Parton Distributions and the Relation to LHC and Higgs Physics - PowerPoint PPT Presentation

Sources of Variations/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 α S correlations. Responsible for differences between groups for extraction of fixed-order PDFs. Birmingham – February 2012 13

Variety of PDFs MSTW make available PDFs in a very wide variety of forms. ● At , LO, NLO and NNLO, with some minor approximations at NNLO. ● Also a variety of extensions such as different α S values, heavy quark masses, different flavour numbers. Latter covered tomorrow. ● Older MRST versions of modified LO* and LO** PDFs and of PDFs including QED evolution. Fit data for scales above 2GeV 2 . (most) DIS data for W 2 > 15GeV 2 . Will mention effect of cuts later. Don’t yet include combined HERA cross-section data. Have checked effects of this. In some cases predictions change by a little over 1 σ , in many cases less. Major problems with high-luminosity D0 lepton asymmetry in some binnings. Same for other groups. Birmingham – February 2012 14

Comparison of gluon from fit 100 using combined HERA data to xg(x,Q 2 =10000GeV 2 ) MSTW2008 NNLO versions with 1 − 80 σ , uncertainty shown. 60 Slight difference in details of 40 normalisation treatment compared to previous versions, still preliminary. 20 First times showed uncertainty. 0 x -5 -4 -3 -2 -1 10 10 10 10 10 1 Value of α S ( M 2 Z ) moves slightly, 15 0 . 1171 → 0 . 1178 . percentage difference at Q 2 =10000GeV 2 10 Changes always within 1 − σ , and really less due to correlations with 5 α S . 0 Uncertainty slightly smaller, especially -5 at very small x . -10 -5 -4 -3 -2 -1 x 10 10 10 10 10 Birmingham – February 2012 15

6 6 xu(x,Q 2 =10000GeV 2 ) xd(x,Q 2 =10000GeV 2 ) 4 4 2 2 0 0 x x -5 -4 -3 -2 -1 -5 -4 -3 -2 -1 10 10 10 10 10 1 10 10 10 10 10 1 15 15 percentage difference at Q 2 =10000GeV 2 percentage difference at Q 2 =10000GeV 2 10 10 5 5 0 0 -5 -5 -10 -10 -5 -4 -3 -2 x -1 -5 -4 -3 -2 x -1 10 10 10 10 10 10 10 10 10 10 Most dramatic change for up quark at about x = 0 . 01 . Birmingham – February 2012 16

Impact on Cross Sections. � σ Z (nb)TeV � σ Z (nb)LHC The values of the predicted cross-sections at NNLO for Z and a 120 GeV Higgs boson at the Tevatron and the LHC (latter for 14 TeV centre of mass energy). PDF set B l + l − σ H (pb)TeV B l + l − σ H (pb)LHC MSTW08 0.2507 0.9549 2.051 50.51 Comb HERA +2.1 % +1.2 % +0.9 % +0.7 % For new global fits 2% effect on Z (and W ) cross sections at Tevatron, but small change at LHC. Similar to, or less than 1 − σ uncertainty in former case. Maximum of ∼ 1% for Higgs. Small effect. Birmingham – February 2012 17

Parton Fits and Uncertainties . Two main approaches. Parton parameterization and Hessian (Error Matrix) approach first used by H1 and ZEUS, and extended by CTEQ. χ 2 − χ 2 min ≡ ∆ χ 2 = H ij ( a i − a (0) i )( a j − a (0) � j ) i,j The Hessian matrix H is related to the covariance matrix of the parameters by C ij ( a ) = ∆ χ 2 ( H − 1 ) ij . We can then use the standard formula for linear error propagation. ∂F ∂F (∆ F ) 2 = ∆ χ 2 � ( H ) − 1 , ij ∂a i ∂a j i,j This is now the most common approach. Birmingham – February 2012 18

Can find and rescale eigenvectors of H leading to diagonal form ∆ χ 2 = � z 2 i i Implemented by CTEQ, then MRST/MSTW, HERAPDF. Uncertainty on physical quantity then given by (∆ F ) 2 = � 2 , F ( S (+) ) − F ( S ( − ) � � ) i i i where S (+) and S ( − ) are PDF sets displaced along eigenvector direction. i i Must choose “correct” ∆ χ 2 given complication of errors in full fit and sometimes conflicting data sets. Birmingham – February 2012 19

Determination of best fit and uncertainties All but NNPDF minimise χ 2 and expand about best fit. ● MSTW08 – 20 eigenvectors. Due to incompatibility of different sets and (perhaps to some extent) parameterisation inflexibility (little direct evidence for this) have inflated ∆ χ 2 of 5 − 20 for eigenvectors. ● CT10 – 26 eigenvectors. Inflated ∆ χ 2 of ∼ 50 for 1 sigma for eigenvectors. Use “∆ χ 2 = 1 ′′ . ● HERAPDF2.0 – 10 eigenvectors. Additional model and parameterisation uncertainties. ● ABKM09 – 21 parton parameters. Use ∆ χ 2 = 1 . Also α S , m c , m b . ● GJR08 – 20 parton parameters (8 fixed for uncertainty) and α S . Use ∆ χ 2 ≈ 20 . Impose strong theory constraint on input form of PDFs. Perhaps surprisingly all get rather similar uncertainties for PDFs cross-sections, though don’t all mean the same. Birmingham – February 2012 20

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 of the original data set F exp, ( k ) Carlo replicas F art,k . i,p i,p Where r ( k ) are random numbers following Gaussian distribution, and S ( k ) p,N is the p analogous normalization shift of the of the replica depending on 1 + r ( k ) p,n σ norm . p Hence, include information about measurements and errors in distribution of F art, ( k ) . i,p Fit to the data replicas obtaining PDF replicas q ( net )( k ) (follows Giele et al .) i Mean µ O and deviation σ O of observable O then given by N rep N rep 1 1 O [ q ( net )( k ) ( O [ q ( net )( k ) � � σ 2 ] − µ O ) 2 . µ O = ] , O = i i N rep N rep 1 1 Eliminates parameterisation dependence by using a neural net which undergoes a series of (mutations via genetic algorithm) to find the best fit. In effect is a much larger sets of parameters – ∼ 37 per distribution. Birmingham – February 2012 21

Parameterisations - for the gluon at small x different parameterisations lead to very different uncertainty for small x gluon. 2 2 Gluon distribution at Q = 5 GeV 0.5 0.5 Fractional uncertainty Fractional uncertainty MSTW 2008 NLO (90% C.L.) 0.4 0.4 CTEQ6.6 NLO 0.3 0.3 Alekhin 2002 NLO NNPDF1.0 (1000 replicas) 0.2 0.2 0.1 0.1 0 0 -0.1 -0.1 -0.2 -0.2 -0.3 -0.3 -0.4 -0.4 -0.5 -0.5 -5 -5 -3 -3 -4 -4 -2 -2 -1 -1 10 10 10 10 10 10 10 10 10 10 x x Most assume single power x λ at input → limited uncertainty. If input at low Q 2 λ positive and small- x input gluon fine-tuned to ∼ 0 . Artificially small uncertainty. If g ( x ) ∝ x λ ± ∆ λ then ∆ g ( x ) = ∆ λ ln(1 /x ) ∗ g ( x ) . MRST/MSTW and NNPDF more flexible (can be negative) → rapid expansion of uncertainty where data runs out. CT10 more flexible than previous versions. Birmingham – February 2012 22

