Analysis Note: Precision Extraction of Momentum Transfer in Diffractive Coherent Exclusive Vector Meson Production

epic-AN-AC-2025-001 Precision Extraction of Momentum Transfer in Diffractive Coherent Exclusive Vector Meson Production Principal Author List: Maci Kesler (Kent State University, [email protected]) Ashik Ikbal Sheikh (Kent State University, [email protected]) Rongrong Ma (Brookhaven National Lab, [email protected]v) Zhoudunming Tu (Brookhaven National Lab, [email protected]v) Thomas Ullrich (Brookhaven National Lab, thomas.ullric[email protected]v) Zhangbu Xu (Brookhaven National Lab, Kent State University, [email protected]v) (December 15, 2025) 1 Abstract Exclusive diffractive vector meson production is a critical component of the e+Aprogram at the EIC. The exclusivity of the event makes it an experimentally clean and ideal measurement to study the onset of saturation and other QCD phenomena. Diffractive processes, e+A→e′+A′+V M where V M =J/ψ, ϕ, etc., allow the measurement of momentum transfer (|t|). A Fourier-Bessel transformation of |t|enables us to extract the spatial distribution of the gluons inside the nucleus. This analysis focuses on improving the precision of the extraction of |t|. 2 Contents 1 Introduction 6 2 Simulation Overview 7 2.1 Event Generator Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Event Selection 8 3.1 TheCodeLogic.................................. 8 3.1.1 Breakdown: the analysis . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.1.2 Breakdown: the transformation . . . . . . . . . . . . . . . . . . . . . 10 3.2 Cuts ........................................ 12 3.2.1 Reconstruction Methods . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Systematic Uncertainies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.3.1 DISBackground ............................. 25 3.3.2 ρProduction ............................... 30 3.3.3 Incoherent Production . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.4 AnalysisCode................................... 32 4 Results and Discussion 33 iii List of Figures 1 Diagram of coherent exclusive VM production. . . . . . . . . . . . . . . . . . 6 2 Diagram of exclusive VM production. The blue arrow indicates the normal direction, ˆn, from the electron scatting plane. The red arrow show the spin direction of the emitted virtual photon. The VM production and decay planes arealsoshown. .................................. 9 3 The leftmost plot shows the reconstruction of θein comparison with the truth. The middle plot shows the resolution and the rightmost plot shows the response. 11 4 QA plots for θe................................... 11 5 The leftmost plot shows the reconstruction of the energy of e′in comparison with the truth. The middle plot shows the resolution and the rightmost plot showstheresponse................................. 11 6 QA plots for the energy of e′. .......................... 12 7 The leftmost plot shows the reconstruction of Q2in comparison with the truth. The middle plot shows the resolution and the rightmost plot shows the response. 14 8 QA plots for the Q2variable. .......................... 14 9 The leftmost plot shows the reconstruction of yin comparison with the truth. The middle plot shows the resolution and the rightmost plot shows the response. 14 10 QA plots for y. .................................. 15 11 The leftmost plot shows the reconstruction of xin comparison with the truth. The middle plot shows the resolution and the rightmost plot shows the response. 15 12 QA plots for x. .................................. 16 13 The leftmost plot shows the reconstruction of pTfor the VM in comparison with the truth. The middle plot shows the resolution and the rightmost plot showstheresponse................................. 17 14 QA plots for the VMs pTdistribution. ..................... 18 iv 15 The leftmost plot shows the reconstruction of the E−pzdistribution of the VM in comparison with the truth. The middle plot shows the resolution and the rightmost plot shows the response. . . . . . . . . . . . . . . . . . . . . . 18 16 QA plots for the VMs E−pzdistribution. . . . . . . . . . . . . . . . . . . . 19 17 |t|=q2distribution in 2-dimensions. The wedge cut is shown by the angle θmax to demonstrate the projection technique. . . . . . . . . . . . . . . . . . 20 18 Distribution of cluster positions in EMcal with and with geometric cut. . . . 21 19 The leftmost plot shows the reconstruction of the E−pzdistribution from e′ in comparison with the truth. We expect this to peak around 2Ee= 20 GeV. The middle plot shows the resolution and the rightmost plot shows the response. 21 20 QA plots for the E−pzdistribution from e′................... 