Tensor Meson Pole contributions to the HLbL piece of $a_μ$ within R$χ$T

TL;DR

This work computes tensor-meson pole contributions to the HLbL piece of $a_\mu$ within Resonance Chiral Theory, using the lightest tensor nonet and two vector nonets in the chiral limit. In the minimal setup, only $\mathcal{F}_1^T$ is nonzero and is fixed by short-distance QCD constraints and radiative widths, yielding a negative total $a_\mu^{T\text{-poles}}$ of about $-4.3\times 10^{-11}$. Extending the Lagrangian to generate $\mathcal{F}_3^T$ flips the sign and yields $a_\mu^{T\text{-poles}}=+1.7(4.4)\times 10^{-11}$, broadly compatible with holographic results within uncertainties, though dominated by the uncertain normalization of $\mathcal{F}_3^T(0,0)$. The authors emphasize that a complete $a_\mu^{\rm T-poles}$ within this framework awaits the inclusion of $\mathcal{F}_{2,4,5}^T$ and urgent double-virtual tensor-transition data to constrain $\mathcal{F}_3^T(0,0)$, underscoring the need for targeted measurements of $T\to\gamma^*\gamma^*$.

Abstract

We compute the tensor meson pole contributions to the Hadronic Light-by-Light piece of $a_μ$ 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 $a_μ^{a_2+f_2+f_2^\prime \rm -pole}=-\left(4.3^{+0.3}_{-0.5}\right)$, 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 $a_μ^{a_2+f_2+f_2^\prime \rm - pole}=+1.7(4.4)$, which agree with these recent determinations within uncertainties (dominated by the $\mathcal F_3^T$ normalization). We point out that $RχT$ generates all form factors, the contributions to $a_μ$ of $\mathcal F_{2,4,5}$ cannot be evaluated in the current basis, preventing for the moment a complete calculation of $a_μ^{\rm T-poles}$ within our framework.

Figures (8)

• Figure 1: Tensor Meson transition FF, which gives the radiative decay with both photons on-shell.
• Figure 2: Comparison between the simple quark model (QM) Schuler:1997yw used in the dispersive determination Hoferichter:2024bae, the holographic QCD (hQCD) hard-wall model Cappiello:2025fyf, the $R \chi T$ model used in this work and the $C_2^0$ of eq. (\ref{['eq_defCA']}) for the single and symmetric double virtual form factor $\mathcal{F}_1^{T}$, for all 3 tensor mesons: $a_2$(1320), $f_2$(1270) and $f_2^\prime$(1525). Our one $\sigma$ uncertainties are displayed by the green band.
• Figure 3: Comparison between the (hQCD) hard-wall model Cappiello:2025fyf and the $R \chi T$ model used in this work for the single and symmetric double virtual form factor $\mathcal{F}_3^{T}$, for $a_2$(1320). The same shape is found for $f_2,f_2'$ (including uncertainties), with a relative factor of $c_T \frac{M_T^3}{M_a^3}$ with respect to the one shown. Our one $\sigma$ uncertainties are displayed by the green band, which, in contrast to $\mathcal{F}_1^T$, is dominated by the large error of $\mathcal{F}_3^T(0,0)$
• Figure 4: Comparison between the simple quark model (QM) Schuler:1997yw used in ref. Hoferichter:2024bae, the hard-wall model (hQCD) Cappiello:2025fyf and this work ($R\chi T$) with the Belle data Belle:2015oin in the helicity basis of eq. (\ref{['eq:helicitybasis']}) for the $f_2$(1270) tensor meson, normalized by $\mathcal{F}^{f_2}(0,0)=\sqrt{\frac{5 \Gamma_{T\gamma\gamma}}{\pi \alpha^2 M_T}}$. Our one $\sigma$ uncertainties are displayed by the green band.
• Figure 5: Comparison of the full asymmetry range of the asymptotic behavior of $f_1^{a_2}(w)$ given by ref. Hoferichter:2020lap using the Light-Cone-Expansion, the quark model (QM) Schuler:1997yw used in ref. Hoferichter:2024bae, the hard-wall model of ref. Cappiello:2025fyf (hQCD), this work ($R\chi T$) (left plot), and the Canterbury Approximant used to correct our asymptotic behavior ($C_2^0$). The same comparison for Light-Cone-Expansion, hQCD and this work is shown in the right plot for $f_3^{a_2}(w)$.