Experimental Aspects of Jet Physics at a Future EIC

B.S. Page,1 Xiaoxuan Chu,1 and E.C. Aschenauer1

1Physics Department, Brookhaven National Laboratory, Upton, NY 11973, U.S.A.(Dated: February 5, 2020)

In this work, we present an overview of experimental considerations relevant to the utilizationof jets at a future Electron-Ion Collider (EIC), a subject which has been largely overlooked up tothis point. A comparison of jet-finding algorithms and resolution parameters is presented alongwith a detailed analysis of basic jet quantities, such as multiplicities and kinematic distributions.A characterization of the energy in the event not associated with a jet is also made. In addition,detector requirements and the effects of realistic detector resolutions are discussed. Finally, anexample analysis is presented in which dijets are used to access the gluon helicity contribution tothe spin of the proton.

I. INTRODUCTION

Jet observables have proven to be powerful tools for theexploration of the subatomic world in high energy colliderenvironments (see for example [1]). Early jet measure-ments at e+e− colliders established the spin-1/2 natureof quarks as well as the existence and spin propertiesof gluons and helped solidify Quantum Chromodynam-ics (QCD) as the correct theory of the strong interaction.At hadron-hadron colliders (as well as the lepton-protoncollider HERA), jets have become indispensable tools forstudies ranging from the determination of both polarizedand unpolarized parton distribution functions (PDFs)to the exploration of the electroweak sector to beyondthe standard model searches, for examples please see[2–11] and references therein. Advances in backgroundsubtraction and substructure techniques have also madejets attractive probes of the hot dense medium createdin heavy ion collisions [12, 13]. Jet data coming frommodern colliders such as the LHC are being matched byever more sophisticated theoretical understanding, withcalculations for many channels reaching next-to-next-to-leading order (NNLO) in αs and including the resum-mation of relevant large logarithms. The combination ofadvanced experimental techniques and theoretical powermake jet observables true precision probes.

Given the utility of jets in other collider settings, itis logical to explore their potential applications at theproposed Electron-Ion Collider (EIC). Several topics im-portant to the physics goals of an EIC have been iden-tified which may benefit from jet analyses, including ac-cessing the gluon Wigner distribution [14, 15], probingthe linearly polarized Weizsaecker-Williams gluon trans-verse momentum dependent distributions [16–19] and thegluon Sivers function [20], exploring the (un)polarizedhadronic structure of the photon [21], constraining(un)polarized quark and gluon PDFs at moderate tohigh momentum fraction (x) values [22], and studies ofhadronization and cold nuclear matter properties [23, 24].There are two features inherent to jets which make themattractive probes for the above topics, the first being thatjets are good surrogates for the scattered partons due tothe fact that they contain many of the final state parti-

cles that arise as the parton hadronizes and thus moreaccurately represent the parton kinematics than singleparticle observables. The second property is that jestshave substructure, which characterizes the distributionof energy within the jet in a rigorous way [25, 26]. Study-ing how substructure is modified between e+p and e+Acollisions could provide information about how partonsloose energy in the cold nuclear medium [27].

Although several studies of the impact jets may have atan EIC for specific topics have been performed, no dedi-cated exploration of the experimental issues surroundingjet finding at an EIC has been done. And, while jets werestudied extensively at HERA, the lower center-of-massenergy envisioned for an EIC means that many of thejet properties observed at HERA will be different at anEIC. Therefore, this paper systematically details severalexperimental aspects of jet physics as they are expectedto manifest at an EIC, as well as outlining an exampleanalysis. We hope this paper will serves as a resource forthose interested in exploring the use of jets at a futureEIC for a wide range of physics topics.

The remainder of the paper is organized as follows:Sec. II describes the Monte Carlo setup used to gener-ate the e+p events we analyze. In Sec. III, we discussesseveral aspects of jet finding, such as the choice of algo-rithm and recombination parameter as well as some basickinematic properties of jets at an EIC. Section IV charac-terizes energy and particle distributions from the under-lying event, as provided by our Monte Carlo, which willadd additional energy to the reconstructed jets. In addi-tion the simulated underlying event activity is comparedto similar measures from p+p events at

√s = 200 GeV

performed by the STAR experiment. The effect that arealistic detector will have on the energy resolution of re-constructed jets is explored in Sec. V by smearing themomentum and energy of incoming particles accordingto a specific detector model. Special attention is given tothe role of hadronic calorimetry at mid-rapidity. SectionVI presents a method for accessing gluon polarization bymeasuring the longitudinal double-spin asymmetry fordijet final states. Strategies for isolating the appropriatepartonic subprocesses utilizing cuts on the dijet kinemat-ics are discussed and the value and statistical precision

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of the expected asymmetries are shown and comparedwith our current knowledge of the polarized parton dis-tribution functions. Finally, we summarize our findingsin section VII.

II. MONTE CARLO SAMPLE

To facilitate the studies presented in this paper, alarge sample of e+p pseudo-data was generated usingPYTHIA-6.4 [28]. Two proton PDFs were used as in-put, depending on the Q2 range being simulated. ForQ2 greater than 1.0 GeV2, the CTEQ6.1 PDF [29] setwas used, while the CTEQ5m set [30] was used for10−5 < Q2 < 1.0 GeV2. The SAS PDF set [31, 32] wasused for cases when the partonic structure of the photonwas relevant. The choice of the lower Q2 limit was drivenby the acceptance of a proposed lowQ2-tagger, which willreside near the beampipe, outside of the central detector.The CTEQ5m PDF is used for the proton, because con-trary to modern PDFs (i.e., CT, NNPDF, HERAPDF,MSTW) its value is not frozen at its input scale Q2

0, butallows description of the partonic structure of the protonat Q2 ≤ Q2

0. In addition, this set reproduces the low Q2

HERA cross sections quite well (see Figure 2 in [21]).Events were generated at center-of-mass energies (

√s)

of 45 GeV and 141 GeV, corresponding to (electron xproton) beam energies of 10 x 50 GeV and 20 x 250 GeV,respectively. These energies represent lower and higherranges generally considered for an EIC [33]. As of thiswriting, EIC machine designs [34] are still being finalized,so the ultimate maximum and minimum beam energiesmay deviate somewhat from those assumed in this paper,but should not greatly affect the conclusions drawn here.It is also envisioned that the EIC will be able to operateat a number of

√s values between these bounds.

In this paper, we only consider neutral current (NC)events, a first study on the capabilities of an EIC forcharged current (CC) events can be found in [35]. TheNC events fall into two categories: resolved and direct.Resolved processes (see Fig. 1(a)) are those in whichthe virtual photon interacts via the hadronic componentof its wavefunction, contributing a quark or gluon to ahard-scattering with a parton from the nucleon. Theresolved category includes the qq → qq, qq → qq, qq →gg, qg → qg, gg → qq, gg → gg subprocesses and plays asignificant role in the production of high-pT particles atlow Q2 . In direct processes, the photon interacts as apoint-like particle with the partons of the nucleon. Thesubprocess which comprise the direct category includeleading order DIS 1(b) (L.O. subprocess), photon-gluonfusion (PGF) 1(c), and QCD Compton (QCDC) 1(d).Direct processes contribute more at high Q2 values whileresolved processes dominate at Q2 < 1 GeV2. The higherorder resolved, QCDC, and PGF subprocesses (hereafterreferred to as the H.O. subprocesses) involve two high-pTpartons separated in azimuth and thus often give rise todijet events.

III. JET FINDING AND JET PROPERTIES

The jets used in this study were formed using stablefinal state particles generated by the PYTHIA MonteCarlo described in section II. Here, stable refers to par-ticles which would not normally decay in the volume ofa detector, such as charged pions, kaons, protons, andneutrons. Neutral pions were allowed to decay and theresulting (predominately) photons were passed to the jetfinder. To match expected detector acceptances, onlyparticles having transverse momenta with respect to thebeam greater than 250 MeV/c and pseudorapidity be-tween ±4.0 were considered candidates for inclusion injets; the scattered lepton was not allowed to be part ofthe jet. The effects of using a higher minimum particlepT cut were also explored (see Sec. III E).