22 21 The leftmost plot shows the reconstruction of E/|p|for e′in comparison with the truth. We expect this to peak at one to demonstrate good matching of the EMcal with the track. The middle plot shows the resolution and the rightmost plotshowstheresponse. ............................. 22 22 QA plots for E/|p|for e′.............................. 23 23 |t|distribution where the black solid line represent the MC truth, the curve with blue dots corresponds to method L reconstruction, and the open pink stars correspond to |t|reconstruction using the projection method with a wedge cut of θmax =π/12. ............................ 26 24 |t|distribution for different θmax values. θmax =π/2 is equivalent to method L. As we decrease the amount of phase space that we keep in our measurement, the diffractive pattern becomes more prominent. The solid black curve corresponds to the MC truth, purple open double diamonds for θmax =π/2, open blue circles for θmax =π/3, gray stars for θmax =π/6, open pink stars for θmax =π/12, and yellow octagons with a cross for θmax =π/24. . . . . . . 27 25 QA plots for the |t|distribution reconstructed using the projection method derived from Eqs. 26 and 28. Here we have used a θmax =π/12 wedge cut. . 28 26 Resolution of the |t|distribution. The inset shows the 2D resolution where the x-axis is |t|MC and the y-axis is given by (|t|MC − |t|proj)/|t|MC....... 28 v 27 |t|distribution with all systematic uncertainties studied in the analysis without any vetoes, cuts, or PID applied. The black solid curve represents the MC truth distribution, the open pink stars correspond to the projection method, light blue plus signs for the DIS distribution, solid orange squares for the incoherent ϕproduction, and the open gray cross for the ρproduction. . . . 29 28 |t|distribution with DIS background shown as the light blue plus signs, the MC truth is the black solid line and the projection method corresponds to the open pink stars. We see that the DIS curve only allows us to resolve the first minima. ...................................... 30 29 |t|distribution with all vetoes and cuts applied. The solid black line corresponds to the MC truth, open pink stars to the projection method, light blue plus signs for DIS with no vetoes, solid green squares for DIS with only OMD vetoes, open red squares for DIS with only RP vetoes, orange stars for DIS with all detector vetoes, open purple circles for DIS with only ηand HFS cuts, and the gray open double diamonds represent DIS with all cuts and vetoes applied. ...................................... 31 30 The leftmost plot shows the mass distributions of ρand ϕwith no PID while the rightmost plot shows the same distributions with PID. The orange dashed curve represents the ρMC mass distribution, the dashed blue curve is for ρ reconstruction, the green long dash is for ρreconstruction before the ϕmass selection is applied, the solid black curve is for the ϕMC distribution, and the dashed purple curve is for the reconstructed ϕ.................. 31 31 |t|distribution with ρproduction. The solid black curve represents the ϕ MC truth distribution, the open pink stars for the ϕdistribution projection method, the gray open crosses represent the ρdistribution with no PID, and the blue open double diamonds represent the ρdistribution with PID. . . . . 32 32 |t|distribution with incoherent production. No vetoes or cuts have been applied. The black solid curve represents the coherent MC truth, the open pink stars for the projection method for coherent ϕreconstruction, and the filled orange squares is the incoherent production distribution. . . . . . . . . . . . 33 vi 33 |t|distribution with incoherent production and individual detector vetoes and cuts. The black solid curve represents the coherent ϕMC distribution, the open pink stars are for the coherent ϕreconstruction using the projection method, the open four triangle x is for incoherent with no vetoes, the red star is for incoherent with only RP vetoes, the blue full cross x is for incoherent with only OMD vetoes, the green full diamond is for incoherent with only ZDC vetoes, the open gray circles are for incoherent with all detector vetoes, the black full down triangle is for incoherent with ηand HFS cuts, and the filled orange squares are for incoherent production with all vetoes applied. . 34 34 |t|distribution after applying a cut of θ=π/12. This plot shows the improvement of the |t|measurement by comparing the coherent truth (MC reconstruction, black curve) with the new projection method (open pink stars). 