A. Zee, Quantum Field Theory in a Nutshell · Problem IV.6.3

Field Theory Redux

Physics — QFT

Problem

The gauge group

G

does not have to be simple; it could be of the form

G_1 \otimes G_2 \otimes \ldots \otimes G_k

with coupling constants

g_1, g_2, \ldots, g_k

. Consider, for example, the case

G = SU(2) \otimes U(1)

and a Higgs field

\phi

transforming like the doublet under

SU(2)

and like a field with charge

\frac{1}{2}

under

U(1)

, so that

D_\mu \phi = \partial_\mu \phi - i[gA^a_\mu(\tau^a/2) + g'B_\mu \frac{1}{2}] \phi

. Let

\langle\phi\rangle = \begin{pmatrix} 0 \\ v \end{pmatrix}

. Determine which linear combinations of the gauge bosons

A^a_\mu

and

B_\mu

acquire mass.