1. Sept 2003Phystat1 Uncertainties of Parton Distribution Functions Daniel Stump Michigan State University & CTEQ

2. Sept 2003Phystat2 High energy particles interact through their quark and gluon constituents – the partons. Asymptotic freedom : the parton cross sections can be approximated by perturbation theory. Factorization theorem : Parton distribution functions in the nucleon are the link between the PQCD theory and measurements on nucleons.

3. Sept 2003Phystat3 Parton distribution functions are important.

4. Sept 2003Phystat4 The goals of QCD global analysis are to find accurate PDF’s; to know the uncertainties of the PDF’s; to enable predictions, including uncertainties.

5. Sept 2003Phystat5 The systematic study of uncertainties of PDF’s developed slowly. Pioneers… J. Collins and D. Soper, CTEQ Note 94/01, hep-ph/9411214. C. Pascaud and F. Zomer, LAL-95-05. M. Botje, Eur. Phys. J. C 14, 285 (2000). Today many groups and individuals are involved in this research.

6. Sept 2003Phystat6 CTEQ group at Michigan State (J. Pumplin, D. Stump, WK. Tung, HL. Lai, P. Nadolsky, J. Huston, R. Brock) and others (J. Collins, S. Kuhlmann, F. Olness, J. Owens) MRST group (A. Martin, R. Roberts, J. Stirling, R. Thorne) Fermilab group (W. Giele, S. Keller, D. Kosower) S. I. Alekhin V. Barone, C. Pascaud, F. Zomer; add B. Portheault HERA collaborations ZEUS – S. Chekanov et al; A. Cooper-Sarkar H1 – C. Adloff et al Current research on PDF uncertainties

7. Sept 2003Phystat7 Outline of this talk (focusing on CTEQ results) General comments; CTEQ6 Our treatment of experimental systematic errors Compatibility of data sets Uncertainty analysis 2 case studies inclusive jet production in pp bar or pp strangeness asymmetry

8. Sept 2003Phystat8 … of short-distance processes using perturbative QCD (NLO) The challenge of Global Analysis is to construct a set of PDF’s with good agreement between data and theory, for many disparate experiments. Global Analysis

9. Sept 2003Phystat9 The program of Global Analysis is not a routine statistical analysis, because of systematic differences between experiments. We must sometimes use physics judgement in this complex real-world problem.

10. Sept 2003Phystat10 Parametrization At low Q 0, of order 1 GeV, P(x) has a few more parameters for increased flexibility. ~ 20 free shape parameters Q dependence of f(x,Q) is obtained by solving the QCD evolution equations (DGLAP).

11. Sept 2003Phystat11 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 : d 2  /d  dpT

12. Sept 2003Phystat12 Global Analysis data from many disparate experiments

13. Sept 2003Phystat13 The Parton Distribution Functions

14. Sept 2003Phystat14 Different ways to plot the parton distributions Linear Logarithmic Q 2 = 10 (solid) and 1000 (dashed) GeV 2

15. Sept 2003Phystat15 In order to show the large and small x regions simultaneously, we plot 3x 5/3 f(x) versus x 1/3. {Integral = momentum fraction}

16. Sept 2003Phystat16 Comparison of CTEQ6 and MRST2002 blue curves : CTEQ6M black dots : MRTS2002 gluon and u quark at Q 2 = 10 GeV 2

17. Sept 2003Phystat17 Our treatment of systematic errors

18. Sept 2003Phystat18 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

19. Sept 2003Phystat19 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.

20. Sept 2003Phystat20 We use  2 minimization with fitting of systematic errors. Minimize  2 w. r. t. {a  }  optimal parameter values {a 0  }. All this would be based on the assumption that D i = T i (a 0 ) +  i r i For statistical errors define T i = T i (a 1, a 2,..,, a d ) a function of d theory parameters (S. D.)

21. Sept 2003Phystat21 Treatment of the normalization error In scattering experiments there is an overall normalization uncertainty from uncertainty of the luminosity. We define where f N = overall normalization factor Minimize  2 w. r. t. both {a  } and f N.

