A modified in-medium evolution equation with color coherence

JHEP05(2021)148 Published for SISSA by Springer Received:February 5, 2021 Revised:April 19, 2021 Accepted:May 6, 2021 Published:May 18, 2021 A modified in-medium evolution equation with color coherence João Barata,aFabio Domínguez,aCarlos A. Salgadoaand Víctor Vilab aInstituto Galego de Física de Altas Enerxías IGFAE, Universidade de Santiago de Compostela, E-15782 Galicia-Spain bCPHT, CNRS, École Polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau, France E-mail: [email protected],[email protected], [email protected],[email protected] Abstract: QCD jets produced in heavy-ion collisions at LHC or RHIC energies partially evolve inside the produced hot and dense quark gluon plasma, offering unique opportunities to study QCD splitting processes in different backgrounds. Induced (modified) splittings are expected to be the most relevant mechanism driving the modifications of in-medium jets compared to vacuum jets for a wide sets of observables. Although color coherence among different emitters has been identified as an essential mechanism in studies of the QCD antenna radiation, it is usually neglected in the multi-gluon medium-induced cascade. This independent gluon emission approximation can be analytically proved to be valid in the limit of very large media, but corrections or modifications to it have not been computed before in the context of the evolution (or rate) equation describing the gluon cascade. We propose a modified evolution equation that includes corrections due to the interference of subsequent emitters. In order to do so, we first compute a modified splitting kernel following the usual procedure of factorizing it from the subsequent Brownian motion. The calculation is performed in the two-gluon configuration with no overlapping formation times, that is expected to provide the first correction to the completely independent picture. Keywords: Heavy Ion Phenomenology, Jets ArXiv ePrint: 2101.12135 Open Access,c The Authors. Article funded by SCOAP3.https://doi.org/10.1007/JHEP05(2021)148 JHEP05(2021)148 Contents 1 Introduction 1 2 Theoretical set up 3 2.1 Gluon emission spectrum off a parton 3 2.2 The generating functional and the shower building blocks 6 3 Computing the interference term in the soft regime 7 4 Introducing color coherence effects into the rate equation 12 5 Conclusion and outlook 13 A Propagation of an energetic parton on a classical field 15 B The medium averages 16 C Outline of double in-medium gluon emission computation 19 1 Introduction One of the strongest evidences for the creation of the Quark Gluon Plasma (QGP) at RHIC [1,2] and LHC [3–6] is jet quenching: the modification of jets due to the interaction with the dense QCD medium created in high-energy collisions of heavy atomic nuclei. The most direct observable consequence of this effect is the suppression of the yields of particles and jets at large transverse momentum — the quenching. However, jet quenching is nowadays a generic name that embraces the modern technology of jet studies, originally developed for jets in vacuum (i.e. in proton-proton or simpler colliding systems), including a plethora of global or sub-jet observables with different degrees of sophistication. These new observables pose a challenge on present theoretical descriptions of in-medium jet cascades that are stimulating advances towards a more precise implementation of the underlying physics. Jets in heavy-ion collisions develop partly inside the surrounding QCD matter and partly outside of it, with quantum interference between the two possibilities. Moreover, the total shower contains both medium-induced radiation as well as (angular-ordered, infrared and soft divergent) vacuum contributions. One of the main theoretical difficulties to write a consistent cascade is to understand how to order the subsequent splittings of the two kinds. Under some circumstances, for which color coherence between the different emitters in the cascade plays a central role, the vacuum and medium contributions to the cascade can be factorized [7,8]. Before a more complete description is available, a usual approximation – 1 – JHEP05(2021)148 is to evolve both cascades independently. For soft gluons, which in the medium have small formation time tf, it can be shown that interference between subsequent emitters can be neglected for large enough media, tf/L 1[9,10]. This independence ensures a probabilistic picture in which evolution equations (known as rate equations) can be easily computed for different jet properties [11,12]. The goal of the present paper is to go beyond this approximation, taking into account the first correction to the completely independent subsequent gluon emission and to propose a modification of the rate equations that takes into account color coherence. The single gluon production, the building block for the in-medium cascade, has been extensively studied along the past few decades [9,10,13–21] and although full numerical solutions are well established [22–24], a fully analytic formulation has not been achieved.1 Nonetheless, for sufficiently large and dense media, the spectrum is well described by the Baier-Dokshitzer-Mueller-Peigné-Schiff-Zakharov (BDMPS-Z) framework [13–16], which encapsulates the propagation of an energetic parton that exchanges multiple soft gluons with the medium. In this regime, the main mechanism for energy loss consists in the emission