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

Abstract

Abstract:We calculate the energy spectrum of a confining 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.

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

“…These flux tubes are stabilised by being wound around a spatial torus and we calculate the energies of the lightest few eigenstates as a function of the flux tube length l, for various quantum numbers. This work greatly extends and supersedes that published in our earlier brief letter [1], and is part of a larger project which has included the calculation of the spectrum in D = 3 + 1 [2], as well as the spectrum and string tensions [3,4] of flux tubes with flux in some higher representations, e.g. k-strings.…”

“…We have kept the discussion in this paper relatively brief, since most of the relevant issues are discussed at greater length in our recent paper on the flux tube spectrum in D = 3 + 1 [2] and in a recent set of lectures by one of the authors [9], to which we refer the interested reader.…”

Introductory lectures to large- QCD phenomenology and lattice results

“…In fact, on the lattice it is also possible to accurately study the corrections to the linear dependence of the torelon energy, which become non-negligible at intermediate values of the torelon length L. This issue has been addressed in many studies [12,16,[109][110][111][112][113][114][115][116][117], including also for flux tubes in higher representations, or for excited string states. The results indicate that the leading correction to the linear dependence of m on L can be expressed in terms of a "Lüscher term", due to the quantum fluctuations of the torelon [118],…”

“…[115], can be accurately compared with the predictions from an effective Nambu-Gotō string model. theories is also confirmed by some recent, high-precision studies of excited string states, like those reported in ref.…”

“…theories is also confirmed by some recent, high-precision studies of excited string states, like those reported in ref. [115] (see fig. 8).…”