The $\gamma \pi \to \pi \pi$ anomaly from lattice QCD and dispersion relations

Abstract

We propose a formalism to extract the $\gamma \pi \to \pi \pi$ chiral anomaly $F_{3\pi }$ from calculations in lattice QCD performed at larger-than-physical pion masses. To this end, we start from a dispersive representation of the ${\gamma }^{(\ast )}\pi \to \pi \pi$ amplitude, whose main quark-mass dependence arises from the $\pi \pi$ scattering phase shift and can be derived from chiral perturbation theory via the inverse-amplitude method. With parameters constrained by lattice calculations of the $P$-wave phase shift, we use this combination of dispersion relations and effective field theory to extrapolate two recent ${\gamma }^{(\ast )}\pi \to \pi \pi$ calculations in lattice QCD to the physical point. Our formalism allows us to extract the radiative coupling of the $\rho (770)$ meson and, for the first time, the chiral anomaly $F_{3\pi }=38(16)(11){\text{GeV}}^{-3}$. The result is consistent with the chiral prediction albeit within large uncertainties, which will improve in accordance with progress in future lattice-QCD computations.

29 pages, 5 figures; version published in JHEP: high-energy continuation of phase explained in more detail

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• Dispersion relations for the hadronic VVA correlator
• Radiative decay of the resonant $K^{\ast }$ and the $\gamma K\to K\pi$ amplitude from lattice QCD
• Anomalous thresholds in $B\to (P,V){\gamma }^{\ast }$ form factors
• Analysis of rescattering effects in $3\pi$ final states
• The semileptonic decays ${\eta }^{(\prime )}\to {\pi }^0{\ell }^{+}{\ell }^{-}$ and ${\eta }^{\prime }\to \eta {\ell }^{+}{\ell }^{-}$ in the standard model
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• Extracting the chiral anomaly from $e^{+}{e}^{-}\to 3\pi$
• Dispersive analysis of the $J/\psi \to {\pi }^0{\gamma }^{\ast }$ transition form factor with $\rho$-$\omega$ mixing effects
• Tests of Three-Flavor Chiral Perturbation Theory
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• Dispersive Analysis of $D$- and $B$-Meson Form Factors with Chiral and Heavy-Quark Constraints