Corrections to Nambu-Goto energy levels from the effective string action

Abstract

The effective action on long strings, such as confining strings in pure Yang-Mills theories, is well-approximated by the Nambu-Goto action, but this action cannot be exact. The leading possible corrections to this action (in a long string expansion in the static gauge), allowed by Lorentz invariance, were recently identified, both for closed strings and for open strings. In this paper we compute explicitly in a Hamiltonian formalism the leading corrections to the lowest-lying Nambu-Goto energy levels in both cases, and verify that they are consistent with the previously computed effective string partition functions. For open strings of length R the leading correction is of order 1/R 4 , for excited closed strings of length R in D > 3 space-time dimensions it is of order 1/R 5 , while for the ground state of the closed string in any dimension it is of order 1/R 7 . We attempt to match our closed string corrections to lattice results, but the latter are still mostly outside the range of convergence of the 1/R expansion that we use.

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

“…This was also shown [32,33], at much the same time, and with a stronger result in D = 3 + 1, using the Polchinski-Strominger conformal gauge approach [24]. More recently there has been further progress [5][6][7][8] in both D = 2 + 1 and D = 3 + 1. (See also [34][35][36].)…”

“…An especially interesting result for us is the demonstration that all the operators that appear in the derivative expansion of the Nambu-Goto action appear with precisely the same coefficients in the general effective string action [5][6][7][8]. This provides a motivation for regarding S eff [h] as being given, in a non-trivial sense, by the full Nambu-Goto action plus a series of 'corrections': in particular at small l where the expansion of the NambuGoto energy diverges and needs to be resummed as in eq.…”

“…The main purpose of these calculations is to learn about the effective string theory that describes closed flux tubes at N = ∞, and possibly at smaller N as well. The details of the analysis in our earlier D = 2 + 1 calculation [1] have been rendered out of date by a great deal of recent analytic progress [5][6][7][8] towards determining the universal terms in the derivative expansion of this string action, which makes new predictions for the low-lying spectrum of long flux tubes, l √ σ ≫ 1. Our lattice calculations are largely complementary in that they concentrate on flux tubes that range from the very short to the moderately long, i.e.…”

“…Again in [39] the corresponding term in the finite temperature expansion of the string tension in a gauge dual of d3 random percolation is found not to take the universal Nambu-Goto value. (Note that this paper predates [5][6][7][8] and so does not comment on the expected universality of this term.) Our expectation that there should be massive modes is closely linked to the idea that the flux tube has an intrinsic width, and there have been papers calculating that at both zero and non-zero T in some confining field theories as well as ideas how to go about doing so [40][41][42][43][44][45][46].…”

“…The lattice paths used in the construction of Polyakov loops in this work. Our set of operators can be divided into three subsets: (a) the simple line operator (1) in several smearing/blocking levels; (b) the wave-like operator (2) whose number depends upon L x , L ⊥ , and the smearing/blocking level; (c) the pulse-like operators (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) in several different smearing/blocking levels. In addition the extent of the transverse deformations is varied.…”