When combining Run5 and Run6 physics results,
the correlation in uncertainties between Run5 and Run6 should be considered.
Below we reveal the maximal common part and uncorrelated part in global
systematic uncertainties (in %) in Run5 and Run6 polarization measurements.
"Maximal common part" means that this error can be considered neither
100% correlated nor uncorrelated, so the recommendation is to take
the worst case: for example, when combining Run5 and Run6 measurements,
the maximal error in the result will be obtained if consider
"maximal common part" as 100% correlated between Run5 and Run6;
when comparing Run5 and Run6 results (making difference or ratio)
the maximal error in the result will be obtained if consider
"maximal common part" as uncorrelated between Run5 and Run6
(we do not consider negatively correlated errors).
Uncorrel Common Total
Run5 Blue 5.4 2.4 5.9
Run6 Blue 4.0 2.4 4.7
Run5 Yell 5.7 2.5 6.2
Run6 Yell 4.1 2.5 4.8
Run5 Blue*Yell 8.1 4.8 9.4
Run6 Blue*Yell 6.8 4.8 8.3
Total errors are taken from
http://www4.rcf.bnl.gov/~cnipol/pubdocs/Run05Offline/NOTE 2005 Polarizations for RHIC.doc
http://www4.rcf.bnl.gov/~cnipol/pubdocs/Run06Offline/NOTE_2006_Polarizations_RHIC.txt
The only source of correlated uncertainties in Run5 and Run6 polarization
measurements is systematic errors in HJet measurements, which include
1. Molecular hydrogen background
2. "Other" background (inelastic background, background from the other beam, etc.)
For molecular (unpolarized) background in polarized atomic hydrogen jet we
used the same correction and uncertainty (2%) in all our measurements
(which was defined in Run4), so it can be considered as a common error
in polarization measurements in Runs 4,5,6, for both blue and yellow beams.
The effect for "other" background has not been corrected for
(the effect was considered negligible) in all our measurements.
However methods for the uncertainty estimation (and analyzers)
were different in different Years. The unceratainties
in Run5 (Run6) were 2.1%/2.1% (1.3%/1.5%) for Blue/Yellow.
Since no correction for "other" background was considered, so it's effect
most probably will lead to the shift in the same direction in measured
polarization values, hence the unceratinties for the "other" background
effect are at least partially correlated in Run5 and Run6.
The maximal overlap in uncertainties between Run5 and Run6
(the maximal common part) is the minimal uncertainty between
Run5 and Run6, so 1.3% and 1.5% for blue and yellow, respectively.
So, the (maximal) common part in polarization uncertainties
between Run5 and Run6 are
sqrt(2.0^2+1.3^2)=2.4% for Blue
sqrt(2.0^2+1.5^2)=2.5% for Yell
And similarly for a product of two beam polarizations:
sqrt((2*2.0)^2+(2*1.3)^2)=4.8% for Blue*Yell
which takes into account molecular and "other" background effects, which are
considered correlated between blue and yellow beams, the maximal overlap
for "other" background being the minimal between blue and yellow, so 1.3%.