



ω2n +k2 +M2

th)


IIth (ωnk)GII

th (ωnk) (D20)

whereint β








) (D21)

where exp(iFψ [ϕ]














i

α0 +int


12


2 (xy)ϕε2(y))




ε1ε22 (xy) =


ε1ε22 (k)


i

α0 +αε0 ϕε(0)+

12


2 (k)ϕε2(minusk))

(D25)



++2 =minusα


+minus2 =minusα


ε1εnn


+minus2 = α

minus+2 Thus αε



φ(x)|x0=0 = Tr(

ρ Φ(x))∣∣∣∣

x0=0 φ(x)|x0=0 = Tr

(ρ partx0Φ(x)

)∣∣∣∣x0=0

(D26)


G(xy)|x0=y0=0 =

Tr(


∣∣∣∣x0=y0=0


Tr(

ρ


])(D27)


)∣∣x0=y0=0



)minus φ(x)φ(y)

∣∣x0=y0=0



2k=0 sum

ε=plusmniαε

1 (D28)

φ(x)|x0=0 =intDϕ

minusipartpartϕ(x)



=12i

(sum

ε=plusmnε iαε


iαε1

)



G(x0y0k)∣∣x0=y0=0 = ξ

2k



2k +

σ2k

4ξ 2k

with

1ξ2k = minus sum


2 (k)







2 (k)

σ2kξ

2k = minus sum


2 (k)

(D30)


k = 1



Γ[φ G] = S[φ ]+i2

Tr lnGminus1 +i2

Tr(Gminus1

0 G)+Γ2[φ G] (D31)




2φ(x)2



=iδ 4S[φ ]


=iδ 3S[φ ]




8

intCd4xG(xx)2 +

(minusiλ )2

12


φ(y)

+(minusiλ )2

48










(D36)


Π(xy) =minusiλ


(minusiλ )2

2φ(x)G(xy)2

φ(y)+(minusiλ )2

6G(xy)3 +O(λ 3) (D37)







λ

2φ(x)2














M(x)2 = m2 +λ

2φ


2φ

2(x)+λ

2G(xx) (D43)






2φ(x)

(GF(xy)2minus 1

4Gρ(xy)2

)φ(y)

+(minusiλ )2

6

(GF(xy)3minus 3

4GF(xy)Gρ(xy)2

)+O(λ 3)


2φ(x)

(2GF(xy)Gρ(xy)

)φ(y) (D45)

+(minusiλ )2

6


4Gρ(xy)3

)+O(λ 3)



∣∣x0=y0 = δ

(3)(xminusy) (D46)


part2x0G(xy) = part

2x0GF(xy)minus i


2x0Gρ(xy)



]= part

2x0GF(xy)minus i




2x +M2(x))

GF(xy) =int y0


int x0

0d4zΠρ(xz)GF(zy)

(2x +M2(x)

)Gρ(xy) =

int y0

x0




G(x0y0k) =int



GF(x0y0k)∣∣x0=y0=0 = ξ

2k



2k +

σ2k

4ξ 2k

Gρ(x0y0k)∣∣x0=y0=0 = 0

partx0Gρ(x0y0k)∣∣x0=y0=0 = 1 (D50)



ξ2k =

n(t = 0k)+ 12

ω(t = 0k) ηk = 0

σ2k

4ξ 4k

= ω2(t = 0k) (D51)


Appendix E





S0[φ ] =int

d4x(

12(partφ)2minus 1

2m2

Bφ2)


d4xλB

4φ(x)4 (E1)


Γ[φ G] = S0[φ ]+i2

Tr lnGminus1 +i2





Trpart 2Sint











B = m2R +δm2 Z2




intxy



S0R[φR] =int

d4x(

12(partφR)2minus 1

2m2

Rφ2R

) (E7)




Tr lnGminus1R +

i2

TrGminus10RGR +Γ

Rint [φRGR] (E8)



ΓRint [φRGR] =

12

intxy


i2









0



ΓRint [φRGR] =

12

intxy


i2




2)δ (xminus y)





4

intxφ


4

intxφ

2R(x)GR(xx)

minus λR +δλ0

8

intxG2



δλ4 δλ2 δλ0




ΠR(xy) =2iδΓR

int

δGR(yx) ΠR(xy) =

iδ 2ΓRint









2δ 3Γint





intabcd


(E16)


intabcd









G(xy) =int



(2π)4δ

(4)(p1 + p2 + p3 + p4)Λ(p1 p2 p3 p4) =int

x1x2x3x4



ΠR(q = q) = ΠR(q = q) = 0

ddq2 ΠR(q = q) =

ddq2 ΠR(q = q) = 0 (E17)



(4)R equiv Z2Γ(4)

Γ(4)(xyuv) =


(E18)




R d

dq2 Gminus1R (q = q) = +1 (E19)









ΠR(xy) = minusi(

δZ02x +δm20 +

λR +δλ0

2GR(xx)

)δ (xminus y)

ΠR(xy) = minusi(

δZ22x +δm22 +

λR +δλ2

2GR(xx)





Z2Γ






R +δZ0k2 +δm20 +

λR +δλ0

2

intq

GR(q)



2

intq

GR(q) (E22)



R (E23)




2

intq




2

intq

G2R(q)VR(q) (E24)



0)

Z = 1

m2B = m2

RminusλB

2

intq

G(q) (E25)

λminus1B = λ

minus1R minus

intq

G2(q)




δλ4 = 3δλ0 (E26)




12

intxy


(minusiλR)2

48

intxy

GR(xy)4 (E27)


ΠR(xy) = minusi(

δZ02x +δm20 +

λR +δλ0

2GR(xx)


R

6G3

R(xy)

(E28)

ΠR(xy) = minusi(

δZ22x +δm22 +

λR +δλ2

2GR(xx)


R

6G3

R(xy)








Z2Γ








R +δZ0k2 +δm20 +

λR +δλ0

2

intq

GR(q)

minus λ 2R

6

intpq




intq


2

intq

G2R(q)VR(q)

+λ 2

R

2

intpq



B =Zminus1(m2

R +δm20)


λB

2

intq

G(q)

minus Zminus4λ 2R

6

intpq



2R

intq


2

intq

G2(q)V (q)

+Zminus4λ 2

R

2

intpq











intq

G2R(q) (E35)

Appendix F



Pdtp f (tp) =

binta

dudtp(u)

duf (tp(u))



d4x =intP

dx0p

intd3x





Cdtc f (tc) = 0



t1intt0

dt f (t)


t1intt3

dt f (t)


Danksagung














Bibliography





1120[hep-ph]











176 BIBLIOGRAPHY



















BIBLIOGRAPHY 177


















178 BIBLIOGRAPHY



















BIBLIOGRAPHY 179




















180 BIBLIOGRAPHY



07070875

















BIBLIOGRAPHY 181



















182 BIBLIOGRAPHY



















BIBLIOGRAPHY 183




















184 BIBLIOGRAPHY




















BIBLIOGRAPHY 185














Introduction



Tracking Solutions







Self-Interactions

Matter Couplings


Summary




Stability












Summary

Conclusions

Conventions

















Acknowledgements

Bibliography

Contents

1 Introduction 1








9 Conclusions 131

viii CONTENTS

A Conventions 135







Bibliography 175

Chapter 1

Introduction


2 1 Introduction




1 Introduction 3



4 1 Introduction



Chapter 2




pl M4GUT M4

SUSY or M4elweak




pl












S =int

d4xradicminusg(x)

(minus R

16πG+

12


) (21)


Rmicroν minusR2


microν) (22)



δgmicroν and


(12(partφ)2minusV







2φ +dV (φ)

dφ= 0 (25)



partmicro

radicminusgpart

micro




3M2plH

2 = ρφ +ρB (26)

3M2pl

aa











1 = Ωφ +ΩB


2 = aminus3partt a3



φ +3Hφ +dV (φ)

dφ= 0 (27)















ωφ =pφ

ρφ

=φ 22minusVφ 22+V

(211)


minus1le ωφ le+1






0 M2pl A


pl 10minus10






Γminus1Γminus 1

2

(1+ωB) (212)




φlt


V (φ) =

M4

pl exp(minusλ

φ

Mpl

)for Γ = 1












Λ =O

((H0

Mpl

) 24+α

Mpl

)=O

(10minus

1224+α Mpl

)


Self-adjusting mass






92

Γ

(1minusω

lowastφ

2)

H(t)2 (213)







LYuk =minussumi

Fi(φ) ψiψi (214)




r (215)




