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SMALL X EVOLUTION WITH IMPACT PARAMETER

DEPENDENCE

A Dissertation in The Physics Department of Penn State by Jeﬀrey J. Berger

c 2012 Jeﬀrey J. Berger

Submitted in Partial Fulﬁllment of the Requirements for the Degree of

Doctor of Philosophy

December 2012 The dissertation of Jeﬀrey J. Berger was reviewed and approved∗ by the following:

Anna M. Sta´sto Assistant Professor of Physics Dissertation Advisor, Chair of Committee

Irina Mocioiu Associate Professor of Physics

Mark Strikman Professor of Physics

Pe´ter Me´sza´ros Professor of Physics Eberly Chair of Astronomy and Astrophysics

Richard W. Robinett Professor of Physics Director of Physics Graduate Studies

∗Signatures are on ﬁle in the Graduate School. Abstract

The small Bjorken x regime is becoming more accessible with the higher energies of modern (LHC) and potential future (LHeC,EIC) colliders. In this regime the total cross section is governed by the so-called Pomeron exchange. In Quantum Chromodynamics (QCD) this object is described by the Balitsky-Fadin-Kuraev- Lipatov (BFKL) evolution equation. At high energy and for dense targets the BFKL equation is superseded by the Balitsky-Kovchegov (BK) evolution equation which includes important effects which are non-linear in the parton density. In this dissertation the numerical solution of the BFKL and BK equations will be presented with full impact parameter dependence. The inclusion of impact parameter dependence in the evolution introduces novel and non-trivial dynamics. The analysis was done in the framework of the dipole model, which is discussed in the introduction of this dissertation. The BFKL and BK equations are then derived in the dipole model and discussed. The numerical solution to the BK equation is presented for both the leading logarithm (LL) approximation and in the running coupling case. Kinematic effects which are beyond the LL approximation are taken into account and found to be non-trivial when impact parameter dependence is considered. The behaviors of the dipole scattering amplitude are discussed in detail. Confinement effects are modelled by including an effective gluon mass into the dipole evolution kernel. This mass regulates the splitting of large dipoles. The solution to the BK equation with a mass scale and running coupling was used to calculate the F2 and FL proton structure functions. Phenomenological corrections are included in order to take into account non-perturbative effects in the structure functions. The model presented is compared to the inclusive DIS data taken from HERA and reasonable agreement is found. Exclusive elastic vector meson production in DIS is also considered. The solution of the BK evolution equation is used to compute the differential cross section for exclusive production of J/Ψ, φ, and ρ, vector mesons. This allows a comparison to a wide range of experimental data from HERA and good agreement is found between our model and the experimental data.

iii Table of Contents

List of Figures vii

List of Tables x

Acknowledgments xi

Chapter 1 Introduction 1 1.1 DeepInelasticScattering...... 3 1.1.1 ThePartonModel ...... 6 1.1.2 DGLAPevolutionandscalingviolation ...... 9 1.2 Small x physics ...... 20 1.2.1− Pre-QCDdevelopments: ReggeTheory ...... 20 1.2.2 TheFroissartBound ...... 23

Chapter 2 Background 26 2.1 BFKLPhysics...... 26 2.1.1 The BFKL Pomeron: Regge theory meets QCD ...... 26 2.1.2 TheDipoleModel...... 30 2.1.3 Mueller’s description of the BFKL equation ...... 32 2.1.4 SymmetriesoftheBFKLequation ...... 43 2.1.5 SolutiontotheBFKLequation ...... 44 2.1.5.1 Special case : dipole-dipole scattering...... 47 2.2 Saturationphysics ...... 50 2.2.1 SaturationintheColorGlassCondensate ...... 51 2.2.2 Multiple scatterings - the Balitsky-Kovchegov equation ... 53

iv 2.2.2.1 SolutionsoftheBKequation ...... 58

Chapter 3 NumericalsolutionoftheBKequation 61 3.1 Numericalmethods ...... 62 3.1.1 Kinematicvariables...... 67 3.1.2 Parallelizationofthecode ...... 69 3.2 Solution without impact parameter dependence ...... 71 3.2.1 Saturationscale...... 73 3.3 The solution with impact parameter dependence ...... 76 3.3.1 Dependenceonthedipolesize ...... 77 3.3.2 Dependenceonimpactparameter ...... 80 3.3.3 Angular dependence and enhancement at r =2b ...... 80 3.3.4 Analysisofamodiﬁedkernel...... 87 3.3.5 The saturation scale with impact parameter dependence .. 91 3.3.5.1 Saturationscaleatsmalldipolesize ...... 94 3.3.5.2 Saturation scale at large dipole size ...... 98 3.3.6 Inclusionoftherunningcoupling ...... 100 3.3.6.1 Comparison with the minimum dipole prescription 104 3.3.6.2 Applying the running coupling to the modiﬁed kernel 107 3.3.6.3 Regularization sensitivity of the Balitsky kernel . . 108 3.3.7 Growthofthecrosssection ...... 113

Chapter 4 Phenomenological application of the solution to the BK equation120 4.1 Regulationoflargedipolesizes ...... 121 4.1.1 Confinementinthefixedcouplingcase ...... 121 4.1.2 Confinementintherunningcouplingcase ...... 127 4.2 InclusiveDISmeasurements ...... 128 4.2.1 Corrections at small Q2 ...... 131 4.3 Exclusivevectormesonproduction ...... 137 4.3.1 Phenomenologicalcorrections ...... 142 4.3.1.1 Q2 + M 2 dependence of the cross section ...... 144 4.3.1.2 W dependenceofthecrosssection ...... 145 4.3.1.3 Ratio of the transverse and longitudinal cross section 148 4.3.1.4 Momentum transfer dependence of the differential crosssection ...... 148 4.3.1.5 Slope of the differential cross section ...... 149

v Chapter 5 Conclusion and Outlook 153

Appendix A The Mellin transform 156

Appendix B Interpolation 157

Bibliography 160

vi List of Figures

1.1 KinematicdiagramofinclusiveDIS ...... 5 1.2 CartoonofDISwithinthepartonmodel ...... 6 1.3 DiagramofDISintheBreitframe ...... 8 1.4 Plot of F2 structure function data from various experiments . . . . 10 1.5 Cartoon of DIS with gluon and sea quark emission...... 11 1.6 Diagrams for single gluon emission from a quark in DIS...... 12 1.7 AmplitudesforsinglegluonemissioninDIS ...... 14 1.8 Squares of Feynman diagrams for real single gluon emissioninDIS. 15 1.9 Cartoon of the DGLAP evolution in the Q2 and x plane...... 17 1.10 Diagrams of various DGLAP splitting functions ...... 17 1.11H1andZEUSPDFﬁt ...... 19 1.12 A generic 2 2scatteringprocess...... 21 → 1.13 An exchange of a Reggeon between two particles...... 21 1.14 A plot of particle mass squared versus spin ...... 22

2.1 Twogluonexchangebetweentwoparticles ...... 27 2.2 BreakdownoftheLipatovvertex ...... 28 2.3 Emissionofnsoftgluons...... 29 2.4 Lowest order of the Pomeron in QCD, two gluon exchange . . . .. 30 2.5 DISinthedipolemodel ...... 30 2.6 Diagram for initial qq¯ statewithoutgluonemission ...... 32 2.7 Two Feynman diagrams of single gluon emission from a q q¯ pair. . . 32 2.8 Diagrams of single dipole emission in position and momentum space. 33 2.9 Diagramofmanyemittedcolordipoles...... 34 2.10 Pictorial representation of the dipole generating function Z ..... 38 2.11 Cartoon of the BFKL evolution in the Q2 and x plane...... 42 2.12 Cartoon two dipoles exchanging a Pomeron...... 48 2.13 Variables used in Lipatov’s solution to the BFKL equation ..... 50 2.14 CartoonillustratingtheCGC ...... 52 2.15 Convolutions of n(r,Y, r′) and γ(r′) being summed to N(r,Y ) ... 55 2.16 Multipledipolescatterings ...... 57

vii 2.17 Pomeronfandiagram...... 58 2.18 Cartoon of the BK evolution in the Q2 and x plane...... 59

3.1 Diagramofadipole-targetsystem ...... 62 3.2 Kinematicvariablesinpositionspace ...... 67 3.3 Scattering amplitude with LL kernel without impact parameter . . 72 3.4 LL kernel saturation scale Qs without impact parameter ...... 75 3.5 LO kernel solution with second angular degree of freedom φ .... 77 3.6 LO kernel solution versus r for fixed b ...... 78 3.7 Cartoon of a large dipole failing to interact ...... 79 3.8 LO kernel solution versus b for fixed r ...... 81 3.9 r =2b peak of the scattering amplitude in r fortheLOkernel . . . 82 3.10 Dipole orientation contributions to the r =2b peak ...... 83 3.11 Scattering amplitude vs. r highlighting angular dependence . . . . . 84 3.12 LO kernel solution versus the angle between r and b ...... 85 3.13 r =2b peakinimpactparameterspace ...... 86 3.14 Modified kernel without impact parameter ...... 90 3.15 Modified kernel solution vs b and r ...... 92 3.16 Illustration of the two solutions to the saturation scaleequation . . 93 3.17 Modified and LL kernel saturation scale Qs at small r vs. y ..... 94 3.18 Modified and LL kernel saturation scale Qs at small r vs. b ..... 95 3.19 Modified and LL kernel saturation scale QsL at large r ...... 98 3.20 Modified and LL kernel saturation scale QsL vs. impact parameter . 101 3.21 Running coupling solution vs. r and b ...... 103 3.22 Running coupling saturation scales Qs and QsL vs. Y ...... 105 3.23 Minimum dipole and Balitsky prescription for running coupling . . 107 3.24 Mod and LL kernels with min dipole coupling vs. r and b ...... 109 3.25 Mod and LL kernels with min dipole coupling for Qs and QsL . . . 110 3.26 Comparison of two methods of freezing the running coupling.. . . . 111 3.27 Bal prescription without b dependence, varying µ in the coupling . . 112 3.28 Interaction regions between the target and the dipole probe. . . . . 114 3.29 Bs vs y for fixed r and kernels with running and fixed coupling. . . 118 4.1 BK evolution with LL kernel with Bessel function cutoff ...... 123 4.2 BK evolution with LO kernel with split theta function cutoff . . . . 124 4.3 BK evolution with LO kernel with full theta function cutoff. . . . . 125 4.4 Impact parameter tails with a massive cutoff ...... 126 4.5 Minimum dipole and Balitsky prescription for running coupling . . 128 4.6 Contribution differences between Min and Bal prescriptions. . . . . 129 4.7 DiagramofinclusiveDISinthedipolemodel...... 129

viii 4.8 F2 structure function plot with dipole and a soft contribution plot 1 133 4.9 F2 structure function plot with dipole and a soft contribution plot 2 134 4.10 F2 structure function plot with dipole and VMD contributions plot 1 136 4.11 F2 structure function plot with dipole and VMD contributions plot 2 137 4.12 The amplitude for exclusive vector meson production in the dipole model ...... 138 4.13 σ for production of J/Ψ with skewed gluon distribution...... 143 2 2 4.14 σ for ρ production plotted as a function of (Q + MV )...... 144 2 2 4.15 σ for J/Ψ production plotted as a function of (Q + MV )...... 145 4.16 W dependence of the cross section for ρ production ...... 146 4.17 W dependence of the cross section for φ production ...... 146 4.18 W dependence of the cross section for JΨproduction ...... 147 4.19 R = σL for ρ φ and J/Ψ ...... 147 σT 4.20 dσ/dt for J/Ψ production with ﬁxed W in bins of Q2 vs. t . . . . 149 4.21 dσ/dt for J/Ψ production with ﬁxed Q2 in bins of t vs. W| | . . . . 149 2 2 | | 4.22 BD vs. Q + MV for ρ,φ,J/Ψ...... 150 4.23 BD vs W for J/ψ and ρ production ...... 151 4.24 Comparison between models in b space ...... 152

B.1 Numericalinterpolationdiagram...... 157

ix List of Tables

3.1 λs for various values ofα ¯s andkernels...... 98 3.2 λsL for various values ofα ¯s andkernels...... 100 3.3 λs and λsL forrunningcouplingkernels...... 108 3.4 λBD for various values ofα ¯s andkernels...... 116 3.5 λsB for various values ofα ¯s andkernels...... 119 4.1 Free parameters used in the initial condition...... 131 4.2 Free parameters used in the vector meson wave functions...... 141 4.3 Free parameters used in the photon wavefunction correction (4.46). 144

x Acknowledgments

I would like to thank my advisor Anna Sta´sto who has worked with me over the last three years. She has mentored me in small x physics as well as programming. I am also grateful for the many opprotunities to− travel to conferences and summer schools which broadened my view of where my research ﬁts into the physics community. I would also like to thank Emil Avsar for the many discussions I have had on small x physics, as well as Henry Kowalski for allowing our group the usage of his Fortran− code used to compute exclusive vector meson production observables. Thanks goes to John Collins who not only taught me the quantum ﬁeld theory that I know but for letting me use his computer system, which unfortunely perished in the line of duty. Finally I would like to thank my friends and family who endured my graduate school with me for keeping it bearable. Especially to all my friends and family who went through the thesis writing with me: my mother Olympia [1],Ross Martin- Wells [2] and his wife Kassandra Martin-Wells [3],Evelyn Duran [4], Jing Liang [5], Stephen Bongiorno [6], and Joshua Wickman [7].

xi Chapter 1

Introduction

If you wish to make an apple pie from scratch you must ﬁrst invent the universe.

Carl Sagan

This dissertation is based on the work that I have done with Dr. Anna Sta´sto from 2009-2012 on evolution equations at small Bjorken x. The work presented is based on the following publications1 and proceedings:

Jeﬀrey Berger and Anna M. Sta´sto Numerical solution of the nonlinear evo- • lution equation at small x with impact parameter and beyond the LL approximation, Phys.Rev. D83 (2011) 034015 arXiv:1010.0671 [hep-ph].

Jeﬀrey Berger and Anna M. Sta´sto Small x nonlinear evolution with impact • parameter and the structure function data, Phys.Rev. D84 (2011) 094022 arXiv:1106.5740 [hep-ph].

Jeﬀrey Berger and Anna M. Sta´sto Exclusive vector meson production and • small-x evolution, submitted to JHEP arXiv:1205.2037 [hep-ph].

1At the time of writing this dissertation much of the work presented here has been published in Phys Rev D. The appropriate citations are given on this page as well as in the bibliography at the end of this work. The copyright to this material and all ﬁgures that have been published in these papers and have been reproduced in this dissertation, in part or in full, belongs to the APS. The work found in these papers has been reused under the open permission for use in dissertations given by the APS. 2

Jeﬀrey Berger Diﬀractive Vector Meson Cross Sections from BK evolution • with Impact Parameter, Deep Inelastic Scattering 2012, Bonn Germany, to be published.

Jeﬀrey Berger Numerical analysis in BK evolution with impact parameter, • QCD Evolution Workshop: from collinear to non collinear case, Newport News VA, To be published by World Scientiﬁc.

Jeﬀrey Berger Nonlinear evolution at small x with impact parameter, Deep • Inelastic Scattering 2011, Newport News VA, To be published by the Amer- ican Institute of Physics.

The kinematic regime of very high center of mass energy has become the fore- front of collider physics, especially in the era of the LHC (Large Hadron Collider) and possible future colliders such as LHeC [8] and the EIC [9]. This regime is characterized by small values of the Bjorken x variable. The small-x region has been studied at DESY with the HERA collider in electron-proton collisions as well as heavy ion collisions at RHIC. The thrust of this dissertation shall be to present the numerical work done on the Balitsky-Fadin-Kuraev-Lipatov (BFKL) and Balitsky-Kovchegov (BK) evolution equations. The comparison in this dissertation with data has been restricted to electron-proton deep inelastic scattering (DIS), but there is nothing preventing further application of this work to other processes. I shall attempt to give a sufficient background in this dissertation such that a student who is familiar in quantum field theory may become acquainted with the terminology and mathematics of this specific area in high energy particle physics. In the first chapter a brief description of some basic background on the kinematics of DIS are discussed as well as outlining the derivation of the Dokshitzer-Gribov- Lipatov-Alterelli-Parisi (DGLAP) evolution equation. Then we will motivate the study of small x physics with a discussion of the pre-QCD Regge theory and some − implications from works of that era. In chapter 2 QCD will be applied to the small x region, first with a discussion − of the construction of the BFKL Pomeron. From there the BFKL equation will be derived in the dipole model. I will also discuss the symmetries and solution of 3 the BFKL equation. The BK evolution equation will be derived which extends the BFKL dynamics into the saturation regime. Chapter 3 is devoted to the numerical solution of the BFKL and BK equations, both with and without impact parameter dependence. The difference between the dynamics of the system with and without impact parameter dependence is nontrivial and is discussed in depth as well as the numerical methods used to solve the BFKL and BK equations. In chapter 4 the solution of the BK equation is applied to various DIS processes, specifically to the description of the proton structure functions as well as exclusive production of vector mesons at HERA. Conclusions are stated in chapter 5.

1.1 Deep Inelastic Scattering

Deep inelastic scattering (DIS) is one of the most important and powerful probes of the internal structure of the nucleon. In this section we will describe DIS initially within the framework of the naive parton model, and then in more rigor by invoking QCD. We shall discuss the DGLAP evolution equation as well as violation of Bjorken scaling behavior. In DIS an incoming electron impacts a target nucleon and interacts via the exchange of a virtual photon. The virtual photon is able to probe the constituents of the target with very high resolution as with high energies the wavelength of this virtual photon is very small, on the order of 10−16 cm. DIS experiments have been done with both ﬁxed targets (such as at SLAC [10] and BCDMS [11]) where an electron (or muon) beam impacts a stationary target, as well as with counter- propagating proton and electron (or positron) beams (such as at HERA [12,13]). At HERA these counter-propagating electron and proton beams collided at center of mass energies 318 GeV, producing the highest energy DIS data available today. Consider a charged lepton (e.g. electron) interacting with the nucleon via the exchange of a virtual photon, as in Fig. 1.1. The incoming charged lepton has momentum k and the outgoing lepton has momentum k′ after exchanging a virtual photon with the nucleon. We concern ourselves only with neutral currents in this dissertation and ignore the relatively small contribution from exchange of the Z boson. The virtual photon carries a momentum of 4

q = k k′ . (1.1) µ µ − µ We may now deﬁne the following variables that describe the kinematics of the event,

Q2 q2 0 , (1.2) ≡ − ≥ Q2 Q2 x = , (1.3) ≡ 2p q Q2 + W 2 · W 2 (p + q)2 , (1.4) ≡ q p y · , (1.5) ≡ k p · s (k + p)2 . (1.6) ≡

The photon virtuality is denoted as Q2, and x is the Bjorken scaling variable. The inelasticity y is deﬁned as the fraction of the incoming electron’s energy that is transfered to the proton. W is the center of mass energy of the virtual photon- proton system and s is the square of the electron-proton center of mass energy. Note that the proton mass has been neglected in (1.3). The photon virtuality Q2 determines the typical distance that can be resolved by the virtual photon probe, which is on the order of 1/Q. At the HERA collider ∼ the kinematic range extended from 10−1 GeV2 up to 104 GeV2 in Q2 and from 10−5 to greater than 10−1 in x. As can be seen from (1.3) when W 2 the value of → ∞ x becomes very small. The diﬀerential cross section for the process in Fig. 1.1 can be written as

dσ α2 =2π em W Lµν , (1.7) dxdQ2 x2Q2s2 µν where α 1 is the ﬁne structure constant. Note that we have treated the em ≈ 137 interaction as being composed of two parts, the leptonic vertex Lµν and the interaction with the target denoted by Wµν. The leptonic vertex can be computed using Quantum Electrodynamics (QED) and has the following form2

2In (1.8) the electron mass is neglected. 5

Incoming electron Outgoing electron

k k′ q Virtual photon

p X Target

Figure 1.1. Kinematic diagram of inclusive DIS where an incoming electron exchanges a virtual photon with a proton. As this is an inclusive process the proton may transition to any ﬁnal state denoted by X.

1 Lµν = Tr[kγ/ µk/′γν]=2(k′µkν + kµk′ν gµνk k′) . (1.8) 2 − ·

The hadronic component is deﬁned by the tensor Wµν given by

1 W (q,p)= P j (0)† X X j (0) P (2π)4δ(4)(p + q p ) , (1.9) µν 4π h | µ | ih | ν | i − x X X where jµ(y) represents the electromagnetic current, px is the momentum carried by the ﬁnal state X, P is the initial state of the proton. The summation is performed over all ﬁnal states X. Conservation of the electromagnetic current µ leads to q Wµν = 0. This condition combined with the conservation of parity

Wµν = Wνµ further restricts the functional form of the hadronic tensor. The form of Wµν which is consistent with Lorentz invariance can be parameterized as

q q p q p q W = F g µ ν + F µ µ ν ν (p q) . (1.10) µν − 1 µν − q2 2 p q − q2 p q − q2 · · · 2 2 The scalar functions F1 and F2 are functions of Q and x so F1 = F1(Q ,x) 2 and F2 = F2(Q ,x). They describe the internal structure of the nucleon and are referred to as the structure functions. It is important to note here that as we have 2 2 2 neglected the weak force contributions the results here are valid for Q < MW ,MZ . 6

Incoming electron Outgoing electron

k k′ q l p

Quarks l′

Figure 1.2. Cartoon of DIS within the parton model, where the proton is composed of three partons which are noninteracting. The virtual photon scatters oﬀ of a single parton with initial momentum l and ﬁnal momentum l′.

1.1.1 The Parton Model

We continue our analysis of inclusive DIS by considering the parton model. The parton model assumes that the structure of the target is an assembly of point-like constituents known as partons [14] (in QCD these partons are the quarks and gluons). The interactions between the partons are neglected as are any emissions from these partons, and although this model is rather simplified it is nevertheless very instructive. In the data from SLAC it was found that there was an independence from the momentum scale probed by the photon Q2 and that the data depended only on the Bjorken x variable [15]. As we shall see, the parton model successfully describes this feature of the data. A diagram of DIS in the parton model is illustrated in Fig 1.2 where the incoming proton is assumed to be composed of free point-like partons and the virtual photon scatters off of a single parton. Resolving a single parton is impossible if the interactions between partons occur on the order of the same timescale as the interaction between the probe and the struck parton. We shall work in the infinite momentum frame, where the target is moving extremely fast and is time dilated. The interaction time between the constituent partons is very long compared to the interaction time between the photon and the struck parton (provided that the value of Q2 is large). In the infinite momentum frame the four momentum of the struck parton in Fig 1.2 is 7

lµ =(l, 0, 0,l) . (1.11)

In this frame the scaling variable x can be interpreted in the parton model as being the fraction of the proton’s longitudinal momentum that is carried by the parton interacting with the virtual photon.3 To prove that in the inﬁnite momentum frame this is a valid interpretation of x we take the partons in Fig 1.2 to be on shell

0= l′2 =(l + q)2 =2l q + q2 =2ξp q Q2 , (1.12) · · − where ξ is the momentum fraction of the proton’s momentum carried by the struck parton. Combining equations (1.12) and (1.3) we ﬁnd

Q2 2ξp q x = = · = ξ . (1.13) 2p q 2p q · · This shows that in the chosen frame the deﬁned variable x and the fraction of the proton’s momentum carried by the parton, ξ, are the same. The diﬀerential cross section for DIS (1.7) can be computed by contracting the hadronic (1.9) and leptonic components (1.8) and can be cast into the following form

dσ 2πα2 = em (1+(1 y)2)F (x,Q2) y2F (x,Q2) . (1.14) dxdQ2 xQ4 − 2 − L We have deﬁned here the longitudinal structure function FL

F = F 2xF . (1.15) L 2 − 1 ∗ FL is related to the γ p cross section for the longitudinally polarized photon

2 Q γ∗p FL = σL , (1.16) 4παem and in this way one can deﬁne a transverse structure function FT which is related

3It is important to note that this is a valid interpretation only when viewed in the inﬁnite- momentum frame and is not generally a correct way of thinking of the deﬁnition of the Bjorken x. 8

q xp

Figure 1.3. Diagram of DIS in the Breit frame where a constituent quark impacts the virtual photon head-on. to the γ∗p cross section for transversely polarized photons. It is instructive to compute the differential cross section in QED and compare it with (1.14). For this purpose we shall assume that the virtual photon scatters off a free point-like quark. In this case the double differential cross section becomes

d2σ 2πα2 = em 1+(1 y)2 e2δ(x ξ) , (1.17) dxdQ2 Q4 − q − where ξ is the longitudinal momentum fraction of the proton carried by the quark and eq is the charge of the quark in units of elementary charge. Comparing equations (1.14) and (1.17) the following relation can be obtained

FL =0 . (1.18)

This is known as the Callan-Gross relation [16] and it stems from the fact that quarks are fermions. Let us investigate this slightly further, by considering the scattering in what is known as the Breit (or ’brick wall’) frame where the scattered parton collides with the virtual photon head-on as in Fig 1.3. Helicity (the projection of the particle spin along the momentum vector of the particle) is conserved in high energy interactions. If the parton was spin 1/2 then due to helicity conservation it could only absorb a photon of spin 1 and could not absorb ± a photon of spin λ = 0 without violating helicity conservation [17]. The fact that the longitudinal structure function is measured to be much smaller than F2 is the evidence of the spin-1/2 nature of quarks as they cannot absorb longitudinally polarized vector bosons4. Through the same comparative analysis we can also ﬁnd a relation for F2,

F = xe2δ(x ξ)=2xF . (1.19) 2 q − 1 4 In QCD the longitudinal structure function FL is small, but nonzero. 9

Notice that in our calculation (1.19) the structure function F2 depends only on x and not on Q2. This is the Bjorken scaling property [15] which was mentioned earlier. A compilation of F2 structure function data can be seen in Fig 1.4 which includes H1 and ZEUS data as well as older ﬁxed target data from BCDMS [11],NMC [18] and E665 [19]. The data are shown as a function of Q2 in bins of x. For large values of x, such as x & 0.1, the data exhibits essentially no dependence on Q2. In this region Bjorken scaling is manifest and F (x,Q2) F (x). 2 ≈ 2 As x becomes small the scaling behavior of F2 is clearly violated and F2 grows with increasing value of Q2. This breakdown of scaling behavior is directly caused by QCD emissions from the constituent partons in the proton.

1.1.2 DGLAP evolution and scaling violation

Scaling violations occur in the structure functions due to QCD corrections originating from gluon emissions and qq¯ pair production. These effects are schematically illustrated in Fig 1.5. The difference between the QCD picture and the parton model can easily be seen, as now the partons may interact with each other as well as emit gluons. These gluons may emit further gluons or can produce quark antiquark pairs. These emitted pairs are known as sea quarks and they can in principle have any flavor. Therefore the proton structure can contain strange and charm contributions as well as contributions from even heavier flavors (the contribution of these more massive quarks decreases with increasing mass). These emissions can be calculated perturbatively and their effect on the structure functions can be evaluated. Let us attempt to illustrate these scaling violations by calculating the structure function for a single quark. By calculating the interaction of the virtual photon and the quark we effectively compute using QCD the perturbative component of the hadronic tensor W µν introduced in the previous section. By comparing this calculation and extracting F2 from the amplitude we can see the effect of gluon emissions on the structure functions. There is an issue however, as it is not inherently obvious that this is possible as W µν is a nonperturbative quantity. As it turns out it is possible to factorize W µν into a nonperturbative part and a hard scattering part (at the scale of Q2) which can be computed perturbatively. 10

2 Figure 1.4. Plot of F2 structure function versus Q for data from HERA [12, 13], BCDMS [11] , NMC [18] and E665 [19] experiments for a wide range of x values. The scaling region is clearly seen at large x and the violation of Bjorken scaling can be seen at smaller x values. This ﬁgure can be found in [20]. 11

Incoming electron Outgoing electron Virtual photon Gluon radiation

Nucleon

Valence quarks

Sea quarks

Figure 1.5. Cartoon of DIS with gluon and sea quark emission leading to scaling violations.

This is known as a collinear factorization and such factorization theorems are vital in the computation of cross sections in QCD (a reader is encouraged to investigate some excellent treatments of factorization [21,22]). The factorization theorems state that the structure function (or in general, a cross section) can be decomposed into two parts [21] given by

1 dξ W µν(q,p)= f (ξ,µ)Ca(q,ξp,µ,α (µ))+remainder (1.20) ξ a/A ξ s a x X Z

The function fa/A(ξ,µ) is the parton distribution function (PDF) which gives the probability of finding a parton a (where a is either a gluon or it is a flavor index for u, u,...¯ quarks) in a hadron A carrying a momentum fraction ξ of the hadron’s a momentum. The function Cξ (q,ξp,µ,αs(µ)) is determined only by the hard scattering portion of the process. It is important to note that we have factorized the dependence on the hadron A out of this quantity and it only depends on the parton a a. The coefficient function Cξ can be computed within perturbation theory as an expansion in powers of αs. To this end we begin by considering the scattering of the virtual photon on a 12

q, µ q, µ l l p p

(a) Amplitude of a virtual photon interact- (b) Square of the photon-quark interaction ing with a quark in DIS without gluon emis- in DIS without gluon emission, dotted line sion. represents the ﬁnal state cut.

Figure 1.6. Diagrams for single gluon emission from a quark in DIS. free quark. As illustrated in Fig 1.6 a quark with momentum p is impacted by the photon of momentum q and exits with momentum l. The amplitude for this process can be directly written down from the QED Feynman rules:

M µ = ie u¯(l)γµu(p) , (1.21) − q which when squared, averaged over initial spin, color, and summed over ﬁnal spins gives

( M 2)µν =2e2(pµlν + pνlµ gµνp l) , (1.22) | | q − · where µ and ν are photon indices. The phase space factor can be found from simply including the term for energy- momentum conservation of the vertex and the delta function

d4l dφ = (2π)4δ(4)(p + q l)(2π)δ(l2) (2π)4 − Z = 2πδ((p + q)2) . (1.23)

In order to evaluate this delta function it is useful to recast these vectors using what is known as Sudakov parameters. All motion in this frame can be decomposed 13 into a sum of three vectors : a transverse vector, the lµ vector introduced (1.11), and an additional light-like vector

1 1 nµ =( , 0, 0, ) . (1.24) 2p −2p

This vector has the property p n = 1. In this way we may represent any arbitrary · vector k as

µ µ µ µ k = c1p + c2n + kT , (1.25) with c1 and c2 as coeﬃcients. Decomposing vectors in this way is known as the Sudakov parameterization and we will make extensive use of this technique. At high energies the mass of the target can be neglected and the photon momentum vector in this process can be written as

Q2 qµ = nµ + qµ , (1.26) 2x T where q is the transverse momentum of the virtual photon and q2 = Q2. It is T − T useful to note that dotting either the lµ or nµ vector with any transverse momentum vector kµ gives zero, n k = p k = 0, as does dotting the vectors with themselves T · T · T l2 = n2 = 0. This means that the property q n = 0 allows only the terms l n =1 · · to survive, so by using the nµ vector it is possible to immediately project out of µν the hadronic tensor W the structure function F2 in the following way

2x n n W µν = F . (1.27) µ ν Q2 2 Drawing the analogy to our squared matrix element and this hadronic tensor we seek to project out the F2 component by the same combination of vectors

n n ( M 2)µν =4e2 . (1.28) µ ν | | q With this decomposition we reduce the delta function in (1.23)

1 δ((l + q)2)= δ((l + νn + q )2)= δ(2ν Q2)= δ(1 x) , (1.29) T − 2ν − 14

q, µ q, µ

l k′ l

k′ p p

k k (a) (b)

Figure 1.7. Amplitude diagrams for single gluon emission from a quark in a DIS processes. where x is the Bjorken x and ν = Q2/2x. Combining the reduced phase space term and the projection of the square of the scattering matrix we are able to arrive at the ﬁrst order expression for F2

dφ F = νnµnνW = νnµnν M 2 = e2δ(1 x) (1.30) 2 µν | |µν 4π q − which is identical to the one presented earlier (1.19) with the momentum fraction of the quark ξ = 1. Note that this is the structure function for the quark and not for the entire proton (hence the omission of the factor of x as well). It can be seen that if the proton was made of a single quark then the PDF for it would be simply a delta function at x = 1. Once again the scaling behavior of F2 is obvious as there is no dependence on the momentum scale Q2 in this expression. The violation of this scaling behavior can be seen at the next order by taking gluon radiation into account. The real gluon emissions from the quark are shown in Fig 1.7. These amplitudes can be assembled into four interactions as in Fig 1.8 which represents the amplitude separated from its complex conjugate by the dashed line. For the purpose of illustration, we will focus on evaluating the diagram Fig 1.8(a) and see the implications of the gluon emission on the behavior of F2. However, in order to accurately compute the O(αs) contribution to F2, all diagrams in Fig 1.8 must be summed. First we write the matrix element for the diagram Fig 1.7(a) which is straight- 15

q, µ q,ν q, µ q,ν q, µ q,ν q, µ q,ν

l l l l

k p p p k p k k

(a) (b) (c) (d)

Figure 1.8. Possible squares of Feynman diagrams for real single gluon emission in DIS processes. The dashed line separates the amplitude from its complex conjugate forward from the QCD and QED Feynman rules

1 M µ =( ig )( ie )¯u(l)γµ /ǫτ au(p), (1.31) − s − q k/ where τ a is the QCD color matrix and once again µ is the photon index.

