Closed flux tubes and their string description in D =3+1 SU(N) gauge theories

We calculate the energy spectrum of ac onfining flux tube that is closed around a spatial torus, as a function of its length l. We do so for various SU(N) gauge theories in 3+ 1 dimensions, and for various values of spin, parity and longitudinal momentum. We are able to present usefully accurate results for about 20 of the lightest such states, for a range of l that begins close to the (finite volume) deconfining phase transition at l√σ ∼1.6, and extends up to l√σ ∼6 (where σ is the string tension). We find that most of these low-lying states are well described by the spectrum of the Nambu-Goto free string theory in flat space-time. Remarkably, this is so not only at the larger values of l, where the gap between the ground state energy and the low-lying excitations becomes small compared to the mass gap, but also down to much shorter lengths where these excitation energies become large compared to √σ, the flux-tube no longer ‘looks’ anything like a thin string, and an expansion of the effective string action in powers of 1/l no longer converges. All this is for flux in the fundamental representation. We also calculate the k = 2 (anti)symmetric ground states and these show larger corrections at small l. So far all this closely resembles our earlier findings in 2+ 1 dimensions. However, and in contrast to the situation in D = 2+ 1, we also find that there are some states, with J P=0−quantum numbers, that show large deviations from the Nambu-Goto spectrum. We investigate the possibility that (some of) these states may encode the massive modes associated with the internal structure of the flux tube, and we discuss how the precocious free string behaviour of most states constrains the effective string action, on which much interesting theoretical progress has recently been made.