Static quark potential and effective string corrections in the (2+1)-d SU(2) Yang-Mills theory

TL;DR

This study tests the effective string description of the static quark potential in (2+1)-dimensional SU(2) Yang-Mills theory by analyzing Polyakov-loop correlators at zero and finite temperature. It confirms the presence of the Lüscher term in the short-distance regime and reveals nonuniversal subleading corrections, with a NG-like second-order correction providing a better fit at long distances and high temperatures. The results show significant gauge-group dependence of higher-order corrections, with SU(2) and Z2 deviating from the NG prediction while SU(3) aligns more closely with free-string expectations; no universal subleading behavior is observed across the three theories. Methodologically, the work combines precise lattice simulations with the Lüscher–Weisz multilevel algorithm to extract the static potential and its derivatives, offering nuanced insights into the validity and limits of effective string pictures in non-Abelian gauge theories.

Abstract

We report on a very accurate measurement of the static quark potential in SU(2) Yang-Mills theory in (2+1) dimensions in order to study the corrections to the linear behaviour. We perform numerical simulations at zero and finite temperature comparing our results with the corrections given by the effective string picture in these two regimes. We also check for universal features discussing our results together with those recently published for the (2+1)-d Z(2) and SU(3) pure gauge theories.

Figures (6)

• Figure 1: Deviation of the results of our numerical simulations at $L=42$, 48,54, 60 and of the Nambu-Goto string model truncated at the second order (continuous line) from the free bosonic string approximation.
• Figure 2: Comparison of the results for $\Delta(R)$ as a function of the scaling variable $\widetilde{R}\sqrt{\sigma}$ for the two samples: $\beta=9,~ L=48$ and $\beta=7.5,~ L=48$
• Figure 3: Comparison of the numerical results for $c(R)$ at $L=8$, $\beta =9$, with the expectations for the free bosonic string (dashed-dotted line) and with the Nambu-Goto string model truncated at the second order (dashed line). The continuous line is what one would find assuming a correction to the linear behaviour like the Lüscher term $\pi /24 R$.
• Figure 4: Deviation of the results of our numerical simulation at $L=8$ and of the Nambu-Goto string model truncated at the second order (continuous line) from the free bosonic string approximation.
• Figure 5: Comparison of the numerical results for $(Q(R)-\sigma)$ at $L=8$, $\beta =9$, with the expectations for the free bosonic string (continuous line) and for the Nambu-Goto string truncated at the second order (dashed line).