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We have studied the $P\to\gamma^\star\gamma^\star$ form factor in ResonanceChiral Theory, with $P = \pi^0,\eta,\eta'$, to compute the contribution of thepseudoscalar pole to the hadronic light-by-light piece of the anomalousmagnetic moment of the muon. In this work we allow the leading $U(3)$ chiralsymmetry breaking terms, obtaining the most general expression for the formfactor of order $\mathcal{O}(m_P^2)$. The parameters of the Effective FieldTheory are obtained by means of short distance constraints on the form factorand matching with the expected behavior from QCD. Those parameters that cannotbe fixed in this way are fitted to experimental determinations of the formfactor within the spacelike momentum region of the virtual photon. Chiralsymmetry relations among the transition form factors for $\pi^0,\eta$ and$\eta'$ allow for a simultaneous fit to experimental data for the three mesons.This shows an inconsistency between the BaBar $\pi^0$ data and the rest of theexperimental inputs. Thus, we find a total pseudoscalar pole contribution of$a_\mu^{P,HLbL}=(8.47\pm 0.16)\cdot 10^{-10}$ for our best fit (neglecting theBaBar $\pi^0$ data). Also, a preliminary rough estimate of the impact of NLO in$1/N_C$ corrections and higher vector multiplets (asym) enlarges theuncertainty up to $a_\mu^{P,HLbL}=(8.47\pm 0.16_{\rm stat}\pm0.09_{N_C}{}^{+0.5}_{-0.0_{\rm asym}})$

Pseudoscalar pole contribution to the hadronic light-by-light piece of $a_\mu$