Dynamical collisional energy loss and transport properties of on- and off-shell heavy quarks in vacuum and in the quark gluon plasma

In this study we evaluate the dynamical collisional energy loss of heavy quarks, their interaction rate, as well as the different transport coefficients (drag and diffusion coefficients, $\widehat{q}$, etc). We calculate these different quantities for (i) perturbative partons (on-shell particles in the vacuum with fixed and running coupling) and (ii) for dynamical quasi-particles (off-shell particles in the QGP medium at finite temperature $T$ with a running coupling in temperature as described by the dynamical quasiparticles model). We use the perturbative elastic $\left[q\right(g)Q\to q(g\left)Q\right]$ cross section for the first case and the infrared enhanced hard thermal loop cross sections for the second. The results obtained in this work demonstrate the effects of a finite parton mass and width on the heavy-quark transport properties and provide the basic ingredients for an explicit study of the microscopic dynamics of heavy flavors in the QGP–as formed in relativistic heavy-ion collisions–within transport approaches developed previously by the authors.

Figure 1

Masses and widths of light quarks and gluons in the DQPM [43] and masses in the HTL as a function of $T/T_c$ ($T_c=0.158$ GeV) (a). The infrared regulator ${\mu }^2$ as a function of the temperature $T/T_c$ given in HTL-GA as well as in the DpQCD model (b). The parameters $\xi$, $\kappa$, and $\alpha$ in the HTL-GA model are described in the introduction.

Figure 2

The total elastic interaction rate $R$ of $c$ quarks in the plasma rest frame for the three different approaches as a function of the heavy-quark momentum for $T=2T_c$ ($T_c=0.158$ GeV) (a) and as a function of the temperature $T$ for a heavy-quark momentum $p=10$ GeV (b). The parameters $\xi$, $\kappa$, and $\alpha$ in the HTL-GA model are described in the introduction.

Figure 3

The relaxation time ${\tau }_c$ (a) and the thermal transition rate ${\omega }_{gc}\left(T\right)$ (b) for different models as a function of $T/T_c$ with $T_c=0.158$ GeV at $\mu =0$ for off-shell (IEHTL, red solid line) and on-shell (DpQCD, yellow dashed line) partons. We consider the DQPM pole masses for the on-shell partons in the DpQCD model and the DQPM spectral functions for the off-shell IEHTL approach. Also shown is a comparison with results in the HTL-GA approaches with constant and running coupling.

Figure 4

The drag force $A$ of $c$ quarks in the plasma rest frame (4.15) and (4.17) for the three different approaches as a function of the heavy-quark momentum $p$ for $T=2T_c$ (a) and as a function of the temperature $T/T_c$ ($T_c=0.158$ GeV) for an intermediate heavy-quark momentum, $p=10$ GeV (b).

Figure 5

The average loss of longitudinal momentum $\langle \Delta p_L\rangle$ per collision for the three different approaches as a function of the heavy-quark momentum and for $T=2T_c$ (a) and as a function of the medium temperature $T/T_c(T_c=0.158 \mathrm{GeV})$ for intermediate heavy momentum $p=10$ GeV (b).

Figure 6

The total cross section ${\sigma }^{qQ}$ for the elastic $qQ$ scattering (a) and $\langle \sigma \left|t\right|\rangle =\int \frac{d\sigma }{dt}\left|t\right|dt$ (b) for the models HTL-GA with running $\alpha$ and DpQCD as a function of $q$ which is related to $s$ by $s=M_Q^2+m_q^2+2M_Q{E}_q$.

Figure 7

${\eta }_D$, the drag force $A$ over $p$ for $c$ quarks for different models as a function of the heavy-quark momentum. The temperature of the scattering partners is 300 MeV. M&T refers to Moore and Teaney (equation (B.31) with $\alpha =0.3$ of Ref. [11]), R&R,min (R&R,max) to the minimum (maximum) value of ${\eta }_D$ given by Riek and Rapp [27] (with the $T$-matrix approach using $U$ potential), P&P to Peshier and Peigne [34], and AdS-CFT to the drag force calculated in the framework of the anti de Sitter–conformal field theory by Gubser [49, 51]. HTL-GA (with $\alpha =0.3$) and HTL-GA (with $\alpha$ run) refer to the two parameter sets of the HTL-GA model for constant and running coupling constant defined in Ref. [18].

