Future Experimental Improvement for the Search of LNV Process in $e\mu$ Sector

Exploring the leptonic sector in frontier experiments is more of importance nowadays, since the conservation of lepton flavor and total lepton number are not guaranteed anymore in the Standard Model after the discovery of neutrino oscillations. $\mu^- + N(A,Z) \rightarrow e^+ + N(A,Z-2)$ conversion in a muonic atom is one of the most promising channels to investigate the lepton number violation process, and the measurement of this process is planned in future $\mu^--e^-$ conversion experiments with a muonic atom in a muon-stopping target. This paper discusses how to maximize the experimental sensitivity of the $\mu^--e^+$ conversion by introducing the new requirement of the mass relation of $M(A,Z-2)<M(A,Z-1)$, where $M(A,Z)$ is the mass of the muon-stopping target nucleus, to get rid of the background from radiative muon capture. The sensitivity of the $\mu^--e^+$ conversion is anticipated to have four orders of magnitude of improvement in forthcoming experiments using a proper target nucleus, which satisfies the mass relation. The most promising isotopes found are $^{40}$Ca and $^{32}$S.