A. Reference Frames

At hadron colliders, analyses are carried out almostexclusively in the reference frame of the detector, thelaboratory frame, as the kinematics of the interactingpartons are not generally known. For DIS, however, thescattering kinematics are known event-by-event whichmakes it possible to boost to other frames. A particu-larly useful frame for jet analyses is the Breit or ‘brickwall’ frame [37]. The Breit frame is oriented such thatfor the lowest order DIS process γ∗q → q′, the virtualphoton and interacting quark collide head-on along thez-axis and is boosted such that the only non-zero compo-nent of the virtual photon four-momentum is pz = −Q.A consequence of this boost is that the z-momentum ofthe incoming quark is Q/2 while the scattered quarkhas z-momentum −Q/2 (hence the name ‘brick wall’frame) and the proton remnant has a z-momentum of(1−x)Q/(2x). This leads to a natural separation betweenjets associated with the struck quark and those associatedwith the proton remnant. Working in the Breit framehas the effect of suppressing contributions from the L.O.subprocess, as the scattered quark has zero transversemomentum by construction (although, as will be seen inIII C, jets can arise from the L.O. process at large Q2

due to final state radiation). A dedicated study of L.O.jets in the laboratory frame can be found in [24]. TheBreit frame is also advantageous for the study of jetsarising from the H.O. subprocesses as their transversemomenta will be taken with respect to the photon-quarkaxis, which is the relevant quantity in these interactions.This is especially important at large values of Q2 wherethe angle between the photon-quark and beam axes issignificant. Boosts to other frames such as the photon-quark center-of-mass or proton rest frames can also beperformed, but are not covered here.

For this analysis, all particle four-momentum wereboosted into the Breit frame and then passed to the jetfinder for clustering. This avoids any changes in the par-ticle content which may arise due to variations in clus-

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Fusion (PGF) and (d) QCD Compton scattering (QCDC). Figure is from [36].

tering between the two frames. When it is necessary topresent a jet quantity with respect to the detector, theBreit frame thrust axis of the jet was simply boosted backto the laboratory frame.

B. Jet Definitions

While the idea of a jet as a collimated spray of par-ticles is conceptually easy to grasp and jets are ofteneasy to identify ‘by eye’ in event displays, a well definedmethod of mapping a set of particles into a set of jets isrequired for jets to be useful in experimental and theo-retical analyses. The collection of rules that determinehow particles are grouped into jets is known as a jet al-gorithm, while the prescription for merging the momentaof individual particles to form the overall jet momentumis known as a recombination scheme. The combination ofjet algorithm, recombination scheme, and any additionalparameters controlling the behavior of the jet algorithmis known as a jet definition and, as the name implies,fully defines a jet for purposes of an analysis [38].

There are a number of jet algorithms on the market[38, 39] and the determination of which algorithm to uti-lize will depend on the requirements of the specific analy-sis being performed. As this manuscript is meant to givea general overview of jets at an EIC, the choice of a spe-cific algorithm is not critical as long as there are not sig-nificant differences between algorithms for basic jet prop-erties. To confirm this is the case, a number of jet quan-tities including yields, particle content, energy profile,transverse momenta, and rapidities were compared usingthe anti-kT [40], kT [41], and SISCone [42] algorithms.

These were chosen as they represent the two broad cate-gories of algorithms, sequential recombination and cone,and have seen use at both HERA and hadron colliderssuch as the LHC. Differences in the studied quantitiesranged from negligible to a couple of percent, indicatingthat the choice of algorithm will have little impact on theconclusions drawn here. Because it produces jets withregular boundaries that are slightly more collimated, theanti-kT algorithm will be used for all subsequent studiesin this paper. It should be noted that several recombi-nation schemes also exist, but only the E Scheme [43],in which particles are combined simply by adding theirfour-momenta, will be considered here.

The other major component of the jet definition whichneeds to be determined is the resolution parameter, R,which sets the effective size of the jet. As with jet algo-rithm, the optimal choice for R will depend on require-ments driven by a particular analysis. Often, the chosenR is a compromise between large values that will capturemore energy from the hadronizing partons and small val-ues, which limit the contamination from underlying event(see Sec. IV for a discussion of expected underlying eventactivity at an EIC). The resolution parameter can be ex-pected to affect the jet yield and particle content of jets,and because it influences how much of the energy fromthe hadronizing parton ends up in the jet, R should alsoaffect how well the jet represents the underlying partonicbehavior.

The jet multiplicity and number of particles within ajet are presented in the left and right panels, respectively,of Fig. 2 for R values of 1.0, 0.7, and 0.4. The jetswere required to have transverse momenta greater than5 GeV/c in the Breit frame and were taken from events

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FIG. 2. [color online] Comparison of jet multiplicity (leftpanel) and particle multiplicity within the jet (right panel)for the anti-kT algorithms and three resolution parameters R= 1.0, 0.7, and 0.4. The Q2 range is between 10−5 GeV2 and500 GeV2 and the Resolved, QCDC, PGF, and leading orderDIS subprocesses have been combined.

with Q2 between 10−5 and 500 GeV2. The L.O. and H.O.subprocesses have been combined. It is seen that thetotal number of found jets increases with increasing R,which is due to the fact that larger R values admit moreparticles (as seen in the right panel), meaning more jetswill pass the 5 GeV/c transverse momentum threshold.

One benefit of jet observables is that they can serveas proxies for hard-scattered partons because the jetwill capture many of the particles emitted as the partonhadronizes. Since the size of R will affect the amount ofpartonic radiation included in the jet, it is necessary toassesses how changes in R impact how well jets reproducethe underlying partonic kinematics. This was done bycomparing the reconstructed dijet invariant mass to theparton level invariant mass. Dijets were reconstructedfrom H.O. events by selecting the two jets with the high-est transverse momentum in the Breit frame, requiringthat they be greater than 120 degrees apart in azimuth,and necessitating that one jet have pT greater than5 GeV/c while the other has pT greater than 4 GeV/c.The comparison between reconstructed dijet and par-tonic invariant mass can be seen in Fig. 3 forR = 1.0, 0.7,and 0.4. Only Q2 values between 10 GeV2 and 100 GeV2

are shown as the conclusions are the same for all Q2 val-ues. The best agreement is seen for the largest R valueand degrades as R decreases. The events at low partonicand large reconstructed dijet invariant mass arise whenthe one or both of the jets comprising the dijet do notmatch the true outgoing partons. With the exception ofthese ‘false’ dijets, the fact that the reconstructed mass isconsistently smaller than the partonic mass for R < 1.0indicates that the smaller cones are not capturing the fullenergy associated with the hard-scattered partons.

Another way to assess how well jets represent the un-derlying partons is to measure the degree to which the jetthrust axes correspond to the directions of the partonswhich give rise to them. This is quantified using the dis-tance measure ∆R, which is the quadrature sum of thedifference between jet and parton rapidity and azimuthal

angle (∆R ≡√

∆η2 + ∆φ2). For each jet comprising the

dijet, ∆R between that jet and the two hard scatteredpartons is found and the minimum is taken. Figure 4presents this minimum ∆R for both jets from the H.O.subprocesses combined for two Q2 bins. The ∆R distri-butions are larger at low Q2 and peak closer to zero as Q2

increases. For Q2 between 1 and 10 GeV2, over 85% ofthe jets are within a ∆R of 0.5 to their matching partonand this jumps to 95% for Q2 between 100 and 500 GeV2.Again, the tails at large ∆R arise when one or both jets inthe dijet do not correspond to one of the true hard scat-tered partons. Surprisingly, at low Q2 , the jet-to-partonmatching is seen to improve as R decreases. This maybe due to the effect of underlying event activity whichwill be more prevalent in larger cones and could pull thejet thrust axis slightly away from the parton direction,or it may be a selection bias effect wherein jets foundwith a smaller R tend to fragment in a more collimatedway. Regardless, Fig. 3 shows that any benefit from theslightly better matching between the jet and parton di-rections at small R is overwhelmed by the parton energymissed by the smaller jet cone.

As it results in the most jets found with the largestparticle content and best reproduces the partonic kine-matics, R will be set to 1.0 for all subsequent studies pre-sented in this paper. Together with the choice of anti-kTfor the jet algorithm and E Scheme recombination, thejet definition is fully quantified.

C. Inclusive Jet Kinematics

While jets arising from e+p collisions were studiedextensively at HERA, the lower center-of-mass energies(maximum

√s of 141 GeV for EIC compared to 320 GeV

for HERA) envisioned for an EIC make a detailed inves-tigation of jet properties and kinematics warranted. Jetswere found using the jet definition outlined in sectionIII B as implemented in FastJet-3.3.1 [44] and were re-quired to have transverse momenta greater than 5 GeV/cin the Breit frame.

