Volume 118B, number 4, 5, 6 PHYSICS LETTERS 9 December 1982 PRECOCIOUS SCALING IN LATTICE GAUGE THEORIES F. GLIOZZI, F. RAVANINI 1 and S. SCIUTO Istituto di Fisica Teorica dell'universitd di Torino, Turin, Italy and Sezione di Torino dellTstituto Nazionale di Fisica Nucleare, Turin, Italy Received 26 July 1982 We propose a method to evaluate numerically and in some cases analytically the two-loop contributions to.physical quantities without computing Feynman graphs. Such contributions are negligible for the SU(2) Wilson action, which shows a precocious scaling; on the contrary they are important for other actions (including Manton and heat kernel ones) and account for the observed violations of universality. I. Introduction. Many different lattice actions can be used to regularize the same continuum field theory. Universality [ 1 ] assures that the ratios among physical quantities have the same values in any formulation as long as one is close enough to the continuum limit, so that the correlation length is large compared to the lattice spacing a. Near their critical point (vanishing g2) the whole set of lattice actions corresponding to an asymptotic- ally free theory are described in terms of only two parameters: the bare coupling constant g2 and the mass scale A, which depends on all the remaining pa- rameters of the lattice action. In this region a physical quantity M with the di- mension of a mass can be written in the following way: (Ma) = (rn/A)exp(-1/bog2)(bog2) -b¢2b~ X [1 + O(g2)] . (I) In eq. (1) b 0 and b 1 are the first two coefficients of the Callan-Symanzik beta function 13(g) defined by: (J(g) = a(d/da)g = bo g3 + bl g5 + b2 g7 + .... (2) It is well known that only b 0 and b 1 have a universal meaning; instead the other coefficients b n (n >~ 2) de- pend on the renormalization scheme, that is they de- 1 Scuola di Specializzazione in Fisica Nucleare dell'Universith di Torino, Turin, Italy. 402 pend on the specific form of the lattice action. The range ofg where eq. (1) can be applied neglecting the O(g 2) corrections is called scaling or critical region. Out of this region the lattice theory looses its univers- al character as the O(g 2) corrections depend on the non-universal coefficients of the beta function. Some years ago Creutz [2] showed that in the SU(2) and SU(3) lattice gauge theories with Wilson action the scaling region is large enough to be reached by Monte Carlo simulations in microscopic crystals stored in a computer. Similarly it has been recently shown [3] that in the associated SU(2) X SU(2) spin system the strong coupling expansion of the mass gap nicely fits to the scaling law (1) for rather large values ofg. Universality provides us with an important consis- tency check of these numerical results: dimensional quantities obtained in different renormalization schemes can be compared if the ratio between the cor. responding A's is known. Such a ratio has been calcu- lated by different methods [4-9] and for various lat- tice actions [10,11] ; the outcome of this comparison is at first sight rather intriguing. Although in some cases, like in the asymmetric lattices [3,11,12], the results are in good agreement with universality, in many others, like for the Manton, heat kernel [13] and mixed fundamental-adjoint actions [ 14-17 ], there is a large discrepancy between the ratios of A's measured by Monte Carlo experiments and the exact values calculated at g = 0. 0 031-9163/82/0000-0000/$02.75 © 1982 North-HoUand