Comment on the Prediction of Two-loop Standard Chiral Perturbation Theory for Low-Energy \pi\pi Scattering

Abstract

Four of the six parameters defining the two-loop \pi\pi scattering amplitude have been determined using Roy dispersion relations. Combining this information with the Standard \chi PT expressions, we obtain the threshold parameters, low-energy phases and the O(p^4) constants $l_1^r,l_2^r$. The result $l_2^r(M_{\rho })=(1.6\pm 0.4\pm 0.9)\times 10^{-3}({\bar{l}}_2=4.17\pm 0.19\pm 0.43$) reproduces the correct D-waves but it is incompatible with existing Standard \chi PT analyses of $K_{l4}$ form factors beyond one loop.

Comments: 14 pages, Latex209, 2 figures