Radiative energy loss of neighboring subjets

We compute the in-medium energy loss probability distribution of two neighboring subjets at leading order, in the large-$N_c$ approximation. Our result exhibits a gradual onset of color decoherence of the system and accounts for two expected limiting cases. When the angular separation is smaller than the characteristic angle for medium-induced radiation, the two-pronged substructure lose energy coherently as a single color charge, namely that of the parent parton. At large angular separation the two subjets lose energy independently. Our result is a first step towards quantifying effects of energy loss as a result of the fluctuation of the multi-parton jet substructure and therefore goes beyond the standard approach to jet quenching based on single parton energy loss. We briefly discuss applications to jet observables in heavy-ion collisions.


Introduction
First observed at RHIC [1,2], then at LHC [3,4,5], the large suppression of high-p T particle spectra in nucleus-nucleus collisions, referred to as "jet quenching", is commonly regarded as a signature of the formation of the quark-gluon plasma (QGP). This striking phenomenon is attributed to medium-induced radiative energy loss of high momentum partons as they propagate through a hot and spatially extended medium [6,7,8]. Measurements of fully reconstructed jets allowed the investigation of new jet quenching observables covering not only measurements of the suppression of inclusive jet spectra [9,10,11,12] and photon-jet correlations [13], but also including information about the redistribution of jet energy [14] within and outside the jet [15,16,17]. In consequence, jet observables, and jet substructure measurements in particular, have shed new light on the mechanism of jet-medium interactions and the details of energy loss while posing new challenges to the theoretical description of in-medium fragmentation at the same time.
In-medium jet quenching have so far been treated at the level of single parton energy loss whose radiation pattern is affected by multiple scattering. This dynamical process gives rise to the Landau-Pomeranchuk-Migdal (LPM) intereference [18,19,20,21,22,23,24], that causes the suppression of the radiation spectrum at large frequencies. The dominant soft emissions take place at time-scales parametrically smaller than the medium size and can therefore be treated as quasi-instantaneous [25,26]. This permits a probabilistic treatment of multiple emissions [27]. For soft gluon emissions, the resulting cascade is governed by turbulence [28] which efficiently transports energy to large angles [29,30,31,32].
The probability of emitting a total energy off the leading parton during the passage through a medium, P 1 ( ), was first discussed in [33]. In the absence of any angular constraints, Figure 1: Radiative energy loss for a single particle (left) and a two-pronged color object (right).
this probability distribution is governed by a single energy scale ω s ∼ α 2 sq L 2 , in a medium of size L and charaterized by the jet quenching parameterq, which is a diffusion coefficient in transverse momentum space. It follows that the related fluctuations of energy loss are also of the same order [34,35]. The resultant single-parton spectrum is shifted towards decreasing energies as compared to the primordial one leading to an overall suppression (see also [36,37,38]). See also [39,40,41,42] for related work on medium-induced multi-gluon emissions.
Extending the calculations of radiative energy loss from single partons to jets proved to be a difficult problem. In point of fact, there is a large probability for nascent partons to branch due to the infrared and mass singularities in QCD splittings, and most of the crucial properties of jets, such as the effect of the cone size, jet substructure and jet mass, appear only on the level of at least one splitting. Moreover, it is difficult to argue that all splittings take place outside of the medium, that typically extends over several fm's, which would justify a treatment where only the parent parton suffers energy loss. This implies that jets are multiparton quantum states whose energy loss pattern is expected to differ from that of independent partons. Besides, it is well know that color coherence plays an important role and leads to angular ordering of subsequent emissions [43,44]. These probabilistic features arise due to the active role of interferences. The treatment of jet fragmentation vertices inside the medium remains therefore an open question. Nevertheless, several Monte-Carlo prescriptions, based on heuristic arguments, exist in the literature, see e.g. [45,46].
The purpose of this paper is to address this outstanding problem from first principles. The key point of our formalism is the treatment of interferences and their active role in governing radiation in the QGP. The processes discussed above are depicted in Fig. 1. In contrast to single-particle energy loss (left), having two participants (right) begs the question of how energy is collectively removed from the system. The main part of this work will consider this problem in detail and arrive at a description of how a system of two partons, formed early in the medium, are affected by radiative energy-loss. In consequence, this extends the scope of [33] and provides a novel tool for dealing with jet observables. We call the resulting distribution the two-prong quenching weight, which will be analyzed in the large-N c limit.
Having in mind a phenomenological observable, our calculation deals with the spectrum of two-pronged substructures inside a jet, both prongs being significantly more energetic than any medium scale. This corresponds to relatively symmetric splitting configurations with small formation times. For the moment, we treat the medium-induced energy loss as a small correction compared to the final subjet energies. This leaves the kinematics of the splitting unaltered within our approximations, while the two-pronged yield is affected due to the steeply falling spectrum of the hard process. Such a measurement could be facilitated by a wide range of jet substructure techniques, for reviews see [47,48,49], and was already considered as a measure of medium effects in [50].
Our findings substantiate an intuitive picture for the two-pronged energy loss. Confirming earlier expectations [51,52,53], the dynamical effects due to the finite resolution power of medium fluctuations play an important role. Initially after formation, the pair remain close in transverse distance and is not resolved by the medium. This implies that energy is taken coherently away from the pair and this process is governed by the total color charge, i.e. the medium sees the pair as the parent parton. There are two effects that break the coherence of the pair. The first is related to quantum decoherence and is common to radiative processes in vacuum and medium [51,53]. Consider gluon emission, with energy ω and transverse momentum k ⊥ , off a parent dipole with opening angle θ 12 . Only gluons that are sufficiently hard to resolve the parent dipole when they are formed at time t f = ω/k 2 ⊥ , that is k ⊥ (θ 12 t f ) −1 , are emitted independently off both legs. This conditions translates into the wellknown angular constraint θ < θ 12 . Softer gluons, with k ⊥ (θ 12 t f ) −1 , are emitted coherently. 1 The second effect, related to in-medium color decoherence is accumulative [54] and introduces a characteristic decoherence time t d ∼ (qθ 12 ) − 1 /2 , which denotes the time when the linearly growing antenna transverse size x 12⊥ = θ 12 t is of order of the medium resolution scale Q −1 s ≡ (qt) −1/2 , when multiple scattering become important. At later times, t > t d , the antenna constituents react independently to medium interactions. The vacuum splitting is therefore non-local from the point of view of medium interactions, but it can still be included in a probabilistic setup.
