Comparison of recoil polarization in the 12 C( e → , e ′ p → ) process for protons extracted from s and p shell
T. Kolar, S. Paul, T. Brecelj, P. Achenbach, R. Böhm, D. Bosnar, E.O. Cohen, M.O. Distler, A. Esser, C. Giusti, M. Hoek, D. Izraeli, S. Kegel, Y. Kohl, I. Korover, J. Lichtenstadt, I. Mardor, H. Merkel, M. Mihovilovič, J. Müller, U. Müller, M. Olivenboim, E. Piasetzky, J. Pochodzalla, B.S. Schlimme, C. Sfienti, S. Širca, R. Spreckels, S. Štajner, M. Thiel, A. Weber, I. Yaron
CComparison of recoil polarization in the C( (cid:126)e, e (cid:48) (cid:126)p ) process for protons extracted from s and p shell T. Kolar a, ∗ , S. Paul b , T. Brecelj a , P. Achenbach c , R. Böhm c , D. Bosnar d , E.O. Cohen b , M.O. Distler c , A. Esser c ,C. Giusti e,f , M. Hoek c , D. Izraeli b , S. Kegel c , Y. Kohl c , I. Korover g,b , J. Lichtenstadt b , I. Mardor b,h , H. Merkel c ,M. Mihovilovič a,c,i , J. Müller c , U. Müller c , M. Olivenboim b , E. Piasetzky b , J. Pochodzalla c , B.S. Schlimme c , C. Sfienti c ,S. Širca i,a , R. Spreckels c , S. Štajner a , M. Thiel c , A. Weber c , I. Yaron b ,(A1 Collaboration) a Jožef Stefan Institute, 1000 Ljubljana, Slovenia b School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel c Institut für Kernphysik, Johannes Gutenberg-Universität, 55099 Mainz, Germany d Department of Physics, Faculty of Science, University of Zagreb, HR-10000 Zagreb, Croatia e Dipartimento di Fisica, Università degli Studi di Pavia, 27100 Pavia, Italy f Istituto Nazionale di Fisica Nucleare sezione di Pavia, 27100 Pavia, Italy g Department of Physics, NRCN, P.O. Box 9001, Beer-Sheva 84190, Israel h Soreq NRC, Yavne 81800, Israel i Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia
Abstract
We present first measurements of the double ratio of the polarization transfer components ( P (cid:48) x /P (cid:48) z ) p / ( P (cid:48) x /P (cid:48) z ) s for knock-outprotons from s and p shells in C measured by the C( (cid:126)e, e (cid:48) (cid:126)p ) reaction in quasi-elastic kinematics. The data are comparedto theoretical predictions in relativistic distorted-wave impulse approximation. Our results show that differences between s -and p -shell protons, observed when compared at the same initial momentum (missing momentum) largely disappear whenthe comparison is done at the same proton virtuality. We observe no density-dependent medium modifications for protonsfrom s and p shells with the same virtuality in spite of the large differences in the respective nuclear densities.
1. Introduction
The effects of the nuclear medium on the structure ofbound nucleons and their dependence on the nuclear av-erage density are subject to theoretical and experimentalinvestigations [1–24]. The C nucleus is a very attrac-tive target to study nuclear density-dependent differencesin bound nucleons. Its nuclear structure is well known, withnucleons in the s and p shells, and the average local nucleardensity in these shells differs by about a factor of two [1].Studying quasi-elastic processes on protons, which are sen-sitive to the proton form-factors, should be a good tool toobserve any density dependence arising from the differencesbetween the protons extracted from the two shells. Goodtheoretical calculations for this nucleus [2, 3] further helpwith the interpretation of the experimental observations.The free nucleon structure is characterized by its elec-tromagnetic form-factors (EM FFs) G E and G M . In theone-photon exchange approximation, the ratio between thetransverse ( x ) and longitudinal ( z ) polarization-transfercomponents, P (cid:48) x /P (cid:48) z , measured by elastic polarized elec-tron scattering is proportional to G E /G M [25]. In quasi-elastic A( (cid:126)e, e (cid:48) (cid:126)p ) reactions, the sensitivity of the P (cid:48) x /P (cid:48) z to ∗ Corresponding author
