Srednicki — Quantum Field Theory · Problem 84.1

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · spontaneous-breaking-gauge-symmetries

Problem

Consider a theory with gauge group

\mathrm{SU}(N)

, with a real scalar field

\Phi

in the adjoint representation, and potential

V(\Phi) = \tfrac{1}{2}m^2\,\mathrm{Tr}\,\Phi^2 + \tfrac{1}{4}\lambda_1\,\mathrm{Tr}\,\Phi^4 + \tfrac{1}{4}\lambda_2\,(\mathrm{Tr}\,\Phi^2)^2 . \tag{84.17}

This is the most general potential consistent with

\mathrm{SU}(N)

symmetry and a

\mathbb{Z}_2

symmetry

\Phi \leftrightarrow -\Phi

, which we impose to keep things simple. We assume

m^2 < 0

. We can work in a basis in which

\Phi = v\,\mathrm{diag}(\alpha_1,\ldots,\alpha_N)

, with the constraints

\sum_i \alpha_i = 0

and

\sum_i \alpha_i^2 = 1

. a) Extremize

V(\Phi)

with respect to

v

. Solve for

v

, and plug your result back into

V(\Phi)

. You should find

V(\Phi) = \frac{-\tfrac{1}{4}(m^2)^2}{\lambda_1 A(\alpha) + \lambda_2 B(\alpha)} , \tag{84.18}

where

A(\alpha)

and

B(\alpha)

are to be determined.