Medium-induced gluon radiation with full resummation of multiple scatterings for realistic parton-medium interactions

The new precision era of jet quenching observables at both RHIC and the LHC calls for an improved and more precise description of in-medium gluon emissions. The development of new theoretical tools and analytical calculations to tackle this challenge has been hampered by the inability to include the effects of multiple scatterings with the medium using a realistic model for the parton-medium interactions. In this paper, we show how the analytical expressions for the full in-medium spectrum, including the resummation of all multiple scatterings, can be written in a form where the numerical evaluation can be easily performed without the need of the usually employed harmonic or single hard approximations. We present the transverse momentum and energy-dependent medium-induced gluon emission distributions for known realistic interaction models to illustrate how our framework can be applied beyond the limited kinematic regions of previous calculations.


Introduction
The analysis of hard probes observables in high-energy nucleus-nucleus collisions has been proven to be one of the main tools at our disposal for the understanding and characterization of the properties of the quark-gluon plasma formed in these collisions. Recently, it has been recognized that besides the familiar studies of jet and hadron suppression, the theory of in-medium gluon emissions also plays a central role in the accurate description of a varied number of observables which can provide valuable information on the different stages of the evolution of the plasma created after the collision [1,2].
Given the importance of medium-induced radiation in the description of heavy-ion collisions, it is necessary to revisit and improve the current implementations of the evaluation of the gluon emission spectrum in a hot plasma. The level of accuracy achieved by the experimental data needs to be matched by the theoretical calculations. Therefore it is of the utmost importance to relax approximations and include all the physical effects in the formalism under which the in-medium radiation spectrum is computed.
The main difficulty encountered when an analytic approach is employed to calculate the emission spectrum off a hard parton is the appropriate inclusion of an arbitrary number of scatterings with the medium. It has long been known that multiple scatterings act coherently over high-energy particles modifying the emission spectrum through the largely studied Landau-Pomeranchuk-Migdal (LPM) effect [3,4]. Going beyond the single particle spectrum, it has been shown that properly accounting for the effects of multiple scatterings is crucial for the description of processes with several particles, where color correlations are important and can be broken through interactions with the medium [5][6][7].
The usual approach to include the effect of multiple scatterings is to use the BDMPS-Z formalism [8][9][10][11], where a formal resummation can be achieved by considering only the non-relativistic dynamics in the transverse plane. When an expansion in terms of the number of scatterings is considered, this formalism has been shown to reproduce the outcomes obtained by different methods in which only a few scatterings are computed [12]. The problem with such approach lies in the fact that the formal expressions for the all-order resummation can be analytically computed only under very specific approximations which may miss some important physical effects. This is the case of the harmonic or multiple soft scattering approximation, which assumes a Gaussian profile for the transverse momentum transfers and does not reproduce the perturbative tails at high transverse momentum k ⊥ . Given the impossibility of having fully analytical expressions, numerical implementations would be highly desirable, but so far, the only thorough attempts to numerically evaluate the emission spectrum and its k ⊥ -dependence have been through computationally costly Monte Carlo implementations [13].
In this paper, following the direction of ref. [14], we show how to numerically evaluate the formal expressions of the medium-induced gluon emission spectrum for an arbitrary number of scatterings and realistic parton-medium interactions, without any further approximations. In order to do so, we numerically solve the appropriate differential equations defining the in-medium propagators which enter in the spectrum expression. In this way, we are able to successfully consider the full transverse momentum dependence and calculate the full emission spectrum, accounting for the relevant kinematic constraints.
The manuscript is organized as follows: in section 2 we outline the fundamental assumptions of the BDMPS-Z approach as well as the limitations of its current approximate analytical evaluations. In section 3 we present our framework, which allows us to calculate exactly the in-medium spectrum including the resummation of all multiple scatterings for realistic parton-medium interactions. Readers not interested in all the details of the derivation can go directly to section 3.2.1 and section 3.3, which contain all the equations needed to compute the full resummed medium-induced k ⊥ -differential and energy spectra. In section 4 we present the numerical results of our approach for two models of parton-medium interaction: Yukawa-type and hard thermal loop (HTL) interaction. Finally, we summarize and conclude in section 5.

