G0 Collaboration

Two-photon exchange (TPE) is a higher-order radiative effect that may explain the discrepancy between different methods of measuring the ratio of the electric and magnetic form factors of the proton ($G_{E}$ and $G_{M}$) Guichon and Vanderhaeghen (2003). Model calculations have shown that including the real part of TPE effects brings the unpolarized cross-section measurements into closer agreement with polarization transfer measurements Arrington et al. (2007). The contribution of higher-order processes, such as TPE effects, must be understood as the precision of electron scattering experiments continues to improve Tjon et al. (2009). The single-spin asymmetry in electron-nucleon scattering (the left-right analyzing power measured in the p($\vec{e}$,e)p reaction, with the sign chosen as in the Madison convention Barschall and Haeberli (1970)) as measured in this work is a parity-conserving asymmetry which gives access to the imaginary part of TPE, providing a valuable test of the theoretical framework for such higher-order processes. It can be measured with either the target or the beam polarized perpendicular (transverse) to the scattering plane Pasquini and Vanderhaeghen (2004).

The asymmetry arises due to the imaginary part of the interference of the two-photon exchange amplitude with that of single-photon exchange. The beam-normal single-spin (BNSSA) asymmetry, $B_{n}$, can be written

where Im denotes the imaginary part, $\sigma_{\uparrow}$ and $\sigma_{\downarrow}$ are the cross-sections for the beam polarized parallel or antiparallel to the normal to the scattering plane, $\hat{n}=\frac{\vec{k}\times\vec{k^{\prime}}}{\left|\vec{k}\times\vec{k^{\prime}}\right|}$ where $\vec{k}$ and $\vec{k^{\prime}}$ are the momenta of the incoming and outgoing electron, and $M_{\gamma}$ and $\left|M_{\gamma\gamma}\right|$ are the amplitudes for single- and two- photon exchange. The BNSSA is linear in $\alpha_{EM}$, the electromagnetic (EM) coupling constant, because the single-spin asymmetry is zero in the Born approximation. In addition, the BNSSA has an order $m_{e}/E$ suppression relative to a target-normal single-spin asymmetry, where $m_{e}$ is the mass of the electron and E is the energy, because the polarized electron is ultra-relativistic. The BNSSA are expected to be on the order of $10^{-6}-10^{-5}$ Pasquini and Vanderhaeghen (2004).

The techniques developed for measuring the small parity-violating (PV) asymmetries ($\sim$ $10^{-5}$), using an electron beam spin polarized in the same direction as the momentum (longitudinal), are also useful for measuring the BNSSA. Precision electroweak electron scattering experiments, such as HAPPEx Kaufman (2007), PVA4 Maas et al. (2005), E158 Vanderhaeghen (2007) and G0Armstrong et al. (2007) have measured the BNSSA at various kinematic settings at forward angles, and backward angle measurements have been made in PVA4 Capozza (2007); Baunack (2010) (preliminary), SAMPLE Wells et al. (2001), and this work. These experiments deliberately polarize the beam in the transverse direction in order to estimate the systematic uncertainty on the PV asymmetries that could be caused by a residual transverse component of the beam and also directly provide access to the imaginary part of the TPE amplitude.

Calculations of the BNSSA are sensitive to the treatment of the intermediate hadronic state in the two-photon exchange amplitude, and different models have been used in the different kinematic regimes. The cross section, $\sigma$, can be parameterized using 6 complex invariant amplitudes ($\tilde{G}_{E}(\nu,Q^{2})$, $\tilde{G}_{M}(\nu,Q^{2})$ and $\tilde{F}_{i}(\nu,Q^{2})$ where $i=3-6$) which are functions of the four-momentum transfer $Q^{2}$ and the Lorentz-invariant, $\nu=(s-u)/4$, where $s$ and $u$ are the standard Mandelstam variables Vanderhaeghen et al. (2004). In the Born approximation, two of the complex form factors reduce to the familiar electric and magnetic form factors $\tilde{G}_{E}(\nu,Q^{2})\rightarrow G_{E}(Q^{2})$, $\tilde{G}_{M}(\nu,Q^{2})\rightarrow G_{M}(Q^{2})$ while the remaining form factors, which originate from processes involving the exchange of at least two photons, vanish ($\tilde{F}_{i}(\nu,Q^{2})\rightarrow 0$). The most relevant models for the backward-angle measurements, including this work, are those which model the nucleon intermediate states, $X$, including elastic ($X=N$) and inelastic ($X=\pi N$) states.

