Post on 29-Mar-2021
Outline Introduction CTP Formalism Flavoured Leptogenesis Numerical Results Conclusions
Flavoured Leptogenesis from Nonequilibrium QFT
Matti Herranena
in collaboration with
M. Benekea B. Garbrechta C. Fidlera P. Schwallerb
Institut für Theoretische Teilchenphysik und Kosmologie,RWTH Aachen Universitya
Institut für Theoretische Physik,Universität Zürichb
Bielefeld, 5.5.2011
M. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 1 / 25
Outline Introduction CTP Formalism Flavoured Leptogenesis Numerical Results Conclusions
Outline
Introduction
Closed Time Path (CTP) Formalism of Noneq. QFT
CTP Approach to Flavoured Leptogenesis
Numerical Results
Conclusions and outlook
M. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 2 / 25
Outline Introduction CTP Formalism Flavoured Leptogenesis Numerical Results Conclusions
Why baryogenesis?
To explain the excess of matter over antimatter in the universe:[WMAP 2003]
nB
nγ=(6.1+0.3−0.2
)× 10−10
Why is there something rather than nothing?
M. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 3 / 25
Outline Introduction CTP Formalism Flavoured Leptogenesis Numerical Results Conclusions
Leptogenesis [Fukugita, Yanagida (1986)]
The Standard Model is extended by adding heavy right-handedMajorana neutrinos Ni, i = 1, 2, . . . (eg. see-saw models)
L ⊃ −h∗abφ ψ`bPRψRa − Y∗iaψ`a(εφ)†ψNi −12ψNiMiψNi + h.c.
Lepton asymmetry is generated through out-of-equilibrium L- andCP-violating Yukawa decays: N1 → `φ
M. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 4 / 25
Outline Introduction CTP Formalism Flavoured Leptogenesis Numerical Results Conclusions
Nonequilibrium QFT Approach to Leptogenesis1
1related / complementary aspects in:Buchmüller, Fredenhagen (2000); De Simone, Riotto (2007);Garny, Hohenegger, Kartavtsev, Lindner (2009 & 2010);Anisimov, Buchmüller, Drewes, Mendizabal (2010);Anisimov, Besak, Bödeker (2010)
M. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 5 / 25
Outline Introduction CTP Formalism Flavoured Leptogenesis Numerical Results Conclusions
Closed Time Path∗ (CTP) Formalism∗ a.k.a. Schwinger-Keldysh Formalism [Schwinger (1961); Keldysh (1964); Calzetta, Hu (1988)]
Usually in QFT matrix elements are computed within the in-outframework:
〈out|S|in〉 ←− time-ordered correlators: 〈T[ψ(x)ψ(y)]〉
In leptogenesis we want to calculate expectation values in a finitedensity medium, e.g.
j0(x) = 〈ψ(x)γ0ψ(x)〉 ≡ tr[ρ ψ(x)γ0ψ(x)
]ρ is an (unknown) quantum density operator
M. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 6 / 25
Outline Introduction CTP Formalism Flavoured Leptogenesis Numerical Results Conclusions
−→ In-In generating functional on a Closed Time Path:
Im t
Re t
Path-ordered correlators:
iSC(x, y) = 〈TC [ψ(x)ψ(y)]〉
Four propagators with respect to real time variable:
iS+−(u, v) = iS<(u, v) = −〈ψ(v)ψ(u)〉iS−+(u, v) = iS>(u, v) = 〈ψ(u)ψ(v)〉iS++(u, v) = iST(u, v) = 〈T(ψ(u)ψ(v))〉
iS−−(u, v) = iST(u, v) = 〈T(ψ(u)ψ(v))〉
M. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 7 / 25
Outline Introduction CTP Formalism Flavoured Leptogenesis Numerical Results Conclusions
Kadanoff-Baym Equations
Generic Schwinger-Dyson equations for (CTP) propagators:
= +G G0 Σ
Kadanoff-Baym equations are the <,> components:
(i∂/− m)S<,> − ΣH � S<,> − Σ<,> � SH =12(Σ> � S< − Σ< � S>
)(A� B)(u, v) ≡
∫d 4w A(u,w)B(w, v) denotes convolution
Renormalization, thermal corrections (thermal masses etc.)
