Selfconsistent Evaluation of Charm and Charmonium in the Quark-Gluon Plasma

A selfconsistent calculation of heavy-quark (HQ) and quarkonium properties in the Quark-Gluon Plasma (QGP) is conducted to quantify flavor transport and color screening in the medium. The main tool is a thermodynamic $T$-matrix approach to compute HQ and quarkonium spectral functions in both scattering and bound-state regimes. The $T$-matrix, in turn, is employed to calculate HQ selfenergies which are implemented into spectral functions beyond the quasiparticle approximation. Charmonium spectral functions are used to evaluate eulcidean-time correlation functions which are compared to results from thermal lattice QCD. The comparisons are performed in various hadronic channels including zero-mode contributions consistently accounting for finite charm-quark width effects. The zero modes are closely related to the charm-quark number susceptibility which is also compared to existing lattice"data". Both the susceptibility and the heavy-light quark $T$-matrix are applied to calculate the thermal charm-quark relaxation rate, or, equivalently, the charm diffusion constant in the QGP. Implications of our findings in the HQ sector for the viscosity-to-entropy-density ratio of the QGP are briefly discussed.


I. INTRODUCTION
In recent years rather remarkable properties of the Quark-Gluon Plasma (QGP) have been discovered, notably a liquid-like behavior deduced from the collective expansion of the medium formed in collisions of heavy nuclei at the Relativistic Heavy Ion Collider (RHIC). Hydrodynamic expansion models, based on the assumption of local thermal equilibrium, require very short equilibration times, of the order of τ therm ≃ 1 fm/c or less, to account for the experimental findings. The microscopic origin of the rapid thermalization remains one of the outstanding problems raised by the RHIC data. It is further complicated by the fact that the very notion of equilibrium erases information about its origin. A phenomenologically suitable probe of thermalization in heavy-ion collisions should therefore interact strongly at the thermal scale but with a relaxation time on the order of the system's lifetime (at RHIC τ QGP ≃ 5 fm/c). The logical choice are thus heavy quarks, i.e., charm and bottom (Q = c, b), for which thermalization is expected to be delayed by a factor m Q /T , quite comparable to τ QGP /τ therm (see, e.g., Ref. [1] for a recent review).
Bound states of heavy quarks (charmonia and bottomonia) are believed to probe the QGP from a somewhat different angle [2][3][4][5][6]. The dissolution pattern of the quarkonium spectrum in matter is possibly the most direct way of diagnosing color-Debye screening of the basic QCD force, f QQ = −∇VQ Q (r), as a function of temperature (and/or density). In-medium quarkonium spectroscopy therefore reveals insights into the deconfin- * e-mail: friek@comp.tamu.edu † e-mail: rapp@comp.tamu.edu ing hadron-to-parton phase transition in QCD. In particular, the use of potential models has been revived recently, largely triggered by the prospect that an inmedium potential can be extracted model-independently from lattice QCD (lQCD) computations. Furthermore, lQCD generates euclidean-time (τ ) correlation functions of quarkonia with good accuracy in different hadronic channels, which provide useful constraints on model calculations of spectral functions in the timelike regime of physical excitations. The extraction of reliable information from such comparisons requires a realistic modeling of the continuum part (scattering regime) of the quarkonium spectral functions, i.e., not only its bound-state part. The account of interactions in the near-threshold region is particularly important to describe situations were bound states dissolve into the continuum, as in the case at hand. In connection with correlator analyses, a comprehensive treatment of bound-state and continuum regimes has been performed using Schrödinger phase shifts [7], a thermodynamic T -matrix [8] or a nonrelativistic Green's function [9] approach. Another important ingredient to a realistic description of quarkonium spectral functions in the QGP are medium effects in the single-particle properties, i.e., in the heavyquark (HQ) propagation. These are closely related to HQ transport [10,11] and the so-called zero-mode contributions to quarkonium spectral and correlation functions [12,13]. In particular, finite-width effects on both quarkonia and heavy quarks have received little attention thus far [8,14].
In the present paper we further develop our previous study of heavy quarks and quarkonia in the QGP using a thermodynamic T -matrix approach [10]. First, we go beyond the quasi-particle approximation for heavy quarks and perform a selfconsistent calculation of the HQ selfenergy and the in-medium heavy-light T -Matrix. The improved HQ spectral functions are then implemented into an off-shell calculation of the quarkonium spectral functions. Second, we expand our comparison with Euclidean correlator ratios to the scalar, vector and axialvector channels. In these channels the presence of zero modes [12,13], resulting from scattering of a single heavy quark on medium constituents ("particle-hole" excitations), is known to impact the large-τ behavior of the in-medium correlators appreciably. Thus far, these contributions have been estimated in quasiparticle approximation neglecting finite width effects [9,12,13,15,16]. In a treatment of heavy quarks consistent with the T -Matrix we evaluate the zero-mode contribution accounting for their full spectral function, in particular finitewidth effects. This automatically yields predictions for the charm-quark number susceptibility, which can be compared to lattice data and enable a more reliable extraction of in-medium quasiparticle masses for heavy quarks.
