A stability study of the Gribov-Levin-Ryskin equation, and its phenomenological implications : Estudo da estabilidade da equação Gribov-Levin-Ryskin e suas implicações fenomenológicas

GUILLERMO GERARDO RIVERA GAMBINI

A STABILITY STUDY OF THE GRIBOV-LEVIN-RYSKIN EQUATION, AND ITS PHENOMENOLOGICAL IMPLICATIONS

ESTUDO DA ESTABILIDADE DA EQUAC¸ ˜AO GRIBOV-LEVIN-RYSKIN E SUAS IMPLICAC¸ ˜OES FENOMENOL ´OGICAS

CAMPINAS 2016

A STABILITY STUDY OF THE GRIBOV-LEVIN-RYSKIN EQUATION, AND ITS PHENOMENOLOGICAL IMPLICATIONS

ESTUDO DA ESTABILIDADE DA EQUAC¸ ˜AO GRIBOV-LEVIN-RYSKIN E SUAS IMPLICAC¸ ˜OES FENOMENOL ´OGICAS

Dissertation presented to the Institute of Physics Gleb Wataghin of the University of Campinas in partial fullfilment of the require-ments for the degree of Master in the area of phenomenology of heavy-ion collisions. bla bla bla bla bla bla bla bla bla Dissertation presented to the Institute of Physics Gleb Wataghin of the University of Campinas in partial fullfilment of the requillllllllllllllllrlllll-llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll

Dissertation presented to the Institute of Physics Gleb Wataghin of the University of Campinas in partial fullfilment of the require-ments for the degree of Master of Physics. Disserta¸c˜ao apresentada ao Instituto de F´ısica Gleb Wataghin da Universidade Estad-ual de Campinas como parte dos requisitos exigidos para a obten¸c˜ao do t´ıtulo de Mestre em F´ısica.

Supervisor/Orientador: Prof. Dr. DONATO GIORGIO TORRIERI ESTE EXEMPLAR CORRESPONDE A`

VERS ˜AO FINAL DA DISSERTAC¸ ˜AO DE-FENDIDA PELO ALUNO GUILLERMO GER-ARDO RIVERA GAMBINI, E ORIENTADA PELO PROF. DR. DONATO GIORGIO TOR-RIERI.

Dissertation presented to the Institute of Physics Gleb Wataghin of the University of Campinas in partial fullfilment of the re-quirements for the degree of Master in the area of phenomenology of heavy-ion colli-sions.sdfsdfsdfffffffffffffffffffffffffffffffffffffffffffffffffffffffffff

CAMPINAS 2016

Ficha catalográfica

Universidade Estadual de Campinas Biblioteca do Instituto de Física Gleb Wataghin Lucimeire de Oliveira Silva da Rocha - CRB 8/9174

Rivera Gambini, Guillermo Gerardo,

R524s RivA stability study of the Gribov-Levin-Ryskin equation, and its phenomenological implications / Guillermo Gerardo Rivera Gambini. – Campinas, SP : [s.n.], 2016.

RivOrientador: Donato Giorgio Torrieri.

RivDissertação (mestrado) – Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin.

Riv1. Colisões entre íons pesados. I. Torrieri, Donato Giorgio,1975-. II. Universidade Estadual de Campinas. Instituto de Física Gleb Wataghin. III. Título.

Informações para Biblioteca Digital

Título em outro idioma: Estudo da estabilidade da equação Gribov-Levin-Ryskin e suas

implicações fenomenológicas

Palavras-chave em inglês:

Heavy ion collisions

Área de concentração: Física Titulação: Mestre em Física Banca examinadora:

Donato Giorgio Torrieri [Orientador] Jorge José Noronha Júnior

José Augusto Chinellato

Data de defesa: 29-06-2016

Programa de Pós-Graduação: Física

APROVADA AO INSTITUTO DE FÍSICA “GLEB WATAGHIN”, DA UNIVERSIDADE ESTADUAL DE CAMPINAS, EM 29 / 06 / 2016.

COMISSÃO JULGADORA:

- Prof. Dr. Donato Giorgio Torrieri – Orientador – DRCC/IFGW/UNICAMP

- Prof. Dr. Jorge José Noronha Júnior – IF/USP

- Prof. Dr. José Augusto Chinellato – DRCC/IFGW/UNICAMP

OBS.: Informo que as assinaturas dos respectivos professores membros da banca

constam na ata de defesa já juntada no processo vida acadêmica do aluno.

CAMPINAS

2016

First, I would like to thank my mom Wilda, my siblings Walter, Waldo and Jessica, and my grandfather Wenceslao for giving me their love and constant support.

Next, I want to show my gratitude to my advisor Giorgio Torrieri for entrusting me this project, guiding me through the ups and downs during the past two years, and always pushing me forward.

Then, I feel like giving many words of gratitude to Ju Brasil, Eder “el baby”, Dennisberg, Hernan “le master”, Ronael, Marvyn, Fiorella, Erick, José Luis, Caroline “la jefa”, and Miauricio1_.

Jeff and Annie, of course, have their own paragraph. Thank you for the jokes, the drama, the constant support, and your time. PhD is just starting, what should I do? Best answer: alligator.

Just as J&A, RM has her own paragraph. Thanks for an interesting 2015. It was fun. Best wishes.

Special thanks to the CPG team, you guys do a great job.

Last but not least, I thank CAPES and CNPq for economic support.

Marshall D. Teach (One Piece)

A equação de Gribov, Levin e Ryskin (GLR) é uma equação de evolução para as funções de distribuição dos gluons que respeita o limite de Froissart. Motivado pelo fato do fluxo elíptico 𝑣2 apresentar scaling no momento transversal, na rapidez, no tamanho do sistema,

etc. e pela similaridade que esses scalings possuem com o scaling de Bjorken, modificamos a equação GLR adicionando uma dependência angular a ela. Dessa maneira, temos uma equação diferencial parcial não linear do tipo 2+1 que pode apresentar instabilidades nas soluções, gerando 𝑣2. A forma geral das soluções foi inspirada pela descomposição de

Fourier para a distribuição de momentos das partículas produzidas em colisões entre íons pesados. Duas soluções para os modos instáveis são obtidas e seus significados físicos discutidos.

The Gribov-Levin-Ryskin (GLR) equation is an evolution equation for gluon distribution functions which respects Froissart’s bound. Motivated by the non-obviousness of elliptic flow 𝑣2 scaling in transverse momentum, rapidity, system size, etc. being explained by

Hydrodynamics, and the similarity they have with Bjorken scaling, the GLR equation is modified by adding an angular dependence to it; therefore, turning it into a 2+1 nonlinear partial differential equation, which can present instabilities in its solutions, generating 𝑣2.

The general form of the solutions was inspired by the Fourier decomposition for the momentum distribution of produced particles in heavy-ion collisions. Two solutions for the unstable modes are found and their physical meaning is discussed.

Figure 1 – The SM explains how the building blocks of matter (quarks and leptons)

interact via three of the four fundamental forces. . . 19

Figure 2 – Experimental rate of hadron production to muon production for

electron-positron annihilation at high energies. Image taken from (SWARTZ,

1996). . . 22

Figure 3 – Left: s-channel diagram for elastic meson-meson scattering. Right:

t-channel diagram for elastic meson-meson scattering. . . 23

Figure 4 – A schematic view of the confinement mechanism. Here a energetic 𝑢𝑑

pair stretches the color string until the potential energy creates another

𝑢𝑑 pair. The process continues forming 𝑢𝑑 pairs until the kinetic energy

is low enough for them to form clusters of quarks and glons. . . 24

Figure 5 – Spin J vs. s(=𝑀2_{) for rho-mesons. Image taken from (ZWIEBACH,}

2004). . . 24

Figure 6 – A gluonic flux tube from a quark to an antiquark. . . 25

Figure 7 – A linear rising potential is an evidence of confinement. . . 25

Figure 8 – The more strange quarks are contained in the hadron, the stronger its

enhancement. Image taken from (ELIA; COLLABORATION et al.,

2013). . . 27

Figure 9 – A relativistic heavy-ion collision. Left: the two nuclei collide at high

energies with impact parameter b. Here, they are seen almost flat due

to length contraction from special relativity. Right: not all partons

as the distance between the centers of the colliding heavy ions. Image

taken from (CHATRCHYAN et al., 2013) . . . 30

Figure 11 – A non-central collision of two nuclei. The reaction plane is defined by

the impact parameter b and the direction Z of the beam. Image taken

from (SNELLINGS, 2011) . . . 30

Figure 12 – Radial flow . . . 32

Figure 13 – Left: Directed flow 𝑣1. Here we can see a shift of about 10% in the

positive x-direction. Being the reaction plane defined as the XZ plane,

we have on the right: positive (in-plane particle emissions) and negative

(out-of-plane particle emissions) 𝑣2. . . 33

Figure 14 – Left: Dust particles present independent paths. Right: fluid particles’s

expansion is determined by density gradients (white arrows). . . 34

Figure 15 – The hydrodynamic explanation for elliptic flow based on the perfect

fluid hypothesis.. . . 35

Figure 16 – Hydrodynamic simulations of 𝑣𝑛 for different centralities at low 𝑝𝑇.