35 35 2-dimensional Fourier-Bessel transformation of the |t|distribution with wedge cut of θ=π/12. This shows the gluon spatial distribution with a comparison from the transforms of the truth, method L, and the projection method distributions. The black solid curve represent the truth distribution, the blue filled circles correspond to |t|reconstruction using method L, and the pink open stars represent the reconstruction of |t|using the new projection technique. . 36 vii List of Tables 1 Cross sections for simulated datasets. . . . . . . . . . . . . . . . . . . . . . . 8 2 Summary of cuts applied at MC and reconstruction level. . . . . . . . . . . . 13 viii 1 Introduction The goal of this analysis is to advance the measurement of the nuclear momentum transfer distribution (|t|) through coherent exclusive vector meson (VM) production. Coherent exclusive VM production is considered a golden channel for imaging the gluon structure in nuclei and testing QCD properties, such as saturation [1]. Figure 1 shows the process of exclusive e A V M e′ A′ t γ∗(Q2) q ¯q Figure 1: Diagram of coherent exclusive VM production. VM production. The incoming electron (e) emits a virtual photon (γ∗) which then fluctuates into a quark-antiquark pair (qand ¯qrespectively) which interacts with the target (A) via a Pomeron and then recombines to form the final state VM (ϕ→K+K−in this analysis). The momentum transfer distribution forms a diffractive pattern that encodes information about the spatial distribution of gluons in the hadron wave function [2]. Since |t|is conjugate to the impact parameter, a Fourier-Bessel (Hankel) transformation of the distribution enables us to obtain the spatial profile of the gluon density. We define the Mandelstam variable |t|as |t|=−(PA′−PA)2,(1) where pA′and pAdenote the four-momenta of the outgoing and incoming nucleus. However, the |t|distribution is a challenging measurement, as it depends on the outgoing nucleus’ momentum, which, for heavy nuclei, we cannot access precisely. Consequently, we have to use different methods to reconstruct this measurement. There are two main complications with this: limited precision and a large background from incoherent production where the nucleus breaks up. This analysis utilizes a projective technique, as demonstrated in [3], to reconstruct the |t|distribution. After constructing the momentum transfer profile, we can take a Fourier-Bessel transformation to extract the spatial distribution of the gluons [4]. As shown in [5], a 2-dimensional transformation of the transverse |t|distribution gives the transverse distribution of the spatial 6 Table 2: Summary of cuts applied at MC and reconstruction level. Variable Cut Value / Range Level Purpose Q21< Q2<10 GeV2MC + RECO Ensure hard photon, avoid poor resolution y(inelasticity) 0.01 < y < 0.85 MC + RECO Avoid trivial interactions and extreme energy loss VM energy EVM >0 MC + RECO Require produced vector meson VM rapidity |yϕ|<3.5 MC + RECO Select central production region VM mass |mVM −1.02|<0.02 GeV/c2 RECO Select ϕmeson Cluster radius r < 550 mm RECO Ensure electron cluster within detector acceptance E−pz(for e′) 15 < E −pz<25 GeV RECO Ensure exclusivity E/p (for e′) 0.9< E/p < 1.2 RECO Match EMCal energy to track momentum |t|projection angle θ < θmax MC + RECO Minimize VM momentum in electron scattering plane where PeMC ,Pe′ MC are the four-momenta of the incoming and scattered electron respectively. We reconstruct the virtuality using Pe′ RECO which is given by Q2 RECO =−(PeMC −Pe′ RECO )2(4) An additional cut is placed on the fractional energy lost by the electron (inelasticity) expressed as yMC =PAMC ·(PeMC −Pe′ MC ) (PAMC ·PeMC ),(5) where PAMC is the four-momentum of the incoming ion. The inelasticity is reconstructed by the following yRECO =PAMC ·(PeMC −Pe′ RECO ) (PAMC ·PeMC ).(6) We define the Bjorken-xvariable, fraction of the hadrons momentum carried by the parton that the electron scatters from, as xRECO =Q2 2[(PeMC −Pe′ RECO )·PAMC ].(7) By putting constraints on these kinematic variables, we are able to isolate a clean, reliable region. For the virtuality, we have Q2>1 to ensure that the photon is hard 13 1 2 3 4 5 6 7 8 9 10 2 [GeV/c] 2 Q 3 10 4 10 5 10 6 10 counts MC RECO Truth vs Reco 2 Q 1 10 2 10 3 10 4 10 1 2 3 4 5 6 7 8 9 10 2 [GeV/c] MC 2 Q 0.3− 0.2− 0.1− 0 0.1 0.2 0.3 MC 2 )/Q RECO 2 -Q MC 2 (Q Resolution 2 Q 1 10 2 10 3 10 4 10 5 10 1 2 3 4 5 6 7 8 9 10 2 [GeV/c] RECO 2 Q 1 2 3 4 5 6 7 8 9 10 2 [GeV/c] MC 2 Q Response 2 Q Figure 7: The leftmost plot shows the reconstruction of Q2in comparison with the truth. The middle plot shows the resolution and the rightmost plot shows the response. 