A corollary of our calculation is to consider for the first time radiative corrections to the process of color decoherence. It was recently realized that both the processes of transverse momentum broadening and gluon emission will receive logarithmically enhanced radiative corrections due to long-lived fluctuations in the medium [55,56,57,58]. It was shown that those universal corrections can be reabsorbed in a renormalization of the jet quenching parameter.
We structure our paper as follows. In order to set the stage, we give a in-depth discussion on how energy is lost by propagating color charges in the medium in Sec. 2. We recapitulate the energy loss distribution for a single color charge and go on to present our main results for the two-prong energy loss distribution, in particular Eq. (14) for the color singlet antenna and Eq. (20) for the generalization to arbitrary color representation. The details of single parton energy loss, presented in Sec. 3, allows us to introduce our formalism. The full derivation of the two-pronged energy loss distribution is presented in Sec. 4. In particular, the radiative corrections appearing for the process of color decoherence are derived in Sec. 4.3. Finally, we summarize and present an outlook in Sec. 5. Our Feynman rules are listed in Appendix A, and the remaining Appendices contain further details of the calculations. A final remark, we work in light-cone (LC) coordinates, defined as x + = (x 0 + x 3 )/2 and x − = x 0 − x 3 , and will abbreviate the LC +-momentum as p + ≡ E and LC-time as x + ≡ t.

One-parton energy loss probability
Let us first review the problem of radiative energy loss of a single, high energy parton passing through a static, dense QCD medium of length L. For simplicity, let us for the moment assume it being a quark. In this setup, the medium scales are a small correction, hence ω E, where the energy E refers to any projectile parton and ω to the medium-induced gluons. In fact, as we will shortly convince ourselves, in this limit the emissions spectrum does not depend explicitly on the parent energy.
In this limit, the cross-section factorizes as follows: 1 Note that these relations also hold for medium-induced quanta, whose transverse momenta and formation times are affected by broadening, where the average transverse momentum at emissions reads k 2 ⊥ ∼ √q ω and the branching time t f ∼ ω/q, respectively.
where σ vac is the quark cross-section and dI/dω stands for the medium-induced bremsstrahlung spectrum that we will discuss shortly. Here, E = E + ω is the quark energy before radiation. The gluon frequency is neglected everywhere except in σ vac , since the steepness of the spectrum allows a small energy loss to yield a large effect on the final spectrum. Typically, dσ vac /dE ∝ E −n where n 1, hence the relevant scale that probes energy loss is shifted to E/n E [33]. To find the inclusive spectrum of single quarks in heavy-ion collisions, one has to integrate over the gluon frequency and add virtual corrections. The final-state spectrum of quarks in heavy-ion collisions is then given as a convolution of the (vacuum) production spectrum dσ vac /dE with the subsequent probability of losing energy, where have introduced the energy loss probability distribution which, to leading order in α s is given by The complete derivation is presented in Sec. 3. In Eq. (3), the first term corresponds to the probability for the high energy parton not to lose energy, namely, that no radiation takes place. The third term corresponds to real emissions while the second term, inside the parenthesis, corresponds to a virtual correction that ensures probability conservation. The medium-induced spectrum that enters Eq. (3) accounts for multiple scattering in the medium, and can be found from the rate [18,19,22,23], whereᾱ ≡ α s C F /π. The rate of emissions off a gluon projectile is found by substituting the color factor C F → C A . The gluon formation time in the medium is affected by the LPM interference and becomes t f ∼ ω/q, which is limited by the size of the medium. Hence Eq. (4) is only valid for gluon energies ω ω c ≡qL 2 /2. Finite-size corrections, neglected so far in this discussion, strongly suppresses the spectrum for ω > ω c [18,19,22,23]. The probability in Eq. (3) corresponds to a single emission. Higher order corrections are important as can be seen by computing the gluon multiplicity above ω, In effect, N (ω) > 1 for ω < ω s where ω s ∼ᾱ 2q L 2 ω c . In this regime, therefore, multiple emissions ought to be resummed. Nonetheless, soft gluon emissions can be regarded as quasiinstantaneous, t f L for ω ω c , and treated as independent [25,26]. As we shall see in more details in the next section, this amounts to writing the single parton energy loss probability as Poisson distribution, where corresponds to the total energy emitted and the Sudakov factor, represents the probability of not radiating between 0 and L. This leads to the famous expression for the radiative energy loss probability where P 1 ( ) ≡ P 1 ( , L), which is only sensitive to the soft energy scale ω s ≡ᾱ 2 ω c /2 [33], see also [36,37,38]. The probability distribution in Eq. (8) is usually referred to as the quenching weight and forms the basis of most theoretical studies of jet quenching, see e.g. [59,60]. The main underlying assumption of these studies is that the initial projectile does not split inside the medium with a vacuum probability. For high-energy jets, this requirement neglects the contributions coming from hard radiation that can be formed early in the medium. This novel contribution will be discussed in the next subsection.

Two-parton energy loss probability at large-N c
We shall now address the main question of this article, namely, the energy loss probability for a neighboring pair of hard partons that originate from the same vertex. For the moment, we disregard any modification to the formation of the pair and study their subsequent evolution. Hence, we shall assume that the splitting occurs quasi-instantaneously so that one can factor out the Born level from the rest of the process. Such a setup extends the concept of the quenching weights, described in the previous subsection, and is a step toward investigating the problem of energy loss of full-fledged jets. The two partons are color connected and formed in an arbitrary (singlet, triplet or octet) color representation of SU (3).