Images taken from (GALE et al., 2013) and (HEINZ; SNELLINGS,

2013). . . 36

Figure 17 – 𝑣2 scaling with respect to rapidity 𝜂 (BUSZA; COLLABORATION et

al., 2009). The rapidity dependence can be factorized as can be seen

here from different center of mass energies and 0-40% centrality for

et al., 2013). The CMS and PHOBOS collaborations give us elliptic

flow divided by eccentricity as a function of the multiplicity density per

unit transverse overlap area in A+A collisions for different energies. . . 37

Figure 19 – 𝑣2 scaling with respect to the transverse momentum 𝑝𝑇 (SHI;

COL-LABORATION et al., 2013). Here, results for √𝑠=7.7 to 200 GeV are

for Au+Au collisions, while those for 2.76 TeV are for Pb+Pb collisions. 37

Figure 20 – 𝑣3scaling with respect to the number of charged particles (CHATRCHYAN

et al., 2013). As the initial geometry is different for PbPb and pPb

colli-sions, it is natural that their 𝑣2 are different. However, the fluctuations

𝑣3 are the same, so we wonder how it can be that there is the same

“Hydrodynamics” for large and small systems. . . 38

Figure 21 – Photon 𝑣2 and hadron 𝑣2 are comparable. The ratio of one to the other

is close to one at low 𝑝𝑇. (ADARE et al., 2012) . . . 38

Figure 22 – The cross section for the scattering of two protons producing a Higgs 𝐻

and it decaying into two photons can be factorized in terms of the cross

section for scattering of two gluons 𝜎𝑔𝑔→𝐻→𝛾𝛾 and the gluon distribution

functions 𝑓𝑔/𝑝(𝑥1,2, 𝑀𝐻). . . 41

Figure 23 – Parton distribution functions (PDF) are the same in ep, pp, eA, pA,

and AA collisions to leading order. Here we illustrate this by showing

a deep inelastic scattering process on the left and a heavy-ion collision

on the right. See appendix C for more on DIS. Also, in this figure

𝑓 (𝑥, 𝑄2_{) = 𝑥𝐺(𝑥, 𝑄}2_{) for gluons. . . . . 42}

Figure 24 – The Zeus data for the gluon structure functions. Image taken from

interacting region. . . 44

Figure 26 – At large values of x, the valence quarks dominate; however, at small x,

it is gluons and sea quarks the most abundant ones. Clearly, the gluon

density is the largest. Image taken from (KOVCHEGOV; LEVIN, 2012) 45

Figure 27 – The DGLAP evolution takes 𝑥𝐺(𝑥, 𝑄2

0) and evolves it to 𝑥𝐺(𝑥, 𝑄2), i.e.

if we measure the distribution function at 𝑄2

0 ≫ Λ𝑄𝐶𝐷, DGLAP will

give its value at a larger 𝑄2_{. . . . . 46}

Figure 28 – The BFKL evolution takes 𝑥0𝐺(𝑥0, 𝑄2) and evolves it to 𝑥𝐺(𝑥, 𝑄2), i.e.

if we measure the distribution function at some 𝑥0, BFKL will give its

value at a larger 𝑥. . . . 47

Figure 29 – DGLAP and BFKL evolution for gluon distribution functions. As we go

to higher energies, BFKL generates gluons faster than DGLAP. Image

taken from (KOVCHEGOV; LEVIN, 2012) . . . 48

Figure 30 – Spontaneous breaking of azimuthal symmetry. . . 52

Figure 31 – Gluon distribution functions’ asymptotics . . . 53

Figure 32 – 𝑄, 𝑥𝐵𝑗, and 𝜑 as analogs of the cylindrical coordinates 𝜌, 𝑧, and 𝜙,

respectively. Image taken from (ARFKEN; WEBER, 2005). . . 55

Figure 33 – Small x means high energy physics, so having 𝑥 → 0 as an initial

condition is the wrong way to start the evolution of our system as we

go down on energies. . . 60

Figure 34 – 𝑢2 vs. Q. Here, the plots overlap for three different 𝑥𝐵𝑗 and impact

parameters. . . 61

Figure 35 – 𝑢2 vs. x. We can see how 𝑢2 has almost null slope. Also, its value

x=0.03, 0.03-10−4, and 0.03+10−4. A very fast growth in the slope is

observed. Also, the growth of 𝑢2 is faster for higher Q. . . 66

Figure 37 – 𝑢2 vs. x for initial configuration 𝑢2(starting x)=10−4, with starting

x=0.003, 0.003-10−6, and 0.003+10−6. This means that, if we have

𝑢2(0.003) = 10−4 as an initial configuration, then our solution quickly

becomes unphysical, i.e 𝑢2>1. . . 66

Figure 38 – Parton 𝑣2 (dashed line) is obtained by means of the 𝑘𝑇-factorization

formula. Now using a fragmentation function, we get hadron 𝑣2 (solid

line).Error bars come from (CHATRCHYAN et al., 2012). . . 71

Figure 39 – A typical dN/d𝑦 distribution of gluons inside a hadron. . . . 80

Figure 40 – A typical dN/d𝑦 distribution of produced particles in a hadronic

colli-sion. The leading particles, clustered around the projectile and target

rapidities, are shown in blue and in red we can see the distribution of

produced mesons. . . 81

Figure 41 – Electron-proton scattering as seen in the proton rest frame. Figure

taken from (KOVCHEGOV; LEVIN, 2012). . . 82

Figure 42 – Experiments around the world show that structure function 𝐹2becomes

𝑄2_{-independent in the Bjorken limit, as can be seen in the lower part}

List of Figures . . . 10

Contents . . . 15

1 INTRODUCTION . . . 18

1.1 Setting the stage . . . 18

1.1.1 The Standard Model . . . 18

1.1.2 Quantum Chromodynamics. . . 19

1.1.3 Quark-Gluon plasma . . . 26

1.2 Elliptic flow . . . 28

1.2.1 Flow. . . 28

1.2.2 Elliptic flow 𝑣2 . . . 29

1.2.3 Common interpretation of 𝑣2: Hydrodynamics . . . 34

1.2.4 Scaling of 𝑣2 . . . 36

2 PARTON DISTRIBUTION FUNCTIONS AND THE ANGULAR DEPENDENCE PROPOSAL . . . 40

2.1 Factorization . . . 40

2.2 Evolution equations for parton densities . . . 42

2.2.1 Parton distribution functions . . . 42

2.2.2 Froissart’s bound . . . 43

equation. . . 49

2.3 Azimuthal dependence . . . 51

2.3.1 Motivation . . . 51

2.3.2 A simple model for 𝐺0(𝑥, 𝑄2) . . . 52

2.3.3 Adding the angular dependence . . . 55

2.3.4 Stability structure of 𝐺(𝑥, 𝑄2_{, 𝜑) . . . . 56}

3 RESULTS . . . 59

3.1 The polynomial solution . . . 59

3.1.1 Derivation . . . 59

3.1.2 Interpretation . . . 61

3.2 The Bessel function solution . . . 62

3.2.1 Derivation . . . 62

3.2.2 Interpretation . . . 66

3.3 Connecting 𝑢2 to 𝑣2 . . . 67

3.3.1 The 𝑘𝑇-factorization formula . . . 68

3.3.2 Fragmentation . . . 68

3.3.3 Hadron 𝑣2 from parton 𝑣2 . . . 69

4 CONCLUSIONS . . . 72

BIBLIOGRAPHY . . . 75

APPENDIX

78

APPENDIX B – DEEP INELASTIC SCATTERING (DIS) . . . 82 APPENDIX C – BJORKEN SCALING . . . 84

1 Introduction

The effort to understand the universe is one of the very few things which lifts human life a little above the level of farce and gives it some of the grace of tragedy.

Steven Weinberg

1.1

Setting the stage

In this section we’ll briefly review the Standard Model of particle physics, Quan-tum Chromodynamics, and Quark-gluon plasma to set the basis for this work.

1.1.1

The Standard Model

In the late 1970s the Standard Model (SM) of particle physics, created by Wein-berg, Salam and Glashow, began to set as the basic theory of matter. Here, the fun-damental constituents are spin-1₂ particles called quarks and leptons which interact by exchanging spin-1 particles (photons, gluons, and the W and Z bosons).

Up until a decade before the SM, protons, neutrons, pions, kaons, and other strongly interacting particles (hadrons) were thought to be elementary. However, the consensus that emerged in 1979 was that hadrons were composed of more basic building blocks called quarks and held together by gluons exchange (HODDESON, 1997).

Figure 1 – The SM explains how the building blocks of matter (quarks and leptons) in-teract via three of the four fundamental forces.

