The factors \( p_{1}-p_{1}'=p_{2}'-p_{2} \) are both the four-momentum of the exchanged meson and come from the derivative of the meson field in the interaction Lagrangian. The Dirac spinors obey $$ \begin{eqnarray*} \gamma_{\mu}p^{\mu}u(p)&=&mu(p) \nonumber \\ \overline{u}(p)\gamma_{\mu}p^{\mu}&=&m\overline{u}(p). \nonumber \end{eqnarray*} $$