Keywords: Probing gluon saturation in semi-hard γ Quantum chromodynamics (QCD), Saturation, Colour Glass Condensate, Shockwave formalism

At high-energy colliders, QCD processes involve at least two scales: the centreof-mass energy s and a hard scale Q 2 that allows the factorization of processes into a hard part computable using the perturbative methods of Feynman diagrams and a non-perturbative part such as parton distribution functions (PDFs). However, large logarithmic corrections can arise, compensating the smallness of the coupling constant.

They have to be resummed, leading to the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations at moderate energies. In the semi-hard limit or Regge-Gribov limit characterized by s ≫ Q 2 ≫ Λ 2 QCD , the large energy logarithms are resummed using the Balitsky-Fadin-Kuraev-Lipatov (BFKL) approach.

In this high-energy domain, gluons dominate the inner structure of nucleons. In both BFKL and DGLAP formalisms, a rise of the gluon distribution is predicted. This rise cannot indefinitely continue as it leads to a growth of cross-sections with energy that eventually violate unitarity. At very high energy, non-linear effects such as gluon recombination have been theoretically predicted, leading to gluon saturation. Strong hints of gluon saturation have been found in the currently available experimental data but they have not been conclusive.

It has been clear that its unambiguous discovery requires the identification and precise beyond leading order computations of observables that are sensitive to saturation physics.

Titre: Sonder la saturation des gluons dans les processus semi-durs γ ( * ) + p/A Mots clés: Chromodynamique quantique (QCD), Saturation, Condensat de verre de couleur, Formalisme des ondes de choc Résumé: Dans les collisionneurs à haute énergie, les processus QCD impliquent au moins deux échelles: l'énergie dans le centre de masse s et une échelle dure Q 2 qui permet la factorisation des processsus en une partie dure calculable via les méthodes perturbatives des diagrammes de Feynman et une partie non-perturbative comme par exemple les fonctions de distribution de partons (PDFs).

Cependant, des grands logarithmes peuvent apparaître compensant la petitesse de la constante de couplage.

Ces corrections doivent être resommées et cela mène aux équations d'évolution Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) aux énergies modérées. Dans la limite semi-dure, appelée aussi limite de Regge-Gribov, characterisée par s ≫ Q 2 ≫ Λ 2 QCD , des grands logarithmes de l'énergie doivent être resommés via l'approche Balitsky-Fadin-Kuraev-Lipatov (BFKL).

Dans cette limite des hautes énergies, les gluons dominent la structure interne des nucléons et que ce soit dans le formalisme de BFKL ou de DGLAP, la distribution de gluon augmente avec l'énergie du centre de masse, ce qui ne peut pas continuer indéfiniment car cela mène à une augmentation des sections efficaces avec l'énergie qui éventuellement viole l'unitarité. Aux très hautes énergies, des effets non-linéaires comme la recombinaison de gluons ont été prédits, menant à la saturation gluonique. Des signes de cette saturation ont été trouvés dans les données expérimentales actuelles, mais aucune n'est concluante.

Il est clair qu'une découverte sans ambiguïté de cette saturation gluonique nécessite l'identification et le calcul précis au-delà de l'ordre dominant d'observables sensible à la physique de la saturation.

Cette thèse participe aux récents efforts menés par la communauté des hautes énergies afin d'atteindre ce but.

Deux théories effectives équivalentes, qui sont toutes deux introduites dans le manuscrit, sont utilisées dans les calculs analytiques d'observables variées pouvant être étudiées expérimentalement au Grand collisionneur de hadrons (LHC) via des collisions ultrapériphériques ou dans des collisionneurs futurs, principalement le collisionneur électron-ion (EIC) et le Grand collisionneur électron-hadron (LHeC) via des diffusions profondément inélastiques inclusives ou diffractives.

Nous proposons et étudions tout particulièrement deux nouvelles classes de processus dans le but de mettre en évidence la saturation gluonique. Tout d'abord, l'approche des ondes de choc est utilisée dans le calcul à l'ordre sous-dominant en perturbation des sections efficaces de production diffractive d'une paire de hadrons et de production diffractive d'un seul hadron dans une collision γ ( * ) + p/A, le photon pouvant être réel ou virtuel. Nous démontrons d'abord que les différentes divergences (molles, colinéaires, en rapidité et ultra-violettes) s'éliminent entre elles puis nous extrayons analytiquement le terme fini. Par ailleurs, nous calculons à l'ordre dominant, à l'aide du formalisme du condensat de verre de couleur, la section efficace de production inclusive de quarkonium + gluon dans une collision profondément inélastique.

This thesis contributes to the efforts, in recent years, of the high-energy physics community towards such a goal. Two equivalent effective field theories -that will be both introduced in the thesis-have been employed to analytically compute various observables that can be studied experimentally at the Large Hadron Collider (LHC) with ultra-peripheral collisions or at future colliders, mainly the Electron-Ion Collider (EIC) and the Large Hadron electron Collider (LHeC) in diffractive or inclusive Deep Inelastic Scattering (DIS).

We propose and study two novel types of processes to investigate saturation physics. First, we use the shockwave approach to calculate at the next-to-leading order the cross-section for dihadron and single-hadron diffractive production in γ ( * ) + p/A collision, i.e. in electroor photoproduction.

We show the explicit cancellation of the different divergences (soft, collinear, in rapidity and ultraviolet) and extract the finite term. Second, we compute the crosssection for the inclusive quarkonium + gluon production in DIS at leading order using the Colour Glass Condensate (CGC) formalism.

First of all, I would like to thank my two supervisors, Lech Szymanowski and Samuel Wallon, for their expert guidance (especially for taking the time to answer all the numerous questions I could have) and support during this 3-year journey.

I would like to thank my collaborators Michael Fucilla, who played a big part in the making of this thesis and Andrey Grabovsky and Zhong-Bo Kang for the help when needed to move forward. I would also like to express my gratitude to Farid Salazar for finding time for the many meetings we have had. It was a pleasure to work with you.

I am thankful to Ronan McNulty, Michael Winn, Renaud Boussarie, Tolga Altinoluk, Guillaume Beuf, Heikki Mäntysaari, Tuomas Lappi and all the other people from the theoretical and experimental community that I had the pleasure to meet and discuss with for insightful and enlightening discussions.

I am grateful to the administrative team at IJCLab, EDPIF and Paris-Saclay University for their availability and their support, which make my work a pleasant experience.

I would like to thank the referees, Krzysztof Golec-Biernat and Dimitri Colferai, for taking the time to review my thesis. Thank you to Asmaa Abada-Zeghal, Jamal Jalilian-Marian, Charlotte Van Hulse, and François Gelis for agreeing to be part of my Jury. I look forward to our discussion at the actual defence.

I am grateful to NCBJ Warsaw for welcoming me and for their hospitality. Finally, I would like to thank my family for always supporting me and my friends for encouraging me, especially Hugo. Thank you for always being there for me.

Contents

General introduction I Unveiling saturation effects at past, present and future colliders

General introduction

Strong interaction is one of the four known fundamental interactions of nature. It governs the dynamics of the fundamental constituents of nuclear matter, the quarks and gluons commonly called partons. The quantum field theory that describes this interaction is Quantum Chromodynamics (QCD). It is based on the non-Abelian gauge group SU pN c q where N c " 3 is the number of quark colours. QCD successfully describes the strange properties of the strong interaction, one of them being colour confinement: isolated partons can never be detected, and only colour-neutral objects can be observed.

Despite this, particle colliders have proven to be particularly useful for the study of QCD properties. One main reason for that is the fact that strong interaction possesses the property of asymptotic freedom proven by Wilczek, Politzer and Gross [START_REF] Gross | Ultraviolet Behavior of Nonabelian Gauge Theories[END_REF][START_REF] Gross | Asymptotically Free Gauge Theories -I[END_REF][START_REF] Politzer | Reliable Perturbative Results for Strong Interactions?[END_REF]. It states that the interaction between colour sources becomes asymptotically weaker as the energy increases and the corresponding length scale decreases. This property means that processes containing large energy scales allow the use of perturbative QCD (pQCD) to describe them. Their study relies upon factorization theorems or schemes which allow the separation of the long-and short-distance dynamics in physical observables which are then convolution of a hard part computable using perturbative methods of Feynman diagrams representing the scattering of quasi-free partons and a nonperturbative part, encoding the long-distance dynamics. This long-distance dynamics cannot be described analytically, and one has to employ methods such as lattice QCD or to fit to experimental data, when dealing with spatial and momentum distributions of partons in proton and nuclei. The most basic distributions are parton distributions functions (PDFs) which give the 1D distributions of specific species of partons that carry a fraction x of the longitudinal momentum fraction of the proton or nucleus in question 1 . We also have the generalized parton distribution functions (GPD) that include the spatial information and the transverse momentumdependent parton distribution functions (TMD) that include the transverse momentum dependence on top of the x dependence. The Wigner distribution is the 5D ´x, ⃗ b, ⃗ k ¯parton distribution from which all the aforementioned distributions ensued from. Another way to approach the long-distance part, apart from the ones mentioned above, is to build phenomenological models from physically credible arguments.

Concerning the hard part, for any scattering process, at least two scales are usually involved: the center-of-mass scale energy squared s and a hard scale satisfying the condition

Q 2 " Λ 2 QCD
where Λ QCD is the QCD mass scale. This justifies the perturbative treatment in α s `Q2 ˘! 1. Although the perturbative approach is very powerful for computing processes, some issues can arise. Among them, one of the most important is the possible appearance of large logarithms of the involved kinematic scales and entering the perturbative series with power increasing along with the order. In these cases, a resummation to all orders of such contributions becomes mandatory. One example of the origin of such logarithms is related to the cancellation of infrared (IR) divergences in IR-safe observables. Using the dimensional regularization procedure, one can have, with ϵ " pD ´4q{2,

1 ϵ `Q2 ˘ϵ " 1 ϵ `ln `Q2 ˘`Opϵq (0.0.1)
which appear when considering collinear gluon emission in the massless quark limit. The pole cancels in IR-safe observables but the logarithm remains at any order in perturbation theory, and corrections of the type α n s ln p `Q2 ˘appear. In the Bjorken limit Q 2 Ñ 8, with and x Bj fixed to non parametrically small values, the resummation of such logarithms leads to the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) [START_REF] Altarelli | Asymptotic Freedom in Parton Language[END_REF][START_REF] Dokshitzer | Calculation of the Structure Functions for Deep Inelastic Scattering and e+ e-Annihilation by Perturbation Theory in Quantum Chromodynamics[END_REF][START_REF] Gribov | Deep inelastic e p scattering in perturbation theory[END_REF][START_REF] Lipatov | The parton model and perturbation theory[END_REF] evolution equations.

In the Regge-Gribov limit or semi-hard region characterized by

s " Q 2 " Λ 2 QCD ,
the logarithmic corrections are of the form lnps{Q 2 q coming from the above hierarchy of scales and related to the occurrence of large rapidity intervals. The Balitsky-Fadin-Kuraev-Lipatov (BFKL) approach is the formalism used to resum to all orders such corrections in the leading logarithmic approximation (LLA) [START_REF] Balitsky | Operator expansion for diffractive high-energy scattering[END_REF][START_REF] Balitsky | Factorization for high-energy scattering[END_REF][START_REF] Balitsky | Factorization and high-energy effective action[END_REF][START_REF] Balitsky | Effective field theory for the small x evolution[END_REF][START_REF] Kovchegov | Small x F(2) structure function of a nucleus including multiple pomeron exchanges[END_REF][START_REF] Kovchegov | Unitarization of the BFKL pomeron on a nucleus[END_REF] meaning resumming corrections of the form α n s ln n ps{Q 2 q and at next-leading-logarithmic approximation (NLLA) [START_REF] Balitsky | Quark contribution to the small-x evolution of color dipole[END_REF][START_REF] Balitsky | Next-to-leading order evolution of color dipoles[END_REF][START_REF] Lappi | Next-to-leading order Balitsky-Kovchegov equation beyond large N c[END_REF][START_REF] Kovchegov | Triumvirate of Running Couplings in Small-x Evolution[END_REF] i.e. resumming corrections of the form α n`1 s ln n ps{Q 2 q. In this approach, cross-sections of hadronic collisions are the convolution of two impact factors related to the transition from each colliding particle to their respective final state and a process-independent Green's function. This approach has been widely accepted as it predicts, for example, the observed rapid growth of the total inclusive deep inelastic scattering (DIS) cross-section with increasing energy. It is also powerful as it can be applied to a wide range of contexts such as in Mueller-Navelet jets production [START_REF] Ducloué | Evidence for high energy resummation effects in mueller-navelet jets at the lhc[END_REF][START_REF] Ducloue | Confronting Mueller-Navelet jets in NLL BFKL with LHC experiments at 7 TeV[END_REF][START_REF] Colferai | Mueller Navelet jets at LHC -complete NLL BFKL calculation[END_REF][START_REF] Ducloué | Probing BFKL dynamics in Mueller-Navelet jet production at the LHC[END_REF]. At last, it is consistent with pre-QCD results derived from Regge theory.

However, the power-like growth with s of cross-sections from BFKL formalism is incompatible with the Froissart-Martin bound [START_REF] Froissart | Asymptotic behavior and subtractions in the mandelstam representation[END_REF][START_REF] Martin | Unitarity and high-energy behavior of scattering amplitudes[END_REF], which stipulates that cross-sections cannot grow faster than A ln 2 psq with A a constant. This eventual violation of the bound is physically interpreted as an infinite growth of the gluon density in the Regge-Gribov region, growth that cannot go untamed. The inner structure of hadrons appears to become a denser and denser gluon system. At some critical gluon density per transverse area, non-linear effects like gluon recombination will slow the growth of the gluon density, leading to gluon saturation. The non-linear generalization of the BFKL equation then describes evolution in energy. The most general one is the Balitsky hierarchy of equations derived in the so-called shockwave approach [START_REF] Balitsky | Operator expansion for diffractive high-energy scattering[END_REF][START_REF] Balitsky | Factorization for high-energy scattering[END_REF][START_REF] Balitsky | Factorization and high-energy effective action[END_REF][START_REF] Balitsky | Effective field theory for the small x evolution[END_REF] or its equivalent formulation the Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov and Kovner (JIMWLK) equation [START_REF] Jalilian-Marian | The BFKL equation from the Wilson renormalization group[END_REF][START_REF] Jalilian-Marian | The Wilson renormalization group for low x physics: Gluon evolution at finite parton density[END_REF][START_REF] Jalilian-Marian | The Wilson renormalization group for low x physics: Towards the high density regime[END_REF][START_REF] Jalilian-Marian | Unitarization of gluon distribution in the doubly logarithmic regime at high density[END_REF][START_REF] Kovner | Relating different approaches to nonlinear QCD evolution at finite gluon density[END_REF][START_REF] Weigert | Unitarity at small Bjorken x[END_REF][START_REF] Iancu | Nonlinear gluon evolution in the color glass condensate. 1[END_REF][START_REF] Ferreiro | Nonlinear gluon evolution in the color glass condensate. 2[END_REF][START_REF] Iancu | The Renormalization group equation for the color glass condensate[END_REF] derived in the so-called Colour Glass Condensate (CGC) formalism. Such dense systems can be achieved in a highly energetic proton but more easily in a heavy nucleus.

While gluon saturation is theoretically well-motivated, only hints of it have been observed in past and present collider experiments, namely at the Hadron-Electron Ring Accelerator (HERA), at the Relativistic Heavy Ion Collider (RHIC) and at the Large Hadron Collider (LHC). No unambiguous signal has been found within the uncertainties of the theory and the experiments. Clearer signals will hopefully be observed at future electron-hadron collider, in particular at the Electron-Ion Collider (EIC) [START_REF] Boer | Gluons and the quark sea at high energies: Distributions, polarization, tomography[END_REF][START_REF] Aschenauer | The electron-ion collider: assessing the energy dependence of key measurements[END_REF][START_REF] Accardi | Electron Ion Collider: The Next QCD Frontier: Understanding the glue that binds us all[END_REF][START_REF] Khalek | Science Requirements and Detector Concepts for the Electron-Ion Collider: EIC Yellow Report[END_REF][START_REF] Brüning | Electron-Hadron Colliders: EIC, LHeC and FCC-eh[END_REF]. In recent years, the high-energy theoretical community has been actively collaborating to improve the theoretical understanding of processes sensitive to saturation and to make the theoretical predictions as accurate as possible by promoting calculations to at least next-to-leading order (NLO) with a view of matching the experimental endeavours for precise data.

This thesis is a contribution to this effort, focusing on γ p˚q P collisions where the photon could be either virtual as in Deep Inelastic Scattering (DIS) or real as in Ultra-Peripheral Collisions (UPC) and P is a nucleon or nucleus target. This work is divided into three parts. In the first part, a motivation and brief review for the research on gluon saturation is presented. The second part is dedicated to a rather detailed presentation of the different effective field theories which have been developed for the study of saturation physics (in the eikonal approximation). The third and final part are original contributions. We present several processes computed within these effective theories that are theoretically sensitive to gluon saturation. More specifically, in chapter 1, we introduce the concept of DIS and UPC as well as a subclass of processes that is diffractive processes that are also of interest in this work. The discussion on DIS will allow for a brief description of the two main kinematic regimes of QCD, i.e. the Bjorken and the Regge-Gribov limit and the different factorization schemes to use depending on these kinematic regimes. We also present the dipole picture, which forms the basis of calculations in the highenergy limit. In chapter 2, we review non-exhaustively the search of saturation at past and present colliders HERA, LHC and RHIC as well as at future colliders, focusing on the EIC and the Large Hadron electron Collider (LHeC). We explain why the past and current experimental signals are unclear and various predictions for future electron-hadron colliders. Chapter 3 is a brief introduction to the shockwave formalism, while chapter 5 is devoted to the Colour Glass Condensate (CGC) effective field theory. In chapter 4, we report schematically and for completeness on the derivation of the γ p˚q Ñ q q diffractive impact factors and γ p˚q P Ñ q q diffractive cross-sections at one-loop accuracy and the γ p˚q Ñ q qg diffractive impact factors and diffractive γ p˚q P Ñ q qgP 1 at Born level, computed within the shockwave formalism. These hard diffractive cross-sections are used in chapters 6 and 7 to compute the diffractive doubleand single-hadron electro-or photoproduction on a nucleon or nucleus at NLO accuracy. The cancellation of divergences is explicitly demonstrated, and we also extract the finite parts of the differential cross-section. The di-hadron work is the topic of [START_REF] Fucilla | NLO computation of diffractive di-hadron production in a saturation framework[END_REF] while the single-hadron result is based on [START_REF] Fucilla | Diffractive single hadron production in a saturation framework at the NLO[END_REF]. Finally, in chapter 8, we discuss the first computation of the short-distance coefficients for direct quarkonium + gluon production in electron-nucleus DIS at small x within the joint CGC + NRQCD framework. This is based on [START_REF] Kang | Direct quarkonium-plus-gluon production in DIS in the Color Glass Condensate[END_REF]. [START_REF] Kovchegov | Unitarization of the BFKL pomeron on a nucleus[END_REF] Chapter 1

Introduction

The class of processes that we are interested in this work are of the form γ p˚q P i.e. involve a projectile photon interacting with a nucleon or nucleus target P . The photon could be real, as in ultra-peripheral collisions (UPC), or virtual as in Deep Inelastic Scattering (DIS) 1 . We will introduce these two type of collisions in this chapter focusing, in the case of DIS, on scattering off a proton. This collision will enable us to understand the different physical pictures of the internal structure of the proton that can emerge depending on the kinematic regimes DIS is studied in.

Light-cone coordinates and conventions

Before discussing the different collisions, we start this work by introducing light-cone variables customary used when studying high-energy particles collisions with particles travelling ultrarelativistically meaning practically at the speed of light and these variables simplify the computations. For a great pedagogical summary on them, see [START_REF] Collins | Light cone variables, rapidity and all that[END_REF]. In this intent, we introduce two light-like vectors associated with the `and ´directions:

n 1 " 1 ? 2 p1, 0 K , 1q, n 2 " 1 ? 2 p1, 0 K , ´1q, n 1 ¨n2 " 1 . (1.1.1)
We can then define the two light-cone components of any arbitrary vector p " pp `, p ´, p K q as p `" p ´" n 2 ¨p " p 0 `p3 ? 2 (1.1.2)

p
´" p `" n 1 ¨p " p 0 ´p3 ? 2 .

(1. 1.3) such that

p µ " p `nµ 1 `p´nµ 2 `pµ K " p `nµ 1 `p´nµ 2 `⃗ p µ .
(1.1.4)

1 Hence, the notation of γ p˚q .

The scalar product of two vectors rewrites into p ¨k " p `k´`p´k``p K ¨kK (1.1.5)

" p `k´`p´k`´⃗ p ¨⃗ k .

(1.1.6)

The metric tensor becomes g `´" g ´`" 1,

g ij " ´δij (1.1.7)
where here i, j are transverse momentum indices. All other components of the metric are equal to 0. The Heisenberg uncertainty principle becomes

x ¯p˘ą 1 .

(1.1.8)

For a particle moving in the `direction, we refer to its p `as its (light-cone) longitudinal momentum and x ´as its longitudinal coordinate. p ´is its energy and x `its light-cone time.

Similar appellations exist for a particle moving in the ´direction, only interchanging `and ´. We also introduce two new gamma matrices:

γ `" γ ´" n 2 { , γ ´" γ `" n 1 { , (1.1.9)
which has the following properties ␣ γ `, γ ´( " 2 , ␣ γ ˘, γ i ( " 0 , `γ˘˘2 " 0 .

(1.1.10)

Finally, a passive boost in the z-direction with velocity β on a vector z µ has the form

x µ " `Λ´1 z `, Λz ´, z K ˘(1.1.11)

Λ " d 1 `β 1 ´β .
(1. 1.12) In this manuscript unless explicitly stated that the space-time dimension is four, we will work with dimensional regularization D " 2 `d, with d " 2 `2ϵ the dimension for the transverse components. We also will introduce a regularization scale µ, with the dimension of a mass, to keep the coupling constant dimensionless:

g 0 " gµ 2´D 2 " gµ 1´d 2 " gµ ´ϵ .
Throughout the rest of this manuscript, we will denote by z ij " z i ´zj , p ij " p i ´pj .

(1. 1.13) Any integral written without bound is an integral from ´8 to `8.

1.2 Deep Inelastic scattering and the phase-diagram for QCD evolution

Deep inelastic scattering: a definition

Let us consider the scattering of a lepton off a nucleus or nucleon target P . For definiteness, let's focus on e ´pk e q`pppq Ñ e ´pk 1 e q`p{X collision. The collision can be studied in different regimes. The first one is the Mott regime where the energy of the outgoing electron is approximately the same as the one of the incoming electron E 1 « E and Q 2 " ´pk ´k1 q 2 almost vanishes. The collision is predominantly elastic and the proton remains intact after the interaction. Only general information on the proton as a whole can be known from these collisions. The second type is the deep scattering regime where Q 2 " m 2 n with m n the mass of the proton, deep meaning that one can probe the internal structure of the proton with a precise (transverse) resolution of the order of 1{Q. The proton has a high probability to break up and so theses collisions are called Deep Inelastic Scatterings (DIS). For Q 2 ! M 2 Z 0 , the collision is mediated by a virtual photon and we only consider this case. This is illustrated in fig. 1.2.1.

The best known measurements of DIS were performed at the Hadron-Electron Ring Accelerator (HERA) located at DESY. This collider operated from 1992 to 2007. e ˘p was studied with center-of-mass energy from 225 to 318 GeV. The proton was probed down to x Bj " 10 ´4 and 1{Q could reach 10 ´18 m which is smaller than the size of the proton measured already in the fifties to be around 10 ´15 m.

X P (p)

e -(k e )

e -(k ′ e )

γ * (q) Figure 1.2.1: e+p deep inelastic scattering via the exchange of a virtual photon γ ˚and its kinematics.

To describe DIS, one usually defines the following variables S " pk e `pq 2 (1.2.1) • S is the center-of-mass energy squared of the e ´p system. For a theorist, its only role is to set an upper limit to

Q 2 "
• s the center-of-mass energy squared of the γ ˚p system2 .

• y is the inelasticity.

• Q 2 is the virtuality of the virtual photon and determine the wavelength of it and gives the (transverse) resolution scale at which the contents of the proton are probed at which is of the order of 1{Q.

• x Bj is the Bjorken-x.

One also define Y " lnp1{x Bj q the total rapidity of the process. In the expression of the different variables, the mass of the proton m 2 n is always neglected. The virtuality Q 2 of the photon can be determined by measurements as one can show that in the rest ftame of the target, it is equal to

Q 2 " 4EE 1 sin 2 ˆθ 2 ˙(1.2.7)
where E, E 1 the energy of the incoming and outgoing electron and θ the scattering angle of the three momenta of the incoming and outgoing electron. The cross-sections for the unpolarized DIS reaction is parameterized in terms of inclusive structure functions:

dσ e ´pÑe ´X dx Bj dy " 2πα 2 em x Bj yQ 2 ␣" 1 `p1 ´yq 2 ‰ F 2 px Bj , Q 2 q ´y2 F L px Bj , Q 2 q ( " 2πα 2 em x Bj yQ 2 " 1 `p1 ´yq 2 ‰ " F 2 px Bj , Q 2 q ´y2 1 `p1 ´yq 2 F L px Bj , Q 2 q * " 2πα 2 em x Bj yQ 2 " 1 `p1 ´yq 2 ‰ σ r px Bj , Q 2 , yq , (1.2.8)
where σ r is the reduced cross-section. We can also define the structure function F T " 2x Bj F 1 with F L " F 2 ´2x Bj F 1 . For Q 2 " Λ 2 QCD , perturbative QCD (pQCD) can be used to study this process. One also employ some factorization scheme and theorem to separate the long and short distance dynamics in the physical observables which are then convolution of a hard part computable using perturbative methods as α s pQ 2 q ! 1 and the non-perturbative part that has to be fitted to experimental data, modellized or evaluated using methods such as lattice QCD. The scheme to use will depend on kinematic regimes DIS is studied in, kinematic regimes defined by x Bj , Q 2 . Different physical pictures of the structure of the proton will then emerge as illustrated in fig. 1

DIS in the Bjorken limit

The first regime is the Bjorken limit where x Bj is fixed at a moderately small value and Q 2 Ñ 8.

The usual description of DIS is done in an infinite momentum frame -a frame where the proton is moving in the ´direction ultra-relativistically with momentum of the from p µ « p0, p ´, 0 K q with p ´Ñ 8. The proton is then described in the QCD-Improved parton model. Partons is the common name for the constituent (anti-) quark and gluons of nucleons (or nucleus). In the parton model, the proton is a dilute system made of the valence partons and wee partons3 i.e. sea quarks and gluons. The partons behave like quasi-free particles during the hard interaction as their interactions with their co-moving partons are suppressed due to time dilatation compared to the timescale of the interaction " 1{Q.

One employs the collinear factorization theorem proven to all orders in α s at leading twist (i.e. at leading power term in the expansion of m 2 n {Q 2 ) to factorize the structure functions collectively labelled as F i :

F i `xBj , Q 2 ˘" n f `1 ÿ f"1 ż 1 x Bj dx x C i f ´xBj x , α s `Q2 ˘¯f p f `x, Q 2 ˘(1.2.9)
where n f is the number of active quark flavours and the `1 is for the gluon, C i f is the partonic scattering coefficient functions computed as a power series in α s i.e. perturbatively and f p f px, Q 2 q is the unpolarized proton Parton Distribution Functions (PDFs) which can be interpreted as the (quasi-) probability distribution to find in the proton a parton of type f and with fraction of light-cone longitudinal momentum of the proton x " k ´{p ´where k is the momentum of the parton. x is also called the Feynman-x.

The leading order diagram (LO) diagram in this framework is to have the virtual photon strikes one parton from the proton. This is illustrated in fig. 1.2.3 and corresponds to DIS in the so-called naive parton model. This four momentum of the parton (necesseraly a quark) is parallel to the proton momentum i.e. k " xp where x is the Feynman variable. The parton has no transverse momentum. The quark after being struck by the photon is an asymptotic final state and is assumed to be on-shell. With the assumption, one has, for massless partons, 0 " pq `xpq 2 " ´Q2 `2xp ¨q `x2 m 2 n ùñ x "

X P (p) e -(k e ) e -(k ′ e ) γ * (q) k = xp k + q
Q 2 2p
¨q " x Bj .

(1.2.10) Therefore, at LO where the struck quark moves completely collinearly to the proton, we have that the Feynman-x, the longitudinal momentum fraction of the parton is equal to Bjorken-x which is a well-defined Lorentz-invariant.

From the naive parton model, one arrives to the Bjorken scaling which is that the structure functions F 2 , F 1 only depend on x Bj and are independent of Q 2 . One also has the Callan-Gross relation F 2 px Bj q " 2x Bj F 1 px Bj q (1.2.11) or F L px Bj q " F 2 px Bj q ´2x Bj F 1 px Bj q " 0 .

(1.2.12)

The scaling behavior was observed at the SLAC-MIT experiment (1968)(1969) in the experimental x Bj -range probed (intermediate x Bj " 0.1´0.2) however this behaviour is violated (see fig. 1.2.4) by QCD collinear corrections such as the quark emitting a gluon, called gluon bremsstrahlung, Figure 1.2.4: F p 2 measured in e ˘`p with electromagnetic interaction. The structure function has been multiplied by 2 ix where i x is the number of the x bin. It ranges from i x " 1px " 0.85q to i x " 24px " 0.00005q. Figure taken from [START_REF] Tanabashi | Review of Particle Physics[END_REF].

that happen before interacting with the photon. These corrections lead to large logarithms of Q 2 after cancelling collinear divergences in physical observables. These logarithms compensate the smallness of the coupling constant with α s lnpQ 2 q " 1. These have to be resummed to all orders and this is done by the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations [START_REF] Altarelli | Asymptotic Freedom in Parton Language[END_REF][START_REF] Dokshitzer | Calculation of the Structure Functions for Deep Inelastic Scattering and e+ e-Annihilation by Perturbation Theory in Quantum Chromodynamics[END_REF][START_REF] Gribov | Deep inelastic e p scattering in perturbation theory[END_REF][START_REF] Lipatov | The parton model and perturbation theory[END_REF]. These evolution equations mean that if one knows the PDFs at some initial scale Q 2 0 , one can find the value of them at some other scale Q 2 (as long as we don't have lnpQ 2 q ! Y ). DGLAP equations can also be regarded as a renormalization group equations which renormalize PDFs with respect to Q 2 .

The dominant logarithmic corrections of the form `αs lnpQ 2 q ˘n with n P N are found when there is strong ordering of the transverse momenta of the partons in the ladder (when working in axial gauge) represented in fig. 1

.2.5 ⃗ k 2 1 ! ⃗ k 2 2 ! ¨¨¨! ⃗ k 2 n ! Q 2 .
(1.2.13)

This gives the DGLAP equations in the Leading Logarithmic Approximation (LLA). With DGLAP, the number of partons inside the proton increases. However, because their transverse size " 1{Q 2 decreases too, as illustrated in fig 1 .2.2, the phase-space density decreases then and the proton remains as a dilute system of partons.

In fig. 1.2.6, the x-dependence of PDFs measured at HERA is shown at scale Q 2 " 10 GeV 2 . The valence quark contributions are dominant at large-x, peaking around x " 1{3. The up quark xu v distribution is twice the one of the down valence quark, which matches the naive expectations that the total momentum of the proton is equally shared among the three valence quarks. For lower values of x, the sea quarks xS and the gluons increase steeply and start to dominate (both distributions are scaled down by a factor of 20 for visibility in the figure). At low value of x, the proton is dominated by gluons. Figure 1.2.6: Proton Parton distribution functions extracted from HERA DIS data at Q 2 " 10 GeV 2 . The sea quark distribution is defined as xS " 2xpū `dq. Figure from [START_REF] Aaron | Combined Measurement and QCD Analysis of the Inclusive e+-p Scattering Cross Sections at HERA[END_REF].

DIS in the Regge-Gribov or high-energy limit

The Regge-Gribov or semi-hard limit is characterized by

s " Q 2 " Λ 2 QCD . (1.2.14)
In this high-energy limit, we have 2p ¨q " pp `qq 2 ´q2 ´p2 " s `Q2 ´m2 p » s (1.2.15)

and so x Bj " Q 2 s in this limit. Therefore, the Regge-Gribov limit is also the small-x Bj limit where x Bj Ñ 0

Regge theory and the Pomeron

Before the advent of QCD, high-energy processes were described by Regge theory. The vocabulary and notions in this theory are still of use in modern literature. We therefore give a brief presentation of it, following [START_REF] Forshaw | Quantum Chromodynamics and the Pomeron[END_REF][START_REF] Fucilla | Probing the high-energy dynamics of QCD: selected theoretical and phenomenological studies[END_REF]. This theory predicts that for a great variety of processes, the high-energy behaviour of a scattering amplitude is Aps, tq " sÑ8 s αptq .

(1.2.16)

The basic underlying idea behind this behaviour comes from the physicist T. Regge who showed that it was useful to think of angular momenta as complex variables when discussing solutions of the Schrödinger equation for non-relativistic potential scattering [START_REF] Regge | Introduction to complex orbital momenta[END_REF]. He demonstrated that for a wide class of potentials, the only singularities of the scattering amplitude in the complex l-plane are poles called Regge poles. Poles at positive integer values of l correspond to bound states or resonances and are important for determining properties of the amplitude. This basic idea is applicable to scattering amplitudes in high-energy particle physics. To get the behaviour eq. (1.2.16), one starts from the partial-wave expansion of the amplitude which expresses it as a series of Legendre polynomials Aps, tq " 8 ÿ l"0 p2l `1q a l ptqP l ˆ1 `2s t ˙, (1.2.17)

where a l ptq are the partial-wave amplitudes and P l pzq are the Legendre polynomials. Sommerfeld [START_REF] Sommerfeld | Partial Differential Equations in Physics: Translated by Ernst G. Straus[END_REF] rewrote the above expansion in terms of a contour integral in the complex angular momentum l plane as Aps, tq " 1 2i

¿ C dl p2l `1q sin πl ÿ η"˘1 η `e´iπl 2 a pηq pl, tqP ˆl, 1 `2s t ˙, (1.2.18)
where a pηq pl, tq and P `l, 1 `2s t ˘are the analytical continuation in l of a l ptq and of the Legendre polynomials. The contour C surrounds the positive real axis as shown in fig. 1.2.7. η is called the signature and assumes only two possible values ˘1. The factor η`e ´iπl 2 is the signature factor and a p˘1q pl, tq are the even-or odd-signature partial-wave functions. One performs then the Sommerfeld-Watson transformation i.e. deforms the contour C into C 1 which is a contour parallel to the imaginary axis and with Re l " ´1{2 and encircles any poles 4 that the functions a pηq pl, tq may have at l " α nη ptq which the lead to picking up residue associated to these poles. After this transformation and assuming that there is only simple poles α nη ptq called Regge poles and using that the amplitude is studied in the high-energy limit with s " |t|, one has Aps, tq " sÑ8 η `e´iπαptq 2 βptqs αptq , (1.2. [START_REF] Ducloue | Confronting Mueller-Navelet jets in NLL BFKL with LHC experiments at 7 TeV[END_REF] where αptq is the leading Regge pole in the l-plane. It is not a fixed value but a function of t. In the more general case, when there are possibly non-simple poles and cuts, the asymptotic expression in s will be different. To arrive to eq. (1.2. [START_REF] Ducloue | Confronting Mueller-Navelet jets in NLL BFKL with LHC experiments at 7 TeV[END_REF]), one uses that in the high-energy limit, the Legendre polynomial is dominated by its leading term P l ˆ1 `2s t ˙" s"|t| Γ p2l `1q Γ 2 pl `1q ´s 2t ¯l .

The expression in eq. (1.2. [START_REF] Ducloue | Confronting Mueller-Navelet jets in NLL BFKL with LHC experiments at 7 TeV[END_REF]) can be viewed as the exchange in the t-channel of an object with effective angular momentum αptq called a Reggeon, and αptq is the Regge trajectory. A Reggeon-exchange amplitude can be considered as a superposition of amplitudes for the exchange of all possible particles in the t-channel. The amplitude can be factorized as shown in fig. 1 where γ ac ptq is the coupling between the Reggeon and the particles a, c and similarly γ bd ptq is the coupling between the Reggeon and particles b, d, the other terms are associated to the universal contribution from the Reggeon exchange. The factor Γpαptqq is explicitly extracted in the above expression to cancel poles of the amplitude at negative integer values of αptq. Poles of the amplitude at positive integer values of αptq can be interpreted as the exchange in the t-channel of a resonance with integer spin. The coupling functions γ are only functions of t, the behaviour on s of the whole amplitude is hence completely determined by the Reggeon exchange. We will now give a definition of reggeization. A particle of mass M and spin J is said to reggeize if the scattering amplitude for a process with exchange in the t-channel of the quantum numbers of the particle behaves asymptotically as Aps, tq 9 s αptq , (1.2.21) where αptq is the Regge trajectory and the particle itself lies on the trajectory i.e. αpM 2 q " J.

Another key concept of the Regge theory is the notion of Pomeron. The optical theorem states that 2 Im A aa ps, t " 0q " 2Kpsqσ aÑX (1. 2.22) with σ aÑX the total cross-section a Ñ X and 2Kpsq the flux factor. In the case of two-particle scattering and for s higher than any other scales and in particular the masses of the initial particles, 2Kpsq " 2s From the asymptotic behaviour of scattering amplitude in eq. (1.2.16) and from the optical theorem in the large-s limit, total cross-sections of processes with one (leading) Reggeon exchange behaves like σ tot 9 s αp0q´1 .

(1.2.23)

The Pomeranchuk theorem derived from general assumptions states that in any scattering process in which there is charge exchange, the cross-section vanishes asymptotically with energy. Foldy and Peierls proved the converse: a scattering process with cross-section that does not fall as s increases is a process dominated by vacuum quantum number exchange. Such behaviour of cross-sections has been observed experimentally and they have been found to even rise with increasing s. Looking at eq. (1.2.23), this means that the Reggeon exchanged has vacuum quantum numbers and intercept α P ptq ą 1. This Reggeon is called Pomeron, named after Pomeranchuk.

Regge theory has proven to be phenomenologically successful for various hadronic collisions [START_REF] Donnachie | Total cross-sections[END_REF]. From these Regge fits, the intercept for the soft Pomeron is found to be α P p0q " 1.08. Regge fit has also been found to be valid for DIS with [START_REF] Forshaw | Quantum Chromodynamics and the Pomeron[END_REF] F 2 px Bj , Q 2 q " Ax ´0.08 `Bx 0. 45 (1.2.24)

for Bjorken-x not too small 0.01 À x Bj À 0.1. However at smaller values of Bjorken-x, the above expression does not fit data well. A steeper x-dependence is found. This deviation can be understood from pQCD computations done in the Regge-Gribov limit or semi-hard limit. In particular, using the BFKL approach which is the subject of the next section, one can define a hard (or BFKL) Pomeron, which if computed at LLA meaning here resumming all orders terms of the form α n s ln n psq, gives a growth in s of cross-sections with vacuum quantum number exchanged of the form σ LLA tot 9 s ω 0 , ω 0 " 4N c lnp2q π α s .

(1.2.25)

BFKL picture

We start by introducing the eikonal approximation, following [START_REF] Lappi | High energy scattering in QCD[END_REF]. The high-energy limit can be used to simplify interaction with the target through gluon exchange. To understand this, consider a qq Ñ qq scattering, as shown in fig. 1.2.9. The "probe" quark moves in the direction with momentum p 1 « pp 1 , 0, 0 K q and the "target" quark moves in the ´direction with momentum p 2 « p0, p 2 , 0 K q. The quark mass is neglected.

The two quarks collide with a center-of-mass energy squared s " pp 1 `p2 q 2 « 2p 1 p 2 higher than any other scales in particular |t| and p 1 " p 2 " a s{2. The exchanged gluon

p 1 , j, σ 1 p 1 ′ , i, σ 1 ′ p 2 , l, σ 2 p 2 ′ , k, σ 2 ′ q Figure 1.
2.9: One gluon exchange for quark-quark scattering.

has momentum q with t " q 2 . Demanding that initial and final quarks are on-shell leads to 0 " p 2 1 1 " pp 1 ´qq 2 " ´2p 1 q ´`t ùñ q

´" t 2p 1 " t s p 2 (1.2.26)

0 " p 2 2 1 " pp 2 `qq 2 " t `2p 2 q `ùñ q `" ´t 2p 2 " ´t s p 1 .

(1.2.27)

Thus, |q ´| ! p 2 and q `! p 1 and t » ´⃗ q 2 . All components of the exchanged gluons are much smaller than ? s. This shows already some key features of scattering in the high-energy limit. The momenta are ordered: the light-cone longitudinal momentum of the colliding particles are larger than the components of the exchanged particles. The momentum transfer is dominated by transverse momentum. Using the Feynman gauge for the propagator of the gluon, the amplitude for the qq Ñ qq scattering reads A " p´iqigt a ij ūσ 1 1 pp 1 1 q γ µ u σ 1 pp 1 q ˆ´ig µν δ ab q 2 ˙igt b kl ūσ 2 1 pp 2 1 q γ ν u σ 2 pp 2 q .

(1.2.28)

To proceed, one uses the Gordon identity:

ūσ 1 pp 1 qγ µ u σ ppq " ūσ 1 pp 1 q "`p `p1 ˘µ `iσ µν `p ´p1 ˘ν‰ u σ ppq .

(1.2.29)

Applying this identity to the upper vertex and using p 1 1 " p 1 ´q » p 1 with the spinor normalization ūσ 1 ppqu σ ppq " δ σ 1 σ , we have

ūσ 1 1 pp 1 1 qγ µ u σ 1 pp 1 q " 2p µ 1 δ σ 1 1 σ 1 .
(1.2.30)

Eq. (1.2.30) is the so-called eikonal vertex. In practice, this specific simplified expression can only be used in covariant gauge and the form of the interactions in light-cone gauge is more involved. However, its properties are quite general and are satisfied in high-energy scattering (at eikonal approximation): the light-cone momenta of the particle interacting with the target is conserved and helicities are conserved (up to correction in 1{s). The amplitude eq. (1.2.28) becomes

A " gt a ij 2p µ 1 δ σ 1 1 σ 1 g µν t gt a kl 2p ν 2 δ σ 2 1 σ 2 " gt a ij δ σ 1 1 σ 1 ˆ2s t ˙gt a kl δ σ 2 1 σ 2 .
(1.2.31)

Another thing to point is the so-called Gribov trick. The metric tensor g µν can be decomposed as g µν " g Kµν `pµ 1 p ν

These effective polarizations are called non-sense polarizations 5 .

The eikonal aproximation and the Gribov trick can be used to define the hard Pomeron. This is encoded in the universal BFKL Green's function G s ´⃗ k 1 , ⃗ k 2 , ⃗ ∆ ¯which at LLA corresponds to the sum of (cut) gluon effective ladders of the form of fig. 1.2.10 represented between the same "probe" and "target" quark of fig. 1.2.9. The sum runs over the number of rungs.

The momenta on the ladder follow the multi-Regge kinematics (MRK):

p 1 " k 1 » q 1 " ¨¨¨" k ǹ » q ǹ " 0 p 2 " ´kń `1 » q ń " ¨¨¨" ´k1 » q 0 " 0 .

(1.2.36)

All t-channel momenta k i are mostly transverse too, i.e. k 2 i » ´⃗ k 2 i and can be assumed to be of the same order with magnitude much smaller than s. The same goes for the momentum transfer with t " ∆ 2 » ´⃗ ∆ 2 . The MRK is the kinematic tha lead to the dominant logartihmic corrections of the form pα s ln sq n , n P N that have to be resummed.

The zig-zag lines in fig. 1.2.10 represents reggeized gluons with a Regge trajectory of the form αptq " 1 `ωptq (1.2.37)

where

ωptq " ω ´⃗ ∆ 2 ¯" α s N c ż d D´2 ⃗ k p2πq D´2 p´⃗ ∆ 2 q ⃗ k 2 p ⃗ ∆ ´⃗ kq 2 .
(1.2.38) The gluon lies on the trajectory i.e. αp0q " 1. The propagator for the reggeized gluons with momentum k i is a modified version of the usual propagator (in Feynman gauge here):

p 1 p 2 p 2 ′ p 1 ′ k 1 -∆ k 1 k i k i+1 k i+1 -∆ k i -∆ k n+1 -∆ k n+1 q n+1 q n q 0 q 1 q i → → → → → → → →
G ab µν pk i q " ig µν δ ab ⃗ k 2 i ˜si ⃗ k 2 i ¸ωp ⃗ k 2 i q " ig µν δ ab ⃗ k 2 i ˜pq i ´qi`1 q 2 ⃗ k 2 i ¸ωp ⃗ k 2 i q
.

(1.2.39)

The reggeization of the gluons in the ladder account for the leading logarithmic contributions of virtual corrections to all orders. The vertex in the ladder between the reggeized gluons and emitted gluons on the s-channel is the so-called Lipatov vertices which is a non-local vertex and is a way to combine the effect of five real emission diagrams into one, as shown in fig. 1.2.11.

In the BFKL approach, the imaginary part of the scattering amplitude for A `B Ñ A 1 `B1 with vacuum quantum number exchanged reads, using k t -factorization theorem 6 :

Im A ABÑA 1 B 1 ps, tq " s p2πq D´2 ż d D´2 ⃗ k 1 ⃗ k 2 1 p ⃗ k 1 ´⃗ ∆ q 2 d D´2 ⃗ k 2 ⃗ k 2 2 p ⃗ k 2 ´⃗ ∆ q 2 Φ A 1 A ´⃗ k 1 , ⃗ ∆ ¯ΦB 1 B p´⃗ k 2 , ⃗ ∆q ˆGs ´⃗ k 1 , ⃗ k 2 ; ⃗ ∆ ¯, (1.2

.40)

6 Two steps justify this factorization. The first one is that with the MRK of eq. (1.2.36), one can set k 1 " 0 on the projectile side and k ń`1 " 0 on the target side i.e. these blob do not depend on these variables. The integrations over k1 and kn`1 can then be reorganized with the integrals over k 1 (k ǹ`1 ) being associated of the projectile (target) side and the integrals over k 1 and k ń`1 are then part of the Green's function. The second step is to use the Gribov trick eq. (1.

G s k 1 , k 2 ; ∆ Φ A ′ A k 1 , ∆ Φ B ′ B -k 2 , ∆ p A p B p B ′ p A ′ k 1 k 2 k 2 -∆ k 1 -∆ -→ -→ -→ -→ Figure 1.2.
12: Representation of the factorized amplitude in the BFKL approach.

The BFKL Green's function evolves according to the BFKL equation [START_REF] Kuraev | Multi -Reggeon Processes in the Yang-Mills Theory[END_REF][START_REF] Fadin | On the Pomeranchuk Singularity in Asymptotically Free Theories[END_REF][START_REF] Kuraev | The Pomeranchuk Singularity in Nonabelian Gauge Theories[END_REF][START_REF] Balitsky | The Pomeranchuk Singularity in Quantum Chromodynamics[END_REF] more commonly found in Mellin space. The Mellin transform of the Green's function is

G ω ´⃗ k 1 , ⃗ k 2 ; ⃗ ∆ ¯" ż 8 1 d ˆs s 0 ˙ˆs s 0 ˙´ω´1 G s ´⃗ k 1 , ⃗ k 2 ; ⃗ ∆ ¯, (1.2.41)
and the BFKL equation then reads

ωG ω ´⃗ k 1 , ⃗ k 2 ; ⃗ ∆ ¯" ⃗ k 2 1 ´⃗ k 1 ´⃗ ∆ ¯2 δ ´⃗ k 1 ´⃗ k 2 ¯`ż d D´2 ⃗ k ⃗ k 2 p ⃗ k ´⃗ ∆ q 2 K BFKL ´⃗ k 1 , ⃗ k; ⃗ ∆ ¯Gω ´⃗ k, ⃗ k 2 ; ⃗ ∆ (1.2.42)
where K BFKL ´⃗ k 1 , ⃗ k; ⃗ ∆ ¯is the BFKL kernel. A schematic representation of this equation can be found in fig. 1.2.13. It is essentially a recursive equation for effective gluon ladder with additional rung.

G ω G ω k 1 k 2 k 1 k 1 -∆ k 2 -∆ k 1 -∆ k 2 k 2 -∆ k 1 -∆ k 1 k k -∆ -→ -→ -→ -→ = -→ -→ + -→ -→ -→ -→ -→ -→ Figure 1.2.
13: Schematic representation of the BFKL equation.

From the factorization eq. (1.2.40) and the optical theorem eq. (1.2.22), the total inclusive DIS cross-section can be factorized into a target proton impact factor, a photon impact factor and the universal BFKL Green's function

σ γ ˚p " 1 p2πq D´2 ż d D´2 ⃗ k 1 p ⃗ k 2 1 q 2 Φ γ ˚γ˚p ⃗ k 1 q ż d D´2 ⃗ k 2 p ⃗ k 2 2 q 2 Φ P P p´⃗ k 2 qG s p ⃗ k 1 , ⃗ k 2 q . (1.2.43)
The convolution in transverse momentum ⃗ k 2 of the proton impact factor, which is the nonperturbative part in the above expression, with the Green's function defines Fpx, ⃗ k q which is the unintegrated gluon density (UGD) distribution. This is related to the usual gluon PDF via

gpx, Q 2 q " ż d d ⃗ k p2πq d Fpx, ⃗ k qθpQ 2 ´⃗ k 2 q . (1.2.44)
From the expression of the solution of the BFKL equation at t " 0, the DIS cross-section is found then to grow with energy like

σ γ ˚ppxq " ˆ1 x ˙ω0 , ω 0 " 4N c lnp2q π α s . (1.2.45)
This growth violates the Froissart-Martin bound [START_REF] Froissart | Asymptotic behavior and subtractions in the mandelstam representation[END_REF][START_REF] Martin | Unitarity and high-energy behavior of scattering amplitudes[END_REF] that stipulates that

σ tot ă A ln 2 s (1.2.46)
with A a constant. This violation cannot be solved by calculations of radiative corrections to a fixed NNNN...NL order [START_REF] Ioffe | Quantum chromodynamics: Perturbative and nonperturbative aspects[END_REF]. This growing of the gluon density cannot go unchecked. Non-linear effects are expected to become important and compete with gluon bremsstrahlung ultimately resulting in a taming of the growth of gluon density. This phenomenon is called gluon saturation 7 . This idea was pioneered by Gribov, Levin and Ryskin [START_REF] Gribov | Singlet Structure Function at Small x: Unitarization of Gluon Ladders[END_REF][START_REF] Gribov | Semihard Processes in QCD[END_REF] where they considered dense proton, dense here meaning various colour sources (sea quarks and gluons) are contained in the target wave-function. For these dense systems, multiple BFKL ladders exchange are then important and they are connected to different colour sources, each ladder interacting with the target independently. Those ladders can also merge together via the so-called triple Pomeron vertices. These mechanisms are represented in the so-called fan diagrams, an example of it is shown in fig. 1.2.14. Those diagrams lead to Gribov, Levin, Ryskin (GLR) evolution equation that resums double logarithmic corrections -corrections of the form `αs lnpQ 2 q lnpY q ˘nfor the UGD. Mueller and Qiu (MQ) [START_REF] Mueller | Gluon Recombination and Shadowing at Small Values of x[END_REF] developed the idea and found an equivalent of the GLR evolution equation for the integrated gluon distribution. A more recent approach to saturation in DIS off a heavy nucleus target with mass number A " 1 is given by the Glauber-Gribov -Mueller (GGM) model formulated in the dipole picture of DIS presented in the next section. In this model, the photon splits into a q q pair or colour dipole before interacting with the nucleus. The nucleons in the target are assumed to be independent and are modelled as an onium (a pair of heavy q q). The colour dipole multiple scatters with the nucleons via 2-gluon exchange. Quantum corrections are then added to this model: the initial colour dipole may develop a cascade of gluons before interacting with the target. In the large N c limit, this cascade is described by Mueller's dipole model [START_REF] Mueller | Soft gluons in the infinite momentum wave function and the BFKL pomeron[END_REF][START_REF] Mueller | Single and double BFKL pomeron exchange and a dipole picture of high-energy hard processes[END_REF][START_REF] Mueller | Unitarity and the BFKL pomeron[END_REF][START_REF] Chen | The Dipole picture of high-energy scattering, the BFKL equation and many gluon compound states[END_REF] and by computing the light-cone wavefunction of an onium using light-cone perturbation theory. This model was applied by Kovchegov to derive the Balitsky-Kovchegov (BK) equation [START_REF] Balitsky | Operator expansion for diffractive high-energy scattering[END_REF][START_REF] Balitsky | Factorization for high-energy scattering[END_REF][START_REF] Balitsky | Factorization and high-energy effective action[END_REF][START_REF] Balitsky | Effective field theory for the small x evolution[END_REF][START_REF] Kovchegov | Small x F(2) structure function of a nucleus including multiple pomeron exchanges[END_REF][START_REF] Kovchegov | Unitarization of the BFKL pomeron on a nucleus[END_REF]. For a more in depth discussion on Mueller's dipole model and the GGM model, see [START_REF] Kovchegov | Quantum Chromodynamics at High Energy[END_REF].

The more refined version of evolution equations incorporating non-linear effects not present in BFKL is the Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov and Kovner (JIMWLK) in the Colour Glass Condensate effective field theory (EFT) [START_REF] Jalilian-Marian | The BFKL equation from the Wilson renormalization group[END_REF][START_REF] Jalilian-Marian | The Wilson renormalization group for low x physics: Gluon evolution at finite parton density[END_REF][START_REF] Jalilian-Marian | The Wilson renormalization group for low x physics: Towards the high density regime[END_REF][START_REF] Jalilian-Marian | Unitarization of gluon distribution in the doubly logarithmic regime at high density[END_REF][START_REF] Kovner | Relating different approaches to nonlinear QCD evolution at finite gluon density[END_REF][START_REF] Weigert | Unitarity at small Bjorken x[END_REF][START_REF] Iancu | Nonlinear gluon evolution in the color glass condensate. 1[END_REF][START_REF] Ferreiro | Nonlinear gluon evolution in the color glass condensate. 2[END_REF][START_REF] Iancu | The Renormalization group equation for the color glass condensate[END_REF] or equivalently the Balitsky hierachy of equations derived in the shockwave approach [START_REF] Balitsky | Operator expansion for diffractive high-energy scattering[END_REF][START_REF] Balitsky | Factorization for high-energy scattering[END_REF][START_REF] Balitsky | Factorization and high-energy effective action[END_REF][START_REF] Balitsky | Effective field theory for the small x evolution[END_REF]. The BK equation is the closed form of the Balitsky hierarchy obtained by taking its mean-field approximation, valid in the large-N c limit. These different evolution equations and their derivation will be explained later on, in section 3.5 and section 5.2.

The momentum below which the non-linear effects are important is the saturation scale Q s and it can be estimated via the concept of recombination probability [START_REF] Iancu | Physics of the Color Glass Condensate[END_REF]. The proton is considered an homogeneous disk of radius R p . The cross-section for gg Ñ g can be estimated as

σ ggÑg `x, Q 2 ˘" α s N c N 2 c ´1 xg `x, Q 2 Q2 . (1.2.47)
The recombination probability for a gluon is obtained dividing the above cross-section by the transverse area of the proton

Γ `x, Q 2 ˘" α s N c N 2 c ´1 xg `x, Q 2 πR 2 p Q 2 .
(1.2.48)

The saturation scale is the scale at which the recombination probability becomes of order one:

Γ `x, Q 2 s ˘" 1 ùñ Q 2 s pxq " α s N c N 2 c ´1 xg `x, Q 2 s πR 2 p . (1.2.49)
From the 1{x-dependence of the gluon distribution coming from LLA BFKL evolution equation, the saturation scale grows as a power-law with 1{x:

Q 2 s pxq9 ˆ1 x ˙ω0 ùñ ln Q 2 s pY q " ω 0 Y `K (1.2.50)
where K is a constant. lnpQ

xg `x, Q 2 s ˘by xg A `x, Q 2 s ˘" xAg `x, Q 2 s ˘" A p1{xq ω 0 and R p by R A " A 1{3 R p .
With these modifications, one has then

Q 2 s pxq " A 1{3 x ´ω0 . (1.2.

51)

The factor A 1{3 is the nuclear oomph factor. This factor implies that saturation effects are more important in heavy nucleus with mass number A " 1.

The dipole picture for DIS

We finish this section by introducing the dipole picture which is the usual picture used to study γ p˚q P processes in this high-energy limit. This picture is built upon the ideas of James D.

Bjorken, John B. Kogut, and Davison E. Soper in [START_REF] Bjorken | Quantum electrodynamics at infinite momentum: Scattering from an external field[END_REF] who conceptualized scattering at highenergy as the scattering from a force-field, originally the electromagnetic field. We work in a frame where the γ p˚q moves in the `direction and the proton8 moves in the ´direction and with momenta

q " ˆq`, ´Q2 2q `, ⃗ 0 ṗ " ˆm2 n 2p ´, p ´, ⃗ 0 ˙, (1.2.52)
where Q 2 " ´q2 is the photon virtuality and m n is the nucleon mass. The center-of-mass energy s of the γ p˚q p system is larger than any other scale and is then given by

s " pq `pq 2 " 2q `p´´Q 2 m 2 n 2q `p´´Q 2 `m2 n » 2q `p´, (1.2.53)
and q `" p

´" a s{2

In the dipole picture, the photon fluctuates into a QCD Fock state before the interaction. This is necessary as at high-energy the interaction is expected to happen with the target gluonic field dominating the inner structure of proton and the photon is a colour-neutral object. This fluctuation has to happen before the interaction. This is because the interaction time is given by

∆x ìnt " 1 p ´" c 2 s (1.2.54)
which is much smaller than the lifetime of the virtual photon

∆x γ " 1 |q ´| " 2q Q 2 . (1.2.55)
The photon is therefore much more likely to split before the collision. The leading order Fock state that the photon fluctuates to is a pair of quark and antiquark forming then a colour dipole, hence the name of this picture. This is represented in fig. 1.2.15.

One of the first papers to depict DIS within the dipole picture were done by N.N. Nikolaev and B.G. Zakharov in [START_REF] Nikolaev | Color transparency and scaling properties of nuclear shadowing in deep inelastic scattering[END_REF]. From the dipole picture, the total inclusive DIS cross-sections can be factorized into three different subparts. By optical theorem, σ γ ˚p is equal to the imaginary part of the forward elastic γ ˚p scattering amplitude. This scattering amplitude at LO can be separated into three stages as shown in fig. 1.2.16: the incoming virtual γ ˚of polarisation λ P p`1, ´1, 0q fluctuates into a q q pair of flavour f , helicities h, h " ˘1{2 and longitudinal momentum fraction z and 1 ´z with respect to the longitudinal momentum of the photon. Then this colour dipole of transverse size ⃗ r scatters elastically with the target and finally the dipole recombines to give a virtual photon. P γ * P ′ q q Figure 1.2.15: γ p˚q p process in the dipole picture at leading order. The gray blob represents the non-perturbative interaction with the target.

P (p) z 1 -z γ * (Q 2 ) P (p)
M γ ˚p T,L `x, Q 2 , ∆ ˘" ÿ f ż d 2 ⃗ r ż 1 0 dz 4π ´ψ˚f γ ˚ψf γ ˚¯T ,L p⃗ r, z, Q 2 q M q qp el px, ⃗ r, ∆q e ip1´zq⃗ r¨⃗ ∆ , (1.2.56)
where ∆ 2 " ´t is the momentum transfer and x is the scale at which the target is probed, here it is Bjorken-x. M q qp el px, ⃗ r, ∆q is the elastic scattering amplitude of the colour dipole. ψ f,h, h,λ γ is the forward photon wave-function for γ λ Ñ q q. They can be determined using light-cone perturbation theory (see for example [START_REF] Kovchegov | Quantum Chromodynamics at High Energy[END_REF]):

ψ f,h, h,λ"0 γ ˚p⃗ r, z, Qq " e f e a N c δ h,´h 2Qzp1 ´zq K 0 `Q|⃗ r | 2π (1.2.57)
and

ψ f,h, h,λ"˘1 γ ˚p⃗ r, z, Qq " ˘ef e a 2N c ˆ´ie ˘iθ ⃗ r " zδ h,˘δh ,¯´p 1 ´zqδ h,¯δh ,˘B r `mf δ h,˘δh ,¯‰ ¯K0 `Q|⃗ r | 2π .
(1.2.58) m f is the quark mass, e f is quark fractional charge, e is the elementary charge, θ ⃗ r is the angle between the dipole size vector ⃗ r and the x-axis in the transverse plane. Q is defined as

Q2 " zp1 ´zqQ 2 `m2 f . (1.2.59)
The non-forward wave functions can be expressed as a product of the forward ones times exp

! ˘i 2 p1 ´zq ⃗ r ¨⃗ ∆ ) [68]
, which is the origin of the exponential factor in eq. (1.2.56). In eq. (1.2.56), the photon wave function are combined to form photon overlap functions:

´ψ˚f γ ˚ψf γ ˚¯T `⃗ r, z, Q 2 ˘" 1 2 ÿ λ"˘1 ÿ h, h"˘1{2 ψ ˚f,h, h,λ γ ˚p⃗ r, z, Qqψ f,h, h,λ γ ˚p⃗ r, z, Qq " 2N c π α em e 2 f ␣" z 2 `p1 ´zq 2 ‰ Q2 K 2 1 p Qrq `m2 f K 2 0 `Qr ˘( , (1.2.60)
and

´ψ˚f γ ˚ψf γ ˚¯L `⃗ r, z, Q 2 ˘" ÿ h, h"˘1{2
ψ ˚f,h, h,λ"0 γ ˚p⃗ r, z, Qqψ f,h, h,λ"0 γ ˚p⃗ r, z, Qq

" 8N c π α em e 2 f Q 2 z 2 p1 ´zq 2 K 2 0 `Qr ˘. (1.2.61)
The dominant contributions in these overlap are for dipole sizes such that Qr ! 1 as the asymptotic behavior of K ν pxq for x Ñ 8 is of the form

K ν pxq " e ´x ?
x .

Following [START_REF] Kowalski | Exclusive diffractive processes at HERA within the dipole picture[END_REF], the dipole elastic scattering amplitude is related to the S-matrix element for the scattering of a dipole of size ⃗ r and at fixed ⃗ b:

M q qp el px, ⃗ r, ∆q " ż d 2 ⃗ b Ă M q qp el ´x, ⃗ r, ⃗ b ¯e´i ⃗ b¨⃗ ∆ " i ż d 2 ⃗ b 2 " 1 ´S ´⃗ r, ⃗ b, x ¯ı e ´i⃗ b¨⃗ ∆ . (1.2.62)
Making the substitution in eq. (1.2.56), one has then

M γ ˚p T,L `x, Q 2 , ∆ ˘" i ÿ f ż d 2 ⃗ r ż 1 0 dz 4π ´ψ˚f γ ˚ψf γ ˚¯T ,L `⃗ r, z, Q 2 ˘ż d 2 ⃗ b 2 " 1 ´S ´⃗ r, ⃗ b, x ¯ı e ´i ⃗ ∆¨r ⃗ b´p 1 2 ´zq⃗ rs .
(1.2.63) From the optical theorem (still following [START_REF] Kowalski | Exclusive diffractive processes at HERA within the dipole picture[END_REF] for the overall factor10 )

σ tot " Im M el p∆ " 0q , (1.2.64)
the DIS inclusive cross-section is

σ γ ˚p T,L px, Q 2 q " ÿ f ż d 2 ⃗ r ż 1 0 dz 4π ´ψ˚f γ ˚ψf γ ˚¯T ,L p⃗ r, z, Qq ż d 2 ⃗ b 2 " 1 ´Re S ´⃗ r, ⃗ b, x ¯ı " ÿ f ż d 2 ⃗ r ż 1 0 dz 4π ´ψ˚f γ ˚ψf γ ˚¯T ,L p⃗ r, z, Qq ż d 2 ⃗ b 2 N ´⃗ r, ⃗ b, x " ÿ f ż d 2 ⃗ r ż 1 0 dz 4π ´ψ˚f γ ˚ψf γ ˚¯T ,L p⃗ r, z, Qqσ q qp p⃗ r, xq , (1.2.65)
where we have used that total dipole cross-section expression is, from applying the optical theorem to eq. (1.2.62):

σ q qp p⃗ r, xq " Im M q qp el px, ⃗ r, ∆ " 0q " ż d 2 ⃗ b 2 " 1 ´Re S ´⃗ r, ⃗ b, x ¯ı " ż d 2 ⃗ b 2 N ´⃗ r, ⃗ b, x " ż d 2 ⃗ b dσ q qp d 2⃗ b .
(1.2.66)

N ´⃗ r, ⃗ b, x ¯is the so-called dipole amplitude. It is the non-perturbative part in the factorized expression of the inclusive DIS cross-section eq. (1.2.65). This will later be identified as depending on correlator of light-like Wilson lines. It evolves according to the Balitsky hierachy of equation or to the BK evolution equation in the large N c -limit. The structure functions are related to the γ ˚p system via the relations below valid in the high-energy limit

F T,L px, Q 2 q " Q 2 4π 2 α em σ γ ˚p T,L , (1.2.67)
F 2 px, Q 2 q " F T px, Q 2 q `FL px, Q 2 q " Q 2 4π 2 α em σ γ ˚p tot (1.2.68) σ γ ˚p tot " σ γ ˚p T `σγ ˚p L . (1.2.69)
They define the reduced cross-section:

σ r px, Q 2 , yq " F 2 px, Q 2 q ´y2 1 `p1 ´yq 2 F L px, Q 2 q . (1.2.70)
In eq. (1.2.65), the photon overlap depends on quark flavours so the photon cross-section and therefore the structure function and reduced cross-section has different contribution from different quark flavours.

Ultra-peripheral collisions

The second type of collision where γ p˚q P can be found is in Ultra-Peripheral Collision (UPC) [START_REF] Baltz | The Physics of Ultraperipheral Collisions at the LHC[END_REF][START_REF] Glaenzer | Mapping the proton using J/ψ photoproduction with ALICE and development of a novel structure of gaseous particle detector[END_REF]. It is a collision of two ultra-relativistic nuclei with impact parameter-separating distance in the transverse plane -much larger that the sum of their radii. A sketch of it is drawn in fig. 1.3.1. These collisions are predominantly mediated by the long-range electromagnetic interaction as the impact parameter minus the sum of the radii is larger than the range of the strong interaction (and the weak and gravitational interactions are weak compared to other forces). Because the nuclei are travelling at speed close to the speed of light, they are Lorentz contracted hence the thin pancake shape of the nuclei in the mentioned figure. The EM fields surrounding the nuclei are concentrated in the direction perpendicular to the direction of motion.

The equivalent photon approximation (EPA) was proposed by Enrico Fermi in 1924 [START_REF] Fermi | On the theory of collisions between atoms and electrically charged particles[END_REF][START_REF] Fermi | On the Theory of the impact between atoms and electrically charged particles[END_REF]. EPA states that moving EM fields of a charged particle is equivalent to a flux of virtual photons. Weizsäcker and Williams applied the approximation to relativistic ion [START_REF] Weizsacker | Radiation emitted in collisions of very fast electrons[END_REF][START_REF] Williams | Nature of the high-energy particles of penetrating radiation and status of ionization and radiation formulae[END_REF][START_REF] Williams | Correlation of certain collision problems with radiation theory[END_REF]. The EM field of the nuclei are then considered as clouds of virtual photons surrounding the nuclei. The number of photon in the cloud is proportional to the square of the atomic number Z which is the number of protons in the nucleus.

The photons are produced coherently by the whole nucleus, i.e. produced by all its constituents and not just a single proton of the nucleus. This imposes that the photon wavelength should be greater than the nuclear radius R A » 1.2A 1{3 rfms where A is the mass number or number of nucleons. Hence, by the uncertainty principle the components of the momentum of the photons go up to the inverse of the nuclear radius. Their transverse momentum are limited to ⃗ p 2 À 1{R 2

A

(no Lorentz contraction in the transverse plane) and the longitudinal component k is limited by the (Lorentz-contracted) longitudinal size R A {γ L with γ L the Lorentz factor i.e.

k À γ L R A . (1.3.1)
The virtuality of the photons Q 2 is also limited by the nucleus radius

Q 2 ă 1 R 2 A « 10 ´3 GeV 2 (1.3.2)
for A ą 16. It is therefore negligible and the photons are then referred as quasi-real photons. The flux at distance r in the transverse plane away from a charge Z nucleus is [69]

d 3 N γ dkd 2 r " Z 2 αw 2 π 2 kr 2 " K 2 1 pwq `1 γ 2 L K 2 0 pwq ȷ (1.3.3)
where w " kr{γ L . Different scenarios in UPC are possible. The first case is to have photon-photon collisions meaning the radiated photons interact with each other and these interactions can lead to the production of lepton pair, of η c or χ c0 , or even, pairs of bosons W ˘. The second case is to have a photonuclear collisions where a photon from one of the nuclei probing the other nucleus, just like for DIS. These two scenarios are represented by the first two diagrams in fig. 1.3.2. We note that in the diagrams the nucleus that emits the photon remain intact after the collision. However, the scenario where the emitter nucleus breaks up is also possible and this breakup may occur through the exchange of an additional photon as drawn in the last diagram of fig. 1.3.2. The physics of UPC is very broad and reviews on the subject can be found in [START_REF] Baltz | The Physics of Ultraperipheral Collisions at the LHC[END_REF][START_REF] Bertulani | Physics of ultra-peripheral nuclear collisions[END_REF][START_REF] Contreras | Ultra-peripheral heavy-ion collisions at the LHC[END_REF].

The cross-section with one photon factorizes as

σ X " ż dk dN γ dk σ γ X pkq (1.3.4)
where σ γ X pkq is the photonuclear cross section. We see now that for the search of gluon saturation, UPC are useful type of collisions as they provide a photon source and so one can study experimentally high-energy processes that involve a photon interacting with a hadronic target just like those studies later on in this thesis. Moreover, with the photon flux eq. (1.3.3) being proportional to Z 2 then for UPC of proton with heavy nucleus and in a photonuclear collsiion scenario, one is practically guarantee that the photon will be emitted from the nuclei and will probe then the proton target.

Diffractive processes

One subclass of events in γ p˚q P collisions that we are interested in are the diffractive processes.

Diffractive processes are defined as reactions characterized by a large, non-exponentially suppressed rapidity gap in the final state [START_REF] Barone | High-Energy Particle Diffraction[END_REF]. The rapidity gap can be understood as no quantum number being exchanged between the colliding particles in a high-energy reaction, in particular it means that the interaction is colour-neutral and that the final states on the projectile and target side are in a colour-singlet state. Indeed, if one has a colour-octet final states, they would radiate gluons as their separation grows and this would produce a plethora of soft particles filling the gap. Diffraction is described naturally though Pomeron exchange in the t-channel. There are three different types of diffractive processes:

• Elastic ones: a `b Ñ a 1 `b1 , • Single diffraction dissociation a `b Ñ X `b1 , • Double diffractive dissociation a `b Ñ X `Y .
Interest in diffractive processes started in the 1990s with the observation for the first item that about 5-10% of DIS events at HERA collider presented a large rapidity gap in the distribution of the final state particle [START_REF] Derrick | Observation of events with a large rapidity gap in deep inelastic scattering at HERA[END_REF][START_REF] Ahmed | Deep inelastic scattering events with a large rapidity gap at HERA[END_REF]. This subset of events are called diffractive deep inelastic scattering (DDIS) 11 . They are of the type γ ˚p Ñ XY where X is the diffractive final state in the fragmentation region of the initial photon and Y is the outgoing proton or one of its low-mass excited states [START_REF] Ahmed | First measurement of the deep inelastic structure of proton diffraction[END_REF][START_REF] Derrick | Measurement of the diffractive structure function in deep elastic scattering at HERA[END_REF][START_REF] Breitweg | Measurement of the diffractive cross-section in deep inelastic scattering using ZEUS 1994 data[END_REF][START_REF] Adloff | Inclusive measurement of diffractive deep inelastic ep scattering[END_REF][START_REF] Aktas | Diffractive deep-inelastic scattering with a leading proton at HERA[END_REF][START_REF] Aktas | Measurement and QCD analysis of the diffractive deep-inelastic scattering cross-section at HERA[END_REF][START_REF] Chekanov | Dissociation of virtual photons in events with a leading proton at HERA[END_REF][START_REF] Chekanov | Study of deep inelastic inclusive and diffractive scattering with the ZEUS forward plug calorimeter[END_REF][START_REF] Chekanov | Deep inelastic scattering with leading protons or large rapidity gaps at HERA[END_REF][START_REF] Aaron | Combined inclusive diffractive cross sections measured with forward proton spectrometers in deep inelastic ep scattering at HERA[END_REF][START_REF] Aaron | Measurement of the cross section for diffractive deep-inelastic scattering with a leading proton at HERA[END_REF][START_REF] Aaron | Inclusive Measurement of Diffractive Deep-Inelastic Scattering at HERA[END_REF]. This is represented on fig. 1.4.1. For measurements where the outgoing proton is not explicitly measured, cuts on M Y are made to ensure relative small contribution from diffractive events with proton dissociation. In particular, for M 2 Y « 1 GeV 2 , these events are hardly distinguishable from single-diffractive events with intact proton. The ratio of diffractive to inclusive total cross-sections has been found to remain roughly constant with γ ˚p center-of-mass energy for Q 2 fixed [START_REF] Breitweg | Measurement of the diffractive cross-section in deep inelastic scattering using ZEUS 1994 data[END_REF][START_REF] Adloff | Inclusive measurement of diffractive deep inelastic ep scattering[END_REF]. DDIS is described by the following variables, some already used in the description of DIS:

S " pk e `pq 2 (1.4.1)

s " pq `pq 2 (1.4.2)

Q 2 " ´q2 " ´pk e ´k1 e q 2 (1.4.3) x Bj " Q 2 2q
¨p "

Q 2 s `Q2 ´m2 n » Q 2 s `Q2 (1.4.4) y " p ¨q p ¨ke " s `Q2 ´m2 n S ´m2 n » s `Q2 S " Q 2 x Bj S (1.4.5)
t " pp ´p1 q 2 (1.4.6)

β " Q 2 2q ¨pp ´p1 q " Q 2 Q 2 `M 2 X ´t
(1.4.7)

x P " q ¨pp ´p1 q

q ¨p " Q 2 `M 2 X ´t s `Q2 ´m2 n » Q 2 `M 2 X ´t s `Q2 (1.4.8)
where k e , k 1 e is the momenta of the incoming and outgoing electron, p 1 the momentum of the Y system and p the incoming proton and M X the mass of the X system. As usual, the mass of the proton is neglected. Y " lnp1{x Bj q is the rapidity gap between the virtual photon and the proton, lnp1{βq is the rapidity gap between the virtual photon and X and Y P " lnp1{x P q is the gap between X and the target. x P can be considered as the fraction of the proton longitudinal momentum carried by the Pomeron. x Bj , β, x P are related via

x Bj " βx P .

(1.4.9)

The cross-section for the DDIS process with intact proton is functions. The transverse and longitudinal diffractive structure functions are defined as

dσ epÑXp D dx Bj dQ 2 dx P dt " 2πα 2 em x Bj Q 4 " 1 `p1 ´yq 2 ‰ " F Dp4q 2 `xBj , Q 2 , x P , t ˘´y 2 1 `p1 ´yq 2 F e -(k e ) e -(k ′ e ) X(p X ) Y (p ′ ) P (p) q t x P p
x P F Dp4q T,L " Q 2 4π 2 α em Q 2 β dσ γ T,L pÑXp dM 2 X dt . (1.4.11)
When the outgoing proton is not directly detected, no measurement of t is possible and only the cross-section integrated in t is obtained [START_REF] Beuf | Diffractive deep inelastic scattering at NLO in the dipole picture: The q qg contribution[END_REF] dσ epÑXp

D dx Bj dQ 2 dx P " 2πα 2 em βQ 4 " 1 `p1 ´yq 2 ‰ " F Dp3q 2 `β, Q 2 , x P ˘´y 2 1 `p1 ´yq 2 F Dp3q L `β, Q 2 , x P ˘ȷ " 2πα 2 em βQ 4 " 1 `p1 ´yq 2 ‰ σ Dp3q r `β, Q 2 x P ˘(1.4.12)
where we also changed form x Bj to β dependence. Prior to DDIS detection, Ingelman and Schlein (IS) described them theoretically based on Regge theory [START_REF] Ingelman | Jet Structure in High Mass Diffractive Scattering[END_REF]. In the IS model, the proton contains intrinsically a Pomeron whose partonic content is probed by the virtual photon. Diffractive structure functions F Dp4q 2 pβ, Q 2 , x P , tq factorizes into a Pomeron flux factor f P{p px P , tq (describing the flux of Pomeron in the proton and might be assumed to be universal) and a universal structure function of the Pomeron, containing the Pomeron parton distribution functions with longitudinal momentum fraction β of the Pomeron. They even added some non-linear effects with the GLR and GLR-MQ on the Pomeron gluon densities in [START_REF] Ingelman | The Pomeron structure in DIS and gluon recombination effects[END_REF]. This model was refuted experimentally, for example [START_REF] Aaron | Inclusive Measurement of Diffractive Deep-Inelastic Scattering at HERA[END_REF] as the dependence on x P was found to depend on β.

In the collinear framework, a QCD factorization theorem was derived in [START_REF] Collins | Proof of factorization for diffractive hard scattering[END_REF], justified by the presence of the hard scale Q 2 . The diffractive structure functions are defined as the convolution of hard coefficient functions C 2i with the diffractive parton distribution functions f D i . The diffractive parton distribution function can be interpreted as describing the partonic content of the Pomeron in this resolved Pomeron model, just like usual proton PDFs in DIS. At higher energies, the diffractive events are described with a direct Pomeron contribution coupling to the diffractive state X of invariant mass M X and within the dipole picture. One of the first papers using the dipole picture for DDIS are done by Nikolai Nikolaev N. and B. G. Zakharov [START_REF] Nikolaev | Pomeron structure function and diffraction dissociation of virtual photons in perturbative QCD[END_REF][START_REF] Nikolaev | The Triple pomeron regime and the structure function of the pomeron in the diffractive deep inelastic scattering at very small x[END_REF]. Depending on M X or equivalently on the value of β , X could be q q or q q with additional gluons emissions. More precisely, as explained in [START_REF] Kowalski | Nuclear enhancement and suppression of diffractive structure functions at high energies[END_REF], x P ă 0.01 is required in practice for the rapidity gap between X and the outgoing hadron to be detectable and to be in the domain of validity of the dipole model. This means that lnp1{βq is not too large for the kinematic at HERA and at the EIC and so taking into account only q q and q qg should be enough. More precisely, as discussed in [START_REF] Bartels | An Analysis of diffraction in deep inelastic scattering[END_REF], in the limit β Ñ 1 or small M X , the dominant component of the proton diffractive structure function is F D L,q q. At intermediate β " 0.5, the dominant component is F D T,q q. In the limit β Ñ 0 or large M X , the q qg Fock state dominates. At LO, the Pomeron is a two-gluon exchange in a colour-singlet state that couples to X. This means that the diffractive processes are highly sensitive to gluon distribution. A good description of the HERA data with Pomeron as two-gluon exchange was achieved in [START_REF] Bartels | An Analysis of diffraction in deep inelastic scattering[END_REF].

Apart from inclusive DDIS, one could focus on more exclusive observable like diffractive jets or meson production. For the production of a vector meson, it is conventional to distinguish two types of events: the coherent or exclusive production where the proton remains intact and the incoherent or dissociative production where the proton breaks up. The coherent events dominate at low t À 1{R 2 with R the size of the target while the incoherent events dominate at large t. The amplitude for exclusive diffractive production of a vector meson in light-cone perturbation theory is [START_REF] Kowalski | Exclusive diffractive processes at HERA within the dipole picture[END_REF]

M γ ˚pÑV p T,L `xP , Q 2 , ∆ ˘" ż d 2 ⃗ r ż 1 0 dz 4π `ψV ψ γ ˚˘T ,L `⃗ r, z, Q 2 ˘Mqqp el px P , ⃗ r, ∆q e ip1´zq⃗ r¨⃗ ∆ " i ż d 2 ⃗ r ż 1 0 dz 4π ż d 2 ⃗ b `ψV ψ γ ˚˘T ,L `⃗ r, z, Q 2 ˘2 " 1 ´S ´⃗ r, ⃗ b, x P ¯ı e ´i ⃗ ∆¨r ⃗ b´p1´zq⃗ rs .
The proton is probed at scale

x P " Q 2 `M 2 V ´t Q 2 `s (1.4.13)
where X " V so that M X " M V in eq. (1.4.8).

The cross-section for coherent vector meson production is [START_REF] Miettinen | Diffraction scattering and the parton structure of hadrons[END_REF] dσ

γ ˚pÑV p T,L dt " 1 16π ˇˇAM γ ˚pÑV p T,L px P , Q 2 , ∆q Eˇˇˇ2 . (1.4.14)
The above expression implies that coherent production is sensitive to the average colour density profile of the target and one could use this cross-section to find the S-matrix element for the dipole-proton scattering [START_REF] Munier | Impact parameter dependent S matrix for dipole proton scattering from diffractive meson electroproduction[END_REF]. The incoherent cross-section for vector meson production can be written as a variance [START_REF] Miettinen | Diffraction scattering and the parton structure of hadrons[END_REF] dσ

γ ˚pÑV X T,L dt " 1 16π ˆBˇˇˇM γ ˚pÑV p T,L px P , Q 2 , ∆q ˇˇ2 F ´ˇˇA M γ ˚pÑV p T,L px P , Q 2 , ∆q Eˇˇˇ2 ˙(1.4.15)
and is therefore sensitive to event-by-event fluctuations.

It is often assumed that the S-matrix element for the dipole scattering is mostly real, i.e. S " Re S (or equivalently Ă M q qp el is predominately imaginary). Thus, one replaces 1 Ś ´⃗ r, ⃗ b, x P ¯in exclusive vector meson production by the dipole amplitude N ´⃗ r, ⃗ b, x P ¯introduced in eq. (1.2.66). One could also replace 2

" 1 ´S ´⃗ r, ⃗ b, x P ¯ı by dσ q qp d 2⃗ b
. This assumption has been used for example in [START_REF] Kowalski | An Impact parameter dipole saturation model[END_REF][START_REF] Kowalski | Exclusive diffractive processes at HERA within the dipole picture[END_REF][START_REF] Mäntysaari | Confronting impact parameter dependent JIMWLK evolution with HERA data[END_REF][START_REF] Mäntysaari | Revealing proton shape fluctuations with incoherent diffraction at high energy[END_REF][START_REF] Mäntysaari | Evidence of strong proton shape fluctuations from incoherent diffraction[END_REF][START_REF] Mäntysaari | Review of proton and nuclear shape fluctuations at high energy[END_REF].

Chapter 2

Saturation at past, present and future colliders

This chapter aims to review non-exhaustively the searches of gluon saturation at past, present and future colliders in γ p˚q P processes where P could be a nucleon or nucleus target. A large bulk of the information originate from [START_REF] Morreale | Mining for Gluon Saturation at Colliders[END_REF][START_REF] Salazar | Exploring the small-x gluon structure of protons and nuclei in high energy deep inelastic scattering[END_REF].

Gluon saturation at past and existing colliders: HERA, LHC, RHIC

Hints of gluon satuation at HERA, the Relativistic Heavy-ion Collider (RHIC) located at Brookhaven National Laboratory (BNL) complex and the Large Hadron Collider (LHC) at CERN have been found in the inclusive DIS measurements (structure functions, reduced crosssection...) and the diffractive measurements but competing mechanisms and uncertainties hinder its unambiguous experimental discovery as we will explain.

Inclusive DIS

Comprehensive analysis of HERA inclusive data have been performed in the saturation framework. The first analysis with saturation in mind was performed with the Golec-Biernat-Wusthoff (GBW) model in [START_REF] Golec-Biernat | Saturation effects in deep inelastic scattering at low Q**2 and its implications on diffraction[END_REF]. This model assumes the following form for the dipole cross-section

σ q qp GBW p⃗ r, xq " σ 0 " 1 ´exp ˆ´⃗ r 2 Q 2 s pxq 4 ˙* (2.1.1)
where the saturation scale is defined as

Q 2 s pxq " Q 2 0 ´x0 x ¯λ . (2.1.2) Q 0 " 1
GeV is for dimensional purposes. σ 0 {2 corresponds to a normalization constant coming from the integration over ⃗ b, i.e. σ 0 {2 " ş R d 2 ⃗ b with R the interaction region and λ is a fit parameter controlling the energy dependence of the saturation scale. There are only three parameters to be fitted: σ 0 , λ, x 0 . This mainly phenomenological model has the basic wanted properties of dipole-target cross-section, namely on one side colour transparency [START_REF] Frankfurt | Interaction of small size wave packet with hadron target[END_REF] i.e. for small |⃗ r | Ñ 0 the dipole cross-section goes to 0 too (this is clear after expanding the exponential to the first order in ⃗ r 2 ). This property can be understood as follow: for small dipole size, the dipole behaves as a colour neutral object so the S-matrix is closed to unity i.e. no scattering with target and therefore the cross-section goes to 0. For large |⃗ r | values, the dipole cross-section goes to one which implies that the DIS cross-section from eq. (1.2.65) goes to a constant value. Thus, saturation tames the growth of the DIS cross-section. The GBW model gives reasonable agreement for inclusive DIS for not too large values of Q 2 for old and more recent HERA measurements. Indeed, in [START_REF] Golec-Biernat | Saturation model of DIS : an update[END_REF], the model is found to give a good description of

F 2 , F c 2 , F b 2
HERA recent data for Q 2 À 10 GeV 2 . It has led to the observation of geometric scaling in [START_REF] Stasto | Geometric scaling for the total gamma* p cross-section in the low x region[END_REF] for data in x ă 0.01 and 0.045 GeV 2 ă Q 2 ă 450 GeV 2 . Geometric scaling is the fact that the σ γ ˚p depends on only one variable τ " Q 2 {Q 2 s pxq instead of Q 2 and x separately. With empirical observables such as geometric scaling , x ď 0.01 is conventionally taken to be the valid regime of the dipole picture or the region to start study of saturation physics1 .

The Bartels, Golec-Biernat, Kowalski (BGBK) parametrisation [START_REF] Bartels | A modification of the saturation model: DGLAP evolution[END_REF] is the DGLAP-improved version of the GBW model. One of the issue of the GBW model is that it does not match with the pQCD result at large Q 2 or small values of |⃗ r |. In [START_REF] Frankfurt | Interaction of small size wave packet with hadron target[END_REF], the cross-section for a dipole of small size ⃗ r in the double logarithmic approximation is found to be of the form

σ q qp p⃗ r, xq " π 2 3 ⃗ r 2 α s pµ 2 qxgpx, µ 2 q , (2.1.3)
where µ 2 behaves as C{⃗ r 2 with C a constant not determined. The above equation is a manifestation of colour transparency. The dipole-target cross-section falls down quadratically for |⃗ r | Ñ 0. From eqs. (2.1.1, 2.1.3), we get a gluon distribution of the form

xgpx, µ 2 q " 3 4π 2 α s pµ 2 q σ 0 Q 2 s pxq . (2.1.4)
If α s is fixed then this gluon distribution is scale-independent, which contradicts with DGLAP evolution equation. To incorporate the DGLAP evolution equation in the small-|⃗ r | behaviour of the dipole cross-section while keeping the large-|⃗ r | behaviour of the GBW model that encodes the idea of saturation, the following modification of the GBW model was proposed:

σ q qp BGBK p⃗ r, xq " σ 0 " 1 ´exp ˆ´π 2 ⃗ r 2 α s pµ 2 qxgpx, µ 2 q 3σ 0 ˙* (2.1.5) µ 2 " C ⃗ r 2 `µ2 0 . (2.1.6)
The assumed initial form of the gluon distribution at the initial scale

Q 2 0 " 1 GeV 2 :
xgpx, Q 2 0 q " A g x ´λg p1 ´xq 5.6 .

(2.1.7)

There are a total of five parameters to fit to data in the BGBK model: A g , λ g , σ 0 , µ 0 , C. The 5.6 exponent that determines the behaviour of the gluon distribution at large x is motivated by gluon PDFs fit.

To go from the initial scale to µ 2 , one uses the DGLAP evolution equation of the gluon distribution at LLA without the quark contribution (as at small-x gluon dominates):

Bg `x, µ 2 B ln µ 2 " α s `µ2 2π ż 1 x dz z P gg pxqg `x{z, µ 2 ˘(2.1.8)
For small-|⃗ r |, µ 2 » C{⃗ r 2 and expanding eq. (2.1.5) to the first order term in ⃗ r 2 , the obtained result is exactly equal to eq. (2.1.3). For large dipole size |⃗ r |, µ 2 » µ 2 0 , one find back the GBW dipole cross-section with

Q 2 s pxq " 4π 2 3σ 0 α s `µ2 0 ˘xg `x, µ 2 0 ˘, (2.1.9)
with the x-dependence given by the gluon distribution at scale µ 2 0 and the cross-section saturates to σ 0 for very large dipole sizes. The BGBK model has been found to describe well inclusive DIS data for Q 2 À 650 GeV 2 in [START_REF] Golec-Biernat | Saturation model of DIS : an update[END_REF].

Another model was constructed by Iancu, Itakura and Munier (IIM) in [START_REF] Iancu | Saturation and BFKL dynamics in the HERA data at small x[END_REF] which is based on universal properties of the solutions of the BK equation. In this model, the dipole amplitude has the form

N p⃗ r, xq " 2 ˆ$ & % N 0 ´|⃗ r |Qs 2 ¯2´γ s`1 κλY ln 2 |⃗ r |Qs ¯: |⃗ r |Q s ď 2 1 ´e´A ln 2 pB|⃗ r |Qsq : |⃗ r |Q s ą 2 (2.1.10)
where Y " lnp1{xq is the total rapidity of the process,

A " ´N 2 0 γ 2 s p1 ´N0 q 2 ln p1 ´N0 q , B " 1 2 p1 ´N0 q ´p1´N 0 q N 0 γs (2.1.11)
and the saturation scale Q s pxq is defined exactly like in the GBW model i.e. as in eq. (2.1.2). The form in eq. (2.1.10) has been found by smoothly interpolating two limiting behavoirs that are analytically under control: the solutions to the LL BFKL equation in the vicinity of the saturation line obtained using the saddle point approximation for small dipole size |⃗ r | ! 1{Q s and the Levin-Tuchin solution [START_REF] Levin | Solution to the evolution equation for high parton density QCD[END_REF] of the BK equation deep inside the saturation region for larger dipole sizes |⃗ r | " 1{Q s . γ s is the saddle point in the vicinity of the saturation regime and κ " χ 2 pγ s q {χ 1 pγ s q with χpγq " 2ψp1q ´ψpγq ´ψp1 ´γq the LL BFKL characteristic function and ψpγq " d ln Γpγq{dγ.

The GBW, BGBK and IIM models have one shortcoming though, which is to not have any ⃗ b dependence which is needed to be able to describe exclusive processes since the total momentum transfer with the target is measured. One would also expects the saturation scale to depend on this impact parameter as more saturation is expected at the center of the proton than at its edge.

The b-CGC model [START_REF] Kowalski | Exclusive diffractive processes at HERA within the dipole picture[END_REF] is the IIM model with a additional ⃗ b-dependence. It consists only in replacing the GBW saturation scale by

Q s " Q s ´x, ⃗ b ¯" ´x0 x ¯λ 2 « exp ˜´⃗ b 2 2B CGC ¸ff 1 2γs
.

(2.1.12)

B CGC is fitted to t-distribution of exclusive photo-production of J{ψ . This model has been used for analysis and comparison to a wide range of HERA data in [START_REF] Watt | Impact parameter dependent color glass condensate dipole model[END_REF][START_REF] Rezaeian | Impact-parameter dependent Color Glass Condensate dipole model and new combined HERA data[END_REF].

The impact parameter saturation (IP-Sat) model [START_REF] Kowalski | An Impact parameter dipole saturation model[END_REF] can be considered as the BGBK model with additional ⃗ b-dependence

N IPSat ´⃗ r, ⃗ b, x ¯" 1 ´exp ˆ´π 2 2N c ⃗ r 2 α s pµ 2 qxgpx, µ 2 qT ´⃗ b ¯˙, (2.1.13)
where T ´⃗ b ¯is the transverse profile function of the target. In the IP-Sat model, the initial condition eq. (2.1.7) is defined at µ 2 0 instead of Q 2 0 and the parameters are µ 2 0 , C, A g , λ g and the parameters for the profile function (instead of σ 0 in the BGBK model). The model was fitted with more recent and diverse HERA data in for example [START_REF] Rezaeian | Analysis of combined HERA data in the Impact-Parameter dependent Saturation model[END_REF][START_REF] Mäntysaari | In depth analysis of the combined HERA data in the dipole models with and without saturation[END_REF].

Roughly speaking then, these two models introduce a ⃗ b dependence in the saturation scale via the addition of a proton transverse profile function. The parameters of this profile function are typically constrained with the exclusive production of vector meson in DIS. All the above models are a parametrization of the dipole amplitude or cross-section and they do not incorporate proper small-x evolution.

