The influence of Fermi motion on the comparison of the polarization transfer to a proton in elastic $\vec ep$ and quasi-elastic $\vec eA$ scattering

TL;DR

This study introduces new expressions for polarization transfer when the proton is in motion and compares them to experimental data, revealing that Fermi motion accounts for up to 20% correction but does not significantly affect the observed deviations.

Contribution

The paper presents the first derivation of polarization-transfer components for a moving proton and compares these to experimental data, clarifying the role of Fermi motion.

Findings

Proton motion correction is up to 20% at high missing momentum.

Double ratio of polarization components largely cancels out motion effects.

Kinematic effects are not the main cause of deviations in previous measurements.

Abstract

A comparison between polarization-transfer to a bound proton in quasi-free kinematics by the A$(\vec{e},e'\vec p)$ knockout reaction and that in elastic scattering off a free proton can provide information on the characteristics of the bound proton. In the past the reported measurements have been compared to those of a free proton with zero initial momentum. We introduce, for the first time, expressions for the polarization-transfer components when the proton is initially in motion and compare them to the $^2$H data measured at the Mainz Microtron (MAMI). We show the ratios of the transverse ($P_x$) and longitudinal ($P_z$) components of the polarization transfer in $^2\textrm{H}(\vec{e},e'\vec p)\textrm{n}$, to those of elastic scattering off a "moving proton", assuming the proton's initial (Fermi) momentum equals the negative missing momentum in the measured reaction. We found that the correction due to the proton motion is up to 20\% at high missing momentum. However the effect on the double ratio $\frac{(P_x/P_z)^A}{(P_x/P_z)^{^1\!\textrm{H}}}$ is largely canceled out, as shown for both $^2$H and $^{12}$C data. This implies that the kinematics is not the primary cause for the deviations between quasi-elastic and elastic scattering reported previously.

Figures

Equations

$${\left(\frac{P_x}{P_z}\right)}_{\mathrm{H}}=-\frac{2M}{(k_0+k_0^{\prime })\tan ({\theta }_{k,k^{\prime }}/2)}\frac{G_E}{G_M},$$

$$p_M=\left(p_{0M},p_x,p_y,p_z\right),$$

$$k_M=\left(k_0,\frac{Q\sqrt{4k_0(k_0-{\omega }_M)-Q^2}}{2q_{zM}},0,\frac{2k_0{\omega }_M+Q^2}{2q_{zM}}\right),$$

$$p_M^{\prime }=p_M+q_M,$$

$$k_M^{\prime }=k_M-q_M,$$

$$q_M=\left({\omega }_M,0,0,q_{zM}\right),$$

$${\omega }_M=\frac{Q^2{p}_{0M}+p_{z}Q\sqrt{4E_{pt}^2+Q^2}}{2E_{pt}^2},$$

$$q_{zM}=\sqrt{Q^2+{\omega }_M^2},$$

$$P_x=-C_P{G}_M\left(G_E({\alpha }_{\ell x}-{\beta }_{\ell x})+G_M{\beta }_{\ell x}\right),$$

$$P_z=-C_P{G}_M\left(G_E({\alpha }_{\ell z}-{\beta }_{\ell z})+G_M{\beta }_{\ell z}\right),$$

$$C_P=\frac{2}{(1+\tau )M^2{\Sigma }_0},$$

$${\Sigma }_0=(G_E^2+\tau G_M^2)\left(\frac{(K\cdot p{)}^2}{M^2{Q}^2(1+\tau )}-1\right)+2\tau G_M^2,$$

$${\alpha }_{\ell x}=\frac{1+\tau }{2}\left(\frac{p_x}{p_0^{\prime }+M}\left(M{K}_0+2p\cdot k-\frac{Q^2}{2}\right)-M{K}_x\right),$$

$${\beta }_{\ell x}=\frac{-1}{4M^2}\left(2k\cdot p-\frac{Q^2}{2}\right)\frac{p_x}{p_0^{\prime }+M}\left(M\omega -\frac{Q^2}{2}\right),$$

$${\alpha }_{\ell z}=\frac{1+\tau }{2}\left(\frac{p_z+q_z}{p_0^{\prime }+M}\left(M{K}_0+2p\cdot k-\frac{Q^2}{2}\right)-M{K}_z\right),$$

$${\beta }_{\ell z}=\frac{-1}{4M^2}\left(2k\cdot p-\frac{Q^2}{2}\right)\left(\frac{p_z+q_z}{p_0^{\prime }+M}\left(M\omega -\frac{Q^2}{2}\right)-M{q}_z\right),$$

$$\frac{P_x}{P_z}=\frac{G_E({\alpha }_{\ell x}-{\beta }_{\ell x})+G_M{\beta }_{\ell x}}{G_E({\alpha }_{\ell z}-{\beta }_{\ell z})+G_M{\beta }_{\ell z}}.$$

