Polarization transfer to bound protons measured by quasi-elastic electron scattering on 12 C
Tilen Brecelj, Sebouh J. Paul, Tim Kolar, Patrick Achenbach, Adi Ashkenazi, Ralph Böhm, Erez O. Cohen, Michael O. Distler, Anselm Esser, Ronald Gilman, Carlotta Giusti, David Izraeli, Igor Korover, Jechiel Lichtenstadt, Israel Mardor, Harald Merkel, Miha Mihovilovič, Ulrich Müller, Mor Olivenboim, Eli Piasetzky, Guy Ron, Björn S. Schlimme, Matthias Schoth, Florian Schulz, Concettina Sfienti, Simon Širca, Samo Štajner, Steffen Strauch, Michaela Thiel, Adrian Weber, Israel Yaron, A1 Collaboration
PPolarization transfer to bound protonsmeasured by quasi-elastic electron scattering on C Tilen Brecelj ‡ , Sebouh J. Paul ‡ , ∗ Tim Kolar, Patrick Achenbach, Adi Ashkenazi, Ralph B¨ohm, Erez O.Cohen, Michael O. Distler, Anselm Esser, Ronald Gilman, Carlotta Giusti, David Izraeli, Igor Korover,
6, 2
Jechiel Lichtenstadt, Israel Mardor,
2, 7
Harald Merkel, Miha Mihoviloviˇc,
8, 1, 3
Ulrich M¨uller, MorOlivenboim, Eli Piasetzky, Guy Ron, Bj¨orn S. Schlimme, Matthias Schoth, Florian Schulz, ConcettinaSfienti, Simon ˇSirca,
8, 1
Samo ˇStajner, Steffen Strauch, Michaela Thiel, Adrian Weber, and Israel Yaron (A1 Collaboration) Joˇzef Stefan Institute, 1000 Ljubljana, Slovenia. School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel. Institut f¨ur Kernphysik, Johannes Gutenberg-Universit¨at, 55099 Mainz, Germany. Rutgers, The State University of New Jersey, Piscataway, NJ 08855, USA. Dipartimento di Fisica, Universit`a degli Studi di Pavia and INFN,Sezione di Pavia, via A. Bassi 6, I-27100 Pavia, Italy. Department of Physics, NRCN, P.O. Box 9001, Beer-Sheva 84190, Israel. Soreq NRC, Yavne 81800, Israel. Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia. Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel. University of South Carolina, Columbia, South Carolina 29208, USA. (Dated: April 30, 2020)We report the measurements of the transverse ( P (cid:48) x ) and longitudinal ( P (cid:48) z ) components of thepolarization transfer to a bound proton in carbon via the quasi-free C( (cid:126)e, e (cid:48) (cid:126)p ) reaction, over awide range of missing momenta. We determine these polarization-transfers separately for protonsknocked out from the s - and p -shells. The electron-beam polarization was measured to determinethe individual components with systematic uncertainties which allow a detailed comparison withtheoretical calculations. I. INTRODUCTION
Polarization transfer from a polarized electron to a pro-ton in elastic scattering has become a recognized methodto measure the proton’s elastic electromagnetic form fac-tors, G E and G M [1–9]. Assuming the one-photon ex-change approximation, the ratio of the transverse ( P (cid:48) x )to longitudinal ( P (cid:48) z ) polarization-transfer components isproportional to the ratio of these form factors, G E /G M [10]. This provides a direct measurement of the form fac-tor (FF) ratio, even under conditions where one of theFFs is much larger than the other, and eliminates manysystematic uncertainties [11].Measuring the ratio of the components of the polariza-tion transfer to a bound proton in quasi-free kinematicson nuclei, which is sensitive to the electromagnetic FFratio, has been suggested as a method to study differ-ences between free and bound protons [4, 5]. As suchit can be used as a tool to identify medium modifica-tions in the bound proton’s internal structure, reflectedin the FFs and thereby in the polarization transfer. De-viations between measured polarization ratios in quasi-free A ( (cid:126)e, e (cid:48) (cid:126)p ) and elastic (cid:126)ep scattering can be interpreted ∗ Corresponding author: [email protected]; ‡ These two authors contributed equally to this work. only by comparing the measurements with realistic cal-culations of nuclear effects such as final-state interactions(FSI).Polarization-transfer experiments have been carriedout on H and C target nuclei at the Mainz Microtron(MAMI) [12–15], as well as on H, He and O at Jeffer-son Lab (JLab) [16–18], in search of medium modificationin the proton internal structure. These experiments wereperformed to study deeply bound nucleons, characterizedby high missing momentum which is equivalent (neglect-ing FSI) to protons with high initial momentum. It wasshown for the H measurements that the deviations in P (cid:48) x /P (cid:48) z from that of elastic (cid:126)ep scattering can be explainedby nuclear effects without the necessity of introducingmodified FFs [12–14]. Furthermore, when comparing thequasi-elastic polarization transfer to that of elastic (cid:126)ep scattering, the double ratio ( P (cid:48) x /P (cid:48) z ) A / ( P (cid:48) x /P (cid:48) z ) H exhibitsa very similar behavior for H [12, 14, 17], He [18], and C [15], suggesting a universality.While the ratio of the components is better deter-mined experimentally than the individual components(smaller systematic uncertainties), it is insensitive to pos-sible common effects to P (cid:48) x and P (cid:48) z which cancel in theratio. To test calculations in better detail, and furthercorroborate their reliability, measurements of individualcomponents of the polarization transfer are required.The polarization-transfer components have previouslybeen measured for H at MAMI [13, 14, 19], as well as a r X i v : . [ nu c l - e x ] A p r He at JLab [18]. For H, statistically significant but rel-atively small deviations were observed between the mea-sured components and the predicted values, but not intheir ratio [13, 14].The C nucleus is a particularly appealing target forsuch studies as one can selectively probe protons fromspecific nuclear shells, s and p . The average local densi-ties in these shells differ by about a factor of two, whichwas predicted to impact the polarization transfer to s -and p -shell protons differently [20]. If modifications tothe bound-proton structure by the nuclear medium exist,and are reflected in the proton FFs, then they may de-pend either on the off-shellness of the