Parton Distributions Functions and Uncertainties

Transcription

1 Parton Distributions Functions and Uncertainties Robert Thorne July 13th, 1 University College London Thanks to Alan Martin, James Stirling and Graeme Watt Banff July 1

2 Strong force makes it difficult to perform analytic calculations of scattering processes involving hadronic particles. The weakening of αs(µ ) at higher scales the Factorization Theorem. e e γ Q perturbative calculable coefficient function C i P (, α s(q )) Hadron scattering with an electron factorizes. Q Scale of scattering = Q mν Momentum fraction of Parton (ν=energy transfer) nonperturbative incalculable parton distribution fi(, Q, αs(q )) P Banff July 1 1

3 The coefficient functions C i P (, α s(q )) are process dependent (new physics) but are calculable as a power-series in αs(q ). P fi(i, Q, αs(q )) C P i (, αs(q )) = k C P,k i ()α k s(q ). C P ij ( i, j, αs(q )) Since the parton distributions fi(, Q, αs(q )) are processindependent, i.e. universal, and evolution with scale is calculable, once they have been measured at one eperiment, one can predict many other scattering processes. P fj(j, Q, αs(q )) Banff July 1

4 General procedure. Start parton evolution at low scale Q 1GeV. In principle 11 different partons to consider. u, ū, d, d, s, s, c, c, b, b, g mc, mb ΛQCD so heavy parton distributions determined perturbatively. Leaves 7 independent combinations, or 6 if we assume s = s starting not to. uv = u ū, dv = d d, sea = (ū + d + s), s + s d ū, g. Input partons parametrised as, e.g. (MRST/MSTW) f(, Q ) = (1 ) η (1 + ɛ.5 + γ) δ. Evolve partons upwards using LO, NLO (or NNLO) DGLAP equations. dfi(, Q, αs(q )) d ln Q = j Pij(, αs(q )) fj(, Q, αs(q )) Banff July 1 3

5 Fit data for scales above 1GeV. Need many different types of eperiment for full determination. Lepton-proton collider HERA (DIS) small- quarks. Also gluons from evolution, and FL(, Q ). Also, jets moderate- gluon. Fied target DIS higher leptons (BCDMS, NMC,...) up quark (proton) or down quark (deuterium) and neutrinos (CHORUS, NuTeV, CCFR) valence or singlet combinations. Di-muon production in neutrino DIS strange quarks and neutrino-antineutrino comparison asymmetry. Drell-Yan production of dileptons quark-antiquark annihilation (E65, E866) high- sea quarks. Deuterium target ū/ d asymmetry. High-pT jets at colliders (Tevatron) high- gluon distribution. W and Z production at colliders (Tevatron) different quark contributions to DIS. Banff July 1 4

6 Banff July 1 5 Various choices of PDF MSTW, CTEQ, NNPDF, Alekhin, HERA, H1, Jimenez- Delgado et al etc.. All LHC cross-sections rely on our understanding of these partons c,c. s,s u d s,s. d u.4 d d.4 c,c.6 u.6 b,b u.8.8 g/1 g/1 f(,q ) 1 Q = 1 GeV 1. f(,q ) 1 4 Q = 1 GeV 1. MSTW 8 NLO PDFs (68% C.L.) This procedure is generally successful and is part of a large-scale, ongoing project. Results in partons of the form shown.

7 Banff July Y(Z).5 D Run II Preliminary /σ.5 d σ/dy.3 NNLO, MRST1 Data * Z/γ Rapidity Ecellent predictive power comparison of MRST prediction for Z rapidity distribution with preliminary data.

8 Parton Fits and Uncertainties. Two main approaches. Parton parameterization and Hessian (Error Matri) approach first used by H1 and ZEUS, and etended by CTEQ. χ χ min χ = i,j Hij(ai a () i )(aj a () j ) The Hessian matri H is related to the covariance matri of the parameters by Cij(a) = χ (H 1 )ij. We can then use the standard formula for linear error propagation. ( F ) = χ i,j F (H) 1 F ij, ai aj This is now the most common approach (sometimes Offset method). Problematic due to etreme variations in χ in different directions in parameter space. Banff July 1 7

9 Solved by finding and rescaling eigenvectors of H leading to diagonal form χ = i z i Implemented by CTEQ, then others. Uncertainty on physical quantity then given by ( (+) ( F ) = 1/ F (S i ) F (S ( ) i ) ), where S (+) i and S ( ) i are PDF sets displaced along eigenvector direction. i Question of choosing correct χ given complication of errors in full fit and sometimes conflicting data sets. CTEQ use χ 4 and MRST/MSTW use more complicated approach results in χ 5, for one σ. Other fits less global, keep to χ = 1. Banff July 1 8

10 The inappropriateness of using χ = 1 when including a large number of sometimes conflicting data sets is shown by eamining the best value of σw and its uncertainty using χ = 1 for individual data sets as obtained by CTEQ using Lagrange Multiplier technique. Banff July 1 9

11 Banff July Fairly consistent. some difference in central values (other possible reasons) and similar errors..1 Compare rigorous treatment of all systematic errors (Alekhin) with simple quadratures approach (MRST), both with χ = 1.. Very conservative fit. d V (,Q =) Eercise for HERA LHC meeting. Fit proton and deuteron structure function data from H1, ZEUS, NMC and BCDMS, for Q > 9GeV using ZM V F NS and same form of parton inputs at same Q = 1GeV..3 Alekhinbench MRSTbench.4 Also from comparison of partons.