The paper is organized as follows: In Sec. II we give a short overview of the T -Matrix formalism employed in our calculations, specifically discussing its extension to off-shell dynamics in the charm-quark propagation. Section III is devoted to properties of cahrmonia in the QGP, i.e., their spectral functions including zero modes (Sec. III A), euclidean correlator ratios in various mesonic channels (Sec. III B) and a comparison of numerical results using two different input potentials to lQCD data (Sec. III C). In Sec. IV we first analyze the selfconsistently calculated charm-quark selfenergies (Sec. IV A) and then apply those to obtain the thermal relaxation rate and a schematic estimate of η/s (Sec. IV B), as well as the charm-quark number susceptibility (Sec. IV C). We conclude in Sec. V.

II. OFF-SHELL T -MATRIX AT FINITE TEMPERATURE
The formalism for calculating the T -Matrix for quarkquark and quark-antiquark scattering and bound states in the QGP, using two-body potentials estimated from heavy-quark free energies computed in lattice QCD (lQCD), has been developed in Refs. [8,10,17]. Here we recollect the main elements while referring to Ref. [10] for further details (e.g., a discussion of relativistic corrections and constraints from vacuum spectroscopy and the highenergy perturbative limit). Starting point is the Bethe-Salpeter Equation which after a 3-dimensional (3D) reduction and partial-wave expansion turns into a 1D integral equation for the scattering amplitude (T -Matrix), in a given color channel (a) and partial wave (l); the relative 3-momentum moduli of the initial, final and intermediate 2-particle state are denoted by q = |q |, q ′ = |q ′ | and k = |k |, respectively (we restrict ourselves to vanishing total 3-momentum, P = 0, of the 2-particle system). The explicit form of the intermediate 2-particle propagator, G 12 , depends on the 3D reduction scheme [18][19][20][21] (in slight deviation from Refs. [8,10,17] we here absorb the Pauli blocking factor, (1 − 2f F ), into G 12 ).
In the following, we focus on the Thompson scheme, since the Blankenbecler-Sugar scheme was found to generate some overbinding in the vacuum quarkonium spectrum [10]. The T -matrix will be applied in both heavyheavy and heavy-light quark channels for which a static (potential) approximation can be justified (i.e., the energy transfer is parameterically suppressed compared to the 3-momentum transfer, ∆q 0 ∼ (∆q) 2 /m Q << ∆q, where ∆q = q ′ − q). For the two-body potential, V l,a , we follow our previous work [10] using either the heavy-quark free (F ) or internal (U ) energy computed in lQCD, implemented into a field-theoretical model for color-Coulomb and confining terms [22] with relativistic corrections (e.g., the Breit interaction [23][24][25] to account for color-magnetic effects). To ensure the convergence of the Fourier transform from coordinate to momentum space we subtract the infinite-distance limit of the potential according to and interpret X ∞ (T )/2 ≡ X(r → ∞, T )/2 as a temperature-dependent correction to the bare HQ mass (real part of selfenergy) induced by the condensate associated with the confining force term. The use of either U or F as potential is believed to bracket the uncertainties in this identification.

A. 2-Particle Propagator
In all previous applications of potential models in the QGP the quarks have been treated as quasiparticles with either vanishing or constant [8,10] width. However, as pointed out in Ref. [10], this approximation implies that quarkonium spectral functions, ρ α (E) (α: quantum numbers of the composite mesonic (or diquark) state), do not possess the proper low-energy limit, ρ α (E → 0) ∝ E. While this is not expected to significantly impact the mass and binding energy of the bound states (for which the total energy is of order twice the HQ mass), the euclidean correlators calculated below do actually involve an integration over ρ α (E) starting from E = 0 with a thermal weight which diverges for E → 0. In previous studies this problem has been evaded by introducing a low-energy cutoff on the spectral function (sufficiently below the lowest bound state as to not affect the correlator). However, recent studies of quark spectral functions [26] find considerable low-energy strength in quarkonium spectral functions, e.g., due to particlehole like structures in the HQ propagator. Thus a more elaborate treatment in the T -Matrix equation, properly accounting for off-shell dynamics, is in order.
