Jet quenching and heavy ion collisions

TESE DE DOUTORAMENTO JET QUENCHING AND HEAVY ION COLLISIONS Víctor Vila Pérez ESCOLA DE DOUTORAMENTO INTERNACIONAL PROGRAMA OFICIAL DE DOUTORAMENTO EN FÍSICA NUCLEAR E DE PARTÍCULAS SANTIAGO DE COMPOSTELA 2020 DECLARACIÓN DO AUTOR DA TESE [JET QUENCHING AND HEAVY ION COLLISIONS] Para defensas telemáticas D. Víctor Vila Pérez Presento a miña tese, seguindo o procedemento axeitado ao Regulamento, e declaro que: 1) A tese abarca os resultados da elaboración do meu traballo. 2) De selo caso, na tese faise referencia ás colaboracións que tivo este traballo. 3) A tese é a versión definitiva presentada para a súa defensa e coincide coa versión enviada en formato electrónico. 4) Confirmo que a tese non incorre en ningún tipo de plaxio doutros autores nin de traballos presentados por min para a obtención doutros títulos. E comprométome a presentar o exemplar impreso da tese no prazo dun mes dende que a EDIUS mo requira, así como o Compromiso Documental de Supervisión no caso de que o orixinal non estea na Escola. En Santiago de Compostela, a 10 de Xullo de 2020 Asdo. Víctor Vila Pérez AUTORIZACIÓN DO DIRECTOR / TITOR DA TESE [JET QUENCHING AND HEAVY ION COLLISIONS] D. Carlos A. Salgado López INFORMA: Que a presente tese, correspóndese co traballo realizado por D. Víctor Vila Pérez, baixo a miña dirección, e a utorizo a súa presentación , considerando que reúne os r equisitos esixidos no R egulamento de Estudos de Doutoramento da USC, e que como director desta non incorre nas causas de abstención establecidas na Lei 40/2015. En Santiago de Compostela, a 10 de Xullo de 2020 Asdo. Carlos A. Salgado López A mis padres, mi hermano y mis abuelos. Agradecimientos En primer lugar, quiero expresar mi profundo agradecimiento a Carlos Salgado, por dirigir mis trabajos desde las etapas previas a esta tesis doctoral hasta la conclusi´on de la misma, d´andome as´ı la oportunidad de poder aprender de ´el en el d´ıa a d´ıa desde un trato personal muy cercano y de confianza. Su dedicaci´on a la F´ısica por encima de todas sus responsabilidades ha sido una gran inspiraci´on, y su sabidur´ıa y gran intuici´on una fuente constante de aprendizaje durante todos estos a˜nos. Gracias por todo. Ser´ıa injusto no dedicar unas l´ıneas de agradecimiento al resto de personas que codirigieron los proyectos que componen este manuscrito. A Konrad Tywoniuk, con el que tuve la oportunidad de trabajar desde mi estancia en el CERN. Su pasi´on y ambici´on inagotables, sus grandes conocimientos y su buena metodolog´ıa han sido un est´ımulo motivacional muy fuerte. A Fabio Dom´ınguez, supervisor de mis estudios durante la mayor parte de esta tesis doctoral, por su disposici´on a discutir y proponer soluciones ingeniosas a los problemas en los que trabajamos juntos. Y a Cyrille Marquet, por recibirme en la ´ Ecole Polytechnique de Par´ıs e iniciarme en un ´ambito de estudio pr´acticamente nuevo para m´ı aportando todo su conocimiento. Por encima de todo les agradezco su sencillez y naturalidad en el trato. Quiero dar las gracias tambi´en al resto de personas con las que he colaborado o que han aportado su grano de arena a esta tesis doctoral a trav´es de discusiones fruct´ıferas, como Guilherme Milhano, Jo˜ao Barata, Anderson Kendi o N´estor Armesto, entre otros. A mis amigos de toda la vida, con los que llegu´e a Santiago de Compostela hace ya una d´ecada, y a todas las personas que desde entonces han formado parte de mi d´ıa a d´ıa. No habr´ıa sido lo mismo sin todas ellas. Tampoco ser´ıa justo no destinar una l´ınea a mencionar otra de mis pasiones, el tenis, incluyendo el entorno de gente tan saludable que rodea a este deporte, ya que creo que indirectamente me ha ayudado a ordenar y aclarar muchas ideas en la cabeza. Por ´ultimo, es dif´ıcil encontrar las palabras adecuadas para agradecer a mis padres y a mi hermano su apoyo en cada una de las decisiones que me han llevado hasta este punto. Siempre preocupados porque disponga de la atm´osfera adecuada para centrarme en mis propios intereses sin pedir nada a cambio, sobra decir que esta tesis doctoral no ser´ıa posible ni tendr´ıa sentido sin ellos a mi lado. No me olvido de mis abuelos, siempre pendientes de mis actividades con esa curiosidad que les caracteriza. Gracias de coraz´on. I acknowledge the financial support given by the European Research Council under the projects HotLHC ERC-2011-StG-279579 and YoctoLHC ERC-2018-ADG-835105, Xunta de Galicia (Conseller´ıa de Educaci´on) under the fellowship ED481A-2017/089 and Ministerio de Educaci´on, Cultura y Deporte under the fellowship FPU16/02236. Contents Motivation 1 1 Introduction 5 1.1 BasicsofQCD ................................. 5 1.2 TheQCDLagrangian.............................. 6 1.3 Asymptoticfreedom .............................. 9 1.4 Confinement................................... 10 1.5 Lepton-hadron deep-inelastic scattering (DIS) . . . . . . . . . . . . . . . . 11 1.6 Thepartonmodel................................ 12 1.7 HotanddenseQCD .............................. 14 2 The formalism framework 19 2.1 Fundamentals of jet evolution . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.1.1 Jet evolution in vacuum . . . . . . . . . . . . . . . . . . . . . . . . 20 2.1.2 In-medium jet evolution . . . . . . . . . . . . . . . . . . . . . . . . 25 2.1.3 Parton showers from a phenomenological insight . . . . . . . . . . . 29 2.2 Radiative energy loss in the BDMPS-Z/ASW approach . . . . . . . . . . . 34 2.2.1 High-energy parton propagation through a medium . . . . . . . . . 35 2.2.2 The medium averaging procedure . . . . . . . . . . . . . . . . . . . 37 2.2.3 The medium-induced gluon radiation spectrum . . . . . . . . . . . 42 2.2.4 The radiation spectrum off a q¯q-antenna in a dense medium . . . . 45 2.3 Small-xphysicsinpQCD............................ 52 2.3.1 BFKLdynamics............................. 53 2.3.2 BK equation and gluon saturation . . . . . . . . . . . . . . . . . . . 55 2.3.3 The Color Glass Condensate . . . . . . . . . . . . . . . . . . . . . . 56 3 Factorized picture of color coherence for gluon radiation off a double QCD antenna 59 3.1 A few remarks on the physical scenario . . . . . . . . . . . . . . . . . . . . 60 3.2 A problem of multiple emitters . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.A Deriving the tilted Wilsonline......................... 70 3.B Technical details on color algebra . . . . . . . . . . . . . . . . . . . . . . . 72 x CONTENTS 4 Finite formation time effects for in-medium parton splittings 77 4.1 Introducing the parton splitting setup . . . . . . . . . . . . . . . . . . . . . 78 4.2 Derivingthespectrum ............................. 81 4.3 The characteristic time-scales of the process . . . . . . . . . . . . . . . . . 84 4.4 Mapping out the phase-space onto the Lund plane . . . . . . . . . . . . . . 87 4.5 Numerical evaluation of the phase-space . . . . . . . . . . . . . . . . . . . 90 4.6 Summaryandoutlook ............................. 92 4.A Considering colored splittings . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.B Beyond the classical picture . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5 Embedding color coherence into the probabilistic picture of a partonic cascade 97 5.1 Thepartonicsetup ............................... 97 5.2 Computingthespectrum............................ 99 5.3 A color coherence correction to the medium-induced parton evolution picture104 5.3.1 Implementing parton showering from a probabilistic perspective . . 104 5.3.2 Embedding color coherence into the medium-modified parton shower evolutionequations...........................108 5.4 Discussion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6 Deciphering the origin of angular correlations in high-energy collisions111 6.1 Generalremarks.................................111 6.2 Revisiting the double inclusive gluon production probability . . . . . . . . 113 6.3 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Summary 127 Resumen 131 xi List of Figures 1.1 The basic interaction vertices in QCD. . . . . . . . . . . . . . . . . . . . . 8 1.2 Charge renormalization in QCD. . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 A compilation of data regarding the behaviour of the strong coupling constant αswith the momentum Q∝1/r (taken from [11]). . . . . . . . . . . 10 1.4 Hadron production from e+e−collisions. ................... 11 1.5 Schematic depiction of a lepton-hadron deep-inelastic scattering. . . . . . . 12 1.6 A sketch of the QCD phase diagram (extracted from [26]). . . . . . . . . . 14 1.7 Jet quenching in a head-on nucleus-nucleus collision (illustration from [32]). 16 1.8 Phase-space gluon density in a hadron (picture from [40]). . . . . . . . . . 17 2.1 Diagrammatic representation of the γ∗→q¯qg process. . . . . . . . . . . . . 21 2.2 Soft radiation confining cone around leading partons iand j(sketch from [2]). 23 2.3 Diagrammatic depiction of angular-ordered emissions in both QED and QCD (drawn out from [50]). . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4 The Lund diagram filled by means of eq. (2.18). Here Ris the jet opening angle(courtesyof[99]).............................. 32 2.5 Diagram showing a multiple scattering eikonal trajectory. . . . . . . . . . . 36 2.6 Diagrammatic representation of the different contributions to the two-point correlator within the Gaussian approximation framework. . . . . . . . . . . 39 2.7 The medium-induced gluon radiation diagram within the soft approximation. 43 2.8 Schematic view of the diagrams corresponding to the three terms that emerge when computing the squared amplitude of the medium-induced gluon radiation in the soft limit. The dashed line sets the amplitude apart from its complex conjugate. . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.9 A numerical evaluation of the medium-induced gluon radiation spectrum (2.47) for a high-energy quark propagating through a static medium as a function of the variables (2.51) (results from [35]). . . . . . . . . . . . . . . 45 2.10 The three diagrams contributing to the gluon spectrum off the antenna in the medium case. Both the in-in and in-out components are sketched in the panels at top, whereas the bottom one is identified with the vacuum contribution (diagrams taken from [116]). . . . . . . . . . . . . . . . . . . . 47 2.11 A color dipole splitting into two new dipoles in the large-Nclimit. . . . . . 54 2.12 Different gluon showers originated from valence quarks within a nucleon (figurefrom[127]). ............................... 56 xii LIST OF FIGURES 3.1 Diagrammatic sketch of the out-out component to the gluon spectrum off aq¯q-antenna propagating within a medium under the soft limit (ω→0). . 62 3.2 Diagrammatic depiction of the probability amplitudes for the general case of three emitters. The green gluon is set outside the medium in order to hold the control of the stage of color coherence through the whole dynamics scenario...................................... 65 3.3 Representation of the squared amplitudes for two in-medium hard splittings. Note that t=t0is assumed, meaning that finite formation time effects for the parton splitting processes are discarded. . . . . . . . . . . . 67 3.4 Schematic view of one of the interference processes under consideration (left). The corresponding large-Nclimit is shown on the right pane. Recall that t=t0is implemented in order to get eq. (3.18). . . . . . . . . . . . . . 67 3.5 Diagrammatic illustration of a probability inteference contributing to the three emitters scenario (left). The large-Nclimit is implemented on the right scheme. Again, t=t0is assumed to obtain eq. (3.19). . . . . . . . . . 68 4.1 Final-state kinematics of the splitting process under consideration. In terms of angular quantities, the antenna opening angle is n12 =n1−n2, with |n1|=θ1,|n2|=θ2and |n12|=θ..................... 78 4.2 The in-in component to the spectrum. The amplitude, in black, and its complex-conjugate, in grey, are sketched out on top of each other to emphasize the character of the splitting. The dipole spreads over the region (¯ t, t), while the quadrupole does it within the region (L, ¯ t). ......... 82 4.3 The Lund diagram for one soft and collinear vacuum splitting. The main phase-space regimes, bounded by the straight lines depicted on the figures, are interpreted as follows (starting from the uppermost line and going down): (i) the first line (magenta-blue on the left, purely magenta on the right) stands for the boundary in which the quantum-mechanical formation time is of the order of the medium length, tf=L; (ii) the magenta-green line represents the boundary where the formation time is of the order of the decoherence time, tf=td; (iii) the green-red line accounts for the boundary that matches the formation time and the time-scale for medium-induced broadening; (iv) finally, the vertical red-blue line (only in the left pane) represents the critical angle θ=θc. ...................... 88 4.4 Numerical evaluation of the medium modification function Fmed for the high- (E > ωc, left pane) and low-energy regimes (E < ωc, right panel). Notice that the scale of the color coding on the right-hand side is rescaled by a factor 10 with respect to the left one. The shaded area corresponds to the available phase-space given the constraint k⊥> Q0. The boundaries for the different regions are the same as those depicted in Fig. 4.3. . . . . . 91 xiii V ´ ıctor Vila P´ erez 5.1 Diagrammatic representation of the relevant contributions to the process set forth above. The leading-eikonal quark, which emanates at t= 0 as a result of the hard process that materializes inside the medium, splits at t=x+in probability amplitude, whilst the splitting takes place at t=x+ cin complex-conjugate amplitude. For its part, eikonality restrictions are eased for the emerging gluon inside the medium – hereafter referred to as the BDMPS-Z gluon, this way allowing the parton to spread over the transverse plane. In a similar fashion as to the interference components corresponding to the processes derived in Chapter 3, the green gluon, laid again outside the medium, serves as means of tracking the coherence history of the partonic system, what, as will be seen later, is essential for the ultimate aim of this study. The plus component of the light-cone momentum for this soft gluon is fixed to the same in both amplitude and its complex-conjugate. As noted above, here the emphasis is on deriving the interference between the probability amplitudes MqM† g(left pane), while the direct component MgM† g(right panel) does not have much to say when looking towards embodying coherence effects into the rate equation for describing mediuminduced partonic splittings. . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 xiv LIST OF FIGURES xv Motivation There is a growing interest and knowledge in heavy-ion collisions (HICs) since it gave a completely new way of understanding the successful theory of strong interactions under conditions that become more and more extreme, the hot and dense Quantum Chromodynamics (QCD). This theory underwent numerous tests in particle accelerators over more than 40 years regarding the behaviour of hadrons in vacuum. More recently, strong interactions have started to be probed in a medium as well, reaching critical values of temperature and density which bring us to an energy scale that favours the deconfinement of quarks and gluons. These conditions make way to the formation of the so-called Quark-Gluon Plasma (QGP), a new state of matter that can behave in unexpected ways due to collective effects. According to the current description of the evolution of the early universe, the Big Bang theory, 10−5safter the explosion a soup of QGP went through a phase transition to confined hadrons, the first hint of a QCD transition. Deepening and broadening knowledge on this transition involves studying the inner structure of matter under extreme conditions of temperature and density. From a theoretical perspective, the overriding challenge is to achieve a robust formulation of in-medium quantum field theory using QCD. However, the theory is non-perturbative at the relevant scales, thus constraining analytic methods. One way to make progress in the knowledge of its properties is to address the QGP as an almost perfect liquid whose ingredients interact strongly. An intense activity of research is being carried out in this respect. In fact, exploring the plasma has long been established as one of the main aims of the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory (BNL) and the Large Hadron Collider (LHC) program at the Conseil Europ´een pour la Recherche Nucl´eaire (CERN). In order to characterize this novel phase of matter under laboratory conditions, various probes are handled in HICs. Hard probes (HP) are created in the early stages of the collision after hard scattering processes and are therefore key probes for infering the medium properties by means of quantifying their modifications. Jets are natural hard probes: due to the hard scale of these high-transverse momentum sprays of collimated particles, the production cross-sections can be addressed purely within the perturbative domain of QCD (pQCD). While jets propagate throughout the deconfined QCD medium, the multiple scattering interaction of the hard partons that make them up with the medium constituents results into the modification of their dynamics and leads to parton energy loss, commonly referred to as jet quenching. The successes of jet quenching testify to the drawing power of hard probes when describing the QGP. The measurement of the di-jet asymmetry at the LHC was one of the clearest experimental evidences of this suppression, which is expected to be enhanced with the medium temperature and the path length of the parton within the medium. V ´ ıctor Vila P´ erez With the final goal of establishing a fully comprehensive theoretical framework behind experimental observations, one of the proposed ideas was that this suppression is due to the partonic energy loss in the QCD matter formed in the collisions. Despite the fact that several approaches try to provide a complete understanding of jet propagation in matter, much still remains to be done in order to accomplish a deeper understanding of the underlying mechanisms responsible for in-medium jet interactions. Generally speaking, this thesis aims precisely to shed light on the mechanisms responsible for jet quenching and medium response to jet propagation as well. One of the key sources responsible for the modifications suffered by an energetic jet as a result of its interaction with the medium ingredients is medium-induced radiation, which can be computed in the framework of different general approaches within pQCD as the BDMPS-Z/ ASW formalism, GLV or AMY, among others. These approaches are nothing more than the generalization of the Landau-Pomeranchuk-Migdal (LPM) effect to pQCD. Moreover, jet quenching Monte Carlo (MC) generators happened to be extremely effective tools toward a better linking between theoretical implementations and experiments. Due to their perturbative calculability, high-transfer momentum processes are computed from first principles in the absence of medium effects. In the presence of a medium the formulation is not so different, which motivates one to make use of jet substructure techniques to provide further details on the interplay between essential qualitative features as energy loss, medium-induced radiation and color coherence. In fact, one of the basic aspects of the parton shower process is color coherence in multi-parton splittings, a central challenge present in the current status of the theory and phenomenology of jet evolution and radiative processes in matter. Many attemps to make progress in this respect have been actively addresed in the last few years, but only considering a frozen configuration for the emitters. In this context, a number of works on the quark-antiquark antenna radiation set up a relevant insight on the description of partonic in-medium propagation. In particular, they have highlighted the importance of interference effects when addressing jet showering inside the medium. The notion of loss of memory of color connections due to color screening between the parent partons and the offspring is proven by introducing decoherence in a dense medium, which results in independent radiation off the emitters. The main body of this manuscript gives a wider and novel insight into the phenomenon of color coherence in multi-gluon radiation for different parton branching processes and its effects on multi-jet final states by putting forward an extension of previous derivations performed in this context, which plays a central role in the description of the quenched jets. Handling multiple emissions configurations hinder the discussions since interferences among different processes might become important. 