Quantum Chromodynamics and the parton model. Module 6: Global analysis of parton distributions

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1 Quantum Chromodynamics and the parton model Module 6: Global analysis of parton distributions Pavel Nadolsky Department of Physics Southern Methodist University (Dallas, TX) June 23, 2017 Pavel Nadolsky (SMU) TMD Collaboration School

2 Where do the PDFs come from? Pavel Nadolsky (SMU) TMD Collaboration School

3 Practical answer: from the Les Houches Accord PDF library (LHAPDF) Almost all recent PDFs are included in the LHAPDF C++ library available at lhapdf.hepforge.org. Thousands of PDF sets are provided and can be linked to your computer code. Which one should you use? Pavel Nadolsky (SMU) TMD Collaboration School

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5 ] T Parton distribution functions in a nucleon At NNLO QCD, general-purpose PDF parametrizations are available from ABM, CT, HERA, MMHT, NNPDF groups [ GeV / p / M Q NMCPD NMC SLAC BCDMS HERA1AV CHORUS FLH108 NTVDMN ZEUSH2 ZEUSF2C H1F2C DYE886 DYE605 CDFWASY CDFZRAP D0ZRAP ATLASWZRAP D0R2CON CDFR2KT ATLASR04JETS CMSWEASY LHCBW NNPDF2.3 Dataset 3 10 x Typical PDFs are constrained by DIS at HERA Vector boson production at low Q, Tevatron Inclusive jet production early LHC data Pavel Nadolsky (SMU) TMD Collaboration School

6 ] T Parton distribution functions in a nucleon Fixed-target data sets are critical at x > 0.01, but may be replaced in the future by collider measurements in t t, direct-γ,w c,... production NNPDF2.3 Dataset [ GeV / p / M Q NMCPD NMC SLAC BCDMS HERA1AV CHORUS FLH108 NTVDMN ZEUSH2 ZEUSF2C H1F2C DYE886 DYE605 CDFWASY CDFZRAP D0ZRAP ATLASWZRAP D0R2CON CDFR2KT ATLASR04JETS CMSWEASY LHCBW Any PDF set makes assumptions about poorly constrained PDF combinations, e.g., sea PDFs at x < 0.01 and x > 0.3.Photon PDF is largely unknown x Pavel Nadolsky (SMU) TMD Collaboration School

7 CT10 NNLO PDFs [arxiv: ] Next-to-the last set from CTEQ-TEA group; superceded by CT14 NNLO (arxiv: ) x f x, Q versus x u val d val 0.1 g 0.1 sea Q 2 GeV u val d val 0.1 g 0.1 sea Q 3.16 GeV u val d val 0.1 g 0.1 sea Q 8 GeV u val d val 0.1 g 0.1 sea Q 85 GeV Pavel Nadolsky (SMU) TMD Collaboration School

8 CT10 NNLO vs. fitted experiments 1.6 HERA : e P e X Q 2 = 60 GeV 2 CT10 NNLO CT10 NLO H1 data D0 Run 2 inclusive jet data reduced cross section F 2 cc (x,q) * i Q 2 = 35 GeV 2 Q 2 = 18 GeV 2 Q 2 = 12 GeV 2 i=4 i=3 i=2 cross section [fb] Q [GeV] 0.5 Q 2 = 6.5 GeV x i=1 i= p [GeV] T Good general agreement Pavel Nadolsky (SMU) TMD Collaboration School

9 CT10 NNLO describes well LHC 7 TeV experiments + X) [pb] l ν + (pp W lep dσ/d η + X) [pb] l ν (pp W lep dσ/d η RESBOS CT10 NNLO 1 ATLAS 35 pb η lep RESBOS CT10 NNLO 1 ATLAS 35 pb η lep A ch RESBOS CT10 NNLO 1 ATLAS 35 pb η lep 0 y y y P T GeV A ch 0.26 p e > 35 GeV T 0.24 data CT10NNLO PDF err CMS electron charge asymmetry, 1 L dt=840 [pb], S=7 TeV ηη lep ATLAS inc. jet 2010, R 0.6 Ratio w.r.t. CT10 NNLO Shifted Data sta.&unc. using FASTNLOv2 Scale unc. PDF unc y y P T GeV Pavel Nadolsky (SMU) TMD Collaboration School