(1+8πM2

plyi

mi

y j

m j

) (216)




nm2n sim y2




LGauge =12










16πGminus f (φ)R+

)equivint


16πGeff+

)

where1

16πGeff=

116πG

+ f (φ)






(Geff)today

∣∣∣∣le 02

Chapter 3



S[φ ] =int

d4x(


) (31)






ρ = sumn

pn|n〉〈n| (33)


〈O〉= Tr(ρO) (34)


ρ = |0〉〈0| (35)


H(x0) =int

d3x(

12(Φ(x))2 +


)(36)




HJ(x0) =int

d3x(

12(Φ(x))2 +


)(37)


ρ =1Z

exp(minusβH) (38)




ρ = |0〉J J〈0| (39)



(310)


exp(

iW [J])

=intDϕ exp

(iS[ϕ]+ i

intd4xJ(x)ϕ(x)

) (311)



d4xJ(x)φ(x) (312)


δΓ[φ ]δφ(x)

= 0 (313)





Γ[φ ] = S[φ ]+i2


iΓ1[φ ] = + +

=18

intd4x [minusiV (4)

cl (φ(x))]G0(xx)2 +112

intd4xint


+


Gminus10 (xy) =








(316)




G0(xy) = G0(xy)+int


= G0(xy)+int


+int

d4vint


+



V (4)cl (0)φ(x)2 + (317)




Γ[φ ] =int

d4x(minusVeff (φ)+

Z(φ)2

(partφ)2 +

) (318)



int d4k(2π)4 ln


k2

)+V1(φ)

minusV1(φ) = + +

=18

[minusV (4)

cl (φ)][int d4k

(2π)41


]2

+1


]2 int d4k(2π)4

int d4q(2π)4


+







12

m2φ

2 +13

gφ3 +

14

λφ4 (319)





exp(

iW [JK])

=intDϕ exp

(iS[ϕ]+ iJϕ +

i2

ϕKϕ

) (320)


Jϕ =int


d4xint




δJ(x) (322)


minusφ(x)φ(y)




intd4xint



δΓ[φ G]δφ(x)

=minusJ(x)minusint


=minus12

K(xy) (324)


δΓ[φ G]δφ(x)


= 0 (325)


Γ[φ G] = S[φ ]+i2

Tr lnGminus1 +i2

Tr(Gminus1

0 G)+Γ2[φ G] (326)

iΓ2[φ G] = + + (327)

=18

intd4x [minusiV (4)

cl (φ(x))]G(xx)2 +112

intd4xint


+



δΓ[φ G]δG(yx)


δG(yx) (328)





(330)


Γ[φ G[φ ]] = Γ[φ ] (331)




trunc(xyG) (332)








exp(

iW [J1 Jn])

=intDϕ exp

(iS[ϕ]+ i

n

sumk=1

1k


) (333)







sumk=1




δΓ

δφ(x)= 0

δΓ

δG(xy)= 0

δΓ









G(x1x2) = G12 =2δW [KL]

δK12 V4(x1x2x3x4) =

4δW [KL]δL1234




Γ[GV4] =i2

Tr lnGminus1 +i2

Tr(Gminus1

0 G)+Γ2[GV4] (337)

iΓ2[GV4] =

=18


+1

24

intd4x1234


minus 148

intd4x1234




V4(x1x2x3x4) =int



δΓ[GV4]δV4



iΓ2[G] = iΓ2[GV4] =18


148

intd4xint





Chapter 4







(minusλφMpl




radicΩde3 (41)


Λ2M2 simO(1)








radicGF






M = = 2GMint

k2ltΛ2

d4k(2π)4

1k2 +M2 = 2GM

Λ2

16π2 f1(M2Λ2) (42)

with f1(M2Λ2) = 1+ M2

Λ2 ln(

M2

Λ2+M2






FΛ(k) =


(43)




Λk

(2π)41

k2 +m2 =Λ2

16π2 f1(m2Λ2) (44)


Λk

(2π)4

int d4Λq

(2π)41

(k2 +m2)(q2 +m2)((q+ k)2 +m2)=

Λ2

(16π2)2 f2(m2Λ2)



can be evaluated

=18

V (4)cl (φ)

[int d4Λk

(2π)41


]2

(45)

=18

V (4)cl (φ)

[Λ2


asymp 18

V (4)cl (φ)

[Λ2

16π2 f1(0)]2

= Vcl(φ) middot

λ 4

8M4pl

[Λ2

16π2 f1(0)]2


=1


]2 intint d4Λkd4

Λq

(2π)81



]2 Λ2



]2 Λ2

(16π2)2 f2(0)



middot

λ 6

12M2pl

Λ2

(16π2)2 f2(0)






pl asymp 10minus120










Λ2M2L


Λ

2M2Lminus1







pl(Hinf 1014GeV)2



iΓ2[φ G] =infin

sumL=2

12LL

intd4x(minusiV (2L)

cl (φ(x)))G(xx)L (48)



δG(yx)(49)


sumL=2

L2LL


δ4(xminus y)



4(xminus y) (410)


eff (x)


infin

sumL=2

L2LL



M2eff (x) = exp

[12

G(xx)d2

dφ 2




M2eff = exp

[12

(int d4Λk

(2π)41

k2 +M2eff

)d2

dφ 2

]V primeprimecl(φ)




M2eff (φ) = exp

[12

(int d4Λk

(2π)41

k2 +m2

)d2

dφ 2

]V primeprimecl(φ)

∣∣∣∣∣m2=M2

eff (φ)

(412)


M2eff (xφ) = exp

[12

G(xx)d2

dφ 2


∣∣∣∣∣G(xx)=G(xx M2




2x +M2(x))

G(xy M2(middot)







Γhf [φ ] = Γ[φ G[φ ]]

=int

d4x(


)+

i2

Tr[

ln(2x +M2

eff (xφ))minus iM2

eff (xφ)G[φ ]]

where


12

G(xxφ)d2

dφ 2

]Vcl(φ(x)) (414)



12

int d4Λk

(2π)4

[ln

(k2 +M2

eff (φ)

k2

)minus

M2eff (φ)

k2 +M2eff (φ)

] (415)


Vhf (φ) = exp[

12

(int d4Λk

(2π)41

k2 +m2

)d2

dφ 2

]Vcl(φ)

∣∣∣∣∣m2=M2

eff (φ)

= V (φ m2)∣∣m2=M2

eff (φ)


V (φ m2)equiv exp[

12

(int d4Λk

(2π)41

k2 +m2

)d2

dφ 2

]Vcl(φ) (416)



M2eff (φ) =


∣∣∣∣m2=M2

eff (φ) (417)








=112

[minusV (3)(φ)

]2 int d4Λk

(2π)4

int d4Λq

(2π)41

(k2 +V (2))(q2 +V (2))((q+ k)2 +V (2))+





eff (φ) = k2 +V (2)(φ)


eff (φ)


∣∣∣∣m2=M2

eff (φ) (419)





M2eff (φ) = exp

[12

(Λ2

16π2 f1(m2Λ2))

d2

dφ 2

]V primeprimecl(φ)

∣∣∣∣∣m2=M2

eff (φ)



M2eff (φ) exp

[Λ2

32π2 f1(0)d2

dφ 2


1+O

(M2

eff

Λ2

)



Λk

(2π)4

[ln(

k2 +m2

k2

)minus m2

k2 +m2

]=

m4

16π2 f0(m2Λ2) (420)




Veff (φ) exp[

Λ2

32π2 f1(0)d2

dφ 2

]Vcl(φ) middot

1+O

(M2

eff

Λ2

)+O

(Veff

M4

) (421)







12

M4eff (φ)


2)









infin

suml=2

12(l +1)

[V (l+1)(φ)

]2 Λ2(lminus1)


2)



M2eff (φ) Λ







32π2d2

dφ 2

]Vcl(φ) (423)






to first order

V1minusloop(φ)[

1plusmn Λ2

32π2d2

dφ 2

]Vcl(φ)


Λ2LM2L L = 234




S[φ ] =int

ddx(


) (424)







12

int ddk(2π)4

[ln

(k2 +M2

eff (φ)

k2

)minus

M2eff (φ)

k2 +M2eff (φ)

] (425)


M2eff (φ) =


∣∣∣∣m2=M2


∣∣m2=M2

eff (φ) (426)


V (φ m2)equiv exp[

12

(int ddk(2π)4

1k2 +m2

)d2

dφ 2

]Vcl(φ) (427)




