^{1}

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^{1}

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”.

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 [

The most fundamental way to compute the properties of strongly interacting matter is provided by the lattice quantum chromodynamics [

In this work, we extend the previous classical nonrelativistic simulations [

The basic assumptions of the model are similar to those of Ref. [

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 [

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 [

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 temperature (T) and quark chemical potential ( μ q ) as an input:

a) quasiparticle masses, for quarks m q and gluons m g , and

b) the coupling constant g 2 , or α s = g 2 / 4 π .

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. [

We consider a multi-component QGP consisting of N ˜ color quasiparticles: N g gluons, N q quarks and N ¯ q antiquarks. The Hamiltonian of this system is H ^ = K ^ + U ^ C with the kinetic and color Coulomb interaction parts

K ^ = ∑ i p ^ i 2 + m i 2 ( T , μ i ) , U ^ C = 1 2 ∑ i ≠ j g 2 ( T , μ i ) ( Q i ⋅ Q j ) 4 π | x i − x j | . (1)

Here i and j summations run over quark and gluon quasiparticles, μ i are their chemical potentials, i , j = 1 , ⋯ , N ˜ , N ˜ = N q + N ¯ q + N g , N q = N u + N d + N s and N ¯ q = N ¯ u + N ¯ d + N ¯ s are total numbers of quarks and antiquarks of all flavours (up, down and strange), respectively, 3D vectors x i are quasiparticle dimensionless spatial coordinates, g 2 ( T , μ i ) / 4 π is coupling constant, the Q i denote the Wong’s quasiparticle color variable (8D-vector in the group S U ( 3 ) ), ( Q i ⋅ Q j ) denote scalar product of color vectors. Nonrelativistic approximation for potential energy is used, while for kinetic energy we still keep relativistic form as the quasiparticle masses are not negligible as compared with temperature. The eigenvalue equation of this Hamiltonian is usually called the “spinless Salpeter equation”.

The grand canonical ensemble with given temperature, net-quark-number ( μ q ) and strange ( μ s ) chemical potentials, and fixed volume V are fully described by the grand partition function

Z ( μ q , μ s , β , V ) = ∑ { N } exp { μ q ( N q − N ¯ q ) / T } exp { μ s ( N s − N ¯ s ) / T } N u ! N d ! N s ! N ¯ u ! N ¯ d ! N ¯ s ! N g ! Z ( { N } , V , β ) . (2)

Here { N } = { N u , N d , N s , N ¯ u , N ¯ d , N ¯ s , N g } . In Equation (2) we explicitly wrote sum over different quark flavors (u,d,s). Below the sum over quark degrees of freedom is understood in the same way. Usual choice of the strange chemical potential is μ s = − μ q (nonstrange matter), such that the total factor in front of ( N s − N ¯ s ) is zero. Therefore, below we omit μ s from the list of variables.

The partition function in canonical ensemble Z ( { N } , V , β ) and the related thermodynamic properties of many particle systems of particles are defined by the diagonal matrix elements of the density-matrix operator ρ ^ = exp ( − β H ^ )

Z ( { N } , V , β ) = ∑ σ , σ ′ ∫ d x d x ′ d μ Q d μ Q ′ δ σ , σ ′ δ ( x − x ′ ) δ ( Q − Q ′ ) 〈 x | e − β H ^ ( Q ) | x ′ 〉 = ∑ σ ∫ d x d μ Q ρ ( x , Q , σ ; { N } ; β ) , (3)

where x, σ and Q denote the multi-dimensional vectors related to spatial, spin and color degrees of freedom of { N } quasiparticles with related flavor indexes respectively. The σ summation and spacial ( d x ≡ d 3 x 1 ⋯ d 3 x N ) and color ( d μ Q ≡ d μ Q 1 ⋯ d μ Q N ) integrations run over all individual degrees of freedom of the particles, d μ Q i denotes integration over SU(3) group Haar measure [

