A Study of Quasi-Real Photon Structure at a Future Electron Ion Collider

A Dissertation for Doctor of Philosophy in Physics

A study of quasi-real photon structure at a future electron ion collider

by Xiaoxuan Chu

Supervisor: Prof. Xu Cai, Dr. E.C. Aschenauer, Dr. J.H. Lee, Prof. Zhongbao Yin Specialty: Experimental Physics Research Area: Strong-Interaction Matter (Quark and Nuclear Matter) Physics

College of Physical Science and Technology Central China Normal University June 2018 Abstract

A photon is a fundamental ingredient of our present understanding of the interactions of quarks and leptons. Being the gauge boson of the theory of quantum electrodynamics (QED), the photon mediates the electromagnetic force between charged objects as a massless particle. While in the framework of quantum chromodynamics (QCD), the quanta themselves can develop a structure. Due to the Heisenberg uncertainty principle written as ∆E∆t > 1, the high energy photon can have an energy ﬂuctuation by an amount of ∆E for a short period of time ∆t. During such a ﬂuctuation, the photon is allowed to split into a charged fermion-antifermion pair, ff¯ , carrying the same quantum numbers as the photon. Based on this, the photon naturally becomes an important object for research on particle physics.

According to QCD, the photon is a superposition of a bare photon state which interacts only with electric charges, and a hadronic photon state which is called the resolved photon. If probed experimentally at very short distances, the intrinsic structure of the resolved photon is recognized as a ﬂux of quark and gluon components, quasi-free according to asymptotic freedom in QCD. The interaction between the resolved photon and other particles is therefore interpreted as an incoherent superposition of scatterings on the fundamental constituent quarks and gluons, leading us to a universal parton distribution function (PDF) describing the quark and gluon densities of the photon. The PDFs of the photon are functions of the momentum fraction, x, carried by the partons within the photon and the scale, Q2, at which the densities are probed.

A future electron ion collider (EIC), named eRHIC at Brookhaven National Labora- tory, will be able to provide collisions of polarized electrons with protons and heavy ions over a wide range of center-of-mass energies (20 GeV to 140 GeV) at an instantaneous luminosity of 1033 −1034 cm−2s−1. Measuring di-jets in quasi-real photoproduction events, one can eﬀectively access the underlying parton dynamics of the photons. In this thesis, we present a di-jet method applied on the EIC to investigate the structure of photon experimentally. We discuss ﬁrstly how jets are reconstructed at the EIC. The basic jet

i kinematics, background due to underlying events and quark/gluon jet discrimination are studied as well. Secondly we discuss the possibility for the photon to interact either directly or in a resolved manner. We provide an expermental method to tag resolved photon processes and measure the di-jet cross section as a function of jet transverse mo-

rec mentum in the range of 0.01 < xγ < 1 at the EIC. It will be shown that both unpolarized and polarized parton distributions in the photon can be extracted, and that the ﬂavor of the parton can be tagged at the EIC.

Keywords: electron ion collider(EIC), quantum chromodynamics (QCD), resolved photon, di-jet, quark jet, gluon jet, ﬂavor tagging.

ii Table of Contents

1 Introduction ...... 1

2 Theoretical framework ...... 9

2.1 Fundamentals of QCD ...... 9

2.1.1 Parton model ...... 9

2.1.2 The running coupling constant ...... 10

2.1.3 Asymptotic freedom and conﬁnement ...... 11

2.2 DIS in QCD framework ...... 12

2.2.1 Basic kinematics in DIS ...... 12

2.2.2 Parton distributions ...... 18

2.2.3 DGLAP evolution equations ...... 20

2.2.4 Initial and ﬁnal state radiation ...... 21

2.2.5 Beam remnants and multiple interactions ...... 22

2.2.6 Hadronization ...... 23

2.3 Jet ...... 24

2.3.1 Jet evolution ...... 24

2.3.2 Jet algorithms ...... 26

2.4 Parton Distribution Function in the photon ...... 28

3 Possible realizations of an EIC ...... 31

3.1 The eRHIC design concept ...... 31

3.1.1 An eRHIC model detector ...... 33

3.1.2 Low-Q2 tagger ...... 37

3.2 The designing requirements for a future EIC ...... 37 4 Experimental aspect of jet physics at EIC ...... 39

4.1 Jet ﬁnding and jet properties ...... 40

4.1.1 Input and reference frames ...... 40

4.1.2 Jet deﬁnitions ...... 41

4.1.3 Jet kinematics ...... 46

4.2 Detector requirements ...... 47

4.3 Underlying event ...... 48

4.4 Quark jet and gluon jet ...... 56

4.5 Summary ...... 60

5 Photon structure at EIC ...... 61

5.1 Monte Carlo set up ...... 61

5.2 Validation with HERA data ...... 63

5.3 Unpolarized photon structure at EIC ...... 66

5.3.1 Separation between direct and resolved processes ...... 66

5.3.2 Unpolarized di-jet cross section ...... 67

5.3.3 Flavor tagging ...... 68

5.4 Polarized photon structure at EIC ...... 73

5.4.1 Polarized di-jet cross section at EIC ...... 75

5.4.2 Asymmetry ...... 75

5.4.3 Advantages compared with HERA ...... 78

5.5 Summary ...... 80

6 Summary and outlook ...... 81

References ...... 82

List of presentations and publications ...... 90 Acknowledgment ...... 92 List of Figures

1.1 The leading order diagrams corresponding to the six main event classes

in the process γγ → hadrons ...... 3 1.2 Di-jet cross sections for the photon-proton scattering from HERA . . . .5

1.3 Examples of diagrams for direct (left) and resolved (right) processes in electron-proton scattering...... 6

1.4 Leading order eﬀective parton distribution function of the photon from H17

1.5 Measurements of the gluon distribution function of the photon from H1 .7

2.1 αs versus Q ...... 11

2.2 The schematic diagram of a DIS process ...... 13

2.3 Combined proton structure function F2 distribution from diﬀerent experiments ...... 16

0 1 2.4 Feynman diagrams for the DIS processes at O(αs) and O(αs) ...... 17 2.5 Feynman diagrams of virtual corrections to LO DIS ...... 18

2.6 The parton distributions in a proton extracted from the combined H1 and ZEUS data by the HERAPDF program ...... 19

2.7 Feynman diagrams corresponding to diﬀerent splitting functions . . . . . 20

2.8 The parton shower with hadronization model in e+e− collisions ...... 22

2.9 Gluon emission ...... 25

2.10 u, d, s and g distribution of photon depends on the momentum fraction x 30

3.1 A layout of the eRHIC collider design ...... 32

3.2 A layout of JLEIC collider design ...... 32

3.3 A schematic view of the tracking system in the eRHIC model detector design ...... 34

3.4 Low Q2-tagger ...... 36 4.1 Jet proﬁle for anti-kT , kT and SISCone algorithms ...... 42

4.2 Jet multiplicity and particle multiplicity inside the jet for anti-kT , kT and SISCone algorithms ...... 43

4.3 Radius comparison multiplicity ...... 44

4.4 Di-jet mass versuss ˆ ...... 45

4.5 Jet pT spectrum in diﬀerent frames, combined diﬀerent subprocesses . . . 46

4.6 Jet η spectrum in diﬀerent frames, combined diﬀerent subprocesses . . . . 47

4.7 Detector requirements for jet η and hadron ...... 48

4.8 Illustration of underlying events ...... 49

4.9 Illustration of “toward”, “away” and “transverse” regions in azimuthal angle ∆φ relative to the direction of the trigger jet in the event ...... 50

4.10 Average number of charged stable particles as a function of ∆φ ...... 51

4.11 Average scalar pT sum of charged stable particles as a function of ∆φ .. 51

4.12 Average number of charged stable particles as a function of ∆φ ...... 52

4.13 The average number and the average scalar pT sum of charged particles in “toward”, “transverse” and “away” regions ...... 53

4.14 The illustration of two oﬀ-axis cones relative to a jet ...... 53

4.15 The number of charged particles density and the average scalar pT sum density ...... 55

4.16 Comparison with the results from STAR ...... 55

4.17 How to match parton with jet ...... 56

4.18 Quark jet and gluon jet kinematics ...... 58

4.19 Quark jet and gluon jet discrimination ...... 59

4.20 Quark jet and gluon jet discrimination ...... 60

γp 2 5.1 The σ (xBj,Q ) simulated with PYTHIA-6 using CTEQ5m and SAS 1D-LO as proton and photon PDFs, respectively ...... 62 5.2 Comparison of the di-jet cross section extracted from the PYTHIA simulation with the HERA data ...... 64

gen rec 5.3 Correlation between xγ and xγ ...... 65

rec 5.4 xγ distributions in resolved and direct processes ...... 65

5.5 The unpolarized di-jet cross section dependence on the average transverse momentum of the jets ...... 67

5.6 Subprocess fraction ...... 69

5.7 Separation of jets from the photon side and proton side ...... 69

5.8 Flavor tagging ...... 70

5.9 Flavor tagging ...... 71

5.10 The input photon PDFs ...... 73

5.11 The input proton PDFs ...... 75

5.12 The measured di-jet cross section in polarized ep collision as a function of the squared jet transverse momentum for the range of the reconstructed parton fractional momentum ...... 76

5.13 ALL ...... 77

5.14 The di-jet cross section at EIC and HERA ...... 78

5.15 The quark (left) and gluon (right) distribution functions of the photon for three diﬀerent sets ...... 79 List of Tables

2.1 Six types of quarks ...... 10 CHAPTER 1

Introduction

The seek of knowledge and determination to explore our universe have been an important factor in the evolution of the human beings. Till today, we have overcome innumerable difficulties to try to understand this world better. Particle physicists play an crucial role in the effort to propose description of the nature of matter, trying to answer some fundamental questions, like: “What are the fundamental components of matter? How can they be defined or observed from the experiments?” In 1969, Richard Feynman proposed the parton model pointing out that hadrons, such as protons, are made up of some point-like constituents termed “partons”. Later, it was proved with experimental observation of Bjorken scaling, partons were matched to quarks and gluons. QCD is the theory of the strong interactions between quarks and gluons.

According to QCD, the photon is a superposition of a bare photon state which interacts only with electric charges, and a hadronic photon state. The ﬁrst idea that energy can be emitted and absorbed only in discrete portions came from Planck and was presented in 1901 in his successful theory describing the energy spectrum of black body radiation [1]. Soon after Planck made his heuristic assumption of abstract ele- ments of energy, Einstein proposed that light can be considered as a ﬂux of particles in 1905 [2]. Many further experiments, beginning with the phenomenon of Compton scattering, validated Einstein’s hypothesis that light itself is quantized. In 1926 the optical physicist Frithiof Wolfers and the chemist G. N. Lewis referred to these particles as the notion of “photon”. Over the last century we have witnessed tremendous progress in our understanding of photons. In QED, the photon mediates the electromagnetic force between charged objects. As the gauge boson of QED, the photon is considered to be a massless and chargeless particle [3] having no internal structure. QED also incorporates

1 the electron, which was the first elementary particle correctly identified as such. The understanding of reactions involving these two particles spawned the theory of gauge interactions, now thought to describe all observed (electroweak, strong and gravitational) interactions. In spite of this long, distinguished history, there is one large class of photonic interactions about which only relatively little is known. In any quantum field theory, the existence of interactions implies that the quanta themselves can develop a structure.

If experimentally probed at very short distances, the intrinsic structure of the photon is recognized as a ﬂux of quark and gluon components, quasi-free according to asymptotic freedom in QCD and described by the photon structure functions [4,5,6]. For example, the photon can ﬂuctuate for a short period of time into a charged fermion-antifermion pair, ff¯, carrying the same quantum numbers as the photon. The lifetime of this

ﬂuctuation increases with the energy of the parent photon (Eγ) and decreases with the

2 2 square of the invariant mass of the pair (Mpair: ∆t ≈ 2Eγ~Mpair). As a result, the photons will interact with hadrons (or other real photons) via two quite different ways. The photon, as a whole, can couple directly to a quark in the struck hadron (direct process). Alternatively, the photon can undergo a transition into a (virtual) hadronic state before encountering the target hadron (resolved process). In this case a quark or gluon “in” the photon can react, via strong interactions, with partons in the struck hadron. Then we can refer to the photon structure, which is a consequence of quantum fluctuations of the field theory.

On the experimental side, in the past decades we have seen large progress on the constraint of photon QCD structure with data obtained from the LEP experiments [7,8] and HERA experiments [9, 10]. Photon structure functions are traditionally measured in e+e− collisions, such as the experiments at LEP, at which two photon physics is one of the most active ﬁelds of the research. The classical way to investigate the structure of the photon at e+e− colliders is to measure the following process:

+ − + − ⋆ ⋆ + − e e → e e γ γ → e e X, (1.1) proceeding via the interaction of two photons, which can be either quasi-real (γ) or

2 Figure 1.1: (a) direct direct; (b) direct point-like; (c) point-like point-like; (d) direct

VMD; (e) VMD VMD;× (f) point-like VMD.× × × × × virtual(γ⋆) emitted by a deeply inelastically scattered electron, where X represents a pair of leptons or a hadronic ﬁnal state.

The six main event classes in two photon physics are listed below and schematically shown in Fig 1.1:

● direct × direct - the photons directly produce a quark pair,

● direct × point-like - the point-like photon splits into a qq¯ pair and one of them (or a daughter thereof) interacts directly with the other photon,

● point-like × point-like - both photons perturbatively split into qq¯ pairs, and subsequently one parton from each photon takes part in hard scattering and produce a quark pair,

● direct × Vector Meson Dominance (VMD) - a direct photon interacts with the partons of VMD photon,

● VMD × VMD - both photons turn into hadrons ﬁrstly and then interact like hadron-hadron scattering,

● point-like × VMD - the point-like photon perturbatively splits into a qq¯ pair and

3 and one of these (or a daughter parton thereof) interacts with a parton from VMD photon.

The structure of the quasi-real photon has been studied at LEP in terms of total cross sections [11], jet production [12], and heavy quark production [13] in the case where none of the scattered beam electrons are observed in detector (anti-tagged). In the case where only one electron is observed (single-tagged), the process can be described as deep inelastic electron scattering off a quasi-real photon. Measuring QED and QCD photon structure functions as well as QCD structure functions of the electron were studied through these class of events. If both electrons are observed (double-tagged), the dynamics of highly virtual photon collisions is probed. The QED and QCD structure of the interactions of two highly virtual photons has also been studied at LEP in terms γ of the effective structure functions of the virtual photon F2 and total cross sections. In the collider HERA at DESY, 820 or 920 GeV protons collided with 27.5 GeV electrons or positrons, with two general purpose detectors, H1 and ZEUS, positioned at opposite interaction regions. The high flux of almost on-shell photons which accompanied the lepton beam also provide a unique opportunity to study the nature of the photon and its interactions. Unlike in eγ scattering, the photon structure is probed by the partons from the proton in the so-called photoproduction events in ep collisions. By tagging high transverse energy (ET ) jets [14, 15, 16, 17, 18, 19, 20, 21], high-pT charged particles [22] or heavy quarks [23] in photoproduction reactions, Parton Distribution Functions (PDFs) of the photon can be constrained. The interaction of electrons and protons at low virtuality is dominated by quasi-real photoproduction processes where the electrons scatter at small angles. Such reactions proceed via two classes of processes, the so-called “resolved” and “direct” processes. Examples of Feynman diagrams of these two processes are shown in Fig. 1.3.

The measured di-jet cross sections as a function of average transverse energy squared of the jets in bins of xγ is presented from Ref. [24]. Jets are found using kT -clustering jets algorithm. The resolved process is selected by the cut xγ < 0.75. The di-jet cross section is corrected for detector eﬀects only, and compared to the predictions of the leading order

4 Figure 1.2: The data are shown as a function of the average transverse energy squared of the jets for several bins in xγ. The data are compared to the leading order prediction from the PYTHIA generator (dash), and to analytical next-to-leading calculations using the GRV (full) and the GS (dot) parton distribution functions of the photon. The plot is from [24].

5 e e γ γ q q q q q q

P P

(a) (b)

Figure 1.3: Examples of diagrams for direct (left) and resolved (right) processes in electron-proton scattering.

PYTHIA Monte Carlo and to the next-to-leading order parton level predictions using the GRV [85] and the GS [83] parton distribution functions of the photon as shown in Fig. 1.2. This cross section is then used to determine an eﬀective parton distribution function of the photon. The evolution of the extracted leading order eﬀective parton

2 distribution function xγfγ~α as a function of the factorisation scalep ˆT is shown in

Fig. 1.4 taken from Ref. [24] for two regions of xγ, 0.2 < xγ < 0.4 and 0.4 < xγ < 0.7. The data are compared with three predictions based on the GRV parametrisation of the parton distribution functions of the photon.

The results for parton distributions as a function of xγ are shown in Fig. 1.5. The two results, obtained from single particles and jets respectively, are consistent and the gluon distribution function is found to be large at small values of xγ and to decrease towards large values of xγ. The measured leading order gluon distribution function from the left ﬁgure of Fig. 1.5 is consistent with the existing parametrisation from SAS [64] and GRV by analyzing di-jet events, and GRV describes the gluon distributions best by analyzing high pT tracks. In the comparison, LAC set doesn’t describe the gluon distributions of the photon very well, especially at small xγ region. From both ﬁgures, we can see that the measured parton distribution functions are still with high uncertainties. Thus it is desirable to measure them with high precision facility.

We will demonstrate that in this thesis, at a future EIC [26] such as eRHIC at

6 2 Figure 1.4: The data are shown as a function of the factorisation scalep ˆT , averaged over xγ in the ranges, left: 0.2 xγ 0.4 and right: 0.4 xγ 0.7. The inner error bar indicates the statistical and the outer< error< bar the full error.< The data< are compared to several theoretical predictions explained in the text. The plot is from [24].

Figure 1.5: The leading order gluon distribution function as a function of xγ, as obtained from 2 2 the charged particle cross section is shown, for an average factorization scale Q pT 38 2 (GeV/c) , together with the measurements obtained from di-jet cross sections.a Thef = innera f error= bar indicates the statistical and the outer error bar the full error. The plot is from [22] and [25].

7 Brookhaven National Laboratory (BNL), it is feasible to do a high precision extraction

−1 of photon PDFs with an integrated luminosity of L = 1 fb . More importantly, the EIC also allows to study the polarized photon PDFs, as both the electron and proton beam can be polarized.

In this thesis, we discuss ﬁrstly the general framework of QCD formulated to understand the successive experimental results in Chpt.2, paving us the way to the understanding of the fundamental constituents for our universe, secondly we brieﬂy describe the framework of jet physics and photon structure study. An introduction of the EIC project and its realization will be described in Chpt.3. Based on the current EIC con- ceptual design, we will discuss our simulation studies of the jets in Chpt.4 and the measurement of di-jet cross sections in quasi-real photoproduction events in (un)polarized ep collisions in Chpt.5. In the end, we will give our summery in Chpt.6.

8 CHAPTER 2

Theoretical framework

2.1 Fundamentals of QCD

The genius of the theory which we now call quantum chromodynamics (QCD) was the result of the development of many theoretical ideas and experimental results. QCD is the theory of strong interaction [27]. The QCD surrogate of electric charge respect to quantum electrodynamics (QED) is a property called color. The heart of this theory is the evidence for the color degree of freedom. The fundamental content of QCD is that hadronic matter is made of quarks. Gluons are the force carrier of the theory, like photons are for the electromagnetic force in QED.

