Uncertainties of Parton Distribution Functions. Daniel Stump. Michigan State University & CTEQ

Transcription

1 Uncertainties of Parton Distribution Functions Daniel Stump Michigan State University & CTEQ Sept 003 Phystat 1

2 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. Sept 003 Phystat

3 Parton distribution functions are important. Sept 003 Phystat 3

4 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. Sept 003 Phystat 4

5 The systematic study of uncertainties of PDF s developed slowly. Pioneers J. Collins and D. Soper, CTEQ Note 94/01, hep-ph/ C. Pascaud and F. Zomer, LAL M. Botje, Eur. Phys. J. C 14, 85 (000). Today many groups and individuals are involved in this research. Sept 003 Phystat 5

6 Current research on PDF uncertainties 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 Sept 003 Phystat 6

7 Outline of this talk (focusing on CTEQ results) General comments; CTEQ6 Our treatment of experimental systematic errors Compatibility of data sets Uncertainty analysis case studies inclusive jet production in pp bar or pp strangeness asymmetry Sept 003 Phystat 7

8 Global Analysis 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. Sept 003 Phystat 8

9 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. Sept 003 Phystat 9

10 Parametrization At low Q 0, of order 1 GeV, f ( a 1 x, Q0 ) = a0 x (1 x) P( x) a P(x) has a few more parameters for increased flexibility. ~ 0 free shape parameters Q dependence of f(x,q) is obtained by solving the QCD evolution equations (DGLAP). Sept 003 Phystat 10

11 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 σ/dη dpt Sept 003 Phystat 11

12 Global Analysis data from many disparate experiments Sept 003 Phystat 1

13 The Parton Distribution Functions Sept 003 Phystat 13

14 Different ways to plot the parton distributions Linear Logarithmic Q = 10 (solid) and 1000 (dashed) GeV Sept 003 Phystat 14

15 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} Sept 003 Phystat 15

16 Comparison of CTEQ6 and MRST00 blue curves : CTEQ6M black dots : MRTS00 gluon and u quark at Q = 10 GeV Sept 003 Phystat 16

17 Our treatment of systematic errors Sept 003 Phystat 17

18 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 Sept 003 Phystat 18

19 Imagine that two experimental groups have measured a quantity θ, with the results shown. OK, what is the value of θ? 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. Sept 003 Phystat 19

20 We use χ minimization with fitting of systematic errors. For statistical errors define χ N = i= 1 ( D T ) i σ i i Di :data value Ti : theoretical value σ i :statistical error (S. D.) T i = T i (a 1, a,..,, a d ) a function of d theory parameters Minimize χ w. r. t. {a µ } optimal parameter values {a 0µ }. All this would be based on the assumption that D i = T i (a 0 ) + σ i r i dp = e r / π dr Sept 003 Phystat 0

21 Treatment of the normalization error In scattering experiments there is an overall normalization uncertainty from uncertainty of the luminosity. We define χ = σ ( a, f 1 N ) f N norm + N i= 1 ( f ) N D i T i σ i where f N = overall normalization factor Minimize χ w. r. t. both {a µ } and f N. Sept 003 Phystat 1

22 A method for general systematic errors D i χ i K 0) + αir i + j= 1 = T ( a β rˆ Define = N i= 1 ( ) D β s T i j α ij i j ij j i + K j= 1 s j α i : statistical error of D i β ij : set of systematic errors (j=1 K) of D i quadratic penalty term Minimize χ with respect to both shape parameters {a µ } and optimized systematic shifts {s j }. Sept 003 Phystat

23 Because χ depends quadratically on {s j } we can solve for the systematic shifts analytically, s s 0 (a). Then let, χ ( ) ( a) = χ global a, s ( a) 0 experiments and minimize w.r.t {a µ }. The systematic shifts {s j } are continually optimized [ s s 0 (a) ] Sept 003 Phystat 3

24 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 χ global (1 f ({ a},{ fn}) = weχ e ({ a},{ fn}) + wn, e e e σ N, 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!) e e ) Sept 003 Phystat 4

25 Quality How well does this fitting procedure work? Sept 003 Phystat 5

26 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. Sept 003 Phystat 6

27 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. Sept 003 Phystat 7

28 How large are the optimized normalization factors? Expt BCDMS H1 (a) H1 (b) ZEUS NMC CCFR E605 D0 CDF f N Sept 003 Phystat 8

29 We must always check that the systematic shifts are not unreasonably large. 10 systematic shifts NMC data j sj systematic shifts ZEUS data j sj Sept 003 Phystat 9

30 Comparison to NMC F i = ( D s β T ) / α i i i i without systematic shifts Sept 003 Phystat 30

31 A study of compatibility Sept 003 Phystat 31

32 N χ χ/n Table of Data Sets The PDF s are not exactly CTEQ6 but very close a no-name generic set of PDF s for illustration purposes BCDMS Fp BCDMS Fd H1 (a) H1 (b) H1 (c ) ZEUS CDHSW F NMC Fp NMC d/p CCFR F N tot = 91 χ global = E605 E866 pp E866 d/p D0 jet CDF jet CDHSW F3 CCFR F3 CDF W Lasy Sept 003 Phystat 3

