Appendix E
Non-abelian Higgs Mechanism
At the basis of the Brout–Englert–Higgs (BEH) mechanism, briefly Higgs mechanism, there is the spontaneous breaking of a gauge symmetry and the issue has been
raised about the meaning and the non-perturbative control of this phenomenon.
A first negative result comes from the Elitzur theorem
38 according to which if
the Lagrangian density (or the Action) is invariant under the group G 0 ⊂ G of gauge
transformations with gauge functions of compact support in space and time, the
corresponding global gauge group G cannot be broken. The argument exploits the
non-perturbative criterion of symmetry breaking discussed in Chap. 20, by which in
the thermodynamical limit the non-invariance of the Euclidean correlation functions
arises from their dependence on the boundary conditions imposed on the functional
integral with a space cutoff V and an ultraviolet cutoff K ; local gauge invariance
precludes the coupling with the boundary conditions, so that the boundary term A ∂V
in Eq. (20.5) becomes ineffective.
In fact, if β
λ , λ ∈ R is a one-parameter subgroup of G, for checking the invariance
of the correlation functions of the Euclidean fields at the points x 1 , x 2 , ..., x n , with
boundary condition ϕ(y) = ¯
ϕ, for y ∈ ∂V , one may consider a sphere O of radius
R, which contains the Euclidean points x 1 , ...x n , well inside V , and a gauge function
Λ(x) ∈ G 0 , such that Λ(x) = λ, for x ∈ O and Λ(x) = 0, whenever |x| > R(1 + ε),
(so that β
Λ
(ϕ(x)) = ϕ(x) for |x| > R(1 + ε)). Then, by Eq. (20.5), one has
< β
λ
(ϕ(x 1 )...ϕ(x n )) > V,K , ¯
ϕ =< β
Λ
(ϕ(x 1 )...ϕ(x n )) > V,K , ¯
ϕ =
< ϕ(x 1 )...ϕ(x n )) > V,K , (β Λ ) −1 ( ¯
ϕ) =< ϕ(x 1 )...ϕ(x n )) > V,K , ¯
ϕ ,
(E.1)
38 S. Elitzur, Phys. Rev. D 2, 3978 (1975); the rigorous proof of the theorem is due to G.F. De
Angelis, D. De Falco and F. Guerra, Phys. Rev. D 17, 1624 (1978).
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0
257
Non-abelian Higgs Mechanism
At the basis of the Brout–Englert–Higgs (BEH) mechanism, briefly Higgs mechanism, there is the spontaneous breaking of a gauge symmetry and the issue has been
raised about the meaning and the non-perturbative control of this phenomenon.
A first negative result comes from the Elitzur theorem
38 according to which if
the Lagrangian density (or the Action) is invariant under the group G 0 ⊂ G of gauge
transformations with gauge functions of compact support in space and time, the
corresponding global gauge group G cannot be broken. The argument exploits the
non-perturbative criterion of symmetry breaking discussed in Chap. 20, by which in
the thermodynamical limit the non-invariance of the Euclidean correlation functions
arises from their dependence on the boundary conditions imposed on the functional
integral with a space cutoff V and an ultraviolet cutoff K ; local gauge invariance
precludes the coupling with the boundary conditions, so that the boundary term A ∂V
in Eq. (20.5) becomes ineffective.
In fact, if β
λ , λ ∈ R is a one-parameter subgroup of G, for checking the invariance
of the correlation functions of the Euclidean fields at the points x 1 , x 2 , ..., x n , with
boundary condition ϕ(y) = ¯
ϕ, for y ∈ ∂V , one may consider a sphere O of radius
R, which contains the Euclidean points x 1 , ...x n , well inside V , and a gauge function
Λ(x) ∈ G 0 , such that Λ(x) = λ, for x ∈ O and Λ(x) = 0, whenever |x| > R(1 + ε),
(so that β
Λ
(ϕ(x)) = ϕ(x) for |x| > R(1 + ε)). Then, by Eq. (20.5), one has
< β
λ
(ϕ(x 1 )...ϕ(x n )) > V,K , ¯
ϕ =< β
Λ
(ϕ(x 1 )...ϕ(x n )) > V,K , ¯
ϕ =
< ϕ(x 1 )...ϕ(x n )) > V,K , (β Λ ) −1 ( ¯
ϕ) =< ϕ(x 1 )...ϕ(x n )) > V,K , ¯
ϕ ,
(E.1)
38 S. Elitzur, Phys. Rev. D 2, 3978 (1975); the rigorous proof of the theorem is due to G.F. De
Angelis, D. De Falco and F. Guerra, Phys. Rev. D 17, 1624 (1978).
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
F. Strocchi, Symmetry Breaking, Theoretical and Mathematical Physics,
https://doi.org/10.1007/978-3-662-62166-0
257