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Abstract

We report our results for the tensor meson pole contributions to the Hadronic Light-by-Light piece of $a_\mu$ in the purely hadronic region, using Resonance Chiral Theory. Given the differences between the dispersive and holographic results for it and the resulting discussion of the corresponding uncertainty estimate for the Hadronic Light-by-Light section of the muon \ensuremath{g - 2} theory initiative second White Paper, we consider timely to present an alternative evaluation. In our approach, in addition to the lightest tensor meson nonet, two vector meson resonance nonets are considered. We work in the chiral limit, where all parameters are determined by imposing short-distance QCD constraints, and the radiative tensor decay widths. Within the minimal setting rendering smooth short-distance behaviour, only the form factor $\mathcal{F}_1^T$ is non-vanishing, in agreement with the result put forward by the dispersive study. In this case, we obtain the following results for the different contributions (in units of $10^{-11}$): $a_\mu^{\rm a_2-pole}=-\left(1.09(10)_{\rm stat}(^{+0.11}_{-0.00})_{\rm syst}\right)$, $a_\mu^{\rm f_2-pole}=-\left(3.4(3)_{\rm stat}(^{+0.0}_{-0.4})_{\rm syst}\right)$ and $a_\mu^{\rm f_2^\prime-pole}=-\left(0.046(14)_{\rm stat}(^{+0.000}_{-0.008})_{\rm syst}\right)$, which add up to $a_\mu^{a_2+f_2+f_2^\prime \rm - pole}=-\left(4.5^{+0.3}_{-0.5}\right)$, in close agreement with the holographic result. However, with an extended Lagrangian, that also generates $\mathcal{F}_3^T$, we have found: $a_\mu^{\rm a_2-pole}=+3.3(1.4)_{\rm stat}(^{+0.00}_{-0.07})_{\rm syst}$, $a_\mu^{\rm f_2-pole}=+6(5)_{\rm stat}(^{+0.12}_{-0.29})_{\rm syst}$ and $ a_\mu^{\rm f_2'-pole}=$ $+0.8(5)_{\rm stat}(^{+0.001}_{-0.000})_{\rm syst}$, summing to $a_\mu^{a_2+f_2+f_2^\prime \rm - pole}=+10(5)$, which agree in sign with the holographic values, exceeding them in magnitude, however.