Lepton-flavor violation (LFV) has been discovered in the neutrino sector by neutrino oscillation experiments. The minimal extension of the Standard Model (SM) to include neutrino masses allows LFV in the charged sector (CLFV) at the loop level, but at rates that are too small to be experimentally observed. Lepton-number violation (LNV) is explicitly forbidden even in the minimally extended SM, so the observation of an LNV process would be unambiguous evidence of physics beyond the SM. The search for the LNV and CLFV process ๐โ+๐(๐ด,๐)โ๐++๐โฒ(๐ด,๐โ2) (referred to as ๐โโ๐+) complements 0๐๐ฝ๐ฝ decay searches, and is sensitive to potential flavor effects in the neutrino mass-generation mechanism. A theoretical motivation for ๐โโ๐+is presented along with a review of the status of past ๐โโ๐+ experiments and future prospects. Special attention is paid to an uncertain and potentially dominant background for these searches, namely, radiative muon capture (RMC). The RMC high energy photon spectrum is theoretically understudied and existing measurements insufficiently constrain this portion of the spectrum, leading to potentially significant impacts on current and future ๐โโ๐+ work.
