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Quark propagators at finite temperature with the clover action

We study properties of the finite temperature quark propagator by using the SU(3) quenched lattice simulation in the Landau gauge and report numerical results of the standard Wilson quark case as well as the improved clover one. The mass function in the deconfinement phase is different from that of the confinement phase, especially at low momentum regions.


Introduction
Quark propagator is an essential ingredient of QCD and should bring us important information on the dynamics of the quark-gluon plasma (QGP). The lattice QCD calculation is a unique tool to investigate the temperature dependence of the quark propagator, since even after the QGP phase transition, the quark-gluon interaction is still strong, i.e., the QCD coupling constant is O (1). Therefore, we cannot use perturbative calculations. The RHIC data suggest that the QGP is almost a perfect fluid indicating that the interaction is very strong.
We first pay an attention to the quark thermal mass, which is an effective mass proportional to temperature and a consequence of the preferred reference frame of the heat bath. It is studied by perturbative QCD at the region that temperature is much higher than critical temperature. The results are reviewed in Ref. [1].
Quark propagator at zero temperature is studied by Bowman et al. using lattice QCD simulation [2]. The case of clover fermion action is studied by Skullerud and Williams [3]. Yet, the quark propagator at finite temperature has not been studied.

Quark propagator
The quark propagator in coordinate space is given by where W f is the fermion matrix, We use the clover fermion action given by [4] Here S W is Wilson fermions and S C is the clover term. It is Fourier transformed to momentum space to calculate the quark mass function and the quark thermal mass: where p µ is the discrete lattice momentum given by where anti-periodic (periodic) boundary condition at temporal (space) direction is imposed.

At finite temperature, where O(4) invariance is broken and O(3)
is invariant, the quark propagator is given by three parameters. Consequently, the general form of the quark propagator at finite temperature is .
We define the mass function as [3] The thermal mass is independent of momentum. The mass is then defined as The quark thermal mass at high temperature is given by perturbative calculation [1], g is the coupling constant and C F (= 4/3) is a group factor.

Numerical results
We have calculated the quark propagator using the Wilson fermion action and the clover fermion action. The SU (3) quenched lattice simulation in the Landau gauge is performed on a 16 3 ×8 lattice.

Mass function
In Figure 1, the mass function is plotted using the Wilson fermion action. Blue dots correspond to the case of the confinement phase (T = 0.78T c ). Other three color dots are cases of the deconfinement phase (1.09T c , 1.37T c and 1.86T c ). In the high momentum region, the mass function decreases for both the cases. This behavior represents asymptotic freedom. In the low momentum region, in contrast, mass function of both cases has different behavior for the two cases. The mass function of the confinement phase increases rapidly as momentum decreases, while that of the deconfinement phase increases slowly as momentum decreases.
Our first result is that the quark mass function of the deconfinement phase is different from that of the confinement phase in which the quark propagator should not have a physical pole.

Thermal mass
In Figure 2, the thermal mass is plotted using the Wilson fermion action. It is evaluated from the mass function at zero momentum. The thermal mass increases linearly as a function of temperature(T). This is consistent with perturbative calculation (2.10).

Clover fermion action versus Wilson fermion action
In Figure 3, we compare the mass function based on the clover fermion action with that on the Wilson fermion action in order to study finite lattice space effect at T = 1.37T c . Red dots stand for the case of Wilson fermion action and blue dots are the case of clover fermion action. The mass function has qualitatively same behavior for the two cases. However, the mass function based on the Wilson fermion action overestimates that on the clover fermion action by about 20%.

Summary
We study the finite temperature quark propagator by using the SU (3) quenched lattice simulation in the Landau gauge. We have focused an attention on the quark mass function. First, the quark mass function of confinement phase is different from that of deconfinement phase at low momentum region. It suggest that in QGP quark behaves on a single particle. Second, the quark thermal mass is shown as a linear function of temperature. Third, the mass function based on clover fermion action is qualitatively same behavior with and quantitatively different behavior from that on Wilson fermion action.

Acknowledgements
The simulation has been done on a supercomputer (NEC SX-5) at Research Center for Nuclear Physics, Osaka University. This work is partially supported by Grant-in-Aid for Scientific Research by Monbu-Kagakusho, No. 13135216, 17340080 and 18540280.