F λ A (x, m Q, M Q ) = a

Transcription

1 Parton Distributions and their Uncertainties Jon Pumplin DPF22 Williamfburg 5/25/2 CTEQ6 PDF analysis (J. Pumplin, D. Stump, W.K. Tung, J. Huston, H. Lai, P. Nadolsky [hep-ph/21195]) include new data sets include correlated systematic experimental errors evaluate uncertainties of the result: Eigenvector PDF sets to map uncertainties Lagrange multiplier results Universal PDF interface: Les Houches Accord Results: W and Z production parton-parton luminosities gluon and quark distributions Measures of uncertainty Measurement of α s Statistical bootstraps Outlook 1

2 Overview of QCD Global Analysis F A H = ^ F a a H Hard Scattering: (Perturbatively Calculable) A A f A a Experimental Input Parton Distributions: Nonperturbative parametrization at Q DGLAP Evolution to Q F λ A (x, m Q, M Q ) = a f a A (x, m µ ) ^F λ a (x, Q µ, M Q ) + O((Λ Q )2 ) Sources of uncertainty: 1. Experimental errors included in χ 2 2. Unknown experimental errors 3. Parametrization dependence 4. Higher-order corrections & Large Logarithms 5. Power Law corrections ( higher twist") Fundamental difficulties: 1. Good experiments run until systematic errors dominate; and the magnitude of systematic errors involves guesswork. 2. Systematic errors of the theory and their correlations cannot even be guessed. 2

3 Experimental Input 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 *, W, Z E65, E772 NA51 E866 Dir.Ph. q q *, W, Z (dir) CDF, D p N N k N p N g q WA7, UA6 E76 CDF, D Jet Inc. p p CDF, D 3

4 Kinematic region covered by data A wide variety of data are tied together by the DGLAP renormalization group evolution equation. Consistency or lack thereof between the experiments can be observed only by applying QCD to tie them together in a global fit. All experiments that use hadrons in the initial state Tevatron, LHC, and non-accelerator experiments require the parton distributions for their analysis. 4

5 Selection of Data CTEQ5 CTEQ6 # sys # sys BCDMS µp 168 no BCDMS µp 339 yes BCDMS µd 156 no BCDMS µd 251 yes H1 ep 172 no H1a ep 14 yes H1b ep 126 yes ZEUS ep 186 no ZEUS ep 229 yes NMC µp 14 no NMC µp 21 yes NMC µp/µn 123 no NMC µp/µn 123 yes CCFR F 2 νn 87 no CCFR F 2 νn 159 yes CCFR F 3 νn 87 no CCFR F 3 νn 87 no E65 pp DY 119 no E65 pp 119 no NA51 pd/pp DY 1 no NA51 pd/pp 1 no E866 pd/pp DY 15 no E866 pd/pp 15 no CDF W 11 no CDF W 11 no CDF jet 33 yes CDF jet 33 yes Dχjet 24 yes DχJet 9 yes New Data (Direct photon data are not used because of uncontrolled systematic k T " effects, which need resummation) 5

6 CTEQ6 Global analysis Input from Experiment: 2 data points with Q>2GeV from e, µ, ν DIS; lepton pair production (DY); lepton asymmetry in W production; high p T inclusive jets; α s (M Z ) from LEP Input from Theory: NLO QCD evolution and hard scattering Parametrize at Q : A x A 1 (1 x) A 2 (1 + A 3 x A 4) s = μs =.4(μu + μ d)/2 at Q ; no intrinsic b or c Construct effective χ 2 global = expts χ 2 n: χ 2 global includes the known systematic errors Minimizing χ 2 global yields Best Fit" PDFs. Variation of χ 2 global in neighborhood of the minimum defines uncertainty limits. Estimate uncertainty as region of parameter space where χ 2 <χ 2 (BestFit) + T 2 with T 1. (Quite different from Gaussian statistics because of unknown correlated systematic errors in theory and experiments as measured by inconsistency between experiments). 6

7 Comment on Parametrization For d val, u val, or g, we use xf(x, Q ) = A x A 1 (1 x) A 2 e A 3x (1 + e A 4x) A 5 This corresponds to d dx ln (xf) = A 1 x A 2 1 x + c 3 + c 4 x 1 + c 5 x i.e., we add a 1:1 Padé form to the singular terms of the traditional A x A 1 (1 x) A 2 parametrization. A sufficiently flexible parametrization is important; but for convergence, there must not be too many flat directions." For that reason, some of the parameters are frozen for some flavors. (To measure a set of continuous PDF functions at Q on the basis of a finite set of data points would appear to be an ill-posed mathematical problem. However, this difficulty is not so severe as might be expected since the actual predictions of interest that are based on the PDFs are discrete quantities. In particular, fine-scale structure in x in the PDFs at Q tend to be smoothed out by evolution in Q. They correspond to flat directions in χ 2 space, so they are not accurately measured; but they have little effect on the applications of interest.) 7

