Diffusion coefficient matrix of the strongly interacting quark-gluon plasma

We study the diffusion properties of the strongly interacting quark-gluon plasma (sQGP) and evaluate the diffusion coefficient matrix for the baryon (B), strange (S) and electric (Q) charges—${\kappa }_{q{q}^{\prime }}$ ($q,q^{\prime }=\mathrm{B},\mathrm{S},\mathrm{Q}$) and show their dependence on temperature $T$ and baryon chemical potential ${\mu }_{\mathrm{B}}$. The nonperturbative nature of the sQGP is evaluated within the dynamical quasiparticle model (DQPM) which is matched to reproduce the equation of state of the partonic matter above the deconfinement temperature $T_c$ from lattice QCD. The calculation of diffusion coefficients is based on two methods: (i) the Chapman-Enskog method for the linearized Boltzmann equation, which allows to explore nonequilibrium corrections for the phase-space distribution function in leading order of the Knudsen numbers as well as (ii) the relaxation time approximation (RTA). In this work we explore the differences between the two methods. We find a good agreement with the available lattice QCD data in case of the electric charge diffusion coefficient (or electric conductivity) at vanishing baryon chemical potential as well as a qualitative agreement with the recent predictions from the holographic approach for all diagonal components of the diffusion coefficient matrix. The knowledge of the diffusion coefficient matrix is also of special interest for more accurate hydrodynamic simulations.

Figure 1

The scaled baryon diffusion coefficient, ${\kappa }_{\mathrm{BB}}/T^2$ (left), and the scaled electric conductivity, ${\sigma }_{\mathrm{el}}/T$ (right), for a partonic system with geometric cross sections, ${\sigma }_{\mathrm{tot}}=10\mathrm{mb}$ at vanishing baryon-chemical potential as a function of temperature from different approaches. We compare the DQPM RTA (${\tau }_i({\mathbf{k}}_i,T,\{{\mu }_q\})$) results from Eq. (27) (for a system obeying quantum statistics, i.e., $a_i=\pm 1$ in Eq. (2) with (blue dashed line) and without (red dashed-double-dotted line) inelastic, flavor-changing channels to the CE (DQPM) results either in RTA (${\tau }_0(T,\{{\mu }_q\})$) (green dashed-dotted line) from Eq. (27) [for a system obeying classical statistics, i.e., $a_i=0$ in Eq. (2)] or for “full” linearized collision term (orange solid line) from Eq. (18) [$a_i=0$ in Eq. (2)].

Figure 2

Left: scaled electric conductivity, ${\sigma }_{\mathrm{el}}/T={\kappa }_{\mathrm{QQ}}/T^2$, as a function of the scaled temperature $T/T_c$ at vanishing chemical potentials, ${\mu }_q=0$, from various approaches. The results from the CE (DQPM) is shown by the red solid line, and from DQPM RTA—by the black dashed line with crossed points. These are compared to results from the lattice QCD calculations (various shaped points with error bars: quenched: orange circle-shaped points [39], light green rhombus-shaped points [86], $N_f=2:$ light cyan circle-shaped points [29], magenta rhombus-shaped points [38], and $N_f=2+1:$ dark cyan circle-shaped points [34] and blue stars [87]), the kinetic partonic cascade model BAMPS (dark-green solid line with triangular-shaped points) [35], and from nonconformal holographic models (blue dotted line from Ref. [14] and violet dashed-dotted line from Ref. [31]). For temperatures below $T_c=0.158\mathrm{GeV}$ we show evaluations from hadronic models: SMASH [19, 42, 88] (grey short-dashed line with squared points), effective field theory (EFT) [30] (blue dashed-dotted line), and CE tuned to a hadron gas [CE (HRG)] from Refs. [15, 17, 40] (dark-red dashed line). Right: scaled electric conductivity of the QGP at fixed scaled temperature, $T=2T_c({\mu }_{\mathrm{B}})$, and vanishing ${\mu }_{\mathrm{Q}}$ and ${\mu }_{\mathrm{S}}$ are shown for varying baryon chemical potential ${\mu }_{\mathrm{B}}$ from the DQPM RTA (black dashed line with cross-shaped points) and the CE (DQPM) (red solid line with circle-shaped points) evaluation.

Figure 3

Scaled cross-electric conductivities, ${\sigma }_{\mathrm{QB}}/T^2$ (left) and ${\sigma }_{\mathrm{QS}}/T^2$ (right), from SMASH [19, 88] (grey short-dashed line with square-shaped points), the DQPM RTA (black dashed line with cross-shaped points), and the CE (DQPM) (red solid line with circle-shaped points) and CE (HRG) [15, 17] (dark-red dashed line) evaluation at vanishing chemical potentials, ${\mu }_q=0$, as a function of scaled temperature in the range from 0.5 to $3T_c$.

