EIC Physics from An All-Silicon Tracking Detector
John Arrington, Reynier Cruz-Torres, Winston DeGraw, Xin Dong, Leo Greiner, Samuel Heppelmann, Barbara Jacak, Yuanjing Ji, Matthew Kelsey, Spencer R. Klein, Yue Shi Lai, Grazyna Odyniec, Sooraj Radhakrishnan, Ernst Sichtermann, Youqi Son, Fernando Torales Acosta, Lei Xia, Nu Xu, Feng Yuan, Yuxiang Zhao
EEIC Physics from An All-Silicon Tracking Detector
John Arrington, Reynier Cruz-Torres, Winston DeGraw, Xin Dong, Leo Greiner, SamuelHeppelmann,
3, 1
Barbara Jacak,
1, 2
Yuanjing Ji, Matthew Kelsey,
4, 1
Spencer R. Klein, Yue Shi Lai, Grazyna Odyniec, Sooraj Radhakrishnan,
5, 1
Ernst Sichtermann, YouqiSong, Fernando Torales Acosta, Lei Xia, Nu Xu, Feng Yuan, and Yuxiang Zhao Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA University of California, Berkeley, Berkeley CA 94720, USA University of California, Davis, Davis CA 95616, USA Wayne State University, Detroit, MI 48202, USA Kent State University, Kent, OH 44242, USA University of Science and Technology of China, Hefei, Anhui Province 230026, China Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, Gansu Province 730000, China (Dated: February 17, 2021)The proposed electron-ion collider has a rich physics program to study the internal structureof protons and heavy nuclei. This program will impose strict requirements on detector design.This paper explores how these requirements can be satisfied using an all-silicon tracking detector,by consideration of three representative probes: heavy flavor hadrons, jets, and exclusive vectormesons.
CONTENTS
I. Introduction, and Physics at the EIC 2II. All-Silicon Tracker Concept Design 2A. Requirements on EIC Tracker 2B. Geometry 2C. Performance 4D. Further momentum resolution optimization 5E. Pointing resolution 8III. Physics Studies and Performance Requirements 10A. Heavy Quarks 101. Physics Introduction 102. Charm-Hadron Reconstruction 123. Charm Structure Function 184. Gluon TMDs through Reconstruction of DD Pairs 215. Gluon Helicity ∆ g/g through Charm-Hadron Double-Spin Asymmetry 246. Charm Baryon Λ ± c Production and Hadronization 287. Heavy Quark Conclusions 31B. Jets 321. Physics Introduction 322. Charged Jet Reconstruction Performance 333. Jet Observables 36C. Exclusive Vector Mesons 371. Physics Introduction 372. ρ production 393. φ production 414. J/ψ production 425. The Υ family 436. Vector meson conclusions 43IV. Summary and Conclusions 44V. Acknowledgements 45References 46 a r X i v : . [ nu c l - e x ] F e b I. INTRODUCTION, AND PHYSICS AT THE EIC
The Electron Ion Collider (EIC) is a planned US-based facility that will make precision measurements of thecollisions of electrons with polarized protons and ions over a large mass range to study Quantum Chromodynamics(QCD) [1, 2]. The EIC will explore a very wide range of physics topics, including - among others - the spin structureof protons and light nuclei, the partonic structure of light and heavy ions, parton transport in nuclear matter, andthe hadronization process. Electrons with energies up to 18 GeV will collide with protons up to 275 GeV and ionswith energies up to 110 GeV per nucleon, at luminosities up to 10 cm − s − [3].The broad physics program and high energies and luminosities impose significant requirements on the EIC detectors.This paper describes requirements for charged-particle tracking, and studies of how the requirements can be met withstate-of-the-art silicon pixel detectors. The requirements discussed here are driven by three promising physics probesat the EIC: heavy quarks, jets and exclusive vector mesons. High precision measurements addressing the physicsquestions listed above will require wide pseudorapidity coverage, excellent momentum resolution, and low mass tolimit bremsstrahlung from electrons and positrons. We have developed a conceptual design for an all-silicon trackingsystem and simulated its implementation, including a first approximation of detector support structures and services.The paper explores the three probes and quantifies how well the design meets the requirements.The paper is organized as follows: Section II describes the proposed all-silicon tracker. Section III presents thephysics goals, detector requirements and simulation studies demonstrating the performance for heavy quarks, jets andexclusive vector mesons. Section IV presents the summary and conclusions of these studies. II. ALL-SILICON TRACKER CONCEPT DESIGNA. Requirements on EIC Tracker
EIC detectors have been conceptualized as general-purpose instruments surrounding the interaction point (IP) andembedded in a solenoidal magnetic field with maximum field strengths of either 1.4 or 3.0 T. Approximately 2.5 m alongthe z axis and a radial extent of 80 cm are allocated for the innermost tracking system, which will be surrounded byother sub-detectors including particle identification (PID) detectors and electromagnetic and hadronic calorimeters.Tracking and vertexing systems at the EIC must have wide kinematic coverage, good momentum resolution andsecondary vertex separation capabilities, in order to carry out the EIC physics programs. There are, in general, twotypes of tracking/vertexing detector designs being considered: 1) a hybrid system composed of silicon-pixel layersfor vertexing plus outer gas detectors ( e.g. Time Projection Chamber (TPC) and Micro-Pattern Gaseous Detectors(MPGD)) and 2) an all-silicon tracker for both momentum and vertexing measurements. Recent R&D work showedthat the all-silicon tracker can deliver a comparable or better momentum resolution than hybrid concepts, whilekeeping the radial dimension quite compact. The compact all-silicon tracker leaves much more open space for theouter PID detectors to enhance their PID performance. If even more space were needed, this could be achieved bymaking the all-silicon tracker even more compact, albeit at the expense of some resolution loss.
B. Geometry
FIG. 1. All-silicon tracker geometry. Left: GEANT4 schematic of the tracker cross section. The barrel, disks, and supportstructure correspond to the green, dark-gray, and yellow components, respectively. The beryllium section of the beam pipe isshown in cyan. The rest of the beam pipe, which takes into account the expected electron-hadron-beam crossing angle is shownin light-gray. Right: detector schematic (side view). The barrel layers, disks, and support structure are represented in blue,red, and yellow, respectively. See text for details.
A schematic of the all-silicon tracker concept considered in this work is shown in Fig. 1. This detector, whichhas been designed within a generic EIC-detector R&D effort [4], corresponds to a cylindrical tracker with radius of43.2 cm and length of 242 cm along the z direction, wrapped around the beam pipe and centered at the nominal IP(which corresponds to ( x, y, z ) = (0 , , − . < z < . µ m. Outside of this region, the beampipe fans out to take into account the beam-crossing angle of ≈
25 mrad. Inside the beam pipe, a vacuum is simulated.The rest of the geometry is embedded in an air volume.The tracker coverage for low values of pseudorapidity, η ≡ − ln (cid:0) tan( θ/ (cid:1) (where θ is the polar angle in a coordinatesystem with the z axis aligned along the beam pipe) is provided by a barrel with 6 layers. The radii at which theselayers are located and their corresponding lengths along z are summarized on Table I and illustrated in Fig. 1 (right).They are laid out in three double layers to provide redundancy, and the middle double layer is placed equidistantlybetween the inner and outer double layers to measure hits in the vicinity of the sagitta to optimize the momentumresolution. The pairing of the barrel layers also has the benefit of reducing the number of stave designs. The vertexingcapabilities of the detector are driven primarily by the first two layers. The innermost layer was placed as close tothe beam pipe as possible, and the position of the second layer was varied until the optimal vertexing performancewas found.Coverage at larger absolute values of pseudorapidity is provided by 5 disks in each direction. These disks areassembled by adding rectangular staves in parallel and giving each a length that satisfies that fits within a circle orradius R , the disk outer radii presented in Table II, along with their z positions and inner radii; see Fig. 1 (right).Given the rectangular geometry of the staves, the hole through which the beam pipe passes is shaped as a squareof side equal to twice the inner radii presented in Table II. While the disks on either side of the x − y plane arepositioned at the same distance from the center of the detector and their outer radii are the same, their inner radiiare optimized to be as close to the beam pipe as possible. Thus, the acceptance limit at high | η | is given by thebeam-pipe geometry. Given the asymmetric nature of the EIC collisions ( i.e. electrons colliding with protons or nucleiwith different lab-frame energies), a potential future improvement is to optimize the disk layout separately for theforward and backward regions. An odd number of disks is favored to measure hits in the vicinity of the sagitta, thusachieving a better resolution. The transition between the barrel and the disks occurs at | η | ≈ . h ) r ad i a t i on l eng t h ( X / X beampipebarreldisk stavesservices and supportTotal FIG. 2. Detector material budget. Left: schematic of the ALICE ITS2 inner-barrel staves used in the all-silicon trackerdesign presented here. Schematic taken from Fig. 1.3 in [5]. Right: all-silicon tracker material scan. The dashed magenta linecorresponds to the material from the beam pipe. The barrel and disk contributions are shown in red and blue, respectively.The aluminum support structure is shown in yellow. The total contribution is shown in black. See text for details.
Both the barrel layers and the disks are made up of realistic staves modeled after the ALICE-ITS2-upgrade inner-barrel staves [5–7] and shown in Fig. 2 (left). Besides the active silicon volume, each stave includes components suchas carbon-fiber support structures and water cooling pipes, which combined correspond to an average material budgetof 0.3% X per stave. The total amount of material that these staves contribute to the all-silicon tracker geometryis shown in Fig. 2 (right) Since the staves create a periodic but φ -varying structure (where φ corresponds to theazimuth), the geometry is scanned around the azimuth for a fixed η , and the minimum and maximum amounts ofmaterial found define the boundaries of the uncertainty band. With the current configuration, the material budget TABLE I. Barrel-layer radii and lengths.Barrel radius length along zlayer [cm] [cm]1 3.30 302 5.70 303 21.00 544 22.68 605 39.30 1056 43.23 114 TABLE II. Disk z position and inner and outer radii.Disk z position outer innernumber [cm] radius [cm] radius [cm]-5 -121 43.23 4.41-4 -97 43.23 3.70-3 -73 43.23 3.18-2 -49 36.26 3.18-1 -25 18.50 3.181 25 18.50 3.182 49 36.26 3.183 73 43.23 3.504 97 43.23 4.705 121 43.23 5.91 contributed by the barrel and disk staves is < X .The attributes of the sensor used in the simulations are taken from the eRD25 and EIC Silicon Consortium [8]descriptions of the projected properties of an EIC specific Monolithic Active Pixel Sensor (MAPS) currently underdevelopment. The sensor silicon pixels have a pitch of 10 × µ m (corresponding to a point resolution of 10/ √ µ m)and silicon thickness of 50 µ m. While this simulation effort uses 0.3% X for the inner two tracking layers, there areongoing R&D efforts to use stitched, thinned and bent air-cooled silicon to allow the vertexing layers to become asthin as 0.05% X [9].As part of the EIC Yellow Report effort, projections were generated for both the radiation length of staves anddiscs [10] based on the eRD25 EIC specific sensor and for the services (location and composition) and mechanicalsupports [11] that would be required to complete a tracking detector. These projections, which were only availableafter most of the work presented here, are referenced for completeness and are reasonably consistent with what isused in the shown simulation for material in the tracking detectors acceptance.The detector is complemented with a simplistic conical aluminum support structure with a thickness of 5 mm whichis tapered for z >
58 cm. As shown in Fig. 2 (right), this support structure adds a significant amount of material to thedetector. However, the projective design concentrates this material into a narrow pseudorapidity range at | η | = 1 . C. Performance
This geometry was implemented in GEANT4 and studied within the full Monte-Carlo framework for detector sim-ulation, Fun4All [13–15]. Performance studies were carried out by generating charged particles ( e.g. pions, electrons,protons, and muons) from the nominal IP in the momentum range 0 < p <
30 GeV /c and over the entire detectoracceptance ( i.e. | η | < < φ < π ). The two magnetic fields considered for the simulations correspond tosolenoidal field maps for the BeAST [16, 17] and BaBar [18] magnets, with peak intensities of 3.0 T and 1.4 T,respectively. The hits resulting from the interaction between the generated particles and the detector (which wassetup with a hit efficiency at 100%) were combined into tracks, and differences between the variables generated andreconstructed in the simulation (labeled ‘truth’ and ‘reco’, respectively) were used to characterize various detectorresolutions. Pattern recognition for combining hits into reconstructed tracks is seeded using truth-track information.Thus, final efficiency studies are not feasible at the moment and will be carried out when more realistic seedingalgorithms are implemented.The momentum resolution, dp/p , is defined as the standard deviation of the ( | (cid:126)p truth | − | (cid:126)p reco | ) / | (cid:126)p truth | distribution,where | (cid:126)p | is the absolute value of the particle momentum. Figure 3 (left) shows the momentum resolution as afunction of momentum for charged pions, electrons, muons, and protons in the pseudorapidity range 0 . < η < . c , significantlyworsening the resolution at low momentum. For other particles and for high-momentum protons, the rise withincreasing momenta originates primarily from the decreasing sagitta for stiffer tracks, with the electron momentumresolution systematically above that for other particles in most of the studied range. Figure 3 (right) shows themomentum-resolution results as a function of pseudorapidity in the momentum range 5 . < p < . /c . Themomentum resolution is approximately constant up to η ≈
2, and then quickly rises. It is worth emphasizing that this dp / p [ % ] - p + p e- - m p < 0.5 h h dp / p [ % ] c < 7.5 GeV/ p FIG. 3. Momentum resolution for different particles in the 3.0 T magnetic field. Left: dp/p as a function of momentum in the0 < η < . dp/p as a function of pseudorapidity in the 5 . < p < . c range. See text for details. pseudorapidity value corresponds to θ ≈ ◦ , implying that the rising part of the resolution represents less than 17%of the polar-angle acceptance. Overall, the performance is very similar for the studied particles, and further resultswill be shown only for pions.More detailed momentum-resolution studies for pions are shown for both magnetic-field settings in Fig. 4. Asexpected from the leading-order ∼ /B dependence of the momentum resolution, doubling the magnetic-field intensityimproves the momentum resolution by a factor of ≈
2. Momentum resolutions are typically parametrized by thefunction dp/p = A · p ⊕ B , where A and B are fit parameters and ⊕ indicates sum in quadrature. The resulting fitsand the fit parameters are shown in the figure. Also shown as gray lines are the requirements determined by thephysics working groups in the EIC Yellow Report effort [19]. In the case of the 3.0 T field, the initial tracker designsatisfies the physics requirements over the entire 0 < p <
30 GeV /c range for − . < η < . p < /c are shown in Fig. 5. D. Further momentum resolution optimization
To study whether the momentum resolution could be improved at forward and backward pseudorapidities, wherethere is tension between the tracker performance and physics requirements, the detector was complemented withadditional tracking stations taking into consideration the space currently projected to be available according to theCentral Detector/Integration & Magnet Working Group [20]. Placing such complementary trackers away from theinteraction point increases the field integral, (cid:82) B · d l , thus improving the momentum resolution, and no specificdetectors are planned to be installed between the backward station and the all-silicon tracker. We examined theimpact of adding methane-based gas electron multiplier (GEM) detectors or additional silicon disks in the backwardand forward regions at z = −
180 and 300 cm, respectively. The resulting momentum resolutions in the backwardregion are shown in Fig. 6. Complementing the all-silicon tracker with a 50 µ m-resolution GEM station yields asmall improvement, mainly for low-momentum particles, while a 10 µ m-pixel silicon disk significantly improves themomentum resolution in the far-backward region (negative η ), by a factor of two or more for high-momentum particles.In the forward region, the complementary station is placed behind a Ring Imaging Cherenkov (RICH) detector.The RICH material budget was provided by the PID detector working group [21] and it corresponds to a dual-radiator(aerogel and C F ) device. The resulting momentum resolutions are shown in Fig. 7. In this region, a 50 µ m-resolutionGEM provides a momentum-resolution enhancement comparable to that of a 10 µ m-pixel silicon disk except in thehighest momentum bin. Both provide a small improvement in the resolution in the far-forward region for momentaabove 5 GeV/ c . In practice, the magnetic-field lines are expected to be shaped in such a way that bending insidethe ∼ | η | > .
