Kinematics

The proposed kinematics are shown in table 1.

   

$$\epsilon \theta_p \sigma_p^{SLAC} \Delta \sigma \Delta \sigma$$ E_e Q² Proton Proton

(GeV) (GeV)² (deg) K.E.(GeV) Momentum (cm²/msr) %/deg %/(%E_e)

1.162 0.50 .762 50.407 0.266 0.756 4.540e-32 13.26 4.22

1.162 1.45 .081 12.540 0.773 1.431 1.640e-33  3.65 4.00

2.262 0.50 .939 60.075 0.266 0.756 6.373e-32 18.56 4.67

2.262 1.45 .746 40.175 0.773 1.431 2.334e-33 14.68 5.36

2.262 3.20 .131 12.525 1.705 2.471 1.227e-34  5.51 4.43

3.362 0.50 .973 63.191 0.266 0.756 7.238e-32 20.93 4.83

3.362 3.20 .610 28.048 1.705 2.471 1.429e-34 14.08 5.48

3.362 4.90 .181 12.664 2.611 3.423 2.617e-35  7.34 4.58

5.562 0.50 .990 65.664 0.266 0.756 8.050e-32 23.16 4.97

5.562 1.45 .963 50.864 0.773 1.431 3.227e-33 20.71 6.13

5.562 3.20 .871 36.255 1.705 2.471 1.751e-34 19.17 6.30

5.562 4.90 .722 26.942 2.611 3.423 3.143e-35 17.48 5.91

$$\epsilon \theta_e \sigma_e^{Hall A} \sigma_e^{SLAC} \Delta \sigma \Delta \sigma$$ E_e Q² Electron

$$\sigma_e^{SLAC}$$ (GeV) (GeV)² (deg) K.E.(GeV) (cm²/msr) %/deg %/(%E_e)

1.162 0.50 .762  40.557 0.896 0.973 4.064e-32 14.70  8.69

1.162 1.45 .081 127.062 0.389 0.991 1.185e-34  2.40 10.07

2.262 0.50 .939  19.158 1.996 0.969 2.217e-31 33.19  8.98

2.262 1.45 .746  38.299 1.489 0.934 1.933e-33 18.20 11.77

2.262 3.20 .131 105.695 0.557 0.988 6.077e-36  3.62 11.49

3.362 0.50 .973  12.584 3.096 0.969 5.479e-31 51.20  9.05

3.362 3.20 .610  44.541 1.657 0.949 5.667e-35 15.20 13.06

3.362 4.90 .181  88.315 0.751 0.987* 1.228e-36  4.92 12.08

5.562 0.50 .990   7.470 5.296 0.969 1.629e-30 87.01  9.09

5.562 1.45 .963  13.398 4.789 0.919 2.282e-32 59.89 12.59

5.562 3.20 .871  22.269 3.857 0.930 3.438e-34 37.05 14.24

5.562 4.90 .722  31.710 2.951 0.950* 1.751e-35 23.69 14.20 4r*assumes G_E/G_M = 0.45 at Q²=4.90

For this measurement, proton detection has several advantages over electron detection. Protons at moderately large angles correspond to forward angle electrons. Detecting the proton allows us to go to lower values of electron scattering angle, down to $\sim7$$^\circ$ for this proposal, than would normally be possible. It also reduces the effect of uncertainty in the measured scattering angle. The cross section for forward angle electrons varies rapidly with scattering angle, while the cross section dependence for the corresponding protons is smaller by a factor of 2-3 for the kinematics where this is a dominant source of uncertainty (as seen in Table 1). The reverse is true for the backwards angle electrons: the angular variation of the cross section is greater for the protons. However, it is a much smaller effect than for the forwards angle electrons, and therefore the increased kinematical dependence does not increase the overall uncertainty in the extracted form factors. The limitation for backwards angle electrons is the reduced cross section within the angular acceptance of the spectrometer. Again, measuring the proton leads to a significant improvement, as the forward angle protons are in a narrow angular range compared to the corresponding backwards angle electrons. Finally, detection of the proton reduces the cross section variation with beam energy for all kinematics.

In addition to the kinematic advantages of detecting the proton, the systematic uncertainties in the Rosenbluth separation are smaller when the proton is measured. This is due to the fact that the proton momentum is the same for all $\epsilon$ values at a given Q². Thus the magnets are not set to different currents when $\epsilon$ is changed (which could lead to small modifications to the optics), and momentum dependence corrections such as detector efficiency and multiple scattering will be nearly identical for the forwards and backwards angle measurements. While there will be small angle-dependent differences in multiple scattering due to the target geometry, the proton momentum will be identical, making the differences smaller than the would be if the electron were detected. The effect of any rate dependent efficiencies on the separation will also be reduced because the difference in cross sections between forward and backwards angles is much smaller than in the case where the electron is detected. Finally, because the scattered electron is not detected, the radiative corrections are significantly smaller (on average by a factor of two).

There will be corrections to the absolute cross section that are larger for measured protons, but for the most part these cancel in the ratios. Proton absorption in the target and spectrometer leads to a correction of a few percent. However, the absorption in the spectrometer will completely cancel when comparing the different $\epsilon$ values, as the proton momentum is identical at all kinematics. There will be a difference absorption in the target because the amount of target material seen by the outgoing proton depends on the scattering angle. For the standard 'beer can' target (4cm length, 6.35cm diameter), the path length through the target varies between 2cm and 3.76cm, which gives a maximum difference of 0.26% in the proton absorption. This will be even smaller if improved target cells (with a smaller diameter) or 'tuna can' cells are used. The target geometry is taken into account in the simulation, and this small difference in absorption can be taken into account in the analysis, with a negligible uncertainty in the final result.

Protons are not always stopped by the HRS collimator, so one can not rely on the collimator to define the solid angle for the measurement. We will define the solid angle using cuts on the reconstructed scattering angles, in a region where the HRS has nearly complete acceptance. While any error in the angular reconstruction will lead to an uncertainty in the absolute solid angle, identical cuts will be used at forwards and backwards angle, and so much of the uncertainty in the solid angle will cancel. As the spectrometer angle changes, the length of the target as seen by the HRS also changes. Thus, any target position dependence of the reconstruction can lead to a change in solid angle. We will measure this position dependence with sieve slit and elastic runs on a 15cm target and the variable z-position optics targets, and use this to correct for target length variation with scattering angle. The high Q² data is taken at smaller scattering angles, where the target length dependence should not be a large problem. The Q²=0.5 GeV² normalization points are taken at larger angles but the angle difference between the high and low $\epsilon$ points are small. The change in target length as seen by the HRS is <20% for the Q²=1.45 point, and $\mathrel{\raise.3ex\hbox{$<$}\mkern-14mu \lower0.6ex\hbox{$\sim$}}$5% for the other kinematics). The Q²=1.45 GeV² point does have a large change in scattering angle (from 12.5$^\circ$ to 51$^\circ$), and we will have to rely on measurement of the solid angle dependence to determine the size of the correction. While we will have to measure the position dependence before we know the size of the correction (and uncertainty), we will assume an additional uncertainty of 0.5% to the acceptance for the Q²=1.45 GeV² point to take into account the larger potential solid angle variation. Once we have measured the solid angle dependence on target position, we may well be able to correct to better than 0.5%, but we would like to note that even if the uncertainty turns out to be twice as large, the uncertainty on the extracted value of G_E/G_M at this Q² is still $\pm$0.03, smaller than the uncertainty in the polarization transfer measurement, and 6 standard deviations from unity if the Hall A result is correct.