Dispersive description of the $K \to π\ell^+ \ell^-$ radiative amplitudes

TL;DR

The paper addresses how to describe the radiative kaon decay form factors $W_+$ and $W_S$ in $K\to\pi\ell^+\ell^-$ using a dispersive framework grounded in analyticity and unitarity. It develops Muskhelishvili–Omnès representations for the isospin components $W^{[1/2]}$ and $W^{[3/2]}$, using KT solutions for $K\to3\pi$ and $\pi\pi$, $K\pi$ phase input to constrain the discontinuities, resulting in a minimal two-parameter model with $a_+$ and $a_S$ linked to $W_+(0)$ and $W_S(0)$. The analysis fixes the sign of $W_+$, describes the energy dependence of $|W_+|^2$ in agreement with data, and makes concrete predictions for $|W_S|^2$ that can be tested experimentally (e.g., at LHCb) and for how the still-unknown $\Delta I=1/2$ piece in $K_S\to\pi^+\pi^-\pi^0$ can be inferred from $W_+(0)+W_S(0)$, with discussion of potential isoscalar resonance effects. Overall, the work provides a data-driven, nonperturbative description of rare kaon radiative amplitudes with implications for SM tests and CP-violating modes.

Abstract

We propose a description of the $K^+$, $K_S$ radiative decay form factors $W_+$, $W_S$ based on general properties of analyticity and unitarity. Starting from the simple consideration of the asymptotic behaviour of the two combinations $2W_+-W_S$ and $W_+ +W_S$ we derive a dispersive representation involving only two parameters. Using the rich experimental information on the $K\to3π$ amplitudes, extended beyond the low energy region using the Khuri-Treiman formalism, we show that the sign of the $W_+$ form factor is unambiguously determined and its energy dependence can be well reproduced. We also show that the yet unknown $Δ{I}=1/2$ part of the $K_S \to π^+π^-π^0$ can be determined from the value of $W_+(0)+W_S(0)$. The possibility of fixing the sign of $W_S$ from experiment is discussed.

Figures (2)

• Figure 1: Real and imaginary parts of the $J=1$ projection of the ${K^+}\to{\pi^+}{\pi^-}{\pi^+}$ amplitude from the KT formalism. The red and blue lines correspond to taking the central values of the KT subtraction parameters which are fitted to the experimental $K\to3\pi$ data. The error bands are generated by varying these parameters. One notices the presence of a singularity which occurs at the endpoint of the left hand (complex) cut at $s=(m_K-m_\pi)^2$, which is integrable (e.g. Kambor:1995yc ).
• Figure 2: Left: Results for $|W_+|^2$ from the dispersive representations corresponding to different choices of signs for $a_{+,S}$: $a_+=\mp 0.575$, $a_S=\pm1.06$, compared to the experimental data from refs. Appel:1999yqBatley:2009aaNA62:2022qes. Right: Results for $|W_S|^2$ with $a_S>0$ and $a_S<0$.