Global Structure of Conformal Theories in the SU(3) Gauge Theory

We investigate SU(3) gauge theories in four dimensions with Nf fundamental fermions, on a lattice using the Wilson fermion. Clarifying the vacuum structure in terms of Polyakov loops in spatial directions and properties of temporal propagators using a new method local analysis, we conjecture that the conformal region exists together with the confining region and the deconfining region in the phase structure parametrized by beta and K, both in the cases of the large Nf QCD within the conformal window (referred as Conformal QCD) with an IR cutoff and small Nf QCD at T/Tc>1 with Tc being the chiral transition temperature (referred as High Temperature QCD). Our numerical simulation on a lattice of the size 16^3 x 64 shows the following evidence of the conjecture. In the conformal region we find the vacuum is the nontrivial Z(3) twisted vacuum modified by non-perturbative effects and temporal propagators of meson behave at large t as a power-law corrected Yukawa-type decaying form. The transition from the conformal region to the deconfining region or the confining region is a sharp transition between different vacua and therefore it suggests a first order transition both in Conformal QCD and in High Temperature QCD. Within our fixed lattice simulation, we find that there is a precise correspondence between Conformal QCD and High Temperature QCD in the temporal propagators under the change of the parameters Nf and T/Tc respectively. In particular, we find the correspondence between Conformal QCD with Nf = 7 and High Temperature QCD with Nf=2 at T ~ 2 Tc being in close relation to a meson unparticle model. From this we estimate the anomalous mass dimension gamma* = 1.2 (1) for Nf=7. We also show that the asymptotic state in the limit T/Tc --> infty is a free quark state in the Z(3) twisted vacuum.

Conformal Behavior in QCD

We give a new perspective on the properties of quarks and gluons at finite temperature T in N_f = 2 ~ 6 QCD. We point out the existence of an IR fixed point for the gauge coupling constant at T>T_c (T_c is the chiral phase transition temperature). Based on this observation we predict theoretically and verify numerically that the correlation functions of a meson G(t) at T/T_c > 1 decay with a power-law corrected Yukawa-type decaying form, G(t)=c exp(-m t)/t^alpha in the conformal region defined by m < c Lambda_IR, where Lambda_IR is the IR cutoff, m is the characteristic scale of the spectrum in the meson cannel and c is a constant of order 1. The decaying form is the characteristics of conformal theories with an IR cutoff. We discuss in detail how the resulting hyper scaling relation of physical observables may modify the existing argument about the order of the chiral phase transition in the N_f=2 case.

Conformal Theories with an IR cutoff

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 Lambda_{IR}, we conjecture that the propagator of a meson G_H(t) on a lattice behaves at large t as a power-law corrected Yukawa-type decaying form G_H(t) = c_H exp{(-m_H t)}/t^{alpha_H} instead of the exponentially decaying form c_Hexp{(-m_H t)}, in the small quark mass region where m_H le c Lambda_{IR}: m_H 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.