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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 CTEQ Summer School
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
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
Global Analysis data from disparate experiments CTEQ Summer School
Our treatment of systematic errors CTEQ Summer School
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
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
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
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
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 c2 with respect to both shape parameters {am} and optimized systematic shifts {sj}. CTEQ Summer School
Because c2 depends quadratically on {sj} we can solve for the systematic shifts analytically, ss0(a). Then let, and minimize w.r.t {am}. The systematic shifts {sj} are continually optimized [ ss0(a) ] CTEQ Summer School
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
4/ Comparisons of data and CTEQ6 CTEQ Summer School
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
“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
“Pull” distribution, comparing theory and data with optimal systematic shifts Comparing theory and data without systematic shifts CTEQ Summer School
Inclusive jet production at the Tevatron Collider (Run 1) CTEQ Summer School
The CDF and D0 jet data can be fit with a hard gluon distribution. CTEQ Summer School
CCFR and NuTeV measurements! CTEQ Summer School
5/ The CTEQ6.1 Parton Distribution Functions CTEQ Summer School
History of the CTEQ u-quark distribution CTEQ Summer School
History of the CTEQ u-quark distribution CTEQ Summer School
History of the CTEQ gluon distribution CTEQ Summer School
History of the CTEQ gluon distribution CTEQ Summer School
CTEQ and Others u quark at Q = 3.16 GeV CTEQ Summer School
CTEQ and Others u quark CTEQ and Others u quark at Q = 100 GeV CTEQ Summer School
CTEQ and Others gluon at Q = 3.16 GeV CTEQ Summer School
CTEQ and Others gluon at Q = 100 GeV CTEQ Summer School
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
CTEQ6.1 sea quark distributions CTEQ Summer School
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