Measurements of the induced polarization in the quasi-elastic A(e, e ′ p → ) process in non-coplanar kinematics
Sebouh J. Paul, Tim Kolar, Tilen Brecelj, Patrick Achenbach, Hartmuth Arenhövel, Adi Ashkenazi, Jure Beričič, Ralph Böhm, Damir Bosnar, Ethan Cline, Erez O. Cohen, Luka Debenjak, Michael O. Distler, Anselm Esser, Ivica Friščić, Ronald Gilman, Carlotta Giusti, Matthias Heilig, Matthias Hoek, David Izraeli, Simon Kegel, Pascal Klag, Yvonne Kohl, Igor Korover, Jechiel Lichtenstadt, Israel Mardor, Harald Merkel, Duncan G. Middleton, Miha Mihovilovič, Julian Müller, Ulrich Müller, Mor Olivenboim, Eliezer Piasetzky, Josef Pochodzalla, Guy Ron, Björn S. Schlimme, Matthias Schoth, Florian Schulz, Concettina Sfienti, Simon Širca, Rouven Spreckels, Samo Štajner, Steffen Strauch, Michaela Thiel, Alexey Tyukin, Adrian Weber, Israel Yaron
MMeasurements of the induced polarization in the quasi-elastic A ( e, e (cid:48) (cid:126)p ) process innon-coplanar kinematics. S.J. Paul a,1, ∗ , T. Kolar b , T. Brecelj b , P. Achenbach c , H. Arenhövel c , A. Ashkenazi a , J. Beričič b , R. Böhm c , D. Bosnar d ,E. Cline e , E.O. Cohen a , L. Debenjak b , M.O. Distler c , A. Esser c , I. Friščić d,2 , R. Gilman e , C. Giusti f , M. Heilig g , M. Hoek c ,D. Izraeli a , S. Kegel c , P. Klag c , Y. Kohl c , I. Korover h,a , J. Lichtenstadt a , I. Mardor a,i , H. Merkel c , D.G. Middleton c ,M. Mihovilovič k,b,c , J. Müller c , U. Müller c , M. Olivenboim a , E. Piasetzky a , J. Pochodzalla c , G. Ron j , B.S. Schlimme c ,M. Schoth c , F. Schulz c , C. Sfienti c , S. Širca k,b , R. Spreckels c , S. Štajner b , S. Strauch l , M. Thiel c , A. Tyukin c , A. Weber c ,I. Yaron a , (A1 Collaboration) a School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel. b Jožef Stefan Institute, 1000 Ljubljana, Slovenia. c Institut für Kernphysik, Johannes Gutenberg-Universität, 55099 Mainz, Germany. d Department of Physics, University of Zagreb, HR-10002 Zagreb, Croatia. e Rutgers, The State University of New Jersey, Piscataway, NJ 08855, USA. f Dipartimento di Fisica, Università degli Studi di Pavia and INFN, Sezione di Pavia, via A. Bassi 6, I-27100 Pavia, Italy. g Universität Konstanz, Fachbereich Physik, Universitätsstraße 10, 78464 Konstanz, Germany. h Department of Physics, NRCN, P.O. Box 9001, Beer-Sheva 84190, Israel. i Soreq NRC, Yavne 81800, Israel. j Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israel. k Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia. l University of South Carolina, Columbia, South Carolina 29208, USA.
Abstract
We report measurements of the induced polarization (cid:126)P of protons knocked out from H and C via the A ( e, e (cid:48) (cid:126)p ) reaction.We have studied the dependence of (cid:126)P on two kinematic variables: the missing momentum p miss and the “off-coplanarity”angle φ pq between the scattering and reaction planes. For the full 360 ◦ range in φ pq , both the normal ( P y ) and, for thefirst time, the transverse ( P x ) components of the induced polarization were measured with respect to the coordinate systemassociated with the scattering plane. P x vanishes in coplanar kinematics, however in non-coplanar kinematics, it is on thesame scale as P y .We find that the dependence on φ pq is sine-like for P x and cosine-like for P y . For carbon, the magnitude of the inducedpolarization is especially large when protons are knocked out from the p / shell at very small p miss . For the deuteron, theinduced polarization is near zero at small | p miss | , and its magnitude increases with | p miss | . For both nuclei such behavior isreproduced qualitatively by theoretical results, driven largely by the spin-orbit part of the final-state interactions. However,for both nuclei, sizeable discrepancies exist between experiment and theory. Within the shell model a spin-orbit term is required in themean-field potential of atomic nuclei in order to explain theenergy splitting of the single-particle levels for reproducingthe magic numbers [1–3]. The spin-orbit interaction alsoplays an important role in optical potentials which describescattering processes. These have a strong influence in thefinal-state interactions (FSI) affecting quasi-free A ( e, e (cid:48) (cid:126)p ) scattering [4–8], as well as various other types of scatteringprocesses [9–12].In elastic ep scattering, within the one-photon-exchangeapproximation, the induced polarization of the proton van- ∗ Corresponding author