2 In the vacuum shower color is preserved, what leads to angular ordering. However, when considering an in-medium partonic setup, motivated by medium-induced gluon radiation, interference patterns might gradually alter the color coherence of the system over a time scale known as the decoherence time, which governs the decoherence regime. As will be shown later, it follows that in this regime the antennas experience a prompt color decorrelation as they travel over the medium, thus loosing memory of their origin. As stated above, this thesis is primarily intended to look into the final state picture of HICs via the study of effects such as the abovementioned. However, initial stages have also their place in this work. The relation between the two types of effects concerning the collisions is discussed before anything else. The understanding of the structure of the wave functions of the colliding nuclei is of paramount importance, typically encoded in their nuclear parton distribution functions (nPDFs), which characterize the partonic structure and dynamics of the nuclei. At high energies, nuclei do not behave as a mere incoherent superposition of their constituents. Instead, coherence effects become important. Not only do they modify their partonic content, but also the underlying dynamics of particle production in scattering processes. PDFs (nPDFs) are not computable within the perturbative approach since they carry a long distance (low energy) part of the cross-section. They are usually fitted from experimental data. There is thus a distinction between such initial state effects (IS), which are a prelude to thermalization, from those giving rise to the presence of a QGP, the final state effects (FS). Stressing the difference between both is of vital importance for an appropriate characterization of the matter produced in HICs, as sometimes they may lead to qualitative similar phenomena with regard to interesting observables. Another important subject line is the analysis of those collective phenomena framed in the context of the Color Glass Condensate effective theory (CGC), which is currently the most universally accepted approach to describe small Bjorken-xdegrees of freedom. It incorporates non-linear recombination effects both at the level of particle production and also in the quantum evolution of hadronic wave functions. Hence, disentangling CGC from other effects demands the analysis of more exclusive observables. One of the first and captivating observations in this respect was the finding of long-range two-particle correlations in pp collisions at the LHC. Similar correlations had been noticed before at RHIC. An accepted explanation for the observed events is related to collective flow in QGP. Nonetheless, this is still an open question. The long-range rapidity correlations may not be linked to the origin of the QGP as they spread out over a very large rapidity interval, concluding that they may originate at very early stage of the collision. In support of this view, as a final piece of this thesis an analysis of angular correlations between gluons produced in high-energy hadronic collisions within the CGC is performed as well, proving that the emitted gluons are necessarily correlated both in rapidity and the emission angle. V ´ ıctor Vila P´ erez QCD α (Μ ) = 0.1184 ± 0.0007 sZ 0.1 0.2 0.3 0.4 0.5 αs (Q) 1 10 100 Q [GeV] Heavy Quarkonia e+e– Annihilation Deep Inelastic Scattering July 2009 Figure 1.3: A compilation of data regarding the behaviour of the strong coupling constant αswith the momentum Q∝1/r (taken from [11]). Asymptotic freedom is in addition the basic principle behind jet formation. The process e+e−→hadrons takes place via the diagram pictured in Fig. 1.4, where the intermediate photon produces a quark-antiquark pair. When these partons pull away, they neutralize their color via gluon radiation which, in turn, will produce more gluons and q¯qpairs, recombining amongst themselves to result in the hadrons that are measured in the final state. These hadrons are not uniformly distributed in all possible directions but instead they are mostly clustered in two directions, forming two jets of collimated hadrons. The demonstration of the existence of jets is credited to G. Sterman and S. Weinberg [12]. Today, jet physics is a topic of great interest given both its importance for QCD studies and Beyond the Standard Model physics searches. 1.4 Confinement Quarks and gluons, the Lagrangian fields of QCD, in contrast to what occurs in other gauge field theories, are not observed in free states – at least at normal temperature and pressure – but that they are caught up within hadrons. This is referred to as confinement. 10 1 Introduction e e q q γ, Z _ 0 _ + Figure 1.4: Hadron production from e+e−collisions. Though this central feature of the theory of strong interactions has so far not been proven from an analytical point of view, there exist numerical lattice calculations which enable to reproduce some of its consequences, namely the hadron masses [13]. 1.5 Lepton-hadron deep-inelastic scattering (DIS) The evidence of the existence of quarks and gluons as elemental constituents of matter precedes the formulation of QCD and has its roots in the lepton-hadron collision experiments where the outgoing lepton is detected at large angles. These experiments involve a highly virtual photon exchange and are called deep-inelastic scattering experiments (DIS), pictured in Fig. 1.5. The total cross-section can be written in terms of two Lorentz invariant quantities known as structure functions, F1and F2. These functions depend on two variables, currently chosen as Q2=−q2>0, x =Q2/(2p·q).(1.17) Both functions were first measured in the late 1960s, resulting into a couple of substantial findings. Amongst other important relations, F1(x, Q2) = F1(x), F2(x, Q2) = F2(x),(1.18) 11 V ´ ıctor Vila P´ erez that is to say, the structure functions only depend on x, the so-called Bjorken scaling [14], and 2xF1(x, Q2) = F2(x, Q2),(1.19) the Callan-Gross relation, which evidences that the hadron constituents probed by the photon have spin 1/2. Subsequently, these constituents were identified with the quarks, giving rise to the parton model. 1.6 The parton model In the parton model – see [15, 16] for more detail, a parton is described, in a given reference frame in which its momentum becomes very large, as an incoherent superposition of elemental constituents called partons. The variables xand 1/Q can be interpreted respectively as the hadron’s momentum fraction that carries the parton probed by the photon, and the transverse resolution with which the hadron is probed. Then, the previously introduced structure function F2can be written as sum – weighted according to the momentum fraction and the squared charge e2 f– of the probabilities f(x, Q2) to find a parton fwith momentum fraction xwhen the hadron is probed with a resolution 1/Q. These are no more than the so-called parton distribution functions (PDFs): LEPTON PROTON k p q = k − k’ k’ xp + q Figure 1.5: Schematic depiction of a lepton-hadron deep-inelastic scattering. 12 1 Introduction F2(x) = X f e2 fxf(x, Q2).(1.20) The parton model quite naturally finds its deepest meaning in the QCD context. Electrically charged partons are identified with quarks. Furthermore, gluons turn up as a quark from the hadron with momentum fraction zcan radiate a gluon, thus giving rise to the quark with momentum fraction x < z which is probed by the photon. This radiation leads to a soft evolution – logarithmic in the PDFs [17]. At present, the determination of the PDFs in hadrons and nuclei is actively on course – see, for example, [18] and [19]. It is of key importance for both its interest in getting to know the hadron’s structure and its practical usefulness for calculating the particle production in collisions which involve hadrons via the well-known factorization theorems [20]. Since the PDFs hold information about shortand large-distance scales, it is not possible to address the situation in a perturbative way, that is to say, it is not yet known how to compute them from first principles owing to the non-perturbative nature of the hadron as QCD bound state. Instead, factorization theorems, valid at any pQCD order, enable to split the problem in two clearly separate parts. The first is perturbative and hence calculable in the pQCD work frame. In particular, it describes the lepton-hadron scattering process. The second part, of a non-perturbative nature, provides the distribution of a parton iwithin the hadron Nat a given energy scale, namely the already introduced PDFs. In both DIS processes and pp collisions at high energies the initial state must be described by using PDFs. They can be determined by global fits to all available data from both DIS and hard scatterings. Global fits can be performed at leading order (LO), next-to-leading order (NLO) or next-to-next-to-leading order (NNLO) in the strong coupling constant αs. In recent years, precision has been optimized and an extension in the kinematic range of experimental measurements for many of these processes have been carried out as well. Moreover, new theoretical developments increase the reliability of global fits. The precision of contemporary experiments demands the use of theoretical implementations at least at NLO and preferably at NNLO in comparisons between theory and experiments. Nevertheless, PDFs hold the major advantage of being completely independent of the process under study in such a way that once extracted for a particular process it is not required to perform the calculation once again. Additionally, known at a given energy scale, PDFs can be derived for any other value of Q2through the use of QCD-based series of evolution equations, the so-called Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations [17,21,22]. 13 V ´ ıctor Vila P´ erez 1.7 Hot and dense QCD It has been more than four decades since T. D. Lee and G. Wick pointed to the possibility of exploring new physics by distributing a large nuclear matter density or a large energy density within a relatively large volume to create new states of dense nuclear matter [23]. It soon became apparent that asymptotic freedom implies the existence of a highly dense form of nuclear matter made up by deconfined quarks and gluons [24], later renamed Quark-Gluon Plasma (QGP). The energy density has been studied and discussed in detail following finite temperature lattice QCD calculations. The results show a rapid increase from a low energy density state to a high energy density one, as one would expect in the phase transition from a state of confined quarks and gluons to deconfined [25]. Indeed, only at temperatures much higher than the critical temperature the (almost free) QGP predicted by aysmptotic freedom will pop up. For moderate temperatures, lattice QCD points towards a QGP with strong residual interactions, the strongly coupled QGP (scQGP), where the phase transition to QGP may seem to be a cross-over. Lattice QCD calculations also reckon with a first order phase transition for high densities of baryons µβ and low temperature, whilst at high temperature and low baryonic density would lead to the above-mentioned cross-over. Accordingly, the phase diagram would present a critical point, as shown in Fig. 1.6. Figure 1.6: A sketch of the QCD phase diagram (extracted from [26]). 14 1 Introduction The quest for the QGP is not just important because it is the QCD state of matter at high temperature or high baryonic density that was present during the first microseconds after the Big Bang. What is more, it can exist in neutron stars, and even brings unique information on the origin of most of the ordinary mass and the confinement of quarks and gluons [27]. During the last decades considerable effort has been expended in experiments focused on achieving the QGP in the laboratory. Firstly, the Intersecting Storage Rings (ISR) experiments at the Conseil Europ´een pour la Recherche Nucl´eaire (CERN) explored light-ion collisions, just as the Alternating Gradient Synchroton (AGS) program at Brookhaven National Laboratory (BNL) was focused on looking deeply into the low energy regime. Later on, both Super Proton Synchrotron (SPS) experiments at CERN and Relativistic Heavy Ion Collider (RHIC) tests at BNL fully studied a wide range of centralities and observables at √s≈20 GeV and √s≈200 GeV per nucleon, respectively. Since then, experiments at CERN’s Large Hadron Collider (LHC) have taken the lead in HICs studies. RHIC data [28–31] showed phenomena such as a strong suppression of high-transverse momentum particles produced in collisions between gold ions to the number that would be expected, provided by the product of the number of nucleon-nucleon collisions with the proton-proton production relying on theoretical considerations. Indeed, STAR and PHENIX experiments at BNL found the first evidence for jets. These showed a significant difference from those seen in simpler collisions. STAR observed that one of the two back-to-back jets was invariably quenched, sometimes weakened and sometimes wholly extinguished. In this way, jets are hard probes (HP), by nature interacting through the strong force but moving so fast and with so much energy that they are often not completely absorbed by the surrounding quarks and gluons in the QGP. The degree of jet quenching – sketched in Fig. 1.7, a phenomenon that emerges in data from millions of collisions events, makes it possible to unveil some key information on the QGP, making it possible to deliver its properties. The most frequent explanation put forward for this suppression is medium-induced radiative energy loss [33–35], by far the most recurrent topic addressed in the ensuing chapters of this thesis. Nevertheless, radiative energy loss models are not the only ones. Models also exist that consider collisional energy loss, approaches that assume an inmediate hadronization of partons and a few others. In order to differentiate between the different types of models it appears necessary to have a closer look at observables such as correlations between the produced particles within a reconstructed jet. High-transverse momentum particle suppression as a result of energy loss owing to inmedium interactions had been foretold by Bjorken for nearly 40 years. More recently, the ALICE, ATLAS and CMS experiments at CERN’s LHC have confirmed the existence of jet quenching in HICs. The much greater collision energies at the LHC push measurements to much higher jet energies, giving way to more detailed characterization of the QGP. 15 V ´ ıctor Vila P´ erez Figure 1.7: Jet quenching in a head-on nucleus-nucleus collision (illustration from [32]). Theoretical understanding of these measurements is challenging, however, and is one of the most important problems in QCD today. With regard to the initial stage in relativistic nuclear collisions, it might be described by parton saturation models such as the so-called Color Glass Condensate (CGC) [36– 39], a state hydrodynamically described as a perfect fluid with very low viscosity. The CGC effective theory has been constructed for organizing the calculation of processes characterized by a large gluon occupation number by means of considering non-linear recombination effects, which factually turn out to be significant when nuclei are proven at sufficiently small values of Bjorken-x. In the small-xregime, the occupation number of gluons becomes large in such a way that gluon self-interactions are of considerable significance, logically leading to gluon saturation – the gluon density proves to be inversely proportional to the strong coupling constant αs. Note that as depicted in Fig. 1.8, there is an increase in the number of gluons on account of the rising of the energy, thus making smaller values of xkinematically accessible. This approach provides a framework for computing particle distributions and their dynamics prior to thermalization. As stated above, all these detailed on-the-ground findings point to the creation of high-density matter. Due to the presence of this medium consisting on a near perfect partonic fluid with high density, high-transverse momentum particle propagation undergones significant changes with regard to vacuum propagation. LHC data available so far appear to be in conformity with these conclusions [41–45]. 16 1 Introduction 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 00000 00000 00000 00000 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 11111 11111 11111 11111 00000 00000 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 11111 11111 00000 00000 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 11111 11111 000 000 000 000 111 111 111 111 000 000 000 000 111 111 111 111 000 000 000 000 000 000 000 111 111 111 111 111 111 111 00 00 00 00 11 11 11 11 00 00 00 00 11 11 11 11 00 00 00 00 00 11 11 11 11 11 00000 00000 00000 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 11111 11111 11111 000 000 000 000 000 000 111 111 111 111 111 111 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 00000 00000 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 11111 11111 000 000 000 000 000 000 111 111 111 111 111 111 000 000 000 000 000 000 111 111 111 111 111 111 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 00 00 00 00 11 11 11 11 000 000 000 000 111 111 111 111 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 000 000 000 000 000 111 111 111 111 111 000 000 000 000 000 000 111 111 111 111 111 111 000 000 000 000 111 111 111 111 000 000 000 000 000 111 111 111 111 111 00 00 00 00 00 11 11 11 11 11 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 00 00 00 00 00 11 11 11 11 11 000 000 000 000 000 000 111 111 111 111 111 111 000 000 000 000 000 111 111 111 111 111 000 000 000 000 000 111 111 111 111 111 000 000 000 111 111 111 00 00 00 00 00 11 11 11 11 11 00 00 00 00 11 11 11 11 00 00 00 00 11 11 11 11 000 000 000 000 000 111 111 111 111 111 00 00 00 11 11 11 000 000 000 000 111 111 111 111 000 000 000 111 111 111 000 000 000 111 111 111 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 000 000 000 000 111 111 111 111 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 000 000 000 000 000 000 111 111 111 111 111 111 000 000 000 000 111 111 111 111 00 00 00 00 11 11 11 11 000 000 000 000 111 111 111 111 000 000 000 000 111 111 111 111 000 000 000 000 111 111 111 111 00 00 00 00 00 11 11 11 11 11 00 00 00 00 11 11 11 11 00 00 00 00 00 11 11 11 11 11 000 000 000 000 111 111 111 111 000 000 000 000 111 111 111 111 000 000 000 000 000 111 111 111 111 111 Low Energ y High Energ y Gluon Density Grows Figure 1.8: Phase-space gluon density in a hadron (picture from [40]). 17 V ´ ıctor Vila P´ erez 18 2 The formalism framework This chapter provides an explicit basis and back-up for describing the nature and challenges associated with the phenomena of interest covered in the chapters that follow. Both the theoretical tools that underpin the presented developments, together with existing models rooted in these analytic methods, and the assumptions and expectations to critically evaluate the topics raised in this thesis are discussed. The main purpose is therefore not only to create a clear framework upon which to build the analyses and studies carried out, but also about highlighting how the undertaken approaches might differ and under what circumstances, eventually toward testing the validity of the interpretations set out in order to make broader generalizations at later stages. In short, this chapter fulfills the primary purpose of articulating the overarching theoretical environment guided by well-established models that best fit the physical scenarios to be dealt with. As it was formerly mentioned in the motivation of this manuscript, the three subsequent chapters are encompassed within the in-medium jet evolution field. To get a sense of the in-medium evolution-based framework and provide extended discussions between energy loss approaches linked back to these theoretical schemes, it is fitting to start this episode by conducting thorough reviews both on vacuum and in-medium jet propagation, putting emphasis on parton energy loss models within the pQCD frame. In this context, theoretical and methodological tools as the Lund jet plane, which provides a powerful visual representation of the radiation within any given jet, thus allowing for an exhaustive comparison of jet quenching models, are also introduced. Furthermore, since the kinematical Lund diagram has a broad scope for constraining Monte Carlo simulations, it naturally leads to discuss how to exploit sophisticated jet substructure observables. V ´ ıctor Vila P´ erez This is called radiative energy loss. Assuming a dominance of medium-induced radiation, experimental findings tend to be interpreted in terms of purely radiative processes. Collisional energy loss Collisional energy loss of hard partons was initially proposed by Bjorken in the early eighties [51]. Later on, several attemps at developing a more thorough analysis beyond the starter model were made by taking notice of novel considerations and effects [52–56]. As is discussed below in more depth, when addressing in-medium parton propagation, in most current approaches for radiative energy loss the medium is modeled as an assembly of heavy static scattering centers in such a way as recoil effects are neglected, hence omitting any reference to collisional energy loss for the leading partons. Indeed, collisional and radiative energy loss are added separately in these frameworks. When comparing collisional and radiative effects, collisional energy loss usually shows less impact for light flavor leading partons – particularly for high jet energies [57], whereas it is meant to be the controlling mechanism for heavy quark energy loss within the low and intermediate energy regimes [58]. In the high-energy regime, heavy quarks become ultrarelativistic and radiative effects become more significant [59]. All in all, collisional energy loss plays a subdominant role with respect to radiative energy loss and will be disregarded throughout this thesis. Radiative energy loss Radiative energy loss has been deemed as the major ingredient when addressing both parton energy loss and jet quenching in relativistic HICs. Due to the presence of the QGP, parton splitting processes are altered with regard to vacuum propagation since they experience ongoing rescatterings with the medium constituents. As mentioned earlier, interference effects play a crucial role when dealing with parton evolution, and within them coherence phenomena are some of the most visible. In this respect, the Landau-Pomeranchuk-Migdal (LPM) effect, first demonstrated experimentally at SLAC [60], constitutes a marked effect when handling medium-induced gluon radiation in the course of parton propagation. In the fifties, Landau, Pomeranchuk and Migdal predicted that the cross-section for bremsstrahlung from highly relativistic particles in dense media is suppressed due to interference caused by multiple in-medium scatterings [61,62]. More specifically, for the radiation process to take place it is required a finite time, marked by the formation time of the producing gluon, tf≃2ω/k2 ⊥, where ωand k⊥are the energy and transverse momentum of the collinear radiated gluon. By comparing the two scales, 26 2 The formalism framework it is appraised that whenever the gluon formation time tfis larger than the mean free path λof the propagating parton, the rescatterings experienced by the hard parton can no longer be considered as independent. This quantum interference effect amongst consecutive scatterings resulting from the LPM effect gives rise to a suppression in the radiation spectrum as against the Bethe-Heitler (BH) bremsstrahlung spectrum, which assumes incoherent rescatterings1[63]. Several different model frameworks for studying medium-induced gluon radiation in hot and dense matter within pQCD have been developed over the past two and a half decades. Among them the following stand out: BDMPS-Z [64–68], ASW [69–73], GLV [74–77], AMY [78–80] and HT [81–85]. All the models listed above rely on the standard assumption that interactions among the leading parton and the emitted gluon with the medium can be computed within pQCD, whether or not the medium properties can be addressed perturbatively. It is within this context that diverse approaches arise with the final goal of calculating the radiative energy loss of the energetic parton. Brief comments on the basic information of these formalisms are provided further below. However, first of all it is worth stressing the common assumptions and approximations amongst them concerning the branching of the leading parton, the medium model or the kinematical considerations for the partonmedium interactions.  