Finite width effects
Collision term
M. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 8 / 25
Outline Introduction CTP Formalism Flavoured Leptogenesis Numerical Results Conclusions
Approximations
Wigner representation:
S(k, x) =
∫d 4r eik·r S
(x +
r2, x− r
2),
and gradient expansion to the lowest order in
x-derivatives: ∂xS(k, x), ∂xΣ(k, x), etc.
coupling constants in Σ(k, x)
−→ Constraint and Kinetic Equations:
2(k0 − k · γγ0)iγ0S<,>` −{
Σ/H` γ
0, iγ0S<,>`
}−{
iΣ/<,>` γ0, γ0SH`
}= −1
2
(iC` − iC†`
)i∂ηiγ0S<,>` −
[Σ/H` γ
0, iγ0S<,>`
]−[iΣ/<,>` γ0, γ0SH
`
]= −1
2
(iC` + iC†`
)η is conformal time variable: dη = dt/a(t)
M. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 9 / 25
Outline Introduction CTP Formalism Flavoured Leptogenesis Numerical Results Conclusions
The zeroth order solutions to Contraint Equation:
Flavour covariant free propagators for `:
iS<`ab(k, η) = −2πδ(k2)k/[ϑ(k0)f+`ab(k, η)− ϑ(−k0)(1ab − f−`ab(−k, η))
]iS>`ab(k, η) = −2πδ(k2)k/
[−ϑ(k0)(1ab − f+`ab(k, η)) + ϑ(−k0)f−`ab(−k, η)
]f±`ab(k, η) are time-dependent distribution functions
a, b are flavour indices
Similar (unflavoured) free propagators for N1 and φ
M. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 10 / 25
Outline Introduction CTP Formalism Flavoured Leptogenesis Numerical Results Conclusions
Contributions to Lepton Collision term
Y-Yukawa interactions:
` `
φ
N
MSM h-Yukawa and gauge interactions:
` `
`
φ
` `
`
A
M. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 11 / 25
Outline Introduction CTP Formalism Flavoured Leptogenesis Numerical Results Conclusions
Kinetic Equation for Lepton Number Densities[Beneke, Garbrecht, Fidler, MH, Schwaller (2010)]
Substituting free propagators into Kinetic Equation
−→ First order equation for n±`ab(η) =∫ d3k
(2π)3 f±`ab(k, η):
∂δn±`ab
∂η=∑
c
[Ξeffac δn±`cb − δn±`acΞ
effcb ]∓ i∆ωeff
`abδn±`ab
−∑
c
[Wacδn±`cb + δn±∗`ca W∗bc]± Sab − Γbl(δn+`ab + δn−`ab)− Γ±fl
`ab
Gradients of the mixing matrices: Ξeff ∼ U†∂ηU
Flavor oscillations by the thermal masses: ∆ωeff` ∼ h2T , Y2T
Collision terms: Washout, CP-violating source, Flavour-blinddamping, Flavour-sensitive damping
M. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 12 / 25
Outline Introduction CTP Formalism Flavoured Leptogenesis Numerical Results Conclusions
Fast Flavour-blind Gauge Interactions
` `
`
A
Blue cut←→ tree-level pair creation andannihilation and 1↔ 2 scatterings
Kinematically forbidden for on-shellexcitations −→ Γbl ∼ g4
2T
Force kinetic equilibrium with generalized chemical potentialsµ±ab(η):
f±`ab(k, η) =
(1
eβ|k|−βµ± + 1
)ab
Flavour and momentum dependence factorizes to first order in µ±ab:
δn±`ab = µ±abT2
12=⇒ δf±`ab(k, η) = δn±`ab
12β3eβ|k|
(eβ|k| + 1)2
−→ Simple interaction terms in the kinetic equationM. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 13 / 25
Outline Introduction CTP Formalism Flavoured Leptogenesis Numerical Results Conclusions
Suppression of Flavour Oscillations