2. Nevertheless, it can be seen in Figs. 6 and 7 that their impact dp / p [ % ] ForwardBackward | < 0.5 h dp / p [ % ] | < 1.0 h dp / p [ % ] | < 1.5 h dp / p [ % ] | < 2.0 h dp / p [ % ] | < 2.5 h dp / p [ % ] | < 3.0 h dp / p [ % ] | < 3.5 h dp / p [ % ] | < 4.0 h FIG. 4. Momentum resolution as a function of momentum for several pseudorapidity bins. The markers correspond to theresolutions extracted from the simulations, and the lines correspond to fits to such resolution curves. The orange (filled) andblue (open) circles correspond to simulations carried out with the BeAST (3.0 T) and BaBar (1.4 T) field maps respectively.The functional form used in the fits is dp/p = Ap ⊕ B , and the parameters A [% / (GeV /c )] and B [%] are given in the plots. TheEIC physics requirements [19] are shown as gray lines for | η | < .
5. In the cases where the forward and backward requirementsare different, the backward requirements are shown as dashed lines. dp / p [ % ] dp / p [ % ] h ( 0.0 , 0.5 )( 0.5 , 1.0 ) ( 1.0 , 1.5 )( 1.5 , 2.0 ) ( 2.0 , 2.5 )( 2.5 , 3.0 ) ( 3.0 , 3.5 ) FIG. 5. Momentum resolution as a function of momentum for several pseudorapidity bins in the low-momentum region. Left:1.4 T magnetic field. Right: 3.0 T magnetic field. is negligible for | η | < .
5. Thus, smaller tracking stations can be constructed to complement the tracker in the EIC. - - - -h dp / p [ % ] c All-Si m) GEM m (50 s All-Si + m pixel Si disk m All-Si + 10 at z = -180 cmauxiliary tracking station - - - -h dp / p [ % ] c - - - -h dp / p [ % ] c
10 < p < 15 GeV/ - - - -h dp / p [ % ] c
15 < p < 20 GeV/ 3.5 - - - -h dp / p [ % ] c
20 < p < 25 GeV/ 3.5 - - - -h dp / p [ % ] c
25 < p < 30 GeV/
FIG. 6. Momentum resolution as a function of pseudorapidity in the backward region. The all-silicon tracker standaloneperformance is shown in black squares. The purple triangles and orange inverted triangles describe the momentum resolutionachieved by complementing the all-silicon tracker with a 50 µ m-resolution GEM station or a 10 µ m-pixel silicon disk at z = −
180 cm, respectively. In this region, the all-silicon tracker complemented with a silicon disk offers a significantly betterperformance. See text for details. h dp / p [ % ] c All-Si m) GEM m (50 s All-Si + m pixel Si disk m All-Si + 10 at z = 300 cmauxiliary tracking station h dp / p [ % ] c h dp / p [ % ] c
10 < p < 15 GeV/ h dp / p [ % ] c
15 < p < 20 GeV/ 2 2.5 3 3.5 h dp / p [ % ] c
20 < p < 25 GeV/ 2 2.5 3 3.5 h dp / p [ % ] c
25 < p < 30 GeV/
FIG. 7. Momentum resolution as a function of pseudorapidity in the forward region. The all-silicon tracker standaloneperformance is shown in black squares. The purple triangles and orange inverted triangles describe the momentum resolutionachieved by complementing the all-silicon tracker with a 50 µ m-resolution GEM station or a 10 µ m-pixel silicon disk at z =300 cm, respectively. In this region, a RICH detector is placed in the space between the all-silicon tracker and the complementarytracking station. The all-silicon tracker complemented with a GEM offers a performance comparable to that of the trackercomplemented with a silicon disk (except in the highest momentum bin shown). See text for details. E. Pointing resolution
In addition to measuring the momenta of particles, the silicon tracker must be able to reconstruct secondary verticesand project track trajectories to the outer detector systems. The Distance of Closest Approach (DCA) is defined asthe spatial separation between the primary vertex and the reconstructed track projected back to the z axis (DCA z )or to the x − y plane (DCA rφ ). The DCA resolutions were determined as the standard deviation of normal functionsfitted to the DCA z and DCA rφ distributions. DCA-resolution results as a function of transverse momentum ( p T )for pions are shown in Figs. 8 and 9. The resulting distributions were characterized via fits with the functional form σ (DCA) = A/p T ⊕ B . The fits and fit parameters are presented in the figures. It is clear that the DCA resolutionsare insensitive to the magnetic field. T p0510152025303540 m ] m ) [ z ( DC A s | < 0.5 h T p020406080100 m ] m ) [ z ( DC A s | < 1.0 h T p020406080100 m ] m ) [ z ( DC A s | < 1.5 h T p050100150200 m ] m ) [ z ( DC A s | < 2.0 h T p0100200300400500600700 m ] m ) [ z ( DC A s | < 2.5 h T p020040060080010001200 m ] m ) [ z ( DC A s | < 3.0 h T p05001000150020002500 m ] m ) [ z ( DC A s | < 3.5 h T p01000200030004000 m ] m ) [ z ( DC A s | < 4.0 h FIG. 8. Longitudinal DCA resolution vs. transverse momentum for several pseudorapidity bins. The circles correspond to theresolutions extracted from the simulations, and the lines correspond to fits to the resolution of the form σ (DCA z ) = A/p T ⊕ B ,with the parameters A [ µ m · GeV /c ] and B [ µ m] given in the figure. The orange, filled and blue, open circles correspond tosimulations using the BeAST (3.0 T) and BaBar (1.4 T) field maps. The polar and azimuthal angular resolutions are determined as the standard deviation of normal functions fitted tothe ∆ θ ≡ θ truth − θ reco and ∆ φ ≡ φ truth − φ reco distributions, respectively. Figure 10 shows the polar and azimuthalangular resolutions at the vertex as a function of momentum for several pseudorapidity bins. While these graphs wereextracted from simulations with the BeAST (3.0 T) magnetic field, the angular resolutions are largely insensitive tothe magnetic field.An important function of an EIC general-purpose tracker is aiding in particle identification (PID). Specifically, agood angular resolution is needed at the spatial coordinates corresponding to the entrance of Cherenkov detectors,since these detectors rarely measure the trajectory of tracks. To study this resolution, the reconstructed momentawere projected onto a cylindrical surface of radius equal to 50 cm and length along the z axis of 260 cm and werecompared to the truth information at the same location. Figure 11 shows the resulting angular resolutions.Primary vertex resolutions in three dimensions and a first look at tracking efficiency are determined by generatingPYTHIA e + p events at 18 ×
275 GeV collisions, reconstructing the final-state particles in the full-simulation detector,and fitting the vertex residual distribution with respect to the generated truth vertex with a Gaussian function.Figure 12, left plot, shows the resulting primary vertex resolution as a function of track multiplicity in events with Q > . One can see the event primary vertex resolution is typically ∼ µ m at a multiplicity of ∼
5, the averagenumber of charged particles produced within the acceptance of tracking detectors in these collisions. Figure 12 rightplot shows the charged pion tracking efficiency in different η regions, which shows reasonable tracking efficienciesover a broad kinematic region. As stated before, pattern recognition is seeded using truth-track information. As a T p0510152025303540 m ] m ) [ f r ( DC A s | < 0.5 h T p01020304050607080 m ] m ) [ f r ( DC A s | < 1.0 h T p020406080100 m ] m ) [ f r ( DC A s | < 1.5 h T p020406080100120140 m ] m ) [ f r ( DC A s | < 2.0 h T p020406080100120140 m ] m ) [ f r ( DC A s | < 2.5 h T p050100150200 m ] m ) [ f r ( DC A s | < 3.0 h T p050100150200250 m ] m ) [ f r ( DC A s | < 3.5 h T p050100150200250300 m ] m ) [ f r ( DC A s | < 4.0 h FIG. 9. Transverse DCA resolution vs. transverse momentum for several pseudorapidity bins. The circles correspond to theresolutions extracted from the simulations, and the lines correspond to fits to the resolution of the form σ (DCA rφ ) = A/p T ⊕ B ,with the parameters A [ µ m · GeV /c ] and B [ µ m] given in the figure. The orange, filled and blue, open circles correspond tosimulations using the BeAST (3.0 T) and BaBar (1.4 T) field maps. -
10 110 [ m r ad ] q d h ( 0.0 , 0.2 ) ( 0.2 , 0.5 ) ( 0.5 , 0.8 )( 0.8 , 1.0 ) ( 1.0 , 1.2 ) ( 1.2 , 1.5 )( 1.5 , 1.8 ) ( 1.8 , 2.0 ) ( 2.0 , 2.2 )( 2.2 , 2.5 ) ( 2.5 , 2.8 ) ( 2.8 , 3.0 )( 3.0 , 3.2 ) ( 3.2 , 3.5 ) ( 3.5 , 3.8 ) -
10 110 [ m r ad ] f d FIG. 10. Polar (left) and azimuthal (right) angular resolutions at the vertex as a function of momentum for several pseudora-pidity bins for pions in the BeAST (3.0 T) magnetic field. These distributions are largely insensitive to the magnetic-field. result, the extracted quantity constitutes a best-case-scenario and is expected to describe the detector performanceonly in very-low-multiplicity events. These tracking efficiencies obtained from the full simulation were applied in thefollowing performance projection studies through fast simulation.The p T threshold can be estimated from p T [GeV /c ] = 0 . · B [T] · r/ r corresponds to the radius of curvature of the track, following a circular trajectory in the magnetic field ofintensity B . The threshold for a track to reach the outer layer of the detector ( r ≥ .
23 cm) in a 3.0 T (1.4 T) uniformsolenoidal magnetic field corresponds to p T ≥
195 (90) MeV/ c in this estimate. We can consider a lower thresholdcorresponding to particles that reach the third barrel layer ( r ≥ .
00 cm), since three is the minimum number of hitsneeded for a momentum reconstruction. Clearly, not reaching the outer layers has a negative impact on the resolutionof such particles. In this case, the threshold in a 3.0 T (1.4 T) uniform solenoidal magnetic field would correspond to p T ≥
95 (44) MeV/ c . However, energy-loss and multiple scattering, in particular for non-relativistic particles, lead to0 -
10 110 [ m r ad ] q d h ( 0.0 , 0.5 ) ( 0.5 , 1.0 ) ( 1.0 , 1.5 )( 1.5 , 2.0 ) ( 2.0 , 2.5 ) ( 2.5 , 3.0 )( 3.0 , 3.5 ) ( 3.5 , 4.0 ) -
10 110 [ m r ad ] f d FIG. 11. Polar (left) and azimuthal (right) angular resolutions at the location of PID detectors as a function of momentum forseveral pseudorapidity bins for pions in the BeAST (3.0 T) magnetic field. These distributions are overall insensitive to themagnetic-field intensity. m ) m V e r t e x R e s o l u ti on ( xyz ) c (GeV/ p Track T r ac k i ng E ff i c i e n c y |<1 h h h FIG. 12. Left: primary vertex resolution determined in the full simulation setup with PYTHIA e + p events at 18 ×
275 GeVcollisions with an event level selection of Q > . Right: Tracking efficiency determined in the full simulation for threedifferent η regions. Both figures incorporate events generated in a 3.0 T magnetic field. higher thresholds than the values estimated above. We have incorporated the efficiencies from full Fun4All simulationsin our heavy quark studies and conservative values for the thresholds in the studies of jets, while the vector mesonstudies are based directly on full simulations albeit in the older EICroot framework [22].In this section, simulations were carried out with magnetic-field maps incorporating a gradual decrease in themagnetic-field strength with increasing distance from the nominal interaction point in the z direction. Furthermore,the BaBar magnet, which is a candidate solenoid for the EIC, peaks at B = 1 . B = 1 . B = 1 . . ∼
10% differences at high- | η | . These differences should not affectthe conclusions reached in each section. III. PHYSICS STUDIES AND PERFORMANCE REQUIREMENTSA. Heavy Quarks
1. Physics Introduction
Heavy quarks are produced through photon-gluon fusion (PGF) at the leading order in high-energy e + p /A deepinelastic collisions: γ ∗ + g → Q + Q , see Fig. 13. Therefore, heavy-quark production via deep inelastic scattering (DIS)has unique sensitivity to the gluon distributions in the nucleon or nucleus, and can elucidate the QCD dynamics of1heavy compared to light flavor quarks. e e' B x p/A p'/A* cc g x DD FIG. 13. Leading-order diagram for charm-anti-charm pair production in e + p /A deep inelastic scatterings (DIS). There have been prior studies of heavy-flavor measurements to investigate gluon distributions at the EIC [23–25]. Inthe study in Ref. [23], charm events were tagged by secondary charged kaon tracks while Ref. [24] reported secondaryvertex reconstruction of the decays of charm mesons. In the past few years, there have been rapid developmentsin both the EIC machine design/performance projections as well as experimental detector R&D, including trackingand vertexing. In this section, we include these new developments in the charm-meson simulations to quantify thephysics reach offered by the all-silicon tracker concept presented in Sec. II. We report secondary vertex reconstructionperformance for charm-hadron decays to hadronic channels. We also discuss QCD background contributions in EICcollisions. The study sets a list of detector performance requirements, especially on the vertexing/tracking detectors.The GEANT-based simulation of the all-silicon-tracker described in Sec. II is used to benchmark a fast simulation,which then provides performance projections for the following physics objectives: • Inclusive heavy-flavor hadron production in unpolarized e + p /A collisions to constrain gluon (nuclear) partondistribution functions (PDFs) in nucleons and nuclei, especially in the large Bjorken- x ( x B ) region ( x B (cid:38) . • Heavy-flavor hadron pair (e.g. D + D ) production to constrain gluon transverse momentum dependent (TMD)PDFs in both unpolarized and transversely-polarized experiments. • Heavy-flavor hadron double spin asymmetry ( A LL ) measurement to constrain the gluon helicity distributions(∆ g/g ). • Heavy-flavor hadrochemistry (abundance between different heavy-flavor hadron states) studies to better under-stand heavy-quark hadronization as well as the impact of cold nuclear matter effects in e + A collisions.Our analysis re-affirms the potential impact of heavy-flavor measurements to constrain the gluon distributionfunctions in nuclear targets reported in Refs. [23, 24]. However, we now include detailed detector response simulationswhen evaluating the physics capabilities of the measurements. The charm structure function of protons measured atthe HERA e + p collider [26] demonstrated the powerful reach of heavy-flavor measurements to constrain the gluondistribution in the proton. Measurements at the EIC will also probe the gluon distributions in nuclei, as also shownin Refs. [23, 24].Heavy-quark (hadron) pair production in DIS has attracted great attention in the last few years [27–35]. Recon-structing the total transverse momentum of the pair can probe the TMD gluon distribution in the nucleon/nucleus.The TMD parton distribution provides an important aspect of the nucleon/nucleus tomography [36], revealing themomentum distribution of partons not only in the longitudinal direction but also in the transverse direction. Amongthe TMD gluon distributions, previous studies have focused on the so-called linearly-polarized gluon distribution ina unpolarized nucleon [27, 32] and the gluon Sivers function in a transversely polarized nucleon [1]. The key forthese proposed measurements at the EIC relies on the precision of the total transverse momentum of the hadronpair reconstructed from the decay products, along with the sensitivity to the gluon TMD from the nucleon/nucleustargets. This will be complementary to the dijet production at the EIC, which has also been explored [37, 38].2The gluon helicity distribution is one of the major topics for the EIC. It can be well constrained from inclusivepolarized structure function measurements over a wide kinematic range. Its impact on the gluon-spin contribution tothe proton spin has been well documented in the EIC White Paper [1]. Extensive studies in the literature indicatethat inclusive single-jet and dijet production provide complementary constraints on the gluon helicity distribution [39–43]. Our analysis of heavy-flavor production in DIS at the EIC shows that this measurement will improve an earliermeasurement from the COMPASS experiment [44] and provide complementary constraints on the gluon helicitydistribution function, in particular, in the moderate x B region.Heavy-quark hadronization is a critical component in understanding the experimental heavy-flavor hadron pro-duction data. Recent experimental data on charm baryon Λ + c production in hadronic collisions at RHIC and LHCshow the Λ + c /D is considerably larger than the fragmentation baseline constrained by the e + + e − and e + p data inthe low to intermediate p T region [45–47]. Several Monte Carlo hadronization models, e.g. color reconnection inPYTHIA [48, 49], rope in DIPSY [50] etc. have been studied in order to understand the Λ + c /D ratio data. e + p /Acollisions at high luminosity EIC experiments will allow us to have a better control on the initial condition comparedto hadronic collisions, and enable detailed investigation in Λ + c baryon production and how hadronization plays a rolefrom e + e − to hadronic collisions.