10 The flow of the PDF analysis PDFs are not measured directly, but some data sets are sensitive to specific combinations of PDFs. By constraining these combinations, the PDFs can be disentangled in a combined (global) fit. Pavel Nadolsky (SMU) TMD Collaboration School

11 The flow of the PDF analysis Data sets and χ 2 /d.o.f. in CT10 NNLO and CT10W NLO analyses Modern fits involve up to 40 experiments, data points, and 100+ free parameters Pavel Nadolsky (SMU) TMD Collaboration School

12 The flow of the PDF analysis We are interested not just in one best fit, but also in the uncertainty of the resulting PDF parametrizations and theoretical predictions based on them. Pavel Nadolsky (SMU) TMD Collaboration School

13 Origin of differences between PDF sets 1. Corrections of wrong or outdated assumptions lead to significant differences between new ( post-2014) and old ( pre-2014) PDF sets inclusion of NNLO QCD, heavy-quark hard scattering contributions relaxation of ad hoc constraints on PDF parametrizations improved numerical approximations Pavel Nadolsky (SMU) TMD Collaboration School

14 Origin of differences between PDF sets 2. PDF uncertainty a range of allowed PDF shapes for plausible input assumptions, partly reflected by the PDF error band is associated with the choice of fitted experiments experimental errors propagated into PDF s handling of inconsistencies between experiments choice of factorization scales, parametrizations for PDF s, higher-twist terms, nuclear effects,... leads to non-negligible differences between the newest PDF sets Pavel Nadolsky (SMU) TMD Collaboration School

15 Stages of the PDF analysis 1. Select valid experimental data 2. Assemble most precise theoretical cross sections and verify their mutual consistency 3. Choose the functional form for PDF parametrizations 4. Implement a procedure to handle nuisance parameters (>200 sources of correlated experimental errors) 5. Perform a fit 6. Make the new PDFs and their uncertainties available to end users Pavel Nadolsky (SMU) TMD Collaboration School

16 1. Selection of experimental data Neutral-current ep DIS data from HERA are the most extensive and precise among all data sets In addition, their systematic errors were reduced recently by cross calibration of H1 and ZEUS detectors However, by their nature they constrain only a limited number of PDF parameters Thus, two complementary approaches to the selection of the data are possible Pavel Nadolsky (SMU) TMD Collaboration School

17 Two strategies for selection of experimental data DIS-based analyses focus on the most precise (HERA DIS) data NC DIS, CC DIS, NC DIS jet, c and b production (ABM; HERAPDF1.0,1.5, 2,0; JR) Global analyses (CT10, MSTW 2008, NNPDF) focus on completeness, reliable flavor decomposition all HERA data + fixed-target DIS data notably, CCFR and NuTeV νn DIS constraining s(x,q) low-q Drell-Yan (E605, E866), W lepton asymmetry, Z rapidity (CT10, MSTW 08, NNPDF2) Tevatron Run-2 and LHC jet production, t t production Pavel Nadolsky (SMU) TMD Collaboration School

18 2. Available theoretical cross sections Process Number of Mass QCD loops scheme Neutral current 2 ZM Moch, Vermaseren, Vogt DIS 2 GM Riemersma, Harris, Smith, van Neerven Buza, Matiounine, Smith, van Neerven Charged current 2 ZM Moch, Vermaseren, Vogt DIS 2 GM Berger et al. pn γ,w,z ll ) X 2,3 ZM Anastasiou et al.; Boughezal et al; Gehrmann p p jx 2 ZM Gehrmann et al. ep jjx 2 ZM pp t tx 2 ZM Csakon, Mitov et al. *ZM/GM: zero-mass/general-mass approximation for c, b contributions Breath-taking progress in (N)NNLO calculations! Pavel Nadolsky (SMU) TMD Collaboration School

