Simulation of measured DVCS cross-section on STAR Detector

Simulation of measured DVCS cross-section on STAR Detector XXX October 0, 01 Abstract Cross-section measurements of DVCS process in a proposed electron ion collider (EIC at RHIC are simulated. The acceptance of recent STAR detector have been taken into account. Results show the potential of doing DVCS at EIC with STAR Detector. 1 Introduction Exclusive measurements of electron and proton scattering processes have been taken at the Hadron Electron Ring Accelerator (HERA and the Thomas Jefferson National Accelerator Facility (JLAB lately in the last decades. The physics significance of doing these measurements are to navigate the structure of nucleon, which is described by the Parton Model [1]. While inclusive electron nucleon process (Deeply Inelastic Scattering [1] delivers the information of Parton Distribution Functions (PDFs and semi-inclusive process delivers the Transverse Momentum Distributions (TMDs of hadrons, the General Parton Distributions (GPDs can be obtained by measuring hard exclusive process [1,]. In addition to only containing the information of the number density of partons with longitudinal momentum fraction x in a fast-moving proton by PDFs and the transverse image of the nucleon by TMDs, GPDs will also give the orbital motion and angular momentum distribution (by a Fourier Transformation of GPDs of quarks in a nucleon. In specific, GPDs (denoted as H and E [,] and will be explained later in Section will help us answer the question how spins of quarks (S q and angular momentums of quarks (L q contribute the total spin of a nucleon. By Ji s sum rule [], we have: J q = 1 dxx[h + E] = Lq + Sq (1 where L q and S q are the angular momentums and spins for quarks. This is the main motivation of studying GPDs. One important example of hard exclusive e+p scattering process is Deeply Virtual Compton Scattering (DVCS, which will gives a clean measure channel to GPDs. The measurement of DVCS process (Fig.1 left always interface with Bethe-Heitler (BH process (Fig.1 middle/right, which has the same initial and final states as DVCS. A new proposed experiment on Relativistic Heavy Ion Collider (RIHC at Brookhaven National Laboratory (BNL has the ability of further 1

explore those processes than those experiments have done in HERA and JLAB by reaching a larger range of kinematics and wide sort of beam polarizations []. This work will merely focus on simulation of DVCS unpolarized cross-section measurements in that proposed Electron Ion Collider (EIC at RHIC with consideration of the Solenoid Tracker At RHIC (STAR detector s acceptance and event reconstruction process of those events. The results will show an expected availability of measuring DVCS unpolarized cross-section of today s version of STAR. It is necessary to mention that other measurements of DVCS, such as charge asymmetry and transverse polarization asymmetry can be obtained in EIC as well [,]. However, this report only concentrate on the unpolarized cross-section. The theory of kinematics and unpolarized beam cross-section of DVCS will be introduced briefly in Section. Section and Section will contain information of the Monte Carlo generation tool and the STAR detector kinematics acceptance respectively. And finally, results will be presented and commented in Section. Figure 1: Amplitudes contributing to the photon lepton prduction cross section of QED: the DVCS amplitude (left while the remaining two diagrams (middle and right represent the Bethe-Heitler amplitudes. cross-section of deeply virtual Compton scattering Compton scattering is a classical experiment of scattering off a proton and this experiment has been well studied over almost 0 years. Deeply Virtual Compton Scattering, an exclusive channel of e+p measurement as mentioned before, is more complicated than the classic Compton scatter process. In DVCS, Proton scatters off a photon in a QED process by absorbing a virtual photon at the beginning and emit a real photon in the final state. As the virtual photon is normally provided by a high energy electron beam, DVCS only can be studied with high energy colliders. Now, let us look at the kinematics of this process.

.1 DVCS kinematics The kinematics related to the cross-section measurement can be seen in Fig. Where, k and k denote the momentum of the electron before and after the collision and p andp denote the momentum of the proton before and after collision. q is the momentum of the virtual photon emitted by the high energy electron beam. In the collider frame, a proton with energy E p and an election with energy E q collide head on. The mass of proton and electron are neglected. Figure : DVCS kinematics The Lorentz invariants are listed in the following: The squared e+p collision energy s = (p + k = E p E e The squared momentum transfer to the electron Q = q = (k k, equal to the virtuality of the exchanged photon. The Bjorken variable x B = Q /p q, often simply denoted by x. It determines the momentum fraction of the parton on which the photon scatters. A big beam energy always give a small x. x + ξ and x ξ are longitudinal parton momentum fractions with respect to the average proton momentum (p + p / before and after the scattering, as shown in Fig. Whereas x is integrated over in the scattering amplitude, ξ is fixed by the process kinematics. For DVCS one has ξ = x B /( x B. The squared momentum transfer to the proton t = (p p. The inelasticity y = (qṗ/(kṗ, thus y can be also expressed as Q /xs. Q, x and s are related by Q = xys. The available phase space is often represented in the plane of x and Q. For a given e+p collision energy, lines of constant y are then lines with a slope of degrees in a double logarithmic x-q plot. These kind of plot can be seen in Section.