In figure 1 we can see the six flavors 𝑓 for quarks, i.e. f= u (up), d(down), c(charm), s(strange), t(top), and b(bottom) in purple, the six leptons: the electron (e), the muon (𝜇), and the tau (𝜏 ), plus their associated neutrinos (𝜈𝑒, 𝜈𝜇, 𝜈𝜏) in green, and the force carriers: the photon (𝛾), the gluon (g), and the Z and W bosons in red.

In this work we are concerned with strong interactions, so now we will briefly see what Quantum Chromodynamics (QCD) is, and study two of its main properties, i.e. color confinemet and asymptotic freedom.

1.1.2

Quantum Chromodynamics

As mentined before, the theory that governs the interactions between quarks and gluons to our present knowledge is called Quantum Chromodynamics, and it is contained

in the SM. Let’s see the lagrangian of the QCD sector ℒ𝑄𝐶𝐷 = − 1 4 ∑︁ 𝑎 𝐹_𝜇𝜈𝑎 𝐹_𝑎𝜇𝜈+ 𝑁𝑓 ∑︁ 𝑓 Ψ𝑓 (︃ 𝑖𝛾𝜇𝜕𝜇− 𝑔𝛾𝜇 ∑︁ 𝑎 𝐴𝑎_𝜇𝜆 𝑎 2 − 𝑚𝑓 )︃ Ψ𝑓, (1.1)

which is written in terms of the quark fields Ψ𝑓, the gluon fields 𝐴𝑎𝜇, the strong coupling constant g, the Dirac matrices 𝛾𝜇_{, and the Gell-Mann matrices 𝜆}𝑎_.

Here, quarks appear in six flavors, each one of them with its own charge and mass, and each of these six flavors can be in any of three different color states (red, blue, and green).

Table 1 – Quarks’ basic properties. Symbol, mass (MeV), and electric charge (in units of the absolute value for the electric charge of the electron) for the six quark flavors.

Name Symbol Mass (MeV) Electric charge (e)

Up u 2.3 2/3 Down d 4.8 -1/3 Charm c 1275 2/3 Strange s 95 -1/3 Top t 173 210 2/3 Bottom b 4180 -1/3

It is important to notice that QCD has been proven to be a very successful theory. As an example, we will see the simplest 𝑒+_𝑒−_{→ ℎ𝑎𝑑𝑟𝑜𝑛𝑠 process, i.e.}

𝑒+𝑒−→ 𝑞𝑞. (1.2)

For us to study this process, let’s see some interesting properties of the theory. The two main properties are known as asymptotic freedom and confinement. So, in order

to understand the former, first we need to see that the QCD coupling constant (running coupling) depends on the momentum 𝑞 of the gluons (HALZEN; MARTIN,2008); i.e.

𝑔2(𝑞2) = 16𝜋

2

(11₃𝑁𝑐− 2₃𝑁𝑓) ln(𝑞2/Λ2𝑄𝐶𝐷)

(1.3)

with QCD scale parameter Λ𝑄𝐶𝐷 ranging from approximately 200 to 300 MeV. Usually we work with

𝛼𝑠(𝑞2) = 𝑔2 4, (1.4) so we have 𝛼𝑠(𝑞2) = 4𝜋 𝛽0ln(𝑞2/Λ2𝑄𝐶𝐷) (1.5)

where 𝛽0 = 11𝑁₃𝑐 − 2₃𝑁𝑓, 𝑁𝑐 is the number of colors, and 𝑁𝑓 is the number of flavors.

If we ask ourselves what happens when quarks exchange highly energetic (𝑞2 _≫

Λ2

𝑄𝐶𝐷) gluons, we will see from equation 1.5 that the running coupling constant 𝛼𝑠 be-comes very small, i.e.

𝛼𝑠(𝑞2) → 0, 𝑎𝑠 𝑞 → ∞. (1.6)

This way, quarks and gluons behave as nearly free particles, so they are asymp-totically free. In this case, as the running coupling is a small number, perturbative QCD (pQCD) is an ideal tool for solving problems.

Now, as we have seen in the high-energy limit, the strong interaction can be neglected, then it can be shown that (PESKIN; SCHROEDER, 1995)

𝜎(𝑒+𝑒−→ ℎ𝑎𝑑𝑟𝑜𝑛𝑠) → 3(∑︁

𝑖

𝑄2_𝑖)𝜎(𝑒+𝑒− → 𝜇+_𝜇−

) (1.7)

in the 𝐸𝑐.𝑚. → ∞ limit for both processes. Here 𝑄𝑖 are the quark charges, and the sum runs over all quarks whose masses are < 𝐸𝑐.𝑚./2. Let’s rewrite 1.7 as

𝜎(𝑒+_𝑒−_{→ ℎ𝑎𝑑𝑟𝑜𝑛𝑠)}

𝜎(𝑒+_𝑒− _{→ 𝜇}+_𝜇−₎ → 3

∑︁

𝑖

𝑄2_𝑖, (1.8)

so that all we have to do is sum the electric charge of the quarks involved. From table 1 and 𝐸𝑐.𝑚. = 3GeV, we have

𝜎ℎ𝑎𝑑𝑟𝑜𝑛𝑠 𝜎𝜇+_𝜇− = 3 (︂₆ 9 )︂ = 2 (1.9)

for up, down, and strange quarks (the lightest ones).

Figure 2 – Experimental rate of hadron production to muon production for electron-positron annihilation at high energies. Image taken from (SWARTZ, 1996).

From the first solid line in figure2we can see how theory agrees with experiment in the high energy limit. Further QCD higher-order corrections give us the dotted line.

So now we wonder what happens when 𝑞 → Λ𝑄𝐶𝐷 in equation 1.5. In this case, the running coupling becomes large, so we can no longer use pQCD. This region is known as non-perturbative QCD and the main property of the theory here is known as confinement.

In the late 1960’s an interesting model known as the dual resonance model de-scribed hadrons as strings. This behavior was later interpreted in terms of scattering of strings. The basic idea is that for an elastic scattering of mesons 𝑝1 + 𝑝2 → 𝑝3 + 𝑝4, we

have the following lowest-order Feynman diagrams 𝑝3 𝑝𝐽 𝑝4 𝑝1 𝑝2 𝑝3 𝑝4 𝑝𝐽 𝑝1 𝑝2

Figure 3 – Left: s-channel diagram for elastic meson-meson scattering. Right: t-channel diagram for elastic meson-meson scattering.

so that the s-channel and the t-channel amplitudes are the same, i.e. they give two alternatives (or dual) descriptions of the same physics.

This duality can be properly realized assuming mesons to be extended objects instead of point particles. The structure given to them is that of a string connecting a quark and an antiquark as shown in figure 4.

Figure 4 – A schematic view of the confinement mechanism. Here a energetic 𝑢𝑑 pair stretches the color string until the potential energy creates another 𝑢𝑑 pair. The process continues forming 𝑢𝑑 pairs until the kinetic energy is low enough for them to form clusters of quarks and glons.

The scattering amplitude for the t-channel (s-channel) is the sum of all the amplitudes from all the t-channel (s-channel) diagrams with different spin (J) values.

Figure 5 – Spin J vs. s(=𝑀2_{) for rho-mesons. Image taken from (ZWIEBACH, 2004).}

So, an excited 𝑞𝑞 system is described as a string whose quark and antiquark are located at its ends and move with large momentum. This excited string fragments into two hadrons with new 𝑞𝑞 pairs created between the separating quark and antiquark. The process continues until the kinetic energy is low enough for them to form clusters of quarks and gluons, so that we cannot observe free quarks; i.e. they are confined into hadrons.

This model is interesting for phenomenology, but it can not be considered as the basis for fundamental theories. More elegant models use non-Abelian gauge fields (see equation1.1) where confinement is a conjecture that a system of coupled massless charged gluons is unstable in vacuum, so they move to a state where only massive excitations can propagate. In such a state a gluonic flux around quarks forms into tubes needed for linear confinement.

Figure 6 – A gluonic flux tube from a quark to an antiquark.

A flux tube is a real physical object and it is the storage medium for the linearly rising interquark potential (CREUTZ, 1983). As can be seen in lattice computations, QCD must be confining (see (FRASCA,2011) )

1.1.3

Quark-Gluon plasma

Let’s stop talking about confinement, and let’s go back to the other QCD main property; i.e. asymptotic freedom, since it is more relevant for this work. It is known that the discovery of asymptotic freedom by Gross, Politzer and Wilczek opened the door for the study of matter at conditions of pressure and temperature as extreme as the one found in neutron-star cores, and in the early Big Bang Universe. This super dense matter is thought to be made of quarks instead of hadrons. For this reason, the name of Quark-Gluon plasma (QGP) was given to it.

