Chiral two-loop pion-pion scattering parameters from crossing-symmetric constraints

Constraints on the parameters in the one- and two-loop pion-pion scattering amplitudes of standard chiral perturbation theory are obtained from explicitly crossing-symmetric sum rules. These constraints are based on a matching of the chiral amplitudes and the physical amplitudes at the symmetry point of the Mandelstam plane. The integrals over absorptive parts appearing in the sum rules are decomposed into crossing-symmetric low- and high-energy components and the chiral parameters are finally related to high-energy absorptive parts. One application uses a simple model of these absorptive parts. The sensitivity of the results to the choice of the energy separating high and low energies is examined with care. A weak dependence on this energy is obtained as long as it stays below $\sim 560\mathrm{MeV}.$ Reliable predictions are obtained for three two-loop parameters.