Application of $K \to 3π$ amplitudes to semileptonic kaon decays

TL;DR

The work addresses the long-distance uncertainties in the rare kaon decay $K^+ \to \pi^+ \nu \bar{\nu}$ by developing dispersive representations of nonlocal form factors $W_\gamma$, $W_V$, and $W_A$ that connect to the local $K \to \pi$ form factor via the OPE. By exploiting unitarity, the $K^+ \pi^- \to \pi^+ \pi^-$ $P$-wave amplitude and the pion vector form factor, the authors build a data-driven framework in which the $K^+ \to \pi^+ \ell^+ \ell^-$ spectrum constrains the discontinuities of the electromagnetic form factor, which in turn anchors the $K^+ \to \pi^+ \nu \bar{\nu}$ amplitude. A Khuri-Treiman–based isospin analysis of $K \to 3\pi$ provides a controlled, model-independent way to extract the needed $K^+ \pi^- \to \pi^+ \pi^-$ input via a four-subtraction dispersive setup with isobars $F_I^{\alpha_i}(s)$, $H_I(s)$. The proposed framework enables a cohesive, data-driven determination of long-distance effects in the neutrino mode, though isospin-breaking corrections and WW-exchange contributions remain key sources of uncertainty to be quantified.

Abstract

We study dispersive representations of nonlocal form factors in $K^+ \to π^+ \ell^+ \ell^-$ and $K^+ \to π^+ ν\barν$ decays, with an aim of improving the theoretical description of the spectrum and decay rate of the neutrino mode. Based on unitarity, these representations invoke the $K^+ π^- \to π^+ π^-$ amplitude in $P$-wave and the pion vector form factor. The $P$-wave amplitude can be effectively parameterized within the dispersive Khuri-Treiman framework, and constrained by experimental information on the CP-conserving $K^+ \to π^+ π^+ π^-$ and $K^+ \to π^0 π^0 π^+$ decays. We also emphasize certain relations between charged-lepton and neutrino non-local form factors based on the Operator Product Expansion, which can be used to impose further phenomenological constraints.

Figures (2)

• Figure 1: Illustration of the local (a) and nonlocal (b-e) contributions to $K^+ \to \pi^+ \nu \bar{\nu}$. Only diagrams (b) and (d) contribute to $K^+ \to \pi^+ \ell^+ \ell^-$ (after replacing the local neutrino coupling with a photon propagator).
• Figure 2: Expanded view of the real (red) and imaginary (blue) parts of all isobars after one (dotted) three (dashed) and six (solid) iterations of the KT equations. The projections of the $K \to 3\pi$ Dalitz region are indicated by the thin gray bands. Here we have taken $M_K=M_{K^+}$ and $M_\pi=M_{\pi^+}$.