19 3. Requirements for PDF parametrizations A. A valid set of f a/p (x,q) must satisfy QCD sum rules Valence sum rule 1 0 [u(x,q) ū(x,q)]dx = [s(x,q) s(x,q)]dx = 0 0 [ d(x,q) d(x,q) ] dx = 1 A proton has net quantum numbers of 2 u quarks + 1 d quark Momentum sum rule [proton] a=g,q, q 1 0 xf a/p (x,q)dx = 1 momenta of all partons must add up to the proton s momentum Through this rule, normalization of g(x,q) is tied to the first moments of quark PDFs Pavel Nadolsky (SMU) TMD Collaboration School

20 3. Requirements for PDF parametrizations B. A valid PDF set must not produce unphysical predictions for observable quantities Example Any concievable hadronic cross section σ must be non-negative: σ 0 this is typically realized by requiring f a/p (x,q) > 0 Any cross section asymmetry A must lie in the range 1 A 1 this constrains the range of allowed PDF pararametrizations Pavel Nadolsky (SMU) TMD Collaboration School

21 Requirements for PDF parametrizations PDF parametrizations for f a/p (x,q) must be flexible just enough to reach agreement with the data, without violating QCD constraints (sum rules, positivity,...) or reproducing random fluctuations F 2(x, Q 2 ) good fit x Pavel Nadolsky (SMU) TMD Collaboration School

22 Requirements for PDF parametrizations PDF parametrizations for f a/p (x,q) must be flexible just enough to reach agreement with the data, without violating QCD constraints (sum rules, positivity,...) or reproducing random fluctuations Pavel Nadolsky (SMU) TMD Collaboration School

23 Requirements for PDF parametrizations PDF parametrizations for f a/p (x,q) must be flexible just enough to reach agreement with the data, without violating QCD constraints (sum rules, positivity,...) or reproducing random fluctuations F 2(x, Q 2 ) x good fit Traditional solution Theoretically motivated functions with a few parameters f i/p (x,q 0 ) = a 0 x a 1 (1 x) a 2 F(x; a 3,a 4,...) x 0: f x a 1 Regge-like behavior x 1: f (1 x) a 2 quark counting rules Pavel Nadolsky (SMU) TMD Collaboration F(a 3 School,a 4,...) affects intermediate

24 Requirements for PDF parametrizations PDF parametrizations for f a/p (x,q) must be flexible just enough to reach agreement with the data, without violating QCD constraints (sum rules, positivity,...) or reproducing random fluctuations F 2(x, Q 2 ) x Radical solution Neural Network PDF collaboration Generate N replicas of the experimental data, randomly scattered around the original data in accordance with the probability suggested by the experimental errors Divide the replicas into a fitting sample and control sample Pavel Nadolsky (SMU) TMD Collaboration School

25 Requirements for PDF parametrizations PDF parametrizations for f a/p (x,q) must be flexible just enough to reach agreement with the data, without violating QCD constraints (sum rules, positivity,...) or reproducing random fluctuations F 2(x, Q 2 ) x Radical solution Neural Network PDF collaboration Parametrize f a/p (x,q) by ultra-flexible functions neural networks A statistical theorem states that any function can be approximated by a neural network with a sufficient number of nodes (in practice, of order 10) Pavel Nadolsky (SMU) TMD Collaboration School

26 Requirements for PDF parametrizations PDF parametrizations for f a/p (x,q) must be flexible just enough to reach agreement with the data, without violating QCD constraints (sum rules, positivity,...) or reproducing random fluctuations F 2(x, Q 2 ) x new good fit old good fit Radical solution Neural Network PDF collaboration Fit the neural nets to the fitting sample, while demanding good agreement with the control sample Smoothness of f a/p (x,q) is preserved, despite its nominal flexibility Pavel Nadolsky (SMU) TMD Collaboration School

27 Hessian error PDFs (CT10) x f(x,q) vs. x u val d val 0.1 g 0.1 sea Q 2 GeV Neural network PDFs xg(x, Q ) 3 10 x NNPDF2.3 NLO replicas NNPDF2.3 NLO mean value NNPDF2.3 NLO 1σ 1 10 error band NNPDF2.3 NLO 68% CL band