Email address: [email protected] (T. Kolar) Present address: MIT-LNS, Cambridge, MA 02139, USA. the G E /G M ratio persists and, hence, the measurement ofpolarization transfer to the knocked-out proton has beensuggested as a tool to investigate nuclear-medium modifi-cations of the bound proton [26]. Good theoretical calcula-tions, which give a reliable account of the nuclear processessuch as final-state interactions (FSI), allow conclusions tobe made from these experiments, by distinguishing betweenthe effects of such nuclear processes and possible modifica-tions of the bound nucleon.Theoretical calculations suggest that comparing the po-larization transfer to knocked-out protons from the s and p shells should result in measurable differences in the ratioof the polarization-transfer components [1]. We study thedouble ratio ( P (cid:48) x /P (cid:48) z ) p / ( P (cid:48) x /P (cid:48) z ) s , which is sensitive to the de-viation of the form-factor ratio, G E /G M , in each shell. Wenote that this is equivalent to ( P (cid:48) sz /P (cid:48) pz ) / ( P (cid:48) sx /P (cid:48) px ) where,based on calculations discussed below, one may expect thatdifferences in FSI for knockout protons from s and p shells(as well as between the longitudinal and transverse compo-nents) will largely cancel out.Almost all theoretical calculations characterize the boundnucleons by their initial internal momentum which, in theabsence of FSI, is equivalent to the measured missing mo-mentum in the reaction. But it has been shown that de-viations of the ratio P (cid:48) x /P (cid:48) z obtained through the quasi-freereaction from that of the free nucleon as a function of thebound-proton virtuality (see Eq. (2)) are in overall good Preprint submitted to Physics Letters B July 30, 2020 a r X i v : . [ nu c l - e x ] J u l greement between different nuclei, and at different mo-mentum transfers and kinematics. This suggests that thenucleon’s virtuality, which is a measure of its “off-shellness”,might be a better variable to characterize the bound nucleon[4]. Since virtuality depends also on the nucleon binding en-ergy, one cannot compare the polarization transfer to thenucleons from the different shells at the exact same missingmomentum and virtuality. However, we chose the kinemat-ics for the measurements so that there is an overlap between s - and p -shell removal for both the missing momentum andthe virtuality.We present here polarization-transfer measurements tothe protons extracted from the s and p shell in C in searchof nuclear-density-dependent modifications of the boundproton. We study the transverse-to-longitudinal compo-nents ratio, and compare the results from the two shellsby the aforementioned double ratios. The data are alsocompared to calculations in relativistic distorted-wave im-pulse approximation (RDWIA) [2] which use free-nucleonelectromagnetic form-factors. We present the comparisonin both missing momentum and bound-proton virtuality,and demonstrate the advantage of using the latter as a pa-rameter for such comparisons.
2. Experimental Setup and Kinematics
The experiment was carried out in A1 Hall at the MainzMicrotron (MAMI) using a
600 MeV continuous-wave (CW)polarized electron beam of about µ A . The measurementswere performed at Q = 0 .
175 GeV /c . The beam polar-ization, P e , was measured periodically using the standardMøller [27, 28] and Mott [29] polarimeters. The polariza-tion range was . < P e < . . The polarizationwas increasing at the beginning of the experiment with thedecrease of the quantum efficiency towards the end-of-lifeof the strained GaAs crystal used as the beam source. Itdropped after the annealing process of the crystal. To ac-count for the variations in P e we used a rolling average ofthe measurements (resetting it after the refreshing process),which was applied in the analysis of the data.We used a C target consisting of three . mm-thickfoils, which were rotated ◦ relative to the beam. This waywe minimized the path of the outgoing proton through the Table 1: Central kinematics of the C( (cid:126)e, e (cid:48) (cid:126)p ) data presented in thiswork. E beam [MeV] 600 Q [ GeV /c ] 0.175 p e [ MeV /c ] 368 θ e [ ◦ ] − p p [ MeV /c ] 665 θ p [ ◦ ] 37.8 p miss [ MeV /c ] −
270 to − ν [ MeV /c ] −
