A. Zee, Quantum Field Theory in a Nutshell · Problem IV.6.2

Field Theory Redux

Physics — QFT

Problem

In general, there may be several Higgs fields belonging to various representations labeled by

\alpha

. Show that the mass squared matrix for the gauge bosons generalize immediately to

(\mu^2)_{ab} = \sum_\alpha g^2(T^a_\alpha v_\alpha \cdot T^b_\alpha v_\alpha)

, where

v_\alpha

is the vacuum expectation value of

\phi_\alpha

and

T^a_\alpha

is the

a

th generator represented on

\phi_\alpha

. Apply this to the situations described in exercises IV.6.1 and IV.6.2 and work out the mass spectrum of the gauge bosons.