Data-driven analysis of the $\gamma\gamma^*\rightarrow\pi^0$ system using mathematical models and the role of feedback-loop dynamics

TL;DR

This paper presents a nonlinear mathematical model for the pion-photon transition form factor, demonstrating how feedback-loop dynamics lead to asymptotic saturation consistent with QCD, and improves data fitting with a novel relation and conformity protocol.

Contribution

It introduces a new mathematical framework linking feedback mechanisms to TFF saturation and enhances data analysis with a novel relation and standardized conformity criteria.

Findings

Model's inhibition leads to TFF saturation consistent with QCD

New fundamental relation between half-saturated TFF and asymptotic limit

Improved fit to Belle data with stronger parameter constraints

Abstract

The data behavior of the pion-photon transition form factor (TFF) is discussed using a nonlinear mathematical model with two parameters $B$ and $C$. Analysis shows that the model's inherent inhibition provides a precondition for the asymptotic saturation of the TFF in agreement with perturbative QCD. Integral to this derivation is the use of a novel fundamental relation between the half-saturated TFF and its asymptotic limit given by $B$. This helps avoid the determination of the location of the TFF at infinite momentum by estimating instead the maximum slope of the curve at $Q^2=C$. In conjunction with the $1\sigma(B,C)$ confidence ellipse, this framework provides an improved fit to the Belle data by setting stronger constraints on the fitting parameters and taking into account the sensitivity to third-party events below Belle's kinematic coverage. Without inhibition, the slope of the TFF curve would continue to grow so that saturation towards the asymptotic QCD limit would not be achieved. In order to compare the key features of TFF fit models to upcoming high-precision data in a standardized way, a conformity protocol in terms of QCD-based criteria is worked out. Adopting a broader perspective, we show that the asymptotic saturation of an inhibited TFF fit curve is analogous to a mechanical system driven by a feedback-loop mechanism in control theory.