Cusps in K_L --> 3 pi decays

Abstract

The pion mass difference generates a pronounced cusp in K → 3π decays, the strength of which is related to the ππ S-wave scattering lengths. We apply an effective field theory framework developed earlier to evaluate the amplitudes for K L → 3π decays in a systematic manner, where the strictures imposed by analyticity and unitarity are respected automatically. The amplitudes for the decay η → 3π are also given.

The cusp effect in eta' --> eta pi pi decays

“…What makes this channel particularly apt for an investigation of the cusp, apart from the enormous statistics collected by the NA48/2 collaboration [3], is the significantly larger branching fraction of K + → π + π + π − compared to K + → π 0 π 0 π + , such that the perturbation of the decay spectrum of the latter is very sizeable. This is in marked contrast to two other decays that have been studied subsequently and that display, in principle, cusp structures generated by the same mechanism: K L → 3π 0 [2,4,8,9], and η → 3π 0 [8,[10][11][12][13][14]. In both of these, the weaker coupling to the charged-pion final state diminishes the cusp to a mere 1-2% effect on the decay spectrum.…”

“…It is set up in such a way that the results are manifestly covariant, with the correct analytic structure of the decay amplitude fully reproduced in the lowenergy region. A consistent power counting scheme can be constructed in a very similar manner to the one in [5,8]: we introduce the formal non-relativistic parameter ǫ and count the pion and η 3-momenta (in the η ′ rest frame) as O(ǫ), the kinetic energies T i = p 0 i − M i as O(ǫ 2 ), and the masses of the particles involved as O(1). Let us remark, however, that as opposed to the case of ππ-scattering, where the mass difference between the two scattering particles amounts to a small contribution in isospin breaking at maximum, for πη-scattering we count M η −M π = O(1).…”

“…for n = a, b, c, d. Analogous definitions hold for derivatives on the η field, with Φ → η and P n → P η . The ππ finalstate Lagrangian can be written as [8] L…”

“…1 Indeed, in the isospin limit BR(η ′ → ηπ + π − ) = 2 BR(η ′ → ηπ 0 π 0 ), so one expects a sizeable effect on the π 0 π 0 decay spectrum. In this article, we adapt the nonrelativistic effective field theory (NREFT) formalism developed in [5,6,8] to these channels. Compared to K → 3π and η → 3π decays, there is a different "secondary" finalstate rescattering channel πη to take into account.…”

Photoproduction of neutral pions

“…Along the lines of previous work which analyzed the cusps in K → 3π and η ′ → ηππ decays [2][3][4], we construct a nonrelativistic theory which provides a rigorous framework to describe the structure of the cusp order by order in a perturbative expansion in terms of small momenta and effective range parameters of pion-nucleon scattering as well as threshold parameters of the photoproduction reaction [5]. By construction, the theory correctly reproduces the low-energy singularities in the Mandelstam plane.…”