18π

M2eff (φ) =

(V (φ m2)+

18π


)m2=M2

eff (φ) (428)


int ddk(2π)d

1k2 +m2 =


microminus2ε

4π

(1ε


m2 +O(ε))

(429)






V Bcl (φ) = V R

cl (φ)+δVcl(φ)



V (φ m2) = exp[

18π

(1ε


m2 +O(ε))

d2

dφ 2

]V B

cl (φ)


δVcl(φ)equiv(

exp[minus 1

8π

1ε

d2

dφ 2

]minus1)

V Rcl (φ) (430)


V (φ m2) = exp[

18π

lnmicro2

m2d2

dφ 2

]V R

cl (φ) (431)




M2eff (φ) = exp

[1

8πln

micro2

m2d2

dφ 2

]V R

clprimeprime(φ)

∣∣∣∣∣m2=M2

eff (φ)

(432)









=1

(8π)2



6)54

(V (3)(φ)

)2

V (2)(φ) (433)





F =1

(8π)L g(F)prod

kge3

(V (k)(φ)

)Vk

(V (2)(φ)

)Vminus1 (434)







=(

V (φ)+1

8πV (2)(φ)

)+ sum

F

g(F)(8π)L

prodkge3

(V (k)(φ)

)Vk

(V (2)(φ)

)Vminus1







∣∣∣∣m2=M2

eff (φ)







M2eff = exp

[λ 2

8πln

micro2

M2eff

]V R


2V0 exp

[λ 2

8πln

(micro2

M2eff

)minusλφ

] (437)



ln

(M2

eff

micro2

)= ln

(λ 2V0

micro2

)+

λ 2

8πln

(micro2

M2eff

)minusλφ

rArr ln

(M2

eff (φ)

micro2

)=

11+λ 2(8π)

[ln(

λ 2V0

micro2

)minusλφ

]


ln

(M2

eff (φ)

micro2

)= ln

(M2

r

micro2

)minus φ

λminus1 +λ(8π)


Vhf (φ) = exp

[λ 2

8πln

(micro2

M2eff (φ)

)]V R


λminus1 +λ(8π)

]




18π

M2eff (φ) =

(Vr +

18π

M2r

)exp[minusλ φ

]




λminus1 = λ

minus1 +λ(8π)



partφ k

∣∣∣∣m2=M2


[minusλ φ

]


F =1

(8π)L g(F)prod

kge3


)Vk

(λ 2Vhf (φ)

)Vminus1 =1



λ 2

8π

)L

Vhf (φ)



[minusλ φ

] (438)


VR = Vr

(1+

λ 2

8π+sumF

g(F)(

λ 2

8π

)L)

= Vr

(1+

λ 2

8π+0781

(λ 2

8π

)2

+

)




exp[plusmn Λ2

32π2d2

dφ 2

](439)





Vcl(φ) = sumj

Vj exp(minusλ j

φ

Mpl

) (440)


Vj rarr Vj exp

[plusmn

λ 2j Λ2

32π2M2pl

] (441)



Veff (φ) Λ2M2

pl M4pl



quintessence models



Vcl(φ) = sumα

cαφminusα (442)


Veff (φ) = sumα

cαφminusα

Γ(α)

infin

sumL=0

Γ(α +2L)L

(plusmnΛ2

32π2φ 2

)L

= sumα

cαφminusα

Γ(α)

intinfin

0dt tαminus1 exp

(minustplusmn Λ2

32π2φ 2 t2)

(443)



1

101

102

103

104

105

106

107

108

10-4 10-3 10-2 10-1 1

Vef

f(φ

) V

(φ0)

φ φ0


Veff(φ)

Vone-loop(φ)

Vtwo-loop(φ)







cαφminusα

Γ(α)12

Γ(α

2)(

Λ2

32π2φ 2

)minus α

2

= sumα

Γ(α

2 )2Γ(α)

cα

(Λ

4πradic

2

)minusα

= const (445)




Γ(α+22 )

2Γ(α)cα

(Λ

4πradic

2

)minus(α+2)

(446)


M sim

Λ for φ Λ

φ for φ Λ (447)


1

103

106

109

1012

110-110-210-310-410-510-6

0

10

102

103

104

105

106

Vef

f(φ

) V

(φ0)

z (f

or tr

acke

r so

lutio

n)

φ φ0

today

CMB

Λ = φ0 asymp Mpl

V(φ) prop φ- α








α+2α+4




-1

-09

-08

-07

-06

-05

0 02 04 06 08 1

ωφ

Ωφ

toda

y (z

=0)ΛMpl =

0103

05

07

09

11 1315

29




Λmaxα

c1(α+4)α


c1(α+4)α sim

(H2

0 Mα+2pl


pl

)1(α+4)



α

ΛM

pl

02 04 06 08 1 12 14 16 18 2 0

05

1

15

2

25

3

-09

8-0

95

-09

-08 -07

-06



zquant sim

(Mpl10

Λ(4πradic

2α(α +1))

) α+23(1+ωB)



radicG) the track-
























dφφ (450)




V1L(φ) =12

int d4k(2π)4

(sumB


gF ln(k2 +mF(φ)2)

) (451)


I0(m2) equivint d4k

(2π)4 ln(k2 +m2) (452)

Il(m2) equivint d4k

(2π)41


(lminus1)dl

(dm2)l I0(m2)



I0(m2) = 2int m2

dm23

int m23dm2

2

int m22dm2

1 I3(m21) + D0 +D1m2 +D2m4 (453)



0

Ifinite0 (m2m2

0) = 2int m2

m20

dm23

int m23

m20

dm22

int m22

m20

dm21 I3(m2

1)

=1

32π2

(m4(

lnm2

m20minus 3

2

)+2m2m2

0minus12

m40

) (454)


V1L(φ) =12

(sumB


FgF Ifinite

0 (mF(φ)2mF(φ0)2)

) (455)


V1L(φ)Vcl(φ) (456)


V1L(φ) asymp 13


132π2 sum

j

(minus1)2s j g j

m j(φ0)2

(dm2

j

dφ(φ minusφ0)

)3

asymp 196π2 sum


(d lnm2

j



)3

(457)








d lnV primeprimecl

∣∣∣∣∣ 13ln(1+ zmax)


) 13

(459)



2Ωde

3H20

8πG


d lnV primeprimecl

∣∣∣∣∣(

1minusωde

2Ωde

07

) 13 1

3radicg j

(13meVm j(φ0)

) 43

(460)





∆m j

m jasymp

d lnm2j



3ln(1+ z)d lnm2

j



ν

emicro

p b Zt


100

10-5

10-10

10-15

10-20

101210910610310010-3

∆mm

meV




∆me


(1minusωde

2Ωde

07

) 13

(462)



φsim Hminus1


y j =dm j(φ)

dφ



y j =dm j

dφ=

12


d lnm2j


m j

2M

d lnm2j

d lnV primeprimecl



Mpl

M

)(1GeVmpn

) 13(

1minusωde

2Ωde

07

) 13

(463)


β j equivy j

m jMpl=

d lnm j

d(φMpl)


β j ∆m j

m j 4

(meVm j

)43

sim 10minus11(

me

m j

)43

(464)









ξ Rφ22


ξ (micro0) = 0


pl (see section 23)

116πGeff (φ)

=1

16πG+

12

ξ φ2


∆Geff

Geffequiv


=minusξ

2(φ

2(t)minusφ2(t0)

)16πGeff (φ(t))








d4xradicminusg(

12


) (465)


Vcl(φ) = Λ+m2φ

22+λφ44




=int

d4xradicminusg(

12


)

where

V (φ R) = Vcl(φ)+12

ξ Rφ2 +

R16πG

+ ε0R2 (467)

B(φ R) = ε3φ2 + ε4R





d4xradicminusg(

12







2φ

2 +λ (micro)

4φ

4 +12

ξ (micro)Rφ2 +

R16πG(micro)

+ ε0(micro)R2



partVLL

part t=

dΛ

dt+

12

dm2

dtφ

2 +14

dλ

dtφ

4 +12

dξ

dtRφ

2minus R16πG2

dGdt

+dε0

dtR2

=1

64π2

(part 2VLL

partφ 2 minusR6

)2

=1

64π2

(m(micro)2 +

λ (micro)2

φ2 +(

ξ (micro)minus 16

)R)2

part2BLL

part t=

dε3

dt2φ

2 +dε4

dt2R

=1

192π2

(part 22VLL

partφ 2 minus 2R5

)=

1192π2

(λ (micro)

