S. Sakai

We study $\mathrm{SU}(3)$ gluon electric and magnetic masses at finite temperatures using quenched lattice QCD on a ${20}^{2}\ifmmode\times\else\texttimes\fi{}32\ifmmode\times\else\texttimes\fi{}6$ lattice. We focus on temperature regions between ${T=T}_{c}$ and ${6T}_{c},$ which are realized in BNL Relativistic Heavy Ion Collider and CERN Large Hadron Collider experiments. Stochastic quantization with a gauge-fixing term is employed to calculate gluon propagators. The temperature dependence of the electric mass is found to be consistent with the hard-thermal-loop perturbation, and the magnetic mass has finite values in the temperature region of interest. Both screening masses have little gauge parameter dependence. The behavior of the gluon propagators is very different in confinement or deconfinement physics. The short distance magnetic part behaves like a confined propagator even in the deconfinement phase. A simulation with a larger lattice, ${32}^{2}\ifmmode\times\else\texttimes\fi{}48\ifmmode\times\else\texttimes\fi{}6,$ shows that the magnetic mass has a stronger finite size effect than the electric mass.

The problem of whether there is a constraint on the number of flavors for quark confinement in QCD is numerically investigated on a lattice with Wilson fermions as quarks. It is shown that even in the strong coupling limit, when the number of flavors exceeds 7, quarks are not confined and chiral symmetry is not spontaneously broken for light quarks.

We investigate the phase structure of lattice QCD for the general number of flavors in the parameter space of gauge coupling constant and quark mass, employing the one-plaquette gauge action and the standard Wilson quark action. Performing a series of simulations for the number of flavors ${N}_{F}=6--360$ with degenerate-mass quarks, we find that when ${N}_{F}>~7$ there is a line of a bulk first order phase transition between the confined phase and a deconfined phase at a finite current quark mass in the strong coupling region and the intermediate coupling region. The massless quark line exists only in the deconfined phase. Based on these numerical results in the strong coupling limit and in the intermediate coupling region, we propose the following phase structure, depending on the number of flavors whose masses are less than ${\ensuremath{\Lambda}}_{d}$ which is the physical scale characterizing the phase transition in the weak coupling region: When ${N}_{F}>~17,$ there is only a trivial IR fixed point and therefore the theory in the continuum limit is free. On the other hand, when $16>~{N}_{F}>~7,$ there is a nontrivial IR fixed point and therefore the theory is nontrivial with anomalous dimensions, however, without quark confinement. Theories which satisfy both quark confinement and spontaneous chiral symmetry breaking in the continuum limit exist only for ${N}_{F}<~6.$

The nature of finite temperature transitions in lattice QCD with Wilson quarks is studied near the chiral limit for the cases of two, three, and six flavors of degenerate quarks (${N}_{F}=2, 3, \mathrm{and} 6$) and also for the case of massless up and down quarks and a light strange quark (${N}_{F}=2+1$). Our simulations mainly performed on lattices with the temporal direction extension ${N}_{t}=4$ indicate that the finite temperature transition in the chiral limit (chiral transition) is continuous (or at most very weakly first order) for ${N}_{F}=2$, while it is of first order for ${N}_{F}=3 \mathrm{and} 6$. We find that the transition is of first order for the case of massless up and down quarks and the physical strange quark where we obtain a value of $\frac{{m}_{\ensuremath{\varphi}}}{{m}_{\ensuremath{\rho}}}$ consistent with the physical value. This result is different from the previous result with staggered quarks at ${N}_{t}=4$ which suggests that the transition in the real world is a crossover. Since the deviation from the continuum limit is large in both studies at ${N}_{t}=4$, a calculation with larger ${N}_{t}$ or with an improved action would be needed in order to obtain a definite conclusion about the nature of the QCD transition. We also discuss the phase structure at zero temperature as well as that at finite temperatures.