Measurement of the Ar(e,e ′ p) and Ti(e,e ′ p) cross sections in Jefferson Lab Hall A
L. Gu, D. Abrams, A. M. Ankowski, L. Jiang, B. Aljawrneh, S. Alsalmi, J. Bane, A. Batz, S. Barcus, M. Barroso, O. Benhar, V. Bellini, J. Bericic, D. Biswas, A. Camsonne, J. Castellanos, J.-P. Chen, M. E. Christy, K. Craycraft, R. Cruz-Torres, H. Dai, D. Day, S.-C. Dusa, E. Fuchey, T. Gautam, C. Giusti, J. Gomez, C. Gu, T. Hague, J.-O. Hansen, F. Hauenstein, D. W. Higinbotham, C. Hyde, C. Keppel, S. Li, R. Lindgren, H. Liu, C. Mariani, R. E. McClellan, D. Meekins, R. Michaels, M. Mihovilovic, M. Murphy, D. Nguyen, M. Nycz, L. Ou, B. Pandey, V. Pandey, K. Park, G. Perera, A. J. R. Puckett, S. N. Santiesteban, S. ?irca, T. Su, L. Tang, Y. Tian, N. Ton, B. Wojtsekhowski, S. Wood, Z. Ye, J. Zhang
JJLAB-PHY-21-3197SLAC-PUB-17571
Measurement of the Ar ( e, e (cid:48) p ) and Ti ( e, e (cid:48) p ) cross sections in Jefferson Lab Hall A L. Gu, D. Abrams, A. M. Ankowski, L. Jiang, B. Aljawrneh, S. Alsalmi, J. Bane, A. Batz, S. Barcus, M. Barroso, O. Benhar, V. Bellini, J. Bericic, D. Biswas, A. Camsonne, J. Castellanos, J.-P. Chen, M. E. Christy, K. Craycraft, R. Cruz-Torres, H. Dai, D. Day, S.-C. Dusa, E. Fuchey, T. Gautam, C. Giusti, J. Gomez, C. Gu, T. Hague, J.-O. Hansen, F. Hauenstein, D. W. Higinbotham, C. Hyde, C. Keppel, S. Li, R. Lindgren, H. Liu, C. Mariani, ∗ R. E. McClellan, D. Meekins, R. Michaels, M. Mihovilovic, M. Murphy, D. Nguyen, M. Nycz, L. Ou, B. Pandey, V. Pandey, K. Park, G. Perera, A. J. R. Puckett, S. N. Santiesteban, S. ˇSirca,
24, 23
T. Su, L. Tang, Y. Tian, N. Ton, B. Wojtsekhowski, S. Wood, Z. Ye, and J. Zhang (The Jefferson Lab Hall A Collaboration) Center for Neutrino Physics, Virginia Tech, Blacksburg, Virginia 24061, USA Department of Physics, University of Virginia, Charlottesville, Virginia 22904, USA SLAC National Accelerator Laboratory, Stanford University, Menlo Park, California 94025, USA North Carolina Agricultural and Technical State University, Greensboro, North Carolina 27401, USA King Saud University, Riyadh 11451, Kingdom of Saudi Arabia The University of Tennessee, Knoxville, Tennessee 37996, USA College of William and Mary, Williamsburg, Virginia 23187, USA The College of William and Mary, Williamsburg, Virginia 23187, USA Georgia Institute of Technology, Georgia 30332, USA INFN and Dipartimento di Fisica, Sapienza Universit`a di Roma, I-00185 Roma, Italy INFN, Sezione di Catania, Catania, 95123, Italy Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606, USA Hampton University, Hampton, Virginia 23669, USA Florida International University, Miami, Florida 33181, USA Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA University of Connecticut, Storrs, Connecticut 06269, USA Dipartimento di Fisica, Universit`a degli Studi di Pavia and INFN, Sezione di Pavia, I-27100 Pavia, Italy Duke University, Durham, North Carolina 27708, USA Kent State University, Kent, Ohio 44242, USA Old Dominion University, Norfolk, Virginia 23529, USA University of New Hampshire, Durham, New Hampshire 03824, USA Columbia University, New York, New York 10027, USA Jozef Stefan Institute, Ljubljana 1000, Slovenia University of Ljubljana, Ljubljana 1000, Slovenia Shandong University, Shandong, 250000, China Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
The E12-14-012 experiment, performed in Jefferson Lab Hall A, has collected exclusive electron-scattering data ( e, e (cid:48) p ) in parallel kinematics using natural argon and natural titanium targets.Here, we report the first results of the analysis of the data set corresponding to beam energy2,222 MeV, electron scattering angle 21 . −
50 deg. The differentialcross sections, measured with ∼
4% uncertainty, have been studied as a function of missing energyand missing momentum, and compared to the results of Monte Carlo simulations, obtained from amodel based on the Distorted Wave Impulse Approximation.