160 to − Figure 1: Kinematics of the measured A( (cid:126)e, e (cid:48) (cid:126)p ) reaction. The scat-tering plane is determined by the ingoing and outgoing-electron mo-mentum, (cid:126)k and (cid:126)k (cid:48) , respectively. The reaction plane is spanned by thetransferred momentum, (cid:126)q , and the outgoing proton’s momentum, (cid:126)p (cid:48) .We choose to represent the polarization components in the scatteringplane by using a right-handed coordinate-system with its axes being: ˆ z parallel to the momentum transfer (cid:126)q , ˆ y along the vector product ofthe ingoing and outgoing-electron momentum, (cid:126)k × (cid:126)k (cid:48) , and ˆ x = ˆ y × ˆ z .Another often-used reference frame is ˆ L ˆ N ˆ S where ˆ L points along theoutgoing proton’s momentum, (cid:126)p (cid:48) , ˆ N is along the vector product (cid:126)p (cid:48) × (cid:126)q ,and ˆ S = ˆ N × ˆ L . There are three important angles that help char-acterize the reaction above. Electron scattering angle, θ e , togetherwith the energy of an ingoing electron, k , determines the momentumtransfer. The azimuthal angle between (cid:126)q and (cid:126)p (cid:48) , φ pq , represents theangle between the scattering and reaction plane, whereas θ pq is thecorresponding polar angle. target, and hence, reduced the energy loss and the probabil-ity of multiple scattering. The two A1 high-resolution spec-trometers [30] were used to analyze the scattered electron(Spectrometer C) and the knock-out proton (SpectrometerA). In Spectrometer A we installed a focal-plane polarime-ter (FPP) [31] in which the polarized protons experiencesecondary scattering on a carbon analyzer, resulting in anangular asymmetry due to the spin-orbit part of the nuclearforce. Its angular distribution is given by σ ( ϑ, ϕ ) σ ( ϑ ) = 1 + A C ( ϑ, E p (cid:48) )( P F P Py cos ϕ − P F P Px sin ϕ ) , (1)where σ ( ϑ ) is the polarization independent part, A C is theanalyzing power of the carbon scatterer, ϑ is the polar angle, ϕ is the azimuthal angle, and P FPP x and P FPP y are the trans-verse polarization components of the proton at the focalplane. The analyzing power depends on the energy of theoutgoing proton E p (cid:48) and was adopted from [32, 33]. To mea-sure this distribution, horizontal drift chambers (HDCs)[34] were placed behind the scatterer.Table 1 and Figure 1 show the kinematic setting we usedand the important kinematic variables, respectively. Weuse a convention where the sign of the missing momentum, (cid:126)p miss = (cid:126)q − (cid:126)p (cid:48) , is determined by the sign of (cid:126)p miss · (cid:126)q . We2efine the virtuality of the embedded nucleon as: ν ≡ (cid:16) m A c − (cid:113) m A − c + p (cid:17) − p − m p c , (2)where m p , m A , and m A − ≡ (cid:112) ( ω − E p (cid:48) + m A c ) − p are the masses of the proton, target nucleus ( C ) and resid-ual nucleus ( B , not necessarily in its ground state), respec-tively. Here, ω = k − k (cid:48) is the energy transfer and E p (cid:48) isthe total energy of the outgoing proton.We chose the kinematic setting shown in Table 1 to ac-cess protons with high missing momentum from both s and p shells. This corresponds to Setting B in previous mea-surements reported in [5, 6]. In previous measurements weexplored regions of positive and negative missing momentato study the general behavior of polarization transfer andcompared it between different nuclei. We now present adedicated measurement performed in 2017 with improvedstatistics and a focus on a missing-momentum range wherethere is an overlap between protons knocked out from s and p shells in both the missing momentum and the virtuality.The present results were obtained from the combined datasets. P m i ss [ G e V / c ] [GeV /c ] C o un t [ ] Overlapregions shellp shell p-shell count s-shell count Figure 2: Upper panel: the missing-momentum-versus-virtualityphase space covered by this experiment. Lower panel: phase-spaceprojection on the virtuality axis. The gray band shows the virtuality-overlap region for protons extracted from carbon’s s and p shell. We distinguished between the protons extracted from the s and p shell based on their measured missing energy, E miss ,defined as E miss = ω − T p (cid:48) − T B , (3)where T p (cid:48) is the kinetic energy of the detected proton and T B is the calculated kinetic energy of the recoiling B nu-cleus. Following [5] and [35], protons with < E miss < correspond primarily to proton removal from the p / shell, while those with < E miss <
60 MeV originate from the s / shell. The missing-momentum-versus-virtualityphase space for protons from both shells is shown in Fig. 2.The shaded area indicates the virtuality range common toboth shells, and the distribution obtained from each shell isprojected in the bottom panel of Fig. 2.