The parton splitting is affected by the medium in such a way that interactions between both the high-energy parton and the offspring with the medium become important. As the transverse momentum of the daughter parton k⊥is quite less than its energy ω, only collinear radiation off the emitters takes place. In the wake of this condition, the two partons involved in each splitting take similar paths, giving way to the LPM effect previously introduced in this section.  It is assumed that the momentum transfer between the hard parton and the medium is localized, meaning that the mean free path of the incoming parton λis much larger than the color screening length of the medium, which is proportional to the inverse of the Debye screening mass µD.  The eikonal picture is implemented: the interaction of the projectile partons with the medium is assumed to be eikonal, that is to say, they propagate through the medium without changing their transverse position. Kinematically speaking, what all this adds up to is that the energy of the leading parton Eis far greater than the transverse momentum exchanged with the medium q⊥. 1The independent Bethe-Heitler gluon spectrum reads ωdIrad/dω ≃L2/ω, with ωthe energy of the radiated parton and Lthe medium length. 27 V ´ ıctor Vila P´ erez Despite common assumptions, in a practical sense each model framework description conducts a particular set of approximations with regard to certain points about the inmedium parton branching or the medium model. However, as each formalism allude to a great number of common ingredients in order to fully model this complex physical scenario, addressing each of them separately does not come easily. Following is a small report on each one – see [86] for a more thorough balance at comparing the specific implementations amongst themselves. Attention is being paid to the BDMPS-Z/ASW formalisms as they constitute the building block that enables to perform the calculations presented in the three main strands of this manuscript. The BDMPS-Z and ASW formalisms [Baier-Dokshitzer-Mueller-Peign´e-Schiff-Zakharov / Armesto-Salgado-Wiedemann] BDMPS-Z was the very first of the in-medium radiative energy loss formalisms. It is developed in a purely path-integral formulation that accounts for the resummation of the scatterings on multiple heavy static colored centers. U. A. Wiedemann enclosed the interference between vacuum and medium-induced radiation in the existing framework at a later stage [70, 71]. The most common model approach for analytical and numerical studies is the saddle point approximation. This involves assuming that the high-energy parton interacts with the medium by way of multiple soft scattering processes, commonly known as either multiple soft scattering or dipole approximation. Numerical analysis implementing this approximation within the BDMPS-Z approach were undertaken by C. A. Salgado and U. A. Wiedemann on the basis of calculating the in-medium splitting gluon probabilities, the so-called quenching weights [72]. Alternatively, M. Gyulassy, P. Levai and I. Vitev (GLV) and U. A. Wiedemann (independently) [70, 76] presented another approximation based on a systematic expansion over the number of scatterings experienced by the incoming parton, the opacity expansion, characterized by the density ρof scattering centers – or the mean free path of the leading parton λ– and the Debye screening mass µD. In some phenomenological calculations the single hard scattering limit is implemented, that is to say, only the leading term of the summation (N= 1) is taken since it is assumed that there is only one dominant hard scattering. Multiple gluon emission scenarios are addressed by iterating the single gluon emission kernel by means of a Poisson convolution, which assumes that the number of radiated partons in a given path responds to a Poisson distribution. The GLV formalism [Gyulassy-Levai-Vitev] The GLV frame is based on the opacity expansion – as just stated, two different model implementations (independently developed) exist for this approach. As also noted above, some phenomenological studies restrict themselves to the single hard scattering limit. Higher orders in opacity were, however, considered by GLV and S. Wicks [77,87]. 28 2 The formalism framework The AMY formalism [Arnold-Moore-Yaffe] The AMY approach was developed to study parton energy loss in a weakly-coupled medium being in thermal equilibrium. While it was primarily formulated on the assumption that the leading parton passes through an infinite size QCD medium, including the effect of finite medium length was a recent addition by S. Caron-Huot and C. Gale [88]. With regard to the case of multiple gluon emissions, AMY uses a set of rate evolution equations so designed as to the radiation probability evolves with the primary hard parton energy loss. The HT formalism [Higher-Twist] The original development of the higher-twist approach was led by X.-f. Guo and X.-N. Wang [81,82]. In this formalism, higher-twist matrix elements analytically characterize the medium properties. Typically, nuclear PDFs factorize out from those matrix elements, thus enabling the latter to fully describe the high-energy parton-medium interaction. As is the case for the most of the other approaches, HT was first developed for single hard scattering until A. Majumder extrapolated the work frame to the case of multiple scatterings later on [85]. Parton branching evolution is performed via medium-modified DGLAP evolution equations. As seen in the main approaches conceived from the perspective of understanding the suppression of high-pThadrons in ultrarelativistic HICs, parton energy loss in QCD matter proves to be an intriguing task which requires the modeling at various stages, just as identifying the dominant mechanism responsible for the observed modifications. In the initial stage of each energy loss formalism high-energy partons are formed in a primary collision, while in the final state they ultimately fragment into hadrons. Between these two stages parton-medium interactions take place, giving rise to the parton cascade. As can be guessed, the evolution of scales that leads to parton showering is suitable to computer simulation using Monte Carlo techniques, which is the focus of the following piece of this section. 2.1.3 Parton showers from a phenomenological insight The parton showering scenario is a Markov process that Monte Carlo (MC) techniques can cover very efficiently. A Markov chain MC allows to estimate a posteriori branching probabilities by assuming that the next branching depends only on the previous one and is randomly generated. Nevertheless, improving the treatment of each regime is required as higher-order perturbative calculations and more sophisticated hadronization models emerge. 29 V ´ ıctor Vila P´ erez Not only is a MC event generator surely able to run simulations suitable for a wide scope of processes taking place in particle physics experiments, but also provides wellfounded information for designing new experimetal setups. In the context of QCD, once the partonic structure of hadrons was known, MC generators started to surface. Indeed, important tools to test QCD as DIS can be understood in terms of parton interactions. Set forth below a more detailed discussion of the physics behind parton showering involved in MC event generators. Getting back to eq. (2.6), it seems to be something that might be implemented and iterated in a MC algorithm. As a matter of fact, it constitutes the starting point to write an iterative algorithm since it provides a general expression for a hard process together with a collinear splitting. The picture is becoming even clearer: it is therefore possible to iterate an expression for the soft emission probability in order to generate a collinear splitting and then prompt it to handle the final state of that splitting as a new hard process, generating an even more collinear splitting from it and so forth, that is to say, the building block to iteratively attach extra partons one by one. In short, a collinear parton shower algorithm is capable of encapsulating soft gluon effects as long as the opening angle is set as the evolution scale. While in the absence of medium effects the MC technique is extensively used for simulating final state parton showers [89,90], perturbative-based jet quenching MC event generators are more challenging as they constitute the baseline for characterizing mediummodified multi-particle final states, as well as a basic tool to describe parton-medium interactions and test medium properties. In recent years, several MC generators implementing partonic energy loss have been developed. Prominent amongst them are PQM [91], using the BDMPS quenching weights, PYQUEN [92], which modifies the standard PYTHIA 6.2 jet events [90] via both a reduction in the energy of the partons involved in the cascade and by adding additional gluons to that shower, Q-PYTHIA [93], including medium-induced gluon radiation within the BDMPS-Z approach, JEWEL [94], whose interest lies in going deeper into the relationship between parton energy loss and some of its specific features as p⊥-broadening or recoil momentum, or MARTINI [95,96], taking advantage of the set of rate evolution equations from AMY [80]. Discussions on MC parton shower algorithms encourage to dip into the kinematical Lund diagram [97] – see [98] to find out more about the standards of the scheme, which constitutes a structured representation of the phase-space within jets and thus make application possible for radiation pattern implementations run by MC event generators. The Lund diagram easily fits in the context of in-medium parton splittings as it renders an important starting point for having an overall vision of what regimes react more to medium effects. Details on the construction of this diagram can be found below since this tool is used to map the radiation spectrum of the splitting process addressed in one of the most relevant chapters of this thesis. 30 2 The formalism framework The Lund plane encompasses the emission kinematics as a function of two variables: the Y-axis represents the emission tranverse momentum k⊥with respect to the jet axis, whereas the inverse of the radiation angle θwith respect to the jet axis is plotted along the X-axis – the natural logarithmic scale is used for both axis. To be clearer, the Lund diagram can be exploited in terms of different variables, but in any case, taking advantage of the fact that ordinary 1 →2 QCD splittings lead to the infrared and collinear divergencies, both encapsulated in eq. (2.6). For the purpose of describing how to proceed further with the filling of the Lund diagram it is helpful to handle the most convenient expression for the splitting probability. A little algebra reshapes eq. (2.6) to an equivalent form, dS= ¯α·dω ω dθ θ,(2.15) where ¯αis defined as ¯α:= 2αsCi/π (with Ci=CFfor a quark splitting). After this, it is also appropriate to define certain quantities that directly fall under the Lund map, namely, the jet invariant mass, an important observable in jet physics since it represents the prototype of a jet substructure observable, which in the collinear limit reads M2=z(1 −z)p2 ⊥θ2,(2.16) where zrepresents the energy fraction carried away by one of the emitted partons and p⊥ accounts for the total momentum of the dipole jet, or the formation time, the characteristic time-scale of the splitting taking place, tf=2z(1 −z)p⊥ k2 ⊥ ,(2.17) with k⊥=z(1 −z)p⊥θ. Then, in the double-logarithmic approximation (DLA) – where O(1 −z) corrections can be neglected2, the Lund parton splitting map can be chart in terms of kinematical variables such as zθ versus 1/θ in logarithmic scaling – or ln(1/z) versus ln(1/θ) and the like, depending on which suits best. In this way, the Lund map can be filled by virtue of eq. (2.15) for the probability that one splitting takes place, dS= ¯α·dln zθ d ln 1 θ,(2.18) or also dS= ¯α·dln 1 zdln 1 θ.(2.19) 2This makes it possible to approximate p⊥≃zEθ,k⊥≃zp⊥θand such. 31 V ´ ıctor Vila P´ erez Figure 2.4 shows a sketch of the Lund map expanded accordingly to eq. (2.18)3, where both soft and large-angle emissions and hard and collinear radiation regimes are explicitly identified within the plane. A couple of clarifications with regard to Fig. 2.4 are necessary. It is noticeable that, according to eqs. (2.18) and (2.19), the Lund diagram is uniformly filled with a density ρ≃¯α. Moreover, higher-order corrections in the strong coupling αsemerge when multiple splittings effects are taken into account. Figure 2.4: The Lund diagram filled by means of eq. (2.18). Here Ris the jet opening angle (courtesy of [99]). The process for populating the map consists on matching each branching with a point within the Lund map, which consequently displays lines of constant formation times with a slope one parametrized by solving ln zθ = ln 1 θ+ ln 1 p⊥t(2.20) for t=tf. The next step is to discuss the Lund diagram for in-medium splittings, a study thoroughly conducted in Chapter 4 for a real and collinear parton splitting inside a medium. 3Note that eqs. (2.18) and (2.19) are restricted to the singlet case. For the octet case, CFmust be replaced by CA=Nc. 32 2 The formalism framework Based on the introduced background, it is easy to deduce that medium modifications are expected to arise from the baseline tf=Lupwards, that is to say, medium effects events show up, by all means, inside the medium of length L. This line can be estimated by imposing t=Lin eq. (2.20). Furthermore, as will be seen in greater depth in Chapter 4, other characteristic time scales related to decoherence and broadening of partons in the medium arise in the addressed physical scenario. They are equally represented as straight lines within the graph, thus dividing the plane in different regimes. Given the presence of various relevant scales, this naturally leads to a careful discussion rooted in the different ordering of the time scales themselves, eventually gaining a clear insight into medium modifications regarding the nature of the splittings and the coherence of the system. As can be seen, Lund diagrams are suitable as a theoretical representation of the phase-space within jets, and not just that but also have been raising attention as a powerful tool in discussions of MC parton shower algorithms since they carry wide-ranging information on jets, including the potential for exploiting their internal structure [99,100]. The MC technique is extensively used for the simulation of final state parton showers in vacuum, whereas in the presence of medium effects, the generalization to multiple splittings is a challenge that remains open whilst some steps forward have been achieved in this respect – [101] is one of the latest. A jet definition, understood as a well-defined procedure to reconstruct jets from the set of hadrons in the final state of the collision, is a means of connecting theoretical models with experimental observables. As a first step, one might think of jets as being in one to one correspondence with the produced partons in a hard scattering. Nevertheless, in a hadron collider environment the actual identification of jets is non-trivial since it is not always true that hadrons are associated with a particular jet. This situation may occur, for example, when jets partially overlap. In view of this situation, several algorithms for identifying jets on a hard scattering have been currently developed. An important requirement for jet algorithms is to be infrared and collinear (IRC) safe, that is, for any partonic configuration, replacing any parton with a collinear set of partons with the same total momentum, or adding any number of infinitely soft partons in any directions, should produce identical result. The two main types are cone algorithms and sequential recombination algorithms. The older cone algorithms [12] define a jet as those hadrons lying within some distance from the jet center. However, not all cone algorithms fulfill IRC safety4. Most LHC studies use sequential recombination algorithms as the generalized-kt algorithm [102], the ktalgorithm [103,104], the Cambridge-Aachen (C/A) algorithm [105] or the anti-ktalgorithm [106], which stem from the approach that, from the pQCD point of view, jets are essentially the product of successive parton branchings. 4In fact, IRC safety is not so much required to find jets, but to deal with them within the pQCD framework. Away from the purpose of comparing theoretical models with experimental data, an IRCunsafe jet algorithm fits just as well. 33 V ´ ıctor Vila P´ erez So, what the algorithm basically do is to invert the process outlined above by successively recombining two particles into one. Moreover, the process can be repeated again, for example, in jet substructure studies. In a first step, the anti-ktalgorithm is used to reconstruct jets, while another sequential-recombinational algorithm reclusters the constituents of that jet to study its internal structure. Finally, it is also important to perform numerical implementations of the algorithms. At the moment, the standard package for jet clustering is FastJet since it provides an implementation of all the cited recombination algorithms [102]. Novel jet substructure variables have been recently defined. Many of them are IRC safe, thus enabling to use perturbative methods. Jet substructure techniques are designed in such a way that the constituents of a jet are reorganized into a hierarchical tree where the nodes account for the successive splitting processes, which is a starting point for the future analysis using techniques such as jet grooming or tagging algorithms. The former concerns a reorganization of the tree by discarding radiation that fails to pass a given criteria, while the latter is undertaken in order to identify the very first splitting that passes a given requirement, thereby splitting a jet into two sub-jets. Over the last years, much has been accomplished on using these techniques for many jet substructure observables, which in turn open new doors of opportunity to explore unknown physics domains – see [100] for a very detailed compilation of standards and resources in jet algorithms and jet substructure phenomenology. 2.2 Radiative energy loss in the BDMPS-Z/ASW approach Just as was stated earlier in this section, a number of formalisms for radiative energy loss of an energetic parton inside hot and dense matter have been put forward over recent decades. This piece is now intended to delve deeply into the details referring to the BDMPS-Z/ASW approach, as this formalism is the backbone of the studies developed in the succeeding chapters. At the beginning, general considerations for analytically describing the hard partonic interaction with the colored medium are introduced, as well as the relevant assumptions about the branching of the incoming parton, the nature of the medium or the kinematical approximations for the projectile parton. Then, the single-inclusive particle spectrum is revisited as the prelude to a review of the medium modification undergone by the gluon spectrum off a quark-antiquark antenna along the same line followed in the vacuum case. All these contribute to set an overall knowledge regarding in-medium parton energy loss as also unravels the basic procedures and techniques which prove necessary to support further studies on parton showering presented in this manuscript. 34 2 The formalism framework 2.2.1 High-energy parton propagation through a medium In an attempt to answer the question of how an energetic parton created in the early stages of heavy-ion collisions traverses a dense medium the most widespread picture used is presented here. Noteworthy is that it is built on the basis of a semi-classical approach that neglects the modifications undergone by the medium configuration as a result of the passage of the incoming parton, namely the recoil of the scattering centers. In particular, to put this into practice the medium is modeled as a background field understood as a collection of static colored scattering centers with a specified density distribution along the trajectory of the projectile. For its part, the propagation of the fast parton throughout this medium field A−(x+,x⊥) is fittingly implemented in terms of Wilson lines in the fundamental representation, W(L+, x+ 0;x⊥) = Pexp "ig ZL+ x+ 0 dx+A−(x+,x⊥)#,(2.21) where x⊥is the transverse position of the hard parton and the boundaries are set in the light-cone space, which is very useful in the context of high-energy collisions. In particular, the light-cone variables are defined as x±=1 √2(x0±x3), p±=1 √2(p0±p3).(2.22) The scalar product reads p·x=p+x−+p−x+−p⊥·x⊥,(2.23) while the rapidity is defined to be y=1 2ln p0+p3 p0−p3=1 2ln p+ p−.