The AAMS and AAMQS global fits [START_REF] Albacete | Non-linear QCD meets data: A Global analysis of lepton-proton scattering with running coupling BK evolution[END_REF][START_REF] Albacete | AAMQS: A non-linear QCD analysis of new HERA data at small-x including heavy quarks[END_REF] were the first fits to use the BK equation for the x-dependence of the dipole amplitude and started the era of using BK in numerical predictions of processes potentially sensitive to saturation. This was an important step forward for rigorous description of data and theory-based numerical predictions as the only non-perturbative input was the dipole amplitude at the initial Bjorken-x typically chosen to be x 0 " 0.01. In these two papers, two initial conditions were chosen. They are inspired from the GBW and McLerran-Venugopalan (MV) models [START_REF] Mclerran | Computing quark and gluon distribution functions for very large nuclei[END_REF][START_REF] Mclerran | Gluon distribution functions for very large nuclei at small transverse momentum[END_REF][START_REF] Mclerran | Green's functions in the color field of a large nucleus[END_REF] but modified by the incorporation of an anomalous dimension γ that controls the steepness of the fall-off of the dipole amplitude with decreasing |⃗ r | and controls the slope of the scattering amplitude in the transition from the dilute region to the black disk region:

N GBW γ p⃗ r, x " x 0 q " 1 ´exp " ´ˆ⃗ r 2 Q 2 s0 4 ˙γȷ , (2.1.14)
and

N MV γ p⃗ r, x " x 0 q " 1 ´exp « ´`⃗ r 2 Q 2 s0 ˘γ 4 ln ˆ1 |⃗ r |Λ `e˙ff . (2.1.15)
The original MV which will be presented in section 5.1.2 corresponds to γ " 1 but in the MV γ model, it is taken as a free parameter to fit to data. The dipole amplitude here are assumed to be impact-parameter independent, which lead to the presence of a fit parameter σ 0 at cross-section level to account for the integration over ⃗ b. They also incorporated running coupling corrections to the evolution i.e. used the running coupling BK (rcBK) equation. The impact-parameter independent rcBK following the Balitsky prescription [START_REF] Balitsky | Quark contribution to the small-x evolution of color dipole[END_REF] reads

BN p⃗ r, xq B ln p1{xq " ż d⃗ r 1 K run p⃗ r, ⃗ r 1 , ⃗ r 2 q ˆrN p⃗ r 1 , xq `N p⃗ r 2 , xq ´N p⃗ r, xq ´N p⃗ r 1 , xq N p⃗ r 2 , xqs , (2.1.16)
with

K run p⃗ r, ⃗ r 1 , ⃗ r 2 q " N c α s `⃗ r 2 2π 2 « ⃗ r 2 ⃗ r 2 1 ⃗ r 2 2 `1 ⃗ r 2 1 ˜αs `⃗ r 2 1 αs `⃗ r 2 2 ˘´1 ¸`1 ⃗ r 2 2 ˜αs `⃗ r 2 2 αs `⃗ r 2 1 ˘´1 ¸ff . (2.1.17)
In both the AAMS and AAMQS papers, they obtained a good description of HERA inclusive structure functions. In the AAMQS compared to the AAMS paper, they included the heavy quark (c, b) contributions into the reduced cross-section and structure functions. These fits are at LO and at this order it is not possible to describe simultaneously massless and massive quark contributions to the DIS cross-sections σ γ ˚p T,L px, Qq with the same parameters when using the BK equation for the energy evolution of the target. Indeed, the authors in [START_REF] Albacete | AAMQS: A non-linear QCD analysis of new HERA data at small-x including heavy quarks[END_REF] introduced two different σ 0 , necessary to have a good description of data at this order:

σ γ ˚p T,L `x, Q 2 ˘" σ 0 ÿ f "u,d,s ż 1 0 dz d⃗ r ˇˇΨ f T,L `ef , m f , z, Q 2 , ⃗ r ˘ˇˇ2 N light p⃗ r, xq `σheavy 0 ÿ f "c,b ż 1 0 dz d⃗ r ˇˇΨ f T,L `ef , m f , z, Q 2 , ⃗ r ˘ˇˇ2 N heavy p⃗ r, xq . (2.1.18)
But it is not physical for the dipole amplitude to depend on the quark masses as they should be negligible in the high-energy interaction with the target and one expects then that the contributions from massless and massive quarks to the structure functions are given by the same dipole amplitude. This is why it is important to go to higher-order to see if this problem persists. It is indeed not the case anymore at NLO as shown by the numerical study in [START_REF] Hänninen | Proton Structure Functions at Next-to-Leading Order in the Dipole Picture with Massive Quarks[END_REF]. This study has been made possible thanks to the recent computations of the NLO light-cone wave function for γ λ Ñ q q and the LO light-cone wave function γ ˚Ñ q qg , including quark mass which are then used to compute the massive contribution to DIS cross-sections at NLO [START_REF] Beuf | Massive quarks in NLO dipole factorization for DIS: Longitudinal photon[END_REF][START_REF] Beuf | Massive Quarks at One Loop in the Dipole Picture of Deep Inelastic Scattering[END_REF][START_REF] Beuf | Massive quarks in NLO dipole factorization for DIS: Transverse photon[END_REF]. What the authors have done in [START_REF] Hänninen | Proton Structure Functions at Next-to-Leading Order in the Dipole Picture with Massive Quarks[END_REF] is that they have taken fits of the dipole amplitude from [START_REF] Beuf | Color Glass Condensate at next-to-leading order meets HERA data[END_REF], numerically computed predictions for massive contributions to inclusive structure functions from these fits and see how well they describe data. The fits from [START_REF] Beuf | Color Glass Condensate at next-to-leading order meets HERA data[END_REF] are at NLO and are only for massless contrubutions as at that time, the massive ones have not been yet computed. The initial condition used is the MV γ model whose expression is eq. (2.1.15). At NLO, there is more freedom in the fitting procedure compared to the leading order. The most important differences are [START_REF] Penttala | Diffractive Processes at Next-to-Leading Order in the Dipole Picture[END_REF]:

1. Two sets of data are used for the fit. The first one is the full HERA data [START_REF] Aaron | Combined Measurement and QCD Analysis of the Inclusive e+-p Scattering Cross Sections at HERA[END_REF][START_REF] Abramowicz | Combination of measurements of inclusive deep inelastic e ˘p scattering cross sections and QCD analysis of HERA data[END_REF] and the second one is the pseudo-data associated to only the light-quark contribution. They are constructed from the full HERA data where the massive quark contribution predicted from the IP-Sat model in [START_REF] Mäntysaari | In depth analysis of the combined HERA data in the dipole models with and without saturation[END_REF] is subtracted. The pseudo-data makes physically more sense as the computations of the structure functions in [START_REF] Beuf | Color Glass Condensate at next-to-leading order meets HERA data[END_REF] uses only light-quark.

2. Three different versions of BK evolution are used corresponding to different approximations of the full NLO BK evolution. To be completely consistent with the perturbative computations, one would need to use the full NLO BK equation [START_REF] Balitsky | Quark contribution to the small-x evolution of color dipole[END_REF][START_REF] Balitsky | Next-to-leading order evolution of color dipoles[END_REF][START_REF] Lappi | Next-to-leading order Balitsky-Kovchegov equation beyond large N c[END_REF][START_REF] Kovchegov | Triumvirate of Running Couplings in Small-x Evolution[END_REF] but it is not feasible because of its numerical complexity. The three approximations are: the kinematically constrained BK (KCBK) [START_REF] Beuf | Improving the kinematics for low-x QCD evolution equations in coordinate space[END_REF] which introduces a kinematical constraint that forces an explicit time ordering between subsequent gluon emission, the resummed BK (resumBK) [START_REF] Iancu | Collinearly-improved BK evolution meets the HERA data[END_REF][START_REF] Iancu | Resumming double logarithms in the QCD evolution of color dipoles[END_REF] which resums the large single and double transverse logarithms that appear at higher order into the BK kernel and finally the target rapidity BK (TBK) where the target rapidity is used in the evolution instead of the projectile rapidity. The target rapidity η is calculated from the projectile one with

η " Y ´max " 0, ln ˆ1 ⃗ r 2 Q 2 0 ˙* (2.1.19)
where

Q 2 0 " 1 GeV 2 .
3. The expression of the strong coupling constant in coordinate space used is

α s `⃗ r 2 ˘" 4π
β 0 ln « ˆµ2 0 Λ 2 QCD ˙1{c `ˆ4C 2 ⃗ r 2 Λ 2 QCD ˙1{c ff c (2.1.20)
with β 0 " p11N c ´2N F q {3, N F " 3 and Λ QCD " 0.241GeV. µ 0 " 2.5Λ QCD and c " 0.2 regulate the running of the coupling in the infra-red region. C 2 from Fourier analysis is expected to have value e ´2γ E [START_REF] Kovchegov | Triumvirate of Running Couplings in Small-x Evolution[END_REF][START_REF] Lappi | On the running coupling in the JIMWLK equation[END_REF] but in [START_REF] Beuf | Color Glass Condensate at next-to-leading order meets HERA data[END_REF] is left as a free parameter determined from fit in order to "absorb missing non-perturbative or higher order contributions in the modified evolution speed". It controls how coordinate scales relate to the momentum ones.

Two schemes for the running of the coupling are used. The first scheme named the Balitsky + smallest dipole (BSD) is to use the Balitsky prescription [START_REF] Balitsky | Quark contribution to the small-x evolution of color dipole[END_REF] in the BK evolution kernel as in eq. (2.1.17) and then set the scale of the coupling constant in the NLO impact factor and in the large transverse logarithms in the ResumBK evolution equation to the smallest dipole ⃗ r 2 " min

␣ ⃗ x 2 01 , ⃗ x 2 20 , ⃗ x 2 21 (
. The second scheme is the parent dipole scheme where eq. (2.1.20) is used in both the evolution equation and the impact factor and the coupling is also always at the scale ⃗ r 2 " ⃗ x 2 01 i.e. the dipole size of the quark-antiquark dipole.

4. Two different values of the initial scale are used, namely Y 0,BK " 0 (so x 0 " 1) and Y 0,BK " lnp1{0.01q. The latter corresponds to x 0 " 0.01 the typical value used for the initial scale in dipole amplitude fits as it is not clear if the assumptions for deriving the MV model and the BK equation are valid for larger values of x (so smaller values of Y ).

These different setups results in 24 fits. Each fits have fours parameters. Three are related to the initial MV γ model: the saturation scale at the initial rapidity Q s0 , the anomalous dimension γ and the transverse area of the proton σ 0 {2. The last parameter is C 2 in eq. (2.1.20). All the fits describe extremely well the data for structure functions with only massless quark contribution and one can no distinguish one fit from another. Comparing the prediction for massive quark contribution from these fits to experimental data from [START_REF] Aaron | Combined Measurement and QCD Analysis of the Inclusive e+-p Scattering Cross Sections at HERA[END_REF][START_REF] Abramowicz | Combination of measurements of inclusive deep inelastic e ˘p scattering cross sections and QCD analysis of HERA data[END_REF][START_REF] Abramowicz | Combination and QCD analysis of charm and beauty production cross-section measurements in deep inelastic ep scattering at HERA[END_REF], only three fits recapitulated in table 2.1 give a simultaneous good description of the charm reduced cross-section and the inclusive total reduced cross-sections as observed in fig. 2.1.1. Results were also compared to the bottom quark production data [START_REF] Abramowicz | Combination and QCD analysis of charm and beauty production cross-section measurements in deep inelastic ep scattering at HERA[END_REF] but this does not provide further constraints on the fits because of the large data uncertainties. With only the charm and inclusive total reduced cross-sections, it is not possible to further constrain the three fits and thus data comparisons with other processes are required such as F L measurements. F L is not completely independent of σ r but is more sensitive to the saturation region and can hopefully give additional constraints. But for now, with the current HERA data for F L [START_REF] Andreev | Measurement of inclusive ep cross sections at high Q 2 at ? s " 225 and 252 GeV and of the longitudinal proton structure function F L at HERA[END_REF], the three fits do not show any differences. [START_REF] Beuf | Color Glass Condensate at next-to-leading order meets HERA data[END_REF] that are found to be compatible with the massive quark DIS HERA data. The "light-q" refers to using pseudo-data consisting of only the light-quark contributions. This is done by taking the full HERA data and subtracting the prediction for the massive quark contribution from the IP-sat parametrization done in [START_REF] Mäntysaari | In depth analysis of the combined HERA data in the dipole models with and without saturation[END_REF]. "PD" and "BSD" refers to the parent dipole and Balitsky + smallest dipole schemes for α s . The value χ 2 tot {N refers to the total structure functions including both light and massive quark. Table from [START_REF] Penttala | Diffractive Processes at Next-to-Leading Order in the Dipole Picture[END_REF].

# Data BK equation α s Y 0,BK m c rGeVs χ 2 c {N m b rGeVs χ 2 b {N χ 2 tot {N 1 Light-q ResumBK PD 0 1.
Figure 2.1.1: Charm reduced DIS cross-section and total reduced cross-section from the three fits of table 2.1 that are compatible with HERA data [START_REF] Aaron | Combined Measurement and QCD Analysis of the Inclusive e+-p Scattering Cross Sections at HERA[END_REF][START_REF] Abramowicz | Combination of measurements of inclusive deep inelastic e ˘p scattering cross sections and QCD analysis of HERA data[END_REF][START_REF] Abramowicz | Combination and QCD analysis of charm and beauty production cross-section measurements in deep inelastic ep scattering at HERA[END_REF][START_REF] Abramowicz | Combination and QCD Analysis of Charm Production Cross Section Measurements in Deep-Inelastic ep Scattering at HERA[END_REF]. Figures from [START_REF] Hänninen | Proton Structure Functions at Next-to-Leading Order in the Dipole Picture with Massive Quarks[END_REF].

Apart from using approximations of the full NLO BK, another way to improve precision is to use BK including the impact parameter dependence, just like in [START_REF] Berger | Small x nonlinear evolution with impact parameter and the structure function data[END_REF]. The ⃗ b-dependent dipole amplitude obtained from BK at LLA has very different behaviour than the ⃗ b-independent dipole amplitude as shown numerically in [START_REF] Golec-Biernat | On solutions of the Balitsky-Kovchegov equation with impact parameter[END_REF]. First, at fixed ⃗ b, the ⃗ b-dependent dipole amplitude goes to 0 for large dipole sizes whereas the ⃗ b-independent dipole amplitude goes to unity i.e. to the black disk limit for large dipole sizes. This drop to 0 is explained by the fact at large ⃗ r and fixed ⃗ b, the end points of the dipole i.e. the quark and antiquark are beyond the interaction region in ⃗ b space and so the dipole amplitude vanishes. Second, for finite ⃗ r, the dipole amplitude, at large ⃗ b, has a power-like tail

N ´⃗ b ¯" 1 | ⃗ b | γ , γ ą 0 . (2.1.21)
As explained in [START_REF] Kovner | Nonlinear QCD evolution: Saturation without unitarization[END_REF][START_REF] Kovner | No Froissart bound from gluon saturation[END_REF], this power tail leads to violation of the Froissart-bound for total cross-sections even though the dipole amplitude is saturated locally in ⃗ b space, i.e. respects locally the black disk limit. Kovner and Wiedemann argued that this power-tail issue is related to the fact that there is no mass scales in BK LL. This is why in [START_REF] Berger | Small x nonlinear evolution with impact parameter and the structure function data[END_REF], the BK kernel was modified to add some mass cut-off to regularize the long-range Coulomb type interaction which also correspond to adding some confinement effect (the mass cut-off should be of the order of Λ QCD ). This scheme is one that is often used in numerical studies. The ⃗ b-dependence has also been kept in [START_REF] Bendova | Solution to the balitsky-kovchegov equation with the collinearly improved kernel including impact-parameter dependence[END_REF] where ResumBK has also been used.

Structure functions analysis that incorporate saturation compared to HERA data have been influential, but some doubt remains on the signals of saturation in inclusive DIS data. In [START_REF] Mäntysaari | In depth analysis of the combined HERA data in the dipole models with and without saturation[END_REF], the authors have performed a comparison between the impact parameter model with and without saturation, namely the IP-Sat and IP Non-Sat which is the linearized version of the IP-Sat model and concretely corresponds to the first non-zero term after expanding the exponential in eq. (2.1.13). They have found that they are both capable of describing the available data when both models are being fitted independently.

Another element that makes the signals of saturation in inclusive DIS data inconclusive is the large non-perturbative contributions at low and moderate Q 2 . More specifically, in [START_REF] Mäntysaari | Confronting impact parameter dependent JIMWLK evolution with HERA data[END_REF][START_REF] Mäntysaari | In depth analysis of the combined HERA data in the dipole models with and without saturation[END_REF], the structure functions F 2 and F L have been shown to have have significant contribution from large dipoles |⃗ r | ą 1{Λ QCD , see fig. 2.1.2. This causes issue as in this region non-perturbative physics should dominate but this is not well-incorporated in the numerics and models. Large dipole contributions arise in the aligned jet configuration where either the quark or the antiquark carries most of the photon longitudinal momentum (z Ñ 0, 1). The aligned jet configuration has more effect in F 2 than in F L because of the differences in structure between the longitudinal and transverse photon wave-function overlaps, see eq. (1.2.60,1.2.61). One needs to go to high Q 2 to suppress the contributions from large dipole but saturation should be less visible in this region of Q 2 .

Diffractive measurements

DDIS have been extensively studied at HERA. The first hints of saturation were observed in the analysis of diffractive data using the GBW model which gives a correct agreement [START_REF] Golec-Biernat | Saturation effects in deep inelastic scattering at low Q**2 and its implications on diffraction[END_REF][START_REF] Golec-Biernat | Saturation in diffractive deep inelastic scattering[END_REF] with the ratio of diffractive to inclusive cross-section found to be almost constant with γ ˚p center-of-mass energy [START_REF] Breitweg | Measurement of the diffractive cross-section in deep inelastic scattering using ZEUS 1994 data[END_REF][START_REF] Adloff | Inclusive measurement of diffractive deep inelastic ep scattering[END_REF]. These results remain true in the BGBK model [START_REF] Bartels | A modification of the saturation model: DGLAP evolution[END_REF]. Just like for inclusive DIS cross-section, there is observation of geometric scaling for the diffractive inclusive cross-section [START_REF] Marquet | Geometric scaling in diffractive deep inelastic scattering[END_REF].

In [START_REF] Marquet | A Unified description of diffractive deep inelastic scattering with saturation[END_REF], an analysis of diffractive structure functions with the IIM has been done. A more refined study of the diffractive structure functions with the IP-Sat and the b-CGC models has been executed in [START_REF] Kowalski | Nuclear enhancement and suppression of diffractive structure functions at high energies[END_REF]. In the latter paper, an interpolation function for the q qg contribution has been constructed to get a better description of it as at the time the complete calculation of this contribution has not been achieved yet. This interpolation function is built out of the two expressions of this contribution in two particular limits. The first limit is the large Q 2 limit where the q qg scatters with the target as an effective gg dipole as the q q transverse distance is much smaller than the qg transverse distance [START_REF] Wusthoff | Large rapidity gap events in deep inelastic scattering[END_REF][START_REF] Gotsman | Diffractive leptoproduction of small masses in QCD[END_REF]. The second approach is derived for β Ñ 0 or large M X limit where the q qg system is treated as two q q dipoles with different sizes. The two models does not coincide for β Ñ 0 at moderate Q 2 [START_REF] Marquet | A Unified description of diffractive deep inelastic scattering with saturation[END_REF] hence the construction of an interpolation function.

Diffractive production of vector particles (DVCS, vector meson production) has also been extensively studied at HERA. Predictions based on the GBW and the AAMQS models for the dependence on Q 2 and on the energy of polarized cross-sections σ T and σ L of the diffractive leptoproduction of the ρ meson at high energy have been found to agree with HERA data for Q 2 ą 5 GeV 2 . Moreover, the t-spectra and energy dependence for coherent production of various vector particles have been compared to HERA data in the IP-Sat model [START_REF] Kowalski | An Impact parameter dipole saturation model[END_REF][START_REF] Rezaeian | Analysis of combined HERA data in the Impact-Parameter dependent Saturation model[END_REF], in the b-CGC model [START_REF] Watt | Impact parameter dependent color glass condensate dipole model[END_REF][START_REF] Rezaeian | Impact-parameter dependent Color Glass Condensate dipole model and new combined HERA data[END_REF] or in both models [START_REF] Armesto | Exclusive vector meson production at high energies and gluon saturation[END_REF]. An analysis using BK, more precisely the three approximation of NLL BK (TBK, ResumBK, KCBK) has been done in [START_REF] Mäntysaari | Exclusive production of light vector mesons at next-to-leading order in the dipole picture[END_REF]. The authors have found good agreement in the Q 2 and s dependence of the NLO cross-section for exclusive ϕ and ρ production in the limit of high photon virtuality, the three types of BK giving similar numerical results. These studies has been applied to LHC for photoproduction too [START_REF] Armesto | Exclusive vector meson production at high energies and gluon saturation[END_REF][START_REF] Goncalves | Investigation of diffractive photoproduction of J{Ψ in hadronic collisions[END_REF][START_REF] Gonçalves | Exclusive Υ photoproduction in hadronic collisions at CERN LHC energies[END_REF][START_REF] Goncalves | Exclusive ρ and J/Ψ photoproduction in ultraperipheral pO and OO collisions at energies available at the CERN Large Hadron Collider[END_REF].

In contrast to coherent production, the incoherent vector meson production is sensitive to event-by-event fluctuations as shown from eq. (1.4.15). One such event-by-event fluctuations shown to be necessary to get a good description of HERA data is the geometric fluctuations or target shape fluctuations in the form of hot spot model [START_REF] Mäntysaari | Revealing proton shape fluctuations with incoherent diffraction at high energy[END_REF][START_REF] Mäntysaari | Evidence of strong proton shape fluctuations from incoherent diffraction[END_REF]. On top of these fluctuations, some implementation of saturation scale fluctuations [START_REF] Bzdak | Probing proton fluctuations with asymmetric rapidity correlations[END_REF][START_REF] Mclerran | Intrinsic Fluctuations of the Proton Saturation Momentum Scale in High Multiplicity p+p Collisions[END_REF] might also be incorporated within the hot spot model. In [START_REF] Kumar | Investigating the structure of gluon fluctuations in the proton with incoherent diffraction at HERA[END_REF], a model of hot spots within hot spots within hot spots (adding a structure of ten hot spots inside each of the original three and then further substructure with around sixty smaller hot spots inside each of the ten hot spots) has been implemented in order to describe the |t|-spectra of J{ψ photoproduction for |t| ă 30 GeV 2 . These studies have been extended to UPC at RHIC and at the LHC. In [START_REF] Mäntysaari | Probing subnucleon scale fluctuations in ultraperipheral heavy ion collisions[END_REF], the addition of event-by-event fluctuations such as the geometric ones has been shown to lead to an increases of the ratio of incoherent to coherent cross-section by a factor of 2. In [START_REF] Cepila | Energy dependence of dissociative J{ψ photoproduction as a signature of gluon saturation at the LHC[END_REF][START_REF] Cepila | Mass dependence of vector meson photoproduction off protons and nuclei within the energy-dependent hot-spot model[END_REF][START_REF] Kumar | Energy dependence of the proton geometry in exclusive vector meson production[END_REF], an energy dependence for the number of hot spots (number increasing with energy) has been added on top of the basic hot spot model. A comprehensive review on the subject of proton and nuclear shape fluctuations can be found in [START_REF] Mäntysaari | Review of proton and nuclear shape fluctuations at high energy[END_REF].

Just like for inclusive DIS, some doubts remain about the detection of saturation in diffractive processes. In [START_REF] Kowalski | An Impact parameter dipole saturation model[END_REF], both the the IP-Sat and IP Non-Sat both describe well the |t|-spectra of exclusive electroproduction of J{ψ. This is also seen in [START_REF] Mäntysaari | In depth analysis of the combined HERA data in the dipole models with and without saturation[END_REF] where on top of the the |t|-spectra, the s dependence of the total cross-section for the exclusive production of J{ψ is well-described by both models.

There is also the fact that in vector meson production, a significant source of theoretical uncertainty arises from the light-cone wave function of the vector meson on top of the uncertainty from the model used for the dipole amplitude. Recent developments on relativistic corrections to vector meson light-cone wave-function can be found in [START_REF] Lappi | Relativistic corrections to the vector meson light front wave function[END_REF].

Gluon saturation at future collider: EIC and LHeC

New generations of electron-hadron colliders will appear in the future, in particular the Electron-Ion Collider (EIC) [START_REF] Boer | Gluons and the quark sea at high energies: Distributions, polarization, tomography[END_REF][START_REF] Aschenauer | The electron-ion collider: assessing the energy dependence of key measurements[END_REF][START_REF] Accardi | Electron Ion Collider: The Next QCD Frontier: Understanding the glue that binds us all[END_REF][START_REF] Khalek | Science Requirements and Detector Concepts for the Electron-Ion Collider: EIC Yellow Report[END_REF][START_REF] Brüning | Electron-Hadron Colliders: EIC, LHeC and FCC-eh[END_REF] which will be built upon the infrastructures of RHIC and which construction is planned to start in 2024. It will collide proton, light and heavy ion beams to electron beam. Compared to HERA, the EIC will have also unpolarized heavy ion beams which will be important for uncovering saturation effects as they should be more important with the nuclear oomph factor in the expression of the saturation scale (see eq. (1.2.51)) despite the fact that the planned maximum center-of-mass energy squared is a little bit lower than at HERA. The EIC will also have a higher luminosity thus high statistics. Combined with high precision theoretical computation and thorough analysis of physical observables of diverse processes, clearer signals will hopefully be observed. There are also a new potential electronhadron colliders at CERN, the Large Hadron electron Collider (LHeC) [START_REF] Brüning | Electron-Hadron Colliders: EIC, LHeC and FCC-eh[END_REF][START_REF] Agostini | The Large Hadron-Electron Collider at the HL-LHC[END_REF][START_REF] Fernandez | A Large Hadron Electron Collider at CERN: Report on the Physics and Design Concepts for Machine and Detector[END_REF][START_REF] Lhec | Electron-Ion Collisions at the LHeC and FCC-he[END_REF] which will go down to values x as low as 10 ´6 and is at an advanced stage of design and awaiting approval.

Inclusive measurements

At HERA, the finding of gluon saturation have major obstacles for an irrefutable sign of it as discussed in section 2.1.1, namely the contribution from large dipoles at low and moderate Q 2 and the small values of the saturation scale Q 2 s as working with proton target. We have also shown that HERA inclusive data could be well-described by both saturated and non-saturated models. At the EIC, one expects higher values of the saturation scale, the saturation scale being enhanced with the nuclear oomph factor. This could lead to more pronounced differences between the saturation framework and leading twist formalism in structure functions, the higher twist corrections (enhanced in DIS on a nucleus) being a sign of saturation physics. The plot in fig. 2.2.1 represents the ratio

F L ´FL pleading twitq F L (2.2.1)
as a function of Q 2 and x with F L obtained in the saturation framework [START_REF] Bartels | Twist expansion of the nucleon structure functions, F(2) and F(L), in the DGLAP improved saturation model[END_REF]. The ratio is negative in both plots implying that higher twists tend to decrease structure functions. The largest difference is observed in the small px, Q 2 q corner. The effect is also higher in e `Au that in e `p as expected. Fig. 2.2.1 demonstrates that F L is rather sensitive to parton saturation. This observable can then be used to further constrain fits of the dipole amplitude as explained in [START_REF] Hänninen | Proton Structure Functions at Next-to-Leading Order in the Dipole Picture with Massive Quarks[END_REF]. Indeed, in the mentioned article the authors have made predictions at EIC kinematics of F L from the three fits of table 2.1. They have found that the fit 3, the one based on from TBK, gives a different prediction as shown in fig. 2.2.2. We also note that at the EIC, one would also be able constraint the nuclear dipole amplitude.

In [START_REF] Marquet | Unveiling saturation effects from nuclear structure function measurements at the EIC[END_REF], the authors generated pseudodata for the structure functions at electron-gold collisions using the rcBK with the MV model at initial scale and compared them to nuclear structure functions from extracted and extrapolated nuclear PDFs (nPDFs). They found that the latter is smaller that the pseudodata at small-x as shown in fig. 2.2.3 and argued that this incompatibility means possible signal of saturation but they do not exclude the possibility that Predictions are computed from the three fits of table 2.1. The inclusive as well as the charm and bottom contribution are shown. Figure from [START_REF] Hänninen | Proton Structure Functions at Next-to-Leading Order in the Dipole Picture with Massive Quarks[END_REF].

refitting nPDFs could successfully describe future nuclear structure functions as at the moment the nPDFs are not well-constrained at small x. The high statistics at the EIC with more precise theoretical computations hopefully will lead to a clear distinction between both scenarios.

A similar study to the one above has been done for the LHeC in [START_REF] Agostini | The Large Hadron-Electron Collider at the HL-LHC[END_REF]. Pseudodata for the inclusive reduced DIS cross-sections have been generated based on either DGLAP-based model or saturation-based model (GBW). They then have fitted the pseudodata with a DGLAP fit. The pseudo data are then fitted by a DGLAP calculation. The fit is excellent for the DGLAPgenerated pseudodata while the DGLAP fit on the GBW generated pseudodata has significant tension. The DGLAP fit manages to absorb some effects into a redefinition of the nPDFs but not completely so if gluon saturation is present at the LHeC, a DGLAP fit might not be able to fully conceal it and gluon saturation could then be observed.

Diffractive measurements

We briefly discuss the potential of diffractive measurements at EIC for the search of gluon saturation.

In [START_REF] Kowalski | Nuclear enhancement and suppression of diffractive structure functions at high energies[END_REF], the authors have predicted using the IP-Sat model and the b-CGC model that the measurements of the single diffractive dissociation in high-energy eA collisions encoded in terms of nuclear diffractive structure functions have interesting features. There is nuclear suppression of the total diffractive structure function in the small β region, a region where the q qg component dominates and there is a nuclear enhancement in the moderate and large β region where the q q components dominate. This nuclear suppression and enhancement are amplified with increasing A in the very small and large β region. They have found a disagreement in their predictions of the nuclear modification factor F D 2A {AF D 2p and predictions based on the leading twist formalism [START_REF] Frankfurt | Leading twist coherent diffraction on nuclei in deep inelastic scattering at small x and nuclear shadowing[END_REF]. They have also argued that the ratio of the diffractive to inclusive total cross-sections is expected to be larger in nuclei than in protons. This study needs to be updated with the recently fully calculated q qg contribution to diffractive structure function [START_REF] Beuf | Diffractive deep inelastic scattering at NLO in the dipole picture: The q qg contribution[END_REF] though.

Another possible diffractive measurement is the coherent vector meson production off nuclei. Predictions for this process using the IP-Sat and IP Non-Sat extended to nucleus target have been computed in [START_REF] Toll | Exclusive diffractive processes in electron-ion collisions[END_REF] and are shown in fig. 2.2.4. There is a clear distinction between the predictions from the saturated model to the non-saturated model for ϕ production, making this measurement one that might be quite sensitive to non-linear effects. To quantify saturation effects, it is necessary to compare these results to predictions from competing mechanisms such as the leading twist nuclear shadowing framework [START_REF] Frankfurt | Leading Twist Nuclear Shadowing Phenomena in Hard Processes with Nuclei[END_REF]. Instead of a heavy nucleus, the EIC offers the possibility to study the small-x gluon in light ions and predictions for exclusive vector meson production in the saturation framework have been performed [START_REF] Mäntysaari | Accessing the gluonic structure of light nuclei at a future electron-ion collider[END_REF].

Some recent theoretical advances for exclusive vector meson production have been made. The NLO computations in the shockwave approach and covariant perturbation theory of exclusive light vector meson production is presented in [START_REF] Boussarie | Next-to-Leading Order Computation of Exclusive Diffractive Light Vector Meson Production in a Saturation Framework[END_REF] and in the light-cone perturbation theory and CGC framework in [START_REF] Mäntysaari | Exclusive production of light vector mesons at next-to-leading order in the dipole picture[END_REF]. The exclusive heavy vector meson production at NLO has been computed in [START_REF] Mäntysaari | Exclusive heavy vector meson production at next-to-leading order in the dipole picture[END_REF][START_REF] Mäntysaari | Complete calculation of exclusive heavy vector meson production at next-to-leading order in the dipole picture[END_REF].

Some other diffractive measurements theoretically computed deemed promising for uncovering Figure 2.2.4: |t|-spectra for the diffractive coherent production of a vector meson in electron-gold collisions. On the left is J{ψ production and on the right is ϕ production. Figure from [START_REF] Toll | Exclusive diffractive processes in electron-ion collisions[END_REF].

saturation are the diffractive production of hard dijet accompanied by a soft jet [START_REF] Iancu | Probing Parton Saturation and the Gluon Dipole via Diffractive Jet Production at the Electron-Ion Collider[END_REF][START_REF] Iancu | Gluon dipole factorisation for diffractive dijets[END_REF] and exclusive diffractive dijet production [START_REF] Boussarie | Impact factor for high-energy two and three jets diffractive production[END_REF][START_REF] Boussarie | On the one loop γ p˚q Ñ qq impact factor and the exclusive diffractive cross sections for the production of two or three jets[END_REF][START_REF] Boussarie | Towards a complete next-to-logarithmic description of forward exclusive diffractive dijet electroproduction at HERA: real corrections[END_REF][START_REF] Hatta | Probing the Small-x Gluon Tomography in Correlated Hard Diffractive Dijet Production in Deep Inelastic Scattering[END_REF][START_REF] Hatta | Probing the Small-x Gluon Tomography in Correlated Hard Diffractive Dijet Production in Deep Inelastic Scattering[END_REF][START_REF] Altinoluk | Diffractive Dijet Production in Deep Inelastic Scattering and Photon-Hadron Collisions in the Color Glass Condensate[END_REF][START_REF] Salazar | Diffractive dijet production in impact parameter dependent saturation models[END_REF][START_REF] Mäntysaari | Diffractive Dijet Production and Wigner Distributions from the Color Glass Condensate[END_REF].

Semi-inclusive measurements

At the LHC and RHIC, measurements in ppdq ´ppAq collisions of two (hadro-produced) hadrons correlations [START_REF] Abdallah | Evidence for Nonlinear Gluon Effects in QCD and Their Mass Number Dependence at STAR[END_REF][START_REF] Braidot | Two Particle Correlations at Forward Rapidity in STAR[END_REF][START_REF] Adare | Suppression of back-to-back hadron pairs at forward rapidity in d`Au Collisions at ? s N N " 200 GeV[END_REF][START_REF] Adams | Forward neutral pion production in p+p and d+Au collisions at s(NN)**(1/2) = 200-GeV[END_REF] or nuclear modification factors [START_REF] Arsene | On the evolution of the nuclear modification factors with rapidity and centrality in d + Au collisions at s(NN)**(1/2) = 200-GeV[END_REF][START_REF] Aaij | Measurement of the Nuclear Modification Factor and Prompt Charged Particle Production in p ´P b and pp Collisions at ? s N N =5 TeV[END_REF][START_REF] Abelev | Transverse momentum distribution and nuclear modification factor of charged particles in p-Pb collisions at ? s N N " 5.02 TeV[END_REF] have been studied and seem to be a sensible observables for saturation [START_REF] Albacete | Single and double inclusive particle production in d+Au collisions at RHIC, leading twist and beyond[END_REF][START_REF] Albacete | Single Inclusive Hadron Production at RHIC and the LHC from the Color Glass Condensate[END_REF][START_REF] Albacete | Azimuthal correlations of forward di-hadrons in d+Au collisions at RHIC in the Color Glass Condensate[END_REF]. It has not yet lead to a clear signal of saturation, as, for example, in the case of di-hadron correlation, other mechanisms could explain the data [START_REF] Kang | Dihadron momentum imbalance and correlations in d+Au collisions[END_REF]. Recent data on nuclear modification factor for the single inclusive production of π 0 [START_REF] Aaij | Nuclear Modification Factor of Neutral Pions in the Forward and Backward Regions in p-Pb Collisions[END_REF] has also shown some tensions between data and CGC predictions [START_REF] Lappi | Single inclusive particle production at high energy from HERA data to proton-nucleus collisions[END_REF] in the forward region.

Even though these semi-inclusive measurements at RHIC and at the LHC have not given clear signals of saturation, they inform the program of future electron-ion colliders and have motivated for example, studies on the forward di-hadron azimuthal correlations in e`ppAq Ñ e 1 `h1 `h2 `X process. The correlation functions are measured in the plane transverse to the beam axis, and as a function of ∆ϕ, the azimuthal angle between the momenta of the two produced hadrons. The peak at ∆ϕ " π corresponds to the back-to-back or away-side peak. Computations in the saturation framework lead to a suppression of this peak in nuclear DIS, a suppression that increases with increasing mass number A. The first feasibility study for this process at the EIC has been carried out in [START_REF] Zheng | Probing Gluon Saturation through Dihadron Correlations at an Electron-Ion Collider[END_REF] where the non-perturbative part of the di-hadron cross-section has been modelized with the GBW model. Their results are displayed on the right plot of fig. 2.2.5. We observe a clear depletion of the back-to-back peak with saturation, and on the left, the suppression clearly depends on the mass number. It is also necessary to compare those results to other competing mechanisms that lead to a suppression of the away side-peak such as cold nuclear matter energy loss or higher twist effect as presented in [START_REF] Xing | Transverse momentum imbalance of back-to-back particle production in p+A and e+A collisions[END_REF].

Some other processes that have been studied theoretically in the saturation framework are the single and di-hadron inclusive production [START_REF] Bergabo | Single inclusive hadron production in DIS at small x: next to leading order corrections[END_REF][START_REF] Bergabo | One-loop corrections to dihadron production in DIS at small x[END_REF][START_REF] Iancu | Dihadron production in DIS at NLO: the real corrections[END_REF], the inclusive dijet production [START_REF] Caucal | Dijet impact factor in DIS at next-to-leading order in the Color Glass Condensate[END_REF][START_REF] Caucal | Back-to-back inclusive dijets in DIS at small x: Sudakov suppression and gluon saturation at NLO[END_REF][START_REF] Caucal | Back-to-back inclusive dijets in DIS at small x: Gluon Weizsäcker-Williams distribution at NLO[END_REF][START_REF] Caucal | Back-to-back inclusive dijets in DIS at small x: Complete NLO results and predictions[END_REF] or photon + dijet inclusive production [START_REF] Roy | NLO impact factor for inclusive photon`dijet production in e `A DIS at small x[END_REF].

The study of saturation physics is therefore very topical. It is important to improve the theoretical understanding of processes sensitive to saturation to be able to distinguish between saturation effects and competing mechanisms and to compute a variety of them in order to find a "smoking gun" process ideally. This is what the high-energy theoretical community has been aiming at in recent years, and the rest of the thesis participates in this endeavour. We will first start by introducing two effective field theories for saturated gluons.

61

Part II

Effective field theories for the study of saturation The shockwave formalism: an introduction

This chapter and the next one are based on [START_REF] Boussarie | On the one loop γ p˚q Ñ qq impact factor and the exclusive diffractive cross sections for the production of two or three jets[END_REF][START_REF] Boussarie | Perturbative study of selected exclusive QCD processes at high and moderate energies[END_REF].

Physical picture

Shockwave formalism is a semi-classical effective theory applicable to semi-hard processes where a dilute projectile scatters off a dense target. The process also needs to be inside the saturation region

Q 2 ă Q 2 s pxq where Q 2
s pxq is the saturation scale. The working frame of this picture is an infinite momentum frame where the target and projectile are moving ultra-relativistically in the ´and `direction, respectively, and with

p t " p p " c s 2 . (3.1.1)
This formalism is a k t -factorization (see section 1.2.3): Scattering amplitudes are the convolutions, in transverse momentum space, of a target impact factor and the probe impact factor. The interaction between the probe and the target happens via the exchange in the t-channel of eikonal gluons -gluons with an effective non-sense polarization 9n µ 2 and with negligible p compared to the p `of the partons in the projectile side. The factorization is based on the separation of the gluon fields in the rapidity space with respect to an arbitrary cut-off η pη ă 0q, illustrated in fig. 3

Boosted gluon field and Lagrangian

The boosted gluonic field

We denote b µ 0 pzq a gluon field in the target rest frame. We want to boost it to the infinite momentum frame described by eq. (3.1.1). This is a passive boost of velocity β in the longitudinal direction.

To find the value of the boost, we use the expression of the target energy in this new frame

E " m t a 1 ´β2 " p t `pt ? 2 " p t ? 2 , (3.2.1)
where m t is the target mass. From the above equation, we can easily find

Λ " ? s m t (3.2.2)
where Λ is defined in eq. (1.1.12). We observe that being in the Regge kinematic, the boost is enormous.

In the new frame, the boosted gluonic field reads, using eq. (1.1.11),

b `px `, x ´, ⃗ x q " Λ ´1b 0 pΛz `, Λ ´1z ´, ⃗ z q , b ´px `, x ´, ⃗ x q " Λb 0 pΛz `, Λ ´1z ´, ⃗ z q , b i px `, x ´, ⃗ x q " b i 0 pΛz `, Λ ´1z ´, ⃗ z q .
Assuming the field vanishes at infinity, the i and `components vanish as they are evaluated close to x `" 8. Thus, up to Λ ´1 " 1{ ? s correction, the above equations become b `px `, x ´, ⃗ x q " 0 , b ´px `, x ´, ⃗ x q " Λb 0 pΛz `, ⃗ z q , b i px `, x ´, ⃗ x q " 0 , where the only surviving component b ´loses its dependence on x ´because Λ is large and so it is always evaluated in x ´" 0.

Exploiting the fact that for any integrable function F pxq ,

lim ΛÑ8 ΛF pΛxq9δpxq , (3.2.3)
we can finally write

b µ px `, x ´, ⃗ x q " b ´px `, ⃗ x qn µ 2 " δpx `qBp⃗ x qn µ 2 (3.2.4)
Eq. (3.2.4) showcases why the terminology of "shockwave" is used: from the point of view of partons in the probe side, all the external gluon fields are localized at the origin of light-cone time x `" 0 and form a shockwave. We recognise the non-sense polarization (n µ 2 ) mentioned before, and the independence in x ´leads to the conservation of the p `component for partons crossing the shockwave, as shown shortly after.

Shockwave Lagrangian

We start from the QCD Lagrangian, which is split between the interacting and the free parts,

L " ´1 4
F aµν F aµν `i ψ { Dψ " L f ree `Lint L int " ´gf abc A b µ A c ν B µ A νa ´1 4 g 2 f abc f ade A b µ A c ν A µd A νe `i ψp´igt a A a { qψ .
The total gluon field A is separated into the external slow field b and the internal fast field A (always from the point of view of the probe) :

A a µ " A a µ `ba µ . (3.2.5)
Using b 2 9 n 2 2 " 0, the interacting part becomes

L int " ´gf abc "´A b µ B µ A νa `bb µ B µ A νa ¯pA c ν `bc ν q `Ab µ B µ b νa A c ν `bb µ B µ b νa A c ν ı ´1 4 g 2 f abc f ade "´A b µ A µd `Ab µ b µd ¯pA νe A c ν `Aνe b c ν `bνe A c ν q `bb µ A µd pA e ν A νc `Ae ν b νc `be ν A νc q ı `i ψ " ´igt a `{ A a `{ b a ˘‰ ψ . (3.2.6)
Working in the light-cone gauge n 2 ¨A " A `" 0, we have b ¨A 9 n 2 ¨A " 0 and the above expression simplifies to

L int " ´gf abc A b µ pB µ A νa q A c ν ´1 4 g 2 f abc f ade A b µ A µd A νe A c ν `ψpgt a { A a qψ ´gf abc b b µ pB µ A νa q A c ν `ψpgt a { b a qψ . (3.2.7)
The first line is the usual interaction terms in QCD, while the last line corresponds to the interactions with the shockwave field and is denoted as

L S int " ´gf acb b ´cg αβ ˆBA a β Bx ´˙A b α `ψpgt a { b a qψ " ´gf acb b ´cg αβ ˆBA a β Bx ´˙A b α `ψgb ´γ`ψ " gf cab b ´cg αβ K ˆBA a β Bx ´˙A b α `ψgb ´γ`ψ " igT c ab b ´cg αβ K ˆBA a β Bx ´˙A b α `ψgb ´γ`ψ , (3.2.8)
where b ´pxq " t a b ´apxq, T c ab " ´if cab . To go from the second to third line in eq. (3.2.8), we used the metric decomposition

g αβ " g αβ K `nα 1 n β 2 `nβ 1 n α 2 (3.2.9)
with our gauge choice n 2 ¨A " 0 to end up with only the tranverse part of the metric for the first term of L S int .

Quark propagator through the shockwave

In this section, I will rederive the Feynman rule for the quark line through the shockwave, following closely the steps in [START_REF] Boussarie | Perturbative study of selected exclusive QCD processes at high and moderate energies[END_REF].

Interaction with two external gluons

Using eq. (3.2.8), a quark line propagating from z 0 to z 3 with z 0 ă 0 and z 3 ą 0 and interacting with two shockwave fields has the form

G p2q pz 3 , z 0 q| z 3 ą0ąz 0 " ż d D z 2 d D z 1 G 0 pz 32 q " igb ´pz 2 qγ `‰ G 0 pz 21 q " igb ´pz 1 qγ `‰ G 0 pz 10 qθpz 3 qθp´z 0 q " ż d D z 2 d D z 1 " igb ´pz 2 , ⃗ z 2 qigb ´pz 1 , ⃗ z 1 q ‰ θpz 3 qθp´z 0 q ż d D p 3 p2πq D d D p 2 p2πq D d D p 1 p2πq D ˆG0 pp 3 qγ `G0 pp 2 qγ `G0 pp 1 q e ´ip 3 ¨z32 e ´ip 2 ¨z21 e ´ip 1 ¨z10 (3.3.1)
where G 0 corresponds to the free quark propagator.. We observe first that the propagators G 0 pp 1 q and G 0 pp 3 q are next to a γ `matrix and G 0 pp 2 q is sandwiched between two γ `matrices. Using eq. (1.1.10), the numerators of these free quark propagators can be greatly simplified. Moreover, the field b ´pzq is independent of z ´; the only dependence on the z í is, therefore, in the exponentials. Integrating over them gives delta functions that impose conservation of the longitudinal momentum i.e., p 3 " p 2 " p 1 " p `.

These lead to

G p2q pz 3 , z 0 q| z 3 ą0ąz 0 " p2πq 2 ż dz 2 dz 1 d d ⃗ z 2 d d ⃗ z 1 " igb ´pz 2 , ⃗ z 2 qigb ´pz 1 , ⃗ z 1 q ‰ θpz 3 qθp´z 0 q ˆż dp `e´ip `z3 0 ż d d ⃗ p 3 p2πq D d d ⃗ p 2 p2πq D d d ⃗ p 1
p2πq D e i⃗ p 3 ¨⃗ z 32 e i⃗ p 2 ¨⃗ z 21 e i⃗ p 1 ¨⃗ z 10 ˆż dp

3 i pp `γ´`p { 3K q 2p `´p 3 ´⃗ p 2 3 ´iε 2p `¯e ´ip 3 z 32 ˆγ`ż dp 2 i ´p2 ´⃗ p 2 2 ´iε 2p `¯e ´ip 2 z 21 ˆż dp 1 i pp `γ´`p { 1K q 2p `´p 1 ´⃗ p 2 1 ´iε 2p `¯e ´ip 1 z 10 . (3.3.2)
To compute the integrals over p í , we use Jordan's lemma to close the contour integral and then use the residue theorem. We note that to have convergent integrals, we need to have the argument of the complex exponential Impp 3 qz 32 ă 0 and so z 32 p `ą 0 with a similar thing for the other two exponentials. The only possible choice is to have p `, z 32 , z 21 , z 10 ą 0 as then z 30 " z 32 `z2

1 `z1 0 ą 0 which is coherent with the condition z 3 ą 0 ą z 0 . Closing the three contours in the lower half plan which leads by residue theorem to a factor of p´2πiq for each integral over p í , one gets We now use the fact that up to correction Λ ´1 " 1{ ? s , b ´pzq 9 δpz `q to set z 2 " z 1 " 0 in the integrand of ⃗ p 2 . The Gaussian integral over ⃗ p 2 shrinks into a delta function p2πq d δp⃗ z 21 q. This shows explicitly that all interactions with the external gluons happen at the same transverse coordinate, which is important to have at the end the resummation of all interactions with the external field into a single Wilson line. with a similar manipulation for the exponential of ⃗ p 3 . We then shift the transverse momentum integrals to get, also using the delta function p2πq d δ p⃗ z 21 q,

G p2q pz 3 , z 0 q| z 3 ą0ąz 0 " p2πq 5 p2πq 3D ż dz 2 dz 1 d d ⃗ z 2 d d ⃗ z 1 " igb ´pz 2 , ⃗ z 2 qigb ´pz 1 , ⃗ z 1 q ‰ θpz 3 qθp´z 0 q ˆż dp `e´ip
G p2q pz 3 , z 0 q| z 3 ą0ąz 0 " 1 4p2πq 2D´3 ż dz 2 dz 1 d d ⃗ z 1 " igb ´pz 2 , ⃗ z 1 qigb ´pz 1 , ⃗ z 1 q ‰ θpz 3 qθp´z 0 q ˆż dp `e´ip `z3 0 θpp `qθpz 32 qθpz 21 qθpz 10 q ˆ`z 32 γ ´`z { 31K z3 2 γ ``z 10 γ ´`z { 10K z1 0 ˆż d d ⃗ p 3 exp « ´i z 32 2p `˜⃗ p 2 3 ´ˆp z 32 ˙2 ⃗ z 2 31 ´iε ¸ff ˆż d d ⃗ p 1 exp « ´i z 10 2p `˜⃗ p 2 1 ´ˆp z 10 ˙2 ⃗ z 2 10 ´iε ¸ff . (3.3.5)
We also have to make those integrals convergent. We note first that The last integral is done using the Schwinger parametrization for denominators

G p2q pz 3 , z 0 q| z 3 ą0ąz 0 " 1 4p2πq 2D´3 ż dz 2 dz 1 d d ⃗ z 1 " igb ´pz 2 , ⃗ z 1 qigb ´pz 1 , ⃗ z 1 q ‰ θpz 3 qθp´z 0 q ˆθpz 32 qθpz 21 qθpz 10 q `z3 2 γ ´`z { 31K ˘γ``z1 0 γ ´`z { 10K z3
1 pA ˘iεq n " 1 p˘iq n Γpnq ż `8 0 dα α n´1 e ˘iαpA˘iεq , (3.3.8)
and one finally obtains

G p2q pz 3 , z 0 q| z 3 ą0ąz 0 " 1 4p2πq D´1 i d ż d d ⃗ z 1 ż dz 2 dz 1 " igb ´pz 2 , ⃗ z 1 qigb ´pz 1 , ⃗ z 1 q ‰ θpz 32 qθpz 21 qθpz 10 q ˆθpz 3 qθp´z 0 q `z3 2 γ ´`z { 31K ˘γ``z1 0 γ ´`z { 10K pz 32 z 10 q D 2 i d`1 Γpd `1q ´´z 30 `⃗ z 2 31 2z 32 `⃗ z 2 10 2z 10 `iε ¯d`1 " iΓpD ´1q 4p2πq D´1 ż d d ⃗ z 1 V p2q p⃗ z 1 q `z3 γ ´`z { 31K ˘γ``´z0 γ ´`z { 10K p´z 3 z 0 q D 2 ´´z 30 `⃗ z 2 31 2z 32 `⃗ z 2 10 2z 10
`iε ¯D´1 θpz 3 qθp´z 0 q (3.3.9)

where we define V p2q p⃗ z q " pigq 2 ż dz 2 dz 1 θpz 32 qθpz 21 qθpz 10 qb ´pz 2 , ⃗ z qb ´pz 1 , ⃗ z q .

(3.3.10)

We will show now that G p2q pz 3 , z 0 q| z 3 ą0ąz 0 is equal to

Gp2q pz 3 , z 0 q| z 3 ą0ąz 0 " ż d D z 1 δpz 1 qG 0 pz 31 qγ `Gpz 10 qθpz 3 qθp´z 0 qV p2q p⃗ z 1 q . (3.3.11)
The physical picture is clearer in eq. (3.3.11) than in eq. (3.3.9): the quark line propagates freely from z 0 to z 1 where it interacts with the two shockwave fields. The interaction with those two fields happens at exactly the light-cone time z 1 " 0 and at the same transverse coordinates ⃗ z 1 .

The quark line freely pursues its propagation from z 1 to z 3 .

Using eq. (3.4.2) and the Schwinger parametrization eq. (3.3.8) for both denominators, one has

Gp2q pz 3 , z 0 q| z 3 ą0ąz 0 " ´Γ2 `D 2 4π D ż dz 1 d d ⃗ z 1 `z3 γ ´`z { 31K ˘γ``´z0 γ ´`z { 10K 2z 3 z 31 `⃗ z 2 31 `iε ˘D 2 p2z 0 z 10 `⃗ z 2 10 `iεq D 2 V p2q p⃗ z 1 qθpz 3 qθp´z 0 q " ´1 i D 4π D ż d d ⃗ z 1 `z3 γ ´`z { 31K ˘γ``´z0 γ ´`z { 10K ˘V p2q p⃗ z 1 qθpz 3 qθp´z 0 q ˆż `8 0 dα α D 2 ´1e iαp´2z 3 z 3 `⃗ z 2 31 `iεq ż `8 0 dβ β D 2 ´1e iβp´2z 0 z 0 `⃗ z 2 10 `iεq ˆż dz 1 e iz 1 p2αz 3 `2βz 0 q " ´1 i D 4π D´1 ż d d ⃗ z 1 `z3 γ ´`z { 31K ˘γ``´z0 γ ´`z { 10K p´z 0 q V p2q p⃗ z 1 qθpz 3 qθp´z 0 q ˆˆ´z 3 z 0 ˙d 2 ż `8 0 dα α d e iα ˆ´2z 3 z 30 `⃗ z 2 31 ´z3 z 0 ⃗ z 2 10 `iε ˙.
(3.3.12)

In going from the second to the third line of eq. (3.3.12), the term proprotional to ε in the exponentials is under control as we have β " p´z 3 {z 0 qα with z 3 , p´z 0 q ą 0. The different positive factors can then be reabsorbed to arrive to only iε in the third line.

Integrating over the remaining Schwinger parameter gives

Gp2q pz 3 , z 0 q| z 3 ą0ąz 0 " ´1 i d`2 4π D´1 ż d d ⃗ z 1 `z3 γ ´`z { 31K ˘γ``´z0 γ ´`z { 10K ˘V p2q p⃗ z 1 qθpz 3 qθp´z 0 q pz 3 q d 2 p´z 0 q d 2 `1 ˆid`1 Γpd `1q ´´2z 3 z 30 `⃗ z 2 31 ´z3 z 0 ⃗ z 2 10 `iε ¯d`1 " iΓpD ´1q 4p2πq D´1 ż d d ⃗ z 1 `z3 γ ´`z { 31K ˘γ``´z0 γ ´`z { 10K z3 0 `⃗ z 2 31 2z 3 ´⃗ z 2 10 2z 0 `iε ¯D´1 V p2q p⃗ z 1 qθpz 3 qθp´z 0 q 1 p´z 3 z 0 q D 2 . (3.3.13)
The above equation is equal to eq. (3.3.9), proving

Gpz 3 , z 0 q| z 3 ą0ąz 0 " Gpz 3 , z 0 q| z 3 ą0ąz 0 (3.3.14)

Generalization to all possible number of interactions

We want to generalize the expression eq. (3.3.11) to have one where all possible numbers of interactions are resummed.

By observing that V p2q p⃗ z q is the pigq 2 -term in the ig-expansion of the Wilson line

" z 3 , z 0 ‰ ⃗ z " P `exp « ig ż z 3 z 0 dz `b´p z `, ⃗ z q ff , (3.3.15)
where P `is the path ordering operator along the z `direction, the logical generalization of eq. (3.3.11) would therefore be

Gpz 3 , z 0 q| z 3 ą0ąz 0 " ż d D z 1 δpz 1 qG 0 pz 31 qγ `G0 pz 10 q " z 3 , z 0 ‰ ⃗ z 1 θpz 3 qθp´z 0 q . (3.3.16)
Again, the physical picture is that the quark line propagates freely before and after the interaction with the shockwave. The interaction at ⃗ z 1 , z `" 0, in the eikonal approximation, is just a colour rotation encoded by the Wilson line " z 3 , z 0 ‰ ⃗ z 1 . Let's prove that this is a correct guess by mathematical induction. We denote by G pnq pz 3 , z 0 q| z 3 ą0ąz 0 the propagator with n interactions and rz 3 , z 1 s pnq ⃗ z 1 the pigq n term in the expansion of eq. (3.3.15). The n " 2 has been shown in the previous section, the n " 1 case is immediate. We should prove the n " 0 case (corresponding to the non-interaction case). Just like before, using the Schwinger representation for the denominators of the free propagators and integrating over z 1 , z 1 and one of the Schwinger parameters, we have

G p0q pz 3 , z 0 q| z 3 ą0ąz 0 " ż d D z 1 δpz 1 qG 0 pz 31 qγ `G0 pz 10 q " z 3 , z 0 ‰ p0q ⃗ z 1 " Γ 2 `D 2 4π D ż d D z 1 δpz 1 q iz { 31 p´z 2 31 `iεq D 2 γ `iz { 10 p´z 2 10 `iεq D 2 " ´1 i D 4π D ż d d ⃗ z 1 `z3 γ ´`z { 31K ˘γ``´z0 γ ´`z { 10K ż `8 0 dα α d 2 e iαp´2z 3 z 3 `⃗ z 2 31 `iεq ż `8 0 dβ β d 2 e iβp´2z 0 z 0 `⃗ z 2 10 `iεq ż dz 1 e iz 1 p2αz 3 `2βz 0 q " 1 i D 4π D´1 z 0 ˆ´z 3 z 0 ˙d 2 ż d d ⃗ z 1 `z3 γ ´`z { 31K ˘γ``´z0 γ ´`z { 10K ż `8 0 dα α d e iα ˆ´2z 3 z 30 `⃗ z 2 31 ´z3 z 0 ⃗ z 2 10 `iε ˙(3.3.17)
We want to integrate over ⃗ z 1 . First, we shift the variable by

⃗ z 1 `z0 z 30 ˆ⃗ z 3 ´z3 z 0 ⃗ z 0 ˙Ñ ⃗ z 1 , (3.3.18)
to have a Gaussian integral. This gives

G p0q pz 3 , z 0 q| z 3 ą0ąz 0 " 1 i D 4π D´1 z 0 ˆ´z 3 z 0 ˙d 2 ż `8 0 dα α d e iα ˆ´2z 3 z 30 `z3 z 30 ⃗ z 2 30 `iε ˆż d d ⃗ z 1 e ´iα z 30 z 0 ⃗ z 2 1 p1`iεq " ´2z 3 z 0 γ ´´2 z 3 z 0 z 30 z { 30K ´z3 z 0 pz 30 q 2 ⃗ z 2 30 γ `´⃗ z 2 1 γ `* . (3.3.19)
Integrating over ⃗ z 1 yields

G p0q pz 3 , z 0 q| z 3 ą0ąz 0 " i d 2 i D 4π D 2 z 0 ˆz3 z 30 ˙d 2 ż `8 0 dα α d 2 e iα ˆ´2z 3 z 30 `z3 z 30 ⃗ z 2 30 `iε "´2z 3 z 0 γ ´´2 z 3 z 0 z 30 z { 30K ´z3 z 0 pz 30 q 2 ⃗ z 2 30 γ ``i d 2 z 0 αz 30 γ `* . (3.3.20)
We now do the last integral

G p0q pz 3 , z 0 q| z 3 ą0ąz 0 " i d 2 i D 4π D 2 z 0 ˆz3 z 30 ˙d 2 "" ´2z 3 z 0 γ ´´2 z 3 z 0 z 30 z { 30K ´z3 z 0 pz 30 q 2 ⃗ z 2 30 γ `ȷ ˆż `8 0 dα α d 2 e iα ˆ´2z 3 z 30 `z3 z 30 ⃗ z 2 30 `iε ˙`i d 2 z 0 z 30 γ `ż `8 0 dα α d 2 ´1 e iα ˆ´2z 3 z 30 `z3 z 30 ⃗ z 2 30 `iε ˙, .
- To go from the first to the second line in the above formula, we have used the Schwinger parametrization eq. (3.3.8). In the last line, we recognize the free propagator in coordinates space eq. (3.4.2), as expected when we have the quark that propagates with no interaction with the external fields. We, therefore, proved that

" i d`1 Γ `D 2 ȋD 4π D 2 z 0 ˆz3 z 30 ˙d 2 $ ' ' & ' ' % ´2z 3 z 0 γ ´´2 z 3 z 0 z 30 z { 30K ´z3 z 0 pz 30 q 2 ⃗ z
G p0q pz 3 , z 0 q| z 3 ą0ąz 0 " ż d D z 1 δpz 1 qG 0 pz 31 qγ `G0 pz 10 q " z 3 , z 0 ‰ p0q ⃗ z 1 " G 0 pz 30 q . (3.3.22)
The next step of the proof is to show that, assuming the result holds for the n case, i.e.

G pnq pz 3 , z 0 q| z 3 ą0ąz 0 " ż d D z 1 δpz 1 qG 0 pz 31 qγ `G0 pz 10 qθpz 3 qθp´z 0 q " z 3 , z 0 ‰ pnq ⃗ z 1 (3.3.23)
it should also hold for the n `1 case.

We start from

G pn`1q pz 3 , z 0 q | z 3 ą0ąz 0 " ż d D z 2 G 0 pz 32 qigγ `b´p z 2 , ⃗ z 2 qG pnq pz 2 , z 0 qθpz 3 qθp´z 0 q " ż d D z 2 d D z 1 δpz 1 qG 0 pz 32 qigγ `b´p z 2 , ⃗ z 2 qG 0 pz 21 qγ `G0 pz 10 q " z 2 , z 0 ‰ pnq ⃗ z 1 θpz 3 qθpz 2 qθp´z 0 q " ż d D z 1 δpz 1 q ż d D z 2 igb ´pz 2 , ⃗ z 2 q " z 2 , z 0 ‰ pnq ⃗ z 1 G 0 pz 32 qγ `G0 pz 21 qγ `G0 pz 10 qθpz 3 qθpz 2 qθp´z 0 q . (3.3.24)
We will follow the same steps as in section 3.3.1.

G pn`1q pz 3 , z 0 q | z 3 ą0ąz 0 " ż d D z 1 δpz 1 q ż d D z 2 igb ´pz 2 , ⃗ z 2 q " z 2 , z 0 ‰ pnq ⃗ z 1 ż d D p 2 p2πq D ż d D p 1 p2πq D e ´ip 2 ¨z32 e ´ip 1 ¨z21 ˆG0 pp 2 qγ `G0 pp 1 qγ `G0 pz 10 qθpz 3 qθpz 2 qθp´z 0 q " ż d D z 1 δpz 1 q ż d D z 2 igb ´pz 2 , ⃗ z 2 q " z 2 , z 0 ‰ pnq ⃗ z 1 ż d D p 2 p2πq D ż d D p 1 p2πq D e ´ip 2 ¨z32 e ´ip 1 ¨z21 ˆi `p2 γ ´`p { 2K 2p 2 ´p2 ´⃗ p 2 2 2p 2 ¯γ`i `p1 γ ´`p { 1K 2p 1 ´p1 ´⃗ p 2 1 ´iε 2p 1 ¯γ`G 0 pz 10 qθpz 3 qθpz 2 qθp´z 0 q (3.3.25)
Because b ´pz 2 q is independent of z 2 , we easily integrate over z 2 to get p2πqδpp 2 ´p1 q, leading to, after simplifying the numerator,

G pn`1q pz 3 , z 0 q | z 3 ą0ąz 0 " ż d D z 1 δpz 1 q ż dz 2 d d ⃗ z 2 igb ´pz 2 , ⃗ z 2 q " z 2 , z 0 ‰ pnq ⃗ z 1 p2πq ż dp `e´ip `z3 1 ˆż d d ⃗ p 2 p2πq D d d ⃗ p 1
p2πq D e i⃗ p 2 ¨⃗ z 32 e i⃗ p Once again, Jordan's lemma and the residue theorem are used for the p í integrals. As before Impp 2 qz 32 ă 0 ùñ p `z3 2 ą 0 and similarly p `z2 1 ą 0. Because z 1 " 0 and z 2 ą 0 with the theta function then we have immediately p `ą 0 which implies then z 32 ą 0.

Closing the contour on the lower plane for both p ´integrals yields As b ´pz 2 , ⃗ z 2 q9δpz 2 q, we can set z 2 " 0 in the integrand of the two transverse momenta. We also have z 1 " 0. The integrals over ⃗ p 1 gives then a delta of ⃗ z 21 :

G pn`1q pz 3 , z 0 q | z 3 ą0ąz 0 " ż d D z 1 δpz 1 q ż dz 2 d d ⃗ z 2 igb ´pz 2 , ⃗ z 2 q " z 2 , z 0 ‰ pnq ⃗ z 1 p2πq 3 p2πq 2D ż dp `e´ip `z3
G pn`1q pz 3 , z 0 q | z 3 ą0ąz 0 " ż d D z 1 δpz 1 q ż dz 2 d d ⃗ z 2 igb ´pz 2 , ⃗ z 2 q " z 2 , z 0 ‰ pnq ⃗ z 1 1 p2πq 2D´3 ż dp `e´ip `z3 1 θpp `qθpz 32 q ˆθpz 21 q ż d d ⃗ p 2 e i⃗ p 2 ¨⃗ z 32 e ´iˆ⃗ p 2 2 ´iε 2p `˙z 3 p `γ´`p { 2K 2p `γ`G 0 pz 10 qθpz 3 qθpz 2 qθp´z 0 qp2πq d δp ⃗ z 21 q " ż d D z 1 δpz 1 q ż dz 2 igb ´pz 2 , ⃗ z 1 q " z 2 , z 0 ‰ pnq ⃗ z 1 θpz 32 qθpz 21 q 1 p2πq D´1 ż dp `e´ip `z3 1 θpp `q ˆż d d ⃗ p 2 exp " ´i z 3 2p `ˆ⃗ p 2 2 ´2⃗ p 2 ¨pz 3 ⃗ z 31 ´iε ˙ȷ p `γ´`p { 2K 2p `γ`G 0 pz 10 qθpz 3 qθpz 2 qθp´z 0 q
The integral over ⃗ p 2 is done by completing the square in the exponential and rescaling the Feynman prescription, then shifting the vector and integrating the obtained Gaussian integral. This leads to:

G pn`1q pz 3 , z 0 q | z 3 ą0ąz 0 " ż d D z 1 δpz 1 q ż dz 2 igb ´pz 2 , ⃗ z 1 q " z 2 , z 0 ‰ pnq ⃗ z 1 θpz 32 qθpz 21 q 1 i d 2 2p2πq D 2 pz 3 q D 2 ˆż `8 0 dp `pp `q d 2 e ip `ˆ´z 31 `⃗ z 2 31 2z 3 `iε ˙`z 3 γ ´`z { 31K ˘γ`G 0 pz 10 qθpz 3 qθpz 2 qθp´z 0 q
Integrating over p `gives

G pn`1q pz 3 , z 0 q | z 3 ą0ąz 0 " ż d D z 1 δpz 1 q ż dz 2 igb ´pz 2 , ⃗ z 1 q " z 2 , z 0 ‰ pnq ⃗ z 1 θpz 32 qθpz 21 q 1 i d 2 2p2πq D 2 pz 3 q D 2 ˆi d 2 `1Γ `d 2 `1z 31 `⃗ z 2 31 2z 3 `iε ¯d 2 `1 `z3 γ ´`z { 31K ˘γ`G 0 pz 10 qθpz 3 qθpz 2 qθp´z 0 q " ż d D z 1 δpz 1 qGpz 31 qγ `G0 pz 10 qθpz 3 qθp´z 0 q ż dz 2 igb ´pz 2 , ⃗ z 1 q " z 2 , z 0 ‰ pnq ⃗ z 1 θpz 32 qθpz 21 qθpz 2 q (3.3.28)
From eq. (3.3.15), we can then write

G pn`1q pz 3 , z 0 q | z 3 ą0ąz 0 " ż d D z 1 δpz 1 qGpz 31 qγ `G0 pz 10 qθpz 3 qθp´z 0 q " z 3 , z 0 ‰ pn`1q ⃗ z 1 (3.3.29)
and this finishes our proof. In eq. (3.3.15), we note that, because b ´pzq9δpz `q

' if z 3 z 0 ą 0 then the integration does not the cross the z `" 0 line and so it is null. Then rz 3 , z 0 s ⃗ z " 1.

' if z 3 z 0 ă 0, the integration crosses the z `" 0 line and the Wilson line is non trivial. It is clear only the point z `" 0 matters in the entire integration domain, and so the latter can be extended from ´8 to z 0 and from z 3 to `8 for free.

We, therefore, define the light-like Wilson line in the fundamental representation of SU pN c q V p⃗ z q " P `exp

" ig ż dz `b´a pz `, ⃗ z qt a ȷ (3.3.30)
and its Fourier transform

V p⃗ p q " ż d d ⃗ z e ´i⃗ p¨⃗ z V p⃗ z q . (3.3.31)
We have a similar definition for the Wilson line in the adjoint representation

U p⃗ z q " P `exp " ig ż dz `b´a pz `, ⃗ z qT a ȷ . (3.3.32)
t a , T a are the generators of SU pN c q in the fundamental and adjoint representations. Those Wilson lines have the following properties

V p⃗ z qV : p⃗ z q " 1 Nc (3.3.33) U p⃗ z qU : p⃗ z q " 1 N 2 c ´1 (3.3.34)
U ab p⃗ z q " U åb p⃗ z q " U : ba p⃗ z q (3.3.35)

U ab p⃗ z qt b " V : p⃗ z qt a V p⃗ z q (3.3.36) V p⃗ z qt a V : p⃗ z q " U : ab p⃗ z qt b " U ba p⃗ z qt b . (3.3.37)
In the Feynman rules that will be listed in the next section, we work with V p⃗ z q or U p⃗ z q instead of rz 3 , z 0 s ⃗ z .

Feynman rules with the shockwave field

i, j and a, b are colour indices for the fundamental and the adjoint representation, respectively.

Free propagators

Free propagator of the quark

G 0 ppq " ip { p 2 `iε (3.4.1) G 0 pxq " Γ `D 2 2π D 2 ix { p´x 2 `iεq D 2 (3.4.2)
Free propagator for the gluon

G µν 0 ppq " ´i p 2 `iε " g µν ´pµ n ν 2 `pν n µ 2 p `ȷ " ´i p 2 `iε d µν ppq (3.4.3)
We can rewrite d µν ppq using eq. (3.2.9):

d µν ppq " d µν 0 ppq ´nµ 2 n ν 2 pp `q2 p 2 (3.4.4) d µν 0 ppq " g µν K ´pµ K n ν 2 `pν K n µ 2 p `´n µ 2 n ν 2 pp `q2 ⃗ p 2 (3.4.5)

Propagators through the shockwave field

Quark line through the shockwave:

G ij px 2 , x 0 q| x 2 ą0ąx 0 " ż d D x 1 δpx 1 qV ij p⃗ x 1 qG 0 px 21 qγ `G0 px 10 qθpx 2 qθp´x 0 q (3.4.6) " iΓpd `1q 4p2πq d`1 ż d d ⃗ x 1 V ij p⃗ x 1 q `x2 γ ´`x { 21K ˘γ``´x0 γ ´`x { 10K p´x 0 x 2 q D 2 ´´x 20 `⃗ x 2 21 2x 2 ´⃗ x 2 10 2x 0 `iε ¯d`1 θpx 2 qθp´x 0 q (3.4.7) " ż d d ⃗ p 2 dp 2 p2πq d`1 ż d d ⃗ p 1 dp 1 p2πq d`1 e ´ip 2 x 2 e i⃗ p 2 ¨⃗ x 2 e ip 1 x 0 e ´i⃗ p 1 ¨⃗ x 1 2πδpp 21 qθpp 2 q ˆe´i ⃗ p 2 2 ´iε 2p 2 x 2 e i ⃗ p 2 1 ´iε 2p
1 x 0 p 2 γ ´`p { 2K 2p 2 γ `Vij p⃗ p 21 q p 1 γ ´`p { 1K 2p 1 θpx 2 qθp´x 0 q , (3.4.8)
The derivations for expressions eqs. (3.4.7,3.4.8) can be found in appendix A.1 and A.2.

x 1 x 0 x 2 -→ -→ p 1 p 2 i j
G ij px 2 , x 0 q| x 0 ą0ąx 2 " ´ż d D x 1 δpx 1 qV : ij p⃗ x 1 qG 0 px 21 qγ `G0 px 10 q θpx 2 qθp´x 0 q (3.4.9) " ´iΓpd `1q 4p2πq d`1 ż d d ⃗ x 1 V : ij p⃗ x 1 q `x2 γ ´`x { 21K ˘γ``´x0 γ ´`x { 10K p´x 0 x 2 q D 2 ´x2 0 ´⃗ x 2 21 2x 2 `⃗ x 2 10 2x 0 `iε ¯d`1 θpx 2 qθp´x 0 q (3.4.10) " ´ż d d ⃗ p 1 dp 1 p2πq d`1 ż d d ⃗ p 2 dp 2 p2πq d`1 e ´ip 2 x 0 e i⃗ p 2 ¨⃗ x 0 e ip 1 x 2 e ´i⃗ p 1 ¨⃗ x 2 2πδpp 12 qθpp 2 q ˆe´i ⃗ p 2 2 ´iε 2p 2 x 0 e i ⃗ p 2 1 ´iε 2p 1 x 2 p 1 γ ´`p { 1K 2p 1 γ `V : ij p⃗ p 12 q p 2 γ ´`p { 2K 2p 2 θpx 2 qθp´x 0 q , (3.4.11) x 1 x 0 x 2 -→ p 2 -→ p 1 i j
G ab µν px 2 , x 0 q ˇˇx 2 ą0ąx 0 " ´Γpdq 2p2πq d`1 ż d d ⃗ x 1 " x 2 g α Kµ ´xα 21K n 2µ ı U ab p⃗ x 1 q " ´x0 g Kαν ´x10Kα n 2ν ‰ `´x 0 x 2 ˘D 2 ´´x 20 `⃗ x 2 21 2x 2 ´⃗ x 2 10 2x 0 `iε ¯d " ´ż dp 2 d d ⃗ p 2 p2πq d`1 ż dp 1 d d ⃗ p 1 p2πq d`1 e ´ip 2 x 2 e ip 1 x 0 e i⃗ p 2 ¨⃗ x 2 e ´i⃗ p 1 ¨⃗ x 0 π δ `p2 1 ˘θ `p2 p1 ˆe´i ⃗ p 2 2 ´iε 2p 2 x 2 e i ⃗ p 2 1 `´iε 2p 1 x 0 d 0µα `p2 , p 2K ˘U ab p⃗ p 21 q g αδ K d 0δν `p1 , p 1K ˘,(3.4.12) x 1 x 0 x 2 -→ -→ p 1 p 2 a b
ū pp, x 0 q| x 0 ą0 " θ `p`˘ū ppq a 2p `eipp¨x 0 q (3.4.13)
v pp, x 0 q| x 0 ą0 " θ `p`˘v ppq a 2p

`eipp¨x 0 q (3.4.14) ε μ pp, x 0 q ˇˇx 0 ą0 " θ `p`˘ε μppq a 2p

`eipp¨x 0 q (3.4.15)

External lines through the shockwave contains gluons with `momenta between e η p p and e η`∆η p p . We now consider a Wilson line at rapidity η 1 defined from the light-cone time x `to y `.

rū pp q , x 0 qs ij 0ąx 0 " p´iq d 2 2p2πq d 2 ˆpq ´x0 ˙d 2 θ `pq ˘θ `´x 0 ˘ż d d ⃗ x 1 ūpp q q b 2p q γ `V ij p⃗ x 1 q ˆ´x 0 γ ´`x { 10K ´x0 exp " ip q ˆx0 ´⃗ x 2 10 2x 0 `iε ˙´i p⃗ p q ¨⃗ x 1 q ȷ (3.4.16) " θ `pq ˘θ `´x 0 b2p q ż d d ⃗ p 1 p2πq d exp « ip q x 0 `ix 0 ˜⃗ p 2 q1 ´iε 2p q ¸´i p⃗ p q1 ¨⃗ x 0 q ff ˆūpp q qV ij p⃗ p 1 q γ `´p q γ ´`p { q1K 2p q (3.4.17) x 0 x 1 -→ p q -→ p 1 i j
rv pp q, x 0 qs ij 0ąx 0 " ´p´iq d 2 2p2πq d 2 ˜pq ´x0 ¸d 2 θ `pq ˘θ `´x 0 ˘ż d d ⃗ x 1 V ij: p⃗ x 1 q ˆexp " ip q ˆx0 ´⃗ x 2 10 2x 0 `iε ˙´i p⃗ p q ¨⃗ x 1 q ȷ ´x0 γ ´`x { 10K ´x0 γ `vpp qq b 2p q (3.4.18) " ´θ `pq ˘θ `´x 0 ˘ż d d ⃗ p 1 p2πq d V ij: p´⃗ p 1 q ˆexp « ip q x 0 `ix 0 ˜⃗ p 2 q1 ´iε 2p q ¸´i p⃗ p q1 ¨⃗ x 0 q ff ´pq γ ´`p { q1K 2p q γ `vpp qq b 2p q (3.4.19) x 1 x 0 -→ p q -→ p 1 j i
rε ν pp g , x 0 qs ab 0ąx 0 " p´iq d 2 p2πq d 2 ˆpg ´x0 ˙d 2 θ `pg ˘θ `´x 0 ˘εσ pp g q b 2p g ż d d ⃗ x 1 U ab p⃗ x 1 q ˆ" ´x0 g σ Kν ´xσ 10K n 2ν ´x0 ȷ exp " ip g ˆx0 ´⃗ x 2 10 2x 0 `iε ˙´i p⃗ p g ¨⃗ x 1 q ȷ (3.4.20) " θ `pg ˘θ `´x 0 ˘εσ pp g q b 2p g ż d d ⃗ p 1 p2πq d U ab p⃗ p 1 q ˆgσ Kν ´pσ g1K p g n 2ν ėxp " ip g `x0 `iε ˘`i x 0 2p g ⃗ p 2 g1 ´i p⃗ p g1 ¨⃗ x 0 q ȷ (3.4.21) x 1 x 0 -→ p g -→ p 1 a b
From eq. (3.3.15), it can be written as

" y `, x `‰η 1 ⃗ z " 1 ``8 ÿ N "1 ż y x`d z 1 igb ή1 pz 1 , ⃗ zq ż z 1 x `dz 2 igb ή1 pz 2 , ⃗ z q . . . ż z Ǹ ´1 x `dz Ǹ igb ή1 pz Ǹ , ⃗ z q (3.5.3) where z 0 " y `.
Expanding this equation up to second order in igb ∆η , we find

" y `, x `‰η 1 ⃗ z " " y `, x `‰η ⃗ z `ig ż y x`d z 1 " y `, z 1 ‰ η ⃗ z b ∆η pz 1 , ⃗ z q " z 1 , x `‰η ⃗ z (3.5.4) `pigq 2 ż y x`d z 2 dz 1 " y `, z 2 ‰ η ⃗ z b ∆η pz 2 , ⃗ z q " z 2 , z 1 ‰ η ⃗ z b ∆η pz 1 , ⃗ z q " z 1 , x `‰η ⃗ z θpz 21 q .
With this, we get:

V η 1 p⃗ z 1 q " V η p⃗ z 1 q `ig ż dz 1 " `8, z 1 ‰ η ⃗ z 1 b ∆η pz 1 , ⃗ z 1 q " z 1 , ´8‰ η ⃗ z 1 (3.5.5) `pigq 2 ż dz 2 dz 1 " `8, z 2 ‰ η ⃗ z 1 b ∆η pz 2 , ⃗ z 1 q " z 2 , z 1 ‰ η ⃗ z 1 b ∆η pz 1 , ⃗ z 1 q " z 1 , ´8‰ η ⃗ z 1 θpz 21 q and V η 1 : p⃗ z 2 q " V η: p⃗ z 2 q ´ig ż dz 1 " ´8, z 1 ‰ η ⃗ z 2 b ´: ∆η pz 1 , ⃗ z 2 q " z 1 , `8‰ η ⃗ z 2 `p´igq 2 ż dz 2 dz 1 " ´8, z 1 ‰ η ⃗ z 2 b ´: ∆η pz 1 , ⃗ z 2 q " z 1 , z 2 ‰ η ⃗ z 2 b ´: ∆η pz 2 , ⃗ z 2 q " z 2 , `8‰ η: ⃗ z 2 θpz 21 q .
We used that the hermitian conjugate is the same as reversing the path (and anti-path ordering) as written in section 4.3.3 in [START_REF] Cherednikov | Parton Densities in Quantum Chromodynamics[END_REF].

These expansions lead to an expansion in order of ∆η of the form

Tr " V η 1 p⃗ z 1 qV η 1 : p⃗ z 2 q ‰ " Tr " V η p⃗ z 1 qV η: p⃗ z 2 q ‰ `pI R `IV q `Opp∆ηq 2 q (3.5.6)
I R `IV is the sum of all contributions from the ten Feynman diagrams that generate one step of the evolution; they give the linear term in the expansion of ∆η of Tr " V η 1 p⃗ z 1 qV η 1 : p⃗ z 2 q ‰ . The diagrams represent two eikonal lines in the fundamental representation moving through the shockwave formed by the classical background field b ή pzq. This interaction is in the colour singlet representation. On top of that, there is a quantum correction of the form of emission and absorption of a b ∆η gluon. I R refers to the "real" diagrams of fig. 3.5.1, where the gluon crosses the shockwave introducing, therefore, an additional adjoint Wilson line rU η p⃗ z 3 qs ab while I V refer to the "virtual" diagrams of fig. 3.5.2 where the gluon does not cross the shockwave.

The propagator built from b ∆η reads (neglecting the dependence in the minus space-time coordinate in the argument inside the bracket) The above expression can be understood qualitatively as follow. Starting from eq. (3.4.12) without the explicitly written colour indices, we have

´Gµν ∆η ¯ab pz 2 , ⃗ z 2 ; ⃗ z 1 , ⃗ z 1 q " p´iq d 2p2πq d`1 ż d d ⃗ z 3 rU η p⃗ z 3 qs ab ż e η`∆η p p e η p p dp `pp `qd´1 p´z 2 z 1 q d 2 `1 ˆexp " ´ip `"z 21 ´⃗ z 2 23 2z 2 `⃗ z 2 31 2z 1 ´iε ȷ* p⃗ z 23 ¨⃗ z 31 qpn µ 2 n ν 2 q (3.5.7) -∞ + +∞ + -∞ + -∞ + -∞ + +∞ + +∞ + +∞ + A B C D z + 1 , b z + 2 , a z 1 z 3 z 2 z + 2 , a z + 1 , b z 1 z 3 z 2 z 1 z 1 z 2 z 2 z 3 z 3 z + 2 , a z + 1 , b z + 2 , a z + 1 , b
G µν pz 2 , z 1 q| z 2 ą0ąz 1 " ´Γpdq 2p2πq d`1 ż d d ⃗ z 3 U p⃗ z 3 q " z 2 g α Kµ ´zα 23K n 2µ ı " ´z1 g Kαν ´z31Kα n 2ν ‰ p´z 2 z 1 q d 2 `1 ´´z 21 `⃗ z 2 23 2z 2 ´⃗ z 2 31 2z 1 `iε ¯d
Making a parallel between the above equation and the quark propagator through the shockwave eq. (3.4.7) and looking at the derivation of the latter in appendix A.1, more specifically eqs. (A.1.3, A.1.4), we can understand that the denominator with a power d is coming from the integration of the conserved longitudinal momentum of the gluon propagating through the shockwave treated as

-∞ + +∞ + -∞ + -∞ + -∞ + +∞ + +∞ + +∞ + z1 z2 z1 z2 z1 z1 z2 z2 z1 z2 z1 z2 -∞ + +∞ + -∞ + +∞ +
G µν pz 2 , z 1 q| z 2 ą0ąz 1 " p´iq d 2p2πq d`1 ż d d ⃗ z 3 U p⃗ z 3 q ż `8 0 dp `pp `qd´1 p´z ´2`z 1 q d 2 `1 ˆexp " ´ip `"z 21 ´⃗ z 2 23 2z 2 `⃗ z 2 31 2z 1 ´iε ȷ* ␣ z 2 z 1 g Kµν `z2 z 31Kµ n 2ν ´z1 z 23Kν n 2µ `⃗ z 23 ¨⃗ z 31 n 2µ n 2ν ( .
Now, we note that we are trying to built a propagator from b ∆η that are eikonal gluons with an effective non-sense polarization proportional to n µ 2 and with `momenta between e η p p and e η`∆η p p . The integral on p `needs to include these two limitations and we also expect the propagator to be proportional to n 2µ n 2ν , remembering Gribov's trick eq. (1.2.33) but with the gluon being attached to only particles on the projectile side. From this discussion, we therefore have a propagator of the form eq. (3.5.7).

Let's start now the computations with the real contributions. The contribution from diagram A in fig. 3.5.1 reads

I A " `µ2 ˘1´d 2 ż 0 ´8 dz 1 ż `8 0 dz 2 G ab ∆η pz 2 , ⃗ z 2 ; z 1 , ⃗ z 1 q ˆTr ´"`8, z 1 ‰ η ⃗ z 1 ´igt b ¯"z 1 , ´8‰ η ⃗ z 1 " ´8, z 2 ‰ η ⃗ z 2 p´igt a q " z 2 , `8‰ η ⃗ z 2 ¯(3.5.8) " g 2 `µ2 ˘1´d 2 p´iq d 2p2πq d`1 ż d d ⃗ z 3 Tr ´V η p⃗ z 1 qt b V η: p⃗ z 2 qt a ¯rU η p⃗ z 3 qs ab ż e η`∆η p p e η p p dp `pp `qd´1 ˆp⃗ z 23 ¨⃗ z 31 q ż 0 ´8 dz 1 p´z 1 q d 2 `1 ż `8 0 dz 2 pz 2 q d 2 `1 exp " ´ip `"z 21 ´⃗ z 2 23 2z 2 `⃗ z 2 31 2z 1 ´iε ȷ* (3.5.9)
Making the change of variable α " ´1{z 1 , β " 1{z 2 and integrating over them, we obtain Thus,

I A " g 2 `µ2 ˘1´d 2 2 d Γ 2 `d 2 2p2πq d`1 ż d d ⃗ z 3 Tr ´V η p⃗ z 1 qt b V η: p⃗ z 2 qt a ¯rU η p⃗ z 3 qs ab ⃗ z 23 ¨⃗ z 31 p⃗ z 2 23 q d 2 p⃗ z ;2 31 q d 2 ˆż e η`
I A " ∆η α s `µ2 ˘1´d 2 Γ 2 `d 2 πd ż d d ⃗ z 3 Tr ´V η p⃗ z 1 qt b V η: p⃗ z 2 qt a ¯rU η p⃗ z 3 qs ab ⃗ z 23 ¨⃗ z 31 p⃗ z 2 23 q d 2 p⃗ z ;2 31 q d 2 (3.5.11)
The contribution from diagram B reads

I B " `µ2 ˘1´d 2 ż 0 ´8 dz 1 ż `8 0 dz 2 G ab ∆η pz 2 , ⃗ z 1 ; z 1 , ⃗ z 2 q ˆTr ´"`8, z 2 ‰ η ⃗ z 1 pigt a q " z 2 , ´8‰ η ⃗ z 1 " ´8, z 1 ‰ η ⃗ z 2 p´igt b q " z 1 , `8‰ η ⃗ z 2
Comparing to eq. (3.5.8), we observe that I B can be obtained form I A by doing ig Ø ´ig for the vertex between the eikonal lines and the gluon, by interchanging ⃗ z 1 Ø ⃗ z 2 and by putting a hermitian conjugate only on the argument of the trace.

Doing the above manipulation on eq. (3.5.11) leads to

I B " I A . (3.5.12)
Diagram C is computed similarly as for diagram A and yields

I C " `µ2 ˘1´d 2 ż 0 ´8 dz 1 ż `8 0 dz 2 G ab ∆η pz 2 , ⃗ z 2 ; z 1 , ⃗ z 2 q ˆTr ´r`8, ´8s η ⃗ z 1 " ´8, z 1 ‰ η ⃗ z 2 p´igt b q " z 1 , z 2 ‰ η ⃗ z 2 p´igt a q " z 2 , `8‰ η ⃗ z 2 " ´g2 `µ2 ˘1´d 2 p´iq d 2p2πq d`1 ż d d ⃗ z 3 Tr ´V η p⃗ z 1 qt b V η: p⃗ z 2 qt a ¯rU η p⃗ z 3 qs ab ż e η`∆η p p e η p p dp `pp `qd´1 ˆe´ip `pz 21 ´iεq p⃗ z 23 ¨⃗ z 32 q ż 0 ´8 dz 1 p´z 1 q d 2 `1 ż `8 0 dz 2 pz 2 q d 2 `1 exp " ´ip `"´⃗ z 2 32 2z 2 ´⃗ z 2 32 2p´z 1 q ´iε ȷ* " ∆η α s `µ2 ˘1´d 2 Γ 2 `d 2 πd ż d d ⃗ z 3 Tr ´V η p⃗ z 1 qt b V η: p⃗ z 2 qt a ¯rU η p⃗ z 3 qs ab 1 p⃗ z 2 32 q d´1 . (3.5.13)
Diagram D is obtained from diagram C by doing the same manipulation as for diagram B, leading to

I D " ∆η α s `µ2 ˘1´d 2 Γ 2 `d 2 πd ż d d ⃗ z 3 Tr ´V η p⃗ z 1 qt b V η: p⃗ z 2 qt a ¯rU η p⃗ z 3 qs ab 1 p⃗ z 2 31 q d´1 . (3.5.14)
The total real contribution then reads

I R " I A `IB `IC `IB " ∆η α s `µ2 ˘1´d 2 Γ 2 `d 2 πd ż d d ⃗ z 3 Tr ´V η p⃗ z 1 qt b V η: p⃗ z 2 qt a ¯rU η p⃗ z 3 qs ab ˆ# 2 p⃗ z 23 ¨⃗ z 31 q p⃗ z 2 23 q d 2 p⃗ z ;2 31 q d 2 `1 p⃗ z 2 23 q d´1 `1 p⃗ z 2 31 q d´1 + . (3.5.15)
Now we should proceed with the virtual contributions of fig. 3.5.2. Instead of a direct computation like for the real case, one can obtain I V , the sum of all six virtual diagrams, using an elegant trick. First, we observe that if we consider the case of zero interaction with the target or equivalently setting all Wilson lines to 1, then any dependence on η disappears, and so we have Tr "

V η 1 p⃗ z 1 qV η 1 : p⃗ z 2 q ‰ˇˇV p⃗ z 1,2 q,U p⃗ z 3 qÑ1 " Tr " V η p⃗ z 1 qV η: p⃗ z 2 q ‰ˇˇV p⃗ z 1,2 q,U p⃗ z 3 qÑ1
" N c and then eq. (3.5.6) reduces to

I R `IV | V p⃗ z 1,2 q,U p⃗ z 3 qÑ1 " 0 ñ I V | V p⃗ z 1,2 q,U p⃗ z 3 qÑ1 " ´IR | V p⃗ z 1,2 q,U p⃗ z 3 qÑ1 . (3.5.16)
Moreover, it is clear that I V has the form

I V " C V Tr " V η p⃗ z 1 qV η: p⃗ z 2 q ‰ ùñ I V | V p⃗ z 1,2 q,U p⃗ z 3 qÑ1 " C V N c . (3.5.17)
From eq. (3.5.15), we have

I R | V p⃗ z 1,2 q,U p⃗ z 3 qÑ1 " ∆η α s `µ2 ˘1´d 2 Γ 2 `d 2 πd N 2 c ´1 2
ż d d ⃗ z 3 # 2 p⃗ z 23 ¨⃗ z 31 q p⃗ z 2 23 q d 2 p⃗ z ;2 31 q d 2 `1 p⃗ z 2 23 q d´1 `1 p⃗ z 2 31 q d´1 + , (3.5.18) then C V " ´∆η α s `µ2 ˘1´d 2 Γ 2 `d 2 πd N 2 c ´1 2N c ż d d ⃗ z 3 # 2 p⃗ z 23 ¨⃗ z 31 q p⃗ z 2 23 q d 2 p⃗ z ;2 31 q d 2 `1 p⃗ z 2 23 q d´1 `1 p⃗ z 2 31 q d´1 + (3.5.19)
and thus, the virtual contribution reads

I V " ´∆η α s `µ2 ˘1´d 2 Γ 2 `d 2 πd ż d d ⃗ z 3 N 2 c ´1 2N c Tr `V η p⃗ z 1 qV η: p⃗ z 2 q # 2 p⃗ z 23 ¨⃗ z 31 q p⃗ z 2 23 q d 2 p⃗ z ;2 31 q d 2 `1 p⃗ z 2 23 q d´1 `1 p⃗ z 2 31 q d´1 + . (3.5.20)
From eq. (3.5.6), (3.5.15) and (3.5.20) and taking the limit ∆η Ñ 0, we get

B Tr `V η p⃗ z 1 qV η: p⃗ z 2 q Bη " α s `µ2 ˘1´d 2 Γ 2 `d 2 πd ż d d ⃗ z 3 # 2 p⃗ z 23 ¨⃗ z 31 q p⃗ z 2 23 q d 2 p⃗ z ;2 31 q d 2 `1 p⃗ z 2 23 q d´1 `1 p⃗ z 2 31 q d´1 + ˆ"Tr ´V η p⃗ z 1 qt b V η: p⃗ z 2 qt a ¯rU η p⃗ z 3 qs ab ´N 2 c ´1 2N c
Tr `V η p⃗ z 1 qV η: p⃗ z 2 q ˘ȷ .

(3.5.21)

We introduce the dipole operator, defined as

U η ij " 1 ´1 N c Tr `V η p⃗ z i qV η: p⃗ z j q ˘. (3.5.22)
Using the property eq. (3.3.36) and the colour Fierz identity

t a ij t a kl " 1 2 δ kj δ il ´1 2N c δ ij δ kl , (3.5.23)
we find that

Tr ´V η p⃗ z 1 qt b V η: p⃗ z 2 qt a ¯rU η p⃗ z 3 qs ab ´N 2 c ´1 2N c Tr `V η p⃗ z 1 qV η: p⃗ z 2 q " V η ij p⃗ z 1 q rU η p⃗ z 3 qs ab t b jk `V η: p⃗ z 2 qt a ˘ki ´´N 2 c ´1 2N c Tr `V η p⃗ z 1 qV η: p⃗ z 2 q " V η ij p⃗ z 1 q " V η: p⃗ z 3 qt a V η p⃗ z 3 q ‰ jk V η: kl p⃗ z 2 qt a li ´N 2 c ´1 2N c Tr `V η p⃗ z 1 qV η: p⃗ z 2 q " t a pq t a li V η ij p⃗ z 1 qV η: jp p⃗ z 3 qV η qk p⃗ z 3 qV η: kl p⃗ z 2 q ´N 2 c ´1 2N c Tr `V η p⃗ z 1 qV η: p⃗ z 2 q " ˆ1 2 δ lq δ pi ´1 2N c δ pq δ li ˙"V η p⃗ z 1 qV η: p⃗ z 3 q ‰ ip " V η p⃗ z 3 qV η: p⃗ z 2 q ‰ ql ´N 2 c ´1 2N c Tr `V η p⃗ z 1 qV η: p⃗ z 2 q " 1 2 Tr `V η p⃗ z 1 qV η: p⃗ z 3 q ˘Tr `V η p⃗ z 3 qV η: p⃗ z 2 q ˘´1 2N c Tr `V η p⃗ z 1 qV η: p⃗ z 3 qV η p⃗ z 3 qV η: p⃗ z 2 q N 2 c ´1 2N c Tr `V η p⃗ z 1 qV η: p⃗ z 2 q " N 2 c 2 p1 ´Uη 13 q p1 ´Uη 32 q ´N 2 c 2 p1 ´Uη 12 q " N 2 c 2 rU η 12 ´Uη 32 ´Uη 13 `Uη 13 U η 32 s . (3.5.24)
The evolution equation for the dipole operator takes thus the form

BU η 12 Bη " α s N c `µ2 ˘1´d 2 Γ 2 `d 2 2π d ż d d ⃗ z 3 # 2 p⃗ z 23 ¨⃗ z 31 q p⃗ z 2 23 q d 2 p⃗ z ;2 31 q d 2 `1 p⃗ z 2 23 q d´1 `1 p⃗ z 2 31 q d´1 + ˆrU η 13 `Uη 32 ´Uη 12 ´Uη 13 U η 32 s . (3.5.25)
Taking D " 4, one recovers the usual B-JIMWLK equation for the dipole operator In physical scattering amplitudes, operators built out of Wilson lines are only present sandwiched between the target in and out state, corresponding to the target impact factor, as mentioned before. It is in this sense that we say that the evolution equation for the dipole operator involves a double dipole operator U η 13 U η 32 composed of four Wilson lines. The latter has its own evolution equation that involves operators with higher numbers of Wilson lines, and those have their own evolution equations too. We have, therefore, an infinite cascade of equations called the Balitsky hierarchy of equations.

BU η 12 Bη " α s N c 2π 2 ż d 2 ⃗ z 3 ⃗ z 2 12 ⃗ z 2 13 ⃗ z 2
A closed form of eq. (3.5.25), the Balitsky-Kovchegov (BK) equation, can be obtained in the large-N c limit which was introduced by 't Hooft [START_REF] Hooft | A Planar Diagram Theory for Strong Interactions[END_REF] (or equivalently in the mean-field approximation). In this limit, the matrix element of the double dipole operator becomes the product of matrix elements of the dipole operator up to correction of order 1{N 2 c . In most cases, BK is used in numerical studies. Numerical studies of the B-JIMWLK evolution equation have found only small differences between BK and B-JIMWLK, differences of the order of 0.1%, much smaller than the simple estimate of 1{N 2 c " 10% from the large-N c approximation [START_REF] Lappi | Next-to-leading order Balitsky-Kovchegov equation beyond large N c[END_REF][START_REF] Kovchegov | Subleading-N(c) corrections in non-linear small-x evolution[END_REF]. Finally, neglecting the non-linear term in eq. (3.5.25), which is the same as taking the dilute approximation, leads to the BFKL equation in its colour-dipole version (see eq.( 9.212) in [START_REF] Ioffe | Quantum chromodynamics: Perturbative and nonperturbative aspects[END_REF]).

