Short distance behaviour of the effective string

TL;DR

This paper probes the short-distance behavior of the interquark potential in a (2+1)D $\mathbb{Z}_2$ lattice gauge model to test the effective string description. Using high-precision Polyakov-loop correlators, it shows that the leading Lüscher term is insufficient at short distances and that the full bosonic-string corrections are required to describe the data, with an absolute test confirming agreement beyond a certain $R$. It further investigates higher-order corrections, finding evidence for $1/R^3$-type terms that resemble Nambu-Goto predictions but with coefficients that differ in sign and magnitude, suggesting additional operators or boundary effects. The boundary term $b$ is derived and tested, with Ising-model data indicating $b$ is compatible with zero at large distances, while cross-model comparisons with SU(2) and SU(3) reveal varying levels of higher-order corrections, highlighting both universal and model-dependent aspects of effective string descriptions. Overall, the work clarifies the regime of validity of the Nambu-Goto/string picture and illuminates the roles of boundary and higher-order terms in finite-temperature LGTs.

Abstract

We study the Polyakov loop correlator in the (2+1) dimensional Z_2 gauge model. An algorithm that we have presented recently, allows us to reach high precision results for a large range of distances and temperatures, giving us the opportunity to test predictions of the effective Nambu-Goto string model. Here we focus on the regime of low temperatures and small distances. In contrast to the high temperature, large distance regime, we find that our numerical results are not well described by the two loop-prediction of the Nambu-Goto model. In addition we compare our data with those for the SU(2) and SU(3) gauge models in (2+1) dimensions obtained by other authors. We generalize the result of Lüscher and Weisz for a boundary term in the interquark potential to the finite temperature case.

Figures (7)

• Figure 1: Comparison between our values of $H(R,20)$, the bosonic string prediction, eq. (\ref{['zsmalltot']}) (continuous line) and the $1/R$ approximation to eq. (\ref{['zsmalltot']}) (dashed line) for the data at $\beta=0.75180$, $L=80$.
• Figure 2: Same as fig. \ref{['bnew1']}, but with a larger resolution.
• Figure 3: Histogram of $\Delta(R_1,R_2)$ for all the values of $R_1,R_2\geq 20$. Notice the very small scale of the $\Delta$ axis.
• Figure 4: Values of $D(R)$ versus $R\sqrt{\sigma}$ for the sample at $\beta=0.65608$ (white squares), the one at $\beta=0.75180$ (black squares) and the $L=40$ one at $\beta=0.73107$ (crosses). The continuous line is the Nambu--Goto expectation, the two dashed lines are the best fit results for $\beta=0.73107$ and $\beta=0.75180$ discussed in the text.
• Figure 5: Same as fig. \ref{['fignew1']} but with a higher resolution.