On the spectrum of closed k=2 flux tubes in D=2+1 SU(N) gauge theories

TL;DR

<p>We study the spectrum of closed k=2 flux tubes winding around a spatial torus in D=2+1 SU(N) gauge theories (N=4,5) using a large operator basis and variational methods. The results show that low-lying states align with SU(N) irreducible representations (k=2A, k=2S) rather than solely with the Z_N center, and many low-lying levels are remarkably well described by the Nambu-Goto free string spectrum for moderate lengths, with corrections of order unity in natural units consistent with an effective string description. The analysis reveals detailed phonon-interaction patterns: corrections are tiny when phonons share the same momentum and sizable when their momenta are opposite, providing insight into the effective-string interactions; a search for extra massive (non-string) states yields no conclusive evidence within the accessible spectrum. The study also identifies unbound ω=2 states in an extended operator basis and demonstrates near-orthogonality between the k=2A and k=2S sectors, supporting a stringy, representation-resolved picture of the k=2 flux-tube spectrum. </p>

Abstract

We calculate the energy spectrum of a k=2 flux tube that is closed around a spatial torus, as a function of its length l. We do so for SU(4) and SU(5) gauge theories in 2 space dimensions. We find that to a very good approximation the eigenstates belong to the irreducible representations of the SU(N) group rather than just to its center, Z_N. We obtain convincing evidence that the low-lying states are, for l not too small, very close to those of the Nambu-Goto free string theory (in flat space-time). The correction terms appear to be typically of O(1) in appropriate units, much as one would expect if the bosonic string model were an effective string theory for the dynamics of these flux tubes. This is in marked contrast to the case of fundamental flux tubes where such corrections have been found to be unnaturally small. Moreover we find that these corrections appear to be particularly small when the `phonons' along the string have the same momentum, and large when their momentum is opposite. This provides information about the detailed nature of the interactions in the effective string theory. We have searched for, but not found, extra states that would arise from the excitation of the massive modes presumably associated with the non-trivial structure of the flux tube.