Analogously to single-parton quenching, we shall compute the medium modification of the two-parton system spectrum. In the collinear limit, p ⊥ E, the quark-gluon spectrum in vacuum is given by dσ vac dzdE dp 2 to leading order in perturbation theory and at leading logarithm, where we denote p ⊥ ≡ |p| throughout the paper. In Eq. (9), P gq (z) stands for the quark-gluon Altarelli-Parisi splitting function and dσ vac dE is the Born-level quark spectrum. For a hard instantaneous splitting the two-parton distribution in the presence of a medium reads dσ med dzdE dp 2 = ∞ 0 d P 2 ( ) dσ vac dz dE dp 2 , where P 2 ( ) ≡ P R 2 ( , θ 12 , L) stands for the energy loss probability distribution for a two-prong structure in the color representation R. In this case, the opening angle is θ 12 = p ⊥ z(1−z)E . The only dependence on in the spectrum appears in the parent parton spectrum, where E = E + . The splitting function P (z) is unmodified because the correction goes as ∆P (z) = P (z) /E [50], which is suppressed as long as the energies of the daughter partons are large enough. However, due to the steeply falling spectrum, which we assume to be governed by a power law behavior, as above, the ratio of medium and vacuum spectra is more sensitive to modifications so the typical energy loss should be compared to E/n E and hence can not be neglected [33].
For reasons that will shortly become clear, the singlet antenna is the irreducible color configuration that one needs to consider in the large-N c limit in order to construct the general result for arbitrary color states. Hence, we shall focus on the decay of a boosted massive object in a singlet color state such as a virtual photon decaying into a quark-antiquark (qq) state, neglecting for the moment the quark masses. Finally, we consider the same set of approximations as for the single quark case in the previous subsection. In particular with regards to the ω 0 ω 1 ω 2 Figure 2: Feynman diagram depiction of the process of decoherence and subsequent energy loss of two partons created early in the medium. In the presence of the two-prong structure, we integrate over all medium-induced emissions, depicted by wavy lines, that can occur as direct contributions (e.g., such as the gluons labeled by ω1 and ω2), interferences (e.g., the gluon labeled by ω0) and virtual contributions. treatment of radiation, we shall work in the soft gluon approximation corresponding to quasiinstantaneous emission, where overlapping formation times are neglected, see the discussion in Sec. 3. Then, as a final step we will then generalize this description to account for a generic total charge of the parent parton.
The cross section of the generic process we are interested in is depicted in the standard form in Fig. 2, where the amplitude (and its complex conjugate) is depicted on the left (right) side of the cut, represented by the dashed line, and we integrate over all medium-induced gluons, depicted by wavy lines. The crucial observation that reduces the complexity of the task at hand consists in realizing that there can only be one gluon connecting the quark (antiquark) in the amplitude and the antiquark (quark) in the complex conjugate amplitude in the large-N c limit [61], as represented by a thick wavy line labeled by ω 0 in Fig. 2. This emission constitutes at the same time a modification of the color structure of the antenna, see Sec. 4.2, which is why we refer to it as a "flip". Hence, to the left of the flip, only virtual diagrams contribute which, as we will see in Sec. 4.3, will lead to the renormalization of the quenching parameterq inside the decoherence parameter. To the right of the flip radiation off the quark and the antiquark factorize, resulting in independent radiative energy loss, as denoted by the thin, gluon lines labeled ω 1 and ω 2 in Fig. 2.
This intuitive picture emerges within our set of approximations that make away with more complicated situations and topologies that are sub-leading. Two such interference contributions are depicted in Fig. 3. The diagram on the left constitutes a non-planar contribution, which is explicitly neglected in the large-N c . The diagram on the right is leading in the N c counting. Nevertheless, it corresponds to two-gluon emission with overlapping formation times. This is sub-leading in the limit of soft gluon emissions, due to the limited phase space for overlapping formation times in a large medium, see e.g. [27].
Let us currently report the result of our calculation for the color singlet energy loss distribution, leaving a full derivation to Sec. 4 (see also Sec. 3 for an introduction to the general formalism). A generalization to arbitrary color representation will follow shortly. We define a regularized splitting rate, acting on the propagating quark-antiquark system as where the index i (j) denotes the leg that is emitting (absorbing) the medium-induced gluon (regardless whether it is in the amplitude or complex conjugate amplitude). In the following, the index "1" ("2") will refer to the quark (antiquark). For example, refers to the direction of propagation of the quark (antiquark), and so on. We find that, for the direct and interference spectra, respectively, where x 12 (t) ≡ n 12 t and n 12 ≡ n 1 − n 2 (|n 12 | ∼ θ 12 is the opening angle of the pair). The function F(x) ∼ e −x , for details see Eq. (71), incorporates the effect of quantum decoherence, and suppresses the interference term for hard Putting all the pieces together, the two-prong energy loss probability of a color singlet dipole in the large-N c limit reads where P sing ( ) ≡ P sing ( , θ 12 , L). This is one of the main results of this paper. Here, P 1 ( , L) describes the independent energy loss of the antenna legs, see Eq. (8), and ∆ med (t) is the socalled decoherence parameter that incorporates the effect of color decoherence, and reads (for a homogeneous medium) see Sec. 4.3 for further details. It constitutes the probability for the medium to resolve the color structure of the pair after traversing a distance t in the medium. Note, that in contrast to the instantaneous nature of quantum decoherence, color decoherence accumulates along the trajectory. It is sensitive to characteristic angle ∼ (qt 3 ) − 1 /2 , above which color coherence is wiped out. One can check that Eq. (14) contains two limiting cases, corresponding to coherent and decoherent antennas. In order to illustrate this point, let us for the moment focus exclusively on the effect of color decoherence, contained in the decoherence parameter Eq. (15). First, we deal with the incoherent case. Letting ∆ med (L) = 1, i.e. θ 12 θ c where θ c ∼ (qL 3 ) − 1 /2 , suppresses the second term and therefore the total energy loss probability of the singlet twoprong structure is given by the product of one-prong energy loss probabilities, In the opposite case θ 12 = 0, so that ∆ med (t) = 0. In this limit the interferences cancel the direct contributions, dI 12 − dI 11 . 2 In the interference term, proportional to Γ 11 (ω, t), we can shift 1 → 1 − ω to allow the integral over ω to act on P 1 ( 1 − ω, L − t). Then one can use that, cf. Eq. (49), and similarly with P 1 ( 2 , L − t). We reconstruct in this way the total derivative acting on the product of energy-loss probabilities, such that the interference term gives rise to where the second term cancels exactly the first term in Eq. (16). Therefore, the energy loss probability of the infinitely narrow antenna vanishes, that is when θ 12 θ c , as expected.