2.1.1 Parton model

The idea of quarks arose from the need to have a physical manifestation for the SU(3) of ﬂavor observed in the spectrum of the lowest-mass mesons and baryons. The parton model was proposed by Richard Feynman in 1960s [28] as a way to analyze high-energy hadron collisions, which indicates any hadron can be considered as a composition of a group of point-like constituents, termed “partons”. For example, a very naive picture is that the proton is made up of two “up” quarks and one “down” quark and neutron is made up of two “down” quarks and one “up” quark. The concept of “gluon” is introduced later.

The interactions of hadrons are due to the interactions of partons which are referred to quarks and gluons is the basic assumption of parton model. The quarks, which are supposed to be fermions, carrying the fractional charge. They are forced to have

9 Table 2.1: Six types of quarks Quark Mass Charge Baryon Number Isospin

2 1 1 u ∼ 4 MeV + 3 3 + 2 1 1 1 d ∼ 7 MeV − 3 3 − 2 2 1 c ∼ 1.5 GeV + 3 3 0 1 1 s ∼ 135 MeV − 3 3 0 2 1 t ∼ 175 GeV + 3 3 0 1 1 b ∼ 5 GeV − 3 3 0 half-intergral spin in order to account for the spins of the lowest-mass baryons. The properties of the six known quarks are listed in Table 2.1. The gluon, which is gauge boson, is the mediator of strong force. The theory postulates the color degree of freedom with three possible values, red, green and blue, and correspondingly the anti-colors. The gluons couple to all particles carrying color charges. Since the gluons themselves carry color charges, it is possible to have gluon self-interactions. The crucial outcome of QCD framework is that it provides the feature of color conﬁnement at long distances keeping color charges binded in the hadrons and asymptotic freedom (detailed in Sec. 2.1.3) while allowing to have essentially free quark interactions at short distances.

2.1.2 The running coupling constant

The strength of the strong interactions is described by the QCD coupling constant αs. The dependence of a coupling on the energy scale is known as running of the coupling, which is given by the renormalization group, it is deﬁned as a β function

∂α Q2 s β α , (2.1) ∂Q2 = ( s)

In QCD, the β function has the perturbative expansion,

2 ′ ′′ 2 β(αs) = −bαs(1 + b αs + b αs + ...), (2.2) where, 33 2n b ( − f ), (2.3) = 12π 10 Figure 2.1: Measurments of αs versus Q. The plot is from the complication of the particle Data Group (PDG) [30]. nf is the number of active light ﬂavors. This leads to the following form of the strong coupling scale dependency 12π α , (2.4) s = Q2 (11nc − 2nf )ln( 2 ) ΛQCD where an arbitrary scale Q0 = ΛQCD ≈ 220 MeV is introduced typically for QCD. nc is the number of color degrees of freedom. The divergence of αs is predicted in Eq. 2.4. As we see from Fig. 2.1, this coupling is predicted to be small at large Q2. Since most of the QCD problems are hard or impossible to solve directly, a perturbation theory is often used in these occasions [29]. ΛQCD here represents an infrared cut-oﬀ for the perturbation theory. Therefore, the perturbation theory is valid only in a region where

αs ≪ 1, which corresponds to Q ≫ ΛQCD.

2.1.3 Asymptotic freedom and conﬁnement

Asymptotic freedom is known as the phenomenon that the coupling constant decreases logarithmically. This property can be described that interactions between particles become asymptotically weaker as the energy scale increases and the corresponding 11 length scale decreases, which means quarks and gluons are allowed to act like free particles at very small distance. In fact, there is no simple intuitive explanation of property of asymptotic freedom in QCD. This is in contrast to QED [31, 32], where we assume the vacuum is consisted of virtual electron-positron pair fluctuations, in the vicinity of an electron charge, the polarization of these virtual fluctuations in vacuum becomes important and partially cancels out the net charge sitting at the center. It would be good to be able to extend this argument to QCD. The new feature in QCD is the presence of the gluons. There are two types of arguments which aim to give a simple explanation of asymptotic freedom [33, 34, 35, 36, 37], describing the phenomenon either as a dielectric or paramagnetic effect.

2.2 DIS in QCD framework

2.2.1 Basic kinematics in DIS

As we know, there is no way for us to detect any isolated fractional electric charge experimentally because of the color confinement, we can not directly observe an elementary constituent either, like partons. However, we learnt that the constituent information can be extracted by scattering a structureless lepton off a structured hadron from the famous Rutherford experiment method. This type of study is firstly performed at the experiment from Stanford Linear Accelerator Center (SLAC), with the target proton being scattered by 20 GeV incoming electron beams [38].

The ep collisions provide interactions between electron beams and proton beams through the exchange of a virtual photon between them. The process can be illustrated by Fig. 2.2 with the notation of e(k) + p(P ) → e(k′) + X, where X represents the ﬁnal hadronic systems. Considering the scattering of a high energy charged lepton oﬀ a hadron target, we label the incoming and outgoing lepton four-momenta by k and k′ respectively. The momentum of the target hadron (assumed hereafter to be a proton) is represented by P . The proton mass is denoted by Mp. The process can be mediated by a virtual vector boson (here it’s a virtual photon), with four-momentum given by

12 Figure 2.2: Schematic diagram of DIS scattering via virtual photon exchange. q = k − k′. The kinematics description remains the same for the exchange of a Z or W boson at high momentum transfer.

Depending on diﬀerent physics situations, the DIS process can be discussed in diﬀer- ent reference frames corresponding which physics aspect you are looking for:

1. the collider frame, where a proton with energy Ep and an electron with energy Ee collide head-on.

2. the rest frame of the hadronic system X, i.e. the center-of-mass of the γp collisions.

3. the rest frame of the proton.

If not speciﬁed, all the plots in this thesis follow this beam direction convention: the hadron beam goes to +z and the positive rapidities are often referred to as “forward” direction, while the electron beam goes to −z, corresponding to negative rapidities. The selection of Lorentz invariant variables describing the process in Fig. 2.2 is only a matter of convention, but the following set of the kinematics variables is commonly used:

1. The squared center-of-mass (CM) energy

2 s = (k + P ) = 4EeEp. (2.5)

2. The magnitude of momentum transfer mediated by the virtual boson

2 2 ′ 2 Q = −q = −(k − k ) , (2.6)

equal to the virtuality of the exchanged photon. 13 3. Bjorken scaling variable Q2 xBj = , (2.7) 2P · q which means the momentum fraction of struck quark taken from the proton.

4. The inelasticity P · q y = , (2.8) P · k giving fraction of the electron energy transfered to the hadronic system. In the

′ 2 collider frame, the energy of the scattered electron is Ee = Ee(1 − y) − (Q ~(4Ee).

5. The energy current transfered from lepton in the target rest frame

P · q ν = . (2.9) Mp

6. The invariant mass of the ﬁnal state hadronic system

2 2 W = (P + q) . (2.10)

2 2 The introduced variables can be related by Q = (s − Mp )xBjy, ν = ys~(2Mp) and W 2 M 2 Q2 1−xBj . Here, ν is the energy lost by the lepton in the proton rest frame, W = p + xBj is the mass of the hadronic system. Through all above equations in this section, there are only two variables are independent in the process at ﬁxed center-of-mass energy. Additionally, the kinematic variable x, which is the momentum fraction of the parton involved in the hard interaction, is equal to xBj if we focus on the leading order (LO)

DIS process. For higher oder DIS process, x is larger than xBj as shown in the following discussions.

In DIS, the resolution power of the exchanged virtual photon is deﬁned by Q2 and

2 xBj. With large momentum transfer Q , small objects are allowed to be resolved. Their 1 transverse momenta is less than Q and localized within a transverse area ∆r ∼ Q . When xBj is small, the proton ﬂuctuation modes with shorter life time can be probed with a

2xBj P 2 2 time resolution ∆t ∼ Q2 . DIS regime is often referred as the regime of Q ≫ Mp and 2 2 2 W ≫ Mp . Therefore, the proton mass term can be ignored in the relations for Q and W 2 in DIS collisions.

14 When Q2 is much smaller than the mass of Z0 or W boson, the DIS process can be written in the one photon exchange assumption. Focusing on the LO DIS, the cross section for neutral current DIS on unpolarized nucleons can be expressed as (neglecting electroweak eﬀects):

d2σ 2πα2 1 1 y 2 F x, Q2 y2F x, Q2 , (2.11) dxdQ2 = xQ4 [( + ( − ) ) 2( ) − L( )] where α represents the ﬁne structure constant. F2 and FL are commonly parameterized frame invariant structure functions for protons. F2 describes the sum over the electromagnetic contribution. FL is the longitudinal structure function, it is deﬁned as

2 2 2 FL(x, Q ) = F2(x, Q ) − 2xF1(x, Q ). F1 is the structure function describing the pure magnetic part of the interaction compared with F2. For longitudinally polarized proton and electron beams, the neutral current cross section for deep inelastic scattering can be written in terms of one structure function g1: 1 d2σ⇆ d2σ⇉ 4πα2 y 2 y g x, Q2 , (2.12) 2[dxdQ2 − dxdQ2 ] ≈ Q4 ( − ) 1( ) where the superscript arrows represent electron and proton longitudinal spin directions.

F2 is defined as: 2 2 2 F2(x, Q ) = Q ei xfi(x, Q ), (2.13) i=q,q¯ g1 corresponding to: 2 1 2 2 g1(x, Q ) = Q ei ∆fi(x, Q ), (2.14) 2 i=q,q¯ 2 where fi(x, Q ) is known as the unpolarized PDF of the proton, used to describe the 2 density of the partons in the proton. ∆fi(x, Q ) is the polarized PDF of the proton. Detailed information about PDFs is discussed in the following sections. Through the inclusive scattering where the final hadronic system doesn’t need to be analyzed, F2, FL and g1 can be practically measured experimentally. By applying a global fit method, the unknown initial PDF can be extracted. In DIS process, the kinematic variables x, Q2 and y can be reconstructed by measuring the outgoing electron.

2 As seen in Fig. 2.3, the structure function F2 shows no dependence on Q (when x is around 0.25), this phenomena is the so called Bjorken scaling. The quark parton model was introduced to explain the scaling behavior observed in the SLAC data. In 15 Figure 2.3: The combined proton structure function F2 data from HERA and ﬁxed target experiments versus Q2 in diﬀerent x bins. The plot is from the compilation of the PDG [30].

16 * * * γ γ q γ q

q g q q g

(a) (b) (c)

0 Figure 2.4: Feynman diagrams for the hard processes based on point-like photons: (a) αs 1 1 LO DIS, αs Photon-Gluon Fusion (PGF) (b) and αs QCD Compton scattering (QCDC)O( ) (c). O( ) O( )

2 this model, without the dependence on Q , the structure function F2 can be changed from the former equation into:

2 F2(x) = Q ei xfi(x), (2.15) i=q,q¯

Although the quark model has been proved to be very successfully describing a lot of experimental data, there are still some paradoxes remaining to be understood. Seen from Fig. 2.3, there is a clear breaking of the Bjorken scaling in the F2 data at large and small x. This phenomena cannot be incorporated in the quark parton model. Besides that, the study in momentum sum rule suggests that the momentum carried by quarks and antiquarks does not add up to the total momentum of protons. All of these facts are telling that there are some other important components in the proton except for quarks and antiquarks. These non-negligible important components are referred to the introduced gluons. In the following sections, we will explain how QCD theory of strong interactions explain the experimental data.

In perturbative QCD (pQCD) framework, the cross section can be calculated by expanding amplitudes in perturbation series of αs for a given physical quantity. The

LO DIS diagram is of zeroth order of αs corresponding to Fig. 2.4(a). The next leading order (NLO) level processes: Photon-Gluon Fusion (PGF) process (Fig. 2.4(b)) and QCD

1 Compton (QCDC) (Fig. 2.4(c)) process are corresponding to the order of O(αs).

17 Figure 2.5: Illustration of higher order virtual corrections to LO DIS.

2.2.2 Parton distributions

a The function fi (x) is the parton distribution function, describing the probability to ﬁnd a parton i inside beam particle a, with parton i carrying a fraction x of the total momentum. The PDF is assumed to be universal and can be used for diﬀerent target particles with various combinations in a wide range of physics processes. A demonstration of the extracted the proton PDF at HERA is shown in Fig. 2.6.

Although some progresses are being made in lattice QCD studies, since we do not fully understand QCD, a derivation from first principles of hadron parton distributions does not yet exist. The PDF fi is still not a priori calculable quantity and must be constrained by fits to experimental data [39]. It is therefore necessary to rely on parameterizations, where experimental data are used in conjunction with the evolution equations for the Q2 dependence. Therefore, several different groups have produced their own fits based on slightly different sets of data, and with some variations in the theoretical assumptions.

Parton distributions are most familiar for hadrons, such as the proton, which are inherently composite objects, made up of quarks and gluons. Meanwhile, they are also familiar for some fundamental particles, such as photon. The resolved photon can be regarded containing quarks and gluons. The parton distribution functions of the photon will be discussed in Sec. 2.4.

18 Figure 2.6: The parton distribution functions for u, d valance quark, sea quark (¯u d¯) and

2 2 gluons as a function of x at Q 10 GeV extracted from the combined H1 and ZEUS+ data. Note that gluon and sea quark distributions= are scaled by a factor of 20. The plot is from [39].

19 Figure 2.7: Illustration of diagrams corresponding to the splitting functions. The plot is from Ref. [46].

2.2.3 DGLAP evolution equations

Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) [43, 44, 45] evolution functions are widely used in global determinations of parton distributions, describing the scale dependence of Q2 for PDF. We’ve leart that one parton can radiate gluons, which will radiate quark-antiquark pairs in turn, the new partons travel collinear with their parent partons. DGLAP eveolution functions are used to discribe these processes. The DGLAP approach is in nature a perturbative treatment of splitting functions. The splitting function Pij(z) is used to describe the probability of a mother parton i splitting into a daughter partons j with momentum fraction z by emitting a parton k with fraction 1−z of the mother parton’s momentum, as demonstrated in Fig. 2.7.

The LO expression for splitting functions are given by

4 1 + z2 3 Pqq(z) = [ + δ(1 − z)], (2.16) 3 (1 − z)+ 2 1 P z z2 1 z 2 , (2.17) qg( ) = 2[ + ( − ) ] 4 1 1 z 2 P z + ( − ) , (2.18) gq( ) = 3[ z ]

20 1 − z z 33 − 2Nf Pgg(z) = 6[ + + z(1 − z)] + [ δ(1 − z)], (2.19) z (1 − z)+ 6 in which f z f z δ 1 z 1 f t dt. The DGLAP equation is of the form [ ( )]+ = ( ) − ( − ) ∫0 ( )

2 2 1 ′ ∂f x, Q α Q dx ′ i( ) s( ) f x ,Q2 P z , (2.20) 2 = Q S ′ i( ) ji( ) ∂ ln Q 2π i x x

x where z ′ . This equation gives a rigorous formalism for calculating the changes of = x 2 2 PDF as Q varies, the PDF shape and size at an initial scale Q0 has to come either from non-perturbative methods or parameterization in x with parameters determined by QCD ﬁts to data.

As mentioned before, the scaling violation phenomena of F2 on HERA data was observed. With the scheme of DGLAP evolution functions, this violations can be explained.

As illustrated in Fig. 2.3: in the large x (x > 0.4) region where valance quarks dominate, struck quarks with large momentum fraction are more likely to radiate gluons through the process of q → qg. When Q2 increases, more radiation is going to be resolved, the contribution to F2 from gluons will be shifted to small x. While in small x region where gluons or sea quarks dominate, higher Q2 promotes the process of a gluon splitting into

2 a pair of sea quarks g → qq¯, which explains the rise of F2 as Q increases.

2.2.4 Initial and ﬁnal state radiation

Every process which contains colored objects in the initial (before the hard scattering) or ﬁnal state (after the hard scattering) in QCD, gluon radiation as the available energies is going to increase. Hard emission of this kind is increasingly important, relative to fragmentation, in determining the event structure. To model this radiation, one method is the parton-shower method, the parton branched into two (or more) partons, the full matrix-element expressions are not used, but only approximations derived by simplifying the kinematics, the interference and helicity structure. The parton shower with a hadronization model (to be discussed in Sec. 2.2.6) in e+e− collision is shown in Fig. 2.8.

21 Figure 2.8: The parton shower with hadronization model in e+e− collision

2.2.5 Beam remnants and multiple interactions

Considering a hadron-hadron collision where not only a single parton-parton interaction occurs, the background due to underlying events can not be ignored. The initial- state radiation algorithm reconstructs one shower initiator in each beam. This initiator only takes some fraction of the total beam energy, leaving behind a beam remnant which takes the rest. For a proton beam, a u quark initiator would leave behind a ud diquark beam remnant, with an antitriplet color charge.

One would also expect the ep collisions have beam remnants, as a result some of the original energy doesn’t enter in the hard interaction. If parton distributions are used to resolve the electron and photon structure. Then both of the beam particles contain a multitude of partons, and so the probability for several interactions in one and the same event need not be negligible. In addition, it is likely that the partons from one beam remnant scatters against several diﬀerent partons from the other beam. These additional interactions called multiple parton interactions, along with the beam remnant will together contribute as the non-ignorable underlying events.

22 2.2.6 Hadronization

After the parton shower has terminated, we are left with a set of partons. The process that partons convert into the observed hadrons is called hadronization. The process of hadronization occurs after high-energy collisions in a particle collider where quarks or gluons are created. In the Standard Model these quarks or gluons combine with quarks and antiquarks spontaneously created from the vacuum to form hadrons. The QCD of the hadronization process are not yet fully understood, but are modeled and parameterized in a number of phenomenological studies, including the Lund string model and so on.

Here I will give a brief introduction of three diﬀerent models for the mechanism of hadronization.

1. Independent fragmentation: the simplest mechanism is to suppose that each parton fragments independently. The fragmenting quark is combined with an anti-quark from a qq¯ pair created out of the vacuum, to give the first-generation meson with energy fraction z. The leftover quark, with the energy fraction 1 − z, is fragmented in the same way, and so on until the leftover energy below some cut-off. Gluon is first spiting into a qq¯ pair, either assigning all the gluon’s momentum to one or the other (z = 0 or 1) with equal possibility, so the gluon behaves as a quark of random flavor. The weakness of independent fragmentation scheme is that the fragmentation of a parton depends on its energy rather that its virtuality. Indeed, the fragmenting parton is assumed to remain on mass-shell, leading to violations of momentum conservation that have to be corrected by rescaling momenta after hadronization is completed.

2. String model: the string model of hadronization is most easily described for e+e− annihilation. Neglecting for the moment the possibility of gluon bremsstrahlung, the produced quark and antiquark move out in opposite directions, losing energy to the color field, which is supposed to collapse into a stringlike configuration between them. The string has a uniform energy per unit length, corresponding to a linear quark confining potential, which is consistent with quarkonium spectroscopy. The string breaks upinto hadron-size pieces through spontaneous qq¯ pair production in its intense color

23 field. In practice, the string fragmentation approach does not look very different from independent fragmentation for the simple quark-antiquark system. The string may be broken up starting at either the quark or the antiquark end, or both simultaneously, and it proceeds iteratively by qq¯ pair creation, as in independent fragmentation. What one gains is a more consistent covariant picture, together with some constraints on the fragmentation function, to ensure independence of whether one starts at the quark or the antiquark, and on the transverse momentum distribution, which is now related to the tunneling mechanism by which qq¯ pairs are created in the color field of the string.

