Breakdown of the expansion of finite-size corrections to the hydrogen Lamb shift in moments of charge distribution

We quantify a limitation in the usual accounting of the finite-size effects, where the leading $\left[{\left(Z\alpha \right)}^4\right]$ and subleading $\left[{\left(Z\alpha \right)}^5\right]$ contributions to the Lamb shift are given by the mean-square radius and the third Zemach moment of the charge distribution. In the presence of any nonsmooth behavior of the nuclear form factor at scales comparable to the inverse Bohr radius, the expansion of the Lamb shift in the moments breaks down. This is relevant for some of the explanations of the “proton size puzzle.” We find, for instance, that the de Rújula toy model of the proton form factor does not resolve the puzzle as claimed, despite the large value of the third Zemach moment. Without relying on the radii expansion, we show how tiny, milli-percent (pcm) changes in the proton electric form factor at a MeV scale would be able to explain the puzzle. It shows that one needs to know all the soft contributions to the proton electric form factor to pcm accuracy for a precision extraction of the proton charge radius from atomic Lamb shifts.

Figure 1

One-photon exchange graph with FF dependent electromagnetic vertex.

Figure 2

Integrand of the 1st-order contribution to the Lamb shift [cf. Eq. (19a) with Eq. (17)] of $e\mathrm{H}$ (blue dash-dotted line) and of $\mu \mathrm{H}$ (red solid line) for the dipole FF. The dotted vertical lines indicate the inverse Bohr radius of the two hydrogens, while the vertical dashed line indicates the onset of the $ep$ data.

Figure 3

Parameters of ${\tilde{G}}_E$ for which the $e\mathrm{H}$ and $\mu \mathrm{H}$ Lamb shifts of Eq. (24) are reproduced, given $Q_0$ set by Eq. (22).