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 [

W ( p , x , Q ; { N } ; β ) = Z ( { N } , V , β ) − 1 ∑ σ ∫ d ξ e i 〈 p | ξ 〉 〈 x − ξ / 2 | e − β H ^ ( Q ) | x + ξ / 2 〉 , (4)

To avoid discussed in literature problems with definition of the relativistic Wigner function [

Further we are going to obtain a new representation of the Wigner functions in the path integral form in the color phase space, which allows the numerical Monte Carlo simulations of the strongly coupled quantum systems of particles in canonical ensemble [

〈 A ^ 〉 = ∫ d μ Q d p d x ( 2 π ) 6 N ˜ A ( p , x , Q ) W ( p , x , Q ; { N } ; β ) , (5)

where the Weyl’s symbol of operator A ^ is:

A ( p , x , Q ) = ∫ d ξ e − i 〈 ξ | p 〉 / ℏ 〈 x − ξ / 2 | A ^ ( Q ) | x + ξ / 2 〉 . (6)

Weyl’s symbols for usual operators like p ^ , x ^ , p ^ 2 , x ^ 2 , H ^ , H ^ 2 etc. can be easily calculated directly from definition (6).

^{1}For the sake of notation convenience, we ascribe superscript ^{(0)} to the original variables.

The exact density matrix ρ = e − β H ^ of interacting quantum systems can be constructed using a path integral approach [^{1} the integral in Equation (3) as follows

∑ σ ∫ d x d μ Q ρ ( x , Q , σ ; { N } ; β ) = ∑ σ ∫ d x d μ Q ∫ d x ( 1 ) d μ Q ( 1 ) ⋯ d x ( M − 1 ) d μ Q ( M − 1 ) ρ ( 1 ) ⋅ ρ ( 2 ) ⋯ ρ ( M − 1 ) × ∑ P q ∑ P q ¯ ∑ P g ( − 1 ) κ P q + κ P q ¯ ∑ σ ′ S ( σ , P q q ¯ g σ ′ ) δ σ ′ , σ × ∫ d x ( M ) d μ Q ( M ) δ ( x − P q q ¯ g x ( M ) ) δ ( Q − P q q ¯ g Q ( M ) ) ρ ( M ) (7)

where x ≡ x ( 0 ) , Q ≡ Q ( 0 ) , spin gives rise to the spin part of the density matrix (S) with exchange effects accounted for by the permutation operators P ^ q , P ^ q ¯ and P ^ g acting on the quark, antiquark and gluon degrees of freedom and the spin projections σ ′ . The sum runs over all permutations. In Equation (7) κ P q and κ P q ¯ are permutation parity, while

ρ ( m ) ≡ ρ ( x ( m − 1 ) , Q ( m − 1 ) ; x ( m ) , Q ( m ) ; { N } ; ε β ) = 〈 x ( m − 1 ) | e − ε H ^ | x ( m ) 〉 δ ( Q ( m − 1 ) − Q ( m ) ) , (8)

is the off-diagonal element of the density matrix. Since the color charge is treated classically, we keep only diagonal terms ( δ ( Q ( m − 1 ) − Q ( m ) ) ) in color degrees of freedom. Accordingly each quasiparticle is represented by a set of coordinates { x i ( 0 ) , ⋯ , x i ( M ) } (“beads”) and an 8-dimensional color vector Q i ( 0 ) in the S U ( 3 ) group. Thus, all “beads” of each quasiparticle are characterized by the same spin projection, flavor and color charge. Notice that masses and coupling constant in each ρ ( m ) 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 ρ ( m ) , which is applicable due to smallness of artificially introduced factor 1 / ( M + 1 ) . Each factor ρ ( m ) should be calculated with the accuracy of order of 1 / ( M + 1 ) θ with θ > 1 , as in this case the error of the whole product in the limit of large M will be equal to zero. In the limit ( M + 1 ) → ∞ ρ l can be approximated by a product of two-particle density matrices ρ i , j ( m ) . Generalizing the electrodynamic plasma results [