Breit frame inclusive jet pT spectra are shown in Fig.5 for four Q2 ranges between 10−5 and 500 GeV2 andfor√s values of 45 and 141 GeV. The spectra have been

scaled to the number of counts expected for 1 fb−1 ofintegrated luminosity. As their behaviors are similar, theH.O. subprocesses have been combined and are comparedwith the L.O. spectra. It is seen that high-pT jet produc-tion is dominated by the H.O. subprocesses while theL.O. spectra are much softer. This is expected becauseat leading order in the Breit frame, the scattered par-ton moves along the z-axis and transverse momentum isgenerated primarily via final state radiation. The preva-lence of this radiation decreases with decreasing Q2, soleading order DIS jets are basically absent for Q2 below10 GeV2. Note that the L.O. jet yield can increase upto two orders of magnitude for Q2 > 1 GeV2 when jet-finding is performed in the lab frame, meaning this framewill be preferred for measurements focusing on L.O. jet

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FIG. 3. [color online] Dijet invariant mass compared to the partonic invariant mass√s for R = 1.0 (left), 0.7 (middle), and

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FIG. 4. [color online] Distance in rapidity-azimuth space be-tween the jets comprising a dijet and the corresponding hard-scattered partons with Q2 between 1 GeV2 and 10 GeV2 (up-per panel) and 100 GeV2 and 500 GeV2 (lower panel) for jetradii of 1.0, 0.7, and 0.4.

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FIG. 5. [color online] Breit frame inclusive jet pT spectra for√s = 141 and 45 GeV in Q2 bins of 10−5 − 1.0 GeV2 (upper

left), 1-10 GeV2 (upper right), 10-100 GeV2 (bottom left),and 100-500 GeV2 (bottom right). The Resolved, QCDC, andPGF subprocesses have been combined and are compared tothe leading order DIS spectra and the histograms have beenscaled to the counts expected for an integrated luminosity of1 fb−1.

production [24]. Figure 5 also demonstrates the criti-cal importance of higher center-of-mass energies for jetstudies as the yields can be orders of magnitude largerfor√s = 141 GeV compared to

√s = 45 GeV. This is

most pronounced at large pT where yield differences areso great, no practical increase in luminosity could com-pensate.

In addition to their typical transverse momenta, it isimportant to understand where jets are located in the

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detector and how that correlates to the xB and Q2 of theevent. Figure 6 shows the inclusive jet pseudorapiditydistributions, in the laboratory frame, as a function of xBfor four Q2 bins ranging from 10−5 GeV2 to 500 GeV2

and√s values of 45 GeV and 141 GeV, for the H.O.

subprocesses. Figure 7 shows the same for the L.O. sub-process in the Q2 ranges 10-100 GeV2 and 100-500 GeV2.As expected from basic DIS kinematics, smaller xB val-ues are probed by larger center-of-mass energies and visaversa. Comparing Figs. 6 and 7, it is seen that the lead-ing order DIS jet pseudorapidity is more strongly corre-lated with xB then that of the Resolved, QCDC, or PGFjets. Because there is only one outgoing parton in DIS,jet pseudorapidity should be strongly determined by theevent xB and Q2 with the observed width of the distri-butions in Fig. 7 due to the finite Q2 ranges and, moreimportantly, final state radiation altering the trajectoryof the outgoing quark. The presence of a second hardparton in H.O. events breaks the strong relationship be-tween xB, Q2, and η and allows the resultant jets to fillthe kinematically allowed phase space. The importanceof hadron beam energy to jet position can also be seenin Figs. 6 and 7 by contrasting the distributions at thetwo√s values for given xB and Q2. Larger hadron beam

momenta impart more of a boost to final state particles,so jets at

√s = 141 GeV will be pushed to higher pseu-

dorapidities compared to jets from collisions at lower√s.

Thus, good forward tracking and calorimetry capabilitieswill be needed to utilize jets at large

√s.

D. Dijet Kinematics

So far, only inclusive jet quantities have been consid-ered, yet as stated above, the H.O. subprocesses natu-rally give rise to correlated two jet final states (dijets).By measuring the properties of both jets in coincidence,dijets can provide information on the leading order kine-matics of the hard scattering event, such as the momen-tum fraction contributed by the virtual photon. Severalstudies have already explored the utility of dijet measure-ments at the EIC [16, 20, 21] and a further study willbe presented in Sec. VI. As before, dijets were selectedby identifying the two jets with the largest transversemomenta in the Breit frame and requiring them to begreater than 120 degrees apart in azimuth. It was fur-ther required that one jet have pT greater than 5 GeV/cwhile the other have pT greater than 4 GeV/c.

The scale relevant for a dijet is its invariant mass,which is simply

√(P1 + P2)2 where P1 and P2 are the

four-momenta of the two jets. The dijet invariant massspectra are shown in Fig. 8 for four Q2 ranges and center-of-mass energies of 45 and 141 GeV. The H.O. subpro-cesses are combined and compared to the L.O. spectra.While the leading order DIS subprocess results in onlyone outgoing quark, at high Q2 , the proton remnantcan receive enough transverse momentum to produce asecond jet which will satisfy the dijet conditions. These

L.O. dijets can be effectively separated from the H.O.dijets via a cut on the ratio of dijet mass over Q (seeFig. 22). As was the case with inclusive jet pT, the di-jet cross section is significantly larger for

√s of 141 GeV

than for 45 GeV and the spectra extend to much highermass values.

To characterize the location of a dijet in the detec-tor, the pseudorapidities of both jets need to be recordedsimultaneously as in Fig. 9. As before, the H.O. subpro-cesses have been combined and now dijets arising fromthe L.O. subprocess are not shown. Only

√s = 141 GeV

events are shown as the√s = 45 GeV distributions

are just shifted to lower pseudorapidity for a given xB- Q2 bin due to the smaller boost from the less ener-getic hadron beam. As was the case for inclusive jets, jetpseudorapidities increase as xB is increased at a fixed Q2

, and for a fixed xB bin, jet pseudorapidities decrease asQ2 is increased.

While the absolute pseudorapidities of the two jetscomprising a dijet depend on the event xB and Q2 asshown in Fig. 9, the relative pseudorapidity is connectedto the dijet invariant mass. Expanding the four-vectorexpression given above, the dijet invariant mass can beapproximated (ignoring the individual jet masses) as:

M ≈√

2pT1pT2 (cosh(∆η)− cos(∆φ)), (1)

where pT1 and pT2 are the transverse momenta of thetwo jets and ∆η and ∆φ are the pseudorapidity and az-imuthal angle differences, respectively, between the twojets. In this form, it is apparent that dijets can acquirea large mass if their constituent jets have large trans-verse momenta, and/or if the pseudorapidity differencebetween the two jets is large. The interplay betweenjet pT and ∆η is made explicit in Fig. 10 which showsthe difference in pseudorapidity and average jet trans-verse momenta of the constituent jets for five invari-ant mass bins and Q2 ranges of 10−5 − 1.0 GeV2 and100.0 − 500.0 GeV2. At low Q2, the average jet pT re-mains small even for the largest mass bins, meaning thatlarger invariant masses are driven by greater pseudora-pidity separations. As Q2 increases, the average jet pTincreases and ∆η does not need to be as large to pro-duce a high invariant mass dijet. Thus, even for highmass dijets, it will be important to have good jet energyresolution to low pT and large detector acceptance.

E. Minimum Particle Transverse Momentum

The jets used in the above discussion were created fromparticles which had a transverse momentum of at least250 MeV/c with respect to the beam. This value waschosen as it is slightly higher than the typical cutoff usedin p+p jet finding at STAR (see for example [45]), whichwhen running at

√s = 200 GeV should have relatively

similar particle pT spectra as can be expected at an EIC.

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(middle-left column), 10-100 GeV2 (middle-right column), and 100-500 GeV2 (right column) for center-of-mass energies of45 GeV (upper row) and 141 GeV (bottom row). The resolved, QCDC, and PGF subprocesses are shown. Note that the topand bottom rows are separately scaled to the counts expected for 1 fb−1 of integrated luminosity.

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FIG. 7. [color online] Inclusive jet laboratory pseudorapid-ity vs xB for Q2 bins of 10-100 GeV2 (left column) and100-500 GeV2 (right column) and center-of-mass energies of45 GeV (upper row) and 141 GeV (bottom row). Only theleading order DIS subprocess is shown. Note that the top andbottom rows are separately scaled to the counts expected for1 fb−1 of integrated luminosity.