(2.24) As mentioned above, adequately addressing the picture of parton eikonal propagation in matter involves the use of Wilson lines, which in this context can be straightforwardly derived from first principles by assuming the leading parton experiences multiple scatterings as it travels across the medium. This approach is displayed in Fig. 2.5, where x1to xnaccount for the position of the scattering centers within the background field5. Now, having fixed everything with regard to the passage of the parton through the medium field, the scattering matrix for one kick reads as [35] S1(p0, p) = Zd4x ei(p0−p)·x¯u(p0)igAa µ(x)Taγµu(p),(2.25) 5The movement of the energetic quark is set in positive x3-direction. 35 V ´ ıctor Vila P´ erez 2.2.3 The medium-induced gluon radiation spectrum Following the introduction of the BDMPS-Z/ASW approach, the medium-induced radiation amplitude corresponding to the splitting of a quark into a quark-gluon antenna is now considered. As a result of the hard process which materializes within the medium the energetic quark that splits is produced, then experiencing multiple scatterings together with the emitted gluon, which takes away a minor fraction of energy off the parent parton –p+k+. In accordance to the soft limit, the path followed by the incoming quark both before and after the splitting is eikonal and can therefore be described by the classical Wilson line (2.21), while the trajectory undergone by the radiated parton is encapsulated in the Green function (2.29) since it includes higher-order transverse corrections to the eikonal trajectory. The raised scenario is depicted in Fig. 2.7, where hrepresents the hard process giving rise to the leading parton, and the multiple scatterings suffered by the propagating partons are also shown. Two leading contributions, allowing the gluon to exhibit vacuum-like propagation or in-medium rescattering, have to be considered in order to compute the total radiation amplitude [35]. They read, respectively, Mq≃ −2gTak⊥·⊥ k2 ⊥Zdx⊥e−ik⊥·x⊥W(L+, x+ 0;x⊥),(2.43) where x+ 0and L+determine the longitudinal boundaries of the medium, and k⊥and ⊥ accounts for, respectively, the momentum and polarization of the gluon radiated outside the medium, and Mg≃ −2g k+Zdx+d2x⊥e−ik⊥·x⊥W(x+, x+ 0;0)Tb ×⊥·∂ ∂y⊥Gba(L+,x⊥;x+,0|k+)W(L+, x+;0). (2.44) Here y⊥has been set to zero to ensure the eikonality of the leading quark. The total amplitude is therefore the sum of eqs. (2.43) and (2.44), Mrad =Mq+Mg.(2.45) Squaring eq. (2.45) gives way to three different contributions according to the position of the radiation vertex in both amplitude and its complex conjugate, as can be seen from the diagrams presented in Fig. 2.8, where the contribution from vacuum radiation, k+dIvac dk+d2k⊥ =αsCF π2 1 k2 ⊥ ,(2.46) pops up as well as expected. Singling out the remainder purely medium-induced gluon radiation contributions to the squared amplitude produces the result [35] 42 2 The formalism framework hp+~ p+ k+ Figure 2.7: The medium-induced gluon radiation diagram within the soft approximation. k+dImed dk+d2k⊥ =αsCF (2π)2k+2Re(1 k+Zdx+d¯x+d2x⊥e−ik⊥·x⊥e−1 2RL+ x+dξ n(ξ)σ(x⊥) ×∂ ∂y⊥·∂ ∂x⊥K(¯x+,x⊥;x+,y⊥=0|k+) +Zdx+d2x⊥e−ik⊥·x⊥2k⊥ k2 ⊥·∂ ∂y⊥K(L+,x⊥;x+,y⊥=0|k+)), (2.47) with x+and ¯x+referring to the longitudinal coordinates in amplitude and complexconjugate amplitude respectively, and Kencapsulating the medium-averages to be performed, K(y+,y⊥;x+,x⊥|k+) = 1 N2−1TrDG(y+,y⊥;x+,x⊥|k+)W†(y+, x+;0)E,(2.48) which in this case will entail solving the two-dimensional path-integral + + Figure 2.8: Schematic view of the diagrams corresponding to the three terms that emerge when computing the squared amplitude of the medium-induced gluon radiation in the soft limit. The dashed line sets the amplitude apart from its complex conjugate. 43 V ´ ıctor Vila P´ erez K(y+,y⊥;x+,x⊥|k+) = Zr⊥(y+)=y⊥ r⊥(x+)=x⊥Dr⊥(ξ) ×exp (Zy+ x+ dξ ik+ 2˙ r2 ⊥−1 2n(ξ)σ(r⊥)). (2.49) Then, using eq. (2.41), in the MSSA the preceding path-integral is equivalent to that of a harmonic oscillator, K(y+,y⊥;x+,x⊥|k+) = Zr⊥(y+)=y⊥ r⊥(x+)=x⊥Dr⊥(ξ) ×exp (ik+ 2Zy+ x+ dξ ˙ r2 ⊥+iˆq(ξ) 2√2k+r2 ⊥). (2.50) Both [35] and [113] provide an instructive discussion on the solution of eq. (2.50) and the corresponding spectrum (2.47). Numerical results on the double-differential mediuminduced gluon radiation spectrum for a quark crossing a static medium are shown in Fig. 2.9. A brief discussion on these results is featured below from a qualitative perspective. The results contained therein are presented on the basis of the variables ωcand κ, defined in terms of the medium properties – the medium transport parameter ˆqand the length of the medium L– as follows, ωc:= 1 2ˆqL2, κ := k2 ⊥ ˆqL,(2.51) where ωccorresponds to the maximal possible value for the gluon frequency. For its part, the formation time of the emitted gluon within the medium, tf≃2ω k2 ⊥ ,(2.52) can be read off on the propagator (2.50) describing non-eikonal corrections. It is instructive to compare this time scale with the medium length for the proper interpretation of the big picture. In cases where the formation time is much shorter than the length of the medium, tfL, consecutive incoherent scatterings take place as the spectrum is proportional to L/tf. In the opposite limit, tfL, the entire medium acts as a single scattering center in a coherent way, thereby leading to a significant reduction of the medium-induced gluon radiation, which is nothing more than a manifestation of the already discussed LPM effect [66]. This can be noted clearly in Fig. 2.9 as a suppression of the spectrum for small values of κ. As a consequence, the examined spectrum shows neither collinear nor infrared divergencies, as opposed to the vacuum case one (2.15). 44 2 The formalism framework Figure 2.9: A numerical evaluation of the medium-induced gluon radiation spectrum (2.47) for a high-energy quark propagating through a static medium as a function of the variables (2.51) (results from [35]). In closing, it is important to state that in real-life heavy-ion collisions the created collective medium is dynamically evolving and spreading out itself. Such a scenario can be transferred directly into the framework via assuming a time-dependent longitudinal density of scattering centers. A straightforward way of modeling the evolution of the medium is by parameterizing this density in terms of a power-dependence regarding the diluteness of the medium as it expands, n(ξ)≃1/ξα, with αthe dilution factor. [35] also presents the corresponding spectra for different values of α. 2.2.4 The radiation spectrum off a q¯q-antenna in a dense medium As has been presented in the previous subsection, the medium-induced gluon radiation spectrum off a single parton does not exhibit any of the divergencies evidenced in the vacuum case. Additionally, the spectrum is strongly suppressed for formation times of the emitted partons larger than the length of the medium, thus showing a major contribution at a characteristic energy and radiation angle on grounds of the medium properties. This proves certainly well that the dense medium is responsible for the energy loss undergone by the leading parton attributable to the radiative process. However, the role of the physical scenario revisited above is limited since it does not incorporate the interference effects that emerge when studying medium-induced radiation off multiple emitters, the block for building up the partonic cascade in the presence of a medium. 45 V ´ ıctor Vila P´ erez As earlier noted too when discussing the different models for studying mediuminduced radiation in hot and dense matter, several approaches have been developed for addressing multiple emissions setups. For example, HT includes the medium-induced radiation spectrum as a modification of the standard Altarelli-Parisi splitting function in vacuum, the higher-twist medium-modified DGLAP evolution equation [82]. Nevertheless, an accurate description for jet evolution in matter requires the analysis of coherence effects between subsequent emitters, as the vacuum case makes clear inasmuch as interference contributions are responsible for angular ordering. With the above picture in view, some key points on a well-known study that seeks to mitigate the discussed limitations are presented in this piece [114–116]. More specifically, the gluon emission off a quark-antiquark antenna in a medium is reviewed to set the stage for the development of the studies which constitute the heart of this thesis. As will be seen, they are heading in exactly the same direction but somehow one step further in terms of getting close to a real-life parton shower. Along the same line of argument proposed in subsection 2.1.1 for studying the radiation spectrum off a q¯q-dipole propagating in vacuum, medium effects on the gluon radiation spectrum off the pair are discussed here. The physical scenario consists of a color singlet or octet antenna with an opening angle θq¯qtravelling through a static deconfined medium as a result of the decay of a highly virtual photon or gluon, respectively. Returning to the vacuum case that has been introduced earlier – eq. (2.12), the spectrum of radiated gluons with energy ωand transverse momentum kreads, for a singlet dipole, ωdNvac sing d3k=αsCF (2π)2ω2Wq¯q,(2.53) with Wq¯q=Rq+R¯q−2J, so that the spectrum is given by the independent radiation off the antenna constituents and the interferences between them. Incorporating the medium into this picture allows for the possibility of both the leading and the emitted partons to interact with the background field. In a similar fashion to the medium-induced gluon radiation spectrum off a single parton, three different contributions have to be taken into account – displayed in Fig. 2.10. Below is an anticipation of the outcome for the interference spectrum assuming eikonality for the leading partons whilst relaxing the eikonal approximation for the emitted gluon [114,116], J= Re Z∞ 0 dt0Zt0 0 dt [1 −∆med(t)] ×Zd2zexp −iκ·z−1 2Z∞ t0 dξ n(ξ)σ(z) + iω 2δn2t ×(∂y−iω δn)·∂zK(t0,z;t, y|ω)|y=δnt+ sym., (2.54) 46 2 The formalism framework where κ=k−xp(¯ κ=k−¯x¯ p), x=ω/p+, is the gluon transverse momentum with respect to the quark (antiquark) and δn:= δk/ω, with δk:= κ−¯ κthe relative transverse momentum of the q¯q-antenna, being closely associated with the opening angle of the pair as |δn| ≃ sin θq¯q. The symmetrical term comes from switching κ→¯ κand, accordingly, δn→ −δn. The independent components, Rqand R¯q, can be traced from the preceding expression for the interference spectrum via taking δn→0. As an example, the quark component is given by Rq= 2Re Z∞ 0 dt0Zt0 0 dt Zd2zexp −ik·z−1 2Z∞ t0 dξ n(ξ)σ(z) ×∂y·∂zK(t0,z;t, y|ω)y=0, (2.55) in this way supporting the result for the medium-induced gluon radiation spectrum shown in the previous subsection, eq. (2.47). An analogous expression holds for the antiquark component, R¯q. 0tLt !0L 0t L 0Lt! 0tL 0Lt! Figure 2.10: The three diagrams contributing to the gluon spectrum off the antenna in the medium case. Both the in-in and in-out components are sketched in the panels at top, whereas the bottom one is identified with the vacuum contribution (diagrams taken from [116]). 47 V ´ ıctor Vila P´ erez Eq. (2.54) characterizes the radiation off the quark-antiquark pair at various stages of its progress through the medium, as further described herein. At the beginning, the two-parton system originated from the splitting of a parent parton in a color singlet state, γ∗→q¯q, is assumed to form at the origin of the coordinate space, spreading over the medium until the gluon emission takes place. In the interim, the antenna-medium interaction is described by the survival probability [116] S(t,0) := 1 −∆med(t, 0) = exp −1 2Zt 0 dξ n(ξ)σ(δnξ),(2.56) where σ(δnξ) is the dipole-medium cross-section. The preceding equation describes the rate of color decoherence of the pair as a result of color interactions with the deconfined medium before the emission occurs. The decoherence parameter ∆med(t, 0), implicit in the definition of the survival probability S(t,0), determines the decoherence time td, a physical time-scale that governs the color decoherence of the dipole inside the medium. In this way, ttdimplies that the q¯q-system decoheres in color and the interference contribution is therefore suppressed. At this stage, the gluon is produced at time tin the amplitude and t0in its complex conjugate. Within the time-like separation between those times, better known as the formation time, the dynamics of the emerging dipole system is characterized by K(t0,z;t, y|ω) = ZD[r] exp (Zt0 t dξ iω 2˙ r2(ξ)−1 2n(ξ)σ(r)),(2.57) the same two-dimensional path-integral arising when performing the medium averages for computing the medium-induced gluon radiation spectrum, eq. (2.49), which encompasses the Brownian motion exhibited by the gluon in the transverse plane from time tto t0. It arises as a result of the accumulation of momentum from the medium and leads to a change of the dipole size from r(t) = yto r(t0) = z. Moving forward on the dynamics picture, the characterization of further propagation from time t0onward is present in the middle term of the second line in eq. (2.54). More specifically, it accounts for the additional momentum accumulated from the medium by the gluon. Finally, the last term in the second line of eq. (2.54) proves to be independent of the medium attributes. Before going deeper in discussing the uprising regimes for in-medium gluon emissions, it is particularly important to define the previous quantities in the context of the already introduced harmonic oscillator approximation (2.41), n(t)σ(r)≃1 2ˆq(t)r2,(2.58) valid within the multiple soft scattering approximation framework. In this approximation, 48 2 The formalism framework the decoherence parameter (2.56) is given by ∆med(t) = 1 −exp −1 12 ˆq δn2t3,(2.59) since the antenna is assumed to linearly grow with time. It is possible now to read the decoherence time tdoff the equation above as looking back at eq. (2.56) it unambiguously stems from the solution of 1 2Ztd 0 dξ n(ξ)σ(δnξ) = 1,(2.60) so for this specific case the time-scale for decoherence of the pair takes the form td≃1 ˆqθ2 q¯q1/3 .(2.61) Likewise, the formation time of a medium-induced gluon can be identified in eq. (2.57). Further, it can be heuristically estimated in terms of the medium transport parameter ˆqsince within a time tf, the gluon acquires a transverse momentum squared k2 f≃ˆqtf, where tf≃2ω/k2 ⊥, thus deducing tf≃2ω ˆq1/2 ,(2.62) so that the formation time in vacuum entered in subsection 2.1.1, tf≃2/(ωθ2), is greater for soft gluons emitted at a fixed angle θ. Indeed, recalling the heuristic interpretation raised throughout the bottom part of the 2.1.1 piece, conducted to properly understand and reflect on the gluon radiation spectrum in vacuum, encourages a similar discussion based on the relevant scales of the current problem. In particular, the characteristic scales that matter in this context are the transverse size of the antenna, r⊥=θq¯qL, where θq¯qis the antenna opening angle, and the tranverse color screening length of the medium, given by the inverse of the characteristic medium scale, Q−1 s, which, by definition, is the maximal transverse momentum that can be accumulated by an induced parton when crossing the whole medium. Intuitively, this scale is also known as the saturation momentum, and it is achieved for a gluon with a frequency ωc, the maximal gluon frequency already introduced in (2.51) as ωc= (1/2)ˆqL2. For a dense medium, it reads Q2 s= ˆqL. Following closely the discussion conducted in [116], it is therefore natural to consider two different regimes on the grounds of the following expression for the medium decoherence parameter (2.56), 1−∆med ≃e−Q2 sr2 ⊥.(2.63) 49 V ´ ıctor Vila P´ erez To start with, in the cases of small antenna sizes, r⊥< Q−1 s, the medium cannot resolve the inner structure of the system, thus interacting as a coherent ensemble. This is referred to as the dipole regime. In terms of the characteristic time scales, this condition is equivalent to say that the decoherence time is much greater than the medium length, td L. It is even possible to decipher the puzzle on the basis of angular scales. Specifically, the aforementioned conditions can also be translated into the context in which the opening angle of the pair is smaller than some critical angle θc, θc=Qs ωc =2 (ˆqL3)1/2,(2.64) the formation angle for a gluon with the maximal frequency ωc, in accordance to the relations above. In short, in this regime jet-medium interactions induce coherent radiation because the antenna probes the medium, being the interference spectrum directly proportional to ∆med =1 12Q2 sr2 ⊥.(2.65) Alternatively, if r⊥> Q−1 sis fulfilled, the medium probes the inner structure of the antenna. In such a situation, very little time passes until the antenna constituents decorrelate themselves from the moment they enter the medium. This is the so-called decoherence regime, which occurs for td< L, meaning that the two-parton system decoheres at a finite distance within the medium. The spectrum therefore consists of independent radiation off each of the emitters since interferences are strongly suppressed in a way that the decoherence parameter reaches its highest value, ∆med = 1. A precise derivation of the three different components which constitute the radiation spectrum off the two color charges crossing the medium has been carried out in [116] within the harmonic oscillator approximation. A numerical study also supports the tworegimes picture commented on above. In order to complete this piece and with a view towards linking with the physical scenario addressed in the chapter that follows the out-out component, responsible for the decoherence of the vacuum radiation, is discussed. Emissions taking place outside the medium correspond to the diagram illustrated in the panel at bottom of Fig. 2.10. Within the harmonic oscillator approximation, it reads [116] Jout−out = [1 −∆med(L, 0)] 4ω2κ·¯κ κ2¯κ2cos (κ+¯ κ)·r 2,(2.66) where r=δnL. 50 2 The formalism framework It is of upmost importance to notice that the preceding expression is directly proportional to the vacuum interference emission pattern encapsulated in the radiation function (2.7), which displays the angular ordering property between subsequent emissions, Wq¯q= 2ω2p1·p2 (p1·k)(p2·k),(2.67) bearing in mind that p1and p2are the four-momenta of the outgoing quark and antiquark, respectively, and ωis the energy of the gluon with momentum k. To prove that, it is enlightening to rewrite the preceding expression in terms of the variables present in eq. (2.66) as follows, Wq¯q= 4ω2δκ2 κ2¯κ2= 4ω21 κ2+1 ¯κ2−2κ·¯κ κ2¯κ2,(2.68) recalling that κ(¯κ) is the gluon transverse momentum with respect to the quark (antiquark), and δκis the relative transverse momentum of the pair. This arrangement is exactly the decomposition of the spectrum (2.12) into the independent components and the interferences, Wij =Rq+R¯q−2J,(2.69) as has been shown in detail in subsection 2.1.1. Here both Rq:= 4ω2/κ2and R¯q:= 4ω2/¯κ2account for the radiation spectrum off the independent constituents of the antenna, and J:= 4ω2κ·¯κ κ2¯κ2describes the two-parton system interference. Accordingly, it can be said that the out-out component therefore controls the decoherence of the vacuum radiation as it only entails the two-parton interaction with the medium. In fact, one might reasonably expect that in view of the corresponding diagram – at the bottom of Fig. 2.10. Most remarkably, it constitutes the leading term in the soft limit, as will be shown in greater detail in the next chapter. Finally, albeit limited to the color singlet configuration as it greatly simplifies the color algebra, it is possible to extrapolate these findings to other color configurations of the final state by proceeding in the same way. As an example, the creation of the q¯qantenna from an octet source is physically a more prominent scenario. In pure vacuum, the cross-section for the color octet case reads [1] ωdNvac octet d3k=αs (2π)2ω2(CFRsing +CAJ),(2.70) where the radiation function Wij given by eq. (2.69) has been renamed as Rsing to clearly trace the singlet component back. Contrary to the singlet case (2.53), the additional contribution, with Jdefined just below eq. (2.69), gives rise to large-angle emissions in what is seen as the radiation off the total charge of the pair, meaning radiation off the parent parton imagined to be on-shell – zero for a virtual photon or CAfor a virtual gluon, 51 V ´ ıctor Vila P´ erez This motivates calculations specifically for this purpose within the CGC framework discussed in this piece. The final chapter of this thesis is precisely devoted to discuss the origin of correlations via studying the more straightforward process containing two-parton correlation functions. 