The general off-shell expression of the uncorrelated 2fermion propagator at finite temperature, G 12 , figuring into the T -matrix equation, can be derived, e.g., within the Matsubara formalism. One has where (i = 1, 2) denotes the (scalar part of) the 1-particle propagator, and z ν = i(2ν+1)πT (Ω λ ) are fermionic (bosonic) Matsubara frequencies. Using the spectral representations of the in-medium (retarded) single-particle propagators, the Matsubara summation in Eq. (3) can be performed explicitly. After analytic continuation to the real axis (Ω λ → E), the positive-energy contributions to the 2particle propagator take the form where denotes the positive-energy part of the quark spectral functions. The factor m 1 m 2 in Eq. (6) is specific to the Thompson reduction scheme, rendering the appropriate expression for G 12 in the limit of on-shell quarks [21]. For the in-medium HQ mass, as mentioned above (recall Eq. (2)), the infinite-distance value of the HQ potential is identified with a "mean-field" contribution, i.e., as a real part of a selfenergy, The additional selfenergy contribution figuring into the quark propagator in Eq. (7) is generated from interactions with thermal light quarks and antiquarks and is therefore distinct from X ∞ (T ). It will be computed from the heavy-light quark T -matrix, as discussed below. For calculating the latter, we need to specify light-quark masses which we model by a thermal mass proportional to gT with a coefficient to approximately match lQCD computations of the energy density of the QGP in quasiparticle approximation. In addition, we supplement a moderate imaginary part for the light-quark selfenergy, Im Σ q = −0.05 GeV, which is in the range of values obtained in Ref. [17] in a similar approach for light quarks, and comparable to what we obtain for heavy quarks (our results for quarkonium spectral function and HQ transport are largely insensitive to the light-quark width). The evaluation of the Matsubara sum in Eq. (3) also generates a contribution that solely arises from thermal excitations (see, e.g., Ref. [27]). It involves one positiveand one negative-energy part of the quark spectral functions which is why we referred to it as a "particle-hole" contribution above. For mesonic channels of the Tmatrix, 12 = QQ, its imaginary part can be cast into the form Here, the negative-energy part of theQ spectral function has been turned into the positive-energy part of the Q spectral function, i.e., theQ line in the original QQ propagator has been "turned around". Together with the Fermi distributions, the interpretation of this contribution to G 12 becomes apparent: an incoming Q, preexisting in the heat bath according to f Q (ω, k), scatters into a Q with energy E + ω, Pauli-blocked according to f Q (E + ω, k). G ph 12 is, in fact, precisely the zero-mode contribution discussed in the context of quarkonium correlators [9,12,13,15]. We return to its evaluation in Sec. III A below.

B. Single-Quark Selfenergy
The HQ selfenergy, Σ Q (ω, k), due to interactions with light anti-/quarks in the heat bath is calculated from the heavy-light (Qq) T -Matrix by closing its light-quark line with a quark propagator weighted with a Fermi distribution function. Using the Matsubara formalism one can express its imaginary part as [17,27] The spin-isospin factor d SI = 4N f counts the degeneracy of available meson (or diquark) states, e.g., with total spin-0 and -1 for S-wave heavy-light scattering [14]. In the expression for the HQ selfenergy, Eq. (10), the thermal light quarks are treated as zero-width quasi particles so that their spectral functions can be replaced by δ functions (but with thermal mass ∼ gT ) 1 . We recall that the gluonic contributions are entirely attributed to the meanfield type condensate term (for HQ transport, we include heavy-light interactions to leading order in perturbation theory which does not generate an imaginary part in the scattering amplitude). The real part of the selfenergy is obtained by a dispersion relation which is a preferred procedure in a selfconsistent prescription since the normalization of the spectral functions can easily be guaranteed. Our framework is similar to the one utilized in Ref. [17] in the light-quark sector. However, in there the calculations of the T -matrix were restricted to on-shell selfenergies while here we use the full off-shell selfenergy of the heavy quark which, in particular, enables to establish the correct low-energy behavior of the mesonic spectral functions (in addition to refinements in the implementation of the potential as developed in Ref. [10] the non-potential corrections are expected to be significantly larger for light-quark interactions). Let us finally comment on the issue of imaginary parts in the potential and selfenergy. Using effective field theories at finite temperature it has been found that the two-body potential operator figuring into a Schrödinger equation acquires an imaginary part [28][29][30][31]. Diagrammatically, this implies that the potential possesses onshell cuts corresponding to dissociation processes of the composite (bound) state. Let us inspect two commonly discussed ("leading-order") processes for the case of charmonium bound states. For large binding energy, E B ≥ T , the dominant process is gluon dissociation, g + J/ψ → c +c, first analyzed by Bhanot and Peskin [32]. In the language of effective field theory, this corresponds to the so-called color-singlet to -octet transition mechanism (up to final-state interactions in the octet channel, which, however, are suppressed