B-JIMWLK evolution equation for the dipole operator in the momentum space

The Fourier transform of the dipole operator and the double dipole operator are defined as:

r U η ij p⃗ p i , ⃗ p j q " ż d d ⃗ z i d d ⃗ z j e ´i⃗ p i ¨⃗ z i e ´i⃗ p j ¨⃗ z j U η ij (3.5.27) Č U η ik U η kj p⃗ p i , ⃗ p j , ⃗ p k q " ż d d ⃗ z i d d ⃗ z j d d ⃗ z k e ´i⃗ p i ¨⃗ z i e ´i⃗ p j ¨⃗ z j e ´i⃗ p k ¨⃗ z k U η ik U η kj . (3.5.28)
In deriving the B-JIMWLK evolution equation for the dipole operator in momentum space, we will write the Fourier tranform of dipole amplitude as depending on three momentum variables instead of just two as in the above definition. This dependence in the third variable should be understood as

r U η ij p⃗ p i , ⃗ p j , ⃗ p k q " r U η ij p⃗ p i , ⃗ p j q p2πq d δ p⃗ p k q , (3.5.29)
such that for the case of i, j " 1, 3, we have

U η 13 " ż d d ⃗ q 1 d d ⃗ q 2 d d ⃗ q 3 p2πq 3d r U η 13 p⃗ q 1 , ⃗ q 2 , ⃗ q 3 q e i⃗ q 1 ¨⃗ z 1 e i⃗ q 2 ¨⃗ z 2 e i⃗ q 3 ¨⃗ z 3 " ż d d ⃗ q 1 d d ⃗ q 2 d d ⃗ q 3 p2πq 3d r U η 13 p⃗ q 1 , ⃗ q 3 q p2πq d δp⃗ q 2 qe i⃗ q 1 ¨⃗ z 1 e i⃗ q 2 ¨⃗ z 2 e i⃗ q 3 ¨⃗ z 3 " ż d d ⃗ q 1 d d ⃗ q 3 p2πq 2d r U η 13 p⃗ q 1 , ⃗ q 3 q e i⃗ q 1 ¨⃗ z 1 e i⃗ q 3 ¨⃗ z 3 .
This notation is therefore coherent with the definition eq. (3.5.27) and is used below in order to have all Wilson-line operators written in one single block.

Fourier transforming eq. (3.5.25) leads to

B r U η 12 Bη " α s N c `µ2 ˘1´d 2 Γ 2 `d 2 2π d ż d d ⃗ z 1 d d ⃗ z 2 d d ⃗ z 3 e ´i⃗ p 1 ¨⃗ z 1 e ´⃗ p 2 ¨⃗ z 2 rU η 13 `Uη 32 ´Uη 12 ´Uη 13 U η 32 s ˆ# 2 p⃗ z 23 ¨⃗ z 31 q p⃗ z 2 23 q d 2 p⃗ z 2 31 q d 2 `1 p⃗ z 2 23 q d´1 `1 p⃗ z 2 31 q d´1 + B r U η 12 Bη " α s N c `µ2 ˘1´d 2 Γ 2 `d 2 2π d ż d d ⃗ q 1 d d ⃗ q 2 d d ⃗ q 3 p2πq 3d " r U η 13 `r U η 32 ´r U η 12 ´Č U η 13 U η 32 ı p⃗ q 1 , ⃗ q 2 , ⃗ q 3 q ˆż d d ⃗ z 1 d d ⃗ z 2 d d ⃗ z 3 # 2 p⃗ z 23 ¨⃗ z 31 q p⃗ z 2 23 q d 2 p⃗ z 2 31 q d 2 `1 p⃗ z 2 23 q d´1 `1 p⃗ z 2 31 q d´1
+ e ´i⃗ p 1 ¨⃗ z 1 e ´i⃗ p 2 ¨⃗ z 2 e i⃗ q 1 ¨⃗ z 1 e i⃗ q 2 ¨⃗ z 2 e i⃗ q 3 ¨⃗ z 3 .

Changing variables with ⃗ u " ⃗ z 13 and ⃗ v " ⃗ z 23 , the above equation becomes

B r U η 12 Bη " α s N c `µ2 ˘1´d 2 Γ 2 `d 2 2π d ż d d ⃗ q 1 d d ⃗ q 2 d d ⃗ q 3 p2πq 3d " r U η 13 `r U η 32 ´r U η 12 ´Č U η 13 U η 32 ı p⃗ q 1 , ⃗ q 2 , ⃗ q 3 q ˆż d d ⃗ u d d ⃗ v d d ⃗ z 3 e ip⃗ q 1 ´⃗ p 1 q¨⃗ u e ip⃗ q 2 ´⃗ p 2 q¨⃗ v e ´ip⃗ p 1 `⃗ p 2 ´⃗ q 1 ´⃗ q 2 ´⃗ q 3 q¨⃗ z 3 ˆ#´2 ⃗ u ¨⃗ v p⃗ u 2 q d 2 p⃗ v 2 q d 2 `1 p⃗ v 2 q d´1 `1 p⃗ u 2 q d´1 + " α s N c `µ2 ˘1´d 2 Γ 2 `d 2 2π d ż d d ⃗ q 1 d d ⃗ q 2 d d ⃗ q 3 p2πq 2d " r U η 13 `r U η 32 ´r U η 12 ´Č U η 13 U η 32 ı p⃗ q 1 , ⃗ q 2 , ⃗ q 3 q ˆδp⃗ p 1 `⃗ p 2 ´⃗ q 1 ´⃗ q 2 ´⃗ q 3 q # ´2 ż d d ⃗ u d d ⃗ v ⃗ u ¨⃗ v p⃗ u 2 q d 2 p⃗ v 2 q d 2 e ip⃗ q 1 ´⃗ p 1 q¨⃗ u e ip⃗ q 2 ´⃗ p 2 q¨⃗ v `p2πq d δp⃗ p 1 ´⃗ q 1 q ż d d ⃗ v e ip⃗ q 2 ´⃗ p 2 q¨⃗ v p⃗ v 2 q d´1 `p2πq d δ p⃗ p 2 ´⃗ q 2 q ż d d ⃗ u e ip⃗ q 1 ´⃗ p 1 q¨⃗ u p⃗ u 2 q d´1 + (3.5.30)
Using the following results, which can be found using the Schwinger parametrization eq. (3.3.8),

ż d d ⃗ z e i⃗ p¨⃗ z p⃗ z 2 q n " π d 2 Γ `d 2 ´nΓ pnq ˆ4 ⃗ p 2 ˙d 2 ´n (3.5.31) ż d d ⃗ z ⃗ z i p⃗ z 2 q n e i⃗ p¨⃗ z " iπ d 2 Γ `d 2 ´n `1Γ pnq ⃗ p i 2 ˆ4 ⃗ p 2 ˙d 2 ´n`1 , (3.5.32)
one finally gets the B-JIMWLK evolution equation for the dipole operator in momentum space:

B r U η 12 Bη " 2α s N c `µ2 ˘1´d 2 ż d d ⃗ q 1 d d ⃗ q 2 d d ⃗ q 3 p2πq 2d δp⃗ p 1 `⃗ p 2 ´⃗ q 1 ´⃗ q 2 ´⃗ q 3 q ˆ" r U η 13 `r U η 32 ´r U η 12 ´Č U η 13 U η 32 ı p⃗ q 1 , ⃗ q 2 , ⃗ q 3 q ˆ$ ' ' & ' ' % 2 p⃗ p 1 ´⃗ q 1 q ¨p⃗ p 2 ´⃗ q 2 q p⃗ p 1 ´⃗ q 1 q 2 p⃗ p 2 ´⃗ q 2 q 2 `π d 2 Γ `1 ´d 2 ˘Γ2 `d 2 Γpd ´1q ¨δ p⃗ p 1 ´⃗ q 1 q " p⃗ p 2 ´⃗ q 2 q 2 ı 1´d 2 `δ p⃗ p 2 ´⃗ q 2 q rp⃗ p 1 ´⃗ q 1 q 2 s 1´d 2 ‹ ‹ ' , / / . / / - (3.5.33) 91 Chapter 4
Diffractive cross-section for γ p˚q P Ñ q qP 1 at NLO and γ p˚q P Ñ q qgP 1 at LO In this chapter, we report schematically and for completeness on the derivation of the γ p˚q Ñ q q diffractive impact factors and γ p˚q P Ñ q q diffractive cross-sections at one-loop accuracy and the γ p˚q Ñ q qg diffractive impact factors and diffractive γ p˚q P Ñ q qgP 1 at Born level. This is based on the work presented in [START_REF] Boussarie | On the one loop γ p˚q Ñ qq impact factor and the exclusive diffractive cross sections for the production of two or three jets[END_REF][START_REF] Boussarie | Perturbative study of selected exclusive QCD processes at high and moderate energies[END_REF]. P here is the initial nucleon or nucleus target remaining intact after the interaction, generically called proton from now on. The photon could either be real or virtual, hence the notation γ p˚q . The direct Pomeron here is a colour singlet QCD shockwave, either built from Balitsky's high energy operator expansion [START_REF] Balitsky | Factorization for high-energy scattering[END_REF][START_REF] Balitsky | Factorization and high-energy effective action[END_REF][START_REF] Balitsky | Effective field theory for the small x evolution[END_REF][START_REF] Balitsky | Operator expansion for high-energy scattering[END_REF], or from the color glass condensate formulation [START_REF] Jalilian-Marian | The BFKL equation from the Wilson renormalization group[END_REF][START_REF] Jalilian-Marian | The Wilson renormalization group for low x physics: Gluon evolution at finite parton density[END_REF][START_REF] Jalilian-Marian | The Wilson renormalization group for low x physics: Towards the high density regime[END_REF][START_REF] Jalilian-Marian | Unitarization of gluon distribution in the doubly logarithmic regime at high density[END_REF][START_REF] Kovner | Relating different approaches to nonlinear QCD evolution at finite gluon density[END_REF][START_REF] Weigert | Unitarity at small Bjorken x[END_REF][START_REF] Iancu | Nonlinear gluon evolution in the color glass condensate. 1[END_REF][START_REF] Ferreiro | Nonlinear gluon evolution in the color glass condensate. 2[END_REF][START_REF] Iancu | The Renormalization group equation for the color glass condensate[END_REF]. The results depend on the total available centre-of-mass energy and can have many applications as the shockwave framework is quite general. In the non-saturated regime, the t-channel exchange might be described in the linear BFKL regime [START_REF] Kuraev | Multi -Reggeon Processes in the Yang-Mills Theory[END_REF][START_REF] Fadin | On the Pomeranchuk Singularity in Asymptotically Free Theories[END_REF][START_REF] Kuraev | The Pomeranchuk Singularity in Nonabelian Gauge Theories[END_REF][START_REF] Balitsky | The Pomeranchuk Singularity in Quantum Chromodynamics[END_REF], here at NLL precision [START_REF] Fadin | BFKL pomeron in the next-to-leading approximation[END_REF][START_REF] Ciafaloni | Energy scale(s) and next-to-leading BFKL equation[END_REF][START_REF] Fadin | Non-forward BFKL pomeron at next-to-leading order[END_REF][START_REF] Fadin | Non-forward NLO BFKL kernel[END_REF] to be consistent with the computations of impact factors at NLO. At higher energies, in the saturation region, the operators built out of Wilson lines evolve with rapidity according to the Balitsky hierarchy of equations. The crosssections obtained from the impact factors depend on the target via the matrix element of the dipole operator U ij , defined in eq. (3.5.22) or the double dipole operator U ik U kj between the in and out target states. These matrix elements are to be described by some models or calculated as solutions of the NLL BK and LL double dipole evolution equation/NLL B-JIMWLK equation with the initial condition at the rapidity of the target. In the BFKL linear limit, these solutions are known analytically for forward scattering with a running coupling [START_REF] Chirilli | Solution of the NLO BFKL Equation and a Strategy for Solving the All-Order BFKL Equation[END_REF][START_REF] Grabovsky | On the solution to the NLO forward BFKL equation[END_REF].

Kinematics

We denote p γ , p q , p q, p g the momentum of the real or virtual photon, of the massless outgoing quark and antiquark and of the emitted gluon for the γ p˚q Ñ q qg impact factors. p 0 , p 0 1 are the momentum of the incoming and outgoing proton respectively.

We are in semi-hard kinematics, in which the centre-of-mass energy s " pp γ `p0 q 2 " 2p γ p 0 is larger than any other scales.

The working frame is the one of the shockwave formalism, presented in section 3.1, with photon and particles on the projectile side moving in the n 1 (i.e. `) direction while particles on the target side move in the n 2 (i.e. ´) direction and such that

p γ " p 0 " c s 2 . (4.1.1)
We work in the light-cone gauge A ¨n2 " 0.

We consider a photon with virtuality Q and no transverse momentum

⃗ p γ " ⃗ 0, p µ γ " p γ n µ 1 ´Q2 2p γ n µ 2 , Q 2 " ´p2 γ ą 0 . (4.1.2)
Its transverse polarization vector will be denoted as ε T . Its longitudinal polarization vector reads

ε α L " p γ Q n α 1 `Q 2p γ n α 2 , ε L " p γ Q , ε Ĺ " Q 2p γ . (4.1.3)
The outgoing quark, antiquark and emitted gluon are on-shell. Defining their longitudinal momentum fraction with respect to the photon as:

x q " p q p γ , x q " p q p γ , x g " p g p γ , (4.1.4)
then their momenta have the form:

p µ q " x q p γ n µ 1 `⃗ p 2 q 2x q p γ n µ 2 `pµ qK p µ q " x qp γ n µ 1 `⃗ p 2 q 2x qp γ n µ 2 `pµ qK p µ g " x g p γ n µ 1 `⃗ p 2 g 2x g p γ n µ 2 `pµ gK . (4.1.5)
The operators built out of Wilson lines are to be evaluated between the proton in and out states. For simplicity of notations, the operator form is used instead of the matrix elements when discussing the impact factors. One comes back to the latter at the cross-section level. Dimensional regularization d " 2 `2ϵ " D ´2 is used to regularize the integration over transverse components while an infrared cut-off αp γ is used to regularize integration over longitudinal component.

γ p˚q Ñ q q LO impact factors

Using the building blocks of section 3.4 and the projector over the colour singlet δ ln ? Nc with n, l the respective colours of the quark and the antiquark, the LO S-matrix element of fig. 4

.2.1 reads:

-

→ -→ -→ p 1 p γ p 2 -→ p q -→ p q y 0 Figure 4.2.1: LO impact factor. S 0 " ż d D y 0 rūpp q , y 0 qs nk p´iee q q γ α θpp γ q ε α b 2p γ e ´ip¨y 0 rvpp q, y 0 qs kl δ ln ? N c " ´1 ? N c 1 b 2p q 1 b 2p q ε α b 2p γ θpp q qθpp q qθpp γ qp´iee q q ż d d ⃗ p 1 p2πq d d d ⃗ p 2 p2πq d Tr " V p⃗ p 1 qV : p´⃗ p 2 q ‰ ˆż d D y 0 θp´y 0 q exp `iy 0 pp q `pq ´pγ q ˘exp ˜iy 0 « ⃗ p 2 q1 ´iε 2p q `⃗ p 2 q2 ´iε 2p q ´pγ ffȩ xp p´i⃗ y 0 ¨p⃗ p q1 `⃗ p q2 ´⃗ p γ qq ūpp q qγ `´p q γ ´`p { q1K 2p q γ α ´pq γ ´`p { q2K 2p q γ `vpp qq .
Integrating with respect to ⃗ y 0 , using the fact that ⃗ p γ " 0, see eq. (4.1.2), and with respect to y 0 which gives the conservation of the longitudinal momentum lead to

S 0 " ´1 ? N c 1 b 2p q 1 b 2p q ε α b 2p γ θpp q qθpp q qθpp γ qp´iee q q 1 p2πq d´1 δpp q `pq ´pγ q ˆż d d ⃗ p 1 d d ⃗ p 2 δp⃗ p q1 `⃗ p q2 q Tr " V p⃗ p 1 qV : p´⃗ p 2 q ‰ ūpp q qγ `pq γ ´`p { q1K 2p q γ α p q γ ´`p { q2K 2p q γ `vpp qq ˆż dy 0 θp´y 0 q exp ˜iy 0 « ⃗ p 2 q1 ´iε 2p q `⃗ p 2 q2 ´iε 2p q ´pγ ff¸.
The integral over y 0 gives, employing the delta function on the transverse and longitudinal momenta and the Schwinger parametrization and also simplifying expressions based on the parametrisation eq. ( 4.1.4),

ż dy 0 θp´y 0 q exp ˜iy 0 « ⃗ p 2 q1 ´iε 2p q `⃗ p 2 q2 ´iε 2p q ´pγ ff" ż `8 0 dy 0 exp ˜´iy 0 « p⃗ p 2 q1 ´iεqp q `p⃗ p 2 q1 ´iεqp q 2p q p q `Q2 2p γ ff" ż `8 0 dy 0 exp ˜´iy 0 `⃗ p 2 q1 `Q2 x q x q ´iε ˘pp γ q 2 2p q p q p γ " ´i ⃗ p 2 q1 `Q2 x q x q ´iε 2x q x qp γ .
Finally, we get

S 0 " ´1 ? N c 1 b 2p q 1 b 2p q ε α b 2p γ θpp q qθpp q qθpp γ qp´iee q q 1 p2πq D´3 δpp q `pq ´pγ q ˆż d d ⃗ p 1 d d ⃗ p 2 δp⃗ p q1 `⃗ p q2 q Tr " V p⃗ p 1 qV : p´⃗ p 2 q ‰ ūpp q qγ `pq γ ´`p { q1K 2p q γ α p q γ ´`p { q2K 2p q γ `vpp qq ˆ´i ⃗ p 2 q1 `Q2 x q x q 2x q x qp γ , (4.2.1)
Removing the non-interacting part, which is equivalent to setting all Wilson lines to 1, then the scattering amplitude has the form

M α 0 " 1 ? N c 1 b 2p q 1 b 2p q 1 b 2p γ θpp q qθpp q qθpp γ qp´iee q q 1 p2πq D´3 p´iqδpp q `pq ´pγ q ˆż d d ⃗ p 1 d d ⃗ p 2 δp⃗ p q1 `⃗ p q2 qN c " 1 ´1 N c Tr " V p⃗ p 1 qV : p´⃗ p 2 q ‰ * ˆūpp q qγ `pq γ ´`p { q1K 2p q γ α p q γ ´`p { q2K 2p q`γ `vpp qq 1 ⃗ p 2 q1 `Q2 x q x q 2x q x qp γ . (4.2.
2)

The reduced matrix element T α is defined as:

M α " 1 ? N c 1 b 2p q 1 b 2p q ´iee q b 2p γ θpp q qθpp q qθpp γ q 1 p2πq D´3 iδpp q `pq ´pγ qT α . (4.2.3)
It is parameterized as follows

T α 0 " ´Nc ż d d ⃗ p 1 d d ⃗ p 2 δp⃗ p q1 `⃗ p q2 q r U 12 Φ α 0 p⃗ p 1 , ⃗ p 2 q . ( 4
Φ α 0 p⃗ p 1 , ⃗ p 2 q " ūpp q qγ `pq γ ´`p { q1K 2p q γ α p q γ ´`p { q2K 2p q`γ `vpp qq 1 ⃗ p 2 q1 `Q2 x q x q 2x q x qp γ . (4.2.5)
Now replacing α by `or i, one obtains using eq. (1.1.10):

Φ 0 " 2x q x qp γ ⃗ p 2 q1 `xq x qQ 2 ūpp q qγ `vpp qq (4.2.6) Φ i 0 " ūpp q q ´p1 ´2x q qp i q1K `1 2 " p { q1K , γ i ı¯γ `vpp qq ⃗ p 2 q1 `xq x qQ 2 . (4.2.7)
The minus component of the LO and NLO impact factors is related to the plus component by QED gauge invariance:

p γ ¨Φ " 0 ùñ Φ ´" Q 2 2pp γ q 2 Φ `. (4.2.8)
4.3 γ p˚q Ñ q q NLO impact factors

Colour factors and Wilson line operators

Eight one-loop diagrams represented in fig. 4.3.1 contribute to the virtual reduced matrix element T α 1 related to the NLO scattering amplitude via eq. (4.2.3). Diagrams are of two kinds as the loop gluon may or may not cross the shockwave. In both cases, the singlet projector is the same as for the LO case δ ln ? Nc (the factor 1{ ?

N c is not present in the calculation of the colour factors below, as it is put in the factor between T α and M α ). Diagrams of the same kind will all have the same colour factors and Wilson line operators. We will compute below those colour factors.

For diagrams where the gluon does not cross the shockwave, we have1 :

C 1 " δ ln V nk p⃗ z 1 qt a kp t a pq V : ql p⃗ z 2 q ´δln δ nk t a kp t a pq δ ql (4.3.1) " C F V lk p⃗ z 1 qδ kq V : ql p⃗ z 2 q ´CF δ nk δ kq δ ql δ ln " C F Tr " V p⃗ z 1 qV : p⃗ z 2 q ‰ ´CF N c " ´CF N c U 12 , (4.3.2)
where

C F " N 2 c ´1
2Nc is the Casimir in the fundamental representation. The first term of eq. ( 4.3.1) corresponds to the contribution with the Wilson lines and the gluon that does not cross the shockwave as well as the singlet projector δ ln and the subtraction term is when we set the Wilson line to 1 (it is the colour factor for the non-interacting part).

For diagrams where the gluon crosses the shockwave, a Wilson line in the adjoint representation is added and one has to use eq. (3.3.36) and the Fierz identity eq. (3.5.23) to compute the colour factor:

C 2 " δ ln V nk p⃗ z 1 qt b kp V : pq p⃗ z 2 qt a ql U ab p⃗ z 3 q ´δln δ nk t b kp δ pq t a ql δ ab " V lk p⃗ z 1 q " V : p⃗ z 3 qt a V p⃗ z 3 q ‰ kp V : pq p⃗ z 2 qt a ql ´δlk C F δ kl " V lk p⃗ z 1 qV : kj p⃗ z 3 qt a jm V mp p⃗ z 3 qV : pq p⃗ z 2 qt a ql ´CF N c " ˆ1 2 δ qm δ jl ´1 2N c δ jm δ ql ˙"V p⃗ z 1 qV : p⃗ z 3 q ‰ lj " V p⃗ z 3 qV : p⃗ z 2 q ‰ mq ´CF N c " 1 2 Tr " V p⃗ z 1 qV : p⃗ z 3 q ‰ Tr " V p⃗ z 3 qV : p⃗ z 2 q ‰ ´1 2N c Tr " V p⃗ z 1 qV : p⃗ z 3 qV p⃗ z 3 qV : p⃗ z 2 q ‰ ´CF N c " 1 2 " Tr " V p⃗ z 1 qV : p⃗ z 3 q ‰ Tr " V p⃗ z 3 qV : p⃗ z 2 q ‰ ´1 ´N 2 c `N 2 c N c Tr " V p⃗ z 1 qV : p⃗ z 2 q ‰ * ´CF N c " 1 2 ␣ N 2 c p1 ´U13 q p1 ´U32 q ´N 2 c p1 ´U12 q ( ´CF N c U 12 " ´N 2 c 2 pU 13 `U32 ´U12 ´U13 U 32 q ´CF N c U 12 . (4.3.3)
From those colours factors, the virtual reduced matrix element can be split into two terms associated to the dipole or double dipole operator:

T α 1 " ´αs N c Γp1 ´ϵq p4πq 1`ϵ ż d d ⃗ p 1 d d ⃗ p 2 " δ p⃗ p q1 `⃗ p q2 q ˆN 2 c ´1 N c ˙r U 12 p⃗ p 1 , ⃗ p 2 q Φ α 1 p⃗ p 1 , ⃗ p 2 q `Nc ż d d ⃗ p 3 p2πq d δ p⃗ p q1 `⃗ p q2 ´⃗ p 3 q " r U 13 `r U 32 ´r U 12 ´Č U 13 U 32 ı p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q Φ α 2 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q * (4.3.4)
The dipole impact factor has contributions from both types of diagrams, while the double dipole impact factor only gets contributions from diagrams where the gluon crosses the shockwave. The µ parameter, which is an arbitrary parameter introduced by dimensional regularization, is included in the definition of Φ α 1 , Φ α 2 .

Computational steps for the virtual diagrams

Just like for the LO impact factor, only Φ 1,2 and Φ i 1,2 have been computed explicitly as Φ 1,2 is related to Φ 1,2 via eq. (4.2.8) from gauge invariance.

Moreover, only five diagrams in fig. 4.3.1 have to be computed as diagrams 6, 7 and 8 can be found from diagrams 2, 3, 4 by doing the following substitution, p q Ø p q, p 1 Ø p 2 and the reversal of the order of the gamma matrices, which we will denote by pq Ø qq IF .

The computational steps are the same for each of the five diagrams:

' Write the S-matrix element of the diagram as an integral over the coordinate of every vertex, using the building blocks of section 3.4 and projecting on the colour singlet.

' Replace the building blocks by their expressions in mixed space representation, as a function of x `and p `, ⃗ p, e.g. a quark line though the shockwave will be replaced by eq. (3.4.8). For a free quark, the propagator will be written as

G 0 pxq " ż dp `dd ⃗ p 2p `p2πq d`1 exp " ´ix `⃗ p 2 ´iε 2p `´ip `x´`i ⃗ p ¨⃗ x ȷ (4.3.5) ˆ"ˆp `γ´`⃗ p 2 2p `γ``p { K ˙`θpp `qθpx `q ´θp´p `qθp´x `q˘`i δpx `qγ `* ,
and or a free gluon, it will be written as

G µν 0 pxq " ż dl `dd ⃗ l 2l `p2πq d`1 exp « ´ix `⃗ l 2 ´iε 2l `´il `x´`i ⃗ l ¨⃗ x ff (4.3.6) ˆ#˜g µν K ´lµ K n ν 2 `lν K n µ 2 l `´⃗ l 2 pl `q2 n µ 2 n ν 2 ¸`θpl `qθpx `q ´θp´l `qθp´x `q˘´2 iδpx `q l `+ .
' Integrate over the ´and transverse components of the vertex coordinates. This gives the explicit conservation of the longitudinal momenta, characteristic of the eikonal interactions. It also gives a delta function of the transverse momentum components, including the momentum kick ⃗ p 1 , ⃗ p 2 , ⃗ p 3 from the t-channel shockwave to the q, q, g lines.

' With those delta functions, the integrals over the momenta of the intermediate particles can be trivially done, apart from one momentum, which is always chosen to be the momentum of the loop gluon.

' One then integrates over the `component of the vertex coordinates, leading to an expression with only an integral over l `and ⃗ l.

' The last step is the integration over ⃗ l and l `. Integration over ⃗ l is regularized by dimensional regularization and done using the Schwinger parametrization eq. (3.3.8) or equivalently using the Feynman's parametrization

1 A α 1 1 ¨¨¨A αn n " Γpα 1 ¨¨¨α n q Γpα 1 q ¨¨¨Γpα n q ż 1 0 dx 1 ¨¨¨x n δ ˜1 ´n ÿ i"1 x i ¸xα 1 ´1 1 ¨¨¨x αn´1 n rA 1 x 1 `¨¨¨A n x n s α 1 `¨¨¨`αn .
(4.3.7) The longitudinal z " l `{p γ integral is regularized by an infrared cut-off α. To extract the divergent part, the following `prescription has been introduced, defined here as

ż z 0 α dz ϕpzq " ż z 0 α dz ϕ 0 pzq `ż z 0 α dz rϕpzq ´ϕ0 pzqs (4.3.8) " ż z 0 α dz ϕ 0 pzq `ż z 0 α dz rϕpzqs `,
where ϕ 0 pzq is the divergent part of ϕpzq when z Ñ 0.

A parallel can be drawn with computation using light-cone perturbation theory when the diagram is expressed as integral over l `, ⃗ l. The numerators of the propagators of the intermediate particles contain three terms, as seen in eq. (4.3.6). Two correspond to the non-instantaneous part, while the last one, proportional to a delta function, is the instantaneous part. One term for the non-instantaneous part is the propagation backwards in time and will give zero unless it is the loop gluon, thanks to the gamma structure or the `momentum ordering. The other term is the propagation of the intermediate particle forward in time of the on-shell particle. At each vertex, `and transverse components of the momenta are conserved, but the light-cone energy or p ´are not. Denominators obtained from integrating the `component of vertex coordinates correspond to the so-called energy denominators in light-cone perturbation theory.

Total dipole virtual impact factors

The sum of the contributions of all diagrams of fig. 4.3.1 for the virtual dipole impact factor reads:

Φ α 1 p⃗ p 1 , ⃗ p 2 q " S V 2 Φ α 0 p⃗ p 1 , ⃗ p 2 q `Φα 1R p⃗ p 1 , ⃗ p 2 q , (4.3.9)
where

S V 2 " " ln ´xq x q α 2 ¯´3 2 ȷ " ln ˆxq x qµ 2 px q ⃗ p q ´xq ⃗ p q q 2 ˙´1 ϵ ȷ `iπ ln ´xq x q α 2 ¯`1 2 ln 2 ´xq x q α 2 ¯´π 2 6 `3 , (4.3.10)
and the finite part

Φ 1R p⃗ p 1 , ⃗ p 2 q " 3 2 Φ 0 p⃗ p 1 , ⃗ p 2 q ln ¨x2 q x 2 q µ 4 Q 2 px q ⃗ p q ´xq ⃗ p q q 2 ´⃗ p 2 q1 `xq x qQ 2 ¯2 ‹ '`ūppqqpC 4 } `C5 1} `C6 1} qvpp qq , (4.3.11)
and

Φ i 1R p⃗ p 1 , ⃗ p 2 q " 3 2 Φ i 0 p⃗ p 1 , ⃗ p 2 q « ln ˆxq x qµ 4 px q ⃗ p q ´xq ⃗ p q q 2 p ⃗ p q1 2 `xq x qQ 2 q ˙´x q x qQ 2 ⃗ p 2 q1 ln ˜xq x qQ 2 ⃗ p 2 q1 `xq x qQ 2 ¸ff `ūpp q qpC 4i K `C5i 1K `C6i 1K qvpp qq . (4.3.12)
The functions C are finite expressions. Their detailed expressions will not be given in this manuscript; only their contributions at cross-sections will be given as it is what interests us.

Total double dipole virtual contribution and rapidity divergences

The double dipole virtual impact factor from diagrams where the gluon crosses the shockwave of fig. 4.3.1 contains rapidity divergences of the form ln α. These are cancelled by adding a subtraction term constructed from the LO reduced matrix element T α 0 eq. ( 4.2.4) with the dipole operator evolved from the non-physical rapidity cut-off ln α to the rapidity divide η via the B-JIMWLK evolution equation in momentum space eq. (3.5.33). The renormalization equation for the dipole operator reads:

r U α 12 " r U e η 12 ´ż e η α dρ B r U 12 Bρ (4.3.13)
with ρ " e η 1 . We have that

B r U 12 Bρ " B r U 12 Be η 1 " 1 ρ B r U 12 Bη 1 (4.3.14) and B r U 12
Bη 1 given by eq. (3.5.33). The evolution equation is of order α s so one sets the operators in the evolution equation at the scale e η without concern and similarly for the Wilson-line operators in the NLO reduced matrix elements. Then, writing r U e η 12 as r U η 12 , we have

r U α 12 " r U η 12 ´ż e η α dρ ρ 2α s N c `µ2 ˘1´d 2 ż d d ⃗ q 1 d d ⃗ q 2 d d ⃗ q 3 p2πq 2d δp⃗ p 1 `⃗ p 2 ´⃗ q 1 ´⃗ q 2 ´⃗ q 3 q ˆ" r U η 13 `r U η 32 ´r U η 12 ´Č U η 13 U η 32 ı p⃗ q 1 , ⃗ q 2 , ⃗ q 3 q ˆ$ ' ' & ' ' % 2 p⃗ p 1 ´⃗ q 1 q ¨p⃗ p 2 ´⃗ q 2 q p⃗ p 1 ´⃗ q 1 q 2 p⃗ p 2 ´⃗ q 2 q 2 `π d 2 Γ `1 ´d 2 ˘Γ2 `d 2 Γpd ´1q ¨δ p⃗ p 1 ´⃗ q 1 q " p⃗ p 2 ´⃗ q 2 q 2 ı 1´d 2 `δ p⃗ p 2 ´⃗ q 2 q rp⃗ p 1 ´⃗ q 1 q 2 s 1´d 2 ‹ ‹ ' , / / . / / - .
And so, plugging the above equation into the LO reduced matrix element T α 0 eq. (4.2.4), the subtraction term is

T α BK " 2α s N 2 c `µ2 ˘1´d 2 ln ˆeη α ˙ż d d ⃗ p 1 d d ⃗ p 2 δp⃗ p q1 `⃗ p q2 qΦ α 0 p⃗ p 1 , ⃗ p 2 q ż d d ⃗ q 1 d d ⃗ q 2 d d ⃗ q 3 p2πq 2d ˆδp⃗ p 1 `⃗ p 2 ´⃗ q 1 ´⃗ q 2 ´⃗ q 3 q " r U η 13 `r U η 32 ´r U η 12 ´Č U η 13 U η 32 ı p⃗ q 1 , ⃗ q 2 , ⃗ q 3 q ˆ$ ' ' & ' ' % 2 p⃗ p 1 ´⃗ q 1 q ¨p⃗ p 2 ´⃗ q 2 q p⃗ p 1 ´⃗ q 1 q 2 p⃗ p 2 ´⃗ q 2 q 2 `π d 2 Γ `1 ´d 2 ˘Γ2 `d 2 Γpd ´1q ¨δ p⃗ p 1 ´⃗ q 1 q " p⃗ p 2 ´⃗ q 2 q 2 ı 1´d 2 `δ p⃗ p 2 ´⃗ q 2 q rp⃗ p 1 ´⃗ q 1 q 2 s 1´d 2 ‹ ‹ ' , / / . / / - " 2α s N 2 c `µ2 ˘1´d 2 ln ˆeη α ˙ż d d ⃗ p 1 d d ⃗ p 2 d d ⃗ p 3 p2πq d " r U η 13 `r U η 32 ´r U η 12 ´Č U η 13 U η 32 ı p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q ˆż d d ⃗ k 1 d d ⃗ k 2 p2πq d δp⃗ p q ´⃗ k 1 `⃗ p q ´⃗ k 2 qΦ α 0 ´⃗ k 1 , ⃗ k 2 ¯δp ⃗ k 1 `⃗ k 2 ´⃗ p 1 ´⃗ p 2 ´⃗ p 3 q (4.3.15) ˆ$ ' ' ' ' & ' ' ' ' % 2 ´⃗ k 1 ´⃗ p 1 ¯¨´⃗ k 2 ´⃗ p 2 ⃗ k 1 ´⃗ p 1 ¯2 ´⃗ k 2 ´⃗ p 2 ¯2 `π d 2 Γ `1 ´d 2 ˘Γ2 `d 2 Γpd ´1q ¨δ ´⃗ k 1 ´⃗ p 1 "´⃗ k 2 ´⃗ p 2 ¯2ȷ 1´d 2 `δ ´⃗ k 2 ´⃗ p 2 "p ⃗ k 1 ´⃗ p 1 q 2 ı 1´d 2 ‹ ‹ ‹ ‹ ' , / / / / . / / / / - Integrating over ⃗ k 2 trivially with ⃗ k 2 " ⃗ p 1 `⃗ p 2 `⃗ p 3 ´⃗ k 1 leads to T α BK " 2α s N 2 c `µ2 ˘1´d 2 ln ˆeη α ˙ż d d ⃗ p 1 d d ⃗ p 2 d d ⃗ p 3 p2πq d " r U η 13 `r U η 32 ´r U η 12 ´Č U η 13 U η 32 ı p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q ˆż d d ⃗ k 1 p2πq d δp⃗ p q1 `⃗ p q2 ´⃗ p 3 qΦ α 0 ´⃗ k 1 , ⃗ p 1 `⃗ p 2 `⃗ p 3 ´⃗ k 1 ¯$ ' & ' % 2 ´⃗ k 1 ´⃗ p 1 ¯¨´⃗ p 1 `⃗ p 3 ´⃗ k 1 ⃗ k 1 ´⃗ p 1 ¯2 ´⃗ p 1 `⃗ p 3 ´⃗ k 1 ¯2 `π d 2 Γ `1 ´d 2 ˘Γ2 `d 2 Γpd ´1q ¨δ ´⃗ k 1 ´⃗ p 1 "´⃗ p 1 `⃗ p 3 ´⃗ k 1 ¯2ȷ 1´d 2 `δ ´⃗ p 1 `⃗ p 3 ´⃗ k 1 "p ⃗ k 1 ´⃗ p 1 q 2 ı 1´d 2 ‹ ‹ ‹ ‹ ' , / / / / . / / / / - . (4.3.16) Changing variables ⃗ p " ⃗ k 1 ´⃗ p 1 yields T α BK " 2α s N 2 c `µ2 ˘1´d 2 ln ˆeη α ˙ż d d ⃗ p 1 d d ⃗ p 2 d d ⃗ p 3 p2πq d " r U η 13 `r U η 32 ´r U η 12 ´Č U η 13 U η 32 ı p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q ˆż d d ⃗ p p2πq d δp⃗ p q1 `⃗ p q2 ´⃗ p 3 qΦ α 0 p⃗ p `⃗ p 1 , ⃗ p 2 `⃗ p 3 ´⃗ p q ˆ$ ' ' & ' ' % 2⃗ p ¨p⃗ p 3 ´⃗ p q ⃗ p 2 p⃗ p 3 ´⃗ p q 2 `π d 2 Γ `1 ´d 2 ˘Γ2 `d 2 Γpd ´1q ¨δ p⃗ p q " p⃗ p 3 ´⃗ p q 2 ı 1´d 2 `δ p⃗ p 3 ´⃗ p q r⃗ p 2 s 1´d 2 ‹ ‹ ' , / / . / / - " 2α s N 2 c ln ˆeη α ˙ż d d ⃗ p 1 d d ⃗ p 2 d d ⃗ p 3 p2πq d δp⃗ p q1 `⃗ p q2 ´⃗ p 3 q " r U η 13 `r U η 32 ´r U η 12 ´Č U η 13 U η 32 ı p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q ˆ"`µ 2 ˘´ϵ ż d d ⃗ p p2πq d Φ α 0 p⃗ p `⃗ p 1 , ⃗ p 2 `⃗ p 3 ´⃗ p q 2⃗ p ¨p⃗ p 3 ´⃗ p q ⃗ p 2 p⃗ p 3 ´⃗ p q 2 `π d 2 Γ `1 ´d 2 ˘Γ2 `d 2 p2πq d Γpd ´1q pµ 2 q ´ϵ p⃗ p 2 3 q 1´d 2 pΦ α 0 p⃗ p 1 , ⃗ p 2 `⃗ p 3 q `Φα 0 p⃗ p 3 `⃗ p 1 , ⃗ p 2 qq + " ´αs N 2 c Γ `2 ´d 2 p4πq 1`ϵ ż d d ⃗ p 1 d d ⃗ p 2 d d ⃗ p 3 p2πq d δp⃗ p q1 `⃗ p q2 ´⃗ p 3 q " r U η 13 `r U η 32 ´r U η 12 ´Č U η 13 U η 32 ı p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q ˆp´2q ln ˆeη α ˙p4πq 1`ϵ Γ `2 ´d 2 ˘"´2 `µ2 ˘´ϵ ż d d ⃗ p p2πq d ⃗ p ¨p⃗ p ´⃗ p 3 q ⃗ p 2 p⃗ p 3 ´⃗ p q 2 Φ α 0 p⃗ p `⃗ p 1 , ⃗ p 2 `⃗ p 3 ´⃗ p q `Γ `1 ´d 2 ˘Γ2 `d 2 p4πq 1`ϵ Γpd ´1q ˆ⃗ p 2 3 µ 2 ˙ϵ pΦ α 0 p⃗ p 1 , ⃗ p 2 `⃗ p 3 q `Φα 0 p⃗ p 3 `⃗ p 1 , ⃗ p 2 qq + (4.3.17)
Comparing eq. ( 4.3.17) with eq. (4.3.4) and more specifically the part associated to the double dipole operator (as we are computing the subtraction term for the double dipole virtual impact factor), we infer that the subtraction impact factor is

Φ α BK p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q " ´2 ln ˆeη α ˙#´2 pµ 2 q ´ϵp4πq 1`ϵ Γp2 ´d 2 q ż d d ⃗ p p2πq d ⃗ p ¨p⃗ p ´⃗ p 3 q ⃗ p 2 p⃗ p 3 ´⃗ p q 2 Φ α 0 p⃗ p `⃗ p 1 , ⃗ p 2 `⃗ p 3 ´⃗ p q ´ˆ1 ϵ `ln ˆ⃗ p 2 3 µ 2 ˙˙pΦ α 0 p⃗ p 1 , ⃗ p 2 `⃗ p 3 q `Φα 0 p⃗ p 3 `⃗ p 1 , ⃗ p 2 qq * (4.3.18)
where we have also expanded the second term of the curly bracket of eq. (4.3.17) up to order ϵ 0 . Using the expression for the LO impact factors eq. (4.2.6, 4.2.7), the conservation of transverse momentum ⃗ p q1 `⃗ p q2 ´⃗ p 3 " ⃗ 0 and integrating over ⃗ p gives Φ BK p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q " ´4x q x qp γ `ūpp q qγ `vpp qq ˘ln ˆeη α

˙(4.3.19) ˆ«ˆl n ˆ⃗ p 2 3 µ 2 ˙`1 ϵ ˙˜´1 ⃗ p 2 q1 `xq x qQ 2 `´1 ⃗ p 2 q2 `xq x qQ 2 p 2 3 ´⃗ p 2 q1 ´⃗ p 2 q2 ´2x q x qQ 2 p⃗ p 2 q1 `xq x qQ 2 qp⃗ p 2 q2 `xq x qQ 2 q ´xq x q ⃗ p 2 3 Q 2 ln ˜p⃗ p 2 q1 `xq x qQ 2 qp⃗ p 2 q2 `xq x qQ 2 q x q x q ⃗ p 2 3 Q 2 ¸ff ,
in the longitudinal case, and

Φ i BK p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q " ´2 ln ˆeη α ˙"ūpp q q ˆpi q1K p1 ´2x q q `1 2 rp { q1K , γ i K s ˙γ`v pp qq (4.3.20) ˆ« ´1 ⃗ p 2 q1 `xq x qQ 2 ˆln ˆ⃗ p 2 3 µ 2 ˙`1 ϵ ˙`1 ⃗ p 2 q1 ln ˜⃗ p 2 q1 `xq x qQ 2 x q x qQ 2 p 2 q2 `xq x qQ 2 p⃗ p 2 q1 `xq x qQ 2 qp⃗ p 2 q2 `xq x qQ 2 q ´xq x q ⃗ p 2 3 Q 2 ˆln ˜p⃗ p 2 q1 `xq x qQ 2 qp⃗ p 2 q2 `xq x qQ 2 q x q x q ⃗ p 2 3 Q 2 ¸ff `pq Ø qq IF + ,
in the transverse case. Combining these terms with the results for Φ α 2 obtained from diagrams where the gluon crosses the shockwave2 , one gets cancellation of the rapidity divergences of the form ln α and gets the finite double dipole virtual impact factor Φ 1α

2 " Φ α 2 `Φα BK :

Φ 12 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q " 2p γ `ūpp q qγ `vpp qq ˘$ & % x q x q `⃗ p 2 3 ´⃗ p 2 q2 ´⃗ p 2 q1 ´2x q x qQ 2 ⃗ p 2 q2 `xq x qQ 2 ¯´⃗ p 2 q1 `xq x qQ 2 ¯´x q x qQ 2 ⃗ p 2 3 ˆln ´xq x q e 2η ¯ln ˜p⃗ p 2 q2 `xq x qQ 2 q `⃗ p 2 q1 `xq x qQ 2 xq x qQ 2 ⃗ p 2 3 xq x q ⃗ p 2 q1 `xq x qQ 2 " 2 ln ´xq e η ¯ˆ1 ϵ `ln ˆ⃗ p 2 3 µ 2 ˙˙´3 2ϵ ȷ `pq Ø qq IF ¸+ `ūpp q q ´C5 2} `C6 2} ¯vpp qq (4.3.21)
in the longitudinal case, or

Φ 1i 2 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q " # ūpp q q ˆpi q1K p1 ´2x q q `1 2 rp { q1K , γ i K s ˙γ`v pp qq ˜´1 ⃗ p 2 q1 `xq x qQ 2 ˆ"2 ln ´xq e η ¯ˆ1 ϵ `ln ˆ⃗ p 2 3 µ 2 ˙˙´3 2ϵ ȷ `ln ´xq x q e 2η » -1 ⃗ p 2 q1 ln ˜⃗ p 2 q1 `xq x qQ 2 x q x qQ 2 ¸´⃗ p 2 q2 `xq x qQ 2 ´⃗ p 2 q1 `xq x qQ 2 ¯´⃗ p 2 q2 `xq x qQ 2 ¯´x q x qQ 2 ⃗ p 2 3 ˆln ˜`⃗ p 2 q1 `xq x qQ 2 ˘`⃗ p 2 q2 `xq x qQ 2 xq x qQ 2 ⃗ p 2 3 ¸ff¸`p q Ø qq IF + `ūpp q q `C5i 2K `C6i 2K ˘vpp qq (4.3.22)
in the transverse case. The above equations seem to contain some remaining 1 ϵ divergent terms. However, these poles are artificial poles and originate from the definition of the Fourier transform of the double dipole operator eq. (3.5.28) which does not take into account its property of vanishing for z i " z k or z j " z k . These poles cancel when convoluting the impact factor with the Wilson-line operators. Defining the part of Φ 1α 2 that contain these poles as

F p⃗ p 1 q " 1 ⃗ p 2 q1 `xq x qQ 2 " 2 ln ´xq e η ¯1 ϵ ´3 2ϵ ȷ , (4.3.23)
which depends only on ⃗ p 1 and is independent of ⃗ p 3 and ⃗ p 2 (up to the ⃗ p 1 Ø ⃗ p 2 permutation), we have then when convoluting as in eq. (4.3.4)

ż d d ⃗ p 1 d d ⃗ p 2 d d ⃗ p 3 p2πq d δp⃗ p q1 `⃗ p q2 ´⃗ p 3 q " r U η 13 `r U η 32 ´r U η 12 ´Č U η 13 U η 32 ı F p⃗ p 1 q " ż d d ⃗ p 1 d d ⃗ p 2 d d ⃗ p 3 p2πq d δp⃗ p q1 `⃗ p q2 ´⃗ p 3 q ż d d ⃗ z 1 d d ⃗ z 2 d d ⃗ z 3 e ´i⃗ p 1 ¨⃗ z 1 e ´i⃗ p 2 ¨⃗ z 2 e ´i⃗ p 3 ¨⃗ z 3 ˆrU 13 `U32 ´U12 ´U13 U 32 s F p⃗ p 1 q " ż d d ⃗ p 1 d d ⃗ p 3 p2πq d ż d d ⃗ z 1 d d ⃗ z 2 d d ⃗ z 3 e ´i⃗ p 1 ¨⃗ z 1 e ´ip⃗ p q1 `⃗ pq´⃗ p 3 q¨⃗ z 2 e ´i⃗ p 3 ¨⃗ z 3 F p⃗ p 1 q rU 13 `U32 ´U12 ´U13 U 32 s " ż d d ⃗ p 1 d d ⃗ z 1 d d ⃗ z 2 d d ⃗ z 3 e ´i⃗ p 1 ¨⃗ z 1 e ´ip⃗ p q1 `⃗ pqq¨⃗ z 2 δp⃗ z 32 qF p⃗ p 1 q rU 13 `U32 ´U12 ´U13 U 32 s " 0 .
Therefore, hereafter, we will write the finite double dipole virtual impact factor as

Φ 12 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q " 2p γ `ūpp q qγ `vpp qq ˘$ & % x q x q `⃗ p 2 3 ´⃗ p 2 q2 ´⃗ p 2 q1 ´2x q x qQ 2 ⃗ p 2 q2 `xq x qQ 2 ¯´⃗ p 2 q1 `xq x qQ 2 ¯´x q x qQ 2 ⃗ p 2 3 ˆln ´xq x q e 2η ¯ln ˜`⃗ p 2 q2 `xq x qQ 2 ˘`⃗ p 2 q1 `xq x qQ 2 xq x qQ 2 ⃗ p 2 3 2x q x q ⃗ p 2 q1 `xq x qQ 2 ln ´xq e η ¯ln ˆ⃗ p 2 3 µ 2 ˙`pq Ø qq IF ¸+ `ūpp q q ´C5 2} `C6 2} ¯vpp qq , (4.3.24)
and

Φ 1i 2 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q " # ūpp q q ˆpi q1K p1 ´2x q q `1 2 rp { q1K , γ i K s ˙γ`v pp qq ˜´2 ⃗ p 2 q1 `xq x qQ 2 ln ´xq e η ¯ln ˆ⃗ p 2 3 µ 2 ln ´xq x q e 2η ¯» -1 ⃗ p 2 q1 ln ˜⃗ p 2 q1 `xq x qQ 2 x q x qQ 2 ¸´⃗ p 2 q2 `xq x qQ 2 ´⃗ p 2 q1 `xq x qQ 2 ¯´⃗ p 2 q2 `xq x qQ 2 ¯´x q x qQ 2 ⃗ p 2 3 ˆln ˜`⃗ p 2 q1 `xq x qQ 2 ˘`⃗ p 2 q2 `xq x qQ 2 xq x qQ 2 ⃗ p 2 3 ¸ff¸`p q Ø qq IF + `ūpp q q `C5i 2K `C6i 2K ˘vpp qq . (4.3.25)

Treatment of the UV divergences in the virtual amplitudes

In the shockwave approach, the only UV-divergences at NLO are associated with the dressing of external states e.g. quark self-energy. But the author showed in [START_REF] Boussarie | Perturbative study of selected exclusive QCD processes at high and moderate energies[END_REF] that by choosing ϵ U V " ϵ IR " ϵ, such diagrams for an external on-shell massless particle line gives 0. Therefore, they concluded that there is no need to renormalize coloured fields and that what would have been UV divergences in the virtual amplitudes if one kept ϵ U V ‰ ϵ IR are now indistinguishable from the IR divergences with the choice of one unique ϵ and cancel then with the IR divergences from the real amplitudes. They also stated that the β-function correction is unnecessary at this order and will appear only starting from next-to-next to leading order. The proof for the specific case of quark self-energy will be shown below. The quark selfenergy diagram with the loop gluon with momentum k and the quark line with momentum p line reads:

Σppq " ż d D k p2πq D pigt a nm q γ µ G 0 pp ´kq pigt a mn γ ν q G 0µν pkq " pigq 2 C F N c ż d D k p2πq D γ µ ipp { ´k {q pp ´kq 2 `iε γ ν ´i k 2 `iε " g µν ´kµ n 2ν `kν n 2µ k `θ `´k 0 `|k `|˘ȷ " pigq 2 N 2 c ´1 2
ż d D k p2πq D γ µ pp { ´k {qγ ν rpp ´kq 2 `iεs rk 2 `iεs " g µν ´kµ n 2ν `kν n 2µ k `θ `´k 0 `|k `|˘ȷ " pigq 2 N 2 c ´1 2
ż d D k p2πq D 1 rk 2 `iεs rpp ´kq 2 `iεs ˆ"γ µ pp { ´k {qγ µ ´k {pp { ´k {qγ ``γ `pp { ´k {qk { k `θ `´k 0 `|k `|˘* " pigq 2 N 2 c ´1 2
ż d D k p2πq D 1 rk 2 `iεs rpp ´kq 2 `iεs ˆ"p2 ´Dqpp { ´k {q ´k {pp { ´k {qγ ``γ `pp { ´k {qk { k `θ `´k 0 `|k `|˘* .
We introduced a cut-off k 0 to deal with the spurious light-cone gauge pole pk `q´1 . This can be written as After evaluation of the Dirac traces and using Feynman's parametrization eq. (4.3.7), one gets:

Σppq " Σ p p { `Σn n 2 { , (4.3
Σ p " 1 4p `Tr " γ `Σppq ‰ " pigq 2 N 2 c ´1 2 1 4p `ż d D k p2πq D 1 rk 2 `iεs rpp ´kq 2 `iεs ˆ"p2 ´Dq Tr " γ `pp { ´k {q ‰ ´Tr rγ `k {pp { ´k {qγ `s `Tr rγ `γ`p p { ´k {qk {s k `θ `´k 0 `|k `|˘* " pigq 2 N 2 c ´1 2p `p2 ´Dq ż d D k p2πq D n 2 ¨pp ´kq rk 2 `iεs rpp ´kq 2 `iεs " pigq 2 N 2 c ´1 2p `p2 ´Dq ż 1 0 dx ż d D k p2πq D n 2 ¨pp ´kq rpk ´xpq 2 `xp1 ´xqp 2 `iεs 2 " pigq 2 N 2 c ´1 2p `p2 ´Dq ż 1 0 dx ż d D k p2πq D n 2 ¨pp ´xpq rk 2 `xp1 ´xqp 2 `iεs 2 . (4.3.29)
Remembering that this computation is to be applied on an external on-sell massless quark with p 2 " 0, the integral becomes then a scaleless integral 9

ş d D k p2πq D 1
´k2 ´iε which in dimensional regularization is set to be zero as shown below.

The scaleless integral becomes after Wick's rotation to the Euclidean space:

ż d D k p2πq D 1 p´k 2 ´iεq " i ż d D k E p2πq D 1 k 4 E " i S D´1 p2πq D ż `8 0 d|k E ||k E | D´5
where S D´1 " 2π

D 2 Γp D 2 q
is the surface of the D-sphere. We introduce an arbitrary cut-off Λ to separate the IR and UV sectors. We also introduce ϵ U V ă 0 and ϵ IR ą 0 as needed for dimensional regularization D " 4 `2ϵ. Then:

ż `8 0 d|k E ||k E | D´5 " lim k IR Ñ0,k U V Ñ`8 "ż Λ k IR d|k E ||k E | D´5 `ż k U V Λ d|k 2 ||k E | D´5 ȷ " lim k IR Ñ0,k U V Ñ`8 " 1 2ϵ IR ´Λ2ϵ IR ´k2ϵ IR IR ¯`1 2ϵ U V ´k2ϵ U V U V ´Λ2ϵ U V ¯ȷ " Λ 2ϵ IR 2ϵ IR ´Λ2ϵ U V 2ϵ U V Ñ 1 2ϵ IR ´1 2ϵ U V .
By analytical continuation, one can drop the conditions ϵ U V ă 0 and ϵ IR ą 0. Writing ϵ U V " ϵ IR " ϵ, then the scaleless integral can be set to 0. This leads therefore to Σ p " 0 .

(4.3.30)

From eq. (4.3.27) and the above equation, we have:

Σ n " 1 4p `Tr rp {Σppqs " pigq 2 N 2 c ´1 2p `ż d D k p2πq D 1 rk 2 `
`|˘* " pigq 2 N 2 c ´1 2p `ż d D k p2πq D 1 rk 2 `iεs rpp ´kq 2 `iεs ˆ"p2 ´Dq `p2 ´p ¨k˘´1 k ``2p ¨kpp `´k `q ´2p ¨pp ´kqk ``2p `k ¨pp ´kq ˘θ `´k 0 `|k `|˘* " pigq 2 N 2 c ´1 2p `ż d D k p2πq D 1 rk 2 `iεs rpp ´kq 2 `iεs ˆ"p2 ´Dqpp 2 ´p ¨kq ´4p ¨kp `´2p 2 k `´2k 2 p k`θ `´k 0 `|k `|˘*
Using the scaleless integral argument (or that p 2 " 0 as we want the quark to be on-shell), one can cancel the terms without k `in the denominator. Thus,

Σ n " pigq 2 pN 2 c ´1q ż d D k p2πq D 1 rk 2 `iεs rpp ´kq 2 `iεs k 2 ´2p ¨k k `θ `´k 0 `|k `|" pigq 2 pN 2 c ´1q ż d D k p2πq D 1 rk 2 `iεs rpp ´kq 2 `iεs pp ´kq 2 ´p2 k `θ `´k 0 `|k `|˘.
Dropping the p 2 term due to wanting the on-shellness of the massless quark, then:

Σ n " pigq 2 N 2 c ´1 p2πq D ż d D k k `pk 2 `iεq θp´k 0 `|k `|q " ´pigq 2 N 2 c ´1 p2πq D ż dk k`θ p´k 0 `|k `|q ż dk ´ż d d ⃗ k 1 ´⃗ k 2 ´2k `k´´i ε " ´pigq 2 N 2 c ´1 p2πq D ż dk k`θ p´k 0 `|k `|q ż dk ´ż `8 0 dα ż d d ⃗ k 1 p´iq e ´iαr ⃗ k 2 ´2k `k´´i εs " ´pigq 2 N 2 c ´1 p2πq D ż dk k`θ p´k 0 `|k `|q ż dk ´ż `8 0 dα 1 p´iq e ´iαp´2k `k´´i εq ´π iα ¯d 2 " ´pigq 2 π d 2 N 2 c ´1 p2πq D ż dk k`θ p´k 0 `|k `|q ż dk ´ż `8 0 dα 1 p´iqi d 2 α p´d 2 `1q´1 e ´iαp´2k `k´´i εq " ´pigq 2 π d 2 N 2 c ´1 p2πq D ż dk k`θ p´k 0 `|k `|q ż dk ´1 p´iqi d 2 p´iq ´d 2 `1Γ `1 ´d 2 p´2k `k´´i εq 1´d 2 " g 2 N 2 c ´1 p2πq D π d 2 Γ ˆ1 ´d 2 ˙ż dk k`ż dk ´θ `´k 0 `|k `|˘`´2 k `k´˘d 2 ´1 " g 2 N 2 c ´1 p2πq D π d 2 Γ ˆ1 ´d 2 ˙#ż `8 k 0 dk k`ż `8 0 dk ´p´1q d 2 ´1 `2k `k´˘d 2 ´1 `ż `8 k 0 dk k`ż 0 ´8 dk ´`´2k `k´˘d 2 ´1 `ż ´k0 ´8 dk k`ż `8 0 dk ´p´2k `k´q d 2 ´1 `ż ´k0 ´8 dk k`ż 0 ´8p´1q d 2 ´1 `2k `k´˘d 2 ´1+ (4.3.31)
By performing the changes of variable such that all the integrals over pk `, k ´q P rk 0 , `8rˆr0, `8r

for every of the four terms, we have cancellation of them two by two:

Σ n " g 2 N 2 c ´1 p2πq D π d 2 Γ ˆ1 ´d 2 ˙#ż `8 k 0 dk k`ż `8 0 dk ´p´1q d 2 ´1 `2k `k´˘d 2 ´1 `ż `8 k 0 dk k`ż `8 0 dk ´`2k `k´˘d 2 ´1 `ż `8 k 0 dk k`ż `8 0 dk ´p2k `k´q d 2 ´1 ż `8 k 0 dk k`ż `8 0 p´1q d 2 ´1 `2k `k´˘d 2 ´1+ " 0 (4.3.32)
Therefore, the quark self-energy diagram applied to an external on-shell massless quark line gives 0 even in the non-covariant gauge:

Σppq " 0 (4.3.33)
4.4 γ p˚q Ñ q qg LO impact factors The colour projector this time to have the whole q qg system in a colour singlet state is b

2 N 2 c ´1 t a
ln with a, n, l the colour index of the gluon, quark and antiquark. The reduced matrix element is defined as, analogous to eq. (4.2.3),

M α " d 2 N 2 c ´1 ´iee q b 2p γ θpp g qθpp q qθpp q qθpp γ q 1 p2πq D´3 p´iqδpp q `pq `pg ´pγ q b 2p g b 2p q b 2p q r T α . (4.4.1)
They are four diagrams contributing to the reduced matrix element r T . Two of them, the ones where the gluon is emitted by the quark line are shown in fig. 4.4.1. The remaining two where the gluon is emitted by the antiquark line can be found by doing the substitution denoted pq Ø qq IF : p q Ø p q, p 1 Ø p 2 and the reversal of the order of the gamma matrices.

The reduced matrix element can be parameterized as:

r T α " ´gµ ´ϵN c ż d d ⃗ p 1 d d ⃗ p 2 " δ p⃗ p q1 `⃗ p q2 `⃗ p g q ˆN 2 c ´1 N c ˙r U 12 p⃗ p 1 , ⃗ p 2 q Φ α 3 p⃗ p 1 , ⃗ p 2 q (4.4.2) `Nc ż d d ⃗ p 3 p2πq d δp⃗ p q1 `⃗ p q2 `⃗ p g3 q " r U 13 `r U 32 ´r U 12 ´Č U 13 U 32 ı p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q Φ α 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q * . (4.4.3)
The `component of the dipole and double dipole impact factors have the following expressions

Φ 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q " p γ ūpp q q " 2x q g µν K `xg `γν K γ µ K ˘‰ γ `vpp qqε gKµ px g p q1Kν ´xq p g3Kν q x q x g px q `xg q ˆQ2 `⃗ p 2 q2 xqp1´xqq ˙ˆQ 2 `⃗ p 2 q1 xq `⃗ p 2 q2 xq `⃗ p 2 g3 xg ˙´pq Ø qq IF , (4.4

.4) and

Φ 3 p⃗ p 1 , ⃗ p 2 q " Φ 4 ´⃗ p 1 , ⃗ p 2 , ⃗ 0 ¯`» - - - ´xg p γ ūpp q qε { g pp { q `p { g qγ `vpp qq x q ´⃗ p g ´xg xq ⃗ p q ¯2 ˆQ2 `⃗ p 2 q2 xqp1´xqq ˙`pq Ø qq IF fi ffi ffi fl (4.4.5) " Φ 4 ´⃗ p 1 , ⃗ p 2 , ⃗ 0 ¯`r Φ 3 p⃗ p 1 , ⃗ p 2 q . (4.4.6)
The transverse component of the double dipole impact factor reads:

Φ i 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q " ε gKµ ūpp q qγ 2x q x qpx q `xg q ˆQ2 `⃗ p 2 q2 xqp1´xqq ˙ˆQ 2 `⃗ p 2 q1 xq `⃗ p 2 q2 xq `⃗ p 2 g3 xg "x q x qQ 2 `γµ K γ i K ˘`´p { q1K γ µ K γ i K p { q2K ¯`4 x q x g p { q2K ´xq p µ g3K ´xg p µ q2K 2 x q x g p µ g3K ´γi K p { q2K ¯`2x q p i q2K ´p { q2K γ µ K ¯´2x qp i q2K ´p { q1K γ µ K ¯ȷ vpp qq ´pq Ø qq IF ,
and the one for the dipole impact factor reads

Φ i 3 p⃗ p 1 , ⃗ p 2 q " Φ i 4 ´⃗ p 1 , ⃗ p 2 , ⃗ 0 ¯`» - - - ´xg ūpp q qε { g ´p { q `p { g ¯γ`´γi K p { q2K ´2x qp i q2K ¯vpp qq 2x q x qp1 ´xq q ´⃗ p g ´xg xq ⃗ p q ¯2 ˆQ2 `⃗ p 2 q2 xqp1´xqq ˙`pq Ø qq IF fi ffi ffi fl (4.4.7)
" Φ i 4.5 γ p˚q P Ñ q qP 1 cross-section at one-loop accuracy

Since the photon in the initial state can appear with different polarizations, the following density matrix is defined:

dσ JI " ˆdσ LL dσ LT dσ T L dσ T T ˙, dσ LT " dσ T L . (4.5.1)
J will represent the photon polarization in the complex conjugated amplitude, while I is the photon polarization for the amplitude.

The authors in [START_REF] Boussarie | On the one loop γ p˚q Ñ qq impact factor and the exclusive diffractive cross sections for the production of two or three jets[END_REF][START_REF] Boussarie | Perturbative study of selected exclusive QCD processes at high and moderate energies[END_REF] parameterized the matrix elements of Wilson-line operators between the in and out proton states as

B P 1 pp 0 1 q ˇˇˇT ˆTr " V ˆ⃗ z 2
˙V : ˆ´⃗ z 2 ˙ȷ ´Nc ˙ˇˇˇP pp 0 q F " 2πδpp 01 0 qF ⃗ p 0 1 ⃗
p 0 p⃗ z q " 2πδpp 01 0 qF p⃗ z q , (4.5.2) and

B P 1 pp 0 1 q ˇˇˇT ˆTr " V ˆ⃗ z 2 ˙V : p⃗ x q ȷ Tr " V p⃗ x qV : ˆ´⃗ z 2 ˙ȷ ´Nc Tr " V ˆ⃗ z 2
˙V : ˆ´⃗ z 2 ˙ȷ˙ˇˇˇˇP pp 0 q F " 2πδpp 01 0 q r F ⃗ p 0 1 ⃗ p 0 p⃗ z, ⃗ x q " 2πδpp 01 0 q r F p⃗ z, ⃗ x q . ( 4.5.3)
The dependence on ⃗ p 0 1 , ⃗ p 0 has been dropped for simplicity of notation, and they assumed the following proton state normalization @ P 1 pp 0 1 q |P pp 0 q D " p2πq d`1 δpp 01 0 qδp⃗ p 0 1 0 qδ s P s P 1 . (4.5.4)

Their Fourier transforms are ż d d ⃗ z e ´i⃗ z¨⃗ p F p⃗ z q " Fp⃗ p q (4.5.5)

ż d d ⃗ z d d ⃗
x e ´i⃗ z¨⃗ q e ´i⃗ p¨⃗ x r F p⃗ z, ⃗ x q " r Fp⃗ q, ⃗ p q . (4.5.6)

These above matrix elements appear when the Wilson-line operators present in eq. (4.2.4, 4.3.4, 4.4.3) i.e. the dipole and double dipole operators in momentum space, are inserted between the proton in and out state, and the overall transverse momentum conservation delta function is extracted.

Each element of the density matrix in eq. (4.5.1) for the γ p˚q P Ñ q qP 1 process at one-loop accuracy can be divided into different contributions

dσ JI " dσ 0JI `dσ 1JI `dσ 2JI , (4.5.7)
where dσ 0JI is the LO contribution, dσ 1JI is the NLO contribution involving two dipole operators and dσ 2JI is the NLO contribution involving a dipole operator and a double dipole operator.

LO contribution

The leading cross-section reads 3

dσ 0JI " α em Q 2 q 2 p2πq 4d N c 1 4pp γ q 2 1 2x q x q dx q dx q d d ⃗ p q d d ⃗ p q δ p1 ´xq ´xq q `εIβ ε Jγ ż d d ⃗ p 1 d d ⃗ p 2 d d ⃗ p 1 1 d d ⃗ p 2 1 δp⃗ p q1 `⃗ p q2 q δp⃗ p 11 1 `⃗ p 22 1 q ˆΦβ 0 p⃗ p 1 , ⃗ p 2 qΦ γ: 0 p⃗ p 1 1 , ⃗ p 2 1 q F ˆ⃗ p 12 2 ˙F˚ˆ⃗ p 1 1 2 1 2 ˙, (4.5.8)
where we introduce the shorthand notation by suppressing summation over helicities of partons

Φ β 0 p⃗ p 1 , ⃗ p 2 qΦ γ: 0 p⃗ p 1 1 , ⃗ p 2 1 q " ÿ λq,λq Φ β 0 p⃗ p 1 , ⃗ p 2 qΦ γ: 0 p⃗ p 1 1 , ⃗ p 2 1 q . (4.5.9)
From eqs. (4.2.6, 4.2.7) and summing over the quark and antiquark helicities, one gets

Φ 0 p⃗ p 1 , ⃗ p 2 qΦ `: 0 p⃗ p 1 1 , ⃗ p 2 1 q " 32pp γ q 4 x 3 q x 3 q ´⃗ p 2 q1 `xq x qQ 2 ¯´⃗ p 2 q1 1 `xq x qQ 2
¯, (4.5.10)

Φ 0 p⃗ p 1 , ⃗ p 2 qΦ i: 0 p⃗ p 1 1 , ⃗ p 2 1 q " 16pp γ q 3 x 2 q x 2 q p i q1K p1 ´2x q q ´⃗ p 2 q1 `xq x qQ 2 ¯´⃗ p 2 q1 1 `xq x qQ 2 ¯, (4.5.11)
Φ i 0 p⃗ p 1 , ⃗ p 2 qΦ k: 0 p⃗ p 1 1 , ⃗ p 2 1 q " 8pp γ q 2 x q x q " p1 ´2x q q 2 g ri K g lk K ´grk K g li K `grl K g rl K g ik K ‰ p q1Kr p q1 1 Kl ´⃗ p 2 q1 `xq x qQ 2 ¯´⃗ p 2 q1 1 `xq x qQ 2
¯. (4.5.12)

When computing the LL and T L elements for the different contributions of cross-section, one uses that

ε Lβ Φ β " ε L Φ ´`ε Ĺ Φ `" Q p γ Φ `, (4.5.13)
which comes from the gauge invariance relation eq. (4.2.8) and the form of the photon longitudinal polarization vector eq. (4.1.3).

Dipole ˆdipole NLO cross-section dσ 1JI

The dipole ˆdipole NLO cross section is given by

dσ 1JI " α s 2π Γ p1 ´ϵq p4πq ϵ ˆN 2 c ´1 2N c ˙αem Q 2 q 2 p2πq 4d N c 1 4pp γ q 2 1 2x q x q dx q dx q d d ⃗ p q d d ⃗ p q δ p1 ´xq ´xq q `εIβ ε Jγ 3All
the partonic cross-sections presented here and below have an added factor of 1 2p2πq 4 compared to those in [START_REF] Boussarie | On the one loop γ p˚q Ñ qq impact factor and the exclusive diffractive cross sections for the production of two or three jets[END_REF][START_REF] Boussarie | Perturbative study of selected exclusive QCD processes at high and moderate energies[END_REF]. This normalization factor is explained in [START_REF] Boussarie | Towards a complete next-to-logarithmic description of forward exclusive diffractive dijet electroproduction at HERA: real corrections[END_REF]. Γp1 ´ϵq

ˆż d d ⃗ p 1 d d ⃗ p 2 d d ⃗ p 1 1 d d ⃗ p 2 1 δ p⃗ p q1 `⃗ p q2 q δ p⃗ p 11 1 `⃗ p 22 1 q F ˆ⃗ p 12 2 ˙F˚ˆ⃗ p 1 1 2 1 2 "Φ β 1 p⃗ p 1 , ⃗ p 2 q Φ γ: 0 p⃗ p 1 1 , ⃗ p 2 1 q `Φβ 0 p⃗ p 1 , ⃗ p 2 q Φ γ: 1 p⃗ p 1 1 , ⃗ p 2 1 q ı . ( 4
p4πq ϵ C F ˆSV `SV 2 ˙dσ 0LL `αs Q 2 4π ˆN 2 c ´1 N c ˙αem Q 2 q 2 p2πq 8 N c dx q dx q d 2 ⃗ p q d 2 ⃗ p q δp1 ´xq ´xq q ˆż d 2 ⃗ p 1 d 2 ⃗ p 2 d 2 ⃗ p 1 1 d 2 ⃗ p 2 1 δp⃗ p q1 `⃗ p q2 q δp⃗ p 11 1 `⃗ p 22 1 q F ˆ⃗ p 12 2 ˙F˚ˆ⃗ p 1 1 2 1 2 # 1 ⃗ p 2 q1 1 `xq x qQ 2 « 6x 2 q x 2 q ⃗ p 2 q1 `xq x qQ 2 ln ˜x2 q x 2 q µ 4 Q 2 px q ⃗ p q ´xq ⃗ p q q 2 p⃗ p 2 q1 `xq x qQ 2 q 2 ż xq 0 dz ˜rpϕ 4 q LL s `ÿ n"5,6 rpϕ n q LL s `|⃗ p 3 " ⃗ 0 ¸`pq Ø qq ¸ff `h.c.| p 1 ,p 2 Øp 1 1 ,p 2 1 + . (4.5.15)
Here, pq Ø qq stands for

p q Ø p q, p 1 p1q Ø p 2 p1q , x q Ø x q .
The `prescription is defined in eq. (4.3.9). All the divergences are inside the contribution

S V `SV 2 " 1 ϵ « ´4ϵ lnpαq ln ˜x2 q x 2 q µ 2 px q ⃗
p q ´xq ⃗ p q q 2 ¸`4 lnpαq `4ϵ ln 2 pαq ´2 lnpx q x qq `3 `2ϵ ln ˜xq x qµ 2 px q ⃗ p q ´xq ⃗ p q q 2 ¸lnpx q x qq `ϵ ln 2 px q x qq ´3ϵ ln

˜xq x qµ 2 px q ⃗ p q ´xq ⃗ p q q 2 ¸´π 2 3 ϵ `6ϵ ff , (4.5.16)
where α is an infrared cut-off imposed on the longitudinal fraction of gluon momenta in order to regularize rapidity divergences. This is why for the terms not proportional to dσ 0JL , ϵ has been set to 0. The T L element reads Finally, the T T element reads

dσ 1T L " α s 2π Γp1 ´ϵq p4πq ϵ C F ˆSV `SV 2 ˙dσ 0T L `αs Q 4π ˆN 2 c ´1 N c ˙αem Q 2 q 2p2πq 8 N c dx q dx q d 2 ⃗ p q d 2 ⃗ p q δp1 ´xq ´xq q ε T i ˆż d 2 ⃗ p 1 d 2 ⃗ p 2 d 2 ⃗ p 1 1 d 2 ⃗ p 2 1 δp⃗ p q1 `⃗ p q2 q δp⃗ p 11 1 `⃗ p 22 1 q F ˆ⃗ p 12 2 ˙F˚ˆ⃗ p 1 1 2 1 2 ˆ$ ' ' ' ' ' & ' ' ' ' ' % «˜ż xq 0 dz ˜"pϕ 4 q i T L ‰ ``ÿ n"5,6 " pϕ n q i T L ‰ `|⃗ p 3 " ⃗ 0 ¸¸`pq Ø qq ff ⃗ p 2 q1 `xq x qQ 2 `3x q x qp1 ´2x q qp i q1 1 K p⃗ p 2 q1 `xq x qQ 2 qp⃗ p 2 q1 1 `xq x qQ 2 q ˆ˜ln ˜x3 q x 3 q µ 8 Q 2 px q ⃗ p q ´xq ⃗ p q q ´4 p⃗ p 2 q1 `xq x qQ 2 q 2 p⃗ p 2 q1 1 `xq x qQ 2 q ¸´x q x qQ 2 ⃗ p 2 q1 1 ln ˜xq x qQ 2 ⃗ p 2 q1 1 `xq x qQ 2 ¸« ˜ż xq 0 dz ˜"pϕ 4 q i LT ‰ ``ÿ n"5,6 " pϕ n q i LT ‰ `|⃗ p 3 " ⃗ 0 ¸¸`pq Ø qq ff 2x q x q ´⃗ p 2 q1 1 `xq x qQ 2
dσ 1T T " α s 2π Γp1 ´ϵq p4πq ϵ C F ˆSV `SV 2 ˙dσ 0T T `αs 4π ˆN 2 c ´1 N c ˙αem Q 2 q 2 p2πq 8 N c dx q dx q d 2 ⃗ p q d 2 ⃗ p q δp1 ´xq ´xq q pε T i ε T k q ˆż d 2 ⃗ p 1 d d ⃗ p 2 d 2 ⃗ p 1 1 d 2 ⃗ p 2 1 δp⃗ p q1 `⃗ p q2 q δp⃗ p 11 1 `⃗ p 22 1 q F ˆ⃗ p 12 2 ˙F˚ˆ⃗ p 1 1 2 1 2 # 3 2 p q1Kr p q1 1 Kl p⃗ p 2 q1 `xq x qQ 2 qp⃗ p 2 q1 1 `xq x qQ 2 q " p1 ´2x q q 2 g ri K g lk K ´grk K g li K `grl K g ik K ı ˆ«ln ˜xq x qµ 4
px q ⃗ p q ´xq ⃗ p q q 2 p⃗ p 2 q1 `xq x qQ 2 q ¸´x q x qQ 2 The expressions for the ϕ functions can all be found in appendix B.1.

⃗ p 2 q1 ln ˜xq x qQ 2 ⃗ p 2 q1 `xq x qQ 2 ¸ff `«˜ż xq 0 dz ˜"pϕ 4 q ik T T ı ``ÿ n"5,6 " pϕ n q ik T T ı `|⃗ p 3 " ⃗ 0 ¸¸`pq Ø qq ff x q x q ´⃗ p 2 q1 1 `xxQ 2 ¯`h.c.|p1,p2Øp

Dipole ˆdouble dipole NLO cross-section dσ 2JI

dσ 2JI has the following expression

dσ 2JI " α s Γ p1 ´ϵq p4πq 1`ϵ α em Q 2 q 2 p2πq 4d N c 1 4pp γ q 2 1 2x q x q dx q dx q d d ⃗ p q d d ⃗ p q δ p1 ´xq ´xq q `εIβ ε Jγ ˆż d d ⃗ p 1 d d ⃗ p 2 d d ⃗ p 1 1 d d ⃗ p 2 1 d d ⃗ p 3 d d ⃗ p 3 1 p2πq d δ p⃗ p q1 `⃗ p q2 ´⃗ p 3 q δ p⃗ p 11 1 `⃗ p 22 1 `⃗ p 33 1 q ˆ"Φ 1β 2 p⃗ p 1 , ⃗ p 2 , ⃗ p 3K q Φ γ: 0 p⃗ p 1 1 , ⃗ p 2 1 q r F ˆ⃗ p 12 2 , ⃗ p 3 ˙F˚ˆ⃗ p 1 1 2 1 2 ˙δ p⃗ p 3 1 q `Φβ 0 p⃗ p 1 , ⃗ p 2 q Φ 1γ: 2 p⃗ p 1 1 , ⃗ p 2 1 , ⃗ p 3 1 q F ˆ⃗ p 12 2 ˙r F ˚ˆ⃗ p 1 1 2 1 2 , ⃗ p 3 1 ˙δ p⃗ p 3 q ȷ . (4.5.19)
This contribution is constructed from eqs. (4.3.24, 4.3.25) and is therefore completely finite. One can then set ϵ " 0.

One finds for the LL element The TL element reads

dσ 2LL " α s Q 2 4π α em Q 2 q 2 p2πq 8 N c dx q dx q d 2 ⃗ p q d 2 ⃗ p q δp1 ´xq ´xq q ˆż d 2 ⃗ p 1 d 2 ⃗ p 2 d 2 ⃗ p 1 1 d 2 ⃗ p 2 1 ż d 2 ⃗ p 3 p2πq 2 δp⃗ p q1 `⃗ p q2 ´⃗ p 3 q δp⃗ p 11 1 `⃗ p 22 1 `⃗ p 3 q ˆ# 1 ⃗ p 2 q1 1 `xq x qQ 2 r F ˆ⃗ p 12 2 , ⃗ p 3 ˙F˚ˆ⃗ p 1 1 2 1 2 4x q x q # x q x qp⃗ p 2 3 ´⃗ p 2 q2 ´⃗ p 2 q1 ´2x q x qQ 2 q p⃗ p 2 q2 `xq x qQ 2 qp⃗ p 2 q1 `xq x qQ 2 q ´xq x qQ 2 ⃗ p 2 3 ˆln ´xq x q e 2η ¯ln ˜p⃗ p 2 q2 `xq x qQ 2 q `⃗ p 2 q1 `xq x qQ 2 xq x qQ 2 ⃗ p 2 3 2x q x q ⃗ p 2 q1 `
dσ 2T L " α s Q 4π α em Q 2 q 2 p2πq 8 N c dx q dx q d 2 ⃗ p q d 2 ⃗ p q δp1 ´xq ´xq q ż d 2 ⃗ p 1 d 2 ⃗ p 2 d 2 ⃗ p 1 1 d 2 ⃗ p 2 1 (4.5.21) ˆż d 2 ⃗ p 3 d 2 ⃗ p 3 1 p2πq 2 δp⃗ p q1 `⃗ p q2 ´⃗ p 3 q δp⃗ p 11 1 `⃗ p 22 1 `⃗ p 33 1 q ˆεT i « δp⃗ p 3 1 q ⃗ p 2 q1 1 `xq x qQ 2 r F ˆ⃗ p 12 2 , ⃗ p 3 ˙F˚ˆ⃗ p 1 1 2 1 2 2p1 ´2x q qp i q1 1 K # x q x qp⃗ p 2 3 ´⃗ p 2 q2 ´⃗ p 2 q1 ´2x q x qQ 2 q p⃗ p 2 q2 `xq x qQ 2 qp⃗ p 2 q1 `xq x qQ 2 q ´xq x qQ 2 ⃗ p 2 3 ln ´xq x q e 2η ln ˜p⃗ p 2 q2 `xq x qQ 2 q `⃗ p 2 q1 `xq x qQ 2 xq x qQ 2 ⃗ p 2 3 ¸115 ´˜2x q x q ⃗ p 2 q1 `xq x qQ 2 ln ´xq e η ¯ln ˆ⃗ p 2 3 µ 2 ˙`pq Ø qq ¸+ `1 2x q x q «˜ż xq 0 dz ÿ n"5,6 " pϕ n q i T L ‰ `¸`pq Ø qq ffδ p⃗ p 3 q ⃗ p 2 q1 `xq x qQ 2 F ˆ⃗ p 12 2 ˙r F ˚ˆ⃗ p 1 1 2 1 2 , ⃗ p 3 1 #2x q x qp1 ´2x q qp i q1 1 K ˜´2 ⃗ p 2 q1 1 `xq x qQ 2 ln ´xq e η ¯ln ˆ⃗ p 2 3 1 µ 2 ln ´xq x q e 2η ¯«´⃗ p 2 q2 1 `xq x qQ 2 p⃗ p 2 q1 1 `xq x qQ 2 qp⃗ p 2 q2 1 `xq x qQ 2 q ´xq x qQ 2 ⃗ p 2 3 1 ˆln ˜p⃗ p 2 q1 1 `xq x qQ 2 qp⃗ p 2 q2 1 `xq x qQ 2 q x q x qQ 2 ⃗ p 2 3 1 1 ⃗ p 2 q1 1 ln ˜⃗ p 2 q1 1 `xq x qQ 2 x q x qQ 2 ¸ff¸`p q Ø qq + `«˜ż xq 0 dz ÿ n"5,6 " pϕ n q i LT ‰ `¸`pq Ø qq ff ˚¸ff .
The T T reads

dσ 2T T " α s 4π α em Q 2 q 2 p2πq 8 N c dx q dx q d 2 ⃗ p q d 2 ⃗ p q δp1 ´xq ´xq q (4.5.22) ˆż d 2 ⃗ p 1 d 2 ⃗ p 2 d 2 ⃗ p 1 1 d 2 ⃗ p 2 1 ż d 2 ⃗ p 3 p2πq 2 δp⃗ p q1 `⃗ p q2 ´⃗ p 3 q δp⃗ p 11 1 `⃗ p 22 1 `⃗ p 3 q pε T i ε T j q ˆ«r F ˆ⃗ p 12 2 , ⃗ p 3 ˙F˚ˆ⃗ p 1 1 2 1 2 ˙1 ⃗ p 2 q1 1 `xq x qQ 2 ˆ˜# p q1 1 Kl p q1Kk " p1 ´2xq 2 g ki K g lj K ´gkj K g li K `gkl K g ij K ı ˜´2 ⃗ p 2 q1 `xq x qQ 2 ˆln ´xq e η ¯ln ˆ⃗ p 2 3 µ 2 ˙`ln ´xq x q e 2η ¯« 1 ⃗ p 2 q1 ln ˜⃗ p 2 q1 `xq x qQ 2 x q x qQ 2 p 2 q2 `xq x qQ 2 p⃗ p 2 q1 `xq x qQ 2 qp⃗ p 2 q2 `xq x qQ 2 q ´xq x qQ 2 ⃗ p 2 3 ˆln ˜p⃗ p 2 q1 `xq x qQ 2 qp⃗ p 2 q2 `xq x qQ 2 q x q x qQ 2 ⃗ p 2 3 ¸ff¸`p q Ø qq + `1 x q x q «˜ż xq 0 dz ÿ n"5,6 " pϕ n q ij T T ı `¸`pq Ø qq ff¸`h .c.|p1,p2Øp 1 1 ,p 2 1 iØj ȷ .
4.6 γ p˚q P Ñ q qgP 1 LO cross-section

Just like for the γ p˚q P Ñ q qP 1 cross-section, the element of the dendity matrix can be separated into different contributions depending on the Wilson-line operators

dσ JI " dσ 3JI `dσ 4JI `dσ 5JI (4.6.1)
where dσ 3JI is the contribution with two dipole operators, dσ 4JI contains a dipole operator and a double dipole operator and dσ 5JI has two dipole operators.

The dipole ˆdipole contribution reads

dσ 3JI " α s µ 2ϵ ˆN 2 c ´1 N c ˙αem Q 2 q 2p2πq 4d N c 1 4pp γ q 2 1 x q x q `εIα ε Jβ dx q dx q d d ⃗ p q d d ⃗ p q dx g d d ⃗ p g x g p2πq d δp1 ´xq ´xq ´xg q ˆż d d ⃗ p 1 d d ⃗ p 2 d d ⃗ p 1 1 d d ⃗ p 2 1 δp⃗ p q1 `⃗ p q2 `⃗ p g q δ p⃗ p 11 1 `⃗ p 22 1 q ˆΦα 3 p⃗ p 1 , ⃗ p 2 qΦ β: 3 p⃗ p 1 1 , ⃗ p 2 1 q F ˆ⃗ p 12 2 ˙F˚ˆ⃗ p 1 1 2 1 2 ˙, (4.6.2)
where we introduce the shorthand notation by suppressing summation over helicities of partons

Φ α 3 p⃗ p 1 , ⃗ p 2 qΦ β: 3 p⃗ p 1 1 , ⃗ p 2 1 q " ÿ λq,λg,λq Φ α 3 p⃗ p 1 , ⃗ p 2 qΦ β: 3 p⃗ p 1 1 , ⃗ p 2 1 q . ( 4.6.3)
The impact factor has the form Φ α 3 " Φ α 4 | ⃗ p 3 "0 `Φ α 3 , as seen in eqs. (4.4.6, 4.4.8). Therefore, its square has the form

Φ α 3 p⃗ p 1 , ⃗ p 2 qΦ β: 3 p⃗ p 1 1 , ⃗ p 2 1 q " r Φ α 3 p⃗ p 1 , ⃗ p 2 q r Φ β: 3 p⃗ p 1 1 , ⃗ p 2 1 q `´Φ α 4 ´⃗ p 1 , ⃗ p 2 , ⃗ 0 ¯r Φ β: 3 p⃗ p 1 1 , ⃗ p 2 1 q `r Φ α 3 p⃗ p 1 , ⃗ p 2 q Φ β: 4 ´⃗ p 1 1 , ⃗ p 2 1 , ⃗ 0 ¯Φ α 4 ´⃗ p 1 , ⃗ p 2 , ⃗ 0 ¯Φβ: 4 ´⃗ p 1 1 , ⃗ p 2 1 , ⃗ 0 ¯. (4.6.4)
Only the first term in the right-hand side of the equality contains divergences. The LL contribution reads

r Φ 3 p⃗ p 1 , ⃗ p 2 q r Φ `: 3 p⃗ p 1 1 , ⃗ p 2 1 q " 8x q x qpp γ q 4 `dx 2 g `4x q px q `xg q Q2 `⃗ p 2 q2 xqp1´xqq ˙ˆQ 2 `⃗ p 2 q2 1 xqp1´xqq ˙px q ⃗ p g ´xg ⃗ p q q 2 ´8x q x qpp γ q 4 `2x g ´dx 2 g `4x q x q˘p x q ⃗ p g ´xg ⃗ p q q ¨px q ⃗ p g ´xg ⃗ p qq ˆQ2 `⃗ p 2 q2 1 xqp1´xqq ˙ˆQ 2 `⃗ p 2 q1 xqp1´xqq ˙px q ⃗ p g ´xg ⃗ p q q 2 px q ⃗ p g ´xg ⃗ p qq 2 `8x q x qpp γ q 4 `dx 2 g `4x qpx q `xg q Q2 `⃗ p 2 q1 xqp1´xqq ˙ˆQ 2 `⃗ p 2 q1 1 xqp1´xqq ˙px q ⃗ p g ´xg ⃗ p qq 2 ´8x q x qpp γ q 4 `2x g ´dx 2 g `4x q x q˘p x q ⃗ p g ´xg ⃗ p q q ¨px q ⃗ p g ´xg ⃗ p qq ˆQ2 `⃗ p 2 q1 1 xqp1´xqq ˙ˆQ 2 `⃗ p 2 q2 xqp1´xqq ˙px q ⃗ p g ´xg ⃗ p q q 2 px q ⃗ p g ´xg ⃗ p qq 2 . (4.6.5)
The T L contribution is

r Φ 3 p⃗ p 1 , ⃗ p 2 q r Φ i: 3 p⃗ p 1 1 , ⃗ p 2 1 q " 4x q `pγ ˘3 px q `xg q ˆQ2 `⃗ p 2 q2 1 xqp1´xqq ˙ˆQ 2 `⃗ p 2 q1 xqp1´xqq
˙˜px q p gK ´xg p qK q µ px qp gK ´xg p qK q ν px q ⃗ p g ´xg ⃗ p q q 2 px q ⃗ p g ´xg ⃗ p qq 2 "x

g p4x q `xg d ´2q ´pµ q2 1 K g iν K ´pν q2 1 K g µi K ¯´p2x q ´1q p4x q x q `xg p2 ´xg dqq g µν K p i q2 1 K ı ´4x q `pγ ˘3 p2x q ´1q `x2 g d `4x q px q `xg q ˘pi q2 1 K px q `xg q ˆQ2 `⃗ p 2 q2 1 xqp1´xqq ˙ˆQ 2 `⃗ p 2 q2 xqp1´xqq ˙px q ⃗ p g ´xg ⃗ p q q 2
`pq Ø qq , (4.6.6)

and finally, the T T contribution reads

r Φ i 3 p⃗ p 1 , ⃗ p 2 q r Φ k: 3 p⃗ p 1 1 , ⃗ p 2 1 q " ´2 `pγ ˘2 px q `xg q px q `xg q ˆQ2 `⃗ p 2 q2 xqp1´xqq ˙ˆQ 2 `⃗ p 2 q1 1
xqp1´xqq ṗx q p gK ´xg p qK q µ px qp gK ´xg p qK q ν px q ⃗ p g ´xg ⃗ p q q 2 px q ⃗ p g ´xg ⃗ p qq 2 ¸!x g ppd ´4qq x g ´2q

" p ν q1 1 K ´pµ q2K g ik K `pk q2K g µi K ḡµν K ´`⃗ p q1 1 ¨⃗ p q2 ˘gik K `pi q1 1 K p k q2K ¯´g νk K p i q1 1 K p µ q2K ´gµi K g νk K `⃗ p q1 1 ¨⃗ p q2 ˘ı ´gµν K ˆ"p2x q ´1q p2x q ´1q p k q1 1 K p i q2K p4x q x q `xg p2 ´xg dqq `4x q x qpp⃗ p q1 1 ¨⃗ p q2 qg ik K `pi q1 1 K p k q2K q ı `´p µ q1 1 K p ν q2K g ik K ´pµ q1 1 K p k q2K g νi K ´pi q1 1 K p ν q2K g µk K ´gµk K g νi K p⃗ p q1 1 ¨⃗ p q2 q xg ppd ´4qx g `2q `xg p2x q ´1qpx g d `4x q ´2q ´gµk K p ν q1 1 K ´gνk K p µ q1 1 K ¯pi q2K `xg p2x q ´1qp k q1 1 K p4x q `xg d ´2q ´gνi K p µ q2K ´gµi K p ν q2K ¯) ´2x q pp γ q 2 px 2 g d `4x q px q `xg qq ´p⃗ p q2 ¨⃗ p q2 1 qg ik K ´p1 ´2x qq 2 p i q2K p k q2 1 K `pi q2 1 K p k q2K xq px q `xg q 2 ˆQ2 `⃗ p 2 q2 xqp1´xqq ˙ˆQ 2 `⃗ p 2 q2 1 xqp1´xqq ˙px q ⃗ p g ´xg ⃗ p q q 2
`pq Ø qq . (4.6.7)

The dipole ˆdouble dipole contribution reads

dσ 4JI " α s µ 2ϵ α em Q 2 q 2p2πq 4d N c 1 4pp γ q 2 1 x q x q pε Iα ε Jβ q dx q dx q d d ⃗ p q d d ⃗ p q dx g d d ⃗ p g x g p2πq d δp1 ´xq ´xq ´xg q ˆż d d ⃗ p 1 d d ⃗ p 2 d d ⃗ p 1 1 d d ⃗ p 2 1 d d ⃗ p 3 d d ⃗ p 3 1 p2πq d δp⃗ p q1 `⃗ p q2 `⃗ p g3 q δp⃗ p 11 1 `⃗ p 22 1 `⃗ p 33 1 q ˆ"Φ α 3 p⃗ p 1 , ⃗ p 2 qΦ β: 4 p⃗ p 1 1 , ⃗ p 2 1 , ⃗ p 3 1 q F ˆ⃗ p 12 2 ˙r F ˚ˆ⃗ p 1 1 2 1 2 , ⃗ p 3 1 ˙δp⃗ p 3 q `Φα 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 qΦ β: 3 p⃗ p 1 1 , ⃗ p 2 1 q r F ˆ⃗ p 12 2 , ⃗ p 3 ˙F˚ˆ⃗ p 1 1 2 1 2 ˙δp⃗ p 3 1 q ȷ . (4.6.8)
The double dipole ˆdouble dipole contribution is given by

dσ 5JI " α s µ 2ϵ α em Q 2 q 2p2πq 4d 1 4pp γ q 2 1 x q x q pε Iα ε Jβ q N 2 c ´1 dx q dx q d d ⃗ p q d d ⃗ p q dx g d d ⃗ p g x g p2πq d δp1 ´xq ´xq ´xg q ˆż d d ⃗ p 1 d d ⃗ p 2 d d ⃗ p 1 1 d d ⃗ p 2 1 d d ⃗ p 3 d d ⃗ p 3 1 p2πq 2d δp⃗ p q1 `⃗ p q2 `⃗ p g3 q δp⃗ p 11 1 `⃗ p 22 1 `⃗ p 33 1 q ˆΦα 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q Φ β: 4 p⃗ p 1 1 , ⃗ p 2 1 , ⃗ p 3 1 q r F ˆ⃗ p 12 2 , ⃗ p 3 ˙r F ˚ˆ⃗ p 1 1 2 1 2 , ⃗ p 3 1 ˙. (4.6.9)
The finite parts of the squared impact factors are presented in appendix B.2.

To summarize, we presented in this chapter the computations in the shockwave formalism of the γ p˚q Ñ q q diffractive impact factors and γ p˚q P Ñ q q diffractive cross-sections at one-loop accuracy and the γ p˚q Ñ q qg diffractive impact factors and diffractive γ p˚q P Ñ q qgP 1 at Born level. We ave explicited the cancellation of the rapidity divergences and how the UV divergences are treated in the NLO impact factor for γ p˚q Ñ q q. These results will be used in chapters 6 and 7 to compute the diffractive production of a pair of hadron or a single hadrons in γ p˚q P collisions. This chapter closes the discussion of the shockwave formalism. The next chapter will be about the second effective field theory used in this thesis to compute processes sensitive to gluon 119 Chapter 5

The Colour Glass condensate effective field theory

In this section, we discuss the Colour Glass condensate (CGC) semi-classical effective field theory (EFT) for hadrons in the saturation regime. This theory is equivalent to the shockwave formalism described before. Comprehensive reviews on the subject can be found in references [START_REF] Venugopalan | The Color glass condensate: A Summary of key ideas and recent developments[END_REF][START_REF] Mclerran | The Color glass condensate and small x physics: Four lectures[END_REF][START_REF] Gelis | The Color Glass Condensate[END_REF][START_REF] Gelis | Color Glass Condensate and Glasma[END_REF][START_REF] Iancu | The Color glass condensate and high-energy scattering in QCD[END_REF][START_REF] Iancu | The Color glass condensate: An Introduction[END_REF][START_REF] Albacete | Gluon saturation and initial conditions for relativistic heavy ion collisions[END_REF][START_REF] Weigert | Evolution at small x(bj): The Color glass condensate[END_REF] and in the book [START_REF] Kovchegov | Quantum Chromodynamics at High Energy[END_REF] that discuss various aspects of CGC and its varied application in detail. We aim to give the necessary notions to understand the work presented in chapter 8. We will be working in D " 4 when discussing and using CGC.

