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3/ Methods of Global Analysis

3/ Methods of Global Analysis. Parameterization of f i (x,Q 2 ). At low Q 0 , of order 1 GeV,. P(x) has a few more parameters for increased flexibility. ~ 20 free shape parameters. CTEQ6 gluon. The Q dependence of f(x,Q 2 ) is obtained by solving the QCD evolution equations (DGLAP).

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3/ Methods of Global Analysis

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  1. 3/ Methods of Global Analysis CTEQ Summer School

  2. Parameterization of fi(x,Q2) At low Q0 , of order 1 GeV, P(x) has a few more parameters for increased flexibility. ~ 20 free shape parameters CTEQ6 gluon The Q dependence of f(x,Q2) is obtained by solving the QCD evolution equations (DGLAP). CTEQ Summer School

  3. CTEQ6 -- Table of experimental data sets H1 (a) 96/97 low-x e+p data ZEUS 96/97 e+p data H1 (b) 98/99 high-Q e-p data D0 : d2s/d dpT CTEQ Summer School

  4. Global Analysis data from disparate experiments CTEQ Summer School

  5. Our treatment of systematic errors CTEQ Summer School

  6. What is a systematic error? “This is why people are so frightened of systematic errors, and most other textbooks avoid the subject altogether. You never know whether you have got them and can never be sure that you have not – like an insidious disease… The good news, however, is that despite popular prejudices and superstitions, once you know what your systematic errors are, they can be handled with standard statistical methods.” R. J. Barlow Statistics CTEQ Summer School

  7. Imagine that two experimental groups have measured a quantity  , with the results shown. OK, what is the value of  ? This is very analogous to what happens in global analysis of PDF’s. But in the case of PDF’s the systematic differences are only visible through the PDF’s. CTEQ Summer School

  8. We use 2 minimization with fitting of systematic errors. For statistical errors define Di : data value Ti : theoretical value si : statistical error Ti = Ti(a1, a2, ..,, ad)a function of d theory parameters Minimize 2 w. r. t. {am}  optimal parameter values {a0m}. All this would be based on the assumption that Di = Ti(a0) + i ri CTEQ Summer School

  9. Treatment of the normalization error In scattering experiments there is an overall normalization uncertainty from uncertainty of the luminosity. We define where fN = overall normalization factor Minimize 2 w. r. t. both {am} and fN. CTEQ Summer School

  10. A method for general systematic errors ai : statistical error of Di bij : set of systematic errors (j=1…K) of Di Define quadratic penalty term Minimize c2 with respect to both shape parameters {am} and optimized systematic shifts {sj}. CTEQ Summer School

  11. Because c2 depends quadratically on {sj} we can solve for the systematic shifts analytically, ss0(a). Then let, and minimize w.r.t {am}. The systematic shifts {sj} are continually optimized [ ss0(a) ] CTEQ Summer School

  12. So, we have accounted for … • Statistical errors • Overall normalization uncertainty (by fitting {fN,e}) • Other systematic errors (analytically) We may make further refinements of the fit with weighting factors Default : we and wN,e = 1 The spirit of global analysis is compromise – the PDF’s should fit all data sets satisfactorily. If the default leaves some experiments unsatisfied, we may be willing to reduce the quality of fit to some experiments in order to fit better another experiment. (However, we use this trick sparingly!) CTEQ Summer School

  13. 4/ Comparisons of data and CTEQ6 CTEQ Summer School

  14. CTEQ6 -- Table of experimental data sets H1 (a) 96/97 low-x e+p data ZEUS 96/97 e+p data H1 (b) 98/99 high-Q e-p data D0 : d2s/d dpT CTEQ Summer School

  15. CTEQ Summer School

  16. CTEQ Summer School

  17. CTEQ Summer School

  18. CTEQ Summer School

  19. “Pull” distribution, comparing theory and data with optimal systematic shifts If we ignore the systematic errors, we observe a systematic difference between D and T. CTEQ Summer School

  20. “Pull” distribution, comparing theory and data with optimal systematic shifts Comparing theory and data without systematic shifts CTEQ Summer School

  21. Inclusive jet production at the Tevatron Collider (Run 1) CTEQ Summer School

  22. The CDF and D0 jet data can be fit with a hard gluon distribution. CTEQ Summer School

  23. CCFR and NuTeV measurements! CTEQ Summer School

  24. 5/ The CTEQ6.1 Parton Distribution Functions CTEQ Summer School

  25. History of the CTEQ u-quark distribution CTEQ Summer School

  26. History of the CTEQ u-quark distribution CTEQ Summer School

  27. History of the CTEQ gluon distribution CTEQ Summer School

  28. History of the CTEQ gluon distribution CTEQ Summer School

  29. CTEQ and Others u quark at Q = 3.16 GeV CTEQ Summer School

  30. CTEQ and Others u quark CTEQ and Others u quark at Q = 100 GeV CTEQ Summer School

  31. CTEQ and Others gluon at Q = 3.16 GeV CTEQ Summer School

  32. CTEQ and Others gluon at Q = 100 GeV CTEQ Summer School

  33. The asymmetric sea  Where do sea quarks come from?  Might expect usea  dsea > s > c > b  Can we explain the rather large dsea to usea asymmetry? CTEQ Summer School

  34. CTEQ6.1 sea quark distributions CTEQ Summer School

  35. Slide copied from James Stirling talk at Oxford • MRST: Q02 = 1 GeV2,Qcut2 = 2 GeV2 xg = Axa(1–x)b(1+Cx0.5+Dx) – Exc(1-x)d • CTEQ6: Q02 = 1.69 GeV2,Qcut2 = 4 GeV2 xg = Axa(1–x)becx(1+Cx)d CTEQ Summer School

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