Systematic Uncertainties

Because of the high precision required for this measurement, we have to insure that we take into account many small corrections that are often ignored. Dead time corrections, bin centering, and uncertainties in the kinematic quantities all have to be corrected precisely.

Computer dead time corrections are measured in the standard data acquisition system in Hall A. The number of triggers generated and the number of events actually written to tape are recorded, and the cross section is corrected by the fraction of events sampled, with a very small associated uncertainty. Electronic dead time cannot be measured in the same direct way. For our experiment, we estimate the total trigger rate of the hodoscopes to stay below 200-300kHz. The current electronics use gate widths of $\sim$200 ns, which would lead to dead times of up to 4-6%. Recent modifications to the electronics allow a measurement of the electronic dead time throughout the run. In addition, there is a plan to change the electronics so that the trigger signals will use 40 ns gate widths, which will reduce the maximum electronic dead time to $\sim$1%, and the typical uncertainty on this correction should be <0.1%.

Our estimates assume that we will accept protons over a $\pm$10mr angular range in scattering angle. Because the cross section is not constant over this range, we will need to apply a bin centering correction to extract the cross section at the central scattering angle. The cross sections can vary significantly over the measured range, but the bin centering correction is quite small because over the 20mr acceptance, the cross section is nearly linear and the acceptance is flat. Simulations of elastic scattering at each of the kinematics including both the cross section variation and geometric acceptance show no significant bin centering corrections, and indicate that the uncertainty in the bin centering correction is <0.1%.

The largest systematic uncertainties come from uncertainty in the scattering kinematics. For a beam energy uncertainty of 0.1%, the cross sections vary by about 0.5%. However, the high and the low Q² measurements at each beam energy have a similar energy dependence (table 1), and so the effect of a 0.1% energy uncertainty is only $\sim$0.1% in the final measured ratio. In addition, if the errors at different energies are correlated (i.e. if linac scaling holds), then there will be additional cancellation between the forwards and backwards measurements. Uncertainty in the scattering angle will have a larger effect on the extracted ratio. There are two contributors to the scattering angle uncertainty: The incoming beam angle, and the scattered proton angle. There are two BPMs just upstream of the the target. They can be surveyed (relative to the nominal beamline) to within 0.5mm, and are separated by 6 meters, giving an uncertainty in the beam angle of 0.12mr. However, the change in angle from run to run (measured with the BPM) can be measured much more accurately, so the 0.12mr offset will be the same for both the forwards and backwards angle runs at each Q² point, and the effect of the angle offset will partially cancel between the high and low epsilon points. Because of this, the 0.12mr offset corresponds to an uncertainty in the final ratio of <0.2%.

The uncertainty in the angle of the scattered proton also breaks down into an overall offset (identical for both forward and backwards angles) and an offset that can vary randomly as the spectrometer angle is changed. The overall offset comes from offsets in the VDC positions relative to the optical axis and from errors in the scattering angle reconstruction. Because we will define the scattering angle acceptance with software cuts rather than with a collimator, an error in the angle reconstruction will modify the size and central angle for the defined angular acceptance. We will use the same cuts for all data and so the uncertainty in the total solid angle will largely cancel, but there can still be an overall offset in the central scattering angle of the software restricted window. Offsets that vary randomly with changing scattering angle come from any shifts of the VDC position during the run, as well as uncertainty in the pointing of the spectrometer. As we will survey the HRS pointing at each setting, the pointing uncertainty will be relatively small. While the survey gives the mechanical axis (as determined by the magnet positions) rather than the true optical axis, the difference is the same at all angles, and is taken as part of the constant offset in the angle reconstruction uncertainty. We estimate a random uncertainty of 0.13mr, and a constant offset uncertainty of 0.20mr. A 0.20mr offset gives an uncertainty in the extracted ratios of $\sim$0.2%, while an 0.13mr random uncertainty contributes $\sim$0.24% to the uncertainty.

Our total uncertainty in the scattering angle (combining the fixed offsets, random offsets, and beam angle offsets) is 0.27mr. This is slightly smaller than the quoted uncertainties used by previous Hall A measurements, but is consistent with the assumptions used. The typical scattering angle uncertainty used in the analysis of the completed Hall A analyses has been 0.3-1.0mr (usually 0.3-0.5mr). Experiments that have done complete surveys of the spectrometer pointing and used kinematic checks of the angle have routinely achieved 0.3mr uncertainty, but we have an additional advantage in that we do not limit the solid angle with the collimator. As it is located $\sim$1 meter from the target, even a small offset of the collimator from the optical axis will offset the central angle of the acceptance. If survey or data can determine its position to 0.1-0.2mm, this is still a 0.10-0.20mr contribution to the uncertainty in the scattering angle, in addition to the uncertainty coming from the spectrometer pointing and incoming beam angle. Because we use software cuts to define the scattering angle acceptance, uncertainties in the central angle come from the HRS pointing and uncertainties in the VDC position, which have a much smaller effect than a comparable uncertainty in the collimator position. With careful surveys at each point, monitoring of the HRS pointing with the LVDTs (linear voltage differential transformers), and several kinematic checks on the spectrometer angle (as discussed in the run plan), we should be able to achieve the assumed angular uncertainty. Table 2 shows the projected uncertainty for the measurement at each value of Q².

 

$$\delta \delta \delta$$
Source Size

(Q²=1.45) (Q²=3.20) (Q²=4.90)

Statistics 0.1-0.3% 0.32% 0.39% 0.45%

Beam Energy 0.05% 0.06% 0.04% 0.05%

Beam Angle 0.12 mr 0.18% 0.13% 0.11%

$$\theta_p$$ 0.13 mr 0.22% 0.27% 0.27%

$$\theta_p$$ 0.20 mr 0.22% 0.17% 0.12%

Bin Centering 0.06% 0.06% 0.06%

Dead Time 0.10% 0.10% 0.10%

Dummy Subtraction 0.1% 0.20% 0.20% 0.20%

Radiative Corrections 2.5% 0.22% 0.34% 0.28%

Q²=0.5 G_E/G_M value 2.0% 0.27% 0.05% 0.02%

*Acceptance 0.1% 0.50% 0.20% 0.20%

*Efficiency 0.1% 0.20% 0.20% 0.20%

*Luminosity 0.1% 0.20% 0.20% 0.20%

Total 0.86% 0.75% 0.75%

Error on G_E/G_M 2.1% 3.9% 8.2%