Improved $\pi^0,\eta,\eta^{\prime}$ transition form factors in resonance chiral theory and their $a_\mu^{\rm{HLbL}}$ contribution

Working with Resonance Chiral Theory, within the two resonance multiplets saturation scheme, we satisfy leading (and some subleading) chiral and asymptotic QCD constraints and accurately fit simultaneously the $\pi^{0},\eta,\eta^{\prime}$ transition form factors, for single and double virtuality. In the latter case, we supplement the few available measurements with lattice data to ensure a faithful description. Mainly due to the new results for the doubly virtual case, we improve over existing descriptions for the $\eta$ and $\eta^\prime$. Our evaluation of the corresponding pole contributions to the hadronic light-by-light piece of the muon $g-2$ read: $a_\mu^{\pi^{0}\text{-}\rm{pole}}=\left(61.9\pm0.6^{+2.4}_{-1.5}\right)\times10^{-11}$, $a_\mu^{\eta\text{-}\rm{pole}}=\left(15.2\pm0.5^{+1.1}_{-0.8}\right)\times10^{-11}$ and $a_\mu^{\eta^\prime\text{-}\rm{pole}}=\left(14.2\pm0.7^{+1.4}_{-0.9}\right)\times10^{-11}$, for a total of $a_\mu^{\pi^0+\eta+\eta^{\prime}\text{-}\rm{pole}}=\left(91.3\pm1.0^{+3.0}_{-1.9}\right)\times10^{-11}$, where the first and second errors are the statistical and systematic uncertainties, respectively.