Charge dependent nucleon-nucleon potentials in covariant chiral effective field theory

The charge-dependent nucleon-nucleon ($NN$) interactions are crucial in understanding the nuclear structure and reaction problems. This work explores the charge-dependent $NN$ interaction in covariant chiral effective field theory. By incorporating the isospin-breaking contributions, we derive the charge-dependent covariant chiral $NN$ potential up to the next-to-next-to-leading order (NNLO). The calculated $np$ and $pp$ phase shifts are in satisfactory agreement with the PWA93 partial wave analysis. Our results contribute to a deeper understanding of the isospin-breaking effects in nuclear forces and provide a solid foundation for future studies of nuclear structure and reactions in the context of a covariant framework.