bound proton (vir-tuality), or on the local nuclear density, or both. There-fore, it is important to measure the components of thepolarization-transfer in both shells over a large missing-momentum range and to have reliable calculations forthem in order to identify modifications which can be re-lated to the FFs.In this work, we present measurements of the C( (cid:126)e, e (cid:48) (cid:126)p ) reaction at MAMI over a wide range of missingmomentum of the struck proton. The beam-polarizationdetermination had a sufficient accuracy for extracting theindividual polarization-transfer components P (cid:48) x and P (cid:48) z and for allowing a meaningful comparison with theoreti-cal calculations. The results of the measured ratio P (cid:48) x /P (cid:48) z have been reported in [15]. The new analysis presentedhere includes improvements in the corrections for the en-ergy loss of the particles when exiting the target, as wellas the polarization-extraction procedure.Section II describes the experimental setup, includingthe beam-line, target and spectrometers. Details of themeasured reaction and its kinematics are given in SectionIII. The data analysis and determination of the polariza-tion components are described in Section IV. Finally, inSection V, we interpret the data, compare them to a setof calculations, and explore the sensitivity to the protonFFs. II. EXPERIMENTAL SETUP
The experiment was performed at MAMI using thebeam-line and spectrometers of the A1 Collaboration[21]. We used a 600 MeV continuous-wave polarized elec-tron beam with a current of about 10 µ A and an ∼ - - - - - - - - - - - - - - - - - - - - Date74767880828486 B e a m P o l a r i z a t i o n ( % ) Setting A avg. beam pol.= 82.6%
Setting B avg. beam pol.= 79.1% M ø llerMott FIG. 1: The measured beam polarizations using the Møller(black circles) and Mott (square, green online) polarimeters.The average beam polarizations for each dataset are shownas horizontal lines. The error bars shown are statistical er-rors, and do not include systematic errors, estimated to bearound 2% for both the Møller and Mott polarimeters. Theshaded regions represent the data-taking periods for the twokinematic settings of the experiment. larized beam source. This resulted in a drop in the beampolarization from 82.6% to 79.1%. These values reflectthe average beam-polarization measurements before andafter the crystal was refreshed. During the data-taking,the beam polarization remained constant within error.The target consisted of three carbon (graphite) foilsof 0.8 mm thickness each, separated by about 15 mmand tilted at an angle of 40 ◦ with respect to the beam.Their transverse dimensions were 4 mm ×
20 mm, asshown in Fig. 2. This design reduced the distance thatthe protons would travel through the target foils, both byusing multiple foils (rather than a single thick foil) and byrotating each foil such that the detected protons wouldexit nearly normal to the foil. This reduced their energyloss in the target. It also improved the resolution forthe reaction-vertex determination, consequently reducingthe systematic uncertainty in the measured polarization-transfer components at the reaction point.Two high-resolution, small-solid-angle spectrometerswith momentum acceptances of 20-25% were used todetect the scattered electrons in coincidence with theknocked-out protons. Each of these spectrometers con-sists of a magnet system with a quadrupole-sextupole-dipole-dipole configuration, followed by vertical driftchambers (VDC) for tracking, and a scintillator systemfor triggering and the timing coincidence between the twospectrometers. The electron spectrometer also includesa ˇCerenkov detector for identifying electrons and distin-guishing them from background particles such as π − andcosmic rays.The proton spectrometer was equipped with a focal-plane-polarimeter (FPP) consisting of a 7 cm thick car-bon analyzer [21, 25] and horizontal drift chambers(HDC) to measure the secondary scattering of the proton FIG. 2: Schematic view of the carbon target, consisting ofthree graphite foils. The thick arrow (orange online) indicatesthe direction of the electron beam. The dimensions shownare h × l × w = 20 mm × × d = 15 mm, and φ t = 40 ◦ . in the analyzer, as shown in Fig. 3. The spin-dependentscattering of the polarized proton by the carbon ana-lyzer enables the determination of the proton’s transversepolarization components at the focal plane [25]. Thepolarization-transfer components at the reaction pointwere obtained by transforming the measured componentsusing the known spin precession in the magnetic field ofthe spectrometer. HorizontalDrift ChambersCarbonAnalyzerScintillatorsVerticalDrift Chambers protons
0 50 100cm shielding house
FIG. 3: Sideview of the detector system of the proton spec-trometer. The standard detector system (VDCs and scintilla-tors) is supplemented by the FPP, which consists of a carbonanalyzer and two double-planes of HDCs for proton trackingafter scattering in the carbon analyzer. Some proton trajec-tories are indicated within the region of acceptance (shadedarea). Adapted from [25].
III. MEASURED REACTION ANDKINEMATICS
The kinematics of the measured reaction are shown inFig. 4. The electron’s initial and final momenta are (cid:126)k and (cid:126)k (cid:48) respectively, and they define the scattering plane of thereaction. The proton’s initial momentum introduces, inaddition, the reaction plane defined by the momentumtransfer (cid:126)q = (cid:126)k − (cid:126)k (cid:48) and the exiting proton’s momentum (cid:126)p (cid:48) . The angle between the scattering plane and the re-action plane is denoted by φ pq .Following the convention of [16], we express the com-ponents of the polarization transfer (cid:126)P (cid:48) in the scattering-plane coordinate system, where ˆ z is along the directionof the momentum transfer (cid:126)q , ˆ y is along the direction of (cid:126)k × (cid:126)k (cid:48) , and ˆ x = ˆ y × ˆ z , forming a right-handed coordinatesystem. ϕ FIG. 4: Kinematics of the measured reaction.
