Volume 128B, number 6 PHYSICS LETTERS 8 September 1983 AN IMPROVED RENORMALIZATION GROUP TRANSFORMATION IN FOUR DIMENSIONS R. CORDERY, R. GUPTA 1 and M.A. NOVOTNY Physics Department, Northeastern University, Boston, MA 02115, USA Received 19 May 1983 A new transformation for Monte Carlo renormalization group analysis of four-dimensional gauge theories is develop- ed. This RG transformation has a scale factor of x,/3. In addition no gauge fixing is required in the construction of the link variables on the renormalized lattice. The Z (2) theory is analyzed as a test case. Introduction. A new transformation for Monte Carlo renormalization group analysis of lattice gauge theories in four dimensions is introduced. The advan- tages of this transformation are two-fold. The scale factor for the transformation proposed is b = X/~, which is smaller than the scale factor b = 2 commonly used. The second important feature is that no gauge fixing is required in the construction of the links on the renormalized lattice. These two features make this transformation ideal for numerical simulations on four-dimensional hypercubic lattices. To illustrate tiffs transformation, we have applied it to the study of the Z(2)lattice gauge theory. The extensions required to apply this transformation to abelian theories such as U(1) and to non-abelian theories such as SU(N) are discussed. The theory. The technique of the Monte Carlo Renormalization Group (MCRG) has been developed by Wilson, Kadanoff, Swendson and Ma [1-4]. The scope of this method for asymptotically free field theories like QCD is to map out the/3-function and to find an improved lattice action. The/3-function gives the phase structure of the theory and the scaling properties of physical quantities having dimensions, such as masses. The results obtained using an improved action have a weaker dependence on the lattice spa- 1 Research supported in part by the National Science Founda- tion under Grant No. 80-8333. cing, so agreement with the perturbative scaling be- havior can be observed on coarser lattices. For theo- ries with a critical point at a non-zero value of the coupling [like compact QED i.e. U(1)], the critical exponents can be determined directly using the MCRG at the critical point. The pure gauge Z(2) theory in four dimensions exhibits a first order transi- tion [5] and a Renormalization Group (RG) analysis does not add much additional information. However, since our goal is to perform the MCRG analysis on U(1) and SU(N)gauge theories, we shall reproduce the essential features of the MCRG using the x/~ scale factor. Readers unfamiliar with the subject are directed to the original sources and excellent reviews [1-4,6,7]. In order to investigate the fixed point behavior of the theory consider the generalized hamiltonian, 3£, parametrized by the infinite set of couplings (Ks). Under the RG transformation, T, the new theory is described by the renormalized hamiltonian The block variables, t, are constructed from the ori- ginal degrees of freedom, s, with a normalized proba- bility distribution P(t, s) such that fDt e(t, s) = 1. (2) These block variables are distributed according to 0.031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland 425