1 10 2 10 3 10 4 10 5 10 1 2 3 4 5 6 7 8 9 10 2 [GeV/c] MC 2 Q 1 2 3 4 5 6 7 8 9 10 2 [GeV/c] RECO 2 Q Bin Migration 0 1 2 3 4 5 6 7 8 9 10 2 [GeV/c] reco bin 2 Q 0 0.2 0.4 0.6 0.8 1 Purity 2 /c 2 bin width=0.1 GeV Purity 0 1 2 3 4 5 6 7 8 9 10 2 [GeV/c] true bin 2 Q 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Stability 2 /c 2 bin width=0.1 GeV Stability 0 1 2 3 4 5 6 7 8 9 10 2 [GeV/c] 2 Q 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Acceptance Acceptance 0 1 2 3 4 5 6 7 8 9 10 2 [GeV/c] 2 Q 0 0.2 0.4 0.6 0.8 1 Efficiency Efficiency 0 1 2 3 4 5 6 7 8 9 10 2 [GeV/c] 2 Q 3 10 4 10 5 10 6 10 Corrected Acceptance Corrected Figure 8: QA plots for the Q2variable. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8y 4 10 5 10 counts MC RECO y Truth vs Reco 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 MC y 1− 0.8− 0.6− 0.4− 0.2− 0 0.2 0.4 0.6 0.8 1 MC )/y RECO -y MC (y 1 10 2 10 3 10 y Resolution 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 RECO y 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 MC y 1 10 2 10 3 10 4 10 5 10 y Response Figure 9: The leftmost plot shows the reconstruction of yin comparison with the truth. The middle plot shows the resolution and the rightmost plot shows the response. 14 1 10 2 10 3 10 4 10 5 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 MC y 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 RECO y Bin Migration 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 reco bin y 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Purity bin width=0.0084 Purity 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 true bin y 0.1 0.2 0.3 0.4 0.5 Stability bin width=0.0084 Stability 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8y 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Acceptance Acceptance 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8y 0.988 0.99 0.992 0.994 0.996 0.998 1 Efficiency Efficiency 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8y 4 10 5 10 6 10 Corrected Acceptance Corrected Figure 10: QA plots for y. 0 0.05 0.1 0.15 0.2 0.25 0.3 x 1 10 2 10 3 10 4 10 5 10 6 10 counts MC RECO x Truth vs Reco 0 0.02 0.04 0.06 MC x 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 MC )/x RECO -x MC (x 1 10 2 10 3 10 4 10 x Resolution 1 10 2 10 3 10 4 10 5 10 6 10 0 0.05 0.1 0.15 0.2 RECO x 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 MC x x Response Figure 11: The leftmost plot shows the reconstruction of xin comparison with the truth. The middle plot shows the resolution and the rightmost plot shows the response. 15 1 10 2 10 3 10 4 10 5 10 6 10 0 0.02 0.04 0.06 0.08 0.1 MC x 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 RECO x Bin Migration 0 0.01 0.02 0.03 0.04 reco bin x 0 0.2 0.4 0.6 0.8 1 Purity bin width=0.005 Purity 0 0.05 0.1 0.15 0.2 true bin x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Stability bin width=0.005 Stability 0 0.05 0.1 0.15 0.2 x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Acceptance Acceptance 0 0.05 0.1 0.15 0.2 x 0 0.2 0.4 0.6 0.8 1 Efficiency Efficiency 0 0.05 0.1 0.15 0.2 x 3 10 4 10 5 10 6 10 Corrected Acceptance Corrected Figure 12: QA plots for x. enough to resolve the partons and Q2<10 to aviod events with poor resolution. For the inelasticity, we have y > 0.01 to avoid events where the electron barely interacts and y < 0.85 to avoid events where the electron loses almost all of it’s energy. The resulting Q2,y, and xdistributions are shown in Figs. 7, 9, and 11. For reference and quality assurance (QA), Figs. 8, 10, and 12 show the results from this analysis. We define the QA checks in the following way (note that unfolding will be done in future work as we plan to integrate deep learning models to perform this task): –Bin migration: Events in true bin ireconstructed in bin j. Ideally this would be a diagonal to show that events are properly reconstructed. –Purity: Ratio of correctly reconstructed events in MC bin and the total reconstructed events. High purity corresponds to less contamination from neighboring bins. We want values to be close to 1. Purity = Nj→j ΣiNi→j (8) –Stability: Ratio of events in a given MC bin that are reconstructed into the same bin. High stability corresponds to events staying in their bin. Stability = Ni→i ΣjNi→j (9) –Acceptance: Fraction of true events that are reconstructed anywhere and survive the selection cuts. This should be a smooth distribution whereas sharp fluctuations indicate a cut boundary or detector geometry edge. We want values to be close to 1. Acceptance = ΣjNi→j Ntrue i (10) 16 –Efficiency: Ratio of true events that are reconstructed and pass final selection criteria. Efficiency = Nreco bin i Ntrue i (11) –Corrected: Acceptance-corrected yield per true bin. This should be comparable to the MC truth distribution. Corrected = Nreco bin i Acceptance (12) •Vector Meson: We reject events where the VM has zero energy to confirm that we skip events without a produced VM. Furthermore, we want to look at regions where the VM is produced centrally to verify that we have a distinct diffractive event. This is done by placing a cut on the VM rapidity which defines how forward or backward a particle is moving in the beam direction. For this reason, we place a constraint on the VM rapidity to be |yϕ|<3.5. Lastly, we use the constraint |mVM −1.02|GeV/c2<0.02 GeV/c2to ensure that we are selecting ϕsince mϕ≈1.02 GeV/c2. VM reconstruction is discussed in more detail in Section 3.2.1 and is shown in Figs. 13 and 15 while the QA plots are presented in Figs. 14 and 16. 