I. INTRODUCTION
Jefferson Lab experiment E12-14-012 was primarilyaimed at obtaining the proton spectral function (SF) ofthe nucleus Ar from a measurement of the cross sectionof the ( e, e (cid:48) p ) reaction e + A → e (cid:48) + p + ( A − ∗ , (1)in which the scattered electron and the knocked out pro-ton are detected in coincidence. Here A denotes the tar- ∗ [email protected] get nucleus in its ground state, while the recoiling ( A − e, e (cid:48) p ) exper-iments have clear advantages, because they are largelyunaffected by strong initial and final state interactions(FSI) between the beam particle and the target, andgive access to the properties of deeply bound protonsin medium-mass and heavy nuclei [3].Under the basic assumption that the scattering pro- a r X i v : . [ nu c l - e x ] D ec cess involves individual nucleons, and neglecting FSI be-tween the outgoing proton and the spectator nucleons,the momentum and removal energy of the knocked outparticle, p and E , can be reconstructed from measuredkinematical variables, and the cross section of the pro-cess is written in simple factorized form in terms of thespectral function of the target nucleus, P ( p , E ), triviallyrelated to the nucleon Green’s function, G ( p , E ), through P ( p , E ) = iπ Im G ( p , E ) . (2)As a consequence, the spectral function—yielding theprobability to remove a proton with momentum p fromthe target nucleus leaving the residual system with exci-tation energy E − E thr , with E thr being the proton emis-sion threshold—can be readily obtained from the data.Significant corrections to the somewhat oversimplifiedscheme outlined above—referred to as Plane Wave Im-pulse Approximation, or PWIA—arise from the occur-rence of FSI. The large body of work devoted to theanalysis of ( e, e (cid:48) p ) data has provided convincing evidencethat the effects of FSI can be accurately included by re-placing the plane wave describing the motion of the out-going proton with a distorted wave, eigenfunction of aphenomenological optical potential accounting for its in-teractions with the mean field of the residual nucleus.In general, the ( e, e (cid:48) p ) cross section computed withinthis approach, known as Distorted Wave Impulse Ap-proximation, or DWIA, involves the off-diagonal spectralfunction, and cannot be written in factorized form [4].However, an approximate procedure restoring factoriza-tion, referred to as factorized DWIA, has been shown toyield accurate results in the case of parallel kinematics,in which the momentum of the outgoing proton and themomentum transfer are parallel [5]. In this kinematicalsetup, the spectral function can still be reliably obtainedfrom ( e, e (cid:48) p ) data after removing the effects of FSI.Additional corrections to the PWIA arise from the dis-tortion of the electron wave functions resulting from in-teractions with the Coulomb field of the nucleus. How-ever, it has been shown that, for nuclei as heavy as Ca,this effect can be accurately taken into account using aneffective momentum transfer [6].Systematic measurements of ( e, e (cid:48) p ) cross sections inthe kinematical regime in which the recoiling nucleusis left in a bound state, performed at Saclay [7] andNIKHEF-K [8], have allowed the determination of thespectral functions of a broad set of nuclei. These studieshave provided a wealth of information on the energies andmomentum distributions of shell-model states belongingto the Fermi sea of the target nuclei, showing at the sametime the limitations of the mean-field description and theimportance of correlation effects [1].Besides being a fundamental quantity of nuclear many-body theory, containing important dynamical informa-tion, the spectral function is a powerful tool, allowingto obtain the cross sections of a variety of nuclear scat-tering processes in the kinematical regime in which the beam particles primarily interact with individual nucle-ons, and FSI can be treated as corrections. Applicationsto inclusive electron-nucleus scattering have offered vastevidence that the formalism based on spectral functionsprovides a comprehensive and consistent framework forthe calculation of nuclear cross sections in a broad kine-matical region, extending from quasielastic (QE) scat-tering to resonance production and deep-inelastic scat-tering [9–11].Over the past several years, a great deal of work hasbeen devoted to applying the spectral function formal-ism to the study of neutrino-nucleus interactions, whosequantitative understanding is needed for the interpreta-tion of accelerator-based searches of neutrino oscillations,see, e.g., Refs. [12, 13]. In this context, it should be notedthat the capability to describe a variety of reaction chan-nels within a unified approach is a critical requirement,because the energy of the beam particles is distributedaccording to a broad flux, typically ranging from a fewhundreds of MeV to a few GeV. Moreover, the knowledgeof the spectral function greatly improves the accuracy ofreconstruction of the neutrino energy, a key quantity inthe oscillation analysis [14, 15].Realistic models of the nuclear spectral functions havebeen obtained from the approach based on the local den-sity approximation, or LDA, in which the information onthe shell-model structure extracted from ( e, e (cid:48) p ) data iscombined to the results of accurate calculations of uni-form nuclear matter at various densities [10]. The ex-isting calculations of neutrino-nucleus cross sections em-ploying LDA spectral functions [11, 14, 16–26], however,are limited to the isospin-symmetric p -shell targets Oand C. Therefore, the results of these studies are ap-plicable to experiments using water- ˇCerenkov detectors,e.g. Super-Kamiokande [27], and mineral oil detectors,e.g. MiniBooNE [28].The analysis of the data collected by the ongoing andfuture experiments using liquid-argon time-projectionchambers, notably the Fermilab Short-Baseline Neutrinoprogram (SBN) [29] and the Deep Underground NeutrinoExperiment (DUNE) [30], will require the extension ofthis approach to the case of a heavier target with largeneutron excess. Moreover, in DUNE the proton and neu-tron spectral functions will both be needed, to extract theDirac phase δ CP from a comparison of neutrino and an-tineutrino oscillations, and achieve an accurate descrip-tion of pion production on protons and neutrons.In the absence of direct measurements, information onthe neutron momentum and removal energy distributionin Ar can be inferred from Ti( e, e (cid:48) p ) data, exploitingthe correspondence between the proton spectrum of ti-tanium, having charge Z = 22, and the neutron spec-trum of argon, having A − Z = 22. The viability ofthis procedure is supported by the results of Ref. [31],whose authors have performed a calculation of the in-clusive Ar( e, e (cid:48) ) and Ti( e, e (cid:48) ) cross sections withinthe framework of the self-consistent Green’s function ap-proach. The aim of Jlab experiment E12-14-012, is thedetermination of the proton spectral functions of argonand titanium from the corresponding ( e, e (cid:48) p ) cross sec-tions.In this article, we present the first results of our analy-sis. In Sec. II we discuss the kinematic setup, the detec-tors and their resolutions, and our definitions of signaland backgrounds. In Sec. III we introduce the missingenergy and the missing momentum, which are the fun-damental variables of our analysis, and discuss the mainelements of the Monte Carlo (MC) simulations employedfor event simulation. Sec. IV is devoted to the uncer-tainties associated with our analysis, while in Sec. V themeasured missing energy and missing momentum distri-butions are compared with the MC predictions. Finally,in Sec. VI we summarize our work and draw the conclu-sions. II. EXPERIMENTAL SETUP
The experiment E12-14-012 was performed at Jeffer-son lab in Spring 2017. Inclusive ( e, e (cid:48) ) and exclusive( e, e (cid:48) p ) electron scattering data were collected on targetsof natural argon and natural titanium, as well as on cali-bration and background targets of carbon and aluminum.The average neutron numbers calculated according to thenatural abundances of isotopes are 21.98 for argon and25.92 for titanium [32]. Therefore, from now on we willrefer to the targets considered here as Ar and Ti, forbrevity.The E12-14-012 experiment used an electron beam ofenergy 2,222 MeV provided by the Continuous ElectronBeam Accelerator Facility (CEBAF) at Jefferson Lab.The average beam current was approximately 15 µ A forthe Ar target and 20 µ A for the Ti target. The scat-tered electrons were momentum analyzed and detectedin the left high-resolution spectrometer (HRS) in Hall Aand the coincident protons were similarly analyzed in theright HRS. The spectrometers are equipped with two ver-tical drift chambers (VDCs) providing tracking informa-tion [34], two scintillator planes for timing measurementsand triggering, double-layered lead-glass calorimeter, agas ˇCerenkov counter used for particle identification [35],pre-shower and shower detectors (proton arm only) [35]and pion rejectors (electron arm only) [35]. The HRSswere positioned with the electron arm at central scatter-ing angle θ e = 21 . θ p (cid:48) = −
50 deg. The beam current and position, thelatter being critical for the electron-vertex reconstruc-tion and momentum calculation, were monitored by res-onant radio-frequency cavities (beam current monitors,or BCMs [35]) and cavities with four antennae (beam po-sition monitors, or BPMs [35]), respectively. The beamsize was monitored using harp scanners, which consistsof a thin wire which moves through the beam. We useda raster of 2 × area to spread the beam and avoidoverheating the target.The experiment employed also an aluminum target and a set of carbon targets, used to evaluate backgrounds andmonitor the spectrometers optics. The aluminum targetwas made of two identical foils of the Al-7075 alloy witha thickness of 0 . ± .