Medium-induced gluon spectrum
The medium-induced gluon radiation spectrum in the high-energy limit has been derived in several formalisms [8,10,15,16]. For clarity, we will summarize the common assumptions entering these derivations and cite the result in the BDMPS-Z framework. Details about the derivations can be found in [8][9][10][11][12]17]. The basic assumptions which play an important role in the successful resummation of multiple scatterings into a compact formula are as follow: • The opening angle of the radiation is small and the emission vertices are given by leadingorder DGLAP splitting functions. Parent and daughter partons can pick up some momentum transverse to the direction of propagation of the initial parton, but their magnitudes are always much smaller than their respective energies.
• The main contribution to radiation comes from elastic scatterings. The interaction between parton and medium is mediated by soft gluons, which are regarded as carrying only transverse momenta of the order of the characteristic medium scale.
• At high-energy, the time scale of any single interaction is much shorter than the formation time of emitted gluons or the time scale for medium evolution. Therefore, the interactions are considered as instantaneous and are calculated for a fixed, but arbitrary, medium configuration which will be averaged over.
The specific details of the parton-medium interaction are given as a phenomenological input through the elastic collision rate V (q), which then enters the calculation through the dipole cross No further assumptions are made on the form of V (q), although in all realistic models it must have the power behavior V (q) ∼ 1/q 4 , which is a direct consequence of having point-like interactions with a Coulomb potential at short distances. For simplicity, we assume the emitted gluon is soft with ω/E 1 where ω is the energy of the emitted gluon and E the energy of the initial parton. This is not a general assumption for the derivation of the formula for the spectrum and will be relaxed in a subsequent publication. 2 In this limit, the medium-induced gluon spectrum off a high-energy parton reads: where k is the two-dimensional transverse momentum of the emitted gluon. The variables t and t correspond to the emission times 3 in the amplitude and conjugate amplitude, respectively, K(t , q; t, p) is the emission kernel in momentum space, and P(∞, k; t , q) is the momentum broadening factor. The radiation off hard quarks or gluons differs by the Casimir factor The Green's function K(t , q; t, p) can be explicitly written in coordinate space as the following path integral while the momentum broadening factor is given by with n(s) the linear medium density. The numerical evaluation of the path integral in eq. (2.3) including all the multiple scatterings for a realistic collision rate V (q) -such as a Yukawa-like interaction -has always posed technical problems, which could not be overcome until very recently with advanced Monte Carlo techniques [13]. For this reason, the spectrum in eq. (2.2) has historically been treated in two limiting cases in which an analytical expression for the kernel is possible: multiple soft and single hard momentum transfer.
Within a multiple soft in-medium scatterings approach, the dipole cross section can be approximated by its leading logarithmic behavior whereq is the transport coefficient that characterizes the average transverse momentum squared transferred from the medium to the projectile per unit path length. This approximation is valid for an opaque media, when configurations where the transverse distance r is large are strongly suppressed. By replacing eq. (2.5) in eq. (2.3), an analytical solution for the path integral is straightforward to obtain in the static case. 4 This result is also known as the harmonic oscillator (HO) or Gaussian approximation. One of the major drawbacks of this method is the strong suppression of the high transverse momentum part of the spectrum where it has an exponential behavior instead of the power-like tails characteristic of Coulomb interactions at short distances. The other scenario corresponds to the radiation pattern resulting from an incoherent superposition of just a few single hard scattering processes. This limit can be obtained by expanding the integrand of eq. (2.3) in powers of the density of scattering centers (n(s)σ(r)) N [12,15]. This approach is usually known as opacity expansion. The first order (N = 1) in this procedure is typically referred to as the Gyulassy-Levai-Vitev (GLV) or first opacity approximation and is applicable for dilute media. When the number of scattering centers is large, resuming the contributions from all orders in opacity is needed, a process that is both analytically and computationally demanding. The above mentioned differences between these two approaches have a direct essential consequence: the energy spectrum produced by the Gaussian approximation is much softer than the one produced by the opacity expansion. This is mainly due to two factors: (1) the absence of destructive LPM interferences in the latter, which naturally appear in the former; and (2) the inclusion of the power-law tails of the interaction cross-section in the first opacity result that otherwise are neglected in the harmonic approximation. 5 The use of these approximate solutions have led to conflicting results when extracting medium properties from measurements taken at RHIC and at the LHC [24,25], while recent studies may indicate that when these approximations are not employed this centrality/energy puzzle seems to disappear [26].
In this manuscript, we thus attempt to provide a framework that naturally includes and goes beyond both limits above by avoiding any assumption on the interaction nature of the parton with the medium. In the following section, we will explain the logical setup that allows us to derive an analytical expression for the medium-induced gluon radiation in the most general case.

