Nuclear-level effective theory of μ→e conversion: Formalism and applications

Over the next decade new $\ensuremath{\mu}\ensuremath{\rightarrow}e$ conversion searches at Fermilab (Mu2e) and J-PARC (COMET, DeeMe) are expected to advance limits on charged lepton flavor violation (CLFV) by more than four orders of magnitude. By considering the consequence of $P$ and $CP$ on elastic $\ensuremath{\mu}\ensuremath{\rightarrow}e$ conversion and the structure of possible charge and current densities, we show that rates are governed by six nuclear responses and a single scale, $q/{m}_{N}$, where $q\ensuremath{\approx}{m}_{\ensuremath{\mu}}$ is the momentum transferred from the leptons to the nucleus. To relate this result to microscopic formulations of CLFV, we construct in nonrelativistic effective theory (NRET) the CLFV nucleon-level interaction, pointing out the relevance of the dimensionless scales $y={(\frac{qb}{2})}^{2}>|{\stackrel{P\vec}{v}}_{N}|>|{\stackrel{P\vec}{v}}_{\ensuremath{\mu}}|>|{\stackrel{P\vec}{v}}_{T}|$, where $b$ is the nuclear size, ${\stackrel{P\vec}{v}}_{N}$ and ${\stackrel{P\vec}{v}}_{\ensuremath{\mu}}$ are the nucleon and muon intrinsic velocities, and ${\stackrel{P\vec}{v}}_{T}$ is the target recoil velocity. We discuss previous work, noting the lack of a systematic treatment of the various small parameters. Because the parameter $y$ is not small, a proper calculation of $\ensuremath{\mu}\ensuremath{\rightarrow}e$ conversion requires a full multipole expansion of the nuclear response functions, an apparently daunting task with Coulomb-distorted electron partial waves. We demonstrate that the multipole expansion can be carried out to high precision by introducing a simplifying local momentum ${q}_{\mathrm{eff}}$ for the electron. Previous work has been limited to simple charge or spin interactions, thereby treating the nucleus effectively as a point particle. We show that such formulations are not compatible with the general form of the $\ensuremath{\mu}\ensuremath{\rightarrow}e$ conversion rate, failing to generate three of the six allowed nuclear response functions. The inclusion of the nucleon velocity ${\stackrel{P\vec}{v}}_{N}$ yields an NRET with 16 operators and a rate of the general form. Consequently, in the current discovery era for CLFV, it provides the most sensible starting point for experimental analysis, defining what can and cannot be determined about CLFV from the highly exclusive process of $\ensuremath{\mu}\ensuremath{\rightarrow}e$ conversion. Finally, we expand the NRET operator basis to account for the effects of ${\stackrel{P\vec}{v}}_{\ensuremath{\mu}}$, associated with the muon's lower component, generating corrections to the CLFV coefficients of the point-nucleus response functions. Using advanced shell-model methods, we compute $\ensuremath{\mu}\ensuremath{\rightarrow}e$ conversion rates for a series of experimental targets, deriving bounds on the coefficients of the CLFV operators. These calculations are the first to include a general basis of CLFV operators, full evaluation of the associated nuclear response functions, and an accurate treatment of electron and muon Coulomb effects. We discuss target selection as an experimental ``knob'' that can be turned to probe the microscopic origins of CLFV. We describe two types of coherence that enhance certain CLFV operators and selection rules that blind elastic $\ensuremath{\mu}\ensuremath{\rightarrow}e$ conversion to others. We discuss the matching of the NRET onto higher level effective field theories, such as those constructed at the light quark level, noting opportunities to build on existing work in direct detection of dark matter. We discuss the relation of $\ensuremath{\mu}\ensuremath{\rightarrow}e$ conversion to $\ensuremath{\mu}\ensuremath{\rightarrow}e+\ensuremath{\gamma}$ and $\ensuremath{\mu}\ensuremath{\rightarrow}3e$, showing how MEG II and Mu3e results will complement those of Mu2e and COMET. Finally we describe a accompanying script---in Mathematica and Python versions---that can be used to compute $\ensuremath{\mu}\ensuremath{\rightarrow}e$ conversion rates in various nuclear targets for the full set of NRET operators.