of induced soft radiation with frequency ωabove the Bethe-Heitler bound, ωωBH ∼µ4 ˆq, but below the critical frequency, ωωc∼ˆqL2, with a typical formation time tf(ω) = 2ω/k2, where kis the transverse momentum of the gluon, Lthe medium length, µthe Debye screening mass and ˆqthe averaged square transverse momentum acquired by a particle propagating in the medium during a time t, i.e. hk2i= ˆqt. Using the previous estimates, one has that tf∼ω/(ˆqtf)∼pω/ˆq, with the gluon acquiring a transverse momentum k2∼ˆqtf∼√ˆqω during the branching process. In the limit where multiple gluon emissions are observed (i.e. ωωc2), we have that gluon radiation is formed almost instantly since tf(ω)tf(ωc) = L, while the final transverse momentum of the gluon k2∼ˆqL √ˆqω µ2, as the gluon still has to propagate until the end of the medium after its formation. Thus, to leading order in inverse powers of the medium length, soft gluons are produced decoherently and almost instantaneously, and the probability to emit a gluon is proportional to L−tf(ω)≈L. This discussion can also be formulated in terms of the angular structure of the emission spectrum. Defining the emission angle θ2=k2 ω2, we have that θ2∼1 ˆqt3 f(ω)1 ˆqL3≡θ2 c; in contrast, the measured angle gets its main contribution from final state broadening, as can be verified by noticing that accumulated transverse momentum is proportional to the traversed length Ltf, while the energy is conserved. This justifies a picture of time localized splittings (as tfL) producing decoherent partons, with the overall transverse structure being determined by individual momentum broadening of the final states [9,30]. More recently, a lot of effort has been put into studying multiparticle interference effects absent from the BDMPS-Z picture. One of such effects is color coherence between emitters, that has been considered in studies exploring the physics of the QCD antenna with an extra in-medium gluon emission [7,30–33]. The main conclusions of such studies were that for short time scales and emission angles, partons keep color coherent and 1See [25–29] for recent efforts. 2See the discussion on the multiple soft emission region conducted in [9,11]. – 2 – JHEP05(2021)148 splittings are not immediately resolved by the medium, while for long time intervals or large emission angles, the medium randomizes the color fields of each parton such that the system evolves decoherently. Generalizing such a picture for a full in-medium jet [7], it was argued that the color coherence between emitters within the in-medium shower might lead to a significant modification of the expected gluon spectrum for relevant experimental conditions. In addition, it is also well-known that in vacuum, color coherence between emitters needs to be taken into account in order to properly describe experimental data [34]. In this paper we include, for the first time, color coherence effects in the resummation of multiple gluon emissions. In particular, color coherence is included by allowing partons to take a finite time to be resolved by the medium after the splitting, followed by decoherent final state broadening.3Since splittings are still sharply localized when compared to the scale L, we can take the single gluon branching process as the building block for a probabilistic gluon shower, similarly to the totally decoherent case [11,37]. The introduction of color coherence however leads the final evolution equation for the gluonic shower to become, in general, non-local in time, since it takes a finite amount of time for partons to color decohere. The details of the color coherence dynamics are obtained by studying the emission of a soft gluon (the same set up as BDMPS-Z) followed by the emission of a vacuum soft gluon which serves as a probe of the color coherence of the two outgoing states. An effective coherence factor is then extracted and applied as a correction to the emission kernel used in the totally decoherent case. Although this ansatz approach does not strictly follow directly from a first principle calculation, it allows us to gauge the effects of including color coherence effects at the level of each splitting. The present paper is divided as follows. Section 2gives an overview of previous works on the resumation of multiple decoherent in-medium gluon emissions [9,11,38]; section 3 presents the computation of the interference due to including a soft vacuum emission, while section 4provides the derivation of the new evolution equation for the gluon shower. The conclusions are presented in section 5. Further details are provided in three appendices. 2 Theoretical set up In this section we briefly review the main results from [9,11], where the double-differential medium-induced gluon emission spectrum was computed and simplified in order to provide a probabilistic picture for the production of medium-induced radiation.4 2.1 Gluon emission spectrum off a parton When an energetic parton propagates in a dense QCD medium, it exchanges multiple soft gluons with the medium. The leading order effect of such interactions is to transversely kick the hard parton. The probability P1(k;t, t0)that the parton acquires a transverse momentum |k|  p+ 0, with p+ 0the parton energy, due to such interactions during a time 3See [35] for a qualitative similar idea introduced within the context of the Hybrid model [36]. 