Jet quenching for measuring the QCD collectivity temporal structure

INTERNATIONAL DOCTORAL SCHOOL OF THE USC Marcos González Martínez PhD Thesis Jet quenching for measuring the QCD collectivity temporal structure Santiago de Compostela, 2023 Doctoral Programme in Nuclear and Particles Physics TESE DE DOUTORAMENTO JET QUENCHING FOR MEASURING THE QCD COLLECTIVITY TEMPORAL STRUCTURE Autor Marcos González Martínez Director: Carlos Alberto Salgado López Carlota Andrés Casas Titor/a: Carlos Alberto Salgado López PROGRAMA DE DOUTORAMENTO EN FÍSICA NUCLEAR E DE PARTÍCULAS SANTIAGO DE COMPOSTELA Acknowledgments First of all, I want to thank Carlos for being my supervisor since my last year of undergraduate degree until now. Despite always being a very busy person, whenever I needed it he found time to teach me a little bit of the deep knowledge about high energy physics he holds. However, for me the most important quality about Carlos is not related to physics. He has always been really kind to me, considering me a person before a PhD student and worrying about my well-being first and foremost. Working on the thesis during the pandemic was very hard for me, and Carlos’ support has been one of the most important things that allowed me to reach the end. I am also really grateful to my collaborators that have been taking care of me almost since the beginning of my PhD: Carlota Andres, Liliana Apolin´ario and Fabio Dominguez. It is fair to say that they contributed to this thesis more than myself, as I would not have been able to write it without all of the help and support I received from them in these four years. I learned a lot of things from them during these years, even more than from Carlos. Furthermore, I want to emphasize that the nice environment of this small working group was very important to me during the hardest moments of the PhD. I am particularly grateful to Carlota, the co-supervisor of the current thesis and who has worked with me in all of the publications and proceedings that I have done during these years. I want to thank too all of the IGFAE staff that make everything go smoother. The essential work done by management unit provides all of the researchers with great help regarding the bureaucratic procedures, cluster and computers maintenance, assistance with travelling to conferences, financial matters... Without them our lives would be much harder. The people of the QCD group has been very important to me as well during these years. I could enjoy many fantastic discussions with first level researchers that helped me improving my knowledge. I am really glad to have been part of our meetings all this time. Besides the persons that have accompanied me throughout my thesis, I want to thank Cyrille Marquet and the people from the CPHT. My short stay in Paris was one of the best experiences I had thanks to them. Leaving work aside, I am also very grateful to all of the people that made my everyday life more fun. To every single colleague, office mate, coworker, friend... that I ever had and with whom I enjoyed some quality time, thank you. Para rematar, quero agradecer especialmente ´a mi˜na familia, aos meus pais, ao meu irm´an, ´a mi˜na cu˜nada e ´a mi˜na sobri˜na por todo o apoio e cari˜no que me te˜nen dado, son iii as persoas m´ais prezadas para min e ´as que m´ais quero. Pero, por riba de todo, qu´erolle dar as grazas a Luca Devesa, o amor da mi˜na vida. The project that led to the current thesis was supported by Ministerio de Universidades of Spain through the National Program FPU (grant number FPU18/01966). This project has received funding from Ministerio de Ciencia e Innovaci´on of Spain under projects FPA2017-83814-P and PID2020-119632GB-I00; Unidad de Excelencia Mar´ıa de Maetzu under projects MDM-2016-0692 and CEX2020-001035-M; Xunta de Galicia under projects ED431C 2017/07, Centro singular de investigaci´on de Galicia accreditation 2019- 2022 and CIGUS Network of Research Centers; and European Research Council under project ERC-2018-ADG-835105 YoctoLHC. Abstract In this thesis we use the medium-induced gluon radiation spectrum to study the quark gluon plasma generated at the LHC. In particular, we employ a new computational algorithm that exactly resums the QCD potential without truncations — either in the opacity expansion or the perturbative tails. First, we compare the fully resummed spectrum with different analytical approaches, improving our understanding of the physics behind the emission process. Then, we find an efficient way of computing the radiation spectrum in a realistically evolving media described by a hydrodynamical simulation, a crucial step to implement the framework in phenomenology. Finally, we analyze the impact that the propagation during the initial stages of the collision, which is usually neglected, has on jet quenching observables. Keywords: Heavy ion collisions (HICs), quark gluon plasma (QGP), jet quenching, initial stages. v List of Figures 1.1 Schematic representation of eikonal propagation off a highly energetic parton travelling through a medium. x1, . . . , xndenote the positions of the scattering centers the parton interacts with, p1, . . . , pnare intermediate momenta, pis the initial momentum and p′the final one. Figure extracted from [1] under a Creative Commons license, CC-BY 4.0. . . . . . . . . . . 8 1.2 Schematic representation of the different contributions to the dipole cross section at leading contribution. Figure extracted from [1] under a Creative Commons license, CC-BY 4.0. . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3 Schematic representation of a gluon emission off a high energy parton induced by the medium. The parent particle is considered to be eikonal, thus does not lose energy and only experiences color rotations due to interactions with the medium. The daughter gluon, which has energy ωand transverse momentum k, is assumed to be soft, therefore its propagation through the medium is next-to-eikonal. . . . . . . . . . . . . . . . . . . . . 16 1.4 Schematic representation of the soft gluon emission in the amplitude (upper part of the picture) and in the complex conjugated amplitude (bottom part ofthepicture). ................................. 17 2.1 Schematic representation of the three different contributions to the mediuminduced gluon radiation spectrum. On the left side of each picture it is represented the emission in the amplitude and on the right side in the complex conjugatedamplitude. ............................. 24 3.1 Left: full resummed in-medium energy spectrum (solid lines), its asymptotically small energy limit (3.16) (dotted lines) and the GLV first opacity result (dashed dotted lines) for the Yukawa potential plotted as a function of the rescaled gluon energy ω/¯ωc= 2ω/(µ2L) for several values of n0L. Right: same as left panel but employing the HTL collision rate and plotting the spectra as a function of ω/¯ωH c= 2ω/(m2 DL) for different TL values. Figure extracted from [2] under a Creative Commons license, CC-BY 4.0. . 47 x LIST OF FIGURES 3.2 Left plot: HO (dotted lines) and HO+NLO (dashed lines) contributions to the energy spectrum as a function of the rescaled gluon energy ˆx=ω/ωBH employing χ= 3.4 (or n0L= 1) and for several values of the parameter a which sets the matching scale Q2=a Q2 c. Central plots: same as previous case using χ= 17.1 (or n0L= 5). Right plot: same as previous cases with χ= 41.8 (or n0L= 12.2). Figure extracted from [2] under a Creative Commons license, CC-BY 4.0. . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3 All-order in-medium radiation energy spectrum for the Yukawa (solid lines) and HTL (dashed dotted) potentials plotted as a function of the rescaled energy x=ω/¯ωc. Different colors represent different values for n0L: magenta corresponds to n0L= 20 (equivalent to TL = 8), green corresponds to n0L= 12.2 (or TL = 5), and red corresponds n0L= 5 (or TL = 2). Figure extracted from [2] under a Creative Commons license, CC-BY 4.0. . 53 3.4 Medium-induced energy spectrum with all-order resummation of multiple scatterings for the Yukawa collision rate (magenta solid lines) compared to the IOE at HO+NLO accuracy (green dashed) and GLV N= 1 approximation (blue dashed dotted) shown as a function of the rescaled gluon energy x=ω/¯ωcemploying different values of the opacity n0L(or χ). Figure extracted from [2] under a Creative Commons license, CC-BY 4.0. . . . . . 55 3.5 In-medium emission rates for the full resummed numerical evaluation of the spectrum (magenta solid lines) and the IOE at HO+NLO approximation (green dashed) as a function of χfor several values of the rescaled gluon energy ˆx=ω/ωBH: ˆx= 17 (left panel), ˆx= 219 (central panel) and ˆx= 1415 (right panel). Figure extracted from [2] under a Creative Commons license,CC-BY4.0................................ 57 3.6 Asymptotic χ→ ∞ limit of the emission rates for the fully resummed evaluation (magenta solid line) and the IOE at HO+NLO accuracy (green dashed line) as a function of the energy ˆx=ω/ωBH. Figure extracted from [2] under a Creative Commons license, CC-BY 4.0. . . . . . . . . . . 60 3.7 Fully resummed in-medium gluon energy distribution (magenta solid line) compared to the analytical expansion at HO+NLO approximation (green dashed line), the first opacity N= 1 GLV spectrum (blue dashed dotted line), and the low energy limit of the all-order evaluation described in section 3.2 (black dotted line) as a function of the gluon rescaled energy ˆx=ω/ωBH. Figure extracted from [2] under a Creative Commons license, CC-BY4.0. ................................... 62 xi Marcos Gonz´ alez Mart´ ınez 4.1 Top panels: all order in-medium energy spectrum for a Yukawa-type collision rate with ¯ R→ ∞, and χ= 5 (left) or χ= 10 (right) as a function of the rescaled gluon energy ω/¯ωc. Purple curves correspond to the spectra computed along the trajectory whose temperature profile is shown in the inset figure, which was sampled with a central production point over the 0−10% centrality class in √sNN = 5.02 TeV Pb-Pb collisions at the LHC. Blue dashed curves correspond to the static spectrum evaluated employing average values for the medium parameters (see eqs. (4.9) and (4.10)). Bottom panels: ratio of the static spectrum w.r.t. the spectrum computed with the hydrodynamic model. Figure extracted from [3] under a Creative Commons license, CC-BY 4.0. . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2 Top panels: all order in-medium energy spectrum for the Yukawa collision rate for χ= 5, and ¯ R→ ∞ (left panel) or ¯ R= 1000 (right panel) as a function of the rescaled gluon energy ω/¯ωc. Purple curves correspond to the spectra computed along the trajectory whose temperature profile is shown in the inset figure, which was sampled with a central production point over the 0 −10% centrality class in √sNN = 5.02 TeV Pb-Pb collisions at the LHC. Blue dashed curves correspond to the static spectrum evaluated employing the scaling laws (4.19)-(4.21). Bottom panels: ratio of the static scenario w.r.t. the spectrum computed with the hydrodynamic model along the selected trajectory. Figure extracted from [3] under a Creative Commons license, CC-BY 4.0. . . . . . . . . . . . . . . . . . . . . 74 4.3 Top panels: all order in-medium energy spectrum for the Yukawa collision rate for χ= 5, and ¯ R→ ∞(left panel) or ¯ R= 1000 (right panel) as a function of the rescaled gluon energy ω/¯ωc. Purple solid lines correspond to the spectra computed along the trajectory whose temperature profile is shown in the inset figure, which was sampled with a central production point over the 0 −10% centrality class in √sNN = 5.02 TeV Pb-Pb collisions at the LHC. Blue dashed lines correspond to the spectrum evaluated using the static scenario (4.19)-(4.21). The dotted curves correspond to the powerlaw scaling laws (4.23)-(4.25) for t0= 0.1 and several values of α:α= 0.3 (orange), α= 0.5 (green), and α= 1.0 (brown). Bottom panels: ratio of the power-law and static spectra w.r.t. the spectrum evaluated along the path with temperature profile showcased in the inset figure. Figure extracted from [3] under a Creative Commons license, CC-BY 4.0. . . . . . 76 xii LIST OF FIGURES 4.4 Top panels: all order in-medium energy spectrum for the Yukawa collision rate for χ= 5, and ¯ R→ ∞ (left panel) or ¯ R= 1000 (right panel) as a function of the rescaled gluon energy ω/¯ωc. Purple solid lines correspond to the spectra computed along the trajectory whose temperature profile is shown in the inset figure, which was sampled with a central production point over the 0 −10% centrality class in √sNN = 5.02 TeV Pb-Pb collisions at the LHC. Blue dashed lines correspond to the spectrum evaluated using the static scenario (4.19)-(4.21). The dotted curves correspond to the power-law scaling laws (4.23)-(4.25) for α= 0.5 and several values of t0:t0= 0.05 (orange), t0= 0.1 (green), and t0= 0.2 (brown). Bottom panels: ratio of the power-law and static spectra w.r.t. the spectrum evaluated along the path with temperature profile showcased in the inset figure. Figure extracted from [3] under a Creative Commons license, CC-BY 4.0. . 78 4.5 Top panels: all order in-medium energy spectrum for the Yukawa collision rate for χ= 5, and ¯ R→ ∞ (left panel) or ¯ R= 1000 (right panel) as a function of the rescaled gluon energy ω/¯ωc. Purple solid lines correspond to the spectra computed along the trajectory whose temperature profile is shown in the inset figure, which was sampled with a central production point over the 0 −10% centrality class in √sNN = 5.02 TeV Pb-Pb collisions at the LHC. Blue dashed and green dotted lines correspond, respectively, to the spectrum evaluated using the static (4.19)-(4.21) and power-law (4.23)-(4.25) scaling laws. Bottom panels: green dotted (blue dashed) lines represent the ratio of the spectrum computed with the powerlaw (static) scaling laws w.r.t. the spectrum evaluated along the path with temperature profile showcased in the inset figure. Figure extracted from [3] under a Creative Commons license, CC-BY 4.0. . . . . . . . . . . . . . . . 79 4.6 Top panels: all order in-medium energy spectrum for the Yukawa collision rate for χ= 3.3, and ¯ R→ ∞ (left panel) or ¯ R= 462 (right panel) as a function of the rescaled gluon energy ω/¯ωc. Purple solid lines correspond to the spectra computed along the trajectory whose temperature profile is shown in the inset figure, which was sampled with a central production point over the 0−10% centrality class in √sNN = 5.02 TeV Pb-Pb collisions at the LHC with an off-central production point. Blue dashed and green dotted lines correspond, respectively, to the spectrum evaluated using the static (4.19)-(4.21) and power-law (4.23)-(4.25) scaling laws. Bottom panels: green dotted (blue dashed) lines represent the ratio of the spectrum computed with the power-law (static) scaling laws w.r.t. the spectrum evaluated along the path with temperature profile showcased in the inset figure. Figure extracted from [3] under a Creative Commons license, CC-BY 4.0. . 80 xiii Marcos Gonz´ alez Mart´ ınez 4.7 Top panels: all order in-medium energy spectrum for the Yukawa collision rate for χ= 7.6, and ¯ R→ ∞ (left panel) or ¯ R= 2300 (right panel) as a function of the rescaled gluon energy ω/¯ωc. Purple solid lines correspond to the spectra computed along the trajectory whose temperature profile is shown in the inset figure, which was sampled with a central production point over the 20 −30% centrality class in √sNN = 5.02 TeV Pb-Pb collision at the LHC. Blue dashed and green dotted lines correspond, respectively, to the spectrum evaluated using the static (4.19)-(4.21) and power-law (4.23)-(4.25) scaling laws. Bottom panels: green dotted (blue dashed) lines represent the ratio of the spectrum computed with the powerlaw (static) scaling laws w.r.t. the spectrum evaluated along the path with temperature profile showcased in the inset figure. Figure extracted from [3] under a Creative Commons license, CC-BY 4.0. . . . . . . . . . . . . . . . 81 4.8 Top panels: all order in-medium energy spectrum for the Yukawa collision rate for χ= 6.6, and ¯ R→ ∞ (left panel) or ¯ R= 2600 (right panel) as a function of the rescaled gluon energy ω/¯ωc. Purple solid lines correspond to the spectra computed along the trajectory whose temperature profile is shown in the inset figure, which was sampled with a non-central production point over the 0−10% centrality class in √sNN = 5.02 TeV Pb-Pb collisions at the LHC. Blue dashed and green dotted lines correspond, respectively, to the spectrum evaluated using the static (4.19)-(4.21) and power-law (4.23)-(4.25) scaling laws. Bottom panels: green dotted (blue dashed) lines represent the ratio of the spectrum computed with the power-law (static) scaling laws w.r.t. the spectrum evaluated along the path with temperature profile showcased in the inset figure. Figure extracted from [3] under a Creative Commons license, CC-BY 4.0. . . . . . . . . . . . . . . . . . . . . 83 4.9 Left: average temperature profile with standard deviation (black points with vertical lines) sampled from the 0−10% centrality class in √sNN = 2.76 TeV Pb-Pb collisions at the LHC, together with a sample of temperature profiles corresponding to off-central production points (colored dashed lines). Right: average temperature profile with standard deviation (black points with vertical lines) sampled from the 10 −30% centrality class in √sNN = 2.76 TeV Pb-Pb collisions at the LHC together with a sample of temperature profiles corresponding to central production points (colored dashed lines). Inset panels display the full sample of temperature profiles in each centrality class. Figure extracted from [3] under a Creative Commons license,CC-BY4.0................................ 85 xiv LIST OF FIGURES 4.10 Top: all order in-medium energy spectrum for the Yukawa collision rate for χ= 5, and ¯ R→ ∞ (left) or ¯ R= 1000 (right) as a function of the rescaled gluon energy ω/¯ωc. Purple solid lines correspond to the spectra computed along the trajectory whose temperature profile is shown in the inset figure, which was sampled from an event corresponding to the 10−30% centrality class in √sNN = 2.76 TeV Pb-Pb collisions at the LHC. Blue dashed and green dotted lines correspond, respectively, to the spectrum evaluated using the static (4.19)-(4.21) and power-law (4.23)-(4.25) scaling laws. Bottom: green dotted (blue dashed) lines represent the ratio of the spectrum computed with the power-law (static) scaling laws w.r.t. the spectrum evaluated along the path with temperature profile showcased in the inset figure. Figure extracted from [3] under a Creative Commons license,CC-BY4.0................................ 86 4.11 Top: all order in-medium energy spectrum for the Yukawa collision rate for χ= 4.4, and ¯ R→ ∞(left) or ¯ R= 1100 (right) as a function of the rescaled gluon energy ω/¯ωc. Purple solid lines correspond to the spectra computed along the trajectory whose temperature profile is shown in the inset figure, which was sampled over a central event in √sNN = 2.76 TeV Pb-Pb collisions at the LHC. Blue dashed and green dotted lines correspond, respectively, to the spectrum evaluated using the static (4.19)-(4.21) and power-law (4.23)-(4.25) scaling laws. Bottom: green dotted (blue dashed) lines represent the ratio of the spectrum computed with the power-law (static) scaling laws w.r.t. the spectrum evaluated along the path with temperature profile showcased in the inset figure. Figure extracted from [3] under a Creative Commons license, CC-BY 4.0. . . . . . . . . . . . . . . . 87 4.12 Top: all order in-medium energy spectrum for the HTL elastic collision rate with χH= 2, and ¯ RH→ ∞ (left) or ¯ RH= 2700 (right) as a function of the rescaled gluon energy ω/¯ωH c. Purple solid lines correspond to the spectra computed along the trajectory whose temperature profile is shown in the inset figure, which was sampled with a central production point over the 0 −10% centrality class in √sNN = 5.02 TeV Pb-Pb collisions at the LHC. Blue dashed and green dotted lines correspond, respectively, to the spectrum evaluated using the static (4.29)-(4.31) and power-law (4.33)- (4.35) scaling laws. Bottom: green dotted (blue dashed) lines represent the ratio of the spectrum computed with the power-law (static) scaling laws w.r.t. the spectrum evaluated along the path with temperature profile showcased in the inset figure. Figure extracted from [3] under a Creative Commons license, CC-BY 4.0. . . . . . . . . . . . . . . . . . . . . . . . . . 90 xv Marcos Gonz´ alez Mart´ ınez 4.13 Top: all order in-medium energy spectrum for the HTL elastic collision rate with χH= 2.7, and ¯ RH→ ∞ (left) or ¯ RH= 7070 (right) as a function of the rescaled gluon energy ω/¯ωH c. Purple solid lines correspond to the spectra computed along the trajectory whose temperature profile is shown in the inset figure, which was sampled with a non-central production point over the 0 −10% centrality class in √sNN = 5.02 TeV Pb-Pb collisions at the LHC. Blue dashed and green dotted lines correspond, respectively, to the spectrum evaluated using the static (4.29)-(4.31) and power-law (4.33)- (4.35) scaling laws. Bottom: green dotted (blue dashed) lines represent the ratio of the spectrum computed with the power-law (static) scaling laws w.r.t. the spectrum evaluated along the path with temperature profile showcased in the inset figure. Figure extracted from [3] under a Creative Commons license, CC-BY 4.0. . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.1 Diagrammatic representation of the three different cases studied in this work: scenario A: τp=τm= 0 (top panel); scenario B: τp=τm>0 (middle panel); and scenario C: τm> τp= 0 (bottom panel). In the last panel, corresponding to case C, the radiated gluon represented illustrates the new kind of contributions we account for only in this scenario in comparison with A and B. They correspond to gluons radiated at times prior to τm which interfere with the later generated medium. Of course, in this case we also take in account emissions produced inside the medium, which are the only ones considered in the previous two cases A and B. Figure extracted from [4] under a Creative Commons license, CC-BY 4.0. . . . . . . . . . . 97 5.2 All-order in-medium energy distribution with full resummation over multiple scatterings for the three different scenarios analyzed: τp=τm= 0 (green solid lines); τp=τm= 0.1L(blue dashed lines); and τp= 0 and τm= 0.1L(red dashed dotted lines) as a function of ω/¯ωcfor n0L= 5. Different panels correspond to different values of the kinematical constraint ¯ R: ¯ R→ ∞ (left), ¯ R= 5000 (central), and ¯ R= 1000 (right). Figure extracted from [4] under a Creative Commons license, CC-BY 4.0. . . . . . . . . . . 100 5.3 All-order in-medium energy distribution with full resummation over multiple scatterings for the three different scenarios analyzed: τp=τm= 0 (green solid lines); τp=τm= 0.3L(blue dashed lines); and τp= 0 and τm= 0.3L(red dashed dotted lines) as a function of ω/¯ωcfor n0L= 5. Different panels correspond to different values of the kinematical constraint ¯ R: ¯ R→ ∞ (left), ¯ R= 5000 (central), and ¯ R= 1000 (right). Figure extracted from [4] under a Creative Commons license, CC-BY 4.0. . . . . . . . . . . 101 xvi LIST OF FIGURES 5.4 In-medium energy distribution for the GLV limit for τp=τm= 0 (green solid lines); τp=τm= 0.1L(blue dashed lines); and τp= 0 and τm= 0.1L (red dashed dotted lines) as a function of the rescaled gluon energy x= ω/¯ωcfor n0L= 5. Different panels correspond to different values of the kinematical cut-off ¯ R:¯ R→ ∞ (left), ¯ R= 5000 (central), and ¯ R= 1000 (right). Figure extracted from [4] under a Creative Commons license, CCBY4.0. .....................................105 5.5 In-medium energy distribution for the GLV limit for τp=τm= 0 (green solid lines); τp=τm= 0.3L(blue dashed lines); and τp= 0 and τm= 0.3L (red dashed dotted lines) as a function of the rescaled gluon energy x= ω/¯ωcfor n0L= 5. Different panels correspond to different values of the kinematical cut-off ¯ R:¯ R→ ∞ (left), ¯ R= 5000 (central), and ¯ R= 1000 (right). Figure extracted from [4] under a Creative Commons license, CCBY4.0. .....................................106 5.6 In-medium energy distribution for the HO approach as a function of the rescaled gluon energy ω/ωcfor the three different scenarios analyzed: τp= τm= 0 (green solid lines); τp=τm= 0.1L(blue dashed lines); and τp= 0 and τm= 0.1L(red dashed dotted lines). Different panels correspond to different values of the kinematical constraint R:R→ ∞ (left), R= 15000 (central), and R= 3000 (right). Figure extracted from [4] under a Creative Commons license, CC-BY 4.0. . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.7 In-medium energy distribution for the HO approach as a function of the rescaled gluon energy ω/ωcfor the three different scenarios analyzed: τp= τm= 0 (green solid lines); τp=τm= 0.3L(blue dashed lines); and τp= 0 and τm= 0.3L(red dashed dotted lines). Different panels correspond to different values of the kinematical constraint R:R→ ∞ (left), R= 15000 (central), and R= 3000 (right). Figure extracted from [4] under a Creative Commons license, CC-BY 4.0. . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.8 Left: QAA, our proxy for the single-inclusive particle suppression RAA, as a function of pT. Right: w2, our proxy for the high-pTazimuthal asymmetry v2, as a function of pT. In each panel we show the results obtained employing the HO approach for the three different scenarios discussed in the thesis: τp=τm= 0 fm (green solid lines), τp=τm= 1 fm (blue dashed lines); and τp= 0 fm and τm= 1 fm (red dashed dotted lines). All of the results are for the 20 −30% centrality class in √sNN = 2.76 TeV Pb-Pb collisions at the LHC. Figure extracted from [4] under a Creative Commons license,CC-BY4.0................................119 xvii Marcos Gonz´ alez Mart´ ınez 5.9 Left: QAA, our proxy for the single-inclusive particle suppression RAA, as a function of pT. Right: w2, our proxy for the high-pTazimuthal asymmetry v2, as a function of pT. In each panel we show the results obtained employing the BDMPS-Z spectrum with full resummation over multiple scatterings for the three different scenarios discussed in the thesis: τp=τm= 0 fm (green solid lines), τp=τm= 1 fm (blue dashed lines); and τp= 0 fm and τm= 1 fm (red dashed dotted lines). All of the results are for the 20 −30% centrality class in √sNN = 2.76 TeV Pb-Pb collisions at the LHC. Figure extracted from [4] under a Creative Commons license, CC-BY 4.0. . . . . . 121 xviii Extended Abstract QCD has been proved in all kind of kinematic regimes since it was formulated in the last century. Over the last decades one of the most active research areas is the study of QCD under extreme density and temperature conditions, where new states of the phase diagram can be found. In particular, at high energies the quarks and gluons inside the nuclear matter deconfine and create the QGP, a hot and dense medium that filled the early universe a few microseconds after the Big Bang. Nowadays we study the QGP and its properties in the laboratory by employing heavy ion collisions in colliders all around the world, like LHC and RHIC. Highly energetic partons travelling through the QGP will experience modifications w.r.t. the vacuum propagation. These modifications are usually encompassed under the term jet quenching. The main objective of this thesis is to study the QGP employing jet quenching techniques. To be more precise, the jet quenching tool we will employ throughout the whole manuscript is the medium-induced gluon radiation spectrum. In this thesis we analyze the following specific topics. First, we discuss the improvement of the computation of in-medium gluon emission. This radiation spectrum is the building block of a great number of phenomenological studies. However, the difficulties that arise when we try to compute it (having to solve a path integral with imaginary potential) force us to usually employ approximations, such as the Harmonic Oscillator (HO) or the opacity expansion. In recent years, a lot of theoretical effort has been put into improving this computation. In this thesis we make use of a previously developed formalism that evaluates the radiation spectrum with full resummation over multiple scatterings, which is then employed in several original studies. The evaluation of the spectrum with this method requires solving numerically the Schwinger-Dyson recursive equations satisfied by the emission kernel and momentum broadening, instead of computing them directly. This approach has many advantages compared to others, among which we highlight the flexibility to use any elastic collision rate model and/or medium profile. After introducing the method for evaluating the spectrum, we use it to improve the current understanding of the physics behind the emission process. As stated in the previous paragraph, it is common to employ approximations when computing the in-medium radiation spectrum. Different approaches rely on distinct approximations, encoding different physical interpretations of the emission process. By using our full spectrum as a reference scenario w.r.t. which compare several analytical formalisms, we can validate the range of applicability of their assumptions across all energy regimes. We will pay special attention to the low energy region (also known as Bethe-Heitler regime), where our understanding of the physics governing the emission process is more limited. As there is no analytical framework that successfully describes the spectrum in this region, here, for the first time, we derive the low energy limit of the all-order spectrum which approximates the full result with high accuracy in the Bethe-Heitler regime. With these results we are able to achieve a proper physical interpretation of the emission process for all gluon energies. 1 Introduction isolate the effects induced by the medium, which can be used to extract information about its properties. Among all of the available hard probes in this thesis we will focus on jets, as previously stated. Roughly speaking, a jet is a parton which propagates (either in medium or vacuum) at high energy radiating many particles in a cone-shape. These particles later hadronize and are experimentally measured by detectors. Jets are also one kind of hard probe which can resolve the time evolution of the medium [32,33]. Moreover, we have high precision data of vacuum jets measured in p-p collisions which can be used to compare with, so we can accurately isolate medium contributions to extract QGP properties [34, 35]. Due to these reasons, we choose jets as the tool that we employ to investigate the QGP. These jets when traveling trough a dense medium like the QGP experience modifications with respect to vacuum propagation. We use jet quenching as an umbrella term that includes all of the possible effects that jets undergo due to interactions with the medium. Among these modifications here we focus on the ones that make jets lose energy compared to the case where these interactions do not exist. There are two kinds of energy loss processes: radiative energy loss and collisional energy loss. Radiative energy loss refers to the emission of quarks and gluons by a high energy parton which, due to interactions with the medium, go outside the jet cone-shape. This leads to these particles not been measured by the detector, thus resulting in a smaller energy assigned to the jet when comparing w.r.t. the case where we have no medium. On the other hand, collisional energy loss denote parton-medium interactions which result in the medium directly absorbing some of the parton energy. Radiative energy loss is dominant for light quarks and gluons, whereas that collisional energy loss is only relevant for heavy quarks, thus it will not be discussed in this thesis. Jet quenching was first conjectured in 1982 by Bjorken [36] and measured experimentally at RHIC 20 years later [37]. Currently, jet quenching is a well-established hard probe measured in several kinds of conditions. For instance, we have many experimental data about the single-inclusive particle spectra [38–43] and two-particle correlations [44–47]. Moreover, other reflections of jet quenching have been seen in several jet analyses [48–51]. Jet quenching is the main tool that we are going to employ in this thesis to investigate the QGP and its properties. We use the following section 1.3 to introduce some of the jet quenching techniques that we will employ in subsequent chapter of this manuscript. 1.2 Objectives and methodology After this more general part of the introduction we include here, to comply with the regulations for doctoral studies of the University of Santiago de Compostela, the main goals of the thesis and a summary of the methodology that we will employ as two separated sections. 5 Marcos Gonz´ alez Mart´ ınez 1.2.1 Objectives and hypothesis The main assumption in which the current thesis relies on is the fact that we consider that the propagation of high energy partons through the QGP is well-described employing jet quenching techniques. This thesis has three main objectives: •Upgrade our current knowledge about the gluon emission process inside the medium. We do so by comparing a recently developed approach which computes the mediuminduced gluon radiation with full resummation over multiple scatterings to several analytical formalisms that rely on different assumptions regarding the radiation process. •Improve the computation of the medium-induced radiation spectrum by extending the before-mentioned exact formalism beyond the static case, allowing the evolution of the medium to be described by a hydrodynamic simulation. •Analyze the impact that initial stages may have on jet quenching. We achieve this goal by evaluating the in-medium emission spectrum for different initial stages scenarios and using the results to describe simultaneously the single-inclusive particle suppression RAA and the high-pTazimuthal asymmetry v2in every scenario. 1.2.2 Methodology This thesis is focused on jet quenching, more precisely on the computation of the mediuminduced radiation spectrum in the soft limit. The theory we employ to perform the calculations is the before-mentioned pQCD. Moreover, in the following section we properly explain the jet quenching techniques that will be useful throughout the whole thesis. 1.3 Jet quenching As previously stated, jet quenching is an umbrella term that assembles several different processes which have in common the fact that they modify jets when passing through a strongly interacting medium. In this section we describe some basic features of jet quenching, emphasizing the most relevant ones for medium-induced gluon radiation, which is going to be the central part of this thesis. We remark that we will not provide rigorous derivations for most of the results, focusing on physical discussions rather than technical details. Proper deductions of the results that we introduce here can be found in literature, see for instance the following reviews [1,52,53]. 6 1 Introduction Propagation through medium. Eikonal approximation and Wilson lines We start by describing how we treat the propagation of a high energy parton inside a medium like the QGP employing jet quenching techniques. This section will be mainly based on [1] and we refer the reader there for further details. First of all, we need to make several assumptions to be able to analyze this problem analytically in an easy way. The most important ones refer to how we treat the medium. Essentially, we assume that the QGP can be considered like a classical background field A(x), being xthe position in coordinate space, that experiences no recoil due to the parton propagation, i.e, the medium does not feel any modification when particles travel through it. At microscopic level we suppose that the medium is composed by a collection of scattering centers which can interact with our jet. Regarding the traveling parton, we assume that its energy is very large. In fact, we will use the eikonal limit, in which the only modification the particle can experience from interactions with the medium is a rotation of its color, but not longitudinal momentum changes. Hence, the parton will propagate in a straight trajectory, which is described by a Wilson line that we briefly derive in this section. As eikonal is an expression employed in many different fields and formalisms to denote high energy limit but with different connotations in each one of them, let us rigorously state what we mean here by eikonal limit. In the following we are going to employ light cone coordinates, which are defined as x±=1 √2(x0±x3), p±=1 √2(p0±p3),(1.5) being xthe position and pthe momentum. With these variables the scalar product is defined as xµpµ≡x·p=x+p−+x−p+−x·p,(1.6) being the bold coordinates 2D vectors in the transverse plane. In particular this implies that in the massless case p2= 0 →p−=p2/(2p+), a very useful relation we will use. In these light cone coordinates the plus component of xplays the role of time, thus we will denote it by x+≡t. Moreover, the longitudinal momentum p+plays the role of the energy, hence we use p+≡ωfrom now on. In the eikonal limit all components of the momentum can be neglected except the longitudinal one ω. Assuming that the parton is a right mover propagating in the positive direction of x3, its large momentum component is ω≫ |p|and |p| ≫ p−≃0, which can be seen employing p−=p2/(2ω). Thus, in the high energy limit, the minus component of the momentum of the parton is suppressed p−≈0, and hence for the high energy parton the only relevant physics happens locally at x−= 0. This allows us to simplify the dependencies of the background field Aa µ(x)≃Aa µ(t, 0,x)≡Aa µ(t, x) and write pµAa µ(x)≃ ωAa −(t, x) when dealing with field insertions in the Feynman diagrams. 7 Marcos Gonz´ alez Mart´ ınez Now that we have stated what eikonal limit means for us, let us go back to the original problem we want to discuss, the propagation of a high energy parton inside a strongly interacting medium. Consider that we have a trajectory like the one shown in Fig. 1.1, where a quark1travels inside a medium interacting with it an arbitrary number of times n. The parton initially had a given 4-momentum pand, after all of the scatterings, it has a different 4-momentum p′. We assume that these parton-medium interactions are mediated by soft gluons and, hence, the scatterings are elastic and changes in transverse momentum coming from them are of the order of the characteristic medium scale. We are going to describe the parton propagation by computing the resummed Smatrix, that accounts for all possible gluon exchanges between medium and particle Figure 1.1: Schematic representation of eikonal propagation off a highly energetic parton travelling through a medium. x1, . . . , xndenote the positions of the scattering centers the parton interacts with, p1, . . . , pnare intermediate momenta, pis the initial momentum and p′the final one. Figure extracted from [1] under a Creative Commons license, CC-BY 4.0. S= ∞ X n=0 Sn,(1.7) with Snthe Smatrix that takes in account nscatterings. Given that n= 0 is the vacuum scenario (no scatterings with the medium) the first non trivial contribution comes from the n= 1 term. Considering only one scattering in Fig. 1.1 gives us the diagram that we need to evaluate to compute S1. Using standard Feynman rules we can write S1 as [1,20,21] S1=ˆd4x ei(p′−p)·x¯u(p′)igAa µ(x)taγµu(p),(1.8) where pand p′are, respectively, initial and final 4-momentum, xare the spacetime coordinates of the scattering center, u(¯u) is the Dirac spinor (antispinor) of the quark 1The gluon case is completely analogous, only needing to change color factors from fundamental representation to adjoint. 8 1 Introduction (antiquark), tais the color matrix and Aa µis the classical field that encodes the medium information. Using the eikonal limit in which all components of the momentum can be neglected except p+we have p+≡ω≃ω′≡p′+, which highly simplifies the Dirac structure of the problem 1 2Pλ¯uλ(p′)γµuλ(p) = 2pµwhere we have averaged over initial state spin configuration and summed over final state spin, as usual. Moreover, we recall that in the eikonal limit we can simplify the Aa µ(x) dependencies on xand write pµAa µ(x)≃ωAa −(t, x) . With all of this we can express eq. (1.8) as S1= 2ω(2π)δ(ω′−ω)ˆd2xe−i(p′−p)·xˆdt igAa −(t, x)ta,(1.9) here the Dirac delta ensures energy conservation in the plus component of the momentum and we have neglected the minus components of the momentum in phases. Now let us continue with the following element of the scattering matrix S2. Using again Feynman rules in Fig. 1.1 with 2 scatterings we have S2=ˆd4p1 (2π)4d4x1d4x2ei(p1−p)·x1ei(p′−p1)·x2¯u(p′)igAa1 µ1(x1)ta1γµ1i/ p1 p2 1+iϵigAa2 µ2(x2)ta2γµ2u(p), (1.10) where we must integrate over the intermediate momentum p1. Using the usual commutation relation of the gamma matrices and the massless Dirac equation, in the eikonal limit the Dirac structure of the problem is again very easy 1 2Pλ¯uλ(p′)γµ1/ p1γµ2uλ(p) = (2pµ1)(2pµ2) and we only have to perform the momentum integrals. The ones over p+ 1≡ω1 and p1are trivial, as they are the definition of Dirac delta in 1 and 2 dimensions, respectively. The remaining integral over p− 1is not immediate, but it can be computed using basic complex integration techniques ˆdp− 1 2π e−ip− 1(t2−t1) p2 1+iϵ ≃ˆdp− 1 2π e−ip− 1(t2−t1) 2ω1p− 1+iϵ =−i 2ωθ(t2−t1),(1.11) with θ(x) being the step function. Finally, the integrations over the x−components are done exactly as before in the case of S1, and they are Dirac deltas that ensure the energy conservation. Putting all of this together and neglecting again the p−components on phases eq. (1.10) can be written as S2= 2ω(2π)δ(ω′−ω)ˆd2xe−ix·(p′−p)1 2!Pig ˆdt Aa −(t, x)ta2 ,(1.12) where we have used the path order operator Pto deal with the medium fields and the step function ˆdx1dx2. . . dxnθ(x2−x1)θ(x3−x2). . . θ(xn−xn−1) = 1 n!Pˆdxn .(1.13) 9 Marcos Gonz´ alez Mart´ ınez We can compute higher order terms of the Smatrix following the previous steps. The generalization to any number of scatterings is straightforward and it is obvious that, in general, the Smatrix contribution corresponding to have nscatterings with the medium is given by Sn= 2ω(2π)δ(ω′−ω)ˆd2xe−ix·(p′−p)1 n!Pig ˆdt Aa −(t, x)tan ,(1.14) hence, the fully resummed Smatrix is S= 2ω(2π)δ(ω′−ω)ˆd2xe−ix·(p′−p)W(x),(1.15) where we have defined the Wilson line in the fundamental representation as W(x) = Pexp ig ˆdt Aa −(t, x)ta.(1.16) This equation describes the propagation of a high energy quark through a medium with a given field configuration. The only effect that the medium induces in the particle is color rotation, whereas that the amount of energy that the parton loses is negligible. The path followed by the particle travelling though the medium is a straight line trajectory. Beyond eikonal approximation. Propagators We have seen in the previous section that in the eikonal approximation a high energy parton propagating inside a medium follows a trajectory described by a Wilson line. However, if we want to describe processes where this highly energetic parton radiates gluons, we must relax this eikonal approximation. We can think about it in the following way: if the parent parton has huge energy then when it radiates a gluon its energy is not going to change significantly, thus, the gluon energy must be small compared with the parent particle energy and its propagation can not be described by a Wilson line. As our goal is to employ medium-induced radiation to describe energy loss, we have to go beyond the eikonal approximation and properly describe the propagation of emitted gluons through a medium. In this section we are going to briefly comment the key details about the derivation of the propagation in this situation, the final result and its consequences, as doing a formal derivation is a bit tedious and not so physically relevant for our purposes. The main difference that we have with respect to the previous case is that now we can not neglect the subdominant element in the denominator of the integrant in eq. (1.11). Thus, we have to keep it, resulting in a phase factor after the integration is performed ˆdp− 1 2π e−ip− 1(t2−t1) p2 1+iϵ =ˆdp− 1 2π 1 2ω1 e−ip− 1(t2−t1) p− 1−p12 2ω1+iϵ =−i 2ω1 θ(t2−t1)e−ip12 2ω1(t2−t1),(1.17) 10 1 Introduction this modification implies that the integration over transverse momentum p1is also modified, now been a Gaussian instead of a delta2 ˆp1 e−ip1·(x1−x2)e−ip12 2ω1(t2−t1)=ω1 2πi(t1−t2)exp −iω1 2 (x1−x2)2 t1−t2,(1.18) and this is the free Feynman propagator of a particle that goes from spacetime point (x1, t1) to (x2, t2) with ω1playing the role of mass in the usual path integral formulation of Quantum Mechanics, which we denote as G0(x2, t2;x1, t1), where the subscript 0 represents that is the vacuum propagator. As we know, Feynman propagators can be expressed using the path integral formulation, thus G0can be written as [54] G0(x2, t2;x1, t1) = ˆr(t2)=x2 r(t1)=x1Dr(t) exp (iω 2ˆt2 t1 dtdr dt2),(1.19) where r(t) represents all of the possible paths that connect the two endpoints of the trajectory. Therefore, now we have one of these propagators between two consecutive field insertions. In this example we have seen that in the S2case the modification with respect to the eikonal limit is that we have eq. (1.19) connecting A−(t1,x1) and A−(t2,x2), and this will hold for higher orders of the expansion. Consequently, when we resum all of the contributions of the scattering matrix, these propagators between intermediate steps combine, given as result the propagator that connects the endpoints of the trajectory, which essentially can be read as a modification of the Wilson line previously defined S= 2ω(2π)δ(ω′−ω)ˆd2xe−ix·(p′−p)ˆDr(t) exp (iω 2ˆdtdr dt2)W(r).(1.20) This equation describes the propagation of a high energy particle trough a medium when it is allowed to have changes in the transverse position. Now the path followed by the parton is a straight trajectory in the longitudinal direction, described by the Wilson line, plus a Brownian, random motion in the transverse plane, described by the path integral. With this we can define the next-to-eikonal propagator of a particle with energy ωbetween (t1,x) and (t2,y) which replaces the Wilson line when we relax the eikonal limit G(y, t2;x, t1|ω) = ˆr(t2)=y r(t1)=xDr(t) exp (iω 2ˆt2 t1 dtdr dt2)W(r;t2, t1),(1.21) where we are now explicitly specifying the endpoints of the Wilson line. Given the path integral representation that we are using to describe this propagator, it is trivial to realize that it satisfies a simple yet useful composition property G(y, t2;x, t1|ω) = ˆd2zG(y, t2;z, t|ω)G(z, t;x, t1|ω),(1.22) 2Throughout this manuscript we will employ the shorthand ´p=´d2p/(2π)2for integrals over transverse momentum. 11 Marcos Gonz´ alez Mart´ ınez that in many cases allows us to split a propagator in two (or more) propagators to simplify calculations. Even though this formulation in terms of the path integral is rigorous and formally correct, it leads to difficulties when trying to evaluate formulas analytically, as path integrals are very complicated mathematical objects that can only be solved in particular cases (for instance, vacuum and harmonic oscillator) and, accordingly, we need to employ approximations to achieve results. We will see later some of the most wellknown used approximations and recent frameworks that improve classical computations in this regard (for example, by solving the path integral numerically instead of analytically, as we will do in chapter 2). Finally, we finish this section mentioning, for completeness, that the next-to-eikonal propagator Gdefined in eq. (1.21) is the Green’s function of the Schrodinger equation i∂t+∂2 y 2ω+gAa −(t, y)taG(y, t;x,0) = iδ(t)δ(2)(y−x),(1.23) where we used the notation ∂t≡∂/∂t and ∂2 y≡∂2/∂2y. In this formulation the eikonal limit ω→ ∞is obtained by neglecting the term that contains the information of the propagation in the transverse plane, ∂y. Performing this modification, Gnaturally becomes W. Medium averages The standard procedure to compute the in-medium gluon emission spectrum consists on drawing the Feynman diagram of the process, compute its associate amplitude employing the Feynman rules and, finally, square the amplitude to obtain the spectrum. It is important to notice then that in all of the work that we did in previous sections our results are for a given field configuration of the medium A−(t, x). Therefore, when computing the squared amplitude we have to average over all possible medium configurations. These medium averages are one of the main points in the thesis, as we will discuss in detail how to improve their computation in realistic conditions. Assuming that the medium is formed by a set of independent scattering centers described by Gaussian distribution, as commonly done, the two-point correlator function is given by (see [52] and appendix B of [55]) Aa −(t, x)Ab −(t′,y)=g2n(t)δa,bδ(t−t′)e V(x−y),(1.24) where n(t) is the linear density of the medium that has the information of how are distributed the scattering centers along the trajectory and e V(r) is the elastic collision rate in configuration space which contains the microscopic details about the parton-medium interaction. Eq. (1.24) has a straightforward interpretation: the correlator between fields is local in color, time and space, having an isotropic profile in the transverse plane. It is specially relevant the fact that the interaction is instantaneous, which allows us to compute the amplitude for a given field configuration and then average over all possible medium configurations. 12 1 Introduction Regarding the potential e V(r) which describes the parton-medium interactions, in general we prefer to work with it in momentum space, thus we have to employ the Fourier transform to switch the representation from configuration space to momentum e V(r) = ˆq eiq·rV(q).