The Electron-Ion Collider: Assessing the Energy Dependence of Key Measurements
E.C. Aschenauer, S. Fazio, J.H. Lee, H. Mäntysaari, B.S. Page, B. Schenke, T. Ullrich, R. Venugopalan, P. Zurita
aa r X i v : . [ nu c l - e x ] S e p NL Formal ReportBNL-114111-2017September 12, 2017
The Electron-Ion Collider:Assessing the Energy Dependenceof Key Measurements isclaimer
This report was prepared as an account of work sponsored by an agency ofthe United States Government. Neither the United States Government norany agency thereof, nor any of their employees, nor any of their contractors,subcontractors, or their employees, makes any warranty, express or implied, orassumes any legal liability or responsibility for the accuracy, completeness, orany third party’s use or the results of such use of any information, apparatus,product, or process disclosed, or represents that its use would not infringeprivately owned rights. Reference herein to any specific commercial product,process, or service by trade name, trademark, manufacturer, or otherwise,does not necessarily constitute or imply its endorsement, recommendation,or favoring by the United States Government or any agency thereof or itscontractors or subcontractors. The views and opinions of authors expressedherein do not necessarily state or reflect those of the United States Governmentor any agency thereof. uthors
E.C. Aschenauer, S. Fazio, J.H. Lee, H. M¨antysaari, B. S. Page, B. Schenke, T. Ullrich ∗ ,R. Venugopalan ∗ , P. Zurita Brookhaven National Laboratory, USA
Acknowledgments
We thank the following colleagues who made valuable contributions to this document and the studiesit is based on: M. Diehl (DESY, Germany), A. Dumitru (Baruch), A. Kiselev (BNL), H. Paukkunen(Jyv¨askyl¨a, Finland), R. Sassot (Buenos Aires, Argentina), V. Skokov (RBRC/BNL), M. Stratmann(T¨ubingen, Germany), L. Zheng (CCNU, China).The authors are indebted to the following colleagues for critical and crucial comments in the prepara-tion of this report: P. Bond (BNL), A. L. Deshpande (BNL/Stony Brook), M. Diehl (DESY, Germany),B. Jacak (LBNL), D. Kharzeev (BNL/Stony Brook), D. Lissauer (BNL), B. M¨uller (BNL), P. Newman(Birmingham, UK), G. Sterman (Stony Brook), R. Tribble (BNL), W. Vogelsang (T¨ubingen, Gemany),W. A. Zajc (Columbia).We are grateful to T. Bowman (BNL) for designing the cover for this report and D. Arkhipkin (BNL)and A. Kiselev (BNL) for generating the event display featured on the back cover.Notice: This manuscript has been authored by employees of Brookhaven Science Associates, LLCunder Contract No. DE-AC02-98CH10886 and DE-SC0012704 with the U.S. Department of Energy.The United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license topublish or reproduce the published form of this manuscript, or allow others to do so, for United StatesGovernment purposes. ∗ Editors i bstract We provide an assessment of the energy dependence of key measurements within the scope of themachine parameters for a U.S. based Electron-Ion Collider (EIC) outlined in the EIC White Paper.We first examine the importance of the physics underlying these measurements in the context of theoutstanding questions in nuclear science. We then demonstrate, through detailed simulations of themeasurements, that the likelihood of transformational scientific insights is greatly enhanced by makingthe energy range and reach of the EIC as large as practically feasible.ii ontents iii
Introduction
An Electron-Ion Collider (EIC) is a key component of the future program of the nuclear physics commu-nity in the US. The EIC will be the world’s first electron-nucleus collider and the world’s first collider toscatter polarized electrons off polarized protons. EIC-based science is extremely broad and diverse. Itruns the gamut from detailed investigation of hadronic structure with unprecedented precision to explo-rations of new regimes of strongly interacting matter. This deep and varied study will greatly expandour knowledge and understanding of Quantum Chromodynamics (QCD), the fundamental quantumtheory of the quark and gluon fields making up nearly all the visible matter in the universe. EIC sciencecan be characterized by a few distinguishing themes that reflect the major challenges facing modernscience today, and that have deep links to cutting edge research in other sub-fields of physics.The EIC White Paper [1] released at the end of 2012, and updated in 2014, presents in detail thescience case of an Electron-Ion Collider. It lays out how the EIC’s ability to collide high-energy electronbeams with high-energy ion beams will provide access to kinematic regions in the nucleon and nucleiwhere their structure is dominated by gluons and how polarized beams will give unprecedented accessto the spatial and spin structure of gluons in the proton. Thus, the machine design needs to aim atachieving highly polarized (70%) electron and proton beams, ion beams from deuteron to the heaviestnuclei, high collision luminosity of 10 − cm − sec − , and center-of-mass energies in a wide rangeup to 140 (cid:112) Z/A
GeV. Several of the studies in the White Paper that demonstrate the physics reachand potential of an EIC assumed the higher energy range, and typical integrated luminosities of 10 fb − .Now that the EIC has been embraced by the nuclear physics community in the 2015 Nuclear Sci-ence Advisory Committee (NSAC) Long Range Plan and the technology to build an EIC is becomingavailable, it is timely to review the arguments that led to the proposed center-of-mass energy rangeconsidered initially. The purpose of this document is not to repeat arguments put forward in the EICWhite Paper. We instead build upon these arguments and scrutinize more closely the energy dependenceof key measurements that are essential to ensure a compelling EIC science program. To be specific,for e + p collisions we define the low energy range of center-of-mass energies to span √ s = 22-63 GeVand the high energy range to span √ s = 45-141 GeV. The corresponding center-of-mass energies for e +A collisions off heavy nuclei are √ s = 15-40 GeV and √ s = 32-90 GeV. We add an important studyomitted in the EIC White Paper, that of jets. Jets provide a highly precise characterization of the finalstate in deep inelastic scattering (DIS) that complements the precision provided by the electron probe.To motivate a deeper appreciation of why a high energy EIC may be needed, we present in Section2, a “big picture” case for EIC science and how the proposed studies open new opportunities, generateexcitement, and pose challenges. We note the lessons provided by past DIS experiments and thoselearned from discoveries at other colliders. In Section 3, we examine in detail the energy dependence ofseveral key measurements in order to guide discussions on the required energy reach of an EIC. Mostof the simulations presented were carried out with the same or improved procedures and programs thatwere used in the EIC White Paper. We conclude this document with a brief summary of our principalobservations. To further advance the compelling scientific case for a large scale EIC project, we will begin by placingits energy requirements in the broader context of the role of QCD and the strong interactions withinthe Standard Model of physics and by articulating why QCD matters in the big picture. We will pointto outstanding challenges in our understanding of how the many-body dynamics of quark and gluonfields give rise to the confined structure and dynamics of the strongly interacting matter that constitutes1early all the mass of the visible universe. More specifically, we will sketch out the landscape of theQCD dynamics inside hadrons and nuclei that is probed by varying the energy and resolution of theelectron probe. While a discovery can rarely be predicted, lessons from the past are instructive. We willexamine lessons provided by completed DIS experiments and the discovery of the quark-gluon plasma toguide our study of the discovery potential of an EIC within the scope of machine parameters discussedin the EIC White Paper.
Quantum Chromodynamics (QCD) representsthe apogee of a quantum field theory. It is anearly self-contained fundamental theory of quarkand gluon fields that is rich in symmetries; theonly external parameters in QCD are the quarkmasses generated by the Higgs mechanism in theStandard Model [2]. Strongly interacting phenom-ena emerge from the interactions generated by thesymmetries of QCD and from the breaking of thesesymmetries by the QCD vacuum and by the quarkmasses. The generation of the mass of strongly in-teracting matter is a striking example of an emer-gent phenomenon. Gluons carry no mass and thelight quarks carry masses roughly a hundredththat of the proton. Despite this lightness of being,quarks and gluons, through their interactions witheach other and the QCD vacuum, generate themass of protons and neutrons, and other stronglyinteracting particles. The dynamics of quark andgluon fields are therefore responsible for nearly allthe mass of visible matter in the universe.Another striking emergent phenomenon is thatcolored quarks and gluons are permanently con-fined within hadrons on a length scale on the or-der of a Fermi. This emergent scale, nowhere ev-ident in the QCD Lagrangian, dictates the maxi-mal distance out to which the chromo-electric andchromo-magnetic fields of quarks and gluons canspread. At such separations, the forces betweenquarks and gluons are so large that it is energet-ically favorable for the QCD vacuum to sponta-neously create quark-antiquark pairs, and gluons,which assist fundamentally in assembling coloredquarks and gluons into colorless hadrons. Howthis phenomenon called “confinement” occurs isstill not understood and is one of the outstandingmysteries of physics.The essence of all of the remarkable and con- founding phenomena of QCD is its non-Abeliannature whereby, unlike photons, gluon fields canself-interact. QCD is therefore an intrinsicallynonlinear theory, and much of the complexity ofits dynamics can be traced to this feature of thetheory. This nonlinear essence is also what makesthe extraction of physics fiendishly difficult. In the40 odd years since the discovery of QCD, power-ful techniques have been developed to elucidatephenomena of nature’s strong interaction. One ofthese is perturbative QCD (pQCD), which exploitsyet another emergent phenomenon of the theory:asymptotic freedom. Quarks and gluons interactweakly at small separations, or equivalently, largemomentum transfers. In this asymptotic limit, theQCD coupling constant is weak and manifestlynonlinear interactions are suppressed; as a con-sequence, computations can be undertaken thatare highly precise and carry predictive power. Inthe opposite strongly interacting regime of QCD,effective field theories (EFT) ingeniously take ad-vantage of symmetries and the separation of softand hard emergent scales to extract physical infor-mation. A noteworthy example is chiral pertur-bation theory, that exploits the chiral symmetryof the QCD Lagrangian, and its breaking by thevacuum, for systematic computations at energiesbelow the emergent QCD scale.The most powerful method of all is latticegauge theory which discretizes and solves QCD inall its complexity on the fastest computers avail-able. Lattice computations have by now accu-rately reproduced ab initio the mass spectrumof hadrons, as well as other important quanti-ties. Most importantly, lattice QCD has estab-lished beyond reasonable doubt that QCD is thecorrect fundamental theory of the strong interac-tions. Furthermore, lattice investigations of QCD2ymmetries, and their breaking, illustrate beauti-fully how complex phenomena such as the mass,spin, and structure of the lightest hadrons to theheaviest nuclei emerge out of seemingly nothing inQCD.Experimental progress has been no less impres-sive. Since the discovery of quarks in DIS experi-ments at SLAC, and of gluons in electron-positroncollisions at DESY, a large number of experiments,across a wide range of energies, have uncoveredfundamental information about QCD. The run-ning of the QCD coupling predicted by asymptoticfreedom is an established fact, in good agreementwith theory predictions. QCD has become pre-cision physics at high momentum transfers; devi-ations of theory from experiment in this regimewould be a harbinger of physics beyond the Stan-dard Model. QCD studies continually reveal newdiscoveries. Candidates for novel tetraquark andpentaquark states have been identified, and thepossibility of exotic hybrid and glueball hadrons is under investigation. The spin of hadrons, now un-derstood as an emergent many-body phenomenon,is being quantified in terms of the spins of the un-derlying quarks and gluons. The discovery of thequark-gluon plasma (QGP) highlights the role ofQCD in cosmology, from the QGP of the early uni-verse, to Big Bang nucleosynthesis, the synthesisof heavy elements, and the high baryon densityregime of neutron stars. Significant insight intothe latter will be provided by the upcoming ex-periments with rare isotope beams. Explorationsof the QCD phase diagram in temperature andbaryon density, to uncover the phases of stronglyinteracting matter, are the focus of extensive ex-perimental effort.As the advances in theory and experimentdemonstrate, QCD is clearly a mature subject.It is therefore important to clearly articulate, inthis context, the urgency of the construction of anElectron-Ion Collider.
