Polarization transfer observables in elastic electron-proton scattering at $Q^2=2.5$, 5.2, 6.8, and $8.5 {\mathrm{GeV}}^2$

Background: Interest in the behavior of nucleon electromagnetic form factors at large momentum transfers has steadily increased since the discovery, using polarization observables, of the rapid decrease of the ratio $G_E^p/G_M^p$ of the proton's electric and magnetic form factors for momentum transfers $Q^2\gtrsim 1$ ${\mathrm{GeV}}^2$, in strong disagreement with previous extractions of this ratio using the traditional Rosenbluth separation technique.

Purpose: The GEp-III and $\mathrm{GEp}\text{−}2\gamma$ experiments were carried out in Jefferson Laboratory's (JLab's) Hall C from 2007 to 2008, to extend the knowledge of $G_E^p/G_M^p$ to the highest practically achievable $Q^2$ given the maximum beam energy of 6 GeV and to search for effects beyond the Born approximation in polarization transfer observables of elastic $\vec{e}p$ scattering. This article provides an expanded description of the common experimental apparatus and data analysis procedures, and reports the results of a final reanalysis of the data from both experiments, including the previously unpublished results of the full-acceptance dataset of the $\mathrm{GEp}\text{−}2\gamma$ experiment.

Methods: Polarization transfer observables in elastic $\vec{e}p\to e\vec{p}$ scattering were measured at central $Q^2$ values of 2.5, 5.2, 6.8, and 8.54 ${\mathrm{GeV}}^2$. At $Q^2=2.5{\mathrm{GeV}}^2$, data were obtained for central values of the virtual photon polarization parameter $\epsilon$ of 0.149, 0.632, and 0.783. The Hall C High Momentum Spectrometer detected and measured the polarization of protons recoiling elastically from collisions of JLab's polarized electron beam with a liquid hydrogen target. A large-acceptance electromagnetic calorimeter detected the elastically scattered electrons in coincidence to suppress inelastic backgrounds.

Results: The final GEp-III data are largely unchanged relative to the originally published results. The statistical uncertainties of the final $\mathrm{GEp}\text{−}2\gamma$ data are significantly reduced at $\epsilon =0.632$ and 0.783 relative to the original publication.

Conclusions: The final GEp-III results show that the decrease with $Q^2$ of $G_E^p/G_M^p$ continues to $Q^2=8.5{\mathrm{GeV}}^2$, but at a slowing rate relative to the approximately linear decrease observed in earlier Hall A measurements. At $Q^2=8.5{\mathrm{GeV}}^2$, $G_E^p/G_M^p$ remains positive but is consistent with zero. At $Q^2=2.5{\mathrm{GeV}}^2$, $G_E^p/G_M^p$ derived from the polarization component ratio $R\propto P_t/P_{\ell }$ shows no statistically significant $\epsilon$ dependence, as expected in the Born approximation. On the other hand, the ratio $P_{\ell }/P_{\ell }^{\mathrm{Born}}$ of the longitudinal polarization transfer component to its Born value shows an enhancement of roughly 1.7% at $\epsilon =0.783$ relative to $\epsilon =0.149$, with $\approx 2.2\sigma$ significance based on the total uncertainty, implying a similar effect in the transverse component $P_t$ that cancels in the ratio $R$.

Figure 1

$G_E^p/G_D$ extracted from cross section measurements vs $Q^2$. The data from before 1980 are shown by open triangle [6], multiplication sign [8], open circle [9], filled diamond [10], filled square [11], crossed diamond [12], crossed square [13], and open square [14]. The SLAC data from the 1990s are shown by filled star [15] and open diamond [16]. The JLab data are shown by asterisk [17] and filled triangle [18]. Figure adapted from Fig. 3 of Ref. [4].