28 Multi-dimensional error analysis χ 2 χ 2 0 a i0 a i Minimization of a likelihood function (χ 2 ) with respect to 30 theoretical (mostly PDF) parameters {a i } and > 100 experimental systematical parameters Pavel Nadolsky (SMU) TMD Collaboration School

29 Multi-dimensional error analysis χ 2 Establish a confidence region for {a i } for a given tolerated increase in χ 2 χ 2 0 a i a a + i0 i χ 2 a i In the ideal case of perfectly compatible Gaussian errors, 68% c.l. on a physical observable X corresponds to χ 2 = 1 independently of the number N of PDF parameters See, e.g., P. Bevington, K. Robinson, Data analysis and error reduction for the physical sciences Pavel Nadolsky (SMU) TMD Collaboration School

30 Multi-dimensional error analysis χ 2 Pitfalls to avoid Landscape disagreements between the experiments a i In the worst situation, significant disagreements between M experimental data sets can produce up to N M! possible solutions for PDF s, with N reached for only about 200 data sets Pavel Nadolsky (SMU) TMD Collaboration School

31 Multi-dimensional error analysis χ 2 Pitfalls to avoid Flat directions unconstrained combinations of PDF parameters a i dependence on free theoretical parameters, especially in the PDF parametrization impossible to derive reliable PDF error sets Pavel Nadolsky (SMU) TMD Collaboration School

32 Multi-dimensional error analysis χ 2 The actual χ 2 function shows a well pronounced global minimum χ 2 0 weak tensions between data sets in the vicinity of χ 2 0 (mini-landscape) a i some dependence on assumptions about flat directions Pavel Nadolsky (SMU) TMD Collaboration School

33 Multi-dimensional error analysis χ 2 The actual χ 2 function shows a well pronounced global minimum χ 2 0 weak tensions between data sets in the vicinity of χ 2 0 (mini-landscape) a i some dependence on assumptions about flat directions Pavel Nadolsky (SMU) TMD Collaboration School

34 Multi-dimensional error analysis χ 2 The actual χ 2 function shows a well pronounced global minimum χ 2 0 weak tensions between data sets in the vicinity of χ 2 0 (mini-landscape) a i some dependence on assumptions about flat directions The likelihood is approximately described by a quadratic χ 2 with a revised tolerance condition χ 2 T 2 Pavel Nadolsky (SMU) TMD Collaboration School

35 Multi-dimensional error analysis χ 2 The actual χ 2 function shows a well pronounced global minimum χ 2 0 weak tensions between data sets in the vicinity of χ 2 0 (mini-landscape) a i some dependence on assumptions about flat directions The likelihood is approximately described by a quadratic χ 2 with a revised tolerance condition χ 2 T 2 Pavel Nadolsky (SMU) TMD Collaboration School

36 Multi-dimensional error analysis χ 2 χ 2 The actual χ 2 function shows a well pronounced global minimum χ 2 0 weak tensions between data sets in the vicinity of χ 2 0 (mini-landscape) a i a i0 a + i a i some dependence on assumptions about flat directions The likelihood is approximately described by a quadratic χ 2 with a revised tolerance condition χ 2 T 2 Pavel Nadolsky (SMU) TMD Collaboration School

37 Confidence intervals in global PDF analyses Monte-Carlo sampling of the PDF parameter space χ 2 Ú Ö È ÍÒÛ Ø È a i A very general approach that realizes stochastic sampling of the probability distribution (Alekhin; Giele, Keller, Kosower; NNPDF) can parametrize PDF s by flexible neural networks (NNPDF) does not rely on smoothness of χ 2 or Gaussian approximations Pavel Nadolsky (SMU) TMD Collaboration School

38 Correlated systematic errors in the PDF analysis At high luminosities, statistical errors in the collider data, such as jet production, are much smaller than correlated systematic errors. The latest data sets are provided with many ( ) sources of correlated errors. Special methods are developed to include the effect of correlated systematic errors into the PDF uncertainties. Pavel Nadolsky (SMU) TMD Collaboration School