The generalization to arbitrary color representation of the parent parton is straightforward in the large-N c limit, see Sec. 4.4 for details. Let us for the moment consider parton splittings via a gluon emission. 3 Hence, for a parent parton with color representation R, the two-pronged energy loss probability reads, where P sing ( 2 ) denotes the color-singlet two-prong quenching weight in Eq. (14). Keep however in mind that the color factor in the interference spectrum entering P sing ( ) has to be substituted by C F ≈ N c /2 and the one-prong quenching weight P R 1 ( ) becomes sensitive to the relevant color charge through the generalized couplingᾱ R = α s C R /π. This is the second main result of the paper.
We plot the two-prong energy loss distribution for a color singlet antenna, evaluated according to Eq. (14), and for a quark splitting, evaluated according to Eq. (20), in Fig. 4 for two different opening angles, see figure caption for details. For comparison, we also evaluate a two-pronged incoherent energy loss distribution, which simply is defined by the first line in Eq. (14). Finally, for the quark we also plot the single-prong energy loss distributions that corresponds to a completely coherent splitting. The color singlet distribution interpolates between a distribution peaked around small values of for small opening angles to the incoherent case at large angles. We notice immediately the sensitivity to the opening angle. At sufficiently high energy, all distributions fall off as − 3 /2 , due to the incoherent nature of hard emissions To summarize, for arbitrary color representation in the large-N c limit, the two-pronged energy loss distribution is therefore a convolution of the quenching weight of the total charge along the whole length of the medium with the two-pronged color singlet distribution that we will derive in full detail in the subsequent sections. (ϵ) Figure 4: The two-prong energy loss distribution for a color singlet antenna (left) and for a quark splitting (right). Medium parameters were chosen to beq = 1 GeV 2 /fm and L = 2 fm, and αs = 0.3. We plot the corresponding distributions for two opening angles, θ12 = 0.2 (solid curves) and θ12 = 0.8 (dashed curves) in addition to the completely incoherent case (dotted curves) and, for the quark, the one-prong energy loss distribution corresponding to a completely coherent splitting (dash-dotted curve).

Single parton energy loss
In this section we shall detail the calculation of the single parton energy loss probability P 1 discussed in Sec. 2.1. It will serve to introduce the formalism that we shall use in the next section where we derive the main result of this work, namely, the two-prong energy loss probability distribution.
The parton, of energy E and transverse momentum p 4 , is assumed to be produced in a hard process, such as an e + e − collision. We will keep the initial angle finite, but small |p|/E < 1, although one can choose a frame where it is zero, in order to introduce the most general formulation that will be used in the case of two color charges. In fact, as we will show below, the radiation spectrum will not depend on the angle. In this case, emissions off the recoiled quark can be ignored by choosing the light-cone gauge A + = 0, which ensures that soft gluons can only be radiated by a forward-moving quark (antiquark). In the present analysis, we also ignore bremsstrahlung radiation associated with the creation of the back-to-back pair and focus solely on medium-induced radiation.
We are interested in medium effects within the following set of approximations: (i) we neglect the transverse momentum broadening of the quarks, and (ii) we assume soft gluon emission, ω E. In this limit, the amplitude (up to irrelevant phase factors) reads, where s and λ are the spin and polarization of the quark and the radiated gluon, respectively. For details see Appendix B. In this expression, G A describes the propagation of the gluon and stands for a Wilson-line in the fundamental representation evaluated along the classical trajectory x, given by x(s) = ns where n ≡ p/E, see Eq. (25). Note that the initial point of the eikonal gluon propagator G ab A (k, L; x, t) is evaluated in coordinate space while the endpoint in momentum space that is, Apart from the color structure, G is equivalent to the Green's function of a non-relativistic particle of mass ω in 2+1 dimensions propagating through a background field, described by the Schrödinger equation where A ≡ T · A − and T is the relevant color matrix. The solution can be cast in the form of a path integral, where the Wilson line reads, and P + implements path ordering. The differential cross section is obtained by squaring the amplitude, averaging over initial and summing over the final state quantum numbers (helicity, color, flavor) and background field configurations. Since we are only interested in the inclusive energy spectrum, we also can integrate out the transverse momentum of the emitted gluons. The cross section then reads where n f is the number of quark flavors and the factor 1/N c corresponds to an average of initial quark colors, and the gluon measure reads, and similarly for the quark. Here, ... stands for ensemble average over medium configurations, which is assumed to be Gaussian with the 2-point correlator given by where dσ el /d 2 q g 4 /q 4 is the small angle 2 to 2 elastic cross-section, and n is the density of medium color charges. The soft radiation spectrum can be extracted from the soft gluon radiation cross-section associated with the quark, E ω, as where we notice that the Born cross-section of quark production, dσ vac /dΩ p = n f s |M (i) (s) (p)| 2 , factorizes.
Let us now discuss briefly the effect of the medium average on the soft emission spectrum. As a consequence of the sum over color indices in the final state together with medium average, t t ′ amplitude c.c.
where we assumed that the gluon emission time in the amplitude to be larger than that in the complex conjugate amplitude, i.e. t > t. A further simplification occurs when applying the Fierz identity where is a Wilson line in the adjoint representation. Using the fact that tr(t a t c ) = δ ac /2, Eq. (30) yields Furthermore, we can make use of the following property of the gluon in-medium propagator, where L < t < t, in order to separate out the dynamics acting between the emission time in the complex-conjugate amplitude and the end of the medium. It is then possible to show that where k ≡ d 2 k/(2π) 2 . Hence, integrating out the transverse momentum forces the gluon, that was emitted in the amplitude at time t and transverse position x, to be absorbed at time t and transverse position y in the complex conjugate amplitude as depicted in Fig. 5. Keeping finite-angle corrections in Eq. (34) leads to a more involved structure of the spectrum.