58 3 Factorized picture of color coherence for gluon radiation off a double QCD antenna In this chapter it is put forward an extension of the derivations presented in subsection 2.2.4 to multiple emissions with the ultimate goal of extrapolating the results to the subsequent parton branchings that constitute the basis of the full medium-modified parton shower. Turning to substance, the radiation off two antennas – a quark-antiquark and a quark-gluon (antiquark-gluon) dipoles, originated after a first hard splitting assumed to be instantaneous at the origin of the coordinate system within a colored medium, is computed. This study focuses on the interference phenomena that arise from the ensemble of medium-induced modifications after the very first splitting during the branching process. Calculations are performed in its high-energy limit, namely eikonality is assumed in order to fix the trajectories of the partons in the transverse position as they propagate away. The usual small-angle approximation also applies here. Both approximations are expected to be justified for this configuration, thus simplifying enormously the complexity of the problem. The final results provide an extension to previous studies of multi-gluon radiation in an attempt to stress the relevance of interference effects when characterizing the QCD branching process. V ´ ıctor Vila P´ erez 3.1 A few remarks on the physical scenario As was pointed out in the previous piece, dealing with multiple emissions configurations encumbers the discussions mainly owing to the emergence of interferences amongst different processes. On one side is the vacuum shower, where color coherence is preserved in such a way as to lead to angular ordering. In this study, interference patterns may gradually alter the color coherence of the whole system over the decoherence time-scale, what is primarily due to medium-induced radiation. In particular, the focus is now on the process γ∗→q¯qg1g2, where both the splitting of the photon into the quark-antiquark antenna, γ∗→q¯q, and that of the leading partons, q→qg1(¯q→¯qg1), are taken to be very hard, whilst the second gluon emission, g2, is assumed to be soft – in concrete terms, the energies of all partons involved are much larger than any medium scale except for the last, very soft gluon. These kinematical choices allow for neglecting finite formation time effects for the splittings inside the medium, meaning that dipoles originate at a fixed point in both amplitude and complex-conjugate amplitude, what streamlines cumbersome algebra. By way of summary, the setup raised here is the soft radiation off a QCD antenna consisting of a quark-gluon (antiquark-gluon) after one of the energetic partons radiates the mentioned hard gluon within a dense medium. This calculation attempts to universalize prior studies conducted on the impact of multiple parton emissions into the final state of a medium-modified parton cascade. More precisely, it aims to go beyond existing standards by seeking to track the coherence record between the consecutive emerging dipoles throughout the entire medium, something that can be controlled by means of the last radiation process naturally occuring in vacuum, just as all partons have already phased out the static medium. As this scenario works in much the same way that the radiation pattern off a quarkantiquark antenna in the strict soft limit [116, 117], it is instructive to recall the full derivation of that radiation spectrum, which has already been introduced very briefly when reviewing the gluon emission spectrum off the antenna beyond the soft limit (see Fig. 2.10). This considers a color-singlet quark-antiquark pair that emanates from the splitting of a virtual photon occurring on very short-time scales within a medium of size L, namely the antenna is assumed to be created at the origin of the coordinate system in both amplitude and its complex-conjugate. In this regard, the non-radiation probability amplitude simply reads Mq¯q≃¯ui(p)ieqγµWij(L, 0; r1)W† jk(0, L;r2)vk(¯p),(3.1) where ¯u(p) and v(¯p) are the spinors for the outgoing quark and antiquark, eqis the quark’s electric charge and γµare the Dirac matrices. The fundamental Wilson lines at transverse positions r1and r2describe the propagation of the two-parton system from its origin to the end of the medium. 60 3 Factorized picture of color coherence for gluon radiation off a double QCD antenna Now, what needs to be noticed is that the actual expression for the medium-modified probability amplitude (3.1) is not required right after since the goal is to factor out the Born-level production amplitude in order to purely discriminate the radiation spectra. There is, however, a need to backtrack the color coding, that is to say, the Wilson lines which account for the restoration of the color coherent structure among the leading partons. The square of eq. (3.1) averaged over all color profiles is given by |Mq¯q|2med =|Mq¯q|2,(3.2) since it is noticeable that the correlator of the resulting four Wilson lines collapses to unity, Tr W(L, 0; r1)W†(0, L;r2)W(L, 0; r2)W†(0, L;r1)=I, thus breaking down the phase space for radiation. It is also important to stress that the square of the corresponding vacuum probability amplitude matches with the previous result. Indeed, as is to be expected, it can be explicitly retrieved by squaring eq. (3.1) straight after taking the Wilson lines to be unity. The following chapter sheds further light on this process via allowing the in-medium splitting to display a finite formation time, this way studying the medium modifications that arise as compared to this baseline. Turning now to the evaluation of the inclusive spectrum for gluon radiation leads to a focus on the radiative interferences between those two emitters. In the strict soft limit, ω→0, when the contribution from in-medium radiated gluons can be neglected due to LPM suppression1, the interference spectrum to be computed is the one pictured in Fig. 3.1. First and foremost, the gluon emission probability amplitudes off each of the partons constituting the singlet antenna are given by Mq¯qg(qg)≃¯ui(p)ieqγµta ijWjk(L, 0; r1)W† kl(0, L;r2)vl(¯p)·gsp· p·k,(3.3) Mq¯qg(¯qg)≃¯ui(p)ieqγµWij(L, 0; r1)W† jk(0, L;r2)ta klvl(¯p)·gs¯p· ¯p·k,(3.4) where the emitted gluon has energy ωand transverse momentum k. In order to proceed with the evaluation of the resulting cross-section, the squared amplitudes (3.3) and (3.4) have to be averaged over the whole ensemble of arbitrary medium color configurations and summed over gluon polarizations. 1This limit is remarkably convenient for the purpose of this study as it provides highly relevant information about the degree of color coherence between the two emitters that make up the antenna (see, e.g., [117]), which then is extensively generalized to the case of three emitters. 61 V ´ ıctor Vila P´ erez Accordingly, the direct terms, Rqand R¯q, turn out to be trivial since the medium averages yield unity – just as for the non-radiation case. As an example, the quark component reads DMq¯qg(qg) 2Emed =Mq¯qg(qg) 2=|Mq¯q|2CFg2 s 4 κ2,(3.5) which factorizes into the the non-radiation probability amplitude once again, in the same vein as for the vacuum case (for a pedagogical derivation of the vacuum emission pattern off a two-parton system go back to subsection 2.1.1). Here it has also been introduced κas the transverse momentum of the gluon with respect to the quark, κ=k−xp. The derivation of the antiquark component follows exactly the same steps as the former amplitude. It only implies the swap κ→¯κ, where ¯κis now the transverse momentum of the gluon with respect to the other leg of the antenna, ¯ κ=k−¯x¯ p. Dealing now with the interferences amongst both amplitudes (Fig. 3.1), they prove to be the dominant terms for interpreting the radiation pattern off the emitters, and hence also the pieces that reveal the inner coherence picture of the radiation process. As an illustration, DMq¯qg(qg)M∗ q¯qg(¯qg)Emed =|Mq¯q|21 NX λDTrW†(0, L;r1)taW(L, 0; r1)W†(0, L;r2) ×taW(L, 0; r2)E·g2 sp· p·k¯p·∗ ¯p·k, (3.6) which displays a similar structure to the vacuum result, with the fundamental difference that a color factor different from unity shows up now. 0L0L Figure 3.1: Diagrammatic sketch of the out-out component to the gluon spectrum off a q¯q-antenna propagating within a medium under the soft limit (ω→0). 62 3 Factorized picture of color coherence for gluon radiation off a double QCD antenna After simple manipulations using both eq. (3.26) and the Fierz identity (3.27) introduced in Appendix 3.B, eq. (3.6) reduces to Mq¯qg,1M∗ q¯qg,2med =−|Mq¯q|2CF 1 N2−1DTr W† A(r2)WA(r1)E(L,0) g2 s4κ·¯κ κ2¯κ2,(3.7) which proves to be directly proportional to the two-parton system interference for the vacuum case, J:= 4ω2κ·¯ κ κ2¯ κ2. One can now define the survival probability S(t2,t1):= 1 N2−1DTr W† A(r2)WA(r1E(t2,t1),(3.8) which as its name suggests accounts for the probability of the coherence effects surviving the passage across the medium, as previously discussed – see eq. (2.56). It is indeed a redefinition of the decoherence parameter, ∆med(t2, t1) := 1 −S(t2,t1),(3.9) a parameter that characterizes the rate of color decoherence of the pair as a result of color interactions with the deconfined medium [115–117]. In this way, since its origin as a result of the very first splitting inside the medium, the two-parton system spreads over the medium until the very end of it, with the Wilson lines reflecting the color rotation in the wake of an arbitrary number of kicks off the background field2. In this interim, the decoherence parameter, implicit in the definition of the survival probability, determines the decoherence time td, a physical time-scale that governs the color decoherence of the dipole inside the medium. A more detailed discussion on these ideas will take place a little further down this piece. At the moment the resulting cross-section is at readiness to be written down, dNmed γ∗→q¯q=dσmed γ∗→q¯q dσnon-rad =X λMtot 2d3k (2π)32ω,(3.10) where d3k=ω2dω d cos θdφ, with θand φthe polar and azimuthal angles of the gluon with respect to the emitter, respectively. A glance at eqs. (3.5) and (3.7) suggests that there is scope for casting the spectrum of emitted gluons in the same fashion as is decomposed for the vacuum case, ωdNmed γ∗→q¯q d3kω→0=αsCF (2π)2ω2hRq+R¯q−2S(L,0)Ji,(3.11) 2A straightforward derivation of tilted Wilson lines for in-medium parton propagation is provided in Appendix 3.A. 63 V ´ ıctor Vila P´ erez where Rq:= 4ω2/κ2and R¯q:= 4ω2/¯κ2are the independent contributions to the radiation off both the quark and the antiquark, respectively, and the last term accounts for the quark-antiquark interference spectrum, with S(L,0) defined according to eq. (3.8) and J:= 4ω2κ·¯κ κ2¯κ2. Each of the terms of eq. (3.11) clearly comes from the results derived in eqs. (3.5), (3.5) (κ→¯κ) and (3.7), respectively. The generalization of the resulting spectrum to the octet case is straightforward and is given by [117] ωdNmed g∗→q¯q d3kω→0=αs (2π)2ω2hCFRsinglet +CAS(L,0)Ji,(3.12) where, in line with the vacuum derivation, Rsing := Rq+R¯q−2S(L,0)Jis defined to track the singlet component down. It is worthwhile to recover the corresponding vacuum spectrum to the process under consideration by merely imposing on the survival probability of the coherence effects to tend to unity, so replicating the absence of the medium, ωdNvac γ∗→q¯q d3k=ωdNmed γ∗→q¯q d3kS(L,0)→1 =αsCF (2π)2ω2Rq+R¯q−2J,(3.13) as explicitly computed in subsection 2.1.1, eq. (2.12). Recapturing the vacuum outcome from the in-medium picture in the dilute limit – S(L,0) →1, and accordingly ∆med →0 – constitutes a simple test to verify the resulting spectrum, eq. (3.11). This can be done in exactly the same way for the octet case, eq. (3.12). To close this section, an intelligible interpretation of the physical picture behind eqs. (3.11) and (3.12) is provided. The factorized form of the spectrum allows for tackling two clear limits. The former was already outlined above. It is based on the case when the color correlation length of the medium is larger than the transverse size of the pair, so that the medium cannot resolve the single emitters, thus acting like an individual object with the total charge of the pair. In other words, the antenna radiates coherently, as one would expect in the dilute limit, so the vacuum spectrum is indeed restored as shown above. In a similar spirit, when the medium is able to resolve the antenna, it breaks the color correlation of the pair, then behaving as two independent emitters. The latter limit can be deemed as the opaque limit owing to coherence effects being deeply sensitive to the passage through the medium, ∆med →1, in this way leading to total decoherence of the spectrum and breaking the angular ordering pattern. Thus far it has been introduced an oversimplified picture of color correlations in a few respects. Keeping in mind the above scenario, from here on out the focus is very much on further deeping the understanding of color coherence by working over a multiple medium-induced emissions picture, whereby both two in-medium hard splittings and a very soft gluon emission take place. 64 3 Factorized picture of color coherence for gluon radiation off a double QCD antenna 3.2 A problem of multiple emitters As previously highlighted, interference effects as color coherence have significant implications for achieving a greater knowledge on parton cascade standards. Interest is now focusing on the establishment of a more involved QCD partonic evolution picture. The diagrams to be computed are those presented in Fig. 3.2 alongside the analogous ones when making the switch q↔¯q. Just to be clear, here it is derived the spectrum of gluon radiation off two-parton QCD antennas originated after two hard splittings, occurring at times 0 and t, within a static color-deconfined medium of size L. The stress is put on the role of color coherence effects by considering an additional soft gluon emission when the leading emitters have already crossed the entire medium length, as a means of tracing the coherence history of the full picture back. Yet again, neither non-eikonal corrections to parton propagation nor finite formation time effects of the splitting processes are taken into consideration. From here, the same steps as outlined for the above calculation apply. The partial amplitude for each of the processes presented in the diagrams sketched out in Fig. 3.2 below can be written as Mq¯qgg(qg)≃¯ui(p1)ieqγµtb ijWjk(L, t;r1)ta klWlm(t, 0; r1)W† mi(0, L;r2)vi(p2)Waα A(L, t;r3) ·g2 sp1·(p3) p1·p3p1·(k) p1·k,(3.14) Mq¯qgg(¯qg)≃¯ui(p1)ieqγµWij(L, t;r1)ta jkWkl(t, 0; r1)W† lm(0, L;r2)tb mivi(p2)Waα A(L, t;r3) ·g2 sp1·(p3) p1·p3p2·(k) p2·k,(3.15) 0L t 0L t 0L t Figure 3.2: Diagrammatic depiction of the probability amplitudes for the general case of three emitters. The green gluon is set outside the medium in order to hold the control of the stage of color coherence through the whole dynamics scenario. 65 V ´ ıctor Vila P´ erez Mq¯qgg(gg)≃¯ui(p1)ieqγµWij(L, t;r1)ta jkWkl(t, 0; r1)W† li(0, L;r2)vi(p2)Waα A(L, t;r3) ×(−ifαyz)·g2 sp1·(p3) p1·p3p3·(k) p3·k,(3.16) up to the phases that are implicit in the definition of the tilted Wilson line that generalises eq. (2.21), explicitly derived in Appendix 3.A . Here p1and p2represent the quark and antiquark momenta (which follow eikonal paths r1and r2, respectively). The momenta of the hard and soft gluons are labelled as p3(following the straight trajectory r3) and k (propagating through the vacuum), respectively. Moving forward on this, it is necessary to both square the preceding probability amplitudes and compute the radiative interferences. For the purpose of unravelling the coherence physics behind the evolution picture, it is essential to implement the large-Nc limit via keeping only the leading terms in the expansion, as the averaging procedure over the possible configurations entails non-trivial correlation functions among the fields retained within the Wilson lines. This is why from here onwards it suffices to focus on purely developing the color algebra3. Technical details on the squaring procedure and the computation of the probability interferences are fully provided in Appendix 3.B. Following the same reasoning as in the derivation of the squared amplitudes Rqand R¯q– eq. (3.5), the direct terms, explicitly schematized in Fig. 3.3, turn out to be trivial under the limit of large number of colors, DMqqgg(qg) 2Emed := Rqg∼C2 F, DMqqgg(¯qg) 2Emed := R¯qg∼C2 F, DMqqgg(gg) 2Emed := Rgg∼CFCA, (3.17) meaning that no medium modifications stand out since they are only proportional to the relevant Casimir factors for each radiation process. Noteworthy is the fact that this result is perfectly compatible with expectations as these configurations are indeed not very different from those introduced in Fig. 3.1 when computing the independent components – the Wilson lines collapse to unity once again when finite formation time effects are overlooked. Likewise, more interesting results are yielded when computing the interference components. Two of the most prominent contributions in the face of identifying a coherence pattern are depicted in Figs. 3.4 and 3.5. In contrast to the direct terms outlined above, 3In any case, the kinematic terms are identical to the ones already computed in both the vacuum and the radiation off the antenna in the soft limit derivations. 66 3 Factorized picture of color coherence for gluon radiation off a double QCD antenna 0L0L tt’ 0L0L tt’ 0L0L tt’ Figure 3.3: Representation of the squared amplitudes for two in-medium hard splittings. Note that t=t0is assumed, meaning that finite formation time effects for the parton splitting processes are discarded. now the survival probabilities are explicitly reflected in the probability interferences, thus providing valuable insights into the color correlation trend of the emitters. A more elaborated discussion on this point is presented further below. Again, restricted to color algebra, the interference contribution resulting from combining eqs. (3.14) and (3.16), sketched out in Fig. 3.4 (left), reads DMq¯qgg(qg)M∗ q¯qgg(gg)Emed ∼ S(L,t),(3.18) where S(L,t)is now the survival probability from the second hard splitting, just as the hard gluon is radiated off the quark, to the end of the medium – see eq. (3.8). In order to properly understand the findings for the interferences, it is important to underline the relationship between the survival probability and the decoherence parameter, eq. (3.9). Proceeding analogously for the process presented in Fig. 3.5 (left), this time associating eqs. (3.15) and (3.16), it appears that DMq¯qgg(¯qg)M∗ q¯qgg(gg)Emed ∼ S(t,0)S(L,t),(3.19) 0L0L tt’ 0L0L tt’ Figure 3.4: Schematic view of one of the interference processes under consideration (left). The corresponding large-Nclimit is shown on the right pane. Recall that t=t0is implemented in order to get eq. (3.18). 67 V ´ ıctor Vila P´ erez To end with, the derivation of eq. (3.19) is presented here as an interference contribution example. Proceeding in the same way as for Rgg, it reads DMq¯qgg(¯qg)M∗ q¯qgg(gg)Emed ∼1 NDTrW1(t, 0) taW1(L, t)W† 1(t0, L)tbW† 1(0, t0)W2(L, 0) tcW† 2(0, L) ×Waα 4(L, t) (−ifαβc)W†βb 4(t0, L)E. (3.30) Now, performing the algebra of the aside from the trace expression gives Waα 4(L, t) (−ifαβc)W†βb 4(t0, L) = 2 TrtaW4(L, t)tαW† 4(L, t) ×h−2 Trtαhtβ, tcii2 TrtβW† 4(t0, L)tbW4(t0, L) =−8 TrtaW4(L, t)tαW† 4(L, t)Trtαhtβ, tciTrtβW† 4(t0, L)tbW4(t0, L) =−4 TrW† 4(L, t)taW4(L, t)htβ, tciTrtβW† 4(t0, L)tbW4(t0, L) =−4 TrW† 4(L, t)taW4(L, t)tβtcTrtβW† 4(t0, L)tbW4(t0, L) + 4 TrW† 4(L, t)taW4(L, t)tctβTrtβW† 4(t0, L)tbW4(t0, L) =−2 TrtcW† 4(L, t)taW4(L, t)W† 4(t0, L)tbW4(t0, L) + 2 TrW† 4(L, t)taW4(L, t)tcW† 4(t0, L)tbW4(t0, L), where it has been used that −ifαβc = (−i)−2iTrtαhtβ, tci=−2 Trtαhtβ, tbi.(3.31) Merging now the preceding result with the trace term present in eq. (3.30), and also repeatedly implementing the Fierz identity produces TrW1(t, 0) taW1(L, t)tbW† 1(t0, L)tcW† 1(0, t0)Waα 4(L, t) (−ifαbβ)W†βc 4(t0, L) =−2 TrW1(t, 0) taW1(L, t)W† 1(t0, L)tbW† 1(0, t0)W2(L, 0) tcW† 2(0, L)TrtcW† 4(L, t)taW4(L, t)W† 4(t0, L)tbW4(t0, L) + 2 TrW1(t, 0) taW1(L, t)W† 1(t0, L)tbW† 1(0, t0)W2(L, 0) tcW† 2(0, L)TrW† 4(L, t)taW4(L, t)tcW† 4(t0, L)tbW4(t0, L) =−1 2TrW† 1(0, t0)W2(L, 0) tcW† 2(0, L)W1(t, 0) W4(L, t)W† 4(t0, L)TrW4(t0, L)tcW† 4(L, t)W1(L, t)W† 1(t0, L) +1 2TrW† 1(0, t0)W2(L, 0) tcW† 2(0, L)W1(t, 0) W4(L, t)tcW† 4(t0, L)TrW4(t0, L)W† 4(L, t)W1(L, t)W† 1(t0, L) =−1 4TrW† 2(0, L)W1(t, 0) W4(L, t)W† 4(t0, L)W1(0, t0)W2(L, 0) W† 4(L, t)W1(L, t)W† 1(t0, L)W4(t0, L) +1 4TrW† 2(0, L)W1(t, 0) W4(L, t)TrW† 4(t0, L)W† 1(0, t0)W2(L, 0)TrW4(t0, L)W† 4(L, t)W1(L, t)W† 1(t0, L). 74 3 Factorized picture of color coherence for gluon radiation off a double QCD antenna At this point, the probability interference is DMq¯qgg(¯qg)M∗ q¯qgg(gg)Emed ∼1 4NDTrW1(t, 0) W4(L, t)W† 2(0, L)TrW† 1(0, t0)W2(L, 0) W† 4(t0, L) ×TrW1(L, t)W† 1(t0, L)W4(t0, L)W† 4(L, t) −TrW† 2(0, L)W1(t, 0) W4(L, t)W† 4(t0, L)W1(0, t0) W2(L, 0) W† 4(L, t)W1(L, t)W† 1(t0, L)W4(t0, L)E, and, in a similar fashion as the above-derived Rgg, undertaking t=t0gives DMq¯qgg(¯qg)M∗ q¯qgg(gg)Emed ∼DTrW1(t, 0) W4(L, t)W† 2(0, L) TrW† 1(0, t)W2(L, 0) W† 4(t, L)−1E. (3.32) As a final step, assuming medium averages are local in time in order to hold separately each time-like region, and keeping only the leading terms in the expansion yields DMq¯qgg(¯qg)M∗ q¯qgg(gg)Emed ∼1 NcDTrW(r1)W†(r2)E(t,0) DTrW(r4)W†(r2)E(L,t) ×1 NcDTrW(r2)W†(r1)E(t,0) DTrW(r2)W†(r4)E(L,t), (3.33) where the convolution of two survival probabilities is clearly identifiable, DMq¯qgg(¯qg)M∗ q¯qgg(gg)Emed ∼ S(t,0)S(L,t),(3.34) this way bringing back the corresponding probability interference presented in Section 3.2, eq. (3.19). 