Engineering Run Data Systematic Error

Overview

The formulae for converting measured asymmetries to physics asymmetries for the forward angle measurements A_F are shown in Figure 1.

Figure 1: Definition of terms relevant to calculating the physics asymmetry *A_F*.

I assume that elastic and inelastic asymmetries can be broken down into three parts as shown in Figure 2.

Figure 2: Composition of the corrected asymmetry *A^C*.

It is a straightforward exercise to derive the final error on A_F from these equations. The statistical error on A_F comes from the statistical error on A_e^C. This error is assigned to A_e^R and additional errors due to linear regression (LR) and dead time (DT) corrections assigned to the other terms of A^C.

Linear Regression Corrections

The data reveal that the correction to the asymmetry is quite small for both sets of electronics (see the last page). The maximum correction is about 8.5% for two out of 28 detector-electronics data channels. The average is closer to 1-2%. For the purposes of the engineering run data it is probably sufficient to assign an error equal to 100% of the average correction; say 1-2% of the corrected asymmetry in each data channel. Here are the raw data from the analysis database, source code used to analyze the data, and final helicity-averaged (IHWP=Out - IHWP=In) results.

Only runs in Julie's very good runs list were used in this analysis. Further cuts were applied such that runs with the following characteristics were removed:

IHWP in indeterminate state (two runs),
position differences greater than 50 nm,
beam current less than 35 microAmps,
and charge asymmetry greater than 10 ppm.

This left a total of 33 runs out of 122 in the list.

It is interesting to note how stable the corrections are when the charge asymmetry is small. The average charge asymmetry for the IHWP OUT state was 0.31 ± 1.07 ppm whereas it was -3.10 ± 1.11 for the IN state. The bottom graphs on pages 1 and 2 of the PostScript file show the correction as a function of detector number for each electronics type sorted by IHWP state. The IHWP IN data are clearly more erratic. The IHWP OUT data appear relatively stable at about -0.3 ppm for both sets of electronics.

Beam Polarization

There is only one set of beam polarization measurements for the entire data taken during January 2003. The results show a measured polarization of 76.68 ± 0.358 %. The relative statistical error is 0.5%. The relative systematic uncertainty associated with the polarimeter is quoted as 0.47% by the maker but published results have not quoted anything better than about 1.5%. Reality probably lies somewhere in between. The fact that there is only one set of measurements for a couple of weeks worth of asymmetry running is another source of error. Results from all the polarimetry performed during the 2002-2003 engineering run indicates that variations in the beam polarization over time do not appear to be significant at the level of the statistics of the measurement. To play it safe let us set the systematic error on the polarization measurement due to polarimeter systematics and potential beam polarization variation at 2%. Combined with the statistical error on the measurement we get a total relative systematic error on the beam polarization of 2.1%.

Dilution Factor

Details of the extraction of the dilution factor and inelastic asymmetry are discussed in a pair of reports (1, 2). A relative error of 20% is assigned to the extraction of R in the first paper. This report shows the dilution factor as a function of detector number for the three different extraction methods discussed in the report. It varies from 10% in lower detectors to 30% in higher detectors.

Things to do:

Obtain table of R values.

Summary

There is still work to be done on this subject but the upshot is that we are overwhelmingly statistics limited at this point. For a modest choice of elastic and inelastic asymmetries (-12 ppm and -17 ppm) and dilution factor (0.15) approximately 95% of the error comes from the statistics on the determination of the elastic asymmetry. This estimation uses the errors listed in the error table. The relative error on the physics asymmetry A_F approaches 50% for this choice of values. If there were an absolute error of 2 ppm on the dead time correction, then the contribution of the statistical error would fall to about 85% and the absolute final error would increase slightly.

There are essentially three tasks remaining to nail this error down:

assign an error due to the dead time correction,
update inelastic asymmetries and errors, and
obtain table of dilution factors.

Table of Errors
Source	Form	Relative Error	Absolute Error	Comments
LR Correction	dA_e^LR/A_e^C	2.0%	approximately 0.24 ppm	Relative error is pretty firm.
Beam Polarization	dP_B/P_B	2.1%	1.61%	Firm
Dilution Factor	dR/R	20%	approximately 0.03	Relative error is pretty firm.
Dead Time Correction	dA_e^DT/A_e^C	??
Inelastic Asymmetry	dA_ie^C/A_ie^C	approximately 29% (??)	approximately 5 ppm (??)	Need to get these numbers updated for 33 runs meeting stringent beam property requirements.
Elastic Asymmetry (Statistical Error)	dA_e^R/A_e^R	approximately 38%	approximately 4.5 ppm	Based on 33 runs meeting stringent beam property requirements.

An error budget (PostScript ) reveals that the final error for a mid-range detector is strongly statistics dominated.