Toy equations for flavour oscillations
∂δn±`ab∂η
= ∓i∆ωeff`abδn±`ab − Γbl(δn+
`ab + δn−`ab)
Parametrically Γbl � ∆ωeff`
−→ Two over-damped solutions with short and long time scales:
Short mode with τshort = 1/(2Γbl) forces an effective constraint:
δn−ab = −δn+ab
Flavour oscillations of δn+ab are over-damped with a long decay time:
τdamp ∼ 2Γbl/(∆ωeff`ab)2
M. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 14 / 25
Outline Introduction CTP Formalism Flavoured Leptogenesis Numerical Results Conclusions
Suppression of Flavour Oscillations
Toy equations for flavour oscillations
∂δn±`ab∂η
= ∓i∆ωeff`abδn±`ab − Γbl(δn+
`ab + δn−`ab)
Parametrically Γbl � ∆ωeff`
−→ Two over-damped solutions with short and long time scales:
Short mode with τshort = 1/(2Γbl) forces an effective constraint:
δn−ab = −δn+ab
Flavour oscillations of δn+ab are over-damped with a long decay time:
τdamp ∼ 2Γbl/(∆ωeff`ab)2
M. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 14 / 25
Outline Introduction CTP Formalism Flavoured Leptogenesis Numerical Results Conclusions
Final Kinetic Equations
Lepton asymmetries q ≡ δn+ − δn− (L- and R-handed):
∂q`ab
∂η=∑
c
[Ξacq`cb − q`acΞcb −Wacq`cb − q`acWcb]− Γfl`ab + 2Sab
∂qRab
∂η= −Γfl
Rab
Majorana neutrino N1:
∂fN1(kcom)
∂η= −2|Y1|2
kcomµ
2kcom 0Σµ
N(kcom)[fN1(kcom)− f eq
N1(kcom)]
Higgs field φ remains in thermal equilibrium
M. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 15 / 25
Outline Introduction CTP Formalism Flavoured Leptogenesis Numerical Results Conclusions
Lepton Asymmetries: Y`ab =n+`ab−n−`ab
s , Fully Flavoured case
hΤ=0.030
0.1 0.2 0.5 1.0 2.0 5.0 10.0 20.0
10-11
10-10
10-9
10-8
z
ÈYÈ
dark blue: Y`11
light blue: Y`22
brown dotted: Re[Y`12]
red dashed: Im[Y`12]
M1 = 1012 GeV, M2 = 1014 GeV
YYuk =
(1.4× 10−2 1× 10−2
i× 10−1 10−1
)
Y`12 = Y∗`21 strongly suppressed before freeze-out at z = M1T ≈ 10
−→ Flavour off-diagonals Y`12,21 can be negleted
M. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 16 / 25
Outline Introduction CTP Formalism Flavoured Leptogenesis Numerical Results Conclusions
Lepton Asymmetries: Y`ab =n+`ab−n−`ab
s , Unflavoured case
hΤ=0.001
0.1 0.2 0.5 1.0 2.0 5.0 10.0 20.0
10-11
10-10
10-9
10-8
z
ÈYÈ
dark blue: Y`11
light blue: Y`22
brown dotted: Re[Y`12]
red dashed: Im[Y`12]
M1 = 1012 GeV, M2 = 1014 GeV
YYuk =
(1.4× 10−2 1× 10−2
i× 10−1 10−1
)
Y`12 = Y∗`21 decay away long after freeze-out at z ≈ 10
−→ Flavour damping by Γfl` can be neglected
M. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 17 / 25
Outline Introduction CTP Formalism Flavoured Leptogenesis Numerical Results Conclusions
Lepton Asymmetries: Y`ab =n+`ab−n−`ab
s , Intermediate regime
hΤ=0.007
0.1 0.2 0.5 1.0 2.0 5.0 10.0 20.0
10-11
10-10
10-9
10-8
z
ÈYÈ
dark blue: Y`11
light blue: Y`22
brown dotted: Re[Y`12]
red dashed: Im[Y`12]
M1 = 1012 GeV, M2 = 1014 GeV
YYuk =
(1.4× 10−2 1× 10−2
i× 10−1 10−1
)
Full kinetic equations need to be solved!
M. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 18 / 25
Outline Introduction CTP Formalism Flavoured Leptogenesis Numerical Results Conclusions
Total Lepton Asymmetry as a function of Γfl`
Scale M1,2 and the couplings YYuk such that Γfl` is varying while the
washout and source terms remain constant
Scenario A
1010 1011 1012 1013 10140
1.´10-9
2.´10-9
3.´10-9
4.´10-9
5.´10-9
6.´10-9
M1@GeVD
ÈY11+
Y22È
fully flavoured
full kinetic equations
unflavoured
M. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 19 / 25
Outline Introduction CTP Formalism Flavoured Leptogenesis Numerical Results Conclusions
Conclusions
First principle description of leptogenesis within the CTPframework
RIS subtraction procedure not required
Simple kinetic equations for lepton number densities, including
Quantum statistical corrections in loops and external states
Sizable for weak washout
Flavour effects
Flavour oscillations are overdamped by fast gauge interactions
Full flavoured equations are needed between fully flavoured andunflavoured regimes
M. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 20 / 25
Outline Introduction CTP Formalism Flavoured Leptogenesis Numerical Results Conclusions
Outlook
Systematic inclusion of thermal effects
Finite widths
Thermal masses
Spectator processes
Resonant leptogenesis
Flavour coherence effects between Neutrinos Ni
M. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 21 / 25
Washout contribution
` `
φ
N
Blue cut←→ tree-level decays andinverse decays N1 ↔ `φ
12
tr∫
d3k(2π)3
∫ ∞0
dk0
2πCY`ab = −
∑c
Wacδn+`cb
with
Wac =12
Y∗1aY1c
∫d3k
(2π)32|k|d3k′
(2π)32√
k′2 + (a(η)M1)2
d3k′′
(2π)32|k′′| (2π)4δ4(k′ − k − k′′)
× 2k · k′(fN1(k′) + fφ(k′′)
) 12β3 eβ|k|
(eβ|k| + 1)2
M. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 22 / 25
CP-violating Source Sab: Wave-function Contribution
Orange cut←→ Interference betweentwo s-channel scatterings
Blue cut←→ Interference betweenloop and tree-level decays
Swfab = i
∑c
[Y∗1aY∗1cY2cY2b − Y∗2aY∗2cY1cY1b]
×(−M1
M2
)∫d3k′
(2π)32√
k′2 + (a(η)M1)2
ΣNµ(k′)ΣµN(k′)gw
δfN1(k′)
where the thermal decay rate is
ΣµN(k) = gw
∫d3p
(2π)32|p|d3q
(2π)32|q| (2π)4δ4(k − p− q) pµ(
1− f eq` (p) + f eq
φ (q))
M. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 23 / 25
CP-violating Source Sab: Vertex Contribution
Orange cut←→ Interference betweens- and t-channel scatterings
Blue cut←→ Interference betweenloop and tree-level decays
In the strongly hierarchical case, M1 � M2:
SVab =
12
Swfab =⇒ Sab =
32
Swfab
ΣµN(k)
MN�T−−−−→ gwkµ
16π recover standard approximation
M. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 24 / 25
Flavour-sensitive MSM Yukawa Interactions
` `
`
φ
Blue cut←→ tree-level pair creation andannihilation and 1↔ 2 scatterings
Kinematically forbidden for on-shellexcitations
Γfl`ab =Γan
([h†h]acq`cb + q†`ac[h
†h]cb − h†acqRcdhdb − h†adq†Rdchcb
)+Γsc
([h†h]acq`cb + q†`ac[h
†h]cb − h†acqRcdhdb − h†adq†Rdchcb
)
Example: 2 flavours, charged lepton basis
Γfl` = (Γan + Γsc) h2
τ
[(1 00 0
)q` + q`
(1 00 0
)− 2
(qR11 0
0 0
)]
Γan,sc ∼ g22T need to be calculated including (thermal) finite width
corrections for ` and φ propagators
M. Herranen (RWTH Aachen) Bielefeld, 5.5.2011 25 / 25