002 g/cm . One of the aluminumfoils was positioned to match the entrance and the otherto match the exit windows of the argon gas target cell.The two thick foils were separated by a distance of 25 cm,corresponding to the length of the argon gas cell and theAl foil’s thickness.The analysis presented here uses data collected withthe settings given in Table I. All of our data were takenin parallel kinematics, in which the momentum transfer, q , and the momentum of the outgoing proton, p (cid:48) , areparallel. The only difference of data collection settingfor Ar and Ti is the scattered electron energy.The VDCs’ tracking information was used to deter-mine the momentum and to reconstruct the direction(in-plane and out-of-plane angles) of the scattered elec-tron and proton, and to reconstruct the interaction ver-tex at the target. We used both the electron and protonarm information separately to reconstruct the interactionvertex and found them in very good agreement. Thetransformation between focal plane and target quanti-ties was computed using an optical matrix, the accuracyof which was verified using the carbon multi-foil targetdata and sieve measurements as described in previouspapers [32, 36, 37]. Possible variations of the optics andmagnetic field in both HRSs are included in the analysisas systematic uncertainties related to the optics.Several different components were used to build thetriggers: the scintillator planes on both the electron andproton spectrometers, along with signals from the gasˇCerenkov (GC) detector, the pion rejector (PR), the pre-shower and the shower detector (PS). Table II lists thetrigger configurations, including details on how the sig-nals from the various detector components are combinedto form a trigger.The triggers used for identifying electron and protoncoincidence events were T1 and T2, where T2 was used toprovide a data sample to calculate the overall T1 triggerefficiency and we were able to compute the efficiency ofT1 using also the product of T3 and T4 efficiencies. Ifthe proton and electron observations from the same eventwere perfectly paired, these values would be the same asT1 trigger efficiency.Electrons and protons were selected in their corre-sponding HRS requiring only one reconstructed goodtrack. For the electron we required also an energy depositof at least 30% in the lead calorimeter ( E cal /p > .
3) anda signal in the ˇCerekov detector of more than 400 analog-digital-converter (ADC) counts. Furthermore, the trackswere required to be within ± ± dp/p of ± β for the proton arm between 0.6 and 0.8 tofurther isolate protons. We only included in our analysis TABLE I. Kinematics settings used to collect the data analyzed here. E (cid:48) e θ e Q | p (cid:48) | T p (cid:48) θ p (cid:48) | q | p m E m (MeV) (deg) (GeV /c ) (MeV/ c ) (MeV) (deg) (MeV/ c ) (MeV/ c ) (MeV)Ar 1,777 21.5 0.549 915 372 − . − . S && S ) and (GC || PR) [LEFT] and (S0&&S2) [RIGHT]T2 ( S || S ) and (GC || PR) [LEFT] and ( S || S ) and not (PS) [RIGHT]T3 ( S && S ) and (GC || PR) [LEFT]T4 ( S && S ) [RIGHT]T5 ( S || S ) and (GC || PR) [LEFT]T6 ( S || S ) and not (PS) [RIGHT] events in which both the electron and the proton wererecorded in a T1 trigger timing window and for whichthe difference in the start time of the individual triggerswas of just few ns (time coincidence cut). For the ar-gon target we also required that the events originatedwithin the central ±
10 cm of the target cell to excludecontamination from the target entry and exit windows.By measuring events from the thick Al foils, positioned atthe same entry and exit window of the target, we deter-mined that the target cell contributions to the measuredcross section was negligible ( < TABLE III. Summary of the efficiency analysis for the argonand titanium targets. Ar target Ti targeta. Live time 98.0% 98.9%b. Tracking 98.3% 98.3%c. Trigger 92.3% 96.9%d. ˇCerenkov cut 99.9% 96.6%e. Calorimeter cut 97.8% 98.1%f. β cut 95.6% 95.3%g. Coincidence time cut 54.8% 55.5% tum p m , defined as in Eq. (3), from lower to higher. Wefound that the electron arm dp/p distributions showedslight variations. We then decided not to use the elec-tron arm dp/p as a kinematical cut in our analysis. Thetrigger efficiencies were computed using the other avail-able trigger as described above. The time coincidence cutefficiency was evaluated selecting a sample of more puresignal events (using a tighter β cut) and looking at theratio of events with and without the time coincidencecuts. The overall efficiency (between 39.6% and 48.9%across all kinematic regions for the Ar target, and be-tween 46.8% and 48.1% for the Ti target) includes cutson the coincidence triggers, calorimeters, both the leadand the ˇCerenkov counter, track reconstruction efficiency,live-time, tracking and β cut. III. DATA ANALYSISA. The ( e, e (cid:48) p ) cross section In electron-nucleus scattering an incident electron,with energy E e , is scattered from a nucleus of mass M A at rest. Electron scattering is generally described in theone-photon exchange approximation, according to whichthe incident electron exchanges a space-like photon, ofenergy ω and momentum q , with the target nucleus.In ( e, e (cid:48) p ) experiments the scattered electron and a pro-ton are detected in coincidence in the final state, andtheir momentum and energy are completely determined.If, in addition, the kinematics is chosen such that theresidual nucleus is left in a specific bound state, the re-action is said to be exclusive.In the following, p (cid:48) , T p (cid:48) , and M will denote the mo-mentum, kinetic energy, and mass of the outgoing pro-ton, while the corresponding quantities associated withthe recoiling residual nucleus will be denoted p R , T R , and M R . The missing momentum and missing energy are ob-tained from the measured kinematical quantities usingthe definitions p m = q − p (cid:48) = p R , (3)and E m = ω − T p (cid:48) − T R . (4)Exploiting energy conservation, implying ω + M A = M + T p (cid:48) + M R + T R , (5)and writing the mass of the residual nucleus in the form M R = M A − M + E thr + E x = M A − + E x , (6)where E thr and M A − denote the proton emission thresh-old and the mass of ( A − E m = E thr + E x . (7)The usual description of the exclusive ( e, e (cid:48) p ) reactionin the QE region assumes the direct knockout mecha-nism, which naturally emerges within the impulse ap-proximation (IA). According to this picture, the elec-tromagnetic probe interacts through a one-body currentwith the quasi-free knocked out proton, while all othernucleons in the target act as spectators. In addition, ifFSI between the outgoing nucleon and the spectators isnegligible, PWIA can be applied, and the ( e, e (cid:48) p ) crosssection reduces to the factorized form d σdωd Ω e (cid:48) dT p (cid:48) d Ω p (cid:48) = Kσ ep P ( − p m , E m ) , (8)where K = | p (cid:48) | E p (cid:48) , with E p (cid:48) = (cid:113) p (cid:48) + M . Here, σ ep isthe differential cross section describing electron scatter-ing off a bound moving proton, stripped of the flux factorand the energy conserving delta-function [40, 41], while P ( − p m , E m ) is the proton spectral function of Eq. (2).Note that Eqs. (3) and (4) imply that the arguments ofthe spectral function can be identified with the initial mo-mentum and the removal energy of the struck nucleon,respectively. Therefore, Eq. (8) shows that within PWIAthe nuclear spectral function, describing the proton mo-mentum and energy distribution of the target nucleus,can be readily extracted from the measured ( e, e (cid:48) p ) crosssection.When FSI are taken into account, and the outgoingproton is described by a distorted wave function as pre-scribed by DWIA, the initial momentum of the strucknucleon is not trivially related to the measured missingmomentum, and the cross section can no longer be writ-ten as in Eq. (8). However, the occurrence of y -scalingin inclusive electron-nucleus scattering [42, 43]—whoseobservation in the analysis of the Ar( e, e (cid:48) ) and Ti( e, e (cid:48) ) data is discussed in Refs. [36, 37]—indicates that the for-malism based on factorization is still largely applicablein the presence of FSI.In principle, within the approach of Refs. [44–46], thebound and scattering states are both derived from anenergy dependent non-Hermitian optical-model Hamil-tonian. While being fully consistent, however, this treat-ment involves severe difficulties. In practice, the bound-state proton wave functions are generally obtained fromphenomenological approaches—although a few studiesbased on realistic microscopic models of the nuclearHamiltonian have been carried out for light and medium-heavy nuclei [47, 48]—while the scattering states areeigenfunctions of phenomenological optical potentials,the parameters of which are determined through a fitto elastic proton-nucleus scattering data.The PWIA description provides a clear understand-ing of the mechanism driving the ( e, e (cid:48) p ) reaction, andthe ensuing factorized expression of the coincidence crosssection, Eq. (8), is essential to obtain from the data anintrinsic property of the target, such as the spectral func-tion, independent of kinematics. As pointed out above,however, the occurrence of FSI leads to a violation offactorization, and makes the extraction of the spectralfunction from the measured cross section more compli-cated [45, 49]. Additional factorization-breaking correc-tions arise from the distortion of the electron wave func-tions, resulting from interactions with the Coulomb fieldof the target [6, 50, 51].The general conditions to recover a factorized expres-sion of the cross section are discussed in Refs. [5, 44, 45,52, 53]. If these requirements are fulfilled, the DWIAcross section can be written in terms of a distorted spec-tral function according to d σdωd Ω e (cid:48) dT p (cid:48) d Ω p (cid:48) = Kσ ep P D ( p (cid:48) , − p m , E m ) . (9)Note, however that, unlike the spectral function appear-ing in Eq. (8), the distorted spectral function is not anintrinsic property of the target, because it depends ex-plicitly on the momentum of the outgoing nucleon, whichin turn depends on the momentum transfer. The mostprominent effects of the inclusion of FSI within the frame-work of DWIA are a shift and a suppression of the miss-ing momentum distributions, produced by the real andimaginary part of the optical potential, respectively. B. Data analysis details
The measured cross sections are usually analyzed interms of missing-energy and missing-momentum distri-butions. For a value of E m corresponding to a peak inthe experimental missing-energy distribution, the dataare usually presented in terms of the reduced cross sec-tion as a function of p m = | p m | . The reduced cross sec-tion, obtained from the measured cross section dividingout the kinematic factor K and the electron-proton crosssection σ ep can be identified with the spectral function inPWIA and with the distorted spectral function in the fac-torized DWIA of Eq. (9). The off-shell extrapolation ofde Forest [40, 41] is generally used to describe the boundnucleon cross section.The experimental reduced cross sections can be com-pared with the corresponding reduced cross section cal-culated using different theoretical models. The compar-ison of the results obtained from the un-factorized andfactorized approaches allows one to make an estimate ofthe accuracy of the factorization scheme, as well as thesensitivity to the different factorization-breaking contri-butions.The six-fold differential cross section as a function of p m and E m was extracted from the data using the ( e, e (cid:48) p )event yield Y for each p m and E m bin d σdωd Ω e (cid:48) dT p (cid:48) d Ω p (cid:48) = Y ( p m , E m ) B × lt × ρ × BH × V B × C rad , (10)where B is the total accumulated beam charge, lt is thelive-time of the detector (fraction of time that the de-tector was able to collect and write data to disk), ρ isthe