Comparison of recoil polarization in the $^{12}{\rm C}(\vec{e},{e}'\vec{p})$ process for protons extracted from $s$ and $p$ shell

We present first measurements of the double ratio of the polarization transfer components $(P^{\prime}_{\!x} \!/ P^{\prime}_{\!z} )_p/ (P^{\prime}_{\!x} \!/ P^{\prime}_{\!z} )_s$ for knock-out protons from $s$ and $p$ shells in $^{12}{\rm C}$ measured by the $^{12}{\rm C}(\vec{e},{e}'\vec{p}\,)$ reaction in quasi-elastic kinematics. The data are compared to 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 when the comparison is done at the same proton virtuality. We observe no density-dependent medium modifications for protons from $s$ and $p$ shells with the same virtuality in spite of the large differences in the respective nuclear densities.


Introduction
The effects of the nuclear medium on the structure of bound nucleons and their dependence on the nuclear average density are subject to theoretical and experimental investigations . The 12 C nucleus is a very attractive target to study nuclear density-dependent differences in bound nucleons. Its nuclear structure is well known, with nucleons in the s and p shells, and the average local nuclear density in these shells differs by about a factor of two [1]. Studying quasi-elastic processes on protons, which are sensitive to the proton form-factors, should be a good tool to observe any density dependence arising from the differences between the protons extracted from the two shells. Good theoretical calculations for this nucleus [2,3] further help with the interpretation of the experimental observations.
The free nucleon structure is characterized by its electromagnetic form-factors (EM FFs) G E and G M . In the one-photon exchange approximation, the ratio between the transverse (x) and longitudinal (z) polarization-transfer components, P x /P z , measured by elastic polarized electron scattering is proportional to G E /G M [25]. In quasielastic A( e, e p ) reactions, the sensitivity of the P x /P z to the G E /G M ratio persists and, hence, the measurement of polarization transfer to the knocked-out proton has been suggested as a tool to investigate nuclear-medium modifications of the bound proton [26]. Good theoretical calculations, which give a reliable account of the nuclear processes such as final-state interactions (FSI), allow conclusions to be made from these experiments, by distinguishing between the effects of such nuclear processes and possible modifications of the bound nucleon.
Theoretical calculations suggest that comparing the polarization transfer to knocked-out protons from the s and p shells should result in measurable differences in the ratio of the polarization-transfer components [1]. We study the double ratio (P x /P z ) p /(P x /P z ) s , which is sensitive to the deviation of the form-factor ratio, G E /G M , in each shell. We note that this is equivalent to (P s z /P p z )/(P s x /P p x ) where, based on calculations discussed below, one may expect that differences in FSI for knockout protons from s and p shells (as well as between the longitudinal and transverse components) will largely cancel out.
Almost all theoretical calculations characterize the bound nucleons by their initial internal momentum which, in the absence of FSI, is equivalent to the measured missing momentum in the reaction. But it has been shown that deviations of the ratio P x /P z obtained through the quasi-free reaction from that of the free nucleon as a function of the bound-proton virtuality (see Eq. (2)) are in overall good agreement between different nuclei, and at different momentum transfers and kinematics. This suggests that the nucleon'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 energy, one cannot compare the polarization transfer to the nucleons from the different shells at the exact same missing momentum and virtuality. However, we chose the kinematics for the measurements so that there is an overlap between s-and p-shell removal for both the missing momentum and the virtuality.
We present here polarization-transfer measurements to the protons extracted from the s and p shell in 12 C in search of nuclear-density-dependent modifications of the bound proton. We study the transverse-to-longitudinal components ratio, and compare the results from the two shells by the aforementioned double ratios. The data are also compared to calculations in relativistic distorted-wave impulse approximation (RDWIA) [2] which use free-nucleon electromagnetic form-factors. We present the comparison in both missing momentum and bound-proton virtuality, and demonstrate the advantage of using the latter as a parameter for such comparisons.