Full text

††journal: Physics Letters B

1 Introduction

Polarization transfer from a polarized electron to a proton in an elastic scattering reaction has become a recognized method to measure the ratio of the proton’s elastic electromagnetic form factors $G_{E}/G_{M}$ [1, 2, 3, 4, 5, 6, 7, 8, 9]. For a proton initially at rest, assuming the one-photon exchange approximation, the ratio of the transverse ($P_{x}$) to longitudinal ($P_{z}$) polarization-transfer components, with respect to the momentum transfer $\vec{q}$, is proportional to $G_{E}/G_{M}$ [10]:

[TABLE]

where $k_{0}$, $k^{\,\prime}_{0}$ are respectively the initial and final electron energies, $\theta_{k,k^{\prime}}$ is the electron scattering angle in the lab frame, and $M$ is the proton’s mass. This provides a direct measurement of the form factor (FF) ratio and eliminates many systematic uncertainties [11].

Measuring the ratio of the components of the polarization transfer to a knock-out proton in quasi-free kinematics on nuclei, which is sensitive to the electromagnetic FF ratio, has been suggested as a method to study differences between free and bound protons [4, 5]. As such it can be used as a tool to identify medium modifications in the bound proton’s internal structure [12], reflected in the FFs and thereby in the polarization transfer. Deviations between polarization ratios in quasi-free and elastic scattering can be interpreted only with realistic calculations of the nuclear effects, such as final state interactions (FSI), meson exchange currents (MEC), isobar currents (IC), and relativistic corrections on the outgoing proton polarization components [13, 14, 15]. However, a comparison to the polarization transfer on a free proton should also consider the Fermi motion of the struck proton, rather than comparing to a reaction with the proton at rest.

Polarization-transfer experiments have been carried out recently on 2H and 12C target nuclei at the Mainz Microtron (MAMI) [13, 14, 16], as well as on 2H, 4He and 16O at Jefferson Lab [17, 18, 19, 20, 21], in search of medium modification in the bound proton’s internal structure. In particular, these experiments were performed to study deeply bound nucleons, characterized by high missing momentum, which is equivalent (neglecting FSI) to protons with high initial momentum. It was shown in [13, 14] that for 2H at low momentum transfer222These data were taken at $Q^{2}$ = 0.18 and 0.4 (GeV/$c$)2., the deviations can be explained by nuclear effects without the necessity of introducing modified FFs. In these experiments, the comparison to the free proton was done either by measurements of the polarization transfer ratio to 1H at similar kinematics [18, 17], or by calculations using Eq. 1 and a fit to the world data of proton FFs [11, 22], which were used in [13, 14, 16].

Kinematically, quasi-elastic scattering differs from elastic scattering in that the bound proton is off-shell, and in that it moves relative to the nucleus with Fermi motion. However, using Eq. 1 to calculate the polarization ratio is only valid if the proton is both on-shell and at rest.

In this work we introduce an alternative approach: comparing the polarization transfer in quasi-free scattering to that of a free proton (on-shell) with a finite initial momentum. The general expressions for polarization transfer to a free moving proton have been developed. We applied this prescription to the 2H and 12C data measured at MAMI over relatively large missing momentum ranges. The elastic kinematics are described in Section 2, while the polarization transfer formulae are given in Section 3. The application of this prescription to MAMI data is shown in Section 4.

2 Kinematics

The kinematics are defined by an electron with initial four-momentum $k$ scattering off a proton with initial four-momentum $p$, through exchange of a virtual photon with four-momentum $q$, resulting in the two particles having final four-momenta $k^{\prime}$ and $p^{\prime}$ respectively.

