In this appendix, we discuss three related topics: the Goldstone theorem, chiral symmetry breaking in the strong interactions, and the extraordinary application of the axial anomaly to the decay of the neutral pion. These topics draw heavily on nonperturbative reasoning, and therefore stand somewhat outside our main line of discussion. At the same time, they are so central to our understanding of the standard model as to bear at least brief description.

The Goldstone theorem

As defined in Section 5.4, a symmetry is spontaneously broken when it does not leave the vacuum state invariant. According to the Goldstone theorem, a spontaneously broken symmetry implies the existence of zero-mass scalar particles. The proof of the Goldstone theorem given by Goldstone, Salam & Weinberg (1962) is very accessible, and the reader is referred to their paper for further details. We give a variant of their argument here.

Consider a set of generators Qa, which commute with the Hamiltonian, [Qa, H] = 0, but for which Qa∣0〉 = ∣a〉 ≠ ∣0〉. That is, each Qa acting on the vacuum gives a new, nonzero state. Evidently the vacuum is not unique. We are familiar with this situation from the standard-model Higgs Lagrange density with a wrong-sign mass term, eq. (5.110).