target density (for argon, corrected for the nominaldensity of gas in the target cell), BH is the local den-sity change due to the beam heating the gas cell timesthe gas expansion due to boiling effects (this correctionis not included in the case of Ti), V B is the effect of theacceptance and kinematical cuts, and C rad is the effectof the radiative corrections and bin center migration.We used the SIMC spectrometer package [54] to simu-late ( e, e (cid:48) p ) events corresponding to our particular kine-matic settings, including geometric details of the targetcell, radiation correction, and Coulomb effects. SIMCalso provided the V B and C rad corrections as in Eq. (10).To simulate the distribution of missing energies and mo-menta of nucleons bound in the argon and titanium nu-clei, SIMC was run with a test SF described in detail inthe following subsection.In Table IV we summarize the energies of the shellmodel states comprising the ground states of Ar and Ti. In our analysis, in case two orbitals overlap in E m ,we set the energy range for the orbital to be the same,and we assumed the probability of emission of an electronto be the same. Table IV also lists energies derived fromprevious data sets, as well as the energy used in the cal-culation of FSI effects according to the model describedin Sec. IV A.SIMC generates events for a broad phase-space, andpropagates the events through a detailed model of theelectron and proton spectrometers to account for accep-tances and resolution effects. Each event is weighted bythe σ cc cross section of de Forest [41] and the SF. Thefinal weighted events do not contain any background. Aspointed out above, SIMC does not include FSI correc-tions other than for the nuclear transparency.The data yield corrected for the above-mentioned fac-tors is then integrated over E m to get the cross section TABLE IV. Parametrization of the missing energy distribu-tions of
Ar and
Ti assumed in this analysis. The centralpeak position E α , its width σ α , and the lower (upper) boundon the considered energy range, E α low ( E α high ) are shown foreach level α . All values are given in units of MeV. α E α σ α E α low E α high argon1 d / .
53 2 8 142 s / .
93 2 8 141 d / .
23 4 14 201 p / . p / . s / . f / .
45 2 8 142 s / .
21 2 14 301 d / .
84 2 14 301 d / .
46 4 14 301 p / . p / . s / . as function of p m . We collected 29.6 (12.5) hours of dataon Ar (Ti), corresponding to ∼
44k (13k) events.We estimated the background due to accidentals to be2% (3%) for Ar (Ti), performing analysis for each bin of E m and p m . First, we selected events in T1 trigger inanti-coincidence between the electron and proton arms.This region corresponds to 100 times the nominal coin-cidence time window width ( ∼ E m and p m distributions. C. Test spectral functions
The spectral function employed to simulate events inSIMC is based on the simplest implementation of thenuclear shell model, P ( p m , E m ) = (cid:88) α | φ α ( p m ) | f α ( E m − E α ) , (11)where the sum runs over all occupied states. In the aboveequation, φ α ( p m ) is the momentum-space wave functionof the state α , normalized to unity, and f α ( E m − E α ) rep-resents the distribution of missing energy peaked at thevalue E α , reflecting the width of the corresponding state.As a consequence of deviations from this mean-field pic-ture originating from nucleon-nucleon correlations, weexpect the Monte Carlo simulations typically to over-estimate the data, due to the partial depletion of theshell-model states and to the correlated contribution tothe nuclear spectral function. TiAr p m (GeV/ c ) π p m n ( p m )( c / G e V ) FIG. 1. Missing momentum distributions of protons in argonand titanium assumed in this analysis.
We compared the momentum distribution, defined as n ( p m ) = (cid:90) P ( p m , E m ) dE m , (12)obtained using the wave functions of Refs. [55, 56] andRef. [57], and found that the differences between themare negligible for both argon and titanium. As shown inFig. 1, the momentum distributions for argon and tita-nium also turn out not to differ significantly. This find-ing suggests that nuclear effects in argon and titaniumare similar.The missing energy distributions are assumed to beGaussian f α ( E m − E α ) = 1 √ πσ α exp (cid:20) − ( E m − E α ) σ α (cid:21) . (13)We obtain the missing energies of the least-bound va-lence orbital for protons—corresponding to the residualnucleus being left in the ground state, with an additionalelectron and the knocked-out proton at rest—from themass difference of the residual system and the target nu-cleus [58]. These values of missing energy, correspondingto the 1 d / (1 f / ) state for Ar (
Ti) in Table IV, aregiven by E thr = M A − + M + m − M A , where m stands for the electron mass.In principle, the energies of other valence levels of Arand
Ti could be obtained from the excitation spectraof
Cl [59] and
Sc [60]. However, the fragmentationof shell-model states induced by long-range correlationsmakes this information difficult to interpret within theindependent-particle model, assumed in Eq. (11), be-cause a few spectroscopic lines typically correspond toa given spin-parity state. To overcome this issue andidentify the dominant lines, we rely on the spectroscopic (a)0.080.060.040.020.00 (b) total1 s / p / p / d / s / d / f / E m (MeV) S ( E m )( M e V ) FIG. 2. Missing energy distribution of protons in (a) argonand (b) titanium assumed in this analysis. strengths determined in past direct pick-up experimentssuch as A ( H , He) for argon [61] and titanium [62].The heavily fragmented 1 d / shell [61, 62]—with over10, densely packed, spectroscopic lines contributing—canbe expected to lend itself well to the approximation by asingle distribution of finite width. To determine its peakposition, in addition to the experimental data [61, 62], weuse the theoretical analyses of Refs. [63, 64] as guidance.More deeper-lying shells—1 p / , 1 p / , and 1 s / —were not probed by the past experiments [61, 62]. Their E α values, as well as the widths σ α for all shells, aredetermined to provide a reasonable description of themissing-energy distributions obtained in this experiment.The resulting parametrization is detailed in Table IV,and presented in Fig. 2. IV. UNCERTAINTY ANALYSIS
