Sum rule for the backward spin polarizability of the nucleon from a backward dispersion relation

A new sum rule for ${\gamma }_{\pi },$ the backward spin polarizability of the nucleon, is derived from a backward-angle dispersion relation. Taking into account single-pion and multipion photoproduction in the s channel up to the energy ${\omega }_{\max }=1.5\mathrm{}\hspace{0.5em}\mathrm{GeV}$ and resonances in the t channel with mass below $1.5\hspace{0.5em}\mathrm{GeV},$ it is found for the proton and neutron that $[{\gamma }_{\pi }{]}_p=-39.5\pm 2.4$ and $[{\gamma }_{\pi }{]}_n=52.5\mathrm{}\pm 2.4,$ respectively, in units of ${10}^{-4}\hspace{0.5em}{\mathrm{fm}}^4.$