The Das-Mathur-Okubo sum rule for the charged pion polarizability in a chiral model

The Das-Mathur-Okubo (DMO) sum rule for the polarizability of charged pions is evaluated for the Nambu-Jona-Lasinio model Lagrangian in both its minimal and extended forms. A comparison is made with the results obtained using the same sum rule from chiral perturbation theory (CHPT), approximate QCD sum rule calculations, explicit calculations on the lattice by Wilcox, and using the semi-empirical Kapusta-Shuryak spectral densities. The $\chi$PT results from Compton scattering are also given. We point to a delicate cancellation between the intrinsic and recoil contributions to $\alpha_{\pi^\pm}$ in the DMO sum rule approach that can lead to calculated polarizabilities of either sign.