22φ

2 +(

ξ (micro)minus 15

)2R)



dλ

dt=

3λ 2

32π2 dm2

dt=

λm2

32π2

dGdt


6)32π2

dΛ

dt=

m4

64π2

dξ

dt=

λ (ξ minus 16)

64π2 dε0

dt=

(ξ minus 16)2

64π2

dε1

dt= minus 1


dt= minus 1

360 middot32π2

dε3

dt=

λ

12 middot32π2 dε4

dt=

ξ minus 15

6 middot32π2

(469)





116πGeff (φ)

=1

16πG+ f1(φ)




V (φ R) = sumnm

cnmφnRm (470)

B(φ R) = sumnm

cnmφnRm




c00 = Λ c20 =m2

2 c40 =

λ

4 c21 =

ξ

2 c01 =

116πG

c02 = ε0 c20 = ε3 c01 = ε4



32π2


2

(ln


2

)(471)

minus(

1120

Cminus 1360

Gminus 130

2R+162X)

lnXminusR6

micro2 +infin

sumj=3




2V (φ R)partφ2



part


164π2


partφ 2 minus R6

)2


part


1192π2


partφ 2 minus 2R5





part


164π2


partφ 2

)2


part


132π2



partφ 2 minus 16








116πGeff (φ(t))

=1




(ξ0minus

16

)R (474)

=[

92

Γ

(1minusω

lowastφ

2)

+9(

ξ0minus16

)(ωBminus

13

)]H2

prop H2







+bφ +12

ξlowastφ

2










64π2

(V primeprimecl(φ)

)2+O

( t32π2

)2 (475)




)2

+O( t

32π2

)2




ξ0minus16

)2

+O( t

32π2

)2 (476)


∆Geff

Geff=


= minus(


16πGeff (φ(t))

= minus ξ0

2

(φ

2(t)minusφ2(t0)

)16πGeff (φ(t))

minus 132π2 ln

(micro2(φ(t))

micro20


(ξ0minus

16

)2

16πGeff (φ(t))




∆Geff


∆φ 2

M2plminus 1

32π2 ln(

H20

H2BBN


H20

(ξ0minus

16

)2 16πH20

M2pl











part


164π2

[2




partφ 2 minus 16

)2]

f2(φ micro0) = f2(φ) (477)







64π2

(ξ0minus

16

)2

+12

( t32π2

)2V (4)

cl (φ)(

ξ0minus16

)3

+O( t

32π2

)3






f3(φ micro) =13

( t32π2

)3V (6)

cl (φ)(

ξ0minus16

)4

+O( t

32π2

)4


pl since

V (6)cl (φ)R3(RM2



Kinetic term



32π2

[minusV1L(φ)minus

(162X)

lnXmicro2

+(minus 1

12partmicroXpart

microXminus 160

22X)

1X

+infin

sumj=4


]

=int d4x

32π2

[minusV1L(φ)+

112X


(part

4)]=

int d4x32π2

[minusV1L(φ)+

12


4)]






44 Summary






Chapter 5





Toy Model








χ)minus 12

micro2χ |χ|2

minus12





V (|φ |) = V0


)(52)









10-2

10-1

100

101

102

103

|φ(t

)|H

Inf

ΓasympHφ0HInf =

10010101001


01sdot10-3

0 02 04 06 08 1

θ φ =

arg

(φ)

t HInf




lowastχ

2


lowastφ

2




0 φ0χ30 ) it





infin

0dt a3




32ρ340

prop Ainfin (55)



0

02

04

06

08

1

0 02 04 06 08 1 12 14 16

|χ(t

)| a

32 H

Inf

t HInf

exact

WKB





κ asympminusN2

sin(2α0)(

χ0

Hinf

)2

middot


1013GeV

(Hinf

1012GeV

)12

if φ30 χ


17 middot10minus8(

ξ1

7V0

ρ0

)13(

Hinf

1012GeV

)76


30






-4sdot10-10

-2sdot10-10

0

2sdot10-10

4sdot10-10

0 02 04 06 08 1 12 14 16 18

n φχ a

3 [3

2 ρ 0

34 ]

t HInf

g=05

Ainfin

nφsdota3

nχsdota3

nχsdota3 WKB


φ0 HInf

V0

ρ0

10-3 10-2 10-1 100 101 102 103 104 105 106

10-810-710-610-510-410-310-210-1

10-10

10-9

10-8 10-7 10-6 10-5

2sdot10-10

5sdot10-10



53 Stability


|θφ |H

=|nφ |


45glowastS(T )

T 3


2H|φ |2 10minus8 (58)


θ2φ lt

3H +2ϕϕ



Chapter 6



S[φ ] =int

d4x(

12(partφ)2minus 1

2m2

φ2minus λ

4φ

4)

(61)





〈OH(t)〉= Tr(


(62)

= Tr(

ρ T[

exp(

+iint t

tinit


[exp(minusiint t

tinit





)(63)

=

T[exp(minusiint t


for t gt t prime

T[exp(+iint t prime


for t lt t prime


〈OH(t)〉= Tr(

ρ TC

[exp(minusiint

CdtHI(t)

)OI(t)

]) (64)








Φ(0x)|ϕ0〉= ϕ(x)|ϕ0〉 (66)

is given by

Zρ [JK] = Tr(

ρ TC

[exp(

iint

Cd4xJ(x)Φ(x)+

i2

intCd4xint

Cd4yΦ(x)K(xy)Φ(y)

)])=

intDϕ+


langϕminus0

∣∣∣∣TC

[exp(

i JΦ+i2

ΦKΦ

)]∣∣∣∣ϕ+0rang



Zρ [JK] =intDϕ+



Dϕ exp(

iS[ϕ]+ i Jϕ +i2

ϕ Kϕ

)


(iS[ϕ]+ i Jϕ +

i2

ϕ Kϕ

) (67)





iα0 + iα1ϕ +i2

ϕα2ϕ

) (68)






Γ[G] =i2

Tr lnGminus1 +i2

Tr(Gminus1

0 G)+Γ2[G] (69)

iΓ2[G] =minusiλ

8

intCd4xG(xx)2 +

(minusiλ )2

48

intCd4xint

Cd4yG(xy)4





δ


2K(xy) (610)




=minusiλ

2G(xx)δ 4


6G(xy)3 (612)





λ

2G(xx)








)GF(xy) =

int y0

0d4zΠF(xz)Gρ(zy)

minusint x0


(2x +M2(x)

)Gρ(xy) =

int y0

x0

d4zΠρ(xz)Gρ(zy)





6G(xy)3 (616)



ω2(tk) =


GF(x0y0k)

)∣∣∣∣∣x0=y0=t


(617)
















Dϕ exp(

iint

Id4xL(x)

) (618)


Zβ [JK] = Tr(

ρ TC+I

[exp(

iint

C+Id4xJ(x)Φ(x)+

i2

intC+I

d4xint


)])=

intDϕ exp

(iint


2

intC+I

d4xint


) (619)






Gth(xy) =int



The meaning ofint






section D12)





intC+I





=minusiλ

2Gth(xx)δ 4


6Gth(xy)3 (623)





S[φ ] =int

d4x(

12(partφ)2minus 1

2m2

Bφ2minus λB

4φ

4)

(624)



B = m2R +δm2 Z2

λB = λR +δλ Z = 1+δZ (625)





λR +δλ0

2

intqGthR(q)+

λ 2R

6

intpq




Gminus1T0

(k = k) = k2 +m2R

ddk2 Gminus1

T0(k = k) = +1 (627)





T0(k)minus∆Π(k)


2

[int T

qGT (q)minus

int T0

qGT0(q)

](628)

+λ 2

R

6

[int T


int T0


]





int Tq equiv

int T0q +



that

∆Π(k) =12

[int T0

q∆G(q)+

int∆T

qGT (q)

]ΛT0(qk)+F(k) (630)





q∆Π(q)


2G2

T0(q)ΛT0(qk)

]=

12

[int T0

q∆

2G(q)+int

∆T

qGT (q)

]ΛT0(qk)+F(k)



int T0

q∆Π(q)


2G2


2

int T0

kΛT0(qk)G2

T0(k)VT0(k p)

]=

=12

int T0

k

[int T0

q∆

2G(q)+int

∆T

qGT (q)

]ΛT0(qk)+F(k)

(δ (kminus p)+

12

G2T0

(k)VT0(k p))




intkΛR(qk)G2

thR(k)VR(k p) (632)


∆Π(p) =12

int T0

q

[∆

2G(q)+F(q)G2T0

(q)]VT0(q p)+

12

int∆T

qGT (q)VT0(q p)




