Confinement and the effective string theory in SU(N->oo) : a lattice study

TL;DR

This work provides lattice evidence for linear confinement in SU(6) by measuring the energy of flux loops winding around a spatial torus across a range of lengths, and compares to SU(4) for cross-checks. The data show that the long-distance correction to the linear potential matches a bosonic string with central charge $c\approx1$, and the ground and first excited string energies are broadly consistent with the Nambu-Goto predictions, including for $k$-strings whose onset of string-like corrections shifts with $N$. The results support an effective string description that remains close to Nambu-Goto in the large-$N$ limit, and indicate that finite-$l$ corrections grow with $k$ and decrease as $N$ increases. These findings have implications for the string/gauge duality picture and for understanding confinement in the $N\to\infty$ limit, motivating further high-$N$ lattice studies to refine the approach to NG behavior.

Abstract

We calculate in the SU(6) gauge theory the mass of the lightest flux loop that winds around a spatial torus, as a function of the torus size, taking care to achieve control of the main systematic errors. For comparison we perform a similar calculation in SU(4). We demonstrate approximate linear confinement and show that the leading correction is consistent with what one expects if the flux tube behaves like a simple bosonic string at long distances. We obtain similar but less accurate results for stable (k-)strings in higher representations. We find some evidence that for k>1 the length scale at which the bosonic string correction becomes dominant increases as N increases. We perform all these calculations not just for long strings, up to about 2.5`fm' in length, but also for shorter strings, down to the minimum length, lc = 1/Tc, where Tc is the deconfining temperature. We find that the mass of the ground-state string, at all length scales, is not very far from the simple Nambu-Goto string theory prediction, and that the fit improves as N increases from N=4 to N=6. We estimate the mass of the first excited string and find that it also follows the Nambu-Goto prediction, albeit more qualitatively. We comment upon the significance of these results for the string description of SU(N) gauge theories in the limit of infinite N.

Figures (5)

• Figure 1: The masses of the lightest, $\bullet$, and first excited, $\circ$, $k=1$ flux loops that wind around a spatial torus of length $l$ in the SU(6) calculation at $\beta=25.05$. The dotted lines are the predictions of the Nambu-Goto string action, as in eqn(\ref{['eqn_NGE']}). The dynamical lower bound on the string length is $l_{min} = 1/aT_c \simeq 6.63$.
• Figure 2: The effective coefficient of the $1/l$ universal string correction, calculated from eqn(\ref{['eqn_ceff']}), for the range of lengths indicated. For $k=1$, $\bullet$, and $k=2$, $\circ$, strings in SU(6).
• Figure 3: The masses of the lightest $k=2$, $\bullet$, and $k=3$, $\circ$, flux loops that wind around a spatial torus of length $l$ in the SU(6) calculation at $\beta=25.05$. The dotted lines are the best fits with a bosonic string correction, as in eqn(\ref{['eqn_corr']}) with $c=1$.
• Figure 4: The masses of the lightest, $\bullet$, and first excited, $\circ$, $k=1$ flux loops that wind around a spatial torus of length $l$ in the SU(4) calculation at $\beta=10.9$. The solid line is the best fit with a bosonic string correction, as in eqn(\ref{['eqn_corr']}) with $c=1$. The dotted lines are the predictions of the Nambu-Goto string action, as in eqn(\ref{['eqn_NGE']}). The dynamical lower bound on the string length is $l_{min} = 1/aT_c \simeq 6.66$.
• Figure 5: The finite temperature effective string tension $\sigma_{eff}(T)$ plotted as a function of $T/T_c$ for SU(6), $\bullet$, and SU(4), $\circ$, at a fixed lattice spacing $a \simeq 0.15/T_c$.