12 Banff July Errors similar. Enormous difference in central values. d V (,Q =) Compare to MRST1 partons with uncertainty from χ = 5..3 MRST1 MRSTbench However, how do partons from very conservative, structure function only data compare to global partons?.4

13 Banff July 1 1 Using similar sort of reasoning MRST used χ 5 for 9% confidence level on partons. Still same basic idea but more sophisticated. 3 1 distance 3 1 BCDMSp BCDMSd H1a H1b ZEUS NMCp NMCr CCFR CCFR3 E65 CDFw E866 Djet CDFjet 4 Eigenvector 4 These limits shown for CTEQ6 eigenvector 4 as function of T = χ. Some sets somewhat outside 9% confidence limits for T = 1 Previous reasoning, allow χ to take a value such that every data set remains roughly within its 9% confidence limit compared to the χ at best global fit.

14 Eplained below (Watt DIS8) Banff July 1 13

15 Banff July 1 14 Distance = χ global Distance = χ global % C.L. 68% C.L. χ χ n - χ 3 n n, 4 NuTeV νn F 3 - χ n, % C.L. 9% C.L. CHORUS νn F 3 Distance = χ global Distance = χ global χ n - χ n, % C.L. χ 9% C.L. n NuTeV νn F - χ n, % C.L. CHORUS νn F 9% C.L. Distance = χ global Distance = χ global χ n - χ n, % C.L. χ 9% C.L. n E866/NuSea pp DY - χ n, % C.L. For each determine the point in χ global at which the appropriate confidence level limit is reached in each direction. E866/NuSea pd/pp DY 9% C.L. Distance = χ global Distance = χ global χ n - χ n, % C.L. 5 68% C.L. χ 1 9% C.L. n SLAC ed F - χ 15 NMC/BCDMS/SLAC F L 9% C.L. For eigenvector 13, for eample, the change in χ for the most sensitive data sets is shown. n, MSTW 8 NLO PDF fit Eigenvector number 13

16 Banff July 1 15 BCDMS µp F BCDMS µd F NMC µp F NMC µd F NMC µn/µp E665 µp F E665 µd F SLAC ep F SLAC ed F NMC/BCDMS/SLAC F L E866/NuSea pp DY E866/NuSea pd/pp DY NuTeV νn F CHORUS νn F NuTeV νn F 3 CHORUS νn F 3 CCFR νn µµx NuTeV νn µµx NC H1 ep 97- σ r NC ZEUS ep 95- σ r CC H1 ep 99- σ r CC ZEUS ep 99- σ r charm H1/ZEUS ep F H1 ep 99- incl. jets ZEUS ep 96- incl. jets D II pp incl. jets CDF II pp incl. jets D II W lν asym. CDF II W lν asym. D II Z rap. CDF II Z rap. 5 Distance = % C.L. 9% C.L. 9% C.L. 68% C.L. χ global 15 Eigenvector number 13 MSTW 8 NLO PDF fit Eigenvector 13 constrained in one direction by E866 Drell-Yan data and in the other direction by NuTeV F p 3 (, Q ) data. In this case the best fits for the two sets are highly inconsistent. χ = 1 well outside 9% confidence level for each. Plot this for all data sets for a given eigenvector.

17 This eigenvector contributes most to the high- sea quark uncertainty, but also a variety of other quarks. MSTW 8 NLO PDF fit (68% C.L.) Fractional contribution to uncertainty from eigenvector number g u v d v S u + d + s + s d - u s + s s - s At input scale Q = 1 GeV Banff July 1 16

18 Banff July 1 17 BCDMS µp F BCDMS µd F NMC µp F NMC µd F NMC µn/µp E665 µp F E665 µd F SLAC ep F SLAC ed F NMC/BCDMS/SLAC F L E866/NuSea pp DY E866/NuSea pd/pp DY NuTeV νn F CHORUS νn F NuTeV νn F 3 CHORUS νn F 3 CCFR νn µµx NuTeV νn µµx NC H1 ep 97- σ r NC ZEUS ep 95- σ r CC H1 ep 99- σ r CC ZEUS ep 99- σ r charm H1/ZEUS ep F H1 ep 99- incl. jets ZEUS ep 96- incl. jets D II pp incl. jets CDF II pp incl. jets D II W lν asym. CDF II W lν asym. D II Z rap. CDF II Z rap. 5 Distance = % C.L. 68% C.L. 68% C.L. 9% C.L. χ global 15 Eigenvector number 9 MSTW 8 NLO PDF fit As a simpler eample, eigenvector 9 constrained most by H1 and ZEUS data on F p (, Q ). 9% confidence limit determining by ZEUS in up direction and H1 in down direction. Both χ 5.

