The $γ^\ast\to ηγ$ and $γ^\ast\to η'γ$ form factors to NNLO accuracy in perturbative QCD

TL;DR

The paper achieves NNLO precision for the flavor-singlet axial-vector sector by computing the full three-loop anomalous-dimension matrix using conformal symmetry at the Wilson-Fisher critical point, enabling NNLO predictions for the hard exclusive $\gamma^\ast\gamma\to \eta,\eta'$ form factors at large momentum transfer. It also implements a consistent variable-flavor-number scheme for charm quarks and provides a model-independent presentation of the results, including several lattice-inspired LCDAs and four light-quark models. The key contributions are the explicit three-loop ADs (available in the Appendix/ancillary file) and the NNLO evolution framework, which together reduce the perturbative uncertainty to about 1% and illuminate the sensitivity of the $\eta'$ channel to a possible gluonium component and SU(3) flavor breaking in the LCDAs. The work lays groundwork for precise comparisons with future high-precision data (e.g., Belle II) and informs B-decay phenomenology involving $\eta$ and $\eta'$ in hard processes.

Abstract

We use conformal symmetry to calculate the NNLO anomalous dimension matrix (three loops) for flavor-singlet axial-vector QCD operators for spin $N \le 8$ from a set of gauge-invariant two-point correlation functions. Combining this result with the recent calculation of the two-loop coefficient functions, we carry out the calculation of the $γγ^\ast\to η$ and $γγ^\ast\to η'$ form factors at large momentum transfers to the NNLO accuracy in perturbative QCD.

Figures (3)

• Figure 1: Model shapes of the $u,d$-quark LCDAs at the scale $\mu_0^2 = 2\,\text{GeV}^2$ used in the numerical analysis. Model I: solid curve (blue); Model II: short dashes (orange); Model III: long dashes (purple); Model IV: dash-dotted curve (brown). The asymptotic LCDA $\Phi(u) \sim 6 u(1-u)$ is shown by dots (green) for comparison.
• Figure 2: The $\gamma^\ast \to \eta'\gamma$ vs. $\gamma^\ast \to \eta\gamma$ form factor at $q^2=112\, \text{GeV}^2$. The experimental data point is from Ref. BaBar:2006ash with statistic and systematic errors added in quadrature. The three sets of points (from top to bottom) are obtained for the values of the gluon LCDA parameter \ref{['LCDA']}$b_2(\mu_0) = -0.2$ (red), $b_2(\mu_0) = 0$ (black) and $b_2(\mu_0) = 0.2$ (blue). The results of the calculation with the four models of quark LCDAs specified in Table \ref{['table:1']} (see also Fig. \ref{['figure:LCDAs']}) are shown with circles (I), triangles (II), diamonds (III) and squares (IV), respectively.
• Figure 3: Absolute values of the form factors $|q^2F_\eta(q^2)|$ (left panel) and $|q^2F_{\eta'}(q^2)|$ (right panel) for time-like $q^2>0$ (solid curves) and space-like $q^2<0$ (dashed) photon virtualities. The three pairs of curves correspond to the different choices of the gluon LCDA: $b_2(\mu_0)=0$ (black), $b_2(\mu_0)= -0.2$ (red), and $b_2(\mu_0)=0.2$ (blue).