3. Cluster model: an important property of the parton branching process is the preconﬁnement of color. Preconﬁnement implies that the pairs of color-connected neighboring parton have an asymptotic mass distribution that falls rapidly at high masses and is asymptotically Q2-dependent and universal. This suggests a calss of cluster hadronization models, in which color-singlet clusters of partons form after the perturbative phase of jet development and then decay into observed hadrons. The simplest way for color- singlet clusters to form after parton branching is through non-perturbative splitting of gluons into qq¯ pairs. Neighboring quarks and antiquarks can then combine into singlets. The resulting cluster mass spectrum is again universal and steeply falling at high masses.

Its precise form is determined by the QCD scale Λ, the cut-oﬀ scale t0, and to a lesser extent the gluon-splitting mechanism. Typical cluster masses are normally two or three √ times of t0.

2.3 Jet

2.3.1 Jet evolution

We saw in the former sections that the color charged objects - quarks and gluons cannot be observed individually. A high energy ep collision can result in high momentum transfers between colliding partons, such scattering will have high transverse momentum and high virtuality. They will reduce their virtuality by emitting gluons and producing quark-antiquark pairs. And during the process of emission, it is with high possibility to

24 Figure 2.9: Two divergences in the process of emitting gluons. get a spray of collimated gluons with high momentum and low momentum ones at wide range seen from Fig. 2.9. After reaching the value of Q = 1 GeV the cascade stops and all quarks form colorless hadrons. Retaining the total momentum of the initial hadron, the whole cascade moves as a narrow spray of baryons and mesons until eventually reaching the detector and being measured. The ﬁnal spray of particles is called a jet.

Jet evolution can be divided into several logical steps, the inclusive jet cross section can be calculated as a convolution of 3 independent functions: PDF, hard scattering cross section and fragmentation function. PDF gives the initial condition of the jets. Hard scattering means a parton-parton scattering with high transferred momentum. Therefore such a process can be calculated in pQCD as ˆ dσ 1 2 Sa+b→c+d = SMS , (2.21) dtˆ 16πsˆ2 where all the variables are in the CMS frame, thus denoted with a hat, tˆ ands ˆ are Mandelstam variables, the last term is the scattering amplitude of the speciﬁc process.

From the experimental point of view, jets are defined as the output of the jet reconstruction algorithm. The jet cross section calculations depend on the algorithm used to find jets. The algorithm needs to be chosen carefully to avoid divergence in the cross section calculations. So different reconstruction algorithms should produce the same results at the parton level when applied on theoretical calculations, hadron level when applied on MC simulations and also at the detector level when up to the hilt considering the detector response collected during the experiments.

25 2.3.2 Jet algorithms

The collection of rules which determine how particles are grouped into jets is known as a jet algorithm, while the prescription for merging the momenta of individual particles to form the overall jet momentum is known as a recombination scheme. The combination of jet algorithm, recombination scheme, and any additional parameters controlling the behavior of the jet algorithm is known as a jet definition and as the name implies, fully defines a jet for purposes of an analysis. Several jet finding algorithms have been developed during the last two decades, each with their own behaviors and characteristics. In general, these algorithms are required to satisfy Collinear- and Infrared-Safe as the physics property of jets, which represents collinear splitting and soft emissions shouldn’t change jets. The determination of which algorithm to utilize will depend on requirements of the specific analysis being performed and in practice, finding the optimal algorithm can be a subtle and time-consuming task. There are a number of jet algorithms on the market, they can roughly be divided into two categories based on how particles are grouped: cone algorithm and kT algorithm. The cone algorithm is based on finding stable cones that encapsulate particles within certain area around their centroid. The centroid of a cone which has N particles is defined by,

i i c ∑i ET η η = c , (2.22) ET i i c ∑i ET φ φ = c , (2.23) ET c c ET = Q ET , (2.24) i

i i i where η , φ , and ET are the pseudo-rapidity, azimuthal angle and transverse energy of the i-th particle of the N particles, respectively. There are different types of cone algorithms, for example, Iterative cone algorithm and SISCone algorithm. These algorithms all require a stable cone finding first [48]. Iterative cone algorithm defines the most energetic in each event to be the seed, and then a cone of radius R is put around the seed, the summation of all the particles enveloped by cone is regarded as a trial jet. If identical within precision, this trial jet will be called a stable cone, and then a jet candidate is found; if not, we will proceed with the next energetic particle till no seeds 26 are above a certain threshold. While SISCone algorithm is a seedless Infrared-Safe cone algorithm. The stable cones are defined if there are at least two particles are on the top of the circle with radius being R. Then a jet is made of all the particles inside this circle.

Usually what we do in clustering algorithms is to select a starting particle and then sequentially add other particles that are close enough to the arising jet. In cone algorithms, the jet candidates have to be round. However, the formed jets in clustering algorithms have no fixed shape. The direction information of particles are not recorded in cone algorithms, on the contrary, clustering algorithms are going to reflect physical the way jets are originated. The kT algorithm tries to find jets on a list of pre-clusters which could be particles or partons. For each pre-cluster in the list, the energy E and momentum p⃗ are known. At first, we define the distance,

2 di = pT,i (2.25) and

∆R2 y y 2 φ φ 2 d min p2 , p2 ij min p2 , p2 ( i − j) + ( i − j) , (2.26) ij = ( T,i T,j) × R2 = ( T,i T,j) × R2 where pT,i(j), yi(j), and φi(j) are transverse momentum, rapidity and azimuthal angle of the i-th and j-th pre-cluster respectively, and R is the jet parameter. Then the we loop all the particles to calculate all the di and dij between two particles, then the algorithm

ﬁnds the minimum value between di and dij (dmin = min(di, dij)). If dmin is one of the dij, we combine the i-th and j-th pre-cluster together by Eij = Ei + Ej and p⃗ij = p⃗i + p⃗j, then replace them with the combined pre-cluster with Eij and p⃗ij and re-calculate the di and dij for the new list. Otherwise, if dmin is one of the di, we remove the i-th pre-cluster from the list as a jet found. The process continues until the pre-cluster list is empty. 2 1 1 1 ∆Rij The anti-kT algorithm deﬁnes di = 2 and dij = min( 2 , 2 ) × R2 , the next steps are pT,i pT,i pT,j the same as kT algorithm.

The major diﬀerence between the kT algorithm and anti-kT algorithm is the way they deal with the soft background. Without any soft background, suppose we only consider jets produced in hard scattering, we reconstruct jets and take this set of jets as J. Then we add the soft background into the process, the constituent particle inside 27 the jets, the shape of the jets, probably the number of the jets will all change compared with the former situation. The phenomena is so called back reaction. According to the

Ref. [94], the back reaction sensitivity is highly suppressed by using anti-kT algorithm in comparison with kT algorithm, especially when the jets are clustered from some hard particles coming from the hard scattering with some soft particles not coming from the hard scattering, the anti-kT algorithm is less susceptible to the diﬀusion of soft radiation and underlying events because those events tend have smaller pT . So the following analysis based on jet method are applied by anti-kT algorithm.

2.4 Parton Distribution Function in the photon

The parton distribution functions of the photon obey the following evolution equations:

γ dqi α αs γ γ γ γ γ = Pq ⊗ Γ + Σ[Pqiqk ⊗ qk + Pqiq¯k ⊗ q¯k ] + Pqig ⊗ g , (2.27) dlnQ2 2π i 2π γ d¯qi α αs γ γ γ γ γ = Pq¯ ⊗ Γ + Σ[Pq¯iqk ⊗ qk + Pq¯iq¯k ⊗ q¯k ] + Pq¯ig ⊗ g , (2.28) dlnQ2 2π i 2π γ dg α γ αs γ γ γ P γ Γ Σ P q P q¯ P g , (2.29) dlnQ2 = 2π g ⊗ + 2π [ gqk ⊗ k + gq¯k ⊗ k ] + gg ⊗ dΓγ α P Γγ Σ P qγ P q¯γ P gγ , (2.30) dlnQ2 = 2π γγ ⊗ + [ γqk ⊗ k + γq¯k ⊗ k ] + γg ⊗

The Altarelli-Parisi splitting kernels describe the parton branching, as illustrated in Fig. 2.7. In leading order they have the following form:

2 2 Pq γ z 3e 2 z 1 z , (2.31) i ( ) = qi [ + ( − ) ] 4 1 + z2 Pqiqk (z) = δik[ + 2δ(1 − z)], (2.32) 3 1 − z

Pqiq¯k (z) = 0, (2.33) 1 P z z2 1 z 2 , (2.34) qig( ) = 2[ + ( − ) ]

Pgγ = 0, (2.35) 4 1 1 z 2 P z + ( − ) , (2.36) gqk ( ) = 3[ z ]

Pgq¯k (z) = Pgqk (z), (2.37) 28 1 − z z 11 nf Pgg(z) = 6[ + + z(1 − z) + ( − )δ(1 − z)], (2.38) z (1 − z)+ 12 18

Because of the arise of the term Pqiγ which is describing the coupling of the photon to quarks, the evolution equations are inhomogeneous. If it were not for this term, the evolution equations would be identical to the evolution equations for parton distribution functions of hadrons like the proton as discussed in the above section. This is why the solution of the homogeneous evolution equations can be identiﬁed with the hadron-like part of the photon structure function, and its x and Q2 behavior is just as in the hadron case. A particular solution to the inhomogeneous evolution equations can be identiﬁed with the point-like part of the photon structure functions.

There are several parton distribution functions for real, and also for virtual photons in the market. Based on the full evolution equations discussed above, both leading and next-to-leading order exist. These properties are very similarly to the parton distribution functions of the proton. The various parton distribution functions of the photon diﬀer

2 in the assumptions made according to the starting scale Q0, which represents the input distribution scale. Different sets include different amount of data used in fitting their parameters (Ref. [49, 50, 51, 52, 53, 54, 55]).

The distributions basically fall into three classes depending on the theoretical concepts used. The first class in using purely phenomenological fits to the data. For example, Drees and Grassie (DG) [81], Levy, Abramowicz and CharchulaL (LAC) [82]. In particully, the first parton distribution functions were obtained by DG, it is using the evolution equations in leading order with Λ = 0.4 GeV. The input distributions of DG 2 2 at Q0 = 1 GeV is parametrized by 13 parameters and fitted to the preliminary PLUTO data at Q2 = 5.3 GeV2 from Ref. [57] which is the only available at that time. Due to the limited amount of data available, further assumptions will need to make, for example, the quark distribution functions for quarks carrying the same charge are assumed to be equal, while the gluon distribution function is generated purely dynamically with the gluon input distribution function being set to be zero. LAC uses essentially the same procedure as the ones from DG, 12 parameters is evolved using the leading order evolution equations for four massless quarks, where Λ is fixed to 0.2 GeV. Compared to DG,

29 Q2=0.1 GeV2 P2=0.1 GeV2 •2 Q2=0.1 GeV2 P2=0.1 GeV2 •2 10 xd 10 DO•GLO xu DO•GLO LAC•G/GAL•GLO LAC•G/GAL•GLO GS•GLO GS•GLO GS•G•96LO GS•G•96LO GRV•G/GRS•GLO GRV•G/GRS•GLO ACFGP/AFG•GNLO ACFGP/AFG•GNLO WHIT•GLO WHIT•GLO SAS•G(v1/v2)LO SAS•G(v1/v2)LO

10•3

10•4 10•3 10•2 10•1 1 10•4 10•3 10•2 10•1 1 x x

2 2 2 2 •2 Q =0.1 GeV P =0.1 GeV Q2=0.1 GeV2 P2=0.1 GeV2 xs 10 xg 10 DO•GLO DO•GLO LAC•G/GAL•GLO LAC•G/GAL•GLO GS•GLO GS•GLO GS•G•96LO GS•G•96LO GRV•G/GRS•GLO GRV•G/GRS•GLO ACFGP/AFG•GNLO 1 ACFGP/AFG•GNLO WHIT•GLO WHIT•GLO SAS•G(v1/v2)LO SAS•G(v1/v2)LO 10•1

•3 10 10•2

10•3

10•4 10•3 10•2 10•1 1 10•4 10•3 10•2 10•1 1 x x

Figure 2.10: u, d, s and g distribution of the photon depends on the momentum fraction x at a ﬁxed Q2 0.1 GeV2. P 2 means the virtuality of the other photon from e+e− scattering.

= LAC is based on much more data, and therefore no assumptions on the relative sizes of the quark input distribution functions are made for LAC. The second class of parametri- sations are based on their input distribution functions from theoretical predictions and obtained from the measured pion structure function using VMD arguments. This class includes Gl¨uck, Reya and Vogt (GRV) [85], AFG [86] and GS [83], all the photon PDFs are named with initials, I will not list all of them in this thesis. GRV is constructed using basically the same strategy which is successfully used for the description of the

2 2 2 proton and pion structure, with Q0 = 0.25 GeV in leading order and from Q0 = 0.30 GeV2 in next-to-leading order. The third class contains both ideas of the two classes above, and in addition is related with the input distribution functions to the measured photon-proton cross section. The mentioned SAS set belongs to the third class. To be clear, the x dependence of diﬀerent photon PDFs are shown in Fig. 2.10. x is need to be noticed with the meaning of momentum fraction on the parton from the photon.

30 CHAPTER 3

Possible realizations of an EIC

3.1 The eRHIC design concept

A number of reports (see Ref. [26, 95, 96, 97]) contain the proposal of a future EIC in the United States. Two independent designs: the eRHIC design (see Fig. 3.1) at Brookhaven National Laboratory (BNL) and the Jefferson Lab Electron Ion Collider (JLEIC) design (see Fig. 3.2) at Thomas Jefferson National Laboratory (JLab), are both using the existing infrastructure and facilities available to the US nuclear science community. The eRHIC design utilizes a new electron beam facility with the existing polarized proton and nuclear beams from Relativistic Heavy Ion Collide (RHIC). The JLEIC design employs a new electron and ion collider ring complex together with the √ upgraded 12 GeV CEBAF. The maximum center of mass energy of EIC is up to s ≈ 141 GeV in the present designs, which is close to half of HERA. However, the EIC will have a luminosity of 102 to 103 times larger than that of HERA. It will be the first polarized electron ion collider in the world.

The EIC machine designs are aimed to achieve highly polarized electron beams (≈ 80%) and nucleon beams (≈ 70%), ion beams from deuteron to the heaviest nuclei, with variable center of mass energies from 20 to 140 GeV, high luminosity (≈ 1033−1034cm−2s−1) and more than one interaction region. Although the collision parameters from the two realizations at BNL and JLab are similar, depending on different design realizations and specific detectors, some differences still exist. I will focus on the eRHIC design of EIC.

31 Figure 3.1: The collider design for eRHIC. The plot is from Ref. [59].

Figure 3.2: The collider design for JLEIC. The plot is from Ref. [59].

32 3.1.1 An eRHIC model detector

The eRHIC design must satisfy the requirements of the science program, which has acceptable technical risk, reasonable cost, and a clear path to achieving design performance after a short period of initial operating time. The eRHIC accelerator is shown in Fig. 3.1, taking full advantages of recent advances in accelerator technology. An electron storage ring, referred to as a Ring-Ring design ( RR-design), is used to meet the physical requirements for eRHIC. The storage ring based design meets or even exceeds the requirements referenced in the Long Range Plan including the upgraded energy reach:

● Center-of-mass energy (Ecm) of (29-140) GeV.

● A luminosity of up to 1034 cm−2s−1.

● High polarization of electron and ion beams with frequent changes to the spin direction as determined by the physics requirements with polarizations well above 50%.

● Beam divergences at the interaction point and apertures of the interaction region magnets that are compatible with the acceptance requirements of the colliding beam detector.

● Collisions of electrons with a large range of light to heavy ions (protons to Gold ions).

● Two interaction regions.

Polarized electron bunches are generated in the polarized electron source. Then the beam is accelerated to 400 MeV by a linear accelerator (LINAC). Once per second, an electron bunch is accelerated in a rapid cycling synchrotron (RCS), which is also located in the RHIC tunnel, to a beam energy of up to 18 GeV and is then injected into the electron storage ring where it is brought into collisions with the hadron beam. The spin orientation of half of the bunches is anti-parallel to the magnetic guide ﬁeld. The other half of the bunches have a spin which is parallel to the guide ﬁeld in the arcs.

33 Figure 3.3: Schematic view of the tracking system used in the model detector design for eRHIC. This plot is from Ref. [59].

High luminosity requires small emittance of hadron beams, which can only be achieved by implementing strong cooling of the ion and proton beams to counteract emittance growth by intrabeam scattering (IBS). The highest luminosity of L = 1 × 1034 cm−2s−1 is achieved with 10 GeV electrons colliding with 275 GeV protons (Ecm = 105 GeV).

A generic model detector design has been developed as shown schematically in Fig. 3.3. This eRHIC model detector dedicated to EIC physics consists of several major parts.

● Tracking detectors. The tracking system of the baseline eRHIC detector will consist of a time projection chamber (TPC), gas electron multiplier (GEM) and silicon

trackers spanning a range −4 < η < 4, shown in Fig. 3.3. It is very important to have excellent momentum resolution over a wide-rapidity range. 34 – Backward/Forward Silicon Tracker: 2 < SηS < 4, designed with 5-7 monolithic active sensor pixels (MAPS) technology discs.

– Backward/Forward GEM Tracker: 1.5 < SηS < 3, designed with 2-3 GEM tracker discs.

– TPC: SηS < 1.5, deliver hits for pseudorapidity up to SηS ∼ 2, but the range with suﬃcient number of hits is indicated as above.

– Vertex Silicon Tracker: SηS < 1, to be equipped with 4-6 layers of MAPS silicon sensors.

● Electromagnetic Calorimeter (ECal). The end-cap and barrel regions of the detector will be equipped with electromagnetic calorimeters covering −4 < η < 4. The different electromagnetic calorimeters have different technologies to account for the different requirements.

– Forward ECal: 1 < η < 4, the requirements for the forward ECal are relatively moderate as its main function is to detect leptons from the decay of vector mesons (VM) and photons from dominately π0 decay. Currently the idea is to have a scintillating ﬁber tungsten powder sampling calorimeter.

– Barrel ECal: −1 < η < 1, this calorimeter needs to provide PID for the scattered lepton at high Q2 and leptons from VM-decays, the energy of these leptons will be determined from the tracking detectors. The same technology as the forward ECal has been considered for the barrel ECal.

– Backward ECal: −4 < η < −1, this calorimeter needs to provide PID for the scattered lepton. It is especially important for the scattered lepton at low Q2. At higher center-of-mass energies photons from π0 decays, the deeply virtual Compton scattering and Bethe-Heitler process are in the acceptance of the backward ECal. Since the requirements in energy and angular resolution are most demanding, it is advised to have a lead tungstate (PWO) crystal calorimeter.