Solving for F2 in the same manner as was done previously we reach an expression which now depends on Q2

α 2ν F (x)= e2 s xP (x)ln . (1.32) 2 q 2π κ2 Here κ is used as an infrared scale. We can see that now there are logarithmic corrections to the structure function F2 which depend on the probing momentum Q2 as ν = p q = Q2/2x. These logarithmic corrections originate from the fact · dk2 2 that the expression for F2 involves the integral k2 , where 1/k comes from the gluon propagator. When the transverse momentumR of the gluon becomes small the structure function diverges, this is known as a collinear divergence and occurs when the emitted gluon and the quark are nearly parallel. We have deﬁned a function P (x) which contains the color factor as well as the kinematic relations which are speciﬁc to the splitting. P (x) is known as the splitting function and it has its own expansion in powers of αs

α P (x,α )= P (0)(x)+ s P (1)(x)+ ... . (1.33) s 2π

Here the superscripts refer to the order of the splitting function in powers of αs which can be calculated in perturbation theory. The sum of all diagrams in Fig

1.8 and the lowest order diagram Fig 1.7(b) lead to the following expression for F2 16

α Q2 F (x,Q2)= e2x δ(1 x)+ s P (x)ln + C(x) . (1.34) 2 q − 2π κ2 Here C(x) is a finite function (known as the coefficient function) of x and the term in the brackets can be identified as the perturbative quark distribution to first order for real gluon emission. In order to compute the structure function of a hadron, such as the proton, the distribution of a quark within the proton needs to be convoluted with our perturbative result and summed over flavors. Calling this quark distribution q0 one arrives at an expression for F2:

1 dξ α x Q2 x F (x,Q2)= xe2 q (x)+ q (ξ) s P ln + C . 2 q 0 ξ 0 2π ξ κ2 ξ q Zx X (1.35) Drawing analogy to (1.19) one deﬁnes the parton distribution function5 as

α 1 dξ x Q2 x q(x,Q2)= q (x)+ s q (ξ) P ln + C . (1.36) 0 2π ξ 0 ξ κ2 ξ Zx This can be thought of as a ’renormalized’ distribution q(x,Q2) where the ’bare’ 2 distribution q0(x) absorbs the collinear divergences discussed earlier. Here Q provides a scale much in the same way a renormalization scale would. Taking the derivative with respect to Q2 we arrive at (part of) the Dokshitzer- Gribov-Lipatov-Alterelli-Parisi (DGLAP) evolution equation :

∂ α (Q2) 1 dξ x Q2 q(x,Q2)= s P q(ξ,Q2) (1.37) ∂Q2 2π ξ ξ Zx The complete derivation of the DGLAP equation requires the calculation of

5The reader should be aware that the derivation presented here was just an outline and that the treatment that was given here was somewhat oversimpliﬁed. A priori we should not expect that we can apply perturbative techniques to a non-perturbative quantity, and why should we be able to? Perhaps the non-perturbative physics dominates the behavior of this object to any scale: once non-perturbative always non-perturbative. Thankfully this isn’t the case as again we may invoke the factorization theorems as we did earlier. The factorization theorems are key to high energy scattering calculations in QCD and an excellent treatment of the collinear factorization in DIS can be found in [21,22]. 17

DGLAP

1 x NON − PERTURBATIVE

2 1 Q ≈ r 2

Figure 1.9. Cartoon of the direction of DGLAP evolution in the Q2 and x plane. Shown is a hadron composed of a number of partons, the size of each parton shrinks as the resolution Q2 increases.

q z g z

g q Pqg Pgq

q¯ 1 z q 1 z − − (a) (b)

g z q z

g q Pgg Pqq

g 1 z g 1 z − − (c) (d)

Figure 1.10. A graphical summary of the Pqg, Pgq, Pqq, and Pgg splitting functions at leading order. 18 additional diagrams. These additional diagrams are not just higher order diagrams of the splitting in Fig 1.7 but of diﬀerent splitting modes as well. The splitting function in (1.37) is denoted as Pqq in the full DGLAP equation and other splittings are possible. These other splitting functions are denoted as Pqg, Pgq, Pgg and are illustrated in Fig 1.10. The full DGLAP evolution equation [23–26] involves all of these splitting functions in the following set of integro-diﬀerential equations

∂ q (x,Q2) α (Q2) 1 dξ Q2 i = s ∂Q2 2 2π ξ g(x,Q )! q ,q¯ x Xj j Z x 2 x 2 2 Pqi,qj ,αs(Q ) Pqi,g ,αs(Q ) q (ξ,Q ) ξ ξ j . ×  x 2 x 2  g(ξ,Q2) Pg,qj ξ ,αs(Q ) Pg,g ξ ,αs(Q ) !   (1.38)

2 2 Here the quark and gluon distribution functions are labeled qi(x,Q ) and g(x,Q ) respectively, where i is a quark flavor index (u,¯u,d,d,s,¯ etc.). The DGLAP evolution equation is a coupled set of differential equations which mix the gluon and quark distribution functions and takes a distribution from an initial value of Q2 and evolves it towards higher Q2, as in Fig 1.9. The initial quark and gluon distributions at low Q2 are non-perturbative and must be modeled from the data before being perturbatively evolved. They describe the probability of finding a parton with longitudinal momentum fraction x at momentum transfer Q2 inside the target. The equations (1.38) and (1.20) provide a framework within which one can compute structure functions and compare them with experimental data. An example of this calculation comparing with collider data can be seen as the solid 2 2 lines in Fig 1.4. The parton distribution functions qi(x,Q ) and g(x,Q ) can be constrained by experiment. It is important to note that, thanks to the collinear factorization theorem, the PDFs are universal objects. This means that they can be measured in one process, for example in DIS, and subsequently used for the prediction of the cross section in another process, for example in hadron-hadron collisions. The PDFs extracted at HERA have been instrumental for making precise predictions for many processes 19

H1 and ZEUS HERA I+II PDF Fit 1

xf Q2 = 10 GeV2

HERAPDF1.5 NNLO (prel.) 0.8 exp. uncert. model uncert. xuv parametrization uncert. 0.6

0.4 xg (× 0.05) xdv

0.2 xS (× 0.05)

0 HERAPDF Structure Function Working Group March 2011 -4 -3 -2 -1 10 10 10 10 x 1

Figure 1.11. A graph of the PDF fits from HERAPDF1.5 with NNLO corrections using H1 and ZEUS data [27]. The u and d valence quark distributions are shown independently and all other quark contributions are summed into the sea quark distributions S. The gluon distribution g is also shown. at the LHC measured recently. In Fig 1.11 the parton distribution functions extracted from the fit to HERA data on structure function are shown as a function of x for a fixed value of Q2. This set of the PDFs is one particular parametrization known as HERAPDF1.5 [27] but there are many such parameterizations, for example MSTW [28], CTEQ [29] and NNPDF [30] among others. The curves denoted by xuv and xdv in Fig 1.11 correspond to the valence quark distributions. These dominate at large values of x 0.1 0.3. The sea quarks, denoted by xS, originate mainly from QCD ≃ − 20 evolution and become important at small values of x, below 0.01, as seen in Fig 1.11. Also shown in this figure is the gluon distribution denoted by xg which dominates at small x (note that the distribution has been rescaled by a factor 0.05 for clarity). This very strong rise of the gluon distribution towards small x is a peculiarity of QCD and it will be the main focus of this dissertation.

1.2 Small x physics − The DGLAP evolution equation has been very successful in describing collider data but at small x large logarithms of the form ln(1/x) begin to appear. These large logarithms are potentially dangerous to the applicability of the DGLAP equation and they must be resummed in order to better deal with this high energy region. The small x regime has been the focus of study for a long time as x 1 , the − ∝ s small x limit coincides with the large s limit. When the collision energy s is much − greater than the momentum transfer t we enter the kinematic regime known as the Regge regime. This region governs the asymptotic behavior of the total cross section where the interactions between hadrons can be modelled by the exchange of quasi-particles known as Reggeons. Before turning our path towards QCD a detour to the past is in order as many of these results are still important in the small x ﬁeld. −

1.2.1 Pre-QCD developments : Regge Theory

In the days before QCD there was no usable theory that could apply perturbative techniques to strong interactions. Instead, gallant eﬀorts were made to apply very general but powerful principles in a non-perturbative manner in order to understand these interactions. By making assumptions about the scattering matrix such as Lorentz invariance, crossing symmetry, and unitarity conditions progress was made in describing high energy interactions. A pre-QCD method of calculating strong interactions known as Regge theory was developed [31,32]. One of the main results of Regge theory was that the scattering amplitude at a center of mass energy s and a momentum transfer t =(p p )2 for a 2 2 process (Fig 1.12) has the 1 − 3 → functional form of 21

p1 p3

p2 p4

t Figure 1.12. A generic 2 2 scattering process. →

p1 p3

p2 p4

Figure 1.13. An exchange of a Reggeon between two particles.

A(s,t) sα(t) , (1.39) ≈ for asymptotically large s, such as s t, where α(t) is a function of the momentum ≫ transfer t. The interactions in Regge theory that yield an amplitude of the form (1.39) come from exchange of quasi-particles called Reggeons. A typical diagram in Regge theory that gives rise to an amplitude of the form (1.39) is depicted in Fig 1.13. These quasi-particles have a spin which is a function of the momentum transfer t. In fact the spin of these quasi-particles is the function α(t) which appears in the equation for the amplitude (1.39). These Reggeons are not real particles, but there are resonances at (half) integer spins that correspond to real particles of mass m and spin j when j = α(m2). These trajectories can be seen by plotting the square 22

6 a (2450) 6 ρ (2350) 5 ) 2

a (2040) 4 4

ρ (1690) 3

Mass Squared (GeV 2 a (1320) 2

ρ(770)

0 0 2 4 6 Angular Momentum (units of h/2π)

Figure 1.14. Plot of particle mass squared (in GeV2) versus spin (in units of ~). It can be seen that the particles plotted lie along a linear trajectory, data taken from [33]. of the masses of various particles versus their spin, as in Fig 1.14, and it can be seen that they lie along straight lines. These lines are the Regge trajectories

α(t)= α(0) + α′t, (1.40) and they correspond to the various quasi-particles in Regge theory. Here α(0) is the intercept of the trajectory and α′ is the slope. These trajectories can be used to deduce the asymptotic s dependence of the diﬀerential cross section (obtained from the imaginary part of the forward scattering amplitude)

dσ ′ s2(α(0)+α t−1). (1.41) dt ∝ The singularity in α(t) with the largest real part (known as the leading singularity) determines the asymptotic behavior of the scattering amplitude. 23

From the amplitude one can obtain the expression for the total cross section at large s where s t. In this regime the total cross section can be shown to behave ≫ as

σ sα(0)−1. (1.42) tot ∝ It is clear that in order for the cross section to grow the intercept α(0) has to be greater than one. The exchange which leads to this growth in cross section, which is seen experimentally, can not be from a charged exchange as this would cause the cross section to vanish asymptotically [34,35]. Instead the exchange has to have the quantum numbers of the vacuum: no charge, no isospin, and a parity of +1. The experimental data on proton-proton scattering clearly indicates a growth of the total cross section with energy and it was found that this behavior could be described by an exchange of a Reggeon with the trajectory [36] 6

α(t)=1.08+(0.25GeV−2)t. (1.43)

Such an exchange that satisfies this trajectory with the quantum numbers of the vacuum is known as the Pomeron. Specifically this is referred to as the ’soft’ Pomeron to differentiate it from a ’hard’ Pomeron, which will be discussed in Sec 2.1.1. No particle resonances have been observed on the Pomeron trajectory, however a possible particle that may lie along this path is the glueball [39].

1.2.2 The Froissart Bound

In 1961 it was derived that there was a limit to the growth of total cross sections in high energy particle collisions, which is known as the Froissart bound [40]. The result was derived in a very general way in the framework of partial wave amplitudes, where the amplitude A(s,z)ofa2 2 process is decomposed into a → series of partial waves

6Alternative ﬁts to the p p¯ (and pp) cross section suggest a ln2(s) dependence, see [37,38], which is consistent with the Froissart bound (see Sec 1.2.2) 24

√s ∞ A(s,z)= a (s)(2l + 1)P (z) . (1.44) πq l l Xl=0 2 Here s is the Mandelstam variable s = (p1 + p2) , the kinematic factor q is 2 s−4 t deﬁned by q = 4 , z is related to the momentum transfer t by z =1+ 2q2 , and

Pl(z) are the Legendre polynomials. In addition to the Mandelstam representation there are two central assumptions made in the derivation of the Froissart bound

[41]. The ﬁrst is that the scattering matrix S, which gives the probability Pfi for a transition from an initial state i into ﬁnal state f | i | i

P = f S i 2 , (1.45) fi |h | | i| is unitary. This can be derived from probability conservation which states that the probability of a transition from an initial state i to any ﬁnal state f is equal | i | i to 1. P = i S† f f S i = i S†S i =1. (1.46) fi h | | ih | | i h | | i Xf Xf This is equivalent to requiring the S matrix to be unitary

S†S = SS† =1 , (1.47) where we have assumed that the states from a complete basis

a a =1 . (1.48) | ih | a X The other assumption that is made is that the strong interaction has a ﬁnite range. The potential of the exchange of a meson between two nucleons goes as

e−mr V = g2 . (1.49) r Here g2 is the coupling, r is the distance of the interaction and m is the mass of the exchanged particle. This is a screened Coulomb potential (i.e. Yukawa potential) which has a range of interaction proportional to the inverse of the mass parameter in the interaction. For the strong force this range is approximately the mass of the pion 25

1 Rstrong . (1.50) ≈ mπ By utilizing these assumptions it is possible to show with generality that the forward scattering amplitude is bounded by:

q2 A(s, 1) < ln2 B(s) , (1.51) | | κ2 where B(s) is some polynomial function in s and κ is a dimensionful scale [40,42]. This leads to the amplitude being bounded by s ln2(s) and by application of the optical theorem one ﬁnds that

π . 2 σtot 2 ln (s) , (1.52) mπ as s . This result has been very important for decades in particle physics, → ∞ providing a bound on phenomenological models. The assumptions of the Froissart bound are very general and apply to the entire energy range of QCD. Therefore the Froissart bound is not restricted to the perturbative regime of QCD. This means it takes into account the nonperturbative regime as well. The Froissart bound is also only valid asymptotically, so at smaller energies the total cross section is allowed to grow at a faster rate. Eventually as the cross-section grows larger the Froissart bound must be obeyed and over the years diﬀerent methods of enforcing this constraint have been proposed [43]. The common theme of these studies has been that as the cross section becomes large one needs to take into account multiple Pomeron exchanges. We shall discuss this eﬀect in detail in Sec 2.2.2. Chapter 2

Background

Don’t ask me, ask the universe. It is way smarter than I am.

Kevin Luschen

2.1 BFKL Physics

As we saw at the end of the last chapter, at large s the total cross section is governed by the exchange of a Reggeon known as the Pomeron. As a theory of strong interactions Regge theory has been, in general, superseded by QCD. In this chapter we shall discuss the description of the Pomeron in QCD in terms of gluon exchanges, which leads to the Balitsky-Fadin-Kuraev-Lipatov (BFKL) evolution equation. This equation is valid for small values of Bjorken x and sums the large logarithms ln(1/x). It can also be derived in the dipole model, which will be done explicitly. After a discussion of BFKL solutions and its symmetries we extend the BFKL equation by the inclusion of multiple scatterings. These multiple scatterings lead to the Balitsky-Kovchegov (BK) evolution equation which is applicable to the kinematic region of high parton density, leading to parton saturation.

2.1.1 The BFKL Pomeron: Regge theory meets QCD

We shall now consider a description of the Pomeron (discussed earlier in Sec 1.2.1) using the framework of QCD (we follow the discussion in [46]). Within QCD the 27

Figure 2.1. Two gluon exchange between two particles, giving the lowest order approximation for the Pomeron in QCD. This is the Low-Nussinov model of the Pomeron [44,45]. only way an exchange of vacuum quantum numbers can occur is by multiple gluon exchange, and at lowest order this corresponds to two gluon exchange as in Fig 2.1. This forms the lowest order approximation to the Pomeron in QCD. The diagram in Fig 2.1 leads to a constant cross section at high energy. Let us investigate some higher order corrections to this model. Restricting our investigation only to the Regge region (where t s) we are | | ≪ interested in diagrams that give terms proportional to αs ln(s). These logarithms in energy can give large contributions in the perturbative regime. The diagrams which give these contributions are ones in which an additional gluon is emitted which is 1 1 separated in rapidity , which is defined as y = ln z , from the emitting quark (or gluon). Consider the situation where two quarks with initial momentum p1 and p2 exchange a gluon as well as emitting a gluon. The final momenta of the quarks becomes p k and p +k where the emitted gluon has momentum k k . Thefive 1− 1 2 2 1− 2 potential diagrams for this exchange are shown in Fig 2.2(b)-2.2(f). If the emitted gluon is assumed to be longitudinally soft (where the longitudinal momentum fraction of the emitted gluon is much less than the parton which emitted it) then the eikonal approximation can be used. The five diagrams can be summed into one, shown in Fig 2.2(a), where the three-gluon vertex has been replaced by an σ effective vertex Γµν. This combined vertex is known as the Lipatov vertex. The Lipatov vertex is a non-local vertex, this is because it contains information about the exchanged gluon in Fig 2.2(c) - 2.2(f) but the emitted gluon does not directly interact with the exchanged gluon.

1While this is referred to as rapidity in small x physics it is not the same rapidity as used p+ − in the rest of particle physics where y = ln p− . As this dissertation only uses the deﬁnition 1 y = ln z when rapidity is referred to it should be unambiguously thought of the latter deﬁnition rather than the former 28

p p k p1 p1 k1 p1 p1 k1 1 1 − 1 − − k1 k1 k1 k2 k1 k2 σ − Γµν − k k 1 − 2 k2 k2 k2 p2 p2 + k2 p2 p2 + k2 p2 p2 + k2

(a) (b) (c)

p1 p1 k1 p1 p1 k1 p1 p1 k1 − − − k1 k1 k k k1 k2 1 − 2 − k k 1 − 2 k2 p2 p2 + k2 p2 p2 + k2 p2 p2 + k2

(d) (e) (f)

Figure 2.2. Diagrams of two quarks emitting a gluon into the final state. There are no self energy or vertex interactions because these do not contribute any αs ln(1/x) terms. These diagrams are summed into the effective non-local Lipatov vertex. The indices µ and ν are Lorentz indices for the two quarks and σ is the Lorentz index of the gluon in the final state.

The procedure outlined above can be generalized to emission of an arbitrary number of gluons in the final state. This is illustrated in Fig 2.3. The vertices in all emissions are the non-local Lipatov vertices, and this is valid because for soft emissions the contributions from emission by a quark or a gluon have identical forms (by virtue of the structure of the eikonal approximation). Each emission is soft compared to the previous one in the ladder. This leads to a strong ordering in longitudinal momentum in the gluon emissions and a large gap in rapidity between each emission. Squaring this graph leads to a ladder of gluon exchanges with effective vertices. It is important to also consider loop corrections to the diagram Fig 2.3. The inclusion of these virtual corrections (described in more depth in [46]) leads to an object known as the Reggeized gluon. These Reggeized gluons have a modified propagator which has a term proportional to a power of s. This term gives us the power-like behavior of the amplitude in s. The Pomeron is constructed as a ladder of two Reggeized gluons exchanging QCD gluons between them as in Fig 2.4. The BFKL equation was originally derived as the integral equation which describes the exchange of a single Pomeron in QCD. It is this QCD Pomeron which is referred to as the BFKL Pomeron or 29

Large rapidity gap

Strongly ordered emissions

Figure 2.3. An amplitude diagram where two quarks interact and emit n gluons into the ﬁnal state. The vertices are eﬀective Lipatov vertices. Each emitted gluon is longitudinally soft so there is a strong ordering in the longitudinal momentum of the emitted gluons. the ’hard’ Pomeron. By calculating the Pomeron in this manner it is possible to extract the intercept (in the leading logarithmic approximation) of the exchange and compare it to the trajectory found in (1.43):

α(0) = 4¯αs ln(2) + 1. (2.1)

In the rest of this work we shall use the rescaled strong coupling, which is given by

α N α¯ = s c , (2.2) s π with Nc as the number of colors. This is not the same Pomeron as in (1.43), the intercept is much larger than the value for the soft Pomeron even for reasonably small values of αs. The larger intercept leads to a much faster growth in the cross-section at large values of s. 30

Reggeized gluon

Lipatov vertices

QCD gluons

p′

Figure 2.4. Diagram where a single Pomeron is exchanged between a qq¯ pair from a virtual photon ﬂuctuation and a nucleon target. Here the vertical lines are Reggeized gluons, which interact with each other via exchange of QCD gluons. The resummation of this kind of gluon ladder leads to the BFKL equation.

γ∗ r γ∗

Figure 2.5. DIS in the dipole model where the virtual photon ﬂuctuates into a q q¯ pair with a characteristic separation r. The dipole interacts with the target and the information about this interaction is encoded in the scattering amplitude N(x, r, b).

2.1.2 The Dipole Model

In the previous subsection we have sketched the construction of the BFKL Pomeron using gluon exchanges. We shall continue to discuss the small x regime using a − somewhat different picture from this point onwards, known as the dipole model. The reason for doing this is that the dipole model is an intuitive and very useful framework for doing analysis in the small x regime. In particular, it is very − convenient for analyzing the effects of gluon saturation, which is the main focus 31 of this dissertation and will be discussed in detail in chapters 3 and 4. The dipole model also allows us to rederive the BFKL equation in the rather intuitive manner, following the original dipole-model derivation done by Mueller [47]. In this model the virtual photon exchanged between the electron and the proton fluctuates into a q q¯ pair far from the interaction point. It is this q q¯ pair which then interacts with the nucleon with a scattering amplitude2 N(x, r, b) as illustrated in 2 Fig 2.5. It is possible to calculate the structure function F2(Q ,x) in this model by the formula

Q2 1 F (x,Q2)= d2r dz d2b( Ψ (r,z,Q2) 2 + Ψ (r,z,Q2) 2) 2 4π2α | T | | L | em Z Z0 Z N(x, r, b) . (2.3) ×

Here x is3 the Bjorken x and Q2 is the virtuality of the virtual photon. The − wavefunction of the virtual photon is denoted by ΨL for longitudinal polarization and ΨT for the transverse polarization. The photon ﬂuctuates into a dipole of size r where the quark carries a longitudinal momentum fraction z of the virtual photon’s momentum. The dipole is at an impact parameter b from the target. The scattering amplitude N(x, r, b) can be related to an object known as the dipole number density n(x, r, r′, b) which will be useful in the formalism introduced later in this chapter. The dipole number density describes the probability of a dipole of size r′ to be emitted at a Bjorken x from an initial dipole r at impact parameter b. Following [48] the equation relating these two quantities is

1 N(x, r, b)= dz d2r′n(x, r, r′, b)γ(r′). (2.4) Z0 Z The factor γ(r′) represents the propagator of a dipole of size r′ through the target which is being scattered on. The exact functional form of γ(r′) is not necessary for the calculations presented in this chapter and the role of the propagator shall

2In this dissertation we shall use the notation of bold faced font to represent a two dimensional vector from here onward, otherwise it should be assumed that the quantity is the magnitude of the associated vector (e.g. x = x ). 3A note on notation : in this dissertation| | any x which is without a subscript is to be understood as the Bjorken x. Any x with a subscript is to be understood as a two dimensional vector which represents a position in transverse coordinate space. 32

Figure 2.6. Diagram for initial qq¯ state without gluon emission

l k l k 1 − 1 − P P k

l2 l2

(a) (b)

Figure 2.7. Two Feynman diagrams of single gluon emission from a q q¯ pair. be discussed in more depth in Sec 2.2.2. The dipole number density is inherently a non-perturbative quantity which cannot be computed from ﬁrst principles. In much the same manner as was done with the DGLAP equation we are able to use perturbative techniques to derive an evolution equation for the dipole number density which allows us to make meaningful predictions of small x interactions. −

2.1.3 Mueller’s description of the BFKL equation

We follow the method of Mueller [47] in his derivation of the BFKL equation using the dipole model. We begin with the light-cone wavefunction of a q q¯ pair4 which forms a color dipole as in Fig 2.6. This color dipole may emit gluon radiation as in

Fig 2.7. In the large Nc limit these gluons can be represented as a double line in

4This is sometimes referred to as an onium state, which is a quark-antiquark pair which has a large mass such that there is a perturbative scale. The name onium derives from quarkonium, which is a ﬂavorless meson state such as J/Ψ ’charmonium’ (c c¯) or Υ ’bottomonium’ (b ¯b). 33

Dipole x02

Dipole x01 l k ; x Dipole splitting 1 − 0 x2 p ; x01 Dipole x02

k ; x2

Dipole x12 Dipole x12

x1 l2 ; x1

(a) (b)

Figure 2.8. Emission of a single color dipole in transverse coordinate space (fig a) and in the pseudo-Feynman diagram in momentum space (Fig b). color degrees of freedom [49], which allows the parent dipole to split into further dipoles as in Fig 2.8(b). In the high energy limit the longitudinal and transverse degrees of freedom become independent of one another and it is convenient to split the wavefunction into what is known as a mixed representation. In this mixed representation the longitudinal degree of freedom is left in momentum space while the transverse degrees of freedom are taken in coordinate space. This separation allows the dynamics to be thought of in a more geometric manner. This can be seen in Fig 2.8(a) where the initial (parent) dipole is given by the two dimensional coordinates x0 and x1 in transverse space. It splits into two (daughter) dipoles by emission and this emission creates a third coordinate x2. This new coordinate forms two dipoles with coordinates x1 and x2 as well as x0 and x2. These coordinates label the positions in transverse space of the color charges. The goal here shall be to construct a wavefunction which contains arbitrarily many dipoles, as in Fig 2.9. We shall do this by first describing the two dipole wavefunction in terms of a one dipole wavefunction. This gives a basis for an iterative procedure where a splitting kernel may be applied to the wavefunction to emit another dipole. Through this procedure a wavefunction of many dipoles will be constructed and we shall find that the equation governing this many dipole state is indeed equivalent to the BFKL equation. 34

Emitted gluons

Color dipoles

Figure 2.9. Diagram of an onium wavefunction consisting of many emitted color dipoles. The gluons are represented as double-lines because they carry two color indices but the color dipoles form between two color charges. In this picture the color dipoles form between the emitted gluons.

In this derivation we assume that every gluon emitted is longitudinally soft compared to the color charge which emitted it. This is equivalent to the leading logarithmic approximation (LLA) where we only keep terms proportional to n n αs ln (1/x). As αs is taken to be a small parameter at high energies, the terms with ln(1/x) compensate for the smallness of αs and must be summed. Terms with an additional power of αs come in at next to leading logarithm (NLL) accuracy. The ﬁrst step is to evaluate the one gluon emission wavefunction in terms of the wavefunction without any gluons. There are two graphs for single gluon emission, as seen in Fig 2.7, and the corresponding wavefunction is computed by summing the terms from each graph [46]. Here the superscript of the wavefunction (0) denotes the number of gluons emitted. The wavefunction Ψ (z0,lT ) represents the wavefunction of a q q¯ pair without any gluons, where the quark has a longitudinal momentum fraction z and the antiquark has 1 z . Similarly the quark has 0 − 0 transverse momentum 5 lµ and the antiquark lµ . T − T For the graphs with gluon emission it is useful to represent the momentum µ µ µ of the quark l1 , the antiquark l2 , and the emitted gluon k in terms of Sudakov parameters. Let us consider the speciﬁc case of Fig 2.7(a):

l2 lµ = z pµ + T pµ + lµ , (2.5) 1 0 1 2p p z 2 T 1 · 2 0 5 µ While the transverse momentum vectors lT noted in this section do only have two non-zero components they are still four-vectors and hence shall not be bolded. Later when only the two non-zero components are considered the transverse momentum shall be denoted by lT. 35

l2 lµ = (1 z z)pµ + T pµ lµ kµ , (2.6) 2 − 0 − 1 2p p (1 z ) 2 − T − T 1 · 2 − 0 k2 kµ = zpµ + T pµ + kµ . (2.7) 1 2p p z 2 T 1 · 2

Here p1 is a momentum vector moving along the positive z axis and p2 is moving along the negative z axis. It is assumed that the emitted gluon is longitudinally soft which means z z , 1 z , this allows us to utilize the eikonal approximation at ≪ 0 − 0 the gluon emission vertex. Using this simpliﬁcation we can extract the contribution from Fig. 2.7(a)

a 2l1 ǫ (0) igτ · Ψ (z , lT) , (2.8) − 2l k 0 1 · with τ a as the QCD color matrix. The eikonal approximation modiﬁes the propagator of the hard quark line and the emission vertex of the soft gluon [50] such that the propagator becomes 1 2l1·k and the interaction vertex becomes igτ a2l ǫ. Here ǫ is the polarization vector − 1 · of the soft gluon which by gauge invariance can be chosen to have no component in the p1 direction

ǫ k ǫµ = T · T pµ + ǫµ , (2.9) p p z 2 T 1 · 2 µ with ǫT being the transverse gluon polarization component. In the soft approximation we are able to evaluate the dot products as follows

z ǫ k l ǫ = 0 T · T , (2.10) 1 · z z k2 l k = 0 T . (2.11) 1 · 2z

As the emission of the gluon is soft we keep only the 1/z terms in the dot products. We may now simplify the contribution from the ﬁrst diagram (2.8) to

a kT ǫT (0) ig2τ ·2 Ψ (z0,lT ). (2.12) − kT The second diagram can be worked out similarly, where a minus sign is gained 36 from the fact that the emission occurs from an anti-quark

a kT ǫT (0) ig2τ ·2 Ψ (z0,lT + kT ). (2.13) kT (0) Notice that the momentum argument in Ψ here is lT + kT and the overall minus sign in the transverse momentum from (2.6) has been dropped. Combining these two terms (and neglecting an overall minus sign) we arrive at the wavefunction for a color dipole splitting into two color dipoles

(1) a ǫT kT (0) (0) Ψ (z0,lT ,z,kT )=2igτ ·2 Ψ (z0,lT ) Ψ (z0,lT + kT ) . (2.14) kT − It is convenient to place the wavefunction in what is known as a mixed representation where the longitudinal degrees of freedom are still in momentum space (in the form of the momentum fraction z) and the transverse degrees of freedom are in coordinate space. In order to bring (2.14) into the mixed representation we must make a Fourier transform from kT to x which are coordinates in transverse space. We use the notation xij = xi xj where xi,xj and xij are all two component − vectors in transverse space. The Fourier transform of the wave function is then

2 2 d lT d kT Φ(1)(z , x ,z, x )= eilT·x01+ikT·x02 Ψ(1)(z , l ; z, k ) . (2.15) 0 01 02 (2π)2 (2π)2 0 T T Z Combining (2.14) and (2.15) we ﬁnd that it is possible to split the expression into two integrals. Shifting the second integral by making the substitution lT → lT kT allows the wavefunctions to be combined. This shift is absorbed by the − integral as it is over all momenta.

(1) Φ (z0, x01,z, x02) 2l 2k k a d T d T ilT·x01 ǫT T ikT·x02 ikT·x12 (0) = 2igτ e · e e Ψ (z , lT) . (2π)2 (2π)2 k2 − 0 Z T (2.16) 37

Solving the integral yields

a (1) gτ ǫT x20 ǫT x21 (0) Φ (z , x01,z, x02)= − · · Φ (z , x01), (2.17) 0 π x2 − x2 0 20 21

(0) where we deﬁne the mixed representation of Ψ (z0, lT) as

2 d lT Φ(0)(z , x )= eilT·x01 Ψ(0)(z , l ) (2.18) 0 01 4π2 0 T Z Squaring the wavefunction (utilizng the identites τ aτ a N /2 and α = g2/4π) ≈ c s with a single emitted gluon and tracing over colorsP we arrive at

2 (1) 2 x01 (0) 2 Φ (z0, x01,z, x02) = 2¯αs 2 2 Φ (z0, x01) . (2.19) | | x02x12 | | It can be seen now that the single gluon wavefunction can be expressed in terms of the zero gluon wavefunction with a branching kernel which is given by

2 x01 2 2 . (2.20) x02x12 However, this is only for single gluon emission which corresponds to Fig 2.8. We would like to allow for emission of an arbitrary number of soft gluons and derive an equation for Fig 2.9. In order to do arrive at an equation for the emission of multiple soft gluons it is useful to write down the expression for a generating functional. It can be shown that [47] this functional Z has the property

δ δ δ (0) ... Z(x01,y,u) Φ (x01,y) δ(u(x )) δ(u(x )) δ(u(x )) 1 2 n u=0 (n−1) = Φ (x01, x02, x03,..., x0n,y).