8 MSU/CTEQ uncertainty methods 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 Hessian Matrix Method: eigenvectors of error matrix yield 4 sets {S i ± } that are displaced up" or down" by χ 2 = 1 from the best fit. Get error by sum of squares and construct extreme PDFs for any observable; or simply look at extremes from the 4 sets. Lagrange Multiplier Method: Track χ 2 as function of F (e.g. σ W ) by minimizing χ 2 + λf. Yields special-purpose PDFs that give extremes of σ W, or y for rapidity distribution of W, or σ for tμt production; or σ t μt( s = 14 TeV)/σ t μt( s = 2TeV), or M W mass measurement error,... 8

9 Hessian (Error Matrix) method Classical error formulae χ 2 = ij (a i a () i )(H) ij (a j a () j ) F (H 1 F ) ij a i a j ( F ) 2 = χ 2 ij Hessian matrix H is inverse of error matrix. Direct application fails because of extreme differences in variation of χ 2 for different directions in the space of fitting parameters ( steep" and flat" directions), as shown by a huge range of eigenvalues of H: Eigenvalues of Hessian matrix 9

10 Convergence problems are solved by an iterative method that finds and rescales the eigenvectors of H, leading to a diagonal form χ 2 = i z 2 i ( F ) 2 = i ( F (S (+) i ) F (S ( ) ) 2 i ) where S (+) i and S (+) i are PDF sets that are displaced along the eigenvector directions. The iterative procedure is available in FORTRAN at pumplin/iterate/ 1

11 χ 2 and Systematic Errors The simplest definition N χ 2 (D = i T i ) 2 i=1 σ 2 i D i = data T i = theory σ i = expt. error" is optimal for random Gaussian errors, D i = T i + σ i r i with P (r) = e r2 /2 2π. With systematic errors, D i = T i (a)+α i r stat,i + K r k β ki. k=1 The fitting parameters are {a λ } (theoretical model) and {r k } (corrections for systematic errors). Published experimental errors: α i is the `standard deviation' of the random uncorrelated error. β ki is the `standard deviation' of the k th (completely correlated!) systematic error on D i. 11

12 To take into account the systematic errors, we define χ 2 (a λ,r k ) = N i=1 ( Di k r kβ ki T i ) 2 α 2 i + k and minimize with respect to {r k }. The result is where r k = k ( A 1 ) kk B k, (systematic shift) A kk = δ kk + B k = N i=1 N i=1 β ki β k i α 2 i β ki (D i T i ) α 2. i The r k 's depend on the PDF model parameters {a λ }. We can solve for them explicitly since the dependence is quadratic. We then minimize the remaining χ 2 (a) with respect to the model parameters {a λ }. {a λ } determine f i (x, Q 2 ). { r k } are are the optimal corrections" for systematic errors; i.e., systematic shifts to be applied to the data points to bring the data from different experiments into compatibility, within the framework of the theoretical model. r 2 k, 12

13 Comparison to Data Comparison of the CTEQ6M fit to data with correlated systematic errors. data set N e χ 2 e χ 2 e /N e Other data sets: BCDMS p BCDMS d H1a H1b ZEUS NMC F2p NMC F2d/p Dχ jet CDF jet CCFR ν DIS (15/156) E65 Drell-Yan (95/119) E866 Drell-Yan (6/15) CDF W-lepton asymmetry (1/11) 13

14 CTEQ6M fit to ZEUS data at low x x.161 x.253 x.4 x.632 x.8 x.13 ZEUS data low x values F2 x,q 2 offset 1.5 x.12 x.161 x.21 x.32 x.5 1 x.253 x Q 2 GeV 2 The data points include the estimated corrections for systematic errors. That is to say, the central values plotted have been shifted by an amount that is consistent with the estimated systematic errors, where the systematic error parameters are determined using other experiments via the global fit. The error bars are statistical errors only. 14

15 CTEQ6M fit to ZEUS data at high x 1.75 x.13 ZEUS data high x values 1.5 x.21 F2 x,q 2 offset x.32 x.5 x.8 x.13 x.18 x x.4 x Q 2 GeV 2 The data points include the estimated corrections for systematic errors. The error bars are statistical errors only. 15

16 N ZEUS i (a) Histogram of residuals for the ZEUS data. The curve is a Gaussian of width ZEUS N D i T i Α i (b) A similar comparison but without the corrections for systematic errors on the data points. 16

17 8 6 NMC F2p N i (a) Histogram of residuals for the NMC data. 8 6 NMC F2p N D i T i Α i (b) A similar comparison but without the corrections for systematic errors on the data points. 17

18 4 3 ZEUS shifts r Systematic shifts for the ZEUS data (1 systematic errors) NMC shifts r Systematic shifts for the NMC data (11 systematic errors) 18

19 CDF inclusive jet cross section (D-T)/T 1 5 CTEQ4M Statistical Errors only CTEQ4HJ MRST GeV Jet Transverse Energy corrected data theory theory CDF inclusive jet pt GeV Recall that these inclusive jet cross section measurements provided the first major stimulus to the study of PDF uncertainties in particular, the uncertainties associated with choices made in the form of parametrizations at Q. 19

