Nuclear-Level Effective Theory of $\mu\rightarrow e$ Conversion

The Mu2e and COMET $\mu \rightarrow e$ conversion experiments are expected to significantly advance limits on new sources of charged lepton flavor violation (CLFV). Almost all theoretical work in the field has focused on just two operators. However, general symmetry arguments lead to a $\mu \rightarrow e$ conversion rate with six response functions, each of which, in principle, is observable by varying nuclear properties of targets. We construct a nucleon-level nonrelativistic effective theory (NRET) to clarify the microscopic origin of these response functions and to relate rate measurements in different targets. This exercise identifies three operators and their small parameters that control the NRET operator expansion. We note inconsistencies in past treatments of these parameters. The NRET is technically challenging, involving 16 operators, several distorted electron partial waves, bound muon upper and lower components, and an exclusive nuclear matrix element. We introduce a trick for treating the electron Coulomb effects accurately, which enables us to include all of these effects while producing transition densities whose one-body matrix elements can be evaluated analytically, greatly simplifying the nuclear physics. We derive bounds on operator coefficients from existing and anticipated $\mu \rightarrow e$ conversion experiments. We discuss how similar NRET formulations have impacted dark matter phenomenology, noting that the tools this community has developed could be adapted for CLFV studies.