Even for unpolarized patons, one has a nontrivial spin structure, parameterized by two functions for each parton type. H(x, ξ, t is relevant for the case where the helicity of the proton is the same before and after the scattering, whereas E(x, ξ, t describes a proton helicity flip. H(x, ξ, t and E(x, ξ, t are defined as General Parton Distributions. In the kinematics limit of H(x, 0, 0 and E(x, 0, 0, the GPD just reduced to the PDFs, which can be extracted from DIS process. [1,]. DVCS cross-section In a measurement of DVCS cross-section, one should always extract the DVCS cross-section with Bethe-Heitler process amplitude. The Bethe-Heitler process, as shown in Fig 1, has the same initial and final states e + N e + N + γ and should be treated as the background of DVCS measurement. We have the total cross-section: dσ T OT (x B, t, Q y dt = dσdv CS (x B, t, Q y dt ± dσint (x B, t, Q y dt + dσbh (x B, t, Q y dt ( where, σ int means the interference term of DVCS and Bethe Heitler amplitudes. While Bethe- Heitler process cross-section is determined by a normal QED calculation and the DVCS crosssection will related to the GPDs, the interference term is quite a trouble to deal with. This term will have different approximations in difference kinematics. To make the physics significance of DVCS measurement more clearly, a little bit more about theory of DVCS cross-section will be given here to show how it related to the GPDs, although the extraction of GPDs from DVCS cross-section is not going to be covered in this study. The leading order calculation of DVCS cross-section explicitly involved in the variable R defined as the ratio of the imaginary part for the amplitudes for the DIS and DVCS processes. This model with R = 0. agree with the HERA data very well. However, even if this effective LO prediction reproduces correctly the experimental data, it is not sufficient as it does not provide a direct insight of the rich information present in GPDs. From conclusion of theorists [,, 6], the General Parton Distribution (H and E evolved in the next to leading order of the differential cross-section of DVCS process. GPDs are denoted here as H(x, ξ = η, t, Q and E(x, ξ = η, t, Q for the unpolarized beam case as explained in Section.1. DVCS cross-sections can be exactly evaluated in terms of the helicity CFFs [6]. To express them in terms of GPDs in a systematically improvable manner, it is may be appropriate to utilize a conventionally defined GPD-inspired CFFs basis, for H and E introduced in the last section, we have the CFFs H and E respect to H and E. This CFFs are related with the GPDs by the following equation [,6]: H (ξ, Q, t = 1 e i [ 1 x/ξ iɛ ± {ξ ξ}]h i(x, ξ, Q, tdx ( 1 Here, summation is over all quarks and thus the subscript i equals to different flavors of quarks. GPDs will be extracted from DVCS cross-section in this way.

. EIC and estar A design for a future Electron Ion Collider (EIC has evolved. At BNL, the erhic design utilizes a new electron beam facility to be build inside the RHIC tunnel to achieve the e+p collide. EIC is proposed to be able to reach different beam energies for electron and proton with x0 GeV, x0 GeV, x0 GeV and 0x0 GeV and this means it allows us to detect DVCS process in a small x B region. From the definition of kinematic variables in section.1, it is easy to conclude the kinematic coverage of EIC. In Fig, one can see the expected distribution of DVCS events in kinematic bins of x and Q of EIC []. Where in Fig, the number of events correspond to Compton scattering, i.e. the contribution of the Bethe-Heitler process has been subtracted. Figure : Expected distribution of DVCS events in bins of x and Q. Events numbers correspond to Compton scattering, i.ee. the contribution of the Bethe-Heitler process to the process ep epγ has been subtracted. Eligibility of subtraction of BH events in EIC kinematics come from that the interference term is negligible in the EIC kinematics, i.e.: dσ INT dσ BH + dσ DV CS ( is very small at small x B region. The details of the derivation of this equation can be seen in reference []. Now the intricate interference term in the expression total cross-section (equation. is ignored. MILOU MILOU is a events generation code according to GPD models that give a good description to the current experimental data [7]. From the GPDs, the cross section for DVCS and BH- DVCS interference are calculated according to the formula of reference [7]. During the iterative

