Probing the $γγ^*\to η^{(\prime)}$ Transition Form Factors with Newly Derived $η^{(\prime)}$-Meson Light-Cone Distribution Amplitudes

TL;DR

This work addresses the problem of predicting the transition form factors for the process $\gamma\gamma^*\to \eta^{(\prime)}$ by employing a quark-flavor mixing framework and leading-twist light-cone distribution amplitudes derived from light-cone sum rules. It combines a transverse-momentum–dependent pQCD approach with a light-cone harmonic oscillator–based description of the leading-twist LCWFs, and systematically examines both valence and non-valence contributions to the form factors. A key contribution is the inclusion of intrinsic charm and gluonic components through $\eta$–$\eta'$–$\eta_c$ and $\eta$–$\eta'$–$G$–$\eta_c$ mixing schemes, with quantitative assessment via $\chi^2$ per degree of freedom and $p$-values, showing improved agreement in many kinematic regions and especially enhanced sensitivity of $Q^2F_{\eta'\gamma}(Q^2)$ to charm at high $Q^2$. The findings provide constraints on the charm content and gluonic admixtures in the η and η′ mesons and offer predictions for upcoming high-precision experiments (e.g., Belle II) to test nonperturbative QCD dynamics in exclusive processes.

Abstract

In the present work, we analyze the properties of the transition form factors (TFFs) for the $γγ^*\to η^{(\prime)}$ process, employing the $η^{(\prime)}$-meson light-cone distribution amplitude (LCDA) derived within the light-cone sum rule framework. To this end, we adopt the quark-flavor mixing scheme for the $η^{(\prime)}$ meson, and compute the TFFs by systematically incorporating transverse-momentum corrections and contributions beyond the leading Fock state. We utilize light-cone harmonic oscillator models to parameterize the longitudinal and transverse behavior of the leading-twist light-cone wavefunction, for which the corresponding LCDA exhibits a unimodal profile. We further examine the potential contributions of intrinsic charm components to the scaled TFFs $Q^2 F_{ηγ}(Q^2)$ and $Q^2 F_{η^\prime γ}(Q^2)$. Leveraging a range of values for the decay constant $f_{η_{c_0}}$ and implementing the $η$-$η'$-$η_c$ and $η$-$η^\prime$-$G$-$η_c$ mixing mechanisms accordingly, together with the recently updated mixing angles, we investigate the impact of the intrinsic $c\bar{c}$ and gluonic component on these observables. In high-$Q^2$ regime, $Q^2 F_{η^\primeγ}(Q^2)$ exhibits a marked increase in sensitivity to the charm quark component, whereas $Q^2F_{ηγ}(Q^2)$ becomes notably stabilized. A detailed discussion of $χ^2/d.o.f$ and $p$-values indicates that the intrinsic charm quark component is important and yields a substantial, non-negligible contribution across the entire $Q^2$ range.

Figures (5)

• Figure 1: The $\eta_q$ and $\eta_s$ mesons' leading-twist LCDAs $\phi_{2;\eta_q}(x,\mu_0) (a)$ and $\phi_{2;\eta_s}(x,\mu_0) (b)$ with $\mu_0=1~{\rm GeV}$. As a comparison, the curves from the nonlocal NJL model GomezDumm:2016bxp, the BS model Ding:2018xwy, the CZ-form and the asymptotic form have also been presented, respectively.
• Figure 2: The $\eta-\gamma$ and $\eta'-\gamma$ TFFs $Q^2 F_{\eta\gamma}(Q^2)$ (a) and $Q^2 F_{\eta'\gamma}(Q^2)$ (b). For comparison, the experimental results from CLEO CLEO:1997fho and BABAR'11 BaBar:2011nrp are also represented.
• Figure 3: The $\eta-\gamma$ and $\eta'-\gamma$ TFFs $Q^2F_{\eta\gamma}(Q^2)$ (a) and $Q^2F_{\eta'\gamma}(Q^2)$ (b) are calculated for different mixing angles. The experimental results of CLEO CLEO:1997fho and BABAR'11 BaBar:2011nrp are also represented.
• Figure 4: The $\eta-\gamma$ and $\eta'-\gamma$ TFFs $Q^2F_{\eta\gamma}(Q^2)$ (a) and $Q^2F_{\eta'\gamma}(Q^2)$ (b) are obtained through $\eta-\eta'-\eta_c$ mixing mechanism. The experimental results from CLEO CLEO:1997fho, BABAR'11 BaBar:2011nrp and BABAR'06 BaBar:2006ash are also represented.
• Figure 5: The $\eta-\gamma$ and $\eta^\prime-\gamma$ TFFs $Q^2F_{\eta\gamma}(Q^2)$ (a) and $Q^2F_{\eta^\prime\gamma}(Q^2)$ (b) are obtained through $\eta-\eta^\prime-G-\eta_c$ mixing mechanism, which the shaded band indicates the uncertainty arising from different choices of light constituent quark masses (e.g. $m_q=0.30\pm0.05~{\rm GeV}$ and $m_s=0.45\pm0.05~{\rm GeV}$). The experimental results from CLEO CLEO:1997fho, BABAR'11 BaBar:2011nrp and BABAR'06 BaBar:2006ash are also represented.