(1.25) At leading order in the coupling constant this elastic collision rate behaves like V(q)∼ 1/q4. This dependence on the momentum implies that for large momentum (at short distances) the collision rate must have the behavior of the Coulomb potential for any realistic interaction model. On the opposite limit, for low momentum q→0 (large distancies) the divergence of the potential should be regulated by the medium, typically at its Debye mass. In this thesis we will use two different models for the collision rate. The first one, the most used by us, is the Yukawa potential, also known as Gyulassy-Wang model [56] VY(q) = 8πµ2 (q2+µ2)2,(1.26) where µis the medium screening mass that regularizes the behavior of the potential at large distances (short momentum). It is related to the Debye mass in a thermal medium, µ2∼m2 D. The second model we will use is the hard thermal loop (HTL) interaction [57], which provide a more precise description of thermal interactions 1 2n(t)VH(q) = g2Ncm2 DT q2(q2+m2 D),(1.27) with Tthe temperature of the medium. Physical observables involve color-less objects, which are built by traces in color space. The simplest of them is the correlation of two Wilson lines Tr W†(x)W(y) Nc =1 Nc Tr exp −ig ˆdt A†(t, x)exp −ig ˆdt A(t, y), (1.28) where we have omitted the color matrices for simplicity. If we had a gluon instead of a quark then the Wilson lines should be in adjoint representation and we have to change the color factor Nc, corresponding to average over initial color configurations, by N2 c−1. Expanding the exponents and taking the terms which only account for one scattering center we get the following leading order contribution Tr W†(x)W(y) Nc≃ P 1 Nc Tr *1 + 1 2(ig)2ˆdt A†(t, x)2 +1 2(ig)2ˆdt A(t, x)2 −(ig)2ˆdt A†(t, x)ˆdt A(t, x)+, (1.29) 13 Marcos Gonz´ alez Mart´ ınez Figure 1.2: Schematic representation of the different contributions to the dipole cross section at leading contribution. Figure extracted from [1] under a Creative Commons license, CC-BY 4.0. which is quadratic in the fields because linear terms vanished due to the color trace. These terms can be schematically interpreted as the two-scattering diagrams shown in Fig. 1.2. The three diagrams illustrated in Fig. 1.2 represent the dipole (pair quark-antiquark, being in the amplitude and conjugated amplitude respectively) interacting twice with the medium. In the first two pictures the transverse dimension of the dipole is not resolved and they are sometimes called contact (or virtual) terms. The last diagram is able to resolve the dipole and these kind of contributions are called real terms. Eq. (1.29) just accounts for the leading contribution in an opacity expansion of the medium to the average (1.28). Computing subleading terms it can be shown that, after some manipulations, the full expression of (1.28) is Tr W†(x)W(y) Nc = exp −CFˆdt n(t)σ(y−x),(1.30) where CFis the Casimir color factor in fundamental representation, 4/3 (3 for gluons in the adjoint) and we defined the dipole cross section σ(r) = ˆq1−e−iq·rV(q).(1.31) This dipole cross section is one of the most important objects, because it allows us to go from diagram amplitudes to physical quantities like spectra. Here it is written in configuration space, but we can also use the Fourier transform to switch to momentum representation σ(q) = −V(q) + (2π)2δ(2)(q)ˆl V(l),(1.32) by doing so we can see explicitly the interpretation in terms of real-virtual contributions discussed in Fig.1.2. The first term in eq. (1.32) represents real contributions to the dipole cross section, whereas that the second one accounts for virtual interactions. 14 1 Introduction soft and single hard scatterings at the same time. This last framework is introduced in detail in chapter 3. In this thesis we employ a new way to compute the BDMPS-Z spectrum numerically exactly, without employing any of the analytical approximations explained in this section. This new method, which was originally introduced in [58], is properly presented in chapter 2. Once that we describe how to compute the in-medium radiation, we will employ this approach for several original studies. First, in chapter 3 we compare the full result with some analytical approaches, checking in that way the validity of the approximations they rely on, which allow us to gain knowledge about the physics behind the emission process. Then, in chapter 4 we extend the computation of the spectrum originally done for the static case in [2, 58] to realistically evolving media, which is crucial to allow the implementation of the framework in phenomenology. Finally, in chapter 5 we use the radiation spectrum to qualitatively analyze the effects that initial stages may have on jet quenching observables. 21 Marcos Gonz´ alez Mart´ ınez 22 2 Medium-induced gluon radiation with full resummation of multiple scatterings Based on the discussion in chapter 1, the evaluation of the general BDMPS-Z spectrum (1.38) is a difficult task, so approximations are usually employed to simplify the procedure. In this chapter we discuss in detail a formalism recently developed that allows to compute the spectrum including a realistic potential with resummation over multiple scatterings. The main idea is to find a numerical solution for the differential equations satisfied by the emission kernel (1.34) and the momentum broadening (1.36) instead of directly solving them, which is a very difficult task in the general case. This framework was originally introduced in Ref. [58]. But, as the formalism plays a crucial role in the rest of the manuscript, which represents the novel work done by the PhD student, we use this chapter to explain the approach, summarizing the most important details. Therefore, Ref. [58] is the guide we use in this chapter and we refer the interested reader there for further details about the method, numerical results of the evaluation of the spectrum, discussions and comparisons with another approaches. When computing the gluon energy distribution without kinematical constraint, which is the integral of the k-differential spectrum over all kspace, this formalism is analogous to the ones presented in [88,89]. The main difference is that here we are able to compute the radiation spectrum differential in k, whereas in [88, 89] only the energy distribution is considered. Also, this approach employs the soft limit as an assumption (to simplify the evaluation of the BDMPS-Z spectrum in the last region of Fig. 1.4), but in [88, 89] Marcos Gonz´ alez Mart´ ınez the authors computed the energy spectrum beyond this limit (which is easy to do if one only cares about the energy distribution). Recently, in [96] it was computed the full k-dependent radiation spectrum beyond the soft limit, however, for the moment the implementation of the method was only achieved for the γ→q¯qsplitting with few numerical results. 2.1 Difficulties of evaluating the BDMPS-Z expression Let us start by explaining in detail which are the main complications we encounter when evaluating the BDMPS-Z medium-induced radiation spectrum (1.38). As we mentioned before, the emission kernel is given by a path integral (eqs. (1.34) and (1.35)) with imaginary potential, which is very difficult to compute in a general case. This is the main reason why, in practice, analytical approaches like the ones we mentioned in the previous section are widely used when calculating the spectrum. Moreover, achieving the solution of the emission kernel for a general medium profile n(t) is quite challenging. Therefore, it is common to employ the so-called “brick” approximation, which relies on considering that the medium is described by a collection of static scatterings centers n(t) = n0during the whole propagation of the parton. This is an strong assumption, because nowadays we know that the QGP is an evolving media which is well-described by hydrodynamic simulations. However, this consideration simplifies the calculations a lot (sometimes it is almost impossible to achieve results without this approximation) and we can use it as a first step to gain some intuition about the most important physics. Consequently, this assumption is widely used, even though we know it is not correct. As the title of this thesis states, here we aim to describe the temporal structure of the QCD collectivity, therefore, we must surpass the brick approximation and be able to evaluate the spectrum for a generic medium profile n(t). The framework we present here allow us to go beyond the brick and evaluate the spectrum in realistic conditions. Figure 2.1: Schematic representation of the three different contributions to the mediuminduced gluon radiation spectrum. On the left side of each picture it is represented the emission in the amplitude and on the right side in the complex conjugated amplitude. The other problem we encounter when evaluating the BDMPS-Z spectrum (1.37) is the fact that the upper limit of time integrals is infinite, consequently, the emission of 24 2 Medium-induced gluon radiation with full resummation of multiple scatterings the gluon can take place anywhere. From a formal perspective this is not a problem at all, everything is well formulated and the resulting spectrum is finite. However, when it comes to numerical computations, things become more intricate. The common approach consists on dividing the whole spectrum in three different contributions, as illustrated in Fig. 2.1. In each picture we are representing the diagram of the amplitude (left) and the corresponding diagram of the conjugated amplitude (right). These three contributions represent the following scenarios: first, the emission of the gluon occurring in the medium for both the amplitude and conjugated amplitude (left picture); second, emission inside the medium for the amplitude and outside the medium for the conjugated amplitude (central picture); and finally, emissions occurring outside the medium for both the amplitude and conjugated amplitude (right picture). Therefore, they are usually called in-in, inout and out-out contributions, respectively. The out-out term, which solely accounts for vacuum radiation, is typically subtracted when analyzing emissions within the medium. This method is useful because it allows for some analytical simplifications in the in-out contribution employing the fact that the final part of the propagation of the gluon in the conjugated amplitude is through vacuum. However, even though these manipulations do not modify the spectrum, which remains finite, when we compute separately the in-in and in-out terms they are both divergent. Thus, what happens is that there are precise cancellations between these two terms, resulting in a finite, non-divergent spectrum. But these cancellations are very difficult to achieve numerically and, hence, the computation time of the spectrum is very large. The new way to compute the spectrum originally proposed in [58] which we use in this thesis overcomes the two limitations: first, it does not need the employment of any further approximations to simplify the path integral (we can even completely ignore the brick assumption, which is done in chapter 4); and, second, the in-in and in-out contributions are merged together in a single term which has a finite integration domain over time. 2.2 Rearranging the spectrum Following the original work done in [58], we consider the emission kernel e Kand momentum broadening Pas propagators which satisfy some particular differential equation that can be solved numerically. By employing these differential equations, we will also be able to perform the time integral over t′in eq. (1.38) analytically, which facilitates the numerical implementation of the method. For convenience, from now on we will work in momentum space. The first step is to write the Schwinger-Dyson type differential equations that describe the evolution of the emission kernel e K(1.35) and the momentum broadening P(1.36) P(t′′,k;t′,q) = (2π)2δ(2)(k−q)−1 2ˆt′′ t′ ds n(s)ˆk′ σ(k′−q)P(t′′,k;s, k′),(2.1) 25 Marcos Gonz´ alez Mart´ ınez e K(t′,q;t, p) = (2π)2δ(2)(q−p)e−ip2 2ω(t′−t)−1 2ˆt′ t ds n(s) ׈k′ σ(q−k′)e−iq2 2ω(t′−s)e K(s, k′;t, p).(2.2) It can be easily checked that both kernel and broadening satisfy these equations just by plugging (1.35) and (1.36) into (2.2) and (2.1), respectively. Given that in these equations the time dependence on t′is explicit, we can employ them to analytically perform the integration over t′in eq. (1.38). Replacing eqs. (2.1) and (2.2) into eq. (1.38) we get ωdI dωd2k=2αsCR (2π)2ω2Re ˆ∞ 0 dtˆ∞ t dt′hk2e−ik2 2ω(t′−t) −1 2ˆt′ t ds n(s)ˆpk1 p·ke−ik2 2ω(t′−s)σ(k−k1)e K(s, k1;t, p) −1 2ˆ∞ t′ ds n(s)ˆpk1 p2e−ip2 2ω(t′−t)σ(k1−p)P(∞,k;s, k1) +1 4ˆt′ t ds1n(s1)ˆ∞ t′ ds2n(s2)ˆpqk1k2 p·qe−iq2 2ω(t′−s1) ×σ(q−k1)σ(k2−q)e K(s1,k1;t, p)P(∞,k;s2,k2)i,(2.3) the first term accounts for the vacuum contribution (is what we called out-out term before), thus we neglect it from now on, as we are only interested in analyzing mediuminduced radiation. Changing the order of integration and performing the integral over t′ for the rest of the terms we get ωdI dωd2k=2αsCR (2π)2ωRe ˆ∞ 0 dtˆ∞ t ds n(s)ˆpk1 ip·k k2σ(k−k1)e K(s, k1;t, p) −ˆ∞ t ds n(s)ˆpk1 i e−ip2 2ω(s−t)σ(k1−p)P(∞,k;s, k1) +1 2ˆ∞ t ds1n(s1)ˆ∞ s1 ds2n(s2)ˆpqk1k2 ip·q q2e−iq2 2ω(s2−s1)−1 ×σ(q−k1)σ(k2−q)e K(s1,k1;t, p)P(∞,k;s2,k2)i.(2.4) Here we can combine the first term and the contribution without phase of the last term employing eq. (2.1); and the part which has phase with the second term using eq. (2.2) reaching the following result for the rearranged spectrum ωdI dωd2k=2αsCR (2π)2ωRe ˆ∞ 0 dtˆ∞ t ds n(s)ˆpql ip·l l2−q q2(2.5) ×σ(l−q)e K(s, q;t, p)P(∞,k;s, l). 26 2 Medium-induced gluon radiation with full resummation of multiple scatterings Now we change the integration order in tand sand, since n(s) is non-zero only when we are inside the medium 0 < s < L, the whole contribution of the spectrum vanishes for any other svalue. The same thing happens in the momentum broadening (see eq. (1.36)), thus we can take Las its endpoint. The final result is then ωdI dωd2k=2αsCR (2π)2ωRe ˆL 0 ds n(s)ˆs 0 dtˆpql ip·l l2−q q2(2.6) ×σ(l−q)e K(s, q;t, p)P(L, k;s, l). With these changes we have already solved one of the problems we had when evaluating the spectrum: now the integration domain of both sand tis finite, and as we have no separation between in-in and in-out terms we do not have to worry about keeping the interference between both contributions with huge numerical precision, which is highly costly in terms of computational power and slows down the evaluation of the spectrum. Moreover, this new expression of the BDMPS-Z medium-induced radiation spectrum has another advantage regarding the numerical evaluation: there is one explicit σfactor in eq. (2.6). We mentioned in section 1.3 that, for any realistic parton-medium interaction model, the collision rate behaves like the Coulomb one at large momentum p,V(p)∼ 1/p4. When we translate this in terms of the dipole cross section, the behavior is exactly the same, σ(p)∼1/p4at large p. This fact ensures that both integrals over qand l are convergent and we have no vacuum contributions. Also, when we take Pand e Kas propagators, this feature provides an initial condition which can be evolved. 2.3 Employing differential equations to compute the in-medium radiation spectrum Now that we have reorganized the spectrum in a nicer way (2.6), the next step done in [58] is to explain how to compute it exactly by solving the differential equations that satisfy e Kand P. We start by realizing that the momentum broadening Psatisfies the following differential equation (which can be easily derived by differentiating (2.1)) ∂τP(τ, k;s, l) = −1 2n(τ)ˆk′ σ(k−k′)P(τ, k′;s, l),(2.7) with initial condition P(s, k;s, l) = (2π)2δ(2)(k−l).(2.8) For convenience, instead of solving this differential equation we define ϕ(τ, k;s, q) = n(s)ˆll l2−q q2σ(l−q)P(τ, k;s, l),(2.9) 27 Marcos Gonz´ alez Mart´ ınez thus, employing (2.7) and (2.8), it is obvious that ϕ(τ, k;s, q) satisfies the same differential equation as P∂τϕ(τ, k;s, q) = −1 2n(τ)ˆk′ σ(k−k′)ϕ(τ, k′;s, q),(2.10) having as initial condition ϕ(s, k;s, q) = n(s)k k2−q q2σ(k−q).(2.11) Therefore, once that we solve this differential equation, we have a solution for the broadening factor P. Moving on to the emission kernel, we realize that it satisfies the following differential equation (again, it is easy to check that the derivative of (2.2) gives this result) ∂te K(s, q;t, p) = ip2 2ωe K(s, q;t, p) + 1 2n(t)ˆk′ σ(k′−p)e K(s, q;t, k′),(2.12) which has as initial condition e K(s, q;s, p) = (2π)2δ(2)(q−p).(2.13) We define now ψ(s, k;t, p) = ˆq ϕ(L, k;s, q)e K(s, q;t, p),(2.14) hence, employing (2.12), it is easy to see that this operator satisfies the following equation ∂tψ(s, k, t, p) = ip2 2ωψ(s, k, t, p) + 1 2n(t)ˆk′ σ(k′−p)ψ(s, k, t, k′),(2.15) with initial condition ψ(s, k, s, p) = ϕ(L, k;s, p),(2.16) this equation may have some problems when solving it numerically, due to the first term in the r.h.s. of eq. (2.15), which causes oscillations. Switching to interaction picture we can alleviate the computational effort needed to achieve the solution ψI(s, k;t, p) = eip2 2ω(s−t)ψ(s, k;t, p).(2.17) Consequently, the differential equation we have to solve is ∂tψI(s, k;t, p) = eip2 2ω(s−t)1 2n(t)ˆk′ σ(k′−p)e−ik′2 2ω(s−t)ψI(s, k;t, k′),(2.18) having as initial condition ψI(s, k;s, p) = ϕ(L, k;s, p).(2.19) 28 2 Medium-induced gluon radiation with full resummation of multiple scatterings Taking all of these considerations in account, we can rewrite the full spectrum (2.6) in the following way ωdI dωd2k=2αsCR (2π)2ωRe ˆL 0 dsˆs 0 dtˆp ie−ip2 2ω(s−t)p·ψI(s, k;t, p).(2.20) Formally the problem is already well formulated: to evaluate the spectrum we start by computing ϕsolving its differential equation, which then is employed as initial condition for the resolution of the differential equation that allows us to get ψI. Finally, once that we have computed ψI, we plug the result in eq. (2.20) and perform the remaining integrations. However, some of these integrals can be evaluated analytically, thus we are going to continue manipulating the spectrum to further simplify the problem before computing it numerically. In practice, for the majority of cases of interest we encounter the direction of momentum kis not relevant, therefore, we can write everything as function of its magnitude kand integrate over its direction. This allows us to employ the rotational symmetry to perform the angular integrals analytically. Moreover, it is convenient to rescale all of the variables to make them dimensionless, as it is more suitable for numerical evaluations. We rescale time variables with the length of the medium Lin the following way s→Ls and dummy momentum variables as p→p2ω/L p. The transverse momentum of the emitted gluon is rescaled through the typical transverse momentum transfer µ, which is usually set to the Debye mass of the medium, and the relevant scale to rescale the gluon energy is the characteristic gluon frequency ¯ωc=µ2L/2. The procedure to perform the angular integral and rescale the variables of the spectrum does not provide any relevant information to our discussion and it was already done in [58], thus, to facilitate the reading we skip the intermediate steps and write directly the final result. We referrer the interested reader to Ref. [58] to learn about the details. The expression of the full medium-induced gluon radiation spectrum we use to compute the numerical results is xdI dxdκ2=αsCR π2Re ˆ1 0 dsˆs 0 dtˆ∞ 0 dp ip e−ip2(s−t)fx(s, κ;t, p),(2.21) where we defined ψI(s, k;t, p) = p p2˜ ψI(s, k;t, p),(2.22) fx(s, κ;t, p) = 1 2µ2Lˆ2π 0 dθκ 2π˜ ψI(sL, µκ;tL, pp2ω/L),(2.23) gx(τ, l;s, q) = 2ω˜ ϕ(τL, lp2ω/L;sL, qp2ω/L).(2.24) The function fxobeys the following differential equation ∂tfx(s, κ;t, p) = 1 2˜n(t)Lˆ∞ 0 dq 2πhq˜ V1(q, p;x)fx(s, κ;t, p) −e−i(q2−p2)(s−t)p˜ V2(q, p;x)fx(s, κ;t, q)i,(2.25) 29 Marcos Gonz´ alez Mart´ ınez having as initial condition fx(s, κ;s, p) = 1 xgx(1, κ/√x;s, p),(2.26) and the other function gxsatisfies ∂τgx(τ, l;s, p) = −1 2˜n(τ)Lˆ∞ 0 qdq 2π˜ V1(l, q;x) [gx(τ, l;s, p)−gx(τ, q;s, p)] ,(2.27) with the initial condition given by gx(s, l;s, p) = 1 2˜n(s)Lh˜ V1(l, p;x)−p l˜ V2(l, p;x)i.(2.28) Here we have also defined the following integrals of the potential V1(k, q)≡V1(|k|,|q|) = ˆ2π 0 dθkq 2πV(k−q),(2.29) V2(k, q)≡V2(|k|,|q|) = ˆ2π 0 dθkq 2πcos θkq V(k−q),(2.30) and the rescaled versions of Viand n(t) ˜ Vi(l, q;x) = 2ω LVi lr2ω L, qr2ω L!,and ˜n(t) = n(tL).(2.31) As we will see, for the models of the collision rate that we consider in this thesis the angular integrals in Vican be done analytically, further simplifying the numerical labor needed. Those models, eqs. (1.26) and (1.27), are the most commonly employed in phenomenological studies. Finally, the dimensionless variables that we will employ to evaluate and plot the spectrum are κ2=k2 µ2, x =ω ¯ωc =2ω µ2L.(2.32) 2.4 Energy distribution In several situations we are more interested in the medium-induced energy distribution rather than the transverse momentum dependent spectrum. It is computed simple by integrating the k-differential spectrum over k ωdI dω=ˆω 0 dk k ˆ2π 0 dθkωdI dωd2k,(2.33) 30 2 Medium-induced gluon radiation with full resummation of multiple scatterings 1 2n V1(q, p) = g2NcT 1 |p2−q2|−1 p(p2+q2+m2 D)2−4p2q2!,(2.58) 1 2n V2(q, p) = g2NcT 2pq p2+q2 |p2−q2|−p2+q2+m2 D p(p2+q2+m2 D)2−4p2q2!,(2.59) where we have computed the angular integrations through usual contour techniques. Equivalently, their rescaled versions (2.31) are given by 1 2˜n˜ V1(q, p;x) = g2NcT 1 |p2−q2|−1 p(p2+q2+ 1/x)2−4p2q2!,(2.60) 1 2˜n˜ V2(q, p;x) = g2NcT 2pq p2+q2 |p2−q2|−p2+q2+ 1/x p(p2+q2+ 1/x)2−4p2q2!.(2.61) We notice here that the outcome of the integration of the HTL collision rate yields divergent results for p=q. However, when solving the differential equations and initial conditions the divergences always cancel, being the final result finite. Even so, we must take special care in numerical calculations to not evaluate the momentum space points with divergences and discontinuities. Let us consider, for instance, the initial condition of the gxfunction. Introducing (2.60) and (2.61) in (2.28) we get gx(s, l;s, p) = g2 sNcTL 2l2"sgn(l−p) + −l2+p2+ 1/x p(l2+p2+ 1/x)2−4l2p2#,(2.62) which has an evident discontinuity for p=l. Nevertheless, the discontinuity does not generate singularities in further steps of the evaluation, and in actual numerical computations we set the discontinuous part to 0. This initial condition is integrated over l2 between l= 0 and l→ ∞. A careless look at eq. (2.62) would tell us that gx(s, l;s, p) behaves like 1/l2in those endpoints, but if we analyze it in more detail we realize that for those two particular points the term inside brackets goes to 0. Then, even though the integration of gx(s, l;s, p) seems to be problematic, the final result is actually convergent. After computing the initial condition for gxthe next step is to evolve it through the correspondent differential equation (2.27). Similar to our previous encounter, we face a related issue in this equation as ˜ V1(l, q;x) exhibits a divergence when l=q. However, it is worth noting that this singularity is not problematic because, at this point, the term enclosed in brackets vanishes. Consequently, the integrant becomes discontinuous at l=q, and in practice we set its value to the average between the left-handed and right-handed limits. Finally, gxis used as initial condition in the differential equation that describes fx(2.25), which also presents divergences at p=qfor both ˜ V1and ˜ V2. Nevertheless, when considering the sum inside the brackets, an exact cancellation occurs 37 Marcos Gonz´ alez Mart´ ınez between the divergent terms. Naturally, this result introduces a new discontinuity at that specific point, which we address in the same manner as we did for gx. Regarding the parameter dependence of the full spectrum employing the HTL potential, we can see in eq. (1.27) that the collision rate depends on the temperature of the thermalized medium Tand the Debye screening mass mD, which takes the role of µ. Therefore, it may seem that we have one more parameter than before, as with the Yukawa model we had dependencies on n,Land µ(2.57). However, the number of parameters is the same for both approaches, as when introducing the HTL collision rate in the spectrum the dependence on the medium density nis replaced by T(2.60)-(2.61). Following (2.57) we define the next quantities as the free parameters we will use for numerical evaluations TL , ¯ωH c=1 2m2 DL , and ¯ RH= ¯ωH cL . (2.63) 38 3 Full spectrum vs. analytical approximations: the role of multiple scatterings To comply with the regulations for doctoral studies of the University of Santiago de Compostela we include here the full information of the article on which this chapter is based: Marcos Gonz´ alez Mart´ ınez From soft to hard radiation: the role of multiple scatterings in mediuminduced gluon emissions, J. High Energ. Phys. 03 (2021) 102 Authors Carlota Andres,aFabio Dominguez,bMarcos Gonzalez Martinez,b aLIP, Av. Prof. Gama Pinto, 2, P-1649-003 Lisboa, Portugal bInstituto Galego de F´ısica de Altas Enerx´ıas (IGFAE), Universidade de Santiago de Compostela, E-15782 Santiago de Compostela (Galicia-Spain) PhD Student Contribution Active participation in all the calculations presented in the paper, as well as in the discussions, meetings and writing of the paper. All of the authors have the same contribution to the publication. Chapter of the thesis in which it is used: 3 Journal and Article Information Journal name: Journal of High Energy Physics Publisher: Springer ISSN: 1029-8479 (print) Year of publication: 2021 DOI:https://doi.org/10.1007/JHEP03(2021)102 Impact factor in 2021: 6.376 Reproduced in this thesis with standard author permissions from Journal of High Energy Physics and Springer (articles distributed under the Creative Commons Attribution license CC-BY-4.0) In this chapter, we study the different regimes of the gluon energy distribution employing numerical results of the spectrum with full resummation over multiple scatterings in distinct approximations (namely, Yukawa and HTL). The evaluation of the all-order spectrum will be compared with several analytical approximations in the regions where they can be applied. This comparative analysis allows us to uncover the underlying physics governing the radiation process, as different approximations rely on distinct assumptions, thus the closest one to the full result will be the framework which better encodes the physical description of the gluon emission. The findings of this chapter are published in [2]. Essentially, we can identify three different regions in the spectrum regarding the energy of the radiated gluon: low, medium and high energy regimes. Our results will corroborate that the large energy regime is well described by the single hard scattering 40 3 Full spectrum vs. analytical approximations: the role of multiple scatterings approximation, as it was already checked in [58]. In the intermediate energy region we will show explicitly that accounting for multiple scatterings is essential, as the GLV result overpredicts our full spectrum and the IOE approach, whose leading order term is the HO approximation, describes it with high precision. Finally, we will derive the correct ω→0 limit of the BDMPS-Z spectrum. With this result we are able to find, for the first time, the effect of multiple scatterings in the ultra soft region of the spectrum (also called Bethe-Heitler regime), which results in a multiplicative suppression factor with respect to the single radiation. As our goal here is just to compare different theoretical frameworks without describing experimental data, for convenience, we decide to perform the comparisons in the simplest scenario, the energy distribution without kinematical constraint ¯ R→ ∞. The computation of this spectrum employing the GLV approximation and the full resummation over multiple scatterings is done following the procedure explained in the previous chapter. 