Email address: [email protected] (S.J. Paul) Present address: UC-Riverside, Riverside, CA 92521, USA. Present address: MIT-LNS, Cambridge, MA 02139, USA. ishes. Consequently, it is the FSI which in the A ( e, e (cid:48) (cid:126)p ) reaction gives rise to a non-vanishing induced polarizationof the knocked-out proton. In view of the fact that it islargely insensitive to details of the nucleon electromagneticform factors, the induced polarization serves as an effec-tive probe of FSI effects in quasi-elastic A ( e, e (cid:48) (cid:126)p ) . Here wepresent measurements of the induced polarization of quasi-elastic protons from H and C over a wide range in themissing momentum, p miss .Previous measurements of the normal component, P y , ofthe induced polarization have been performed at MIT-Bateson H [13] at low p miss and C with large p miss in coplanarkinematics [14]. Measurements of P y were also performedon He at Jefferson Lab (JLab) [15, 16] over a wide p miss range.In Ref. [14], it was found that the induced polarizationsof protons knocked out from the s shell of C show a dif-ferent behavior from those knocked out of the p shell. This Preprint submitted to Physics Letters B August 28, 2020 a r X i v : . [ nu c l - e x ] A ug ifference was attributed to the spin-orbit ( L · S ) interac-tion. The measured values of P y for H at low p miss in [13]were much smaller than those measured for other nuclei in[14–16]. All measurements prior to those reported here wererestricted to almost-coplanar geometry.The induced polarization measurements presented herewere performed at the Mainz Microtron (MAMI), duringfour run periods from 2012-2017. Our measurements forboth nuclei cover a large range in missing momentum, p miss ,and the full 360 ◦ range in the off-coplanarity angle, φ pq (See Fig. 1). Calculations predict a dependence of the in-duced polarization on φ pq largely due to the L · S interactionwhich hitherto has been unexplored. Our H data greatlyextend the range in p miss compared to the previous mea-surements in [13], while our C data significantly improvethe statistical precision and the range in φ pq compared tothe existing data [14]. Furthermore, we measure not onlythe normal component, P y , but also, for the first time, thetransverse component, P x , which vanishes in coplanar kine-matics ( φ pq = 0 ◦ or ◦ ). The transferred polarizationsmeasured in our experiments were reported in [17–19] for H and [20–22] for C.Section 2 describes the experimental setup, the measuredreaction, and the kinematic settings. The data analysisand extraction of the induced polarization are described inSec. 3. The details of theoretical calculations, to which wecompare our data, are given in Sec. 4. We then present thedata for both nuclei and their dependence on p miss and φ pq in Secs. 5 and 6, and conclude in Sec. 7. The experiments were performed at MAMI using the A1beamline and spectrometers [23]. For these measurements,a 600-690 MeV polarized continuous-wave electron beamwas used. The beam current was ≈ µ A. Due to the fre-quent flipping of the beam helicity (about 1 Hz), the averagebeam polarization in our event sample is zero, as verifiedby internal checks on the data.The targets used for the H and C measurements werea 50 mm long oblong cell filled with liquid deuterium [17–19] and a set of three 0.8 mm-thick graphite foils [20–22],respectively. We also performed calibration runs using aliquid hydrogen target.Two high-resolution, small-solid-angle spectrometerswith momentum acceptances of 20-25% were used to de-tect the scattered electrons and knocked-out protons incoincidence. Each of these spectrometers consists of anmomentum-analysing magnet system followed by a set ofvertical drift chambers (VDCs) for tracking, and a scintil-lator system for triggering and defining the time coincidencebetween the two spectrometers.The proton spectrometer was equipped with a focal-planepolarimeter (FPP) with a 3-7 cm thick carbon analyzer anda set of horizontal drift chambers (HDCs) [23, 24]. The spin-dependent scattering of the polarized proton by the carbonanalyzer allows the determination of the proton polarizationat the focal plane. The polarization at the interaction point is then determined by correcting for the spin precession inthe spectrometer’s magnetic field [24]. More details of theexperiment can be found in [17–22].The kinematics of the measured reactions are shown inFig. 1. The electron’s initial and final momenta are (cid:126)k and (cid:126)k (cid:48) respectively, which define the scattering plane of the re-action. The reaction plane is defined by the momentumtransfer (cid:126)q = (cid:126)k − (cid:126)k (cid:48) and the recoiling proton’s momentum (cid:126)p (cid:48) . We refer to the angle between the scattering plane andthe reaction plane as the “off-coplanarity” angle of the re-action, denoted by φ pq .Following the convention of [15], we express the compo-nents of the induced polarization (cid:126)P in the scattering-planecoordinate system, such that ˆ y is normal to the scatteringplane (along the direction of (cid:126)k × (cid:126)k (cid:48) ), ˆ z is along the direc-tion of the momentum transfer (cid:126)q , and ˆ x = ˆ y × ˆ z , forming aright-handed coordinate system. ϕ pq miss Figure 1: Kinematics of the reaction with the definitions of thekinematic variables.