35 Figure 3.4: Left: The current design of the low Q2-tagger. Right: The range of acceptance in Q2 and scattering angle, θ, of electrons from PYTHIA events.

● Hadron Calorimeter (HCal). The resolution requirements for HCal are relatively moderate, therefore standard HCal techniques are totally applicable.

– Forward HCal: 1 < η < 4, this HCal is mainly for jet physics in DIS and diﬀractive events. It helps deﬁne a clean rapidity gap.

– Backward HCal: −4 < η < −1, this HCal is designed for the jet physics and will be useful to identify scattered lepton at low Q2 when ECal is not enough to separate leptons from hadrons.

Particle identification for different purpose relies on different combinational method.

π, K, p separation can be achieved in central rapidity region η < 1 with the detection of internally reﬂected Cherenkov light (DIRC) or proximity focusing Aerogel-RICH plus the speciﬁc energy loss dE~dx from TPC. The separation in 1 < SηS < 3 is obtained in RICH, where a very good meomentum resolution from the tracking is needed.

Lepton identiﬁcation mainly relies on the E~p in the region −3 < η < 3. For 1 < SηS < 3, additional HCal response and γ suppression via tracking can be applied to ﬁnd electrons.

In SηS > 3, combinational ECal and HCal responses and γ suppression via tracking will provide a clean access to the scattered lepton.

36 3.1.2 Low-Q2 tagger

A dedicated low Q2-tagger is planed at eRHIC, it will measure the scattered electrons from low Q2 events. These electrons will miss the main detector, and so installing an auxiliary device is essential for low Q2 physics. The current concept of the low Q2-tagger is shown in Fig. 3.4. It consists of three tracking layers followed by an electromagnetic calorimeter. Initial simulation studies show that this type of device can help to reconstruct the scattering angle and the energy of the electron and hence measure the Q2 of the event. The detector is placed near the outgoing electron beam roughly 15m from the IP after the ﬁrst set of bending dipole magnets where the scattered electrons will separate from the main beam. The right panel of Fig. 3.4 shows the Q2 acceptance. Current designs for an EIC low Q2-tagger assume a lead tungstate (PbWO4) crystal √ calorimeter with a energy resolution of 2%~ E ⊕1% preceded by Silicon detector planes for a high precision measurement of the incident scattered electron angle. The current design of the low Q2-tagger essentially covers the region of Q2 above 10−5 GeV2.

3.2 The designing requirements for a future EIC

The EIC is a multi-purpose collider designed to answer a wide range of the most com- pelling science questions concerning to our fundamental understanding of QCD physics. The most intriguing questions that an EIC will address include:

● Proton spin: EIC would be able to deliver decisive measurements on how much intrinsic spin of quarks and gluons contribution to the proton.

● The motion of quarks and gluons in the proton: Semi-inclusive measurements with polarized beams would enable us to selectively probe with precision the correlation between the spin of a fast moving proton and the conﬁned transverse motion of both quarks and gluons within.

● The tomographic images of the proton: By measuring exclusive processes, the EIC, with its unprecedented luminosity and detector coverage, would create

37 detailed images of the proton gluonic matter distribution.

● QCD matter at an extreme gluon density: By measuring the diﬀractive cross sections together with the total DIS cross sections, the EIC would provide the ﬁrst unambiguous evidence for the novel QCD matter of saturated gluons.

● Quark hadronization: By measuring pion and D0 meson, the EIC would provide the ﬁrst measurement of the quark mass dependence of the hadronization along with the response of nuclear matter to a fast moving quark.

The targeted physics programs put some requirements on the EIC machine designs.

To deliver enough physics statistics in a promising time, a high luminosity ∼ 1033cm−2s−1 is required. Extracting the structure function and exploring the nuclear time-space evolution needs a ﬂexible beam energy varying in a wide range. To study the spin distribution, electrons and proton/light nuclei must be highly polarized. A large variety of nuclear beams are needed to study the nuclear size dependence. In the semi-inclusive DIS studies, a wide acceptance detector with good particle identiﬁcation (PID) is in- evitable. For some exclusive measurements, it is demanding to have the acceptance for protons generated in the very forward region.

38 CHAPTER 4

Experimental aspect of jet physics at EIC

The high-energy ep interactions provide a powerful way to test the prediction of QCD. In this theory, the interactions between electrons and protons produce partons with large transverse momenta, which manifest themselves as a cluster of particles traveling in roughly the same direction. These clusters are referred to as “jets”. This chapter presents a selection of the latest results about jets at the future EIC in both photoproduction and deep inelastic scattering (DIS) regimes. The low Q2 pseudo-data are constrained to the region of interest for photoproduction by requiring that the scattered electron remains in the beam pipe, undetected in the main detector, the photon virtuality is therefore restricted to Q2 < 0.1 GeV2, and Q2 > 10−5 GeV2 according to the lower limit of the low Q2-tagger acceptance, whereas the relatively high Q2 pseudo-data, corresponding to

10 < Q2 < 100 GeV2. In this chapter, we present an experimental overview of topics and considerations relevant to jet-ﬁnding at a future EIC. Electron-proton deep inelastic scattering (DIS) events with center-of-mass collision energies of 141 GeV were generated from PYTHIA, with the unpolarized PDF input from the LHAPDF library. The CTEQ5m PDF [62] is used for proton, since the cross section of low Q2 events is well described by applying this PDF. The SAS PDF is used for the photon. And jets were subsequently found using the anti-kT algorithm (recombination parameter R = 1.0) as implemented in the FastJet software package [94]. A detailed overview of basic jet quantities, such as multiplicities and kinematic distributions, are presented along with a characterization of the energy in the event not associated with a jet. Detector requirements are also discussed. Finally, the prospects of identifying jets originating from quarks vs gluons are explored.

39 4.1 Jet ﬁnding and jet properties

4.1.1 Input and reference frames

The jets used in this study are formed from the stable ﬁnal state particles. Here, stable refers to particles which would not normally decay in the volume of a detector, such as charged pions, kaons, and neutrons. The decayed photons are also taken into candidates for jets. To align with detector design constraints of eRHIC, only particles with momenta greater than 250 MeV transverse to the beam and pseudorapidity between

±4.5 are considered candidates for inclusion in jets. At hadron colliders, analyses are carried out almost exclusively in the reference frame of the detector, the laboratory frame, as the kinematics of the interacting partons are not generally known. At an EIC, however, the scattering kinematics are known event- by-event which makes it possible to boost to other frames. A particularly useful frame for jet analyses is the Breit or “brick wall” frame. The Breit frame is oriented such that for the lowest order DIS process γ∗q → q′, the virtual photon and interacting quark collide head-on along the z-axis and is boosted such that the only non-zero component of the virtual photon four-momentum is pz = −Q. A consequence of this boost is that the z-momentum of the incoming quark is Q~2 while the scattered quark has z-momentum −Q~2 (hence the name ‘brick wall’ frame) and the proton remnant has a z-momentum of (1 − x)Q~(2x). This leads to a natural separation between jets associated with the struck quark and those associated with the proton remnant. Note that this z-momentum separation does not necessarily hold for higher order processes such as photon-gluon fusion. Another merit of working in the Breit frame is that the transverse momentum of the jet is measured with respect to the virtual photon direction.

While it would be possible to ﬁnd jets from particles in the lab frame and then boost these jet thrust axes into the Breit frame, we ﬁrst transforms each particle into the Breit frame and then uses these boosted particles in the jet clustering. This avoids any changes in particle content which may arise due to variations in clustering in the two frames. However, when it is necessary to present some jet quantity with respect to the

40 detector, the thrust axis of the Breit frame jet is simply boosted back to the laboratory frame.

4.1.2 Jet deﬁnitions

While the idea of a jet as a collimated spray of particles is conceptually easy to grasp and jets are often easy to identify “by eye” in event displays, a well deﬁned method of mapping a set of particles into a set of jets is required to be useful in physics analysis. As this chapter is meant to give an overview of jets at an EIC as opposed to focusing on a single analysis, a comparison of the general behavior of several representative algorithms will be performed before selecting one for the studies in later sections. It should be noted that several recombination schemes also exist, but only the E Scheme, in which particles are combined simply by adding their four-momenta, will be considered here.

The deﬁnition of sequential recombination and cone algorithms have been discussed in Sec. 2.3.2. For the comparisons presented here, two sequential recombination algorithms

(kT and anti-kT ) and one cone algorithm (SISCone) were selected. These algorithms all include a radius or recombination parameter R which sets the eﬀective size of a jet and several values are compared for the anti-kT algorithm. The SISCone algorithm has several other parameters which control its behavior, including a split-merge fraction and choice of split-merge scale. The split-merge fraction was taken to be 0.75 and other parameters were set to the default values from FastJet and not varied.

The general properties compared between algorithms are presented in Fig. 4.1 and Fig. 4.2, including the energy distribution within the jet, jet multiplicity, and particle multiplicity within a jet. The event sample was conﬁned to Q2 values between 10 and

100 GeV2. All algorithms were run with R = 1.0 and jets were required to have transverse momenta greater than 4 GeV in the Breit frame. The resolved, QCDC, PGF, and DIS subprocesses are combined.

Fig. 4.1(a) shows the deﬁnition of jet proﬁle, which means the the fractional transverse momentum carried by particles within a radius r of the jet thrust axis to that of the

41 1

0.8 Profile(r)

0.6

R = 1.0 0.4 anti-kT kT SISCone 0.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 r

(a) (b)

Figure 4.1: (a) Illustration of jet profile. (b) Comparison of jet profile for anti-kT , kT and SISCone algorithms. total jet, it can be defined with the formula:

i Q pT ri

i between the particle i and the central axis of jet, pT is the transverse momentum of the particle i. The jet profile in Fig. 4.1(b) measures the momentum fraction depending on r. The kT and anti-kT curves track each other to r ∼ 0.8 at which point kT merges with the SISCone curve, which always sits somewhat below anti-kT . Lower values of Profile(r) indicate that SISCone jets are somewhat more diffuse in their energy distribution while the slower rise toward unity at large r for kT and SISCone signal that the jet boundaries are somewhat more susceptible to soft radiation. This susceptibility is a known feature of both the kT and SISCone algorithms and leads to irregular jet boundaries as well as the possibility of particles lying outside the nominal jet radius. The anti-kT boundaries on the other hand are not susceptible to soft radiation and are always smooth.

Fig. 4.2 shows the jet multiplicity and particle multiplicity within a jet, respectively.

Again, little diﬀerence is seen between algorithms with the kT algorithm more likely to ﬁnd more jets in an event and to cluster more particles into a given jet due to its propensity for including soft particles at large radii in the clustering. Note that the 42 105 104 counts counts 4 10 103

103 102

102 R = 1.0, resolved process R = 1.0, resolved process anti-k anti-k 10 T T k k T 10 T SISCone SISCone 1 1 0 1 2 3 4 5 6 7 0 5 10 15 20 25 30 Number of jets Number of particles

(a) (b)

Figure 4.2: (a) Comparison of jet multiplicity for anti-kT , kT and SISCone algorithms. (b)

Comparison of particle multiplicity inside the jet for anti-kT , kT and SISCone algorithms. slight enhancement seen for two-jet events over single-jet events is due to the inclusion of the resolved, QCDC, and PGF subprocesses which naturally produce di-jets. The comparisons in Fig. 4.2 show that while there are diﬀerences in the general behavior of the kT , anti-kT , and SISCone algorithms, they are relatively minor. It should also be noted that there is almost no dependence on algorithm for kinematic quantities such as transverse momentum or jet direction. This indicates that, at least for the overview presented here, the choice of algorithm will have little impact on the conclusions drawn. Due to the fact that it produces the most collimated jets with regular boundaries, the anti-kT algorithm will be used for all subsequent studies in this study.

The other major component of the jet definition which needs to be determined is the resolution parameter R. As with jet algorithm, the optimal choice for R may depend on requirements driven by a specific analysis. Often, the chosen R is a compromise between large values which will capture the full energy of the hadronizing partons and small values which limit the contamination from underlying event. Although not investigated as a function of R, underlying event properties are studied in Sec. 4.3. The quantities which are explored as a function of R include the jet multiplicity and in-jet particle multiplicity. An additional study was performed to quantify how well jets with different R values represent the underlying partonic behavior.

The jet multiplicity and number of particles within a jet are presented in the left

43 anti-k , resolved T anti-kT, resolved 105 R = 1.0 104 R = 1.0

counts R = 0.7 counts R = 0.7 R = 0.4 R = 0.4 4 10 103

103 102

102 10 10

1 1 0 1 2 3 4 5 6 7 8 9 10 0 5 10 15 20 25 30 Number of jets Number of particles

Figure 4.3: Comparison of jet multiplicity and particle content within a jet for the diﬀerent resolution parameter R 1.0, R 0.7 and R 0.4.

= = = and right panels, respectively, of Fig. 4.3 for R values of 1.0, 0.7, and 0.4. As with the algorithm studies, the jets were required to have transverse momentum greater than 4 GeV in the Breit frame and were taken from events with Q2 between 10 and 100 GeV2.

The resolved, QCDC, PGF, and DIS subprocesses have been combined and the anti-kT algorithm is used. It is seen that the total number of found jets increases with increasing R. This is due to the fact that larger R values admit more particles (as seen in the right- hand panel) which mean more jets will pass the 4 GeV transverse momentum threshold. This ordering is maintained for one-, two-, three-, and 4-jet events but is inverted for the (highly suppressed) ﬁve-jet events at which point geometric phase space constraints discourage ﬁnding numerous jets with large areas.

One merit of jet observables is that they can serve as proxies for the hard-scattered partons. It is therefore important to investigate if the size of R aﬀects how well jets reproduce the partonic kinematics. To test this, the reconstructed di-jet invariant mass √ is compared to sˆ, which is the Mandelstam variable for the invariant mass of the partonic hard-scattering process ab → cd. The di-jet invariant mass can be approximated as (ignoring the individual jet masses):

» M ≈ 2pT 1pT 2 (cosh(∆η) − cos(∆φ)), (4.2) where pT 1 and pT 2 are the transverse momenta of the two jets, ∆η and ∆φ are the pseu-

44 100 100

90 90 102 R = 1.0, resolved R = 0.7, resolved 80 102 80 70 70 60 60 10 50 10 50 di-jet mass [GeV] di-jet mass [GeV] 40 40 30 30 20 20 1 1 10 10 0 0 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 s [GeV] s [GeV]

(a) (b)

100 anti-k , resolved 90 3 T R = 0.4, resolved 10 R = 1.0 80 counts R = 0.7 70 10 R = 0.4 2 60 10 50 di-jet mass [GeV] 40 1 10 30 20 10 1 −1 0 10 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 s [GeV] di-jet mass [GeV]

Figure 4.4: (a, b, c) Di-jet mass versuss ˆ for R 1.0, R 0.7 and R 0.4, and (d) di-jet mass distribution for diﬀerent R. = = = dorapidity and azimuthal angle diﬀerences, respectively, between the two jets. Di-jets were found by selecting the two jets in an event with the highest transverse momentum in the Breit frame and requiring that the two jets are greater than 120 degrees apart in azimuth and that one jet have pT greater than 5 GeV while the other has pT greater than 4 GeV. The comparison between reconstructed di-jet and partonic invariant mass in resolved process can be seen in Fig. 4.4 for R = 1.0, 0.7, and 0.4. The agreement is best seen for R = 1.0, however degrades as R becomes smaller. The fact that the reconstructed di-jet mass is consistently smaller than the partonic invariant mass for

R < 1.0 indicates that these smaller cones can not capture all the energy associated with the hard-scattered partons.

As it results in the best correlation for the di-jet mass, R will be set to 1.0 for all

45 subsequent studies presented in this section. Together with the choice of anti-kT for the jet algorithm and E Scheme recombination, the jet deﬁnition is fully quantiﬁed.

4.1.3 Jet kinematics

9 9 9 10 10 10 Lab Frame 8 8 8 10 resolved process10 10

counts 7 VMD 107 107 10 γ -proton Frame Breit Frame 6 QCDC & PGF 6 6 10 10 10 LO DIS 5 5 5 10 10 10

104 104 104

3 3 3 10 10 10

102 102 102

10 10 10

1 1 1 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25 p [GeV] p [GeV] p [GeV] T T T

Figure 4.5: The inclusive jet pT distribution in three diﬀerent frames based on diﬀerent subprocesses. Jets are reconstructed with stable particles, the particles inside the jets require

−5 2 2 2 to be pT 250 MeV and 4.5 η 4.5. Events are selected with 10 GeV Q 0.1 GeV .

> − < < < < While jets arising from ep collisions were studied extensively at HERA, the lower beam energies envisioned for an EIC make a detailed investigation of jet properties and kinematics warranted. We discuss some general kinematics of inclusive jets in this section. Diﬀerent frames (lab frame, γ-proton frame and breit frame) and subprocesses (resolved process, QCDC, PGF, LO DIS and VMD) are discussed as well.

The inclusive jet pT distribution and η distribution is shown in Fig. 4.5 and Fig. 4.6, respectively. Jets produced from hard processes usually carry high pT . As shown in Fig. 4.5, the three higher order subprocesses (the resolved, QCDC and PGF) dominate at high pT region in all the frames, and they show roughly the same behavior while the leading-order DIS process and soft VMD spectra are much softer, only reaching one third to one fourth the pT of the higher-order processes. By applying a high pT cut on jets, these three types of processes can be well selected. In the following sections, di-jet method is widely used. Ideally, the two jets selected as a di-jet pair in resolved, QCDC, and PGF events will accurately reﬂect the kinematics of the two hard-scattered partons.

46 10 10 10 10 10 10

9 9 9 10 10 10

8 8 8 counts 10 10 10 107 107 107

6 6 6 10 10 10

5 5 5 10 10 10 104 104 104 Lab Frame 3 3 3 10 resolved process 10 10 VMD 2 102 102 Breit Frame 10 QCDC & PGF γ-proton Frame 10 LO DIS 10 10 1 1 1 −6 −4 −2 0 2 4 2 4 6 8 10 12 14 −4 −2 0 2 4 6 8 10 12 14 16 18 ηjet ηjet ηjet

Figure 4.6: The inclusive jet η distribution in three diﬀerent frames based on diﬀerent subprocesses. The kinematics cuts are the same with FIG. 4.5.

Apparently, this is not always the case, as radiation and jet reconstruction features can cause the energy and/or direction of the jet to deviate somewhat from that of the parton from which it originated. It can also be the case for events with more than two jets, that one of the two highest pT jets did not arise directly from the hard-scattered parton meaning the experimentally chosen dijet will not represent the hard scattering event. The higher order subprocesses which naturally produce di-jets could be enhanced by high pT selection of the jets. In Fig. 4.6, the inclusive jet pseudorapidity spectrum is presented. We deﬁne in the ep collision, the incoming electron beam is in negative η direction, the incoming proton beam is in positive η direction. The value of ηjet changes sharply if boosted into diﬀerent frames.