On February 10th, 2000, it was announced by CERN that compelling evidence for the existence of a new state of matter in which quarks, instead of being bound up into more complex particles such as protons and neutrons, are liberated to roam freely. The collisions created temperatures over 100 000 times as hot as the centre of the sun, and energy densities twenty times that of ordinary nuclear matter, densities which have never before been reached in laboratory experiments. This state of matter found in heavy-ion collisions features many of the characteristics of the theoretically predicted quark-gluon plasma (CERN, 2000).

There are many important experimental findings supporting the discovery of QGP. See, for example, (HEINZ; JACOB, 2000). Let’s comment on a very popular one; i.e. strangeness enhancement.

In 1982, Johann Rafelski and Berndt M¨𝑢ller showed us that enhanced abun-dances of rare, strange hadrons (Ω, Λ, etc.) can be used as indicators for the formation of QGP (RAFELSKI; MÜLLER, 1982).

Strangeness enhancements are defined as ratios1.10, so, as the ALICE collabo-ration worked with Pb-Pb collisions, they defined strangeness enhancement as

𝑠𝑡𝑟𝑎𝑛𝑔𝑒 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑦𝑖𝑒𝑙𝑑𝑠 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑖𝑛 𝑃 𝑏 − 𝑃 𝑏 𝑐𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛𝑠 𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑 𝑡𝑜 ⟨𝑁𝑝𝑎𝑟𝑡⟩

𝑠𝑡𝑟𝑎𝑛𝑔𝑒 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 𝑦𝑖𝑒𝑙𝑑𝑠 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑖𝑛 𝑝𝑝 𝑐𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛𝑠 𝑎𝑡 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑒𝑛𝑒𝑟𝑔𝑦 , (1.10)

where ⟨𝑁𝑝𝑎𝑟𝑡⟩ is the mean number of participant nucleons (ELIA;

COLLABO-RATION et al.,2013).

Figure 8 – The more strange quarks are contained in the hadron, the stronger its enhance-ment. Image taken from (ELIA; COLLABORATION et al., 2013).

Figure8shows us the strangeness enhancement results for Λ(|S|=1), Ξ−(|S|=1), and Ω−+Ω+(|S|=1) in Pb-Pb collisions at center of mass energy of 2.76 TeV, as functions of the mean number of participant nucleons ⟨𝑁𝑝𝑎𝑟𝑡⟩. Here we can see that the enhancements

are >1 and increase with ⟨𝑁𝑝𝑎𝑟𝑡⟩. Also, they increase with strangeness content of produced particles.

All in all, if strangeness enhancement disappears in AA collisions at energies where the QGP phase is nonexistent or negligible, then this would mean that strangeness enhancement is a real deconfinement signature (TORRIERI,2011).

So far we have given evidence for the existence of QGP, so in the next section we will study an important property of it known as elliptic flow. We will define elliptic flow, discuss a little bit about its usual interpretation in terms of Hydrodynamics, and see experimental data elliptic flow scaling is based on.

1.2

Elliptic flow

1.2.1

Flow

As mentioned before, it is thought that matter in the first microseconds after the Big Bang and in the core of dense neutron stars is composed of quarks and gluons in a deconfined state. For this reason, it is of major interest for QCD phenomenology the study of matter at extreme densities and temperatures. So, one question arises: how can we study matter at such extreme conditions here on Earth? The answer relies on relativistic heavy-ion collisions, since they are a unique tool to create and study such states of hot QCD matter under controlled conditions (SNELLINGS, 2011).

Spectators

Participants b

before collision after collision

Figure 9 – A relativistic heavy-ion collision. Left: the two nuclei collide at high ener-gies with impact parameter b. Here, they are seen almost flat due to length contraction from special relativity. Right: not all partons participate in the collision. Image taken from CERN (CERN,2000).

As in the early Universe, the hot and dense system of quarks and gluons created in a heavy-ion collision will expand and cool down forming matter, e.g. protons and neutrons. The passing of this matter from QGP to hadrons is done through a collective expansion which is known as flow.

1.2.2

Elliptic flow 𝑣

In order to better understand this flow, let’s see some definitions first.

Impact parameter

If we think of heavy ions as extended objects, as the ones shown in the left side of figure 9, the impact parameter b is defined as the distance between their centers in a plane transverse to the beam axis. Despite not being directly observed, this quantity is important for defining two other concepts which are the reaction plane and the centrality.

Figure 10 – Geometry of a heavy-ion collision. The impact parameter 𝑏 is defined as the distance between the centers of the colliding heavy ions. Image taken from (CHATRCHYAN et al., 2013)

Reaction plane

Figure 11 is how we imagine a non-central collision of, say, two nuclei. As we can see, the x-axis is drawn parallel to the impact parameter b and the z-axis is parallel to the beam direction (black arrows). This way the zx-plane defines the reaction plane (RP).

Figure 11 – A non-central collision of two nuclei. The reaction plane is defined by the impact parameter b and the direction Z of the beam. Image taken from (SNELLINGS, 2011)

Centrality

As seen in figures 9, 10, and 11, not all partons participate in the collisions; never-theless, we would like to have an idea of how many of them do. Unfortunately, the number of participants 𝑁𝑝𝑎𝑟𝑡 and/or the number of spectators 𝑁𝑠𝑝𝑒𝑐 are not directly measurable (ABELEV et al., 2013). For this reason, we define a useful quantity called centrality c, which is expressed as a fraction of the total cross section 𝜎𝑡𝑜𝑡𝑎𝑙. For example, in an nucleus-nucleus collision (AA collision) with impact parameter b, c is defined integrating the b-distribution 𝑑𝜎/𝑑𝑏 as follows

𝑐(𝑏) = ∫︀𝑏 0(𝑑𝜎/𝑑𝑏 ′ )𝑑𝑏′ ∫︀∞ 0 (𝑑𝜎/𝑑𝑏 ′ )𝑑𝑏′ ≤ 1 (1.11)

In conclusion, central collisions (small b) mean there is a large participating zone (also known as fireball); i.e. large 𝑁𝑝𝑎𝑟𝑡. On the contrary, peripheral collisions (large b) are characterized by their small number of participants (large number of spectators 𝑁𝑠𝑝𝑒𝑐).

Now that we are studying the geometrical aspects of flow, we would like to distin-guish between radial flow and anisotropic flow.

Let’s imagine a perfectly central collision (c(b=0)=1) where the fireball experiences a radially symmetric pressure gradient (figure 12a). In this scenario, the produced particles develop a radial velocity component (figure 12b).

This expansion is what is known as radial flow. Of course, this is an extreme scenario of zero transverse momentum produced particles. Consequently, we go beyond radial flow and study the anisotropy in particle momentum distributions correlated with the reaction plane. This anisotropy is called anisotropic flow and we can see it in figure 11, where we notice that the fireball is almond-shaped.

(a) The pressure gradient is radially symmetric.(b) The produced particles get a radial velocity component.

Figure 12 – Radial flow

It was the work of Voloshin and Zhang (VOLOSHIN; ZHANG, 1996) to translate these ideas into mathematical form by making a Fourier expansion of the momentum distribution with respect to the reaction plane (CHATRCHYAN et al., 2013)

𝐸𝑑 3_𝑁 𝑑3_𝑝 = 𝑑3𝑁 𝑝𝑇𝑑𝑝𝑇𝑑𝑦𝑑𝜑 = 1 2𝜋 𝑑2𝑁 𝑝𝑇𝑑𝑝𝑇𝑑𝑦 [︃ 1 + ∞ ∑︁ 𝑛=1 2𝑣𝑛(𝑝𝑇, 𝑦)𝑐𝑜𝑠[𝑛(𝜙 − Ψ𝑅𝑃)] ]︃ (1.12)

Here we have the particle energy 𝐸, the transverse momentum 𝑝𝑇, the rapidity 𝑦, the azimuthal angle 𝜙, the angle of the reaction plane Ψ𝑅𝑃 (as defined in figure10), the number of participants N, and the Fourier coefficients 𝑣𝑛1.

These 𝑣𝑛 are of special interest for us, since they reflect the different types of anisotropies. Let’s take a look at the first two of them.

1 _{See Appendix A for definitions of 𝑝}

𝑥 𝑦 1 0.5 -1 -0.5 0 1 0.5 -0.5 -1 𝑣1 = 10% 𝑥 𝑦 1 0.5 -1 -0.5 0 1 0.5 -0.5 -1 𝑣2 = 10% 𝑣2 = −10%

Figure 13 – Left: Directed flow 𝑣1. Here we can see a shift of about 10% in the positive

x-direction. Being the reaction plane defined as the XZ plane, we have on the right: positive (in-plane particle emissions) and negative (out-of-plane particle emissions) 𝑣2.

The first harmonic 𝑣1 is called directed flow and it represents an overall shift of the

distribution in the transverse plane.

The second harmonic 𝑣2 is called elliptic flow and it is the most dominant

modula-tion of the azimuthal particle producmodula-tion.

