We derive now the non-relativistic one-pion exchange interaction.

Here \( p_{1} \), \( p_{1}' \), \( p_{2} \), \( p_{2}' \) and \( k=p_{1}-p_{1}' \) denote four-momenta. The vertices are given by the pseudovector Lagrangian $$ {\cal L}_{pv}=\frac{f_{\pi}}{m_{\pi}}\overline{\psi}\gamma_{5}\gamma_{\mu} \psi\partial^{\mu}\phi_{\pi}. $$ From the Feynman diagram rules we can write the two-body interaction as $$ V^{pv}=\frac{f_{\pi}^{2}}{m_{\pi}^{2}}\frac{\overline{u}(p_{1}')\gamma_{5} \gamma_{\mu}(p_{1}-p_{1}')^{\mu}u(p_{1})\overline{u}(p_{2}')\gamma_{5} \gamma_{\nu}(p_{2}'-p_{2})^{\nu}u(p_{2})}{(p_{1}-p_{1}')^{2}-m_{\pi}^{2}}. $$