Precision Sum Rule for Nucleon Isovector Polarizabilities and the Proton-Neutron Mass Difference

Abstract

The precision of the electromagnetic proton-neutron mass difference $δm_\mathrm{QED}$ extracted from the Cottingham formula hinges on a subtraction function whose low-energy normalization $\bar S(0)$ is fixed by the isovector combination of the proton and neutron polarizabilities, $(α_{E1}-β_{M1})^{p-n}$. We derive a dispersive sum rule for this combination in terms of $s$-channel photoabsorption cross sections and the product of $t$-channel $γγ\toπη/K\bar K_{I_t=1}$ and $πη/K\bar K_{I_t=1}\to N\bar N$ amplitudes, without invoking Reggeon dominance. Combining empirical pion-photoproduction multipoles with coupled-channel Muskhelishvili--Omnès representations of the scalar-isovector $πη/K\bar K$ amplitudes, we obtain $(α_{E1}-β_{M1})^{p-n}=-2.26(73)\times10^{-4}\,\mathrm{fm}^3$, fixing its sign and reducing the uncertainty by a factor of 4 compared with the previously known value. This result yields $\bar S(0)=-1.76(61)\,\mathrm{GeV}^{-2}$, leading to $δm_\mathrm{QED}=0.71^{+0.03}_{-0.06}\,\mathrm{MeV}$, substantially more precise than previous Cottingham determinations. The negative $\bar S(0)$ also provides a stringent low-energy test of Reggeon dominance in the subtraction function.