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

Abstract:

We investigate $SU(3)$ gauge theories in four dimensions with $N_f$ fundamental fermions, on a lattice using the Wilson fermion. We first introduce a new concept ``conformal theories with an IR cutoff', after pointing out that the following two categories in $SU(3)$ gauge theories with fundamental $N_f$ fermions possess an IR fixed point: Large $N_f (N_f^{c} \le N_f \le 16)$ QCD within the conformal window (referred as Conformal QCD) and small $N_f (1 \le N_f \le N_f^{c}-1)$ QCD at temperature $T/T_c > 1$ (referred as High Temperature QCD). In the case of Conformal QCD in the continuum limit, the compact space and/or time gives an IR cutoff. In the case of High Temperature QCD, the temperature $T$ plays a role of an IR cutoff together with a cutoff due to possible compact space, depending on how to take the continuum limit. We note any lattice calculation is performed on a finite lattice. Thus any calculation on a lattice possesses an IR cutoff. In the conformal theories with an IR cutoff there exists the ``conformal region'' together with the confining region and the deconfining region. We verify numerically on a lattice of the size $16^3\times 64$ the existence of the conformal region and the non-trivial $Z(3)$ structure of the vacuum and the Yukawa-type decay form of meson propagators in the conformal region. We stress that High Temperature QCD is intrinsically accompanied with an IR cutoff. Therefore the understanding the vacuum structure and the property of correlation functions is the key to resolve long standing issues in High Temperature QCD. We further argue 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: the one boundary is close to meson states and the other is close to free quark states. In particular, we find the correspondence between Conformal QCD with $N_f = 7$ and High Temperature QCD with $N_f=2$ at $T\sim 2\, T_c$ being 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 investigate $SU(3)$ gauge theories in four dimensions with $N_f$ fundamental fermions, on a lattice using the Wilson fermion. We first introduce a new concept ``conformal theories with an IR cutoff', after pointing out that the following two categories in $SU(3)$ gauge theories with fundamental $N_f$ fermions possess an IR fixed point: Large $N_f (N_f^{c} \le N_f \le 16)$ QCD within the conformal window (referred as Conformal QCD) and small $N_f (1 \le N_f \le N_f^{c}-1)$ QCD at temperature $T/T_c > 1$ (referred as High Temperature QCD). In the case of Conformal QCD in the continuum limit, the compact space and/or time gives an IR cutoff. In the case of High Temperature QCD, the temperature $T$ plays a role of an IR cutoff together with a cutoff due to possible compact space, depending on how to take the continuum limit. We note any lattice calculation is performed on a finite lattice. Thus any calculation on a lattice possesses an IR cutoff. In the conformal theories with an IR cutoff there exists the ``conformal region'' together with the confining region and the deconfining region. We verify numerically on a lattice of the size $16^3\times 64$ the existence of the conformal region and the non-trivial $Z(3)$ structure of the vacuum and the Yukawa-type decay form of meson propagators in the conformal region. We stress that High Temperature QCD is intrinsically accompanied with an IR cutoff. Therefore the understanding the vacuum structure and the property of correlation functions is the key to resolve long standing issues in High Temperature QCD. We further argue 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: the one boundary is close to meson states and the other is close to free quark states. In particular, we find the correspondence between Conformal QCD with $N_f = 7$ and High Temperature QCD with $N_f=2$ at $T\sim 2\, T_c$ being in close relation to a meson unparticle model. From this we estimate the anomalous mass dimension $\gamma^* = 1.2 (1)$ for $N_f=7$.