$\gamma W$-box Inside-Out: Nuclear Polarizabilities Distort the Beta Decay Spectrum

I consider the $\gamma W$-box correction to superallowed nuclear $\beta$-decays in the framework of dispersion relations. I address a novel effect of a distortion of the emitted electron energy spectrum by nuclear polarizabilities and show that this effect, while neglected in the literature, is sizable. I estimate its size in the approximation of a linear energy dependence, and using two models that are expected to give the lower and the upper bound. The respective correction to the $\beta^+$ spectrum is estimated to be $\Delta_R(E)=(1.6\pm1.6)\times10^{-4}{E}/{\rm MeV}$ assuming a conservative 100\% uncertainty. The effect is positive-definite and can be observed if a high-precision measurement of the positron spectrum is viable. If only the full rate is observed, it should be included in the calculated ${\cal F}t$-values of nuclear decays. I argue that this novel effect should be evaluated in state-of-the-art nuclear models and included in the analyses of nuclear beta decay experiments to ensure the correct extraction of $V_{ud}$ from decay rates, and of the Fierz interference term from precision measurements of decay spectra.