The behavior of QCD at very short distancesis well understood and can be used to predict thehard scattering of partons, where “partons” de-note quarks, antiquarks, and gluons. However,the mechanisms by which QCD produces the bulkof the visible world remains mysterious and needscontinuing and varied efforts to resolve. A pro-found subset of outstanding questions concerns theinternal structure of hadrons and nuclei at high en-ergies. With increasing energy, gluons (and the ac-companying “sea” of quark-antiquark pairs) pro-liferate and the “veil” that hides their dynamicsat low energies is lifted. All emergent structureof hadrons such as mass and spin are a result ofthe confined many-body dynamics of partons. InFig. 1, we show the landscape of hadron structurethat opens up at high energies. The vertical axisrepresents varying resolution Q and the horizon-tal axis represents varying parton density, an in-creasing function of 1 /x , where x is the momentumfraction of the proton carried by a parton. Whenthe parton density is fixed to be small (large x ),increasing the energy of the probe increases Q , allowing one to finely resolve the partons insideprotons and nuclei.At low Q and large x , strong interactionphysics is described by hadrons and their interac-tions. In this regime, the QCD coupling is large,the fields are nonlinear, and the physics is nonper-turbative. Moving up the vertical axis at large x ,to very large Q , the coupling becomes weak dueto asymptotic freedom, and perturbative QCD de-scribes well the interactions of quarks and gluons.How the transition from low to high Q occurs,and what the degrees of freedom describing thistransition region are, is not understood. Confine-ment and chiral symmetry breaking play a keyrole but how they influence many-body dynam-ics in this regime is unclear. At large Q , as onemoves towards higher parton density, many-bodycorrelations between quarks and gluons become in-creasingly important. When the coupling is weak,it is possible to quantify these, and describe theirevolution using the linear evolution equations ofperturbative QCD. How such many-body correla-tions are modified in nuclei, or when the proton is3 omeronsRegge trajectories H i gh - D en s i t y G l uon M a tt e r Q S ( x ) Quarks and GluonsHadrons C on f i n e m e n t , C h i r a l S y mm e t r y B r eak i ng Strongly CorrelatedQuark-Gluon Dynamics L i n e a r e v o l u t i o n N o n - li n e a r e v o l u t i o n Q (GeV ) 1/x non - pe r t u r ba t i v epe r t u r ba t i v e s t r ong c oup li ng w ea k c oup li ng R es o l u t i on Parton Density
Figure 1: Landscape of QCD. The vertical axis represents varying resolution Q and the horizontal axis varyingparton density, an increasing function of 1 /x . polarized, is not known.At very large parton densities, or small x ,gluon degrees of freedom become dominant. Totalcross-sections in high energy scattering are dom-inated by the physics of small x and low Q .Nonperturbative QCD dynamics in this kinematicregime has historically been parametrized in termsof effective color singlet degrees of freedom called“Pomerons” and “Reggeons” and their interac-tions. How they are constituted fundamentallyin terms of the dynamics of the underlying quarkand gluon degrees of freedom, and that of the vac-uum, is terra incognita . Remarkably, this cornerof the QCD landscape, which contributes the mostto the high energy scattering of hadrons, is per-haps the least understood. The takeaway messagehere is that even though there are aspects of theregimes discussed in Fig. 1 that are known, essen-tial elements in constructing dynamical pictures of hadrons and nuclei that are important for a firstprinciples understanding of strong interaction phe-nomena are missing.A novel regime of QCD may exist in the up-per right corner of Fig. 1, where parton densitiesare high. For high but fixed values of the res-olution Q , the density of gluons saturates withdecreasing x . In this gluon saturation regime, thechromo-electric and chromo-magnetic fields are asstrong as allowed in QCD–the strongest fields innature– even though asymptotic freedom ensuresthat the QCD coupling is weak . The feature ofweak coupling is key because it allows, for thefirst time, systematic computations of the many-body dynamics of quarks and gluons in an in-trinsically nonlinear regime of QCD. Note thatlattice QCD primarily studies static quantities–dynamical quantities, especially at high partondensities, are mostly outside its purview.4he structure of QCD, and indeed the require-ment that strongly interacting matter be stable,suggests that these saturated gluons do not forma Bose-Einstein condensate. Instead, they gen-erate a unique form of strongly interacting mat-ter, called a Color Glass Condensate (CGC) [3].The strong interplay of attractive and repulsivemany-body forces amongst gluons ensures thattheir typical momenta are peaked not at zero mo-mentum as in a Bose-Einstein condensate, but in-stead at an emergent saturation momentum scale Q s . Further, the interactions amongst gluons car-rying larger or smaller fractions of the proton’smomentum are time dilated to much longer thanstrong interaction time scales. This is typical forglassy material, and indeed, the equations that de-scribe the CGC are similar in this respect to thosefor glasses. A prediction of the CGC descriptionis that the saturation scale grows with energy and nuclear size and, for large values of both, can bemuch larger than the intrinsic QCD scale. As in-dicated in Fig. 1, for resolutions Q (cid:29) Q s , thenonlinear dynamics of gluons goes over smoothlyinto the linear evolution of quarks and gluons thatis characteristic of pQCD.Properties of the CGC in the gluon saturationregime can be computed in a weak coupling effec-tive theory. The QCD evolution equations in thisCGC effective theory are intrinsically nonlinear,and generate a hierarchy of many-body correlationfunctions whose renormalization group evolutionis analogous to that of other strongly correlatedsystems. In particular, because interactions aretime dilated, and the coherence length of probesis large relative to hadron sizes, the longitudinaland transverse dynamics of gluons nearly decou-ples in the upper right corner of Fig. 1. The many-body physics of the QCD land-scape provides a robust non-Abelian counterpartto those of other strongly correlated systems,where one observes remarkable emergent phenom-ena with universal features. Indeed, condensedmatter and cold-atom physicists are exploringways to mimic the strongly interacting dynamicsof gauge theories [4]. The structure of QCD sug-gests strong discovery potential both by deep in-vestigation of many-body dynamics in QCD andsimultaneous attempts to reproduce it in otherstrongly correlated systems.An outstanding question is whether there ex-ists matter inside protons and complex nuclei thatis universal or whether the “doping” caused byquantum features that are unique to each hadron(such as their spin, flavor and baryon number)makes it impossible to isolate universal features oftheir dynamics. The conventional wisdom is thatsuch quantum numbers are carried mostly by par-tons that possess large fractions of the hadron’smomentum. In this picture, partons with smallerand smaller momentum fractions x are increas-ingly like the spinless, flavorless and baryon-freeQCD vacuum. Alternately, it is conceivable that the long arm of confinement ensures “memory” ofvacuum doping down to the smallest values of x .Models of high parton density matter predictthat the strong chromo-fields of gluon saturationare achieved precociously in large nuclei at largervalues of x than in the proton. In particular,the saturation scale has a nuclear enhancement: Q s ∼ A / . However, at very small x , if gluon in-teractions were universal, the values of Q s ( x ) inthe proton and in more complex nuclei would beidentical. This is only likely to be achieved forenergies much larger than those at an EIC. Nev-ertheless, the deviations of Q s ( x ) from asymptoticexpectations may have a specific structure that isaccessible for the EIC range of energies. The ob-servation of such systematics would therefore pro-vide indirect evidence for the universality of satu-rated gluon matter.Besides this question of obvious interest, nu-clei are an important ingredient towards a deeperunderstanding of our big picture landscape. Anapparently elementary question of how the gluondistribution in the nucleus deviates from a simpleconvolution of nucleon and gluon distributions isunresolved. More detailed questions about how5onfinement operates in nuclei versus nucleons –an example being the role of gluons and quarksin the short range nuclear forces that bind nuclei-have no clear answer at present.Another fascinating window into the dynam-ics of confinement is the dynamical conversion ofquarks and gluons into hadrons. Nuclei are a QCDlaboratory to examine this hadronization questionwith “fresh eyes”. While QCD in nuclei has beenexplored for nearly 40 years, it has never beenstudied previously with deep inelastic scatteringprocesses in collider mode. Energy reach is cru-cial in this regard because it allows i) wide explo-ration of the nuclear landscape of Fig. 1 ii) noveltools such as heavy quark and jet observables, thatwere previously out of reach, and iii) access to awider phase space to study the QCD dynamics ofhadronization.How the spin of the proton emerges fromthe many-body dynamics of quarks and gluonspresents several puzzles that high energies at anEIC can help resolve. The first of these is to un-derstand how the relative contributions of the par-ton spins to the spin sum rule (see Eq. 2) holdacross the QCD landscape despite the wide changein energy and resolution. Since the dynamics ofspin also involves the orbital motion of quarks andgluons, this motivates developing tools that allowone to image the transverse spatial and momen-tum structure of the proton. The imaging of the3-D spin structure of the proton in the gluon dom-inated regime will be an important first for theEIC; this topic has been extensively discussed inthe EIC White Paper. As noted there, uncertain-ties in the contributions to the quark and gluon spins at small x are significant. Reducing theseuncertainties, while maintaining the large Q val-ues essential for controlled computations, requireshigh energies: this is because Q ∼ xs , where s isthe squared center-of-mass energy.An important question, related to the previousdiscussion on the universality of the vacuum, isto understand how spin is transmitted to small x .So-called quantum anomalies, which cause symme-tries satisfied by the QCD equations of motion tonot be satisfied by the QCD amplitudes, may playan important role at small x that is not transpar-ent in the x -integrated quantities that contributeto the sum rule. The flavor structure of the QCDvacuum in polarized protons extracted from semi-inclusive DIS is an important ingredient in build-ing a more complete picture. The high energiesat an EIC will also enable precision DIS studieswith charged current probes that can further helpresolve the flavor structure of the proton.The precision study of the QCD landscape en-abled by exploring, for the first time in DIS, nu-clei and polarized protons in collider mode at highenergies and luminosities will generate novel re-sults. One such example is diffractive scatteringoff nuclei, where one can probe, with fine resolu-tion, Pomeron color singlet exchanges that carrythe quantum numbers of the vacuum. The widerange of such innovative measurements, a few ofwhich we will focus on in this report, promise fun-damental insight into the many-body nature of theoutstanding mystery of confinement. Because sev-eral of these are first measurements in unchartedregions of the QCD landscape, discoveries that re-solve fundamental questions are highly likely. The discovery of rapidly growing gluon dis-tributions in DIS experiments at the HERA col-lider gave a clear experimental indication that theproton at high energies is a complex many-bodysystem where gluon degrees of freedom are dom-inant. This discovery gave impetus to the ideathat there exists a novel saturation regime in QCDwhere many-body dynamics is intrinsically nonlin-ear. Precision measurements of quark and gluon distributions at HERA played a major role in thediscovery of the Higgs boson and the characteriza-tion of its properties, and are a key tool in ongoingsearches at the LHC for physics beyond the Stan-dard Model [5].At the EIC, the focus is on parton distributionsin nuclei and on polarized parton distributions inspin polarized protons. The kinematic reach inBjorken x and the momentum resolution Q for6 Q ( G e V ) Current polarized DIS ep data:
CERN DESY JLab-6 SLAC
Current polarized RHIC pp data:
PHENIX π STAR 1-jet W bosons JLab-12
101 10 -4 -3 -2 -1 E I C √ s = − G e V , . ≤ y ≤ . E I C √ s = − G e V , . ≤ y ≤ . Measurements with A ≥ 56 (Fe): eA/μA DIS (E-139, E-665, EMC, NMC) JLAB-12 ν A DIS (CCFR, CDHSW, CHORUS, NuTeV)
DY (E772, E866)DY (E906) x -4 -5 -3 -2 -1 Q ( G e V ) E I C √ s = − G e V , . ≤ y ≤ . E I C √ s = − G e V , . ≤ y ≤ . perturbativenon-perturbative Figure 2:
Left:
The range in x vs. Q , accessible with an EIC in polarized e + p collisions compared to past(CERN, DESY, SLAC) and existing (JLAB) facilities as well as to polarized p + p collisions at RHIC. Two differ-ent energy ranges from 22–63 GeV (hatched) and from 45–141 GeV (beige) are indicated. Right:
The kinematicacceptance in x vs. Q of completed lepton-nucleus(DIS) and Drell-Yan (DY) experiments, as well as JLAB-12(all fixed target) compared to the EIC acceptance in two energy ranges, 15–40 GeV (hatched) and from 32–90GeV (beige). DIS for a range of EIC energies in e + p collisions(with and without polarized protons) is shown inFig. 2 (left). The kinematic reach in e +A colli-sions is shown in Fig. 2 (right). For e + p the twoenergy ranges depicted are, i) a high energy rangeof center-of-mass range of √ s = 45-141 GeV, andii) a lower energy range of √ s = 22-63 GeV. In e +A collisions off heavy nuclei, the correspond-ing low energy center-of-mass range is √ s = 15-40GeV and the higher energy range is √ s = 32-90GeV. Diagonal lines on the plot represent lines ofconstant “inelasticity” y . In the rest frame of theproton (or nucleus), the inelasticity is the ratio ofthe energy carried by the virtual photon dividedby the energy of the incoming electron. Figure 2(left) also shows the x - Q values for which data areavailable from fixed target DIS polarized e + p ex-periments as well as from polarized p + p collisionsat RHIC. Correspondingly, Fig. 2 (right) shows the x - Q values for which data are available from fixedtarget e +A collisions. In both cases, for Q > , there are no data below x ∼ · − . Alter-nately, for Q = 1 GeV , the kinematic reach ofthe EIC would exceed extant world data by nearlytwo orders of magnitude for polarized e + p scatter-ing and a factor of 50 for e +A collisions. Thus,a region that is currently terra incognita for theextraction of gluon distributions and for the study of gluon saturation will become available for pre-cision measurements at the EIC. − Q (GeV ) x g ( x , Q ) CTEQ14 NNLO x = . x = - x = - x = - no DIS data for given x Figure 3: Proton PDFs of gluons as functions of Q forvarious x values as derived by the CTEQ collaborationin NNLO [6].The bands indicate the uncertainties inour knowledge of gluon PDFs. They are colored in therange where the relevant DIS data (HERA) is available. Even though gluons, unlike quarks, do notcouple directly to electromagnetic probes, we canlearn about their properties from “scaling vio-lations”. These in particular describe changesin quark distributions with Q and Bjorken x .The evolution of gluon distributions with Q ex-tracted from these scaling violations is described7
665 CollaborationPhys. Rev. D54, 3006 Q (GeV ) F -2
101 10 -1 -1 mrs98 x = . ( × ) . ( × ) . ( × ) . ( × ) . ( × ) . ( × ) . ( × ) . ( × ) . ( × ) . ( × ) . ( × ) . ( × ) . ( × ) . ( × ) . ( × ) cteq4 pe r t u r ba t i v enon - pe r t u r ba t i v e η η G ( η ) E665 result (±1σ contour)at Q = 8 GeV E665 CollaborationZ. Phys. C71, 391-403 (1996)
CTEQ14 NNLOCTEQ14 NLOCTEQ14 LO η ≈ x
Figure 4:
Left:
Structure function F of the proton as a function of Q for various x values measured by the E665experiment at √ s = 31 GeV. Right:
Gluon distribution derived from the E665 F data (depicted on the left) asa function of x for Q = 8 GeV compared to results from state-of-the-art gluon PDFs [6]. by the DGLAP renormalization group equations(RGE) [7–9] of perturbative QCD (pQCD). Therenormalization group flow of information is to-wards smaller x and larger Q . A wide lever armin Q is essential for the extraction of parton dis-tributions while a wide coverage in x is mandatoryto access a broad dynamical regime.We see from Fig. 2 that the difference betweenthe high and low energy ranges shown correspondsto a factor of 5 increase in x reach for a fixed Q ,and likewise, a factor of 5 increase in Q reach forfixed x . DIS measurements with data collected inthis additional area can further constrain signif-icantly nuclear gluon distributions and their ex-trapolations (via DGLAP evolution) to small x .An example of lessons from electron-protoncollisions at HERA for the EIC is depicted inFig. 3. The gluon distribution is parametrizedat a low momentum resolution scale using theHERA electron-proton inclusive reduced cross- section (see Eq. 4) data and evolved, using theDGLAP RGE, from this low Q scale towardshigher Q at fixed x . We see that the next-to-next-to leading order (NNLO) DGLAP evolution per-formed by the CTEQ collaboration [6] generatesgluon distributions to good accuracy for x = 0 . x = 0 .
01 in the entire Q range plotted. How-ever, at the smaller x = 10 − and x = 10 − values,the gluon distribution shows larger uncertainties, especially at the small Q where high quality dataexist . The precision of low Q data in this case isineffectual due to the lack of data at the larger Q where the DGLAP RGE is initialized and evolvedfrom. In contrast, the gluon distribution at larger x values is well constrained over the range shownby virtue of the larger Q lever arm. This exampleillustrates why a greater EIC energy will not onlyimprove our knowledge of the gluon distributionover a wider Q range but also more precisely inthe range that is already accessible at lower ener-8ies.The lesson drawn, of the importance of ex-panded reach in x and Q , is starker and morepertinent for e +A collisions at the EIC. In thiscase, the parametrization of the data at the initialscale will not have the x - Q reach of e + p collisionsat HERA. To illustrate this, Fig. 4 (left) showsthe structure function F extracted for the x - Q reach of the fixed target E665 data at √ s = 31GeV. Though it is for e + p data, it holds an im-portant lesson for the lower e +A center-of-massenergy of 40 GeV. Scaling violations, the varia-tion of F p ( x, Q ) with Q , are clearly visible onlyfor x ≤ .
01. The small x and large Q region therefore governs the precision with which gluondistributions can be extracted from these scalingviolations.The extracted proton gluon distribution isshown in Fig. 4 (right). It has large uncertain-ties at Q = 8 GeV and x = 0 .