19 Not surprising this eigenvector contributes most to the gluon uncertainty. MSTW 8 NLO PDF fit (68% C.L.) Fractional contribution to uncertainty from eigenvector number g u v d v S u + d + s + s d - u s + s s - s At input scale Q = 1 GeV Banff July 1 18

20 Banff July 1 19 Eigenvector number NC H1 ep 97- σ r NuTeV ν N µµx CCFR ν N µµx E866/NuSea pd/pp DY NuTeV ν N F 3 NuTeV ν N µµx BCDMS µd F - 1 (CTEQ) - 5 (MRST) -5 BCDMS µp F NC ZEUS ep 95- σ r NC ZEUS ep 95- σ r SLAC ed F E866/NuSea pd/pp DY NuTeV ν N F 3 D II W lν asym. NuTeV ν N F E866/NuSea pd/pp DY CCFR ν N µµx E866/NuSea pd/pp DY NC H1 ep 97- σ r NuTeV ν N µµx Tolerance T = (MRST) (CTEQ) χ global 15 NC H1 ep 97- σ r NMC µd F NuTeV ν N µµx E866/NuSea pd/pp DY NuTeV ν N µµx NuTeV ν N µµx D II W lν asym. BCDMS µd F NC H1 ep 97- σ r BCDMS µd F NC H1 ep 97- σ r E866/NuSea pd/pp DY E866/NuSea pp DY NMC µd F NC H1 ep 97- σ r CCFR ν N µµx NuTeV ν N µµx D II W lν asym. NC H1 ep 97- σ r NuTeV ν N F 3 MSTW 8 NLO PDF fit Approach repeated for all eigenvectors to determine uncertainty on each. On average χ = 4 for 9% and χ = 15 for 1 σ, but large variations, and asymmetries.

21 Banff July Fractional uncertainty Fit without any Tevatron jets 1.3 Fit with Tevatron Run I jet data 1.4 MSTW 8 NLO (68% C.L.) Gluon distribution at Q = 1 GeV Even though one data set constrains each eigenvector limit, doesn t mean others do not contribute.

22 Normalisation Uncertainties Previously the normalization of each data set was determined by the best fit and then fied. Technical difficulties in including this feature in uncertainties. Now implement procedure of allowing normalisations of all sets to vary in best fit and scan over eigenvectors, with penalty term for each set ) 4 (1 N χ N = σ N Quartic penalty avoids very large deviations. Still shift down at LO (fit failure) and slightly at NLO. Rescale errors with normalization to avoid bias (D Agostini). Banff July 1 1

23 Banff July 1 χ for benchmark data 458/589 in reduced fit 56/589 within global fit. Still lack of compatibility some places, e.g high- gluon. Latter have greater uncertainty. Compatibility using dynamical tolerance uncertainty approach, but not using χ = Ratio to MSTW 8 NLO Can investigate by repeating HERA-LHC Workshop eercise of obtaining PDFs by fitting to DIS data with conservative cuts only. 1. Ratio to MSTW 8 NLO.9 Comparison of normal and benchmark sets shown Up valence distribution at Q = GeV Down valence distribution at Q = GeV u v (, Q = GeV ).3 Dashed lines: χ = Fit to reduced dataset.6 MSTW 8 NLO (68% C.L.).7 d v (, Q = GeV ).15. Dashed lines: χ = 1 In practice should give a conservative estimation of uncertainties..5 Fit to reduced dataset.3.35 MSTW 8 NLO (68% C.L.).4 Up valence distribution at Q = GeV Down valence distribution at Q = GeV

24 Banff July Ratio to MSTW 8 NLO CTEQ6.6 NLO 1.8 MSTW 8 NLO (9% C.L.) Down quark distribution at Q = 1 GeV Ratio to MSTW 8 NLO Only reasonable agreement between groups despite inflated tolerance Uncertainty on MSTW u and d distributions, along with CTEQ CTEQ6.6 NLO 1.8 MSTW 8 NLO (9% C.L.) Up quark distribution at Q = 1 GeV

25 Different PDF sets MSTW8 fit all previous types of data. Most up-to-date Tevatron jet data. Not most recent HERA combination of data. PDFs at LO, NLO and NNLO. CTEQ6.6 very similar. Not quite as up-to-date on Tevatron data. PDFs at NLO. NNPDF. include all above ecept HERA jet data (not strongest constraint) and heavy flavour structure functions. Include HERA combined data. PDFs at NLO. HERAPDF. based entirely on HERA inclusive structure functions, neutral and charged current. Use combined data. PDFs at LO, NLO. ABKM9 fit to DIS and fied target Drell-Yan data. PDFs at NLO and NNLO. GJR8 fit to DIS, fied target Drell-Yan and Tevatron jet data. PDFs at NLO and NNLO. Banff July 1 4

26 Determination of best fit and uncertainties All but NNPDF minimise χ and define eigenvectors of parameter combinations epanding about best fit. MSTW8 eigenvectors. Due to incompatibility of different sets and (perhaps to some etent) parameterisation infleibility (little direct evidence for this) have inflated χ of 5 for eigenvectors. CTEQ6.6 eigenvectors. Inflated χ of 5 for 1 sigma for eigenvectors (no normalization uncertainties in CTEQ6.6). HERAPDF. 9 eigenvectors. Use χ = 1. Additional model and parameterisation uncertainties. ABKM9 1 parton parameters. Use χ = 1. Also αs, mc, mb. GJR8 parton parameters and αs. Use χ. Impose strong theory constraint on input form of PDFs. Perhaps surprisingly all get rather similar uncertainties for PDFs cross-sections. Banff July 1 5

27 Neural Network group (Ball et al.) limit parameterization dependence. Leads to alternative approach to best fit and uncertainties. First part of approach, no longer perturb about best fit. Construct a set of Monte Carlo replicas F art,k i,p of the original data set F ep,(k) i,p. Where r (k) p are random numbers following Gaussian distribution, and S (k) p,n analogous normalization shift of the of the replica depending on 1 + r (k) p,nσ norm p. is the Hence, include information about measurements and errors in distribution of F art,(k) i,p. Fit to the replicas of the data obtaining a set of PDF replicas q (net)(k) i (follows Giele et al.) Mean µo and deviation σo of observable O then given by µo = 1 Nrep Nrep 1 O[q (net)(k) i ], σ O = 1 Nrep Nrep 1 (O[q (net)(k) i ] µo). Banff July 1 6