The final momenta of the proton and electron, $\vec{p}^{\,\prime}$ and $\vec{k}^{\,\prime}$, are measured in the spectrometers. The “missing momentum”, defined as $\vec{p}_{\rm miss}=\vec{q}-\vec{p}^{\,\prime}$, is calculated using the momentum transfer defined by $\vec{q}=\vec{k}-\vec{k}^{\,\prime}$. In the absence of FSI, the initial proton momentum is given by the missing momentum, $\vec{p}=-\vec{p}_{\rm miss}$. By convention, the missing momentum is considered positive (negative) if $\vec{p}_{\rm miss}\cdot\vec{q}$ is positive (negative).

These measured kinematics are shown in the top diagram in Figure 1, and are compared, in the middle and bottom diagrams, to the kinematics of elastic scattering off a proton initially at rest, and off a proton with initial momentum $\vec{p}=-\vec{p}_{\rm miss}$, respectively.

Each measured quasi-elastic event is compared to an elastic event that has the same incident energy ($k_{0}$), the same magnitude of the four-momentum transfer ($Q^{2}=-q^{2}$) and the same initial proton momentum ($\vec{p}$) as the measured quasi-elastic event. Also, the struck proton is on-shell in its initial and final state in the elastic event333This is equivalent to the Bjorken condition, $\left[\frac{Q^{2}}{2p\cdot q}\right]_{M}=1$.. These criteria uniquely define a set of kinematics, here referred to as “moving-proton” kinematics. We use a subscript “M” to distinguish between the kinematic variables in the moving-proton kinematics from the measured quantities.

In the standard coordinate system for polarization transfer, where $\vec{q}\,||\,\hat{z}$ and the scattering is in the $xz$ plane, the moving-proton kinematics are444For the derivation, see the supplementary material.:

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

using $p_{0M}$ and $E_{pt}$ as shorthand for $\sqrt{M^{2}+|\vec{p}|^{2}}$ and $\sqrt{M^{2}+p_{x}^{2}+p_{y}^{2}}$, respectively. Note that $p_{x}$, $p_{y}$ and $p_{z}$ are the coordinates of $\vec{p}=-\vec{p}_{\rm miss}$ in the $\vec{q}\,||\,\hat{z}$ coordinate system.

In previous publications [13, 14, 16], we used a different type of free-proton kinematics (“resting proton”, denoted with a subscript “R” in Fig. 1 and elsewhere in this paper) where instead of having the same $\vec{p}$ as in the measured kinematics, we had $\vec{p}_{R}=\vec{0}$. The resting-proton kinematics may therefore be evaluated by substituting $\vec{p}=0$ in Eqs. 2-8.

3 Polarization transfer

The general expressions for the polarization transfer from a longitudinally polarized electron to an initially moving nucleon with momentum $\vec{p}$ in the scattering plane are presented below, with more details in [23]:

[TABLE]

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

using $\tau=\frac{Q^{2}}{4M^{2}}$ and $K=k+k^{\prime}$. The factors of $-C_{P}G_{M}$ in Eq. 9 and 10 cancel out in the ratio:

[TABLE]

Equation 17 reduces to Equation 1 if the proton is initially at rest. It should be noted that the formulae in this section are only valid for elastic scattering in the one-photon approximation.

4 Application to MAMI A$(\vec{e},e^{\prime}\vec{p})$ data

Polarization-transfer measurements on 2H and 12C have been carried out at MAMI [24, 25] at momentum transfers of $Q^{2}$ = 0.18 and 0.40 (GeV/c)2 [13, 14, 16]. The kinematic settings of the measurements are given in Tables 1 and 2 of the supplemental material. The double ratios $\frac{(P_{x}/P_{z})^{A}}{(P_{x}/P_{z})^{{}^{1}\!\textrm{H}}}$ were presented for both nuclei as functions of the struck proton’s virtuality (“off-shellness”), $\nu\equiv(p^{\prime}-q)^{2}-M^{2}$, which proved to be a useful parameter for a unified description of the deviation of the bound proton from a free one. In addition, Ref. [14] presents the ratios $\frac{(P_{x})^{A}}{(P_{x})^{{}^{1}\!\rm H}}$ and $\frac{(P_{z})^{A}}{(P_{z})^{{}^{1}\!\rm H}}$ for the deuteron data, and showed that they were in agreement with the theoretical calculation, indicating that the deviations from the free proton were primarily due to FSI.

Applying the moving-proton prescription to the deuteron data [13, 14], we notice a good correspondence between the measured and evaluated kinematic variables in the left side of Fig. 2. These differences are much smaller than those between the resting-proton kinematics and the measured kinematics, shown on the right side of Fig. 2.