2. Charm-Hadron Reconstruction a. PYTHIA Event Generator:
Electron-proton ( e + p ) collisions are generated using the PYTHIA v6.4 eventgenerator [51] with the explicit parameters used in these studies documented in [52]. Events are generated withvector-meson diffractive and resolved processes, semi-hard QCD 2 → e + p collisions. Radiative corrections are not included for these studies except where explicitly mentioned. - - - - ) B (x log )] ( G e V [ Q l og · e+p 18 |<3 h Charm Events | 4 - - - - ) B (x log )] ( G e V [ Q l og · e+p 10 |<3 h Charm Events |4 - - - - ) B (x log )] ( G e V [ Q l og · e+p 5 |<3 h Charm Events |
FIG. 14. Two-dimensional log ( Q ) versus log ( x B ) coverage for three beam-energy configurations in which each eventcontains at least one charm hadron with pseudorapidity | η | <
3: 18 ×
275 GeV, 10 ×
100 GeV and 5 ×
41 GeV, respectively.
In the one-photon-exchange approximation, an incoming electron of four momentum e scatters into a final state e (cid:48) via the emission of a virtual photon of four momentum q = e − e (cid:48) , which subsequently interacts with the hadronbeam with four momentum p . We follow the “HERA convention”, in which the hadron beam momentum is alongthe positive z direction. Several kinematical variables are typically used to characterize the scattering process. The3Bjorken scaling variable is defined as x B ≡ Q / (2 p · q ) and Q ≡ − q is minus the square of the four momentumtransfer. The inelasticity is defined as y ≡ p · q/ ( p · e ).The EIC program will run at multiple e + p (A) beam energy configurations and we have simulated several center-of-mass (CM) energies. Figure 14 shows the Q and x B reach for e + p collisions in the 18 ×
275 GeV (electron andproton beam energies with head-on collisions), 10 ×
100 GeV, and 5 ×
41 GeV beam configurations in which at leastone charm hadron is produced within pseudorapidity | η | < Q , the kinematic reach shifts to larger values of x B .Figure 15 shows the x B and Q coverage in slices of D η . Comparing to the middle plot of Fig. 14 one can seethat the largest Q events appear at lower | η | while the large- x B events at fixed Q correspond to large values of η .The fact that high- x B D mesons tend to be produced at forward rapidity will, in effect, truncate the measurablecross section at low Q and high x B . Therefore, careful planning of which beam energies are suited for particularphysics goals are needed. For example, as demonstrated in Ref. [23], high– x B charm structure function measurementshave the strongest constraint on the gluon nPDFs compared to inclusive measurements. Therefore, charm structurefunction measurements at lower collision energies would have a higher impact on the gluon nPDFs. Moreover, sinceat least two beam energies are needed to extract the charm structure functions at fixed Q and x B , good kinematicoverlap in Q and x B between the two energies would be needed to enable the measurements over a broad kinematicrange. More details are discussed in Sec. III A 3. - - - - ) B (x log )] ( G e V [ Q l og · e+p 10 [-1.0,1.0] ˛ h D 4 - - - - ) B (x log )] ( G e V [ Q l og · e+p 10 [2.5,3.0] ˛ h D FIG. 15. Two-dimensional log ( Q ) versus log ( x B ) coverage for e + p collisions at 10 ×
100 GeV with events containing D mesons in − < η < . < η < . D mesons (left panel) and decay pions (middle and right panels) in18 ×
275 GeV electron-proton collisions generated with PYTHIA 6. Each red semi-circle shows the absolute momentum scale ateach order of magnitude as indicated by the x -axis intercept. The z -axis denotes the yield scaled to 10 fb − . The right panelshows the decay-pion distributions after applying an event-level cut requiring x B > . Charm hadrons and their decay particle distributions are studied in 18 ×
275 GeV electron-proton collisions. Weshow here those of D mesons and D decayed pions that are needed for reconstructing D mesons experimentally; thegeneral features of other charm hadrons (and bottom hadrons) and decay particles are similar. Figure 16 shows themomentum versus polar angle for both D and decay pions. We additionally show the decay pion distributions aftera event level cut of x B > x B cut also show the correlation described in the text above, i.e. high– x B eventsproduce more charm hadrons in the forward region. b. Fast Simulation for Charm-Hadron Reconstruction: The analyses and projections done in the next sectionsmake use of a fast simulation to implement the detector responses and to allow generation of sufficient statistics to carryout detailed studies. Our simulation of the heavy-flavor observables focuses primarily on the detector performance inthe | η | < D mesons are reconstructed through the decay channel D → K − π + with a branching ratio of ∼ TABLE III. Smearing parameters used in fast simulation in different η bins: momentum resolution with two sets of magnetic-field configurations, DCA rφ pointing resolution and particle identification (PID) momentum upper limits. All p and p T valuesare in the unit of GeV/ c . η σ p /p - 3.0 T (%) σ p /p - 1.5 T (%) σ (DCA r φ ) ( µ m) p PIDmax (GeV/ c )(-3.0,-2.5) 0.1 · p ⊕ · p ⊕ p T ⊕
15 10(-2.5,-2.0) 0.02 · p ⊕ · p ⊕ p T ⊕
15 10(-2.0,-1.0) 0.02 · p ⊕ · p ⊕ p T ⊕
10 10(-1.0,1.0) 0.02 · p ⊕ · p ⊕ p T ⊕ · p ⊕ · p ⊕ p T ⊕
10 50(2.0,2.5) 0.02 · p ⊕ · p ⊕ p T ⊕
15 50(2.5,3.0) 0.1 · p ⊕ · p ⊕ p T ⊕
15 50
The distribution of different variables characterizing the D -decay topology is shown in Fig. 17, for tracks fromsignal (unlike sign K − π + and K + π − pairs) and background (like sign K + π + and K − π − pairs) candidates from within3 σ of the D mass peak. The distributions show a clear separation between the signal and background, allowing foran improvement in the signal-to-background ratio ( S/B ) and signal significance by placing cuts on the variables. Thecuts used in the analyses in following subsections are shown in Table IV.
TABLE IV. Cuts on the decay-topology variables used for different D p T bins. p T Pair DCA rφ ( µ m) DecayLength ( µ m) cos θ rφ < p T < c <
120 - -1.0 < p T < c < >
40 - p T > c < > > Figure 18 shows the reconstruction of D mesons using tracks passed through the fast simulation without (left)and with selection cuts on the D -decay-topology variables with two momentum resolution requirements (middleand right). In all p T bins, applying cuts on D -decay-topology variables improves S/B by a factor of ∼
10 for p T > c . The signal significance, defined as S/ √ S + B , is also improved for most bins, except for the lowest p T bin. For p T > c , the improvement in significance is about 50%, while the improvement is around 25% for the p T -integrated signal.The impact of different PID scenarios in D reconstruction is also studied. The S/B ratio increases by a factor ofabout 3.5 (2.5) and the signal significance by about 65% (50%) when going from a PID capability up to a momentumof 5 GeV/ c within − < η < − < η <
3) to the PID scenario used in this simulation. The improvement formidrapidity | η | < − < η < − m] µ [ φ r DCA π − − − − A r b . U n i t s e + p, 18 x 275 GeV < 3.0 GeV T SignalBackground 0 200 400 600 800 100 m] µ [ φ r K DCA − − − −
10 0 200 400 600 800 100 m] µ pair DCA [ − − − − m] µ [ φ r Decay Length − − − − A r b . U n i t s ) φ r θ cos( − − FIG. 17. The distribution of variables characterizing the D -decay topology in the transverse direction, for signal (unlike sign)and background (like sign) candidates within 3 σ of the D mass peak. The different panels from top left show distributions ofDCA of pions to PV, DCA of kaons to PV, DCA between the Kπ pairs, Decay Length and the cosine of the pointing angle tothe PV, respectively. The candidate D mesons have 2.0 < p T < . c . ] [GeV/c π K M C oun t s = 2881 T L = 0.013 fb ] [GeV/c π K M = 1447 T ] [GeV/c π K M = 1378 T ] [GeV/c π K M C oun t s = 22805 T pWithout Vertexing3.0T B field ] [GeV/c π K M = 14990 T pWith Vertexing3.0T B field ] [GeV/c π K M = 13299 T pWith Vertexing1.5T B field FIG. 18. The D meson invariant mass distributions, reconstructed through K, π daughter pairs, for 2 < p T < c (toprow) and p T > D signal with two momentum resolutionrequirements (see Table III). As anticipated, the D mass width with the 3.0 T resolution parameter is about half ofthat with 1.5 T resolution parameter. This results in the p T -integrated D significance with the 3.0 T configurationbeing about 50% larger than the 1.5 T configuration.Given that the all-silicon tracker has an outer radius of r = 43 cm, charged tracks must have at least p T ∼ . c to reach the last tracking layer in the central barrel. The impact of this threshold on the D reconstruction is negligible,leading to a <
1% reduction in D significance. The p T -threshold effect is implicitly included in the following physicssimulation by applying the tracking acceptance obtained from the full simulation (right panel of Fig. 12). FIG. 19. Kinematic distributions, in polar coordinates, of D ∗ + mesons (left), decay D (middle) and pions (right) in 18 × x -axis intercept. The z -axis denotes the yield scaled to 10 fb − . ) c ) (GeV/ )-m(D + p m(D C oun t s · Pseudo-DataPseudo-BKGData-BKGTruth Signal ) c ) (GeV/ )-m(D + p m(D C oun t s · ) c [0,0.5] (GeV/ ˛ ) *+ (D T p ) c ) (GeV/ )-m(D + p m(D C oun t s · ) c [0.5,1] (GeV/ ˛ ) *+ (D T p ) c ) (GeV/ )-m(D + p m(D C oun t s · ) c [1,2] (GeV/ ˛ ) *+ (D T p FIG. 20. m ( D π + ) − m ( D ) distributions for D ∗ + candidates reconstructed using the method described in the text. The openblack circles show all correct sign D π + combinations, and the red histogram shows a background estimation using like-sign D candidates. The blue closed circles show the difference between the two, and the green histogram shows the truth-level D ∗ + distribution. The top panel shows the p T integrated distribution, and the bottom three show three low D ∗ + p T bins. Thesimulated integrated luminosity shown here is 0.056 fb − . D ∗ + → D π + . This decay is near threshold, and thereforethe decay pion generally has a relatively low momentum (and is denoted as a slow pion ). Due to this feature, thequantity ∆ m ≡ m ( D π + ) − m ( D ) is typically used to extract the D ∗ + yield as the signal peaks slightly above thepion mass, while combinatorial backgrounds peak at higher ∆ m . In this channel, the signal can be separated frombackground without the need for secondary vertex reconstruction. Conversely, the p T = 200 MeV/ c threshold fora track to hit all layers yields a 60% efficiency for the slow pions from D ∗ + decays. Figure 19 shows the kinematicdistributions of D ∗ + mesons (left), decay D (middle) and pions (rights) in e + p ×
275 GeV collisions generated withPYTHIA 6.We study the viability of reconstructing this channel in a scenario where the slow pion can be reconstructed belowthe 200 GeV/ c p T threshold using hits only within the first three barrel layers, which would lower the p T thresholddown to about 100 MeV/ c and equate to a 90% slow-pion acceptance efficiency. There are no current studies of themomentum resolution for such a scenario and for these studies we chose a conservative 10%. D ∗ + candidates arereconstructed in the simulation by first selecting D → Kπ combinations after the nominal fast-simulation smearingwith decay tracks having | η | < p T >
200 MeV/ c , and with a pair invariant mass within 30 MeV of the D PDGmass. D candidates are paired with correct-sign slow pions to form D ∗ + candidates. Slow pions with p T between100 and 200 MeV/ c are smeared with the aforementioned 10% momentum resolution. For slow pions with p T > c , the nominal values are used. Figure 20 shows the m ( D π + ) − m ( D ) distributions after the fast-simulationsmearing for a sample size corresponding to an integrated luminosity of 0.056 fb − . A background distribution isestimated using like-sign Kπ pairs when reconstructing the D , and is shown as the red histogram. The differencebetween the signal and background distributions, shown as the blue data points, is compared to the true D ∗ + decaysshown as the green histogram. Besides a small residual background present within the peak of the distribution, thesignal is well isolated. Also shown in Fig. 20 are the distributions in three low D ∗ + p T bins, and it is observed that atvery low p T there is still good separation between signal and backgrounds. The signal significance of the D ∗ + channelis comparable to that of the inclusive D → Kπ channel with secondary vertex reconstruction when scaled to thesame integrated luminosity, and will therefore be a viable channel for some charm-hadron measurements at the EIC. Radial Decay Length (mm) A r b . U n it s - - - |<3.0 h , | c [0.0,1.0] GeV/ ˛ T p Full Sim.Fast Sim. (mm) f r DCA D A r b . U n it s - - |<3.0 h , | c [0.0,1.0] GeV/ ˛ T p Full Sim.Fast Sim. (mm) f r DCA p K- A r b . U n it s - - - |<3.0 h , | c [0.0,1.0] GeV/ ˛ T p Full Sim.Fast Sim. (mm) f r DCA p A r b . U n it s - - - |<3.0 h , | c [0.0,1.0] GeV/ ˛ T p Full Sim.Fast Sim. (mm) f r K DCA A r b . U n it s - - - |<3.0 h , | c [0.0,1.0] GeV/ ˛ T p Full Sim.Fast Sim. f r vtx (cid:215) T p D A r b . U n it s - - - |<3.0 h , | c [0.0,1.0] GeV/ ˛ T p Full Sim.Fast Sim. FIG. 21. Comparison of the reconstructed D topological variables in the GEANT4-based all-silicon simulation (data points)and in fast simulation (blue histograms). All distributions are normalized to have unit area. The D candidates shown hereare required to have a | η | < p T < c . c. Fast-Simulation Validation The fast simulation procedure used for subsequent physics studies is validatedby performing the same D topological reconstruction in both the full simulation described in Sec. II, and the fastsimulation using the single-track resolutions determined in the full simulation as input to the smearing routine.8In the full-simulation setup, PYTHIA 8 e + p events are embedded into the full simulator described in Sec. II. Forthese studies, we simulate the highest beam energies, e + p ×
275 GeV, and apply an event-level cut of Q > in both fast- and full-simulation setups. ) c (GeV/ T p T opo . E ff i c i e n c y D Full Sim.<0.5 h h h h h ) c (GeV/ T p F u ll/ F a s t FIG. 22. (Top) Comparison of the reconstructed D topological reconstruction efficiency in the GEANT4-based all-silicon sim-ulation (closed points) and fast simulation (open points). (Bottom) Ratio of efficiencies obtained from full and fast simulations. The smearing routine is applied in a similar way as described in Sec. III A 2 b, with some minor differences. Toaccount for tails on the DCA distributions as well as charge-dependent differences, the smearing factors used areexplicitly drawn from charge-dependent DCA distributions instead of assuming a pure Gaussian distribution. Weadditionally apply a pseudo-tracking efficiency as determined in the full simulation with truth particle seeding to keepthe low- p T thresholds between full and fast simulation consistent (see Fig. 12 right plot). We determine our trackmultiplicities in the fast simulation (from which we determine the primary vertex smearing factor) by counting alltracks within | η | < | η | < D topological variables reconstructed in the full and fast simulation setups with D candidatesrequired to have | η | < p T < c . It can be seen that all topological distributions agree quite well betweenthe fast and full simulations. To further quantify the comparison across phase space we apply the topological cutsdescribed in Sec. III A 2 b and plot the efficiency as a function of D η and p T in Fig. 22. Within the statisticaluncertainties of each respective sample there is general agreement between the efficiencies determined in the full andfast simulations, thus, validating that the fast-simulation smearing procedure gives an adequate description of a fullGEANT4-based simulation.