4See [10] for related work. – 3 – JHEP05(2021)148 L−t0is given by (see [9,11] and references therein) P1(k;L, t0) = Zr e−ir·kP(r;L, t0) = Zr e−ir·ke−CA 2RL t0dt n(t)σ(r),(2.1) where P(r)is the dipole operator in the adjoint representation, n=n(t)the density of scattering centers, and σthe in-medium elastic cross-section (see appendix B). Taking the derivative with respect to Lone obtains the following evolution equation for this probability distribution, ∂LP1(k;L, t0) = ZlC(l, L)P1(k−l;L, t0),(2.2) where for the momentum space integrals we use the shorthand Rq=R(2π)−2d2q, and for the position space integrals we use Rr=Rd2r. Here the broadening kernel is given by C(l, t) = −CA 2n(t)σ(l).(2.3) In addition to momentum broadening, the multiple interactions with the medium also induce the production of soft radiation. In a similar fashion, one can construct the probability P2(k,q;L, t0)of observing two outgoing partons, with transverse momentum kand qrespectively, from an initial state with momentum −→ p0= (p+ 0,p0), for a process happening between times t0and L. Taking into account that the soft modes have typical formation times inside the medium much smaller than the medium length tfL, one can effectively ignore the formation time of radiation when compared to any other time scale.5In this approximation, the two outgoing states evolve decoherently at late times and one can write [9,11] P2(k,q, z;L, t0)=2g2z(1 −z)ZL t0 dt Zm,Q,lK(Q,l, z, p+ 0;t) ×P1(m−p0;t, t0)P1(k−p;L, t)P1(q−(m+l−p); L, t). (2.4) Here lis the transverse momentum acquired during the branching process, mthe momentum of the initial parton before splitting, pthe transverse momentum of the outgoing parton with energy zp+ 0just after the splitting, and Q=p−z(m+l)the relative transverse momentum of the system after branching — see figure 1. Noticing that Qand lare the only momenta scales directly entering the branching process, they can be neglected in a regime that makes it possible to build a probabilistic picture for the emission process. The observation that such a region exists can be argued as follows. The scale lis generated by transverse momentum broadening during the branching process and thus l2∼ˆqtfˆqL, which can be neglected with respect to k2∼q2∼ ˆqL. Then, disregarding such a scale in the single particle broadening contributions and integrating Kover lwe get P2(k,q, z;L, t0) = 2g2z(1 −z)ZL t0 dt Zm,QK(Q, z, p+ 0;t) ×P1(m−p0;t, t0)P1(k−p;L, t)P1(q−(m−p); L, t), (2.5) with Q=p−zm. 5See [9,11] for a detailed discussion of this approximation, in a notation close to the one implemented in this manuscript. More details can also be found in the references therein and follow the qualitative discussion in the previous section. – 4 – JHEP05(2021)148 Figure 1. Diagrammatic representation of eq. (2.4). The labels used follow the notation in the main text and the gray blob denotes the underlying QCD medium. The relative momentum Qis a purely kinematical scale, i.e. it captures how noncollinear the outgoing parton’s momentum are after the branching. Therefore, in this sense, there is a priori no constraint on the values it can take. However, the magnitude of Qis determined by the BDMPS-Z splitting kernel K, which, as we will show below, is peaked around Q2∼qˆqzp+ 0ˆqL (for small z), with smaller momentum scales blocked by the LPM coherence effect and larger values being exponentially harder to obtain via multiple soft scattering [9,11]. As a consequence, one can further simplify the above relation to P2(k,q, z;L, t0)=2g2z(1 −z)ZL t0 dt Zm,QK(Q, z, p+ 0;t) ×P1(m−p0;t, t0)P1(k−zm;L, t)P1(q−(1 −z)m;L, t) ≡2g2z(1 −z)ZL t0 dt ZmK(z, p+ 0;t) ×P1(m−p0;t, t0)P1(k−zm;L, t)P1(q−(1 −z)m;L, t), (2.6) where we used the fact that neglecting the momentum exchanges during branching leads to a collinear splitting. The splitting kernel is given by [11] K(z, p+ 0;t) = Pgg(z) 2πsˆq(t)(1 −z+z2) p+ 0z(1 −z),(2.7) where purely gluonic degrees of freedom are assumed. Here Pgg is the Altarelli-Parisi vacuum kernel multiplied by a factor of CAand the time dependence can be dropped as long as one assumes that the medium is static and homogeneous (plasma brick model). – 5 – JHEP05(2021)148 2.2 The generating functional and the shower building blocks Using the results from the previous section, we now derive an evolution equation for the single gluon inclusive distribution resumming multiple medium-induced gluon emissions. We make use of the generating functional method [39–41], although this is not crucial. We first consider the functional Zp0(u;t, t0)within the time interval t0≤t≤L, Zp0(u;t, t0) = ∞ X n=1 1 n!ZΩn Pn(−→ k1,...,−→ kn;t, t0)u(−→ k1). . . u(−→ kn),(2.8) where the integration is performed over all the individual phase-spaces Ωnon each term of the sum, and u(−→ k)is a test function that will eventually drop out via functional differentiation.6All possible physical processes are stored in Zvia the elementary probabilities Pn(−→ k1,...,−→ kn), which correspond to the probability of measuring nfinal-state gluons with the momentum assigned to each one at time t. For the case at hand, the evolution is defined by the single particle broadening probability P1and the branching probability P2. These are related to the probabilities Pnintroduced in the previous section by Pn(−→ k1,...,−→ kn;t, t0)=2p+ 0(2π)δ n X i=1 k+ i−p+ 0!Pn(k1,...,kn;t, t0),(2.9) where the energy fractions zihave been omitted for the sake of clarity. This relation makes it explicit that the dynamics are constrained to the transverse plane, since p+ 0=Pn i=1 k+ i and the remaining freedom in the k+ iis fixed by the splitting energy fractions zi. In order to obtain an evolution equation for the functional, an evolution law for P2is needed. However, as this probability already includes broadening contributions associated to in/out-going legs (see figure 1), we truncate such terms by introducing the associated branching probability e P2(k,q, z;L, t0)=2g2z(1 −z)ZL t0 dt K(z, p+ 0;t)(2π)4δ(2)(k−zp0)δ(2)(q−(1 −z)p0), (2.10) where we have already performed the integration over the initial delta function. Notice that by doing this, double counting contributions already included in P1is avoided. The time-evolution equation is then given by ∂Le P2(k,q, z;L, t0)=2g2z(1 −z)K(z, p+ 0;L)(2π)4δ(2)(k−zp0)δ(2)(q−(1 −z)p0).