28 NNPDF approach additionally (largely) eliminates parameterisation dependence by using a neural net which undergoes a series of evolutions (mutations via genetic algorithm) to find the best fit, rather than a fied parameterisation. In effect is a much larger sets of parameters 37 per distribution. Includes pre-processing eponents as 1 and to aid convergence of fit, f(, Q ) = A(1 ) m n NN() where n, m are in fairly narrow ranges, so overall behaviour guided at these etremes where data constraints vanish. Data included constantly increasing. Recently NNPDF., first global fit of this type. Freedom in parameterisation means best fit to all data would tend to reproduce data fluctuations (as far as this is possible). Must guard against this. Banff July 1 7

29 Split data sets randomly into equal size training and validation sets. NMC-pd 3.5 E tr E val E target Fit until quality of fit to validation set starts to go up, even though training set still (hopefully slowly) improving. 3.5 Criterion for stopping the fit not simply value of error function (analogous to χ ) for full global data set, but split into different data sets GA generations In earlier versions weighted error function for different data sets in early stages to try to give all sets a similar quality fit. Banff July 1 8

30 Difficult to know when to stop (analogous to variable χ in other approaches?). Some evidence not at best fit in previous versions (Forte, DESY Oct 9). Quality of global fit for both training and validation decreasing significantly after stopping point. Fluctuations in sets smaller with longer stopping. Banff July 1 9

31 Statistical behaviour (arguably) more like epected for longer stopping? Number of data points 3 /Ndata.5. Banff July 1 3

32 Uncertainty from overlearnt fits (green) was (normally) rather smaller than default (blue). Arguable if lack of smoothness becomes a problem. significant improvements in NNPDF.. Banff July 1 31

33 Banff July 1 3 Is there a definitive set of stopping criteria? I would suggest uncertainty now more analogous to smaller χ, but actual value very difficult to ascertain. Fluctuations in error function (and χ ) still arguably a bit larger than naively epected. g (, Q ) 1 Significant reductions (usually) in uncertainty in latest version, and changed central values, just due to change in stopping and fitting procedures. 3 NNPDF. DIS(HERAold) + t = 4 NNPDF Criterion for increase in fit to validation sets relative to decrease in training sets made more strict...4 V (, Q ).6 Weighted training in early stages according to a target (determined iteratively), so stopping for global fit more in line with individual sets..8 1 NNPDF. DIS(HERAold) + t = 1. NNPDF1.

34 Also reductions in uncertainty due to inclusion of new data. (Improved treatment of normalisations generally increases uncertainty slightly.) NNPDF uncertainties pretty similar to other groups, with some particular eceptions. Banff July 1 33

35 Banff July 1 34 Uncertainties on valence quarks not notably different to other groups at all V (, Q ) MSTW 8 1. NNPDF. CTEQ6.6

36 Banff July 1 35 MRST/MSTW and NNPDF more fleible (can be negative) rapid epansion of uncertainty where data runs out. If g() λ± λ then g() = λ ln(1/) g(). Most assume single power λ at input limited uncertainty. If input at low Q λ positive and small- input gluon fine-tuned to. Artificially small uncertainty Fractional uncertainty Alekhin NLO NNPDF1. (1 replicas) CTEQ6.6 NLO.4 MSTW 8 NLO (9% C.L.).5 Gluon distribution at Q = 5 GeV Gluon Parameterisation - small different parameterisations lead to very different uncertainty for small gluon.

37 Banff July 1 36 CTEQ find more compatibility between Run I and Run II fits. Fit by MSTW and CTEQ and now also NNPDF. Former found gluon much softer for Run II. Fits not very consistent between runs Fit with Tevatron Run I jet data.7 Fit without any Tevatron jet data Ratio to MSTW 8 NLO.8 MSTW 8 NLO (9% C.L.) Gluon distribution at Q = 1 GeV Constrained indirectly, but quite accurately, by DIS data, and directly by Tevatron high-pt jets, now Run I and Run II available. Slightly confusing picture. Gluon Distribution - large.

38 Generally high- PDFs parameterised so will behave like (1 ) η as 1. More fleibility in CTEQ. Very hard high- gluon distribution (more-so even than NNPDF uncertainties). However, is gluon, which is radiated from quarks, harder than the up valence distribution for 1? Banff July 1 37

39 Banff July 1 38 Difference in region of data! Effect of nuclear corrections and/or heavy quark treatment? MSTW assumes shape of strange given by theory assumption that suppression of form of massive quarks. Significantly different to CTEQ fitting to same data assuming only same small- power for strange as light quarks MRST 1 NLO CTEQ6.6 NLO.4 MSTW 8 NLO (9% C.L.).6.8 (s+s)/(u+d) 1 1. (s+s)/(u+d) distribution at Q = 5 GeV Direct fit to s, s from dimuon data leads to significant uncertainty increase compared to assumption of fied fraction of sea used until recently. Constraint for.1. Strange Quarks

40 Banff July 1 39 Overestimate of uncertainty? Impacts on small- light quarks s + (, Q.5 ) MSTW 8 NNPDF. CTEQ6.6 NNPDF., which includes dimuon data, have no theoretical constraint on strange quark distribution at all at small.