ρ ˜ ( x , x ( 1 ) , ⋯ , x ( M ) ; Q ; { N } ; β ) = exp { − β U } per ‖ ϕ _ ˜ ( M ) , ( 0 ) ‖ N g λ ˜ g 3 N g det ‖ ϕ ˜ ( M ) , ( 0 ) ‖ N q λ ˜ q 3 N q det ‖ ϕ ˜ ( M ) , ( 0 ) ‖ N q λ ˜ q ¯ 3 N ¯ q ∏ l = 1 M ∏ i = 1 N ϕ i i ( m ) . (9)

In Equation (9) the effective total color interaction energy

U = 1 M + 1 ∑ l = 1 M + 1 1 2 ∑ i , j ( i ≠ j ) N Φ i , j ( | x i ( m − 1 ) − x j ( m − 1 ) | , | x i ( m ) − x j ( m ) | , Q i , Q j ) (10)

is described in terms of the off-diagonal elements of the effective potential approximated by the diagonal ones by means of

ε Φ i , j ( x i , j ( m − 1 ) , x i , j ( m ) ; , Q i , Q j ) ≈ ε [ Φ i , j ( x i , j ( m − 1 ) , x i , j ( m − 1 ) , Q i , Q j ) + Φ i , j ( x i , j ( m ) , x i , j ( m ) , Q i , Q j ) ] / 2 ∝ − ln ( ρ i , j ( m ) ) . Here the diagonal two-particle effective quantum Kelbg potential is

Φ i , j ( x i , j ( m ) , x i , j ( m ) , Q i , Q j , ε ) = g 2 ( T , μ q ) 〈 Q i | Q j 〉 4 π | x i ( m ) − x j ( m ) | { 1 − e − ( x i , j ( m ) ) 2 + π x i , j ( m ) [ 1 − erf ( x i , j ( m ) ) ] } , (11)

with x i , j ( m ) = | x i ( m ) − x j ( m ) | / Δ λ i , j , Δ λ i , j = 2 π ε / m i , j , m i , j = m i m j / ( m i + m j ) Other quantities in Equation (9) are defined as follows: λ ˜ a 3 = λ a 3 0.5 π / ( β m ) 5 with

λ a = 2 π β / m a 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.

Functions ϕ i , i ( m ) ≡ K 2 ( z i ( m ) ) / ( z ( m ) ) i 2 ( z i ( m ) = ε m i ( T , μ q ) 1 + | η i ( m ) | 2 / ε 2 are defined by modified Bessel functions. Gluon matrix elements are ϕ _ ˜ i , j ( M ) , ( 0 ) = K 2 ( z i , j ( M ) , ( 0 ) ) / ( z i , j ( M ) , ( 0 ) ) 2 δ ( Q i − Q j ) , while quark and antiquark matrix elements ϕ ˜ i , j ( M ) , ( 0 ) = K 2 ( z i , j ( M ) , ( 0 ) ) / ( z i , j ( M ) , ( 0 ) ) 2 δ ( Q i − Q j ) δ f i , f j δ σ i , σ j depend additionally on spin variables σ i and flavor index f i of the particle, which can take values “up”, “down” and “strange”, δ σ i , σ j and δ f i , f j are the Kronecker’s deltas. These functions allow to exclude the Pauli blocking for particles with different spins, flavors and colors. Here arguments of modified Bessel functions are z i , j ( M ) , ( 0 ) = ε m i ( T , μ q ) 1 + | x i ( 0 ) − x j ( M ) | 2 / ε 2 .

The coordinates of the quasiparticle ‘‘beads” x i ( m ) = x i ( 0 ) + y i ( m ) , ( l > 0 ) are expressed in terms of x i ( 0 ) and vectors between neighboring beads of an i particle, defined as y i ( m ) = ∑ k = 1 l η i ( k ) , while η i ( 1 ) , ⋯ , η i ( M ) are vector variable of integration in Equation (7).

