Improved ${\pi }^0,\eta ,{\eta }^{\mathrm{\prime }}$ transition form factors in resonance chiral theory and their $a_{\mu }^{\mathrm{H}\mathrm{L}\mathrm{b}\mathrm{L}}$ 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 ${\pi }^0,\eta ,{\eta }^{\mathrm{\prime }}$ 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 ${\eta }^{\mathrm{\prime }}$. Our evaluation of the corresponding pole contributions to the hadronic light-by-light piece of the muon $g-2$ read: $a_{\mu }^{{\pi }^0\text{-}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{e}}=(61.9\pm 0.6_{-1.5}^{+2.4})\times {10}^{-11}$, $a_{\mu }^{\eta \text{-}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{e}}=(15.2\pm 0.5_{-0.8}^{+1.1})\times {10}^{-11}$ and $a_{\mu }^{{\eta }^{\mathrm{\prime }}\text{-}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{e}}=(14.2\pm 0.7_{-0.9}^{+1.4})\times {10}^{-11}$, for a total of $a_{\mu }^{{\pi }^0+\eta +{\eta }^{\mathrm{\prime }}\text{-}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{e}}=(91.3\pm 1.0_{-1.9}^{+3.0})\times {10}^{-11}$, where the first and second errors are the statistical and systematic uncertainties, respectively.