Anticipating our application of these techniques to multiple projectiles in Sec. 4, we presently introduce the building block where n i ≡ p i /E i and x i (t) = n i t and we have suppressed the explicit dependence on the gluon energy ω. Often we will use the following shorthand W(x j , t ; x i , t) ≡ W ij . It describes the propagation of a gluon between the emission at position x i (t) and absorption at position x j (t ). Note that the emission (absorption) can occur both in the amplitude or in its complex conjugate, due to the property of the quark-gluon vertex described in Appendix A.
The contribution to the spectrum, depicted by the leftmost diagram in Fig. 5, reads then where α s = g 2 /4π and we have labeled the direction of the quark with the index "1" (n 1 ≡ n).
The medium average of the two-point function is performed in Appendix C, where Eq. (C.10) yields where ∆x 1 = x 1 (t )−x 1 (t) and the 3-point functionS (3) (x, y, v) is explicitly given in Eq. (C.7). For a vanishing dipole size, i.e. v = 0, we introduce the common notation, Thus, the contribution from the first time-ordering reads, The This spectrum was first derived by BDMPS-Z [18,19,20,21,22,23], see also [24,62,63] for equivalent formulations. 5 Similarly, the contributions from diagrams (c) and (d) in Fig. 5, corresponding to virtual corrections, simply read as expected. This ensures that real and virtual contributions cancel for inclusive quantities, i.e., dI a+b+c+d = 0. The expression in Eq. (40) can be further simplified by noticing that the dominant contribution involves a strong correlation of the time-integrations. Introducing the variable τ = t −t, we note that its range is bounded by the coherence time t f ∼ ω/q, which for soft emissions τ < t f L − t ∼ L. Hence, in the limit of large medium one can approximate the time integration over τ as follows, Formally, this allows to treat multiple radiation as independent, see Fig. 6, with a constant rate where the 2-point function lives in the time interval [t + τ, t]. Note that letting x = y = 0 before integrating over τ in Eq. (43) generates a spurious ∼ τ −2 divergence that is regulated by integrating over τ before integrating over the soft gluon transverse momentum that yields the condition x = y = 0, see [25].
In order to proceed, we must specify what we mean by the medium interaction potential. As stated earlier, we will currently only account for diffusive broadening in the plasma, corresponding to the harmonic oscillator approximation, Eq. (D.1). We find the rate of direct emissions by taking the limit x 12 → 0 of Eq. (70), where we used the results in Appendix D, in particular Eq. (D.4). Indeed, we recover the well-kown LPM rate, as in Eq. (4). It is worth to highlight the probabilistic nature of multiple emissions that arise in this case. Since, in the small formation time limit t f L, we can treat them as independent, we easily realize that the spectrum for radiating n gluons reads This leads directly to defining an energy loss probability in the form of a Poisson distribution, as in Eq. (6), that was first introduced in Ref. [33], see also [36,37,38]. However, in line with the derivations of Sec. 4, it is convenient to use an alternative way of expressing this probability in terms of a rate equation. As depicted in Fig. 7, one of the corrections to the probability at time t L reads simply To obtain the full correction we should sum the contribution from all the diagrams in Fig. 5, and we will also perform the same set of approximations as described in Eq. (42), in particular regarding the branching time. The total correction then reads Hence, the total probability obeys the following equation where the first term corresponds to the absence of energy loss. By taking a derivative with respect to the final time, we obtain the following evolution equation for the energy-loss probability, where we have used the notation from Eq. (11), with Γ(ω, t) ≡ Γ 11 (ω, t). It is a straightforward exercise to check, that we also obtain this equation by acting with a time derivative directly on Eq. (6). The solution to Eq. (49), or equivalently Eq. (6), can be found using a Laplace transform of the energy loss probability, where the contour C runs parallel to the imaginary axis to the right of any singularity of P 1 (ν, L) in the complex-ν plane. Inserting this into the evolution equation, we get where γ(ν, t) = ∞ 0 dω Γ(ω, t)e −νω = − πνᾱ 2q is the Laplace transform of the (regularized) splitting rate. The solution to Eq. (51), with initial conditionP 1 (ν, 0) = 1, is simplỹ Taking the inverse Laplace transform, we recover the simple form of the energy loss probability, given in Eq. (8) [33].  Fig. 2 and Fig. 3 is obvious, and is obtained by deforming the Wilson lines in the complex conjugate amplitude to appear below the ones in the amplitude.

Two-parton energy loss and derivation of Eq. (14)
In Sec. 2.2, we anticipated the final result for the two-pronged energy loss distributions that is shown to take a rather simple form. For completeness, we shall in the following present a complete derivation of Eq. (14). In order to help the reader walk through the steps, we explicitly restate our working assumptions at the outset: (i) The energetic antenna forms instantaneously at t = 0, for a discussion see Appendix B.
(ii) The radiated gluons are very soft such that the total energy loss is much smaller than the energies of either of the two prongs. (iii) We focus on medium induced radiation and implicitly subtract the vacuum part from each radiation, see Appendix C for details. (iv) We neglect overlapping emissions owing to the fact that their formation time is much smaller than the medium length. (v) Finally, we focus on the leading-N c contribution, see Appendix E for a discussion.
