Momentum Distribution Functions in Quark-Gluon Plasma

Based on the constituent quasiparticle model of the quark-gluon plasma (QGP), the Wigner function is presented in the form of a color path integral. The Monte Carlo calculations of the quark and gluon densities, pair correlation functions and the momentum distribution functions for strongly coupled QGP plasma in thermal equilibrium at barion chemical potential equal to zero have been carried out. Analysis of the pair correlation functions points out on arising glueballs and related gluon bound states. Comparison results between the momentum distribution functions and Maxwell-Boltzmann distributions show the significant influence of the interparticle interaction on the high energy asymptotics of the momentum distribution functions resulting in the appearance of quantum “tails”.


Introduction
Studying the quark-gluon plasma (QGP) is nowadays one of the most important goals in high-energy physics. In recent years, experiments at the Relativistic Heavy-Ion Collider (RHIC) at Brookhaven National Laboratory [1] and the Large Hadron Collider (LHC) at CERN have provided a wealth of data. The most striking result, obtained from analysis of these experimental data [1] [2], is that the deconfined quark-gluon matter behaves as an almost perfect fluid rather than a perfect gas, as it could be expected from the asymptotic freedom.
The most fundamental way to compute the properties of strongly interacting matter is provided by the lattice quantum chromodynamics [2] [3] [4]. However interpretation of these very complicated numerical computations requires the application of various quantum chromodynamics (QCD) motivated, albeit sche-matic, models simulating various aspects of the full theory and allowing for a deeper physical understanding. Moreover, such models are needed in cases when the lattice QCD fails, e.g. at large quark chemical potentials and out of equilibrium. It is, therefore, crucial to devise reliable and manageable theoretical tools for a quantitative description of non-Abelian QGP both in and out of equilibrium.
Kinetic theory for the QGP can be formulated in two ways, namely: the color degrees of freedom are treated quantum mechanically and the distribution function of plasma constituents is a matrix in color space; in the second approach, on the other hand, the color may be considered as a continuous classical variable [5]. Here we use the latter approach that describes a particle carrying a classical color charge interacting with the chromodynamic field. The used approach is based on a quasiparticle picture and is motivated by the expectation that the main features of non-Abelian plasmas can be understood in simple semi-classical terms without the difficulties inherent to a full quantum field-theoretical analysis.
In this work, we extend the previous classical nonrelativistic simulations [1] to take into account quantum and spin effects. This is done in the frame of quan-

Basics of the Model
The basic assumptions of the model are similar to those of Ref. [6] [7]: 1) Quasiparticles masses (m) are of order or higher than the mean kinetic energy per particle. This assumption is based on the analysis of QCD lattice data [8] [9] [10]. For instance, at zero net-baryon density it amounts to m T , where T is a temperature.
2) In view of the first assumption, interparticle interaction is dominated by a color-electric Coulomb potential. Magnetic effects are neglected as subleading ones.
3) Relying on the fact that the color representations are large, the color operators are substituted by their average values, i.e. by Wong's classical color vectors [eight-dimensional (8D) in SU(3)] with the quadratic and cubic Casimir conditions [11]. 4) We consider the 3-flavor quark model. For the sake of simplicity we assume the masses of "up", "down" and "strange" quarks to be equal. As for the gluon quasiparticles, we allow their mass to be different (heavier) from that of quarks. Thus, this model requires the following quantities as a function of tempera- Input quantities should be deduced from lattice QCD data or from an appropriate model simulating these data. Applicability of such approach was discussed in Refs. [5] [6] in detail. Our approach differs from that of Ref. [5] [6] by a quantum treatment to quasiparticles instead of the classical one.
We consider a multi-component QGP consisting of N  color quasiparticles: g N gluons, q N quarks and q N antiquarks. The Hamiltonian of this system is ˆˆˆC H K U = + with the kinetic and color Coulomb interaction parts Here i and j summations run over quark and gluon quasiparticles, i µ are their chemical potentials, , 1, , In Equation (2)  ; , where x, σ and Q denote the multi-dimensional vectors related to spatial, spin N quasiparticles with related flavor indexes respectively. The σ summation and spacial (

