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Context Relativity - gauge theories The Standard Model of particle physics Beyond the Standard Model Summary

The Standard Model of particle physics and beyond.

Benjamin Fuks & Michel Rausch de Traubenberg

IPHC Strasbourg / University of [email protected] & [email protected]

Slides available on http://www.cern.ch/fuks/esc.php

6-13 July 2011

The Standard Model of particle physics and beyond. Benjamin Fuks & Michel Rausch de Traubenberg - 1


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Outline.1 Context.2 Special relativity and gauge theories.

Action and symmetries.Poincare and Lorentz algebras and their representations.Relativistic wave equations.Gauge symmetries - Yang-Mills theories - symmetry breaking.

3 Construction of the Standard Model.Quantum Electrodynamics (QED).Scattering theory - Calculation of a squared matrix element.Weak interactions.The electroweak theory.Quantum Chromodynamics.

4 Beyond the Standard Model of particle physics.The Standard Model: advantages and open questions.Grand unied theories.Supersymmetry.Extra-dimensional theories.String theory.

5 Summary.



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Building blocks describing matter.

Atom compositness.

* Neutrons.* Protons.* Electrons .

Proton and neutron compositness .

* Naively: up and down quarks .* In reality: dynamical objects made of

Valence and sea quarks. Gluons [see below...] .

Beta decays .

* n p + e + e .* Needs for a neutrino .



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Three families of fermionic particles[Why three?].

Quarks:Family Up-type quark Down-type quark

1st generation up quark u down quark d2nd generation charm quark c strange quark s3rd generation top quark t bottom quark b

Leptons:

Family Charged lepton Neutrino

1st generation electron e electron neutrino e2nd generation muon muon neutrino

3rd

generation tau tau neutrino In addition, the associated antiparticles.

The only difference between generations lies in the (increasing) mass.

Experimental status [Particle Data Group Review] .* All these particules have been observed.

* Last ones: top quark (1995) and tau neutrino (2001).The Standard Model of particle physics and beyond. Benjamin Fuks & Michel Rausch de Traubenberg - 4


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Fundamental interactions and gauge bosons.

Electromagnetism .* Interactions between charged particles (quarks and charged leptons).* Mediated by massless photons (spin one).

Weak interaction .* Interactions between the left-handed components of the fermions.* Mediated by massive weak bosons W and Z 0 (spin one).* Self interactions between W and Z 0 bosons (and photons) [see below...] .

Strong interactions .* Interactions between colored particles (quarks).* Mediated by massless gluons g (spin one).* Self interactions between gluons [see below...] .* Hadrons and mesons are made of quarks and gluons.* At the nucleus level: binding of protons and neutrons.

Gravity .* Interactions between all particules.* Mediated by the (non-observed) massless graviton (spin two).* Not described by the Standard Model.* Attempts: superstrings, M -theory, quantum loop gravity, ...



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The Standard Model of particle physics - framework (1).

Symmetry principles elementary particles and their interactions.* Compatible with special relativity .Minkowski spacetime with the metric = diag(1 , 1, 1, 1).Scalar product x y = x y = x y = x 0y 0 x y .Invariance of the speed of light c .Physics independent of the inertial reference frame .

* Compatible with quantum mechanics .Classical elds : relativistic analogous of wave functions.* Quantum eld theory .

Quantization of the elds: harmonic and fermionic oscillators.* Based on gauge theories [see below...] .

Conventions.

* = c = 1 and , = diag (1 , 1, 1, 1).* Raising and lowering indices : V = V and V = V .* Indices.

Greek letters: , . . . = 0 , 1, 2, 3.Roman letters: i , j , . . . = 1 , 2, 3.



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What is a symmetry?

* A symmetry operation leaves the laws of physics invariant .e.g. , Newtons law is the same in any inertial frame: F = m d

2 x d t 2 .

Examples of symmetry.

* Spacetime symmetries : rotations, Lorentz boosts, translations.* Internal symmetries : quantum mechanics: | e i | .

Nther theroem.

* To each symmetry is associated a conserved charge.* Examples : electric charge, energy, angular momentum, . . .



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Dynamics is based on symmetry principles.

* Spacetime symmetries (Poincare).Particle types: scalars, spinors, vectors, ...Beyond: supersymmetry, extra-dimensions.

* Internal symmetries (gauge interactions).Electromagnetism, weak and strong interactions.Beyond: Grand Unied Theories.

Importance of symmetry breaking and anomalies [see below...] .* Masses of the gauge bosons .* Generation of the fermion masses .* Quantum numbers of the particles.



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5 Summary.



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Euler-Lagrange equations - theoretical concepts.

We consider a set of elds (x ).

* They depend on spacetime coordinates (relativistic).

A system is described by a Lagrangian L(, ) where = x .* Variables : the elds and their rst-order derivatives .

Action .

* Related to the Lagrangian S = R d 4x L.Equations of motion .

* Dynamics described by the principle of least action .* Leads to Euler-Lagrange equations:

L

L ( )

= 0 where and are taken independent .



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Euler-Lagrange equations - example.

The electromagnetic potential A (x ) =

V (t , x ), A(t , x )

.

External electromagnetic current: j (x ) = (t , x ), (t , x ).The system is described by the Lagrangian L(the action S = R d 4x L).

L= 14 F

F A j

with F = A A =0

BBBB@

0 E 1 E 2 E 3E 1 0

B 3 B 2

E 2 B 3 0 B 1E 3 B 2 B 1 0

1

CCCCA.

[Einstein conventions: repeated indices are summed.]

Equations of motion .* The Euler-Lagrange equations are

L A

L ( A )

= 0 F = j ( E = B = + E t .* The constraint equations come from

F + F + F = 0

( B = 0

E

= B

t

.


C R l i i h i Th S d d M d l f i l h i B d h S d d M d l S


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Symmetries.

What is an invariant Lagrangian under a symmetry? .* We associate an operator (or matrix) G to the symmetry:

(x ) G (x ) and L L+ (. . . ) .* The action is thus invariant .

Symmetries in quantum mechanics .

* Wigner: G is a (anti)-unitary operator.* For unitary operators, g , hermitian, so that

G = exp ig = exp i i g i . i are the transformation parameters .g i are the symmetry generators .

* Example : rotations R ( ) = exp[

i

J ] ( J

angular momentum).

Symmetry group and algebra .* The product of two symmetries is a symmetry {G i }form a group .* This implies that {g i }form an algebra .

g i , g j

g i g j g j g i = if ij k g k .

* Rotations : [J i , J j ] = iJ k with ( i , j , k ) cyclic.The Standard Model of particle physics and beyond. Benjamin Fuks & Michel Rausch de Traubenberg - 13

C t t R l ti it g g th i Th St d d M d l f ti l h i B d th St d d M d l S


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5 Summary.


Context Relativity gauge theories The Standard Model of particle physics Beyond the Standard Model Summary


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The Poincare group and quantum eld theory.

Quantum mechanics is invariant under the Galileo group.

Maxwell equations are invariant under the Poincare group.

Consistency principles.

