Measurement of the 12C(e,e′) Cross Sections at Q2=0.8GeV2/c2

Few-Body Syst (2024) 65:78 https://doi.org/10.1007/s00601-024-01944-y M. Mihoviloviˇc·L. Doria ·P. Achenbach · A. M. Ankowski ·S. Bacca · D. Bosnar ·A. Denig ·M.O. Distler · A. Esser ·I. Frišˇci´c·C. Giusti ·M. Hoek ·S. Kegel · M. Littich ·G.D. Megias ·H. Merkel ·U. Müller · J. Pochodzalla ·B. S. Schlimme ·M. Schoth ·C. Sfienti · S. Širca ·J. E. Sobczyk ·Y. Stöttinger ·M. Thiel Measurement of the 12C(e,e)Cross Sections at Q2=0.8GeV2/c2 Received: 20 June 2024 / Accepted: 1 July 2024 / Published online: 1 August 2024 © The Author(s) 2024 Abstract We present the findings of a study based on a new inelastic electron-scattering experiment on the 12C nucleus focusing on the kinematic region of Q2=0.8GeV2/c2. The measured cross section is sensitive to the transverse response function and provides a stringent test of theoretical models, as well as of the theoretical assumptions made in Monte-Carlo event-generator codes developed for the interpretation of neutrino-nucleus experiments,suchasDUNEandHyperK.WefindthatmoderngeneratorssuchasGENIEandGiBUUreproduce our new experimental data within 10%. M. Mihoviloviˇc·S. Širca Jožef Stefan Institute, 1000 Ljubljana, Slovenia M. Mihoviloviˇc·S. Širca Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia L. Doria ·S. Bacca ·A. Denig ·M. O. Distler ·A. Esser ·M. Hoek ·S. Kegel ·M. Littich ·H. Merkel·U. Müller · J. Pochodzalla ·B. S. Schlimme ·M. Schoth ·C. Sfienti ·J. E. Sobczyk ·Y. Stöttinger ·M. Thiel Institut für Kernphysik, Johannes Gutenberg-Universität Mainz, 55128 Mainz, Germany P. Achenbach Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA A. M. Ankowski Institute of Theoretical Physics, University of Wrocław, pl. Maxa Borna 9, 50-204 Wrocław, Poland L. Doria ·S. Bacca ·A. Denig ·H. Merkel (B )·B. S. Schlimme ·C. Sfienti ·J. E. Sobczyk ·M. Thiel PRISMA+ Cluster of Excellence, Johannes Gutenberg-Universität Mainz, 55128 Mainz, Germany E-mail: [email protected] D. Bosnar ·I. Frišˇci´c Department of Physics, University of Zagreb, HR-10002 Zagreb, Croatia C. Giusti INFN, Sezione di Pavia, 27100 Pavia, Italy G. D. Megias Departamento de Física Atómica, Molecular y Nuclear, Universidad de Sevilla, 41080 Seville, Spain 78 Page 2 of 9 M. Mihoviloviˇcetal. 1 Introduction Electrons represent a very precise probe for the investigation of the atomic nucleus [1]. In the past decades, experiments with electrons have provided increasingly accurate information on the structure of nuclei and their constituents [2–9]. At the heart of this effort are the inelastic scattering experiments on nuclear targets at energies below 1 GeV, which give insight into the properties and dynamics of nucleons embedded in the nuclear medium. In such scattering processes, an electron with energy E0interacts with the nucleus at rest by exchanging a virtual photon transferring energy ωand momentum q, such that Q2=q2−ω2>0. Using the nucleon mass mNas a scale, the energy and momentum transfer variables can be rewritten in dimensionless form as λ=ω 2mN ,κ=q 2mN ,τ=Q2 4m2 N=κ2−λ2. The differential cross section describing the inclusiveinteraction of the electron with the nucleus can be written as d2σ ddω=σM[vLRL(ω, q)+vTRT(ω, q)],(1) where σMis the Mott cross Sect. [1], while vLand vTare kinematic factors given by vL=τ κ22 ,v T=τ 2κ2+tan2θe 2, and θeis the angle of the scattered electron. The cross section depends on two nuclear responses, the longitudinal and the transverse response functions, which are both functions of ωand q=|q|. The longitudinal response function, RL(ω, q), depends on the charge operator and carries information on the nucleon-nucleon correlations, while the transverse response function, RT(ω, q), is driven by the magnetic currents [1]. Various cross section measurements were performed, mostly before 2000, but the acquired data were dominated by the longitudinal (charge) part of nuclear response. Historically, the most extensively studied nucleus has been carbon. For this nucleus, the richest sample of (e,e)data exists. It consists of almost 3500 data points from 12 experiments [10,11] for energies between 0.12 and 17.3 GeV and scattering angles up to 145◦. These data have been used to study the structure of this nucleus and to develop models describing its electromagnetic response. In the past, theoretical calculations for 12C were often limited to the quasi-elastic (QE) region, and the most demanding part was the description of the transverse response, which has been for a long time incomplete [1]. A more comprehensive description of the 12C(e,e)cross section was developed in the microscopic calculation byGiletal. [12], and more recently Megias et al. [13] proposed a superscaling model, called SuSAv2-MEC, which considers the complete inelastic spectrum. The model shows quite good agreement with data over a broad range of energy transfer. However, the