A Dispersive Treatment of $K_{\ell 4}$ Decays

Abstract

$K_{\ell 4}$ decays offer several reasons of interest: they allow an accurate measurement of $\pi \pi$-scattering lengths; they provide the best source for the determination of some low-energy constants of ChPT; one form factor is directly related to the chiral anomaly, which can be measured here. We present a dispersive treatment of $K_{\ell 4}$ decays that provides a resummation of $\pi \pi$- and $K\pi$-rescattering effects. The free parameters of the dispersion relation are fitted to the data of the high-statistics experiments E865 and NA48/2. The matching to ChPT at NLO and NNLO enables us to determine the LECs $L_1^r$, $L_2^r$ and $L_3^r$. With recently published data from NA48/2, the LEC $L_9^r$ can be determined as well. In contrast to a pure chiral treatment, the dispersion relation describes the observed curvature of one of the form factors, which we understand as a rescattering effect beyond NNLO.

Comments: 86 pages, 21 figures. Draws on and extends arXiv:1412.5171 [hep-ph] and arXiv:1209.0755 [hep-ph]