Spectral Density Functions and Their Sum Rules in an Effective Chiral Field Theory

Abstract

The validity of Weinberg's two sum rules for massless QCD, as well as the six additional sum rules introduced into chiral perturbation theory by Gasser and Leutwyler, are investigated for the extended Nambu-Jona-Lasinio chiral model that includes vector and axial vector degrees of freedom. A detailed analysis of the vector, axial vector and coupled pion plus longitudinal axial vector modes is given. We show that, under Pauli-Villars regularization of the meson polarization amplitudes that determine the spectral density functions, all of the sum rules involving inverse moments higher than zero are automatically obeyed by the model spectral densities. By contrast, the zero moment sum rules acquire a non-vanishing right hand side that is proportional to the quark condensate density of the non-perturbative groundstate. We use selected sum rules in conjunction with other calculations to obtain explicit expressions for the scale-independent coupling constants $\bar l_i$ of chiral perturbation theory in the combination $\bar l_i+\ln(m^2_\pi/\mu^2)$, to evaluate the Das-Guralnik-Mathur-Low-Young current algebra expression for the electromagnetic mass difference of the pion, and to compute the pion electrical polarizability $\alpha_E$. The sum rule calculations %of $\alpha_E$ set an upper limit of $\alpha_E\leq \alpha/(8\pi^2m_\pi f^2_\pi) \approx 6\times10^{-4}$fm$^3$ on this quantity, that is independent of the quark mass up to ${\cal O}(m^2_\pi/m^2)$, and only depends on the fundamental constants: pion mass $m_\pi$ and pion decay constant $f_\pi$, in addition to fine structure constant $\alpha\approx 1/137$.