Abstract
We present and develop a general dispersive framework allowing us to construct representations of the amplitudes for the processes $P\pi\to\pi\pi$, $P=K,\eta$, valid at the two-loop level in the low-energy expansion. The construction proceeds through a two-step iteration, starting from the tree-level amplitudes and their S and P partial-wave projections. The one-loop amplitudes are obtained for all possible configurations of pion masses. The second iteration is presented in detail in the cases where either all masses of charged and neutral pions are equal, or for the decay into three neutral pions. Issues related to analyticity properties of the amplitudes and of their lowest partial-wave projections are given particular attention. This study is introduced by a brief survey of the situation, for both experimental and theoretical aspects, of the decay modes into three pions of charged and neutral kaons and of the eta meson.
Highlights
Our experimental knowledge of the Dalitz-plot structures of the amplitudes for the processes P → πππ has substantially improved during the last decades, for P 1⁄4 KÆ [1–7], P 1⁄4 KL [8], or P 1⁄4 η [9–18]
KÆ and K0L decays into three pions. These decay amplitudes were computed using the framework of chiral perturbation theory up to next-toleading order (NLO) already in 1991 [58], at that time ignoring all the isospin-breaking effects
The primary purpose of this study was to present a detailed account of the dispersive approach to the construction of Pπ → ππ, P 1⁄4 KÆ; KL; KS; η, scattering amplitudes that possess all the correct analytic properties at the order two loops in the low-energy expansion
Summary
Our experimental knowledge of the Dalitz-plot structures of the amplitudes for the processes P → πππ has substantially improved during the last decades, for P 1⁄4 KÆ [1–7], P 1⁄4 KL [8], or P 1⁄4 η [9–18]. We give the full isospin-breaking result for all the amplitudes only at the one-loop level, while at the first stage, the expressions of the two-loop amplitudes are only worked out in the limit where the masses of the neutral and charged pions are equal. This allows describing some general features of our construction in a simpler framework without having to deal, in addition, with several kinematic complications that arise only when the intermediate- and final-state pions have unequal masses. A comprehensive account with more details on some of the technical aspects can be found in Ref. [55]
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