Conformal Window and Correlation Functions in Lattice Conformal QCD*
*Based on the work in Collaboration with K.-I. Ishikawa, Yu Nakayama and T. Yoshie
We discuss various aspects of Conformal Field Theories on the Lattice. We mainly investigate the SU(3) gauge theory with Nf degenerate fermions in the fundamental representation, employing the one-plaquette gauge action and the Wilson fermion action.
First we make a brief review of our previous works on the phase structure of lattice gauge theories in terms of the gauge coupling constant and the quark mass. We thereby clarify the reason why we conjecture that the conformal window is 7 ≤ Nf ≤ 16.
Secondly, we introduce a new concept, “conformal theories with IR cutof” and point out that any numerical simulation on a lattice is bounded by an IR cutoff ∧IR. Then we make predictions that when Nf is within the conformal window, the propagator of a meson G(t) behaves at large t, as G(t) = c exp (−mHt)/tα, that is, a modified Yukawa-type decay form, instead of the usual exponential decay form exp (−mHt), in the small quark mass region. This holds on an any lattice for any coupling constant g, as far as g is between 0 and g*, where g* is the IR fixed point. We verify that numerical results really satisfy the predictions for the Nf = 7 case and the Nf = 16 case.
Thirdly, we discuss small number of flavors (Nf = 2 ∼ 6) QCD at finite temperatures. We point out theoretically and verify numerically that the correlation functions at T/Tc > 1 exhibit the characteristics of the conformal function with IR cutoff, an exponential decay with power correction.
Investigating our numerical data by a new method which we call the “local-analysis” of propagators, we observe that the Nf = 7 case and the Nf = 2 at T ∼ 2Tc case are similar to each other, while the Nf = 16 case and the Nf = 2 at T = 102 ∼ 105Tc cases are similar to each other.
Further, we observe our data are consistent with the picture that the Nf = 7 case and the Nf = 2 at T ∼ 2Tc case are close to the meson unparticle model. On the other hand, the Nf = 16 case and the Nf = 2 at T = 102 ∼ 105Tc cases are close to a free state in the Z(3) twisted vacuum. All results are consistent with naive physical intuition and give clues for long standing issues at high temperatures such as why the free energy at high temperatures does not reach the Stefan-Boltzmann ideal gas limit even at T = 100Tc.
