Nonperturbative Effects on Radiative Energy Loss of Heavy Quarks

The radiative energy loss of fast partons traveling through the quark-gluon plasma (QGP) is commonly studied within perturbative QCD (pQCD). Nonperturbative (NP) effects, which are expected to become important near the critical temperature, have been much less investigated. Here, we utilize a recently developed $T$-matrix approach to incorporate NP effects for gluon emission off heavy quarks propagating through the QGP. We set up four cases that contain, starting from a Born diagram calculation with color-Coulomb interaction, an increasing level of NP components, by subsequently including (remnants of) confining interactions, resummation in the heavy-light scattering amplitude, and off-shell spectral functions for both heavy and light partons. For each case we compute the power spectra of the emitted gluons, heavy-quark transport coefficients (drag and transverse-momentum broadening, $\hat{q}$), and the path-length dependent energy loss within a"QGP brick"at fixed temperature. Investigating the differences in these quantities between the four cases illustrates how NP mechanisms affect gluon radiation processes. While the baseline perturbative processes experience a strong suppression of soft radiation due to thermal masses of the emitted gluons, confining interactions, ladder resummations and broad spectral functions (re-)generate a large enhancement toward low momenta and low temperatures. For example, for a 10 GeV charm quark at 200 MeV temperature, they enhance the transport coefficients by up to a factor of 10, while the results smoothly converge to perturbative results at sufficiently hard scales.


Introduction
In ultra-relativistic heavy-ion collisions (URHICs) heavy quarks are mostly produced in primordial hard collisions of the incoming nucleons, on a short timescale, τ prod ∼ 1/2m Q , governed by the heavy-quark (HQ) mass, m Q ; therefore, they probe the entire evolution of the fireball formed in these reactions. The large mass of heavy quarks also enables theoretical simplifications in the description of their transport through the medium [1], allowing for rather direct connections between their microscopic interactions and pertinent observables in experiment. Heavy-flavor (HF) hadrons are therefore excellent probes of the properties of the QGP and its hadronization in URHICs [1][2][3][4][5].
One generally distinguishes two processes that control the dynamics of heavy quarks in the QGP -collisional and radiative ones. The former are mainly responsible for thermalization of heavy quarks at low momenta, p Q 3M Q T , while the latter become increasingly relevant at higher HQ momenta [6][7][8]. Radiative processes, i.e., medium-induced gluon emission, in the relativistic limit have been widely studied over the past two decades. Various formalisms, such as AMY [9], ASW [10], BDMPS-Z [11,12], DGLV [13,14], higher-twist [15,16], and SCET [17,18], have been developed for the phenomenology of light-flavor jet quenching in URHICs. Among them, some have been extended to massive quarks [10,[19][20][21][22][23] and implemented into numerical simulations for their transport in URHICs [7,[24][25][26][27][28][29], see also Refs. [30] and [31,32] for recent comparisons between different energy loss formalisms for light-flavor jets and heavy quarks, respectively. Gluon emission within these approaches is generally described using perturbative-QCD (pQCD) methods. Within different expansion schemes, these approaches concentrate on the regions of the phase space where nonperturbative (NP) effects are expected to be small or can be absorbed into a transport parameter that encodes the medium properties, most notably the jet transport coefficient,q . So far, few studies of NP effects in radiative processes have been carried out at a microscopic level (see, e.g., Ref. [26]), especially in the relatively lowmomentum and temperature regimes where standard expansion and factorization schemes are no longer reliable. The objective of the present work is to utilize a NP T -matrix approach to provide insights into mechanisms of radiative energy loss of heavy quarks which are not easily assessed in pQCD approaches.
The phenomenology of HF observables in URHICs, especially the measured elliptic flow, requires heavy quarks to have an interaction strength with the medium that goes well beyond pQCD [3,31,33], especially at low momentum and temperature. As rigorous QCD calculations are challenging in this regime, one has to resort to model approaches. An essential step in improving their reliability is to root the model parameters in fits to lattice-QCD (lQCD) data as much as possible. In our previous work we have developed a quantum many-body approach with a QCD-inspired model Hamiltonian [34][35][36] that turns out to be an effective tool in describing a wide variety of QGP properties with a relatively small number of parameters, including its equation of state (EoS), HQ free energies, and correlation functions, while predicting spectral properties and transport coefficients (e.g., shear viscosity and HQ diffusion coefficient). Key features of this approach are the inclusion of remnants of the confining force in the QGP, ladder resummed amplitudes leading to the dynamical generation of bound states as the pseudo-critical temperature, T pc , is approached from above, and a full off-shell treatment of propagators and scattering amplitudes which is, in fact, mandated by the large widths that develop for low-momentum partons. In this work, we will deploy this approach to calculate the radiative energy loss of heavy quarks. The off-shell transport theory underlying this framework is based on the Kadanoff-Baym equation following the same logic as used in our previous study of the HQ collisional energy loss [36]. To illustrate how NP physics affects gluon radiation, we set up a systematic comparison between four different scenarios. Case (1) is the perturbative baseline in our approach that only includes screened Coulomb interactions at the Born level. We add confining interactions in case (2) and the resummation of t-channel diagrams in case (3). In case (4), we further include the off-shell effects for parton and HQ propagators, i.e., their broad spectral functions, representing the full results in our current framework. We analyze the differences between these four setups for several quantities, such as emission power spectra, transport coefficients, and fractional energy loss within a QGP brick medium.
