From unal@pngs02.hep.upenn.edu  Sun Oct  3 15:02:44 1993
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From: unal@pngs02.hep.upenn.edu (unal)
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To: liss@fnald.fnal.gov, claudioc@fnald.fnal.gov, yagil@fnald.fnal.gov,
        dan@fnald.fnal.gov, bed@fnald.fnal.gov, hughes@fnald.fnal.gov,
        winer@fnald.fnal.gov, williams@williams.hep.upenn.edu,
        pekka@cepheid.physics.utoronto.ca
Cc: unal@pngs02.hep.upenn.edu
Subject: SVX/SLT background correlations
Date: Sun, 03 Oct 93 15:02:34 -0500
Status: R


 Hi,

  I have been trying to investigate the background
correlation between SVX and SLT in W+jets and its impact
on the significance of the excess of events in a combined
analysis.

*  The first ingredient needed is the amount of
true heavy flavor tags and fake tags for each tagger.Here
is what I have done to estimate these various contributions.
Some of the numbers are not really more than educated guesses.

 I started from the mistag rate for SVX, defined from the
-DL rate from jets => 0.73 +- 0.09 events in W+3,4 jets.
To define the number of Wbbbar,Wccbar I took the
average between the raw MC value and the value from the (+dl)-(-dl)
rate from jets, with the difference as systematic error.
Since to reproduce the +dl background estimate, one has
to multiply by 2.8 the raw MC estimates of Wbbbar,ccbar, I used
as central value 1.9*MC value with an error of +-0.9*MC value
>From this, the number of Wbbbar before tagging is 2.63+-1.25
and the number of Wccbar before tagging is 4.29+-2.03
The number of Wc before tagging is 4.16+-1.24
 For all these W+heavy flavor events, I assumed that the
tagging efficiencies of SVX and SLT are uncorrelated, as indicated
by top MC and used the following estimates

  eff (Wbbbar SVX)=0.20+-0.04    eff(Wbbbar SLT)=0.15+-0.05 (???)
  eff (Wccbar SVX)=0.04+-0.016   eff(Wccbar SLT)=0.052+-0.02
  eff (Wc SVX)=0.035+-0.012      eff(Wc SLT)=0.020+-0.004

 For the fake rate of SLT, I used the value from
generic jets * 0.75+-0.15 (from the estimated purity of the tagger)
 Finally, I added the other background sources (0.13+-0.05 for SVX
and 0.33+-0.12 for SLT assuming that they are uncorrelated).

 From the previous numbers, the total backgrounds are

          1.70+-0.41 for SVX and 2.94+-0.51 for SLT.

 The W+b/c content is ~ 0.85 events for SVX and ~ 0.70 events for SLT

 The fact that I get a SLT background close to the direct
estimate (3.0 events) is (if I used the correct efficiencies)
a  cross-check of the overall consistency of the
SVX/SLT efficiencies, the heavy flavor purities and the
overall background rates.

* To compute the probability of a background fluctuation, I ran
a small monte-carlo to simulate a large number of
experiments with only background. 
This MC generates the total number of
Wbbbar,Wccbar and Wc each with Poisson statistic. Each event
is tagged by SVX, SLT of both according to the estimated
efficiencies by generating random numbers.
The number of fakes is computed in a similar way, starting from
the total (52) number of W+3,4 jets. To account for
the (small) fake correlation from the track multiplicity
dependance of the fake rates, I assumed that the probability
of the double fake is two times higher than the product of
the individual probabilities (the factor 2 is the factor that
I get in generic jet events).
Finally, I accounted for the systematic uncertainties by
smearing the various parameters with gaussian fluctuations.
The Wbbbar and Wccbar errors are taken to be 100% correlated
as well as the various SVX tagging efficiency errors.

* From this MC, I get the following probabilities

 P (Number of SVX tags >=6) = 1.15 %
 P (Number of SLT tags >=7) = 3.57 %
 P (Number of SVX + SLT tags >=13)=0.29 %
 
The average number of overlaps for all the experiments is 0.15
(3.3 % of the average total number of tags. Naively, one would
have expected a little less than twice the jet by jet overlap
in a jet sample which is 2 %).

For the experiments with 13 tags or more, the average number of overlaps
is ~ 1.0. Clearly, these background fluctuations are
"dominated" by Wbbbar,ccbar fluctuations which produce
at the same time a larger number of overlaps. Around 7% of these
background experiments have 3 or more overlaps.
Given the fact that 3 overlaps is already more than we expect from
top, I dont think that we should put to much quantitative emphasis
on this 7%. However this large number of overlaps is certainly
a good indication that we are really seing an excess of Wbbbar event.

* Adding the dilepton background (0.61+-0.13) and assuming that
it is completely uncorrelated with the others, the
probability to get 15 "tags" or more from all the background
sources is 0.119 %


                           Guillaume