Short distance behaviour of the effective string

We study the Polyakov loop correlator in the (2+1) dimensional Z2 gauge model. An algorithm that we have presented recently, allows us to reach high precision results for a large range of distances and temperatures, giving us the opportunity to test predictions of the effective Nambu–Goto string model. Here we focus on the regime of low temperatures and small distances. In contrast to the high temperature, large distance regime, we find that our numerical results are not well described by the two loop-prediction of the Nambu–Goto model. In addition we compare our data with those for the SU(2) and SU(3) gauge models in (2+1) dimensions obtained by other authors. We generalize the result of Lüscher and Weisz for a boundary term in the interquark potential to the finite temperature case. Recently a renewed interest has been attracted by the effective string description of the interquark potential in lattice gauge theories (LGT) [1]-[11]. On one hand, in [2, 4] high precision simulations of the interquark potential were run, using algorithms [1, 4, 5] that allow to measure the Polyakov loop correlation function with highly reduced variance. On the other hand, in [9] the lowest excited states in the spectrum of the confining flux tube in the (2+1) dimensional Z2 gauge model were studied as well, using a variant of earlier methods. These works enabled one to test some longstanding conjectures on the effective string description of the interquark potential, and also triggered some new theoretical effort toward a better understanding of the effective string model itself. While it seems by now understood that the first order correction to the interquark potential is given by the so called Lüscher term [12], several other issues are still open and require further investigation. A tentative list of these open problems includes: • The determination of higher order terms (and possibly the full functional form) of the effective string action. The presence of the Lüscher term in the interquark potential only tells us that, at leading order, the effective string action is simply a two-dimensional quantum field theory of d − 2 free bosonic fields (one for each transverse degree of freedom of the fluctuating string). It does not help to identify the higher order terms in the action, which should describe the string self-interaction. To this end, one has to evaluate higher order corrections (i.e. higher order powers in 1/R, R being the interquark distance) in the interquark potential. We addressed this problem in two recent papers [3, 4], testing the simplest possible effective action, i.e. the Nambu–Goto string (see below for a detailed discussion) in the large distance, finite temperature regime of the 3d Ising gauge model. We found an impressive agreement between Monte Carlo data and the prediction from the Nambu–Goto action truncated at second order , but apparently no room was left for higher order corrections, which should necessarily be present if the Nambu–Goto string is the correct picture. • The universality issue. As we mentioned above, the first order correction to the interquark potential (i.e. the Lüscher term) shows an impressive degree of universality and has been detected with the precise numerical value predicted by the effective string theory in all of the models studied up to now (ranging from the 3d gauge Ising model to the SU(2) and SU(3) models, both in d=3 and in d=4). Which is the situation for higher order corrections? Do they also show the same universal behaviour? Preliminary results suggest that this is not the case [2], however these evidences were obtained in the short distance regime only (at present only in the case of the 3d Ising model one can reach the large distance regime), where non-universal boundary terms are present (see last item below) and make the analysis much more difficult. A similar agreement was observed in [13] where the large distance regime of the interface potential in the 3d Ising model was studied