29.1 Higgs Mechanism: Problems of the Perturbative Approach
205
not invariant under the U (1) global group (symmetry breaking). Thus, the expansion
can be seen as an expansion around a (symmetry breaking) mean field ansatz, and
it is very important that a renormalized perturbation theory based on it exists and
yields a non-vanishing symmetry breaking order parameter < ϕ > = 0 at all orders.
This is the standard (perturbative) analysis of the Higgs mechanism.
The extraordinary success of the standard model motivates an examination of the
Higgs mechanism from a general non-perturbative point of view. In this perspective,
one of the problems is that mean field expansions may yield misleading results about
the occurrence of symmetry breaking and the energy spectrum (see Chaps. 20, 21
above).
As a matter of fact, a non-perturbative analysis of the possible existence of a symmetry breaking order parameter, by using the Euclidean functional integral approach
defined by the Lagrangian (29.1), gives symmetric correlation functions and in particular < ϕ >= 0 (Elithur-De Angelis-De Falco-Guerra (EDDG) theorem).
193 This
means that the mean field ansatz is incompatible with the non-perturbative quantum
effects and the approximation leading to (29.3) is not correct.
The same negative conclusion would be reached if, (as an alternative to the transformation which leads to (29.2)), by means of a gauge transformation one reduces
ϕ(x) to a real, not necessarily positive, field ϕ r (x). This means that the local gauge
invariance has not been completely eliminated and the corresponding Lagrangian,
of the same form (29.2) with ρ replaced by ϕ r , is invariant under a residual Z 2 local
gauge group. Then, an easy adaptation of the proof of the EDDG theorem gives
< ϕ >= 0 and no symmetry breaking.
In order to avoid the vanishing of a symmetry breaking order parameter, one
must reconsider the problem by adding to the Lagrangian (29.1) a gauge fixing L G F
which breaks local gauge invariance. Then, the discussion of the Higgs mechanism
necessarily becomes gauge fixing dependent; this should not appear strange, since
the vacuum expectation of ϕ is a gauge dependent quantity.
194
A non-perturbative analysis of the Higgs mechanism shall be discussed in the
following subsections in the prototypic cases of local gauges and of the physical
Coulomb gauge; in the latter case we shall get a complete characterization of the
Higgs mechanism, namely both the absence of Goldstone bosons and the related
absence of massless vector bosons.
193 S. Elitzur, Phys. Rev. D 12, 3978 (1975); G.F. De Angelis, D. De Falco and F. Guerra, Phys. Rev.
D 17, 1624 (1978). The crux of the argument is that gauge invariance decouples the transformations
of the fields inside a volume V (in a Euclidean functional integral approach) from the transformation
of the boundary, so that the boundary conditions are ineffective and cannot trigger non- symmetric
correlation functions. For a simple account of the argument, see, e.g. F. Strocchi, Elements of
Quantum Mechanics of Infinite Systems, World Scientific 1985, Part C, Sect. 2.5.
194 The above problem of non-perturbative consistency arises also for gauge fixings involving a
mean field ansatz, as for the case of the unitary gauge; see S. Weinberg, The Quantum theory of
Fields, Vol. II, Sect. 21.1.
205
not invariant under the U (1) global group (symmetry breaking). Thus, the expansion
can be seen as an expansion around a (symmetry breaking) mean field ansatz, and
it is very important that a renormalized perturbation theory based on it exists and
yields a non-vanishing symmetry breaking order parameter < ϕ > = 0 at all orders.
This is the standard (perturbative) analysis of the Higgs mechanism.
The extraordinary success of the standard model motivates an examination of the
Higgs mechanism from a general non-perturbative point of view. In this perspective,
one of the problems is that mean field expansions may yield misleading results about
the occurrence of symmetry breaking and the energy spectrum (see Chaps. 20, 21
above).
As a matter of fact, a non-perturbative analysis of the possible existence of a symmetry breaking order parameter, by using the Euclidean functional integral approach
defined by the Lagrangian (29.1), gives symmetric correlation functions and in particular < ϕ >= 0 (Elithur-De Angelis-De Falco-Guerra (EDDG) theorem).
193 This
means that the mean field ansatz is incompatible with the non-perturbative quantum
effects and the approximation leading to (29.3) is not correct.
The same negative conclusion would be reached if, (as an alternative to the transformation which leads to (29.2)), by means of a gauge transformation one reduces
ϕ(x) to a real, not necessarily positive, field ϕ r (x). This means that the local gauge
invariance has not been completely eliminated and the corresponding Lagrangian,
of the same form (29.2) with ρ replaced by ϕ r , is invariant under a residual Z 2 local
gauge group. Then, an easy adaptation of the proof of the EDDG theorem gives
< ϕ >= 0 and no symmetry breaking.
In order to avoid the vanishing of a symmetry breaking order parameter, one
must reconsider the problem by adding to the Lagrangian (29.1) a gauge fixing L G F
which breaks local gauge invariance. Then, the discussion of the Higgs mechanism
necessarily becomes gauge fixing dependent; this should not appear strange, since
the vacuum expectation of ϕ is a gauge dependent quantity.
194
A non-perturbative analysis of the Higgs mechanism shall be discussed in the
following subsections in the prototypic cases of local gauges and of the physical
Coulomb gauge; in the latter case we shall get a complete characterization of the
Higgs mechanism, namely both the absence of Goldstone bosons and the related
absence of massless vector bosons.
193 S. Elitzur, Phys. Rev. D 12, 3978 (1975); G.F. De Angelis, D. De Falco and F. Guerra, Phys. Rev.
D 17, 1624 (1978). The crux of the argument is that gauge invariance decouples the transformations
of the fields inside a volume V (in a Euclidean functional integral approach) from the transformation
of the boundary, so that the boundary conditions are ineffective and cannot trigger non- symmetric
correlation functions. For a simple account of the argument, see, e.g. F. Strocchi, Elements of
Quantum Mechanics of Infinite Systems, World Scientific 1985, Part C, Sect. 2.5.
194 The above problem of non-perturbative consistency arises also for gauge fixings involving a
mean field ansatz, as for the case of the unitary gauge; see S. Weinberg, The Quantum theory of
Fields, Vol. II, Sect. 21.1.