The particle pT cutoff is largely driven by detector con-siderations, with the magnetic field strength often the

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2100 GeV

= 141 GeVsH.O.:

= 141 GeVsL.O.:

= 45 GeVsH.O.:

= 45 GeVsL.O.:

FIG. 8. [color online] Breit frame dijet invariant mass spectrafor√s = 141 and 45 GeV inQ2 bins of 10−5−1.0 GeV2 (upper

left), 1-10 GeV2 (upper right), 10-100 GeV2 (bottom left),and 100-500 GeV2 (bottom right). The Resolved, QCDC, andPGF subprocesses have been combined and are compared tothe leading order DIS spectra and the histograms have beenscaled to the counts expected for an integrated luminosity of1 fb−1.

dominant factor. While detector designs for the EIC

8

4− 2− 0 2 4

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2100 GeV

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)T

(High pLab

ηJet

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1−10

1

10

210

FIG. 9. [color online] Laboratory frame jet η− η correlations for dijets in select xB bins for Q2 ranges of 1-10 GeV2 (top row),10-100 GeV2 (middle row), and 100-500 GeV2 (bottom row). Only the 141 GeV center-of-mass energy is shown. The resolved,QCDC, and PGF subprocesses have been combined and leading order DIS events are not included. Note that all panels sharethe same scale, which has been set to represent the number of counts expected for an integrated luminosity of 1 fb−1.

are still in active development, many include a relativelylarge solenoidal magnetic field of 2 to 3 Tesla in order toprovide good pT resolution over the full pseudorapidityrange. This will limit the acceptance for low pT chargedparticles as they will bend so severely in the magneticfield that they will not reach the calorimeters and will bedisplaced significantly from any neutral particles whicharise from the hadronizing parton.

To study the effect that the loss of low pT particleswill have on jet quantities, the jet finding was rerun withthe low pT particle cutoff doubled to 500 MeV/c. Themost obvious effects of raising the cutoff are a reductionin jet/dijet yields and the average number of particles ina jet. The jet and dijet yields are reduced by roughly37% for 10−5 < Q2 < 1 GeV2 and 20% for 10−5 < Q2 <500 GeV2. The effect of the minimum pT cut on jetparticle content can be seen in Fig. 11 for all particles aswell as charged hadrons only.

Removing low pT particles may also affect how well jets

reproduce the kinematics of the underlying partons dueto the loss of energy contributed by these particles. Theimpact of this loss was studied in the same way as the Rdependence in Sec. III B, by comparing the reconstructeddijet mass to the di-parton invariant mass and by mea-suring ∆R between the jet and parton directions. Thehigher minimum particle pT cut slightly reduces the re-constructed dijet mass versus the true di-parton invariantmass, much like what was seen when reducing R in Fig.3, although the magnitude of the effect is not as great.There was no visible change in the ∆R distributions. Asdetector designs become more advanced, further studieswill need to be made to ensure that there is sufficientacceptance for low pT particles.

9

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1η|

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Mass = 20­30 GeV

Mass = 30­40 GeV

Mass = 40­50 GeV

Mass > 50 GeV

FIG. 10. [color online] Average jet pT (left column) and ra-pidity difference (right column) for jets comprising a dijetseparated in bins of dijet invariant mass for Q2 ranges of 10-100 GeV2 (top row) and 100-500 GeV2 (bottom row). Onlythe 141 GeV center-of-mass energy is shown. The resolved,QCDC, and PGF subprocesses have been combined and lead-ing order DIS events are not included. Histograms have beenscaled to the counts expected for an integrated luminosity of1 fb−1.

5 10 15 20 25 30 35 40 [GeV/c]

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art

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> 250 MeV/cT

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> 500 MeV/cT

All Particles, p

> 500 MeV/cT

Charged Particles, p

RMS: All Particles (>250 MeV/c)

RMS: Charged Particles (>500 MeV/c)

FIG. 11. [color online] The average number of particles in ajet as a function of the transverse momentum of the jet forall stable particles and only charged particles for minimumparticle pTs of 250 and 500 MeV/c. Also shown are the RMSvariations for all particles with pT > 250 MeV/c and chargedparticles with pT > 500 MeV/c.

IV. UNDERLYING EVENT PROPERTIES

The underlying event activity, which contributes back-ground energy to jet signals is quantified in this sectionfor jets produced from the H.O. subprocesses in the Breitframe. ‘Underlying event’ (UE) refers to those particles,which do not arise from the outgoing hard-scattered par-tons and can contain contributions from initial and finalstate radiation (ISR, FSR), beam remnants, and multipleparton interactions (MPI). As the QCDC, and PGF sub-processes proceed via a direct γ + parton interaction,there is no contribution from MPI and the only beamremnant arises from the hadron side. On the other hand,the resolved subprocess is defined by parton + partonscattering where one parton is supplied by the photon(see Fig 1), and because the photon behaves hadronically,it will contribute to the beam remnant and allow for MPI.Because MPI effects are expected to be small for the EICkinematics and because they are difficult to model accu-rately, the MPI contribution has been disabled in theMonte Carlo.

Toward

Away

Transverse Transverse

Direction

Trigger Jet

φ∆

φ

0

π2

η

Trigger

Jet

Away

Transverse

Toward

Transverse

Away

FIG. 12. Illustration of the ‘Toward’, ‘Away’, and ‘Trans-verse’ regions as defined relative to the highest pT jet in theevent. The angle ∆φ ≡ φ − φref jet is the azimuthal anglebetween a charged particle and the highest pT jet, from −πto π. The Toward region is defined as |∆φ| < 60◦, while theAway region is |∆φ| > 120◦. The Transverse region is definedas 60◦ < |∆φ| < 120◦. The plot is from [46].

We utilize two methods to analyze UE effects in e+pcollisions, the ‘region method’ [46] and the ‘off-axis conemethod’ [47]. In the region method, the azimuthal angleof the highest pT jet in each dijet event is selected as thereference angle and particles are grouped into one of threeregions based on their azimuthal angle relative to this ref-erence, ∆φ ≡ (φ − φref jet). The particle candidate poolis identical to that used in the jet-finding. The ‘Toward’region is defined as |∆φ| < 60◦ and contains the referencejet, while the ‘Away’ region has |∆φ| > 120◦ and gener-ally contains the lower pT , or associated, jet of the dijet.The ‘Transverse’ region is defined as 60◦ < |∆φ| < 120◦

and the activity here is dominated by the UE. Figure 12illustrates the definition of the three regions.

Three observables are used to characterize UE activ-ity: the average charged particle multiplicity (〈Nch〉), the

10

average charged particle scalar pT sum (〈sum pT〉), andaverage charged particle pT (〈pT〉). Figure 13 presents〈Nch〉 (top) and 〈sum pT〉 (bottom) as a function of|∆φ| for particles with pT > 250 MeV/c and −4 <η < 4. Reference jets with pT > 5 GeV/c (associ-ated jet pT > 4 GeV/c) and pT > 8 GeV/c (associ-ated jet pT > 7 GeV/c) were compared along with jetsfrom two Q2 regions: 1 GeV2 < Q2 < 10 GeV2 and10 GeV2 < Q2 < 100 GeV2. There is a strong depen-dence on jet pT in the Toward and Away regions for both〈Nch〉 and 〈sum pT〉, which is expected as these regionsare dominated by the jets. In the Transverse region, nodependence on jet pT is seen for the average number ofcharged particles while a mild difference is seen in the pTsum for the higher Q2 range. Interestingly, it is the lowerpT jets which show a higher pT sum in the transverse re-gion, seemingly indicating that jets with higher pT leaveless energy available for underlying event activity. NoQ2 dependence is seen for 〈Nch〉 while larger Q2s lead togreater sum pT s in all regions.

The dependence of 〈Nch〉 and 〈pT〉 on the trigger jetpT in the Toward, Away, and Transverse regions is mademore explicit in Fig. 14 for Q2 between 1 GeV2 and10 GeV2. It is seen that both the particle density (top)and average pT (bottom) in the Toward and Away regiondepend strongly on the trigger jet pT as is to be expected.Conversely, both quantities show a weak anti-correlationwith trigger jet pT in the transverse region, in agreementwith Fig. 13. The effect of initial and final state radi-ation (ISR/FSR) on the observables can be seen in theright hand column of Fig. 14 where the radiation effectshave been disabled. The presence of ISR/FSR leads toan increase in 〈Nch〉 for all three regions while surpris-ingly, the average charged particle pT is seen to increasesomewhat without ISR/FSR effects.

Unlike the identical species configurations that are of-ten run at colliders, the collisions at an EIC will be asym-metric in both particle type and beam energy. This willlead to an asymmetric η dependence in particle produc-tion and UE activity as seen in Fig. 15. Here, the av-erage charged particle multiplicity densities and averagecharged particle pT sum densities in the Transverse re-gion are shown as a function of reference jet pT for parti-cles in Backward (−4 < η < −1), Mid (−1 < η < 1), andForward (1 < η < 4) pseudorapidity ranges (as definedin the laboratory frame) from the region method (filledsymbols). The reference jet was required to be withinthe same pseudorapidity range as the particles with theadded restriction that the jet η must be 0.4 units awayfrom a range boundary in order to facilitate comparisonsto the off-axis cone UE characterization method. It isseen that the UE charged particle density is higher inthe Forward (hadron-going) direction which is expectedas any beam remnant contribution will generally followthe struck hadron. Also, particle density in the forwardregion will be higher due to the boost from the moreenergetic hadron beam.