The total systematic uncertainty in this analysis wasestimated by summing in quadrature the contributionslisted in Table V. We determined the kinematic and ac-ceptance cuts ensuring that there are no dependencieson kinematic variables and input theory model, in thisway all uncertainties are uncorrelated bin to bin. All thekinematic and acceptance cuts were varied by the res-
TABLE V. Contributions to systematical uncertainties for ar-gon and titanium average over all the E m and p m bins.Ar Ti1. Total statistical uncertainty 0.53% 0.78%2. Total systematic uncertainty 2.75% 2.39%a. Beam x & y offset 0.56% 0.48%b. Beam energy 0.10% 0.10%c. Beam charge 0.30% 0.30%d. HRS x & y offset 0.72% 0.69%g. Optics (q1, q2, q3) 1.10% 0.34%h. Acceptance cut ( θ, φ, z ) 1.23% 1.39%i. Target thickness/density/length 0.2% 0.2%j. Calorimeter & ˇCerenkov cut 0.02% 0.02%k. Radiative and Coulomb corr. 1.00% 1.00%l. β cut 0.63% 0.48%m. Boiling effect 0.70% —n. Cross section model 1.00% 1.00%o. Trigger and coincidence time cut 0.99% 0.78% olution of the variable under consideration. Except forthe transparency corrections, the MC used to evaluatethose uncertainties did not contain effects due to FSI,such as a quenching of the strength of the cross sectionand a modification of the kinematic of the outgoing par-ticles. A priori the MC simulation could depend on theunderlying theoretical model. However, we repeated theanalysis of systematic uncertainties varying its ingredi-ents, and did not observe any substantial variations ofthe obtained results. As the obtained results depend onthe Monte Carlo calculation, it is important to estimateuncertainties resulting from its inputs. To determine theuncertainties related to the target position, we performedthe simulation with the inputs for the beam’s and spec-trometer’s x and y offsets varied within uncertainties, andwe recomputed the optical transport matrix varying thethree quadrupole magnetic fields, one at the time. Eachof these runs was compared to the reference run, andthe corresponding differences were summed in quadra-ture to give the total systematic uncertainty due to theMonte Carlo simulation. The uncertainties related tothe calorimeter and ˇCerenkov detectors were determinedby changing the corresponding cut by a small amountand calculating the difference with respect to the nomi-nal yield value. The uncertainty due to the acceptancecuts on the angles was calculated using the same method.We included an overall fixed uncertainty for both thebeam charge and beam energy, as in the previous workon C, Ti, Ar, and Al [32, 36, 37]. We evaluated the sys-tematic uncertainties related to the trigger efficiency bydetermining variations across multiple runs, as well as byapplying different acceptance cuts. A fixed uncertaintywas assigned to take care of those variations.The time-coincidence cut efficiency, as other accep-tance cuts, was evaluated by changing the cut by ± σ .SIMC generates events including the effects from ra-diative processes: vacuum polarization, vertex correc-tions, and internal bremsstrahlung. External radiative (MeV) m E ) M e V s r c m - ( p ' W d p ' d T e ' W d w d s d Ar exp. dataMC predictionBackground x10 (MeV) m E ) M e V s r c m - ( p ' W d p ' d T e ' W d w d s d Ti exp. dataMC predictionBackground x10
FIG. 3. Six-fold differential cross section as a function of miss-ing energy for argon (top panel) and titanium (bottom panel).The background estimate (line connecting the experimentaldata points) is multiplied by 10 for purpose of presentation.The MC predictions, based on the mean-field SF, include acorrection for the nuclear transparency, while other FSI effectsare not accounted for. processes refer to electrons losing energy while passingthrough material in the target. Radiative correction inSIMC are implemented following the recipe of Dasu [65],using the Whitlow’s approach [66, 67]. We considered afixed 1% uncertainty due to the theoretical model for theradiative corrections over the full kinematic range as inour previous work. We generated different MC wherethe radiative corrections were re-scaled by (cid:112) ( Q ) / Q being the four-momentum transfer squared, and re-analyzed the data and looked for variations. Coulombcorrections were included in the local effective momen-tum approximation [68]. A 10% uncertainty associatedwith the Coulomb potential was included as systematicuncertainty. Finally, we included a target thickness un-certainty and an uncertainty due to the boiling effectcorrection [33].The measured and MC predicted differential cross sec-tions d σ/dωd Ω e dpd Ω p are presented in Fig. 3 as a func-tion of E m and in Fig. 4 as a function of p m , integratedover the full range of E m , for Ar (top panel) and Ti - - (MeV/c) m p ) M e V s r c m - ( p ' W d p ' d T e ' W d w d s d Ar exp. data stat error onlyAr exp. data sys+stat errorsMC predictionBackground x10 - - (MeV/c) m p ) M e V s r c m - ( p ' W d p ' d T e ' W d w d s d Ti exp. data stat error onlyTi exp. data stat+sys errorsMC predictionBackground x10
FIG. 4. Same as Fig. 3 but for the cross section as a functionof missing momentum. The inner (outer) uncertainty bandscorrespond to statistical (total) uncertainties. (bottom panel) targets.The MC simulation clearly overestimates the extractedcross sections. As the nuclear model underlying the sim-ulation neglects the effects of FSI other than the nucleartransparency and all correlations between nucleons, thisdifference is by no means surprising. Both FSI and par-tial depletion of the shell-model states require furtherstudies, base on all five datasets collected by the JLabE12-14-012 experiment, which will be reported elsewhere.
