Confining strings inSU(N)gauge theories

Abstract

We calculate the string tensions of k-strings in $\mathrm{SU}(N)$ gauge theories in both 3 and 4 dimensions. We do so for SU(4) and SU(5) in $D=3+1,$ and for SU(4) and SU(6) in $D=2+1.$ 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 $2\ensuremath{\sigma}$ level, with both the M(-theory) QCD-inspired conjecture that ${\ensuremath{\sigma}}_{k}\ensuremath{\propto}\mathrm{sin}(\ensuremath{\pi}k/N)$ and with ``Casimir scaling,'' ${\ensuremath{\sigma}}_{k}\ensuremath{\propto}k(N\ensuremath{-}k).$ In $D=2+1,$ where our results are very precise, 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 of the unstable strings, which show up as resonant states in the string mass spectrum. While in $D=3+1$ our results are not accurate enough to extract the string tensions of unstable strings, our more precise calculations in $D=2+1$ show that there the ratios between the tensions of unstable strings and the tension of the fundamental string are in reasonably good agreement with (approximate) Casimir scaling. We also investigate the basic assumption that confining flux tubes are described by an effective string theory at large distances, and we attempt to determine the corresponding universality class. We estimate the coefficient of the universal L\"uscher correction from periodic strings that are longer than 1 fm, and find ${c}_{L}=0.98(4)$ in the $D=3+1$ SU(2) gauge theory and ${c}_{L}=0.558(19)$ in $D=2+1.$ These values are within $2\ensuremath{\sigma}$ of the simple bosonic string values, ${c}_{L}=\ensuremath{\pi}/3$ and ${c}_{L}=\ensuremath{\pi}/6,$ respectively, and are inconsistent with other simple effective string theories such as the fermionic, supersymmetric, or Neveu-Schwartz theory.