Computation of the string tension in four-dimensional Yang-Mills theory using large N reduction

Continuum reduction and Monte Carlo simulation are used to calculate the heavy quark potential and the string tension in large N Yang-Mills theory in four dimensions. The potential is calculated out to a separation of nine lattice units on a lattice with extent six in each direction.


I. INTRODUCTION
Through dimensional transmutation, the dimensionless classical coupling constant of QCD is transformed to a running coupling controlled by a physical scale Λ QCD . Roughly speaking, weak coupling perturbative approximations to processes with momentum transfer Q are expansions in 1/ln(Q/Λ QCD ). This gives excellent results when Q/Λ QCD is large.
When one cannot rely on that, an alternative is the large N approach. The gauge group is generalized to SU (N ), and the expansion is in 1/N .
Although the large N limit of Yang-Mills theory is still out of reach, it is known that it enjoys several simplifications. Among them is the possibility of reducing the space-time volume without affecting certain physical results [1]. Continuum reduction [2,3], i.e. reduction to a physical size of order 1/Λ QCD avoids some of the difficulties [4,5,6] with reduction to a single space-time point. The addition of double trace terms to the action [7] is an alternative approach.
It is now practical to obtain good results by combining continuum reduction with numerical simulations needing only modest resources. In previous numerical work [8,9,10,11], it has been shown that not only bulk quantities but also physical quantities based on Wilson loops are accessible. The previous results based on Wilson loops have been in three dimensions. In the work described here, we show that the method is still practical in four dimensions.
We have used continuum reduction and Monte Carlo simulation to calculate the heavy quark potential and the string tension in large N Yang-Mills theory in four dimensions. An important aspect of the method is that reduction allows the calculation of infinite volume, infinite N Wilson loops that are larger than the reduced lattice. In particular, in this work, the heavy quark potential is calculated out to a separation of nine lattice units on a lattice with extent six in each direction. The results for the string tension are compatible with those obtained on large lattices at smaller N .

II. METHODS
The standard Wilson Yang-Mills action with gauge group SU (N ) is used. In the large N limit, g 2 is taken to zero with the inverse 't Hooft coupling b = 1 g 2 N held fixed.
We report results with N = 37, 47, and 59 on a lattice of size 6 4 . This is large enough so that in the range of available b, the lattice is not too coarse but is still small enough to give a manageable computational cost.
In four dimensions, the useful range of couplings and physical lattice spacings a on a given lattice is more limited than in three dimensions. For 6 4 , the system becomes unstable to the bulk transition for small b. The smallest b we have used is 0.3450, which is just above the unstable point. For sufficiently large b, the center symmetry breaks in one lattice direction, and reduction no longer holds in that direction. This is the large N limit of the finite temperature phase transition [3,12] . Nevertheless, we encountered no bulk transitions during the simulations.
As described in previous work [3], we update the gauge field configurations with heat bath and over-relaxation methods. In this work, one update will mean one heat bath sweep to the fluctuations and simplifies the t behavior of the loops [13]. The smearing is a fourdimensional version of the method used in [10]. One lattice direction is arbitrarily chosen as the "time" direction. Links in the remaining three spatial directions (but not in the time direction) are smeared. After the Wilson loops in these "time"-space planes are measured, the process is repeated with each of the other lattice directions chosen as "time." When smearing the links in spatial directions, only staples in spatial planes are used.
One step in the iteration takes one from a set U There are two parameters, namely, the smearing factor f and the number of smearing steps n. We use f = 0.45 and n = 5 so that τ = f n = 2.25 and the associated length scale is √ τ = 1.5.
The use of finer smearing steps f = 0.1, n = 25 or a larger length scale f = 0.45, n = 10 was more costly and did not lead to further improvement.
Data were collected on planar, rectangular Wilson loops of size k ×j with k and j ranging from 1 to 9 and with j the extent in the "time" direction. The j decay of the loops is fit to a simple exponential. To avoid a possible distortion from a combination of smearing and very small error bars at the shortest separation, loops that are 1 × j and k × 1 are not included in the fit. The rate of the exponential decay is taken as the static quark potential at the separation k. The k dependence of the exponential is fit to obtain the string tension. With all quantities in lattice units, the three parameter potential that is used in the fits is An example of fits to Wilson loop data is given in Fig. 1, which is for b = 0.3480 and N = 47. At a fixed k, the decay in j of loop data is fit to the form Ae −m(k)j using k and j from 2 through 9, inclusive. This gives the potential m(k) at separation k which is then plotted as a function of k in Fig. 2. A fit of the potential for the b = 0.3480 and N = 47 data to the form of Eq. 1 gives To verify that N is sufficiently large, we have the results for N = 37, 47, and 59 in Fig. 3.
The string tensions from fits to the N = 47 and N = 59 data agree.