Revisiting Quark Confinement in the Proton through the Force on Quarks

Technical Summary: Revisiting Quark Confinement in the Proton through the Force on Quarks

Problem Statement
Quark confinement, the phenomenon where colored quarks are permanently bound within color-neutral hadrons, remains a central unsolved problem in the Standard Model. While recent progress has been made in understanding hadron mechanical properties via pressure and shear force distributions (encoded in Energy-Momentum Tensor form factors), these characterize the stress balance of the composite system rather than the direct color force acting on individual quarks. A previous study by Ji et al. [1] proposed a framework to define and measure the transverse color-Lorentz force on quarks in the proton by relating it to the quark scalar form factor, $G_{s,q}(q^2)$. However, that analysis faced two primary limitations:

1. Sparse Data at Large Momentum Transfer: The available experimental and lattice QCD data for the relevant form factors were restricted to low spacelike momentum transfers ($|q^2| \lesssim 2 \text{ GeV}^2$), leaving the large-$|q^2|$ region unconstrained. This gap is critical for the Fourier-Bessel reconstruction of the force at short and intermediate transverse distances.
2. Parametrization Dependence: The reconstruction in Ref. [1] relied on specific phenomenological functional parametrizations for the form factors. This introduced model dependence, particularly in regions where data is sparse, making the determination of the force less robust.

Methodology
This work addresses these limitations through two main methodological improvements:

1. Incorporation of Light-Cone QCD Sum Rules (LCSR): To supplement the limited experimental and lattice data in the large-$|q^2|$ region, the authors incorporate LCSR results for the proton's quark Energy-Momentum Tensor (EMT) form factors. These calculations provide complementary information in the range $1 \lesssim |q^2| \lesssim 10 \text{ GeV}^2$, extending the kinematic coverage significantly beyond previous inputs.

2. Regularized Inverse Problem Formulation: Instead of assuming a specific functional form for the form factors $A_q(q^2)$, $B_q(q^2)$, and $C_q(q^2)$, the authors formulate the reconstruction of the transverse force $F_q(\vec{r}_\perp)$ as a regularized inverse problem.

   • The relationship between the scalar form factor and the force is treated as a Hankel transform (Eq. 6).
   • Recognizing that the available data consists of a finite, uncertain set of momentum-space constraints, the authors discretize the integral equation (Eq. 12) to form a linear system.
   • Due to the severe ill-conditioning of the transformation matrix (condition number $\sim 10^{16}$), a direct inversion is unstable. The authors apply Tikhonov regularization (Eq. 14), imposing a smoothness constraint via a third-derivative operator to suppress unphysical oscillations while preserving slowly varying features.
   • The regularization parameter $\alpha$ is determined systematically using the L-curve method to balance data fidelity and solution smoothness.

Key Contributions

• Model-Independent Reconstruction: By solving the inverse problem directly, the authors eliminate the need for prescribed functional parametrizations of the form factors, providing a determination of the quark force that is less dependent on specific model assumptions.
• Extended Kinematic Coverage: The integration of LCSR inputs fills the gap in the large-$|q^2|$ region, allowing for a more reliable reconstruction of the force at short transverse distances ($r_\perp \lesssim 1 \text{ fm}$).
• Uncertainty Quantification: The study explicitly quantifies how uncertainties in the input data (experimental, lattice, and LCSR) propagate to the reconstructed force, identifying specific kinematic gaps (e.g., $0.35 \le -q^2 \le 1.0 \text{ GeV}^2$) that limit precision at larger transverse distances.

Results

• Force Profile: The reconstructed transverse force $F_q(r_\perp)$ exhibits a smooth, stable profile. In the intermediate region ($0.7 \le r_\perp \le 1.1 \text{ fm}$), the force is approximately constant and attractive, with an average value of about $-0.3 \text{ GeV/fm}$.
• Consistency with Confinement: The results are consistent with the findings of Ref. [1] and support the physical picture implied by a linear QCD potential, confirming the existence of a net confining force.
• Reduced Uncertainty: The inclusion of LCSR data significantly reduces the uncertainty in the reconstructed force for the region $0.5 \le r_\perp \le 1.0 \text{ fm}$ compared to the previous analysis.
• Sensitivity to Input Precision: The study demonstrates that while the inverse method reduces parametrization dependence, the precision of the result remains sensitive to the accuracy of the input data.
  • With 10% uncertainty assigned to LCSR inputs, the force is well-constrained in the intermediate region.
  • At larger distances ($r_\perp > 1.1 \text{ fm}$), uncertainties grow rapidly due to the lack of data in the transition region ($0.35 \le -q^2 \le 1.0 \text{ GeV}^2$).
  • Prospective tests suggest that reducing input uncertainties to 5% and filling the transition kinematic gap with synthetic constraints could yield a tightly constrained force distribution across the full range.

Significance and Outlook
The authors claim that this work provides a complementary, less parametrization-dependent determination of the quark confinement force, reinforcing the robustness of the framework proposed in Ref. [1]. By directly reconstructing the force from momentum-space constraints without assuming a specific functional form, the study offers a more rigorous test of the confinement mechanism.

The paper modestly frames this analysis as an "initial attempt" toward a quantitatively conclusive determination. It identifies several necessary future developments:

1. Consistent propagation of uncertainties from the transverse quark-number density (derived from electromagnetic form factors).
2. Systematic comparison of different regularization schemes to assess methodological dependence.
3. Formulating the determination of the EMT form factors themselves as an inverse problem to reduce model dependence in LCSR calculations.
4. Obtaining more precise inputs in the currently unconstrained kinematic region ($0.35 \le -q^2 \le 1.0 \text{ GeV}^2$).

Ultimately, the framework presented serves as a tool for studying confinement from a force-distribution perspective and can be extended to mesons, exotic hadrons, and gluon force distributions, potentially contributing to a more complete understanding of how partons are bound by the strong interaction.