Standard Model and Beyond · Problem 4.5

The Strong Interactions and QCD

Physics — Particle Physics · non-abelian-gauge-transformation

Problem

The gauge transformation for a non-abelian gauge field is given in (4.24). (a) Verify that (4.24) reduces to (4.26) for small

|\beta^i|

. (b) Verify the transformation (4.22) for a matter field. (c) Use (4.24) to prove (4.30) for all

\vec{\beta}

(i.e., not just small

|\beta^i|

). This implies that

\mathcal{L}_{KE_A}

is invariant. (d) Rederive (4.30) by the simpler method of first proving

ig\vec{F}_{\mu\nu} \cdot \vec{L} = [D_\mu, D_\nu]

, with

D_\mu

from (4.17), and then showing that

UD_\mu U^{-1} = D'_\mu \equiv \partial_\mu + ig\vec{A}^{\prime}_\mu \cdot \vec{L}