Nucleon polarizabilities revisited: insights from the Bernabéu-Ericson-Ferro Fontan-Tarrach sum rule

Abstract

We present a complete dispersive evaluation of the Bernabéu-Ericson-Ferro Fontan-Tarrach (BEFT) sum rule for the nucleon dipole polarizability difference $α_{E1}-β_{M1}$. The analysis is based on hyperbolic dispersion relations, which separate the $s$- and $t$-channel contributions without double counting and are particularly suited to backward real Compton scattering. This framework makes it possible to precisely determine the polarizability difference from independent dispersive input rather than from fits to low-energy Compton scattering data. The $s$-channel part is updated with modern single-pion photoproduction multipoles and model estimates of non-$πN$ inelastic channels. For the $t$-channel, we use contemporary dispersive input for $γγ\toππ/K\bar K_{I_t=0}$ together with Roy-Steiner equation solutions for $ππ/K\bar K_{I_t=0}\to N\bar N$, and we extend the calculation to the isovector $πη/K\bar K_{I_t=1}$ channel associated with the $a_0(980)$. The resulting treatment resolves the long-standing puzzle of earlier BEFT evaluations driven by the scalar-isoscalar two-pion channel and gives a controlled estimate of the subleading isovector contribution. Combining the resulting BEFT values with the Baldin sum rule, we obtain, in units of $10^{-4}\,\mathrm{fm}^3$, \[ α_{E1}^p=11.17(52),\quad β_{M1}^p=2.84(52),\quad α_{E1}^n=12.18(59),\quad β_{M1}^n=3.03(59). \] For the proton, these values provide a benchmark with a precision comparable to state-of-the-art dispersive and chiral effective field theory extractions. For the neutron, whose polarizabilities are much less directly constrained experimentally, the result provides an independent dispersive benchmark with an uncertainty comparable to that of the proton.