Progress On Studying Uncertainties Of Parton Distributions And Their Physical Predictions

Transcription

1 Progress On Studying Uncertainties Of Parton Distributions And Their Physical Predictions Brief summary of traditional global QCD analysis What s uncertain about parton distribution functions? Studying uncertainties of parton distributions and their physical predictions: Ideal approaches in a world with perfect experiments Progress on a realistic global analysis of uncertainties General approach and techniques two complementary methods: (J. Pumplin, D. Stump) The Hessian method: orthonormal basis PDF sets, and a general master equation for calculating uncertainties of any quantity dependent on PDFs The Lagrange multiplier method: robust and optimized PDFs for quantifying uncertainties of specific physical variables Results and physical applications W/Z cross-sections at Tevatron/LHC/RHIC W rapidity distribution, (W-mass measurement) Parton Luminosities at Tevatron/LHC/RHIC/VLHC (Higgs-, top-x-sections, high p T jets,... etc.) Outlook

2 Global QCD Analysis in a Nutshell Master Equation for QCD Parton Model the Factorization Theorem F A (x; m Q ; M Q ) = X a f a A (x; m μ ) Ω ^F a (x; Q μ ; M Q )+O((Λ Q )2 ) F A H F ^ a a H A Experimental Input Parton Dist. Fn. Non-Perturbative Parametrization at Q0 GLAP Evolution to Q A f A a Hard Cross-section perturbative calculable (may contain s n Log n (M/Q)) Sources of Uncertainties and Challenges: Experimental errors; (and uncertainties on errors!) Parametrization dependence; Higher-order corrections; Large Logarithms; Power-law (higher twist) corrections.

3 Experimental Input: Phyical Processes & Experiments DIS e N N *, W, Z q SLAC BCDMS NMC, E665 H1, ZEUS N N g q CDHS, CHARM CCFR CHORUS DY p N N k N p N q q q q *, W, Z *, W, Z E605, E772 NA51 E866 CDF, D0 Dir.Ph. (dir) p N N k N p N g q WA70, UA6 E706 CDF, D0 Jet Inc. p p CDF, D0

4 The kinematic range in the (x, Q) plane of data points included in a typical global QCD analysis CTEQ DIS (fixed target) HERA ( 94) DY W-asymmetry Direct-γ Jets /X Q (GeV)

5 Global QCD Fit (CTEQ) * Parametrization of the non-perturbative PDFs: (at Q 0 = 1 GeV), e.g. f i (x, Q 0 )=a i 0 xai 1(1 x) ai 2(1 + a i 3 xai 4). * The fitting is done by minimizing a global effective chi-square function, χ 2 global, overall figure of merit of the fit regulates trade-off s between the many expts. χ 2 g = n χ 2 n (n labels the experiments) ( ) 2 χ 2 1 Nn n = σn N + w n I ( Nn D ni T ni (a) σ D ni D ni : data point σni D : combined error T ni (a): theory value (dependent on {a i }) for the I th data point in experiment n. N n : Normalization para. for expt. n. w n : a prior based on physics considerations (1: inclusion; 0: exclusion; or other). * This effective χ 2 function does not have the full probabilistic significance of an ideal statistical analysis. (cf. uncertainty section.) ) 2

6 Overview of Parton Distribution Functions of the Proton CTEQ5M Q = 5 GeV Gluon / 15 d bar u bar s c u v d v (d bar -u bar ) * x x f(x,q)

7

8 CTEQ5M t to E866 : pd 2 pp (x2) E866 CTEQ5M Xsec(pd)/2Xsec(pp) X2 Fit to CDF W-lepton asymmetry: CDF W-lepton Asymmetry CTEQ5M CDF data Y

