Comparing the Nambu-Goto string with LGT results

TL;DR

The study tests whether the full Nambu-Goto effective string can reproduce the interquark potential observed in (2+1)D lattice gauge theories across temperatures and gauge groups. It combines a spectrum-based, fully summed NG partition function with perturbative LO/NLO results and contrasts them against high-precision Monte Carlo data for $\\mathbb{Z}_2$, $SU(2)$, and $SU(3)$. The main findings show strong agreement between NG predictions and MC data at large interquark separations and near the deconfinement transition, but persistent discrepancies at short distances in the low-temperature regime, indicating a breakdown of the NG description there and pointing to the need for alternative effective models. Overall, NG provides a robust, near-universal description for large-distance behavior and high-temperature regimes, while a crossover to non-NG string dynamics emerges as $L$ or $R$ decreases, with the details depending on the gauge group. This highlights the NG action as a useful mean-field-like effective string for certain regimes but not a universal description of the interquark potential across all scales.

Abstract

We discuss a way to evaluate the full prediction for the interquark potential which is expected from the effective Nambu-Goto string model. We check the correctness of the prescription reproducing the results obtained with the zeta function regularization for the first two perturbative orders. We compare the predictions with existing Monte Carlo data for the (2+1) dimensional Z(2), SU(2) and SU(3) gauge theories: in the low temperature regime, we find good agreement for large enough interquark distances, but an increasing mismatch between theoretical predictions and numerical results is observed as shorter and shorter distances are investigated. On the contrary, at high temperatures (approaching the deconfinement transition from below) a remarkable agreement between Monte Carlo data and the expectations from the Nambu-Goto effective string is observed for a wide range of interquark distances.

Figures (4)

• Figure 1: Differences between the values of $\Gamma(R)$ for the full Nambu-Goto action (dashed line), the NLO approximation (solid line) and the Monte Carlo results (crosses) with respect to the LO approximation for the sample at $\beta=0.75180$, $L=80$ in the Ising gauge model (data of table \ref{['tab1']}). Notice that for this value of $L$ the NLO and full Nambu-Goto results almost coincide and cannot be separated in the figure.
• Figure 2: Same as figure \ref{['fig1']} but for the data at $L=12$ (see table \ref{['tab4']}). In this case the difference between NLO and full Nambu-Goto predictions is perfectly detectable.
• Figure 3: Same as figure \ref{['fig1']}, but for the data at $R=32$ (see table \ref{['tabrfix']}). As mentioned in the text, in this case the Monte Carlo results appear to interpolate between the full Nambu-Goto behaviour (dashed line) at low temperature and the NLO one (solid line) at high temperature.
• Figure 4: Values of $Q(R)$ for the full Nambu-Goto action (dashed line), the NLO approximation (solid line), the LO approximation (dotted line) and the Monte Carlo results (crosses) for the sample at $\beta=9.0$, $L=8$ in the $SU(2)$ model (data of table \ref{['tab6']}).