Generally high- x PDFs parameterised so will behave like (1 − x ) η as x → 1 . More flexibility in CTEQ. Very hard high- x gluon distribution (more-so even than NNPDF uncertainties). However, is gluon, which is radiated from quarks, harder than the up valence distribution for x → 1 ? Birmingham – February 2012 23

0.6 NNPDF2.1 0.5 CT10 0.4 MSTW08 0.3 ) 2 0 (x, Q 0.2 + 0.1 xs 0 -0.1 -0.2 -0.3 -3 -2 -1 10 10 10 x MSTW has theory assumption on strange at small x , CT10 less strong and NNPDF fully flexible. Variation near x = 0 . 05 where data exists likely due to heavy flavour definitions/nuclear corrections. Birmingham – February 2012 24

Heavy Quarks – Essential to treat these correctly. Two distinct regimes: Near threshold Q 2 ∼ m 2 H massive quarks not partons. Created in final state. Described using Fixed Flavour Number Scheme (FFNS). n f F ( x, Q 2 ) = C F F ( Q 2 /m 2 k ( Q 2 ) H ) ⊗ f k Does not sum ln n ( Q 2 /m 2 H ) terms, and not calculated for many processes beyond LO. Used by AB(K)M and (G)JR. Sometimes final state details in this scheme only. Alternative, at high scales Q 2 ≫ m 2 H heavy quarks like massless partons. Behave like up, down, strange. Sum ln( Q 2 /m 2 H ) terms via evolution. Zero Mass Variable Flavour Number Scheme (ZM-VFNS). Normal assumption in calculations. Ignores O ( m 2 H /Q 2 ) corrections. No longer used. n f +1 F ( x, Q 2 ) = C ZMV F ( Q 2 ) . ⊗ f j j Advocate a General Mass Variable Flavour Number Scheme (GM-VFNS) interpolating between the two well-defined limits of Q 2 ≤ m 2 H and Q 2 ≫ m 2 H . Used by MRST/MSTW and more recently (as default) by CTEQ, and now also by HERAPDF and NNPDF. Birmingham – February 2012 25

H1 _ Various definitions possible. Versions H1 F b b (x,Q 2 ) 2 used by MSTW (RT) and CTEQ × 6 i x=0.0002 (ACOT) have converged somewhat. i=5 _ b F b 2 x=0.0005 i=4 Various significant differences still 10 exist as illustrated by comparison x=0.0013 i=3 to most recent H1 data on bottom x=0.005 1 i=2 production. -1 x=0.013 10 i=1 H1 Data x=0.032 MSTW08 NNLO i=0 -2 MSTW08 10 CTEQ6.6 2 3 10 10 10 Q 2 / GeV 2 Birmingham – February 2012 26

1.1 1.1 GMVFNSa/2008 at NNLO for g(x,Q 2 ) GMVFNSa/2008 at NLO for g(x,Q 2 ) MSTW08 MSTW08NNLO GMVFNS1 GMVFNS1 GMVFNS2 GMVFNS2 1.05 GMVFNS3 1.05 GMVFNS3 GMVFNSopt 1 1 0.95 0.95 GMVFNS4 GMVFNS4 GMVFNS5 GMVFNS5 GMVFNS6 GMVFNS6 ZMVFNS GMVFNSopt 0.9 0.9 -5 -4 -3 -2 -1 -5 -4 -3 -2 -1 10 10 10 10 10 10 10 10 10 10 1.1 1.1 GMVFNSa/2008 at NNLO for u(x,Q 2 ) GMVFNSa/2008 at NLO for u(x,Q 2 ) MSTW08 MSTW08NNLO GMVFNS1 GMVFNS1 GMVFNS2 GMVFNS2 1.05 GMVFNS3 1.05 GMVFNS3 GMVFNSopt 1 1 0.95 0.95 GMVFNS4 GMVFNS4 GMVFNS5 GMVFNS5 GMVFNS6 GMVFNS6 ZMVFNS GMVFNSopt 0.9 0.9 -5 -4 -3 -2 -1 -5 -4 -3 -2 -1 10 10 10 10 10 10 10 10 10 10 Variations in partons extracted from global fit due to different choices of GM-VFNS at NLO and at NNLO. Birmingham – February 2012 27

PDF correlation with α S . Can also look at PDF changes and uncertainties at different α S ( M 2 Z ) . Fully included (difficult to disentangle) in ABKM, (G)JR), but often only for one fixed α S ( M 2 Z ) . MSTW produce sets for limits of α S uncertainty – PDF uncertainties reduced since quality of fit already worse than best fit. 2 2 2 Gluon at Q = M = (120 GeV) H 1.05 1.05 Ratio to MSTW 2008 NNLO Ratio to MSTW 2008 NNLO 1.04 1.04 MSTW 2008 NNLO (68% C.L.) α Fix at +68% C.L. limit 1.03 1.03 S α Fix at - 68% C.L. limit 1.02 1.02 S 1.01 1.01 1 1 0.99 0.99 0.98 0.98 0.97 0.97 y = 0 y = 0 at LHC at Tevatron 0.96 0.96 0.95 0.95 -3 -3 -4 -4 -2 -2 -1 -1 10 10 10 10 10 10 10 10 x x Expected gluon– α S ( M 2 Z ) small– x anti-correlation → high- x correlation from sum rule. Birmingham – February 2012 28

NNLO predictions for Higgs ( 120GeV ) production for different allowed α S ( M 2 Z ) values and their uncertainties. Higgs (M = 120 GeV) with MSTW 2008 NNLO PDFs H Tevatron, s = 1.96 TeV LHC, s = 14 TeV 6 6 (%) (%) α 2 2 (M ) S H gg luminosity 4 4 NNLO NNLO H H σ σ 2 2 ∆ ∆ 0 0 −2 −2 −4 −4 68% C.L. uncertainties −6 −6 σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ −1 −1 − − /2 /2 0 0 + + /2 /2 +1 +1 −1 −1 − − /2 /2 0 0 + + /2 /2 +1 +1 ∆ α 2 ∆ α 2 (M ) (M ) S S Z Z Increases by a factor of 2 − 3 (up more than down) at LHC. Direct α S ( M 2 Z ) dependence mitigated somewhat by anti-correlated small- x gluon (asymmetry feature of minor problems in fit to HERA data). At Tevatron intrinsic gluon uncertainty dominates. Birmingham – February 2012 29

Other sources of Uncertainty. Also other sources which (mainly) lead to inaccuracies common to all fixed-order extractions. ● Standard higher orders NNLO. Many sets available here, soon all of them. ● QED and Weak (comparable to NNLO ?) ( α 3 s ∼ α ). Sometime enhancements. ● Nuclear/deuterium corrections to structure functions. s ln n − 1 (1 /x ) ), or large x ( α n s ln 2 n − 1 (1 − x ) ). ● Resummations, e.g. small x ( α n ● low Q 2 (higher twist), saturation. Birmingham – February 2012 30

