Determination of coupling patterns by parallel searches for $\mu^-\to e^+$ and $\mu^-\to e^-$ in muonic atoms

We investigate a possibility that the $\mu^-\to e^+$ conversion is discovered prior to the $\mu^-\to e^-$ conversion, and its implications to the new physics search. We focus on the specific model including the mixing of the $SU(2)_L$ doublet- and singlet-type scalar leptoquarks, which induces not only the lepton flavor violation but also the lepton number violation. Such a structure is motivated by R-parity violating (RPV) supersymmetric models, where a sbottom mediates the conversion processes. We formulate the $\mu^-\to e^+$ rate in analogy with the muon capture in a muonic atom, and numerically evaluate it using several target nuclei. The lepton flavor universality test of pion decay directly limits the $\mu^-\to e^+$ rate, and the maximally allowed $\mu^-\to e^+$ branching ratio is $\sim 10^{-18}$ under the various bounds on RPV parameters. We show that either $\mu^-\to e^-$ or $\mu^-\to e^+$ signals can be discovered in near future experiments. This indicates that parallel searches for these conversions will give us significant information on the pattern of coupling constants.