* Relativistic quantum mechanics.Relativistic equations (Klein-Gordon, Dirac, Maxwell, ...)

* Quantum eld theory

The eld are quantized: second quantization.(harmonic and fermionic oscillators).




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The Poincare algebra and the particle masses.

The Poincare algebra reads ( , = 0 , 1, 2, 3)

hL , L i = i L L + L L ,hL , P i = i P P ,

hP , P

i= 0 ,

where* L = L is antisymmetric ..* Lij = J k rotations ; (i , j , k ) is a cyclic permutation of (1 , 2, 3).* L0i = K i boosts (i = 1 , 2, 3).* P spacetime translations .

Beware of the adopted conventions(especially in the literature).

The particle masses .* A Casimir operator is an operator commuting with all generators.

quantum numbers .* The quadratic Casimir Q2 reads Q2 = P P = E 2 p p = m 2 .* The masses are the eigenvalues of the Q2 operator.The Standard Model of particle physics and beyond. Benjamin Fuks & Michel Rausch de Traubenberg - 15



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Reminder: the rotation algebra and its representations.

The rotation algebra reads

hJ i , J j i= i ij k J k = iJ k with ( i , j , k ) a cyclic permutation of (1 , 2, 3).iJ k with ( i , j , k ) an anticyclic permutation of (1 , 2, 3).The operator J J .

* Dening J = ( J 1, J 2, J 3), we have

J

J , J i

= 0.

* J J is thus a Casimir operator (commuting with all generators).Representations.

* A representation is characterized byTwo numbers: j

12

N and m { j , j + 1 , . . . , j 1, j }.* The J i matrices are (2 j + 1)

(2 j + 1) matrices .

j = 1 / 2: Pauli matrices (over two).j = 1: usual rotation matrices (in three dimensions).

* A state is represented by a ket j , m such thatJ j , m = p j ( j + 1) m (m 1) j , m 1 ,J 3 j , m = m j , m and J J j , m = j ( j + 1) j , m .




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The Lorentz algebra and the particle spins.

The Lorentz algebra reads

hL , L i= i L L + L L ,* We dene N i = 12 (J

i + iK i ) and N i = 12 (J i iK i ).

One gets

Ni, N j

= iNk ,

Ni, N j

= iNk and

Ni, N j

= 0 .

Denition of the spin.

N i N i = sl (2)sl (2) so (3)so (3) .The representations of so (3) are known:

N i

S

N i

S J i

= N i

+ N i

spin = S + S .

The particle spins are the representations of the Lorentz algebra .* (0 , 0) scalar elds.* (1 / 2, 0) and (0 , 1/ 2) left and right spinors .* (1 / 2, 1/ 2)

vector elds.




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y g g p p y y y

Representations of the Lorentz algebra (1).

The (four-dimensional) vector representation (1 / 2, 1/ 2).* Action on four-vectors X .* Generators : a set of 10 4 4 matrices

(J ) = i .* A nite Lorentz transformation is given by

( 12 , 12 ) = exp hi 2 J

i,where R are the transformation parameters.* Example 1: a rotation with = 12 = 21 ,

R ( ) = exp

hi J 12

i=

0BBBB@

1 0 0 00 cos sin 0

0 sin cos 00 0 0 1

1CCCCA

.

* Example 2: a boost of speed v = tanh with = 01 = 10 ,

B () = exp hi J 01i=0BBBB@

cosh sinh 0 0sinh cosh 0 0

0 0 1 00 0 0 1

1CCCCA

=0BBBB@

0 0 0 00 0 1 0

0 0 0 1

1CCCCA

.




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y g g p p y y y


Pauli matrices in four dimensions .

* Conventions:

0 = 1 00 1, 1 = 0 11 0, 2 = 0 i i 0 , 3 = 1 00 1,* Denitions:

= ( 0 , i) , = ( 0 , i)

with = 1 , 2 and = 1, 2.The (un)dotted nature of the indices is related to Dirac spinors [see below...] .

Beware of the position (lower or upper, rst or second) of the indices.Beware of the types (undotted or dotted) of the indices.




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The left-handed Weyl spinor representation (1 / 2, 0).* Action on complex left-handed spinors ( = 1 , 2).* Generators : a set of 10 2 2 matrices

( ) = i 4 .

* A nite Lorentz transformation is given by

( 12 ,0) = exp hi 2 i.The right-handed Weyl spinor representation (0 , 1/ 2).* Action on complex right-handed spinors ( = 1, 2).* Generators : a set of 10 2 2 matrices

( ) = i 4

.

* A nite Lorentz transformation is given by

(0 , 12 ) = exp hi 2 i.Complex conjugation maps left-handed and right-handed spinors .




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Lowering and raising spin indices .* We can dene a metric acting on spin space [Beware of the conventions] ,

= 0 11 0 and = 0 11 0 . = 0 11 0 and = 0 11 0 .

* One has:

= , = , = , = .

Beware of the adopted conventions for the position of the indices(we are summing on the second index).

= .




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Four-component fermions: Dirac and Majorana spinors (1).

Dirac matrices in four dimensions (in the chiral representation) .

* Denition: =

0

0

.

* The (Clifford) algebra satised by the -matrices reads

, = + = 2 .* The chirality matrix , i.e. , the fth Dirac matrix is dened by

5 = i 0 1 2 3 = 1 00 1 and 5 , = 0 .




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Four-component fermions: Dirac and Majorana spinors (2).

A Dirac spinor is dened as

D = ,which is a reducible representation of the Lorentz algebra.

* Generators of the Lorentz algebra : a set of 10 4 4 matrices

= i 4

,

= 0

0 * A nite Lorentz transformation is given by( 12 , 0)(0 ,

12 )

= exp hi 2 i= ( 12 ,0) 00 (0 , 12 )! .A Majorana spinor is dened as

M = ,a Dirac spinor with conjugate left- and right-handed components .

= with =

.




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Summary - Representations and particles.

Irreducible representations of the Poincare algebra vs. particles.Scalar particles (Higgs boson ).

* (0 , 0) representation .

Massive Dirac fermions (quarks and leptons after symmetry breaking).* (1 / 2, 0)(0 , 1/ 2) representation .* The mass term mixes both spinor representations.

Massive Majorana fermions (not in the Standard Model dark matter ).* (1 / 2, 0)(0 , 1/ 2) representation .* A Majorana eld is self conjugate (the particle = the antiparticle).* The mass term mixes both spinor representations.

Massless Weyl fermions (fermions before symmetry breaking).* (1 / 2, 0) or (0 , 1/ 2) representation .* The conjugate of a left-handed fermion is right-handed.

Massless and massive vector particles (gauge bosons ).* (1 / 2, 1/ 2) representation .




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Outline.1 Context.

2 Special relativity and gauge theories.Action and symmetries.Poincare and Lorentz algebras and their representations.Relativistic wave equations.Gauge symmetries - Yang-Mills theories - symmetry breaking.


4 Beyond the Standard Model of particle physics.

The Standard Model: advantages and open questions.Grand unied theories.Supersymmetry.Extra-dimensional theories.String theory.

5 Summary.




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Relativistic wave equations: scalar elds.