3.1 Energy regimes and energy scales We are going to use this short section to explain in detail the different energy regions of the energy distribution and which are the scales that separates them. The number of scatterings a radiated gluon is expected to encounter in a medium characterized by a length Land a mean free path λis of the order of L/λ. For every scattering the gluon may acquire a transverse momentum of the order of the typical momentum transfer ˜µcoming from the scattering center, being larger momentum transfers unusual. Then, gluons which have not undergone an uncommon hard scattering with the medium have a transverse momentum between ˜µ2≲k2≲L˜µ2 λ.(3.1) It is also reasonable to assume that medium length represents an upper limit for the value of the formation time of the gluons (1.39) tf∼2ω/k2. Therefore, gluons with an energy ω≳˜µ2L2 2λ≡ωc,(3.2) must have experienced one or more momentum transfers much bigger than the typical one ˜µ. Consequently, during the gluon emission at least one hard scattering occurred. This single hard scattering dominates the whole emission process, as is well illustrated, for instance, in Fig. 2 of Ref. [98]. Hence, in the high energy limit of the spectrum the opacity expansion at first order, which only encodes the information of one single hard scattering, gives a good description of the spectrum with full resummation over multiple scatterings, as it was already checked in [58]. Considering the opposite scenario, if the gluon formation time is smaller than the mean free path λ, then it is impossible to undergo two or more scatterings during the 41 Marcos Gonz´ alez Mart´ ınez emission process. Thus, gluons which possess an energy ω≲1 2˜µ2λ≡ωBH ,(3.3) are also expected to be well described considering just one scattering. This region of the spectrum is called Bethe-Heitler (BH) regime [99]. We are left then with the intermediate energy region of the spectrum ωBH < ω < ωc where multiple soft scatterings are expected to dominate the gluon propagation. Therefore, in this regime the HO approximation and extensions like the IOE should provide a good description of the full medium-induced radiation spectrum. We can relate the parameters from this section with the ones we defined in chapter 2 in the following way by introducing a constant factor of order 1 C: ˜µ2=Cµ2and λ=C/n0. With this choice we can now establish a correspondence between the previous fundamental parameters of the spectrum n0Land ¯ωc, and the two energy scales we defined in this section employing the following relations ¯ω2 c=ωBH ωc C2,and (n0L)2=C2ωc ωBH .(3.4) For a general case the energy distribution is perfectly described just by using two dimensionless combinations of the energies ω,ωcand ωBH, as we showed in chapter 2 (in the ¯ R→ ∞ limit the kinematic constraint is not a parameter anymore, thus we only need two of the three quantities defined in eq. (2.57) to determine the spectrum). This important feature will be employed in section 3.3, first defining the energy scales in terms of the parameters used in the HO approach and then setting a relation between the parameters employed in both formalisms, HO approximation and fully resummed spectrum. 3.2 Single scattering regimes In this section we will analyze the asymptotic energy limits of the spectrum, focusing on the regimes where one scattering should provide a good description of the full result. We already know that the high energy region is well described by the GLV approach, thus, here we will pay special attention to the BH regime. Nonetheless, the derivation we follow here can be applied to both limits, hence we are able to provide analytical proof, not just comparing numerical results, that the N= 1 order in opacity expansion corresponds to the high energy limit of the spectrum with full resummation over multiple scatterings. As our goal is to check if one single scattering is enough or not to describe the emission process in the low and large energy limits, the approach we follow is to derive the first opacity contributions N= 1 and N= 2, and see explicitly if the former dominates the latter in the asymptotic limits of the energy distribution. Different terms in the opacity expansion are easily build by replacing iteratively the recursive equation of the emission kernel (2.2) into the complete spectrum (2.6) (the ¯ R→ ∞limit is easily obtained 42 3 Full spectrum vs. analytical approximations: the role of multiple scatterings by integrating (2.6) over all kspace. By doing so, the contribution of the momentum broadening vanishes, as we discussed in chapter 2. We also notice that, since e Kdoes not depend on l, the integral over lof the second term in brackets is zero). With this method the first two contributions N= 1,2 yield ωdImed dωN=1 =2αsCR ωn0Re ˆL 0 dsˆs 0 dtˆpq ip·q q2σ(q−p)e−ip2 2ω(s−t),(3.5) ωdImed dωN=2 =2αsCR ωn0Re ˆL 0 dsˆs 0 dtˆpql ip·q q2σ(q−l)e−il2 2ω(s−t) ×−n0 2ˆs t ds′σ(l−p)e−ip2−l2 2ω(s′−t).(3.6) Now let us focus on the integration over transverse momentum pin the second term N= 2. Replacing the dipole cross section by the collision rate through eq. (1.32) we get the following expression for the integral ˆp pσ(l−p)e−ip2−l2 2ω(s′−t)=ˆp V(l−p)l−pe−ip2−l2 2ω(s′−t).(3.7) The r.h.s. phase factor becomes trivially 1 for p−lsmall enough, therefore, we are remaining with the integral over pof (l−p)V(l−p), which vanishes due to symmetry. On the other hand, if pis very big the phase factor oscillates rapidly and its contribution to the integration is negligible, thus the integral is also 0 in this case. Hence, in the asymptotic limits we are interested in the whole contribution of the phase factor can be considered as imposing a lower cut-off in the momentum integration and approximate ˆp pσ(l−p)e−ip2−l2 2ω(s′−t)≃lˆq2>M2 V(q)≡lΣM2,(3.8) where M2is the undetermined IR cut-off. With this consideration the second opacity contribution is then given by ωdImed dωN=2 ≃2αsCR ωn0Re ˆL 0 dsˆs 0 dtˆql il·q q2σ(q−l)e−il2 2ω(s−t) ×−n0 2ˆs t ds′ΣM2.(3.9) In the large energy limit ωis huge, then the phase factor in the pintegral (3.7) will be one unless pis also very large. This sets an approximate value for the cut-off M2∼2ω/L, as if p2is much smaller than that factor the integration is zero by symmetry. In this case the Σ factor we defined (3.8) only has access to the high momentum tail of the potential. We discussed already several times that in this limit the collision rate must 43 Marcos Gonz´ alez Mart´ ınez behave like V(q)∼µ2/q4for any realistic parton-medium interaction model. Thus, for this asymptotic case we have Σ = ˆq2>M2 V(q)∼ˆ∞ M2 2qdqµ2 q4∼2µ2 M2=µ2L ω,(3.10) where we ignored most of constant numerical factors for simplicity. Therefore, we have explicitly shown that the second term in the opacity expansion has an additional factor 1/ω compared to the N= 1 contribution, which suppress it for large energies. Higher order terms in the expansion would be suppressed by additional powers of the energy ω, resulting in the GLV approximation N= 1 first opacity result being the asymptotic limit of the full spectrum for large gluon energies ω. On the other hand, in the small energy limit the phase factor we have in (3.7) has rapid oscillations unless (p−l)2is also very small, resulting in a negligible contribution to the integral. Therefore, the contribution of this factor is to cut some possible divergence when we have no-momentum transfer. As we have to keep the phase in the first line of (3.9), now the IR cut-off is set to M2∼l2. If there is no divergence in the IR sector, then M2is no longer necessary and the cut-off can be ignored M2= 0, in which case Σ is just the total cross section of the collision. The interpretation of these results can be made in terms of real-virtual cancellations. We remind that the two terms of the dipole cross section (1.32) represent the contribution of real and virtual terms to the scattering, respectively. For the high energy limit the cancellation between both contributions is almost exact, resulting in single hard scatterings controlling the radiation probability, whereas that subsequent soft scatterings do not play a role in the emission process and just contribute to momentum broadening. In the opposite limit, for small gluon energies the cancellation between both contributions does not occur. The restriction we discussed earlier based on formation time arguments, which limits the number of scatterings the gluon can undergo, only affects the real terms. In contrast, virtual contributions remain unconstrained. This results in real scatterings being limited just to small transverse momentum transfers, but virtual contributions are not restricted at all. As it is common in these kind of calculations, it is possible to formally resumme higher order contributions, resulting in an exponentiation of the virtual terms. Hence, it can be interpreted as a probability factor which encodes the probability of not experiencing further scatterings. Taking in account the exponentiation of the virtual contributions of higher order terms in the opacity expansion N= 3,4. . . the fully resummed spectrum for low energies is given by ωdImed dωω→0 =2αsCR ωRe ˆL 0 ds n0ˆs 0 dtˆpq ip·q q2σ(q−p)e−ip2 2ω+1 2n0Σ(p2)(s−t),(3.11) comparing this equation with eq. (3.5), it is evident that it is simply the N= 1 result multiplied by a no-scattering probability factor. 44 3 Full spectrum vs. analytical approximations: the role of multiple scatterings These probability factors that suppress the spectrum for low energies were already present in sections 5 and 6 of Ref. [75]. That reference aims to compute arbitrary orders in the opacity expansion rather that analyzing different energy regimes, but it can be shown that the “incoherent limit” presented there, characterized by L→ ∞ and n0Lfixed, can be resummed giving an expression similar to our low energy limit (3.11). Taking the x→0 limit, with xgiven by (2.32), it can be proved that both cases are equivalent. To conclude our study of the small-ωlimit of the spectrum, we provide now numerical comparisons between the recently derived eq. (3.11) and the spectrum with full resummation over multiple scatterings, evaluated as we discussed in chapter 2. We start by simplifying the low energy result, performing all of the analytical integrals we can before computing it numerically. Given the dependence of (3.11) on q,sand t, we can perform independently the integration over transverse momentum qand time variables sand t. Starting with q S(p2) = ˆq p·q q2σ(q−p) = ˆq1−p·q q2V(q−p),(3.12) here we employed again eq. (1.32) to express the dipole cross section in terms of V. As the collision rate V(q) is only function of q2(see (1.26) and (1.27)), V(q) = ˆ V(q2), we can compute the angular integral S(p2) = 1 (2π)2ˆ2π 0 dθˆ∞ 0 dq(q−pcos θ)ˆ Vp2+q2−2pq cos θ =1 4πˆ∞ p2 duˆ V(u) = Σ(p2),(3.13) where, to ensure the convergence of the remaining integral over transverse momentum at large p, Σ must be taken by its full expression, even if there is no IR divergence in the potential. Here we used the previous definition of Σ Σ(M2) = ˆq2>M2 V(q) = 1 (2π)2ˆ2π 0 dθˆ∞ M2 dq q ˆ V(q2) = 1 4πˆ∞ M2 duˆ V(u).(3.14) Regarding the time integrals, for the brick approximation we are using they can be easily performed, as we just have to integrate exponentials T(p2) = ˆL 0 dsˆs 0 dt i e−ip2 2ω+1 2n0Σ(p2)(s−t) =L p2 2ω−i 2n0Σ(p2)−i hp2 2ω−i 2n0Σ(p2)i2"e−ip2 2ω+1 2n0Σ(p2)L−1#,(3.15) we should notice here that in the ω→0 limit the second term is subdominant compared with the first one (it has an extra ωpower), hence we can just neglect it. 45 Marcos Gonz´ alez Mart´ ınez Plugging these results together in the equation of the spectrum (3.11) yields ωdImed dωω→0 =2αsCR ωRe n0ˆp iΣ(p2)T(p2),(3.16) where we just need to perform numerically the last remaining integral over p. To continue the evaluation we have to specify a collision rate model, so we can compute Σ. As in the previous chapter, we are going to employ the same models for the potential, Yukawa-type and HTL. Regarding the Yukawa collision rate (1.26), it is IR safe for zero momentum transfer q=0. Integrating it to obtain Σ gives ΣY(M2) = 2µ2 M2+µ2,(3.17) and we clearly see we can take M= 0, as there is no divergence, and employ ΣY= 2 in the asymptotically small energy limit. With respect to the HTL case (1.27), it presents a divergence in the IR sector. The integration of the potential yields 1 2n0ΣH(M2) = αsNcTln 1 + m2 D M2,(3.18) being now the low momentum cut-off M2essential to achieve a finite result. We remind that, for illustration purposes, in numerical evaluations the strong coupling constant value is taken to αs= 0.3 and the parent parton is assumed to be a quark CR=CF= 4/3. We compare in Fig. 3.1 results obtained for the full resummed energy spectrum (solid lines), the asymptotically small energy limit (3.16) (dotted lines) and the GLV first opacity result (dashed dotted lines) plotted as function of the rescaled gluon energy x=ω/¯ωc. On the left we have spectra employing the Yukawa potential for several values of n0Land on the right using the HTL model with TL as parameter (following the discussion done in chapter 2). The values of n0Land TL were chosen to be equivalent when employing the correspondence relations shown in [58]. It is evident that for all cases the low energy limit coincide with the full result if ωis small enough, thus corroborating that we derived the correct small energy limit of the full BDMPS-Z spectrum. Comparing left and right plots in Fig. 3.1 we see that the shape of the spectrum for small energies strongly depends on the details of the collision rate, but the full spectrum and the low-ωlimit curves do not experience additional discrepancies depending on the chosen model. Therefore, the physical picture we described here holds unalterable for any collision rate employed, independently of the IR problems it may have. Furthermore, comparing solid curves (full spectrum) and dashed-dotted ones (GLV) we explicitly see that one single scattering is not enough to properly describe the emission process in this energy regime, as GLV curves largely overestimate the correct result. Thus, 46 3 Full spectrum vs. analytical approximations: the role of multiple scatterings contributions to the spectrum which would provide a more accurate description of the gluon emission. These terms were computed in [92] but here we are not going to consider them for two reasons. First, they are given by more complex formulae, which require longer computation time, thus going against the aim of using approximations in order to get simple equations that are easy to evaluate instead of computing the full solution. Second, and most importantly, in [92] the authors discuss in detail the relative contribution of the NNLO correction compared with respect to HO+NLO, claiming that its effect is small. Therefore, the HO+NLO spectrum should provide a good approximation of the full result, which we will explicitly check by comparing both spectra. Finally, we also remark that the introduction of Q2 cwas done to regularize and UV divergent integration (it plays the role of qmax in (3.20)), thus, it has to be big enough to fulfill this duty. This feature imposes an even more stringent limitation on the minimum energy value ˆxwhere we can employ this formalism compared to the constraint arising from the inability to solve (3.23) and (3.32) for ˆx < 2e. We have found that results can only be trusted for energies larger than ˆx > 10, which we will employ as lower limit in all of the numerical results that we will provide. 10−210−11 10 102 ω/¯ ωc 0 2 4 6 8 10 12 14 ωdImed/dω n0L= 20 n0L= 12.2 n0L= 5 Figure 3.3: All-order in-medium radiation energy spectrum for the Yukawa (solid lines) and HTL (dashed dotted) potentials plotted as a function of the rescaled energy x=ω/¯ωc. Different colors represent different values for n0L: magenta corresponds to n0L= 20 (equivalent to TL = 8), green corresponds to n0L= 12.2 (or TL = 5), and red corresponds n0L= 5 (or TL = 2). Figure extracted from [2] under a Creative Commons license, CCBY 4.0. To establish comparisons between both formalisms, IOE at HO+NLO accuracy and the spectrum fully resummed over multiple scatterings, it is mandatory to provide a set 53 Marcos Gonz´ alez Mart´ ınez of relations between the parameters of each framework. The parameters of the spectrum depend on the particular collision rate model employed. In section 3.2 we used both HTL and Yukawa-type interaction potentials to discuss the low energy regime of the spectrum. However, in the intermediate energy region we are working now, the outcome of both models is very similar, as the biggest differences in their behavior are located for small momentum transfers. We show this explicitly in Fig. 3.3, where we illustrate comparison between spectra computed with the Yukawa and HTL collision rates for different values of n0Land TL. The mapping between parameters of both models was done employing the leading-logarithmic mapping derived in [92] for small dipole sizes, which was also used in [58] m2 D=e µ2, TL =n0L e αsNc ,¯ωH c=e¯ωc,and xH=x e.(3.39) We clearly see in Fig. 3.3 that for moderate and high energies ω≥2·10−2¯ωcthe agreement between spectra computed using the Yukawa and HTL collision rates is excellent. Therefore, in the region of interest for this section we can just employ one of them and the results and conclusions will hold for the other as well. We chose, for simplicity, to use only the Yukawa interaction model in numerical evaluations. Then, we have to provide a relation between the parameters employed in the full numerical evaluation of the spectrum and the IOE at HO+NLO accuracy for the Yukawa collision rate. We follow again Ref. [92], which tells us that we should identify ˆq0=n0µ2,and µ⋆2=Cµ2,(3.40) with C=e−1+2γE 4≃0.29 .(3.41) These relations allow us to establish the following correspondence between the variables employed to evaluate the full spectrum, n0Land x=ω/¯ωc(2.57), and those entering the computation of the HO+NLO approximation, χand ˆx(3.30) ˆx=n0L C2x , and χ=n0L C.(3.42) Unlike the HO approximation, with this approach we are able to employ this one-to-one mapping of the parameters (3.42) to establish quantitative comparisons between different formalisms. In Fig. 3.4 we illustrate comparison between our all-order fully resummed approach (magenta solid lines), the IOE at HO+NLO precision (green dashed lines) and the first opacity N= 1 approximation (blue dashed dotted lines), computed through eq. (2.56). From left to right we present increasing values of n0L(or χ), being the smallest value just above the border for which the HO+NLO result is independent of the chosen matching scale (see Fig. 3.2) and the biggest value is set to coincide with the value of the parameters 54 3 Full spectrum vs. analytical approximations: the role of multiple scatterings 0.2 1 10 ω/¯ ωc 0 0.25 0.5 0.75 1 1.25 ωdImed/dω n0L= 5.0 (χ= 17.1) Full HO + NLO GLV N= 1 0.1 1 10 ω/¯ ωc 0 0.75 1.5 2.25 3 n0L= 12.2 (χ= 41.8) 0.1 1 10 50 ω/¯ ωc 0 1.5 3 4.5 6 n0L= 22.0 (χ= 75.4) Figure 3.4: Medium-induced energy spectrum with all-order resummation of multiple scatterings for the Yukawa collision rate (magenta solid lines) compared to the IOE at HO+NLO accuracy (green dashed) and GLV N= 1 approximation (blue dashed dotted) shown as a function of the rescaled gluon energy x=ω/¯ωcemploying different values of the opacity n0L(or χ). Figure extracted from [2] under a Creative Commons license, CC-BY 4.0. employed in Ref. [92]. The energy range of every panel was selected to show the region where the IOE framework can be employed (ˆx > 10) and multiple scatterings are thought to be essential for the emission process. We see in all of the three plots that in the high energy region there is a perfect agreement between all formalisms, as it should. This is especially relevant because the IOE encodes the correct high energy behaviour of the spectrum, which other approximations that account for multiple scatterings, like the HO, can not reproduce. In [58] it is performed a comparison between the HO and full result where this feature is well illustrated. Focusing now on the intermediate energy regime that we are studying in this section, we see in Fig. 3.4 that for all cases the HO+NLO approach is a good approximation of the all-order full resummed spectrum, whereas that the first opacity result highly overestimates it. Rising the value of n0Lincreases the range of applicability of the IOE approximation (remember that n0Land χare essentially the same (3.42) and χdirectly measures how big the multiple scattering region is (3.30)) at the same time that reduces differences between the HO+NLO and full result. These results not only check the validity of the IOE approach, but also expose the crucial importance of correctly accounting for multiple scatterings below the high energy region of the spectrum to achieve a precise description of the gluon emission process. Our numerical results explicitly illustrate the key role that multiple scatterings play in in-medium gluon radiation. 55 Marcos Gonz´ alez Mart´ ınez 3.3.2 In-medium emission rates We conclude the study of the multiple scattering region by adopting an inverted perspective compared to our previous approach. Rather than fixing the opacity (n0Lor χ) while varying the energy (xor ˆx), we now set ˆxas constant and treat χas the variable of interest. Equivalently, we are interested in studying the spectrum as function of the medium length L, which is easier to achieve by employing the emission rates instead of the spectrum itself. These rates are commonly defined as the derivative of the spectrum with respect to L ωd2I dωdL.(3.43) Instantaneous emission rates inside the medium can be interpreted as the probability that a soft gluon is radiated off a hard parton at a given time. However, as we mentioned before, gluons have an extended formation time that is not zero (1.39). Therefore, their emission can not be attributed to any given time coordinate [89]. Thus, this particular interpretation of the emission rates is only valid for formation times much smaller than the medium length, which is an assumption employed in the AMY formalism, as we discussed in section 1.3. Using this framework it is possible to include multiple emissions through rate equations, as it is done, for example, in MARTINI [103], a Monte Carlo event generator. Interpreting emission rates as the probability of radiating gluons at concrete times would be possible in the low energy region of the spectrum too, as in this regime the formation times are very small. Nevertheless, this is the part of the spectrum that has the smallest contribution in phenomenological studies, so it has not been considered physically interesting until now. Medium-induced emission rates provide us with very interesting information about the importance of multiple scatterings. In the case of short media, where a single scattering should work well, the first opacity result predicts a linear growth of the emission rates, whereas that for the opposite limit, in the asymptotic case of an infinitely large medium L→ ∞ the rates saturate to the AMY result computed in [104]. These two properties of the emission rates have already been confirmed in [89]. Here we are going to compute the rates of the all-order fully resummed spectrum, which are equivalent to the ones evaluated in [89], and compare them to the rates obtained from the IOE at HO+NLO accuracy. Employing the full resummed result explained in chapter 2 the computation of the emission rates is very simple. As in eq. (2.35) the medium length only appears in the upper limit of the sintegration (remember that the svariable is rescaled by L), performing the derivative of this expression with respect to Lcorresponds to just not computing the integral over the position of the last scattering center s. For the IOE we only need to derive the analytical expressions of the LO and NLO contributions. There are two equivalent ways of proceeding: either deriving with respect to Lthe spectrum before we rescale it (eqs. (3.25) and (3.27)) or taking the rescaled spectrum (eqs. (3.34) and (3.37)) and deriving it with respect to χ. If expressed in terms of the same variables the outcome of both methods is exactly the same. We follow the 56 3 Full spectrum vs. analytical approximations: the role of multiple scatterings latter approach, as it is further easy and quick than the former. Deriving (3.34) with respect to χyields for the HO contribution (recall that we are working with a= 1) ˆxd2IHO dˆxdχ=2αsCR πRe ((i−1)rΛ 2ˆxtan (1 −i)χrΛ 2ˆx!),(3.44) whereas that considering the derivative of (3.37) leads to the following expression for the NLO term ˆxd2INLO dˆxdχ=√2αsCR πRe i−1 √ˆxΛˆ1 0 ds K2(s)ln −(1 + i) 4√2K2(s)+γE +χ K2(s) dK2(s) dχ1−ln −(1 + i) 4√2K2(s)−γE,(3.45) with K2(s) already defined in eq. (3.38) and its corresponding derivative with respect to the variable χcan be written as dK2(s) dχ=−(1 −i)rΛ 2ˆx  s sin2(1 −i)χqΛ 2ˆxs+1−s cos2(1 −i)χqΛ 2ˆx(1 −s)  .