Finally, all medium averages are approximated by the harmonic approximation, we refer to Appendix D for detailed calculations of all relevant two-and three-point functions. We shall proceed by constructing an evolution equation for the two-prong probability of a color singlet dipole, depicted in Fig. 8, that resums multiple missions, including for the first time interferences and virtual contributions between the two prongs. We note that at zeroth order, that is in the absence of radiation, the energy loss probability must be given by δ( ). Then, as within the Dyson-Schwinger construction, we will evaluate the correction to the energy-loss probability at a late time, where the building block in the intermediate evolution is the color matrix M ijjī (1221). In this way, we derive a new rate equation for two-prong structures that propagate through the medium. We organize our discussion into considering direct emissions, interferences and contributions to the decoherence parameter separately.

Direct emissions from the qq pair
At some late time, let us consider the direct emission of a real gluon, i.e. emitted and absorbed by the quark (leg "1"). By late, we mean that there are no more emissions between the absorption time and the evaluation time t L . The emission couples to a matrix element with a generic color structure, which we denote by M ijjī (1221), as mentioned above, see the left diagram in Fig. 9. The color structure of the contribution in Fig. 9 reads where x 1 (t) (x 1 (t )) is the position of the quark at the time of emission (absorption) and we have used the Fierz identity in Eq. (31). The generalized Green's function W ab is defined in Eq. (35). Due to color conservation, which is insured by the correlator Eq. (28), we can anticipate that, after performing the medium averaging, the correlator becomes which leads to δ ca [t c t a ]ī i = C F δī i . This diagonalizes the matrix M ijji (1221), thus reproducing the original color structure of P sing ( , t L ). After a direct emission we are left with the same color structure as we started out with times a direct emission off the quark, see the two diagrams on the right side of the arrow in Fig. 9 where it is made explicit that the correction appears with a factor 1/N c . This particular contribution leads to a correction to the two-pronged probability that reads This correlator is identical to the one we considered for the one-particle energy loss for a particular time ordering, cf. Eq. (46), and we can therefore simply refer to Sec. 3 for further details. After summing real and virtual diagrams, as in Fig. 5, we obtain the full correction for the quark emissions, where Γ 11 (ω, t) is defined in Eq. (11). Similarly, for emissions off the antiquark (leg "2") we can readily write Together these expressions simply resum independent soft gluon emissions off each leg, and constitute the limit of completely incoherent energy loss.

Interference emissions off the qq pair
Let us now turn our attention to the interference contributions, considering first the first diagram in Fig. 10. Proceeding as in the previous subsection, we write down its color structure, Once again, we can anticipate the relevant color structure based on the fact that we are left with two Wilson lines in the adjoint representation whose average generates a delta function in color space, Using this observation in the last line of Eq. (58), we find the following color structure where we have used the Fierz identity where t a is the color matrix in the fundamental representation. Usually, we will also define (T b ) ac ≡ if abc . At leading-N c only the first term in Eq. (60) contributes as depicted in Fig. 10. At this level the matrix does not diagonalize and this contribution to P 2 involves a new object S 2 that reduces to in the absence of further emissions. This object governs the color decoherence of the antenna, see [54]. We will calculate radiative corrections to this object in Sec. 4.3. Thus, the contribution to the splitting probability from the interference contributions reads where the overall minus-sign arises from charge conjugation of the quark-gluon vertex. We again encounter a 2-point function, connecting the quark and the antiquark. Using Eq. (35) and Eq. (C.10), we find that where ∆x 2 = x 2 (t ) − x 2 (t). Note that, in comparison with Eq. (C.10), we have shifted the coordinates with respect to the Wilson line at coordinate x 2 due to the presence of U 2 . Finally, enforcing the boundary conditions y = x 1 (t) and x = x 2 (t ), we find with x 12 ≡ n 12 t. Inserting the above expression in Eq. (63), and accounting for all topologies (as in Fig. 5, where the upper leg is denoted "1", and has support in the amplitude, and the lower leg "2" that has support in the c.c. amplitude), yields where we have applied the approximation Eq. (42) and Γ ij (ω, t) is defined according to Eq. (11) with and, analogously, dI 21 (dω dt) with x ↔ y. However, note thatS (3) is symmetric in the two first arguments, see Eq. (D.4). We recover with Eq. (67) the leading order antenna radiation spectrum [51,64,53].
In the limit of τ t, one can neglect the phase and the ωn 12 in the derivative, see Eq. (81) in the next section for a parametric argument. Hence, one can approximate the interference spectrum as follows where we have used Eq. (38). In the harmonic oscillator approximation, Eq. (D.1), we find that ∂ x · ∂ y K(x, y)| x=0,y=x 12 = 1 2π ωΩ sinh Ωτ where Ω = (1 + i)/2 q/ω. Integrating over τ yields The interference spectrum therefore reads, where The function F appearing above introduces a suppression mechanism, namely that of quantum-mechanical decoherence, of emitted gluons with transverse momentum k 2 ⊥ ∼ √q ω [51,53]. Hard gluons which are emitted at time t, i.e. those with momentum k 2 ⊥ x −2 12⊥ , can resolve the internal structure of the antenna. Since interferences are suppressed, the direct components are dominating the total spectrum. Sufficiently soft gluons, on the other hand, cannot resolve the antenna. In this case, the interferences are not suppressed and contribute with a negative weight to the total spectrum. For t < t d thus, whenever the local medium scale dominates, x −2

12⊥
√q ω, the contribution from the interferences can be neglected. Hence, for our purposes and numerical evaluation, it is sufficient to consider the overall suppression F(x) ∼ e −x , as expected from the discussion above.
In the next section, we shall encounter another suppression mechanism related to the color decoherence of the pair.

Radiative corrections to the decoherence parameter
We are now left with evaluating radiative contributions to S 2 , which is proportional to the non-diagonal matrix S 2 ∝ M iijj (1221). At first glance, this appears to be a formidable task involving a hierarchy of higher-order color structures. However, a significant simplification of the problem emerges in the large-N c limit where S 2 factorizes into the product of two independent dipoles [61], see Fig. 11. We have provided some further discussion about this crucial point in Appendix E, where we argue in general terms that any radiative (real) contribution to S 2 are subleading in 1/N c or, as depicted in Fig. 3 (right), subleading in t br /L due to overlapping formation time. Due to the latter point, gluons that cross the cut are subleading, and thus S 2 does not contribute to energy loss. It can also be shown that S 1 is real valued. The correction to S 1 due to a single emission, see Fig. 12, is given by where we recognize the dipole cross-section, and Σ 12 (t) denotes the exchange of a (virtual) gluon. In the harmonic-oscillator approximation, employed in Appendix D, we have where x 12 ≡ n 12 t is the transverse size of the dipole andq F is the jet quenching parameter in the fundamental representation. This factor leads to a suppression of the interference contributions at times larger than a characteristic decoherence time, which parametrically goes as t d ∼ (qθ 12 ) − 1 /3 .