Experimental Setup and Kinematics
The experiment was carried out in A1 Hall at the Mainz Microtron (MAMI) using a 600 MeV continuous-wave (CW) polarized electron beam of about 10 µA. The measurements were performed at Q 2 = 0.175 GeV 2 /c 2 . The beam polarization, P e , was measured periodically using the standard Møller [27,28] and Mott [29] polarimeters. The polarization range was 80.5% < P e < 88.7%. The polarization was increasing at the beginning of the experiment with the decrease of the quantum efficiency towards the end-of-life of the strained GaAs crystal used as the beam source. It dropped after the annealing process of the crystal. To account for the variations in P e we used a rolling average of the measurements (resetting it after the refreshing process), which was applied in the analysis of the data.
We used a 12 C target consisting of three 0.8 mm-thick foils, which were rotated 40 • relative to the beam. This way we minimized the path of the outgoing proton through the  The scattering plane is determined by the ingoing and outgoing-electron momentum, k and k , respectively. The reaction plane is spanned by the transferred momentum, q, and the outgoing proton's momentum, p . We choose to represent the polarization components in the scattering plane by using a right-handed coordinate-system with its axes being: z parallel to the momentum transfer q,ŷ along the vector product of the ingoing and outgoing-electron momentum, k × k , andx =ŷ ×ẑ.
Another often-used reference frame isLNŜ whereL points along the outgoing proton's momentum, p ,N is along the vector product p × q, andŜ =N ×L. There are three important angles that help characterize the reaction above. Electron scattering angle, θe, together with the energy of an ingoing electron, k 0 , determines the momentum transfer. The azimuthal angle between q and p , φpq, represents the angle between the scattering and reaction plane, whereas θpq is the corresponding polar angle.
target, and hence, reduced the energy loss and the probability of multiple scattering. The two A1 high-resolution spectrometers [30] were used to analyze the scattered electron (Spectrometer C) and the knock-out proton (Spectrometer A). In Spectrometer A we installed a focal-plane polarimeter (FPP) [31] in which the polarized protons experience secondary scattering on a carbon analyzer, resulting in an angular asymmetry due to the spin-orbit part of the nuclear force. Its angular distribution is given by where σ 0 (ϑ) is the polarization independent part, A C is the analyzing 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 transverse polarization components of the proton at the focal plane. The analyzing power depends on the energy of the outgoing proton E p and was adopted from [32,33]. To measure this distribution, horizontal drift chambers (HDCs) [34] were placed behind the scatterer. Table 1 and Figure 1 show the kinematic setting we used and the important kinematic variables, respectively. We use a convention where the sign of the missing momentum, p miss = q − p , is determined by the sign of p miss · q. We define the virtuality of the embedded nucleon as: where m p , m A , and are the masses of the proton, target nucleus ( 12 C) and residual nucleus ( 11 B, not necessarily in its ground state), respectively. Here, ω = k 0 − k 0 is the energy transfer and E p is the total energy of the outgoing proton. We chose the kinematic setting shown in Table 1 to access protons with high missing momentum from both s and p shells. This corresponds to Setting B in previous measurements reported in [5,6]. In previous measurements we explored regions of positive and negative missing momenta to study the general behavior of polarization transfer and compared it between different nuclei. We now present a dedicated measurement performed in 2017 with improved statistics and a focus on a missing-momentum range where there 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 data sets.  We distinguished between the protons extracted from the s and p shell based on their measured missing energy, E miss , defined as where T p is the kinetic energy of the detected proton and T11 B is the calculated kinetic energy of the recoiling 11 B nucleus. Following [5] and [35], protons with 15 < E miss < 25 MeV correspond primarily to proton removal from the p 3/2 shell, while those with 30 < E miss < 60 MeV originate from the s 1/2 shell. The missing-momentum-versus-virtuality phase space for protons from both shells is shown in Fig. 2. The shaded area indicates the virtuality range common to both shells, and the distribution obtained from each shell is projected in the bottom panel of Fig. 2.