Setting up the evaluation
Apart from the difficulties that arise when attempting to compute the kernel given by eq. (2.3) without any further approximations, there are several obstacles in numerically evaluating the spectrum in eq. (2.2). Hence, it is convenient to rearrange eq. (2.2) to put it in a more appropriate form.
One issue is that the gluon emission can occur anywhere after the initial parton is created, that is, either inside or outside the medium. In consequence, the interferences between these two types of emissions need to be properly taken into account. The common way of addressing this is by splitting the semi-infinite integration in the emission times into two terms bound by the length of the medium, which correspond to the pure in-medium emissions (usually denoted as "in-in" contributions) and the medium-vacuum interference (usually named "in-out"). While this is not a limiting factor, it demands precise cancellations between these two types of contributions, which might involve an additional level of precision in the numerical evaluation, thus making it inefficient. To overcome this issue, we analytically perform the integration in t in eq. (2.2). This introduces in turn an integration over the position of one of the scatterings. Since no scatterings occur outside the medium, the resulting time integrations are naturally bound by its length and thus, separating them into "in-in" and "in-out" pieces is no longer necessary.
On the other hand, as it was explained in the previous section, the expressions of the broadening factor P and the kernel K are naturally written in coordinate space, the latter involving a complicated path integral. To avoid the difficulties that arise when attempting to numerically compute this path integral, we work directly in momentum space by considering these objects as propagators that satisfy specific differential equations which can be numerically solved by conventional methods.

Reorganization of the spectrum
We start by performing the t integration in the in-medium spectrum given by eq. (2.2). Even though t is an argument of both K and P in eq. (2.2), it can be integrated out without knowing the explicit form of either of these two factors. For that end, we only need to notice that both K and P are propagators that satisfy the following Schwinger-Dyson type equations: Here, the dependence in t is confined to phase factors and limits of integration. Now we can replace eqs. (3.1) and (3.2) in eq. (2.2) and perform the t -integration analytically and, after some manipulations (shown in appendix A), arrive at where the vacuum contribution has already been subtracted. In this expression there is still one time integral running up to infinity, but with the main difference that it represents the position of one of the scatterings and hence the integrand is zero outside of the medium. This allows us to use the length of the medium L as the upper limit for this integral and for the end point of the momentum broadening as well. The effect of emissions outside of the medium has already been integrated out (or subtracted in the case of the purely vacuum emissions) and there is no need to deal with interferences separately or rely on precise cancellations between different terms. This new expression for the in-medium spectrum has another advantage with regard to its numerical evaluation. The dipole cross section behaves, for any realistic parton-medium interaction, as σ(q) ∼ 1/q 4 at large q, which guarantees the convergence of the integrals over q and l, while also providing a convenient initial condition which can be evolved when K and P are taken as propagators.
It is worth noticing that extracting the first order in opacity result from eq. (3.3) is straightforward. We only need to take the vacuum versions of P and K, which can be read off directly from the first term in the r.h.s. of eqs. (3.1) and (3.2).

Recasting the evaluation of the spectrum as a first order linear differential equations problem
In this section we will describe in detail how to evaluate the differential spectrum of eq. (3.3) using evolution equations instead of finding explicit expressions for the kernel K and the broadening factor P.
First, we recognize that P satisfies the following differential equation: with initial condition P(s, k; s, l) = (2π) 2 δ (2) (k − l). Instead of trying to solve this equation for P, we define (based on eq.
Now, we perform a similar manipulation for the kernel K. First, we recognize that it satisfies the following differential equation which also satisfies with initial condition ψ(s, k; s, p) = φ(L, k; s, p). The first term in the right-hand side of eq. (3.10) causes oscillations which guarantee the convergence of the p-integrals, but it might become a problem when attempting to numerically solve the differential equation. It is then convenient to switch to the interaction picture, with with initial condition The full k-dependent spectrum can then be written as 14) The procedure to evaluate the spectrum is then clear. First, we start by computing eq. (3.7), then numerically solve eq. (3.6) to get the r.h.s. of eq. (3.13), which is the starting point to numerically solve eq. (3.12). Once we have a solution for ψ I , it can be plugged into eq. (3.14) and integrated numerically to obtain the spectrum. Before we can use this procedure for the numerical evaluation a few more manipulations are needed. Let us recall the form of the dipole cross section σ, in momentum space, in terms of the collision rate V , (3.15) Then, the differential equation eq. (3.6) and its initial condition eq. (3.7) take the form while the differential equation for ψ I given in eq. (3.12) is now For most of the cases of interest the direction of k is irrelevant, thus, we can focus on the spectrum as a function only of its magnitude. We can therefore integrate over the direction of k, which allows us to use rotational symmetry to analytically perform all angular integrals. The spectrum to evaluate is then which will be written in terms of the functions