It is worth noting that 𝑣2 is the asymmetry that characterizes out-of-plane or

in-plane particle emissions, as can be seen in figure 13. Here 𝑣2 = 10% means there is

more flow in the reaction plane and 𝑣2 = −10% means there is more flow out of the

reaction plane. Also, 𝑣2 is commonly thought to be a consequence of the azimuthal

anisotropy in the transverse plane of the fireball, and that it is not created at an early stage of the process, so that it may be described by hydrodynamics.

1.2.3

Common interpretation of 𝑣

: Hydrodynamics

Hydrodynamics is a classical theory which plays an important role in science, as this theoretical framework allows us to describe the motion of fluids (SOUZA; KOIDE; KODAMA,2016). Ideal hydrodynamics is characterized by the system being much larger than its infinitesimal elements, and the latter being much larger than the constituents. For example, if our system is the atmosphere of the Earth, a volume of one squared meter can be taken as the infinitesimal element, and the air as the constituent. Also, the system must expand slowly with respect to the microscopic evolution. This means that we are working in local thermodynamic equilibrium. As we are interested in the study of relativistic heavy-ion collisions, the fluid’s parameters to be taken into account are the energy density 𝑒 and pressure 𝑝(𝑒), the flow velocity of the fluid 𝑢𝜇and the chemical potential. A small deviation from ideal hydrodynamics gives us a scenario where the dynamics depend on the gradients 𝜕𝑢 and 𝜕𝑒, so that new parameters arise: shear (𝜂) and bulk viscosity (𝜁). As shown in figure 14, dust particles ignore each other, so a particle’s path is independent of its neighbors. On the other hand, fluid particles continuously interact among them, hence their expansion is determined by density gradients.

Figure 14 – Left: Dust particles present independent paths. Right: fluid particles’s ex-pansion is determined by density gradients (white arrows).

The common interpretation of 𝑣2 relies on the hypothesis that matter in

heavy-ion collisheavy-ions behaves like a perfect fluid (extremely low viscosity). Here, initial anisotropies in the collision area produce anisotropies in the collective flow of matter (LUZUM; PETERSEN, 2014). This idea can be better understood with the help of figure 15where we can see how two nuclei collide. One is going in-plane and the other out-of-plane, and when they collide the collision region is mainly elliptical, this is the initial anisotropy. In this fireball, higher density gradients are present in the reaction plane, leading to a larger flow in this plane than out of it.

Hydrodynamic evolution Larger Larger flow flow smaller flow smaller flow Nucleus (Going out−of−plane) (Going in−plane) Nucleus Collision region (size~multiplicity)

Figure 15 – The hydrodynamic explanation for elliptic flow based on the perfect fluid hypothesis.

In fact, hydrodynamical simulations nicely fit experimental data (GALE et al.,

2013) at low 𝑝𝑇 (figure 16). Now, one question arises: does hydrodynamics allow for the simple observed 𝑣𝑛 scaling across sizes and energies? This theory involves many parameters such as initial temperature, equation of state, transport coefficients, freezeout condition, etc. which vary non-trivially with system size and contribute significantly to 𝑣𝑛 (TORRIERI; BETZ; GYULASSY, 2012; HATTA et al., 2014). Let’s take a look at some experimental puzzles of heavy ion collisions which challenge hydrodynamics as the origin of 𝑣𝑛.

0 0.05 0.1 0.15 0.2 0 0.5 1 1.5 2 〈vn 2 〉 1/2 p_T [GeV] ATLAS 10-20%, EP η/s =0.2 v₂ v₃ v₄ v₅ 0 0.05 0.1 0.15 0.2 0 0.5 1 1.5 2 〈vn 2 〉 1/2 pT [GeV] ATLAS 20-30%, EP narrow: η/s(T) wide: η/s=0.2 v₂ v3 v4 v₅

Figure 16 – Hydrodynamic simulations of 𝑣𝑛 for different centralities at low 𝑝𝑇. Images taken from (GALE et al., 2013) and (HEINZ; SNELLINGS,2013).

1.2.4

Scaling of 𝑣

We have just seen that hydrodynamic simulations impressively fit experimental data from RHIC and LHC. However, as we just pointed out in the previous subsection, the large number of parameters which contribute non-trivially to 𝑣2 contrasts with the

fact that data across sizes and energies looks remarkably simple. In this subsection it will be shown different scalings of 𝑣2 and 𝑣3. 2.

η-y

Au+Au 0-40%

Figure 17 – 𝑣2 scaling with respect to rapidity 𝜂 (BUSZA; COLLABORATION et al.,

2009). The rapidity dependence can be factorized as can be seen here from different center of mass energies and 0-40% centrality for Au-Au collisions.

2 _{See appendixAfor definitions of the variables of 𝑣} 2

) -2 ( fm η /d ch (1/S) dN 0 10 20 30 40 50 60 ε /₂ v 0.05 0.1 0.15 0.2 0.25 0.3 0.35 PbPb 2.76 TeV AuAu 200 GeV AuAu 62.4 GeV CuCu 200 GeV CuCu 62.4 GeV CMS PHOBOS

Figure 18 – 𝑣2 scaling with respect to transverse multiplicity density (CHATRCHYAN et

al.,2013). The CMS and PHOBOS collaborations give us elliptic flow divided by eccentricity as a function of the multiplicity density per unit transverse overlap area in A+A collisions for different energies.

0

0.05

0.1

0.15

0.2

Fit to 200 GeV data

0.1 0.2 (b1) 20 - 30% 2.76 TeV 200 GeV 62.4 GeV 39 GeV 0.1 (c1) 30 -40% 27 GeV 19.6 GeV 11.5 GeV 7.7 GeV

0

1

2

3

0

1

2

3

0

1

2

3

(GeV/c)

T

p

2

v

{

4

}

Figure 19 – 𝑣2 scaling with respect to the transverse momentum 𝑝𝑇 (SHI;

COLLABO-RATION et al., 2013). Here, results for √𝑠=7.7 to 200 GeV are for Au+Au collisions, while those for 2.76 TeV are for Pb+Pb collisions.

offline trk N 0 100 200 300 3

v

v

Figure 20 – 𝑣3 scaling with respect to the number of charged particles (CHATRCHYAN

et al.,2013). As the initial geometry is different for PbPb and pPb collisions, it is natural that their 𝑣2 are different. However, the fluctuations 𝑣3 are the

same, so we wonder how it can be that there is the same “Hydrodynamics” for large and small systems.

A very important experimental result is photon 𝑣2 which is comparable

quan-titatively to hadron 𝑣2 (see image 21), and nearly equal to it at low 𝑝𝑇. Also, since photons are created at every stage of the process, i.e. they are not just a final state product. Therefore, photon 𝑣2 tells us that elliptic flow, in general, can be created at an

early stage. This is the last experimental puzzle that will be shown in this work. See (GAMBINI; TORRIERI, 2016) for more.

Figure 21 – Photon 𝑣2 and hadron 𝑣2 are comparable. The ratio of one to the other is

As we have just seen, asymptotic freedom, one of the main properties of Quantum Chromodynamics, allows the existence of a new state of matter composed of deconfined quarks and gluons at very high temperatures. This hot QCD matter can be studied by colliding heavy ions at high energies in particle accelerators. The main property of the produced particles after the fireball created in HIC, studied in this work, is that of elliptic flow, which is commonly interpreted as the hydrodynamic evolution of a perfect fluid which has initial anisotropies. Also, 𝑣2 is believed to be a final state product.

So far we have seen how scalings of elliptic flow 𝑣2 in energy, rapidity, etc.,

scalings of 𝑣3 in system size, and photon 𝑣2 being comparable to hadron 𝑣2 are not

obvious from the hydrodynamic picture (TORRIERI; BETZ; GYULASSY, 2012). This puzzles make us wonder if hydrodynamics gives the correct description for the origin of 𝑣𝑛. So, in the next chapter we will study an alternative to it given by instabilities of the initial state. We will explain how adding an angular dependence to parton distribution functions, in the form of a small perturbation, could be used to produce elliptic flow and explain its scaling.

2 Parton distribution functions and the

an-gular dependence proposal

I think that only daring speculation can lead us further and not accumulation of facts.

Albert Einstein

In the last chapter we saw that asymptotic freedom, one of the main properties of QCD, allows quarks and gluons to behave as nearly free particles, due to the small value of the running coupling constant at high values of the virtuality 𝑞2 of the probe (see equation 1.5). In this section, we will see that a consequence of this property is that hadronic collisions become relatively simple at high energies, so that perturbation theory can be applied to a subset of the process, leading to factorization of the total cross sections.