75 V ´ ıctor Vila P´ erez 76 4 Finite formation time effects for in-medium parton splittings This piece of work, based on [145], seeks to explore the set of medium-induced modifications that arise from allowing a parton splitting such as those presented in the preceding chapter to occur at a finite formation time. Though the discussion still focuses in the context of a final-state color-singlet splitting, γ→q¯q, it shall remain valid for generic splitting processes involving a total color charge. Once again, the high-energy limit is implemented throughout the whole derivation, thereby considerably reducing the complexity of the problem to a semi-classical picture of parton propagation along well-defined trajectories. Contrary to the physical scenarios covered so far in this manuscript, this time the time-like separation of the splitting vertices in amplitude and complex-conjugate amplitude will actually lead to twoand four-point correlation functions of Wilson lines that resum parton-medium interactions. In fact, the dipole and the quadrupole account for the medium’s ability to disrupt the color correlation amongst the leading partons at various stages of the process being observed. As evidenced earlier when computing the gluon radiation spectrum off a singlet antenna in the strict soft limit, in the absence of this separation these correlators collapse to unity, this way preventing the possibility to drag the effects of the splitting process throughout the full parton evolution picture. Given that, as previous setups conducted in this thesis have not taken account on a matter as fundamental toward taking notice of accumulated effects of parton-medium interactions over long distances, the formation of the antenna itself is now under examination. To end this study, the findings for the emission spectrum in the presence of a medium are collectively mapped on the already entered Lund plane. V ´ ıctor Vila P´ erez 4.1 Introducing the parton splitting setup The physical scenario considered here consists of the splitting of a parent parton with momentum ~p0= [E, p0] into two daughter partons with the following final-state kinematics, ~p1= [zE, p1], ~p2= [(1 −z)E, p2],(4.1) with p0=p1+p2according to transverse momentum conservation. Again, the notation refers to light-cone kinematics, with t:= x+standing for light-cone time and E:= p+the light-cone energy. The minus component of the momenta is integrated out throughout the derivation, thus producing an explicit time dependence. As stated above, for the moment focus is put on a final-state singlet antenna. More specifically, the splitting of a transversely polarized photon into a quark-antiquark dipole is addressed, as shown in Fig. 4.1. The partial amplitude for the chosen in-medium splitting reads Mγ→q¯q=e Eeip2 1 2zE +ip2 2 2(1−z)ELZ∞ 0 dt Zk1,k2G(p1, L;k1, t|zE)¯ G(p2, L;k2, t|(1 −z)E)ij ×γλ,s,s0(z)k·∗ λG0(k1+k2, t|E), (4.2) where γλ,s,s0(z) = iδs,−s0[zδλ,s −(1 −z)δλ,−s]/pz(1 −z) is the quark-gluon splitting vertex, with λ=±1 and k:= (1 −z)k1−zk2. Here Gand ¯ Grepresent the propagators for the quark and the antiquark, respectively. As usual for high-energy processes, partonmedium interactions are assumed to only exchange transverse momentum, so that k1 and k2correspond to the transverse momentum shared immediately after splitting. For the sake of alleviating notation, the shortcuts Rx:= Rd2xfor transverse coordinate and Rk:= Rd2k/(2π)2for transverse momentum integrations are implemented. [E, p0] [zE, p1] [(1 −z)E, p2] θ1θ2 Figure 4.1: Final-state kinematics of the splitting process under consideration. In terms of angular quantities, the antenna opening angle is n12 =n1−n2, with |n1|=θ1,|n2|=θ2 and |n12|=θ. 78 4 Finite formation time effects for in-medium parton splittings The fully dressed propagator in momentum space transforms to configuration space as G(p1, t1;p0, t0) = Zx1,x2 e−ip1·x1+ip0·x0G(~x1, ~x0),(4.3) where color and energy indices have been suppressed. In configuration space, G(~x1, ~x0) is described by the Green’s function propagator, which has been widely introduced through this manuscript, eq. (2.29). For propagation outside the medium, the propagator reduces to G(p1, L;k1, t|E)t>L = (2π)2δ(k1−p1)G0(p1, L −t|E),(4.4) where G0(k, t|E) = e−ik2 2Et, which also corresponds to the photon propagator in eq. (4.2). The expression for the antiquark propagator is analogous. As the focus of the present study is on the limit of hard splittings inside the medium, meaning that the formation time of the splittings are meant to be much shorter than the typical medium time-scales, a semi-classical picture is expected to dominate the crosssection on general grounds. This situation contrasts with the limit of medium-induced branchings [64,67,76,80,82,146,147], where one considers emissions with transverse momenta dominated by interactions with the medium. The high-energy limit is implemented as well. This in turn means that the energy of the leading partons is infinite, E→ ∞, but one can nevertheless trace the finite momentum sharing fraction zback. A two-step approximation process has to be carried out toward establishing this correspondence. The first move fixes the trajectories of the energetic partons, so that they follow classical trajectories determined by the kinematics of the process, whilst the second stage attaches the dipole to a common reference point in transverse space. In fact, both incoming partons are intended to have large enough energy in such a way that fluctuations in transverse position due to scattering with the medium constituents can be overlooked. This contribution can factually be isolated by considering the so-called eikonal expansion of the propagator (2.29) [148,149]. Its zeroth-term, which neglects further transverse momentum broadening in the medium, reads G(0)(~x1, ~x0) = G0(x1−x0, t1−t0)W(t1, t0; [xcl]),(4.5) where xcl(t) = t1−t t1−t0x0+t−t0 t1−t0x1is the classical trajectory. Now, according to the Fourier transformation (4.3), the propagator in mixed representation turns out to be G(0)(p1, t1;p0, t0) = e−ip2 1 2E(t1−t0)Zy0,y1 e−i(p1−p0)·y0(t1−t0)E 2πi ei(t1−t0)E 2(y1−n)2 ×W(t1, t0; [y0+ (t−t0)y1]), (4.6) 79 V ´ ıctor Vila P´ erez where n:= p1/E. Imposing EL−1by virtue of the semi-classical limit reduces the kernel of the preceding equation to a Delta function of its argument, this way ensuring that the particles propagate along the classical path. This delivers the following result for the propagator1, G(0)(p1, t1;p0, t0) = e−ip2 1 2E(t1−t0)Zx e−i(p1−p0)·xW(t1, t0; [x+ (t−t0)n]),(4.7) which corresponds to the S-matrix of a high-energy parton crossing the medium that has been introduced earlier in eq. (2.28). So far, this procedure removes all non-eikonal broadening effects after the dipole has been formed inside the medium. However, the initial position of the trajectory that describes each leg of the antenna is not yet firmly set, what leads to a smearing of the dipole initial position in terms of transverse coordinates2. In fact, for physical processes occurring at large time lengths with respect to the initial position, the origin of the Wilson line can be treated as a small correction, thus giving rise to the following result for the quark propagator, G(0)(p1, t1;p0, t0) = (2π)2δ(p0−p1)e−ip2 1 2E(t1−t0)W(t1, t0; [nt]),(4.8) making sure that the Wilson lines are initiated at the same initial transverse position and time. Now, entering the preceding expression for the propagator into the partial amplitude (4.2), and assuming the splitting takes place within the medium of size L, 0 < t < L, gives Min γ→q¯q=e Eγλ,s,s0(z)p·∗ λZL 0 dt exp −iL−t tfhW1(L, t)W† 2(L, t)iij ,(4.9) where p:= (1−z)p1−zp2is only related to the final-state momenta. Noteworthy is that the quantum-mechanical formation time can be read off this expression, tf=2z(1 −z)E p2.(4.10) This arrangement completes the implementation of the semi-classical approximation, in which the leading partons propagate along certain trajectories determined by their kinematics. This way both Wilson lines correspond to each of the legs of the antenna, such that Wi(¯ t, t) := W(¯ t, t; [ri(s)]), with ri(s) = nis(n1=p1/(zE) and n2=p2/[(1 −z)E]). 1A similar derivation is performed in [150]. 2Appendix 4.B provides insight in this respect. 80 4 Finite formation time effects for in-medium parton splittings As a matter of fact, the partial amplitude for emissions taking place outside the medium, t > L, can be straightforwardly deduced via integrating over the splitting time. It explicitly delivers Mout γ→q¯q=iδij z(1 −z) Eeγλ,s,s0(z)p·∗ λ p2.(4.11) The total amplitude will simply be the sum of eqs. (4.9) and (4.11), Mγ→q¯q= Min γ→q¯q+Mout γ→q¯q. 4.2 Deriving the spectrum Turning to the computation of the spectrum of the splittings, it is instructive to recall the following quantities, SIJ (t1, t0) := 1 NDTr WIW† JE,(4.12) the two-point function, already discussed in subsection 2.2.2, which corresponds to the dipole cross-section. Here the extent of the Wilson lines is implicit from the left-hand side of the equation, and {I, J}={1,2,¯ 1,¯ 2}. Note that the trajectories in amplitude, r1and r2, and complex-conjugate amplitude, r¯ 1and r¯ 2, are only shifted due the difference in the splitting time, and this varies independently, see Fig. 4.2. For its part, the four-point function, Q(t1, t0) = 1 NDTr W1W† 2W¯ 2W† ¯ 1E,(4.13) is also referred to as a quadrupole. In the harmonic oscillator approximation (2.41), and for the case of a static medium of size L, the two-point correlator reads SIJ (t1, t0) = exp −1 4ˆqZt1 t0 dsr2 IJ (s),(4.14) with ˆq:= CFˆ ¯qthe transport coefficient for fundamental degrees of freedom. rIJ := rI−rJ describes here the separation between the eikonal trajectories in transverse space. Under the large number of colors approach, the quadrupole can be written as follows [146,147], Q(t1, t0) = S1¯ 1(t1, t0)S2¯ 2(t1, t0) + Zt1 t0 ds S1¯ 1(t1, s)S2¯ 2(t1, s)T(s)S12(s, t0)S¯ 1¯ 2(s, t0),(4.15) where T(s) is the transition amplitude, T(s) = −1 2ˆqr2 12(s) + r2 ¯ 1¯ 2(s)−r2 1¯ 2(s)−r2 ¯ 12(s)=−ˆqr1¯ 1(s)·r2¯ 2(s).(4.16) 81 V ´ ıctor Vila P´ erez r1 r¯ 1 r¯ 2 r2 Lt ¯ t Figure 4.2: The in-in component to the spectrum. The amplitude, in black, and its complex-conjugate, in grey, are sketched out on top of each other to emphasize the character of the splitting. The dipole spreads over the region (¯ t, t), while the quadrupole does it within the region (L, ¯ t). Each of the terms in the equation above are the so-called factorizable and nonfactorizable pieces of the quadrupole. They describe the propagation of two possible color configurations under the large-Nclimit, such that color is conserved at any given time. In particular, the first term in eq. (4.15) accounts for the propagation of two antennas that separately correlate particles 1 and 2 in amplitude and complex-conjugate amplitude, so that they evolve independently of each other. This is why it is known as the factorizable piece of the quadrupole. The non-factorizable piece, on the other hand, involves a gluon exchange, encapsulated in the transition amplitude, which alters the color correlation of the system from a two-parton correlated state for t < s, to the uncorrelated system at times t > s. By virtue of the undertaken approximations, the separations between the leading parton trajectories are either constant or grow linearly with time. Indeed, in its high-energy limit, the separation of the splitting time in amplitude, t, and complex-conjugate amplitude, ¯ t, dictates the precise straight trajectories of the Wilson lines. More specifically, r1¯ 1(s) = n1(¯ t−t),r2¯ 2(s) = n2(¯ t−t), r12(s) = n12(s−t),r¯ 1¯ 2(s) = n12(s−¯ t),(4.17) where n12 := n1−n2. Noteworthy is to bear in mind its relation with the relative transverse momentum of the pair, n12 =p/[z(1 −z)E], with p= (1 −z)p1−zp2. Now, assuming a vanishing initial momentum p0produces n1= (1 −z)n12 and n2=−zn12. In angular terms, θ:= |n12|and θ1(2) := |n1(2)|. In this way, the transition amplitude (4.16) takes the simple form T(s) = −ˆqn1·n2(¯ t−t)2=−ˆqz(1 −z)n2 12(¯ t−t)2.(4.18) 82 4 Finite formation time effects for in-medium parton splittings Having set all these aspects, the inclusive spectrum for the splitting process under consideration can be written as dNmed dzdp2=1 4(2π)2z(1 −z)|Mγ→q¯q|2=1 4(2π)2z(1 −z)|Min γ→q¯q+Mout γ→q¯q|2,(4.19) where the averaging procedure over the ensemble of medium configurations is implicit. Here, the total spectrum in the presence of a medium can be computed as the sum of three different components. The in-in one, which corresponds to the splitting taking place inside the medium in both amplitude and complex-conjugate amplitude; the in-out contribution, in the sense of an interference between an emission taking place within the medium in amplitude and outside the medium in its complex-conjugate (or vice versa); and the last contribution corresponding to an emission outside the medium in amplitude and complex-conjugate amplitude. It is therefore intuitive to define the vacuum crosssection from the out-out component, h|Mout|2i. It reads dNvac dzdθ =αem π Pqγ(z) θ,(4.20) where it has been used that p2= [z(1−z)Eθ]2, and the definition of the relevant AltarelliParisi splitting function, Pqγ(z) = nfN[z2+(1−z)2], with nfthe number of quark flavors. A little algebra gives rise to the following expression for the in-in spectrum, dNin-in dzdθ =dNvac dzdθ 2Re ZL 0 dt tfZL t d¯ t tf e−i¯ t−t tfQ(L, ¯ t)S12(¯ t, t),(4.21) where the quadrupole Q(L, ¯ t) := Q(L, ¯ t;t) explicitly depends on the splitting time in amplitude. For its part, the in-out interference spectrum reads dNin-out dzdθ =−dNvac dzdθ 2Im ZL 0 dt tf e−iL−t tfS12(L, t).(4.22) Summing up all three components, the total spectrum can be written in the following manner, dNmed dzdθ =dNvac dzdθ [1 + Fmed(z, θ)] ,(4.23) where the factor Fmed encompasses the medium modifications. This function reads Fmed = 2 ZL 0 d¯ t tf cos ¯ t−t tfS12(¯ t, t)Q(L, ¯ t)−sin L−t tfS12(L, t),(4.24) 83 V ´ ıctor Vila P´ erez As is to be expected, the total phase-space for emissions inside the medium is the sum of the four contributions, (PS)tf<L = 4 X i=1 (PS)i=1 4ln2ER2L. (4.36) Hence, there is a relatively large probability for a splitting taking place inside the medium, i.e. (PS)tf<L/(PS)tot ∼1/2 asymptotically when E→ ∞. In the context of Monte Carlo simulations, the ratio is slowly varying and lies close to ∼40 −45% [152]. To end with, it is also enlightening to consider the low-energy regime, i.e. E < ωc. Here, the logarithmic contributions are automatically restricted and significant corrections to the leading-logarithmic approximation should be included. Nevertheless, it is interesting to point out that the region bounded by the condition tf< tbroad < td, corresponding to the (A.1) area, scales like (PS)tf<tbroad(E<ωc)=1 2ln2ER Qs .(4.37) This regime will not be discussed further than this, however the impact of the remaining phase-space regimes can be systematically implemented following the steps above. 4.5 Numerical evaluation of the phase-space As the main result of this work is to demonstrate the factorization property of the medium spectrum given by eq. (4.23), a numerical evaluation of the medium modification function onto the kinematical Lund plane is presented here. In particular, Q0= 0.2 GeV and ˆq= 1.5 GeV2/fm are taken as reference values. The following jet and medium parameters are used for two different evaluations: for the high-energy regime, E= 1000 GeV and L= 2 fm, while for the low-energy regime E= 240 GeV and L= 8 fm4. The final results are shown in Fig. 4.4, with Fmed defined in eq. (4.24). The plot at the left displays the result of evaluating the medium modification function into the Lund diagram defined in Fig. 4.3 for the high-energy regime, whereas the one on the right side represents the scheme within the low-energy regime. Lines are equivalent in both maps. As stated earlier when developing the Lund diagram, the full shaded area corresponds to the available phase-space given the constraint k⊥> Q0, and the lines correspond to the result of imposing tf=L,tf=tdand tf=tbroad (from top to bottom). 4The reason behind this choice is to map the same in-medium phase-space tf< L, so that the characteristic angle θcin the high-energy regime is approximately located at the same absolute angle as θbroad in the low-energy regime. 90 4 Finite formation time effects for in-medium parton splittings It is therefore worthwhile to trace the medium modification function for different limits. Looking first at the high-energy regime (see Fig. 4.4, left pane), in the region where vacuum-like emissions are expected to take place, tf< tbroad < L with td> L (θ < θc) and tf> L, medium effects are negligible. Indeed, the onset of medium modifications follow the line tf=td, and they are more pronounced within the regime bounded by tbroad < tf< td, as predicted in Section 4.3. This is of little surprise as tbroad is related to the transverse momentum broadening along the medium length L, this way making sense for larger media. In the low-energy regime (Fig. 4.4, right panel), on the other hand, medium effects are much larger. In fact, the scaling behaviour stated above can only be thought to hold in a parametric sense, and care has to be taken with the assumptions regarding the importance of the transverse momentum broadening. In fact, this parameter choice illustrates that part of the jet, i.e. large-angle and soft emissions, happen to reach the non-perturbative scale while still being inside the medium (in terms of time-scales, tf< L). This constitutes a new category of in-medium modifications that goes beyond the scope of this work. � � � � � � � � � � � � �� �/θ �� �/� ���� ��=����� ��� � =��� ����/�� L=2. fm � � � � � � � � � � � � � � � � � � � � �� �/θ �� �/� ���� ��=���� ��� � =��� ����/�� L=8. fm �� �� �� �� �� �� �� Figure 4.4: Numerical evaluation of the medium modification function Fmed for the high- (E > ωc, left pane) and low-energy regimes (E < ωc, right panel). Notice that the scale of the color coding on the right-hand side is rescaled by a factor 10 with respect to the left one. The shaded area corresponds to the available phase-space given the constraint k⊥> Q0. The boundaries for the different regions are the same as those depicted in Fig. 4.3. 91 V ´ ıctor Vila P´ erez 4.6 Summary and outlook This work is intended to extend studies on the production of hard radiation in the presence of hot and dense matter. However, two regimes of vacuum-like emissions are found inside the medium. These schemes include the regime of short formation times (in particular tf< tbroad < L, corresponding to the region (A.1) in Fig. 4.3), and small angles (θ < θc, region (A.4) in Fig. 4.3). Even so, the fate of these two types of emissions is different as only in the former scheme do the splitting products lose their color coherence before reaching the end of the medium. As a result of this prompt decoherence, the splitted partons should therefore become subject to independent energy loss processes. For its part, emissions in region (A.4) are color coherent when they exit the medium and should behave as an individual object. The border at which long-distance medium effects start to play a role in the splitting process was also identified. This takes place at tf≥tbroad and is also recovered in the numerical evaluation of the medium modification function onto the kinematical Lund plane, shown in Fig. 4.4. This study therefore confirms the notion of purely vacuum-like emissions inside the medium, meaning that the in-medium splitting function is equal to the one in vacuum, Fmed ≈0. Finally, computing higher-order splitting processes goes beyond the scope of this analysis. Yet despite this, the discussion presented around the time-scales further corroborates the validity of the undertaken assumptions for the regions of vacuum-like emissions. 4.A Considering colored splittings Generalizing the nature of the addressed splitting process to arbitrary color representations is straightforward, and does not modify the general structure of eq. (4.23). One simply has to replace the coupling constant αem →αs, and the Altarelli-Parisi splitting function for the relevant one, e.g. Pqγ(z)→Pij(z) in the vacuum spectrum, eq. (4.20). The laborious part is rather incurred by working out the color algebra, which this time turns out to be more involved when computing the medium modification function Fmed. Taken as a concrete example the splitting process q→q+g, the expressions for the dipole and the quadrupole have to be replaced by the following ones, S12(¯ t, t)→1 N2−1DTr W† 2(t, ¯ t)W1(¯ t, t)Tr W† 0(t, ¯ t)W2(¯ t, t) −1 NTr W† 0(t, ¯ t)W1(¯ t, t)E, (4.38) 92 4 Finite formation time effects for in-medium parton splittings Q→1 N2−1DTr W† ¯ 1(¯ t, L)W1(L, ¯ t)W† 2(¯ t, L)W¯ 2(L, ¯ t)Tr W† ¯ 2(¯ t, L)W2(L, ¯ t) −1 NTr W† ¯ 1(¯ t, L)W1(L, ¯ t)E. (4.39) Now, the second term in both expressions above can be neglected under the large-Nc limit, thus remaining S12(¯ t, t)→S12(¯ t, t)S20(¯ t, t),(4.40) Q→Q(L, ¯ t)S2¯ 2(L, ¯ t).(4.41) The emergence of these new dipole structures, S20(¯ t, t) and S2¯ 2(L, ¯ t), does not preclude transferring the qualitative insight regarding color-singlet dipole splittings to colorcharged ones. In the harmonic oscillator approximation, they read S20(¯ t, t) = e−1 12 ˆq(1−z)2θ2τ3,(4.42) S2¯ 2(L, ¯ t) = e−1 4ˆqz2θ2(L−t0)τ2,(4.43) where now the differences amongst the trajectories described by the Wilson lines are given by r20(s) = n1(s−t) and r2¯ 2=n2(t0−t). Focusing on the dynamics during formation, in the large-Nclimit the scheme S12S20 only depends on the jet quenching parameter through the combination ˆqeff = ˆq[1 + (1 − z)2]≈[(1 −z)N+z2CF]ˆ ¯q, with ˆq=CFˆ ¯q. Indeed, this effective ˆqeff has been previously identified for medium-induced quark-gluon splitting, see e.g. [147]. Similar conclusions can be reached as regards the correction to the quadrupole. Therefore, on balance, though the general outcome of this study is based on the calculation of a color-singlet splitting, the discussion above establishes that it can be straightforwardly extrapolated to splittings involving a non-zero total color charge via carefully considering the color dependence of ˆqeff, together with the expected replacements in the vacuum spectrum, what also validates the discussion of soft and collinear radiation applied to the building of the Lund plane performed in Section 4.4. 4.B Beyond the classical picture By implementing the high-energy limit, E→ ∞, and using the propagator previously introduced in eq. (4.7), the amplitude is given by 93 V ´ ıctor Vila P´ erez Min γ→q¯q=e Eγγ→q¯q λ,s,s0(z)ZL 0 dt e−iL−t tf ×hp+i[(1 −z)∂x1−z∂x2]i·λhW1(tL, t)W† 2(tL, t)iij x1=x2=n0t,(4.44) up to factors that cancel out when computing the cross-section. Here n0= (p1+p2)/E, and the trajectories followed by the antenna constituents are described by ri:= xi+ (s− t)ni, whereas in the derivation presented in Section 4.2 it was assumed that xi= 0. The in-in spectrum then becomes dNin-in dzdp2=dNvac dzdp22Re ZL 0 dt tfZL t d¯ t tf e−i¯ t−t tfˆ V1hQ(L, ¯ t)S12(¯ t, t)ix2=x1=n0t ¯ x2=¯ x1=n0t ,(4.45) and, in the same spirit, the in-out spectrum reads dNin-out dzdp2=dNvac dzdp22Im ZL 0 dt tf e−iL−t tfˆ V2hS12(L, t)ix2=x1=n0t,(4.46) where tf= 2z(1 −z)E/p2. Here the following operators have also been introduced, ˆ V1=1 p2p+i[(1 −z)∂x1−z∂x2]·p−i[(1 −z)∂¯x1−z∂¯x2],(4.47) ˆ V2=p p2p+i[(1 −z)∂x1−z∂x2].