Chapter 7













F [ϕ] = α0 +int

Cd4xα1(x)ϕ(x)+

12


+13





0n) (73)


F [ϕ] = α0 +int


12

intd3xint






n (x1 xn))lowast

(75)



int d3k1



(2π)3δ

3(k1 + +kn) iαε1εnn (k1 kn) (76)



(i

S[ϕ]+ Jϕ +12






Γ[φ G] = S[φ ]+i2

Tr lnGminus1 +i2

Tr(Gminus1

0 Gminus1)+ Γ2[φ G] (77)



Γ[φ G] = S[φ ]+i2

Tr lnGminus1 +i2

Tr(Gminus1





ΓnG[φ Gα3α4 ] = F3[φ α3α4 ]+12

Tr(

δ 2F3

δφδφG)

+Γ2nG[φ Gα3α4 ] (79)

The 2PI functional




= iαn(x1 xn)+int


+12





δφ(x)δφ(y) (712)




0n) (713)




ΓnG[φ Gα3 = 0α4 = 0 ] = 0 (714)


iΓ2[Gα4] =18

intC

d4x1234


]G(x1x2)G(x3x4)

+148

intC

d4x1234d4x5678


]G(x1x5)times (715)


]










δG(yx)=



δG(yx) (717)


Π(xy) = ΠG(xy)+Π

nG(xy) (718)


Gnonminusloc(xy)

ΠnG(xy) = Π


surface(xy)









where


Πλα(xy) = Πε










6


4 (x1x2x3y)

+minusiλφ(x)

2


3 (x1x2y) (721)






M(x)2 = m2 +λ

2φ


λ

2φ

2(x)+λ

2G(xx) (723)



(minusiλ )2

6G(xy)3 M(x)2 = m2 +

λ

2G(xx)



6




+16




3 (x1x2x3) = 0







Cd4z[Π

Gnonminusloc(xz)+Π


]G(zy)





(part

2x0 +k2 +M2(x0)

)GF(x0y0k) =

int y0

0dz0


minusint x0

0dz0



+14


2x0 +k2 +M2(x0)

)Gρ(xy) =

int y0

x0


where


(x0k)+Πminusλα

(x0k)


+λα


(x0k))

(727)


(2x +M2(x)

)GF(xy) =

int y0

0d4zΠF(xz)Gρ(zy)

minusint x0


minus λ

6V nG


V nG3 (xxy) (

2x +M2(x))

Gρ(xy) =int y0

x0

d4zΠρ(xz)Gρ(zy)

where


intC



intC




















iGminus10th(xy) =

(minus2xminusm2)







2ωk

cos(ωk(x0minus y0)

) (731)

G0ρ(x0y0k) =1


)for x0y0 isin C+I


nBE(ωk) =1

eβωk minus1 ωk =

radicm2 +k2


0th GCI0th GIC



GCC0th(xy) = GCI

0th(xy) =

GII0th(xy) = GIC

0th(xy) =

minusiλint

Cd4x = minusiλ

intId4x = minusiλ

intC+I

d4x =

(732)



6

intC+I

d4uint


=










GCC0F(0y0k) =

intCdz0

δs(z0)GCC0th(z

0y0k) (733)


intCdz0

δa(z0)GCC0th(z

0y0k) (734)

where


)


) (735)



intCdz0


0y0k)

=(736)




=

(GII


)δs(z0)+


δa(z0)

= (738)


GCI0th(y

0minusiτk) =int

Cdz0 GCC

0th(x0z0k)∆T

0 (z0minusiτk)

= (739)





∆0(vz) =int d3k




∆0(ωnz0k) =

(GII

0th(ωnk)G0th(00k)

)δs(z0)+

2iωnGII0th(ωnk)

δa(z0) (740)


time then becomes

GIC0th(ωny0k) =

intCdz0


0y0k) (741)


= equiv equiv th

0L



(minusiλ )2

6

intCd4uint


=(minusiλ )2

6

intCd4uint

Cd4z1

intCd4z2

intCd4z3


Id4v∆

T0 (z1v)∆T

0 (z2v)∆T0 (z3v)∆0(vz4)

]G0th(z4y)

equiv minusiλ6

intCd4uint

Cd4z1

intCd4z2

intCd4z3


αth40L(z1z2z3z4)

]G0th(z4y)




th

0L= equiv








th41L(z1z2z3z4) +

th=

th

0L+

th

1L+





+int

Cdw0

intCdz0




0(minusiτ primek)

= +

(742)


= equiv =

= + + +

= + + +





S0(xy) = =

+ + +

+ + +









GCCth (xy) = GCI

th (xy) =

GIIth (xy) = GIC

th (xy) =

minusiλint

Cd4x = minusiλ

intId4x = minusiλ

intC+I

d4x =

(744)




1I(x0) =














mth GCImth GIC



GIImth(xy) = GII




2x +m2B)

GCImth(xy) =

(2x +m2

B)

GCCmF(xy) =

(2x +m2

B)

GCCmρ(xy) = 0 (750)









th)


intβ

0dτprimeΠ




β

0dτ eminusiωnτ

part2τ GIC



mth)(y0k)



mth)(y0k) =

[(iωn GIC


]τ=β

τ=0


2n +k2 +M2

th)




mth)(y0k)


2y0 +k2 +m2

B

)disc(GIC

mth)(y0k) =

(part

2y0 +k2 +m2

B

)disc(partτGIC

mth)(y0k) = 0












disc(GICmth)(y


∣∣τ=β




∣∣τ=β

τ=0 =GCC

mF(0y0k)Gth(00k)


GICmth(ωny0k) =

(GII

th (ωnk)Gth(00k)

)GCC

mF(0y0k)minus(

iωnGIIth (ωnk)

)GCC

mρ(0y0k)

=int

Cdz0


0y0k) (753)



∆m(ωnz0k) =(

GIIth (ωnk)

Gth(00k)

)δs(z0)+

(2iωnGII

th (ωnk))

δa(z0)



=










C+Id4uint

C+Id4vGth(xu)

[1minus1I(u0)1I(v0)




mth(x0ωnk)+

intC+I

du0int

Cdv0(


0ωnk))

minus iint

Cdu0 GCC


Next GCImth(x

0ωnk) and GCImth(v


GCIth (x0ωnk)

=int

Cdz0[

GCCmth(x

0z0k)+int

C+Idu0

intCdv0(


0z0k))]

∆Tm(z0ωnk)

minus iint

Cdu0 GCC


=int

Cdz0[

GCCth (x0z0k)minus

intCdu0int

Idv0(


mth(v0z0k)

)]∆

Tm(z0ωnk)

minus iint

Cdu0 GCC


=int

Cdz0 GCC

th (x0z0k)

∆


th (ωnk)

minusint

Cdu0int

Idv0

Πth(z0v0k)GICmth(v

0u0k)∆Tm(u0ωnk)




Cdu0int

Idv0

Πth(z0v0k)GICmth(v


= minusiT suml

intCdu0


m(u0ωnk)

= minusiT suml


m(ωnk)

= minusiT suml


GIIth (ωnk)

Gth(00k)


GCIth (x0ωnk) =

intCdz0 GCC

th (x0z0k)

∆


th (ωnk)

minus iT summ


GIIth (ωnk)

Gth(00k)

=

intCdz0 GCC

th (x0z0k)∆T (z0ωnk) (756)


∆T (z0ωnk) = ∆


mΠth(z0ωmk)

[δnm

TGII

th (ωnk)minusGII


Gth(00k)

]equiv ∆




D(ωnωmk) =δnm

TGII

th (ωnk)minusGII


Gth(00k)(758)

=δnm

TGII


m(ωmk)

=δnm

TGII

th (ωnk)minusint

Cdw0

intCdz0


=δnm

TGII

th (ωnk)minusint

Cdw0

intCdz0





D(ωnωmk) = 0 (759)



th + Πnlth(z


T summ


Πnlth(z

0ωmk)D(ωmωnk)






intCdz0 GCC


=


intCdz0


=




th(v0z0k) (761)




th(v0z0k)

= = +




sm(minusiτk) =







intCdw0

intCdz0




= +

= +







= +





nlth(v

0z0k)∣∣GIC


(765)


= + +

+

+










6

intC+I

d4uint


=

= + + +

+ + +




=

+

+




th

0L 2PI= equiv (767)


intId4vint

Id4w∆

Tm(z1v)∆T



th

0L 2PI= equiv (768)





part2x0 +k2 +M2

th)


minus iint

Cdz0 [

ΠGthnl(x

0z0k)+ΠnGthnl(x


Gth(z0y0k)

where ΠGthnl(x






part2x0 +k2 +M2

th)