In the limit of M → ∞ functions ϕ i i ( m ) describe the new relativistic measure of developed color path integrals. This measure is created by relativistic operator of kinetic energy K = p 2 + m 2 ( T , μ q ) . 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:

W ( p , x , Q ; { N } ; β ) = C ( M ) Z ( { N } , V , β ) ∑ σ ∑ P ^ q P ^ q ¯ P ^ g ( − 1 ) κ P ^ q + κ P ^ q ¯ S ( σ , P ^ q q ¯ g σ ′ ) | σ ′ = σ ∫ d μ Q ( m ) δ ( Q − P q q ¯ g Q ( m ) ) × ∫ d ξ ∫ d q ( 1 ) ⋯ d q ( M − 1 ) d μ Q exp { − π 〈 ξ | P ^ q q ¯ g + E | ξ 〉 2 M + i 〈 ξ | p 〉 − π | P ^ q q ¯ g x − x | 2 M − ∑ m = 0 M − 1 [ π | q ( m ) − q ( m + 1 ) | 2 + ε U ( ( P ^ q q ¯ g x − x ) m M + x + q ( m ) − ( M − m ) ξ 2 M + m P ^ q q ¯ g ξ 2 M ) ] } (12)

where C ( M ) = M 6 N ˜ ( M − 1 ) / 2 is constant and q ( M ) = q ( 0 ) . The antisymmetrization for quarks and symmetrization for gluons takes into account quantum statistics. Here we have replaced variables of integration x ( m ) for any given permutation P ^ q P ^ q ¯ P ^ g by relation

x ( m ) = ( P ^ q P ^ q ¯ P ^ g x − x ) m M + x + q ( m ) − ( M − m ) ξ 2 M + m P ^ q q ¯ g ξ 2 M . (13)

In Equation (12) E is the unit matrix, while the matrix presenting permutation P ^ q P ^ q ¯ P ^ g is equal to unit matrix with appropriately transposed columns.

The expression (12) for the Wigner function contains complex-valued function and so is inconvenient for Monte Carlo simulations, usually making use real valued functions. To overcome this difficulty we have to get explicit analytical real valued expression for the Wigner function. This can be done for free particles ( U ( x ) ≡ 0 ) and linear or harmonic potentials U ( x ) . For general U ( x ) this integral cannot be analytically calculated but proper approximation of the Wigner function for any potential U ( x ) can be obtained from the Taylor expansion up to the first or second order in the variables ξ :

U ( ( P ^ q P ^ q ¯ P ^ g x − x ) m M + x + q ( m ) − ( M − m ) ξ 2 M + m P ^ q P ^ q ¯ P ^ g ξ 2 M ) ≈ U ( ( P ^ q q ¯ g x − x ) m M + x + q ( m ) ) − 〈 ( M − m ) ξ 2 M − m P ^ q q ¯ g ξ 2 M | ∂ U ( ( P ^ q q ¯ g x − x ) m M + x + q ( m ) ) ∂ x 〉 + 1 2 〈 ( M − m ) ξ 2 M − m P ^ q q ¯ g ξ 2 M | ∂ 2 U ( ( P ^ q q ¯ g x − x ) m M + x + q ( m ) ) ∂ x 2 | ( M − m ) ξ 2 M − m P ^ q P ^ q ¯ P ^ g ξ 2 M 〉 . (14)

Here P ^ q q ¯ g = P ^ q P ^ q ¯ P ^ g . 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.

This approximation for the Wigner function takes the form of the gaussian integral and can be calculated analytically. Here for simplicity let us consider expressions related to the linear approximation accounting for the linear term in this expansion. It is straightforward to account for the quadratic term. In the linear approximation, the Wigner function can be written in the form:

W ( p , x , Q ; { N } ; β ) = C ( M ) Z ( { N } , V , β ) ∑ σ ∑ P ^ q q ¯ g ( ± 1 ) κ P ^ q + κ P ^ q ¯ S ( σ , P ^ q q ¯ g σ ′ ) | σ ′ = σ × ∫ d μ Q ( M ) δ ( Q − P q q ¯ g Q ( M ) ) exp { − π | P ^ q q ¯ g x − x | 2 M } ∫ d q ( 1 ) ⋯ d q ( M − 1 )