by 1/N 2 c and thus numerically negligible). In our T -matrix formalism, the inclusion of this process would require a coupled-channel treatment, with a cc-gluon intermediate state, whose cut exactly produces the corresponding decay channel. Such a calculation is beyond the scope of the present work. For small binding energies, E B < T (as relevant for excited states or the case of reduced in-medium J/ψ binding energies close to its dissolution temperature), gluodissociation is phase-space suppressed and thus superseded by the Landau-damping phenomenon in the (spacelike) one-gluon exchange of the potential. This is precisely the imaginary part of the two-body potential discussed in Refs. [28,30,31]. Physically, it corresponds to "quasifree" dissociation [33], p + J/ψ → p + c +c, induced by thermal partons p = g, q,q (which generate the imaginary part of the one-loop correction to the exchanged gluon). In the limit of small binding (or large charmonium size), the incoming thermal partons with energy ∼T do not sense the size of the bound state and thus the scattering effectively happens on the level of an individual charm (or anti-charm) quark. Thus, in the T -matrix formalism this process is encoded in the selfenergy of a single (anti-) charm quark, which is included in our calculations. For large binding, quasifree dissociation ceases since the thermal parton does not resolve the substructure of the color neutral bound state [28,30]. On the other hand, when formulated as a potential contribution, the large-distance limit of its imaginary part coincides with twice the imaginary part of the charm-quark selfenergy [28]. Therefore our evaluation of the correlators within the T -Matrix approach using a real potential is well in line with the use of a complex potential in a Schrödinger treatment. In the T -matrix, imaginary parts are generated via unitarization of intermediate (on-shell) states including single-particle selfenergy contributions.

III. QUARKONIA
In this section we apply the selfconsistent T -matrix to charmonia with special consideration of the zero-mode contribution (Sec. III A) to the Euclidean correlators in different mesonic channels, α = S, P S, V, AV , i.e., scalar, pseudoscalar, vector and axialvector, respectively (Sec. III B), followed by a discussion of numerical results using either U or F as underlying two-body potential (Sec. III C).

A. Spectral Functions and Zero Modes
With the cc T -Matrix from Eq. (1) we proceed to determine the charmonium spectral function including both bound and scattering states as in Ref. [10]. The T -matrix signifies the rescattering contribution to the correlation function which is schematically given by [8] where G 0 denotes the free 2-particle loop. This can be be represented diagrammatically as where compared to Eq. (6) the loop now also includes an integration over relative momentum, k, as well as vertices (denoted by dots) specifying the quantum-number channel α. The spectral function is then defined as usual by ρ = −2ImG.
Quantitative comparisons to euclidean correlator ratios as "measured" in lQCD [34,35] require the inclusion of zero-mode contributions [12,13] which turn out to be different for different meson channels α. This, in particular, lifts the spin degeneracy within S-wave (P S − V ) and P -wave (S − AV ) states of the T -matrix. This is not surprising since HQ symmetry is not expected to be valid for zero modes. A relativistic evaluation of the vertices figuring into ρ zm α ≡ −2Im G ph α is therefore in order. To be consistent with the treatment of the T -Matrix we evaluate this contribution beyond the zero-width quasiparticle approximation (as applied in the literature to date) by taking into account the finite width of the quark. As in previous literature, we focus on the leading part of the zero-mode contribution which does not involve a twobody potential contribution 2 . Augmenting Eq. (9) with relativistic vertices one obtains [12] ρ zm where we introduced the notation with r, k and P = (E, P) denoting 4-vectors for the purposes of the above two and the following equation only. The kinematic factors arising from the different Dirac structures take the form a summation over spatial indices i is implied for the vector and axialvector case. In the previously adopted zerowidth approximation the zero-mode spectral function can be simplified to [12] ρ zm α (E, P → 0) = 2πE δ(E)χ α (T ) (18) where denotes a generalized susceptibility with coefficients corresponding to different quantum number channels according to In our numerical calculations reported below we evaluate ρ zm α directly from Eq. (15) using the positive-energy charm-quark spectral function, Eq. (7), with the selfconsistently determined off-shell selfenergy. This puts the treatment of the zero-mode contribution on the same level as the cc scattering and bound-state part using the 2-particle propagator G 12 in Eq. (6).

B. Euclidean Correlator Ratios
The transformation of the spectral function to the euclidean correlator is given by As before, we focus an vanishing total 3-momentum, P = 0, of the QQ. The temperature kernel K imprints an exponential τ dependence on the correlator. To better exhibit the in-medium modifications of G α (τ, T ) it is therefore common to analyze the correlator ratio where the denominator is the so-called reconstructed correlator which is evaluated with the kernel K at temperature T but with a "reference" spectral function typically taken as the vacuum one (or at low temperature T * where no significant medium modifications are expected).