The missing momentum (cid:126)p miss ≡ (cid:126)q − (cid:126)p (cid:48) is the recoil mo-mentum of the residual nuclear system. Neglecting FSI, − (cid:126)p miss is equal to the initial momentum of the emittedproton, (cid:126)p i . We conventionally define positive and negativesigns for p miss by the sign of (cid:126)p miss · (cid:126)q .Our H measurements were performed at six kinematicsettings, labeled A through F, with varying ranges of p miss and invariant four-momentum transfers Q = − q . SettingsA and F were both centered at p miss = 0 , and have Q =0 .
40 (GeV /c ) . Settings B and E covered large positive p miss ,at Q = 0 . and 0.65 (GeV /c ) , respectively. Settings Cand D covered small and large negative p miss , respectively,and were both at Q = 0 .
18 (GeV /c ) . Details are given inTable 1.Our C measurements were taken at two kinematic set-tings. The first is the same Setting A of the deuteron mea-surements (centered near p miss = 0 , at Q = 0 .
40 (GeV /c ) ).The second is Setting G, which covered a region of large neg-ative p miss at Q = 0 .
18 (GeV /c ) ; this is similar to SettingD of the deuteron measurements, except with a differentbeam energy and the other kinematic variables modifiedaccordingly . In our earlier publications [20–22] on C, this setting is referred toas Setting B. We refer to this setting as Setting G in this work in orderto distinguish it from the Setting B of the deuteron measurements. able 1: The kinematic settings in the H( (cid:126)e, e (cid:48) (cid:126)p ) and C( (cid:126)e, e (cid:48) (cid:126)p ) measurements. The angles and momenta represent the central values for thetwo spectrometers: p p and θ p ( p e and θ e ) are the knocked out proton (scattered electron) momentum and scattering angles, respectively. Thenumber of events passing the event selection cuts are also given. Kinematic settingA B C D E F G E beam [MeV] 600 600 630 630 690 690 600 Q [ (GeV /c ) ] 0.40 0.40 0.18 0.18 0.65 0.40 0.18 p miss [MeV /c ] −
80 to 75 75 to 175 −
80 to − −
220 to −
130 60 to 220 −
70 to 70 −
250 to − p e [MeV /c ] 384 463 509 398 464 474 368 θ e [deg] 82.4 73.8 43.4 49.4 90.9 67.1 52.9 p p [MeV /c ] 668 495 484 665 656 668 665 θ p [deg] − − − − − − × ) H
68 19 438 201 10 232 — C s /
268 — — — — — 274 C p /
160 — — — — — 436In each of the kinematic settings presented in this work,the spectrometers’ reference trajectories form a parallel re-action ( (cid:126)p (cid:48) (cid:107) (cid:126)q ). However, due to the spectrometer accep-tance, our data sample included reactions with θ pq (the an-gle between (cid:126)p and (cid:126)q ) up to ≈ ◦ , with the full 360 ◦ rangein the off-coplanarity angle φ pq . Software cuts were applied to the data, in order to ensuregood tracking, time coincidence, and event quality. Thesecuts applied here are identical to those of the earlier publi-cations [17–22] on the transferred polarization, unless oth-erwise noted below.We applied additional tracking cuts to the proton’s tra-jectory, requiring it to be within the part of the spec-trometer where the precession of the proton’s spin is wellknown and the false asymmetry could be determined us-ing dedicated elastic ep measurements. In order to reducefalse asymmetries, we also removed events where the pro-ton would either be outside of the geometric acceptance ofthe detector, or produce a hit on a malfunctioning channelof the HDCs, if it had scattered in the azimuthally oppo-site direction. Additionally, the polar angle Θ FPP of thesecondary scattering was required to be greater than ◦ inorder to avoid spin-independent Coulomb-scattering events,and less than ◦ in order to improve the stability of thefalse-asymmetry determination.Following [17–19], we required the missing mass of the H ( e, e (cid:48) (cid:126)p ) reaction to be consistent with the mass of a neu-tron. For the C sample, we distinguish between protonsknocked out from the s and p shells, following [20–22], byusing cuts on the missing energy, E miss in the reaction, de-fined as [25]: 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 is thecalculated kinetic energy of the recoiling residual system,assuming it is B in the ground state. For the s -shell sam-ple, we used the cut < E miss < MeV, while for the p -shell sample, we used < E miss < MeV [20–22].The p -shell cut accepts events in which the residual A − system is left in one of several discrete states, including theground-state of B as well as a few excited states. The s -shell selection cut is much wider, comprising a broad rangewithin the continuum of unbound residual A − states. Before extracting the values of P x and P y for H and C, we first determined the false asymmetry using elastic ep events (for which the induced polarization is expectedto be zero). This was accomplished by maximizing the loglikelihood log L = (cid:88) events log (cid:126)A T · − sin Φ FPP cos Φ
FPP , (2)for the ep event sample, where Φ FPP is the azimuthal angleof the secondary scattering and (cid:126)A is the false asymmetry inthe focal plane coordinate system, parameterized as (cid:126)A = a x + a x φ vth a y + a y θ vth , (3)where θ vth and φ vth are the incident angles of the protontrajectory extrapolated from the VDCs to the HDCs. a x , a x , a y , and a y are the fitted coefficients. We then extractedthe induced polarization for H and C by maximizing the3og likelihood log L = (cid:88) events log a S · (cid:126)P + (cid:126)A ) T · − sin Φ FPP cos Φ
