Spin-Independent Two-Neutrino Exchange Potential with Mixing and $CP$-Violation

We develop a new approach for calculating the spin-independent 2-neutrino exchange potential (2-NEP) between non-relativistic fermions which places emphasis on the neutrino vacuum state, an area of theoretical interest in recent years. The 2-NEP is a natural probe of fundamental issues of neutrino physics such as neutrino masses, flavor mixing, the number of neutrino flavors, neutrino nature (Dirac or Majorana), $CP$-violation, and the neutrino vacuum state. We explore the dependence of the 2-NEP on the mixing of neutrino mass states assuming normal and inverted mass ordering for nucleon-nucleon, nucleon-lepton, and lepton-lepton interactions, and the $CP$-violation phase in the neutrino mixing matrix.