× exp { − ∑ m = 0 M − 1 [ π | q ( m ) − q ( m + 1 ) | 2 + ε U ( ( P ^ q q ¯ g x − x ) m M + x + q ( m ) ) ] } × ∫ d ξ exp { − π 〈 ξ | P ^ q q ¯ g + E | ξ 〉 2 M + 〈 ξ | i p + ∑ m = 0 M − 1 | ( M − m ) 2 M E − m 2 M P ^ q q ¯ g | ∂ ε U ( x ¯ ) ∂ x ¯ 〉 | x ¯ = ( P ^ q q ¯ g x − x ) m M + x + q ( m ) } . (15)

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 [

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 [

So the Wigner function is determined by the path integral over all closed trajectories ( q ( M ) = q ( 0 ) ) and can be presented in the form

W ( p , x , Q ; { N } ; β ) ≈ C ( M ) Z ( { N } , V , β ) ∑ σ ∫ d q ( 1 ) ⋯ d q ( M − 1 ) × exp { − ∑ m = 0 M − 1 [ π | q ( m ) − q ( m + 1 ) | 2 + ε U ( x + q ( m ) ) ] } × exp { M 4 π | i p + ε 2 ∑ m = 0 M − 1 M − 2 m M ∂ U ( x + q ( m ) ) ∂ x | 2 } × ∑ σ q { 1 − ∑ l < t N q δ σ l , q σ t , q δ f l , q f t , q δ ( ( Q l , q − Q t , q ) z ) exp ( − 2 π | x l , q − x t , q | 2 M ) δ ( ( p ˜ l , q − p ˜ t , q ) M 2π ) }

× ∑ σ q ¯ { 1 − ∑ l < t N ¯ q δ σ l , q ¯ σ t , q ¯ δ f l , q ¯ f t , q ¯ δ ( ( Q l , q ¯ − Q t , q ¯ ) z ) exp ( − 2 π | x l , q ¯ − x t , q ¯ | 2 M ) δ ( ( p ˜ l , q ¯ − p ˜ t , q ¯ ) M 2π ) } × ∑ σ g { 1 + ∑ l < t N g δ ( ( Q l , g − Q t , g ) z ) exp ( − 2 π | x l , g − x t , g | 2 M ) δ ( ( p ˜ l , g − p ˜ t , g ) M 2π ) } , (16)

where

p ˜ t , a = p t , a + ε 2 ∑ m = 0 M − 1 ∂ U ( x + q ( m ) ) ∂ x t , a .

Here in the delta function of color vectors δ ( Q i − Q j ) we have introduced a fit parameter z and δ σ i , σ j and δ f i , f j are the Kronecker’s deltas depending on spin σ i and flavor indexes f i of quasiparticle. The later can take values “up”, “down” and “strange”. These functions allow to realize the exchange interaction for particles with the same spins, flavors and colors.

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 [

W H ( p , x , Q ) = 1 ( p , x , Q ) sph ∫ − ∞ ∞ d p 1 d x 1 d Q 1 exp ( − p 1 2 Δ p 2 − x 1 2 Δ x 2 − Q 1 2 Δ Q 2 ) W ( p + p 1 , x + x 1 , Q + Q 1 ) ,

with the equivalent cell area in the color phase space

( p , x , Q ) sph = ∫ − ∞ ∞ d p 1 d x 1 d Q 1 exp ( − p 1 2 Δ p 2 − x 1 2 Δ x 2 − Q 1 2 Δ Q 2 ) .