Determination of the Transferred Polarization and Uncertainties
We followed the convention of [7] to express individual components of the outgoing polarization in the scattering plane, P . Our coordinate system convention is also shown in Fig. 1.
To obtain the polarization components we utilized the maximum-likelihood estimation where we optimized the outgoing-proton polarization. Because our kinematics are close to parallel, we assumed only one induced component, P y , [36] and two transferred components, P x and P z , to be non-zero. Therefore, the total polarization of the outgoing proton at the target is where h is electron helicity. The contributions from the rest of the components are either very small in (anti)parallel kinematics or cancel out because of their anti-symmetric dependence on the angle between the scattering and reaction planes, φ pq [3]. Protons travel through magnetic fields of the spectrometer before reaching the FPP, where we measure their polarization components, P FPP x and P FPP y . Therefore, before we evaluate the likelihood function, we propagate the proposed estimates of target components from Eq. (4) through the spectrometer with the spin transfer matrix S which was calculated with the QSPIN program [37]. To determine the target polarization components that best fit the measured distribution from Eq. (1), we maximize the following loglikelihood function where is determined per-event. It includes trajectory dependent spin-transfer coefficients, S ij , and the measured azimuthal angle ϕ after the secondary scattering of the proton. The uncertainties of the extracted components and their ratios were estimated through the numerical second-order partial derivative of the log-likelihood function and, besides the numerical error, include a part of the systematic spintransfer error as well. As can be seen in Table 2, the beam polarization and the analyzing power are the largest contributors to the uncertainty in the polarization components P x and P z , while their effect largely cancels out when we form either a single or a double ratio. The uncertainties in the beam energy and the central kinematics affect the basis The sources contributing to the systematic uncertainties of individual components, P x , P z , single ratios, (P x /P z )s,p, and the double ratio, (P x /P z )s/(P x /P z )p. All values are in percent. vectors of the scattering-plane coordinate system and influence the binning of the events. Another important contributor to the uncertainty when determining the secondaryscattering distribution is the quality of the alignment between the tracks extrapolated from the VDC to the HDC plane and those measured by the HDCs themselves. The above three sources of uncertainty (beam energy, central kinematics, and detector alignment) were studied through the repetition of the analysis with modified values. We modified each contributor separately by its uncertainty value, and determined how much this affected the extracted polarizations. Similarly, we determined the contributions from various software cuts that are employed in the analysis, by setting each of them slightly tighter and looking for the average effect of the modified cut over all of the bins. Because a modification of the cut always impacts a number of events that we consider, we performed a parallel re-analysis, where we left the chosen cut unchanged but reduced the number of events through a random selection.
Another possible source of the systematic uncertainty is the separation of the protons from s and p shell by the missing-energy cut. Although the neighboring boundaries of the two E miss ranges sit 5 MeV apart, each of them contain a small amount of protons coming from the other shell. To estimate the magnitude of this cross-contamination, we evaluated the amount of overlap by performing separate fits over the s-and p-shell peaks in the available 12 C structure function. We found that for our p miss range, the p-shell cut includes around 5% of protons coming from the s shell, whereas the amount of protons coming from the p shell that are included in the s-shell cut is negligible. We multiplied these cross-contamination estimates with relative differences between the individual components for two shells, in order to obtain the corresponding uncertainty. Since the difference is positive for one component and negative for the other, we added the uncertainties in quadrature for the single ratio, whereas the uncertainty on the double ratio, although in principle vanishing, is dominated by the p-shell single-ratio uncertainty.
The last two items from the Table 2 correspond with the quality of the spin-precession evaluation in our maximumlikelihood algorithm. We started by comparing the results obtained from employing the spin-transfer matrix to those calculated using the QSPIN program which is more precise but considerably slower. The second contribution arises from the finite resolution of the proton's trajectory parameters (e.g. vertex position). Here we again used QSPIN to evaluate average dispersion from analysis of 100 trajectories with normally distributed variations in each parameter, where the parameter's resolution was used as a standard deviation of the sampling function. Finally, we obtain the total estimated systematic uncertainty by adding contributions from each source in quadrature.