The missing momentum (cid:126)p miss = (cid:126)q − (cid:126)p (cid:48) is the recoil mo-mentum of the residual nucleus. Neglecting FSI, − (cid:126)p miss is equal to the initial momentum of the emitted proton.We conventionally define positive and negative signs for p miss as the sign of (cid:126)p miss · (cid:126)q .Our data were taken at two kinematic settings, A andB, covering different ranges in p miss at different invari-ant four-momentum transfers, Q = − q . Setting Awas centered near p miss = 0, at Q = 0 .
40 (GeV /c ) .Setting B covered a region of large negative p miss , at Q = 0 .
18 (GeV /c ) . The details of these kinematic set-tings are given in Table I.Following [26], we distinguish between protons knock-out from the s - and p -shells using cuts on the missingenergy, E miss in the reaction, defined as: E miss ≡ ω − T p − T B , (1)where ω = k − k (cid:48) is the energy transfer, T p is the mea-sured kinetic energy of the outgoing proton, and T B isthe calculated kinetic energy of the recoiling B nucleus.The measured E miss spectrum is shown in Fig. 5. For the s -shell sample, we used the cut 30 MeV < E miss < p -shell sample, we used 15 MeV < E miss <
25 MeV [15]. Also shown in Fig. 5 are the p miss spectra for the two shells. TABLE I: The kinematic settings in the C( (cid:126)e, e (cid:48) (cid:126)p ) experi-ment. The angles and momenta represent the central valuesfor the two spectrometers: p p and θ p ( p e and θ e ) are theknocked out proton (scattered electron) momentum and scat-tering angles, respectively. Kinematic SetupA B E beam [MeV] 600 600 Q [(GeV /c ) ] 0.40 0.18 p miss [MeV] −
130 to 100 −
250 to − p e [MeV/ c ] 385 368 θ e [deg] 82.4 52.9 p p [MeV/ c ] 668 665 θ p [deg] − −
10 20 30 40 50 60 70 80 90 E miss [MeV]0.00.20.40.60.81.0 c o un t s B(g.s.) B (2.125 MeV) B (5.020 MeV) B (6.742 MeV) s p -9 -8 -7 -300 -200 -100 0 100 200 300 p miss [MeV] ∫ S d E m i ss [ M e V - ] FIG. 5: The measured proton missing-energy spectrum for C( (cid:126)e, e (cid:48) (cid:126)p ) (data shown are for Setup A). The distinct peakscorrespond to removal of p / -shell protons in C result-ing in B ground state and excited states as noted. The E miss ranges considered in the analysis for p / and s / pro-tons are marked in red and blue, respectively (color online).The inset shows the momentum distribution predictions of theindependent-particle shell model for p / and s / protons in C, adapted from [26].
When protons are knocked out from the p -shell of thecarbon nucleus, the A − B). However, knockout from the s -shell leaves theresidual A − p = 0 has aminimum for the p -state and a maximum for the s -state(see inset of Fig. 5).An important quantity characterizing the proton prior to its knock-out is its “off-shellness”. We quantify thisusing the virtuality, ν , a variable defined as [12] ν ≡ (cid:16) M A c − (cid:113) M A − c + p (cid:17) − p − M p c , (2)where M A is the mass of the target nucleus, M A − ≡ (cid:113) ( ω − E p + M A c ) − p is the mass of the residualnucleus (not necessarily in its ground state) determinedevent by event, and E p is the total energy of the outgoingproton. We note that the virtuality depends not only on p miss , but also on M A − . The minimum value of | ν | (fora given target nucleus) is | ν | min = (cid:16) M p − (cid:0) M A − M g . s .A − (cid:1) (cid:17) c , (3)where M g . s .A − is the ground-state mass of the the resid-ual nucleus (in this case, B). For the C( (cid:126)e, e (cid:48) (cid:126)p ) Breaction, | ν | min = 0.0297 (GeV /c ) . Protons knocked outfrom the s -shell are generally further off-shell than thosein the p -shell (even in events at the same p miss , due totheir larger E miss ). IV. DATA ANALYSISA. Event reconstruction and selection
For the event reconstruction, we used the
Cola++ re-construction framework [27], developed by the A1 col-laboration. The reconstructed angles and positions ofthe tracks at the VDC plane are used in a fit to findthe initial momentum, angle and vertex position of eachparticle. A multivariate polynomial fit is then used toreconstruct the proton’s spin-transfer matrix (STM) foreach event.Software cuts were then applied to the data. We re-quire the time-coincidence between the scintillators of thetwo spectrometers to be within a ± ◦ . We also required the fitted position of thescattering point in the FPP (determined by the HDC andthe extrapolated trajectory from the VDC) to be consis-tent with the actual position of the analyzer. B. Polarization fitting
