$\pi$-$\eta$ mixing and charge symmetry violating NN potential in matter

We construct density dependent Class $III$ charge symmetry violating (CSV) potential due to mixing of $\pi$-$\eta$ meson with off-shell corrections. The density dependence enters through the non-vanishing $\pi$-$\eta$ mixing driven by both the neutron-proton mass difference and their asymmetric density distribution. The contribution of density dependent CSV potential is found to be appreciably larger than the contribution of vacuum CSV potential.