ar X iv : 1 50 3 . 02 35 9 v 1 [ he p - la t ] 9 M ar 2 01 5 IR fixed points in SU ( 3 ) gauge Theories

We propose a novel RG method to specify the location of the IR fixed point in lattice gauge theories and apply it to the SU(3) gauge theories with Nf fundamental fermions. It is based on the scaling behavior of the propagator through the RG analysis with a finite IR cut-off, which we cannot remove in the conformal field theories in sharp contrast with the confining theories. The method also enables us to estimate the anomalous mass dimension in the continuum limit at the IR fixed point. We perform the program for Nf = 16, 12, 8 and Nf = 7 and indeed identify the location of the IR fixed points in all cases. Scale invariance, or more precisely conformal invariance has become a fundamental concept in understanding the universal aspects of the nature from the Planck scale to the Hubble scale. They appear not only in critical phenomena of condensed matter physics, but also in quantum gravity, high energy particle phenomenology, and all the way up to cosmology [1]. Many conformal field theories, however, are strongly coupled, and much remains unsolved in their theoretical understanding. In particular, when realized by gauge theories, the constructive approaches to the conformal fixed points are still rudimentary [2]. The aim of this article is to clarify some important aspects of these constructive approaches and offer one simple criterion on conformal invariance. Obviously, the central question is to locate the IR fixed point within a given class of theories. In this article we propose a novel and simple RG method to specify the location of the IR fixed point in lattice gauge theories by studying the scaling behavior of the propagator. We will apply the technique to the SU(3) gauge theories with Nf fundamental fermions (within the conformal window), and estimate the anomalous mass dimension. We perform this program for Nf = 16, 12, 8 and Nf = 7, and indeed identify the location of the IR fixed points in all cases. We constructively define gauge theories on Euclidean plane R as the continuum limit of lattice gauge theories on the Euclidean lattice of the size Nx = Ny = Nz = N and Nt = rN (r being an aspect ratio, which is fixed as r = 4 throughout the article), taking the limit of the lattice space a → 0 and N → ∞, with L = N a and Lt = Nt a fixed. When L and/or Lt are finite, the system is bounded by an IR cutoff ΛIR ∼ 1/L. We impose an anti-periodic boundary condition in the time direction for fermion fields and periodic boundary conditions otherwise. In conformal field theories the IR cutoff is an indispensable ingredient because there is no other natural scale to compare, which will be further elucidated in this article. Our general argument that follows can be applied to any gauge theories with fermions in arbitrary (vector-like) representations, but to be specific, we focus on SU(3) gauge theories with Nf fundamental Dirac fermions. For the lattice regularization of the action, we employ the Wilson quark action and the RG improved gauge action[3] (also known as the Iwasaki gauge action in the literature). Given the regularized action, the theory is defined by two parameters; the bare coupling constant g0 and the bare degenerate quark mass m0 at ultraviolet (UV) cutoff. We also use, instead of g0 and m0, β = 6/g 2 0 and the hopping parameter K = 1/2(m0a+ 4). As for observables, together with the plaquette and the Polyakov loop in each space-time direction, we measure the quark massmq defined through Ward-Takahashi identities mq = 〈0|∇4A4|PS〉 2〈0|P |PS〉 , (1) Preprint submitted to Elsevier March 10, 2015 where P is the pseudo-scalar density and A4 the fourth component of the local axial vector current, with renormalization constants being suppressed. The quark mass mq defined in this way only depends on β and K up to order 1/N corrections. One of the most important observables we will study is the t dependence of the propagator of the local meson operator in the H channel: