Measurement of polarization-transfer to bound protons in carbon and its virtuality dependence

We measured the ratio $P_{x}/P_{z}$ of the transverse to longitudinal components of polarization transferred from electrons to bound protons in $^{12}\mathrm{C}$ by the $^{12}\mathrm{C}(\vec{e},e'\vec{p})$ process at the Mainz Microtron (MAMI). We observed consistent deviations from unity of this ratio normalized to the free-proton ratio, $(P_{x}/P_{z})_{^{12}\mathrm{C}}/(P_{x}/P_{z})_{^{1}\mathrm{H}}$, for both $s$- and $p$-shell knocked out protons, even though they are embedded in averaged local densities that differ by about a factor of two. The dependence of the double ratio on proton virtuality is similar to the one for knocked out protons from $^{2}\mathrm{H}$ and $^{4}\mathrm{He}$, suggesting a universal behavior. It further implies no dependence on average local nuclear density.

Deviations of quasi-elastic measurements on nuclei from those performed on protons or from calculations using freeproton form-factors (FFs) reflect various many-body effects, potentially including medium modifications of the bound proton structure in the nuclear field [1,2]. The ratio of the transverse (P x ) to longitudinal (P z ) polarization transfer components measured in the elastic double-polarized process 1 H( e, e p) is proportional to the ratio of the electric to magnetic FFs of the free proton, R1 H ≡ (P x /P z )1 H ∝ G p E /G p M [3]. In nuclei, the ratio of the polarization transfer components to a bound proton, R A ≡ (P x /P z ) A , can be determined from the analogous quasi-free proton knockout process A ( e, e p). Measurements of R A eliminate many systematic uncertainties and thus constitute a sensitive and precise tool to study possible deviations of a bound proton properties from a free one.
Previous double polarized proton knock-out experiments on light nuclei, 2 H and 4 He, were found to be in agreement when compared in terms of the proton virtuality, which is a measure of the "off-shellness" of the bound proton (see Eq. (2)). The measurements showed no dependence on the average nuclear density nor on momentum transfer [4]. For the deuteron, detailed calculations [5] explained the devia-tions from the free proton by final state interactions (FSI). It is thus interesting to extend the measurements to heavier nuclei were FSI effects are expected to be different.
The 12 C nucleus is a particularly appealing target for such studies as one can selectively probe protons from specific nuclear shells, s and p. The average local densities in these shells differ by about a factor of two, which was predicted to impact the polarization transfer to s-and p-shell protons differently [6]. Previous measurements on s-and p-shell protons in 16 O were limited in statistics and the kinematical range covered [7].
In this paper we report on the measurements of the P x /P z ratio for protons bound in carbon, 12 C ( e, e p), and present the double ratio R12 C /R1 H . In terms of virtuality, our results exhibit consistency between s-and p-shell protons as well as with measurements obtained on other light nuclei. Thus, they confirm the absence of average nuclear density dependence even in the heavier nucleus 12 C.
The experiment was performed at the Mainz Microtron (MAMI) accelerator using the A1 beam-line and spectrometers [8]. We used a 600 MeV continuous-wave polarized electron beam with a current of about 10 µA. The average beam polarization was about 80%, measured with a Møller polarimeter and verified by Mott polarimetry. The uncertainty in the beam polarization was less than 5%. The beam helicity was flipped at a rate of 1 Hz. Two highresolution, small solid-angle spectrometers with momentum acceptances of 20 − 25 % were used to detect the scattered electrons in coincidence with the knocked-out protons. The target consisted of three carbon foils of 0.8 mm thickness each, separated by about 1.5 cm and tilted at an angle of 40 • with respect to the beam. The usage of three tilted foils reduced the proton energy loss in the target and improved the resolution for the reaction-vertex determination. This reduced the systematic uncertainty in the determined polarization transfer components at the reaction point. The proton spectrometer was equipped with a polarimeter placed behind its focal-plane (FPP) using a 7 cm thick carbon analyzer [8,9]. The spin-dependent scattering of the polarized proton by the carbon analyzer enables the determination of the proton transverse polarization components at the focal plane [9]. The polarization-transfer components at the reaction point were obtained by correcting the measured components for the spin precession in the magnetic field of the spectrometer. Following the convention of [2], both P z and P x were determined in the scattering plane, defined by the incident and scattered electron momenta, where P z is along and P x is perpendicular to the momentum transfer vector, q.