9 Comparison of CDF inclusive jet cross-section measurement with NLO QCD calculations with CTEQ5M and CTEQ5HJ PDFs Ratio: Prel. data / NLO QCD (CTEQ5M CTEQ5HJ) CTEQ5M : 1.00 norm. facor : CTEQ5HJ: 1.04 CDF Incl. Jet : p t * dσ/dpt (10-14 nb GeV 6 ) Data / CTEQ5M CTEQ5HJ / CTEQ5M CDF Data ( Prel. ) CTEQ5HJ CTEQ5M (Error bars: statistical only) 14% < Corr. Sys. Err. < 27% p T (GeV)

10 Inclusive Jet Production Data of D0 -- compared to NLO QCD using CTEQ4HJ (CTEQ4M)

11 What's Uncertain about PDFs? Quite a lot! Strange and anti-strange Quarks Details in the {u,d} quark sector: Up/Down differences and ratios (important for precision W/Z physics at colliders) The Gluon Distribution (important for most SM and New Physics processes at very high energies) Heavy Quark Distributions: Very little is known! Are heavy quarks radiatively-generated" exclusively; or are there intrinsic components of charm and/or bottom? It is important to quantify the uncertainties of the parton distributions functions (PDF s) and, more importantly, physical predictions which depend on PDF s not just the bands, but a systematic knowledge of the correlated variations.

12 !" #$&%&'( )+*," -/.$0/ &78./" 8 9 :, ;< )=",-/.$0/12>?-/.$0/1;@9 Ä (% >" ", B< )C" :39;D.E Ä ;FG H B HI(JK7 L

13 Percentage Uncertainties of Parton Distributions (By M. Botje) Sources of uncertainties: ffl Experimental statistical and systematic errors; ffl ffl ffl input": various theoretical corrections; analysis": phenom. analysis procedure; Scale": PQCD renor. and fact. scale-dependence

14 Figure 9: The parton momentum densities xg, xs (both divided by a factor of 20), xu v and xd v versus x at Q 2 =10GeV 2. The full curves show the results from the QCD fit with the errors drawn as shaded bands. The dashed curves are from the CTEQ4 [8] parton distribution set. 25

15 Global Analyses using an Effective χ 2 Function Stress inclusion of all relevant (global) experimental data to constraint the PDFs; Intend to be logical & useful extension of conventional, practical global QCD analyses. Use an effective χ 2 -analysis with function χ 2 global Use a collection of DIS experiments S. Alekhin, hep-ph/ V. Barone, C. Pascaud and F. Zomer, Eur. Phys. J. C12, 243 (2000) [hep-ph/ ]. M. Botje, Eur. Phys. J. C14, 285 (2000) [hep-ph/ ]. Use full global analysis data set of CTEQ5, + some key advances in methodology J. Pumplin, D. Stump, R. Brock, D. Casey, J. Huston, J. Kalk, WKT (MSU) hep-ph/ , ,

16 Bird's Eye View of the MSU Study (Pumplin, Stump, WKT, et al.) Inputs: Experimental Inputs: (i) full global data set of CTEQ5; (ii) effective global 2 function: Choice of priors: NLO QCD ; other theory models; Parametrization of nonperturbative PDFs. 2-dim illustration of the neighborhood of the global minimum in the 16-dim parton parameter space a j L X 2 - contours a i "Sampling" 2 complementary optimized systematic methods: (i) Lagrange multiplier; (ii) Hessian matrix. Error Estimates 2 distinct steps: Output: (i) calc. CL in each expt.; (ii) estimate overall uncertainty. For physical variable X: Std.Min. set S 0 : X Alternate hypotheses sets {S ± i } i=1,..2n: X

17

18 First Test of whether the effective global 2 method makes any sense: look at the distribution of the fluctuations of the 1300 data points included in the global fit. Comparison of data and CTEQ5m èëtheory"è 800 all data 600 dn dx x m t Σ The histogram includes all data used in the æt except jet production. The curve has no adjustable parameters; it's just N expè,x 2 è= p 2ç where N is the number of data points. The area under the curve èor histogramè is N. Diæerences m i,t i errors. are within the published measurement At least globally the distribution of æuctuations is Gaussian with the right width.