Figure 2

$G_M^p/{\mu }_p{G}_D$ extracted from cross section measurements vs $Q^2$. The symbols are the same as in Fig. 1. Additional data points at the highest $Q^2$, open square [19] and open star [20], were extracted from cross sections assuming ${\mu }_p{G}_E^p/G_M^p=1$. The solid (dashed) line is a fit by Ref. [21] (Ref. [22]). Figure adapted from Fig. 4 of Ref. [4].

Figure 3

The ratio ${\mu }_p{G}_E^p/G_M^p$ from the first two JLab experiments are shown as filled circle [28, 29] and filled square [30, 31], compared to Rosenbluth separation results, shown as open diamond [16], open circle [17], filled diamond [18], and open square [6, 9, 10, 11, 12, 13, 14, 15, 32]. The curve shows the linear fit to the polarization data from Ref. [30]. Figure adapted from Fig. 9 of Ref. [4].

Figure 4

Design drawing of the FPP installed in the HMS detector package, with the HMS drift chambers and the trigger planes.

Figure 5

BigCal calorimeter with its front aluminum shielding plates removed, exposing the stack of 1744 lead-glass blocks.

Figure 6

Simplified illustration of elastic event selection for the GEp-III kinematics: $Q^2=5.2{\mathrm{GeV}}^2$ (top row), $Q^2=6.8{\mathrm{GeV}}^2$ (middle row) and $Q^2=8.5{\mathrm{GeV}}^2$ (bottom row). Exclusivity cut variables are $\delta p_p\equiv 100\times \frac{p_p-p_p\left({\theta }_p\right)}{p_0}$ (left column), $\delta p_e\equiv 100\times \frac{p_p-p_p\left({\theta }_e\right)}{p_0}$ (middle column), and $\delta \varphi \equiv {\varphi }_e-{\varphi }_p-\pi$ (right column). The distribution of each variable is shown for all events (red empty circles), events selected by applying $\pm 3\sigma$ cuts of fixed width to both of the other two variables (black filled squares), and events rejected by these cuts (blue empty triangles). Vertical dotted lines indicate the $\pm 3\sigma$ cut applied to each variable. Similar plots for the $\mathrm{GEp}\text{−}2\gamma$ kinematics can be found in Ref. [52]. Note that the horizontal axis range in each plot is a fixed multiple of the elastic peak width, which varies with $Q^2$ and $E_e$.

Figure 7

Example of the Gaussian sideband fit of the $\delta p_e$ distribution used to estimate the residual background contamination of the final elastic event selection cuts at $Q^2=8.5{\mathrm{GeV}}^2$. Data (black filled circles) are shown after applying $\pm 3\sigma$ cuts to both $\delta p_p$ and $\delta \varphi$. In this example, the estimated fractional background contamination, integrated within the $\pm 3\sigma$ cut region (black vertical lines), is $f\equiv \frac{B}{S+B}=(4.89\pm 0.01)\%$, where $S$ and $B$ refer to the signal and the background, respectively, and the quoted uncertainty is statistical only. See text for details.

Figure 8

Fractional background contamination $f$ as a function of $\delta p_p$ at $Q^2=8.5{\mathrm{GeV}}^2$, with $\pm 3\sigma$ cuts applied to $\delta p_e$ and $\delta \varphi$. The horizontal error bars represent the rms deviation from the mean of the $\delta p_p$ values of all events in each bin. The uncertainties $\mathrm{\Delta }f$ are statistical only and are smaller than the data points.

Figure 9

Focal-plane helicity difference/sum ratio asymmetry $(f_{+}-f_{-})/(f_{+}+f_{-})$, defined as in Eq. (20), for the $\mathrm{GEp}\text{−}2\gamma$ ($Q^2=2.5{\mathrm{GeV}}^2$) kinematics, for single-track events selected according to the criteria discussed in Sec. 3b2. $〈\epsilon 〉$ is the acceptance-averaged value of $\epsilon$. The left (right) column shows the asymmetries for events scattering in the first (second) analyzer. Asymmetries are shown for $〈\epsilon 〉=0.153$ (top), $〈\epsilon 〉=0.638$ (middle), and $〈\epsilon 〉=0.790$ (bottom). Red curves are fits using $(f_{+}-f_{-})/(f_{+}+f_{-})=ccos\left(\phi \right)-ssin\left(\phi \right)$. Asymmetry fit results are shown in Table 5.