3. Charm Structure Function
To extract the statistical projections for the charm structure function, F c ¯ c , we first calculate the reduced charmcross sections in a two-dimensional grid of log ( Q ) and log ( x B ) for 10 ×
100 GeV and 5 ×
41 GeV electron+protoncollision configurations in PYTHIA 6 fast simulations. These two energies are chosen as they provide good coverageat high x B and low Q (contrasted with higher energies such as e + p ×
275 GeV), and have relatively good overlap in Q and x B , as illustrated in Fig. 23. Here, Q and x B are taken directly from the generator level. As will be discussedbelow, to extract F c ¯ c at fixed Q and x B , at least two energies are needed. Therefore, good overlap is desired andachievable with the selected beam configurations. We scale all uncertainties to an equivalent 10 fb − worth of datafor each beam-energy configuration as a baseline.9 ) B (x log - - - - )] ( G e V [ Q l og
275 GeV · e+p 18 100 GeV · e+p 1041 GeV · e+p 5 | < 3 h Charm Events |
FIG. 23. Q vs. x B coverage of charm events with | η | < The reduced cross section is explicitly defined as σ c ¯ cr ( x B , Q ) = dN ( D + D ) / L · ε · B ( D → Kπ ) · f ( c → D ) · d x B d Q × x B Q πα [1 + (1 − y ) ] , (1)where y is the inelasticity, L is the integrated luminosity, ε is the total efficiency (tracking, PID, reconstruction andacceptance), B ( D → Kπ ) is the D branching ratio to Kπ , and f ( c → D ) is the D fragmentation fraction inPYTHIA (56.6%). As can be observed from the latter quantity, for the purposes of these calculations we scale themeasured D yield to get the total charm cross section. The binning in log( Q ) and log( x B ) is chosen to be five equalbins per decade along each dimension.The number of D + D candidates is determined by counting the number of true D → Kπ decays with invariantmass within ± σ of the peak, daughter tracks within | η | <
3, and that pass all topological reconstruction requirementsoutlined in Sec. III A 2 b. An additional tracking efficiency is applied using the efficiency curves shown in Fig. 12(right) according to the decay daughter kinematics to simulate a realistic low- p T threshold. The background yieldsare counted similarly within the same mass window, and are composed of any unlike-charge-sign Kπ pair when bothhadrons have p < c or any K/π/p unlike-charge-sign combination when at least one hadron has as a p > c . With this definition, background counts include combinatorial backgrounds, partially-reconstructed charmdecays, and scenarios with a true D decay hadron combined with a random hadron. We then take the statisticaluncertainty of the counts as (cid:112) N ( S ) + N ( B ).Figure 24 shows the reduced charm cross sections in 10 ×
100 GeV and 5 ×
41 GeV electron+proton collisions withan integrated luminosity of 10 fb − . At low Q , the cross section becomes truncated around x B ≈ D η acceptance, as also illustrated in Fig. 15.To calculate the charm structure function F c ¯ c , we take the cross sections at 10 ×
100 GeV and 5 ×
41 GeV at fixed x B and Q and fit the linear form: σ c ¯ cr ( x B , Q ) = F c ¯ c ( x B , Q ) − y Y + F c ¯ cL ( x B , Q ) , (2)where Y + = 1 + (1 − y ) . An example fit for four slices of Q and x B are shown in Fig. 25 (left). The extracted F c ¯ c from the fits are shown in Fig. 25 (right) and the relative statistical uncertainties in Fig. 26.In the example linear fits in Fig. 25 left panel, the data points closest to y /Y + = 0 correspond to the higher10 ×
100 GeV e + p collision cross section. We have estimated the uncertainty on the F c ¯ c in scenarios where we varythe sample sizes of the 10 ×
100 GeV and 5 ×
41 GeV data sets. We find that reducing the lower energy integratedluminosity by up to a factor of ten has a small impact on the extracted F c ¯ c . Conversely, reducing the higher energysample size by any factor increases the relative uncertainty on F c ¯ c by the square-root of the scale factor, as this datapoint provides the best constraint on the y -axis intercept. Therefore, any EIC beam usage request could prioritize alarger integrated luminosity for the larger of the two beam energies.Compared to the work in Refs. [23, 24], our simulation studies represent a more accurate description of charm-reconstruction capabilities with the EIC detector as we have included PID, momentum and single track pointingresolutions guided by ongoing detector development/requirements and a full GEANT-based simulation. Furthermore,0 ) B log(x - - - C · ) , Q B ( x cc r s - - - - - , . , . , . , . , . , . , . , . PYTHIA6 e+p -1
100 GeV 10 fb ·
10 (x1.2) -1
41 GeV 10 fb · ), C (GeV Q FIG. 24. The reduced charm cross section in bins of log ( x B ) and log ( Q ) for 10 ×
100 GeV (closed black circles) and 5 × Q bin are scaled by the constant terms C defined in the plot, and the 5 ×
41 GeV arefurther scaled up by twenty percent for clarity. The uncertainties shown are calculated using the signal significance as describedin the text scaled to an integrated luminosity of 10 fb − at each energy. The 10 ×
100 GeV data is placed at the x -axis bincenters while the 5 ×
41 GeV is displaced along the x -axis for clarity. + /Y y ) , Q B ( x cc r s = 3.1 GeV Q x = 0.005x = 0.008x = 0.012x = 0.019 = 3.1 GeV Q x = 0.005x = 0.008x = 0.012x = 0.019 = 3.1 GeV Q x = 0.005x = 0.008x = 0.012x = 0.019 = 3.1 GeV Q x = 0.005x = 0.008x = 0.012x = 0.019 ) B log(x - - - C · ) , Q B ( x cc F - - - - » sPYTHIA6 e+p , . , . , . , . , . , . , . , . ), C (GeV Q FIG. 25. (Left) Example linear fits to the reduced charm cross sections versus y /Y + in four different slices of Q and x B .(Right) The measured F c ¯ c in bins of log( x B ) and Q . The data points in each Q bin are scaled by a factor of C for clarity.The statistical uncertainties are scaled to 10 fb − . The lines are added to the data to aid the reader. we have included for the first time the primary vertex resolution in the topological reconstruction of D → Kπ decaysin an EIC simulation.In Ref. [24] the longitudinal charm structure functions F ccL are derived from simulation using charm events taggedby the identification of a displaced kaon vertex, and contain background levels that are less than 2%. Comparing thekinematic coverage in Q and x B of F ccL , our derived F cc has slightly better coverage, particularly in the high- x B region ( > F ccL , as opposed to two energies used for F cc . However, it should benoted in Ref. [24] single track and primary vertex resolutions are not folded into the kaon distributions. Incorporatingthese resolutions would significantly smear the charm and background kaon vertex distributions and in turn reducethe charm event purity and limit the kinematic coverage. Therefore, our studies show that now even with a realisticdetector response measurements of F cc are possible across a broad kinematic range.1 ) B log(x - - - ( S t a t . ) cc ) / F cc ( F s - -
10 1 = 1 GeV Q = 12 GeV Q = 76 GeV Q FIG. 26. The projected F c ¯ c relative statistical uncertainties scaled to 10 fb − for three representative Q bins.
4. Gluon TMDs through Reconstruction of DD Pairs
Production of charm anti-charm hadron pairs offers an unique opportunity to study gluon Transverse MomentumDependent (TMD) distributions. Since charm production in DIS proceeds primarily via the photon-gluon fusion(PGF) process (see Fig. 13), reconstruction of both the charm and anti-charm hadron allows to reconstruct thekinematics of the initial gluon, up to corrections from hadronization, initial- and final-state radiation effects. GluonTMDs are hardly constrained with the present experimental data and are of fundamental interest to the physics ofthe EIC.The gluon Sivers asymmetry and TMDs of linearly polarized gluons can be linked to azimuthal anisotropies of theproduced charm anti-charm hadron pair. The Sivers asymmetry can be extracted from the measurements of transversesingle-spin asymmetry, A UT , as a function of the azimuthal angle of the c ¯ c hadron pair relative to the direction ofproton spin. A UT ( p T ) is defined as = [ σ L ( p T ) − σ R ( p T )] / [ σ L ( p T ) + σ R ( p T )], where σ L ( R ) are the cross sections forparticle production of interest with spin polarized in the direction opposite to (same as) the spin of the proton, and p T is the transverse momentum of the heavy hadron pair. The A UT is directly related to the Sivers asymmetry [37], A UT ( p T ) ∝ ∆ f g/p ↑ ( x g , k T ) f g ( x g , k T ) , (3)where x g is the momentum fraction, k T is the transverse momentum of the gluon and ∆ f g/p ↑ and f g are the gluonSivers function and the unpolarized gluon TMD respectively. Note that the notation here is different from the Trentoconvention [53].The transverse momentum distribution of linearly-polarized gluons is related to the azimuthal distribution of themomentum of the c ¯ c hadron pair and can be accessed in unpolarized e + p or e + A collisions [32], |(cid:104) cos(2 φ T ) (cid:105)| ∝ q T M p h ⊥ g ( x g , k T ) f g ( x g , k T ) , (4)where q T is the sum of momenta of the heavy quarks in the pair, φ T is the azimuthal angle of q T with respect to theleading charm meson, M p the proton mass, while h ⊥ g and f g are the linearly polarized gluon TMD and the unpolarizedgluon TMD respectively.In this section we present studies on charm hadron pair reconstruction at an EIC experiment with the detectorsimulation settings described in Section III A 2. The input distribution at the gluon level for the Sivers asymmetry istaken from previous simulation studies that use D meson pair to study the gluon Sivers asymmetry at the EIC [37],while that for the linearly polarized gluon TMDs is taken from [32]. The simulations in this section were carried outfor electron beam at 18 GeV and proton beam at 275 GeV.The x g - Q distribution of events with a D D hadron pair is shown in the left panel of Fig. 27. The parton x g shown is the gluon momentum fraction, instead of the Bjorken- x . The middle and right panels of the figure show the p T - η distributions of D mesons, and of decay pions from D mesons, that are part of a D D pair. From the middle2 − − η D ( G e V / c ) T p D − − e + p 18 x 275 GeV pair D part of a D D − − η Daughter pion ( G e V / c ) T D augh t e r p i on p − e + p 18 x 275 GeV pair D Daughter pions from a D
FIG. 27. Event distributions based on a PYTHIA 6 simulation: (Left) Q - x g distribution of events with a D D hadron pair.(Middle) p T - η distribution of D mesons that are part of a D D hadron pair and (Right) p T - η distribution of pions fromdecay of D mesons that are part of a D D hadron pair. and right panels of the figure it can be seen that the p T of both the D meson and its decay daughters are rathersmall, up to ∼ c , and most of the daughter pions are have | η | <
3, which is within our detector coverage.As noted above, to probe the intrinsic transverse-momentum dependence in the gluon distribution we need to takeinto account initial- and final-state radiation and hadronization effects in our analysis. The initial-state radiation canbe included through the scale evolution of the TMD parton distribution [36]. Similarly, a relevant evolution can becarried out for the final-state gluon radiation by studying the soft factor associated with final state c ¯ c pair [31, 34].For the hadronization effects, on the other hand, we have to rely on Monte-Carlo simulations. The left panel of Fig. 28shows the correlation between the azimuthal-angle directions of the c ¯ c pair momentum and that of the corresponding D D hadron pair momentum, demonstrating that the angular correlations are well preserved during hadronization.The middle panel of Fig. 28 shows the evolution of the signal strength for the transverse single-spin asymmetry ( A UT )from the initial gluon to the gluon reconstructed from the c ¯ c pair and the D D pair. The impact of hadronizationis small, about a 30% reduction in signal strength from the c ¯ c to the D D level. A larger dilution is seen whengoing from the initial gluon to the c ¯ c level and may be specific to the Monte Carlo used (PYTHIA v6.4). This impactis found not to arise from initial-state radiation. However, in this PYTHIA simulation, excluding events where thephoton first splits to a c ¯ c pair, and then one of the charm quarks scatters off the gluon, results in a very small dilutionfrom the initial gluon to c ¯ c . This can be seen in the right panel of Fig 28. − cc φ − DD φ − kS φ − − ) k S φ ( U T A initial gluonccbarDDbar e + p, 18 x 275 GeV < 10 GeV/c T PYTHIA 6.4 − kS φ − − ) k S φ ( U T A initial gluonccbarDDbar e + p, 18 x 275 GeV < 10 GeV/c T PYTHIA 6.4Excluding events where photon splits to a c quark pair andone of the c quarks scatters off the gluon
