The closed string spectrum of SU(N) gauge theories in 2+1 dimensions

TL;DR

We study the closed flux-tube spectrum of $SU(N)$ gauge theories in $2+1$ dimensions on the lattice across $N=2$–$8$ and string lengths $l\sim 0.45$–$3$ fm. Ground-state energies are extracted with high precision from smeared Polyakov-loop correlators and used to determine the string tension $\sigma$, while the lowest ~30 excited states are resolved using a large operator basis and a variational transfer-matrix analysis. The results show unambiguous evidence that the spectrum lies in the universality class of the Nambu-Goto free bosonic string, with $E_n(l)$ well described by the NG prediction (and a covariant string description with small corrections) down to short lengths and possibly at all scales in the large-$N$ limit. This supports the view of confinement in these theories as governed by a covariant effective string theory, with implications for string/gauge dual descriptions.

Abstract

We use lattice techniques to study the closed-string spectrum of SU(N) gauge theories in 2+1 dimensions. We calculate the energies of the lowest lying ~30 states for strings with lengths between l ~ 0.45 fm and l ~ 3 fm, and compare to different theoretical predictions. We obtain unambiguous evidence that the closed-strings are in the universality class of the Nambu-Goto free bosonic string. Moreover, we clearly see that our data can be described by a covariant string theory with a small/moderate correction down to very short distance scales, and possibly on all distance scales at large-N.

Figures (5)

• Figure 1: The ground state energy per unit length, in lattice units, vs. the string length for $SU(3)$ and $\beta=14.7172$ (left panel) and for $SU(6)$ and $\beta=90.00$ (right panel). All the data points are the result of the single exponential fits, except when $l>5/\sqrt{\sigma}$, where we use the results from double exponential fits (see Section \ref{['contamination']}). The blue lines are the results of using the fitting ansatz Eq. (\ref{['fit']}). Using the values we obtain for the string tensions we plot the NG predictions of Eq. (\ref{['NG0']}) (magenta lines) and of the Luscher formula in Eq. (\ref{['luscher0']}) (black lines).
• Figure 2: The effective central charges $C^{(1,2)}_{\rm eff}$ vs. the string length for the 'S' data points, for $SU(3)$ and $\beta=14.7172$ in the left panel, and for $SU(5)$ and $\beta=80.00$ in the right panel.
• Figure 3: The energy of the 1st excited state, divided by $\sigma l$ (See Tables \ref{['fits-gs-partA']}-\ref{['fits-gs-partB']} for the values of the string tensions). Left panel: Comparing $SU(3)$ and $SU(6)$ for a similar lattice spacing of $a^2\sigma \simeq 0.03$. Right panel: Comparing different lattice spacings for $SU(3)$. The black line is the NG prediction, while the dotted(dashed) lines are the NG prediction expanded to leading and next-to-leading order.
• Figure 4: The energies of the lowest $7$ states in units of $\sqrt{\sigma}$ as a function of $l \sqrt{\sigma}$. The three lines are the NG predictions for the ground state(red), $1^{st}$ excited state(blue) and $2^{nd}$ excited state(black). Left panel: We present the results for the case of $SU(3)$ for two different values of $\beta$ (two different lattice spacings). Right panel: Results for $SU(6)$ with $\beta=90.000$. In both panels we denote the degeneracy of the states in the legends.
• Figure 5: $\sqrt{E^2/\sigma - \left( 2 \pi q/ \sqrt{\sigma} l \right)^2}$ as a function of $l\sqrt{\sigma}$ for the ground state (g.s.) and excited states (e.s.) of the non-zero $q$ sector. The four lines present the NG prediction Eq. (\ref{['NG0']}). For $N_R=3, N_L=1$ and $q=2$, the expected degeneracy is three, which agrees with our excited state data, i.e. two(one) states with positive(negative) parity in red(blue).