= Chiral fermion trace contraction
{title2=$H^{\mu\nu}(a,b)H_{\mu\nu}(c,d)=4(a\cdot c)(b\cdot d)$}
With <Minkowski metric> signature $+---$ and $\operatorname{tr}(\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma\gamma_5)=4i\epsilon^{\mu\nu\rho\sigma}$, define $H^{\mu\nu}(a,b)=a^\mu b^\nu+a^\nu b^\mu-g^{\mu\nu}a\cdot b+i\epsilon^{\mu\nu\rho\sigma}a_\rho b_\sigma$. Then $\operatorname{tr}[\not a\gamma^\mu(1-\gamma_5)\not b\gamma^\nu(1-\gamma_5)]=8H^{\mu\nu}(a,b)$. The symmetric-tensor contraction gives twice the sum of the two possible dot-product pairings; the <Levi-Civita symbol> contraction, including the two factors of $i$, gives twice their difference. Their sum is the displayed identity, explaining the single pairing in left-chiral weak rates.
Back to article page