The Wigner Function for Canonical Ensemble
The Wigner function of the multiparticle system in canonical ensemble is defined as the Fourier transform of the off-diagonal matrix element of density matrix in coordinate representation [13] where the Weyl's symbol of operator Â is: Weyl's symbols for usual operators like

Path Integral Representation of the Matrix Elements of the Density Matrix Operator
The exact density matrix 1 For the sake of notation convenience, we ascribe superscript (0) to the original variables.
, spin gives rise to the spin part of the density matrix (S) with exchange effects accounted for by the permutation operators ˆq P , ˆq P and ˆg P acting on the quark, antiquark and gluon degrees of freedom and the spin projections σ ′ . The sum runs over all permutations. In Equation (7) q P κ and q P κ are permutation parity, while is the off-diagonal element of the density matrix. Since the color charge is treated classically, we keep only diagonal terms grees of freedom. Accordingly each quasiparticle is represented by a set of coor- ("beads") and an 8-dimensional color vector ( ) group. Thus, all "beads" of each quasiparticle are characterized by the same spin projection, flavor and color charge. Notice that masses and coupling are the same as those for the original quasiparticles, i.e. these are still defined by the actual temperature T.
The main advantage of decomposition (7) is that it allows us to use perturbation theory to obtain approximation for density matrices ( ) m ρ , which is applicable due to smallness of artificially introduced factor ( ) should be calculated with the accuracy of order of ( ) θ > , as in this case the error of the whole product in the limit of large M will be equal to zero. In the limit ( ) 1 M + → ∞ l ρ can be approximated by a product of two-particle density matrices ( ) , m i j ρ . Generalizing the electrodynamic plasma results [7] to the quark-gluon plasma case, we write approximate ρ  In Equation (9) the effective total color interaction energy is described in terms of the off-diagonal elements of the effective potential approximated by the diagonal ones by means of Here the diagonal two-particle effective quantum Kelbg potential is Other quantities in Equation (9) are defined as follows: being a thermal wavelength of an a type quasiparticle ( , , a q q g = ). The antisymmetrization and symmetrization takes into account quantum statistics and results in appearing permanent for gluons and determinants for quarks/antiquarks.
defined by modified Bessel functions. Gluon matrix elements are The coordinates of the quasiparticle ''beads" ( ) and vectors between neighboring beads of an i particle, defined as (7).
In the limit of M → ∞ functions ( ) m ii φ describe the new relativistic measure of developed color path integrals. This measure is created by relativistic operator of kinetic energy ( ) 2 2 , q K p m T µ = + . Let us note that in the limit of large particle mass the relativistic measure coincide with the Gaussian one used in Feynman and Wiener path integrals.
According to Equation (4) the antisymmetrized Wigner function can be written as the Fourier transform of the off-diagonal matrix element of the density matrix: where ( ) ( ) In Equation (12) E is the unit matrix, while the matrix presenting permutation ˆˆq q g P P P is equal to unit matrix with appropriately transposed columns.