(4.48) The dipole term, which encompasses the additional shift of the initial positions of the Wilson lines, is now S12(t1, t0) = exp (−1 4ˆq∆t"x12 +1 2∆tn122+1 12∆t2n2 12#) (4.49) for generic time intervals, ∆t:= t1−t0, and x12 := x1−x2, while the missing terms in the quadrupole, eq. (4.15), read SI¯ I(t1, t0) = exp "−1 4ˆq∆txI¯ I+nIτ2#(4.50) by virtue of eq. (4.12). As a result of the constraints on the initial transverse position in amplitude and complex-conjugate amplitude, what leads to both x1=x2and ¯ x1=¯ x2, respectively, 94 4 Finite formation time effects for in-medium parton splittings the resulting spectra are similar to the terms derived to obtain eq. (4.23), except for a unique prefactor emerging under the integrals of the in-in and in-out components that arises from the more involved vertices presented just above. Taking a closer look at these terms for the factorizable piece of the in-in component, the correction factor appearing under the integral reads 1−iˆqτ2 4z(1 −z)E−iˆqτLτξ z(1 −z)E−ˆqτLτξ 2z(1 −z)E21 + τ 2ξτL+ˆqτLξ [z(1 −z)Eθ]2,(4.51) with τL:= L−¯ tand ξ:= (1 −z)2+z2. For the in-out component, the correction factor is 1 −iˆq(L−t)2/[4z(1 −z)E], which closely resembles the two first terms in the preceding expression when exchanging τby L−t. Neglecting all finite-zcorrections, and assuming short times, t, ¯ tL, these terms scale as 1−itf tdτ td2−itf tbroad  τ tbroad −tf tbroad 2τ tbroad 2+tf tbroad 2,(4.52) what explicitly proves that corrections due to the vertex start to play a role whenever the kinematical formation time ceases to be the shortest time-scale to which compare the difference of emission times τ. In particular, this is starting to occur when τ≤tbroad < tf. 95 V ´ ıctor Vila P´ erez 96 5 Embedding color coherence into the probabilistic picture of a partonic cascade This chapter, the last one in this thesis devoted to the study of final state effects, is set up as a first attempt to embed the notion of color coherence into the evolution equation describing the branching processes that make up a medium-induced parton shower. As shall be seen later, this implementation of coherence effects into the rate equations scene takes, as a baseline, a parton configuration not so different from those addressed in Chapters 3 and 4. Then, after computing the relevant diagrams within the BDMPS-Z approach, the latest step towards eliciting an approachable probabilistic cascading picture which now encompasses color connections amongst the leading partons is to manipulate the emission kernel so that one can factor the coherence scheme out. 5.1 The partonic setup As repeatedly stated throughout this manuscript, the intrinsic motivation of the studies presented in this thesis comprises a more detailed description and knowledge regarding a real-life parton shower. This means going a step beyond the main approximations undertaken in previous studies under this subject, as laid down in both Chapter 3 and 4. 97 V ´ ıctor Vila P´ erez These chapters accomplish this via handling multiple emissions configurations and taking notice of finite formation time effects for the splittings inside the medium, respectively. The partonic setup addressed here is not much different from those referred to. In particular, the double inclusive gluon production process occurring in a dense medium is raised now, where the extra gluon compared to the BDMPS-Z baseline is assumed to be very soft and, thus, the last splitting naturally takes place outside the medium – otherwise the gluon could no longer survive. In the same vein as the above calculation, the formation time of the quark-gluon dipole inside the medium is taken as finite. As is clear given the above choices for the starting partonic configuration, only the two diagrams shown in Fig. 5.1 become significant for this study. What is more, as the right-hand diagram in Fig. 5.1 is nothing more than a mere color charge correction to the in-in component of the BDMPS-Z gluon radiation spectrum, the focus here is put on computing the interference contribution depicted on the left panel. The following section provides a detailed account of the computation of the spectrum according to these considerations. Lx+ 0 0L x+ c + + Lx+ 0 0L x+ c + + Figure 5.1: Diagrammatic representation of the relevant contributions to the process set forth above. The leading-eikonal quark, which emanates at t= 0 as a result of the hard process that materializes inside the medium, splits at t=x+in probability amplitude, whilst the splitting takes place at t=x+ cin complex-conjugate amplitude. For its part, eikonality restrictions are eased for the emerging gluon inside the medium – hereafter referred to as the BDMPS-Z gluon, this way allowing the parton to spread over the transverse plane. In a similar fashion as to the interference components corresponding to the processes derived in Chapter 3, the green gluon, laid again outside the medium, serves as means of tracking the coherence history of the partonic system, what, as will be seen later, is essential for the ultimate aim of this study. The plus component of the light-cone momentum for this soft gluon is fixed to the same in both amplitude and its complexconjugate. As noted above, here the emphasis is on deriving the interference between the probability amplitudes MqM† g(left pane), while the direct component MgM† g(right panel) does not have much to say when looking towards embodying coherence effects into the rate equation for describing medium-induced partonic splittings. 98 5 Embedding color coherence into the probabilistic picture of a partonic cascade 5.2 Computing the spectrum As discussed above, emphasis is put on the computation of the interference contribution displayed on the left-hand side of Fig. 5.1. By following the same standards as for deriving the spectrum of the processes presented under the preceding chapters, the probability amplitudes for the two branches making up this component can be written as Mq=−2g2 ωZz⊥,x+ e−ik⊥·z⊥∂x⊥|0Gab(L+,z⊥;x+,x⊥|k+)tc miWij(L+, x+;0) ×tb jkWkl(x+,0; 0)k0 ⊥·0 ⊥ k02 ⊥ , (5.1) Mg=−2ig2 ωZz⊥,x+ e−ik⊥·z⊥∂x⊥|0fcdaGab(L+,z⊥;x+,x⊥|k+)Wij(L+, x+;0) ×tb jkWkl(x+,0; 0)k0 ⊥·0 ⊥ k02 ⊥ , (5.2) where the shortcuts Rx+:= RL 0dx+and Rx:= Rd2xhave been implemented. Thus, the interference between the probability amplitudes above is given by DMqM† gE=4ig4 ω2k02Zz⊥,z0 ⊥,x+,x+ c e−ik⊥·(z0 ⊥−z⊥)∂x⊥·∂x0 ⊥Dtc miWijta jkGabWkl ×fdcbW† l¯ kG†d¯at¯a ¯ k¯ jW† ¯ j¯mEx⊥=x0 ⊥=0, (5.3) where all the position boundaries have been removed in order to shorten the expression. Now performing the color algebra over the objects inside the brackets delivers Tr W† (L+,x+ c)tcW(L+,x+ c)ta0t¯aWa0a (x+ c,x+)Gab(L+,z⊥;x+,x⊥)fdcbG†d¯a(L+,z0 ⊥;x+ c,x0 ⊥),(5.4) which eventually reduces to i 4fa0¯aAfbcdWcA (L+,x+ c)Wa0a (x+ c,x+)Gab (L+,x+)(z⊥,x⊥)G†d¯a (L+,x+ c)(z0 ⊥,x0 ⊥) =i 4Zy⊥ fa0¯aAfbcdWcA (L+,x+ c)Wa0a (x+ c,x+)Gal (x+ c,x+)(y⊥,x⊥)Glb (L+,x+ c)(z⊥,y⊥,)G†d¯a (L+,x+ c)(z0 ⊥,x0 ⊥). (5.5) This last step involves the use of the identity ta ijtb jk =1 2Nδabδij +1 2dabctc ij +i 2fabctc ik.(5.6) 99 V ´ ıctor Vila P´ erez describing the parton propagation from t0until the point the branching process takes place. The splitting, assumed to be time independent, is given by2[143] K(z, p+ 0) = Pgg 2πsˆq(1 −z+z2) z(1 −z)p+ 0 ,(5.33) where Pgg is the leading-order Altarelli-Parisi gluon splitting function. It is important to stress that the derivation above was carried out under the assumption that the energetic parton does not experience a branching process during the time interval (L, t0), so that one is only left with the broadening piece. During this interim, however, the transverse momentum of the high-energy parton can undergo broadening, and then a 1 →2 parton splitting can take place, eventually to be followed by an additional broadening course experienced by the daughter partons. Hence, in the latter case, there is a need to enter a change into eq. (5.32). In particular, the second building block now reads ˜ P2(k,q, z;L, t0) = 2g2z(1 −z)ZL t0 dt K(z, p+ 0)(2π)4δ(2)(k−zp0)δ2(q−(1 −z)p0),(5.34) where it has been set the initial momentum of the incoming parton as the momentum just after it was created. This produces the following expression for the time evolution, ∂t˜ P2(k,q, z;t, t0) = 2g2z(1 −z)K(z, p+ 0)(2π)4δ(2)(k−zp0)δ(2)(q−(1 −z)p0),(5.35) while the linear variation of ˜ P2is given by ˜ P2(k,q, z;t0+dt, t0) = ˜ P2(t0, t0) + 2g2z(1 −z)K(z, p+ 0)(2π)4δ(2)(k−zp0) ×δ(2)(q−(1 −z)p0)dt, (5.36) with ˜ P2(t0, t0) = 0. Combining now this result with the time evolution equation derived for P1, and writing down the linear variation of the generating functional as Zp0(t0+dt, t0) = ZdΩkP1(~ k, t0+dt, t0)u(~ k) +1 2ZdΩk1dΩk2P2(~ k1,~ k2;t0+dt, t0)u(~ k1)u(~ k2), (5.37) 2As mentioned earlier in this manuscript, the splitting kernel can be generalized to any other flavors. 106 5 Embedding color coherence into the probabilistic picture of a partonic cascade it is possible to infer the master equation for the generating functional. It reduces to ∂tZp0(t, t0|u) = ZlC(l, t0)u(p+ 0,p0+l) +αsZzK(z, p+ 0) [u(z~p0)u((1 −z)~p0)−u(~p0)] , (5.38) where the last term, proportional to u(~p0), ensures conservation of probability during the evolution. At this point, the master equation enables one to straightforwardly determine the inclusive one-gluon distribution D(x, k, t) that accounts for the probability of finding a gluon with energy fraction xand transverse momentum kproduced at time tvia the decay of an ancestor parton. Explicitly, D(x, k, t) := k+δZp0[t, t0|u] δu(~ k)u=1 .(5.39) Finally, entering eq. (5.38) into the preceding expression for the inclusive one-gluon distribution and taking its time derivative produces [143] ∂tD(x, k, t) = ZlC(l)D(x, k−l, t) +αsZz2 z2Kz, x zp+ 0Dx z,k zΘ(z−x)−K(z, xp+ 0)D(x, k, t), (5.40) a rate equation that favours a friendly interpretation in the light of the meaning of each of the terms that make it up. In particular, the first term accounts for the transverse momentum broadening via medium rescattering between successive in-medium splittings, while the two terms within the square brackets correspond to the production of a new gluon with energy fraction xand transverse momentum kemitted off a parton with energy fraction x/z and transverse momentum k/z, and the absortion of a gluon with energy fraction xand transverse momentum kvia an in-medium splitting, respectively. To complete this study, the following piece explores the possibility of embodying the notion of color coherence amongst the leading partons into the rate equation sketched at the end of this subsection. 107 V ´ ıctor Vila P´ erez 5.3.2 Embedding color coherence into the medium-modified parton shower evolution equations To recapitulate, after computing the spectrum of the addressed scenario, an estimation of the corresponding coherence factor was accomplished at the bottom of Section 5.2, which provides a sound basis to somehow link the coherence piece to the building blocks that shape the evolution equation for the inclusive one-gluon spectrum outlined right above. Indeed, it seems intuitively plausible to embed this image as a correction into one of the basic elements that are used to develop the resulting rate equation. More specifically, the decoherence parameter shown in eq. (5.24) naturally enters into the basic skeleton as a correction to the splitting kernel. Explicitly, the coherence-modified kernel reads K(z, p+ 0)→ K0(z, p+ 0;t−t0) = ZlK(l)∆med(l),(5.41) where ∆med(l, z, p+ 0;L−t) := 1 −exp −ˆq 12θ2(l, z, p+ 0)(L−t)3.(5.42) Here the emission angle is given by θ(l, z, p+ 0) = l z(1 −z)p+ 0 ,(5.43) this way depending on the transferred momentum land both the energy fraction zand the energy of the leading parton p+ 0. Given the coherence-modified kernel estimated above, it will now be possible to take a further step towards achieving an evolution equation of the same substance as eq. (5.40). In fact, the same steps undertaken above for the Markov shower must be followed, namely deriving the time evolution equation for the two-particle broadening piece as well as the coherence-modified master equation for the generating functional, which ultimately leads to the desired evolution equation for the one-gluon distribution. Some estimates developed via comparing the relevant time-scales involved in the big picture suggest that there is broad agreement with established principles on jet propagation in a dense medium, many of which have been widely discussed throughout this manuscript. 108 5 Embedding color coherence into the probabilistic picture of a partonic cascade 5.4 Discussion and outlook This study represents a first attempt to implement coherence effects into the standard resummation formula for a medium-induced QCD cascade. In particular, under the framework of the BDMPS-Z approach, a novel configuration involving an extra soft gluon emission with respect to the standard BDMPS-Z scheme is computed as a baseline for quantifying color coherence, with the ultimate aim of achieving a coherence-modified evolution equation for the inclusive one-gluon distribution. Not surprisingly, the computed emission spectrum displays a friendly structure to be fitted into the splitting kernel K since the coherence piece can be straightforwardly factored out. After introducing the basic elements for the derivation of the standard evolution equation some insights on the expected outcome are discussed. Specifically, the coherencemodified rate equation features an only major change that rests upon swapping the original splitting kernel by a modified branching kernel that is composed by the coherence piece – deduced when computing the total radiation spectrum. Preliminary evidences are very much in line with baseline studies in this respect. Lastly, it should be noted that the fact that the computation is framed in the context of the Markovian approximation may lead to confusion as this approach disregards interference effects. For this reason, certain corrections concerning the branching and broadening pieces are being performed in order to alleviate this mismatch. However, further studies will be necessary to get closer to a robust coherence-modified evolution equation for the medium-induced partonic showering. Numerical tests will also evidence the reliability of the final outcome. 109 V ´ ıctor Vila P´ erez 110 6 Deciphering the origin of angular correlations in high-energy collisions This final piece is aiming to supplement already existing studies on rapidity and angular correlations between produced particles in high-energy collisions. More specifically, considering a high-energy scattering of a hadronic projectile on a stationary target in the lab frame, one seeks to provide a simple picture around the origin of final-state particle correlations via studying a two-gluon final state. 6.1 General remarks One of the main observations of the CMS collaboration was the detection of long-range correlations in pp collisions at √s= 7 TeV between two charged hadrons emerging from the collision collimated in their azimuthal angle φ, while at the same time being separated in rapidity η[156]. Remarkably, these correlations recall those seen in relativistic heavyion data from RHIC [157,158]. The phenomenon was called ridge because of its shape on the φ−ηdata graph. As correlations spread over a wide rapidity interval, arguments around causality issues lead to the conclusion that they have their roots in the very early stages of the collision. 111 V ´ ıctor Vila P´ erez The thinking behind this is that relativistic particles propagating out of the interaction region would otherwise be causally disconnected and, eventually, uncorrelated [141,142]. As becomes clear, particle correlations might provide support to the development of better knowledge in order to disentangle initial and final state effects. Additionally, albeit longrange rapidity correlations may not be associated with the formation of the QGP at all, they could indeed compete to go beyond to enhance the dynamics underlying final-state jet quenching [159,160]. A number of mechanisms designed for justifying the ridge in pp and AA collisions have been proposed ( [161–164], among others). In particular, it is thought that a hydrodynamical evolution starting in the aftermath of the scattering leads to a collective behaviour of the final state of the gluons cloud, which will then be accountable for the rise of this phenomenon in the context of HIC. Although there is broad consensus on the origin of the correlations in heavy-ion collisions, the puzzle mechanism remains outstanding when it comes to pp and pA collisions. This logic above has also been proposed as an attemp to understand the source of the correlations in small systems [165]. Based on the aforementioned causality argument, there is also a great deal of debate around the socalled glasma approximation, founded on the dilute-dilute limit of the earlier introduced CGC framework [141,142,166]. The framework of this study is exactly the same to that presented in [167], where the focus is not placed on the physics behind the glasma approach, but instead a straight logical reasoning is proposed with no reference being made to any specific model of highenergy evolution. Considering the high-energy scattering of a hadronic projectile on a stationary target in the lab frame, one can observe two final-state projectile gluons with rapidities Y1and Y2that scattered on the target. As is well known according to the BFKL dynamics entered in subsection 2.3.1, at high energies the hadronic wave function is gluon dominated and, in consecuence, their Lorentz transformation properties are homogeneously distributed in rapidity in a boost invariant state. This means the same projectile gluon distribution at rapidities Y1and Y2, so that, given an impact parameter and a target field configuration, a gluon is as likely to emerge at Y1as it is at Y2, what is reminiscent of the rise of particle correlations. Similarly, if one of the gluons is likely to acquire a transverse momentum q, the same holds true for the other. Hence, one clearly expects forward gluon correlations to be very generic within this scenario. This is particularly true for a projectile wave function dominated by the classical Weizsacker-Williams (WW) field, as in this context fluctuations in the wave function are small and the gluon density is almost the same at all rapidities. More explicitly, using light-cone Hamiltonian perturbation theory, the hadronic wave function at high energy reads [168,169] 112 6 Deciphering the origin of angular correlations in high-energy collisions |Ψi= exp niZd2x ba i(x)Zdη a†a i(x, η) + aa i(x, η)oB(a, a†)|ψi,(6.1) where ψis the wave function of valence charges that determines the distribution of fast partons responsible for the color charge density ρ. Here Bis a Bogolyuvov-type operator of the soft gluon fields a, and baccounts for the WW field depending on ρvia classical Yang-Mills equations of motion. In the dilute projectile limit is given by ba i=g πZy fi(x−y)ρa(y),with fi(x−y) = (x−y)i (x−y)2.(6.2) Though the above debate on the roots of angular correlations is greatly simplified in a few respects, it paves the way for performing realistic calculations when computing high-energy multigluon amplitudes as a way to unlock valuable information about the source of such correlations. As an example, both gluons do not scatter off the target with the exact same momentum transfers as, even setting both the same impact parameter and a fixed target fields configuration, this leads to a non-trivial probability distribution of momentum transfers. Besides that, observing a gluon in the final state not only requires that it acquires some transverse momentum, but it must also decorrelate from the source valence charge of the incoming wave function. Therefore, and with a view to better understanding the expected features of angular correlations within the high-energy QCD work frame discussed above, the two-gluon inclusive cross-section is recapped on a very detailed level. 6.2 Revisiting the double inclusive gluon production probability Although calculating multigluon amplitudes has already been the subject of much debate, there still seems to be no definite results regarding high-energy pp collisions. Both the dense-dense [170] and dense-dilute [171,172] configurations for the colliding hadrons have already been covered, but the situation which comes closer to the LHC collisions scenario is probably one where the density in the proton wave function is neither parametrically large nor perturbatively small, so that the scope of [172] is the one that fits best with this line. However, this in depth analysis takes as its starting point the two-gluon probability amplitude provided by [171]. To be more specific, the most general expression for the double-gluon probability amplitude displayed in [171], which is built upon the singlegluon production amplitude in a dense-dilute scattering, is given by Aab ij (k, p) = Zu,z eikz+ipu Zx1,x2nfi(z−x1)[Sz−Sx1]acfj(u−x2)[Su−Sx2]bd ˆρd x2ˆρc x1+ +fi(z−x1)[Sz−Sx1]acfj(u−x2)[Suδux1−Sx2δx2x1]bmTc md ˆρd x2o, (6.3) 113 V ´ ıctor Vila P´ erez where ˆρa(x) is the operator of the color charge density, and Sab(x) is the eikonal scattering matrix determined by the target color fields. Then, performing the symmetrization of the operators within the preceding amplitude, ˆρaˆρa1=1 2{ˆρa,ˆρa1}+1 2Tc aa1ˆρc,(6.4) leads to get the two-gluon probability amplitude outlined in [172] back. Explicitly, Aab ij (k, p) = Zu,z eikz+ipu Zx1,x2nfi(z−x1)[Sz−Sx1]acfj(u−x2)[Su−Sx2]bd 1 2{ρd x2, ρc x1} +fi(z−x1)[Sz−Sx1]acfj(u−x2)[Su−Sx2]bd 1 2Te dcρe x1δ(x1−x2) +fi(z−x1)[Sz−Sx1]acfj(u−x2)[Suδux1−Sx2δx2x1]bmTc mdρd x2o. (6.5) The first term of the above equation straightforwardly produces Aab ij (k, p)[1] =Zu,z eikz+ipu Zx1,x2nfi(z−x1)[Sz−Sx1]acρc x1onfj(u−x2)[Su−Sx2]bdρd x2o.(6.6) Now integrating once the third line of eq. (6.5) and renaming x2→x1gives Aab ij (k, p)[3] =Zu,z eikz+ipu Zx1nfi(z−u)[Sz−Su]acfj(u−x1)Sbm uTc mdρd x1 −fi(z−x1)[Sz−Sx1]acfj(u−x1)Sbm x1Tc mdρd x1o, (6.7) where the first term can be reorganised in a more compact manner as follows, Aab ij (k, p)[3(i)] =Zu,z eikz+ipu Zx1 fi(z−u)fj(u−x1)n[Sz−Su]¯ρ(x1)S† uoab,(6.8) with ¯ρ:= Taρa. Hence, all that remains is to work out the second line of both eq. (6.5) and (6.7). Combining the two delivers Aab ij (k, p)[2+3(ii)] =Zu,z eikz+ipu Zx1nfi(z−x1)[Sz−Sx1]acfj(u−x1)[Su−Sx1]bd 1 2Te dcρe x1 −fi(z−x1)[Sz−Sx1]acfj(u−x1)Sbm x1Tc mdρd x1o =Zu,z eikz+ipu Zx1hfi(z−x1)fj(u−x1)in−1 2[Sz−Sx1]acTe cdρe x1[S† u−S† x1]db −[Sz−Sx1]acTd cmρd x1Smb† x1o =−Zu,z eikz+ipu Zx1hfi(z−x1)fj(u−x1)in[Sz−Sx1]¯ρx11 2[S† u−S† x1] + S† x1o, 114 6 Deciphering the origin of angular correlations in high-energy collisions which finally gives rise to Aab ij (k, p)[2+3(ii)] =−1 2Zu,z eikz+ipu Zx1hfi(z−x1)fj(u−x1)in[Sz−Sx1]¯ρx1[S† u+S† x1]oab.