Srednicki — Quantum Field Theory · Problem 83.7

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · chiral-symmetry-breaking-u1

Problem

Suppose that the

\mathrm{U}(1)_A

symmetry is not anomalous, so that we must include a ninth Goldstone boson. We can write

U(x) = \exp[2i\pi^a(x)T^a/f_\pi + i\pi^9(x)/f_9] . \tag{83.33}

The ninth Goldstone boson is given its own decay constant

f_9

, since there is no symmetry that forces it to be equal to

f_\pi

. We write the two-derivative terms in the lagrangian as

\mathcal{L} = -\tfrac{1}{4}f_\pi^2\,\mathrm{Tr}\,\partial^\mu U^\dagger\partial_\mu U - \tfrac{1}{4}F^2\,\partial^\mu(\det U^\dagger)\partial_\mu(\det U) . \tag{83.34}

a) By requiring all nine Goldstone fields to have canonical kinetic terms, determine

F

in terms of

f_\pi

and

f_9

. b) To simplify the analysis, let

m_u = m_d \equiv m \ll m_s

. Find the masses of the nine pseudogoldstone bosons. Identify the three lightest as the pions, and call their mass

m_\pi

. Show that another one of the nine has a mass less than or equal to

\sqrt{3}\,m_\pi

. (The nonexistence of such a particle in nature is the

U(1)

problem; the axial anomaly solves this problem.)