Dispersive construction of two-loop $P\to \pi \pi \pi$ $(P=K,\eta )$ amplitudes

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.

38 pages, 11 figures; minor reformulations for clarification, 4 references added

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• Cusps in K --> 3 pi decays
• Search for direct CP violating charge asymmetries in $K^{\pm }\to {\pi }^{\pm }{\pi }^{+}{\pi }^{-}$ and $K^{\pm }\to {\pi }^{\pm }{\pi }^0{\pi }^0$ decays
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• Measurement of the Slope Parameter for the eta->3pi0 Decay in the pp->pp eta Reaction
• Dispersive Approach to Chiral Perturbation Theory
• Measurement of the $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0$ Dalitz plot distribution
• Detailed Study of the KL -> 3pi0 Dalitz Plot
• Measurement of the matrix elements for the decays ${\eta }^{\prime }\to \eta {\pi }^{+}{\pi }^{-}$ and ${\eta }^{\prime }\to \eta {\pi }^0{\pi }^0$
• Determination of $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0$ Dalitz plot slopes and asymmetries with the KLOE detector
• $\eta$ and ${\eta }^{\prime }$ Physics at BES-III
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Cited by in corpus (6)

• Precision tests of fundamental physics with $\eta$ and ${\eta }^{\prime }$ mesons
• Decay amplitudes to three hadrons from finite-volume matrix elements
• What can we learn about light-meson interactions at electron-positron colliders?
• Patterns of $C$- and $CP$-violation in hadronic $\eta$ and ${\eta }^{\prime }$ three-body decays
• Chiral Perturbation Theory Reflections on Effective Theories of the Standard Model
• Subthreshold parameters of $\pi \pi$ scattering revisited