FIG. 28. (Left) Correlation between the φ angle of the transverse momentum of the c ¯ c pair and that of the D D pair. (Middle)The evolution of signal strength for A UT from initial gluon (input) to gluons reconstructed from c ¯ c pair and from D D pair.(Right) Same as middle panel, but excluding events in which the photon first split to a c ¯ c pair and then one of the charmquarks scatter off the gluon. The D D meson pair is reconstructed using the azimuthal angle difference, ∆ φ DD between the D and D mesonin the pair. The distribution is expected to be predominantly back-to-back. Figure 29 shows the ∆ φ DD distributionfrom D and D meson candidates within 3 σ of the D mass peak in solid black circles and that from D and D meson candidates in a 12 σ mass window outside of the mass peak in solid red squares. The two panels are for twodifferent PID scenarios, a perfect PID case and a PID case corresponding to that from the Detector Matrix. Thesignal pairs show significant excess of candidates above the background. The number of signal D D pairs is obtainedby subtracting the integrated counts over the full ∆ φ DD range from the side-band distribution to that from the signal3distribution. The signal-to-background ratio ( S/B ) as well as the signal significance ( S/ √ S + B ) are also indicatedin the figure. DD φ∆ e + p, 18 x 275 GeV, 0.5 fbDet.Matrix PIDSignalSide band DD φ∆ e + p, 18 x 275 GeV, 0.5 fbPerfect PID FIG. 29. (Left) The ∆ φ DD distributions from D and D meson candidates within 3 σ of the D mass peak (solid black circles)and from those within a 12 σ mass window outside of the mass peak (solid red squares) for a PID scenario corresponding tothat from the Detector Matrix (left) and a perfect-PID scenario (right). The luminosity corresponding to the statistics used in Fig. 29 is 0.5 fb − . Projections of statistical uncertaintiescan be estimated for A UT and (cid:104) cos(2 φ T ) (cid:105) from these significance numbers. As the A UT is given by A UT = ( N L − N R ) / ( N L + N R ), where N L and N R are the total number of signal D D pairs with momentum vector opposite to andaligned with the proton spin, respectively, we have for the statistical uncertainty in A UT is equivalent to the inverse ofthe projected significance ( σ ) for the combined N L + N R candidates at the projected luminosity times the polarizationfraction (P), 1/ σ P. The estimation of statistical uncertainty for (cid:104) cos(2 φ T ) (cid:105) also follows a similar procedure and goesas 1 /σ with σ being the significance for the D D pair signal at the projected luminosity.The D D meson-pair reconstruction is studied for different pair p T , event Q and Bjorken x B bins, using asimulated sample of 0.5 fb − luminosity. The projected uncertainty on A UT as a function of φ kS is shown in Fig. 30,along with the A UT signals at the parton level and for reconstructed D D pairs. The uncertainties are scaled by 1/ P ,where P is the proton-beam polarization at the EIC experiment and is taken to be 70%. The projected uncertaintieson the φ kS integrated A UT as a function of the D D meson pair p T is shown in Fig. 30, while those with projectionsfor uncertainties in different Q and x B bins for A UT as a function of φ kS are shown in Fig. 30 left plot. Theprojected uncertainty on the φ kS integrated (cid:104) A UT (cid:105) is 0.57%, which implies a 7 σ measurement for the projected signalcorresponding to the 10% positivity bound. The uncertainty on (cid:104) cos(2 φ T ) (cid:105) does not get the scale contribution frompolarization fraction, and is ∼ − luminosity. For a projected signal of 2%, this thenimplies a 5 σ measurement.The above analyses only use D D pairs, which is about 1 / − for the 18 ×
275 GeV collisions is feasible. For a 3% signal this would imply a 9 σ and 7 σ measurementsrespectively for the two PID scenarios, making it possible to measure the signal associated with gluon TMDs. Thesemeasurements would be valuable to constrain the gluon TMDs along with measurements from di-jets [37, 38] while DD reconstruction offers a cleaner access to the initial gluon distributions.To study the impact of these measurements on the gluon Sivers function and the linearly-polarized gluon distribu-tion, we can combine the above simulations with the initial- and final-state radiation effects by applying the soft gluonresummation formalism [31, 34]. The final-state gluon radiation can contribute to a nonzero cos(2 φ T ) asymmetry inheavy-quark pair production in DIS [54, 55]. All these effects should be included in the final results to extract thesenovel gluon TMD distributions.4 kS φ − ) k S φ ( U T A > 1 GeV Qparton D De + p 18 x 275 GeV Projected Luminosity 100 fb > 0 GeV/c D DT p (GeV/c) D DT p − − U T U n c e r t a i n t y on A e + p 18 x 275 GeV 1Projected Luminosity 100 fb > 1 GeV Q FIG. 30. (Left) Statistical-uncertainty projections for A UT in bins of azimuthal angle of the pair momentum of the D D pair relative to the spin of the proton ( φ kS ). The two curves indicate the signal strength at parton and D D levels. (Right)Statistical uncertainty projections for (cid:104) A UT (cid:105) in bins of D D meson pair p T . A 70% proton-beam polarization is included theuncertainty projections. kS φ − ) k S φ ( U T A > 1 GeV Q > 3 GeV Q > 5 GeV Qparton D De + p 18 x 275 GeV Projected Luminosity 100 fb > 0 GeV/c D DT p kS φ − ) k S φ ( U T A < 0.00032 B x < 0.001 B B D De + p 18 x 275 GeV Projected Luminosity 100 fb > 0 GeV/c D DT p FIG. 31. Statistical-uncertainty projections for A UT in bins of azimuthal angle of the pair momentum of the D D pair relativeto the spin of the proton ( φ kS ), for different Q (left) and x (right) selections. The two curves indicate the signal strength atparton and D D levels. A 70% proton beam polarization is included the uncertainty projections.
5. Gluon Helicity ∆ g/g through Charm-Hadron Double-Spin Asymmetry In electron-proton DIS processes, if both the electron and proton beams are longitudinally polarized, one canmeasure double-spin asymmetries A LL in the inclusive or semi-inclusive channels, and thus extract the polarizedstructure function g . With extensive measurements in a broad kinematics region, a comprehensive QCD fit can beperformed to extract quark and gluon helicity distributions.In addition to the aforementioned classic way to extract the gluon polarization within a longitudinally polarizednucleon, the heavy-flavor production can also contribute to the study. For instance, A LL measurements in the e + p → e (cid:48) D + X process can be linked to ∆ g/g assuming Photon-Gluon Fusion. At leading order, it can be writtenas A LL = a LL · ∆ g/g , where a LL is double-spin asymmetry of partonic kinematics of the hard-scattering process. Thishas been studied in detail in references [44, 56].The COMPASS collaboration performed a pioneering study on the charm-hadron A LL measurement in polarized µp collisions [44]. Because of the lack of a vertex detector for the D -decay-topology study, the signal-to-background ratiofor the D sample is low. In addition, the luminosity and acceptance are limited at COMPASS relative to the EIC.The study can be dramatically enhanced at the EIC with a good vertex detector to allow topological reconstruction5of charm-hadron decays along with the high luminosity and large acceptance. This provides a direct way, in somesense, to measure ∆ g/g over a broad kinematic range in ( x B , Q ). In the following, a simulation study at the EICwill be described in detail. ] c [GeV/ p K M · ] c E v en t s / . [ M e V / <0.00032 B x ] c [GeV/ p K M · ] c E v en t s / . [ M e V / <0.00157 B x ] c [GeV/ p K M · ] c E v en t s / . [ M e V / <0.42543 B x FIG. 32. Fits to the Kπ invariant-mass distributions in a few different Bjorken- x bins for 18 ×
275 GeV e + p collisions. The redand green dashed curves are the signal (Gaussian) and background (linear) fits, and the blue curve is the sum. The data was generated by pythiaeRHIC (PYTHIA v6.4) and then smeared according to the resolution listed inTable III via the fast-simulation setup described above. To identify D with a good signal-to-background ratio, anoptimization on D -decay topology cuts was performed. Three topological distributions (pair-DCA, Decay-Length rφ ,and cos θ rφ ) were investigated by using “truth” D to Kπ two-body decay and a background sample with Kπ invariantmass outside of the D mass window ( ± σ ), shown in Fig. 32. The yield of the background sample chosen for thestudy depends on the selection range of the M Kπ distribution outside the D peak, in order to be avoid of this artificialeffect, the signal and background samples were separately self-normalized for the three topological distributions andthe crossing points between these two samples were chosen to be the analysis cuts. Table V lists the choices of thesecuts for the different beam energy configurations used in the following projection estimation. In addition to the D -decay topology cuts, the following kinematic cuts were used during the analysis: Q > , 0 . < y < . W > . TABLE V. D decay topology cuts for different beam-energy configurations for A LL projection calculation.Selection criteria 18 ×
275 GeV 5 ×
100 GeV 5 ×
41 GeV Kπ pair-DCA < µ m < µ m < µ mDecay-Length rφ > µ m > µ m > µ mcos θ rφ > . > . > . D candidate and background (estimated through a linear function fit) events within 3 σ ofthe peak in different Bjorken- x bins. The numbers included here correspond to 0.24 fb − ×
41 GeV e + p collisions. x min B x max B N Signal N Background . . +39 − +79 − . . +36 − +74 − . . +39 − +76 − . . +37 − +68 − . . +37 − +68 − . . +39 − +70 − . . +39 − +72 − . . +42 − +73 − . . +43 − +79 − . . +55 − +110 − The data were binned in Bjorken- x after all selection requirements. In each bin, the reconstructed Kπ invariant-mass spectrum was fitted to a gaussian function for signal plus a linear background to extract the number of D signal and background, as shown in Fig. 32. In general, one can see that the signal is quite significant for all the bins.The fit results, as well as the binning information, for 5 ×
41 GeV e + p collisions are summarized in Table VI. These6numbers were scaled to 10 fb − of integrated luminosity for each collision energy configuration in order to obtain theprojected statistical uncertainties on the A LL experimental observable, as shown in Fig. 33. -4 -3 -2 -1
10 0.00.050.10.150.20.250.3
Bjorken x ) ( G e V Q A b s . U n c e r t. LL A
18 GeV x 275 GeV5 GeV x 100 GeV5 GeV x 41 GeV
Polarizations:Int. Luminosity: e: 80%, p: 70% -1 Absolute uncertaintyCharm hadrons + x e' + D fi e + p FIG. 33. Projections on double-spin asymmetry A LL in the e + p → eD + X process for different beam-energy configurations.The data was binned in Bjorken- x . The position of each data point in the plot is defined by the weighted center of Bjorken- x and Q for this particular bin. The uncertainty indicated for each data point should be interpreted using the scale shown onthe right-side vertical axis of the plot. After obtaining the projections on A LL , a LL was calculated event by event using equations 5.8 and 5.9 at leadingorder from reference [56]. Unlike the COMPASS data analysis, where a neural network was employed to map the a LL to measurable kinematic variables, we take event-generator-level information directly to calculate a LL . The meanvalue in each bin was used to extract the ∆ g/g uncertainty from the A LL uncertainty.Due to the nature of the photon-gluon fusion process, as shown in Fig. 13, the value of Bjorken- x ( x B ) obtainedfrom the scattered electron differs from the gluon- x ( x g ), which describes the momentum fraction carried by thegluon inside the proton. A conversion from x B to x g is needed to interpret the measurement in terms of the partonicstructure of gluon inside the proton, unlike the situation in virtual photon-quark scattering where x B is equal to x ofthe quark. The event generator PYTHIA was used for the conversion bin-by-bin: in each x B bin, the photon-gluonfusion subprocess was identified, and then the target parton x (here, it is x g ) distribution was drawn, from which themean value can be obtained. As an example, Figure 34 shows the relation between Bjorken- x and gluon- x both atevent level (left) and at bin-by-bin level (right) for the energy configuration 18 ×
275 GeV.Figure 35 shows the projections on ∆ g/g as a function of x g and Q for different energy configurations at the EIC.In addition, the calculations using NNPDF unpolarized and polarized PDFs at certain x g and Q values are alsoshown as colored bands [57, 58]. The only existing measurement in this channel from the COMPASS collaborationis also shown. As one can see from the plot, ∆ g/g can be measured at high precision by taking advantage of open-charm production at the EIC. Although the precision is lower compared to the g structure function measurements,this physics channel can provide an opportunity to access gluon distribution at a different angle, thus providing acrosscheck on the complicated QCD fits in order to extract the gluon information. Moreover, due to the shift between x B and x g , the measurements at the EIC allow us to study gluon polarization in a relatively high- x g region, whichis unique compared to the g measurement at the EIC. Especially, in the overlap region 0 . < x g < . g/g can be significantly reduced by combining measurements indifferent beam-energy configurations.We would like to emphasize that the inclusive measurements of the polarized structure functions over a wide range ofkinematics at the EIC will play the dominant role in constraining the gluon helicity distribution and its contributionto the proton spin [1]. Nevertheless, the longitudinal double-spin asymmetries of heavy flavor production in DISprocesses will provide complementary constraints on the gluon helicity distribution. As we show in the above, in some7 ) g (Gluon x log - - - - - - - ( B j o r k en x ) l og - - - - - - - ) g (Gluon x log - - - - - - - ( B j o r k en x ) l og - - - - - - - FIG. 34. The relation between Bjorken- x ( x B ) and gluon- x ( x g ) at event level (left) and average x g values in different x B bins(right) for energy configuration of 18 ×