(2.11) Now, combining this result with the time-evolution equation for P1and taking into account that for an infinitesimal time step dt Zp0(t0+dt, t0) = ZdΩkP1(−→ k , t0+dt, t0)u(−→ k) +1 2ZdΩk1dΩk2P2(−→ k1,−→ k2;t0+dt, t0)u(−→ k1)u(−→ k2), (2.12) 6We define the functional derivative as δu(−→ p) δu(−→ q)=δ(p+−q+)δ(p−q). Inclusive distributions (as we are interested in computing) can be obtained from functional Zby taking the functional derivative with respect to uat u= 1 [41]. – 6 – JHEP05(2021)148 one can eventually write the evolution law for the functional ∂tZp0(t, t0|u)t=t0 =Zl C(l, t)u(p+ 0,p0+l) +αsZzK(z, p+ 0;t) [u(z−→ p0)u((1 −z)−→ p0)−u(−→ p0)] , (2.13) where the last term in the O(αs)bracket comes from probability conservation (so that when u= 1 the right-hand side vanishes). This relation can be extended to the full shower ∂tZp0(t, t0|u) = Zq+ql C(l, t)u(q+,q+l)δZp0(t, t0|u) δu(~q) +αsZzZq+qK(z, q+;t) [u(z−→ q)u((1 −z)−→ q)−u(−→ q)] δZp0(t, t0|u) δu(~q). (2.14) Finally, the inclusive one-gluon distribution D(x, k, t), which represents the probability of observing a gluon at time twith momentum fraction xand transverse momentum k, can be obtained from the functional [11] D(x, k, t)≡k+ δZp0(t, t0|u) δu(−→ k)!u=1 .(2.15) Finally, using eq. (2.14) and noticing that K(z, p+ 0;t) = K(1 −z, p+ 0;t), we obtain ∂tD(x, k, t) = Zl C(l, t)D(x, k−l, t)(2.16) +αsZz2 z2Kz, x zp+ 0;tDx z,k z;tΘ(z−x)−Kz, xp+ 0;tD(x, k, t). This is the well-known rate equation taking into account multiple soft medium-induced gluon production derived in [11,37]. It has a very simple interpretation: the O(α0 s)term corresponds to the broadening in momentum space occurring between in-medium splittings; the first term at αsorder corresponds to the production of a gluon with energy fraction x and momentum kfrom a parton of the same kinematics enhanced by a 1 zfactor; and the last term corresponds to a gluon with momentum fraction xand transverse momentum kbeing displaced to another energy and momentum mode via a splitting inside the medium, such that the creation and annihilation rates are balanced — and thus probability is conserved. 3 Computing the interference term in the soft regime In order to gauge the role of color coherence, we study the vacuum emission of a soft gluon after the single in-medium gluon emission considered in the previous sections. The process under consideration is depicted in figure 2: the diagram in the right hand panel corresponds to the direct term, while the diagram on the left one corresponds to the interference contribution. Although both pieces have to be taken into account, the interference term is the only one that carries new physical information as it resolves the in-medium quark-gluon antenna. The equivalent BDMPS-Z contribution is obtained by removing the extra vacuum gluon from both diagrams. – 7 – JHEP05(2021)148 t=0 t=Lt=x+t=x+ c t=L t=0 xgxg ´ Figure 2. Left: the interference diagram MqM† gcomputed in this paper with the soft gluon highlighted in blue. The remaining gluon leg is referred in the text as BDMPS-Z gluon. Right: the direct diagram MgM† gthat is not directly computed since it is proportional to the BDMPSZ result. On the left hand side we have explicitly indicated all the time intervals present in the problem, with the emission time not matching in the amplitude (x+) and its complex-conjugate (x+ c). The medium is assumed to be static and homogeneous of length L, and the initiating quark is produced in a hard process (black blob) at time t=t0= 0 in both amplitude and complex-conjugate amplitude. Note that the initial hard process can be factorized from the rest of the diagram and is henceforth disregarded. With that in mind, we start by computing the eikonal emission amplitude of a gluon from either a quark or a gluon in vacuum, Mpure vac q= 2gtak0·ε0 ⊥ k02,(3.1) Mpure vac g= 2igfabc k0·ε0 ⊥ k02.(3.2) Using these results and the rules introduced in appendix A, one can write the amplitudes for the case in which the two gluons are emitted either from the eikonal quark line (Mq) or when the vacuum gluon comes from the gluon line (Mg), Mq=−2g2 ωZxgx+e−ik·xg∂x|0Gab(L, x+|ω)·ε⊥tc miUij(L, x+)tb jkUkl(x+,0)k0·ε0 ⊥ k02,(3.3) Mg=−2ig2 ωZxgx+e−ik·xg∂x|0fcdaGab(L, x+|ω)·ε⊥Uij(L, x+)tb jkUkl(x+,0)k0·ε0 ⊥ k02,(3.4) where k(ω) and k0(ω0) correspond to the transverse momentum (energy) of the in-medium BDMPS-Z gluon and the vacuum gluon, respectively. Their transverse polarization vectors are given by ε⊥and ε0 ⊥. We denote Rx+≡RL 0dx+and we have suppressed the dependence on transverse positions for clarity. We also implement the approximation in which the frequency of the vacuum gluon matches in amplitude and complex-conjugate amplitude. The interference