4.2 Detector requirements

The tracking system of the baseline eRHIC detector will consist of a TPC, GEM and silicon detectors spanning a range of −4 < η < 4 in pseudorapidity. The end-cap and barrel region on the detector will be equipped with electromagnetic calorimeters covering

−4.5 < η < 4.5. Hadronic calorimeter will be used mostly for jet physics at full energy in the forward (hadron beam going direction) and backward (electron beam going direction) rapidities spanning 2 < SηS < 4.5. Projected momentum and energy resolutions of these

47 8 4 10 10 Lab Frame 107 ±

counts p Lab Frame ± counts 3 K 10 106 π± 105 102 104

10 103

102 1 10

-4 -2 0 2 4 6 8 10 12 1 jet 0 1 2 3 4 5 6 7 8 9 10 η p [GeV] T

(a) (b)

± ± ± Figure 4.7: (a) Inclusive jet η spectra, with jets pT 4 GeV, (b)π , K and p production in jets for all processes. Jets are reconstructed with stable> particles, the particles inside the jets −5 2 2 require to be pT 250 MeV and 4.5 η 4.5. Events are selected with 10 GeV Q 0.1 2 GeV . > − < < < < devices are better than a few percent, which extends the capability of this detector to a large variety of physics topics.

In this section, we present the detector requirement for successful jet measurement. The spectra shown in the Lab frame represents the distribution of jets within the detector can be observed in Fig. 4.7(a). Since the pseudorapidity of particle candidates within the jets are requested to be −4.5 < η < 4.5 according to the coverage from the electromagnetic calorimeters, now it’s seen that the reconstructed jets all locate in the η region from -

4.5 to 4.5. Inside the jets, the charged hadron pT spectrum in lab frame is shown in Fig. 4.7(b). The charged pions are the major component for the ﬁnal state particles.

4.3 Underlying event

The underlying event contribution, as one of the background contributions to jet signals, is measured through a di-jet method in this section. High pT di-jet events are selected by requiring that the pT of the trigger jet is above 5 GeV and the associate one is above 4.5 GeV. These events belong to resolved, QCDC and PGF processes. In resolved processes, the QCD Monte Carlo models simulate a electron-proton collision

48 Figure 4.8: Illustration of underlying events. through a hard parton+parton → parton+parton scattering. In QCDC and PGF, the collision occurred through a γ+parton → parton+parton scattering. From both two types of scatterings, the resulting event contains particles from the two outgoing partons and those coming from the breakup of the incoming particles (beam remnants). From the Feynman diagrams of these two types of processes shown in Fig. 2.4, although the hardest 2-to-2 scattering is of the most interest, the other soft scatterings as background to the hard processes are also important to be studied. The “hard scattering” component consist of the outgoing two jets plus initial- and finial-state radiation. The background generated due to these multiple soft scatterings is classified as an underlying events. The underlying event is everything except the two outgoing hard scattered jets and consists of the beam remnant plus possible contributions from the “hard scattering” arising from the initial and final-state radiation.

There are several methods to measure the underlying event effects in the jet analysis. Two different methods are discussed in this section to measure the underlying events in ep collision. The first method is called the “region method”. In each high pT di-jet event, the highest pT jet is called the trigger jet, the second highest pT jet is called the associate jet, these two jets are almost back-to-back in φ direction. We use the direction of the trigger jet in each event to divide the particles into three regions in η − φ space, where η is the pseudorapidity measured along the beam axis and ∆φ(φ−φtrig jet) is the azimuthal angle relative to the trigger jet. The particle candidate pool is the exact same input as used for the jet finding algorithm. The “toward” region contains the trigger jet, while 49 2π Trigger Jet Away Direction ∆φ Transverse

Toward φ Trigger Jet Transverse Transverse Toward

Transverse Away

Away 0 η

Figure 4.9: Illustration of “toward”, “away” and “transverse” regions in azimuthal angle ∆φ relative to the direction of the trigger jet in the event. The angle ∆φ φ φtrig jet is the relative azimuthal angle between charged particles and the trigger jet. The “toward”= − region is deﬁned by ∆φ 60○ (includes particles inside the trigger jet), while the “away” region is ∆φ 120○.

○ ○ TheS “transverse”S < region is deﬁned by 60 ∆φ 120 . The plot is from [99]. S S > < S S < the “away” region, on the average, contains the associate jet. The “transverse” region is perpendicular to the plane of the hard 2-to-2 scattering and is very sensitive to the “underlying event” component of the QCD Monte Carlo models. Fig. 4.9 illustrates the way we deﬁne the three regions. The average number of charged particles, ⟨Nch⟩, we also call it “multiplicity”, and the average scalar pT sum of charged particles, ⟨pT sum⟩, in each region are measured versus ∆φ. A di-jet is reconstructed in each event with stable particles, the particles inside the jets require to be pT > 250 MeV and −4.5 < η < 4.5. −5 2 2 2 The di-jet events are selected with 10 GeV < Q < 0.1 GeV . When we plot ⟨Nch⟩ and

⟨pT sum⟩ as a function of ∆φ, we include all the charged particles. Fig. 4.10 and Fig. 4.11 show the results on the charged multiplicity distribution and the sum of transverse momentum distribution, respectively, in the azimuthal angle ∆φ

50 1.2 Trigger jet p >5 GeV 1 T Trigger jet p >8 GeV T 0.8

0.6 in 3.6 degree bin 〉 0.4 ch N 〈 0.2

0 Toward Transverse Away 0 0.5 1 1.5 2 2.5 3 ∆φ [rad]

Figure 4.10: Average number of charged stable particles as a function of the azimuthal angle,

trig jet 2 2 ∆φ, between the particle and the trigger jet for pT 5 GeV and 8 GeV, Q 0.1 GeV . ○ Each point corresponds to the Nch in a 3.6 bin. The> “toward”, “transverse”< and “away” regions deﬁned in Fig. 4.9 are labeled.⟨ ⟩

3 Trigger jet p >5 GeV 2.5 T Trigger jet p >8 GeV 2 T

1.5

in 3.6 degree bin 1 〉 0.5 sum T p

〈 0 Toward Transverse Away -0.5 0 0.5 1 1.5 2 2.5 3 ∆φ [rad]

Figure 4.11: Average scalar pT sum of charged stable particles as a function of the az- trig jet imuthal angle, ∆φ, between the particle and the trigger jet for pT 5 GeV and 8 GeV, 2 2 ○ Q 0.1 GeV . Each point corresponds to the pT sum in a 3.6 bin. The> “toward”, “transverse”< and “away” regions deﬁned in Fig. 4.9 are⟨ labeled.⟩

51 0.9 Trigger jet p >5 GeV, all processes 0.8 T 10 GeV2 < Q2 < 100 GeV2 0.7 1 GeV2 < Q2 < 10 GeV2 0.6 Q2 < 0.1 GeV2 0.5 0.4 in 3.6 degree bin

〉 0.3 ch

N 0.2 〈 0.1 0 Toward Transverse Away -0.1 0 0.5 1 1.5 2 2.5 3 ∆φ [rad]

Figure 4.12: Average number of charged stable particles (middle pseodurapidity particles: 1 η 1) as a function of the azimuthal angle, ∆φ, between the particle and the trigger jet for

trig jet trig jet 2 2 2 2 2 2 −pT < < 5 GeV, 1 η 1, in Q 0.1 GeV , 1 Q 10 GeV and 10 Q 100 GeV bins. > − < < < < < < < relative to the trigger jet for pT > 5 GeV and 8 GeV. The “toward” region contains the highest charged particle multiplicity and summation of charged particle transverse momentum. The associated jet is not exactly back-to-back with the trigger jet in φ direction, which results in the long tails in the “away” region for both plots. In the “transverse” region where we put our interest of the underlying events, a relatively ﬂat distribution is presented. Then the background eﬀect due to the underlying events can be removed by subtraction of the multiplicity and summation of the transverse momentum in the “transverse” region.

As illustrated in Fig. 4.12, we calculate the average of number of charged particles in middle psedorapidity in three diﬀerent Q2 bins to show the Q2 dependence. High Q2 shows a much higher number of charged particles than low Q2, especially in “toward” and “away” regions, in the “transverse” region, a very small increase is observed when Q2 is high.

Fig. 4.13(a) shows the average number of charged particles in 1 GeV bin as a function of the trigger jet pT (pT > 5 GeV) in three regions. Fig. 4.13(b) shows the average scalar

52 14 Toward 16 12 Away 14 Transverse 10 12 10 8 in 1 GeV/c bin 〉

ch 6

N 6 〈 (GeV) in 1 GeV/c bin 4 〉

T 4 2 2 0 0 Sum p

0 2 4 6 8 10 12 14 16 18 20 〈 0 2 4 6 8 10 12 14 16 18 20 Trigger jet p [GeV] Trigger jet p [GeV] T T

(a) (b)

Figure 4.13: (a) The average number of charged particles in “toward”, “transverse” and

“away” regions as a function of the transverse momentum of the trigger jet (pT 5 GeV). (b)

The average scalar pT sum of charged particles in “toward”, “transverse” and “away”> regions as a function of the transverse momentum of the trigger jet (pT 5 GeV).

> pT sum of charged particles (pT > 250 MeV and −4.5 < η < 4.5) in 1 GeV bin as a function of the trigger jet pT (pT > 5 GeV) in three regions.

Figure 4.14: The illustration of two oﬀ-axis cones relative to a jet. The plot is from Ref. [100].

The second method is called off-axis cone method [101], which is developed by the ALICE experiment. The off-axis cone method is a method to study underlying event on the level of jet by jet, different from the “region method” which is on the level of event. For every reconstructed jet, we draw two off-axis cones (cone(-) and cone(+)), 53 each of which is centered at the same η as the jet but ±π~2 away in φ from the jet φ as shown in Fig. 4.14. Then we collect particles falling inside the two cones. The particle candidate pool consists of the input as used for the jet finding algorithm (pT > 250 MeV and −4.5 < η < 4.5). We choose the cones radius to be 0.4. The multiplicity density is defined as the average number of charged particles inside each cone ⟨Nch⟩ divided by the 2 cone area πR . The ⟨pT sum⟩ density is defined as the average off-axis cone pT divided by the cone area.

We compare the results from the two methods in Fig. 4.15(a) and Fig. 4.15(b). Results from the region method is measured with all the charged particles in the “transverse” region with each particle in the pseudorapidity range from -1 to 1 and with the trigger jet pseudorapidity is also from -1 to 1, so the multiplicity density (⟨dN⟩~dφdη) and transverse momentum density (⟨dpT sum⟩~dφdη) are defined as the average number 2π and the average sum pt of all the charged particles divided by η − φ space area (2 × 3 ), respectively. The trigger jet pT > 5 GeV. The region method takes the underlying events uniformly distribute in η direction, so we did the calculations for each observable in an integrated η range. However, the η dependence appears in the off-axis method, since we define the pseudorapidity of two cones the same as η of jets. From the comparison, results from both methods can roughly be consistent. So we can conclude that the effects from underlying events show small dependence on η.

We can also compare the results with that measured from STAR by using the region √ method. The underlying events are studied in p + p collision at sNN = 200 GeV from √ STAR [102], the c.m.s energy is comparable with our energy at EIC ( sNN = 141 GeV). Compared with Fig. 7 of [102], the multiplicity density of our results is shown Fig. 4.16, which is slightly lower than the STAR results. The average pT sum which can be obtained 2π from Fig. 4.15(b) if multiplied by the η − φ area factor (2 × 3 ), is close to the results from STAR.

54 0.6 0.6 region method 0.5 0.5 off•axis method 0.4 0.4 in 1 GeV bin in 1 GeV bin η η

d 0.3 d 0.3 φ φ /d /d 〉 〉

0.2 0.2 ch N 〈 0.1 sum 0.1 d T p 〈

0 d 0 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 Trigger jet p [GeV] Trigger jet p [GeV] T T

(a) (b)

Figure 4.15: (a) The average number of charged particles (region method: pT 250 MeV, ○ ○ 1 η 1 and 60 ∆φ 120 , the trigger jet 1 η 1; oﬀ-axis method: pT 250> MeV and −1 < η < 1) density< asS a functionS < of the transverse− < momentum< of the trigger jet> (pT 5 GeV). −(b)< The< average scalar pT sum density of charged particles (region method: pT >250 MeV, ○ ○ 1 η 1 and 60 ∆φ 120 ; oﬀ-axis method: pT 250 MeV and 1 η 1) as> a function −of the< transverse< momentum< S S < of the trigger jet (pT 5> GeV). − < < >

0.7 η

d 1.4

φ 2 2 0.6 Q2 < 0.1 GeV2 Q < 0.1 GeV /d 〉

1.2 2 2 2 1 GeV2 < Q2 < 10 GeV2 ch 0.5 1 GeV < Q < 10 GeV

N 1 〈 0.4 0.8

0.3 0.6

0.2 0.4 of charged tracks [GeV]

0.1 T 0.2

0 0 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14

Trigger Jet p [GeV] mean p Trigger Jet p [GeV] T T

(a) (b)

Figure 4.16: (a) The number density of charged particles versus the trigger jet pT . (b) The mean pT of the charged particles versus the trigger jet pT .

55 e

γ 104 trigger jet and its parton counts

103 associte jet and its parton beamparton:q q 102 q tgtparton:q 10

1 P 0 1 2 3 4 5 6 ∆ R

(a) (b)

Figure 4.17: (a) Example of Feynman diagram in resolved process. (b) The ∆R distribution between jets and its matched partons.

4.4 Quark jet and gluon jet

Jets originated from quark and gluon have different properties. Designing an method to effectively separate quark- and gluon-initiated jets is a longstanding open question. It is usually done via jet substructure observables like jet shapes which exploit differences in the radiation pattern of quarks and gluons. In this section, we are going to present the discrimination of quark and gluon jets at EIC. In general, we are interested in developing quark/gluon jet discrimination in resolved processes. Since quarks and gluons can branch into one another, are ill-defined concepts beyond the lowest order of the perturbative series, and are not directly observed in the final state of the collisions, the concept of a quark and a gluon jet might itself seem ill-defined at first sight. Rather than trying to determine a truth definition of a quark or a gluon, our approach is to consider the geometric distance between the final partons and final jets. Here, we obtain the quark/gluon jet samples by reconstructing jets in each event, and then by applying a geometric match we define two types of jets originated from quark or gluon.

The geometric match method can be explained as: in resolved process, di-jets are produced from “final partons”. Final partons are defined as the output partons of hard scattering, then fragment into di-jets, for example in Fig. 4.17(a), the final partons can be the out going quark and antiquark in hard scattering. “Initial partons” are defined as

56 the incoming partons of hard scattering, for example in Fig. 4.17(a), initial partons can be the quark from the photon side(beamparton in PYTHIA) and antiquark from proton side(tgtparton in PYTHIA). Firstly, we determine the final two partons in each event, one is “from” beamparton, another is “from” the tgtparton. Secondly, in each event we calculate the pseudorapidity difference ∆ηjet−parton and the azimuthal angle ∆φjet−parton » between trigger jet and the two final partons. By applying ∆R = ∆η2 + ∆φ2, so we know ∆R1 for trigger jet and one parton and ∆R2 for trigger jet and another parton. Select the minimum value of two R, trigger jet matches the parton with the smaller value of R. Then the associate jet matches the other parton. ∆R distribution between trigger jet and its original parton, also ∆R between associate jet and its original parton, is shown in Fig. 4.17. To avoid losing a lot of statistic, we choose ∆R < 0.5 to do the match. This method is also applied in the following Sec. 5.3.3.

With the diﬀerent color interaction and hadronization, the physical basic for distin- guishing between gluon jets and quark jets is the fact that that branching of gluons is stronger than the branching of quarks according to QCD. This can be directly seen by comparing the lowest-order elementary branching probability [103] for gluons, g → g + g and g → q+q¯, with that for quarks, q → q+g. Therefore, the gluon jets are expected to be wider and with higher multiplicities than quark jets of similar ET . Here, the multiplicity means the total number of particle ﬂow candidates reconstructed within the jet. Seen from Fig. 4.18(a) and Fig. 4.18(b), the energy distribution shows gluon jets energy is a little higher than quark jet, however the transverse momentum of gluon jets is a little lower than that of quark jets. The particle multiplicity is higher in gluon jets than quark jets 4.18(c). The ∆R of inside particles relative to the central axis of the gluon jets tends to dominate at larger value compared with those of quark ones from Fig. 4.18(d). Gluon jets are less collimated than quark jets.

The jet proﬁle for quark and gluon jets is shown in Fig. 4.19(a). The radius of inner cone can be chosen to be any one smaller than the jet radius. At a ﬁxed r, a quark jet inner cone always contains more energy than that of gluon jet. The leading hadron is

L the hadron with highest pT inside the jet. The leading hadron pT (pT ) fraction is deﬁned as the pT of the leading hadron divided by the pT of the jet. The leading hadron’s pT

57 ×10•3 quark jet 0.1 quark jet 50

fraction gluon jet fraction 0.08 gluon jet 40 0.06 30

20 0.04

10 0.02

0 0 5 10 15 20 25 30 4 6 8 10 12 14 16 18 20 E GeV p GeV jet T

(a) (b)

0.18 quark jet 0.35 quark jet 0.16 0.3 fraction gluon jet fraction gluon jet 0.14 0.12 0.25 0.1 0.2 0.08 0.15 0.06 0.1 0.04 0.02 0.05 0 0 0 2 4 6 8 10 12 14 16 18 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Numbers of Particles inside jet ∆ R

Figure 4.18: (a) Energy distribution of jets originated from quark and gluon. (b) Multiplicity.

2 2 (c) ∆R ∆η ∆φ distribution for each particles inside jet regarding to the jet. (d) pT » distribution= of jets.+

58 ×103

2 Quark jet 1.8 Gluon jet counts 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 pL/pjet T T

(a) (b)

Figure 4.19: (a) The jet proﬁle for quark and gluon jets. (b) Leading hadron pT fraction of quark and gluon jets. fraction corresponding to the whole jet is shown in Fig. 4.19(b). The peak is at diﬀerent value of the fraction, a quark jet tends to contain leading hadron with higher pT fraction than that of a gluon jet. The meaning of the y axis (fraction) is the number of counts divided by the number of jets, which can be regarded as the possibility.

The jet mass was also found to be useful for identifying the partonic origin of a jet, the jet mass distribution is presented in Fig. 4.20(a), As mentioned before, the larger color factor associated with gluon results in the production of a larger number of particles and also a softer hadron pT spectrum after the shower. The jet girth is also to good variable to distinguish quark/gluon jet, deﬁned as

i pT 2 girth2 = Q jet SRiS (4.3) i∈cone pT where i represents particle inside the jet, Ri is the distance between the particle i and the central axis of the jet.

59 0.25 Quark jet 0.09 Quark jet 0.08 fraction fraction 0.2 Gluon jet Gluon jet 0.07

0.06 0.15 0.05

0.04 0.1 0.03

0.05 0.02 0.01

0 0 0 2 4 6 8 10 12 14 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Mass [GeV] Girth2

(a) (b)

Figure 4.20: (a) The jet mass for quark and gluon jets. (b) Jet girth2 distribution for quark jet and gluon jets. Blue curves represent gluon jets, the red ones represent quark jets.

4.5 Summary

Jets are widely used in the analysis of particle physics. Due to carrying more information on the original partons than hadrons, jets are measured in particle detectors and studied to determine the properties of the original partons. In this chapter, we present the basic kinematics of jets and the detector requirements of measuring jets at EIC. In the di-jet events, we estimate the back ground eﬀect coming from the underlying events. The possibility of discriminating jets originate from quarks and gluons is discussed. Since these two types of jets overlap in the distribution of multiplied variables, to some extend, quark jet and gluon jet are hard to be separated. So further study is still required then.