39 4. Statistical aspects J. Pumplin et al., JHEP 0207, 012 (2002), and references therein; J. Collins & J. Pumplin, hep-ph/ Suppose there are N PDF parameters {a i }, N exp experiments, M k data points and N k correlated systematic errors in each experiment Each systematic error is associated with a random parameter r n, assumed to be distributed as a Gaussian distribution with unit dispersion The best external estimate of syst. errors corresponds to {r n = 0}; but we must allow for r n 0 The most likely combination of {a} and {r} is found by minimizing χ 2 = N exp k=1 w k χ 2 k w k > 0 are the weights applied to emphasize or de-emphasize contributions from individual experiments (default: w k = 1) Pavel Nadolsky (SMU) TMD Collaboration School

40 4. Statistical aspects J. Pumplin et al., JHEP 0207, 012 (2002), and references therein; J. Collins & J. Pumplin, hep-ph/ χ 2 for one experiment is ( ) M k χ 2 k = R 2 1 k R k D i T i ({a}) r n β ni + σ 2 i=1 i n=1 n=1 D i and T i are data and theory values at each point σ i = σstat 2 +σ2 syst,uncor is the total statistical + systematical uncorrelated error n β nir k are correlated systematic shifts β ni is the correlation matrix; is provided with the data or theoretical cross sections before the fit n r2 n is the penalty for deviations of r n from their expected values, r n = 0 r 2 n Pavel Nadolsky (SMU) TMD Collaboration School

41 4. Statistical aspects J. Pumplin et al., JHEP 0207, 012 (2002), and references therein; J. Collins & J. Pumplin, hep-ph/ Each χ k can be analytically minimized with respect to the Gaussian r n, with the result R k r n ({a}) = (A 1 ) nn B n ({a}) M k A nn = δ nn + i=1 n =1 β ni β n M k i β ni (D i T i ) σi 2 ; B n ({a}) = σ 2 i=1 i M k χ 2 k = 1 σ 2 i=1 i (D ki T ki ({a})) 2 + R k n,n =1 B n (A 1 ) nn B n (1) Numerical minimization of k w kχ 2 k (a,r(a)) (with χ k from Eq. (1)) then establishes the region of acceptable {a}, which includes the largest possible variations of {a} that are allowed by the systematic effects Pavel Nadolsky (SMU) TMD Collaboration School

42 Key formulas of the Hessian method The Hessian method is used by CTXX PDFs to propagate the PDF uncertainty into QCD predictions. It provides several simple formulas for computing the PDF uncertainties and PDF-induced correlations. Pavel Nadolsky (SMU) TMD Collaboration School

43 Tolerance hypersphere in the PDF space 2-dim (i,j) rendition of N-dim (22) PDF parameter space a j u l p(i) contours of constant c 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 0 rescaling by the iterative method s 0 a i (a) Original parameter basis Hessian eigenvector basis sets u l z k (b) Orthonormal eigenvector basis A hyperellipse χ 2 T 2 in space of N physical PDF parameters {a i } is mapped onto a filled hypersphere of radius T in space of N orthonormal PDF parameters {z i } Pavel Nadolsky (SMU) TMD Collaboration School

44 Tolerance hypersphere in the PDF space 2-dim (i,j) rendition of N-dim (22) PDF parameter space X z m (b) Orthonormal eigenvector basis A symmetric PDF error for a physical observable X is given by X = X z m = X = 1 N ( ) 2 i=1 X (+) i X ( ) 2 i Asymmetric PDF errors can be also computed. Pavel Nadolsky (SMU) TMD Collaboration School

45 Tolerance hypersphere in the PDF space 2-dim (i,j) rendition of N-dim (22) PDF parameter space X ϕ Y (b) Orthonormal eigenvector basis Correlation cosine for observables X and Y : ( X (+) i X ( ) i cosϕ = X Y X Y = 1 4 X Y N i=1 )( Y (+) i ) Y ( ) i Pavel Nadolsky (SMU) TMD Collaboration School

46 Sensitivity of PDFs to input cross sections It is often interesting to know which specific data sets included in the PDF analysis constrain a given f a/p (x,q); or which PDF flavors drive the PDF uncertainty in a given QCD observable X This question can be addressed in several ways: by computing the correlation cosine cosφ in the Hessian method (Nadolsky et al., ) by a Lagrange multiplier scan of X (Stump et al., hep-ph/ ) or by introducing effective Gaussian variables S n dependent on X (Dulat et al., , ; Lai et al., ) Pavel Nadolsky (SMU) TMD Collaboration School