(3.46) 0 20 40 60 χ 0 0.02 0.04 0.06 0.08 ˆ xd2Imed/dˆ xdχ ˆ x= 17 0 20 40 60 χ 0.00 0.01 0.02 0.03 ˆ x= 219 Full HO + NLO 0 20 40 60 χ 0 0.0025 0.005 0.0075 0.01 ˆ x= 1415 Figure 3.5: In-medium emission rates for the full resummed numerical evaluation of the spectrum (magenta solid lines) and the IOE at HO+NLO approximation (green dashed) as a function of χfor several values of the rescaled gluon energy ˆx=ω/ωBH: ˆx= 17 (left panel), ˆx= 219 (central panel) and ˆx= 1415 (right panel). Figure extracted from [2] under a Creative Commons license, CC-BY 4.0. We show in Fig. (3.5) the results of the evaluation of the emission rates for both formalisms, full resummed spectrum (magenta solid lines) and HO+NLO approximation 57 Marcos Gonz´ alez Mart´ ınez (green dashed lines) as a function of χand for several gluon rescaled energies (from left to right increasing values of ˆx). We confirm in these plots that the rates have the expected behavior: they start growing linearly for small values of χ(dilute/small media) and saturate to a constant value for large χ(dense/large media). However, the performance of the IOE is more limited compared to the full result, exhibiting oscillations in the transition region between the low and large χlimits. The amplitude of these oscillations diminishes as the energy increases, as expected. For large energies, the IOE converges towards the first opacity result, which lacks such oscillations. These results are in agreement with Fig. 3.4, where in the left panel we saw that for low energies and small χvalue the IOE starts to break. The second important discrepancy we can see in Fig. 3.5 is that, even though both approaches reach saturation at (more or less) the same time, they saturate to different values. The difference between both asymptotic values of the emission rates is bigger for smaller rescaled energies, corroborating again that IOE results in the low energy regime should not be trusted. The agreement between the two frameworks is much better for larger energies, however, in this case the asymptotic value of the rates is reached for very large χvalues, as it is illustrated in the right plot. We can compute the asymptotic value of the emission rates in both formalisms and compare them. We start with the rates of the all-order fully resummed result. Following the discussion done in chapter 2, the full BDMPS-Z medium-induced energy distribution without kinetic cut-off ( ¯ R→ ∞) for a brick, eqs. (2.35)-(2.37), can be written in a more compact way as (see [2] for details) xdImed dx= 4αsCRRe ˆ1 0 dsˆs 0 dtˆp ip·Fx(t;p),(3.47) with Fx(t;p) a 2-dimensional function satisfying the following differential equation ∂tFx(t;p) = −ip2Fx(t;p)−1 2n0Lˆp′ ˜σ(p′−p;x)Fx(t;p′),(3.48) having as initial condition Fx(0; p) = n0Lˆq q q2˜σ(q−p;x),(3.49) where ˜σ(q;x) is the rescaled version of the dipole cross section ˜σ(q;x) = xµ2σ(qpxµ2),(3.50) and with xdenoting the previously defined rescaled gluon energy x=ω/¯ωc. Performing the derivative of eq. (3.47) with respect to Land employing our dictionary (3.42) we see that the rates of the full resummed spectrum in terms of χand ˆxare given by ˆxd2I dˆxdχ= 4αsCRRe ˆ1 0 dtˆp ip·Fˆx(t;p)1 C,(3.51) 58 3 Full spectrum vs. analytical approximations: the role of multiple scatterings satisfying Fˆxthe following differential equation ∂tFˆx(t;p) = −ip2Fˆx(t;p)−1 2C2ˆp′ ˜σ(p′−p; ˆx)Fˆx(t;p′),(3.52) with initial condition Fˆx(0; p) = C2ˆq q q2˜σ(q−p; ˆx),(3.53) and ˜σ(q; ˆx) defined as ˜σ(q; ˆx) = ˆxµ2σ(qpˆxµ2).(3.54) To compute the L→ ∞ asymptotic limit of the rates we just need to perform the t integration from 0 to ∞(remember that tis rescaled as tL). The only dependence on t is contained in Fˆx(t;p), so we simply integrate its differential equation (3.52) Fˆx(∞;p)−Fˆx(0; p) = −ip2ˆ∞ 0 dtFˆx(t;p)−1 2C2ˆp′ ˜σ(p′−p; ˆx)ˆ∞ 0 dtFˆx(t;p′),(3.55) employing Fˆx(∞;p) = 0, eq. (3.53) and defining Gˆx(p) = ˆ∞ 0 dtFˆx(t;p)1 C,(3.56) the asymptotic limit of the full in-medium emission rates is given by lim χ→∞ ˆxd2I dˆxdχ= 4αsCRRe ˆp ip·Gˆx(p),(3.57) with Gˆx(t;p) satisfying the following integral equation ˆq q q2˜σ(q−p; ˆx) = ip2 CGˆx(p) + 1 2Cˆp′ ˜σ(p′−p; ˆx)Gˆx(p).(3.58) This equation can be computed with numerical methods and it is in agreement with the soft limit of the AMY result, which can be seen comparing with eqs. 6 and 7 of Ref. [104]. On the other hand, for the analytical expansion we just need to take the χ→ ∞ limit of the previously calculated rates (3.44) and (3.45). Regarding the LO contribution, the χ→ ∞ limit of the tan function yields −i, thus, after taking the real part we get lim χ→∞ ˆxd2IHO dˆxdχ=αsCR πr2Λ ˆx.(3.59) Now focusing on the NLO term, all of the dependence on χis contained in the K2(s) function (3.38). Taking the χ→ ∞ limit of this expression we get lim χ→∞ K2(s) = 2i , (3.60) 59 Marcos Gonz´ alez Mart´ ınez 10 102103 ω/ωBH 0 0.02 0.04 0.06 0.08 Asymptotic rate Full HO + NLO Figure 3.6: Asymptotic χ→ ∞ limit of the emission rates for the fully resummed evaluation (magenta solid line) and the IOE at HO+NLO accuracy (green dashed line) as a function of the energy ˆx=ω/ωBH. Figure extracted from [2] under a Creative Commons license, CC-BY 4.0. and, therefore, the second line of (3.45) vanishes, as K2(s) is constant, and we only have the contribution of the first line. The integral over sis trivially 1 and we take the real part of the logarithm, yielding the final result lim χ→∞ ˆxd2INLO dˆxdχ=αsCR π 1 √2ΛˆxhγE−ln 2 + π 4i.(3.61) We plot in Fig. 3.6 results for the asymptotic χ→ ∞ limit value of the rates for both approaches, all-order evaluation (magenta solid line) and HO+NLO approximation (green dashed line) as a function of the gluon rescaled energy ˆx=ω/ωBH. These results are in agreement with our previous findings: the analytical expansion of the spectrum agrees with the full result for large gluon energies (ˆx > 200). The situation corresponding to this high energy regime was shown in center and right plots of Fig. 3.5, where we see that the agreement between both frameworks is reached at very large values of χ(or n0L). 3.4 Full picture of the the emission process The final section of this chapter serves as a summary of the results presented in previous sections 3.2 and 3.3. Whereas that in these sections we focused on analyzing every energy 60 3 Full spectrum vs. analytical approximations: the role of multiple scatterings region separately, here we provide the complete description that we have nowadays of the emission process across the different energy regimes. Starting with the high energy region of the spectrum (ω > ωc), we showed explicitly in an analytical way in section 3.2 that the first opacity result is indeed the high energy limit of the full BDMPS-Z spectrum, as it was already numerically confirmed in [58]. One single hard scattering is enough to properly describe the whole emission process. In this regime the agreement between the GLV spectrum and full solution is excellent, and we can simply employ the former as it is really fast to compute. Moving on to lower energies, in the intermediate energy regime (ωBH < ω < ωc) the common assumption used to be that the LPM effect would gain importance, resulting in a suppression of the spectrum and, therefore, accounting for multiple scatterings would be essential in this region. We checked this statement by comparing the full spectrum with the IOE in section 3.3, resulting in a great agreement between both frameworks (on the range of applicability of the IOE: ω > 10 ωBH and χ > 14). This is very important because, until now, we only could perform qualitative comparisons with formalisms that take in account multiple scatterings (see [58]), but here we compared both approaches in a quantitative way. Finally, for the small energy BH regime (ω < ωBH) we derived for the first time the correct low energy limit of the complete spectrum. Comparisons between our newly derived result (3.11) and the full solution were performed in section 3.2 employing different parton-medium interaction models with different small momentum behaviours, resulting in an excellent agreement for energies low enough. The physical interpretation of the radiation process in this region is that we have the first opacity N= 1 spectrum, i.e., just one scattering, multiplied by a no-more scattering probability factor coming from the resummation of the contribution of virtual terms, which suppresses the GLV limit. We show in Fig. 3.7 a summary of all of these results. Here we plot the full mediuminduced gluon radiation spectrum (magenta solid line) as a function of the rescaled gluon energy ω/ωBH for a given value of n0L(or χ) compared with the GLV N= 1 result (blue dashed dotted line), the HO+NLO approximation (green dashed line) and the low energy limit of the spectrum (black dotted line) for a Yukawa-type interaction model. We clearly see the features we just mentioned: the correct description of the spectrum employing one single hard scattering in the large energy limit, the importance of accounting for multiple scatterings for intermediate energies and the good performance achieved by our new result in the small energy region. We notice, however, that the analytical understanding of the whole spectrum is not complete yet, there is a small region among ω∼3ωBH and ω∼10 ωBH where the IOE can not be applied due to fundamental problems and the efficiency of our small energy result is not good enough. This problem was partially solved in [105] where the authors derived a better expression of the spectrum for low ω, which they call Resummed Opacity Expansion (ROE). The first term in this expansion coincides with our result presented here (3.11) and higher orders correct its behaviour for not-so-low energies. Our results are in agreement with findings in Ref. [105], as it can be seen by comparing Fig. 3.7 with Figure 61 Marcos Gonz´ alez Mart´ ınez 0.2 1 10 100 103 ω/ωBH 0 2 4 6 8 10 12 ωdImed/dω χ= 41.8 (n0L= 12.2) Full HO + NLO GLV N= 1 Low-ωlimit Figure 3.7: Fully resummed in-medium gluon energy distribution (magenta solid line) compared to the analytical expansion at HO+NLO approximation (green dashed line), the first opacity N= 1 GLV spectrum (blue dashed dotted line), and the low energy limit of the all-order evaluation described in section 3.2 (black dotted line) as a function of the gluon rescaled energy ˆx=ω/ωBH. Figure extracted from [2] under a Creative Commons license, CC-BY 4.0. 7 of [105]. However, there are some details one needs to be careful with. First, in [105] for dealing with the problematic border between the small and intermediate energy regions the authors extrapolate the results of the IOE below its range of applicability, where it is not well defined, to be able to match the result for the small ωregime. In numerical evaluations they also employ only the first order in the ROE, which is just our low-ω limit (3.11), instead of trying to achieve a better agreement with the full spectrum. This results in having discrepancies with respect to the all-order solution, as it can be seen in the ratio plots of Figure 7 [105]. 62 4 Beyond the brick: full spectrum in evolving media The next sections will address the issue of using scaling laws on our all-order full resummed spectrum over multiple scatterings introduced in chapter 2 for a Yukawa-type parton-medium interaction model, with the goal of making our framework suitable for phenomenological studies. We will analyze different matching schemes, finding the one that provides the more accurate description of the spectrum for arbitrary conditions. As always in this theses, for illustration purposes we set αs= 0.3 and assume the initial hard parton to be a highly energetic quark CR=CF= 4/3. 4.3 First attempt of finding scaling laws: static case with average values First of all, we start by explaining how to implement a hydrodynamical simulation of the medium within our numerical approach. As mentioned in previous sections, we need to relate the parameters n(t) and µ2(t) with local properties of the medium, as the temperature, along the trajectory followed by the initial hard parton ξ(t). We employ the common procedure used on the field and establish the following relationship nhydro(t) = k1Thydro(ξ(t)) ,and µ2 hydro(t) = k2T2 hydro(ξ(t)) ,(4.8) being Thydro the temperature of the medium along the path described by the emitter, which is taken from a hydrodynamic model. The first equation in (4.8) is based on the Stefan-Boltzmann limit of the QCD equation-of-state, which establishes the medium volume density to be proportional to T3. The second relation is motivated by the leading order result of the hard thermal loop perturbative calculation of the thermal mass of the medium [57]. The values of the proportionality parameters k1and k2are determined in phenomenological studies by fitting the experimental data for a particular centrality class and, hence, to achieve consistent results here we will assume that for a given centrality class the numerical values of k1and k2remain constant for every trajectory. The hydrodynamic simulation we employ to extract the local temperature profiles Thydro(ξ(t)) is the smooth-averaged 2+1 viscous hydrodynamic model described in Refs. [111,112]. The starting point of this hydrodynamic model is at proper time τhydro = 1 fm, its initial condition is an energy density proportional to the density of binary collisions and fixes the ratio of shear viscosity to entropy density η/s to η/s = 0.08, being constant for the whole evolution. The simulation considers the medium to be in chemical equilibrium described by an equation of state inspired by Lattice QCD calculations and it sets a freeze-out temperature at Tf= 140 MeV. Further details concerning the description of the hydrodynamic model can be found in [111,112]1. The extraction of the temperature profile 1We notice that in these references it is also provided another hydrodynamical model which employs η/s = 0.16 and the initial condition is based on a Color-Glass-Condensate model. We have checked that using this different model does not affect the results we present here. Also, this model will be used in a phenomenological study in chapter 5. 69 Marcos Gonz´ alez Mart´ ınez employed to evaluate the medium-induced emission spectrum will be done by considering different representative straight lines for various centrality classes at √sNN = 5.02 TeV Pb-Pb collisions [113,114]. Moreover, in appendix 4.A we provide results for the radiation spectrum computed along trajectories extracted considering straight lines from the EKRT event-by-event (EbyE) hydrodynamic simulations in [115]. The static radiation spectrum has been computed in a wide variability of conditions, therefore it is the first natural candidate to be taken as reference scenario to find some relation between the values of the parameters of the spectrum that allow us to approximate the full result in an arbitrary case. The employment of the average values of the medium parameters along the path followed by the hard parton was the first option taken in account for this purpose, and it is used in several phenomenological studies. We consider just the ¯ R→ ∞ case (4.6) for simplicity, which needs only two parameters to be evaluated (4.7). Taking average values corresponds to identifying χ=ˆL+τhydro τhydro dt nhydro(t),(4.9) ¯ωc=1 2ˆL+τhydro τhydro dt µ2 hydro(t),(4.10) with Lthe length of the path and τhydro the initial proper time of the hydrodynamical model. We show in Fig. 4.1 results for the full spectrum evaluated along a representative trajectory (whose temperature profile is showcased in the inset figure) with a central production point sampled over the 0 −10% centrality class in √sNN = 5.02 TeV Pb- Pb collisions at the LHC (solid purple curve) and the correspondent equivalent static scenario (blue dashed curve) computed employing (4.9) and (4.10). On top panels it is represented the spectrum as a function of the rescaled gluon energy ω/¯ωc, with ¯ωc computed by (4.10), and on bottom panels it is plotted the ratio of the equivalent static scenario with respect to the spectrum computed along the actual path shown in the inset figure. The difference between left and right plots is the value of k1employed, which is k1= 0.5 for left and k= 1.0 for right. Employing (4.9) for the selected trajectory, they correspond, respectively, to χ= 5 and χ= 10 in terms of the variables of the static spectrum. Our results clearly show that the equivalent static scenario computed using average values for the parameters of the medium (4.9) and (4.10) overestimates the spectrum evaluated along the trajectory. One could argue that we are studying the spectrum without kinematical constraint ¯ R→ ∞ and, therefore, in the physical energy distribution the results would be better. However, the greatest discrepancies in Fig. 4.1 correspond to the high energy region of the spectrum, which remains unmodified after the kinematic cut-off is imposed. The analysis of different trajectories sampled with other production points and for different centrality classes in √sNN = 5.02 TeV Pb-Pb collisions at the LHC for many different values of the medium parameters gave as result outcomes in agreement with those 70 4 Beyond the brick: full spectrum in evolving media 10−210−11 10 0 1 2 3 4 ωdImed/dω χ= 5 hydro average 10−210−11 10 0 1 2 3 4 5 6 7 χ= 10 10−210−11 10 ω/¯ ωc 1.00 1.25 1.50 1.75 ratio 10−210−11 10 ω/¯ ωc 1.00 1.25 1.50 1.75 2.5 5.0 7.5 10.0 t(fm) 200 300 400 Thydro (MeV) Figure 4.1: Top panels: all order in-medium energy spectrum for a Yukawa-type collision rate with ¯ R→ ∞, and χ= 5 (left) or χ= 10 (right) as a function of the rescaled gluon energy ω/¯ωc. Purple curves correspond to the spectra computed along the trajectory whose temperature profile is shown in the inset figure, which was sampled with a central production point over the 0 −10% centrality class in √sNN = 5.02 TeV Pb-Pb collisions at the LHC. Blue dashed curves correspond to the static spectrum evaluated employing average values for the medium parameters (see eqs. (4.9) and (4.10)). Bottom panels: ratio of the static spectrum w.r.t. the spectrum computed with the hydrodynamic model. Figure extracted from [3] under a Creative Commons license, CC-BY 4.0. shown in Fig. 4.1, thus including them here would hinder the discussion unnecessarily. Therefore, the conclusion is evident: the employment of average values with the static spectrum does not correctly reproduce the spectrum computed along the real path followed by the emitter, being the largest disagreements located at high gluon energies, which was already noticed for the HO approach in [70]. We discuss in the following sections how to properly match the high energy tails of spectra computed in different medium profiles. 4.4 Matching the high energy tail of the spectrum Focusing now on the high energy regime of the energy distribution, as in this region the kinematic condition is not relevant we can use the simpler expression (4.6). Moreover, we showed explicitly in section 3.2 that the large energy limit of the full spectrum corresponds 71 Marcos Gonz´ alez Mart´ ınez to the first order in opacity expansion result. Therefore, we can substitute the full inmedium emission kernel in (4.6) by the vacuum kernel e K(s, q;t, p)≈(2π)2δ(2)(q−p)e−ip2 2ω(s−t),(4.11) yielding ωdImed dω≈2αsCR ωRe ˆL 0 ds n(s)ˆpl 2ω p21−e−ip2 2ωsp·l l2σ(s;l−p).(4.12) The l-integral can be done in the same way as we did in section 3.2 ˆl p·l l2σ(s;l−p) = ˆl V(s;l)θ(l2−p2)≡Σ(s;p2).(4.13) Plugging this into the spectrum and rescaling the last dummy momentum variable we get ωdImed dω≈2αsCR ωRe ˆL 0 ds n(s)ˆp 2ω p21−e−ip2 2ωsΣ(s;p2) =αsCR πRe ˆL 0 ds n(s)ˆ∞ 0 dr r1−e−irΣ(s; 2ωr/s).(4.14) Σ can be analytically computed for some collision rate models, as for the Yukawa interaction that we are employing now, obtaining (3.17) Σ(s;p2) = 2µ2(s) p2+µ2(s).(4.15) Thus, for the large energy ωlimit that we are studying now we get Σ(s; 2ωr/s)−−−→ ω→∞ s µ2(s) ωr ,(4.16) hence, the last two remaining integrals in (4.14) factorize and we can write ωdImed dω≈αsCR πω Re ˆL 0 ds s n(s)µ2(s)ˆ∞ 0 dr r21−e−ir =αsCR 2ωˆL 0 ds s n(s)µ2(s).(4.17) With this result we clearly see that the spectrum for a general case where the medium is not static but described by an arbitrary density n(s) and with a generic thermal screening mass µ2(s) behaves as 1/ω in the large energy limit, which coincides with the behavior of the static result as we discussed in section 3.2 and which can be written as ωdImed dωstatic ≈αsCR 2 χ¯ωc ω.(4.18) 72 4 Beyond the brick: full spectrum in evolving media Analyzing (4.17) we realize that the dependence on the time variable sof the parameters of the medium is just through a particular integration, which gives us the form of the scaling law we must employ to match the high energy tails of spectra computed for different medium configurations. Moreover, here we employed the Yukawa-type interaction model, but this result is more general. In fact, as long as the collision rate presents the large momentum functional form V(s;p)∝µ2(s)/p4, like the HTL model, then the same derivation can be done, because Σ only catches the high momentum tail behavior of the potential for large argument variables. 4.5 Scaling laws using as reference the static scenario We revisit the use of the static result as the reference scenario for the scaling laws, but employing this time an enhanced version compared to the average values case. The improvement lies in the ability to accurately match the high energy tail of the spectrum, which was the worst described region in Fig. 4.1. The comparison between (4.17) and (4.18) indicates that the high energy tails will match if we define the following combination of parameters as scaling law χ¯ωc=ˆL+τhydro τhydro dt t nhydro(t)µ2 hydro(t),(4.19) here we shifted the limits of the integral by τhydro, because this is the initialization time of the hydrodynamic simulation and for earlier times we have no results for nhydro(t) or µ2 hydro(t). This scaling law was already proposed for the HO approximation with ˆq(t)∼ n(t)µ2(t) and for the GLV N= 1 opacity result with constant screening mass in Ref. [70]. A well-defined set of scaling laws needs to establish enough relationships to determine values for all of the medium parameters used to evaluate the radiation spectrum. For the static result we have dependence on three parameters (4.7), thus we have to include two more scaling laws to establish a proper matching between the spectrum computed for a general trajectory and the static scenario. Using [70,116] as a guide we choose to set χ=ˆL+τhydro τhydro dt nhydro(t),(4.20) χ¯ R=3 2ˆL+τhydro τhydro dt t2nhydro(t)µ2 hydro(t).