The second technique used to investigate UE effects is

the off-axis cone method [47], developed by the ALICEcollaboration. The off-axis cone method studies the UEon a jet-by-jet level, as opposed to the region methodwhich is designed to study the UE on the event level.For every reconstructed jet, two off-axis cones (cone(-)and cone(+)) are defined, each of which is centered atthe same η as the jet but ±π/2 away in φ from the jetφ, as shown in Fig. 16, and the particles which fall insidethese cones are used to characterize the underlying event.The cone radius was chosen to be 0.4 so as not to overlapwith the primary jet. The multiplicity density is definedas the average number of charged particles inside eachcone, 〈Nch〉, divided by the cone area while the 〈pT sum〉density is defined as the average off-axis cone pT dividedby the cone area.

The UE results from the off-axis method (open sym-bols) are compared to the region method (closed sym-bols) in Fig. 15. As with the reference jets from theregion method, the jets from the off-axis method wererequired to be more than 0.4 away from a range bound-ary in η so that the full off-axis cone would fit in theindicated pseudorapidity bin. With the jet and particlepseudorapidities defined in this way, a faithful compari-son between the region and off-axis methods can be madeand the agreement is seen to be very good. The pseu-dorapidity dependence of the UE activity seen in Fig. 15means the off-axis cone method will be important whencorrecting jet quantities for underlying event contamina-tion as it will be necessary to measure that componentat the pseudorapidity of the jet.

The results described above characterize the expectedunderlying event at an EIC as generated in our MonteCarlo. To get a better feeling for the size of these effectsand provide a sanity check on the simulation, it is instruc-tive to compare the simulated e+p results with p+p dataat a similar center-of-mass energy. The STAR experi-ment [48] at the Relativistic Heavy Ion Collider (RHIC)has performed a similar analysis of UE activity as pre-sented here on p+p data taken at

√s = 200 GeV, which

provides an opportunity for such a comparison. TheSTAR analysis [49] measured both 〈Nch〉 and 〈pT〉 for allcharged particles in the mid-rapidity region (−1 < η < 1)with pT > 0.2 GeV/c. The STAR average charged parti-cle density varies from 0.8 to 0.5 for reference jets with pTof 5 GeV/c to 40 GeV/c. This is a factor of roughly twoto four greater than what is observed at mid or forwardrapidity in Fig. 15. This is not surprising as the STARresult involves the collision of two protons. However, theaverage charged particle pT measured by STAR is rela-tively flat with a value of 0.6 GeV/c, which is at least afactor of two lower than the result presented in Fig. 14.The larger 〈pT〉 in the Transverse region at the EIC is dueto the the boost into the Breit frame. While the parti-cles which participate in the hard scattering processesinitially move along the photon-parton axis, the underly-ing event particles arise largely from the proton and arethus more aligned with the beam axis. When measuredwith respect to the photon-parton axis, as is done in the

11

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FIG. 13. [color online] Average number of charged particles (top) and average charged particle scalar pT sum (bottom) as a

function of the azimuthal angle, ∆φ, between the particle and the reference jet for pref jetT > 5 GeV or 8 GeV, and Q2 < 10 GeV2

or 10 < Q2 < 100 GeV2. Each point corresponds to the 〈Nch〉 in a 3.6◦ bin.

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FIG. 14. [color online] Charged particle density (top row) and mean pT (bottom row) as a function of the transverse momentumof the reference jet for the Toward, Away, and Transverse regions. The left column is the standard simulation, while the rightcolumn has had initial and final state radiation disabled. Reference jets were required to have pT > 5 GeV/c and laboratorypseudorapidity between ±4 while Q2 was selected to be between 1 GeV2 and 10 GeV2.

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FIG. 15. [color online] Average charged particle density (top) and charged particle pT sum densities (bottom) in the Transverseregion as a function of trigger jet pT for three pseudorapidity ranges and Q2 between 1 GeV2 and 10 GeV2. Results from theregions method (closed circles) and off-axis cone method (open squares) are compared. The displayed pseudorapidity rangesapply to the particles used in the analysis while the reference jets were required to be more than 0.4 units of pseudorapidityfrom a boundary.

13

FIG. 16. [color online] Illustration of two off-axis cones rela-tive to a jet.

Breit frame, these underlying event particles acquire, onaverage, larger transverse momenta. When analyzed inthe laboratory frame, 〈pT〉 in the Transverse region isroughly 0.6 GeV/c, in agreement with the STAR result.

V. DETECTOR EFFECTS

The jets used for the results presented in Secs. IIIand IV were reconstructed at ‘particle level’, taking asinput the exact four-momenta of all generated final stateparticles. These jets do not include distortions whichwill arise from the finite energy and momentum reso-lutions and inefficiencies of any real detector. Becausethe entirety of the EIC physics program requires highresolution calorimetry and tracking over a wide accep-tance range, the induced distortions are expected to besmall. Nevertheless, it is important to investigate howjets will be affected by a realistic detector environment.In order to quantify how these distortions will affect jetreconstruction, the energy and momenta of input parti-cles were smeared based on a model EIC detector beforebeing clustered into jets. These smeared jets were thencompared to the corresponding particle level jets to studydetector effects.

A. Smearing Generator and Detector Model

Generally, detector effects are investigated by prop-agating simulated events through a detailed detectormodel which reproduces the relevant energy and mo-mentum resolutions, efficiencies, material budgets, andreadout responses of the actual device. As such detailedmodels for prospective EIC detectors are only startingto be developed, and key detector technology choices arestill in flux, the effects of finite resolution and acceptanceon jet-finding were explored using a smearing generator,

which alters a particle’s energy or momentum based ona specific resolution function. While not a substitute fora full detector simulation, this smearing method has thebenefit of being much faster computationally, making iteasy to investigate different sub-detector configurationsand resolutions.

The smearing generator used allows a user to define‘devices’ which encode the behavior of individual or col-lections of detector subsystems. A single device willsmear the energy, momentum, or direction of all par-ticles which fall into its acceptance. Here, acceptancenot only refers to the spatial extent of the device, butalso to particle properties such as charge and how theparticle interacts with the detector material (hadroni-cally or electromagnetically). Three particle charge andinteraction types are used in the smearing performedhere: charged hadronic, neutral hadronic, and electro-magnetic. Charged hadronic particles (assumed to bedetected using a tracker) have their momenta and tra-jectories smeared while neutral hadronic and electromag-netic particles (assumed to be detected with calorime-ters) have their energies smeared. Because a device willonly smear either the energy or momentum componentof a particle, the energy-momentum-mass relationship ofthe smeared 4-vector will be broken. To address this, af-ter the smearing was performed, the charged hadron en-ergies were altered to match their momenta assuming theparticles had a pion mass. Similarly, the momenta of neu-tral hadrons and particles interacting electromagneticallywere set equal to the smeared energy, which is equivalentto assuming the particle was massless. This simplisticcompensation will be inadequate for particles with signif-icant mass, such as protons and neutrons, but is sufficientfor the purpose of this study. When more complete de-tector simulations are developed, efforts should be madeto determine the utility of the calorimeter systems as wellas particle identification for more accurate particle four-momentum reconstruction.

For this study, the smearing generator devices weredefined such that they would reproduce the projectedbehavior of BeAST, Brookhaven’s ‘green field’ detectorproposal. BeAST is built around a 3 Tesla solenoidalmagnet and will include high precision tracking detec-tors spanning a pseudorapidity range of |η| < 3.5, elec-tromagnetic calorimetry covering the range |η| < 4.0, andhadron calorimetry in the forward and backward regions1 < |η| < 4.0. BeAST will also have good vertex de-tection and particle identification capabilities as well asinstrumentation to detect particles scattered at small an-gles, both in the hadron and lepton beam direction, suchas Roman pots and a system to tag low Q2 electrons.However, these systems do not directly affect jet recon-struction and were therefore not included in this simu-lation. The calorimeter resolutions assumed for differentdetector regions can be found in Tab. I while the trackingresolution for different particle momenta as a function ofpseudorapidity can be seen in Fig. 17. Several modifica-tions to the baseline BeAST configuration were also con-

14

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Track Momentum = 10 GeV/c

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Track Momentum = 50 GeV/c

FIG. 17. [color online] Track momentum resolution assumedfor the smearing generator as a function of track pseudora-pidity. The points represent extractions of resolution at spe-cific momenta and pseudorapidity from simulation of a modelBeAST tracking detector and the curves are instances of thefunction used to fit the points that was passed to the smearinggenerator.

sidered, including the introduction of a track finding in-efficiency factor of 5% and the addition of a mid-rapidityhadron calorimeter assuming a high and low energy res-olution.

