Low Energy Constants from K l 4 Form - Factors

We have calculated the form-factors F and G in K → ππℓν decays (K ℓ4) to two-loop order in Chiral Perturbation Theory (ChPT). Combining this together with earlier two-loop calculations an updated set of values for the L r i , the ChPT constants at O(p 4), is obtained. We discuss the uncertainties in the determination and the changes compared to previous estimates. 1. The theoretical study of K → ππlν decays, K ℓ4 decays, provides an interesting possibility to test our understanding of the long-distance dynamics of the strong sector in the Standard Model. Perturbative QCD can not be applied at energies lower than the spontaneous symmetry scale, Λ χ ∼ m ρ. One has to resort to an effective approach to the full theory, Chiral Perturbation Theory (ChPT) [1], to obtain a reliable model independent description of physical processes. The main purpose in the study of K ℓ4 decays is twofold: i) This decay is one of the cleanest sources of ππ pairs at low-energies and thus provides us with the possibility to check ππ-scattering near threshold. ii) The form-factors themselves are also directly of interest. They provide a direct test of our understanding of the three-flavour sector and are in addition one of the major inputs to determine the needed constants to predict other quantities. The early history can be found in the review [2]. K ℓ4 have been treated in the context of ChPT at one-loop [3] and in a dispersively improved one-loop representation [4]. These studies delivered the standard values of the low-energy constants L r 1 , L r 2 and L r 3. On the other hand, ππ-scattering has been studied at two-loop order in two-flavour ChPT, direct [5] and via dispersive methods [6]. The main remaining uncertainty on the prediction of ππ-scattering is the value of the low-energy constants. In [5] the standard values from [4] were used. A Roy equation analysis using high energy ππ-data, gave significantly different values for these constants [7]. There are several possible sources for this discrepancy: the Omnès representation of [4] is not sufficient for K ℓ4 ; the presence of large systematic effects in the ππ-scattering data base and Roy analysis; or large corrections to the one-loop relation between two-and three-flavour ChPT constants. Therefore, and since new experiments are underway, a full two-loop calculation of K ℓ4 is necessary. A first step was the estimate …