levan babukhadia joint run i analysis group & editorial board meeting, fermilab, august 4, 2000...
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Levan BabukhadiaJoint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000
Levan BabukhadiaLevan Babukhadia
Joint Run I Analysis Group & Editorial Board #121 MeetingJoint Run I Analysis Group & Editorial Board #121 Meeting
Rapidity Dependence ofInclusive Jet Cross Section
( Final error analysis - 2 studies )
Rapidity Dependence ofInclusive Jet Cross Section
( Final error analysis - 2 studies )
Fermilab, DZeroAugust 4, 2000Fermilab, DZeroAugust 4, 2000
http://www-d0.fnal.gov/~blevan/my_analysis/analysis.html
Levan BabukhadiaJoint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000
ET (GeV)
d2
(dE
T d)
(fb
/GeV
) 0.0 0.5 0.5 1.0 1.0 1.5 1.5 2.0 2.0 3.0
DØ PreliminaryDØ Preliminary
1
pb 92 Ldt
1
pb 92 LdtRun 1BRun 1B
Nominal cross sections & statistical errors only
Rapidity Dependence of Inclusive Jet Cross SectionRapidity Dependence of Inclusive Jet Cross Section
Levan BabukhadiaJoint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000
ET (GeV)
Fra
ctio
nal E
rror
s (%
)
0.0 0.5
0.5 1.0
1.0 1.5
ET (GeV)
1.5 2.0
2.0 3.0
Sources of Systematic UncertaintiesSources of Systematic Uncertainties
DØ PreliminaryDØ Preliminary
1
pb 92 Ldt
1
pb 92 LdtRun 1BRun 1B
Luminosity Jet Energy Scale Selection efficiency Resolutions & Unfolding
Total
Levan BabukhadiaJoint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000
• NLO pQCD predictions (s3):
- Ellis, et al., Phys. Rev. D, 64, (1990) EKS - Aversa, et al., Phys. Rev. Lett., 65, (1990) - Giele, et al., Phys. Rev. Lett., 73, (1994) JETRAD
• Choices (hep-ph/9801285, EPJ C5, 687, 1998): - Renormalization Scale (~10%) - PDFs (~20% with ET dependence) - Clustering Alg. (~5% with ET dependence)
Uncertainties in Theoretical PredictionsUncertainties in Theoretical Predictions
2R
1.3R
DØ uses: JETRAD, , Rsep= 1.3.DØ uses: JETRAD, , Rsep= 1.3.2maxTE 2maxTE
CDF uses: EKS, , Rsep= 2.0.CDF uses: EKS, , Rsep= 2.0.2jetTE 2jetTE
Levan BabukhadiaJoint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000
0.5 1.0
0.0 0.5
1.0 1.5
ET (GeV)
DØ PreliminaryDØ Preliminary
DØ PreliminaryDØ Preliminary
DØ PreliminaryDØ Preliminary
1.5 2.0
2.0 3.0
DØ PreliminaryDØ Preliminary
DØ PreliminaryDØ Preliminary
ET (GeV)
Comparisons to JETRAD with:
PDF: CTEQ4M
Rsep= 1.3
2ETmax
( D
a ta
- T
he o
r y )
/ T
he o
r yComparisons to Theoretical PredictionsComparisons to Theoretical Predictions
Deviations from QCD at highest ET not significant within errors.
Deviations from QCD at highest ET not significant within errors.
Good agreement with theory over ~seven orders!
Good agreement with theory over ~seven orders!
Levan BabukhadiaJoint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000
0.5 1.0
0.0 0.5
1.0 1.5
ET (GeV)
DØ PreliminaryDØ Preliminary
DØ PreliminaryDØ Preliminary
DØ PreliminaryDØ Preliminary
1.5 2.0
2.0 3.0
DØ PreliminaryDØ Preliminary
DØ PreliminaryDØ Preliminary
ET (GeV)
Comparisons to JETRAD with:
PDF: CTEQ4HJ
Rsep= 1.3
2ETmax
( D
a ta
- T
he o
r y )
/ T
he o
r yComparisons to Theoretical PredictionsComparisons to Theoretical Predictions
CTEQ4HJ appears to produce better agreement with the data. Work is underway to obtain a quantitative measure
of agreement.
CTEQ4HJ appears to produce better agreement with the data. Work is underway to obtain a quantitative measure
of agreement.
Levan BabukhadiaJoint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000
0.5 1.0
0.0 0.5
1.0 1.5
ET (GeV)
DØ PreliminaryDØ Preliminary
DØ PreliminaryDØ Preliminary
DØ PreliminaryDØ Preliminary
1.5 2.0
2.0 3.0
DØ PreliminaryDØ Preliminary
DØ PreliminaryDØ Preliminary
ET (GeV)
Comparisons to Theoretical PredictionsComparisons to Theoretical Predictions
Comparisons to JETRAD with:
PDF: CTEQ3M
Rsep= 1.3
2ETmax
Deviations from QCD at highest ET not significant within errors.
