Assessment of the $a_{2}(1320)$ tensor-meson contribution to $η/η^{\prime}\toπ^{0}γγ$ decays

TL;DR

The paper investigates the rare decay $\eta\to\pi^{0}\gamma\gamma$ by extending the established vector-meson dominance (VMD) and linear sigma model (L$\sigma$M) frameworks to include the tensor resonance $a_{2}(1320)$ within a chiral context. The authors construct the $a_{2}$ contribution using $a_{2}\to P P$ and $a_{2}\to \gamma\gamma$ couplings, a tensor propagator with a width $\Gamma_{a_{2}}(s)$, and fix the overall sign of the tensor–photon coupling from $\gamma\gamma\to\pi^{0}\eta$ data, which implies destructive interference with vector exchanges. Their results show that while the $a_{2}$ is individually small, its interference with vector resonances reduces the total width to $\Gamma(\eta\to\pi^{0}\gamma\gamma)=0.154(22)$ eV and $BR=1.17(17)\times 10^{-4}$, about $14\%$ below the previous estimate and in excellent agreement with the KLOE-2 measurement, though still below the PDG value. For $\eta^{\prime}\to\pi^{0}\gamma\gamma$, the process remains dominated by on-shell $\omega$ exchange, making the $a_{2}$ contribution negligible. The study emphasizes the significance of tensor–vector interference for precision tests and advocates future high-precision measurements (e.g., JEF, REDTOP, HIAF) to further probe tensor dynamics in $\eta/\eta^{\prime}$ decays and related semileptonic processes.

Abstract

In light of the recent measurement of the $η\toπ^{0}γγ$ decay by the KLOE-2 Collaboration, a previous analysis including vector- and scalar-meson exchange contributions using the VMD and L$σ$M frameworks, respectively, is extended in the present study to incorporate the effects of the $a_{2}(1320)$ tensor meson within a chiral context. Although the individual contribution of the $a_{2}$ is negligible, its destructive interference with the vector-meson resonances is found to be significant, representing approximately $18\%$ of the total signal and substantially affecting the diphoton invariant-mass distribution, especially at low $m_{γγ}^{2}$ values. The total decay rate is calculated to be $Γ(η\toπ^{0}γγ)=0.154(22)$ eV, which corresponds to a branching ratio of ${\rm BR}(η\toπ^{0}γγ)=1.17(17)\times10^{-4}$. This result is approximately $5σ$ below the reported value of the PDG, $2.55(22)\times10^{-4}$, while it is in very good agreement with the KLOE-2 measurement, $0.98(11_{\rm stat})(14_{\rm syst})\times10^{-4}$. In contrast, the total contribution of the $a_{2}$ is found to be negligible in $η^{\prime}\toπ^{0}γγ$, as this process is completely dominated by the exchange of an on-shell $ω$ vector resonance.

Figures (6)

• Figure 1: Schematic diagram for the $a_{2}$-exchange contribution to the decay $\eta^{(\prime)}\to\pi^{0}\gamma\gamma$.
• Figure 2: Total (black solid) and individual contributions to the $\eta\to\pi^{0}\gamma\gamma$ invariant-mass distribution $d\Gamma/dm_{\gamma\gamma}^{2}$.
• Figure 3: Dalitz plots for the full squared amplitude $|\mathcal{A}_{{\rm{V+S+}}a_{2}}|^{2}$ of the decay $\eta\to\pi^{0}\gamma\gamma$ (left) and for the ratio $|\mathcal{A}_{{\rm{V+S+}}a_{2}}|^{2}/|\mathcal{A}_{{\rm{V+S}}}|^{2}$ (right).
• Figure 4: Left: Comparison between the A2 (blue circles) A2atMAMI:2014zdf, Crystal Ball (red squares) Prakhov:2008zz and KLOE-2 (green triangles) KLOE-2:2025ggc measurements of the $\eta\to\pi^{0}\gamma\gamma$ invariant-mass distribution, $d\Gamma/dm_{\gamma\gamma}^{2}$, and our prediction (black solid line) based on vector-, scalar-, and tensor-meson exchanges. The orange dashed line is the prediction from Ref. Escribano:2018cwg, where the contribution of the $a_{2}(1320)$ was not included. Right: Our prediction (black solid) compared to data, and to the theoretical results from Refs. Danilkin:2017lyn (magenta dotted), Lu:2020qeo (green dash-dotted) and Oset:2008hp (brown dashed).
• Figure 5: Left: Comparison between the BESIII (blue circles) BESIII:2016oet measurement of the $\eta^{\prime}\to\pi^{0}\gamma\gamma$ invariant-mass distribution, $d\Gamma/dm_{\gamma\gamma}^{2}$, and our prediction (black solid line) based on vector-, scalar-, and tensor-meson exchanges. The orange dashed line is the prediction from Ref. Escribano:2018cwg, where the contribution of the $a_{2}(1320)$ was not included. Right: Dalitz plot for the decay $\eta^{\prime}\to\pi^{0}\gamma\gamma$.