(2.21)

Here we have deﬁned an auxiliary variable u in the same way as [47,51]. Thus by repeated diﬀerentiation of the generating functional one can arrive at a wavefunction containing an arbitrary number of soft gluons. In (2.21) we have switched from the longitudinal momentum fraction z as our longitudinal coordinate to rapidity 38

x01 Y x01 e− ∝ no dipole splitting

dipole generated by Z(x02,y,u) x01 x01

x02 (Y y) 2 e− − x01 ∝ Splitting 2 2 ∝ x02x12 x12

dipole generated by Z(x12,y,u)

0 Y y Y Rapidity−

Figure 2.10. A pictorial representation of the terms in the dipole generating function Z. The top path describes the ﬁrst term in (2.23) where the initial dipole x01 does not split. The bottom path describes the second term in (2.23) where after Y y rapidity 2 − x01 x01 splits into two dipoles x02 and x12 with a splitting probability 2 2 . x02x12 y, which is deﬁned as 1/z. To complete the calculation of Z it is necessary to include the virtual graphs. One manner of doing this is by enforcing the probability conservation condition

2 (0) d x01 dyZ(x01,y,u) Φ (x01,y)=1. (2.22) |u=1 Z Z With u = 1 the generating functional in (2.21) can be interpreted as the inclusive probability of finding n 1 soft gluons in the wavefunction. − This method was used in Mueller’s first work [47] in order to come to the form of the generating function, including the virtual corrections. The virtual corrections can also be computed explicitly in perturbation theory, as done in [52]. Here we will simply state the final result for the evolution equation of the generating function :

x01 −2¯αsY ln( ) Z(x01,Y,u)= e ρ u(x01)

2 2 Y x α¯s d x2x − 2αNc (Y −y) ln 01 + 01 dye π ( ρ )Z(x ,y,u)Z(x ,y,u). (2.23) 2π x2 x2 02 12 ZR 02 12 Z0 39

Here Y is the final rapidity of the dipoles. Both R and ρ act as a minimum dipole size (R is actually a two dimensional area R(x01,x02) which is bounded by x02 > ρ and x01 > ρ). This minimum dipole size is a cut on real emissions in the ultra-violet regime. Let us take a moment and see what exactly (2.23) is representing. It is instructive to draw an analogy between the rapidity in (2.23) and time, so that we are looking at the evolution of dipole configurations between the ’initial time’ Y = 0 and the ’final time’ of Y . The first term corresponds to the initial dipole x01 remaining intact over the entire rapidity range 0 Y (i.e. → not splitting). The second term is where the splitting comes in. The exponential factor which depends on Y y represents the lifetime of the initial dipole x01. − After this distance in rapidity the x01 dipole splits into two dipoles x02 and x12, which is represented by the two generating functionals Z(x02,y,u)Z(x12,y,u). The schematic representation of (2.23) can be seen in Fig 2.10. The splitting 2 x01 probability is determined by the splitting kernel 2 2 . x02x12 From (2.23) one can obtain an equation for the number density of dipoles 6 of a given size r from the initial dipole x01. The number density of dipoles can be expressed in terms of the generating functional [51]

2 δZ(x01,Y,u) n(x01, r,Y )= r 2π . (2.24) δu(r) |u=1 Using (2.23) and (2.24) we are able to derive the evolution equation for the dipole number density

x01 −2¯αsY ln( ) n(x01, r,Y ) = rδ(r x )e ρ − 01 2 2 Y x α¯s x d x2 −2¯α (Y −y) ln 01 + 01 dye s ( ρ )n(x , r,y) . (2.25) π x2 x2 12 ZR 12 02 Z0 The above equation can be rewritten as a diﬀerential equation in rapidity7

6 It should be remembered that the number density n(xij,Y ) also depends on the impact parameter of the dipole in question, n(xij,Y )= n(xij, bij,Y ). In coordinate space we deﬁne the xi+xj impact parameter to be bij = 2 . In this section we shall suppress the dependence on impact parameter in our notation. 7Taking the derivative of (2.25) with respect to rapidity one arrives at

− x01 d x01 2¯αsY ln( ) n(x01, r,Y ) = 2¯αs ln rδ(r x )e ρ dY − ρ − 01 h 40

∞ ∞ d 2αNc 2 ′ 2 ′ n(x01, r,Y )= d x12 d x2δ(x x ) dY 2π2 12 − 12 Z0 Z0 x2 x 01 2πδ(x x )ln 01 n(x′ , r,Y ). (2.26) × x2 x2 − 2 − 0 ρ 12 12 02 which is the dipole equation in rapidity. It is more desirable to change (2.26) further from an evolution of the dipole number density n(x01, r,Y ) to the scattering amplitude N(x01,Y ). It is this latter quantity which is directly used in the calculation of the structure functions (2.3). By utilizing (2.4) and multiplying through by γ(r) as well as integrating over r it is possible to recast (2.26) into

d 2αN ∞ ∞ x c 2x ′ 2x ′ N( 01,Y )= 2 d 12 d 2δ(x12 x12) dY 2π 0 0 − 2 Z Z x01 x01 ′ 2πδ(x x )ln N(x12 ,Y ). (2.27) × x2 x2 − 2 − 0 ρ 12 02 In the literature and in the following chapters of this dissertation an equivalent form of (2.27) is used which has the form

2 ∂N(x01,Y ) α¯s 2 x01 = d x2 [N(x12,Y )+ N(x02,Y ) N(x01,Y )] . (2.28) ∂Y 2π x2 x2 − Z 02 12 In order to show the equivalence between (2.27) and (2.28) consider the dissent- ing terms one by one. The N(x02,Y ) term is placed into the evolution equation in order to keep the symmetry between emission of dipole x12 and x02. The

N(x01,Y ) term comprises the virtual corrections to the evolution. Consider only the N(x01,Y ) term in (2.28)

Y 2 2 x α¯s d x2x01 −2¯α (Y −y)ln 01 + dye s ( ρ )n(x , r,y) π x2 x2 12 ZR 12 02 Z0 # α¯ d2x x2 + s 2 01 n(x , r,y) . π x2 x2 12 ZR 12 02

Identifying the bracketed quantity as n(x01, r,Y ) from (2.25) we arrive at (2.26) 41

x2 d2x 01 N(x ,Y ). (2.29) 2 x2 x2 01 Z 12 02 We can rewrite the kernel in the following manner:

2 x01 1 1 2x12 x02 2 2 = 2 + 2 2 · 2 , (2.30) x02x12 x12 x02 − x12x02 The virtual contribution can thus be rewritten in the form of

π 1 1 dx 2 + N(x ,Y ) , (2.31) 2 2 x2 x2 01 Z 12 02 Here we have used the fact that the contributions to the integral from the third term in (2.30) vanishes when integrated over the angle between the dipoles x01 and x12. 2 2 Splitting up the integral and making the respective shift from dx2 to dx02 or 2 dx12, depending on the term allows the integrals to be solved for

π x01 dx 2 x01 dx 2 x N(x ,Y ) = 12 + 02 =2πN(x ,Y )ln 01 2 01 x2 x2 01 ρ Zρ 12 Zρ 02 2 x01 = d x22πδ(x x )N(x12,Y )ln . 0 − 2 ρ Z (2.32)

Here our integration limits go from ρ, which is a minimum dipole size, to the parent dipole size x01. The minimum dipole size ρ corresponds to an ultraviolet cutoff in our integral. It can be seen that the term in (2.32) matches the last term in (2.27). It is interesting to note that there is a necessity for the introduction of a cutoff ρ in (2.27) but in (2.28) there is no need for this cutoff. In the form (2.28) we find that when any dipole size in the system becomes very small there is a cancellation between the virtual term (given by N(x01,Y )) and the other terms in the equation.

When x 0, then x x and N(x01,Y ) cancels with N(x12,Y ) leaving only 02 → 12 → 01 N(x02,Y ) in the integrand. As the scattering amplitude of a very small dipole goes to zero the entire integrand vanishes, avoiding any divergences in the equation. 42

1 BFKL x

DGLAP NON − PERTURBATIVE

2 1 Q ≈ r 2

Figure 2.11. Cartoon of the direction of the BFKL evolution equation compared with DGLAP evolution in the Q2 and x plane. Shown is a hadron composed of a number of partons, the size of each parton shrinks as the resolution Q2 increases. In contrast as the x decreases further emissions of gluons begin to cause the number of partons in the hadron to grow.

This eﬀect of the scattering amplitude being zero in the limit of r 0 is known → as color transparency and is a characteristic feature of perturbative QCD [53–55]. The equation (2.28) was derived by Mueller [47] and can be shown as equivalent to the BFKL equation derived by [56–59] in earlier works. As with DGLAP, BFKL is a diﬀerential equation and gives us no information about the scattering amplitude N(x01,Y ) at the initial rapidity Y = Y0. It is an important distinction that while the DGLAP initial condition is strictly non-perturbative, this is not true for the BFKL initial condition. As the initial condition for BFKL is at a 43

fixed x it contains a wide range of Q2, which encompasses both perturbative and nonperturbative physics. Also in contrast to the DGLAP equation, which is an evolution equation in the momentum scale Q2, the BFKL equation is an evolution equation in rapidity Y = ln(1/x). Evolving to higher rapidities is equivalent to evolving to smaller values of x (and higher energies). A picture of these two different evolution equations and their behavior in the 1/x and Q2 plane is shown in Fig 2.11. While the BFKL equation effectively resums the large ln(1/x) terms it is still vulnerable to acquiring terms proportional to ln(Q2) and at large Q2 the evolution cannot be relied upon.

2.1.4 Symmetries of the BFKL equation

The BFKL equation contains several interesting and useful symmetry properties. Let us consider the measure of the integral in (2.28), which contains the dipole splitting kernel

α N x2 s c 2x 01 2 d 2 2 2 . (2.33) 2π x02x12 There are several symmetries of the measure which can be stated explicitly as in [60]. It is clear that the measure is invariant to any rescaling xi λxi, where → λ is a complex number. Further symmetries can be seen better by rewriting the measure in expanded form,

α N (x x )2 s c 2x 0 1 2 d 2 −2 2 . (2.34) 2π (x0 x2) (x1 x2) − − In this form it can be seen that shifting all coordinates by a constant vector ′ xi xi + c also leaves the measure invariant. Inserting a scale x into the kernel → and converting to radial coordinates we get

2 2 ′4 αsNc x2 (x0 x1) x 2 d ′2 dθ −2 2 ′2 . (2.35) 2π 2x (x0 x2) (x1 x2) x − − ′ ′ It can be shown that the transformation xi/x x /xi leaves the measure → invariant. The symmetries that have been mentioned thus far form a M¨obius symmetry. Taking the coordinates as complex numbers x = z +iz and x¯ = z iz 1 2 1− 2 as in [60,61] the transformation is 44

x a(x/x′)+ b , (2.36) x′ → c(x/x′)+ d with a similar transformation for the complex conjugate. This transformation can also be formulated in the framework of the symmetry group PSL(2, C) which is discussed more in [62]. The Möbius symmetry of the leading order BFKL equation was exploited by Lipatov [61,63] in order to find a solution using conformal eigenvalue functions. This solution of the BFKL equation will be discussed more in Sec 2.1.5. The measure (2.33) is scale invariant, the only dimensionful quantities are the dipole sizes involved and there is no external scale. This is because at leading order the BFKL equation has a frozen coupling constantα ¯s. At higher order the running of the coupling introduces a scale in the problem ΛQCD. The introduction of this scale will break the scale invariance of the measure as well as the other symmetries that are listed in this section. It is interesting however, to note that in N = 4 Super Yang-Mills (SYM) field theory the conformal (Möbius) symmetry of the kernel holds at the next to leading order as well [64].

2.1.5 Solution to the BFKL equation

The BFKL equation is solvable analytically in the leading order8. This solution can give us insight into the behavior of the dipole scattering amplitude. It is more instructive to begin with the BFKL equation in the form of (2.27)

∂N 01 = 2¯α K(x ,x )N d2x . (2.37) ∂Y s 12 01 12 12 Z We have simpliﬁed our notation here to Nij = N(xij, bij,Y ). For simplicity we assume that the dipole scattering amplitude is independent of impact parameter. The kernel in (2.37) is

α¯ ∞ K(x ,x ) = s dx′ δ(x x′ ) 12 01 π 12 12 − 12 Z0 8It is also possible to solve the BFKL equation with the running coupling but we shall not consider this case here. 45

x2 x 01 2πδ(x x )ln 01 . (2.38) × x2 x2 − 2 − 0 ρ 12 02 This kernel has eigenfunctions9 of the form x1+2iν. This yields the following eigenvalue equation for the BFKL kernel

2 1+2iν 1+2iν d x12K(x01,x12)x12 = χ(ν)x01 , (2.39) Z which can be solved and yields the LL BFKL eigenvalue

1 1 1 1 χ(ν)= ψ(1) ψ( iν) ψ( + iν). (2.40) − 2 2 − − 2 2 Here ψ(z) is the digamma function which is deﬁned by ψ(z)=Γ′(z)/Γ(z). Sepa- rating the variables in (2.37) and integrating over Y we arrive at

Y 2 N(Y, x01) N(0, x01)=2¯α K(x ,x )N(y, x12)d x12dy . (2.41) − s 12 01 Z0 Transforming the scattering amplitude with the equation

∞ −ωY N˜(ω, x01)= e N(Y,x01)dY , (2.42) Z0 and inserting (2.41) into the right hand side of (2.42) one arrives at

∞ Y 2 −ωY N˜(ω, x01)= N˜(0, x01)+2¯αs d x12K(x12,x01) e N(Y,x12)dydY . 0 0 Z Z Z (2.43) By changing the order of integration and introducing a factor of eω(y−y) into the integrand one can solve the integral and the equation becomes

2¯α N˜(ω, x )= N˜(0, x )+ s d2x K(x ,x )N˜(ω, x ) . (2.44) 01 01 ω 12 12 01 12 Z In order to decouple the branching kernel from the scattering amplitude within the integrand we invoke the Mellin transform (which is discussed further in ap-

9We are using ν here not to mean the variable ν = Q2/2x as in the previous chapter but instead as a complex number. 46 pendix A). This transformation has the form

N˜(ω,ν)= x1−2iνN˜(ω, x)dx . (2.45) Z Inserting (2.44) into the right hand side of (2.45) one arrives at

2¯α N˜(ω,ν)= N˜(0,ν)+ s x1−2iνK(x ,x )N˜(ω,x )d2x d2x . (2.46) ω 01 01 12 12 12 01 Z Z The Mellin transform allows the kernel to be decoupled from the scattering amplitude by invoking (2.39) in (2.46). The equation now has the form

2¯α N˜(ω,ν)= N˜(0,ν)+ s χ(ν)N˜(ω,ν) . (2.47) ω Rearranging the equation gives us

ω N˜(ω,ν)= N˜(0,ν) . (2.48) −2¯α χ(ν) ω s − This leads to a branch cut along the line at ω = 2¯αsχ(ν). Invoking the inverse transformations one arrives at the scattering amplitude

dν x 1−2iν N(Y, x )= 01 e2¯αsχ(ν), (2.49) 01 2π x′ Z This equation is in the correct form to apply the saddle point method (the method of steepest descent [65]) for solving the integral. We use ν = 0 as a saddle point of χ(ν) and expand χ(ν) around ν = 0 to second order. The integral can be analytically solved at that point. One ﬁnds χ(0) = 2ln(2) and χ′′(0) = 14ζ(3) − where ζ(n) is the Riemann zeta function.

1−2iν dν x01 2¯α (χ(0)+ 1 ν2χ′′(0))Y N(Y, x ) = e s 2 01 2π x′ Z dν x 1−2iν 2 ′′ = e2¯αsχ(0)Y 01 eα¯sν χ (0)Y 2π x′ Z x e4 ln(2)¯αsY ln2(x /x′) = 01 exp − 01 (2.50) 2x′ 14ζ(3)¯α Y α¯s14ζ(3)πY s p 47

Let us analyze the behavior of (2.50) term by term. The term e4 ln(2)¯αsY is responsible for the growth of the scattering amplitude. Since the amplitude grows exponentially in rapidity this yields a power-like increase in the energy (as Y = ln(1/x) and x 1/s). While a power law growth in energy is observed for the total ∝ cross section (discussed in Sec 1.2.1), the growth seen in (2.50) is too fast to explain the data. The Regge trajectory may be extracted from (2.50) and compared to the trajectory found in the data for the soft Pomeron (1.43). The trajectory from (2.50) is larger than what is found in the data for the soft pomeron. 2 ′ The term exp − ln (x01/x ) in (2.50) is responsible for the diﬀusion of the so- α¯s14ζ(3)Y lution. As the rapidityh increasesi the initial condition, which is localized in dipole size, begins to spread to dipole sizes beyond the initial condition. This causes dipole sizes which were non-interacting at low rapidities to become interacting. There has been no mention of a target thus far in the calculation and this is because all the information about the target is actually encoded in the initial condition. The BFKL evolution is the same for a proton as a heavy ion. As the target (proton or heavy ion) is a complicated object the initial condition must be modelled and can be found by comparing the results to data. Additionally this solution of the BFKL equation has been obtained by neglecting the impact parameter dependence of the system. The impact parameter dependence is hidden within the scattering amplitude N(xij,Y ) = N(xij, bij,Y ). Neglecting impact parameter is equivalent to the assumption that the target has inﬁnite size. It is possible to consider a special case in which the target and the projectile are both dipoles. By exploiting the conformal invariance of the LL BFKL equation discussed in Sec 2.1.4 an analytical solution with impact parameter can be found.

2.1.5.1 Special case : dipole-dipole scattering

In the instance where both projectile and target are dipoles one can ﬁnd an analytic solution. This solution was found in [66] and was based on the original solution derived by Lipatov [63]. This dipole-dipole interaction is shown in Fig 2.12. In transverse coordinate space the interaction looks like Fig 2.13 where the dipoles are separated from each other by an impact parameter b. In this case the scattering amplitude for two dipoles can be written as [66] 48

Figure 2.12. Two dipoles interacting via Pomeron exchange. Here one dipole is labeled with transverse size r and the other dipole has transverse size r′. The two dipoles interact at an impact parameter b.

2 2 ′ 1 d r1 d r2 2 2 2 ′ ′ F (r1, r2,Y = Y1 + Y2, b) = 2′ 2′ d b1d b2d (b2 b1) −2 2πr1 r2 − 2 Z ′ ′ ′ δ (b1 b2 b + b b)n(r1, r ,Y , b1) × − − 1 2 − 1 1 ′ ′ ′ ′ ′ n(r2, r ,Y , b2)σ (r , r , b b ) . (2.51) × 2 2 DD 1 2 1 − 2

The dipole-dipole cross section is given by

′ ′ ′ ′ 2 ′ ′ r1+r2 ′ ′ r1+r2 ′ ′ ′ ′ b1 b2 + 2 b1 b2 2 r r b b | − ′ ′ || − − ′ ′ | σDD( 1, 2, 1 2)= αs ln r −r r −r , (2.52) − " b′ b′ + 1 2 b′ b′ 1 2 !# | 1 − 2 2 || 1 − 2 − 2 | 49 and the dipole number density has the form

+∞ dν d2w n2 n(r, r′,Y, b)= 16 ν2 + e2¯αsχ(n,ν)Y (2π)3 r′2 4 n=−∞ Z Z ∗ r Xr r r En,ν 0 w, 0 w En,ν 0 + b w, 0 + b w . (2.53) × 2 − − 2 − 2 − − 2 − The summation is over the conformal spin n, the variable ω is the conjugate variable to Y in the Mellin transform (see Sec 2.1.5), and the coordinate w is a center of mass coordinate. The functions Eνn(ρ10,ρ20) are the conformal eigenfunctions given by

h ∗ h¯ n ρ12 ρ12 E (ρ10,ρ20)=( 1) . (2.54) νn − ρ ρ ρ∗ ρ∗ 10 20 10 20 We again are using complex notation where the spatial coordinates ρa = xa+iya ∗ and ρa = x iy and ρ = ρ ρ . The conformal weights h and h¯ are deﬁned a − a ij i − j as

1+ n h = + iν, (2.55) 2 1 n h¯ = − + iν. (2.56) 2 with n being the value of conformal spin. The BFKL dynamics are contained within the χ(ν,n) term of (2.53) which is again the LL eigenvalue of the BFKL kernel with non-zero conformal spin

1 n +1 1 n +1 χ(ν,n)= ψ(1) ψ | | + iν ψ | | iν . (2.57) − 2 2 − 2 2 − This can be compared with (2.39). This solution (2.53) contains explicitly the information about the impact parameter dependence of the dipole number density. In this way it allows some analytical insight into the evolution with impact parameter. This will prove useful in Sec 3.3. 50

ρ1′

ρ1 r b

r0 ρ2′

ρ2

Figure 2.13. Geometry of two dipoles interacting in the transverse position plane. This is the coordinate space version of Fig 2.12, the coordinate w is not shown.

2.2 Saturation physics

Looking back at (2.50) we can extract some key properties of BFKL dynamics. The most striking feature is that the dependence on rapidity is very strong which causes a rapid growth in the dipole scattering amplitude N(r, b,Y ) (as well as the dipole number density). The growth of the scattering amplitude is driven by the dipole splitting (which was described earlier in terms of gluon emission). As we evolve the scattering amplitude to larger rapidities the x value decreases and the amplitude increases due to the increased density of gluons [29]. This can be seen in Fig 1.11 where the gluon distribution grows rapidly at small x values. It is reasonable to expect that once the gluon density (and thus the dipole scattering amplitude) becomes large that additional effects are manifest. At large densities the assumption of single-scattering events breaks down and multiple interactions must be taken into account. These multiple scatterings change the dynamics of the system and shall be the main focus of the rest of this dissertation. Without any additional effects the dipole scattering amplitude is not bounded at large Y , i.e. N > 1. Multiple scatterings serve to limit the growth of the scattering amplitude. Due to these effects the scattering amplitude eventually levels off to a constant (thus, saturating). This effect is known as parton (or gluon) saturation. 51

Saturation physics has become an increasingly interesting field as this kinematic region begins to be probed more deeply by colliders. HERA has probed x values well below 10−3 and in this region it is not unreasonable to expect to see these effects and it is hoped that even smaller x values will be explored by the potential future electron-proton collider (such as the LHeC [8] and EIC [9]). It is not at all clear that saturation has been observed and the DGLAP equation does an excellent job describing the data from HERA. There has been some evidence that the fit deteriorates for very small x and moderate Q2 [67] (although this is does not show that BFKL evolution or saturation was observed or that DGLAP is not applicable). There are also potential signs of saturation in the diffractive cross section data from HERA [68–71] and while all of this is interesting it is by no means definite. Parton saturation is also expected to play an important role in heavy ion collisions where the increased density of partons in the initial condition causes a saturated state at a lower rapidity than in ep collisions. Parton saturation has been initially considered in [72]. Since then two main approaches to saturation have been developed. The first is to approach saturation physics through the consideration of multiple scatterings of dipoles with the target. Multiple scatterings shall be the focus of the rest of the this dissertation in the form of the Balitsky-Kovchegov (BK) equation. The second approach is within the framework of the Color Glass Condensate (CGC). First we will briefly discuss the CGC model before treating multiple scattering effects.

2.2.1 Saturation in the Color Glass Condensate

The Color Glass Condensate (CGC) model seeks to treat a very fast moving hadron as a universal state which describes the properties of hadrons at very high energies. The CGC will not be discussed in depth in this dissertation and we shall give below only a brief outline of the main features of the CGC. Very good reviews on the CGC [73–76] are present in the literature if the reader is looking for a more in depth discussion. In the CGC framework a very high energy hadron is moving very quickly and it becomes length contracted and time dilated. The interactions between the gluons in the hadron slow down due to the time dilation eﬀect which gives the eﬀect 52

Large x : ′Frozen′

Large rapidity gaps

Small x : ′Dynamic′

Figure 2.14. Cartoon of a quark emitting a cascade of gluons which are strongly ordered in longitudinal momentum. The partons with a large x value are time dilated and frozen while further down the cascade the small x partons are not severely time dilated. of the gluons being ’frozen’ in the hadron. However not all gluons are frozen in this manner because as the energy increases there are emissions of small x gluons − which carry a lower momentum and are not severely dilated, which is illustrated in Fig 2.14. This is where the term color glass comes from, as the large x gluons − are colored and over a short time scale they are frozen. On a long time scale the large x gluons are allowed to evolve. This is because the lifetime of a ﬂuctuation − is proportional to the x value, so large x gluons have a long lifetime while the − small x gluons have a short lifetime. This separation of scales at the leading − logarithm allows a separation of phases between these small x gluons and the − longer lived frozen large x gluons. − The term condensate is used as the packing of the phase space by the high density of gluons leads to a very high occupation number of gluons which have their transverse momenta peaked at the saturation scale Qs given by the equation [75]

2 1 xg(x,Qs) 2 2 α¯s 1. (2.58) R Qs ≈ Here our gluon distribution is represented by xg with the saturation momentum 53

given by Qs and the radius of the hadron in transverse space as R. The CGC state may be used to model the highly energetic hadron (or nucleus) in many interactions such as ep,pp,pA and AA collisions. It is worth noting that there is no mention of color dipoles in this construction and the CGC construction instead involves classical color fields. Even though different frameworks are used similar conclusions can be reached in some instances. The dynamics of the CGC model are contained in the evolution equation known as the Jalilian-Marian - Iancu - McLerran - Weigert - Leonidov - Kovner (JIMWLK) equation. The BK equation and the JIMWLK equation both cover the same kinematic regime and in this sense they are closely related. The JIMWLK equation takes into account finite Nc corrections as compared to the BK equation [77–81] and is equivalent to Balitsky’s infinite equation hierarchy [82]. Bal- itsky’s equation hierarchy can be simplified when the mean field limit10 is taken. As a result a single closed equation is obtained which is the BK equation. There have been numerical studies which compare the JIMWLK evolution equation to the BK equation , see for example [83,84].

2.2.2 Multiple scatterings - the Balitsky-Kovchegov equation

The BFKL equation only takes into account single hard Pomeron exchange. This is a reasonable assumption when the system is dilute. As the density of dipoles grows the probability of multiple scatterings increases, and these eﬀects can become large. Multiple scatterings have been investigated from the perspective of Regge theory since the late 50s in the form of Glauber-Gribov multiple scattering theory [85]. Taking multiple scatterings into account aﬀects the structure of the evolution equation itself. The derivation of the BK equation in this dissertation follows closely the work [86] and utilizes the Mueller dipole method. There are other ways of arriving at the BK equation, such as the derivation by Balitsky [82] or within the CGC framework (as discussed in Sec 2.2.1). For our purposes it will be useful to go back to the

10 The mean ﬁeld limit is often used synonymously with the large Nc limit. It should be noted that these two limits are not necessarily equivalent. 54 derivation of the BFKL equation which was done in the dipole framework in Sec 2.1.3. We will use the generating functional method again. The number density of dipoles (2.24) was obtained from the generating functional in (2.23). We can deﬁne the number density for two dipoles in a similar manner

1 δ δ n (x , x , x ; Y )= x2x24π2 Z(x ,Y,u) , (2.59) 2 01 1 2 2! 1 2 δu(x ) δu(x ) 01 1 2 u=1

as well as an arbitrary number of dipoles

k 1 2 δ nk(x01, x1, x2,..., xj; Y )= xi 2π Z(x01,Y,u) . (2.60) k! δu(xi)! i=1 u=1 Y

Again note that the dependence of the number density on impact parameter is suppressed. Diﬀerentiating the generating functional gives us the dipole number density for various numbers of dipoles. Using (2.23) along with (2.60) one arrives at a generalized expression for the density of any number of dipoles

Y x 2 α¯s −2¯α ln 01 (Y −y) x n (x , x , x ,..., x ; Y )= dye s ( ρ ) d2x 01 i 01 1 2 i 2π 2 x2 x2 Z0 Zρ 02 12

2ni(x02, x1, x2,..., xi; y)+ nj(x01, x1, x2; y,..., xj)nk(x01, x1, x2,..., xk; y) . × ! jX+k=i (2.61)

This equation is very much in the same spirit as (2.23) and can be interpreted in the same way. As (2.61) was derived from (2.23) the fact that they have the same structure and interpretation is not surprising. The first term on the right hand side of (2.61) corresponds to the group of i dipoles not splitting, and the second term corresponds to i dipoles splitting into j and k dipoles (such that j + k = i, and similar to the figure 2.10). The goal of these manipulations is to compute the scattering amplitude of a projectile (which contains many dipoles) on a target in which each dipole may multiply scattering off of the target. This requires that we take into account a 55

n1(x01,Y, x1) n2(x01,Y, x1, x2) x1 x1 x2

γ(x1) γ(x1) γ(x2)

Target Target

Figure 2.15. A diagram which illustrates the convolution of the dipole number density ni(x01,Y, x1,..., xi) with the propagators γ(xi). Summing over all such interactions gives the dipole scattering amplitude N(x01,Y ) at a rapidity Y for an incoming parent dipole x01. This is shown mathematically in (2.63). dipole interacting (exchanging a Pomeron) with the target any number of times. In order to take into account multiple exchanges we need to sum over all numbers of these exchanges. This requires accounting for the propagation of the dipole through the nucleus. We shall call this propagator γ(r, b) = γ(r), once again suppressing the impact parameter dependence. For a large target this propagator can be taken to be

N(r,Y =0)= γ(r) . (2.62) − This propagator is a non-perturbative quantity and it can be related to the scattering amplitude at rapidity Y = 0 (this quantity can be ﬁtted to experimental data). However, it is fortunate that the exact form of the propagator is not necessary for the rest of the following calculation. The number of propagators that must be convoluted with the dipole number density is equal to the number of dipoles (for the density of two dipoles n2(x01, x1, x2; Y ) we convolute it with two propagators γ(x1) and γ(x2)). By summing over all number of exchanges we arrive at the scattering amplitude

∞ i 2 d xj N(x01,Y )= n (x01,Y, x1,..., xi) γ(xj) . (2.63) − i 2πx2 i=1 j=1 j X Z Y This summation of multiple scatterings is illustrated Fig 2.15. 2 i d xj By multiplying (2.61) by γ(xj) 2 , and integrating over all xj one can j=1 2πxj Q 56 utilize (2.63) to arrive at

x01 −2¯αs ln( )Y N(x01,Y )= γ(x01)e ρ Y − 2 x01 α¯s −2¯αs ln( )(Y −y) 2 x01 + dye ρ d x2 (2N(x02,y) N(x02,y)N(x12,y)). π x2 x2 − Z0 Zρ 02 12 (2.64)

Taking the derivative with respect to rapidity, in a similar manner to what was done in Sec 2.1.3, results in the following diﬀerential equation

2 ∂N(x01,Y ) α¯s 2 2x01 x01 2 = d x2 4π ln δ (x01 x02) N(x02,Y ) ∂Y π x2 x2 − ρ − Zρ 02 12 2 α¯s 2 x01 d x2 N(x02,Y )N(x12,Y ). (2.65) − π x2 x2 Zρ 02 12 We may recast the form of this equation (in the same manner as before in Sec 2.1.3) to obtain

∂N(x ,Y ) α¯ x2 01 = s d2x 01 ∂Y 2π 2 x2 x2 Zρ 02 12 [N(x02,Y )+ N(x12,Y ) N(x01,Y ) N(x02,Y )N(x12,Y )] . × − − (2.66)

This equation is known as the Balitsky-Kovchegov (BK) evolution equation [82, 86,87]. The terms linear in N on the right hand side of (2.66) correspond to the linear BFKL equation (2.28) derived earlier. The nonlinear term N(x02,Y )N(x12,Y ) in (2.66) takes into account multiple scatterings between the incoming projectile and the target. This is illustrated in Fig 2.16. The leading order kernel for the BK equation is identical to the leading order kernel in the BFKL equation and they have the same symmetries as described in Sec (2.1.4). It should be noted that the scatterings that have been considered here are uncorrelated. The derivation is strictly valid for the case of a nuclear target but will be used for a proton as well. The interpretation of (2.66) in terms of Pomerons is that it allows Pomeron 57

γ∗ γ∗

p p

Figure 2.16. Diagram where a virtual photon ﬂuctuates into a dipole which then emits many dipoles. There are multiple scatterings between the dipoles and the target where each double line represents a color dipole, in this formulation a gluon in the large Nc limit. splitting in the interaction, thereby resumming all so-called Pomeron ’fan diagrams’11, as in Fig 2.17 [86]. This view of the BK equation identical to the one presented in Fig 2.16 (imagine ’pulling down’ on the interactions in Fig 2.16 and dragging the gluon emissions and a diagram similar to 2.17 emerges). The nonlinear term becomes important in the regime where the scattering amplitude N becomes large, which corresponds to the dense partonic system. At small values of the scattering amplitude the system is dilute and the nonlinear term can be neglected. As the scattering amplitude approaches unity the integrand

N(x02,Y )+ N(x12,Y ) N(x01,Y ) N(x02,Y )N(x12,Y ) (2.67) − − in (2.66) goes to zero, cutting oﬀ the growth of the amplitude. The point at which the nonlinear term becomes important signiﬁes a distinct change in the dynamics of the system as the evolution shifts from being governed by the BFKL equation to the BK equation. This shift is due to the fact that the number density of dipoles

11Pomeron loops however, are still a diﬃcult topic to tackle, a brief review can be found in [88]. 58

γ∗ γ∗

p p

Figure 2.17. Diagram where multiple Pomerons split and are exchanged between a qq¯ pair from a virtual photon fluctuation and a proton. These diagrams are resummed in equation (2.66) leading to the nonlinear contribution. Note that no correlations are included so each Pomeron scattering with the target is independent, so that the scattering is incoherent. increases to the point that multiple scatterings can no longer be neglected. The state in which the dipole density is large is usually referred to as parton saturation. This effect is illustrated schematically in Fig 2.18 along with the different regions of validity of the various evolution equations introduced so far in this dissertation. Multiple scatterings serve to enforce the unitarity of the scattering amplitude, which states that the dipole scattering amplitude should be bounded (i.e. N 1). ≤

2.2.2.1 Solutions of the BK equation

Let us now investigate some of the work done in solving the BK equation. There is currently no closed analytic solution to the BK equation as we had with the BFKL equation. There is however some work in the same vein as the conformal method used earlier [66]. In most analyses the BK equation is solved using simplifying assumptions. Typically the impact parameter is neglected, which reduces the number of variables in the problem. 59

BK 1 x Saturation Boundary

BFKL DGLAP NON − PERTURBATIVE

2 1 Q ≈ r 2

Figure 2.18. Cartoon of the direction of the BK evolution equation compared with BFKL and DGLAP evolution in the Q2 and x plane. Shown is a hadron composed of a number of partons, the size of each parton shrinks as the resolution Q2 increases. In contrast as the x decreases further emissions of gluons begin to cause the number of partons in the hadron to grow. As the number of hadrons grows they begin to form a dense system and multiple scatterings are important to take into account, this is the saturated region.

It is possible to investigate the solution of the BK equation without any spatial degrees of freedom, and while this is not physically motivated it does provide some insight into the basic features of the solution. Without spatial dependence the equation12 becomes

12This is known as the Malthus-Verhulst equation [89] (also sometimes referred to as a logistic function) and it can be used as a simple model for population dynamics which grow exponentially before saturating to a no-growth situation. 60

dN(Y ) = ω(N(Y ) N 2(Y )) (2.68) dY − where ω is a multiplicative constant. Separating variables (and letting N = N(Y )) the diﬀerential equation becomes

dN dN dN = + = ωdY . (2.69) N(1 N) N 1 N − − Integrating both sides yields

N ln(N) ln(1 N)=ln = ωY + C . (2.70) − − 1 N − This equation can be solved for N, giving the following result

eωY N = C′ . (2.71) 1+ eωY The solution has the property of saturating to unity for large values of Y . The limit of small scattering amplitude N(Y ) = 0 is a solution, however it is not a stable solution. This instability is due to the linear portion which will tend for the amplitude to grow. The only stable solution for in the equation is N(Y ) = 1, and for a suﬃciently long evolution the system will reach this saturated state. A more physically motivated solution is to assume that the scattering amplitude depends on only one spatial dimension, the dipole size r. This lack of dependence on the impact parameter b corresponds to a target which is very large spatially. For dipoles which are smaller than the size of the target and at small impact parameter this is a good approximation, but as the spatial extent of the proton is not large this is a rather crude approximation. There are multiple works on the BK equation without impact parameter dependence that were done both numerically [90–96] as well as some analytical work [97–101]. The BK equation lends itself well to numerical solution in both the case with one spatial degree of freedom or more. In the next section we shall present the numerical solution to the BK equation both with and without impact parameter dependence. The inclusion of impact parameter introduces novel and nontrivial behaviors into the system as we shall see. Chapter 3

Numerical solution of the BK equation

Solving the BK equation without impact parameter is like drinking non-alcoholic beer, there is something important missing

Anna Stasto

In this chapter I will present my numerical work on the BK evolution equation. Several studies of the BK equation have been done by other groups [90–92,94,102]. In these works the BK equation was solved using the assumption that the only relevant spatial degree of freedom is the dipole size. This assumption makes the solution of the BK equation relatively easy to find numerically. The justification for this approximation is that one assumes the target is very large, neglecting the impact parameter dependence. This brings a large scale into the problem and results in symmetries of the evolution being broken [92,103]. If this is viewed from momentum space, neglecting impact parameter is equivalent to solving for the forward limit of the scattering in the evolution where the triple Pomeron vertex has zero momentum transfered through it as well as zero momentum in both outgoing reggeized gluons [104]. On the other hand the full (unapproximated) kernel is Möbius invariant and one expects the solution of the BK equation to reflect this symmetry (at LL level with fixed coupling). In order to respect Möbius invariance 62

θ b φ

T arget

Figure 3.1. Diagram of a dipole-target system with all spatial degrees of freedom: dipole size r, impact parameter b, dipole orientation θ and orientation with respect to the target φ. the inclusion of the impact parameter is necessary. Impact parameter dependence is also important in order to accurately compare results from BK evolution models with experimental results, such as exclusive vector meson production (which will be discussed in chapter 4). In this chapter I will discuss the method that I used to find the numerical solution to the BK equation in cases with and without impact parameter dependence (Sec 3.1), following our work in [105]. The behavior and properties of the numerical solution are discussed for cases with (Sec 3.3) and without (Sec 3.2) impact parameter. I shall discuss in detail the differences between the solutions with and without impact parameter as well as the effects of further corrections, such as kinematical constraints (Sec 3.3.4), and inclusion of the running coupling (Sec 3.3.6)

3.1 Numerical methods

The first step to solve the BK equation numerically with impact parameter is to define the variables that are inherent to each dipole in the problem and discretized these degrees of freedom. The scattering amplitude depends on five variables : the dipole size r, the impact parameter b, the angle between the impact parameter 63 and the dipole size θ, the angle between the target and the impact parameter φ, and the rapidity Y . The spatial variables are illustrated in Fig 3.1. We shall assume that the target is cylindrically symmetric, this means that the scattering amplitude does not depend on the angle φ. We discretized the spatial coordinates in terms of logarithmic variables log10(r) and log10(b) with the angular coordinate for our calculation being cos(θ), which varies from 1 to 1. The number of points − in each dimension determines the resolution of the solution. A grid which is too sparse is prone to numerical artifacts and missing important dynamics, while a dense grid quickly becomes very computationally intensive. The grid chosen for our calculation is 200 200 20 giving a total of 800, 000 functional evaluations r × b × θ per step in rapidity. Several other grid sizes were tested and the chosen size was settled on as a balance of accuracy and computational speed. For the case of the solution without impact parameter there is only one spatial variable r. Therefore we see that the computation with impact parameter is much more demanding as far as computing time is concerned compared to the calculation without impact parameter. The limits on the grid size are of vital importance to the validity of the numerical calculation. When the calculation becomes non-zero at the boundaries then the entire solution can no longer be trusted for accuracy. The requirement that the boundary be far from the non-zero (or fully saturated) region of the solution is compounded with the problem of grid density. Increasing the size of the area being numerically evaluated will require more points in order to maintain the density of the grid. The grid size is essentially what dictates to what rapidity the system may be evolved. This is because an equation being solved on a small grid cannot be evolved far in rapidity before the solution encounters the boundary, or the ’box wall’ as it may be called (the ’box’ being a region in r,b and θ that is being numerically evaluated). To compound the difficulty, it is not possible to separate the limits of the dipole size r and the impact parameter b as these variables are correlated (as we shall see in detail in Sec 3.3.3). The grid limits chosen are 10−8 108 for both the dipole size r and the impact parameter b and 1 1 for → − → the angle cos(θ). It is not necessary the hold the density of grid points between r and b equal, so one could evaluate fewer points in impact parameter space than in dipole size as long as the limits of evaluation were equal. However, in all of 64 this work the density of the grid in impact parameter and in dipole size were held equal. The BK equation is an evolution equation which requires a non-perturbative initial condition at some initial rapidity. We choose the initial condition of the scattering amplitude N(r, b,Y = 0) to be of the Glauber-Mueller form [51,60,106]

2 N(r, b,Y =0)=1 e−10r T (b), (3.1) − where T (b) is the impact parameter proﬁle of the target and is taken to be Gaus- sian:

2 − b T (b)= e 2 . (3.2)

The constants in this particular initial condition have been chosen rather arbitrarily. In this chapter we are focusing on the mathematical properties of the solution. The physical values that have been ﬁt to experimental data shall be introduced in chapter 4. For small dipole sizes the scattering amplitude is zero, corresponding to the property of color transparency. For large values of impact parameter the scattering amplitude goes to zero as well. This corresponds to the situation where the dipole is very far from the target and barely interacts with it. The initial size of the target is dictated by the proﬁle T (b). We note here that the initial condition does not have θ dependence. As we shall see in Sec 3.3.3 that the evolution generates an angular dependence. With the initial condition set, one can now evolve this scattering amplitude from Y0 to Y0 +∆Y by the BK equation (2.66). After evolving by a small step in rapidity ∆Y the equation becomes

Y0+∆Y N(x01, b01,Y0 +∆Y ) = N(x01, b01,Y0)+ Zx2 ZY0 [N(x02, b02,y)+ N(x12, b12,y) N(x01, b01,y) × − N(x02, b02,y)N(x12, b12,y)] dy , (3.3) −

2 where = d x2K(x ,x ,x ). The splitting kernel K(x ,x ,x ) depends x2 01 02 12 01 02 12 only onR theR dipole sizes in the problem, but it can potentially depend on the 65 rapidity Y (as in Sec 3.3.4). Taking the ﬁrst approximation to the rapidity integral in (3.3) we ﬁnd

(1) (0) N (x01, b01,Y0 +∆Y )= N (x01, b01,Y0)

(0) (0) + ∆Y N (x02, b02,Y0)+ N (x12, b12,Y0) Zx2 (0) (0) (0) N (x01, b01,Y ) N (x02, b02,Y )N (x12, b12,Y ) . − 0 − 0 0 (3.4)

Here the superscripts denote the approximation used, where the ﬁrst approximation to the solution is N (1). In this situation the initial condition is the same as the zeroth approximation N (0). The approximation (3.4) can be reﬁned by a linear interpolation procedure

(1) (0) Y (N (x01, b01,Y +∆Y ) N (x01, b01,Y )) N(x , b ,y) = 0 − 0 01 01 ∆Y (0) + N (x01, b01,Y0), (3.5) as in [60,105]. Inserting (3.5) into the right-hand side of (3.3) we reach the expression for the second approximation to our integral:

(2) (0) N (x01, b01,Y0 +∆Y )= N (x01, b01,Y0)

∆Y (0) (0) (0) + N (x02, b02,Y )+ N (x12, b12,Y ) N (x01, b01,Y ) 2 0 0 − 0 Zx2 (0) (0) N (x02, b02,Y )N (x12, b12,Y ) − 0 0 ∆Y + N (1)(x , b ,Y +∆Y )+ N (1)(x , b ,Y +∆Y ) 2 02 02 0 12 12 0 Zx2 (1) (1) (1) N (x01, b01,Y +∆Y ) N (x02, b02,Y +∆Y )N (x12, b12,Y +∆Y ) − 0 − 0 0 ∆Y (1) (0) + N (x02, b02,Y +∆Y ) N (x02, b02,Y ) 6 0 − 0 Zx2 (1) (0) N (x12, b12,Y +∆Y ) N (x12, b12,Y ) . × 0 − 0 (3.6) 66

This method of approximation can be continued by applying further linear inter- polations, such as

(n−1) (0) Y (N (x01, b01,Y +∆Y ) N (x01, b01,Y )) N(x , b ,y) = 0 − 0 01 01 ∆Y (0) + N (x01, b01,Y0). (3.7)

By inserting this into the right-hand side of (3.3) one can arrive at the nth approximation to the scattering amplitude, which has a similar form to (3.6). In order to gauge the accuracy of each approximation a value ǫ was deﬁned such that

(n) (n−1) N (x01, b01,Y +∆Y ) N (x01, b01,Y +∆Y ) ǫ = | (n−1) − |. (3.8) N (x01, b01,Y +∆Y )

If ǫ is greater than some pre-defined accuracy parameter ǫ0 then it is deemed necessary to go to a further approximation order, and we invoke (3.7) to solve for n+1 N (x01,Y +∆Y ). If ǫ<ǫ0 then the desired accuracy has been reached and the scattering amplitude at Y0 +∆Y has been solved for. The number of iterations of (3.7) generally varies greatly during the calculation. At low values of rapidity, when the solution is very close to the initial condition, a high number of iterations are required. As the equation ’forgets’ the initial condition very quickly (becomes independent of the initial condition) the order of the approximation needed to satisfy the accuracy condition drops dramatically. Additionally, the exact form of the splitting kernel K(x01,x02,x12) factors into how many iterations are needed to reach the predetermined accuracy limit. For these reasons we have implemented a dynamical number of iterations in our solving program such that it checks the accuracy for every point in the r,b, and θ grid against the previous order. If any points are found to be above the allowed accuracy limit then another iteration of the linear interpolation scheme is calculated. When implementing the cutoff check it is important to not simply compute ǫ blindly as if the previous order n−1 N (x01, b01,Y +∆Y ) 0 then the ǫ value will always be extremely high. In ≈ order to prevent this from happening a large cutoff was placed on the calculation of ǫ in our program as well as a large upper limit on the number of iterations (this 67

β x02 θ02

b02 x2 θ01 b 01 x01

b12 θ Target 12 x12

Figure 3.2. The labeling of the kinematic variables in position space for a dipole x01 splitting into two dipoles x02 and x12.

prevents the program from falling into an inﬁnite loop). We have chosen ǫ0 =0.02 in our computations. A scheme of ﬁxed number of iterations is possible but this is unnecessary for large rapidities and causes the calculation of asymptotic behaviors to take an exceedingly long time.

Once the scattering amplitude N(x01, b01,Y0 +∆Y ) has been solved for the results are then output to a ﬁle and the process is started over again, but this time solving for N(x01,Y0 + 2∆Y ) and using N(x01,Y0 +∆Y ) as the initial scattering amplitude. The rapidity steps used in our calculation were set to be ∆Y = 0.2. Smaller steps were found to have no eﬀect on the solution. In general it was possible to reach very large rapidities (on the order of 50, which generates a total of approximately 5 GBs of data) with the limits of the grid that were set. The computation time of our C code was around one hour per unit of rapidity when computed on 32 processing cores each with a clock speed of 3 GHz.

3.1.1 Kinematic variables

The biggest diﬃculty with the inclusion of impact parameter dependence is the number of degrees of freedom in the system. Not only is the impact parameter b 68 included but the inclusion of the angle between the dipole size r and the impact parameter b is required. As the evolution requires all of these variables for each dipole there are nine variables that must be kept track of (three variables per dipole and three dipoles: one parent dipole and the two daughter dipoles which it splits into). These variables can be seen labeled in Fig 3.2.

As the product of the program is a scattering amplitude N(x01,b01,θ01,Y ) the values x01,b01,θ01 are ﬁxed for the evaluation of a single point. The value of x02 is integrated over as well as the angle β, which is the angle between dipole x01 and x02. The rest of the angles, dipole sizes, and impact parameters in the problem may be solved in terms of these ﬁxed quantities and these integrated variables.

For example for the kinematic variables for dipole x12 become

2 2 2 2 x = (x1 x2) =(x1 x0) +(x0 x2) + 2(x1 x0) (x0 x2) 12 − − − − · − 2 2 = x01 + x02 +2x01x02 cos(β), (3.9)

2 1 2 1 2 2 b = (x1 + x2) = (x0 + x1) +(x2 x0) + 2(x1 + x0) (x2 x0) 12 4 4 − · − x2 = b2 + 02 + b x cos(β θ ), (3.10) 01 4 01 02 − 01 and

x12 b12 1 cos(φ12) = · = ((x1 x0 + x0 x2) (x1 + x0 + x2 x0)) x12b12 2x12b12 − − · − 1 = ([(x1 x0)+(x0 x2)] [(x1 + x0)+(x2 x0)]) 2x12b12 − − · − 1 2 = 2x01b01 cos(φ01)+ x01x02 cos(β)+2x02b01 cos(β θ01)+ x02 . 2x12b12 − (3.11)

The variables for x02 can be solved for in an identical manner. As the values of these variables in general do not land exactly on our grid an interpolation scheme is required to arrive at the scattering amplitude at these values. A linear interpolation procedure was used and is described in further detail 69 in Appendix B.

3.1.2 Parallelization of the code

Due to the large number of points at which the integrals need to be evaluated the parallelization of the code was essential. There are many options open to parallelize computations in C. We have chosen the MPI [107] library for the method of parallelization as it is compatible with many large computing clusters. Other methods (such as threads) allowed the utilization of all the cores on a single machine (node) however cross-node talk was not supported, which placed an upper bound on the number of cores in the 2 6 range, which was insuﬃcient for our − computational needs. There are two obvious ways to go about dividing the calculation of (3.6) into smaller pieces that are able to be distributed between various processing cores. Let us simplify the notation and rewrite (3.6) into the form:

xmax π N(x01, b01,Y0 +∆Y )= N(x01, b01,Y0)+ dx2dθ2Z, (3.12) Zxmin Z0 where Z contains the integrand. The ﬁrst method is to divide the grid into regions and give each region to a diﬀerent core. Each core would have less functional evaluations to do but it would have the entire range of the integral to compute: x2 R

xmax π CORE1 N(x01, b01,Y0 +∆Y ) = N(x01, b01,Y0)+ dx2dθ2Z x 0 Z min Z xmin

Here we have not broken the integration over θ up and have assumed only four cores in this example. The other method1 is to allow each core the full range of r, b, and θ but to break the integral up amongst these cores so that each core computes only a x2 fraction of the integral:R

a1 π CORE1 N(x01, b01,Y0 +∆Y )= N(x01, b01,Y0) + dx2dθ2Z Zxmin Z0 a2 π CORE2 + dx2dθ2Z Za1 Z0 a3 π CORE3 + dx2dθ2Z Za2 Z0 xmax π CORE4 + dx2dθ2Z (3.14) Za3 Z0 It is the latter method that we have chosen to use to break up the computation, note that we do not divide up the integration over the angle θ2 but only over the dipole size x2. It was found that the number of points on the grid was not as important to the accuracy of the solution as the resolution of the integral in x2. The integrals were computed using Gaussian quadrature [108] with the number of evaluations per integral as 12 in both the x2 and θ2 dimensions. The fewer cores that are conscripted into the computation the more evaluations per integral are required in the x2 integral and the slower the computation became. With our grid (and 12 functional evaluations per integral) we found that at approximately 20 integration regions were needed to provide a good accuracy for the solution. We had access to a maximum of 32 cores so our integral was broken into this many regions.

1If one had many cores at their disposal these two methods could be combined. 71

3.2 Solution without impact parameter dependence

We discuss ﬁrst the solution of the BK equation without impact parameter dependence. The only spatial degree of freedom is the dipole size r and we assume the target to be inﬁnitely large. This analysis is similar to what was done in other works [90–92,105,109] and evaluating this case provides a good check of the code. Without impact parameter our initial condition is taken simply to be

2 N(x ,Y )=1 e−x01 . (3.15) 01 0 − When the dipole x is small (3.15) is also small, i.e. N(x ,Y ) x2 , and 01 01 0 ≈ 01 saturates to unity when x01 is large. There are no physical scales in the problem and all constants are set arbitrarily. When evaluating without impact parameter we may have a much larger and denser grid due to a much smaller number of functional evaluations being required. In this case our grid in log10(r) is set to consist of 5000 points and goes from 10−10 1010. → Let us first consider the LL case with fixed couplingα ¯s = 0.1. The solution using the initial condition (3.15) is shown in Fig 3.3 for different values of rapidity (Y = 1, 2,..., 10 in Fig 3.3(a) and Y = 10, 20,.., 80 in Fig 3.3(b)). We can see in Fig 3.3(a) that after a short evolution the shape of the solution is no longer the same as the initial condition. This is referred to as the initial condition being forgotten, illustrating that the solution is not very sensitive to the form of the initial condition. At large dipole sizes the solution of the scattering amplitude is constant and equal to N = 1. The region in which the amplitude is equal to unity is referred to as the saturation regime. Evolving to larger rapidities, as in Fig 3.3(b), it can be seen that a wave-front is formed which begins propagating towards smaller dipole sizes (the ultraviolet regime). This wave-front travels with a constant velocity, which is best analyzed from the perspective of the saturation scale, discussed below. 72

∆Y: 1.0 | max Y: 10.0

1.0

0.8

0.6 N(Y,r)

0.4

0.2

0.0 -2 -1 0 1 2 10 10 10 10 10 Dipole Size (Arbitrary Units) (a) ∆Y: 10.0 | max Y: 80.0

1.0

0.8

0.6 N(Y,r)

0.4

0.2

0.0 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 10 10 10 10 10 10 10 10 10 10 10 Dipole Size (Arbitrary Units) (b)

Figure 3.3. Solution of the BK equation without impact parameter with the LL kernel, and couplingα ¯s = 0.1. Fig a shows how the initial condition (rightmost curve) is quickly forgotten after a very small number of steps in rapidity, where each curve past the initial condition represents an additional one unit of rapidity. Fig b shows the movement of the wave-front towards smaller dipole size where once again the rightmost curve is the initial condition and each curve afterwards is 10 additional units of rapidity. 73

3.2.1 Saturation scale

A saturated system is deﬁned by a momentum scale which characterizes the transition between linear (BFKL) and non-linear (BK) dynamics. This occurs when the scattering amplitude becomes large (when multiple scatterings are important), and we shall choose a (somewhat arbitrary) demarcation κ to signify when the scattering amplitude becomes large. When N >κ the system is said to be dense and the dynamics are governed by the BK equation. When N <κ the system is dilute and the linear BFKL equation is applicable, since the multiple scattering term is proportional to N 2 and is negligible at small N. The boundary between these states is deﬁned by the equation

κ = N(rs(Y ),Y ) , (3.16) where rs(Y ) is the saturation radius. It is convenient to deﬁne the saturation scale as the inverse of the saturation radius

1 = rs(Y ) . (3.17) Qs(Y ) 1 In this dissertation κ will consistently chosen to be κ = 2 . The saturation scale

Qs grows with an increase of rapidity. This momentum scale is the characteristic momentum for the gluons within the system, and the system is dilute for a dipole probe above this momentum and dense for a probe below this momentum. It was found that the form of the solution at high rapidity is largely independent of the initial condition. The only eﬀect of the initial condition on the solution is to shift the solution towards larger (or smaller) dipole sizes. This can be seen in Fig 3.3(a) where the initial condition is quickly forgotten and the shape of the solution is determined by the evolution equation. With increasing rapidity the shape of the solution does not change, but is merely shifted towards smaller dipole sizes (as seen in Fig 3.3(b)). The property that the shape is universal, regardless of initial condition, and all that changes is the position of this wavefront in dipole size is known as geometric scaling. The realization of geometric scaling in mathematics is that the scattering amplitude does not independently depend on the rapidity Y and the dipole size r but on only one combined variable rQs(Y ). 74

N(r, Y ) N(rQ (Y )). (3.18) → s Here we see that the saturation scale appears in the argument, which is not surprising as Qs is a dynamically generated quantity that gives an internal dimensionful scale to the system. Geometric scaling behavior was observed in the DIS data gathered at HERA [110,111]. The dipole size at which saturation occurs decreases as rapidity increases, and this can be seen in Fig 3.4(a), where the intersection between the horizontal bar and the scattering amplitude corresponds to the solution of (3.16). How much the saturation scale changes per unit of rapidity directly corresponds to the velocity of the wave-front of the scattering amplitude. The logarithm of the saturation momentum Qs =1/rs versus rapidity shows a linear dependence at large rapidities on the rapidity and is plotted in Fig 3.4(b).

As the behavior of Qs is approximated by a straight line on a logarithmic plot we can parameterize it by

2 2 λsα¯sY Qs = Q0e . (3.19)

The value of λs can be extracted from the solution in Fig 3.4(b). This value can be compared with analytic results. It has been shown in [100, 103] that the form of the saturation scale is as follows

χ(γ ) 3 Q2(Y )= Q2 exp α¯ c Y ln(Y ) , (3.20) s 0 s 1 γ − 2(1 γ ) − c − c where γc = 0.62. For the LL kernel γc is the value which solves the implicit ′ equation γcχ (γc) = χ(γc), where χ(γ) is the BFKL eigenvalue. The contribution from the second term in the exponent of (3.20) is less than 10 percent compared ∼ to the ﬁrst term, which is linearly dependent on Y for the rapidities that are being considered here. So we expect by (3.20) that the slope of λs is given analytically by

χ(γ ) λ = c =4.88. (3.21) s 1 γ − c Extracting the λs from the graph Fig 3.4(b) we found λs =4.4 for bothα ¯s =0.1 75

∆Y: 10.0 | max Y: 80.0

1.0

0.8

0.6 N(Y,r)

0.4

0.2

0.0 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 10 10 10 10 10 10 10 10 10 10 10 Dipole Size (Arbitrary Units) (a)

20 ) 2 s ln(Q

0 20 40 60 80 Rapidity

(b)

Figure 3.4. Results for the computation of the saturation scale for the LL kernel with frozen couplingα ¯s = 0.1. The horizontal bar on Fig a shows the boundary between the dilute system (below) and the dense system (above). The intersection of the scattering amplitude curves and this boundary gives the result for the saturation size rs which is related to the saturation scale by Qs = 1/rs. Fig b shows these points plotted on a logarithmic vertical axis. 76

andα ¯s = 0.2, which is within 10 percent of the expected theoretical value. We observed that the exponential in (3.19) is linear inα ¯s, which is expected in the LL approximation. The value for λs is extracted only from the intermediate rapidities in Fig 3.4(b). At low values of rapidity the solution still depends on the initial condition and has not reached an asymptotic form. This can be seen by an increase in the slope of the saturation scale which occurs between rapidity units 0 and 35 . 2 We extract the value of λs from the rapidity region of 35 to 50, where the slope of ln(Qs) is approximately constant. At very large rapidities ( 50+) we begin to encounter eﬀects from the box wall which introduces numerical artefacts into the evolution.

3.3 The solution with impact parameter dependence

Inclusion of impact parameter dependence in the dynamics of the system changes the evolution dramatically. There are many features which are unique to the case with impact parameter which greatly affect the interpretation and the predictions of the solution to the BK equation. We shall go through these features and attempt to give a physical interpretation of why this arises when the target is localized in transverse space as opposed to the assumption of a target of infinite size. We do assume the target is cylindrically symmetric by neglecting the second φ angle in Fig 3.1. A separate numerical analysis was done with this additional angle included and it was found that the dependence on this angle was removed very quickly by the evolution (even when the initial conditions depended on this angle φ). This effect can be seen in Fig 3.5 where the scattering amplitude is plotted versus the azimuthal angle φ. The initial condition is shown by the dotted and dashed line and it depended on φ, however within five units of rapidity there is no dependence on φ. In this section we use the initial condition (3.1) for all calculations. It is important to note that in this chapter we are only interested in the mathematical properties of the solution. Therefore, the computations in this chapter include large dipoles and no physical scales. These large dipoles need to be regu-

2 The rapidities at which these transitions occur is not ﬁxed and will vary withα ¯s as well as the exact form of the splitting kernel used. 77

Dipole Size: 1.000 | Impact parameter: 1.000 | cos(θ) = 0.0 | Delta Y: 5.0 | max Y: 25.0

1.0

0.8

0.6 N(Y,r,b,θ,Φ) 0.4

0.2

0.0 -1.0 -0.5 0.0 0.5 1.0 Cos(Φ)

Figure 3.5. Graph of the scattering amplitude N(Y, r, b, θ, φ) versus the azimuthal angle φ. The initial condition is represented by the dashed-dotted line and has a dependence on φ. Each line corresponds to a change in rapidity ∆Y = 5.0 to illustrate how the scattering amplitude does not depend on the angle φ after several steps in rapidity. This shows the ’washing out’ of the angle φ, and it will be assumed that the scattering amplitude is independent of φ in the rest of this dissertation. lated to arrive at a physical solution. The introduction of a physical scale and a realistic solution to the BK equation will be presented in chapter 4.

3.3.1 Dependence on the dipole size

The single most striking feature of the solution to the BK equation with impact parameter dependence is the behavior of the scattering amplitude at large dipole sizes. As can be seen in Fig 3.6 the amplitude at large dipole sizes, which starts saturated by the initial condition (3.1), quickly drops to zero. This can be interpreted as follows: a very large dipole, even at very small impact parameter, will miss the target as the color charges are very far from the interaction region. Without impact parameter this dipole would always interact as the target had inﬁnite size. As a result, the amplitude in that case was always equal to unity for 78

Impact parameter: 1.00 | cos(θ): 0.0 | ∆ Y: 1.0 | max Y: 10.0

1.0

0.8

0.6 N(Y,r,b,θ) 0.4

0.2

0.0 -3 -2 -1 0 1 2 3 10 10 10 10 10 10 10 Dipole Size (Arbitrary Units) (a) Impact parameter: 1.00 | cos(θ): 0.0 | ∆ Y: 10.0 | max Y: 50.0

1.0

0.8

0.6 N(Y,r,b,θ) 0.4

0.2

0.0 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 10 10 10 10 10 10 10 10 10 10 10 10 10 Dipole Size (Arbitrary Units) (b)

Figure 3.6. Graphs of the scattering amplitude N(r, b, θ, Y ) versus the dipole size r. The initial condition is represented by the dashed-dotted line. Fig a : each line corresponds to a small change in rapidity ∆Y = 1.0 to illustrate the rapid departure from the initial condition where the curves in the large dipole regime are dropping and moving towards the small dipole regime while the curves in the small dipole regime proceed to evolve towards smaller dipoles. Fig b : larger intervals in rapidity ∆Y = 10.0 illustrate the propagation of the wave-front to both the small dipole regime as well as the turn-around in the propagation of the wavefront in the large dipole regime and the evolution towards larger dipoles. 79

Figure 3.7. Cartoon of a very large dipole failing to interact with the target as neither of the color charges composing the dipole impinge on the interaction region of the localized target. arbitrarily large dipole sizes. With a localized target it is possible for a dipole to be too large and miss the target (this is illustrated in Fig 3.7) in the case with impact parameter dependence. The solution possesses two fronts, one for small dipole sizes (ultraviolet region) and the second one for large dipole sizes (infra-red region). The solution evolves towards both the small dipole and large dipole sizes. This evolution in the large dipole size region has many implications for the dynamics of the system. First, the dynamics in the infra-red regime is no longer hidden underneath the veil of saturation, as was the case when impact parameter was neglected. We shall see that when the running coupling is taken into account in (Sec 3.3.6) the regularization of the coupling in the infra-red region becomes very important to the dynamics of the system. The strong coupling αs is large in the large-dipole regime and the dynamics are non-perturbative. Without the impact parameter dependence the specifics of the behavior of the coupling at large dipole sizes did not matter as in this region the system was completely saturated. In the case with impact parameter dependence care must be taken to cut off these large dipoles. The introduction of a confinement scale will be discussed in Sec 4.1 and is key to reproducing the dynamics seen in collider data. Finally the form of the scattering amplitude now admits two solutions to the condition which defines the saturation scale (3.16), one for small dipole sizes and one for large dipole sizes. The behavior of this second saturation scale shall be the focus of Sec 3.3.5.2. 80

3.3.2 Dependence on impact parameter

The dependence of the scattering amplitude on the impact parameter can be seen in Fig 3.8 in both linear and logarithmic scales. The initial condition is steeply falling as a function of impact parameter (3.2), as can be seen by the dotted-and- dashed line in Fig 3.8, and the solution quickly departs from the initial condition and evolves towards larger impact parameter. This corresponds to the diffusion of the interaction regions in transverse space as dipoles are created at larger values of impact parameter. The increase in the interaction area will directly affect the value and growth of the cross section and the black disc radius, which will be discussed in Sec 3.3.7. It can be seen in Fig 3.8(b) that there is a change in behavior from the steeply falling profile of the initial condition (given by a Gaussian) to a power-like behavior. This leads to power-like tails in impact parameter space and there are several important effects from these tails that will be discussed throughout the rest of this dissertation. These tails violate the Froissart bound when the cross section is calculated (as will be discussed in Sec 3.3.7) and these long-range effects need to be stemmed in order to bring the growth of the cross section under control. The inclusion of a mass scale cuts off the long distance effects damps these long tails and helps to control the growth of the cross section, as will be seen in Sec 4.1.

3.3.3 Angular dependence and enhancement at r =2b

There is an interesting and nontrivial behavior of the scattering amplitude at large dipole sizes which is related to the correlation between dipole size and impact parameter. There is an enhancement when the dipole size is equal to twice the impact parameter. This can be seen in Fig 3.9 where at r = 2b there is a sharp increase in the scattering amplitude when viewed as a function of the dipole size. This is only apparent for relatively large dipole size and impact parameters because for small dipole sizes and impact parameters the orientation and configuration of the dipole matters much less. For configurations of large dipoles we reach scenarios as in Fig 3.10 where one configuration of the dipole will interact weakly as the color charges fail to hit the interaction region of the target and another configuration will cause a strong 81

Dipole Size: 1.00 | cos(θ): 0.0 | ∆ Y: 10.0 | max Y: 50.0

1.0

0.8

0.6 N(Y,r,b,θ) 0.4

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0.0 -1 0 1 2 3 10 10 10 10 10 Impact Parameter (Arbitrary Units) (a) Dipole Size: 0.11| cos(θ): 0.0 | ∆ Y: 5.0 | max Y: 30.0 0

-2

-4 ln(N(Y,r,b,θ)) -6

-8

-1 0 1 2 3 10 10 10 10 10 Impact Parameter (Arbitrary Units) (b)

Figure 3.8. Graphs of the scattering amplitude N(r, b, θ, Y ) versus the impact parameter b. The initial condition is shown by the dashed-dotted line. Fig a : linear graph of the scattering amplitude N(r, b, θ, Y ) versus impact parameter, the departure from the steeply falling initial proﬁle in impact parameter can be seen. Fig b : logarithmic plot of N(r, b, θ, Y ) versus impact parameter, showing a clear change in slope from the steeply falling proﬁle to a power-like tail. 82

Impact parameter: 100.00 | cos(θ): 0.0 | ∆ Y: 10.0 | max Y: 50.0

1.0

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0.6 N(Y,r,b,θ) 0.4

0.2

0.0 -2 -1 0 1 2 3 4 5 6 7 8 10 10 10 10 10 10 10 10 10 10 10 Dipole Size (Arbitrary Units)

Figure 3.9. Plot of the r = 2b peak of the scattering amplitude in dipole size using the LO kernel. The initial condition is very close to zero in this plot and it is not visible. scattering. This implies that there exists a strong angular dependence in this system. The amplitude for the parallel arrangement (as in Fig 3.10(a)) possesses a much sharper peak compared to the perpendicular configuration (as in Fig 3.10(b)). This is illustrated in Figs 3.9 and 3.11 where the amplitude for the perpendicular and the parallel configurations are shown respectively. This effect can also be seen by plotting the scattering amplitude versus the angle θ as in Fig 3.12. When r = 2b there is an enhancement in the regions of cos(θ)= 1, 1 which is generated by the − evolution. Away from the r =2b region one finds that the equation is not sensitive to dipole orientation. It is interesting to note that in Fig 3.13 where the scattering amplitude is plotted against the impact parameter the peak completely vanishes when the dipole size is oriented perpendicular to the impact parameter. However, if the scattering amplitude is plotted versus the dipole size, as in Fig 3.11, even away from the 83

Target Target

r b r

(a) (b)

Figure 3.10. Diagrams of two possible orientations of the projectile dipole with regards to the target, with the dipole size parallel (Fig a) and perpendicular (Fig b) to the impact parameter. It can be seen that when r = 2b the orientation where the dipole is parallel to the impact parameter gains an enhancement as the color charge impacts the target directly. parallel conﬁguration the peak remains, albeit less pronounced. This diﬀerence can be better seen if we refer back to the solution presented in Sec 2.1.5 which utilizes conformal eigenfunctions. Consider the conformal eigenfunctions that are found in the solution from Sec 2.1.5.1

∗ r r r r En,ν 0 w, 0 w En,ν 0 + b w, 0 + b w , (3.22) 2 − − 2 − 2 − − 2 − where we have written the arguments in the same geometric form as [66]. Here b is the impact parameter and r0,r are the sizes of the target and the projectile. The coordinate w is the center of mass of one of the dipoles and is integrated over. The coordinates r and b (compared with the ones used in Sec 2.1.5) can be seen in Fig 2.13, note that w is not shown as it is integrated over and will not play a part in this analysis. The largest contribution from (3.22) comes from the point where w = 0 so we shall consider only this region of small w. Writing the second eigenfunction out explicitly, according to (2.54), yields 84

Impact parameter: 100.00 | cos(θ): 1.0 | ∆ Y: 10.0 | max Y: 50.0

1.0

0.8

0.6 N(Y,r,b,θ) 0.4

0.2

0.0 -2 -1 0 1 2 3 4 5 6 7 8 10 10 10 10 10 10 10 10 10 10 10 Dipole Size (Arbitrary Units)

Figure 3.11. Graph of the scattering amplitude versus dipole size for cos(θ) = 1 for large dipole size b = 100.0. The peak is much more deﬁned than in Fig 3.9 due to the angular orientation of the dipole, showing in certain cases strong dependence on θ. In this plot the initial condition is near zero and it is not visible on the graph.

− 1 n +iν 1+n +iν 2 ∗ 2 n,ν∗ r0 r0 n r r E , ( 1) ∗ ∗ . 2 − 2 − b + r b r b + r b r 2 − 2 ! 2 − 2 ! (3.23) Taking n = 0 (which gives the largest contribution) we are able to combine the two terms in (3.23) into

1 2 +iν 2 ∗ r r r En,ν 0 , 0 | | . (3.24) 2 ∗ 2 2 − 2  b2 r (b∗)2 r  − 2 − 2   Going to polar coordinates b = b eiθb and r = r eiθr as well as deﬁning ∆θ = θ θ | | | | r − b (3.24) becomes 85

Dipole Size: 1.00| Impact parameter: 1.000 | ∆ Y: 10.0 | max Y: 50.0

1.0

0.8

θ) 0.6 N(Y,r,b, 0.4

0.2

0.0 -1.0 -0.5 0.0 0.5 1.0 Cos(θ)

(a)

Dipole Size: 100.00| Impact parameter: 47.863 | ∆ Y: 10.0 | max Y: 50.0

1.0

0.8

θ) 0.6 N(Y,r,b, 0.4

0.2

0.0 -1.0 -0.5 0.0 0.5 1.0 Cos(θ)

(b)

Figure 3.12. Graphs of the scattering amplitude versus cos(θ) for medium dipole size (Fig a) and for large dipole size (Fig b). In Fig b the dipole size is set to r = 2b and it can be seen that there is an enhancement at cos(θ)= 1. The initial condition in these ± plots is saturated to unity across all angles for these values of r and b. 86

Dipole Size: 100.00 | cos(θ): 0.0 | ∆ Y: 10.0 | max Y: 50.0

1.0

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0.6 N(Y,r,b,θ) 0.4

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0.0 0 1 2 3 4 10 10 10 10 10 Impact Parameter (Arbitrary Units) (a) Dipole Size: 100.00 | cos(θ): 1.0 | ∆ Y: 10.0 | max Y: 50.0

1.0

0.8

0.6 N(Y,r,b,θ) 0.4

0.2

0.0 0 1 2 3 4 10 10 10 10 10 Impact Parameter (Arbitrary Units) (b)

Figure 3.13. Graphs of the scattering amplitude versus impact parameter for cos(θ) = 0 (perpendicular conﬁguration, Fig a) and cos(θ)= 1 (parallel conﬁguration, Fig b). The ± initial condition is shown as the dotted and dashed line. The peak that is seen when the dipole size and impact parameter are parallel disappears when the orientation is perpendicular. 87

1 2 +iν 2 ∗ r0 r0 r En,ν ,  | |  . (3.25) 2 − 2 2 2 b 2 e2i∆θ |r| b 2 e−2i∆θ |r|  2 2   | | − | | −    When the denominator is very small there is a large enhancement in the scattering amplitude. This occurs when ∆θ =0,π and b = r/2, which is what is observed in the numerical result as well, see Fig 3.11. If we consider the case for perpendicularly oriented dipoles, where ∆θ = π/2 the expression becomes

1+2iν

2 ∗ r0 r0 r En,ν ,  | |  . (3.26) 2 − 2 2 b 2 + |r|  2   | |    The form of (3.26) is responsible for the fact that even when the dipoles are perpendicular a peak is visible when the impact parameter is fixed, as in Fig 3.9. However, when the dipole size is held fixed and the impact parameter varied no peak is visible for such perpendicular configurations, as in Fig 3.8(a).