The results presented here are from backward angle ($\theta_{lab}$=108^∘) measurements of the BNSSA in elastic electron scattering on hydrogen and quasi-elastic electron scattering on deuterium, taken as part of the G0 experiment Androić et al. (2011, 2010). Two incident beam energies, 362 MeV and 687 MeV, were used for each target, corresponding to $Q^{2}\sim$ 0.22 GeV${}^{2}/c^{2}$ and 0.63 GeV${}^{2}/c^{2}$, respectively. Beam currents ranged from 20 $\mu$A to 60 $\mu$A, for a total of about 50 hours of beam. The experimental apparatus consisted of a 20 cm aluminum target cell which was used to hold either liquid hydrogen or deuterium, and a toroidal-field magnetic spectrometer which was used to separate the (quasi-)elastically and inelastically scattered electrons. The apparatus had eight-fold azimuthal symmetry around the beamline, with three sets of main detectors in each octant. Focal plane detectors (FPDs), consisting of 14 scintillator arcs in each octant, and cryostat exit detectors (CEDs), an array of 9 scintillator paddles, were used for kinematic separation by looking at the coincidence of individual CED and FPD pairs. An aerogel Čerenkov detector in each octant was used to distinguish electrons and pions. A set of synthetic quartz Čerenkov luminosity (LUMI) monitors placed symmetrically around the beamline at low angles (high incident rate) was used as a beam diagnostic, as will be discussed below.

The polarized electrons were produced from circularly polarized light incident on a strained GaAs photocathode Sinclair et al. (2007). Rapid helicity reversal, at 30 Hz using a Pockels cell, ensured that the conditions for which an asymmetry is measured do not change. A slow helicity reversal ($\sim$ several days), using an insertable half-wave plate, was also employed to reduce helicity-correlated effects. A Wien filter was used to produce a transversely polarized electron beam. The electron polarization vector was rotated by 90^∘ to beam left from what it would be for longitudinal polarization (as determined by Moller polarimeter measurements at various Wien angle settings). The magnitude of the polarization of the beam was 85.8$\%$ with uncertainties of $\pm$2.0$\%$ and $\pm$1.4$\%$ for the 362 MeV and 687 MeV energies, respectively.

The direction of the beam polarization was flipped in a quartet pattern ($+--+$ or $-++-$) with plus (minus) corresponding to beam left (right) looking downstream. The measured asymmetry in each octant was formed from the difference in normalized yields, $Y$, for the plus and minus states in each quartet over the sum

where $\vec{p}_{e}$ is the beam polarization. The normal to the scattering plane, $\hat{n}$, is transverse to the beam and to the scattered electron momentum, thus the measured asymmetry varies as a function of octant. To extract the value of the transverse asymmetry, we fit the data as a sinusoidal function of azimuthal scattering angle, $\phi$, and obtained the amplitude, $B_{n}$, corrected for the magnitude of the beam polarization, $\left|\vec{p}_{e}\right|$.

As the beam passes through the magnetic elements of the accelerator, the electron spin precesses in the horizontal plane. The out-of-plane component of the polarization induced by these elements is small and thus is not expected to contribute significantly to the phase of the asymmetry. The phases as determined from the main detectors are consistent from dataset to dataset, though with large uncertainties, especially in the high energy data. To confirm the stability of the out-of-plane phase from dataset to dataset, we used the high precision LUMI data, which are dominated by Møller (electron-electron) scattering. We discovered that although the LUMI phases are consistent from dataset to dataset, they were not consistent with the phases determined from the main detector data, indicating that there was a geometrical offset between the two sets of detectors Mammei (2010). In the final fits to the main detector data (see Figure 1) only the amplitude was allowed to vary and the phases were fixed to the weighted average of the phases of the main detectors at 362 MeV, or $-2.3^{\circ}\pm 1.6^{\circ}$, where the data are more precise. The results are summarized in Table 1. The uncertainties in the 362 MeV data are small compared to the size of the asymmetries and the quality of the data is apparent in the plots. The hydrogen data at 362 MeV were missing an electronics channel in octant 2, so that octant has been omitted from the final results. At 687 MeV both the expected values of the asymmetries and the rates were smaller, and there were very few data taken with a deuterium target, resulting in much larger relative uncertainties. Both the hydrogen and deuterium asymmetry at 687 MeV are consistent with zero.