C+Idz0

Πnlth(x

0z0k)Gth(z0y0k)


intC+I

dz0Π

nlth(x

0z0k)Gth(z0y0k) =


Πnlth(x


Idv0

Πnlth(x

0v0k)int

Cdz0


=int

Cdz0[int

Idv0

Πnlth(x

0v0k)(

∆m(v0z0k)+int

Idw0 D(v0w0k)Πnl

th(w0z0k)

)]Gth(z0y0k)


ΠGthnl(x

0z0k) = Πnlth(x


ΠnGthnl(x

0z0k) =int

Idv0int

Idw0

Πnlth(x




Idv0

Πnlth(x






ΠGthnl(x

0z0k) =

ΠnGthnl(x

0z0k) = (771)

iΠthλα(x0z0k) =


Πthλα =infin

sumk=0

Π(k)thλα

ΠnGthnl =

infin

sumk=0

Π(k)nGthnl (772)

where

Π(0)thλα



Π(k)thλα




(x0z0k)



iΠ(0)thλα

(x0z0k) = =th

0L 2PI(773)


iΠ(1)thλα

(x0z0k) =

iΠ(2)thλα

(x0z0k) =




th

0L2PI



iΠ(0)thλα

(xz) =minusiλ

6


40L2PI(x1x2x3z) (774)


part2x0 +k2 +M2

th)



∣∣∣x0=y0=0

(part

2x0 +k2 +M2

th)

Gthρ(x0y0k)∣∣x0=y0=0 = 0 (775)


Πthλα(x0z0k)


ΠnGthnl(x

0z0k)



(k)thλα










R +δm20 +

λR +δλ0

2GthR(xx)


C(xminus y) (777)

minus iint

Cd4z[Π

GthnlR(xz)+Π


]GthR(zy)



ΠGthnlR(xz) = Z Π

Gthnl(xz) = Π

nlthR(xz)



nGthnl(xz) =

intId4vint

Id4wΠ

nlthR(xv)DR(vw)Πnl



Id4vΠ

nlthR(xv)∆m(vz)



ΠnlthR(xy) =

(minusiλR)2

6GthR(xy)3


iΠ(0)thλαR(xz) =

minusiλR

6


40L2PIR(x1x2x3z) (779)





Chapter 8







Gminus1R (xy) = i

(2x +m2

R)


ΠR(xy) = minusi(

δZ02x +δm20 +

λR +δλ0

2GR(xx)

)δ

4C(xminus y)+

(minusiλR)2

6GR(xy)3










(iS[ϕ]+

i2

ϕ Kϕ +i

4L1234ϕ1ϕ2ϕ3ϕ4

) (82)



iα0 +i2

ϕα2ϕ +i

4(α4)1234ϕ1ϕ2ϕ3ϕ4

) (83)




εn=plusmnα




intrarrintC



δ


14

L(xyzw) (85)


δ

δG(xy)Γ

L[G] =minus12

K(xy) (86)


ΓL[G] = Γ[GV4]+

14




δΓ

δV4=minusα4


4C(x1minus x2)δ 4





V4(x1x2x3x4) =int


4 + iAnG4 )(y1y2y3y4)]


4 (x1x2x3x4) (87)


(2x +M2(x)

)GF(xy) =

int y0


int x0

0d4zΠρ(xz)GF(zy)

minus λ

6V nG

4 (xxxy) (88)(2x +M2(x)

)Gρ(xy) =

int y0

x0

d4zΠρ(xz)Gρ(zy)


823 Renormalization



(2x +δZ02x +m2

R +δm20 +

λR +δλ0

2GR(xx)

)GFR(xy)

=int y0


int x0


minus λR

6V nG

4R(xxxy) (89a)

(2x +δZ02x +m2

R +δm20 +

λR +δλ0

2GR(xx)

)GρR(xy)

=int y0

x0






6GR(xy)3



V nG4R(xxxy) =



V nG4R(xxxy) =







minus λR

6V nG

4R(xxxy)equivint


where

iΠλαR(xy) =16








0001

001

01

1

0 5 10 15 20 25 30 35 40 45 50

G(t

0k

=0)

t mR

GF(t0k)

Gρ(t0k)

-06-04-02

0 02 04 06 08

1 12 14 16

0 05 1 15 2 25 3 35 4

Πλα

(tk

=0)

t mR

ΠλαF(tk)

Πλαρ(tk)














V nG4R(x1x2x3x4)|x0

i =0 = minusiλR


i =0




(δm2

0 +m2R +

λR +δλ0

2GthR(xx)

)]GthR(xy)|x0y0=0

minus Zminus1 λR

6(minusiλR)




neiωnτ

[k2 +Zminus1

(δm2

0 +m2R +

λR +δλ0

2

intqGthR(q)

minus λ 2R

6

intpq


GthR(ωnk)|τrarr0

= minusT sumn





neiωnτ

[k2 +Zminus1

(Gminus1

thR(ωnk)minusZk2)]

GthR(ωnk)|τrarr0

= minusT sumn


2n GthR(ωnk)

]τrarr0




Outlook








(part

2x0 +k2 +M2(x0)

)GF(x0y0k) =

int y0

0dz0



minusint x0

0dz0







m2B = m2

Rminusλ

2

int d3p(2π)3

1

2radic

m2R +p2

reg


M2(x0) = m2B +

λ

2



6



)


6



)



(2π)3d3q










λB

2

intq

G(q)minus λ 2R

6Z4

intpq


V (k) = λBminusλB

2

intq

G2(q)V (q) (814)

minus λ 2R

Z4

intq

G(q)G(kminusq)+λ 2

R

2Z4

intpq






Zd



R (815)

Z2Vvac(k = 0) = λR






4 αaaaa4 αssaa




M2 (x0) = m2B +

λB

2

int d3p(2π)3 GF(x0x0p) (816)



6Z4



)


6Z4



)



6Z2



4](x00k)

)


6Z2


4](x00k)minus 1


4](x00k)

)



=int d3p

(2π)3d3q








1

12

14

16

18

2

22

24

0 10 20 30 40 50 60 70 80 90 100

mR

min

it

iteration

14

15

16

17

18

19

20

21

1 10

V(k

)

kminit

0th iteration

10th iteration

20th iteration

30th iteration




m2th = Z Gminus1

th (k = 0) (817)





001

01

1

10

100

1000

0 05 1 15 2

δλ

λR24


asmR = 025

asmR = 0125 01

1

10

100

1000

0 05 1 15 2

|δm

2 mR

2 |

λR24

Nonpert 3-loop

Pert 1-loop







partx0GF(x0y0k)∣∣x0=y0=0 = 0 (818)



where


nGth(ωnk)

ωth(k)2 =(



=T sumn

(1minusω2

n Gth(ωnk))

Gth(k) (819)




Z2

βint0





Gth(k)=






int β

0 rarr 2int β2







03

04

05

06

07

08

001 01 1 10 100

GF(t

tk)

t mR

k = 0

k = mR

k = 2mR

500 1000 1500 2000t mR

KB Gauss

KB Non-Gauss

ThQFT



Etotal(t) =int d3k

(2π)3

[12


B +λB

4


)GF(x0y0k)|x0=y0=t

minus 14

int t

0dz0 (


minus 14


14


+ const (820)





s



-03-02-01

0 01 02 03 04

001 01 1 10 100

microm

R

t mR

2

21

22

23T

mR

500 1000 1500 2000t mR

KB Gauss

KB Non-Gauss

Thermal Eq

0

1

0 8

f BE(n

)

ωmR0

1

0 8

f BE(n

)

ωmR



nfit(tk) =[

exp(


)minus1]minus1





=14



24Z2 V nG4 (xxxx)

∣∣x0=0

=λ 2

R

24

intkpq








15

155

16

165

17

175

18

185

19

001 01 1 10 100

mth

(t)

mR

t mR


18

10 100



mAth ω(tk = 0) =


GFR(x0y0k)

∣∣∣x0=y0=t

ωth(k = 0) =radic


∣∣∣τ=0

mBth

(limsrarrinfin


2 tminus sprime2 k)