3. Determination of the Transferred Polarizationand Uncertainties
We followed the convention of [7] to express individualcomponents of the outgoing polarization in the scatteringplane, P . Our coordinate system convention is also shownin Fig. 1.To obtain the polarization components we utilized themaximum-likelihood estimation where we optimized theoutgoing-proton polarization. Because our kinematics areclose to parallel, we assumed only one induced component, P y , [36] and two transferred components, P (cid:48) x and P (cid:48) z , to benon-zero. Therefore, the total polarization of the outgoingproton at the target is P = ( hP e P (cid:48) x , P y , hP e P (cid:48) z ) T , (4)where h is electron helicity. The contributions from therest of the components are either very small in (anti)parallelkinematics or cancel out because of their anti-symmetric de-pendence on the angle between the scattering and reactionplanes, φ pq [3].Protons travel through magnetic fields of the spectrom-eter before reaching the FPP, where we measure their po-larization components, P FPP x and P FPP y . Therefore, beforewe evaluate the likelihood function, we propagate the pro-posed estimates of target components from Eq. (4) throughthe spectrometer with the spin transfer matrix S which wascalculated with the QSPIN program [37]. To determine thetarget polarization components that best fit the measureddistribution from Eq. (1), we maximize the following log-likelihood function log L = (cid:88) k log(1 + A C ( ϑ, E p (cid:48) ) λ · P ) , (5)where λ = S yx cos ϕ − S xx sin ϕS yy cos ϕ − S xy sin ϕS yz cos ϕ − S xz sin ϕ (6)is determined per-event. It includes trajectory dependentspin-transfer coefficients, S ij , and the measured azimuthalangle ϕ after the secondary scattering of the proton.The uncertainties of the extracted components and theirratios were estimated through the numerical second-orderpartial derivative of the log-likelihood function and, besidesthe numerical error, include a part of the systematic spin-transfer error as well. As can be seen in Table 2, the beampolarization and the analyzing power are the largest con-tributors to the uncertainty in the polarization components P (cid:48) x and P (cid:48) z , while their effect largely cancels out when weform either a single or a double ratio. The uncertainties inthe beam energy and the central kinematics affect the basis3 able 2: The sources contributing to the systematic uncertaintiesof individual components, P (cid:48) x , P (cid:48) z , single ratios, ( P (cid:48) x /P (cid:48) z ) s,p , and thedouble ratio, ( P (cid:48) x /P (cid:48) z ) s / ( P (cid:48) x /P (cid:48) z ) p . All values are in percent. P (cid:48) x P (cid:48) z ( P (cid:48) x /P (cid:48) z ) s,p ( P (cid:48) x /P (cid:48) z ) s ( P (cid:48) x /P (cid:48) z ) p Beam pol. 2.0 2.0 0.0 0.0Analyzing power 1.0 1.0 0.0 0.0Beam energy 0.2 0.6 0.8 < < E miss cut s shell < < < p shell 0.2 0.5 0.6Precession (STM fit) 0.3 0.3 < < < < s and p shell by themissing-energy cut. Although the neighboring boundariesof the two E miss ranges sit apart, each of them con-tain a small amount of protons coming from the other shell.To estimate the magnitude of this cross-contamination, weevaluated the amount of overlap by performing separate fitsover the s - and p -shell peaks in the available C structurefunction. We found that for our p miss range, the p -shellcut includes around of protons coming from the s shell,whereas the amount of protons coming from the p shellthat are included in the s -shell cut is negligible. We multi-plied these cross-contamination estimates with relative dif-ferences between the individual components for two shells,in order to obtain the corresponding uncertainty. Since thedifference is positive for one component and negative forthe other, we added the uncertainties in quadrature for thesingle ratio, whereas the uncertainty on the double ratio,although in principle vanishing, is dominated by the p -shellsingle-ratio uncertainty.The last two items from the Table 2 correspond with the quality of the spin-precession evaluation in our maximum-likelihood algorithm. We started by comparing the resultsobtained from employing the spin-transfer matrix to thosecalculated using the QSPIN program which is more precisebut considerably slower. The second contribution arisesfrom the finite resolution of the proton’s trajectory param-eters (e.g. vertex position). Here we again used
QSPIN toevaluate average dispersion from analysis of trajecto-ries with normally distributed variations in each parameter,where the parameter’s resolution was used as a standarddeviation of the sampling function. Finally, we obtain thetotal estimated systematic uncertainty by adding contribu-tions from each source in quadrature.