Component Pseudorapidity Range Resolution

Back EMCal −4.0 < η < −2 1.5%√E⊕ 1%

Mid-Back EMCal −2 < η < −1 7%√E⊕ 1%

Mid EMCal −1 < η < 1 10%√E⊕ 1%

Fwd EMCal 1 < η < 4.0 10%√E⊕ 1%

Fwd/Back HCal 1 < |η| < 4.0 50%√E⊕ 10.0%

Lo Res Mid Hcal −1 < η < 1 75%√E⊕ 15%

Hi Res Mid Hcal −1 < η < 1 35%√E⊕ 2%

TABLE I. Assumed energy resolutions and psuedorapidityranges for the electromagnetic and hadron calorimeters in-cluded in the detector smearing model.

B. Smearing Results

Using the smearing procedure and resolution parame-ters presented above, individual particle 4-momenta werealtered and then passed to the jet-finder to be clusteredinto jets using the same procedure as for unaltered par-ticles. Thus, for each event, there will be a set of un-altered particle level jets and a set of smeared jets. In

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FIG. 18. [color online] Correlation between particle level andsmeared jet pT for the BeAST detector setup (upper left),BeAST assuming a 5% track finding inefficiency (upper right),and BeAST assuming a mid-rapidity hadron calorimeter withresolution 15% ⊕ 75%√

E(lower left) and resolution 2% ⊕ 35%√

E

(lower right).

order to evaluate how the smearing procedure has mod-ified the properties of a given particle level jet, an as-sociation must be made between that particle level jetand a particular smeared jet. This is done by findingthe smeared jet which minimizes the quantity ∆R =√

(yParticle − ySmeared)2 + (φParticle − φSmeared)2 for eachparticle level jet, with y being the rapidity and φ the az-imuthal angle of the jet. Particle level and smeared jetswere required to have ∆R < 1.0 in order to be consideredassociated.

The relationship between the transverse momenta ofassociated particle level and smeared jets can be seenin Fig. 18 for the baseline BeAST design, as well asthe 5% track finding inefficiency and mid-rapidity hadroncalorimeter scenarios. As the baseline design does not in-clude a hadron calorimeter at mid-rapidity, neutrons andK0

L’s in this region are not detected, meaning smearedjets will tend to have lower transverse momentum thantheir corresponding particle level jets. The populationand extent of this tail depends on the average num-ber of neutral hadrons in the event sample and theamount of transverse momentum they carry. Removing5% of charged particles increases somewhat the numberof events which populate the off-diagonal tail. The inclu-sion of a hadron calorimeter at mid-rapidity captures theremaining neutral energy making the correlation betweenparticle level and smeared jet pT more symmetric aroundthe diagonal. The width of the distribution is then de-termined by the resolution of the hadron calorimeter.

A more detailed comparison of the relationships shown

15

in Fig. 18 can be obtained by taking projections ontothe particle level axis for narrow slices of smeared jetpT (or vice versa). Figure 19 presents three such pro-jections with smeared jet transverse momenta of 7, 10,and 13 GeV/c for the baseline BeAST and BeAST withtwo mid-rapidity hadron calorimeter configurations andessentially shows how different particle level pT valuescontribute to a given smeared jet pT . It is evident thatthe high resolution hadron calorimeter (green) substan-tially improves the jet resolution, however, it is less clearthat the low resolution calorimeter (red) provides muchadvantage over the baseline design (blue). Smeared jetsfound with the low resolution calorimeter option have lesscontribution from particle level jets with larger pT thanin the baseline design, however, these smeared jets obtaina large contribution from lower pT particle level jets dueto the large energy distortion introduced by the calorime-ter. Implications of this observation will be discussed inthe next section.

In addition to transverse momentum, smearing of therapidity and azimuthal angle of the jet thrust axes werealso investigated and very good agreement between par-ticle level and smeared jets was seen. It should be noted,however, that the position resolutions inherent to thecalorimeters were not considered in this exercise as theydepend on details such as material, tower size, and read-out which have not been finalized. This should be revis-ited when more complete detector simulations are avail-able and will be critically important for future work in-vestigating the utility of jet shape observables.

C. Hadron Calorimetry

The decision not to include a mid-rapidity hadroncalorimeter in the BeAST design was based on severalconsiderations including the low energies of producedparticles, the modest fraction of total energy carried byneutral hadrons, the use of streaming readouts whichdo not require a trigger, and finally, the significant costof such a detector. Figure 19 makes it clear that ahadron calorimeter with sufficiently high resolution canmarkedly improve jet energy measurements. Unfortu-nately, calorimeter cost increases with resolution, mean-ing the inclusion of such a high resolution mid-rapidityhadron calorimeter (which must cover a large volume)may be infeasible. While the corrections to jet energyneeded in the absence of a hadron calorimeter will bemodest, it is worth considering the benefits that could beprovided by a more economical lower resolution calorime-ter.

One such benefit would be the ability to implementan unbiased jet trigger. While the current plan calls fora data acquisition system capable of recording all inter-actions, this ability has not been demonstrated, whichmeans that the capability of triggering on events withjets in an unbiasd way could be necessary. Even if sucha streaming readout is possible, a traditional trigger sys-

tem including a hadron calorimeter may be more eco-nomically feasible. A hadron calorimeter would also pro-vide in situ measurements of neutral hadron abundancesand energies, which would reduce the uncertainty in anyMonte Carlo based corrections to the jet energy. Figure18 shows that even a low resolution hadron calorimeterwould reduce the number of jets reconstructed at signif-icantly lower transverse momenta, which would mitigatethe loss of jets which would otherwise fail a minimum pTcut. The largest benefit, though, would likely come fromthe ability to differentiate between jets which do and donot contain neutral hadrons.

A hadron calorimeter should make it possible to sep-arate jets containing neutral hadrons from those whichdon’t by identifying energy deposits which do not have acorresponding charged particle track. The energy resolu-tions of the roughly 65% of jets which do not contain aneutral hadron will be dominated by the high precisiontracker and electromagnetic calorimeters. The superiorresolution for jets which do not contain neutral hadronsversus those which do, can be seen in Fig. 20. Separat-ing jets in this way would allow a much smaller correctionto be applied to the majority of jets while reserving thelarger corrections for the 35% of jets which contain en-ergy from neutral hadrons. This scheme should improveoverall jet energy resolution much more than what wouldbe possible considering only the energy recorded by thecalorimeter.

VI. JET APPLICATION: TAGGINGPHOTON-GLUON FUSION

Previous sections have focused on technical aspects ofjet finding at an EIC without discussion of potential ap-plications. As stated above, the utility of dijets at an EIChas been explored recently in the context of accessing thegluon Sivers function [20] and Weizacker-Williams gluondistributions [16] as well as determining polarized andunpolarized photon structure functions [21]. This sec-tion will present a related measurement in which dijetsare used to tag photon-gluon fusion events for the pur-pose of exploring the gluon contribution to the spin of theproton, ∆G , via the longitudinal double spin asymmetryALL at leading order.

A. Kinematics and Tagging

One of the signatures of the PGF process is the pro-duction of particles with large momenta transverse to thephoton-proton interaction axis which are back-to-back inazimuth, meaning the observation of a dijet in the Breitframe can be used to tag possible PGF events. Unfortu-nately, both the resolved and QCD compton processes,which are background to a ∆G measurement, also pro-duce such dijets. While a global analysis could likely han-dle these background contributions in a consistent way,

16

0 5 10 15 20 25 30

1

10

210

310

410

510­1

Counts

/fb = 7 GeV/c

TSmeared Jet p

No Mid­Rap HCal

E 75%/⊕HCal Res: 15%

E 35%/⊕HCal Res: 2%

0 5 10 15 20 25 30 [GeV/c]

TParticle Level Jet p

= 10 GeV/cT

Smeared Jet p

0 5 10 15 20 25 30

= 13 GeV/cT

Smeared Jet p

FIG. 19. [color online] Particle jet pT spectra for smeared jet pT values of 7 (left), 10 (middle), and 13 GeV/c (right).The baseline BeAST design (blue) and BeAST plus two mid-rapidity hadron calorimeter configurations (red and green) arecompared.