amplitude MqM† gis then given by MqM† g=−4g4i ω2(k0)2Zxgx0 gx+x+ c eik·(x0 g−xg)∂x·∂x0htc miUijta jkGabUklfdcbU† lkG†data kjU† jmix=x0=0 , (3.5) where h i indicates the medium average over all possible configurations of the background field — see appendix B. The color structure with the space-time arguments given explicitly – 8 – JHEP05(2021)148 vestigación de Galicia accreditation 2019-2022); from the European Union ERDF; from the Spanish Research State Agency by “María de Maeztu” Units of Excellence program MDM-2016-0692 and project FPA2017-83814-P and from the European Research Council project ERC-2018-ADG-835105 YoctoLHC. The work of V.V. is supported by the Agence Nationale de la Recherche under the project ANR-16-CE31-0019-02. J.B. is supported by a fellowship from “la Caixa” Foundation (ID 100010434) — fellowship code LCF/BQ/ DI18/11660057, and by funding from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No. 713673. A Propagation of an energetic parton on a classical field When traversing a dense QCD medium, a parton with transverse momentum pand energy p+ 0 |p|keeps a straight trajectory in transverse space and only its color field gets rotated. In this high-energy regime, the in-medium propagator reduces to a Wilson line [43,48] W(x;L, t0) = Pexp ig ZL t0 dx+A−(x;x+)!,(A.1) so that the propagation is confined to a medium of length L−t0. Here A−is the − light-cone component of the classical background field describing the medium, while xis the transverse position at which the parton is located along the future light cone.9For simplicity, the gauge field’s color indices are implicitly contracted with the generators of the color algebra. In the main text, we reserve the Wsymbol for adjoint Wilson lines and Ufor the fundamental representation case. In addition, we omit the transverse position as an argument when x=0. A useful relation between fundamental and adjoint Wilson lines is given by [10,42,43,48,49] W†ab(x) = Wba(x) = 2 Tr[tbU†(x)taU(x)] ,(A.2) where we have made the (adjoint) color indices explicit and traced in the fundamental representation. Two other useful identities are Wba(x)ta=U(x)tbU†(x), tbWba(x) = U†(x)taU(x).(A.3) Easing the eikonal restriction and including sub-eikonal corrections O(p/p+ 0), yields the more general in-medium propagator [43,48,49] G(x, L;y, t0) = Zx yDrexp ip+ 0 2ZL t0 dξ ˙ r2(ξ)!×W(L, t0;r(ξ)) ,(A.4) where now the trajectory in transverse space is not fixed, with the eikonal propagator recovered in the limit p+ 0→ ∞. This propagator obeys the simple composition law G(x, L;y, t0) = Zz G(x, L;z, t)G(z, t;y, t0),(A.5) used in the main text in order to explore the x+locality of the medium averages. 9We use two interchangeable notations for the light-cone time dependence: either it is given as an argument or it appears as a lower index. – 15 – JHEP05(2021)148 Finally, we also make use of the results presented in [45,46], where a gradient expansion around the classical trajectory of G(x, L;y, t0)is performed. The leading result of such an expansion was referred to, in the main text, as the tilted Wilson and it is given by G(x, L;y, t0) = G0(x, L;y, t0)W(xclassical)L,t0+O L−t0 p+ 0 ∂2 xclassical !.(A.6) Here G0is the vacuum propagator (i.e. eq. (A.4) with the gauge field term removed), and xclassical is the classical trajectory in transverse space between positions yand xover the time interval L−t0, xclassical(s) = x+x−y L−t0 (s−L), t0≤s≤L . (A.7) Note that eq. (A.6) is derived under the assumption that p p+ 0 is finite. B The medium averages This appendix outlines how to perform the medium averages present in our calculation. For a more detailed discussion on this topic, references such as [48] should elucidate the reader. The basic object we wish to compute is the following two-point function in the adjoint representation10 within a time interval L−t0, TrhW(x)W†(y)i N2 c−1=1 N2 c−1Tr exp ig Zx+A−(x;x+)exp −ig Zy+A† −(y;y+), (B.1) where we have used the explicit form of the Wilson line from appendix A. In order to proceed, one expands in the coupling gup to the first non-trivial order and models the field correlator as hAa −(x;x+)Ab −(y;y+)i=δabδ(x+−y+)Bγ(x,y;x+).(B.2) Here Bγcorresponds to the Fourier transform of the elastic in-medium scattering potential [50], Bγ(x,y;x+) = g2n(x+)Zk eik·(x−y)γ(k) = Bγ(x−y;x+),(B.3) where n(x+)is the longitudinal density of scattering centers inside the medium and γthe bare in-medium scattering potential, which in UV behaves as γ(k)∼1/k4. This connects to the dipole cross-section as follows, σ(x, t) = σ(x)n(t)=2g2Zk (1 −eik·x)Bγ(x, t)≈ˆq(t) 2CA x2log 1 µ2x2≈ˆq(t) 2CA x2log Q2 c µ2,(B.4) where in the next to last expression we have used the UV behavior of γand introduced ˆq(t) = 4πα2 sCAn(t). In the last expression we have also used the harmonic oscillator approximation and regulated the logarithm by introducing a large momentum scale Q2 c∼ 10An analogous result can be derived in the fundamental representation. – 16 – JHEP05(2021)148 1/x2µ2, with µthe Debye mass. Expanding the Wilson line to first non-trivial order produces Wab(x) = δab +iAc(x)fcab −CA 2δabBγ(0).