Isospin breaking in pion and Ke4 form factors

Abstract

Isospin breaking in the Kℓ4 form factors induced by the difference between charged and neutral pion masses is discussed within a framework built on suitably subtracted dispersion representations. The Kℓ4 form factors are constructed in an iterative way up to two loops in the low-energy expansion by implementing analyticity, crossing, and unitarity due to two-meson intermediate states. Analytical expressions for the phases of the two-loop form factors of the K± → π+π−e±νe channel are presented, allowing one to connect the difference of form-factor phase shifts measured experimentally (out of the isospin limit) and the difference of S- and P-wave ππ phase shifts studied theoretically (in the isospin limit). The dependence with respect to the two S-wave scattering lengths a00 and a20 in the isospin limit is worked out in a general way, in contrast 0 0 to previous analyses based on one-loop chiral perturbation theory. The results on the phases of the K± → π+π−e±νe form factors obtained by the NA48/2 collaboration at the CERN SPS are reanalysed including isospin-breaking correction to extract values for the scattering lengths a00 and a02