Deuterium corrections. 1.15 on-shell convolution + off-shell (mKP) 1.1 density N d / F 2 1.05 F 2 1 0.95 0 0.2 0.4 0.6 0.8 1 x Variation in W + /W − ratio probably partially related to the issue of deuterium corrections. Recent study (Accardi et al ) suggests these may be large. Uncertainty in correction as large as PDF uncertainty, but size of corrections can be larger. Birmingham – February 2012 31

1.05 MSTW found improvement in fit to both global data set and lepton asymmetry with deuterium corrections, but < 1 for all but very high x . Also find significant improvement 1 with rather more plausible deuterium correction factor corrections. Ongoing study for MSTW. Simple model constrained model 0.95 D0II electron combined E T weighted D0II electron combined E T 0.9 -2 -1 10 10 x Birmingham – February 2012 32

PDFs at NNLO NNLO splitting functions (Moch, Vermaseren and Vogt) allow essentially full NNLO determination of partons now being performed, though heavy flavour not fully worked out in the fixed-flavour number scheme (FFNS) and jet cross-sections are only approximate. Improves consistency of fit very slightly, and reduces α S . Surely this is best, i.e. most accurate. Yes, but ...... only know some hard cross-sections at NNLO. Processes with two strongly interacting particles largely completed DIS coefficient functions and sum rules p ) → γ ⋆ , W, Z (including rapidity dist.), H, A 0 , WH, ZH . pp (¯ But for many other final states NNLO not known. NLO still more appropriate. Birmingham – February 2012 33

How do NNLO PDFs compare to NLO? 30 xg(x,Q 2 =100GeV 2 ) 20 5 MSTW08 NNLO MSTW08 NLO 10 xg at Q 2 =2GeV 2 x 0 -4 -3 -2 -1 10 10 10 10 15 percentage difference at Q 2 =100GeV 2 0 10 5 MSTW08 NNLO 0 MSTW08 NLO -5 -5 -10 x -5 -4 -3 -2 -1 -4 -3 -2 -1 x 10 10 10 10 10 1 10 10 10 10 Gluons different at NLO and NNLO at low Q 2 . Largely washed out by evolution, but only because of different α S . Birmingham – February 2012 34

Sometimes vital to use NNLO xu(x,Q 2 =100GeV 2 ) 1.5 PDFs if calculating at NNLO. MSTW08 NNLO Systematic difference between 1 PDF defined at NLO and at NNLO. MSTW08 NLO 0.5 Due to large (negative) gluon coefficient function at not too small x . 0 -4 -3 -2 -1 10 10 10 10 x 15 Systematic difference between percentage difference at Q 2 =100GeV 2 PDF defined at NLO and at 10 NNLO. MSTW08 NNLO 5 0 MSTW08 NLO -5 -10 -4 -3 -2 -1 x 10 10 10 10 Birmingham – February 2012 35

Considerations of differences and of NNLO MSTW 2008 K-factor for Drell-Yan Cross-section ~ (x,Q 2 ) HERA (F 2 (x,Q 2 ) FixedTarget ) + c σ 2 H1 ZEUS NMC 2 BCDMS x=0.0005 (c=0.35) SLAC x=0.00016 (c=0.4) M=4GeV x=0.0013 (c=0.3) NNLO 1.5 NLO x=0.0032 (c=0.25) 1.5 x=0.008 (c=0.2) 1 LO x=0.013 (c=0.15) 1 x=0.05 (c=0.1) 0.5 x=0.18 (c=0.05) 0.5 x=0.35 (c=0.0) NNLO NLO LO 0 2 3 1 10 10 Q 2 (GeV 2 ) 10 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 x F In general NNLO corrections either positive for cross sections, e.g. Drell Yan, or for evolution in structure functions. Automatically leads to lower α S ( M 2 Z ) at NNLO than at NLO, i.e. 0 . 1171 rather than 0 . 1202 . Difference between two quite stable. Birmingham – February 2012 36

Converging on general agreement that the NNLO values of α S are 0 . 0002 − 0 . 0003 smaller than the NLO values of α S ? MSTW08 – α S ( M 2 Z ) = 0 . 1202 → 0 . 1171 . ABKM09 – α S ( M 2 Z ) = 0 . 1179 → 0 . 1135 . GJR/JR – α S ( M 2 Z ) = 0 . 1145 → 0 . 1124 . NNPDF2.1 – α S ( M 2 Z ) = 0 . 1191 → 0 . 1174 . CT10.1 – α S ( M 2 Z ) = 0 . 1196 → 0 . 1180 (both prelim – PDF4LHC, DESY July). HERAPDF1.6 – α S ( M 2 Z ) = 0 . 1202 at NLO and general preference for ∼ 0 . 1176 at NNLO. Central values differ far more than NLO → NNLO trend. Birmingham – February 2012 37

NLO → NNLO PDF differences Σ Σ (q q ) luminosity at LHC ( s = 7 TeV) (q q ) luminosity at LHC ( s = 7 TeV) q q 1.2 1.2 1.2 1.2 Ratio to MSTW 2008 NLO (68% C.L.) Ratio to MSTW 2008 NLO (68% C.L.) Ratio to MSTW 2008 NNLO (68% C.L.) Ratio to MSTW 2008 NNLO (68% C.L.) MSTW08 NNLO MSTW08 NLO 1.15 1.15 1.15 1.15 HERAPDF1.0 HERAPDF1.0 HERAPDF1.5 HERAPDF1.5 ABKM09 1.1 1.1 1.1 1.1 ABKM09 JR09 GJR08 NNPDF2.1 G. Watt (September 2011) G. Watt (September 2011) 1.05 1.05 1.05 1.05 1 1 1 1 0.95 0.95 0.95 0.95 0.9 0.9 0.9 0.9 0.85 0.85 0.85 0.85 W Z W Z 0.8 0.8 0.8 0.8 -3 -3 -2 -2 -1 -1 -3 -3 -2 -2 -1 -1 10 10 10 10 10 10 10 10 10 10 10 10 s s / s / s s s / s / s Luminosity differences for quarks largely the same at NNLO as at NLO, except for HERAPDF1.5 at large x . Differences between different sets not likely to be due to theory choices which would diminish at higher orders, or approx. at NNLO which would change relative NLO and NNLO differences. Birmingham – February 2012 38

gg luminosity at LHC ( s = 7 TeV) gg luminosity at LHC ( s = 7 TeV) 1.2 1.2 1.2 1.2 Ratio to MSTW 2008 NLO (68% C.L.) Ratio to MSTW 2008 NLO (68% C.L.) Ratio to MSTW 2008 NNLO (68% C.L.) Ratio to MSTW 2008 NNLO (68% C.L.) MSTW08 NLO MSTW08 NNLO 1.15 1.15 1.15 1.15 HERAPDF1.0 HERAPDF1.0 HERAPDF1.5 HERAPDF1.5 ABKM09 1.1 1.1 1.1 1.1 ABKM09 JR09 GJR08 NNPDF2.1 G. Watt (September 2011) G. Watt (September 2011) 1.05 1.05 1.05 1.05 1 1 1 1 0.95 0.95 0.95 0.95 0.9 0.9 0.9 0.9 0.85 0.85 0.85 0.85 0.8 0.8 0.8 0.8 120 180 240 120 180 240 -3 -3 -3 -3 -2 -2 t t -1 -1 -2 -2 t t -1 -1 10 10 10 10 10 10 10 10 10 10 10 10 M (GeV) M (GeV) H H s s / s / s s s / s / s Luminosity differences for the gluon also largely the same at NNLO as at NLO, except for HERAPDF1.5 again. Birmingham – February 2012 39

