2 Gauge Theories and the Standard Model
31
Also for ξ → ∞ the unitary gauge description is recovered in that the Goldstone
propagator vanishes and the gauge propagator reproduces that of the unitary gauge
in Eq. (2.62). All ξ dependence, including the unphysical singularities of the ζ and
A μ propagators at k 2 = ξM 2 , present in individual Feynman diagrams, must cancel
in the sum of all contributions to any physical quantity.
An additional complication is that a Faddeev-Popov ghost is also present in R ξ
gauges (while it is absent in an unbroken abelian gauge theory). In fact under an
infinitesimal gauge transformation with parameter θ(x):
A μ → A μ − ∂ μ θ
φ → (1 − ieθ)[v + h(x)/
√
2 − iζ(x)/
√
2] ,
(2.71)
so that:
δA μ = −∂ μ θ, δh = −eζ θ, δζ = eθ
√
2(v + h/
√
2) .
(2.72)
The gauge fixing condition ∂ μ A μ − ξMζ = 0 undergoes the variation:
∂ μ A
μ
− ξMζ → ∂ μ A
μ
− ξMζ − [∂
2
+ ξM
2 (1 + h/v
√
2)]θ ,
(2.73)
where we used M =
√
2ev. From this, recalling the discussion in Sect. 2.6, we
see that the ghost is not coupled to the gauge boson (as usual for an abelian gauge
theory) but has a coupling to the Higgs field h. The ghost lagrangian is:
Ghost = ¯
η[∂
2
+ ξM
2 (1 + h/v
√
2)]η .
(2.74)
The ghost mass is seen to be m gh =
√
ξM and its propagator is:
iD gh =
i
k 2 − ξM 2 + ii
.
(2.75)
The detailed Feynman rules follow from all the basic vertices involving the gauge
boson, the Higgs, the would be Goldstone boson and the ghost and can be easily
derived, with some algebra, from the total lagrangian including the gauge fixing
and ghost additions. The generalization to the non abelian case is in principle
straightforward, with some formal complications involving the projectors over the
space of the would be Goldstone bosons and over the orthogonal space of the Higgs
particles. But for each gauge boson that takes mass M a we still have a corresponding
would be Goldstone boson and a ghost with mass
√
ξ M a . The Feynman diagrams,
both for the abelian and the non abelian case, are listed explicitly, for example, in
the Cheng and Li textbook in ref.[17].
We conclude that the renormalizability of non abelian gauge theories, also in
presence of spontaneous symmetry breaking, was proven in the fundamental works
of t’Hooft and Veltman [21] and discussed in detail in [22].
31
Also for ξ → ∞ the unitary gauge description is recovered in that the Goldstone
propagator vanishes and the gauge propagator reproduces that of the unitary gauge
in Eq. (2.62). All ξ dependence, including the unphysical singularities of the ζ and
A μ propagators at k 2 = ξM 2 , present in individual Feynman diagrams, must cancel
in the sum of all contributions to any physical quantity.
An additional complication is that a Faddeev-Popov ghost is also present in R ξ
gauges (while it is absent in an unbroken abelian gauge theory). In fact under an
infinitesimal gauge transformation with parameter θ(x):
A μ → A μ − ∂ μ θ
φ → (1 − ieθ)[v + h(x)/
√
2 − iζ(x)/
√
2] ,
(2.71)
so that:
δA μ = −∂ μ θ, δh = −eζ θ, δζ = eθ
√
2(v + h/
√
2) .
(2.72)
The gauge fixing condition ∂ μ A μ − ξMζ = 0 undergoes the variation:
∂ μ A
μ
− ξMζ → ∂ μ A
μ
− ξMζ − [∂
2
+ ξM
2 (1 + h/v
√
2)]θ ,
(2.73)
where we used M =
√
2ev. From this, recalling the discussion in Sect. 2.6, we
see that the ghost is not coupled to the gauge boson (as usual for an abelian gauge
theory) but has a coupling to the Higgs field h. The ghost lagrangian is:
Ghost = ¯
η[∂
2
+ ξM
2 (1 + h/v
√
2)]η .
(2.74)
The ghost mass is seen to be m gh =
√
ξM and its propagator is:
iD gh =
i
k 2 − ξM 2 + ii
.
(2.75)
The detailed Feynman rules follow from all the basic vertices involving the gauge
boson, the Higgs, the would be Goldstone boson and the ghost and can be easily
derived, with some algebra, from the total lagrangian including the gauge fixing
and ghost additions. The generalization to the non abelian case is in principle
straightforward, with some formal complications involving the projectors over the
space of the would be Goldstone bosons and over the orthogonal space of the Higgs
particles. But for each gauge boson that takes mass M a we still have a corresponding
would be Goldstone boson and a ghost with mass
√
ξ M a . The Feynman diagrams,
both for the abelian and the non abelian case, are listed explicitly, for example, in
the Cheng and Li textbook in ref.[17].
We conclude that the renormalizability of non abelian gauge theories, also in
presence of spontaneous symmetry breaking, was proven in the fundamental works
of t’Hooft and Veltman [21] and discussed in detail in [22].