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arxiv: 1809.06399 · v2 · submitted 2018-09-17 · ✦ hep-ph

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Solving integral equations in ηto 3π

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classification ✦ hep-ph
keywords amplitudesdispersivepathalonganalysisangulardifficultiesequations
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A dispersive analysis of $\eta\to 3\pi$ decays has been performed in the past by many authors. The numerical analysis of the pertinent integral equations is hampered by two technical difficulties: i) The angular averages of the amplitudes need to be performed along a complicated path in the complex plane. ii) The averaged amplitudes develop singularities along the path of integration in the dispersive representation of the full amplitudes. It is a delicate affair to handle these singularities properly, and independent checks of the obtained solutions are demanding and time consuming. In the present article, we propose a solution method that avoids these difficulties. It is based on a simple deformation of the path of integration in the dispersive representation (not in the angular average). Numerical solutions are then obtained rather straightforwardly. We expect that the method also works for $\omega\to 3\pi$.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Dispersive analysis of the $J/\psi\to\pi^0 \gamma^\ast$ transition form factor with $\rho$-$\omega$ mixing effects

    hep-ph 2025-11 unverdicted novelty 7.0

    Dispersive analysis with ρ-ω mixing produces a two-parameter fit describing BESIII data on the J/ψ→π⁰γ* form factor from 0 to 2.8 GeV and extracts a (62 ± 21)° relative phase between strong and electromagnetic modes.

  2. Improved Standard-Model predictions for $\eta^{(\prime)}\to \ell^+ \ell^-$

    hep-ph 2025-12 accept novelty 5.0

    Updated SM predictions yield Br(η→e⁺e⁻)=5.37(4)(2)[4]×10⁻⁹, Br(η→μ⁺μ⁻)=4.54(4)(2)[4]×10⁻⁶, Br(η'→e⁺e⁻)=1.80(2)(3)[3]×10⁻¹⁰, and Br(η'→μ⁺μ⁻)=1.22(2)(2)[3]×10⁻⁷, with a mild 1.6σ tension in the η→μ⁺μ⁻ channel.