Denition:* (0 , 0) representation of the Lorentz algebra.* Lorentz transformation of a scalar eld

(x ) (x ) = (x ) .Correspondence principle .

* P i .* Application to the mass-energy relation: the Klein-Gordon equation .P 2 = m 2 ( + m 2) = 0.

* The associated Lagrangian is given by LKG = ( )( ) m2,cf. Euler-Lagrange equations:

L

L ( ) = 0 where and are taken independent .Scalar elds in the Standard Model.

The only undiscovered particle is a scalar eld: the Higgs boson .

Remark: in supersymmetry, we have a lot of scalar elds [see below...] .




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Relativistic wave equations: vector elds (1).

Denition:* (1 / 2, 1/ 2) representation of the Lorentz algebra.* Lorentz transformation of a vector eld A

A (x ) A (x ) = `( 12 , 12 ) A (x ) .Maxwell equations and Lagrangian .

* The relativistic Maxwell equations are

F = j .

* F = A A is the eld strength tensor .* j is the electromagnetic current .* The associated Lagrangian is given by

LEM = 14

F F A j .* This corresponds to an Abelian U (1) (commutative) gauge group .




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Relativistic wave equations: vector elds (2).

The non-abelian (non-commutative) groupSU

(N

).* The group is of dimension N 2 1.* The algebra is generated by N 2 1 matrices T a (a = 1 , . . . , N 2 1),

T a , T b= if ab c T c ,where f ab c are the structure constants of the algebra.Example: SU (2): f ab

c = ab

c .

* Usually employed representations for model building.Fundamental and anti-fundamental: N N matrices so that

Tr (T a ) = 0 and Ta = T a .Adjoint: ( N 2 1) (N 2 1) matrices given by

(T a )b c = if ab c .* For a given representation R:

Tr `T a Tb= Rab ,where R is the Dynkin index of the representation.The Standard Model of particle physics and beyond. Benjamin Fuks & Michel Rausch de Traubenberg - 28



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Relativistic wave equations: vector elds (3).Application to physics .

* We select a gauge group (here: SU (N )).* We dene a coupling constant (here g ).* We assign representations of the group to matter elds.* The N 2 1 gauge bosons are given by A = A a T a .* The eld strength tensor is dened by

F = A A ig[A , A ]= h Ac Ac + g f ab cAa AbiT c .

* The associated Lagrangian is given by [Yang, Mills (1954)]

LYM = 1

4 R

Tr (F F ) .

* Contains self interactions of the vector elds.

Vector elds in the Standard Model.The gauge group is SU (3) c SU (2) L U (1) Y [see below...] .The bosons are the photon, the weak W and Z 0 bosons, and the gluons .




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Relativistic wave equations: Dirac spinors.

Denition:* (1 / 2, 0)(0 , 1/ 2) representation of the Poincare algebra.* Lorentz transformations of a Dirac eld D

D (x ) D (x ) = ( 12 ,0)(0 , 12 )D (x ) .Diracs idea.

* The Klein-Gordon equation is quadratic

particles and antiparticles.* A conceptual problem in the 1920s .* Linearization of the dAlembertian:

(i m)D = 0 LD = D (i m)D ,where

D = D 0. ( )2 = { , }= 2 .

Fermionic elds in the Standard Model.

Matter Dirac spinors after symmetry breaking .Matter Weyl spinors before symmetry breaking [see below...] .




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Summary - Relativistic wave equations.

Relativistic wave equations.General properties .

* The equations derive from Poincare invariance .

Scalar particles (Higgs boson ).* Klein-Gordon equation .

Massive Dirac and Majorana fermions (quarks and leptons ).

* Dirac equation .

Massless and massive vector particles (gauge bosons ).* Maxwell equations (Abelian case).* Yang-Mills equations (non-Abelian case).




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Outline.1 Context.


3 Construction of the Standard Model.

Quantum Electrodynamics (QED).Scattering theory - Calculation of a squared matrix element.Weak interactions.The electroweak theory.Quantum Chromodynamics.



5 Summary.




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Global symmetries for the Dirac Lagrangian.

Toy model.* We select the gauge group SU (N ) with a coupling constant g .* We assign the fundamental representations to the fermion elds ,

= 0B@

1...

N

1CA

, =

`1 N

.

* The Lagrangian reads

L= i m .The global SU (N ) invariance.

* We dene a global SU (N ) transformation of parameters a ,

( x ) (x ) = exp h+ ig a T funda i U ,( x ) (x ) = exp h ig a T funda i U .

* The Lagrangian is invariant ,

L L.




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Gauge symmetries for the Dirac Lagrangian (1).

Local (internal) SU (N ) invariance.* Promotion of the global invariance to a local invariance.* We dene a local SU (N ) transformation of parameters a (x ),

( x ) (x ) = U (x ) , ( x ) (x ) = U (x ) .

* The Lagrangian is not invariant anymore .

( x ) U (x ) ( x )L= i m L.

* Due to :The spacetime dependence of U (x ).The presence of derivatives in the Lagrangian.

* Idea : modication of the derivative.Introduction of a new eld with ad hoc transformation rules.Recovery of the Lagrangian invariance.




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Gauge symmetries for the Dirac Lagrangian (2).

Local (internal) SU (N ) invariance.

* Local invariance is recovered after:The introduction of a new vector eld A = A a T funda with

A (x ) A (x ) = U (x )hA (x ) + i g iU (x ) ,F (x )

U (x )F (x )U (x )

Tr

`F F

Tr

`F F

.

The modication of the derivative into a covariant derivative ,

( x ) D ( x ) = h igA (x )i( x ) .Transformation laws :

D ( x ) U (x ) D ( x ) L L.This holds (and simplies) for U (1) gauge invariance. In particular:A (x ) A (x ) = A (x ) + (x ) .

Example: Abelian U (1) e . m . gauge group for electromagnetism .




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Symmetry breaking - theoretical setup.

Let us consider a U (1) X gauge symmetry.* Gauge boson X gauge coupling constant gX .

Matter content.* A set of fermionic particles j of charge q j X .* A complex scalar eld with charge q .

Lagrangian.* Kinetic and gauge interaction terms for all elds .

Lkin = 14

F F + j i D j + ( D )(D )

= 14

X X

X X

+ j

i + g X q j X X

j + h + ig X q X ih ig X q X i.* A scalar potential ( LV = V scal ) and Yukawa interactions .

V scal = 2 + ()2 with > 0 , 2 > 0 ,LYuk = y j j j + h .c . with y j being the Yukawa coupling.




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Symmetry breaking - minimization of the scalar potential.

The system lies at the minimum of thepotential.

d V scald

= 0 =1

2s 2 e i 0 .v = 2 is the vacuum expectation value(vev) of the eld .We dene such that 0 = 0.

We shift the scalar eld by its vev

=1

2hv + A + i B i,where A and B are real scalar elds.

V scal = 2 + ()2 .




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Symmetry breaking - mass eigenstates (1).

We shift the scalar eld by its vev.

=1

2hv + A + i Bi.Scalar mass eigenstates.