41 Banff July 1 4 In fact NNPDF now smallest uncertainty on this by some way (no data above =.). All tend towards positive momentum asymmetry, but all fairly consistent with zero, or with enough to remove (or seriously) reduce NuTeV anomaly on sin θw s - (, Q.1 )..3.4 MSTW 8.5 NNPDF. CTEQ6.6.6 Most recent sets obtain s s for first time from differences in ν, ν dimuon production. Strange asymmetry.

42 Banff July 1 41 Epected gluon αs(m Z ) small anti-correlation high- correlation from sum rule Ratio to MSTW 8 NNLO.97 y = at LHC y = at Tevatron Fi α S at - 68% C.L. limit 1.3 Fi α S at +68% C.L. limit 1.4 MSTW 8 NNLO (68% C.L.) 1.5 Gluon at Q = M H = (1 GeV) PDF uncertainties reduced since quality of fit already worse than best fit. at NLO and αs(m Z ) = at NNLO from MSTW. only for one fied αs(m Z ). Can be determined from fit, e.g. α S(M Z ) = Can also look at PDF changes and uncertainties at different αs(m Z ). Latter usually PDF correlation with αs.

43 Banff July 1 4 Quarks roughly opposite to gluons. Strong anti-correlation at high- due to evolution and positive coefficient functions Ratio to MSTW 8 NNLO.97 y = at LHC y = at Tevatron Fi α S at - 68% C.L. limit 1.3 Fi α S at +68% C.L. limit 1.4 MSTW 8 NNLO (68% C.L.) 1.5 Up quark at Q = M Z Gluon feeds into evolution of quarks, but change in αs(m Z ) just outweighs gluon change, i.e. larger αs(m Z ) slightly more evolution.

44 Banff July 1 43 Total uncertainty envelope of set of uncertainties. Increases by up to 5% at LHC. Largely due to effect of PDFs. α S (M ) Z α S (M ) Z 1 σ σ / + σ / +1 σ 1 σ σ / + σ / +1 σ σ B(Z l l ) (nb) Z % Free α S unc. 1.8% +1.9% Fied α S unc. 1.6% + σ B(Z l l ) (nb) Z % Free α S unc..1% +1.7% Fied α S unc. 1.6%.6.14 Tevatron, s = 1.96 TeV LHC, s = 14 TeV Z cross sections with MSTW 8 NNLO PDFs NNLO predictions for Z production for allowed αs(m Z ) values and their uncertainties. Additional uncertainty from αs(m Z ) variation for quantities depends on how PDFs and coupling are correlated.

45 Banff July 1 44 Increases by a factor of 3 (up more than down) at LHC. Direct αs(m Z ) dependence mitigated somewhat by anti-correlated small- gluon (asymmetry feature of minor problems in fit to HERA data). At Tevatron intrinsic gluon uncertainty dominates. α S (M ) Z α S (M ) Z 6 1 σ σ / + σ / +1 σ 6 1 σ σ / + σ / +1 σ 68% C.L. uncertainties 4 4 NNLO σ H 4 NNLO σ H 4 (%) (%) α (M ) S H gg luminosity 6 6 Tevatron, s = 1.96 TeV LHC, s = 14 TeV Higgs (M = 1 GeV) with MSTW 8 NNLO PDFs H NNLO predictions for Higgs (1GeV) production for different allowed αs(m Z ) values and their uncertainties.

46 Other Sources of Uncertainty It is vital to consider theoretical/assumption-dependent uncertainties: Methods of determining best fit and uncertainties. Underlying assumptions in procedure, e.g. parameterisations and data used. Treatment of heavy flavours. PDF and αs correlations. Responsible for differences between groups for etraction of fied-order PDFs. Banff July 1 45

47 Also other sources which (mainly) lead to inaccuracies common to all fied-order etractions. QED and Weak (comparable to NNLO?) (α 3 s α). Sometime enhancements. Standard higher orders (NNLO may sets available here.) Resummations, e.g. small (α n s ln n 1 (1/)), or large (α n s ln n 1 (1 )) low Q (higher twist), saturation In fact probably does lead to some of the difference it PDFs observed. Banff July 1 46

48 Banff July 1 47 Also H + t t at ŝ/s.1. Cross-section for t t almost identical in PDF terms to 45GeV Higgs. s / s Ratio to MSTW 8 NLO (68% C.L.) MH (GeV) tt HERAPDF NNPDF. CTEQ MSTW8 1. gg luminosity at LHC ( s = 7 TeV) Predictions by various groups - parton luminosities NLO. Plots by G. Watt.

49 Banff July 1 48 Clearly some distinct variation between groups. Much can be understood in terms of previous differences in approaches. s / s Ratio to MSTW 8 NLO (68% C.L.) MH (GeV) tt GJR8 1.1 ABKM9 CTEQ MSTW8 1. gg luminosity at LHC ( s = 7 TeV)

50 Banff July 1 49 Many of the same general features for quark-antiquark luminosity. Some differences mainly at higher. s / s Ratio to MSTW 8 NLO (68% C.L.) W Z HERAPDF NNPDF. CTEQ MSTW8 1. Σ q (qq) luminosity at LHC ( s = 7 TeV)

51 Banff July 1 5 All plots and more at Canonical eample W, Z production, but higher ŝ/s relevant for W H or vector boson fusion. s / s Ratio to MSTW 8 NLO (68% C.L.) W Z GJR8 1.1 ABKM9 CTEQ MSTW8 1. Σ q (qq) luminosity at LHC ( s = 7 TeV)

