Improved $π^0,η,η^{\prime}$ transition form factors in resonance chiral theory and their $a_μ^{\rm{HLbL}}$ 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: $a_\mu^{\pi^{0}\text{-}\rm{pole}}=\left(60.4\pm0.5^{+3.2}_{-1.8}\right)\times10^{-11}$, $a_\mu^{\eta\text{-}\mathrm{pole}}=\left(15.2\pm0.5^{+1.1}_{-0.7}\right)\times10^{-11}$ and $a_\mu^{\eta^\prime\text{-}\rm{pole}}=\left(14.4\pm0.8^{+1.4}_{-1.0}\right)\times10^{-11}$, for a total of $a_\mu^{\pi^0+\eta+\eta^{\prime}\text{-}\rm{pole}}=\left(90.0\pm1.1^{+3.7}_{-2.2}\right)\times10^{-11}$, where the first and second error are the statistical and systematic uncertainties, respectively.