0

10

20

30

40

[G

eV

/c]

TD

ete

cto

r L

eve

l Je

t p

1

10

210

310

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510Without Neutral Hadrons

0 10 20 30 40

[GeV/c]T

Particle Level Jet p

0

10

20

30

40

[G

eV

/c]

TD

ete

cto

r Level Jet p

1

10

210

310

410

510With Neutral Hadrons

FIG. 20. [color online] Relationship between particle level andsmeared jet transverse momenta for jets which do not (upperpanel) and do (lower panel) contain neutral hadrons.

it is worth exploring what can be done experimentally toisolate the PGF process.

Because the dijet kinematics approximate those of theoutgoing partons, they can be used to reconstruct prop-erties of the event which will help separate the PGF pro-cess from resolved and QCDC events. In this analysis,the two variables used for this purpose are xγ and xp,which are the momentum fractions carried by the par-ton originating from the photon and the parton comingfrom the proton, respectively. These quantities are re-constructed from the dijet kinematics as follows:

xγ =1

2Eey

(mT1e

−Y1 +mT2e−Y2)

(2)

xp =1

2Ep

(mT1e

Y1 +mT2eY2), (3)

where Ee and Ep are the energies of the incoming electronand proton beams, respectively, y is the inelasticity, mT

is the jet transverse mass defined as the quadrature sumof the jet mass and pT , and Y is the jet rapidity, in thelaboratory frame. The correlation between generated andreconstructed xγ and xp is quite good (see [21]). Dijetswere reconstructed using the same method as describedin Sec. III D.

As they are largely a low Q2 phenomenon, a significantfraction of resolved events can be eliminated simply byrequiring that Q2 > 1 GeV2. However, because requir-ing two high-pT jets significantly biases the event sampleagainst leading order DIS and toward higher-order pro-cesses, a non-negligible resolved contribution remains forQ2 > 1 GeV2 (see Fig. 21). This remaining resolvedcomponent can be greatly reduced by requiring that thereconstructed xγ be close to unity. As explained in [21],the virtual photon behaves as a point particle for thePGF and QCDC processes, meaning it contributes 100%of its momentum to the interaction and thus should havexγ = 1. Conversely, for a resolved event, the photon be-haves as a composite particle and only a fraction of itsmomentum contributes to the interaction, meaning xγwill have a broad distribution of values less than unity.

17

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1A

rbitra

ry U

nits

2 < 10 GeV

2 < Q

21 GeV

Resolved

QCD­Compton

Photon­Gluon Fusion

DIS

0 0.2 0.4 0.6 0.8 1γx

2 < 100 GeV

2 < Q

210 GeV

0 0.2 0.4 0.6 0.8 1

2 < 500 GeV

2 < Q

2100 GeV

15

1x

FIG. 21. [color online] Reconstructed xγ for the resolved, QCDC, PGF , and DIS subprocesses in Q2 bins of 1-10 GeV2 (left),10-100 GeV2 (middle), and 100-500 GeV2 (right). Note that each panel has been scaled separately an arbitrary amount andin the right panel, the DIS curve has been scaled down by an additional factor of 15.

Figure 21 presents the reconstructed xγ distributions fordijets from the resolved, QCDC, PGF, and leading or-der DIS processes for three Q2 ranges. It is clear that acut on xγ can effectively remove a large fraction of theresolved contribution while preserving most of the directevents. Cuts on xγ of 0.75 and 0.60 were placed for theQ2 ranges 1− 10 GeV2 and 10− 100 GeV2, respectively,which reduce the resolved component to less than 10%of the remaining PGF contribution. No cut is placed onthe 100-500 GeV2 bin. It should be noted that the dijetsfrom leading order DIS which appear at larger Q2 valuesarise when the target remnant receives a large enoughtransverse kick to form a jet which passes the selectioncriteria. Such events can be eliminated, with minimal lossto PGF and QCDC yields, by cutting on the ratio of dijetmass to Q (the ratio was required to be greater than 2.0)as shown in Fig. 22. For the following, residual contri-butions from resolved and leading order DIS events wereomitted for simplicity as neither subprocess was found tocontribute significantly to the expected asymmetry.

Removing the QCDC contribution is not as straight-forward as eliminating resolved events because the PGFand QCDC processes have very similar event topologies.However, as can be seen in Fig. 23, the PGF crosssection peaks at lower values of xp relative to QCDCevents. Thus, at least at Q2 below 100 GeV2 where thethe QCDC cross section is small compared to PGF, xpcan be used to select regions of high or low signal-to-background. It should be noted that placing an upperxp cut will restrict the maximum accessible dijet mass asseen in Fig. 24. The relationship between xp and dijetmass (at leading order) is given by the Eq. 4:

xp = xB +M2jj

sy, (4)

where xB is Bjorken-x, Mjj is the invariant mass of thedijet system, s is the center-of-mass energy, and y is theinelasticity. Equation 4 shows that the xP values accessi-

ble to this measurement are driven by the center-of-massenergy and that for a minimum dijet mass of 10 GeV2, sof 20000 GeV2, and a maximum inelasticity of 0.95, thelowest xp available is roughly 5×10−3. Figure 24 presentsaccessible xP values as a function of dijet invariant massas well as curves delineating the available phase-space forcenter-of-mass energies of 141 and 45 GeV.

B. Expected Asymmetry

The method used here to determine the behavior ofALL is the same as in [21], which was adapted from [50].For each simulated event, a weight was calculated usingthe subprocess and kinematic information from PYTHIAas well as external (un)polarized PDFs. The asymmetryis then found as the average over these weights. Theweights are calculated according to:

w = a(s, t, µ2, Q2)∆fγ

∗

a (xa, µ2)

fγ∗

a (xa, µ2)

∆fNb (xb, µ2)

fNb (xb, µ2), (5)

where a(s, t, µ2, Q2) is the subprocess dependent partonlevel asymmetry, (∆)fγ

∗

a (xa, µ2) is the (polarized) PDF

for the virtual photon, and (∆)fNb (xb, µ2) is the (polar-

ized) PDF for the proton. The leading order expres-sions for a were taken from [50] and include the ap-propriate depolarization factors. The DSSV14 [51] andNNPDFpol1.1 [3] sets were used to describe the polar-ized proton and were normalized by the MSTW2008[52] and NNPDF2.3 [53] unpolarized PDFs, respec-tively. Because only direct events were considered, the∆fNb (xb, µ

2)/fNb (xb, µ2) term is identically unity.

Figure 25 shows ALL as a function of dijet invariantmass for the QCDC and PGF subprocesses obtained us-ing the DSSV set (NNPDF is similar) for Q2 between10 and 100 GeV2. The error bars represent the RMS ofthe distribution of weights in each mass bin. The width

18

0.2

0.4

0.6

0.8

1

Arb

itra

ry U

nits

2 < 100 GeV

2 < Q

210 GeV

0 2 4 6 8 10

Dijet Mass / Q

0

0.2

0.4

0.6

0.8

1

Arb

itra

ry U

nits

2 < 500 GeV

2 < Q

2100 GeV

30

1x

Resolved

QCD­Compton

Photon­Gluon Fusion

DIS

FIG. 22. [color online] Ratio of dijet invariant mass over Qfor the resolved, QCDC, PGF, and DIS subprocesses in Q2

bins of 10-100 GeV2 (upper panel), and 100-500 GeV2 (lowerpanel). This ratio allows the separation of dijet events arisingfrom the leading order DIS subprocess from all others. Notethat each panel has been scaled an arbitrary amount and inthe lower panel, the DIS curve has been scaled down by anadditional factor of 30.

of the weight distribution is larger for the QCDC processbecause the sign of the weight can change due to differentasymmetry signs for up and down quarks. For the PGFprocess on the other hand, the gluon asymmetry is posi-tive everywhere in the relevant kinematics and the a termis always negative, meaning the weight is always negativeand therefore the spread in weights is smaller. It is seenthat the asymmetries for each subprocess grow with di-jet mass and become sizable, reaching values of 20% forthe highest masses. However, because the QCDC andPGF asymmetries are roughly equal in magnitude butopposite in sign, one can expect that the total asymme-try will be significantly smaller than the asymmetry foreither individual subprocess.

The combined QCDC and PGF ALL obtained usingboth the DSSV and NNPDF PDFs can be seen in Fig 26

as a function of dijet invariant mass in three Q2 ranges.The error bars represent expected statistical uncertain-ties calculated according to:

σ =1

PePp

√1

N−A2LL

N, (6)

where Pe and Pp are the electron and proton beam po-larizations (taken as 80% and 70%, respectively) and N isthe number of expected events assuming an integrated lu-minosity of 10 fb−1 or 50 fb−1. The green bands representthe uncertainty on the NNPDFpol1.1 polarized PDF. Asexpected, the QCDC and PGF asymmetries cancel to alarge degree, resulting in maximum asymmetries of a fewpercent.

