A Dispersive Treatment ofK_ℓ4Decays

Abstract

K 4 are for several reasons an especially interesting decay channel of K mesons:K 4 decays allow an accurate measurement of a combination of S -wave ππ scattering lengths, one form factor of the decay is connected to the chiral anomaly and the decay is the best source for the determination of some low energy constants of ChPT. We present a dispersive approach to K 4 decays, which takes rescattering effects fully into account. Some fits to NA48/2 and E865 measurements and results of the matching to ChPT are shown. MotivationK 4 , the semileptonic decay of a kaon into two pions and a lepton-neutrino pair, plays a crucial role in the context of low energy hadron physics, because it provides almost unique information about some of the O(p 4 ) low energy constants (LECs) of Chiral Perturbation Theory (ChPT), the effective low energy theory of QCD. The physical region of K 4 starts already at the ππ threshold, thus it happens at lower energies than e.g. elastic Kπ scattering. Since ChPT is an expansion in the masses and momenta, it is expected to converge better at lower energies. Therefore K 4 is a particularly interesting process to study.Besides, as the hadronic final state contains two pions, K 4 is also one of the best sources of information on the ππ scattering lengths a 0 0 and a 2 0 [1]. On the experimental side, we are confronted with impressive precision from high statistics measurements. During the last decade, the process has been measured in the E865 experiment at BNL [2] and in the NA48/2 experiment at CERN [1]. Very recently, the NA48/2 collaboration has published the results on the branching ratio and form factors of K 4 , based on more than a million events [3].Here, we present preliminary results of a new dispersive treatment of K 4 decays. Dispersion relations are an interesting tool to treat low energy hadronic processes. They are based on the very general principles of analyticity and unitarity. The derivation of the dispersion relation applies a chiral power counting and is valid up to and including O(p 6 ). The dispersion relation is parametrised by five subtraction constants. As soon as these constants have been fixed, the energy dependence is fully determined by the dispersion relation. The presented method implements a summation of final state rescattering effects, thus we expect it to incorporate the most important contributions beyond O(p 6 ). a

A dispersive treatment ofKℓ4decays

“…The obtained values are close to recent results of a global two-loop χPT fit [18]. Compared to our former treatment [6,7], the new determination of the LECs has a much better precision.…”

“…The current framework represents a substantial improvement and extension of our former treatment [6,7]. The dispersion relation provides a resummation of ππ-and Kπ-rescattering effects.…”

Bl4decays and the extraction of|Vub|

“…For its application to the closely related process of K l4 decays, see Refs. [33,34]. Note, however, that everything discussed in the following is to be understood at fixed s l : dispersion theory as applied here does not allow us to improve on the form factor dependence on the dilepton invariant mass, beyond what the chiral representation in the previous section includes.…”

“…The dispersive method using inhomogeneities as described above has by now been used for a variety of lowenergy processes, such as η → 3π [32,38], ω/φ → 3π [39], K → ππ [40], K l4 [33,34], γγ → ππ [41,42], or γπ → ππ [43,44]. In several of those cases, the inhomogeneities (given in terms of hat functions), which incorporate left-hand-cut structures, and the amplitudes given in terms of Omnès-type solutions with a right-hand cut only are calculated iteratively from each other, until convergence is reached.…”

Isospin breaking effects in $$K_{\ell 4}$$ K ℓ 4 decays

“…The motivation for doing such a transformation is that in an isospin symmetric world, calculations may become much easier. The isospin breaking corrections for K 4 will be used in a forthcoming dispersive treatment of this decay [26][27][28], which is performed in the isospin limit.…”