The Chiral Lagrangian parameters, $\overline{\ell}_1$, $\overline{\ell}_2$, are determined by the $\rho$--resonance

Abstract

The all--important consequence of Chiral Dynamics for $\pi \pi$ scattering is the Adler zero, which forces $\pi \pi$ amplitudes to grow asymptotically. The continuation of this subthreshold zero into the physical regions requires a $P$--wave resonance, to be identified with the $\rho$. It is a feature of $\pi \pi$ scattering that convergent dispersive integrals for the $I=1$ channel are essentially saturated by the $\rho$--resonance and are much larger than those with $I=2$ quantum numbers. These facts predict the parameters $\overline{\ell}_1$, $\overline{\ell}_2$ of the Gasser--Leutwyler Chiral Lagrangian, as well as reproducing the well--known KSFR relation and self-consistently generating the $\rho-$resonance.