Set of equations to solve numerically
For convenience, we change our variables to make them dimensionless: dummy momentum variables are rescaled as p → 2ω/L p and time variables as s → Ls. The typical transverse momentum transfer µ, usually taken as the Debye mass of the screened interactions, sets the scale for the transverse momentum and the energy of the emitted gluons. The dimensionless variables in which we will evaluate the spectrum are 6 and then write the differential medium-induced gluon emission spectrum as

25)
6ω c is usually known as characteristic gluon frequency and, as we will see later, the emission of gluons with ω >ωc is suppressed.
where f x satisfies the differential equation with initial condition f x (s, κ; s, p) = 1 x g x (1, κ/ √ x; s, p) , (3.27) and g x satisfies the differential equation with initial condition Hereñ(s) = n(sL), whileṼ 1 andṼ 2 are the two first angular moments of the rescaled collision rateṼ where θ qp is the angle between q and p, andṼ (q; x) = 2ω L V (q 2ω/L). For the usual models for the collision rate used in phenomenology, these angular integrals can be performed analytically, as will be shown in section 4.

Energy spectrum
We can also integrate over transverse momentum to obtain the energy spectrum, always keeping in mind that the integration must respect the kinematical constraint k ≤ ω. We get while g x is still obtained by solving eq. (3.28) with initial condition eq. (3.29). The case where the kinematical condition is removed, i.e.R → ∞, is much simpler since the momentum broadening of the emitted gluon is irrelevant and therefore there is no need to solve eq. (3.28). This can be seen directly from the equations by integrating l from 0 to ∞ in eq. (3.28) and noticing the right hand side vanishes.

Numerical results
In this section, we present the results of our numerical analysis. For simplicity, we perform our calculations in a static thermally equilibrated quark-gluon plasma and leave the extension to expanding media for subsequent publications. The linear density of scatterings is then a constant n(t) = n 0 . For illustration purposes, we always consider the case where the parent parton is a quark. Therefore we take C R = C F = 4/9. The strong coupling is fixed to α s = 0.3. The numerical implementation of equations (3.25)-(3.29) involves momentum integrations that run up to infinity. The high-momentum tail of V (q) ∼ 1/q 4 guarantees that all the integrands in eqs. (3.25)-(3.29) approach zero as 1/q 4 (or faster), with increasing momenta. As such, the numerical evaluation uses an upper cut-off for these integrals, and we have carefully checked the stability of the result when this cut-off is changed.
In the following, we study the results of our approach for two collision rate models. We first consider a Yukawa-type interaction and make straightforward comparisons with the respective first order in opacity. We also attempt to compare our results with the multiple soft Gaussian approximation, but always keeping in mind that there are subtle complications when attempting a direct correspondence between the parameters involved in both evaluations. Finally, we consider the case where the interaction is modeled through the collision rate calculated perturbatively in a hard thermal loop (HTL) formalism [27].

