Determination of the $\eta$-$\eta^{\prime}$ Mixing Angle from $F_{\eta^{(\prime)}\gamma}(Q^2)$ within the Self-consistent Light-front Quark Model

Technical Summary: Determination of the $\eta$-$\eta'$ Mixing Angle from $F_{\eta(\prime)\gamma}(Q^2)$ within the Self-consistent Light-front Quark Model

Problem Statement
The $\eta$ and $\eta'$ mesons serve as critical probes of nonperturbative Quantum Chromodynamics (QCD), particularly regarding spontaneous chiral symmetry breaking, $SU(3)$ flavor breaking, and the axial $U_A(1)$ anomaly. Unlike pure quark-antiquark flavor eigenstates, these physical states are mixtures of nonstrange ($|u\bar{u} + d\bar{d}\rangle/\sqrt{2}$) and strange ($|s\bar{s}\rangle$) components. While the quark flavor mixing scheme (Feldmann-Kroll-Stech or FKS) successfully describes these states using a single mixing angle $\theta$ and two decay constants, precise determination of $\theta$ remains a subject of investigation. Recent measurements, such as those by LHCb in $B^0_{(s)} \to J/\psi \eta^{(\prime)}$ decays, provide precise reference values, but independent determinations using transition form factors (TFFs) are necessary to validate the mixing scheme across different observables.

Methodology
The author employs the self-consistent Light-front Quark Model (LFQM) combined with the quark flavor mixing scheme. Key methodological features include:

• Framework: The model utilizes a common set of light-front wave functions for the nonstrange ($\eta_q$) and strange ($\eta_s$) components. The formulation is "self-consistent" based on the replacement $M \to M_0$, which improves the treatment of covariance and zero-mode contributions in decay constants and distribution amplitudes.
• Wave Functions: The radial wave functions are modeled using a Gaussian form, with constituent quark masses ($m_q, m_s$) and Gaussian parameters ($\beta_{q\bar{q}}, \beta_{s\bar{s}}$) fixed from previous studies of pseudoscalar and vector mesons.
• Transition Form Factors (TFFs): The author calculates the TFFs $F_{\eta\gamma}(Q^2)$ and $F_{\eta'\gamma}(Q^2)$ as functions of the mixing angle $\theta$. The TFFs are derived from the overlap of the light-front wave functions with the electromagnetic current, separating nonstrange and strange contributions which are then mixed via $\theta$.
• Data Analysis: A combined $\chi^2$ scan is performed using experimental TFF data from the CLEO and BABAR collaborations (covering $2 < Q^2 < 40$ GeV$^2$). The quantity $Q^2 F_{P\gamma}(Q^2)$ is compared between theory and experiment to extract the optimal mixing angle.

Key Contributions and Results

1. Extraction of the Mixing Angle: The combined $\chi^2$ analysis yields a best-fit mixing angle of $\theta = 41.5^\circ$. This value is in close agreement with the recent LHCb result of 41.6^{+1.0}_{-1.2}^\circ, validating the quark flavor mixing scheme within the LFQM framework.
2. Decay Constants: Using $\theta = 41.5^\circ$, the nonstrange and strange decay constants are calculated as $f_q = 0.131 \pm 0.006$ GeV and $f_s = 0.174 \pm 0.004$ GeV. The strange constant $f_s$ aligns well with FKS expectations, while $f_q$ is slightly smaller than the FKS value, attributed to the model's use of isospin-symmetric nonstrange quark masses.
3. Light-Cone Distribution Amplitudes (LCDAs): The leading-twist LCDAs for both mesons are found to be symmetric around $x=1/2$ and slightly narrower than the asymptotic form ($6x(1-x)$), with suppressed endpoint regions.
4. Moments and Structure:
   • Gegenbauer Moments: The second Gegenbauer moment for the $\eta$ meson is calculated as $a_2^\eta = -0.034^{+0.023}_{-0.024}$, consistent with BABAR and Kroll analyses. The $\eta'$ moment is $a_2^{\eta'} = -0.059^{+0.022}_{-0.023}$.
   • $\xi$-moments: The second $\xi$-moment is $\langle \xi^2 \rangle_\eta = 0.188 \pm 0.008$, agreeing with BABAR and Kroll fits.
   • Transverse Momentum: The $\eta'$ meson exhibits slightly larger transverse momentum moments than the $\eta$, reflecting its larger strange quark component and more localized distribution.
5. Electromagnetic Properties:
   • Radii: The electromagnetic interaction radii are predicted as $r_\eta = 0.666^{+0.030}_{-0.029}$ fm and $r_{\eta'} = 0.610^{+0.028}_{-0.025}$ fm. These values show excellent agreement with measurements from the A2 and BESIII collaborations.
   • Two-Photon Decay Widths: The predicted widths are $\Gamma(\eta \to \gamma\gamma) = 0.43(3)$ keV and $\Gamma(\eta' \to \gamma\gamma) = 3.08(28)$ keV. These values are somewhat lower than experimental data ($0.516(18)$ keV and $4.35(14)$ keV, respectively).

Significance and Claims
The paper demonstrates that TFF data provide strong constraints on the light-front description of the nonperturbative properties of the $\eta$-$\eta'$ system. The consistency between the mixing angle derived from TFFs and that derived from heavy-meson decays supports the applicability of the quark flavor mixing scheme in the self-consistent LFQM.

The author modestly notes a discrepancy in the two-photon decay widths. They attribute this to the lack of direct experimental constraints on the real photon value $F_{\eta(\prime)\gamma}(0)$ in the fitting data, forcing the model to rely on extrapolation. Consequently, the predicted widths are lower than experimental values. The paper concludes that while the self-consistent LFQM offers a coherent description of the system's internal structure (radii, moments, and TFFs), the deviation in decay widths may indicate limitations of the single-angle mixing scheme or require further examination via future measurements.