$T$-matrix approach to quark-gluon plasma

A self-consistent thermodynamic $T$-matrix approach is deployed to study the microscopic properties of the quark-gluon plasma (QGP), encompassing both light- and heavy-parton degrees of freedom in a unified framework. The starting point is a relativistic effective Hamiltonian with a universal color force. The input in-medium potential is quantitatively constrained by computing the heavy-quark (HQ) free energy from the static $T$-matrix and fitting it to pertinent lattice-QCD (lQCD) data. The corresponding $T$-matrix is then applied to compute the equation of state (EoS) of the QGP in a two-particle irreducible formalism, including the full off-shell properties of the selfconsistent single-parton spectral functions and their two-body interaction. In particular, the skeleton diagram functional is fully resummed to account for emerging bound and scattering states as the critical temperature is approached from above. We find that the solution satisfying three sets of lQCD data (EoS, HQ free energy, and quarkonium correlator ratios) is not unique. As limiting cases we discuss a weakly coupled solution, which features color potentials close to the free energy, relatively sharp quasiparticle spectral functions and weak hadronic resonances near $T_{\mathrm{c}}$, and a strongly coupled solution with a strong color potential (much larger than the free energy), resulting in broad nonquasiparticle parton spectral functions and strong hadronic resonance states which dominate the EoS when approaching $T_{\mathrm{c}}$.

Figure 1

$T$-matrix resummation for ladder diagrams.

Figure 2

Examples of diagrams that are resummed by the generalized $T$-matrix for EoS.

Figure 3

Results of a weakly coupled solution for the self-consistent fit to extract the static HQ potential: single HQ and $Q\overline{Q}$ self-energies, ${\mathrm{\Sigma }}_X(\omega ,\infty )$ (first row), and spectral functions, ${\rho }_X(z,\infty )$ (second row), potential $\tilde{V}\left(r\right)$ and free energies (third row), and interference function, $\varphi \left(x_{e}r\right)$ (fourth row), in the first four columns corresponding to different temperatures. The last column shows the temperature dependence of the fitted screening masses (top panel) and the scale factor, $x_e$ (bottom panel), figuring in the interference function. The free-energy lQCD data are from Ref. [48].

Figure 4

Weakly coupled solution for charmonium (${\eta }_c$, left panels) and bottomonium (${\eta }_b$, right panels) spectral functions (upper panels) and correlators ratios (middle panels) with (first and third column) and without (second and fourth column) interference effects in the imaginary part of the potential. The lQCD data for ${\eta }_c$ [51] and ${\eta }_b$ [52] correlator ratios are shown in the first and third bottom panel, respectively, while the second and fourth bottom panel display the temperature dependence of the charm- and bottom-quark mass, respectively.

Figure 5

Weakly coupled solution for the QGP bulk medium: fit results for the input masses for quarks and gluons (left panel), the QGP pressure in comparison to lQCD data [46] (middle panel; solid line: total, dashed line: LWF contribution), and the ratio of LWF contribution to total pressure (right panel).

Figure 6

Weakly coupled solution for parton spectral properties of the QGP. The figure is organized into four 3-by-4 panels of 12 plots, with each panel for a given temperature (upper left: $T$ = 0.194 GeV, upper right: $T$ = 0.258 GeV, lower left: $T$ = 0.320 GeV and lower right: $T$ = 0.400 GeV). Each panel contains four rows corresponding to different parton species (light quarks ($q$), gluons ($g$), charm quarks ($c$), and bottom quarks ($b$) in the first, second, third, and fourth row of each panel, respectively). Each row contains three panels showing (from left to right) the energy dependence of the pertinent real and imaginary part of the self-energy and the resulting spectral functions, for four different values of the single-parton 3-momentum ($p$) in the thermal frame.

