Non-Abelian kinetic effects are taken into account for the study of the relaxation of collective motion in a quark-gluon plasma(QGP). An explicit Compton scattering is considered to calculate the relaxation time. It is shown that the new non-Abelian relaxation has a physical mechanism from the non-linear dynamics and the meaning of the relaxation is discussed. PACS number: 12.38.Mh The quark-gluon plasma(QGP) has been predicted to be produced in high energy nuclear collisions. The properties of QGP are of importance for understanding experimental results[1]. Collective motion plays an importance role in a plasma. Remarkably, the collective behaviors of a QGP have difference from a electromagnetic plasma due to color degrees of freedom[2]. It means that some new transport problems emerge, which is associated with color degrees of freedom[3, 4]. Obviously, the relaxation processes from collisional term used to be easily realized since we are familiar with the old knowledge. The collisional terms of QGP thus were given by Selikhov and Gyulassay[5] under the consideration of the quantum fluctuations and the color relaxation from the color collisional term which is only relevant to color degrees of freedom was studied as well as momentum relaxation[6]. However, we can’t help but ask such questions: Is there a relaxation related to both color and momentum Supported by the National Natural Science Foundation of China Grant No. 19805003 Email address space? If there is, what is the relaxation? In this letter, we will try to solve this problem. Firstly, we need to review the development of QGP kinetic-theory approach. It is wellknown that kinetic theory[7] is believed to describe correctly the quark-gluon plasma(QGP) physics as well as temperature field[8] and kinetic-theory approach to transport coefficients is of advantage of convenience. In fact, more and more attention is recently concentrated on the applications of kinetic theory to hot QCD[6, 9]. However, the treatment of nonAbelian counterparts in kinetic equations had ever been a difficult task[10, 11] for long time. We just believed that the solution of the problem had been made improvement until the non-Abelian mean-field dynamics[11] and ’double perturbation’ approach[12] were presented recently. These progresses are also of importance for understanding color relaxation physics. We roughly give the analysis before new physics are studied in detail: Apart from collisional terms, non-Abelian covariant derivative enters the formalism of the kinetic equations. This includes color self-coupling contribution in the covariant derivative. Here it is clear that the self-coupling term represents the collective behaviors due to color degrees of freedom instead of dynamic behaviors described in collisional term in color space. Then we think that a relaxation process from the non-linear dynamics shall also be produced even if for a collisionless QGP. Now we start our studies from the kinetic equations of a collisionless QGP[13], pDμQ±(p, x) ± g 2 p∂ p{Fμν(x), Q±(p, x)} = 0, (1) pD̃μG(p, x) + g 2 p∂ p{F̃μν(x), G(p, x)} = 0, (2) where the letters with ̃ represent the corresponding operators in adjoint representation of SU(3). Here we consider the density fluctuations ∆Q±, ∆G deviating from equilibrium distribution functions Q (0) ± and G (0) and replace the induced field A by a. Assuming the fluctuations to be weak, p ∼ gT, aμ ∼ T, i∂μ ∼ gT . Thus we write the equations for the fluctuations from ’double perturbation’ approach[12] p∂μ∆Q± + ig ∑ λ p[aμ, ∆Q (λ) ± ] ± gp fμν∂ ν p Q (0) ± = 0 (3)

At sufficiently high temperature and density, quantum chromodynamics (QCD) predicts phase transition from the hadronic phase to the quark-gluon plasma phase. Lattice QCD is the most useful tool to investigate this critical phenomenon, which status is briefly reviewed. The usual problem in the Lagrangian formulation at finite density is either an incorrect continuum limit or its complex action and a premature onset of the transition as the chemical potential is raised. We show how the difficulties are overcome in our Hamiltonian approach.

The theoretical renormalization-group approach is applied to the study of the short-time critical behavior of the Ginzburg-Landau model with long-range random impurities which have a power-like correlation r(-(d-p)). The system initially at a high temperature is firstly quenched to the critical temperature T-c and then released to an evolution with a model A dynamics. The asymptotic scaling laws are studied in the frame of a double expansion in epsilon = 4 - d and delta = epsilon + p with delta of the order of epsilon. The initial slip exponent theta' for the order parameter is calculated up to the first order in epsilon = 4 - d or in epsilon (1/2) corresponding to different fixed points, respectively. For d < 4, the short-time behavior of the order parameter is investigated in one loop. Two different logarithmic and exponential-logarithmic corrections to short-time behaviors of both the autocorrelation and the order parameter are also solved in d = 4 dimension. Crossover between the nonrandom behavior and quenched behavior is found for n = 4 (where n is the spin dimensionality) in d = 4 dimension.

The theoretic renormalization-group approach is applied to the study of short-time dynamics of the d-dimensional n-component spin systems with long-range interactions r -(d+σ) and quenched disorder which has long-range correlations r -(d-ρ) . Asymptotic scaling laws are obtained in a frame of double expansions in ∊=2σ-d and ρ with ρ of the order ∊. The static exponents are obtained exactly to all the order. The initial slip exponents θ′ for the order parameter and θ for the response function, as well as the dynamic exponent z, are calculated upto the first order in ∊. In d=2σ, in contrast to the unique logarithmic decay in the long-time regime which does not depend on σ, ρ, n and the disorder, we find rich scaling structures including logarithmic and exponential-logarithmic scalings in the short-time regime. Non-universal critical scalings of Ising systems are also discussed for d=2σ.