One can also write the corresponding Dyson-Schwinger equation, which is solved by The additional exponent in Eq. (77) accounts for radiative corrections to the jet quenching parameter, as will become clear from the discussion below.
One of the four contribution to Σ 12 (t) can be read directly off from Fig. 12 and reads where we already have employed our standard approximations in the limits of the integral over τ ≡ t − t. The remaining contributions can easily be read off from Fig. 5, where the upper leg is now denoted "1" while the lower leg is "2" (i.e. both have support in the amplitude). The quantity we need is a 3-point function that lives during the time interval [t + τ, t]. It is explicitly calculated in Eqs. (C.7) and (C.8), and reads where ∆x 1 = x 1 (t )−x 1 (t) and v(s) = n 12 s describes the evolution of the size of the antenna in the relevant time interval. In our case, the boundary conditions are y = x 1 (t) and x = x 2 (t ), which leads to At this point we can make use of the time separation τ t to simplify the above formula further. The frequency of the emitted gluon is parametrically ω ∼ τ /r 2 ⊥ , where r ⊥ is the size of the virtual fluctuation, hence, the phase yields Therefore, unless the radiated gluon is collinear to either the quark or the gluon, the phase in Eq. (80) is negligible. Likewise, based on the same parametric estimates. Moreover, one can neglect the variation of v during the exchange, v(t) ≈ v(t ) ≡ x 12 . These approximation were also employed to simplify the interference spectrum in Sec. 4.2.
The remaining three contributions can be read off from Fig. 5, where the upper leg is "1" and the lower leg is "2". Hence, within the approximation described above, the sum over all four contributions reads where . (83) where the three-point functionS (3) (x, y, x 12 ) can be found in Eq. (D.7). Applying the Fourier transform, this corresponds to where this time the relevant 3-point function can be read off from Eq. (D.8). Expanding in the opening angle, this term can be written as where we explicitly denote that the jet quenching parameter is proportional to the color factor of the fundamental representation, with Here we have stated the integration limits of the double logarithmic phase-space and regulated the double divergence. As expected, there is no vacuum contribution to this quantity. In Eq. (86) τ 0 ∼ m −1 D is related to the in-medium correlation length (where m D is the Debye screening mass) and τ max is related to the time t. However, we have to make sure that the phase space for the integral over the transverse momentum is available, at any given time t (τ max is always limited by L from above). Since t ≤ t d (where t d ∼ 1 (x 2 12q ) is the decoherence time), we can safely set τ max = t, and The double-log contributions are the largest at t = t d , which leads to the constraint In this suggestive form, it becomes clear that Σ 12 (t) accounts for a radiative correction to the dipole cross-section in Eq. (75), that can be absorbed into the jet quenching parameter q →q (1) ≡q + ∆q(x 12 , t). Those double logarithmic corrections in Eq. (87) are universal and have already been encountered in the context of transverse momentum broadening or radiative energy loss [55,56,57]. We note that the radiative corrections which will accompany the (general) 3-pointS (3) , e.g. in Eq. (43) (and also for the interferences in Eq. (68)), that were calculated for the first time in [56], can be implemented in our work by a similar redefinition ofq in the respective spectra.
The full dipole cross-section reads then such that the decoherence parameter is where we used that q A ≈ 2q F in the large-N c limit in order to compare to the leading-order decoherence parameter [54]. This factor accounts for an accumulative process of color decoherence as the pair propagates in the medium. In summary, calculating the off-diagonal color structure S 2 at large-N c , we have recovered the decoherence parameter describing an antenna traversing the medium and found how radiative corrections contribute. These novel ingredients adds to the program of computing radiative corrections to in-medium processes (transverse momentum broadening, medium-induced emissions and in-medium decoherence).

Final answer and generalization to arbitrary color representation
Combining the results obtained in Secs. 4.1, 4.2 and 4.3, we find that the two-prong energy loss probability obeys the equation Taking the derivative with respect to t we obtain which is identical to the expression one obtains after applying a time derivative to Eq. (14).
Let us now consider the generalization of Eq. (14) to arbitrary color representation of the initial projectile. From the outset we will neglect g → q +q splitting for two main reasons. First, because in this case all interferences between the outgoing quarks are suppressed in the large-N c limit and, second, because this splitting does not involve a soft divergence and will not contribute at leading-double-log accuracy in vacuum emissions. Hence, we are left with the processes q → g + q and g → g + g, both involving gluon emission off a total charge. Both also have soft and collinear divergences. We have depicted these fundamental splitting processes in Fig. 13 as Feynman diagrams and in the double-line notation that embodies the large-N c limit. The resulting diagrams for the two-pronged energy loss probability in the fundamental (quark splitting) and adjoint (gluon splitting) representations are depicted in Fig. 14.
The generalization is straightforward in the large-N c limit. Both diagrams in Fig. 14 consists of two (for the fundamental) or three (for the adjoint) separately propagating color structures. Interferences can only arise from the central structure common to both and also to the color singlet, see Fig. 8, that has been extensively analysed above. Hence, additional diagrams arise only as direct terms involving the outer-most (for the fundamental antenna) and outer-most and inner-most (for the adjoint antenna) structures that only contribute to the direct emissions. Hence, in contrast to the color singlet case, a colored antenna can emit via the total color charge up to the flip.
