Isospin breaking in the phases of the Ke4 form factors

Understand

Isospin breaking in the Kl4 form factors induced by the difference between charged and neutral pion masses is studied.

• Starting from suitably subtracted dispersion representations, the 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\pm -> pi^+ pi^- e^\pm nu_e channel are given, 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 pi-pi phase shifts studied theoretically (in the isospin limit).
• The isospin-breaking correction consists of the sum of a universal part, involving only pi-pi rescattering, and a process-dependent contribution, involving the form factors in the coupled channels.