Neutral pion to two-photons transition form factor revisited

Based upon a combined formalism of Schwinger-Dyson and Bethe-Salpeter equations in quantum chromodynamics (QCD), we propose a QCD kindred algebraic model for the dressed quark propagator, for the Bethe-Salpeter amplitude of the pion and the electromagnetic quark-photon interaction vertex. We then compute the $\gamma^{*}\pi^0\gamma$ transition form factor $G^{\gamma^{*}\pi^0\gamma}(Q^2)$ for a wide range of photon momentum transfer squared $Q^2$. The quark propagator is expanded out in its perturbative functional form but with dynamically generated dressed quark mass. It has complex conjugate pole singularities in the complex-momentum plane which is motivated by the solution of the quark gap equation with rainbow-ladder truncation of the infinite set of Schwinger-Dyson equations. This complex pole singularity structure of the quark propagator can be associated with a signal of confinement which prevents quarks to become stable asymptotic states. The Bethe-Salpeter amplitude is expressed without a spectral density function, which encapsulate its low and large momentum behaviour. The QCD evolution of the distribution amplitude is also incorporated into our model through the direct implementation of Efremov-Radyushkin-Brodsky-Lepage evolution equations. We include the effects of the quark anomalous magnetic moment in the description of the quark-photon vertex whose infrared enhancement is known to dictate hadronic properties. Once the QCD kindred model is constructed, we calculate the form factor $G^{\gamma^{*}\pi^0\gamma}(Q^2)$ and find it consistent with direct QCD-based studies as well as most available experimental data. It slightly exceeds the conformal limit for large $Q^2$ which might be attributed to the scaling violations in QCD. The associated interaction radius and neutral pion decay width turn out to be compatible with experimental data.