(4.21) If we are interested in computing the no-kinematic constraint energy distribution ¯ R→ ∞, then the last equation is not needed, as it happened in the previous section where we just defined two scaling laws. We illustrate in Fig. 4.2 the performance of the static spectrum as reference scenario (blue dashed lines) when trying to estimate the spectrum computed along a trajectory described by a hydrodynamic simulation (purple solid lines). Here we plot the spectra as 73 Marcos Gonz´ alez Mart´ ınez 10−210−11 10 0 1 2 3 4 5 ωdImed/dω ¯ R→ ∞ hydro static 10−210−11 10 0.0 0.2 0.4 0.6 0.8 ¯ R= 1000 10−210−11 10 ω/¯ ωc 0.9 0.95 1 ratio 10−210−11 10 ω/¯ ωc 0.5 0.75 1 2.5 5.0 7.5 10.0 t(fm) 200 300 400 Thydro (MeV) Figure 4.2: Top panels: all order in-medium energy spectrum for the Yukawa collision rate for χ= 5, and ¯ R→ ∞ (left panel) or ¯ R= 1000 (right panel) as a function of the rescaled gluon energy ω/¯ωc. Purple curves correspond to the spectra computed along the trajectory whose temperature profile is shown in the inset figure, which was sampled with a central production point over the 0−10% centrality class in √sNN = 5.02 TeV Pb-Pb collisions at the LHC. Blue dashed curves correspond to the static spectrum evaluated employing the scaling laws (4.19)-(4.21). Bottom panels: ratio of the static scenario w.r.t. the spectrum computed with the hydrodynamic model along the selected trajectory. Figure extracted from [3] under a Creative Commons license, CC-BY 4.0. function of the rescaled gluon energy ω/¯ωcwith ¯ωcdefined through (4.19) and the static case is computed following the scaling laws (4.19)-(4.21). The trajectory employed is the same one used in Fig. 4.1 and its temperature profile is showcased in the inset figure. Top panels show the spectra and bottom plots the ratio of the static case w.r.t. the scenario computed along the actual path. Left panels represent the energy distribution without kinematic condition ¯ R→ ∞ and right panels ¯ R= 1000 (corresponding to k2= 45), and in both cases we employ χ= 5 (corresponding to k1= 0.5). As expected, results in Fig. 4.2 using the new scaling laws correctly describe the high energy tail of the spectrum, in contrast to Fig. 4.1 where we employed average values of the medium parameters. Nevertheless, for smaller gluon energies discrepancies between the spectra appear and for the case with finite kinematical cut-off (right panel) the scaling breaks down, resulting in disagreements of more than 50%. 74 4 Beyond the brick: full spectrum in evolving media Fig. 4.2 correctly represents the general performance of the static scaling laws (4.19)- (4.21). In the next section we propose another matching scheme which the aim of improving the performance of the static scaling in the low energy region, providing results for a wide variety of conditions and always showing the static scenario as well. Thus, displaying now more results for the static scaling laws is pointless. 4.6 Scaling laws using as reference a power-law expansion Scaling laws are mainly employed because they allow us to pre-tabulate the mediuminduced radiation spectrum for a large range of the values of the parameters, thus we are able to estimate the spectrum for almost any realistic description of the QGP. In the case of the static scenario we just described, we could pre-evaluate the spectrum for different combinations of the free parameters (4.7) and in phenomenological studies use the relations (4.19)-(4.21) to determine the values of the parameters along every selected trajectory, instead of having to compute the spectrum for each path. Nonetheless, as we illustrated in Fig. 4.2, using the static spectrum as reference scenario for the scaling laws yields significant errors when attempting to describe the all-order fully resummed spectrum along a realistic trajectory sampled from a hydrodynamical simulation. Therefore, here we propose the employment of another scenario as reference which allow us to describe the full spectrum more accurately for any combination of the values of the medium parameters: a power-law expanding medium, which can be used under the same conditions as the static case for pre-tabulating the spectrum. With our newly proposed reference scenario we assume the following functional forms for the medium linear density of scattering centers n(t) and the collision rate screening mass µ2(t) n(t) = ¯n0 (t+¯ t0)α,and µ2(t) = ¯µ2 0 (t+¯ t0)2α,(4.22) being the constants ¯n0and ¯µ0proportional to the maximum values of the density and screening mass at the initial time ¯ t0, respectively. If we set the power αto be α= 1 then we recover the well-known Bjorken expansion, which describes an evolving medium with free streaming along the longitudinal direction. The scaling laws analogous to (4.19)-(4.21) for this scenario can be written as ˆ¯ L 0 dt n(t) = ˆL+τhydro τhydro dt nhydro(t),(4.23) ˆ¯ L 0 dt t n(t)µ2(t) = ˆL+τhydro τhydro dt t nhydro(t)µ2 hydro(t),(4.24) 75 Marcos Gonz´ alez Mart´ ınez 10−210−11 10 0 1 2 3 4 5 ωdImed/dω ¯ R→ ∞ hydro α= 1 α= 0.5 α= 0.3 static 10−210−11 10 0.0 0.2 0.4 0.6 0.8 ¯ R= 1000 10−210−11 10 ω/¯ ωc 0.9 0.95 1 1.05 ratio 10−210−11 10 ω/¯ ωc 0.5 1 1.5 2 2.5 5.0 7.5 10.0 t(fm) 200 300 400 Thydro (MeV) Figure 4.3: Top panels: all order in-medium energy spectrum for the Yukawa collision rate for χ= 5, and ¯ R→ ∞ (left panel) or ¯ R= 1000 (right panel) as a function of the rescaled gluon energy ω/¯ωc. Purple solid lines correspond to the spectra computed along the trajectory whose temperature profile is shown in the inset figure, which was sampled with a central production point over the 0 −10% centrality class in √sNN = 5.02 TeV Pb-Pb collisions at the LHC. Blue dashed lines correspond to the spectrum evaluated using the static scenario (4.19)-(4.21). The dotted curves correspond to the power-law scaling laws (4.23)-(4.25) for t0= 0.1 and several values of α:α= 0.3 (orange), α= 0.5 (green), and α= 1.0 (brown). Bottom panels: ratio of the power-law and static spectra w.r.t. the spectrum evaluated along the path with temperature profile showcased in the inset figure. Figure extracted from [3] under a Creative Commons license, CC-BY 4.0. ˆ¯ L 0 dt t2n(t)µ2(t) = ˆL+τhydro τhydro dt t2nhydro(t)µ2 hydro(t),(4.25) with ¯ Lthe endpoint of the trajectory along the power-law case. In this set of equations (4.24) plays the same role as (4.19) before, ensuring the correct matching of the high energy tails of both spectra following the discussion of section 4.4. With the scaling formulated in its current configuration the number of parameters we need to specify is larger than for the static scaling, as now the density and screening mass profiles (4.22) depend on two additional parameters: αand t0. One could simply treat them as free parameters and pre-tabulate the spectrum varying them. However, this assumption would complicate our scaling laws, which we want to keep as simple as 76 4 Beyond the brick: full spectrum in evolving media possible. Following that spirit, we analyzed the performance of the power-law scaling for several values of αand t0in different trajectories, to check if there is some fixed value of the parameters which provides good results in any conditions. We remember that all of the results we are going to show are computed with a Yukawa collision rate, being the HTL case treated in appendix 4.B. We start by comparing in Fig. 4.3 the energy distribution computed along the same typical trajectory sampled over the 0 −10% centrality class in √sNN = 5.02 TeV Pb-Pb collisions at the LHC that we used in Figs. 4.1 and 4.2, whose temperature profile is showcased in the inset figure, and the results arising from the employment of the powerlaw scaling laws (4.23)-(4.25) for several values of αwith t0fixed to t0= 0.1. The spectra is represented as a function of ω/¯ωc, with ¯ωcdefined through (4.19). Here we employ the same parameter values as in previous section: k1= 0.5 and k2= 45, which corresponds to χ= 5 and ¯ R= 1000, respectively (for the spectrum with kinematic constraint shown in the right panels). The performance of the power-law scaling compared with respect to the static scenario will be analyzed later, after we have discussed the role of αand t0. In Fig. 4.3 we clearly see that the best approximation of the spectrum computed along the trajectory coming from the hydrodynamic simulation (purple solid curves) is achieved by the power-law expansion with α= 0.5 (green dotted lines) for both spectra, either with (left) or without (right) kinetic condition imposed. We have computed these spectra for different values of the fitting parameters k1and k2for several sampled trajectories and, overall, the results shown in Fig. 4.3 constitute a good representation of the results we obtained, being the α= 0.5 scenario the one which always provides the best description of the spectrum along the actual path. Thus, we decided to fix αto α= 0.5 for the rest of the manuscript instead of being considered as a free parameter. Moving on to the opposite scenario, we illustrate in Fig. 4.4 the effect on the scaling laws of varying t0for fixed α. We still represent the spectra as a function of ω/¯ωc, with ¯ωcdefined through (4.19), and use the same trajectory as in previous cases with the same values for the free parameters: k1= 0.5 and k2= 45. We clearly see that changing the value of t0produces minimal effects in the outcomes of the scaling, being the best description of the spectrum computed along the actual trajectory (purple solid lines) reached by the power-law scenario with t0= 0.1. Therefore, we choose to set t0= 0.1 as a fixed value instead of treating it as a free parameter, thus reducing in one the total number of parameters required to determine the spectrum with the approach we propose at the same time that the precision of the results remains almost unchanged. It is important to notice that by setting both αand t0to fixed values the total number of parameters we need to specify with the power-law scaling is exactly the same as for the static scenario, three, which can be determined through the three scaling laws we already established (4.23)-(4.25). With this the complexity of the power-law scaling reduces significantly while maintaining the same levels of accuracy in the description of the spectra along realistic hydrodynamic trajectories, which will greatly help its implementation in future phenomenological works. Once that we have selected fixed values for αand t0we proceed to analyse in detail 77 Marcos Gonz´ alez Mart´ ınez 10−210−11 10 0 1 2 3 4 5 ωdImed/dω ¯ R→ ∞ hydro α= 0.5t0= 0.05 α= 0.5t0= 0.1 α= 0.5t0= 0.2 static 10−210−11 10 0.0 0.2 0.4 0.6 0.8 ¯ R= 1000 10−210−11 10 ω/¯ ωc 0.9 0.95 1 ratio 10−210−11 10 ω/¯ ωc 0.5 0.75 1 2.5 5.0 7.5 10.0 t(fm) 200 300 400 Thydro (MeV) Figure 4.4: Top panels: all order in-medium energy spectrum for the Yukawa collision rate for χ= 5, and ¯ R→ ∞ (left panel) or ¯ R= 1000 (right panel) as a function of the rescaled gluon energy ω/¯ωc. Purple solid lines correspond to the spectra computed along the trajectory whose temperature profile is shown in the inset figure, which was sampled with a central production point over the 0 −10% centrality class in √sNN = 5.02 TeV Pb-Pb collisions at the LHC. Blue dashed lines correspond to the spectrum evaluated using the static scenario (4.19)-(4.21). The dotted curves correspond to the power-law scaling laws (4.23)-(4.25) for α= 0.5 and several values of t0:t0= 0.05 (orange), t0= 0.1 (green), and t0= 0.2 (brown). Bottom panels: ratio of the power-law and static spectra w.r.t. the spectrum evaluated along the path with temperature profile showcased in the inset figure. Figure extracted from [3] under a Creative Commons license, CC-BY 4.0. the performance of our proposed power-law scaling in comparison with the classical static scenario. We show in Fig. 4.5 the full resummed medium-induced energy spectrum evaluated along the same path as in previous figures (purple solid line), whose temperature profile is showcased in the inset figure, and for the same values of the parameters k1= 0.5 and k2= 45, which correspond, respectively, to χ= 5 and ¯ R= 1000, when the kinematic constraint is imposed. The spectrum is plotted as a function of ω/¯ωc, with ¯ωcdefined through (4.19), and it is compared to the spectrum resulting from the application of the static scaling laws (4.19)-(4.21) (blue dashed line) and the power-law scenario scaling computed through (4.23)-(4.25) (green dotted curve). We clearly see that the performance of 78 4 Beyond the brick: full spectrum in evolving media 0 2 4 6 8 10 12 14 t (fm) 0.0 0.1 0.2 0.3 0.4 0.5 (GeV) hydro T 0 2 4 6 8 10 12 14 t (fm) 0.0 0.1 0.2 0.3 0.4 0.5 (GeV) hydro T [0-10]% Event-by-event Off-central production Average Event-by-event 0 2 4 6 8 10 12 14 t (fm) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 (GeV) hydro T 0 2 4 6 8 10 12 14 t (fm) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 (GeV) hydro T [10-30]% Event-by-event Central production Average Event-by-event Figure 4.9: Left: average temperature profile with standard deviation (black points with vertical lines) sampled from the 0 −10% centrality class in √sNN = 2.76 TeV Pb-Pb collisions at the LHC, together with a sample of temperature profiles corresponding to off-central production points (colored dashed lines). Right: average temperature profile with standard deviation (black points with vertical lines) sampled from the 10 −30% centrality class in √sNN = 2.76 TeV Pb-Pb collisions at the LHC together with a sample of temperature profiles corresponding to central production points (colored dashed lines). Inset panels display the full sample of temperature profiles in each centrality class. Figure extracted from [3] under a Creative Commons license, CC-BY 4.0. to different scenarios: purple solid lines represent the spectrum computed along the actual path whose temperature profile is shown in the inset figure, blue dashed lines correspond to the employment of the scaling laws using the static scenario as reference (4.19)-(4.21), and green dotted lines are computed with the power-law matching given by (4.23)-(4.25). Here we have used as values of the parameters k1= 0.5 and k2= 50, which for this trajectory correspond to χ= 5 and ¯ R→ ∞(left); and χ= 5 and ¯ R= 1000 (right), in terms of static parameters. Observing the results we conclude that, regardless of the evident fluctuations that the sampled trajectory shown in the inset figure presents, the scaling laws which use the power-law scenario as reference provide an accurate approximation of the spectrum even for small energies either with the kinetic condition lifted (left) or imposed (right). Finally, we show in Fig. 4.11 the spectra computed for a trajectory sampled over the 0−10% centrality class in √sNN = 2.76 TeV Pb-Pb collisions at the LHC with a noncentral production point. As before, the purple solid line represents the spectrum evaluated along the actual path whose temperature profile is showcased in the inset figure, the blue dashed curve corresponds to the employment of the static scaling laws (4.19)-(4.21), and the green dotted line is obtained using the power-law scenario as reference (4.23)- (4.25). We have used the same parameter values as in the previous figure k1= 0.5 and 85 Marcos Gonz´ alez Mart´ ınez 10−210−11 10 0 1 2 3 4 5 ωdImed/dω ¯ R→ ∞ hydro α= 0.5 static 10−210−11 10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ¯ R= 1000 10−210−11 10 ω/¯ ωc 0.9 0.95 1 ratio 10−210−11 10 ω/¯ ωc 0.5 0.75 1 0 2 4 6 8 10 t(fm) 100 200 300 400 Thydro (MeV) Figure 4.10: Top: all order in-medium energy spectrum for the Yukawa collision rate for χ= 5, and ¯ R→ ∞ (left) or ¯ R= 1000 (right) as a function of the rescaled gluon energy ω/¯ωc. Purple solid lines correspond to the spectra computed along the trajectory whose temperature profile is shown in the inset figure, which was sampled from an event corresponding to the 10 −30% centrality class in √sNN = 2.76 TeV Pb-Pb collisions at the LHC. Blue dashed and green dotted lines correspond, respectively, to the spectrum evaluated using the static (4.19)-(4.21) and power-law (4.23)-(4.25) scaling laws. Bottom: green dotted (blue dashed) lines represent the ratio of the spectrum computed with the power-law (static) scaling laws w.r.t. the spectrum evaluated along the path with temperature profile showcased in the inset figure. Figure extracted from [3] under a Creative Commons license, CC-BY 4.0. k2= 50, which in this trajectory correspond to χ= 4.4 and ¯ R→ ∞ (left panel); and χ= 4.4 and ¯ R= 1100 (right panel). We clearly see that even in such an extreme scenario the power-law matching provide a really good description of the spectrum computed along the actual path, greatly improving the performance of the static scaling. The results shown in these two extreme scenarios allow us to conclude that accounting for EbyE fluctuations do not affect the performance of the power-law scaling in a sizeable way. Thus, we can still use the power-law scenario as reference when employing scaling laws and the resulting spectrum should provide a precise description of the spectrum computed along the path. 86 4 Beyond the brick: full spectrum in evolving media 10−210−11 10 0 1 2 3 4 ωdImed/dω ¯ R→ ∞ hydro α= 0.5 static 10−210−11 10 0.0 0.2 0.4 0.6 0.8¯ R= 1100 10−210−11 10 ω/¯ ωc 0.95 1 1.05 ratio 10−210−11 10 ω/¯ ωc 0.5 0.75 1 0 2 4 6 8 10 t(fm) 100 150 200 Thydro (MeV) Figure 4.11: Top: all order in-medium energy spectrum for the Yukawa collision rate for χ= 4.4, and ¯ R→ ∞ (left) or ¯ R= 1100 (right) as a function of the rescaled gluon energy ω/¯ωc. Purple solid lines correspond to the spectra computed along the trajectory whose temperature profile is shown in the inset figure, which was sampled over a central event in √sNN = 2.76 TeV Pb-Pb collisions at the LHC. Blue dashed and green dotted lines correspond, respectively, to the spectrum evaluated using the static (4.19)-(4.21) and power-law (4.23)-(4.25) scaling laws. Bottom: green dotted (blue dashed) lines represent the ratio of the spectrum computed with the power-law (static) scaling laws w.r.t. the spectrum evaluated along the path with temperature profile showcased in the inset figure. Figure extracted from [3] under a Creative Commons license, CC-BY 4.0. 4.B Results for the HTL interaction model In this appendix we provide results of the performance of the power-law matching when employing a different collision rate model, the HTL, in order to illustrate the flexibility of our approach. We start by briefly remembering the expression of the HTL potential 1 2n(s)VH(s;q) = g2 sNcm2 D(s)T(s) q2(q2+m2 D(s)) ,(4.26) where we have explicitly written the temporal dependence on the Debye mass mD(s) and medium temperature T(s). We have already explained in chapter 2 how to evaluate the fully resummed medium-induced radiation spectrum using this collision rate. The 87 Marcos Gonz´ alez Mart´ ınez interested reader can also see Ref. [58] for further details. We also discussed in chapter 2 that the parameters we need to specify to compute the spectrum with this model are T, m2 Dand L, but for convenience we define (2.63) χH=TL , ¯ωH c=1 2m2 DL , and ¯ RH= ¯ωH cL . (4.27) When computing the spectrum in realistically evolving media we need to specify a relationship between the Debye mass and the local properties of the medium. This relation at LO in the HTL approach is given by [57] m2 D,hydro(t) = c2T2 hydro(ξ(t)) ,(4.28) where Thydro is the temperature of the medium along the path followed by the emitter ξ(t) which will be extracted from the hydrodynamical model [111, 112]. We notice that at LO in the HTL approach the c2parameter is given by c2=g2 s(1 + Nf/6). However, for convenience and in analogy to the Yukawa potential we decide to leave c2as a free parameter. 4.B.1 Scaling laws using the static scenario as reference In analogy with the work done for the Yukawa-type interaction, we can define the following scaling laws which relate a spectrum computed along a hydrodynamic path with an equivalent static scenario χH=c1ˆL+τhydro τhydro dt Thydro(t),(4.29) χH¯ωH c=c1ˆL+τhydro τhydro dt t Thydro(t)m2 D,hydro(t),(4.30) χH¯ RH=3c1 2ˆL+τhydro τhydro dt t2Thydro(t)m2 D,hydro(t),(4.31) with c1a constant parameter which plays the role of k1for the Yukawa case (4.8). 4.B.2 Scaling laws using the power-law scenario as reference We can also follow the work done in section 4.6 for the Yukawa model and define for the HTL potential a set of relationships between the spectrum computed in a general path and an equivalent power-law expanding scenario. In this case we parametrize the Debye mass mDand the medium temperature Twith the following relations T(t) = ¯ T0 (t+¯ t0)α,and m2 D(t) = ¯m2 D 0 (t+¯ t0)2α,(4.32) 88 4 Beyond the brick: full spectrum in evolving media here ¯ T0and ¯mD 0 are, respectively, proportional to the maximum value of the temperature and Debye mass, which are reached at the initial time ¯ t0. As done for the Yukawatype interaction, we set the following values for the extra free parameters α= 0.5 and t0=¯ t0/¯ L= 0.1. Then, now we need to establish three scaling laws, as we have the same amount of parameters as in the static scenario. For the power-law matching we choose ˆ¯ L 0 dt T(t) = ˆL+τhydro τhydro dt Thydro(t),(4.33) ˆ¯ L 0 dt t T(t)m2 D(t) = ˆL+τhydro τhydro dt t Thydro(t)m2 D,hydro(t),(4.34) ˆ¯ L 0 dt t2T(t)m2 D(t) = ˆL+τhydro τhydro dt t2Thydro(t)m2 D,hydro(t),(4.35) with (4.34) ensuring the correct matching of the high energy tails of both spectra (see section 4.4). Fig. 4.12 illustrates the results obtained for the gluon energy spectrum evaluated along the same path employed in Fig. 4.5 whose temperature profile is showcased in the inset figure, i.e., a straight-line trajectory sampled with a central production point for the 0 −10% centrality class in √sNN = 5.02 TeV Pb-Pb collisions at the LHC. Here we plot the different spectra as function of the rescaled gluon energy ω/¯ωH c, with ¯ωH cdefined through (4.30): the purple solid line stands for the spectrum computed along the path, the blue dashed curve correspond to the spectrum coming from the static scaling laws (4.29)- (4.31), and the green dotted line represents the spectrum evaluated for the power-law matching (4.33)-(4.35). The values of the parameters employed here were c1= 0.2 and c2= 121, which for this particular trajectory correspond to χH= 2 and ¯ RH→ ∞ (left), and χH= 2 and RH= 2700 (right) in terms of static variables. The outcomes presented in Fig. 4.12 are essentially the same we already got in Fig. 4.5 for the Yukawa model: the static scenario has an acceptable performance describing the spectrum computed along realistic paths when the kinematic constraint is lifted (left), but it gives large errors for small gluon energies when it is imposed (right). The power-law approach greatly improves these results, providing an excellent approximation of the spectrum along the trajectory for any condition and energy. We conclude by showing in Fig. 4.13 the performance of the power-law scaling with the HTL collision rate in an extreme scenario which possess a non-monotonically decreasing temperature, illustrated in the inset figure. The chosen trajectory is the same we previously employed for the Yukawa collision rate in Fig. 4.8. Here we have set the same parameter values as in the previous figure c1= 0.2 and c2= 121, which for this path correspond to χH= 2.7 and ¯ RH→ ∞ (left), and χH= 2.7 and ¯ RH= 7070 (right). Again, the results obtained for the HTL parton-medium interaction model align perfectly with previous findings for the Yukawa-type collision rate in Fig. 4.8: the power-law scaling 89 Marcos Gonz´ alez Mart´ ınez 10−210−11 10 0 1 2 3 4 5 ωdImed/dω ¯ RH→ ∞ hydro α= 0.5 static 10−210−11 10 0.0 0.2 0.4 0.6 0.8 ¯ RH= 2700 10−210−11 10 ω/¯ ωH c 0.9 0.95 1 ratio 10−210−11 10 ω/¯ ωH c 0.25 0.5 0.75 1 2.5 5.0 7.5 10.0 t(fm) 200 300 400 Thydro (MeV) Figure 4.12: Top: all order in-medium energy spectrum for the HTL elastic collision rate with χH= 2, and ¯ RH→ ∞ (left) or ¯ RH= 2700 (right) as a function of the rescaled gluon energy ω/¯ωH c. Purple solid lines correspond to the spectra computed along the trajectory whose temperature profile is shown in the inset figure, which was sampled with a central production point over the 0 −10% centrality class in √sNN = 5.02 TeV Pb-Pb collisions at the LHC. Blue dashed and green dotted lines correspond, respectively, to the spectrum evaluated using the static (4.29)-(4.31) and power-law (4.33)-(4.35) scaling laws. Bottom: green dotted (blue dashed) lines represent the ratio of the spectrum computed with the power-law (static) scaling laws w.r.t. the spectrum evaluated along the path with temperature profile showcased in the inset figure. Figure extracted from [3] under a Creative Commons license, CC-BY 4.0. provides a more accurate description of the spectrum computed along the path than the static scenario. For energies smaller than the characteristic gluon energy the deviations observed always remain below 15%, being significantly smaller than the ones observed in the static result. Taking all of this in account, we can assure that the power-law approach we presented here provides an accurate description of the spectrum across the entire range of gluon energies for realistic trajectories sampled over hydrodynamic simulations, even for the most extreme conditions regarding the temperature behaviour during the initial times. Moreover, the agreement between the results obtained employing the Yukawa and HTL models manifest the robustness of the approach in capturing the behavior of the full medium- 90 4 Beyond the brick: full spectrum in evolving media 10−210−11 10 0 1 2 3 4 5 6 7 ωdImed/dω ¯ RH→ ∞ hydro α= 0.5 static 10−210−11 10 0.0 0.2 0.4 0.6 0.8 1.0 1.2 ¯ RH= 7070 10−210−11 10 ω/¯ ωH c 0.9 1 1.1 ratio 10−210−11 10 ω/¯ ωH c 0.5 0.75 1 1.25 2.5 5.0 7.5 10.0 12.5 15.0 t(fm) 150 200 250 Thydro (MeV) Figure 4.13: Top: all order in-medium energy spectrum for the HTL elastic collision rate with χH= 2.7, and ¯ RH→ ∞(left) or ¯ RH= 7070 (right) as a function of the rescaled gluon energy ω/¯ωH c. Purple solid lines correspond to the spectra computed along the trajectory whose temperature profile is shown in the inset figure, which was sampled with a noncentral production point over the 0 −10% centrality class in √sNN = 5.02 TeV Pb-Pb collisions at the LHC. Blue dashed and green dotted lines correspond, respectively, to the spectrum evaluated using the static (4.29)-(4.31) and power-law (4.33)-(4.35) scaling laws. Bottom: green dotted (blue dashed) lines represent the ratio of the spectrum computed with the power-law (static) scaling laws w.r.t. the spectrum evaluated along the path with temperature profile showcased in the inset figure. Figure extracted from [3] under a Creative Commons license, CC-BY 4.0. induced gluon energy distribution, regardless of the specific parton-medium interaction model employed. 