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We quantify a limitation in the usual accounting of the finite-size effects,where the leading $[(Z\alpha)^4]$ and subleading $[(Z\alpha)^5]$ contributionsto the Lamb shift are given by the mean-square radius and the third Zemachmoment of the charge distribution. In the presence of any non-smooth behaviourof the nuclear form factor at scales comparable to the inverse Bohr radius, theexpansion of the Lamb shift in the moments breaks down. This is relevant forsome of the explanations of the "proton size puzzle". We find, for instance,that the de R\'ujula toy model of the proton form factor does not resolve thepuzzle as claimed, despite the large value of the third Zemach moment. Withoutrelying on the radii expansion, we show how tiny, milli-percent (pcm) changesin the proton electric form factor at a MeV scale would be able to explain thepuzzle. It shows that one needs to know all the soft contributions to protonelectric form factor to pcm accuracy for a precision extraction of the protoncharge radius from atomic Lamb shifts.

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