275 GeV calculated from the PYTHIA v6.4 simulation.
Gluon x ) ( G e V Q g / g A b s . U n c e r t. D
18 GeV x 275 GeV5 GeV x 100 GeV5 GeV x 41 GeV
Polarizations:Int. Luminosity: e: 80%, p: 70% -1 Absolute uncertaintyCharm hadronsGlobal fit
COMPASS data = 160 GeV m FXT: E + x e' + D fi e + p FIG. 35. Projections on ∆ g/g as a function of gluon- x ( x g ) and Q . The position of each data point is according to the meanvalue of x g and Q of the particular bin. The uncertainty for the data points corresponds to the scale shown in the verticalaxis on the right side of the plot. The colored band behind the data set of each beam energy configuration is the uncertaintycalculated using NNPDF unpolarized and polarized PDFs [57, 58]. The red triangle marker shows the existing measurementfrom COMPASS [44]. kinematics, e.g. the moderate- x g region, heavy flavor production may play a unique role. More importantly, this canbe compared to similar measurements of inclusive jet and dijet production [40–43]. All of these studies should becarried out systematically in EIC experiments to answer the nucleon spin puzzle.8
6. Charm Baryon Λ ± c Production and Hadronization
The hadronization process remains a challenging problem that is yet to be understood in QCD. Fragmentationfunctions (FFs) have been widely applied under the collinear factorization and are constrained via experimental datafrom e + + e − or e + p collisions and are expected to be universal and thus directly applicable to hadronic collisions.Many Monte Carlo event generators utilize the same or similar schemes for partons hadronizing into hadrons, e.g.Lund string fragmentation used in the PYTHIA generator.Recently, data from p + p , p + A , and A + A collisions at RHIC and LHC showed that the Λ + c /D ratio is consider-ably larger than the fragmentation baseline [45, 46]. The new color reconnection (CR) scheme implemented in thePYTHIA8 generator [49], together with the baryon-junction scheme, increases the Λ + c /D ratio at low p T and iscomparable to the experimental data. A detailed investigation of the Λ + c production at high-luminosity EIC collisionswill offer an opportunity to enable detailed investigations to understand how the hadronization plays a role from e + e − to hadronic collisions.The Λ + c baryon has an extremely short lifetime with a proper decay length cτ ∼ µ m (a factor of 2 smallerthan D and an order of magnitude smaller than B hadrons). The decay vertex reconstruction of Λ + c will place verystringent requirements on the detector pointing/vertexing/PID capabilities. In this section, we will describe how asilicon tracker can improve the Λ + c signal significance and statistical uncertainties on physics observables at the EIC.The e + p collision events are generated with PYTHIA v6.4 using the EIC tune with 18 ×
275 GeV beam energies,and processed through the fast-simulation framework described in Sec. III A 2 b. Similar to the D simulation, DCAand momentum resolutions based on the parametrizations listed in Table III are used in the fast simulation. Weapply the primary vertex resolution as well as its multiplicity-dependence, and the tracking efficiency, evaluated fromfull GEANT4 simulation, as shown in Fig. 12. We assume that p/K/π tracks can be separated perfectly below themomentum limits listed in Table III. Momentum (GeV/c)) Momentum vs Theta ( = 1 fb +c Λ→ p e ←
18 GeV ×
275 |<3 η ~ 3.5M in | +c Λ −
10 1 10 − η =2 η =3 η ℒ Momentum (GeV/c)) Momentum vs Theta ( = 1 fb c Λ→ p e ←
18 GeV ×
275 |<3 η ~ 3.4M in | c Λ −
10 1 10 − η =2 η =3 η ℒ FIG. 36. Kinematic distributions of Λ + c (Left) and Λ − c (right) from PYTHIA v6.4 with EIC tune in e + p (18 ×
275 GeV) collisionsas a function of momentum and polar angle. The total counts of Λ c within | η | < × at a integrated luminosityof 1 fb − in this calculation. Fig. 36 shows kinematic distributions of produced Λ + c and Λ − c hadrons from PYTHIA v6.4 in e + p ×
275 GeVcollisions as a function of momentum and θ (angle with respect to the beam line) in polar coordinates. There aremore Λ + c produced in the very forward region compared to Λ − c in PYTHIA v6.4, which is due to processes in whichcharm quarks re-combine with the beam remnants. In the central acceptance region, e.g. | η | <
3, the Λ − c / Λ + c ratio isclose to 1.In this simulation study, final state Λ + c (Λ − c ) hadrons are reconstructed via the decays of Λ + c → K − pπ + (and itscharge-conjugate channel), which include one non-resonant channel and three resonant channels [59]. In PYTHIAv6.4, branching ratios (B.R.) for these channels are not up-to-date and one resonant decay channel ( π + Λ(1520))is missing. Table VII compares the branching-ratio values of various channels used in PYTHIA v6.4 and in PDG2020 [59]. In this study, only the non-resonant channel is used for signal reconstruction while the final statistics forthe Λ + c signals are scaled to the total B.R. (6.28%) for the pK − π + channel.We combine all three-track triplets pK − π + and ¯ pK + π − with the right-sign combination for signal reconstruction.If they are not from the decay of the same Λ ± c , the combinations are regarded as background. Figure 37 (left plot)shows a sketch of the Λ c -decay topology, and the right plot shows the multiplicity distributions of identified particles( p/K/π ) in e + p events. Given that the multiplicity of produced particles from such kind of events is small, especially9 Daughterpair DCA D e c a y l e n g t h Primary Vertex Λ ! DCA p 𝜋 K p DCA K DCA 𝜋 DCA 𝜃 Multiplicity0 2 4 6 8 10 12 14 16 18 20 C oun t s p Kp
275 GeV PYTHIA v6.4 · ep 18 FIG. 37. (Left) A schematic cartoon of Λ c -decay topology. (Right) Final-state identified hadron ( p/K/π ) multiplicity distri-butions in e + p (18 ×
275 GeV) collisions from a PYTHIA v6.4 simulation.TABLE VII. Λ + c → pK − π + decay channels including intermediate resonance channels and their branching ratios in PYTHIAv6.4 and PDG 2020. Decay Channel B.R. (PYTHIA 6) B.R. (PDG-2020) pK ∗ → pK − π + × K − ∆(1232) ++ → K − pπ + × π + Λ(1520) → π + pK − N/A 2.20% × K − pπ + for protons, it is expected that the combinatorial background level is lower than p + p collisions at similar energies.Figure 38 shows the normalized distributions of selected topological variables in the − < η < − < η < < η < p T region of 2 < p T < c . The signal and background distributions are different inthese variables, allowing the topological separation to enhance the Λ c -signal reconstruction. A set of loose topologicalcuts is applied to keep a high reconstruction efficiency, which is listed in Table VIII. There is no minimum- p T cutapplied on daughter tracks in the current calculation. The projected invariant-mass distributions of Λ + c signals at indifferent η ranges are shown Fig. 39. TABLE VIII. Decay topology cuts for Λ ± c signal reconstruction.Selection criteria 0
10 GeV/ c Pair-DCA rφ < µ m < µ m < µ mΛ c -DCA rφ < µ m < µ m < µ mDecay-Length rφ > µ m > µ m > µ m Figure 40 shows projected statistical uncertainties of Λ + c /D and Λ − c /D as a function of p T in | η | < η in 2
10 GeV/ c (right) with 10 fb − e + p (18 ×
275 GeV) collisions. Figure 41 shows the projectionsof Λ + c /D as a function of p T for two η regions ( | η | < < η < + c cross section used in Fig. 40 and41 is based on the recent PYTHIA v8.3 calculation which includes the latest development on the color reconnectionscheme for baryon production at high energy p + p collisions. Also shown in Fig. 41 are the existing measurementsin p + p collisions from ALICE [46] and e + p DIS and γp collisions from ZEUS [60, 61]. The projection shows thatmeasurements at EIC e + p DIS collisions would allow us to systematically investigate the Λ c production over a broadkinematic region, which will shed detail insights on charm hadrochemistry and charm-quark hadronization.Recent measurements in open and closed charm hadron production in high multiplicity p + p collisions at RHICand LHC attracted lots of interests [62–64]. Various models including multi-parton interaction, color reconnectionimplemented in PYTHIA, and coherent production [65] have been exercised while the exact production mechanismis still under investigation. Measurements of various charm hadrons including D and Λ + c in e + p /A collisions at theEIC will give us an opportunity to study these non-perturbative features in detail.High statistics will enable to perform the double-spin transfer D LL measurement of Λ c baryon similar to Λ inpolarized e + p collisions. Early prediction on Λ c D LL for polarized p + p collisions at RHIC suggested the sensitivity0 ) f (r q cos0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 A r b . U n i t - -
10 <4.0 GeV/c T h -3.0 10 <4.0 GeV/c T h -1.0 10 <4.0 GeV/c T h ) f (r Pair-DCA0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 A r b . U n i t - - 10 <4.0 GeV/c T h -3.0 10 <4.0 GeV/c T h -1.0 10 <4.0 GeV/c T h FIG. 38. cos θ rφ (upper row) and Pair-DCA rφ (lower row) distributions of signal and background in Λ c reconstruction for − < η < − < η < < η < ) Mass (GeV/c2.2 2.25 2.3 2.35 2.4 C oun t s pe r . M e V × < -1 η -3< <10 GeV/c T +c Λ ) Mass (GeV/c2.2 2.25 2.3 2.35 2.4 C oun t s pe r . M e V × < 1 η -1< <10 GeV/c T +c Λ ) Mass (GeV/c2.2 2.25 2.3 2.35 2.4 C oun t s pe r . M e V × < 3 η 1< <10 GeV/c T +c Λ FIG. 39. Invariant-mass distributions of pK − π + triplets in Λ + c reconstruction in 2 < p T < c from DIS e + p (18 × − < η < − − < η < < η < to the parton spin structure (gluon helicity) inside proton as well as polarized fragmentation function [66]. Whilethe measurement turned out to be challenging at RHIC due to experimental limitations, a high-luminosity EIC willenable the Λ c D LL measurement with the desired instrumentation capability for secondary vertex reconstruction.1 (GeV/c) T p0 1 2 3 4 5 6 7 8 9 10 B a r y on / M e s on R a t i o /D +c L PYTHIA8 D/ -c L PYTHIA8 /D +c L Proj. D/ -c L Proj. 275 GeV · ep 18 -1 L = 10 fb (cid:242) |<3 h | h - - - - B a r y on / M e s on R a t i o /D +c L PYTHIA8 D/ -c L PYTHIA8 /D +c L Proj. D/ -c L Proj. 275 GeV · ep 18 -1 L = 10 fb (cid:242) <10 GeV/c T FIG. 40. Projected statistical uncertainty of the Λ + c /D and Λ − c /D ratios as a function of p T for | η | < η in 2 < p T < 10 GeV/ c (right) in 10 fb − e + p × 275 GeV collisions. The central values for the projected pointsare extracted from PYTHIA v8.3 simulations. (GeV/c) T p / D + c L ep@141 GeV -1 Proj. 10 fbPYTHIA8 ep@141 GeVALICE pp@7 TeV |y|<0.5PYTHIA8 pp@7 TeV |y|<0.5 <3 h 1< |<1 h | > ) T Z E U S D I S ( p | < . h = / G e V , | s > . G e V / c ) T p ( p g Z E U S | < . h W = - G e V , | FIG. 41. The projected statistical uncertainty of Λ + c /D as a function of p T in | η | < < η < e + p × 275 GeVcollisions. The mean values for the projected points are from PYTHIA8.3 calculations. Open circle and open square points arethe measurements in p + p √ s = 7 TeV collisions from ALICE [46] and e + p collisions from ZEUS [60, 61]. 7. Heavy Quark Conclusions We presented detailed simulation studies of heavy flavor measurements, focusing on charm hadron reconstructionusing the all-silicon tracker in electron-proton collisions. Precision measurements must separate charm hadron decayvertices from the collision vertex, which imposes stringent requirements on the tracking/vertexing capabilities for EICexperiments. The tracker concept presented here meets those requirements.Our simulations of the inclusive charm structure function F c ¯ c measurement in various ( x B , Q ) bins are veryencouraging. High luminosity e + p /A collisions at EIC will enable unprecedented precision measurement of F c ¯ c ,covering a x B region of 10 − up to > D D pairs allows study of the gluon TMD functions including the gluon Sivers function inelectron scattering on transversely polarized protons. The linearly polarized Boer-Mulders function in unpolarized e + p collisions is also accessible. With 100 fb − integrated luminosity for 18 × 275 GeV e + p collisions, the projecteduncertainties on (cid:104) A UT (cid:105) and (cid:104) cos(2 φ T )) (cid:105) are 0.57% and 0.4% respectively using D D pair reconstructed via the Kπ channel.Charm hadron double-spin asymmetry ( A LL ) in longitudinally polarized e + p collision provides a unique sensitivityto the gluon helicity ∆ g/g contribution to the proton spin. We evaluated statistical uncertainties of D hadron A LL g/g based on a leading order calculation. Compared to the existing measurement from COMPASS and currentQCD analysis uncertainties, future measurements at the EIC will provide significantly improved precision in A LL ,complementary to inclusive spin-dependent structure function measurements.Λ + c production has attracted significant interest, inspired by recent findings in hadronic collisions at RHIC andthe LHC. We investigated the charm hadron Λ + c reconstruction capability and projected Λ + c physics performance, inparticular the measurement of Λ + c /D over a wide kinematic region. This will open a great opportunity to characterizethe Λ + c production mechanism and gain insights into hadronization, along with using Λ + c as a tool to investigate thestructure of nucleons and nuclei. B. Jets 1. Physics Introduction Partons from initial hard scatterings cannot be observed directly as final-state particles. Instead, they hadronizeinto a directional spray of final-state particles. Consequently, jets are composite objects that relate such final-stateparticles measured in the detector to an initial parton, and so can serve as a powerful tool for probing QCD. Jets aremeasured experimentally by clustering the observed particles using a particular clustering algorithm (anti- k T [67] inthis work) within a chosen jet resolution parameter. Q qA jet e FIG. 42. Leading order deep inelastic scattering diagram. The struck quark is observed as a jet of final state hadrons andserves as an excellent probe of the nucleus. Earlier studies of e + p collisions utilized jets, but in a limited fashion [25]. An extensive jet program has beenproposed for the EIC, as follows: • Jets from electroproduction in DIS can be used to study parton energy loss and interactions in cold nuclearmatter [68]. • Inclusive jet production in polarized electron-proton collisions constrains the helicity-dependent parton distri-bution function (PDF) of the proton at low x , complementary to existing measurements at high- x [40]. • Inclusive jet production in DIS off nuclei [69] and dijet quasi-real photoproduction [70] can be used to advanceour knowledge of nuclear PDFs. • Dijet photoproduction gives access to the photon PDF [71]. • Single-inclusive lepton scattering resulting in jets, where the scattered electron is not observed, has been proposedas an EIC measurement to study transverse spin effects in the nucleon [39, 72].3 • Jets complement measurements of the three-dimensional structure of hadrons, encoded in transverse momentum-dependent (TMD) PDFs and fragmentation functions (TMD FFs). Unlike in the semi-inclusive DIS case, jetmeasurements allow the extraction of these two quantities separately. Specifically, jet measurements at the EIChave been proposed to constrain the quark Sivers function, transversity distribution, and the Collins FF [73]. • Dijet production can be used to access gluon TMD functions at the EIC [37, 38]. • Charm-jet cross section measurements can be used to resolve the tension between different experimental resultsregarding the strangeness content of the nucleon [74]. • Substructure measurements can be used to tune parton-shower event generators and study cold nuclear mattereffects as explored in [75].In this section we study the jet momentum and angular resolutions achievable with the all-silicon tracker describedabove. Then, we will focus on two specific observables, the azimuthal difference between jets produced in DIS andthe scattered electron and the jet fragmentation function, to illustrate physics with jets reconstructed in the silicontracker. 2. Charged Jet Reconstruction Performance Charged jets have been measured extensively in p + p collisions with the ALICE detector at the large hadroncollider [76]. Track-only jets can often offer greater experimental precision, but are traditionally harder to compare totheoretical calculations. However, there has been progress in connecting experimental track-only jet observables withtheoretical studies [77]. To quantify the jet reconstruction performance of the silicon tracker described in section II,electron-proton collisions are simulated with the PYTHIA8 Monte-Carlo generator and a full GEANT4 simulationwith a 1.4 T and 3.0 T solenoidal magnetic field. Jets are reconstructed using the anti- k T algorithm, with a largeresolution parameter of R = 1 . 0. This is feasible due to the relatively low multiplicity of particles produced in e + p collisions. Reconstructed jets are required to have 4 or more constituents and a minimum total energy of 4 . | η | < . 