4. Results and Discussion
We show in the top two panels of Fig. 3 the polarization-transfer components P (cid:48) x and P (cid:48) z to protons knocked-out fromthe s and p shells, as a function of the missing momen-tum, p miss , and virtuality, ν . As in Fig. 2, the gray bandin the plots indicates the virtuality-overlap region betweenthe protons extracted from s and p shells. The solid linesrepresent calculations in relativistic distorted-wave impulseapproximation (RDWIA), where we use the average demo-cratic optical potential from [38], relativistic bound-statewave functions obtained with the NL-SH parametrization[39], and free-proton electromagnetic form-factors from [40].Because the original RDWIA program from [2] was writtenfor use with co-planar kinematics only, we modified it to in-clude the remainder of the 18 hadronic structure functionspresent in A( (cid:126)e, e (cid:48) (cid:126)p ) reaction under one-photon-exchange ap-proximation [8, 41].The effects of FSI can be appreciated by comparing theRDWIA (solid lines) and PWIA (dashed lines) calculations.To explore the sensitivity of the polarization componentsto the ratio G E /G M we repeated the calculation with aform-factor ratio modified by ± . The impact of thisvariation on the results of the calculation is shown as a bandaround the respective calculation with no modification. Wenote that in this kinematic region, varying the form-factorratio has a very small effect on the transverse component, P (cid:48) x , while the longitudinal component, P (cid:48) z , shows a lineardependence on the G E /G M , as can be seen in Fig. 3. Thebehavior of the individual components is translated to thelinear dependence of their ratio, P (cid:48) x /P (cid:48) z , on the form-factorratio.Nuclear effects can not only differ for protons from the s and p shell, but may also have different effects on thetransverse ( x ) and longitudinal ( z ) polarization componentswhen we consider protons from a single shell. This can beseen as a deviation from unity in the bottom panel of Fig. 3,where we show P (cid:48) x /P (cid:48) z for each shell separately, as well as inFig. 4, which includes component ratios, P (cid:48) si /P (cid:48) pi ( i = x, z ,)for the two shells. Such differences are also foreseen by thetheoretical calculations. To minimize these differences whensearching for medium modifications in the proton structure,we examine the double ratio ( P (cid:48) x /P (cid:48) z ) p / ( P (cid:48) x /P (cid:48) z ) s . The dou-ble ratio is shown for the measured components as a func-4 .500.450.400.35 P x s-shellp-shell Overlap regionPWIADWIA P z P miss [GeV/c] | P x / P z | [GeV /c ] Figure 3: The measured polarization components P (cid:48) x (top), P (cid:48) z (middle), and their ratio P (cid:48) x /P (cid:48) z (bottom) as a function of missing momentum(left) and virtuality (right). Lines represent RDWIA and PWIA calculations for the corresponding shell obtained using a slightly modified programfrom [2] (see text). The shaded colored regions correspond to RDWIA calculations with the form-factor ratio, G E /G M , modified by up to ± . P miss [GeV/c] C o m p o n e n t r a t i o ( P x ) s ( P x ) p ( P z ) s ( P z ) p [GeV /c ] PWIA DWIA
Figure 4: Ratios of given polarization-transfer components ( P (cid:48) x or P (cid:48) z ) for each shell in C ( s or p ) as a function of missing momentum (left) andvirtuality (right). We note that here the virtuality range is narrower since the ratios, which compare the two shells, can be calculated only in theoverlap region. The solid and dashed lines represent the RDWIA and the PWIA calculations, respectively. Following the components, only theratio ( P (cid:48) z ) s / ( P (cid:48) z ) p is sensitive to the electromagnetic form-factor ratio’s modification, and hence, has a visible band around the calculation. Sincewe are searching for differences between the two shells, we modified the electromagnetic form-factor ratio only for one of them. . ± . al-though we could not exclude a slight dependence on p miss asindicated by the RDWIA calculation (solid line). This de-pendence is absent in the PWIA calculation (dashed line).The data from the two shells (in the overlap region) areshown as a function of the proton virtuality in the bottompanel of Fig. 5, where the knockout protons from the twoshells can be compared at the same ν . The weighted aver-age is . ± . , suggesting no difference between s and p protons. P miss [GeV/c] ( P x / P z ) p / ( P x / P z ) s Weighted Avg. = 1.15 ± 0.03Weighted Avg. = 1.15 ± 0.03 [GeV /c ] ( P x / P z ) p / ( P x / P x ) s Weighted Avg. = 1.05 ± 0.05Measurement PWIA DWIA
Figure 5: The polarization transfer double ratio as a function of miss-ing momentum (top) and virtuality (bottom). The solid and dashedgray lines represent the RDWIA and the PWIA calculation, respec-tively, whereas the colored line and band correspond to the weightedaverage of the measurements and its uncertainty.
