Pion-photon transition form factor in QCD. Theoretical predictions and topology-based data analysis

We discuss the evaluation of the transition form factor (TFF) $F^{\gamma*\gamma\pi^0}(Q^2)$ by means of QCD theory and by state-space reconstruction from topological data analysis. We first calculate this quantity in terms of quark-gluon interactions using light cone sum rules (LCSRs). The spectral density includes radiative corrections in leading, next-to-leading, and next-to-next-to-leading-order of perturbative QCD. Besides, it takes into account the twist-four and twist-six terms. The hard-scattering part in the LCSR is convoluted with various pion distribution amplitudes with different morphologies in order to obtain a wide range of predictions for the form factor, including two-loop evolution which accounts for heavy-quark thresholds. We then use nonlinear time series analysis to extract information on the long-term $Q^2$ behavior of the measured scaled form factor in terms of state-space attractors embedded in $\mathbb{R}^3$. These are reconstructed by applying the Packard-Takens method of delays to appropriate samplings of the data obtained in the CLEO, \textit{BABAR}, and Belle single-tagged $e^+e^- \to e^+e^-\pi^0$ experiments. The corresponding lag plots show an aggregation of states around the value $Q^2F^{\gamma*\gamma\pi^0}(Q^2) \approx 0.165\pm 0.005$ GeV pertaining to the momentum interval $Q^2\in [9-11]$~GeV$^2$. We argue that this attractor portrait is a transient precursor of a distribution of states peaking closer to the asymptotic limit $Q^2F^{\gamma*\gamma\pi^0}(Q^2\to \infty)=\sqrt{2}f_\pi$~GeV. More data with a regular increment of 1~GeV$^2$ in the range between 10 and 25~GeV$^2$ would be sufficient to faithfully determine the terminal portrait of the attractor.