There are two types of polarization observables thatcan be obtained in this experimental setup: the inducedpolarization, (cid:126)P , (which is the proton polarization for anunpolarized electron beam) and the polarization transfer, (cid:126)P (cid:48) , (which determines the beam polarization-dependentpart of the proton polarization). The total outgoing pro-ton polarization is related to these observables via: (cid:126)P p, tot = (cid:126)P + hP e (cid:126)P (cid:48) , (4)where h and P e are the helicity and polarization of theelectron beam.We perform a fit to obtain (cid:126)P and (cid:126)P (cid:48) by maximizingthe log-likelihoodlog L = (cid:88) k log(1 + (cid:126)P p, tot · (cid:126)λ k ) , (5)where the (cid:126)λ k for each event are given by (cid:126)λ k = a S − − sin φ FPP cos φ FPP , (6)and φ FPP is the measured azimuthal scattering angle inthe FPP. S is the calculated spin-transfer matrix forthe event and a is the analyzing power of the event (asdetermined in [28, 29]).Three of these components ( P x , P (cid:48) y and P z ) are ex-pected to be very small for individual events and anti-symmetric in their dependence on φ pq [30]. Hence, theywould average to zero when considering event samplesthat have symmetric distributions in φ pq , as is nearlythe case in our dataset. Therefore, in order to improvethe stability of our fit, we fix these parameters to zero ,leaving us with (cid:126)P p, tot = hP e P (cid:48) x P y hP e P (cid:48) z . (7)We partitioned both the s - and p -shell knockout datainto bins by p miss , and performed the above procedureon each bin separately, the results of which are shown inFig. 6(a). Likewise, we binned the data by the virtuality,and show the results in Fig. 6(b). We present the results We have also performed the fit with all six parameters included,and have found that this does not strongly affect the outcome ofthe fit except in bins with poor statistics. binned by both variables to show how the polarizationmay vary with the protons’ motion ( p miss bins) and alsotheir off-shellness (virtuality bins). This avoids conflatingeffects related to one variable with those related to theother, as the two are correlated. C. Systematic uncertainties in the measurements
The systematic errors in these measurements are dueto a few sources, which are presented in Table II. Thelargest contribution to the uncertainty in the polariza-tion components P (cid:48) x and P (cid:48) z is due to the beam polariza-tion, which was determined with an estimated accuracyof 2%. These components are also sensitive to the analyz-ing power of the carbon secondary scatterer, which in thiskinematic region is known to about 1% [25, 28, 29]. How-ever, the ratio P (cid:48) x /P (cid:48) z is independent of both the beampolarization and the analyzing power.The uncertainties in the beam energy and the centralkinematics of the spectrometers in each dataset affect thebasis vectors that define the scattering-plane coordinatesystem, as well as which bin an event goes into. Thealignment between the HDC coordinate system and thetracks extrapolated from the VDC to the HDC planealso affected the polarization measurement, since thesemeasurements depend on the distribution of the anglesof the secondary scattering.These three sources of uncertainty (beam energy, cen-tral kinematics, and detector alignment) were determinedin the following manner. We modified each of the vari-ables one-by-one by their uncertainty values, and re-peated the analysis, and then determined how much thisaffected the extracted polarizations. The errors fromeach source were then added in quadrature. TABLE II: Systematic uncertainties of P (cid:48) x , P (cid:48) z and P (cid:48) x /P (cid:48) z .All values are in percent. See text for details. dP (cid:48) x /P (cid:48) x dP (cid:48) z /P (cid:48) z d [ P (cid:48) x /P (cid:48) z ] P (cid:48) x /P (cid:48) z Beam pol. 2.0 2.0 0.0Analyzing power 1.0 1.0 0.0Beam energy 0.2 0.6 0.8Central kinematics 0.6 0.8 0.9Alignment < < The systematics due to software cuts were studied byreanalyzing the data with each cut slightly tighter thanthe actual value that is used in the final analysis andtaking the average of the effects of the modified cut overall of the bins. The values in the row labeled “softwarecuts” in Table II are the quadratic sums of the effectsfrom each of the different cuts. P x Set A, s Set B, s Set A, p Set B, p P z -300 -250 -200 -150 -100 -50 0 50 100 p miss [MeV/ c ]1.61.41.21.00.80.60.4 P x / P z p miss < 0 p miss > 0 (a) Polarization-transfer components and their ratioversus missing momentum. P x Set A, s Set B, s Set A, p Set B, p P z -0.150 -0.100 -0.050 0.000 -0.050 -0.100 -0.150 [(GeV/ c ) ]1.61.41.21.00.80.60.4 P x / P z p miss < 0 p miss > 0 (b) Polarization-transfer components and their ratio versusvirtuality.