The theoretic renormalization group approach is applied to the study of short-time critical behavior of the Ginzburg–Landau model with weakly long-range interactions . The system initially at a high temperature is firstly quenched to the critical temperature and then released to an evolution with a model A dynamics. A double expansion in and with of order is employed, where is the spatial dimension. The asymptotic scaling laws and the initial slip exponents and for the order parameter and the response function respectively are calculated to the second order in for close to 2.

The random phase approximation is applied to the coupled-cluster expansions of lattice gauge theory (LGT). Using this method, wavefunctions are approximated by linear combination of graphs consisting of only one connected Wilson loop. We study the excited state energy and wavefunction in (2+1)-D SU(3) LGT up to the third order. The glueball mass shows a good scaling behavior.

An improved lattice Hamiltonian is applied to the SU(3) lattice gauge theory in 2+1 dimensions. The glueball masses of 0(+) and 0(-) have been calculated by using the coupled-clusters expansion with the truncated eigenvalue equations in a scheme preserving the continuum limit. The calculations up to third-order are carried out, and the results show better scaling behavior than that given by non-improved one.

At sufficiently high temperature and density, quantum chromodynamics (QCD) is expected to undergo a phase transition from the confined phase to the quark-gluon plasma phase. In the Lagrangian lattice formulation the Monte Carlo method works well for QCD at finite temperature; however, it breaks down at finite chemical potential. We develop a Hamiltonian approach to lattice QCD at finite chemical potential and solve it in the case of free quarks and in the strong coupling limit. At zero temperature, we calculate the vacuum energy, chiral condensate, quark number density and its susceptibility, as well as mass of the pseudoscalar, vector mesons and nucleon. We find that the chiral phase transition is of first order, and the critical chemical potential is mu(C) = m(dyn)((0)) (dynamical quark mass at mu = 0). This is consistent with mu(C) approximate to M-N((0))/3 (where M-N((0)) is the nucleon mass at mu = 0).

The kinetic spherical model with long-ranged interactions and an arbitrary initial order m0 quenched from a very high temperature to T(≤ Tc) is solved. In the short-time regime, the bulk order increases with a power law in both the critical and phase-ordering dynamics. To the latter dynamics, a power law for the relative order \({m_r} \sim {t^{ - k}}\) is found in the intermediate time-regime. The short-time scaling relations of small m0 are generalized to an arbitrary m0 and all the time larger than tmic. The characteristic functions \(\varphi (b,{m_0})\) for the scaling of m0 and ε(b,T′) for TT′ = T/Tc are obtained. The crossover between scaling regimes is discussed in detail.

The self-charging model of two-dimensional Josephson-junction arrays. at T = 0 is studied by the coupled cluster expansion method. The calculated results for the mass gap converge nicely. Besides the critical point, we also determine the critical parameters at T = 0. The system displays a second-order phase transition and the results are consistent with those obtained by other methods.

The truncated eigenvalue equation of SU(N) lattice gauge theory is studied by using improved lattice gauge Hamiltonian with a proper truncation scheme that preserves the continuum limit. The calculations of vacuum state wavefunction and glueball mass of (2+1)-dimensional SU(2) theory up to third order are carried out, the results show the improvement of scaling behavior in deep weak coupling region.

The renormalisation group approach is applied to the study of the short-time critical behaviour of the d-dimensional Ginzburg-Landau model with long-range interaction of the form in momentum space. Firstly the system is quenched from a high temperature to the critical temperature and then relaxes to equilibrium within the model A dynamics. The asymptotic scaling laws and the initial slip exponents and of the order parameter and the response function respectively, are calculated to the second order in .

The duality transformation is carried out for an n-species Ising spin system interacting with Z(2) gauge fields. For n > 1, we find that the dual model has topological terms when the surface is topologically nontrivial. The plaquette interaction of the gauge fields is dual to an n-spin coupling in the dual model.

The truncated eigenvalue equation of SU(N) lattice gauge theory is studied by using improved lattice gauge Hamiltonian with a proper truncation scheme that preserves the continuum limit. The calculations of vacuum state wavefunction and glueball mass of (2+1)-dimensional SU(2) theory up to third order sere carried out, the results show the improvement of scaling behavior in deep weak coupling region.

Using coupled cluster method, we calculate the vacuum wave function and the mass gaps of (2 + 1)-dimensional U(1) lattice gauge theory with improved Hamiltonian up to the seventh order. The results are compared with those from the unimproved Hamiltonian.

Non-Abelian kinetic effects are taken into account for the study of the relaxation of collective motion in a quark-gluon plasma(QGP). An explicit Compton scattering is considered to calculate the relaxation time. It is shown that the new non-Abelian relaxation has a physical mechanism from the non-linear dynamics and the meaning of the relaxation is discussed.

We study QCD in 2 dimensions using the improved lattice fermionic Hamiltonian proposed by Luo, Chen, Xu and Jiang. The vector mass and the chiral condensate are computed for various SU(Nc) gauge groups. We do observe considerable improvement in comparison with the Wilson quark case.

We present an effective relaxation-time theory to study the collisionless quark-gluon plasma. Applying this method we calculate the damping rate to be of order $g^2T$ and find plasmon scattering is the damping mechanism. The damping for the transverse mode is stronger than the longitudinal one.