3.3.4 Analysis of a modiﬁed kernel

It is possible to investigate the behavior of the BK equation beyond the LL approximation. A modified kernel was derived in [112], which we shall rederive here. This kernel (partially) includes important kinematical effects which have a non-negligible impact on the numerical solution. An alternative method to the derivation of the branching kernel presented in Sec 2.1.3 is to proceed by utilizing the rules of light-cone perturbation theory and evaluate energy denominators [113–116]. These energy denominators come from the intermediate states of the infinite momentum wave function of the dipole cascade. The diagram in Fig 2.7(a) can be described in this framework as

a 2gτ kT ǫ Φ(1)(z , l ,z, k )= · Φ(0)(z , l ) , (3.27) 0 T T D 0 T 88 where we have once again used the eikonal approximation for the coupling of the emitted gluon. The energy denominator D is given by

D = P − [(P l k)− + l− + k−] , (3.28) − − − − 2 + + − 2 + + where k = kT /2k ,k = zP and l = lT /2l ,l = z0P . If the LL approximation is taken then the emitted gluon is assumed to be longitudinally soft. This means k+ is small and k− is large, becoming the dominant term in the energy de- − nominator D. Computing (3.27) with DLL = k one would retrieve the LL kernel as found in Sec 2.1.3. However, if l− k− both terms should be kept in the denominator ≈

− − Dmod = l + k . (3.29)

Using this modiﬁed energy denominator in (3.27) we arrive at

a 2gτ ǫT kT · , (3.30) 2 2 kT + Q¯

2 2 z where we introduce the notation Q¯ = lT . z0

(1) a ǫT kT (0) (0) Ψ (z , lT,z, kT)=2gτ · Ψ (z , lT) Ψ (z , lT + kT) 0 2 2 0 0 kT + Q¯ − Fourier transforming this as before

(1) Φ (z0, x01,z, x02)= 2l 2k k a d T d T ikT·x01+ikT·x02 ǫT T (0) (0) 2gτ e · Ψ (z0, lT) Ψ (z0, lT + kT) (2π)2 (2π)2 k 2 + Q¯2 − Z T (3.31)

Taking only the ﬁrst term and continuing the calculation

2l 2k k (1) a d T d T ilT·x01+ikT·x02 ǫT T (0) Φ (z0, x01,z, x02)=2gτ e · Ψ (z0, lT) (2π)2 (2π)2 k 2 + Q¯2 Z T 89

2l 2 a d T d kT ilT·x01 ǫT cos φ kT (0) = 2igτ | |e | | | | J ( k x )Ψ (z , lT) 2 2 2 1 T 02 0 (2π) (2π) kT + Q¯ | | Z 2 d l 1 ǫT x02 = 2igτ a T eilT ·x01 · QK¯ (Qx¯ )Ψ(0)(z ,l ) (2π)2 (2π) x 1 02 0 T Z 02 a 1 ǫT x02 (0) = 2igτ · QK¯ 1(Qx¯ 02)Φ (z0, x01) (3.32) (2π) x02

Here K1 is the modiﬁed Bessel function of the second kind. The second term is solved in a very similar manner yielding a wavefunction for single gluon emission of the form

(1) Φ (z0, x01,z, x02)= a 2igτ ǫT x02 ǫT x12 (0) · QK¯ (Qx¯ )+ · QK¯ (Qx¯ ) Φ (z , x01). (3.33) 2π x 1 02 x 1 12 0 02 12 Comparing to (2.19) we can square the wavefunction and extract the corrected splitting kernel

K(x01,x02,x12; Y = ln(1/z)) = x02 x12 Q¯2 K2(Qx¯ )+ K2(Qx¯ ) 2K (Qx¯ )K (Qx¯ ) · . (3.34) 1 02 1 12 − 1 02 1 12 x x 02 12 This modification introduces into the branching kernel a rapidity dependence. First let us investigate the effect of the modified kernel on the evolution without impact parameter dependence. The scattering amplitude obtained from the BK equation (2.66) with the modified kernel (3.34) can be seen in Fig 3.14(a) as a function of dipole size. The saturation scale obtained in this calculation is shown in Fig 3.14(b). Very little difference can be seen between the solution using the modified kernel and the LL kernel. This effect can be seen through the functional form of (3.34). For small values of the argument of the Bessel functions in (3.34) the modified kernel will reduce down to the LL kernel. The two kernels differ in the region where

1 x ,x . (3.35) 12 02 ≫ Q¯ 90

∆Y: 10.0 | max Y: 80.0

1.0

0.8

0.6 N(Y,r)

0.4

0.2

0.0 -6 -5 -4 -3 -2 -1 0 1 2 10 10 10 10 10 10 10 10 10 Dipole Size (Arbitrary Units) (a)

20 ) 2 s ln(Q 10

0 20 40 60 80 Rapidity

(b)

Figure 3.14. Graphs of the scattering amplitude calculated without impact parameter dependence with the modified kernel (3.34), which are the dashed lines, versus the LL result which are the solid lines. The initial condition for both runs is identical and given by (3.1). Fig a : The dipole size dependence of the modified kernel versus the LL kernel, very little difference is shown. Fig b : A comparison on the saturation scales of the modified kernel and the LL kernel, again very little difference is seen. 91

In this large dipole regime the system is saturated and the scattering amplitude is at unity. The right hand side of the BK equation (2.66) goes to zero, giving no contribution from this region. Additionally when the system is evolved to very large rapidities we again recover the behavior of the LL kernel (as Q¯ becomes very small). With these kinematical constraints the modified kernel provides almost no difference in dynamics versus the LL kernel, yielding almost identical λs values. Without impact parameter dependence the inclusion of the non-linear term in the BK equation kills the infra-red contribution of the modified kernel. The dynamics of the modified kernel (3.34) are markedly different when impact parameter is taken into account. In this case the contribution from the region of large dipole sizes is non-negligible and one can expect a larger sensitivity to the modification of the kernel. This can be seen in Fig 3.15 where the modified kernel is compared to the evolution with the LL kernel when impact parameter dependence is included. The evolution of the modified kernel is markedly slower than that of the LL kernel in the small dipole regime. This is due to the contributions from the large-dipole regime (3.35) which is no longer hidden by saturation. The large dipole regime evolves nearly identically between the LL and modified kernels because in this region x01 is large which forces Q¯ to be small, reducing the modified kernel to the LL kernel.

3.3.5 The saturation scale with impact parameter dependence

As in Sec 3.2.1 we deﬁne the impact parameter dependent saturation scale to satisfy the following equation

N(r =1/Q (b,Y ),b,θ,Y ) = κ (3.36) h s i where once again we have chosen κ =0.5. The brackets denote averaging over the dipole orientation angle θ. Clearly the saturation scale obtained from (3.36) will depend on impact parameter. One expects that as a more central collision (small b) probes a denser region the saturation momentum will be larger than for a more peripheral collision (large b). Additionally if we plot the boundary between the dilute and the 92

Impact parameter: 1.00 | cos(θ): 0.0 | ∆ Y: 10.0 | max Y: 50.0

1.0

0.8

0.6 N(Y,r,b,θ) 0.4

0.2

0.0 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 10 10 10 10 10 10 10 10 10 10 10 10 10 Dipole Size (Arbitrary Units) (a) Dipole Size: 1.00 | cos(θ): 0.0 | ∆ Y: 10.0 | max Y: 50.0

1.0

0.8

0.6 N(Y,r,b,θ) 0.4

0.2

0.0 -1 0 1 2 3 10 10 10 10 10 Impact Parameter (Arbitrary Units) (b)

Figure 3.15. Plots of the scattering amplitude computed with the LL kernel (solid line) versus the modified kernel (3.34) (dashed line) in the case with impact parameter dependence. The initial condition is given by the dotted and dashed line for both kernels. Fig a : comparison of the scattering amplitude versus the dipole size showing that the modified kernel evolves much slower in the small dipole region than the LL kernel but comparably fast in the large dipole regime. Fig b : comparison of the scattering amplitude versus impact parameter showing a slowdown in the evolution of the modified kernel in impact parameter. 93

Impact parameter: 1.00 | cos(θ): 0.0 | ∆ Y: 10.0 | max Y: 50.0

1.0

0.8

0.6 N(Y,r,b,θ) 0.4

0.2

0.0 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 10 10 10 10 10 10 10 10 10 10 10 10 10 Dipole Size (Arbitrary Units)

Figure 3.16. A graph of the scattering amplitude with a horizontal line at N = 0.5. The intersections of this horizontal line and the scattering amplitude indicate the solutions of the saturation condition (3.36). It can be seen that with impact parameter dependence there are now two solutions for any given rapidity, one in the ultraviolet and one in the infra-red region. saturated regimes with impact parameter dependence it is clear that the condition (3.36) now admits two solutions. One solution is for small dipole sizes. The saturation scale extracted in this regime will increase with rapidity, similar to what was found in the case without impact parameter dependence. The second solution is for large dipole sizes. As is clear from Fig 3.16 this second ’saturation scale’ will decrease with increasing rapidity. This second evolution front is only present in the case with impact parameter dependence. After a long evolution very large dipoles, which are non-perturbative, start to play an important role in the dynamics. These large dipoles need to be regulated by the inclusion of conﬁnement eﬀects, which will be discussed in Sec 4.1. For now let us investigate the properties of both saturation scales individually. Our notation shall be to consider Qs to be the solution to (3.36) for the small dipole regime and QsL shall be the solution for 94

Impact parameter: 1.000

15 ) 2 s

ln(Q 10

0 10 20 30 40 50 Rapidity

Figure 3.17. Logarithmic plot of the saturation scale Qs versus rapidity for the modiﬁed kernel (dashed line) compared to the LL kernel (solid line) showing the slowdown of the evolution with the modiﬁed kernel compared the LL kernel. the large dipole regime.

3.3.5.1 Saturation scale at small dipole size

The saturation scale for small dipole sizes behaves very similarly to the case without impact parameter dependence. The parameterization of the saturation scale is similar to (3.19),

2 2 α¯sλsY Qs(Y,b)= Q0e S(b), (3.37) where S(b) is the proﬁle of the saturation scale in impact parameter space. Again

Q0 is a normalization constant and λs is a constant governing the rapidity dependence of the saturation scale.

The constant λs can be extracted from the numerical solutions in the region of large rapidities. In this region the value of λs was found to be approximately constant. For small rapidities the preasymptotic contributions aﬀect the saturation scale, causing λs to grow. At extremely large rapidities the saturation scale is again 95

Rapidity: 25.000

5 ) 2 s ln(Q

-1 0 1 10 10 10 Impact Parameter (Arbitrary Units)

Figure 3.18. Logarithmic plot of the saturation scale Qs versus impact parameter for the modified kernel (dashed line) compared to the LL kernel (solid line) for two different rapidities, showing the presence of power tails in impact parameter space. The range of impact parameter plotted is only for dipole sizes which are saturated (as at large impact parameter the scattering amplitude is small and not saturated). affected by the maximum size of the evaluation region in the numerical analysis, causing λs to decrease. The graph of the saturation scale as a function of rapidity for both the LL and modified kernel can be seen in Fig 3.17.

The value of λs for the LL kernel was found to be λs =4.4. This is consistent with what was found in Sec 3.2.1 for the rapidity dependence of the saturation scale without impact parameter. For fixed impact parameter the behavior in this region of the LL kernel is unchanged, which is not surprising. Extracting the λs for the modified kernel we find λs = 3.6, which corresponds to a 20% decrease in the evolution speed of the modified kernel versus the LL kernel. Clearly the subleading kinematical effects contained in the modified kernel are important and strongly modify the contributions from large dipole sizes (as discussed in Sec 3.3.4). The impact parameter dependence of the saturation scale can be seen in Fig 96

3.18 for both the modiﬁed and LO kernel. For ﬁxed rapidity the saturation scale has a clear change in behavior as one goes to large impact parameter, shifting from steeply falling to a power-like behavior. This is expected from conformal invariance and can be seen analytically by considering the conformal representation of the scattering amplitude. Taking the form of the dipole number density from [66]

2 ′ 1+i∞ b2 α rr dλ λ 1 α¯ χ(λ)Y +2(1−λ) ln ′ n(r, r′,Y, b)= s − e s rr , (3.38) b2 2π [ν2 +1/4]2 Z1−i∞ where we are only concerned with the exponential in the integrand. We have changed variables from [66] (here λ =1+iν) and taken only the contribution from zero conformal spin. Now we follow the method of [103] to uncover the dynamics on the saturation boundary, which will lead to an expression for the saturation scale. On the saturation boundary it is expected that the dipole number density obeys the saddle point condition and that it also vanishes at the saddle point. This gives two conditions on the system

b2 0 =α ¯ Yχ′(λ )+2ln , and (3.39) s 0 rr′ b2 0 =α ¯ χ(λ )Y + 2(1 λ )ln . (3.40) s 0 − 0 rr′

The λ which satisﬁes this boundary line is denoted by λ0. By combining equations (3.39) and (3.40) we arrive at

χ′(λ ) 1 0 = , (3.41) χ(λ ) 1 λ 0 − 0 which yields a value of λ 0.37. The line deﬁned by (3.39) and (3.40) leads to 0 ≈ a nearly constant amplitude, but not exactly. The inclusion of a correction into the boundary conditions gives us an expression for the saturation boundary, which we will denote by λc (where the subscript stands for critical). The new conditions become 97

b2 0 =α ¯ Yχ′(λ )+2ln , (3.42) s c rr′ 3 b2 ln(¯α Yχ′′(λ )) =α ¯ χ(λ )Y + 2(1 λ )ln . (3.43) 2 s c s c − c rr′

3 ′′ The additional factor of 2 ln(¯αsYχ (λc)) comes from the presence of saturation eﬀects. The necessity of this term was argued in [103] by considering the BFKL equation as a diﬀusion equation in the presence of an absorptive boundary. Solving (3.43) for r in order to determine the behavior of the saturation boundary we arrive at

α¯sYχ(λc) 2 1−λc 1 2 r0 e 2 = Qs(r0,b,Y )= 4 3 . (3.44) r b ′′ 2(1−λ ) (¯αsYχ (λc)) c It is useful to notice that the dependence on rapidity is the same as the case without impact parameter (3.20). The saturation scale depends on impact parameter by a power 1/b4. This factor is expected due to the conformal symmetry of the evolution. When rapidity is large

λc approaches λ0.

Varying the couplingα ¯s is expected to leave the exponent λs constant in the parametrization (3.37). We found that when changingα ¯s to 0.2 the value of λs is constant for the LL kernel. This is expected in the LL approximation as the exponent should be linear inα ¯s. For the modified kernel we found λs =2.5 forα ¯s. The variation of the exponent with the coupling show that for the modified kernel the evolution is non-linear inα ¯s, which is as expected since the modified kernel contributions are proportional to higher powers of αs. These exponents were also tested for robustness against varying initial conditions and it was found that the extracted values of λs are consistent for a variety of initial conditions. The compilation of the extracted λs values is shown in Table 3.1. 98

Kernel λs LL Kernelα ¯s =0.1 4.4 LL Kernelα ¯s =0.2 4.4 Modiﬁed Kernelα ¯s =0.1 3.6 Modiﬁed Kernelα ¯s =0.2 2.5

Table 3.1. Values of λs for various values ofα ¯s and kernels.

Impact parameter: 1.000 0

-5

-10 ) 2 sL

ln(Q -15

-20

-25 0 10 20 30 40 50 Rapidity

Figure 3.19. Logarithmic plot of the saturation scale QsL versus rapidity for the modiﬁed kernel (dashed line) compared to the LL kernel (solid line).

3.3.5.2 Saturation scale at large dipole size

The saturation scale at large dipole sizes is parameterized similarly to the saturation scale at small dipole sizes

2 2 −α¯sλsLY QsL = Q0Le SL(b). (3.45)

Note that there is now a minus sign in the exponent as compared to (3.36) as (3.45) evolves towards large dipole sizes. Hence this saturation scale will decrease with rapidity. 99

The graph of the saturation scale for large dipole size can be seen in Fig 3.19 for both the LL and modified kernel. At low (Y < 10) rapidities the preasymtotpic behavior is quite strong. This is because the difference between the initial condition and the solution is quite significant in this region and so it takes many steps in rapidity for the evolution equation to lose the dependence on the initial condition. Initially the saturation scale is not defined (as the initial condition provides no solution in the large dipole region to (3.36)).

At higher rapidities it is found that λsL is constant at high rapidities, similar to λs. At extremely high rapidities the limitations of the region being numerically calculated again impose themselves spoiling the constant nature of λsL. Extracting

λsL from Fig 3.19 it is found that λsL =6.0 for the LL kernel and λsL =5.8 when

α¯s = 0.1. The evolution in the infra-red regime is faster than in the ultraviolet regime by a significant margin for both the modified and the LL kernel. This was an unexpected result as conformal invariance would lead one to conclude that the evolution in both directions should be of equal speed. The mechanism behind this difference in evolution speeds is, at this point, not well understood. The slowdown of the growth of the solution to the BK equation (2.66) with the modified kernel (3.34) is much less significant in the infra-red region, being only 3% compared to the 20% slowdown seen in the small dipole size region. ∼ ∼ The reason for this becomes clear from the functional form of the modified kernel (3.34). In the large dipole region x is large which makes Q¯ 1. This forces 01 ≪ the arguments of the Bessel functions to be small, which reduces the kernel to the LL case, unless the daughter dipoles are large. This restricts the kinematic regime from which these contributions come and is the reason that the difference between the modified and LL kernel is so modest, especially in comparison with the differences observed in the evolution towards small dipole sizes. It is important to note that there are other corrections in addition to the kinematic effects which led to the derivation of (3.34). It is expected that these kinematic effects on the kernel preserve conformal invariance and that including further effects would restore the invariance, which is broken in the modified kernel. This would give an equal effect to the large and the small dipole sizes and further slow the speed of the evolution. In the LL approximation it is possible to derive an analytic expression for the 100

Kernel λsL LO Kernelα ¯s =0.1 6.0 LO Kernelα ¯s =0.2 5.9 Modiﬁed Kernelα ¯s =0.1 5.8 Modiﬁed Kernelα ¯s =0.2 5.2

Table 3.2. Values of λsL for various values ofα ¯s and kernels. saturation scale at large dipole sizes, similar to the one for small dipole sizes in (3.44). For large dipole sizes the solution to (3.43) becomes

− α¯sYχ(λc) 1−λc 2 1 e QsL(r0,b,Y )= 2 3 . (3.46) r ′′ 2(1−λ ) 0 (¯αsYχ (λc)) c The impact parameter dependence has dropped out in this form of the solution.

This can be seen in Fig 3.20 where the saturation scale QsL is plotted versus impact parameter. The evolution speed is expected to be the same comparing with (3.44), however our solution does not obey this conformal symmetry. Again the origin of this asymmetry is currently not understood.

The value of λsL was extracted from the solution forα ¯s = 0.2 and was found to be 5.9 for the LL kernel. This is very close to the value found in the case with

α¯s = 0.1. For the modiﬁed kernel the value of λsL = 5.2 forα ¯s = 0.2. The value of λsL depends onα ¯s which is due to the higher order eﬀects.

The values of the exponent λsL were found to be robust for diﬀerent initial conditions in much the same way as the values of λs in Sec 3.3.5.1. A list of the extracted values for λsL can be found in Table 3.2.

3.3.6 Inclusion of the running coupling

The running coupling in the BFKL [56–59] and BK [82,86,87] equations appear at next-to-leading order. It is an important eﬀect which needs to be taken into account in order to compare these calculations to the experimental data. The kernel with running coupling has been derived in two independent calculations. Kovchegov and Weigert [117] performed the calculation, obtaining the splitting kernel with running coupling which has the form 101

Rapidity: 25.000 -10

-11 ) 2 sL -12 ln(Q

-13

-14 -4 -3 -2 -1 0 1 10 10 10 10 10 10 Impact Parameter (Arbitrary Units)

Figure 3.20. Logarithmic plot of the saturation scale QsL versus impact parameter for the modiﬁed kernel (dashed line) compared to the LL kernel (solid line). It can be seen that there is no dependence in impact parameter. The range of impact parameter plotted is only for impact parameters which are saturated (as at large impact parameter the scattering amplitude is small and not saturated).

2 2 2 2 KW Nc αs(x02) αs(x12) αs(x02)αs(x12) x02 x12 K (x01,x02,x12)= 2 2 + 2 2 2 2 · 2 , 2π x02 x12 − αsR x02x12 (3.47) where

2 2 2 2 x02+x12 x02x12 1 −2 · x x2 −x2 x02 x12 x2 −x2 R2 = x x 12 02 12 02 12 . (3.48) 02 12 x 02 The expression for the running coupling that is used is

2 Nc α¯s(r )= 1 (3.49) πβ0 ln 2 2 r ΛQCD 102

where Nc is the number of colors and β0 is taken to be

33 2n β = − f . (3.50) 0 12π with nf as the number of active quark ﬂavors. There was a separate derivation by Balitsky [118] which resulted in a splitting kernel of the form

Bal K (x01,x02,x12) α¯ (x2 ) x2 1 α (x2 ) 1 α (x2 ) = s 01 01 + s 02 1 + s 12 1 . (3.51) 2π x2 x2 x2 α (x2 ) − x2 α (x2 ) − 02 12 02 s 12 12 s 02 Initially it seems that the the kernels (3.51) and (3.47) are incompatible. This is not the case and it has been shown that these two kernels are compatible at the order that they were calculated, diﬀering only by a higher order term known as the subtraction term [119]. The subtraction term will not be considered in the numerical analysis in this work. The KW kernel (3.47) is more numerically taxing than (3.51). For this reason we have chosen to conduct our analysis with the Balitsky kernel (3.51) only. The solution using the running coupling prescription (3.51) can be seen in Fig 3.21, where it is compared to the solution using the LL kernel with ﬁxed coupling

α¯s = 0.1. The running coupling solution can be seen to transition from unity to zero over a smaller range of dipole sizes than the LL kernel in the small dipole regime. The shape is the same in the large dipole regime because in this region the running coupling becomes ﬁxed and the Balitsky kernel reduces to the LL kernel. Setting the coupling to be ﬁxed at large dipole sizes regulates the large dipoles in our numerical analysis, so we introduce a freezing of the coupling such that

2 α¯s(rcut)=¯αcut =0.3. (3.52)

The coupling is set to beα ¯cut when r>rcut. The value at which the running coupling is frozen determines the evolution speed in this region. The choice of

α¯cut = 0.3 sets the evolution speed in this region to be nearly three times that of the LL kernel (withα ¯s = 0.1 in Fig 3.21). The impact parameter dependence 103

Impact parameter: 1.00 | cos(θ): 0.0 | ∆ Y: 5.0 | max Y: 20.0

1.0

0.8

0.6 N(Y,r,b,θ) 0.4

0.2

0.0 -4 -3 -2 -1 0 1 2 3 4 5 6 10 10 10 10 10 10 10 10 10 10 10 Dipole Size (Arbitrary Units) (a) Dipole Size: 100.00| cos(θ): 0.0 | ∆ Y: 5.0 | max Y: 20.0 0

-2

-4 ln(N(Y,r,b,θ)) -6

-8

0 1 2 3 4 10 10 10 10 10 Impact Parameter (Arbitrary Units) (b)

Figure 3.21. Solution of the BK equation with the Balitsky prescription for running coupling (solid line) and the LL kernel (dashed line) with ﬁxed couplingα ¯s = 0.1. Fig a shows the scattering amplitude versus dipole size. Fig b is a plot of the scattering amplitude versus the impact parameter. The initial condition for both kernels is given by the dashed and dotted line. 104 can also be seen in Fig 3.21 and again the behavior in the large dipole regime is the same as the LL kernel, only with a faster evolution, due to the coupling being large and frozen in this region. The saturation scale from the calculation using the Balitsky kernel can be evaluated and the plots of both the large and small dipole saturation scales are presented in Fig 3.22. The parameterization here has been changed from (3.37) and (3.45) to

Q2 eλsY , and (3.53) s ≈ Q2 e−λsLY . (3.54) sL ≈

This parameterization is chosen due to the dependences of the saturation scale being almost exponential, as can be seen in Fig 3.22. This is because of the strong dependence of the saturation scale on the infra-red regime, leading to exponential behavior. For the Balitsky running coupling prescription these exponents are found to be λs =0.29 and λsL =1.68. The BK equation with running coupling is solved to a lower value of rapidity than the ﬁxed coupling case. This is due to the fact that in the running coupling case the evolution towards the infra-red regime is very fast and the solution quickly encounters the limits of the numerical grid.

3.3.6.1 Comparison with the minimum dipole prescription

We have also evaluated the solution with the minimum dipole prescription. Al- though this prescription is rather ad-hoc it does provide a valuable check of our calculation. By taking the LL kernel with ﬁxed coupling and replacing the ﬁxed 2 2 2 coupling withα ¯s(min(x01,x02,x12)) we arrive at

2 min 2 2 2 x01 K =α ¯s(min(x01,x02,x12)) 2 2 . (3.55) x02x12 The motivation for this choice is that the Balitsky kernel reduces to this sim- pliﬁed minimum dipole prescription in the limit of strongly ordered dipole sizes. Consider the following cases: 105

Impact parameter: 1.000 8

6 ) 2 s

ln(Q 5

0 5 10 15 20 Rapidity

(a)

Impact parameter: 1.000 0

-10 ) 2 sL ln(Q -20

-30

0 5 10 15 20 Rapidity

(b)

Figure 3.22. Logarithmic plot of the saturation scales Qs (Fig a) and QsL (Fig b) versus rapidity for the Balitsky kernel (solid) and the LL with ﬁxed coupling (dashed) ofα ¯s = 0.1. 106

Case 1 : x x x In this case (3.51) reduces to • 01 ≈ 02 ≫ 12 α¯ (x2 ) α (x2 ) α¯ (x2 ) s 01 s 01 1 + s 12 . (3.56) x2 α (x2 ) − x2 02 s 12 12

With x12 being small, the dominant contribution comes from the second term

2 α¯s(x12) 2 . (3.57) x12

This is the minimum dipole prescription (3.55) when x12 is the smallest dipole size in the problem.

Case 2 : x x x In this case (3.51) reduces to • 01 ≈ 12 ≫ 02 α¯ (x2 ) α (x2 ) α¯ (x2 ) s 01 s 01 1 + s 02 . (3.58) x2 α (x2 ) − x2 01 s 02 02

With x12 being small the dominant contribution comes from the second term

2 α¯s(x02) 2 . (3.59) x02

That is the minimum dipole prescription (3.55) when x02 is the smallest dipole size in the problem.

Case 3 : x x x In this case (3.51) reduces to • 02 ≈ 12 ≫ 01 2 2 α¯s(x01)x01 2 2 . (3.60) x12x02

This is the minimum dipole prescription (3.55) when x01 is the smallest dipole size in the problem (3.55).

If the dipole sizes are large then the coupling is ﬁxed and both (3.55) and (3.51) reduce to the LL kernel. In general this would lead us to believe that the evolution with the minimum dipole prescription is similar to that of the Balitsky kernel. As can be seen in Fig 3.23 the evolution of the Balitsky kernel and the minimum dipole prescription are indeed very similar. 107

Impact parameter: 1.00 | cos(θ): 0.0 | ∆ Y: 5.0 | max Y: 20.0

1.0

0.8

0.6 N(Y,r,b,θ) 0.4

0.2

0.0 -4 -3 -2 -1 0 1 2 3 4 5 6 10 10 10 10 10 10 10 10 10 10 10 Dipole Size (Arbitrary Units)

Figure 3.23. A comparison between the scattering amplitude using the minimum dipole prescription (solid) and the Balitsky prescription (dashed) for running coupling versus dipole size.

3.3.6.2 Applying the running coupling to the modiﬁed kernel

The previous treatments of the running coupling in the BK equation have been considered only for the LL kernel. There have been no derivations which include the running coupling and kinematic corrections on the kernel. However, one can include running coupling corrections in the modiﬁed kernel (3.34) by using the minimum dipole prescription. By following the procedure of replacing the constant

αs term in the kernel with a running coupling which is determined by the smallest 2 2 2 dipole in the problem, namely αs(min(x01,x02,x12)), one can arrive at a modiﬁed kernel with running coupling corrections,

2 2 2 K(x01,x02,x12; Y )=¯αs(min(x01,x02,x12)) 108

Kernel λs λsL LL Kernel with minimum dipole prescription 0.28 1.7 Modiﬁed Kernel with minimum dipole prescription 0.1 0.46 Balitsky Kernel 0.26 1.8

Table 3.3. Values of λs and λsL for running coupling kernels.

x02 x12 Q¯2 K2(Qx¯ )+ K2(Qx¯ ) 2K (Qx¯ )K (Qx¯ ) · . (3.61) × 1 02 1 12 − 1 02 1 12 x x 02 12 This procedure is very similar to that which was used to arrive at (3.55). A comparison between the minimum dipole prescription at LL and the minimum dipole prescription with kinematic corrections can be found in Fig 3.24. It is clear that the modiﬁed kernel again exhibits a slower evolution than the LL kernel. The speed of the evolution can be seen in the comparison of the saturation scale shown in Fig 3.25. In both saturation scales it can be seen that the evolution speed is dramatically reduced. The exponents λs and λsL for the modiﬁed kernel with running coupling can be seen in Table 3.3.

3.3.6.3 Regularization sensitivity of the Balitsky kernel

It was found, and warrants a small discussion, that the presence of terms proportional to the inverse coupling constant in (3.51), leading to unexpected behavior. Let us consider an alternative method to regularize the running coupling at large dipole sizes. Instead of freezing the coupling at an arbitrary ﬁxed value when the dipole size becomes large let us set a mass scale µ. By including this mass scale in the running coupling itself we are able to regulate the coupling at large dipole sizes, giving the coupling the following form

1 αs(r)= . (3.62) 1 1 2 β0 ln 2 2 + µ ΛQCD r This caps the coupling at a cutoﬀ value for arbitrarily large dipoles of

1 αcut = . (3.63) µ2 β0 ln 2 ΛQCD 109

Impact parameter: 1.00 | cos(θ): 0.0 | ∆ Y: 5.0 | max Y: 15.0

1.0

0.8

0.6 N(Y,r,b,θ) 0.4

0.2

0.0 -3 -2 -1 0 1 2 3 4 5 6 10 10 10 10 10 10 10 10 10 10 Dipole Size (Arbitrary Units) (a) Dipole Size: 100.00| cos(θ): 0.0 | ∆ Y: 5.0 | max Y: 15.0 0

-2

-4 ln(N(Y,r,b,θ)) -6

-8

-1 0 1 2 3 4 10 10 10 10 10 10 Impact Parameter (Arbitrary Units) (b)

Figure 3.24. Comparison between the modiﬁed kernel with the minimum dipole prescription (3.61)(dashed) and the LL with the minimum dipole size prescription (3.55)(solid). Fig a is of the scattering amplitude versus the dipole size. Fig b is a logarithmic plot of the scattering amplitude versus the impact parameter. The initial condition is given by the dotted and dashed line. 110

Impact parameter: 1.000

6 ) 2 s 5 ln(Q

0 5 10 15 Rapidity

(a)

Impact parameter: 1.000 0

-5

) -10 2 sL ln(Q -15

-20

0 5 10 15 Rapidity

(b)

Figure 3.25. Comparison between the modiﬁed kernel with the minimum dipole prescription (3.61)(dashed) and the LL with the minimum dipole size prescription (3.55)(solid). Fig a compares the saturation scale Qs versus rapidity and Fig b is a plot of QsL versus rapidity. 111

0.30

0.25 s α

0.20

0.15 -2 -1 0 1 2 10 10 10 10 10 r (Arbitrary Units)

Figure 3.26. A plot of the frozen coupling where αs freezes at αs = 0.3 (solid) and the regulation method with the inclusion of the µ parameter (dashed). Here we use µ = 2.52GeV2 such that both coupling prescriptions freeze out at the same value for comparison.