The differences between the measured and the resting-proton kinematics are more pronounced for settings B and D than in setting A, due to larger $|\vec{p}_{\rm miss}|$ and virtuality in the former.

There are larger differences between the moving-proton kinematics and the measured kinematics in the 12C data [16], as shown in the supplementary material, due to the larger Fermi motion in the carbon nucleus. However, they are still much smaller than the difference between the resting-proton kinematics and the measured kinematics.

In Figure 3, we compare the polarization-transfer components calculated for the resting- and the moving-proton kinematics. We show in this figure the ratios $(P_{x})^{{}^{1}\!\rm H}_{\rm moving}/(P_{x})^{{}^{1}\!\rm H}_{\rm resting}$, $(P_{z})^{{}^{1}\!\rm H}_{\rm moving}/(P_{z})^{{}^{1}\!\rm H}_{\rm resting}$, and $(P_{x}/P_{z})^{{}^{1}\!\rm H}_{\rm moving}/(P_{x}/P_{z})^{{}^{1}\!\rm H}_{\rm resting}$, as calculated for the kinematics of the 2H events in Ref. [13]. The difference between resting and moving for the individual components is small around $\nu=0$, and it increases up to 20% (15%) at large virtuality555This is expected, since the $|\vec{p}_{\rm miss}|$ is near zero at small $\nu$ and increases monotonically with $\nu$ for any given target and residual nuclear masses. with negative (positive) $p_{\rm miss}$. The difference between moving and resting in the ratio $P_{x}/P_{z}$ is small: up to 3% for $p_{\rm miss}<0$ and up to 10% at $p_{\rm miss}>0$.

The ratios of the polarization observables measured for the deuteron to the values calculated event-by-event for a moving proton using Equations 9 and 10 are shown in Figure 4. These are contrasted with the corresponding ratios for the resting proton, which were reported in [14]. The double ratio $\frac{(P_{x}/P_{z})^{{}^{2}\!\textrm{H}}}{(P_{x}/P_{z})^{{}^{1}\!\textrm{H}}}$, calculated event-by-event, is also presented in Fig. 4, using a procedure described in detail in the supplementary material. As expected from Fig. 3, the moving effect on the double ratio is small.

A good agreement was found between the double ratios $\frac{(P_{x}/P_{z})^{A}}{(P_{x}/P_{z})^{{}^{1}\!\textrm{H}}}$ measured for 12C and 2H as functions of virtuality using resting-proton kinematics [16]. In order to test if this agreement is preserved using moving-proton kinematics, we calculated the double ratios for the 12C datasets from MAMI [16] in the same manner as for 2H. The double ratios for the two nuclei are compared with one another in Figure 5, showing that the good agreement is maintained.

5 Conclusions

We observe that using the moving-proton kinematics described in this paper reduces, but does not eliminate, the difference between measured components of the polarization transfer in quasi-free scattering and those in elastic scattering. The deviations from the free-proton in the transverse ($P_{x}$) component become smaller (around 7%) when using the moving kinematics. Similarly, the deviations in the longitudinal ($P_{z}$) component are reduced, but remain significantly large, up to 50% (Fig. 4). Since using the moving proton kinematics has a similar effect on both components, the effect on the double ratio largely cancels out. Thus the double ratio is less sensitive to the kinematics than the separate components. This implies that the choice of elastic kinematics is not the primary cause of the deviations in $P_{x}/P_{z}$ between quasi-free and elastic scattering.

It was previously noted that the virtuality is a useful parameter for comparing the double ratio $\frac{(P_{x}/P_{z})^{A}}{(P_{x}/P_{z})^{{}^{1}\!\textrm{H}}}$ of different nuclei at different $Q^{2}$. Here we show that the double ratio has only a weak sensitivity to the initial momentum of the proton, which continues to make the double ratio a preferred observable for studying the differences between bound and free protons.

Acknowledgements

This work is supported by the Israel Science Foundation (Grant 390/15) of the Israel Academy of Arts and Sciences, by the PAZY Foundation (grant 294/18), by the Deutsche Forschungsgemeinschaft (Collaborative Research Center 1044), by the Slovenian Research Agency (research core funding No. P1–0102), by the U.S. National Science Foundation (PHY-1505615), and by the Croatian Science Foundation Project No. 8570.

Appendix A: Supplemental material

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.physletb.2019.04.004.