22. Sept 2003Phystat22 A method for general systematic errors Minimize  2 with respect to both shape parameters {a  } and optimized systematic shifts {s j }. quadratic penalty term  i : statistical error of D i  ij : set of systematic errors (j=1…K) of D i Define

23. Sept 2003Phystat23 and minimize w.r.t {a  }. The systematic shifts {s j } are continually optimized [ s  s 0 (a) ] Because  2 depends quadratically on {s j } we can solve for the systematic shifts analytically, s  s 0 (a). Then let,

24. Sept 2003Phystat24 So, we have accounted for … Statistical errors Overall normalization uncertainty (by fitting {f N,e }) Other systematic errors (analytically) We may make further refinements of the fit with weighting factors Default : w e and w N,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. (We use this sparingly!)

25. Sept 2003Phystat25 How well does this fitting procedure work? Quality

26. Sept 2003Phystat26 Comparison of the CTEQ6M fit to the H1 data in separate x bins. The data points include optimized shifts for systematic errors. The error bars are statistical only.

27. Sept 2003Phystat27 Comparison of the CTEQ6M fit to the inclusive jet data. (a) D0 cross section versus p T for 5 rapidity bins; (b) CDF cross section for central rapidity.

28. Sept 2003Phystat28 How large are the optimized normalization factors? Exptf N BCDMS0.976 H1 (a)1.010 H1 (b)0.988 ZEUS0.997 NMC1.011 CCFR1.020 E6050.950 D00.974 CDF1.004

29. Sept 2003Phystat29 We must always check that the systematic shifts are not unreasonably large. jsj 11.67 2-0.67 3-1.25 4-0.44 50.00 6-1.07 71.28 80.62 9-0.40 100.21 jsj 10.67 2-0.81 3-0.35 40.25 50.05 60.70 7-0.31 81.05 90.61 100.26 110.22 10 systematic shifts NMC data 11 systematic shifts ZEUS data

30. Sept 2003Phystat30 Comparison to NMC F 2 without systematic shifts

31. Sept 2003Phystat31 A study of compatibility

32. Sept 2003Phystat32 The PDF’s are not exactly CTEQ6 but very close – a no-name generic set of PDF’s for illustration purposes. Table of Data Sets 1 BCDMS F2p 339 366.11.08 2 BCDMS F2d 251 273.61.09 3 H1 (a) 104 97.80.94 4 H1 (b) 126 127.31.01 5 H1 (c ) 129 108.90.84 6 ZEUS 229 261.11.14 7 CDHSW F2 85 65.60.77 8 NMC F2p 201 295.51.47 9 NMC d/p 123 115.40.94 10 CCFR F2 69 84.91.23 11E60511994.70.80 12E866 pp184239.21.30 13E866 d/p155.00.33 14D0 jet9062.60.70 15CDF jet3356.11.70 16CDHSW F39676.40.80 17CCFR F38726.80.31 18CDF W Lasy118.70.79 N  2  2 /N N tot = 2291  2 global = 2368.

33. Sept 2003Phystat33 The effect of setting all normalization constants to 1. 1BCDMS F2p186.5 2BCDMS F2d27.6 3H1 (a)7.3 4H1 (b)10.1 5H1 (c )24.0 8NMC F2p4.0 11E60513.3 12E866 pp95.7  2  2 (opt. norm) = 2368.  2 (norm 1) = 2742.  2 = 374.0

34. Sept 2003Phystat34 Example 1. The effect of giving the CCFR F2 data set a heavy weight. By applying weighting factors in the fitting function, we can test the “compatibility” of disparate data sets. 3H1 (a)8.3 7CDHSW F26.3 8NMC F2p18.1 10CCFR F2  19.7 12E866 pp5.5 14D0 jet23.5  2  2 (CCFR) =  19.7  2 (other) = +63.3 Giving a single data set a large weight is tantamount to determining the PDF’s from that data set alone. The result is a significant improvement for that data set but which does not fit the others.

35. Sept 2003Phystat35 Example 1b. The effect of giving the CCFR F2 data weight 0, i.e., removing the data set from the global analysis. 3H1 (a)  8.3 6ZEUS6.9 8NMC F2p  10.1 10CCFR F240.0  2  2 (CCFR) = +40.0  2 (other) =  17.4 Imagine starting with the other data sets, not including CCFR. The result of adding CCFR is that  2 global of the other sets increases by 17.4 ; this must be an acceptable increase of  2.

