Chapter 29
Symmetry Breaking in Gauge Theories
The standard model of elementary particle physics crucially relies on the description
of elementary particle interactions by gauge field theories and the unification of electromagnetic and weak interactions is made possible by the mechanism of spontaneous
symmetry breaking. The intriguing fact that such a breaking of a continuous symmetry (the SU (2) × U (1) group) is not accompanied by massless Goldstone bosons, in
apparent contrast with the conclusions of the Goldstone theorem, demands a general
(possibly non-perturbative) understanding and control of such a phenomenon, the
so-called Higgs mechanism.
29.1 Higgs Mechanism: Problems of the Perturbative
Approach
The standard discussion of this mechanism is based on the perturbative expansion
and, in particular, the evasion of the Goldstone theorem is checked at the tree level
with the disappearance of the massless Goldstone bosons and with the vector bosons
becoming massive.
191 This is clearly displayed by the Higgs–Kibble (abelian) model
of a (complex) scalar field ϕ interacting with a real gauge field A μ , defined by the
following Lagrangian (ρ(x) ≡ |ϕ(x)|)
L = −
1
4
F μν
2
+
1
2
|D μ ϕ|
2
− U (ρ) D μ = ∂ μ − ie A μ .
(29.1)
191 P.W. Higgs, Phys. Lett. 12, 132 (1964); Phys. Rev. Lett. 13, 508 (1964); Phys. Rev. 145, 1156
(1966); Spontaneous Symmetry Breaking, in Phenomenology of particle physics at high energy:
Proc. 14th Scottish Univ. Summer School in Physics 1973, R.L. Crawford and R. Jennings eds.,
Academic Press 1974, p. 529; G.S. Guralnik, C.R. Hagen and T.W. Kibble, Phys. Rev. Lett. 13, 585
(1964); Broken Symmetries and the Goldstone Theorem, in Advances in Particle Physics, Vol. 2,
R.L. Cool and R.E. Marshak eds., Interscience 1968, p. 567; T.W. Kibble, Broken Symmetries, in
Proc. Oxford Int. Conf. on Elementary Particles, 1965, Oxford Univ. Press 1966, p. 19; Phys. Rev.
155, 1554 (1966); F. Englert and R. Brout, Phys. Rev. Lett. 13, 321 (1964).
© 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_29
203
Symmetry Breaking in Gauge Theories
The standard model of elementary particle physics crucially relies on the description
of elementary particle interactions by gauge field theories and the unification of electromagnetic and weak interactions is made possible by the mechanism of spontaneous
symmetry breaking. The intriguing fact that such a breaking of a continuous symmetry (the SU (2) × U (1) group) is not accompanied by massless Goldstone bosons, in
apparent contrast with the conclusions of the Goldstone theorem, demands a general
(possibly non-perturbative) understanding and control of such a phenomenon, the
so-called Higgs mechanism.
29.1 Higgs Mechanism: Problems of the Perturbative
Approach
The standard discussion of this mechanism is based on the perturbative expansion
and, in particular, the evasion of the Goldstone theorem is checked at the tree level
with the disappearance of the massless Goldstone bosons and with the vector bosons
becoming massive.
191 This is clearly displayed by the Higgs–Kibble (abelian) model
of a (complex) scalar field ϕ interacting with a real gauge field A μ , defined by the
following Lagrangian (ρ(x) ≡ |ϕ(x)|)
L = −
1
4
F μν
2
+
1
2
|D μ ϕ|
2
− U (ρ) D μ = ∂ μ − ie A μ .
(29.1)
191 P.W. Higgs, Phys. Lett. 12, 132 (1964); Phys. Rev. Lett. 13, 508 (1964); Phys. Rev. 145, 1156
(1966); Spontaneous Symmetry Breaking, in Phenomenology of particle physics at high energy:
Proc. 14th Scottish Univ. Summer School in Physics 1973, R.L. Crawford and R. Jennings eds.,
Academic Press 1974, p. 529; G.S. Guralnik, C.R. Hagen and T.W. Kibble, Phys. Rev. Lett. 13, 585
(1964); Broken Symmetries and the Goldstone Theorem, in Advances in Particle Physics, Vol. 2,
R.L. Cool and R.E. Marshak eds., Interscience 1968, p. 567; T.W. Kibble, Broken Symmetries, in
Proc. Oxford Int. Conf. on Elementary Particles, 1965, Oxford Univ. Press 1966, p. 19; Phys. Rev.
155, 1554 (1966); F. Englert and R. Brout, Phys. Rev. Lett. 13, 321 (1964).
© 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_29
203