Standard Model and Beyond · Problem 4.2

The Strong Interactions and QCD

Physics — Particle Physics · discrete-gauge-symmetry-spontaneous-breaking

Problem

As mentioned in Section 3.2.5 it often occurs that the spontaneous breaking of a continuous symmetry leaves a discrete subgroup unbroken. If the original symmetry was local, the remaining subgroup is known as a discrete gauge symmetry (e.g., Ibáñez and Ross, 1992). As a simple example, consider a model involving three complex scalars

\phi_i

i = 1 \cdots 3

, with a

U(1)

gauge symmetry. Suppose the potential is of the form

V(\phi_1, \phi_2, \phi_3) = \sum_{i=1}^3 V_i(\phi_i^\dagger\phi_i) + V_{\rm cubic},

where

V_i(\phi_i^\dagger\phi_i)

are quadratic functions of

\phi_i^\dagger\phi_i

(i.e., quadratic and quartic in

\phi_i

and

\phi_i^\dagger

) and

V_{\rm cubic} = \sigma_1\phi_1\phi_2\phi_3 + \sigma_2\phi_1^2\phi_2^\dagger + {\rm h.c.}

(a) Find the

U(1)

charges of the

\phi_i

for which the theory is

U(1)

invariant. (b) Show that

\sigma_{1,2}

can be taken to be real w.l.o.g. (c) Suppose the

V_i

are such that the minimum of

V

occurs for

\langle 0|\phi_3|0\rangle \neq 0

but

\langle 0|\phi_{1,2}|0\rangle = 0

. Show that a discrete

Z_3

symmetry remains unbroken.

Related theory

Spontaneous Symmetry Breaking Particle

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