Muon-to-positron conversion from light Majorana neutrino exchange: nuclear responses and pion-pole-improved multipoles

We study muon-to-positron conversion in nuclei that is mediated by light Majorana neutrinos, which probes lepton-number violation in the electron-muon sector. As the second paper in a series, this work extends our study of local dimension-nine interactions to the mass mechanism. We derive the conversion amplitude from the second-order effective four-Fermi interaction and express it in terms of nuclear responses in the one-body impulse approximation. Using phenomenological nucleon form factors, we retain convection, axial charge, weak magnetism, and the longitudinal pion-pole current in a specified pion-pole-improved prescription. The two physical time orderings and the distinct momentum transfers are derived in the intermediate-state response. We complete its angular reduction and derive the corresponding closure operator, including spin-gradient terms. Both the symmetric (scalar) and antisymmetric (vector) lepton-hadron Lorentz contractions contribute to the spin-averaged, angle-integrated rate. For $0^+\!\to0^+$ transitions we give explicit formulas in terms of two coherent radial amplitudes, without restricting the intermediate nuclei to spin zero. This provides an interface to nuclear calculations of the required nuclear matrix elements. However, in this work, no isotope-specific numerical predictions are attempted. The present treatment uses approximate lepton wavefunctions and recoil-free nuclear propagation. Two-body weak currents and systematic chiral effective-field-theory matching are left for future work.