In the analysis, cuts were applied to identify coincident electrons and protons that originate from the carbon target, and to ensure good reconstruction of tracks in the spectrometers and the FPP. To remove Coulomb scattering events by the carbon analyzer, we selected only events that scattered by more than 8 • in the FPP.
The polarization transfer components P x and P z were first determined as a function of the proton missing momentum defined as p miss = q − p p , where p p is the outgoing proton momentum. We define the scalar missing momentum, p miss ≡ ± | p miss |, where the sign is taken to be positive (negative) if the longitudinal component of p miss is parallel (anti-parallel) to q. The measurements were performed in two kinematical settings that covered two ranges in p miss and two ranges in the invariant four-momentum transfer Q 2 = q 2 − ω 2 , where ω is the energy transfer. Details of the kinematics are summarized in Table 1.
The protons knocked out from the s and p shells were identified by their missing energy. The missing energy is defined as sured kinetic energy of the outgoing proton, and T11 B is the calculated kinetic energy of a recoiling 11 B nucleus (g.s.). The missing-energy spectrum of setting A is shown in Fig. 1. The sharp peaks correspond to the ground state and the lowest excited states of the recoiling 11 B. Following Dutta et al. [10] we present the polarizationtransfer results for two ranges of E miss shown in the figure: the first (15 < E miss < 25 MeV) corresponds to proton removal primarily from the 12 C p 3/2 shell; the second (30 < E miss < 60 MeV) corresponds predominantly to proton removal from the s-shell. The missing energy cut allows some s-shell strength in the p-shell region and vice versa. In addition, we show the combined data from the entire E miss range (10 < E miss < 90 MeV) covering proton removal from both s-and p-shells. The inset in Fig. 1 (adapted from [10]), shows the predicted momentum distributions of p-and sshell protons in 12 C obtained from an independent particle shell model spectral function (S) [10]. The difference between the s-and p-shell proton momentum distributions around p miss = 0, may impact the polarization transfer in this region. Helicity-independent uncertainties in the measured ratios (acceptance, detector efficiency, target density, etc.) largely cancel out due to frequent flips of the beam helicity. The uncertainties in beam polarization, carbon analyzing power and efficiency are reduced well below the statistical uncertainty by taking the P x /P z ratio. The total systematic uncertainty in R12 C , dominated by the vertex position reconstruction in the target, does not exceed 2% and is about 25% of the statistical uncertainty. In the following figures, only the statistical uncertainties are shown.
The measured helicity-dependent ratios R12 C for both settings are presented in Fig. 2 (top) as a function of p miss . The difference in R12 C between s-and p-shell proton re- We removed some contributions to the differences between data at the same p miss , which are due to the different kinematics (or momentum transfer), by dividing R12 C by the hydrogen ratio where E is the incident electron energy, and M p is the proton mass. The scattered electron energy (E = E E, Q 2 ) and scattering angle (θ e = θ e E, Q 2 ) are calculated assuming elastic electron-proton scattering. R1 H was calculated on an event by event basis using the proton FFs parametrized by Bernauer et al. [11] and averaged over the bin. The double-ratio of the 12 C data to 1 H, R12 C /R1 H , is shown in the bottom panel of Fig. 2. However, even after division by R1 H , the differences between sand p-shell results at the same missing momenta are still significant.
The bound nucleon can be characterized also by its virtuality, i.e. its "off-shellness". There is no unique way to define virtuality. Following [4] we define the virtuality, ν, of a bound proton as where (ω − E p + M A ) 2 − p 2 miss (determined event by event), and E p is the total energy of the outgoing proton. We note that the virtuality (Eq. (2)) is not a unique function of p miss . This is demonstrated in the two dimensional event distribution of ν vs p miss shown in the supplementary material [12]. Equation (2) implies that the struck proton is off-shell ( = M 2 p c 2 ) and the recoil system is on-shell. The virtuality dependence of R12 C /R1 H is shown in Fig. 3. The double ratios are shown separately for positive and negative missing momenta due to possible differences, as observed in 4 He [2] and calculated for 2 H due to FSI [4].