Investigation to stability under changes in cuts. cut to 20GeV 2 , but no real Raise W 2 1.1 partons/MSTW2008 at NLO for g(x,Q 2 ) changes. MSTW08 NLO Q 2 =5GeV 2 , W 2 =20GeV 2 Q 2 =10GeV 2 , W 2 =20GeV 2 1.05 cut to 5GeV 2 and then Also raise Q 2 10GeV 2 . 1 At NLO some movement just outside 0.95 Preliminary default error bands at general x . Q 2 =10,000GeV 2 0.9 Find α S ( M 2 Z ) = 0 . 1202 → 0 . 1193 → -5 -4 -3 -2 -1 10 10 10 10 10 0 . 1175 , though for Q 2 = 10GeV 2 cut 1.1 partons/MSTW2008 at NLO for u(x,Q 2 ) MSTW08 NLO error has roughly doubled to about Q 2 =5GeV 2 , W 2 =20GeV 2 Q 2 =10GeV 2 , W 2 =20GeV 2 1.05 0 . 0025 . 1 0.95 Preliminary Q 2 =10,000GeV 2 0.9 -5 -4 -3 -2 -1 10 10 10 10 10 Birmingham – February 2012 40

At NNLO most movement outside 1.1 partons/MSTW2008 at NNLO for g(x,Q 2 ) default error bands at low x , where MSTW08 NNLO Q 2 =5GeV 2 , W 2 =20GeV 2 constraint vanishes as Q 2 cut raises. Q 2 =10GeV 2 , W 2 =20GeV 2 1.05 cut = 10GeV 2 no points below x = For Q 2 1 0 . 0001 , and little lever arm for evolution constraint for a bit higher. 0.95 Preliminary Find α S ( M 2 Z ) = 0 . 1171 → 0 . 1171 → 0.9 -5 -4 -3 -2 -1 0 . 1164 , i.e. no change of significance. 10 10 10 10 10 1.1 partons/MSTW2008 at NNLO for u(x,Q 2 ) MSTW08 NNLO Q 2 =5GeV 2 , W 2 =20GeV 2 Q 2 =10GeV 2 , W 2 =20GeV 2 1.05 1 0.95 Preliminary 0.9 -5 -4 -3 -2 -1 10 10 10 10 10 Birmingham – February 2012 41

The % change in the cross sections after cuts ( M H = 165GeV ). NLO NNLO 5GeV 2 10GeV 2 5GeV 2 10GeV 2 Q 2 cut W Tev 0.0 -2.4 -0.7 -0.4 0.0 -0.8 -0.4 0.0 Z Tev W LHC (7TeV) -0.2 -0.1 -0.2 -0.2 Z LHC (7TeV) -0.2 -0.3 -0.4 -0.5 W LHC (14TeV) -0.6 -1.1 0.3 0.8 Z LHC (14TeV) -0.6 -1.5 0.2 0.4 -1.1 -1.5 -1.2 -3.2 Higgs TeV Higgs LHC (7TeV) -0.8 -2.5 0.4 -1.8 Higgs LHC (14TeV) -0.9 -1.9 1.0 -0.8 More variation at NLO than at NNLO, i.e. 7 changes of > 1% compared to 4 . However, both small, and changes with change in Q 2 cut slow. Does not suggest significant higher twist or problem with default cuts. Birmingham – February 2012 42

Small- x Theory At each order in α S each splitting function and coefficient function obtains an extra power of ln(1 /x ) (some accidental zeros in P gg ), i.e. P ij ( x, α s ( Q 2 )) , C P i ( x, α s ( Q 2 )) ∼ s ( Q 2 ) ln m − 1 (1 /x ) . α m Summed using BFKL equation (and a lot of work – Altarelli-Ball-Forte, Ciafaloni- Colferai-Salam-Stasto and White-RT) H1 Preliminary F L Comparison to H1 prelim data on 1 1 ) 2 (x, Q F L ( x, Q 2 ) at low Q 2 , only within H1 (Prelim.) MSTW NLO MSTW NNLO E = 460, 575, 920 GeV White-RT approach, suggests WT NLO + NLL(1/x) p L F resummations may be important. 0.000059 0.000087 0.00013 0.00017 0.00021 0.00029 0.00040 0.00052 0.00067 0.00090 0.0011 0.0015 0.0023 x Could possibly give a few percent 0.5 0.5 effect on Higgs cross sections. 0 0 2 10 10 2 2 Q / GeV Birmingham – February 2012 43

H1 Collaboration F L 0.28 0.43 0.59 0.88 1.29 1.69 2.24 3.19 4.02 5.40 6.86 10.3 14.6 x 10 4 0.4 0.2 H1 Data 0 HERAPDF1.0 NLO MSTW08 NNLO CT10 NLO GJR08 NNLO NNPDF2.1 NLO ABKM09 NNLO -0.2 2 10 50 Q 2 / GeV 2 However, quite a large PDF uncertainty (in general) and even larger spread, at fixed order. Birmingham – February 2012 44

Fits to Jet Data and relation to NNLO ∅ D Run II inclusive jet data (cone, R = 0.7) σ µ (Ratio w.r.t. NLO with = p using MSTW08 NNLO PDFs) NNLO approx. jet corrections. T 1.4 1.4 T T = p = p 1.3 1.3 JET JET µ µ 0.0 < |y | < 0.4 0.4 < |y | < 0.8 with with 1.2 1.2 Shape of corrections as function 1.1 1.1 σ σ Ratio w.r.t NLO Ratio w.r.t NLO 1 1 of p T at NLO and also at approx. 0.9 0.9 µ µ × = = 0.5 p R F 0.8 T 0.8 µ µ × σ = = 1.0 p Solid lines: NLO R F T 0.7 µ µ × 0.7 σ = = 2.0 p Dashed lines: NLO + 2-loop threshold NNLO in inclusive case. R F T 0.6 0.6 2 2 10 10 JET JET p (GeV) p (GeV) T T 1.4 1.4 T T NNLO uses threshold (Kidonakis = p = p 1.3 1.3 JET JET µ µ 0.8 < |y | < 1.2 1.2 < |y | < 1.6 with with 1.2 1.2 and Owens) approx. for Tevatron 1.1 1.1 σ σ Ratio w.r.t NLO Ratio w.r.t NLO 1 1 0.9 0.9 jets. 0.8 0.8 0.7 0.7 0.6 0.6 2 2 10 10 JET JET p (GeV) p (GeV) T T NNLO approximation not large 1.4 1.4 T T = p = p and aids stability – always worst 1.3 1.3 JET JET µ µ 1.6 < |y | < 2.0 2.0 < |y | < 2.4 with 1.2 with 1.2 1.1 1.1 at high- p T i.e. high- x . Includes σ σ Ratio w.r.t NLO Ratio w.r.t NLO 1 1 0.9 0.9 large ln( p T /µ ) terms predicted by 0.8 0.8 0.7 0.7 0.6 0.6 2 2 renormalisation group. 10 10 JET JET p (GeV) p (GeV) T T Birmingham – February 2012 45

de Florian and Vogelsang result for inclusive jet K-factor for dσ/dp T at order α 2+ n compared to NLO. S Birmingham – February 2012 46