FIG. 6: The measured polarization-transfer components, P (cid:48) x (top), P (cid:48) z (middle), and their ratio, P (cid:48) x /P (cid:48) z (bottom). These areplotted versus the missing-momentum (a) and virtuality (b). In the virtuality plot (b), the data with positive (negative) p miss are shown separately. The grey band shows the kinematically forbidden region at | ν | < /c ) (see Eq. 3). Theuncertainties shown are statistical only. Systematic errors are discussed in Section IV C. Triangles (circles) refer to kinematicSetting A (B). Symbols that are open on the left (right) side refer to s - ( p -) shell removals, and are colored blue (red) online.The legend is common to all panels in the figure. The uncertainty in the spin-precession evaluation wasestimated by comparing the STM calculated internally bythe
Cola++ software in the event reconstruction (whichuses a polynomial fit) and a more precise (but muchslower) calculation using the
QSPIN program [25]. Thefit was able to reproduce the spin-precession with an ac-curacy of 0.3% [25].Furthermore, the finite resolution of the proton’s tra-jectory parameters, especially the vertex position, addsadditional systematic uncertainty to the precession. Tobegin determining this part of the systematic uncertainty,we used
QSPIN to calculate the spin-transfer matrix, S ref ,for a reference trajectory. We then produced 100 othertrajectories with normally distributed variations in eachparameter, where the standard deviation of each param-eter equals the resolution of that parameter, to produceprecession matrices S i . The resulting uncertainty on the measured polarization due to the trajectory is then δ (cid:126)P = ( I − avg i [ S i ] S − ) (cid:126)P , (8)where I is the 3 × i [ S i ] repre-sents the average of the matrices S i . The average is per-formed because the measured polarization is calculatedusing an ensemble of trajectories, rather than a singletrajectory.The total estimated systematic uncertainties are cal-culated by adding the effects of each of the individualsources in quadrature. The systematic uncertainties forthe individual transfer components P (cid:48) x and P (cid:48) z are ∼ ∼ V. INTERPRETATION OF RESULTSA. General observations
As shown in Fig. 6, the P (cid:48) x components are less sen-sitive to the different kinematic variables, p miss and ν ,than P (cid:48) z . We find that P (cid:48) x is nearly identical for s - and p - shells at the same p miss . The P (cid:48) z component for p -shellknockout dips down at small negative p miss , while thisdoes not seem to be the case in s -shell knockout.We observe that where the two kinematic settings over-lap in virtuality, there is no regularity in P (cid:48) z , and that P (cid:48) z is much larger at Setting A than at Setting B.In order to further interpret our polarization-transferdata, we compare them to dedicated calculations of C( (cid:126)e, e (cid:48) (cid:126)p ) described in Section V B below. B. Calculations of the polarization transfer for C We compared the measured polarization transfer torelativistic distorted-wave impulse approximation (RD-WIA) calculations [31] where the FSI between the out-going nucleon and the residual nucleus are described bya phenomenological relativistic optical potential.In the calculations, the so-called “democratic” opti-cal potential [32] has been used, which has been ob-tained using a global fit to over 200 datasets using elas-tic proton-nucleus scattering over a broad range of nu-clei from helium to lead. The differences in the calcu-lated polarization transfer due to the choice of the opti-cal potential were estimated at about 2-4%. The evalu-ation was done by comparing the results obtained withthe democratic and the energy-dependent and atomic-number-independent (EDAI) relativistic optical poten-tial [33], which is a single-nucleus parameterization, con-structed to reproduce elastic proton-scattering data juston C.The relativistic bound-state wave functions used in thecalculations have been obtained by solving the relativisticHartree-Bogoliubov Equations using the program
ADFX [34]. The model is applied in the mean-field approxi-mation to the description of ground-state properties ofspherical nuclei, using a Lagrangian containing the σ , ω and ρ mesons and the photon field [35–38]. Moreover,finite-range interactions are included to describe pair-ing correlations and the coupling to particle continuumstates. The Lagrangian parameters are usually obtainedby a fitting procedure to some bulk properties of a set ofnuclei. The wave-functions used in our calculations wereobtained with the NL-SH parametrization [39]. The re-sults of the calculations using different parameterizationsdiffer by about 0.5-0.8%. The Coulomb distortion of the electron wave functionsis considered using the effective-momentum approxima-tion. Our calculation uses the parameterization of thefree-proton FFs from [40], which are known to within0.5% in the kinematic region of our experiment.In coplanar kinematics, a set of 8 structure functionscontribute to the polarization transfer [30, 41]. In non -coplanar kinematics, an additional structure function,¯ h (cid:48) N , contributes to P (cid:48) x , but not P (cid:48) z .The RDWIA program [31] was written to perform cal-culations only in the coplanar kinematics of the usual A ( (cid:126)e, e (cid:48) (cid:126)p ) experiments. Therefore, it calculates only thestructure functions that contribute in coplanar kinemat-ics and not ¯ h (cid:48) N . We performed calculations in non-coplanar kinematics using the approximation¯ h (cid:48) N = − ¯ h (cid:48) S , (9)where ¯ h (cid:48) S is one of the structure functions calculated in[31]. Eq. 9 is exactly true in parallel kinematics ( θ pq = 0)[30], and we assume that it provides a valid approxima-tion at small θ pq , which dominate the kinematics of ourdata.We averaged the polarizations calculated for the kine-matics of a sample of events in each p miss bin and showthe results in Fig. 7 as solid curves, compared to our data.In order to examine the sensitivity of the calculations tothe proton FFs, we show as long-dashed (dotted) curvesthe results obtained with G E /G M rescaled by +( − ) 10%.For Setting B, the curves for P (cid:48) x at ±
