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Momentum Flow Mechanisms and Color-Lorentz Forces on Quarks in the Nucleon

Xiangdong Ji, Chen Yang

TL;DR

This work reframes nucleon momentum conservation in terms of a continuous momentum current density derived from the QCD energy–momentum tensor, decomposed into quark kinetic, gluon tensor, and trace anomaly components. By combining lattice QCD results and experimental fits for gravitational form factors, it maps the coordinate-space momentum flow and color-Lorentz forces, showing that the trace anomaly contributes a sizable negative-pressure term that drives confinement with an average force of about 1 GeV/fm. The analysis clarifies that momentum flow is not simply a mechanical pressure but a balance of kinetic transport and interaction forces, highlighting the crucial role and current uncertainties of the anomaly sector through the scalar form factor $G_s(q^2)$ (and related $C/D$ form factors). It also discusses frame-dependent simplifications in the infinite-momentum limit, where $T^{++}$ dominates, and outlines paths to reduce uncertainties in future measurements and analyses.

Abstract

Momentum conservation in the nucleon is examined in terms of continuous flow of the momentum current density (or in short, momentum flow), which receives contributions from both kinetic motion and interacting forces involving quarks and gluons. While quarks conduct momentum flow through their kinetic motion and the gluon scalar (anomaly) contributes via pure interactions, the gluon stress tensor has both effects. The quarks momentum flow encodes the information of the color-Lorentz force density on them, and the momentum conservation allows to trace its origin to the gluon tensor and anomaly (a ``negative pressure'' potential). From the state-of-the-art lattice calculations and experimental fits on the form factors of the QCD energy-momentum tensor, we exhibit pictures of the momentum flow and the color-Lorentz forces on the quarks in the nucleon. In particular, the anomaly contributes a critical attractive force with a strength similar to that of a heavy-quark confinement potential.

Momentum Flow Mechanisms and Color-Lorentz Forces on Quarks in the Nucleon

TL;DR

This work reframes nucleon momentum conservation in terms of a continuous momentum current density derived from the QCD energy–momentum tensor, decomposed into quark kinetic, gluon tensor, and trace anomaly components. By combining lattice QCD results and experimental fits for gravitational form factors, it maps the coordinate-space momentum flow and color-Lorentz forces, showing that the trace anomaly contributes a sizable negative-pressure term that drives confinement with an average force of about 1 GeV/fm. The analysis clarifies that momentum flow is not simply a mechanical pressure but a balance of kinetic transport and interaction forces, highlighting the crucial role and current uncertainties of the anomaly sector through the scalar form factor (and related form factors). It also discusses frame-dependent simplifications in the infinite-momentum limit, where dominates, and outlines paths to reduce uncertainties in future measurements and analyses.

Abstract

Momentum conservation in the nucleon is examined in terms of continuous flow of the momentum current density (or in short, momentum flow), which receives contributions from both kinetic motion and interacting forces involving quarks and gluons. While quarks conduct momentum flow through their kinetic motion and the gluon scalar (anomaly) contributes via pure interactions, the gluon stress tensor has both effects. The quarks momentum flow encodes the information of the color-Lorentz force density on them, and the momentum conservation allows to trace its origin to the gluon tensor and anomaly (a ``negative pressure'' potential). From the state-of-the-art lattice calculations and experimental fits on the form factors of the QCD energy-momentum tensor, we exhibit pictures of the momentum flow and the color-Lorentz forces on the quarks in the nucleon. In particular, the anomaly contributes a critical attractive force with a strength similar to that of a heavy-quark confinement potential.

Paper Structure

This paper contains 4 sections, 17 equations, 2 figures.

Table of Contents

  1. Introduction
  2. QCD Mechanisms for the Momentum Flow in the Nucleon
  3. Color-Lorentz Forces on Quarks
  4. Conclusion and Discussion

Figures (2)

  • Figure 1: Trace of the momentum current distribution in proton and its decomposition into three components in Eqs. (\ref{['eq:Trace-qg']}) and (\ref{['eq:Trace-a']}), illustrated using the latest phenomenological fits to both lattice QCD calculations and experimental data Guo:2025jiz. The total current (blue dashed) satisfying the virial theorem in Eq. (\ref{['eq:Virial']}) is decomposed into the positive quark kinetic (black solid) and gluon tensor (green solid) contributions and the negative trace anomaly (red dashed) contribution. Uncertainties of 90% confidence interval are shown as shaded areas.
  • Figure 2: Force density distributions acting on quarks in the proton, visualized using the latest phenomenological fits to lattice QCD calculations and experimental data Guo:2025jiz: the large attractive force from the anomaly (red dashed) and the repulsive force from the gluon tensor (green dashed) combine to produce the total confining force (black solid) on quarks. The uncertainties are shown as shaded areas.