description of the cross sections at large scattering angles remains incomplete. An important motivation for new studies of inclusive cross sections comes also from the neutrino physics community [14,15]. Shortand long-baseline neutrino experiments detect neutrinos through their interactions withnucleiandaimattheprecisemeasurementofneutrinomasses,mixingangles,andCP-violatingphaseinthe lepton sector. These measurements represent one of the highest priorities of contemporary fundamental physics and hinge on the ability of the experiments to reconstruct the neutrino energy and on the precise knowledge of the neutrino-nucleus cross sections. Although a vigorous experimental program for the measurement of such cross sections is in progress [16–20], neutrino experiments are mostly limited by statistical uncertainties and the lack of knowledge of the neutrino flux. Electron scattering experiments, with a precisely determined beam energy and the possibility to perform inclusive as well as exclusive measurements with different final states, have the potential to provide very precise data for testing the nuclear models employed in neutrino experiments. Indeed, it has been demonstrated that the interpretation of the measured neutrino oscillations requires extensive theoretical and experimental support from the nuclear physics community. In this context the 12C(e,e) reaction has played an important role in the development of reliable models describing cross Sect. [13,21]in experiments like MiniBooNE [22], MINERvA [23], and T2K [24] that use carbon-based materials (mineral Measurement of the 12C(e,e)Cross Sections Page 3 of 9 78 oils, plastic scintillators) as detector medium. To ensure further involvement of modern neutrino event generators like GENIE [25]andGiBUU[26], the advances in the built-in theoretical models must be complemented by the new experimental data on relevant nuclear targets and in relevant kinematics [27,28]. In this paper we focus on 12C and present new data at Q2=0.8GeV2/c2for two reasons. On the one hand carbon is an interesting target for neutrino experiments as mentioned above, and on the other hand new data at large Q2in a kinematic dominated by the transverse response will boost further theoretical progress [29–38]. 2 Experiment The measurement of the inclusive cross section on 12C was performed at the Mainz Microtron (MAMI) facility using the spectrometer setup of the A1 Collaboration [39]. In the experiment, an electron beam with energy E0=855MeV was used in combination with a 43 mg/cm2thick carbon foil target. For measuring the cross section as a function of energy of scattered electron Ewe employed a magnetic spectrometer (spectrometer A) with 20% momentum acceptance and 28 msr angular acceptance. The spectrometer was positioned at a fixed angle of 70◦, while its momentum settings were adjusted to measure the cross section as a function of ω=E0−E. The measurements were made for seven different momentum settings between 310 MeV/c and 650 MeV/cin order to collect data in the region of the QE peak and the -resonance. For each setting we collected 1.8 million events. The central momentum of each setting was measured to a relative accuracy of 8 ×10−5. The spectrometer was equipped with a detector package consisting of two layers of vertical drift chambers (VDCs) for tracking, two layers of plastic scintillation detectors for triggering, and a threshold Cherenkov detector for electron identification. The beam current was between 2 and 3 μA and was limited by the maximum data acquisition rate, resulting in a raw rate of about 500 Hz. The current was determined by a non-invasive fluxgate-magnetometer with an accuracy of <0.2%. The experiment provided new cross sections measurements in the region of beam energies and scattering angles, where the existing measurements are very sparse, see Fig. 1. The quasi-elastic peak is centered at |q|=0.84 GeV/c, thus nicely complementing previous measurements at |q|≈0.8GeV/cperformed at 560 MeV and 1299 MeV [40,41]. The experimental cross sections for the 12C(e,e)reaction were extracted from the data by dividing the measured distributions of counts by the integrated luminosity and the solid angle accepted by the spectrometer. The accepted solid angle was simulated using a dedicated simulation for the three spectrometers facility of the A1 Collaboration [39], Simul++. To ensure a reliable comparison with the data, the simulation included realistic momentum and spatial resolutions of the spectrometer. The relative momentum, angular, and vertex resolutions (FWHM) were 2.4×10−4,4.7mrad,and9.4 mm, respectively. The simulation also considered the This work Sealock et al. 1989 Barreau et al. 1983 Existing data q [ GeV / c ] ω[GeV] 10.80.60.40.20 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Fig. 1 Kinematic configurations of the 12C(e,e)cross section data in terms of energy