C. Numerical Results
Before turning to the numerical results for the finite-T spectral functions and correlator ratios let us briefly summarize our input quantities, largely as given in Ref. [10]  (where more details can be found, including extensive analysis of the associated uncertainties). Our potentials (F or U ) are based on fits to the lattice results for (2+1)-flavor QCD from Refs. [36][37][38] ("potential 1" in Ref. [10]). In vacuum our off-shell calculations reproduce the results of Ref. [10] since in the limit of vanishing cquark width the propagator G 12 reduces to the standard Thompson form. The bare charm-quark mass is set to m 0 c = 1.264 GeV, which, together with a vacuum selfenergy of X vac ∞ /2 = 0.6 GeV, gives a fair description of the spin-average masses of J/ψ-η c , ψ ′ and χ c states.
The full off-shell treatment with more realistic c-quark propagators in the present work leads to moderate but significant changes for the in-medium results. In addition, the zero-mode contributions in the V , S and AV channels (recall Eq. (15)) have a marked impact on the corresponding correlator ratios. For a comparison to lQCD correlator ratios we choose the results of the N f = 2 computations from Ref. [35], which correspond more closely to our input potentials than quenched calculations. The critical temperature in the simulations of the lQCD correlator ratios [35] is T c ≃ 210 MeV, which is not far from the one underlying our potential [36], T c ≃ 196 MeV. Therefore, rather than normalizing to the different T c 's, we compare our results for the correlator ratios with the lQCD data in absolute units of euclidean time, τ . Each correlator ratio is plotted up to the midpoint, τ = 1/2T , of the total τ range (the correlators are symmetric about this point). Thus, comparable temperatures are easily identified by the same τ range in the plotted ratio (if one were to normalize all results to T c , lQCD data at given absolute temperature would be compared to T -matrix calculations at slightly lower T ).
The in-medium results using the internal energy, U , as potential are summarized in Fig. 1 in terms of the Sand P -wave spectral functions in the upper panels (degenerate for η c -J/ψ and for χ c states) and the correlator ratios in the middle and lower panels (where the degeneracies are lifted by the zero modes). Since the (magnitude of the) imaginary part of the selfconsistently calculated c-quark selfenergy turns out to be around 0.050-0.100 GeV for low-momentum on-shell charm quarks (cf. Fig. 3 below), the closest comparison to our earlier quasiparticle results is for the case of a constant (energy and 3-momentum independent) imaginary part of Im Σ Q = −0.05 GeV (Fig. 14 in Ref. [10]). In the Swave spectral function we find slightly more attraction for the ground-state peak for temperatures below 1.5 T c (for higher T it dissolves, as in Ref. [10]). Despite the larger on-shell width in our present treatment, which is up to twice as large as in our previous quasiparticle calculations, the width of the J/Ψ-η c peaks is very similar. This is a direct consequence of the energy and momentum dependence of the selfenergy which decreases considerably off-shell (cf. Fig. 3 below) and thus reduces the "operative" width of c-quarks in cc bound states. It becomes apparent in the η c correlator ratio (where no zero-mode is active) which at the lowest considered temperature (1.2 T c ) drops to about 0.85 compared to 0.9 in Ref. [10], even though the peak position of the bound state is shifted to slightly lower energies in the present calculation (which tends to increase the large-τ correlator ratio). This finding shows that a proper off-shell treatment is warranted to correctly account for the low-energy strength in the spectral function which has significant impact on the correlator ratio at large τ . We are not too concerned that the drop to 0.85 is noticeably larger than in the lQCD data since we have neglected several effects which will contribute further low-energy strength to the spectral function, e.g., imaginary parts in the c-quark selfenergy from scattering off thermal gluons, or coupled channels in the cc T -matrix such as DD and ccg (inducing singlet-to-octet transitions). These are expected to be especially relevant close to T c (D-meson states will form close to T c and gluo-dissociation is efficient for large charmonium binding, E B ≥ T ). On the other hand, we note that the P S correlator ratio is remarkably independent of temperature and closer to one for intermediate and small τ than in Ref. [10] (e.g., no more than ∼2% above one), which improves the agreement with lQCD. In the V , S and AV channels the zero-mode contributions lead to a marked enhancement of the correlator ratios at large τ , especially for the P -wave channels where the c 1 coefficient is non-zero (for the vector channel, we sum over the spatial components only, corresponding to Γ = γ i in Eq. (15). Compared to a zero-width treatment of the zero-mode contribution, corresponding to Eq. (19), the inclusion of a finite quark width increases the correlator ratios at large τ by ca. 0.1-0.15. Overall, the agreement of the the calculated correlator ratios with N f = 2 lQCD data [35] included in the middle and lower panels of Fig. 1 is fair. The largest discrepancies of ca. ∼30% at both intermediate and large τ occur in the AV (χ c1 ) channel where the zero-mode contribution is the strongest. The extracted melting temperatures of the charmonium bound states are close to our earlier determination with finite but constant width [10], i.e., ca. 1.5 T c for the J/Ψ and below 1.2 T c for all other states (Ψ ′ , χ c ).
