Systematic calculation of s-wave pion and kaon self-energies in asymmetric nuclear matter

We calculate the pion self-energies in asymmetric nuclear matter in the two-loop approximation of chiral perturbation theory. We find three types of corrections beyond the well-known linear density approximation. The resulting s-wave potential (or equivalent mass-shift) of a $\pi^-$ in the center of a heavy nucleus like Pb turns out to be $U_{\pi^-}=\Delta m_{\pi^-} \simeq 14$ MeV, about half of what is needed to form the deeply bound and narrow pionic atom-states recently observed at GSI. The potential for a $\pi^+$ under the same conditions is $U_{\pi^+}=\Delta m_{\pi^+} \simeq -1$ MeV. In the same way we calculate the mass-shifts of $K^+$-mesons in symmetric nuclear matter and neutron matter and find an increase of the $K^+$-mass by 9% and 5%, respectively. As a further application we give an analytical result for the third order term in the scattering length expansion of the in-medium self-energy of an interacting light-boson/heavy-fermion system.