Figure 10

Focal plane helicity difference/sum ratio asymmetry $(f_{+}-f_{-})/(f_{+}+f_{-})$, defined as in Eq. (20), for the GEp-III kinematics, for FPP1 and FPP2 data combined, for single-track events selected according to the criteria discussed in Sec. 3b2. Asymmetry fit results are shown in Table 5. The asymmetry at $Q^2=5.2{\mathrm{GeV}}^2$ is also shown separately for events with precession angles $\chi <\pi$ and $\chi \ge \pi$, illustrating the expected sign change of the $sin\left(\phi \right)$ term.

Figure 11

Azimuthal angular distributions for FPP1 (left column) and FPP2 (right column) for the $\mathrm{GEp}\text{−}2\gamma$ kinematics, for events selected according to the criteria discussed in Sec. 3b2. The helicity-sum distribution (black filled triangles) cancels the asymmetry due to the proton's transferred polarization. The raw $\phi$ distributions for the $+$ (pink empty circles) and $-$ (green empty squares) helicity states include contributions from the transferred polarization and the false asymmetry. The corrected $\phi$ distributions $2f_{+}/(f_{+}+f_{-})$ (red filled circles) and $2f_{-}/(f_{+}+f_{-})$ (blue filled squares) exhibit pure sinusoidal behavior and include only contributions from the transferred polarization terms, assuming the induced polarization terms are small.

Figure 12

Normalized focal-plane asymmetry $\frac{A_x^{FPP}}{〈A_y〉P_e{P}_{\ell }}$, integrated over the accepted $\vartheta$ range, vs precession angle $\chi \equiv \gamma {\kappa }_p{\theta }_{\mathrm{bend}}$, for all six kinematics. The black solid curve is $-sin\chi$, which is the expected value of the normalized asymmetry in the ideal dipole approximation to the spin transport in the HMS.

Figure 13

$Q^2$ dependence of $R$ (a), $P_t$ (b), and $P_{\ell }$ (c) for the $\mathrm{GEp}\text{−}2\gamma$ kinematics. Uncertainties shown are statistical only. $R$ is compared to $R\left(Q^2\right)$ from Global Fit II described in the appendix, which includes the $\mathrm{GEp}\text{−}2\gamma$ results reported in this work. Transferred polarization components are compared to their Born approximation values, which are also computed in each bin from Global Fit II. Note that for $〈\epsilon 〉=0.153$, $P_t$ and $P_{\ell }$ are equal to their Born approximation values by definition, because this setting is used to determine the analyzing power $A_y$. See text for details.

Figure 14

Analyzing power $A_y$ vs $p_T\equiv p_{p}sin\vartheta$ for the four different $Q^2$ values from GEp-III/2$\gamma$. Data are from both polarimeters combined. Curves are the fits to the data, used to estimate the position and value of the maximum in $A_y\left(p_T\right)$. See text for details.

Figure 15

Maximum ($A_y^{max}$) and average (${\overline{A}}_y$) analyzing power as a function of ${〈p_p〉}^{-1}$, compared to existing data from Refs. [42] (GEp-II) and [54] (Azhgirey 2005). The average $A_y$ values for GEp-III/$\mathrm{GEp}\text{−}2\gamma$ and GEp-II are computed for $0.06\le p_T\left(\text{GeV}\right)\le 1.2$. Curves are linear fits to the data. See text for details.

Figure 16

Transferred polarization components of the inelastic background vs $\delta p_p$ for the $\mathrm{GEp}\text{−}2\gamma$ kinematics. Panel (a) shows the transverse component $P_t^{inel}$, while panel (b) shows the longitudinal component $P_{\ell }^{inel}$. Vertical lines illustrate the approximate final cut regions for $\epsilon =0.153$ (black dotted), $\epsilon =0.638$ (red dot-dashed), and $\epsilon =0.790$ (blue double dot-dashed).

