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The Pennsylvania State University The Graduate School 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 At high energy and for dense targets the BFKL equation is superseded by the Balitsky-Kovchegov (BK) evolution equation which includes important effects which are non-linear in the parton density.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 param- eter 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 run- ning 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 produc- tion 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.