Numerical Studies of Lattice Loop Equations in Pure Gauge Theory

Peter D. Anderson

Publication: Loop equations and bootstrap methods in the lattice

Abstract: Monte Carlo lattice simulations of pure Yang-Mills theory, such as the one proposed here, have been shown to be ideally suited for GPU computations since all changes in the action are local (as opposed to dynamical fermions that are non-local in the lattice simulation). The action is written in terms of link variables and only depends on nearest neighbors. Thus any lattice site with even (or odd) parity can be run simultaneously. The link is an element of the gauge group SU(NC) and acts as a parallel transporter from one site to the next site. A Wilson Loop is defined as the trace of a product of links associated with a closed path in the lattice. Its expectation value is measured by averaging over a large number of statistically independent configurations that are obtained in the simulation. After such independent configurations are obtained and saved, the Wilson loops averages can be measured off-line.

Currently, we have implemented the Monte Carlo simulation for NC = 3 using CUDA on two Tesla C2070 GPUs using the Wilson Action, which is the simplest action that reproduces the correct continuum limit. Each link is updated by multplying it by a matrix randomly chosen from a pool of randomly generated matrices in the gauge group. These updates are accepted or rejected via the Metropolis algorithm to minimize the action. To optimize parallelizability, the updates are performed several times at each link to bring it into a state of “thermal equilibrium” with its neighboring links. We have performed calculations of glueball mass spectrums which took advantage of improved actions that involved the next smallest Wilson Loop in the action as well.

We wish to extend our studies to larger NC and larger lattice sizes so we can determine the region where the loop equations are numerically satisfied and where large-NC simplifications are applicable. Since the loop equations are the Schwinger-Dyson equations for the Wilson Loops, they are dependent on the form of the action and need to be adjusted accordingly. However, we should find that they still hold. In order to investigate the phase space, we need access to a larger number of GPUs than we currently have. Our goal is to rewrite the existing CUDA code for NC = 3 in OpenCL and allow for generic NC. In particular we would like to verify the approach to large-NC behavior at NC = 8.

Finally we would like to emphasize that the loop equations are theoretically exact equations, valid for any value of NC and for the lattice, independently of the continuum limit. Therefore they are an ideal tool to evaluate the Monte Carlo algorithm.

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