The paper is organized as follows. In Sec. 2 we lay out our formalism for calculating the HQ radiative energy loss within the T -matrix formalism (Sec. 2.1), disucss its realtion to pQCD diagrams (sec. 2.2), and define four different model setups with an increasing content of NP components (Sec. 2.3). In Sec. 3, we compare the numerical results for typical energy loss quantities from these four cases. A summary and outlook is given in Sec. 4.

Gluon Radiation in the T -Matrix Formalism
In this section, we first develop the formalism for computing the radiative energy loss in the context of the in-medium T -matrix (Sec. 2.1), discuss the relation of the NP calculation to the diagrams in pQCD including the issue of gauge invariance (Sec. 2.2), and then introduce four different model cases of increasing levels of NP effects to illustrate how these influence radiative processes of charm quarks (Sec. 2.3).

Many-body formalism of heavy-quark radiative energy loss
The derivation of the formalism for evaluating radiative processes is similar to that in our previous work [36]. We first recall the calculation for collisional processes and then derive the equations for radiative processes.
We start with a Kadanoff-Baym equarion for the heavy quark (Q) in terms of its Green's function, G Q and selfenergy, Σ Q , where the >, < superscripts denote the fixed-ordered quantities in real-time many-body field theory (sometimes referred to as Wightman functions). These quantities depend on Wigner coordinates ω and p with their Fourier conjugates t 1 − t 2 and x 1 − x 2 , respectively. By putting the incoming heavy quark on-shell [36], Eq. (2.1) can be reduced to a semi-classical Boltzmann equation for its phase space distribution where the transition rate w(p, k) encodes the quantum many-body information of the system, which can be used to obtain transport coefficients [2]. The first and second terms on the right hand side of Eq. (2.2) correspond to the first and second terms of Eq. (2.1) respectively, as discussed in Ref. [36]. For the collisional 2→2 scattering processes, Σ > Q in Eq. (2.1) can be schematically expressed via a HQ scattering amplitude off thermal partons, M Qi→Qi , as where p (p ) and q (q ) are the 4-momenta of the incoming (outgoing) heavy quark and thermal parton, i, respectively. We use the notation d 4p = d 4 p/(2π) 3 2 (p) for the 4momentum phase space elements (with on-shell energy Q (p)) and δ (4) for the 4-momentum conserving Dirac delta function. The Green's functions can be represented by the spectral functions (ρ) with Bose/Fermi factors, n i,Q , as The transition rate, w(p, k), derived from Σ > Q can then be expressed as where ω is the energy of the outgoing heavy quark. The off-shell spectral functions of both thermal partons and heavy quarks (x = Q, i), which encode the non-trivial quantum many-body physics of the system, can be expressed as where the M xj→xj are "elastic" heavy-light or light-light scattering amplitudes. The summation is over all internal degrees of freedom with spin (s), color (c) and flavor (f ) degeneracy factors and averaging procedure. The second line of Eq. (2.7) is related to its first line through the optical theorem. We utilize the second line to calculate the elastic selfenergy as detailed in Ref. [35]. The real part of the selfenergy is obtained from a dispersion relation.