2.1

Factorization

Because of asymptotic freedom, the strong interactions effectively become weak (small 𝛼𝑠(𝑞2)) over short distances (large energy exchange 𝑞2), so that hadrons can be treated as a system of nearly free quarks and gluons, collectively known as partons in this context.

scat-tering cross sections in hadronic processes can be divided into three parts. The hadron is described in terms of incoherent constituents (partons), these partons undergo the hard scattering, and finally they form into hadrons again. The first and third parts can not be treated using perturbation theory, but they are universal, i.e. the description of hadrons in terms of quarks and gluons is the same in all processes, only having predictable de-pendence on energy (BURGESS; MOORE, 2006). On the other hand, the second part can be treated by means of perturbation theory, arising from asymptotic freedom. As an example, we have factorization for the 𝐻 → 𝛾𝛾 process

Figure 22 – The cross section for the scattering of two protons producing a Higgs 𝐻 and it decaying into two photons can be factorized in terms of the cross section for scattering of two gluons 𝜎𝑔𝑔→𝐻→𝛾𝛾 and the gluon distribution functions 𝑓𝑔/𝑝(𝑥1,2, 𝑀𝐻).

Here the cross section 𝜎𝑝𝑝→𝐻→𝛾𝛾 for the production and decay of H, for example via 𝑔 + 𝑔 → 𝐻, at lowest order in 𝑔𝑆 can be factorized

𝜎𝑝𝑝→𝐻→𝛾𝛾 = 𝜎𝑔𝑔→𝐻→𝛾𝛾𝑓𝑔/𝑝(𝑥1, 𝑀𝐻)𝑓𝑔/𝑝(𝑥2, 𝑀𝐻) + ... (2.1)

The cross section 𝜎𝑔𝑔→𝐻→𝛾𝛾 for scattering of two gluons can be computed as a perturbation series, while the gluon distribution functions 𝑓𝑔/𝑝(𝑥, 𝜇2) can not.

2.2

Evolution equations for parton densities

The main purpose of this work is to give an alternative to Hydrydynamics for the explanation of 𝑣2 scaling. To do so, we argue that elliptic flow must be an initial state

property of the system of deconfined quarks and gluons. For this reason, in this section we will define what parton distributions are, and see that the BFKL evolution presents a paradigm shift with respect to the DGLAP evolution. As important as it is, BFKL can’t avoid to break Froissart’s bound; for this reason, we move to the GLR evolution which corrects BFKL’s divergence problem, but keeps the same evolution in 𝑥𝐵𝑗.

2.2.1

Parton distribution functions

In the sixties, Feynman and Bjorken argued that high-energy experiments should reveal the existence of particles that are parts of hadrons. Those hadrons’ constituents were named partons. The experimental verification arrived in the deep inelastic scattering of electrons on protons accomplished at the Standford Linear Accelerator (SLAC) in 1969. This way, partons were identified with quarks and gluons.

000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111

f(x,Q)

f(x,Q)

Figure 23 – Parton distribution functions (PDF) are the same in ep, pp, eA, pA, and AA collisions to leading order. Here we illustrate this by showing a deep inelastic scattering process on the left and a heavy-ion collision on the right. See appendix C for more on DIS. Also, in this figure 𝑓 (𝑥, 𝑄2_{) = 𝑥𝐺(𝑥, 𝑄}2₎

As we are interested in how these partons are distributed with respect to their momenta inside hadrons, we define parton distribution functions 𝑥𝐺(𝑥, 𝑄2) as the prob-ability density for finding a particle with a momentum 𝑝 = 𝑥𝐵𝑗𝑃 at resolution scale 𝑄2 (see figure 23).

Figure 24 – The Zeus data for the gluon structure functions. Image taken from ( MCLER-RAN, 2002)

2.2.2

Froissart’s bound

An important consequence of the unitarity of the S-matrix is the Froissart’s bound, which is a limit for the growth of total particle scattering cross sections at high energies. In this subsection we will see an intuitive derivation of it, originally attributed to Heisenberg 1_{. Consider the following hadron-hadron scattering 2 → 2 at high energies.} This reaction can occur only if there is enough energy in the interaction area to produce at least a pair of pions.

It is reasonable to think that the strength of the interaction should fall off as the impact parameter b increases. This can be seen in figure 25, as the interacting area

b

Figure 25 – A Hadron-hadron collision at high energies. The shaded area is the interacting region.

decreases with the impact parameter b. Also, taking into account that the probability for this reaction to happen is likely to decrease for the production heavier particles, we have

𝐼𝑛𝑡𝑒𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑒𝑛𝑒𝑟𝑔𝑦 ∝ (𝑖𝑛𝑡𝑒𝑟𝑎𝑐𝑡𝑖𝑛𝑔 𝑎𝑟𝑒𝑎) × (𝑡𝑜𝑡𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦) (2.2)

𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑖𝑛𝑡𝑒𝑟𝑎𝑐𝑡𝑖𝑜𝑛 ∝ 𝑒−𝑚𝑏×√𝑠, (2.3)

where 𝑠 is the total squared center of mass energy. Since the minimum we need to produce is a pair of pions, then the average energy of two pions (𝑘0) gives us a lower

limit for the interaction energy. Moreover, taking 𝑚 = 𝑚𝜋 (the lightest mass for a QCD bound state) gives us a higher bound for the interaction energy. Therefore, we get

𝑘0 ≤ 𝑘𝑒−𝑚𝜋𝑏

√

𝑏𝑚𝑎𝑥 = 1 𝑚𝜋 ln𝑘 √ 𝑠 𝑘0 . (2.5)

So, a rough estimate of the total cross section reads

𝜎 ≃ 𝜋𝑏2_𝑚𝑎𝑥 = 𝜋 𝑚2 𝜋 ln2 𝑘 √ 𝑠 𝑘0 ∝ ln2_{𝑠 (2.6)}

This limit for the growth of total cross sections was first derived by Marcel Froissart in 1961 and it’s known as the Froissart’s bound (FROISSART, 1961).

2.2.3

Linear evolution equations for gluon distribution functions

Now that we have briefly reviewed what parton distribution functions are and given an intuitive derivation of the Froissart’s bound, let’s study the linear evolution in Q and x of these PDFs.

Figure 26 – At large values of x, the valence quarks dominate; however, at small x, it is gluons and sea quarks the most abundant ones. Clearly, the gluon density is the largest. Image taken from (KOVCHEGOV; LEVIN, 2012)

As can be seen in figure26, quark contributions are subleading. Thus, as we are working in high energy (small x) scenarios, we will work with gluon distribution functions only.

Resummation of series in powers of 𝛼𝑠ln 𝑄2, also known as leading logarith-mic approximation (LLA) in 𝑄2_{, were first considered by Dokshitzer, Gribov, Lipatov,}

Altarelli, and Parisi to give us an evolution equation (DGLAP) for gluon distribution functions: 𝑄2 𝜕 𝜕𝑄2[𝑥𝐺(𝑥, 𝑄 2_{)] =} 𝛼𝑠(𝑄2)𝐶𝐴 𝜋 ∫︁ 1 𝑥 𝑑𝑧 𝑧 [𝑧𝐺(𝑧, 𝑄 2_{)]. (2.7)}

An important feature of these types of equations is that they don’t allow the complete calculation of the gluon distribution function 𝑥𝐺(𝑥, 𝑄2_{), but only its evolution}

from some starting point, e.g. some 𝑄0 ≫ Λ𝑄𝐶𝐷 for DGLAP. DGLAP

ln 1/x

𝑄2 𝑄2₀ 𝑄2

Figure 27 – The DGLAP evolution takes 𝑥𝐺(𝑥, 𝑄2

0) and evolves it to 𝑥𝐺(𝑥, 𝑄2), i.e. if we

measure the distribution function at 𝑄2₀ ≫ Λ𝑄𝐶𝐷, DGLAP will give its value at a larger 𝑄2_.

As we said before, high energy QCD is the theory of small Bjorken’s x. In this limit, resummations of series in the LLA-in-1/x give us the Balitsky-Fadin-Kuraev-Lipatov (BFKL) evolution equation for the unintegrated2 _{gluon distribution function}

𝜕𝜙(𝑦, ⃗𝑘) 𝜕𝑦 = 𝛼𝑠𝐶𝐴 2𝜋2 ∫︁ 𝑑2𝑝⃗ (2𝜋)2 ⃗𝑘2 ⃗ 𝑝2_{(⃗𝑘 − ⃗𝑝)}2 [︁ 2𝜙(𝑦, ⃗𝑝) − 𝜙(𝑦, ⃗𝑘)]︁, (2.8) where 𝑦 = ln(1/𝑥) is the rapidity of the gluon. The most remarkable feature of this evolution is that it allows for an exponential growth for the gluon distribution function

𝜙(𝑦, ⃗𝑘) ∼ 𝑒𝜔 ¯𝛼𝑠𝑦_{, (2.9)}

where 𝜔 = 4 ln 2 in leading order.

BFKL ln 1/x

ln1/𝑥

ln1/𝑥0

𝑄2

Figure 28 – The BFKL evolution takes 𝑥0𝐺(𝑥0, 𝑄2) and evolves it to 𝑥𝐺(𝑥, 𝑄2), i.e. if we

measure the distribution function at some 𝑥0, BFKL will give its value at a

larger 𝑥.