(B.5) Under the above assumptions, the medium average at linear order, i.e. the leading contribution for N= 1 in the opacity expansion, becomes TrhW(x)W†(y)iN=1 N2 c−1= 1 −CA 2Zx+n(x+)σ(x−y)=1−Zx+ ˆq(x+) 4(x−y)2,(B.6) with logarithmic contributions absorbed in ˆq. Re-exponentiating produces TrhW(x)W†(y)i N2 c−1= exp −CA 2Zx+n(x+)σ(x−y)≡ P(x−y;L, t0),(B.7) where we have introduced the dipole operator P(r), whose Fourier transform gives the single particle broadening probability, as already stated in section 2. More complex averages can be computed following the same procedure. A relevant example, possible when sub-eikonal corrections are included, is the following two-point function, which under the harmonic oscillator approximation reads [43,48] TrhG(x,y)W†(0)ix+ c,x+ N2 c−1=Zx yDrexp "iω 2Zx+ c x+dt ˙ r2−Zx+ c x+dt ˆq(t) 4r2#,(B.8) where on the left-hand side we have suppressed the dependence on the gluon frequency ω. Here G(x,y)is the full gluon propagator, and we have absorbed log Q2 c/µ2in the definition of ˆq, as done in the previous example. Within the harmonic approximation implemented above, one can give an explicit solution [10,42] 1 N2 c−1TrhG(x,y)W†(0)ix+ c,x+≡ K(x,y)x+ c,x+ =A(x+ c, x+) iπ exp iA(x+ c, x+)(B(x+, x+ c)x2+B(x+ c, x+)y2−2x·y). (B.9) In particular, for a static and homogeneous medium the parametric functions Aand Bare given by A(x+ c, x+) = ωΩ 2 sin(Ω(x+ c−x+)) , B(x+ c, x+) = 2 cos(Ω(x+ c−x+)) ,(B.10) while for other medium profiles we refer the reader to [51]. Here Ωis the harmonic oscillator frequency, Ω = 1−i 2sˆq ω.(B.11) Another example is the three-point function appearing in eq. (3.10). This correlator has already been explicitly computed in [9] using the same techniques outlined above (see – 17 – JHEP05(2021)148 next appendix for some details). Explicitly, it can be written within the harmonic oscillator approximation as follows, Zxg zDr1Zx0 g x0Dr2exp iω 2ZL x+ c dt ˙ r2 1−˙ r2 2−ˆq 8ZL x+ c dt r2 1+r2 2+ (r1−r2)2! =Zxg zDr1Zx0 g x0Dr2exp iω 2ZL x+ c dt ˙ r2 1−˙ r2 2−ˆq 4ZL x+ c dt r2 1+r2 2−r1·r2!. (B.12) When writing the dynamical terms we assumed that gluons’ energies match in both amplitude and its complex-conjugate (otherwise the calculation is slightly more evolved, but still feasible). At this point, the unitary transformation r1= Γ(r0 1+βr0 2),r2= Γ(r0 2+βr0 1),(B.13) is boost invariant and leaves the kinetic part of the Lagrangian unchanged, while the potential gives a diagonal term (with Γ−1=p1−β2) Γ2hr2 1(1 −β+β2) + r2 2(1 −β+β2)i= Γ2(1 −β+β2)(r2 1+r2 2).(B.14) Imposing that the cross-terms vanish results in 1−4β+β2= 0 =⇒β= 2 ±√3,(B.15) so that choosing the convergent solution β= 2 −√3, one gets ZΓ(xg−βx0 g) Γ(z−βx0) Dr1ZΓ(x0 g−βxg) Γ(x0−βz) Dr2exp ZL x+ c dtiω 2˙ r2 1−ˆq√3 8r2 1! ×exp ZL x+ c dt −iω 2˙ r2 2−ˆq√3 8r2 2!≡ J(bg,b)L,x+ cJ†(cg,c)L,x+ c. (B.16) In this way, the present medium average is factorized into a product of the previous twopoint correlator, with an effective diffusion coefficient ˆqeff = ˆq√3 2. Note that we use Jto denote Kwith ˆq→ˆqeff, while the †denotes that one should use the conjugate harmonic oscillator frequency. Here we have also introduced bg= Γ(xg−βx0 g),b= Γ(z−βx0),cg= Γ(x0 g−βxg),c= Γ(x0−βz).(B.17) Finally, these results can be used to simplify the spectrum in eq. (3.10) to the form ωω0dI d2kd2k0dωdω0=−ω0dI dω0d2k0g2CFαs (2π)2ω2Re"Zxgx0 gx+x+ cz eik(x0 g−xg) ×∂x·∂x0K(z,x)x+ c,x+J(bg,b)L,x+ cJ†(cg,c)L,x+ c#x=x0=0 . (B.18) – 18 – JHEP05(2021)148 t=0 t=L t=x+t=x+ c t=L t=0 t=y+c t=y+ 0 x=0 y=0 000 xg xg ´ xg − x=0 y’ − ´ xg Figure 3. Leading color interference diagram for the case of double in-medium gluon emission. We identify the different time scales (above) and the transverse position of each splitting (below). In addition, we indicate the position in transverse space for the outgoing states. C Outline of double in-medium gluon emission computation In this appendix, we study the interference diagram in the particular case where the second gluon emission happens inside the medium. We focus solely on the color structure of the squared amplitude, which can be directly read off from the diagram. The analysis of the full double in-medium gluon emission constitutes an extremely challenging problem in the literature (see [52,53] for more in-depth discussions for both dense and dilute systems). In what follows, we will still assume that any formation time is much smaller than the medium length and we will not discuss the problem of overlapping formation times. In figure 3, we depict the leading color interference diagram associated to this process. As pointed out before, there are many ways one can order the time scales x+,x+ c,y+and y+ c, but for the purposes of this appendix we will take y+ c> y+> x+ c> x+. We also use that the hard quark loses energy ωdue to the gluon emission, with the first emission (in amplitude) having energy ξω and the second (1 −ξ)ω. In particular, the color structure of the depicted process reads hUni(L, y+)tc ijGcd(L, ¯ xg;y+,0|(1 −ξ)ω)Ujk(y+, x+)ta klUlm(x+,0)Gab(L, xg;x+,0|ξω) ×G†b¯ b(L, x0 g;y+ c,y0|ξω)G†d¯ d(L, ¯ x0 g;y+ c,y0|(1 −ξ)ω)f¯c¯ d¯ bG†¯c¯a(y+ c,y0;x+ c,0|ω) ×U† m¯ l(x+ c,0)t¯a ¯ l¯ kU† ¯ kn(L, x+ c)i,(C.1) where eikonality is assumed for the leading parton, such that x=y≡0.11 After some algebraic manipulations and assuming ξ→1as done in the main text, such that (1−ξ)ω→ 11In the triple gluon vertex at y0one would need to introduce dummy variables in order to evaluate the derivative operators appearing in the full squared amplitude, and only then set all positions to y0. However, for the current discussion this subtlety does not become relevant since we are only interested in the color structure. – 19 – JHEP05(2021)148 ω0and ξω →ω, it results Zz1z2z3z4z5 TrhG(z1;0|ω)W†(0)ix+ c,x+hfl¯ahWhc(y+,x+ c)Gcα(y+ c,z2;y+,0|ω0)Gαd(L,¯ xg;y+ c,z2|ω0) ×G†d¯ d(L,¯ x0 g;y+ c,y0|ω0)G†b¯ b(L,x0 g;y+ c,y0|ω)Glβ(y+,z3;x+ c,z1|ω)Gβσ(y+ c,z4;y+,z3|ω) ×Gσb(L,xg;y+ c,z4|ω)f¯c¯ d¯ bG†¯cγ(y+ c,y0;y+,z5|ω)G†γ¯a(y+,z5;x+ c,0|ω)i.