The key step to extend the above formalism from collisional to radiative processes is to replace the collisional amplitude M 2→2 in Eq. (2.3) with radiative amplitude M 1→2 and remove one incoming Green's function, G < i . This leads to where k = (ν, k) is he outgoing gluon 4-momentum. The amplitude for the Q → Qg process is calculated using the pQCD diagrams for gluon emission, (2.10) We choose the standard Dirac spinor (u) and the polarization tensor ( µ ) in Coulomb gauge where the sum of the polarization tensors is s ε µ ε ν = δ ij − k i k j /k 2 . The amplitude is evaluated using on shell energies where the in-medium HQ mass is determined by the manybody formalism [36], while, for reasons of gauge invariance, we restrict the gluon polarization tensor to its transverse components, i.e., In leading order for the bare Q → Qg splitting process, a restriction to on-shell energies, ω = ε Q (p ) and ν = ε g (k) in the spectral functions leads to a vanishing rate w(p, k), since the δ-functions for energy and momentum conservation cannot be simultaneously satisfied. However, at the leading dressed order (skeleton order), w(p, k) is finite since the spectral functions for outgoing heavy quark and gluons allow for off-shell energies, opening up phase space that simultaneously satisfies energy and momentum conservation. As shown in Eqs. (2.6) and (2.7), the information on elastic scatterings between the outgoing partons (Q, g) and the medium, and the mean free path of the outgoing partons, are encoded in the off-shell properties of the spectral functions. Following Ref. [2], the transition rate w(p, k) can be used to evaluate the drag coefficient (as used in a Langevin simulation below), where k L ≡ (k · p/p) is the longitudinal momentum transfer. For high-energy (or large k) radiative processes, k L ≈ k is a good approximation. Thus, it is useful to define the power function k w(p, k) so that the drag coefficient can be approximately expressed as A(p) ≈ d 3 kk w(p, k)/p. If we further assume an azimuthal symmetry of the emission, we can define the power function per k T and k L phase space as The factor 2πk T originates from integrating over the azimuth angle. This power function characterizes how the radiated energy is distributed over the phase space. We can integrate over k T to obain where x = (k L + ε g (k))/(p + ε Q (p))) (x ≈ k L /p at high momentum) is the longitudinal momentum fraction of the emitted gluon taken from the parent heavy quark. The variable x is often referred to as a light-front coordinate and commonly used in existing literature for radiative energy loss.

Relation to pQCD diagrams
To obtain a better understanding of how the NP framework employed here relates to pQCD calculations, we carry out a comparison of relevant diagrams in this section. This will also allow us to address the issue of gauge invariance for the NP case, which is a rather challenging one in the presence of resummed interactions. + + + + Figure 1. The first row displays the square of the Born diagrams that can be generated from the cuts of the diagrams shown in the lower two rows.
We start with the tree-level pQCD diagrams [22,37] shown in Fig. 1. The diagrams displayed in the first row represent the (naive) order-g 2 contributions to the amplitude for two incoming quarks going into two quarks and a gluon. The sum of these amplitudes is transverse (k µ M µ = 0) and thus gauge invariant in the usual sense. If we contract the external legs between terms in the left and right parentheses, we obtain the selfenergy diagrams in the second and third rows, which relate to the diagrams in the first row by cutting rules (optical theorem). As discussed in the previous section, the key quantities to evaluate the radiative processes are the selfenergies, which explicitly figure in Eq. (2.1) or in the spectral functions as shown in Eq. (2.6). The NP selfenergies can be calculated through a series of coupled Dyson-Schwinger equations (DSEs); our approach falls into this category. The structure of the equations is controlled by the skeleton diagram expansion where an example is shown in Fig. 2. In this example, by inserting the dressed vertex (first row) and the dressed gluon (second row) into the equation in third row, we can generate the pQCD selfenergy diagrams shown in second and third rows in Fig. 1. With a specific choice of the skeleton expansion, the diagrams generated by DSEs can encompass any finite-order perturbative diagrams. In other words, using the DSEs perturbatively is an alternative and equivalent diagram expansion method for the perturbative diagram expansion. With a specific truncation scheme, the DSEs form a closed system of equations that allow NP solutions. These solutions can be obtained through selfconsistent iterations. They resum the diagrams with specific patterns (such as ladder diagrams, rainbow diagrams and ring diagrams) to infinite order, which goes beyond any fixed-order perturbative expansion and thus can provide meaningful results at large coupling strength. However, these truncation schemes usually compromise explicit gauge invariance since they select a particular subset of diagrams to resum. There is no general solution to this problem yet in the strongly interacting regime, although progress has been made in resolving this issue in vacuum [38][39][40].
In the present article we focus on NP features at large interaction strength. We compromise on exact gauge invariance by carrying out the t-channel ladder resummation but restrict ourselves to projecting out the gauge invariant part by applying the transverse projector, M µ T = P µν T M ν 1 , i.e., we only keep the bare radiation vertex with emission of the gauge invariant transverse mode, while in principle also a gauge invariant longitudinal mode is available in medium [41,42]. Thereby we also neglect the vertex corrections which would be required in connection with an in-medium emission vertex. More rigorous methods are beyond the scope of this work.
