The time-reversal invariance violating nucleon-nucleon potential in the large $N_C$ expansion

The violation of time reversal invariance (T) in nuclear systems, such as a non-zero neutron electric dipole moment, can serve as a sensitive probe of beyond the standard model (BSM) physics. While the QCD $\theta$ term violates T, its effects are small, thus opening a potential window for the detection of BSM physics. In addition to the neutron electric dipole moment, T violation could also be observed in systems of two and more nucleons. We analyze the T-violating nucleon-nucleon potential in terms of the large $N_C$ expansion of QCD, and identify a hierarchy of terms. This hierarchy can then be mapped onto existing T-violating potentials, such as those derived from meson-exchange models, thereby delineating the terms that should be most important in phenomenological applications.