* The scalar potentiel reads now

V scal = v 2A2 +

h14

A4 +14

B 4 +12

A2B 2 + vA3 + vAB 2

i.

* One gets self interactions between A and B .

* A is a massive real scalar eld, m2A = 22 , the so-called Higgs boson .

* B is a massless pseudoscalar eld, the so-called Goldstone boson .




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We shift the scalar eld by its vev. =

1 2hv + A + i Bi.

Gauge boson mass m X .

* The kinetic and gauge interaction terms for the scalar eld read now

D D = h + ig X q X ih ig X q X i=

12

A A +12

B B +12

g2Xv2X X + . . .

* One gets kinetic terms for the A and B elds.

* The dots stand for bilinear and trilinear interactions of A, B and X .

* The gauge boson becomes massive, mX = gXv.

* The Goldstone boson is eaten the third polarization state of X .* The gauge symmetry is spontaneously broken.




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=1

2hv + A + i Bi.Fermion masses m j .

* The Yukawa interactions read now

LYuk = y j j j 1

2 y jv j j +

1 2 y j (A + i B) j

j .

* One gets Yukawa interactions between A, B and j .

* The fermion elds become massive, m j = y jv.




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Summary - Nther procedure.

Nther procedure to get gauge invariant Lagrangians.

1 Choose a gauge group .2 Setup the matter eld content in a given representation.

3 Start from the free Lagrangian for matter elds .

4 Promote derivatives to covariant derivatives .

5 Add kinetic terms for the gauge bosons ( LYM or LM ).

Some remarks:* The Nther procedure holds for both fermion and scalar elds.* This implies that the interactions are dictated by the geometry .

* The gauge group and matter content are not predicted .* The symmetry can be eventually broken .* The theory must be anomaly-free .* This holds in any number of spacetime dimensions .* This can be generalized to superelds (supersymmetry, supergravity).




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Outline.1 Context.






5 Summary.



Th i l


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Theoretical setup.

The electromagnetism is the simplest gauge theory .

We consider an Abelian gauge group, U (1) e . m . .* Gauge boson: the photon A .* Gauge coupling constant: the electromagnetic coupling constant e.

We relate e to = e 2

4 .

Both quantities depend on the energy (cf. renormalization): (0)

1137

and (100GeV) 1

128.

Matter content.

NameField

Electric charge q 1st gen. 2nd gen. 3 rd gen.

Charged lepton e 1Neutrino e 0

Up-type quarks u c t 2/ 3Down-type quarks d s b 1/ 3



L i


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Lagrangian.

We start from the free Lagrangian ,

Lfree = X j = e , e , u , d ,... j i j .The Nther procedure leads to

LQED = X j = e , e , u , d ,... j i D j 14 F F

with ( D = ieqA ,F = A A .

The electromagnetic interactions are given by

Lint =

X j = e , u , d ,... j eq A j .

photon-fermion-antifermion vertices :* the fermions couple through their spin .* q the fermions couple through their electric charge .



O tli


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Outline.1 Context.





5 Summary.



F L g gi t ti l t ti (1)


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From Lagrangians to practical computations (1).

Scattering theory .* Initial state i (t ) at a date t .* Evolution to a date t * Transition to a nal state f (t ) (at the date t ).* The transition is related to the so-called S -matrix :

S = Df (t )

i (t )E= Df (t )

S

i (t )E.

Perturbative calculation of S .* S is related to the path integral

Z d`eldse i R d 4x L(x ) ,* S can be perturbatively expanded as :

S = + i Z d4x L(x ) 12Z d

4x d 4x T nL(x )L(x )o + . . .= no interaction + one interaction + two interactions + . . .= + iT .

We need to calculate T .



From Lagrangians to practical comp tations (2)


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Example in QED with one interaction: the e + e

process .

* The Lagrangian is given by

LQED = X j = e , e , u , d ,... j i D j 14 F F with ( D = ieqA ,F = A A .* Initial state i = e + e and nal state: f = .

* One single interaction term containing the e , e and A elds.

LQED e e A e .* The corresponding contribution to S reads

i

Z d 4x

L(x )

f i

= i

Z d 4x

he e A e

i.

More than one interaction .

* Intermediate, virtual particles are allowed .e.g. : e + e + e + e + .

* Same principles , but accounting in addition for chronology.



From Lagrangians to practical computations (3)


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We consider the specic process i 1(p a ) + i 2(p b )

f 1(p 1) + . . . + f n (p n ).

* The initial state is i (t ) = i 1(p a ), i 2(p b ) (as in colliders).

* The n-particle nal state is f (t ) = f 1(p 1), . . . , f n (p n ).

* p a , p b , p 1 ,. . . , and p n are the four-momenta.

We solve the equations of motion and the elds are expanded as plane waves .

= Z d 4p h(. . . )e ip x + ( . . . )e + ip x i. . .* The unspecied terms correspond to annihilation/creation operators of

(anti)particles (harmonic and fermionic oscillators).

We inject these solutions in the Lagrangian .* Integrating the exponentials leads to momentum conservation .

Z d 4x he ip a x e ip b x Y j e ip j x i= (2 )4 (4)p a + p b X j p j .The Standard Model of particle physics and beyond. Benjamin Fuks & Michel Rausch de Traubenberg - 48




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We dene the matrix element .

iT = (2 )4 (4)p a + p b X j p j iM .By denition, the total cross section :

* Is the total production rate of the nal state from the initial state.

* Requires an integration over all nal state congurations .

* Requires an average over all initial state congurations .

=1F

Z dPS (n)

M

2 .

The differential cross section with respect to a kinematical variable is

d d

=1F Z dPS (n)M 2 (p a , p b , p 1 , . . . , p n ).





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Total cross section.

= 1F Z dPS

(n)

M2 .

The integration over phase space ( cf. nal state) reads

Z dPS (n) =

Z (2 )4 (4)

pa + pb

X j

p j

Y j

d 4p j

(2 )4 (2 )(p

2 j

m2 j )(p

0 j )

.

* It includes momentum conservation .* It includes mass-shell conditions .* The energy is positive .* We integrate over all nal state momentum congurations.

The ux factor F (cf. initial state) reads

1F

=1

4q (p a p b )2 m 2a m 2b .* It normalizes with respect to the initial state density by surface unit.





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The squared matrix element

|M

|2

* Is averaged over the initial state quantum numbers and spins.* Is summed over the nal state quantum numbers and spins.

* Can be calculated with the Feynman rules derived from the Lagrangian.External particles : spinors, polarization vectors, . . . .Intermediate particles : propagators.Interaction vertices .

External particles .

* Rules derived from the solutions of the equations of motion .

Propagators .* Rules derived from the free Lagrangians .

Vertices .

* Rules directly extracted from the interaction terms of the Lagrangian.





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Feymman rules for external particles (spinors, polarization vectors).

u s ( p )

v s ( p )

u s ( p )

v s ( p )

( p )

( p )

* Obtained after solving Dirac and Maxwell equations .

= Z d 4p h(. . . )e ip x + ( . . . )e + ip x i. . .* They are the physical degrees of freedom (included in the dots).* We do not need their explicit forms for practical calculations [see below...] .





