2 Gauge Theories and the Standard Model
29
of the previous example. The Higgs mechanism implies the absence of propagation
of massless phonons (states with dispersion relation ω = kv with constant v).
Also the mass term for A is manifested by the exponential decrease of B inside the
superconductor (Meissner effect).
2.8 Quantization of Spontaneously Broken Gauge Theories:
R ξ Gauges
We have discussed in Sect. 2.6 the problems arising in the quantization of a
gauge theory and in the formulation of the correct Feynman rules (gauge fixing
terms, ghosts etc.). Here we give a concise account of the corresponding results
for spontaneously broken gauge theories. In particular we describe the R ξ gauge
formalism [13, 17, 20]: in this formalism the interplay of transverse and longitudinal
gauge boson degrees of freedom is made explicit and their combination leads to the
cancellation from physical quantities of the gauge parameter ξ . We work out in
detail an abelian example that later will be easy to generalize to the non abelian
case.
We restart from the abelian model of Eq. (2.56) (with Q = −1). In the treatment
presented there the would be Goldstone boson ζ(x) was completely eliminated
from the lagrangian by a non linear field transformation formally identical to a
gauge transformation corresponding to the U(1) symmetry of the lagrangian. In that
description, in the new variables we eventually obtain a theory with only physical
fields: a massive gauge boson A μ with mass M =
√
2ev and a Higgs particle h with
mass m h =
√
2μ. This is called a “unitary” gauge, because only physical fields
appear. But for a massive gauge boson the propagator:
iD μν (k) = −i
g μν − k μ k ν /M 2
k 2 − M 2 + ii
,
(2.62)
has a bad ultraviolet behaviour due to the second term in the numerator. This
choice does not prove to be the most convenient for a discussion of the ultraviolet
behaviour of the theory. Alternatively one can go to an alternative formulation where
the would be Goldstone boson remains in the lagrangian but the complication of
keeping spurious degrees of freedom is compensated by having all propagators with
good ultraviolet behaviour (“renormalizable” gauges). To this end we replace the
non linear transformation for φ in Eq. (2.58) with its linear equivalent (after all
perturbation theory deals with the small oscillations around the minimum):
φ(x) → [v + h(x)/
√
2] exp[−iζ(x)/v
√
2] ∼ [v + h(x)/
√
2 − iζ(x)/
√
2] .
(2.63)
Here we have only applied a shift by the amount v and separated the real and
imaginary components of the resulting field with vanishing vacuum expectation
29
of the previous example. The Higgs mechanism implies the absence of propagation
of massless phonons (states with dispersion relation ω = kv with constant v).
Also the mass term for A is manifested by the exponential decrease of B inside the
superconductor (Meissner effect).
2.8 Quantization of Spontaneously Broken Gauge Theories:
R ξ Gauges
We have discussed in Sect. 2.6 the problems arising in the quantization of a
gauge theory and in the formulation of the correct Feynman rules (gauge fixing
terms, ghosts etc.). Here we give a concise account of the corresponding results
for spontaneously broken gauge theories. In particular we describe the R ξ gauge
formalism [13, 17, 20]: in this formalism the interplay of transverse and longitudinal
gauge boson degrees of freedom is made explicit and their combination leads to the
cancellation from physical quantities of the gauge parameter ξ . We work out in
detail an abelian example that later will be easy to generalize to the non abelian
case.
We restart from the abelian model of Eq. (2.56) (with Q = −1). In the treatment
presented there the would be Goldstone boson ζ(x) was completely eliminated
from the lagrangian by a non linear field transformation formally identical to a
gauge transformation corresponding to the U(1) symmetry of the lagrangian. In that
description, in the new variables we eventually obtain a theory with only physical
fields: a massive gauge boson A μ with mass M =
√
2ev and a Higgs particle h with
mass m h =
√
2μ. This is called a “unitary” gauge, because only physical fields
appear. But for a massive gauge boson the propagator:
iD μν (k) = −i
g μν − k μ k ν /M 2
k 2 − M 2 + ii
,
(2.62)
has a bad ultraviolet behaviour due to the second term in the numerator. This
choice does not prove to be the most convenient for a discussion of the ultraviolet
behaviour of the theory. Alternatively one can go to an alternative formulation where
the would be Goldstone boson remains in the lagrangian but the complication of
keeping spurious degrees of freedom is compensated by having all propagators with
good ultraviolet behaviour (“renormalizable” gauges). To this end we replace the
non linear transformation for φ in Eq. (2.58) with its linear equivalent (after all
perturbation theory deals with the small oscillations around the minimum):
φ(x) → [v + h(x)/
√
2] exp[−iζ(x)/v
√
2] ∼ [v + h(x)/
√
2 − iζ(x)/
√
2] .
(2.63)
Here we have only applied a shift by the amount v and separated the real and
imaginary components of the resulting field with vanishing vacuum expectation