and momentum transfers ωand |q|. Black points represent available cross section measurements with relative uncertainties smaller than 10%. The kinematics covered by this work are presented with red circles. The complementary measurements of Barreau et al. [40], and Sealock et al. [41], also at |q|∼0.8GeV/c, are shown with blue diamonds and green squares, respectively 78 Page 4 of 9 M. Mihoviloviˇcetal. Table 1 Detection efficiencies of the setup Contribution Efficiency factor Uncertainty Scintillator efficiency 0.990 0.003 Cherenkov efficiency 0.999 0.001 VDC efficiency 0.999 0.001 Particle identification 0.983 0.017 The correction factors for the cross sections are given by their inverses Fig. 2 Left: Measured cross section compared to the QE calculations of Megias et al. [13], Giusti et al. [45,46], Sobczyk et al. [47] and Benharet al. [10]. The gray band shows the envelope of the quasi-elastic cross-section calculations and represents a measure of the differences between different models. Center: Measured cross section compared to the full theoretical calculations of Megias et al. [13] based on the SuSAv2-MEC model and predictions of Ankowski [48] based on the global Fermi gas model (FG) and local Fermi gas model (LFG). Right: Comparison of the new data with the results of the Monte-Carlo generators GiBUU [26] and GENIE (version 3.4) [25] employing LFG and SuSAv2-MEC [21] nuclear cross-section models electron energy corrections due to multiple scattering and radiation losses. The internal and external radiative corrections were included using the formalism of Mo and Tsai [42]. The accompanying multiple scattering corrections in the target and surrounding material were approximated by a Landau distribution [43]. Altogether, the energy corrections have less than 4% effect on the measured cross section. The integrated luminosity was determined from the product of the accumulated charge and the surface densityofthetargetmaterial,correctedfordead-timeandDAQprescalefactors.Theluminositywasdetermined separately for each collected data-sample to ensure that dead-time and prescale corrections were consistently considered when weighting the measured spectra. The measured spectra were corrected for the inefficiencies of the detection system, see Table 1.The efficiencies of the scintillation detector and the Cherenkov detector were evaluated in a past experiment [44] and were determined to be 99.0% and 99.85 %, respectively. The efficiency of the track reconstruction in the VDCs was determined to be (99.98 ±0.05)%. All three corrections were considered as multiplicative correction factors. Several cuts were applied to both data and simulation. First, a cut on the Cherenkov signal was applied to identify electrons and minimize the background arising from cosmic particles and by negatively charged pions from the 12C(e,π−)reaction. This cut was followed by cuts on the nominal momentum and angular acceptance of the spectrometer in order to remove the artefacts at its edges, caused by the inefficient parts of the detectors, fringe fields in the spectrometers, and secondary particles rescattered from parts of the collimator. The extracted cross sections are presented in Fig. 2. The systematic uncertainties of the extracted cross sections are a combination of various contributions. The uncertainties related to the detector efficiencies are collected in Table 1. The uncertainty of the luminosity is given by the uncertainty of the absolute beam current calibration, which amounts to 3.3 nA at 855 MeV and the fluctuations of the beam current were related to the instabilities of accelerator operation. The latter were smaller than 5 nA, resulting in the total systematic uncertainty smaller than 0.16 %. The dominant contribution Measurement of the 12C(e,e)Cross Sections Page 5 of 9 78 to the systematic uncertainty is related to the misidentification of particles in the Cherenkov detector and cuts applied to distinguish electrons from pions and muons. This uncertainty was estimated to be 1.7%.Thelast relevant contribution to the systematic uncertainty can be evaluated by the formalism of Mo and Tsai [42], employed to describe radiative and multiple scattering corrections to the cross section. These corrections add 0.2 % to the total uncertainty of the measured cross sections. Finally, the uncertainty of the position of the extracted cross sections on the energy scale is related to the ambiguities in the absolute energy calibration of the accelerator and spectrometer and amounts to 2.7 MeV, which is less than 1/3 of the employed energy bin size. 3 Comparison to Models and Event Generators The extracted cross section is first compared to QE calculations of Giusti et al. [45,46], Sobczyk et al. [47], Megias et al. [13] and Benhar et al. [10]. Figure 2(Left) shows that the calculations agree with each other at the level of 4 % at the top of the QE peak. The