Template for Electronic Submission to ACS Journals 35 © 2023 Robbins, Jocelyn & Yuan, Melinda. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits the user to copy, distribute, and transmit the work provided that the original authors and source are credited. Inside the Nucleon: Tomographic Interpretations and Uni- versality of GPDs with DDVCS Jocelyn Robbins1, Melinda Yuan2, Marie Boër3 1University of Colorado, Boulder 2Columbia University in the City of New York 3Virginia Polytechnic Institute and State University KEYWORDS: Generalized Parton Distributions (GPDs), Double Deeply Virtual Compton Scattering (DDVCS), Nucleon Structure, Quantum Chromodynamics (QCD), Tomographic Interpretations ABSTRACT: The goal of Double Deeply Virtual Compton Scattering (DDVCS) experiments is to better understand the internal structure of the nucleon. Previous attempts to resolve the internal structure of nucleons have resulted in electromagnetic form factors and parton distribution functions for elastic scat- tering and deep inelastic scattering processes, respectively. Generalized Parton Distributions (GPDs) are the latest attempt to unify these models of nucleon structure. The GPDs of DDVCS give us ability to investigate off of the diagonal where x ̸= ±ξ. The main goal of our analysis is to determine the best ex- perimental setup in order to deduce the kinematic variables on which GPDs depend from the lab ob- servables. The effectiveness of our data collection in the laboratory is by determined the physical kine- matics, Q2, Q′2,t, xi,ϕLM, ϕCMV , and θCMV . We can then run DDVCS experiments and collect data on ob- servables to improve upon the current models for GPDs of the nucleon. INTRODUCTION The fundamental aim of particle physics is to discover and understand matter down to its smallest possible constituents. The discov- ery of the quark in 1964 revolutionized this process, breaking apart nucleons, which were previously believed to be elementary particles[6]. These subatomic particles were detected using scattering, a process in which a high energy particle beam is directed at a stationary object, resulting in collisions be- tween the particles in the beam and object. Data is collected from these collisions, such as cross sections or scattering byproducts, which is then used to infer information about the internal structure of the object. The mod- el that we are attempting to explore is Gen- eralized Parton Distributions, currently the most detailed model of nucleon structure to exist. GPDs are a hybrid of its predecessors, form factors (produced through elastic scat- tering) and parton distributions (produced through deep inelastic scattering)[8]. The particular scattering process that we are in- vestigating is Double Deeply Virtual Comp- ton Scattering (DDVCS), in which an elec- tron beam is scattered off a proton, exchang- ing a virtual photon in the process. The out- going virtual photon will then decay into a detectable muon-antimuon pair as seen in Figure 1[7]. Columbia Undergraduate Science Journal Vol. 17, 2023 Robbins, Yuan, et. al. 36 The current reactions studied are Timelike Compton Scattering (TCS) and Deeply Vir- tual Compton Scattering (DVCS). Both of these collisions include one virtual photon, with TCS containing an outgoing virtual pho- ton and DVCS an incoming virtual photon. In DDVCS experiments, both the initial and fi- nal photons are virtual. The virtuality of the photons impacts the matrix elements that describe the reaction. The matrix element refers to the probability amplitude of finding a quark at a space-time point in a nucleon, then finding the same quark at another space-time point in the nucleon which has now changed its momentum [5]. The in- volvement of 2 space-time points means that the matrix element is non-local, and the dif- fering momenta of the initial and final nucle- on makes the matrix element non-forward. GPDs depend on the following kinematic variables: Q2, the virtuality of the exchanged photon, t, the momentum transfer to the nu- cleon, Bjorken x (xBj), the fraction of the total nucleon momentum, and three reaction an- gles. These kinematics be used to calculate ξ and ξ’, components of light cone frame momentum, using the following equations [3]. Using ξ and ξ’, the longitudinal momentum transfer fraction of incoming spacelike pho- ton, -2(ξ−ξ’), and the longitudinal momentum transfer fraction of outgoing timelike photon, (2ξ’−ξ), can be calculated. These momentum transfers make DDVCS unique because in a DDVCS reaction, ξ ̸= ξ’ unlike TCS and DVCS reactions where ξ = ±ξ’ [3]. When ξ ̸= ξ’, xBj and ξ dependence can be decoupled, allowing access to ‘off-diagonal elements’ of GPD regions. Currently, with TCS and DVCS, only elements on the diagonal xBj = ±ξ are accessible due to the dependence of x and ξ on each other. However, x and