FPP , (4)where S is the calculated spin-transfer matrix for the protontrajectory of the event, and a is the analyzing power ofthe event (as determined by [26, 27]). (cid:126)P is the inducedpolarization. We constrain P z to be zero in order to improvethe stability of our fit. This constraint has a negligibleeffect on the fitted P x and P y except in bins with very poorstatistics.The corrections for false asymmetry are larger for P x thanfor P y ; the r.m.s. values of these corrections are ≈ ≈ | A y | is generally largerthan | A x | , and the off-diagonal terms of the spin-transfermatrix, S xy and S yx , dominate over the much smaller di-agonal terms S xx and S yy . Details of the false-asymmetrydetermination, and the checks we used to validate its long-term stability, may be found in the supplementary material. The systematic errors in these measurements are due toa few sources, which are presented in Table 2. They aredominated by the uncertainty on the false asymmetry ofthe FPP. This is due to the limited statistics of the elastic ep sample used to determine this false asymmetry, and itcontributes 0.012 to the systematic error on our correctedresults for H and C.The analyzing power of the carbon secondary scatterer isknown to about 1% in this kinematic region [24, 26, 27]. Itleads to a relative error of the same size on each componentof (cid:126)P . The uncertainty on the precession of the proton’s spinintroduces an additional 0.4% relative error. The system-atic error due to the uncertainty of the alignment betweenthe HDC and the VDC detector systems was investigatedto be less than 0.001, absolute, for both components. Thiswas determined by repeating the analysis with each of thealignment parameters modified by plus or minus its uncer-tainty. In a similar manner, we estimate the uncertainty onboth components due to the uncertainty on the kinematicsettings (i.e. the beam energy and the two spectrometers’angles and momenta) to be about 0.001.The software cuts, described in Sec. 3.1, introduce an ad-ditional absolute uncertainty of ≈ For comparison, theoretical calculations of the inducedpolarization for H and C have been performed. For Hwe have used a non relativistic calculation [28] including a
Table 2: Sources of systematic errors on P x and P y . We distinguishbetween sources of systematic errors that do not scale with P y (abso-lute errors), and those that do (relative errors). The total systematicerrors are then ∆ P x = (cid:113) ∆ P x, abs + (∆ P x, rel P x ) and similarly for ∆ P y . ∆ P x, abs ∆ P y, abs ∆ P xy, rel False asymmetry 0.010 0.012 —Software cuts 0.005 0.006 —Detector alignment < . < . —Kinematic setting .
001 0 . —Precession — — 0.4%Analyzing power — — 1.0%Total 0.011 0.013 1.1% H( e, e (cid:48) (cid:126)p ) p miss [MeV/ c ]0.050.000.050.10 P x P x P y P y Q = 0.18 (GeV/ c ) p miss [MeV/ c ]0.150.100.050.000.05 P y Q = 0.40 (GeV/ c ) MIT (Milbrath et al , 1998)
200 100 0 100 200 300 p miss [MeV/ c ]0.100.050.000.05 P y Q = 0.65 (GeV/ c ) Set. ASet. BSet. CSet. D Set. ESet. Fcalc (no
L S )calc (full)
Figure 2: The measured components of the induced polarizationfor H, P x (top panel) and P y (three lower panels for each Q ), asfunctions of the missing momentum. Different symbols (color online)represent different kinematic settings as shown in the inset of thelowest panel. Grey open triangles indicate the measurements fromMIT-Bates [13], taken at Q = 0.38 and 0.50 (GeV /c ) , both with p miss centered at zero. The uncertainties shown are statistical only.Systematic errors are discussed in Sec. 3.3. Theoretical results with(without) the L · S part of the interaction are shown as solid black(dashed, green online) curves. C( e, e (cid:48) (cid:126)p ) p miss [MeV/ c ]0.100.050.000.050.100.15 P x P x s
200 150 100 50 0 50 100 p miss [MeV/ c ]0.200.150.100.050.000.050.10 P x p p miss [MeV/ c ]0.100.050.000.050.100.15 P y P y s Set. A, s Set. G, s calc (no L S ) Set. A, p Set. G, p calc (full)200 150 100 50 0 50 100 p miss [MeV/ c ]0.200.150.100.050.000.050.10 P y p Figure 3: The measured induced-polarization components, P x (left panel) and P y (right panel) for C as functions of the missing momentumcompared to theory. Triangles (circles) refer to kinematic Setting A (G). Symbols that are open on the left (right) side refer to s -shell ( p -shell)removals, and are colored blue (red) online. The calculations with (without) the L · S potential are shown in solid black (dashed, green online)curves. realistic N N -potential, meson-exchange (MEC) and isobar(IC) currents, and relativistic contributions (RC) of lead-ing order. For the bound and scattering states the realis-tic Argonne V potential [29] has been taken. As nucleonelectromagnetic form factors we used the parameterizationsfrom [30].For C, calculations were performed using a program[7] based on the relativistic distorted-wave approximation(RDWIA) where the FSI between the outgoing proton andthe residual nucleus are described by a phenomenologicalrelativistic optical potential. The original program [7] wasmodified [22] in order to account for non-coplanar kinemat-ics, by including all relevant structure functions [6]. In theRDWIA calculations, only the one-body electromagneticnuclear current is included. We chose the current oper-ator corresponding to the cc2 definition [31], and we usedthe same parametrization of the nucleon form factors [30] asin the H calculations. The relativistic proton bound-statewave functions were obtained from the NL-SH parametriza-tion [32] and the scattering states from the so-called “demo-cratic” parameterization of the optical potential [33]We found that for both nuclei, the calculated inducedpolarization has very little sensitivity to the