QED of Bernabéu-Tarrach sum rule for electric polarizability and its implication for the Lamb shift

We attempt to rehabilitate a sum rule (proposed long ago by Bernabéu and Tarrach) which relates the electric polarizability of a particle to the total photoabsorption of quasireal longitudinally polarized photons by that particle. We discuss its perturbative verification in quantum electrodynamics (QED), which is largely responsible for the skepticism about its validity. The failure of the QED test can be understood via the Sugawara-Kanazawa theorem and is due to the nonvanishing contour contribution in the pertinent dispersion relation. We show another example where this contribution is absent and the perturbative test works exactly. On the empirical side, we show that the sum rule gives a reasonable estimate of the $\pi N$-channel contribution to the proton electric polarizability. If this sum rule is valid indeed, there should be a sum rule for the so-called "subtraction function" entering the data-driven calculations of the polarizability effects in the Lamb shift. We have written down a possible sum rule for the subtraction function and verified it in a perturbative calculation.