0 1 2 3 4 5 6 7 8 9 10 [GeV/c] T,VM p 1 10 2 10 3 10 4 10 5 10 6 10 counts MC RECO Truth vs Reco (VM) T p 0 0.5 1 1.5 2 2.5 3 [GeV/c] T,VM,MC p 0.1− 0.08− 0.06− 0.04− 0.02− 0 0.02 0.04 0.06 0.08 0.1 T,MC )/p T,MC -p T,RECO (p 1 10 2 10 3 10 4 10 Resolution (VM) T p 0 0.5 1 1.5 2 2.5 3 [GeV/c] T,VM,RECO p 0 0.5 1 1.5 2 2.5 3 [GeV/c] T,VM,MC p 1 10 2 10 3 10 4 10 5 10 Response (VM) T p Figure 13: The leftmost plot shows the reconstruction of pTfor the VM in comparison with the truth. The middle plot shows the resolution and the rightmost plot shows the response. •Momentum Distribution: For the new method presented in this analysis (projection method), we want to find the region of phase space where the four-momentum of the VM is dominate in the direction normal (ˆn) to the electron scattering plane [3]. The reason for this is that |t|component along the ˆndirection is unaffected by the momentum resolution e′. Therefore, we apply a cut on the angle between ˆnand the direction of the scattered electron (ˆx). This minimizes the VMs four-momenta in the direction of the electron scattering plane. We do this by defining the angle θmax = tan−1(qx/qy) where |t|⊥=q2 ⊥=q2 x+q2 y qx=q⊥sin θ qy=q⊥cos θ, (13) 17 1 10 2 10 3 10 4 10 5 10 0 0.5 1 1.5 2 2.5 3 [GeV/c] T,VM,MC p 0 0.5 1 1.5 2 2.5 3 [GeV/c] T,VM,RECO p Bin Migration 0 0.5 1 1.5 2 2.5 3 [GeV/c] T,VM,reco bin p 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Purity bin width=0.05 GeV/c Purity 0 0.5 1 1.5 2 2.5 3 [GeV/c] T,VM,true bin p 0 0.2 0.4 0.6 0.8 1 Stability bin width=0.05 GeV/c Stability 0 0.5 1 1.5 2 2.5 3 [GeV/c] T,VM p 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Acceptance Acceptance 0 0.5 1 1.5 2 2.5 3 [GeV/c] T,VM p 0 0.2 0.4 0.6 0.8 1 Efficiency Efficiency 0 0.5 1 1.5 2 2.5 3 [GeV/c] T,VM p 3 10 4 10 5 10 6 10 Corrected Acceptance Corrected Figure 14: QA plots for the VMs pTdistribution. 0 2 4 6 8 10 12 14 16 18 20 [GeV] VM ) z (E-p 2 10 3 10 4 10 5 10 counts MC RECO Truth vs Reco (VM) z E-p 0 2 4 6 8 10 12 14 16 18 20 [GeV] VM,MC ) z (E-p 0.1− 0.08− 0.06− 0.04− 0.02− 0 0.02 0.04 0.06 0.08 0.1 VM,MC ) z ]/(E-p VM,MC ) z - (E-p VM,RECO ) z [(E-p 1 10 2 10 3 10 4 10 Resolution (VM) z E-p 1 10 2 10 3 10 4 10 5 10 0 2 4 6 8 10 12 14 16 18 20 [GeV] VM,RECO ) z (E-p 0 2 4 6 8 10 12 14 16 18 20 [GeV] VM,MC ) z (E-p Response (VM) z E-p Figure 15: The leftmost plot shows the reconstruction of the E−pzdistribution of the VM in comparison with the truth. The middle plot shows the resolution and the rightmost plot shows the response. 18 1 10 2 10 3 10 4 10 5 10 0 1 2 3 4 5 6 7 8 9 10 [GeV] VM,MC ) z (E-p 0 1 2 3 4 5 6 7 8 9 10 [GeV] VM,RECO ) z (E-p Bin Migration 0 2 4 6 8 10 12 14 16 18 20 [GeV] VM,reco bin ) z (E-p 0 0.2 0.4 0.6 0.8 1 Purity bin width=0.1 GeV Purity 0 2 4 6 8 10 12 14 16 18 20 [GeV] VM,true bin ) z (E-p 0 0.2 0.4 0.6 0.8 1 Stability bin width=0.1 GeV Stability 0 2 4 6 8 10 12 14 16 18 20 [GeV] VM ) z (E-p 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Acceptance Acceptance 0 2 4 6 8 10 12 14 16 18 20 [GeV] VM ) z (E-p 0 0.2 0.4 0.6 0.8 1 Efficiency Efficiency 0 2 4 6 8 10 12 14 16 18 20 [GeV] VM ) z (E-p 4 10 5 10 Corrected Acceptance Corrected Figure 16: QA plots for the VMs E−pzdistribution. choosing the optimal angle needed for analysis. Figure 17 shows the 2-dimensional |t| distribution where a cut of θmax is shown. The following cuts have been made only on the reconstruction level of the events from the detector information (in addition to the cuts mentioned above). •Geometric acceptance: We place a constraint on the radius of clusters (r < 550 mm) that are too far from the detector to ensure that the electron lands within the detector’s sensitive region. The resulting distribution is shown in the rightmost plot of Figure 18. •Event selection: Here, we choose 15 < E −pz<25 GeV of the scattered electron to ensure exclusivity. Additionally, we implement a selection of 0.9< E/p < 1.2 to confirm that the EMCal energy matches the track momentum and that we select true electrons. The results of these cuts are shown in Figures 21 and 19 while the QA plots are shown in Figs. 22 and 20. 