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

TL;DR

This work shows that the chiral Lagrangian parameters ${ar extell}_1$ and ${ar extell}_2$ can be determined from ππ scattering by exploiting the Adler zero and the dominance of the ${ ho}$ resonance in the ${I=1}$ channel. Through fixed-$t$ dispersion relations with two subtractions and a continuity argument for zero contours, the authors connect resonance dynamics to the KSFR relation and to the low-energy constants, achieving a self-consistent generation of the ${ ho}$ resonance. They derive expressions for the ${D}$-wave scattering lengths in terms of ${ar extell}_1$ and ${ar extell}_2$ and obtain ${ar extell}_1 = -0.3 \,\pm\, 1.1$ and ${ar extell}_2 = 4.5 \,\pm\, 0.5$, in agreement with Gasser–Leutwyler. A zero-tracking method further provides a predictive ${P}$-wave phase shift that aligns with experimental ππ phase shifts, reinforcing the consistency of the extracted chiral parameters with resonance physics.

Abstract

The all--important consequence of Chiral Dynamics for $ππ$ scattering is the Adler zero, which forces $ππ$ amplitudes to grow asymptotically. The continuation of this subthreshold zero into the physical regions requires a $P$--wave resonance, to be identified with the $ρ$. It is a feature of $ππ$ scattering that convergent dispersive integrals for the $I=1$ channel are essentially saturated by the $ρ$--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 $ρ-$resonance.