Impact on Higgs at Tevatron. Plots (Watt) show the gluon luminosities at the Tevatron NLO and NNLO gg luminosity at Tevatron ( s = 1.96 TeV) gg luminosity at Tevatron ( s = 1.96 TeV) 1.4 1.4 1.4 1.4 Ratio to MSTW 2008 NLO (68% C.L.) Ratio to MSTW 2008 NLO (68% C.L.) Ratio to MSTW 2008 NNLO (68% C.L.) Ratio to MSTW 2008 NNLO (68% C.L.) MSTW08 NLO MSTW08 NNLO 1.3 1.3 1.3 1.3 CTEQ6.6 HERAPDF1.0 CT10 ABKM09 1.2 1.2 1.2 1.2 JR09 NNPDF2.1 1.1 1.1 1.1 1.1 1 1 1 1 0.9 0.9 0.9 0.9 0.8 0.8 0.8 0.8 0.7 0.7 0.7 0.7 0.6 0.6 0.6 0.6 -3 -3 -2 -2 -1 -1 -3 -3 -2 -2 -1 -1 10 10 10 10 10 10 10 10 10 10 10 10 120 180 240 120 180 240 s s / s / s s s / s / s M (GeV) M (GeV) H H Similar to the LHC, but deviations with high- x PDF origin persist to lower ˆ s . Differences in α S ( M 2 Z ) generally increase effect of discrepancy. Birmingham – February 2012 47

Higgs production via gluon fusion at the Tevatron and LHC at NLO and NNLO. → → NNLO gg H at the LHC ( s = 7 TeV) for M = 120 GeV NNLO gg H at the Tevatron ( s = 1.96 TeV) for M = 120 GeV H H 1.2 1.2 (pb) (pb) (pb) (pb) 17 17 1.1 1.1 16 16 H H H H σ σ σ σ 1 1 15 15 0.9 0.9 G. Watt (April 2011) 14 14 0.8 0.8 13 13 68% C.L. PDF 68% C.L. PDF MSTW08 MSTW08 12 12 0.7 0.7 HERAPDF1.0 HERAPDF1.0 11 11 0.6 0.6 ABKM09 ABKM09 GJR08/JR09 GJR08/JR09 Vertical error bars Vertical error bars 10 10 0.5 0.5 Inner: PDF only Inner: PDF only Closed symbols: NNLO Closed symbols: NNLO α α Outer: PDF+ Outer: PDF+ S Open symbols: NLO S Open symbols: NLO 9 9 0.4 0.4 0.11 0.11 0.115 0.115 0.12 0.12 0.125 0.125 0.13 0.13 0.11 0.11 0.115 0.115 0.12 0.12 0.125 0.125 0.13 0.13 α α 2 2 α α 2 2 (M (M ) ) (M (M ) ) S S S S Z Z Z Z → → NNLO gg H at the LHC ( s = 7 TeV) for M = 240 GeV NNLO gg H at the Tevatron ( s = 1.96 TeV) for M = 240 GeV H H 0.12 0.12 (pb) (pb) (pb) (pb) 3.8 3.8 0.11 0.11 3.6 3.6 H H H H 0.1 0.1 σ σ σ σ 3.4 3.4 0.09 0.09 3.2 3.2 G. Watt (April 2011) 0.08 0.08 3 3 68% C.L. PDF 68% C.L. PDF 0.07 0.07 2.8 2.8 MSTW08 MSTW08 0.06 0.06 HERAPDF1.0 HERAPDF1.0 2.6 2.6 ABKM09 ABKM09 0.05 0.05 2.4 2.4 GJR08/JR09 GJR08/JR09 Vertical error bars Vertical error bars Inner: PDF only Inner: PDF only 0.04 0.04 2.2 2.2 Closed symbols: NNLO Closed symbols: NNLO α α Outer: PDF+ Outer: PDF+ S Open symbols: NLO S Open symbols: NLO 2 2 0.03 0.03 0.11 0.11 0.115 0.115 0.12 0.12 0.125 0.125 0.13 0.13 0.11 0.11 0.115 0.115 0.12 0.12 0.125 0.125 0.13 0.13 α α α α 2 2 2 2 (M (M ) ) (M (M ) ) S S S S Z Z Z Z Larger deviation at the Tevatron. NNLO pattern very similar to NLO. Birmingham – February 2012 48

High- x gluon, at least to some extent, constrained by comparison to Tevatron jet data. However, important point, CDF Z -rapidity data, or cross sections, sets Tevatron normalisation in a fit. Only allows a few percent variation in normalisation. Different PDF predictions for W and Z cross sections at the Tevatron compared to data. ± ± → ν 0 → + - NNLO W l at the Tevatron ( s = 1.96 TeV) NNLO Z l l at the Tevatron ( s = 1.96 TeV) 3 3 0.3 0.3 ) (nb) ) (nb) ) (nb) ) (nb) Thicker lines: central value -1 -1 CDF, L = 72 pb CDF, L = 2.1 fb Thinner lines: stat.+syst. 2.95 2.95 0.29 0.29 Thicker lines: central value Shaded bands: stat.+syst.+lumi. Thinner lines: stat.+syst. ∅ -1 Shaded bands: stat.+syst.+lumi. D , L = 1 fb 2.9 2.9 0.28 0.28 ν ν - - l l + + ± ± l l l l 2.85 2.85 0.27 0.27 → → → → 2.8 2.8 0.26 0.26 0 0 ± ± B(Z B(Z B(W B(W 2.75 2.75 0.25 0.25 ⋅ ⋅ 68% C.L. PDF 68% C.L. PDF 2.7 2.7 0.24 0.24 ⋅ ⋅ 0 0 Z Z ± ± MSTW08 MSTW08 W W σ σ 2.65 2.65 0.23 0.23 σ σ HERAPDF1.0 HERAPDF1.0 2.6 2.6 0.22 0.22 ABKM09 ABKM09 Vertical error bars Vertical error bars Inner: PDF only Inner: PDF only 2.55 2.55 JR09 0.21 0.21 JR09 α α Outer: PDF+ Outer: PDF+ S S 2.5 2.5 0.2 0.2 0.11 0.11 0.115 0.115 0.12 0.12 0.125 0.125 0.13 0.13 0.11 0.11 0.115 0.115 0.12 0.12 0.125 0.125 0.13 0.13 α α 2 2 α α 2 2 (M (M ) ) (M (M ) ) S S S S Z Z Z Z Everyone ok or a bit high. Normalisation no room to move down. Birmingham – February 2012 49

