Confining strings in SU(N) gauge theories

TL;DR

This work computes $k$-string tensions in SU($N$) gauge theories across D=3+1 and D=2+1, testing MQCD and Casimir-scaling predictions and probing the universality class of confining flux tubes. By analyzing flux loops winding around spatial tori, the authors extract $\sigma_k$ and examine universal Luscher-type corrections, finding results consistent with a simple bosonic string in both dimensions. The data show near Casimir scaling for stable $k$-strings, with strong binding observed (i.e., $\sigma_k$ significantly less than $k$ times the fundamental tension) and width measurements indicating flux-tube cross-sections roughly independent of the carried flux. High-temperature spatial string tensions also follow approximate Casimir scaling, reinforcing the scenario in which confinement arises from a bosonic-string-like flux tube and the flux cross-section is nearly flux-independent in a deep-London dual-superconductor picture.

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 $2 σ$ 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 Lüscher 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 $2 σ$ of the simple bosonic string values and are inconsistent with other simple effective string theories.

Figures (11)

• Figure 1: The D=2+1 effective string correction coefficient estimated from the masses of flux loops of different lengths (indicated by the span of the horizontal error bar) using eqn(\ref{['eqn_polypair']}). The solid line is what one expects for a simple bosonic string.
• Figure 2: The D=2+1 string correction coefficient estimated by fitting the masses of all flux loops with length greater than $L$, as a function of $L$.
• Figure 3: The D=3+1 effective string correction coefficient estimated from the masses of flux loops of different lengths (indicated by the span of the horizontal error bar) using eqn(\ref{['eqn_polypair']}). The solid line is what one expects for a simple bosonic string. For comparison the dashed line indicates the value for the Neveu-Schwartz string. We use masses from the third column in Table \ref{['table_d4su2_cstring1']}.
• Figure 4: The ratio of $k=2$ and $k=1$ string tensions in D=3+1 SU(4) at $\beta=10.7$ extracted from flux loops of length $l=aL$. We show values extracted using a bosonic string correction, ($\bullet$), and no string correction at all ($\circ$).
• Figure 5: The ratio of $k=2$ and $k=1$ string tensions in our D=3+1 SU(4) ($\bullet$) and SU(5) ($\circ$) lattice calculations plotted as a function of $a^2\sigma$. Extrapolations to the continuum limit, using a leading $O(a^2)$ correction, are displayed.