10% are very closeto the calculations for unmodified FFs, indicating that P (cid:48) x is not sensitive to the FF ratios in this region. This makesthe P (cid:48) x at large p miss useful for testing the agreementbetween the calculations and the data. By contrast, P (cid:48) z is strongly sensitive to the FF scaling. At Setting A,on the other hand, both P (cid:48) x and P (cid:48) z are sensitive to thescaling of the FF ratio.For the s -shell knockout at both kinematic settings,there is excellent agreement between the calculations andthe data, except in P (cid:48) z at large negative p miss (Setting B),where they differ by about 10%. One may consider thepossibility of scaling G E /G M in order to obtain a betterfit to the data.For the p -shell knockout, the calculations and the dataare in decent agreement with each other, but the agree-ment is not as good as for the s -shell. Specifically, atSetting A, P (cid:48) z is overestimated by the calculations, whilethe magnitude of P (cid:48) x appears to be somewhat underesti-mated by the calculations.Before quantifying the agreement between the calcula-tions and the data, we discuss in Sec. V C the uncertain-ties and limitations of the calculations. C. Uncertainties of the calculations
We have adopted a model (RDWIA) which is basedon some assumptions and approximations which may af- P x calc (unmodified)calc ( G E / G M +10%)calc ( G E / G M s Set B, s P z -300 -200 -100 0 100 p miss [MeV/ c ]1.61.41.21.00.80.60.4 P x / P z p miss < 0 p miss > 0 (a) s -shell knockout P x calc (unmodified)calc ( G E / G M +10%)calc ( G E / G M p Set B, p P z -300 -200 -100 0 100 p miss [MeV/ c ]1.61.41.21.00.80.60.4 P x / P z p miss < 0 p miss > 0 (b) p -shell knockout FIG. 7: DWIA calculations of the polarization observables, with unmodified FFs (solid curves), and with G E /G M scaled by+10% ( − s -shell knockout (a) (blue online) and p -shell knockout (b) (red online). fect the comparison with the data. Within the modelthere are uncertainties due to the choice of the differentingredients (bound-state wave functions, optical poten-tials, and proton form factors) that are adopted in thecalculations. We evaluated the combined contribution tothe uncertainty in the calculation due to the parameteri-zations of these ingredients to be 2.2%, 3.8% and 4.0% for P (cid:48) x , P (cid:48) z , and P (cid:48) x /P (cid:48) z , respectively. These were obtained byadding the estimated contributions, as discussed in Sec-tion V B, from those three sources in quadrature. By farthe largest contribution to this uncertainty comes fromthe choice of the optical potential.In the calculations for the p -shell knockout, it is as-sumed that the B residual nucleus is left in its groundstate. The data include also the excited states, as shownin the measured missing-energy spectrum of Fig. 5. How-ever, the majority of the p -shell contribution comes from the ground state and we have checked that the excitedstates do not strongly affect the data for p -shell knock-out. Furthermore, the wave-function of the p -shell hasa minimum at p = 0, possibly reducing the numericalaccuracy of the calculations at low p miss for this shell.The s -shell wave-function does not have this problem,but the fact that in this case the A − D. Quantifying the agreement of the calculationswith the data
To quantify the agreement between the calculationsand the data, we use a bin-by-bin comparison, employingthe χ criterion: χ = (cid:88) α ∈ { x,z } i ∈ bins ( P (cid:48) meas α,i − P (cid:48) calc α,i ) ( dP (cid:48) meas α,i ) + ( dP (cid:48) calc α,i ) , (10)where P (cid:48) meas α,i and P (cid:48) calc α,i are the measured and calcu-lated values of the polarization transfer for a given axis, α ( x and z ), in the i th bin. The uncertainty in the mea-surement, dP (cid:48) meas α,i , includes the statistical error of thefit and the systematic uncertainty added in quadrature.For dP (cid:48) calc α,i , we only included the errors of the calcula-tion due to the parameterizations of the optical poten-tial, wave-function, and free-proton FFs (see Sec. V C),since the impacts of other effects on the calculations areundetermined. The χ values are then converted to p -values, denoted by p val [ χ ], where the number of degreesof freedom is twice the number of bins, as there is a mea-surement of P (cid:48) x and P (cid:48) z for each bin.Scaling the FF ratio G E /G M has been suggested abovein order to obtain a better fit to the data. In order todetermine if this scaling improves the agreement betweenthe calculations and the data, we varied the value of a mod ≡ (cid:18) G E G M (cid:19) bound (cid:30)(cid:18) G E G M (cid:19) free − , (11)and then reevaluated χ and the p -values. We deter-mined the optimal values of a mod which minimized the χ separately for the s - and p -shells in each kinematicsetting. The uncertainty on the fitted values of a mod aregiven by δa mod = (cid:20) ∂ ∂a [ χ ] (cid:21) − / , (12)evaluated where χ is at its minimum. The results ofthese comparisons are given in Table III.This comparison shows that the s -shell calculations atboth kinematic settings agree with the data, even with-out scaling the FFs. At Setting B the optimized scaling a mod = − .
6% improves the agreement for P (cid:48) z , while atSetting A there is no need of FF scaling.The p -shell calculations, without and even with FFscaling, are generally in worse agreement with the datathan those for the s -shell. However, as shown in Table III, about 3% for both P (cid:48) x and P (cid:48) z , see Table II. TABLE III: The p -values for the fits with unscaled and opti-mally scaled FFs. We also include the a mod for the optimalscaling. The number of degrees of freedom for the compari-son with scaled FFs is one fewer than for those with unscaledFFs.shell/ w/ unmod. FFs a mod w/ mod. FFssetting n dof p val [ χ ] (%) n dof p val [ χ ] s / A 24 0.91 0 . ± . − . ± . p / A 24 0.017 10 . ± . − . ± . a scaling of a mod = 10 .
4% at Setting A greatly improvesthe agreement with the data. This is an indication ofeither genuine modifications to the FFs, or of other pos-sible effects which mimic a modification to the FFs. AtSetting B, no FF scaling improves the agreement withthe data.
VI. CONCLUSIONS
Measurements of the transverse and longitudinalcomponents of the polarization transfer in quasi-free C( (cid:126)e, e (cid:48) (cid:126)p ) reaction have been presented and comparedto RDWIA calculations. The comparison gives an over-all good agreement, but some discrepancies are observedin s -shell knockout at large negative p miss and in p -shellknockout.For s -shell knockout, both P (cid:48) x and P (cid:48) z are in very goodagreement with the data at low p miss . P (cid:48) x is in goodagreement with the data also at high p miss . The only no-table discrepancy for the s -shell is that the calculationsunderestimate P (cid:48) z by about 10% at large negative p miss .The s -shell knockout at large p miss is a region of partic-ular interest when searching for medium modificationsto the form factors [20], because it has both the largestvirtuality and the largest nuclear density in our dataset.We note that in this region, P (cid:48) x is insensitive to G E /G M ,and therefore the agreement between the calculated andmeasured P (cid:48) x gives further credibility to the calculations.Modifying the form-factor ratio in the calculations by − .