Yukawa-type interaction
The collision rate V for a Yukawa-type elastic scattering center is given by: where the screening mass µ is related to the Debye mass in a thermal medium, µ 2 ∼ m 2 D . Its rescaled versionṼ isṼ (q, x) = 8π where x = 2ω µ 2 L . Performing the angular integrations of eqs. (3.30) and (3.31), yields In fact, V is normalized in this way to ensure that n 0 L is the correct parameter for an opacity expansion. 7 We can now use the above expressions to numerically solve the medium-induced gluon emission spectrum. This collision rate clearly depends only on one parameter µ 2 , but the full evaluation x −1 = 32 x −1 = 10 Figure 1. Full medium-induced gluon radiation k ⊥ -differential spectrum for a medium with n0L = 1 for the Yukawa collision rate (left column) compared to the GLV first opacity approximation (right column). The upper panels show these spectra as function of the rescaled gluon energy x = ω/ωc for fixed values of the rescaled gluon transverse momentum κ = k/µ. The lower panels show these spectra versus κ 2 for fixed values of x.
of the spectrum also depends on n 0 and L. We compute first the full resummed medium-induced k ⊥ -differential spectrum for this interaction. A comparison of our results to the first term in the opacity expansion (GLV N = 1) 8 is shown in figures 1 and 2, for two different values of n 0 L = 1, 5, respectively. Note that, though the plots are labeled by their value of n 0 L only, it has to be kept in mind that the two other parameters involved in the evaluation are implicit in the definition of κ 2 = k 2 /µ 2 and x = 2ω/(µ 2 L).
As expected, the full spectrum (left panel) is smaller than the GLV approximation (right panel), given that the latter does not account for the LPM suppression. Nonetheless, for smaller values of n 0 L and large κ the opacity expansion is justified and thus the GLV result should be a good approximation. As seen in figures 1 and 2, the tails of the spectra (large ω and κ) in both panels are indeed quite similar. This is expected since in those kinematical regions (ω >ω c ), where the spectrum is known to be suppressed, the interaction is supposed to be dominated by a single hard scattering. As we move away from this kinematic regime, the differences between the two approaches start to become more visible. Figure 1 shows that for n 0 L = 1 there is already a substantial discrepancy between the full resummation and the first opacity approximation. This difference increases with n 0 L, as can be seen in figure 2. One should take into account the scale of the vertical axis between the two figures: both spectra increase with n 0 L, and so their differences. x −1 = 32 Figure 2. Full medium-induced gluon radiation k ⊥ -differential spectrum for a medium with n0L = 5 for the Yukawa-type interaction (left column) compared to the GLV first opacity approximation (right column). The upper panels show these spectra as function of the rescaled gluon energy x = ω/ωc for fixed values of the rescaled gluon transverse momentum κ = k/µ. The lower panels show these spectra versus κ 2 for fixed values of x.
We now present the numerical evaluation of the full resummed medium-induced energy spectrum given by eq. (3.32) for the Yukawa collision rate. Even though this energy distribution depends on the same three parameters as the transverse momentum spectrum, we can also employ instead the following: n 0 L ,ω c = µ 2 L 2 /2 , andR =ω c L , (4.6) where the latter can be seen as a dimensionless kinematic constraint on the transverse momentum phase space of the emitted gluon ensuring that k ≤ ω. Indeed, the limitR → ∞ -which removes this kinematic constraint -can be viewed as the limit of infinite in-medium path length since it corresponds to L → ∞ forω c fixed. In figure 3 we show the comparison of the full resummed medium-induced gluon energy distribution for the Yukawa-type interaction (left panel) with the GLV first opacity (right panel) assuming n 0 L = 5. As previously mentioned, due to the lack of LPM suppression in the GLV approach, for a fixed value ofR, the full result is smaller than the first opacity evaluation.
A comparison fixing the value of the linear density of scattering centers, n 0 = 1 fm −1 , and µ = 0.6 GeV while varying the medium path lengths, L = 2, 3, 4, 5 fm, is shown in figure 4. Since the opacity expansion is justified for small values of n 0 L, for a fixed linear density, the smaller the value of the path length the smaller the discrepancy between the first opacity approximation and the full resummed result. Furthermore, at large gluon energies, the GLV energy distribution is very similar to the full resummed solution, since in this kinematical region the process is dominated by   a single hard scattering. Now we turn our attention to the comparison between our full resummed results for the Yukawa collision rate and the multiple soft Gaussian approximation. As it was already mentioned, a direct comparison between the two evaluations is not straightforward, since the parameters they employ  are different. In fact, the harmonic oscillator spectrum depends only on two parameters: the jet quenching parameterq -defined through eq. (2.5) -and the length of the medium L. In principle, the jet quenching parameter should be directly related to the parameters of the interaction (n 0 and µ in this case), but given the long high-momentum tail of the elastic cross section it also receives logarithmic contributions related to the available phase space. 9 By expanding the dipole cross section in eq. (2.1), it can be shown that where q max is the upper cut-off of the q-integral. We will not attempt to determine the most precise value for this logarithm. Instead, we fixqL = 1.3 (n 0 L) µ 2 in order to make qualitative comparisons between the shapes of the resulting curves.
The results for the k ⊥ -differential in-medium spectrum for the full resummed solution (solid) and the HO approximation (dotted) for a path length L = 6 fm are shown in figure 5. The left panel shows their evolution with energy at fixed transverse momentum, while on the right, their evolution with transverse momentum for two different values of the energy is presented. As expected, at large k 2 the HO fails to reproduce the tails of the full resummed result, as opposed to the GLV approximation in figures 1 and 2.
In figure 6 we show the medium-induced gluon energy distribution for the full resummed solution and the Gaussian approximation for a medium of L = 6 fm. As in the previous figures, the solid lines correspond to the full resummed result for n 0 L = 5, µ = 1.6 GeV (in blue) and µ = 0.9 GeV (in red). The dotted lines represent the HO evaluation for the corresponding values ofq according toqL = 1.3 (n 0 L) µ 2 , i.e.,q = 2.8 GeV 2 /fm (blue) andq = 0.9 GeV 2 /fm (red). As expected, the two different calculations are similar at low energies, where the HO approximation is well justified. The agreement in this region is indeed better than for the GLV case, as can be seen in figure 4.  We have checked that this observation is independent of our choice of the numerical value of the logarithmic factor used to fix the parameters in eq. (4.7). Increasing (decreasing) this factor moves the peak of the HO curve upwards (downwards) and to the right (left), but always staying located at lower energies than the full resummed peak, while keeping the form of the low-ω tail. It is then clear that the full resummed energy distribution is much harder than the multiple soft scattering approximation.
It is worth noting that no conclusions can be drawn from the fact that the magnitude of the HO spectrum is larger than that of the full resummed one in figure 5. In figure 6 can be clearly seen that the choice of the value of ω determines which of the curves goes above the other in the right panel of figure 5.

