On the proton charge extensions

We examine how corrections to $S$-state energy levels, $E_{nS}$, in hydrogenic atoms due to the finite proton size are affected by moments of the proton charge distribution. The corrections to $E_{nS}$ are computed moment by moment. The results demonstrate that the next-to-leading order term in the expansion is of order $r_p / a_B$ times the size of the leading order $\langle r_p^2 \rangle$ term. Our analysis thus dispels any concern that the larger relative size of this term for muonic hydrogen versus electronic hydrogen might account for the current discrepancy of proton radius measurements extracted from the two systems. Furthermore, the next-to-leading order term in powers of $r_p / a_B$ that we derive from a dipole proton form factor is proportional to $\langle r_p^3 \rangle$, rather than $\langle r_p^4 \rangle$ as would be expected from the scalar nature of the form factor. The dependence of the finite-size correction on $\langle r_p^3 \rangle$ and higher odd-power moments is shown to be a general result for any spherically symmetric proton charge distribution. A method for computing the moment expansion of the finite-size correction to arbitrary order is introduced and the results are tabulated for principal quantum numbers up to $n=7$.