6% in this region improves their agreement with thedata (by scaling P (cid:48) z while keeping P (cid:48) x unchanged).For the p -shell knockout, the agreement with the datais not as good as for the s -shell. At low p miss , scaling theform-factor ratio by +10.4% greatly improves the agree-ment with the data, which however, remains worse thanfor the s -shell. At high p miss , scaling the form-factor ratiodoes not improve the agreement with the data. Furthertheoretical work is required to understand the source ofthe discrepancies for p -shell knockout.No global scaling to G E /G M , common to all kinematicregions in our dataset, would solve all the discrepancies.Scaling the form-factor ratio differently for each kine-matic region leads to a good agreement between the data0and calculations. We note, however, that form-factormodification is not the only possible solution. Other ex-planations can be envisaged and deserve further explo-ration.From the experimental point of view, the large statis-tical errors at the large negative p miss setting for bothshells will be reduced by combining our data with thoseof a recent measurement from MAMI on C at the samekinematic setting. The combined dataset could eitherimprove the statistical significance of the deviations orshow them to be statistical fluctuations.Furthermore, the study of s -shell knockout in futureexperiments may be useful in search of medium modi-fications of the nucleon form factors in nuclei. Such ameasurement has been proposed at MAMI on Ca [42].Since the 1 s -shell has large virtuality in Ca (similar tothat of C), it would be a suitable nucleus to extend the searches for medium modifications.
Acknowledgments
We would like to thank the Mainz Microtron opera-tors and technical crew for the excellent operation of theaccelerator. This work is supported by the Israel Sci-ence Foundation (Grants 390/15, 951/19) of the IsraelAcademy of Arts and Sciences, by the PAZY Founda-tion (Grant 294/18), by the Israel Ministry of Science,Technology and Space, by the Deutsche Forschungsge-meinschaft (Collaborative Research Center 1044), by theU.S. National Science Foundation (PHY-1205782). Weacknowledge the financial support from the Slovenian Re-search Agency (research core funding No. P1–0102). [1] M. K. Jones et al. (Jefferson Lab Hall A), G Ep /G Mp ratio by polarization transfer in polarized (cid:126)ep → e(cid:126)p , Phys.Rev. Lett. , 1398 (2000), arXiv:nucl-ex/9910005 [nucl-ex] .[2] O. Gayou et al. (Jefferson Lab Hall A), Measurement of G Ep /G Mp in (cid:126)ep → e(cid:126)p to Q = 5.6 GeV , Phys. Rev.Lett. , 092301 (2002), arXiv:nucl-ex/0111010 [nucl-ex].[3] V. Punjabi et al. , Proton elastic form-factor ratios toQ = 3.5-GeV by polarization transfer, Phys. Rev. C , 055202 (2005), [Erratum: Phys. Rev. C 71, 069902(2005), doi:10.1103/PhysRevC.71.069902], arXiv:nucl-ex/0501018 [nucl-ex] .[4] B. D. Milbrath, J. I. McIntyre, C. S. Armstrong, D. H.Barkhuff, W. Bertozzi, J. Chen, and others (BatesFPP Collaboration) (Bates FPP), A Comparison of po-larization observables in electron scattering from the pro-ton and deuteron, Phys. Rev. Lett. , 452 (1998), [Er-ratum: Phys. Rev. Lett. 82, 2221 (1999)], arXiv:nucl-ex/9712006 [nucl-ex] .[5] D. H. Barkhuff et al. , Measurement of recoil proton po-larizations in the electrodisintegration of deuterium bypolarized electrons, Phys. Lett. B , 39 (1999).[6] T. Pospischil et al. (A1), Measurement of G Ep /G Mp viapolarization transfer at Q = 0.4 (GeV /c ) , Eur. Phys.J. A , 125 (2001).[7] O. Gayou, K. Wijesooriya, et al. (The Jefferson Lab HallA Collaboration), Measurements of the elastic electro-magnetic form factor ratio µ p G ep /G mp via polarizationtransfer, Phys. Rev. C , 038202 (2001).[8] G. MacLachlan et al. , The ratio of proton elec-tromagnetic form factors via recoil polarimetry at q =1.13(gev /c ) , Nucl. Phys. A , 261 (2006).[9] M. K. Jones et al. (Resonance Spin Structure Collabo-ration), Proton G E /G M from beam-target asymmetry,Phys. Rev. C , 035201 (2006).[10] A. I. Akhiezer and M. Rekalo, Polarization effects in thescattering of leptons by hadrons, Sov. J. Part. Nucl. ,277 (1974), [Fiz. Elem. Chast. Atom. Yadra 4, (1973)662].[11] C. F. Perdrisat et al. , Nucleon electromagnetic form fac- tors, Prog. Part. Nucl. Phys. , 694 (2007).[12] I. Yaron, D. Izraeli, et al. , Polarization-transfer measure-ment to a large-virtuality bound proton in the deuteron,Phys. Lett. B , 21 (2017).[13] D. Izraeli, I. Yaron, et al. (A1), Components ofpolarization-transfer to a bound proton in a deuteronmeasured by quasi-elastic electron scattering, Phys. Lett.B , 107 (2018), arXiv:1801.01306 [nucl-ex] .[14] S. Paul, D. Izraeli, T. Brecelj, I. Yaron, et al. (A1),Polarization-transfer measurements in deuteron quasi-elastic anti-parallel kinematics, Phys. Lett. B , 599(2019), arXiv:1905.05594 .