52 Banff July 1 51 Again plots by G Watt using PDF4LHC benchmark criteria. Dotted lines show how central PDF predictions vary with αs(m Z ). Z α S (M ) 1.5 Vertical error bars Inner: PDF only Outer: PDF+α S GJR8 ABKM9 11 HERAPDF1. NNPDF. CTEQ % C.L. PDF MSTW8 1 σ H 1.5 (pb) 13 NLO gg H at the LHC ( s = 7 TeV) for M H = 1 GeV Variations in Cross-Section Predictions NLO

53 Banff July 1 5 Clearly much more variation in predictions than uncertainties claimed by individual groups. Z α S (M ) 4.4 Vertical error bars Inner: PDF only Outer: PDF+α S GJR8 ABKM9 4.6 HERAPDF1. NNPDF. CTEQ % C.L. PDF MSTW8 5 σ H 5. (pb) 5.4 NLO gg H at the LHC ( s = 7 TeV) for M H = 18 GeV

54 Banff July 1 53 Ecluding GJR8 amount of difference due to αs(m Z ) variations 3 4%. Z α S (M ).3 GJR8.4 Vertical error bars Inner: PDF only Outer: PDF+α S ABKM9 HERAPDF1..5 NNPDF. CTEQ % C.L. PDF MSTW8.7 σ H.8 (pb).9 3 NLO gg H at the LHC ( s = 7 TeV) for M H = 4 GeV

55 Banff July 1 54 CTEQ6.6 now heading back towards MSTW8 and NNPDF.. Z α S (M ) GJR8 13 Vertical error bars Inner: PDF only Outer: PDF+α S ABKM9 HERAPDF1. 14 NNPDF. CTEQ % C.L. PDF MSTW σ tt (pb) NLO tt cross sections at the LHC ( s = 7 TeV)

56 Banff July 1 55 W + + W cross-section. αs(m Z ) dependence now more due to PDF variation with αs(m Z ). Z α S (M ) 9. Vertical error bars Inner: PDF only Outer: PDF+α S GJR8 9.4 ABKM9 HERAPDF1. σ W ± 9.6 NNPDF. B(W 9.8 CTEQ % C.L. PDF MSTW8 ± 1. l 1.4 ± ν) (nb) ± ± NLO W l ν at the LHC ( s = 7 TeV)

57 Banff July 1 56 Roughly similar variation for ŝ up to a few times higher. Again variations somewhat bigger than individual uncertainties. Z α S (M ).84 Vertical error bars Inner: PDF only Outer: PDF+α S GJR8.86 ABKM9 HERAPDF1. σ.88 NNPDF. Z.9 CTEQ6.6 B(Z.9 68% C.L. PDF MSTW8.94 l l ) (nb) NLO Z l l at the LHC ( s = 7 TeV)

58 Banff July 1 57 All plots and more at Quite a variation in ratio. Shows variations in flavour and quark-antiquark decompositions. Z α S (M ) Vertical error bars Inner: PDF only Outer: PDF+α S GJR8 1.4 ABKM9 HERAPDF1. NNPDF CTEQ6.6 68% C.L. PDF MSTW8 R ± 1.46 σ W / σ W NLO W /W ratio at the LHC ( s = 7 TeV)

59 Deviations In predictions clearly much more than uncertainty claimed by each. In some cases clear reason why central values differ, e.g. lack of some constraining data, though uncertainties then do not reflect true uncertainty. Sometimes no good understanding, or due to difference in procedure which is simply a matter of disagreement, e.g. gluon parameterisation at small affects predicted Higgs cross-section. What is true uncertainty. Task asked of PDF4LHC group. Interim recommendation take envelope of global sets, MSTW, CTEQ NNPDF (check other sets) and take central point as uncertainty. Not very satisfactory, but not clear what would be an improvement, especially as a general rule. Usually not a big disagreement, and factor of about epansion of MSTW uncertainty. Banff July 1 58

60 Sometimes rather worse than this for special case, e.g. Warsinsky at recent Higgs- LHC working group meeting. mb values bring CTEQ and MSTW together but eaggerate NNPDF difference. Banff July 1 59

61 Conclusions One can determine the parton distributions and predict cross-sections at the LHC, and the fit quality using NLO or NNLO QCD is fairly good. Various ways of looking at uncertainties due to errors on data. All give roughly the same value for uncertainties on PDFs and predictions 1 5% for most LHC quantities. All should be, if anything, an overestimate, i.e. inflated tolerance, or missing data sets which would have an effect, and uncertainty undoubtedly related to the choice of stopping in NNPDF. Effects from input assumptions e.g. selection of data fitted, cuts and input parameterisation can shift central values of predictions significantly. Different groups do not always agree very well despite generous uncertainties. Some improvements if effects of heavy flavour treatments and αs accounted for. αs and PDFs correlated. Now being dealt with properly for in general. Reduces but does not remove discrepancies. Errors from higher orders/resummation potentially large. Imperfect theory used to fit data. Banff July 1 6

62 Etraction of PDFs from eisting data and use for LHC far from a straightforward procedure. Lots of theoretical issues to consider for real precision. Relatively few cases where Standard Model discrepancies will not require some significant input from PDF physics to determine real significance. Banff July 1 61

63 Banff July 1 6 Significant differences in central values sometimes, and in shape of small- gluon uncertainty. More consistent data sets χ = 1 for uncertainties. Nevertheless similar to CTEQ/MSTW d v. d v S (.5) S (.5).4 u v.4 u v.6 (ep+model+param) CTEQ6.6 9% CL.6 (ep+model+param) MSTW8 9% CL g (.5) HERAPDF.(prel.) g (.5) HERAPDF.(prel.).8.8 f 1 Q = 1 GeV f 1 Q = 1 GeV HERA fits are only to HERA data, but averaged H1/ZEUS data (reduced correlated errors) not yet used by others so far (in NNP DF. - effect slightly unclear).