3.2.1 Reconstruction Methods The MCParticles branch is used to reconstruct the MC particle momentum, generator status, mass, and PDG arrays for each MC particle. To reconstruct the EMCal endcap clusters, we use the position and energy arrays from the EcalEndcapNClusters and EcalEndcapNRecHits branches. The EcalEndcapNClusterAssociations branch is used for the rec and sim ID arrays. We also reconstruct calorimeter tracks using the _CalorimeterTrackProjections_points branch for the position and momentum arrays. The ReconstructedChargedParticles branch is used to reconstruct the 19 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 |t|x[GeV/c] 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 |t|y[GeV/c] 1 10 102 103 2D |t| Distribution θ max Figure 17: |t|=q2distribution in 2-dimensions. The wedge cut is shown by the angle θmax to demonstrate the projection technique. 20 1 10 2 10 3 10 4 10 Cluster Position Without Cut 800−600−400−200−0 200 400 600 800 x [mm] 800− 600− 400− 200− 0 200 400 600 800 y [mm] Cluster Position Without Cut 1 10 2 10 3 10 4 10 Cluster Position With Cut 800−600−400−200−0 200 400 600 800 x [mm] 800− 600− 400− 200− 0 200 400 600 800 y [mm] Cluster Position With Cut Figure 18: Distribution of cluster positions in EMcal with and with geometric cut. 15 16 17 18 19 20 21 22 23 24 25 ) [GeV] z (E - p 0 500 1000 1500 2000 2500 3 10× counts MC RECO ) Truth vs. Reco (e'+HFS) z (E-p 19.4 19.5 19.6 19.7 19.8 19.9 20 20.1 [GeV] MC ) z (E-p 0.3− 0.2− 0.1− 0 0.1 0.2 0.3 MC ) z )/(E-p MC ) z -(E-p REC ) z ((E-p 1 10 2 10 3 10 4 10 5 10 ) Resolution (e'+HFS) z (E-p 1 10 2 10 3 10 4 10 5 10 16 18 20 22 24 ) [GeV] z,trk - p EMCal (E 19.4 19.5 19.6 19.7 19.8 19.9 20 20.1 ) [GeV] z,MC -p MC (E ) Response (e'+HFS) z (E-p Figure 19: The leftmost plot shows the reconstruction of the E−pzdistribution from e′in comparison with the truth. We expect this to peak around 2Ee= 20 GeV. The middle plot shows the resolution and the rightmost plot shows the response. 21 19 19.2 19.4 19.6 19.8 20 20.2 20.4 MC ) z (E-p 16 18 20 22 24 RECO ) z (E-p 1 10 2 10 3 10 4 10 5 10 Bin Migration 18 18.5 19 19.5 20 20.5 21 21.5 22 [GeV] reco bin ) z (E-p 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 Purity bin width=0.45 GeV Purity 18 18.5 19 19.5 20 20.5 21 21.5 22 [GeV] true bin ) z (E-p 0 0.02 0.04 0.06 0.08 0.1 0.12 Stability bin width=0.45 GeV Stability 18 18.5 19 19.5 20 20.5 21 21.5 22 ) [GeV] z (E-p 0 0.1 0.2 0.3 0.4 0.5 6− 10× Acceptance Acceptance 18 18.5 19 19.5 20 20.5 21 21.5 22 ) [GeV] z (E-p 0 0.2 0.4 0.6 0.8 1 Efficiency Efficiency 18 18.5 19 19.5 20 20.5 21 21.5 22 ) [GeV] z (E-p 9 10 10 10 11 10 12 10 Corrected Acceptance Corrected Figure 20: QA plots for the E−pzdistribution from e′. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 E/|p| 4 10 5 10 6 10 7 10 counts MC RECO E/|p| Truth vs. Reco 0.98 0.99 1 1.01 1.02 1.03 1.04 MC (E/|p|) 0.2− 0.15− 0.1− 0.05− 0 0.05 0.1 trk /|p| EEMC E 1 10 2 10 3 10 4 10 5 10 E/|p| Resolution 1 10 2 10 3 10 4 10 5 10 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 trk /|p| EEMC E 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 MC (E/|p|) E/|p| Response Figure 21: The leftmost plot shows the reconstruction of E/|p|for e′in comparison with the truth. We expect this to peak at one to demonstrate good matching of the EMcal with the track. The middle plot shows the resolution and the rightmost plot shows the response. 22 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 2 |t| [GeV/c] 1− 10 1 10 2 10 3 10 4 10 5 10 ] 2 /d|t| [nb(GeV/c)σd : MCφ /12π= max θ: φ DIS /12π= max θ: φIncoherent /12π= max θ: ρ ePIC Simulation 25.10.2/3, 10x100 GeV - K + K→ φ φ e'Au'→eAu /A -1 = 10 fb int L Figure 27: |t|distribution with all systematic uncertainties studied in the analysis without any vetoes, cuts, or PID applied. The black solid curve represents the MC truth distribution, the open pink stars correspond to the projection method, light blue plus signs for the DIS distribution, solid orange squares for the incoherent ϕproduction, and the open gray cross for the ρproduction. 29 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 2 |t| [GeV/c] 1− 10 1 10 2 10 3 10 4 10 5 10 ] 2 /d|t| [nb/(GeV/c)σd : MCφ /12π= max θ: φ DIS ePIC Simulation 25.10.2, 10x100 GeV - K + K→ φ φ e'Au'→eAu /A -1 = 10 fb int L Figure 28: |t|distribution with DIS background shown as the light blue plus signs, the MC truth is the black solid line and the projection method corresponds to the open pink stars. We see that the DIS curve only allows us to resolve the first minima. 