A dispersive approach to the CP conserving $K\toπ\ell^+\ell^-$ radiative decays

TL;DR

This work develops a dispersive, model-independent framework to describe CP-conserving radiative kaon decays K^+ -> pi^+ l^+ l^- and K_S -> pi^0 l^+ l^- by constructing minimal two-parameter representations for the form factors W_+(s) and W_S(s). It combines updated K -> 3 pi amplitudes from Khuri–Treiman equations with explicit K pi intermediate states, yielding coupled Muskhelishvili–Omnès representations controlled by the parameters a_+ and a_S, and constraining the Delta I = 1/2 piece through a linear relation with tilde mu_1. The analysis fixes a_+ to be negative and shows the sign of a_S can be probed via the s-dependence of |W_S|^2 and the joint behavior of |W_+|^2, with two viable a_S solutions in agreement with current data; it also demonstrates that omega/phi and higher-mass resonances have subleading impact. The framework provides a data-driven path to extract both magnitudes and relative phases of radiative kaon amplitudes and offers a cross-check for short-distance CP-violating contributions relevant to precision tests of the Standard Model.

Abstract

We reconsider the constraints on the form factors $W_+ (s)$ and $W_S (s)$, describing the radiative decay modes $K^+\toπ^+ \ell^+\ell^-$ and $K_S\toπ^0 \ell^+\ell^-$, associated with the general properties of analyticity and unitarity. Starting from the simple consideration of the asymptotic behaviours of the two combinations $2 W_+ (s) - W_S (s)$ and $W_+ (s) + W_S (s)$, we derive a minimal pair of dispersive representations which involves only two free parameters. An important input for these representations consists of the $K\to3π$ decay amplitudes, for which we use a set of solutions of the Khuri-Treiman equations obtained recently. These solutions provide an extrapolation from the physical $K\to3π$ decay region up to the resonant $Kπ\toππ$ scattering regions. We show that the experimental energy dependence of $|W_+|^2$ can be well reproduced and that the sign of $W_+$ is unambiguously determined. We also show that the yet unknown $Δ{I}=1/2$ part of the $K_S\to π^+ π^- π^0$ amplitude can be determined from the value of $W_+(0) + W_S(0)$. The possibility of fixing the sign of $W_S(0)$ using experimental data on both $|W_+|^2$ and $|W_S|^2$ is discussed.

Figures (13)

• Figure 1: Contributions to the unitarity relation from the $\pi\pi$ and the $K\pi$ states. A black dot represents an electromagnetic or strong vertex, a green square denotes a weak $\Delta{S}=1$ vertex
• Figure 2: Graph giving rise to an anomalous threshold
• Figure 3: Complex left-hand cut of the $K\pi\to \pi\pi$ partial-wave amplitudes (red line). The dotted line represents the unitarity cut. The sub-figure shows an enlarged view of the vicinity of the positive real axis illustrating the effect of performing an infinitesimal imaginary shift of $m_K$ which separates the complex cut from the unitarity cut. The black dot shows the position of the singularity.
• Figure 4: The phase of the $\pi\pi$ form factor used in the Omnès-type representation (right) and the resulting modulus (left).
• Figure 5: Phase (right) and corresponding modulus (left) of the $K\pi$ vector form factor $f_+^{K\pi}(s)$. The phase is compared with the $K\pi$$P$-wave scattering phase-shift from refs. Estabrooks:1977xeAston:1987ir. The modulus is compared at low energy to a quadratic representation of experimental $K_{l3}$ decay dataKLOE:2006kms.