To highlight model-independent information obtained from our apporach, we conduct calculations for four setups with different inputs representing different NP effects and compare their differences. Since they are all calculated in one model, potential gauge artifacts are expected to be similar in the different setups so that the differences between them give insights into model-independent features.

Four cases of NP effects
Within the formalism laid out above the NP effects that we study in this work are essentially encoded in the spectral functions (selfenergies) of Eq. (2.7) which figure in the emission rate, Eq. (2.11). These effects are mainly from 3 sources: confining interactions, ladder resummations, and off-shell interactions, each of them playing a critical role in our manybody approach to heavy quarks in the QGP [35]. In the following we will define 3 scenarios, in addition to the full results, which allow us to exhibit their effects on gluon radiation from a heavy quark traveling through the QGP.
2nd-order 2nd-order All order All order Medium quasi-particle quasi-particle quasi-particle off-shell spectra Table 1. Four cases are labeled by numbers or abbreviations in the first row where their differences in interaction, resummation scheme, medium content are listed in the second, third, and fourth rows respectively.
X X X X Figure 3. Self-energy diagrams for the four different cases discussed in Tab. 1. Helical lines denote V C , helical lines with double dashed lines denote V CS , thin solid lines with arrows denote on-shell quasi-particle partons (q, g, HQ), thick solid lines with arrows denote off-shell dressed partons.
. The four cases are summarized in Tab. 1, illustrated in Fig. 3 and defined in detail in the following.
Case-1, "V C Born", contains none of the three NP effects mentioned above, and as such represents the perturbative baseline in our approach. It only includes color-Coulomb interactions with coupling constant and Debye mass taken from the strongly coupled solution (SCS) in Ref. [36]. It does not contain a t-channel ladder resummation, and we only keep the second-order Born diagrams as shown in Fig. 3, which are the leading-order diagrams that generate an imaginary part of the selfenergy and consequently a finite width of the spectral functions of the outgoing heavy quark and gluon in Eq.(2.11) as required for a finite radiation rate. These diagrams are related to the amplitude squared of the Born scattering amplitude using the cutting rule. The thermal-medium partons are taken as zero-width quasiparticles with their masses fitted to the QGP EoS as discussed in our previous work [36]. This case is most closely related to the treatment in typical pQCD calculations, although some differences persist, most notably the large thermal masses of the emitted gluon.
Case-2, "V CS Born" adds a NP effect to case-1, namely the confining interaction (potential) again with parameters taken from the SCS of Ref. [36], illustrated in the second panel by the extra bars in the gluon exchange lines. This is still done in leading-order Born approximation and with the same medium as in case-1.
Case-3, "T-matrix Onshell", additionally includes the t-channel ladder resummation in the heavy-light and light-light T -matrices (used to compute the outgoing HQ and gluon selfenergies) compared to case-2, which is illustrated in the third panel of Fig. 3. The other components are the same as in case (2).
Finally, case-4, "T-matrix Offshell", uses finite width off-shell spectral functions for the thermal-medium partons. As shown in the last panel of Fig. 3, we now dress all internal lines in a selfconsistent caculation of selfenergies and T -matrices. In addition to the medium partons in the upper part of the diagram, the incoming and internal HQ lines are also dressed. The off-shell spectral functions are taken from the full solution of the SCS as discussed in Ref. [35]. Case-4 is the most complete and consistent many-body theory calculation for the rate within our current approach, containing all three NP effects as highlighted at the beginning of this section.
Let us briefly reiterate on the role of the thermal-medium partons in the HQ transport calculation. For cases-1, -2 and -3, we use a zero-width quasiparticle medium. For case-4, the broad spectral functions that represent the off-shell medium are the predictions of the SCS in Ref. [36]. Both scenarios describe the EoS of the QGP, implying that the effective density of "scattering centers" is quite similar, in order to focus on the genuine effects of the NP ingredients. Also note that the resummations carried out in the present work refer to t-channel ladder diagrams for a single (dynamic) scattering center. Higher orders due to multiple scattering centers and multiple gluon emission are implicitly included in the Langevin simulation, which is, of course, incoherent and thus neglects the Landau-Pomeranchuk-Migdal (LPM) effect [43][44][45]. While the latter is essential for radiation off light-flavor partons, it is mass suppressed for heavy quarks due to reduced formation times. A rigorous implementation of coherence effects in transport approaches remains challenging, see, e.g., Ref. [24] for an approximate treatment in the HQ context.

Numerical Analysis
In this section, we present and analyze the numerical results for the four cases defined above. We first study various microscopic quantities and then discuss how they manifest themselves in transport coefficients. We start by presenting the pertinent spectral functions for heavy quarks and gluons in Sec. 3.1, followed by the corresponding power spectra in Sec. 3.2. The resulting drag coefficients and fractional energy loss are calculated in Sec. 3.3. In Sec. 3.4 we discuss our results in comparison to a pQCD calculation from the literature.