20 CDF Inclusive jets systematic errors Percentage change in cross section 2 (a) High P T Hadron response -2 2 (c) Energy Scale Stability -2 2 (e) Underlying Event -2 2 (g)calorimeter Resolution 2 (b) Low P T Hadron response -2 2 (d) Fragmentation -2 2 (f) Neutral Pion Response -2 2 (h) Normalization Transverse Energy (GeV) k r k xxx 2

21 W rapidity distributions Our methods allow us to calculate the extreme predictions due to PDF uncertainty for whatever quantity is of experimental interest. For example, extremes of σ W, y, y 2 for W production at FNAL relevant for M W measurement: Same curves after subtracting central values... 21

22 Uncertainty of the gluon distribution Uncertainty bands (envelope of possible fits) for the gluon distribution at Q 2 = 1 GeV 2. The curves correspond to CTEQ5M1 (solid) CTEQ5HJ (dashed) MRST21 (dotted) Ironically, the differences between these is comparable to the estimated uncertainty! The uncertainties of quark distributions (not shown) are smaller than this gluon uncertainty, because the DIS measurements are sensitive to the square of the quark charge in leading order. The uncertainties of all PDFs decrease with increasing Q convergent evolution" 22

23 Measurement of α s We find that the CTEQ6 analysis is nicely consistent with the World Average determination of α s (M Z ). But it is not precise enough to improve that value. PDG 2 summary Y decay Average Hadronic Jets e e e e event shap Fragmentation Z width ep event shap Polarized DIS DIS Lattice tau decays Α s M Z 23

24 χ 2 versus α S (M Z ) for individual data sets in CTEQ BCDMSp BCDMSd H1a H1b ZEUS NMCp NMCr NMCrx CCFR CCFR E NA CDFw E Djet CDFjet

25 Measurment of α S (M Z ): If assume χ 2 = 1 criterion in each experiment, the experiments are inconsistent. Our error estimate (T = 1) is α S (M Z ) =.1165 ±.65 This corresponds to somewhat conservative assumptions perhaps to be thought of as an effective 2 σ" limit. Hence it is comparable to the MRST limit based on T = 5. Χ 2 1 ranges from GA BCDMSp BCDMSd H1a ZEUS NMCp H1b NMCr CCFR2 CCFR3 E65 CDFw E866 Djet CDFjet MRST value alpha S 25

26 Similar situation for W and Z cross sections When a strict χ 2 = 1 criterion was applied to self-consistent subsets of the experiments, the subsets were not consistent with each other. The true error is therefore considerably larger than χ 2 = 1 would imply. 26

27 New ways to measure consistency of fit (Work in progress with John Collins) Key idea: In addition to the Hypothesis-testing criterion χ 2 2N we use the stronger Parameter-fitting criterion χ 2 1 The parameters here are relative weights assigned to various experiments, or to results obtained using various experimental methods. Examples: Plot minimum χ 2 i vs. χ 2 tot χ2 i, where χ2 i is one of the experiments, or all data on nuclei, or all data at low Q 2,... or Plot both as function of Lagrange multiplier u where (1 u)χ 2 i + (1 + u)(χ 2 tot χ2 i ) is the quantity minimized. Can obtain quantitative results by fitting to a model with a single common parameter p: χ 2 i = A + ( p ) 2 p = ± sin θ sin ) θ 2 χ 2 not i = B + ( p S cos θ p = S ± cos θ These differ by S ± 1, i.e., by S standard deviations" 27

28 NMC D2/H2 NMC D2/H2 S = 2.6 BCDMS D2 BCDMS D2 S = 7.6 Fits to 8 of the experiments in the CTEQ5 analysis Expt S tan φ

29 Application: Uncertainties of luminosity functions at LHC.2 Luminosity function at LHC.1 Fractional Uncertainty Q-Q --> W + G-G ± Q-Q --> W s ^ (GeV) Note that one component of the uncertainty in predicting the Higgs production cross section at LHC is an uncertainty of 8 % due to PDF uncertainty. 29

30 Outlook Parton distributions of the proton are increasingly well measured. Useful tools are in place to estimate the uncertainty of PDFs and to propagate those uncertainties to physical predictions. The Les Houches Accord interface makes it easy to handle the large number of PDF solutions that are needed to characterize uncertainties. (hep-ph/24316) Work on refining the knowledge of the Tolerance Parameter" T is underway Collins & Pumplin [hep-ph/1527] Statistical bootstrap methods Improvements in the treatment of heavy quark effects are in progress. Fermilab run II data and HERA II data will provide the next major experimental steps forward. Parton Distribution Functions are a major avenue toward understanding the fundamental nonperturbative physics of the proton. They are also a crucial prerequisite for precision Standard Model studies and New Physics searches at hadron colliders and experiments with hadron targets. 3