When we perform similar non-relativistic expansions for scalar and vector mesons we obtain for the $\sigma$ meson $$V^{\sigma}= g_{\sigma NN}^{2}\frac{1}{\mathbf{k}^{2}+m_{\sigma}^{2}}\left (-1+\frac{\mathbf{q}^{2}}{2M_N^2} -\frac{\mathbf{k}^{2}}{8M_N^2}-\frac{\mathbf{LS}}{2M_N^2}\right).$$ We note an attractive central force and spin-orbit force. This term has an intermediate range. We have defined $1/2(p_{1}+p_{1}')=\mathbf{q}$. For the full potential add the exchange part and the isospin dependence as well.