Figure 17

Transferred polarization components of the inelastic background vs $\delta p_p$ for the GEp-III kinematics. Panel (a) shows the transverse component $P_t^{inel}$, while panel (b) shows the longitudinal component $P_{\ell }^{inel}$. Vertical lines illustrate the approximate final cut regions for $Q^2=5.2{\mathrm{GeV}}^2$ (black dotted), $Q^2=6.8{\mathrm{GeV}}^2$ (red dot-dashed), and $Q^2=8.5{\mathrm{GeV}}^2$ (blue double dot-dashed).

Figure 18

Final results of GEp-III (black filled triangles) for ${\mu }_p{G}_E^p/G_M^p$, with selected existing data from cross section and polarization measurements. The error bars shown are statistical. The band below the data shows the final, one-sided systematic uncertainties for GEp-III. The originally published results [31] (black empty triangles) are shown for comparison, offset slightly in $Q^2$ for clarity. The final weighted-average result of $\mathrm{GEp}\text{−}2\gamma$ for $R$ at $Q^2=2.5{\mathrm{GeV}}^2$ is shown as the pink empty star. Existing polarization transfer data are from Refs. [29] (blue filled circles) and [30, 42] (red filled squares). Rosenbluth separation data are from Refs. [17] (green empty circles), [16] (green empty diamonds), and [18] (green filled diamonds).

Figure 19

Final, acceptance-averaged results of the $\mathrm{GEp}\text{−}2\gamma$ experiment, without bin-centering corrections, as a function of $\epsilon$, for the ratio $R\equiv -{\mu }_p\frac{P_t}{P_{\ell }}\sqrt{\frac{\tau (1+\epsilon )}{2\epsilon }}$ (a), and the ratio $P_{\ell }/P_{\ell }^{\mathrm{Born}}$ (b), compared to the originally published results [47] (Meziane11), and the GEp-I result [29] (Punjabi05) at $Q^2=2.47{\mathrm{GeV}}^2$. Error bars on the data points are statistical only. For $R$, the (one-sided) total and point-to-point (relative to $\epsilon =0.79$) systematic uncertainty bands are shown, while only the point-to-point (relative to $〈\epsilon 〉=0.153$) systematic errors are shown for $P_{\ell }/P_{\ell }^{\mathrm{Born}}$ (also one-sided). The originally published points from Ref. [47] have been offset by −0.03 in $\epsilon$ for clarity. Note that $P_{\ell }/P_{\ell }^{\mathrm{Born}}\equiv 1$ at $〈\epsilon 〉=0.153$.

Figure 20

Bin-centering-corrected results for the $\epsilon$ dependence of the ratio $R$ at the common central $Q^2$ of 2.5 ${\mathrm{GeV}}^2$ (red filled squares), with statistical uncertainties only. The red solid line is the linear fit to the corrected data reported in Table 13. The red shaded region indicates the pointwise, $1\sigma$ uncertainty band of the linear fit (68% confidence level). The blue dashed horizontal line is the weighted average of the three measurements assuming no $\epsilon$ dependence of $R$. The blue hatched region indicates the 68% confidence interval ($1\sigma$) for the weighted average. The results of the constant fit are also quoted in Table 13. The GEp-I result [29] (empty triangle), corrected to 2.5 ${\mathrm{GeV}}^2$ using the same approach as the $\mathrm{GEp}\text{−}2\gamma$ data, is shown for comparison.

Figure 21

Several VMD fits compared to the JLab $G_E^p/G_M^p$ data. The solid curve (black) is the fit of Lomon [71], the dashed curve (blue) is that of Iachello et al. [68], and the dotted curve (red) is that of Bijker and Iachello [73]. Data are from Refs. [29] (blue circles), [42] (red squares), and the present work (black triangles for GEp-III and pink star for $\mathrm{GEp}\text{−}2\gamma$). The $\mathrm{GEp}\text{−}2\gamma$ result shown is the weighted average of the three $\epsilon$ points without bin-centering corrections (see Table 13). Figure adapted from Fig. 23 of Ref. [4].