2 _{The unintegrated gluon distribution function is definded as 𝜙(𝑥, 𝑘}

As is shown in figure28, BFKL turns out to be the product of a paradigm shift in resummation parameters. However, the BFKL evolution has a problem: it generates gluons at higher energies (smaller 𝑥𝐵𝑗) faster than the DGLAP evolution (see figure 29)

Figure 29 – DGLAP and BFKL evolution for gluon distribution functions. As we go to higher energies, BFKL generates gluons faster than DGLAP. Image taken from (KOVCHEGOV; LEVIN,2012)

It can be shown that this rapid growth of the number of gluons leads to the violation of the Froissart’s bound (KOVCHEGOV; LEVIN, 2012), as it predicts

𝜎𝐵𝐹 𝐾𝐿_𝑡𝑜𝑡 ∼ 𝑠𝛼𝑃−1_{, (2.10)}

and Froissart’s bound tells us that the total cross section 𝜎𝑡𝑜𝑡 in QCD cannot grow faster than the logarithm of the energy squared

𝜎𝑡𝑜𝑡 ≤ 𝐶 ln2𝑠, (2.11)

where C is a constant. As a consequence, we need to find a way to solve this problem keeping the evolution in 𝑥𝐵𝑗 (BFKL evolution).

2.2.4

The nonlinear Gribov-Levin-Ryskin and Mueller-Qiu (GLR-MQ)

evolu-tion equaevolu-tion

As an alternative to the BFKL evolution equation, the GLR equation for the unintegrated gluon distribution funtions was first derived in (GRIBOV; LEVIN; RYSKIN,

1983), using the same kernel as the BFKL’s. In this work we will use the equation for the integrated gluon distribution functions 𝑥𝐺(𝑥, 𝑄2_{) which is (KOVCHEGOV; LEVIN,}

2012) 𝑄 2 𝜕 𝜕𝑄 𝜕[𝑥𝐺(𝑥, 𝑄2_)] 𝜕𝑙𝑛(1/𝑥) = 𝛼𝑠𝑁𝑐 𝜋 [𝑥𝐺(𝑥, 𝑄 2_)] ⏟ ⏞ 𝐵𝐹 𝐾𝐿 − 𝛼 2 𝑠𝑁𝑐𝜋 2𝐶𝐹𝑆⊥ 1 𝑄2[𝑥𝐺(𝑥, 𝑄 2_)]2 ⏟ ⏞ 𝑛𝑜𝑛−𝑙𝑖𝑛𝑒𝑎𝑟 𝑡𝑒𝑟𝑚 . (2.12)

Here 𝐶𝐹 = (𝑁𝑐2− 1)/(2𝑁𝑐) is the Casimir operator in the fundamental represen-tation of 𝑆𝑈 (𝑁𝑐) and 𝑆⊥ = 𝜋𝑅2 is the cross sectional area of our target; e.g. a proton.

Also, the strong coupling constant 𝛼𝑠 is a small parameter (𝛼𝑠< 1) since we are working in pQCD.

Notice the quadratic term in the GLR-MQ evolution equation (equation 2.12). This is the fundamental correction made to the BFKL equation which is the product of gluons merging3_{. This process of fusioning gluons slows down and then stops the growth} of gluon distribution functions when going into smaller and smaller 𝑥𝐵𝑗 (higher energies), saturating the hadron (see figure 29).

The saturation region is defined by the saturation scale 𝑄𝑠, which is the value for Q when the gluon distribution function stops growing. Mathematically, this region is defined by 𝑄 2 𝜕 𝜕𝑄 𝜕[𝑥𝐺(𝑥, 𝑄2_)] 𝜕𝑙𝑛(1/𝑥) ⃒ ⃒ ⃒ ⃒ ⃒_𝑄=𝑄 𝑠(𝑥) = 0, (2.13)

which is the same as (see equation2.12)

𝛼𝑠𝑁𝑐 𝜋 [𝑥𝐺(𝑥, 𝑄 2 𝑠(𝑥))] − 𝛼2 𝑠𝑁𝑐𝜋 2𝐶𝐹𝑆⊥ 1 𝑄2 𝑠(𝑥) [𝑥𝐺(𝑥, 𝑄2_𝑠(𝑥))]2 = 0 (2.14)

Notice that Froissart’s bound, which is a very general property of total particle cross sections at high energies, breaks in the saturation region for the BFKL evolution, but it doesn’t do it in the GLR-MQ evolution.

To sum up, if we go into smaller and smaller 𝑥𝐵𝑗 regions (higher energies), we reach the saturation region where the growth of gluon distribution functions ceases and the BFKL evolution breaks Froissart’s bound. However, this problem is solved by the nonlinear term of the GLR-MQ evolution equation. In the next section we will modify this equation by adding another degree of freedom in order to turn it into a 2+1 nonlinear partial differential equation that could present instabilities.

2.3

Azimuthal dependence

We will begin this section by motivating the addition of an angular dependence to parton distribution functions. Next, we will model a solution for the azimuthally symmetric gluon distribution function 𝑥𝐺0(𝑥, 𝑄2). After giving a naive derivation for

a modified GLR equation that has an angular dependence, we will perturbate the az-imuthally symmetric (saturation) solution 𝐺0(𝑥, 𝑄2) inspired by the Fourier expansion of

the momentum distribution of 𝑣2, so that an equation for the Fourier coefficients of the

non azimuthally symmetric gluon distribution function can be found.

2.3.1

Motivation

As we have seen in the previous section in equation 2.12, the GLR-MQ evolu-tion equaevolu-tion does not carry the angular variable dependence. Therefore, its soluevolu-tions should be azimuthally symmetric. However, neglecting the angular dependence may have consequences when testing the stability of the solutions.

So, what would happen if the angular dependence in gluon distribution functions were not neglected? Well, our nonlinear differential equation would gain another degree of freedom, turning into a 2+1 nonlinear differential equation, hence becoming susceptible to present instabilities.

To give an example of how important the role of instabilities can be, let’s take a look at the following figure

Figure 30 – Spontaneous breaking of azimuthal symmetry.

As we can see, the more we open the lid, the faster the water flows (it is more energetic) hence passing from a nice symmetric flux of water to an amorphous one. In the same way, we think that rising the energy (going down in 𝑥𝐵𝑗) will have, as a consequence, a spontaneous azimuthal symmetry breaking for gluon distribution functions.

2.3.2

A simple model for 𝐺

(𝑥, 𝑄

)

The full form of parton distribution functions is unknown. Nevertheless, we know their asymptotics for small and large 𝑄 as compared to the saturation scale 𝑄𝑠(𝑥) = 𝛼2

u n k n o w n 𝐺0(𝑥, 𝑄2)𝛼4𝑠/𝑥2𝜆

1

Q 𝑄𝑠(𝑥)

Figure 31 – Gluon distribution functions’ asymptotics

Here 𝛼𝑠 is our running coupling constant in pQCD, so it’s a small number (smaller than 1), Λ𝑄𝐶𝐷 is the QCD scale parameter, and 𝜆 is a parameter related to the pomeron intercept4_.

In order to model azimuthally symmetric gluon distribution functions 𝑥𝐺0(𝑥, 𝑄2),

we will use a series of functions ℎ𝑛 defined by

ℎ𝑛(𝑥) = 1

2(1 + tanh(𝑛𝑥)) , (2.15)

the infinite-n limit of which is the Heaviside step function

Θ(𝑥) = lim 𝑛→∞ℎ𝑛(𝑥) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0, 𝑥 < 0 1, 𝑥 > 0 (2.16)

4 _{In pre-QCD language, hadronic cross sections were described in terms of the exchange of a}

hypothet-ical particle with the quantum numbers of the vacuum known as the pomeron. Here 𝜎𝑡𝑜𝑡∼ 𝑠𝛼(0)−1.

So, being 𝑠 and 𝑡 Mandelstam variables, 𝛼(𝑡 = 0) ≡ 𝛼𝑃 intercepts the second axis in the (𝑡, 𝛼)-plane;

This way we propose 𝐺0(𝑥, 𝑄2) 𝛼4 𝑠 𝑥2𝜆 = 1 2 [︃ (1 − tanh(𝜉)) + (1 + tanh(𝜉))𝑄 2 𝑠(𝑥) 𝑄2 ]︃ (2.17)

as our model function inspired by the ℎ𝑛(𝑥). Here we have defined the dimen-sionless quantity 𝜉 by

𝜉 = 𝑄 − 𝑄𝑠(𝑥)

𝜁 (2.18)

along with “skin depth” 𝜁 given by

𝜁 = (︃ (2𝜆 + 1)(𝛼𝑠𝑁𝑐)𝜋 (𝛼𝑠𝑁𝑐+ (𝜋 + 𝛼𝑠𝑁𝑐)𝑥)𝑥𝜆 )︃_(︂ 𝛼𝑠 𝑁𝑐 )︂ Λ𝑄𝐶𝐷 ∼ 𝛼𝑠 𝑁𝑐 Λ𝑄𝐶𝐷 ≪ Λ𝑄𝐶𝐷 (2.19)

so that we get the desired asymptotic behaviour

𝐺0(𝑥, 𝑄2) 𝛼4_𝑠 𝑥2𝜆 = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 , 𝑄 ≪ 𝑄𝑠(𝑥) (𝑄𝑠(𝑥)/𝑄)2 , 𝑄 ≫ 𝑄𝑠(𝑥) (2.20)

resembling the Heaviside step function (2.16), modeling figure 31.