3.3.2 ρProduction Another source of systematic uncertainty is the production of the ρVM. Figure 31 shows the |t|distribution with the included ρcontamination. After implementing PID using the branch ReconstructedChargedParticles.PDG, we observe that the projection method is able to remove all of the ρproduction. Figure 30 shows the VM mass distributions with and without PID. We see that without PID, the selection on the ϕmass successfully reduces the number of ρparticles but there still remains a significant amount of misidentified ρparticles. Figure 31 shows the |t|distribution of the misidentified ϕ. Again, we can only resolve the first minima. However, upon implementing PID, we are able to remove all of the ρbackground as shown in Figs. 31 and 30. 3.3.3 Incoherent Production The dominate source of systematic uncertainty in this analysis arises from the incoherent production. As shown in Fig. 32, we see that the incoherent background dominates our diffractive pattern in the |t|distribution. After implementing the detector vetoes and cuts as mentioned in Section 3.3.1, we see some surpression, as shown in Fig. 33, but there is still an overwhelming amount of incoherent production remaining. Following the proposed spin-based technique in [3], we plan to statistically remove this background in the future. 30 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 2 |t| [GeV/c] 1− 10 1 10 2 10 3 10 4 10 5 10 ] 2 /d|t| [nb/(GeV/c)σd : MCφ/12π= max θ: φ DIS: no vetoes DIS: OMD only DIS: RP only DIS: only detectors onlyηDIS: DIS: all vetoes ePIC Simulation 25.10.2, 10x100 GeV - K + K→ φ φ e'Au'→eAu /A -1 = 10 fb int L Figure 29: |t|distribution with all vetoes and cuts applied. The solid black line corresponds to the MC truth, open pink stars to the projection method, light blue plus signs for DIS with no vetoes, solid green squares for DIS with only OMD vetoes, open red squares for DIS with only RP vetoes, orange stars for DIS with all detector vetoes, open purple circles for DIS with only ηand HFS cuts, and the gray open double diamonds represent DIS with all cuts and vetoes applied. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ] 2 VM mass [GeV/c 2 10 3 10 4 10 5 10 6 10 counts Mass Distribution (no PID) MCρ RECρ REC before mass cutρ MCφ RECφ Mass Distribution (no PID) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ] 2 VM mass [GeV/c 0 500 1000 1500 2000 2500 3000 3500 3 10× counts Mass Distribution (with PID) MCρ RECρ REC before mass cutρ MCφ RECφ Mass Distribution (with PID) Figure 30: The leftmost plot shows the mass distributions of ρand ϕwith no PID while the rightmost plot shows the same distributions with PID. The orange dashed curve represents the ρMC mass distribution, the dashed blue curve is for ρreconstruction, the green long dash is for ρreconstruction before the ϕmass selection is applied, the solid black curve is for the ϕMC distribution, and the dashed purple curve is for the reconstructed ϕ. 31 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 2 |t| [GeV/c] 1− 10 1 10 2 10 3 10 4 10 5 10 ] 2 /d|t| [nb/(GeV/c)σd : MCφ /12π= max θ: φ /12π= max θ (no PID): ρ /12π= max θ (w. PID): ρ ePIC Simulation 25.10.2/3, 10x100 GeV - K + K→ φ φ e'Au'→eAu /A -1 = 10 fb int L Figure 31: |t|distribution with ρproduction. The solid black curve represents the ϕMC truth distribution, the open pink stars for the ϕdistribution projection method, the gray open crosses represent the ρdistribution with no PID, and the blue open double diamonds represent the ρdistribution with PID. 3.4 Analysis Code –Code found on GitHub [15] where you will find: ∗model simulations: Simulations were developed and tested before the projection technique was implemented using the ePIC software. The files needed to reproduce the model are found in this folder. ∗EICreconOutputReader: An original simple analysis code based on [16]. ∗diffractive phi analysis: The final version of the analysis code (used to write this note) can be found here. This folder contains: ·All necessary header files in the ”header files” folder. Plot generators in the ”plot macros” folder. ·In this folder there are plot macros to generate: 1. Plots in this analysis note labeled ”AnalysisNote plots diffractive phi.C”. 2. Comprehensive plots of the |t|distribution scaled to Early Science and preTDR luminosities, different configurations of systematic uncertainties, and further detector analyses. This macro is labeled ”diffractive t plots.C” 3. Analysis plots that can be used for further investigation labeled ”plot QA.C”. 