33 The effect of setting all normalization constants to 1. χ BCDMS Fp BCDMS Fd H1 (a) H1 (b) H1 (c ) NMC Fp E605 E866 pp χ (opt. norm) = 368. χ (norm 1) = 74. χ = Sept 003 Phystat 33

34 By applying weighting factors in the fitting function, we can test the compatibility of disparate data sets. Example 1. The effect of giving the CCFR F data set a heavy weight. Sept 003 Phystat H1 (a) CDHSW F NMC Fp CCFR F E866 pp D0 jet χ (CCFR) = 19.7 χ (other) = 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. χ

35 Example 1b. The effect of giving the CCFR F data weight 0, i.e., removing the data set from the global analysis. χ H1 (a) ZEUS NMC Fp CCFR F χ (CCFR) = χ (other) = 17.4 Imagine starting with the other data sets, not including CCFR. The result of adding CCFR is that χ global of the other sets increases by 17.4 ; this must be an acceptable increase of χ. Sept 003 Phystat 35

36 Example 5. Giving heavy weight to H1 and BCDMS χ for all data sets χ BCDMS Fd H1 (a) H1 (b) ZEUS CDHSW F NMC Fp CCFR F D0 jet CDHSW F3 CCFR F3 χ (H & B) = 38.7 χ (other) = Sept 003 Phystat 36

37 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 χ must be large. Adding a new data set and refitting may increase the χ s of other data sets by amounts >> 1. Sept 003 Phystat 37

38 Clever ways to test the compatibility of disparate data sets Plot χ versus χ J Collins and J Pumplin (hep-ph/001195) The Bootstrap Method Efron and Tibshirani, Introduction to the Bootstrap (Chapman&Hall) Chernick, Bootstrap Methods (Wiley) Sept 003 Phystat 38

39 Uncertainty Analysis (I) Methods Sept 003 Phystat 39

40 We continue to use χ global as figure of merit. Explore the variation of χ global in the neighborhood of the minimum. a nearby points are also acceptable the standard fit, minimum χ a 1 The Hessian method H µν 1 a µ χ a ν 0 (µ, ν = 1 3 d) Sept 003 Phystat 40

41 Classical error formula for a variable X(a) X a ( X ) = χ ( 1 H ) µ, ν µ µν X a Obtain better convergence using eigenvectors of H µν d ( ) [ ( ) ( )] ( + ) ( ) X = X S X S µ = 1 Master Formula S µ (+) and S µ ( ) denote PDF sets displaced from the standard set, along the ± directions of the µ th eigenvector, by distance T = ( χ) in parameter space. (available in the LHAPDF format : d alternate sets) µ µ ν Sept 003 Phystat 41

42 The Lagrange Multiplier Method for analyzing the uncertainty of PDFdependent predictions. The fitting function for constrained fits ( a, λ) χ ( a ) X ( a ) F + µ = global µ λ Minimization of F [w.r.t {a µ } and λ] gives the best fit for the value X(a min,µ ) of the variable X. Hence we obtain a curve of χ global versus X. µ λ : Lagrange multiplier controlled by the parameter λ Sept 003 Phystat 4

43 The question of tolerance X : any variable that depends on PDF s X 0 : the prediction in the standard set χ(x) : curve of constrained fits For the specified tolerance ( χ = T ) there is a corresponding range of uncertainty, ± X. What should we use for T? Sept 003 Phystat 43

44 Estimation of parameters in Gaussian error analysis would have T = 1 We do not use this criterion. Sept 003 Phystat 44

45 Aside: The familiar ideal example Consider N measurements {θ i } of a quantity θ with normal errors {σ i } θ = θ + σ r i true Estimate θ by minimization of χ, i= 1 i i i N ( θi θ ) χ ( θ ) = θcombined = σ The mean of θ combined is θ true, the SD is and χ ( θ c ± c c θ ) χ ( θ ) = 1. i θ / σ i i 1/ σ i i θ c = 1/ σ i i 1/ The proof of this theorem is straightforward. It does not apply to our problem because of systematic errors. dp = e -r ( = σ / N ) / π Sept 003 Phystat 45

46 Add a systematic error to the ideal model θ = θ + σ r + βi r ~ i true i Estimate θ by minimization of χ i N ( θi βis θ ) θ, s) = + i= 1 σ i χ ( s θ combined = i θ / σ i i 1/ σ i i and (for simplicity suppose β i = β ) ( s : systematic shift, θ : observable ) 1 ( θ c) = + β 1/ σ i i Then, letting χ θ ) χ [ θ, s ( θ )], again ( 0 ( = σ /N + β ) χ ( θ c ± θc ) χ ( θc) = 1. Sept 003 Phystat 46

