Chiral projector ($P_{L,R}=(1\mp {\gamma }^5)/2$)

The projectors $P_L=(1-{\gamma }^5)/2$ and $P_R=(1+{\gamma }^5)/2$ select the two chirality eigenspaces of a Dirac field. They satisfy $P_L+P_R=I$, $P_L{P}_R=0$, and $P_L^2=P_L$, $P_R^2=P_R$.

A Lorentz scalar Dirac bilinear connects opposite chiralities. In the stated on-shell normalization ${\bar{u}}_R{u}_L=m_{\mathrm{spinor}}$, whereas ${\bar{u}}_L{u}_L=0$. Such an overlap verifies the allowed Lorentz structure but does not determine an independent coefficient multiplying a different mass operator.

With Minkowski metric signature $+---$ and $\mathrm{tr}({\gamma }^{\mu }{\gamma }^{\nu }{\gamma }^{\rho }{\gamma }^{\sigma }{\gamma }_5)=4i{ϵ}^{\mu \nu \rho \sigma }$, define $H^{\mu \nu }(a,b)=a^{\mu }{b}^{\nu }+a^{\nu }{b}^{\mu }-g^{\mu \nu }a\cdot b+i{ϵ}^{\mu \nu \rho \sigma }{a}_{\rho }{b}_{\sigma }$. Then $\mathrm{tr}[/a{\gamma }^{\mu }(1-{\gamma }_5)/b{\gamma }^{\nu }(1-{\gamma }_5)]=8H^{\mu \nu }(a,b)$. 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.