Figure 7

Weakly coupled solution for the imaginary part of the color-singlet $S$-wave $T$-matrices (without interference effects) in the bottomonium ($b\overline{b}$; first column), charmonium ($c\overline{c}$; second column), $D$-meson ($c\overline{q}$; third column), light-quark ($q\overline{q}$; fourth column), and glueball ($gg$, last column) channels. The four rows correspond to different temperatures, $T=0.194$ GeV, $T=0.258$ GeV, $T=0.320$ GeV, and $T=0.400$ GeV from top down; in each panel, the $T$-matrix is displayed for four different values of the single-parton 3-momentum ($p_{\text{cm}}$) in the two-body CM frame.

Figure 8

Results of a strongly coupled solution for the self-consistent fit to extract the static HQ potential: single-quark and $Q\overline{Q}$ self-energies, ${\mathrm{\Sigma }}_X(\omega ,\infty )$ (first row), and spectral functions, ${\rho }_X(z,\infty )$ (second row), potential $\tilde{V}\left(r\right)$ and free energies (third row), and interference function, $\varphi \left(x_{e}r\right)$ (fourth row), in the first four columns corresponding to different temperatures. The last column shows the temperature dependence of the fitted screening masses (top panel) and the scale factor, $x_e$ (bottom panel), figuring in the interference function. The free-energy lQCD data are from Ref. [48].

Figure 9

Strongly coupled solution for charmonium (${\eta }_c$, left panels) and bottomonium (${\eta }_b$, right panels) spectral functions (upper panels) and correlators ratios (middle panels) with (first and third column) and without (second and fourth column) interference effects in the imaginary part of the potential. The lQCD data for ${\eta }_c$ [51] and ${\eta }_b$ [52] correlator ratios are shown in the first and third bottom panel, respectively, while the second and fourth bottom panel display the temperature dependence of the charm- and bottom-quark mass, respectively.

Figure 10

Strongly coupled solution for the QGP bulk medium: fit results of the input masses for quarks and gluons (left panel), the QGP pressure in comparison to lQCD data [46] (middle panel: solid line, total; dashed line, LWF contribution), and the ratio of LWF contribution to total pressure (right panel).

Figure 11

Strongly coupled solution for parton spectral properties of the QGP. The figure is organized into four 3-by-4 panels of 12 plots, with each panel for a fixed temperature (upper left: $T$ = 0.194 GeV, upper right: $T$ = 0.258 GeV, lower left: $T$ = 0.320 GeV and lower right: $T$ = 0.400 GeV). Each panel contains four rows corresponding to different parton species (light quarks ($q$), gluons ($g$), charm quarks ($c$), and bottom quarks ($b$) in the first, second, third, and fourth row of each panel, respectively). Each row contains three panels showing (from left to right) the energy dependence of the pertinent real and imaginary part of the self-energy and the resulting spectral functions, for four different values of the single-parton 3-momentum ($p$) in the thermal frame.

Figure 12

Strongly coupled solution for the imaginary part of the color-singlet $S$-wave $T$-matrices (without interference effects) in the bottomonium ($b\overline{b}$; first column), charmonium ($c\overline{c}$; second column), $D$-meson ($c\overline{q}$; third column), light-quark ($q\overline{q}$; fourth column), and glueball ($gg$, last column) channels. The four rows correspond to different temperatures, $T=0.194$ GeV, $T=0.258$ GeV, $T=0.320$ GeV, and $T=0.400$ GeV from top down; in each panel, the $T$-matrix is displayed for four different single-parton momenta ($p_{\mathrm{cm}}$) in the two-body CM frame.

Figure 13

The first row depicts $\mathcal{M}·{\mathcal{M}}^{†}$ including interference effects that can be obtained by cutting the diagrams as shown in the second row. The third row is the $T$-matrix generalization of the diagrams in the second row.

Figure 14

The left panel shows the diagram corresponding to the BSE implementation of loop effects in the potential, while the right panel is based on a Faddeev equation for the $Q\overline{Q}$+light-parton interaction with the thermal light-parton line being closed off.