The results obtained using the free energy, F , as potential are compiled in Fig. 2. As in previous work [9,10,39], we find a much stronger suppression of the bound states compared to using U , with a melting temperature for the S-wave ground state of 1.2 T c or even lower, whereas the P -wave spectral function is already structureless at this temperature. Nevertheless, the correlator ratio in the η c channel (no zero mode) is surprisingly T -stable and close to one: the loss of low-energy strength due to the dissolved bound state is compensated by a reduced cc threshold in connection with a nonperturbative rescattering strength in the threshold region generated through the T -matrix [8]. Indeed, the in-medium charm-quark mass (correction) following from Eq. (8) for F ∞ is significantly smaller than for U ∞ . This, however, generates a markedly enhanced zero-mode contribution to the correlator ratios in the V , S and AV channels, compared to the case with U as potential (the effect of the selfconsistently calculated finite quark width in the zero modes is a ca. 10% enhancement at large τ ). In particular in the P -wave channels, the large-τ is quite a bit off the N f =2 lQCD data. Apparently a potential closer to the internal energy is favored (with larger m * c but stronger cc binding).

IV. OPEN-CHARM TRANSPORT
We now turn to examining the numerical results for the single charm-quark properties in the QGP, specifically its selfenergy (mass correction and scattering rate; Sec. IV A), thermal relaxation rate (Sec. IV B) and number susceptibility (Sec. IV C). We recall that these are selfconsistently computed via numerical iteration with the T -matrix in all heavy-light quark channels, Eq. (1).

A. Selfenergy
The real and imaginary parts of the charm-quark selfenergy following from the T -Matrix are displayed in Fig. 3 as a function of quark energy for vanishing 3momentum and for three temperatures. The general structure is that of a maximum around the on-shell en- ergy, ω ≃ m * c , in the imaginary part, associated with a typical zero crossing in the real part. The peak structure is caused by Feshbach type resonances (or threshold enhancements) around the heavy-light quark threshold in the T -matrix [10,14]. These features are the reason that, upon (off-shell) integration over ω in the two-particle propagator (T -matrix), the contributions from the real part largely cancel while the width effects drop significantly faster (and thus on average are smaller) than for a fixed (constant) quasiparticle particle width as employed in previous work [8,10,14].
More quantitatively, in the case of U as potential (left panel in Fig. 3), the on-shell width of the charm quark reaches a value of up to Γ c = −2 ImΣ c ≃ 250 MeV at the lowest considered temperature (1.2 T c ), quite similar to what has been calculated in Ref. [14] using different lQCD inputs for U . The magnitude decreases with increasing temperature (not as much as in Ref. [14]), which is quite remarkable given the fact that the thermal densities of the thermal anti-/quarks decrease appreciably (in the "T -̺" approximation, the selfenergy would be directly proportional to the light-quark density, Σ c ∼ T cq n q ). It clearly reflects the weakening of the two-body interaction as temperature increases, due to color-charge screening in both the Coulomb and confining parts of the potential. In other words, the maximal interaction strength is realized at the lowest temperature, just above T c .
For the F -potential the c-quark selfenergy is reduced by about a factor of 6 at the lowest temperature, which reduces to a factor of ∼2-3 at 2 T c . In this scenario, no Feshbach resonances form, but rescattering embodied in the T -matrix still produces a notable threshold enhancement which induces an on-shell width of ca. 40-70 MeV.

