Past exam of the particle physics course of the University of Oxford / 2021 / 4 / b / Solution

For a massless fermion, chirality is fixed by helicity. For a massless antifermion, the field-chirality label is opposite to its physical helicity: a left-chiral field creates negative-helicity particles and positive-helicity antiparticles. At $E\gg m$, wrong-chirality amplitudes are suppressed by $m/E$, so chirality-preserving vector vertices conserve helicity along a physical fermion line up to these corrections.

For electron-positron annihilation into a muon pair through a vector mediator, each annihilating or created pair therefore has opposite physical helicities. Writing each pair in the order (negative particle, positive particle), the nonzero possibilities areHere $L,R$ refer to physical helicity along each particle's own momentum, not to the sign of its spin projected on a single common axis. The incoming pair has total spin projection $m=\pm 1$ along the electron beam; the outgoing pair has $m^{\prime }=\pm 1$ along the muon direction. Same-helicity pairs vanish in the strictly massless vector-current limit.