Probing Instanton Dynamics in the Pion Vector Form Factor with Wilson Flow

TL;DR

The paper addresses whether the instanton liquid model can capture nonperturbative QCD effects in hadron structure by examining the pion electromagnetic form factor $F_{\pi}(q^2)$ under Wilson flow, which isolates instanton-dominated vacuum contributions. The authors compute $F_{\pi}(q^2,t)$ from lattice two-point and three-point correlators on a $N_f=2+1$ ensemble, using ratios and a massive renormalization scheme with $Z_V(t)$ and a flow-tuned $\kappa(t)$ to keep $m_{\pi}$ stable as $t$ varies. Preliminary results show $F_{\pi}(q^2,t)$ stays near its $t=0$ value and only slowly approaches the tree-level limit as $t$ increases, while $Z_V(t)$ and $\kappa(t)$ rapidly converge to their tree-level values, suggesting the vacuum structure evolves more slowly than ultraviolet fluctuations. These insights set the stage for direct comparison with instanton-liquid model predictions and motivate future studies across multiple lattice spacings and pion masses to enable chiral and continuum extrapolations.

Abstract

Instanton liquid model is believed to capture the main features of vacuum QCD dynamics. Recently, multiple predictions for hadron structure functions have been derived and compared with experimental measurements and lattice QCD calculations, finding a general agreement. In order to explore the precision of the instanton liquid model, one has to compare its predictions with non-perturbative simulations in a regime dominated by instanton dynamics. This has been performed for two gluon-sensitive observables: the gluon Green's function and the strong running coupling constant. In this contribution, we propose to study a fermionic observable, the pion electromagnetic form factor, for which instanton liquid model predictions have been discussed in Phys.Rev.D 109, 074029. We use the Wilson flow to single out the dominant contribution from the instantons out of a lattice QCD configuration ensemble. We describe the details of our numerical setup, and some first, preliminary results.

Figures (3)

• Figure 1: The pion effective mass at different Wilson flow time $t$ after tuning the $\kappa$ parameter. Although the structure of excited states is different, the ground state energies are matched within the statistical uncertainties. The range of $t$ corresponds roughly to the range investigated in Ref. Athenodorou:2016gsa.
• Figure 2: The value of the $\kappa$ parameter $\kappa^{am_{\pi}=0.9668}(t)$, for which the pion mass in lattice units remains constant, as a function of the Wilson flow time $t$. The violet, horizontal dashed line denotes the value of $\kappa$ of the sea quarks in the ensemble, whereas the black, horizontal dotted line represents the tree-level value estimation. Vertical lines denote specific values of the Wilson flow time $t$ for which the pion electromagnetic form factor was estimated as shown in Fig. \ref{['fig:form_factor']}. It shows that the measurements of the form factor were performed in the regime where $\kappa$ was already close to its tree-level value. The statistical uncertainties are smaller than the symbol size.
• Figure 3: The pion electromagnetic form factor $F_{\pi}(q^2,t)$ as a function of the momentum transfer $q^2$ in GeV for different values of the Wilson flow $t$. The solid black line corresponds to the tree-level estimation obtained by calculating the two- and three-point functions on a unit field.