60 CHAPTER 5

Photon structure at EIC

5.1 Monte Carlo set up

In this chapter, we use pseudo-data generated by the Monte Carlo generator PYTHIA- 6, with the unpolarized PDF input from the LHAPDF library. In PYTHIA, depending on the wave function components for the incoming virtual photon, the major hard processes are divided into three classes: the direct processes, the soft VMD processes and the resolved processes (hard VMD and anomalous). The direct photon interacts as a point-like particle with the partons of the nucleon, with major subprocesses in the direct category: LO DIS, Photon-Gluon Fusion (PGF) and QCD Compton (QCDC). The VMD and anomalous components interact through their hadronic structure. Resolved photon processes play a signiﬁcant role in the production of hard high-pT processes at

Q2 ≈ 0. The following hard subprocesses are grouped in the resolved processes category: qq → qq, qg → qg, gg → gg, qq¯ → qq,¯ gg → qq¯ and qq¯ → gg. The CTEQ5m [62] PDF is used for the proton, because contrary to modern PDFs (i.e., CT, NNPDF, HERAPDF,

2 MSTW) its PDF is not frozen at its input scale Q0, but allows description of the partonic 2 2 structure of the proton at Q ≤ Q0. The simulation used SAS 1D-LO [64] as photon-PDF. This was for several reasons. Most currently existing photon PDFs as we discussed in Sec. 2.4 (DG-G, LAC-G/GAL- G, GS-G, GS-G-96, GRV-G/GRS-G, ACFGP/AFG-G, WHIT-G and SAS-G(v1/v2)) are only constrained by ﬁts to the sparse F2 data from electron-positron colliders [65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80] before LEP; the older DO-G PDF is based on low energy photon-proton data. None of the existing Photon PDFs has HERA H1 [89]

61 220 b] Q2 = 0.2 GeV2 b] Q2 = 0.5 GeV2 µ µ 140 [ 200 [ PYTHIA p PYTHIA p γ γ

σ HERA Data σ HERA Data 180 120 160

140 100 120

100 80

80 60 60

0.8 PYTHIA 10-5 10-4 0.8 10PYTHIA-5 10-4 10-3 resolved resolved 0.6 VMD 0.6 VMD direct direct 0.4 0.4

subprocess fraction 0.2 subprocess fraction 0.2

0 0 10-5 10-4 10-3 10-5 10-4 10-3 xBj xBj

γp 2 Figure 5.1: The σ xBj,Q simulated with PYTHIA-6 using CTEQ5m and SAS 1D-LO as γp 2 proton and photon PDFs,( respectively,) in comparison with the σ xBj,Q as extracted from + 2 2 2 2 the HERA e p data [63]. Left: Q 0.2 GeV , Right: Q 0.5 GeV( . Bottom:) the diﬀerent subprocess fractions for resolved, VMD= and direct photon processes= as a function of xBj.

62 or ZEUS [92, 93] data sensitive to the partonic structure of the photon included in the ﬁts. Ref. [22, 89, 91] discuss that the H1 data are best described by the SAS and GRV Photon PDFs. But as none of the Photon PDFs provide an evaluation of an uncertainty band as is standard for the current Proton PDFs, and with the statistical precision of the HERA data remaining limited, no real quantitative preference for any of the photon PDF sets can be determined as we learn from Sec. 2.4. The SAS PDF is best suited for use in PYTHIA, since the vector meson and anomalous photon components are unfolded, thus avoiding double counting of resolved photon subprocesses. Fig. 5.1 (upper plots) shows an good agreement between the PYTHIA-6 simulation using CTEQ5m as the Proton PDF and SAS 1D-LO as the Photon PDF and the low Q2 data from HERA [63]. The lower parts of Fig. 5.1 show the fraction of direct, resolved (hard VMD and anomalous) photon and VMD processes.

FastJet [94] is used for jet reconstruction. The kinematics are constrained to the region of interest for photoproduction by requiring that the scattered electron remains in the beam pipe, undetected in the main detector. The photon virtuality is therefore restricted to Q2 < 0.1 GeV2 and Q2 > 10−5 GeV2, according to the lower limit of the low Q2-tagger acceptance.

5.2 Validation with HERA data

The variable xγ can be reconstructed from the momenta and angles of di-jets as 1 −η1 −η2 xγ = (pT,1e + pT,2e ), (5.1) 2Eey where Ee is the electron beam energy and y is the energy fraction taken by the photon from the electron (y Eγ ). Eq. 5.1 is valid in the lab frame in LO. = Ee The di-jet cross section measured by H1 at HERA [24] is shown in Fig. 5.2 as a jets function of the squared jet transverse energy ET in ranges of reconstructed xγ. Here jet1 jet2 jets jets ET +ET ET is the average transverse energy of the two highest pT jets: (ET = 2 ). jets ET is required to be above 10 GeV. The ratio of the diﬀerence and the sum of the jet1 jet2 SET −ET S transverse energies of the jets is required to satisfy jet1 jet2 < 0.25, and the transverse (ET +ET ) 63 107 PYTHIA [nb] ) 2 ) 106 H1 Data jets T rec 6 xγ =0.75•1 (×10 ) σ

2 5

d 10 xrec=0.6•0.75 (×105)

dlog((E γ rec γ 4

dx 10

rec 4 xγ =0.5•0.6 (×10 ) 103

rec 3 xγ =0.4•0.5 (×10 ) 102 rec 2 xγ =0.3•0.4 (×10 )

10 rec xγ =0.2•0.3 (×10)

rec xγ =0.1•0.2 1 103 jets (E )2 [GeV2] T

Figure 5.2: Comparison of the di-jet cross section extracted from the PYTHIA simulation jet1 jet2 jets SET −ET S with the HERA data. The kinematics cuts are from HERA: ET 10 GeV, jet1 jet2 0.25, (ET +ET ) 2 2 the photon virtuality Q 4 GeV , the fractional photon energy is> between 0.2 y 0.83,< and ηjet1+ηjet2 jets the average of pseudo-rapidity< of the two jets is restricted to 0 2 2 and< <∆η 1. The H1 data is from [24]. < < S S < energy of individual jets is required to be above 7.5 GeV. The fractional photon energy is restricted to 0.2 < y < 0.83. The average of the pseudo-rapidity of the two jets is ηjet1+ηjet2 restricted to 0 < 2 < 2, and the difference of the jet pseudo-rapidities is required to be within S∆ηjetsS < 1. The simulation results are obtained for 27 GeV electrons colliding with protons of 820 GeV, and the comparison of our simulation with the H1 data shows that the simulation reproduces the measured data well. Some of the observed difference is due to the use of the anti-kT algorithm [94] for the jet finding instead of the cone algorithm used for the HERA results.

64 1 gen γ x 0.8 102

0.6

0.4 10

0.2 1 0 0 0.2 0.4 0.6 0.8 1 rec xγ

gen rec Figure 5.3: Correlation between xγ and xγ . A di-jet is reconstructed in each event with stable particles, the particles inside the jets require to be pT 250 MeV and 4.5 η 4.5. −5 2 2 2 jets jet1 The di-jet events are selected with 10 GeV Q 0.1 GeV>, N 2, trigger− jet< pT< 5 jet2 GeV, and associated jet pT 4.5 GeV. < < ≥ > > 3 ×10 1.4

1.2 counts 1

0.8

0.6 Resolved Direct 0.4

0.2

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 xrec γ

rec Figure 5.4: xγ distributions in resolved and direct processes.

65 5.3 Unpolarized photon structure at EIC

5.3.1 Separation between direct and resolved processes

In this analysis, jets are reconstructed with the anti-kT algorithm, which is based on the energy distribution of final state particles in the angular space. All the stable and visible particles produced in the collision with pT > 250 MeV and −4.5 < η < 4.5 in the laboratory system are taken as input. The jet cone radius parameter has been set to R = 1 in the jet finding algorithm. This simulation is performed for the planned EIC electron and proton beam energy configuration of 20 GeV × 250 GeV. We consider jet events with two or more jets; 85% of all events have exactly two jets (pT > 3 GeV).

In each event, the jet with the highest pT is referred to as the trigger jet, and the jet with the second highest pT the associated jet. Events are selected with the requirement

jet jet −5 that the trigger jet has pT > 5 GeV, the associated jet has pT > 4.5 GeV, and 10 GeV2 < Q2 < 0.1 GeV2. The average transverse momentum of the trigger and associated jet1 jet2 di-jet pT +pT 2 jets is pT = 2 . In this analysis, the event kinematic variables Q and y are obtained directly from PYTHIA simulations without reconstructing them from the event information. The variable y = Eγ~Ee can be experimentally reconstructed in two ways. The scattered lepton, if detected in the low Q2-tagger as described in Sec. 3.1.2, provides a direct measurement of y. The other possibility to reconstruct y is through the Jacquet- Blondel method [98], which utilizes the hadronic ﬁnal state. Reference [104] discussed this method and its performance for charged current events at an EIC.

We reconstruct xγ in di-jet events according to Eq. 5.1. The strong correlation be-

rec gen tween the reconstructed xγ and the true xγ in PYTHIA is shown in Fig. 5.3. It clearly rec shows that the di-jet observable is ideal for this measurement. The xγ distribution for rec the resolved (direct) process dominates in the low (high) xγ regime (see Fig. 5.4), which provides good separation of the two types of processes. For example, by selecting events

rec rec with xγ ∼ 0 or xγ ∼ 1, one can divide the di-jet events into subsamples in which rec the resolved and direct processes dominate, respectively. As a smaller xγ cut is chosen higher purity for the resolved process is obtained. Considering the balance between

66 109 108 [nb] ) 2 ) 107

di-jet T 106

σ 5 2 10 d 4

dlog((p 10 rec γ 103 dx 102 10

-1 1 ∫Ldt = 1 fb rec 7 xγ =0.7-1 (×10 ) rec 6 -1 xγ =0.6-0.7 (×10 ) 10 rec 5 xγ =0.5-0.6 (×10 ) rec × 4 -2 xγ =0.4-0.5 ( 10 ) rec 3 10 xγ =0.3-0.4 (×10 ) rec × 2 -3 xγ =0.2-0.3 ( 10 ) rec 10 xγ =0.1-0.2 (×10) rec xγ =0.01-0.1 10-4 102 di-jet (p )2 [GeV2] T

Figure 5.5: The unpolarized di-jet cross section dependence on the average transverse mo- jet1 jet2 di-jet pT +pT mentum of the jets pT 2 and the reconstructed xγ for an integrated luminosity −1 2 −5 2 2 2 jets of 1 fb . Low Q events are= selected: 10 GeV Q 0.1 GeV , N 2. The anti-kT jet1 jet2 algorithm is used with R 1. For the trigger jet pT < 5 GeV,< and the associated≥ jet pT 4.5 GeV. = > >

rec statistics and purity, xγ < 0.6 is chosen; with this cut the fraction of the resolved process

(Nres~(Nres + Ndir)) is up to 91.2%.

5.3.2 Unpolarized di-jet cross section

Fig. 5.5 shows the resulting high precision double diﬀerential di-jet cross section over a wide kinematic range with an integrated luminosity of 1fb−1. With a global ﬁt the unpolarized photon PDFs can be extracted from the cross section.

67 5.3.3 Flavor tagging

The resolved process has several types of subprocesses, divided into 7 types: qq → qq

(which means qiqj → qiqj, with qiqj standing for both quark or anti-quark), q(photon)g → q(photon)g, g(photon)q → g(photon)q, gg → gg, qq¯ → qq¯ (which means qiq¯i → qkq¯k), gg → qq¯ and qq¯ → gg. Since the first four types of subprocess account for more than 96% of the resolved process, we mainly discuss these four types. The subprocess fraction depends on the average transverse momentum of the di-jet, as shown in the left of Fig. 5.6. di-jet The process qq → qq is more likely to dominate in the large pT region. Gluon jets produced in gg → gg process are softer. As shown in the right of Fig. 5.6, the fraction of the different subprocesses depends on the average pseudo-rapidity. The qg → qg process dominates in the negative ηdi-jet region. In order to precisely determine the photon PDF for different parton components, it is important to devise an experimental handle on the flavor of the parton involved in the hard interaction from the photon side. Tagging the parton flavor through identified hadrons in jets has recently become an important tool, especially in pp collisions, to study PDFs and fragmentation functions. (For theoretical details and first experimental results see Ref. [105, 106, 107, 108] and [109, 110, 111]. In the following, we apply this method to tag the parton flavor content of the photon at an EIC.

We demonstrate in this section that the outgoing jet close to the electron beam pseudo-rapidity is more likely to take the incoming parton flavor from the photon side. The leading hadron species inside those jets are found to be strongly correlated to the underlying parton flavors. Then a straightforward strategy is to tag the parton flavor of the photon through the leading hadron type inside the photon side jet. We find a high cut on the transverse momentum fraction of the leading hadron will enhance the sensitivity to the parton flavor even further.

In the following we call the parton coming from the photon the “beam parton” and the one coming from the proton the “target parton”. In a leading order 2 → 2 scattering process in quasi-real photoproduction events, the beam parton is converted to the jet from the photon side, while the target parton to the jet from the proton side. In our

68 1 1 qg→qg qg→qg 0.9 qq→qq 0.9 qq→qq 0.8 gq→gq 0.8 gq→gq

fraction → fraction → 0.7 gg gg 0.7 gg gg 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 6 8 10 12 14 16 18 20 -1 -0.5 0 0.5 1 1.5 2 2.5 3 pdi-jet [GeV] ηdi-jet T

Figure 5.6: Left: the fraction of the major subprocesses of the resolved process dependence

di-jet di-jet ηjet1+ηjet2 on pT . Right: the fraction distribution dependence on η 2 . =

103 103 1.8 × × Jet from the photon side Jet from the proton side 1.4 qg→qg qg→qg qg→qg 1.6 qq→qq qq→qq qq→qq 1.2 gq→gq counts 1.4 gq→gq gq→gq counts gg→gg gg→gg 1.2 gg→gg 1

1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 •3 •2 •1 0 1 2 3 4 5 •4 •2 0 2 4 jet η •η η γ p

Figure 5.7: Left: the pseudo-rapidity distribution of jets from the photon side and proton side in diﬀerent subprocesses. Right: ∆η distribution between jets from the photon side and jets from the proton side on an event-by-event analysis.

69 aoso h htnfrtredffrn uso the on cuts different three for photon the of flavors 5.8 Figure

P P 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.1 rbblt ffidn edn hre arndpnigo ieetparton different on depending hadron charged leading a finding of P Probability : 0 1 01 02 035 30 25 20 15 10 5 0 u quark u p quark • K • π • π + p p no cut T T fraction>0.7 fraction>0.4 K + p + 0.7 0.8 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.9 0.2 0.3 0.4 0.5 0.6 0.9 0.1 0.1 0 0 1 1 01 02 035 30 25 20 15 10 5 0 d d quark p quark • K • 70 π • p T π + fraction. K + p + 0.7 0.8 0.7 0.8 0.2 0.3 0.4 0.5 0.6 0.9 0.2 0.3 0.4 0.5 0.6 0.9 0.1 0.1 0 0 1 1 p 01 02 035 30 25 20 15 10 5 0 s quark T s quark p rcini esrdas measured is fraction • K • π • π + K + p + p p T jet T L . simulation, it is possible to apply a geometric match between the outgoing partons and the jets. Therefore, in each di-jet event, the jet from the photon side and the jet from the proton side are accessible in the simulation. We find in our simulation that the pseudo-rapidity distribution of the outgoing jets from the photon and proton sides are distinguishable, as shown in the left of Fig. 5.7; an exception is that gluon jets from both sides overlap with each other in the gg → gg process. If we define the proton beam direction as the positive pseudo-rapidity direction, jets from the photon side dominate at more negative pseudo-rapidities compared with jets from the proton side.

On an event-by-event basis, the pseudo-rapidity difference between the two outgoing jet jet jets ∆η = ηγ − ηp can be used to identify the photon and proton side jets. As shown in the right of Fig. 5.7, the value of ∆η is mostly negative. The gg → gg process is the only exception, so the quark jet from the photon side can be well identified. For 82.0% of the events the jets from the photon side take the more negative pseudo-rapidity than those from the proton side, which provides an experimental way to separate jets from the photon side and jets from the proton side: in each event, we take the jet with more negative pseudo-rapidity as the jet from the photon. To determine the involved parton flavor of the photon, we need to use the information from the charged hadron with the highest pT (leading hadron) inside the photon side jet. The correlation between the leading hadron type and the underlying parton flavor is shown in Fig. 5.8. The photon side jet from a u quark in the initial state has most likely a π+ as leading hadron. Similarly, the photon side jet from an s quark is more likely to contain a leading K−. The sensitivity of the parton flavor to the leading hadron type can be enhanced further L pT with a higher leading hadron pT fraction ( jet ) cut. This relation is particularly strong pT for strange quarks and kaons.

In conclusion, it is possible to tag the parton ﬂavor of the photon by selecting the outgoing jet with the more negative pseudo-rapidity, and applying a cut on the leading hadron type with the requirement of the leading hadron carrying a high pT fraction. In Fig. 5.9, we present the ﬂavor distribution of the beam parton after selecting those jets with the leading hadron to be a pion or kaon. If the leading particle of the jet is a π+ (π−), the most likely scenario is that this jet originated from a u (¯u) quark. For K+

71 arntp ntepoo iejtfrtredﬀrn uso the on cuts diﬀerent three for jet side photon the in type hadron 5.9 Figure

rbblt ftepro ao ntepoo e eedn nteleading the on depending jet photon the in flavor parton the of P Probability : P P 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.1 0 1 01 02 03 045 40 35 30 25 20 15 10 5 0 π π s • + u d d p p p T T T s u fraction>0.3 fraction>0.6 fraction>0.8 72 g 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.1 0 0 1 1 01 02 03 045 40 35 30 25 20 15 10 5 0 K K s + • u p d T fraction. d s u g ijtcossection cross di-jet EIC at structure photon Polarized 5.4 gluon and quark 4.4 . separate Sec. to in method discussed a has gluons; EIC by an initiated at jets also be can jet side photon ihteplrzdcosscindfie as defined section cross polarized the with ( EIC, at events di-jet selected the of scenarios 5.10 Figure oaie htnPF n rtnPF,rsetvl.Terlvn rs eto asym- section cross relevant The respectively. PDFs, proton and PDFs photon polarized and K −

h ogtdnlyplrzdpoo Dscnb xrce esrn h polarized the measuring extracted be can PDFs photon polarized longitudinally The γ γ ), + As Au , •0.8 •0.4 •0.8 •0.4 •0.6 •0.2 •0.6 •0.2 u 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 •1 − 10 0 0 1 (¯ •2 u eoigtehlct ftesatrn atce.∆ particles. scattering the of helicity the denoting n ¯ and ) htncpro asymmetries parton Photonic : min.pol. max.pol. s dx ( d s urshv h ihs rbblt fbigteiiilqak The quark. initial the being of probability highest the have quarks ) 2 γ ∆ dQ γ γ σ 10 2 •1 = ∆ γ flux ∆ σ ⊗ = ∆ 1 2 ( f x σ γ rec γ ( ( 1 x ++ A 73 i γ i γ = Q ,

) γ γ ≡ ,d ,g s, d, u, A A − g d 2 ∆ •0.4 •0.8 •0.4 •0.8 •0.2 •0.6 •0.2 •0.6 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 µ , σ •1 10 0 0 f 1 ( i γ ) +− •2 ~ ⊗ f . )) i γ ∆ , o h mxml n “minimal” and “maximal” the for f p ( x p µ , f γ ) n ∆ and ⊗ 10 •1 σ ij f p ersn the represent x γ rec (5.2) (5.3) 1 metry measured experimentally is ALL = ∆σ~σ. Because PYTHIA does not incorporate spin dependent cross sections, this information needs to be constructed externally. A relatively straightforward way of doing this is to calculate asymmetries on an event-by- event basis and then apply the asymmetry as an event weight. For this analysis, event weights were calculated depending on the kinematics and subprocesses as generated in PYTHIA, and applied in an external analysis of the PYTHIA output. The event information available from PYTHIA is the kinematics (x, Q2). The asymmetry weight for a given process can be constructed as

∆f γ x , µ2 ∆f p x , µ2 w aˆ s,ˆ t,ˆ µ2,Q2 a ( a ) b ( b ) (5.4) = ( ) × γ 2 × p 2 fa (xa, µ ) fb (xb, µ ) wherea ˆ is the hard subprocess asymmetry. The leading-order formulas for helicity- dependent and helicity-averaged cross sections for scattering of partons in the PGF, QCDC and DIS subprocess are taken from [113] and the lowest order equations [114] for the resolved subprocesses asymmetries are obtained from [115].