64 Parameterisations MSTW predictions for W+ and W- cross-sections for LHC with common fied order QCD and vector boson width effects, and common branching ratios. Quoted uncertainty for ratio very small, i.e..8%. Prediction sensitive to u and d quarks. σ(w + ) σ(w ) u() d() d()ū() u() d(), If ū() d(),, which data implies, and most parameterisations assume. Fit includes most recent neutrino DIS and Tevatron vector boson data. Uncertainties should account for this. Significantly more difference than uncertainty from other PDFs, including MRST (effect noted for W -asymmetry by Cooper-Sarkar). Very interesting for early data. Banff July 1 63

65 Banff July 1 64 σ W + B(W + l ν) (nb) GJR8 NLO 3.8 ABKM9 NLO HERAPDF1. NNPDF band shrinks dramatically with new data. 3.9 NNPDF. (1) NNPDF. (1) NNPDF1. (1) 4 CTEQ6.6 NLO Some of the differences not well understood. MSTW8 NLO MSTW8 NNLO % C.L. PDF σ W - NLO NNLO hardly affects ratio. B(W l ν) (nb) Again comparing comparing more groups even get even more discrepancies between them R(W /W ) = W and W total cross sections at the LHC ( s = 7 TeV)

66 Difficult to know when fit to validation set has started increasing significantly for some sets. DYE E tr E val E target GA generations Banff July 1 65

67 Banff July GMVFNS4 GMVFNS5 GMVFNS6 ZMVFNS.95 1 Some changes in PDFs large compared to one-sigma uncertainty. GMVFNSa/8 at NLO for u(,q ) 1.5 MSTW8 GMVFNS1 GMVFNS GMVFNS3 GMVFNSopt Better fit for GMVFNS1, GMVFNS3 and GMVFNS Converges to at most about 15 of original. GMVFNSa/8 at NLO for g(,q ) GMVFNS4 GMVFNS5 GMVFNS6 ZMVFNS.95 Initial χ can change by 5. 1 Variations in partons etracted from global fit due to different choices of GM-VFNS at NLO. 1.5 MSTW8 GMVFNS1 GMVFNS GMVFNS3 GMVFNSopt 1.1

68 Banff July GMVFNS4 GMVFNS5 GMVFNS6 GMVFNSopt.95 1 Biggest variation in high- gluon, which has large uncertainty. GMVFNSa/8 at NNLO for u(,q ) 1.5 MSTW8NNLO GMVFNS1 GMVFNS GMVFNS3 1.1 At worst changes approach uncertainty Converge to about 1. None a marked improvement. GMVFNSa/8 at NNLO for g(,q ) GMVFNS4 GMVFNS5 GMVFNS6 GMVFNSopt.95 Initial changes in χ <. 1 Variations in partons etracted from global fit due to different choices of GM-VFNS at NNLO. 1.5 MSTW8NNLO GMVFNS1 GMVFNS GMVFNS3 1.1

69 Banff July Tevatron Ratio to MSTW 8 NNLO.9 LHC14 LHC MSTW 8 NNLO (9% C.L.) mc = 1.15 GeV mc = 1.65 GeV mc = 1.15 GeV, fi α S (M Z ) = 1.65 GeV, fi α S (M ) mc Z 1. Light quark singlet distribution at Q = M Z Tevatron Ratio to MSTW 8 NNLO LHC7.9 LHC MSTW 8 NNLO (9% C.L.) mc = 1.15 GeV mc = 1.65 GeV mc = 1.15 GeV, fi α S (M Z ) = 1.65 GeV, fi α S (M ) mc Z Ratio of partons when mc is varied either with or without varying αs 1. Gluon distribution at Q = M H = (1 GeV)

70 Systematic difference between PDF defined at NLO and at NNLO MSTW8 NNLO u(,q =1GeV ) MSTW8 NLO percentage difference at Q =1GeV 1 MSTW8 NNLO 5-5 MSTW8 NLO Banff July 1 69

71 Parameterisation dependence reason for inflated χ = 1 Tolerance? Proposal by Pumplin that this may be the case. Simple model, fit depends on one parameter, min at z = and χ = z. Add second parameter y, could get χ profile as shown. For long narrow ellipse can get shift in best fit z such that value corresponds to χ in original model with magnitude much greater than improvement in best fit quality. (1/R R), where R is ratio of minor to major ais. Banff July 1 7

72 Why this doesn t apply to global fits 1. In MSTW/MRST and CTEQ global fits there are more free parameters in obtaining best fit minimum than in determining eigenvectors. Even if correct argument, doesn t directly apply since main effect in change of minimum already accounted for.. This very elliptical profile only occurs if two of the parameters are very correlated. This is in fact why we do not leave all our parameters free in eigenvectors. Along major ais very flat direction always suddenly turns up due to quartic and higher terms in χ distribution. Two parameters compensate almost eactly near minimum, then compensation suddenly breaks. Argument based on quadratic terms breaks down. 3. If z and y highly correlated a large change in z is likely not a large change in a PDF distribution (eplaining small improvement is χ ). 4. If a new parameter is introduced which is not highly correlated with one already there the R is not small and change in old parameters in new best fit is commensurate with the improvement in χ Basic arguments seem to be validated by a variety of checks. Banff July 1 71