This prescription for the cutoff of the running coupling is the one that will be used in chapter 4 when comparing these solutions to HERA data. A comparison between the two running coupling prescriptions can be seen in Fig 3.26 where the curve which flattens is the plot of the running coupling which freezes while the other is of the coupling with µ. The inclusion of µ in the running coupling provides a smooth transition from the perturbative region to the frozen region. At small dipole sizes the couplings are identical and it is only at large dipole sizes that the difference is noticeable. It was found that the Balitsky kernel was very sensitive to this change between the prescriptions of how the running coupling is frozen. This sensitivity was not only in the region near the freezing of the coupling, but at small dipole sizes as well. Even when separated from the transition region by orders of magnitude the solution of the scattering amplitude with the Balitsky kernel had large differences (factors of two or more) between the prescriptions in this region. It was found that this effect is not unique to the case with impact parameter dependence but it is 112

∆Y: 10.0 | max Y: 10.0

1.0

0.8

μ2=2.0 0.6 μ2=1.0 μ2=0.5 N(Y,r)

0.4

0.2 Initial condition

0.0 -1 0 10 10 Dipole Size (Arbitrary Units)

Figure 3.27. A plot of the solution to the BK equation using the Balitsky prescription for running coupling solved without impact parameter for various values of µ. It can be seen that as µ increases (and the coupling decreases) the scattering amplitude increases for a range of µ. At large µ the scattering amplitude does decrease as expected (not shown). While the effect is small it does illustrate the sensitivity of (3.51) with respect to the regulation of the coupling. The minimum dipole prescription does not show this behavior. present without impact parameter dependence as well. Let us investigate how this large dipole regime may affect the entire evolution of the scattering amplitude when kernel (3.51) is used. To do this we varied µ for various evolutions without impact parameter dependence. This can be seen in Fig 3.27 where the scattering amplitude at fixed rapidity for different µ is graphed, and it is clear that there is some unexpected behavior. For a time as µ is increased the scattering amplitude is enhanced, indicating it has received a larger contribution from the evolution, even though a larger µ means a smaller coupling. Eventually at extremely large µ values the scattering amplitude does decrease, but only for very large values of µ. If the same analysis is done with the minimum dipole prescription one finds 113 the evolution behaves as expected, decreasing the scattering amplitude as µ is increased. Additionally the minimum dipole prescription is not sensitive to the prescription of which the running coupling is frozen. The reason that the Balit- sky prescription behaves in such a markedly different way is the presence of terms proportional to the inverse of the coupling. These terms probe different scales and provide enhancements when the coupling decreases, manifesting as these unexpected behaviors of the evolution when using the (3.51) as the splitting kernel for the evolution. While this would point to the Balitsky kernel being unstable we shall see that this kernel is in fact key to fitting the data in chapter 4. This is because the BK equation is known to have an evolution which is too rapid to explain the data and, as will be seen in Sec 4.1.2 when a cutoff is introduced on the large dipoles the Balitsky kernel provides a much slower evolution. More work is needed to understand exactly the nature of these instabilities and behavior of the BK equation with running coupling.

3.3.7 Growth of the cross section

The dipole cross section can be obtained by integrating the dipole scattering amplitude over the impact parameter

σ(r, Y )=2 d2bN(r, b,Y ). (3.64) Z The dipole cross section therefore is a function of the dipole size and the rapidity only. It was shown that the solution to the BK equation at leading order does not obey the Froissart bound and, in fact, generates power-like tails in impact parameter space [120]. These power tails were seen in our analysis earlier in Sec 3.3.2. Let us investigate this result and the implications by considering the target of the scattering as consisting of two regions. The region most central to the target is the ’black disk’ region, where the parton density is large and the scattering is strong (N 1). Outside of this region is the ’grey disk’, where the density is lower and ≈ 0

N01 0 x0 ≈

N02 1 ≈ N 1 12 ≈

′W hite′ region N 0 ≈ x2 ′Grey Disk′ region 0

′Black Disk′ region N 1 ≈ Figure 3.28. Cartoon of the various interaction regions between the target and the dipole probe. Shown is a dipole x01 at large impact parameter which (almost) does not interact forming two dipoles x12 and x02 which do interact strongly with the target, leading to power-like tails.

Consider a parent dipole splitting into two daughter dipoles at large impact parameter, the dipoles will only scatter strongly if one of the color charges impinges on the central dense region of the target. This situation is seen in Fig 3.28 and leads to the relation

x x b, (3.65) 02 ≈ 12 ≈ where b = b01. This changes the form of the LL splitting kernel to

2 2 x01 x01 2 2 4 . (3.66) x02x12 ≈ b With the color charges of the daughter dipoles impacting the black disk of the target the scattering amplitude is approximately unity for N02 and N12 with

N01 being close to zero. This transforms the BK equation (2.66) at large impact parameters into

2 2 ∂ r 2 r 2 N(r, b,Y ) d x2 = πR (Y ) . (3.67) ∂Y ≈ b4 b4 0 ZR The integration was done over the interaction region, giving the area of the black disc where R0(Y ) is the radius of the black disc region, which depends on rapidity. By treating the dipoles which are interacting in Fig 3.28 as colored probes (as 115 a single color charge is impacting the target) the dependence of the black disc radius on rapidity is exponential [121–123]. Using R (Y ) R (Y )eY in (3.67) it 0 ∝ 0 0 is possible to compute the dipole cross section :

σ = d2bN(r, b,Y ) x−λ , (3.68) ≈ Z where λ is some positive exponent. This is a power law growth in x which, as x 1 , violates the Froissart bound. ∼ s The rapid growth of the scattering amplitude at large rapidities is expected to be dampened by conﬁnement eﬀects, which will be discussed more in Chapter 4. Let us consider the contribution coming from the black disc region of the interaction (as in Fig 3.28) where the scattering is strong. The equation for the black disc cross section is given by

σ(r, Y ) =2 d2bN(r, b,Y )Θ(N(r, b,Y ) κ). (3.69) BD − Z The value κ is a parameter which describes the transition to ’black’ or ’strong scattering’ and is taken to be 0.5. The black disc cross section can thus be approximated as

σ(r, Y ) 2πR2 (r, Y ) , (3.70) BD ≈ BD where RBD(r, Y ) is the radius of the black disc region. At large dipole sizes the black disc cross section depends exponentially on rapidity. Parameterizing the radius of the black disc as an exponential in rapidity yields the expression

2 2 λBDα¯sY RBD(r, Y )= R0e . (3.71)

We may extract the slope λBD from the graphs of σBD, where R0 is a normalization constant. These extracted values of λBD are found in Table 3.4. The exponential growth in rapidity corresponds to a power-like growth in the energy (as Y ln(s)), ∝ which violates the Froissart bound.

Again in the running coupling case σBD behaves as an exponential. Therefore, the black disc cross section with running coupling can be parameterized by 116

Kernel λBD LL Kernel,α ¯s =0.1 2.4 LL Kernel,α ¯s =0.2 2.0 Modified Kernel,α ¯s =0.1 2.6 Modified Kernel,α ¯s =0.2 1.6 LL Kernel with minimum dipole prescription 0.76 Modified Kernel with minimum dipole prescription 0.1 Balitsky Kernel 0.76

Table 3.4. Values of λBD for various values ofα ¯s and kernels.

2 2 λBDY σBD =2πRBD(r, Y )=2πR0e . (3.72)

The values of λBD for the black disc cross section for the running coupling cases can also be found in Table 3.4. Again, as the cross section behaves as an exponential in rapidity it has a power-like dependence in energy, violating the Froissart bound. Further corrections are required to bring the calculation into agreement with this bound on the growth of the cross section, such as confinement effects. It is possible to investigate the growth of the black disc region through the same formalism that was used to describe saturation scales. A ’saturation scale’ can be defined for the impact parameter in a similar way as for the dipole size.

Let this ’saturation impact parameter’ be Bs and be deﬁned by the equation

N(r, b = B ,θ,Y ) = κ. (3.73) h s i

Again κ is arbitrary which is chosen to a value κ = 0.5. The plots for Bs can be found in Fig 3.29 which includes the following cases : fixed coupling LL kernel, fixed coupling modified kernel, LL kernel with minimum dipole prescription for running coupling, and the Balitsky prescription for the running coupling. Again, the large rapidity behavior of the saturation impact parameter Bs appears to be linear on a logarithmic plot, which leads to the parameterization

2 2 λBsα¯sY Bs = B0 e , (3.74) for the ﬁxed coupling case. The slope is given by the parameter λBs and the nor- 117

malization by B0. For the case of running coupling we choose the parameterization

2 2 λBsY Bs = B0 e . (3.75)

These values of the slope λBs can be found in Table 3.5. It can be seen that the values of λBD and λBs are nearly identical. This is because they are measures of a related quantity. Both quantities Bs and λBs are a measure of the rate of expansion of the interaction area, and so it is not surprising that they are quite similar.

In the same way that the saturation scale for Qs and QsL have been solved for analytically in Sec 3.3.5.1 and 3.3.5.2 it is possible to solve for the saturation radius in impact parameter. By solving (3.43) for b2 we are able to arrive at an analytical solution for the saturation impact parameter given by

α¯sYχ(λc) 2(1−λc) 2 e Bs (r0,r,Y )= rr0 3 . (3.76) ′′ 4(1−λ ) (¯αsYχ (λc)) c We see that compared to (3.44) and (3.46) the equation (3.76) for the saturation impact parameter grows at half of the speed. This is due to the conformal invariance of the BK evolution equation. The relation of the saturation exponent

λBs being half that of λLs or λs is shown to hold nearly true, for the LL kernel where the conformal invariance is manifest (this is also numerically confirmed in Table 3.5). This completes the discussion on the mathematical properties of the solutions to the BK equation with impact parameter with various splitting kernels. With a description of the solution to the BK equation one can now apply this solution to various processes measured at particle colliders. There are several processes in which the numerical work presented in this chapter can be applied to. In the next chapter we shall focus on improving these solutions by including confinement effects and comparing theoretical predictions calculated with our model to data gathered by the H1 and ZEUS collaborations at HERA. 118

Dipole Size: 1.000 5

4 )

s 3 ln(B

0 10 20 30 40 50 Rapidity

Dipole Size: 1.000

4 ) s ln(B 3

0 5 10 15 20 Rapidity

Figure 3.29. Plots of the black disc cross section versus rapidity for various fixed dipole sizes and kernels with the coupling fixed atα ¯s = 0.1 (Fig a) and running coupling (Fig b). In Fig a the solid curve is the LL kernel and the dashed curve is the modified kernel, both withα ¯s = 0.1. In Fig b the solid line is the Balitsky prescription for the running coupling, and the dashed line is the LL kernel with the minimum dipole prescription. 119

Kernel λsB LL Kernelα ¯s =0.1 2.6 LL Kernelα ¯s =0.2 2.6 Modified Kernelα ¯s =0.1 2.2 Modified Kernelα ¯s =0.2 2.0 LL Kernel with minimum dipole prescription 0.72 Modified Kernel with minimum dipole prescription 0.1 Balitsky Kernel 0.74

Table 3.5. Values of λsB for various values ofα ¯s and kernels. Chapter 4

Phenomenological application of the solution to the BK equation

You can’t drink good beer all the time.

Theodore A. Corcovilos

It is easy to forget in the midst of all this formalism and numerics that we are attempting to explain phenomenological events. The real test of a model comes in its power to describe measurable phenomena and make testable predictions. To this end let us turn our attention to applying the solutions of the BK equation with impact parameter dependence to actual collider data. In this chapter we shall ﬁrst apply the solution for the BK equation to inclusive

DIS measurements, namely the computation of the structure functions F2 and FL. These calculations will then be compared to the combined HERA measurements from ZEUS and HI. Then we shall shift to the calculation of exclusive production for the ρ,φ and J/Ψ vector mesons. Various non-perturbative and phenomenological corrections shall be discussed along the way towards a consistent description of these exclusive and inclusive DIS measurements. Before any observables are actually calculated however, more needs to be done with the solution to the BK equation. The large dipole sizes must be regulated as conﬁnement cuts oﬀ these long distances in QCD. 121

4.1 Regulation of large dipole sizes

In order to properly fit the data the large dipole sizes must be regularized. The large dipole size regime is equivalent to the low momentum, infra-red region as r ∝ 1 k . In this region confinement effects are important and these effects cut off the large dipoles. Confinement itself is a non-perturbative effect and the exact mechanism is still not understood. In order to take these non-perturbative effects into account we introduce a massive parameter m. This scale determines the distance at which these confinement effects cut off the large dipoles. Let us consider some physical motivation to determine the form of the cutoff that must be introduced. First, it is clear that confinement effects must be included in the evolution and not simply in the initial condition. Cutting off the large dipoles in the initial condition will not fulfill the role of confinement as the evolution will again generate large dipoles. These large dipoles then diffuse towards the infra-red regime and populate the region that was initially cutoff, so that the confinement effect inserted into the initial condition is not preserved through the evolution. This effect was studied in [60]. The only way to implement the cutoff in the solution is to include confinement effects in the evolution equation itself, specifically in the branching kernel of the equation. Initially, we shall consider the effects of the confinement scale m on the fixed coupling case before proceeding to the case with running coupling.

4.1.1 Conﬁnement in the ﬁxed coupling case

One manner of including confinement effects is to include a gluon mass m in the evolution kernel. This will limit the propagation of the strong force to a distance scale r 1/m. Lattice simulations have suggested that there exists an effective max ≈ gluon mass [124,125] and this would serve to regulate the large dipole sizes. With this motivation one may include a mass for the gluon propagator in the branching kernel of the BK equation [126–129]. Let us derive this new kernel by recalling (2.16) in the derivation of the LL kernel. We rewrite this integral by utilizing the identity in [129] 122

(1) Φ (z0, x01,z, x02) 2l 2k ˆ a d T ilT·x01 d T kT ikT·x02 ikT·x12 (0) = 2gτ e − e e ǫTΨ (z , lT) . (2π)2 ∇ i(2π)2 k2 − · 0 Z Z T ! (4.1)

Here kˆ is the normalized unit vector of the two-dimensional kT vector and T ∇ 1 acts on x2. By replacing the Coulomb propagator 2 with the propagator from a kT 1 2 2 screened Yukawa potential 2 2 (where m =1/rmax) the gluon mass is included kT +m and the branching kernel becomes

(1) Φ (z0, x01,z, x02) 2l 2k ˆ a d T ilT·x01 d T kT ikT·x02 ikT·x12 (0) = 2gτ e − e e ǫTΨ (z , lT) . (2π)2 ∇ i(2π)2 k2 + m2 − · 0 Z Z T ! (4.2)

2 Solving the d kT integral and transforming to the mixed representation brings us to the equation

a (1) igτ (0) Φ (z , x01,z, x02)= [K (mx )ˆx K (mx )ˆx ] ǫTΨ (z , x01) . 0 π ∇ 0 02 02 − 0 12 12 · 0 (4.3) Taking the derivative we arrive at

a (1) igτ m (0) Φ (z , x01,z, x02)= − [K (mx )ˆx K (mx )ˆx ] ǫTΨ (z , x01) . 0 π 1 02 02 − 1 12 12 · 0 (4.4) and squaring the modulus of this wavefunction yields the following branching kernel with a gluon mass m

2 2 2 x02 x12 K =α ¯sm K1 (mx02)ˆx02 + K1 (mx12)ˆx12 2K1(mx02)K1(mx12) · . − x02x12 (4.5) In the limit of a small mass (m 0) the Bessel functions become K (mr) 1 . → 1 → mr This brings us back to the limit of the LL kernel for the BFKL and BK equation. The solution to the BK equation using the kernel (4.5) can be seen in Fig 4.1. 123

Impact parameter: 1.00 | cos(θ): 0.0 | ∆ Y: 1.0 | max Y: 8.0

1.0

0.8

0.6 N(Y,r,b,θ) 0.4

0.2

0.0 -2 -1 0 1 2 10 10 10 10 10 Dipole Size (GeV-1)

Figure 4.1. Plot of the scattering amplitude versus dipole size for the kernel (4.5). Note that the evolution in the large dipole size is not stiﬂed. The initial condition is shown by the dashed-dotted line and contains the cutoﬀ on dipoles larger than rmax.

The evolution in the large dipole region is suppressed, but not entirely cut-off by the Bessel functions in the kernel. This is because the Bessel functions, while giving an exponential drop off at large values, are not a hard cutoff. Additionally large contributions can occur if a large parent dipole splits into a large and a small daughter dipole. Consider the case when x01 is large and it splits into a large dipole x02 and a small dipole x12. The Bessel functions cutoff all terms with K1(mx02) 2 in them but the term K1 (mx12) remains. This contribution from the small dipole drives the evolution in the (non-perturbative) large dipole regime. We have also tried another approach in order to test whether the Bessel functions provide a cutoff which is too soft. A more stringent cutoff was applied in the same spirit as (4.5). By substituting mK (mr) with 1 Θ( 1 r2) the kernel 1 r m2 − reduces to the LL kernel for small dipole sizes but is completely cut-off for large 124

Impact parameter: 1.00 | cos(θ): 0.0 | ∆ Y: 1.0 | max Y: 8.0

1.0

0.8

0.6 N(Y,r,b,θ) 0.4

0.2

0.0 -2 -1 0 1 2 10 10 10 10 10 Dipole Size (GeV-1)

Figure 4.2. Plot of the scattering amplitude versus dipole size for the kernel (4.6). Note that the evolution in the large dipole size is not stifled in this case either. The cutoff in the initial condition here is cutoff for dipoles larger than 1/m and is shown by the dashed and dotted line. dipole sizes. The form of this kernel is

K = 1 1 1 1 x02 x12 1 1 α¯ Θ( x2 )+ Θ( x2 ) 2 · Θ( x2 )Θ( x2 ) . s x2 m2 − 02 x2 m2 − 12 − x2 x2 m2 − 12 m2 − 02 02 12 02 12 (4.6)

The solution to the evolution equation using this kernel can be seen in Fig 4.2, and again the diffusion in the infra-red region is not stopped by this cutoff. Even though the cutoffs are very rigid the emission of a small daughter dipole will give non-vanishing contributions. This effect is identical to what was discussed earlier with respect to (4.5) and underscores the fact that cutoffs on separate terms in the branching kernel will allow contributions in the infra-red region to remain. In order to cut all contributions in the infra-red region the entire kernel must 125

Impact parameter: 1.00 | cos(θ): 0.0 | ∆ Y: 1.0 | max Y: 8.0

1.0

0.8

0.6 N(Y,r,b,θ) 0.4

0.2

0.0 -2 -1 0 1 2 10 10 10 10 10 Dipole Size (GeV-1)

Figure 4.3. Plot of the scattering amplitude versus dipole size for the kernel (4.7). This time the evolution in the large dipole size regime is halted. The cutoﬀ in the initial condition here is cutoﬀ for dipoles larger than 1/m and is shown by the dashed and dotted line. be set to zero at large dipole sizes. Because of this we introduce an alternative prescription to the previous two splitting kernels where splittings are not allowed when either daughter dipole has a size greater than the distance scale 1/m. This ansatz has the form

2 x01 1 2 1 2 K =α ¯s 2 2 Θ( 2 x12)Θ( 2 x02) , (4.7) x02x12 m − m − 1 and whenever any daughter dipole greater than the size m is emitted the entire kernel is set to zero. The solution to the BK equation with kernel (4.7) can be seen in Fig 4.3 where the diffusion into the large dipole sizes is halted. It is interesting to note that the cutoff in the initial condition that sets the 1 scattering amplitude to zero for x01 > m is not respected by the evolution. A new 2 cutoff is generated by the evolution, which acts on dipoles with sizes x01 > m . 126

Dipole Size: 1.00| cos(θ): 0.0 | ∆ Y: 1.0 | max Y: 8.0 0

-2

-4 ln(N(Y,r,b,θ)) -6

-8

-2 -1 0 1 2 10 10 10 10 10 Impact Parameter (GeV-1)

Figure 4.4. Logarithmic plot of the scattering amplitude versus impact parameter for the kernel (4.7). The tails in impact parameter space can be seen to have been damped.

This is because the kernel is cutoff not on the parent dipole x01, but the daughter 2 dipoles x02 and x12. Until the parent dipole is at the size x01 = m it is possible to 1 emit two dipoles which are both underneath the cutoff m , allowing contributions in 2 this region. Once the parent dipole reaches m it is no longer possible to emit both 1 daughter dipoles underneath the cutoff m , and the evolution is halted. For this reason in the rest of the calculation we have implemented a cutoff on the initial condition of the form

2 N (Y =0)=0 for x > . (4.8) 01 01 m It was previously discussed in Sec 3.3.2 that there exist power-like tails in impact parameter space. These tails originate from the perturbative evolution and result in a growth of the cross-section which violates the Froissart bound. Examining the behavior in impact parameter space of the evolution with our cutoﬀ we ﬁnd that these impact parameter tails have been damped. This can be seen in 127

Fig 4.4 where the power-like behavior is no longer present due to the cutoﬀ and the steeply falling proﬁle in impact parameter space is instead present.

4.1.2 Conﬁnement in the running coupling case

To compute physical observables it is vital to include the running of the coupling. Once again we are presented with a choice between two prescriptions as discussed in Sec 3.3.6, and again we shall choose to use Balitsky’s (3.51) prescription as it is easier to numerically evaluate than the Kovchegov-Weigert (3.47) prescription. The Balitsky kernel (3.51) has the form

Bal K (x01,x02,x12) α¯ (x2 ) x2 1 α (x2 ) 1 α (x2 ) = s 01 01 + s 02 1 + s 12 1 . (4.9) 2π x2 x2 x2 α (x2 ) − x2 α (x2 ) − 02 12 02 s 12 12 s 02 The inclusion of the gluon mass m, as was done in the LL approach (4.5), is not straightforward and trivial to do when including the running coupling. In order to take confinement effects into account for the Balitsky kernel (4.9) the simplest scenario was chosen, in which the entire kernel is set to zero if either daughter dipole is larger than the mass scale. This is analogous to the fixed coupling case (4.7). The form of the Balitsky kernel with these cutoffs is

α¯ (x2 ) x2 1 α (x2 ) 1 α (x2 ) KBal(x ,x ,x ) = s 01 01 + s 02 1 + s 12 1 01 02 12 2π x2 x2 x2 α (x2 ) − x2 α (x2 ) − 02 12 02 s 12 12 s 02 1 1 Θ( x2 )Θ( x2 ). (4.10) × m2 − 12 m2 − 02

As was seen Sec 3.3.6.1 the Balitsky kernel was similar to the minimum dipole prescription (3.55). The same method can be used to apply conﬁnement eﬀects to the minimum dipole prescription at LL, yielding a kernel of the form

2 min 2 2 2 x01 1 2 1 2 K (x01,x02,x12)=¯αs(min(x01,x02,x12)) 2 2 Θ( 2 x12)Θ( 2 x02). (4.11) x02x12 m − m − 128

1.0

0.8

0.0.6 N(Y,r,b,θ) 00.4.

0.2

0.0 1 2 10-2 10-1 100 101 10 Dipole Size (GeV-1)

Figure 4.5. A comparison between the scattering amplitude using the minimum dipole prescription (solid) and the Balitsky prescription (dashed) for running coupling versus dipole size.

While without confinement effects it was found that the Balitsky and minimum dipole prescriptions were very similar in evolution speed, with the inclusion of a cutoff they now vary dramatically. As can be seen in Fig 4.5 the Balitsky prescription now has a much slower evolution than the minimum dipole prescription. This effect is due to the dipole configurations seen in Fig 4.6 where cos(θ) = 1. ± The slight difference in sizes of the dipoles gives a nontrivial contribution to the difference in the evolution speed of the two kernels in the presence of a cutoff. We shall use the kernel (4.10) for the calculations in the rest of this chapter.

4.2 Inclusive DIS measurements

Inclusive DIS in the dipole model is shown in Fig 4.7. The equation for the diﬀerential cross-section for inclusive DIS was given way back in chapter 1, equation 129

x0 x1

x0 x2 x1 (a)

x0 x1

x0 x1 x2 (b)

Figure 4.6. Diagrams of dipole emissions that yield a diﬀerence between (4.9) and (4.11). Fig a is when cos(θ)=1 and Fig b cos(θ)= 1. −

γ∗ r γ∗

Figure 4.7. Diagram of inclusive DIS in the dipole model. A proton in the initial state interacts with a virtual photon which has ﬂuctuated into a dipole pair.

(1.14). As we can see the cross-section is directly related to the structure functions

F2 and FL, and the structure functions have been measured very precisely at HERA (see Fig 1.4). Our analysis here shall follow that in [130]. The structure functions

F2 and FL can be computed in the dipole model by the formulae

Q2 1 F (Q2,Y ) = d2r dz Ψ (r,z,Q2) 2 2 4π2α | L | em Z Z0 + Ψ (r,z,Q2) 2 σ (r, Y ) (4.12) | T | dip Q2 1 F (Q2,Y ) = d2r dz Ψ (r,z,Q2) 2σ (r, Y ). (4.13) L 4π2α | L | dip em Z Z0 130

Here r is the dipole size. The dependence on the Bjorken x is held within the rapidity dependence Y = ln(1/x). The momentum Q2 is the virtuality of the 2 probing photon, σdip(r, Y ) is the dipole scattering cross section, and ΨT/L(r,z,Q ) represents the wavefunction for the photon splitting. The photon splitting into a q q¯ pair is calculable in QED and is well known to be

3α Ψ (r,z,Q2) 2 = em e2 z2 + (1 z)2 Q¯2 K2(Q¯ r)+ m2 K2(Q¯ r) | T | 2π2 f − f 1 f f 0 f f X (4.14) 3α Ψ (r,z,Q2) 2 = em e2 4Q2z2(1 z2)K2(Q¯ r) , (4.15) | L | 2π2 f − 0 f f X where the summation f is over the quark ﬂavors, ef the charge of the quark, z the longitudinal momentumP fraction of the photon carried by the quark, and

Q¯2 = z(1 z)Q2 + m2 . (4.16) f − f

The dipole cross section is found by integrating the dipole scattering amplitude N(r, b,Y ) over impact parameter b as in 3.64. The amplitude N(r, b,Y ) is taken from our solution to the BK equation with (4.10) as the splitting kernel. When dealing with physical measurements the initial condition is very important as it −2 must ﬁt the data at the initial rapidity. We take x0 = 10 and our initial rapidity is Y0 = ln(1/x0). The initial condition is taken from [131], which is a Glauber- Mueller initial condition parameterized as

π2 N(r, b,Y )=1 exp r2x g(x ,µ2)T (b) . (4.17) 0 − −2N 0 0 c The term xg(x,µ2) is the integrated gluon distribution at a scale µ2. The scale µ is 2 C2 2 2 2 set equal to µ = r2 + µ0 with C = 2 and µ0 =1.16GeV . The gluon distribution is taken initially to have the form

xg(x,η2)= Axλg (1 x)5.6 . (4.18) − In [131] the gluon distribution was evolved using the leading-order DGLAP evolu- 131

µ0 1.16547 GeV C 2 A 2.55042 λg 0.01980 −2 BG 3.65 GeV m 0.37 GeV Table 4.1. Values of the free parameters used in the initial condition. tion equation (without quark contributions). In contrast to [131] we shall only take the initial condition of the gluon distribution and instead of utilizing the DGLAP equation to evolve it we shall allow the BK equation to evolve the dipole scattering amplitude. The impact parameter proﬁle T (b) is again taken to be steeply falling in b as

1 b2 T (b)= exp − , (4.19) 8π 2B G with BG as a initial scale parameter which determines the initial distribution of the interaction region in impact parameter space. There are several parameters in the initial condition (4.17) that can be varied, such A or λg in the integrated gluon distribution. As running the program to evolve each initial condition with impact parameter dependence takes a long time (on the order of a day) most of these parameters were not varied. The only parameter that was varied was BG which sets the scale in impact parameter space for the interaction region. This parameter was related to the the mass parameter m in 1 (4.10) by the relation BG = 2m2 . Although these two scales need not be ﬁxed together in exactly this manner there is a strong correlation in the BK equation between dipole size and impact parameter. Because of this the value of BG and m cannot be seen as completely independent. The values of these parameters are summarized in Table 4.1

4.2.1 Corrections at small Q2

The initial condition used in our calculations is cut at large dipole sizes as described in (4.8). This leads the calculations with our model to underestimate the data at values of x 0.01. This is because the initial model (4.17) was fitted in [131] to ≈ 132 the experimental data without any such cuts on the infra-red regime. It was found that for the model in [131] when m =0.35GeV and Q2 =4.5 GeV2 dipoles larger than the cutoff rcut =1/m correspond to approximately 30% of the contributions to F , and dipoles larger than r = 2/m are 10% of F . This shows that F 2 cut ∼ 2 2 has a large sensitivity to the large dipole regime even for moderate Q2 values and contains important non-perturbative contributions at these Q2 values. These contributions correspond to the existence of aligned jet configurations in the transverse structure function [132,133]. This is evident in (4.14) where 2 ΨT (r,z,Q ) has large contributions from the endpoints z = 0, 1. This yields a widely spread distribution of dipoles which contribute to F2. It is possible to mit- igate this effect of the cutoff by decreasing the value of the cutoff mass. However, this is not an option as the impact parameter profile and the cutoff mass are linked. Changing the cutoff mass very far from m =0.35GeV would make our model in- consistent with the diffractive J/Ψ production data (which will be discussed in Sec 4.3.1.5). It is important to note that this property of the solution originates from the fact that in the evolution the impact parameter and dipole size are not independent. This is in contrast to (4.17) where the dipole sizes and impact parameter are decoupled. Rather than shift the cutoff further we instead add in the non-perturbative contribution in a separate term to F2. This contribution is important at low and intermediate Q2 values (< 15GeV2) and falls off at Q2 increases. In the large- dipole region we assume the dipole cross section is replaced by a constant σ0 which modifies (4.12) into the following soft contribution:

Q2 soft 2r 2 2 2 2 F2 = 2 σ0 d dz ΨL(r,z,Q ) + ΨT (r,z,Q ) . (4.20) 4π αem 2 | | | | Z m Z

The constant σ0 is ﬁt to the data. Combining the soft contribution (4.20) with the dip perturbatively calculated dipole portion from (4.12), which we shall label F2 , we arrive at the expression F2

tot dip soft F2 = F2 + F2 . (4.21)

2 2 The parameter σ0 is ﬁt to the data at Q =4.5GeV (which is the lowest bin of 133

Q2 = 4.50 Q2 = 6.50 Q2 = 8.50

1.6 1.6 1.6 1.4 1.4 1.4 1.2 1.2 1.2 2 2 2 F 1.0 F 1.0 F 1.0 0.8 0.8 0.8 0.6 0.6 0.6 10-4 10-3 10-2 10-4 10-3 10-2 10-3 10-2 x x x

Q2 = 10.00 Q2 = 12.00 Q2 = 15.00

1.6 1.6 1.6 1.4 1.4 1.4 1.2 1.2 1.2 2 2 2 F 1.0 F 1.0 F 1.0 0.8 0.8 0.8 0.6 0.6 0.6 10-3 10-2 10-3 10-2 10-3 10-2 x x x

Q2 = 18.00 Q2 = 22.00 Q2 = 27.00

1.6 1.6 1.6 1.4 1.4 1.4 1.2 1.2 1.2 2 2 2 F 1.0 F 1.0 F 1.0 0.8 0.8 0.8 0.6 0.6 0.6 10-3 10-2 10-3 10-2 10-3 10-2 x x x

Figure 4.8. First set of plots of the combined HERA data for F2 across various xbj values compared with the calculation for F2 without corrections (dashed) including the soft-physics (4.20) contribution (solid). Each graph shows a diﬀerent value of Q2. The F2 contribution with the soft correction is better able to describe the data but once again the slope of the calculation is too steep.

2 soft Q considered) and x =0.01. The integral in F2 only involves dipoles which are equal to the cutoff rcut =2/m or larger, as can be seen in the modified integration soft 2 limits. F2 is slowly varying with Q and compensates for the non-perturbative effects well. Adding in separate contributions for ultra-violet and the infra-red regimes is similar to what was done in [134,135].

The plots of F2, including the soft corrections, can be seen in Figs 4.8 and 4.9. The slope of the calculation is still too steep owning to the overly-fast evolution from the BK equation with the splitting kernel (4.10). It is also seen that the contribution from the soft correction drops oﬀ as Q2 becomes large, which is to be expected. 134

Q2 = 35.00 Q2 = 45.00 Q2 = 60.00

1.6 1.6 1.6 1.4 1.4 1.4 1.2 1.2 1.2 2 2 2 F 1.0 F 1.0 F 1.0 0.8 0.8 0.8 0.6 0.6 0.6 10-3 10-2 10-3 10-2 10-2 x x x

Q2 = 70.00 Q2 = 90.00 Q2 = 120.00

1.6 1.6 1.6 1.4 1.4 1.4 1.2 1.2 1.2 2 2 2 F 1.0 F 1.0 F 1.0 0.8 0.8 0.8 0.6 0.6 0.6 10-2 10-2 10-2 x x x

Q2 = 150.00 Q2 = 200.00 Q2 = 250.00

1.6 1.6 1.6 1.4 1.4 1.4 1.2 1.2 1.2 2 2 2 F 1.0 F 1.0 F 1.0 0.8 0.8 0.8 0.6 0.6 0.6 10-2 10-2 10-2 x x x

Figure 4.9. Second set of plots of the combined HERA data for F2 across various xbj values compared with the calculation for F2 without corrections (dashed) including the soft-physics (4.20) contribution (solid). Each graph shows a diﬀerent value of Q2. The F2 contribution with the soft correction is better able to describe the data but once again the slope of the calculation is too steep.

