Nuclear Physics B243 (1984) 261-272 © North-Holland Publishing Company MONTE CARLO CALCULATIONS WITH DYNAMICAL FERMIONS BY A LOCAL STOCHASTIC PROCESS Pietro ROSSII and Daniel ZWANZIGER2 Physics Department, New York University, New York, New York 10003, USA Received 17 January 1984 (Revised 9 April 1984) We develop and test numerically a Monte Carlo method for fermions on a lattice which accounts for the effect of the fermionic determinant to arbitrary accuracy. It is tested numerically in a 4-dimensional model with SU(2) color group and scalar fermionic quarks interacting with gluons. Computer time grows linearly with the volume of the lattice and the updating of gluons is not restricted to small jumps. The method is based on random location updating, instead of an ordered sweep, in which quarks are updated, on the average, R times more frequently than gluons. It is proven that the error in R is only of order l/R instead of l/R 1/2 as one might naively expect. Quarks are represented by pseudofermionic variables in M pseudoflavors (which requires M times more memory for each physical fermionic degree of freedom) with an error in M of order l/M. The method is tested by calculating the self-energy of an external quark, a quantity which would be infinite in the absence of dynamical or sea quarks. For the quantities measured, the dependence on R -~ is linear for R I>8, and, within our statistical uncertainty, M = 2 is already asymptotic. 1. Introduction In order to make calculations in lattice gauge theories and also of electrons in crystals, it is necessary to include efficiently the determinant which represents the effects of closed fermionic loops. We present here a new method and the results of numerical calculations to test it, in a lattice gauge theory model with SU(2) as local gauge group and scalar fermionic quarks. In particular, we evaluate the self-energy of an external quark. This quantity would be infinite in the absence of dynamical quarks or sea quarks, as they are sometimes called. However, the external quark attracts dynamical quarks or anti-quarks from the vacuum to form a hadron, and a finite self-energy results. Good agreement with an analytic calculation in the strong coupling regime is obtained by the present method. Monte Carlo calculations with quarks were originally done in the quenched or valence approximation which neglects the influence of the quarks on the gluon distribution [1-3]. Methods [4-1 1] which do account for the fermionic determinant either contain a systematic error of unknown size [12] or, if exact, suffer from various drawbacks such as requiring an amount of computer time which grows faster than Dean's Dissertation Fellow. 2 This research was supported in part by the National Science Foundationunder grant no. PHY-8116102. 261