Deviations from QCD at highest ET not significant within errors.
Good agreement with theory over seven orders!
Good agreement with theory over seven orders!
( D
a ta
- T
he o
r y )
/ T
he o
r y
Levan BabukhadiaJoint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000
0.5 1.0
0.0 0.5
1.0 1.5
ET (GeV)
DØ PreliminaryDØ Preliminary
DØ PreliminaryDØ Preliminary
DØ PreliminaryDØ Preliminary
1.5 2.0
2.0 3.0
DØ PreliminaryDØ Preliminary
DØ PreliminaryDØ Preliminary
ET (GeV)
Comparisons to JETRAD with:
PDF: MRST
Rsep= 1.3
2ETmax
( D
a ta
- T
he o
r y )
/ T
he o
r yComparisons to Theoretical PredictionsComparisons to Theoretical Predictions
PDF’s of MRST family appear to have worst agreement with the data in overall normalization.
PDF’s of MRST family appear to have worst agreement with the data in overall normalization.
Levan BabukhadiaJoint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000
0.5 1.0
0.0 0.5
1.0 1.5
ET (GeV)
DØ PreliminaryDØ Preliminary
DØ PreliminaryDØ Preliminary
DØ PreliminaryDØ Preliminary
1.5 2.0
2.0 3.0
DØ PreliminaryDØ Preliminary
DØ PreliminaryDØ Preliminary
ET (GeV)
Comparisons to JETRAD with:
PDF: MRTSg
Rsep= 1.3
2ETmax
( D
a ta
- T
he o
r y )
/ T
he o
r yComparisons to Theoretical PredictionsComparisons to Theoretical Predictions
PDF’s of MRST family appear to have worst agreement with the data in overall normalization.
PDF’s of MRST family appear to have worst agreement with the data in overall normalization.
Levan BabukhadiaJoint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000
0.5 1.0
0.0 0.5
1.0 1.5
ET (GeV)
DØ PreliminaryDØ Preliminary
DØ PreliminaryDØ Preliminary
DØ PreliminaryDØ Preliminary
1.5 2.0
2.0 3.0
DØ PreliminaryDØ Preliminary
DØ PreliminaryDØ Preliminary
ET (GeV)
Comparisons to JETRAD with:
PDF: MRSTg
Rsep= 1.3
2ETmax
( D
a ta
- T
he o
r y )
/ T
he o
r yComparisons to Theoretical PredictionsComparisons to Theoretical Predictions
PDF’s of MRST family appear to have worst agreement with the data in overall normalization.
PDF’s of MRST family appear to have worst agreement with the data in overall normalization.
Levan BabukhadiaJoint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000
Building Full Covariance Matrix Building Full Covariance Matrix
αj
αi
αji,
αji, σσρCov
For any error subcomponent , define a covariance matrix:
Then a Full Covariance (or Error) matrix is given bysumming the covariance matrices of all
error subcomponents, i.e.:
β
βj
βi
βji,
Fullji,ji, σσρCovCov
with correlation coefficients [-1,1] and standard errors ;and we take all five regions together: i, j = 1, 90.
In case of Rapidity Dependence analysis, in additionto error correlations in ET, one should also address
error correlations in pseudorapidity .
Levan BabukhadiaJoint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000
Building Full Covariance Matrix(all but JES errors)
Building Full Covariance Matrix(all but JES errors)
Source of Error Correlation in ET Correlation in
Statistical(Inclusive Jets Data)
uncorrelated uncorrelated
Luminosity correlated correlated
Selection Efficiencies correlated uncorrelated
-Bias correlated correlated
Resolutions & Unfolding: Resolutions Paramaterization Resolutions Closure Ansatz Fit
correlatedcorrelatedcorrelated
uncorrelatedcorrelated
uncorrelated
Levan BabukhadiaJoint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000
Building Full Covariance Matrix (JES)Building Full Covariance Matrix (JES)
Source of Error Correlation in ET Correlation in
Jet Energy Scale:
Statistical uncorrelated uncorrelated
Offset: Underlying Event* Noise, Zero Suppression*
correlatedcorrelated
correlatedcorrelated
Response: Fit Background Sys. Bias L/H(incl punchthr) Low ET bias** Detector Scale (Fcryo) Detector Scale (-depen.) Kt_error**
partially correlatedcorrelatedcorrelatedcorrelatedcorrelatedcorrelatedcorrelated
partially correlatedcorrelatedcorrelatedcorrelatedcorrelated
uncorrelatedcorrelated
Showering: Statistical Jet Limit MC Closure 2%
correlatedcorrelated
(un)correlated (?)
uncorrelatedcorrelated
(un)correlated (?)