73 Parameterisation used in MSTW fits. Only those in red appear in eigenvectors. Of others only Av, Ad, Ag and fied by sum rules and δs correlation. fied due to total Banff July 1 7

74 Banff July 1 73 Trace to eigenvectors 14 and 18 in direction such that χ 1 for uncertainty. change in PDF twice that epected from change in χ for global fit Main change in PDFs in valence quarks shown. Worst change 1.8 bigger than uncertainty in dv. Size of change in PDF not well correlated to relative size of change in parameters. check1/8 at NLO for u V (,Q ) Move each by about 1% and refit. Like having y away from best fit. Refit 8 worse. Some parameters left free move by 3 times quoted uncertainty check1/8 at NLO for d V (,Q ).9 1 Try checking by setting all the parameters not in eigenvectors equal to zero, or if this is not sensible a round value. Fit quality awful. Need these parameters in global fit

75 Banff July 1 74 This time change in PDF pretty much what deterioration in fit quality suggests Again relevant eigenvectors suggest uncertainty corresponds to χ Biggest change in PDFs shown. At most variation about 1.8 uncertainty, in uv. check/8 at NLO for u V (,Q ) Refit with usual eigenvector parameters free. χ is 3 worse Set ɛ.5 term in uv to zero. Usual magnitude is about twice the uncertainty. check/8 at NLO for d V (,Q ).9 1 Try removing parameter which is not highly correlated, i.e. one in eigenvector definition

76 Banff July MSTWp/8 at NLO for u V (,Q ) Change in partons negligible. MSTWp/8 at NLO for d V (,Q ).9 Fit quality improved by units. 1 Also tried adding terms to polynomial in two valence parameterisations

77 Banff July 1 76 However, used non-optimum choice of parameters in eigenvectors for dv (, Q ) in MRST1. Correction of this lead to automatic increase in uncertainty of about 5% at =.5, with no new free parameters. Factor of up to 5% too low for F CC (e + p). Ecellent agreement between two for F CC (e p). Uncertainty using Hessian approach was % for F CC (e p) and 1% for F CC (e + p) Per cent change in e + cross section -6 Plot shown for Lagrange multiplier method for charged current HERA structure functions at =.5 (Red curve fied αs). -4 V Per cent change in e - cross section 1 * V * U * U 5 * + W W * Recall study in first MRST uncertainties paper comparing the Hessian approach with 15 parameters and Lagrange multiplier with parameters and same χ = 5 for both. 5 T* 1 T and e - cross sections are varied at HERA χ increase in global analysis as the e +

78 Banff July 1 77 Etra parameter in eigenvectors for gluon increases uncertainty by about the amount epected Per cent change in W cross section Slightly smaller in latter case. Using 3 parameters lead to narrowing uncertainty in gluon at. affects Tevatron uncertainty. S -5 1 * S * 5 * R * R Per cent change in H cross section P * + P Uncertainty using Hessian approach is 1.% for W and 3% for Higgs. Q * Q Lagrange multiplier method for W and 1GeV Higgs at the Tevatron χ increase in global analysis as the W and H cross sections are varied at the TEVATRON

79 Banff July 1 78 If this was clearly more than 1% the limitations in parameters were addressed and the problem solved. In all cases introduction of etra parameters in the Lagrange multiplier method led to at most a moderate increase in uncertainty Per cent change in W cross section -6 Also looked at uncertainties on moments of u d using Hessian and Lagrange multiplier approaches. Very similar and latter could be slightly smaller. -4 D Per cent change in H cross section D * 1 * A 5 * A * + 5 C * C Uncertainty using Hessian approach about 1% smaller B B * 4 Lagrange multiplier method for W and 1GeV Higgs at the LHC. 6 χ increase in global analysis as the W and H cross sections are varied at the LHC

80 Not looking for χ = 1 anyway Majority of eigenvectors correspond to χ 3. More types of data and weaker cuts than CTEQ. Even more discrepancy? Banff July 1 79

81 Banff July 1 8 Not applied by CTEQ. Part of the reason for large tolerance? Fractional uncertainty MSTW 8 NLO (9% C.L.) Fied data set normalisations Norms. in eigenvectors.1 Fractional uncertainty Use of normalisation uncertainties increases uncertainties on partons significantly Fied data set normalisations Norms. in eigenvectors.8 MSTW 8 NLO (9% C.L.).1 4 Strange quark distribution at Q = 1 GeV 4 Gluon distribution at Q = 1 GeV Difficult to account for in tolerance for eigenvectors some very sensitive (size of quarks) others insensitive (ū d determined from ratios) Fractional uncertainty Fied data set normalisations Norms. in eigenvectors.8 MSTW 8 NLO (9% C.L.).1 Fractional uncertainty Fied data set normalisations Norms. in eigenvectors.8 MSTW 8 NLO (9% C.L.).1 4 Up antiquark distribution at Q = 1 GeV 4 Down antiquark distribution at Q = 1 GeV Normalization uncertainty 1 1.5%, for all partons Fractional uncertainty MSTW 8 NLO (9% C.L.) Fied data set normalisations Norms. in eigenvectors.1 Fractional uncertainty Comparison of full uncertainty and that from no normalization uncertainties (ecept in best fit) MSTW 8 NLO (9% C.L.) Fied data set normalisations Norms. in eigenvectors.5 4 Up valence distribution at Q = 1 GeV 4 Down valence distribution at Q = 1 GeV

82 Banff July 1 81