[Paper Review] Parton Distributions from Boosted Fields in the Coulomb Gauge

Xiang Gao, Wei-Yang Liu+1 more|arXiv (Cornell University)|Jun 26, 2023

Particle physics theoretical and experimental studiesPhysics and Astronomy3 citations

TL;DR

This paper proposes a new method to calculate parton distribution functions (PDFs) using correlations of boosted quarks and gluons in the Coulomb gauge, avoiding gauge-invariant Wilson lines. The approach eliminates linear power divergences, enables larger off-axis momenta, and reduces computational cost by 50% while achieving comparable precision to standard quasi-PDFs, as demonstrated for the pion valence quark PDF at |p| ~ 2.2 GeV on a 0.06 fm lattice.

ABSTRACT

We propose a new method to calculate parton distribution functions (PDFs) from lattice correlations of boosted quarks and gluons in the Coulomb gauge. Compared to the widely used gauge-invariant Wilson-line operators, these correlations greatly simplify the renormalization thanks to the absence of linear power divergence. Besides, they enable access to larger off-axis momenta under preserved 3D rotational symmetry, as well as enhanced long-range precision that facilitates the Fourier transform. We verify the factorization formula that relates this new observable to the quark PDF at one-loop order in perturbation theory. Moreover, through a lattice calculation of the pion valence quark PDF, we demonstrate the aforementioned advantage and features of the Coulomb gauge correlation and show that it yields consistent result with the gauge-invariant method. This opens the door to a more efficient way to calculate parton physics on the lattice.

Motivation & Objective

• To develop a more efficient and precise method for calculating parton distribution functions (PDFs) from lattice QCD.
• To address the computational challenges of gauge-invariant quasi-PDFs, including linear power divergences and high noise at large separations.
• To enable the use of large off-axis momenta in lattice simulations, improving momentum resolution and signal-to-noise ratios.
• To verify the validity of the Coulomb gauge approach through next-to-leading-order perturbative matching.
• To demonstrate the method’s feasibility and efficiency by computing the pion valence quark PDF on a 0.06 fm lattice with |p| ~ 2.2 GeV.

Proposed method

• The method defines a quasi-PDF using quark and gluon correlations in the Coulomb gauge, bypassing the need for gauge-invariant Wilson lines.
• It leverages large-momentum effective theory (LaMET) to match the Coulomb gauge quasi-PDF to the physical PDF at next-to-leading order (NLO).
• The absence of Wilson lines removes linear power divergences and associated renormalons, simplifying renormalization to a multiplicative Z(a) factor.
• Momentum smearing is applied in the Coulomb gauge to access high-momentum states, and both Coulomb gauge and gauge-invariant quasi-PDFs are computed simultaneously using shared quark propagators.
• The method uses a hybrid scheme to compute matrix elements, with improved long-range precision due to reduced exponential noise growth.
• The approach is validated through NLO matching and comparison with global fits and gauge-invariant quasi-PDFs on the same lattice ensemble.

Experimental results

Research questions

• RQ1 Can parton distribution functions be reliably extracted from Coulomb gauge correlations without gauge-invariant Wilson lines?
• RQ2 Does the absence of linear power divergence in Coulomb gauge correlations lead to improved signal-to-noise ratios at large spatial separations?
• RQ3 Can large off-axis momenta be effectively used in lattice simulations with Coulomb gauge quasi-PDFs to reduce computational cost?
• RQ4 How does the NLO matching between Coulomb gauge quasi-PDFs and physical PDFs compare to the standard gauge-invariant approach?
• RQ5 Is the Coulomb gauge method consistent with phenomenological global fits at moderate to large x values?

Key findings

• The Coulomb gauge quasi-PDF method achieves comparable precision to the gauge-invariant quasi-PDF method while reducing computational cost by approximately 50%.
• The method enables the use of large off-axis momenta (|p| ~ 2.2 GeV), which is not feasible with standard gauge-invariant approaches due to rotational symmetry and Wilson line issues.
• The absence of linear power divergence and associated renormalons leads to stable, z-independent renormalization via a multiplicative Z(a) factor.
• The matched PDFs from the Coulomb gauge method agree with those from the gauge-invariant method within 1σ errors at NLO, confirming consistency in momentum space.
• The PDFs extracted from the Coulomb gauge method show good agreement with global fits at moderate to large x, validating its phenomenological relevance.
• The hybrid-scheme matrix elements demonstrate significantly improved long-range precision in the Coulomb gauge compared to the gauge-invariant case, especially at large |z|.