NLO PDF (with NLO ˆ σ ) µ = p T / 2 µ = p T µ = 2 p T MSTW08 0.75 (0.30) 0.68 (0.28) 0.91 (0.84) CTEQ6.6 1.25 (0.14) 1.66 (0.20) 2.38 (0.84) CT10 1.03 (0.13) 1.20 (0.19) 1.81 (0.84) NNPDF2.1 0.74 (0.29) 0.82 (0.25) 1.23 (0.69) HERAPDF1.0 2.43 (0.39) 3.26 (0.66) 4.03 (1.67) HERAPDF1.5 2.26 (0.40) 3.05 (0.66) 3.80 (1.66) ABKM09 1.62 (0.52) 2.21 (0.85) 3.26 (2.10) GJR08 1.36 (0.23) 0.94 (0.13) 0.79 (0.36) NNLO PDF (with NLO+2-loop ˆ σ ) µ = p T / 2 µ = p T µ = 2 p T MSTW08 1.39 (0.42) 0.69 (0.44) 0.97 (0.48) HERAPDF1.0, α S ( M 2 Z ) = 0 . 1145 2.64 (0.36) 2.15 (0.36) 2.20 (0.46) HERAPDF1.0, α S ( M 2 Z ) = 0 . 1176 2.24 (0.35) 1.17 (0.32) 1.23 (0.31) ABKM09 2.55 (0.82) 2.76 (0.89) 3.41 (1.17) JR09 0.75 (0.37) 1.26 (0.41) 2.21 (0.49) Table 1: Values of χ 2 /N pts . for the CDF Run II inclusive jet data using the k T jet algorithm with N pts . = 76 and N corr . = 17 , for different PDF sets and different scale choices At most a 1- σ shift in normalisation is allowed. Birmingham – February 2012 50

NLO PDF (with NLO ˆ σ ) µ = p T / 2 µ = p T µ = 2 p T MSTW08 0.75 ( + 0.32) 0.68 ( − 0.88) 0.63 ( − 2.69 ) CTEQ6.6 1.03 ( − 2.47 ) 1.04 ( − 3.49 ) 0.99 ( − 4.75 ) CT10 0.99 ( − 1.64 ) 0.92 ( − 2.69 ) 0.86 ( − 4.10 ) NNPDF2.1 0.74 ( − 0.33) 0.79 ( − 1.60 ) 0.80 ( − 3.12 ) HERAPDF1.0 1.52 ( − 4.07 ) 1.57 ( − 5.21 ) 1.43 ( − 6.22 ) HERAPDF1.5 1.48 ( − 3.85 ) 1.52 ( − 5.00 ) 1.39 ( − 6.03 ) ABKM09 1.03 ( − 3.49 ) 1.01 ( − 4.53 ) 1.05 ( − 5.80 ) GJR08 1.14 ( + 2.47 ) 0.93 ( + 1.25 ) 0.79 ( − 0.50) NNLO PDF (with NLO+2-loop ˆ σ ) µ = p T / 2 µ = p T µ = 2 p T MSTW08 1.39 ( + 0.35) 0.69 ( − 0.45) 0.97 ( − 1.30 ) HERAPDF1.0, α S ( M 2 Z ) = 0 . 1145 2.37 ( − 2.65 ) 1.48 ( − 3.64 ) 1.29 ( − 4.12 ) HERAPDF1.0, α S ( M 2 Z ) = 0 . 1176 2.24 ( − 0.48) 1.13 ( − 1.60 ) 1.09 ( − 2.23 ) ABKM09 1.53 ( − 4.27 ) 1.23 ( − 5.05 ) 1.44 ( − 5.65 ) JR09 0.75 ( + 0.13) 1.26 ( − 0.61) 2.20 ( − 1.22 ) Table 2: Values of χ 2 /N pts . for the CDF Run II inclusive jet data using the k T jet algorithm No restriction is imposed on the shift in normalisation and the optimal value of “ − r lumi . ” is shown in brackets. Birmingham – February 2012 51

Comparisons to LHC data CMS results very similar. Birmingham – February 2012 52

NNLO W and Z cross sections at the LHC ( s = 7 TeV) Differences in predictions at ) (nb) ) (nb) NNLO compared to NLO (Watt). 1.02 1.02 R(W/Z) = -1 CMS, L = 36 pb 10.8 - - Differences very much the same l l 10.9 + + -1 ATLAS, L = 33-36 pb l l 1 1 11.0 → → as they are comparing at NLO. 0 0 B(Z B(Z 0.98 0.98 G. Watt (September 2011) ⋅ ⋅ 0 0 Z Z σ σ 0.96 0.96 68% C.L. PDF 0.94 0.94 MSTW08 NNPDF2.1 HERAPDF1.0 0.92 0.92 HERAPDF1.5 ABKM09 JR09 0.9 0.9 Inner error bars: PDF only α Outer error bars: PDF+ S 0.88 0.88 9.8 9.8 10 10 10.2 10.2 10.4 10.4 10.6 10.6 10.8 10.8 11 11 ± ± ± ± σ σ ⋅ ⋅ → → ν ν B(W B(W l l ) (nb) ) (nb) ± ± W W Birmingham – February 2012 53

Differential data on rapidity is becoming very constraining – on both shapes and on normalisations of predictions. Would be particularly interesting to see for γ ⋆ at low masses (LHCb). Birmingham – February 2012 54

Clearly some of this information lost in ratios and asymmetries. Ideally want individual distributions, with full correlations. Birmingham – February 2012 55

Details from single charged-lepton cross sections and asymmetries – Stirling for low p T main boost from W decay to leptons. 0.5 LHC 7 TeV 0.4 Dip towards − 1 for lower p T cuts MSTW2008 NLO from preferential forward production 0.3 0 from d V ( x 1 )¯ u ( x 2 ) due to axial vector 0.2 10 20 nature of coupling. 30 35 0.1 Eventual turn-up when/if u V ( x 1 ) ¯ d ( x 2 ) ≫ 0.0 d V ( x 1 )¯ u ( x 2 ) A +- (y) -0.1 The larger the lepton p T the earlier -0.2 (in terms of increasing y ℓ ) this will -0.3 happen, and for p T → m W / 2 there -0.4 is no V ± A dominance at all. W asymmetry -0.5 So asymmetry at large y ℓ in terms of lepton asymmetry, -0.6 variable p Tlep (min) p T tells us about d/u at large x . -0.7 0 1 2 3 4 5 y lep or y W Birmingham – February 2012 56

MSTW comparison better if p T cut at 20GeV 2 . Birmingham – February 2012 57

LHCb (with p T (min) = 20GeV ) already testing dip. With higher p T (min) could potentially see upturn. Birmingham – February 2012 58