For small Δ x 2 and Δ p 2 making use of the mean value theorem for integrals the Wigner function can be written in the form:

W H ( p , x , Q ; { N } ; β ) ≈ C ( M ) Z ( { N } , V , β ) ∑ σ ∫ d q ( 1 ) ⋯ d q ( M − 1 ) × exp { − ∑ m = 0 M − 1 [ π | q ( m ) − q ( m + 1 ) | 2 + ε U ( x + q ( m ) ) ] } × exp { M 4π | i p + ε 2 ∑ m = 0 M − 1 M − 2 m M ∂ U ( x + q ( m ) ) ∂ x | 2 } × ∑ σ exp ( − β ∑ l < t N q v l t q ) exp ( − β ∑ l < t N ¯ q v l t q ¯ ) exp ( − β ∑ l < t N g v l t g ) , (17)

where the final expression for the phase space pair pseudopotentials accounting for the quantum statistical effects look like:

v l t a = − k T ln { 1 − δ σ l , a σ t , a δ f l , a f t , a exp ( − | Q l , a − Q t , a | 2 2 Δ ˜ Q 2 ) × exp ( − 2 π | x l , a − x t , a | 2 ( 1 − Δ ˜ a , x 2 / λ a 2 1 + Δ ˜ a , x 2 / λ a 2 ) λ a 2 ) exp ( − | ( p ˜ l , a − p ˜ t , a ) | 2 λ a 2 ( 2 π ℏ ) 2 ( Δ ˜ p 2 / λ a 2 ) ) }

v l t g = − k T ln { 1 + exp ( − | Q l , g − Q t , g | 2 2 Δ ˜ Q 2 ) × exp ( − 2 π | x l , g − x t , g | 2 ( 1 − Δ ˜ g , x 2 / λ g 2 1 + Δ ˜ g , x 2 / λ g 2 ) λ g 2 ) exp ( − | ( p ˜ l , g − g ˜ t , g ) | 2 λ g 2 ( 2 π ℏ ) 2 ( Δ ˜ p 2 / λ g 2 ) ) } . (18)

Here a = q , q ¯ , Δ a , x 2 = 2 π 2 − 2 and z = 1 / 2 . To extent the region of applicability

of obtained phase space pair pseudopotential the Δ ˜ p 2 and Δ ˜ Q 2 can be considered as a fit function small in comparison with unity. Our test calculations [

The main idea of the Monte Carlo simulations of QGP consists in constructing a Markov chain of different quasiparticle states in the color configuration or color phase spaces [

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 [

The input parameters of the model should be deduced from the lattice QCD data or can be taken as HTL values of m g and m q . In the present simulations we take only a possible set of parameters [

m g 2 ( { μ q } , T ) = 1 12 ( ( 2 N c + N f ) T 2 + 3 π 2 ∑ q = u , d , s μ q 2 ) g 2 ( { μ q } , T ≫ T C ) (19)

m q 2 ( { μ q } , T ) = N g 16 N c ( T 2 + μ q 2 π 2 ) g 2 ( { μ q } , T ≫ T C ) , (20)

where N f is the number of quark flavors which can be excited, N c = 3 for SU(3) group, and g 2 is the QCD running coupling constant squared, generally depending on T and all μ q . All masses depend on combinations

z g = ( T 2 + 3 π 2 ( 2 N c + N f ) ∑ q = u , d , s μ q 2 ) 1 / 2 and z q = ( T 2 + μ q 2 π 2 ) 1 / 2 rather than on two independent variables T and μ q .

It is also reasonable to assume that g 2 is a function of this single variable z g . This choice is done because g 2 like gluons is related to the whole system rather than one specific quark flavor. Then we can use the same “one-loop analytic coupling”

α s ( Q 2 ) = 4 π 11 − ( 2 / 3 ) N f [ 1 ln ( Q 2 / Λ QCD 2 ) + Λ QCD 2 Λ QCD 2 − Q 2 ] (21)

and substitute Q by 2 π z g to use this coupling in our simulations.

Here we present preliminary results of our QGP simulations.

