Uncertainty quantification for $\mu \to e$ conversion in nuclei: charge distributions

Predicting the rate for $\mu\to e$ conversion in nuclei for a given set of effective operators mediating the violation of lepton flavor symmetry crucially depends on hadronic and nuclear matrix elements. In particular, the uncertainties inherent in this non-perturbative input limit the discriminating power that can be achieved among operators by studying different target isotopes. In order to quantify the associated uncertainties, as a first step, we go back to nuclear charge densities and propagate the uncertainties from electron scattering data for a range of isotopes relevant for $\mu\to e$ conversion in nuclei, including $^{40,48}$Ca, $^{48,50}$Ti, and $^{27}$Al. We provide as central results Fourier-Bessel expansions of the corresponding charge distributions with complete covariance matrices, accounting for Coulomb-distortion effects in a self-consistent manner throughout the calculation. As an application, we evaluate the overlap integrals for $\mu\to e$ conversion mediated by dipole operators. In combination with modern ab-initio methods, our results will allow for the evaluation of general $\mu\to e$ conversion rates with quantified uncertainties.