91 Marcos Gonz´ alez Mart´ ınez 92 5 The role of initial stages on jet quenching To comply with the regulations for doctoral studies of the University of Santiago de Compostela we include here the full information of the article on which this chapter is based: Marcos Gonz´ alez Mart´ ınez Medium-induced radiation with vacuum propagation in the prehydrodynamics phase, J. High Energ. Phys. 03 (2023) 189 Authors Carlota Andres,aLiliana Apolin´ario,bc Fabio Dominguez,dMarcos Gonzalez Martinez,d Carlos A. Salgado,d aCPHT, CNRS, ´ Ecole polytechnique, IP Paris, F-91128 Palaiseau, France bLIP, Av. Prof. Gama Pinto, 2, P-1649-003 Lisboa, Portugal cInstituto Superior T´ecnico (IST), Universidade de Lisboa, Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal dInstituto Galego de F´ısica de Altas Enerx´ıas (IGFAE), Universidade de Santiago de Compostela, Santiago de Compostela 15782, Spain PhD Student Contribution Active participation in all the calculations presented in the paper, as well as in the discussions, meetings and writing of the paper. All of the authors have the same contribution to the publication. Chapter of the thesis in which it is used: 5 Journal and Article Information Journal name: Journal of High Energy Physics Publisher: Springer ISSN: 1029-8479 (print) Year of publication: 2023 DOI:https://doi.org/10.1007/JHEP03(2023)189 Impact factor in 2022 (current value, 2023 not available yet): 5.4 Reproduced in this thesis with standard author permissions from Journal of High Energy Physics and Springer (articles distributed under the Creative Commons Attribution license CC-BY-4.0) The final chapter of the thesis is dedicated to addressing a very important goal of this manuscript: what is the role (if any) that initial stages of heavy ion collisions play in jet quenching? For the first time, the role of vacuum radiation within the short period of time before thermalization is here studied in detail. This contribution has been overlooked in previous works, and its study allow us to obtain information about the temporal structure of the QCD collectivity in the first instants, which is one of the main goals of this thesis, as its title states. 94 5 The role of initial stages on jet quenching with CR=CF= 4/3 and αs= 0.3, for illustration purposes. We also choose n0L= 5 and plot the different spectra as function of the rescaled gluon energy ω/ωc. The colors of the different curves were chosen to match the scenario they correspond to in Fig. 5.1: green solid curves for case A) τp=τm= 0, blue dashed lines correspond to scenario B) τp=τm>0, and red dashed dotted lines for case C) τm> τp= 0. In Fig. 5.2 we set τm/L = 0.1, whereas that in Fig. 5.3 we employed τm/L = 0.3, except for the green curves which always have τm= 0. In both figures from left to right we have decreasing values of the kinematical cut-off: on the left plot it is shown the energy distribution with the kinetic constraint removed ( ¯ R→ ∞), on the central panel we set ¯ R= 5000, and on the right panel ¯ R= 1000. 10−310−210−11 10 50 ω/¯ ωc 0.0 1.5 3.0 4.5 6.0 ωdIFull/dω ¯ R→ ∞ τp=τm= 0 τp=τm= 0.3L τp= 0, τm= 0.3L 10−310−210−11 10 50 ω/¯ ωc 0.0 0.2 0.4 0.6 0.8 1.0 ¯ R= 5000 10−310−210−11 10 50 ω/¯ ωc 0.0 0.2 0.4 0.6 ¯ R= 1000 Full, n0L= 5 Figure 5.3: All-order in-medium energy distribution with full resummation over multiple scatterings for the three different scenarios analyzed: τp=τm= 0 (green solid lines); τp=τm= 0.3L(blue dashed lines); and τp= 0 and τm= 0.3L(red dashed dotted lines) as a function of ω/¯ωcfor n0L= 5. Different panels correspond to different values of the kinematical constraint ¯ R:¯ R→ ∞ (left), ¯ R= 5000 (central), and ¯ R= 1000 (right). Figure extracted from [4] under a Creative Commons license, CC-BY 4.0. The first important feature we want to comment is the fact that, under the same conditions, the spectrum corresponding to completely neglect the initial stages τp=τm> 0 (blue dashed curves) is always smaller than the one which considers τp=τm= 0 (green solid lines), with the difference between curves increasing for bigger values of τm. This is clearly seen by comparing Figs. 5.2 and 5.3, and is something that we anticipated. As we previously discussed, both cases are formally equivalent, being the only difference between them the length of the medium, which is Lfor τp=τm= 0, but in the τp=τm>0 case it corresponds to L−τm, being a shorter length than for the former scenario when τm= 0. Hence, if we employ the same medium parameters in both cases to compute the spectrum, it is obvious that the one corresponding to τp=τm>0 is always going to be smaller. Analyzing now the scenario which includes the initial stages radiation in vacuum τm> τp= 0 (red dashed dotted curves) we see in Figs. 5.2 and 5.3 that this case is 101 Marcos Gonz´ alez Mart´ ınez always somewhere in between the other two. When considering a small τm/L, as in Fig. 5.2, the differences between this scenario (red dashed dotted lines) and the one in which the medium is formed at τm= 0 (green solid curves) are almost negligible, only been noticeable on the left panel of Fig. 5.2 for very small gluon energies. However, when increasing the value of τm/L these differences are more notorious, as it can be seen in Fig. 5.3, where both spectra only have a good agreement in the high energy region for all panels. The fact that highly energetic gluons seem to be insensitive to the subtraction of part of the medium during the early times of the propagation can be understood in terms of the formation time (1.39). As we discussed in section 3.1, harder gluons need larger transverse momentum transfers to decorrelate from the emitter, thus having bigger formation times. Gluons which have a formation time larger than τmcan be radiated before the creation of the medium and still undergo its effects before being completely decorrelated from the emitter. However, gluons with a small formation time only contribute to the mediuminduced radiation spectrum if they are emitted after the medium is created. In following sections, 5.3 and 5.4, we will prove how the relevant energy scales that separate both regimes appear when we use analytical approaches, like the single hard scattering (GLV, section 5.3) and multiple soft scatterings (HO, section 5.4) approximations. 5.3 GLV limit As we already discussed in chapter 2, the single scattering contribution to the full spectrum can be derived by replacing the emission kernel e Kand the momentum broadening Pby their vacuum versions (2.49) and (2.48), respectively. Introducing eqs. (2.48) and (2.49) in the general spectrum (5.2) we can perform the integral over time t, as all of the dependence on this variable is in the exponential of the kernel (2.49). By doing so we get that the GLV transverse momentum dependent spectrum is given by ωdIGLV dωd2k=αsCR π2Re ˆL τm ds n(s)ˆp 1 p2p·k k2−11−e−ip2 2ω(s−τp)σ(k−p).(5.10) With this equation we can see that the generalized GLV medium-induced radiation spectrum where we allow the emitter to be produced before the hydrodynamization of the medium (τp< τm, illustrated in the bottom picture of Fig. 5.1), can be written as two different contributions off hard partons produced inside the medium (top picture of Fig. 5.1) with different longitudinal extensions ωdIGLV dωd2kτm>τp =ωdIGLV dωd2kτm=τp−αsCR π2Re ˆτm τp ds n(s)ˆp 1 p2p·k k2−1 ×1−e−ip2 2ω(s−τp)σ(k−p), (5.11) 102 5 The role of initial stages on jet quenching where the last term is just the emission spectrum of an emitter travelling inside a medium from τpto τm. The difference between the initial propagation being either in vacuum or in medium is precisely the amount of radiation one gets from an in-medium propagation during that initial slab (τm−τp). This is a particular result that holds only for the GLV spectrum and not for the all-order result presented in the previous section, because in the full spectrum case multiple scatterings can occur anywhere and, hence, we can not separate between emissions interacting with the medium before or after τm. We recall that it was proven in section 3.2 (see also [2,58]) that the GLV limit provides the correct description of the fully resummed spectrum over multiple scatterings in the large energy regime. Therefore, this relation which is specific for the first opacity result is expected to hold for the complete BDMPS-Z spectrum at high energies. Moving on to the energy spectrum, we can perform the integration over transverse momentum kin (5.10) employing the relation between the dipole cross section σand the elastic collision rate V(1.32) and the Yukawa potential (1.26). Integrating also the angular part of the transverse momentum pwe reach (see chapter 2 for details about simplifying the GLV spectrum for the Yukawa collision rate) ωdIGLV dω=4αsCR πRe ˆL τm ds n(s)ˆ∞ 0 dp p"µ2 µ2+p2−µ2 p(µ2+p2+ω2)2−4ω2p2# ×1−e−ip2 2ω(s−τp), (5.12) where we integrated kin a finite domain 0 ≤k < ω in order to keep the kinematic cut-off. We can remove the kinematical condition by allowing kto go up to infinity, which results in the fading of the second term in square brackets in (5.12), thus yielding ωdIGLV dω=4αsCR πRe ˆL τm ds n(s)ˆ∞ 0 dp p µ2 µ2+p21−e−ip2 2ω(s−τp).(5.13) The relation between the spectrum with initial vacuum propagation and the scenario where the emitter is produced inside the medium obtained in (5.11) holds as well in both energy distributions (5.12) and (5.13). 5.3.1 GLV limit for the brick As we did in the previous section, to give numerical results of the evaluation of the spectrum we restrict ourselves, for simplicity, to the most basic scenario where the medium is described by a brick of static linear density of scattering centers n(s) = n0for s∈ [τm, L], and with a constant screening mass µ(s) = µfor s∈[τm, L]. With this simple consideration the integration over sin (5.10) is trivial to perform, yielding the following 103 Marcos Gonz´ alez Mart´ ınez result for the GLV transverse momentum dependent spectrum in a brick ωdIGLV dωd2k=αsCRn0 π2ˆp 1 p2p·k k2−1σ(k−p)(L−τm −2ω p2sin p2(L−τp) 2ω−sin p2(τm−τp) 2ω). (5.14) We can also simplify the energy distribution employing the brick approximation. First we focus on the energy spectrum with kinematical constraint (5.12). Integrating over s we find that the energy distribution is given by ωdIGLV dω=4αsCRn0 πˆ∞ 0 dp p"µ2 µ2+p2−µ2 p(µ2+p2+ω2)2−4ω2p2# ×L−τm−2ω p2sin p2(L−τp) 2ω−sin p2(τm−τp) 2ω.(5.15) As we have done throughout the whole thesis, we work with dimensionless quantities to reduce the number of free parameters. Following that spirit, we rescale the dummy momentum variable p→pp2ω/L in (5.15), getting ωdIGLV dω=4αsCRn0L πˆ∞ 0 dp p"1 1 + xp2−1 p(1 + xp2+x2¯ R/2)2−2x3¯ Rp2# ×1−τm L−1 p2sin p2(1 −τp L)−sin p2 L(τm−τp),(5.16) with x=ω/¯ωc, and ¯ωcand ¯ Rdefined in (5.9). In this case we will also work always with τpbeing either τp= 0 or τp=τm. Therefore, the general GLV energy distribution (5.16) is function of the same variables as the full result (5.9): n0L,x,¯ Rand τm/L. Let us now move on to the case where the kinematical constraint is removed. With this consideration, the second term inside the square brackets in (5.15) vanishes, hence we get ωdIGLV dω=4αsCRn0 πˆ∞ 0 dp p µ2 µ2+p2 ×L−τm−2ω p2sin p2(L−τp) 2ω−sin p2(τm−τp) 2ω.(5.17) Setting τp= 0 in (5.17) we clearly see that this expression corresponds to the difference between the GLV spectra for an emitter created in the medium travelling through bricks of longitudinal extensions Land τm, respectively, as it occurred in (5.11). We can 104 5 The role of initial stages on jet quenching rewrite (5.17) by separating the two different terms and rescaling the momentum variables accordingly, obtaining ωdIGLV dω=4αsCR π n0Lˆ∞ 0 dpp2−sin p2 p3 1 1 + ω ¯ωcp2 −n0τmˆ∞ 0 dpp2−sin p2 p3 1 1 + ω ¯ωmp2!,(5.18) where we have defined, in analogy with ¯ωc, the following energy scale ¯ωm≡µ2τm/2 = ¯ωcτm/L. When ω≫¯ωmthe second term in (5.18) becomes small compared with the first one. Therefore, we explicitly see in an analytical way that the initial part of the propagation is not relevant for radiated gluons with large energies, obtaining identical spectra for hard partons produced inside or outside the medium. 10−310−210−11 10 50 ω/¯ ωc 0.0 1.5 3.0 4.5 6.0 7.5 ωdIGLV/dω ¯ R→ ∞ τp=τm= 0 τp=τm= 0.1L τp= 0, τm= 0.1L 10−310−210−11 10 50 ω/¯ ωc 0.0 0.4 0.8 1.2 1.6 2.0 ¯ R= 5000 10−310−210−11 10 50 ω/¯ ωc 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 ¯ R= 1000 GLV, n0L= 5 Figure 5.4: In-medium energy distribution for the GLV limit for τp=τm= 0 (green solid lines); τp=τm= 0.1L(blue dashed lines); and τp= 0 and τm= 0.1L(red dashed dotted lines) as a function of the rescaled gluon energy x=ω/¯ωcfor n0L= 5. Different panels correspond to different values of the kinematical cut-off ¯ R:¯ R→ ∞ (left), ¯ R= 5000 (central), and ¯ R= 1000 (right). Figure extracted from [4] under a Creative Commons license, CC-BY 4.0. We show in Figs. 5.4 and 5.5 results of the numerical evaluation of the GLV mediuminduced energy distribution off a hard quark as a function of the rescaled gluon energy ω/¯ωcfor the three different scenarios illustrated in Fig. 5.1, which correspond to different considerations regarding the description of the initial stages. We employ the same line styles as in section 5.2: green solid curves for τp=τm= 0; blue dashed lines for τp= τm>0; and red dashed dotted curves for τm> τp= 0. Different panels in each figure correspond to different values of the kinematical cut-off, chosen to math the same ones employed with the full spectrum: ¯ R→ ∞ (no kinematical condition imposed) on the left, ¯ R= 5000 on the central panel and ¯ R= 1000 on the right. The difference between 105 Marcos Gonz´ alez Mart´ ınez 10−310−210−11 10 50 ω/¯ ωc 0.0 1.5 3.0 4.5 6.0 7.5 ωdIGLV/dω ¯ R→ ∞ τp=τm= 0 τp=τm= 0.3L τp= 0, τm= 0.3L 10−310−210−11 10 50 ω/¯ ωc 0.0 0.4 0.8 1.2 1.6 2.0 ¯ R= 5000 10−310−210−11 10 50 ω/¯ ωc 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 ¯ R= 1000 GLV, n0L= 5 Figure 5.5: In-medium energy distribution for the GLV limit for τp=τm= 0 (green solid lines); τp=τm= 0.3L(blue dashed lines); and τp= 0 and τm= 0.3L(red dashed dotted lines) as a function of the rescaled gluon energy x=ω/¯ωcfor n0L= 5. Different panels correspond to different values of the kinematical cut-off ¯ R:¯ R→ ∞ (left), ¯ R= 5000 (central), and ¯ R= 1000 (right). Figure extracted from [4] under a Creative Commons license, CC-BY 4.0. Figs. 5.4 and 5.5 is the value of τm/L employed to compute the curves where it is not zero (blue dashed and red dashed dotted). In Fig. 5.4 we use a small value of that parameter τm/L = 0.1, whereas that in Fig. 5.5 we increase it and set its value to τm/L = 0.3. The results presented in Fig. 5.4 and 5.5 are in agreement with the findings obtained in the previous section for the fully resummed spectrum: the scenario where initial stages are completely neglected (τp=τm>0, blue dashed lines) is always below the case where the initial propagation is done through medium (τp=τm= 0, green solid curves), and the scenario with vacuum propagation before entering the medium (τm> τp= 0, red dashed dotted lines) is somewhere in between those two. We have an agreement between τm> τp= 0 and τp=τm= 0 cases for high energies, showing again the insensitivity of large energy gluons to the details of the propagation during the initial stages. However, we have differences between both spectra at lower energies. Comparing Figs. 5.4 and 5.5 we can see that the transition region is located at ω/¯ωc∼τm/L (or, equivalently, ω∼¯ωm), as we anticipated in (5.18) when we defined the new energy scale ¯ωm=µ2τm/2. 5.4 HO approach As we previously mentioned in chapter 3, the HO approximation, which accounts for multiple soft scatterings during the emission of the gluon, relies on taking just the leading logarithmic behavior of the dipole cross section in coordinate space σ(r) (3.19) n(s)σ(r)≡n(s)ˆq eiq·rσ(q)≈1 2ˆq(s)r2+O(r2ln r2).(5.19) 106 5 The role of initial stages on jet quenching The natural way of expressing the HO medium-induced radiation spectrum is in coordinate space, thus we take the Fourier transform of the full spectrum written in momentum space (5.2) ωdI dωd2k=2αsCR (2π)3ωRe ˆL τm ds n(s)ˆs τp dtˆd2xd2yi[σ(x)−σ(y)] ×x−y (x−y)2·∂rK(s, x;t, r)|r=0 e−1 2´L sdξn(ξ)σ(y)e−ik·y, (5.20) where we employed the following relations P(L, k;s, l) = ˆd2ye−i(k−l)·ye−1 2´L sdξn(ξ)σ(y),(5.21) e K(s, q;t, p) = ˆd2xd2re−i(q·x−p·r)K(s, x;t, r).(5.22) We proceed as we did for the full spectrum and isolate the contribution coming from the radiation of the early stages in the scenario where the initial hard parton is created before the hydrodynamization of the medium τm≥τpby diving the time integral over tin two terms at τm ωdI dωd2k=2αsCR (2π)3ωRe ˆL τm ds n(s)ˆτm τp dtˆd2xd2yi[σ(x)−σ(y)] ×x−y (x−y)2·∂rK(s, x;t, r)|r=0 e−1 2´L sdξn(ξ)σ(y)e−ik·y +2αsCR (2π)3ωRe ˆL τm ds n(s)ˆs τm dtˆd2xd2yi[σ(x)−σ(y)] ×x−y (x−y)2·∂rK(s, x;t, r)|r=0 e−1 2´L sdξn(ξ)σ(y)e−ik·y. (5.23) In this expression we can simplify the first term employing the convolution property of the kernel in coordinate space K(s, x;t, r) = ˆd2zK(s, x;τm,z)K0(τm,z;t, r),(5.24) where K0is the vacuum kernel in coordinate space, given by the Fourier transform of eq. (2.49) K0(τm,z;t, r) = −iω 2π(τm−t)exp (iω 2 (z−r)2 (τm−t)).(5.25) 107 Marcos Gonz´ alez Mart´ ınez Plugging (5.24) and (5.25) into the first term of (5.23) we get ωdI dωd2k=2αsCRω (2π)4Re ˆL τm ds n(s)ˆτm τp dt (τm−t)2ˆd2xd2yd2zi[σ(x)−σ(y)] ×z·(y−x) (x−y)2exp iω 2 z2 τm−tK(s, x;τm,z)e−1 2´L sdξn(ξ)σ(y)e−ik·y +2αsCR (2π)3ωRe ˆL τm ds n(s)ˆs τm dtˆd2xd2yi[σ(x)−σ(y)] ×x−y (x−y)2·∂rK(s, x;t, r)|r=0 e−1 2´L sdξn(ξ)σ(y)e−ik·y. (5.26) As it happened with the fully resummed spectrum (5.6), the above expression has two contributions, the second term is just the emission spectrum in the situation where the emitter is created at the same time as the medium τp=τm, whereas that the first contribution accounts for the additional radiation coming from the propagation through vacuum during the initial stages. 5.4.1 HO for the brick Now we restrict ourselves to the brick scenario where the medium is considered to be static with a constant jet quenching parameter ˆq(s) = ˆq0for s∈[τm, L]. The static case is one of the few in which the HO emission kernel can be computed analytically, and it is given by [70,106] KHO(s, x;t, r) = −iωΩ 2πsin [Ω(s−t)] exp iωΩ 2 sin [Ω(s−t)] cos [Ω(s−t)] x2+ cos [Ω(s−t)] r2−2x·r, (5.27) with Ω defined as Ω≡(1 −i) 2rˆq0 ω.(5.28) We will show in the next section the expressions of the kernel and spectrum obtained employing the HO approach in the case where the jet quenching parameter has a powerlaw dependence on s. These expressions will be used later in section 5.5 when discussing the simultaneous description of RAA and high-pTv2. For other profiles we refer the reader, for instance, to Refs. [106,107]. Introducing (5.19) and (5.27) in (5.26) we can perform the integrals over position variables xand y. By doing so we get the following expression for the HO transverse 108 5 The role of initial stages on jet quenching momentum dependent spectrum ωdIHO dωd2k=αsCR π2ωRe ˆL τm dsˆτm τp dtiˆq0 he A(s, t)i2 1 ˆq0(L−s)−2iωΩe A(s,t) e B(s,t) ×exp   −k2 ˆq0(L−s)−2iωΩe A(s,t) e B(s,t)    +αsCR π2ωRe ˆL τm dsˆs τm dtiˆq0sec2[Ω(s−t)] ˆq0(L−s)−2iωΩ cot[Ω(s−t)] ×exp −k2 ˆq0(L−s)−2iωΩ cot[Ω(s−t)], (5.29) where we defined the following functions e A(s, t)≡cos[Ω(s−τm)] −Ω(τm−t) sin[Ω(s−τm)] ,(5.30) e B(s, t)≡sin[Ω(s−τm)] + Ω(τm−t) cos[Ω(s−τm)] .(5.31) Now we have to integrate the k-dependent HO spectrum (5.29) over transverse momentum to get the energy distribution. Starting with the case where the kinematical condition is kept 0 ≤k < ω we get (after rescaling time variables t→tL and s→sL) ωdIHO dω= −2αsCR πRe ˆ1 τm/L dsˆτm/L τp/L dtiωc ωA2(s, t) exp   −Rω2 2ω2 c1−s−iωΩLA(s,t) ωcB(s,t)  −1  −2αsCR πRe ˆ1 τm L dsˆs τm L dtiωc ωcos2[ΩL(s−t)]  exp   −R ω2 2ω2 ch1−s−cot[ΩL(s−t)] ΩLi  −1 , (5.32) with Aand B, respectively, the rescaled versions of the previously defined e Aand e B functions A(s, t)≡cos [ΩL(s−τm/L)] −ΩL(τm/L −t) sin [ΩL(s−τm/L)] ,(5.33) B(s, t)≡sin [ΩL(s−τm/L)] + ΩL(τm/L −t) cos [ΩL(s−τm/L)] ,(5.34) being ωcand R ωc≡1 2ˆq0L2,and R≡ωcL . (5.35) 109 Marcos Gonz´ alez Mart´ ınez Here Rplays the role of the kinematical constraint, as ¯ Rdid before. Thus, performing the integration over all transverse momentum space is equivalent to taking the R→ ∞ limit with ωcfixed. Imposing R→ ∞ in (5.32) we obtain ωdIHO dω=2αsCR πln cos hΩL1−τm Li−ΩLτm−τp Lsin hΩL1−τm Li.(5.36) One should notice that the quantity ΩLcan be written as a function of ω/ωconly ΩL= (1 −i)rωc 2ω,(5.37) thus, if τpis set to τp=τmor τp= 0, as we always do in this manuscript, then the energy distribution without kinematic cut-off depends just on ω/ωcand τm/L. If we keep the kinematical condition finite then Ris also a free parameter. As we did with the GLV result, we can employ the general HO spectrum without kinematical constraint where the emitter is created before the medium (bottom panel in Fig. 5.1) (5.36) to derive a new energy scale which will determine the border between the regions where the details about the propagation during the initial stages are relevant or not. We just need to realize that (5.36) can be written in terms of the spectrum where the initial hard parton is produced inside the medium (scenario A) in the top panel of Fig. 5.1) in the following way ωdIHO dωτm>τp=0 =ωdIHO dωτm=τp=0 −2αsCR πln  cos (ΩL) cos [ΩL(1 −τm/L)] −Ωτmsin [ΩL(1 −τm/L)], (5.38) where, unlike what happened for the GLV result (5.11), now the last term is not the medium-induced radiation spectrum for the case where the emitter is created at the same time as the medium, through which it propagates from τp= 0 to τm. Once that we have (5.38), we rewrite the numerator inside the logarithm of the last term cos (ΩL) = cos [ΩL(1 −τm/L)] cos (ΩL τm/L)−sin [ΩL(1 −τm/L)] sin (ΩL τm/L), (5.39) in the |Ωτm| ≪ 1 limit this equation becomes cos (ΩL)|Ωτm|≪1 −−−−−→ cos[ΩL(1 −τm/L)] −Ωτmsin [ΩL(1 −τm/L)] ,(5.40) which is exactly the expression of the denominator of the second term on the r.h.s. of eq. (5.38). Therefore, this term goes to 0 in the |Ωτm| ≪ 1 limit, setting a new energy scale ωmdefined by ωm≡1 2ˆq0τ2 m=ωc τ2 m L2,(5.41) 110 5 The role of initial stages on jet quenching 5.5.2 HO approach We decided to employ the HO framework in our analysis because the QWs formalism and its phenomenological applications have been mostly developed in this approximation. As we have been considering during this section, the initial hard quark is created at time τp, and then propagates following a straight line. This trajectory can be either along the in-plane or the out-of-plane directions of the QGP created in the heavy ion collision. We can see in (5.53) that the only parameters we need to specify to compute the HO energy distribution for an expanding medium following a power-law behaviour are ωand ˆq. The gluon energy ωis also used as a variable for evaluating the QWs (5.56), so we compute the spectrum for a large range of ωvalues. However, regarding the jet quenching parameter ˆq(ξ), we need to specify the relation between this local transport coefficient and the temperature T(ξ) for a given proper time of the trajectory ξ. Motivated by [142] we set this relation between the two quantities in the QGP phase to be5 ˆq(ξ) = K1T3(ξ),(5.60) and, as ˆq(ξ) needs to follow a power-law dependence with respect to ξto be able to have an analytical solution for the energy spectrum, we establish the following relation for the temperature T(ξ) = T0ξ0 ξ+ξ0c ,(5.61) with ξ0,T0and csome constant parameters that we determine by comparing (5.61) with temperature profiles extracted from taking straight lines within the 2+1 viscous hydrodynamic model for √sNN = 2.76 TeV Pb-Pb collisions described in Refs. [111,112]. We considered several trajectories and the power-law decay was a good approximation for the behaviour of the temperature in all of them. As mentioned in chapter 4, the starting point of this hydrodynamic model is at proper time τhydro = 1 fm, its initial condition is given by a factorized Kharzeev-Levin-Nardi model [143] and fixes the ratio of shear viscosity to entropy density η/s to η/s = 0.16, being constant for the whole evolution. The simulation considers the medium to be in chemical equilibrium described by an equation of state inspired by Lattice QCD calculations and it sets a freeze-out temperature at Tf= 140 MeV. Further details concerning the description of the hydrodynamic model can be found in [111, 112]. Taking this in account, we decided to set ξ0=τhydro = 1 fm and then determine the values of the other two constants cand T0by identifying (5.61) to the value of the temperature at initial and final proper time of the trajectory. Employing this procedure we get cin = 0.72 and Tin 0= 610 MeV for the in-plane direction, and cout = 0.67 and Tout 0= 589 MeV for the out-of-plane direction. With this consideration we can compute the HO energy distribution (5.53) for the three initial stages scenarios that we are analyzing (see Fig. 5.1) in both paths. Once that we have the gluon energy spectrum, we calculate the corresponding QWs P(ϵ), as 5In this study we do not take in account any quenching in the hadronic phase. 