Global structure of conformal theories in the SU(3) gauge theory

We investigate SU(3) gauge theories in four dimensions with N-f 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 that we call "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 N-f QCD within the conformal window (referred as conformal QCD) with an IR cutoff and small N-f QCD at T/T-c > 1 with T-c being the chiral transition temperature (referred to 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 that the vacuum is the nontrivial Z(3) twisted vacuum modified by nonperturbative effects and that temporal propagators of mesons 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 high-temperature QCD. To confirm the conjecture and distinguish it from the possibility of crossover phenomena, we need to take the continuum/thermodynamic limit, which we do not attempt in this work. 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 N-f and T / T-c, respectively: one boundary is close to meson states, and the other is close to free quark states. In particular, conformal QCD with N-f = 7 corresponds to high-temperature QCD with N-f = 2 at T similar to 2T(c), both of which are in close relation to a meson unparticle model. From this, we estimate the anomalous mass dimension gamma* = 1.2(1) for N-f = 7. We also show that the asymptotic state in the limit T / T-c -> infinity is a free quark state in the Z(3) twisted vacuum. The approach to a free quark state is very slow; even at T / Tc similar to 10(5), the state is affected by nonperturbative effects. This is possibly connected with the slow approach of the free energy to the Stefan-Boltzmann ideal gas limit.