CERN-THESIS-2012-153 26/07/2012 - CERN Document Server

gauge invariant field strength tensor:
The final QED Lagrangian is:
Fµν = ∂µAν − ∂νAµ. (2.7)
LQED = ¯ ψ (iγ µ ∂µ − m)ψ + e ¯ ψγ µ Aµψ − 1
4 FµνF µν . (2.8)
From there, it is clear that the addition of a mass term, 1
2 m2 AµA µ is prohibited by gauge invariance. Thus,
the gauge particle for this field, the photon, must be massless.
An important feature derived from renormalization is the fact that coupling ‘constants’ are not constants
but rather depend logarithmically on the energy scale, q 2 , of the measurements. This “running” coupling
constant, α(q 2 ) = e 2 (q 2 )/4π, is:
α(q 2 ) =
α(µ 2 )
1 − α(µ2 )
3π log
q2 µ 2
(2.9)
where µ is the reference or renormalization momentum. α(q 2 ) describes how the effective charge depends on
the separation of the two charged particles, this is a sort of “charge screening”. As q 2 increases, the photon
sees more charge until α(q 2 ) becomes infinity, at a very large q 2 .
2.2.2 Quantum Chromodynamics
Quantum Chronodynamics (QCD) is the gauge field theory that describes the strong interactions of colored
quarks and gluons. It is the SU(3) component of the SU(3)×SU(2)×U(1) SM. In a similar way to the
LQED derivation, the QCD Lagrangian is inferred from local gauge invariance. In this case the U(1) gauge
group is replaced by the SU(3) group of phase transformations on the quark color fields. From there, strong
interactions of quark fields ψq and gluon fields Aµ are described in the SM Lagrangian by the following:
LQCD =
q
¯ψq,a
µ
iγ ∂µδab − gsγ µ t C abA C
µ − mqδab ψq,b − 1
4 F A µνF Aµν , (2.10)
where repeated indices are summed over. The γ µ are the Dirac γ-matrices. ψq,a, are the quark-field spinors
for a quark flavor q and mass mq carry a color index a that runs from a = 1 to Nc = 3. That is, quarks
come in three colors. Quarks are the fundamental representation of the SU(3) group. The index C of the
gluon field A C µ runs from C = 1 to N 2 c − 1 = 8, which means that there are eight kinds of gluons. The t C ab
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