Tree scattering in Yukawa theory (${\mathcal{L}}_{\mathrm{int}}=-g\bar{\psi }\varphi \psi$)

For a real scalar coupled through $-g\bar{\psi }\varphi \psi$, the scalar-fermion Feynman vertex is $-ig$. Two-fermion scattering has scalar exchange in the $t$ and $u$ channels with a relative minus from exchanging identical external fermions. Fermion-antifermion scattering has $t$ exchange and $s$ annihilation, again with a relative fermionic minus for a fixed state ordering. Fermion-scalar scattering has the two orderings along one open fermion line, using internal momenta $p_1+p_2$ and $p_1-p_4$; these have the same sign. The Mandelstam variables specify the propagator denominators.

The pseudoscalar Yukawa interaction with coefficient $-g$ has the displayed Feynman vertex. Its two-fermion tree-level Feynman diagrams are the $t$ and $u$ exchanges; antisymmetry of the final identical fermions requires their relative minus. Fermion-antifermion scattering has $t$ exchange and $s$ annihilation, again with a relative minus. With initial and final states ordered as $b_1^{†}{d}_2^{†}|0⟩$ and $b_3^{†}{d}_4^{†}|0⟩$, the normal-ordered current contributes $-d^{†}d\bar{v}{\gamma }^{5}v$ in the exchange channel but $db$ and $b^{†}{d}^{†}$ in the annihilation channel. These fermionic signs fix the overall sign as well as their interference; changing an external-state phase changes both terms together.