We obtain $$ V^{pv}(r)=-\frac{f_{\pi}^{2}}{m_{\pi}^{2}}\sigma_{1}\cdot{\nabla}\sigma_{2}\cdot{\nabla}\frac{e^{-m_{\pi}r}}{r}. $$

Carrying out the differentation of $$ V^{pv}(r)=-\frac{f_{\pi}^{2}}{m_{\pi}^{2}}\sigma_{1}\cdot{\nabla}\sigma_{2}\cdot{\nabla}\frac{e^{-m_{\pi}r}}{r}. $$ we arrive at the famous one-pion exchange potential with central and tensor parts $$ V(\mathbf{r})= -\frac{f_{\pi}^{2}}{m_{\pi}^{2}}\left\{C_{\sigma}\mathbf{\sigma}_1\cdot\mathbf{\sigma}_2+ C_T \left( 1 + \frac{3}{m_\alpha r} + \frac{3}{\left(m_\alpha r\right)^2}\right) S_{12}(\hat r)\right\}\frac{\exp{-m_\pi r}}{m_\pi r}. $$ For the full potential add the exchange part and the \( \mathbf{\tau}_1\cdot\mathbf{\tau}_2 \) term as well. (Subtle point: there is a divergence which gets cancelled by using cutoffs) This leads to coefficients \( C_{\sigma} \) and \( C_T \) which are fitted to data.