integration step, the code provides an output with the estimated cross-section and the accuracy of the integration. In this project, MILOU is used with the following criteria: The generating kinematics range, according to Figure, are from to 1.0 for x B and from 0 to 0 for Q. This whole kinematic region is then divided logarithmically into 00 bins. Although MILOU have the ability of calculate all of BH, DVCS, interference and total cross-section, only Bethe-Heitler and DVCS events were generated. At the end of day, BH events will be counted as background. For a fully exclusive DVCS process, all the kinematic variable listed in Section.1 should be considered. Besides x and Q, t and y sere set according to the acceptance of the STAR Detector and the acceptance will be describe in details in the next section. Here, just set t range to be [0.00, 1.] and y range to be [0.01, 0.8]. Based on the purpose of extracting GPDs for future work, the cross-sections were calculated to the next to leading order (NLO. For each kinematic bin, 000000 events were generated. The t dependence of the DVCS cross-section had been chosen to an exponential form: dσ dt = exp(b(q t and B(Q was fixed to be.6 in this study. Beam energies included x0 GeV, x0 GeV, x0 GeV and 0x0 GeV. STAR detector cuts To measure truly exclusive processes, it is essential to detect all final state particles. Protons are measured by Roman Pots. The most challenge part of the proton measurement is that the recoil proton can hardly be measured. The STAR Detector can measure the proton energy above 17MeV and this is the reason why t is cut at 0.0 GeV in MILOU. The relation between the proton recoil angle to the detecting energy is that the less the recoil angle is, the less the energy the proton carries. y is measured using a hadronic method and depends on the sum over the energy minus the longitudinal momentum of all the hadronic final-state particles. Electrons are detected in both calorimeter (EMC and tracker (TRK. In this project, the expected method of measurement will be considered as either EMC or TRK whichever has a better energy resolution. A plot on the pseudo-rapidity gaps on x and Q plane of the STAR Detector for beam energy x0gev and 0x0GeV can be seem in Fig, [], from which we can see that the at some pseudo-rapidity range, the detector have no acceptance. Cuts also appear in the detecting energy of electrons and photons: only electrons and photons greater than 1GeV can be detected. In conclusion, the acceptance cuts are following: 6

Figure : Expected distribution of detected DVCS at STAR Detector with consideration of pseudo-rapidity cuts. No coverage at pseudo-rapidity: η: < η < 1, < η <.6 E γ and E e should be greater than 1GeV. As already fixed in the generation part, 0.0 < t < 1., 0.01 < y < 0.8. Simulation results All the cross-sections in different kinematics bins and beam energies described in section were generated and calculated to their measured value. Only the results for beam energy x0 GeV and 0x0 GeV were plotted. In the following figures Fig to Fig 0. You can see all crosssection vs t diagram. The expectation cross-sections were calculated not only with the generated events number and acceptance, but also with a 90 percent detector efficiency. The statistic errors were estimated with respect to the number of accepted events rather than generated events. percent of BH cross-section, percent of systematic error and 90 percent of efficiency were added to the total systematic error. The luminosity was assumed 00 pico barn and lepton method was applied for event reconstruction. From the cross-sections, one can conclude first of all, the kinematic coverage of EIC with consideration of acceptance of STAR Detector. For some bins, there are no acceptance cross-sections after the application of those kinematic cuts described in Section, although there supposed to have generated cross-sections. Those bins which have no acceptance match those pseudo-rapidity gaps showed in figure. Secondly, for those bins who have low acceptance, the error bars are very big because errors proportional to the inverse square root of the number of accepted events. Note that, the bins which have infinite error bars imply that there are no acceptance of the 7

detector and those bins are not included in the graphs. Further more, conclusion can be made that at large Q and x value, the events are all have low acceptance than those for low Q and x. These errors are the contributed by both low acceptance of the detector and high BH background. Last but not least, the errors on very low t and very high t region are big in almost every kinematic bins. One can see that in the following plots, those values with big errors were plotted in values multiple times of their original value. This was did in two reasons: 1, as it makes the plots looks not nice when the range of error exceed the expectation value, amplifying the expectation value will ensure that the expectation values will always appear much larger that the errors in the graphs. this will indicate which bins have good acceptance and which have not by just looking at if the expectation values were amplified. In general, the more times the cross-section was amplified, the lower acceptance it has. 8