19 Lagrange Multiplier Method: (specific but robust) Let X be a physical quantity, then minimize Ψ = χ 2 g + λx to probe the neighborhood of the minimum. 2-dim (i,j) rendering of d-dim PDF parameter space a j L X X: physics variable contours of c 2 global a i MC sampling LM method Sample variation of χ 2 g as a function of X W by {SX m 1350 },m=1, 2...; 1320 Obtain PDF sets 1290 T 2 {S X ± 1260 } which extremize 1230 the variation of X for a given tolerance χ of χ 2 global W nb 2 global W production at the Tevatron Uncertainty of X: X = 1 2 (X(S+ X ) X(S X )), Apply to, e.g. σ W/Z, σ Top, σ Higgs,..., at the Tevatron, RHIC and LHC.

20 Error Estimate: 1. Use all information provided by each individual experiments to calculate the (90%) CL w.r.t. that experiment. Example, H1 expt. (with full correlated error matrix): This 2 function does have statistical significance! 1.1 The dashed horizontal line is the % CL level The error bar w.r.t. the H1 expt. is shown by the red line with arrows. H1 2 N nb Tevatron 2. Make the 90% error bar calc. for all expts. W 25 Tevatron W nb BCDMSp BCDMSd H1 ZEUS NMCp NMCr NMCrx CCFR2 CCFR3 E605 NA51 CDFw E866 D0jet CDFjet W In view of the problem with interpretation of the absolute 2 for some real experiments, use relative w.r.t. the best estimate solution S 0.

21 Optimized Sampling of PDF Parameter Space Alternate Hypotheses in uncertainty study Hessian Matrix: (general but approx.) Diagonalize the Hessian matrix calculated from χ 2 g, then move along each of the eigenvectors, i, to get up/down PDF sets {S i ± }. 2-dim (i,j) rendition of d-dim (~16) PDF parameter space a j u l p(i) contours of constant 2 global u l : eigenvector in the l-direction p(i): point of largest a i with tolerance T s 0 : global minimum z l p(i) diagonalization and T s rescaling by s 0 0 the iterative method a i (a) Original parameter basis Hessian eigenvector basis sets u l z k (b) Orthonormal eigenvector basis PDF sets {S i ± }, i =1,...,n, spanning the full PDF parameter space in the neighborhood of the global mimimum; The uncertainty of any physical variable X can be calculated as: (the master eq.) X = T 2t i (X(S + i ) X(S i ))2

22 7KH+HVVLDQ0HWKRGRITXDQWLI\LQJXQFHUWDLQWLHVE\D FRPSOHWHVHWRIRUWKRQRUPDO HLJHQYHFWRU3')V

23 Challenges to Numerical Calculation of the Hessian Matrix in 16-dimensional PDF parameter space Global landscape:- physics * gradients vary widely: steep / flatness measured by eigen values of Hessian: ~ 10 6 a j a i Local bumpiness: unphysical various reasons: theory model is NOT smooth!; finite steps, MC methods,... It is well known that general purpose minimization programs do not provide reliable error estimates for global analysis. (Step size choices not designed to deal with these problems)

24 Eigenvalues of the Hessian Matrix for two different choices of parametrization of the PDF parameter space

25 Iterative Method to generate Eigenvectors: (and dramatically improve numerical reliability) the χ 2 = const. ellipsoid a j 0 Physical parameters a i 0 a j n X 2 2 X = χ ij ( H a i 1 ) ij X a j nth iteration a i n Nth iteration: Eigenvector orthonormal basis X 2 a j N = z j S i S j 2 = χ ( X i ) z i S j + 2 a i N = z i + i = [ X ( S ) X ( S i S i + i )] 2

26 * This iterative method has been shown by Botje and Zomer to greatly improve their own global analyses, and it is now being considered for adoption in MINUIT (as an option) by its author F. James.