Instead of the soft correction (4.20) the non-perturbative contributions could come from the vector meson dominance (VMD) model. For a more in-depth description of this model please see [136–138]. Following [134] the form of the vector meson dominance term is

2 4 2 2 2 2 2 2 VMD Q mvσv(W ) Q mvσv(W ) mv F2 = 2 2 2 2 + ζ0 2 2 2 2 2 2 . (4.22) 4π γv (Q + mv) γv (Q + mv) Q + mv ! v=Xρ,ω,φ

Here mv is the mass of the vector meson,ζ0 =0.7, and γv is related to the leptonic decay width of the vector meson Γv→e+e− by 135

2 2 παemmv γv = . (4.23) 3Γv→e+e− The leptonic decay widths are taken from [139]. The hadron-vector meson cross section σv is a function of energy and is taken to be

1 σ = σ = σ(π+p)+ σ(π−p) , (4.24) ρ ω 2 1 σ = σ(K+p)+ σ(K−p) σ(π+p)+ σ(π−p) . (4.25) φ − 2 The cross-sections σ(π±p) and σ(K±p) are taken to be the same parameterization as [140], which is reproduced here (σ in (mb) and s in GeV2) :

σ(π+p) = 13.63s0.0808 + 36.025s−0.4525, (4.26) σ(π−p) = 13.63s0.0808 + 27.56s−0.4525, (4.27) σ(K+p) = 11.82s0.0808 + 26.36s−0.4525, (4.28) σ(K−p) = 11.82s0.0808 +8.15s−0.4525. (4.29)

dip Combining the perturbative dipole contribution F2 with the VMD contribu- VMD tion F2 the total F2 structure function is

tot dip VMD F2 = F2 + F2 . (4.30)

tot The plot of F2 with the VMD contribution can be seen in Figs 4.10 and 4.11. It is clear that even at the lowest Q2 the contribution from the VMD model is not enough to bring the calculation into agreement with the data. The VMD term is only substantial for Q2 < 4 GeV2 and an additional contribution would still be needed from Q2 = 4.5 GeV2 to as much as Q2 = 45 GeV2. This is not surprising as this has been seen in [135]. It can be seen that the slope of our calculations is slightly too steep in x (for all Q2 bins), this implies that the LL evolution with the running coupling has a faster evolution in rapidity than the data. One possible solution is to begin varying the value of ΛQCD and take it as a ﬁtting parameter. This was done in [141,142], 136

Q2 = 4.50 Q2 = 6.50 Q2 = 8.50

1.6 1.6 1.6 1.4 1.4 1.4 1.2 1.2 1.2 2 2 2 F 1.0 F 1.0 F 1.0 0.8 0.8 0.8 0.6 0.6 0.6 10-4 10-3 10-2 10-4 10-3 10-2 10-3 10-2 x x x

Q2 = 10.00 Q2 = 12.00 Q2 = 15.00

1.6 1.6 1.6 1.4 1.4 1.4 1.2 1.2 1.2 2 2 2 F 1.0 F 1.0 F 1.0 0.8 0.8 0.8 0.6 0.6 0.6 10-3 10-2 10-3 10-2 10-3 10-2 x x x

Q2 = 18.00 Q2 = 22.00 Q2 = 27.00

1.6 1.6 1.6 1.4 1.4 1.4 1.2 1.2 1.2 2 2 2 F 1.0 F 1.0 F 1.0 0.8 0.8 0.8 0.6 0.6 0.6 10-3 10-2 10-3 10-2 10-3 10-2 x x x

Figure 4.10. First set of plots of the combined HERA data for F2 across various x values compared with the calculation for F2 without the VMD (4.22) correction (dashed) and including (solid). Each graph shows a diﬀerent value of Q2. The VMD contribution is insuﬃcient to give good agreement with the data.

but by our estimates it would take a value of ΛQCD well below the pion mass (on the order of tens of MeV) to ﬁt the data. The fact that the LL evolution with running coupling has a faster evolution is not unexpected as NLL corrections to the BK equation need to be accounted for. While this have been derived [143], the BK equation has not been numerically solved with NLL corrections. However, an analysis has been performed [144] using the saturation-boundary method of [103] and this analysis indicates the NLL corrections to the BK equation are substantial. Full analysis of the BK equation with NLL corrections is thus left for future investigation. 137

Q2 = 35.00 Q2 = 45.00 Q2 = 60.00

1.6 1.6 1.6 1.4 1.4 1.4 1.2 1.2 1.2 2 2 2 F 1.0 F 1.0 F 1.0 0.8 0.8 0.8 0.6 0.6 0.6 10-3 10-2 10-3 10-2 10-2 x x x

Q2 = 70.00 Q2 = 90.00 Q2 = 120.00

1.6 1.6 1.6 1.4 1.4 1.4 1.2 1.2 1.2 2 2 2 F 1.0 F 1.0 F 1.0 0.8 0.8 0.8 0.6 0.6 0.6 10-2 10-2 10-2 x x x

Q2 = 150.00 Q2 = 200.00 Q2 = 250.00

1.6 1.6 1.6 1.4 1.4 1.4 1.2 1.2 1.2 2 2 2 F 1.0 F 1.0 F 1.0 0.8 0.8 0.8 0.6 0.6 0.6 10-2 10-2 10-2 x x x

Figure 4.11. Second set of plots of the combined HERA data for F2 across various x values compared with the calculation for F2 without the VMD (4.22) correction (dashed) and including (solid). Each graph shows a diﬀerent value of Q2. The VMD contribution is insuﬃcient to give good agreement with the data.

4.3 Exclusive vector meson production

Exclusive diﬀractive vector meson production has been measured at HERA for a variety of vector mesons. In this work we shall focus speciﬁcally on the ρ,φ and J/Ψ vector meson, our analysis following that in [145]. Vector meson production measurements are able to access information about the electron-proton interaction at small x. Through this interaction information on the size of the proton − in impact parameter space can be extracted. Additionally, this process can be used as a tool to constrain the vector meson wavefunctions. Summaries of these experimental measurements can be found in [146,147]. The ability to measure the momentum transfer t in vector meson production is what gives access to the information in impact parameter space of the interaction 138

p p

Figure 4.12. Diagram of exclusive vector meson production in the dipole model. A proton in the initial state interacts with a virtual photon which has ﬂuctuated into a dipole pair. A vector meson is produced in the ﬁnal state along with the initial proton.

(as ∆ and b are conjugate variables, ∆2 = t). This provides information about − the transverse size of the interaction region [148]. While numerous other studies have used the dipole model to describe exclusive diffractive vector meson production [128,131,149–153], without the inclusion of impact parameter as a variable the information encoded in the momentum transfer t is lost. The novel aspect of the model presented in this dissertation is that the inclusion of impact parameter as a dynamic degree of freedom in the evolution equation allows for new predictive power of the dependence of the cross section on the momentum transfer. The diffusion in impact parameter space leads to an increase in the interaction area, and thus the cross section, as t decreases (b 1/√t). Diffusion in transverse space is also seen in Monte-Carlo simula- ∝ tions of dipole branchings [128,154–156]. In this section it will be shown that the numerical solution of the BK evolution equation with impact parameter dependence is compatible with experimental data on exclusive diffractive vector meson production. In the process γ∗p Vp the proton scatters elastically with the virtual photon, → producing a vector meson in the interaction. A diagram of the amplitude of this interaction in the dipole picture can be seen in Fig 4.12. The amplitude of such an interaction is given by

2 2 V A(x, ∆,Q)= d r dzΨh,h¯ (r,z,Q ) (x, r, ∆)Ψh,h¯ (r, z) , (4.31) ¯ N Xh,h Z Z 139 where h (h¯) is the helicity of the quark (anti-quark). The diﬀerential cross section is given by dσ 1 = A(x, ∆,Q) 2 , (4.32) dt 16π | | where ∆2 = t. − 2 The photon wavefunction Ψh,h¯ (r,z,Q ) is the same as given in (4.14) and (4.15). V The vector meson wavefunction Ψh,h¯ (r, z) has been parameterized in many diﬀerent models in the literature [150,157–160]. While the method of parameterizing the vector meson wave function varies, all are constrained by some model independent requirements. The wavefunction mus satisfy the normalization condition

2 h,h¯ 2 1= d r dz ψV (z, r) . (4.33) ¯ | | Xh,h Z In (4.33) only contributions from the valence quarks are taken into account. Ad- ditionally the leptonic decay width Γ(V e+e−) must match with experiment. → This second constraint comes from the value of the wave function at r = 0 and has the form

1 π f dz ψ (z,r =0)= V , where 0 J µ (0) V e f M εµ . (4.34) V N 2ˆe h | em | i≡ q V V Z0 r c V

Here fV is the electromagnetic coupling of the meson to the current ande ˆV is the eﬀective charge of the quarks in units of e, the elementary charge. In the case of the ρ meson it is the charge of the combination (uu¯ dd¯)/√2 (i.e.e ˆ =1/√2). − V The method chosen here is the NNPZ wavefunction [149,151]. This model assumes the meson is composed of a quark and anti-quark in a harmonic oscillator potential. This gives the wave-function a Gaussian dependence on the distance between the quarks in transverse coordinate space. The long distance physics is constrained by information from spectroscopic models. The NNPZ wavefunction also includes short-distance physics originating from the exchange of hard gluons between the valence quarks of the vector mesons. The wave-function in this model is also relativized, giving the wavefunction 140

h,h¯ Nc 1 2 2 2 ψ (z, r) = δ ¯ [M z(1 z) + m ]φ(r, z) , (4.35) V h,−h 4π M z(1 z) V − −∇r q r V − where MV is the mass of the vector meson. The function φ(r, z) is the radial wavefunction and it has the form

φ(z,r) = Ψ0L/T (1S) m2R2 2z(1 z)r2 m2R2 4z(1 z)√2πR2 exp q exp − exp q × − −8z(1 z) − R2 2 − 16a3(r) + C4 rK (r A(z,r)/B(z,r)) . (4.36) A(z,r)B(z,r)3 1 · The functions A and B are given by

C2a2(r)m2 A2(z,r) = 1+ q 4C2a2(r)m2 (4.37) z(1 z) − q − C2a2(r) B2(z,r) = , (4.38) z(1 z) − and

3 a(r) . (4.39) ≡ 8mqαs(r) The C term (sometimes referred to as a Coulombic term) in (4.36) is set to zero in this analysis, in much the same way as [131]. This term is set to zero due to it having poor functional behavior at the origin. While for the light mesons ρ and φ this approximation is not unreasonable there is slightly less justiﬁcation for the heavier J/Ψ.

Through the two conditions, (4.33) and (4.34), the normalization Ψ0L/T (1S) 2 and the parameter R are ﬁxed. The normalization Ψ0L/T (1S) is diﬀerent for the longitudinal and transverse wave functions of each vector meson. On the other hand R is taken to be the same for both the transverse and longitudinal polarizations of each vector meson. R is determined by the lepton decay width 141

ρ φ J/Ψ Ψ0(1S)T 0.025 0.028 0.039 Ψ0(1S)L 0.024 0.025 0.039 R2 12.77 11.0 2.188 Table 4.2. Values of the free parameters used in the vector meson wave functions. of the transverse polarization, this means that the longitudinal decay width varies somewhat. The parameter values used can be found in Table 4.2.

In (4.31) the dipole scattering amplitude N(x, r, ∆) can be translated into coordinate space by a Fourier transform

(x, r, ∆,z)=2 d2bN(x, r, b)eiδ·b. (4.40) N Z This formula was derived with the assumption that the target is much larger than the dipole. If the dipole is not taken to be inﬁnitely small but of ﬁnite size then (4.40) must be corrected. This was done in [161] and it was found that for the non-forward case (∆ = 0) the Fourier transformation becomes 6

(x, r, ∆,z)=2 d2bN(x, r, b)eiδ·(b−(1−z)r). (4.41) N Z This correction was applied in [162] and was shown to give a non-trivial contribution to the cross-section. This was especially prevalent in the dependence on the momentum transfer t of the cross section versus Q2 + M 2 . | | V The initial condition used in the evolution of the scattering amplitude is the same as was used in (4.17), with the same parameters (see Table 4.1). With all of the parts of the cross section deﬁned we may once again use our solution to the BK equation with a splitting kernel of the form (4.10) to arrive at a physical observable and compare this to actual data. In this section the data from HERA for vector meson production of ρ [163–165], φ [165,166] and J/Ψ [167,168] is used. Plotting 2 2 the cross section of J/Ψ versus Q + MV (where MV is the vector meson mass) in Fig 4.13 it can be seen that the calculation falls quite short of the experimental data. It would have been naive to believe that we would not need to include any corrections to our dipole calculation in order to ﬁt the data well, so let us discuss the corrections that are necessary. 142

4.3.1 Phenomenological corrections

The ﬁrst correction that we shall consider is a skewed gluon distribution. This is needed because the longitudinal momentum fractions of the two exchanged gluons in vector meson production need not be identical (unlike the case for F2 and FL). This skewed distribution introduces substantial corrections at small energies and the correction goes to zero as the energy becomes large. Following [169] the form of the skewed gluon distribution is

(2λsk+3) 2 2 2 Γ(λsk +5/2) (xg(x,η ))sk = xg(x,η ) , (4.42) √π Γ(λsk + 4) where ∂ ln(xg(x,η2)) λ . (4.43) sk ≡ ∂ ln(1/x) There is no term in the evolution of the scattering amplitude N(r, b,Y ) for the integrated gluon distribution, so this skewedness correction does not aﬀect the evolution at all. The gluon distribution does come into the initial condition (4.17). Modifying the initial condition changes the normalization of the solution substantially, see Fig 4.13. The skewed gluon distribution is suﬃcient to bring the J/Ψ calculations into agreement with the data. Still this correction is not enough for the lighter mesons ρ and φ. The next correction is including a contribution for the real part of the scattering amplitude. We follow [151,153,169] in the method of accounting for this contribution by multiplying the scattering amplitude (4.31) by a factor (1+ β2). The term β is the ratio of the real scattering amplitude to the imaginary part and is given by

β = tan(πλr/2) , (4.44) with ∂ ln(Aγ∗p→Vp) λ = T,L . (4.45) r ∂ ln(1/x) This contribution is not suﬃcient to account for the diﬀerence between the calculation and the data for the ρ and φ mesons as (4.44) is a rather modest correction. A third and more substantial correction is needed, especially at low Q2. This 143

γ* p => J/ψ p 102

101 (nb) σ

100

10-1 101 102 Q2 + M2 (GeV2)

Figure 4.13. A graph of the cross-section for J/Ψ production as a function of Q2 comparing the calculation without a skewed gluon distribution (solid line) to one including a skewed gluon distribution (dashed line) given by (4.42). comes in the form of a correction to the photon wavefunction given in (4.15) and (4.14). These wavefunctions were derived assuming that there is a hard scale Q2 which corresponds to 1/r2, and while the wavefunction of a longitudinally polarized photon is peaked at this value a transversely polarized photon yields a much broader distribution. This is due to the end-point contributions at z =0, 1. This corresponds to aligned jet configurations in the transverse structure function (further literature on this effect can be found in [132,133]). This broadening of the photon distribution for the transverse polarization leads to a contribution at low Q2 values, which correspond to large non-perturbative dipoles (which we have cut off). In this region the photon behaves as if it has a hadronic component which is taken into account in a manner following [153] and [128]:

2 2 −ω (x01−R) 2 2 1+ Be Ψγ Ψγ −ω2R2 . (4.46) | | →| | 1+ Be ! The factors B, ω2, and R2 are varied to obtain the best ﬁt. 144

R 6.8 GeV−1 B 6.0 ω2 0.2 GeV2 Table 4.3. Values of the free parameters used in the photon wavefunction correction (4.46).

γ* p => ρ p γ* p => ρ p

3 3 10 W = 75 GeV 10 W = 90 GeV

102 102 (nb) (nb) σ 101 σ 101

100 100

10-1 10-1 100 101 100 101 Q2 + M 2 (GeV2) Q2 + M 2 (GeV2) v v

Figure 4.14. Cross section σ(Q2,W 2) for ρ production plotted as a function of (Q2 + 2 MV ). The experimental data is from [163–165].

While calculating the cross section from the solution of the BK equation does not take as long as calculating the solution for the BK equation it is still time con- suming (over an hour) which makes a proper ﬁt not realistic in terms of computing time. Instead, a section of parameter space was explored and the best ﬁtting parameters were chosen. These parameters can be found in Table 4.3. All exclusive measurements in the rest of this section are done including the three corrections mentioned here. The parameters for these corrections do not vary between vector mesons.

4.3.1.1 Q2 + M 2 dependence of the cross section

Let us ﬁrst compare our calculation to the experimental data for the cross section integrated over the momentum transfer t. Figs 4.14 and 4.15 plot the cross section 2 2 2 as a function of (Q + MV ), where MV is the mass squared of the vector meson 2 2 2 being evaluated. The variable (Q + MV ) is used instead of Q in order to provide an inherent scale for each vector meson so that the plots may be more comparable to each other. We have integrated over the momentum transfer range that has been 145

γ* p => φ p γ* p => J/ψ p 102 2 10 W = 75 GeV W = 90 GeV

101 101 (nb) (nb) σ σ

100 100

10-1 10-1 100 101 101 102 Q2 + M 2 (GeV2) Q2 + M 2 (GeV2) v v

Figure 4.15. Cross section σ(Q2,W 2) for the production of φ and J/Ψ vector mesons 2 2 is plotted as a function of (Q + MV ). The experimental data is from [165–168]. probed by the experimental data. The data on ρ production [163] is integrated in t for t < 0.5 GeV2, t < 1.0 GeV2 [164], and t < 3 GeV2 [165]. For φ production | | | | | | the range of t is t < 0.6 GeV2 [166] and t < 3 GeV2 [165]. Finally for J/Ψ | | | | t < 1.2 GeV2 [168] and t < 1.0 GeV2 [167]. The W values were computed at | | | | W = 90 GeV for J/Ψ production and W = 75 GeV or W = 90 GeV for ρ and φ depending on the data used. It can be seen that the data is well reproduced by our calculations, the only signiﬁcant deviations from the data come from the low Q2 regime of the ρ meson.

4.3.1.2 W dependence of the cross section

The integrated cross section can be plotted as a function of W for diﬀerent bins of Q2. This can be seen in Figs 4.16,4.17, and 4.18 for ρ,φ, and J/Ψ production respectively. For production of J/Ψ and φ mesons the data is well described by our calculations. In the case of ρ production the data is poorly described. While the trends of the data in ρ are well described the normalization is consistently low, and this is exacerbated in the low Q2 region. This is the region which is not well described perturbatively and the corrections here are large. There may be an additional non-perturbative correction that needs to be taken into account, or the wavefunction for ρ may need to be treated more carefully. 146

γ* p => ρ p γ* p => ρ p

103 103 Q2 = 2.0 GeV2

Q2 = 2.4 GeV2 Q2 = 3.1 GeV2 Q2 = 3.7 GeV2 2 2 2 2 10 Q = 4.8 GeV 10 Q2 = 6.0 GeV2 Q2 = 7.2 GeV2

(nb) (nb) Q2 = 8.3 GeV2 σ Q2 = 10.9 GeV2 σ

Q2 = 13.5 GeV2 101 101 Q2 = 19.7 GeV2

Q2 = 32.0 GeV2 100 100 0 50 100 150 0 50 100 150 W (GeV) W (GeV)

γ* p => ρ p

103

Q2 = 3.3 GeV2

102 Q2 = 6.6 GeV2 (nb) σ Q2 = 11.9 GeV2

101 Q2 = 19.5 GeV2

Q2 = 35.6 GeV2 100 0 50 100 150 W (GeV)

Figure 4.16. W dependence of the vector meson cross section for elastic production of ρ. The experimental data is from [163–166].

γ* p => φ p γ* p => φ p

102 102 Q2 = 2.4 GeV2

Q2 = 3.8 GeV2 Q2 = 3.3 GeV2 (nb) (nb)

σ Q2 = 6.5 GeV2 σ Q2 = 6.6 GeV2 101 101

Q2 = 13.0 GeV2

Q2 = 15.8 GeV2

100 100 0 50 100 150 0 50 100 150 W (GeV) W (GeV)

Figure 4.17. W dependence of the vector meson cross section for elastic production of φ. The experimental data is from [165,166]. 147

γ* p => J/ψ p γ* p => J/ψ p

2 2 2 Q = 0.05 GeV 2 2 2 10 10 Q = 0.4 GeV Q2 = 3.2 GeV2 Q2 = 3.1 GeV2

Q2 = 7.0 GeV2 Q2 = 6.8 GeV2

2 2 (nb) (nb) Q = 16.0 GeV σ σ 101 101 Q2 = 22.4 GeV2

100 100 0 100 200 300 0 100 200 300 W (GeV) W (GeV)

Figure 4.18. W dependence of the vector meson cross section for elastic production of J/Ψ. The experimental data is from [167,168].

γ* p => ρ p γ* p => ρ p

8 8

6 6 T T σ σ / / L L σ σ 4 4 R= R=

2 2

0 0 0 5 10 15 20 25 0 5 10 15 20 25 Q2 (GeV2) Q2 (GeV2)

γ* p => φ p γ* p => J/ψ p 3

6 2 T T σ σ / / L L σ σ 4 R= R=

0 0 0 5 10 15 20 25 0 5 10 15 20 25 Q2 (GeV2) Q2 (GeV2)

Figure 4.19. The ratio of R = σL cross section for longitudinally to transversely σT polarized vector mesons. The plots are for ρ (top left W = 75GeV and right W = 90GeV), φ (bottom left W = 90GeV), and J/Ψ (bottom right W = 75GeV). Data taken from [163–168]. 148

4.3.1.3 Ratio of the transverse and longitudinal cross section

The ratio of the longitudinal cross section to the transverse part is defined to be R = σL , which has been measured at HERA. This ratio provides a sensitivity to σT the exact form of the vector meson wavefunctions which are used. These ratios are plotted in Fig 4.19 for ρ,φ, and J/Ψ. The experimental data averages over large bins in the energy W and our calculations correspond to the central value of these bins. It can be seen that these ratios are well described by our calculations except for the φ meson, whose value of R is larger across the entire evaluated range. This seems to be a common property of the φ calculations in the dipole model and a similar result can be seen in [131]. The formula for R suggests that the ratio should be approximately independent of W , as this dependence would cancel between σT and σL. This is what is seen in the data as R is independent of W . Looking closer at the equations for σT and σL we find that this exact cancellation does not seem to hold. The different components of the cross section are sensitive to the dipole distributions derived from the photon wavefunctions. These wavefunctions are peaked differently for the longitudinal and transverse components (as was described in Sec 4.3.1). In our calculations it was found that the ratio R does not depend strongly on the energy W which is consistent with what is seen in the data.

4.3.1.4 Momentum transfer dependence of the diﬀerential cross section

Integrating over the momentum transfer t removes the information about the impact parameter profile of this interaction. Because the impact parameter and momentum transfer are conjugate to each other (by the Fourier transform (4.41)) the t distribution is of key interest. It can be seen in Fig 4.20 the differential cross dσ 2 section dt for J/Ψ plotted as a function of t for fixed W and Q describes the HERA data well. Similarly plotting the differential cross section as a function of W with fixed t and Q2 in Fig 4.21 also describes the data well. The t dependence of the differential cross section can be seen to be linear when the differential cross section is plotted logarithmically. This implies we may parameterize the differential cross section as 149

γ* p => J/ψ p γ* p => J/ψ p 103 103

102 W = 100 GeV 102 W = 90 GeV ) ) 2 2

1 1 2 10 Q = 0.05 GeV 10 2 2 Q = 3.2 GeV /dt (nb/GeV /dt (nb/GeV 2 σ Q 2 σ Q 2 d = 7.0 GeV d = 3.1 GeV 2 2 2 Q = 6.8 GeV 100 100 2 2 2 Q = 22.4 GeV Q = 16.0 GeV 2 2

10-1 10-1 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 |t| (GeV2) |t| (GeV2)

Figure 4.20. Diﬀerential cross section for J/Ψ production for a ﬁxed W in bins of Q2 as a function of momentum transfer t . Calculations were done with W = 100GeV and | | W = 90GeV. The experimental data is from H1 experiment [168].

γ* p => J/ψ p γ* p => J/ψ p 103 103 Q2 = 0.05 GeV2 Q2 = 8.9 GeV2

|t| = 0.03 GeV2 2 2 2 10 |t| = 0.10 GeV 10 ) ) 2 2 |t| = 0.22 GeV2 |t| = 0.05 GeV2

|t| = 0.43 GeV2 |t| = 0.19 GeV2 /dt (nb/GeV /dt (nb/GeV σ σ d 101 d 101 |t| = 0.83 GeV2

|t| = 0.64 GeV2

100 100 102 102 W (GeV) W (GeV)

Figure 4.21. The diﬀerential cross section for J/Ψ production as a function of W for ﬁxed Q2 in bins of momentum transfer t , data from H1 [168]. | |

dσ e−BD|t|. (4.47) dt ∝

The dimensionful slope parameter BD has been measured at HERA and can be compared to our calculation.

4.3.1.5 Slope of the diﬀerential cross section

The slope of the diﬀractive cross section is described by the dimensionful parameter

BD, as in (4.47). The information about the growth of the interaction region of the 150

γ* p => ρ p γ* p => φ p

8 8 ) )

-2 6 -2 6 (GeV (GeV d d B B

4 4

2 2 101 101 Q2 + M2 (GeV2) Q2 + M2 (GeV2)

γ* p => J/ψ p

8 )

-2 6 (GeV d B

2 101 Q2 + M2 (GeV2)

2 2 Figure 4.22. Dependence of the slope parameter BD on combined variable (Q + MV ) for ρ,φ,J/Ψ vector mesons. Data taken from [163–168] scattering process is contained within this parameter. It can be seen in Fig 4.22 that our calculation using impact parameter dependence does reproduce the data well in both slope and normalization (except for the ρ data which is again slightly 2 below in normalization). At low Q the BD values are inherent to each vector meson involved. However, as the Q2 value increases it can be seen that all vector mesons converge to a similar value of B 4 GeV−2. This is consistent with our D ≈ view that BD is a measure of the interaction area, but it is not simply a measure of the interaction area of the target but a convolution of the interaction area of the target and the projectile. At low Q2 the projectile is large in coordinate space 2 and BD probes this convolution. However when Q is high the spatial distribution of the projectile is very small. These small probes eﬀectively measure the impact parameter distribution of the target proton, so it is expected that at high Q2 all processes to reach a common value, which is what is seen. 151

γ* p => J/ψ p γ* p => J/ψ p 6 6

Q2 = 0.05 GeV2 Q2 = 8.9 GeV2

5 5 ) ) -2 -2 (GeV (GeV d d B B 4 4

3 3 102 102 W (GeV) W (GeV)

γ* p => ρ p γ* p => ρ p 8 7

7 Q2 = 3.5 GeV2 6 Q2 = 11.0 GeV2 ) ) -2 -2 (GeV (GeV d d B B 6 5

5 4 102 102 W (GeV) W (GeV)

Figure 4.23. Dependence of the slope parameter BD versus W for J/ψ and ρ production. Data is from H1 experiment [164,168].

The dependence of BD on the energy W illustrates the diffusion in impact parameter space of the interaction region. Other models, such as [131], have found that BD is nearly independent of W . The error bars from experiment are quite large and while this does not preclude BD being independent of W the data indicates a growing trend. With the inclusion of impact parameter dependence in the solution for the dipole scattering amplitude we find that the slope of the data is reproduced, as can be seen in Fig 4.23. The slope of BD in W is determined by the cutoff m in the splitting kernel of the BK equation. The intercept is determined by BG, which is the initial distribution in impact parameter space for the dipole scattering 1 amplitude. It is useful to note that in this work BG = 2m2 , but while the impact parameter distribution and the dipole size cutoff are related they need not be directly related in this manner. It is key to emphasize that this slope comes from 152

Dipole Size: 1.00 | cos(θ): 0.0 | ∆ Y: 4.0 | max Y: 8.0 0.6

0.4 N(y)

0.2

0.0 -1 0 1 10 10 10 Impact Parameter (GeV-1)

Figure 4.24. A graph comparing the impact parameter profiles of the model used in [131] (dashed line) versus the model presented in this dissertation (solid line). It can be seen that our model shows a diffusion in impact parameter space which is not present in [131], which drives the growth of the slope of BD. the inherent inclusion of impact parameter dependence in our model, which allows us to describe the behavior of the interaction region in this manner. This can be seen in Fig 4.24 which compares the impact parameter profile in our model to the model presented in [131]. Evolving with impact parameter dependence allows the solution to diffuse in impact parameter space, leading to the slope seen in BD.

The value of m (and thus BG) was set in order to bring the normalization of BD for J/Ψ into agreement with the data. The normalization for ρ is low compared to the data, but the trends of the data are well reproduced. It is possible to change the m parameter to bring the ρ calculation in BD into agreement with measurement but this would bring J/Ψ out of agreement. This again suggests that a correction that is speciﬁc to ρ is needed. Chapter 5

Conclusion and Outlook

Whiskey today, space tomorrow

Ross Martin-Wells

In this dissertation the subject of dipole scattering at small Bjorken x has been the focus, specifically in DIS processes. The purpose of this chapter is to give a summary of the various conclusions drawn throughout the other four chapters. The first two chapters were introduction and background material which cul- minated in the derivation of the BFKL equation (following Mueller’s method [47]) and the BK equation (following Kovchegov’s method [86]). The nonlinear BK equation was the focus of the rest of this dissertation. The BK equation has no analytic solution and must be solved numerically. This has been done by many other groups [90–92,94,102] and the properties of the BK equation are well known. However, this has usually been done by assuming that the dipole scattering amplitude is only a function of the dipole size. This neglects the dependence on impact parameter, which greatly simplifies the numerical solution, but at the same time removes many important dynamical effects from the solution. In chapter 3 the solution of the BK evolution equation with (and without) impact parameter dependence was presented. Several novel features were seen seen in the solution with impact parameter dependence which are summarized here:

The scattering amplitude at large dipole sizes goes to zero. This corresponds • to a large dipole missing the target, as in Fig 3.7. 154

There is an enhancement of the scattering amplitude when the dipole size • is twice the impact parameter. This corresponds to a geometric eﬀect (Fig 3.9), which was described in Fig 3.10

Without inclusion of any dependence in the initial condition on the angle be- • tween the dipole size and the impact parameter, a dependence on this angle is dynamically generated by the evolution equation. This is seen most promi- nently when the dipole size is twice the impact parameter (Figs 3.11,3.9) and in the dependence on cos(θ) of the scattering amplitude (Fig 3.12).

A second evolution front in the large dipole size region was created. This • introduced a contribution from the infra-red regime which must be regulated before the solution can be applied to any physical observables.

Power-like tails in impact parameter space were found to exist, which drive • the violation of the Froissart bound when the total cross section is calculated. These must be tamed by the introduction of a mass scale which models conﬁnement eﬀects.

Higher order kinematical effects were incorporated in the form of a modified splitting kernel (3.34). It was found that these effects were small without impact parameter dependence but with the inclusion of impact parameter dependence the kinematic corrections significantly slowed the evolution. This is directly related to the presence of a second evolution front in the large dipole size regime which does not exist without impact parameter dependence. The inclusion of the running coupling was done by consideration of the Balitsky prescription (3.51) as well as considering the minimum dipole prescription (3.55). The Balitsky prescription was found to be very similar to the minimum dipole prescription, which is expected. The second wavefront in the infra-red regime, which is unique to the solution with impact parameter dependence, forces the issue of regularization of the running coupling. Without impact parameter this region was regulated by the saturation scale. The method of regulating the coupling was found to have a dramatic effect on the behavior of the Balitsky kernel, showing an inherent instability even deep in the small-dipole region far away from the infra- red region. This behavior is related to the fact that in the functional form of 155 the Balitsky prescription there are terms proportional to the inverse of the strong coupling αs. This effect is present even in the case without impact parameter dependence and needs more investigation to fully understand it. In chapter 4 we utilized the solution with impact parameter dependence to compute observables in DIS and compare them to data from HERA. Before any physical observables could be computed the large non-perturbative dipoles had to be regulated by the inclusion of confinement effects. It was found that by application of several theta functions to the branching kernel the behavior of the infra-red regime was most consistent with the data. The inclusion of confinement effects introduce a mass parameter m into the evolution which is the scale at which dipoles of size r 1 and above are cut. ≈ m The solution to the BK evolution equation computed with the Balitsky prescription for running coupling and the theta function cutoffs on the kernel was used to calculate both inclusive and exclusive DIS data from HERA. With the inclusion of a non-perturbative contribution the structure functions F2 and FL were well described by this calculation. The slope of the calculation is steeper than the data for F2 suggests, which is due to the rapid evolution of the BK equation. The FL data has large error bars and was not very discriminatory. Applying the solution to exclusive diffractive vector meson production at HERA for the ρ,φ and J/Ψ mesons good agreement was found with the data. The ρ meson was consistently low in normalization, but the trends of the data were still well described. Of particular interest is the energy dependence of the slope of the differential cross section. This slope describes the distribution of the interaction region in impact parameter space. It was found in our model that this slope is not constant with energy, which is a trend seen in the HERA data (Fig 4.23). The solution with impact parameter naturally produces a dependence of this slope on the energy that is not present in other models. There are further applications of the solution of the BK equation with impact parameter to phenomenology, such as to hadronic and heavy ion collision. Addi- tionally there are corrections to the BK equation which are large. Kinematical and energy conservation effects in the same spirit as (3.34) allow the calculation to go beyond the LL. Additionally it would be interesting to investigate the solution to the BK equation using the NLL kernel and compare to the experimental data. Appendix A

The Mellin transform

The Mellin transform is similar to a Laplace transform only with logarithmic variables. Let us take the Laplace transform

f(s)= F (µ)e−sµdµ (A.1) Z and make the change of variables s ln(x/x′). We have introduced some dimen- →− sionful constant x′ such that the argument of the logarithm remains dimensionless. This changes our transform into the form

′ s f(x) = F (µ)eln(x/x ) dµ (A.2) Z = f(µ)(x/x′)sdµ (A.3) Z which is precisely the Mellin transform. The inverse Mellin transform is given by

1 x x −1−µ F (µ)= d f(x). (A.4) 2πi x′ x′ Z This transformation is useful for the terms in the transform are also the eigenfunctions of the BFKL kernel. Appendix B

Interpolation

The interpolation procedure for the numerical analysis is an important step in the procedure. The method described here is merely for linear interpolation. Interpo- lation utilizing higher order (as high as third order) was used and the diﬀerence in the numerical result was found to be negligible. Non-local methods of interpolation (such as splines) were not used due to the large computational time.

N(x12)

N(x01)

log(x01)log(x12) log(x02) /log(x) log(xa) log(xb)

Figure B.1. A cartoon illustrating the need for interpolation where x12 is a value that is not on a grid point. The dotted line represents the interpolated value for the scattering amplitude and from this it is possible to extract the scattering amplitude N(x12). 158

By discretizing the degrees of freedom of the scattering amplitude and evaluating it only on some certain points (grid points) we introduce the possibility of requiring the value of the scattering amplitude at a point that is not a grid point. Let us consider a case without impact parameter dependence to illustrate. Con- sider Fig B.1, where there is a cartoon of the scattering amplitude computed on a one dimensional grid. The dashed lines represent the grid positions in dipole size and the black dots represent the scattering amplitude N at these positions. In this depiction x01 is on a grid point (which, by construction, x01,b01 and θ01 will always be on a grid point) and so is x02 (this is not necessarily true, and in general is not). The evaluation of (2.66) requires we know the scattering amplitude N(x12) as well, and in Fig B.1 we see that x12 is not on a grid point. The value for N(x12) is unknown as only at these grid points is the scattering amplitude evaluated. In order to extract the value of N(x12) we ﬁnd a line between the two closest values for the scattering amplitude and use the equation of this line to extract the scattering amplitude from. Let us assign the values xa and xb to the two points surrounding the area of interest (labeled by a dashed circle). The extracted value of N(x12) from the dotted line of the interpolated data is

N(x ) N(x ) N(x )= N(x ) + (log(x ) log(x )) b − a . (B.1) 12 a 12 − a log(x ) log(x ) b − a Note that here we are working in the logarithmic variables of dipole size because it is in logarithmic variables that the grid spacing is uniform. This can be recast in a form which is slightly more complicated but will be more helpful in the full three-dimensional interpolation case

N(x )=(1 X(x ))N(x )+ X(x )N(x ). (B.2) 12 − 12 a 12 b where the function X(x12) is deﬁned by

log x log x X(x )= 12 − a . (B.3) 12 log x log x b − a This can be generalized to the three-dimensional case where we utilize similar notation in the b and θ dimensions as well with 159

log(b ) log(b ) B(b )= 12 − a , (B.4) 12 log(b ) log(b ) b − a and cos θ cos θ Φ(θ )= 12 − a . (B.5) 12 cos θ cos θ b − 12 We use cos(θ) here as the grid is uniform in cos(θ), not θ. In the same method as (B.2) it is possible to arrive at an interpolation formula in three dimensions:

N(x ) = (1 X(x ))(1 B(x ))(1 φ(x ))N(x ,b ,θ ) 12 − 12 − 12 − 12 a a a + (1 X(x ))(1 B(x ))φ(x )N(x ,b ,θ ) − 12 − 12 12 a a b + (1 X(x ))B(x )(1 φ(x ))N(x ,b ,θ ) − 12 12 − 12 a b a + (1 X(x ))B(x )φ(x )N(x ,b ,θ ) − 12 12 12 a b b + X(x )(1 B(x ))(1 φ(x ))N(x ,b ,θ ) 12 − 12 − 12 b a a + X(x )(1 B(x ))φ(x )N(x ,b ,θ ) 12 − 12 12 b a b + X(x )B(x )(1 φ(x ))N(x ,b ,θ ) 12 12 − 12 b b a + X(x12)B(x12)φ(x12)N(xb,bb,θb). (B.6)

It is important to note that when programming the interpolation procedure it must be rugged for points near the grid boundaries. Care should also be taken as the variables in the interpolation procedure are logarithmic variables as the grid is uniform only in logarithmic variables. Bibliography

[1] Pinto, O. (2012) Variation in blood pressure in people with dementia who wander and non-wandering populations., Ph.D. thesis, State University of New York at Binghamton.