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Interactions and propagators.

QED Lagrangian.

LQED =

X j= e, e ,u,d,...

j i ieqA j 14 F F .

( p2 )

( p1 )

A ieq ( p )

i p + mp2 m2 A( p)

ip2

We need to x the gauge to derive the photon propagator. Feynman gauge: A = 0 .

Any other theory would lead to similar rules.



Example of a calculation:e + e + (1)


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Example of a calculation: (1).

Drawing of the Feynman diagram, using the available Feynman rules .

A

e+ ( pa )

e ( pb )

+ ( p1 )

( p2 )

Amplitude iM from the Feynman rules (following reversely the fermion lines).

iM = hv s a (p a ) (ie ) u s b (p b )ihu s 2 (p 2) (ie ) v s 1 (p 1)i i (p a + p b )2 .



Example of a calculation:e + e + (2).


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Derivation of the conjugate amplitude iM .iM = hv s a (p a ) (ie ) u s b (p b )ihu s 2 (p 2) (ie ) v s 1 (p 1)i i (p a + p b )2 ,

iM = hu s b (p b ) ( ie ) v s a (p a )ihv s 1 (p 1) ( ie ) u s 2 (p 2)i i (p a + p b )2 .* Denitions: u = u

0and v = v

0.

* We remind that ( ) = 0 0.* We remind that 0 0 = 1 and ( 0) = 0.

Computation of the squared matrix element |M |2 .

|M |2 =12

12 `iM iM .

* We average over the initial electron spin 1/ 2.

* We average over the initial positron spin 1/ 2.





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Computation of the squared matrix element

|M

|2 .

|M |2 =e 4

4(p a + p b )4Tr h ( /p b + m e ) ( /p a m e )iTr h ( /p 1 m ) ( /p 2 + m )i.

* We have performed a sum over all the particle spins .

* We have introduced /p = p , the electron and muon masses me and m .

* We have used the properties derived from the Dirac equation

Xs us(p)us(p) = /p + m and Xs vs(p)vs(p) = /p m .* For completeness, Maxwell equations tell us that

X (p) (p) = . [This relation is gauge-dependent.]The Standard Model of particle physics and beyond. Benjamin Fuks & Michel Rausch de Traubenberg - 56




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p ( )

Simplication of the traces, in the massless case.

|M |2 =8e 4

(p a + p b )4h(p b p 1)( p a p 2) + ( p b p 2)( p a p 1)i.* We have used the properties of the Dirac matrices

Tr h 1 . . . 2k+ 1i= 0 ,Tr h i= 4 ,

Tr

h

i= 4

+

,

Tr

h 5

i= 0 ,Tr h 5 i= 0 ,Tr h 5 i= 4i with 0123 = 1 .





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p ( )

Mandelstam variables and differential cross section.

|M |2 =2e 4

s 2 t 2 + u 2d d t = e 4

8 s 4t 2 + u 2.* We have introduced the Mandelstam variables

s = (pa + pb)2 = ( p1 + p2)2 ,

t = (pa p1)2 = ( pb p2)2 ,u = (pa p2)2 = ( pb p1)2 .

Remark: sub-processes names according to the propagator.

p a + p b

p a

p b

p 1

p 2

p a p 1

p a

p b

p 1

p 2

p a p 2

p a

p b

p 1

p 2

s -channel t -channel u -channel



Summary - Matrix elements from Feynman rules.


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y y

Calculation of a matrix element.1 Extraction of the Feymman rules from the Lagrangian.

2 Drawing of all possible Feynman diagrams for the considered process.

3 Derivation of the transition amplitudes using the Feyman rules.

4 Calculation of the squared matrix element .

* Sum/average over nal/initial internal quantum numbers .* Calculation of traces of Dirac matrices .

* Possible use of the Mandelstam variables .



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The Fermi model of weak interactions (1).


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( )

Proton decay (Hahn and Meitner, 1911).

p n + e + .

* Momentum conservation xes nal state energies to a single value(depending on the proton energy).

* Observation: the energy spectrum of the electron is continuous .Solution (Pauli, 1930): introduction of the neutrino .

p n + e + + e u d + e+ + e at the quark level .

* Reminder: p = uud (naively).* Reminder: n = udd (naively).* 1 3 particle process: continuous electron energy spectrum .

How to construct a Lagrangian describing beta decays? .





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Phenomenological model based on four-point interactions (Fermi, 1932) .

LFermi = 2 2G F hd 1 52 u ih e 1 52 e i+ h .c . .

G F

u

d

e +

e

* Phenomenological model reproducing experimental data.* Based on four-fermion interactions .* The coupling constant G F is measured .* G F = 1 .1637105 GeV2 is dimensionful .





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The Fermi Lagrangian can be rewritten as

LFermi = 2 2G F hd 1 5

2u ih e 1

5

2e i+ h .c .= 2 2G F H L + h .c . .

* It contains a leptonic piece L and a quark piece H .* Both pieces have the same structure .

The structure of the weak interactions* The leptonic piece L has a V A structure:

L = e 1 5

2e =

12

e e 12

e 5e .

* Similarly, the quark piece H has a V A structure.* The Fermi Lagrangian contains thus VV , AA and VA terms .

Behavior under parity transformations .* Under a parity transformation: V V and A A.* The VA terms (and thus weak interactions) violate parity .* Parity violation has been observed experimentally (Wu et al. , 1956).





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Analysis of the currents L and H .

L = e 1 5

2e and H = d

1 52

u .

* Presence of the left-handed chirality projector P L = (1 5)/ 2.Projectors and their properties .

* The chirality projectors are given by

P L =1 5

2and P R =

1 + 5

2.

* They fulll the properties

P L + P R = 1 , P 2L = P L and P 2R = P R .

* If is a Dirac spinor, left and right associated spinors are recovered by

D = LR , L = P LD = L0 , R = P R D = 0R .Only left-handed fermions are sensitive to the weak interactions .





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Introducting the left-handed chirality projector P L = 1 / 2(1 5):L = e P Le = e , L e , L and ( L ) = e P L e = e , L e , L ,H = d P Lu = d , L u , L and ( H ) = u P Ld = u , L d , L .

Behavior of the elds under the weak interacions .* Left-handed electron and neutrino behave similarly.* Up and down quarks behave similarly.

Idea: group into doublets the left-handed components of the elds :

Le = e , Le , L and Q = u , Ld , L.The currents are then rewritten as :

L = e , L e , L = Le 0 10 0Le ,

(L ) = e , L e , L = Le 0 01 0Le .[Similar expressions hold for the quark piece].



From Fermi model toSU (2)L gauge theory (1).


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Problems of the Fermi model.

LFermi = 2 2G F H L + h .c . .* Issues with quantum corrections, i.e. , non-renormalizability .* Effective theory valid up to an energy scale E m w 100 GeV.* Fermi model is not based on gauge symmetry principles .

Solution: a gauge theory (Glashow, Salam, Weinberg, 60-70, [Nobel prize, 1979] ).* Four fermion interactions can be seen as a s -channel diagram .* Introduction of a new gauge boson W .* This boson couples to fermions with a strength g w .

