Confinement and the effective string theory in $SU(N \rightarrow \infty)$: A Lattice study

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