The string tension from smeared Wilson loops at large N

We present the results of a high statistics analysis of smeared Wilson loops in 4 dimensional SU(N) Yang-Mills theory for various values of N. The data is used to analyze the behaviour of smeared Creutz ratios, extracting from them the value of the string tension and other asymptotic parameters. A scaling analysis allows us to extrapolate to the continuum limit for N=3,5,6 and 8. The results are consistent with a $1/N^2$ approach towards the large N limit. The same analysis is done for the TEK model (one-point lattice) for N=841 and a non-minimal symmetric twist with flux of $k=9$. The results match perfectly with the extrapolated large N values, confirming the validity of the reduction idea for this range of parameters.

There is considerable interest in gauge theories at large N for their simplicity, proximity to phenomenologically interesting field theories and their presumed connection to string theory. Lattice gauge theory has proved to be a fundamental tool in deriving the non-perturbative properties of Yang-Mills theories at small N. In approaching large N, the standard pathway is to study the theory at increasing values of N and to extrapolate the results to infinite N. This is, no doubt, a costly procedure, with the additional risks involved in any extrapolation procedure. Nevertheless, results point towards a somewhat fast approach to the large N limit in many of its observables [1]- [2]. An alternative would be to use the simplifications involved in the large N theory to find a way to simulate it directly. An idea going in this direction is that of reduction or volume independence [3]- [4]- [5]- [6]. This allows the possibility of trading the space-time degrees of freedom with those of the group. The essential ingredient for the idea to work is invariance under Z 4 (N) symmetry, which is broken in the original proposal [3]- [4]. In the twisted Eguchi-Kawai model (TEK) [5], introduced by the present authors, an invariance subgroup is preserved at sufficiently weak coupling, enabling reduction to work. Recently, it was reported in Ref. [7]- [8]- [9] that symmetry-breaking takes place at intermediate couplings and N > 100. To circumvent this problem we proposed a slight variation of the model [10]. It exploits the freedom associated with an integer parameter entering the formulation, and representing the chromomagnetic flux through each two-dimensional plane.
Traditionally this parameter was kept fixed when taking the large N limit, while we advocated the need to scale it with √ N in order to avoid symmetry-breaking phase transitions. In practice, the modification involves no additional technical or computational cost. Our initial tests [10] were free from the problems reported earlier. To further test the validity of this idea demanded performing state of the art computations of the large N observables and comparing them with those obtained for the TEK model. Furthermore, even if reduction operates at the level of the lattice model, our ultimate goal is the continuum theory, so a scaling analysis is necessary. These were our original motivations for embarking in the present work.
Although other observables are possible, we have focused upon the string tension. This can be obtained as the slope of the linear quark-antiquark potential. Lately, the best determinations of the potential and of the string tension have been obtained by compactifying one dimension, and studying the connected correlation function of Polyakov lines [2]. In the large N limit this is subleading with respect to the disconnected term, and it is unclear how to make the connection. Thus, we stick to the traditional way in which the string tension is obtained from the expectation value of Wilson loops W(T, R). Here one meets a technical but severe difficulty, since large Wilson loops are very noisy quantities. Furthermore, the Wilson loops themselves are affected by ultraviolet divergences so that we will rather focus on the traditional Creutz ratios: which are defined for half-integer R and T . In the limit R << T these quantities are lattice approximants to the force F(R) among quarks separated by a distance R. Although, Creutz ratios get rid of the constant and perimeter divergences in Wilson loops, they do so through a cancellation, which makes them even more numerically challenging. To reduce the errors we resort to the well-known Ape-smearing procedure [11] for the ordinary theory. The corresponding smearing for the TEK model is given by with z µν the twist tensor. Proj N stands for the operator that projects onto SU(N) matrices. This process can be iterated several times and produces a considerable noise reduction in the data. One could extract the string tension from the force F(R) obtained through Creutz ratios for R << T smeared in the three directions transverse to T . This is, however, very impractical in our case.
It is much more effective to employ four-dimensional smearing and values R ≈ T . The problem that arises in this approach is that not only the error, but also the value of the Wilson loops and Creutz ratios vary with the number of smearing steps. This could be an important source of systematic uncertainties, which might prevent a precision determination of the string tension from this source. To circumvent this problem, our strategy has been to use the smeared Creutz ratio values to extrapolate back and obtain un-smeared values. The extrapolated Creutz ratios do not depend on the number of smearing steps, and the errors are considerably smaller than the original un-smeared ratios. Having explained the main observables that we will be using, let us summarize in the next paragraph the goals and methodology used in this work.
Our main goal is the determination of the string tension for large N Yang-Mills theory by means of the study of smeared Creutz ratios on the lattice. The large N value will be obtained by extrapolation of data taken at N=3,5,  Table I. The analysis of data and presentation of results follows the following steps: obtained, but dismissed for the analysis for being more sensitive to lattice artifacts. Larger loops can also be obtained but are too noisy and/or the number of smearing steps falls too short for them.