G F =

W

gw gw

G F g 2w

p 2m 2w g 2w m 2w

* Prediction: g w O(1) m w 100 GeV.The Standard Model of particle physics and beyond. Benjamin Fuks & Michel Rausch de Traubenberg - 66




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Choice of the gauge group: suggested by the currents :

L = Le 0 10 0Le = Le 1 + i 22 Le ,(L ) = Le

0 01 0

Le . = Le

1 i 22

Le .

[Similar expressions hold for the quark piece].

* Two Pauli matrices appear naturally.* i / 2 are the generators of the SU (2) algebra

(in the fundamental (dimension 2) representation ).

We choose the SU (2) gauge group to describe weak interactions .





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We choose the SU (2) L gauge group to describe weak interactions .

1/ 2 i are the generators of the fundamental representation .

h12

i ,12

j

i= i ij k

12

k ,

The left-handed doublets lie in the fundamental representation 2

e.

* The left-handed elds are the only ones sensible to weak interactions.* A doublet is a two-dimensional object .* The Pauli matrices are 2 2 matrices.* This explains the L-subscript in SU (2) L.

The right-handed leptons lie in the trivial representation 1

e.

* Non-sensible to weak interactions.

SU (2) L three gauge bosons W i with i = 1 , 2, 3.



The SU (2)L gauge theory for weak interactions (1).


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How to construct the SU (2) L Lagrangian?

We start from the free Lagrangian for fermions .* Simplication-1: no quarks here .* Simplication-2: no right-handed neutrinos .

Lfree = Le i Le + e R i e R .* A mass term mixes left and right-handed fermions .* The mass term are forbidden since Le 2

eand e R 1

e.

We make the Lagrangian invariant under SU (2) L gauge transformations .* SU (2) L gauge transformations are given by

Le

exp

hig w i (x )

i

2 iLe = U (x )Le and e R

e R .

* Gauge invariance requires covariant derivatives ,

Le D Le = h ig w W i i 2 iLe and e R D e R = e R .* We have introduced one gauge boson for each generator three W i .





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The matter sector Lagrangian reads then .

Lweak , matter = Le i D Le + e R i

D e R .withD Le = h ig w W i i 2 iLe and D e R = e R .

We must then add kinetic terms for the gauge bosons :

Lweak , gauge = 14

W i W i .

* The eld strength tensor reads:

W i = W i W i + g w i jk W j W k .* Gauge invariance implies the transformation laws :

i

2W i U h i 2 W i + i g w iU .





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The weak interaction Lagrangian for leptons.

Lweak , e = Lei DLe + eRi

DeR 14 Wi Wi .

withD Le = h ig w W i i 2 iLe ,D e R = e R ,

W i = W i W i + g w i jk W j W k .

Observation of the weak W i -bosons :* The experimentally observed W -bosons are dened by

W =12`W

1

iW 2

.* The W 3-boson cannot be identied to the Z 0 or :Both couple to left-handed and right-handed leptons.SU (2) L gauge theory cannot explain all data...



Summary - A gauge theory for weak interactions.


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A gauge theory for weak interactions.Based on the non-Abelian SU (2) L gauge group.

Matter (1): doublets with the left-handed component of the elds .

* Fundamental representation .* Generators : Pauli matrices (over two).

Matter (2): the right-handed component of the elds are singlet .

Three massless gauge bosons .

* (W 1

,W 2

) =(W +

,W

).* W 3 = Z 0 , A need for another theory: the electroweak theory .



Outline.


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1 Context.2 Special relativity and gauge theories.




5 Summary.



The electroweak theory (1).


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Electromagnetism and weak interactions :

* SU (2) L: what is the neutral boson W 3?* How to get a single formalism for electromagnetic and weak interactions?

Idea: introduction of the hypercharge Abelian group :

* U (1) Y : we have a neutral gauge boson B B = B B .* U (1) Y : we have a coupling constant g Y .

* SU (2) L U (1) Y : W 3 and B mix to the Z 0-boson and the photon.Quantum numbers under the electroweak gauge group :

* SU (2) L: left-handed quarks and leptons

2

e.

* SU (2) L: right-handed quarks and leptons 1

e.

* U (1) Y : xed in order to reproduce the correct electric charges.





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Nther procedure leads to the following Lagrangian.

LEW = 14 B B 14 W i W i

+3

Xf =1 hLf i D Lf + e Rf i D e f R i+

3

Xf =1 hQ f

i D

Q f + u Rf

i D

u f R + d Rf

i D

d f R

i.

* We have introduced the left-handed lepton and quark doublets

L1 = e , Le , L , L2 = , L, L , L3 = , L, L ,Q 1 =

u , Ld , L, Q 2 =

c , Ls , L, Q 3 =

t , Lb , L.* We have introduced the right-handed lepton and quark singletse 1R = e , R , e

2R = , R , e

3R = , R ,

u 1R = u , R , u 2R = c , R , u

3R = t , R , d

1R = d , R , d

2R = s , R , d

3R = b , R .





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Nther procedure leads to the following Lagrangian.

LEW = 14

B B 14

W i W i

+3

Xf =1 hLf i D Lf + e Rf i D e f R i+

3

Xf =1 hQ f i D Q

f + u Rf i D u

f R + d Rf i

D d f R i.

* The covariant derivatives are given by

D = ig Y YB ig w TiW i

Y is the hypercharge operator (to be dened).

The representation matrices T i are i

2 and 0 for doublets and singlets.



Gauge boson mixing.


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The neutral gauge bosons mix as

AZ != cos w sin w sin w cos w ! B W 3! .

where the weak mixing angle w will be dened later [see below...] .

The neutral interactions (for the electron) are given by

Lint = Le g Y Y Le B + g w 3

2W 3Le + e Rf g Y Y e R B e Rf

= Le cos w g Y Y Le + sin w g w 32 A Le + e Rf cos w g Y Y e R A e Rf + Le

sin w g Y Y Le + cos w g w

3

2

Z Le e Rf sin w g Y Y e R Z e Rf .

To reproduce electromagnetic interactions, we need

e = gYcos w = gwsin w and Q = Y + T 3 .

This denes the hypercharge quantum numbers.



Field content of the electroweak theory.


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Field SU (2) L rep.Quantum numbers

Y T 3 Q

Lf = 0@

e f , L

e f , L1A

2

e1212

12

120

1

e f R 1

e1 0 1

Q f =

0@u f , L

d f , L1A2

e16

16

12

1223

13

u f R 123 0

23

d f R 1 13 0 13The Standard Model of particle physics and beyond. Benjamin Fuks & Michel Rausch de Traubenberg - 78 Context Relativity - gauge theories The Standard Model of particle physics Beyond the Standard Model Summary

Electroweak symmetry breaking (1).


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The weak W -bosons and Z 0-bosons are observed as massive .* The electroweak symmetry must be broken.* The photon must stay massless.

Breaking mechanism: we introduce a Higgs multiplet .* We need to break SU (2) L cannot be an SU (2) L-singlet.* The Z 0-boson is massive U (1) Y must be broken Y = 0.