[2] Martin-Wells, R. (2012) The mapping of resonant structures of ultracold cesium collisions in a juggling atomic fountain, Ph.D. thesis, The Pennsyl- vania State University.

[3] Martin-Wells, K. (2012) Radar polarization properties and lunar sec- ondary cratering, Ph.D. thesis, Cornell University.

[4] Duran Urrea, E. (2012) A community-based study of social, prosodic, and syntactic factors in code-switching, Ph.D. thesis, The Pennsylvania State University.

[5] Liang, J. (2012) Hybrid Magnetic/semiconductor NWs: synthesis and characterization, Ph.D. thesis, The Pennsylvania State University.

[6] Bongiorno, S. (2012) Characterization and Applications of Hybrid CMOS Detectors in X-ray Astronomy, Ph.D. thesis, The Pennsylvania State Uni- versity.

[7] Wickman, J. (2012) Inﬂation and Primordial Gravity Waves in the Era of Precision Cosmology, Ph.D. thesis, University of Deleware.

[8] Abelleira Fernandez, J. et al. (2012) “A Large Hadron Electron Col- lider at CERN: Report on the Physics and Design Concepts for Machine and Detector,” 1206.2913.

[9] Boer, D., M. Diehl, R. Milner, R. Venugopalan, W. Vogelsang, et al. (2011) “Gluons and the quark sea at high energies: Distributions, polarization, tomography,” 1108.1713. 161

[10] Miller, G., E. D. Bloom, G. Buschhorn, D. Coward, H. DeStae- bler, et al. (1972) “Inelastic electron-Proton Scattering at Large Momen- tum Transfers,” Phys.Rev., D5, p. 528.

[11] Benvenuti, A. et al. (1989) “A High Statistics Measurement of the Pro- ton Structure Functions F(2) (x, Q**2) and R from Deep Inelastic Muon Scattering at High Q**2,” Phys.Lett., B223, p. 485.

[12] Ahmed, T. et al. (1995) “A Measurement of the proton structure function f2 (x, Q**2),” Nucl.Phys., B439, pp. 471–502, hep-ex/9503001.

[13] Derrick, M. et al. (1995) “Measurement of the proton structure function F2 from the 1993 HERA data,” Z.Phys., C65, pp. 379–398.

[14] Feynman, R. P. (1969) “Very high-energy collisions of hadrons,” Phys.Rev.Lett., 23, pp. 1415–1417.

[15] Bjorken, J. (1969) “Asymptotic Sum Rules at Inﬁnite Momentum,” Phys.Rev., 179, pp. 1547–1553.

[16] Callan, J., Curtis G. and D. J. Gross (1969) “High-energy electroproduction and the constitution of the electric current,” Phys.Rev.Lett., 22, pp. 156–159.

[17] Halzen, F. and A. D. Martin (1984) “Quarks and Leptons : An Intro- ductory Course in Modern Particle Physics,” .

[18] Amaudruz, P. et al. (1992) “Proton and deuteron f2 structure functions in deep inelastic muon scattering,” Phys.Lett., B295, pp. 159–168.

[19] Adams, M. et al. (1996) “Proton and deuteron structure functions in muon scattering at 470-GeV,” Phys.Rev., D54, pp. 3006–3056.

[20] Targett-Adams, C. (2005) “Structure functions and parton distribution functions at the HERA ep collider,” hep-ex/0507024.

[21] Collins, J. C., D. E. Soper, and G. F. Sterman (1988) “Factorization of Hard Processes in QCD,” Adv.Ser.Direct.High Energy Phys., 5, pp. 1– 91, to be publ. in ’Perturbative QCD’ (A.H. Mueller, ed.) (World Scientiﬁc Publ., 1989), hep-ph/0409313.

[22] Collins, J. (2011) “Foundations of perturbative QCD,” .

[23] Lipatov, L. (1975) “The parton model and perturbation theory,” Sov.J.Nucl.Phys., 20, pp. 94–102. 162

[24] Gribov, V. and L. Lipatov (1972) “Deep inelastic e p scattering in perturbation theory,” Sov.J.Nucl.Phys., 15, pp. 438–450.

[25] Altarelli, G. and G. Parisi (1977) “Asymptotic Freedom in Parton Lan- guage,” Nucl.Phys., B126, p. 298.

[26] Dokshitzer, Y. L. (1977) “Calculation of the Structure Functions for Deep Inelastic Scattering and e+ e- Annihilation by Perturbation Theory in Quan- tum Chromodynamics.” Sov.Phys.JETP, 46, pp. 641–653.

[27] Cooper-Sarkar, A. (2011) “PDF Fits at HERA,” 1112.2107.

[28] Watt, G. (2012) “MSTW PDFs and impact of PDFs on cross sections at Tevatron and LHC,” 1201.1295.

[29] Lai, H.-L. et al. (2010) “New parton distributions for collider physics,” Phys. Rev., D82, p. 074024, 1007.2241.

[30] Ball, R. D., V. Bertone, F. Cerutti, L. Del Debbio, S. Forte, et al. (2011) “Impact of Heavy Quark Masses on Parton Distributions and LHC Phenomenology,” Nucl.Phys., B849, pp. 296–363, 1101.1300.

[31] Regge, T. (1959) “Introduction to complex orbital momenta,” Nuovo Cim., 14, p. 951.

[32] Collins, P. (1977) “An Introduction to Regge Theory and High-Energy Physics,” .

[33] Beringer, J. et al. (2012) “Review of Particle Physics (RPP),” Phys.Rev., D86, p. 010001.

[34] Pomeranchuk, I. I. (1956) Sov. Phys. JETP, 3, p. 306.

[35] Okun, L. B. and I. I. Pomeranchuk (1956) Sov. Phys. JETP, 3, p. 307.

[36] Donnachie, A. and P. Landshoff (1984) “p p and anti-p p Elastic Scat- tering,” Nucl.Phys., B231, p. 189.

[37] Block, M. and F. Halzen (2005) “New evidence for the saturation of the Froissart bound,” Phys.Rev., D72, p. 036006, 19 pages, 8 ﬁgures, hep-ph/0506031.

[38] Block, M., E. Gregores, F. Halzen, and G. Pancheri (1999) “Pho- ton - proton and photon-photon scattering from nucleon-nucleon forward amplitudes,” Phys.Rev., D60, p. 054024, 37 pages, Latex2.09, uses eps- ﬁg.sty, 27 postscript ﬁgures Report-no: MADPH-98-1041, NUHEP 501, hep-ph/9809403. 163

[39] Kaidalov, A. and Y. Simonov (2000) “Glueball masses and Pomeron trajectory in nonperturbative QCD approach,” Phys.Lett., B477, pp. 163– 170, hep-ph/9912434.

[40] Froissart, M. (1961) “Asymptotic behavior and subtractions in the Man- delstam representation,” Phys.Rev., 123, pp. 1053–1057.

[41] Stasto, A. M. (2004) “Nonlinear evolution equations in QCD,” Acta Phys.Polon., B35, pp. 3069–3102, hep-ph/0412084.

[42] Martin, A. (1963) “Unitarity and high-energy behavior of scattering amplitudes,” Phys.Rev., 129, pp. 1432–1436.

[43] Gribov, V. (1968) “A REGGEON DIAGRAM TECHNIQUE,” Sov.Phys.JETP, 26, pp. 414–422.

[44] Nussinov, S. (1975) “Colored Quark Version of Some Hadronic Puzzles,” Phys.Rev.Lett., 34, pp. 1286–1289.

[45] Low, F. (1975) “A Model of the Bare Pomeron,” Phys.Rev., D12, pp. 163– 173.

[46] Forshaw, J. R. and D. Ross (1997) “Quantum chromodynamics and the pomeron,” Cambridge Lect.Notes Phys., 9, pp. 1–248.

[47] Mueller, A. H. (1994) “Soft gluons in the inﬁnite momentum wave function and the BFKL pomeron,” Nucl. Phys., B415, pp. 373–385.

[48] Mueller, A. H. and B. Patel (1994) “Single and double BFKL pomeron exchange and a dipole picture of high-energy hard processes,” Nucl.Phys., B425, pp. 471–488, hep-ph/9403256.

[49] ’t Hooft, G. (1971) “Renormalization of Massless Yang-Mills Fields,” Nucl.Phys., B33, pp. 173–199.

[50] Ioffe, e., Boris Lazarevich, e. Fadin, Victor Sergeevich, and e. Lipatov, Lev Nikolaevich (2010) “Quantum chromodynamics: Per- turbative and nonperturbative aspects,” .

[51] Mueller, A. H. (1999) “Small x physics, high parton densities and parton saturation in QCD,” , pp. 180–209hep-ph/9911289.

[52] Chen, Z. and A. H. Mueller (1995) “The Dipole picture of high- energy scattering, the BFKL equation and many gluon compound states,” Nucl.Phys., B451, pp. 579–604. 164

[53] Kopeliovich, B., L. Lapidus, and A. Zamolodchikov (1981) “DY- NAMICS OF COLOR IN HADRON DIFFRACTION ON NUCLEI,” JETP Lett., 33, pp. 595–597.

[54] Bertsch, G., S. J. Brodsky, A. Goldhaber, and J. Gunion (1981) “Diﬀractive Excitation in QCD,” Phys.Rev.Lett., 47, p. 297.

[55] Nemchik, J., N. N. Nikolaev, and B. Zakharov (1994) “Color transparency after the NE18 and E665 experiments: Outlook and perspectives at CEBAF,” nucl-th/9406005.

[56] Fadin, V. S., E. A. Kuraev, and L. N. Lipatov (1975) “On the Pomer- anchuk Singularity in Asymptotically Free Theories,” Phys. Lett., B60, pp. 50–52.

[57] Balitsky, I. I. and L. N. Lipatov (1978) “The Pomeranchuk Singularity in Quantum Chromodynamics,” Sov. J. Nucl. Phys., 28, pp. 822–829.

[58] Kuraev, E., L. Lipatov, and V. S. Fadin (1976) “Multi - Reggeon Pro- cesses in the Yang-Mills Theory,” Sov.Phys.JETP, 44, pp. 443–450.

[59] Kuraev, E., L. Lipatov, and V. S. Fadin (1977) “The Pomeranchuk Singularity in Nonabelian Gauge Theories,” Sov.Phys.JETP, 45, pp. 199– 204.

[60] Golec-Biernat, K. J. and A. M. Stasto (2003) “On solutions of the Balitsky-Kovchegov equation with impact parameter,” Nucl. Phys., B668, pp. 345–363, hep-ph/0306279.

[61] Lipatov, L. (1997) “Small x physics in perturbative QCD,” Phys.Rept., 286, pp. 131–198, hep-ph/9610276.

[62] Gubser, S. S. (2011) “Conformal symmetry and the Balitsky-Kovchegov equation,” 1102.4040.

[63] Lipatov, L. N. (1986) “The Bare Pomeron in Quantum Chromodynamics,” Sov. Phys. JETP, 63, pp. 904–912.

[64] Balitsky, I. and G. A. Chirilli (2009) “Conformal kernel for NLO BFKL equation in N = 4 SYM,” Phys.Rev., D79, p. 031502, 0812.3416.

[65] Arfken, G. B. and H. J. Weber (2005) “Mathematical Methods for Physicists Sixth Edition,” .

[66] Navelet, H. and S. Wallon (1998) “Onium onium scattering at ﬁxed impact parameter: Exact equivalence between the color dipole model and the BFKL pomeron,” Nucl. Phys., B522, pp. 237–281, hep-ph/9705296. 165

[67] Abramowicz, H. and A. Caldwell (1999) “HERA collider physics,” Rev.Mod.Phys., 71, pp. 1275–1410, hep-ex/9903037.

[68] Golec-Biernat, K. J. and M. Wusthoff (1999) “Saturation in diﬀrac- tive deep inelastic scattering,” Phys.Rev., D60, p. 114023, hep-ph/9903358.

[69] Gotsman, E., E. Levin, and U. Maor (1997) “Diffractive leptoproduction of small masses in QCD,” Nucl.Phys., B493, pp. 354–396, 37 pages, LaTeX, 16 figures in seperate ps gzip file Report-no: TAUP 2338/96 : CBPF 021/96, hep-ph/9606280.

[70] Bartels, J. and M. Wusthoff (1996) “A Perturbative approach to diﬀractive deep inelastic scattering,” J.Phys.G, G22, pp. 929–936.

[71] Bartels, J., J. R. Ellis, H. Kowalski, and M. Wusthoff (1999) “An Analysis of diﬀraction in deep inelastic scattering,” Eur.Phys.J., C7, pp. 443–458, hep-ph/9803497.

[72] Gribov, L., E. Levin, and M. Ryskin (1983) “Semihard Processes in QCD,” Phys.Rept., 100, pp. 1–150.

[73] Gelis, F., E. Iancu, J. Jalilian-Marian, and R. Venugopalan (2010) “The Color Glass Condensate,” Ann.Rev.Nucl.Part.Sci., 60, pp. 463–489, 1002.0333.

[74] McLerran, L. (2008) “A Brief Introduction to the Color Glass Condensate and the Glasma,” , pp. 3–180812.4989.

[75] Iancu, E., A. Leonidov, and L. McLerran (2002) “The Color glass condensate: An Introduction,” , pp. 73–145hep-ph/0202270.

[76] Venugopalan, R. (2005) “The Color glass condensate: An Overview,” Eur.Phys.J., C43, pp. 337–344, hep-ph/0502190.

[77] Jalilian-Marian, J., A. Kovner, L. D. McLerran, and H. Weigert (1997) “The Intrinsic glue distribution at very small x,” Phys.Rev., D55, pp. 5414–5428, hep-ph/9606337.

[78] Jalilian-Marian, J., A. Kovner, and H. Weigert (1999) “The Wilson renormalization group for low x physics: Gluon evolution at ﬁnite parton density,” Phys. Rev., D59, p. 014015, hep-ph/9709432.

[79] Jalilian-Marian, J., A. Kovner, A. Leonidov, and H. Weigert (1999) “The Wilson renormalization group for low x physics: Towards the high density regime,” Phys. Rev., D59, p. 014014, hep-ph/9706377. 166

[80] Weigert, H. (2002) “Unitarity at small Bjorken x,” Nucl. Phys., A703, pp. 823–860, hep-ph/0004044.

[81] Kovner, A., J. G. Milhano, and H. Weigert (2000) “Relating diﬀerent approaches to nonlinear QCD evolution at ﬁnite gluon density,” Phys.Rev., D62, p. 114005, hep-ph/0004014.

[82] Balitsky, I. (1996) “Operator expansion for high-energy scattering,” Nucl. Phys., B463, pp. 99–160, hep-ph/9509348.

[83] Rummukainen, K. and H. Weigert (2004) “Universal features of JIMWLK and BK evolution at small x,” Nucl.Phys., A739, pp. 183–226, hep-ph/0309306.

[84] Kovchegov, Y. V., J. Kuokkanen, K. Rummukainen, and H. Weigert (2009) “Subleading-N(c) corrections in non-linear small-x evolution,” Nucl.Phys., A823, pp. 47–82, 0812.3238.

[85] Barone, V. and E. Predazzi (2002) “High-energy particle diﬀraction,” .

[86] Kovchegov, Y. V. (1999) “Small-x F2 structure function of a nucleus including multiple pomeron exchanges,” Phys. Rev., D60, p. 034008, hep-ph/9901281.

[87] Kovchegov, Y. V. (2000) “Unitarization of the BFKL pomeron on a nucleus,” Phys. Rev., D61, p. 074018, hep-ph/9905214.

[88] Lublinsky, M. (2006) “High energy evolution with Pomeron loops,” hep-ph/0605025.

[89] Garnier, J. and A. Quetelet´ (1838) Correspondance math´ematique et physique, v. 10, Impr. d’H. Vandekerckhove. URL http://books.google.com/books?id=8GsEAAAAYAAJ

[90] Golec-Biernat, K. J., L. Motyka, and A. M. Stasto (2002) “Dif- fusion into infra-red and unitarization of the BFKL pomeron,” Phys. Rev., D65, p. 074037, hep-ph/0110325.

[91] Lublinsky, M., E. Gotsman, E. Levin, and U. Maor (2001) “Non- linear evolution and parton distributions at LHC and THERA energies,” Nucl. Phys., A696, pp. 851–869, hep-ph/0102321.

[92] Lublinsky, M. (2001) “Scaling phenomena from non-linear evolution in high energy DIS,” Eur. Phys. J., C21, pp. 513–519, hep-ph/0106112. 167

[93] Braun, M. (2000) “Structure function of the nucleus in the perturbative QCD with N(c) goes to inﬁnity (BFKL pomeron fan diagrams),” Eur.Phys.J., C16, pp. 337–347, hep-ph/0001268.

[94] Armesto, N. and M. A. Braun (2001) “Parton densities and dipole cross-sections at small x in large nuclei,” Eur. Phys. J., C20, pp. 517–522, hep-ph/0104038.

[95] Albacete, J. L., N. Armesto, A. Kovner, C. A. Salgado, and U. A. Wiedemann (2004) “Energy dependence of the Cronin eﬀect from nonlinear QCD evolution,” Phys.Rev.Lett., 92, p. 082001, hep-ph/0307179.

[96] Albacete, J. L., N. Armesto, J. G. Milhano, C. A. Salgado, and U. A. Wiedemann (2005) “Numerical analysis of the Balitsky-Kovchegov equation with running coupling: Dependence of the saturation scale on nuclear size and rapidity,” Phys. Rev., D71, p. 014003, hep-ph/0408216.

[97] Levin, E. and K. Tuchin (2001) “New scaling at high-energy DIS,” Nucl.Phys., A691, pp. 779–790, hep-ph/0012167.

[98] Levin, E. and K. Tuchin (2000) “Solution to the evolution equation for high parton density QCD,” Nucl.Phys., B573, pp. 833–852, hep-ph/9908317.

[99] Levin, E. and K. Tuchin (2001) “Nonlinear evolution and saturation for heavy nuclei in DIS,” Nucl.Phys., A693, pp. 787–798, hep-ph/0101275.

[100] Munier, S. and R. B. Peschanski (2004) “Traveling wave fronts and the transition to saturation,” Phys. Rev., D69, p. 034008, hep-ph/0310357.

[101] Bartels, J., V. Fadin, and L. Lipatov (2004) “Solution of the fan diagram equation in (2+1)-dimensional QCD,” Nucl.Phys., B698, pp. 255–276, hep-ph/0406193.

[102] Braun, M. A. (2001) “High-energy interaction with the nucleus in the perturbative QCD with N ,” hep-ph/0101070. c → ∞ [103] Mueller, A. H. and D. N. Triantafyllopoulos (2002) “The energy dependence of the saturation momentum,” Nucl. Phys., B640, pp. 331–350, hep-ph/0205167.

[104] Bartels, J. and K. Kutak (2008) “A momentum Space Analysis of the Triple Pomeron Vertex in pQCD,” Eur. Phys. J., C53, pp. 533–548, 0710.3060. 168

[105] Berger, J. and A. Stasto (2011) “Numerical solution of the nonlinear evolution equation at small x with impact parameter and beyond the LL approximation,” Phys. Rev., D83, p. 034015, 1010.0671. [106] Mueller, A. H. (1999) “Parton saturation at small x and in large nuclei,” Nucl. Phys., B558, pp. 285–303, hep-ph/9904404. [107] (2012), “The Message Passing Interface (MPI) standard,” . URL http://www.mcs.anl.gov/research/projects/mpi/ [108] Press, W., S. Teukolsky, W. Vetterling, and B. Flannery (2007) Numerical Recipes 3rd Edition: The Art of Scientific Computing, Cambridge University Press. URL http://books.google.com/books?id=1aAOdzK3FegC [109] Marquet, C. and G. Soyez (2005) “The Balitsky-Kovchegov equation in full momentum space,” Nucl. Phys., A760, pp. 208–222, hep-ph/0504080. [110] Armesto, N., C. A. Salgado, and U. A. Wiedemann (2005) “Relat- ing high-energy lepton-hadron, proton-nucleus and nucleus-nucleus collisions through geometric scaling,” Phys.Rev.Lett., 94, p. 022002, hep-ph/0407018. [111] Freund, A., K. Rummukainen, H. Weigert, and A. Schafer (2003) “Geometric scaling in inclusive eA reactions and nonlinear perturbative QCD,” Phys.Rev.Lett., 90, p. 222002, hep-ph/0210139. [112] Motyka, L. and A. M. Stasto (2009) “Exact kinematics in the small x evolution of the color dipole and gluon cascade,” Phys. Rev., D79, p. 085016, 0901.4949. [113] Weinberg, S. (1966) “Dynamics at infinite momentum,” Phys.Rev., 150, pp. 1313–1318. [114] Bjorken, J., J. B. Kogut, and D. E. Soper (1971) “Quantum Elec- trodynamics at Infinite Momentum: Scattering from an External Field,” Phys.Rev., D3, p. 1382. [115] Lepage, G. P. and S. J. Brodsky (1980) “Exclusive Processes in Pertur- bative Quantum Chromodynamics,” Phys.Rev., D22, p. 2157. [116] Brodsky, S. J., H.-C. Pauli, and S. S. Pinsky (1998) “Quantum chromodynamics and other field theories on the light cone,” Phys.Rept., 301, pp. 299–486, hep-ph/9705477. [117] Kovchegov, Y. V. and H. Weigert (2007) “Triumvirate of running couplings in small-x evolution,” Nucl. Phys., A784, pp. 188–226, hep-ph/0609090. 169

[118] Balitsky, I. (2007) “Quark contribution to the small-x evolution of color dipole,” Phys. Rev., D75, p. 014001, hep-ph/0609105.

[119] Albacete, J. L. and Y. V. Kovchegov (2007) “Solving High Energy Evolution Equation Including Running Coupling Corrections,” Phys. Rev., D75, p. 125021, 0704.0612.

[120] Kovner, A. and U. A. Wiedemann (2003) “No Froissart bound from gluon saturation,” Phys. Lett., B551, pp. 311–316, hep-ph/0207335.

[121] Kovner, A. and U. A. Wiedemann (2002) “Nonlinear QCD evolution: Saturation without unitarization,” Phys. Rev., D66, p. 051502, hep-ph/0112140.

[122] Kovner, A. and U. A. Wiedemann (2002) “Perturbative saturation and the soft pomeron,” Phys.Rev., D66, p. 034031, hep-ph/0204277.

[123] Ferreiro, E., E. Iancu, K. Itakura, and L. McLerran (2002) “Froissart bound from gluon saturation,” Nucl.Phys., A710, pp. 373–414, hep-ph/0206241.

[124] Dudal, D., O. Oliveira, and N. Vandersickel (2010) “Indirect lattice evidence for the Reﬁned Gribov-Zwanziger formalism and the gluon condensate A2 in the Landau gauge,” Phys.Rev., D81, p. 074505, 1002.2374.

[125] Oliveira, O. and P. Bicudo (2011) “Running Gluon Mass from Landau Gauge Lattice QCD Propagator,” J.Phys.G, G38, p. 045003, 1002.4151.

[126] Nikolaev, N. N. and B. Zakharov (1994) “The Triple pomeron regime and the structure function of the pomeron in the diﬀractive deep inelastic scattering at very small x,” Z.Phys., C64, pp. 631–652, hep-ph/9306230.

[127] Nikolaev, N. N., B. Zakharov, and V. Zoller (1994) “The s channel approach to Lipatov’s pomeron and hadronic cross-sections,” JETP Lett., 59, pp. 6–12, hep-ph/9312268.

[128] Flensburg, C., G. Gustafson, and L. Lonnblad (2009) “Elastic and quasi-elastic pp and γ∗p scattering in the Dipole Model,” Eur.Phys.J., C60, pp. 233–247, 0807.0325.

[129] Avsar, E., G. Gustafson, and L. Lonnblad (2007) “Diifractive excitation in DIS and pp collisions,” JHEP, 0712, p. 012, 0709.1368.

[130] Berger, J. and A. M. Stasto (2011) “Small x nonlinear evolution with impact parameter and the structure function data,” * Temporary entry *, 1106.5740. 170

[131] Kowalski, H., L. Motyka, and G. Watt (2006) “Exclusive diﬀractive processes at HERA within the dipole picture,” Phys.Rev., D74, p. 074016, hep-ph/0606272.

[132] Golec-Biernat, K. and M. Wusthoff (1999) “Saturation eﬀects in deep inelastic scattering at low Q**2 and its implications on diﬀraction,” Phys. Rev., D59, p. 014017, hep-ph/9807513.

[133] Ewerz, C., A. von Manteuffel, and O. Nachtmann (2011) “On the Energy Dependence of the Dipole-Proton Cross Section in Deep Inelastic Scattering,” JHEP, 1103, p. 062, 1101.0288.

[134] Kwiecinski, J. and B. M. Badelek (1989) “Analysis of the electroproduction structure functions in the low Q**2 region combining the vector dominance and the parton model with possible scaling violation,” Z.Phys., C43, p. 251.

[135] Martin, A. D., M. Ryskin, and A. Stasto (1999) “The Description of F2 at low Q**2,” Eur.Phys.J., C7, pp. 643–654, hep-ph/9806212.

[136] Brodsky, S. J. and J. Pumplin (1969) “Photon-Nucleus Total Cross- Sections,” Phys.Rev., 182, pp. 1794–1804.

[137] Gribov, V. (1970) “Interaction of gamma quanta and electrons with nuclei at high-energies,” Sov.Phys.JETP, 30, pp. 709–717.

[138] Ritson, D. (1971) “Phenomenological predictions for deep inelastic electron scattering,” Phys.Rev., D3, p. 1267.

[139] Nakamura, K. et al. (2010) “Review of particle physics,” J.Phys.G, G37, p. 075021.

[140] Donnachie, A. and P. Landshoff (1992) “Total cross-sections,” Phys.Lett., B296, pp. 227–232, hep-ph/9209205.

[141] Albacete, J. L., N. Armesto, J. G. Milhano, and C. A. Salgado (2009) “Non-linear QCD meets data: A Global analysis of lepton-proton scattering with running coupling BK evolution,” Phys.Rev., D80, p. 034031, 0902.1112.

[142] Albacete, J., N. Armesto, J. Milhano, P. Arias, and C. Salgado (2010) “AAMQS: A non-linear QCD analysis of new HERA data at small-x including heavy quarks,” 1012.4408.

[143] Balitsky, I. and G. A. Chirilli (2008) “Next-to-leading order evolution of color dipoles,” Phys. Rev., D77, p. 014019, 0710.4330. 171

[144] Avsar, E., A. M. Stasto, D. N. Triantafyllopoulos, and D. Za- slavsky (2011) “Next-to-leading and resummed BFKL evolution with saturation boundary,” 1107.1252.

[145] Berger, J. and A. M. Stasto (2012) “Exclusive vector meson production and small-x evolution,” 1205.2037.

[146] Levy, A. (2007) “Exclusive vector meson electroproduction at HERA,” , pp. 2–90711.0737.

[147] Bunyatyan, A. (2008) “Exclusive vector mesons and DVCS at HERA,” Nucl.Phys.Proc.Suppl., 179-180, pp. 69–77.

[148] Frankfurt, L., M. Strikman, and C. Weiss (2011) “Transverse nucleon structure and diagnostics of hard parton-parton processes at LHC,” Phys.Rev., D83, p. 054012, 1009.2559.

[149] Nemchik, J., N. N. Nikolaev, and B. Zakharov (1994) “Scanning the BFKL pomeron in elastic production of vector mesons at HERA,” Phys.Lett., B341, pp. 228–237, hep-ph/9405355.

[150] Frankfurt, L., W. Koepf, and M. Strikman (1996) “Hard diﬀractive electroproduction of vector mesons in QCD,” Phys.Rev., D54, pp. 3194– 3215, hep-ph/9509311.

[151] Nemchik, J., N. N. Nikolaev, E. Predazzi, and B. Zakharov (1997) “Color dipole phenomenology of diﬀractive electroproduction of light vector mesons at HERA,” Z.Phys., C75, pp. 71–87, hep-ph/9605231.

[152] Munier, S., A. Stasto, and A. H. Mueller (2001) “Impact parameter dependent S matrix for dipole proton scattering from diﬀractive meson electroproduction,” Nucl.Phys., B603, pp. 427–445, hep-ph/0102291.

[153] Forshaw, J. R., R. Sandapen, and G. Shaw (2004) “Color dipoles and rho, phi electroproduction,” Phys.Rev., D69, p. 094013, hep-ph/0312172.

[154] Avsar, E., G. Gustafson, and L. Lonnblad (2005) “Energy conservation and saturation in small-x evolution,” JHEP, 07, p. 062, hep-ph/0503181.

[155] Avsar, E., G. Gustafson, and L. Lonnblad (2007) “Small-x dipole evolution beyond the large-N(c) limit,” JHEP, 01, p. 012, hep-ph/0610157.

[156] Avsar, E. (2007) “On the Dipole Swing and the Search for Frame Indepen- dence in the Dipole Model,” JHEP, 11, p. 027, 0709.1371. 172

[157] Dosch, H. G., T. Gousset, G. Kulzinger, and H. Pirner (1997) “Vector meson leptoproduction and nonperturbative gluon ﬂuctuations in QCD,” Phys.Rev., D55, pp. 2602–2615, hep-ph/9608203.

[158] Kulzinger, G., H. G. Dosch, and H. Pirner (1999) “Diﬀractive pho- toproduction and leptoproduction of vector mesons rho, rho-prime and rho- prime-prime,” Eur.Phys.J., C7, pp. 73–86, hep-ph/9806352.

[159] Brodsky, S. J., L. Frankfurt, J. Gunion, A. H. Mueller, and M. Strikman (1994) “Diﬀractive leptoproduction of vector mesons in QCD,” Phys.Rev., D50, pp. 3134–3144, hep-ph/9402283.

[160] de Teramond, G. F. and S. J. Brodsky (2005) “Hadronic spectrum of a holographic dual of QCD,” Phys.Rev.Lett., 94, p. 201601, hep-th/0501022.

[161] Bartels, J., K. J. Golec-Biernat, and K. Peters (2003) “On the dipole picture in the nonforward direction,” Acta Phys.Polon., B34, pp. 3051–3068, hep-ph/0301192.

[162] Kowalski, H. and D. Teaney (2003) “An Impact parameter dipole saturation model,” Phys.Rev., D68, p. 114005, hep-ph/0304189.

[163] Adloff, C. et al. (2000) “Elastic electroproduction of rho mesons at HERA,” Eur.Phys.J., C13, pp. 371–396, hep-ex/9902019.

[164] Chekanov, S. et al. (2007) “Exclusive rho0 production in deep inelastic scattering at HERA,” PMC Phys., A1, p. 6, 0708.1478.

[165] Aaron, F. et al. (2010) “Diﬀractive Electroproduction of rho and phi Mesons at HERA,” JHEP, 1005, p. 032, arXiv:0910.5831.

[166] Chekanov, S. et al. (2005) “Exclusive electroproduction of phi mesons at HERA,” Nucl.Phys., B718, pp. 3–31, hep-ex/0504010.

[167] Chekanov, S. et al. (2004) “Exclusive electroproduction of J/psi mesons at HERA,” Nucl.Phys., B695, pp. 3–37, hep-ex/0404008.

[168] Aktas, A. et al. (2006) “Elastic J/psi production at HERA,” Eur.Phys.J., C46, pp. 585–603, hep-ex/0510016.

[169] Martin, A. D., M. Ryskin, and T. Teubner (2000) “Q**2 dependence of diﬀractive vector meson electroproduction,” Phys.Rev., D62, p. 014022, hep-ph/9912551. Vita Jeﬀrey J. Berger

EDUCATION Pennsylvania State University, Ph.D Physics, 2012 SUNY Binghamton, B.S. Physics with Honors, B.A. Mathematics, 2007

EXPERIENCE Research Assistant Penn State University Summer 2009 - June 2012 Teaching Assistant Pennsylvania State University Fall 2007 - Fall 2009 Research Assistant Binghamton University. Summer 2005 - Summer 2007 Teaching Assistant Binghamton University Spring 2003 - Summer 2007 JOURNAL ARTICLES Jeffrey Berger, Anna Stasto, Sub. JHEP May 2012 — ARXIV 1205.2037v1 • Jeffrey Berger, Anna Stasto, Physical Review D 84:094022 2011 — ARXIV • 1106.5740v3 Jeffrey Berger, Anna Stasto, Physical Review D 83:034015 2011 — ARXIV • 1010.0671v1 Charles A. Nelson, Jeffrey J. Berger, Joshua R. Wickman, Euro Phys J C • 46:385-402 2006 — ARXIV hep-ph/0510348v4 Charles A. Nelson, Eric Gasparo Barbagiovanni, Jeffrey J. Berger, Elisa K. • Pueschel, Joshua R. Wickman, Euro Phys J C 45:121-138 2006 — ARXIV hep-ph/0506240v4

PROCEEDINGS Deep Inelastic Scattering 2012, To be published • QCD Evolution Workshop: from collinear to non collinear case, To be pub- • lished by World Scientiﬁc Deep Inelastic Scattering 2011, To be published by the American Institute • of Physics TALKS Deep Inelastic Scattering 2012 • QCD Evolution Workshop: from collinear to non collinear case 2011 • Deep Inelastic Scattering 2011 • Syracuse Unviersity 2007 • University of Rochester 2007 • Houghton College 2006 •