Results and Discussion
We show in the top two panels of Fig. 3 the polarizationtransfer components P x and P z to protons knocked-out from the s and p shells, as a function of the missing momentum, p miss , and virtuality, ν. As in Fig. 2, the gray band in the plots indicates the virtuality-overlap region between the protons extracted from s and p shells. The solid lines represent calculations in relativistic distorted-wave impulse approximation (RDWIA), where we use the average democratic optical potential from [38], relativistic bound-state wave functions obtained with the NL-SH parametrization [39], and free-proton electromagnetic form-factors from [40]. Because the original RDWIA program from [2] was written for use with co-planar kinematics only, we modified it to include the remainder of the 18 hadronic structure functions present in A( e, e p) reaction under one-photon-exchange approximation [8,41].
The effects of FSI can be appreciated by comparing the RDWIA (solid lines) and PWIA (dashed lines) calculations. To explore the sensitivity of the polarization components to the ratio G E /G M we repeated the calculation with a form-factor ratio modified by ±5%. The impact of this variation on the results of the calculation is shown as a band around the respective calculation with no modification. We note that in this kinematic region, varying the form-factor ratio has a very small effect on the transverse component, P x , while the longitudinal component, P z , shows a linear dependence on the G E /G M , as can be seen in Fig. 3. The behavior of the individual components is translated to the linear dependence of their ratio, P x /P z , on the form-factor ratio.
Nuclear effects can not only differ for protons from the s and p shell, but may also have different effects on the transverse (x) and longitudinal (z) polarization components when we consider protons from a single shell. This can be seen as a deviation from unity in the bottom panel of Fig. 3, where we show P x /P z for each shell separately, as well as in Fig. 4, which includes component ratios, P s i /P p i (i = x, z,) for the two shells. Such differences are also foreseen by the theoretical calculations. To minimize these differences when searching for medium modifications in the proton structure, we examine the double ratio (P x /P z ) p /(P x /P z ) s . The double ratio is shown for the measured components as a func- Component ratio PWIA DWIA Figure 4: Ratios of given polarization-transfer components (P x or P z ) for each shell in 12 C (s or p) as a function of missing momentum (left) and virtuality (right). We note that here the virtuality range is narrower since the ratios, which compare the two shells, can be calculated only in the overlap region. The solid and dashed lines represent the RDWIA and the PWIA calculations, respectively. Following the components, only the ratio (P z )s/(P z )p is sensitive to the electromagnetic form-factor ratio's modification, and hence, has a visible band around the calculation. Since we are searching for differences between the two shells, we modified the electromagnetic form-factor ratio only for one of them.
tion of missing momentum in the top panel of Fig. 5   The results suggest that proton virtuality is a good parameter to characterize the properties of a bound proton. Indeed, differences that were suggested and might have been observed focusing on the missing momentum of the reaction are much reduced or disappear when protons at the same ν are compared. This is further corroborated by the calculations when events of the similar ν are considered rather than p miss bins in which protons of a larger virtuality range are combined. It can be further deduced that there is no statistically significant difference between polarization ratios for s-and p-shell protons. The small deviation of double ratio from unity can be already accounted for with the unmodified electromagnetic form-factor ratio and simple PWIA calculations, while measurements are also in agreement with RDWIA calculations (reduced χ 2 = 0.48, p = 0.89). Thus, we found no evidence of density-dependent modifications.
The ratios of the polarization-transfer components P x /P z to deeply bound protons were measured for several nuclei. It was shown that a comparison of this ratio to that of a free proton, (P x /P z ) A /(P x /P z ) H , at given ν shows the same deviations for 2 H, 4 He, and 12 C despite different kinematic conditions. The agreement of the results when the proton is bound in 2 H, which is a slightly-bound two-body system and often used as an effective neutron target, with those when bound in nuclei with a high average nuclear density (like 4 He and 12 C) also supports our observation. While FSI and the local nuclear density may differ between these nuclei, their effect on the polarization transfer is similar, and no nuclear-density-dependent modifications are observed. Clearly, our results suggest virtuality to be a better parameter to characterize the bound proton than p miss .

Conclusions
We presented measurements of the polarization transfer to deeply bound protons in the s and p shells of 12 C by polarized electrons with the 12 C( e, e p ) reaction. To investigate nuclear-density dependence and possible in-medium modification of the proton's EM FF, we utilized the fact that the ratio of the transverse to longitudinal components is sensitive to the EM FF ratio. The measured polarization ratios for protons extracted from the two shells were studied and compared as a function of either the missing momentum or the bound-proton virtuality. We concluded that the bound proton is better characterized by its virtuality rather than the missing momentum. Although according to some theories, there is a large difference in the nuclear density between the two shells in 12 C, the measurements show no significant differences between the s-and p-shell protons to the level of 5% when compared at the same virtuality. Furthermore, the observed slight deviation from unity is expected from both PWIA and RDWIA calculations.

Acknowledgments
We would like to thank the Mainz Microtron operators and technical crew for the excellent operation of the accelerator. This work is supported by the Israel Science Foundation (Grants 390/15, 951/19) of the Israel Academy of Arts and Sciences, by the PAZY Foundation (Grant 294/18), by the Israel Ministry of Science, Technology and Space, by the Deutsche Forschungsgemeinschaft (Collaborative Research Center 1044), by the U.S. National Science Foundation (PHY-1205782). We acknowledge the financial support from the Slovenian Research Agency (research core funding No. P1-0102) and from the Croatian Science Foundation (under the project 8570).