36. Sept 2003Phystat36 2BCDMS F2d  15.1 3H1 (a)  12.4 4H1 (b)  4.3 6ZEUS27.5 7CDHSW F219.2 8NMC F2p8.0 10CCFR F254.5 14D0 jet22.0 16CDHSW F311.0 17CCFR F35.9 Example 5. Giving heavy weight to H1 and BCDMS  2 for all data sets  2  2 ( H & B ) =  38.7  2 ( other ) = +149.9

37. Sept 2003Phystat37 Lessons from these reweighting studies Global analysis requires compromises – the PDF model that gives the best fit to one set of data does not give the best fit to others. This is not surprising because there are systematic differences between the experiments. The scale of acceptable changes of  2 must be large. Adding a new data set and refitting may increase the  2 ‘s of other data sets by amounts >> 1.

38. Sept 2003Phystat38 Clever ways to test the compatibility of disparate data sets Plot  2 versus  2 J Collins and J Pumplin (hep-ph/0201195) The Bootstrap Method Efron and Tibshirani, Introduction to the Bootstrap (Chapman&Hall) Chernick, Bootstrap Methods (Wiley)

39. Sept 2003Phystat39 Uncertainty Analysis (I) Methods

40. Sept 2003Phystat40 We continue to use  2 global as figure of merit. Explore the variation of  2 global in the neighborhood of the minimum. The Hessian method (  = 1 2 3 … d) a1a1 a2a2 the standard fit, minimum  2 nearby points are also acceptable

41. Sept 2003Phystat41 Classical error formula for a variable X(a) Obtain better convergence using eigenvectors of H  S  (+) and S  (  ) denote PDF sets displaced from the standard set, along the  directions of the  th eigenvector, by distance T =  (  2) in parameter space. ( available in the LHAPDF format : 2d alternate sets ) “Master Formula”

42. Sept 2003Phystat42 Minimization of F [w.r.t {a  } and ] gives the best fit for the value X(a min,  ) of the variable X. Hence we obtain a curve of  2 global versus X. The Lagrange Multiplier Method … for analyzing the uncertainty of PDF- dependent predictions. The fitting function for constrained fits  : Lagrange multiplier controlled by the parameter

43. Sept 2003Phystat43 The question of tolerance X : any variable that depends on PDF’s X 0 : the prediction in the standard set  2 (X) : curve of constrained fits For the specified tolerance (  2 = T 2 ) there is a corresponding range of uncertainty,   X. What should we use for T?

44. Sept 2003Phystat44 Estimation of parameters in Gaussian error analysis would have T = 1 We do not use this criterion.

45. Sept 2003Phystat45 Aside: The familiar ideal example Consider N measurements {  i } of a quantity  with normal errors {  i } Estimate  by minimization of  2, The mean of  combined is  true, the SD is The proof of this theorem is straightforward. It does not apply to our problem because of systematic errors. and ( =  /  N )

46. Sept 2003Phystat46 Add a systematic error to the ideal model… Estimate  by minimization of  2 ( s : systematic shift,  : observable ) Then, letting, again and (for simplicity suppose  i =  ( =  2 /N +  2 )

47. Sept 2003Phystat47 Reasons We keep the normalization factors fixed as we vary the point in parameter space. The criterion  2 = 1 requires that the systematic shifts be continually optimized versus {a  }. Systematic errors may be nongaussian. The published “standard deviations”  ij may be inaccurate. We trust our physics judgement instead. Still we do not apply the criterion  2 = 1 !

48. Sept 2003Phystat48 To judge the PDF uncertainty, we return to the individual experiments. Lumping all the data together in one variable –  2 global – is too constraining. Global analysis is a compromise. All data sets should be fit reasonably well -- that is what we check. As we vary {a  }, does any experiment rule out the displacement from the standard set?

49. Sept 2003Phystat49 In testing the goodness of fit, we keep the normalization factors (i.e., optimized luminosity shifts) fixed as we vary the shape parameters. End result e.g., ~100 for ~2000 data points. This does not contradict the  2 = 1 criterion used by other groups, because that refers to a different  2 in which the normalization factors are continually optimized as the {a  } vary.