details of thenucleon form factors. A change of 10% to the form-factorratio G E /G M in the calculations affects the induced polar-ization by less than 0.005.In order to examine the influence of the L · S interaction on the induced polarization, we repeated these calculationswhile switching off this part of the potential. As we willshow in Secs. 5 and 6, the L · S interaction is the dominantsource of the induced polarization. We partitioned the H data for each kinematic settinginto bins by p miss and extracted P x and P y for each binseparately, and present the results in Fig. 2. The resultsfor P x (top panel) are consistent with zero within error.The extracted values of P y , on the other hand, are nearzero at small p miss and deviate from zero at large | p miss | .This is consistent with earlier measurements of P y fromMIT [13] (grey pentagons in Fig. 2), which show that for | p miss | < MeV/ c , the values of P y are consistent withzero. The reason for this feature lies in the fact that for p miss = 0 , which is the ideal quasi-free case, the influence ofFSI is very small, almost zero.In order to compare with theory, we calculated P x and P y for a sample of the H events which reflect the distributionof events in the phase space at each kinematic setting andtook the average value by bins in p miss , as shown in Fig. 2by solid black curves. The calculations for P x vanish, whilethose for P y are near zero at p miss ≈ , but they becomeincreasingly negative at increasing | p miss | . The P y calcu-5 H( e, e (cid:48) (cid:126)p ) P x P x
180 135 90 45 0 45 90 135 180 pq [deg]0.150.100.050.000.050.100.15 P x P y P y
180 135 90 45 0 45 90 135 180 pq [deg]0.150.100.050.000.050.100.15 P y Set. CSet. Dcalc (no
L S )calc (full)
Figure 4: Comparison of the φ pq dependence of the measured induced polarization components P x (left panels) and P y (right panels) for H tothe theory for selected kinematic settings. Symbols are explained in the inset of the right bottom panel. The calculations with (without) the L · S potential are shown as the solid black (dashed, green online) curves. lations match the data very well except at large negative p miss (Setting D).The theoretical results obtained by switching off the L · S interaction are presented as dashed (green online) curves inFig. 2. Similar to the complete calculation, the calculated P x without the L · S interaction is essentially zero. For P y ,the corresponding difference between the full and no- L · S calculation is large, indicating the important influence ofthe L · S interaction on P y . The theory appears to describethe data quite well except for P y in Settings C and D.For C we performed the same polarization-fitting proce-dure and present the results in Fig. 3. Unlike in our resultsfor H, the measured P x for C is non-zero when binned in p miss . While a non-zero P x at coplanar kinematics is theo-retically forbidden (see [5]), our data are not restricted tocoplanarity, and therefore we do not require P x to be zero.The non-zero values of P x in the left panels of Fig. 3 reflectthe distributions of other kinematic variables (such as φ pq )within each p miss bin. We will discuss this in further detailin Sec. 6, where we will show that P x is highly dependenton φ pq .The component P y for the s -shell data (upper part ofthe right panel of Fig. 3) appears to be nearly constant at P y ≈ − . as function of p miss , while the p -shell data showa significant variation with p miss .A comparison of our C data for P y with the earlier MITmeasurements [14] is shown in the supplementary material.We found that our data are consistent with the latter whenrestricting the range of φ pq in our data sample to match that of [14] (i.e., near ◦ ) and comparing them at thesame | p miss | .The calculations for the C events using samples of s -shell and p-shell data from both kinematic sets, with (with-out) the L · S part of the optical potential, as functions ofthe binned p miss , are shown in Fig. 3 as solid black (dashed,green online) curves. The results including the L · S termshow much larger deviations from zero than the resultswithout it, indicating that most of the deviation from zerooriginates from the L · S term.The P x curves are non-zero, and reflect the asymmetriesof the kinematics of the accepted events. For P y , the full-potential calculation (solid black) curves are above the mea-sured data points by up to 0.05, and in some cases, thecalculations have the opposite sign of the measured P y . In order to examine the φ pq dependence of P x and P y , weperformed the fits to the data in bins of φ pq .The results for H at the negative- p miss kinematic set-tings (C and D) are shown in Fig. 4. These are the onlytwo settings for which we observe a significant φ pq depen-dence in the data (for completeness, the results for the othersettings are shown in the supplementary material). In bothof these two settings, the P x have a sine-shaped dependenceon φ pq , albeit with opposite signs for the two settings. In allsix of the H settings, P y shows no statistically significantvariation with respect to φ pq .6 C( e, e (cid:48) (cid:126)p ) P x s (Sets A and G) P x P x p (Set. A)
180 135 90 45 0 45 90 135 180 pq [deg]0.150.100.050.000.050.100.150.200.25 P x p (Set. G) P y s P y Set. A, s Set. G, s calc (no L S ) Set. A, p Set. G, p calc (full)0.300.250.200.150.100.050.000.050.100.150.200.250.30 P y p (Set. A)