The results suggest that proton virtuality is a good pa-rameter to characterize the properties of a bound proton.Indeed, differences that were suggested and might have beenobserved focusing on the missing momentum of the reactionare much reduced or disappear when protons at the same ν are compared. This is further corroborated by the calcula-tions when events of the similar ν are considered rather than p miss bins in which protons of a larger virtuality range arecombined. It can be further deduced that there is no statis-tically significant difference between polarization ratios for s - and p -shell protons. The small deviation of double ratiofrom unity can be already accounted for with the unmodi-fied electromagnetic form-factor ratio and simple PWIA cal-culations, while measurements are also in agreement withRDWIA calculations (reduced χ = 0 . , p = 0 . ). Thus, we found no evidence of density-dependent modifications.The ratios of the polarization-transfer components P (cid:48) x /P (cid:48) z to deeply bound protons were measured for several nuclei.It was shown that a comparison of this ratio to that ofa free proton, ( P (cid:48) x /P (cid:48) z ) A / ( P (cid:48) x /P (cid:48) z ) H , at given ν shows thesame deviations for H , He , and C despite different kine-matic conditions. The agreement of the results when theproton is bound in H , which is a slightly-bound two-bodysystem and often used as an effective neutron target, withthose when bound in nuclei with a high average nucleardensity (like He and C ) also supports our observation.While FSI and the local nuclear density may differ betweenthese nuclei, their effect on the polarization transfer is simi-lar, and no nuclear-density-dependent modifications are ob-served. Clearly, our results suggest virtuality to be a betterparameter to characterize the bound proton than p miss .
5. Conclusions
We presented measurements of the polarization transferto deeply bound protons in the s and p shells of C bypolarized electrons with the C( (cid:126)e, e (cid:48) (cid:126)p ) reaction. To inves-tigate nuclear-density dependence and possible in-mediummodification of the proton’s EM FF, we utilized the factthat the ratio of the transverse to longitudinal componentsis sensitive to the EM FF ratio. The measured polarizationratios for protons extracted from the two shells were studiedand compared as a function of either the missing momen-tum or the bound-proton virtuality. We concluded that thebound proton is better characterized by its virtuality ratherthan the missing momentum. Although according to sometheories, there is a large difference in the nuclear densitybetween the two shells in C , the measurements show nosignificant differences between the s - and p -shell protonsto the level of 5% when compared at the same virtuality.Furthermore, the observed slight deviation from unity isexpected from both PWIA and RDWIA calculations.
6. Acknowledgments
We would like to thank the Mainz Microtron operatorsand technical crew for the excellent operation of the acceler-ator. This work is supported by the Israel Science Founda-tion (Grants 390/15, 951/19) of the Israel Academy of Artsand Sciences, by the PAZY Foundation (Grant 294/18),by the Israel Ministry of Science, Technology and Space,by the Deutsche Forschungsgemeinschaft (Collaborative Re-search Center 1044), by the U.S. National Science Founda-tion (PHY-1205782). We acknowledge the financial supportfrom the Slovenian Research Agency (research core fundingNo. P1-0102) and from the Croatian Science Foundation(under the project 8570).
