1. Levan Babukhadia Joint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000 Levan Babukhadia Joint Run I Analysis Group & Editorial Board #121 Meeting Joint Run I Analysis Group & Editorial Board #121 Meeting Rapidity Dependence of Inclusive Jet Cross Section ( Final error analysis -  2 studies ) Rapidity Dependence of Inclusive Jet Cross Section ( Final error analysis -  2 studies ) Fermilab, DZero August 4, 2000 Fermilab, DZero August 4, 2000 http://www-d0.fnal.gov/~blevan/my_analysis/analysis.html

2. Levan Babukhadia Joint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000 E T (GeV)  d 2  (dE T d  )  (fb/GeV) l 0.0    0.5 H 0.5    1.0 s 1.0    1.5 n 1.5    2.0 t 2.0    3.0 DØ Preliminary Run 1B Nominal cross sections & statistical errors only Rapidity Dependence of Inclusive Jet Cross Section

3. Levan Babukhadia Joint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000 E T (GeV) Fractional Errors (%) 0.0    0.5 0.5    1.0 1.0    1.5 E T (GeV) 1.5    2.0 2.0    3.0 Sources of Systematic Uncertainties DØ Preliminary Run 1B  Luminosity  Jet Energy Scale  Selection efficiency  Resolutions & Unfolding  Total

4. Levan Babukhadia Joint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000 NLO pQCD predictions (  s 3 ): - 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 E T dependence) - Clustering Alg. (~5% with E T dependence) Uncertainties in Theoretical Predictions 2R 1.3R DØ uses: JETRAD,, R sep = 1.3. CDF uses: EKS,, R sep = 2.0.

5. Levan Babukhadia Joint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000 l 0.5    1.0 l 0.0    0.5 l 1.0    1.5 E T (GeV) DØ Preliminary l 1.5    2.0 l 2.0    3.0 DØ Preliminary E T (GeV) Comparisons to JETRAD with: PDF: CTEQ4M R sep = 1.3 ( Data - Theory ) / Theory Comparisons to Theoretical Predictions Deviations from QCD at highest E T not significant within errors. Good agreement with theory over ~seven orders!

6. Levan Babukhadia Joint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000 l 0.5    1.0 l 0.0    0.5 l 1.0    1.5 E T (GeV) DØ Preliminary l 1.5    2.0 l 2.0    3.0 DØ Preliminary E T (GeV) Comparisons to JETRAD with: PDF: CTEQ4HJ R sep = 1.3 ( Data - Theory ) / Theory Comparisons to Theoretical Predictions CTEQ4HJ appears to produce better agreement with the data. Work is underway to obtain a quantitative measure of agreement.

7. Levan Babukhadia Joint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000 l 0.5    1.0 l 0.0    0.5 l 1.0    1.5 E T (GeV) DØ Preliminary l 1.5    2.0 l 2.0    3.0 DØ Preliminary E T (GeV) Comparisons to Theoretical Predictions Comparisons to JETRAD with: PDF: CTEQ3M R sep = 1.3 Deviations from QCD at highest E T not significant within errors. Good agreement with theory over seven orders! ( Data - Theory ) / Theory

8. Levan Babukhadia Joint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000 l 0.5    1.0 l 0.0    0.5 l 1.0    1.5 E T (GeV) DØ Preliminary l 1.5    2.0 l 2.0    3.0 DØ Preliminary E T (GeV) Comparisons to JETRAD with: PDF: MRST R sep = 1.3 ( Data - Theory ) / Theory Comparisons to Theoretical Predictions PDF’s of MRST family appear to have worst agreement with the data in overall normalization.

9. Levan Babukhadia Joint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000 l 0.5    1.0 l 0.0    0.5 l 1.0    1.5 E T (GeV) DØ Preliminary l 1.5    2.0 l 2.0    3.0 DØ Preliminary E T (GeV) Comparisons to JETRAD with: PDF: MRTSg  R sep = 1.3 ( Data - Theory ) / Theory Comparisons to Theoretical Predictions PDF’s of MRST family appear to have worst agreement with the data in overall normalization.

10. Levan Babukhadia Joint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000 l 0.5    1.0 l 0.0    0.5 l 1.0    1.5 E T (GeV) DØ Preliminary l 1.5    2.0 l 2.0    3.0 DØ Preliminary E T (GeV) Comparisons to JETRAD with: PDF: MRSTg  R sep = 1.3 ( Data - Theory ) / Theory Comparisons to Theoretical Predictions PDF’s of MRST family appear to have worst agreement with the data in overall normalization.

11. Levan Babukhadia Joint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000 Building Full Covariance Matrix For any error subcomponent , define a covariance matrix: Then a Full Covariance (or Error) matrix is given by summing the covariance matrices of all error subcomponents, i.e.: 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 addition to error correlations in E T, one should also address error correlations in pseudorapidity .

12. Levan Babukhadia Joint 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)

13. Levan Babukhadia Joint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000 Building Full Covariance Matrix (JES) * - Also have “stat” components treated as correlated in E T but not in  ; ** - Negligibly small or zero for jet E T greater than ~50-60 GeV;

14. Levan Babukhadia Joint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000 Origin of the Showering Closure 2 % Error

15. Levan Babukhadia Joint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000  2 Calculation Standard definition with full error matrix: Is biased in case of large correlated errors. If redefined to associate fractional experimental errors to Theory, bias is removed: As demonstrated in jets PRD in preparation.

16. Levan Babukhadia Joint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000 Error 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

17. Levan Babukhadia Joint 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 Case

18. Levan Babukhadia Joint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000 Details on Our Error Matrix First, consider showering error correlations in E T (  ET ) and  (   ): Let me set   = 0 The Error matrix is then not +def for  ET = 1 but becomes +def if  ET < ~ 0.95 in principle, regardless of   value Of course, it is hard to justify any one value of  ET in this approach … more so that we expect  ET ~ 1. Perhaps MATH can help?

19. Levan Babukhadia Joint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000 … and Same for Individual  Regions Looks hopeful but, again, the approach is hard to justify...

20. Levan Babukhadia Joint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000 Forcing Matrix to be Positive Definite -- 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...

21. Levan Babukhadia Joint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000  2 as a Function of the Fix Here considered is the case with  ET = 1 and   = 0 and the  2 is calculated using the Standard (not jets PRD) method We see comforting behavior and the results are somewhat surprising!

22. Levan Babukhadia Joint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000 … and Same for Individual  Regions Relative independence of fudge constant, once f > 1, is also observed in individual  regions

23. Levan Babukhadia Joint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000 A More Realistic Example Here we consider more realistic case with  ET = 1 and   = 1 and the  2 calculated using the unbiased method, i.e. method used in jets PRD. Again, nice behavior is observed but results are still somewhat unexpected

24. Levan Babukhadia Joint Run I Analysis Group & Editorial Board meeting, Fermilab, August 4, 2000 … and Same for Individual  Regions Again, everything seems to hold for individual  regions

25. Levan Babukhadia Joint 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)  2 s The remaining question is HOW TO QUANIFY these differences in  2  2 -less draft of the PRL now exists. It has passed through 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 Issues