A. Final state interactions
Within DWIA, FSI between the outgoing proton andthe spectator nucleons are described by a complex,energy dependent, phenomenological optical potential(OP). The OPs available for calculations were deter-mined by fitting a set of elastic proton-nucleus scatteringdata for a range of target nuclei and beam energies. Dif-ferent parametrizations, yielding equivalently good de-scriptions of the data, can give differences and theoreti-cal uncertainties when “equivalent” OPs are used in kine-matical regions for which experimental data are not avail- - - - [MeV/c] m p ] - [ ( G e V / c ) r ed s DWIAPWIA
FIG. 5. Reduced cross section as a function of missing mo-mentum for the 1 p / proton knockout from argon. We com-pare the PWIA and DWIA results obtained for the parallelkinematics considered in this analysis. able, or when they are extended to inelastic scatteringand to calculation of the cross section of different nuclearreactions.Nonrelativistic and relativistic OPs are available for( e, e (cid:48) p ) calculations within nonrelativistic and relativis-tic DWIA frameworks. However, nonrelativistic phe-nomenological OPs are available for energies not largerthan 200 MeV. It is generally believed that above ∼ e, e (cid:48) p ) reactions the differences between the non-relativistic and relativistic DWIA results depend on kine-matics and increase with the outgoing proton energy, andfor proton energies above 200 MeV a relativistic calcula-tion is necessary.We have used the so-called “democratic”(DEM) rela-tivistic OP [70], obtained from a global fit to over 200 setsof elastic proton-nucleus scattering data, comprised of abroad range of targets, from helium to lead, at energiesup to 1,040 MeV.An example of the comparison between PWIA andDWIA results is given in Fig. 5, where the reduced crosssection as a function of p m is displayed for proton knock-out from the 1 p / argon orbital. Calculations are per-formed within the relativistic model of Ref. [69] for theparallel kinematics of the present experiment. Positiveand negative values of p m indicate, conventionally, casesin which | q | < | p (cid:48) | and | q | > | p (cid:48) | , respectively. The reduc-tion and the shift produced in the reduced cross sectionby FSI in the DWIA calculation can be clearly seen.The two dashed lines drawn in the region of positive p m of the figure indicate the value of p m corresponding tothe peaks of the DWIA and PWIA reduced cross sections.We use the distance between the two dashed lines as ameasure of the shift produced by FSI.The reduction of the calculated cross section produced0 TABLE VI. Shifts between the reduced DWIA and PWIAcross sections, and the DWIA to PWIA cross-section ratios,obtained for proton knockout from various argon orbital us-ing different optical potentials: DEM [70], EDAD3 [71], andEDAD1 [71]. All results are calculated for p m > c ) DWIA/PWIAEDAD1 EDAD3 DEM EDAD1 EDAD3 DEM1 d / . − . . .
58 0 .
57 0 . s / . . . .
78 0 .
78 0 . d / − . − . − . .
57 0 .
57 0 . p / . . . .
43 0 .
39 0 . p / . . . .
47 0 .
44 0 . s / . . . .
42 0 .
38 0 . by FSI can be measured by the DWIA/PWIA ratio,which is defined here as the ratio of the integral over p m of the DWIA and PWIA reduced cross sections. Boththe shift and the DWIA/PWIA ratios are computed sep-arately for the positive and negative p m regions.The theoretical uncertainty of the shift and the reduc-tion produced by FSI has been evaluated investigatingthe sensitivity of the DWIA and PWIA results to differ-ent choices of the theoretical ingredients of the calcula-tion.The uncertainty due to the choice of the OP has beenevaluated by comparing the results obtained with theDEM and other energy-dependent and atomic-numberdependent relativistic OPs, referred to as EDAD1 andEDAD3 [71] . The shift and the DWIA/PWIA ratio inthe positive p m region, computed for proton knock outfrom various argon orbitals using the DEM, EDAD1, andEDAD3 potentials are reported in Table VI. The resultsindicate a slight dependence of FSI effects on the choiceof OP.Note that the three OPs were determined by a fittingprocedure of elastic proton scattering data over a widerange of nuclei, which, however, did not include argon.This means that the ability of the phenomenological OPsto describe elastic proton scattering data on argon is notguaranteed. A test of this ability is presented in Fig. 6,where the Ar( p, p (cid:48) ) cross section calculated at 0.8 GeVwith the three OPs is compared to the correspondingexperimental cross section obtained using the HRS of theLos Alamos Meson Physics Facility [72]. The results ofthe three OPs largely overlap, and their agreement withthe experimental cross section, although not perfect, ismore than reasonable, in particular if we consider that ithas not been obtained from a fit to the data.In the relativistic DWIA and PWIA calculations differ-ent current conserving (cc) expressions of the one-bodynuclear current operator can be adopted. The differentexpressions are equivalent for on-shell nucleons, while dif-ferences can arise for off-shell nucleons. For all the resultsthat we have presented until now, and as a basis for thepresent calculations, we have adopted the cc1 prescrip-tion [41]. We note that, historically, the cc1 cross section (deg) c.m. q -
10 110 ( m b / s r) W d s d DataDEMEDAD1EDAD3
FIG. 6. Differential cross section for elastic proton scatteringon Ar at 0.8 GeV as a function of scattering angle. Resultsfor the DEM, EDAD1, and EDAD3 optical potentials, whichturn out to almost completely overlap, are compared with theexperimental data [72]. has been often used to obtain the reduced cross sectionfrom the experimental and theoretical cross section. Theimpact of using a different cross section—such as the cc2model of Ref. [41]—in the determination of the spectralfunction will be discussed in future analysis.We have also checked that the differences obtained us-ing different proton form factors in the calculation of thenuclear current are always negligible in the kinematic sit-uation of the present experiment.The bound proton states adopted in the calculationsare self-consistent Dirac-Hartree solutions derived withina relativistic mean field approach using a Lagrangiancontaining σ , ω , and ρ mesons, with medium dependentparametrizations of the meson-nucleon vertices that canbe more directly related to the underlying microscopicdescription of nuclear interactions [55, 56]. Pairing ef-fects have been included carrying out Bardeen-Cooper-Schrieffer (BCS) calculations. The