The classical picture of CGC

The basic idea of CGC is to separate the partonic content of a hadron moving ultra-relativistically with large momentum p t (hadron in an infinite momentum frame) into two types of degrees of freedom. This separation is based on the value of their longitudinal momentum k ´" xp t to an arbitrary cut-off Λ ´" x 0 p t . One can also define the associated momentum-space rapidity variable Y " ln `pt {Λ ´˘. The large-x partons, which can be considered as the valence partons in a first simplistic picture, are those with momentum k ´ą Λ ´while the small-x soft partons have k ´ă Λ ´and can be considered as the wee partons. The former are treated as static colour charges localized on the x `-axis and appearing as a thin sheet. In contrast, the latter are the dynamical degrees of freedom described by the usual gauge field A µ . The static and localized properties can be understood using the Heisenberg uncertainty principle eq. (1.1.8). The partons have a lifetime of the order of

∆x ´" 1 k `" 2xP ⃗ k 2 (5.1.1) which means that ∆x ẃ ! ∆x v , (5.1.2)
and so the fast partons are only slowly varying, their dynamics being essentially frozen. Similarly, their longitudinal extent is given by ∆x `" 1 k

´"

xP

´, (5.1.3) with G 0 p⃗ x ´⃗ z q the Green's function associated to the 2-dimensional Laplacian i.e. satisfying ⃗ ∇2 G 0 p⃗ x ´⃗ z q " δ 2 p⃗ x ´⃗ z q and

G 0 p⃗ x ´⃗ z q " ´ż d 2 ⃗ k p2πq 2 e i ⃗ k¨p⃗ x´⃗ zq ⃗ k 2 . (5.1.13)
It is, however, logarithmically divergent. One often used way to regularize it is to introduce an infrared regulator m " Λ QCD so that the Green's function has the form

Gp⃗ x ´⃗ z q " ´ż d 2 ⃗ k p2πq 2 e ´i⃗ k¨p⃗ x´⃗ zq ⃗ k 2 `m2 " ´1 2π K 0 pm|⃗ x ´⃗ z |q .
(5.1.14)

It then vanishes for distances greater than Λ ´1 QCD . From eqs. (5.1.10, 5.1.11), we observe that specifying ρ is equivalent to specifying A µ . The probability of a given configuration of ρ is given by the gauge invariant functional W Λ ´rρs. This statistical weight functional encodes all the spatial correlations of the colour charge density at scale Λ

´.

As explained above, the large-x partons are frozen in time. This implies that ρ is frozen at a specific configuration for a given collision. However, it varies on an event-by-event basis. Thus, to obtain physical observables, one needs to average over all the possible configurations via the CGC average, whose expression is

xOy Y " ż rDρs W Y rρs O rρs , (5.1.15)
where O rρs is the observable of interest computed for a particular configuration ρ or equivalently for a given classical target field. In physical processes, this amounts to using the CGC effective Feynman rules listed below on top of the usual QCD Feynman rules. In writing the above average, we assume that the weight functional is normalized to one:

ż rDρs W rρs " 1 . (5.1.16)
The name "colour glass condensate" can now be easily understood from the above discussion:

• colour: the theory deals with gluons which carry colour charges.

• glass 2 : the ρ colour charge density are stochastic variables, and the fast and slow gluons have well-separated time scales, with the fast gluons being frozen in a given configuration at a given event. high occupation number 1{α s property.

We finish this section with some discussion of the differences between CGC and shockwave formalism. The separation of partons into fast and slow gluons is done on the target side, not on the projectile side. Thus, a physically well-motivated picture of the target and where the classical field comes from is needed from the start. In the shockwave formalism, the classical gauge field eq. (3.2.4) is exactly located at x `" 0, which would also be the case also for the classical gauge field in CGC if the colour current has the form eq. (5.1.6) i.e. the target is Lorentz-contracted to an infinitely thin sheet. The gauge field in CGC also has an explicit expression in terms of the colour charge density in A `" 0 gauge. One key difference is in the computation of scattering amplitudes. In the shockwave formalism, the scattering amplitudes are first written in full D-dimensional coordinates space. After putting the expressions of the Feynman rules, the vertex coordinates are fully integrated out. One gets at the end the expression of the amplitude in pp `, ⃗ p q space. Matrix elements of operators of Wilson lines are written as functions of the transverse momentum transfer to the parton lines by the background shockwave field. In CGC, things are reversed: one starts with the scattering amplitude in full momentum space and then uses the CGC effective vertices (written in a slightly different form than in the shockwave formalism) and integrates out the momentum such as loop momentum to get to scattering amplitude written in mixed space pp `, ⃗ x q. Correlators of operators of Wilson line are functions of the transverse coordinates of the partons the target interact with. The amplitudes in the two formalisms for the same process should be related by non-trivial Fourier transform. This can be explicitly seen comparing computations in chapters 4 and 8.

CGC Feynman rules

In high-energy QCD scattering, the fundamental degrees of freedom are the light-like Wilson lines that resum the multiple interactions of particles with large p `with the background classical field of the target, as already seen in the previous chapters with the shockwave formalism. The CGC effective Feynman rules are in the mixed-space representation pp `, ⃗ x q. They are also within the eikonal approximation: the multiple scatterings only colour rotate the parton lines while conserving the longitudinal momentum p `, the transverse coordinate ⃗ x, and any additional quantum numbers like polarization and helicities. The effect of colour rotation is encoded in the light-like Wilson lines in the fundamental and adjoint representations V p⃗ x q " P `exp ˆig ż dz `A´a pz `, ⃗ x qt a ˙(5.1.17)

U p⃗ x q " P `exp ˆig ż dz `A´a pz `, ⃗ x qT a ˙, (5.1.18) analogously to the Wilson lines in the shockwave formalism and have the same properties found at the end of section 3.3.2.

The CGC effective vertices are shown below where we label below σ, σ 1 the spinor indices and i, j, a, b the colour indices in the SU pN c q fundamental and adjoint representations, respectively.

CGC vertex for quark [START_REF] Mclerran | Fock space distributions, structure functions, higher twists and small x[END_REF]: CGC vertex for the antiquark: CGC vertex for the gluon3 [START_REF] Ayala | The Gluon propagator in nonAbelian Weizsacker-Williams fields[END_REF][START_REF] Gelis | Gluon propagation inside a high-energy nucleus[END_REF]:

T q σσ 1 ,ij pl, l 1 q " 2πδpl `´l 1`q γ σσ 1 ż d d ⃗ z e ´ip ⃗ l´⃗ l 1 q¨⃗ z V ij p⃗ z q (5.1.19) ÝÑ ÝÑ l 1 l σ, i σ 1 , j
T q σσ 1 ,ij pl, l 1 q " ´2πδpl 1`´l`q γ σσ 1 ż d d ⃗ z e ´ip ⃗ l 1 ´⃗ lq¨⃗ z V : ij p⃗ z q (5.1.20) ÝÑ ÝÑ l l 1 σ 1 , j σ, i
T g µν,ab pl, l 1 q " ´p2πqδpl `´l 1`q p2l `qg µν sgnpl `q ż d d ⃗ z e ´ip ⃗ l´⃗ l 1 q¨⃗ z U sgnpl `q ab p⃗ z q (5.1.21) ÝÑ ÝÑ l 1 l µ, a ν, b

The McLerran-Venugopalan (MV) model

We have seen above how the classical gauge field of the target is related to its colour charge density. This is a non-perturbative input and needs to be modelled. One of the most commonly used models is the McLerran-Venugopalan (MV) model [START_REF] Mclerran | Computing quark and gluon distribution functions for very large nuclei[END_REF][START_REF] Mclerran | Gluon distribution functions for very large nuclei at small transverse momentum[END_REF][START_REF] Mclerran | Green's functions in the color field of a large nucleus[END_REF]. The MV model gave the basic ideas from which CGC was then formulated. It is not the same thing as CGC, but a model one makes within the CGC classical field picture.

The model applies to a very large nucleus with nuclear mass number A " 1 and small x 0 but not asymptotically small to have 10 ´2 À x w À 10 ´1, i.e. x 0 should be much larger than the Bjorken x of the external probe that the nucleus is interacting with [START_REF] Taels | Quantum chromodynamics at small Bjorken-x[END_REF].

Let us explain the physical arguments for such a model [START_REF] Lappi | High energy scattering in QCD[END_REF][START_REF] Jeon | Random walks of partons in SU(N(c)) and classical representations of color charges in QCD at small x[END_REF]: Comparing the wee parton longitudinal extension given by eq. ( 5.1.3) with the longitudinal size of the nucleus of mass m A , which is Lorentz contracted into a thin pancake in the IMF with a radius size

2R A γ with γ " p t m A , (5.1.22)
Then for x w ! A ´1{3 , the wee partons resolve the whole nucleus in the longitudinal direction. Those wee partons have, however, | ⃗ k | " Λ QCD , which is a necessary condition for them to be dynamical degrees of freedom. So in the transverse plane, they have a resolution of ∆⃗ x " 1

| ⃗ k | ! Λ ´1
QCD " R p and resolve then only part of the transverse sections of the nucleons instead of the nucleons as a whole and so are sensitive to some colour charges. These charges considered as random variables are uncorrelated in the longitudinal direction since they are confined to different nucleons and do not know about each other. The total colour charge at fixed transverse coordinates comes from a sum of colour charges in only partially overlapping colour charges in the different nucleons. Therefore, the total colour charges at two different transverse coordinates are different. The colour charges at different transverse coordinates can then be assumed to be uncorrelated. Also, at fixed transverse coordinates, one has a large number of colour charges of the order of A 1{3 . Thus, the colour charge density ρ a at specific coordinates arises from the sum of many uncorrelated colour charges. Applying the central limit theorem, this is then a stochastic Gaussian variable local in x `and ⃗ x. This means that

W Y 0 rρs " N exp " ´ż dx `ż d 2 ⃗ x ρ a px `, ⃗ x q ρ a px `, ⃗ x q 2µ 2 px `q * , (5.1.23)
where N a normalisation constant such that the weight function is normalised to 1 and µ 2 px `q is a number density of hard colour charges per unit of volume in dx `d2 ⃗ x and per unit of colour.

It should be thought of as being very narrowly peaked in x `. Its precise form does not really matter, as physics in the high energy limit will only depend on its integral over x `.

The Gaussian property implies 4 @ ρ a `x`, ⃗ x ˘D " 0 , (5.1.25) A ρ a `x`, ⃗ x ˘ρb `y`, ⃗ y ˘E " g 2 δ ab µ 2 `x`˘δ px `´y `qδp⃗ x ´⃗ y q .

(5.1.26)

The first equation ensures that the nucleus is a colour-neutral object on average. With this Gaussian assumption, all odd-point functions of the colour charge density vanish, and all evenpoint functions can be expressed in terms of the above two-point function through Wick's theorem.

In the above two-point correlator, no dependence on the target transverse profile function is present, so it is implicitly assumed that the colour charge density has infinite transverse extension. One simple way to add some transverse coordinate dependence is with a target profile function that is completely factorized from µ 2 px `q as it has been done, for example, in [START_REF] Mäntysaari | Confronting impact parameter dependent JIMWLK evolution with HERA data[END_REF] or add directly some transverse dependence in the number density µ 2 px `q Ñ µ 2 px `, ⃗ x q.

The properties of the colour charge correlators coming from the Gaussian weight function make some average operators doable analytically, such as the dipole correlator

S p2q Y p⃗ x, ⃗ y q " 1 N c @ Tr " V p⃗ x qV : p⃗ y q ‰D Y , (5.1.27)
which is associated with scattering a colour dipole or quark-antiquark pair with the target, the dipole amplitude being defined as N Y p⃗ x, ⃗ y q " 1 ´Sp2q Y p⃗ x, ⃗ y q .

(5.1.28) Indeed, within the Gaussian approximation, the average of two Wilson lines reads [START_REF] Gelis | Probing colored glass via q anti-q photoproduction[END_REF][START_REF] Blaizot | High-energy pA collisions in the color glass condensate approach. 2. Quark production[END_REF]]

@ V p⃗ x qV : p⃗ y q D Y 0 " exp " ´g4 2 t a t a "ż dz `µ2 pz `qȷ ż d 2 ⃗ z rG 0 p⃗ x ´⃗ z q ´G0 p⃗ y ´⃗ z qs 2 * . (5.1.29)
It is translational invariant in the transverse plane

@ V p⃗ x q V : p⃗ y q D Y 0 " exp " ´g4 2 t a t a "ż dz `µ2 pz `qȷ ż d 2 ⃗ z rG 0 p⃗ z q ´G0 p⃗ z ´⃗ r qs 2 * 4
Different conventions exist for where to put g. We follow the conventions in [START_REF] Lappi | High energy scattering in QCD[END_REF][START_REF] Lappi | Wilson line correlator in the MV model: Relating the glasma to deep inelastic scattering[END_REF] where g is explicitly written for the two-point correlator of ρ and µ 2 is interpreted as a number density of colour charges. If g is implicitly in µ 2 px `q, then it is interpreted as the density of colour charged squared and

µ 2 A " ż dx `µ2 px `q " g 2 A 2πR 2 A , (5.1.24)
is the average colour charge squared per unit of transverse area [START_REF] Venugopalan | The Color glass condensate: A Summary of key ideas and recent developments[END_REF] and per colour (or the typical fluctuation of the colour charge density).

"

A V p ⃗ 0 qV : p⃗ y ´⃗ x q E Y 0
with ⃗ r " ⃗ x ´⃗ y. The translational invariance comes from the assumption that the colour charge density has infinite transverse extension.

Introducing the notation Bp⃗ r q " ż d 2 ⃗ z rG 0 p⃗ z q ´G0 p⃗ z ´⃗ r qs 2 , (5.1.30)

and using the Fourier transform representation of G 0 eq. ( 5.1.13), we have

Bp⃗ r q " ż d 2 ⃗ z « ż d 2 ⃗ k p2πq 2 ˜e´i ⃗ k¨⃗ z ⃗ k 2 ´e´i ⃗ k¨p⃗ z´⃗ r q ⃗ k 2 ¸ff2 " ż d 2 ⃗ z ż d 2 ⃗ k p2πq 2 d 2 ⃗ p p2πq 2 1 ⃗ k 2 1 ⃗ p 2 ´e´i ⃗ k¨⃗ z ´e´i ⃗ k¨p⃗ z´⃗ r q ¯´e ´i⃗ p¨⃗ z ´e´i⃗ p¨p⃗ z´⃗ r q " ż d 2 ⃗ k p2πq 2 d 2 ⃗ p p2πq 2 1 ⃗ k 2 1 ⃗ p 2 p2πq 2 δ 2 p ⃗ k `⃗ pq " 1 ´ei⃗ p¨⃗ r ´ei ⃗ k¨⃗ z `eip ⃗ k`⃗ p q¨⃗ r ı " ż d 2 ⃗ k p2πq 2 1 ⃗ k 4 " 2 ´e´i ⃗ k¨⃗ r ´ei ⃗ k¨⃗ r ı " 1 π ż `8 0 d| ⃗ k | 1 ´J0 pk|⃗ r |q | ⃗ k | 3
Adding an infra-red cut-off Λ of order Λ QCD which corresponds to another way to regularize compared to eq. ( 5.1.14), one can approximate Bp⃗ r q by the finite expression

Bp⃗ r q « 1 4π ⃗ r 2 ln ˆ1 |⃗ r |Λ ˙. (5.1.31)
This approximation is valid for |⃗ r | ! Λ ´1 " Λ ´1 QCD . The different ways to regularize the IR divergence lead to different values for the remaining finite term in Bp⃗ r q. The leading log is the universal, though. The dipole correlator eq. (5.1.27) in the MV model is then

S p2q Y 0 p⃗ x, ⃗ y q " 1 N c @ Tr " V p⃗ x qV : p⃗ y q ‰D Y 0 " exp " ´g4 8π C F "ż dz `µ2 pz `qȷ ⃗ r 2 ln ˆ1 |⃗ r |Λ ˙* (5.1.32)
A model-independent definition of the saturation scale often used is that S p2q Y p⃗ x, ⃗ y q " e ´c (5. 1.33) with c a constant often chosen to be 1/2 and r 2 s " 1{Q 2 s . This gives an implicit equation of the saturation scale. Here, we have then

Q 2 s 9 g 4 4π C F "ż dz `µ2 pz `qȷ , (5.1.34)
The exact expression of Q s depends on the precise convention chosen to determine it from the dipole correlator. The saturation scale should have Casimir scaling, i.e., be proportional to the relevant Casimir. For a dipole correlator with the Wilson lines in the adjoint representation, one would have

C A " N c instead of C F .
The MV model dipole correlator can be rewritten as

S p2q Y 0 p⃗ x, ⃗ y q " exp " ´Q2 s ⃗ r 2 4 ln ˆ1 ⃗ r 2 Λ 2 ˙* . (5.1.35)
It has the following asymptotic behaviours:

S p2q Y 0 p⃗ x, ⃗ y q Ñ 1 for |⃗ r | ! 1{Q s (5.1.36) S p2q Y 0 p⃗ x, ⃗ y q Ñ 0 for |⃗ r | " 1{Q s .
(5.1.37)

Expanding eq. ( 5. 1.35) in the small |⃗ r | region, the dipole amplitude in the MV model behaves like N Y 0 p⃗ x, ⃗ y q " 1 ´Sp2q Y 0 p⃗ x, ⃗ y q » ⃗ r 2 Ñ 0 for small |⃗ r | and one has the wanted colour transparency behaviour. For large dipoles, the dipole amplitude goes to 1, i.e. to the black disk limit, which is important for the unitarity of the S-matrix. Thus, saturation tames the growth of the dipole amplitude to the black disk limit.

Other correlators found using the Gaussian approximation are the fount point correlator [START_REF] Blaizot | High-energy pA collisions in the color glass condensate approach. 2. Quark production[END_REF] and the six-point one [START_REF] Dominguez | Universality of multiparticle production in QCD at high energies[END_REF].

The MV model is often used as the non-perturbative initial condition for the evolution equation mentioned below. It is also used for direct phenomenological studies for collisions where the values of x probed are not too small to require evolution.

Evolution equation

The above discussion treats of the classical picture of the CGC EFT that will be used in chapter 8. For the sake of completeness, we will briefly discuss the quantum corrections to this model that must be considered when going to smaller x 0 .

At one step of the evolution, when the cut-off decreases from Λ ´to Λ 1´5 , classical gluons with momentum in the strip Λ ´" k ´" Λ 1´a re now considered as semi-fast gluons, are integrated out and are absorbed into the new classical colour charge density ρ 1 " ρ `δρ generated by the now large-x partons with k ´" Λ 1´. These stochastic fluctuations ultimately lead to the renormalization of the weight functional W Λ ´rρs Ñ W Λ 1´rρ 1 s defined at the new scale Λ 1´. After this procedure, one has again the above classical picture where the static colour sources generate the classical gluon fields with momentum Λ 1´" k ´with the classical Yang-Mills equation. One also needs to average over all configurations to get physical observables with the only change in the expression of the CGC average being W Y Ñ W Y 1 with Y 1 " lnpp t {Λ 1´q . The non-linear evolution equation that governs the evolution of W Λ with the cut-off is the Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov and Kovner (JIMWLK) equation [START_REF] Jalilian-Marian | The BFKL equation from the Wilson renormalization group[END_REF][START_REF] Jalilian-Marian | The Wilson renormalization group for low x physics: Gluon evolution at finite parton density[END_REF][START_REF] Jalilian-Marian | The Wilson renormalization group for low x physics: Towards the high density regime[END_REF][START_REF] Jalilian-Marian | Unitarization of gluon distribution in the doubly logarithmic regime at high density[END_REF][START_REF] Kovner | Relating different approaches to nonlinear QCD evolution at finite gluon density[END_REF][START_REF] Weigert | Unitarity at small Bjorken x[END_REF][START_REF] Iancu | Nonlinear gluon evolution in the color glass condensate. 1[END_REF][START_REF] Ferreiro | Nonlinear gluon evolution in the color glass condensate. 2[END_REF][START_REF] Iancu | The Renormalization group equation for the color glass condensate[END_REF]. A sketch of its derivation can be found in [START_REF] Gelis | The Color Glass Condensate[END_REF][START_REF] Taels | Quantum chromodynamics at small Bjorken-x[END_REF]. As a renormalization group equation, it ensures the independence of physical observables to the arbitrary cut-off Λ ´that separates the gluons into two kinds of degrees of freedom. It has the form

BW Λ ´rρs B ln pΛ ´q " ´H " ρ, δ δρ ȷ W Λ ´rρs . (5.2.1)
It is often expressed in terms of the momentum-space rapidity variable Y " ln `pt {Λ ´˘and in terms of the solution of the Yang-Mills equations A ´a " α a [START_REF] Kovchegov | Quantum Chromodynamics at High Energy[END_REF][START_REF] Taels | Quantum chromodynamics at small Bjorken-x[END_REF]:

BW Y rαs BY " 1 2
ż d 2 ⃗ x d 2 ⃗ y δ 2 δα a Y p⃗ x qδα b Y p⃗ y q W Y rαsχ ab p⃗ x, ⃗ y q ´ż d 2 ⃗ x δ δα a Y p⃗ x q W Y rαsσ a p⃗ x q (5.2.2)
where the induced colour charge σ a p⃗ x q and the charge-charge correlator χ ab p⃗ x, ⃗ y q is

σ a p⃗ xq " i g 2π ż d 2 ⃗ z p2πq 2 1 p⃗ x ´⃗ z q 2
Tr " T a U : p⃗ x qU p⃗ z q ‰ (5.2.3)

χ ab p⃗ x, ⃗ y q " 1 π ż d 2 ⃗ z p2πq 2
p⃗ x ´⃗ z q ¨p⃗ y ´⃗ z q p⃗ x ´⃗ z q 2 p⃗ y ´⃗ z q 2 `1 `U : p⃗ x qU p⃗ y q ´U : p⃗ x qU p⃗ z q ´U : p⃗ z qU p⃗ y q ˘ab .

(5.2.4)

It can be rewritten into a simplified form using

1 2 ż d 2 ⃗ y δχ ab p⃗ x, ⃗ y q δα b Y p⃗ y q " σ a p⃗ x q , (5.2.5)
leading to

BW Y rαs BY " ´HJIMWLK W Y rαs H JIMWLK " ´1 2
ż d 2 ⃗ x d 2 ⃗ y δ δα a Y p⃗ x q χ ab p⃗ x, ⃗ y q δ δα b Y p⃗ y q , (5.2.6)
where H JIMWLK is the JIMWLK Hamiltonian. From eq. (5.2.6), one can gets an evolution equation for a generic observable O, starting from eq. (5.1.15) and taking its derivative with respect to Y :

B BY xOy Y " ż Dα BW Y rαs BY O rαs " ż Dα 1 2 ż d 2 ⃗ x d 2 ⃗ y δ δα a Y p⃗ x q ˆχab p⃗ x, ⃗ y q δ δα b Y p⃗ y q W Y rαs ˙O rαs " ´ż Dα 1 2 ż d 2 ⃗ x d 2 ⃗ y χ ab p⃗ x, ⃗ y q ˆδ δα b Y p⃗ y q W Y rαs ˙δ δα a Y p⃗ x q O rαs " ż Dα ż d 2 ⃗ x d 2 ⃗ y W Y rαs 1 2 δ δα b Y p⃗ y q χ ab p⃗ x, ⃗ y q δ δα a Y p⃗ x q O rαs " ´xH JIMWLK Oy Y . ( 5

.2.7)

To have the last equation, the following result is used χ ab p⃗ x, ⃗ y q " χ ba p⃗ y, ⃗ x q , (5.2.8)

which follows immediately from its definition eq. (5.2.4) and eq. (3.3.35). Substituting in eq. (5.2.7)

O " 1 ´1 N c Tr " V p⃗ x qV : p⃗ y q ‰ (5.2.9)
and doing a considerable amount of algebra using eqs. Diffractive production of a pair of hadrons with large p T in γ p˚q `p{A collisions

In this chapter which is based on the work we realized in [START_REF] Fucilla | NLO computation of diffractive di-hadron production in a saturation framework[END_REF], we present the full NLO computation of diffractve di-hadron production in the high-energy limit.

6.1 Theoretical framework

Hybrid collinear/high-energy factorization

The process of interest is:

γ p˚q pp γ q `P pp 0 q Ñ h 1 pp h 1 q `h2 pp h 2 q `X `P 1 pp 0 1 q (6.1.1)

where P is a nucleon or a nucleus target, generically called proton in the following. The initial photon plays the role of a probe (also named projectile). Our computation applies both to the photoproduction case (including ultraperipheral collisions) and to the electroproduction case (e.g. at EIC). A gap in rapidity is assumed between the outgoing nucleon/nucleus and the diffractive system pXh 1 h 2 q. This is illustrated by fig. 6.1.1.

We will be working in a combination of collinear factorization and small-x factorization, more precisely in the shockwave formalism presented in chapter 3. The reference frame and kinematics for the photon and outgoing on-shell partons in the hard part (before collinear factorization) are the same as in section 4.1. We will write the momentum of the hadrons produced on the projectile side as

p µ h i " x h i p γ n µ 1 `m2 h i `⃗ p 2 h i 2x h i p γ n µ 2 `pµ h i K pi " 1, 2q ; (6.1.2)
where x h i is longitudinal momentum fraction of the hadrons with respect to the photon one. The kinematical region considered here is such that ⃗

p 2 h 1 " ⃗ p 2 h 2 " Λ 2 QCD .
The hadron momenta are the hard scale, making the use of perturbative QCD and collinear factorization possible. We

h 1 h 2 γ ( * ) P | rapidity gap | P X 1 X 2 h 1 h 2 γ ( * ) P | rapidity gap | P ′ X 1 X 2 Figure 6.
1.1: Left: Amplitude of the process (6.1.1) at LO. Right: An example of amplitude contributing to the process (6.1.1) at NLO. The grey blob symbolizes the QCD shockwave. The double line symbolizes the target, which remains intact in the figure, but could just as well break. The quark and antiquark fragment into the systems ph 1 X 1 q and ph 2 X 2 q. The two tagged hadrons h 1 and h 2 are drawn in red and blue. also want lnpQ 2 {µ 2 F q, lnp⃗ p 2 h {µ 2 F q ! lnp1{x Bj q with µ F the fragmentation-function factorization scale to not have dominance of DGLAP effects.

The constraint ⃗ p 2 " ⃗ p 2 h 1,2 , with ⃗ p, the relative transverse momentum of the two hadrons has also been considered. This means that the two produced hadrons have a large enough separation angle (or in other words a large enough invariant mass) so that it will not be necessary to consider the di-hadron unpolarized fragmentation functions: each hadron, typically pion, can be produced by two well-separated fragmentation cascades. The hard cross-section has been computed using the shockwave formalisms and is presented in section 4.5 and section 4.6.

LO order

QCD collinear factorization stipulates that the total cross-section, at leading-twist, reads, see ref [START_REF] Collins | Foundations of perturbative QCD[END_REF] (chapter 12) and ref. [START_REF] Altarelli | Processes Involving Fragmentation Functions Beyond the Leading Order in QCD[END_REF],

dσ q qÑh 1 h 2 JI dx h 1 dx h 2 " ÿ q ż 1 x h 1 dx q x q ż 1 x h 2 dx q x q D h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙dσ JI dx q dx q `ph 1 Ø h 2 q , (6.1.3)
where q specifies the quark flavor types (q " u, d, s, c, b), and J, I " L, T specify the photon polarization for the complex conjugated amplitude and for the amplitude respectively as defined by the density matrix eq. (4.5.1). Here D h qpqq denotes the quark (antiquark) Fragmentation Function (FF) and dσ is the partonic cross-section, i.e. the cross-section for the subprocess γ p˚q pp γ q `P pp 0 q Ñ qpp q q `qpp qq `P 1 pp 0 1 q . (6.1.4)

The graphical convention used in the present chapter for any fragmentation function is given in fig. 6.1.2. All detailed computations will be done considering only the first term in eq. ( 6.1.3), remembering the second term is just simply obtained by doing h 1 Ø h 2 .

The partonic cross-section for (6.1.4) has been computed in the shockwave framework and can be found in section 4.5. The structure of the result for the whole process (6.1.1) at LO is illustrated in fig. 6.1.3.

Collinear factorization means that the produced hadrons should fly collinearly to the fragmenting partons. This means here that the following constraints should be fulfilled

p q " x q x h 1 p h1 , ⃗ p q " x q x h 1 ⃗ p h 1 , (6.1.5)
p q " x q x h 2 p h2 , ⃗ p q " x q x h 2 ⃗ p h 2 . (6.1.6)
Using eq. ( 6.1.3), the Born order hard cross-section of eq. (4.5.8) as well as the explicit expressions of the product Φ β 0 Φ γ: 0 in eqs. (4.5.10-4.5.12), the LO cross-sections are obtained and read for LL:

dσ q qÑh 1 h 2 0LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 " 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1 x h 2 dx q x q x q ˆxq x h 1 ˙d ˆxq x h 2 ˙d ˆδp1 ´xq ´xq qD h 1 q ˆxh 1 x q ˙Dh 2 q ˆxh 2 x q ˙FLL `ph 1 Ø h 2 q , (6.1.7)
where

F LL " ˇˇˇˇˇˇż d d ⃗ p 2 F ´xq 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 xq x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 `xq x qQ 2 ˇˇˇˇˇˇ2 . (6.1.8)
This LO cross-section can be written differently, using transverse momentum conservation, see eq. (4.5.8),

dσ q qÑh 1 h 2 0LL dx h 1 dx h 2 d d ⃗ p h 1 K d d ⃗ p h 2 " 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1 x h 2 dx q x q x q ˆxq x h 1 ˙d ˆxq x h 2 ˙d (6.1.9)
ˆδp1 ´xq ´xq qD h 1 q ˆxh 1 x q ˙Dh 2 q ˆxh 2 x q ˙r F LL `ph 1 Ø h 2 q , (6.1.10)

where

r F LL " ˇˇˇˇˇˇż d d ⃗ p 1 F ´´´x q 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 1 ¯x q x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 `xq x qQ 2 ˇˇˇˇˇˇ2 . (6.1.11)
Both forms can be used interchangeably in the NLO cross-sections that are proportional to the LO cross-sections, i.e when dealing with the soft, virtual, and counter-term contribution to the NLO cross-sections. For the collinear quark-gluon contribution, eq. ( 6.1.7) will be used while for the collinear antiquark-gluon contribution, eq. ( 6.1.10) will be used.

Similarly, the non-diagonal interference term T L can be written in the two equivalent forms

dσ q qÑh 1 h 2 0T L dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 " α em Q p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1 x h 2 dx q ˆxq x h 1 ˙d ˆxq x h 2 ˙d
ˆpx q ´xq qδp1 ´xq ´xq qD h 1 q ˆxh 1 x q ˙Dh 2 q ˆxh 2 x q ˙FT L `ph 1 Ø h 2 q , (6.1.12)

F T L " ż d d ⃗ p 2 F ´xq 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 xq x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 `xq x qQ 2 ˆ» - - ż d d ⃗ p 2 1 F ´xq 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 1 xq x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2 `xq x qQ 2 ˆxq x h 2 ⃗ p h 2 ´⃗ p 2 1 ˙¨⃗ ε T fi ffi fl ˚, (6.1.13)
and

dσ q qÑh 1 h 2 0T L dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 " α em Q p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1 x h 2 dx q ˆxq x h 1 ˙d ˆxq x h 2 ˙d ˆpx q ´xq qδp1 ´xq ´xq qD h 1 q ˆxh 1 x q ˙Dh 2 q ˆxh 2 x q ˙r F T L `ph 1 Ø h 2 q , (6.1.14)
where

r F T L " ż d d ⃗ p 1K F ´´´x q 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 1 ¯x q x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 `xq x qQ 2 ˆ» - - ż d d ⃗ p 1 1 F ´´´x q 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 1 1 ¯x q x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 `xq x qQ 2 ˆxq x h 1 p h 1 K ´p1 1 K ˙¨ε T fi ffi fl ˚.
(6.1.15)

The most complicated contribution T T reads

dσ q qÑh 1 h 2 0T T dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 " α em 2p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 x h 2 dx q x q ˆxq x h 1 ˙d ˆxq x h 2 ˙d ˆδp1 ´xq ´xq qD h 1 q ˆxh 1 x q ˙Dh 2 q ˆxh 2 x q ˙FT T `ph 1 Ø h 2 q , (6.1.16)
where

F T T " " px q ´xq q 2 g ri K g lk K ´grk K g li K `grl K g ik K ı ˆż d d ⃗ p 2 F ´xq 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 xq x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 `xq x qQ 2 ˆxq x h 2 p h 2 K ´p2K ˙r ε T i ˆ» - - ż d d ⃗ p 2 1 F ´xq 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 1 xq x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2 `xq x qQ 2 ˆxq x h 2 p h 2 K ´p2 1 K ˙l ε T k fi ffi fl ˚(6.1.17)
or equivalently

dσ q qÑh 1 h 2 0T T dx h 1 dx h 2 d d p h 1 K d d p h 2 K " α em 2p2πq 4d N c ÿ q ż 1 x h 1 dx q x q ż 1 x h 2 dx q x q ˆxq x h 1 ˙d ˆxq x h 2 ˙d ˆδp1 ´xq ´xq qQ 2 q D h 1 q ˆxh 1 x q ˙Dh 2 q ˆxh 2 x q ˙r F T T `ph 1 Ø h 2 q , (6.1.18)
where

r F T T " " px q ´xq q 2 g ri K g lk K ´grk K g li K `grl K g ik K ı ˆż d d ⃗ p 1 F ´´´x q 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 1 ¯x q x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 `xq x qQ 2 ˆxq x h 1 p h 1 K ´p1K ˙r ε T i ˆ» - - ż d d ⃗ p 1 1 F ´´´x q 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 1 1 ¯x q x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 `xq x qQ 2 ˆxq x h 1 p h 1 K ´p1 1 K ˙l ε T k fi ffi fl ˚. (6.1.19) 1 Q x q ´xq 2x qx q ˆxq x h 2 ⃗ p h 2 ´⃗ p 2 1 ˙¨⃗ ε T (6.1.20) or 1 Q x q ´xq 2x qx q ˆxq x h 1 p h 1 K ´p1 1 K ˙¨ε T . (6.1.21)
The T T cross-section differs from the LL cross-section by a factor of

1 Q 2 1 4x 2 q x 2 q " px q ´xq q 2 g ri K g lk K ´grk K g li K `grl K g ik K ı ˆxq x h 1 p h 1 ´p1 ˙r ε T i ˆxq x h 1 p h 1 ´p1 1 ˙l ε T k (6.1.22) or 1 Q 2 1 4x 2 q x 2 q " px q ´xq q 2 g ri K g lk K ´grk K g li K `grl K g ik K ı ˆxq x h 2 p h 2 ´p2 ˙r ε T i ˆxq x h 2 p h 2 ´p2 1 ˙l ε T k . (6.1.
23) The factors of 1{Q and 1{Q 2 come from the photon polarization while the other modifications come from the expression of the squared of the impact factors. Those modifications and additional factors between T L and T T cross-section wrt LL will remain true when going to NLO, for what concerns the extraction of divergences. This means that no additional detailed calculations are needed for those cases.

NLO computations in a nutshell

Different types of contributions in the dipole picture Just like in eqs. (4.5.7, 4.6.1), it is convenient at NLO to separate the various contributions from the dipole point of view, as illustrated in fig. 6.1.4. In this figure, we exhibit a few examples of diagrams, either virtual or real, as a representative of each 5 classes of diagrams. There are indeed 5 classes of contributions from the dipole point of view, namely dσ iJI pi " 1, ¨¨¨5q, so that the NLO elements of density matrix can be written as dσ JI " dσ 0JI `dσ 1JI `dσ 2JI `dσ 3JI `dσ 4JI `dσ 5JI .

(6.1.24)

Now, we will shortly discuss each of these 5 NLO corrections.

For the virtual diagrams, there are two classes of diagrams: the diagrams in which the virtual gluon does not cross the shockwave, thus contributing to dσ 1IJ , purely made of dipole ˆdipole terms; the diagrams in which the virtual gluon does cross the shockwave, contributing both to dσ 1IJ , made of dipole ˆdipole terms as well as to dσ 2IJ , made of double dipole ˆdipole (and dipole ˆdouble dipole) terms (see section 4.3.1).

For the real diagrams, there are three classes of diagrams: the diagrams in which the real gluon does not cross the shockwave, thus contributing to dσ 3IJ , purely made of dipole ˆdipole terms; the diagrams in which the real gluon crosses exactly once the shockwave, contributing both to dσ 3IJ , made of dipole ˆdipole terms as well as to dσ 4IJ , made of double dipole

Overview of cancellation of divergences

Before providing technical details, let us sketch the way the computation will be done, putting emphasis on the infrared (IR) sector.

When generically decomposing any on-shell parton momentum in the Sudakov basis as

1 p µ " zp `nµ 1 `⃗ p 2 2zp `nµ 2 `pµ K , (6.1.25)
in the IR sector, we face three kinds of divergences:

• Rapidity: z goes to 0 and p K arbitrary.

• Soft: any component of the gluon momentum goes linearly to 0 (obtained with both z and p K " z pK " z going to 0).

• Collinear: parton's p K is proportional to the transverse momentum of the other parton in the splitting, z being arbitrary.

Technically, since the z integration is regulated through a lower cut-off (named α), one should be careful with the fact that the appearance of ln α may have originated from both rapidity or soft divergences.

The calculation goes as follows. First, the rapidity divergences, appearing only in the virtual corrections in the present computation, are taken care of at the amplitude level by absorbing them in the shockwave through one step of B-JIMWLK evolution. This removes part of terms with ln α related to pure rapidity divergences. This has been shown in section 4.3.4.

Next, at the level of cross-section, we separate the soft divergent contribution from the nonsoft divergent terms in the real contribution. Combining real and virtual contributions, these soft divergent terms will disappear as guaranteed by the Kinoshita-Lee-Nauenberg theorem.

Finally, the remaining type of divergences, which are of purely collinear nature, will be absorbed into the fragmentation functions through one step of the DGLAP evolution equation [START_REF] Altarelli | Asymptotic Freedom in Parton Language[END_REF][START_REF] Dokshitzer | Calculation of the Structure Functions for Deep Inelastic Scattering and e+ e-Annihilation by Perturbation Theory in Quantum Chromodynamics[END_REF][START_REF] Gribov | Deep inelastic e p scattering in perturbation theory[END_REF][START_REF] Lipatov | The parton model and perturbation theory[END_REF].

Different fragmentation contributions to the NLO cross-section

At NLO, we have to deal with 5 kinds of contributions to the cross-section, illustrated in fig. 6.1.5:

(a) γ ˚`P Ñ h 1 `h2 `X `P cross-section at one-loop (i.e. virtual contributions), (b) γ ˚`P Ñ h 1 `h2 `g `X `P cross-section at Born level (i.e. real contributions), (c) γ ˚`P Ñ h 1 `h2 `q `X `P cross-section at Born level (i.e. real contributions), (d) γ ˚`P Ñ h 1 `h2 `q `X `P cross-section at Born level (i.e. real contributions), (e) FFs counterterms, 1 Here p `is a large fixed momentum, eg p γ in our present case. where X denotes the remnants of the fragmentation. Contributions (a) and (e) are easy to treat since (a) is simply the convolution of a known one-loop result with fragmentation functions, while (e) is obtained from the Born result when one renormalizes the fragmentation functions. We just split them into finite and divergent parts.

+ + ˜`q Ø q (d) (e)
For the real contributions (b), (c), (d), the treatment is less straightforward even if the partonic real corrections are also already known.

Contribution (b) is the most complicated one, it contains both soft and collinear divergences. When we square the amplitude contributing to (b), see fig. 6.1.6, there is a series of finite contributions plus one, represented by the sum of contributions ( 1), ( 2), ( 3), ( 4) of fig. 6.1.6, that contains all divergences, and that belongs only to the dipole-dipole contribution. We add and subtract to the latter its soft limit to obtain the following structure σpbqdiv " ÿ λq,λg,λq

|A q q,sing´dipole | 2 div " σpbqdiv,1 `σ pbqdiv,2 `σ pbqdiv,3 `σ pbqdiv,4

" σsoft pbqdiv `pσ pbqdiv,1 ´σ sof t pbqdiv,1 q `pσ pbqdiv,2 ´σ sof t pbqdiv,2 q `pσ pbqdiv,3 ´σ sof t pbqdiv,3 q `pσ pbqdiv,4 ´σ sof t pbqdiv,4 q (6. 1.26) with the meaning that each of these four contributions is in one-to-one correspondence to the four diagrams (1), ( 2), ( 3), ( 4) in fig. 6.1.6. Among these various terms, the terms pσ pbqdiv,1 σsoft pbqdiv,1 q and pσ pbqdiv,3 ´σ sof t pbqdiv,3 q are collinearly divergent while the terms pσ pbqdiv,2 ´σ sof t pbqdiv,2 q and pσ pbqdiv,4 ´σ sof t pbqdiv,4 q are finite. In fig. 6 Here, one does not encounter any soft divergence. The only divergence comes from the contribution σpcqdiv,3 which has a collinear divergence when the fragmenting gluon and the antiquark are collinear.

The discussion for the fourth case, see fig. 6.1.8, involving the fragmentation from antiquark and gluon goes along the same line: the only divergence comes from the contribution σpdqdiv,1 which has a collinear divergence when the fragmenting gluon and the quark are collinear.

Counterterms from FFs renormalization

The renormalization equations of FFs express the bare FFs in terms of dressed ones. In MS scheme, at factorization scale µ F , they take the form, following notations of ref. [START_REF] Ivanov | Inclusive production of a pair of hadrons separated by a large interval of rapidity in proton collisions[END_REF] D h q pxq " D h q px, µ F q ´αs 2π ˆ1 ε `ln

µ 2 F µ 2 ˙ż 1 x dz z " D h q ´x z , µ F ¯Pqq pzq `Dh g ´x z , µ F ¯Pgq pzq ı , D h g pxq " D h g px, µ F q ´αs 2π ˆ1 ε `ln µ 2 F µ 2 ˙ż 1 x dz z « ÿ q,q D h q ´x z , µ F ¯Pqg pzq `Dh g ´x z , µ F ¯Pgg pzq ff , (6.2.1) (b) = + (1) (2)
+ +

(3)

+ finite contributions

Figure 6.1.8: NLO cross-section in the case of fragmentation from the gluon and the antiquark. We explicitly isolate the diagram which contains divergences, namely a collinear divergence between the fragmenting gluon and the quark.

where 1 ε " Γp1´ϵq ϵp4πq ϵ " 1 ϵ `γE ´lnp4πq and µ is an arbitrary parameter introduced by dimensional regularization. The LO splitting functions are given by

P qq pzq " C F " 1 `z2 p1 ´zq ``3 2 δp1 ´zq ȷ , (6.2.2)
P gq pzq " C F 1 `p1 ´zq 2 z , (6.2.3)
P qg pzq " T R " z 2 `p1 ´zq 2 ‰ with T R " 1 2 , ( 6
.2.4)

P gg pzq " 2C A " 1 p1 ´zq ``1 z ´2 `zp1 ´zq ȷ `ˆ11 6 C A ´nf 3 ˙δp1 ´zq , (6.2.5)
where the + prescription is defined as

ż 1 a dβ F pβq p1 ´βq `" ż 1 a dβ F pβq ´F p1q p1 ´βq ´ż a 0 dβ F p1q 1 ´β . (6.2.6)
The collinear counterterms due to the renormalization of the bare FFs are calculated by inserting eq. ( 6.2.1) into the contributions (6.1.7, 6.1.12, 6. 1.16). This corresponds to the contribution (e) in fig. 6.1.5.

For LL cross-section, this counterterm takes the form

dσ q qÑh 1 h 2 LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc t " 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1 x h 2 dx q x q x q ˆxq x h 1 ˙dˆx q x h 2 ˙dδp1 ´xq ´xq q ˆFLL ´´α s 2π ¯ˆ1 ε `ln µ 2 F µ 2 ˙#ż 1 x h 1 xq dβ 1 β 1 " P qq pβ 1 qD h 1 q ˆxh 1 β 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F Ṗgq pβ 1 qD h 1 g ˆxh 1 β 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙ȷ `ż 1 x h 2 x q dβ 2 β 2 " P qq pβ 2 qD h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 β 2 x q , µ F Ṗgq pβ 2 qD h 1 q ˆxh 1 x q , µ F ˙Dh 2 g ˆxh 2 β 2 x q , µ F ˙ȷ* `ph 1 Ø h 2 q " dσ h 1 h 2 LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇc t div `dσ h 1 h 2 LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇc t fin . (6.2.7)
We stress here that, for any separate term in the curly bracket, one can indifferently use F LL or r F LL . In particular, for the first two terms we can use F LL and for the last two r F LL . This simple observation is useful when observing the cancellation of divergences at the level of integrands. The same remark applies also for other transitions.

For cross-sections involving other combinations of polarizations, we have

dσ q qÑh 1 h 2 T L dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇc t " α em Q p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1 x h 2 dx q ˆxq x h 1 ˙d ˆxq x h 2 ˙d px q ´xq q δp1 ´xq ´xq q ˆFT L ´´α s 2π ¯ˆ1 ε `ln µ 2 F µ 2 ˙#ż 1 x h 1 xq dβ 1 β 1 " P qq pβ 1 qD h 1 q ˆxh 1 β 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F Ṗgq pβ 1 qD h 1 g ˆxh 1 β 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙ȷ `ż 1 x h 2 x q dβ 2 β 2 " P qq pβ 2 qD h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 β 2 x q , µ F Ṗgq pβ 2 qD h 1 q ˆxh 1 x q , µ F ˙Dh 2 g ˆxh 2 β 2 x q , µ F ˙ȷ* `ph 1 Ø h 2 q " dσ h 1 h 2 T L dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇc t div `dσ h 1 h 2 T L dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇc t fin , (6.2.8)
and

dσ q qÑh 1 h 2 T T dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc t " α em 2p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 x h 2 dx q x q ˆxq x h 1 ˙d ˆxq x h 2 ˙dδp1 ´xq ´xq q ˆFT T ´´α s 2π ¯ˆ1 ε `ln µ 2 F µ 2 ˙#ż 1 x h 1 xq dβ 1 β 1 " P qq pβ 1 qD h 1 q ˆxh 1 β 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F Ṗgq pβ 1 qD h 1 g ˆxh 1 β 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙ȷ `ż 1 x h 2 x q dβ 2 β 2 " P qq pβ 2 qD h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 β 2 x q , µ F Ṗgq pβ 2 qD h 1 q ˆxh 1 x q , µ F ˙Dh 2 g ˆxh 2 β 2 x q , µ F ˙ȷ* `ph 1 Ø h 2 q " dσ h 1 h 2 T T dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇc t div `dσ h 1 h 2 T T dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇc t fin
. (6.2.9)

The divergent parts are the ones containing 1{ε and the finite terms are the ones with lnpµ 2 F {µ 2 q. The dependence on the arbitrary parameter µ disappears at the end when all finite terms are put together.

NLO cross-section: Virtual dipole ˆdipole contribution

By using the collinear factorization formula eq. (6.1.3) and the expressions of the hard dσ 1JI cross-section in eqs. (4.5.15, 4.5.17,4.5.18) with the expression of ´SV `SV 2 ¯in eq. (4.5.16), we obtain the dipole ˆdipole virtual contribution to the full cross-section. We split it into divergent part and finite part.

The divergent part of the LL element reads

dσ q qÑh 1 h 2 1LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇd iv " 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1
x h 2 dx q x q x q ˆxq x h 1 ˙d ˆxq x h 2 ˙d δp1 ´xq ´xq q ˆDh 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙FLL (6.3.1)

ˆαs 2π C F 1 ε » - -´4ϵ lnpαq ln ¨µ2 ´⃗ p h 2 x h 2 ´⃗ p h 1 x h 1 ¯2 ‹ '`4 lnpαq `4ϵ ln 2 pαq ´2 lnpx q x qq `3fi ffi fl `ph 1 Ø h 2 q .
Its finite part reads, putting ϵ " 0,

dσ q qÑh 1 h 2 1LL dx h 1 dx h 2 d 2 ⃗ p h 1 d 2 ⃗ p h 2 ˇˇˇˇfi n " 2α em Q 2 p2πq 8 N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1 x h 2 dx q x q x q ˆxq x h 1 ˙2 ˆxq x h 2 ˙2 δp1 ´xq ´xq q ˆDh 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙FLL ˆαs 2π C F » - -2 ln ¨µ2 x q x q ´⃗ p h 2 x h 2 ´⃗ p h 1 x h 1 ¯2 ‹ 'lnpxqxqq `ln 2 px q x qq ´3 ln ¨µ2 x q x q ´⃗ p h 2 x h 2 ´⃗ p h 1 x h 1 ¯2 ‹ '´π 2 3 `6fi ffi fl `αs Q 2 4π ˆN 2 c ´1 N c ˙αem 2p2πq 8 N c ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 x h 2 dx q x q δp1 ´xq ´xq q ˆˆx q x h 1 ˙2 ˆxq x h 2 ˙2 D h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ż d 2 ⃗ p 2 F ˆxq 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 ˙ż d 2 ⃗ p 2 1 F ˚ˆx q 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 1 $ ' & ' % 1 ´xq x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2 `xq x qQ 2 » - - 6x 2 q x 2 q ´xq x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 `xq x qQ 2 ˆln ¨µ4 Q 2 ´⃗ p h 2 x h 2 ´⃗ p h 1 x h 1 ¯2 ˆ´xq x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 `xq x qQ 2 ˙2 ‹ ‹ ‹ ' `˜ż xq 0 dz ˜rpϕ 4 q LL s ``ÿ n"5,6 rpϕ n q LL s `|⃗ p 3 " ⃗ 0 ¸`pq Ø qq ¸ff `h.c| p 1 ,p 2 Øp 1 1 ,p 2 1 + `ph 1 Ø h 2 q . (6.3.2)
As a reminder, pq Ø qq stands for p q Ø p q, p 1 p1q Ø p 2 p1q , x q Ø x q.

For the TL, we have,

dσ q qÑh 1 h 2 1T L dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 K ˇˇˇˇd iv " α em Q p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1 x h 2 dx q ˆxq x h 1 ˙d ˆxq x h 2 ˙d px q ´xq q ˆδp1 ´xq ´xq qD h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙FT L ˆαs 2π C F 1 ε » - -´4ϵ lnpαq ln ¨µ2 ´⃗ p h 2 x h 2 ´⃗ p h 1 x h 1 ¯2 ‹ '`4 lnpαq `4ϵ ln 2 pαq ´2 lnpx q x qq `3fi ffi fl `ph 1 Ø h 2 q , (6.3.3)
and

dσ q qÑh 1 h 2 1T L dx h 1 dx h 2 d 2 ⃗ p h 1 d 2 ⃗ p h 2 ˇˇˇˇfi n " α em Q p2πq 8 N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1 x h 2 dx q ˆxq x h 1 ˙2 ˆxq x h 2 ˙2 px q ´xq q ˆδp1 ´xq ´xq qD h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙FT L ˆαs 2π C F » - -2 ln ¨µ2 x q x q ´⃗ p h 2 x h 2 ´⃗ p h 1 x h 1
¯2 ‹ 'lnpxqxqq `ln 2 px q x qq ´3 ln ¨µ2

x q x q ´⃗ p h 2

x h 2 ´⃗ p h 1 x h 1 ¯2 ‹ '´π 2 3 `6fi ffi fl `αs Q 4π ˆN 2 c ´1 N c ˙αem 2p2πq 8 N c ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 x h 2 dx q x q δp1 ´xq ´xq q ˆˆx q x h 1 ˙2 ˆxq x h 2 ˙2 D h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 1 x q , µ F ż d 2 ⃗ p 1 d 2 ⃗ p 2 F ˆ⃗ p 12 2 ˙δ ˆxq x h 1 ⃗ p h 1 ´⃗ p 1 `xq x h 2 ⃗ p h 2 ´⃗ p 2 ż d 2 ⃗ p 1 1 d 2 ⃗ p 2 1 F ˚ˆ⃗ p 1 1 2 2 ˙δ ˆxq x h 1 ⃗ p h 1 ´⃗ p 1 `xq x h 2 ⃗ p h 2 ´⃗ p 2 1 ˙εT i ˆ» - - - - - - «˜ż xq 0 dz ˜"pϕ 4 q i T L ‰ ``ÿ n"5,6 " pϕ n q i T L ‰ `|⃗ p 3 " ⃗ 0 ¸¸`pq Ø qq ff xq x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 `xq x qQ 2
`3x q x qpx q ´xq q ´xq

x h 1 p h 1 K ´p1 1 K ¯i ˆ´xq x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 `xq x qQ 2 ˙ˆ´x q x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 `xq x qQ 2 ln ¨Q2 µ 8 ´⃗ p h 2 x h 2 ´⃗ p h 1 x h 1 ¯´4 x q x q ˆ´xq x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 `xq x qQ 2 ˙ˆ´x q x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 `xq x qQ 2 ˙‹ ‹ ' ´xq x qQ 2 ´xq x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 ln ¨xq x qQ 2 ´xq x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 `xq x qQ 2 ‹ ' ‹ ' `«˜ż xq 0 dz ˜"pϕ 4 q i LT ‰ ``ÿ n"5,6 " pϕ n q i LT ‰ `|⃗ p 3 " ⃗ 0 ¸¸`pq Ø qq ff 2x q x q ˆ´xq x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 `xq x qQ 2 ˙fi ffi ffi ffi ffi ffi fl `ph 1 Ø h 2 q . (6.3.4)
In the case of the T T transition, we have

dσ q qÑh 1 h 2 1T T dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇd iv " α em 2p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 x h 2 dx q x q ˆxq x h 1 ˙d ˆxq x h 2 ˙d δp1 ´xq ´xq q ˆDh 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙FT T ˆαs 2π C F 1 ε » - -´4ϵ lnpαq ln ¨µ2 ´⃗ p h 2 x h 2 ´⃗ p h 1 x h 1 ¯2 ‹ '`4 lnpαq `4ϵ ln 2 pαq ´2 lnpx q x qq `3fi ffi fl `ph 1 Ø h 2 q (6.3.5)
while the finite part reads

dσ q qÑh 1 h 2 1T T dx h 1 dx h 2 d 2 ⃗ p h 1 d 2 ⃗ p h 2 ˇˇˇˇfi n " α em 2p2πq 8 N c ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 x h 2 dx q x q ˆxq x h 1 ˙2 ˆxq x h 2 ˙2 δp1 ´xq ´xq q ˆDh 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙FT T ˆαs 2π C F » - -2 ln ¨µ2 x q x q ´⃗ p h 2 x h 2 ´⃗ p h 1 x h 1 ¯2 ‹ 'lnpxqxqq `ln 2 px q x qq ´3 ln ¨µ2 x q x q ´⃗ p h 2 x h 2 ´⃗ p h 1 x h 1 ¯2 ‹ '´π 2 3 `6fi ffi fl `αs 4π ˆN 2 c ´1 N c ˙αem 2p2πq 8 N c ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 h 2 dx q x q δp1 ´xq ´xq q ˆˆx q x h 1 ˙2 ˆxq x h 2 ˙2 D h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ż d 2 ⃗ p 1 d 2 ⃗ p 2 F ˆ⃗ p 12 2 ˙δ ˆxq x h 1 ⃗ p h 1 ´⃗ p 1 `xq x h 2 ⃗ p h 2 ´⃗ p 2 ż d 2 ⃗ p 1 1 d 2 ⃗ p 2 1 F ˚ˆ⃗ p 1 1 2 1 2 ˙δ ˆxq x h 1 ⃗ p h 1 ´⃗ p 1 1 `xq x h 2 ⃗ p h 2 ´⃗ p 2 1 εT i ε T k $ ' ' & ' ' % 3 2 ´xq x h 1 p h 1 K ´p1K ¯r ´xq x h 1 p h 1 K ´p1 1 K ¯l ˆ´xq x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 `xq x qQ 2 ˙ˆ´x q x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 `xq x qQ 2 "px q ´xq q 2 g ri K g lk K ´grk K g li K `grl K g ik K ı ˆ» - - -ln ¨µ4 x q x q ´⃗ p h 2 x h 2 ´⃗ p h 1 x h 1 ¯2 ˆ´xq x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 `xq x qQ 2 ˙‹ ‹ ' ´xq x qQ 2 ´xq x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 ln ¨xq x qQ 2 ´xq x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 `xq x qQ 2 ‹ ' fi ffi fl `«˜ż xq 0 dz ˜"pϕ 4 q ik T T ı ``ÿ n"5,6 " pϕ n q ik T T ı `|⃗ p 3 " ⃗ 0 ¸¸`pq Ø qq ff ˆ´xq x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 `xq x qQ 2 ˙xq x q ``h.c.|p1,p2Øp

NLO cross-section: Real corrections

In this section, we will discuss the real corrections. Since, as explained above, the calculation is almost completely identical in the LL, T L, and T T cases (apart from factors that do not affect the general strategy), we will show the details of the LL case only. For the others, we will just report the final results.

The dipole ˆdipole partonic cross-sections is given by eq (4.6.2) with the squared impact factor Φ α 3 " Φ α 4 | ⃗ p 3 " ⃗ 0 `r Φ α 3 of the form eq. (4.6.4) where only the squared of r Φ α 3 provides divergences in the cross-sections and are given by eqs. (4.6.5-4.6.7). We will focus in this section on this square. From the relevant equations mentioned above, the divergent part of the LL partonic crosssection from real emission is given by

dσ 3LL | div " 2α em Q 2 p2πq 4d N c Q 2 q dx 1 q dx 1 qδp1 ´x1 q ´x1 q ´xg qd d ⃗ p q d d ⃗ p q α s C F µ 2ϵ dx g x g d d ⃗ p g p2πq d ˆż d d ⃗ p 1 d d ⃗ p 2 δ p⃗ p q1 `⃗ p q2 `⃗ p g q F ˆ⃗ p 12K 2 ż d d ⃗ p 1 1 d d ⃗ p 2 1 δ `⃗ p q1 1 `⃗ p q2 1 `⃗ p g ˘F˚ˆ⃗ p 1 1 2 1 2 $ ' ' & ' ' % dx 2 g `4x 1 q px 1 q `xg q ˆQ2 `⃗ p 2 q2 x 1 q p1´x 1 q q ˙ˆQ 2 `⃗ p 2 q2 1 x 1 q p1´x 1 q q ˙px 1 q ⃗ p g ´xg ⃗ p q q 2
´`2x g ´dx 2 g `4x

1 q x 1 q˘`x 1 q ⃗ p g ´xg ⃗ p q ˘¨`x 1 q ⃗ p g ´xg ⃗ p qQ 2 `⃗ p 2 q2 1 x 1 q p1´x 1 q q ˙ˆQ 2 `⃗ p 2 q1 x 1 q p1´x 1 q q ˙`x 1 q ⃗ p g ´xg ⃗ p q ˘2 `x1 q ⃗ p g ´xg ⃗ p q˘2 `dx 2 g `4x 1 qpx 1 q `xg q ˆQ2 `⃗ p 2 q1 x 1 q p1´x 1 q q ˙ˆQ 2 `⃗ p 2 q1 1 x 1 q p1´x 1 q q ˙px 1 q ⃗ p g ´xg ⃗ p qq 2
´`2x g ´dx 2 g `4x 1 q x 1 q˘`x 1 q ⃗ p g ´xg ⃗ p q ˘¨`x

1 q ⃗ p g ´xg ⃗ p qQ 2 `⃗ p 2 q1 1 x 1 q p1´x 1 q q ˙ˆQ 2 `⃗ p 2 q2 x 1 q p1´x 1 q q ˙`x 1 q ⃗ p g ´xg ⃗ p q ˘2 `x1 q ⃗ p g ´xg ⃗ p q˘2 , / / . / / - (6.4.1)
where compared to eq. (4.6.2), we relabelled the longitudinal momentum fraction of the quark and antiquark with a prime. This is because later we will do some necessary change of variable. When two partons labeled i and j become collinear, the variable

⃗ A ij " x i ⃗ p j ´xj ⃗ p i (6.4.2)
vanishes. In the present case, the first term in the bracket of eq. (6.4.1) gives the collinear divergences ( ⃗ A 2 qg Ñ 0, i.e. quark-gluon channel) and the third ( ⃗ A 2 qg Ñ 0, i.e. antiquark-gluon channel).

Fragmentation from quark and antiquark

As explained above, there are several contributions to the final cross-section that contain divergences. In this section, we deal with extracting the soft and collinear divergences associated with the contribution (b) in fig. 6.1.5. This contribution corresponds to the situation in which the quark and the antiquark fragment and there is an additional emission of a gluon with respect to the LO case. Below,

• We compute the collinear divergence of the diagram (1) of fig. 6.1.6 and show that it is removed by the + prescription part of the first term of eq. (6.2.7).

• Similarly, we calculate the collinear divergence of the diagram (3) of fig. 6.1.6 and show that it is removed by the + prescription part of the third term of eq. (6.2.7).

• We extract the soft divergences of diagrams ( 1), ( 2), ( 3), ( 4) of fig. 6.1.6 and discuss the complete cancellation of divergences of this contribution.

The calculations of the collinear divergences is done by Fourier transforming the F ´⃗ p 12 2 ¯, as defined in eq. (4.5.5). We will also be using the identity (see eg ref. [START_REF] Chirilli | Inclusive Hadron Productions in pA Collisions[END_REF]):

1 µ 2ϵ ż d d ⃗ q e ´i⃗ q¨⃗ r ⃗ q 2 " π ˆ4π µ 2 ⃗ r 2 ˙ϵ Γpϵq , (6.4.3)
which leads to

1 µ 2ϵ ż d d ⃗ q p2πq d e ´i⃗ q¨⃗ r ⃗ q 2 " 1 4 1`ϵ π 2`2ϵ π 1`ϵ ˆ4 µ 2 ⃗ r 2 ˙ϵ Γpϵq " 1 4π 1 p4πq ϵ ˆ4 µ 2 ⃗ r 2 ˙ϵ ˆ1 ϵ ´γE `Opϵq " 1 4π p1 ´ϵ lnp4πqq ˆ1 `ϵ ln ˆ22 µ 2 ⃗ r 2 ˙˙ˆ1 ϵ ´γE `Opϵq " 1 4π " 1 ϵ `ln ˆ22 µ 2 ⃗ r 2 ˙´lnp4πq ´γE `Opϵq * " 1 4π " 1 ε `ln ˆ22 µ 2 ⃗ r 2 ˙´2γ E `Opϵq * " 1 4π " 1 ε `ln ˆc2 0 µ 2 ⃗ r 2 ˙`Opϵq * " 1 4π 1 ε ˆc2 0 ⃗ r 2 µ 2 ˙ϵ `Opϵq . (6.4.4)
with c 0 " 2e ´γE . We also have to change variables from the usual fraction of longitudinal photon momentum of the partons px 1 i , x g q with i " q, q, defined in eq. (4.1.4), and as used in eq. (6.4.1), to the variable of the fraction of longitudinal photon momentum of the parent parton x i and longitudinal fraction β with respect to the parent parton and not with respect to the photon anymore:

px 1 i , x g q Ñ px i , βq with x 1 i " βx i . (6.4.5)
Changing variables is necessary to be able to compare with eqs. (6.2.7), (6.2.8), and (6.2.9). Note that to make notations lighter, we will remove the ' when one particle is a spectator, see figs. 6.4.1 and 6.4.2.

The Fourier transform of F ´⃗ p 12 2 ¯is necessary in order to be able to integrate over the transverse momentum of the spectator parton, as it allows for the complete factorization of this momentum from this non-perturbative function.

Collinear contributions: q-g splitting x q , ⃗ p q `⃗ p g x 1 q " β 1 x q , ⃗ p q

x g " p1 ´β1 qx q , ⃗ p g

x q, ⃗ p q Figure 6.4.1: Kinematics for the q ´g splitting contribution. We indicate the longitudinal fraction of momentum carried by the partons as well as their transverse momenta.

We use the kinematics illustrated in fig. 6.4.1. The term in eq. ( 6.4.1) considered for this collinear contribution is the first one in the bracket.

dσ q qÑh 1 h 2 3LL | coll. qg " dx h 1 dx h 2 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx 1 q x 1 q ż 1 x h 2 dx q x q ż 1 α dx g x g δp1 ´x1 q ´xq ´xg q ˆDh 1 q ˆxh 1 x 1 q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙dd ⃗ p q d d ⃗ p q α s µ 2ϵ C F d d ⃗ p g p2πq d ˆż d d ⃗ p 1 d d ⃗ p 2 δ p⃗ p q1 `⃗ p q2 `⃗ p g q F ˆ⃗ p 12 2 ż d d ⃗ p 1 1 d d ⃗ p 2 1 δ `⃗ p q1 1 `⃗ p q2 1 `⃗ p g ˘F˚ˆ⃗ p 1 1 2 1 2 ṗdx 2 g `4x 1 q px 1 q `xg qqx 2 q p1 ´xq q 2 ´xq p1 ´xq qQ 2 `⃗ p 2 q2 ¯´x qp1 ´xq qQ 2 `⃗ p 2 q2 1 ¯px 1 q ⃗ p g ´xg ⃗ p q q 2 `ph 1 Ø h 2 q " dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx 1 q x 1 q ż 1 α dx g x g ż 1 x h 2 dx q x q δp1 ´x1 q ´xq ´xg q ˆˆx 1 q x h 1 ˙d ˆxq x h 2 ˙d D h 1 q ˆxh 1 x 1 q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙αs µ 2ϵ C F d d ⃗ p g p2πq d ˆż d d ⃗ p 1 d d ⃗ p 2K δ ˆx1 q x h 1 ⃗ p h 1 ´⃗ p 1 `xq x h 2 ⃗ p h 2 ´⃗ p 2 `⃗ p g ˙F ˆ⃗ p 12 2 ż d d ⃗ p 1 1 d d ⃗ p 2 1 δ ˆx1 q x h 1 ⃗ p h 1 ´⃗ p 1 1 `xq x h 2 ⃗ p h 2 ´⃗ p 2 1 `⃗ p g ˙F˚ˆ⃗ p 1 1 2 1 2 ˆpdx 2 g `4x 1
q px 1 q `xg qqx 2 q p1 ´xq q 2 ˆxq p1 ´xq qQ 2 `´xq

x h 2 ⃗ p h 2 ´⃗ p 2 ¯2˙ˆx
qp1 ´xq qQ 2 `´xq

x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2˙´x 1 q ⃗ p g ´xg x 1 q x h 1 ⃗ p h 1 ¯2 `ph 1 Ø h 2 q " dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx 1 q x 1 q ż 1 α dx g x g ż 1
x h 2 dx q x q δp1 ´x1 q ´xq ´xg q

ˆˆx 1 q x h 1 ˙d ˆxq x h 2 ˙d D h 1 q ˆxh 1 x 1 q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙αs µ 2ϵ C F d d ⃗ p g p2πq d ˆż d d ⃗ p 2 F ˆx1 q 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 `⃗ p g 2 ż d d ⃗ p 2 1 F ˚ˆx 1 q 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 1 `⃗ p g 2 ṗdx 2
g `4x 1 q px 1 q `xg qqx 2 q p1 ´xq q 2 ˆxq p1 ´xq qQ 2 `´xq

x h 2 ⃗ p h 2 ´⃗ p 2 ¯2˙ˆx
qp1 ´xq qQ 2 `´xq

x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2˙´x 1 q ⃗ p g ´xg x 1 q x h 1 ⃗ p h 1 ¯2 `ph 1 Ø h 2 q .
After performing the change of variable

x 1 q " β 1 x q x g " p1 ´β1 qx q (6.4.6) and using the Jacobian dx 1 q dx g " x q dx q dβ 1 we can rewrite the longitudinal integration in the symbolic form

ż 1 x h 1 dx 1 q x 1 q ż 1 x h 2 dx q x q ż 1 α dx g x g δp1 ´x1 q ´xq ´xg q " ż 1 x h 1 dx 1 q x 1 q ż 1 α dx g x g ż `8
´8 dx q x q θpx q ´xh 2 qθp1 ´xq qδp1 ´x1 q ´xq ´xg q " ż 1

x h 1 dx 1 q x 1 q ż 1 α dx g x g θp1 ´x1 q ´xg ´xh 2 qθpx 1 q `xg q 1 1 ´x1 q ´xg " ż 1´x h 2 x h 1 dx q x q 1 1 ´xq ż 1´α xq x h 1 xq dβ 1 β 1 p1 ´β1 q . (6.4.7)
After this manipulation, we obtain

dσ q qÑh 1 h 2 3LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg " 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1´x h 2
x h 1 dx q x q p1 ´xq q

ˆxq x h 1 ˙d ˆ1 ´xq x h 2 ˙d ˆż 1´α xq x h 1 xq dβ 1 β 1 D h 1 q ˆxh 1 β 1 x q , µ F ˙Dh 2 q ˆxh 2 1 ´xq , µ F ż d d ⃗ p 2 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆβ 1 xq 2x h 1 ⃗ p h 1 `1´xq 2x h 2 ⃗ p h 2 ´⃗ p 2 ˙F p⃗ z 1 q
x q p1 ´xq qQ 2 `´1´xq

x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 ˆż d d ⃗ p 2 1 ż d d ⃗ z 2 e i⃗ z 2 ¨ˆβ 1 xq 2x h 1 ⃗ p h 1 `1´xq 2x h 2 ⃗ p h 2 ´⃗ p 2 1
˙F ˚p⃗ z 2 q

x q p1 ´xq qQ 2 `´1´xq

x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2 ˆ2p1 `β2 1 q `2ϵp1 ´β1 q 2 `4ϵp1 `β2 1 q ln β 1 1 ´β1 ˆe´i ´⃗ z 1 ´⃗ z 2 2 ¯¨p1´β 1 qxq x h 1 ⃗ p h 1 α s µ 2ϵ C F ż d d ⃗ p g p2πq d e ´i´⃗ z 12
2 ¯¨⃗ pg p⃗ p g q 2 `ph 1 Ø h 2 q. (6.4.8)

Integrating over ⃗ p g using eq. ( 6.4.4), yields

dσ q qÑh 1 h 2 3LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg " 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1 x h 2 dx q x q x qδp1 ´xq ´xq q ˆxq x h 1 ˙d ˆxq x h 2 ˙d ˆż d d ⃗ p 2 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆxq 2x h 1 ⃗ p h 1 `1´xq 2x h 2 ⃗ p h 2 ´⃗ p 2 ˙F p⃗ z 1 q
x q p1 ´xq qQ 2 `´1´xq

x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 ˆż d d ⃗ p 2 1 ż d d ⃗ z 2 e i⃗ z 2 ¨ˆxq 2x h 1 ⃗ p h 1 `1´xq 2x h 2 ⃗ p h 2 ´⃗ p 2 1
˙F ˚p⃗ z 2 q

x q p1 ´xq qQ 2 `´1´xq

x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2 ˆż 1´α xq x h 1 xq dβ 1 β 1 D h 1 q ˆxh 1 β 1 x q , µ F ˙Dh 2 q ˆxh 2 1 ´xq , µ F αs C F 2π " 1 ε ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙ϵ 1 `β2 1 1 ´β1 `p1 ´β1 q 2 `2p1 `β2 1 q ln β 1 p1 ´β1 q ȷ `ph 1 Ø h 2 q. (6.4.9)
Here in eq. ( 6.4.9) we have put back the integral in x q using ż 1

x h 1 dx q ż 1 x h 2 dx q δp1 ´xq ´xq q " ż 1 x h 1 dx q ż `8
´8 dx q δp1 ´xq ´xq qθp1 ´xq qθpx q ´xh 2 q " ż 1´x h 2

x h 1 dx q (6.4.10) in order to have the same form as in the LO cross-section (6.1.7). Now, to separate the collinear and soft contribution we introduce the plus prescription, as defined in eq. (6.2.6), and after, we expand the factor 1 ϵ ´4c 2 0 ⃗ z 2 12 µ 2 ¯ϵ within accuracy of order ϵ 0 , only in those terms whose integrand is safe in the limit β 1 Ñ 1.

dσ q qÑh 1 h 2 3LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg " 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1
x h 2 dx q x q x qδp1 ´xq ´xq q

ˆxq x h 1 ˙d ˆxq x h 2 ˙d ˆż d d ⃗ p 2 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆxq 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 ˙F p⃗ z 1 q
x q x qQ 2 `´xq

x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 ˆż d d ⃗ p 2 1 ż d d ⃗ z 2 e i⃗ z 2 ¨ˆxq 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 1
˙F ˚p⃗ z 2 q

x q x qQ 2 `´xq

x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2 ˆ#ż 1 x h 1 xq dβ 1 β 1 D h 1 q ˆxh 1 β 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙αs C F 2π 1 ε 1 `β2 1 p1 ´β1 q ż 1´α xq x h 1 xq dβ 1 D h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙αs C F 2π 1 ε ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙ϵ 2 1 ´β1 ´Dh 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙αs C F 2π 1 ε 2 ln ˆ1 ´xh 1 x q ż 1 x h 1 xq dβ 1 β 1 D h 1 q ˆxh 1 β 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙αs C F 2π " ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙1 `β2 1 p1 ´β1 q p1 ´β1 q 2 `2p1 `β2 1 q ln β 1 p1 ´β1 q ȷ ´Dh 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F αs C F 2π 2 ln ˆ1 ´xh 1 x q ˙ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙* `ph 1 Ø h 2 q " dσ q qÑh 1 h 2 3LL dx h 1 dx h 2 d⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg div `dσ q qÑh 1 h 2 3LL dx h 1 dx h 2 d⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg fin , (6.4.11)
where the term denoted by the label "coll. qg div" corresponds to the sum of the first three terms in the curly bracket, whereas the remaining terms in the curly bracket contribute to the term denoted "coll. qg fin".

This gives the following expression for the divergent part:

dσ q qÑh 1 h 2 3LL dx h 1 dx h 2 d⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg div " 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1 x h 2 dx q x q x q ˆxq x h 1 ˙d ˆxq x h 2 ˙d δp1 ´xq ´xq q ˆż d d ⃗ p 2 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆxq 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 ˙F p⃗ z 1 q
x q x qQ 2 `´xq

x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 ˆż d d ⃗ p 2 1 ż d d ⃗ z 2 e i⃗ z 2 ¨ˆxq 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 1
˙F ˚p⃗ z 2 q

x q x qQ 2 `´xq

x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2 ˆαs 2π 1 ε « ż 1 x h 1 xq dβ 1 β 1 C F 1 `β2 1 p1 ´β1 q `Dh 1 q ˆxh 1 β 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ż 1´α xq x h 1 xq dβ 1 C F 2 1 ´β1 ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙ϵ D h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F 2C F ln ˆ1 ´xh 1 x q ˙Dh 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙ȷ `ph 1 Ø h 2 q . ( 6.4.12)
The first term in the bracket cancels part of the first term in the bracket in eq. ( 6.2.7), i.e. the part involving the `prescription in the spitting function P qq , and the remaining part of the P qq term is cancelled by an analogous contribution in the virtual part. The second term in eq. (6.4.12) has to be removed to avoid double counting as it corresponds to the soft contribution and will be taken into account later in the paper. The third and last term, in eq. (6.4.12), will compensate an analogous term in the soft contribution.

The finite part for the LL contribution takes the form

dσ q qÑh 1 h 2 3LL dx h 1 dx h 2 d⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg fin " 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1 x h 2 dx q x q x qδp1 ´xq ´xq q ˆxq x h 1 ˙d ˆxq x h 2 ˙d ˆż d d ⃗ p 2 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆxq 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 ˙F p⃗ z 1 q
x q x qQ 2 `´xq

x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 ˆż d d ⃗ p 2 1 ż d d ⃗ z 2 e i⃗ z 2 ¨ˆxq 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 1
˙F ˚p⃗ z 2 q

x q x qQ 2 `´xq

x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2 ˆαs C F 2π # ż 1 x h 1 xq dβ 1 β 1 D h 1 q ˆxh 1 β 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F "ln ˆ4c 2 0 ⃗ z 2 12 µ 2
˙1 `β2 1 p1 ´β1 q ``p1 ´β1 q 2 `2p1 `β2

1 q ln β 1 p1 ´β1 q ȷ ´2 ln ˆ1 ´xh 1 x q ˙ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙Dh 1 q ˆxh 1 x q , µ F ˙Dhq q ˆxh 2 x q , µ F ˙* `ph 1 Ø h 2 q . (6.4.13)
Similarly, one gets for the T L case

dσ q qÑh 1 h 2 3T L dx h 1 dx h 2 d⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg " α em Q p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1 x h 2
dx q px q ´xq qδp1 ´xq ´xq q

ˆxq x h 1 ˙d ˆxq x h 2 ˙d ˆż d d ⃗ p 2 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆxq 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 ˙F p⃗ z 1 q
x q x qQ 2 `´xq

x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 ˆż d d ⃗ p 2 1 ż d d ⃗ z 2 e i⃗ z 2 ¨ˆxq 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 1 K ˙F ˚p⃗ z 2 q
x q x qQ 2 `´xq

x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2 ˆxq x h 2 ⃗ p h 2 ´⃗ p 2 1 ˙¨⃗ ε T ˆż 1´α xq x h 1 xq dβ 1 β 1 D h 1 q ˆxh 1 β 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F αs C F 2π " 1 ε ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙ϵ 1 `β2 1 1 ´β1 `p1 ´β1 q 2 `2p1 `β2 1 q ln β 1 p1 ´β1 q ȷ `ph 1 Ø h 2 q " dσ q qÑh 1 h 2 3T L dx h 1 dx h 2 d⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg div `dσ q qÑh 1 h 2 3T L dx h 1 dx h 2 d⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg fin , (6.4.14)
where

dσ q qÑh 1 h 2 3T L dx h 1 dx h 2 d⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg div " α em Q p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1 x h 2 dx q px q ´xq q ˆxq x h 1 ˙d ˆxq x h 2 ˙d δp1 ´xq ´xq q ˆż d d ⃗ p 2 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆxq 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 ˙F p⃗ z 1 q
x q x qQ 2 `´xq

x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 ˆż d d ⃗ p 2 1 ż d d ⃗ z 2 e i⃗ z 2 ¨ˆxq 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 1
˙F ˚p⃗ z 2 q

x q x qQ 2 `´xq

x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2 ˆxq x h 2 ⃗ p h 2 ´⃗ p 2 1 ˙¨⃗ ε T ˆαs 2π 1 ε « ż 1 x h 1 xq dβ 1 β 1 C F 1 `β2 1 p1 ´β1 q `Dh 1 q ˆxh 1 β 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ż 1´α xq x h 1 xq dβ 1 C F 2 1 ´β1 ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙ϵ D h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F 2C F ln ˆ1 ´xh 1 x q ˙Dh 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙ȷ `ph 1 Ø h 2 q , and
dσ q qÑh 1 h 2 3T L dx h 1 dx h 2 d⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg fin " α em Q p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1 x h 2
dx q px q ´xq qδp1 ´xq ´xq q

ˆxq x h 1 ˙d ˆxq x h 2 ˙d ˆż d d ⃗ p 2 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆxq 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 ˙F p⃗ z 1 q
x q x qQ 2 `´xq

x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 ˆż d d ⃗ p 2 1 ż d d ⃗ z 2 e i⃗ z 2 ¨ˆxq 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 1
˙F ˚p⃗ z 2 q

x q x qQ 2 `´xq

x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2 ˆxq x h 2 ⃗ p h 2 ´⃗ p 2 1 ˙¨⃗ ε T ˆαs C F 2π # ż 1 x h 1 xq dβ 1 β 1 D h 1 q ˆxh 1 β 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F "ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙1 `β2 1 p1 ´β1 q ``p1 ´β1 q 2 `2p1 `β2 1 q ln β 1 p1 ´β1 q ȷ ´2 ln ˆ1 ´xh 1 x q ˙ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙Dh 1 q ˆxh 1 x q , µ F ˙Dhq q ˆxh 2 x q , µ F ˙* `ph 1 Ø h 2 q , (6.4.15)
and finally, one obtains for the T T case

dσ q qÑh 1 h 2 3T T dx h 1 dx h 2 d⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg " α em 2p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 x h 2 dx q x q δp1 ´xq ´xq q ˆxq x h 1 ˙d ˆxq x h 2 ˙d ˆ"px q ´xq q 2 g ri K g lk K ´grk K g li K `grl K g ik K ı ˆż d d ⃗ p 2 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆxq 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 ˙F p⃗ z 1 q
x q p1 ´xq qQ 2 `´xq

x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 ˆxq x h 2 p h 2 K ´p2K ˙r ε T i ˆż d d ⃗ p 2 1 ż d d ⃗ z 2 e i⃗ z 2 ¨ˆxq 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 1
˙F ˚p⃗ z 2 q

x q x qQ 2 `´xq

x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2 ˆxq x h 2 p h 2 K ´p2 1 K ˙l ε T k ˆż 1´α xq x h 1 xq dβ 1 β 1 D h 1 q ˆxh 1 β 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F αs C F 2π " 1 ε ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙ϵ 1 `β2 1 1 ´β1 `p1 ´β1 q 2 `2p1 `β2 1 q ln β 1 p1 ´β1 q ȷ `ph 1 Ø h 2 q " dσ q qÑh 1 h 2 3T T dx h 1 dx h 2 dp h 1 K d d p h 2 K ˇˇˇc oll. qg div `dσ q qÑh 1 h 2 3T T dx h 1 dx h 2 dp h 1 K d d p h 2 K ˇˇˇc oll. qg fin , (6.4.16)
where

dσ q qÑh 1 h 2 3T T dx h 1 dx h 2 d⃗ p h 1 d d ⃗ p h 2 ˇˇˇc oll. qg div " α em 2p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 x h 2 dx q x q ˆxq
x h 1 ˙d ˆxq x h 2 ˙d δp1 ´xq ´xq q ˆ"px q ´xq q 2 g ri K g lk

K ´grk K g li K `grl K g ik K ı ˆż d d ⃗ p 2 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆxq 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 ˙F p⃗ z 1 q
x q x qQ 2 `´xq

x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 ˆxq x h 2 p h 2 K ´p2K ˙r ε T i ˆż d d ⃗ p 2 1 ż d d ⃗ z 2 e i⃗ z 2 ¨ˆxq 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 1
˙F ˚p⃗ z 2 q

x q x qQ 2 `´xq

x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2 ˆxq x h 2 p h 2 K ´p2 1 K ˙l ε T k ˆαs 2π 1 ε « ż 1 x h 1 xq dβ 1 β 1 C F 1 `β2 1 p1 ´β1 q `Dh 1 q ˆxh 1 β 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ż 1´α xq x h 1 xq dβ 1 C F 2 1 ´β1 ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙ϵ D h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F 2C F ln ˆ1 ´xh 1 x q ˙Dh 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙ȷ `ph 1 Ø h 2 q , and
dσ q qÑh 1 h 2 3T T dx h 1 dx h 2 d⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg fin " α em 2p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 x h 2 dx q x q δp1 ´xq ´xq q ˆxq x h 1 ˙d ˆxq x h 2 ˙d ˆ"px q ´xq q 2 g ri K g lk K ´grk K g li K `grl K g ik K ı ˆż d d ⃗ p 2 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆxq 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 ˙F p⃗ z 1 q
x q x qQ 2 `´xq

x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 ˆxq x h 2 p h 2 K ´p2K ˙r ε T i ˆż d d ⃗ p 2 1 ż d d ⃗ z 2 e i⃗ z 2 ¨ˆxq 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 1
˙F ˚p⃗ z 2 q

x q x qQ 2 `´xq

x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2 ˆxq x h 2 p h 2 K ´p2 1 K ˙l ε T k ˆαs C F 2π # ż 1 x h 1 xq dβ 1 β 1 D h 1 q ˆxh 1 β 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F "ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙1 `β2 1 p1 ´β1 q ``p1 ´β1 q 2 `2p1 `β2 1 q ln β 1 p1 ´β1 q ȷ ´2 ln ˆ1 ´xh 1 x q ˙ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙Dh 1 q ˆxh 1 x q , µ F ˙Dhq q ˆxh 2 x q , µ F ˙* `ph 1 Ø h 2 q . ( 6.4.17)
Collinear contributions: q-g splitting Here the term in eq. (4.6.5) to consider is the third one. The calculation proceeds in the same way as for the quark-gluon collinear contribution but this time the integration is performed over ⃗ p 2 and ⃗ p 2 1 . To observe the cancellation of these collinear divergences, one has to use the different representations we gave for the LO cross-section, as explained before, see eqs. (6.1.10, 6.1.11) with respect to eqs. (6.1.7, 6.1.8).

This time, the change of variable to be done is

x 1 q " β 2 x q, x g " p1 ´β2 qx q .

(6.4.18)

This kinematics is illustrated in fig. 6.4.2. The boundaries of integration for px q, β 2 q are calculated in the same spirit as in eq. ( 6.4.7). Thus, after changes of variable and integrations, the third term in (6.4.1) takes the form

dσ q qÑh 1 h 2 3LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg " 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1
x h 2 dx q x q x q δp1 ´xq ´xq q

ˆxq x h 1 ˙d ˆxq x h 2 ˙d
x q, ⃗ p q `⃗ p g x q , ⃗ p q

x g " p1 ´β2 qx q, ⃗ p g

x 1 q " β 2 x q, ⃗ p q Figure 6.4.2: Kinematics for the q ´g splitting contribution. We indicate the longitudinal fraction of momentum carried by the partons as well as their transverse momenta.

ˆż d d ⃗ p 1 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆ´x q 2x h 1 ⃗ p h 1 ´xq 2x h 2 ⃗ p h 2 `⃗ p 1 ˙F p⃗ z 1 q
x q x qQ 2 `´xq

x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 ˆż d d ⃗ p 1 1 ż d d ⃗ z 2 e i⃗ z 2 ¨ˆ´x q 2x h 1 ⃗ p h 1 ´xq 2x h 2 ⃗ p h 2 `⃗ p 1 1
˙F ˚p⃗ z 2 q

x q x qQ 2 `´xq

x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 ˆż 1´α x q x h 2 x q dβ 2 β 2 D h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 β 2 x q , µ F αs C F 2π " 1 ε ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙ϵ 1 `β2 2 1 ´β2 `p1 ´β2 q 2 `2p1 `β2 2 q ln β 2 p1 ´β2 q ȷ `ph 1 Ø h 2 q " dσ q qÑh 1 h 2 3LL dx h 1 dx h 2 d⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg div `dσ q qÑh 1 h 2 3LL dx h 1 dx h 2 d⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg fin . (6.4.19)
Adding the `prescription and expanding up to ϵ 0 , just like for the collinear qg contribution, one gets the finite and divergent part of eq. ( 6.4.19).

The divergent part is

dσ q qÑh 1 h 2 3LL dx h 1 dx h 2 d⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg div " 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1 x h 2 dx q x q x q ˆxq x h 1 ˙d ˆxq x h 2 ˙d δp1 ´xq ´xq q ˆż d d ⃗ p 1 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆ´x q 2x h 1 ⃗ p h 1 ´xq 2x h 2 ⃗ p h 2 `⃗ p 1 ˙F p⃗ z 1 q
x q x qQ 2 `´xq

x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 ˆż d d ⃗ p 1 1 ż d d ⃗ z 2 e ´i⃗ z 2 ¨ˆ´x q 2x h 1 ⃗ p h 1 ´xq 2x h 2 ⃗ p h 2 `⃗ p 1 1
˙F ˚p⃗ z 2 q

x q x qQ 2 `´xq

x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 ˆαs 2π 1 ε » - ż 1 x h 2 x q dβ 2 β 2 C F 1 `β2 2 p1 ´β2 q `Dh 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 β 2 x q , µ F ż 1´α xq x h 2 x q dβ 2 C F 2 1 ´β2 ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙ϵ D h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F 2C F ln ˆ1 ´xh 2 x q ˙Dh 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙ȷ `ph 1 Ø h 2 q . (6.4.20)
The first term cancels with the + prescription term in the second P qq in eq. ( 6.2.7). We have to remove the second term to avoid double counting with the soft contribution. The third term is to be removed by the soft contribution.

