Speaker
Description
We implement a dispersive parameterization of the nonlocal form factors (NFFs) governing photon or $Z$-boson exchange in the semileptonic $K^+ \to \pi^+ \ell^+ \ell^-$ $(\ell=e,\mu)$ and $K^+ \to \pi^+ \nu \bar{\nu}$ decays, respectively, to improve the theoretical description of the spectrum and decay rate of the golden neutrino mode. The parameterization links the hadronic $K^+ \pi^- \to \pi^+ \pi^-$ amplitude in P-wave with the pion vector form factor (VFF) through unitarity. The phase of the hadronic amplitude, in turn, can be constrained by a fit to the Dalitz slope parameters of all CP-conserving $K \to 3\pi$ decays, assuming isospin symmetry and elastic rescattering of the pions. We develop the complete basis of reduced isospin amplitudes in the complex plane by iterating the Khuri-Treiman equations, and compare to recent results in the literature. Finally, we emphasize a relation between the charged-lepton and neutrino NFFs in terms of the subtractions in their dispersive representations.