Quarkonia and heavy-quark relaxation times in the quark-gluon plasma

A thermodynamic $T$-matrix approach for elastic two-body interactions is employed to calculate spectral functions of open and hidden heavy-quark systems in the quark-gluon plasma. This enables the evaluation of quarkonium bound-state properties and heavy-quark diffusion on a common basis and thus to obtain mutual constraints. The two-body interaction kernel is approximated within a potential picture for spacelike momentum transfers. An effective field-theoretical model combining color-Coulomb and confining terms is implemented with relativistic corrections and for different color channels. Four pertinent model parameters, characterizing the coupling strengths and screening, are adjusted to reproduce the color-average heavy-quark free energy as computed in thermal lattice QCD. The approach is tested against vacuum spectroscopy in the open ($D$, $B$) and hidden ($\Psi$ and $\Upsilon$) flavor sectors, as well as in the high-energy limit of elastic perturbative QCD scattering. Theoretical uncertainties in the static reduction scheme of the four-dimensional Bethe-Salpeter equation are elucidated. The quarkonium spectral functions are used to calculate Euclidean correlators which are discussed in light of lattice QCD results, while heavy-quark relaxation rates and diffusion coefficients are extracted utilizing a Fokker-Planck equation.

Figure 1

Color-average HQ free energies obtained in thermal lQCD computations by Kaczmarek et al. (left panel) [12, 39] and Petreczky et al. (right panel) [13, 41] compared to our fits using the microscopic model suggested in Refs. [21, 22].

Figure 2

Temperature dependence of the parameters deduced from our fit to the color-average free energies from the two lQCD calculations displayed in Fig. 1. (Left panel) Strong coupling constant ${\alpha }_s$; (right panel) screening masses of the color-Coulomb and confining parts, and dimension-2 condensate “glueball mass,” $m_G$.

Figure 3

Temperature dependence of the dimension-2 condensate $C_2$, Eq. (14), (left panel) and of the infinite-distance limit of the free and internal energies for the two lQCD computations displayed in Fig. 1.

Figure 4

Vacuum HQ potentials in different color configurations of $Q\mathrm{Q̄}$ (singlet and octet) and $\mathit{QQ}$ (triplet and sextet) channels, extracted from model-fits to color-average $Q\mathrm{Q̄}$ free energies in lQCD for $N_f=2+1$ [12, 39, 40] (“potential 1,” left panel) and for $N_f=3$ [13, 41] (“potential 2,” right panel).

Figure 5

In-medium heavy-quark free and internal energies at various temperatures in color-singlet (long-dashed lines), -octet (short-dashed lines), -anti-triplet (dash-dotted lines), -sextet (dotted lines), and -average (solid lines) channels, extracted from $N_f=2+1$ (left column) and $N_f=3$ lQCD computations (right column).

Figure 6

One-gluon exchange diagram for quark-quark scattering; in the center-of-mass system, the relative four-momentum in the incoming (outgoing) state is $q=(p_1-p_2)/2$ [$q^{\prime }=(p_1^{\prime }-p_2^{\prime })/2$].

Figure 7

(Left panel) Perturbative QCD cross section for quark-quark scattering via $t$-channel gluon exchange using a vector interaction. The full pQCD result (solid line) is compared to the $T$-matrix result in Born approximation using (i) a nonrelativistic treatment of the spinor structure (doted line) and (ii) the correction factors $R$ and $B$. (Right panel) The same but for a scalar interaction. The heavy- and light-quark masses are set to 1.7 and 0.4 GeV, respectively, and the strong coupling and Debye mass have been fixed at 0.3 and 0.67 GeV, respectively.

Figure 8

Quarkonium spectroscopy in vacuum for $S$- and $P$-wave charmonia (upper panels) and $S$- and $P$-wave bottomonia (lower panels).

Figure 9

Imaginary part of the charm-light $T$-matrix in the vacuum using $m_q=0.4$ GeV. We show the color singlet $Q\mathrm{q̄}$ (upper left plot), antitriplet $\mathit{Qq}$ (upper right plot), octet $Q\mathrm{q̄}$ (lower left plot), and sextet $\mathit{Qq}$ (lower right plot) channels.

Figure 10

Charmonium spectral functions and Euclidean correlators in the pseudoscalar ($S$-wave) channel at various temperatures using the internal energy ($U$) as potential. Results for two reduction schemes and two different potentials are compared.

Figure 11

Charmonium spectral functions in the scalar ($P$-wave) channel at various temperatures using the internal energy ($U$) as potential. Results for two reduction schemes and two different potentials are compared.

Figure 12

Same as in Fig. 10 but using the free energy ($F$) as potential.

Figure 13

Same as in Fig. 11 but using the free energy ($F$) as potential.

Figure 14

Same as in Fig. 10 but employing a single-quark width of 100 MeV.

Figure 15

Bottomonium spectral functions and Euclidean-correlator ratios, using $U$ as potential, for the pseudoscalar channel at various temperatures. We compare the BbS and Th schemes as well as the two different potentials.

Figure 16

Bottomonium spectral functions using $U$ as potential for the scalar channel at various temperatures using the BbS (upper panels) and Th reduction schemes (lower panels).

Figure 17

Same as in Fig. 15 but using $F$ as potential.

Figure 18

Same as in Fig. 16 but using $F$ as potential.

Figure 19

Imaginary part of the in-medium on-shell $T$ matrix for charm-light quark scattering in the color-singlet (upper left), antitriplet (upper right), octet (lower left), and sextet (lower right) channels. In all cases lQCD potential-1 is used for $U$ within the Thompson scheme. Note the factor 100 difference in the $y$ scales of the upper and lower panels.

Figure 20

Charm-quark relaxation rate as a function of three-momentum calculated in the $T$-matrix approach using $U$ as potential, compared the LO pQCD with ${\alpha }_s=0.4$. A perturbative gluon contribution has been added to the heavy-light $T$-matrix rates.

Figure 21

Same as in Fig. 20 but using $F$ as potential.

Figure 22

Comparison of our results for charm-quark relaxation rates (left plot) using $U$ and $F$ potentials to previous $T$-matrix calculations [6] and estimates from AdS/CFT [72, 73]. The $T$-matrix rates have been augmented by perturbative scattering off thermal gluons. The right panel shows the corresponding spatial diffusion constants.

Figure 23

Bottom-quark relaxation rates as a function of three-momentum calculated in the $T$-matrix approach using $U$ as potential (plus perturbative scattering off thermal gluons) compared to LO pQCD.

Figure 24

Same as in Fig. 23 but using $F$ as potential.

Figure 25

Dependence of the vacuum $J/\Psi$ binding energy on a momentum cutoff introduced in the $T$-matrix integral in Eq. (15).

Figure 26

Temperature dependence of the binding energy (${\epsilon }_B$) of the $J/\Psi$ (or ${\eta }_c$) using $U$ as potential, for various combinations of reduction scheme and lQCD input.

Figure 27

Temperature dependence of the binding energy (${\epsilon }_B$) of the $\Upsilon$ (or ${\eta }_b$) using $U$ as potential (left plot) and $F$ as potential (right plot) for various combinations of reduction scheme and lQCD input.

Figure 28

Comparison of the pseudoscalar spectral functions for charmonium (upper panels) and bottomonium (lower panels) using the full potential (left column) and the color-Coulomb term only (right column). In all cases $U$ has been used as potential together with the Th reduction.