The finite part is

dσ q qÑh 1 h 2 3LL dx h 1 dx h 2 d⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg fin " 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1
x h 2 dx q x q x q δp1 ´xq ´xq q

ˆxq x h 1 ˙d ˆxq x h 2 ˙d ˆż d d ⃗ p 1 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆ´x q 2x h 1 ⃗ p h 1 ´xq 2x h 2 ⃗ p h 2 `⃗ p 1 ˙F p⃗ z 1 q
x q x qQ 2 `´xq

x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 ˆż d d ⃗ p 1 1 ż d d ⃗ z 2 e ´i⃗ z 2 ¨ˆ´x q 2x h 1 ⃗ p h 1 ´xq 2x h 2 ⃗ p h 2 `⃗ p 1 1
˙F ˚p⃗ z 2 q

x q x qQ 2 `´xq

x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 ˆαs C F 2π $ & % ż 1 x h 2 x q dβ 2 β 2 D h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 β 2 x q , µ F "ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙1 `β2 2 p1 ´β2 q ``p1 ´β2 q 2 `2p1 `β2 2 q ln β 2 p1 ´β2 q `ȷ ´2 ln ˆ1 ´xh 2 x q ˙ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙Dh 1 q ˆxh 1 x q , µ F ˙Dhq q ˆxh 2 x q , µ F ˙* `ph 1 Ø h 2 q . (6.4.21)
For the T L transition,

dσ q qÑh 1 h 2 3T L dx h 1 dx h 2 d⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg " α em Q p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1
x h 2 dx q px q ´xq q δp1 ´xq ´xq q

ˆxq x h 1 ˙d ˆxq x h 2 ˙d ˆż d d ⃗ p 1 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆ´x q 2x h 1 ⃗ p h 1 ´xq 2x h 2 ⃗ p h 2 `⃗ p 1 ˙F p⃗ z 1 q
x q x qQ 2 `´xq

x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 ˆż d d ⃗ p 1 1 ż d d ⃗ z 2 e i⃗ z 2 ¨ˆ´x q 2x h 1 ⃗ p h 1 ´xq 2x h 2 ⃗ p h 2 `⃗ p 1 1
˙F ˚p⃗ z 2 q

x q x qQ 2 `´xq

x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 ˆxq x h 1 p h 1 K ´p1 1 K ˙¨ε T ˆż 1´α x q x h 2 x q dβ 2 β 2 D h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 β 2 x q , µ F αs C F 2π " 1 ε ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙ϵ 1 `β2 2 1 ´β2 `p1 ´β2 q 2 `2p1 `β2 2 q ln β 2 p1 ´β2 q ȷ `ph 1 Ø h 2 q " dσ q qÑh 1 h 2 3T L dx h 1 dx h 2 d⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg div `dσ q qÑh 1 h 2 3T L dx h 1 dx h 2 d⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg fin , (6.4.22)
where

dσ q qÑh 1 h 2 3T L dx h 1 dx h 2 d⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg div " α em Q p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1 x h 2 dx q px q ´xq q ˆxq x h 1 ˙d ˆxq x h 2 ˙d δp1 ´xq ´xq q ˆż d d ⃗ p 1 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆ´x q 2x h 1 ⃗ p h 1 ´xq 2x h 2 ⃗ p h 2 `⃗ p 1 ˙F p⃗ z 1 q
x q x qQ 2 `´xq

x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 ˆż d d ⃗ p 1 1 ż d d ⃗ z 2 e i⃗ z 2 ¨ˆ´x q 2x h 1 ⃗ p h 1 ´xq 2x h 2 ⃗ p h 2 `⃗ p 1 1 ˙F ˚p⃗ z 2 q
x q x qQ 2 `´xq

x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 ˆxq x h 1 p h 1 K ´p1 1 K ˙¨ε T ˆαs 2π 1 ε » - ż 1 x h 2 x q dβ 2 β 2 C F 1 `β2 2 p1 ´β2 q `Dh 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 β 2 x q , µ F ż 1´α x q x h 2 x q dβ 2 C F 2 1 ´β2 ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙ϵ D h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ´2C F ln ˆ1 ´xh 2 x q ˙Dh 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙ȷ `ph 1 Ø h 2 q , (6.4.23)
and

dσ q qÑh 1 h 2 3T L dx h 1 dx h 2 d⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg fin " α em Q p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1
x h 2 dx q px q ´xq q δp1 ´xq ´xq q

ˆxq x h 1 ˙d ˆxq x h 2 ˙d ˆż d d ⃗ p 1 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆ´x q 2x h 1 ⃗ p h 1 ´xq 2x h 2 ⃗ p h 2 `⃗ p 1 ˙F p⃗ z 1 q
x q x qQ 2 `´xq

x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 ˆż d d ⃗ p 1 1 ż d d ⃗ z 2 e i⃗ z 2 ¨ˆ´x q 2x h 1 ⃗ p h 1 ´xq 2x h 2 ⃗ p h 2 `⃗ p 1 1
˙F ˚p⃗ z 2 q

x q x qQ 2 `´xq

x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 ˆxq x h 1 p h 1 K ´p1 1 K ˙¨ε T ˆαs C F 2π $ & % ż 1 x h 2 x q dβ 2 β 2 D h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 β 2 x q , µ F "ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙1 `β2 2 p1 ´β2 q ``p1 ´β2 q 2 `2p1 `β2 2 q ln β 2 p1 ´β2 q `ȷ ´2 ln ˆ1 ´xh 2 x q ˙ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙Dh 1 q ˆxh 1 x q , µ F ˙Dhq q ˆxh 2 x q , µ F ˙* `ph 1 Ø h 2 q . ( 6.4.24)
For the TT case, we get

dσ q qÑh 1 h 2 3T T dx h 1 dx h 2 d⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg " α em 2p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 x h 2 dx q x q δp1 ´xq ´xq q ˆxq x h 1 ˙d ˆxq x h 2 ˙d ˆ"px q ´xq q 2 g ri K g lk K ´grk K g li K `grl K g ik K ı ˆż d d ⃗ p 1 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆ´x q 2x h 1 ⃗ p h 1 ´xq 2x h 2 ⃗ p h 2 `⃗ p 1 ˙F p⃗ z 1 q
x q x qQ 2 `´xq

x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 ˆxq x h 1 p h 1 K ´p1K ˙r ε T i ˆż d d ⃗ p 1 1 ż d d ⃗ z 2 e i⃗ z 2 ¨ˆ´x q 2x h 1 ⃗ p h 1 ´xq 2x h 2 ⃗ p h 2 `⃗ p 1 1
˙F ˚p⃗ z 2 q

x q x qQ 2 `´xq

x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 ˆxq x h 1 p h 1 K ´p1 1 K ˙l ε T k ˆż 1´α x q x h 2 x q dβ 2 β 2 D h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 β 2 x q , µ F ˆαs C F 2π " 1 ε ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙ϵ 1 `β2 2 1 ´β2 `p1 ´β2 q 2 `2p1 `β2 2 q ln β 2 p1 ´β2 q ȷ `ph 1 Ø h 2 q " dσ q qÑh 1 h 2 3T T dx h 1 dx h 2 d⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg div `dσ q qÑh 1 h 2 3T T dx h 1 dx h 2 d⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg fin , (6.4.25)
where

dσ q qÑh 1 h 2 3T T dx h 1 dx h 2 d⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg div " α em 2p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 x h 2 dx q x q ˆxq x h 1 ˙d ˆxq x h 2 ˙d δp1 ´xq ´xq q ˆ"px q ´xq q 2 g ri K g lk K ´grk K g li K `grl K g ik K ı ˆż d d ⃗ p 1 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆ´x q 2x h 1 ⃗ p h 1 ´xq 2x h 2 ⃗ p h 2 `⃗ p 1 ˙F p⃗ z 1 q
x q x qQ 2 `´xq

x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 ˆxq x h 1 p h 1 K ´p1K ˙r ε T i ˆż d d ⃗ p 1 1 ż d d ⃗ z 2 e i⃗ z 2 ¨ˆ´x q 2x h 1 ⃗ p h 1 ´xq 2x h 2 ⃗ p h 2 `⃗ p 1 1
˙F ˚p⃗ z 2 q

x q x qQ 2 `´xq

x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 ˆxq x h 1 p h 1 K ´p1 1 K ˙l ε T k ˆαs 2π 1 ε » - ż 1 x h 2 x q dβ 2 β 2 C F 1 `β2 2 p1 ´β2 q `Dh 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 β 2 x q , µ F ż 1´α x q x h 2 x q dβ 2 C F 2 1 ´β2 ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙ϵ D h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F 2C F ln ˆ1 ´xh 2 x q ˙Dh 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙ȷ `ph 1 Ø h 2 q , (6.4.26)
and

dσ q qÑh 1 h 2 3T T dx h 1 dx h 2 d⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg fin " α em 2p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 x h 2 dx q x q δp1 ´xq ´xq q ˆxq x h 1 ˙d ˆxq x h 2 ˙d ˆ"px q ´xq q 2 g ri K g lk K ´grk K g li K `grl K g ik K ı ˆż d d ⃗ p 1 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆ´x q 2x h 1 ⃗ p h 1 ´xq 2x h 2 ⃗ p h 2 `⃗ p 1 ˙F p⃗ z 1 q
x q x qQ 2 `´xq

x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 ˆxq x h 1 p h 1 K ´p1K ˙r ε T i ˆż d d ⃗ p 1 1 ż d d ⃗ z 2 e i⃗ z 2 ¨ˆ´x q 2x h 1 ⃗ p h 1 ´xq 2x h 2 ⃗ p h 2 `⃗ p 1 1
˙F ˚p⃗ z 2 q

x q x qQ 2 `´xq

x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 ˆxq x h 1 p h 1 K ´p1 1 K ˙l ε T k ˆαs C F 2π $ & % ż 1 x h 2 x q dβ 2 β 2 D h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 β 2 x q , µ F "ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙1 `β2 2 p1 ´β2 q ``p1 ´β2 q 2 `2p1 `β2 2 q ln β 2 p1 ´β2 q `ȷ ´2 ln ˆ1 ´xh 2 x q ˙ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙Dh 1 q ˆxh 1 x q , µ F ˙Dhq q ˆxh 2 x q , µ F ˙* `ph 1 Ø h 2 q . (6.4.27)

Soft contribution

To calculate the soft contribution of the divergent part of the real emission cross-section, the soft limit of eq. ( 6.4.1) is taken by setting ⃗ p g " x g ⃗ u, where |⃗ u | " |⃗ p h |, which extracts the divergence on x g .

dσ q qÑh 1 h 2 3LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇs oft " 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx 1 q x 1 q ż 1 x h 2 dx 1 q x 1 q D h 1 q ˆxh 1 x 1 q , µ F ˙Dh 2 q ˆxh 2 x 1 q , µ F ẋ1 q x h 1 ˙d ˆx1 q x h 2 ˙d ż 1 α dx g x 3´d g δp1 ´x1 q ´x1 q ´xg q α s C F µ 2ϵ ż d d ⃗ u p2πq d ˆż d d ⃗ p 1 d d ⃗ p 2 F ˆ⃗ p 12 2 ˙δ ˆx1 q x h 1 ⃗ p h 1 ´⃗ p 1 `x1 q x h 2 ⃗ p h 2 ´⃗ p 2 `xg ⃗ u ż d d ⃗ p 1 1 d d ⃗ p 2 1 F ˚ˆ⃗ p 1 1 2 1 2 ˙δ ˆx1 q x h 1 ⃗ p h 1 ´⃗ p 1 1 `x1 q x h 2 ⃗ p h 2 ´⃗ p 2 1 `xg ⃗ u $ ' ' ' ' ' ' ' & ' ' ' ' ' ' ' % dx 2 g `4x 1 q px 1 q `xg q ¨Q2 `ˆx 1 q x h 2 ⃗ p h 2 ´⃗ p 2 ˙2 p1´x 1 q qx 1 q '¨Q 2 `ˆx 1 q x h 2 ⃗ p h 2 ´⃗ p 2 1 ˙2 p1´x 1 q qx 1 q 'x 12 q ´⃗ u ´⃗ p h 1 x h 1 ¯2 `dx 2 g `4x 1 qpx 1 q `xg q ¨Q2 `ˆx 1 q x h 2 ⃗ p h 2 ´⃗ p 2 `xg⃗ u ˙2 p1´x 1 q qx 1 q '¨Q 2 `ˆx 1 q x h 2 ⃗ p h 2 ´⃗ p 2 1 `xg⃗ u ˙2 p1´x 1 q qx 1 q 'x 12 q ´⃗ u ´⃗ p h 2 x h 2 ¯2
´"2x g ´dx 2

g `4x 1 q x 1 q‰ ´⃗ u ´⃗ p h 1 x h 1 ¯¨´⃗ u ´⃗ p h 2 x h 2 Q2 `ˆx 1 q x h 2 ⃗ p h 2 ´⃗ p 2 1 ˙2 p1´x 1 q qx 1 q '¨Q 2 `ˆx 1 q x h 2 ⃗ p h 2 ´⃗ p 2 `xg⃗ u ˙2 p1´x 1 q qx 1 q 'x 1 q x 1 q ´⃗ u ´⃗ p h 1 x h 1 ¯2 ´⃗ u ´⃗ p h 2 x h 2 ¯2
´"2x g ´dx 2

g `4x 1 q x 1 q‰ ´⃗ u ´⃗ p h 1 x h 1 ¯¨´⃗ u ´⃗ p h 2 x h 2 Q2 `ˆx 1 q x h 2 ⃗ p h 2 ´⃗ p 2 ˙2 p1´x 1 q qx 1 q '¨Q 2 `ˆx 1 q x h 2 ⃗ p h 2 ´⃗ p 2 1 `xg⃗ u ˙2 p1´x 1 q qx 1 q 'x 1 q x 1 q ´⃗ u ´⃗ p h 1 x h 1 ¯2 ´⃗ u ´⃗ p h 2 x h 2 ¯2 , / / / / / / / . / / / / / / / -
`ph 1 Ø h 2 q . (6.4.28)

The limit x g Ñ 0 in the F function and impact factor can be taken safely in the non-divergent terms of the cross-section, as x 1 q and x 1 q are limited from below by x h 1 , x h 2 and so cannot be arbitrary small (i.e. of order x g ). The cross-section in the soft limit becomes:

dσ q qÑh 1 h 2 3LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇs oft " 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx 1 q x 1 q ż 1 x h 2 dx 1 q x 1 q D h 1 q ˆxh 1 x 1 q , µ F ˙Dh 2 q ˆxh 2 x 1 q , µ F ẋ1 q x h 1 ˙d ˆx1 q x h 2 ˙d ż 1 α dx g x 3´d g δp1 ´x1 q ´x1 q ´xg q α s C F µ 2ϵ ż d d ⃗ u p2πq d ˆż d d ⃗ p 1 d d ⃗ p 2 F ˆ⃗ p 12 2 ˙δ ˆx1 q x h 1 ⃗ p h 1 ´⃗ p 1 `x1 q x h 2 ⃗ p h 2 ´⃗ p 2 ż d d ⃗ p 1 1 d d ⃗ p 2 1 F ˚ˆ⃗ p 1 1 2 1 2 ˙δ ˆx1 q x h 1 ⃗ p h 1 ´⃗ p 1 1 `x1 q x h 2 ⃗ p h 2 ´⃗ p 2 1 $ ' ' ' ' ' ' ' & ' ' ' ' ' ' ' % 4 ¨Q2 `ˆx 1 q x h 2 ⃗ p h 2 ´⃗ p 2 ˙2 p1´x 1 q qx 1 q '¨Q 2 `ˆx 1 q x h 2 ⃗ p h 2 ´⃗ p 2 1 ˙2 p1´x 1 q qx 1 q '´⃗ u ´⃗ p h 1 x h 1 ¯2 `4 ¨Q2 `ˆx 1 q x h 2 ⃗ p h 2 ´⃗ p 2 ˙2 p1´x 1 q qx 1 q '¨Q 2 `ˆx 1 q x h 2 ⃗ p h 2 ´⃗ p 2 1 ˙2 p1´x 1 q qx 1 q '´⃗ u ´⃗ p h 2 x h 2 ¯2 ´4 ´⃗ u ´⃗ p h 1 x h 1 ¯¨´⃗ u ´⃗ p h 2 x h 2 Q2 `ˆx 1 q x h 2 ⃗ p h 2 ´⃗ p 2 1 ˙2 p1´x 1 q qx 1 q '¨Q 2 `ˆx 1 q x h 2 ⃗ p h 2 ´⃗ p 2 ˙2 p1´x 1 q qx 1 q '´⃗ u ´⃗ p h 1 x h 1 ¯2 ´⃗ u ´⃗ p h 2 x h 2 ¯2 ´4 ´⃗ u ´⃗ p h 1 x h 1 ¯¨´⃗ u ´⃗ p h 2 x h 2 Q2 `ˆx 1 q x h 2 ⃗ p h 2 ´⃗ p 2 ˙2 p1´x 1 q qx 1 q '¨Q 2 `ˆx 1 q x h 2 ⃗ p h 2 ´⃗ p 1 2 ˙2 p1´x 1 q qx 1 q '´⃗ u ´⃗ p h 1 x h 1 ¯2 ´⃗ u ´⃗ p h 2 x h 2 ¯2 , / / / / / / / . / / / / / / / - `ph 1 Ø h 2 q " 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx 1 q x 1 q ż 1 x h 2 dx 1 q x 1 q D h 1 q ˆxh 1 x 1 q , µ F ˙Dh 2 q ˆxh 2 x 1 q , µ F ẋ1 q x h 1 ˙d ˆx1 q x h 2 ˙d ż 1 α dx g x 3´d g δp1 ´x1 q ´x1 q ´xg q α s C F µ 2ϵ ż d d ⃗ u p2πq d ˆż d d ⃗ p 1 d d ⃗ p 2 F ˆ⃗ p 12 2 ˙δ ˆx1 q x h 1 ⃗ p h 1 ´⃗ p 1 `x1 q x h 2 ⃗ p h 2 ´⃗ p 2 ż d d ⃗ p 1 1 d d ⃗ p 2 1 F ˚ˆ⃗ p 1 1 2 1 2 ˙δ ˆx1 q x h 1 ⃗ p h 1 ´⃗ p 1 1 `x1 q x h 2 ⃗ p h 2 ´⃗ p 2 1 4 ¨Q2 `ˆx 1 q x h 2 ⃗ p h 2 ´⃗ p 2 ˙2 p1´x 1 q qx 1 q '¨Q 2 `ˆx 1 q x h 2 ⃗ p h 2 ´⃗ p 2 1 ˙2 p1´x 1 q qx 1 q '$ ' & ' % 1 ´⃗ u ´⃗ p h 1 x h ¯2 `1 ´⃗ u ´⃗ p h 2 x h 2 ¯2 ´2 ´⃗ u ´⃗ p h 1 x h 1 ¯¨´⃗ u ´⃗ p h 2 x h 2 ⃗ u ´⃗ p h 1 x h 1 ¯2 ´⃗ u ´⃗ p h 2 x h 2 ¯2 , / .
/ -`ph 1 Ø h 2 q . (6.4.29)

Then, the cross-section is divided into two parts in order to do two different changes of variables, which are the same changes as in eqs. (6.4.6, 6.4.18), in each part:

x 1 q " β 1 x q , x g " p1 ´β1 qx q , (6.4.30)

x 1 q " β 2 x q , x g " p1 ´β2 qx q , (6.4.31)

where the integration boundaries are calculated following the steps in eq. ( 6.4.7). This division and changes of variable respect the symmetry between diagrams (1) + (2) on one side, and (3) + (4) on the other side in fig. 6.1.6. The limits β 1,2 Ñ 1, corresponding to x g Ñ 0 are then taken. The choice of splitting the cross-section in this way, comes from the will to observe the cancellation of divergences at integrand level. The first term, after the transformation (6.4.30)

and after taking the limit, gives

dσ q qÑh 1 h 2 3LL dx h 1 dx h 2 d d ⃗ p h 1 d d p h 2 K ˇˇˇˇs oft β 1 " α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1´x h 2 x h 1 dx q x q 1 1 ´xq ˆxq x h 1 ˙d ˆ1 ´xq x h 2 ˙d ˆDh 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 1 ´xq , µ F ż d d ⃗ p 2 F ˆxq 2x h 1 ⃗ p h 1 `1 ´xq 2x h 2 ⃗ p h 2 ´⃗ p 2 ż d d ⃗ p 2 1 F ˚ˆx q 2x h 1 ⃗ p h 1 `1 ´xq 2x h 2 ⃗ p h 2 ´⃗ p 2 1 ż 1´α xq x h 1 xq dβ 1 p1 ´β1 q 1´2ϵ x 1´2ϵ q x q ˆ4p1 ´xq q 2 x 2 q ˆp1 ´xq qx q Q 2 `´1´xq x h 2 ⃗ p h 2 ´⃗ p 2 ¯2˙ˆp 1 ´xq qx q Q 2 `´1´xq x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2α s C F µ 2ϵ ż d d ⃗ u p2πq d $ ' & ' % 1 ´⃗ u ´⃗ p h 1 x h 1 ¯2 `1 ´⃗ u ´⃗ p h 2 x h 2 ¯2 ´2 ´⃗ u ´⃗ p h 1 x h 1 ¯¨´⃗ u ´⃗ p h 2 x h 2 ⃗ u ´⃗ p h 1 x h 1 ¯2 ´⃗ u ´⃗ p h 2 x h 2 ¯2 , / . / - `ph 1 Ø h 2 q . (6.4.32)
In a similar fashion, the transformation (6.4.31) leads to second contribution, which reads

dσ q qÑh 1 h 2 3LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇs oft β 2 " α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1´x h 1 x h 2 dx q x q 1 1 ´xq ˆxq x h 2 ˙d ˆ1 ´xq x h 1 ˙d ˆDh 1 q ˆxh 1 1 ´xq , µ F ˙Dh 2 q ˆxh 2 x q , µ F ż d d ⃗ p 2 F ˆ1 ´xq 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 ż d d ⃗ p 2 1 F ˚ˆ1 ´xq 2x h 1 ⃗ p h 1 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 1 ż 1´α x q x h 2
x q dβ 2 p1 ´β2 q 1´2ϵ x 1´2ϵ q x q ˆ4p1 ´xq q 2 x 2 q ˆp1 ´xq qx qQ 2 `´xq

x h 2 ⃗ p h 2 ´⃗ p 2 ¯2˙ˆp 1 ´xq qx qQ 2 `´xq x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2α s C F µ 2ϵ ż d d ⃗ u p2πq d $ ' & ' % 1 ´⃗ u ´⃗ p h 1 x h 1 ¯2 `1 ´⃗ u ´⃗ p h 2 x h 2 ¯2 ´2 ´⃗ u ´⃗ p h 1 x h 1 ¯¨´⃗ u ´⃗ p h 2 x h 2 ⃗ u ´⃗ p h 1 x h 1 ¯2 ´⃗ u ´⃗ p h 2 x h 2 ¯2 , / . / - `ph 1 Ø h 2 q . (6.4.33)
Next, we integrate over ⃗ u, which gives

I u " α s C F µ 2ϵ ż d d ⃗ u p2πq d $ ' & ' % 1 ´⃗ u ´⃗ p h 1 x h ¯2 `1 ´⃗ u ´⃗ p h 2 x h 2 ¯2 ´2 ´⃗ u ´⃗ p h 1 x h 1 ¯¨´⃗ u ´⃗ p h 2 x h 2 ⃗ u ´⃗ p h 1 x h 1 ¯2 ´⃗ u ´⃗ p h 2 x h 2 ¯2 , / .
/ - Combining both integrals over β eq. (6.4.35) and over ⃗ u eq. ( 6.4.34), reintroducing x q , x q with eq. (6.4.10) and keeping only the divergent terms, eqs. (6.4.32) and (6.4.33) become respectively

" α s C F µ 2ϵ ˆ⃗ p h 1 x h 1 ´⃗ p h 2 x h 2 ˙2 ż d d ⃗ u p2πq d 1 ´⃗ u ´⃗ p h 1 x h 1 ¯2 ´⃗ u ´⃗ p h 2 x h 2 ¯2 " α s C F µ 2ϵ ˆ⃗ p h 1 x h 1 ´⃗ p h 2 x h 2 ˙2 ż d d ⃗ u p2πq d 1 ⃗ u 2 ´⃗ u ´´⃗ p h 1 x h 1 ´⃗ p h 2 x h 2 ¯¯2 " α s C F µ 2ϵ ˆ⃗ p h 1 x h 1 ´⃗ p h 2 x h 2 ˙2 1 p2πq d π 1`ϵ Γp1 ´ϵqβpϵ, ϵq « ˆ⃗ p h 1 x h 1 ´⃗ p h 2 x h 2 ˙2ff ϵ´1 " α s 2π C F 1 ε » - -1 `ϵ ln ¨´⃗ p h 1 x h 1 ´⃗ p h 2 x h 2 ¯2 µ 2 ‹ ' fi ffi fl . ( 6
dσ q qÑh 1 h 2 3LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇs oft β 1 " α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1 x h 2 dx q x q x q ˆxq x h 1 ˙d ˆxq x h 2 ˙d δp1 ´xq ´xq q ˆDh 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙FLL α S C F 2π 4 ε " ´ln α `ln x q `ln ˆ1 ´xh 1 x q ε ln 2 α ´ϵ ln α ln ¨´⃗ p h 1 x h 1 ´⃗ p h 2 x h 2 ¯2 µ 2 ‹ '`ϵ ln x q ˜ln x q `2 ln ˆ1 ´xh 1 x q ln ¨´⃗ p h 1 x h 1 ´⃗ p h 2 x h 2 ¯2 µ 2 ‹ ' ‹ '`ϵ ln ˆ1 ´xh 1 x q ˙¨l n ¨´⃗ p h 1 x h 1 ´⃗ p h 2 x h 2 ¯2 µ 2 ‹ '`ln ˆ1 ´xh 1 x q ˙‹ ' fi ffi fl `ph 1 Ø h 2 q , and
dσ q qÑh 1 h 2 3LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇs oft β 2 " α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1 x h 2 dx q x q x q ˆxq x h 1 ˙d ˆxq x h 2 ˙d δp1 ´xq ´xq q ˆDh 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙FLL α S C F 2π 4 ε " ´ln α `ln x q `ln ˆ1 ´xh 2 x q ε ln 2 α ´ϵ ln α ln ¨´⃗ p h 1 x h 1 ´⃗ p h 2 x h 2 ¯2 µ 2 ‹ '`ϵ ln x q˜l n x q `2 ln ˆ1 ´xh 2 x q ln ¨´⃗ p h 1 x h 1 ´⃗ p h 2 x h 2 ¯2 µ 2 ‹ ' ‹ '`ϵ ln ˆ1 ´xh 2 x q ˙¨l n ¨´⃗ p h 1 x h 1 ´⃗ p h 2 x h 2 ¯2 µ 2 ‹ '`ln ˆ1 ´xh 2 x q ˙‹ ' fi ffi fl `ph 1 Ø h 2 q .
As said above, the total soft contribution expression is found by summing the two above equations. As usual, we split the final result into divergent and finite part. In the LL case, we obtain

dσ q qÑh 1 h 2 3LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇs oft div " 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1 x h 2 dx q x q x q ˆxq x h 1 ˙d ˆxq x h 2 ˙d ˆδp1 ´xq ´xq qD h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙FLL ˆαs C F 2π 1 ε " ´4 ln α `2 ln x q `2 ln ˆ1 ´xh 1 x q ˙´4ϵ ln 2 α ´4ϵ ln α ln ¨´⃗ p h 1 x h 1 ´⃗ p h 2 x h 2 ¯2 µ 2 ‹ '`2 ln x q `2 ln ˆ1 ´xh 2
x q ˙fi ffi fl `ph 1 Ø h 2 q , (6.4.36)

and

dσ q qÑh 1 h 2 3LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇs oft fin " 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1 x h 2 dx q x q x q ˆxq x h 1 ˙d ˆxq x h 2 ˙d ˆδp1 ´xq ´xq qD h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙FLL ˆαs C F π » - -ln x q ¨ln x q `2 ln ˆ1 ´xh 1 x q ˙`ln ¨´⃗ p h 1 x h 1 ´⃗ p h 2 x h 2 ¯2 µ 2 ‹ ' ‹ ' `ln ˆ1 ´xh 1 x q ˙¨l n ¨´⃗ p h 1 x h 1 ´⃗ p h 2 x h 2 ¯2 µ 2 ‹ '`ln ˆ1 ´xh 1 x q ˙‹ ' `ln x q˜l n x q `2 ln ˆ1 ´xh 2 x q ˙`ln ¨´⃗ p h 1 x h 1 ´⃗ p h 2 x h 2 ¯2 µ 2 ‹ ' ‹ ' `ln ˆ1 ´xh 2 x q ˙¨l n ¨´⃗ p h 1 x h 1 ´⃗ p h 2 x h 2 ¯2 µ 2 ‹ '`ln ˆ1 ´xh 2 x q ˙‹ ' fi ffi fl `ph 1 Ø h 2 q . ( 6.4.37)
For the T L case, the calculation leads to

dσ q qÑh 1 h 2 3T L dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇs oft div " α em Q p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1 x h 2 dx q px q ´xq q ˆxq x h 1 ˙d ˆxq x h 2 ˙d ˆδp1 ´xq ´xq qD h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙FT L ˆαs C F 2π 1 ε " ´4 ln α `2 ln x q `2 ln ˆ1 ´xh 1 x q ˙´4ϵ ln 2 α ´4ϵ ln α ln ¨´⃗ p h 1 x h 1 ´⃗ p h 2 x h 2 ¯2 µ 2 ‹ '`2 ln x q `2 ln ˆ1 ´xh 2 x q ˙fi ffi fl `ph 1 Ø h 2 q , (6.4.38)
and

dσ q qÑh 1 h 2 3T L dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇs oft fin " α em Q p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1 x h 2 dx q px q ´xq q ˆxq x h 1 ˙d ˆxq x h 2 ˙d ˆδp1 ´xq ´xq qD h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙FT L ˆαs C F π » - -ln x q ¨ln x q `2 ln ˆ1 ´xh 1 x q ˙`ln ¨´⃗ p h 1 x h 1 ´⃗ p h x h ¯2 µ 2 ‹ ' ‹ ' `ln ˆ1 ´xh 1 x q ˙¨l n ¨´⃗ p h 1 x h 1 ´⃗ p h 2 x h 2 ¯2 µ 2 ‹ '`ln ˆ1 ´xh 1 x q ˙‹ ' `ln x q˜l n x q `2 ln ˆ1 ´xh 2 x q ˙`ln ¨´⃗ p h 1 x h 1 ´⃗ p h 2 x h 2 ¯2 µ 2 ‹ ' ‹ ' `ln ˆ1 ´xh 2 x q ˙¨l n ¨´⃗ p h 1 x h 1 ´⃗ p h 2 x h 2 ¯2 µ 2 ‹ '`ln ˆ1 ´xh 2 x q ˙‹ ' fi ffi fl `ph 1 Ø h 2 q . (6.4.39)
Finally, the results for the T T case read

dσ q qÑh 1 h 2 3T T dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇs oft div " α em 2p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 x h 2 dx q x q ˆxq x h 1 ˙d ˆxq x h 2 ˙d ˆδp1 ´xq ´xq qD h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙FT T ˆαs C F 2π 1 ε " ´4 ln α `2 ln x q `2 ln ˆ1 ´xh 1 x q ˙´4ϵ ln α ´4ϵ ln α ln ¨´⃗ p h 1 x h 1 ´⃗ p h 2 x h 2 ¯2 µ 2 ‹ '`2 ln x q `2 ln ˆ1 ´xh 2
x q ˙fi ffi fl `ph 1 Ø h 2 q , (6.4.40)

and

dσ q qÑh 1 h 2 3T T dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇs oft fin " α em 2p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 x h 2 dx q x q ˆxq x h 1 ˙d ˆxq x h 2 ˙d ˆδp1 ´xq ´xq qD h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙FT T ˆαs C F π » - -ln x q ¨ln x q `2 ln ˆ1 ´xh 1 x q ˙`ln ¨´⃗ p h 1 x h 1 ´⃗ p h x h ¯2 µ 2 ‹ ' ‹ ' `ln ˆ1 ´xh 1 x q ˙¨l n ¨´⃗ p h 1 x h 1 ´⃗ p h 2 x h 2 ¯2 µ 2 ‹ '`ln ˆ1 ´xh 1 x q ˙‹ ' `ln x q˜l n x q `2 ln ˆ1 ´xh 2 x q ˙`ln ¨´⃗ p h 1 x h 1 ´⃗ p h 2 x h 2 ¯2 µ 2 ‹ ' ‹ ' `ln ˆ1 ´xh 2 x q ˙¨l n ¨´⃗ p h 1 x h 1 ´⃗ p h 2 x h 2 ¯2 µ 2 ‹ '`ln ˆ1 ´xh 2 x q ˙‹ ' fi ffi fl `ph 1 Ø h 2 q . (6.4.41)
At this level, we are already able to observe the full cancellation of soft divergences (and hence the disappearance of ln α-terms). Consider, for instance, the longitudinal cross-section.

Combining the divergent soft contribution, coming from the real part, see eq. ( 6.4.36), with the virtual contribution eq. ( 6.3.1) we see the complete cancellation of these ln α-terms and also of 1 ϵ lnpx q x qq-term. Moreover, surviving 1 ϵ divergent terms cancel in combination with:

• Terms proportional to 3 2 δp1 ´βi q appearing inside the splitting functions in eq. ( 6.2.7)

• Term proportional to ln ´1 ´xh 1 xq ¯in eq. ( 6.4.12)

• Term proportional to ln ´1 ´xh 2 xq ¯in eq. ( 6.4.20)

Now, we are only left with collinearly divergent contributions related to the case of fragmentation from quark and gluon or from antiquark and gluon. These should cancel the only two divergent contributions left in eq. ( 6.2.7), i.e., the ones proportional to P gq pβ i q.

Fragmentation from antiquark and gluon

In this section, we deal with extracting the collinear divergences associated with the contribution (d) in fig. 6.1.5. This contribution corresponds to the situation in which the antiquark and the gluon fragment, while the quark plays the role of "spectator" emitted particle. This case is much simpler than before. We do not have to deal with any soft divergence and the only IR divergence that appears is when the fragmenting gluon is emitted by the quark line after the shockwave and the emitted quark and gluon become collinear. Hence, we can directly compute the contribution due to the first term of eq. ( 6.4.1). We emphasize the difference with the contribution calculated in section 6.4.1. Although at the level of hard computation the term that generates the present divergence is the same as the one that generates the collinear divergence in section 6.4.1, the situation is completely different. In the present case, we integrate out the quark kinematic variables and remain differential in the variable of the emitted gluon, while, in section 6.4.1 it is exactly the opposite.

Collinear contribution: q-g splitting According to the above discussion, we should focus on the first term of eq. ( 6.4.1), which exhibits a collinear pole, namely

dσ g qÑh 1 h 2 3LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg " 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx g x 2 g ż 1 0 dx 1 q ż 1 x h 2 dx q x q δp1 ´x1 q ´xq ´xg q ˆˆx g x h 1 ˙d ˆxq x h 2 ˙d D h 1 g ˆxh 1 x g , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙αs µ 2ϵ C F ż d d ⃗ p q p2πq d ˆż d d ⃗ p 2 F ˆ⃗ p q 2 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 `xg 2x h 1 ⃗ p h 1 ż d d ⃗ p 2 1 F ˚ˆ⃗ p q 2 `xq 2x h 2 ⃗ p h 2 ´⃗ p 2 1 `xg 2x h 1 ⃗ p h 1 1 ˆxq p1 ´xq qQ 2 `´xq x h 2 ⃗ p h 2 ´⃗ p 2 ¯2˙ˆx qp1 ´xq qQ 2 `´xq x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2ṗ dx 2 g `4x 1 q px 1 q `xg qqx 2 q p1 ´xq q 2 ´x1 q xg x h 1 ⃗ p h 1 ´xg ⃗ p q ¯2 `ph 1 Ø h 2 q . (6.4.42)
Using a change of variable similar to (6.4.6), here

x g " β 1 x q

x 1 q " p1 ´β1 qx q with the Jacobian dx 1 q dx g " dx q dβ 1 x q and treating the integration over longitudinal fractions as follows

ż 1 x h 1 dx g x 2 g ż 1 x h 2 dx q x q ż 1 0 dx 1 q δp1 ´x1 q ´xq ´xg q " ż 1 x h 1 dx g x 2 g ż 1 0 dx 1 q ż `8
´8 dx q x q θpx q ´xh 2 q θp1 ´xq q δp1 ´x1 q ´xq ´xg q "

ż 1´x h 2 x h 1 dx q 1 x q p1 ´xq q ż 1 x h 1 xq dβ 1 β 2 1 , we get dσ g qÑh 1 h 2 3LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg " 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1´x h 2 x h 1 dx q ż 1 x h 1 xq dβ 1 β 1 x q p1 ´xq q ˆxq x h 1 ˙d ˆ1 ´xq x h 2 ˙d ˆDh 1 g ˆxh 1 β 1 x q , µ F ˙Dh 2 q ˆxh 2 1 ´xq , µ F ż d d ⃗ p 2 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆ1´xq 2x h 2 ⃗ p h 2 ´⃗ p 2 `β1 xq 2x h 1 ⃗ p h 1 ẋq p1 ´xq qQ 2 `´1´xq x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 F p⃗ z 1 q ˆż d d ⃗ p 2 1 ż d d ⃗ z 2 e i⃗ z 2 ¨ˆ1´xq 2x h 2 ⃗ p h 2 ´⃗ p 2 1 `β1 xq 2x h 1 ⃗ p h 1 ẋq p1 ´xq qQ 2 `´1´xq x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2 F ˚p⃗ z 2 q ˆ2pp1 ´β1 q 2 `1q `2ϵβ 2 1 β 1 β d´2 1 α s µ 2ϵ C F ż d d ⃗ p q p2πq d e ´i´⃗ z 1 ´⃗ z 2 2
¯¨⃗ pq

´p1´β 1 qxq x h 1 ⃗ p h 1 ´⃗ p q ¯2 `ph 1 Ø h 2 q . (6.4.43)
After translating ⃗ p q and using eq. ( 6.4.4) in eq. ( 6.4.43) to perform the integration over quark transverse momenta, we obtain

dσ g qÑh 1 h 2 3LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg " 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1´x h 2 x h 1 dx q ż 1 x h 1 xq dβ 1 β 1 x q p1 ´xq q ˆxq x h 1 ˙d ˆˆ1 ´xq x h 2 ˙d D h 1 g ˆxh 1 β 1 x q , µ F ˙Dh 2 q ˆxh 2 1 ´xq , µ F ż d d ⃗ p 2 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆ1´xq 2x h 2 ⃗ p h 2 ´⃗ p 2 `β1 xq 2x h 1 ⃗ p h 1 ẋq p1 ´xq qQ 2 `´1´xq x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 F p⃗ z 1 q ˆż d d ⃗ p 2 1 ż d d ⃗ z 2 e i⃗ z 2 ¨ˆ1´xq 2x h 2 ⃗ p h 2 ´⃗ p 2 1 `β1 xq 2x h 1 ⃗ p h 1 ẋq p1 ´xq qQ 2 `´1´xq x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2 F ˚p⃗ z 2 q ˆ2pp1 ´β1 q 2 `1q `2ϵβ 2 1 `4ϵpp1 ´β1 q 2 `1q ln β 1 β 1 ˆe´i ´⃗ z 1 ´⃗ z 2 2 ¯¨p1´β 1 qxq x h 1 ⃗ p h 1 α s C F 4π ˆ1 ε `ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙˙`ph 1 Ø h 2 q " 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1´x h 2 x h 1 dx q ż 1 x h 1 xq dβ 1 β 1 x q p1 ´xq q ˆxq x h 1 ˙d ˆˆ1 ´xq x h 2 ˙d D h 1 g ˆxh 1 β 1 x q , µ F ˙Dh 2 q ˆxh 2 1 ´xq , µ F ˆż d d ⃗ p 2 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆ1´xq 2x h 2 ⃗ p h 2 ´⃗ p 2 `xq 2x h 1 ⃗ p h 1 ẋq p1 ´xq qQ 2 `´1´xq x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 F p⃗ z 1 q ˆż d d ⃗ p 2 1 ż d d ⃗ z 2 e i⃗ z 2 ¨ˆ1´xq 2x h 2 ⃗ p h 2 ´⃗ p 2 1 `xq 2x h 1 ⃗ p h 1 ẋq p1 ´xq qQ 2 `´1´xq x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2 F ˚p⃗ z 2 q ˆαs 2π C F " 1 ε 1 `p1 ´β1 q 2 β 1 `β1 `2p1 `p1 ´βq 2 q ln β 1 β 1 `1 `p1 ´β1 q 2 β 1 ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙ȷ `ph 1 Ø h 2 q " dσ g qÑh 1 h 2 3LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 K ˇˇˇˇc oll. qg div `dσ g qÑh 1 h 2 3LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll.qg fin , (6.4.44)
where the term labeled with "div" contains the first term of the square bracket. Putting back x q using eq. ( 6.4.10), this divergent term takes the form:

dσ g qÑh 1 h 2 3LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg div " 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1 x h 2 xq dx q x q x q ˆxq x h 1 ˙d ˆxq x h 2 ˙d ˆδp1 ´xq ´xq q F LL α s 2π 1 ε ż 1 x h 1 xq dβ 1 β 1 ˆDh 1 g ˆxh 1 β 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙CF 1 `p1 ´β1 q 2 β 1 `ph 1 Ø h 2 q . (6.4.45)
This is the term needed to cancel the divergent term proportional to P gq pβ 1 q in (6.2.7). Instead, the finite part in eq. ( 6.4.44) reads

dσ g qÑh 1 h 2 3LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg fin " 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1 x h 2 xq dx q x q x q ˆxq x h 1 ˙d ˆxq x h 2 ˙d ˆδp1 ´xq ´xq q ż 1 x h 1 xq dβ 1 β 1 D h 1 g ˆxh 1 β 1 x q , µ F ˙Dh 2 q ˆxh 2 1 ´xq , µ F ż d d ⃗ p 2 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆ1´xq 2x h 2 ⃗ p h 2 ´⃗ p 2 `xq 2x h 1 ⃗ p h 1 ẋq p1 ´xq qQ 2 `´1´xq x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 F p⃗ z 1 q ˆż d d ⃗ p 2 1 ż d d ⃗ z 2 e i⃗ z 2 ¨ˆ1´xq 2x h 2 ⃗ p h 2 ´⃗ p 2 1 `xq 2x h 1 ⃗ p h 1 ẋq p1 ´xq qQ 2 `´1´xq x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2 F ˚p⃗ z 2 q ˆαs 2π C F " β 1 `2p1 `p1 ´βq 2 q ln β 1 β 1 `1 `p1 ´β1 q 2 β 1 ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙ȷ `ph 1 Ø h 2 q . (6.4.46)
In a similar way, in the TL case, we get

dσ g qÑh 1 h 2 3T L dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg " α em Q p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1 x h 2 dx q ˆxq x h 1
˙d ˆxq x h 2 ˙d px q ´xq q ˆδp1 ´xq ´xq q

ż 1 x h 1 xq dβ 1 β 1 D h 1 g ˆxh 1 β 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ż d d ⃗ p 2 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆ1´xq 2x h 2 ⃗ p h 2 ´⃗ p 2 `xq 2x h 1 ⃗ p h 1 ẋq p1 ´xq qQ 2 `´1´xq x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 F p⃗ z 1 q ˆż d d ⃗ p 2 1 ż d d ⃗ z 2 e i⃗ z 2 ¨ˆ1´xq 2x h 2 ⃗ p h 2 ´⃗ p 2 1 `xq 2x h 1 ⃗ p h 1 ẋq p1 ´xq qQ 2 `´1´xq x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2 F ˚p⃗ z 2 q ˆxq x h 2 ⃗ p h 2 ´⃗ p 2 1 ˙¨⃗ ε T ˆαs 2π C F " 1 ε 1 `p1 ´β1 q 2 β 1 `β1 `2p1 `p1 ´βq 2 q ln β 1 β 1 `1 `p1 ´β1 q 2 β 1 ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙ȷ `ph 1 Ø h 2 q " dσ g qÑh 1 h 2 3T L dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg div `dσ g qÑh 1 h 2 3T L dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg fin . (6.4.47)
Finally, in the TT case, we have

dσ g qÑh 1 h 2 3T T dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg " α em 2p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 x h 2 dx q x q ˆxq x h 1 ˙d ˆxq x h 2 ˙d δp1 ´xq ´xq q ˆż 1 x h 1 xq dβ 1 β 1 D h 1 g ˆxh 1 β 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙"px q ´xq q 2 g ri K g lk K ´grk K g li K `grl K g ik K ı ˆż d d ⃗ p 2 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆ1´xq 2x h 2 ⃗ p h 2 ´⃗ p 2 `xq 2x h 1 ⃗ p h 1 ẋq p1 ´xq qQ 2 `´1´xq x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 F p⃗ z 1 q ˆxq x h 2 p h 2 K ´p2K ˙r ε T i ˆż d d ⃗ p 2 1 ż d d ⃗ z 2 e i⃗ z 2 ¨ˆ1´xq 2x h 2 ⃗ p h 2 ´⃗ p 2 1 `xq 2x h 1 ⃗ p h 1 ẋq p1 ´xq qQ 2 `´1´xq x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2 F ˚p⃗ z 2 q ˆxq x h 2 p h 2 K ´p2 1 K ˙l ε T k ˆαs 2π C F " 1 ε 1 `p1 ´β1 q 2 β 1 `β1 `2p1 `p1 ´βq 2 q ln β 1 β 1 `1 `p1 ´β1 q 2 β 1 ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙ȷ `ph 1 Ø h 2 q " dσ g qÑh 1 h 2 3T T dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg div `dσ g qÑh 1 h 2 3T T dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg fin . (6.4.48)
These results conclude the discussion of divergences in the case of fragmentation from antiquark and gluon.

Fragmentation from quark and gluon

In this section, we deal with extracting the collinear divergences associated with the contribution (c) in fig. 6.1.5. This contribution corresponds to the situation in which the quark and the gluon fragment, while the antiquark plays the role of the "spectator" emitted particle.

Collinear contribution: q-g splitting

The term in eq. ( 6.4.1) to consider is the third one. The calculation proceeds in the same way as for the antiquark and gluon fragmentation, but this time the integration is over ⃗ p 2,2 1 in the F function.

For the LL case, we get

dσ qgÑh 1 h 2 3LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg " 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1
x h 2 dx q x q x q δp1 ´xq ´xq q

ˆxq x h 1 ˙d ˆxq x h 2 ˙d ˆż 1 x h 2 x q dβ 2 β 2 D h 2 q ˆxh 1 x q , µ F ˙Dh 2 g ˆxh 2 β 2 x q , µ F ż d d ⃗ p 1 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆ´x q 2x h 2 ⃗ p h 2 `⃗ p 1 ´xq 2x h 1 ⃗ p h 1 ẋq
x qQ 2 `´xq .4.49) This term cancels the divergent term proportional to P gq pβ 2 q in eq. ( 6.2.7). This is the last remaining cancellation of divergences, the rest of the cross-section is now completely finite.

x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 F p⃗ z 1 q ˆż d d ⃗ p 1 1 ż d d ⃗ z 2 e i⃗ z 2 ¨ˆ´x q 2x h 2 ⃗ p h 2 ´⃗ p 1 1 ´xq 2x h 1 ⃗ p h 1 ẋq p1 ´xq qQ 2 `´xq x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 F ˚p⃗ z 2K q ˆαs 2π C F " 1 ε 1 `p1 ´β2 q 2 β 2 `β2 `2p1 `p1 ´β2 q 2 q ln β 2 β 2 `1 `p1 ´β2 q 2 β 2 ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙ȷ `ph 1 Ø h 2 q " dσ qgÑh 1 h 2 3LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg div `dσ qgÑh 1 h 2 3LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg fin . ( 6
For the TL case, we get

dσ qgÑh 1 h 2 3T L dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 K ˇˇˇˇc oll. qg " α em Q p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q ż 1
x h 2 dx q px q ´xq q δp1 ´xq ´xq q

ˆxq x h 1 ˙d ˆxq x h 2 ˙d ˆż 1 x h 2 x q dβ 2 β 2 D h 2 q ˆxh 1 x q , µ F ˙Dh 2 g ˆxh 2 β 2 x q , µ F ż d d ⃗ p 1 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆ´x q 2x h 2 ⃗ p h 2 `⃗ p 1 ´xq 2x h 1 ⃗ p h 1 ẋq
x qQ 2 `´xq

x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 F p⃗ z 1 q ˆż d d ⃗ p 1 1 ż d d ⃗ z 2 e i⃗ z 2 ¨ˆ´x q 2x h 2 ⃗ p h 2 ´⃗ p 1 1 ´xq 2x h 1 ⃗ p h 1 ẋq p1 ´xq qQ 2 `´xq x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 F ˚p⃗ z 2 q ˆxq x h 1 p h 1 K ´p1 1 K ˙¨ε T ˆαs 2π C F " 1 ε 1 `p1 ´β2 q 2 β 2 `β2 `2 ln β 2 p1 `p1 ´β2 q 2 q β 2 `1 `p1 ´β2 q 2 β 2 ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙ȷ `ph 1 Ø h 2 q " dσ qgÑh 1 h 2 3T L dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg div `dσ qgÑh 1 h 2 3T L dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg fin . (6.4.50)
Finally, for the TT case, we get

dσ qgÑh 1 h 2 3T T dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg " α em 2p2πq 4d N c ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 x h 2 dx q x q δp1 ´xq ´xq q ˆxq x h 1 ˙d ˆxq x h 2 ˙d ˆż 1 x h 2 x q dβ 2 β 2 D h 2 q ˆxh 1 x q , µ F ˙Dh 2 g ˆxh 2 β 2 x q , µ F ˙"px q ´xq q 2 g ri K g lk K ´grk K g li K `grl K g ik K ı ˆż d d ⃗ p 1 ż d d ⃗ z 1 e ´i⃗ z 1 ¨ˆ´x q 2x h 2 ⃗ p h 2 `⃗ p 1 ´xq 2x h 1 ⃗ p h 1 ẋq
x qQ 2 `´xq

x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 F p⃗ z 1 q ˆxq x h 1 p h 1 K ´p1K ˙r ε T i ˆż d d ⃗ p 1 1 ż d d ⃗ z 2 e i⃗ z 2 ¨ˆ´x q 2x h 2 ⃗ p h 2 ´⃗ p 1 1 ´xq 2x h 1 ⃗ p h 1 ẋq p1 ´xq qQ 2 `´xq x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 F ˚p⃗ z 2s q ˆxq x h 1 p h 1 K ´p1 1 K ˙l ε T k ˆαs 2π C F " 1 ε 1 `p1 ´β2 q 2 β 2 `β2 `2 ln β 2 p1 `p1 ´β2 q 2 q β 2 `1 `p1 ´β2 q 2 β 2 ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙ȷ `ph 1 Ø h 2 q " dσ qgÑh 1 h 2 3T T dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg div `dσ qgÑh 1 h 2 3T T dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇc oll. qg fin . ( 6
.4.51)

Additional finite terms

Some of the finite terms of our calculation are presented in previous sections. They come as a result of the extraction of divergences. There are many other terms, completely disconnected from divergences, which however contribute to the final result. We proceed to list them, also emphasizing again what their nature is.

Virtual corrections: Dipole ˆdouble dipole contribution

The 1-loop correction to the γ ˚Ñ q q contains a dipole and double dipole terms. The first one receives a contribution from all diagrams, while the second one gets contributions only from diagrams where the virtual gluon crosses the shockwave. At the cross-section level there will therefore be two contributions:

• The one due to the interference between the dipole correction and the Born amplitude. This contains divergences and it is the one that we have completely computed in section 6.3.

• The one due to the interference between the double dipole correction and the Born amplitude. Any rapidity divergence present in this term is completely reabsorbed into the renormalized Wilson operator, at the amplitude level, with the help of the B-JIMWLK evolution (see section 4.3.4) . After this operation, this contribution is finite and can be taken in convolution with FFs without any additional manipulation.

From eq. (4.5.20), we get

dσ q qÑh 1 h 2 2LL dx h 1 d 2 ⃗ p h 1 dx h 2 d 2 ⃗ p h 2 " α em α s Q 2 p2πq 8 N c x 2 h 1 x 2 h 2 ÿ q Q 2 q 8π ż 1 x h 1 dx q ż 1
x h 2 dx q x q x q δp1 ´xq ´xq qD rpϕ n q LL s `¸`pq Ø qq

h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ż d 2 ⃗ p 1 d 2 ⃗ p 2 d 2 ⃗ p 1 1 d 2 ⃗ p 2 1 ż d 2 ⃗ p 3 p2πq 2 δ ˆxq x h 1 ⃗ p h 1 ´⃗ p 1 `xq x h 2 ⃗ p h 2 ´⃗ p 2 ´⃗ p 3 ˙δp⃗ p 11 1 `⃗ p 22 1 `⃗ p 3 q ˆ$ ' & ' % r F ´⃗ p 12 2 , ⃗ p 3 ¯F˚´⃗ p 1 1 2 1 2 xq x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 `xq x qQ 2 ˆ¨4 x q x q $ ' ' & ' ' % x q x q ˆ⃗ p 2 3 ´´xq x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 ´´xq x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 ´2x q x qQ 2 ẋq x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 `xq x qQ 2 ˙ˆ´x q x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 `xq x qQ 2 ˙´x q x qQ 2 ⃗ p 2 3 ˆln ´xq x q e 2η ¯ln ¨ˆ´x q x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 `xq x qQ 2 ˙ˆ´x q x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 `xq x qQ 2 ẋq x qQ 2 ⃗ p 2 3 ‹ ‹ ' ´¨2 x q x q ´xq x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 `xq x qQ 2
ff¸`h .c.| p 1 ,p 2 Øp 1 1 ,p 2 1 + `ph 1 Ø h 2 q . (6.5.1)
Concerning the other transitions, we have for T L, using eq. (4.5.21),

dσ q qÑh 1 h 2 2T L dx h 1 d 2 ⃗ p h 1 dx h 2 d 2 ⃗ p h 2 " α em α s Q p2πq 8 N c x 2 h 1 x 2 h 2 ÿ q Q 2 q 8π ż 1 x h 1 dx q ż 1
x h 2 dx q x q x q δp1 ´xq ´xq q

ˆDh 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙ż d 2 ⃗ p 1 d 2 ⃗ p 2 d 2 ⃗ p 1 1 d 2 ⃗ p 2 1 ˆż d 2 ⃗ p 3 d 2 ⃗ p 3 1 p2πq 2 δ ˆxq x h 1 ⃗ p h 1 ´⃗ p 1 `xq x h 2 ⃗ p h 2 ´⃗ p 2 ´⃗ p 3 ˙δ p⃗ p 11 1 `⃗ p 22 1 `⃗ p 33 1 q ˆεT i » - - δp⃗ p 3 1 q ´xq x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 `xq x qQ 2 r F ˆ⃗ p 12 2 , ⃗ p 3 ˙F˚ˆ⃗ p 1 1 2 1 2 ˙˜2px q ´xq q ˆxq x h 1 p h 1 K ´p1 1 K ˙i ˆ$ ' ' & ' ' % x q x q ˆ⃗ p 2 3 ´´xq x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 ´´xq x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 ´2x q x qQ 2 ẋq x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 `xq x qQ 2 ˙ˆ´x q x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 `xq x qQ 2 ˙´x q x qQ 2 ⃗ p 2 3 ln ´xq x q e 2η ln ¨ˆ´x q x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 `xq x qQ 2 ˙ˆ´x q x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 `xq x qQ 2 ẋq x qQ 2 ⃗ p 2 3 ‹ ‹ ' ´¨2 x q x q ´xq x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 `xq x qQ 2 ln ´xq e η ¯ln ˆ⃗ p 2 3 µ 2 ˙`pq Ø qq ‹ ' , / . / - `1 2x q x q «˜ż xq 0 dz ÿ n"5,6 " pϕ n q i T L ‰ `¸`pq Ø qq ffδ p⃗ p 3 q ´xq x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 `xq x qQ 2 F ˆ⃗ p 12 2 ˙r F ˚ˆ⃗ p 1 1 2 1 2 , ⃗ p 3 1 ˙˜# 2x q x qpx q ´xq q ˆxq x h 1 p h 1 K ´p1 1 K ˙i ˆ¨´2 ´xq x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 `xq x qQ 2 ln ´xq e η ¯ln ˆ⃗ p 2 3 1 µ 2 ln ´xq x q e 2η ¯´xq x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2 `xq x qQ 2 ˆ´xq x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2 `xq x qQ 2 ˙ˆ´x q x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 `xq x qQ 2 ˙´x q x qQ 2 ⃗ p 2 3 1 ˆln ¨ˆ´x q x h 2 ⃗ p h 2 ´⃗ p 2 1 ¯2 `xq x qQ 2 ˙ˆ´x q x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 `xq x qQ 2 ẋq x qQ 2 ⃗ p 2 3 1 ‹ ‹ ' `1 ´xq x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 ln ´xq x q e 2η ¯ln ¨´xq x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 `xq x qQ 2 x q x qQ 2 ‹ ' ‹ '`pq Ø qq , / . / - `«˜ż xq 0 dz ÿ n"5,6 " pϕ n q i LT ‰ `¸`pq Ø qq ff ˚¸ff `ph 1 Ø h 2 q , (6.5.2)
and for T T , using eq. (4.5.22), we have

dσ q qÑh 1 h 2 2T T dx h 1 d 2 ⃗ p h 1 dx h 2 d 2 ⃗ p h 2 " α em α s p2πq 8 N c x 2 h 1 x 2 h 2 ÿ q Q 2 q 8π ż 1 x h 1 dx q ż 1
x h 2 dx q x q x q δp1 ´xq ´xq qD

h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ż d 2 ⃗ p 1 d 2 ⃗ p 2 d 2 ⃗ p 1 1 d 2 ⃗ p 2 1 ż d 2 ⃗ p 3 p2πq 2 δ ˆxq x h 1 ⃗ p h 1 ´⃗ p 1 `xq x h 2 ⃗ p h 2 ´⃗ p 2 ´⃗ p 3 ˙δ p⃗ p 11 1 `⃗ p 22 1 `⃗ p 3 q `εT i ε T j » - -r F ˆ⃗ p 12 2 , ⃗ p 3 ˙F˚ˆ⃗ p 1 1 2 1 2 ˙1 ´xq x h 1 ⃗ p h 1 ´⃗ p 1 1 ¯2 `xq x qQ 2 ˆ"ˆx q x h 1 p h 1 K ´p1 1 K ˙l ˆxq x h 1 p h 1 K ´p1K ˙k ˆ"px q ´xq q 2 g ki K g lj K ´gkj K g li K `gkl K g ij K ı ¨´2 ´xq x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 `xq x qQ 2 ln ´xq e η ¯ln ˆ⃗ p 2 3 µ 2 1 ´xq x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 ln ´xq x q e 2η ¯ln ¨´xq x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 `xq x qQ 2 x q x qQ 2 ‹ ' ´ln ´xq x q e 2η ¯´xq x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 `xq x qQ 2 ˆ´xq x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 `xq x qQ 2 ˙ˆ´x q x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 `xq x qQ 2 ˙´x q x qQ 2 ⃗ p 2 3 ˆln ¨ˆ´x q x h 2 ⃗ p h 2 ´⃗ p 2 ¯2 `xq x qQ 2 ˙ˆ´x q x h 1 ⃗ p h 1 ´⃗ p 1 ¯2 `xq x qQ 2 ẋq x qQ 2 ⃗ p 2 3 ‹ ‹ ' ‹ ‹ ' `pq Ø qq , / / . / / - `1 x q x q «˜ż xq 0 dz ÿ n"5,6 " pϕ n q ij T T ı `¸`pq Ø qq ff¸`h .c|p1,p2Øp 1 1 ,p 2 1 iØj ff `ph 1 Ø h 2 q . (6.5.3)
The ϕ function are defined in appendix B.1.

Real corrections: Fragmentation from quark and antiquark

In this case, we refer to the finite terms related to the contribution (b) of fig. 6.1.5. In order to better understand what these contributions are, we recall that the impact factor for the transition γ ˚Ñ q qg has a double dipole contribution (Φ

p`,iq 4
) and a single dipole (Φ p`,iq 3

) contribution. The finite contributions which we obtain are • Finite terms related to the dipole ˆdipole contribution.

• Dipole ˆdouble dipole contribution.

• double dipole ˆdouble dipole contribution.

Finite part of dipole ˆdipole contribution

In eq. (4.6.4), the first term in the right-hand side is the only one containing divergences and that we have considered in the previous sections. After isolating soft and collinear divergences, finite terms remain. The finite contributions for σpbqdiv,1 and σpbqdiv,3 have been computed respectively in sections 6.4.1 and 6.4.1, see eqs. (6.4.13), (6.4.15), (6.4.17) and (6.4.21), (6.4.24), (6.4.27).

Besides, the terms pσ pbqdiv,2 ´σ sof t pbqdiv,2 q and pσ pbqdiv,4 ´σ sof t pbqdiv,4 q in eq. ( 6.1.26) are finite. Their contribution read

dσ q qÑh 1 h 2 3LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇfi nite, (b) 2,4 " α s C F µ 2ϵ 2α em Q 2 p2πq 4d N c x d h 1 x d h 2 ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 x h 2 dx q x q ż 1 α dx g x 3´d g px q x qq d´1 δp1 ´xq ´xq ´xg q ˆDh 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙ż d d ⃗ u p2πq d ż d d ⃗ p 1 d d ⃗ p 2 F ˆ⃗ p 12 2 ż d d ⃗ p 1 1 d d ⃗ p 2 1 F ˚ˆ⃗ p 1 1 2 1 2 ˙δ p⃗ p 11 1 `⃗ p 22 1 q ˆ$ ' ' ' ' ' ' ' & ' ' ' ' ' ' ' % 8x q x q δ ´xq x h 1 ⃗ p h 1 ´⃗ p 1 `xq x h 2 ⃗ p h 2 ´⃗ p 2 Q2 `ˆx q x h 2 ⃗ p h 2 ´⃗ p 2 1 ˙2 xqp1´xqq '¨Q 2 `ˆxq x h 1 ⃗ p h 1 ´⃗ p 1 ˙2 xqp1´xqq ' ´⃗ u ´⃗ p h 1 x h 1 ¯¨´⃗ u ´⃗ p h 2 x h 2 ⃗ u ´⃗ p h 1 x h 1 ¯2 ´⃗ u ´⃗ p h 2 x h 2 ¯2 ´p2x g ´dx 2 g `4x q x qqδ ´xq x h 1 ⃗ p h 1 ´⃗ p 1 `xq x h 2 ⃗ p h 2 ´⃗ p 2 `xg ⃗ u Q2 `ˆx q x h 2 ⃗ p h 2 ´⃗ p 2 1 ˙2 xqp1´xqq '¨Q 2 `ˆxq x h 1 ⃗ p h 1 ´⃗ p 1 ˙2 xqp1´xqq ' ´⃗ u ´⃗ p h 1 x h 1 ¯¨´⃗ u ´⃗ p h 2 x h 2 ⃗ u ´⃗ p h 1 x h 1 ¯2 ´⃗ u ´⃗ p h 2 x h 2 ¯2 ´p2x g ´dx 2 g `4x q x qqδ ´xq x h 1 ⃗ p h 1 ´⃗ p 1 `xq x h 2 ⃗ p h 2 ´⃗ p 2 `xg ⃗ u Q2 `ˆx q x h 2 ⃗ p h 2 ´⃗ p 2 ˙2 xqp1´xqq '¨Q 2 `ˆxq x h 1 ⃗ p h 1 ´⃗ p 1 1 ˙2 xqp1´xqq ' ˆ´⃗ u ´⃗ p h 1 x h 1 ¯¨´⃗ u ´⃗ p h 2 x h 2 ⃗ u ´⃗ p h 1 x h 1 ¯2 ´⃗ u ´⃗ p h 2 x h 2 ¯2 , / .

/ -

`ph 1 Ø h 2 q , (6.5.4)

in the LL case. The same contribution in the TL and TT cases is, respectively,

dσ q qÑh 1 h 2 3T L dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇfi nite, (b) 2,4 " α s C F µ 2ϵ α em Q p2πq 4d N c x d h 1 x d h 2 ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 x h 2 x q x q ż 1 α dx g x 3´d
g px q x qq d´1 δp1 ´xq ´xq ´xg q

ˆDh 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙ż d d ⃗ u p2πq d ż d d ⃗ p 1 d d ⃗ p 2 F ˆ⃗ p 12 2 ż d d ⃗ p 1 1 d d ⃗ p 2 1 F ˚ˆ⃗ p 1 1 2 1 2 ˙δ p⃗ p 11 1 `⃗ p 22 1 q ε T i ˆ$ ' ' ' ' ' ' ' & ' ' ' ' ' ' ' % δ ´xq x h 1 ⃗ p h 1 ´⃗ p 1 `xq x h 2 ⃗ p h 2 ´⃗ p 2 `xg ⃗ u Q2 `ˆx q x h 2 ⃗ p h 2 ´⃗ p 2 1 ˙2 xqp1´xqq '¨Q 2 `ˆxq x h 1 ⃗ p h 1 ´⃗ p 1 ˙2 xqp1´xqq '´u K ´ph 1 K x h 1 ¯µ ´uK ´ph 2 K x h 2 ¯ν ´⃗ u ´⃗ p h 1 x h 1 ¯2 ´⃗ u ´⃗ p h 2 x h 2 ¯2 ˆ1 x qpx q `xg q " x g p4x q `dx g ´2q ˆˆx q x h 2 p h 2 K ´p2 1 K ˙µ g iν K ´ˆx q x h 2 p h 2 K ´p2 1 K ˙ν g iµ K ṗ2x q ´1qp4x qx q `xg p2 ´xg dqqg µν K ˆxq x h 2 p h 2 K ´p2 1 K ˙iff `δ ´xq x h 1 ⃗ p h 1 ´⃗ p 1 `xq x h 2 ⃗ p h 2 ´⃗ p 2 `xg ⃗ u Q2 `ˆxq x h 1 ⃗ p h 1 ´⃗ p 1 1 ˙2 xqp1´xqq '¨Q 2 `ˆx q x h 2 ⃗ p h 2 ´⃗ p 2 ˙2 xqp1´xqq '´u K ´ph 2 K x h 2 ¯µ ´uK ´ph 1 K x h 1 ¯ν ´⃗ u ´⃗ p h 1 x h 1 ¯2 ´⃗ u ´⃗ p h 2 x h 2 ¯2 ˆ1 x q px q `xg q " x g p4x q `dx g ´2q ˆˆx q x h 1 p h 1 K ´p1 1 K ˙µ g iν K ´ˆx q x h 1 p h 1 K ´p1 1 K ˙ν g iµ K ṗ2x q ´1qp4x qx q `xg p2 ´xg dqqg µν K ˆxq x h 2 p h 1 K ´p1 1 K ˙iff ´8 δ ´xq x h 1 ⃗ p h 1 ´⃗ p 1 `xq x h 2 ⃗ p h 2 ´⃗ p 2 Q2 `ˆx q x h 2 ⃗ p h 2 ´⃗ p 2 1 ˙2 xqp1´xqq '¨Q 2 `ˆxq x h 1 ⃗ p h 1 ´⃗ p 1 ˙2 xqp1´xqq ' ´⃗ u ´⃗ p h 1 x h 1 ¯¨´⃗ u ´⃗ p h 2 x h 2 ⃗ u ´⃗ p h 1 x h 1 ¯2 ´⃗ u ´⃗ p h 2 x h 2 ¯2 ˆˆx q x h 2 p h 2 K ´p2 1 K ˙i px q ´xq q + `ph 1 Ø h 2 q , and
dσ q qÑh 1 h 2 3T T dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 ˇˇˇˇfi nite, (b) 2,4 " α s C F µ 2ϵ α em 2p2πq 4d N c x d h 1 x d h 2 ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 x h 2 dx q x q ż 1 α dx g x 3´d g
δp1 ´xq ´xq ´xg q px q x qq d´1

ˆDh 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ˙ż d d ⃗ u p2πq d ż d d ⃗ p 1 d d ⃗ p 2 F ˆ⃗ p 12 2 ż d d ⃗ p 1 1 d d ⃗ p 2 1 F ˚ˆ⃗ p 1 1 2 1 2 ˙δp⃗ p 11 1 `⃗ p 22 1 qε T i ε T k ˆ$ ' ' & ' ' % » - - - ¨´δ p⃗ p q1 `⃗ p q2 `xg ⃗ u q ˆQ2 `⃗ p 2 q2 xqp1´xqq ˙ˆQ 2 `⃗ p 2 q1 1 xqp1´xqq ˙´u K ´pqK xq ¯µ ´uK ´pqK xq ¯ν ´⃗ u ´⃗ pq xq ¯2 ´⃗ u ´pq xq ¯2
ˆ1 px q `xg qpx q `xg qx q x q ! x g ppd ´4qqx g ´2q

" p ν q1 1 K ´pµ q2K g ik K `pk q2K g µi K ḡµν K ´`⃗ p q1 1 ¨⃗ p q2 ˘gik K `pi q1 1 K p k q2K ¯´g νk K p i q1 1 K p µ q2K ´gµi K g νk K `⃗ p q1 1 ¨⃗ p q2 ˘ı ´gµν K ˆ"p2x q ´1q p2x q ´1q p k q1 1 K p i q2K p4x q x q `xg p2 ´xg dqq `4x q x qpp⃗ p q1 1 ¨⃗ p q2 qg ik K `pi q1 1 K p k q2K q ı `´p µ q1 1 K p ν q2K g ik K ´pµ q1 1 K p k q2K g νi K ´pi q1 1 K p ν q2K g µk K ´gµk K g νi K p⃗ p q1 1 ¨⃗ p q2 q xg ppd ´4qx g `2q `xg p2x q ´1qpx g d `4x q ´2q ´gµk K p ν q1 1 K ´gνk K p µ q1 1 K ¯pi q2K `xg p2x q ´1qp k q1 1 K p4x q `xg d ´2q ´gνi K p µ q2K ´gνk K p ν q1 1 K ¯)¯`p q Ø qq ı `8 δp⃗ p q1 `⃗ p q2 q ˆQ2 `⃗ p 2 q2 xqp1´xqq ˙ˆQ 2 `⃗ p 2 q1 1 xqp1´xqq ˙´⃗ u ´⃗ pq xq ¯¨´⃗ u ´⃗ pq xq ⃗ u ´⃗ pq xq ¯2 ´⃗ u ´⃗ pq xq ¯2 ˆ1 x q x q " ´px q ´xq q 2 g ri K g kl K `gil K g rk K ´grl K g ik K ı p q2Kr p q1 1 Kl * `ph 1 Ø h 2 q .
In the above expression, the following replacement needs to be done:

p qK " x q x h 1 p h 1 K , p qK " x q x h 2 p h 2 K . (6.5.5)
The remaining term in eq. (4.6.4), for arbitrary polarization, is

dσ q qÑh 1 h 2 3JI dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 " α s α em C F µ 2ϵ p2πq 4d N c 1 4pp γ q 2 1 x d h 1 x d h 2 ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 x h 2 dx q x q px q x qq d´1 D h 1 q ˆxh 1 x q , µ F ˆDh 2 q ˆxh 2 x q , µ F ˙ż 1 0 dx g x g ż d d ⃗ p g p2πq d δp1 ´xq ´xq ´xg q ż d d ⃗ p 1 d d ⃗ p 2 d d ⃗ p 1 1 d d ⃗ p 2 1 ˆF ˆ⃗ p 12 2 ˙F˚ˆ⃗ p 1 1 2 1 2 ˙δ ˆxq x h 1 ⃗ p h 1 ´⃗ p 1 `xq x h 2 ⃗ p h 2 ´⃗ p 2 `⃗ p g ˙δp⃗ p 11 1 `⃗ p 22 1 q ˆεIα ε Jβ " Φ α 3 p⃗ p 1 , ⃗ p 2 qΦ β: 3 p⃗ p 1 1 , ⃗ p 2 1 q ´r Φ α 3 p⃗ p 1 , ⃗ p 2 q r Φ β: 3 p⃗ p 1 1 , ⃗ p 2 1 q ı `ph 1 Ø h 2 q . (6.5.6)

Dipole ˆdouble dipole contribution and double dipole ˆdouble dipole contribution

The dipole ˆdouble dipole contribution, for arbitrary polarization, is given by

dσ q qÑh 1 h 2 4JI dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 " α s α em 2µ 2ϵ p2πq 4d N c 1 4pp γ q 2 1 x d h 1 x d h 2 ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 x h 2 dx q x q px q x qq d´1 D h 1 q ˆxh 1 x q , µ F ˙Dh 2 q ˆxh 2 x q , µ F ż 1 0 dx g x g ż d d ⃗ p g p2πq d δp1 ´xq ´xq ´xg q ż d d ⃗ p 1 d d ⃗ p 2 d d ⃗ p 1 1 d d ⃗ p 2 1 d d ⃗ p 3 d d ⃗ p 3 1 p2πq d ˆδ ˆxq x h 1 ⃗ p h 1 ´⃗ p 1 `xq x h 2 ⃗ p h 2 ´⃗ p 2 `⃗ p g3 ˙δp⃗ p 11 1 `⃗ p 22 1 `⃗ p 33 1 qpε Iα ε Jβ q ˆ"Φ α 3 p⃗ p 1 , ⃗ p 2 qΦ β: 4 p⃗ p 1 1 , ⃗ p 2 1 , ⃗ p 3 1 qF ˆ⃗ p 12 2 ˙r F ˚ˆ⃗ p 1 1 2 1 2 , ⃗ p 3 1 ˙δp⃗ p 3 q `Φα 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 qΦ β: 3 p⃗ p 1 1 , ⃗ p 2 1 q r F ˆ⃗ p 12 2 , ⃗ p 3 ˙F˚ˆ⃗ p 1 1 2 1 2 ˙δp⃗ p 3 1 q ȷ `ph 1 Ø h 2 q . (6.5.7)
The double dipole ˆdouble dipole contribution, for arbitrary polarization, is given by

dσ q qÑh 1 h 2 5JI dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 " α s α em pε Iα ε Jβ q 2µ 2ϵ p2πq 4d pN 2 c ´1q 1 4pp γ q 2 1 x d h 1 x d h 2 ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 x h 2 dx q x q px q x qq d´1 D h 1 q ˆxh 1 x q , µ F Ḋh 2 q ˆxh 2 x q , µ F ˙ż 1 0 dx g x g ż d d ⃗ p g p2πq d δp1 ´xq ´xq ´xg q ż d d ⃗ p 1 d d ⃗ p 2 d d ⃗ p 1 1 d d ⃗ p 2 1 ˆż d d ⃗ p 3 d d ⃗ p 3 1 p2πq 2d δ ˆxq x h 1 ⃗ p h 1 ´⃗ p 1 `xq x h 2 ⃗ p h 2 ´⃗ p 2 `⃗ p g3 ˙δp⃗ p 11 1 `⃗ p 22 1 `⃗ p 33 1 q ˆΦα 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 qΦ β: 4 p⃗ p 1 1 , ⃗ p 2 1 , ⃗ p 3 1 q r F ˆ⃗ p 12 2 , ⃗ p 3 ˙r F ˚ˆ⃗ p 1 1 2 1 2 , ⃗ p 3 1 ṗh 1 Ø h 2 q . (6.5.8)
Expression for the squared impact factors can be found in appendix B.2. They are written in terms of p q , p q, p g and the following identification should be done:

p qK " x q x h 1 p h 1 K , p qK " x q x h 2 p h 2 K . ( 6
.5.9)

Real corrections: Fragmentation from antiquark and gluon

Finite part of dipole ˆdipole contribution

When squaring the dipole contribution, we have also finite terms. This time we write separately for each polarization transition. In the LL case, we have

dσ g qÑh 1 h 2 3LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 " α s α em C F µ 2ϵ p2πq 4d N c 1 4pp γ q 2 1 x d h 1 x d h 2 ÿ q Q 2 q ż 1 x h 1 dx g x g ż 1 x h 2 dx q x q px g x qq d´1 D h 1 g ˆxh 1 x g , µ F Ḋh 2 q ˆxh 2 x q , µ F ˙ż 1 0 dx q x q ż d d ⃗ p q p2πq d δp1 ´xq ´xq ´xg q ż d d ⃗ p 1 d d ⃗ p 2 d d ⃗ p 1 1 d d ⃗ p 2 1 ˆF ˆ⃗ p 12 2 ˙F˚ˆ⃗ p 1 1 2 1 2 ˙δ ˆ⃗ p q1 `xq x h 2 ⃗ p h 2 ´⃗ p 2 `xg x h 1 ⃗ p h 1 ˙δp⃗ p 11 1 `⃗ p 22 1 q Q 2 pp γ q 2 ˆ» - - - - - - - - Φ 3 p⃗ p 1 , ⃗ p 2 qΦ `: 3 p⃗ p 1 1 , ⃗ p 2 1 q ´8x q x qpp γ q 4 `dx 2 g `4x q px q `xg q Q2 `ˆx q x h 2 ⃗ p h 2 ´⃗ p 2 ˙2 xqp1´xqq '¨Q 2 `ˆx q x h 2 ⃗ p h 2 ´⃗ p 2 1 ˙2 xqp1´xqq ' ˆ1 ´xq xg x h 1 ⃗ p h 1 ´xg ⃗ p q ¯2 fi ffi fl `ph 1 Ø h 2 q . (6.5.10)
For TL case, we have

dσ g qÑh 1 h 2 3T L dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 " α s α em C F µ 2ϵ p2πq 4d N c 1 4pp γ q 2 1 x d h 1 x d h 2 ÿ q Q 2 q ż 1 x h 1 dx g x g ż 1 x h 2 dx q x q px g x qq d´1 D h 1 g ˆxh 1 x g , µ F Ḋh 2 q ˆxh 2 x q , µ F ˙ż 1 0 dx q x q ż d d ⃗ p q p2πq d δp1 ´xq ´xq ´xg q ż d d ⃗ p 1 d d ⃗ p 2 d d ⃗ p 1 1 d d ⃗ p 2 1 ˆF ˆ⃗ p 12 2 ˙F˚ˆ⃗ p 1 1 2 1 2 ˙δ ˆ⃗ p q1 `xq x h 2 ⃗ p h 2 ´⃗ p 2 `xg x h 1 ⃗ p h 1 ˙δp⃗ p 11 1 `⃗ p 22 1 qε T i Q p γ ˆ» - - - - - - - - Φ 3 p⃗ p 1 , ⃗ p 2 qΦ i: 3 p⃗ p 1 1 , ⃗ p 2 1 q `4x q `pγ ˘3 p2x q ´1q `x2 g d `4x q px q `xg q Q2 `ˆx q x h 2 ⃗ p h 2 ´⃗ p 2 ˙2 xqp1´xqq '¨Q 2 `ˆx q x h 2 ⃗ p h 2 ´⃗ p 2 1 ˙2 xqp1´xqq ' ˆ´xq x h 2 p h 2 K ´p2 1 K ¯i px q `xg q ´xq xg x h 1 ⃗ p h 1 ´xg ⃗ p q ¯2 fi ffi fl `ph 1 Ø h 2 q . (6.5.11)
Finally, for TT case, we obtain

dσ g qÑh 1 h 2 3T T dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 " α s α em C F µ 2ϵ p2πq 4d N c 1 4pp γ q 2 1 x d h 1 x d h 2 ÿ q Q 2 q ż 1 x h 1 dx g x g ż 1 x h 2 dx q x q px g x qq d´1 D h 1 g ˆxh 1 x g , µ F Ḋh 2 q ˆxh 2 x q , µ F ˙ż 1 0 dx q x q ż d d ⃗ p q p2πq d δp1 ´xq ´xq ´xg q ż d d ⃗ p 1 d d ⃗ p 2 ż d d ⃗ p 1 1 d d ⃗ p 2 1 ˆF ˆ⃗ p 12 2 ˙F˚ˆ⃗ p 1 1 2 1 2 ˙δ ˆ⃗ p q1 `xq x h 2 ⃗ p h 2 ´⃗ p 2 `xg x h 1 ⃗ p h 1 ˙δp⃗ p 11 1 `⃗ p 22 1 qε T i ε T k ˆ» - - - - - - - - Φ i 3 p⃗ p 1 , ⃗ p 2 qΦ k: 3 p⃗ p 1 1 , ⃗ p 2 1 q ´2x q pp γ q 2 px 2 g d `4x q px q `xg qq ¨Q2 `ˆx q x h 2 ⃗ p h 2 ´⃗ p 2 ˙2 xqp1´xqq '¨Q 2 `ˆx q x h 2 ⃗ p h 2 ´⃗ p 2 1 ˙2 xqp1´xqq ' ˆ"p1 ´2x qq 2 g ri K g lk K ´gli K g rk K `grl K g ik K ‰ ´xq x h 2 p h 2 K ´p2K ¯r ´xq x h 2 p h 2 K ´p2 1 K ¯l x qpx q `xg q 2 ´xq xg x h 1 ⃗ p h 1 ´xg ⃗ p q ¯2 fi ffi fl
`ph 1 Ø h 2 q . (6.5.12)

Dipole ˆdouble dipole contribution and double dipole ˆdouble dipole contribution

The dipole ˆdouble dipole contribution, for arbitrary polarization, is given by

dσ g qÑh 1 h 2 4JI dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 " α s α em 2µ 2ϵ p2πq 4d N c 1 4pp γ q 2 1 x d h 1 x d h 2 ÿ q Q 2 q ż 1 x h 1 dx g x g ż 1 x h 2 dx q x q px g x qq d´1 D h 1 g ˆxh 1 x g , µ F ˆDh 2 q ˆxh 2 x q , µ F ˙ż 1 0 dx q x q ż d d ⃗ p q p2πq d δp1 ´xq ´xq ´xg q ż d d ⃗ p 1 d d ⃗ p 2 d d ⃗ p 1 1 d d ⃗ p 2 1 ˆdd ⃗ p 3 d d ⃗ p 3 1 p2πq d δ ˆ⃗ p q1 `xq x h 2 ⃗ p h 2 ´⃗ p 2 `xg x h 1 ⃗ p h 1 ´⃗ p 3 ˙δp⃗ p 11 1 `⃗ p 22 1 `⃗ p 33 1 q ˆpε Iα ε Jβ q " Φ α 3 p⃗ p 1 , ⃗ p 2 qΦ β: 4 p⃗ p 1 1 , ⃗ p 2 1 , ⃗ p 3 1 qF ˆ⃗ p 12 2 ˙r F ˚ˆ⃗ p 1 1 2 1 2 , ⃗ p 3 1 ˙δp⃗ p 3 q `Φα 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 qΦ β: 3 p⃗ p 1 1 , ⃗ p 2 1 q r F ˆ⃗ p 12 2 , ⃗ p 3 ˙F˚ˆ⃗ p 1 1 2 1 2 ˙δp⃗ p 3 1 q ȷ `ph 1 Ø h 2 q . (6.5.13)
The double dipole ˆdouble dipole contribution, for arbitrary polarization, is given by

dσ g qÑh 1 h 2 5JI dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 " α s α em 2µ 2ϵ p2πq 4d pN 2 c ´1q 1 4pp γ q 2 1 x d h 1 x d h 2 ÿ q Q 2 q ż 1 x h 1 dx g x g ż 1 x h 2 dx q x q px g x qq d´1 D h 1 g ˆxh 1 x q , µ F Ḋh 2 q ˆxh 2 x q , µ F ˙ż 1 0 dx q x q ż d d ⃗ p q p2πq d δp1 ´xq ´xq ´xg q ż d d ⃗ p 1 d d ⃗ p 2 d d ⃗ p 1 1 d d ⃗ p 2 1 ˆż d d ⃗ p 3 d d ⃗ p 3 1 p2πq 2d δ ˆ⃗ p q1 `xq x h 2 ⃗ p h 2 ´⃗ p 2 `xg x h 1 ⃗ p h 1 ´⃗ p 3 ˙δp⃗ p 11 1 `⃗ p 22 1 `⃗ p 33 1 q ˆpε Iα ε Jβ qΦ α 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 qΦ β: 4 p⃗ p 1 1 , ⃗ p 2 1 , ⃗ p 3 1 q r F ˆ⃗ p 12 2 , ⃗ p 3 ˙r F ˚ˆ⃗ p 1 1 2 1 2 , ⃗ p 3 1 ṗh 1 Ø h 2 q . (6.5.14)
Expression for the squared impact factors can be found in appendix B.2. They are written in terms of p q , p q, p g and the following identification should be done:

p gK " x g x h 1 p h 1 K , p qK " x q x h 2 p h 2 K . ( 6
.5.15)

Real corrections: Fragmentation from quark and gluon

Finite part of dipole ˆdipole contribution

When squaring the dipole contribution, one also gets finite terms. This time we write separately for each polarization transition. In the LL case, we have

dσ qgÑh 1 h 2 3LL dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 " α s α em C F µ 2ϵ p2πq 4d N c 1 4pp γ q 2 1 x d h 1 x d h 2 ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 x h 2 dx g x g px q x g q d´1 D h 1 q ˆxh 1 x q , µ F Ḋh 2 g ˆxh 2 x q , µ F ˙ż 1 0 dx q x q ż d d ⃗ p q p2πq d δp1 ´xq ´xq ´xg q ż d d ⃗ p 1 d d ⃗ p 2 ż d d ⃗ p 1 1 d d ⃗ p 2 1 ˆF ˆ⃗ p 12 2 ˙F˚ˆ⃗ p 1 1 2 1 2 ˙δ ˆxq x h 1 ⃗ p h 1 ´⃗ p 1 `⃗ p q2 `xg x h 2 ⃗ p h 2 ˙δp⃗ p 11 1 `⃗ p 22 1 q Q 2 pp γ q 2 ˆ» - - - - - - - - Φ 3 p⃗ p 1 , ⃗ p 2 qΦ `: 3 p⃗ p 1 1 , ⃗ p 2 1 q ´8x q x qpp γ q 4 `dx 2 g `4x qpx q `xg q Q2 `ˆxq x h 1 ⃗ p h 1 ´⃗ p 1 ˙2 xqp1´xqq '¨Q 2 `ˆxq x h 1 ⃗ p h 1 ´⃗ p 1 1 ˙2 xqp1´xqq ' ˆ1 ´xq xg x h 2 ⃗ p h 2 ´xg ⃗ p q¯2 fi ffi fl `ph 1 Ø h 2 q . (6.5.16)
For TL case, we have

dσ qgÑh 1 h 2 3T L dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 " α s α em C F µ 2ϵ p2πq 4d N c 1 4pp γ q 2 1 x d h 1 x d h 2 ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 x h 2 dx g x g px q x g q d´1 D h 1 q ˆxh 1 x q , µ F Ḋh 2 g ˆxh 2 x g , µ F ˙ż 1 0 dx q x q ż d d ⃗ p q p2πq d δp1 ´xq ´xq ´xg q ż d d ⃗ p 1 d d ⃗ p 2 d d ⃗ p 1 1 d d ⃗ p 2 1 ˆF ˆ⃗ p 12 2 ˙F˚ˆ⃗ p 1 1 2 1 2 ˙δ ˆxq x h 1 ⃗ p h 1 ´⃗ p 1 `⃗ p q2 `xg x h 2 ⃗ p h 2 ˙δp⃗ p 11 1 `⃗ p 22 1 qε T i Q p γ ˆ» - - - - - - - - Φ 3 p⃗ p 1 , ⃗ p 2 qΦ i: 3 p⃗ p 1 1 , ⃗ p 2 1 q `4x q `pγ ˘3 p2x q ´1q `x2 g d `4x q px q `xg q Q2 `ˆxq x h 1 ⃗ p h 1 ´⃗ p 1 ˙2 xqp1´xqq '¨Q 2 `ˆxq x h 1 ⃗ p h 1 ´⃗ p 1 1 ˙2 xqp1´xqq ' ˆ´xq x h 1 p h 1 K ´p1 1 K ¯i px q `xg q ´xq xg x h 2 ⃗ p h 2 ´xg ⃗ p q¯2
fi ffi fl `ph 1 Ø h 2 q . (6.5.17)

Finally, for TT case, we obtain

dσ qgÑh 1 h 2 3T T dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 " α s α em C F µ 2ϵ p2πq 4d N c 1 4pp γ q 2 1 x d h 1 x d h 2 ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 x h 2 dx g x g px q x g q d´1 D h 1 q ˆxh 1 x q , µ F Ḋh 2 g ˆxh 2 x g , µ F ˙ż 1 0 dx q x q ż d d ⃗ p q p2πq d δp1 ´xq ´xq ´xg q ż d d ⃗ p 1 d d ⃗ p 2 ż d d ⃗ p 1 1 d d ⃗ p 2 1 ˆF ˆ⃗ p 12 2 ˙F˚ˆ⃗ p 1 1 2 1 2 ˙δ ˆxq x h 1 ⃗ p h 1 ´⃗ p 1 `⃗ p q2 `xg x h 2 ⃗ p h 2 ˙δp⃗ p 11 1 `⃗ p 22 1 q ε T i ε T k ˆ» - - - - - - - - Φ i 3 p⃗ p 1 , ⃗ p 2 qΦ k: 3 p⃗ p 1 1 , ⃗ p 2 1 q ´2x qpp γ q 2 px 2 g d `4x qpx q `xg qq ¨Q2 `ˆxq x h 1 ⃗ p h 1 ´⃗ p 1 ˙2 xqp1´xqq '¨Q 2 `ˆxq x h 1 ⃗ p h 1 ´⃗ p 1 1 ˙2 xqp1´xqq ' ˆ"p1 ´2x q q 2 g ir K g lk K ´gil K g rl K `gik K g lr K ‰ ´xq x h 1 p h 1 K ´p1K ¯r ´xq x h 1 p h 1 K ´p1 1 K ¯l x q px q `xg q 2 ´xq xg x h 2 ⃗ p h 2 ´xg ⃗ p q¯2 fi ffi fl `ph 1 Ø h 2 q . (6.5.18)
Dipole ˆdouble dipole contribution and double dipole ˆdouble dipole contribution

The dipole ˆdouble dipole contribution is given, for arbitrary polarization, by

dσ q qÑh 1 h 2 4JI dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 " α s α em 2µ 2ϵ p2πq 4d N c 1 4pp γ q 2 1 x d h 1 x d h 2 ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 x h 2 dx g x g px q x g q d´1 D h 1 q ˆxh 1 x q , µ F Ḋh 2 g ˆxh 2 x g , µ F ˙ż 1 0 dx q x q ż d d ⃗ p q p2πq d δp1 ´xq ´xq ´xg q ż d d ⃗ p 1 d d ⃗ p 2 d d ⃗ p 1 1 d d p 2 1 ˆdd ⃗ p 3 d d ⃗ p 3 1 p2πq d δ ˆxq x h 1 ⃗ p h 1 ´⃗ p 1 `⃗ p q2 `xg x h 2 ⃗ p h 2 ´⃗ p 3 ˙δp⃗ p 11 1 `⃗ p 22 1 `⃗ p 33 1 q ˆpε Iα ε Jβ q " Φ α 3 p⃗ p 1 , ⃗ p 2 qΦ β: 4 p⃗ p 1 1 , ⃗ p 2 1 , ⃗ p 3 1 qF ˆ⃗ p 12 2 ˙r F ˚ˆ⃗ p 1 1 2 1 2 , ⃗ p 3 1 ˙δp⃗ p 3 q `Φα 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 qΦ β: 3 p⃗ p 1 1 , ⃗ p 2 1 q r F ˆ⃗ p 12 2 , ⃗ p 3 ˙F˚ˆ⃗ p 1 1 2 1 2 ˙δp⃗ p 3 1 q ȷ `ph 1 Ø h 2 q . (6.5.19)
The double dipole ˆdouble dipole contribution is given, for arbitrary polarization, by

dσ q qÑh 1 h 2 5JI dx h 1 dx h 2 d d ⃗ p h 1 d d ⃗ p h 2 " α s α em 2µ 2ϵ p2πq 4d pN 2 c ´1q 1 4pp γ q 2 1 x d h 1 x d h 2 ÿ q Q 2 q ż 1 x h 1 dx q x q ż 1 x h 2 dx g x g px q x g q d´1 D h 1 q ˆxh 1 x q , µ F Ḋh 2 g ˆxh 2 x g , µ F ˙ż 1 0 dx q x q ż d d ⃗ p q p2πq d δp1 ´xq ´xq ´xg q ż d d ⃗ p 1 d d ⃗ p 2 d d ⃗ p 1 1 d d ⃗ p 2 1 ˆż d d ⃗ p 3 d d ⃗ p 3 1 p2πq 2d δ ˆxq x h 1 ⃗ p h 1 ´⃗ p 1 `⃗ p q2 `xg x h 2 ⃗ p h 2 ´⃗ p 3 ˙δp⃗ p 11 1 `⃗ p 22 1 `⃗ p 33 1 q ˆpε Iα ε Jβ qΦ α 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 qΦ β: 4 p⃗ p 1 1 , ⃗ p 2 1 , ⃗ p 3 1 q r F ˆ⃗ p 12 2 , ⃗ p 3 ˙r F ˚ˆ⃗ p 1 1 2 1 2 , ⃗ p 3 1 ṗh 1 Ø h 2 q . (6.5.20)
Expression for the squared impact factors can be found in Appendix B.2. They are written in terms of p q , p q, p g and the following identification should be done:

p qK " x q x h 1 p h 1 K , p gK " x g x h 2 p h 2 K . ( 6
.5.21)

Conclusions and outlook

In this chapter, we have considered, for the first time at NLO, the diffractive production of a pair of hadrons with large ⃗ p 2 h " Λ 2 QCD , in γ p˚q -nucleon/nucleus scattering for Q 2 , ⃗ p 2 , t arbitrary, the only conditions being that lnpQ 2 {µ 2 F q, lnp⃗ p 2 h {µ 2 F q ! lnp1{xq with µ F the FF-factorization scale and ⃗ p 2 h ! ⃗ p 2 with ⃗ p the relative transverse momentum of the two hadrons. Our main result is the explicit finite result for the cross-section at NLO, obtained after showing explicitly the cancellation of rapidity divergences (through the B-JIMWLK equation), soft divergences and collinear divergences between real, virtual contributions, and DGLAP evolution equation governing fragmentation functions.

Finite contributions and purely divergent contributions have been separated and the sum of the latter has been shown to be zero. Hence, the collection of all terms labeled with "fin" in sections 6.2, 6.3, 6.4, plus all the formulas in section 6.5 give the final and main result of this chapter. These results add a new piece in the list of processes presented in chapter 2 which are very promising to probe gluonic saturation in nucleons and nuclei at NLO.

The evident continuation of this work would be to do some phenomenological studies in order to have some quantitatives values and as a first step, some values for the LO cross-section. In order to do that, the proton matrix element eq. (4.5.2) and its Fourier transform eq. (4.5.5) need to be modelize and the simplest choice is the GBW model [START_REF] Golec-Biernat | Saturation effects in deep inelastic scattering at low Q**2 and its implications on diffraction[END_REF][START_REF] Golec-Biernat | Saturation model of DIS : an update[END_REF][START_REF] Golec-Biernat | Saturation in diffractive deep inelastic scattering[END_REF]. Following [START_REF] Boussarie | Towards a complete next-to-logarithmic description of forward exclusive diffractive dijet electroproduction at HERA: real corrections[END_REF], this would be

F 0 p⃗ z q " A P 1 pp 0 1 q ˇˇT ´Tr " V ´⃗ z 2 ¯V : ´´⃗ z 2 ¯ı ´Nc ¯ˇˇP pp 0 q E 2πδpp 01 0 q ˇˇˇˇˇp 0 Ñp 0 1 " N c σ 0 ˜1 ´e´⃗ z 2 4R 2 0 ¸,
with the saturation radius defined as

R 0 " 1 Q 0 ˆxP x 0 ˙λ{2 , (6.6.1)
where

x P " Q 2 `M 2 ´t Q 2 `s2 ´M 2 p (6.6.2)
and the values of the parameters for 3 active quark flavours [START_REF] Golec-Biernat | Saturation effects in deep inelastic scattering at low Q**2 and its implications on diffraction[END_REF] Q 0 " 1 GeV, λ " 0.288, x 0 " 3.04 ˆ10 ´4, σ 0 " 23.03 mb . (6.6.

3)

The non-forward matrix element in impact parameter space reads

F p⃗ z q " ż d 2 ⃗ b e ´i⃗ b¨⃗ p 0 1 0 F ⃗ b p⃗ z q . (6.6.4)
In [START_REF] Boussarie | Towards a complete next-to-logarithmic description of forward exclusive diffractive dijet electroproduction at HERA: real corrections[END_REF], they choose to have a dependence in the impact parameter variable that is factorized from the GBW model and takes the form of a Gaussian proton profile function:

F ⃗ b p⃗ z q " 1 2πB G e ´⃗ b 2
2B G F 0 p⃗ z q . (6.6.5)

This gives then

Fp⃗ p q " ż d 2 ⃗ z e ´i⃗ z¨⃗ p ż d 2 ⃗ b e ´i⃗ b¨⃗ p 0 1 0 F ⃗ b p⃗ z q " N c σ 0 " p2πq 2 δp⃗ p q ´4πR 2 0 e ´R2 0 ⃗ p 2 ı e ´BG 2 ⃗ p 2 0 1 0 . (6.6.6)
After completing the LO phenomenology, the NLO ones would ideallly need to be done. This might reveal the need of other resummations which has been previously shown to be the case when one computes NLO corrections when studying processes with saturation physics [START_REF] Stasto | Towards the Test of Saturation Physics Beyond Leading Logarithm[END_REF]. Indeed we have for example in eq. ( 6.4.37) the presence of ln ´1 ´xq x h 1 ¯that can blow up in the phase-space region x q Ñ x h 1 . This would mean that some threshold resummation could be necessary.

Finally, one could also extend the results to have dissociative production. The calculations are based of the results from section 4.5 and section 4.6 where the assumption of proton remaining intact is made. The impact factors for γ p˚q Ñ q q and for γ p˚q Ñ q qg in the shockwave background will remain unchanged. The only thing that would change would be the target impact factor. For example, for dipole ˆdipole contributions to the cross-section which are 9 F ´⃗ p 12 2 ¯F˚´⃗ p 1 1 2 1 2 with F defined in eq. (4.5.2) and eq. (4.5.5), we would have instead

ÿ X ż d d ⃗ p X dp X p2πq d`1 2p X A P pp 0 q ˇˇr U : 1 1 2 1 p⃗ p 1 1 , ⃗ p 2 1 q ˇˇX E A X ˇˇr U 12 p⃗ p 1 , ⃗ p 2 q ˇˇP pp 0 q E " A P pp 0 q ˇˇr U : 1 1 2 1 p⃗ p 1 1 , ⃗ p 2 1 q r U 12 p⃗ p 1 , ⃗ p 2 q ˇˇP pp 0 q E .
Chapter 7

Diffractive single hadron in γ p˚q `p{A collisions

As an extension of the work presented in chapter 6, we present here the full NLO computations of the diffractive single-hadron electro-or photoproduction in the high-energy limit. This chapter is based on [START_REF] Fucilla | Diffractive single hadron production in a saturation framework at the NLO[END_REF].

7.1 Theoretical framework

Hybrid factorization

The process of interest in this chapter is:

γ p˚q pp γ q `P pp 0 q Ñ hpp h q `X `P 1 pp 0 1 q (7.1.1)

where P is a nucleon or a nucleus target, generically called proton in the following. A gap in rapidity is assumed between the outgoing proton and the diffractive system pXhq. This is illustrated by fig. 7.1.1.

We will be working in hybrid factorization i.e. a combination of collinear factorization and small-x factorization, more precisely in the shockwave formalism presented in chapter 3. The reference frame and kinematics for the photon and outgoing on-shell partons in the hard part (before collinear factorization) are the same as in section 4.1. We will write the momentum of the hadrons produced on the projectile side as

p µ h " x h p γ n µ 1 `m2 h `⃗ p 2 h 2x h p γ n µ 2 `pµ hK ; (7.1.2)
where x h is the fraction of the photon longitudinal momentum carried by the hadron. The kinematical region considered here is such that ⃗ p 2 h " Λ 2 QCD . The hadron momentum is the hard scale, making the use of collinear factorization possible. We also want lnpQ 2 {µ 2 F q, lnp⃗ p 2 h {µ 2 F q ! lnp1{x Bj q with µ F the fragmentation-function factorization scale to not have dominance of DGLAP effects. The quark and the antiquark fragment into the systems phXq (in the specific diagram h is produced by the quark, but can be as well produced by the antiquark). The tagged hadron h is drawn in red.

LO results

Let's consider the case of fragmentation from the quark as represented in fig. 7.1.2, the case for the antiquark is completely identical. The dashed line is to represent the integration over phase space.

Collinear factorization implies that the produced hadron flies collinearly to the fragmenting parton, we then have the following constraint:

p q " x q x h p h , ⃗ p q " x q x h ⃗ p h . (7.1.3) dσ qÑh 0JI dx h " ÿ q ż 1 x h dx q x q D h q ˆxh 1 x q , µ F ˙dσ 0JI dx q , (7.1.4)
where q specifies the quark flavor types (q " u, d, s, c, b), and J, I " L, T specify the photon polarization since we deal here with a modulus square amplitude (J labels the photon polarization in the complex conjugated amplitude and I in the amplitude). D h q denotes the quark Fragmentation Function (FF) and dσ is the partonic cross-section, i.e. the cross-section for the subprocess γ p˚q pp γ q `P pp 0 q Ñ qpp q q `qpp qq `P 1 pp 01 q .

(7.1.5)

Using eq. ( 7.1.4), the Born order hard cross-section of eq. ( 4.5.8) as well as the explicit expressions of the product Φ β 0 Φ γ: 0 in eqs. (4.5.10-4.5.12), the LO cross-sections are obtained and read dσ qÑh

0JI dx h d d ⃗ p h " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx q x 1`d q p1 ´xq q 2 D h q ˆxh x q ˙fJI , (7.1.6)
where

f LL " ż d d ⃗ p q ż d d ⃗ p 2 F ´xq 2x h ⃗ p h `1 2 ⃗ p q ´⃗ p 2 ⃗ p 2 q2 `xq p1 ´xq qQ 2 ż d d ⃗ p 2 1 F ˚´xq 2x h ⃗ p h `1 2 ⃗ p q ´⃗ p 2 1 ⃗ p 2 q2 1 `xq p1 ´xq qQ 2 , (7.1.7)
f T L " ż d d ⃗ p q ż d d ⃗ p 2 F ´xq 2x h ⃗ p h `1 2 ⃗ p q ´⃗ p 2 ⃗ p 2 q2 `xq p1 ´xq qQ 2 ż d d ⃗ p 2 1 F ˚´xq 2x h ⃗ p h `1 2 ⃗ p q ´⃗ p 2 1 ⃗ p 2 q2 1 `xq p1 ´xq qQ 2 ˆp1 ´2x q q 2x q p1 ´xq qQ `⃗ p q2 1 ¨⃗ ε T ˘˚, (7.1.8)
f T T " ż d d ⃗ p q ż d d ⃗ p 2 F ´xq 2x h ⃗ p h `1 2 ⃗ p q ´⃗ p 2 ⃗ p 2 q2 `xq p1 ´xq qQ 2 ż d d ⃗ p 2 1 F ˚´xq 2x h ⃗ p h `1 2 ⃗ p q ´⃗ p 2 1 ⃗ p 2 q2 1 `xq p1 ´xq qQ 2 ˆ"p1 ´2x q q 2 g ri K g lk K ´grk K g li K `grl K g ik K ı ε T i p q2Kr `εT k p q2 1 Kl ˘4x 2 q p1 ´xq q 2 Q 2 , (7.1.9)
are the three different functions for the LL, T L and T T cross-section respectively. For compactness, we use the short notation

f JI `xq , x h , ⃗ p h , Q 2 ˘" f JI . (7.1.10)
The correct cross section, in the case of antiquark fragmentation, is obtained by including an overall minus sign in the whole argument of the function F, making the sum over q with q " ū, d, s, c, b and performing the re-labelling px q , ⃗ p q , p 2 , p 2 1 q Ø px q, ⃗ p q, p 1 , p 1 1 q 1 . We will call this last operation pq Ø qq re-labelling.

NLO computation in a nutshell

Different mechanisms of fragmentation

At the next-to-leading order, there are six kinds of contributions to the cross-section depending on the fragmentation function used (a) γ ˚`P Ñ h`q`X`P cross-section at one-loop (i.e. virtual contribution and fragmentation from a quark) , (b) γ ˚`P Ñ h`q`X`P cross-section at one-loop (i.e. virtual contribution and fragmentation from an antiquark) , (c) γ ˚`P Ñ h `q `g `X `P cross-section at Born level (i.e. real contribution and fragmentation from a quark) , (d) γ ˚`P Ñ h `q `g `X `P cross-section at Born level (i.e. real contribution and fragmentation from an antiquark) , (e) γ ˚`P Ñ h `q `q `X `P cross-section at Born level (i.e. real contribution and fragmentation from a gluon) , (f) FFs counterterms from the renormalization of the quark or antiquark FF.