5, according to the acceptance ofthe all-silicon tracker. Reconstructed jets within ∆ R = 0 . p T ≥ 70 MeV/ c , with a higherthreshold depending on η of the constituent. The minimum p T for different η regions is shown in Table IX. The valuesin Table IX are extracted from Ref. [19] and exceed the values discussed in section II. These are based on the needfor three or more traversed barrel layers or disks in the all-silicon tracker in order to determine the track curvaturefor a charged particle and hence its transverse momentum. Jets with constituents that hit the conical supports wherethe central barrel meets the forward and backward disks are omitted. Based on Fig. 2, we take this range to be1 . < | η | < . TABLE IX. Minimum p T -threshold (in MeV/ c ) for charged jet constituents. B field [T] | η | < . . < | η | < . . < | η | < . . < | η | < . . < | η | < . Reconstructed jets are matched to truth-level jets by requiring that one or more reconstructed tracks in the jetoriginate from a truth particle. If this truth particle is a constituent of a truth-level jet, the reconstructed and truthjet are matched. A unique reco-truth match is enforced for the following jet performance studies. Once the jets arematched, the neutral components from the truth-level jets are subtracted to obtain charged truth jets. The 4-momentaof the neutral constituents are subtracted from the particle-level generated jets to obtain charged jet 4-vector: p jet, µ charged = p jet, µ total − p jet, µ neutral . (5)Certain aspects of the original jet will be unaltered by the subtraction; the jet area, for example is not recalculatedaccording to Eq. 5. A negligible difference was found in the jet performance studies using particle-level jets that wereoriginally charged-only, versus particle-level jets where the neutral components are subtracted.Figure 43 shows the momentum response matrix for charged jets passing all criteria, with p T ≥ c . There isa strong correlation between the reconstructed jet momentum, p RecoJet , and the charged truth jet momentum, p Truth , ChJet ,indicated by the prominent diagonal line in the histogram. There are, however, jets that fall outside of this strong4 RecoJet p05101520253035404550 [ G e V / c ] T r u t h , C h J e t p momentum_responseEntries 172777Mean x 9.833Mean y 10.24Std Dev x 6.843Std Dev y 7.395 momentum_responseEntries 172777Mean x 9.833Mean y 10.24Std Dev x 6.843Std Dev y 7.395 Charged Jet Momentum Response RecoJet p05101520253035404550 [ G e V / c ] T r u t h , C h J e t p momentum_responseEntries 97982Mean x 13.7Mean y 14.62Std Dev x 8.539Std Dev y 9.274 momentum_responseEntries 97982Mean x 13.7Mean y 14.62Std Dev x 8.539Std Dev y 9.274 Charged Jet Momentum Response FIG. 43. Charged-jet momentum response for jets with N constituent ≥ e + p events at 20 × 100 GeVcollisions in a 1.4 T (left) and 3.0 T (right) magnetic field. linear correlation. The most important variable impacting the energy and position resolution of jets after all otherselections are made is the number of particles not reconstructed as part of the jet (ie “missing” particles). - - - - - N o . M i ss ed J e t C on s t i t uen t s n_missed_vs_dP_PEntries 201662Mean x 0.03709Mean y 1.106Std Dev x 0.08965Std Dev y 0.9407 n_missed_vs_dP_PEntries 201662Mean x 0.03709Mean y 1.106Std Dev x 0.08965Std Dev y 0.9407 No. Missed Jet Constituents VS. dP/P ( p Truthjet p Recojet )/ p Truthjet C o un t s Jets with N Missed < 1Jets with N Missed FIG. 44. (Left) Number of missed jet constituents ( N truthconstituent − N recoconstituent ) vs. Jet d p/p . (Right) d p/p distribution of jetswith less than 1 missing constituent (blue), and jets with with one or more missing constituents (red). Figure 44 shows the energy resolution of jets vs. the difference between number of truth and reconstructed con-stituents, N Missed = N truth , chconstituent − N recoconstituent . The left panel shows a distribution centered at 0 for jets with no missingconstituents. As the number of missed jet constituents increases, this distribution broadens and shifts towards highervalues of d p/p . The right panel shows the d p/p distribution of two populations of jets: jets with less than 1 missingconstituent and jets with one or more missing constituents. Jets in a magnetic field of 1.4 T (3.0 T) with less than 1missing constituent make up approximately 58% (49%) of jets that pass all other selection criteria and have a narrowdistribution centered at 0 that is well described by a gaussian fit. Jets with one or more missing constituents, however,have a much broader d p/p distribution that is shifted toward higher values. These are best described by a Landaudistribution. Consequently, we characterize the performance of the silicon tracker for these two populations of jetsseparately; gaussian fits to the the combined distributions are dominated by the narrow peak at d p/p ≈ 0, with thedetrimental effect of poorly characterizing the non-gaussian “shoulder” shown in red in Fig. 44.Figure 45 shows the momentum and angular resolutions of jets with no missing constituents. The resolutions areshown in bins of jet momentum and η . The resolutions are extracted by fitting the angular or momentum distributionsin each bin to a Gaussian function, and extracting the standard deviation and its uncertainty from the fit. Figure 46displays the resolutions for jets with one or more missing constituents. The angular resolutions shown in Fig. 46 arecalculated using the same method as in Fig. 45 as well as the method described in section II C for single particles. The5 dp / p [ % ] B = 3.0 T N Missed < 1 d [ m r a d ] p <6.0 GeV/ c p <8.0 GeV/ c p <10.0 GeV/ c p <12.0 GeV/ c p <15.0 GeV/ c p <20.0 GeV/ c d [ m r a d ] p [GeV/ c ] dp / p [ % ] p [GeV/ c ] d [ m r a d ] p [GeV/ c ] d [ m r a d ] FIG. 45. Momentum and angular resolutions of charged jets with no missing constituents, reconstructed with the all-Silicontracker simulated in PYTHIA e + p collisions at 20 × 100 GeV with the 3.0 T magnetic-field configuration. momentum resolution for jets with one or more missing particles is more difficult to describe. The d p/p distributionshown in Fig. 44 in red is best fit with a Landau distribution, where σ is undefined. Therefore, instead of a fit, thesimple numerical standard deviation is taken, and reported for the momentum resolution in Fig. 46. Figures 47 and48 show the resulting resolutions of applying this procedure to jets simulated in a 1.4 T magnetic field. dp / p [ % ] B = 3.0 T N Missed p <6.0 GeV/ c p <8.0 GeV/ c p <10.0 GeV/ c p <12.0 GeV/ c p <15.0 GeV/ c p <20.0 GeV/ c d [ m r a d ] d [ m r a d ] p [GeV/ c ] dp / p [ % ] p [GeV/ c ] d [ m r a d ] p [GeV/ c ] d [ m r a d ] FIG. 46. Momentum and angular resolutions of charged jets with one or more missing constituents, reconstructed with theall-Silicon tracker simulated in PYTHIA e + p collisions at 20 × 100 GeV with the 3.0 T magnetic-field configuration. dp / p [ % ] B = 1.4 T N Missed < 1 d [ m r a d ] d [ m r a d ] p <6.0 GeV/ c p <8.0 GeV/ c p <10.0 GeV/ c p <12.0 GeV/ c p <15.0 GeV/ c p <20.0 GeV/ c p [GeV/ c ] dp / p [ % ] p [GeV/ c ] d [ m r a d ] p [GeV/ c ] d [ m r a d ] FIG. 47. Momentum and angular resolutions of charged jets with no missing constituents, reconstructed with the all-Silicontracker simulated in PYTHIA e + p collisions at 20 × 100 GeV with the 1.4 T magnetic-field configuration. dp / p [ % ] B = 1.4 T N Missed p <6.0 GeV/ c p <8.0 GeV/ c p <10.0 GeV/ c p <12.0 GeV/ c p <15.0 GeV/ c p <20.0 GeV/ c d [ m r a d ] d [ m r a d ] p [GeV/ c ] dp / p [ % ] p [GeV/ c ] d [ m r a d ] p [GeV/ c ] d [ m r a d ] FIG. 48. Momentum and angular resolutions of charged jets with no missing constituents, reconstructed with the all-Silicontracker simulated in PYTHIA e + p collisions at 20 × 100 GeV with the 1.4 T magnetic-field configuration. 3. Jet Observables Figure 49 shows full simulation results for the azimuthal difference between jets and the scattered electron, | ϕ jet − ϕ e − π | in a 1.4 and 3.0 T magnetic field. Jets with N Missing < | | = | jet e | N o r m a li z e d C o un t s | Truth | B = 3.0 T| Reco | B = 3.0 T| Truth | B = 1.4 T| Reco | B = 1.4 T FIG. 49. Azimuthal correlation between the scattered electron and jets in e + p collisions simulated with a 1.4 T and 3.0 Tmagnetic field. Dashed lines display the correlation between particle-level scattered electron and jets. The solid lines displaythe correlation between the reconstructed electron and reconstructed jets. shows a peak at zero as expected from LO DIS where the electron and jet are emitted back-to-back. In the limitthat the transverse momentum imbalance, p e T /p jet T , is much smaller than the electron transverse momentum, thisobservable can provide clean access to the quark TMD PDF and the Sivers effect in transversely polarized scatteringsin e + p collisions [78].Figure 50 shows the particle-level and reconstructed-level charged jet fragmentation functions in the two magneticfields. Measurements of the charged-jet fragmentation function in e + p should provide sensitivity to the process ofhadron formation. Comparisons of charged jet fragmentation functions measured in e + p and e +A collisions canelucidate the effects of nuclear matter on the fragmentation process, and yield information on parton transport in anuclear medium. For lower energy jets, which begin to fragment inside a nucleus in e + A collisions, we can use thenucleus as a filter to probe hadronization.The choice of magnetic field is shown to have little effect on both observables shown. While the different magneticfields result in different angular and momentum resolutions, that will in turn affect the the azimuthal correlation andfragmentation function measurements, the magnitude of these changes is quite small. For example, the momentumresolution for reconstructed jets with less than 1 missing constituent is approximately between 0.3% and 0.45% in a3.0 T magnetic field. In a 1.4 T magnetic field, the momentum resolution ranges between 0.5% and 0.75%. This changein the momentum resolution of a few fractions of a percent is not expected to significantly impact the fragmentationfunction measurements, with similar reasoning applying to the differences in angular resolution and their effect onthe electron-jet correlation measurements. C. Exclusive Vector Mesons 1. Physics Introduction Exclusive production of vector mesons is an important channel for imaging light and heavy nuclei. The overallproduction cross section is directly sensitive to the gluon density in the target. The Good-Walker paradigm relatesthe cross sections for coherent and incoherent photoproduction to the spatial distribution of gluons in the nucleus andto event-by-event fluctuations in the nuclear configuration respectively. The Q evolution of these cross sections canprovide a detailed picture of the nuclei over a range of length scales.Exclusive vector-meson production occurs when an incident photon fluctuates to a qq pair which then scatters8 z = p constituent / p jet / N j e t s d N / d z Charged Truth Jets B = 3.0 TReconstructed Jets B = 3.0 TCharged Truth Jets B = 1.4 TReconstructed Jets B = 1.4 T FIG. 50. Truth-level (dashed lines) and reconstructed-level (solid lines) charged jet fragmentation as a function of z = p constituent /p jet in 1.4 T and 3.0 T magnetic fields. elastically from the nuclear target, emerging as a real vector meson. Vector mesons are color singlets, so the scatteringmust involve at least two gluons and is usually described in terms of Pomeron exchange. The Pomeron has the samequantum numbers as the vacuum, so this scattering is elastic. At lower photon energies, Reggeon exchange may alsocontribute; Reggeons represent meson trajectories, so are mostly quarks, so can transfer a wider range of quantumnumbers, including charge.The exact meaning of exclusivity will depend on the analysis under consideration. In incoherent photoproduction,the breakup of the nuclear target produces additional particles. These are generally in the far-forward region, soshould not cause confusion. There can also be parton radiation from the photon before it interacts with the Pomeron;this is the resolved component of the photon. Because of the constraints of spin and color neutrality, this involvesthe gluonic component of the photon; the radiation is smaller for vector meson final states than for other processessuch as jets and open heavy flavor production [79]. Good detector acceptance is required to be able to separate thesenon-exclusive resolved processes from direct production.For the EIC, the greatest interest is to use high-energy photoproduction via Pomeron exchange to probe the gluondistributions in nuclear targets. Because the Pomeron involves the exchange of at least two gluons, the relationshipbetween cross section and the spatial dependence of the gluons in a nucleus is determined in a manner similar to thatused to probe GPDs in a proton [80–83]. The two-dimensional Fourier transform of dσ/dt , where t is the squaredfour-momentum transfer from the target, gives the two-dimensional (transverse to the photon direction) density ofinteraction sites within the nucleus in the infinite-momentum frame: dσdt ∝ (cid:90) t max p T dp T J ( bp T ) (cid:114) dσ coh dt , (6)where b is the impact parameter of the struck parton within the nucleus, and J is a Bessel function.The incoherent cross section on proton and nuclear targets is sensitive to event-by-event fluctuations in the nucleartarget, due to variations in the nucleon positions (for A > dσ/dt for incoherent production. Fluctuations at smalldistance scales are related to the cross section at large | t | . The incoherent cross section should also evolve with collisionenergy [85]. At low energies, the incoherent cross section should rise with increasing collision energies. However, inthe ultra-high energy limit, the nucleus will look like a black disk, with no event-by-event fluctuations. So, as thecollision energy rises, the incoherent cross section should reach a maximum and then decrease [85, 86]. The energy atwhich the cross section is a maximum depends on the vector-meson mass. For the ρ , it might be within the range ofthe EIC. It would manifest itself as a rapidity-dependent variation of the ratio of the incoherent to the coherent cross9section. Although the effects of gluonic hot spots may be somewhat washed out in e + A collisions, they may still bevisible at larger | t | , or in lighter nuclei.The main kinematics variables for exclusive vector-meson production are the x and Q of the struck gluons. Thekinematics is dominated by the case where the two gluons have very different x values [87, 88] and the large- x gluondominates the momentum transfer; the other gluon is often treated as a spectator. Then, the Bjorken- x can bedetermined from the rapidity y of the vector meson. For photoproduction [89], away from threshold, x = M V γm p exp ( − y ) , (7)where M V is the vector-meson mass, m p is the proton mass, and γ is the Lorentz boost of the ion. Large Q willshift these reactions slightly [90]. The reaction Q comes from the hard scales determined by the photon Q and thevector meson mass: Q = ( Q γ + M V ) / . (8)To study production at low and moderate Q , where phenomena like the colored-glass condensate are most visible,lighter mesons are required.Equation 7 highlights a key requirement for a tracking detector at an EIC: wide rapidity coverage, to cover a widerange in x , up to the kinematic limits. Eq. 7 fails near threshold, i.e. as x → 1, because it neglects the proton mass,but more detailed calculations show that, for 18 GeV electrons on 275 GeV protons, the high- x limit is near y = 4.For coherent photoproduction on ions, the maximum x is about 0.03, since coherence must be maintained over acoherence distance l c = 2 k/ ( M V + Q ) which is larger than the nuclear diameter. This limits coherent productionto y < 2, while incoherent production extends up to y ≈ 3; the maximum is lower than for protons because of thelower per-nucleon ion energy. For ions, Fermi momentum leads to incoherent production with x > 1. This is anactive area of study at Jefferson lab [91]. The large x / large | y | region is also critical for studying meson productionvia Reggeon exchange [92]; Reggeon exchange allows for a much wider range of final states (including exotica) thanphoton-Pomeron fusion.The relationship between x and y depends on the ion-beam energy, so it is possible to shift the rapidity of a desired x value by changing the ion energy. This may be of value for studying near-threshold photoproduction, where the lowphoton energies correspond to forward production. This is important for studying production via 3-gluon exchange,and for searches for pentaquarks or similar exotica [92].The minimum x corresponds to the maximum photon energy, which we will take to be the electron beam energy,18 GeV, even though the photon flux is drops rapidly near threshold. For both e + p and e + A collisions this appearsnear y = − 4. Because of the different per-nucleon beam energies, this corresponds to x values of about 6 × − and10 − respectively (at Q ≈ M V / Q > , the rapidity values are slightly shifted.Other detector requirements are more channel specific. To address them, we use events simulated using the eS-TARlight Monte Carlo generator [90]. eSTARlight simulates the production of a variety of vector mesons. It uses e + p cross sections based on parametrized HERA and fixed-target data, including their Q dependence. e + A crosssections are calculated using the e + p cross sections and a quantum Glauber calculation. The angular distributions ofthe decays are also based on HERA data, with s -channel helicity