Tensor Meson Pole contributions to the HLbL piece of $a_{\mu}$ within R$\chi$T

Abstract

We compute 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 groups determinations, we consider timely to present an alternative evaluation. In our approach, the lightest tensor meson nonet and two vector meson resonance nonets are considered in the chiral limit. Disregarding operators with derivatives, only the form factor $\mathcal{F}_1^T$ is non-vanishing, as assumed in the dispersive study. All parameters are determined by imposing a set of short-distance QCD constraints, and the radiative decay widths. In this case, we obtain (in units of $10^{-11}$): $ a_2$-pole: $-\left(1.02(10)_{\rm stat}(^{+0.00}_{-0.12})_{\rm syst}\right)$, $f_2$-pole: $-\left(3.2(3)_{\rm stat}(^{+0.0}_{-0.4})_{\rm syst}\right)$ and $f_2^\prime$-pole: $-\left(0.042(13)_{\rm stat}\right)$, which add up to $a_\mu^{a_2+f_2+f_2^\prime \rm -pole}=-\left(4.3^{+0.3}_{-0.5}\right)$, in close agreement with the holographic result when truncated to $\mathcal F_1^T$ only. However, with an ad-hoc extended Lagrangian, that also generates $\mathcal F_3^T$, as in the holographic approach, we have found: $ a_2$-pole: $+0.47(1.43)_{\rm norm}(3)_{\rm stat}(^{+0.06}_{-0.00})_{\rm syst}$, $f_2$-pole: $+1.18(4.18)_{\rm norm}(12)_{\rm stat}(^{+0.24}_{-0.00})_{\rm syst}$ and $f_2^\prime$-pole: $+0.040(78)_{\rm norm}(2)_{\rm stat}$, summing to $a_\mu^{a_2+f_2+f_2^\prime \rm - pole}=+1.7(4.4)$, which agree with these recent determinations within uncertainties (dominated by the $\mathcal F_3^T$ normalization). We point out that $R\chi T$ generates all form factors, the contributions to $a_\mu$ of $\mathcal F_{2,4,5}$ cannot be evaluated in the current basis, preventing for the moment a complete calculation of $a_\mu^{\rm T-poles}$ within our framework.