Organization of the final cross-section

We can organize the different cross-sections depending on the FF used, the divergences (the different divergences to treat and cancel are the same as in section 6.

Fragmentation from quark

The virtual part of the cross-section can be split into

dσ qÑh JI dx h d d ⃗ p h ˇˇˇv irt. NLO " dσ qÑh 1JI dx h d d ⃗ p h ˇˇˇS V `dσ qÑh 1JI dx h d d ⃗ p h ˇˇˇfi n `dσ qÑh 2JI dx h d d ⃗ p h , (7.1.11)
where the first term contains the divergences of the virtual dipole ˆdipole contribution and the second term is a finite virtual dipole ˆdipole contribution. They are constructing from eqs. (4.5.15, 4.5.17, 4.5.18), to be more exact the first term is built from the term proportional to ´SV `SV 2 ¯and the second term is the remaining part not proportional to ´SV The real part of the cross-section can be split into

dσ qÑh JI dx h d d ⃗ p h ˇˇˇr eal NLO " dσ qÑh 3JI dx h d d ⃗ p h `dσ qÑh 4JI dx h d d ⃗ p h `dσ qÑh 5JI dx h d d ⃗ p h . (7.1.12)
The last two contribution corresponding to dipole ˆdouble dipole and double dipole ˆdouble dipole contributions are finite, while the first one for real dipole ˆdipole contribution can be separated into a finite part and a part that contains the singularities

dσ qÑh 3JI dx h d d ⃗ p h " dσ qÑh 3JI dx h d d ⃗ p h ˇˇˇd iv `dσ qÑh 3JI dx h d d ⃗ p h ˇˇˇfi n . (7.1.13)
This separation comes from the fact that in the partonic cross-sections for dipole ˆdipole real contribution is given by eq. (4.6.2) with the impact factor squared given by eq. (4.6.4). Only the first term in the right-hand side of the equality of eq. (4.6.4) contains divergences. The second term of the above equation is therefore proportional to all the terms in the right-hand side of the equality of eq. (4.6.4) minus the first term r Φ α 3 p⃗ p 1 , ⃗ p 2 q r Φ β: 3 p⃗ p 1 1 , ⃗ p 2 1 q. This first term is the one contributing to the singular part of real dipole ˆdipole contribution. This contribution contains both soft and collinear singularity that can be promptly separated by casting the contribution into the following form The last term in eq. ( 7.1.15) is finite and defined as

dσ qÑh 3JI dx h d d ⃗ p h ˇˇˇd iv " dσ qÑh 3JI dx h d d ⃗ p h ˇˇˇp 1q `dσ qÑh 3JI dx h d d ⃗ p h ˇˇˇp 2q `dσ qÑh 3JI dx h d d ⃗ p h ˇˇˇp 3q `dσ qÑh 3JI dx h d d ⃗ p h ˇˇˇp 4q (7.1.14)
dσ qÑh 3JI dx h d d ⃗ p h ˇˇˇr eal fin. sub. " dσ qÑh 3JI dx h d d ⃗ p h ˇˇˇp 2q ´dσ qÑh 3JI dx h d d ⃗ p h ˇˇˇp 2q soft `dσ qÑh 3JI dx h d d ⃗ p h ˇˇˇp 4q ´dσ qÑh 3JI dx h d d ⃗ p h ˇˇˇp 4q soft . (7.1.19)
The organization for total cross-section for the contribution from the fragmentation from the antiquark follows the same structure.

Fragmentation from gluon

This fragmentation mechanism is only possible when a real gluon is produced, therefore we only deal with real corrections that can be arranged as

dσ gÑh JI dx h d d ⃗ p h ˇˇˇr eal NLO " dσ gÑh 3JI dx h d d ⃗ p h `dσ gÑh 4JI dx h d d ⃗ p h `dσ gÑh 5JI dx h d d ⃗ p h , (7.1.20)
where the last two contributions are finite while the first one can be split as

dσ gÑh 3JI dx h d d ⃗ p h " dσ gÑh 3JI dx h d d ⃗ p h ˇˇˇd iv `dσ gÑh 3JI dx h d d ⃗ p h ˇˇˇfi n. 1 `dσ gÑh 3JI dx h d d ⃗ p h ˇˇˇfi n. 2 . ( 7.1.21)
The singular part contains contributions coming from diagrams (1) and (3) in fig. 7.1.5

dσ gÑh 3JI dx h d d ⃗ p h ˇˇˇd iv " dσ gÑh 3JI dx h d d ⃗ p h ˇˇˇp 1q `dσ gÑh 3JI dx h d d ⃗ p h ˇˇˇp 3q . (7.1.22) 205 (1) (2) (3) (4)
dσ gÑh 3JI dx h d d ⃗ p h ˇˇˇfi n.1 " dσ gÑh 3JI dx h d d ⃗ p h ˇˇˇp 2q `dσ gÑh 3JI dx h d d ⃗ p h ˇˇˇp 4q , (7.1.24)
while the last term eq. ( 7.1.21) contains the rest of the dipole ˆdipole contribution, i.e. is defined similarly to

dσ qÑh 3JI dx h d d ⃗ p h ˇˇˇfi n
in eq. (7.1.13) .

In the sections below, we will focus on the case of fragmentation of the quark (and the necessary term from fragmentation from gluon to have cancellation of divergences) in the explicit calculations and will state how the contribution for fragmentation from the antiquark can be obtained from the results for the quark case. Therefore the cancellation of divergences for the case of antiquark fragmenting will not be explicit but implied by the symmetry between quark and antiquark case.

Counterterms from FF renormalization

The counterterm is constructed by inserting eq. ( 6.2.1) into the LO contribution eq. ( 7.1.6). This produces

dσ qÑh JI dx h d d p hK ˇˇˇc t " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx q x 1`d q p1 ´xq q 2 f JI ˆ´´α s 2π ¯ˆ1 ε `ln ˆµ2 F µ 2 ˙˙ż 1 x h xq dβ β " D h q ˆxh βx q , µ F ˙Pqq pβq `Dh g ˆxh βx q , µ F ˙Pgq pβq ȷ " dσ qÑh JI dx h d d p hK ˇˇˇc t div `dσ qÑh JI dx h d d p hK ˇˇˇc t fin , (7.2.1)
where the `prescription is defined as in eq. (6.2.6), the LO splitting function are shown in eqs (6.2.2, 6.2.3).

It is also useful to split the divergent part as

dσ qÑh JI dx h d d ⃗ p h ˇˇˇc t div " dσ qÑh JI dx h d d ⃗ p h ˇˇˇc t div Pqq `dσ qÑh JI dx h d d ⃗ p h ˇˇˇc t div Pgq , (7.2.2)
where the first term contains the contribution proportional to the quark FF, D h q , while the second contains the one proportional to the gluon FF, D h g .

The counterterm for the antiquark fragmentation case is obtained as before, by including a minus sign in the argument of the function F, making the sum be over q with q one of the five antiquark flavor species (in order to have the FFs of antiquarks) and performing the pq Ø qq re-labelling.

NLO cross-section: Virtual contributions

Dipole ˆdipole contribution

From eq. (4.5.16) and eq. ( 7.1.6), we obtain for the divergent part

dσ qÑh 1JI dx h d d ⃗ p hK ˇˇˇS V div " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx q x 1`d q p1 ´xq q 2 D h q ˆxh x q , µ F ż d d ⃗ p q ż d d ⃗ p 2 F ´xq 2x h ⃗ p h `1 2 ⃗ p q ´⃗ p 2 ⃗ p 2 q2 `xq p1 ´xq qQ 2 ż d d ⃗ p 2 1 F ˚´xq 2x h ⃗ p h `1 2 ⃗ p q ´⃗ p 2 1 ⃗ p 2 q2 1 `xq p1 ´xq qQ 2 δ JI ˆαs 2π C F 1 ε » - -´4ϵ lnpαq ln ¨p1 ´xq q 2 µ 2 ´⃗ p q ´1´xq x h ⃗ p h ¯2 ‹ '`4 lnpαq `4ϵ ln 2 pαq ´2 lnpx q p1 ´xq qq `3fi ffi fl (7.3.1)
and for the finite part

dσ qÑh 1JI dx h d d ⃗ p h ˇˇˇS V fin " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx q x 1`d q p1 ´xq q 2 D h q ˆxh x q , µ F ż d d ⃗ p q ż d d ⃗ p 2 F ´xq 2x h ⃗ p h `1 2 ⃗ p q ´⃗ p 2 ⃗ p 2 q2 `xq p1 ´xq qQ 2 ż d d ⃗ p 2 1 F ˚´xq 2x h ⃗ p h `1 2 ⃗ p q ´⃗ p 2 1 ⃗ p 2 q2 1 `xq p1 ´xq qQ 2 δ JI ˆαs 2π C F 1 ε » - -2ϵ ln ¨p1 ´xq qµ 2
x q ´⃗ p q ´1´xq

x h ⃗ p h ¯2 ‹ 'lnpxqp1 ´xq qq `ϵ ln 2 px q p1 ´xq qq ´3ϵ ln ¨p1 ´xq qµ 2

x q ´⃗ p q ´1´xq x h ⃗ p h ¯2 ‹ '´π 2 3 ϵ `6ϵ fi ffi fl (7.3.2)
where

δ LL " 1 , δ T L " p1 ´2x q q 2x q p1 ´xq qQ `⃗ p q2 1 ¨⃗ ε T ˘˚, δ T T " " p1 ´2x q q 2 g ri K g lk K ´grk K g li K `grl K g ik K ı ε T i p q2Kr `εT k p q2 1 Kl ˘4x 2 q p1 ´xq q 2 Q 2 . ( 7.3.3)
The corresponding contribution in the case of fragmentation from antiquark is obtained by including a minus sign in the argument of the function F, making the sum be over q with q one of the five antiquark flavor species and performing the pq Ø qq re-labelling (also inside δ JI ).

The remaining finite parts from eqs. (4.5.15, 4.5.17, 4.5.18) read

dσ qÑh 1LL dx h d 2 ⃗ p h ˇˇˇfi n " 2α em Q 2 p2πq 8 N c x 2 h ÿ q Q 2 q ż 1 x h dx q x q D h q ˆxh x q , µ F ˙αs 2π C F 4 ˆż d 2 ⃗ p q d 2 ⃗ p 1 d 2 ⃗ p 2 d 2 ⃗ p 1 1 d 2 ⃗ p 2 1 δ ˆxq x h ⃗ p h ´⃗ p 1 `⃗ p q2 ˙δp⃗ p 11 1 `⃗ p 22 1 q F ˆ⃗ p 12 2 ˙F˚ˆ⃗ p 1 1 2 1 2 # 1 ⃗ p 2 q1 1 `xq x qQ 2 « 6x 2 q x 2 q ⃗ p 2 q1 `xq x qQ 2 ln ˜x2 q x 2 q µ 4 Q 2 px q ⃗ p q ´xq ⃗ p q q 2 p⃗ p 2 q1 `xq x qQ 2 q 2 ż xq 0 dz ˜rpϕ 4 q LL s ``ÿ n"5,6 rpϕ n q LL s `|⃗ p 3 " ⃗ 0 ¸`pq Ø qq ¸ff `h.c.| p 1 ,p 2 Øp 1 1 ,p 2 1 + xq"1´xq ⃗ pq" xq x h ⃗ p h , (7.3.4)
in the LL case,

dσ qÑh 1T L dx h d 2 ⃗ p h ˇˇˇfi n " α em Q p2πq 8 N c x 2 h ÿ q Q 2 q ż 1 x h dx q x q D h q ˆxh x q , µ F ˙αs 4π C F ε T i ˆż d 2 ⃗ p q d 2 ⃗ p 1 d 2 ⃗ p 2 d 2 ⃗ p 1 1 d 2 ⃗ p 2 1 δ ˆxq x h ⃗ p h ´⃗ p 1 `⃗ p q2 ˙δp⃗ p 11 1 `⃗ p 22 1 q F ˆ⃗ p 12 2 ˙F˚ˆ⃗ p 1 1 2 1 2 $ ' ' ' ' ' & ' ' ' ' ' % «˜ż xq 0 dz ˜"pϕ 4 q i T L ‰ ``ÿ n"5,6 " pϕ n q i T L ‰ `|⃗ p 3 " ⃗ 0 ¸¸`pq Ø qq ff ⃗ p 2 q1 `xq x qQ 2 `3x q x qp1 ´2x q qp i q1 1 K p⃗ p 2 q1 `xq x qQ 2 q ˆ1 p⃗ p 2 q1 1 `xq x qQ 2 q ˜ln ˜x3 q x 3 q µ 8 Q 2 px q ⃗ p q ´xq ⃗ p q q ´4 p⃗ p 2 q1 `xq x qQ 2 q 2 p⃗ p 2 q1 1 `xq x qQ 2 q ¸´x q x qQ 2 ⃗ p 2 q1 1 ln ˜xq x qQ 2 ⃗ p 2 q1 1 `xq x qQ 2 ¸« ˜ż xq 0 dz ˜"pϕ 4 q i LT ‰ ``ÿ n"5,6 " pϕ n q i LT ‰ `|⃗ p 3 " ⃗ 0 ¸¸`pq Ø qq ff 2x q x q ´⃗ p 2 q1 1 `xq x qQ 2 ¯, / / / / / . / / / / / -xq"1´xq ⃗ pq" xq x h ⃗ p h , (7.3.5)
in the TL case, and

dσ qÑh 1T T dx h d 2 ⃗ p h ˇˇˇfi n " α em 2 p2πq 8 N c x 2 h ÿ q Q 2 q ż 1 x h dx q x q D h q ˆxh x q , µ F ˙pε T i ε T k q ˆż d 2 ⃗ p q d 2 ⃗ p 1 d d ⃗ p 2 d 2 ⃗ p 1 1 d 2 ⃗ p 2 1 δ ˆxq x h ⃗ p h ´⃗ p 1 `⃗ p q2 ˙δp⃗ p 11 1 `⃗ p 22 1 q F ˆ⃗ p 12 2 ˙F˚ˆ⃗ p 1 1 2 1 2 αs 2π C F # 3 2
p q1Kr p q1 1 Kl p⃗ p 2 q1 `xq x qQ 2 qp⃗ p 2 q1 1 `xq x qQ 2 q " p1 ´2x q q 2 g ri K g lk K ´grk K g li K `grl K g ik K ı 209 ˆ«ln ˜xq x qµ 4 px q ⃗ p q ´xq ⃗ p q q 2 p⃗ p 2 q1 `xq x qQ 2 q ¸´x q x qQ 2 ⃗ p 2 q1 ln ˜xq x qQ 2 ⃗ p 2 q1 `xq x qQ 2 ¸ff `«˜ż xq 0 dz ˜"pϕ 4 q ik T T ı ``ÿ n"5,6 " pϕ n q ik T T ı `|⃗ p 3 " ⃗ 0 ¸¸`pq Ø qq ff x q x q ´⃗ p 2 q1 1 `xxQ 2 ¯`h.c.|p1,p2Øp 1 1 ,p 2 1 iØk , / / / / / . / / / / / -xq"1´xq ⃗ pq" xq x h ⃗ p h , (7.3.6)
in the TT case.

Dipole ˆdouble dipole contribution

From eqs. (4.5.20-4.5.22), we have then

dσ qÑh 2LL dx h d 2 ⃗ p h " 2α em Q 2 p2πq 8 N c x 2 h ÿ q Q 2 q ż 1 x h dx q x q D h q ˆxh x q , µ F ˙ż d 2 ⃗ p q d 2 ⃗ p 1 d 2 ⃗ p 2 d 2 ⃗ p 1 1 d 2 ⃗ p 2 1 ˆż d 2 ⃗ p 3 p2πq 2 δ ˆxq x h ⃗ p h ´⃗ p 1 `⃗ p q2 ´⃗ p 3 ˙δp⃗ p 11 1 `⃗ p 22 1 `⃗ p 3 q α s 2π 1 8 # 1 ⃗ p 2 q1 1 `xq x qQ 2 r F ˆ⃗ p 12 2 , ⃗ p 3 Ḟ˚ˆ⃗ p 1 1 2 1 2 ˙«4x q x q « x q x qp⃗ p 2 3 ´⃗ p 2 q2 ´⃗ p 2 q1 ´2x q x qQ 2 q p⃗ p 2 q2 `xq x qQ 2 qp⃗ p 2 q1 `xq x qQ 2 q ´xq x qQ 2 ⃗ p 2 3 ln ´xq x q e 2η ln ˜p⃗ p 2 q2 `xq x qQ 2 q `⃗ p 2 q1 `xq x qQ 2 xq x qQ 2 ⃗ p 2 3 ¸´˜2 x q x q ⃗ p 2 q1 `xq x qQ 2 ln ´xq e η ¯ln ˆ⃗ p 2 3 µ 2 ˙`pq Ø qq ¸ff `Q2 «˜ż xq 0 dz ÿ n"5,6 rpϕ n q LL s `¸`pq Ø qq ffff `h.c.| p 1 ,p 1 Øp 1 1 ,p 2 1 + xq"1´xq ⃗ pq" xq x h ⃗ p h , (7.3.7)
in the LL case,

dσ qÑh 2T L dx h d 2 ⃗ p h " α em Q p2πq 8 N c x 2 h ÿ q Q 2 q ż 1 x h dx q x q D h q ˆxh x q , µ F ˙ż d 2 ⃗ p q d 2 ⃗ p 1 d 2 ⃗ p 2 d 2 ⃗ p 1 1 d 2 ⃗ p 2 1 ˆż d 2 ⃗ p 3 d 2 ⃗ p 3 1 p2πq 2 δ ˆxq x h ⃗ p h ´⃗ p 1 `⃗ p q2 ´⃗ p 3 ˙δp⃗ p 11 1 `⃗ p 22 1 `⃗ p 33 1 q α s 8π ε T i # δp⃗ p 3 1 q ⃗ p 2 q1 1 `xq x qQ 2 r F ˆ⃗ p 12 2 , ⃗ p 3 Ḟ˚ˆ⃗ p 1 1 2 1 2 ˙«2p1 ´2x q qp i q1 1 K « x q x qp⃗ p 2 3 ´⃗ p 2 q2 ´⃗ p 2 q1 ´2x q x qQ 2 q p⃗ p 2 q2 `xq x qQ 2 qp⃗ p 2 q1 `xq x qQ 2 q ´xq x qQ 2 ⃗ p 2 3 ln ´xq x q e 2η ln ˜p⃗ p 2 q2 `xq x qQ 2 q `⃗ p 2 q1 `xq x qQ 2 xq x qQ 2 ⃗ p 2 3 ¸´˜2 x q x q ⃗ p 2 q1 `xq x qQ 2 ln ´xq e η ¯ln ˆ⃗ p 2 3 µ 2 ˙`pq Ø qq ¸ff `1 2x q x q «˜ż xq 0 dz ÿ n"5,6 " pϕ n q i T L ‰ `¸`pq Ø qq ffff `δp⃗ p 3 q ⃗ p 2 q1 `xq x qQ 2 r F ˚ˆ⃗ p 1 1 2 1 2 , ⃗ p 3 1 Ḟ ˆ⃗ p 12 2 ˙«« 2x q x qp1 ´2x q qp i q1 1 K ˜´2 ⃗ p 2 q1 1 `xq x qQ 2 ln ´xq e η ¯ln ˆ⃗ p 2 3 1 µ 2 ˙`ln ´xq x q e 2η p⃗ p 2 q2 1 `xq x qQ 2 q p⃗ p 2 q1 1 `xq x qQ 2 qp⃗ p 2 q2 1 `xq x qQ 2 q ´xq x qQ 2 ⃗ p 2 3 1 ln ˜p⃗ p 2 q1 1 `xq x qQ 2 qp⃗ p 2 q2 1 `xq x qQ 2 q x q x qQ 2 ⃗ p 2 3 1 1 ⃗ p 2 q1 1 ln ˜⃗ p 2 q1 1 `xq x qQ 2 x q x qQ 2 ¸¸¸`p q Ø qq ff `«˜ż xq 0 dz ÿ n"5,6 " pϕ n q i LT ‰ `¸`pq Ø qq ff ˚ff+ xq"1´xq ⃗ pq" xq x h ⃗ p h , (7.3.8)
in the TL case, and

dσ qÑh 2T T dx h d 2 ⃗ p h " α em 2 p2πq 8 N c x 2 h ÿ q Q 2 q ż 1 x h dx q x q D h q ˆxh x q , µ F ˙ż d 2 ⃗ p 1 d 2 ⃗ p 2 d 2 ⃗ p 1 1 d 2 ⃗ p 2 1 ˆż d 2 ⃗ p q ż d 2 ⃗ p 3 p2πq 2 δ ˆxq x h ⃗ p h ´⃗ p 1 `⃗ p q2 ´⃗ p 3 ˙δp⃗ p 11 1 `⃗ p 22 1 `⃗ p 3 q α s 4π pε T i ε T j q # r F ˆ⃗ p 12 2 , ⃗ p 3 ˙F˚ˆ⃗ p 1 1 2 1 2 1 ⃗ p 2 q1 1 `xq x qQ 2 » - - - » - - -p q1 1 Kl p q1Kk " p1 ´2xq 2 g ki K g lj K ´gkj K g li K `gkl K g ij K ı » - - - ´2 ln ´xq e η ¯ln ˆ⃗ p 2 3 µ 2 ⃗ p 2 q1 `xq x qQ 2 `ln ´xq x q e 2η ¯˜1 ⃗ p 2 q1 ln ˜⃗ p 2 q1 `xq x qQ 2 x q x qQ 2 ¸´⃗ p 2 q2 `xq x qQ 2 p⃗ p 2 q1 `xq x qQ 2 qp⃗ p 2 q2 `xq x qQ 2 q ´xq x qQ 2 ⃗ p 2 3 ˆln ˜p⃗ p 2 q1 `xq x qQ 2 qp⃗ p 2 q2 `xq x qQ 2 q x q x qQ 2 ⃗ p 2 3 ¸¸ff `pq Ø qq ff `1 x q x q «˜ż xq 0 dz ÿ n"5,6 " pϕ n q ij T T ı `¸`pq Ø qq ffff `h.c.|p1,p2Øp 1 1 ,p 2 1 iØj + xq"1´xq ⃗ pq" xq x h ⃗ p h , (7.3.9)
in the TT case.

The corresponding finite virtual contributions in the case of antiquark fragmentation are obtained as follows:

1. by changing the integration variables x q , ⃗ p q to x q, ⃗ p q , 2. making the sum be over q with q one of the five antiquark flavor species (in order to have the FFs of the antiquarks),

3. computing the objects in the curly brackets fixing x q " 1 ´xq and ⃗ p q " xq x h ⃗ p h , 4. making the changes x q Ñ x q and ⃗ p q Ñ ⃗ p q in the argument of the first delta function.

NLO cross-section: Real contributions and cancellation of divergences

The detailed calculations in this section will be done for JI " LL but the final results will be presented for general JI as the T L and T T calculations proceed in the same way as for the LL case, the only differences are in the corrective factors eq. (7.3.3) and in the functions eqs. (7.1.7, 7.1.8, 7.1.9). The LL partonic cross-section of interest here is given by eq. ( 6.4.1). We keep the prime notations as some necessary change of variable will be needed and one need to distinguish the longitudinal momentum fraction of the parton before and after splitting.

Fragmentation from quark

Collinear contributions: q-g splitting The term in eq. (6.4.1) to consider is the first term in the bracket, corresponding to the diagram p1q in fig. 7.1.4. This contribution contains both soft and collinear divergences. The soft contribution are treated separately as mentioned above therefore its soft contribution will be removed from the total contribution. This is represented in fig. 7.4.1.

Convoluting this contribution as in eq. ( 7.1.4) as well as using the constraint eq. (7.1.3), we get

dσ qÑh 3LL dx h d d ⃗ p h ˇˇˇp 1q " 2α em Q 2 p2πq 4d N c ÿ q Q 2 q ż 1 x h dx 1 q x 1 q ż 1´x 1 q α dx g x g D h q ˆxh x 1 q , µ F ˙ˆx 1 q x h ˙d ˆp1 ´x1 q ´xg q 2 px g `x1 q q 2 px 1 q q 2 α s C F µ 2ϵ ż d d ⃗ p q ż d d ⃗ p g p2πq d ż d d ⃗ p 2 F ´x1 q 2x h ⃗ p h `⃗ pq 2 `⃗ pg 2 ´⃗ p 2 px 1 q `xg qp1 ´x1 q ´xg qQ 2 `⃗ p 2 q2 ż d d ⃗ p 2 1 F ˚´x 1 q 2x h ⃗ p h `⃗ pq 2 `⃗ pg 2 ´⃗ p 2 1 px 1 q `xg qp1 ´x1 q ´xg qQ 2 `⃗ p 2 q2 1 ¯dx 2 g `4x 1 q px 1 q `xg q ´⃗ p g ´xg x h ⃗ p h ¯2 . (7.4.1)
The cross section in eq. (7.4.1) is expressed in terms of the fractions of longitudinal momenta of the initial photon carried by the quark and gluon produced after the splitting. However, to observe the cancellation of collinear divergences between this contribution and the counterterms coming from the renormalization of FF, it is necessary to perform the change of variables

x 1 q " βx q , x g " p1 ´βqx q ,

where x q is the fraction of longitudinal momenta of the initial photon carried by the quark before the splitting, while β is the fraction of longitudinal momenta of the initial quark carried by the final quark. This is the same kind of change of variables used in the computations of chapter 6, see fig. 6.4.1. Then, the longitudinal integrations become

ż 1 x h dx 1 q ż 1´x 1 q α dx g " ..... ‰ " ż 1 x h dx q x q ż 1´α xq x h xq dβ " ..... ‰ xg"p1´βqxq, x 1 q "βxq , (7.4.3)
where " ..... ‰ represents the whole integrand function in eq. ( 7.4.1) and the factor x q in the right hand side comes the Jacobian of the transformation in eq. (7.4.2). After a bit of algebra, we end up with

dσ qÑh 3LL dx h d d ⃗ p h ˇˇˇp 1q " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx q x 1`d q p1 ´xq q 2 ż 1´α xq x h xq dβ β ˆβd´2 p1 `β2 `ϵp1 ´βq 2 q p1 ´βq D h q ˆxh βx q , µ F ˙ż d d ⃗ p q 2α s C F µ 2ϵ ż d d ⃗ p g p2πq d ż d d ⃗ p 2 F ´βxq 2x h ⃗ p h `⃗ pq 2 `⃗ pg 2 ´⃗ p 2 xq p1 ´xq qQ 2 `⃗ p 2 q2 ż d d ⃗ p 2 1 F ˚´βxq 2x h ⃗ p h `⃗ pq 2 `⃗ pg 2 ´⃗ p 2 1 xq p1 ´xq qQ 2 `⃗ p 2 q2 1 ¯1 ´⃗ p g ´p1´βqxq x h ⃗ p h ¯2 . (7.4.4)
The integral over ⃗ p g is performed by expressing the F in terms of their Fourier transform eq. (4.5.5) and by using eq. (6.4.4), leading to

dσ qÑh 3LL dx h d d ⃗ p h ˇˇˇp 1q " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx q x 1`d q p1 ´xq q 2 ˆż d d ⃗ p q ż d d ⃗ p 2 ż d d ⃗ z 1 e ´i⃗ z 1 ¨´xq 2x h ⃗ p h `⃗ p q 2 ´⃗ p 2 xq p1 ´xq qQ 2 `⃗ p 2 q2 ¯F p⃗ z 1 q ż d d ⃗ p 2 1 ż d d ⃗ z 2 e i⃗ z 2 ¨´xq 2x h ⃗ p h `⃗ p q 2 ´⃗ p 2 xq p1 ´xq qQ 2 `⃗ p 2 q2 1 ¯F ˚p⃗ z 2 q ˆαs 2π C F # ż 1´α xq x h xq dβ β D h q ˆxh βx q , µ F ˙1 ε ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙ϵ 1 `β2 1 ´β `ż 1 x h xq dβ β D h q ˆxh βx q , µ F ˙p1 ´βq 2 `2p1 `β2 q ln β 1 ´β + . (7.4.5)
Introducing the `prescription defined in eq. (6.2.6) and after some algebra, we get

dσ qÑh 3LL dx h d d ⃗ p h ˇˇˇp 1q " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx q x 1`d q p1 ´xq q 2 (7.4.6) ˆż d d ⃗ p q ż d d ⃗ p 2 ż d d ⃗ z 1 e ´i⃗ z 1 ¨´xq 2x h ⃗ p h `⃗ p q 2 ´⃗ p 2 xq p1 ´xq qQ 2 `⃗ p 2 q2 ¯F p⃗ z 1 q ż d d ⃗ p 2 1 ż d d ⃗ z 2 e i⃗ z 2 ¨´xq 2x h ⃗ p h `⃗ p q 2 ´⃗ p 2 xq p1 ´xq qQ 2 `⃗ p 2 q2 1 ¯F ˚p⃗ z 2 q ˆαs 2π C F # 1 ε ż 1 x h xq dβ β D h q ˆxh βx q , µ F ˙1 `β2 p1 ´βq ``ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙ż 1 x h xq dβ β D h q ˆxh βx q , µ F ˙1 `β2 p1 ´βq 1 ε ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙ϵ ż 1´α xq x h xq dβ D h q ˆxh x q , µ F ˙2 1 ´β ´1 ε D h q ˆxh x q , µ F ˙2 ln ˆ1 ´xh x q ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙Dh q ˆxh x q , µ F ˙2 ln ˆ1 ´xh x q ˙`ż 1 x h xq dβ β D h q ˆxh βx q , µ F ˙p1 ´βq 2 `2p1 `β2 q ln β 1 ´β + .
The term proportional to 1 ε ´4c 2 0 ⃗ z 2 12 µ 2 ¯ϵ is the soft contribution of this contribution and is to be removed. Separating this contribution minus its soft part into divergent and finite parts and writing it for general JI photon polarizations, we have

dσ qÑh 3JI dx h d d ⃗ p h ˇˇˇc oll. qg div " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx q x 1`d q p1 ´xq q 2 f JI ˆαs 2π 1 ε « ż 1 x h xq dβ β C F 1 `β2 p1 ´βq `Dh q ˆxh βx q , µ F ˙´2C F ln ˆ1 ´xh x q ˙Dh q ˆxh x q , µ F ˙ff , (7.4.7)
for the divergent part and

dσ qÑh 3JI dx h d d ⃗ p h ˇˇˇc oll. qg fin " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx q x 1`d q p1 ´xq q 2 f JI (7.4.8) ˆαs 2π C F ż 1 x h xq dβ β D h q ˆxh βx q , µ F ˙"p1 ´βq `2 1 `β2 p1 ´βq ln β ȷ `2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1
x h dx q x 1`d q p1 ´xq q 2 (7.4.9)

ˆż d d ⃗ p q ż d d ⃗ p 2 ż d d ⃗ z 1 e ´i⃗ z 1 ¨´xq 2x h ⃗ p h `⃗ p q 2 ´⃗ p 2 xq p1 ´xq qQ 2 `⃗ p 2 q2 ¯F p⃗ z 1 q ż d d ⃗ p 2 1 ż d d ⃗ z 2 e i⃗ z 2 ¨´xq 2x h ⃗ p h `⃗ p q 2 ´⃗ p 2 xq p1 ´xq qQ 2 `⃗ p 2 q2 1 ¯F ˚p⃗ z 2 q ˆδJI α s 2π C F ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙#ż 1 x h xq dβ β D h q ˆxh βx q , µ F ˙1 `β2 p1 ´βq `´D h q ˆxh x q , µ F ˙2 ln ˆ1 ´xh x q

˙+

for the finite part.

The corresponding contributions in the case of fragmentation from antiquark are obtained by including an overall minus sign in the argument of the function F or in the term in bracket in the exponential of ⃗ z i , making the sum be over q with q one of the five antiquark flavor species and performing the pq Ø qq re-labelling.

Collinear contributions: q-g splitting The term in eq. ( 6.4.1) to consider is the third term in the bracket, corresponding to the diagram p3q in fig. 7.1.4. This divergent contribution in the case of quark fragmentation is shown in fig. 7.4.2. The computation is similar to the one of the previous section but no change of variable is performed as the fragmenting particle is not involved in the splitting and one therefore integrate directly over x g . Because no change of variable is needed, we re-label the prime notation x 1 q , x 1 q with x q , x q. The obtained divergence is of collinear nature i.e. a singularity of the form 1{ϵ.

Convoluting this contribution as in eq. ( 7.1.4) as well as using the constraint eq. (7.1.3) and the Fourier transform eq. (4.5.5),

dσ qÑh 3LL dx h d d ⃗ p h ˇˇˇp 3q " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx q D h q ˆxh x q , µ F ˙x1`d q p1 ´xq q 2 ż 1´xq α dx g x g ˆż d d ⃗ p q ż d d ⃗ p 1 ż d d ⃗ z 1 e ´i⃗ z 1 ¨´⃗ p 1 ´xq 2x h ⃗ p h ´⃗ p q 2 xq p1 ´xq qQ 2 `´xq x h ⃗ p h ´⃗ p 1 ¯2 F p⃗ z 1 q ˆż d d ⃗ p 1 1 ż d d ⃗ z 2 e i⃗ z 2 ¨´⃗ p 1 1 ´xq 2x h ⃗ p h ´⃗ p q 2 xq p1 ´xq qQ 2 `´xq x h ⃗ p h ´⃗ p 1 1 ¯2 F ˚p⃗ z 2 q ˆαs C F µ 2ϵ ż d d ⃗ p g p2πq d e ´i⃗ pg¨⃗ z 21 2 ´⃗ p g ´xg 1´xq´xg ⃗ p q¯2
dx 2 g `4p1 ´xq ´xg qp1 ´xq q p1 ´xq ´xg q 2 . (7.4.10)

Using eq. (6.4.4) after shifting the intergral over ⃗ p g , we have

dσ qÑh 3LL dx h d d ⃗ p h ˇˇˇp 3q " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx q D h q ˆxh x q , µ F ˙x1`d q p1 ´xq q 2 ż 1´xq α dx g x g ˆż d d ⃗ p q ż d d ⃗ p 1 ż d d ⃗ z 1 e ´i⃗ z 1 ¨´⃗ p 1 ´xq 2x h ⃗ p h ´1´xq 2p1´xq ´xg q ⃗ pq xq p1 ´xq qQ 2 `´xq x h ⃗ p h ´⃗ p 1 ¯2 F p⃗ z 1 q ˆż d d ⃗ p 1 1 ż d d ⃗ z 2 e i⃗ z 2 ¨´⃗ p 1 1 ´xq 2x h ⃗ p h ´1´xq 2p1´xq ´xg q ⃗ pq xq p1 ´xq qQ 2 `´xq x h ⃗ p h ´⃗ p 1 1 ¯2 F ˚p⃗ z 2 q ˆαs 2π C F 1 ε ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙ϵ x 2 g `2p1 ´xq q 2 ´2x g p1 ´xq q `ϵx 2 g p1 ´xq ´xg q 2 .
Rescaling ⃗ p q to absorb the factor 1´xq 1´xq´xg , we get

dσ qÑh 3LL dx h d d ⃗ p h ˇˇˇp 3q " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx q D h q ˆxh x q , µ F ˙x1`d q p1 ´xq q 2 ˆż d d ⃗ p q ż d d ⃗ p 1 ż d d ⃗ z 1 e ´i⃗ z 1 ¨´⃗ p 1 ´xq 2x h ⃗ p h ´⃗ p q 2 xq p1 ´xq qQ 2 `´xq x h ⃗ p h ´⃗ p 1 ¯2 F p⃗ z 1 q ˆż d d ⃗ p 1 1 ż d d ⃗ z 2 e i⃗ z 2 ¨´⃗ p 1 1 ´xq 2x h ⃗ p h ´⃗ p q 2 xq p1 ´xq qQ 2 `´xq x h ⃗ p h ´⃗ p 1 1 ¯2 F ˚p⃗ z 2 q ˆαs 2π C F 1 ε ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙ϵ ż 1´xq α dx g x g ˆ1 ´xq ´xg 1 ´xq ˙d x 2 g `2p1 ´xq q 2 ´2x g p1 ´xq q `ϵx 2 g p1 ´xq ´xg q 2 " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx q D h q ˆxh x q , µ F ˙x1`d q p1 ´xq q 2 ˆż d d ⃗ p q ż d d ⃗ p 1 ż d d ⃗ z 1 e ´i⃗ z 1 ¨´⃗ p 1 ´xq 2x h ⃗ p h ´⃗ p q 2 xq p1 ´xq qQ 2 `´xq x h ⃗ p h ´⃗ p 1 ¯2 F p⃗ z 1 q ˆż d d ⃗ p 1 1 ż d d ⃗ z 2 e i⃗ z 2 ¨´⃗ p 1 1 ´xq 2x h ⃗ p h ´⃗ p q 2 xq p1 ´xq qQ 2 `´xq x h ⃗ p h ´⃗ p 1 1 ¯2 F ˚p⃗ z 2 q ˆαs 2π C F 1 ε ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙ϵ ż 1´xq α dx g x g # x 2
g `2p1 ´xq q 2 ´2x g p1 ´xq q p1 ´xq q 2 `ϵ x 2 g p1 ´xq q 2 `2ϵ ln ˆ1 ´xg 1 ´xq ˙x2 g `2p1 ´xq q 2 ´2x g p1 ´xq q p1 ´xq q 2

+ .

The soft contribution to remove is the term equal to

2p1´xqq 2
p1´xqq 2 in the first term inside the curly bracket. Removing this term and expanding all the terms up to order ϵ 0 , we have

dσ qÑh 3JI dx h d d ⃗ p h ˇˇˇc oll. qg " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx q D h q ˆxh x q , µ F ˙x1`d q p1 ´xq q 2 ˆż d d ⃗ p q ż d d ⃗ p 1 ż d d ⃗ z 1 e ´i⃗ z 1 ¨´⃗ p 1 ´xq 2x h ⃗ p h ´⃗ p q 2 xq p1 ´xq qQ 2 `´xq x h ⃗ p h ´⃗ p 1 ¯2 F p⃗ z 1 q ˆż d d ⃗ p 1 1 ż d d ⃗ z 2 e i⃗ z 2 ¨´⃗ p 1 1 ´xq 2x h ⃗ p h ´⃗ p q 2 xq p1 ´xq qQ 2 `´xq x h ⃗ p h ´⃗ p 1 1 ¯2 F ˚p⃗ z 2 q ˆαs 2π C F ż 1´xq α dx g x g # 1 ε x 2 g ´2x g p1 ´xq q p1 ´xq q 2 `ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙x2 g ´2x g p1 ´xq q p1 ´xq q 2 `x2 g p1 ´xq q 2
`2 ln ˆ1 ´xg 1 ´xq ˙x2 g `2p1 ´xq q 2 ´2x g p1 ´xq q p1 ´xq q 2 + .

Now we note that in the last term of the bracket, in the limit x g Ñ 0, we have

ż 1´xq 0 dx g x g 2 ln ˆ1 ´xg 1 ´xq ˙2p1 ´xq q 2 p1 ´xq q 2 " 4 ż 1 0 dx g x g ln p1 ´xg q " ´4ζp2q
and so this term has no issue in the region x g Ñ 0. All the other terms inside the curly bracket of eq. (7.4.11) have no issue for x g Ñ 0. Therefore, we can set α " 0 in the integration borne of x g . We also do the following change of variables

⃗ p 2 " ´⃗ p 1 ``⃗ p q `xq x h ⃗ p h ⃗ p 2 1 " ´⃗ p 1 1 `⃗ p q `xq x h ⃗ p h . (7.4.12)
These lead to finally

dσ qÑh 3JI dx h d d ⃗ p h ˇˇˇc oll. qg " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx q D h q ˆxh x q , µ F ˙x1`d q p1 ´xq q 2 ˆż d d ⃗ p q ż d d ⃗ p 2 ż d d ⃗ z 1 e ´i⃗ z 1 ¨´´⃗ p 2 `xq 2x h ⃗ p h `⃗ p q 2 xq p1 ´xq qQ 2 `⃗ p 2 q2 F p⃗ z 1 q ż d d ⃗ p 2 1 ż d d ⃗ z 2 e i⃗ z 2 ¨´´⃗ p 2 1 `xq 2x h ⃗ p h `⃗ p q 2 xq p1 ´xq qQ 2 `⃗ p 2 q2 1 F ˚p⃗ z 2 q ˆαs 2π C F " 1 ε ż 1´xq 0 dx g x g ´2p1 ´xq q p1 ´xq q 2 `ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙ż 1´xq 0 dx g x g ´2p1 ´xq q p1 ´xq q 2 `ż 1´xq 0 dx g x g p1 ´xq q 2 `ż 1´xq 0 dx g x g 2 ln ˆ1 ´xg 1 ´xq ˙x2 g `2p1 ´xg q 2 ´2x g p1 ´xq q p1 ´xq q 2 + " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx q D h q ˆxh x q , µ F ˙x1`d q p1 ´xq q 2 ˆż d d ⃗ p q ż d d ⃗ p 2 ż d d ⃗ z 1 e ´i⃗ z 1 ¨´´⃗ p 2 `xq 2x h ⃗ p h `⃗ p q 2 xq p1 ´xq qQ 2 `⃗ p 2 q2 F p⃗ z 1 q ż d d ⃗ p 2 1 ż d d ⃗ z 2 e i⃗ z 2 ¨´´⃗ p 2 1 `xq 2x h ⃗ p h `⃗ p q 2 xq p1 ´xq qQ 2 `⃗ p 2 q2 1 F ˚p⃗ z 2 q ˆαs 2π C F " ´3 2 1 ε ´3 2 ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙`1 2 `5 2 ´4ζp2q * . (7.4.13)
We therefore obtain two contribution, a divergent and a finite part. The results for general polarization JI reads, for the divergent part,

dσ qÑh 3JI dx h d d ⃗ p h ˇˇˇc oll. qg div " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx q x 1`d q p1 ´xq q 2 f JI D h q ˆxh x q , µ F ˙αs 2π C F " ´3 2 1 ε ȷ (7.4.14)
and

dσ qÑh 3JI dx h d d ⃗ p h ˇˇˇc oll. qg fin " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx q x 1`d q p1 ´xq q 2 f JI D h q ˆxh x q , µ F ˙αs 2π C F r3 ´4ζp2qs `2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx q x 1`d q p1 ´xq q 2 D h q ˆxh x q , µ F ż d d ⃗ p q ż d d ⃗ p 2 ż d d ⃗ z 1 e ´i⃗ z 1 ¨´´⃗ p 2 `xq 2x h ⃗ p h `⃗ p q 2 xq p1 ´xq qQ 2 `⃗ p 2 q2 F p⃗ z 1 q ż d d ⃗ p 2 1 ż d d ⃗ z 2 e i⃗ z 2 ¨´´⃗ p 2 1 `xq 2x h ⃗ p h `⃗ p q 2 xq p1 ´xq qQ 2 `⃗ p 2 q2 1 F ˚p⃗ z 2 q ˆδJI α s 2π C F 3 2 ln ˆ⃗ z 2 12 µ 2 4c 2 0
ḟor the finite part. The corresponding contributions in the case of fragmentation from antiquark are obtained by including an overall minus sign in the argument of the function F or in the term in bracket in the exponential of ⃗ z i , making the sum be over q with q one of the five antiquark flavor species and performing the pq Ø qq re-labelling.

Soft contribution

In this section we deal with the associated soft divergences associated to the four diagrams of fig. 7.4.3. To calculate the soft contribution of the divergent part of the real emission crosssection, the soft limit of eq. (6.4.1) (after convoluting with the quark FF as in eq. ( 7.1.4)) is taken by setting ⃗ p g " x g ⃗ u where |⃗ u | " |⃗ p h | which extracts the divergence on x g and then setting x g to 0 in the integrand apart from the divergent term. This can be done safely in the nondivergent terms of the cross-section, as x 1 q is limited from below by x h and so cannot be arbitrary small (i.e. of order x g ). This gives

dσ qÑh 3LL dx h d d ⃗ p h ˇˇˇs oft " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx 1 q px 1 q q 1`d p1 ´x1 q q 2 D h q ˆxh x 1 q , µ F ˙ż 1´x 1 q α dx g x 3´d g ˆż d d ⃗ p q ż d d ⃗ p 2 F ´x1 q 2x h ⃗ p h `pq 2 ´⃗ p 2 x1 q p1 ´x1 q qQ 2 `⃗ p 2 q2 ż d d ⃗ p 2 1 F ˚´x 1 q 2x h ⃗ p h `pq 2 ´⃗ p 2 1 x1 q p1 ´x1 q qQ 2 `⃗ p 2 q2 1 ˆ4 α s C F µ 2ϵ ż d d ⃗ u p2πq d $ ' & ' % 1 ´⃗ u ´⃗ p h x h ¯2 `1 ´⃗ u ´⃗ pq 1´x 1 q ¯2 ´2 ´⃗ u ´⃗ p h x h ¯¨´⃗ u ´⃗ pq 1´x 1 q ⃗ u ´⃗ p h x h ¯2 ´⃗ u ´⃗ pq 1´x 1 q ¯2 , / . / - . (7.4.15)
Before proceeding with the next step, an observation is necessary. During the calculation of the collinear contribution, we came across two-dimensional integrals in the variables x 1 q and x g , which we treated slightly differently. In particular, in the calculation of the collinear contribution due to the qg splitting (as well as for the virtual contribution), we integrated directly over x g , while, in the contribution due to the splitting qg we first carried out the change of variables in eq. (7.4.2), to obtain a form of divergences similar to that present in the counterterms. Since the integration over the final variable x q is never done explicitly, this can make it difficult to observe the cancellation at the integrand level. We clarify this statement with a toy example. Suppose to have the integral

ż 1 x h dx 1 q ż 1´x 1 q α dx g 1 x g . (7.4.16)
If we rename x 1 q as x q and integrate over x g , we get

I 1 " ż 1 x h dx q ln ˆ1 ´xq α ˙. (7.4.17)
From the other side, if we perform the change of variables in (7.4.2) and then integrate over β, we get

I 2 " ż 1 x h dx q ln ˆxq ´xh α ˙. (7.4.18)
The difference between I 1 and I 2 is obviously zero since they are the same integral, however, the cancellation is only seen by integrating over x q ,

I 1 ´I2 " ż 1 x h dx q ln ˆ1 ´xq x q ´xh ˙" 0 , (7.4.19)
i.e. not at the level of the integrand. This problem can be overcome by treating the soft contribution in a symmetrical way with respect to the two different procedures. That is, separating the soft cross-section into two equal parts and treating the two different contributions as explained before. We will label them as

dσ qÑh 3LL dx h d d ⃗ p h ˇˇˇs oft β and dσ qÑh 3LL dx h d d ⃗ p h ˇˇˇs oft xg
. The total soft contribution will be then equal to

dσ qÑh 3LL dx h d d ⃗ p h ˇˇˇs oft " 1 2
dσ qÑh 3LL dx h d d ⃗ p h ˇˇˇs oft β `1 2 dσ qÑh 3LL dx h d d ⃗ p h ˇˇˇs oft xg . (7.4.20)
We also note that, in the contribution in which the change of variable in eq. ( 7.4.2) is carried out i.e.

dσ qÑh 3LL dx h d d ⃗ p h ˇˇˇs oft β
, since we are in the soft limit β can be set to 1 everywhere, except in the term p1 ´βq ´1`2ϵ , which is clearly singular. The first contribution, after the change of variable reads where

dσ qÑh 3LL dx h d d ⃗ p h ˇˇˇs oft β " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx q x 1`d q p1 ´xq q 2 D h q ˆxh x q , µ F ż d d ⃗ p q ż d d ⃗ p 2 F ˚´xq 2x h ⃗ p h `
I u " α s C F µ 2ϵ ż d d ⃗ u p2πq d $ ' & ' % 1 ´⃗ u ´⃗ p h x h ¯2 `1 ´⃗ u ´⃗ pq 1´xq ¯2 ´2 ´⃗ u ´⃗ p h x h ¯¨´⃗ u ´⃗ pq 1´xq ⃗ u ´⃗ p h x h ¯2 ´⃗ u ´⃗ pq 1´xq ¯2 , / . / - " α s C F µ 2ϵ ˆ⃗ p h x h ´⃗ p q 1 ´xq ˙2 ż d d ⃗ u p2πq d 1 ´⃗ u ´⃗ p h x h ¯2 ´⃗ u ´⃗ pq 1´xq ¯2 " α s C F µ 2ϵ ˆ⃗ p h x h ´⃗ p q 1 ´xq ˙2 ż d d ⃗ u p2πq d 1 ⃗ u 2 ´⃗ u ´´⃗ p h x h ´⃗ pq 1´xq ¯¯2 " α s C F µ 2ϵ ˆ⃗ p h x h ´⃗ p q 1 ´xq ˙2 1 p2πq d π 1`ϵ Γ p1 ´ϵq βpϵ, ϵq « ˆ⃗ p h x h ´⃗ p q 1 ´xq ˙2ff ϵ´1 " α s 2π C F 1 ε » - -1 `ϵ ln ¨´⃗ p h x h ´⃗ pq 1´xq ¯2 µ 2 ‹ ' fi ffi fl . (7.4.22)
Putting eq. (7.4.22) and the result of eq. (6.4.35) into eq. ( 7.4.21), we have finally

1 2 dσ qÑh 3LL dx h d d ⃗ p h ˇˇˇs oft β " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx q x 1`d q p1 ´xq q 2 D h q ˆxh x q , µ F ż d d ⃗ p q ż d d ⃗ p 2 F ´xq 2x h ⃗ p h `pq 2 ´⃗ p 2 xq p1 ´xq qQ 2 `⃗ p 2 q2 ż d d ⃗ p 2 1 F ˚´xq 2x h ⃗ p h `pq 2 ´⃗ p 2 1 xq p1 ´xq qQ 2 `⃗ p 2 q2 1 ˆαs 2π C F 1 ε $ ' & ' % ´2 ln α `2 ln x q `2 ln ˆ1 ´xh x q ˙´2ϵ ln 2 α ´2ϵ ln α ln ¨´⃗ p h x h ´⃗ pq 1´xq ¯2 µ 2 ‹ ' `2ϵ ln 2 ˆxq ˆ1 ´xh x q ˙˙`2ϵ ln ¨´⃗ p h x h ´⃗ pq 1´xq ¯2 µ 2 ‹ 'ln ˆxq ˆ1 ´xh x q ˙˙, / . / - . (7.4.23)
We now compute the second contribution.Coming back to eq. ( 7.4.15) and renaming x 1 q by x q , it reads

dσ qÑh 3LL dx h d d ⃗ p h ˇˇˇs oft xg " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1
x h dx q x 1`d q p1 ´xq q 2 D h q ˆxh x q , µ F ˙(7.4.24)

ˆż d d ⃗ p q ż d d ⃗ p 2 F ´xq 2x h ⃗ p h `pq 2 ´⃗ p 2 xq p1 ´xq qQ 2 `⃗ p 2 q2 ż d d ⃗ p 2 1 F ˚´xq 2x h ⃗ p h `pq 2 ´⃗ p 2 1 xq p1 ´xq qQ 2 `⃗ p 2 q2 1 4 ż 1´xq α dx g x 1´2ϵ g I u .
We have

ż 1´xq α dx g x 1´2ϵ g " ż 1´xq α dx g
x g p1 `2ϵ ln x g q " ln p1 ´xq q ´ln α `ϵ ln 2 p1 ´xq q ´ϵ ln 2 α .

(7.4.25)

Combining the above equation with eq. (7.4.22), we have

1 2 dσ qÑh 3LL dx h d d ⃗ p h ˇˇˇs oft xg " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx q x 1`d q p1 ´xq q 2 D h q ˆxh x q , µ F ż d d ⃗ p q ż d d ⃗ p 2 F ´xq 2x h ⃗ p h `pq 2 ´⃗ p 2 xq p1 ´xq qQ 2 `⃗ p 2 q2 ż d d ⃗ p 2 1 F ˚´xq 2x h ⃗ p h `pq 2 ´⃗ p 2 1 xq p1 ´xq qQ 2 `⃗ p 2 q2 1 ˆαs 2π C F 1 ε $ ' & ' % 2
" 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx q x 1`d q p1 ´xq q 2 D h q ˆxh x q , µ F ż d d ⃗ p q ż d d ⃗ p 2 F ´xq 2x h ⃗ p h `pq 2 ´⃗ p 2 xq p1 ´xq qQ 2 `⃗ p 2 q2 ż d d ⃗ p 2 1 F ˚´xq 2x h ⃗ p h `pq 2 ´⃗ p 2 1 xq p1 ´xq qQ 2 `⃗ p 2 q2 1 ˆαs 2π C F 1 ε $ ' & ' %
´4 ln α `2 ln px q p1 ´xq qq `2 ln ˆ1 ´xh x q ˙´4 ln 2 α ´4ϵ ln α ln

¨´⃗ p h x h ´⃗ pq 1´xq ¯2 µ 2 ‹ ' , / .
/ for the divergent part and ‹ 'ln ˆxq p1 ´xq q ˆ1 ´xh x q ˙˙`2 ln 2 p1 ´xq q , / .

dσ qÑh 3LL dx h d d ⃗ p h ˇˇˇs oft fin " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx q x 1`d q p1 ´xq q 2 D h q ˆxh x q , µ F ż d d ⃗ p q ż d d ⃗ p 2 F ´xq 2x h ⃗ p h `

/ -

for the finite part. Transcribing the above results to the general JI cross-section, the soft divergent part reads then

dσ qÑh 3JI dx h d d ⃗ p h ˇˇˇs oft div " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx q x 1`d q p1 ´xq q 2 D h q ˆxh x q , µ F ż d d ⃗ p q ż d d ⃗ p 2 F ´xq 2x h ⃗ p h `pq 2 ´⃗ p 2 xq p1 ´xq qQ 2 `⃗ p 2 q2 ż d d ⃗ p 2 1 F ˚´xq 2x h ⃗ p h `pq 2 ´⃗ p 2 1 xq p1 ´xq qQ 2 `⃗ p 2 q2 1 δ JI ˆαs 2π C F 1 ε $ ' & ' %
´4 ln α `2 ln px q p1 ´xq qq `2 ln ˆ1 ´xh x q ˙´4 ln 2 α ´4ϵ ln α ln

¨´⃗ p h x h ´⃗ pq 1´xq ¯2 µ 2 ‹ ' , / .
/ - (7.4.27) and the finite part reads

dσ qÑh 3JI dx h d d ⃗ p h ˇˇˇs oft fin " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1
x h dx q x 1`d q p1 ´xq q 2 D h q ˆxh x q , µ F ˙(7.4.28)

ˆż d d ⃗ p q ż d d ⃗ p 2 F ´xq 2x h ⃗ p h `pq 2 ´⃗ p 2 xq p1 ´xq qQ 2 `⃗ p 2 q2 ż d d ⃗ p 2 1 F ˚´xq 2x h ⃗ p h `pq 2 ´⃗ p 2 1 xq p1 ´xq qQ 2 `⃗ p 2 q2 1 δ JI ˆαs 2π C F $ ' & ' % 2 ln 2 ˆxq ˆ1 ´xh x q ˙˙`2 ln ¨´⃗ p h x h ´⃗ pq 1´xq ¯2 µ 2
‹ 'ln ˆxq p1 ´xq q ˆ1 ´xh x q ˙˙`2 ln 2 p1 ´xq q , / .

/ -.

The corresponding contributions in the case of fragmentation from antiquark are obtained by including a minus sign in the argument of the function F, making the sum be over q with q one of the five antiquark flavor species and performing the pq Ø qq re-labelling (also inside δ JI ).

Cancellation of divergences in the quark fragmentation case

We can now show the cancellation of divergences in the quark fragmentation channel. First, we combine the divergent virtual contribution eq. ( 7.3.1) and soft contribution divergent eq. (7.4.27),

dσ qÑh 1JI dx h d d ⃗ p h ˇˇˇS V div `dσ qÑh 3JI dx h d d ⃗ p h ˇˇˇs oft div " 2α em Q 2 p2πq 4d N c x d h ˆÿ q Q 2 q ż 1 x h dx q x 1`d q p1 ´xq q 2 f JI D h q ˆxh x q , µ F ˙αs 2π C F 1 ε " 3 `2 ln ˆ1 ´xh x q ˙ȷ (7.4.29)
and we observe the full cancellation of α-divergent terms. Then, we sum the divergent contribution labelled with P qq in eq. ( 7

" 2α em Q 2 p2πq 4d N c x d h ˆÿ q Q 2 q ż 1 x h dx q x 1`d q p1 ´xq q 2 f JI D h q ˆxh x q , µ F ˙αs 2π C F 1 ε " ´3 ´2 ln ˆ1 ´xh x q ˙ȷ . (7.4.30)
The two contributions in eqs. (7.4.29, 7.4.30) cancel each other, giving a full cancellation in the quark fragmentation case. The cancellation in the case of antiquark fragmentation takes place in the same way. Collinear contributions: q-g splitting The strategy of the computation is identical to that of section 7.4.1, but considerably much simpler because there are no soft divergences involved.

Fragmentation from gluon

Convoluting the first term in the bracket of eq. ( 6.4.1) with a gluon FF as in eq. ( 7.1.4), we have .4.31) This time the correct change of variables to make is

dσ gÑh 3LL dx h d d ⃗ p h ˇˇˇc oll. qg " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx g ż 1´xg 0 dx 1 q 1 x 4 g x d g p1 ´x1 q ´xg q 2 px 1 q `xg q 2 D h g ˆxh x g , µ F αs C F µ 2 ż d d ⃗ p q p2πq d ż d d ⃗ p q ż d d ⃗ p 2 F ´⃗ pq 2 `⃗ pq 2 `xg 2x h ⃗ p h ´⃗ p 2 p1 ´x1 q ´xg qpx 1 q `xg qQ 2 `⃗ p 2 q2 ż d d ⃗ p 2 1 F ˚´⃗ pq 2 `⃗ pq 2 `xg 2x h ⃗ p h ´⃗ p 2 1 p1 ´x1 q ´xg qpx 1 q `xg qQ 2 `⃗ p 2 q2 1 ˆdx 2 g `4x 1 q px 1 q `xg q ´⃗ p q ´x1 q x h ⃗ p h ¯2 . ( 7
x g " βx q , x 1 q " p1 ´βqx q , (7. 4.32) and the longitudinal integrations become

ż 1 x h dx g ż 1´xg 0 dx 1 q " ..... ‰ " ż 1 x h dx q x q ż 1 x h xq dβ " ..... ‰ xg"βxq, x 1 q "p1´βqxq , (7.4.33)
where " ..... ‰ represents the whole integrand function in eq. ( 7.4.31) and the factor x q in the right hand side comes the Jacobian of the transformation in eq. (7.4.32).

Applying the change of variable on eq. ( 7.4.31) and using the Fourier transform eq. (4.5.5), we have

dσ gÑh 3LL dx h d d ⃗ p h ˇˇˇc oll. qg " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx q x 1`d q p1 ´xq q 2 ż 1 x h xq dβ β ˆβ2ϵ " p1 ´βq 2 `1 `ϵβ 2 ‰ β D h g ˆxh βx q , µ F ˙ż d d ⃗ p q ż d d ⃗ p 2 ż d d ⃗ z 1 e ´i⃗ z 1 ¨´βxq 2x h ⃗ p h `⃗ p q 2 ´⃗ p 2 xq p1 ´xq qQ 2 `⃗ p 2 q2 F p⃗ z 1 q ˆż d d ⃗ p 2 1 ż d d ⃗ z 2 e i⃗ z 2 ¨´βxq 2x h ⃗ p h `⃗ p q 2 ´⃗ p 2 1 xq p1 ´xq qQ 2 `⃗ p 2 q2 1 F ˚p⃗ z 2 q 2α s C F µ 2 ż d d ⃗ p q p2πq d e ´i⃗ pq¨⃗ z 12 2 ´⃗ p q ´p1´βqxq x h ⃗ p h ¯2 .
Using eq. ( 6.4.4) and expanding up to ϵ 0 , we finally have

dσ gÑh 3LL dx h d d ⃗ p h ˇˇˇc oll. qg " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1
x h dx q x 1`d q p1 ´xq q 2 (7.4.34)

ˆż d d ⃗ p q ż d d ⃗ p 2 ż d d ⃗ z 1 e ´i⃗ z 1 ¨´xq 2x h ⃗ p h `⃗ p q 2 ´⃗ p 2 xq p1 ´xq qQ 2 `⃗ p 2 q2 F p⃗ z 1 q ż d d ⃗ p 2 1 ż d d ⃗ z 2 e i⃗ z 2 ¨´xq 2x h ⃗ p h `⃗ p q 2 ´⃗ p 2 1 xq p1 ´xq qQ 2 `⃗ p 2 q2 1 F ˚p⃗ z 2 q ˆαs 2π C F ż 1 x h xq dβ β D h g ˆxh βx q , µ F ˙# 1 ε 1 `p1 ´βq 2 β `β2 `2 `1 `p1 ´βq 2 ˘ln β β `1 `p1 ´βq 2 β ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙+ .
Therefore, the divergent part of this contribution for general JI cross-section is

dσ gÑh 3JI dx h d d ⃗ p h ˇˇˇc oll. qg div " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1 x h dx q x 1`d q p1 ´xq q 2 f JI ˆαs 2π 1 ε ż 1 x h xq dβ β C F 1 `p1 ´βq 2 β D h g ˆxh βx q , µ F ˙, (7.4.35)
while the finite part reads

dσ gÑh 3JI dx h d d ⃗ p h ˇˇˇc oll. qg fin " 2α em Q 2 p2πq 4d N c x d h ÿ q Q 2 q ż 1
x h dx q x 1`d q p1 ´xq q 2 (7.4.36)

ˆż d d ⃗ p q ż d d ⃗ p 2 ż d d ⃗ z 1 e ´i⃗ z 1 ¨´xq 2x h ⃗ p h `⃗ p q 2 ´⃗ p 2 xq p1 ´xq qQ 2 `⃗ p 2 q2 F p⃗ z 1 q ż d d ⃗ p 2 1 ż d d ⃗ z 2 e i⃗ z 2 ¨´xq 2x h ⃗ p h `⃗ p q 2 ´⃗ p 2 1 xq p1 ´xq qQ 2 `⃗ p 2 q2 1 F ˚p⃗ z 2 qδ JI ˆαs 2π C F ż 1 x h xq dβ β D h g ˆxh βx q , µ F ˙# β 2 `2 `1 `p1 ´βq 2 ˘ln β β `1 `p1 ´βq 2 β ln ˆ4c 2 0 ⃗ z 2 12 µ 2 ˙+ .
The divergent contribution in eq. (7.4.35) exactly cancels the one proportional to the P gq in eq. (7.2.2) and complete our proof of the cancellation of divergences.

The corresponding contributions for diagram p3q of fig. 7.1.5 are obtained by including a minus sign in the argument of the function F or in the term in bracket in the exponential of ⃗ z i , making the sum be over q with q one of the five antiquark flavor species and performing the pq Ø qq re-labelling. Obviously, the divergences that appear in this case cancel out that proportional to the P gq in the renormalization of the FF of the antiquarks (in the case of antiquark fragmentation).

Additional finite terms for the real contributions

Some of the finite terms of our calculation are presented in previous sections. They come as a result of the extraction of divergences. There are many other terms, completely disconnected from divergences, which however contribute to the final result. We proceed to list them. We also set d " 2 in the below expressions as they are finite contributions.

Fragmentation from quark

Finite remainder in the Φα "

α em p2πq 8 N c x 2 h ÿ q Q 2 q ż 1 x h dx q D h q ˆxh x q , µ F ˙ż d 2 ⃗ p 1 d 2 ⃗ p 2 d 2 ⃗ p 1 1 d 2 ⃗ p 2 1 ˆ1 pp γ q 2 ż d 2 ⃗ p q ż d 2 ⃗ p g p2πq 2 ż 1´xq 0 dx g x g p1 ´xq ´xg q F ˆ⃗ p 12 2 ˙F˚ˆ⃗ p 1 1 2 1 2 ˙εIα ε Jβ α s 4 C F ˆ#" Φα 3 p⃗ p 1 , ⃗ p 2 q Φβ: 3 p⃗ p 1 1 , ⃗ p 2 1 q δ ˆxq x h ⃗ p h ´⃗ p 1 `⃗ p q2 `⃗ p g ˙δ ˆxq x h ⃗ p h ´⃗ p 1 1 `⃗ p q2 1 `⃗ p g ˙ȷp⃗ pgÑxg ⃗ pgq
´" Φα 3 p⃗ p 1 , ⃗ p 2 q Φβ:

3 p⃗ p 1 1 , ⃗ p 2 1 q δ ˆxq x h ⃗ p h ´⃗ p 1 `⃗ p q2 `⃗ p g ˙δ ˆxq x h ⃗ p h ´⃗ p 1 1 `⃗ p q2 1 `⃗ p g ˙ȷp⃗ pgÑxg ⃗ pgq pxgÑ0q
, .

-, (7.5.1) where Φ3 p⃗ p 1 , ⃗ p 2 q Φ`: 3 p⃗ p 1 1 , ⃗ p 2 1 q " 8x q x qpp γ q 4 `dx 2 g ´2x g ´4x q x q˘p x q ⃗ p g ´xg ⃗ p q q ¨px q ⃗ p g ´xg ⃗ p qq ˆQ2 `⃗ p 2 q2 1 xqp1´xqq ˙ˆQ 2 `⃗ p 2 q1 xqp1´xqq ˙px q ⃗ p g ´xg ⃗ p q q 2 px q ⃗ p g ´xg ⃗ p qq 2 ´8x q x qpp γ q 4 `2x g ´dx 2 g `4x q x q˘p x q ⃗ p g ´xg ⃗ p q q ¨px q ⃗ p g ´xg ⃗ p qq ˆQ2 `⃗ p 2 q1 1 xqp1´xqq ˙ˆQ 2 `⃗ p 2 q2 xqp1´xqq ˙px q ⃗ p g ´xg ⃗ p q q 2 px q ⃗ p g ´xg ⃗ p qq 2 , (7.5.2)

Φ3 p⃗ p 1 , ⃗ p 2 q Φi: 3 p⃗ p 1 1 , ⃗ p 2 1 q " 4x q `pγ ˘3 px q `xg q ˆQ2 `⃗ p 2 q2 1 xqp1´xqq ˙ˆQ 2 `⃗ p 2 q1 xqp1´xqq
ṗx q p gK ´xg p qK q µ px qp gK ´xg p qK q ν px q ⃗ p g ´xg ⃗ p q q 2 px q ⃗ p g ´xg ⃗ p qq 2 ¸"x g p4x q `xg d ´2q ´pµ

q2 1 K g iν K ´pν q2 1 K g µi K p2x
q ´1q p4x q x q `xg p2 ´xg dqq g µν

K p i q2 1 K ‰ `pq Ø qq , (7.5.3) Φi 3 p⃗ p 1 , ⃗ p 2 q Φk: 3 p⃗ p 1 1 , ⃗ p 2 1 q " ´2 `pγ ˘2 px q `xg q px q `xg q ˆQ2 `⃗ p 2 q2 xqp1´xqq ˙ˆQ 2 `⃗ p 2 q1 1
xqp1´xqq ṗx q p gK ´xg p qK q µ px qp gK ´xg p qK q ν px q ⃗ p g ´xg ⃗ p q q 2 px q ⃗ p g ´xg ⃗ p qq 2 ¸!x g ppd ´4qq x g ´2q

" p ν q1 1 K ´pµ q2K g ik K `pk q2K g µi K ḡµν K ´`⃗ p q1 1 ¨⃗ p q2 ˘gik K `pi q1 1 K p k q2K ¯´g νk K p i q1 1 K p µ q2K ´gµi K g νk K `⃗ p q1 1 ¨⃗ p q2 ˘ı ´gµν K
ˆ"p2x q ´1q p2x q ´1q p k q1 1 K p i q2K p4x q x q `xg p2 ´xg dqq `4x q x qpp⃗ p q1 1 ¨⃗ p q2 qg ik

K `pi q1 1 K p k q2K q ı `´p µ q1 1 K p ν q2K g ik K ´pµ q1 1 K p k q2K g νi K ´pi q1 1 K p ν q2K g µk K ´gµk K g νi K p⃗ p q1 1 ¨⃗ p q2 q xg ppd ´4qx g `2q `xg p2x q ´1qpx g d `4x q ´2q ´gµk K p ν q1 1 K ´gνk K p µ q1 1 K ¯pi q2K `xg p2x q ´1qp k q1 1 K p4x q `xg d ´2q ´gνi K p µ q2K ´gµi K p ν q2K ¯) `pq Ø qq . (7.5.4)
Here one has to fix x q " 1 ´xq ´xg and ⃗ p q "

xq x h ⃗ p h in the different Φα 3 Φβ: 3 .
For clarity, the notation x g Ñ 0 in the second term of eq. ( 7.5.1) indicates that, after extracting the singularity 1{x 2 g (i.e. after the rescaling ⃗ p g Ñ x g ⃗ p g ), throughout the remaining regular part, x g can be set to zero. Then the subtraction between the two terms will make the divergence of the type 1{x g and this will be fully compensated by the factor x g in the numerator of the second line of the eq. (7.5.1).

The corresponding antiquark contribution is easily obtained by exchanging x q and ⃗ p q with x q and ⃗ p q in eq. (7.5.1) and setting x q " 1 ´xq and ⃗ p q " xq x h ⃗ p h in eqs. (7.5.2, 7.5.3, 7.5.4).

Finite part of the dipole ˆdipole contribution

As explained in the discussion after eq. ( 7.1.13), theire is final finite dipole ˆdipole real contribution which is proportional to all the terms in the right-hand side of the equality of eq. (4.6.4) minus the first term r Φ α 3 p⃗ p 1 , ⃗ p 2 q r Φ β: 3 p⃗ p 1 1 , ⃗ p 2 1 q. Using eq. (4.6.2), it reads

dσ qÑh 3JI dx h d 2 ⃗ p h ˇˇˇfi n " α em p2πq 8 N c x 2 h ÿ q Q 2 q ż 1 x h dx q D h q ˆxh x q , µ F ˙ż d 2 ⃗ p 1 d 2 ⃗ p 2 d 2 ⃗ p 1 1 d 2 ⃗ p 2 1 ˆż d 2 ⃗ p q ż d 2 ⃗ p g p2πq 2 ż 1´xq 0 dx g x g 1 p1 ´xq ´xg q δ ˆxq x h ⃗ p h ´⃗ p 1 `⃗ p q2 `⃗ p g ˙δ p⃗ p 11 1 `⃗ p 22 1 q ˆF ˆ⃗ p 12 2 ˙F˚ˆ⃗ p 1 1 2 1 2 ˙εIα ε Jβ α s C F 4pp γ q 2 " Φ α 3 p⃗ p 1 , ⃗ p 2 q Φ β: 3 p⃗ p 1 1 , ⃗ p 2 1 q ´r Φ α 3 p⃗ p 1 , ⃗ p 2 q r Φ β: 3 p⃗ p 1 1 , ⃗ p 2 1 q ı , (7.5.5)
where one has to fix x q " 1 ´xq ´xg and ⃗ p q " xq x h ⃗ p h .

Dipole ˆdouble dipole contribution

From eq. (4.6.8), the dipole ˆdouble dipole contribution is The expressions for the interferences Φ α 3 Φ β: 4 , Φ α 4 Φ β: 3 , in the various cases, are given in appendix B.2. In those formulas one has to fix x q " 1 ´xq ´xg and ⃗ p q " xq x h ⃗ p h .

dσ qÑh 4JI dx h d 2 ⃗ p h " α em 2p2πq 8 N c x 2 h ÿ q Q 2 q ż 1 x h dx q D h q ˆxh x q , µ F ˙ż d 2 ⃗ p 1 d 2 ⃗ p 2 d 2 ⃗ p 1 1 d 2 ⃗ p 2 1 ˆż d 2 ⃗ p 3 d 2 ⃗ p 3 1 p2πq 2 ż d 2 ⃗ p q ż d 2 ⃗

Double dipole ˆdouble dipole contribution

From eq. (4.6.9), the double dipole ˆdouble dipole contribution is

dσ qÑh 5JI dx h d 2 ⃗ p h " α em 2p2πq 8 pN 2 c ´1q x 2 h ÿ q Q 2 q ż 1 x h dx q D h q ˆxh x q , µ F ˙ż d 2 ⃗ p 1 d 2 ⃗ p 2 d 2 ⃗ p 1 1 d 2 ⃗ p 2 1 ˆż d 2 ⃗ p 3 p2πq 2 d 2 ⃗ p 3 1 p2πq 2 ż d 2 ⃗ p q ż d 2 ⃗ p g p2πq 2 ż 1´xq 0 dx g δ p⃗ p 11 1 `⃗ p 22 1 `⃗ p 33 1 q x g p1 ´xq ´xg q δ ˆxq x h ⃗ p h ´⃗ p 1 `⃗ p q2 `⃗ p g3 ˆεIα ε Jβ α s 4pp γ q 2 Φ α 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q Φ β: 4 p⃗ p 1 1 , ⃗ p 2 1 , ⃗ p 3 1 q r F ˆ⃗ p 12 2 , ⃗ p 3 ˙r F ˚ˆ⃗ p 1 1 2 1 2 , ⃗ p 3 1 ˙. (7.5.7)
The expressions for Φ α 4 Φ β4 , in the various cases, are given in appendix B.2. In those formulas one has to fix x q " 1 ´xq ´xg and ⃗ p q " xq x h ⃗ p h .

The corresponding antiquark contributions of eqs. (7.5.5, 7.5.6, 7.5.7) are easily obtained by exchanging x q and ⃗ p q with x q and ⃗ p q and setting x q " 1 ´xq and ⃗ p q " xq x h ⃗ p h in the various interferences of impact factors present in the formulas of appendix B.2.

Fragmentation from gluon

The finite contributions in the case of gluon fragmentation are obtained in a similar way to what is shown in the case of quark fragmentation.

Finite remainder in the Φ3 part This contribution is represented in fig. 7.5.2 and corresponds to eq. (7.1.24). It reads

dσ gÑh 3JI dx h d 2 ⃗ p h ˇˇˇfi n. 1 " α em p2πq 8 N c x 2 h ÿ q Q 2 q ż 1 x h dx g D h g ˆxh x g , µ F ˙ż d 2 ⃗ p 1 d 2 ⃗ p 2 d 2 ⃗ p 1 1 d 2 ⃗ p 2 1 ˆż d 2 ⃗ p q ż d 2 ⃗ p q p2πq 2 ż 1´xg 0 dx q 1 x q p1 ´xq ´xg q F ˆ⃗ p 12 2 ˙F˚ˆ⃗ p 1 1 2 1 2 ˙εIα ε Jβ α s C F 4pp γ q 2 ˆδ ˆxg x h ⃗ p h `⃗ p q1 `⃗ p q2 ˙δ p⃗ p 11 1 `⃗ p 22 1 q Φα 3 p⃗ p 1 , ⃗ p 2 q Φβ: 3 p⃗ p 1 1 , ⃗ p 2 1 q . (7.5.8)

Finite part of the dipole ˆdipole contribution

The second finite dipole ˆdipole contribution of eq. (7.1.21) reads

dσ gÑh 3JI dx h d 2 ⃗ p h ˇˇˇfi n. 2 " α em p2πq 8 N c x 2 h ÿ q Q 2 q ż 1 x h dx g D h g ˆxh x g , µ F ˙ż d 2 ⃗ p 1 d 2 ⃗ p 2 d 2 ⃗ p 1 1 d 2 ⃗ p 2 1 ˆż d 2 ⃗ p q ż d 2 ⃗ p q p2πq 2 ż 1´xg 0 dx q 1 x q p1 ´xq ´xg q F ˆ⃗ p 12 2 ˙F˚ˆ⃗ p 1 1 2 1 2 ˙εIα ε Jβ α s C F 4pp γ q 2 ˆδ ˆxg x h ⃗ p h `⃗ p q1 `⃗ p q2 ˙δ p⃗ p 11 1 `⃗ p 22 1 q " Φ α 3 p⃗ p 1 , ⃗ p 2 q Φ β: 3 p⃗ p 1 1 , ⃗ p 2 1 q ´r Φ α 3 p⃗ p 1 , ⃗ p 2 q r Φ β: 3 p⃗ p 1 1 , ⃗ p 2 1 q ı .
(7.5.9)

Dipole ˆdouble dipole contribution From eq. (4.6.8), the dipole ˆdouble dipole contribution for the case of gluon fragmentation is From eq. (4.6.9), the double dipole ˆdouble dipole contribution for the case of gluon fragmentation is

dσ gÑh 4JI dx h d 2 ⃗ p h " α em 2p2πq 8 N c x 2 h ÿ q Q 2 q ż 1 x h dx g D h g ˆxh x g , µ F ˙ż d 2 ⃗ p 1 d 2 ⃗ p 2 d 2 ⃗ p 1 1 d 2 ⃗ p 2 1 ˆż d 2 ⃗ p 3 d 2 ⃗ p 3 1 p2πq 2 ż d 2 ⃗ p q ż d 2 ⃗ p q p2πq 2 ż 1´xg 0 dx q x q δ p⃗ p 11 1 `⃗ p 22 1 `⃗ p 33 1 q p1 ´xq ´xg q δ ˆxg x h ⃗ p h `⃗ p q1 `⃗ p q2 ´⃗ p 3 εIα ε Jβ α s 4pp γ q 2 " Φ α 3 p⃗ p 1 , ⃗ p 2 qΦ β: 4 p⃗ p 1 1 , ⃗ p 2 1 , ⃗ p 3 1 qF ˆ⃗ p 12 2 ˙r F ˚ˆ⃗ p 1 1 2 1 2 , ⃗ p 3 1 ˙δp⃗ p 3 q `Φα 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 qΦ β: 3 p⃗ p 1 1 , ⃗ p 2 1 q r F ˆ⃗ p 12 2 , ⃗ p 3 ˙F˚ˆ⃗ p 1 1 2 1 2 ˙δp⃗ p 3 1 q ȷ . ( 7
dσ gÑh 5JI dx h d 2 ⃗ p h " α em 2p2πq 8 pN 2 c ´1q x 2 h ÿ q Q 2 q ż 1 x h dx g D h g ˆxh x g , µ F ˙ż d 2 ⃗ p 1 d 2 ⃗ p 2 d 2 ⃗ p 1 1 d 2 ⃗ p 2 1 ˆż d 2 ⃗ p 3 p2πq 2 d 2 ⃗ p 3 1 p2πq 2 ż d 2 ⃗ p q ż d 2 ⃗ p q p2πq 2 ż 1´xg 0 dx q δ p⃗ p 11 1 `⃗ p 22 1 `⃗ p 33 1 q x q p1 ´xq ´xg q δ ˆxg x h ⃗ p h `⃗ p q1 `⃗ p q2 ´⃗ p 3 εIα ε Jβ α s 4pp γ q 2 Φ α 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q Φ β: 4 p⃗ p 1 1 , ⃗ p 2 1 , ⃗ p 3 1 q r F ˆ⃗ p 12 2 , ⃗ p 3 ˙r F ˚ˆ⃗ p 1 1 2 1 2 , ⃗ p 3 1 ˙. ( 7
.5.11)

Summary

In this chapter, we have considered, for the first time at NLO, the diffractive production of a single hadron at large p T , in γ p˚q nucleon/nucleus scattering, in the most general kinematics which includes both lepto-and photoproduction (and at eikonal approximation). Our main result is the explicit finite result for the cross-section at NLO, obtained after showing explicitly the cancellation of rapidity divergences (through the B-JIMWLK equation), soft divergences and collinear divergences between real, virtual contributions, and counterterms due to the renormalization of FFs. Finite contributions and purely divergent contributions have been separated and the sum of the latter has been shown to be zero. This is a continuation of the work presented in chapter 6 and so a similar oulook as in section 6.6 can be drawn. We will denote p q , p q, p g the momenta of the on-shell massive quark, antiquark, and gluon respectively. The momenta p q , p q will be parameterized as

X k e -→ γ * p A -→ q, λ ← - k ′ e -→ -→ p q = p 2 + k, i, s p q = p 2 -k, ī, s -→ p g , a, λ H p -→
p q " p 2 `k , p q " p 2 ´k , (8.1.4)
where p is the total momentum of the heavy quark pair and k is their relative momentum. The on-shell conditions for the quark (antiquark) are

p 2 q " ´p 2 `k¯2 " m 2 Q , p 2 q " ´p 2 ´k¯2 " m 2 Q , (8.1.5)
with m Q the heavy quark mass. These two conditions imply

p ¨k " 0 , p 2 " 4pm 2 Q ´k2 q . (8.1.6)
We define the longitudinal momentum fraction of the quark, antiquark, and gluon with respect to the photon as:

x q " p q q `,
x q " p q q `, x g " p g q `.

(8.1.7) Furthermore, we introduce ξ " k `{q `. Then we find the following relations between the momentum fractions, using the longitudinal momentum conservation q `" p ``p g :

p q " p 2 `k ñ x q " 1 2 ´xg 2 `ξ , p q " p 2 ´k ñ x q " 1 2 ´xg 2 ´ξ . (8.1.8)
We denote the polarizations of the virtual photon and the gluon as λ and λ respectively. The two transverse polarizations are denoted by λ, λ " ˘1, and for the virtual photon the longitudinal polarization is denoted by λ " 0. The spin indices for the quark and antiquark are indicated as s and s respectively. The colour indices of the quark, antiquark and gluon are i, ī and a respectively.

Our kinematics variables and quantum numbers are summarized in table 8.1.

p A four-momentum of the nucleus p n " p A {A average four-momentum of a nucleon in the nucleus k e incoming electron four-momentum k 1 e outgoing electron four-momentum q " k e ´k1 e virtual photon four-momentum S " pp n `ke q 2 center-of-mass energy squared per nucleon of the electron-nucleus system s " pp n `qq 2 center-of-mass energy squared per nucleon of the virtual photon-nucleus system Q 2 " ´q2 virtuality squared of incoming photon p q , p q, p g quark, antiquark, gluon four-momenta m 2 Q quark (antiquark) mass squared p " p q `pq total momentum of the heavy quark pair k " 1 2 pp q ´pq q relative momentum of the heavy quark pair x q , x q, x g longitudinal momentum fractions of the quark, antiquark and gluon relative to the virtual photon λp λq polarization of virtual photon (gluon) s, s spin of quark and antiquark i, ī, a colour indices of quark, antiquark and gluon

NRQCD + CGC factorizations

The differential cross-section for quarkonium production can be factorized using the NRQCD factorization formalism [START_REF] Bodwin | Rigorous QCD analysis of inclusive annihilation and production of heavy quarkonium[END_REF][START_REF] Zwirner | J{ψ production in deep inelastic scattering[END_REF] :

dσ H " ÿ κ dσ κ xO H κ y , (8.1.9)
At the level of the amplitude, there are four contributions to Q Qg production in DIS:

M " M R 1 `MR 2 `MR 3 `MR 4 , (8.1.22)
where R 1 pR 3 q, R 2 pR 4 q refer to the contributions where the gluon is emitted by the quark (antiquark) before and after the shockwave respectively. The diagrams associated with these contributions are shown in fig. 8.1.2. Thus, at the level of the differential cross-section in eq. (8.1.21) there are 16 contributions. The amplitudes for Q Qg production have the generic form

R 1 R 2 R 3 R 4
M λ, λ,a R 1p3q ,ss,i īpp, k, p g q " ee Q 2π ż d 2 ⃗ r d 2 ⃗ b d 2 ⃗ z e ´i⃗ k¨⃗ r e ´i⃗ p¨⃗ b e ´i⃗ pg¨⃗ z C a R 1p3q ,i ī ˆ⃗ b `⃗ r 2 , ⃗ b ´⃗ r 2 , ⃗ z us ´p 2 `k¯N λ, λ R 1p3q pp, k, p g ; ⃗ r, ⃗ b, ⃗ z qv s ´p 2 ´k¯, (8.1.23)
for the diagrams in which the gluon is emitted before the shockwave, and .24) for the diagrams in which the gluon is emitted after the shockwave. Here s (s) and i ( ī) are the spin and colour index of the quark (antiquark) respectively. As is customary in CGC calculations, we work with transverse spatial coordinates. The quark (antiquark) scatters with the shockwave at transverse location ⃗ x p⃗ y q, and the gluon scatters at location ⃗ z. We then introduce the coordinates .25) which are conjugate to ⃗ p and ⃗ k respectively. C a R i ,i ī and N λ, λ R i stand for the colour structure and the perturbative factor associated to diagram R i . Their explicit expressions will be computed in the section 8.2. The short-distance coefficients for the amplitude for Q Qrκsg are obtained by inserting eq. ( 8.1.23) into eq. (8.1.11), leading to where we specialized on the S-wave state and defined

M λ, λ,a R 2p4q ,ss,i īpp, k, p g q " ee Q 2π ż d 2 ⃗ r d 2 ⃗ b e ´i⃗ k¨⃗ r e ´i⃗ p¨⃗ b e ´i⃗ pg¨p ⃗ b˘⃗ r 2 q C a R 2p4q ,i ī ˆ⃗ b `⃗ r 2 , ⃗ b ´⃗ r 2 us ´p 2 `k¯N λ, λ R 2p4q pp, k, p g ; ⃗ r qv s ´p 2 ´k¯, (8.1
⃗ b " 1 2 p⃗ x `⃗ yq , ⃗ r " ⃗ x ´⃗ y , (8.1
M λ, λ,a,κ,Jz R 1p3q pp, p g q " ee Q 2π ż d 2 ⃗
F λ, λ,κ,Jz R 1p3q pp, p g , Q; ⃗ r, ⃗ b, ⃗ z q " Tr " Π JJz pp, 0qN λ, λ R 1p3q pp, 0, p g ; ⃗ r, ⃗ b, ⃗ z q ı . (8.1.27)
Similar expressions are obtained for M λ, λ,κ,Jz R 2p4q pp, p g q. The functions F will be computed in section 8.3.

Q Qg production in the CGC

In this section, we compute the amplitude for Q Qg production in virtual photon-nucleus collision in the CGC. This calculation has been carried out in [START_REF] Beuf | Massive quarks in NLO dipole factorization for DIS: Longitudinal photon[END_REF][START_REF] Beuf | Massive quarks in NLO dipole factorization for DIS: Transverse photon[END_REF] within light-cone perturbation theory. Here, we follow the approach in [START_REF] Caucal | Dijet impact factor in DIS at next-to-leading order in the Color Glass Condensate[END_REF] by performing our computation using covariant perturbation theory. We employ the standard QCD+QED Feynman rules in momentum space (see appendix C.2) with the effective CGC vertices in eqs. (5.1.19), (5.1.20) and (5.1.21). We work in light-cone gauge for the photon and gluon fields: n 2 ¨AQED " 0 " A QED and n 2 ¨AQCD " 0 " A QCD . In this gauge the polarization vectors are

ε µ pq, λ " 0q " ˆ0, Q q `, 0 K ˙, (8.2.1)
ε µ pq, λ " ˘1q " p0, 0, ϵ λ K q , (8.2.2)
for the virtual photon, and

ε µ pp g , λ " ˘1q " ˜0, ⃗ ϵ λ ¨⃗ p g p g , ϵ λ K ¸, (8.2.3)
for the real gluon. Here we introduced the two dimensional vector

ϵ λ K " 1 ? 2 p1, ˘iq .
It is useful to define ω µν " 1 2 rγ µ , γ ν s. For i, j transverse component indices, we have

γ i γ j " 1 2 ␣ γ i , γ j ( `1 2 " γ i , γ j ‰ " g ij K `ωij . (8.2.4)
We present the explicit computations for the amplitudes corresponding to the diagrams R 1 (gluon emission from quark before the shockwave) and R 2 (gluon emission from quark after the shockwave). The computations for the contributions R 3 and R 4 are almost identical, thus we simply give their results.

8.2.1 Gluon emission from quark before the shockwave

Ñ q ´l1 Ñ l1 Ñ l 2 Ñ q, λ Ñ pq " p 2 `k, i, s Ñ pq " p 2 ´k, ī, s Ñ pg, a, λ
S λ, λ,a R 1 ,ss,i īpp, k, p g q " ż d 4 l 1 p2πq 4
ż d 4 l 2 p2πq 4 ūs pp q qT q in pp q , l 1 ´l2 qG 0 pl 1 ´l2 qigγ µ t b nm G 0 pl 1 qp´iee Q { εpq, λqq ˆG0 pl 1 ´qqT q m īpq ´l1 , p qqv spp qqε ˚αpp g , λqT g αν,ab pp g , l 2 qG νµ 0 pl 2 q . (8.2.5)

Replacing the propagators and the vertices by their expressions and using the relations in eq. (C.2.6) and eq. (3.3.36), then the scattering amplitude reads S λ, λ,a R 1 ,ss,i īpp, k, p g q " p2πqδpp q `pq `pg ´q`q ee Q 2π

ż d 2 ⃗ x d 2 ⃗ y d 2 ⃗ z " V p⃗ xqV : p⃗ z qt a V p⃗ z qV : p⃗ y q ‰ i ī e ´i⃗ pq¨⃗ x
ˆe´i⃗ pq¨⃗ y e ´i⃗ pg¨⃗ z p´2q `q ż d 4 l 1 p2πq 2 ż d 4 l 2 p2πq 3 δpp q ´q``l1 q δpp g ´l2 q e i ⃗ l 1 ¨p⃗ x´⃗ yq e i ⃗ l 2 ¨p⃗ z´⃗ xq ˆg p2p g q 1 2q

`ū s pp q qγ `"p { l 1 ´{ l 2 q `mQ

‰ { ε ˚pl 2 , λqp { l 1 `mQ q{ εpq, λ, q " p { l 1 ´{ qq `mQ ‰ γ `vs pp qq rpl 1 ´l2 q 2 ´m2 Q `iεs " l 2 1 ´m2 Q `iε ı " pl 1 ´qq 2 ´m2 Q `iε ı " l 2 2 `iε ‰ . (8.2.6)
The `components of the loop variables l 1 and l 2 are easily integrated over with the two delta functions thanks to the eikonal interaction of the partons in the projectile with the shockwave. Following the strategy in [START_REF] Caucal | Dijet impact factor in DIS at next-to-leading order in the Color Glass Condensate[END_REF] and using eq. (C.5.1), the Dirac structure can be divided into two parts:

γ `"p { l 1 ´{ l 2 q `mQ ‰ { ε ˚pl 2 , λqp { l 1 `mQ q " ⃗ ϵ λn 2 x q x g # ˆ⃗ l 2 ´xg 1 ´xq ⃗ l 1 ˙m "ˆ1 `xg 2x q ˙δnm ´xg 2x q ω mn ȷ `xq x g `xq ˆxg x q ˙2 m Q 2 γ n + ˆγ`p { l 1 `mQ q ´⃗ ϵ λn γ n γ `ˆx q 1 ´xq ˙pl 2 1 ´m2 Q q . ( 8.2.7)
The first term corresponds to a regular term. The second one, proportional to the factor l 2 1 ´m2 Q , can be identified as an instantaneous term. It is associated with the diagram in light-cone perturbation theory where the quark with momentum l 1 is instantaneous; the factor canceling the quark propagator.

Using eq. (C.1.3), the reduced amplitude for Q Qg with the gluon emitted by the quark before the shockwave reads

M λ, λ,a R 1 ,ss,i īpp, k, p g q " ee Q 2π ż d 2 ⃗ x d 2 ⃗ y d 2 ⃗ z C a R 1 ,i īp⃗
x, ⃗ y, ⃗ z q e ´i⃗ pq¨⃗ x e ´i⃗ pq¨⃗ y e ´i⃗ pg¨⃗ z ˆū s pp q q ´N λ, λ R 1 ,reg pp, k, p g ; ⃗ x, ⃗ y, ⃗ z q `N λ, λ R 1 ,inst pp, k, p g ; ⃗ x, ⃗ y, ⃗ z q ¯vs pp qq , (8.2.8)

where we define the colour structure C a R 1 p⃗ x, ⃗ y, ⃗ z q " V p⃗ x qV : p⃗ z qt a V p⃗ z qV : p⃗ y q ´ta 1 , (8.2.9)

and the perturbative factors

N λ, λ R 1 ,reg pp, k, p g ; ⃗ x, ⃗ y, ⃗ z q " g 2π ż d 2 ⃗ l 1 p2πq 2 d 2 ⃗ l 2
p2πq 2 e i ⃗ l 1 ¨p⃗ x´⃗ yq e i ⃗ l 2 ¨p⃗ z´⃗ xq I R 1 ,reg T λ, λ R 1 ,reg pl 1 , l 2 q ˇˇˇl 1 "q `´q `xq , l 2 "q `xg , (8.2.10)

for the regular, and

N λ, λ R 1 ,inst pp, k, p g ; ⃗ x, ⃗ y, ⃗ z q " g 2π ż d 2 ⃗ l 1 p2πq 2 d 2 ⃗ l 2
p2πq 2 e i ⃗ l 1 ¨p⃗ x´⃗ yq e i ⃗ l 2 ¨p⃗ z´⃗ xq I R 1 ,inst T λ, λ R 1 ,inst pl 1 , l 2 q ˇˇˇl 1 "q `´q `xq , l 2 "q `xg , (8.2.11) for the instantaneous contribution.