References [1] G. Ron, et al., Nuclear density dependence of in-medium polarization, Phys. Rev. C (2013) 028202. doi:10.1103/PhysRevC.87.028202 .62] A. Meucci, C. Giusti, F. D. Pacati, Relativistic correc-tions in ( e, e (cid:48) p ) knockout reactions, Phys. Rev. C (2001) 014604. doi:10.1103/PhysRevC.64.014604 .[3] C. Giusti, F. Pacati, Complete determination of scat-tering amplitudes and nucleon polarization in elec-tromagnetic knockout reactions, Nuclear Physics A (1989) 685 – 711. doi:10.1016/0375-9474(89)90003-1 .[4] S. Paul, D. Izraeli, T. Brecelj, I. Yaron, et al.,Quasi-elastic polarization-transfer measurements onthe deuteron in anti-parallel kinematics, Physics Let-ters B (2019) 599 – 605. doi:10.1016/j.physletb.2019.07.002 .[5] D. Izraeli, T. Brecelj, et al., Measurement ofpolarization-transfer to bound protons in carbon andits virtuality dependence, Physics Letters B (2018)95 – 98. doi:10.1016/j.physletb.2018.03.027 .[6] T. Brecelj, S. J. Paul, T. Kolar, et al., Polarizationtransfer to bound protons measured by quasielasticelectron scattering on C , Phys. Rev. C 101 (2020)064615. doi:10.1103/PhysRevC.101.064615 .[7] S. Strauch, S. Dieterich, et al., Polarization transfer inthe He( (cid:126)e, e (cid:48) (cid:126)p ) H reaction up to q = 2 . /c ) ,Phys. Rev. Lett. (2003) 052301. doi:10.1103/PhysRevLett.91.052301 .[8] S. Boffi, C. Giusti, F. d. Pacati, M. Radici, Electromag-netic Response of Atomic Nuclei, Vol. 20. 20 of OxfordStudies in Nuclear Physics, Clarendon Press, OxfordUK, 1996.[9] J. V. Noble, Modification of the nucleon’s propertiesin nuclear matter, Phys. Rev. Lett. (1981) 412–415. doi:10.1103/PhysRevLett.46.412 .[10] R. Schiavilla, R. B. Wiringa, J. Carlson, Coulombsum and proton-proton correlations in few-body nu-clei, Phys. Rev. Lett. (1993) 3856–3859. doi:10.1103/PhysRevLett.70.3856 .[11] J. Jourdan, Longitudinal response functions: thecoulomb sum revisited, Physics Letters B (1995)189 – 195. doi:10.1016/0370-2693(95)00581-5 .[12] D. Lu, et al., In-medium electron-nucleon scattering,Physics Letters B (1998) 217 – 223. doi:10.1016/S0370-2693(97)01385-3 .[13] A. Thomas, Hadron structure in medium, NuclearPhysics A (2000) 149 – 156, physics of Hadronsand Nuclei. doi:10.1016/S0375-9474(00)00086-5 .[14] J. Ryckebusch, et al., Relativistic formulation ofglauber theory for A( e, e (cid:48) p ) reactions, Nuclear PhysicsA (2003) 226 – 250. doi:10.1016/j.nuclphysa.2003.08.022 . [15] J. Aubert, et al., The ratio of the nucleon struc-ture functions F N2 for iron and deuterium, PhysicsLetters B (3) (1983) 275 – 278. doi:10.1016/0370-2693(83)90437-9 .[16] G. van der Steenhoven, et al., Nuclear-density depen-dence of the electron-proton coupling, Phys. Rev. Lett. (1987) 1727–1730. doi:10.1103/PhysRevLett.58.1727 .[17] D. Eyl, A. Frey, et al., First measurement of thepolarisation transfer on the proton in the reac-tions H( (cid:126)e, e (cid:48) (cid:126)p ) and D( (cid:126)e, e (cid:48) (cid:126)p ) , Zeitschrift für PhysikA Hadrons and Nuclei (1995) 211–214. doi:10.1007/BF01298910 .[18] B. D. Milbrath, J. I. McIntyre, et al., Comparison ofpolarization observables in electron scattering from theproton and deuteron, Phys. Rev. Lett. (1998) 452–455. doi:10.1103/PhysRevLett.80.452 .[19] S. Malov, K. Wijesooriya, et al., Polarization transferin the O( (cid:126)e, e (cid:48) (cid:126)p ) N reaction, Phys. Rev. C (2000)057302. doi:10.1103/PhysRevC.62.057302 .[20] S. Dieterich, et al., Polarization transfer in the He( (cid:126)e, e (cid:48) , (cid:126)p ) H reaction, Physics Letters B (2001)47 – 52. doi:10.1016/S0370-2693(01)00052-1 .[21] B. Hu, M. K. Jones, P. E. Ulmer, et al., Polariza-tion transfer in the H( (cid:126)e, e (cid:48) (cid:126)p )n reaction up to Q =1 .
