Volume 225, number 4 PHYSICS LETTERS B 27 July 1989 GLUON CONDENSATION IN SU(3) LATTICE GAUGE THEORY ~: Masstmo CAMPOSTRINI, Adriano DI GIACOMO and Ylgit GUNDUC Dlpartlrnemo dl Flstca dell Umversltaand INFN, 1-56100 Plsa, Italy Received 12 May 1989 The gluon condensate of &menslon 4 is determined for a pure gauge SU ( 3 ) lattice gauge theory A critical review of the existing procedures to extract the condensate ~s presented 1 Introduction The determination of the gluon condensate param- eters of the QCD vacuum is an important issue m the understanding of strong interactions [ 1,2 ] A determination from first principles of quantities like G2 G2= G~.,G~,. , (1 1) which have non-trivial dimension in mass, is only possible in a non-perturbatlve formulation of the theory Lattice is in this respect a unique theoretical tool For a pure gauge theory with gauge group SU (2) it has been definitely shown by Monte Carlo simula- tion on the lattice that G2 is different from zero, and that it lS of the correct order of magnitude required by experiment #~ Of course, a reahstlc value should be computed with gauge group SU(3) and In the presence of quarks Some pioneering estimates for SU (3) exist in the literature [ 5 ] in the quenched ap- proximation In this paper we present some new re- sults for SU ( 3 ) in the same approximation Extracting G2, or higher condensates, from the lat- ~r Partially supported by MPI (ltahan Ministry for Pubhc Education) a Supported by ICTP Programme for Research and Training m ltahan Laboratories, on leave from Hacenepe Umvers~ty, An- kara, Turkey ~ For a recent determination see ref [3 ] For a review of exist- mg results see ref [4] 0370-2693/89/$ 03 50 © Elsevier Science Pubhshers B V ( North-Holland Physics Publishing Division ) rice ~s not a trivial task The only gauge lnvartant quantities on a lattice are closed Wilson loops These are sums of operators of arbitrarily high dimension, and are dominated by large perturbatlve additive re- normahzatlons in the weak couphng region, where asymptotic scahng is expected to hold A typical method to determine G2 ts expressed by the following formula [ 4 ] (1--W,j)~Z, 7~212j2G2a4+ C~ J 12No ~ff~+O(a 6) (12) W,j is a rectangular Wilson loop of size t X J, fl= 2No~ g2, Ztj= 1 + ~., z~ /fl n is a multiphcatlve finite renor- mahzation with respect to the continuum definmon of G2, the sum Y~c~/fl n is an addmve renormahza- tlon, which is a cut-off version of the quartically di- vergent renormahzation of an operator of &mension 4 At sufficiently large fl asymptotic scaling ts ex- pected to hold, i e 1 (fl)"'/2h°~exp(4-~b~) (13) a~ Z \ ~ / bo and b~ are the first two coefficients of the fl-func- tion of the theory 11{ Nc "] ( Nc "] 2 bo= 3 \16~r2J ' b~= 3---~\16zc2j (14) Eq ( 1 2) is expected to hold for a Wilson loop W,j at values offl at which the physical correlation length of the theory is large compared to l and j, so that the 393