27 2 Considerations in Estimating the Tolerance T of χ global ( χ 2 global < T 2 ) χ 2 global = 1 has no statistical significance in this context. Most experiments only give a combined ( effective ) systematic error; Combining χ 2 s of 15 experiments on very diverse processes and accuracies is an extremely dicey business. Basic assumption of 15 acceptable experiments, in spite of the fact that some are nominally inconsistent in the strict statistical sense, implies T >> 1. Basic assumption of compatibility of the 15 experiments, in spite of the fact that some are nominally incompatible in the strict statistical sense, implies T >> 1. We can measure the acceptable T for compatibility. Quantitative estimate of T from comparison of sample PDFs (the Alternate Hypotheses) obtained from Hessian and Lagrange methods to the individual experiments. o Evaluate the 90% CL range for each experiment based on available error information (using χ 2 normalized to standard fit, if necessary) o Combine these ranges into one estimated tolerance for Τ Bottom Line: 10 < T < 15

28 Gluon Up quark Two extreme parton distributions with T = 10 at Q = 2 GeV and Q = 100 GeV (in dashed lines). The uncertainty band for T < 10 is shaded. (T defines the tolerance: 2 global < T 2 )

29 Gluon Up-quark Ratio of parton distribution to Best Fit distribution for two extreme T = 10 cases at Q = 10 GeV. The region allowed by T < 10 is shaded.

30 Predicted rapidity distribution for W production at the Tevatron Six curves represent the alternate hypotheses of one ± 2 for the physical variables t, <y>, and <y2 >.

31 Predicted rapidity distribution for W production at the Tevatron -- deviation from the best estimate solution S 0 Six curves represent the alternate hypotheses of one ± 2 for the physical variables t, <y>, and <y2 >.

32 Compare the results obtained by the Lagrange and Hessian methods Predicted rapidity distribution for W production at the Tevatron -- deviation from the best estimate solution S 0 Three (of the six) curves for each case representing the "extremes" for the physical variables t, <y>, and <y2>. Since the LM method is fully robust (no approx.), the agreement proves the efficacy of the the Hessian method.

33 Correlated uncertainties in W/Z production Cross-section at the Tevatron CDF ** D0 Ellipse : Allowed region with Tolerance = 10. ** CDF with the same luminosity input as D0

34 Fractional Uncertainties Fractional Luminosity Uncertainties at Tevatron Run II GG QQbar γ QQbar W + QQbar Z GQ γ GQ W W, Z Sqrt (s^)

35 0.3 Fractional Luminosity Uncertainties at Tevatron Run II Fractional Uncertainties GG QQbar γ QQbar W + QQbar Z GQ γ GQ W - 0 Sqrt (s^) W, Z 115 GeV 10 2

36 0.25 Luminosity Uncertainties at LHC Fractional Uncertainties GG QQbar γ QQbar W + GQ W + GQ Z 0 W, Z 115 GeV sqrt (s^)

37 0.6 Luminosity Uncertainties at RHIC 500 GeV 0.5 Fractional Uncertainties GG QQbar γ QQbar W + GQ W + GQ W W, Z sqrt (s^)

38 1 0.9 Luminosity Uncertainties at RHIC 200 GeV Fractional Uncertainties GG QQbar γ QQbar W + QQbar W - GQ γ GQ W W, Z sqrt (s^)

39 Outlook This is only the very beginning of studying uncertainties in global QCD analysis in a quantitative manner It should be regarded more as a demonstration of principles. There is a lot of room for collaboration among theorists and experimentalists Several data sets are about to be updated (CCFR, H1, ZEUS,... ) Specific results of this study will likely change soon. Many other sources of uncertainties in the overall global analysis have not yet been incorporated: Theoretical uncertainties due to higher-order PQCD corrections and resummation; Uncertainties introduced by the choice of parametrization (This has been partially explored by us.) The Hessian eigenvectors provide a systematic way to distinguish flat/steep directions in parameter space; and to suggest ways to improve the choice of parametrization. Continued progress in this venture is of vital importance for our understanding of the parton structure of hadrons (fundamental physics of its own right), for precision SM physics studies at future colliders, and for New Physics searches.