  We have studied the $P\to\gamma^\star\gamma^\star$ form factor in Resonance
Chiral Theory, with $P = \pi^0\eta\eta'$, to compute the contribution of the
pseudoscalar pole to the hadronic light-by-light piece of the anomalous
magnetic moment of the muon. In this work we allow the leading $U(3)$ chiral
symmetry breaking terms, obtaining the most general expression for the form
factor up to $\mathcal{O}(m_P^2)$. The parameters of the Effective Field Theory
are obtained by means of short distance constraints on the form factor and
matching with the expected behavior from QCD. Those parameters that cannot be
fixed in this way are fitted to experimental determinations of the form factor
within the spacelike region. Chiral symmetry 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 the experimental 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 (that neglecting the BaBar $\pi^0$ data). Also, a
preliminary rough estimate of the impact of NLO in $1/N_C$ corrections and
higher vector multiplets (asym) enlarges the uncertainty up to
$a_\mu^{P,HLbL}=(8.47\pm 0.16_{\rm stat}\pm 0.09_{N_C}{}^{+0.5}_{-0.0_{\rm
asym}})10^{-10}$.