01. These aresignificantly larger than the CTEQ evolved gluondistribution shown previously that utilizes HERAdata, which had a much wider coverage in x and Q . For nuclear gluon distributions, the compan-ion plot to Fig. 3, demonstrating the precision towhich these can be extracted at the two represen-tative center-of-mass energies, is shown and dis-cussed in Sec. 3.4. The energy, luminosity and kinematic reachneeded for ground breaking discoveries are in ad-vance uncertain by their very nature. In the pre-vious section, we discussed some lessons from pastDIS experiments. Experience drawn from QCDdiscoveries in related sub-fields can also provideimportant guidance. We will discuss here the ex-perience gained from heavy-ion collisions, in par-ticular the discovery of the Quark Gluon Plasma. − √s NN = 7.7 GeV11.5 GeV14.5 GeV19.6 GeV27 GeV39 GeV62.4 GeV200 GeV STAR Preliminary (statistical errors only) (GeV/c) T p ( - % ) / ( - % ) R C P N pa r t Figure 5: Charged hadron suppression, expressed by R CP , measured at RHIC energies, √ s NN = 11.5–200GeV and at the LHC at √ s NN = 2.76 TeV. The errorbands at unity on the right side of the plot correspondto the p T independent systematic uncertainties. Compelling evidence that there was strikingnew physics occurring in heavy-ion collisions atRHIC was provided by the discovery of “jet quenching” [10–12]. The observable was the ratio R CP of the inclusive hadron spectrum in centralAu+Au relative to that in peripheral Au+Au col-lisions, each of them normalized by the respectivenumber of collisions. The absence of nuclear ef-fects corresponds to R CP = 1. Figure 5 plots R CP for a wide range of center-of-mass energies per nu-cleon at RHIC, from √ s NN = 11.5 GeV to √ s NN = 200 GeV. Also shown in the plot is R CP fromthe LHC at √ s NN = 2.76 TeV.At CERN’s SPS, the highest energy heavy-ion experiments before RHIC, heavy-ion beams onfixed targets had a maximal per nucleon energyof 158 GeV, corresponding to √ s NN = 17.2 GeV.This is close to the √ s NN = 19.6 GeV RHIC curvein Fig. 5 [13–15]. At these energies, one observesa depletion at small p T and an enhancement atlarge p T , a phenomenon often called the “CroninEffect”. An explanation of this effect is that themultiple scattering of partons depletes the lower p T part of the spectrum and shifts their averagemomenta to higher p T . At RHIC, for √ s NN =200 GeV, R CP ∼ . p T partons were suf-fering significant energy loss in their interactionswith the medium and thereby provided strong ev-idence that a QGP had been created. More dif-ferential measurements, fortified by more refinedtheory computations of jet quenching, have subse-9uently reinforced the early role played by R CP asa key measurement in the discovery of the stronglyinteracting QGP.While jet quenching is unambiguous at √ s NN = 200 GeV, it is also noteworthy in retrospect thatquenching is seen marginally at √ s NN = 39 GeVand more clearly at √ s NN = 62.4 GeV. However,the evidence for this result requires significant p T reach; quenching is seen clearly only above p T = 3GeV. Since physics at higher p T is luminosity hun-gry, it took over a decade of further running and abeam energy scan to obtain the beautiful system-atic behavior established by Fig. 5.Increasing the energy of the ion beams beyondthe original design values is prohibitively expen-sive. Consequently, it was important that RHIC’sdesign energy enabled a “day one” discovery. Notethat the suppression of R CP shown in Fig. 5 ismaximal at the highest RHIC energy and is notexceeded by data from the LHC, as also shown inthe figure. The fact that the highest RHIC energysaturates jet quenching is remarkable and was notanticipated in early models of jet quenching.Continual luminosity upgrades, resulting fromexperience with the RHIC accelerator, have ledover time to more than a forty-fold increase in lu-minosity over its design value. This experienceappears to be a generic feature of high energy col-liders. In addition, the RHIC collider has showntremendous versatility in running at a variety ofenergies, employing ion species from the lightestions to Uranium. This versatility, and the imple-mentation of electron cooling of heavy-ion beamsto further increase the luminosity in RHIC’s BESII phase, is key to sustaining its future discoverypotential. RHIC was the world’s first heavy-ion collider;the EIC will be the world’s first electron-nucleusand polarized electron-polarized proton collider.A key lesson from RHIC’s success and future po-tential in exploring the “hot and dense QCD”phase diagram is that the discovery of similarlynovel many-body dynamics in the QCD landscapesketched in Fig. 1 requires a significant energyrange and reach well beyond those of prior fixed-target DIS machines.In a DIS collider, the kinematic equivalent ofvarying the center-of–mass energy in heavy-ioncollisions is the expanded range in the Bjorkenvariable x for fixed Q . In polarized electron-polarized proton collisions, for fixed Q , the reachin x at the highest proposed EIC energy is two or-ders of magnitude greater than at fixed-target DISexperiments. For DIS collisions off heavy nuclei,at fixed Q , the range in x is a factor of 50 greaterthan available at fixed-target machines. This ex-tended reach and range is comparable to that ofRHIC relative to prior fixed-target heavy-ion ex-periments.The success of RHIC suggests that the discov-ery potential of a high energy EIC is high and thatenergy reach and range plays a pivotal role. Verylittle is known about spin and angular momen-tum of quarks and gluons at small x , as well astheir transverse spatial and momentum distribu-tions. Nuclear gluon distributions are terra incog-nita , as is our understanding of the propagation ofjets and heavy quarks in cold nuclear matter. Weshall now demonstrate, with a simple case study,the utmost importance of energy range and reachfor the physics of gluon saturation. In Fig. 6, the saturation scales Q s for e +A col-lisions at an EIC, with two different maximal √ s ,are compared to the values achievable in e + p col-lisions at HERA. The first point to note is thatthe projected saturation scales in e +A collisionsat both EIC energies are significantly larger thanthose in e + p collisions, even though the HERA √ s value is approximately eight times greater than the lower EIC energy. This enhancement of Q s innuclei is a striking consequence of the high energyDIS probe interacting simultaneously with partonsin different nucleons along its path through the nu-cleus.We also observe that the maximal Q s in e +Acollisions at the EIC is approximately 50% largerfor the higher energy of √ s max = 90 GeV, com-10 perturbative regime perturbative regime HERA (ep) EIC √s max = 90 GeV (eAu)x ≤ 0.01 Λ Q S (GeV ) EIC √s max = 40 GeV (eAu)
Figure 6: Accessible values of the saturation scale Q s at an EIC in e +A collisions assuming two different maximalcenter-of-mass energies. The reach in Q s for e + p collisions at HERA is shown for comparison. pared to √ s max = 40 GeV. The difference in Q s may appear relatively mild but we will demon-strate in the following that this difference is suffi-cient to generate a dramatic change in DIS observ-ables with increased center-of-mass energy. Thisis analogous to the message from Fig. 5 where weclearly observe the dramatic effect of jet quench-ing once √ s NN is increased from 39 GeV to 62.4GeV and beyond.To compute observables in DIS events at highenergy, it is advantageous to study the scatteringprocess in the rest frame of the target proton ornucleus. In this frame, the scattering process hastwo stages. The virtual photon first splits intoa quark-antiquark pair (the color dipole), whichsubsequently interacts with the target. This is il-lustrated in Fig. 7. Another simplification in thehigh energy limit is that the dipole does not changeits size r ⊥ (transverse distance between the quarkand antiquark) over the course of the interactionwith the target.Multiple interactions of the dipole with the tar-get become important when the dipole size is of theorder | (cid:126)r ⊥ | ∼ /Q s . In this regime, the imaginarypart of the dipole forward scattering amplitude N ( (cid:126)r ⊥ ,(cid:126)b ⊥ , x ), where (cid:126)b ⊥ is the impact parameter,takes on a characteristic exponentiated form [16]: N = 1 − exp (cid:32) − r ⊥ Q s ( x,(cid:126)b ⊥ )4 ln 1 r ⊥ Λ (cid:33) , (1)where Λ is a soft QCD scale. At high energies, this dipole scattering ampli-tude enters all relevant observables such as the to-tal and diffractive cross-sections. It is thus highlyrelevant how much it can vary given a certain col-lision energy. If a higher collision energy can pro-vide access to a significantly wider range of valuesfor the dipole amplitude, in particular at small x ,it would allow for a more robust test of the satu-ration picture. Figure 7: The forward scattering amplitude for DISon a nuclear target. The virtual photon splits into a q ¯ q pair of fixed size r ⊥ , which then interacts with thetarget at impact parameter b ⊥ . To study the effect of a varying reach in Q , one may, to good approximation, replace r ⊥ in (1) by the typical transverse resolution scale2 /Q to obtain the simpler expression N ∼ − exp (cid:8) − Q s /Q (cid:9) . The appearance of both Q s and Q in the exponential is crucial. Its effect isdemonstrated in Fig. 8, where the dipole ampli-11ude N is plotted as a function of Q for fixed x = 10 − . While the variation of this quantitywithin the Q reach at √ s = 40 GeV is only ap-proximately 20%, it reaches up to a factor of 6within the Q reach at √ s = 90 GeV. To fur-ther draw on the analogy we raised previouslywith RHIC, the 20% suppression in R C P seen at √ s NN = 39 GeV is not robust given the system-atic uncertainties shown in Fig. 5. In contrast, thefactor 5 suppression seen at √ s NN = 200 GeV, isan unambiguous signature of discovery.As also shown in Fig. 8, this effect is furtherenhanced for diffractive events. In that case, the square of the dipole amplitude enters the cross-section. This quantity changes by a factor 25 overthe Q range at √ s = 90 GeV, but only by 1.7 forthe lower center-of-mass energy. See Section 3.6.2for further discussion.In addition to the wider range in Q , the widerreach in Q s (see Fig. 6) will allow for measure-ments with a similar lever arm in Q also at lowervalues of x . Therefore in this simple dipole modelcase study, we have demonstrated that becausethe higher center-of-mass energies provide a sig-nificantly broader reach in x and Q , saturationeffects can be large in DIS even if the Q s values do not differ widely. This is crucial for probing thephysics of gluon saturation and for testing as wellas constraining existing models. √ s = G e V √ s = G e V x=10 -3 , Q S = 1.2 GeV (Au)Q (GeV ) − e x p ( - Q S / Q ) [ − e x p ( - Q S / Q ) ] Figure 8: The curve in blue depicts the variation inthe dipole scattering amplitude as a function of Q .Inclusive DIS cross-sections are sensitive to this quan-tity. The shaded areas indicate the Q reach at twodifferent center-of-mass energies. The curve in red de-picts the variation in the square of the dipole scatter-ing amplitude with Q . This quantity enters diffractivecross-sections – see Section 3.6.2. In the following, we will examine the energy requirements of key measurements within the range ofenergies, and assuming the acceptance of a model detector, as outlined in the EIC White Paper. Sub-sections 3.1 and 3.2 respectively present the case for precision studies of the spin of the proton andthe three dimensional imaging of parton distributions within. Subsections 3.4 and 3.3 focus on partondistributions in the nuclear medium. Measurements that are sensitive to gluon saturation are discussedin subsection 3.6. We conclude in subsection 3.7 by presenting a novel study of jets at an EIC.
Understanding how the spin of the protonemerges from the properties and dynamic interac-tions of its constituents is an outstanding puzzlein hadronic physics and a key motivation for therealization of a polarized EIC. This topic is ad-dressed in Sec. 2 of the EIC White Paper [1]. Theextent to which quarks and gluons with a givenmomentum fraction x have their spins aligned withthe spin direction of a nucleon is encoded in the helicity dependent parton distribution functions.Knowledge of these fundamental quantities, alongwith estimates of their uncertainties, is gatheredfrom comprehensive QCD analyses [17,18,20] of allavailable data taken in spin-dependent DIS andproton-proton collisions, with and without addi-tional identified hadrons in the final state. By in-tegrating these polarised parton distributions overthe momentum fraction x from 0 to 1 at a fixed12 -0.500.511.5 x min x min x min ∫ d x Δ g ( x , Q ) x m i n ∫ d x Δ Σ ( x , Q ) x m i n ½ - ∫ d x [ ½ Δ Σ ( x , Q ) + Δ g ( x , Q ) ] x m i n -6 -5 -4 -3 -2 -1 -6 -5 -4 -3 -2 -1 -6 -5 -4 -3 -2 -1 Q = 10 GeV Q = 10 GeV DSSV 2008
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Figure 9: 90% C.L. uncertainty estimates for the running integrals of the gluon helicity ( left ), quark helicity( middle ), and orbital angular momentum ( right ) distribution at Q = 10 GeV as a function of x min . The gray-shaded band denotes the DSSV08 [17] fit which includes primarily DIS data. The blue-shaded band is based onthe DSSV14 fit [18], which includes polarized p + p data from RHIC collected prior to 2012. The yellow-shadedband is a projection, which accounts for the most recent RHIC data [19]. The region constrained by current datalies to the right of the vertical dashed lines. Q , the spin of the proton can be written in termsof its constituents using the Jaffe–Manohar sumrule [21]12 = 12 (cid:90) d x ∆Σ (cid:0) x, Q (cid:1) + (cid:90) d x ∆ g (cid:0) x, Q (cid:1) + L ( Q ) , (2)where ∆Σ( x, Q ) represents the quark helicitycontribution, and ∆ g ( x, Q ) represents the gluonhelicity contribution to the total spin of the pro-ton. The respective orbital angular momenta ofquarks and gluons are represented by L ( Q ) = (cid:80) q (cid:2) L q ( Q ) + L ¯ q ( Q ) (cid:3) + L g ( Q ).Figure 9 shows an extraction of the integrals ofthe quark and gluon contributions in Eq. 2, run-ning between x = x min and x = 1 with their 90%confidence level (C.L) uncertainties. The gray-shaded band is the outcome of the DSSV08 [17]analysis, which is almost exclusively based onthe existing DIS data. The blue-shaded bandshows the result of the DSSV14 [18] fit, which in-cludes polarized p + p data from RHIC. The yellow-shaded region shows the projected constraints onthe parton distributions once all RHIC data col-lected through 2015 is included. In the plots, theregion to the right of the dashed vertical line isconstrained by current data. It is clear that preci- sion data are needed to determine the parton con-tribution to the proton’s spin, especially at low x . x g ( x , Q ) g (x,Q )uncertainty DSSV 2014 Q =10 GeV √ s = . G e V √ s = . G e V √ s = . G e V -50510 -5 -4 -3 -2 -1 EIC pseudo-data
Figure 10: Present knowledge of the evolution in x ofthe structure function g , based on the DSSV14 ex-traction [19]. The dotted lines show the results foralternative fits that are within the 90% C.L. limit. The fraction of the spin from angular mo-menta can be obtained by subtracting ∆Σ( Q )and ∆ G ( Q ) from the total spin of the proton, us-ing the sum rule in Eq. 2. The right panel in Fig. 913hows how the angular momenta contribution istotally unconstrained at moderate to low x . Akey observable in disentangling the various partoncontributions to the proton spin is the polarizedstructure function g ( x, Q ). It is proportional tothe difference of the neutral current cross-sectionsof DIS events, with the beams polarized paralleland anti-parallel in the longitudinal direction,12 (cid:20) d σ (cid:28) d x d Q − d σ ⇒ d x d Q (cid:21) (cid:39) πα Q y (2 − y ) g ( x, Q ) . (3) Fixed targetDIS data x=5.2 × -5 (+53)8.2 × -5 (+45)1.3 × -4 (+38)2.1 × -4 (+33)3.3 × -4 (+28)5.2 × -4 (+24)8.2 × -4 (+21)1.3 × -3 (+19)2.1 × -3 (+17)3.3 × -3 (+15.5)5.2 × -3 (+14)8.2 × -3 (+13)1.3 × -2 (+12) -2 ×