47 Sensitivity of pp ( ) jet+x to gluon PDF CT10NLO, g x,100gev 0.5 CDF single inc. jetσ bin from arxiv: CT10NLO, g x,100gev ATLAS single inc. jet Σ bin (R=0.6) cosφ 0.0 cosφ 0.0 from arxiv: x cos φ(x) measures the correlation between the PDF uncertainty of d 2 σ/(dp T dy) in inclusive jet production, and g(x,q) at a given x. The curves show cos ϕ between the NLO theory cross sections in experimental {p Tj,y j } bins and g(x,q), for various x values in g(x,q). x Pavel Nadolsky (SMU) TMD Collaboration School

48 Sensitivity of pp ( ) jet+x to gluon PDF CT10NLO, g x,100gev 0.5 CDF single inc. jetσ bin from arxiv: CT10NLO, g x,100gev ATLAS single inc. jet Σ bin (R=0.6) cosφ 0.0 cosφ 0.0 from arxiv: x The CDF (left) and ATLAS (right) jet data are correlated with g(x,q) when cosφ 0.7 or cosφ 0.7. The CDF (ATLAS) jet measurement are directly sensitive to g(x,q) with x 0.1 (0.01), and indirectly at x via the momentum sum rule. In the ATLAS jet data, the bins with p j T < 200 GeV (pink dashed curves) probe g(x,q) in a wider range than at CDF. x Pavel Nadolsky (SMU) TMD Collaboration School

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50 Combination of PDF uncertainties from several groups As the final step, it may be necessary to combine QCD cross sections X i from several PDF ensembles and find the ultimate PDF uncertainty. This combination can be done... (a)...according to the PDF4LHC convention, by combining X i found for every individual error PDF (M. Botje et al., (2011), arxiv: ; R. Ball, et al., ) (b)...by introducing meta-pdfs to first average the input PDFs in the PDF parameter space, before computing X i (J. Gao, P. Nadolsky, ). Σ z in nb W and Z total cross sections, LHC 8 TeV, NNLO Dashed: CT10, MSTW 08, NNPDF2.3 predictions Solid: META PDF prediction Σ W in nb Pavel Nadolsky (SMU) TMD Collaboration School

51 α s (M Z ) and heavy-quark masses in the PDF fits The QCD coupling, α s (M Z ), and MS heavy-quark masses, m c (m c ), m b (m b ), and m t (m t ), can be also varied in the PDF fits. However, the resulting constraints on these parameters are much weaker than from their dedicated measurements. CT10 NNLO assumes a fixed α s (M Z ) = 0.118±0.002 at 90%c.l., which is equal to the world average α s (M Z ). For any X, the α s uncertainty αs X is correlated with the PDF uncertainty PDF X, the two must be combined in the full prediction. CT10 provides PDF error sets to compute the PDF+α s uncertainty with full correlation. Pavel Nadolsky (SMU) TMD Collaboration School

52 Constraints on α s (M Z ) and the MS charm mass m c (m c ) in the CT10 NNLO fit χ 2 CT10.2 NNLO candidate 3340 Total χ PRELIMINARY α s (M Z ) error at 68 C.L. 1. CT10, fit 1 2. fit 2 3. fit 3 4. fit 4 5. FFN Alekhin et al., Χ CT10, withλunc. 7. FFN Alekhin et al., Χ 2 1 world avg.± 1Σ Λ 0 O Α 2 s Estimated O Α 3 s NLO: α s (M Z ) = ± at 90% c.l. NNLO: α s (M Z ) = 0.118± mc mc GeV Gao, Guzzi, Nadolsky, m c (m c ) = GeV at 68% c.l. Pavel Nadolsky (SMU) TMD Collaboration School