We have decomposed the two perturbative factors into Dirac structures according to eq. (8.2.7)

T λ, λ R 1 ,reg pl 1 , l 2 q " ⃗ ϵ λn 4q `xq "ˆ⃗ l 2 ´xg 1 ´xq ⃗ l 1 ˙m "ˆ1 `xg 2x q ˙δnm ´xg 2x q ω mn ȷ `xq x g `xq ˆxg x q ˙2 m Q 2 γ n + 1 2q
`γ`p { l 1 `mQ q{ εpλ, qq

" p { l 1 ´{ qq `mQ ‰ γ `, (8.2.12) T λ, λ R 1 ,inst pl 1 , l 2 q " ´2q `xg
x q 1 ´xq ⃗ ϵ λn 1 2q `γn γ `{ εpλ, qq

" p { l 1 ´{ qq `mQ ‰ γ `, (8.2.13)
and defined the corresponding pole integrals over

l í I R 1 ,reg " ż dl 1 dl 2 ´2q rpl 1 ´l2 q 2 ´m2 Q `iεs " l 2 1 ´m2 Q `iε ı " pl 1 ´qq 2 ´m2 Q `iε ı " l 2 2 `iε ‰ , (8.2.14)
I R 1 ,inst " ż dl 1 d 2 ´2q rpl 1 ´l2 q 2 ´m2 Q `iεs " pl 1 ´qq 2 ´m2 Q `iε ı " l 2 2 `iε ‰ . (8.2.15)
The pole structures are computed using Cauchy's residue theorem, closing the contour on the upper half plane. We finally obtain the following results

I R 1 ,inst " ´p2πq 2 θpx g qθpx q qθpx qqθp1 ´xq q 2q `xq x qx g 1 " Q 2 `⃗ l 2 1 `m2 Q xq `p⃗ l 2 ´⃗ l 1 q 2 `m2 Q xq `⃗ l 2 2 xg ȷ , (8.2.16
)

I R 1 ,reg " p2πq 2 θpx g qθpx q qθp1 ´xq qθpx qq 2q `xg x q ˆ1 " Q 2 x qp1 ´xq q `m2 Q `⃗ l 2 1 ı " Q 2 `⃗ l 2 1 `m2 Q xq `p⃗ l 2 ´⃗ l 1 q 2 `m2 Q xq `⃗ l 2 2 xg ȷ . (8.2.17)

Explicit calculation of the Dirac structures and transverse momenta integrals

We consider first the case in which the photon is longitudinally polarized. From eq. (8.2.1), we have { εpq, λ " 0q " Q q `γ `. We observe easily from eq. (8.2.13) that the corresponding instantaneous term vanishes as pγ `q2 " 0:

T λ"0, λ R 1 ,ins pl 1 , l 2 q " 0 . (8.2.18)
For the regular term, after some elementary algebra, we obtain

T λ"0, λ R 1 ,reg pl 1 , l 2 q " ´8Qq `xq x qp1 ´xq q⃗ ϵ λn "ˆ⃗ l 2 ´xg 1 ´xq ⃗ l 1 ˙m "ˆ1 `xg 2x q ˙δnm ´xg 2x q ω mn ȷ `xq x g `xq ˆxg x q ˙2 m Q 2 γ n + γ `. (8.2.19)
For the transversely polarized photon case, using eq. ( 8.2.2), we have { εpq, λ " ˘1q " ´⃗ ϵ λ i γ i . The regular term takes the form:

T λ"˘1, λ R 1 ,reg pl 1 , l 2 q " ´4q `xq ⃗ ϵ λn ⃗ ϵ λ i "ˆ⃗ l 2 ´xg 1 ´xq ⃗ l 1 ˙m "ˆ1 `xg 2x q ˙δnm ´xg 2x q ω mn ȷ `xq x g `xq ˆxg x q ˙2 m Q 2 γ n + ! ⃗ l 1j " δ ij p1 ´2x qq ´ωij ‰ γ ``m Q γ i γ `) , (8.2.20)
while the instantaneous term reads

T λ"˘1, λ R 1 ,inst pl 1 , l 2 q " 2q `xg x q x q 1 ´xq ⃗ ϵ λn ⃗ ϵ λ i γ n γ i γ `. ( 8
N λ"0, λ R 1 ,inst pp, k, p g ; ⃗ x, ⃗ y, ⃗ z q " 0 , (8.2.22)
N λ"0, λ R 1 ,reg pp, k, p g ; ⃗ x, ⃗ y, ⃗ z q " ´2Q x qp1 ´xq q x g g π θpx g qθpx q qθp1 ´xq qθpx qq⃗ ϵ λn γ

#I 1m p⃗ y ´⃗ x, ⃗ z ´⃗ x q "ˆ1 `xg 2x q ˙δnm ´xg 2x q ω mn ȷ ´mQ I 0 p⃗ y ´⃗ x, ⃗ z ´⃗ x q x 2 g 2p1 ´xq qx q γ n + , (8.2.23)
and for transversely polarized photon

N λ"˘1, λ R 1 ,reg pp, k, p g ; ⃗ x, ⃗ y, ⃗ z q " 1 x g g π θpx g qθpx q qθp1 ´xq qθpx qq⃗ ϵ λn ⃗ ϵ λ i ˆ"I 2mj p⃗ y ´⃗ x, ⃗ z ´⃗ x q "ˆ1 `xg 2x q ˙δnm ´xg 2x q ω mn ȷ " δ ij p1 ´2x qq ´ωij ‰ γ ´mQ I 1m p⃗ y ´⃗ x, ⃗ z ´⃗ x q "ˆ1 `xg 2x q ˙δnm ´xg 2x q ω mn ȷ γ i γ mQ Ĩ1j p⃗ y ´⃗ x, ⃗ z ´⃗ x q x 2 g 2p1 ´xq qx q γ n " δ ij p1 ´2x qq ´ωij ‰ γ m2 Q I 0 p⃗ y ´⃗ x, ⃗ z ´⃗ x q x 2 g 2p1 ´xq qx q γ n γ i γ `+ , (8.2.24)
N λ"˘1, λ R 1 ,inst pp, k, p g ; ⃗ x, ⃗ y, ⃗ z q " ´xg x q x q 2p1 ´xq q g π θpx g qθpx q qθpx qqθp1 ´xq q⃗

ϵ λn ⃗ ϵ λ i QR 1 ,ξ K 1 p QR 1 ,ξ X R 1 ,ξ q X R 1 ,ξ γ n γ i γ `, (8.2.25)
where we introduced the variables

Q2 R 1 ,ξ " Q 2 `m2 Q 1 ´xg x q x q , X 2
R 1 ,ξ " x g x q p⃗ z ´⃗ x q 2 `xq x qp⃗ y ´⃗ x q 2 `xq x g p⃗ y ´⃗ z q 2 . (8.2.26)

The transverse integrals have been absorbed in the definition of the I 0 , I 1m , I 2mj , Ĩ1j functions. Explicit expressions for these functions can be found in appendix C.7 where they have been computed using both Schwinger and Feynman parametrization. They have to be taken with z 1 " x q, z 2 " x g , z 3 " x q . In the massless limit, the perturbative factors in eqs. (8.2.23), (8.2.24) and (8.2.25) match with those presented in [START_REF] Caucal | Dijet impact factor in DIS at next-to-leading order in the Color Glass Condensate[END_REF] (before spinor contraction) up to a factor of 1{p2q `q that comes from our different convention of the perturbative factor N λ, λ R 1 . We have also checked that our results are consistent with those obtained in [START_REF] Beuf | Massive quarks in NLO dipole factorization for DIS: Transverse photon[END_REF] and [START_REF] Beuf | Massive quarks in NLO dipole factorization for DIS: Longitudinal photon[END_REF] within the light-cone perturbation formalism. Now, let us express our results in terms of the variables introduced in eqs. (8.1.4), (8.1.8), and (8.1.25). The reduced amplitude becomes

M λ, λ,a R 1 ,ss,i īpp, k, p g q " ee Q 2π ż d 2 ⃗ r d 2 ⃗ b d 2 ⃗ z C a R 1 ,i ī ˆ⃗ b `⃗ r 2 , ⃗ b ´⃗ r 2 , ⃗ z ˙e´i⃗ p¨⃗ b e ´i⃗ k¨⃗ r e ´i⃗ pg¨⃗ z ˆū s ´p 2 `k¯" N λ, λ R 1 ,reg ´p, k, p g ; ⃗ r, ⃗ b, ⃗ z ¯`N λ, λ R 1 ,inst ´p, k, p g ; ⃗ r, ⃗ b, ⃗ z ¯ı v s ´p 2 ´k¯, (8.2.27)
with z 1 " x q " 1 2 ´xg 2 ´ξ, z 2 " x g and z 3 " x q " 1 2 ´xg 2 `ξ.

Gluon emission from quark after the shockwave

The scattering amplitude for the R 2 diagram (represented in fig. 8.2.2) is given by S λ, λ,a R 2 ,ss,i īpp, k, p g q " ż d 4 l p2πq 4 ūs pp q q `ig{ ε ˚pp g , λqt a in ˘G0 pp q `pg qT q nm pp q `pg , lqG 0 plqp´iee Q { εpq, λqq ˆG0 pl ´qqT q m īpq ´l, p qqv spp qq . (8.2.28) Inserting the corresponding propagators and vertices, we find S λ, λ,a R 2 ,ss,i īpp, k, p g q " p2πqδpp q `pg `pq ´q`q ee Q 2π ż d 2 ⃗ x d 2 ⃗ y e ´ip⃗ pq`⃗ pgq¨⃗ x e ´i⃗ pq¨⃗ y " t a V p⃗ x qV : p⃗ y q ‰ i ī ˆig ż d 4 l p2πq 2 δpp q ´q``l`q e i ⃗ l¨p⃗ x´⃗ yq 1 rpp q `pg q 2 ´m2 Q `iεs ˆū s pp q q { ε ˚pλ , p g qp { p q `{ p 3 `mQ qγ `p{ l `mQ q{ εpλ, qqp { l ´{ q `mQ qγ

Ñ q ´l Ñ l Ñ q, λ Ñ pq " p 2 `k, i, s Ñ pq " p 2 ´k, ī, s Ñ pg, a, λ
rl 2 ´m2 Q `iεsrpl ´qq 2 ´m2 Q `iεs v spp qq . (8.2.29)
The corresponding reduced amplitude takes the form

M λ, λ,a R 2 ,ss,i īpp, k, p g q " ee Q 2π ż d 2 ⃗ x d 2
⃗ y e ´ip⃗ pq`⃗ pgq¨⃗ x e ´i⃗ pq¨⃗ y C a R 2 ,i īp⃗ x, ⃗ y q ˆū s pp q qN λ, λ R 2 pp, k, p g ; ⃗ r qv spp qq , with the colour structure:

C a R 2 p⃗ x, ⃗ y q " " t a V p⃗ x qV : p⃗ y q ´ta 1 ‰ , (8.2.30)
and the perturbative factor:

N λ, λ R 2 pp, k, p g ; ⃗ r q " ig2q `ż d 4 l p2πq 2 δpp q ´q``l`q e i ⃗ l¨⃗ r T λ, λ R 2 plq rl 2 ´m2 Q `iεsrpl ´qq 2 ´m2 Q `iεs . (8.2.31)
The Dirac structure which has been simplified using eq. (C.5.2) reads:

T λ, λ R 2 plq " ⃗ ϵ λn rp⃗ p g ´x⃗ p q q 2 `x2 m 2 Q s ! 2p⃗ p g ´x⃗ p q q m " δ mn ´1 `x 2 ¯`x 2 ω nm ı `x2 m Q γ n ( 1 2q
`γ`p { l `mQ q{ εpλ, qqp { l ´{ q `mQ qγ `, (8.2.32) where x " x g {x q .

As in our calculation for the R 1 contribution, the integration over l `is trivial due to the delta function δpp q ´q``l`q , and the integration over l ´is performed using the residue theorem, closing the contour on the upper plane. We find then N λ, λ R 2 pp, k, p g ; ⃗ r q " ´gθpx qqθp1 ´xq q ż d 2 ⃗ l 2π

e i ⃗ l¨⃗ r Q2 R 2 ,ξ `⃗ l 2 T λ, λ R 2 plq ˇˇˇl `"q `´q `xq , (8.2.33) where Q2 R 2 ,ξ " Q 2 p1 ´xq qx q `m2 Q . (8.2.34)
The explicit expressions for the perturbative factors are:

N λ"0, λ R 2 pp, k, p g ; ⃗ r q " 2Qp1 ´xq qx q g θpx qqθp1 ´xq qK 0 p QR 2 ,ξ |⃗ r |q ⃗ ϵ λn rp⃗ p g ´x⃗ p q q 2 `x2 m 2 Q s ˆ!2p⃗ p g ´x⃗ p q q m " δ mn ´1 `x 2 ¯`x 2 ω nm ı `mQ x 2 γ n ) γ `, (8.2.35)
for longitudinally polarized photon, and

N λ"˘1, λ R 2 pp, k, p g ; ⃗ r q " ´gθpx qqθp1 ´xq q ⃗ ϵ λn ⃗ ϵ λ i rp⃗ p g ´x⃗ p q q 2 `m2 Q x 2 s ˆ!2p⃗ p g ´x⃗ p q q m " δ mn ´1 `x 2 ¯`x 2 ω nm ı `mQ x 2 γ n ) ˆ" i⃗ r j |⃗ r | QR 2 ,ξ K 1 p QR 2 ,ξ |⃗ r |q " δ ij p2x q ´1q `ωij ‰ γ `´m Q γ i γ `K0 p QR 2 ,ξ |⃗ r |q * , ( 8
M λ, λ,a R 2 ,ss,i īpp, k, p g q " ee Q 2π ż d 2 ⃗ b d 2 ⃗ r e ´i⃗ p¨⃗ b e ´i⃗ k¨⃗ r e ´i⃗ pg¨p ⃗ b`⃗ r 2 q C a R 2 ,i ī ˆ⃗ b `⃗ r 2 , ⃗ b ´⃗ r 2 us ´p 2 `k¯N λ, λ R 2 pp, k, p g ; ⃗ r qv s ´p 2 `k¯.
As in the R 1 case, we have checked that eqs. (8.2.35) and (8.2.36) match with those presented in [START_REF] Caucal | Dijet impact factor in DIS at next-to-leading order in the Color Glass Condensate[END_REF] (before spinor contraction) up to a factor of 1{p2q `q due to our convention in the normalization.

Gluon emission from antiquark

The computations for the diagrams R 3 and R 4 where the gluon is emitted by the antiquark follow the same steps as those in R 1 and R 2 . We omit the specific details of the derivation and simply present the results. R 3 diagram: gluon emission before the shockwave

M λ, λ,a R 3 ,ss,i īpp, k, p g q " ee Q 2π ż d 2 ⃗ r d 2 ⃗ b d 2 ⃗ z C a R 3 ,i ī ˆ⃗ b `⃗ r 2 , ⃗ b ´⃗ r 2
, ⃗ z ˙e´i⃗ p¨⃗ b e ´i⃗ k¨⃗ r e ´i⃗ pg¨⃗ z ˆū s ´p 2 `k¯´N λ, λ R 3 ,reg pp, k, p g ; ⃗ r, ⃗ b, ⃗ z q `N λ, λ R 3 ,inst pp, k, p g ; ⃗ r, ⃗ b, ⃗ z q ¯vs ´p 2 ´k¯, (8.2.37) where the colour factor is the identical as in the R 1 contribution

C a R 3 p⃗ x, ⃗ y, ⃗ z q " V p⃗ x qV : p⃗ z qt a V p⃗ z qV : p⃗ y q ´ta 1 , (8.2.38)
and the perturbative factors are3

N λ"0, λ R 3 ,reg pp, k, p g ; ⃗ r, ⃗ b, ⃗ z q " 2Q
x q p1 ´xq q x g g π θpx g qθ px qq θ p1 ´xq q θ px q q⃗ ϵ λn γ

#I 1m p⃗ x ´⃗ y, ⃗ z ´⃗ y q "ˆ1 `xg 2x q ˙δnm `xg 2x q ω mn ȷ ´mQ I 0 p⃗ x ´⃗ y, ⃗ z ´⃗ y q x 2 g 2p1 ´xq qx q γ n + , (8.2.39)
N λ"0, λ R 3 ,inst pp, k, p g ; ⃗ r, ⃗ b, ⃗ z q " 0 , for longitudinally polarized photon, and

N λ"˘1, λ R 3 ,reg pp, k, p g ; ⃗ r, ⃗ b, ⃗ z q " ´1
x g g π θpx g qθ px qq θ p1 ´xq q θ px q q⃗ ϵ λn ⃗ ϵ λ i ˆ"I 2mj p⃗ x ´⃗ y, ⃗ z ´⃗ y qγ `"δ ij p1 ´2x q q `ωij ‰ " δ nm ˆ1 `xg 2x q ˙`x g 2x q ω mn ȷ `mQ

I 1m p⃗ x ´⃗ y, ⃗ z ´⃗ y qγ `γi " δ nm ˆ1 `xg 2x q ˙`x g 2x q ω mn ȷ ´mQ Ĩ1j p⃗ x ´⃗ y, ⃗ z ´⃗ y q x 2 g 2p1 ´xq qx q γ `"δ ij p1 ´2x q q `ωij ‰ γ n ´m2 Q x 2 g p1 ´xq qx q I 0 p⃗ x ´⃗ y, ⃗ z ´⃗ y qγ `γi γ n + , (8.2.40)
N λ"˘1, λ R 3 ,inst pp, k, p g ; ⃗ x, ⃗ y, ⃗ z q " x g x q x q 2p1 ´xq q g π θpx g qθ px qq θ px q q θ p1 ´xq q⃗

ϵ λn ⃗ ϵ λ i QR 3 ,ξ K 1 p QR 3 ,ξ X R 3 ,ξ q X R 3 ,ξ γ `γi γ n , (8.2.41)
for the transversely polarized photon case. The variables Q2 R 3 ,ξ and X 2 R 3 ,ξ are identical to those in eq. (8.2.26). In this case, the I 0 , I 1m , I 2mj , Ĩ1j functions have to be evaluated with z

1 " x q " 1 2 ´xg 2 `ξ, z 2 " x g , z 3 " x q " 1 2 ´xg 2 ´ξ.
R 4 diagram: gluon emission after the shockwave

M λ, λ,a R 4 ,ss,i īpp, k, p g q " ee Q 2π ż d 2 ⃗ r d 2 ⃗ b e ´i⃗ p¨⃗ b e ´i⃗ pg¨p ⃗ b´⃗ r 2 q e ´i⃗ k¨⃗ r C a R 4 ,i ī ˆ⃗ b `⃗ r 2 , ⃗ b ´⃗ r 2 us ´p 2 `k¯N λ, λ R 4 pp, k, p g ; ⃗ r qv s ´p 2 ´k¯,
where the colour factor is

C a R 4 p⃗ x, ⃗ y q " " V p⃗ x qV : p⃗ y qt a ´ta 1 ‰ , (8.2.42)
and the perturbative factors are

N λ"0, λ R 4
pp, k, p g ; ⃗ r q " ´2Qp1 ´xq qx q g θpx q qθ p1 ´xq q K

0 p QR 4 ,ξ |⃗ r |q ⃗ ϵ λn " p⃗ p g ´x⃗ p qq 2 `x 2 m 2 Q ı ˆγ`! 2 p⃗ p g ´x⃗ p qq m " δ mn ´1 `x 2 ¯´x 2 ω nm ı ´mQ x2 γ n ) , (8.2.43)
for longitudinally polarized photon, and

N λ"˘1, λ R 4 pp, k, p g ; ⃗ r q " gθpx q qθ p1 ´xq q ⃗ ϵ λn ⃗ ϵ λ i " p⃗ p g ´x⃗ p qq 2 `x 2 m 2 Q ı ˆ"´i ⃗ r j |⃗ r | QR 4 ,ξ K 1 p QR 4 ,ξ |⃗ r |q " δ ij p2x q ´1q ´ωij ‰ γ `´m Q K 0 p QR 4 ,ξ |⃗ r |qγ i γ `* ˆ!2 p⃗ p g ´x⃗ p qq m " δ mn ´1 `x 2 ¯´x 2 ω nm ı ´mQ x2 γ n ) , (8.2.44)
for transversely polarized photon. Here we defined the variables

x " Having computed the amplitudes for Q Qg production in the previous section, we are now in a position to compute the short-distance coefficients by proper projection to the quantum state κ.

x g x q Q2 R 4 ,ξ " x q p1 ´xq qQ 2 `m2 Q . ( 8

Gluon emission from quark before the shockwave

We project the amplitude R 1 in eq. (8.2.27) using eq. (8.1.11), we find

M λ, λ,κ,Jz R 1 pp, p g q " ee Q 2π ż d 2 ⃗ r d 2 ⃗ b d 2 ⃗ z Tr " C R 1 ˆ⃗ b `⃗ r 2 , ⃗ b ´⃗ r 2 , ⃗ z ˙Crcs ȷ e ´i⃗ p¨⃗ b e ´i⃗ pg¨⃗ z ˆ´F λ, λ,κ,Jz R 1 ,reg pp, p g , Q; ⃗ r, ⃗ b, ⃗ z q `Fλ, λ,κ,Jz R 1 ,inst pp, p g , Q; ⃗ r, ⃗ b, ⃗ z q ¯, (8.3.1)
where we defined the projected perturbative factor:

F λ, λ,κ,Jz R 1 ,reg pp, p g , Q; ⃗ r, ⃗ b, ⃗ z q " Tr " Π JJz pp, 0qN λ, λ R 1 ,reg pp, 0, p g ; ⃗ r, ⃗ b, ⃗ z q ı (8.3.2)
with a similar expression for the instantaneous counterpart. First, we evaluate the perturbative factors in eqs. (8.2.23), (8.2.24) and (8.2.25) at k " 0 (corresponding to the S-wave), we find

N λ"0, λ R 1 ,reg pp, 0, p g ; ⃗ r, ⃗ b, ⃗ z q (8.3.3) " ´Q p1 `xg q x g g 2π ⃗ ϵ λn γ `#I 1m p⃗ y ´⃗ x, ⃗ z ´⃗ x q rδ nm ´xg ω mn s ´mQ I 0 p⃗ y ´⃗ x, ⃗ z ´⃗ x q 2x 2 g 1 `xg γ n + , N λ"˘1, λ R 1 ,reg pp, 0, p g ; ⃗ r, ⃗ b, ⃗ z q " 1 x g p1 ´xg q g π ⃗ ϵ λn ⃗ ϵ λ i (8.3.4) ˆ#I 2mj p⃗ y ´⃗ x, ⃗ z ´⃗ x q rδ nm `xg ω nm s " δ ij x g ´ωij ‰ γ `´m Q I 1m p⃗ y ´⃗ x, ⃗ z ´⃗ x q rδ nm `xg ω nm s γ i γ mQ Ĩ1j p⃗ y ´⃗ x, ⃗ z ´⃗ x q 2x 2 g p1 `xg q γ n " δ ij x g ´ωij ‰ γ `´m 2 Q I 0 p⃗ y ´⃗ x, ⃗ z ´⃗ x q 2x 2 g p1 `xg q γ n γ i γ `+ , N λ"˘1, λ R 1 ,inst pp, 0, p g ; ⃗ r, ⃗ b, ⃗ z q " ´xg p1 ´xg q 2 2p1 `xg q g 2π QR 1 K 1 p QR 1 X R 1 q X R 1 ⃗ ϵ λn ⃗ ϵ λ i γ n γ i γ `, (8.3
.5) with Q2 R 1 ,ξ"0 " Q2 R 1 " Q 2 `m2 Q 4 p1 ´xg q X 2 R 1 ,ξ"0 " X 2 R 1 " x g ˆ1 2 ´xg 2 ˙ˆ⃗ z ´⃗ b ´⃗ r 2 ˙2 `ˆ1 2 ´xg 2 ˙2 ⃗ r 2 `ˆ1 2 ´xg 2 ˙xg ˆ⃗ b ´⃗ r 2 ´⃗ z ˙2 . (8.3.6)
To compute F λ, λ,κ,Jz , we use the projectors in eqs. (8.1.14), (8.1.15), and perform the traces over the gamma matrices with the help of the identities in appendix C.3. We find

F λ"0, λ, 1 S rcs 0 ,Jz R 1 ,reg pp, p g , Q; ⃗ r, ⃗ b, ⃗ z q " ´i 1 a 2m Q Qp1 `xg q gp π ⃗ ϵ λn ϵ nm I 1m p⃗ y ´⃗ x, ⃗ z ´⃗ x q , (8.3.7) F λ"0, λ, 3 S rcs 1 ,Jz R 1 ,reg pp, p g , Q; ⃗ r, ⃗ b, ⃗ z q " ε ρ pJ z q a 2m Q Q p1 `xg q x g g π ⃗ ϵ λn # g ρ`I n 1 p⃗ y ´⃗ x, ⃗ z ´⃗ x q `x2 g 1 `xg I 0 p⃗ y ´⃗ x, ⃗ z ´⃗ x q ˆpg ρ`⃗ p n ´gρn p `q( , (8.3.8)
for the photon longitudinal polarization, and

F λ"˘1, λ, 1 S rcs 0 ,Jz R 1 ,inst pp, p g , Q; ⃗ r, ⃗ b, ⃗ z q " ´i 1 a 8m Q x g p1 ´xg q 2 p1 `xg q gp π QR 1 K 1 p QR 1 X R 1 q X R 1 ⃗ ϵ λn ⃗ ϵ λ i ϵ ni , (8.3.9) F λ"˘1, λ, 3 S rcs 1 ,Jz R 1 ,inst pp, p g , Q; ⃗ r, ⃗ b, ⃗ z q " ´ερ pJ z q c m Q 2 x g p1 ´xg q 2 p1 `xg q g π QR 1 K 1 p QR 1 X R 1 q X R 1 ⃗ ϵ λn ⃗ ϵ λ i g ρ`δni , (8.3.10)
F λ"˘1, λ, 1 S rcs 0 ,Jz R 1 ,reg pp, p g , Q; ⃗ r, ⃗ b, ⃗ z q " ´id 2 m Q 1 x g p1 ´xg q gp π ⃗ ϵ λn ⃗ ϵ λ i ␣ ´I2mj p⃗ y ´⃗ x, ⃗ z ´⃗ x q " ´δnm ϵ ij `x2 g δ ij ϵ nm ‰ `2x 2 g p1 `xg q m 2 Q I 0 p⃗ y ´⃗ x, ⃗ z ´⃗ x qϵ ni + , (8.3.11)
F λ"˘1, λ, 3 S rcs 1 ,Jz R 1 ,reg pp, p g , Q; ⃗ r, ⃗ b, ⃗ z q " ε ρ pJ z q a 2m Q 1 x g p1 ´xg q g π ⃗ ϵ λn ⃗ ϵ λ i ␣ ´2x g I 2mj p⃗ y ´⃗ x, ⃗ z ´⃗ x qg ρ`" δ nm δ ij `ϵij ϵ nm ‰ `I1m p⃗ y ´⃗ x, ⃗ z ´⃗ x q " δ nm pg ρ`⃗ p i ´gρi p `q ´xg δ im pg ρ`⃗ p n ´p`gnρ q `xg δ in pg ρ`⃗ p m ´p`gmρ q ‰ ´2x 2 g p1 `xg q Ĩ1j p⃗ y ´⃗ x, ⃗ z ´⃗ x q " δ ij x g pg ρ`⃗ p n ´gρn p `q ´δjn pg ρ`⃗ p i ´p`giρ q `δin pg ρ`⃗ p j ´p`gjρ q ‰ ´4x 2 g p1 `xg q m 2 Q I 0 p⃗ y ´⃗ x, ⃗ z ´⃗ x qg ρ`δni + , ( 8

Gluon emission from quark after the shockwave

The projected reduced amplitude reads

M λ, λ,κ,Jz R 2 pp, p g q " ee Q 2π ż d 2 ⃗ r F λ, λ,κ,Jz R 2 pp, p g , Q; ⃗ r q ż d 2 ⃗ b e ´i⃗ p¨⃗ b e ´i⃗ pg¨p ⃗ b`⃗ r 2 q ˆTr " C R 2 ˆ⃗ b `⃗ r 2 ; ⃗ b ´⃗ r 2 ˙Crcs ȷ , (8.3.13) with F λ, λ,κ,Jz R 2 pp, p g , Q; ⃗ r q " Tr " Π JJz pp, 0qN λ, λ R 2 pp, 0, p g ; ⃗ r q ı . (8.3.14)
The computations for F λ, λ,κ,Jz R 2 are similar to the ones for F R 1 . We will just give the results below.

F λ"0, λ, 1 S rcs 0 ,Jz R 2 pp, p g , Q; ⃗ r q " i d 2 m Q gp `xg p1 `xg qQK 0 p QR 2 |⃗ r |q ⃗ ϵ λn " ´⃗ p g ´x ⃗ p 2 ¯2 `x2 m 2 Q ȷ ˆ⃗ p g ´x ⃗ p 2 ˙m ϵ nm , (8.3.15)
F λ"0, λ, 3 S rcs 1 ,Jz R 2 pp, p g , Q; ⃗ r q " ´ερ pJ z q c m Q 2 gp1 `xg qp1 ´xg qQK 0 p QR 2 |⃗ r |q ⃗ ϵ λn " ´⃗ p g ´x ⃗ p 2 ¯2 `x2 m 2 Q ȷ ˆ"4 ˆ⃗ p g ´x ⃗ p 2 ˙m δ mn ´1 `x 2 ¯gρ``x2 pg ρ`⃗ p n ´gρn p `q* , (8.3.16)
for longitudinally polarized photon, and

F λ"˘1, λ, 1 S rcs 0 ,Jz R 2 pp, p g , Q; ⃗ r q " d 2 m Q igp `⃗ ϵ λn ⃗ ϵ λ i " ´⃗ p g ´x ⃗ p 2 ¯2 `x2 m 2 Q ȷ " 2i QR 2 K 1 p QR 2 |⃗ r |q ˆ⃗ p g ´x ⃗ p 2 ˙m ⃗ r j |⃗ r | ˆ"´1 `x 2 ¯δmn ϵ ji `x 2 x g δ ij ϵ nm ı `x2 m 2 Q K 0 p QR 2 |⃗ r |qϵ ni ) , (8.3.17) F λ"˘1, λ, 3 S rcs 1 ,Jz R 2 pp, p g , Q; ⃗ r q " ´ερ pJ z q a 2m Q g ⃗ ϵ λn ⃗ ϵ λ i " ´⃗ p g ´x ⃗ p 2 ¯2 `x2 m 2 Q ȷ " 4i ˆ⃗ p g ´x ⃗ p 2 ˙m ⃗ r j |⃗ r | QR 2 K 1 p QR 2 |⃗ r |q ˆgρ`" ´1 `x 2 ¯xg δ ij δ mn `x 2 ϵ ij ϵ nm ı `2 ˆ⃗ p g ´x ⃗ p 2 ˙m K 0 p QR 2 |⃗ r |q ˆ"δ mn ´1 `x 2 ¯`⃗ p i g ρ`´p`gρi ˘´x 2 δ im `⃗ p n g ρ`´p`gρn ˘`x 2 δ in `⃗ p m g ρ`´p`gρm ˘ı `ix 2 ⃗ r j |⃗ r | QR 2 K 1 p QR 2 |⃗ r |q " δ ij x g `⃗ p n g ρ`´gρn p `˘´δ jn `⃗ p i g ρ`´p`giρ ˘`δ in `⃗ p j g ρ`´p`gjρ ˘‰ ´2x 2 m 2 Q K 0 p QR 2 |⃗ r |qg ρ`δin + , (8.3.18)
for transversely polarized photon, where

x "

2x g p1 ´xg q , Q2 R 2 " Q2 R 2 ,ξ"0 " Q 2 p1 ´xg qp1 `xg q 4 `m2 Q . ( 8
F λ"0, λ, 1 S r8s 0 ,Jz R 3 ,reg pp, p g , Q; ⃗ r, ⃗ b, ⃗ z q " ´i 1 a 2m Q Qp1 `xg q gp π ⃗ ϵ λn I 1m p⃗ x ´⃗ y, ⃗ z ´⃗ y qϵ nm , (8.3.20) F λ"0, λ, 3 S rcs 1 ,Jz R 3 ,reg pp, p g , Q; ⃗ r, ⃗ b, ⃗ z q " ´ερ pJ z q a 2m Q Q 1 `xg x g g π ⃗ ϵ λn # g ρ`I n 1 p⃗ x ´⃗ y, ⃗ z ´⃗ y q `x2 g 1 `xg I 0 p⃗ x ´⃗ y, ⃗ z ´⃗ y q ˆ`g ρ`⃗ p n ´gρn p `˘( , (8.3.21)
for photon longitudinal polarization, and

F λ"˘1, λ, 1 S r8s 0 ,Jz R 3 ,inst pp, p g , Q; ⃗ r, ⃗ b, ⃗ z q " ´i 1 a 8m Q x g p1 ´xg q 2 p1 `xg q gp π ˆQ R 3 K 1 p QR 3 X R 3 q X R 3 ˙⃗ ϵ λn ⃗ ϵ λ i ϵ ni , (8.3.22) F λ"˘1, λ, 3 S rcs 1 ,Jz R 3 ,inst pp, p g , Q; ⃗ r, ⃗ b, ⃗ z q " ε ρ pJ z q c m Q 2 x g p1 ´xg q 2 p1 `xg q g π ˆQ R 3 K 1 p QR 3 X R 3 q X R 3 ˙⃗ ϵ λn ⃗ ϵ λ i g ρ`δin , (8.3.23) F λ"˘1, λ, 1 S r8s 0 ,Jz R 3 ,reg pp, p g , Q; ⃗ r, ⃗ b, ⃗ z q " ´id 2 m Q 1 x g p1 ´xg q gp π ⃗ ϵ λn ⃗ ϵ λ i ␣ I 2mj p⃗ x ´⃗ y, ⃗ z ´⃗ y q " ´x2 g δ ij ϵ mn `δnm ϵ ij ‰ ´m2 Q 2x 2 g 1 `xg I 0 p⃗ x ´⃗ y, ⃗ z ´⃗ y qϵ in + , (8.3.24)
F λ"˘1, λ, 3 S 0 "1 rcs ,Jz R 3 ,reg pp, p g , Q; ⃗ r, ⃗ b, ⃗ z q " ε ρ pJ z q a 2m Q 1 x g p1 ´xg q g π ⃗ ϵ λn ⃗ ϵ λ i ␣ 2x g I 2mj p⃗ x ´⃗ y, ⃗ z ´⃗ y qg ρ`" δ ij δ nm `ϵij ϵ nm ‰ `2x 2 g 1 `xg Ĩ1j p⃗ x ´⃗ y, ⃗ z ´⃗ y q " x g δ ij pg ρ`⃗ p n ´gρn p `q ´δjn `gρ`⃗ p i ´p`giρ ˘`δ in `gρ`⃗ p j ´p`gjρ ˘‰ ´I1m p⃗ x ´⃗ y, ⃗ z ´⃗ y q " δ nm pg ρ`⃗ p i ´gρi p `q ´xg δ im `gρ`⃗ p n ´p`gnρ ˘`x g δ in `gρ`⃗ p m ´p`gmρ ˘‰ `m2 Q 4x 2 g 1 `xg I 0 p⃗ x ´⃗ y, ⃗ z ´⃗ y qg ρ`δin + , (8.3.25)
for transversely polarized photon. The variables Q2 R 3 and X 2 R 3 are identical to those in eq. (8.3.6). The integrals I 0 , I 1m , I 2mj , Ĩ1j (defined in appendix C.7) are evaluated at z

1 " 1 2 ´xg 2 , z 2 " x g , z 3 " 1 2 ´xg 2 .
R 4 diagram: gluon emission after the shockwave

F λ"0, λ, 1 S r8s 0 ,Jz R 4 pp, p g , Q; ⃗ r q " i d 2 m Q gp `xg p1 `xg qQK 0 p QR 4 |⃗ r |q ⃗ ϵ λn " ´⃗ p 3 ´x ⃗ p 2 ¯2 `x 2 m 2 Q ȷ ˆ⃗ p 3 ´x ⃗ p 2 ˙m ϵ nm , (8.3.26) F λ"0, λ, 3 S rcs 1 ,Jz R 4 pp, p g , Q; ⃗ r q " ε ρ pJ z q c m Q 2 gp1 ´xg qp1 `xg qQK 0 p QR 4 |⃗ r |q ⃗ ϵ λn " ´⃗ p 3 ´x ⃗ p 2 ¯2 `x 2 m 2 Q ȷ ˆ"4 ˆ⃗ p 3 ´x ⃗ p 2 ˙m δ mn ´1 `x 2 ¯gρ``x2 `gρ`⃗ p n ´gρn p `˘* , (8.3.27)
for longitudinally polarized photon, and

F λ"˘1, λ, 1 S r8s 0 ,Jz R 4 pp, p g , Q; ⃗ r q " i d 2 m Q gp `⃗ ϵ λn ⃗ ϵ λ i " ´⃗ p 3 ´x ⃗ p 2 ¯2 `x 2 m 2 Q ȷ " ´2i QR 4 K 1 p QR 4 |⃗ r |q ⃗ r j |⃗ r | ˆ⃗ p 3 ´x ⃗ p 2 ˙m ˆ" x 2 x g δ ij ϵ nm `´1 `x 2 ¯δmn ϵ ji ı `x 2 m 2 Q K 0 p QR 4 |⃗ r |qϵ ni ) , (8.3.28)
F λ"˘1, λ, 3 S rcs 1 ,Jz R 4 pp, p g , Q; ⃗ r q " ε ρ pJ z q a 2m Q g ⃗ ϵ λn ⃗ ϵ λ i " ´⃗ p 3 ´x ⃗ p 2 ¯2 `x 2 m 2 Q ȷ " ´4i QR 4 K 1 p QR 4 |⃗ r |q ⃗ r j |⃗ r | ˆ⃗ p 3 ´x ⃗ p 2 ˙m g ρ" x g ´1 `x 2 ¯δij δ mn `x 2 ϵ ij ϵ nm ı ´ix 2 QR 4 K 1 p QR 4 |⃗ r |q ⃗ r j |⃗ r | ˆ"δ ij x g pg ρ`⃗ p n ´gρn p `q ´δjn `⃗ p i g ρ`´p`giρ ˘`δ in `gρ`⃗ p j ´p`gjρ ˘‰ ´2x 2 m 2 Q K 0 p QR 4 |⃗ r |qg ρ`δin `2K 0 p QR 4 |⃗ r |q ˆ⃗ p 3 ´x ⃗ p 2 ˙m ˆ"δ mn ´1 `x 2 ¯pg ρ`⃗ p i ´gρi p `q ´x 2 δ im pg ρ`⃗ p n ´p`gnρ q `x 2 δ in pg ρ`⃗ p m ´p`gmρ q ı) , (8.3.29)
for transversely polarized photon. The variables are defined as x and Q2 R 4 are given by

x " 2x g 1 ´xg , Q2 R 4 " Q2 R 4 ,ξ"0 " Q 2 p1 ´xg qp1 `xg q 4 `m2 Q ,
which are the same as in eq. (8.3.19).

We close this section, by pointing out the symmetry between quark-antiquark symmetry in the perturbative functions F λ, λ,κ,Jz

R i : F λ, λ, 1 S rcs 0 ,Jz R 1p2q p⃗ r q " F λ, λ, 1 S rcs 0 ,Jz R 3p4q p´⃗ r q , ( 8
S p2q Y p⃗ x, ⃗ y q " 1 N c @ Tr " V p⃗ x qV : p⃗ y q ‰D Y , (8.4.7) S p2,2q Y p⃗ x, ⃗ y; ⃗ y 1 , ⃗ x 1 q " 1 N 2 c @ Tr " V p⃗ x qV : p⃗ y q ‰ Tr " V p⃗ y 1 qV : p⃗ x 1 q ‰D Y , (8.4.8)
S p4q Y p⃗ x, ⃗ y, ⃗ y 1 , ⃗ x 1 q " 1 N c @ Tr " V p⃗ x qV : p⃗ y qV p⃗ y 1 qV : p⃗ x 1 q ‰D Y (8.4.9) S p2,4q p⃗ x, ⃗ z; ⃗ z, ⃗ y, ⃗ y 1 , ⃗ x 1 q " 1 N 2 c @ Tr " V p⃗ x qV : p⃗ z q ‰ Tr " V p⃗ z qV : p⃗ y qV p⃗ y 1 qV : p⃗ x 1 q ‰D Y (8.4.10) S p6q p⃗ z, ⃗ y, ⃗ x, ⃗ z, ⃗ y 1 , ⃗ x 1 q " 1 N c @ Tr " V p⃗ z qV : p⃗ y qV p⃗ x qV : p⃗ z qV p⃗ y 1 qV : p⃗ x 1 q ‰D Y (8.4.11) S p4,4q p⃗ x, ⃗ z, ⃗ z 1 , ⃗ x 1 ; ⃗ z, ⃗ y, ⃗ y 1 , ⃗ z 1 q " 1 N 2 c @ Tr " V p⃗ x qV : p⃗ z qV p⃗ z 1 qV : p⃗ x 1 q ‰ Tr "
V p⃗ z qV : p⃗ y qV p⃗ y 1 qV : p⃗ z 1 q ‰D Y , (8.4.12)

S p8q p⃗ z, ⃗ y, ⃗ x, ⃗ z 1 , ⃗ x 1 , ⃗ y 1 q " 1 N c @ Tr "
V p⃗ z qV : p⃗ y qV p⃗ x qV : p⃗ z qV p⃗ z 1 qV : p⃗ x 1 qV p⃗ y 1 qV : p⃗ z 1 q ‰D Y . (8.4.13) Noting that C R 1 " C R 3 , we have the following relations among the colour correlators

Ξ rcs R 1 R 1 ,Y " Ξ rcs R 1 R 3 ,Y " Ξ rcs R 3 R 1 ,Y " Ξ rcs R 3 R 3 ,Y Ξ rcs R 1 R 2 ,Y " Ξ rcs R 3 R 2 ,Y Ξ rcs R 1 R 4 ,Y " Ξ rcs R 3 R 4 ,Y (8.4.14)
Furthermore, we can relate correlators using hermitian conjugation:

Ξ rcs R i R j ,Y " Ξ :rcs R j R i ,Y (8.4.15)
with also a re-labeling of the variables ⃗ x, ⃗ y, ⃗ z Ø ⃗ x 1 , ⃗ y 1 , ⃗ z 1 . As a consequence of these relations, it is sufficient to compute 6 colour correlators, whose results we give below.

R 1 R 1 contribution Ξ r1s R 1 R 1 ,Y `⃗ x, ⃗ y, ⃗ z; ⃗ y 1 , ⃗ x 1 , ⃗ z 1 ˘" 1 2 ´Sp8q p⃗ z, ⃗ y, ⃗ x, ⃗ z 1 , ⃗ x 1 , ⃗ y 1 q ´Sp2,2q p⃗ x, ⃗ y ; ⃗ y 1 , ⃗ x 1 q ¯, (8.4.16) Ξ r8s R 1 R 1 ,Y `⃗ x, ⃗ y , ⃗ z; ⃗ y 1 , ⃗ x 1 , ⃗ z 1 ˘" 1 2pN 2 c ´1q ! N 2 c S p4,4q p⃗ x, ⃗ z, ⃗ z 1 , ⃗ x 1 ; ⃗ z, ⃗ y, ⃗ y 1 , ⃗ z 1 q
´Sp8q p⃗ z, ⃗ y, ⃗ x, ⃗ z 1 , ⃗ x 1 , ⃗ y 1 q ´Sp4q p⃗ x, ⃗ y, ⃗ y 1 , ⃗ x 1 q `Sp2,2q p⃗ x, ⃗ y; ⃗ y 1 , ⃗ x 1 q ´N 2 c S p2,2q p⃗ x, ⃗ z; ⃗ z, ⃗ y q `Sp2q p⃗ x, ⃗ y q ´N 2 c S p2,2q p⃗ y 1 , ⃗ z 1 ; ⃗ z 1 , ⃗ x 1 q `Sp2q p⃗ y 1 , ⃗ x 1 q `pN 2 c ´1q

) . (8.4.17)

R 2 R 2 contribution Ξ r1s R 2 R 2 ,Y `⃗ x, ⃗ y; ⃗ y 1 , ⃗ x 1 ˘" 1 2
´Sp4q p⃗ x, ⃗ y, ⃗ y 1 , ⃗ x 1 q ´Sp2,2q p⃗ x, ⃗ y; ⃗ y 1 , ⃗ x 1 q ¯, (8.4.18)

Ξ r8s R 2 R 2 ,Y `⃗ x, ⃗ y; ⃗ y 1 , ⃗ x 1 ˘" 1 2 ˆN 2 c ´2 N 2 c ´1 S p4q p⃗ x, ⃗ y, ⃗ y 1 , ⃗ x 1 q `1 N 2 c
´1 S p2,2q p⃗ x, ⃗ y; ⃗ y 1 , ⃗ x 1 q ´Sp2q p⃗ x, ⃗ y q ´Sp2q p⃗ y 1 , ⃗ x 1 q `1¯. (8.4.19)

R 4 R 4 contribution Ξ r1s R 4 R 4 ,Y `⃗ x, ⃗ y; ⃗ y 1 , ⃗ x 1 ˘" 1 2 ´Sp4q p⃗ x, ⃗ y, ⃗ y 1 , ⃗ x 1 q ´Sp2,2q p⃗ x, ⃗ y; ⃗ y 1 , ⃗ x 1 q ¯, (8.4.20) Ξ r8s R 4 R 4 ,Y `⃗ x, ⃗ y; ⃗ y 1 , ⃗ x 1 ˘" 1 2 ˆN 2 c ´2 N 2 c ´1 S p4q p⃗ x, ⃗ y, ⃗ y 1 , ⃗ x 1 q `1 N 2 c ´1 S p2,2q p⃗ x, ⃗ y; ⃗ y 1 , ⃗ x 1 q ´Sp2q p⃗ x, ⃗ y q ´Sp2q p⃗ y 1 , ⃗ x 1 q `1¯, (8.4.21)
which is the same as the R 2 R 2 contribution.

R 1 R 2 contribution Ξ r1s R 1 R 2 ,Y `⃗ x, ⃗ y, ⃗ z; ⃗ y 1 , ⃗ x 1 ˘" 1 2
´Sp6q p⃗ z, ⃗ y, ⃗ x, ⃗ z, ⃗ y 1 , ⃗ x 1 q ´Sp2,2q p⃗ x, ⃗ y; ⃗ y 1 , ⃗ x 1 q ¯, (8.4.22)

Ξ r8s R 1 R 2 ,Y `⃗ x, ⃗ y, ⃗ z; ⃗ y 1 ; ⃗ y 1 , ⃗ x 1 " 1 2pN 2 c ´1q ! N 2
c S p2,4q p⃗ x, ⃗ z; ⃗ z, ⃗ y, ⃗ y 1 , ⃗ x 1 q ´Sp6q p⃗ z, ⃗ y, ⃗ x, ⃗ z, ⃗ y 1 , ⃗ x 1 q ´Sp4q p⃗ x, ⃗ y, ⃗ y 1 , ⃗ x 1 q `Sp2,2q p⃗ x, ⃗ y; ⃗ y 1 , ⃗ x 1 q ´N 2 c S p2,2q p⃗ x, ⃗ z; ⃗ z, ⃗ yq `Sp2q p⃗ x, ⃗ y q ´pN 2 c ´1qS p2q p⃗ y 1 , ⃗ x 1 q `pN 2 c ´1q

) .

(8.4.23)

R 2 R 4 contribution Ξ r1s R 2 R 4 ,Y `⃗ x, ⃗ y; ⃗ y 1 , ⃗ x 1 ˘" 1 2 ´Sp4q `⃗ x, ⃗ y, ⃗ y 1 , ⃗ x 1 ˘´S p2,2q `⃗ x, ⃗ y; ⃗ y 1 , ⃗ x 1 ˘¯, (8.4.24) Ξ r8s R 2 R 4 ,Y `⃗ x, ⃗ y; ⃗ y 1 , ⃗ x 1 ˘" 1 2 " N 2 c `1 N 2 c ´1 S p2,2q p⃗ x, ⃗ y; ⃗ y 1 , ⃗ x 1 q ´2 N 2 c
´1 S p4q p⃗ x, ⃗ y, ⃗ y 1 , ⃗ x 1 q ´Sp2q p⃗ x, ⃗ yq ´Sp2q p⃗ y 1 , ⃗ x 1 q `1) . (8.4.25)

R 1 R 4 contribution Ξ r1s R 1 R 4 ,Y `⃗ x, ⃗ y, ⃗ z; ⃗ y 1 , ⃗ x 1 ˘" 1 2 ! S p6q `⃗ z, ⃗ y, ⃗ x, ⃗ z, ⃗ y 1 , ⃗ x 1 ˘´S p2,2q `⃗ x, ⃗ y; ⃗ y 1 , ⃗ x 1 ˘) , (8.4.26) Ξ r8s R 1 R 4 ,Y `⃗ x, ⃗ y, ⃗ z; ⃗ y 1 , ⃗ x 1 " 1 2pN 2 c ´1q ! N 2
c S p2,4q p⃗ z, ⃗ y; ⃗ x, ⃗ z, ⃗ y 1 , ⃗ x 1 q ´Sp6q p⃗ z, ⃗ y, ⃗ x, ⃗ z, ⃗ y 1 , ⃗ x 1 q ´Sp4q p⃗ x, ⃗ y, ⃗ y 1 , ⃗ x 1 q `Sp2,2q p⃗ x, ⃗ y; ⃗ y 1 , ⃗ x 1 q ´N 2 c S p2,2q p⃗ x, ⃗ z; ⃗ z, ⃗ yq `Sp2q p⃗ x, ⃗ y q ´pN 2 c ´1qS p2q p⃗ y 1 , ⃗ x 1 q `pN 2 c ´1q

) .

(

Γ λ, 1 S rcs 0 R 1 R 1 p⃗ r, ⃗ r 1 q " Γ λ, 1 S rcs 0 R 3 R 3 p´⃗ r, ´⃗ r 1 q " Γ λ, 1 S rcs 0 R 1 R 3 p⃗ r, ´⃗ r 1 q , Γ λ, 1 S rcs 0 R 2 R 2 p⃗ r, ⃗ r 1 q " Γ λ, 1 S rcs 0 R 4 R 4 p´⃗ r, ´⃗ r 1 q " Γ λ, 1 S rcs 0 R 2 R 4 p⃗ r, ´⃗ r 1 q , Γ λ, 1 S rcs 0 R 1 R 2 p⃗ r, ⃗ r 1 q " Γ λ, 1 S rcs 0 R 3 R 4 p´⃗ r, ´⃗ r 1 q " Γ λ, 1 S rcs 0 R 1 R 4 p⃗ r, ´⃗ r 1 q " Γ λ, 1 S rcs 0 R 3 R 2 p´⃗ r, ⃗ r 1 q , (8.4.28)
and

Γ λ, 3 S rcs 1 R 1 R 1 p⃗ r, ⃗ r 1 q " Γ λ, 3 S rcs 1 R 3 R 3 p´⃗ r, ´⃗ r 1 q " ´Γλ, 3 S rcs 1 R 1 R 3 p⃗ r, ´⃗ r 1 q , Γ λ, 3 S rcs 1 R 2 R 2 p⃗ r, ⃗ r 1 q " Γ λ, 3 S rcs 1 R 4 R 4 p´⃗ r, ´⃗ r 1 q " ´Γλ, 3 S rcs 1 R 2 R 4 p⃗ r, ´⃗ r 1 q , Γ λ, 3 S rcs 1 R 1 R 2 p⃗ r, ⃗ r 1 q " Γ λ, 3 S rcs 1 R 3 R 4 p´⃗ r, ´⃗ r 1 q " ´Γλ, 3 S rcs 1 R 1 R 4 p⃗ r, ´⃗ r 1 q " ´Γλ, 3 S rcs 1 R 3 R 2 p´⃗ r, ⃗ r 1 q . (8.4.29)
In the relations above, all other arguments of the functions Γ λ,κ R i R j are kept fixed, and for the sake of compactness, they are not shown explicitly.

Furthermore, hermitian conjugation implies

Γ λ,κ R i R j pp, p g , Q; ⃗ r, ⃗ b, ⃗ z, ⃗ r 1 , ⃗ b 1 , ⃗ z 1 q " Γ :λ,κ R j R i pp, p g , Q; ⃗ r 1 , ⃗ b 1 , ⃗ z 1 , ⃗ r, ⃗ b, ⃗ zq . (8.4.30)
As a consequence of these relations, it is sufficient to compute the contributions for Γ

λ,κ R 1 R 1 , Γ λ,κ R 2 R 2 and Γ λ,κ R 1 R 2 .
In order to compute the perturbative factors, we use repeatedly:

ÿ λ"˘1 ⃗ ϵ λ i ⃗ ϵ ˚λ j " ´gKij " δ ij , (8.4.31)
when summing over transverse polarizations for the gluon (and photon). We also use ϵ ji ϵ mn " `δjm δ in ´δjn δ im ˘. (8.4.32) For the particular case of Γ λ, 3 S rcs 1 , we also use the completeness relation in eq. (8.1.16) and the identities in appendix C.6.

We present the results separately for longitudinally and transversely polarized photons. They will depend on the transverse integrals I 0 , I 1m , I 2mj , Ĩ1j defined in appendix C.7 and that are to be evaluated at z 1 " 1 2 ´xg 2 , z 2 " x g and z 3 " 1 2 ´xg 2 . They also depend on the different variables defined in eqs. (8.3.6) and (8.3.19).

Longitudinally polarized photon

R 1 R 1 contribution Γ λ"0,κ R 1 R 1 " Γ λ"0,κ R 1 regR 1 reg , (8.4.33)
where

Γ λ"0, 1 S rcs 0 R 1 regR 1 reg ´p, p g , Q; ⃗ r, ⃗ b, ⃗ z, ⃗ r 1 , ⃗ b 1 , ⃗ z 1 " Q 2 2m Q p1 `xg q 2 g 2 pp `q2 π 2 I 1 p⃗ y ´⃗ x, ⃗ z ´⃗ x q ¨I1 `⃗ y 1 ´⃗ x 1 , ⃗ z 1 ´⃗ x 1 ˘, (8.4.34)
for the κ " 1 S 0 state, and

Γ λ"0, 3 S rcs 1 R 1 regR 1 reg ´p, p g , Q; ⃗ r, ⃗ b, ⃗ z, ⃗ r 1 , ⃗ b 1 , ⃗ z 1 " 2m Q 3 Q 2 p1 `xg q 2 x 2 g g 2 pp `q2 π 2 # 1 4m 2 Q I 1 p⃗ y ´⃗ x, ⃗ z ´⃗ x q ¨I1 `⃗ y 1 ´⃗ x 1 , ⃗ z 1 ´⃗ x 1 ˘`2x 4 g p1 `xg q 2 ˆI0 p⃗ y ´⃗ x, ⃗ z ´⃗ x q I 0 `⃗ y 1 ´⃗ x 1 , ⃗ z 1 ´⃗ x 1 ˘( , (8.4.35)
for the κ " 3 S 1 state.

R 2 R 2 contribution Γ λ"0, 1 S rcs 0 R 2 R 2 pp, p g , Q; ⃗ r, ⃗ r 1 q " 2 m Q g 2 pp `q2 x 2 g p1 `xg q 2 QK 0 p QR 2 |⃗ r |qQK 0 p QR 2 |⃗ r 1 |q ´⃗ p g ´x ⃗ p 2 ¯2 " ´⃗ p g ´x ⃗ p 2 ¯2 `x2 m 2 Q ȷ 2 , (8.4.36)
for the κ " 1 S 0 state, and

Γ λ"0, 3 S rcs 1 R 2 R 2 pp, p g , Q; ⃗ r, ⃗ r 1 q " 1 3m Q pp `q2 g 2 p1 `xg q 2 p1 ´xg q 2 QK 0 p QR 2 |⃗ r |qQK 0 p QR 2 |⃗ r 1 |q 1 " ´⃗ p g ´x ⃗ p 2 ¯2 `x2 m 2 Q ȷ 2 ˆ#2 ´1 `x 2 ¯2 ˆ⃗ p g ´x ⃗ p 2 ˙2 `x4 m 2 Q + , (8.4.37)
for the κ " 3 S 1 state.

R 1 R 2 contribution Γ λ"0,κ R 1 R 2 " Γ λ"0,κ R 1 regR 2 , (8.4.38)
where

Γ λ"0, 1 S rcs 0 R 1 regR 2 ´p, p g , Q; ⃗ r, ⃗ b, ⃗ z, ⃗ r 1 " ´1 m Q x g p1 `xg q 2 g 2 pp `q2 π Q 2 K 0 p QR 2 |⃗ r 1 |q ´⃗ p g ´x ⃗ p 2 ¯¨I 1 p⃗ y ´⃗ x, ⃗ z ´⃗ x q " ´⃗ p g ´x ⃗ p 2 ¯2 `x2 m 2 Q ȷ , (8.4.39)
for the κ " 1 S 0 state, and

Γ λ"0, 3 S rcs 1 R 1 regR 2 ´p, p g , Q; ⃗ r, ⃗ b, ⃗ z, ⃗ r 1 " ´2m Q 3 p1 `xg q 2 p1 ´xg q x g g 2 pp `q2 π Q 2 K 0 p QR 2 |⃗ r 1 |q 1 " ´⃗ p g ´x ⃗ p 2 ¯2 `x2 m 2 Q ȷ ˆ# 1 2m 2 Q ´1 `x 2 ¯I1 p⃗ y ´⃗ x, ⃗ z ´⃗ x q ¨ˆ⃗ p g ´x ⃗ p 2 ˙`x 2 x 2 g 1 `xg I 0 p⃗ y ´⃗ x, ⃗ z ´⃗ x q + , (8.4.40)
for the κ " 3 S 1 state.

Transversely polarized photon

Here we present the results averaged over the two transverse polarizations.

R 1 R 1 contribution Γ λ"˘1,κ R 1 R 1 " Γ λ"˘1,κ R 1 instR 1 inst `Γλ"˘1,κ R 1 regR 1 inst `Γλ"˘1,κ R 1 instR 1 reg `Γλ"˘1,κ R 1 regR 1 reg , (8.4.41)
where

ÿ λ"˘1 Γ λ"˘1, 1 S rcs 0 R 1 instR 1 inst ´p, p g , Q; ⃗ r, ⃗ b, ⃗ z, ⃗ r 1 , ⃗ b 1 , ⃗ z 1 " 1 4m Q x 2 g p1 ´xg q 4 p1 `xg q 2 g 2 pp `q2 π 2 QR 1 K 1 p QR 1 X R 1 q X R 1 QR 1 K 1 p QR 1 X R 1 q X R 1 , (8.4.42) ÿ λ"˘1 Γ λ"˘1, 1 S rcs 0 R 1 regR 1 inst ´p, p g , Q; ⃗ r, ⃗ b, ⃗ z, ⃗ r 1 , ⃗ b 1 , ⃗ z 1 " 1 2m Q p1 ´xg q p1 `xg q g 2 pp `q2 π 2 QR 1 K 1 p QR 1 X R 1 q X R 1 ␣ ´I2mj p⃗ y ´⃗ x, ⃗ z ´⃗ x q p1 `x2 g qδ jm `4x 2 g p1 `xg q m 2 Q I 0 p⃗ y ´⃗ x, ⃗ z ´⃗ x q + , (8.4.43) ÿ λ"˘1 Γ λ"˘1, 1 S rcs 0 R 1 instR 1 reg ´p, p g , Q; ⃗ r, ⃗ b, ⃗ z, ⃗ r 1 , ⃗ b 1 , ⃗ z 1 " 1 2m Q p1 ´xg q p1 `xg q g 2 pp `q2 π 2 QR 1 K 1 p QR 1 X R 1 q X R 1 ␣ ´I2 mj `⃗ y 1 ´⃗ x 1 , ⃗ z 1 ´⃗ x 1 ˘p1 `x2 g qδ jm `4x 2 g p1 `xg q m 2 Q I 0 `⃗ y 1 ´⃗ x 1 , ⃗ z 1 ´⃗ x 1 ˘+ , (8.4.44) ÿ λ"˘1 Γ λ"˘1, 1 S rcs 0 R 1 regR 1 reg ´p, p g , Q; ⃗ r, ⃗ b, ⃗ z, ⃗ r 1 , ⃗ b 1 , ⃗ z 1 " 2 m Q 1 x 2 g p1 ´xg q 2 g 2 pp `q2 π 2 ␣ I 2mj p⃗ y ´⃗ x, ⃗ z ´⃗ x q I 2rl `⃗ y 1 ´⃗ x 1 , ⃗ z 1 ´⃗ x 1 "p1 `x4 g qδ rm δ jl ´2x 2 g pδ lm δ jr ´δlr δ jm q ı ´2x 2 g p1 `xg q m 2 Q I 2mj p⃗ y ´⃗ x, ⃗ z ´⃗ x q ˆI0 `⃗ y 1 ´⃗ x 1 , ⃗ z 1 ´⃗ x 1 ˘p1 `x2 g qδ mj ´2x 2 g p1 `xg q m 2 Q I 0 p⃗ y ´⃗ x, ⃗ z ´⃗ x q ˆI2 rl `⃗ y 1 ´⃗ x 1 , ⃗ z 1 ´⃗ x 1 ˘p1 `x2 g qδ rl `8x 4 g p1 `xg q 2 m 4 Q I 0 p⃗ y ´⃗ x, ⃗ z ´⃗ x q ˆI0 `⃗ y 1 ´⃗ x 1 , ⃗ z 1 ´⃗ x 1 ˘( , (8.4.45)
for the κ " 1 S 0 state, and

ÿ λ"˘1 Γ λ"˘1, 3 S rcs 1 R 1 instR 1 inst ´p, p g , Q; ⃗ r, ⃗ b, ⃗ z, ⃗ r 1 , ⃗ b 1 , ⃗ z 1 " 1 12m Q x 2 g p1 ´xg q 4 p1 `xg q 2 g 2 pp `q2 π 2 QR 1 K 1 p QR 1 X R 1 q X R 1 QR 1 K 1 p QR 1 X R 1 q X R 1 , (8.4.46) ÿ λ"˘1 Γ λ"˘1, 3 S rcs 1 R 1 regR 1 inst ´p, p g , Q; ⃗ r, ⃗ b, ⃗ z, ⃗ r 1 , ⃗ b 1 , ⃗ z 1 " 1 3m Q x g p1 ´xg q p1 `xg q g 2 pp `q2 π 2 QR 1 K 1 p QR 1 X R 1 q X R 1 " I 2mj p⃗ y ´⃗ x, ⃗ z ´⃗ x q δ mj `2x g p1 `xg q m 2 Q ˆI0 p⃗ y ´⃗ x, ⃗ z ´⃗ x qu , (8.4.47) ÿ λ"˘1 Γ λ"˘1, 3 S rcs 1 R 1 regR 1 reg ´p, p g , Q; ⃗ r, ⃗ b, ⃗ z, ⃗ r 1 , ⃗ b 1 , ⃗ z 1 " 4m Q 3 1 x 2 g p1 ´xg q 2 g 2 pp `q2 π 2 # x 2 g m 2 Q I 2mj p⃗ y ´⃗ x, ⃗ z ´⃗ x q I 2rl `⃗ y 1 ´⃗ x 1 , ⃗ z 1 ´⃗ x 1 "δ rm δ lj ´δlm δ rj `δjm δ lr ı `2x 3 g p1 `xg q I 2mj p⃗ y ´⃗ x, ⃗ z ´⃗ x q I 0 `⃗ y 1 ´⃗ x 1 , ⃗ z 1 ´⃗ x 1 ˘δjm `I1m p⃗ y ´⃗ x, ⃗ z ´⃗ x q I 1r `⃗ y 1 ´⃗ x 1 , ⃗ z 1 ´⃗ x 1 ˘δrm " 1 `x2 g ‰ `x2 g p1 `xg q I 1m p⃗ y ´⃗ x, ⃗ z ´⃗ x q Ĩ1 l `⃗ y 1 ´⃗ x 1 , ⃗ z 1 ´⃗ x 1 ˘δlm r1 ´xg s 2 `x2 g p1 `xg q Ĩ1j p⃗ y ´⃗ x, ⃗ z ´⃗ x q I 1r `⃗ y 1 ´⃗ x 1 , ⃗ z 1 ´⃗ x 1 ˘δjr r1 ´xg s 2 `4x 4 g p1 `xg q 2 Ĩ1j p⃗ y ´⃗ x, ⃗ z ´⃗ x q Ĩ1 l `⃗ y 1 ´⃗ x 1 , ⃗ z 1 ´⃗ x 1 ˘δjl " x 2 g `1‰ `2x 3 g p1 `xg q I 0 p⃗ y ´⃗ x, ⃗ z ´⃗ x q I 2rl `⃗ y 1 ´⃗ x 1 , ⃗ z 1 ´⃗ x 1 ˘δlr `m2 Q 4x 4 g p1 `xg q 2 I 0 p⃗ y ´⃗ x, ⃗ z ´⃗ x q I 0 `⃗ y 1 ´⃗ x 1 , ⃗ z 1 ´⃗ x 1 ˘+ , (8.4.48) ÿ λ"˘1 Γ λ"˘1, 3 S rcs 1 R 1 instR 1 reg ´p, p g , Q; ⃗ r, ⃗ b, ⃗ z, ⃗ r 1 , ⃗ b 1 , ⃗ z 1 " 1 3m Q x g p1 ´xg q p1 `xg q g 2 pp `q2 π 2 QR 1 K 1 p QR 1 X R 1 q X R 1 " I 2mj p⃗ y 1 ´⃗ x 1 , ⃗ z 1 ´⃗ x 1 qδ mj `2x g p1 `xg q m 2 Q ˆI0 p⃗ y 1 ´⃗ x 1 , ⃗ z 1 ´⃗ x 1 q + , (8.4.49)
for the κ " 3 S 1 state.

R 2 R 2 contribution ÿ λ"˘1 Γ λ"˘1, 1 S rcs 0 R 2 R 2 pp, p g , Q; ⃗ r, ⃗ r 1 q " 2 m Q g 2 pp `q2 1 " ´⃗ p g ´x ⃗ p 2 ¯2 `x2 m 2 Q ȷ 2 ˆ#4 " ´1 `x 2 ¯2 `x2 4 x 2 g ȷ ˆ⃗ p g ´x ⃗ p 2 ˙2 ⃗ r ¨⃗ r 1 |⃗ r ||⃗ r 1 | QR 2 K 1 p QR 2 |⃗ r |q QR 2 K 1 p QR 2 |⃗ r 1 |q `i2x 2 "´1 `x 2 ¯`x 2 x g ı m 2 Q ⃗ r ¨ˆ⃗ p g ´x ⃗ p 2 ˙Q R 2 K 1 p QR 2 |⃗ r |q |⃗ r | K 0 p QR 2 |⃗ r 1 |q ´i2x 2 "´1 `x 2 ¯`x 2 x g ı m 2 Q ⃗ r 1 ¨ˆ⃗ p g ´x ⃗ p 2 ˙Q R 2 K 1 p QR 2 |⃗ r 1 |q |⃗ r 1 | K 0 p QR 2 |⃗ r |q `2x 4 m 4 Q K 0 p QR 2 |⃗ r |qK 0 p QR 2 |⃗ r 1 |q + , (8.4.50)
for the κ " 1 S 0 state, and

ÿ λ"˘1 Γ λ"˘1, 3 S rcs 1 R 2 R 2 pp, p g , Q; ⃗ r, ⃗ r 1 q " 2m Q 3 g 2 pp `q2 1 " ´⃗ p g ´x ⃗ p 2 ¯2 `x2 m 2 Q ȷ 2 # 4 m 2 Q ˆ⃗ p g ´x ⃗ p 2 ˙2 ⃗ r 1 ¨⃗ r |⃗ r 1 ||⃗ r | QR 2 K 1 p QR 2 |⃗ r |q QR 2 K 1 p QR 2 |⃗ r 1 |q " ´1 `x 2 ¯2 x 2 g `´x 2 ¯2ȷ ´i2x 2 ˆ⃗ p g ´x ⃗ p 2 ˙¨⃗ r |⃗ r | QR 2 K 1 p QR 2 |⃗ r |qK 0 p QR 2 |⃗ r 1 |q "´1 `x 2 ¯xg `x 2 ı `8 ˆ⃗ p g ´x ⃗ p 2 ˙2 K 0 p QR 2 |⃗ r |qK 0 p QR 2 |⃗ r 1 |q " ´1 `x 2 ¯2 `´x 2 ¯2ȷ `i2x 2 ˆ⃗ p g ´x ⃗ p 2 ˙¨⃗ r 1 |⃗ r 1 | K 0 p QR 2 |⃗ r |q QR 2 K 1 p QR 2 |⃗ r 1 |q r1 ´xg s ´i2x 2 ˆ⃗ p g ´x ⃗ p 2 ˙¨⃗ r |⃗ r | K 0 p QR 2 |⃗ r 1 |q QR 2 K 1 p QR 2 |⃗ r |q r1 ´xg s `2x 4 ⃗ r ¨⃗ r 1 |⃗ r ||⃗ r 1 | QR 2 K 1 p QR 2 |⃗ r |q QR 2 K 1 p QR 2 |⃗ r 1 |q " x 2 g `1‰ `i2x 2 ˆ⃗ p g ´x ⃗ p 2 ˙¨⃗ r 1 |⃗ r 1 | QR 2 K 1 p QR 2 |⃗ r 1 |qK 0 p QR 2 |⃗ r |q "´1 `x 2 ¯xg `x 2 ı `2x 4 m 2 Q K 0 p QR 2 |⃗ r |qK 0 p QR 2 |⃗ r 1 |q + , (8.4.51)
for the κ " 3 S 1 state.

R 1 R 2 contribution Γ λ"˘1,κ R 1 R 2 " Γ λ"˘1,κ R 1 insR 2 `Γλ"˘1,κ R 1 regR 2 , (8.4.52)
where

ÿ λ"˘1 Γ λ"˘1, 1 S rcs 0 R 1 instR 2 ´p, p g , Q; ⃗ r, ⃗ b, ⃗ z, ⃗ r 1 " ´1 m Q x g p1 ´xg q 2 p1 `xg q g 2 pp `q2 π QR 1 K 1 p QR 1 X R 1 q X R 1 1 " ´⃗ p g ´x ⃗ p 2 ¯2 `x2 m 2 Q ȷ ˆ"´i ˆ⃗ p g ´x ⃗ p 2 ˙¨⃗ r 1 |⃗ r 1 | QR 2 K 1 p QR 2 |⃗ r 1 |q "´1 `x 2 ¯`x 2 x g ı `x2 m 2 Q K 0 p QR 2 |⃗ r 1 |q * , (8.4.53) ÿ λ"˘1 Γ λ"˘1, 1 S rcs 0 R 1 regR 2 ´p, p g , Q; ⃗ r, ⃗ b, ⃗ z, ⃗ r 1 " ´2 m Q 1 x g p1 ´xg q g 2 pp `q2 π 1 " ´⃗ p g ´x ⃗ p 2 ¯2 `x2 m 2 Q ȷ ˆ"2i QR 2 K 1 p QR 2 |⃗ r 1 |qI 2mj p⃗ y ´⃗ x, ⃗ z ´⃗ x q ˆ⃗ p g ´x ⃗ p 2 ˙r ⃗ r 1 l |⃗ r 1 | ˆ"´1 `x 2 `x3 g x 2 ¯δrm δ jl `´x 2 g ´1 `x 2 ¯`x 2 x g ¯´δ rl δ mj ´δrj δ ml ¯ı ´x2 `1 `x2 g ˘m2 Q K 0 p QR 2 |⃗ r 1 |qI 2mj p⃗ y ´⃗ x, ⃗ z ´⃗ x q δ jm ´i 4x 2 g p1 `xg q m 2 Q QR 2 K 1 p QR 2 |⃗ r 1 |qI 0 p⃗ y ´⃗ x, ⃗ z ´⃗ x q ˆ⃗ p g ´x ⃗ p 2 ˙¨⃗ r 1 |⃗ r 1 | "´1 `x 2 ¯`x 2 x g ı `4x 2 g x 2 p1 `xg q m 4 Q K 0 p QR 2 |⃗ r 1 |qI 0 p⃗ y ´⃗ x, ⃗ z ´⃗ xq + , (8.4.54)
for the κ " 1 S 0 state, and

ÿ λ"˘1 Γ λ"˘1, 3 S rcs 1 R 1 instR 2 ´p, p g , Q; ⃗ r, ⃗ b, ⃗ z, ⃗ r 1 " 1 3m Q x g p1 ´xg q 2 p1 `xg q g 2 pp `q2 π QR 1 K 1 p QR 1 X R 1 q X R 1 1 " ´⃗ p g ´x ⃗ p 2 ¯2 `x2 m 2 Q ȷ ˆ"´i ˆ⃗ p g ´x ⃗ p 2 ˙¨⃗ r 1 |⃗ r 1 | QR 2 K 1 p QR 2 |⃗ r 1 |q "´1 `x 2 ¯xg `x 2 ı ´x2 m 2 Q K 0 p QR 2 |⃗ r 1 |q * , (8.4.55) ÿ λ"˘1 Γ λ"˘1, 3 S rcs 1 R 1 regR 2 ´p, p g , Q; ⃗ r, ⃗ b, ⃗ z, ⃗ r 1 " ´2m Q 3 1 x g p1 ´xg q g 2 pp `q2 π 1 " ´⃗ p g ´x ⃗ p 2 ¯2 `x2 m 2 Q ȷ ˆ#´2x g QR 2 K 1 p QR 2 |⃗ r 1 |qI 2mj p⃗ y ´⃗ x, ⃗ z ´⃗ x q ˆ⃗ p g ´x ⃗ p 2 ˙r ´i⃗ r 1 l |⃗ r 1 | 1 m 2 Q " x g ´1 `x 2 ¯`x 2 ı ˆ´δ rm δ lj `δlr δ jm ´δlm δ rj ¯`2x g x 2 K 0 p QR 2 |⃗ r 1 |qI 2mj p⃗ y ´⃗ x, ⃗ z ´⃗ x q δ jm `4K 0 p QR 2 |⃗ r 1 |qI 1 p⃗ y ´⃗ x, ⃗ z ´⃗ x q ¨ˆ⃗ p g ´x ⃗ p 2 ˙"´1 `x 2 ¯`x g x 2 ı `ix 2 p1 ´xg q 2 QR 2 K 1 p QR 2 |⃗ r 1 |qI 1 p⃗ y ´⃗ x, ⃗ z ´⃗ x q ¨⃗ r 1 |⃗ r 1 | `4x 2 g p1 ´xg q p1 `xg q K 0 p QR 2 |⃗ r 1 |q Ĩ1 p⃗ y ´⃗ x, ⃗ z ´⃗ x q ¨ˆ⃗ p g ´x ⃗ p 2 i 4x 2 g x 2 `x2 g `1p 1 `xg q QR 2 K 1 p QR 2 |⃗ r 1 |q Ĩ1 p⃗ y ´⃗ x, ⃗ z ´⃗ x q ¨⃗ r 1 |⃗ r 1 | `i 4x 2 g p1 `xg q QR 2 K 1 p QR 2 |⃗ r 1 |qI 0 p⃗ y ´⃗ x, ⃗ z ´⃗ x q ˆ⃗ p g ´x ⃗ p 2 ˙¨⃗ r 1 |⃗ r 1 | " x g ´1 `x 2 ¯`x 2 ı `4x 2 g x 2 p1 `xg q m 2 Q K 0 p QR 2 |⃗ r 1 |qI 0 p⃗ y ´⃗ x, ⃗ z ´⃗ x q + , (8.4.56)
for the κ " 3 S 1 state.

Summary and Outlook

In this chapter, we presented the first computation of the short-distance coefficients for direct quarkonium + gluon production in electron-nucleus collisions at small x within the joint CGC + NRQCD framework based on [START_REF] Kang | Direct quarkonium-plus-gluon production in DIS in the Color Glass Condensate[END_REF]. We study both colour octet and singlet contributions and focus on the S-wave states. We revisited the computation of Q Qg production in the CGC within covariant perturbation theory with momentum space Feynman rules, and CGC effective vertices which resum coherent multiple interactions of the heavy quark pair and the gluon with the gluon background field of the nucleus to all orders. Then, we projected the amplitudes to the specific quantum state κ for the heavy quark pair. Our final results for the short distance coefficients for the differential cross-section are presented in section 8.4. They are expressed as a sum over 16 contributions dσ λ R i R j ,κ corresponding to all possible locations for the emission of the gluon. We show the results for 6 of these contributions, as the others can be easily obtained by quark-antiquark exchange or hermitian conjugation. Each of these contributions is written as a convolution of perturbatively calculable impact factors Γ λ,κ R i R j with multi-point lightlike Wilson line correlators Ξ rcs R i R j ,Y . The colour correlators are collected in section 8.4.1 for both singlet and octet states. The perturbative factors depend both on the polarization of the virtual photon and the spin state κ and are presented in section 8.4.2. The calculation for the P -wave can be carried out similarly but this contribution is not presented in this chapter as it has not yet been computed at the time of writing.

In the future, we plan to implement the fragmentation of the gluon into a jet (hadron),

to study the azimuthal correlations of direct quarkonium + jet (hadron) at the future EIC. It would be interesting to compare our results with those obtained within the TMD formalism [START_REF] D'alesio | Azimuthal asymmetries in semi-inclusive J{ψ `jet production at an EIC[END_REF]. In particular, we expect a matching between both formalisms in the limit in which the transverse momentum imbalance between the quarkonium and the jet is small compared to the invariant mass of the quarkonium-jet system, as it has been observed to occur for other processes [START_REF] Caucal | Back-to-back inclusive dijets in DIS at small x: Sudakov suppression and gluon saturation at NLO[END_REF][START_REF] Caucal | Back-to-back inclusive dijets in DIS at small x: Gluon Weizsäcker-Williams distribution at NLO[END_REF][START_REF] Caucal | Back-to-back inclusive dijets in DIS at small x: Complete NLO results and predictions[END_REF][START_REF] Dominguez | Universality of Unintegrated Gluon Distributions at small x[END_REF][START_REF] Xiao | Transverse Momentum Dependent Parton Distributions at Small-x[END_REF][START_REF] Taels | Dijet photoproduction at low x at next-to-leading order and its back-to-back limit[END_REF]. Furthermore, by integrating over the phase space of the gluon, we can obtain the next-toleading order real corrections for direct quarkonium production in DIS at small-x. Last, we could focus on the regime where the final state gluon is soft and derive expressions for the associated Sudakov logarithms for direct quarkonium production in the limit of small transverse momentum [START_REF] Mueller | Sudakov Resummation in Small-x Saturation Formalism[END_REF][START_REF] Qiu | Universal Suppression of Heavy Quarkonium Production in pA Collisions at Low Transverse Momentum[END_REF][START_REF] Hatta | Azimuthal angular asymmetry of soft gluon radiation in jet production[END_REF].

Conclusion

In this thesis, we have studied several processes in the high-energy limit where the non-linear effects of QCD start to become relevant. These novel calculations are done using the dipole picture presented in chapter 1 and the shockwave and CGC effective field theories outlined in chapters 3 and 5.

In chapter 6 and 7, we consider for the first time at NLO the diffractive production of a pair of hadrons, which is based on the article [START_REF] Fucilla | NLO computation of diffractive di-hadron production in a saturation framework[END_REF], or of a single hadron with large ⃗ p 2 h " Λ 2 QCD , which is the subject of [START_REF] Fucilla | Diffractive single hadron production in a saturation framework at the NLO[END_REF], in γ p˚q nucleon or nucleus scattering. The computations have been performed within a hybrid factorization, combining collinear factorization and shockwave formalism. They are at eikonal approximation and in the most general kinematics, i.e. Q 2 , ⃗ p 2 , t arbitrary. This makes our results applicable to both the LHC in UPC and at the future EIC. We have also extracted the finite result for the cross-section at NLO. These processes adds a new piece in the list of processes which are very promising to probe gluonic saturation in nucleons and nuclei at NLO, some of which are presented in chapter 2. With these higher-order calculations, it will be interesting to do future phenomenological studies. We could see how the NLO modifies the leading order results. It might then reveal the need for other resummation schemes, such as threshold resummation. Comparison to calculations that do not involve saturation is also crucial.

In chapter 8, we have presented some work based on [START_REF] Kang | Direct quarkonium-plus-gluon production in DIS in the Color Glass Condensate[END_REF] on the cross-section for direct quarkonium + gluon production in inclusive electron-nucleus collisions within the joint CGC + NRQCD framework. We have studied both colour octet and singlet contributions and focus on the S-wave states. We have first computed the Q Qg production in the CGC formalism within covariant perturbation theory, then have projected the different amplitudes to get the amplitude for the Q Qrκsg production. Our results could be used directly later on to study the azimuthal correlations of direct quarkonium + jet (hadron) at the future Electron-Ion Collider. It would be interesting to compare our results with those obtained within the TMD formalism [START_REF] D'alesio | Azimuthal asymmetries in semi-inclusive J{ψ `jet production at an EIC[END_REF]. In particular, we expect a matching between both formalisms in the limit in which the transverse momentum imbalance between the quarkonium and the jet is small compared to the invariant mass of the quarkonium-jet system. Furthermore, by integrating over the phase-space of the gluon, we could obtain the NLO real corrections for direct quarkonium production in DIS at small-x. Last, we could focus on the regime where the final state gluon is soft and derive expressions for the associated Sudakov logarithms for direct quarkonium production in the limit of small transverse momentum [START_REF] Mueller | Sudakov Resummation in Small-x Saturation Formalism[END_REF][START_REF] Qiu | Universal Suppression of Heavy Quarkonium Production in pA Collisions at Low Transverse Momentum[END_REF][START_REF] Hatta | Azimuthal angular asymmetry of soft gluon radiation in jet production[END_REF].

Dans ce manuscript, nous étudions trois processus dans les collisions γ p˚q P aux hautes énergies, dans le domaine non-linéaire de la chromodynamique quantique (QCD) afin d'avoir une meilleure compréhension des effets de saturation gluonique. Le photon peut être réel comme c'est le cas dans les collisions ultra-périphériques (UPC) ou virtuel comme dans les diffusions profondément inélastiques (DIS). P peut-être un nucléon ou un noyau. Ces calculs reposent sur des outils de QCD perturbative notamment sur les théorèmes de factorisation. Ces théoremes permettent dans les processus impliquant des larges échelles d'énergie une séparation des dyamiques longue et courte distance dans les observables physiques qui sont alors une convolution d'une partie dure calculable à l'aide des méthodes habituelles des diagrammes de Feynman et d'une partie non-perturbative non calculable perturbativement. Cette dernière demande alors l'utilisation de méthodes telles que la QCD sur réseau, de modèles phénoménologiques se basant sur des arguments physiques ou encore d'être contrainte par les données expérimentales. La partie dure d'un processus hadronique implique aux moins deux échelles: l'énergie au centre de masse s et une échelle dure Q 2 " Λ 2 QCD qui justifie le traitement perturbative en α s pQ 2 q ! 1.

Dans le chapitre 1, nous introduisons tout d'abord les variables sur le cône de lumière avant de définir les différent types de collision impliquant un photon (réel ou virtual) projectile et un nucléon ou noyau cible.

Tout d'abord, nous présentons les diffusions profondément inélastiques (DIS). Ce sont des collisions e ´pk e q `pppq Ñ e ´pk 1 e q `X où Q 2 " ´pk e ´k1 e q 2 " m 2 n avec m n la masse du proton de telle sorte que la structure interne du proton peut être étudiée avec un résolution de l'ordre de 1{Q. Pour Q 2 " Λ 2 QCD , la QCD perturbative peut être utilisée pour étudier DIS. Selon le régime cinématique défini par Q 2 et Bjorken-x dans lequel DIS est étudiée, le théorème de factorisation à utiliser sera différent, comme recapitulé dans la fig. 8.5.1.

On distingue deux régimes cinématiques principaux: le régime colinéaire caractérisé par Bjorken-x non-paramétriquement petit et Q 2 Ñ 8 et le régime de Regge-Gribov ou limite semi-dure avec s " Q 2 " Λ 2 QCD ou x Bj " Q 2 s Ñ 0. Dans le premier cas, la factorization colinéaire s'applique. Des grands logarithmes α s lnpQ 2 q " 1 apparaissant après compensation des divergences colinéaires sont resommées par l'équation de Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP). Dans le deuxième cas, la factorisation dite k t s'applique et les sections efficaces totales sont des convolutions dans l'espace des moments transverses de facteurs d'impact qui sont process-dépendant (ils dépendent des particules qui collisionnent) et d'une fonction de Green BFKL qui est universel et détermine totalement le comportement en s des sections efficaces. Ainsi, on s'attend à ce que des effets non-linéaires telles que la recombinaison des gluons deviennent importants à partir d'une densité critique de gluons. Ces effets rivalisent avec l'émission de gluons et mène au phénomène de la saturation des gluons. Ce régime de saturation de gluons apparait aux hautes énergies et pour Q 2 en dessous de Q 2 s pxq qui est l'échelle de saturation. Une approximative estimation de cette échelle donne Q 2 s 9 A 1{3 p1{x Bj q ω 0 . Les effets de saturation sont donc plus importants dans un proton très énergétique ou pour un noyau cible lourd. Les extensions non-linéaires de l'équation d'évolution BFKL sont l'equation Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov and Kovner (JIMWLK) et son équivalent, la hiérarchie d'équations Balitsky. Nous introduisons aussi l'image du dipôle utilisée pour les collisions aux hautes énergies impliquant un photon et une cible hadronique.

DGLAP BFKL Saturation ln Q 2 s (Y ) ln Q 2 Y = ln   1 x Bj   Non -pertutbative ln Λ
Nous définissons ensuite UPC qui sont des collisions pp, pA, AA caractérisées par un paramètre d'impact, c'est-à-dire la distance entre les deux noyaux dans le plan transverse de la collision, plus grand que la somme des rayons des deux noyaux qui entrent en collisions. L'interaction est principalement électromagnétique.

La dernière section porte sur les processus diffractives plus précisément sur DIS diffractive et la production cohérente (ou exclusive) ou incohérente (ou dissociative) d'un méson vecteur.

Le chapitre 2 est une revue non-exhaustive de la recherche de la saturation des gluons dans les collisioneurs passés, présents et futurs . Nous montrons que les signaux expérimentaux jusqu'à présent obtenus à HERA,LHC et RHIC ne sont pas concluantes et que potentiellemnt des signaux beaucoup plus clairs peuvent espérer être observés à EIC (et LHeC) du fait qu'en plus de faisceau de protons, des faisceaus d'ions lours seront utilisés et donc il y aura plus d'effets de saturation (l'echelle de saturation étant proportional à A 1{3 ). La saturation des gluons est donc un sujet d'actualité. Une découverte sans ambigïté de la saturation gluonique nécessite l'identification et le calcul précis au-delà de l'ordre dominant d'observables sensible à la physique de la saturation. Le reste de la thèse est une contribution aux récents efforts menés par la communauté des hautes énergies afin d'atteindre ce but.

Pour cela, on introduit deux théories des champs effectives semi-classiques, le formalisme des ondes de chocs et celui du condensat de verre de couleur (CGC) aux chapitres 3 et 5. Dans ces deux formalism, le référentiel est celui d'un référentiel au moment infini de la cible i.e. la cible voyage dans la direction ´avec un large moment p t et le projectile voyage dans la direction àvec un large moment p p . Dans le premier formalisme des ondes de chocs, les gluons son séparés en deux types de degrés de liberté, les gluons rapides avec un moment longitudinal p `ą e η p p et les gluons lents avec p `ă e η p p où η ă 0 est une coupure en rapidité arbitraire. Les gluons rapides correspondent à des corrections quantiques dans le facteur d'impact du projectile alors que les gluon lents sont traités comme des champs externes classiques. Ces champs lents ont la forme suivante b µ pzq " δpz `qBp⃗ z qn µ 2 . L'interaction avec ces champs est donc instantanée (ils sont localisés exactement à x `" 0 formant une onde de choc) et eikonale. Les règles de Feynman effectives sont explicités et nous montrons comment les interactions à tous les ordres avec les champ externes sont resommés dans les lignes de Wilson définies comme V p⃗ x q " P `exp ˆig ż dz `b´a pz `, ⃗ x qt a ˙(8.5.1)

U p⃗ x q " P `exp ˆig ż dz `b´a pz `, ⃗ x qT a ˙. (8.5.2)

Le bloc non-perturbative de ce formalism est constitué d'élements de matrices du type xP 1 |O| P y où P 1 pP q est l'état sortant (entrant) de la cible hadronique et O est un opérateur construit à partir des lignes de Wilson. Un exampe de tel opérateur est le l'opérateur dipolaire défini comme

U ij " 1 ´1 N c Tr " V p⃗ z i qV : p⃗ z j q ‰ . (8.5.3)
Cet opérateur évolue selon l'équation d'évolution BK-JIMWLK dont la dérivation en dimension quelconque est expliquée dans la section 3.5. Le deuxième formalisme discuté est CGC qui est un formalisme équivalent au formalisme des ondes de chocs. Une différence est que la séparation des dégrés de liberté en partons rapides et lents a lieu au niveau de la cible et par rapport à une coupure arbitraire Λ ´" x 0 p t . Les partons rapides sont considés comme des charges de couleurs dont les dynamiques est gelées pendant le temps d'une collision et occupant une extension longitudinale étroite. Ils constituent des sources pour les partons lents via l'équation classique de Yang-Mills rD µ , F µν s " J ν (8.5.4) où le courant de couleur est définie par J µa " δ µ´ρa px `, ⃗ x q . (8.5.5) ρ a px `, ⃗ x q est la densité de charges de couleurs dans la cible définissant la distribution spatial en x `, ⃗ x des charges de couleurs. ´.

A l'aide de règles de Feynman du formalisme des ondes de chocs, nous détaillons brièvement dans le chapitre 4 (en suivant [START_REF] Boussarie | On the one loop γ p˚q Ñ qq impact factor and the exclusive diffractive cross sections for the production of two or three jets[END_REF][START_REF] Boussarie | Perturbative study of selected exclusive QCD processes at high and moderate energies[END_REF]) le calculs de la sections efficace diffractive pour γ p˚q P Ñ q qP 1 à l'ordre sous-dominant et pour γ p˚q P Ñ q qgP 1 à l'ordre dominant. Nous explicitons en particulier dans le facteur d'impact γ p˚q Ñ q q à l'ordre sous-dominant comment les divergences en rapidité s'annulent via l'équation d'évolution dipolaire et comment les divergences ultraviolettes sont traitées.

Avec les théories effectives définies aux chapitre précédents, nous calculons trois nouveaux processus dans le règime non-linéaire de QCD. Aux chapitres 6 et 7, nous considérons la production diffractive d'une pair de hadrons (sujet de l'article [START_REF] Fucilla | NLO computation of diffractive di-hadron production in a saturation framework[END_REF]) ou d'un seul hadron (sujet de [START_REF] Fucilla | Diffractive single hadron production in a saturation framework at the NLO[END_REF]) à large ⃗ p 2 h " Λ 2 QCD dans une collision γ p˚q P . Ces calculs sont réalisés dans une factorisation hybride mélant factorisation colinéare et shockwave formalisme. Ils se basent par ailleurs sur les résultats du chapitre 4 et sont réalisés dans la cinématique la plus générale (mais à l'approximation eikonale). Il serait intéresant de faire une étude phénoménologiques avec cs résultats. Cela peut révéler le besoin d'autres resommations.

Dans le dernier chapitre, nous présentons les résultats pour la section efficace de la production inclusive d'un quarkonium accompagné par un gluon en DIS, se concentrant sur les états S. Ce travail est le sujet de [START_REF] Kang | Direct quarkonium-plus-gluon production in DIS in the Color Glass Condensate[END_REF]. Les calculs sont réalisés à l'aide de la factorisation NRQCD et CGC. Les résultats peuvent être utilisés directement pour étudier la production d'un quarkonium avec un jet ou un hadron et plus particulièrement la corrélation azimutale entre le quarkonium et jet (hadron) qui peut ensuite être comparé aux résultats obtenues avec le formalisme TMD [START_REF] D'alesio | Azimuthal asymmetries in semi-inclusive J{ψ `jet production at an EIC[END_REF]. Nous pouvons aussi intégrer sur l'espace de phase du gluon pour obtenir les corrections réelles à l'ordre sous-dominant pour la production d'un quarkonium dans DIS. On peut aussi se concentrer sur le régime où le gluon à l'état final est mou et dériver ainsi les expressions pour les logarithmes de Sudakov associés à la production directe du quarkonium dans la limite où son impulsion transverse est petite.

En résumé, ce manuscript participe aux récents efforts menés par la communauté théorique des hautes énergies d'améliorer la compréhension des processus sensible à la saturation gluonique afin de matcher la précision des données expérimentaux présentes et futures et de calculer une grande variété de ces processus. We get eq. (3.4.7).

Part IV

Appendices

A.2 Quark line through the shockwave field in momentum space p2πq D e ´ip 2 x 2 e ´ip 2 x 2 e i⃗ p 2 ¨⃗ x 2 e ´i⃗ x 1 ¨⃗ p 21 e ´i⃗ p 1 ¨⃗ x 0 e ip 1 x 0 e ix 1 p 21 e ip The integration over p 2 , p 1 are done using Jordan's lemma and the residue theorem. We remark that to have convergent integrals, we need to have the argument of the complex exponential Impp 2 qx 2 ă 0 and Impp 1 qx 0 ą 0. Because x 2 ą 0 and x 0 ă 0, we then have p 2 ą 0 and p 1 ą 0. Closing the contour in the lower half plane which leads by residue theorem to a factor p´2πiq for each integral over p í , one gets also using eq. ( 3 In what follows, for the ϕ functions, x " x q , x " x q.

B.1.2 ϕ 4

The arguments in the integrals of B.1.1 are ⃗ q 1 " ⃗ p 1 ´ˆx ´z x ˙⃗ p q , ⃗ q 2 " ˆx ´z x ˙px⃗ p q ´x⃗ p q q , ∆ 1 " px ´zq px `zq Q 2 , ∆ 2 " ´x px `zq x px ´zq ⃗ q 2 ´i0 .

Let us write the impact factors in terms of these variables. They read:

(longitudinal NLO) ˆ(longitudinal LO) contribution : pϕ 4 q LL " ´4px ´zqpx `zq z r´xpx ´zqpz `1qI 2 `q2Kk p2x 2 ´p2x ´zqpz `1qqI k 1 s , (B.1.14) (longitudinal NLO) ˆ(transverse LO) contribution : pϕ 4 q j LT " p1 ´2xqp q1 1 j K pϕ 4 q LL ´4px ´zqpx `zqp1 ´2x `zqrp⃗ q ¨⃗ p q1 pϕ 4 q i T L " 2trpx ´x ´zqq i 2K q 1Kk `p´8xx ´6xz `2z 2 `3z `1qq j 1K q 2Kk sI k 1 ´2r4x 2 ´xp3z `5q `pz `1q 2 sq 2Kk I ik `px ´x ´zq p⃗ q 2 ¨⃗ q 1 q I i `I2 rpx ´x ´zqq i 2K `xp2px ´zq 2 ´5x `3z `1qq i 1K s ´xr2px ´zq 2 ´5x `3z `1sI i pϕ 4 q ij T T " " px ´x ´2zqpx ´x ´zqp⃗ q 2 ¨⃗ p q1 1 qq i 1K `pz `1qpp⃗ q 1 ¨⃗ q 2 q p i q1 1 K ´p⃗ q 1 ¨⃗ p q1 1 qq i 2K q ‰ I j 1 `2xrq 2Kk ´px ´zqq 1Kk spp i q1 1 K I jk ´gij K p q1 1 Kl I kl q `2px ´zqrp2x `zqp⃗ q 2 ¨⃗ p q1 1 q ´xp⃗ q 1 ¨⃗ p q1 1 qsI ij `rp1 ´zqpp⃗ q 1 ¨⃗ p q1 1 qq j 2K ´p⃗ q 2 ¨⃗ p q1 1 qq j 1K q ´p1 ´2xqpx ´x `zq p⃗ q 1 ¨⃗ q 2 q p j q1 1 K sI i 1 ´2 " px ´zqpxq j 1K ´p2x `zqq j 2K qp q1 1 Kk `p1 ´2xq `4x 2 ´p3z `5qx `pz `1q 2 ˘q2Kk p q1 1 j K ı I ik ´x px ´xq `2px ´zq 2 ´5x `3z `1˘p j q1 1 K I i 3 `x px `zq pp i q1 1 K I j 3 ´gij K p q1 1 Kk I k 3 q `I2 " g ij K `p1 ´zqp⃗ q 2 ¨⃗ p q1 1 q ´xp1 `x ´zqp⃗ q 1 ¨⃗ p q1 1 q pp1 ´zqq j 2K ´xp1 `x ´zqq j 1K qp q1 1 i K ´px ´xq `px ´x `zqq i 2K ´x `2px ´zq 2 ´5x `3z `1˘q i 1K ˘pq1 1 j K ı `Ik 1 " g ij K `px ´x `zqp⃗ q 1 ¨⃗ p q1 1 qq 2Kk `p1 ´zqp⃗ q 2 ¨⃗ p q1 1 qq 1Kk ´pz `1q p⃗ q 1 ¨⃗ q 2 q p q1 1 Kk qj 1K ppx ´x `zqq 2Kk p i q1 1 K ´pz `1qq i 2K p q1 1 Kk q `qj 2K ppx ´x ´2zqpx ´x ´zqq i 1K p q1 1 Kk `p1 ´zqq 1Kk p q1 1 i K q ´p1 ´2xqpp1 ´2x `zqq i 2K q 1Kk ´p2z 2 `3z ´xp8x `6zq `1qq i 1K q 2Kk qp q1 pϕ 5 q ij T T " ´2px ´zq " z x p⃗ q 1 ¨⃗ p q1 1 q ´p2x `zqp⃗ q 2 ¨⃗ p q1 1 q ı I ij

`"´xpx ´zq 2 Q 2 p i q1 1 K `px ´x `2zqpx ´x `zqp⃗ q 2 ¨⃗ p q1 1 qq i 1K ´p⃗ q 1 ¨⃗ p q1 1 qppz `1qq i 2K ´2 z x p2x ´zqq i 1K q `ppz `1q p⃗ q 1 ¨⃗ q 2 q ´´x `z x ¯⃗ r 2 qp i

q1 1 K ı I j 1 ´2
x x pxq 2Kk `px ´zqq 1Kk q ´gij K p q1 1 Kl I kl ´pq1 1 i K I jk "x px ´xq px ´zq 2 Q 2 p j q1 1 K ´pz ´1qp⃗ q 1 ¨⃗ p q1 1 qq j

2K

`pz ´1qp⃗ q 2 ¨⃗ p q1 1 qq j 1K `x ´x x `px 2 ´zq⃗ q 2 1 `xpx ´x `zqp⃗ q 1 ¨⃗ q 2 q ˘pk `p1 ´zqpg ij K p⃗ q 2 ¨⃗ p q1 1 q `qk 2K p i q1 1 K q `p2x `z ´3qpg ik K p⃗ q 1 ¨⃗ p q1 1 q `qk 1K p i q1 1 K q

q1 1 K ȷ I i 1 `2 " x ´x x `xp4x
ı I 2 `´3x `z ´z x ¯pi q1 1 K I k 3 ´x x p3x ´zqg ij K p q1 1 Kk I k 3 `"px ´xqp j q1 1 K ␣ px ´x `zqq i 2K q 1Kk ´p2z 2 ´6xz `3z ´8xx `1qq 2Kk q i 1K ´2px ´x `2z ´z2 x qq 1Kk q i 1K * `xpx ´zq 2 Q 2 g ij K p q1 1 Kk
`p1 ´zqq 1Kk pg ij K p⃗ q 2 ¨⃗ p q1 1 q `qj 2K p i q1 1 K q `ppx ´x `zqq 2Kk ´2q 1Kk q pg ij K p⃗ q 1 ¨⃗ p q1 1 q `qj 1K p i q1 1 K q `gij K ´´x `z x ¯⃗ q 2 1 ´pz `1qp⃗ q 1 ¨⃗ q 2 q ¯pq1 1 Kk `´px ´x ´2zqpx ´x ´zqq i 1K q j 2K ´pz `1qq i 2K q j 1K `2p2x ´zq z x q i 1K q j 1K ¯pq1 1 Kk ı I k 1

283

`2xx z " px ´xq 2 p j q1 1 K pq 2Kk I ik `Ii 3 q ´pi q1 1 K pq 2Kk I jk `Ik 3 q `gij K p q1 1 Kk pq 2Kl I kl `Ik 3 q `pI 2 `q2Kk I k 1 q ´gij K p⃗ q 1 ¨⃗ p q1 1 q `qj 1K p i q1 1 K ´p1 ´2xq 2 q i 1K p j q1

284

ˆ» ---

x q `dx 2 g `4x q px q `xg q ˘px q ⃗ p g3 ´xg ⃗ p q1 q ¨`x q ⃗ p g3 1 ´xg ⃗ p q1 1 xq px q `xg q 2 ´p⃗ p g3 `⃗ p q1 q 2 xqpxq`xgq `Q2 ¯ˆp⃗ p g3 `⃗ p q1 q 2 xq `⃗ p 2 g3 xg `⃗ p 2 q1 xq `Q2 4x q x q `2x g ´dx 2 g ˘px q ⃗ p g3 ´xg ⃗ p q2 q ¨`x q ⃗ p g3 1 ´xg ⃗ p q1 1 px q `xg q px q `xg q ´p⃗ p q2 `⃗ p g3 q 2

xqpxq`xgq ¯P i K ¯pdx g `4x q ´4q ´´⃗ G ¨⃗ P ¯pi q1K p2x q ´1q `4 px q ´1q x q ´dx 2 g x2 g x q px g `xq q 3 ˆQ2 `⃗ p ¯P i K ¯pdx g `4x q ´2q ´´⃗ H ¨⃗ P ¯pi q1K p2x q ´1q px g p2 ´dx g q `4x q x qq x 2 g x q px g `xq q px g `xq q 2 ˆQ2 `⃗ p 2

Ḣi

K px g px g d `d ´2q `xq p2 ´4x qqq x q x g px g `xq q 2 px g `xq q ˆQ2 `⃗ p 2 Here, G i K " x qp i g3 1 K ´xg p i q2 1 K , H i K " x q p i g3 1 K ´xg p i q1 1 K , P i K " x qp i g3K ´xg p i q2K .