50. Sept 2003Phystat50 Some groups do use the criterion of  2 = 1 for PDF error analysis. Often they are using limited data sets – e.g., an experimental group using only their own data. Then the  2 = 1 criterion may underestimate the uncertainty implied by systematic differences between experiments. An interesting compendium of methods, by R. Thorne CTEQ6  2 = 100 (fixed norms) ZEUS  2 = 50 (effective) MRST01  2 = 20 H1  2 = 1 Alekhin  2 = 1 GKK not using  

51. Sept 2003Phystat51 Uncertainties of Parton Distributions (II) Results

52. Sept 2003Phystat52 Estimate the uncertainty on the predicted cross section for pp bar  W+X at the Tevatron collider. global  2 local  2 ’s

53. Sept 2003Phystat53 Each experiment defines a “prediction” and a “range”. This figure shows the  2 = 1 ranges.

54. Sept 2003Phystat54 This figure shows broader ranges for each experiment based on the “90% confidence level” (cumulative distribution function of the rescaled  2 ).

55. Sept 2003Phystat55 The final result is an uncertainty range for the prediction of  W. Survey of  w  B l  predictions (by R. Thorne) … PDF setenergy  w  B l  nb  PDF uncert AlekhinTevatron2.73  0.05 MRST2002Tevatron2.59  0.03 CTEQ6Tevatron2.54  0.10 AlekhinLHC215.  6. MRST2002LHC204.  4. CTEQ6LHC205.  8.

56. Sept 2003Phystat56 How well can we determine the value of  S ( M Z ) from Global Analysis? For each value of  S, find the best global fit. Then look at the  2 value for each experiment as a function of  S.

57. Sept 2003Phystat57 Each experiment defines a “prediction” and a “range”. This figure shows the  2 = 1 ranges. Particle data group (shaded strip) is 0.117  0.002. The fluctuations are larger than expected for normal statistics. The vertical lines have  2 global =100,  s (MZ)=0.1165  0.0065

58. Sept 2003Phystat58 Uncertainties of the PDF’s themselves (only interesting to the model builders) Gluon and U quark at Q 2 = 10 GeV 2.

59. Sept 2003Phystat59 Comparing “alternate sets” Gluon at Q 2 = 10 GeV 2 U quark at Q 2 = 10 GeV 2 red – CTEQ6.1blue – Fermi2002 (H1, BCDMS, E665)

60. Sept 2003Phystat60 CTEQ error band with MRST2002 superimposed Q 2 = 10 GeV 2

61. Sept 2003Phystat61 Uncertainties of LHC parton-parton luminosities Provides simple estimates of PDF uncertainties at the LHC.

62. Sept 2003Phystat62 Outlook Necessary infrastructure for hadron colliders Tools exist to study uncertainties. This physics is data driven -- HERA II and Fermilab Run 2 will contribute. Ready for the LHC

63. Sept 2003Phystat63 Cases

64. Sept 2003Phystat64 Inclusive jet production and the search for new physics (hep-ph/0303013) Inclusive jet cross section : D0 data and 40 alternate PDF sets Fractional differences

65. Sept 2003Phystat65 Is there room for new physics from Run Ib? Contact interaction model with  = 1.6, 2.0, 2.4 TeV

66. Sept 2003Phystat66 The inclusive jet cross section versus pT for 3 rapidity bins at the LHC. Predictions of all 40 eigenvector basis sets are superimposed.

67. Sept 2003Phystat67 Strangeness asymmetry The NuTeV Collaboration has measured the cross sections for -Fe and -Fe to     X. A significant fraction of the CS comes from  s and bar s bar interactions. We have added this data into the global fit to determine

68. Sept 2003Phystat68 Figure 1. Typical strangeness asymmetry s  (x) and the associated momentum asymmetry S  (x). The axes are chosen such that both large and small x regions are adepquately represented, and that the area iunder each curve equals the correponding integral. [S-] values A : 0.312 x 10  3 B : 0.160 x 10  3 C : 0.103 x 10  3

69. Sept 2003Phystat69 Figure 2. Correlation between  2 values and [S  ] Red: dimuon cross section Blue: other data sensitive to s  s bar (F3)

70. Sept 2003Phystat70 Figure 3. Comparison of the s  (x) and S  (x) functions for three PDF sets: our central fit “B” (dot-dash) BPZ (blue) NuTeV (red)