274
19. Spontaneously Broken Local Symmetry
into (19.67) we obtain
(❗ + M
2 )A ˆ ν − ∂
ν (∂ μ A ˆ μ )(1 − 1/ξ) = q( ˆ
χ 2 ∂
ν χ ˆ 1 − χ ˆ 1 ∂
ν χ ˆ 2 )
2 ˆ χ
2
χ
2
− q A
ν ( ˆ 1 + 2vχ ˆ 1 + ˆ 2 ). (19.73)
The operator appearing on the left-hand side now does have an inverse (see
problem 19.8) and yields the general form for the gauge boson propagator
[
]
(1 − ξ)k
μ k
ν
μν
i −g +
(k
2
− M
2 )
−1 .
(19.74)
k 2 − ξM 2
This propagator is very remarkable
2 . The standard massive vector boson
propagator
μν
i(−g + k
μ k
ν /M
2 )(k
2
− M
2 )
−1
(19.75)
is seen to correspond to the limit ξ → ∞, and in this gauge the high-energy
disease outlined in section 19.1 appears to threaten renormalizability (in fact,
it can be shown that there is a consistent set of Feynman rules for this gauge,
and the theory is renormalizable thanks to many cancellations of divergences).
For any finite ξ, however, the high-energy behaviour of the gauge boson propagator is actually ∼ 1/k
2 , which is as good as the renormalizable theory of
QED (in Lorentz gauge). Note, however, that there seems to be another pole
in the propagator (19.74) at k
2 = ξM
2 : this is surely unphysical since it depends on the arbitrary parameter ξ. A full treatment (’t Hooft 1971b) shows
that this pole is always cancelled by an exactly similar pole in the propagator
for the ˆ
χ 2 field itself. These finite-ξ gauges are called R gauges (since they
are ‘manifestly renormalizable’) and typically involve unphysical Higgs fields
such as ˆ
χ 2 . The infinite-ξ gauge is known as the U gauge (U for unitary) since
only physical particles appear in this gauge. For tree diagram calculations, of
course, it is easiest to use the U gauge Feynman rules: the technical difficulties
with this gauge choice only enter in loop calculations, for which the R gauge
choice is easier.
Notice that in our master formula (19.74) for the gauge boson propagator
the limit M → 0 may be safely taken (compare the remarks about this limit
for the ‘naive’ massive vector boson propagator in section 19.1). This yields
the massless vector boson (photon) propagator in a general ξ-gauge, exactly
as in equation (7.122) or (19.23).
We now proceed with the generalization of these ideas to the non-Abelian
SU(2) case, which is the one relevant to the electroweak theory. The general
non-Abelian case was treated by Kibble (1967).
2 A vector boson propagator of similar form was first introduced by Lee and Yang (1962),
but their discussion was not within the framework of a spontaneously broken theory, so that
Higgs particles were not present, and the physical limit was obtained only as ξ → 0.
19. Spontaneously Broken Local Symmetry
into (19.67) we obtain
(❗ + M
2 )A ˆ ν − ∂
ν (∂ μ A ˆ μ )(1 − 1/ξ) = q( ˆ
χ 2 ∂
ν χ ˆ 1 − χ ˆ 1 ∂
ν χ ˆ 2 )
2 ˆ χ
2
χ
2
− q A
ν ( ˆ 1 + 2vχ ˆ 1 + ˆ 2 ). (19.73)
The operator appearing on the left-hand side now does have an inverse (see
problem 19.8) and yields the general form for the gauge boson propagator
[
]
(1 − ξ)k
μ k
ν
μν
i −g +
(k
2
− M
2 )
−1 .
(19.74)
k 2 − ξM 2
This propagator is very remarkable
2 . The standard massive vector boson
propagator
μν
i(−g + k
μ k
ν /M
2 )(k
2
− M
2 )
−1
(19.75)
is seen to correspond to the limit ξ → ∞, and in this gauge the high-energy
disease outlined in section 19.1 appears to threaten renormalizability (in fact,
it can be shown that there is a consistent set of Feynman rules for this gauge,
and the theory is renormalizable thanks to many cancellations of divergences).
For any finite ξ, however, the high-energy behaviour of the gauge boson propagator is actually ∼ 1/k
2 , which is as good as the renormalizable theory of
QED (in Lorentz gauge). Note, however, that there seems to be another pole
in the propagator (19.74) at k
2 = ξM
2 : this is surely unphysical since it depends on the arbitrary parameter ξ. A full treatment (’t Hooft 1971b) shows
that this pole is always cancelled by an exactly similar pole in the propagator
for the ˆ
χ 2 field itself. These finite-ξ gauges are called R gauges (since they
are ‘manifestly renormalizable’) and typically involve unphysical Higgs fields
such as ˆ
χ 2 . The infinite-ξ gauge is known as the U gauge (U for unitary) since
only physical particles appear in this gauge. For tree diagram calculations, of
course, it is easiest to use the U gauge Feynman rules: the technical difficulties
with this gauge choice only enter in loop calculations, for which the R gauge
choice is easier.
Notice that in our master formula (19.74) for the gauge boson propagator
the limit M → 0 may be safely taken (compare the remarks about this limit
for the ‘naive’ massive vector boson propagator in section 19.1). This yields
the massless vector boson (photon) propagator in a general ξ-gauge, exactly
as in equation (7.122) or (19.23).
We now proceed with the generalization of these ideas to the non-Abelian
SU(2) case, which is the one relevant to the electroweak theory. The general
non-Abelian case was treated by Kibble (1967).
2 A vector boson propagator of similar form was first introduced by Lee and Yang (1962),
but their discussion was not within the framework of a spontaneously broken theory, so that
Higgs particles were not present, and the physical limit was obtained only as ξ → 0.