[15] D. Izraeli, T. Brecelj, et al. (A1), Measurement ofpolarization-transfer to bound protons in carbon andits virtuality dependence, Phys. Lett. B , 95 (2018),arXiv:1711.09680 [nucl-ex] .[16] S. Strauch et al. (Jefferson Lab E93-049), Polariza-tion transfer in the H e( (cid:126)e, e (cid:48) (cid:126)p ) H reaction up to Q =2.6 (GeV/ c ) , Phys. Rev. Lett. , 052301 (2003).[17] B. Hu et al. , Polarization transfer in the H( (cid:126)e, e (cid:48) (cid:126)p )n reac-tion up to Q = 1.61 (GeV/c) , Phys. Rev. C , 064004(2006).[18] M. Paolone, S. P. Malace, S. Strauch, et al. (E03-104Collaboration), Polarization transfer in the He( (cid:126)e, e (cid:48) (cid:126)p ) H reaction at Q = 0 . . /c ) , Phys. Rev. Lett. , 072001 (2010).[19] S. Paul, T. Brecelj, H. Arenh¨ovel, et al. , The influenceof Fermi motion on the comparison of the polarizationtransfer to a proton in elastic (cid:126)ep and quasi-elastic (cid:126)eA scat-tering, Phys. Lett. B , 445 (2019), arXiv:1901.10958[nucl-ex] .[20] G. Ron, W. Cosyn, E. Piasetzky, J. Ryckebusch,and J. Lichtenstadt, Nuclear density dependence of in-medium polarization, Phys. Rev. C , 028202 (2013).[21] K. Blomqvist et al. , The three-spectrometer facility atMAMI, Nucl. Instrum. and Meth. A , 263 (1998).[22] B. Wagner, H. G. Andresen, K. H. Steffens, W. Hart-mann, W. Heil, and E. Reichert, A Moller polarimeterfor CW and pulsed intermediate-energy electron beams,Nucl. Instrum. Meth. A , 541 (1990).[23] P. Bartsch, Aufbau eines Møller-polarimeters f¨ur die drei-spektrometer-anlage und messung der he-lizit¨atsasymmetrie in der reaktion p ( e, e (cid:48) p ) π im bereichder ∆ -resonanz , Ph.D. thesis, Institut f¨ur Kernphysik derUniversit¨at Mainz (2001).[24] V. Tioukine, K. Aulenbacher, E. Riehn, et al. , A Mottpolarimeter operating at MeV electron beam energies,Rev. Sci. Instrum. , 033303 (2011).[25] T. Pospischil et al. , The focal plane proton-polarimeterfor the 3-spectrometer setup at MAMI, Nucl. Instrum.Methods. Phys. Res., Sect. A , 713 (2002).[26] D. Dutta et al. , Quasielastic ( e, e (cid:48) p ) reaction on C, Fe,and
Au, Phys. Rev. C , 064603 (2003).[27] M. Distler, H. Merkel, and M. Weis, Data acquisition andanalysis for the 3-spectrometer-setup at mami, in Pro-ceedings of the 12th IEEE Real Time Congress on Nu-clear and Plasma Sciences, 2001 (2001).[28] E. Aprile-Giboni, R. Hausammann, E. Heer, R. Hess,C. Lechanoine-Leluc, W. Leo, S. Morenzoni, Y. Onel,and D. Rapin, Proton-carbon effective analyzing powerbetween 95 and 570 MeV, Nucl. Instrum. Meth. , 147(1983).[29] M. W. McNaughton et al. , The p-C analyzing power be-tween 100 and 750 MeV, Nucl. Instrum. Meth. A ,435 (1985).[30] C. Giusti and F. D. Pacati, Complete Determination ofScattering Amplitudes and Nucleon Polarization in Elec-tromagnetic Knockout Reactions, Nucl. Phys. A , 685(1989).[31] A. Meucci, C. Giusti, and F. D. Pacati, Relativistic cor-rections in (e, e-prime p) knockout reactions, Phys. Rev.C , 014604 (2001), arXiv:nucl-th/0101034 [nucl-th] .[32] E. D. Cooper, S. Hama, and B. C. Clark, Global diracoptical potential from helium to lead, Phys. Rev. C ,034605 (2009). [33] E. D. Cooper, S. Hama, B. C. Clark, and R. L. Mercer,Global Dirac phenomenology for proton nucleus elasticscattering, Phys. Rev. C , 297 (1993).[34] W. Poschl, D. Vretenar, and P. Ring, RelativisticHartree-Bogolyubov theory in coordinate space: Finiteelement solution for a nuclear system with sphericalsymmetry, Comput. Phys. Commun. , 217 (1997),arXiv:nucl-th/9706031 [nucl-th] .[35] B. D. Serot and J. D. Walecka, The Relativistic NuclearMany Body Problem, Adv. Nucl. Phys. , 1 (1986).[36] B. D. Serot and J. D. Walecka, Recent progress in quan-tum hadrodynamics, Int. J. Mod. Phys. E , 515 (1997),arXiv:nucl-th/9701058 [nucl-th] .[37] P. Ring, Relativistic mean field in finite nuclei, Prog.Part. Nucl. Phys. , 193 (1996).[38] G. A. Lalazissis, J. Konig, and P. Ring, A Newparametrization for the Lagrangian density of relativis-tic mean field theory, Phys. Rev. C , 540 (1997),arXiv:nucl-th/9607039 [nucl-th] .[39] M. Sharma, M. Nagarajan, and P. Ring, Rho meson cou-pling in the relativistic mean field theory and descriptionof exotic nuclei, Phys. Lett. B , 377 (1993).[40] J. C. Bernauer, M. O. Distler, J. Friedrich, T. Walcher,P. Achenbach, C. Ayerbe-Gayoso, et al. (A1), Electricand magnetic form factors of the proton, Phys. Rev. C , 015206 (2014).[41] S. Boffi, C. Giusti, F. d. Pacati, and M. Radici, Electro-magnetic Response of Atomic Nuclei , Oxford Studies inNuclear Physics, Vol. 20. 20 (Clarendon Press, OxfordUK, 1996).[42] P. Achenbach et al. , A1 MAMI Proposal: PolarizationTransfer Measurements in Ca( (cid:126)e, e (cid:48) , (cid:126)p )39