Figure 22

The JLab $G_E^p/G_M^p$ data compared to the results of a selection of constituent quark models. The short dashed curve (blue) is from Boffi et al. [79], the solid (orange) from de Melo et al. [80], the long dashed (magenta) from Gross et al. [81], the dotted (red) from Chung and Coester [82], and the dash-dotted (cyan) from Cardarelli et al. [83]. Data are the same as in Fig. 21. Figure adapted from Fig. 24 of Ref. [4].

Figure 23

Selected data for $Q^2{F}_2^p/F_1^p$ from cross section and polarization observables. Polarization transfer data and symbols are the same as in Fig. 18. Cross section data are from Refs. [16] (filled green circles), [18] (empty green circles), and [17] (empty green triangles). The cross section data show flattening starting at $Q^2\approx 3{\mathrm{GeV}}^2$. However, the polarization transfer data continue to rise up to $Q^2=8.5{\mathrm{GeV}}^2$.

Figure 24

Same data as Fig. 23, plotted as $(Q^2/{ln}^2(Q^2/{\mathrm{\Lambda }}^2))F_2^p/F_1^p$ as proposed by Belitsky et al. [105], for $\mathrm{\Lambda }=0.3$ GeV.

Figure 25

Lattice QCD results for ${\mu }_p{G}_E^p/G_M^p$ obtained using a method based on the Feynman-Hellman theorem [134] (pink filled circles), compared to polarization transfer data from Refs. [28, 29] (blue empty circles), [30, 42] (red empty squares), the final GEp-III data (black empty triangles), and the weighted-average of the final $\mathrm{GEp}\text{−}2\gamma$ data (pink empty star). The solid curve is the fit to the data using Eq. (44) from Ref. [4], and has not been refitted using the final results reported in this work.

Figure 26

Comparison of polarization transfer data for ${\mu }_p{G}_E^p/G_M^p$ with the DSE-based calculation of Ref. [135].

Figure 27

Final, bin-centering-corrected results of $\mathrm{GEp}\text{−}2\gamma$ for the ratio $R$, compared to several theoretical predictions for the $\epsilon$ dependence of $R$ at $Q^2=2.5{\mathrm{GeV}}^2$ due to TPEX corrections. The blue solid horizontal line is the weighted average of the corrected data (see Table 13 and Fig. 20). Curves show calculations from Borisyuk et al. [147] (cyan dashed), Blunden et al. [148] [green dot-dashed ($N$ only) and green dotted ($N+\mathrm{\Delta }$)], Bystritskiy et al. [143] (pink dot-long dashed), Afanasev et al. [45] (black solid), and Kivel et al. [149] [red dotted (BLW) and red dashed (COZ)]. Note that because the ratio $R$ is proportional to the Born value of ${\mu }_p{G}_E^p/G_M^p$, each curve can be renormalized, in principle, by an overall multiplicative factor. See text for details.

Figure 28

Global fit results for the proton form factor ratio ${\mu }_p{G}_E^p/G_M^p$, compared to selected data from measurements of cross sections and polarization observables, including the final results of GEp-III (black solid triangles) and $\mathrm{GEp}\text{−}2\gamma$ (pink empty star, weighted average). Other polarization data are from Refs. [28, 29] (Punjabi05), [30, 42] (Puckett12), [166] (Crawford07), [167] (Ron11), [168] (Zhan11), and [169] (Paolone11). Rosenbluth separation data are from Refs. [16] (Andivahis94), [17] (Christy04), and [18] (Qattan05). Global Fit I includes the data from Refs. [167, 168, 169], while excluding the data from Ref. [166] and the two lowest $Q^2$ points from Ref. [29]. Global Fit II excludes the data from Refs. [167, 168, 169]. Shaded regions indicate $1\sigma$, pointwise uncertainty bands.