Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 50 / 2 / d / Solution

Use the Feynman vertex $-i\lambda$ for the scalar Yukawa coupling, the internal scalar propagator $i/(k^2-{\mu }^2+i0)$, incoming factors $u_r(p)$ and outgoing Dirac adjoints ${\bar{u}}_{r^{\prime }}(p^{\prime })$. Define $i\mathcal{M}$ as the coefficient in the connected S-matrix element after removing its overall momentum-conserving Dirac delta function. A direct pairing contributesThe crossed pairing interchanges the identical outgoing fermions and therefore has a relative fermionic sign of minus. With $t=(p-p^{\prime }{)}^2$ and $u=(p-q^{\prime }{)}^2$, the result isIt changes sign under exchange of the two outgoing momenta and spin labels, as required for identical fermions. No fermion spin sum or identical-particle phase-space factor belongs in this amplitude; those enter a relativistic scattering cross-section calculation.