Experimental Searches for Muon to Electron Conversion in a Nucleus: COMET, DeeMe, and Mu2e. A Contributed paper for Snowmass 21

Searches for charged lepton flavor violation (CLFV) offer deep probes for a range of new physics scenarios, such as super-symmetric models, theories involving scalar leptoquarks or additional Higgs doublets, and models explaining the neutrino mass hierarchy and the matter-antimatter asymmetry of the universe via leptogenesis. The coherent, neutrinoless conversion of a muon to an electron in the field of a nucleus, $\mu^{-}+N\rightarrow e^{-} +N$, is one example of a muonic CLFV process which has sensitivity to this new physics. This paper details three experiments: COMET, DeeMe and Mu2e which will search for $\mu^{-}+N\rightarrow e^{-} +N$ in the coming decade. These experiments offer sensitivity up to an effective new physics mass scale of $\mathcal{O}(10^{4}$ TeV/c$^{2}$), going far beyond what can be achieved in direct, collider-based, searches. The theoretical motivation, designs and anticipated timelines for these three experiments are presented. These experiments are a crucial part of a global search for CLFV. Continued support for all planned experimental searches for muonic CLFV is strongly encouraged.