The s-and p-shell protons have different wave functions as reflected also in their missing-momentum distributions. These differences, such as the behavior at p miss = 0 (see [10] and Fig. 1) and possibly the total angular momentum, may affect the polarization transfer, as predicted by calculations [13,14]. Nevertheless, the corresponding double-ratios have the same smooth behavior, and show the same virtuality dependence, as is clearly shown in Fig. 3. Motivated by the observed good agreement between the s-and p-shell protons as a function of virtuality, we combined the data and obtained R12 C /R1 H for the entire missing energy region, 10 < E miss < 90 MeV.
In Fig. 4, the 12 C double ratios, combined for s-and p-shell proton removals, are compared with those of 2 H obtained at MAMI for the same kinematics [4], as well as to 2 H and 4 He data measured at JLab at Q 2 = 1 and 0.8 GeV 2 /c 2 , respectively [2,15]. Note that the data shown in Fig. 4 are not identical to those in Fig. 3 due to the different E miss range and bins. The new 12 C data almost double the virtuality range covered by the data from light nuclei. The higher values of R12 C /R1 H at |ν| < 0.04 GeV 2 /c 2 are due to p 3/2 protons whose behavior is attributed to the p wave function properties at small |p miss | [13,14], unlike s-shell protons in the other nuclei.
The data suggest that the double-ratio is characterized well by the virtuality of the struck proton. Virtuality seems to be a better parameter than p miss to describe polarization  [4] with essentially the same kinematics as the present work [4]. The light symbols are 2 H and 4 He data measured at JLab at Q 2 = 1 and 0.8 GeV 2 /c 2 , respectively [2,15]. The triangles (circles) refer to setting A (B) as in Fig. 2. transfer to a struck proton in different nuclei.
To test this hypothesis, we compared the 12 C data to the MAMI 2 H data. To enable an event-by-event comparison, we adjusted the theoretical calculations [5] for 2 H to reproduce the measured polarization transfer components of 2 H. The ratio of the 12 C data to the adjusted model is 1.07 ± 0.03. In this comparison we excluded the afore mentioned |ν| < 0.04 GeV 2 /c 2 data, due to its special behavior.
The agreement between the data from the different nuclei suggests that the observed deviations from the free proton ratio have a common origin. We note that Eq. (1) is valid only for a free proton. Thus the double ratio in nuclei does not remove FSI effects. When the deuteron data [4] were compared to the calculations of [5], the theory was in good agreement with the data. This implies that in the deuteron most of the deviation is due to FSI. One may speculate that the same effects dominate in heavier nuclei as well, although this has to be confirmed by detailed calculations. These should take into account the variation of the kinematics over the experimental phase space. Such a task is rather involved and may depend on various parameters. The new 12 C data may provide important information for determination of the mechanisms at work and the validity of using free proton FFs in such calculations.
The data confirm that the virtuality of the struck proton is the preferred parameter for comparing the deviations from a free proton. This implies that further measurements to look for local nuclear density effects should compare polarization transfer to protons at different local densities, like different shells, but with the same virtuality as can be de-duced from Fig. 3. This comparison requires high statistics in order to confirm or reject in medium modification of the proton FFs, which are estimated at a few percent [6].
To summarize, our data of the polarization-transfer ratios for 12 C extend the previous nuclear measurements on 2 H and almost double the virtuality range. The new double ratios R12 C /R1 H agree well with those previously measured on 2 H and 4 He, including those obtained in different kinematics. The double-ratios exhibit a similar shape for nuclei with very different average local density. The new results suggest also that measurements of both 2 H and 4 He over an extended virtuality range are needed. Indeed, such measurements were proposed [16] and approved at JLab.