In order to better isolate the gluon contribution, itwould be helpful to reduce the fraction of QCDC events,which carry information on the quarks. As mentionedabove, the reconstructed momentum fraction carried bythe parton from the proton can be used to select kine-matic regions where the PGF subprocess is dominant.Figure 27 presents dijet ALL as a function of invariantmass for the bin Q2 = 10− 100 GeV2 for three xp slices:0.005 < xp < 0.03, 0.03 < xp < 0.1, and 0.1 < xp < 1.0,with the ratio of PGF to QCDC events decreasing withincreasing xp. Note that only NNPDF1.1 results areshown for clarity. The bars again show expected sta-tistical uncertainties assuming 10 and 50 fb−1 and thegreen bands are the uncertainty on the NNPDFpol1.1polarized PDFs. The effects of slicing in xp are modest,but do shift the asymmetries to more negative values andincrease the ratio of PDF to statistical uncertainties.

The PDF uncertainties presented in Fig. 26 and 27represent the current state of knowledge on the helicitystructure of the proton. These uncertainties will shrinksubstantially with the addition of inclusive g1 measure-ments, which will be the golden channel for the constraintof ∆g(x,Q2). Figures 26 and 27 show that substantialintegrated luminosities will be needed in order for the di-jet measurements to improve on our current knowledge of∆g(x,Q2) meaning it will be unlikely the dijet measure-ment can compete directly with g1 in constraining thegluon contribution to the proton spin. The benefit of thedijet measurement will likely be in its complementarityto g1 as the dijets arise from different subprocesses andwill have different associated systematics than inclusiveobservables.

VII. SUMMARY AND OUTLOOK

Jet observables have proven their utility as probes ofthe subatomic realm at virtually all high energy collidersoperated to date, while recent experimental and theoret-ical advances, spurred by the success of modern colliderssuch as the LHC, have seen jets become true precisionprobes. This success behooves those interested in the

19

3−

10 2−10 1−10 10

0.2

0.4

0.6

0.8

1A

rbitra

ry U

nits

2 < 10 GeV

2 < Q

21 GeV

3−

10 2−10 1−10 1P

x

2 < 100 GeV

2 < Q

210 GeV

3−

10 2−10 1−10 1

2 < 500 GeV

2 < Q

2100 GeV

QCD­Compton

Photon­Gluon Fusion

FIG. 23. [color online] Reconstructed momentum fraction of the parton arising from the proton for the QCDC and PGFsubprocesses in Q2 bins of 1-10 GeV2 (left), 10-100 GeV2 (middle), and 100-500 GeV2 (right).

0 10 20 30 40 50 60 70 80 90 100]2Dijet Mass [GeV/c

3−10

2−10

1−10

1Px

2−10

1−10

1

2 = 10 ­ 100 GeV

2Q

= 141 GeVsKinematic Limit:

= 45 GeVsKinematic Limit:

FIG. 24. [color online] Reconstructed momentum fraction ofthe parton arising from the proton vs the invariant mass ofthe resulting dijet for the QCDC and PGF processes combinedand Q2 between 10 and 100 GeV2. The solid red and dashedblack lines denote the allowed phasespace for

√s = 141 and

45 GeV, respectively.

science an EIC will address to explore the potential ben-efits that jets could provide. To that end, this paper hassystematically explored a number of topics relevant tothe experimental analysis of jets at an EIC.

The first issues addressed were particulars of the ac-tual jet finding. There was no significant dependenceseen on the choice of jet algorithm, but jets with largerradii were found to better reproduce the underlying par-tonic kinematics and the anti-kT algorithm with R = 1.0was chosen for all subsequent studies. Next, jet kine-matic distributions were quantified, comparing inclusivejet pT, dijet mass, and pseudorapidity spectra for both

10 20 30 40 50 60 70 80 90 100]2Dijet Mass [GeV/c

0.4−

0.3−

0.2−

0.1−

0

0.1

0.2

0.3

0.4

LL

A 2 = 10 ­ 100 GeV

2Q

DSSV14 PDF

QCD­Compton

Photon­Gluon Fusion

FIG. 25. [color online] Dijet ALL as a function of dijet in-variant mass for the QCDC and PGF subprocesses in the10-100 GeV2 Q2 bin

inclusive jets and dijets for a range of Q2 values, sub-processes, and center-of-mass energies. It was seen thathigher center-of-mass energies produce greater yields ofjets/dijets, especially at larger pT/mass. The pseudora-pidity of jets was also seen to increase with

√s, driven by

the larger boost imparted by higher hadron beam energy.The energy contribution from underlying event activitywas also studied using two different methods and wasfound to be small, although it will need to be consideredwhen dealing with low pT jets, where even small under-lying event contributions can have a fractionally largereffect. Distortions of the jet pT due to realistic detectorresolutions were investigated using a smearing programtuned to replicate the BeAST detector design and were

20

0.04−

0.03−

0.02−

0.01−

0

0.01LL

A2

< 10 GeV2

< Q2

1 GeV

­1 L dt = 10 fb∫Outer Marks:

­1 L dt = 50 fb∫Inner Marks:

0.04−

0.03−

0.02−

0.01−

0

0.01

LL

A

2 < 100 GeV

2 < Q

210 GeV

DSSV14

NNPDFpol1.1

NNPDFpol1.1 PDF Uncert

10 20 30 40 50 60

]2Dijet Mass [GeV/c

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

LL

A 2 < 500 GeV

2 < Q

2100 GeV

FIG. 26. [color online] Dijet ALL as a function of dijet in-variant mass for the combined QCDC and PGF subprocessesusing the DSSV14 and NNPDF1.1 polarized PDFs in the 1-10 GeV2 (top), 10-100 GeV2 (middle), and 100-500 GeV2

(bottom) Q2 bins. Note that projected statistical uncertain-ties for the DSSV14 points are not shown for clarity, but arenearly identical to those from NNPDF1.1.

found to be minor. Special attention was given to the roleof hadron calorimetry at mid-rapidity with high resolu-tion, low resolution and no calorimeter options explored.A scheme to use a low resolution calorimeter as a neu-tral hadron veto system with the goal of improving the

overall jet energy resolution was also discussed. Finally,an example analysis was presented in which dijets wereused to tag photon-gluon fusion events for the purpose ofconstraining the gluon helicity contribution to the pro-ton spin. Methods for reducing background and isolatingPGF events were demonstrated and the expected asym-metries and their uncertainties were found and comparedto current knowledge of gluon polarizations.

While this paper provides a solid introduction to theexperimental reality of jet physics at an EIC, the topicis still relatively new and more detailed follow-up studieswill be needed to build a robust EIC jet program. Areasof future study include potentially fruitful topics such asjet substructure and the use of jets in e+A collisions,which were not addressed here at all. In addition, morerealistic detector simulation and modeling will be neededin order to inform detector performance requirements.We hope this paper will serve as a valuable resource andjumping off point for both theorists and experimentalistswho wish to further pursue jet topics at the EIC.

ACKNOWLEDGMENTS

We would like to thank Felix Ringer, Kyle Lee, KoljaKauder, Miguel Arratia, Barbara Jacak, and FrankPetriello for helpful discussions. We are grateful toAlexander Kiselev for providing details on proposedtracking resolutions for the BeAST detector design. B.P.is supported by the Program Development program atBrookhaven National Laboratory, while E.C.A and X.C.acknowledge support from the U.S. Department of En-ergy under contract number de-sc0012704.

21

10 12 14 16 18 20 22

0.03−

0.02−

0.01−

0

0.01

0.02

LL

A < 0.03P

0.005 < x

2 < 100.0 GeV

2 < Q

210.0 GeV

10 15 20 25 30 35 40

]2Dijet Mass [GeV/c

< 0.1P

0.03 < x

­1 L dt = 10 fb∫Outer Marks:

­1 L dt = 50 fb∫Inner Marks:

10 20 30 40 50 60

< 1.0P

0.1 < x

NNPDFpol1.1

NNPDFpol1.1 PDF Uncert

FIG. 27. [color online] Dijet ALL as a function of dijet invariant mass for the combined QCDC and PGF subprocesses usingthe NNPDF1.1 polarized PDFs in the 10-100 GeV2 Q2 bin for partonic momentum fractions of 0.005 < xP < 0.03 (left),0.03 < xP < 0.1 (middle), and 0.1 < xP < 1.0 (right).

22

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