Equivalence Between Vector Meson Dominance and Unitarised Chiral Perturbation Theory

Abstract

It is explicitly shown that either the approximate solution of the integral equation for the inverse of the pion form factor, or the result of the Pad\'e approximant method of resumming the one loop Chiral Perturbation Theory (CPTH) are equivalent to the standard vector meson dominance (VMD) models, using the vector meson coupling to two pseudoscalars given by the KSRF relation. Inconsistencies between the one loop CPTH and its unitarised version (or the VMD model) are pointed out. The situation is better for the CPTH calculation of the scalar form factor and the related S-wave $\pi \pi$ scattering. The branching ratios of $\tau \to \pi^+ \pi^0 \nu$, $\tau \to K \pi \nu$, $\tau \to K^+ \eta \nu$ and $\tau \to K^+ \bar{K^0} \nu$ using only two inputs as the $\rho $ and $K^*$ masses, or the two corresponding rms radii, agree with the experimental data. Using the same number of parameters, the corresponding one loop CPTH calculation cannot explain the $\tau$ data.