Hard thermal loop interaction
To illustrate the flexibility of our approach, we also implement the collision rate derived from hard thermal loop calculations which should, in principle, provide a more accurate description of the thermal interactions. For this purpose, we take , (4.8) which was obtained at leading-order in the coupling in thermal field theory in a weakly-coupled medium [27]. The angular integrations given by eqs. (3.30) and (3.31) can be performed analytically for this collision rate giving (4.10) One difficulty with this potential is that it is divergent for p = q. These divergences always disappear when trying to solve both the initial conditions and the differential equations. Nevertheless, care must be taken to avoid a numerical evaluation at that particular point.
Plugging eqs. (4.9) and (4.10) in eq. (3.29) gives The result is clearly discontinuous, but it does not induce any singularities in the subsequent steps of the calculations. In practice, the discontinuous term will be set to zero at p = l. This result will enter the integrations over l 2 from zero to (possibly) infinity, so it is important to note that even though g x (s, l; s, p) may seem to behave like 1/l 2 in both of those endpoints, the factor in brackets goes to zero, thus guaranteeing the integration of g x (s, l; s, p) over l 2 to be convergent. This initial condition must be evolved with eq. (3.28), whereṼ 1 (l, q; x) is singular for l = q, but this singularity does not play any role since, again, the term in brackets goes to zero in that limit. The resulting integrand will have a discontinuity at that point and will be assigned the average value between left-handed and right-handed limits. Similarly to the previous cases, each of the two terms in the right-hand-side of eq. (3.26) has a divergence, but these divergences cancel out when the sum of the two terms is considered. Again, there will be a discontinuity which will be handled in a similar manner as the one appearing in eq. (3.28).
It is worth noticing that the HTL collision rate given by eq. (4.8) depends on the Debye mass m D and the medium temperature T . By replacing eqs. (4.9) and (4.10) in the differential equations of section 3.2.1, it is straightforward to see that the full resummed spectrum for this type of interaction depends on the following three free parameters T , µ 2 = m 2 D , and L. 10 For the energy distribution we will make use instead of T L ,ω c = µ 2 L/2 , andR =ω c L . (4.12) We now show the full resummed transverse momentum and energy-dependent in-medium distributions for the HTL collision rate in figures 7 and 8, respectively. We do not attempt to quantitatively compare these distributions with the equivalent ones for the Yukawa interaction, since the parameters involved in their evaluation are different and a careful analysis on how to best pursue a meaningful comparison is beyond the scope of this paper. However, it is worth noticing that the shapes (and magnitudes) of the transverse momentum spectra in figure 7 are substantially different from those obtained for the Yukawa interaction, see figures 1 and 2. In particular, the differential spectrum for fixed values of the energy (lower panels of figure 7) has a pronounced peak which seems to be smoothed out for larger values of T L. It is indeed expected that the discrepancies between both interactions become more evident for values of the parameters corresponding to a lower number of scatterings. In contrast, as the medium gets larger/denser, the low-energy part of the spectrum should be closer to the HO approximation.

