In the non-relativistic limit we have to lowest order $$ E_{1}=\sqrt{\mathbf{p}_{1}^{2}+m^{2}}\approx m \approx E_{1}' $$ and then \( (p_{1}-p_{1}')^{2}=-\mathbf{k}^{2} \), so we get for the contribution to the NN potential $$ \begin{eqnarray} V^{pv}&=&-\frac{f_{\pi}^{2}}{m_{\pi}^{2}}4m^{2}\frac{1}{\mathbf{k}^{2}+m^{2}} \frac{2m\cdot 2m}{4m^{2}}\frac{\sigma_{1}}{2m}\cdot(\mathbf{p}_{1}-\mathbf{p'}_{1}) \frac{\sigma_{2}}{2m}\cdot (\mathbf{p}_{1}-\mathbf{p'}_{1}) \nonumber \\ &=&-\frac{f_{\pi}^{2}}{m_{\pi}^{2}} \frac{(\sigma_{1}\cdot\mathbf{k})(\sigma_{2}\cdot\mathbf{k})}{\mathbf{k}^{2}+m_{\pi}^{2}}. \nonumber \end{eqnarray} $$ We have omitted exchange terms and the isospin term \( \mathbf{\tau}_1\cdot\mathbf{\tau}_2 \).