180 135 90 45 0 45 90 135 180 pq [deg]0.150.100.050.000.050.100.150.200.25 P y p (Set. G) Figure 5: For C, the φ pq dependence of the measured polarization components P x (left panels) and P y (right panels) compared to theory.These are shown for s -shell knockout for both kinematic settings (top panels), and for p -shell knockout at small p miss (Setting A, middle panels)and large negative p miss (Setting G, bottom panels). Calculations with (without) the L · S part of the optical potential are shown as solid black(dashed, green online) curves. In the top panels, the curves for Setting A are shown in grey (light green online for no- L · S ) in order to contrastwith those of Setting G. P x for Setting C an almost van-ishing φ pq dependence in contrast to the data, while forSetting D a small sine-shaped behavior is obtained, how-ever, with a much smaller amplitude than the much morepronounced data. For P y the agreement with the data ap-pears to be better. For both settings, there is only a smallvariation in P y with respect to φ pq , and its amplitude islarger for Setting D than for Setting C. Furthermore, onenotes for both components a sizeable contribution for thecase where the L · S -part is switched-off.The results for C are shown in Fig. 5. The P x resultshave a negative sine-like shape, with nearly the same am-plitude for s -shell knockout from both kinematic settings aswell as for p -shell knockout at large negative p miss (SettingG); the latter however has a more pronounced asymmetrybetween the regions of positive and negative φ pq . A pos-sible explanation for this asymmetry is that the acceptedevents with positive and negative φ pq have slightly differentkinematics from one another, due to acceptance effects. For p -shell knockout at small p miss (Setting A) the amplitude isconsiderably larger. In near-coplanar kinematics ( φ pq ≈ ◦ or ± ◦ ), the measured P x is consistent with zero.The component P y exhibits a similar φ pq dependence in s -shell knockout (top right panel) and p -shell knockout atlarge | p miss | (Setting G, lowest right panel) being approxi-mately constant at − p -shell knockout atsmall p miss (Setting A, middle right panel), P y has a largecosine-like φ pq dependence.The theoretical results with (without) the L · S part ofthe optical potential are displayed in Fig. 5 as solid black(dashed, green online) curves. The calculations predict incontrast to the data a much smaller P x for the s shell thanshown by the data, moreover with the opposite sign. For the p -shell knockout at Setting G (left lowest panel) P x is almostvanishing. For the p -shell knockout at Setting A (left middlepanel), P x shows a distorted sine-like dependence, similarto the data, but with an about 35% smaller amplitude.For P y (right panels) the theory agrees much better withthe data than for P x . For s -shell knockout there appearsto be an offset by about 0.05 compared to the data. Inparticular for φ pq near ± ◦ , the theory predicts a posi-tive value in contrast to the data. On the other hand, for p -shell knockout (middle and lower right panels) the agree-ment is quite good. For the p -shell knockout in SettingA, the amplitude of P y increases as p miss approaches zero,both in the data and in the calculations, as shown in thesupplementary material.When performing the calculations with the L · S termof the potential switched off (dashed curves, green online),both components of the induced polarization are nearlyzero, except for p -shell knockout at p miss near zero (Set-ting A). In that region, the no- L · S curves are not closeto zero and show a stronger oscillatory behavior but with asmaller amplitude, about one third of the one with the L · S term included. We have measured in A ( e, e (cid:48) (cid:126)p ) the induced polarizationcomponents P x and P y for both C and H, greatly extend-ing the kinematic range of previous P y -only measurementsfor both nuclei. Within the regions where our kinematicsoverlap in p miss and φ pq with the existing experiments, ourdata are consistent with the other experiments. We findthat in the regions where the induced polarization dependson φ pq , the dependence is sine-like for P x and cosine-like for P y .For both nuclei, P x is consistent with zero for near-coplanar events, as expected. However, the theoretical re-sults for P x for both nuclei predict a considerably smaller P x than reflected in the data, moreover in some cases withthe opposite sign.For H, the induced polarization is near zero at small | p miss | . P x shows no significant deviation from zero exceptat the negative- p miss settings, where it shows a dependencyon φ pq . P y becomes increasingly negative at larger | p miss | .This is consistent with the calculations, although the datashow a significantly steeper decrease at negative p miss . P y shows no significant dependence on φ pq at any of the kine-matic settings in this work. In the calculations, most of thedeviation of P y from zero comes from the L · S interaction.For C, the measured φ pq dependence of P x shows apronounced negative sine-like shape in both s - and p -shellknockout which is not reflected in the theoretical results.For the s -shell knockout, P y has no strong dependence oneither p miss nor φ pq . However, for p -shell knockout from C, P y is strongly dependent on φ pq at small p miss , and P x is also more strongly dependent on φ pq than in any other re-gion explored in this work. This behavior at small p miss for p -shell knockout is reproduced by the calculations, whereinthe large dependence on φ pq is mainly due to the spin-orbitterm of the FSI.The new data presented in this work provide a rich op-portunity to further fine-tune the L · S part of the opticaland N N potentials.