Figure 8

The spatial diffusion coefficient $D_s$ for $c$ quarks, for HTL-GA with constant and running coupling [18], for DpQCD and IEHTL in comparison with the results from lQCD [53, 54] as a function of the temperature. For the lQCD points the boxes show the statistical error estimated by the “Jackknife method” while the bars stand for systematic uncertainties from the MEM analyses. M&T refers to Moore and Teaney (Eq. (B.31) with $\alpha =0.3$ of [11]) and R&R,min (R&R,max) to the minimum (maximum) value of $D_s$ given by Riek and Rapp [27] (with the $T$-matrix approach using $U$ potential).

Figure 9

$c$-quark energy loss, $dE/dx$, in the plasma rest frame from the three different approaches as a function of the heavy-quark momentum $p$ for $T=2T_c$ ($T_c=0.158$ GeV) (a) and as a function of the medium temperature $T$ for a heavy-quark momentum of $p=10$ GeV (b).

Figure 10

$\widehat{q}$ for $c$ quarks in the plasma rest frame for the three different approaches. (a) $\widehat{q}$ as a function of the heavy-quark momentum $p$ for $T=2T_c$ ($T_c=0.158$ GeV). (b) $\widehat{q}$ as a function of the temperature $T/T_c$ for an intermediate heavy-quark momentum of $p=10$ GeV.

Figure 11

The transverse-momentum transfer squared $\langle \Delta p_T^2\rangle$ of $c$ quarks per collision in the plasma rest frame for the three different approaches: (a) as a function of the heavy-quark momentum $p$ for $T=2T_c$ ($T_c=0.158$ GeV) and (b) as a function of the temperature $T/T_c$ for an intermediate heavy-quark momentum of $p=10$ GeV.

Figure 12

$\widehat{q}$ for $c$ quarks for different approaches as a function of the heavy-quark momentum. The temperature of the scattering partners is $T=400$ MeV. M&T refers to Moore and Teaney (Eq. (B.32) with $\alpha =0.3$ of Ref. [11]), HTL-B refers to Beraudo et al. [57], with the two choices $\mu =2\pi T$ and $\mu =\pi T$ ($\mu$ is a parameter in the coupling constant), and AdS-CFT to $\widehat{q}$ calculated in the framework of the anti de Sitter–conformal field $\mathcal{N}=4$ SYM theory of Refs. [58, 59, 60]. HTL-GA (with $\alpha =0.3$) and HTL-GA (with running $\alpha$) refer to the two parameter sets of the HTL-GA model for constant and running coupling defined in Ref. [18].

Figure 13

The total transport cross section ${\sigma }_{\mathrm{I}}^{\mathrm{tr}}$ (7.2) for the $q\left(g\right)c\to q\left(g\right)$ processes for the three different approaches as a function of the heavy-quark momentum $p$ for $T=2T_c$ (a) and as a function of the temperature $T/T_c$ ($T_c=0.158$ GeV) for an intermediate heavy-quark momentum, $p=10$ GeV (b).

Figure 14

The total transport cross section ${\sigma }_{\mathrm{II}}^{\mathrm{tr}}$ (7.9) for the $q\left(g\right)c\to q\left(g\right)$ processes for the three different approaches as a function of the heavy-quark momentum $p$ for $T=2T_c$ (a) and as a function of the temperature $T/T_c$ ($T_c=0.158$ GeV) for an intermediate heavy-quark momentum, $p=10$ GeV (b).

Figure 15

The ratio ${\sigma }_{\mathrm{I}}^{\mathrm{tr}}/{\sigma }_{\mathrm{II}}^{\mathrm{tr}}$ as a function of the heavy-quark momentum $p$ for $T=2T_c$ (a) and as a function of the temperature for $p=10$ GeV (b). We note that ${\sigma }_{\mathrm{I}}^{\mathrm{tr}}/{\sigma }_{\mathrm{II}}^{\mathrm{tr}}={\sigma }_{\mathrm{I},qQ}^{\mathrm{tr}}/{\sigma }_{\mathrm{II},qQ}^{\mathrm{tr}}+{\sigma }_{\mathrm{I},gQ}^{\mathrm{tr}}/{\sigma }_{\mathrm{II},gQ}^{\mathrm{tr}}$.