B. Thermal Relaxation Rate
Next we turn to the calculation of the thermal relaxation rate of charm quarks in the QGP, employing a Fokker-Planck treatment following Ref. [40]. The relation of the drag coefficient, A(p), to the T -matrix has been elaborated in Ref. [14], including all color configurations in c-q and c-q scattering in S-and P -waves. For a more realistic evaluation of the total coefficient, we add to the T -matrix contribution the effect of c-quark scattering off thermal gluons which we approximate with the leading order perturbative diagrams (using α s =0.4) including Debye screening masses in the exchange propagators (t-channel gluon exchange gives the dominant contribution). In Fig. 4 the full results for A(p) are compared to perturbative calculations in which scattering off both anti-/quarks and gluons is obtained from the LO diagrams. Compared to the quasiparticle treatment in Ref. [10], the full off-shell treatment leads to a ca. 10% increase of the drag coefficient at low 3-momenta, while the deviations are small at momenta of k ≥ 2 GeV. This is well in line with the finding in Ref. [10] that the dependence of the drag coefficient on the 3D reduction scheme of the T -matrix equation (relating to its off energy-shell behavior via different 2-particle propagators) is small. Thus, we confirm as a robust feature that c-quark thermalization using the nonperturbative T -matrix is accelerated over pQCD calculations by about a factor of 4 (2) when using U (F ) as potential.
In addition to the direct calculation of the relaxation rate following from the collision term in the Boltzmann (or Fokker-Planck) equation with the heavy-light T -matrix [10,14], the off-shell framework set up in the present paper enables an alternative method, namely from the zero-energy limit of the heavy-quarkonium spectral function [11,41], (χ c ≡ χ 00 : charm-quark number susceptibility, D s : spatial diffusion constant). The values extracted from this method correspond to the zero-momentum limit and are compared to the pertinent values from the collision integral using the heavy-light T -matrix in Tab. I (for a consistent comparison the contribution from scattering off thermal gluons is not included). Note that the latter involves the Qq scattering amplitude squared, while the imaginary part of the quarkonium spectral function is basically determined by the imaginary part of the singlequark spectral function which, in turn, is obtained from the imaginary part of the heavy-light scattering amplitude, cf. Eq. (10). We find that the extraction from the low-energy limit of the charmonium spectral function in the vector channel tends to give larger (smaller ) values at lower (higher) for both U or F as two-body potential. One of the uncertainties which is presumably reflected by these deviations is the Fokker-Planck approximation when using the heavy-light T -matrix to evaluate the diffusion coefficient. For example, in Ref. [42] it has been found that a D-meson resonance model in the QGP can lead to a violation of the Einstein relation by underestimating of γ c by up to 10% at temperatures of T ≃ 300 MeV, while the agreement is closer at lower temperatures (less than 5% for T < 250 MeV; note that the D-meson like correlations in the T -matrix dissolve for temperatures above 1.5 T c ≃ 300 MeV). Overall, the discrepancies between the two methods are within ∼30%, which, given the different schemes of evaluating this transport coefficient, is not too bad. The systematic trends are similar when using F or U as potential, and the large difference between these 2 scenarios is robust.
Let us end this section by a schematic evaluation of the widely discussed ratio of viscosity to entropy density, η/s. Using kinetic theory, one can roughly relate this quantity to the spatial HQ diffusion coefficient as [3] The numerical coefficient probably constitutes a lower limit, applicable to a weakly coupled system; it is most likely larger for strongly coupled liquids, e.g. ∼1/2 in AdS/CFT. Nevertheless, the results for U suggest that η/s is not far from the conjectured lower limit of 0.08 for quantum liquids. Moreover, its T -dependence seems to indicate a minimum value when approaching T c from above which would resemble rather generic behavior of substances in the vicinity of a critical point. This feature is also present for the F -potential, albeit the Tdependence is less pronounced; of course, the values are also much larger compared to the U -potential, by about a factor of 2-4. In fact, the η/s value for F at 2 T c is rather close to perturbative calculations [43,44] where η/s ≃ 1 with little T -dependence, indicating that resummation effects in the T -matrix do not play a large role under these conditions.