47 Still we do not apply the criterion χ = 1! Reasons We keep the normalization factors fixed as we vary the point in parameter space. The criterion χ = 1 requires that the systematic shifts be continually optimized versus {a µ }. Systematic errors may be nongaussian. The published standard deviations β ij may be inaccurate. We trust our physics judgement instead. Sept 003 Phystat 47

48 To judge the PDF uncertainty, we return to the individual experiments. Lumping all the data together in one variable χ 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 µ }, does any experiment rule out the displacement from the standard set? Sept 003 Phystat 48

49 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 χ fixed norms >> 1 e.g., ~100 for ~000 data points. This does not contradict the χ = 1 criterion used by other groups, because that refers to a different χ in which the normalization factors are continually optimized as the {a µ } vary. Sept 003 Phystat 49

50 Some groups do use the criterion of χ = 1 for PDF error analysis. Often they are using limited data sets e.g., an experimental group using only their own data. Then the χ = 1 criterion may underestimate the uncertainty implied by systematic differences between experiments. An interesting compendium of methods, by R. Thorne CTEQ6 ZEUS MRST01 H1 Alekhin GKK χ = 100 (fixed norms) χ = 50 (effective) χ = 0 χ = 1 χ = 1 not using χ Sept 003 Phystat 50

51 Uncertainties of Parton Distributions (II) Results Sept 003 Phystat 51

52 Estimate the uncertainty on the predicted cross section for pp bar W+X at the Tevatron collider. global χ local χ s Sept 003 Phystat 5

53 Each experiment defines a prediction and a range. This figure shows the χ = 1 ranges. Sept 003 Phystat 53

54 This figure shows broader ranges for each experiment based on the 90% confidence level (cumulative distribution function of the rescaled χ ). Sept 003 Phystat 54

55 The final result is an uncertainty range for the prediction of σ W. Survey of σ w B lν predictions (by R. Thorne) PDF set energy σ w B lν [nb] PDF uncert Alekhin Tevatron.73 ± 0.05 MRST00 Tevatron.59 ± 0.03 CTEQ6 Tevatron.54 ± 0.10 Alekhin LHC 15. ± 6. MRST00 LHC 04. ± 4. CTEQ6 LHC 05. ± 8. Sept 003 Phystat 55

56 How well can we determine the value of α S ( M Z ) from Global Analysis? For each value of α S, find the best global fit. Then look at the χ value for each experiment as a function of α S. Sept 003 Phystat 56

57 Each experiment defines a prediction and a range. This figure shows the χ = 1 ranges. Particle data group (shaded strip) is 0.117±0.00. The fluctuations are larger than expected for normal statistics. The vertical lines have χ global=100, α s (MZ)=0.1165± Sept 003 Phystat 57

58 Uncertainties of the PDF s themselves (only interesting to the model builders) Gluon and U quark at Q = 10 GeV. Sept 003 Phystat 58

59 Comparing alternate sets red CTEQ6.1 blue Fermi00 (H1, BCDMS, E665) Gluon at Q = 10 GeV U quark at Q = 10 GeV Sept 003 Phystat 59

60 CTEQ error band with MRST00 superimposed Q = 10 GeV Sept 003 Phystat 60

61 Uncertainties of LHC parton-parton luminosities Lum( sˆ) C f ( x ) f ( x ) δ (ˆ s x x s) dx dx = ij i, j i 1 Provides simple estimates of PDF uncertainties at the LHC. Sept 003 Phystat 61 j 1 1

62 Outlook Necessary infrastructure for hadron colliders Tools exist to study uncertainties. This physics is data driven -- HERA II and Fermilab Run will contribute. Ready for the LHC Sept 003 Phystat 6

63 Cases Sept 003 Phystat 63

64 Inclusive jet production and the search for new physics Inclusive jet cross section : D0 data and 40 alternate PDF sets Fractional differences (hep-ph/ ) Sept 003 Phystat 64

65 Is there room for new physics from Run Ib? Contact interaction model with Λ = 1.6,.0,.4 TeV Sept 003 Phystat 65

66 The inclusive jet cross section versus pt for 3 rapidity bins at the LHC. Predictions of all 40 eigenvector basis sets are superimposed. Sept 003 Phystat 66

67 Sept 003 Phystat 67 Strangeness asymmetry The NuTeV Collaboration has measured the cross sections for ν-fe and ν-fe to µ + µ X. A significant fraction of the CS comes from ν s and ν bar s bar interactions. We have added this data into the global fit to determine ± ± ± ± ± ± = ± = ± ) ( ] [ and )} ( ) ( { ) ( ) ( ] [ and ) ( ) ( ) ( ), ( and ), ( dx x S S x s x s x x S dx x s s x s x s x s Q x s Q x s

68 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 : 0.31 x 10 3 B : x 10 3 C : x 10 3 Sept 003 Phystat 68

69 Figure. Correlation between χ values and [S ] Red: dimuon cross section Blue: other data sensitive to s s bar (F3) Sept 003 Phystat 69

70 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) Sept 003 Phystat 70