Comment on the Prediction of Two-loop Standard Chiral Perturbation Theory for Low-Energy ππScattering

Abstract

Four of the six parameters defining the two-loop ππscattering amplitude have been determined using Roy dispersion relations. Combining this information with the Standard χPT expressions, we obtain the threshold parameters, low-energy phases and the O(p^4) constants $l_1^r, l_2^r$. The result $l_2^r ( M_ρ ) = ( 1.6 \pm 0.4 \pm 0.9 ) \times 10^{-3} ( {\bar l_2} = 4.17 \pm 0.19 \pm 0.43$) reproduces the correct D-waves but it is incompatible with existing Standard χPT analyses of $K_{l4}$ form factors beyond one loop.