or $0.5\%$ uncertainty. Planned experiments are expected to reduce this uncertainty [@modkg].

To obtain the charge radius from the Lamb shift requires not only accurate data but also accurate calculation of all corrections that are not hadronic size corrections. Of these, the two-photon correction, which includes the relativistically correct polarizability correction, has drawn continued attention. A deuteron is easily distorted compared to a single proton, and we shall see that the polarizability correction for the $\mu$-$D$ system is about two orders of magnitude larger than for $\mu$-$H$. The requirement that the $\mu$-$D$ polarizability correction be safely smaller than the radius-related energy shift can become quite severe.

Of course, without knowing the underlying reason for the proton radius discrepancy, we cannot with certainty predict what the deuteron discrepancy will be. However, we will give the anticipated energy discrepancy in one scenario, and thereby obtain a working number with the expectation that other scenarios would give results similar within a factor of a few. As a reminder, the main energy shift due to finite hadron (or nuclear) size is $$\Delta E_{\rm finite\ size} = \frac{2\pi Z\alpha}{3} \frac{(m_r Z\alpha)^3}{n^3 \pi} r^2_E(h) \,,$$ for a hydrogen-like atom in the $nS$ state, where $m_r$ is 