Spectrum of confining strings in SU(N) gauge theories

TL;DR

This study tests whether the sine formula for k-string tensions, $R(k,N)= {\sigma_k\over \sigma} = {\sin(k\pi/N)\over \sin(\pi/N)}$, extends to non-supersymmetric four-dimensional SU($N$) gauge theories and explores its universality alongside Casimir scaling. Using lattice Monte Carlo simulations for $N=4$ and $N=6$, the authors extract $k$-string tensions from wall–wall correlators, finding results like $R(2,4) \approx 1.40$–$1.41$, $R(2,6) \approx 1.72$, and $R(3,6) \approx 2.00$, consistent with the sine formula within about 1–4% and disfavoring Casimir scaling as the exact description. A strong-coupling lattice Hamiltonian expansion shows that Casimir scaling holds at leading order but receives explicit $O(1/N)$ corrections at next-to-leading order, illustrating how deviations can arise in nonperturbative regimes. The paper also draws analogies to two-dimensional SU($N$)$\times$SU($N$) chiral models, where the sine formula governs bound-state spectra and Casimir scaling governs short-distance correlators, supporting a broader universality of confining-string spectra across SU($N$) theories. These results constrain confinement models and support a universal pattern for the spectrum of confining strings, while clarifying the circumstances under which Casimir scaling can fail.

Abstract

We study the spectrum of the confining strings in four-dimensional SU(N) gauge theories. We compute, for the SU(4) and SU(6) gauge theories formulated on a lattice, the string tensions sigma_k related to sources with Z_N charge k, using Monte Carlo simulations. Our results are consistent with the sine formula sigma_k/sigma = sin k pi/N / sin pi/N for the ratio between sigma_k and the standard string tension sigma. For the SU(4) and SU(6) cases the accuracy is approximately 1% and 2%, respectively. The sine formula is known to emerge in various realizations of supersymmetric SU(N) gauge theories. On the other hand, our results show deviations from Casimir scaling. We also discuss an analogous behavior exhibited by two-dimensional SU(N) x SU(N) chiral models.