Measurement of the Transverse Beam Spin Asymmetry in Elastic Electron-Proton Scattering and the Inelastic Contribution to the Imaginary Part of the Two-Photon Exchange Amplitude

We report on a measurement of the asymmetry in the scattering of transversely polarized electrons off unpolarized protons, $A_{\perp }$, at two $Q^2$ values of 0.106 and $0.230(\mathrm{G}\mathrm{e}\mathrm{V}/c{)}^2$ and a scattering angle of $30°<{\theta }_e<40°$. The measured transverse asymmetries are $A_{\perp }\mathbf{(}{Q}^2=0.106(\mathrm{G}\mathrm{e}\mathrm{V}/c{)}^2\mathbf{)}=(-8.59\pm {0.89}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}\pm {0.75}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}})\times {10}^{-6}$ and $A_{\perp }\mathbf{(}{Q}^2=0.230(\mathrm{G}\mathrm{e}\mathrm{V}/c{)}^2\mathbf{)}=(-8.52\pm {2.31}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}\pm {0.87}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}})\times {10}^{-6}$. The first errors denote the statistical error and the second the systematic uncertainties. From comparison with theoretical estimates of $A_{\perp }$ we conclude that $\pi N$-intermediate states give a substantial contribution to the imaginary part of the two-photon amplitude. There is no obvious reason why this should be different for the real part of the two-photon amplitude, which enters into the radiative corrections for the Rosenbluth separation measurements of the electric form factor of the proton.

Figure 1

The two-photon exchange diagram. The filled blob X represents the response of the nucleon to the scattering of the virtual photon.

Figure 2

The momentum vector ${\overrightarrow{k}}_e$ is pointing here out of the paper plane. The momentum vector ${\overrightarrow{k}}_{\mathrm{o}\mathrm{u}\mathrm{t}}$ of the outgoing electron can take all possible ${\varphi }_e$ values. Both together define the coordinate system according to the Madison convention [26] with ${\overrightarrow{S}}_n=({\overrightarrow{k}}_e\times {\overrightarrow{k}}_{\mathrm{o}\mathrm{u}\mathrm{t}})/|{\overrightarrow{k}}_e\times {\overrightarrow{k}}_{\mathrm{o}\mathrm{u}\mathrm{t}}|$. The direction of the electron polarization vector ${\overrightarrow{P}}_e$ for the + helicity state is indicated by the arrow. ${\varphi }_e$ and ${\varphi }_{{\overrightarrow{P}}_e}$ are counted as indicated. The elastic scattered electrons are detected in the ${\varphi }_e$-symmetric ${\mathrm{P}\mathrm{b}\mathrm{F}}_2$ calorimeter of the A4 experiment. For the extraction of $A_{\perp }^m$, the detector has been divided into eight sectors as indicated in the figure.

Figure 3

The upper plot shows the result of our measurements of $A_{\perp }$ for a beam energy of 569.31 MeV and at the lower plot for the beam energy of 855.15 MeV. The asymmetry is plotted as a function of the laboratory angle ${\varphi }_e$ as defined in Fig. 2. The 1022 ${\mathrm{P}\mathrm{b}\mathrm{F}}_2$ detectors of the calorimeter have been divided into eight subsets (sectors 1 to 8), each spanning an angular range of $\approx 45°$ in ${\varphi }_e$. For the lower plot only 756 channels of the ${\mathrm{P}\mathrm{b}\mathrm{F}}_2$ detector had been installed (sectors 1, 2, 5, 6, and part of sector 8).

Figure 4

The results of two model calculations [11, 29] are shown with the results of our measurements of $A_{\perp }$ (see text for explanation).