theoretical uncertain-ties on the shift and the DWIA/PWIA ratio due to theuse of wave functions obtained with a different descrip-tion of pairing, based on the relativistic Dirac-Hartree-Bogoliubov (DHB) model [57], turn out to be negligible.In our analysis we assumed the missing energy distri-bution for each of the orbitals in Ar and Ti as shownin Fig. 2. The lower and upper energy bounds assumedin the DWIA analysis of FSI are given for each orbital inTable IV. The FSI correction has been applied event byevent in both the missing energy and missing momentumdistributions. We applied different corrections for eventswith | q | < | p (cid:48) | and | q | > | p (cid:48) | , according to the theoreticalpredictions mentioned before. For each event, we usedthe reconstructed energy and momentum of both elec-tron and proton to determine the orbital involved in theprimary interaction. Then, we applied the FSI correc-tion, based on the p m sign. For orbitals that overlap weuse a simple prescription to determine the most probable1orbital from which the electron was emitted, as describedin Sec. III B. V. DIFFERENTIAL CROSS SECTIONCOMPARISON
Figures 7 and 8 show a comparison between the mea-sured differential cross sections of Ar and Ti andthe MC predictions including full FSI corrections, plot-ted as a function of p m for three different ranges of E m . The missing energy regions for Ar ( Ti) are: E m <
27 MeV ( E m <
30 MeV), 27 < E m <
44 MeV(30 < E m <
54 MeV ) and 44 < E m <
70 MeV(54 < E m <
90 MeV).We estimated the background to be of the order 2%for Ar and 3% for Ti. The MC systematic uncer-tainties from FSI are estimated by varying the followingingredients of the model:(i) the optical potential (DEM, EDAD1, or EDAD3);(ii) the pairing mechanism underlying the determi-nation of the wave functions (the default BCSmodel [55, 56] or the DHB model [57]);(iii) the parametrization of the nucleon form factors.The total systematic uncertainty is obtained by addingin quadrature all the variations, and including an overalluncertainty of the theoretical model of 15%.A prominent feature of both Figs. 7 and 8 is that theagreement between data and MC predictions includingFSI, which turns out to be quite good in the regionof low missing energies, becomes significantly worse atlarger E m . This behavior can be explained consideringthat, according to the shell-model picture employed inMC simulations, missing energies E m >
27 MeV corre-spond to proton knockout from the deeply bound 1 p / ,1 p / , and 1 s / states.As discussed in Sec. III C, the energies and widths ofthese states are only estimated, and not determined fromexperimental data. Underestimating the widths and theassociated overlaps of energy distributions would imply asmaller value for the differential cross section and a shiftin the p m distribution between data and MC. We havetested this hypothesis by varying the width of the high-energy states in the test SF and redoing our full analysis,and noticed an improved agreement between data andMC.More generally, it has to be kept in mind that a clearidentification of single particle states in interacting many-body systems—ultimately based on Landau theory ofnormal Fermi liquids—is only possible in the vicinityof the Fermi surface, corresponding to the lowest valueof missing energy, see, e.g., Ref. [73]. An accurate de-scription of the data at large missing energy will re-quire a more realistic model of the nuclear spectral func-tion, taking into account dynamical effects beyond the - (MeV/c) m p ) M e V s r c m - ( p ' W d p ' d T e ' W d w d s d Ar exp. dataMC prediction (with FSI) (a) 0 < E m <
27 MeV - (MeV/c) m p ) M e V s r c m - ( p ' W d p ' d T e ' W d w d s d Ar exp. dataMC prediction (with FSI) (b) 27 < E m <
44 MeV - (MeV/c) m p ) M e V s r c m - ( p ' W d p ' d T e ' W d w d s d Ar exp. dataMC prediction (with FSI) (c) 44 < E m <
70 MeV
FIG. 7. Six-fold differential cross section for argon as a func-tion of missing momentum integrated over different ranges ofmissing energy. The background estimate is multiplied by 10for presentation. The MC predictions, based on the mean-field SF, include the full FSI corrections. - (MeV/c) m p ) M e V s r c m - ( p ' W d p ' d T e ' W d w d s d Ti exp. dataMC prediction (with FSI) (a) 0 < E m <
30 MeV - (MeV/c) m p ) M e V s r c m - ( p ' W d p ' d T e ' W d w d s d Ti exp. dataMC prediction (with FSI) (b) 30 < E m <
54 MeV - (MeV/c) m p ) M e V s r c m - ( p ' W d p ' d T e ' W d w d s d Ti exp. dataMC prediction (with FSI) (c) 54 < E m <
90 MeV
FIG. 8. Same as Fig. 7 but for titanium. mean-field approximation, notably nucleon-nucleon cor-relations, leading to the appearance of protons in contin-uum states.
VI. SUMMARY AND CONCLUSIONS
In this paper, we report the first results of the analysisof ( e, e (cid:48) p ) data at beam energy E e = 2 ,
222 MeV an elec-tron scattering angle θ e = 21 . < E m <
30 MeVand missing momentum covered by our measurementsappears satisfactory. The larger discrepancies observedat the larger missing energies such as 30 < E m <
44 MeVre likely to be ascribable to the limitations of the theoret-ical model based on the mean-field approximation, em-ployed in MC event generation, which is long known tobe inadequate to describe the dynamics of deeply boundnucleons [1]. Understanding these discrepancies at quan-titative level will require the inclusion of reaction mech-anisms beyond DWIA, such as multi-step processes andmulti-nucleon emission triggered by nucleon-nucleon cor-relations.The missing energy spectra obtained from our analysiscontain valuable new information on the internal struc-ture and dynamics of the nuclear targets, encoded in thepositions and widths of the observed peaks.The determination of these spectra particularly fordeep-lying hole excitations is, in fact, a first step towardsthe derivation of the spectral functions for medium-massnuclei, such as Ar and Ti, within the framework of LDA,that represents the ultimate aim of our experiment.The Ar and Ti measurements discussed in this arti-cle, providing the first ( e, e (cid:48) p ) data in the kinematicalrange relevant to neutrino experiments—most notablyDUNE—comprises the first of five datasets collected bythe JLab E12-14-012 experiment. The combined analy-sis of all data, which is currently under way, will provideinformation of unparalleled value for the development ofrealistic nuclear models, and will allow the extraction ofAr and Ti spectral functions. ACKNOWLEDGMENTS
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