Pseudoscalar pole light-by-light contributions to the muon $(g-2)$ in Resonance Chiral Theory

We have studied the $P\to\gamma^\star\gamma^\star$ transition form-factors ($P=\pi^0,\eta,\eta'$) within a chiral invariant framework that allows us to relate the three form-factors and evaluate the corresponding contributions to the muon anomalous magnetic moment $a_\mu$, through pseudoscalar pole contributions. We use a chiral invariant Lagrangian to describe the interactions between the pseudo-Goldstones from the spontaneous chiral symmetry breaking and the massive meson resonances. We will consider just the lightest vector and pseudoscalar resonance multiplets. Photon interactions and flavor breaking effects are accounted for in this covariant framework. This article studies the most general corrections of order $m_P^2$ within this setting. Requiring short-distance constraints fixes most of the parameters entering the form-factors, consistent with previous determinations. The remaining ones are obtained from a fit of these form-factors to experimental measurements in the space-like ($q^2\le0$) region of photon momenta. The combination of data, chiral symmetry relations between form-factors and high-energy constraints allows us to determine with improved precision the on-shell $P$-pole contribution to the Hadronic Light-by-Light scattering of the muon anomalous magnetic moment: we obtain $a_{\mu}^{P,HLbL}=(8.47\pm 0.16)\cdot10^{-10}$ for our best fit. This result was obtained excluding BaBar $\pi^0$ data, which our analysis finds in conflict with the remaining experimental inputs. This study also allows us to determine the parameters describing the $\eta-\eta'$ system in the two-mixing angle scheme and their correlations. Finally, a preliminary rough estimate of the impact of loop corrections ($1/N_C$) and higher vector multiplets (asym) enlarges the uncertainty up to $a_\mu^{P,HLbL} = (8.47\pm 0.16_{\rm sta}\pm0.09_{1/N_C}{}^{+0.5}_{-0.0}{}_{\rm asym})\cdot 10^{-10}$.