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.
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