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 Israel Ministry of Science, Technol-ogy and Spaces, by the PAZY Foundation (Grant 294/18),by the Deutsche Forschungsgemeinschaft (Collaborative Re-search Center 1044), by the U.S. National Science Founda-tion (PHY-1205782, PHY-1505615), and by the CroatianScience Foundation Project No. 8570. We acknowledge thefinancial support from the Slovenian Research Agency (re-search core funding No. P1–0102).
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FPP , (5)for the ep event sample, where Φ FPP is the azimuthal angle ofthe secondary scattering and (cid:126)A is the false asymmetry in thefocal plane coordinate system, parameterized as (cid:126)A = a x + a x φ vth a y + a y θ vth , (6)where θ vth and φ vth are the incident angles of the proton trajec-tory extrapolated from the VDCs to the HDCs. a x , a x , a y , and a y are the fitted coefficients.To validate the fits for the false asymmetry, we divided the ep event sample into slices by three of the proton kinematic vari-ables: its momentum (divided by the reference momentum of thespectrometer setting), its polar angle θ p , and its azimuthal an-gle φ p , and performed fits for the components P FPP x and P FPP y of the induced polarization in the focal-plane coordinates foreach slice. These fits were performed with (blue filled circles,connected by solid lines) and without (orange open circles, con-nected by dashed lines) the corrections for the false asymmetry,as shown in Fig. S.1. The corrected values of P FPP x (top) and P FPP y (bottom) are consistent with zero. The uncorrected val-ues of P FPP x are slightly higher than zero, whereas those of P FPP y are large, especially when | φ p − ◦ | is large, indicating that thecorrections for the former are small while those of the latter arelarge.The ep sample used for the false-asymmetry determinationwere taken in 2012, just before the H measurements at SettingsA-D were obtained. We used three checks to validate the long-term stability of the false-asymmetry parameterization betweenthe four run periods. First, we compared the C data for atSetting G taken in the 2015 run period with the data taken atthe same setting during the 2017 run period, and found thatthese are consistent within error of each other. Second, the Hdata at Settings A and F (which are very similar kinematics toone another), taken in 2012 and 2016 respectively, are consistentwith one another within error. Finally, we checked that theresults for P x (in interaction-point coordinates) at φ pq near ◦ and ± ◦ in all four run periods are consistent with zero withinerror, as expected. Together, these checks indicate that the falseasymmetry is stable between the run periods. S.2 Comparison of C data to MIT results
In Fig. S.2, we compare our C measurements (half-filledtriangles and circles) to those from MIT [14] (filled diamonds)which were taken at φ pq centered at 180 ◦ . For this comparison,we selected only the events with | φ pq | > ◦ for consistencywith the the kinematics of the MIT measurements. Following[14], we plot our s -shell data separately for two bins in E miss : 29to 39 MeV/ c and 39 to 50 MeV/ c . The data are shown in binsof | p miss | .For the p -shell knockout, both our high- | φ pq | data and thoseof [14] have large negative P y at small | p miss | and small negative P y at large | p miss | . We fitted our data and those of [14] to astraight line and show this fit as a dashed green line in the toppanel of Fig. S.2. Both datasets are consistent with the fit withinerror ( χ = 3 . with 9 degrees of freedom; p val = 0 . ).For the s -shell knockout, we took the weighted average(horizontal-line fit) of P y for our measurements and those of [14]and obtained P avg y = − . . This is shown as a purple dashedline in the middle and bottom panels of Fig. S.2. Both our dataand those of [14] are consistent within uncertainty of this value( χ = 19 . , with 21 d.o.f.; p val = 0 . ). S.3 φ pq -dependence of the induced polar-ization in H In Fig. S.3, we show the dependence of P x and P y on φ pq forall six kinematic settings in both the data and the calculations.For P x , there is no statistically significant variation in the dataexcept in Settings C and D, as noted in the paper. P y shows nosignificant dependence on φ pq , while the calculation-curves arenearly flat for all six settings except Setting D (large negative p miss , at Q = 0 . (GeV/ c ) ). S.4 φ pq -dependence of the induced polar-ization in C at small p miss As noted in the paper, the data at Setting A for p -shell knock-out has a large φ pq -dependence for P y , while the