10 (+11)3.3 × -2 (+10)5.2 × -2 (+9)8.2 × -2 (+8)1.3 × -1 (+7)2.1 × -1 (+6)3.3 × -1 (+5)5.2 × -1 (+4) PresentuncertaintiesEIC projected data: √ s = 44.7 GeV √ s = 63.2 GeV √ s = 141.4 GeV g ( x , Q ) + c on s t ( x ) Q (GeV ) Figure 11: Projections for the structure function g atdifferent √ s , compared with a model extrapolation andits uncertainties [18]. The curves correspond to differ-ent values of x that are specified next to each curve.For clarity, constants are added to g to separate dif-ferent x bins; moreover, multiple data points in thesame x - Q bin are displaced horizontally. The grayarea marks the phase space currently covered by fixedtarget experiments. See text for details. The integral of the structure function over x is sensitive to the contribution from the quarksand the derivative versus Q is sensitive to the gluon distribution. Therefore ∆ g ( x, Q ) can beaccessed in DIS data via scaling violation fits ∼ d g (cid:0) x, Q (cid:1) / dln Q . However, a precise scalingviolation fit requires, depending on the respectiveuncertainties, a sufficiently large lever arm in Q at any given value of x . Figure 10 shows howthe present knowledge of the structure function g rapidly deteriorates and uncertainties explodeat low x . The EIC pseudo-data are depicted bythe red data points. The uncertainties are smallerthan the symbols illustrating the enormous con-straining power an EIC will have on g . -0.3-0.2-0.1-00.10.20.3 -0.2 0 0.2 ½ - ∫ dx (Quarks + Gluons) (10 -3 < x < 1) − ∫ d x ( Q ua r ks + G l uon s ) ( - < x < - ) EIC √ s=44.7 GeV √ s=44.7 − √ s=44.7 − Figure 12: The EIC’s impact on the knowledge of theintegral of the quark and gluon spin contribution in therange 10 − < x < − ( y -axis) versus the contribu-tion from the orbital angular momentum in the range10 − < x < x -axis). Figure 11 shows the structure func-tion g ( x, Q ) in e + p collisions at √ s =44 . , . , . − . The uncertain-ties of the DSSV14 theoretical prediction [18] areshown by the blue bands. It is clear that theassumed sampled luminosity is already enoughto get really precise measurements, whereas thelarger √ s extends greatly the reach to lower x val-ues where present uncertainties are large. Given14he high statistical precision, it will be critical toconstrain experimental systematic uncertaintiesto below a few percent [19].Figure 12 uses simulated data to clearlydemonstrate the EIC’s impact on the knowledgeof the integral of the proton’s quark and gluonspin contributions for 10 − < x < − versus thecontribution to the orbital angular momentum for the range 10 − < x <
1. A dramatic shrinkageof the uncertainties in the parton helicities is seenwith the largest energy reach. The underlying rea-son for this rapid shrinkage can be traced to thevery unstable behavior of g ( x, Q ) due to the lackof data at small x shown in Fig. 10. Data obtainedin the small x region constrain this behavior. The parton structure of the proton changessignificantly across the QCD landscape sketchedin Fig. 1 of Section 2.2. We illustrate schemati-cally in Fig. 13 how varying x from high values( x ∼
1) to low values ( x ∼ − ) at a given res-olution scale Q of a few GeV reveals the com-plex many-body structure of quarks and gluons in-side the proton. The structure revealed by dialingdown in x changes from the valence quark domi-nated regime, to a regime where the proton’s con-stituents are gluons and sea quark-antiquark pairsgenerated through QCD radiation, and finally atsmall x to an intrinsically nonlinear regime wherethe gluon density is so large that the gluons radi-ate and recombine at the same rate. -2 -1 -3 -4 x Figure 13: The development of the internal quark andgluon structure of the proton going from high to low x . Decreasing x corresponds to increasing the center-of-mass energy. High luminosities at the EIC, combined witha large kinematic reach, open up a unique oppor-tunity to go far beyond our present largely onedimensional picture of the proton. It will enableparton “femtoscopy” by correlating informationon parton contributions to the proton’s spin withtheir transverse momentum and spatial distribu-tions inside the proton. Such three dimensional images have the potential to radically impact ourunderstanding of the confining dynamics of quarksand gluons in QCD. This is because one will beable to probe, with fine resolution Q , parton dy-namics as a function of impact parameter in theproton, out to length scales where their interac-tions are no longer weakly coupled but becomeincreasingly strongly coupled generating the phe-nomena of chiral symmetry breaking and confine-ment.The three dimensional parton structure ofhadrons is uncovered in DIS by measurements ofexclusive final states, wherein the proton remainsintact after scattering off the lepton probe. Thetransverse position of the scattered quark or gluonis obtained by performing a Fourier transform ofthe differential cross-section d σ/ d t , where t is thesquared momentum transfer between the incom-ing proton and the scattered proton. Examplesof exclusive processes are deeply virtual Comptonscattering (DVCS) and the exclusive productionof vector mesons. These are illustrated in Fig. 14.The nonperturbative quantities that encodesuch spatial tomographic information are oftenreferred to as Generalized Parton Distributions(GPDs) and are defined at a nonperturbative fac-torization scale that separates the nonperturba-tive information encoded from perturbative dy-namics at short distances. Powerful renormaliza-tion group arguments, analogous to those of theDGLAP equations for the one dimension partondistributions, can be employed to understand howthe three dimensional dynamics encoded in theGPDs changes as this factorization scale is var-ied [22, 23].GPDs provide important insight into the three15 + ξ x − ξp p ′ x + ξ x − ξp p ′ γ ∗ γ ∗ γ V Figure 14: Diagrams depicting deeply virtual Compton scattering ( left ) and exclusive vector meson production( right ) in terms of GPDs, represented by the yellow blobs. The upper filled oval in the right figure represents themeson wave function. The symbol ξ reflects the asymmetry in the longitudinal momentum fraction of the struckparton in the initial and final state. dimensional structure of polarized protons. Fa-mously, the second moment of one set of quarkGPDs gives the total quark angular momentum ofthe proton, and another set of gluon GPDs canidentically be related to the total gluon angularmomentum. From the “Ji sum rule” [24], the pro-ton’s spin can be expressed as the sum of thesetotal angular momenta. In Sec 3.1, we discussedthe Jaffe-Manohar spin sum rule that decomposesthe spin of the proton into the sum of the quarkand gluon helicities, and their respective angularmomenta. Therefore, in principle, GPD measure-ments can be combined with the direct measure-ments of quark and gluon spin helicities, to pro-vide further insight into quark and gluon orbitalmomenta. However, there are a number of subtleissues that need to be resolved before this programcan be realized fully [25, 26].At present, our empirical knowledge aboutGPDs from DVCS data is mostly limited to thevalence quark region, from the HERMES [27–31]experiment at HERA, the Jefferson Lab 6 GeV ex-periments [32–34] and COMPASS [35] at CERN.In the near future, one anticipates results fromthe Jefferson Lab 12 GeV experiments. There isalso limited relatively low precision HERA dataon sea quarks and gluons from the H1 [36, 37] andZEUS [38] experiments, and in the near future, aglimpse into sea quark distributions will be pro-vided by COMPASS. A high energy, high lumi-nosity EIC will extract sea quark and gluon GPDswith unprecedented reach and precision. Trans-verse spatial distribution of quarks and gluons, inboth protons and complex nuclei, will be extracted through precise measurements of the t -dependenceof DVCS and exclusive cross-sections for produc-tion of J/ψ , φ , π , K and other mesons. For pro-tons, the interval 0 ≈ | t | ≤ . will enableone to map out parton distributions down to animpact parameter of ∼ . t . These simulated dataare based on GPDs extracted from a fit to theworld DVCS data. Bearing in mind the x, Q kine-matic coverage shown in Fig. 2, each bin can beaccessed either only at lower center-of-mass energy(blue band) or at higher energy (red band). Thepurple band represents a region typically reachableat both low and high energies. The impact pa-rameter dependent parton distribution functionsobtained show clearly the growth of parton distri-butions at low x and high Q , where sea quarksare important. The evolution in x and Q cantherefore teach us about the relative spatial dis-tributions of valence quarks, the quark sea andgluons. The plot in Fig. 15 demonstrates that awide window in Q resolution is available at thehigh center-of-mass energy for x ∼ − . Such a Q reach at fixed x is important to extract thegluon spatial distribution through the scaling vi-olation of the DVCS cross-section, just as is thecase for the g and F structure functions.From these data, one can also extract the meansquared radius (cid:104) b T (cid:105) of partons in the proton as afunction of Bjorken x . This is shown in Fig. 16.At small x , this dependence is closely related to16 T (fm) Q √s = 140 GeV1.6 × -3 < x < 2.5 × -3 x B F ( x B , b T ) ( f m - )
10 < Q < 17.8 GeV < 5.6 GeV < 3.2 GeV b T (fm) x e + p → e + p + γ ∫ Ldt = 10 fb -1
10 < Q < 17.8 GeV x B F ( x B , b T ) ( f m - ) √s = 140 GeV -3 < x < 2.5×10 -3 √s = 45 GeV -2 < x < 2.5×10 -2 √s = 45 GeV -2 < x < 0.1 Figure 15: The projected precision of the transverse spatial distribution of partons obtained from the Fouriertransform of the measurement of the unpolarized DVCS cross-sections as a function of | t | at an EIC for a targetedluminosity of 10 fb − at each center-of-mass energy. b T is the distance from the center of the proton, known alsoas “impact parameter”. Left plots show the evolution in x at a fixed Q (10 < Q < . ). Right plotshows the evolution in Q at a fixed x (1 . × − < x < . × ). See text for more details. the QCD string tension in the Regge framework.In this framework, the transition from large tosmall x contains important information that al-lows one to deduce how the dynamical degrees offreedom transition from Reggeon exchanges to so-called Pomeron exchanges, or – in parton language– from quark to gluon exchanges, where the lattercarries the quantum numbers of the QCD vacuum.The evolution over a large range in Q can teachus how the the string tension evolves from thisnonperturbative stringy picture to that of QCDbremsstrahlung. One can thus study with un-precedented precision how the dynamics changeswhen going upwards from the lower right cornerin Fig. 1.In Fig. 16, an inelasticity of y ≤ . x donot go below x = 10 − . The analysis of data withhigher y and lower x is possible but more involved.These considerations are also valid at lower √ s .Therefore, at lower energies there is limited reachbeyond the Reggeon exchange dominated region.Another important exclusive channel is that of J/ψ production, which provides unique access tothe unpolarized gluon GPD through the dominantphoton-gluon fusion production mechanism; this mechanism is discussed further in Sec. 3.4 and il-lustrated in Fig. 19. Transverse spatial images ob-tained from Fourier transforming the t -dependent γ ∗ p → J/ψ + p (cid:48) J/ψ cross-section for √ s = 140GeV show that gluon distributions can be accessedacross the entire transverse plane with fine resolu-tion at small x . x B 〈 b T 〉 ( f m ) γ * + p → γ + p √s = 140 GeV Q = 4.08 GeV = 7.28 GeV = 12.9 GeV Figure 16: The average value of the mean squared par-ton radius of the proton, extracted from the DVCScross-section, plotted as a function of Bjorken x . Re-sults are shown for three different values of Q . Plotfrom the EIC White Paper [1]. Incoherent exclusive scattering is characterizedby the breakup of the proton. These processesare unique in that they are sensitive to the colorcharge fluctuations in the proton. This is discussed17ater on page 25. A combined study of the co-herent processes discussed here (where the protonstays intact), with incoherent exclusive reactions,may allow one to reconstruct how gluon saturationsets in through the progressive clumping of gluons in the transverse plane. Conversely, one may beable to reconstruct where the transition to piondegrees of freedom sets in by quantifying the rel-ative distribution in impact parameter of gluonsand sea quarks.
At sufficiently large Q values in DIS, thecross-section for the exchange of virtual W ± bosons becomes comparable to that of the vir-tual photon case. These charged current (CC)events access different combinations of quark andanti-quark flavors than photon-mediated DIS. (Fora recent discussion of the energy requirementsof neutral current mediated measurements at theEIC, see Ref. [40].) As discussed in [41, 42], theseevents can be used to constrain flavor-dependentparton distribution functions. In this context, CCDIS measurements have an advantage over semi-inclusive DIS (SIDIS) studies. The latter can alsoconstrain flavor-dependent PDFs, but only withthe additional input of fragmentation functions re-lating the out-going hadron with the flavor of theprogenitor quark. Charged current measurementsprovide access to flavor-dependent PDFs with- out the uncertainties introduced by fragmentationfunctions. The theoretically clean CC channel willbe complementary to SIDIS, providing stringenttests of SIDIS computations as well as addressingthe universality of the PDFs. While CC DIS wasstudied extensively at HERA [43], the ability of anEIC to provide polarized proton beams will allowaccess to polarized flavor-dependent PDFs (via thesingle target spin asymmetry) which are importantfor solving the proton spin puzzle as well as inves-tigating nonperturbative aspects of proton struc-ture. Note that in this regard W measurements inpolarized p + p collisions at RHIC [44–47] provideimportant complementary information. Models ofPDF behavior at high x based on helicity reten-tion, which for example predict that ∆ d/d → x → x Q ( G e V ) E I C √ s = G e V , . ≤ y ≤ . E I C √ s = G e V , . ≤ y ≤ . E I C √ s = G e V , . ≤ y ≤ .
101 10 -3 -4 -2 -1 Projected CC DIS data:EIC e+p √ s = 141 GeVHERA CC e+p DIS data 10 100 1000 √ s (GeV) NLO, 0.01 ≤ y ≤ min (GeV ) HERAe − pe − n √ s = G e V √ s = G e V √ s = G e V √ s = G e V σ CC ( pb ) Figure 17:
Left:
Kinematic reach in x and Q for three EIC center-of-mass energies with 0 . ≤ y ≤ . √ s = 141 GeV and the shaded region gives theextent of HERA CC data. Right:
Integrated unpolarized CC DIS cross-section for electron-proton and electron-neutron (dashed line) scattering at NLO accuracy as a function of center-of-mass energy for Q > Q and0 . ≤ y ≤ . He beams) can provide in-formation on possible charge symmetry violation,which may be relevant for the EMC effect [49, 50].It should be noted that studies on parity violat-ing single-spin observables which are sensitive toneutral current γ − Z interference have also beenperformed [40].In order to maximize the impact of CC mea-surements at an EIC, it will be important toachieve the largest possible √ s , as both the CCcross-section and available kinematic reach in x and Q increase with √ s . The x − Q cover-age of CC DIS measurements at an EIC is shownon the left-hand panel of Fig. 17 along with theavailable kinematic range for three center-of-massenergies. The available phase space for CC pro-duction is largest for the largest √ s . These mea-surements will constrain flavor-dependent polar-ized PDFs over a wide range in x . In addition,the x − Q coverage afforded by the largest √ s val-ues provides significant overlap with the HERA re- gion [43]. Thus the EIC can improve on the statis-tically limited unpolarized HERA results while ex-tending them to higher- x . Finally, the kinematicreach provided by the largest √ s available at anEIC means that for fixed x ≥ .