(C.2) It is now possible to break down the medium averages for each time interval as follows,          fl¯ahhWhcGlβG†γ¯aiin (x+ c, y+), f¯c¯ d¯ bhGcαGβσG†¯cγ iin (y+, y+ c), hGαdG†d¯ dGσbG†b¯ biin (y+ c, L). (C.3) The color structure of this system is quite involved and cannot be written in a closed form [9,10]. In order to proceed, we take again the late time evolution of the system to be that of two independent color dipoles, so that we obtain          flahhWhcGlβG†γaiin (x+ c, y+), fidbhGcdGβbG†iγiin (y+, y+ c), TrhG(¯ xg;z2|ω0)G†(¯ x0 g;y0|ω0)iTrhG(xg;z4|ω)G†(x0 g;y0|ω)iin (y+ c, L). (C.4) For the intermediate medium average, we extract the overall color factor by making use of the techniques introduced in the previous appendix. Taking eqs. (B.2) and (B.5), we consider the contribution of an extra scattering center to the medium average at time τ, such that the time interval is split into two sub-intervals: the former from (y+, y+ c−τ), while the latter (infinitesimal one) is (y+ c−τ, y+ c). Thus, we can write any Wilson line as Wij(x)y+,y+ c=Wik(x)y+ c−τ,y+δkj 1−CA 2Bγ(0)−ifksj As(x)y+ c,y+ c−τ .(C.5) Finally, by using eq. (C.5) in the second line of eq. (C.4) and after the some color algebra we obtain, in accordance with the results shown in the main text, fidbhWcd(x1)Wβb(x2)W†iγ(x3)iy+,y+ c∼fidbhWcd(x1)Wβb(x2)W†iγ(x3)iy+ c−τ,y+ ×1−3CA 2Bγ(0) + CA 2[Bγ(x1−x2) + Bγ(x3−x2) + Bγ(x1−x3)]y+ c,y+ c−τ .(C.6) The time locality of the medium averages and this result leads to the conclusion that the medium average within the time interval (x+ c, y+)must take the form fhlafijkhWhiGljG†kaix+ c,y+,(C.7) matching the result in the main text. For other orderings of the branching times in amplitude and conjugate amplitude, we find the same result for the earliest time interval. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. – 20 – JHEP05(2021)148 References [1] STAR collaboration, Experimental and theoretical challenges in the search for the quark gluon plasma: the STAR collaboration’s critical assessment of the evidence from RHIC collisions,Nucl. Phys. A 757 (2005) 102 [nucl-ex/0501009] [INSPIRE]. [2] PHENIX collaboration, Formation of dense partonic matter in relativistic nucleus-nucleus collisions at RHIC: experimental evaluation by the PHENIX collaboration,Nucl. Phys. A 757 (2005) 184 [nucl-ex/0410003] [INSPIRE]. [3] ALICE collaboration, Suppression of charged particle production at large transverse momentum in central Pb-Pb collisions at √sNN = 2.76 TeV,Phys. Lett. B 696 (2011) 30 [arXiv:1012.1004] [INSPIRE]. [4] CMS collaboration, Charged-particle nuclear modification factors in PbPb and pPb collisions at √sNN = 5.02 TeV,JHEP 04 (2017) 039 [arXiv:1611.01664] [INSPIRE]. [5] CMS collaboration, Observation and studies of jet quenching in PbPb collisions at nucleon-nucleon center-of-mass energy = 2.76 TeV,Phys. Rev. C 84 (2011) 024906 [arXiv:1102.1957] [INSPIRE]. [6] ALICE collaboration, Measurement of jet suppression in central Pb-Pb collisions at √sNN = 2.76 TeV,Phys. Lett. B 746 (2015) 1 [arXiv:1502.01689] [INSPIRE]. [7] J. Casalderrey-Solana, Y. Mehtar-Tani, C.A. Salgado and K. Tywoniuk, New picture of jet quenching dictated by color coherence,Phys. Lett. B 725 (2013) 357 [arXiv:1210.7765] [INSPIRE]. [8] P. Caucal, E. Iancu, A.H. Mueller and G. Soyez, Vacuum-like jet fragmentation in a dense QCD medium,Phys. Rev. Lett. 120 (2018) 232001 [arXiv:1801.09703] [INSPIRE]. [9] J.-P. Blaizot, F. Dominguez, E. Iancu and Y. Mehtar-Tani, Medium-induced gluon branching, JHEP 01 (2013) 143 [arXiv:1209.4585] [INSPIRE]. [10] L. Apolinário, N. Armesto, G. Milhano and C.A. Salgado, Medium-induced gluon radiation beyond the eikonal approximation,Nucl. Phys. A 932 (2014) 152 [arXiv:1404.7079] [INSPIRE]. [11] J.-P. Blaizot, F. Dominguez, E. Iancu and Y. Mehtar-Tani, Probabilistic picture for medium-induced jet evolution,JHEP 06 (2014) 075 [arXiv:1311.5823] [INSPIRE]. [12] S. Jeon and G.D. Moore, Energy loss of leading partons in a thermal QCD medium,Phys. Rev. C 71 (2005) 034901 [hep-ph/0309332] [INSPIRE]. [13] R. Baier, Y.L. Dokshitzer, A.H. Mueller, S. Peigne and D. Schiff, Radiative energy loss and pTbroadening of high-energy partons in nuclei,Nucl. Phys. B 484 (1997) 265 [hep-ph/9608322] [INSPIRE]. [14] R. Baier, Y.L. Dokshitzer, A.H. Mueller, S. Peigne and D. Schiff, Radiative energy loss of high-energy quarks and gluons in a finite volume quark-gluon plasma,Nucl. Phys. B 483 (1997) 291 [hep-ph/9607355] [INSPIRE]. [15] B.G. Zakharov, Fully quantum treatment of the Landau-Pomeranchuk-Migdal effect in QED and QCD,JETP Lett. 63 (1996) 952 [hep-ph/9607440] [INSPIRE]. [16] B.G. Zakharov, Radiative energy loss of high-energy quarks in finite size nuclear matter and quark-gluon plasma,JETP Lett. 65 (1997) 615 [hep-ph/9704255] [INSPIRE]. – 21 – JHEP05(2021)148 [17] M. Gyulassy, P. Levai and I. Vitev, Reaction operator approach to non-Abelian energy loss, Nucl. Phys. B 594 (2001) 371 [nucl-th/0006010] [INSPIRE]. [18] U.A. Wiedemann, Gluon radiation off hard quarks in a nuclear environment: opacity expansion,Nucl. Phys. B 588 (2000) 303 [hep-ph/0005129] [INSPIRE]. [19] X.-N. Wang and X.