Spectral properties of radiated partons
The spectral functions of the outgoing heavy quark and gluon are the key quantities for calculating the rate of radiation for the Q → Qg process given by Eq. (2.11). These spectral functions are shown in Fig. 4 for our 4 different cases, each one for three HQ and gluon momenta, p=2, 10, 40 GeV and two temperatures , T =0.194, 0.4 GeV, as a function of energy around their on-shell values. We recall that for cases 1-3, these spectral functions differ from those in the internal lines in Fig. 3 where they are zero-width quasiparticles. For all cases, a markedly different feature from the standard pQCD approach are the rather large thermal-parton masses dictated by the constraints from the QGP EoS and microscopically related to the Polyakov loop encoding the nontrivial information of the confinement [35] generated selfconsistently from the confining potential. Similar features from including effects of the Polyakov loop have recently been found from a different perspective, within the Polyakov-quark-meson model [46]. With a gluon mass of around 1 GeV at low temperature, soft radiation is heavily suppressed. With increasing 3-momentum at a fixed temperature (from top to bottom row in Fig. 4), the spectral functions in the "V C -Born" case (left panels) show an increasing width while the opposite trend is found for the other 3 cases which is caused by the presence of the confining interaction whose strength rapidly decreases with increasing momentum (transfer), especially at low temperature. At the same time, at high momentum (bottom row), there is little difference among the different cases, with well defined quasiparticles characteristic for pQCD calculations. These features are a direct reflection of our previous findings that the QGP medium at moderate temperatures is strongly coupled at large wavelengths while recovering a more weakly interacting quasiparticle structure with increasing resolution [35]. For the temperature dependence at a fixed 3-momentum, the "V C Born" case shows a clear increase in width from low to high temperatures, which is a consequence of the increasing thermal-parton density which overwhelms the moderate loss in interaction strength caused by a stronger Debye screening. This remains true for the other three cases at high momenta, while at low momentum this trend is much less pronounced, even slightly inverted for the "T-matrix Offshell" case. At the lowest temperature and momentum (T = 0.194 GeV, p =2 GeV), all 3 cases (2-4) involving the confining interaction show appreciable distortions from a simple Lorentzian shape. In particular, the gluon spectral functions exhibit substantial strength, even collective modes signaled by peaks, in the spacelike region, i.e., for ω < p (for both temperatures), which is the most relevant one for the emission process of the incoming on-shell heavy quark. The spacelike strength tends to increase with the inclusion of more NP effects. At high temperature and low momentum (T = 0.400 GeV, p =2 GeV), the inclusion of the string interaction alone, at the Born level, strongly enhances the strength in the space-like region, compare the dash-dotted blue lines in the 2nd and 1st panel of the upper row in Fig. 4. Although the addition of resummation (case-3) and off-shell (case-4) effects further distorts the spectral functions (cf. 3rd and 4th panel), they do not lead to significant differences for the phase space in the spacelike region. For higher momentum, p=10 GeV, at both temperatures, the change of the spectral properties by adding the NP effects is less pronounced. Still, the largest modification is generated by the inclusion of the string interaction at low tempera-  tures, while the further differences in cases 3 and 4 are not significant. At high temperature and p=10 GeV, even the inclusion of the string interaction is no longer significant.

Power spectra
Inserting the spectral functions discussed in the previous section into the rate, Eq. (2.11), we can evaluate the power spectra of radiation defined in Eqs. (2.13) and (2.14). The former essentially corresponds the radiation spectrum, while the latter, with the transverse momentum transfer integrated, is more readily interpreted as the radiative energy loss of the heavy quark. At the end of this section we compute the transverse-momentum broadening coefficient,q, for the "V C Born" and "V CS Born" cases and illustrate how they relate back to their differences in the power spectra. In Fig. 5 we display contour plots of the power spectra in the k T -k L plane, where each of the 4 interaction cases is represented by each of the four columns, and each row represents a pair of temperature and HQ 3-momentum, for p=10 GeV with T =0.194, 0.4 GeV in the upper 2 rows, and p=40 GeV with T =0.194, 0.4 GeV in the lower 2 rows. The power spectra are projected onto the k L values (i.e., integrated over k T ) and shown in Fig. 6, where the 4 combinations of p and T are combined into a single plot for each of our 4 cases. The generic features of the power spectra are a suppression of very soft radiation, i.e., for small k L , due to the large effective gluon mass, and a suppression of the collinear radiation, i.e., for small angles, k T /k L , which is the well known deadcone effect [47]. As expected, the  2 and 4)). The maximum values, W max , of each the power spectra are labeled as "max" with unit (GeV/fm) in the plots. radiation power increases with both temperature and momentum, roughly by one order of magnitude when going from T =0.194 to 0.4 GeV, and between a factor of 5 (for case-1 at T=0.194 GeV) and 10% (for cases 2-4 at T =0.4 GeV) when going from p=10 GeV to 40 GeV (note that the contour plots have been scaled to their respective maxima in each panel, as quoted in the figure legend). The k T distributions of the power spectra tend to narrow down with increasing p.