32 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 2 |t| [GeV/c] 1− 10 1 10 2 10 3 10 4 10 5 10 ] 2 /d|t| [nb/(GeV/c)σd : MCφCoherent /12 π= max θ: φCoherent /12π= max θ: φIncoherent ePIC Simulation 25.10.2, 10x100 GeV - K + K→ φ φ e'Au'→eAu /A -1 = 10 fb int L Figure 32: |t|distribution with incoherent production. No vetoes or cuts have been applied. The black solid curve represents the coherent MC truth, the open pink stars for the projection method for coherent ϕreconstruction, and the filled orange squares is the incoherent production distribution. ·The ”preTDR” folder contains all the information needed to reproduce the plots in the preTDR. ·The analysis code is found in the ”src” folder labeled ”diffractive vm full analysis.cxx”. Scripts in this folder are for submitting jobs to condor and preparing the file lists to be fed into the analysis code. To submit jobs and run multiple files from a list: 1. From eic-shell: ./prep file list.sh 2. Not in eic-shell: ./run analysis 2 To run the analysis for a single file: ·From the terminal: root ’diffractive vm full analysis.cxx (<input filename.root>,<output filename.root>)’ 4 Results and Discussion The results of the |t|distribution reconstruction are shown in Figs. 34 and 35. First, we can see a significant enhancement in resolving the diffractive pattern of the |t| distribution. Previously, the best method available for us to use was method L as stated in [14]. The projection method enables us to determine the peaks and minima with a much better resolution than before. The price we pay for making the wedge cut in the projection method is a loss of statistics. In this example, we use θmax =π/12 33 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 2 |t| [GeV/c] 1− 10 1 10 2 10 3 10 4 10 5 10 2 /d|t| [nb/(GeV/c)]σd : MCφ Coherent /12 π= max θ: φCoherent : no vetoesφIncoh. : RPφIncoh. : OMDφIncoh. : ZDCφIncoh. : all detectorsφIncoh. cutsη: φIncoh. : all vetoesφIncoh. ePIC Simulation 25.10.2, 10x100 GeV - K + K→ φ φ e'Au'→eAu /A -1 = 10 fb int L Figure 33: |t|distribution with incoherent production and individual detector vetoes and cuts. The black solid curve represents the coherent ϕMC distribution, the open pink stars are for the coherent ϕreconstruction using the projection method, the open four triangle x is for incoherent with no vetoes, the red star is for incoherent with only RP vetoes, the blue full cross x is for incoherent with only OMD vetoes, the green full diamond is for incoherent with only ZDC vetoes, the open gray circles are for incoherent with all detector vetoes, the black full down triangle is for incoherent with ηand HFS cuts, and the filled orange squares are for incoherent production with all vetoes applied. 34 0 0.05 0.1 0.15 2 |t| [GeV/c] 2− 10 1 2 10 4 10 6 10 ] 2 /d|t| [nb/(GeV/c)σd , 0.01 < y < 0.85 2 <10 GeV 2 1<Q | < 0.02 GeV φ M− inv |<3.5, |M φ |y ePIC Simulation 25.10.2 - K + K→ φ , 10x100 GeVφ e'Au'→eAu /A -1 = 10 fb int L MC φSartre /12π = max θ RECO φSartre Figure 34: |t|distribution after applying a cut of θ=π/12. This plot shows the improvement of the |t|measurement by comparing the coherent truth (MC reconstruction, black curve) with the new projection method (open pink stars). 35 10−5−0 5 10 b [fm] 0.05− 0 0.05 0.1 0.15 F(b) db ∫ F(b)/ ePIC Simulation 25.10.2, 10x100 GeV - K + K→ φ φ e'Au'→eAu MC Method L Projection method Figure 35: 2-dimensional Fourier-Bessel transformation of the |t|distribution with wedge cut of θ=π/12. This shows the gluon spatial distribution with a comparison from the transforms of the truth, method L, and the projection method distributions. The black solid curve represent the truth distribution, the blue filled circles correspond to |t|reconstruction using method L, and the pink open stars represent the reconstruction of |t|using the new projection technique. 36 which results in ≈83% of lost coherent VM events. Second, Fig. 35 clearly indicates an improvement from the method L reconstruction. This analysis demonstrates that the projection method used to reconstruct the momentum distribution is an effective approach to overcome the previous challenges of the |t|measurement and allows more precise spatial imaging of the gluons inside the nucleus. In future work we will statistically separate the coherent and incoherent events to reduce the incoherent background. The outcome will be a complete analysis technique for the measurement of the |t|distribution. References [1] A. Accardi et al., Eur. Phys. J. A 52, 268 (2016), 1212.1701. [2] N. Armesto and A. H. Rezaeian, Phys. Rev. D 90, 054003 (2014), 1402.4831. [3] M. Kesler et al., (2025), 2502.15596. [4] T. Toll and T. Ullrich, Phys. Rev. C 87, 024913 (2013), 1211.3048. [5] STAR Collaboration, L. Adamczyk et al., Phys. Rev. C 96, 054904 (2017). [6] Reconstruction campaigns, Accessed: November 10, 2025. [7] S. Rahman, Accessed: December 15, 2025. [8] T. Toll and T. Ullrich, Comput. Phys. Commun. 185, 1835 (2014), 1307.8059. 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