  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 $\alpha\approx 1/137$.
