levan babukhadia joint run i analysis group & editorial board meeting, fermilab, august 4, 2000...

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


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


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


1

pb 92 Ldt

1

pb 92 LdtRun 1BRun 1B

Luminosity Jet Energy Scale Selection efficiency Resolutions & Unfolding

Total


• 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


0.5 1.0

0.0 0.5

1.0 1.5

ET (GeV)




1.5 2.0

2.0 3.0



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.


Good agreement with theory over ~seven orders!

Good agreement with theory over ~seven orders!


0.5 1.0

0.0 0.5

1.0 1.5

ET (GeV)




1.5 2.0

2.0 3.0



ET (GeV)


PDF: CTEQ4HJ

Rsep= 1.3

2ETmax

( D

a ta

- T

he o

r y )

/ T

he o


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.


0.5 1.0

0.0 0.5

1.0 1.5

ET (GeV)




1.5 2.0

2.0 3.0



ET (GeV)

Comparisons to Theoretical PredictionsComparisons to Theoretical Predictions


PDF: CTEQ3M

Rsep= 1.3

2ETmax



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


0.5 1.0

0.0 0.5

1.0 1.5

ET (GeV)




1.5 2.0

2.0 3.0



ET (GeV)


PDF: MRST

Rsep= 1.3

2ETmax

( D

a ta

- T

he o

r y )

/ T

he o


PDF’s of MRST family appear to have worst agreement with the data in overall normalization.



0.5 1.0

0.0 0.5

1.0 1.5

ET (GeV)




1.5 2.0

2.0 3.0



ET (GeV)


PDF: MRTSg

Rsep= 1.3

2ETmax

( D

a ta

- T

he o

r y )

/ T

he o





0.5 1.0

0.0 0.5

1.0 1.5

ET (GeV)




1.5 2.0

2.0 3.0



ET (GeV)


PDF: MRSTg

Rsep= 1.3

2ETmax

( D

a ta

- T

he o

r y )

/ T

he o





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 .


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


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


Response: Fit Background Sys. Bias L/H(incl punchthr) Low ET bias** Detector Scale (Fcryo) Detector Scale (-depen.) Kt_error**

partially correlatedcorrelatedcorrelatedcorrelatedcorrelatedcorrelatedcorrelated

partially correlatedcorrelatedcorrelatedcorrelatedcorrelated


Showering: Statistical Jet Limit MC Closure 2%


(un)correlated (?)


(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;


Origin of the Showering Closure 2 % ErrorOrigin of the Showering Closure 2 % Error


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.


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


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


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?


… and Same for Individual Regions… and Same for Individual Regions

Looks hopeful but,again, the approach is hard to justify ...


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 ...


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!



Relative independenceof fudge constant,

once f > 1, is also observed

in individual regions


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



Again, everythingseems to hold for

individual regions


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

levan babukhadia joint run i analysis group & editorial board meeting, fermilab, august 4, 2000...

Documents

e t gev d preliminary

highest e t

meeting joint run

t d fbgev

levan babukhadia joint

e t dependence uncertainties

e t gev fractional errors

e t dependence clustering