The transverse data have been fully corrected for electronics effects (e.g. deadtime), background asymmetries and helicity-correlated beam parameters. The corrections to the data are performed on an octant-by-octant basis in the same way as in the longitudinal data Androić et al. (2010). The uncertainties associated with each correction (see Table 2) are calculated as the quadrature difference in the uncertainties on the amplitudes of the fits before and after a given correction, except in the case of the linear regression correction in the 687 MeV data, which resulted in a smaller fit uncertainty after the correction. In this case, the uncertainty is approximated by scaling the uncertainties for the 362 MeV data by the square root of the ratios of the number of quartets in each dataset. As there is yet no prescription to calculate the radiative effects for the beam-normal single-spin asymmetry, we have not made any corrections for standard radiative (real photon) effects Tsai (1971).

Our two measurements of the BNSSA for scattering from the proton are shown on a plot with the preliminary PVA4 Capozza (2007); Baunack (2010) and the SAMPLE Wells et al. (2001) backward angle measurements (see Figure 2). The data are shown in comparison to the theoretical prediction Pasquini and Vanderhaeghen (2004). For the first time we have extracted values for the BNSSA of the neutron. In the static approximation, the asymmetry for deuterium is simply the cross-section-weighted average asymmetry for the proton and the neutron

where $\sigma_{p,n}$ is the proton ($p$) or neutron ($n$) cross-section, and $B_{n}^{n,p,d}$ is the measured BNSSA for a neutron ($n$), proton ($p$) or deuteron ($d$) target. Estimates of the proton and neutron cross-sections and the extracted BNSSA for the neutron are given in Table 3 and compared to the theory G0w . The cross-sections were calculated using estimates of the nucleon EM form factors with a relative uncertainty of 5%.

The estimate for the neutron asymmetry for each energy is made by solving for $B_{n}^{n}$ in Eq. 3. The estimate of the neutron BNSSA at 687 MeV has very large uncertainties which prevent us from drawing any conclusions. At 362 MeV, the resulting neutron asymmetry is smaller in magnitude than the proton asymmetry and opposite in sign (positive). In the resonance region the elastic contribution is calculated using the electromagnetic form factors at the vertices, while the contribution from $\pi$N intermediate states depends on both resonant and nonresonant invariant amplitudes for $\pi$N intermediate states, which are taken from phenomenological analysis fitted to available experimental data Pasquini and Vanderhaeghen (2004); Drechsel et al. (1999). The asymmetry at the measured values of $Q^{2}$ is dominated by the term proportional to $G_{M}$ which changes sign between proton and neutron. Furthermore the larger magnitude of the neutron asymmetry for smaller energies follows from the dominance of the quasi-real Compton contribution. It corresponds to the two exchanged photons being quasi-real and the invariant mass of the hadronic intermediate state approaching the value of the e-N center of mass energy. In Fig. 2, the behavior of the proton asymmetry is driven by the increasing contribution of the quasi-real Compton scattering up to energy $E_{e}\approx 0.360$ GeV. At higher energy the resonant structure of the pion electroproduction amplitudes comes into play with a contribution of opposite sign, which leads to a smaller asymmetry in absolute value. In order to make a better estimate of the neutron asymmetry it will be necessary to use a more sophisticated deuterium model, similar to the calculation of Schiavilla Schiavilla (2006); Schiavilla et al. (2004) for the estimate of the longitudinal asymmetries.

Measurements of the BNSSA in the resonance region are valuable tests of the theoretical framework which calculates the radiative corrections for precision electron scattering experiments. This work doubles the world dataset for the BNSSA in elastic electron-proton scattering at backward angles. More importantly, the addition of these data allows us to span the range of energies up to 1 GeV, including the value at 362 MeV which is at the estimated peak of the theoretical prediction. In addition, asymmetries from quasi-elastic deuteron scattering have been used to provide the first estimate of the BNSSA for the neutron, which is in agreement with the predicted value at 362 MeV. The agreement between the theoretical predictions and the measured values clearly shows that it is necessary to take into account the $\pi$N intermediate state contributions in the calculation of the hadronic intermediate state when estimating the effects of the TPE contributions.