08, there will be adecade or more coverage in Q . This will allow thestudy of the Q evolution of both the CC cross-section and single-spin asymmetry over a wider x range than is possible at lower √ s .The dependence of the cross-section on √ s canbe seen in the right-hand panel of Fig. 17 whichshows the CC DIS cross-section integrated over Q > Q for several values of Q . For Q =100 GeV , the drop in the cross-section as a func-tion of √ s is relatively modest. However, as Q increases, the dip in the cross-section at lower √ s values becomes quite dramatic, making the collec-tion of sufficient statistics at high Q difficult evenwith large integrated luminosities. The higher √ s values eliminate the need to compensate for thefalling cross-section with higher instantaneous lu-minosities. The e ± + p DIS experiments at HERA [51]yielded very accurate information on unpolarizedproton structure in a wide kinematic range downto x = 10 − for Q (cid:38)
10 GeV , where perturba-tive computations are reliable. The HERA exper-iments performed numerous measurements withneutral-current, charged-current, as well as jet andheavy-quark tagged cross-sections. These mea-surements provide the main data set to unravelthe proton’s internal structure and form the back-bone of all the present day global fits of PartonDistribution Functions (PDFs) [52, 53] in a pro-ton. These PDFs are crucial in searches for newphysics at the LHC.Similarly, e ± +A DIS scattering experimentsat an EIC [54] will extract the nuclear PDFs(nPDFs) [55, 56], that are important for a deeperunderstanding of heavy-ion collisions. So far,the kinematic reach of the available fixed target e ± +A cross-section measurements is much morerestricted than for protons, with very little dataavailable for low x ≤ − at Q ∼
10 GeV . Fur- thermore, as noted previously, even at intermedi-ate x they do not provide the required lever armto extract information on gluons through scalingviolations. As a consequence, gluon and sea quarknuclear PDFs are widely unconstrained.An important recent development is the releaseof the nPDF global fit, EPPS16 [57], that includesLHC data from the p +Pb run 1. However, at low- x and low- Q , the latest LHC p +Pb data does lit-tle to constrain nPDFs. The reason for this mod-erate impact lies in the need to evolve the data“backwards”, i.e. from low to high Q , which re-quires much higher precision data. In sharp con-trast to PDFs in the proton, the role of gluons innuclei is still terra incognita .As an example, the nuclear gluon distributionfor e +A collisions at an EIC as a function of Q and for different values of x is shown in Fig. 18.The gluon distribution was obtained using theNLO parameters from the CTEQ collaboration [6]multiplied by the corresponding nuclear correctionfactor from EPPS16. The uncertainty bands re-19
10 CTEQ14NLO+EPPS16
Kinematic limits for√s= 40 GeV eA collisions Kinematic limits for√s= 90 GeV eA collisions
110 CTEQ14NLO+EPPS160.10.1 x = - x = . Q (GeV ) Q max ~ 160 GeV x = - Q max ~ 810 GeV x = - x = . Q (GeV ) x g ( x , Q ) x g ( x , Q ) Figure 18: Nuclear Parton Distribution Functions of gluons as functions of Q for various x values obtained bymultiplying the gluon distribution in the proton extracted by the CTEQ collaboration to NLO [6] with the nuclearmodification ratio for gluons extracted by EPPS16 [57]. The PDFs are cut off at the kinematic limits imposed bythe indicated energies of √ s = 40 GeV ( left ) and 90 GeV ( right ), proposed for e +A collisions at an EIC. We willshow later in Fig. 24 that these uncertainties will be greatly constrained by EIC data. For instance, at x = 10 − they are reduced by a factor of ∼ flect the combined uncertainties from both distri-butions. In contrast to the e + p case shown inFig. 3 of Section 2.4, the DGLAP evolution gen-erates gluon distributions to good accuracy onlyfor high values of x . At x ∼ − the limited leverarm in Q complicates the precise extraction of thegluon density at √ s = 40 GeV. It is only when in-creasing the energy by a factor of 2 or more thatone can access the higher Q where the gluon den-sity can be reliably determined. Furthermore, thereach of the insufficiently explored low- x domainis feasible only at √ s = 90 GeV center-of-massenergy. We will show later that the uncertaintiesfrom current world data on nuclear gluon distri-butions will be significantly reduced by EIC data. N, A G
N,A (x)x g cx, Q e e ʹ c Figure 19: Charm pair production via photon-gluonfusion.
Therefore, an EIC with a wide lever arm in x and Q is critical for unambiguous determinationof the parton structure of nuclei. Such a determi-nation is an important first step towards a deeperexamination of outstanding questions regarding i)how color is confined in a nucleus as opposed to aproton, ii) the nature of the residual color forcesthat bind nucleons together at short distances, andiii) the response of the nuclear medium to coloredprobes.In DIS processes, the fully inclusive reducedcross-section can be written in terms of the struc-ture functions F and F L as σ reduced = F ( x, Q ) − y − y ) F L ( x, Q ) , (4)where F is sensitive to the sum of the quark andanti-quark momentum distributions and F L is sen-sitive to the gluon distribution. For EIC kinemat-ics, up to 10 −
15% of the inclusive cross-section isfrom production of charm quarks–the charm struc-ture function can be measured in nuclei for thefirst time. Since the dominant process is the pro-duction of charm-anticharm pairs through photon-gluon fusion (illustrated in Fig. 19) the measure-ment of this cross-section allows for an indepen-dent extraction of the gluon distribution in nuclei.Simultaneous measurements of the F , F L , and F c ¯ cL structure functions are key to uniquely con-strain PDFs. The current theoretical description,20ven in the case of proton PDFs, has an ambi-guity when the heavy quark production thresh-olds are crossed [58]. This ambiguity, usuallycalled a “mass scheme”, has a significant impacton the PDFs extracted and can be resolved bydetermining the heavy quark structure function F c ¯ cL . Such measurements provide precise valuesof heavy quark masses. σ s t a t / ( σ sys t ⊗ σ s t a t ) ( % ) -1 x -4 -3 -2 -1
10 1 ) ( G e V Q
10 e+Au √s = 89.4 GeV ∫ Ldt = 4 fb -1 /ASys. unc. = 1.6 % Figure 20: Fraction of statistical uncertainty over totaluncertainty in measuring the reduced cross-section at √ s = 89 . x and Q . The assumedsystematic uncertainty is 1.6%. For a quantitative estimate of what can beachieved, collisions at three different √ s were sim-ulated with PYTHIA 6.4 generator [59] includ-ing nuclear modifications [60]. A collection of allthe simulated measurements of DIS reduced cross-sections at an EIC, together with the EPPS16 un-certainties, are shown in Fig. 21 for inclusive (left)and charm (right) production. In both cases, thepoints are shifted by − log ( x ) for visibility. Thestatistical and systematic uncertainties are addedin quadrature. This study corresponds to a com-bined integrated luminosity of 10 fb − . For thecorresponding statistics, the experimental uncer-tainties are dominated by systematic errors, asshown in Fig. 20.Figure 21 (left) depicts in the shaded region,for comparision, the current world data from DISoff heavy-ions. There are no charm measurementsin e +Au collisions. The dashed line in both plotscorresponds to the kinematic limit at the lower40 GeV center-of-mass energy. We observe that the current extrapolated uncertainties, depictedby the grey bands [57, 61], become substantiallylarger beyond this dashed line. Thus data fromthe higher center-of-mass energy will significantlyconstrain these uncertainties, and thereby, QCDevolution of PDFs to smaller x .To emphasize the precision achievable at anEIC, two examples of the reduced cross-sectionas a function of x at the Q values of 4.4 GeV and 139 GeV are shown in Fig. 22 for inclusive(left) and charm (right) production. It is clearfrom Fig. 22 that at large values of x , the uncer-tainties are very small. It is only at x < − andsmall Q that the expected experimental errors onthe EIC measurements become much smaller thanthe uncertainties from the EPPS16 parametriza-tion that are largest at the smallest x values; thesewill clearly be significantly constrained by data atthese x values.Measuring the longitudinal structure func-tion F L , poses additional experimental challenges.This observable is typically very small at high val-ues of x and Q but increases with smaller x and,at a fixed value of x , also rises with Q . It can beextracted through a Rosenbluth separation anal-ysis and requires measurements from collisions ata minimum of three different center-of-mass ener-gies. Using Eq. 4, a fit to the data leads to thenegative of the gradient, giving F L . Experimentaldata with a wide range in center-of-mass energygives the lever arm necessary for precise extrac-tion of F L .Figure 23 shows a collection of the possible F L measurements in e +Au collisions at the EICfor both inclusive (left) and charm (right) produc-tion, plotted versus Q for a number of x values.The two sets of three different center-of-mass en-ergies used in each extraction of F L are also in-dicated on the plot by full and open circles re-spectively. For each set, the simulation assumesan integrated luminosity of 10 fb − . The gray-shaded bands indicate the uncertainties in our cur-rent knowledge of F L derived from the EPPS16 nu-clear PDF [57, 61]. With this luminosity, an EICcan perform very precise measurements of F L inseveral x, Q bins. This accuracy is crucial for asignificant measurement of a quantity that is ex-pected to be very small. The comparison of EIC21seudo-data and errors with the current depicteduncertainties (gray band) demonstrate dramati-cally the need for higher energies allowing one toreach lower x values where uncertainties are large.For F L at Q >
10 GeV and for charm F L thelower energy range does not provide any substan- tial improvement. It is also important to notethat EIC can achieve a comparable precision inmeasuring F L for the proton, improving even onthe existing measurements from HERA [62] wherekinematics overlap. ( x ) )- l og ( x , Q r ed σ x = 5.2 × -1 x = 3.2 × -1 x = 2.0 × -1 x = 1.3 × -1 x = 8.2 × -2 x = 5.2 × -2 x = 3.2 × -2 x = . × - x = . × - x = . × - x = . × - x = . × - x = . × - x = . × - x = . × - x = . × - x = . × - e+Au √s = 31.6 GeV√s = 44.7 GeV√s = 89.4 GeV Fe) ≥ World Data (A CT14NLO+EPPS16 ∫ Ldt = 10 fb -1 /A ) (GeV Q √s = 40 GeV kinematic limit ) (GeV Q ( x ) / )- l og ( x , Q r ed cc σ x = 5.2 × -1 x = 3.2 × -1 x = 2.0 × -1 x = 1.3 × -1 x = 8.2 × -2 x = . × - x = . × - x = . × - x = . × - x = . × - x = . × - x = . × - x = . × - x = . × - x = . × - x = . × - x = . × - x = . × - e+Au ∫ Ldt = 10 fb -1 /A √s = 31.6 GeV√s = 44.7 GeV√s = 89.4 GeVCT14NLO +EPPS16√s = 40 GeV kinematic limit Figure 21: Inclusive ( left ) and charm ( right ) reduced cross-sections plotted as functions of Q and x for both EICpseudo-data and the EPPS16 model (gray-shaded curves) [57, 61]. The uncertainties represent statistical andsystematics added in quadrature. Also shown on the left plot is the region covered by currently available data. x -3 -2 ) ( x , Q r ed σ = 4.4 GeV Q = 4.4 GeV Q = 139 GeV Q = 139 GeV Q e+Au ∫ Ldt = 10 fb -1 /A ∫ Ldt = 10 fb -1 /ACT14NLO+EPPS16√s = 31.6 GeV√s = 44.7 GeV√s = 89.4 GeV CT14NLO+EPPS16√s = 31.6 GeV√s = 44.7 GeV√s = 89.4 GeV x ) ( x , Q r ed cc σ e+Au -3 -2 Figure 22: Inclusive ( left ) and charm ( right ) reduced cross-sections as a function of x at the Q values of 4.4 GeV (solid circles) and 139 GeV (open circles) at three different center-of-mass energies. See text for details. = 1.4 GeV QC=0 = 2.5 GeV QC=0.05 = 4.4 GeV QC=0.1 = 7.8 GeV QC=0.15 = 14 GeV QC=0.2 = 25 GeV QC=0.25 = 44 GeV QC=0.3 = 78 GeV QC=0.35 = 139 GeV QC=0.4 = 78 GeV QC=0.35 = 247 GeV QC=0.45 x − − − − Ldt = ∫
10 fb -1 /A e+Au CT14NLO+EPPS16√s = 31.6, 38.7, 44.7 GeV √s = 63.2, 77.5, 89.4 GeV F L ( x , Q ) + c = 1.4 GeV QC=0 = 2.5 GeV QC=0.005 = 4.4 GeV QC=0.01 = 7.8 GeV QC=0.015 = 14 GeV QC=0.02 = 25 GeV QC=0.025 = 44 GeV QC=0.03 = 78 GeV QC=0.035 = 139 GeV QC=0.04 x − − − − Ldt = ∫
10 fb -1 /A e+Au CT14NLO+EPPS16√s = 31.6, 38.7, 44.7 GeV √s = 63.2, 77.5, 89.4 GeV F L cc ( x , Q ) + c Figure 23: Inclusive ( left ) and charm ( right ) F L structure function, plotted as a function of Q and x . Theuncertainties represent statistical and systematics added in quadrature. The gray-shaded bands depict the un-certainties in our current knowledge of F L derived from the EPPS16 nuclear PDF [57, 61]. See text for furtherdetails. The simulations discussed in the previous sec-tion suggest that an EIC will have an enormousimpact on the global extractions of nuclear PDFs,particularly for the gluons. While the LHC datahave achieved a substantial broadening of cover-age in the kinematical space, the newly exploredregions scan a Q range where the DGLAP RGEsignificantly wash away the nuclear effects, leavingthe low x gluon nearly unconstrained.The modification introduced by the nuclear en-vironment can be quantified in terms of the ratiobetween the nucleus A and the free proton PDF( R Af , f = q, g ) for quarks and gluons, with devi-ations from unity being manifestations of nucleareffects. A depletion of this ratio relative to unityis often called shadowing. The impact study ofEIC simulated data shown in Fig. 21 was done byincorporating these data into the EPPS16 fit [57].However, as the parameterization is too stiff in theas yet unexplored low x region, additional free pa-rameters for the gluons have been added to the functional form (EPPS16* [61, 63]). The corre-sponding R Pb g from EPPS16* is shown in Fig. 24.The grey band represents the EPPS16* theo-retical uncertainty. The orange band is the resultof including the EIC simulated inclusive reducedcross-section data in the fit. The lower panel ofeach plot shows the reduction factor in the uncer-tainty (orange curve) with respect to the baselinefit (gray band). It is clear that the higher center-of-mass energy has a significantly larger impact inthe whole kinematical range with the relative un-certainty roughly a factor of 2 smaller than for thelower center-of-mass energy.We also examined the simulated charm quarkreduced cross-section (blue hatched band), forwhich no data currently exist. The impact ofits measurement for nuclear gluon distributions isshown in Fig. 24. While it brings no additionalconstraint on the low - x region, its impact at high- x is remarkable providing up to a factor 8 reduc-tion in uncertainty (blue curve).23 P b R = 1.69 GeV Q = 31.6 - 89.4 GeVs x − − − −
10 1 R ed . f a c t o r = 1.69 GeV Q = 31.6 - 44.7 GeVs x − − − −
10 1 R ed . f a c t o r g P b R EPPS16* + EIC (inclusive + charm)EPPS16* + EIC (inclusive only)EPPS16* = 10 GeV Q = 31.6 - 89.4 GeVs x − − − −
10 1 R ed . f a c t o r EPPS16* + EIC (inclusive + charm)EPPS16* + EIC (inclusive only) g P b R = 10 GeV Q = 31.6 - 44.7 GeVs x − − − −
10 1 R ed . f a c t o r g P b R Figure 24: The ratio R Pb g , from EPPS16*, of gluon distributions in a lead nucleus relative to the proton, for thelow ( left ) and high ( right ) √ s , at Q = 1 .
69 GeV and Q = 10 GeV (upper and lower plots, respectively). Thegrey band represents the EPPS16* theoretical uncertainty. The orange (blue hatched) band includes the EICsimulated inclusive (charm quark) reduced cross-section data. The lower panel in each plot shows the reductionfactor in the uncertainty with respect to the baseline fit. Impact on Heavy-Ion Physics
Measurements over the last two decades, firstat RHIC and later at the LHC, have providedstrong evidence for the formation of a stronglycoupled plasma of quarks and gluons (sQGP) inhigh energy collisions of heavy nuclei. This sQGPappears to behave like a nearly perfect liquid andis well described by hydrodynamics at around 1fm/ c after the initial impact of the two nuclei[64–67]. For reviews, see [68–71].Despite the significant insight accumulated inthe past 17 years, little is understood about how the initial non-equilibrium state, whose propertiesare little known, evolves towards a system in ther-mal equilibrium. A conjectured picture of the ini-tial phase, based on the CGC framework, suggeststhat at leading order the collision can be approxi-mated by the collision of “shock waves” of classicalgluon fields (Glasma fields), [72–74] resulting inthe production of non-equilibrium gluonic matter.Unfortunately, heavy-ion collisions themselvescannot teach us much about the initial state be-cause most of the details are wiped out during the24 PPS16 x EPPS16 x R g P b ( x , Q ) R g P b ( x , Q ) Q = 10 GeV Q = 1.69 GeV -3 -2 -1 -4 -3 -2 -1 -4 E I C √ s = G e V √ s = G e V √ s = G e V √ s = G e V LHC-AA LHC-AARHIC-AA RHIC-AA
Figure 25: EPPS16 ratio of gluon PDF in a Pb nucleus relative to that of the proton ( R Pb g ), and its uncertaintyband at Q = 1 .