Coccccc
* - Also have “stat” components treated as correlated in ET but not in ;** - Negligibly small or zero for jet ET greater than ~50-60 GeV;
Levan BabukhadiaJoint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000
Origin of the Showering Closure 2 % ErrorOrigin of the Showering Closure 2 % Error
Levan BabukhadiaJoint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000
2 Calculation2 Calculation
ji,
jjii2 TDTDχ
-1Fullji,Cov
Standard definition with full error matrix:
Is biased in case of large correlated errors.
If redefined to associate fractional experimentalerrors to Theory, bias is removed:
ji,jj
-1
j
j
i
ijFullji,ii
2 TDT
D
T
DCovTDχ
As demonstrated in jets PRD in preparation.
Levan BabukhadiaJoint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000
Error Matrix in GeneralError Matrix in General
Error Matrix is REAL and SYMMETRIC; as such can ALWAYS be diagonalized (linear Algebra)
Once diagonalized, on the diagonal will have all posi-tive numbers because they will simply be
squares of errors in this “diagonalized space”
Necessary and sufficient condition for positive definiteness is that ALL eigenvalues i > 0.
Since we just showed that eigenvalues of Error matrix must ALL be positive, Error matrix must
be +def ...ALWAYS !
This imposes N nontrivial conditions on correlations
Levan BabukhadiaJoint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000
Our Error (or Full Covariance Matrix) is not +def
We now think this is due to numerical precision (roundoff errors) in dealing with 90 x 90 matrix
(e.g. our JES response fit 11x11 matrix as it appears
in the NIM paper is also not +def, but it’s full version
is nearly +def)
Can we fix our matrix in such a way that the results are independent of the “amount” of fix
( “2 renormalization” ) ?
I will show some developments in this direction …
( a simple re-scaling alone does not help much )
Error Matrix in Our CaseError Matrix in Our Case
Levan BabukhadiaJoint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000
Details on Our Error MatrixDetails on Our Error Matrix
First, consider showering error correlations in ET ( ET ) and ( ):
Let me set = 0
The Error matrix is then not +def for
ET = 1but becomes +def if
ET < ~ 0.95 in principle, regardless
of value
Of course, it is hard tojustify any one value of ET in this approach … more so that
we expect ET ~ 1.
Perhaps MATH can help?
Levan BabukhadiaJoint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000
… and Same for Individual Regions… and Same for Individual Regions
Looks hopeful but,again, the approach is hard to justify ...
Levan BabukhadiaJoint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000
Forcing Matrix to be Positive DefiniteForcing Matrix to be Positive Definite
j,iminj,ij,i 1
j,i 1
j,i
j,ij,ij,i
j,iminj,ij,i
imin
j,iij,i 1
j,i
j,i
fA S A~
A~
S S A~
S A
0A~
A~
A~
fA~
A
0min
S AS A~A
-- start off with a not +def matrix A
-- diagonalize it using simil. transf. S
-- find the smallest eigenvalue
-- form a correction matrix
-- now all eigenvalues must be positive
This method is used for example in famous MINUIT...
In our case, however, we really need to have 2
originating from such a fixed matrix to be independent of the amount of fix, a “fudge” constant f.
This procedure can be called 2 renormalization ...
Levan BabukhadiaJoint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000
2 as a Function of the Fix2 as a Function of the Fix
Here considered is the case withET = 1 and = 0
and the 2 is calculated using the Standard (not jets PRD) method
We see comforting behaviorand the results are somewhat
surprising!
Levan BabukhadiaJoint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000
… and Same for Individual Regions… and Same for Individual Regions
Relative independenceof fudge constant,
once f > 1, is also observed
in individual regions
Levan BabukhadiaJoint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000
A More Realistic ExampleA More Realistic Example
Here we consider more realistic case
withET = 1 and = 1
and the 2 calculated using the unbiased
method, i.e. methodused in jets PRD.
Again, nice behavior is
observed but results are still somewhat
unexpected
Levan BabukhadiaJoint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000
… and Same for Individual Regions… and Same for Individual Regions
Again, everythingseems to hold for
individual regions
Levan BabukhadiaJoint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000
There seems to be some message in renormalized (or so far perhaps only regularized) 2s
The remaining question is HOW TO QUANIFY
these differences in 2
2-less draft of the PRL now exists. It has passedthrough the first round of approval among the Run I/QCD,
the EB, and others as it was being submitted to ICHEP.I incorporated most of the comments and the current
version of the PRL is (and will continue to be) posted at:
http://www-d0.fnal.gov/~blevan/my_analysis/analysis.html
Remaining IssuesRemaining Issues