Chiral perturbation theory in the presence of resonances: Application to π π and π K scattering

Abstract

We consider pion-pion and pion-kaon scattering in the framework of an effective chiral lagrangian which contains the Goldstone pseudoscalars and chirally coupled resonances with spin ⩽ 1. The couplings of these resonances are determined by their transformation properties under the diagonal vectorial subgroup of chiral symmetry. At leading order, the momentum-independent parts of the resonance propagators saturate the low-energy constants which contribute at next-to-leading order in the chiral expansion. The momentum-dependent contributions from the resonance propagators allow for a partial resummation of contributions of yet higher orders (⩾ E 6 in the chiral expansion). We demonstrate that the imaginary part of the resonance propagators can be generated by a K-matrix approach. We evaluate S-, P-, D- and F-waves in ππ scattering. We compare to available data and existing Roy equation fits and find a good description of the various phase shifts up to energies of about √ s = 700 MeV. From the S-wave phase shifts, we can evaluate the phase of the parameter ϵ′ which is related to direct CP violation in K → 2π decays. We find φ( ϵ′) = 44.5°. We also discuss the importance of the unitarity corrections to various partial waves. In the πK system, we work out S- and P-waves. We find a satisfactory description of the available data up to energies of about √ s = 900 MeV. We also point towards the importance of a more accurate determination of the πK scattering amplitudes.