the reduced mass. Experiment shows about a $320\,\mu$eV energy discrepancy for the $\mu$-$H$ $2S$ state, compared to expectations based on the CODATA 2010 [@Mohr:2012tt] electron based proton radius.

In a scenario where the $\mu$-$H$ energy discrepancy is not actually due to a proton size change but rather to the exchange of a new particle that has a special coupling to the muon and [*e. g.*]{} a dark photon coupling (*i.e.,* a squelched electromagnetic coupling) to other particles, the $\mu$-$D$ energy would be the same as in the $\mu$-$H$ except for the reduced-mass-cubed factor. In this case the anticipated $\mu$-$D$ energy discrepancy is $$\Delta E_{\rm discrepant}(\mu{\rm -}D) \approx 380 \ \mu{\rm{eV}}.$$ Hence the two-photon corrections should be calculated within $100\ \mu$eV or less. As the two-photon corrections are of order $2$ meV, this requires a $5\%$ or better accuracy.

There are several currently available polarizability calculations [@friar; @pachuckiD; @rosenfelder; @FriarPayne; @Friar:2013rha; @Ji:2013oba]. Ref. [@pachuckiD] includes the elastic contributions and quotes an uncertainty limit well within the requirement. A more recent calculation of Ref. [@Ji:2013oba] follows the same lines but includes further relativistic corrections. In addition, much of that calculation is supported by a calculation  [@Friar:2013rha] that uses the zero-range approximation which 