53 α s (M Z ) and heavy-quark masses in the PDF fits The QCD coupling, α s (M Z ), and MS heavy-quark masses, m c (m c ), m b (m b ), and m t (m t ), can be also varied in the PDF fits. However, the resulting constraints on these parameters are much weaker than from their dedicated measurements. CT10 NNLO assumes a fixed α s (M Z ) = 0.118±0.002 at 90%c.l., which is equal to the world average α s (M Z ). For any X, the α s uncertainty αs X is correlated with the PDF uncertainty PDF X, the two must be combined in the full prediction. CT10 provides PDF error sets to compute the PDF+α s uncertainty with full correlation. Pavel Nadolsky (SMU) TMD Collaboration School

54 PDF+α s uncertainties of the total gg H cross section Χ 2 14 TeV ΣH Α S Dulat et al., (pb) σ H NNLO gg H at the LHC ( % C.L. PDF MSTW08 CT10 NNPDF2.3 nolhc NNPDF2.3 HERAPDF1.5 ABM11 JR09 ggh@nnlo (v1.4.1), µ R s = 8 TeV) for M H = µ F = M H = 126 GeV PDG α S (M ) Z Forte, Watt, / 2 Pavel Nadolsky (SMU) TMD Collaboration School

55 CT10 estimate of the PDF+α s uncertainty CT10 NNLO set provides 2N +1 = 51 PDF error sets at α s (M Z ) = 0.118, to compute PDF X = 1 25 ( ) X (+) i X ( ) 2 i (the PDF uncertainty); 2 i=1 2 best-fit PDFs for for α s = and 0.120, to compute αs X = X(α s = 0.120) X(α s = 0.116) 2 (the α s uncertainty). The formula for the combined PDF+α s uncertainty with with full correlation is (Lai et al., ) PDF+αs X = ( PDF X) 2 +( αs X) 2 Pavel Nadolsky (SMU) TMD Collaboration School

56 What about s(x, Q) s(x, Q)? This can be tested in subprocesses W + s c and W s c In the experiment, charm quarks can be identified by their semileptonic decays, So one sees c sµ + ν and c sµ ν νn µ cx µ µ + X νn µ + cx µ + µ X SIDIS muon pair production (NuTeV) Pavel Nadolsky (SMU) TMD Collaboration School

57 Total strangeness and strangeness asymmetry Denote [q i ](Q) 1 0 xq i (x,q)dx net moment fraction carried by q i and introduce s ± (x) = s(x)± s(x) (total strangeness and its asymmetry). It is possible that 1 0 s (x)dx = 0 (a proton has no net strangeness), but [S ] 1 (s and s have different x distributions) 0 xs (x)dx 0 A large non-vanishing [S ] might explain the NuTeV weak angle anomaly Pavel Nadolsky (SMU) TMD Collaboration School

58 CCFR (inclusive νn DIS) and NuTeV (SIDIS dimuon production): constraints on strangeness Ratio to CTEQ s at Q 2. GeV PRELIMINARY Blue: CTEQ6.6 Green: CT09 Red: MSTW 08 NLO s + (x,q) is reasonably well constrained at x > 0.01; practically unknown at x < NNPDF estimate (least biased by the parametrization of s (x,q)): x [S ](Q 2 = 20 GeV 2 ) = 0±0.009 No statistically significant [S ]; but the PDF error is large enough to eliminate the NuTeV anomaly (!) Pavel Nadolsky (SMU) TMD Collaboration School

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63 Conclusion: the role of the PDF analysis now and in the future It is a payoff decade for the PDF analysis efforts! A windfall of new data to compare with (LHC, HERA, Tevatron, JLab, RHIC,... ) Tests of QCD factorization at new s, targeting 1% precision New tools (HERA Fitter, APPLGRID, FastNLO,... ) and methods (MC sampling, PDF reweighting) revolutionize the PDF analysis Understanding of the PDFs at the (N)NNLO level will be crucial for the success of the LHC physics program Pavel Nadolsky (SMU) TMD Collaboration School

64 Modern parton distribution functions...are indispensable in computations of inclusive hadronic reactions at CERN and other laboratories Pavel Nadolsky (SMU) TMD Collaboration School

65 PDF analysis continues to be a fertile field open to young researchers! Pavel Nadolsky (SMU) TMD Collaboration School