Inclusive Jets at the LHC ATLAS data compared to various PDF set predictions. Each fit well so far with size of correlated uncertainties limiting discriminative power. Interesting to see jets from LHCb as well. Birmingham – February 2012 59

Top-antitop Cross-section Inclusive cross-section known approximately to NNLO Intrinsic theory uncertainty not very large – for example, recent NNLL calculation by Beneke et al. Data getting precise. Main uncertainty in choice of PDFs, not in individual uncertainty but choice of set. Correlated to Higgs predictions. NLO t t cross sections at the LHC ( s = 7 TeV) NNLO (approx.) t t cross sections at the LHC ( s = 7 TeV) 210 210 220 220 (pb) (pb) (pb) (pb) pole pole m = 171.3 GeV m = 171.3 GeV -1 -1 t t CMS, L = 0.8-1.09 fb CMS, L = 0.8-1.09 fb 200 200 -1 -1 ATLAS, L = 0.7 fb ATLAS, L = 0.7 fb 200 200 t t t t t t t t 190 190 σ σ σ σ 180 180 G. Watt (September 2011) G. Watt (September 2011) 180 180 68% C.L. PDF 170 170 MSTW08 68% C.L. PDF 160 160 160 160 CTEQ6.6 MSTW08 CT10 NNPDF2.1 150 150 NNPDF2.1 HERAPDF1.0 140 140 HERAPDF1.5 HERAPDF1.0 140 140 ABKM09 HERAPDF1.5 Vertical error bars Vertical error bars JR09 ABKM09 Inner: PDF only Inner: PDF only 130 130 GJR08 α 120 120 α Outer: PDF+ Outer: PDF+ S S 120 120 0.114 0.114 0.116 0.116 0.118 0.118 0.12 0.12 0.122 0.122 0.124 0.124 0.111 0.112 0.113 0.114 0.115 0.116 0.117 0.118 0.119 0.12 0.121 0.111 0.112 0.113 0.114 0.115 0.116 0.117 0.118 0.119 0.12 0.121 α α α α 2 2 2 2 (M (M ) ) (M (M ) ) S S S S Z Z Z Z Plots by G. Watt. Differences between groups significant at NLO, and at NNLO. Birmingham – February 2012 60

Uncertainty in t ¯ t , Higgs via gluon fusion and ratios. PDF only uncertainty, but α S uncertainty cancels in ratios. Very strong correlation of top with Higgs production for m H ∼ 400GeV at the LHC. Similar correlation for m H ∼ 400 × 1 . 96 / 7 ∼ 130GeV at the Tevatron. Particularly important at the moment. Birmingham – February 2012 61

What will be the advantages of running WJS 2010 5 at 8TeV ? ratios of LHC parton luminosities: 8 TeV / 7 TeV and 9 TeV / 7 TeV Limited for quark dominated processes 4 up to m X > 1TeV , but more for gluon gg Σ qq dominated processes for M X > 200 − ( Σ q)( Σ q) 3 300GeV . luminosity ratio 2 9/7 8/7 MSTW2008NLO 1 10 100 1000 M X (GeV) Birmingham – February 2012 62

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. Nearly full range of NNLO PDFs now. Comparison between different PDF sets at NLO and NNLO very similar. Various ways of looking at experimental uncertainties. Uncertainties ∼ 1 − 5% for most LHC quantities. Ratios, e.g. W + /W − tight constraint on partons, but don’t want to lose information when taking ratios. Effects from input assumptions e.g. selection of data fitted, cuts and input parameterisation can shift central values of predictions significantly. Also affect size of uncertainties. Want balance between freedom and sensible constraints. Data from the LHC just starting to have some effect on improving the precision of PDFs. Might start to discriminate between PDFs first. Extraction of PDFs from existing data and use for LHC far from a straightforward procedure. Lots of 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. Birmingham – February 2012 63

Different PDF sets ● MSTW08 – 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. ● CT10 – very similar. PDFs at NLO. CT10 include HERA combination and more Tevatron data though also run I jet data. Not large changes from CTEQ6.6. CT10W gives higher weight to Tevatron asymmetry data. ● NNPDF2.1 – include all except HERA jet data (not strong constraint). NNPDF2.1 improves on NNPDF2.0 by better heavy flavour treatment. PDFs at NLO and very recently NNLO and LO . ● HERAPDF1.0 – based on HERA inclusive structure functions, neutral and charged current. Use combined data. PDFs at NLO and (without uncertainties) NNLO. ● ABKM09 – fit to DIS and fixed target Drell-Yan data. PDFs at NLO and NNLO. Less conservative cuts at low W 2 than other groups – fit for higher twist corrections rather than attempt to avoid them. ● GJR08 – fit to DIS, fixed target Drell-Yan and Tevatron jet data (not at NNLO. PDFs at NLO and NNLO. Birmingham – February 2012 64

ν l p > 35 GeV, E > 25 GeV T T 0.3 Lepton charge asymmetry 0.25 0.2 0.15 0.1 0.05 χ χ 2 2 MSTW08: = 530 (8 pts.), = 147 (12 pts.) µ e ∅ χ χ 2 2 Fit new D A : = 97, = 58 µ µ e 0 χ χ 2 2 Weight by 100: = 6, = 88 µ e χ χ 2 2 Cut BCDMS+NMCn/p: = 14, = 55 µ e ∅ χ χ 2 2 Deut. corr., fit old D A : = 190, = 42 -0.05 µ µ e ∅ χ χ 2 2 Deut. corr., fit new D A : = 6, = 75 µ µ e ∅ χ χ 2 2 Deut. corr., fit new D A : = 173, = 23 µ -0.1 e e -1 ∅ D (prel.) A , L = 4.9 fb µ -1 ∅ D (publ.) A , L = 0.75 fb e -0.15 -1 CDF (publ.) A , L = 0.17 fb e -0.2 0 0.5 1 1.5 2 2.5 3 η | | l MSTW (and NNPDF and CTEQ) have difficulty fitting new D0 lepton asymmetry (particularly muon in different E T bins) along with other data. MSTW better when low number of data points sets given (slightly) more weight. Also improved using deuterium corrections. Birmingham – February 2012 65

The inappropriateness of using ∆ χ 2 = 1 when including a large number of sometimes conflicting data sets is shown by examining the best value of σ W and its uncertainty using ∆ χ 2 = 1 for individual data sets as obtained by CTEQ using Lagrange Multiplier technique. Birmingham – February 2012 66

