Srednicki — Quantum Field Theory · Problem 87.3

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · standard-model-gauge-higgs-sector

Problem

In this problem we will work out the generator matrices introduced in section 86 for the case of the Standard Model. a) Write the Higgs field as

\varphi = \frac{1}{\sqrt{2}}\begin{pmatrix}\phi_1 + i\phi_3 \\ \phi_2 + i\phi_4\end{pmatrix} \,, \tag{87.30}

where

\phi_i

is a real scalar field. Express the SU(2) generators

T^a

and the hypercharge generator

Y

4\times 4

matrices

\mathcal{T}^a

and

\mathcal{Y}

that act on

\phi_i

. Hint: see problem 86.1. b) Compute the matrix

F^a{}_i

, defined in eq.\,(86.12). c) Compute the mass-squared matrix for the vector fields,

(M^2)^{ab} = F^a{}_i F^b{}_i

, and find its eigenvalues.