Making use of the quark and gluon densities (see on

First of all for physical analysis of interparticle interaction let us consider spatial arrangement of the quasiparticles in the QGP by studying a pair distribution function (PDF) g a b ( R ) defined as

g a b ( | R 1 − R 2 | ) = 1 Z ∑ σ ∑ i , j , i ≠ j δ a i , a δ b j , b N a N b V 2 ∫ d r d μ Q δ ( R 1 − r i ) δ ( R 2 − r j ) ρ ( r , Q , σ ; { N } ; β ) , (22)

where a / a i and b / b j 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 = | R 1 − R 2 | from each other. The PDF depends only on the difference of coordinates because of the translational invariance of the system. In a non-interacting classical system, g a b ( R ) ≡ 1 , whereas interactions and quantum statistics result in a redistribution of the particles. At temperatures T = 175 , 350, and 525 MeV the PDF averaged over the quasiparticle spin, colors and flavors are shown in

The quasiparticle interaction is dominated by attraction at short distances. Indeed, the QGP lowers its total energy by minimizing the color Coulomb interaction energy via a spontaneous “anti-ferromagnetic”-like ordering of color vectors, i.e. the color vectors of nearest neighbor quasiparticles become anti-parallel. This short-distance attraction is stronger for gluon-gluon and gluon-(anti)quark pairs than for (anti)quark-(anti)quark ones because of the corresponding difference in values of quadratic Casimir invariants, which determine the maximal values of the effective color charge products | Q i ⋅ Q j | in color Kelbg (Coulomb) potentials. For gluon-gluon pairs | Q g ⋅ Q g | = 24 , for gluon-(anti)quark pairs | Q g ⋅ Q q | = | Q g ⋅ Q q ¯ | ≈ 10 , and for (anti)quark-(anti)quark pairs | Q q ⋅ Q q | = | Q q ¯ ⋅ Q q ¯ | = | Q q ⋅ Q q ¯ | = 4 . Stronger gg attraction additionally enhances correlation of the gluon-gluon pairs at short distances. At the same time

the short-distance attraction is the only reason of the gluon-(anti)quark short-distance correlation.

The short-distance correlation implies formation of the bound states of the gluon clusters (glueballs), if the product R 2 g a b ( R ) related to gluons has maximum (not shown here) [

Obtained at the first time the Monte Carlo results for strongly coupled QGP momentum distribution functions have been calculated at the mentioned before second stage of simulation. From physical point of view the momentum distribution function w a ( | p | ) gives a probability density for particle of type a to have momentum p:

W a ( p a ; { N } ) = ∫ d x d p d μ Q δ ( p a − p ) W H ( p , x , Q ; { N } ; β ) (23)

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 exp ( − ( p λ a ) 2 / 4 π ℏ 2 ) even at strong coupling. Here at

strongly coupled QGP plasmas. On these figures the dotted lines present the standard Maxwell distributions, while the dashed lines present Monte Carlo calculations for ideal QGP plasma.

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 available 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 [

Solid lines on

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 c o n s t / p 8 and the Maxwell distributions with effective temperature that exceeds the temperature of medium [

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.

In this paper, we have derived the new color path integral representation of Wigner function for the QGP plasma quasiparticle model for canonical ensemble. We have obtained explicit expression of the Wigner function in linear approximation resembling the Maxwell-Boltzmann distribution on momentum variables, but with quantum corrections. This approximation contains also the oscillatory multiplier describing quantum interference. The Monte Carlo calculations of the quark and gluon densities, pair distribution function and the momentum distribution function for strongly coupled QGP plasma in thermal equilibrium at barion chemical potential equal to zero have been done. Comparison with classical Maxwell-Boltzmann shows the significant influence of the interparticle interaction on the high energy asymptotics of the momentum distribution functions resulting in appearance of quantum “tails”. As a continuation of this work we are going to do detailed calculations of the spacial pair distribution functions, color correlation functions, investigate screening properties of QGP and compare Debye mass with available lattice QCD calculations.

We acknowledge stimulating discussions with A.N. Starostin, D. Blaschke and Yu.B. Ivanov. Authors acknowledge the Presidium RAS program number 6 “New approaches to the creation and study of state of matter under extreme conditions” for financial support.

The authors declare no conflicts of interest regarding the publication of this paper.

Filinov, V. and Larkin, A. (2019) Momentum Distribution Functions in Quark-Gluon Plasma. Journal of Applied Mathematics and Physics, 7, 1997-2014. https://doi.org/10.4236/jamp.2019.79137