)minus12




13

14

15

16

1 15 2 25 3 35

mth

mR

(as mR)-1

T = 15 mR

14

15

16

17

mth

mR

T = 17 mR


16

17

18

19m

thm

RT = 20 mR

001 01 1 10 100t mR











0

05

1

15

2

25

3

35

001 01 1 10 102 103 104

GF(t

tk=

0)

t mR

A24 A18

E18

E24





A18 λ = 18 m2B =minus687 m2

R E18 λR = 18 λB = 3718 m2B =minus1439 m2

R

A24 λ = 24 m2B =minus949 m2

R E24 λR = 24 λB = 6343 m2B =minus2514 m2

R



095

1

105

0 1 2 3 4 5 6 7GF(t

tk)

GF(0

0k

)

kmR

t mR = 2000

095

1

105

GF(t

tk)

GF(0

0k

)

t mR = 10

095

1

105

GF(t

tk)

GF(0

0k

)

t mR = 05

095

1

105

GF(t

tk)

GF(0

0k

)

t mR = 00 KB Gauss

095

1

105

GF(t

tk)

Gth

(k)

t mR = 20

0 1 2 3 4 5 6 7

095

1

105

kmR

t mR = 2000

095

1

105t mR = 10

095

1

105t mR = 05

095

1


095

1

105t mR = 20




09 1

11 12

0 1 2 3 4 5 6 7

GF(t

tk)

Gth

(k)

kmR

t mR = 2000

09 1

11 12

GF(t

tk)

Gth

(k)

t mR = 10

09 1

11 12

GF(t

tk)

Gth

(k)

t mR = 05

09 1

11 12

GF(t

tk)

Gth

(k)

t mR = 00 KB Gauss

09 1

11 12

GF(t

tk)

Gth

(k)

t mR = 20

0 1 2 3 4 5 6 7

09 1 11 12

kmR

t mR = 2000

09 1 11 12

t mR = 10

09 1 11 12

t mR = 05

09 1 11 12


09 1 11 12

t mR = 20



84 Summary 129

84 Summary






Chapter 9

Conclusions




132 9 Conclusions






9 Conclusions 133


Appendix A

Conventions


Rαmicroνλ = +

(partνΓ

α


αmicroν +Γ

η

microλΓ

αην minusΓ

η

microνΓα

ηλ

)


Γαmicroν =

12


)


microαλ


micro


Appendix B




S[φ ψ j] =int

d4x(


) (B1)




fk(φ)OSMk (B2)


exp(

iW [J])

=intDϕ

int (prod

jDψ j

)exp(

iS[ϕψ j]+ iint

d4xJ(x)ϕ(x))

(B3)


exp(


prodjDψ j



exp(

iΓ[φ ])

= exp(

iW [J]minus iint

d4xJ(x)φ(x))

=intDϕ exp

(iSeff [ϕ]+ i


)






radicminusg


d4xradicminusg(


) (B5)






Gminus10 (xy) =


= i

(2x +

δ 2V (φ R)δφ 2

∣∣∣∣φ(x)R(x)

)δ

4(xminus y) (B6)



Tr lnGminus10 (B7)



lndet A (B8)

with the operator


δφ 2

∣∣∣∣φ(x)R(x)

(B9)




mln

λm

micro2 =12i



ipart



K(xys) = summ



ζA(ν) = summ

iΓ(ν)

infinint0


Γ(ν)

infinint0

ds(is)νminus1int

d4xK(xxs) (B11)


infin


K(xys) =i∆12


(minusσ(xy)

2isminus is

(X(y)minus R(y)

6

)) (B12)







partxmicropartyν

] (B13)


infinj=0(is)


ζA(ν) =i

Γ(ν)

int d4x16π2

radicminusg

infin

sumj=0



= iint d4x

16π2

radicminusg



(XminusR6)1minusν

νminus1


sumj=3



)



32π2


X2

2

(ln

Xmicro2 minus

32

)+ g1(xx)X

(ln

Xmicro2 minus1

)minusg2(xx) ln

Xmicro2 +

infin

sumj=3

g j(xx)( jminus3)

X jminus2

]

(B14)


g0(xx) = 1 (B15)

g1(xx) = 0

g2(xx) =1


302R+

162X

=1

120Cminus 1

360Gminus 1

302R+

162X






0 =d


(part


βNpart

partcN+sum

i

intd4xγiφi(x)

δ

δφi(x)

)Γ[φi]






part


(sumN

βN(micro)part

partcN+sum

i


δ

δφi(x)

)ΓLL[φi micro]








part



= minus

(sumN

βNpart

partcN+sum

i

intd4xγiφi(x)

δ

δφi(x)

)S[φi]

= minusint



nmβ nm2(φ nRm)

+ γφ φ


partφ

)+ γgmicroν

gmicroν

(12


δgmicroν

)]



part



= minus part

part ln micro20


=int d4x

32π2


2minus 1

120Cminus 1

360Gminus 1

302R+

162X]

= minusint d4x

32π2

radicminusg

[12 sum

nm

(n

sumk=0

m

suml=0

dkldnminuskmminusl

)φ

nRmminus 1120

Cminus 1360

G

+16 sum

nm


15

δn0δm1

)2(φ nRm)

]


βnm =1

64π2

n

sumk=0

m

suml=0

dkldnminuskmminusl

β nm =1

192π2


15

δn0δm1

) (B17)


1120


1360








part


164π2


partφ 2 minus R6

)2

part


1192π2


partφ 2 minus R5

) (B19)




d4xradicminusg(

12


)







0 ) yields [60]


∣∣∣∣∣t=ln


micro20







Appendix C






(C1)


G(xy) = G0(xy)+int

d4uint



G(0)(xy) = G0(xy)


d4uint


= G0(xy)+int

d4uint


+int

d4uint

d4vint

d4zint






Π(0) = Π[φ G0]

Π(k) = Π[φ G(k)] = Π[φ G0

infin

sumn=0

(Π(kminus1)G0)n]



Π(k)








M2eff (xφ) = exp

[12

G(xx)d2

dφ 2


∣∣∣∣∣G(xx)=G(xx M2




2x +M2(x))

G(xy M2(middot)




Γhf [φ ] = Γ[φ G[φ ]] (C5)

=int

ddx(


)+

i2

Tr[

ln(2x +M2

eff (xφ))minus iM2

eff (xφ)G[φ ]]

where


12

G(xxφ)d2

dφ 2

]Vcl(φ(x)) (C6)



12

G(xx M2(middot)

) d2

dφ 2

]Vcl(φ(x)) (C7)


M2eff (xφ) =



eff (middot φ)






iΓnotad[φ ] = +

=112

intddxint







eff (middot φ)




Derivation






infin

sumn=0











12

Ghf (xx)d2

dφ 2







li=1 N(γi)

F =1

N(F)F (C12)


K(F) =


prodi=1

S(γi)

∣∣∣∣∣ |S(F)| (C13)






Γnotad[φ ] =i2



where


δGhf

(GminusGhf

) (C15)


G = Ghf

infin

sumn=0













Appendix D




D11 Thermal State


ρ =1Z

exp(minusβH)





Dϕ exp(

iint

Id4xL(x)

)C+I

exp

(i

infin

sumn=0


)C+α

(D1)







Zβ [J] = Tr(

ρ TC+I

[exp(

iint

C+Id4xJ(x)Φ(x)

)])=intDϕ exp

(iint


)



∣∣∣∣∣J=0

(D2)




Tr(


)= Tr

(eminusβHeτH


)= Tr

(eminusτH


)= Tr

(eminusβH










iS0[φ0]]

=N0 exp[

iint

Id4x(

12(partφ0)2minus 1

2m2

φ20

)]









ξ2k =

nBE(ωk)+ 12

ωk ηk = 0

σ2k

4ξ 4k


1eβωk minus1

(D4)











ωk sinh(ωkβ )


iFint [φ0] = + O(λ 2)

=minusiλ4

intI

d4x

3D0(xx)2 +6φ0(x)2D0(xx)+φ0(x)4 + O(λ 2)




iα thn (x1 xn) =

(δ iF [φ0]


∣∣∣∣φ0=0

)δC(x0








7rarr (D7)




= + O(λ 2) (D8)




= minusλ

intβ

0dτ ∆






4 (k1k2k3k4) =

= minusλ

intβ

0dτ





+ O(λ 2)

rarr minusλ







iGminus10th(xy) =


2φ

2)











nlth(xy)



=(minusiλ )2

2φ


6Gth(xy)3 (D13)

M2th = m2 +

λ

2φ


λ

2φ

2 +λ

2Gth(xx)




2x +M2th)


C+Id4zΠ



th GCIth GIC



th GCIth GIC



(2x +M2

th)


intCd4zΠ

CCth (xz)GCC

th (zy)

minusiint

Id4zΠ

CIth (xz)GIC

th (zy) (D15)