(B.2.6)

The interference term in the dipole ˆdipole contribution reads ´Φi 4 ´⃗ p 1 , ⃗ p 2 , ⃗ 0 ¯r Φ `: 3 p⃗ p 1 1 , ⃗ p 2 1 q `r Φ i 3 p⃗ p 1 , ⃗ p 2 q Φ `: 4 ´⃗ p 1 1 , ⃗ p 2 1 , ⃗ 0 ¯" 4p `3 γ ¨∆i qK x q x q `dx 2 g `dx g ´2x g `2x q ´4x q x q⃗ ∆ 2 q px g `xq q 2 px g `xq q ˆQ2 `⃗ p 2 pxg`xqqxq ⃗ J ¨⃗ ∆ q ¯pi q2K `dx 2 g `4x q px g `xq q ˘p1 ´2x qq `xg ´´⃗ J ¨⃗ p q2 ¯∆i qK ´´⃗ p q2 ¨⃗ ∆ q ¯Ji K ¯pdx g `4x q ´4q

x g px g `xq q 3 ⃗ ∆ 2 q ˆQ2 `⃗ p ¯Ki K ¯pdx g `4x q ´2q `´⃗ K ¨⃗ ∆ q ¯pi q2K p1 ´2x qq px g pdx g ´2q ´4x q x qq xg px g `xq q 2 x q px g `xq q ⃗ ∆ 2 q ˆQ2 `⃗ p 2 ´xg ´´⃗ p q1 ¨⃗ ∆ q ¯Xi K ´´⃗ X ¨⃗ p q1 ¯∆i qK ¯pdx g `4x q ´2q `´⃗ X ¨⃗ ∆ q ¯pi q1K p1 ´2x q q px g pdx g ´2q ´4x q x qq x g ⃗ ∆ 2 q px g `xq q px g `xq q 2 ˆQ2 `⃗ p where ⃗ ∆ q " x q ⃗ p g ´xg ⃗ p q x q `xg , ⃗ ∆ q " x q ⃗ p g ´xg ⃗ p q x q `xg (B.2.8)

X i K "
x qp i gK ´xg p i q2K "P i K | p 3 "0 , J i K " x q p i gK ´xg p i q1 1 K "

H i K | p 1 3 "0 , K i K " x qp i gK ´xg p i q2 1 K " G i K | p 1 3 "0 . (B.2.9)
The TL transition is obtained from above by complex conjugation and inverting the naming of the different momenta in (B.2.7) and (B.2.5).

The double dipole ˆdipole have, respectively, the form Φ i 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q Φ `: 3 p⃗ p 1 1 , ⃗ p 2 1 q " Φ i 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q Φ `: 4 ´⃗ p 1 1 , ⃗ p 2 1 , ⃗ 0 ¯`Φ i 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q r Φ `: 3 p⃗ p 1 1 , ⃗ p 2 1 q , (B.2.10) Φ 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q Φ i: 3 p⃗ p 1 1 , ⃗ p 2 1 q " Φ 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q Φ i: 4 ´⃗ p 1 1 , ⃗ p 2 1 , ⃗ 0 ¯`Φ 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q r Φ i: 3 p⃗ p 1 1 , ⃗ p 2 1 q , (B.2.11) where Φ i 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q r Φ `: 3 p⃗ p 1 1 , ⃗ p 2 1 q " 4p `3 γ px q `xg q ⃗ ∆ 2 q ˆ⃗ p 2 x q x q∆ i q pdx g px g `1q ´2 p1 ´2x q q px q `xg qq px q `xg q px q `xg q `pdx g `4x q ´2q ´∆i q ⃗ P ¨⃗ p q1 ´P i ⃗ p q1 ¨⃗ ∆ q px q `xg q 2 ˆ⃗ p 2 q1 xqpxq`xgq `Q2 ṗ2x q ´1q p i q1 ⃗ P ¨⃗ ∆ q px g pdx g ´2q ´4x q x qq x g px q `xg q 2 ˆ⃗ p 2 q1 xqpxq`xgq `Q2 ˙´ppd ´4qx g ´4x q q ´W i ⃗ p q2 ¨⃗ ∆ q ´∆i q ⃗ W ¨⃗ p q2 px q `xg q 2 ˆ⃗ p 2 q2 xqpxq`xgq `Q2 ṗ2x q ´1q `dx 2 g `4x q px q `xg q ˘pi q2 ⃗ W ¨⃗ ∆ q

x g px q `xg q 2 ˆ⃗ p x g ⃗ ∆ 2 q px q `xg q 2 ˆQ2 `⃗ p 2 x q x g ppd ´4qx g ´4x q `2q ´P i ´⃗ p q2 1 ¨⃗ ∆ q ¯´∆ i q ´⃗ P ¨⃗ p q2 1 ¯x q px q `xg q ˆ⃗ p 2 q1 xqpxq`xgq `Q2 ẋq px q ´xq `xg q p i q2 1 ´⃗ P ¨⃗ ∆ q ¯px g pdx g ´2q ´4x q x qq x q px q `xg q ˆ⃗ p 2 q1 xqpxq`xgq `Q2 ṗx q ´xq `xg q `dx 2 g `4x q px q `xg q ˘pi q2 1 ´⃗ W ¨⃗ ∆ q px q `xg q

ˆ⃗ p 2 q2 xqpxq`xgq `Q2
ẋg ppd ´4qx g ´4x q q ´∆i q ´⃗ W ¨⃗ p q2 xq » ---´gik K x q x q px g d `d ´2 `2x qq px g `xq q 2 px g `xq q ´2P k K p i q1K p1 ´2x q q

x g px g `xq q 2 ˆQ2 `⃗ p xqpxg`xqq ˙ˆpd ´4qx g ´2x q x g `xq `pd ´2qx g ´2x q x g `xq 1

x 2 g x q px g `xq q 2 x q px g `xq q 2 ˆQ2 `⃗ p 2 ˙!´⃗ H ¨⃗ P ¯"p i q1K p k q2 1 K p1 ´2x q q ˆp1 ´2x qq px g p2 ´dx g q `4x q x qq `´g ik K `⃗ p q1 ¨⃗ p q2 1 ˘`p k q1K p i q2 1 K ¯px g p2 ´pd ´4qx g q `4x q x qq ı `ppd ´4q x g ´2q "

x g ´⃗ H ¨⃗ p q2 ´´⃗ P ¨⃗ p q2 1 ¯Hk K ¯pi q1K p1 ´2x q q pdx g `4x q ´2q ) ´1 x 2 g x q x q px g `xq q 4 ˆQ2 `⃗ p `P k K p i q1K p1 ´2x q q ¯´2 ´⃗ G ¨⃗ p q1 ¯´P k K p i q1 1 K `P i K p k q1 1 K p1 ´2x q q ¯ı `´⃗ G ¨⃗ P ¯"p k q1K p i q1 1 K ´pi q1K p k q1 1 K p1 ´2x q q 2 `gik K `⃗ p q1 ¨⃗ p q1 1 ˘ı `dx 2 g `4x q px g `xq q ˘)ı « ppd ´2q x g ´2x q q x q px g `xq q 3 ´gik K ´⃗ p q2 1 ¨⃗ ∆ q ¯`p i q2 1 K ∆ k qK `pk q2 1 K ∆ i qK p1 ´2x qq xq ´ppd ´4q x g ´2x q q ´gik K ´⃗ p q2 1 ¨⃗ ∆ q ¯`p i q2 1 K ∆ k qK ¯`p k q2 1 K ∆ i qK pdx g `2x q q p1 ´2x qq px g `xq q 2 px g `xq q ´1 x g px g `xq q 2 x q px g `xq q 2 ˆQ2 `⃗ p 2 q1 xqpxg`xqq ˙tx g ppd ´4qx g `2q

`p1 Ø 1 1 , 2 Ø 2 1 , 3 Ø 3
ˆ"p i q1K ´´⃗ p q2 1 ¨⃗ ∆ q ¯Xk K ´´⃗ X ¨⃗ p q2 1 ¯∆k qK ¯p2x q ´1q ´´⃗ X ¨⃗ p q2 1 ¯´g ik K p⃗ p q1 ¨⃗ ∆ q q `pk q1K ∆ i qK Xk K ´´⃗ p q1 ¨⃗ ∆ q ¯pi q2 1 K ´`⃗ p q1 ¨⃗ p q2 1 ˘∆i qK ¯ı `4x q x g p1 ´2x q q p i q1K ´´⃗ p q2 1 ¨⃗ ∆ q ¯Xk K ´´⃗ X ¨⃗ p q2 1 ¯∆k qK xg p1 ´2x qq pdx g `4x q ´2q p k q2 1 K ´´⃗ p q1 ¨⃗ ∆ q ¯Xi K ´´⃗ X ¨⃗ p q1 ¯∆i qK ¯´x g ppd ´4qx g ´2q ˆ"´g ik K ´⃗ X ¨⃗ p q1 ¯`X i K p k q1K ¯´⃗ p q2 1 ¨⃗ ∆ q ¯`´´⃗ X ¨⃗ p q1 ¯pi q2 1 K ´p⃗ p q1 ¨⃗ p q2 1 qX i K ¯∆k qK ı `´⃗ X ¨⃗ ∆ q ¯pi q1K p k q2 1 K p1 ´2x q q p1 ´2x qq px g pdx g ´2q ´4x q x qq ´´⃗ X ¨⃗ ∆ q ¯´g ik K `⃗ p q1 ¨⃗ p q2 1 ˘`p k q1K p i q2 1 K ¯px g p2 ´pd ´4q x g q `4x q x qq ) ´1 x g px g `xq q 4 ˆQ2 `⃗ p 2 q2 pxg`xqqxq ˙xq tx g pdx g `4x q ´4q rp1 ´2x qq « ppd ´2q x g ´2x q q x q px g `xq q 3 ´gik K ´⃗ p q2 1 ¨⃗ ∆ q ¯`p i q2 1 K ∆ k qK `pk q2 1 K ∆ i qK p1 ´2x qq xq ´ppd ´4q x g ´2x q q ´gik K ´⃗ p q2 1 ¨⃗ ∆ q ¯`p i q2 1 K ∆ k qK ¯`p k q2 1 K ∆ i qK pdx g `2x q q p1 ´2x qq px g `xq q 2 px g `xq q ´1 x g px g `xq q 2 x q px g `xq q 2 ˆQ2 `⃗ p 2 q1 xqpxg`xqq ˙tx g ppd ´4qx g `2q

ˆ´p k q2 1 K ´´⃗ p q2 ¨⃗ ∆ q ¯V i K ´´⃗ V ¨⃗ p q2 ¯∆i qK ¯`p i q2K ´´⃗ V ¨⃗ p q2 1 ¯∆k qK ´´⃗ p q2 1 ¨⃗ ∆ q ¯V k K ¯V k K ´´⃗ p q2 ¨⃗ ∆ q ¯pi q2 1 K ´`⃗ p q2 ¨⃗ p q2 1 ˘∆i qK ¯`´`⃗ p q2 ¨⃗ p q2 1 ˘V i K ´´⃗ V ¨⃗ p q2 ¯pi q2 1 K ¯∆k qK `gik K ´´⃗ V ¨⃗ p q2 1 ¯´⃗ p q2 ¨⃗ ∆ q ¯´´⃗ V ¨⃗ p q2 ¯´⃗ p q2 1 ¨⃗ ∆ q ¯¯`p k q2K ´´⃗ V ¨⃗ p q2 1 ¯∆i qK ´´⃗ p q2 1 ¨⃗ ∆ q ¯V i K ¯ı `´⃗ V ¨⃗ ∆ q ¯´p i q2K p k q2 1 K p1 ´2x qq 2 ´gik K `⃗ p q2 ¨⃗ p q2 1 ˘´p k q2K p i q2 1 K ¯`dx 2 g ´4x q px q ´1q ˘)ı `p1 Ø 1 1 , 2 Ø 2
ˆ"p i q1K ´´⃗ p q2 1 ¨⃗ ∆ q ¯P k K ´´⃗ P ¨⃗ p q2 1 ¯∆k qK ¯p2x q ´1q ´´⃗ P ¨⃗ p q2 1 ¯´g ik K ´⃗ p q1 ¨⃗ ∆ q ¯`p k q1K ∆ i qK P k K ´´⃗ p q1 ¨⃗ ∆ q ¯pi q2 1 K `⃗ p q1 ¨⃗ p q2 1 ˘∆i qK ¯ı `4x q x g p1 ´2x q q p i q1K ´´⃗ p q2 ¯pi q1K p k q2 1 K p1 ´2x q q p1 ´2x qq px g pdx g ´2q ´4x q x qq ´´⃗ P ¨⃗ ∆ q ¯´g ik K `⃗ p q1 ¨⃗ p q2 1 ˘`p k q1K p i q2 1 K ¯px g p2 ´pd ´4qx g q `4x q x qq ) . (B.2.19)

As above, the dipole ˆdouble dipole contribution is obtained by complex conjugation and changing the momenta.

291

Trpγ α γ β γ µ γ ν γ ρ γ σ q " 4g αβ pg µν g ρσ ´gµρ g νσ `gµσ g νρ q ´4g αµ pg βν g ρσ ´gβρ g νσ `gβσ g νρ q `4g αν pg βµ g ρσ ´gβρ g µσ `gβσ g µρ q ´4g αρ pg βµ g νσ ´gβν g µσ `gβσ g µν q `4g ασ pg βµ g νρ ´gβν g µρ `gβρ g µν q (C.3.4)

Tr `γ5 γ µ γ ν ˘" 0 (C.3.5)

Tr ´γ5 γ µ γ ν γ α γ β ¯" ´4iϵ µναβ (C.3.6)

Tr ´γ5 γ α ω ij γ β ¯" 1 2

"

Tr `γ5 γ α γ i γ j γ `˘´Tr `γ5 γ α γ j γ i γ `˘‰ " 1 2 " ´4iϵ αijβ ´p´4iϵ αjiβ q ı " ´4iϵ αijβ (C.3.7)

Trpγ α γ β ω ij q " 1 2

" Trpγ α γ β γ i γ j q ´Trpγ α γ β γ j γ i q ı " 1 2 " 4pg αβ g ij ´gαi g βj `gαj g βi q ´4pg αβ g ij ´gαj g βi `gαi g βj q ı " 4pg αj g βi ´gαi g βj q (C.3.8)

Hence, for the last trace, if either α or β is `or ´, the trace is equal to 0.

C.4 Spin projector

The spin projector is defined as For the Π 00 , we have, using eq. (8.1.6), Π 00 pp, kq "

Π SSz pp,
1 b 8m 3 Q ˆ{ p 2 ´{ k ´mQ ˙γ5 ˆ{ p 2 `{ k `mQ " 1 b 8m 3 Q γ 5 ˆ´{ p 2 `{ k ´mQ ˙ˆ{ p 2 `{ k `mQ " 1 b 8m 3 Q γ 5 ˆ´p 2 4 ´{ p 2 { k ´{ p 2 m Q `{ k { p 2 `k2 `mQ { k ´mQ { p 2 ´mQ { k ´m2 Q " 1 b 8m 3 Q γ 5 ˜´4pm 2 Q ´k2 q 4 ´{ p 2 { k `{ k { p 2 ´{ pm Q `k2 ´m2 Q " 1 b 8m 3 Q γ 5 ˆ´2m 2 Q `2k 2 ´1 2 " { p, { k ‰ ´mQ{ p ˙. (C.4.4)
For the Π 1Sz , we have

Π 1Sz pp, kq " 1 b 8m 3 Q ˆ{ p 2 ´{ k ´mQ ˙{ ε ˚pS z q ˆ{ p 2 `{ k `mQ " 1 b 8m 3 Q "ˆ{ p 2 ´{ k ˙{ ε ˚pS z q ´mQ { ε ˚pS z q ȷ ˆ{ p 2 `{ k `mQ " 1 b 8m 3 Q " 2 ´p 2 ´k¯¨ε ˚pS z q ´{ ε ˚pS z q ˆ{ p 2 ´{ k ˙´m Q { ε ˚pS z q ȷ ˆ{ p 2 `{ k `mQ " ´ερ pS z q b 8m 3 Q 2k ρ ˆ{ p 2 `{ k `mQ ˙`ε ρ pS z q b 8m 3 Q γ ρ ˆ´{ p 2 `{ k ´mQ ˙ˆ{ p 2 `{ k `mQ " ´ερ pS z q b 8m 3 Q k ρ `{ p `2{ k `2m Q ˘`ε ρ pS z q b 8m 3 Q γ ρ ˆ´2m 2 Q `2k 2 ´1 2 " { p, { k ‰ ´mQ{ p ˙.
(C.4.5)

Setting k " 0, we obtain eqs. (8.1.14, 8.1.15).

C.5 Dirac structure manipulation for gluon emission

We use the Ward identity l 2 ¨{ ε ˚pl 2 , λq, Eq. (8.2.3), the properties of the Gamma matrices and the Dirac equations ūs pp q qp { p q ´mQ q " 0 and p { p q `mQ qv spp qq in the below manipulations.

Emission of a gluon by the quark before the shockwave γ `"p { l 1 ´{ l 2 q `mQ ‰ { ε ˚pl 2 , λqp { l 1 `mQ q " γ `"2pl 1 ´l2 q ¨ε˚p l 2 , λq ´{ ε ˚pl 2 , λqp { l 1 ´{ l 2 q `mQ { ε ˚pl 2 , λq ‰ p { l 1 `mQ q " γ `«2 ˜l1 ⃗ l 2 ¨⃗ ϵ λl 2 ´⃗ l 1 ¨⃗ ϵ λ˚¸`⃗ ϵ λn γ n p { l 1 ´{ l 2 q ´mQ ⃗ ϵ λn γ n ff p { l 1 `mQ q " γ `"2 l 1 l 2 ˆ⃗ l 2 ´l2 l 1 ⃗ l 1 ˙¨⃗ ϵ λ˚`⃗ ϵ λn γ n ˆ⃗ l 2 ´l2 l 1 ⃗ l 1 ˙m γ m `⃗ ϵ λn γ n ˆ1 ´l2 l 1 ˙p{ l 1 ´mQ `mQ q ´mQ ⃗ ϵ λn γ n ı p { l 1 `mQ q " γ `"2 l 1 l 2 ˆ⃗ l 2 ´l2 l 1 ⃗ l 1 ˙m ⃗ ϵ λn " δ nm `l2 2l 1 γ n γ m ȷ `⃗ ϵ λn γ n ˆ1 ´l2 l 1 ˙p{ l 1 ´mQ q ´l2 l 1 m Q ⃗ ϵ λn γ n ȷ p { l 1 `mQ q (C.5.1)

Emission of a gluon by the quark after the shockwave ūs pp q q{ ε ˚pp g , λqp { p q `{ p g `mQ qγ " ūs pp q q " 2ε ˚pp g , λq ¨pp q `pg q ´p{ p q `{ p g q{ ε ˚pp g , λq `mQ { ε ˚pp g , λq ı γ " ūs pp q q « 2 ˜⃗ p g ¨⃗ ϵ λp g p q ´⃗ ϵ λ˚¨⃗ p q ¸´p { p q ´mQ q{ ε ˚pp g , λq ´{ p g { ε ˚pp g , λq ff γ " ūs pp q q " 2 x p⃗ p g ´x⃗ p q q ¨⃗ ϵ λ˚`´{ p g ´x{ p q `x{ p q ¯⃗ ϵ λn γ n ȷ γ " ūs pp q q " 2 x p⃗ p g ´x⃗ p q q ¨⃗ ϵ λ˚´p ⃗ p g ´x⃗ p q q m γ m ⃗ ϵ λn γ n `xm Q ⃗ ϵ λn γ n ȷ γ " ūs pp q q⃗ ϵ λn " 2 x p⃗ p g ´x⃗ p q q m ´δmn ´x 2 γ m γ n ¯`xm Q γ n ȷ γ " ūs pp q q⃗ ϵ λn " 2 x p⃗ p g ´x⃗ p q q m " δ mn ´1 `x 2

¯`x 2 ω nm ı `xm Q γ n * γ `(C.5.2)
We have defined x " p g {p q .

Emission of a gluon by the antiquark before the shockwave Emission of a gluon by the antiquark after the shockwave γ `p´{ p g ´{ p q `mQ q{ ε ˚pp g , λqv spp qq " γ `"2p´p g ´pq q ¨ε˚p p g , λq ´{ ε ˚pp g , λqp´{ p g ´{ p qq `mQ { ε ˚pp g , λq ı v spp qq " γ `"´2p q ¨ε˚p p g , λq `{ ε ˚pp g , λq { p g `{ ε ˚pp g , λqp { p q `mQ q ı v spp qq for Repβq ą 0.

Ip⃗ r 1 , ⃗ r 2 q " ż 8 where

0 ds ż d 2 ⃗ l 1 d 2 ⃗ l 2 p2πq 2 e i ⃗ l 1 ¨⃗ r 1 e i ⃗ l 2 ¨⃗ r 2 e ´s" Q 2 `⃗ l 2 2 ´1 z 2 `1 z 3 ¯`⃗ l 2 1 ´1 z 1 `1 z 3 ¯`m 2 Q ´1 z 1 `1 z 3 ¯`⃗ l 2 ¨2 z 3 ⃗ l 1 ı " ż 8
Q2 " Q 2 `m2 Q 1 ´z2 z 1 z 3 z 2 z 3 1 ´z1 `ˆ⃗ r 1 ´z2 1 ´z1 ⃗ r 2 ˙2 z 1 p1 ´z1 q " ⃗ r 2 2 z 2 z 3 1 ´z1 `z2 2 z 1 1 ´z1
⃗ r 2 2 `z1 pz 2 `z3 q⃗ r 2 1 ´2z 2 z 1 ⃗ r 1 ¨⃗ r 2 " z 1 z 2 p⃗ r 1 ´⃗ r 2 q 2 `z1 z 3 ⃗ r 2 1 `⃗ r 2 2 ˆz2 z 3 1 ´z1 `z2 2 z 3 1 ´z1 ´z1 z 2 ˙⃗ r 2 2 " z 1 z 2 p⃗ r 1 ´⃗ r 2 q 2 `z1 z 3 ⃗ r 2 1 `⃗ r 2 2 ˆz2 z 3 `z2 2 z 1 ´z2 z 1 pz 2 `z3 q 1 ´z1 ˙⃗ r 2 2 " z 1 z 2 p⃗ r 1 ´⃗ r 2 q 2 `z1 z 3 ⃗ r 2 1 `z2 z 3 ⃗ r 2 2 (C.7.9) (C.7.12)

I 0 p⃗ r 1 , ⃗ r 2 q " ż d 2 ⃗ l 1 d 2 ⃗ l 2 p2πq 2 e i ⃗ l 1 ¨⃗ r 1 e i ⃗ l 2 ¨⃗ r 2 " Q 2 z 1 p1 ´z1 q `⃗ l 2 1 `m2 Q ı " Q 2 `⃗ l 2 1 `m2 Q z 1 `⃗ l 2 2 z 2 `p⃗ l 1 `⃗ l 2 q 2 `m2 Q z 3 ȷ " 1 z 1 p1 ´z1 q ż 8 0 ds ż 8 0 dt ż d 2 ⃗ l 1 d 2 ⃗ l 2 p2πq 2 e ´s« Q 2 `⃗ l 2 1 `m2 Q z 1 p1´z 1 q ff e ´t« Q 2 `⃗ l 2 1 `m2 Q z 1 `⃗ l 2 2 z 2 `p⃗ l 1 `⃗ l 2 q 2 `m2
Finally, the last integral has the following expression Ĩ1j p⃗ r 1 , ⃗ r 2 q " z 2 z 3 4p1 ´z1 q

I 0 p⃗ r 1 , ⃗ r 2 q " ż d 2 ⃗ l 1 d 2 ⃗ l 2 p2πq 2 e i ⃗ l 1 ¨⃗ r 1 e i ⃗ l 2 ¨⃗ r 2 " Q 2 z 1 p1 ´z1 q `⃗ l 2 1 `m2 Q ı " Q 2 `⃗ l 2 1 `m2 Q z 1 `⃗ l 2 2 z 2 `p⃗ l 1 `⃗ l 2 q 2 `m2 Q z 3 ȷ " 1 z 1 p1 ´z1 q ż 1 0 dx ż d 2 ⃗ l 1 d 2 ⃗ l 2 p2πq 2 e i ⃗ l 1 ¨⃗ r 1 e i ⃗ l 2 ¨⃗ r 2 D 2 with D " xQ 2 `p1 ´xqQ 2 `x ˜⃗ l 2 1 `m2 Q z 1 `⃗ l 2 2 z 2 `p⃗ l 1 `⃗ l 2 q 2 `m2 Q z 3 ¸`p1 ´xq ⃗ l 2 1 `m2 Q z 1 p1 ´z1 q " Q 2 `m2 Q xp1 ´z2
qp1 ´z1 q `p1 ´xqz 3 z 1 z 3 p1 ´z1 q `⃗ l 2 1 xp1 ´z2 qp1 ´z1 q `p1 ´xqz 3 z 1 z 3 p1 ´z1 q `⃗ l 2 2 xp1 ´z1 q

z 2 z 3 `⃗ l 2 ¨2x z 3 ⃗ l 1 " Q 2 `m2 Q xz 1 z 2 `z3 z 1 z 3 p1 ´z1 q `⃗ l 2 1 xz 1 z 2 `z3 z 1 z 3 p1 ´z1 q `⃗ l 2 2 xp1 ´z1 q z 2 z 3 `⃗ l 2 ¨2x z 3 ⃗ l 1 .
Using the Schwinger parametrization again, we have I 0 p⃗ r 1 , ⃗ r 2 q " 1 z 1 p1 ´z1 q where I define

Q 2 F pxq " Q 2 `m2 Q xz 1 z 2 `

Figure 1 . 2 . 2 :

 122 Figure 1.2.2: The "phase-diagram" of QCD in DIS showing different regions of applicability of evolution equations. Each colored circles represents a parton with transverse area δS K " 1{Q 2 and Feynman-x of the order of x Bj .

Figure 1 . 2 . 3 :

 123 Figure 1.2.3: e+p deep inelastic scattering in the naive parton model, where the virtual photon couples to a single collinear parton (quark).

Figure 1 . 2 . 5 :

 125 Figure 1.2.5: e `p deep inelastic scattering in the QCD-imporved parton model. The multiple emissions of partons represented by the wiggly lines can involve either gluons and (anti-) quarks. They are resummed by DGLAP equations.

Figure 1 . 2 . 7 :

 127 Figure 1.2.7: Sommerfeld-Watson transformation

Figure 1 . 2 . 8 :

 128 Figure 1.2.8: A Regge exchange diagram.

Figure 1 .

 1 Figure 1.2.10: BFKL effective gluon ladder in a qq Ñ qq scattering.

Figure 1

 1 Figure 1.2.11: Lipatov vertex

Figure 1 . 2 . 14 :

 1214 Figure 1.2.14: Example of fan diagram. The dark circles are Lipatov's effective vertices present in BFKL ladders. Figure from [58].

Figure 1 . 2 . 16 :

 1216 Figure 1.2.16: Feynman diagram for the γ ˚p Ñ γ ˚p scattering in the dipole model. The gray blob depicts the non-perturbative interaction with the target.

Figure 1 . 3 . 1 :

 131 Figure 1.3.1: Schematic view of an ultra-peripheral collision of relativisitc nulcei. Figure from [70].

Figure 1 . 3 . 2 :

 132 Figure 1.3.2: The different possible scenarios for UPC. (a) is photon-photon reaction. (b) is the photon-nucleus reaction without breaking of the nucleus emitter. (c) photonuclear reaction with additional photon exchange leading to breaking of the emitter nucleus. Figure take from [69].

Figure 1 . 4 . 1 :

 141 Figure 1.4.1: Diffractive deep inelastic scattering (DDIS)

Figure 1 . 4 . 2 :

 142 Figure 1.4.2: DDIS mediated by a Pomeron and kinematics.

Figure 1 . 4 . 3 :

 143 Figure 1.4.3: Resolved Pomeron model for inclusive DDIS

Figure 2 . 1 . 2 :

 212 Figure 2.1.2: Contribution to the structure functions F 2 and F L at x " 0.01 (in the IP-Sat model) from dipole sizes smaller than r max for different values of Q 2 . Figure from [121].

Figure 2 . 2 . 1 :

 221 Figure 2.2.1: Relative difference between F L obtained in the saturation framework and the leading twist formalism for proton (left) and Gold (right) . Figure from [35].

Figure 2 . 2 . 2 :

 222 Figure 2.2.2: Predictions for the proton F L at EIC kinematics plotted at constant Bjorken x.Predictions are computed from the three fits of table 2.1. The inclusive as well as the charm and bottom contribution are shown. Figure from[START_REF] Hänninen | Proton Structure Functions at Next-to-Leading Order in the Dipole Picture with Massive Quarks[END_REF].

Figure 2 . 2 . 3 :

 223 Figure 2.2.3: Comparison of F Au 2 pseudodata (black points) obtained from rcBK solutions to the values obtained in the collinear factorization framework with the extrapolated EPS09 and DSSZ nuclear PDFs. Figure from [168].

Figure 2 .

 2 Figure 2.2.5: Left: Di-hadron correlation function predicted in the CGC framework in eA DIS collisions for different nuclei. A depletion of the back-to-back peak is observed. Right: A comparison of the correlation function with and without saturation. The gray band reflects uncertainties in the CGC predictions due to uncertainties on the saturation scale. Figure from [35].

Figure 3 . 4 . 1 :

 341 Figure 3.4.1: Quark propagator through the shockwave

Figure 3 . 4 . 2 :

 342 Figure 3.4.2: Antiquark propagator through the shockwave

Figure 3 . 4 . 3 :

 343 Figure 3.4.3: Gluon propagator through the shockwave

Figure 3 . 4 . 4 :

 344 Figure 3.4.4: External quark line through the shockwave

Figure 3 . 4 . 5 :

 345 Figure 3.4.5: External antiquark through the shockwave

Figure 3 . 4 . 6 :

 346 Figure 3.4.6: External gluon through the shockwave

Figure 3 . 5 . 1 :

 351 Figure 3.5.1: "Real" contributions to the evolution

Figure 3 .

 3 Figure 3.5.2: "Virtual" contributions to the evolution

8 Figure 4 . 3 . 1 :

 8431 Figure 4.3.1: NLO diagrams for γ p˚q Ñ q q.

Figure 4 . 4 . 1 :

 441 Figure 4.4.1: Diagrams for the γ p˚q Ñ q qg amplitude.

Figure 5 .

 5 Figure 5.1.1: CGC vertex for quarks.

Figure 5 .

 5 Figure 5.1.2: CGC vertex for antiquarks.

Figure 5 . 1 . 3 :

 513 Figure 5.1.3: CGC vertex for gluons.

  (3.3.36, 3.3.37) with the Fierz identity eq.(3.5.23) gives back at the end eq. (3.5.26) (with the average of the dipole and double dipole operators).131Part IIIDiffractive di-hadron and single hadron production in γ p˚q `p{A collisions and inclusive quarkonia `g production in DISChapter 6

Figure 6 . 1 . 2 :

 612 Figure 6.1.2: Graphical convention for the fragmentation function of a parton (here a quark for illustration) to a hadron h plus spectators. In the rest of this chapter, we will use the left-hand side of this drawing.

Figure 6 . 1 . 3 :

 613 Figure 6.1.3: Diagram of the LO process at cross-section level. The blob is the shockwave (we do not draw the coupling with the target for clarity) and the squares the FFs, see fig. 6.1.2. The dashed line is to represent the integration over phase space.

Figure 6 . 1 . 4 :

 614 Figure 6.1.4: Illustration of the 5 kinds of contributions to the NLO cross-section from the dipole point of view. Arrows show to which combination of dipole structures each type of diagrams contributes.

Figure 6 . 1 . 5 :

 615 Figure 6.1.5: The 5 kinds of contributions to the NLO cross-section.

Figure 6 . 1 . 6 :Figure 6 . 1 . 7 :

 616617 Figure 6.1.6: NLO cross-section in the case of fragmentation from the quark and the antiquark.We explicitly isolate the diagrams which contain divergences. Diagram (1) contains a collinear divergence between the fragmenting quark and the gluon as well as a soft gluon divergence. Diagram (2) contains a soft gluon divergence. Diagram (3) contains a collinear divergence between the fragmenting antiquark and the gluon as well as a soft gluon divergence. Diagram (4) contains a soft gluon divergence. By "finite terms", here, we mean all diagrams in which the gluon crosses the shockwave at least once.

Figure 7 . 1 . 1 :

 711 Figure 7.1.1: Left: An example of amplitude of the process (7.1.1) at LO. Right: An example of amplitude contributing to the process (6.1.1) at NLO. The grey blob symbolizes the QCD shockwave. The double line symbolizes the target, which remains intact in the figure, but could just as well break. The quark and the antiquark fragment into the systems phXq (in the specific diagram h is produced by the quark, but can be as well produced by the antiquark). The tagged hadron h is drawn in red.

Figure 7 . 1 . 2 :

 712 Figure 7.1.2: Diagram of the LO process at cross-section level. The blob is the shockwave (we do not draw the coupling with the target for clarity) and the rectangle is the FF, see fig. 6.1.2. The dashed line is to represent the integration over phase space.

1 . 3 )

 13 and also on the different contributions from the Wilson-line operator point of view just like in eqs. (4.5.7, 4.6.1). These different contributions depending on the Wilson-line operator are represented in fig. 7.1.3. In this figure, we exhibit a few examples of diagrams, either virtual or real, as a representative of each 5 classes of diagrams.

Figure 7 . 1 . 3 :

 713 Figure 7.1.3: Illustration of the 5 kinds of contributions to the NLO cross-section (with fragmentation from quark) from the Wilson-line operator point of view. Arrows show to which combination of dipole structures each type of diagrams contributes.

Figure 7 . 1 . 4 :

 714 Figure 7.1.4: Real diagrams with the gluon emitted after the shockwave, in the quark fragmentation case.

Figure 7 . 1 . 5 : 23 )

 71523 Figure 7.1.5: Real diagrams with the gluon emitted after the shockwave, in the gluon fragmentation case.

Figure 7 . 2 . 1 :

 721 Figure 7.2.1: Diagrammatic representation of the quark counterterm.

Figure 7 . 4 . 1 :

 741 Figure

Figure 7 . 4 . 2 :

 742 Figure

Figure 7 . 4 . 3 :

 743 Figure

Finally, we can

  have a contribution coming from the fragmentation of a gluon. As already mentioned, the two divergent contributions are diagrams p1q and p3q of fig.7.1.5. The contribution of the diagram p3q is simple to derive once that of the diagram p1q has been calculated.

Figure 7 . 4 . 4 :

 744 Figure 7.4.4: The IR-divergent contribution associated with the quark Ñ quark `gluon splitting, with the gluon fragmenting into the identified hadronic state.

Figure 7 . 5 . 1 :

 751 Figure

Figure 7 . 5 . 2 :

 752 Figure

.5. 10 )

 10 Double dipole ˆdouble dipole contribution

Figure 8 . 1 . 1 :

 811 Figure 8.1.1: Representative diagram for direct quarkonium + gluon production in electronnucleus collisions within the joint CGC + NRQCD framework. The gray oval represents the interaction of the heavy quark pair and gluon with the gluon shockwave of the nucleus. The red oval represents the hadronization of the heavy quark pair to the quarkonium H with momentum p.

Figure 8 . 1 . 2 :

 812 Figure 8.1.2: The four diagrams for heavy quark pair + gluon production in DIS in the CGC formalism. The black circles represent the interaction of the partons with the background field of the nucleus.

r d 2

 2 ⃗ b d 2 ⃗ z e ´i⃗ p¨⃗ b e ´i⃗ pg¨⃗ z Tr " C a R 1p3q p⃗ r, ⃗ b, ⃗ z qC rcs ı ˆFλ, λ,κ,Jz R 1p3q ´p, p g , Q; ⃗ r, ⃗ b, ⃗ z ¯(8.1.26)

Figure 8 . 2 . 1 :

 821 Figure 8.2.1: Real gluon emission from the quark before the shockwave. l 1 and l 2 are loop momenta.

Figure 8 . 2 . 2 :

 822 Figure 8.2.2: Real emission of the gluon after the shockwave and from the quark

  .2.36) for the transversely polarized photon case. The reduced scattering amplitude, in the notations of eqs. (8.1.4), (8.1.8) and (8.1.25) becomes

Starting from eq. ( 3 . 4 . 6 )d D p 1 p2πq D e ´ip 2

 34612 neglecting the i, j colour indices, we haveGpx 2 , x 0 q| x 2 ą0ąx 0 " ż d D x 1 δpx 1 qV p⃗ x 1 qG 0 px 21 qγ `G0 px 10 qθpx 2 qθp´x 0 q " ż d D x 1 δpx 1 qV p⃗ x 1 q ż d D p 2 p2πq D ¨x21 e ´ip 1 ¨x10 G 0 pp 2 qγ `G0 pp 1 qθpx 2 qθp´x 0 q " ż dx 1 d d ⃗ x 1 V p⃗ x 1 q ż d D p 2 p2πq D d D p 1

3 `xxp1 ´2xq z r2q 2Kk I ik `Ii 3 ´qi 1K p2q 2Kk I k 1 `

 331 I2 qsu , (B.1.16) (transverse NLO) ˆ(transverse LO) contribution :

q2 1 pxg`xqqxq ˙ˆQ 2 `⃗ p 2 q1

 122

`⃗ p 2 q2xq ˙ˆQ 2 `⃗ p 2 q2 1

 221

1 xq ˙ˆQ 2 `⃗ p 2 q2pxg`xqqxq ˙ˆQ 2 `

 1222

q2 1 pxg`xqqxq ˙ˆQ 2 `⃗ p 2 q1

 122

z 1 z 3 ¯ż d 2 ⃗ l 1 2π e i ⃗ l 1 ¨⃗ r 1 e ´⃗ l 2 1 sp1´z 2 q z 1 z 3 ż d 2 ⃗ l 2 2π e ⃗ l 2 ¨´i⃗ r 2 ´2s z 3 ⃗ l 1 ¯e´⃗ l 2 2 sp1´z 1 q z 2 z 3 "z 1 z 3 ¯ż d 2 ⃗ l 1 2π e i ⃗ l 1 ¨⃗ r 1 e ´⃗ l 2 1 sp1´z 2 q z 1 z 3 1 2π ˆπz 2 z 3 sp1 ´z1 q ˙e p i⃗ r 2 ´2s z 3 ⃗ l 1q 2 4sp1´z 1 q z 2 z 3 "⃗ l 1 ¨´i⃗ r 1 ´i⃗ r 2 z 1 z 3 ¯e´⃗ r 2 2 z 2 z 3 4sp1´z 1 q 1 2π ˆπz 1 p1 ´z1 q s ˙e´´⃗ r 1 ´z2 1´z 1 ⃗ r 2 ¯2 z 1 p1´z 1 q 4s " z 1 z 2 z 3 1 `´⃗ r 1 ´z2 1´z 1 ⃗ r 2 ¯2z 1 p1´z 1 q ȷ " z 1 z 2 z 3 4 ż 8 0ds s p´1q´1 e ´s Q2 e ´X2 4s " z 1 z 2 z 3 4 2 ˆ2 Q X ˙K1 p QXq " z 1 z 2 z 3 QK 1

 31233233123232323124s31184s231

Q ´s z 1 p1´z 1 q `p1´z 2 qt z 1 z 3 ż d 2 ⃗ l 1 2π e ´⃗ l 2 1 ´s z 1 p1´z 1 q `p1´z 2 qt z 1 z 3 ¯ei ⃗ l 1 ¨⃗ r 1 ż d 2 ⃗ l 2 2π e ´tp1´z 1 q z 2 z 3 ⃗Q ´s z 1 p1´z 1 q `p1´z 2 qt z 1 z 3 ż d 2 ⃗ l 1 2π e ´⃗ l 2 1 ´s z 1 p1´z 1 q `p1´z 2 qt z 1 z 3 ¯ei ⃗ l 1 ¨⃗ r 1 1 2π ˆπz 2 z 3 p1 ´z1 qt ˙e p i⃗ r 2 ´2t z 3 Q ´s z 1 p1´z 1 q `p1´z 2 qt z 1 z 3 ¯e´⃗ r 2 2 z 2 z 3 4tp1´z 1 q ˆż d 2 ⃗ l 1 2π e ´⃗ l 2 1 ´s`t z 1 p1´z 1 q ¯e⃗ l 1 ¨´i⃗ r 1 ´i z 2 1´z 1 ⃗ r 2 "Q ´s z 1 p1´z 1 q `p1´z 2 qt z 1 z 3 ¯e´⃗ r 2 2 z 2 z 3 e ´ps`tqQ 2 e ´m2 Q ´s z 1 p1´z 1 q `p1´z 2 qt z 1 z 3 ˆe´´⃗ r 1 ´z2 1´z 1 ⃗ r 2 ¯2 z 1 p1´z 1 q 4u ż u 0 dv v 2 e ´m2 Q v z 2 z 3 p1´z 1 q e ´⃗ r 2 2 z 2 z 3 4vp1´z 1

 3213321333211232331202231 q .

p1´z 1 q ˜i z 1 p1 ´z1 q 4u 2 ˆ⃗ r 1 ´z2 1 ´z1 ⃗ r 2 ˙jȩ´´⃗ r 1 ´z2 1´z 1 ⃗ r 2 ¯2 z 1 p1´z 1

 2121

Q xz 1 z 2 `z3 z 1 z 3 p1´z 1 q ¯ż d 2 ⃗ l 1 2π e i ⃗ l 1 ¨⃗ r 1 e ´⃗ l 2 1 s xz 1 z 2 `z3 z 1 z 3 p1´z 1 q ˆż d 2 ⃗ l 2 2π e ⃗ l 2 ¨´i⃗ r 2 ´2xs z 3 ⃗ l 1 ¯e´⃗ l 2 2 sxp1´z 1 q z 2 z 3 "Q xz 1 z 2 `z3 z 1 z 3 p1´z 1 q ¯ż d 2 ⃗ l 1 2π e i ⃗ l 1 ¨⃗ r 1 e ´⃗ l 2 1 s xz 1 z 2 `z3 z 1 z 3 p1´z 1 Q xz 1 z 2 `z3 z 1 z 3 p1´z 1 q ¯e´⃗ r 2 2 z 2 z 3 4sxp1´z 1 q ˆż d 2 ⃗ l 1 2π e ⃗ l 1 ¨´i⃗ r 1 ´i z 2 1´z 1 ⃗ r 2 ¯e´⃗ l 2 1 sQ xz 1 z 2 `z3 z 1 z 3 p1´z 1 q ¯e´⃗ r 2 2 z 2 z 3 4sxp1´z 1 q ˆ1 2π ˆπz 1 p1 ´z1 q s ˙e´´⃗ r 1 ´z2 1´z 1 ⃗ r 2 ¯2 z 1 p1´z 1 q 4s "Q xz 1 z 2 `z3 z 1 z 3 p1´z 1 q ¯e´1 4s " ⃗ r 2 2 z 2 z 3 p1´z 1 qx `´⃗ r 1 ´z2 1´z 1 ⃗ r 2 ¯2z 1 p1´z 1

 13231122112124s4s23111 F pxqX F pxqq (C.7.14)

  Ultra-peripheral collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Diffractive processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gluon saturation at future collider: EIC and LHeC . . . . . . . . . . . . . . . . . Physical picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Boosted gluon field and Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 The boosted gluonic field . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Shockwave Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Quark propagator through the shockwave . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Interaction with two external gluons . . . . . . . . . . . . . . . . . . . . . 3.3.2 Generalization to all possible number of interactions . . . . . . . . . . . .

1 Introduction 1.1 Light-cone coordinates and conventions . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Deep Inelastic scattering and the phase-diagram for QCD evolution . . . . . . . . 1.2.1 Deep inelastic scattering: a definition . . . . . . . . . . . . . . . . . . . . 1.2.2 DIS in the Bjorken limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 DIS in the Regge-Gribov or high-energy limit . . . . . . . . . . . . . . . . 1.3 2 Saturation at past, present and future colliders 2.1 Gluon saturation at past and existing colliders: HERA, LHC, RHIC . . . . . . . 2.1.1 Inclusive DIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Diffractive measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 2.2.1 Inclusive measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Diffractive measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Semi-inclusive measurements . . . . . . . . . . . . . . . . . . . . . . . . . II Effective field theories for the study of saturation 3 The shockwave formalism: an introduction 3.1

Table 2 . 1 :

 21 Table of the three dipole amplitude fitted in

	42	1.86	4.83	1.37	1.25

  .1.1: Fast gluons, represented by the red gluons in fig. 3.1.1, have p `ą e η p p and correspond to quantum corrections to the projectile impact factor. They are the quantum modes.

	p + p n 1
	k + > e η p + p
	k + < e η p + p
	p -t n 2
	Figure 3.1.1: Separation in rapidity
	Wilson lines on the target in and out states is what constitute the target impact factor.
	Such operators evolve according to the B-JIMWLK renormalization group equations.
	' Slow gluons, represented in blue in fig. 3.1.1, have p `ă e η p p . They correspond to external
	fields from the point of view of partons in the probe side and they will be resumed into
	Wilson lines. They are the classical modes. The action of operators built out of these

'

1 ¨⃗ z 21

				ż	dp	2p `´p 2 2 i pp `γ´`p { 2K q 2p `¯e ´⃗ p 2 2 ´iε	´ip 2 z 32
	ˆγ`ż dp	1	p 1	i ´⃗ p 2 1 ´iε 2p `e´ip 1

z 21 G 0 pz 10 qθpz 3 qθpz 2 qθp´z 0 q .

	(3.3.26)

  .2.4) From eqs. (4.2.2, 4.2.3, 4.2.4), the LO impact factor reads

  1 1 ,p 2 1

		,
		/
		/
		/
		/
		/
		.
	iØk	/ /	.
		/
		/
		/
		-
		(4.5.18)

  xq x qQ 2 ln

					´xq e η ¯ln	3 µ 2 ˙`pq Ø qq ˆ⃗ p 2 ¸+
	`Q2	«˜ż	xq	dz	ÿ
		0		n"5,6

rpϕ n q LL s `¸`pq Ø qq ff¸`h .c.| p 1 ,p 2 Øp 1 1 ,p 2 1 + . (

4

.5.20)

  [START_REF] Abramowicz | Combination and QCD Analysis of Charm Production Cross Section Measurements in Deep-Inelastic ep Scattering at HERA[END_REF] dipole (and dipole ˆdouble dipole) terms; the diagrams in which the real gluon crosses exactly twice the shockwave, contributing to dσ 3IJ , made of dipole ˆdipole terms, to dσ 4IJ , made of double dipole ˆdipole (and dipole ˆdouble dipole) terms, and to dσ 5IJ , made of double dipole ˆdouble dipole terms.

	virtual contributions
	dσ 1IJ	dipole ˆdipole
	dσ 2IJ	double dipole ˆdipole
	real contributions
	dσ 3IJ	dipole ˆdipole
	dσ 4IJ	double dipole ˆdipole
	dσ 5IJ	double dipole ˆdouble dipole

  .1.7, the contribution with fragmentation from quark and gluon is considered. Again, we have

	σpcqdiv "	ÿ	|A qg,sing´dipole | 2 div " σpcqdiv,3 .	(6.1.27)
		λq,λg,λq

  1 1 ,p 2 1

		,
		/
		/
		/
		/
		/
		.
	iØk	/ /
		/
		/
		/
		-
	`ph 1 Ø h 2 q .	(6.3.6)

  lnp1 ´xq q ´2 ln α ´2ϵ ln α ln

						¨´⃗ p h x h	´⃗ pq 1´xq µ 2	¯2	‹ '´2ϵ ln 2 α
		`2ϵ lnp1 ´xq q ln	¨´⃗ p h x h	´⃗ pq 1´xq µ 2	¯2	‹ '`2ϵ ln 2 p1 ´xq q	/ -, / .	.	(7.4.26)
	From eqs. (7.4.20), (7.4.23) and (7.4.26), we obtain
	3LL dσ qÑh	ˇˇˇs
	dx h d d ⃗ p h	oft div

Table 8 .

 8 1: Summary of kinematic variables (see fig. 8.1.1).

  .3.12) for the transversely polarized photon. The integrals I 0 , I 1m , I 2mj , Ĩ1j (defined in appendix C.7) are evaluated at z 1 " 1

	2	´xg 2 , z 2 " x g and z 3 " 1 2	´xg 2 .

  To compute the colour correlator in eq. (8.4.5) we recall the colour structure C R i were given in eqs. (8.2.9), (8.2.30), (8.2.38) and (8.2.42) and the colour projector in eq. (8.1.12). To calculate our results we repeatedly use the Fierz identity eq.(3.5.23).The colour correlators will be expressed in terms of the multipole light-like Wilson line correlators:

	8.4.1 Colour Correlators
						.3.30)
	and
	F	λ, λ, 3 S 1 ,Jz rcs R 1p2q	p⃗ r q "	´Fλ, λ, 3 S rcs 1 ,Jz R 3p4q	p´⃗ r q .	(8.3.31)

  Cette densité est fixe dans une configuration spécifique lors d'une collision (les partons rapides étant statiques le temps d'une collision). La probabilité pour une configuration spécifique ρ a est donnée par la functionnelle W Λ ´rρs. Comme ρ varie de collisions en collisions, les observables physiques sont obtenues après une moyenne CGC définie par xOy Λ ´" ż DρW Λ ´rρsOrρs . (8.5.6) Une solution de l'equation de Yang-Mills est explicitée en gauge du cône de lumière A `" 0. Nous donnons aussi les différentes règles de Feynman effective en CGC, introduisons le modèle de McLerran-Venugopalan et discutons de l'équation d'évolution JIMWLK pour W Λ ´, donnant l'évolution de W avec la coupure Λ

  ¨⃗ x 21 ´iε ˙* p `γ´`p { 2K 2p ¨⃗ x 10 ´iε ˙ȷ "u . (A.1.2)To compute the transverse momentum integrals, we have to first complete the square in the arguments of the exponentialGpx 2 , x 0 q| x 2 ą0ąxwe then shift the transverse momentum, remove the term that gives 0 by parity and rescale the Feynman prescription in order to have convergent integrals, noting thatGpx 2 , x 0 q| x 2 ą0ąxPerforming the integral over transverse momentum that are now simple Gaussian integral, we haveGpx 2 , x 0 q| x 2 ą0ąxThe integral over p `is done using the Schwinger parametrization eq. (3.3.8) giving thenGpx 2 , x 0 q| x 2 ą0ąx 0 " ⃗ x 1 V p⃗ x 1 q `x2 γ ´`x { 21K ˘γ``´x0 γ ´`x { 10K p´x

			ˆż d d ⃗ p 2 p2πq D exp	" ´i x 2p 2 `ˆ⃗ p 2 2	´2p	x2 ⃗ p 2 `γż
		d d ⃗ p 1 p2πq D exp ż d d ⃗ x 1 V p⃗ x 1 q " ´i p´x 0 q 2p `ˆ⃗ p 2 1 1 i d 4p2πq D´1 ´2p ´´x 20 p´x 0 q ⃗ `⃗ x 2 i D´1 ΓpD ´1q 21 2x 2 ´⃗ x 2 10 2x 0 `iε ¯D´1 θpx 2 qθp´x 0 q p 1 0 d d ⃗ x 1 V p⃗ x 1 qp2πq 3 ż dp `e´ip `x2 ˆ`x 2 γ ´`x { 21K " ż p´x ˘γ``´x0 γ ´`x { 10K D 2 (A.1.4) 2 x 0 q 0 θpp `qθpx 2 qθp´x 0 q ˆż d d ⃗ p 2 p2πq D exp # ´i x 2 2p `«ˆ⃗ p 2 ´px 2 ⃗ x 21 ˙2 ´ˆp x 2 ˙2 ⃗ x 2 21 ´iε ff+ p `γ´`p { 2K 2p `γż " ż iΓpd `1q 4p2πq d`1 d d 2 x 0 q D 2 ´´x 20 `⃗ x 2 21 2x 2 10 2x 0 `iε ´⃗ x 2 ¯d`1 θpx 2 qθp´x 0 q
			d d ⃗ p 1 p2πq D exp	#	´i p´x 0 q 2p `«ˆ⃗ p 1	´pp ´x0 q	⃗ x 2 10 ˙2	´ˆp x0 ˙2 ⃗ x 2 10 ´iε	ff+	p `γ´`p { 1K 2p `,
	leads to										x p `, p´x 0 q 2 `ą 0. This p
													1 4p2πq 2D´3	ż	dp `e´ip `x2 0 θpp `qθpx 2 qθp´x 0 q
						ˆż d d ⃗ p 2 e	´i x 2 p `⃗ p 2 2 p1´iεq e i p	2x	2 p⃗ x 2 21 `iεq x 2 γ ´`x { 21K x 2	γ	ż
							d d ⃗ p 1 e ´iˆ´x 0 2p `˙⃗ p 2 1 p1´iεq	e i	ˆp2	x	0 ˙p⃗ x 2 10 `iεq p´x 0 qγ ´`x { 10K p´x 0 q	.
						0
	"	ż	d d ⃗ x 1 V p⃗ x 1 q	1 4p2πq 2D´3	ż	dp `exp	" ip `"´x 20	`⃗ x 2 21 2x 2	´⃗ x 2 10 2x 0 `iε	ȷ*	θpp	`q
	ˆθpx 2 qθp´x 0 q	`x2 γ ´`x { 21K	˘γ``´x0 γ ``x { 10K 2 x 0 q	p´x ˆ2πp ìx 2 ˙d 2 ˆ2πp ìx	0 ˙d 2
	"	ż	d d ⃗ x 1 V p⃗ x 1 q	1 i d 4p2πq D´1	ż `8 0	dp `pp `qD´2 exp	" ip `"´x 20	`⃗ x 2 21 2x 2	´⃗ x 2 10 2x 0 `iε	ȷ*	θpx 2 qθp´x 0 q
	ˆ`x 2 γ ´`x { 21K	˘γ``´x0 γ ´`x { 10K 2 2 x 0 q D	p´x .	(A.1.3)

0 " ż d d ⃗ x 1 V p⃗ x 1 q

  e ´ip 2 x 2 e i⃗ p 2 ¨⃗ x 2 e ´i⃗ p 1 ¨⃗ x 0 e ip 1 x 0 p2πqδpp 21 qż d d ⃗ x 1 V p⃗ x 1 q e ´i⃗ x 1 ¨⃗ p 21 ˆż dp 2 e ´ip 2 x 2 i `p2 γ ´`p { 2K ¯γ`ż dp 1 e ip 1 x 0 i `p1 γ ``p { 1K

				1 x	0
	ˆi `p2 γ ´`p { 2K 2 ´p2 ´⃗ p 2 2 ´iε 2p 2p 2 ¯γ`i `p1 γ ´`p { 1K 1 ´p1 ´⃗ p 2 1 ´iε 2p 2p 1 ¯θpx 2 qθp´x 0 q
	"	p2πq D ż d d ⃗ p 2 dp 2	ż d d ⃗ p 1 dp
			2p	2p
			2 ´p2 ´⃗ p 2 2 ´iε 2p	1 ´iε 1 ´p1 ´⃗ p 2 2p

1 p2πq D 2 1 ¯θpx 2 qθp´x 0 q .

  .3.31), Gpx 2 , x 0 q| x 2 ą0ąx e ´ip 2 x 2 e i⃗ p 2 ¨⃗ x 2 e ´i⃗ p 1 ¨⃗ x 0 e ip 1 x 0 p2πqδpp 21 qV p⃗ p 21 q eWe get(3.4.8).I 22 " I 11 | 1Ø2 .

								(B.1.13)
					0
	"	ż d d ⃗ p 2 dp p2πq D 2	ż d d ⃗ p 1 dp	´i ⃗ p 2 2 ´iε 2p 2 x	2 e	i	⃗ p 2 1 ´iε 2p 1 x 0 θpp 2 q
	ˆp2 γ ´`p { 2K 2p 2	γ	`p1 γ ´`p { 1K 2p 1	θpx 2 qθp´x 0 q .

1 p2πq D

  K pq j 1K pI 2 `2q 2Kk I k 1 q ´2q 2Kk I jk ´Ij 3 q `gij K pp⃗ q 1 ¨⃗ p q1 1 qpI 2 `2q 2Kk I k 1 q `pq1 1 Kk p2q 2Kl I kl `IkHere the integrals from B.1.1 will have the following arguments : p⃗ p 2 q2 `xxQ 2 q, ∆ 2 " px ´zqpx `zqQ 2 , (B.1.19) With such variables, it is easy to see that the argument in the square roots in (B.1.6) are full squares. In terms of the variables in (B.1.18), the impact factors read:(longitudinal NLO) ˆ(longitudinal LO) :pϕ 5 q LL " 4px ´zqp´2xpx `zq `z2 `zq xz " xpx ´zqI 2 ´pzq 1Kk ´x px `zq q 2Kk q I k 2K px ´x ´zq `qi 1K p8x 3 ´6x 2 pz `2q `xpz `3qp2z `1q ´2z 2 q

	`2q i 2K q 1Kk px ´x ´zqI k 1	`2	x x	pxp8x ´3q ´6xz `2z 2 `zqI i 1
	`2 x	"	xq i						‰	I 2
	´4 x	"	px ´zqpx `zqq 1Kk `xp4x 2 ´xp3z `5q `pz `1q 2 qq 2Kk	‰	I ik
	´4 z		xxpx ´xq	" q 2Kk I ik `Ii 3	´qi 1K ´q2Kk I k 1 `I2 ¯ı ,	(B.1.22)
	(transverse NLO) ˆ(transverse LO) :
										K 1 j	ı
	`xx z	" px ´xq 2 p j q1 1 K p2q 2Kk I ik `Ii 3	´qi 1K pI 2 `2q 2Kk I k 1 qq
	`pi q1 1 3 qq ı	.	(B.1.17)
	B.1.3 ϕ 5
					⃗ q 1 "	ˆx	´z x	˙⃗ p 3	´z x	⃗ p 1 , ⃗ q 2 " ⃗ p q1	´z x	⃗ p q ,	(B.1.18)
				∆ 1 "	zpx ´zq x 2 x 1	ı	, (B.1.20)
	(longitudinal NLO) ˆ(transverse LO) :
	pϕ 5 q j LT " px ´xqp j q1 1 K pϕ 5 q LL	(B.1.21)
				`4px ´zqpx ´x ´zq x	´zq k 1K ´xpx `zqq k 2K ¯pq1 1 Kl	´gj Kk I l 1	`Ij 1	¯,
	(transverse NLO) ˆ(longitudinal LO) :
	pϕ 5 q i T L " 2		" px ´x ´zq p⃗ q 1 ¨⃗ q 2 q ´xpx ´zq 2 Q 2 `p z x	´xq⃗ q 2 1	ı	I i 1
	`2 x	"	xq 2Kk p´8xx ´6xz `2z 2 `3z `1q `2q 1Kk p2xz ´2x 2 `x ´z2 q ‰	q i 1K I k 1
										282

  2 ´p3z `5qx `pz `1q 2 qq 2Kk `px ´zqpx `zqq 1Kk ˘pj

						q1 1 K
	´x	´z x	´xp2x ´z ´2qq j 2K `zq j 1K ¯pq1 1 Kk	ȷ	I ik
	`x px ´xq x	`2z 2 ´6xz `z `xp8x ´3q ˘pj q1 1 K I i 3
	`"px ´xq ˆpx ´x `zqq i 2K `ˆ6pz `2qx ´8x 2 ´pz `3qp2z `1q	`2 z 2 x	˙qi 1K r i K	˙pj q1 1 K

  1 K ¯ı . (B.1.23)´gijK p⃗ p q2 ¨⃗ p q1 1 q ´pi q1 1 K p j Finite part of the squared impact factors for real correctionsB.2.1 LL transitionThe double dipole ˆdouble dipole contribution isΦ 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q Φ `: 4 p⃗ p 1 1 , ⃗ p 2 1 , ⃗ p 3 1 q " 8p `4 γ

	B.1.4 ϕ 6
	We will use the variable			⃗ q "	ˆx	´z x	˙⃗ p 3	´z x	⃗ p 1 .	(B.1.24)
	(longitudinal NLO) ˆ(longitudinal NLO) :
							pϕ 6 q LL " ´4xx 2 J 0 ,	(B.1.25)
	(longitudinal NLO) ˆ(transverse NLO) :
						pϕ 6 q j LT " p1 ´2xqp j q1 1 K pϕ 6 q LL ,	(B.1.26)
	(transverse NLO) ˆ(longitudinal NLO) :
						pϕ 6 q i T L " 2x	"	p1 ´2xqp i q2K J 0	´Ji 1K	‰	,	(B.1.27)
	(transverse NLO) ˆ(transverse NLO) :
		pϕ 6 q ij T T "	x " px ´xq 2 p i q2K p j q1 1 K	q2K	ı	J 0
			`x	" px ´xqp j q1 1 K g i Kk ´pq1 1 Kk g ij K	`pi q1 1 K g j Kk	ı	J k 1K .	(B.1.28)
	We introduced
	J k 1K "	px ´zq 2 x 2	q k K ⃗ q 2 ln	˜⃗ p 2 q2 `xxQ 2 ⃗ p 2 q2 `xxQ 2 `x2 x zpx´zq ⃗ q 2	¸,	(B.1.29)
	and
	J 0 "	z q2 `xxQ 2 q xp⃗ p 2	´2xpx ´zq `z2 xzp⃗ p 2 q2 `xxQ 2 q	ln	˜x2 xµ 2 zpx ´zqp⃗ p 2 q2 `xxQ 2 q `x2 x⃗ q 2 ¸.	(B.1.30)
	B.2 x 2 g	ˆ⃗ p 2 q2 1 xqp1´xqq	`Q2 ˙ˆQ 2 `⃗ p 2 q1 1 xq	`⃗ p 2 q2 1 xq	g3 1 `⃗ p 2 xg

  `Q2 ¯ˆp⃗ p q2 `⃗ p g3 q2 The interference term in the dipole ˆdipole contribution reads´r Φ 3 p⃗ p 1 , ⃗ p 2 q Φ `: 4 ´⃗ p 1 1 , ⃗ p 2 1 , ⃗ 0 ¯`Φ 4 ´⃗ p 1 , ⃗ p 2 , ⃗ 0 ¯r Φ `: 3 p⃗ p 1 1 , ⃗ p 2 1 qThe double dipole ˆdipole contribution has the formΦ 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q Φ `: 3 p⃗ p 1 1 , ⃗ p 2 1 q " Φ 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q Φ `: 4 ´⃗ p 1 1 , ⃗ p 2 1 , ⃗ 0 ¯`Φ 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q r Φ `: 3 p⃗ p 1 1 , ⃗ p 2 1 q ,For the dipole ˆdouble dipole contribution, one just has to complex conjugate (B.2.4) and also invert the name of the momenta i.e. 1 1 , 2 1 Ø 1, 2.

	B.2.2 LT/TL transition
	xq The double dipole ˆdouble dipole contribution is Φ i 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q Φ `: 4 p⃗ p 1 1 , ⃗ p 2 1 , ⃗ p 3 1 q	`⃗ p 2 g3 xg	`⃗ p 2 q2 xq	`Q2	˙fi ffi ffi fl `pq Ø qq. (B.2.1)
	"	ˆQ2 `⃗ p 2 g3 xg	`⃗ p 2 q1 xq	´4p `3 γ q2 `⃗ p 2 xq ˙ˆQ 2 `⃗ p 2 g3 1 xg	`⃗ p 2 q1 1 xq	q2 1 xq `⃗ p 2	" ẋg
		´´⃗ P ¨⃗ p q1	» -¯Gi				8p `4 γ
				--	x g px q `xg q	ˆ⃗ p 2 q2 1 xqpxq`xgq	`Q2 ˙ˆ⃗ p 2 q1 1 xq	`⃗ p 2 q2 1 xq	g3 xg `⃗ p 2	`Q2	$
				' ' &	p4x q x q `xg p2 ´dx g qq ´⃗ p g	´xg xq ⃗ p q¯¨`x	q ⃗ p g ´xg ⃗ p q1 1	⃗
				' ' %			p g	´xg xq ⃗ p q¯2 ˆ⃗ p 2 q1 1 xqpxq`xgq	`Q2	ẋq
									fi
					¯¨´⃗ p g ˙, / ⃗ ´xg xq ⃗ p q1 1 / . ffi `pq Ø qq ffi / -`Q2 p q xq ⃗ ´xg ¯2 ˆ⃗ p 2 g `4x q px q `xg q ˘´⃗ p g `dx 2 p g ´xg xq ⃗ p q q2 xqpxq`xgq / fl
		`p1 Ø 1 1 , 2 Ø 2 1 q.	(B.2.2)
									(B.2.3)
	where
		Φ 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q r Φ `3 p⃗ p 1 1 , ⃗ p 2 1 q "	x g px q `xg q	ˆ⃗ p 2 q2 xqpxq`xgq	8p γ 4 `Q2 ˙ˆ⃗ p 2 q1 xq	`⃗ p 2 q2 xq	g3 xg `⃗ p 2	`Q2	$
					' ' &	p4x q x q `xg p2 ´dx g qq ´⃗ p g	´xg xq ⃗ p q¯¨p x q ⃗ p g3 ´xg ⃗ p q1 q
					' ' %			´⃗ p g	´xg xq ⃗ p q¯2 ˆ⃗ p 2 q1 1 xqpxq`xgq	`Q2	ẋq
							`dx 2 g `4x q px q `xg q ˘´⃗ p g p g ´xg xq ⃗ p q xqpxq`xgq ´xg xq ⃗ p q q2 1 ¯2 ˆ⃗ p 2	¯¨´⃗ p g3 `Q2 ˙, / ⃗ ´xg xq ⃗ p q1 / -/ / . `pq Ø qq. (B.2.4)

K

´´⃗ G ¨⃗ p q1

  ¨⃗ p q1 ¯∆i qK ´´⃗ p q1 ¨⃗ ∆ q¯X i K ¯pdx g `4x q ´4q ´´⃗ X ¨⃗ ∆ q¯p i q1K p2x q ´1q `4 px q ´1q x q ´dx 2

		2 g xg	`⃗ p 2 q1 xq	`⃗ p 2 q2 xq ˙ˆQ 2 `⃗ p 2 q2 1 pxg`xqqxq ˙ˆQ 2 `⃗ p 2 q1 xqpxg`xqq	ẋg
	´´⃗ X g	xg
	px g `xq q 3 ⃗ ∆ 2 q ˆQ2 `⃗ p 2 g xg	`⃗ p 2 q1 xq	`⃗ p 2 q2 xq ˙ˆQ 2 `⃗ p 2 q1 xqpxg`xqq ˙ˆQ 2 `⃗ p 2 xqpxg`xqq q1 1	˙‹ ‹ '
	`pq Ø qq ,				(B.2.7)

  1 ¯´W i ´⃗ p q2 1 ¨⃗ ∆ q

						¯px
		q `xg q	ˆ⃗ p 2 xqpxq`xgq q2	`Q2	˙fi ffi ffi fl `pq Ø qq. (B.2.13)
	Here, we introduced
		W i K " x q p i g3K ´xg p i q1K .	(B.2.14)
	B.2.3 TT transition
	The double dipole ˆdouble dipole contribution is
		¨p`2
	Φ i 4 p⃗ p 1 , ⃗ p 2K , ⃗ p 3 q Φ k: 4 p⃗ p 1 1 , ⃗ p 2 1 , ⃗ p 3 1 q "	ˆQ2 `⃗ p 2 g3 xg	`⃗ p 2 q1 xq	γ xq ˙ˆQ 2 `⃗ p 2 `⃗ p 2 q2 g3 1 xg	`⃗ p 2 q1 1 xq	`⃗ p 2 q2 1

  1 ¯´g ik K ´⃗ P ¨⃗ p q1 ¯`P i K p k q1K ¯`x g H k K ´´⃗ P ¨⃗ p q1 ¯pi q2 1 K ´`⃗ p q1 ¨⃗ p q2 1 ˘P i H i ´´⃗ P ¨⃗ p q2 1 ¯pk q1K ´`⃗ p q1 ¨⃗ p q2 1 ˘P k K ¯`x g ´⃗ H ¨⃗ p q1 ¯´g ik K ´⃗ P ¨⃗ p q2 1 ¯`P k K p i q2 1 K ¯ı `2x g ´´⃗ H ¨⃗ p q2 1 ¯P k

		¯ı
		K
		"
	`ppd ´4q x g `2q	x g K

  ´´⃗ G ¨⃗ p q1 1 ¯´⃗ P ¨⃗ p q1 ¯´´⃗ G ¨⃗ p q1 ¯´⃗ P ¨⃗ p q1 1 ¯¯``⃗ p q1 ¨⃗ p q1 1 ˘´G i K P k

	2 q1 xqpxg`xqq ˙ˆQ 2 `⃗ p 2 q1 1 xqpxg`xqq	˙tx g ppd ´4q x g ´4x qq
	ˆ"g ik K	K	´Gk K P i K	2
	´⃗ G ¨⃗ p q1 1 ¯´P i K p k q1K

  1 , i Ø kq ¯`pq Ø qq. (B.2.15)The interference term in the dipole ˆdipole contribution reads´r Φ i 3 p⃗ p 1 , ⃗ p 2 q Φ k: 4 ´⃗ p 1 1 , ⃗ p 2 1 , ⃗ 0 ¯`Φ i 4 ´⃗ p 1 , ⃗ p 2 , ⃗ 0 ¯r Φ k: 3 p⃗ p 1 1 , ⃗ p 2 1 q

				"
	¨2p `2 γ
	⃗ ∆ 2 q	ˆQ2 `⃗ p 2 g xg	`⃗ p 2 q1 xq	`⃗ p 2 q2 xq ˙ˆQ 2 `⃗ p 2 q2 1 pxg`xqqxq

  1 , i Ø kq ˘`pq Ø qq.The double dipole ˆdipole contribution has the formΦ i 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q Φ k: 3 p⃗ p 1 1 , ⃗ p 2 1 q " Φ i 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q Φ k: 4 ´⃗ p 1 1 , ⃗ p 2 1 , ⃗ 0 ¯`Φ i 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q r Φ k: 3 p⃗ p 1 1 , ⃗ p21 q , (B.2.18) where

						(B.2.16)
	Here,
					V i K " x q p i gK ´xg p i q1K .	(B.2.17)
	Φ i 4 p⃗ p 1 , ⃗ p 2 , ⃗ p 3 q r Φ k: 3 p⃗ p 1 1 , ⃗ p 2 1 q
	"	⃗ ∆ 2 q	ˆQ2 `⃗ p 2 g3 xg	`⃗ p 2 q1 xq	2p `2 γ `⃗ p 2 q2 xq ˙ˆQ 2 `⃗ p 2 q2 1 pxg`xqqxq

  1 ¨⃗ ∆ q ¯P k qq pdx g `4x q ´2q p k q2 1 K ´´⃗ p q1 ¨⃗ ∆ q ¯P i K ´´⃗ P ¨⃗ p q1 ¯∆i qK ¯´x g ppd ´4qx g ´2q ˆ"´g ik K ´⃗ P ¨⃗ p q1 ¯`P i K p k q1K ¯´⃗ p q2 1 ¨⃗ ∆ q ¯`´´⃗ P ¨⃗ p q1 ¯pi q2 1 K ´`⃗ p q1 ¨⃗ p q2 1 ˘P i

		¯∆k	ı
	K	qK
	`´⃗ P ¨⃗ ∆ q

K ´´⃗ P ¨⃗ p q2 1 ¯∆k qK xg p1 ´2x

  , λq ¨pl 2 ´l1 q ´p{ l 2 ´{ l 1 q{ ε ˚pl 2 , λq `mQ {ε ˚pl 2 , λq ‰ γ ´⃗ ϵ λ˚¨⃗ l 1 ¸`p { l 2 ´{ l 1 q⃗ ϵ λn γ n ´mQ ⃗ ϵ λn γ n ´{ l 1 ˙⃗ ϵ λn γ n ´mQ ⃗ ϵ λn γ n ˙¨⃗ ϵ λ˚´ˆ⃗ l 2 ´l2 l 1 ⃗ l 1 ˙m γ m ⃗ ϵ λn γ n `mQ ´mQ ˘⃗ ϵ λn γ n ´mQ ⃗ ϵ λn γ n

	"	" ´{ l 1 `mQ	‰	" ´2 l l	1 2 ˆ⃗ l 2	´l2 l 1 ⃗ l 1 ˙m ⃗ ϵ λn	"	δ nm ˆ1	´l2 2l 1 ˙`l 2l 2 1 ω mn	ȷ	´l2 l 1 m Q ⃗ ϵ λn γ n	*	γ	1
					´l2 l 1 ˙pl 2 1	´m2 Q q⃗ ϵ λn γ n γ	"
		" ´{ l 1 `mQ	‰	#	´2 p g `pq p g	ˆ⃗ l 2	´pg q `´p	q ⃗ l 1 ˙m ⃗ ϵ λn	« δ nm	˜pg `2p 2pp g `pq q q	¸`p 2pp g `pq q g	ω mn	ff
	´pg p g `pq m Q ⃗ ϵ λn γ n	+	γ	``˜1	´pg p g `pq ¸pl 2 1	´m2 Q q⃗ ϵ λn γ n γ	"
		" ´{ l 1 `mQ	‰	#	´2 p q p g ˆ⃗ l 2	´xg 1 ´xq	⃗ l 1 ˙m ⃗ ϵ λn	« δ nm	˜pg `2p q 2p q	¸`p 2p g q ω mn	ff
	´pq p g `pq	p g p q m Q ⃗ ϵ λn γ n	+	γ	``p q `´p q q pl 2 1	´m2 Q q⃗ ϵ λn γ n γ	"
		2	p q p g ⃗ ϵ λn "	´{ l 1 `mQ	‰	γ `#´ˆ⃗ l 2	´xg 1 ´xq	⃗ l 1	˙m « δ nm ˜1	`pg 2p q ¸`p 2p g q ω mn	ff
															,
	`mQ 2	p p g `pq q	˜pg p q ¸2 γ n	. -	`xq 1 ´xq	pl 2 1	´m2
	" ´{ l 1 `mQ	‰	{ ε ˚pl 2 , λq	" p { l 2 ´{ l 1 q `mQ	‰	γ	"
		"	´{ l 1 `mQ	‰ " 2ε ˚pl 2 ff
															γ	"
		"	´{ l 1 `mQ	‰	"	´2 l l	1 2 ˆ⃗ l 2	´l2 l 1 ⃗ l 1 ˙¨⃗ ϵ λ˚`ˆ{ l 2	´l2 l 1 { l 1	`l2 l 1 { l 1 ȷ	γ	"
	" ´{ l 1 `mQ 1 ⃗ l 1 `ˆl ‰ " ´2 l 1 l 2 ˆ⃗ l 2 ´l2 l 2 l 1 ´1˙`{ l 1 ȷ	γ	"
		" ´{ l 1 `mQ	‰	" ´2 l l	1 2 ˆ⃗ l 2	´l2 l 1 ⃗ l 1 ˙m ⃗ ϵ λn	"	δ nm `l2 2l 1 γ m γ n	ȷ	´l2 l 1 m Q ⃗ ϵ λn γ n
	`ˆl l	2 1	´1˙`{ l 1 `mQ ˘⃗ ϵ λn γ n	*	γ	`298

" " ´{ l 1 `mQ ‰ « ´2 ˜⃗ ϵ λ˚¨⃗ l 2 l 2 l 1 Q q⃗ ϵ λn γ n γ `(C.5.3)

  " γ `«´2 ˜pq ⃗ p g ¨⃗ ϵ λp g ´⃗ p q ¨⃗ ϵ λ˚¸´⃗ ϵ λn γ n { p g ff v spp qq " γ `"´2 x p⃗ p g ´x⃗ p qq ¨⃗ ϵ λ˚´⃗ ϵ λn γ n p { p g ´x { p qq ´x⃗ ϵ λn γ n { p qȷ v spp qq " γ `"´2 x p⃗ p g ´x⃗ p qq m ⃗ ϵ λn δ mn `p⃗ p g ´x⃗ p qq m ⃗ ϵ λn γ n γ m `xm Q ⃗ ϵ λn γ n ȷ v spp qq " γ `"´2 x p⃗ p g ´x⃗ p qq m ⃗ ϵ λn ´δmn ´x 2 γ n γ m ¯`xm Q ⃗ ϵ λn γ n ȷ v spp qq" ⃗ ϵ λn γ `"´2 x p⃗ p g ´x⃗ p qq m I 0 p⃗ r 1 , ⃗ r 2 q (C.7.5) z 1 , z 2 , z 3 such that z 1 `z2 `z3 " 1. I will use the following relation to compute them

	"	´i B B⃗ r j 1
		ż 8 0	ds s ν´1 e ´sA 2 e	´B2 {s " 2	ˆA B	˙´ν	K ´ν p2ABq	(C.7.6)
	with A 2 ą 0, B 2 ą 0.
	C.7.1 Schwinger's parametrization
	Schwinger's parametrization	1 D β "	1 Γpβq	0 ż 8	dss β´1 e ´sD	(C.7.7)
									*
				" δ mn ´1	`x 2	¯´x 2	ω nm	ı	`xm Q γ n	v spp qq .	(C.5.4)
	where x " p g {p q .

There are differences between proton PDFs and nuclear PDFs as dynamics from having multiple proton and neutrons arise in nucleus.

s is usually denoted W 2 in the literature.

terminology coined by Feynman

In principle, cuts can also be present.

Because n 2 i " 0 but the square of the polarization vector of gluons should be equal to ´1, this is the reason for the appellation non-sense polarization.

Another indication that the BFKL picture is incomplete is the problem of diffusion in the infra-red region of the virtualities of the gluon on the ladder.

If the target is a nucleus travelling with momentum pA and has mass number A, p is then usually the momentum of the average nucleon with p " pA{A. Such a definition makes it easier to study nuclear effects that modify the simple assumptions that the nucleus consists of a number of free nucleons.

In some references, the factor 1{4π is absorbed in the photon wave function.

This is a different overall factor than what is presented at eq.(1.2.22) where the factor is 1{s in high-energy limit. This is because in eq. (1.2.22), the scattering amplitude A is dimensionless. If one defines a scattering amplitude not dimensionless, the overall factor in the optical theorem is then different.

A comprehensive review on the subject can be found in[START_REF] Hebecker | Diffraction in deep inelastic scattering[END_REF].

For more precise discussion on the validity of the dipole picture, see[START_REF] Ewerz | On the Range of Validity of the Dipole Picture[END_REF].

We work here in coordinate space to determine the colour factors as it is easier to understand this way. A simple Fourier transform allows to go back to momentum space.

These results have not been explicitly written in this manuscript for brevity and because we are more interested in the finite double dipole impact factor after cancellation of the rapidity divergences. For the curious reader though, these results can be found in[START_REF] Boussarie | On the one loop γ p˚q Ñ qq impact factor and the exclusive diffractive cross sections for the production of two or three jets[END_REF][START_REF] Boussarie | Perturbative study of selected exclusive QCD processes at high and moderate energies[END_REF].

´⃗ p 1 , ⃗ p 2 , ⃗ 0 ¯`r Φ i 3 p⃗ p 1 , ⃗ p 2 q . (4.4.8)

Indeed for a hadron moving in the ´direction, the classical current should to be proportional to the momenta of the large-x degrees of freedom. One then expect that tje most important (or only non-zero) component of the current is J´.

A system that is time dependent but whose time dependence is unnaturally slow compared to the timescales that one is probing it with is known as a glass in statistical physics. An actual glass is a glass in the statistical physics sense as glass are non-cristalline like liquids but do not flow. In a loose sense, glass can be considered as a liquid that flows so slowly that in the timescale of a human observant it appear like a solid.

The effective vertex for the gluon propagating in the shockwave can be obtained by iterating the interaction of a gluon with the background gluon field in light-cone gauge n2 ¨A " 0, interaction using the triple gluon vertex. The l `factor comes from the fact that this vertex depends on momenta and using also the conservation of the longitudinal momentum of the propagating gluon (eikonal approximation).

One requires that αs ln `Λ´{ Λ 1´˘! 1 to ensure that the quantum fluctuations are under control.

Since the variables involved are all integration variables, this last operation is not necessary at the LO level and in some NLO contributions, but we will always do it for clarity of notation.

Here we assume the azimuthal angle of the scattered electron is integrated out, and thus we do not consider azimuthal correlations with respect to the electron. Thus, it is sufficient to consider the diagonal elements in the polarization of the virtual photon[START_REF] Mäntysaari | Gluon imaging using azimuthal correlations in diffractive scattering at the Electron-Ion Collider[END_REF].

We leave in the arguments of the transverse integrals ⃗ x, ⃗ y instead of replacing them by ⃗ b ˘⃗ r 2 to simplify the notations. But of course ⃗ x, ⃗ y then should be understood as functions of ⃗ r, ⃗ b.

Acknowledgements

Part I

Unveiling saturation effects at past, present and future colliders thus ∆x v ! ∆x ẁ , (5. 1.4) and so the large-x partons are sharply localized within a longitudinal extent ∆x v À 1{Λ

´.

At leading order, the small-x gluons are treated as strong classical fields A cl " 1{g justified by their high-occupation number xA cl A cl y " 1{α s .

The colour charges generate a static colour current (independent of light-cone time x ´):

J µ a pxq " δ µ´ρa px `, ⃗ x q , (5. 1.5) where ρ a px `, ⃗ x q is the hadron colour charge density describing the spatial distribution of these color charges in x `and ⃗ x. The current lives in the adjoint representation of SU pN c q, i.e. J µ " J µa T a and has only one component, its other component being suppressed by 1{p t 1 . Its support on the x `-axis is narrow, justified with the above discussion using the uncertainty principle. In the asymptotic limit where the hadron is Lorentz-contracted to an infinitely thin pancake, the current then has the form J µ a pxq " δ µ´δ px `qρ a p⃗ x q .

(5.1.6)

The classical small-x gluons are found by solving the classical Yang-Mills equations of motion in the presence of the current given by eq. (5.1.5) rD µ , F µν s " J ν

(5.1.7)

with D µ the covariant derivative D µ " B µ ´igA a µ T a (5. 1.8) and F µν the field strength tensor

(5.1.9)

In the Lorentz gauge, defined as B µ A µ " 0, a solution of the Yang-Mills equations can be found analytically [START_REF] Iancu | Nonlinear gluon evolution in the color glass condensate. 1[END_REF]:

A µa " δ µ´αa px `, ⃗ x q .

(5.1.10) α a px `, ⃗ x q satisfies the Poisson equation ⃗ ∇ 2 α a px `, ⃗ x q " ´ρa px `, ⃗ x q (5. 1.11) where ρ is the colour charge density in the covariant gauge. Looking at the above expressions, we see that the solution in the Lorentz gauge is also valid in the A `" 0 gauge. The Poisson equation can be solved in Fourier space α a px `, ⃗ x q " ´ż d 2 ⃗ z G 0 p⃗ x ´⃗ z qρ a px `, ⃗ z q , (5.1.12)

Chapter 8

Direct quarkonium-plus-gluon production in DIS in the Colour Glass Condensate

This chapter is dedicated to the last process studied in this thesis, the direct quarkonium inclusive production accompanied by a gluon in DIS at small-x. This is based on [START_REF] Kang | Direct quarkonium-plus-gluon production in DIS in the Color Glass Condensate[END_REF]. We employ the Non-Relativistic QCD (NRQCD) factorization framework focusing on the S-wave contribution to the formation of the quarkonium, and include both colour singlet and octet contributions. Our short-distance coefficients for the production of the heavy quark pair are obtained within the CGC effective field theory.

Theoretical framework

Kinematics and notations

We study direct quarkonium production H accompanied by a gluon g in inclusive deep inelastic lepton-nucleus scattering at small-x, epk e q `App A q Ñ epk 1 e q `Hppq `gpp g q `X , (

as shown in fig. 8.1.1. We work in a dipole frame where the target moves ultra-relativistically in the ´light-cone direction while the photon flies close to the `light-cone direction. The average four-momentum of the nucleon in the nucleus is

where we ignored the mass of the nucleon. The four-momentum of the virtual photon is given by

where κ " 2S`1 L rcs J . Here, the symbols S, L J, rcs stand for spin, orbital angular momentum, total angular momentum, and colour state of the heavy quark pair. In this work, we consider both colour singlet r1s and octet contributions r8s. dσ κ are the short-distance coefficients for the production of the heavy quark pair in the quantum state κ, and xO H κ y are the Long-Distance Matrix Element (LDME), which are typically fitted and encode the non-perturbative mechanism for the hadronization of the heavy-quark pair Q Qrκs to the quarkonium.

The short-distance coefficient functions dσ κ are averaged over the degenerate quantum states of the heavy-quark pair

where N colour " 1 for the colour singlet state or N colour " N 2 c ´1 for the colour octet state. Eq. (8.1.10) is obtained from the scattering amplitude projected into a specific κ quantum state of the heavy-quark pair. In this work we focus on the S-wave contribution to the heavy quarkonium production, for which the projection reads 1 :

where we used the fact that x00; SS z |JJ z y " δ SJ δ SzJz for L " 0 (S-wave).

The colour projector is defined as

To carry out the sum over spins, it is useful to introduce the spin projector:

In particular, for the states 1 S 0 and 3 S 1 with k " 0 these projectors are given by [START_REF] Kang | Quarkonium production in high energy proton-nucleus collisions: CGC meets NRQCD[END_REF] (see appendix C.4) where εpJ z q is the polarization vector of the quarkonium which satisfies the relation ÿ Jz ε ρ pJ z qε α pJ z q " ´gρα `pρ p α p 2 " P ρα . (8.1.16)

The scattering amplitude M ss,i īpp, k, p g q is computed with the CGC formalism and in particular using the effective Feynman rules presented in section 5.1.1 and the definition of the CGC reduced amplitude in eq. (C.1.3).

Outline of the computation

Without loss of generality, we compute the differential cross-section for direct quarkonium production + gluon in virtual photon-nucleus collision. The DIS result can be readily obtained from 2 dσ e`AÑe 1 `H`g`X ds dQ

Here λ denotes the polarization of the virtual photon, and f λ `W 2 , Q 2 ˘are the photon flux factors:

with y the inelasticity satisfying y " ps `Q2 q{S. Following eq. (8.1.9), the differential cross-section for direct quarkonium + gluon production reads

where the short-distance coefficients for the differential cross-section for Q Qrκsg are given by, using eqs. (C.1.9) and (8.1.10),

x¨¨¨y Y refers to the CGC average over all possible charge configurations inside the target at rapidity scale Y , see eq. ( 5.1.15). We summed over the two transverse polarizations λ and the colour a of the gluon. Furthermore, following the convention in [START_REF] Kang | Quarkonium production in high energy proton-nucleus collisions: CGC meets NRQCD[END_REF], we denote s ř

Differential cross-section for direct quarkonium + gluon production

As we discussed in section 8.1.3, it is sufficient to compute the short-distance coefficients for differential cross-section for direct quarkonium + gluon production. We show separately the results for longitudinally and transversely polarized photons, as well as the different κ states of the heavy quark pair. As discussed in section 8.1.3 there are 16 contributions:

where

The differential cross-section will have the following schematic form:

where we define the differential element

the colour correlator

.5) and the perturbative factor

(8.4.6) For a κ " 2S`1 L rcs J state, the information on the colour state rcs of the heavy-quark pair is fully encoded in the colour correlator in eq. (8.4.5), while the information about the spin, angular momentum and total momentum 2S`1 L J of the heavy-quark pair, and the polarization λ of the photon is completely encompassed in the perturbative factor in eq. (8.4.6).

Appendix A

Further details for chapter 3

A.1 Quark line through the shockwave field in coordinate space Starting from eq. (3.4.6) neglecting the i, j colour indices, we have

x 2 e ´ip 2 x 2 e i⃗ p 2 ¨⃗ x 21 e ip 1 x 0 e ix 1 pp 2 ´p1 q e ip 1 x 0 e i⃗ p 1 ¨⃗

where the conservation of the longitudinal momentum p `" p 2 " p 1 comes from the integration over x 1 . The integration over p 2 , p 1 are done using Jordan's lemma and the residue theorem. We remark that to have convergent integrals, we need to have the argument of the complex exponential Impp 2 qx 2 ă 0 and Impp 1 qx 0 ą 0. Because x 2 ą 0 and x 0 ă 0, we then have p `ą 0. Closing the contour in the lower half plane which leads by residue theorem to a factor p´2πiq for each integral over p í , one gets

Further details for chapter 4

B.1 Finite parts of virtual corrections

B.1.1 Building blocks integrals

ı , (B.1.3)

The arguments of these integrals will be different for each diagram so we will write them explicitly before giving the expression of each diagram, but we will omit them in the equations for the reader's convenience.

Explicit results for the first 3 integrals in (B.1.1-B.1.4) are obtained by a straightforward Feynman parameter integration. We will express them using the following variables :

where ∆ ij " ∆ i ´∆j . One gets :

and

2 pρ 1 ´ρ2 q p⃗ q 2 12 q 2 ln "ˆ´ρ

Please note that in some cases the real part of ∆ 1 or ∆ 2 will be negative so the previous results can acquire an imaginary part from the imaginary part ˘i0 of the arguments. The last integral in (B.1.4) can be expressed in terms of the other ones by writing

with

Appendix C

Appendices for the quarkonia `gluon production in DIS

We regroup here all the appendices related to chapter 8.

C.1 CGC scattering amplitudes and cross-sections for DIS

We compute in this appendix the cross-section γ ˚pqq `App A q Ñ N `Xpp X q with specific polarization for the photon, same polarization for the scattering amplitude and the hermitian conjugated scattering amplitude.

The formula for the γ ˚pqq `App A q Ñ N `Xpp X q cross-section without averaging over the quantum number of the initial states and summing over the quantum number of the final states is

In the high-energy limit, the flux factor 2Kpsq " 2s " 4q `pÁ . Ă M is the usual scattering amplitude (with the i factor inside though) i.e.

where S| non-int means putting all the Wilson line to 1.

In the CGC formalism, the effective CGC vertices include all possible scatterings off the target, including the possibility of no-scattering which has to be subtracted to obtain the physical amplitude. This can be done systematically by subtracting from the scattering amplitude a term in which all the Wilson lines inside the effective CGC vertices are set to unity. Furthermore, to obtain the "reduced amplitude" one factorizes the overall delta function from "plus" light-cone momentum conservation. Mathematically, this amounts to the following operation:

The relationship between the usual scattering amplitude present in the cross-section and the so-called reduced scattering amplitude is, therefore (written in a nasty way):

From this and using the nucleus state normalization

where the last line is obtained using

Putting eq. (C.1.5) into eq. (C.1.1), one has

For a production of one particle, we have

294

For a production of two particles,

We define the transverse cross-section as dσ T "

C.2 Feynman rules

In this section, we gather the elementary Feynman rules needed for our computation. We remind the reader that we work in light-cone gauge A `" 0. We label σ, σ 1 the spinor indices and i, j, a, b the colour indices in the SU pN c q fundamental and adjoint representations respectively. The free massive quark and gluon propagator are Π αβ plqΠ βδ pl 1 q " ´ÿ λ"˘1 ε α pl, λqε ˚δ pl 1 , λq , (C.2.5)

where λ is the gluon polarization.

C.3 Gamma traces

This section is dedicated to collecting some useful traces of gamma matrices. Here, ω µν " 1 2 rγ µ , γ ν s. These results are used in particular to compute F λ, λ,κ,Jz R i which is given by Eq. (8.1.27). They are:

Trpodd nb of γq " 0 (C.3.1)

Trpγ µ γ ν γ ρ γ σ q " 4pg µν g ρσ ´gµρ g νσ `gµσ g νρ q (C.3.3)

C.6 Lorentz contraction with P αρ

Here, we list a few useful identities that will aid in the calculation of the perturbative factor defined in eq. (8.4.6) when κ " 3 S rcs 1 . The following contractions are encountered when summing over J z and using eq. ( 8 We calculate in this appendix the transverse integrals that appear in diagrams R 1 and R 3 .

`p⃗ l 1 `⃗ l 2 q 2 `m2 We do the following change of variables u " s `t and v " t with s " u ´v ą 0, t " v ą 0 ñ u ą v ą 0 to obtain finally I 0 p⃗ r 1 , ⃗ r 2 q " z 2 z 3 4p1 ´z1 q ˙2 ´⃗ r 1 ´z2 1´z 1 ⃗ r 2 ¯m z 1 p1 ´z1 q 4u `z2 z 3 4vp1 ´z1 q 2⃗ r 2m * " i z 2 z 3 2vp1 ´z1 q ⃗ r 2m , we obtain I 1m p⃗ r 1 , ⃗ r 2 q " pz 2 z 3 q 2 8p1 ´z1 q 2 i⃗ r 2m This leads to I 2mj p⃗ r 1 , ⃗ r 2 q " pz 2 z 3 q 2 8p1 ´z1 q 2 i⃗ r 2m