[Paper Review] Scale r_0 and the static potential from the CLS lattices

Francesco Knechtli, Björn Leder|arXiv (Cornell University)|Dec 6, 2010

Quantum Chromodynamics and Particle Interactions2 references8 citations

TL;DR

This paper presents a precision determination of the QCD scale $ r_0 $ and the static quark-antiquark potential using HYP-smeared Wilson loops on two-flavor CLS lattice ensembles. By applying variational methods and HYP smearing to reduce noise and improve ground state overlap, the authors extract $ r_0 $ at three lattice spacings and extrapolate to the chiral limit, finding $ r_0 = 0.5 \, \text{fm} $ when combined with the $ \Lambda $-parameter, with strong agreement across collaborations near the physical point.

ABSTRACT

We report on the measurement of the static potential and the scale r_0 from HYP-smeared Wilson loops in two flavour QCD. We analyse the quark mass dependence of the potential and r_0 at three lattice spacings. We also compare the QCD static potential around distance r_0 with the static potential obtained from potential models.

Motivation & Objective

• To determine the QCD scale $ r_0 $ from the static potential in two-flavor QCD using CLS lattice ensembles with improved Wilson fermions and Wilson gauge action.
• To reduce statistical and systematic errors in the static potential by applying HYP smearing to gauge links and using a variational method with multiple smearing levels.
• To extrapolate $ r_0 $ to the chiral limit and compare with other collaborations, assessing quark mass dependence and scale-setting consistency.
• To compute a renormalized coupling via the derivative of the static force and compare it with perturbative and effective string theory predictions.
• To evaluate the convergence of $ r_0 $ and $ M_{PS} $ data across collaborations toward the physical point, assessing universality and scaling.

Proposed method

• Measure rectangular Wilson loops with HYP-smeared spatial and temporal links to suppress ultraviolet noise and improve ground state overlap.
• Use a variational method with $ M=4 $ levels of spatial HYP smearing to construct a correlator matrix and solve the generalized eigenvalue problem for effective masses.
• Extract the static potential $ V(r) $ from the exponential decay of the Wilson loop correlator at large time separations.
• Define the scale $ r_0 $ as the distance where $ r^2 F(r) = r^2 \partial_r V(r) = 1.0 $, using finite differences on the lattice.
• Apply linear extrapolation in the quark mass $ am_q $ to the chiral limit for $ r_0/a $, using both HYP1 and HYP2 smearing parameters.
• Compute the renormalized coupling $ c(r) = \frac{1}{2} r^3 F'(r) $ using finite differences and compare with 3-loop perturbation theory and effective string models.

Experimental results

Research questions

• RQ1 How does the scale $ r_0 $ depend on the sea quark mass in two-flavor QCD, and what is its value in the chiral limit?
• RQ2 To what extent do HYP-smeared Wilson loops and variational methods reduce statistical and systematic errors in the static potential?
• RQ3 How does the static potential and its derivative compare with perturbative predictions and effective string theory at intermediate distances?
• RQ4 How do the $ r_0 $ and $ M_{PS} $ data from different collaborations converge toward the physical point?
• RQ5 What is the value of the $ \Lambda $-parameter in $ N_f=2 $ QCD, and how does it compare with previous determinations?

Key findings

• The chiral extrapolation of $ r_0/a $ yields $ r_0/a(5.2) = 6.05(5) $, $ r_0/a(5.3) = 7.05(3) $, and $ r_0/a(5.5) = 9.59(16) $, with HYP2 smearing showing smaller errors and compatibility with HYP1.
• The $ \Lambda $-parameter is determined as $ r_0 \Lambda_{\overline{\text{MS}}}^{N_f=2} = 0.73(3)(5) $, combining statistical and perturbative uncertainties.
• The renormalized coupling $ c(r) $ at $ r_0 M_{PS} \approx 1 $ shows good agreement with 3-loop perturbation theory for $ N_f=2 $, while the $ N_f=0 $ case approaches the universal string theory value $ -\pi/12 $ at large $ r $.
• The data for $ r_0(x)/r_0(1) $ with $ x = r_0^2 M_{PS}^2 $ show narrowing spread toward the physical point, indicating convergence across collaborations.
• Discrepancies with QCDSF data at $ \beta=5.2 $ are attributed to systematic effects from global fitting of the potential, not statistical fluctuations.
• The static potential at $ r \approx r_0 $ shows good agreement with potential models such as the Cornell and Richardson potentials, especially at intermediate distances.