Extrapolation to the un-smeared Creutz ratio with error.
The extrapolation procedure depends on the values of R and T . For small values the smeared Creutz ratio are very well fitted to a dependence a(1 − exp{−b/(n s + δ)}) where n s is the number of smearing steps. This dependence is suggested by perturbation theory. In Fig. 1

Analysis of square R = T Creutz ratios
The square Creutz ratios for large values of R = T are expected to behave as A non-zero lattice string tension κ is the consequence of Confinement. The linear term in 1/R 2 is the predicted behaviour both from perturbation theory and from a string description of the quark-  Table. I.

Scaling analysis
Since our goal is continuum physics we should extrapolate our results to the continuum limit.
Scaling implies that, close enough to the continuum limit, results obtained at different values λ L should coincide once the lattice spacing a(λ L ) is chosen appropriately. In particular, the length of both sides of a rectangular Wilson loop are given in physical units by t = T a(λ L ) and r = Ra(λ L ).
Using this fact and the definition of Creutz ratios one concludes: where the dots contain higher powers of a(λ L ). The continuum functionF(t, r) is given bỹ where W(t, r) is the value of the continuum t × r Wilson loop. Notice that, although the Wilson loop itself has perimeter and corner divergences, these disappear when taking the second derivative with respect to t and r. Thus,F(t, r) is a well-defined continuum function having the dimensions of energy square.
In perturbation theory one getsF (t, r) = γ P (z) 1 where γ P is a given function of the aspect ratio z = r/t. For the full non-perturbative theory, one can study the behaviour of the function as t and r goes to infinity. One expects where σ is the string tension, and the dots represent subleading terms starting with 1/(min(t, r)) 4 .
The expansion is also exactly the same as predicted by an effective string theory description of the Wilson loop expectation value.
This analysis justifies the parametrization used previously for square Creutz ratios with γ = γ(1) and κ(λ L ) = σa 2 (λ L ). In order to compute the continuum string tension we need to determine a(λ L ). For very small values of λ L perturbation theory dictates its form: However, it is well-known that scaling seems to work much beyond the region where Eq. 8 provides a good approximation. There are several proposals in the literature, which have been discussed and tested in many papers, which argue that Eq. 8 can be extended to the whole scaling region using improved couplings λ I (λ L ) in the previous formula, instead of λ L itself. All these proposals can be considered perturbative renormalization prescriptions, and the ratio of the corresponding scales is obtainable by a perturbative calculation, i.e. the ratio of lambda parameters.
A particular proposal that has shown good results in previous studies was done by Parisi [12] and used in the analysis of Ref. [13]. When expressed in Λ MS units it is given by 1). A somewhat different proposal resulted from the analysis of Allton et al. [2]. It is based on a different definition of the effective coupling λ A = λ L /u P (and a somewhat modified expression for f (λ A )).
Scaling then implies that the continuum string tension can be determined in Λ MS units as follows: Another remarkable feature of our result is that the N dependence matches perfectly with that obtained in Ref. [2], which used different observables, techniques and range of 't Hooft couplings.
The actual value of the large N ratio given in that reference was 0.503 (2), which seems inconsistent with our result on statistical grounds. However, the estimated systematic errors quoted in Ref. [2] are as large as 0.04. We should also mention that a recent analysis, largely complementary to ours, has obtained an estimate of the large N string tension which is consistent with our result [14].
In order to give a robust prediction for the large N string tension, we should also estimate our systematic errors. The most important source of these errors arises from an overall scale. If we repeat the procedure replacing the expression of a E by the formula given by Allton et al. [2] our estimate of the large N ratio Λ MS / √ σ becomes 0.525 (2). This is a 2 percent change in the predicted value, which is 5 times bigger than the statistical error.
