Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 40 / 2 / iv / Solution

The Dirac field has the global symmetry $\psi \mapsto e^{i\alpha }\psi$, $\bar{\psi }\mapsto e^{-i\alpha }\bar{\psi }$, while the real scalar is unchanged. Both the free Dirac action and the pseudoscalar Yukawa interaction preserve this symmetry. Therefore Dirac fermion number conservation holds: its charge counts particles minus antiparticles.

The initial state has charge $+1$, whereas a final antiparticle and a neutral scalar have charge $-1$. Since the S-matrix commutes with that charge,This holds at every order, not just tree level. In the Feynman rules, a continuous fermion arrow cannot connect these specified external states. Replacing only the outgoing $\bar{u}$ by a $v$ in the previous expression would not give a physical amplitude. Changing both external fermions to antiparticles would instead produce an allowed process with reversed fermion flow and the appropriate $v$ spinors; moving a leg between initial and final states is a different operation governed by crossing symmetry.