Standard Model and Beyond · Problem 4.9

The Strong Interactions and QCD

Physics — Particle Physics · gauge-boson-mass-matrix

Problem

Prove that the gauge boson mass matrix in (4.48) is real, symmetric, has nonnegative eigenvalues, and that

N - M

eigenvalues are non-zero. Hint: define an eigenvalue and normalized eigenvector of

M^2

\lambda

and

w

, i.e.,

M^2 w = \lambda w

. Use the fact that

\lambda = w^T M^2 w = w_i M^2_{ij} w_j

. Recall that the vectors

L^i\nu

span an

N - M

dimensional space.