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

Abstract

The precision of the electromagnetic proton-neutron mass difference $\delta m_{\mathrm{QED}}$ 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, $({\alpha }_{E1}-{\beta }_{M1}{)}^{p-n}$. We derive a dispersive sum rule for this combination in terms of $s$-channel photoabsorption cross sections and the product of $t$-channel $\gamma \gamma \to \pi \eta /K{\bar{K}}_{I_t=1}$ and $\pi \eta /K{\bar{K}}_{I_t=1}\to N\bar{N}$ amplitudes, without invoking Reggeon dominance. Combining empirical pion-photoproduction multipoles with coupled-channel Muskhelishvili--Omnès representations of the scalar-isovector $\pi \eta /K\bar{K}$ amplitudes, we obtain $({\alpha }_{E1}-{\beta }_{M1}{)}^{p-n}=-2.26(73)\times 10^{-4}{\mathrm{fm}}^3$, 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 $\delta m_{\mathrm{QED}}=0.71_{-0.06}^{+0.03}\mathrm{MeV}$, 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.

10 pages, 2 figures