Slave Equations for Connected Correlation Functions

Abstract

One of the most important objectives of studying euclidean fields on the space-time lattice is the computation of connected correlation functions and the subsequent extraction in the continuum limit of the renormalised quantities characterising the theory. Computer simulation of lattice field theory offers, in principle, a direct method of achieving this aim. However, in practice it is acknowledged that a number of difficulties arise, one of which concerns the size of the statistical error compared with the signal. For example, a natural quantity to measure is the mass-gap of the theory which is obtained from the exponent in the exponential decay of the appropriate correlation function: G (t) ∼ e ^−Mt. It is necessary that the mass be extracted from a region where t is large enough for this simple behaviour to have clearly set in: the contributions of competing exponentials must be negligible. In this region G (t) is itself small compared with G (0), and in many methods for simulation the error on G(t) can be shown to be independent of t and so the signal-to-noise ratio decreases as e ^−Mt. The consequence is a large increase in the required computer time for accurate results. This problem has been tackled by clever choice of the operators appearing in the correlation function. A good choice reduces the overall scale of the error (e.g. by variance reduction techniques), and can also reduce the value of t where pure exponential decay sets in if the operator predominantly projects the state of interest out of the vacuum. I shall present another approach to this problem based on the Langevin or stochastic simulation method. Extra fields are introduced which directly give unbiassed estimators for the connected correlators of interest. The extra fields evolve in Langevin time according to a linear auxiliary, or slave, equation which depends on the original quantum field. There are two advantages with the technique. The first is that the error analysis is straightforward because the signal is computed directly as the mean of an unbiassed estimator. The second advantage is the subject of this talk. It can be shown that the error on G(t) decreases with increasing t and that this decrease is exponential in theories with a mass gap.