comparison of the experimental results to the comprehensive calculations of Megias et al. [13], which are based on the SuSAv2-MEC model, are shown in Fig. 2(Center). The model exhibits very good overall agreement with the data, on average at the level of 7%. Surprisingly the largest inconsistency between the data and the calculations appears at the top of the QE peak, where the discrepancyis 9 %. SincetheQEcalculationsshow aconsistentpicturethere,theobserved discrepancybetween the data and the SuSAv2-MEC model is most likely related to the incomplete or inconsistent description of the processes in the “dip" region, which are the only remaining relevant contributions to the cross section at ω≤400 MeV. An agreement at a similar level has been achieved by Ankowski [48] who calculated cross sections by using both the global Fermi gas model (FG) and the local Fermi gas model (LFG) in combination with the Bosted-Christy [49,50] approach for describing pion production processes in the “dip” and in the -resonance region. The relative deviation of the FG calculation from the data is on average 10%, while the prediction of the LFG model agrees with the data at the level of 8%. Both calculations exhibit a visible inconsistency at the top of the QE peak and in the “dip" region, where the calculated cross-section do not follow the correct trend of the data. Finally, the extracted cross-sections were compared also to the results of the Monte-Carlo generators GiBUU [26] and GENIE [11,25]. Figure2(Right) shows that at the selected kinematic setting the generators describe the data reasonably well. GiBUU agrees with the data at the level of 9%. The accuracy of the GENIE generator depends on the model used for calculating nuclear cross sections. When the local Fermi gas model is used, the calculated cross section agrees on average with the data at the level of 22%. Similarly to the results of Ankowski’s LFG model, the simulated cross section overestimates the QE cross section. Additionally, the GENIE simulation lacks strength in the -region, where the calculated cross section is 17% smaller than the data. The simulated results improve when GENIE uses the SuSAv2 model for describing QE scattering and processes in the “dip" region. In this case we observe much better agreement between the data and simulation at the QE peak. Note that GENIE still uses its default model for describing the cross section in the -resonance region, which is the source of difference with respect to the Megias et al. result shown in the center panel. For a more insightful analysis of the measured cross section and comparison with the existing results obtained under different kinematic conditions, the scaling formalism [51] can be employed. The formalism was first developed within the framework of the relativistic Fermi gas model where the characteristic momentum is the Fermi momentum kF, which can be expressed as a dimensionless scale parameter ξF=1+k2 F/m2 N−1. Building on this formalism, two dimensionless scaling variables ψand ψwere proposed [52]: ψ≡1 √ξF λ−τ (1+λ)τ +κ√τ(τ +1) , ψ≡1 √ξF λ−τ (1+λ)τ+κ√τ(τ+1) .(2) The variable ψis corrected for an empirical energy shift Eshift corresponding to the average of the separation energies of the various shells contributing to the nuclear ground state [51]. With the shift one achieves the center ofthe QE peak to be at ψ=0. Thenecessary shiftis achieved by substituting λand τwith λ=λ−Eshift/2mN and τ=κ2−λ2. 78 Page 6 of 9 M. Mihoviloviˇcetal. The idea of the scaling formalism is to factorize the elastic cross section on a single nucleon, obtaining in this way a universal scaling function which contains information about the nuclear structure. For that purpose, reduced longitudinal and transverse response functions are introduced as [52]: fL=kF RL GL(κ, λ),fT=kF RT GT(κ, λ). The functions GLand GTare expressed as: GL(κ, λ) =(κ2/τ)[˜ G2 E+˜ W2] 2κ[1+ξF(1+ψ2)/2], GT(κ, λ) =2τ˜ G2 M+˜ W2 2κ[1+ξF(1+ψ2)/2], where =ξF(1−ψ2)√τ(1+τ κ+1 3ξF(1−ψ2)τ κ2, ˜ W1=τ˜ G2 M, ˜ W2=1 1+τ˜ G2 E+τ˜ G2 M, ˜ G2 E=ZG p E 2+NG n E 2, ˜ G2 M=ZG p M 2+NG n M 2. Here Zand Nrepresent the number of protons and neutrons in the nucleus, respectively, Gp,n Eand Gp,n M are nucleon electric and magnetic form factors [53]. Using these functions a dimensionless scaling function for the total cross section can be written as: f=kF d2σ/dedω σM[vLGL(κ, λ) +vTGT(κ, λ)]. =fLsin2χTL +fTcos2χTL,(3) where the angle χTL is defined as tan2χTL =vLGL vTGT .