Quark susceptibility in a generalized dynamical quasiparticle model

The quark susceptibility ${\chi }_q$ at zero and finite quark chemical potential provides a critical benchmark to determine the quark-gluon-plasma (QGP) degrees of freedom in relation to the results from lattice QCD (lQCD) in addition to the equation of state and transport coefficients. Here we extend the familiar dynamical quasiparticle model (DQPM) to partonic propagators that explicitly depend on the three-momentum with respect to the partonic medium at rest in order to match perturbative QCD (pQCD) at high momenta. Within the extended dynamical quasiparticle model (${\mathrm{DQPM}}^{*}$) we reproduce simultaneously the lQCD results for the quark number density and susceptibility and the QGP pressure at zero and finite (but small) chemical potential ${\mu }_q$. The shear viscosity $\eta$ and the electric conductivity ${\sigma }_e$ from the extended quasiparticle model (${\mathrm{DQPM}}^{*}$) also turn out to be in close agreement with lattice results for ${\mu }_q=0$. The ${\mathrm{DQPM}}^{*}$, furthermore, allows one to evaluate the momentum $p$, temperature $T$, and chemical potential ${\mu }_q$ dependencies of the partonic degrees of freedom also for larger ${\mu }_q$, which are mandatory for transport studies of heavy-ion collisions in the regime $5<\sqrt{s_{NN}}<10$ GeV.

Figure 1

The ${\mathrm{DQPM}}^{*}$ gluon (a) and light quark (b) masses and widths given by (2.3) using the coupling (2.9)–(2.13) for different quark chemical potentials as a function of the temperature $T$. (c) Gluon and light quark masses as a function of the momentum squared for $T=2T_c$ and ${\mu }_q=0,0.2,0.3$ GeV.

Figure 2

Energy density $\epsilon$, entropy density $s$, pressure density $P$, and trace anomaly $(I=\epsilon -3P)$ as a function of temperature $T$ at ${\mu }_B=0$ (a) and at ${\mu }_B=400$ MeV (b) from ${\mathrm{DQPM}}^{*}$ compared to lQCD data from Ref. [7].

Figure 3

(a) The baryon number density $n_B$ from ${\mathrm{DQPM}}^{*}$ as compared to lattice data from [7] for $N_f=3$ for quark chemical potential ${\mu }_q=0$. (b) The susceptibility ${\chi }_2$ from ${\mathrm{DQPM}}^{*}$ as compared to lattice data from [7] for $N_f=3$ and ${\mu }_q=0$ using Eq. (4.8). The lower (orange) line gives the result from the conventional DQPM, i.e., with momentum-independent masses.

Figure 4

The shear viscosity to entropy density ratio $\eta /s$ from different models as a function of temperature $T$ for ${\mu }_q=0$ (a) and $\eta /s$ given by the ${\mathrm{DQPM}}^{*}$ approach as a function of $(T,{\mu }_q)$ (b). The orange solid line in (a) results from the standard DQPM where the parton masses and widths are independent of momenta [27]. The thick red solid line shows the ${\mathrm{DQPM}}^{*}$ result using Eqs. (5.1) and (5.2), where the parton masses and width are temperature, chemical potential, and momentum dependent. The lattice QCD data for pure $\text{SU}\left(3\right)$ gauge theory are taken from Ref. [46] (red spheres), from Ref. [47] (green pyramid and blue cubic), and from Ref. [48] (black cylinder and pink dodecahedron). The orange dashed line gives the Kovtun-Son-Starinets lower bound [49, 50] ${(\eta /s)}_{KSS}=1/\left(4\pi \right)$. Finally, the black solid line refers to the calculation of $\eta /s$ in Yang-Mills theory from Ref. [45].

Figure 5

${\sigma }_e/T$ following different models as a function of temperature $T$ for ${\mu }_q=0$ (a) and ${\sigma }_e/T$ given by the ${\mathrm{DQPM}}^{☆}$ approach as a function of $(T,{\mu }_q)$ (b). The orange thin solid line in (a) results from the standard DQPM where the parton masses and widths are independent of momenta [27]. The red thick solid line shows the ${\mathrm{DQPM}}^{*}$ result using Eqs. (6.1), where the parton masses and width are temperature, chemical potential, and momentum dependent. The lattice QCD data are taken from Ref. [55] (red spheres), Ref. [56] (pink pentagon), Ref. [57] (blue cubic), Ref. [58] (Cyan pyramid), Ref. [59] (green cone), Ref. [60] (black cylinder), and Ref. [61] (blue disk). Qin, MEM (2013) refers to Ref. [62] where a Dyson-Schwinger approach is used. The electric charge is explicitly multiplied out using $e^2=4\pi /137$. The average charge squared is $C_{EM}=8\pi \alpha /3$ with $\alpha =1/137$. Note that the pQCD result at leading order beyond the leading logarithm [63] is ${\sigma }_e/T\approx 5.97/e^2\approx 65$.