The color force acting on a quark in the pion and nucleon

TL;DR

This work presents a unified semiclassical picture in which instanton–anti-instanton molecular configurations in the QCD vacuum generate sizable twist-3 color Lorentz forces acting on quarks inside hadrons. By deriving emergent form factors that relate the color Lorentz operator to hadronic gravitational and transversity form factors, the authors connect nonperturbative vacuum structure to observable quantities such as $g_T(x)$ and the $d_2$ moment, and show that molecular contributions dominate at low $Q^2$. The nucleon results yield color-force form factors that agree with recent lattice extractions, while the pion exhibits vanishing $d_2$ as expected for a spin-0 state, underscoring a strong spin dependence of the twist-3 color force. These findings illuminate how topological vacuum fluctuations imprint on partonic structure and motivate extensions to GPDs and TMDs for mapping spatial and dynamical force distributions in hadrons.

Abstract

In the Operator Product Expansion (OPE) of hard scattering amplitudes, the twist-3 operators describe local colored Lorentz forces acting on a quark, thereby providing a measure of the strength of the gluon fields. Its value is directly accessible from the nucleon twist-3 polarized $g_2$-parton distribution function. In the semiclassical (instanton-based) QCD vacuum models, the leading non-perturbative contribution stems from correlated instanton-anti-instanton pairs, or molecules. We analyze the magnitude of the color force on a 'struck' quark in light hadrons (pion and nucleon), in the context of the instanton liquid model (ILM). We derive explicitly the pertinent form factors associated to the color Lorenz force, and show that they are intimately related to the pertinent hadronic gravitational and transversity form factors. Using the ILM enhanced by molecules, we detail the ensuing colored force distribution in the transverse plane for luminal pions and nucleons. The results for the nucleons are in good agreement with those reported recently from a lattice collaboration.

Figures (18)

• Figure 1: (a) An instanton as an effective 't Hooft operator with 4 external lines; (b) Quark propagation in the $I \bar{I}$ molecule. Both figures assume two light quark flavors, $N_f=2$
• Figure 2: The points show the numerical results for the dimensionless combination $\rho^2 T(r)$ versus $r/\rho$. The line is our approximate parameterization $0.0257/(r^2 + 1)^{0.75}$
• Figure 3: The points in the upper plot show the numerical results for the zero mode overlap integral $|T_{I\bar{I}}(R) |^2$ versus $r/\rho$. The line is our approximate parameterization $2/(R^2 + 1)^4$. The lower plot shows the same amplitude times $R^3$ from the 4d radial integral. It displays a sharp peak at $R/\rho\approx 1$.
• Figure 4: The upper figure is a sketch of a molecular configuration pierced by a quark path in the time direction. The lower figure is a random example of how three components of the field (m=1,2,3, blue red,green respectively) depend on time along the line.
• Figure 5: The single instanton/anti-instanton vertices with the insertion of the color Lorentz operator (crossed-circle): (a), (b) along a fermion line and (c) with a pair of Fermion lines. The latter is surppessed by $1/N_c$ compared to the former.