rest of the datashow no such dependence. This was observed in both the dataand the calculations. We also found that P x for this subset ofthe data is also larger in magnitude than for the rest of the data.In order to determine if this φ pq dependence is correlated with p miss , we partitioned the p -shell data into slices by p miss , and plotthe φ pq dependence for each slice in Figs. S.4 (Setting G) andS.5 (Setting A). A purple dotted curve represents a fit to thedata in each slice by the equations P x = a x + b x sin φ pq (7)and P y = a y + b y cos φ pq . (8)The values of b x and b y (which consider the φ pq dependence of P x and P y ) are shown in Fig. S.6. The fit parameter b x is small atlarge negative p miss (Setting G), and increases in absolute valueto a negative plateau as p miss approaches 0 in Setting A. The fitparameter b y is near zero for large p miss (Setting G) and reachesa sharp peak at p miss = 0 (middle of Setting A). The calculationscurves for P x at Setting G are flat with respect to φ pq , whereasthe data suggest otherwise, as reflected by the non-zero b x in thefits. H( e, e (cid:48) (cid:126)p ) P F PP x r e s i d u a l p p / p p , ref P F PP y r e s i d u a l beforeafter 1 0 1 2 p p , ref [deg] 4 2 0 2 4 p p , ref [deg] Figure S.1: The residual induced polarizations in focal-plane coordinates, P FPP x (top panels) and P FPP y (bottom panels) measured for ep scattering,without (orange, open circles, connected by a dashed line) and with (blue, filled circles, connected by a solid line) corrections for the false asymmetry.These are presented in slices of the relative momentum of the proton (left), polar angle (middle), and azimuthal angle (right) C( e, e (cid:48) (cid:126)p ) P y P y p E miss < 24 MeV Set. A p Set. G p MIT C Set. A s Set. G s fit0.200.150.100.050.000.050.100.150.20 P y s
29 < E miss < 39 MeV p miss | [MeV/ c ]0.250.200.150.100.050.000.050.100.150.20 P y s
39 < E miss < 50 MeV Figure S.2: Comparison between the measured P y for near-coplanar( φ pq ≈ ◦ ) events in this work and the MIT measurements [14].The cut used in our data for this plot was | φ pq | > ◦ . Following[14], the events have been partitioned in slices by E miss . A purpledotted line indicates a combined fit of our data with those of [14]. H( e, e (cid:48) (cid:126)p ) P x P x P x P x P x P x
180 135 90 45 0 45 90 135 180 pq [deg]0.150.100.050.000.050.100.15 P x P y P y Set. ASet. BSet. CSet. D Set. ESet. Fcalc (no
L S )calc (full)0.150.100.050.000.050.100.15 P y P y P y P y
180 135 90 45 0 45 90 135 180 pq [deg]0.150.100.050.000.050.100.15 P y Figure S.3: The measured induced polarizations, P x (left panels) and P y (right panels) for H, plotted versus φ pq . The data are shown on separatepanels for each kinematic setting. The calculations with (without) the L · S interaction are shown as solid black (green dashed) curves. C( e, e (cid:48) (cid:126)p ) P x P x
240 < p miss < 210 MeV/ c P x
210 < p miss < 180 MeV/ c P x
180 < p miss < 150 MeV/ c
180 135 90 45 0 45 90 135 180 pq [deg]0.40.20.00.20.40.6 P x
150 < p miss < 120 MeV/ c P y P y
240 < p miss < 210 MeV/ c Set. G, p fit to data calc (no L S )calc (full)0.40.20.00.20.40.6 P y
210 < p miss < 180 MeV/ c P y
180 < p miss < 150 MeV/ c
180 135 90 45 0 45 90 135 180 pq [deg]0.40.20.00.20.40.6 P y
150 < p miss < 120 MeV/ c Figure S.4: P x (left panels) and P y (right panels) as functions of φ pq , in several slices of p miss for the p -shell knockout in C at SettingA. The calculations with (without) the L · S part of the optical potential are shown as solid black (green dashed) curves. Fits of the data to P x = a x + b x sin( φ pq ) and P y = a y + b y cos( φ pq ) are shown for each slice (purple dotted curves). C( e, e (cid:48) (cid:126)p ) P x P x
100 < p miss < 60 MeV/ c P x
60 < p miss < 30 MeV/ c P x
30 < p miss < 0 MeV/ c P x p miss < 30 MeV/ c P x
30 < p miss < 60 MeV/ c
180 135 90 45 0 45 90 135 180 pq [deg]0.40.20.00.20.40.6 P x
60 < p miss < 100 MeV/ c P y P y
100 < p miss < 60 MeV/ c Set. A, p fit to data calc (no L S )calc (full)0.40.20.00.20.40.6 P y
60 < p miss < 30 MeV/ c P y
30 < p miss < 0 MeV/ c P y p miss < 30 MeV/ c P y
30 < p miss < 60 MeV/ c
180 135 90 45 0 45 90 135 180 pq [deg]0.40.20.00.20.40.6 P y
60 < p miss < 100 MeV/ c Figure S.5: Same as Fig. S.4, for Setting A. C( e, e (cid:48) (cid:126)p ) b x Set. A p Set. G p p miss [MeV/ c ]0.050.000.050.100.150.200.250.300.350.400.450.50 b y Figure S.6: The parameters b x and b y from the fits P x = a x + b x sin φ pq and P y = a y + b y sin φ pq . These parameters describe the φ pq dependence of P x and P y respectively.respectively.