69 and 10 GeV [57]. The plot for Q = 1 .
69 GeV is indicative for processes that produce morethan 90% of all final state particles in a heavy-ion collision at mid-rapidity. The bands on the top of each panelreflect the referring kinematic acceptance of the typical RHIC and LHC experiment. For details, see text. Thevertical red and blue lines indicate the kinematic limits for different EIC center-of-mass energies. evolution of the plasma. The final observables aresensitive to both, the initial state and the finalstate, whose transport parameters one ultimatelyseeks to extract. Therefore, information on the ini-tial state needs to be extracted from experimentson p +A and ultimately e +A with small and wellunderstood final state effects.It was demonstrated in [75] how e + p data canbe successfully used to understand shape fluctu-ations of the proton. Here, the authors studiedmeasurements of coherent and incoherent diffrac-tive vector meson production at HERA to con-strain the density profile of the proton and themagnitude of event-by-event fluctuations. Work-ing within the CGC picture, they found that thegluon density of the proton has large geometricfluctuations. No such data for e +A collisions ex-ists. Assumptions on initial state fluctuations andanisotropies that govern many aspects of the ob-served collective flow phenomena are rather spec-ulative at present.Data from an EIC can therefore have a pro-found impact on our understanding of the prop-erties of the initial state in heavy-ion collisions,such as the momentum and spatial distributions of gluons and sea quarks. Nuclear effects, suchas shadowing and saturation, can be studied. Byvarying the scale and energy of the collision the in-terplay between the soft non-perturbative and thehard perturbative regimes can be addressed.In order to illustrate how the EIC energy mapsonto the kinematic range in A+A collisions we fo-cus on the longitudinal momentum distributionsin the nucleus, the nPDFs described earlier in thissection. Figure 25 shows the EPPS16 [57] nuclearPDF and it’s uncertainty band at Q = 1 .
69 and10 GeV . The plot for Q = 1 .
69 GeV is indica-tive for processes that produce more than 90% ofall final state particles. The bands on the top ofeach panel reflect the referring kinematic accep-tance of the typical RHIC and LHC experiments.We used x ≈ p T / √ s exp( ± η ) where p T ≈ Q ; wechose for the pseudo-rapidity window η = ±
1, typ-ical for the central barrel acceptance of heavy-ionexperiments. The horizontal red and blue lines in-dicate the EIC kinematic limits for two differentcenter-of-mass energies √ s = 40 and 90 GeV, re-spectively. While data from √ s =40 GeV will pro-vide an important constraint on the RHIC A+Adata, it will not reach into the regime where the25ulk of LHC A+A data comes from. On the otherhand, for √ s =90 GeV, the nPDFs cover the kine-matics of semi-hard processes at both RHIC andthe LHC. This expanded coverage is of great im- portance for a common quantitative framework ofA+A collisions at both colliders in the quest for adeeper understanding of the intial conditions andthe transport properties of the sQGP. A key goal of the EIC is to access the high parton density regime of the QCD landscape depicted inFig. 1. The saturation scale Q s , characterizing the QCD dynamics in this regime, is expected to scale as A / . Hence, DIS off heavy nuclei at large per nucleon center-of-mass energies √ s NN makes it possibleto cleanly access this novel intrinsically nonlinear regime of the theory. When Q s is larger than theintrinsic QCD scale, weak coupling methods may be applicable and enable one to relate the nonlineardynamics of gluons and quarks to experimental measurements.A strong hint in the theory that such a novel regime must exist follows from the unitarity boundon QCD cross-sections. This fundamental bound would be violated if the observed rapid rise of gluondistributions with decreasing x persists at even lower x . Remarkably, there exist weakly coupled albeitstrongly interacting many-body interactions in the theory that cause gluons at small x to recombine intoharder gluons at the same rate at which they like to shed softer gluons. A deeper understanding of thisemergent effect, and the wider framework in which such phenomena are embedded, has the potential toradically transform the study of intrinsically nonlinear dynamics in QCD.Although there is a significant body of data at small x from HERA, RHIC and the LHC that canbe described in saturation models, there are important caveats that stand in the way of a discoveryclaim. While saturation models do an excellent job describing a wide variety of HERA data [76], thecorresponding saturation scales, as shown in Fig. 6, are very small. Larger (nuclear) saturation scalesare accessed in proton-nucleus and nucleus-nucleus collisions at RHIC and the LHC, but interpretationof data in terms of the evolving parton dynamics of the nuclear wavefunction is complicated by strongfinal state effects occuring after the collision. Both of these concerns are mitigated in e +A collisions atthe EIC. Figure 6 clearly shows the significant increase in Q s relative to HERA, and final state effectscan be controlled fully.In the following, we shall review the center-of-mass energy requirements for two observables thatare especially sensitive to saturation effects. We will first discuss in Sec. 3.6.1 the evolution of theback-to-back correlation of the two produced hadrons in double inclusive scattering. Here, the available √ s NN range directly determines the magnitude of the saturation scale that is accessed. In Sec. 3.6.2,we will return to diffractive scattering that was briefly mentioned in the context of imaging in Sec. 3.2.Diffractive measurements have fundamentally impacted physics over the centuries. 21st century diffrac-tion measurements at the EIC hold similar potential for discovery. We will demonstrate in a simplesaturation model, the likelihood that novel physics will first manifest itself in these processes. Multiparton correlations allow us to recon-struct the internal structure of protons far morethan single parton distributions alone permit. Akey measurement of multiparton correlations in e +A is the distribution of the azimuthal anglebetween two hadrons h and h in the process e +A → e (cid:48) + h + h + X . This process was discussedpreviously in the EIC White Paper [1]. Thesecorrelations are sensitive to the transverse mo-mentum dependence of the gluon distribution aswell as gluon correlations for which first principlescomputations are now becoming available [77, 78].26 .5 2 2.5 3 3.5 4 4.5 C ( Δ φ ) Δφ (rad) Δφ (rad) Δφ (rad) √s=40 GeV √s=90 GeV epeAu no saturationeAu saturation √s=63 GeV < Q < 2 GeV p Ttrig > 2 GeV/c1 GeV/c < p
Tassoc < p
Ttrig ∫ L dt = 10 fb -1 Figure 26: Comparison of dihadron correlation functions from a saturation model prediction for e +Au collisions(red curve) with e + p collisions (black curve) and calculations from a conventional non-saturated model (hollowdata points) for three different center-of-mass energies ranging from √ s =40 to 90 GeV. For details see text. The precise measurement of these dihadron cor-relations at an EIC would allow one not only todetermine whether the saturation regime has beenreached, but study as well the nonlinear evolutionof spatial multi-gluon correlations. The A depen-dence of this measurement provides another han-dle to study the nonlinear evolution of such corre-lations, and to ascertain their universal features.The saturation scale Q s for a given nucleus de-pends on the gluon momentum fraction x g . Eventhough x g is not directly accessible experimentally,one can effectively constrain the underlying x g dis-tribution in controlling the experimentally mea-sured x by varying beam energies. Dihadron cor-relations are relatively simpler to study at a col-lider. They are measured in the plane transverseto the beam axis, and are plotted as a function ofthe azimuthal angle ∆ φ between the momenta ofthe produced hadrons in that plane. The near-sidepeak (∆ φ =0) of this ∆ φ distribution is dominatedby the fragmentation from the leading jet, whilethe away-side peak (∆ φ = π ) is expected to be dom-inated by back-to-back jets produced in the hard2 → e + p and e +A collisions respectively would then be aclear experimental signature of such an effect. Thehighest transverse momentum hadron in the di-hadron correlation function is called the “trigger” hadron, while the other hadron is referred to asthe “associated” hadron with p assocT < p trigT . Theselected p T ranges affect the effective Q , that to-gether with x g , are the key parameters that governthe process.In order to elucidate the importance of thecenter-of-mass dependence of this measurement wegenerated dihadron correlations for three differentenergies, √ s = 40, 63, and 90 GeV in e + p and e +Au collisions following the procedures describedin [79]. Only charged pions π ± s were used. Thecalculations were performed for 1 < Q < and include a Sudakov form factor to account forthe radiation generated by parton showers. Thehadrons were selected to have p trigT > c and 1 GeV /c < p assocT < p trigT . Statistical errorbars correspond to 10 fb − /A integrated luminos-ity. The away-side correlation peak for the threedifferent energies is shown in Fig. 26. Each paneldepicts the e + p reference curve in black, as wellas the predictions from saturation models in e +Auin red. It is important to verify how precisely thesuppression of the away-side peak can be stud-ied at an EIC and how the saturation modelpredictions can be clearly distinguished from aconventional leading twist shadowing (LTS) sce-nario [80, 81]. Such scenarios include nonlinear in-teractions only in the initial conditions but not inthe QCD evolution of the distributions.To obtain results for the LTS scenario, weuse a hybrid Monte Carlo generator, consisting ofPYTHIA-6 [59] for parton generation, showering27nd fragmentation, DPMJet-III [82] for the nu-clear geometry, and a cold matter energy-loss af-terburner [83]. We employ the nPDFs from EPS09[60] to describe the shadowed parton distributions.The resulting LTS correlation function is indi-cated in Fig. 26 by black hollow points. As ex-pected, the difference between e + p and the non-saturated e +Au is minuscule. Final state nucleareffects hardly alter the correlation peak, an obser-vation that is in agreement with findings at RHICwhen comparing p + p with p +A collisions at mid-rapidity. The difference between the saturated e +Au case and the e + p reference is already visi-ble at lower energies but becomes striking at thehighest energies, namely, at the lowest- x g rangeaccessible.To better illustrate the energy dependence, weplot in Fig. 27 the ratio of the correlation functionsin e +Au over those in e + p for all three energies.Note that the suppression is stronger by a factorof ∼ √ s = 90 GeV when compared to the low- est simulated energy of 40 GeV. Measuring a sup-pression greater than 20% relative to e + p will becrucial in the comparison of data with saturationmodel calculations that typically carry uncertain-ties of at least in this order [79]. The ability tostudy dihadron suppression over a wide range of x g is of the utmost importance for this observable.Fig. 28 shows the corresponding x g distribu-tions for dihadrons produced at the three differ-ent center-of-mass energies discussed. The largerthe energy, the smaller the x g values one can ac-cess, and the further we reach into the saturationregime. Since the saturation scale is a functionof x g alone, we also show the reference Q s valueson the top of the plot. Only a sufficiently widelever arm will allow one to study the non-linearevolution in x g and Q and extract the satura-tion scale with high precision. As we have shown,this requires center-of-mass energies in the rangeof √ s =90 GeV. Δφ (rad) C ( Δ φ ) e A u / C ( Δ φ ) ep √s=40 GeV√s=63 GeV√s=90 GeV Figure 27: Ratio of the dihadron correlation func-tions in e +Au collisions over those in e + p for the threecenter-of-mass energies. x g Q s2 (GeV ) -3 -2 a . u . √s=90 GeV, 〈 x g 〉 =0.06, x gpeak =0.017√s=63 GeV, 〈 x g 〉 =0.08, x gpeak =0.032√s=40 GeV, 〈 x g 〉 =0.13, x gpeak =0.068 Figure 28: x g distributions probed by the correlatedhadron pairs for different center-of-mass energies, √ s =40, 63, and 90 GeV in e +Au collisions. The averageand peak values for the distributions are shown. Thegluon saturation scales Q s corresponding to x g valuesare displayed on top of the plot. .6.2 Diffraction In diffractive DIS, the incoming electron in-teracts with the target proton or nucleus with-out exchanging net color. Experimentally, sucha scattering manifests itself as a rapidity gap inthe detector between the target remnants and thediffractively produced system. This is an indica-tion of a colorless exchange with “vacuum” quan-tum numbers between the projectile fragmentsand that of the target. In contrast, if the exchangecarried color, confinement dictates that the QCDstring corresponding to this exchange would frag-ment into a shower of hadrons that fill up thisgap in rapidity. Diffractive experiments are there-fore in principle outstanding probes of how con-finement operates. However, as we will discuss,diffraction is also a sensitive probe of saturation.Novel diffractive measurements that are feasibleat the EIC thence offer an opportunity to studyphenomena that are sensitive to the strong colorfields generated by both weak and strong couplingdynamics. √ s = G e V x I P = . √ s = G e V x I P = . √ s = G e V x I P = . qq qqg ( F A D − A F p D ) / ( A F p D ) M x2 (GeV ) Q = 5 GeV x IP = 0.01x IP = 0.001 Figure 29: Relative modification of the diffractivestructure function of the nucleus as a function of the in-variant mass of the diffractive system for two differentvalues of x and Q = 5 GeV . At leading order in perturbative QCD, diffrac-tive scattering can be understood as the color sin-glet exchange of two gluons. The cross-sectionis therefore proportional to the square of thetarget’s gluon distribution function. Diffractiveevents are also sensitive to the geometric struc- ture of hadrons. As noted in Sec. 3.2, the Fouriertransform of the t dependence of diffractive cross-sections gives the spatial distribution of gluon con-figurations inside hadrons.In the dipole picture we discussed previouslyin Sec. 2.6, the strong sensitivity of diffractive DISto gluon saturation manifests itself as a strong de-pendence of the cross-section on the ratio Q s /Q .Specifically, it is proportional to the squared dipoleamplitude, and has the functional form: (cid:104) − e − Q s /Q (cid:105) . (5)Since the cross-section depends on both Q s and Q , a large lever arm in Q , in addition to that in x , can dramatically reveal the onset of saturation.This was illustrated in Fig. 8 of Sec. 2.6, where thekinematically allowed Q range where the dipoleamplitude is probed is shown for the two differentEIC energies. It was also shown there that theimpact of a wider lever arm is much greater fordiffractive processes relative to that for inclusiveDIS final states.A striking result from HERA was that approx-imately 15% of the e + p scattering events werediffractive [84]. As discussed in the White Pa-per, a strong prediction of the saturation pictureis that the ratio of diffractive to total cross-sectionshould be enhanced in a heavy nucleus comparedto that of the proton. This is in stark contrast toleading twist shadowing calculations that predicta suppression.A simple observable to study nuclear effectsis the diffractive structure function, F D , which isproportional to the total diffractive cross-section.At high energy, the dipole picture of diffrac-tion predicts that the virtual photon emitted bythe electron fluctuates into | q ¯ q (cid:105) , | q ¯ qg (cid:105) , or higher“Fock” states, which then scatter elastically off thetarget without exchanging net color. These statessubsequently hadronize to a sytem with invariantmass M x . Smaller values of M x are primarily fromthe fragmentation of the | q ¯ q (cid:105) while large M x valuescorrespond to significant contributions from thehigher Fock states. In saturation model computa-tions [85], the q ¯ q dipole contribution (small M x )29 IP Q ( G e V ) Maximum qq̅g Supression √s=40 GeV -2 -3 x IP Q ( G e V ) Maximum qq̅g Supression at √s=90 GeV -2 -3 Figure 30: Maximum q ¯ qg dipole suppression observable at √ s NN = 40 GeV (left) and at √ s NN = 90 GeV (right). is enhanced in a nucleus relative to a proton. Incontrast, because the strong color field of the nu-cleus absorbs a q ¯ qg dipole more strongly than theproton, the diffractive cross-section at large M x issuppressed in the nucleus relative to a proton.This is demonstrated in Fig. 29 that depictsthe modification of the diffractive structure func-tion in heavy nuclei, F DA , relative to that of theproton, F Dp , as a function of the invariant massof the diffractive system. The corresponding kine-matical coverages of the two proposed EIC ener-gies, √ s = 40 GeV and √ s = 90 GeV are shown.At Q = 5 GeV , where one has perturbative con-trol over the DIS probe, we observe for √ s = 90GeV that one can scan the entire region in M x where one sees a sign change from enhancementto suppression.The variable x P denotes the momentum frac-tion of the colorless exchange with respect to thehadron. The logarithm of x P is proportional tothe size of the rapidity gap; for x P ≤ .
