Conformal theories with an infrared cutoff

Abstract

We give a new perspective on the dynamics of conformal theories realized in the SU(N ) gauge theory, when the number of flavors N f is within the conformal window. Motivated by the RG argument on conformal theories with a finite IR cutoff ΛIR, we conjecture that the propagator of a meson GH (t) on a lattice behaves at large t as a power-law corrected Yukawa-type decaying form GH(t) =cH exp (−mHt)/t α H instead of the exponentially decaying form cH exp (−mHt), in the small quark mass region where mH ≤ c ΛIR: mH is the mass of the ground state hadron in the channel H and c is a constant of order 1. The transition between the "conformal region" and the "confining region" is a first order transition. Our numerical results verify the predictions for the N f = 7 case and the N f = 16 case in the SU(3) gauge theory with the fundamental representation.Conformal field theories are ubiquitous in nature and play important roles not only in particle physics beyond the standard model but also in condensed matter physics. Nonperturbative understanding of their dynamics in four-dimensional space-time is ardently desired. In this article, we give a new perspective on the dynamics of conformal theories realized in the SU(N ) gauge theory when the number of flavors N f is within the conformal window [1] by studying meson propagators with a finite infrared (IR) cutoff. Some preliminary results have been presented in [2].Our general argument that follows can be applied to any gauge theories with arbitrary representations as long as they are in the conformal window, but to be specific, we focus on SU(3) gauge theories with N f fundamental fermions ("quarks"). We define the conformal field theory in a constructive way [3]: We employ the Wilson quark action and the standard one-plaquette gauge action on the Euclidean lattice of the size N x = N y = N z = N and N t = rN with aspect ratio r. We impose periodic boundary conditions except for an anti-periodic boundary condition in the time direction for fermion fields. We eventually take the continuum limit by sending the lattice space a → 0 with N → ∞ keeping L = N a fixed. When L = finite, the continuum limit defines a theory with an IR cutoff. The theory is defined by two parameters; the bare coupling constant g 0 and the bare degenerate quark mass m 0 at ultraviolet (UV) cutoff. We also use, instead of g 0 and m 0 , β = 6/g 2 0 and K = 1/2(m 0 a+4). For a later purpose, we define the quark mass m q through Ward-Takahashi identities with renormalization constants being suppressed [2].Let us quickly remind ourselves of the renormalization group (RG) flow when N f is in the conformal window. One important fact is finite size lattices in computer simulation always introduce an IR cutoff Λ IR ∼ 1/(N a). If the IR cutoff were zero, when quarks have tiny masses, the RG trajectory would stay close to the critical line, approaching the IR fixed point and finally would pass away from the IR fixed point to infinity. Therefore the IR behavior is governed by the "confining region". Only on the mass...

Walking signals inNf=8QCD on the lattice

“…In particular, the above two-loop/ladder studies would suggest that the walking theory if existed might be in between N f = 8 and N f = 12. As to N f = 12 there have been many analyses including those of ourselves which are consistent with the theory being inside the conformal window [10,11,[15][16][17][18][19][20][21][22][23][24], although some works prefer the SχSB phase [25,26]. There were also simulations on N f = 10 [27] consistent with the infrared conformality.…”

“…To date, some groups carried out lattice studies on 8-flavors, with Wilson fermions [10,11,23,24] and with staggered fermions [12,15,25,26,[31][32][33][34]. The Refs.…”

“…The Refs. [10,11,23,24] concluded the N f = 8 is in the conformal window, but Refs. [12,15,25,26,31,32] concluded that the N f = 8 resides on the chiral broken phase.…”

The twisted Polyakov loop coupling and the search for an IR fixed point

“…The scaling of (1/L), where L is a finite lattice extent, tells us whether the theory is conformal or not in the infinite volume limit. The other independent method has also been suggested in the paper [10]. The authors gave a conjecture that the correlation function of the mesonic operator in the finite volume around the IRFP becomes the Yukawa-type function not an exponential form.…”

“…The volume scaling for the Dirac eigenmodes [9] and the shape of the mesonic correlation function [10,23] also show the signal of the IRFP. In the studies of SU(3) gauge theory with N f = 7 and 10 [24,25,26], they found a signal of the conformality for each theory.…”