ξ de- coupling enables newly accessible regions of GPDs, which will allow for a more detailed picture of the distribution of nuclear forces inside a nucleon as well as the determination of parton transverse densities. Such investi- gation will add to the current model of GPDs and facilitate more detailed tomographies of the nucleon. Another focus of the DDVCS experiments is the question about the universality of GPDs. Under current theories, GPDs are presumed to be universal, which means that the calcu- lations of GPDs will be the same regardless the experiment used to measure them. However, there is no concrete experimental proof to support this conclusion. DDVCS re- actions have the unique ability to simultane- ously study spacelike (defined as Q2 > Q’2) and timelike regions (defined as Q’2 < Q2) in order to determine the two regions result in the same leading order and twist. The agreement of results between the spacelike and timelike regions would then in turn pro- vide support of the universality of GPDs Figure 1: DDVCS e− + e−→µ+ + µ−[2] Columbia Undergraduate Science Journal Vol. 17, 2023 Robbins, Yuan, et. al. 37 METHODS In order to simulate DDVCS collisions, we used the event generator DEEPGen, devel- oped by Dr. Marie Boër. These event gener- ators, written in C++ and run through ROOT, generate simulated particle collisions events given a certain set of parameters, such as luminosity, beam energy, and phase space. It can be altered depending on the type of collision it is replicating. The version that we used, DEEPGen 5.0, simulates deep exclu- sive photo- and electro- production of lepton pairs and photons, including DDVCS as well as DVCS and TCS. The parameters of the event generator were set to match experi- ments at Hall C of Jefferson Lab with an 11 GeV electron beam. The generator creates equal weighted events and uses a Monte- Carlo simulation technique to weight the events with an n-differential cross section[4]. The weights are multiplied by a normaliza- tion factor, where L is luminosity and ∆Ω is the dimension of the phase space. There are five weighting options; total unpo- larized, DDVCS, Bethe-Heitler (BH), DDVCS/BH, and beam spin asymmetry. The beam spin asymmetry weighting provides insight into how difficult it will be to measure the asymmetric polarization of the electron beam. Simulated events with the total unpo- larized weight are proportional to the number of measured events in the physical experi- ment. The total unpolarized weight is calcu- lated with the DDVCS weight and BH weight, Wtot = |WDDVCS + WBH|2. Analyzing the data with DDVCS and BH weighting allows their contributions to be distinguished. The BH contribution is precisely known due to it’s dependence on QED calculations and proton form factors[1]. It describes the hard (known) region of DDVCS reactions. The DDVCS contribution, on the other hand, describes the soft region of the reactions and can not be calculated. The ratio of DDVCS and BH weighting al- lows for insights into where the so-called ‘new physics’ can be found. Areas where DDVCS/BH is large are promising in terms of information about the GPDs in the previously inaccessible regions. Figure 2: Illustration of non-local non-forward matrix element[5] Figure 3:The DDVCS and BH contributions to the total unpolarized weight[2] Columbia Undergraduate Science Journal Vol. 17, 2023 Robbins, Yuan, et. al. 38 After generating the simulated and weighted DDVCS events, we analyzed the data to bet- ter understand how the experimental kine- matics variables relate to each other. In addi- tion to the variables of t, ξ, ξ′ displayed in Figure 1 above, the relationship between the various angles in the particle scattering, shown below, were explored. The experi- ment takes place on three planes, with the far left plane being created by the initial and final electrons from the electron beam, the middle plane by the photons and the nucle- on, and the plane on the far right by the mu- ons. The notation CM indicates the meas- urements are taken from the center of mass frame. The angles of interest are θCM, the angle between the incoming and outgoing electrons, ϕL, the angle between the plane of the electrons and the plane of the photons, and ϕCM, is the angle between the plane of the muons and the plane of the photons. The last angle, θCM, is the angle the scattered muon makes from parallel. DISCUSSION While the earlier graphs were created to bet- ter understand the relevant kinematic rela- tionships, the later graphs explored how events can be interpreted. The first set of graphs (Figure 5) traces out the amount of events that exists within a certain angle range (corresponding to the angles in Figure 4). The ϕ angles have a range of π and the θ graph measures entries between 0 and π. Figure 5: θCM (upper left), ϕCM (upper right), and ϕL (lower) Figure 6: ’ Both of the center of mass angles (ϕCM and θCM appear largely symmetric without many irregularities. However, the ϕL graph displays unexpected asymmetries, which may arise due to relations between all three angles and warrants more exploration. The next figure, which compares ’, provides important information about events that need to be excluded from the data set and events that might be useful for physical interpretation. The events along the timelike and spacelike cut line are events that must be excluded. The line corresponds with = 1 which makes the data difficult to interpret. Figure 4:Collision vectors and angles Columbia Undergraduate Science Journal Vol. 17, 2023 Robbins, Yuan, et. al. 39 Figure 7: ξ vs ξ’ Figure 7 is a comparison of ξ and ξ’ with boundaries that define later data cuts in Q2 vs Q’2. By comparing the ξ and ξ’, we are granted the ability to determine how far ’off the diagonal’ we can measure our events. The diagonal, in this case, is defined as x = ±ξ and has previously restricted the investi- gation of GPDs in the ERBL and DGLAP re- gions. The cuts that are made focus on are- as where measured events can be physically interpreted. The cuts on the ξ vs ξ’ graph correspond to selected Q2 and Q’2 bands (Figure 8). The graphs are also restricted in terms t, between −0.15 GeV and −0.55 GeV. t must be con- strained because while there are more events measured between −0.55 GeV and −1.05 GeV, when the momentum transfer becomes too large, approximations are ren- dered invalid and a physical interpretation loses meaning. Thus, the graphs must be constrained in t. The value of these graphs are derived from the fact that Q2 and Q’2 cannot be measured outright. The transfer of the virtual photon’s momentum occurs in the soft region, which is not physically accessible with our current technology, but can still be calculated from the measured kinematics. Using the Q2 vs Q’2 selected bands and the Bjorken x hy- pothesis, the structure of the proton can be determined. The Bjorken x hypothesis states that, for point-like particles, as the limit Q2 → ∞, GPDs lose dependence on Q2. Hence, if the GPD demonstrates dependence on Q2, it’s structure cannot be classified as point- like.[9] SUMMARY AND OUTLOOK Double Deeply Virtual Compton Scattering is the next step when it comes to accessing and analyzing GPDs. The incoming and out- going virtual photons make DDVCS unique from previous experiments, allowing access to regions of Generalized Parton Distribu- tions that were previously unreachable. This is done through taking advantage of the fact that in DDVCS, ξ ̸= ±ξ’ which allows access Figure 8: Q2 vs Q’2 Selected Bands, Spacelike (left) and Timelike (right) Columbia Undergraduate Science Journal Vol. 17, 2023 Robbins, Yuan, et. al. 40 beyond the x = ±ξ diagonal GPDs are cur- rently restricted to. DDVCS also provides opportunity to com- pare measurements of GPDs in spacelike and timelike regions simultaneously, which can be used to evaluate the validity of GPD universality. The DEEPGen Event Generator provided events that could be applied to dif- ferent weighting systems and analyzed. Boundaries were created for Q2 vs Q’2 using ξ vs ξ’, allowing for greater insight into proton GPDs as well as an understanding of the ex- tent to which measurements can be taken ’off of the diagonal.’ The future of this project requires more data analysis and planning for the Jefferson Lab proposal. The kinematics regarding the Bjorken x Hypothesis contains a vast domain of interpretation that has potential to be ex- plored further. Additionally, more detailed data analysis can be done by with regards to the relationship between the experimental angles and the other kinematics variables, as this study lacks the proper angular correc- tions and acceptance cuts from Q2 and other values. Looking forward, there exists a great deal more investigation into DDVCS to be done, as it appears to be an untapped realm when it comes to uncovering more physics regarding proton GPDs. AUTHOR INFORMATION Corresponding Author Melinda Yuan: my2740@columbia.edu Funding Sources This work was made possible by the Nation- al Science Foundation under grant No. PHY- 2149165. ACKNOWLEDGMENTS We would like to thank our mentor, Dr. Marie Boër for the opportunity to take part in her research and for her guidance through the research process. We acknowledge the out- standing support from the National Science Foundation, the Virginia Tech Physics de- partment, and the Virginia Tech Center for Neutrino Physics. ABBREVIATIONS DDVCS: Double Deeply Virtual Compton Scattering GPDs: Generalized Parton Distributions TCS: Timelike Compton Scattering BH: Bethe-Heitler REFERENCES [1] C. Adloff. “Measurement of Deeply Virtual Compton Scattering at HERA”. 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