Peculiar Properties of SU(2) Gauge Field Thermodynamics on a Finite Lattice. Calculation of Beta-function

The new method of nonperturbative calculation of the beta-function in the lattice gauge theory is proposed. The method is based on the finite size scaling hypothesis. Ever since the pioneering work by Creutz [1] the approach to asymptotic scaling, and thus the continuum limit, was one of the central issues in studies of gauge theories on the lattice. Although the first results were promising, the lack of asymptotic scaling of physical observables has been observed in SU(N) gauge theories. One of the main source of the nonperturbative results in the gauge theories today is the Monte-Carlo (MC) lattice calculations. For the SU(N) pure gauge theories on lattices of size Nτ ×N 3 σ MC results are the dimensionless functions of the bare coupling constant g (another form for the coupling, β ≡ 2N/g, is often used). The transformation of these functions to physical quantities are done by multiplying them on lattice spacing a in the corresponding powers. The length scale L and the temperature T are given as L = Nσa, T = (Nτa) −1 . (1) To define the physical quantities one needs a connection between lattice spacing a and bare coupling constant g. Such a connection is formulated in terms of the beta-function βf(g) through the equation βf(g) = −a dg da . (2) The perturbation theory gives the asymptotic expansion of the beta-function β f = −b0g 3 − b1g 5 + O(g), b0 = 11N 48π , b1 = 34 3 ( N 16π )2 , (3)