We get $$ V^{pv}=-\frac{f_{\pi}^{2}}{m_{\pi}^{2}}4m^{2}\frac{\overline{u}(p_{1}') \gamma_{5}u(p_{1})\overline{u}(p_{2}')\gamma_{5}u(p_{2})}{(p_{1}-p_{1}') ^{2}-m_{\pi}^{2}}. $$ By inserting expressions for the Dirac spinors, we find $$ \begin{eqnarray*} \overline{u}(p_{1}')\gamma_{5}u(p_{1})&=&\sqrt{\frac{(E_{1}'+m)(E_{1}+m)} {4m^{2}}}\left(\begin{array}{cc}\chi^{\dagger}&-\frac{\sigma_{1}\cdot{ \bf p_{1}}}{E_{1}' +m}\chi^{\dagger}\end{array}\right)\left(\begin{array}{cc}0&1\\1&0\end{array} \right)\nonumber \\ &&\times \left(\begin{array}{c}\chi\\ \frac{\sigma_{1}\cdot\mathbf{p_{1}}}{E_{1}+m}\chi \end{array}\right) \nonumber \\ &=&\sqrt{\frac{(E_{1}'+m)(E_{1}+m)}{4m^{2}}}\left(\frac{\sigma_{1}\cdot \mathbf{p_{1}}}{E_{1}+m}-\frac{\sigma_{1}\cdot\mathbf{p_{1}'}}{E_{1}'+m}\right) \nonumber \end{eqnarray*} $$