Confining strings in SU(N) gauge theories

TL;DR

This paper calculates and analyzes the tensions of k-strings in SU(N) gauge theories across different dimensions, comparing results with theoretical models and exploring the effective string description of flux tubes.

Contribution

It provides numerical estimates of k-string tensions, tests scaling laws, and investigates the effective string theory description in SU(N) gauge theories in 2+1 and 3+1 dimensions.

Findings

k-string tension ratios are consistent with Casimir scaling in 3+1D within 2 sigma.

Deviations from MQCD formula observed in 2+1D.

Lüscher correction coefficient close to bosonic string value.

Abstract

We calculate the string tensions of $k$-strings in SU($N$) gauge theories in both 3 and 4 dimensions. In D=3+1, we find that the ratio of the $k=2$ string tension to the $k = 1$ fundamental string tension is consistent, at the $2 \sigma$ level, with both the M(-theory)QCD-inspired conjecture and with `Casimir scaling'. In D=2+1 we see a definite deviation from the MQCD formula, as well as a much smaller but still significant deviation from Casimir scaling. We find that in both D=2+1 and D=3+1 the high temperature spatial $k$-string tensions also satisfy approximate Casimir scaling. We point out that approximate Casimir scaling arises naturally if the cross-section of the flux tube is nearly independent of the flux carried, and that this will occur in an effective dual superconducting description, if we are in the deep-London limit. We estimate, numerically, the intrinsic width of $k$-strings in D=2+1 and indeed find little variation with $k$. In addition to the stable $k$-strings we investigate some ofthe unstable strings, finding in D=2+1 that they satisfy (approximate) Casimir scaling. We also investigate the basic assumption that confining flux tubes are described by an effective string theory at large distances. We estimate the coefficient of the universal L\"uscher correction from periodic strings that are longer than 1 fermi, and find $c_L=0.98(4)$ in D=3+1 and $c_L=0.558(19)$ in D=2+1. These values are within $2 \sigma$ of the simple bosonic string values and are inconsistent with other simple effective string theories.