(6.9) Putting the results (6.6), (6.9) and (6.8) all together, the two-gluon production probability amplitude eventually reduces to Aab ij (k, p) = Zu,z eikz+ipu Zx1,x2nfi(z−x1)[Sz−Sx1]acρc x1onfj(u−x2)[Su−Sx2]bdρd x2o −1 2Zx1 fi(z−x1)fj(u−x1)n[Sz−Sx1]¯ρx1[S† u+S† x1]oab +Zx1 fi(z−u)fj(u−x1)n[Sz−Su]¯ρ(x1)S† uoab, (6.10) which reveals a clear physical picture when looking at each of the terms separately. In particular, the first line corresponds to independent production of the two gluons. The second term, on the other hand, accounts for the emission of the two gluons off the same color source in the projectile wave-function. The third one comprises the remaining configuration, the process in which both gluons are subsequently produced in the collision: the softer gluon is emitted in the wave-function on the hands of the former hard gluon. It might be also insightful to interpret them in terms of BFKL ladders – see Fig. 2.12, as the square of the first line would represent a piece of a diagram including two independent ladders, whilst the square of the other two terms would match with the emission of two gluons within the same BFKL ladder. Squaring eq. (6.10) gives rise to several terms in the cross-section, dN d2pd2kdηdξ =σ4+σ3+σ2P,T .(6.11) As a matter of fact, it is possible to identify them according to the physical setup they account for in the same vein as just discussed for the full amplitude. As an example, the square of the first line of eq. (6.10), concerning the independent production of the two gluons, reads σ4=Zu,z,¯u,¯z eik(z−¯z)+ip(u−¯u)Zx1,x2,¯x1,¯x2 ~ f(¯z−¯x1)·~ f(z−x1)~ f(¯u−¯x2)·~ f(u−x2) ×nρx1[S† z−S† x1][S¯z−S¯x1]ρ¯x1onρx2[S† u−S† x2][S¯u−S¯x2]ρ¯x2o. (6.12) 115 V ´ ıctor Vila P´ erez As might be expected, eqs. (6.40) and (6.18) are just the same, as is also its interpretation regarding the appearance of angular correlations, already well-founded earlier. For its part, eq. (6.41) accounts for some extra terms arising when carrying out the symmetrization right after squaring the probability amplitude. Accordingly, the last two terms of eq. (6.41) match with the first two ones obtained in eq. (6.19). The remaining four emanate from labouring the original expression for Σ3. Indeed, below is performed the same procedure followed for drawing σ4and σ3(4) off the original expression of Σ4, eq. (6.21), but now to get the contributions to σ3out of Σ3, eq. (6.22), which, obviously, have to fit with those last four terms derived in eq. (6.19). Explicitly, Σ3=hZu,¯u eip(u−¯u)Fab u¯x1(k)Mmd ux2¯u¯x2Ta mc [1.1] −Fab x2¯x1(k)Gmd x2¯x2(p)Ta mc [1.2]iˆρc x2ˆρd ¯x2ˆρb ¯x1 +hZu,¯u eip(u−¯u)Fab x1¯u(k)Mmc ¯u¯x2ux2Tb md [2.1] −Fab x1¯x2(k)Gmc ¯x2x2(−p)Tb md [2.2]iˆρc x2ˆρa x1ˆρd ¯x2, (6.42) with Σ3 [1.1] =Zu,¯u eip(u−¯u)Fab u¯x1(k)Mmd ux2¯u¯x2Ta cmρc x2ρd ¯x2ρb ¯x1 =−Zu,¯u eip(u−¯u)ρb ¯x1Fba ¯x1u(−k)Tc amρc x2Mmd ux2¯u¯x2ρd ¯x2 =−Zu,¯u eip(u−¯u)ρ¯x1F¯x1u(−k)¯ρx2Mux2¯u¯x2ρ¯x2, (6.43) Σ3 [1.2] =−Fab x2¯x1(k)Gmd x2¯x2(p)Ta cmρc x2ρd ¯x2ρb ¯x1=ρb ¯x1Fba ¯x1x2(−k)Tc amρc x2Gmd x2¯x2(p)ρd ¯x2 =ρ¯x1F¯x1x2(−k)¯ρx2Gx2¯x2(p)ρ¯x2=ρxFxy(−k)¯ρyGyw(p)ρw,(6.44) Σ3 [2.1] =Zu,¯u eip(u−¯u)Fab x1¯u(k)Mmc ¯u¯x2ux2Tb mdρc x2ρa x1ρd ¯x2 =Zu,¯u eip(u−¯u)ρa x1Fab x1¯u(k)Td bmρd ¯x2Mmc ¯u¯x2ux2ρc x2 =Zu,¯u eip(u−¯u)ρx1Fx1¯x1(k)¯ρ¯x2M¯u¯x2ux2ρx2, (6.45) Σ3 [2.2] =−Fab x1¯x2(k)Gmc ¯x2x2(−p)Tb mdρc x2ρa x1ρd ¯x2=−ρa x1Fab x1¯x2(k)Tb mdρd ¯x2Gmc ¯x2x2(p)ρc x2 =−ρx1Fx1¯x2(k)¯ρ¯x2G¯x2x2(p)ρx2=−ρxFxy(k)¯ρyGyw(−p)ρw.(6.46) Taken all together delivers σ3= ˆρxFxy(−k)¯ρyGyw(p)ρw−ρxFxy(k)ˆ ¯ρyGyw(−p)ρw− −Zu,¯u eip(u−¯u)hρ¯x1F¯x1u(−k)¯ρx2Mux2¯u¯x2ρ¯x2−ρx1Fx1¯u(k)¯ρ¯x2M¯u¯x2ux2ρx2i,(6.47) 122 6 Deciphering the origin of angular correlations in high-energy collisions what, as stated above, is in accordance with the last four terms of eq. (6.19). Finally, after this cumbersome algebra procedure, the fully ordered expression for the double inclusive gluon production reads dN d2pd2kdηdξ =σ4+ ¯σ3+ ¯σ2P,T ,(6.48) where σ4= [ρF(k)ρ][ρF(p)ρ],(6.49) ¯σ3=σ3(4) +σ3,(6.50) with σ3(4) =1 2ρxFxy(−k)¯ρyFyw(−p)ρw−1 2ρxFxy(k)¯ρyFyw(p)ρw −1 2Tr[¯ρxFxx(k)][ρyFyw(p)ρw]−1 2[ρxFxy(k)ρy]Tr[¯ρxFxx(p)] −1 2ρxFxy(k)¯ρyFyw(−p)ρw−1 2ρxFxy(−k)¯ρyFyw(p)ρw, (6.51) σ3=ρxFxy(−k)¯ρyGyw(p)ρw−ρxFxy(k)¯ρyGyw(−p)ρw −Zu,¯u eip(u−¯u)hρ¯x1F¯x1u(−k)¯ρx2Mux2¯u¯x2ρ¯x2−ρx1Fx1¯u(k)¯ρ¯x2M¯u¯x2ux2ρx2i.(6.52) For now, contributions involving a combination of two ˆρoperators, ¯σ2, which come from both Σ4and Σ3, as well as from the original expression of Σ2, eq. (6.23), are disregarded, since major cancelations amongst the vast number of terms that arise are not expected, thus making it very difficult to draw relevant conclusions about the origin of the aforementioned correlations. Furthermore, noticing the similarity between the emerging contributions with the ones obtained for ¯σ3, correlations are very likely to belong to the same nature. As a final step, given the similarity between the terms that make up ¯σ3, it is well worth attempting to simplify even more eq. (6.50) in such a way as to streamline the procedure for averaging. More specifically, writing down the expression for ¯σ3above as follows leads to non-trivial cancellations that can make it much easier to compute the remaining correlators arising in the final expression, 123 V ´ ıctor Vila P´ erez ¯σ3=−1 2ρxFxy(k)¯ρyhFyw(p) + Fyw(−p) + 2Gyw(−p)iρw −1 2ρxFxy(−k)¯ρyhFyw(p)−Fyw(−p)−2Gyw(p)iρw −Zu,¯u eip(u−¯u)hρ¯x1F¯x1u(−k)¯ρx2Mux2¯u¯x2ρ¯x2−ρx1Fx1¯u(k)¯ρ¯x2M¯u¯x2ux2ρx2i −1 2Tr[¯ρxFxx(k)][ρyFyw(p)ρw]−1 2[ρxFxy(k)ρy]Tr[¯ρxFxx(p)]. (6.53) Now rewriting Gyw(p) and Gyw(−p) in terms of Mand Fas Gyw(p) = Muy¯uw −Fyw(p), Gyw(−p) = M¯uwuy −Fyw(−p),(6.54) gives rise to ¯σ3=−1 2ρxFxy(k)¯ρyhFyw(p) + Fyw(−p)−2Fyw(−p)iρw −1 2ρxFxy(−k)¯ρyhFyw(p)−Fyw(−p) + 2Fyw(p)iρw −Zu,¯u eip(u−¯u)nρ¯x1hF¯x1u−F¯x1x2i(−k)¯ρx2Mux2¯u¯x2ρ¯x2 −ρx1hFx1¯u−Fx1¯x2i(k)¯ρ¯x2M¯u¯x2ux2ρx2o −1 2Tr[¯ρxFxx(k)][ρyFyw(p)ρw]−1 2[ρxFxy(k)ρy]Tr[¯ρxFxx(p)]. (6.55) Finally, arguing that both contributions within the explicit integral cancel each other out owing to symmetry reasons delivers ¯σ3=−1 2ρxhFxy(k) + Fxy(−k)i¯ρyhFyw(p)−Fyw(−p)iρw −ρxFxy(−k)¯ρyFyw(p)ρw −1 2Tr[¯ρxFxx(k)][ρyFyw(p)ρw]−1 2[ρxFxy(k)ρy]Tr[¯ρxFxx(p)]. (6.56) Eq. (6.56) constitutes a more friendly expression in pursuit of averaging this specific contribution with respect to the projectile color charge distribution via the MV model, earlier introduced in subsection 2.3.3, which establishes that the distribution of color sources can be reproduced with a Gaussian weight function. This last step is planned to be carried out in a future study that could provide an insightful picture regarding the understanding of the correlations this component can lead to. 124 6 Deciphering the origin of angular correlations in high-energy collisions 6.3 Conclusions and outlook This chapter was intended to showcase that, in high-energy collisions, the appearance of angular correlations between emerging particles can be tested under very simple standards. More precisally, for this purpose it is appropriate to address the computation of the double inclusive gluon production probability within the framework in which one of the objects is dense and the other dilute. In this sense, as the cross-section can be split into three different pieces accounting for three very clear configurations, each of these contributions can be addressed independently in the face of interpreting the origin of the correlations. In particular, the final expression for σ4straightforwardly reveals the backstory behind the physics of correlations without going into state-of-the-art approaches. For its part, a cumbersome algebraic workout leads to a large extent simplified expression for one of the remaining terms making up the cross-section, this way setting the stage for a feasible averaging process over the projectile wave function, what might unveil different evidences on the source of these correlations. To finish off, it must be noted that this work seeks to provide a way out of the results presented in [167], study on which this work is fully based. To be quite specific, when calculating the fully ordered expression for the probability via performing the full symmetrization between the color charge density operators after squaring the amplitude, [167] mismatches the origin of the different terms contributing to σ3– coming from both Σ4and Σ3, what prevents a substantial simplification of the resulting formulae and, consequently, severely complicates the averaging procedure. It will be seen if this novel development status gives renewed insights in this regard. 125 V ´ ıctor Vila P´ erez 126 Summary Heavy-ion collisions provide an accessible way to better understand the physics behind the strong interaction sector of the Standard Model of particle physics, as heavy-ion physics encompasses a wide range region from highly energetic partons described by pQCD to strong interactions between partons at lower scales. In particular, strongly interacting matter undergoes a phase transition from a confined phase to a deconfined Quark-Gluon Plasma under extremely high conditions, which is a very lively topic in the research community. Experimental measurements show that the collimated multiparticle sprays called jets lose considerable energy as they propagate through QGP, referred to as jet quenching, the main research topic of this doctoral thesis. Jet quenching can be used to probe the QGP properties by looking at different observables related to parton energy loss. Indeed, a total comprehension of in-medium jet evolution depends largely upon being capable of capturing jets’ behaviour when they spread through hot and dense matter. It is for this reason that this phenomenon is potentially one of the most versatile experimental tools for characterizing the QGP. The broad kinematical reach of the LHC and the much larger luminosities expected for the near future open the door to both completely redesigned and new jet tools which may then be used to somehow isolate the different scales at different stages of the evolution, including initial stages. Here, a key piece for making sense of the data depends on a sound control of the splitting process within the relevant domain. The problem of elementary parton splittings in fact constitutes an important consideration in this context as is one of the key ingredients that enter the formulation of a Monte Carlo parton shower. In a diagrammatic language, final-state in-medium emissions are described via a classical current representing high-energy particles that act as sources of soft gluons and originate from a fixed position in probability amplitude and complex-conjugate amplitude. Likewise, the interference spectrum off multiple emitters has been performed assuming an instantaneous splitting of the current, giving rise to the well-known antenna radiation pattern, one of the cornerstones upon which this thesis was built. Three of the four research studies carried out in this manuscript are indeed further developments in this issue. The first of these was presented in Chapter 3, where the radiation spectrum off twoparton QCD antennas originated after two hard splittings within a static color-deconfined medium was derived. Strong emphasis was placed on the role of the color coherence phenomenon via considering an additional soft gluon emission when the emitters have already traversed the entire medium, this way opening up the possibility of tracing the coherence history of the whole scenario back. Neither non-eikonal corrections to parton propagation nor finite formation time effects of the splitting processes were taken into consideration throughout the entire derivation. As a warm-up exercise, the derivation of the radiation pattern off a q¯q-antenna in the strict soft limit is revisited since the major issue in this piece runs along the same lines, but it does attempt nonetheless to open up some new insight into the description of a final-state medium-induced partonic cascade. V ´ ıctor Vila P´ erez Right from the very beginning, it is easy to realise that the survival probabilities corresponding to the parton configurations concerning the splittings under consideration will play a significant role in the face of conducting a qualitative discussion in terms of twopoint correlation functions, objects that emerge on a recurring basis. More specifically, writing the probability amplitudes as a function of the survival probabilities of the leading dipoles enables one to find a factorization in the large-Nclimit that allows establishing a clear physical picture concerning the full coherence picture of the presented setup. To be more precise, the main qualitative outcome can be summarized as follows within the limit of large number of colors: color coherence factorizes along the parton shower. In other words, if coherence is preserved after the very first in-medium splitting, the subsequent smaller antennas will radiate coherently too. Conversely, when interferences wash out coherence completely, multiple emissions may be considered as a probabilistic shower of independent branchings. It is worth noting, however, that beyond the large-Nclimit the proof of this factorization is spoiled since subleading n-point correlation functions pop up – what would require a more involved implementation. Additionally, a straightforward derivation of tilted Wilson lines is performed, meant for the cases in which a very energetic quark – or a gluon – is produced at a fixed angle in a hard vertex inside the medium. The objectives of the study carried out in Chapter 4 are in line with those described just above. More precisely, it also looks at a collinear parton splitting inside a colordeconfined medium, but now particular focus is put on studying the medium-modifications that emerge from allowing this splitting to take place at a finite distance within the medium. Unlike the preceding work, where the dipole was assumed to be formed quasiinstantaneously close to the origin of the coordinate system, here the study concentrates on looking further at the formation of the antenna itself, for the moment without considering additional radiative processes afterwards. A key challenge here entails deciphering whether the properties of the antenna are set at the moment of formation or whether those still can undergo modifications over large distances until the end of the medium. The findings point to the importance of both regimes. In particular, the main outcome of this study, namely the emission spectrum in a medium, proves to be factorable as a product of the vacuum cross-section and a function as for the medium-induced modification, this way allowing for an ongoing discrimination of medium effects. This function, named Fmed, encapsulates all information regarding the medium modification factor corresponding to a 1 →2 splitting function. As with earlier works on this subject, it is of great help to map the spectrum of a 1 →2 parton splitting inside a medium onto the kinematical Lund plane, what enables one to conduct an straightforward discussion concerning the different regimes and relevant time-scales arising in the map for in-medium splittings. Especially worth mentioning is the appearance of two regimes of vacuum-like emissions within the medium, meaning that the in-medium splitting function is equal to the one in vacuum in these areas. The project was completed with a numerical study that largely verifies the performed analysis and the generalization of the process under consideration to be valid for arbitrary splitting processes. 128 Chapter 5 is the last piece of this manuscript dedicated to the study of final state effects via giving attention to the relevance of color coherence in the description of the partonic showering within the context of HIC. Specifically, the analysis carried out seeks evidences to design a coherence-modified evolution equation describing the medium-induced partonic cascade. To this end, a partonic scheme not dissimilar from those dealt with already in the thesis is developed. This involves the double inclusive gluon production process taking place inside a medium, with the added difficulty of allowing the splitting to display a finite formation time, in close analogy with the preceding study. Explicitly computing the radiation spectrum associated to this process leads to a cumbersome output involving the convolution of three kernel structures that makes standard discussing methods not quite useful. It is possible, however, to implement some approximations prior to find the exact solution which reveals a more favourable analytical environment in order to embed the notion of color coherence into the original rate equations built upon a Markov-like branching approach. In particular, the novel rate equation would only exhibit a significant modification that would affect the splitting kernel, now comprising a coherence piece which is exactly a simplified estimation of the decoherence parameter ∆med. Preliminary findings point out to a close concordance with baseline studies in this respect. Just to be clear on the inconsistency that implies the sense of attempting to insert coherence effects into a pure Markovian showering picture, certain modifications are being carried out through the branching and broadening pieces as a way to lighten this mismatch, whilst further developments will be required for achieving a more reliable description of a real-life parton emission rate. Finally, even though this thesis is mainly focused on in-depth studies as regards final state effects, initial state effects also take part here. In this vein, Chapter 6, the concluding chapter of this manuscript, falls within the CGC effective theory work frame. Its object is to pursue a holistic explanation about the origin of final-state particle correlations, which may hold a key to disentangle initial and final state effects. To achieve this end, the double inclusive gluon production probability is computed within the dense-dilute context, in which some pieces reveal angular correlations whose origin can be straightforwardly argued under basic principles. The next step, planned for a future analysis, includes performing the medium averaging procedure so as to give a renewed perspective concerning the roots of such correlations. As a final remark, the works presented in this thesis have been devised to both reinforce existing studies on jet evolution in matter and provide new insights in the face of deciphering the puzzle of how jet constituents lose energy in a QCD medium. Achieving a sound knowledge in this respect will lead to a complete understanding of the cascading of a jet, this way allowing for the development of reliable Monte Carlo event generators as well. V ´ ıctor Vila P´ erez 130 Resumen En las ´ultimas d´ecadas se ha generado un inter´es creciente alrededor del estudio de las colisiones de iones pesados, ya que estas brindan una forma completamente nueva de entender la exitosa teor´ıa de las interacciones fuertes en condiciones extremas, lo que se conoce como QCD a alta temperatura y densidad. Esta teor´ıa se someti´o a numerosas pruebas en aceleradores de part´ıculas durante m´as de 40 a˜nos con el objeto de estudiar el comportamiento de los hadrones en el vac´ıo. M´as recientemente, la interacci´on fuerte ha comenzado tambi´en a ser probada en un medio, alcanzando valores cr´ıticos de temperatura y densidad que llevan a una escala de energ´ıa que favorece el deconfinamiento de quarks y gluones. Estas condiciones dan paso a la formaci´on del llamado Plasma de Quarks y Gluones (QGP), un nuevo estado de la materia que puede comportarse de manera inesperada debido a efectos colectivos. De acuerdo con la descripci´on actual de la evoluci´on del universo primitivo, la teor´ıa del Big Bang, 10−5sdespu´es de la explosi´on, una sopa de QGP experiment´o una transici´on de fase a hadrones confinados, la primera pista de una transici´on de QCD. Profundizar y ampliar el conocimiento sobre esta transici´on implica estudiar la estructura interna de la materia en condiciones extremas de temperatura y densidad. Desde una perspectiva te´orica, el desaf´ıo principal al que se enfrenta la comunidad cient´ıfica no es otro que lograr una formulaci´on s´olida alrededor de una teor´ıa cu´antica de campos en el medio utilizando QCD. Sin embargo, esta teor´ıa no es perturbativa en torno a las escalas relevantes, lo que limita el uso de m´etodos anal´ıticos. Una forma de avanzar en el conocimiento de sus propiedades es abordar el QGP como un l´ıquido casi perfecto cuyos ingredientes interact´uan fuertemente. El estudio del plasma se ha establecido durante mucho tiempo como uno de los principales objetivos del Relativistic Heavy Ion Collider (RHIC) en el Brookhaven National Laboratory (BNL) y del programa del Large Hadron Collider (LHC) en el Conseil Europ´een pour la Recherche Nucl´eaire (CERN). Para caracterizar esta nueva fase de la materia en condiciones de laboratorio se manejan distintas probes en el contexto de las colisiones de iones de pesados. Las hard probes (HP) se crean en las primeras etapas de la colisi´on despu´es de los procesos de dispersi´on hard y, por lo tanto, son probes clave para inferir las propiedades del medio. Los jets son hard probes naturales ya que las secciones eficaces de producci´on pueden abordarse exclusivamente dentro del dominio perturbativo de la QCD (pQCD). Mientras los jets se propagan por el medio QCD deconfinado, la dispersi´on m´ultiple que sufren los partones hard que los componen con los componentes del medio da lugar a la modificaci´on de su din´amica y conduce a la p´erdida de energ´ıa de los mismos, com´unmente conocida como jet quenching. Concretamente, el jet quenching se puede usar para probar las propiedades del QGP por medio del estudio de distintos observables relacionados con la p´erdida de energ´ıa de los partones.