C. Charm-quark number susceptibility
Quark-number susceptibilities, χ q , which we already utilized in connection with the c-quark diffusion coefficient, Eq. (22), can be computed rather accurately in lQCD [45] and are therefore of great interest to constrain effective models of the QGP. For example, HQ number susceptibilities have been used to extract effective inmedium charm-and bottom-quark mass corrections by fitting a zero-width quasiparticle expression, Eq. (19), to the lQCD results [46]. Here, we carry out a full offshell calculation including finite-width effects through the single c-quark propagators figuring into the charmonium spectral function [47], For high T ≫ m Q , Σ Q , this quantity reduces to In Fig. 5 we summarize our results for the HQ susceptibility and compare to lQCD data as well as the zero-width quasiparticle limit, Eq. (19), used in previous estimates. When plotted as a function of temperature over in-medium c-quark mass (left panel), the results for the U -potential are slightly above the ones for F (note that the in-medium c-quark mass is 10-20% larger for the U -potential, mostly due to the larger mass correction resulting from U ∞ /2 compared to F ∞ /2). The full results using U are significantly above an off-shell calculation where the imaginary parts are put to zero, which in turn agree very well with the zero-width quasiparticle limit (dashed line). When plotted as a function of temperature (right panel in Fig. 5), it turns out that the results for F are somewhat above those for U , due to the smaller m c (T ). On the other hand, the finite-width effects in the full results using U produce an increase in χ c over the no-width limit which corresponds to a c-quark mass decrease of ca. 150-200 MeV in the zero-width quasiparticle expression. Thus, finite widths considerably affect the extraction of the in-medium c-quark mass from the susceptibility. This effect also leads to a significant improvement in the comparison to lQCD results: the full results with U roughly lie in the uncertainty band encompassed by the lQCD results while the full results with F tend to lie at the upper end of that band. At the lowest considered temperature, the results for U seem to lie below the N t =8 lQCD computations, but as indicated in connection with the large-τ limit of the S-wave charmonium correlator ratios, we believe that further width (and coupled-channel) corrections need to be included close to T c before more precise conclusions can be drawn. At least the underestimate of the lQCD data in our calculations for χ c and the large-τ limit of R P S,V is consistent.

V. CONCLUSIONS
We have conducted a study of charmonium and opencharm properties in the Quark-Gluon Plasma using a thermodynamic T -Matrix formalism 3 . For the first time in the context of heavy quarks in the QGP, we have implemented this scheme selfconsistently at the one-and two-body level (i.e., selfenergy and scattering amplitude), including microscopically calculated off-shell effects, in particular imaginary parts. Within the the Matsubara formalism, the two-particle propagator in the T -matrix equation automatically generates "zero-mode" contributions, i.e., scattering off pre-existing charm quarks in the heat bath. Following our earlier work [10] the two-body input potential was constructed using a field-theoretical model for Coulomb and confining forces with its 4 parameters fitted to finite-temperature lattice QCD data of the color-averaged heavy-quark free energy. As limiting cases of the interaction strength we have considered the resulting free and internal energy as an underlying potential. Once the input potential and the bare charmquark mass are fixed (the latter is adjusted to reproduce the charmonium ground-state mass in vacuum), there are no further tunable parameters in the heavy-heavy and heavy-light sector of our approach; heavy-quark interactions with gluons are treated perturbatively but turn out to play a minor role (even with α s =0.4). We are then able to comprehensively compute hadronic spectral functions in different charmonium and D-meson channels, as well as heavy-quark selfenergies and transport properties in the QGP. Specifically, we have applied these quantities to calculate euclidean correlator ratios for charmonia (consistently including zero-modes in scalar and axial/vector channels) and the charm-quark number susceptibility, which have both been "measured" with good accuracy in thermal lattice QCD. Overall, we find an encouraging degree of agreement of our results with those in lQCD, especially at temperatures T ≥ 1.5 T c . (possi-bly with a preference for using U as potential), keeping in mind that neither F nor U , as computed in lQCD, necessarily provide a rigorous definition of a two-body potential, as both quantities are computed from differences of thermal averages. Especially close to T c a more complete description of charm and charmonia remains a challenge, requiring further refinements such as the inclusion of (pre-) hadronic states in the QQ and Qq T -matrices (coupled channels), a nonperturbative treatment of interactions with thermal gluons and a careful assessment of retardation effects. The large entropy contribution in the HQ free energy close to T c indeed suggests many additional states to play a role. Another extension of our approach concerns the 3-momentum dependence of charmonium correlators/spectral functions, for which interesting lQCD results are now becoming available [48].
Our analyses and findings reveal the intricate relation between charmonium bound-state properties and heavyquark diffusion in the QGP, which, in particular, cannot be captured by perturbative treatments. The very same (resummed) force that generates resonance-like correlations (or threshold enhancements) in charmonium spectral functions -crucial for properly describing the lQCD correlator ratios -is operative in low-momentum charmquark scattering (closely related to zero modes), and thus most likely instrumental in reducing the pertinent thermal relaxation times as required in phenomenological applications to RHIC data [14,49]. It remains an open question how large c-quark momenta need to be for radiative processes to become competitive with elastic scattering [1,50].
In conclusion, we believe that studying the many-body physics of heavy-quark systems in the QGP can indeed reveal valuable insights into the medium modifications of the basic QCD force via its effects on bound-state properties and heavy-flavor transport. The nonperturbative nature of the problem, especially in the vicinity of T c , requires effective approaches whose predictive power greatly benefits from constraints obtained from thermal lattice QCD. This will hopefully pave the way toward illuminating and quantifying the properties of the QGP even in the case of rather strong coupling, and help understand some of the fascinating phenomena observed in ultrarelativistic heavy-ion collisions.