The most signficant of the NP effects is the inclusion of the string interaction, when going from case-1 to case-2, and it is more pronounced for lower temperatures, as borne out from the maximum values in the power spectra which increase between a factor of ∼30 for (p,T )=(10,0.194) GeV and factor of ∼1.6 for (p,T )=(40,0.4) GeV. However, at low temperature, the additional NP effects in cases 3 and 4 (resummation and off-shell effects) are still rather significant, especially for rather soft gluons (x < ∼ 0.2), see the first and second  panel in Fig. 6, while they have a small impact at high T , cf. third and fourth panel in Fig. 6. These features directly reflect the discussion of the spectral properties of the outgoing gluon and heavy quark in the previous section.
The high-energy transport parameterq represents the average transverse-momentum transfer per mean-free-path of the heavy quark. It has been related to the elastic part of the commonly calculated coefficients in a Fokker-Planck equation [2], specifically the transverse-momentum diffusion coefficient, B el 0 , asq = 4B el 0 ≈ 4T Q (p)A el (p) [48]; the second approximate equality involving the elastic friction coefficient, A el , follows from the Einstein relation (which is routinely enforced in practical applications). In pQCD based approaches [30],q is usually factorized and used as a fit parameter encoding the NP properties of the QGP. Here we can check whether this factorization also holds at a more differential level, e.g., for the power spectra. Toward this end, we first compute theq for the "V C Born" and "V CS Born" scenarios from B el 0 with our Born amplitudes, to illustrate how the string interaction affects this widely used jet transport coefficient. The results are shown in Fig. 7. We here use a fixed coupling constant which implies thatq increases logarithmically at high momentum, which is similar to the behavior found in Ref [49] but different from results evaluated with a running coupling which typically come out approximately momentum independent. As expected from our preceding discussion of spectral functions and power spectra, the results for "V CS Born" and "V C Born" converge toward each other at high momentum, but the former is significantly enhanced for momenta below about 10 GeV. This scale can thus be identified as the transition regime from perturbative to the nonperturbative. A NP enhancement factor, defined in terms of the ratio of thq's from the 2 scenarios, is shown in the right panel of Fig. 7; while the absolute value of the gluonq is larger than the one for charm quarks by about a factor of 2, the relative NP enhancement factor is actually larger for heavy quarks than for gluons. The temperature dependence of the HQq, scaled by T 3 , is displayed in Fig. 8 for the "V CS Born" case; it reiterates the importance of the string interaction at relatively low momenta, significantly enhancing the coupling strength toward small temperatures. We can the test the "factorization hypothesis" referred to above by multiplying the NP enhancement factor into the power spectra of "V C Born" and compare it to the power spectra of "V CS Born". More concretely, we define the "rescaled V C Born" power spectrum as which accounts for the NP enhancement in both the emitted gluon and the outgoing heavy quark their respective emitted momenta. The results are plotted in Fig. 9 show that this prescription indeed gives an approximate mapping from "V C Born" to "V CS Born", although deviations of up to ±30% or so persist, especially at low temperature.  Figure 9. Comparison of the k T -integrated power spectra (k/p)dN dtd(k L /p) ≈ xdN dtdx , as a function of longitudinal-momentum fraction (k L ) of the emitted gluon for the "V C Born" case (dotted lines), the "V CS Born" case (dashed lines), and the "V C Born" case augmented with theq scaled enhancement factor from the two cases as defined in eq. (3.1) (solid lines), at low (high) temperature in the left (right) two panels, each for two different HQ momenta.
For completeness, we display in Fig. 10 our full result for the temperature dependence of q, i.e., in the "T -matrix Offshell" scenario, as obtained from the elastic friction coefficient, A el (p), by use of the Einstein relation based on Fig. 11 of Ref. [36] (which involves a partial-wave expansions up to l=8 to achieve a decent convergence at high momentum). It's magnitude is further enhanced by close to a factor of 2 at low momenta compared to the "V CS Born" scenario, while a significant temperature dependence mostly arises for momenta below 10 GeV and temperatures below ∼300 MeV.  Figure 10. Dimensionless-scaled HQ transport coefficient,q/T 3 , in the "T -matrix Offshell" scenario as a function of temperature for different HQ momenta.