-F. Guo, Multiple parton scattering in nuclei: parton energy loss,Nucl. Phys. A 696 (2001) 788 [hep-ph/0102230] [INSPIRE]. [20] P.B. Arnold, G.D. Moore and L.G. Yaffe, Photon and gluon emission in relativistic plasmas, JHEP 06 (2002) 030 [hep-ph/0204343] [INSPIRE]. [21] M.D. Sievert, I. Vitev and B. Yoon, A complete set of in-medium splitting functions to any order in opacity,Phys. Lett. B 795 (2019) 502 [arXiv:1903.06170] [INSPIRE]. [22] S. Caron-Huot and C. Gale, Finite-size effects on the radiative energy loss of a fast parton in hot and dense strongly interacting matter,Phys. Rev. C 82 (2010) 064902 [arXiv:1006.2379] [INSPIRE]. [23] X. Feal and R. Vazquez, Intensity of gluon bremsstrahlung in a finite plasma,Phys. Rev. D 98 (2018) 074029 [arXiv:1811.01591] [INSPIRE]. [24] W. Ke, Y. Xu and S.A. Bass, Modified Boltzmann approach for modeling the splitting vertices induced by the hot QCD medium in the deep Landau-Pomeranchuk-Migdal region,Phys. Rev. C100 (2019) 064911 [arXiv:1810.08177] [INSPIRE]. [25] C. Andres, L. Apolinário and F. Dominguez, Medium-induced gluon radiation with full resummation of multiple scatterings for realistic parton-medium interactions,JHEP 07 (2020) 114 [arXiv:2002.01517] [INSPIRE]. [26] C. Andres, F. Dominguez and M. Gonzalez Martinez, From soft to hard radiation: the role of multiple scatterings in medium-induced gluon emissions,JHEP 03 (2021) 102 [arXiv:2011.06522] [INSPIRE]. [27] Y. Mehtar-Tani, Gluon bremsstrahlung in finite media beyond multiple soft scattering approximation,JHEP 07 (2019) 057 [arXiv:1903.00506] [INSPIRE]. [28] Y. Mehtar-Tani and K. Tywoniuk, Improved opacity expansion for medium-induced parton splitting,JHEP 06 (2020) 187 [arXiv:1910.02032] [INSPIRE]. [29] J. Barata and Y. Mehtar-Tani, Improved opacity expansion at NNLO for medium induced gluon radiation,JHEP 10 (2020) 176 [arXiv:2004.02323] [INSPIRE]. [30] Y. Mehtar-Tani, C.A. Salgado and K. Tywoniuk, The radiation pattern of a QCD antenna in a dense medium,JHEP 10 (2012) 197 [arXiv:1205.5739] [INSPIRE]. [31] Y. Mehtar-Tani and K. Tywoniuk, Jet coherence in QCD media: the antenna radiation spectrum,JHEP 01 (2013) 031 [arXiv:1105.1346] [INSPIRE]. [32] Y. Mehtar-Tani, C.A. Salgado and K. Tywoniuk, The radiation pattern of a QCD antenna in a dilute medium,JHEP 04 (2012) 064 [arXiv:1112.5031] [INSPIRE]. [33] J. Casalderrey-Solana and E. Iancu, Interference effects in medium-induced gluon radiation, JHEP 08 (2011) 015 [arXiv:1105.1760] [INSPIRE]. [34] Y.I. Azimov, Y.L. Dokshitzer, V.A. Khoze and S.I. Troyan, Humpbacked QCD plateau in hadron spectra,Z. Phys. C 31 (1986) 213 [INSPIRE]. [35] Z. Hulcher, D. Pablos and K. Rajagopal, Resolution effects in the hybrid strong/weak coupling model,JHEP 03 (2018) 010 [arXiv:1707.05245] [INSPIRE]. – 22 – JHEP05(2021)148 [36] J. Casalderrey-Solana, D.C. Gulhan, J.G. Milhano, D. Pablos and K. Rajagopal, A hybrid strong/weak coupling approach to jet quenching,JHEP 10 (2014) 019 [Erratum ibid. 09 (2015) 175] [arXiv:1405.3864] [INSPIRE]. [37] R. Baier, A.H. Mueller, D. Schiff and D.T. Son, ‘Bottom up’ thermalization in heavy ion collisions,Phys. Lett. B 502 (2001) 51 [hep-ph/0009237] [INSPIRE]. [38] J.-P. Blaizot, E. Iancu and Y. Mehtar-Tani, Medium-induced QCD cascade: democratic branching and wave turbulence,Phys. Rev. Lett. 111 (2013) 052001 [arXiv:1301.6102] [INSPIRE]. [39] R.K. Ellis, W.J. Stirling and B.R. Webber, QCD and collider physics,Camb. Monogr. Part. Phys. Nucl. Phys. Cosmol. 8(1996) 1 [INSPIRE]. [40] Y.L. Dokshitzer, V.A. Khoze, A.H. Mueller and S.I. Troian, Basics of perturbative QCD, Ed. Frontieres, Gif-sur-Yvette, France (1991). [41] P. Cvitanovic, P. Hoyer and K. Zalewski, Parton evolution as a branching process,Nucl. Phys. B 176 (1980) 429 [INSPIRE]. [42] L. Apolinario, N. Armesto and C.A. Salgado, Medium-induced emissions of hard gluons, Phys. Lett. B 718 (2012) 160 [arXiv:1204.2929] [INSPIRE]. [43] J. Casalderrey-Solana and C.A. Salgado, Introductory lectures on jet quenching in heavy ion collisions,Acta Phys. Polon. B 38 (2007) 3731 [arXiv:0712.3443] [INSPIRE]. [44] C.A. Salgado and U.A. Wiedemann, Calculating quenching weights,Phys. Rev. D 68 (2003) 014008 [hep-ph/0302184] [INSPIRE]. [45] T. Altinoluk, N. Armesto, G. Beuf, M. Martínez and C.A. Salgado, Next-to-eikonal corrections in the CGC: gluon production and spin asymmetries in pA collisions,JHEP 07 (2014) 068 [arXiv:1404.2219] [INSPIRE]. [46] T. Altinoluk, N. Armesto, G. Beuf and A. Moscoso, Next-to-next-to-eikonal corrections in the CGC,JHEP 01 (2016) 114 [arXiv:1505.01400] [INSPIRE]. [47] F. Domínguez, J.G. Milhano, C.A. Salgado, K. Tywoniuk and V. Vila, Mapping collinear in-medium parton splittings,Eur. Phys. J. C 80 (2020) 11 [arXiv:1907.03653] [INSPIRE]. [48] Y. Mehtar-Tani, J.G. Milhano and K. Tywoniuk, Jet physics in heavy-ion collisions,Int. J. Mod. Phys. A 28 (2013) 1340013 [arXiv:1302.2579] [INSPIRE]. [49] A. Kovner and U.A. Wiedemann, Gluon radiation and parton energy loss,hep-ph/0304151 [INSPIRE]. [50] A. Kovner and U.A. Wiedemann, Eikonal evolution and gluon radiation,Phys. Rev. D 64 (2001) 114002 [hep-ph/0106240] [INSPIRE]. [51] P.B. Arnold, Simple formula for high-energy gluon bremsstrahlung in a finite, expanding medium,Phys. Rev. D 79 (2009) 065025 [arXiv:0808.2767] [INSPIRE]. [52] P. Arnold and S. Iqbal, The LPM effect in sequential bremsstrahlung,JHEP 04 (2015) 070 [Erratum ibid. 09 (2016) 072] [arXiv:1501.04964] [INSPIRE]. [53] J. Casalderrey-Solana, D. Pablos and K. Tywoniuk, Two-gluon emission and interference in a thin QCD medium: insights into jet formation,JHEP 11 (2016) 174 [arXiv:1512.07561] [INSPIRE]. – 23 –