Conclusions and outlook
In this work, by using Schwinger-Dyson type equations, we derive an analytical expression for the medium-induced gluon radiation spectrum in the soft limit. Our expression contains the full resummation of multiple scatterings and can be used for any realistic parton-medium interaction without further assumptions, thus providing robust results outside the usually employed multiple   soft or single hard scattering approximations. The final outcome is a set of differential equations eqs. (3.25)-(3.29) that can be easily solved and are not as computationally demanding as previous approaches.
In this manuscript, we use the Yukawa and hard thermal loop parton-medium interaction models to compute the full resummed transverse momentum and energy-dependent in-medium gluon emission spectra. We also compare our results with those obtained within the above mentioned limiting cases, finding the differences among them significant. More specifically, the full resummed spectrum, for both types of interactions, is smaller that the corresponding first opacity evaluation, due to the lack of LPM suppression in the latter approach. However, for large gluon transverse momentum and energy, the full resummed result is very close to the GLV one, since in this kinematic region the process is dominated by a single hard scattering. The harmonic approximation also differs substantially from the full resummed calculations. Specifically, at high gluon energies and transverse momenta, the harmonic oscillator spectra go much faster to zero than the full results. Furthermore, the peak of the energy distribution in the HO evaluation always stays to the left of the corresponding one in the full resummed curve, thus, being the full resummed energy distribution harder than the harmonic evaluation. At low gluon energies, both calculations are very similar, since this is the region where the Gaussian approximation is better justified.
The method of evaluation presented here has a great potential to improve the study of jet quenching observables. In future works we plan to expand the reach of our calculations in two main directions: relaxing approximations that will allow us to improve current phenomenological tools, and adapt our formalism to allow precise numerical evaluations of calculations sensitive to the effect of multiple scatterings with the medium.
First, we will relax the soft gluon approximation, thus allowing emitted gluons to take a finite fraction of the energy, then we will explore the case of non-static media. With these two improvements in hand, the formalism can then be used to calculate distributions of energy loss under realistic conditions, which can be used, for instance, to compute the nuclear modification factor and the high transverse momentum azimuthal anisotropies.
On the other hand, we will apply our formalism to evaluate the radiation pattern of an antenna, as a first step towards improving the precision of the evaluations aimed at understanding the role of color coherence in the description of jet and intra-jet observables. First of all, we discard the first term in eq. (A.1) since it is the vacuum contribution and we only want to keep track of the medium-induced radiation. In all the remaining terms, the t -dependence of the integrand is given by just a phase factor, so we change the order of integration to perform this integral first.
With the new order of the integrals, the t -integral in the second term -second line of eq. (A.1) -, goes from s to ∞. It is important to recall now that a regularization procedure to avoid divergences at late times, in the form of a factor e − t , has been omitted in eq. (2.2). When this factor is properly taken into account, the evaluation of the t -integral of this term in the upper limit vanishes. This is equivalent to include the appropriate i prescription in the free propagators. The contribution of the second term of eq. (A.1) after performing t -integral is then given by For the third term in eq. (A.1), the integration limits of the t -integral are finite (from t to s), hence, the integration is straightforward. It is important to notice that, since P is always real, the evaluation of this integral at the lower limit yields a purely imaginary term, thus not contributing to the final result.
The t -integral of the last term in eq. (A.1) is trivial and both terms survive. The final result of integrating in t eq. (A.1), excluding the vacuum contribution, is given by To finalize, the terms with and without phases can be recombined as follows: the contribution without phase in the third term with the first term by using eq. (3.1); and the contribution which do contains a phase in the third term with the second term by using eq. Changing the order of integration in t and s we arrive at eq. (3.3).