Abstract
decays have several features of interest: they allow an accurate measurement of ππ-scattering lengths; the decay is the best source for the determination of some low-energy constants of chiral perturbation theory (χPT); one form factor of the decay is connected to the chiral anomaly.We present the results of our dispersive analysis of decays, which provides a resummation of ππ- and Kπ-rescattering effects. The free parameters of the dispersion relation are fitted to the data of the high-statistics experiments E865 and NA48/2. By matching to χPT at NLO and NNLO, we determine the low-energy constants and . In contrast to a pure chiral treatment, the dispersion relation describes the observed curvature of one of the form factors, which we understand as an effect of rescattering beyond NNLO.
Highlights
Fits to data We perform a fit of the dispersion relation to both, the E865 [7, 8] and NA48/2 data sets [6, 9], corrected for additional isospin-breaking effects that were not taken into account in the experimental analyses [17]
We show the results of the matching at NLO and NNLO for the low-energy constants Lr1, Lr2 and Lr3
The dispersion relation is based on unitarity, analyticity and crossing
Summary
Where ∈ {e, μ} is either an electron or a muon. So far, experimental data is only available on the electron mode. Are defined to contain only the right-hand cut of the partial waves of the form factors F and G in the three channels. E.g. the function M0 contains the right-hand cut of the s-channel S-wave f0 of the form factor F : s2 M0(s) = P (s) + π. Take care of the righthand cuts of S- and P -waves in all channels, such that all the discontinuities are divided up into functions of a single variable They satisfy inhomogeneous Omnes equations with the solution. Numerical solution of the dispersion relation We note that the integral equations are linear in the subtraction constants. We determine the subtraction constants using three sources of information: first, we fit the experimental data on the form factors F and G from the high-statistics experiments NA48/2 [6, 9] and E865 [7, 8]. We fix the subtraction constants that are not well determined by the data with chiral input
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