01, aclean separation in rapidity exists between pro-jectile and target fragments. Observation of sucha sign change, for the kinematics noted, wouldprovide strong evidence that DIS is probing thegluon saturation regime. On the other hand, whilethe nuclear enhancement of the diffractive cross-section can be identified at √ s = 40 GeV, thesign change is out of reach. For the larger gapsof x P = 0 . √ s = 90 GeV, study-ing the QCD evolution of the diffractive structure function with x P is feasible.The maximal suppression of diffractive cross-sections in the nucleus relative to the proton thatcan be measured at both design energies is shownin Fig. 30. One can investigate for how wide a re-gion in x P – Q this suppression is seen. Towardsthis end, we compute the same ratio as the oneplotted in Fig. 29, at the highest kinematically al-lowed M x . Recall that the greatest suppressionof the ratio of cross-sections is at the largest M x .This ratio is only shown in the region where thesign is negative; it is only at the highest EIC en-ergy that a significant window in x P – Q exists.This is especially the case if we demand Q belarge enough for a perturbative treatment of theDIS probe. Kinematic limits fore + Au → e′ + Au′ + J/ ψ Q ( G e V ) x IP Saturation modelsnot reliably applicable E I C √ s = G e V E I C √ s = G e V Figure 31: Kinematical coverage for diffractive J/ Ψproduction.
30n addition to inclusive diffraction, exclusivevector meson production provides an additionalhandle to study small x gluon distributions. Theadvantage is that such a process is experimentallyclean, and it is easy to measure the squared trans-verse momentum transfer t to the target. As notedpreviously, the momentum transfer is Fourier con-jugate to the impact parameter. Therefore, asdiscussed in Sec. 3.2, measuring exclusive vectormeson production differentially in t makes it pos- sible to study the impact parameter profile of thesmall x structure. Recent phenomenological stud-ies have shown that exclusive J/ Ψ production canbe used to construct the average density profile,and its event-by-event fluctuations, for both pro-tons and nuclei [75, 86, 87]. As we show here inFig. 31, √ s NN = 90 GeV is needed in order tohave access to J/ Ψ production in the region wheresaturation model computations are valid.
Since the earliest days of collider physics, jets have been an important tool in the exploration of QCDand have provided important discoveries and insights in all colliding systems, including e + + e − , e + p hadron+hadron, and nucleus+nucleus. (See for example [88].) With the advances in experimental tech-niques, and corresponding advances in theoretical understanding over time, jets have become precisiontools for studying the parton structure of matter. Jets are guaranteed to contribute at the EIC to avariety of key electron-nucleus and electron-hadron physics topics, such as: • The study of hadronization, to shed light on the nature of color neutralization and confinement(see [42]) • Parton shower evolution in strong color fields to measure cold nuclear matter transport coefficients(see [42]) • The study of diffractive dijet production, which can possibly provide direct access to the gluonWigner function (see [89, 90]) • Constraints on high- x quark and gluon PDFs • Precision measurements of (un)polarized hadronic photon structure (see [91]) • Measurement of the gluon helicity distribution in the proton, ∆ g , and its evolution via the photon-gluon fusion process.While jets are familiar objects in high-energy physics analyses, thanks to the modern colliders suchas the LHC [92], those produced at an EIC will have important differences to the ones produced athadron-hadron machines. Figure 32 presents a typical e + p dijet event arising from the photon-gluonfusion process. In contrast to events in hadron-hadron scattering, e + p events are very clean, with littleenergy present that is not associated with the jets. Additionally, the jets themselves contain relativelyfew particles, and the particles have moderate energies. This will make precision tracking essential forcontrol of jet energy scale systematics.There are at least two features inherent to jets which make them attractive alternatives to single-hadron observables. The first such feature is that they are better surrogates for the scattered partons.As jets contain more of the final state particles which arise as a parton fragments, they representmore accurately the energy and momentum of the initial parton. This is important in cases where it isnecessary to reconstruct the partonic kinematics. Examples of these are determining x γ , the momentumfraction of the parton with respect to the exchanged photon, to isolate resolved photon events for studiesof the partonic structure of the photon or when studying the azimuthal correlations discussed in Sec.31 E T ( G e V )
02 -2 -4 -2 0 -2 -4 ϕ η
Figure 32: Visualization of a typical e + p dijet eventshowing the transverse energy and distribution of par-ticles in rapidity-phi space in the Breit frame. e jete ʹ Figure 33: Schematic depiction of a struck parton prop-agating through cold nuclear matter resulting in the for-mation of a single jet. e +A to baseline e + p collisions, as wellas a comparison of jets which form outside and inside the nuclear medium (illustrated in Fig. 33),will provide a wealth of information about the propagation of partons through nuclear matter and thedynamics underlying the emergence of hadrons from colored partons [42]. To facilitate the studies presented in this sec-tion, DIS events were generated using PYTHIA6.4 [59] and stable particles with transverse mo-menta above 250 MeV/ c were clustered into jetsin the Breit frame using the anti-k T algorithm [93]as implemented in the FastJet package [94]. Whenselecting a dijet event, it was required that oneof the jets had p T greater than 5 GeV/ c and theother greater than 4 GeV/ c . It was also requiredthat the jets be more than 120 degrees apart inazimuthal angle.Because jets are expected to be important ob-servables for many physics topics, it is crucial thatan EIC be in position to utilize them to theirfullest potential. There are several aspects of jetmeasurements which benefit from higher center-of-mass energies. Some of these benefits can beseen in Fig. 34, which shows expected dijet yieldsas a function of invariant mass assuming 1 fb − ofintegrated e + p luminosity for two Q ranges andseveral √ s values representing proposed e + p and e +A energies. We see that the production cross-section increases with √ s , compensating for the high luminosities that would otherwise be neededto collect sufficient statistics. We see this to be es-pecially true for high invariant mass, where the di-jet cross-section drops much more rapidly for lowerenergies.High energies will be needed to take advan-tage of the largest jet p T and dijet mass ranges,which will be important for characterizing observ-ables such as cross-sections and spin asymmetries(among others) that depend on these variables[91, 95]. It is worth noting that the correctionof raw jet spectra by means of the standard un-folding procedures is affected by the steepness ofthe spectra. Higher energies imply harder spectra,as illustrated in Fig. 34, and subsequently requiresmaller unfolding factors thus reducing the overalluncertainties in the final spectra.As seen above, access to higher p T ’s enabledby larger collision energies is important for extend-ing the kinematic reach of jet measurements. Inaddition, the significant yield of jets at high p T may be important for certain analyses which uti-lize the substructure of these objects. Figure 3232 oun t s / f b - C oun t s / f b - Dijet Mass (GeV/c ) Dijet Mass (GeV/c ) √s = 141 GeV√s = 63 GeV √s = 90 GeV√s = 40 GeV Q = 10-100 GeV Q = 10-100 GeV Q = 1-10 GeV Q = 1-10 GeV
10 20 30 40 50 60 70 80 90 10 20 30 40 50 60 70 80 90 10011010 (cid:31)(cid:31) aajetjet Figure 34: Dijet yields as a function of invariant mass scaled to a luminosity of 1 fb − for Q = 1 −
10 GeV (leftcolumn) and Q = 10 −
100 GeV (right column). The top row compares proposed e + p center-of-mass energieswhile the bottom row compares e +A energies. shows that a typical jet at an EIC will contain rel-atively few particles. This is quantified in Fig. 35,which shows (for inclusive jets) the average num-ber of particles inside a jet as a function of the jettransverse momentum for two Q ranges and √ s values. It is seen that the particle content of a jetgrows as a function of p T and is largely insensitiveto Q or √ s .Analyses which utilize jet substructure havebecome quite important in both hadron-hadron[96–98] and heavy ion [99] collisions and will cer-tainly be important in both e + p and e +A at anEIC. Substructure will be invaluable to the studyof the nuclear medium, where observables such asthe jet fragmentation function, jet profile, and firstsplitting (among many others) will quantify jetmodifications and thus shed light on how partonslose energy while traversing the medium.Observables sensitive to jet substructure canalso be used to identify and separate jets that arise Jet p T (GeV/c) o f P a r t i c l e s i n J e t = 1 -
10 GeV Q = 10 -
100 GeV o f P a r t i c l e s i n J e t Photon-Gluon Fusion√s = 141 GeV√s = 40 GeV
Figure 35: Average number of particles with trans-verse momentum greater than 250 MeV/ c within ajet as a function of jet transverse momentum (withthe root-mean-square of the distribution representedby the error bars) for Q = 1 −
10 GeV (bottom)and Q = 10 −
100 GeV (top) and two center-of-massenergies. from quarks versus those from gluons as the dif-33erent fragmentations of the two parton types willlead to different energy distributions within a jet.For measurements which take into account jet sub-structure, it will be important that the jet containsenough particles to construct meaningful observ- ables. The larger yield of high p T jets availableat higher center-of-mass energies will ensure thatprecision measurements utilizing jet substructurecan be carried out. Progress has been achieved in developingframeworks to extend our understanding of par-ton structure beyond the one dimensional PDFs.One such example are the GPDs that provide in-formation on the transverse spatial structure ofhadrons. Transverse momentum dependent par-ton distributions (TMDs) provide another pow-erful example [101–103]. The TMDs depend notonly on the longitudinal momentum fraction x ofthe parton but also on its transverse momentum k T and therefore contain much more detailed in-formation on the internal structure of polarizedand unpolarized protons relative to the PDFs.Thus far, the main focus of studies has been onquark TMDs while the available studies of gluonTMDs are rather sparse. Of particular interest,is the distribution of linearly polarized gluons in-side an unpolarized hadron, h (1) ⊥ [104, 105]. It hasbeen shown that this distribution can be accessedthrough measuring azimuthal anisotropies in pro- cesses such as jet pair (dijet) production in e + p and e +A scattering [106–111]. Furthermore, it isrecognized that these gluon distributions play acentral role in small x saturation phenomena [110].Given the important dual role of these measure-ments, we conducted first feasibility studies of di-jet measurements at an EIC in e +A collisions andinvestigated their dependence on √ s .Recent studies in p + p [112] and p +Pb colli-sions [113] at the LHC have revealed long-rangenear-side azimuthal angular cos 2 φ correlations forparticle production in high multiplicity events.Such correlations are commonly referred to as“ridge” correlations. They can also be quanti-fied by measurements of v = (cid:104) cos 2 φ (cid:105) . Since theazimuthal angle correlation in dijet production in e +A at high energies originates from long-rangedeikonal interactions [110], one can make this con-nection explicit by parameterizing the azimuthalstructure arising from the linearly polarized gluon d σ / d φ ( m b ) d σ / d φ ( m b ) d σ / d φ ( m b ) φ (rad) × -9 All photon polarizationsv = -0.04 (-0.04)e+Au √s = 90 GeV 1.25 < q T < 1.75 GeV/c 2.75 < P T < 3.25 GeV/c ∫ Ldt = 1 fb -1 /A10 2 3 4 5 6 φ (rad) × -9 Longitudinal photon polarizationsv = 0.16 (0.14)10 2 3 4 5 6 φ (rad) × -9 Transversal photon polarizationsv = -0.14 (-0.13)10 2 3 4 5 6 x ≈ 5 × -3 Parton levelReconstructed dijets
Figure 36: d σ/ d φ distributions for parton pairs (blue points) generated with the MC-Dijet [100] generator andcorresponding reconstructed dijets (red points) in √ s =90 GeV e +Au collisions for 1 . < q T < .
75 GeV/ c and 2 . < P T < .
25 GeV/ c . The error bars reflect an integrated luminosity of 1 fb − /A. The left plot showsthe azimuthal anisotropy for all virtual photon polarizations, and the middle and right plot for transverse andlongitudinal polarized photons, respectively. For details, see text. v , defined as in the above.However, the azimuthal angle φ is here defined asthe angle between the transverse momentum vec-tor of the jets, (cid:126)P T , and the transverse momentumimbalance, (cid:126)q T .We studied the production of a q ¯ q dijet atleading order in the high energy (small x ) scat-tering of an electron off a gold nucleus. For oursimulations, we used the MC-Dijet event genera-tor [100] to generate the correlated partons. Thegenerator determines the distribution of linearlypolarized gluons of a dense target at small x bysolving the B-JIMWLK renormalization evolutionequation [114–117]. We restrict ourselves to kine-matic configurations where q T < P T , referred to asthe “correlation limit” or TMD regime [107, 118].Only in this limit can the underlying theory be ex-pressed in terms of a specific gluon TMD, part ofwhich is the distribution of linearly polarized glu-ons, h (1) ⊥ that one seeks to determine. The partonsare then passed to fragmentation algorithms fromthe PYTHIA8 event generator [119] for showeringand hadronization into jets. After experimentalacceptance and kinematic cuts, all remaining finalstate particles are used as input to a jet finder al- gorithm (FastJet [94]) and the relevant kinematicvariables are calculated.Figure 36 shows the resulting d σ/ d φ distribu-tions for the original parton pairs (blue points)and the reconstructed dijets (red points) in √ s =90GeV e +Au collisions for 1 . < q T < .
75 GeV/ c and 2 . < P T < .
25 GeV/ c . The error bars re-flect an integrated luminosity of 1 fb − /A. Theleft plot shows the azimuthal anisotropy for allvirtual photon polarizations, and the middle andright plot for transversal and longitudinal polar-ized photons, respectively. The quantitative mea-sure of the anisotropy ( v ) is listed in the figures.The values shown are those for parton pairs; theaccompanying numbers in parenthesis denote thevalues derived from the reconstructed dijets.Note the characteristic phase shift of π/ v due to the dominance of transverslypolarized photons, as shown in the leftmost plot inFig. 36. While the polarization of the virtual pho-ton cannot be measured directly in this process,one will be able to disentangle both contributions P T ( G e V / c ) q T (GeV/c) √ s = G e V , x = . x = . x = . √ s = G e V , x = . x = . x = . Acceptance in Correlation Range √s = 90 GeV √s = 40 GeV e+Au P T ( G e V / c ) q T (GeV/c) L √s = 90 GeV √s = 40 GeV e+Au Figure 37: Kinematic range in q T versus P T of the relevant correlation region, q T < P T , for two EIC energies, √ s =40 and 90 GeV. On the left plot we depict lines of constant x for the referring energies and on the right weshow lines of constant azimuthal anisotropy for longitudinally polarized virtual photons.