* U (1) e . m . is not broken one component of is electrically neutral.We introduce a Higgs doublet of SU (2) L with Y = 1 / 2.

=h1h2! h+1h02! .

The Higgs Lagrangian is given by .

LHiggs = D D + 2`2 = D D V (,) .* The covariant derivative reads D = i 2 g Y B ig w i 2 W i .* The scalar potential is required for symmetry breaking.

The Standard Model of particle physics and beyond. Benjamin Fuks & Michel Rausch de Traubenberg - 79 Context Relativity - gauge theories The Standard Model of particle physics Beyond the Standard Model Summary

Electroweak symmetry breaking (2).


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At the minimum of the potential, the neutral component of gets a vev.

=1

2 0v .We select the so-called unitary gauge .

=1

2 0v + h.* The three Goldstone bosons have been eliminated from the equations.

They have been eaten by the W and Z 0 bosons to get massive.* The remaining degree of freedom is the (Brout-Englert-)Higgs boson .

The Standard Model of particle physics and beyond Benjamin Fuks & Michel Rausch de Traubenberg - 80 Context Relativity - gauge theories The Standard Model of particle physics Beyond the Standard Model Summary

Mass eigenstates - gauge boson masses (1).W hif h l ld b i


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=1

2 0

v + h.The Higgs covariant derivative reads then :D =

1 2 0v + h i 2 0

@

g Y 2 B +

g w 2 W

3

g w 2 W 1 iW 2g w

2

W 1 + iW 2

g Y 2 B

g w 2 W

31A0v + h.

From the kinetic terms, one obtains the mass matrix, in the W 3B basis .D D W 3 B 14 g 2w v 2 14 g Y g w v 214 g Y g w v 2 14 g 2Y v 2 ! W 3B ! .

* The physical states correspond to eigenvectors of the mass matrix.

* The mass matrix is diagonalized after the rotation

AZ = cos w sin w sin w cos w B W 3,with cos2 w =

g 2w g 2w + g 2Y

.


Mass eigenstates - gauge boson masses (2).


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The mass matrix is diagonalized after the rotation

AZ = cos w sin w sin w cos w B W 3.As for the weak theory, we rotate W 1 and W 2 .

W =1

2`W 1

iW 2

.

After the two rotations, the Lagrangian reads

D D =e 2v 2

4sin2 w W + W +

e 2v 2

8sin2 w cos2 w Z Z + . . . .

* We obtain a W -boson mass term , m w = ev 2sin w .* We obtain a Z 0-boson mass term , m z = ev 2sin w cos w .

* The photon remains massless, m = 0 .


Mass eigenstates - Higgs kinetic and interaction terms.


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The Higgs kinetic and gauge interaction terms lead to

D D = 12 h h +

e 2v 2

4sin2 w W + W +

e 2v 2

8sin2 w cos2 w Z Z

+e 2v

2sin2 w W + W h +

e 2v 4sin2 w cos2 w

Z Z h

+ e 24sin2 w

W + W hh + e 28sin2 w cos2 w Z Z hh .

* We obtain gauge boson mass terms .* We obtain a Higgs kinetic term .

* We obtain trilinear interaction terms .* We obtain quartic interaction terms .* Remark: no interaction between the Higgs boson and the photon.


Mass eigenstates - fermion masses (1).


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The fermion masses are obtained from the Yukawa interactions.

LYuk = u R y u `Q d R y d `Q e R y e `L+h .c .

* We have introduced the SU (2) invariant product A B = A1B 2 A2B 1 .* Flavor (or generation) indices are understood:

d R y d `Q 3

Xf , f =1 d Rf (y d )f f `Q

f

* The Lagrangian terms are matrix products in avor space .

The mass matrices read

Lmass = v

2 u R y u u L v

2 d R y d d L v

2 e R y e e L + h .c . ,

where we have performed the shift = 1 2 0v + h,and introduced u f L = u f , L, . . . .


Mass eigenstates - fermion masses (2).


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The fermion mass Lagrangian read :

Lmass = v

2 u R y u u L v

2 d R y d d L v

2 e R y e e L + h .c . .* The physical states correspond to eigenvectors of the mass matrices.* Diagonalization: any complex matrix fulll

y = V R y U L ,

with y real and diagonal and U L, V R unitary .

Diagonalization of the fermion sector : we got replacement rules,

u L = 0@

u Lc Lt L1A

U u L u L , u R = `u R c R t R u R (V u R ) , . . .* The up-type quark mass terms become

v 2 u R y u u L

v 2hu R (V u R )ihV u R y u (U u L )ihU u L u Li= v 2 u R y u u L

where u L, u R are gauge-eigenstates and u L, u R mass-eigenstates.


Mass eigenstates - avor andCP violation.


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The neutral interactions are still diagonal in avor space, e.g. ,

Lint =23

e u L A u L 23

e hu L(U u L )i AhU u L u Li= 23 e u L A u L .due to unitarity of U u L .

The charged interactions are now non-diagonal in avor space, e.g. ,

Lint =e

2sin w u L W + d L

e 2sin w hu L(U u L )i W + hU d L d Li

=e

2sin w u Lh(U u L )U d Li W + d L .* Charged current interactions become proportionnal to the CKM matrix ,

VCKM = ( UuL)UdL [Nobel prize, 2008] .

* One phase and three angles to parameterize a unitary 3 3 matrix.Flavor and CP violation in the Standard Model .


Summary - The electroweak theory.


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The electroweak theory.Based on the SU (2) L U (1) Y gauge group.

* SU (2) L: weak interactions, three W i -bosons acting on left-handedfermions and on the Higgs eld.

* U (1) Y : hypercharge interactions, one B -bosons acting on both left-and right-handed fermions and on the Higgs eld.

The gauge group is broken to U (1) e . m . .

* The neutral component of the Higgs doublet gets a vev .* Hypercharge quantum numbers are chosen consistently.

The elds get the correct electric charge ( Q = T 3 + Y ).

* W 1 and W 2 bosons mix to W .* B and W 3 bosons mix to Z 0 and .

Yukawa interactions with the Higgs eld lead to fermion masses .

Experimental challenge: the discovery of the Higgs boson .

The Standard Model of particle physics and beyond Benjamin Fuks & Michel Rausch de Traubenberg 87 Context Relativity - gauge theories The Standard Model of particle physics Beyond the Standard Model Summary

Outline.1 Context.


88/122

1 Context.2 Special relativity and gauge theories.


3 Construction of the Standard Model.Quantum Electrodynamics (QED).

Scattering theory - Calculation of a squared matrix element.Weak interactions.The electroweak theory.Quantum Chromodynamics.

4 Beyond the Standard Model of particle physics.The Standard Model: advantages and open questions.

Grand unied theories.Supersymmetry.Extra-dimensional theories.String theory.

5 Summary.


The SU (3) c gauge group.


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Discovery of the color quantum numbers. [Barnes et al. (1964)]

* The predicted | = |sss baryon is a spin 3/2 particle.* The | wave function is fully symmetric (spin + avor).* This contradicts the spin-statistics theorem .