To give a more precise prediction we should use a non-perturbative renormalization prescription to fix a(λ L ). It is possible to give a prescription based on Wilson loops and which follows the same philosophy as the one used to define the Sommer scale [15]. Let us consider the dimensionless function G(r) ≡ r 2F (r, r). A scaler can be defined as the one satisfying G(r) =Ḡ. If scaling holds, the choice ofḠ is irrelevant (provided the equation has a solution), since it amounts to a change of units. For our analysis we tookḠ = 1.65, by analogy with Sommer scale. However, we checked that taking other choices (Ḡ = 2. andḠ = 2.5) give consistent results up to a change of units. We recall that the idea of considering Creutz ratios with different aspect ratios z = R/T to define the scale appears in Ref. [16].
One possible way to determine the scale is by solving forR(λ L ) in the equation This gives us a(λ L ) =r/R(λ L ). Although, our data points are defined only for half integer R, it is easy to interpolate and obtain any realR. Interpolation is a much more robust procedure than extrapolation, and one can use different interpolating functions to estimate errors.
The main problem of the previous procedure is that, as explained previously, the Creutz ratios have intrinsic scaling violations given by the second term in Eq. (4). Hence, a much better procedure is to make a simultaneous fit to all the square Creutz ratio data χ(R, R) for a particular value In addition to the determination of the string tension, which sets the long-distance behaviour of Creutz ratios, there is considerable interest in the parameters that determine the approach to this long-distance limit. In particular, our results show that the large N slope parameter γ has a value of 0.272 (5). The slope takes a non-zero value in perturbation theory equal to γ P (1) = (π+2)λ 16π 2 (1−1/N 2 ). Using this formula to define an effective coupling, our data implies λ eff. ≈ 8.4. At long distances, however, a new perspective arises which describes this term as arising from the fluctuation of the chromo-electric flux-tube stretching among the quark and the anti-quark. In the limit in which their separation is large compared to the thickness of this flux tube, an effective string theory description of the dynamics arises. The picture predicts [17] that the coefficient of the 1/r 2 contribution to the force F(r) is γ(0) = π 12 . This prediction has been verified by lattice data. In our present case, it would be possible to study the functionF(r, t) for r t using information of non-square smeared Creutz ratios. In particular the function γ(z) provides interesting information about the properties of the effective string theory. We can use our data to determine the function γ(z) for the large N theory. The value for z = 1 coincides with the parameter γ appearing in Table. II. Since by definition γ(z) = γ(1/z), we can parametrize this function in the vicinity of z = 1 as γ(z) = γ(1)(1 + τ (z−1) 2 2z ). Our data for z > 0.5 allow a determination of τ. For all values of N and λ we get τ = 0.31 (6).
As mentioned previously, the string picture predicts γ(0) = π/12. However, the leading string fluctuation prediction for γ(1) is ≈ 0.16. Our numerical result for γ(1) is far from this value and rather close to π/12. The same happens for the τ coefficient, which is predicted to be close to 2. Remarkably, lowest order perturbation theory also has a prediction for τ = 2/(π+2) ≈ 0.39, which is consistent with our data. The whole issue of string fluctuations for Wilson loops with different aspect ratios is being investigated at present [18].
In summary, we have presented a very precise measurement of the string tension for SU(N) Yang-Mills theory in the large N limit. It is remarkable that the N dependence is consistent with that obtained from correlation of Polyakov lines covering a different range of scales and distances r √ σ [2]. The large N result is also consistent with that obtained from the TEK single-site model, as predicted by the reduction idea.