Understanding the ``Proton Radius Puzzle'': Nuclear Polarizability Correction in $\mu$D

The accuracy of the proton radius was improved ten-fold by new spectroscopic measurements in muonic hydrogen~[1] but it differs by $7 \sigma$ from hydrogen determinations. This discrepancy, has been coined the ``proton radius puzzle". The results of new high-precision experiments on muonic deuterium indicate a new deuterium radius puzzle~[2]. The accuracy of the nuclear charge radius determination from these measurements is limited by the uncertainty in the nuclear structure effects. We have calculated this correction in Ref.~[3] including the first estimate of the nuclear-model dependence. Due to the importance of constraining the uncertainty, we will determine the statistical and systematic uncertainties of the $\chi$EFT potentials by determining the co-variance matrices of our predictions. I will also discuss an alternate method that may reduce the theoretical uncertainty.\\ [][1] R.~Pohl {\it et al.}, Nature 466, 213 (2010).\\ [][2] R.~Pohl {\it et al.}, to appear; ECT* workshop, June 20-24 (2016).\\ [][3] O.J.~Hernandez, {\it et al.}, Phys. Lett. B (2014).