(4) This angle characterizes the ratio between the longitudinal and transverse contributions to the cross section. At χTL ≈0 the inclusive cross section is dominated by the transverse response, while at χTL ≈90◦the cross section is governed by the longitudinal response [51]. Using the scaling variables in Eqs. (2), (3), and (4), the measured cross sections could be compared to the previous measurements at |q|≈0.8GeV/cof Barreau et al. [40] and Sealock et al. [41]. The extracted values of the dimensionless scaling function f(ψ)are shown in Fig. 3. This figure shows the approximate scaling of the measured cross sections which starts to break for ψ>0 when the transverse contributions of the -resonance begin to dominate the cross Sect. [51,52]. At the QE peak the experimental values of this work and Sealock et al. collected at scattering angles θ<90◦, which corresponds to χTL =36◦and 46◦, respectively, agree very well with each other. On the other hand, the scaling function reconstructed from data of Barreau et al. at θ=145◦is over 20 % higher at the top of the QE peak. These data coincide with a much smaller value of χTL =12◦, and are thus dominated by the transverse response RTwhich is known to break scaling due to various nonelastic contributions ranging from final-state-interaction (FSI) effects to contributions from the meson-exchange currents (MEC) [51]. Experimental values were compared also to the full calculations of Megias et al.. Interestingly, at the top of the QE peak the theory is consistent with the data of Barreau et al., but overshoots the experimental values of this work and that of Sealock et al.. The analysis of the QE cross sections has revealed that for these two data sets the QE part matches the strength of the measured cross section, indicating that the discrepancy might be due to the overestimated MEC contributions. Measurement of the 12C(e,e)Cross Sections Page 7 of 9 78 SuSAv2 QE (1299 MeV) SuSAv2 QE (855 MeV) SuSAv2 QE (560 MeV) SuSAv2 (1299 MeV) SuSAv2 (855 MeV) SuSAv2 (560 MeV) Sealock et al. Barreau et al. This work ψ f(ψ) 3210−1−2 1.5 1 0.5 0 Fig. 3 The scaling function f(ψ)at |q|≈0.8GeV/c. The experimental values of this work, Barreau et al. [40], and Sealock et al. [41] are shown together with the results of the SuSAv2-MEC model at corresponding beam energies of 560MeV (Barreau et al.), 855 MeV (this work) and 1299 MeV (Sealock et al.) and presented with the full lines. The dash-dotted lines are used to present contributions of the QE processes to the calculated scaling functions [13] 4 Conclusions We presented the experimental cross section for the inclusive reaction 12C(e,e)at Q2=0.8GeV2/c2.The measurement was made at the kinematics that is relevant for the accelerator based neutrino experiments, but where the available data are scarce. Since the new data set has not been considered in any of the theoretical models, we could use them to challenge the calculations and generators employed in the interpretation of the experiments with neutrinos. We have demonstrated that the event generators in combination with selected nuclear models are capable of describing data at the level of 10 %. For even higher precision of the generators in the future, the built-it nuclear models need to be further refined, especially the description of the transverse part of the interaction, which governs the inclusive cross section in the region of the “dip" and -resonance. To achieve this goal, further theoretical and experimental investigations of cross sections at Q2≈1GeV2/c2 are needed. Kinematics at lower Q2would also be useful to test MEC models. In particular, the prospect of having ab-initio calculations in the light and mid-mass sector with MEC [30,35,54–57] will motivate further experimental activities in the future on various targets. Acknowledgements The authors would like to thank the MAMI accelerator group for the excellent beam quality which made this experiment possible. We also thank U. Mosel for useful discussions. This work is supported by the Federal State of RhinelandPalatinate, by the Deutsche Forschungsgemeinschaft (DFG) through the Cluster of Excellence “Precision Physics, Fundamental Interactions, and Structure of Matter" (PRISMA+EXC 2118/1) funded by the DFG within the German Excellence Strategy (Project ID 390831469), by the DFG grant "Electron Scattering on Nuclei for Neutrino Physics" (Project ID 521414474), by the Slovenian Research Agency under Grants P1-0102 and J1-4383, by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 101026014, by Croatian Science Foundation under the project IP-2018-01-8570 and by the University of Tokyo ICRR’s InterUniversity Research Program FY2024 (Ref. 2024i-J-001), Japan; the Spanish Ministerio de Ciencia, Innovación y Universidades and ERDF (European Regional Development Fund) under contract PID2020-114687GB100 and by the Junta de Andalucía grant No. FQM160. Author Contributions M.M. and L.D. wrote the main manuscript text. S.B., A.A., C.G., G.M. and J.S. provided the theoretical interpretation of the data, all other authors were involved in the data taking. All authors reviewed the manuscript. Data Availability Data can be obtained from M. Mihovilovic <[email protected] >. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. 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