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

TL;DR

This work benchmarks the string-like description of confining flux tubes in D=2+1 SU(N) gauge theories by computing a large set of energies $E_n(l)$ from lattice simulations and comparing them to the Nambu-Goto (NG) spectrum and universal effective-string predictions. The SU(6) results at fine lattice spacing show an exceptional agreement with NG across a wide range of lengths, with universal $O(1/l)$ and $O(1/l^3)$ corrections confirmed and a leading non-NG term consistent with $O(1/l^7)$ for the ground state; excited states generally require a resummed correction rather than a simple power, though single-phonon states follow the $O(1/l^7)$ trend down to small $l$. The analysis finds no evidence for non-stringy massive flux-tube modes, and large-N behavior is already apparent in SU(6), supporting the universality and effectiveness of the stringy description at large $N$. The results bolster the effective string action framework and provide quantitative tests of universal coefficients, with implications for understanding confinement in lower-dimensional gauge theories.

Abstract

We carry out lattice calculations of the spectrum of confining flux tubes that wind around a spatial torus of variable length l, in 2+1 dimensions. We compare the energies of the lowest c.30 states to the free string Nambu-Goto model and to recent results on the universal properties of effective string actions. Our most useful calculations are in SU(6) at a small lattice spacing, which we check is very close to the large-N continuum limit. We find that the energies, En(l), are remarkably close to the predictions of the free string Nambu-Goto model, even well below the critical length at which the expansion of the Nambu-Goto energy in powers of 1/l diverges and the series needs to be resummed. Our analysis of the ground state supports the universality of the O(1/l) and the O(1/l^3) corrections to l.sigma, and we find that the deviations from Nambu-Goto at small l prefer a leading correction that is O(1/l^7), consistent with theoretical expectations. We find that the low-lying states that contain a single phonon excitation are also consistent with the leading O(1/l^7) correction dominating down to the smallest values of l. By contrast our analysis of the other light excited states clearly shows that for these states the corrections at smaller l resum to a much smaller effective power. Finally, and in contrast to our recent calculations in D=3+1, we find no evidence for the presence of any non-stringy states that could indicate the excitation of massive flux tube modes.

Figures (30)

• Figure 1: String tension in units of $g^2N$ for various continuum SU($N$) gauge theories. The curve is a best fit to $N\geq 2$ of the conventional functional form: $\frac{\surd\sigma}{g^2N} = 0.19638 - \frac{0.1144}{N^2}$.
• Figure 2: Effective energies extracted from the correlator $C(t=an_t)$ using eqn(\ref{['eqn_Eeff']}). For the absolute ground state of a flux tube of length $l/a=16,24,32,64$, $\circ$ in ascending order. Also for the $l=32a$ flux tube: the first, second and third excitations with $p=0,\, P=+$, $\bullet$; the ground state with $p=2\pi/l$ and $P=-$, $\star$; the ground and first excited states with $p=0,\, P=-$, $\diamond$, shifted upwards by $\Delta E = 0.1$ for clarity. All from SU(6) at $\beta=171$.
• Figure 3:
• Figure 5: Energy of ground state versus $1/l\surd\sigma \equiv T/\surd\sigma$ for SU(2) with $l=4a(\beta)$, and $\beta$ being varied. Solid line is Nambu-Goto; dashed blue line is $\propto (T_c-T)$ as expected from the universality class of the critical point, and dashed red line is the universal prediction for $E_0$ up to $O(1/l^5)$.
• Figure 6: Energy of absolute ground state for SU(2) at $\beta=5.6$. Compared to full Nambu-Goto (solid curve) and just the Lüscher correction (dashed curve).