The two-pronged energy loss probability for an antenna in arbitrary color representation will then consist of the following elements. In the completely incoherent case, the antenna can radiate off the total color charge or off the additional hard gluon. In term containing the interference contribution, the antenna behaves as the total charge right up to the flip. Hence, the two-pronged energy loss for arbitrary color representation reads, where R = F/A and we define P R 1 ( , t) according to Eq.
F ω s . Finally, the color factor of the interference spectrum, see Eq. (13), entering the rates Γ 12 (ω, t) and Γ 21 (ω, t) has to be transformed as −C F → −N c /2. 6 This more general distribution, follows from the evolution equation which is solved bỹ where the initial condition was set asP R 2 (ν, 0) = 1. Applying that C F N c /2, consistent with the large-N c approximation, we find thatP R 2 (ν, t) =P R 1 (ν, t)P sing (ν, t), which leads to Eq. (20).

Conclusions and outlook
In this work, we have computed the energy loss probability distribution of a boosted pair of partons created promptly in a dense medium. Our main results are given in Eq. (14) and Eq. (20) and are derived in the large-N c approximation. We have identified two processes that delay the independent energy degradation of the individual partons, related to quantum and color decoherence. In effect, the delay is related to the time when the color of the dipole "flips", see Secs. 4.2 and 4.3, induced by an interference exchange. Before the flip, then, the medium only resolves the total color charge of the system and the dipole loses energy as if it were the parent parton. After the flip, on the other hand, energy is lost independently by the dipole constituents.
The two-pronged energy loss distribution is a new tool that can be applied to observables involving vacuum splittings which take place inside the medium, albeit within the approximation that the formation time is much shorter than the decoherence time. It is important for phenomenological applications of jet quenching to capture these effects due to the collinear 6 This is a consequence of color conservation. Consider a total charge Q 0 splitting into two daughter partons with charges Q 1 and Q 2 , respectively, so that Q 0 = Q 1 + Q 2 . Interferences will be proportional to the product of daughter charges, which are found to be Q 1 · Q 2 = (Q 2 0 − Q 2 1 − Q 2 2 )/2, where Q 2 ≡ CR. For photon splitting Q 1 · Q 2 = −CF , while for gluon emission Q 1 · Q 2 = −Nc/2. singularity of vacuum emissions inside a jet. Hence the theoretical uncertainties of jet quenching observables that are affected by such splittings can be greatly reduced. Simply considering the small-and large-angle limits of our expressions, immediately imply a collimation of the jet sample emerging in heavy-ion collisions compared to proton-proton due to the additional suppression of large-angle structures. Our approach may be extended to multiple vacuum splitting to compute radiative energy loss of the fluctuating jet substructure. Phenomenological applications of this tool will be presented in upcoming publications.
As we already have pointed out, the vacuum splitting process at the cause of the twopronged structure is non-local from the point of view of medium interactions (decoherence). However, it is amenable to a probabilistic interpretation where the angular and time-scales are clearly defined from the kinematics of the parent dipole. Our results will therefore also serve as guidance for current and future Monte-Carlo event generators of jet quenching. where the classical trajectory is given by x cl (s) = x 0 + s−t 0 t−t 0 (x 1 − x 0 ) and G 0 is the vacuum propagator in coordinate space, After performing the Fourier transforms with respect to the end-points of the propagator, we obtain G(p, t; p 0 , t 0 ) = e −i p 2 2E (t−t 0 ) y 0 ,y 1 e −i(p−p 0 )·(y 0 +t 0 y 1 ) η ε y 1 − p E U x cl (s) = y 0 + sy 1 , (A. 6) where η ε (x), with ε ≡ i/(E(t − t 0 )), is the heat kernel in two dimensions and embodies the Heisenberg uncertainty principle. It is also a nascent delta function, δ(x) = lim ε→0 η ε (x), so taking the limits E (t − t 0 ) −1 and (p/E)s y 0 , we obtain G(p, t; p 0 , t 0 ) = (2π) 2 δ(p − p 0 ) U t, t 0 ; [x cl (s) = ns] e −i p 2 2E (t−t 0 ) , (A.7) where the two-dimensional vector n = p/E parameterizes the trajectory of the projectile. Below, we list the splitting vertices necessary for the computation as depicted in Fig. A.15. Initial and final momenta are incoming and outgoing, respectively. The quark-gluon vertex reads, V ss λ (q, z) = (ig t a )ū(p , s )/ * λ (k)u(p, s) where q ≡ k − zp. The antiquark-gluon vertex is obtain form the former by the replacement i → −i and s → −s. The gluon-gluon vertex reads Furthermore, we have the gluon-quark vertex Finally, to each vertex one must attach a time integration.
We make now the following change of variables, u ≡ r − x 1 and v = x 1 − x 2 . (C.5) As a result the quantum phase yields (recall that x 1 (s) = n 1 s and x 2 (s) = n 2 s), t t dsṙ 2 (s) = t t ds (u(s) + n 1 ) 2 = t t dsu 2 (s) + 2 z f − x 1 (t )) · n 1 − 2 z i − x 1 (t) · n 1 + n 2 1 (t − t) . (C.6) Then the 3-point function therefore reads v(t) = n 12 t, (C.9) where n 12 = n 1 − n 2 , is the coordinate of the emitting system centre-of-mass. When v is constant (which we will assume when deriving the double logarithmic contribution)S (3) (u f , u i , v) is equivalent to that introduced in Ref. [56]. The 2-point function is deduced from the 3-point function by letting p 1 = p 2 :

Appendix D. The harmonic approximation
The reduced 3-point function Eq. (C.8) is the basic building block to be evaluated. This can be carried out analytically in the harmonic approximation, where we have explicitly denoted the color factor dependence of the jet quenching parameter (throughout the paperq ≡q A , unless explicitly stated otherwise). Using Eq. (D.1) in Eq. (C.8) and assuming v ≈ v(t ) ≈ v(t) to be constant in the interval τ ≡ t f − t i , which is assumed to be small throughout the paper, we find which are both sub-leading compared to virtual emissions (that do not connect the amplitude with the c.c.) in the large-N c limit.