Introduction of the color quantum number .

The SU (3) c gauge group.

* Observed particles are color neutral .* The minimal way to write an antisymmetric wave function for

| is

| = mn |s m s n s .* The quarks lie thus in a 3 of the new gauge group SU(3)c .


Field content of the Standard Model and representation.

Field SU (3) rep SU (2) rep U (1) charge


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Field SU (3) c rep. SU (2) L rep. U (1) Y charge

Lf 1

e2

e 12

e Rf 1

e1

e1

Q f 3

e2

e16

u Rf 3

e1

e23

d Rf 3

e1

e 13

1

e2

e12

B 1

e1

e0

W 1

e

3

e

0g 8

e1

e0

* The matter Lagrangian involves the covariant derivative

D = ig Y YB ig w T i W i ig s T a g a .* We introduce a kinetic term for each gauge boson.

* The Higgs potential and Yukawa interactions are as in the electroweak theory.


QCD factorization theorem (1)


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Quarks and gluons are not seen in nature due to connement .In a hadron collision at high energy, they can however interact .

Predictions can be made thanks to the QCD factorization theorem.

hadr =

Xab

Z d x a d x b f a / A(x a ; F ) f b / B (x b ; F )

d partd

(x a , x b , p a , p b , . . . , F )

where hadr is the hadronic cross section (hadrons any nal state).* Pab all partonic initial states (partons a , b = q , q , g ).* x a is the momentum fraction of the hadron A carried by the parton a .* x b is the momentum fraction of the hadron B carried by the parton b .* If the nal state contains any parton:

Fragmentation functions ( from partons to observable hadrons ).


QCD factorization theorem (2)


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Predictions can be made thanks to the QCD factorization theorem.

hadr = Xab Z d x a d x b f a / A(x a ; F ) f b / B (x b ; F ) d partd (x a , x b , p a , p b , . . . , F )* f a / p 1 (x a ; F ), f b / p 2 (x b ; F ) : parton densities .

Long distance physics .Probability to have a parton with a momentum fraction x in ahadron.

* d part : differential partonic cross section (which you can now calculate).

Short distance physics .

F - Factorization scale .(how to distinguish long and short distance physics).


Parton showering and hadronization

* At high energy initial and nal state partons radiate other partons


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At high energy, initial and nal state partons radiate other partons .

* Finally, very low energy partons hadronize .


Summary - The real life of a collision at the LHC


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[Stolen from Sherpa ]

Th St d d M d l f ti l h i d b d B j i F k & Mi h l R h d T b b g 94 Context Relativity - gauge theories The Standard Model of particle physics Beyond the Standard Model Summary

Outline.1 Context.


95/122






5 Summary.

Th St d d M d l f ti l h i d b d B j i F k & Mi h l R h d T b b g 95 Context Relativity - gauge theories The Standard Model of particle physics Beyond the Standard Model Summary

The Standard Model: advantages and open questions (1).

The Standard Model of particle physics.


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The Standard Model of particle physics.* Is a mathematically consistent theory.* Is compatible with (almost) all experimental results [e.g. , LEP EWWG].

Measurement Fit |O meas O fit|/ meas0 1 2 3

0 1 2 3

had (mZ)(5) 0.02758 0.00035 0.02768

mZ [GeV ]mZ [GeV ] 91.1875 0.0021 91.1874 Z [GeV ] Z [GeV ] 2.4952 0.0023 2.4959 had [nb ]

0 41.540 0.037 41.479R lR l 20.767 0.025 20.742A

fbA0,l 0.01714 0.00095 0.01645Al(P )Al(P ) 0.1465 0.0032 0.1481RbRb 0.21629 0.00066 0.21579RcRc 0.1721 0.0030 0.1723AfbA

0,b 0.0992 0.0016 0.1038AfbA

0,c 0.0707 0.0035 0.0742AbAb 0.923 0.020 0.935AcAc 0.670 0.027 0.668Al(SLD)Al(SLD) 0.1513 0.0021 0.1481sin 2 effsin

2 lept(Q fb) 0.2324 0.0012 0.2314mW [Ge V]mW [Ge V] 80.399 0.023 80.379 W [GeV ] W [GeV ] 2.085 0.042 2.092m t [GeV ]m t [GeV ] 173.3 1.1 173.4

July 2010

Th St d d M d l f ti l h i d b d B j i F k & Mi h l R h d T b b 96 Context Relativity - gauge theories The Standard Model of particle physics Beyond the Standard Model Summary

The Standard Model: advantages and open questions (2).


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Open questions.

* Why are there three families of quarks and leptons?* Why does one family consist of {Q, uR, dR; L, er}?* Why is the electric charge quantized ?* Why is the local gauge group SU(3)c SU(2)L U(1)Y?* Why is the spacetime four-dimensional ?* Why is there 26 free parameters ?* What is the origin of quark and lepton masses and mixings ?* What is the origin of CP violation ?* What is the origin of matter-antimatter asymmetry ?* What is the nature of dark matter ?* What is the role of gravity ?* Why is the electroweak scale (100 GeV) much lower than the Planck

scale ( 1019 GeV) ?

Th St d d M d l f ti l h i d b d B j i F k & Mi h l R h d T b b 97 Context Relativity - gauge theories The Standard Model of particle physics Beyond the Standard Model Summary

Outline.1 Context.


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5 Summary.

Th S d d M d l f i l h i d b d B j i F k & Mi h l R h d T b b 98 Context Relativity - gauge theories The Standard Model of particle physics Beyond the Standard Model Summary

Grand Unied Theories: a unied gauge group (1).

C d h bi i i h S d d M d l?


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Can we reduce the arbitrariness in the Standard Model?

* A single direct factor for the gauge group.* A common representation for quarks and leptons.* Unication of g Y , g w and g s to a single coupling constant.

Unication of the Standard Model coupling constants.

* The coupling constants (at zero energy) are highly different .

Electromagnetism Weak Strong

1/ 137 1/ 30 1

* The coupling strengths depend on the energy due to quantum corrections .

Unication.

There exists a unication scale .

The coupling strengths are identical .



Running of the coupling constants.


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* The coupling constant at rst order of perturbation theory reads

= +

* These calculations lead to the energy dependence of the couplings.

* Unication requires additional matter (e.g. , supersymmetry [see below...] ).



How to choose of a grand unied gauge group


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How to choose of a grand unied gauge group.

* We want to pick up G so that SU(3)c SU(2)L U(1)YG.* Electromagnetism must not be broken .* The Standard Model must be reproduced at low energy .* Matter must be chiral .* Interesting cases are:

G = 8 4SO (4N + 2) with N 2E 6

How to specify representations for the matter elds.

* The Standard Model must be reproduced at low energy .* The choice for the Higgs elds

breaking mechanism .

Specify the Lagrangian.

LGUT = Lkin + Lgauge + LYuk + Lbreaking .





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LGUT = Lkin + Lgauge + LYuk + Lbreaking .

* Lkin : Poincare invariance .* Lkin + Lgauge : gauge invarian

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