Measured cross-section (x0gev 6 Kinematics for 1.0<Q <1.78GeV 0.0018<x<0.001 0.001<x<0.0098 0.0098<x<0.006 0.006<x<0.01 0.01<x<0.018 0.018<x<0.01 0.01<x<0.098 0.098<x<0.06 x0 0.06<x<0.1 x 0.0001<x<0.00018 0 0. 0. 0.6 0.8 1 1. 1. Figure : Simulated cross-sections with error bars to t in different kinematics. 9

Measured cross-section (x0gev Kinematics for 1.78<Q <.16GeV 0.001<x<0.0098 0.0098<x<0.006 0.006<x<0.01 0.01<x<0.018 0.018<x<0.01 0.01<x<0.098 0.098<x<0.06 0.1<x<0.18 x 0.06<x<0.1 1 0 0. 0. 0.6 0.8 1 1. 1. Figure 6: Simulated cross-sections with error bars to t in different kinematics.

Measured cross-section (x0gev Kinematics for.16<q <.6GeV 0.0098<x<0.006 0.006<x<0.01 0.01<x<0.018 0.018<x<0.01 0.01<x<0.098 0.098<x<0.06 x0 0.06<x<0.1 x0 0.1<x<0.18 x0 0.18<x<0.1 x0 0.1<x<0.98 0 0. 0. 0.6 0.8 1 1. 1. Figure 7: Simulated cross-sections with error bars to t in different kinematics. 11

Measured cross-section (x0gev Kinematics for.6<q <.0GeV 0.006<x<0.01 0.01<x<0.018 0.018<x<0.01 0.01<x<0.098 0.098<x<0.06 0.06<x<0.1 x 0.1<x<0.18 x 0.18<x<0.1 x 0.1<x<0.98 x 0.98<x<0.6 x1 0.0098<x<0.006 0 0. 0. 0.6 0.8 1 1. 1. Figure 8: Simulated cross-sections with error bars to t in different kinematics. 1

Measured cross-section (x0gev Kinematics for.0<q <17.78GeV 0.0018<x<0.001 0.001<x<0.0098 0.0098<x<0.006 0.006<x<0.01 x 0.1<x<0.98 x 0.98<x<0.6 x1 0.006<x<0.01 x 0.01<x<0.018 x0 0.18<x<0.1 x 0.1<x<0.18 0 0. 0. 0.6 0.8 1 1. 1. Figure 9: Simulated cross-sections with error bars to t in different kinematics. 1

Measured cross-section (x0gev Kinematics for 17.78<Q <1.6GeV 0.98<x<0.6 0.6<x<0.1 x 0.018<x<0.01 x 0.01<x<0.098 x 0.098<x<0.06 x 0.06<x<0.1 x 0.1<x<0.18 x 0.18<x<0.1 x 0.1<x<0.98 0 0. 0. 0.6 0.8 1 1. 1. Figure : Simulated cross-sections with error bars to t in different kinematics. 1

Measured cross-section (x0gev Kinematics for 1.6<Q <6.GeV 0.98<x<0.6 0.6<x<0.1 x 0.098<x<0.06 x 0.06<x<0.1 x 0.1<x<0.18 x 0.18<x<0.1 x1 0.01<x<0.098 0 0. 0. 0.6 0.8 1 1. 1. Figure 11: Simulated cross-sections with error bars to t in different kinematics. 1

Measured cross-section (x0gev Kinematics for 6.<Q <0.00GeV 0.6<x<0.1 x0 0.098<x<0.06 x0 0.1<x<0.98 x0 0.06<x<0.1 x0 0.1<x<0.18 x0 0.18<x<0.1 0 0. 0. 0.6 0.8 1 1. 1. Figure 1: Simulated cross-sections with error bars to t in different kinematics. 16

Measured cross-section (0x0GeV 6 Kinematics for 1.0<Q <1.78GeV 0.0001<x<0.00018 0.00018<x<0.0001 0.0001<x<0.00098 0.0006<x<0.001 0.001<x<0.018 0.0018<x<0.001 0.001<x<0.0098 0.0098<x<0.006 x 0.00006<x<0.0001 x 0.00098<x<0.0006 x 0.006<x<0.01 0 0. 0. 0.6 0.8 1 1. 1. Figure 1: Simulated cross-sections with error bars to t in different kinematics. 17

Measured cross-section (0x0GeV 6 Kinematics for 1.78<Q <.16GeV 0.00018<x<0.0001 0.0001<x<0.00098 0.00098<x<0.0006 0.001<x<0.0018 0.0018<x<0.001 0.001<x<0.0098 0.0098<x<0.006 0.006<x<0.01 x8 0.0006<x<0.001 x8 0.01<x<0.018 1 0 0. 0. 0.6 0.8 1 1. 1. Figure 1: Simulated cross-sections with error bars to t in different kinematics. 18

Measured cross-section (0x0GeV Kinematics for.16<q <.6GeV 0.0001<x<0.00098 0.00098<x<0.0006 0.0006<x<0.001 0.0018<x<0.001 0.001<x<0.0098 0.0098<x<0.006 0.006<x<0.01 0.01<x<0.018 x 0.001<x<0.0018 x 0.018<x<0.01 x0 0.01<x<0.98 0 0. 0. 0.6 0.8 1 1. 1. Figure 1: Simulated cross-sections with error bars to t in different kinematics. 19

Measured cross-section (0x0GeV Kinematics for.6<q <.0GeV 0.0006<x<0.001 0.0018<x<0.001 0.001<x<0.0098 0.0098<x<0.006 0.006<x<0.01 0.01<x<0.018 0.018<x<0.01 x 0.00098<x<0.0006 x 0.0006<x<0.001 x 0.01<x<0.098 x0 0.098<x<0.06 1 0 0. 0. 0.6 0.8 1 1. 1. Figure 16: Simulated cross-sections with error bars to t in different kinematics. 0

Measured cross-section (0x0GeV 6 Kinematics for.0<q <17.78GeV 0.001<x<0.0098 0.0098<x<0.006 0.006<x<0.01 0.01<x<0.018 0.01<x<0.098 x 0.001<x<0.0018 x 0.098<x<0.06 x0 0.0018<x<0.001 x0 0.06<x<0.1 0 0. 0. 0.6 0.8 1 1. 1. Figure 17: Simulated cross-sections with error bars to t in different kinematics. 1

Measured cross-section (0x0GeV Kinematics for 17.78<Q <1.6GeV x 0.0098<x<0.006 x 0.006<x<0.01 x 0.01<x<0.018 x 0.018<x<0.01 x 0.01<x<0.098 x0 0.01<x<0.098 x 0.06<x<0.1 0 0. 0. 0.6 0.8 1 1. 1. Figure 18: Simulated cross-sections with error bars to t in different kinematics.

Measured cross-section (0x0GeV Kinematics for 1.6<Q <6.GeV x 0.0098<x<0.006 x 0.01<x<0.098 x 0.098<x<0.06 x 0.006<x<0.01 x 0.01<x<0.018 x 0.018<x<0.01 x0 0.1<x<0.18 x0 0.18<x<0.1 x0 0.06<x<0.1 0 0. 0. 0.6 0.8 1 1. 1. Figure 19: Simulated cross-sections with error bars to t in different kinematics.

Measured cross-section (0x0GeV Kinematics for 6.<Q <0.0GeV x 0.006<x<0.01 x 0.01<x<0.018 x 0.018<x<0.01 x 0.01<x<0.098 x0 0.098<x<0.06 x0 0.06<x<0.1 x0 0.1<x<0.18 0 0. 0. 0.6 0.8 1 1. 1. Figure 0: Simulated cross-sections with error bars to t in different kinematics.

6 Summary This results can be used as a reference of how STAR Detector coverage should be improved for DVCS measurements in EIC in the future. The suggestion comes naturally that the detector will have better ability of DVCS cross-section measurements if the pseudo-rapidity gaps were recovered and the acceptance for high Q and high x events was raised. Thanks to Prof Ye s advising and providing of the source code of this project. References [1] Francis Halzen, Alan D. Martin, Quarks and Leptons: An Introductory Course in Modern Particle Physics, John Wiley and Sons, Inc. (199. [] estar collaboration, Electron Ion Collider: The Next QCD Frontier, arxiv:11.170. [] D.C. Aschenauer, S. Fazio, K. Kumericki and D. Muller, arxiv:10.0077v [hep-ph] 1 Aug 01. [] Xiangdong Ji, Deeply Virtual Compton Scattering, arxiv:hep-ph/960981 (1997. [] Zhenyu Ye, Deeply Virtual Compton Scattering at estar, STAR estar meeting, UCLA, 01/8/8. [6] A.Freund, D.Muller and A. Kirchner, Nucl. Phys. B 69, (00; A. Freund, Phys.Rev.D 68 096006 (00. [7] E. Perez, L. Schoeffel, L. Favart, MILOU: a monte-carlo for Deeply Virtual Compton Scattering, arxiv:hep-ph/011189v1 0 Nov 00.