35y either analyzing their dependence on Q , q T ,and P T , or by two-component fits constraining v L and v T using the relation v = ( Rv L + v T ) / (1+ R ),where R is a kinematic factor depending only onknown kinematic quantities such as Q and P T .The reconstructed dijets reflect the originalanisotropy at the parton level remarkably well de-spite the dijet spectra not being corrected for ef-ficiency in this study. The loss in dijet yield,mostly due to loss of low p T particles, is on theorder of ∼ e + p events generatedby the PYTHIA event generator that contains noazimuthal anisotropies effects. We find, with theappropriate cuts, a signal to background level ofaround 4:1. The background contributions showno modulations and can be easily subtracted viatwo-component fits.To illustrate the energy dependence of thismeasurement, we plot in Fig. 37 the kinematic range in q T versus P T of the relevant “correlation”region q T < P T , for two EIC energies, √ s =40 and90 GeV. In the left plot, we depict lines of constant x for these energies, and in the right, we showlines of constant azimuthal anisotropy for longitu-dinally polarized virtual photons ( v L ). It becomesimmediately clear that substantial anisotropies, v L ≥ .
15, can only be observed at the larger en-ergy.From an experimental point of view, even moreimportant is the magnitude of the average trans-verse momentum P T . This is because jet recon-struction requires sufficiently large jet energies tobe feasible. The lower the jet energy, the more par-ticles in the jet cone fall below the typical track-ing thresholds ( p T ∼
250 MeV/ c in collider detec-tors), making jet reconstruction de facto impossi-ble. This point was also emphasized in Sec. 3.7.1.Ultimately, the extraction of the gluon distribu-tion h (1) ⊥ , requires a wide range of coverage in q T , P T and thereby x and v [110], that only will befeasible at the higher EIC energy. In this report, we assessed the case for anEIC with the highest energies discussed within thescope of the EIC White Paper. We began by firsttaking the big picture view of understanding ofthe parton structure of protons and complex nu-clei. We observed that while there are corners ofthe QCD landscape where a great deal is under-stood, there are fundamental questions across thewide swath of this landscape that demand answers.Chief among those is the puzzling confining many-body dynamics of quarks and gluons. Little is un-derstood about how the parton structure of pro-tons and nuclei changes under boosts, how quarksand gluons arrange themselves spatially, and whattheir distributions are in transverse momentum.How is the spin of the proton divided up amongstthe helicities of quarks and gluons and their con-fined orbital motion? Are the quantum numbersof protons and nuclei carried only by valence par-tons with large momentum fractions x ∼ x partons?We know little of the correlations amongst par-tons, and how these change with energy and reso-lution across the landscape. We discussed the con-jectured phenomenon of gluon saturation, wherebymatter inside protons and nuclei is weakly coupledeven though the corresponding color electric andmagnetic fields are amongst the strongest in allnature. If this conjecture is confirmed by exper-iment, the physics of saturated gluons will be aremarkable example of fully nonlinear dynamicalphenomena whose properties can nevertheless becomputed. An understanding of this corner of thelandscape may provide a window to other regimesof confining dynamics where equally strong colorfields exist but where the absence of small param-eters impair the comparison of theory to experi-ment. An example of such a phenomenon is thephysics of how struck quarks and gluons transforminto different species of hadrons.The EIC brings penetrating and varied tools36o obtain answers to the above questions. As theworld’s first DIS collider with polarized protonbeams, it will have a much wider lever arm relativeto previous experiments, to extract the helicitydistributions of quarks and gluons and gain insightinto their confined orbital motion. The high lu-minosities will enable the extraction of transversespatial and momentum distributions of quarks andgluons. Charged-current probes of polarized pro-tons will allow for clean extraction of polarized fla-vor distributions. Furthermore, the wide lever armin x will reveal the extent to which the proton’sspin is transmitted to small x . In general, a sig-nificant contribution from small x to the proton’squantum numbers may lead to novel empirical in-sight into the role of the QCD vacuum in hadronstructure.As the world’s first electron-nucleus collider,the EIC will enable first measurements of nucleargluon distributions in DIS. These will reveal theextent of nuclear shadowing, and how these dis-tributions change with Q . Measurements of nu-clear fragments, in coincidence with hadron finalstates, will provide novel information on the par-ton structure of the composite objects that trans-mit short-range nuclear forces. Nuclei are also alaboratory to understand multiple scattering, en-ergy loss and hadronization of quarks and gluonsin QCD media. How jet showers develop in thenuclear medium, and the propagation of heavyquarks in this medium, will be studied for the firsttime. It is anticipated that gluon saturation sets inat much larger values of x (lower energies) in largenuclei than in the proton. The EIC should be ableto unambiguously extract how the nuclear satura-tion scale evolves with x , and reveal its dependenceon the atomic number. Such studies can also lookfor evidence that the physics of gluon saturationis universal–that phenomena in this regime do notdepend on the quantum numbers that are specificto each nucleus.To fully exploit the novel capabilities of thismachine with its versatile polarized proton and nu-clear beams, and in the high luminosities that willbe available, it is important to assess the center-of-mass energy requirements of the key measure-ments that will fulfill the promise of a deep andvaried exploration of the QCD landscape. To il- lustrate the energy dependence of these measure-ments, we chose two center-of-mass energy scenar-ios, √ s ≈
60 GeV and 140 GeV for e + p and √ s NN ≈
40 and 90 GeV for e +A collisions. Weassumed further that the higher energy machinecan also run at lower energies.In assessing the energy case, we first examinedthe relevant lessons provided by past DIS exper-iments and those provided by heavy-ion experi-ments. We found that large lever arms in energyand resolution are crucial in maximizing the dis-covery potential in both of the cases studied. Inthe DIS case, scaling violation effects from the Q evolution of quark and gluon distributions aremore pronounced at small x . Because of this fea-ture of the theory, further reach towards small x has a bigger role in constraining uncertainties thansimple kinematic considerations alone might sug-gest. As a specific case study, we explored in asimple dipole model, the impact of a larger leverarm in Q for a possible discovery of gluon sat-uration. While the saturation scale Q s does notvary strongly with x from low to high energies, theadditional factor of 5 in Q can have a dramaticeffect. This is because inclusive, and especially, ex-clusive and diffractive measurements have a strongnonlinear dependence on Q s /Q , not on Q s alone.A striking effect of the dependence of a mea-surement on center-of-mass energy was observedin simulations of the helicity distributions of theproton. Our studies showed a rapid shrinkage ofthe uncertainties in the parton helicities at thehigher collision energy relative to the lower energy.This is because the uncertainties in the polarized g structure functions grow very rapidly with de-creasing x . Therefore, even a factor 5 enhancedreach relative to available data will strongly con-strain these uncertainties.The EIC White Paper laid out how high lumi-nosities at an EIC, combined with a large kine-matic reach, open up a unique opportunity togo far beyond our present largely one-dimensional“images” of the proton. Our studies in this con-text illustrate that a high energy EIC will be theideal machine for detailed quantitative studies ofhard exclusive reactions and the unexplored seaquark and gluon GPDs. We showed that a largelever arm in Q and a wide x coverage are essential37n determining the transverse spatial distributionof partons obtained from the measurement of theDVCS cross-sections.The measurements of structure functions inDIS provide the main data set to unravel the in-ternal structure of hadrons and nuclei and formthe basis of all present-day PDFs. While highlyprecise measurements for the proton were per-formed at HERA, our knowledge of structure func-tions of nuclei is poor. We presented in this re-port the results of simulations of the F , F L , aswell as F c ¯ c and F c ¯ cL structure functions of nucleifor the two center-of-mass energies. We showedthat the higher center-of-mass energy has a signif-icantly larger impact on the extraction of the nu-clear gluon distribution over the whole kinematicrange. The relative uncertainties for the higherenergy are approximately a factor 2 smaller com-pared to the lower energy range. While this lowerenergy range does improve our current knowledgeof nuclear PDFs substantially at larger x , it pro-vides only moderate constraining power at lower x , a range that is relevant for our understandingof p +A and A+A collisions at the LHC. We alsostudied the luminosity requirements of these mea-surements. We find that the experimental errorsare dominated by systematic uncertainties overmost of the available x – Q phase space. In con-trast with other measurements, these studies re-quire only modest luminosities; statistics beyonda few fb − do not improve the precision achieved.The ability to reach low x in e +A collisions ismandatory to explore the realm where gluon sat-uration can be measured. To study the energydependence of probes sensitive to saturation weselected the suppression of dihadron productionas an example of a key measurement of satura-tion. We find that only the larger energy providesenough lever arm to study the nonlinear evolutionin x g and Q . At lower energies, the suppressionis rather small and competes with systematic un-certainties in the measurements and in theoreticalmodels.As discussed in the EIC White Paper, a gen-uine prediction from the saturation picture is theenhancement of the ratio of diffractive to total cross-section in the nucleus compared to that ofthe proton. This is in contrast to conventionalleading twist shadowing calculations that predicta suppression in this ratio. We expanded ourstudies to higher Q values than those exploredin the EIC White Paper. We find a transitionfrom an enhancement of the diffractive structurefunction in nuclei to a suppression as a functionof increasing invariant-mass of produced particles.This “sign flip” signals the strong absorption ofquark-antiquark-gluon and higher Fock states bythe saturated gluon medium. Our simulations alsoshowed that this sign flip signature of saturation isnot accessible at the lower center-of-mass energy.We also added in this report the importanttopic of jet measurements that was omitted in theEIC White Paper. Jets have become a precisiontool in the exploration of QCD and have providedimportant discoveries and insights in many collid-ing systems. In addition to performing a generalstudy of the effect of collision energy on jet observ-ables, we looked at the measurement of azimuthalanisotropies in dijets. These conjectured asymme-tries allow one to extract the transverse momen-tum dependent distribution of linearly polarizedgluons in nuclei. The results of our studies are notsurprising for energy hungry hard probes such asjets. We showed that a high energy EIC will be in-deed essential to take advantage of the full poten-tial of jet measurements. We also noted that thelarger yield of high p T jets and the harder jet spec-tra available at the higher collider energies cannotbe compensated by higher luminosities. The po-tential of an EIC for measuring dijet asymmetriesat the lower energy is minimal.In Table 1, we list the key measurements atan EIC for which the energy dependence was as-sessed, either in this document or already in theWhite Paper [1]. The last column ranks the roleof the chosen higher energy, relative to that ofthe lower energy, for each measurement. Based onthese results, we conclude that the greater reachprovided by the higher energy chosen for our studygreatly enhances the physics potential of an EICand amplifies the discovery potential of these mea-surements.38 rocess Observables What we learn Impact ofhigh energyon physics Inclusive DIS Unpolarized d σ d x d Q Gluon momentum distributions g A ( x, Q ), nuclearwave function SignificantCollective nuclear effects at intermediate- x Significant Q evolution: onset of DGLAP violation, satura-tion IndispensableBeyond DGLAP A-dependence of shadowing andantishadowing IndispensableParton distribution functions in nuclei ModeratePolarized structurefunction g Unravel the different parton contributions to thespin of the proton (polarized gluon distribution∆ G , ∆Σ, L q , g ) IndispensableSemi-inclusiveDIS Productioncross-section foridentified hadrons ∗ Polarized quark and antiquark densities, quark con-tribution to proton spin; asymmetries like ∆¯ u − ∆ ¯ d ;∆ s Indispensable(Un)polarizedcross-section in W production ∗ Flavor separation at medium x and contribution ofquarks to the proton spin IndispensableProduction of lightand heavy identifiedhadrons ∗ Transport coefficients in nuclear matter, color neu-tralization: mass dependence of hadronization,multiple scattering and mass dependence of energyloss medium effect of heavy quarkonium production ModerateDihadroncorrelations k T -dependent gluons f ( x, k T ); gluon correlations,non-linear QCD evolution/universality; saturationscale Q s SignificantDifferentialcross-sections andspin asymmetrieswith longitudinaland transversepolarization ∗ Sivers function & unpolarized quark and gluonTMDs, quantum interference & spin-orbital corre-lations, 3D imaging of quark and gluon’s motion,QCD dynamics in an unprecedented Q ( P hT ) range Moderate forquarks;significant forgluonsExclusive anddiffractive DISin e + p & e + A collisions Spin asymmetries,d σ/ d t GPDs, transverse spatial distributions of quarksand gluons; total angular momentum Significant σ diff /σ tot , d σ/ d t ,d σ/ d Q Spatial distribution of gluons in nuclei; non-linearsmall- x evolution; saturation dynamics IndispensableInclusive jets,dijets,photon-jet d σ/ d t for diffractivedijets Direct access to the gluon Wigner function Indispensable(Un)polarized dijetcross-sections Constraints for high- x quark and gluon PDFs, andfor (un)polarized photon PDF IndispensableTable 1: Summary of key measurements at the EIC along with our assessment of the impact of the EIC WhitePaper’s higher center-of-mass energy range on these measurements. ∗ Energy dependence already discussed in the EIC White Paper [1]. ppendix A Kinematic Variables Variable Description
A Atomic Number b T Transverse position of parton inside a nucleon/nucleus. Often referred to as impact parameter.∆ g Gluon helicity contribution to the total spin of the proton. ∆ g is a function of x and Q .∆ G Integrated gluon helicity contribution to the total spin of the proton, i.e. (cid:82) d x ∆ g ( x, Q ). ∆ G isa function of Q only.∆Σ Quark helicity contribution to the total spin of the proton. ∆Σ is a function of x and Q . η Pseudo-rapidity of particle or jet. F Structure function sensitive to the sum of quark and anti-quark momentum distributions in thenucleon/nucleus. F is a function of x and Q . F L Longitudinal structure function dominated by the gluon momentum distribution in the nu-cleon/nucleus at low x . F L is a function of x and Q . F D Diffractive structure function. F D is a function of x and Q . g Polarized structure function. g is a function of x and Q . h (1) T Distribution of linearly polarized gluons inside an unpolarized hadron. k T Intrinsic transverse momentum of partons in the nucleon/nucleus. L Orbital angular momenta of quarks and gluons. L is a function of Q only. M X In diffraction, is the squared mass of the diffractive final state. p T Transverse momentum of a hadron or jet. P T Average transverse momentum of a dijet. q T Difference in momenta of a dijet, or transverse momentum imbalance. Q Squared momentum transfer to the lepton, equal to the virtuality of the exchanged photon. Q can be interpreted as the resolution power of the scatter. Note the relation Q ≈ xys . Q s Saturation scale, indicating the Q value at a given x were gluon saturation starts to dominate. Q s is a function of x only. R A f Ratio of a the parton distribution function in a nucleus, A, over that in the proton. s Squared center-of-mass energy. In e + p collisions s ≈ E e E p . √ s NN Nucleon-nucleon center-of-mass energy in heavy-ion collisions. σ reduced Inclusive DIS cross-section, simplified (reduced) by dividing out the Mott cross-section. t Square of the momentum transfer at the hadronic vertex, ( p in − p out ) . v The second harmonic coefficient of the azimuthal Fourier decomposition of a given momentumdistribution. Measure of azimuthal momentum space anisotropy of particle emission. x In the parton model, is the fraction of the nucleon or nucleus momentum carried by the struckparton (0 < x < e + p ). x g Longitudinal momentum fraction of a gluon involved in hard interactions. x P In diffraction, is the momentum fraction of the colorless exchange (Pomeron) with respect to thehadron. x γ The momentum fraction carried by the parton from a resolved virtual photon, where resolvedrefers to the case when the photon fluctuates into a virtual hadronic state and can contribute aquark or gluon to a hard-scatter with a parton from the target. ξ In Deeply Virtual Compton Scattering (DVCS), reflects the asymmetry in the longitudinal mo-mentum fraction of the struck parton in the initial and final state. y Inelasticity defines as the fraction of the lepton’s energy lost in the nucleon rest frame. It it thusalso the fraction of the incoming electron energy carried by the exchange boson in the rest frameof the nucleon. Note that (0 < y < eferences [1] A. Accardi et al. , Eur. Phys. J. A52 , 268 (2016), arXiv:1212.1701.[2] F. Wilczek, Nucl. Phys.
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