γ γ 2 ∆f ∆fa (xa,µ ) The second term of Eq. 5.4, γ , is the photonic parton asymmetry. γ 2 1 f fa (xa,µ ) = for xa = 1 in direct photon processes (PGF, QCDC, DIS). In resolved processes (hard QCD 2 → 2 processes), f γ is the unpolarized photon PDF; we use the aforementioned SAS 1D-LO PDF. ∆f γ(x, µ2), the parton distributions of longitudinally polarized photons, are experimentally completely unknown. In this analysis, two very diﬀerent scenarios [116, 117, 118, 119] for the polarized photon PDFs were considered, assuming

“maximal” (∆f γ(x, µ2) = f γ(x, µ2)) or “minimal” (∆f γ(x, µ2) = 0) polarization based on the positivity constraints

γ 2 γ 2 S∆f (x, µ )S ≤ f (x, µ ) (5.5) at the input scale µ (also commonly referred to as Qˆ2), where µ is deﬁned to be

1 µ2 pˆ2 Q2 (5.6) = T + 2

To take the u quark as an example, the results of the two assumptions are presented

γ γ γ in the top left of Fig. 5.10 in terms of the photonic parton asymmetries Af ≡ ∆f ~f , for our event selection at Q2 < 0.1 GeV2 in LO. These sets are used in the following 74 p u 1 A 0.9 DSSV 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 10•2 10•1 1 gen xp

p p p Figure 5.11: protonic parton asymmetry Au ∆fu fu for the DSSV polarized proton PDF.

≡ ~ to calculate the di-jet double spin asymmetry ALL. The third term of Eq. 5.4 is the

p p p parton asymmetry in the proton, deﬁned as Af ≡ ∆f ~f . In the simulation, the input for f p is the unpolarized proton PDF of CTEQ5m and we choose DSSV [120, 121] for the polarized proton PDF; an example of u quark is shown in the Fig. 5.11.

5.4.1 Polarized di-jet cross section at EIC

The ﬁnal polarized di-jet cross section is measured with the same selection criteria as for the unpolarized cross section and applying the event weights as described before. Fig. 5.12 shows the polarized di-jet cross section using the “maximal” polarization scenario for the partons in the photon. The polarized di-jet cross section can be measured at an EIC over a wide kinematic range with high accuracy.

5.4.2 Asymmetry

di-jet In Fig. 5.13 ALL as a function of pT is shown. Here, the statistical errors δA are estimated from 1 δA = √ , (5.7) Lσ

75 106

5 [nb]

) 10 2 ) 104 di-jet T

σ 3

∆ 10 2 d

dlog((p 102 rec γ

dx 10

-1 ∫Ldt = 1 fb rec 7 -1 xγ =0.7-1 (×10 ) 10 rec 6 xγ =0.6-0.7 (×10 ) rec 5 xγ =0.5-0.6 (×10 ) -2 rec × 4 10 xγ =0.4-0.5 ( 10 ) rec 3 xγ =0.3-0.4 (×10 ) rec × 2 -3 xγ =0.2-0.3 ( 10 ) rec 10 xγ =0.1-0.2 (×10) rec xγ =0.01-0.1 10-4 102 103 di-jet (p )2 [GeV2] T

Figure 5.12: The measured di-jet cross section in polarized ep collision as a function of the squared jet transverse momentum for the range of the reconstructed parton fractional momentum. The kinematics are the same as in Fig. 5.5. where the integrated luminosity L = 1fb−1. In the top left plot of Fig. 5.13, the asym- rec metry is displayed by applying a cut xγ < 0.6 to select a region where the resolved processes dominate. The “maximal” and “minimal” scenarios for the polarization of the partons in the photon lead to a signiﬁcant diﬀerence in the predicted asymmetry. The

rec structure of the photon plays an important role in this region. Approaching the large xγ region, the direct processes start to dominate. In this kinematic region the asymmetry is dominated by the polarization of the partons in the proton, the photon is mainly a point-like particle, therefore the two scenarios converge to the same ALL.

di-jet The bottom of Fig. 5.13 shows ALL as a function of η . The overall behavior of

di-jet di-jet the asymmetry as a function of η follows the one as a function of pT , showing

76 0.05 0.05 LL LL •1 A L = 1 fb , xrec < 0.6 A xrec > 0.6 0.04 ∫ γ 0.04 γ max.pol.γ 0.03 0.03 min.pol.γ 0.02 0.02

0.01 0.01

0 0

•0.01 •0.01 5 6 7 8 9 10 11 12 5 6 7 8 9 10 11 12 pdi•jet [GeV] pdi•jet [GeV] T T

0.05 0.05 LL rec LL rec A x < 0.6 A x > 0.6 0.04 γ 0.04 γ

0.03 0.03

0.02 0.02

0.01 0.01

0 0

•0.01 •0.01 •0.5 0 0.5 1 1.5 2 2.5 •0.5 0 0.5 1 1.5 2 2.5 ηdi•jet ηdi•jet

di-jet di-jet Figure 5.13: Top: the cross section asymmetry as a function of pT (top) and η (bot- rec tom). The two columns show the asymmetry for two xγ regions to enhance the contribution from the resolved photon processes (left) and the direct photon processes (right). The kinematics are the same as in Fig. 5.5.

77 7 6 jets 101.075 GeV < E < 101.15 GeV 1.15 jets 1.25 T 10 GeV < ET < 10 GeV [nb] [nb] ) 6 H1 data ) H1 data 2 2

) ) 5 µ2 PYTHIA HERA, = 2 PYTHIA HERA, µ2 = 2 jets T PYTHIA HERA, µ2 = 1 jets T 5 PYTHIA HERA, µ2 = 1 PYTHIA HERA, µ2 = 0.5 4 σ σ PYTHIA HERA, µ2 = 0.5 2 -1 2

d EIC, ∫Ldt = 1 fb d -1 4 EIC, ∫Ldt = 1 fb dlog((E dlog((E 3 rec γ 3 rec γ dx dx 2 2

1 1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

EIC 0.08 rec EIC 0.08 rec

HERA xγ HERA xγ 0.06 0.06

0.04 0.04

0.02 0.02

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 rec rec stat. uncertainty xγ stat. uncertainty xγ stat. uncertainty stat. uncertainty

jets Figure 5.14: The di-jet cross section at EIC and HERA for two diﬀerent ET bins as a rec function of xγ . The dashed curves represent the PYTHIA-6 simulation for HERA with varying the renormalization and factorization scale µ2 by 0.5 (magenta) and 2 (red), respectively. The EIC kinematics are the same as in Fig. 5.5. The H1 data are taken from [24]; the inner error bars represent the statistical uncertainties, the outer ones are the quadratic sum

1.075 jets 1.15 of the statistical and systematic uncertainties. Left: 10 GeV ET 10 GeV. Right: 1.15 jets 1.25 10 GeV ET 10 GeV. Bottom panel: The ratio of the statistical< uncertainties< between predicted for< EIC< and the measured HERA data.

rec a signiﬁcant diﬀerence in the xγ region where the resolved processes dominate. In conclusion, the cross section asymmetry is sensitive to the polarization of the partons in the photon in the resolved photon processes, and the polarized photon PDFs can be well constrained by measuring ALL at an EIC.

5.4.3 Advantages compared with HERA

Based on the results discussed in the former sections, the main advantages of an EIC to constrain the unpolarized photon PDFs can be summarized as:

1. The existing world data both from e+e− collisions and HERA are statistically lim- 78 q 2 2 g γ

= 30 GeV γ x T LAC-G/GAL-G LO x 10-1 GRV-LO SAS 1D-LO 10-2 10-2

10-3

10-4

HERA xγ range HERA xγ range -5 EIC xγ range 10 EIC xγ range

10-2 10-1 1 10-2 10-1 1 xγ xγ

Figure 5.15: The quark (left) and gluon (right) distribution functions of the photon for three

2 diﬀerent sets: LAC-G (magenta), GRV-LO (red) and SAS 1D-LO (blue) at the average pT = 2 30 GeV . Indicated are the xγ ranges covered by the HERA data and as anticipated for the EIC data.

ited. As described earlier, the existing photon PDFs do not provide an evaluation of their uncertainty bands, which makes it impossible to reach a quantitative as- sessment of the impact from EIC data. The HERA data are consistent within uncertainties with GRV-LO and SAS 1D-LO (see Fig. 7 in Ref [22]). Fig. 5.14 shows a comparison of the statistical precision of the di-jet cross section for two

jets rec bins in ET with overlapping kinematics at HERA and EIC as a function of xγ . The superior statistical precision of an EIC (bottom panels) will allow a precision determination of the photon PDFs and their uncertainties. The HERA data and PYTHIA-6 simulation are the same as shown in Fig. 5.2. Also shown in Fig. 5.14 (top panels) is the variation of the renormalisation and factorization scale by 0.5 and 2; the scale dependence is small.

2. The high statistical precision will be critical to constrain the photon PDFs at

lower xγ. At EIC xγ > 0.01 can be reached compared with xγ > 0.05 at HERA [91] + − and xγ > 0.01 for the e e data [112] used to constrain the photon PDFs. From

Fig. 5.15, it can be seen that the diﬀerent photon PDFs diverge at xγ < 0.1, a region where a high statistics measurement can diﬀerentiate between them.

3. The current world data do not provide any information to disentangle the different quark flavors. The described tagging method provides a new way to independently 79 constrain the separate (anti-)quarks flavors, which is a significant step forward.

5.5 Summary

The hadronic structure of the photon can be accessed at low Q2 in deep inelastic scattering through tagging resolved photon processes. We have shown in a detailed analysis the capability of a future EIC to perform di-jet measurements to extract (un)polarized photon PDFs: di-jets produced in direct and resolved process can be well separated by reconstructing xγ, which has a strong correlation with the true xγ, and one can effectively extract the underlying photon PDFs by measuring di-jet cross sections in photoproduction events. Jets from the photon side can be identified by selecting the more negative pseudo-rapidity jet in each event. Moreover, it is possible to probe the content of the photon by tagging leading hadrons inside the jets from the photon side; the flavor of the originating quark is highly correlated with the identified hadron. With polarized beams, the polarized photon PDFs, which are totally unknown so far, can be extracted at an EIC.

80 CHAPTER 6

Summary and outlook

In this thesis, jets at a future EIC have been studied. The possibility of measuring jets is explored with our Monto Carlo simulation method. We discussed the diﬀerent jet reconstruction algorithms, jet kinematics and detector requirements for measuring jets in the experiment. By analyzing di-jet events, the background from underlying events are calculated, both the region method and oﬀ-axis method are applied in the analysis. How to distinguish jets originate from quark and gluon still remains a challenge at an EIC. Since some variables shown in the discrimination are not independent, in the further research, the Toolkit for Multivariate Data Analysis with ROOT (TMVA) is expected to be a good method to be applied.

We also use a di-jet method to measure the photon structure in both unpolarized and polarized ep collisions. High pT di-jet events are selected, the photon PDFs can be extracted by calculating the di-jet cross section in the unpolarized case. While in the polarized case, the polarized PDFs can be extracted by measuring the asymmetry. These PDFs are important for the precious input of ILC γγ option.

After all, there are still open questions and ideas for future studies based on the current results. For example, we can think about considering the detector response in the simulation, which includes the detector coverage and resolution. At eRHIC, a fast smearing method implemented in the eic-smear package is a systematic way to study the eﬀects of detector resolution, this method will bring us to measure the detector response.

81 References

[1] M. Planck. On the Law of Distribution of Energy in the Normal Spectrum[J]. An- nalen der Physik. 4 (3), 553 (1901).

[2] A. Einstein. Uber¨ einen die Erzeugung und Verwandlung des Lichtes betreﬀenden heuristischen Gesichtspunkt[J]. Annalen der Physik. 17 (6), 132-148 (1905).

[3] V.V. Kobychev, S.B. Popov. Constraints on the photon charge based on observations of extragalactic sources[J]. Astron. Lett. 31, 147-151 (2005).

[4] T.F. Walsh, P. Zerwas. Two-photon processes in the parton model[J]. Phys. Lett. B 44, 195-198 (1973).

[5] W. Slominski, J. Szwed. On the Electron Structure Function[J]. Acta Phys. Polon. 27, 1887-1914 (1996).

[6] E. Witten. Anomalous cross section for photon-photon scattering in gauge theo- ries[J]. Nucl. Phys. B 120, 189-202 (1977).

[7] Armin B¨ohrer.Photon structure and heavy ﬂavour production in γγ collisions at LEP[DB]. [arXiv:hep-ex/0305029].

[8] A. De Roeck, Measurements of the photon structure function at LEP[J]. Eur. Phys. J. C 33, s394-s397 (2004).

[9] J.M. Butterworth. Photon structure as seen at HERA[C]. DESY 95-043. [arXiv:hep- ex/9503011].

[10] P.J. Bussey. The virtual photon structure at HERA[J]. Nucl. Phys. Proc. Suppl. 126, 17-21(2004).

[11] M. Acciarri et al.. Cross section of hadron production in γγ collisions at LEP[J]. Phys. Lett. B 408, 450-464 (1997). √ [12] G. Abbiendi. et al. Di-jet production in photon-photon collisions at s = 161 and 172 GeV[J]. Eur. Phys. J. C 10, 547-561 (1999).

[13] M. Acciarri. et al. Measurement of inclusive D production in two-photon collisions at LEP[J], Phys. Lett. B 467, 137-146 (1999).

[14] H1 Collaboration. Inclusive parton cross-sections in photoproduction and photon structure[J]. Phys. Lett. B 314, 436 (1993).

[15] H1 Collaboration. Comparison of deep inelastic scattering with photoproduction interactions at HERA[J]. Phys. Lett. B 368, 412-422 (1995).

[16] H1 Collaboration. Inclusive parton cross-sections in photoproduction and photon structure[J]. Nucl. Phys. B 445, 195-218 (1995).

82 [17] H1 Collaboration. Measurement of dijet cross-sections in low Q2 ep collisions and the extraction of an eﬀective parton density for the virtual photon[J]. Eur. Phys. J. C 13, 397-414 (2000).

[18] H1 Collaboration. Measurement of dijet cross sections in photoproduction at HERA[J]. Eur. Phys. J. C 25, 13-23 (2002).

[19] ZEUS Collaboration. Dijet cross-sections in photoproduction at HERA[J]. Phys. Lett. B 348, 665-680 (1995).

[20] ZEUS Collaboration. Dijet angular distributions in direct and resolved photoproduction at HERA[J]. Phys. Lett. B 384, 401 (1996).

[21] ZEUS Collaboration. High E(T ) inclusive jet cross-sections in photoproduction at HERA[J]. Eur. Phys. J. C 4, 591 (1998).

[22] H1 Collaboration. Charged particle cross-sections in photoproduction and extraction of the gluon density in the photon[J]. Eur. Phys. J. C 10, 363-372 (1999).

[23] S. Frixione, P. Nason, G. Ridolﬁ. Diﬀerential Distributions for Heavy Flavour Pro- duction at HERA[J]. Nucl. Phys. B 454, 3-24 (1995).

[24] H1 Collaboration. Measurement of the inclusive dijet cross-section in photoproduction and determination of an eﬀective parton distribution in the photon[J]. Eur. Phys. J. C 1, 97-107 (1998).

[25] J. Cvach. Real and virtual photon structure from dijet events[J]. Nucl. Phys. B, Proc. Suppl. 79, 501-504 (1999).

[26] A. Accardi, et al. Electron Ion Collider: The Next QCD Frontier - Understanding the glue that binds us all[J]. Eur. Phys. J. A 52, 268 (2016).

[27] H. David Politzer. Asymptotic freedom: An approach to strong interactions[J]. Phys. Rep. 14, 129-180 (1974).

[28] M. Gell-mann. A schematic model of baryons and mesons[J]. Phys. Lett. 8, 214-215 (1964).

[29] L.N. Lipatov. The parton model and perturbation theory[J]. Sov. J. Nucl. Phys. 20, 94-102 (1975).

[30] J. Beringer, et al. Review of Particle Physics (RPP)[J]. Phys. Rev. D 86, 010001 (2012).

[31] R.P. Feynman. Mathematical formulation of the quantum theory of electromagnetic interaction[J]. Phys. Rev. 80, 440-457 (2012).

[32] M. Gckeler, R. Horsley, V. Linke, et al. Is there a Landau pole problem in QED[J]. Phys. Rev. Lett 80, 4119-4122 (1998).

[33] V.N. Gribov. Lectures at the 12th Winter School of the Leningrad Nuclear Physics Insititute[R], SLAC-TRANS-176 (1977). 83 [34] S.D. Drell. [C]. Trans. NY Acad. Sci. Series II 40, 76 (1980).

[35] N.K. Nielsen. Asymptotic freedom as a spin eﬀect[J]. Am. J. Phys. 49, 1171 (1981).

[36] R.J. Hughes. Some Comments On Asymptotic Freedom[J]. Phys. Lett. B 97, 246- 248 (1980).

[37] R.J. Hughes. More Comments On Asymptotic Freedom[J]. Nucl. Phys. B 186, 376- 396 (1981).

[38] W.K.H. Panofsky, et al. Proposals for Initial Electron Scattering Experiments Using the SLAC Spectrometer Facilities[R]. SLAC-PROPOSAL-E-004, SLAC- PROPOSAL-E-04, SLAC-PROPOSAL-E-4.

[39] F.D. Aaron, et al. Combined measurement and QCD analysis of the inclusive ep scattering cross sections at HERA[J]. JHEP 1001, 109 (2010).

[40] P. Skands, Introduction to QCD[M]. [arXiv:1207.2389].

