[Paper Review] The static quark potential and scaling behavior of SU(3) l attice Yang-Mills theory

Silvia Necco|arXiv (Cornell University)|Jun 5, 2003

Quantum Chromodynamics and Particle Interactions4 citations

TL;DR

This PhD thesis presents a non-perturbative lattice study of the static quark potential in pure SU(3) Yang-Mills theory across distances from 0.05 to 0.8 fm. Using improved lattice actions (Iwasaki and DBW2), it confirms quadratic leading-order lattice artifacts, validates perturbation theory up to α ≈ 0.3 via the force scheme, and finds excellent agreement with the bosonic string model at r ≳ 0.5 fm, while confirming universality in scaling via r₀ and Tcr₀ across actions.

ABSTRACT

The potential between a static quark and antiquark in pure SU(3) Yang-Mills theory is evaluated non-perturbatively through computations on the lattice in the region from short to intermediate distances (0.05 fm < r < 0.8 fm). In the high energy regime the results are compared with the parameter-free prediction of perturbation theory obtained by solving the Renormalization Group equation at two and three loops. The choice of the renormalization scheme to define a running coupling turns out to be important for the accuracy of the perturbative prediction: by obtaining the running coupling through the force, perturbation theory is applicable up to alpha ~ 0.3, while from the static potential only up to alpha ~ 0.15. In the region where perturbation is supposed to be reliable, no large unexpected non-perturbative term is observed. In the second part of this work, universality and scaling behavior of different formulations of Yang-Mills theory on the lattice are discussed, in particular the Iwasaki and DBW2 actions. The deconfinement temperature T_c*r0 and the glueball masses (m_0++)*r0, (m_2++)*r0 are analyzed. Particular attention is dedicated to the violation of physical positivity which occur in these actions and the consequences in the extraction of physical quantities from Euclidean correlation functions.

Motivation & Objective

• To evaluate the static quark potential in pure SU(3) Yang-Mills theory non-perturbatively on the lattice across short to intermediate distances (0.05–0.8 fm).
• To investigate the continuum limit and scaling behavior of improved lattice actions (Iwasaki, DBW2) by analyzing lattice artifacts and universality via r₀ and Tcr₀.
• To test the validity of perturbation theory and the bosonic string model in describing the static quark potential and force at various energy scales.
• To assess the scaling behavior of glueball masses (0⁺⁺, 2⁺⁺) and the impact of unphysical positivity violations in correlation functions for improved actions.

Proposed method

• Non-perturbative lattice simulations of SU(3) Yang-Mills theory using Wilson, Iwasaki, and DBW2 gauge actions at multiple lattice spacings.
• Extraction of the static quark potential and force from Wilson loops using the standard smearing and correlation function techniques.
• Continuum extrapolation of dimensionless quantities (e.g., r₀/a, V(r), F(r)) using parametrized fits to lattice data.
• Comparison of the static potential and force with perturbative predictions from two- and three-loop running coupling, and with the parameter-free bosonic string model.
• Evaluation of the critical deconfinement temperature Tc via the effective potential method and scaling of Tcr₀ to test universality.
• Analysis of glueball masses using irreducible representations of the cubic group and Euclidean correlation functions, with attention to positivity violations in the propagator.

Experimental results

Research questions

• RQ1 To what extent do improved lattice actions (Iwasaki, DBW2) reduce lattice artifacts in the static quark potential and force compared to the Wilson plaquette action?
• RQ2 How well does perturbation theory describe the static quark potential and force, and what role does the choice of renormalization scheme (q̄q vs. SF) play in this agreement?
• RQ3 Does the bosonic string model accurately describe the long-distance behavior of the static quark potential and force, and from what distance does it become valid?
• RQ4 Is the scaling behavior of Tcr₀ universal across different lattice actions, and does it confirm the universality of non-perturbative observables in Yang-Mills theory?
• RQ5 What is the scaling behavior of glueball masses (0⁺⁺, 2⁺⁺), and how are they affected by unphysical positivity violations in the correlation functions of improved actions?

Key findings

• The leading lattice artifacts in the static quark potential and force are quadratic in the lattice spacing, confirming theoretical expectations.
• Perturbation theory agrees well with non-perturbative data up to α ≈ 0.3 when the running coupling is defined via the force, but only up to α ≈ 0.15 when derived from the potential alone.
• The bosonic string model describes the static quark potential and force with remarkable accuracy for distances r ≳ 0.5 fm, indicating its relevance at intermediate to large distances.
• The critical deconfinement temperature scaled by r₀ (Tcr₀) shows perfect agreement across the Wilson plaquette, Iwasaki, and DBW2 actions in the continuum limit, confirming universality.
• The Iwasaki action exhibits significantly reduced lattice artifacts in Tcr₀ and αq̄q(μ) compared to the Wilson action, especially at finite lattice spacing.
• No clear scaling behavior is observed for the glueball masses m₀⁺⁺r₀ and m₂⁺⁺r₀, and unphysical positivity violations in the propagator are found to affect the extraction of physical masses.