ρ (2x +M2

th)

GCCF (xy) =

int y0

0d4zΠ

CCF (xz)GCC

ρ (zy)

minusint x0

0d4zΠ

CCρ (xz)GCC

F (zy) (D16)

minusint

β

0dτ

intd3zΠ


th ((minusiτz)y)

(2x +M2

th)

GCCρ (xy) =

int y0

x0

d4zΠCCρ (xz)GCC

ρ (zy)



2x +M2th)

GICth (xy) =

int y0

0d4zΠ

ICth (xz)GCC

ρ (zy)

minusint

β

0dτ

intd3zΠ


th ((minusiτz)y)

(2x +M2

th)

GCIth (xy) = minus

int x0

0d4zΠ

CCρ (xz)GCI

th (zy) (D17)

minusint

β

0dτ

intd3zΠ



th)


minusint

β

0dτ

intd3zΠ


th ((minusiτz)y)


Gth(x0y0k) =int


GIIth (k0k) =

intβ



GICth (k0y0k) =

intβ


th (minusiτy0k)



k0 = ωn =2π





th (ωnk)


neiωnτ GIC

th (ωny0k) (D19)



ω2n +k2 +M2

th)


IIth (ωnk)GII

th (ωnk) (D20)

whereint β








) (D21)

where exp(iFψ [ϕ]














i

α0 +int


12


2 (xy)ϕε2(y))




ε1ε22 (xy) =


ε1ε22 (k)


i

α0 +αε0 ϕε(0)+

12


2 (k)ϕε2(minusk))

(D25)



++2 =minusα


+minus2 =minusα


ε1εnn


+minus2 = α

minus+2 Thus αε



φ(x)|x0=0 = Tr(

ρ Φ(x))∣∣∣∣

x0=0 φ(x)|x0=0 = Tr

(ρ partx0Φ(x)

)∣∣∣∣x0=0

(D26)


G(xy)|x0=y0=0 =

Tr(


∣∣∣∣x0=y0=0


Tr(

ρ


])(D27)


)∣∣x0=y0=0



)minus φ(x)φ(y)

∣∣x0=y0=0



2k=0 sum

ε=plusmniαε

1 (D28)

φ(x)|x0=0 =intDϕ

minusipartpartϕ(x)



=12i

(sum

ε=plusmnε iαε


iαε1

)



G(x0y0k)∣∣x0=y0=0 = ξ

2k



2k +

σ2k

4ξ 2k

with

1ξ2k = minus sum


2 (k)







2 (k)

σ2kξ

2k = minus sum


2 (k)

(D30)


k = 1



Γ[φ G] = S[φ ]+i2

Tr lnGminus1 +i2

Tr(Gminus1

0 G)+Γ2[φ G] (D31)




2φ(x)2



=iδ 4S[φ ]


=iδ 3S[φ ]




8

intCd4xG(xx)2 +

(minusiλ )2

12


φ(y)

+(minusiλ )2

48










(D36)


Π(xy) =minusiλ


(minusiλ )2

2φ(x)G(xy)2

φ(y)+(minusiλ )2

6G(xy)3 +O(λ 3) (D37)







λ

2φ(x)2














M(x)2 = m2 +λ

2φ


2φ

2(x)+λ

2G(xx) (D43)






2φ(x)

(GF(xy)2minus 1

4Gρ(xy)2

)φ(y)

+(minusiλ )2

6

(GF(xy)3minus 3

4GF(xy)Gρ(xy)2

)+O(λ 3)


2φ(x)

(2GF(xy)Gρ(xy)

)φ(y) (D45)

+(minusiλ )2

6


4Gρ(xy)3

)+O(λ 3)



∣∣x0=y0 = δ

(3)(xminusy) (D46)


part2x0G(xy) = part

2x0GF(xy)minus i


2x0Gρ(xy)



]= part

2x0GF(xy)minus i




2x +M2(x))

GF(xy) =int y0


int x0

0d4zΠρ(xz)GF(zy)

(2x +M2(x)

)Gρ(xy) =

int y0

x0




G(x0y0k) =int



GF(x0y0k)∣∣x0=y0=0 = ξ

2k



2k +

σ2k

4ξ 2k

Gρ(x0y0k)∣∣x0=y0=0 = 0

partx0Gρ(x0y0k)∣∣x0=y0=0 = 1 (D50)



ξ2k =

n(t = 0k)+ 12

ω(t = 0k) ηk = 0

σ2k

4ξ 4k

= ω2(t = 0k) (D51)


Appendix E





S0[φ ] =int

d4x(

12(partφ)2minus 1

2m2

Bφ2)


d4xλB

4φ(x)4 (E1)


Γ[φ G] = S0[φ ]+i2

Tr lnGminus1 +i2





Trpart 2Sint











B = m2R +δm2 Z2




intxy



S0R[φR] =int

d4x(

12(partφR)2minus 1

2m2

Rφ2R

) (E7)




Tr lnGminus1R +

i2

TrGminus10RGR +Γ

Rint [φRGR] (E8)



ΓRint [φRGR] =

12

intxy


i2









0



ΓRint [φRGR] =

12

intxy


i2




2)δ (xminus y)





4

intxφ


4

intxφ

2R(x)GR(xx)

minus λR +δλ0

8

intxG2



δλ4 δλ2 δλ0




ΠR(xy) =2iδΓR

int

δGR(yx) ΠR(xy) =

iδ 2ΓRint









2δ 3Γint





intabcd


(E16)


intabcd









G(xy) =int



(2π)4δ

(4)(p1 + p2 + p3 + p4)Λ(p1 p2 p3 p4) =int

x1x2x3x4



ΠR(q = q) = ΠR(q = q) = 0

ddq2 ΠR(q = q) =

ddq2 ΠR(q = q) = 0 (E17)



(4)R equiv Z2Γ(4)

Γ(4)(xyuv) =


(E18)




R d

dq2 Gminus1R (q = q) = +1 (E19)









ΠR(xy) = minusi(

δZ02x +δm20 +

λR +δλ0

2GR(xx)

)δ (xminus y)

ΠR(xy) = minusi(

δZ22x +δm22 +

λR +δλ2

2GR(xx)





Z2Γ






R +δZ0k2 +δm20 +

λR +δλ0

2

intq

GR(q)



2

intq

GR(q) (E22)



R (E23)




2

intq




2

intq

G2R(q)VR(q) (E24)



0)

Z = 1

m2B = m2

RminusλB

2

intq

G(q) (E25)

λminus1B = λ

minus1R minus

intq

G2(q)




δλ4 = 3δλ0 (E26)




12

intxy


(minusiλR)2

48

intxy

GR(xy)4 (E27)


ΠR(xy) = minusi(

δZ02x +δm20 +

λR +δλ0

2GR(xx)


R

6G3

R(xy)

(E28)

ΠR(xy) = minusi(

δZ22x +δm22 +

λR +δλ2

2GR(xx)


R

6G3

R(xy)








Z2Γ








R +δZ0k2 +δm20 +

λR +δλ0

2

intq

GR(q)

minus λ 2R

6

intpq




intq


2

intq

G2R(q)VR(q)

+λ 2

R

2

intpq



B =Zminus1(m2

R +δm20)


λB

2

intq

G(q)

minus Zminus4λ 2R

6

intpq



2R

intq


2

intq

G2(q)V (q)

+Zminus4λ 2

R

2

intpq











intq

G2R(q) (E35)

Appendix F



Pdtp f (tp) =

binta

dudtp(u)

duf (tp(u))



d4x =intP

dx0p

intd3x





Cdtc f (tc) = 0



t1intt0

dt f (t)


t1intt3

dt f (t)


Danksagung














Bibliography





1120[hep-ph]











176 BIBLIOGRAPHY



















BIBLIOGRAPHY 177


















178 BIBLIOGRAPHY



















BIBLIOGRAPHY 179




















180 BIBLIOGRAPHY



07070875

















BIBLIOGRAPHY 181



















182 BIBLIOGRAPHY



















BIBLIOGRAPHY 183




















184 BIBLIOGRAPHY




















BIBLIOGRAPHY 185














Introduction



Tracking Solutions







Self-Interactions

Matter Couplings


Summary




Stability












Summary

Conclusions

Conventions

















Acknowledgements

Bibliography

Particle Physics and Dark Energy: Beyond Classical Dynamics

Documents

Transcript of Particle Physics and Dark Energy: Beyond Classical Dynamics