Harmonic and Linear Approximation for the Wigner Function
The expression (12) Here ˆˆˆq qg q q g P P P P = . The second term means scalar product of the vector related to ξ combination with the multidimensional gradient of pseudopotential, while the third term means quadratic form with matrix of the second derivatives. x To test the developed approximations calculations of thermodynamic values and the ground state wave functions for quantum particle in 1D and 3D potential field, which strongly differs from harmonic one have been carried out in [20] [21]. The used approximation gives practically exact results even for potentials, which have no matter with harmonic potential.
In degenerate system, average distance between fermions is less than the thermal wavelength λ and virtual trajectories in path integrals (15) are strongly entangled. This is the reason why permutations cannot significantly affect the potential energy in (15) in comparison with the case of the identical permutation.
So we can replace permutation in the potential energy in (15) by the identical permutation. Now all permutations in (15) are acting only on variables x and ξ and can be taken out of the path integral.
As was done in [25] for electromagnetic plasma it is enough for our purpose to take into account the pair permutations. These permutations cannot significantly affect the potential energy in (15)    q l q t q l q t q q g g N l q t q l q t q f f l q t q l t N l g t g l g t g l g t g l t To regularize integration over momenta and to avoid difficulties arising at Monte Carlo simulations due to the presence of delta-function in expression (16) let us consider the positive Husimi distributions being a coarse-graining Wigner function with a Gaussian smoothing [14]. So the Wigner function has to be averaged over arbitrary "small" phase space cell.
where the final expression for the phase space pair pseudopotentials accounting for the quantum statistical effects look like: Journal of Applied Mathematics and Physics ( ) To extent the region of applicability of obtained phase space pair pseudopotential the 2 p ∆  and 2 Q ∆  can be considered as a fit function small in comparison with unity. Our test calculations [25] have shown that the best fit for 2 p ∆  can be written in the form At the second stage, the fixed number of quarks, antiquarks, and gluons is chosen to be equal to the obtained at the first stage average values of quasiparticles and calculations are performed in the canonical ensemble. However now according to (17) it is necessary do to integration in the color phase space and beside the second and third types of elementary step described above it is necessary to introduce new type of Markovian elementary step changing the momentum of quasiparticle [25].
The input parameters of the model should be deduced from the lattice QCD data or can be taken as HTL values of g m and q m . In the present simulations we take only a possible set of parameters [7]: It is also reasonable to assume that 2 g is a function of this single variable g z . This choice is done because 2 g like gluons is related to the whole system rather than one specific quark flavor. Then we can use the same "one-loop analytic coupling" and substitute Q by 2π g z to use this coupling in our simulations.

The QGP Mometum Distribution Functions
Here we present preliminary results of our QGP simulations. Figure 1 shows dependences of the quark and gluon densities on temperature at barion chemical potential equal to zero ( q q n n = ) obtained according to the Equation (2) at the first stage of Monte Carlo simulation in grand canonical ensemble [7]. Journal of Applied Mathematics and Physics where i a a and j b b are types of the particles ( , q q = or g). The PDF gives a probability density to find a pair of particles of types a and b at a certain distance  R g R related to gluons has maximum (not shown here) [7]. Glueballs and strongly correlated pairs of the other quasiparticles are uniformly distributed in space. The last conclusion comes from the fact that the gg, qq and qq PDF's at distance larger than 0.7 fm are practically equal to unity. Possible existence of the medium-modified bound states was actively discussed some time ago, e.g., in [26] and later in [27] where , , a q q g = .
The non ideal classical systems of particles due to the commutativity of the kinetic and potential energy operators have Maxwell distribution function (MD) proportional to ( ) ( ) 2 2 exp 4π a pλ −  even at strong coupling. Here at Figure 3 and Figure 4 we present results for both ideal QGP plasmas and results for  The momentum distribution functions for quarks and antiquarks practically coincide with each other (as example, the solid and dash-dot-dot lines 1).
Quantum effects can affect the shape of kinetic energy distribution function.
Quantum ideal systems of particles due to the quantum statistics have Fermi or Bose momentum distribution functions. In addition interaction of a quantum particle with its surroundings restricts the volume of configuration space availa-Journal of Applied Mathematics and Physics ble for particles, which, can also affect the shape of momentum distribution function due to the uncertainty relation, i.e., in a rise in the fraction of particles with higher momenta [30].
Solid lines on Figure 3 and Figure 4 present the Monte Carlo results supporting these phenomena for non ideal QGP plasma. Difference in momentum distribution functions for both ideal and non ideal QGP plasma is decreasing with increasing temperature and lowering coupling plasma parameter. As before the main physical reason responsible for difference in behaviour of momentum distribution function of quarks and gluons is that the quadratic Casimir values responsible for interparticle interaction is significantly larger for gluons in comparison to quarks.
The peculiarities in asymptotic region of the quark momentum distribution function relates in arising out "quantum tails" due to the uncertainty relation as we have mentioned above. High energy quantum "tail" (the solid line 4) is approximated by sum of the Maxwell distributions and product of 8 const p and the Maxwell distributions with effective temperature that exceeds the temperature of medium [31]. So in this approximation the two fit constants are made use of.
To verify the relevance of all above discussed trends, a more refined color-, flavor-, spin-resolving analysis of the distribution functions is necessary. This work is presently in progress.

Conclusion
In