Isospin breaking in $K\pi$ vector form-factors for the weak and rare decays $K_{\ell3}$, $K\to\pi\nu\bar\nu$ and $K\to\pi\ell^+\ell^-$

TL;DR

This paper computes isospin-breaking effects in $K o\pi$ vector form-factors at order $p^6$ in Chiral Perturbation Theory, providing numerical results relevant for weak and rare kaon decays.

Contribution

It introduces first-order isospin-breaking corrections to $K o\pi$ vector form-factors within Chiral Perturbation Theory and derives relations valid to all orders in the theory.

Findings

Numerical values for all eight form-factors at $t=0$.

Relations between form-factors valid to first order in quark-mass difference.

Results applicable to weak and rare kaon decay processes.

Abstract

We calculate the two form-factors for the four Kaon to pion transitions via a vector current to order $p^6$ in Chiral Perturbation Theory to first order in isospin breaking via the quark masses. In addition we derive relations between these form-factors valid to first order in the up-down quark-mass difference but to all orders in Chiral Perturbation Theory. We present numerical results for all eight form-factors at $t=0$ and for varying $t$ and for the scalar form-factors at the Callan-Treiman point.