Two-pion exchange contributions to charge asymmetric and charge dependent nuclear forces

We develop a simple but systematic approach to the $2\pi$ exchange contributions to $\mathrm{NN}$ scattering graphs. When applied to $s$-wave $\mathrm{NN}$ scattering lengths, this scheme predicts a $2\pi$ contribution of magnitude ${\left|a_{\mathrm{nn}}\right|}_{2\pi }-{\left|a_{\mathrm{pp}}\right|}_{2\pi }\approx \frac{1}{3}$ fm and ${\left|a_{\mathrm{pn}}\right|}_{2\pi }-{\left|a_{\mathrm{nn}}\right|}_{2\pi }\approx -\frac{1}{6}$ fm. If we then combine this effect with simple ${\pi }^{\pm }$, ${\pi }^0$ and ${\rho }^{\pm }$, ${\rho }^0$ pole graphs, the net charge asymmetric scattering length is $\left|a_{\mathrm{nn}}\mathrm{}\right|-\left|\mathrm{}{a}_{\mathrm{pp}}\right|\approx 1.2$ fm while the net charge dependent result is $\left|a_{\mathrm{pn}}\mathrm{}\right|-\left|\mathrm{}{a}_{\mathrm{nn}}\right|\approx 2.7$ fm. This compares favorably with the respective experimental values of approximately 1.5 and 5 fm.