117 Marcos Gonz´ alez Mart´ ınez Case K1 A: τp=τm= 0 fm 1.5 B: τp=τm= 1 fm 16 C: τp= 0 and τm= 1 fm 8 Table 5.1: K1values obtained by fitting eq. (5.58) to ALICE RAA data [141] with pT> 6 GeV for the 20 −30% centrality class in √sNN = 2.76 TeV Pb-Pb collisions. We present results for the three different initial stages scenarios analyzed here, which are illustrated in Fig. 5.1. it is done in [70], for the in-plane and out-of-plane directions. The QWs are then used to derive the quenching factors (5.57) in both paths employing n= 4, which are later employed to compute QAA (5.58). We repeat the procedure several times for different K1 values and perform a χ2-fit to the ALICE single-inclusive suppression RAA data for the 20−30% centrality class in √sNN = 2.76 TeV Pb-Pb collisions [141] in order to obtain the optimal K1value. This best value of the fitting parameter K1is finally used to compute the azimuthal asymmetry w2(5.59), following [119]. We present in table 5.1 the results obtained for the fitting parameter K1for the three different initial stages analyzed in this manuscript: emitter and medium created at the same time filling the initial stages with medium τp=τm= 0 fm; emitter and medium produced at the same time neglecting the propagation prior to the hydrodynamization τp=τm= 1 fm; and allowing the emitter to propagate through vacuum before the creation of the medium τm= 1 fm and τp= 0 fm. It is important to notice that for the first scenario (top picture in Fig. 5.1) we need to perform some extrapolation of the temperature to have non-zero results before the hydrodynamical model is applicable, which we do by employing (5.61) outside the range where it is derived ξ < τhydro. For the other two cases we simply take τm=τhydro and then apply (5.61) for τm≤ξ≤L. The results in table 5.1 manifest the great discrepancies for the transport coefficient ˆqbetween the three different cases, demonstrating that the jet quenching parameter highly depends on the details of the initial stages modeling (this can also be seen in [15]). As it is obvious, the largest medium (case A), τp=τm= 0) is the one which needs the smallest ˆqvalue to produce the same amount of quenching as the other two scenarios. The difference between having vacuum propagation before the creation of the medium or not (cases C) τp= 0, τm= 1 fm and B) τp=τm= 1 fm, respectively) results in just a factor 2 of difference in the value of K1. With these results it is evident that the treatment of the early stages of the collision has a crucial importance in jet quenching studies which extract the value of the jet quenching parameter by fitting the data of single-inclusive suppression observables. We present in Fig. 5.8 the results for our proxies of RAA (left) and high-pTv2(right) for the three different initial stages scenarios considered as a function of the transverse 118 5 The role of initial stages on jet quenching 10 20 30 40 50 pT(GeV) 0.0 0.2 0.4 0.6 0.8 1.0 QAA τp=τm= 0 τp=τm= 1 fm τp= 0 and τm= 1 fm 10 20 30 40 50 pT(GeV) 0.02 0.04 0.06 0.08 0.10 0.12 w2 HO Figure 5.8: Left: QAA, our proxy for the single-inclusive particle suppression RAA, as a function of pT. Right: w2, our proxy for the high-pTazimuthal asymmetry v2, as a function of pT. In each panel we show the results obtained employing the HO approach for the three different scenarios discussed in the thesis: τp=τm= 0 fm (green solid lines), τp=τm= 1 fm (blue dashed lines); and τp= 0 fm and τm= 1 fm (red dashed dotted lines). All of the results are for the 20 −30% centrality class in √sNN = 2.76 TeV Pb-Pb collisions at the LHC. Figure extracted from [4] under a Creative Commons license, CCBY 4.0. momentum pT. As we saw before, the values of the fitting parameter K1obtained for the different cases, which are shown in table 5.1, presented large discrepancies. However, the left plot of Fig. 5.8 shows that all of the scenarios reproduce (almost) the same quenching QAA, which is in agreement with previous findings [116,119,128,129]. Moreover, even though all of the initial stages scenarios have the same amount of energy loss, they yield different results for the azimuthal asymmetry, as it can be seen on the right panel. These results are in agreement with [119], were the authors showed that delaying the starting point of the hydrodynamic model increased the azimuthal asymmetry, and they evidence the importance of properly describing the medium-induced radiation during the initial stages in phenomenological studies. Nonetheless, the correct way of eliminating the parton-medium interactions during the early stages is not to neglect the propagation until the hydrodynamization time like it is usually done (blue dashed curves in Fig. 5.8), but to allow the hard parton to travel through vacuum before entering the medium, as we propose in this work (red dashed dotted curves), and which results in a smaller decrease of the azimuthal asymmetry with respect to the previous case. 119 Marcos Gonz´ alez Mart´ ınez 5.5.3 Fully resummed spectrum We conclude the study of the effect of initial stages on jet quenching observables by repeating the analysis done in the previous section employing now a better description of the medium-induced energy distribution which includes full resummation over multiple scatterings (5.8). For the all-order spectrum the parameters we need to relate with the the medium temperature are nand µ, and we decide to set the following relations n(ξ) = C1T(ξ),and µ2(ξ) = 6παs eT2(ξ),(5.62) being T(ξ) the local medium temperature, which in this case we take directly from the hydrodynamical model in [111,112], we keep αs= 0.3 constant as we have done throughout the whole thesis and C1is our only fitting parameter. For the second relation in (5.62) we have taken the leading order HTL result for the Debye mass mD[57] and used the leading logarithmic matching between the Yukawa and HTL cross sections employed in previous works [58,92], which results in the following relation between the screening and Debye masses µ2=m2 D/e. We could have not employed the second relation in (5.62) and simply use µ2∝T2(as we did in chapter 4), having then a second fitting parameter to take in account in the analysis. However, as we are only employing a toy model to derive qualitative results, we do not need to achieve highly precise fitting results and we decided to keep things as simple as possible by having just one fitting parameter. The fully resummed BDMPS-Z spectrum always needs to be computed by numerical methods, thus in this case there is no advantage in parametrizing the temperature with a particular profile which allows for analytical simplifications that relieve the computational power needed to perform the evaluation and we just take T(ξ) directly from the hydrodynamical model described in [111,112]. There is, however, one case where the hydrodynamical description is not enough to perform the computation and we need to make some assumption about the temperature. The scenario where we set τm= 0, illustrated in the top picture of Fig. 5.1, has parton-medium interactions before the hydrodynamization time τhydro = 1 fm, thus, we need to extrapolate the temperature before the hydrodynamical simulation is applicable. We decided then to fit the temperature coming from the hydrodynamic model to (5.61) and employ the outcome function for ξ < τhydro. With these considerations we are now capable of reproducing the analysis performed in the previous section: we compute the all-order medium-induced emission spectrum for the three studied scenarios in the in-plane and out-of-plane directions6, with both spectra we compute the QWs and quenching factor for both paths, which are then used to compute QAA (5.58) using n= 4. QAA is evaluated for several values of the fitting parameter C1, and then we perform a χ2-fit to the ALICE RAA data for the 20 −30% centrality class in √sNN = 2.76 TeV Pb-Pb collisions [141] at the LHC. With this fit we obtain the optimal value of the free parameter C1for every scenario described in Fig. 5.1 and, finally, we use this C1value to obtain our proxy for the high-pTazimuthal asymmetry w2(5.59). 6The numerical evaluation of the all-order spectrum for a longitudinally expanding medium is performed in exactly the same way as it was explained in chapter 4, without using any scaling law. 120 5 The role of initial stages on jet quenching Case C1 A: τp=τm= 0 fm 2.2 B: τp=τm= 1 fm 4.5 C: τp= 0 and τm= 1 fm 2.5 Table 5.2: C1values obtained by fitting eq. (5.58) to ALICE RAA data [141] with pT> 6 GeV for the 20 −30% centrality class in √sNN = 2.76 TeV Pb-Pb collisions. We present results for the three different initial stages scenarios analyzed here, which are illustrated in Fig. 5.1. 10 20 30 40 50 pT(GeV) 0.0 0.2 0.4 0.6 0.8 1.0 QAA τp=τm= 0 τp=τm= 1 fm τp= 0 and τm= 1 fm 10 20 30 40 50 pT(GeV) 0.02 0.04 0.06 0.08 w2 Full Figure 5.9: Left: QAA, our proxy for the single-inclusive particle suppression RAA, as a function of pT. Right: w2, our proxy for the high-pTazimuthal asymmetry v2, as a function of pT. In each panel we show the results obtained employing the BDMPS-Z spectrum with full resummation over multiple scatterings for the three different scenarios discussed in the thesis: τp=τm= 0 fm (green solid lines), τp=τm= 1 fm (blue dashed lines); and τp= 0 fm and τm= 1 fm (red dashed dotted lines). All of the results are for the 20 −30% centrality class in √sNN = 2.76 TeV Pb-Pb collisions at the LHC. Figure extracted from [4] under a Creative Commons license, CC-BY 4.0. We show in table 5.2 and Fig. 5.9 the outcomes of our study. In table 5.2 we have listed the vales of the fitting parameter C1for every initial stages model. As in the HO case, C1is different for each scenario so the same amount of quenching is achieved with media of different lengths. However, now the values of the fitting parameter have much smaller discrepancies, pointing out that having a better description of the energy distribution gives more robust results than relying on approximations. Fig. 5.9 contains 121 Marcos Gonz´ alez Mart´ ınez the results for our proxies of RAA (left) and high-pTv2(right) for the three different initial stages scenarios considered as a function of pT. Once more, the results obtained with the full spectrum are in agreement with the ones coming from the HO approach, but now the different scenarios are mode differentiated in the right panel. Again, we have the same amount of energy loss (all curves agree in the left panel) but with different azimuthal asymmetry, and now the case with vacuum propagation before the QGP phase (red dashed dotted line) is closer to the scenario which neglects the radiation coming from the initial stages (blue dashed line) compared to the HO framework (see right plots of Figs. 5.8 and 5.9). This scenario also exhibits more disparities when compared to the case where the medium is created at τm= 0 (green solid line). These results are somehow expected if we take a look at the spectra outcomes obtained in sections 5.2 and 5.4. For the full result the separation between different scenarios is clearer than for the HO approach, having larger discrepancies spectra with different considerations regarding the initial stages propagation. Our results show for the first time the strong dependence of this set of observables on the description of the initial stages after the collision in a formalism with full resummation of multiple scatterings with a realistic medium-probe interaction. Reaching this result within a more accurate calculation of the medium-induced radiation, which does not require the usually employed approximations like HO or GLV, reveals the potential of jet quenching to constraint the dynamics of the initial stages. This is just the first step towards a deep understanding of the description of initial stages in heavy ion collisions and more exhaustive analysis will be done in the future. For example, we used the most simple assumption regarding the propagation during the early stages. We could replace the propagation through vacuum by a detailed model of the interactions in the pre-QGP phase. Presumably, this change would have a large effect in initial stages radiation, according to recent results [121–124]. Moreover, we used a toy model to derive the RAA and v2in the different scenarios, whose results can only be employed to extract qualitative conclusions. The natural step to follow is to perform a more thorough phenomenological analysis, like in [119], using the general fully resummed energy distribution with propagation during the initial stages that we derived here, which will be done in future works. 122 6 Summary, Results and Conclusions In this thesis we employed jet quenching to study the Quantum Chromodynamics (QCD) collectivity. More precisely, we have improved the current knowledge about the emission process, achieving a precise description of the medium-induced gluon radiation spectrum in realistic conditions. This is essential to accurately analyze the properties of the quark gluon plasma (QGP), as many phenomenological works employ this spectrum as basic input. After a general introduction where high-energy propagation and radiation in a background field is briefly explained, we present, in chapter 2, the new computational tool that will be employed in the rest of the chapters. This new tool allows to perform a resummation of arbitrary number of scatterings for a realistic potential. The main idea is to solve the differential equations that satisfy both the emission kernel and the momentum broadening, instead of directly computing them, which is very difficult to perform in a general scenario. This approach is very flexible, allowing to include any elastic collision rate model or medium profile, but it has the disadvantage of being highly computationally demanding. The original contributions of the thesis began in the third chapter, where we used this framework to improve the current knowledge of the physics behind the emission process. We did so by using the medium-induced radiation spectrum with full resummation over multiple scatterings as a baseline w.r.t. which we compared several analytical approximations. As every approach relies on different assumptions regarding the physical interpretation of the emission process, the formalism which provides the most precise description of the full spectrum would be the one which better encodes the information about the gluon radiation. We divided the energy distribution in three different regions: Marcos Gonz´ alez Mart´ ınez high, intermediate and low energy regimes. For the the high energy region, where the gluon energy is higher than the characteristic frequency of the medium, we checked that the Gyulassy-Levai-Vitev (GLV) result is indeed the correct limit of the full spectrum. Therefore, in this regime one single hard scattering is enough to precisely describe the radiation process, and accounting for multiple scatterings is not necessary, as expected. Going a little lower in energy, we reach the multiple scattering regime. In this region it is mandatory to consider more than just one scattering, as in this regime the Landau-Pomeranchuk-Migdal (LPM) effect gains relevance and suppresses the spectrum comparing to the single radiation scenario. This is the region where we typically use the Harmonic Oscillator (HO) approximation for describing the spectrum. However, performing quantitative comparisons between the full spectrum and the Harmonic Oscillator approximation is not possible, as when we define the jet quenching parameter in the HO formalism we introduce and arbitrary upper cut-off. Hence, the mapping between the parameters of both frameworks would depend on this arbitrary choice. Fortunately, in recent years, the Improved Opacity Expansion (IOE) has been developed, an approximation which performs an expansion of the emission kernel using the HO solution as Leading Order (LO) contribution. This approach allows us to perform quantitative comparisons, thus we decided to compare the IOE with our full result. By doing so we found out that the IOE described with high precision (in its region of applicability) the all-order spectrum, whereas the GLV result largely overpredicted it. Therefore, accounting for multiple scatterings in this region is essential to correctly describe the emission process. Finally, in the Bethe-Heitler (BH) region there is no analytical framework which computes the energy distribution. Thus here we derived, for the first time, the low energy limit of the complete spectrum, which gives an accurate approximation of the full result for energies of the gluon small enough. This newly derived result has the advantage of having an straightforward interpretation of the emission process: in this region the spectrum is given by the single scattering approximation multiplied by a suppression factor which accounts for the probability of not having further scatterings. The noscattering probability factor comes from the contribution of virtual terms. Removing this factor leads to a high overestimation of the spectrum, thus the first opacity result does not correctly describe the radiation process in this region, which was generally assumed before. This is the first time that the contribution of multiple scatterings to the emission process in the low energy region is described. Moving on towards the improvement of the framework that we use to compute the spectrum, the following chapter of the thesis focused on computing the spectrum beyond the brick approximation, which assumes that the background medium is a static collection of scattering centres. As we know, this is an unrealistic approximation, as the QGP is an evolving medium well-described by hydrodynamical simulations. However, the brick assumption is commonly employed in theoretical analysis as a first step which could provide us with insightful results before performing more realistic and complicated 124 6 Summary, Results and Conclusions computations. Nevertheless, computing the spectrum in evolving media is a mandatory step to be able to implement our framework in high precision phenomenological studies which describe experimental data. As we mentioned before, our method can be applied to any medium profile, thus the brick approximation can be easily improved, but at the cost of requiring a great amount of computing time. This makes the computation unpractical for phenomenological works which need to evaluate the spectrum several times in different conditions. To overcome this limitation we explored the use of scaling laws, a simple set of relations between parameters that allow us to identify spectra computed in different conditions. Scaling laws where used in the past with high success within the HO framework using as reference scenario the static case. Employing these scaling laws makes it possible to pre-tabulate the spectrum for an extensive range of the values of parameters, and, when we need to compute the spectrum for a specific trajectory, one can apply the scaling laws to extract the pertinent value from the pre-computed results. However, we found that the performance of the scaling laws using as reference the static scenario was not satisfactory for the all-order spectrum, so we decided to explore other options. Given that many hydrodynamical simulations that describe the QGP provide temperature profiles which decay over time as a power-law to a good approximation, we chose to employ a power-law medium profile as a reference to define a new set of scaling laws. The performance of this newly proposed set of scaling laws was excellent for all of the energy range of the spectrum, an even in extreme scenarios, including the effects of fluctuations or changing the model of the collision rate employed. Therefore, we can conclude that the tabulation of the fully resummed spectrum for this power-law expanding profile and the use of our proposed scaling laws will allow a successful implementation of this formalism in phenomenology. After enhancing the computation of the medium-induced radiation spectrum, the concluding chapter of the thesis was devoted to study a very important topic: analyze the role played by the initial stages of the collision on jet quenching observables. Traditionally, the propagation of the high energy parton during the early stages of the collision has been neglected in the jet quenching community, arguing that the short length of the initial stages (∼0.2−1 fm) compared with the total length of the QGP (∼5−10 fm) should make their contribution negligible. However, a few years ago it was pointed out that when describing simultaneously two jet quenching observables (RAA and high-pTv2), different treatments of the initial stages resulted in distinct descriptions of the experimental data. Therefore, finding the correct way of accounting for the contribution of the initial stages is essential to achieve high precision results in phenomenology. With that goal in mind, we compared three scenarios with different descriptions of the propagation of the high energy parton during the early stages. First, we considered the usual model employed in phenomenology, neglecting the whole contribution of the initial stages and setting the parton to be created at the same time as the medium after the collision. Second, we assumed, as before, that parton and medium are both formed simultaneously. However, this time we align this creation moment with the instant the collision occurs. Finally, we propose a new scenario never analyzed before where we consider 125 Marcos Gonz´ alez Mart´ ınez that the parton is produced with the collision and propagates, through vacuum, until the medium is created at the time the hydrodynamical simulation begins. By performing this comparison we were able to isolate the contribution of the initial stages to the mediuminduced gluon radiation. Analyzing the results at the level of the spectrum we found out that neglecting the propagation during the early stages gives an smaller spectrum than for any of the other two cases. Propagating through vacuum or medium during the initial stages provided the same results for the high energy tail of the spectrum. Nevertheless, great discrepancies between these two scenarios appeared in the low energy region, which corresponds to gluons with small formation time. This result can be understood taking in account that highly energetic gluons need larger transverse momentum transfers to decorrelate from the emitter, and, therefore, they have bigger formation times. Gluons which possess a formation time larger than the formation time of the medium τmcan be radiated before the hydrodynamization and still undergo the effects induced by the medium before being completely decorrelated from the emitter. However, gluons with small formation times only contribute to the in-medium radiation spectrum if they are emitted after the medium is created. This is the reason why the high energy tails remain unchanged and disagreements appear in the mid-low energy region. Moreover, by computing the different spectra using analytical approaches, like the HO and GLV, we found the new energy scales that separate the region where the effects of initial stages become relevant for the radiation spectrum. Finally, using the spectra evaluated for the three different initial stages scenarios, we computed proxies for the inclusive suppression of charged particles and the high momentum azimuthal asymmetry. With the spectrum we computed the quenching weights (QWs), which are distributions that encode the probability that the parton losses a certain amount of energy due to independent gluon emissions induced by the medium. These QWs allow us to compute the quenching factor, which is the ratio between the partonic medium and vacuum spectrum, and it can be used to estimate the inclusive suppression and azimuthal asymmetry. To achieve a simultaneous description of both observables, we computed the inclusive suppression for several values of the free parameter and fitted the results to experimental data of the RAA of collisions at the Large Hadron Collider (LHC). Then, we used the optimal value extracted from the fit to estimate the azimuthal asymmetry, having in that way a simultaneous description of both observables. As we have different medium configurations for every scenario, but we always fit the results to the same energy loss (same RAA value), the properties of the medium are different for each case. Consequently, the high momentum azimuthal asymmetry obtained also changes between distinct scenarios. This analysis was performed using the HO approach and the full spectrum, with similar conclusions for both frameworks, thus our results are robust. We conclude that, the treatment of the parton propagation during the initial stages is crucial in jet quenching studies. Finally, we want to remark that some of the current hot topics in the community, such as understanding the apparent lack of jet quenching in small systems [144] and the proper interpretation of jet quenching physics for intermediate A-A collisions, for instance in the upcoming O-O run at the LHC [145] and the proposed 126