Past exam of the particle physics course of the University of Oxford / 2018 / 6 / a / Solution

For negligible fermion masses, the neutral-current vertex connects a fermion and antifermion of opposite helicity. The two incoming choices are $e_L^{-}{e}_R^{+}$ and $e_R^{-}{e}_L^{+}$; the two outgoing choices are $f_L{\bar{f}}_R$ and $f_R{\bar{f}}_L$. Their combinations give four channels, denoted $LL,LR,RL,RR$, where each letter refers to the fermion rather than antifermion.

Take the incoming electron momentum along $+z$ and Positron momentum along $-z$. The initial $L$ pair has both spins along $-z$, giving total projection $-1$; the $R$ pair has both along $+z$, giving projection $+1$. If the outgoing fermion moves along $\hat{\mathbf{n}}$, an outgoing $L$ pair has both spins along $-\hat{\mathbf{n}}$ and an $R$ pair both along $+\hat{\mathbf{n}}$. The figure shows momentum and spin arrows separately for all four channels.