Closed flux tubes in higher representations and their string description in D=2+1 SU(N) gauge theories

TL;DR

This study numerically examines closed confining flux tubes in multiple SU($N$) representations (notably SU($6$) at $eta=171$) in D=$2+1$, comparing their spectra to the Nambu-Goto free-string description. Ground states, including those with nonzero longitudinal momentum, align remarkably well with NG predictions even at short lengths, while excited states often deviate and in many cases are unstable or decaying, complicating clear interpretation. The authors provide evidence that higher-representation flux tubes exist as bound states of fundamental flux tubes, with overlaps indicating limited mixing away from decay thresholds; Casimir scaling of string tensions emerges across representations. The work supports a predominantly stringy, NG-like description for many sectors, highlights the role of binding and decay in the spectrum, and clarifies where additional massive modes might or might not appear, contributing to our understanding of confinement and effective string descriptions in nonfundamental representations.

Abstract

We calculate, numerically, the low-lying spectrum of closed confining flux tubes that carry flux in different representations of SU(N). We do so for SU(6) at beta=171, where the calculated low-energy physics is very close to the continuum limit and, in many respects, also close to N=infinity. We focus on the adjoint, 84, 120, k=2A,2S and k=3A,3M,3S representations and provide evidence that the corresponding flux tubes, albeit mostly unstable, do in fact exist. We observe that the ground state of a flux tube with momentum along its axis appears to be well defined in all cases and is well described by the Nambu-Goto free string spectrum, all the way down to very small lengths, just as it is for flux tubes carrying fundamental flux. Excited states, however, typically show very much larger deviations from Nambu-Goto than the corresponding excitations of fundamental flux tubes and, indeed, cannot be extracted in many cases. We discuss whether what we are seeing here are separate stringy and massive modes or simply large corrections to energy levels that will become string-like at larger lengths.

Figures (34)

• Figure 1: Effective energy of the k=2A, p=0, P=+ (variational) ground state of a flux tube of length $l/a=16,20,24,28,32,36,40,44,48,52,64$. Lines are our plateaux estimates ($\pm 1\sigma$ error bands).
• Figure 2: $k=2A$ ground states with $p=0,2\pi/l,4\pi/l$ and with $P=+$, $\circ$, and $P=-$, $\bullet$. Solid red curves are Nambu-Goto predictions. Dashed red lines are the model in eqn(\ref{['eqn_Epmodel']}). Dashed blue lines denotes lower boundaries of scattering states formed of two fundamental flux tubes with total momentum $p$. Vertical line denotes location of 'deconfinement' transition.
• Figure 3: Energy of $k=2A$ ground state with $p=0$ and $P=+$, minus predictions of various 'models': Nambu-Goto, $\bullet$; linear plus Lüscher correction, $\circ$; and only linear term, $\Box$. The solid curve includes an $O(1/l^7)$ correction to Nambu-Goto. Vertical line denotes location of 'deconfinement' transition.
• Figure 4: Phonon excitation energies, as defined in eqn(\ref{['eqn_exq']}), of $k=2A$ ground states with $p=0, 2\pi/l, 4\pi/l$ and with $P=+$ ($\bullet$) or $P=-$ ($\blacksquare$). Open symbols shown for $p=0, 2\pi/l$ are without the zero-point energy in eqn(\ref{['eqn_exq']}). Horizontal lines are Nambu-Goto predictions. Vertical line denotes location of 'deconfinement' transition.
• Figure 5: Fitting the $p=2\pi/l$ ground state energies to the model in eqn(\ref{['eqn_Epmodel']}) and extracting the excitation mass averaged over $l\geq l_0$. For representations $r=2A$ ($\bullet$) and $r=3A$ ($\circ$).