Standard Model and Beyond · Problem 2.10

Lie Groups, Lie Algebras, and Symmetries

Physics — Particle Physics · gordon-decomposition

Problem

Prove the Gordon decomposition formulas

\begin{aligned} 2m(\bar{u}_2\gamma^\mu u_1) &= \bar{u}_2(p_2+p_1)^\mu u_1 + i\bar{u}_2\sigma^{\mu\nu}(p_2-p_1)_\nu u_1\\ 2m(\bar{u}_2\gamma^\mu\gamma^5 u_1) &= \bar{u}_2(p_2-p_1)^\mu\gamma^5 u_1 + i\bar{u}_2\sigma^{\mu\nu}(p_2+p_1)_\nu\gamma^5 u_1\\ 0 &= \bar{u}_2(p_2-p_1)^\mu u_1 + i\bar{u}_2\sigma^{\mu\nu}(p_2+p_1)_\nu u_1\\ 0 &= \bar{u}_2(p_2+p_1)^\mu\gamma^5 u_1 + i\bar{u}_2\sigma^{\mu\nu}(p_2-p_1)_\nu\gamma^5 u_1, \end{aligned}

where

u_{1,2}

are two Dirac

u

spinors for a particle of mass

m