Drag coefficient and energy loss
The drag (or friction) coefficient, A(p), is essentially an integral of the power spectrum over k T and k L , divided by the HQ momentum, p. It quantifies the fractional energy loss per unit time. Since for high-energy quarks the velocity is near the speed of light, it can be regarded as the fractional energy loss per unit path length. In the following, we discuss how the different cases affect the radiative drag coefficient, and implement them into a Langevin simulation for a charm quark in the background of QGP of finite size and fixed temperature, sometimes referred to as a "QGP brick". Several different energy loss approaches for charm and bottom quarks have been compared in this setup in Ref. [31].
In the first row of Fig. 11, we show the drag coefficient as a function of HQ momentum for the four different cases in each panel for different temperatures. At large momentum (p=40 GeV), the drag coefficients essentially saturate at high p and show a close-to-linear increase with temperature over the considered range, T =0.194-0.4 GeV, in all four cases; the latter is still true at lower momentum, p=10 GeV, except for the "V C Born" case which is closer to a quadratic temperature dependence. The drag coefficient is strongly suppressed at small momentum, as a consequence of the large thermal-gluon mass. To better quantify the enhancement with the subsequent inclusion of NP effects we plot in the second row of Fig. 11 the ratio between adjacent cases. The ratio A (2) /A (1) in the first panel, "V CS Born" relative to "V C Born", shows the largest relative enhancement, most pronounced at lot T and low p, as seen before, reiterating the importance of the string interaction, even at the Born level. The ratio A (3) /A (2) in the second panel exhibits a more moderate, but still substantial enhancement due to the resummation of t-channel ladder diagrams in the heavy-light T -matrix. The ratio A (4) /A (3) in the third panel, indicates that the additional off-shell treatment of the spectral function of the medium partons has a still smaller impact; at high momentum, even the temperature hierarchy of this effect p i =20(GeV) is not definite. The combined impact of the NP effects on our pQCD baseline scenario, quantified by the ratio A (4) /A (1) shown in the lower right panel of Fig. 11, enhances the radiative contribution to the drag coefficient by up to a factor of ∼150 at small momenta, below 3 GeV, and at temperatures close to T pc . This is, however, somewhat academic, since its absolute magnitude is very small compared to the elastic contribution in the same framework [36], e.g., A el (p = 3 GeV, T = 0.194 GeV) = 0.1/fm vs. compared to 0.03/fm for the radiative drag. On the other hand, already at p=5 GeV and T =0.4GeV, where both contributions are near 0.08/fm, the NP enhancement of a factor of ∼4 is certainly relevant. Finally, we implement the radiative transport coefficients into a Langevin simulation given by incremental time steps for the HQ position and momentum, where ρ is a random number determined from a Gaussian distribution function, P (ρ) = (2π) −3/2 e −ρ/2 , and the transport coefficients are We use the pre-point Ito setup in which the relaxation rate is equal to the drag coefficient, and the momentum diffusion coefficients are defined via the Einstein relation and obtained Tmatrix Onshell Figure 13. The quantity (k/p)dN d(k L /p) ≈ xdN dx as a function of k L /p. The initial HQ energy is 20 GeV for a static medium of temperature T =225 MeV and length L=5 fm. The pQCD result is taken from Ref. [50]. by solving the differential equation Γ(p) = 1/ε Q (D(p)/T − dD(p)/dε Q ) with the boundary condition D(∞) > 0. In Fig. 12, we show the results of 3D Langevin simulations for a charm quark inside a static QGP brick with a 10 fm path length. We have verified that the fluctuation term in Eq. (3.2) is not important for the average energy loss and therefore the Langevin equation can be replaced by a 1-D differential equation, dp = −γ(p)ε Q (p)dx. Without the LPM effect [45], the energy loss for small path lengths is near linear. For large path lengths, the negative curvature is caused by the fact that the radiative drag coefficient decreases with decreasing HQ momentum. At low temperature and low momentum, our many-body approach generally predicts a much smaller radiation and energy loss than pQCD-based approaches (cf. Fig. 27 in Ref. [31]). The main reason is the large gluon mass creating a large energy threshold for the emission. However, the thermal masses are a key feature of our many-body approach, essentially constrained by the QGP equation of state, as elaborated in Sec. 3.1. In other words, the emitted gluon can only propagate in the energy momentum modes supported by the ambient medium. This is quite different from approaches where the emitted gluon in the QGP is assumed to be massless. The NP effects recover a good portion of the suppression in the power spectra, as we will discuss in more detail in the following section.