[41] T. Kinoshita. Mass singularities of Feynman amplitudes[J]. J. Math. Phys. 3, 650- 677 (1962).

[42] T.D. Lee, M. Nauenberg. Degenerate Systems and Mass Singularities. Phys. Rev. 133, B1549-B1562 (1964).

[43] Yuri L Dokshitzer. Calculation of the structure functions for deep inelastic scattering and e+e− annihilation by perturbation theory in Quantum Chromodynamics[J]. Sov. Phys. JETP 46, 641-653 (1977).

[44] V.N. Gribov, L.N. Lipatov. Deep inelastic ep scattering in perturbation theory[J]. Sov. J. Nucl. Phys. 15, 438-450 (1972).

[45] G. Altarelli, G. Parisi. Asymptotic freedom in parton language[J]. Nucl. Phys. B 126, 298 (1977).

[46] P. Belov. Combination of the H1 and ZEUS inclusive cross-section measurements at proton beam energies of 460 GeV and 575 GeV and tests of low Bjorken-x phenomenological models[D], (2013). [DESY-THESIS-2013-017].

[47] G.F. Sterman. Partons, factorization and resummation[R]. [hep-ph/9606312]

[48] G.C. Blazey, J. R. Dittmann, S. D. Ellis, et al. Proceedings of the Run II QCD and Weak Boson Physics Workshop[C]. [arXiv:hep-ex/0005012].

[49] H. Abramowicz, K. Charchula, A. Levy. Parametrization of parton distributions in the photon[J]. Phys. Lett. B 269, 458-464 (1991).

[50] AMY Collaboration. Charge asymmetry of hadron jets[J]. Phys. Lett. B 252, 491- 498 (1990).

[51] CELLO Collaboration. Contributed paper to the XXVth International Conference on HEP. Singapore, unpublished (1990). 84 [52] JADE Collaboration. Z. Phys. C 24, 231-245 (1984)

[53] PLUTO Collaboration, Phys. Lett. B 149, 421426 (1984); Z. Phys. C 26, 353358 (1984); Nucl. Phys. B 281, 365380 (1987)

[54] TASSO Collaboration, Z. Phys. C 31, 527535 (1986).

[55] TPC/2γ Collaboration, Z. Phys. C 18, 281287 (1983); Phys. Rev. Lett. 54, 763766 (1985); Phys. Rev. Lett. 58, 97100 (1987).

[56] K. Hagiwara, M. Tanaka, and I. Watanabe, Phys. Rev. D 51, 3197-3219 (1995).

[57] C. Berger, in International Symposium on Lepton Photon Interactions at High Energies, Ithaca, N. Y., 4-9 Aug, 1983, edited by D.G. Cassel and D.L. Kreinick, pages 376404, Lab Nucl. Studies (1983).

[58] M. Cacciari, G. P. Salam, G. Soyez. The Anti-k(t) jet clustering algorithm. JHEP 0804, 063 (2008).

[59] E.C. Aschenauer, et al. eRHIC Design Study - An Electron-Ion Collider at BNL[M]. [arxiv: 1409.1633].

[60] T. Sjostrand, S. Mrenna, P.Z. Skands. PYTHIA 6.4 Physics and Manual. JHEP 0605, 026 (2006).

[61] M.R. Whalley, D. Bourilkov, R.C. Group, The Les Houches accord PDFs (LHAPDF) and LHAGLUE[DB]. [arXiv:hep-ph/0508110].

[62] H.L. Lai, J. Huston, S. Kuhlmann, et al.. Global QCD Analysis of Parton Structure of the Nucleon: CTEQ5 Parton Distributions[J]. Eur. Phys. J. C 12, 375-392 (2000).

[63] I. Abt, A. M. Cooper-Sarkar, B. Foster, et al.. Investigation into the limits of perturbation theory at low Q2 using HERA deep inelastic scattering data[J]. Phys. Rev. D 96, 014001 (2017).

[64] G.A. Schuler, T. Sjoestrand. Low and high mass components of the photon distribution functions[J], Zeit. Phys. C 68, 607 (1995).

γ 2 [65] PLUTO Collaboration. Measurement of the photon structure function F2 (x, Q )[J]. Phys. Lett. B 142, 111 -118 (1984).

[66] PLUTO Collaboration. Measurement and QCD analysis of the photon structure 2 function F2(x, Q )[J]. Nucl. Phys. B 281, 365-380 (1987).

[67] PLUTO Collaboration. First measurement of the photon structure function F2[J]. Phys. Lett. B 107, 168-172 (1981).

[68] PLUTO Collaboration. Measurement of deep inelastic electron scattering oﬀ virtual photons[J]. Phys. Lett. B 142, 119 (1984).

[69] JADE Collaboration. Measurements of Energy Correlations in e+e− →Hadrons[J]. Z. Phys. C 24, 231 (1984). 85 γ [70] TOPAZ Collaboration. Measurement of the photon structure function F2 and jet production at TRISTAN[J]. Phys. Lett. B 322, 477-487 (1994).

γ [71] TPC/2γ Collaboration. Observation of scaling of the photon structure function F2 at low Q2[J]. Phys. Rev. Lett. 58, 97 (1987).

γ 2 [72] TPC/2γ Collaboration. Measurement of the photon structure function F2 (x, Q ) in the region 0.2 < Q2 < 7 GeV2[J]. Z. Phys. C 34, 1 (1987).

[73] AMY Collaboration. A Measurement of the photon structure function F2[J]. Phys. Lett. B 252, 491-498 (1990).

[74] AMY Collaboration. Measurement of the inclusive jet cross-section in photon- photon interactions at TRISTAN[J]. Phys. Lett. B 325, 248-256 (1994).

[75] OPAL Collaboration. Measurement of the photon structure function F2 (gamma) in the reaction e+e− → e+e− + hadrons at LEP[J]. Z. Phys. C 61, 199-208 (1994).

[76] OPAL Collaboration. Measurement of the charm structure function F2(c)(gamma) of the photon at LEP[J]. Phys. Lett. B 539, 13-24 (2002).

γ [77] OPAL Collaboration. Measurement of the photon structure function F2 at low x[J]. Phys. Lett. B 412, 225-234 (1997).

[78] TOPAZ Collaboration. Measurement of the inclusive cross-section of jets in γγ interactions at TRISTAN[J]. Phys. Lett. B 314, 149-158 (1993).

[79] ALEPH Collaboration. An Experimental study of γγ → hadrons at LEP[J]. Phys. Lett. B 313, 509-519 (1993).

[80] DELPHI Collaboration. First evidence of hard scattering processes in single tagged gamma gamma collision[J]. Phys. Lett. B 342, 402 (1995).

[81] M. Dress and K. Grassie, Parametrizations of the Photon Structure and Applications to Supersymmetric Particle Production at HERA[J]. Z. Phys. C 28, 451-462 (1985).

[82] H. Abramowicz, K. Charchuta and A. Levy. Parametrization of parton distributions in the photon[J]. Phys. Lett. B 269, 458-464 (1991).

[83] L. E. Gordon, J. K. Storrow. The parton distribution functions of the photon and 2 the structure function F2(x, Q )[J]. Z. Phys. C 56, 307 (1992). [84] L. E. Gordon, J. K. Storrow. New parton distribution functions of the photon[J]. Nucl. Phys. B 489, 405 (1997).

[85] M. Gluck, E. Reya, A. Vogt. Parton structure of the photon beyond the leading order[J]. Phys. Rev. D 45, 3986 (1992); Phys. Rev. D 46, 1973 (1992) ; M. Gluck, E. Reya, M. Stratmann. Parton content of virtual photons[J]. Phys. Rev. D 51, 3220 (1995).

86 [86] P. Aurenche, J. P. Guillet, M. Fontannaz. Parton distributions in the photon[J]. Z. Phys. C 64, 621 (1994); P. Aurenche, P. Chiappetta, M. Fontannaz. Higher order QCD corrections to the photoproduction of a direct photon at HERA[J]. Z. Phys. C 56, 589 (1992).

[87] G. A. Schuler, T. Sjostrand. Parton distributions of the virtual photon[J]. Phys. Lett. B 376, 193 (1996).

[88] D.W. Duke, J.F. Owens. Quantum-chromodynamic corrections to deep-inelastic compton scattering[J]. Phys. Rev. D 26, 1600-1609 (1982).

[89] H1 Collaboration. Low Q2 jet production at HERA and virtual photon structure[J]. Phys. Lett. B 415, 418-434 (1997).

[90] X. Chu, B. Page and E.C. Aschenauer in preparation.

[91] H1 Collaboration. Measurement of di-jet cross-sections in photoproduction and photon structure[J]. Phys. Lett. B 483, 36-48 (2000)

[92] ZEUS Collaboration. Dijet photoproduction at HERA and the structure of the photon[J]. Eur. Phys. J. C 23, 615-631 (2002).

[93] ZEUS Collaboration. Study of the photon remnant in resolved photoproduction at HERA[J]. Phys. Lett. B 354, 163-177 (1995).

[94] M. Cacciari, G.P. Salam. FastJet.fr[J]. Phys. Lett. B 641, 57 (2006).

[95] D. Boer, M. Diehl, R.Milner, R. Venugopalan, et al. Gluons and the quark sea at high energies: distributions, polarization, tomography[DB]. [arXiv:1108.1713].

[96] S.J. Brodsky, A.L. Deshpande, H. Gao, et al. QCD and Hadron Physics[DB]. [arXiv:1502.05728].

[97] U. Heinz, P. Sorensen, A. Deshpande, et al. Exploring the properties of the phases of QCD matter research opportunities and priorities for the next decade[DB]. [arXiv:1501.06477].

[98] F. Jacquet, A. Blondel. Report From The Study Group On Detectors For Charged Current Events - in: U. Amaldi et al DESY, Hamburg, 1979, pp. 391; For a recent overview of diﬀerent methods to determine kinematic variables in DIS see: J. Blumlein, Prog. Part. Nucl. Phys. 69, 28 (2013).

[99] T. Aﬀolder, et al., (CDF Collaboration). Charged jet evolution and the underlying event in proton-antiproton collisions at 1.8 TeV[J]. Phys. Rev. D 65, 092002 (2002).

[100] Z. Chang, Inclusive jet longitudinal double-spin asymmetry all measurements in 510 GeV polarized pp collisions at STAR[D].

[101] B. Abelev, et al. (ALICE Collaboration). Charged jet evolution and the underlying event in proton-antiproton collisions at 1.8 TeV[J]. Phys. Rev. D 91, 112012 (2015).

87 [102] J. Bielcikova (STAR Collaboration). Jet reconstruction and underlying event studies in p+p and d+Au collisions from STAR[DB]. [arXiv:1107.4891]

[103] G. Altarelli, G. Parisi. Asymptotic Freedom in Parton Languag[J]. Nucl. Phys. B126, 298 (1977).

[104] E.C. Aschenauer, T. Burton, T. Martini, et al. Helicity Parton Distributions at a future electron-ion collider: A Quantitative Appraisal[J]. Phys. Rev. D 88, 114025 (2013).

[105] Z.B. Kang, A. Prokudin, F. Ringer, et al. Collins azimuthal asymmetries of hadron production inside jets[DB]. [arXiv:1707.00913].

[106] Z.B. Kang, X. Liu, F. Ringer, et al. The transverse momentum distribution of hadrons within jets[DB]. [arXiv:1705.08443].

[107] T. Kaufmann, A. Mukherjee, W. Vogelsang. Hadron fragmentation inside jets in hadronic collisions[J]. Phys. Rev. D 92, 054015 (2015).

[108] T. Kaufmann, A. Mukherjee, W. Vogelsang. Access to photon fragmentation functions in hadronic jet production[J]. Phys. Rev. D 93, 114021 (2016).

[109] The ATLAS Collaboration. Measurement of jet fragmentation in 5.02 TeV proton- lead and proton-proton collisions with the ATLAS detector[DB]. [arXiv:1706.02859]. √ [110] J.K. Adkins. Azimuthal single-spin asymmetries of charged pions in Jets in s = 200 GeV in pp collisions at STAR[J]. Int. J. Mod. Phys. Conf. Ser. 40, 1660040 (2016).

[111] J.K. Adkins. Constraining transversity and nucleon transverse-polarization structure through polarized-proton collisions at STAR[C]. PoS DIS2015, 193 (2015).

g [112] L3 Collaboration. The Q2 evolution of the hadronic photon structure function F2 at LEP[J]. Phys. Lett. B 447, 147-156 (1999).

[113] A. Bravar, K. Kurek, R. Windmolders. POLDIS: A Monte Carlo for POLarized (semiinclusive) deep inelastic scattering[J]. Comput. Phys. Commun. 105, 42-61 (1997).

[114] J. Babcock, E. Monsay, D.W. Sivers. Quantum- chromodynamic predictions for inclusive spin-spin asymmetries at large transverse momentum[J]. Phys. Rev. D 19, 1483 (1979).

[115] The HERMES Collaboration. Leading-order determination of the gluon polarization[J]. J. High Energ. Phys. (2010) 2010: 130.

[116] M. Gluck, W. Vogelsang. Polarized parton distributions of the photon[J], Z. Phys. C 55, 353-356 (1992).

[117] M. Gluck, W. Vogelsang. Current conservation and the parton structure of the longitudinally polarized photon[J]. Z. Phys. C 57, 309-310 (1993).

88 [118] M. Gluck, M. Stratmann, W. Vogelsang. Polarized photon structure at e+e− colliders[J]. Phys. Lett. B 337, 373-375 (1994).

[119] J. M. Butterworth, N. Goodman, M. Stratmann, et al. Photoproduction Processes in polarized ep collisions at HERA[R]. CERN-TH/97-310, DTP/97/98, UCL/HEP 97-08 (1997). [arXiv:hep-ph/9711250].

[120] M. Hirai, S. Kumano, N. Saito. Determination of polarized parton distribution functions and their uncertainties[J]. Phys. Rev. D 69, 054021 (2004).

[121] D. de Florian, R. Sassot, M. Stratmann, et al. Extraction of spin-dependent parton densities and their uncertainties[J]. Phys. Rev. D 80, 034030 (2009).

89 List of presentations and publications

Presentations

1. Jan. 2017, oral presentation, Institute of Particle Physics Seminar, Wuhan, Hubei, China

2. Dec. 2016, oral presentation, Early Career Researcher Symposium, Upton, NY, USA

3. Nov. 2016, oral presentation, 7th International Conference on Physics Op- portunities at an Electron-Ion-Collider (POETIC7), Philadelphia, PA, USA

4. Jul. 2016, oral presentation, Electron-Ion Collider Users Meeting, Chicago, IL, USA

5. Jun. 2016, oral presentation and poster presentation, 2016 RHIC & AGS An- nual Users’ Meeting, Upton, NY, USA

6. Apr. 2016, oral presentation, 24th International Workshop on Deep-Inelastic Scattering and Related Subjects (DIS24), Hamburg, Germany

7. Jan. 2016, oral presentation, Electron-Ion Collider Users Meeting, Berkeley, CA, USA

8. Sep. 2015, poster presentation, BNL Young Researcher Symposium, Upton, NY, USA

Publications

1. X. Chu, E.C. Aschenauer and J.H. Lee, Proceedings of Science (DIS2016), 269, 2016

90 2. X. Chu, E.C. Aschenauer, J.H. Lee and L. Zheng, Phys. Rev. D, 96, 074035, 2017

3. X. Chu and E.C. Aschenauer, Proceedings of Science (DIS2018), 168, 2018

91 致致致谢谢谢

博士的求学时光，给人以知识和智慧，也让人收获经历与成熟。六年里，对那些引导我、帮助我、激励我的人，我心中充满了感激。在此博士毕业之际，我想对这些人表示真诚的感谢。

首先我要由衷地感谢我的导师蔡勖教授。蔡老师知识渊博，治学严谨。在我读博初期，他引导我进入粒子物理专业，鼓励我尝试新的研究课题。进入博士课题的研究之后，对遇到困难的我，他总能给予正确的引导，与我讨论，让我在过程中学习、理解知识。蔡老师给我树立了学术目标，教会我对待物理问题的正确的思考方式。正是因为他悉心指导，为我创造了良好的科研条件，我才能顺利完成博士科研课题。

感谢美国布鲁克海文国家实验室的Elke-Caroline Aschenauer研究员。我在BNL的三年的时间里，无论在生活上还是学习上，她都给予了极大的帮助。作为一名女性科学家，她给我树立了榜样。她清晰的思路、敏锐的洞察力，让我受益颇多；同时，她也教会了我如何在团队中与同事相处，如何合作。这些都使我在BNL度过了一段充实和快乐的时光。

感谢美国布鲁克海文国家实验室的Jeong-Hun Lee研究员。在科研过程中，每次我遇到困难，他都会通过不同的方式引导我，很多问题就在讨论中迎刃而解。他教给了我如何透过现象理解物理本质的方法，这样的思维方式使我受益匪浅。

感谢殷中宝教授给我提供了参与EIC项目的机会。在BNL学习期间，殷老师一直督促我在学习上不断努力。在我论文的撰写过程中，殷老师不厌其烦地帮我修改，在他的宝贵建议之下，我的论文才得以顺利完成。

感谢复杂系统小组的李炜老师、池丽萍老师、邓为炳老师和杨纯斌老师在我读博时期给过我很多帮助。感谢参与我答辩的老师们在学术上给我的指导，感谢答辩秘书细心帮我安排答辩事宜。

感谢郑亮师兄对我学习上的耐心指导以及生活上的帮助。感谢组里的各位师兄师姐：朱月英，赵龙峰，粟柱，邹以江，韩继辉，以及各位师弟师妹：张文俊，邓盛峰，肖羽，申建民，韩日旺，陈立菊，马飞，张陶，程智胜，谢容容，李志艳，范锐，庹奎，陈向娜，徐点。感谢你们给了我一个大家庭般的温暖。感谢ALICE小组的常婉师妹，在我回国之后给予了很多帮助。感谢BNL的师兄师姐：王晓蓉，柯宏伟，

92 查王妹，杨帅，张金龙，张正桥，杨钱，周龙，小伙伴：梅金成，涂彪，叶永金，周晨生，还有我的室友刘圳，师妹祝鹤龄。你们友善热情，让我在BNL的生活多姿多彩。

感谢好朋友张彩霞，张花香一直以来对我的关心。感谢挚友彭彭，我们彼此分享生活点滴，艰难时相互鼓励，开心时一齐欢笑。衷心希望你们一直努力，一生快乐。

最后，同时也是最重要的，我要感谢我的父母。你们一直默默地站在背后，含辛茹苦。感谢你们尊重我的每一个决定，支持我选择自己喜欢的生活方式。养育之恩无以为报，只愿你们健康。

没想到六年的时间会过得这么快，随着这篇论文的完成，我的学生时期即将画上句号。父母恩，恩师情，同窗谊，情意深长，一语难表，再次感谢所有亲人，老师，朋友，同窗，因为你们的一路陪伴，我才会更加从容。诗里讲：“今天我要离你而去，去探索新的生活，你把这支木笛交给我，我深深懂得，你不是要我重复过去的歌曲”。希望这篇论文不是我学术的终点，也希望这不仅仅只是一个希望。我会带着你们赠予我的“木笛”继续努力。

褚晓璇

二零一八年六月

于武汉桂子山