The Effective Theory of Muon-to-Electron Conversion

TL;DR

This work develops a complete low-energy, nonrelativistic EFT for muon-to-electron conversion, expressing rates as a product of leptonic LECs and nuclear response functions to capture all possible CLFV operators. By embedding a 16-operator basis into a nuclear framework and accounting for distorted electron waves, the approach identifies nuclear responses as experimental dials that can be tuned via target choice, particularly using $^{27}$Al to access both elastic and inelastic channels. The study argues that inelastic transitions to low-lying $^{27}$Al states provide powerful discriminants of the underlying CLFV operators, enabling not only discovery potential but mechanism identification, with a clear path to matching these low-energy results to high-energy UV theories through an EFT tower (MuonBridge). This framework supports interpreting upcoming Mu2e/COMET results, constraining CLFV sources and their UV scales (potentially exceeding $10^4$ TeV), and guiding joint analyses with other experimental limits.

Abstract

We summarize recent work to develop an effective theory of muon-to-electron conversion, based on a complete set of low-energy effective operators that are developed from a systematic expansion in velocities and momenta. The expansion effectively factors rates into sums of particle physics and nuclear physics terms, where the former are expressed as bilinears in the LECs (the low-energy constants of the effective theory) and the latter are the associated nuclear responses. One can view the nuclear responses as ``dials" that can be adjusted -- for example, by selection of targets with specific properties -- in order to isolate the former. We show that an important dial, in the case of Mu2e and COMET, will be inelastic transitions to certain low-energy nuclear states that are resolvable in 27Al. If these transitions are exploited, the experiments have the potential not only to discover charged lepton flavor violation (CLFV), but to determine the operators responsible for the CLFV. We also discuss how such low-energy results can be ``ported" to higher energies through a tower of matched EFTs, so they can be combined with other experimental limits to further constrain CLFV

Figures (3)

• Figure 1: The tower of linked effective field theories that can be employed to connect low-energy $\mu \rightarrow e$ results to higher energy scales. The new MuonBridge software is described in tower.
• Figure 2: Illustration of the differences in target responses --- C, O, F, Na, Al, Si, S, Ca, Ti, Fe, Cu --- to allowed (left two panels) and velocity-dependent (right panels) operators.
• Figure 3: Expected electron counts (50 keV/c bins) in Mu2e Run I for CLFV mediated by isoscalar charge ($M$), longitudinal spin ($\Sigma^{\prime \prime}$), transverse spin ($\Sigma$ + $\Sigma^\prime$), transverse spin-velocity ($\tilde{\Phi}$ and $\tilde{\Phi}^\prime$), axial charge ($\tilde{\Omega}$), and longitudinal convection ($\tilde{\Delta}^{\prime \prime}$) interactions. The total signal (black) is obtained by combining ground state (blue) and exited state [0.84 MeV (red), 1.0 MeV (purple), and 2.2 MeV (orange)] contributions. The green squares are the estimated DIO background. The gray shading indicates the energy interval where the Mu2e sensitivity is optimized. The error bars are estimates of the combined uncertainty from statistics and from the nuclear physics of the responses.