Static SU(3) potentials for sources in various representations

TL;DR

The paper nonperturbatively determines static potentials and string tensions for static sources in several SU(3) representations using improved lattice actions on isotropic and anisotropic lattices. Potentials follow the confinement form $V(R) \simeq -\frac{A}{R} + KR + C$ at intermediate distances, with string tensions roughly scaling with the quadratic Casimir, providing a quantitative test of confinement models. Through Wilson-loop measurements, smearing, and careful fitting (including covariance-aware error analysis and scale setting via $r_0$), the authors demonstrate good scaling for the fundamental representation and observe Casimir-like trends across higher representations, while discretization errors dominate systematics for larger tensions; no screening is observed within the accessible ranges. The results offer lattice benchmarks for confinement mechanisms across representations, informing theoretical models and guiding future high-precision studies of nonperturbative QCD dynamics.

Abstract

The potentials and string tensions between static sources in a variety of representations (fundamental, 8, 6, 15-antisymmetric, 10, 27 and 15-symmetric) have been computed by measuring Wilson loops in pure gauge SU(3). The simulations have been done primarily on anisotropic lattices, using a tadpole improved action improved to O(a_{s}^4). A range of lattice spacings (0.43 fm, 0.25 fm and 0.11 fm) and volumes ($8^3\times 24$, $10^3 \times 24$, $16^3 \times 24$ and $18^3 \times 24$) has been used in an attempt to control discretization and finite volume effects. At intermediate distances, the results show approximate Casimir scaling. Finite lattice spacing effects dominate systematic error, and are particularly large for the representations with the largest string tensions.

Figures (6)

• Figure 1: $V(2,2,2)$ versus $T$ for $8^3\times24$ lattice and $\beta=2.4$. The fit range is shown by the solid line.
• Figure 2: The static quark potential $V(R)$ in terms of hadronic scale $r_{0}$ for representations fundamental and 8. The double line shows the central value of the fit $\pm$ error. The horizontal dotted line shows the potential energy of glue-lumps. Good scaling behavior is observed. The weighted average string tensions for representations fundamental and 8 are $Kr_{0}^2$=1.324(4)(51) and $Kr_{0}^2$=2.602(9)(119), respectively.
• Figure 3: Potential versus $R$ for an $8^4$ lattice at $\beta_{Pl}=6.8$. $Ka^2$ shows the string tension times the square of the lattice spacing. $A$ indicates the Coulombic coefficient. The off-axis points deviate from the fit by about $3-8\%$. Only on-axis points are used in the fit.
• Figure 4: A typical plot of $a_{t}V(R)$ versus $R$ for various representations $8^3\times24$ lattice at $\beta=2.4$. $Ka_{s}^2$ shows the string tension times the spatial lattice spacing square. The error bars on the points are the sum in quadrature of statistical and systematic errors. The systematic errors are due to a change of the fit range of $V$ versus $T$.
• Figure 5: Same as figure \ref{['rn_ref_2.4a']} but for the $18^3\times24$ lattice at $\beta=2.4$.