Now that we have a function 𝐺0(𝑥, 𝑄2) to model gluon distribution functions

that are azimuthally symmetric, the next step is to modify the equation that governs their evolution in order to test their stability.

2.3.3

Adding the angular dependence

As we have seen before, the GLR-MQ equation reads

𝑄 2 𝜕 𝜕𝑄 𝜕[𝑥𝐺(𝑥, 𝑄2_)] 𝜕𝑙𝑛(1/𝑥) = 𝛼𝑠𝑁𝑐 𝜋 [𝑥𝐺(𝑥, 𝑄 2_{)] −} 𝛼2𝑠𝑁𝑐𝜋 2𝐶𝐹𝑆⊥ 1 𝑄2[𝑥𝐺(𝑥, 𝑄 2_)]2_{. (2.21)}

Adding an azimuthal angular 𝜑 dependence to it will be understood under the following reasoning. Let’s think of our variables as the ones in a cylindrical coordinate system; i.e. 𝑄 ∼ 𝜌, 𝑥𝐵𝑗 ∼ 𝑧, and 𝜑 ∼ 𝜙

Figure 32 – 𝑄, 𝑥𝐵𝑗, and 𝜑 as analogs of the cylindrical coordinates 𝜌, 𝑧, and 𝜙, respec-tively. Image taken from (ARFKEN; WEBER, 2005).

As we know, for a scalar function 𝑓 = 𝑓 (𝜌, 𝑧, 𝜙), we have

𝑑𝑓 = ∇𝑓 · 𝑑r = (︃ 𝜕𝑓 𝜕𝜌𝜌 +^ 1 𝜌 𝜕𝑓 𝜕𝜙𝜙 +^ 𝜕𝑓 𝜕𝑧𝑧^ )︃ · 𝑑r,

so that there exists the unitary vector ^𝑙 defined by

𝜕𝑓 𝜕𝜌𝜌 +^ 1 𝜌 𝜕𝑓 𝜕𝜙𝜙 =^ (︃ 𝜕𝑓 𝜕𝜌 + 1 𝜌 𝜕𝑓 𝜕𝜙 )︃ ^ 𝑙, (2.22)

which allows us to unite 𝜕𝜌 and 𝜕𝜙 into a single derivative (the right hand side of equation 2.22). Bearing this in mind, we propose adding 𝜑 to 𝑄 and 𝑥𝐵𝑗, uniting 𝜕𝑄 and 𝜕𝜑 into a single derivative in the aforementioned form, i.e.

𝜕 𝜕𝑄 → 𝜕 𝜕𝑄 + 1 𝑄 𝜕 𝜕𝜑 (2.23)

This way the non-azimuthally symmetric GLR-MQ evolution equation (nasGLR-MQ) reads 𝑄 2 (︃ 𝜕 𝜕𝑄 + 1 𝑄 𝜕 𝜕𝜑 )︃ 𝜕[𝑥𝐺(𝑥, 𝑄2_{, 𝜑)]} 𝜕𝑙𝑛(1/𝑥) = 𝛼𝑠𝑁𝑐 𝜋 [𝑥𝐺(𝑥, 𝑄 2_{, 𝜑)] −} 𝛼2𝑠𝑁𝑐𝜋 2𝐶𝐹𝑆⊥ 1 𝑄2[𝑥𝐺(𝑥, 𝑄 2_{, 𝜑)]}2 (2.24)

2.3.4

Stability structure of 𝐺(𝑥, 𝑄

, 𝜑)

Now, we want to solve equation2.24. For this purpose, we propose the following ansatz which resembles the Fourier decomposition in the 𝑝𝑇 and 𝑦 distribution of produced particles (see equation 1.12)

𝐺(𝑥, 𝑄2, 𝜑) = 𝐺0(𝑥, 𝑄2) (︃ 1 + ∞ ∑︁ 𝑛=1 𝑢𝑛(𝑥, 𝑄2) cos(𝑛𝜑 + 𝛽𝑛) )︃ . (2.25)

This form for 𝐺(𝑥, 𝑄2, 𝜑) allows us to test its stability structure. How so? Well, as mentioned before, 2+1 nonlinear differential equations can have instabilities. In our case, if the solutions for nasGLR-MQ didn’t present instabilities, i.e. there were no 𝜑-dependence on the gluon distribution functions, then all 𝑢𝑛(𝑥, 𝑄2) would vanish.

Let’s proceed by inserting 2.25 into 2.24, taking into consideration the form of 𝐺0(𝑥, 𝑄2) modeled in 2.17. After some algebra we get to an equation of the form

𝐴 + ∞ ∑︁ 𝑛=1 𝐵𝑛sin(𝑛𝜑) + ∞ ∑︁ 𝑛=1 𝐶𝑛cos(𝑛𝜑) = 0 (2.26)

where the the linear independence of the set {1, sin 𝜑, cos 𝜑, sin 2𝜑, cos 2𝜑...} gives us

𝐴 = 0, 𝐵𝑛 = 0, 𝐶𝑛= 0.

The first one of these equations is

𝐴 = 𝑁𝑐𝜋 16𝛼2 𝑠𝐶𝐹𝑆⊥ 1 𝑄2 (︃ 𝑥2𝜆+1 [︃ (1 − tanh(𝜉)) + (1 + tanh(𝜉))𝑄 2 𝑠(𝑥) 𝑄2 ]︃)︃2 ∞ ∑︁ 𝑛=1 𝑢2_𝑛(𝑥, 𝑄2)𝑐𝑜𝑠(2𝛽𝑛) = 0. (2.27) Therefore, ∞ ∑︁ 𝑛=1 𝑢2_𝑛(𝑥, 𝑄2)𝑐𝑜𝑠(2𝛽𝑛) = 0. (2.28)

Equations 𝐵𝑛 = 0 and 𝐶𝑛 = 0 seem to be much harder to solve, so we will analyze them in a limit of particular interest; i.e. 𝑄 ≪ 𝑄𝑠(𝑥). Here, tanh(𝜉) → −1; as a result, we get two sets of equations

𝑥𝜕𝑢𝑛(𝑥, 𝑄 2₎ 𝜕𝑥 = −(2𝜆 + 1)𝑢𝑛(𝑥, 𝑄 2 ) + 𝑁𝑐𝜋 2𝛼2 𝑠𝐶𝐹𝑆⊥ 𝑥2𝜆+1 𝑄2 1 𝑛 [︃_𝑛−1 ∑︁ 𝑘=1 𝑢𝑘(𝑥, 𝑄2)𝑢𝑛−𝑘(𝑥, 𝑄2) sin(𝛽𝑛− 𝛽𝑘− 𝛽𝑛−𝑘) +2 ∞ ∑︁ 𝑘=1 𝑢𝑘(𝑥, 𝑄2)𝑢𝑛+𝑘(𝑥, 𝑄2) sin(𝛽𝑛+ 𝛽𝑘− 𝛽𝑛+𝑘) ]︃ (2.29) and (2𝜆 + 1)𝑄 2 𝜕𝑢𝑛(𝑥, 𝑄2) 𝜕𝑄 + 𝑄𝑥 2 𝜕2_𝑢 𝑛(𝑥, 𝑄2) 𝜕𝑄𝜕𝑥 = 𝛼𝑠𝑁𝑐 𝜋 𝑢𝑛(𝑥, 𝑄 2₎ + 𝑁𝑐𝜋 2𝛼2 𝑠𝐶𝐹𝑆⊥ 𝑥2𝜆+1 𝑄2 [︃ 2𝑢𝑛(𝑥, 𝑄2) +1 2 𝑛−1 ∑︁ 𝑘=0 𝑢𝑘(𝑥, 𝑄2)𝑢𝑛−𝑘(𝑥, 𝑄2) cos(𝛽𝑛− 𝛽𝑘− 𝛽𝑛−𝑘) + ∞ ∑︁ 𝑘=0 𝑢𝑘(𝑥, 𝑄2)𝑢𝑛+𝑘(𝑥, 𝑄2) cos(𝛽𝑛+ 𝛽𝑘− 𝛽𝑛+𝑘) ]︃ , (2.30)

the latter being of special interest, since it gives us the evolution of the 𝑢𝑛(𝑥, 𝑄2) in 𝑄 and 𝑥𝐵𝑗.