conservation holding for photoproduction, but witha rising fraction of longitudinally-polarized photons as the Q rises.The detector simulations were done for the detector described above, using a GEANT3 model, using the EICROOTsimulation framework. Simulations were done with uniform magnetic fields of 1.5 and 3.0 T.We consider four mesons: the ρ , φ , the J/ψ and the Υ family. Each illustrates different aspects of the detectorrequirements. The ψ (cid:48) has also been studied, but present no specific problems, so we skip them here. These allproduce simple final states, with two leptons or charged mesons from the vector-meson decay, plus the scatteredelectron and the scattered and/or dissociated nuclear target. An intact target is only marginally scattered, so, exceptfor very light nuclei is not detectable. If the target dissociates, it may leave remnants which will be visible in thefar-forward detectors. Observing these remnants is critical for determining if an event was coherent or incoherentphotoproduction. These remnants typically have rapidity near the beam rapidity, so will be studied with a set offar-forward detectors that observe charged and neutral particles with rapidity above 5 [93]. ρ production The ρ is the lightest vector meson, so it allows studies down to the lowest possible x values, albeit at low Q , andat a cost of a somewhat more complicated wave function than the φ . However, it is experimentally much simpler, andwill be of value in regions where the φ may not be detectable.0 FIG. 51. (left) Rapidity distributions for ρ photoproduction at an EIC and (right) pseudorapidity distributions of the daughter π ± from ρ decay for four different x B ranges; in all cases, the highest range is at the left. The top two panels are for collisionsof 18 GeV electrons on 275 GeV protons, while the bottom are for 18 GeV electrons on 100 GeV/n Au nuclei. In both plots,the distributions are broken up based on the x B value of the struck parton, with the largest x B on the right sides of the plots. The left panels of Fig. 51 shows the rapidity distribution for ρ photoproduction in e + A and e + p collisions. Thedistribution is broken up to show subsamples with different x values, clearly showing the relationship between rapidityand x . ρ photoproduction in e + A occurs from rapidity -4 to 2, while the e + p distribution extends from rapidity -4to 4.The right panels show the pseudorapidity distributions of the daughter pions for the same subsamples in x . Mostof the daughter pions are distributed within 1 unit in pseudorapidity of the rapidity of the parent ρ . As a rough ruleof thumb the acceptance in pseudorapidity should be one unit wider than the rapidity of the produced ρ . This arguesfor a pseudorapidity coverage that extends to ± 5. This extended coverage is also important in measurements of the J/ψ polarization, needed to separate the transverse and longitudinal components of the γp or γA cross section [94].Unfortunately, this broad pseudorapidity coverage is precluded by the EIC beampipe design, because the non-zerocrossing angle limits how close one can come to the forward direction. It may be possible to instrument some areas atlarger rapidity, but with incomplete azimuthal coverage, by taking advantage of areas above and below the beampipe;because the crossing angle occurs in the horizontal plane, space is more constricted horizontally than vertically.One way to expand the coverage at large x would be to run at a lower ion beam energy. This will shift the rapiditydistribution, with the x → φ production The φ is the lighter of the two mesons highlighted in the 2012 EIC White Paper [1]. It decays to K + K − Q valueof the decay. The charged kaons are produced with momenta of only 135 MeV/c in the φ rest frame. For an at-rest φ , the kaon velocities are only v ≈ . c , and so they have rather large specific energy loss and are easily stopped.Unless the φ are Lorentz boosted, either longitudinally due to being produced away from y = 0, or because they areproduced at large Q , leading to a large p T , the decays may not be visible. FIG. 52. (top) The detection efficiency for φ → K + K − in ep collisions in (left) a 1.5 T magnetic field and (right) a 3.0 Tmagnetic field, as a function of φ rapidity and p T . The z axis is efficiency, from 0 to 1. The efficiencies at a given p T andrapidity would be very similar for eA collisions. (bottom left) The p T distribution for coherent φ production in e + p (blue) and eAu (red) collisions at an EIC, normalized to contain the same number of events. The eAu production is at much lower p T because of the larger size of the target. This plot is for all Q , but is dominated by photoproduction, with Q near 0. (bottomright) Scatter plot showing the relationship between φ rapidity and kaon daughter pseudorapidity. A kaon pseudorapidity range | η | < | y | < Figure 52 (bottom left) shows the p T spectra for coherently-produced φ in e + p and e + A collisions, again simulatedin eSTARlight. This is for head-on collisions, with no crossing angle. For e + A collisions, the bulk of the productionis at p T < 100 MeV/c, while for e + p , most of the production is in the 100 to 750 MeV/c range.The two top panels show the φ reconstruction efficiency for the all-silicon detector in 1.5 and 3.0 T fields, as a2function of p T and rapidity. The efficiency is uniformly lower in the 3.0 T field, likely because the tracks curl up moretightly. For both, the efficiency generally decreases as p T decreases, because the kaons are too soft to be reconstructed.The low efficiency near y = 0 is because of the very low kaon momentum; at larger | y | , the kaon momenta are boosted;the higher velocity reduces the kaon specific energy loss, dE/dx , so the kaons can more easily penetrate the beampipeand silicon layers, even with the larger column density due to the angle of incidence. The y ≈ e + A p T range for both magnetic fields, while at 3.0 T, it severely affects most production in e + p collisions. For electroproduction, the p T range is higher, but, especially at 3.0 T, the efficiency is likely to be reducedfor electroproduction as well as photoproduction. It would be very difficult to design a detector with better acceptancein this region, due to the required beampipe thickness and large kaon dE/dx . The efficiency also drops at very large | y | and low-to-modest p T , because, as can be seen in Fig. 52 (left), a pseudorapidity acceptance out to | η | < φ rapidity | y | < φ → K L K S (34.0% branching ratio) and l + l − (with branching ratio 3 × − each for ee and µµ ) seem unattractive, due respectively to the difficulty in reconstructing the K L and the low branching ratio. J/ψ production FIG. 53. The reconstructed J/ψ → e + e − mass peak for the all-silicon detector, in (left) 1.5 T and (right) 3.0 T magnetic fields,for e + A collisions with an integrated luminosity of 10 fb − /A. Although most of the peak is well fit by a Gaussian, The highermagnetic field improves the resolution by almost a factor of two. There are also significant shoulders outside the Gaussian,from less well reconstructed events. The shoulder is larger on the low-mass side, because of electron bremsstrahlung in thedetector material. With the higher field, the resolution (sigma) is more than a factor of 2 better, allowing better resolution ofthe peak. The Gaussian fits were done only to the central part of the peak, as shown by the solid part of the red curves. The J/ψ is the other meson highlighted in the 2012 EIC White Paper [1]. It can be reconstructed from its decaysinto either the µ + µ − or e + e − final state. Although the J/ψ reconstruction is straightforward, bremsstrahlung fromelectrons can produce a low-mass shoulder in the e + e − mass spectrum if the detector is too thick. Figure 53 shows thereconstructed J/ψ spectrum expected from the all-silicon detector for 1.5 and 3.0 T magnetic fields. The peaks werefitted to a Gaussian function, with resolutions for the two fields of 22 MeV/ c and 13 MeV/ c respectively. Althoughthe Gaussian describes the peak well, there significant shoulders are visible. The shoulders have two components:events that are less well reconstructed, and events where the electron underwent bremsstrahlung in the beampipe or3detector material. The latter only contributes to the low-mass shoulder. The bremsstrahlung contribution should bethe same for the two fields.Although a low-mass shoulder is visible in both spectra, the peaks stand out clearly. The Gaussians are fitted to thedata above the J/ψ peak, and the portion of the lower mass data where the peak is above the shoulder, as indicatedby the darker black line.The origin of the shoulders is demonstrated in Fig. 53, which shows the dilepton mass M ee vs. p T . In additionto the pileup around M J/ψ , there is a clear diagonal band with lower M ee but higher p T . This band is expected dueto bremsstrahlung. If one of the electrons radiates a photon and loses energy while traversing the detector, the pairwill be reconstructed with lower pair mass, but higher p T [95]. These events can be rejected by cuts on pair p T and M ee , or with a photon veto in the calorimeter. There is a further unsimulated source for these events, the decay J/ψ → e + e − γ . For photon energies above 100 MeV, the branching ratio for this channel is 0.88% [96], or 15% of therate to e + e − . FIG. 54. A scatter plot showing the relationship between the reconstructed J/ψ → e + e − mass and transverse momentum in e + p collisions in a 3.0 T field. The vast majority of events are reconstructed with M ee ≈ M J/ψ , with a p T distribution expectedfor coherent photoproduction. There is also a clear diagonal band extending to lower M ee but higher p T . In these events, oneof the leptons radiated a photon in the detector material, leading to the lower pair invariant mass and higher p T . 5. The Υ family The three Υ states are relatively heavy, ∼ 10 GeV/ c , but with rather small mass splittings - 563 MeV between thefirst and second, and 331 MeV between the second and third. Good momentum resolution is required to effectivelyseparate the three states. Figure 55 shows the e + e − mass spectrum expected from the three Upsilon states, in twodifferent rapidity ranges. Although the Bjorken- x ranges are different for positive and negative rapidity, the detectorresolution should be similar at + y and − y . Table X shows the resolutions extracted from a Gaussian fit to the Υ(1 S )peaks. The resolution is about 40% better in the 3.0 T field and in both cases worsens by about 20% at larger | y | .Nonetheless, either magnetic field option provides adequate separation over the full range in y . 6. Vector meson conclusions A full EIC program will include studies of the vector mesons considered here along with other vector mesons,likely including the ρ (cid:48) and other excited states. In most cases, the proposed silicon detector exceeds the requirements4 FIG. 55. The reconstructed e + e − mass spectrum in the Upsilon region, with the Υ(1 S ), Υ(2 S ) Υ(3 S ) peaks, in the ratiopredicted by eSTARlight, in 10 fb − of data, in (left) 1.5 T and (right) 3.0 T fields. The spectra are divided into two rapidityranges, | y | < | y | > 1. Because the Upsilons are heavy, most of the production occurs in | y | < S ) peak shown in Fig. 55.Rapidity 1.5 T field 3.0 T field | y | < c 40 MeV/ c | y | > c 51 MeV/ c for vector meson reconstruction. There are two problematic areas - inadequate coverage in pseudorapidity to coverlight vector meson over the full range of Bjorken- x , and problematic reconstruction of very soft kaons from φ decays.Rapidity is closely linked to Bjorken- x , so limitations in rapidity will reduce acceptance at large and small x . The large x limitation is problematic both for parton measurements, and for searches for exotica, including the XYZ states [92]and pentaquarks, and for backward production of mesons. However, the pseudorapidity coverage of tracking detectorsis largely limited by the positioning of the beampipe. IV. SUMMARY AND CONCLUSIONS The EIC will be a high-luminosity, variable-energy collider with a broad and compelling physics program. Com-pleting this program will require detectors capable of making precision measurements of many physics channels over awide kinematic range. We have presented a design for an all-silicon EIC tracking detector and shown that it providestracking with high resolution and good vertex reconstruction over a large kinematic range, enabling the EIC’s broadphysics. We have also performed physics simulations for a variety of reactions for which tracking is critical, includingmeasurements of heavy quarks, jets, and exclusive vector mesons. In most cases, the proposed all-silicon trackerprovides the necessary performance, and we have identified cases where improved performance could enhance thekinematic coverage or enable new measurements.Our simulations have demonstrated the feasibility of a broad program of heavy-quark studies that allow for measure-ments of gluonic PDFs, TMDs, and helicity distributions in nucleons and nuclei, as well as cold nuclear matter effects(see Sec. III A). The proposed tracking system based on the ultra-thin (0.3% X per layer) and fine-pitch (10 × µ m )MAPS sensor technology provides the momentum resolution as well as vertex reconstruction performance necessaryto enable these studies, with potential for improved sensitivity in low momentum and/or charm baryon reconstructionwith further reduced detector thickness.The charged-jet energy and angular resolution performance of the all-silicon tracker was studied. Jets can be5reconstructed with a resolution parameter R = 1 . 0, as the multiplicity in the collisions is small. While there is asignificant resolution loss when jet constituents are not all properly included in charged jets, the charged jet resolutionswith the silicon tracker are nevertheless encouraging. Finally, we studied azimuthal differences between jets and thescattered electron, which can provide access to parton transport in nuclear matter, the quark TMD PDF and theSivers effect in transversely polarized e-p collisions, and the charged-jet fragmentation function, which should besensitive to the hadronization process.The silicon detector can likewise reconstruct important vector meson decays. Our simulations considered recon-structions of ρ , φ , J/ψ and Υ decays, but these are representative of other vector mesons decays, including the ψ (cid:48) and ρ ∗ decays. The momentum resolution will allow us to cleanly separate the different Υ states in the dileptonspectrum. The major limitation of the proposed tracker is the limited pseudorapidity coverage. Broad coverage isimportant to reconstruct the photoproduction and electroproduction over the full range of Bjorken − x . Unfortunately,this limitation appears to come primarily from the current beampipe design; the non-zero beam crossing angle leadsto a “X” shaped interaction region, limiting the length of the free (for detector) region in | z | , thereby limiting theacceptance of any tracking detector design. A secondary consideration is that the detector should have low mass, tominimize bremsstrahlung in dielectron events.For D , Λ c and most quarkonia reconstruction, a stronger magnetic field choice would be preferred as it enablesbetter momentum and therefore mass resolution for these resonances. However, the impact on the low p T thresholddue to the increased magnetic field strength is much less than the gain in signal significance due to the better massresolution. For D mesons, the p T -integrated signal significance is improved by ∼ 50% comparing 3.0 T vs. 1.5 Tmagnetic field setting (See Sec. III A 2 b). For D ∗ + → D π + reconstruction, the p T threshold for the soft pion maycause acceptance loss for low p T D ∗ + mesons. Our initial study shows reconstruction using fewer tracking layers forthe soft pions enables D ∗ + to be still a viable channel in the 3.0 T magnetic field configuration (See Sec. III A 2 b).The efficiency for φ → K + K − is significantly higher for the 1.5 T field than at 3.0 T, because the higher field causesthe tracks to curl up more tightly.One limitation, common to all of these analyses is that the Bjorken- x of the struck parton is strongly correlatedwith the final state rapidity, with interactions at very large or very small x corresponding to large | y | , where detectoracceptance may be limited by the interaction region geometry. This, along with other issues identified in the simula-tions presented here, will require further examination of the tracking system or overall spectrometer design to see ifmodifications or optimizations can improve these measurements.The all-silicon tracker geometry presented in this document will be revised as details of the EIC overall detectorare established and silicon-pixel R&D efforts progress, e.g. optimizing an asymmetric tracker geometry in case thenominal interaction point is shifted away from (0 , , φ ≈ z axis aligned along thehadron beampipe), the field integral is smaller, and the momentum resolution degrades. Conversely, for tracks with φ ≈ π the field integral is larger, and the momentum resolution improves. This effect is more significant at higherpseudorapidities and hadron momenta. 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