Comparison to a pQCD calculation
We finally carry out a smore quantitative comparison of our results to a pQCD calculation from the literature [50]. Toward this end we focus on the quantity xdN/dx commonly displayed in pQCD approaches, which characterizes the energy loss spectrum. We obtain this by integrating the power spectrum, xdN/dx = dt xdN/(dtdx), over time as obtained from our Langevin simulation in Sec. 3.3 which accounts for the time dependence of the charm-quark momentum, p(t). In general, xdN/dx is approximately equal to xdN/(dtdx) multiplied by the length of the medium. We choose a QGP brick of length L=5 fm and temperature T =0.225 GeV, and and initial in charm-quark momentum of 20 GeV. For a most direct comparison of our calculations to the pQCD approach, we evaluate our "V C Born" scenario with the same parameters as in Ref. [50], i.e., coupling constant α s = 0.3, Coulomb Debye mass m d = g 1 + N f /6 with N f = 2.5, thermal light-quark and gluon masses of m q = m d / √ 6 and m g = m d / √ 2, respectively, and charm-quark mass m c = 1.2 GeV. At T =0.225 GeV, this amounts to m d =0.51 GeV, m q =0.21 GeV, m g =0. 36 GeV and m c =1.2 GeV. The result of this retuned case agrees with the pQCD result taken from Ref [50] within ∼25%, except for x values close to 1 where our results go to zero faster, see Fig. 13. This comparison gives us further confidence that contributions from gaugeinvariant pieces that are missing in our approach are not large, and that the comparisons of the different NP effects (which are much larger) are meaningful. In particular, if we include in the retuned set-up the string interaction (in Born approximation), we find a rather significant enhancement over the Coulomb-only scenario, as before. On the other hand, if we use our baseline parameters for the "V C Born" case (α s = 0.27, N f =3, m d =0.42 GeV, m q =0.43 GeV, m g =1. 15 GeV and m c =1.76 GeV), the resulting energy loss spectrum is much reduced, cf. Fig. 13. While the interaction strength is quite similar to the retuned scenario, the much larger quasiparticle masses fixed by fitting to the lQCD EoS reduce the color charge density and with it the rate of radiation. In addition, the larger gluon and charm-quark masses (the latter as constrained by the HQ free energy) suppress the phase space for radiation. If we include the NP effects, i.e., the string interactions and resummations, the energy loss spectrum substantially inceases over the "V C Born" result but is still significantly below the pQCD calculation 2 . The large interaction strength cannot compensate the loss caused by the large NP masses, which are largely generated by the (remnants of the) confining force in our approach.

Conclusion
We have analyzed the radiative energy loss of heavy quarks within a thermodynamic Tmatrix approach, on the same footing with earlier studies of collisional energy loss. The most relevant aspects of this calculation are its nonperturbative components, specifically remnants of the confining force above T pc , t-channel ladder resummations and off-shell spectral functions. These have been previously constrained by various sets of lattice-QCD data, most notably the QGP equation of state and heavy-quark free energy. To scrutinize their relevance, we have set up four scenarios with a subsequently increasing degree of NP effects.
We have found that all three NP effects referred to above substantially affect the power spectra and radiative transport coefficients at soft energy scales, i.e., at low HQ and/or gluon momenta and low temperatures. Ranking them by their importance, the inclusion of string interactions (even in Born approximation) generates the largest enhancement, followed by t-channel resummations in the two-body T -matrix, while the off-shell medium induces comparatively small modifications. Furthermore, we explicitly showed that the NP effects become gradually suppressed with increasing resolution scale; e.g., at T =0.4 GeV the NP enhancement in the radiative drag coefficient amounts to less than 40(20)% for charm quarks of momenta of ∼20 (40) GeV. This supports the convergence of pQCD-based approaches at high parton energies. While our calculations are carried out in a fixed gauge, we have checked that, when using color-Coulomb Born interactions and matching the input parameters to state-of-the-art pQCD calculations, our results for energy loss spectra for a high-energy charm quark agree with the latter within ∼25% or so. However, when using the thermal parton masses as dictated by the constraints from lattice-QCD, our baseline "pQCD" calculation results in a strongly suppressed radiation spectrum, mostly due to the large energy cost (mass) of the thermal modes available to the radiated gluons in the ambient QGP medium. The combined enhancement effect of the NP interactions cannot fully recover this suppression. It will thus be important to further srutinize the robustness of this result, and investigate its impact on in the phenomenology of HF observables in URHICs.