Figure 4

Scaled cross-electric conductivities, ${\sigma }_{\mathrm{QB}}/T^2$ (left) and ${\sigma }_{\mathrm{QS}}/T^2$ (right), from the DQPM RTA (black dashed line with cross-shaped points) and the CE (DQPM) evaluation at fixed scaled temperature, $T=2T_c({\mu }_{\mathrm{B}})$, shown over baryon chemical potential ${\mu }_{\mathrm{B}}$ in range 0 to 0.5 GeV. Further, the other chemical potentials are fixed to zero, ${\mu }_{\mathrm{Q}}=0$ and ${\mu }_{\mathrm{S}}=0$.

Figure 5

Scaled strange and strange-baryon diffusion coefficients, ${\kappa }_{\mathrm{SS}}/T^2$ (left) and ${\kappa }_{\mathrm{SB}}/T^2$ (right), as a function of scaled temperature in the range from 0.5 to 3 $T_c$ at vanishing chemical potentials, ${\mu }_q=0$. We compare results from CE (DQPM) (red solid line with circles), DQPM RTA (black dashed-line with crossed-shaped points), the CE (HRG) [15, 17] (dark-red dashed line) and from nonconformal holography [14] (blue dotted line).

Figure 6

Scaled strange and strange-baryon diffusion coefficients, ${\kappa }_{\mathrm{SS}}/T^2$ (left) and ${\kappa }_{\mathrm{SB}}/T^2$ (right), from the DQPM RTA (black dashed line with cross-shaped points) and the CE (DQPM) evaluation at fixed scaled temperature, $T=2T_c({\mu }_{\mathrm{B}})$, shown over baryon chemical potential ${\mu }_{\mathrm{B}}$ in range 0 to 0.5 GeV. Further, the other chemical potentials are fixed to zero, ${\mu }_{\mathrm{Q}}=0$ and ${\mu }_{\mathrm{S}}=0$.

Figure 7

Scaled baryon diffusion coefficient, ${\kappa }_{\mathrm{BB}}/T^2$, (left) at vanishing chemical potentials, ${\mu }_q=0$, as a function of the scaled temperature from various approaches and (right) at fixed scaled temperature, $T=2T_c({\mu }_{\mathrm{B}})$, as a function of baryon chemical potential ${\mu }_{\mathrm{B}}$. The strange and electric potential are fixed to zero: ${\mu }_{\mathrm{S}}=0$ and ${\mu }_{\mathrm{Q}}=0$. We show results from the CE evaluation tuned to DQPM (red solid line), as described above, and tuned to a hadron gas from Refs. [15, 17] (dark-red dashed line). We again compare to the calculation from DQPM RTA [18] (black dashed line with crosses) and to nonconformal holography [13, 14] (blue dotted line) as done for the electric conductivity.

Figure 8

The effective quark (left) and gluon (right) pole-masses $M$ (upper row) and their widths $\gamma$ (lower row) from the actual DQPM as a function of the temperature $T$ and baryon chemical potential ${\mu }_{\mathrm{B}}$ [89]. The strange quark mass was assumed to be $m_s=m_q+\mathrm{\Delta }m$, with $\mathrm{\Delta }m=30\mathrm{MeV}$, and the width is identical with the widths of the light quarks, ${\gamma }_s={\gamma }_q$. In the calculation of the diffusion matrix with the Chapman-Enskog method partons were assumed to be on-shell and thus the widths vanish.

Figure 9

The running coupling ${\alpha }_s=g^2/(4\pi )$ from the actual DQPM as a function of the scaled temperature $T/T_c$ at ${\mu }_{\mathrm{B}}=0$ a) and for moderate values of baryon chemical potential ${\mu }_{\mathrm{B}}<=0.5\mathrm{GeV}$ (b) [89]. The lattice results for quenched QCD, $N_f=0$, (blue circles) are taken from Ref. [48] and for $N_f=2$ (black triangles) are taken from Ref. [80].

Figure 10

DQPM total cross sections between different partons for the on-shell case from Eq. (11) evaluated in the center of mass of the collision system as a function of the collision energy $\sqrt{s}$ for (a) fixed ${\mu }_{\mathrm{B}}=0$, $T=0.19\mathrm{GeV}$ for different elastic channels, (b) for various values of temperature ${\mu }_{\mathrm{B}}=0$, $T=0.19$, 0.316, 0.458 GeV, (c) various values of baryon chemical potential ${\mu }_{\mathrm{B}}=0$, 0.2, 0.5 GeV at fixed $T=0.19\mathrm{GeV}$, and (d) for various values of baryon chemical potential ${\mu }_{\mathrm{B}}=0$, 0.2, 0.5 GeV for $T=2Tc({\mu }_{\mathrm{B}})(T=0.316,0.310,0.275\mathrm{GeV})$. For the minimal allowed center-of-momentum energy of the colliding partons only their pole-masses are taken into account. In the Chapman-Enskog evaluation of the diffusion matrix, the inelastic channels were neglected for simplicity.

Figure 11

Scaled pressure $P_0/T^4$ (a) and scaled energy density $\epsilon /T^4$ (b) from the DQPM (lines) as a function of temperature $T$ at various values of ${\mu }_{\mathrm{B}}=0$, 0.2, 0.4 GeV. The lQCD results obtained by the BMW group are taken from Refs. [69, 70] (circles).