258
Appendix E: Non-abelian Higgs Mechanism
since, by the local gauge invariance, the change of variables ϕ(x j ) → β
Λ
(ϕ(x j )),
x j ∈ V , leaves invariant the action as well as the boundary term (β
Λ
(ϕ(y)) = ϕ(y),
for y ∈ ∂V ).
This implies the invariance of the correlation functions for given cutoffs V and
K and therefore also in the limit of cutoff removal.
As mentioned in Chap. 29, Elitzur theorem contradicts the mean field ansatz at
the basis of the standard cheap explanation of the Higgs mechanism using a local
gauge invariant Lagrangian; a failure of the mean field ansatz was also displayed in
Chap. 20, Sect. 5, for the one-dimensional Ising model.
However, Elitzur theorem does not invalidate the standard perturbative expansion,
which requires the addition of a gauge fixing which breaks G. Clearly, Elitzur theorem
does not apply to the Coulomb gauge of scalar quantum electrodynamics, since the
local gauge group G is reduced to the identity by the gauge fixing. In the Feynman–
Gupta–Bleuler gauge, the gauge functions of the residual group are restricted to
satisfy the wave equation and do not have compact support in time.
One might argue that the question of the spontaneous breaking of the global
gauge group G and the evasion of the Goldstone theorem are annulled by choosing
the unitary gauge or better the (local and renormalizable) ξ-gauges.
39 However, the
corresponding gauge fixing
L G F = −
1
2 F
a F
a
, F
a
≡ ∂ μ A
a
μ − iξ(t
a
) nm < ϕ m > (ϕ n − < ϕ n >)
involves a mean field ansatz on the vacuum expectation value of the Higgs scalar
field ϕ and the evasion of the Goldstone theorem by a mean field ansatz may be in
conflict with the non-perturbative control, as displayed in Sect. 21.5. Thus, from a
non-perturbative point of view the question remains.
A non-perturbative argument for the absence of physical Goldstone bosons in the
abelian Higgs mechanism has been given in Sect. 29.2. A similar proof may also be
given in the non-abelian case, in the BRST gauge.
40
The BRST quantization is formulated in terms of
i) a local covariant field algebra F generated by the sets of standard fields
(fermionic ψ, Higgs ϕ, vector fields A
a
μ , Nakanishi–Lautrup field B
a ) transforming as irreducible representations of a (global gauge) n-dimensional (compact)
Lie group G and by two additional sets of anticommuting (local) Hermitian
fields c
a
, ¯
c
a , called Faddeev–Popov ghosts;
ii) the local BRST transformations, labelled by a parameter θ, which anticommutes
with c
a , ¯
c
a and with the fermionic fields; on the matter and gauge fields the
infinitesimal BRST transformations have the same form of local gauge transformations with “parameter” θ c
a
(x) (e.g. δ
θ
ψ = it
a
θc
a
ψ, δ
θ A
b
μ = θ(D μ c)
a ,
D μ denoting the covariant derivative) and on the additional fields are defined
by
39 For a detailed analysis of the ξ-gauges, see S. Weinberg, The Quantum Theory of Fields, Vol. II,
Cambridge Univ. Press 1996, Sect. 21.1.
40 G. De Palma and F. Strocchi, Ann. Phys. 336, 112 (2013).
Appendix E: Non-abelian Higgs Mechanism
since, by the local gauge invariance, the change of variables ϕ(x j ) → β
Λ
(ϕ(x j )),
x j ∈ V , leaves invariant the action as well as the boundary term (β
Λ
(ϕ(y)) = ϕ(y),
for y ∈ ∂V ).
This implies the invariance of the correlation functions for given cutoffs V and
K and therefore also in the limit of cutoff removal.
As mentioned in Chap. 29, Elitzur theorem contradicts the mean field ansatz at
the basis of the standard cheap explanation of the Higgs mechanism using a local
gauge invariant Lagrangian; a failure of the mean field ansatz was also displayed in
Chap. 20, Sect. 5, for the one-dimensional Ising model.
However, Elitzur theorem does not invalidate the standard perturbative expansion,
which requires the addition of a gauge fixing which breaks G. Clearly, Elitzur theorem
does not apply to the Coulomb gauge of scalar quantum electrodynamics, since the
local gauge group G is reduced to the identity by the gauge fixing. In the Feynman–
Gupta–Bleuler gauge, the gauge functions of the residual group are restricted to
satisfy the wave equation and do not have compact support in time.
One might argue that the question of the spontaneous breaking of the global
gauge group G and the evasion of the Goldstone theorem are annulled by choosing
the unitary gauge or better the (local and renormalizable) ξ-gauges.
39 However, the
corresponding gauge fixing
L G F = −
1
2 F
a F
a
, F
a
≡ ∂ μ A
a
μ − iξ(t
a
) nm < ϕ m > (ϕ n − < ϕ n >)
involves a mean field ansatz on the vacuum expectation value of the Higgs scalar
field ϕ and the evasion of the Goldstone theorem by a mean field ansatz may be in
conflict with the non-perturbative control, as displayed in Sect. 21.5. Thus, from a
non-perturbative point of view the question remains.
A non-perturbative argument for the absence of physical Goldstone bosons in the
abelian Higgs mechanism has been given in Sect. 29.2. A similar proof may also be
given in the non-abelian case, in the BRST gauge.
40
The BRST quantization is formulated in terms of
i) a local covariant field algebra F generated by the sets of standard fields
(fermionic ψ, Higgs ϕ, vector fields A
a
μ , Nakanishi–Lautrup field B
a ) transforming as irreducible representations of a (global gauge) n-dimensional (compact)
Lie group G and by two additional sets of anticommuting (local) Hermitian
fields c
a
, ¯
c
a , called Faddeev–Popov ghosts;
ii) the local BRST transformations, labelled by a parameter θ, which anticommutes
with c
a , ¯
c
a and with the fermionic fields; on the matter and gauge fields the
infinitesimal BRST transformations have the same form of local gauge transformations with “parameter” θ c
a
(x) (e.g. δ
θ
ψ = it
a
θc
a
ψ, δ
θ A
b
μ = θ(D μ c)
a ,
D μ denoting the covariant derivative) and on the additional fields are defined
by
39 For a detailed analysis of the ξ-gauges, see S. Weinberg, The Quantum Theory of Fields, Vol. II,
Cambridge Univ. Press 1996, Sect. 21.1.
40 G. De Palma and F. Strocchi, Ann. Phys. 336, 112 (2013).