61 (GeV /c ) , Phys. Rev. C (2006) 064004. doi:10.1103/PhysRevC.73.064004 .[22] M. Paolone, S. P. Malace, S. Strauch, et al., Polar-ization transfer in the He( (cid:126)e, e (cid:48) (cid:126)p ) H reaction at Q =0 . and . /c ) , Phys. Rev. Lett. (2010)072001. doi:10.1103/PhysRevLett.105.072001 .[23] I. Yaron, D. Izraeli, et al., Polarization-transfer mea-surement to a large-virtuality bound proton in thedeuteron, Physics Letters B (2017) 21 – 24. doi:10.1016/j.physletb.2017.01.034 .[24] D. Izraeli, I. Yaron, B. Schlimme, et al., Compo-nents of polarization-transfer to a bound proton in adeuteron measured by quasi-elastic electron scatter-ing, Physics Letters B (2018) 107 – 111. doi:10.1016/j.physletb.2018.03.063 .[25] A. Akhiezer, M. Rekalo, Polarization effects in the scat-tering of leptons by hadrons, Sov. J. Part. Nucl. (1974) 277.[26] C. Glashausser, et al., Measurement of Recoil Polariza-tion in the O( (cid:126)e, e (cid:48) (cid:126)p ) reaction with 4 GeV electrons,TJNAF Proposal 89-033, 1989.[27] B. Wagner, H. Andresen, K. Steffens, et al., A Møllerpolarimeter for CW and pulsed intermediate energyelectron beams, Nuclear Instruments and Methods inPhysics Research Section A: Accelerators, Spectrome-ters, Detectors and Associated Equipment (1990)541 – 548. doi:10.1016/0168-9002(90)90296-I .728] P. Bartsch, Aufbau eines Møller-Polarimeters für dieDrei-Spektrometer-Anlage und Messung der Helizität-sasymmetrie in der Reaktion p ( (cid:126)e, e (cid:48) p ) π im Bereich der δ -Resonanz, Ph.D. thesis, Institut für Kernphysik derUniversität Mainz (2001).[29] V. Tioukine, K. Aulenbacher, E. Riehn, A mott po-larimeter operating at mev electron beam energies,Review of Scientific Instruments (2011) 033303. doi:10.1063/1.3556593 .[30] K. Blomqvist, et al., The three-spectrometer facility atthe Mainz microtron MAMI, Nuclear Instruments andMethods in Physics Research Section A: Accelerators,Spectrometers, Detectors and Associated Equipment (1998) 263 – 301. doi:10.1016/S0168-9002(97)01133-9 .[31] T. Pospischil, et al., The focal plane proton-polarimeter for the 3-spectrometer setup at MAMI,Nuclear Instruments and Methods in Physics ResearchSection A: Accelerators, Spectrometers, Detectors andAssociated Equipment (2002) 713 – 725. doi:10.1016/S0168-9002(01)01955-6 .[32] E. Aprile-Giboni, R. Hausammann, E. Heer, R. Hess,C. Lechanoine-Leluc, W. Leo, S. Morenzoni, Y. Onel,D. Rapin, Proton-carbon effective analyzing powerbetween 95 and 570 MeV, Nucl. Instrum. Meth. (1983) 147–157. doi:10.1016/0167-5087(83)91302-9 .[33] M. W. McNaughton, et al., The p-C analyzing powerbetween 100 and 750 MeV, Nucl. Instrum. Meth. A (1985) 435–440. doi:10.1016/0168-9002(85)90595-9 .[34] T. Pospischil, et al., The horizontal drift cham-bers for the focal plane proton-polarimeter of the 3-spectrometer setup at MAMI, Nuclear Instruments andMethods in Physics Research Section A: Accelerators,Spectrometers, Detectors and Associated Equipment (2002) 726 – 733. doi:10.1016/S0168-9002(01)01954-4 .[35] D. Dutta, D. van Westrum, et al., Quasielastic ( e, e (cid:48) p ) reaction on C , Fe , and Au , Phys. Rev. C (2003) 064603. doi:10.1103/PhysRevC.68.064603 .[36] S. J. Paul, T. Kolar, T. Brecelj, et al., To be published.[37] T. Pospischil, Aufbau und inbetriebnahme einesprotonen-polarimeters an mami und messung derproton-polarisation in der reaktion p ( (cid:126)e, e (cid:48) (cid:126)p ) π in paral-leler kinematik im bereich der δ (1232)-resonanz, Ph.D.thesis, Institut für Kernphysik der Universität Mainz(2000).[38] E. D. Cooper, S. Hama, B. C. Clark, Global dirac op-tical potential from helium to lead, Phys. Rev. C (2009) 034605. doi:10.1103/PhysRevC.80.034605 . [39] M. Sharma, M. Nagarajan, P. Ring, Rho meson cou-pling in the relativistic mean field theory and descrip-tion of exotic nuclei, Physics Letters B 312 (1993) 377 –381. doi:https://doi.org/10.1016/0370-2693(93)90970-S .[40] J. C. Bernauer, M. O. Distler, J. Friedrich, T. Walcher,et al., Electric and magnetic form factors of the pro-ton, Phys. Rev. C (2014) 015206. doi:10.1103/PhysRevC.90.015206 .[41] A. Picklesimer, J. W. Van Orden, Polarization re-sponse functions and the ( (cid:126)e, e (cid:48) (cid:126)p ) reaction, Phys. Rev. C (1989) 290–303. doi:10.1103/PhysRevC.40.290doi:10.1103/PhysRevC.40.290