Predictions by various groups - parton luminosities – NLO . Plots by G. Watt. gg luminosity at LHC ( s = 7 TeV) 1.2 1.2 Ratio to MSTW 2008 NLO (68% C.L.) Ratio to MSTW 2008 NLO (68% C.L.) MSTW08 NLO 1.15 1.15 CTEQ6.6 CT10 1.1 1.1 NNPDF2.1 1.05 1.05 G. Watt (March 2011) 1 1 0.95 0.95 0.9 0.9 0.85 0.85 0.8 0.8 120 180 240 -3 -3 -2 -2 t t -1 -1 10 10 10 10 10 10 M (GeV) H s s / s / s Cross-section for t ¯ t almost identical in PDF terms to 450GeV Higgs. Also H + t ¯ � t at s/s ∼ 0 . 1 . ˆ Birmingham – February 2012 67

gg luminosity at LHC ( s = 7 TeV) 1.2 1.2 Ratio to MSTW 2008 NLO (68% C.L.) Ratio to MSTW 2008 NLO (68% C.L.) MSTW08 NLO 1.15 1.15 HERAPDF1.0 HERAPDF1.5 1.1 1.1 ABKM09 GJR08 G. Watt (September 2011) 1.05 1.05 1 1 0.95 0.95 0.9 0.9 0.85 0.85 0.8 0.8 120 180 240 -3 -3 -2 -2 t t -1 -1 10 10 10 10 10 10 M (GeV) H s s / s / s Clearly some distinct variation between groups. Much can be understood in terms of previous differences in approaches. Uncertainties not completely comparable. Birmingham – February 2012 68

Σ (q q ) luminosity at LHC ( s = 7 TeV) q 1.2 1.2 Ratio to MSTW 2008 NLO (68% C.L.) Ratio to MSTW 2008 NLO (68% C.L.) MSTW08 NLO 1.15 1.15 CTEQ6.6 CT10 1.1 1.1 NNPDF2.1 1.05 1.05 G. Watt (March 2011) 1 1 0.95 0.95 0.9 0.9 0.85 0.85 W Z 0.8 0.8 -3 -3 -2 -2 -1 -1 10 10 10 10 10 10 s s / s / s Many of the same general features for quark-antiquark luminosity. Some differences mainly at higher x . Birmingham – February 2012 69

Σ (q q ) luminosity at LHC ( s = 7 TeV) q 1.2 1.2 Ratio to MSTW 2008 NLO (68% C.L.) Ratio to MSTW 2008 NLO (68% C.L.) MSTW08 NLO 1.15 1.15 HERAPDF1.0 HERAPDF1.5 1.1 1.1 ABKM09 GJR08 G. Watt (September 2011) 1.05 1.05 1 1 0.95 0.95 0.9 0.9 0.85 0.85 W Z 0.8 0.8 -3 -3 -2 -2 -1 -1 10 10 10 10 10 10 s s / s / s Canonical example W, Z production, but higher ˆ s/s relevant for WH or vector boson fusion. All plots and more at http://projects.hepforge.org/mstwpdf/pdf4lhc Birmingham – February 2012 70

Variations in Cross-Section Predictions – NLO → NLO gg H at the LHC ( s = 7 TeV) for M = 120 GeV H 13 13 (pb) (pb) 12.5 12.5 H H σ σ 12 12 68% C.L. PDF 11.5 11.5 MSTW08 CTEQ6.6 NNPDF2.0 11 11 HERAPDF1.0 ABKM09 Vertical error bars Inner: PDF only GJR08 α 10.5 10.5 Outer: PDF+ S 0.114 0.114 0.116 0.116 0.118 0.118 0.12 0.12 0.122 0.122 0.124 0.124 α α 2 2 (M (M ) ) S S Z Z Dotted lines show how central PDF predictions vary with α S ( M 2 Z ) . Again plots by G Watt using PDF4LHC benchmark criteria. Birmingham – February 2012 71

→ NLO gg H at the LHC ( s = 7 TeV) for M = 240 GeV H 3 3 (pb) (pb) 2.9 2.9 H H σ σ 2.8 2.8 2.7 2.7 68% C.L. PDF 2.6 2.6 MSTW08 CTEQ6.6 2.5 2.5 NNPDF2.0 HERAPDF1.0 2.4 2.4 ABKM09 Vertical error bars Inner: PDF only GJR08 α Outer: PDF+ 2.3 2.3 S 0.114 0.114 0.116 0.116 0.118 0.118 0.12 0.12 0.122 0.122 0.124 0.124 α α 2 2 (M (M ) ) S S Z Z Excluding GJR08 amount of difference due to α S ( M 2 Z ) variations 3 − 4% . Birmingham – February 2012 72

0 - → + NLO Z l l at the LHC ( s = 7 TeV) 1.02 1.02 ) (nb) ) (nb) 1 1 0.98 0.98 - - l l + + l l 0.96 0.96 → → 0.94 0.94 0 0 B(Z B(Z 0.92 0.92 68% C.L. PDF MSTW08 ⋅ ⋅ 0.9 0.9 0 0 Z Z CTEQ6.6 σ σ 0.88 0.88 NNPDF2.0 HERAPDF1.0 0.86 0.86 ABKM09 Vertical error bars Inner: PDF only 0.84 0.84 GJR08 α Outer: PDF+ S 0.82 0.82 0.114 0.114 0.116 0.116 0.118 0.118 0.12 0.12 0.122 0.122 0.124 0.124 α α 2 2 (M (M ) ) S S Z Z α S ( M 2 Z ) dependence now more due to PDF variation with α S ( M 2 Z ) . Again variations somewhat bigger than individual uncertainties. Birmingham – February 2012 73

- + NLO W /W ratio at the LHC ( s = 7 TeV) - - W W 1.5 1.5 σ σ / / + + W W 1.48 1.48 σ σ ≡ ≡ ± ± R R 1.46 1.46 68% C.L. PDF MSTW08 1.44 1.44 CTEQ6.6 NNPDF2.0 HERAPDF1.0 1.42 1.42 ABKM09 Vertical error bars Inner: PDF only GJR08 α Outer: PDF+ S 1.4 1.4 0.114 0.114 0.116 0.116 0.118 0.118 0.12 0.12 0.122 0.122 0.124 0.124 α α 2 2 (M (M ) ) S S Z Z Quite a variation in ratio. Shows variations in flavour and quark-antiquark decompositions. All plots and more at http://projects.hepforge.org/mstwpdf/pdf4lhc Birmingham – February 2012 74

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 x affects predicted Higgs cross-section. What is true uncertainty for comparing to unknown production cross section. 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 2 expansion of MSTW uncertainty. Birmingham – February 2012 75

gg luminosity at LHC ( s = 7 TeV) gg luminosity at LHC ( s = 7 TeV) 1.2 1.2 1.2 1.2 Ratio to MSTW 2008 NLO (68% C.L.) Ratio to MSTW 2008 NLO (68% C.L.) Ratio to MSTW 2008 NLO (68% C.L.) Ratio to MSTW 2008 NLO (68% C.L.) MSTW08 MSTW08 NLO 1.15 1.15 1.15 1.15 CTEQ6.6 CTEQ6.6 NNPDF2.0 CT10 1.1 1.1 1.1 1.1 HERAPDF1.0 NNPDF2.1 1.05 1.05 1.05 1.05 G. Watt (March 2011) 1 1 1 1 0.95 0.95 0.95 0.95 0.9 0.9 0.9 0.9 0.85 0.85 0.85 0.85 0.8 0.8 0.8 0.8 120 180 240 120 180 240 -3 -3 -3 -3 -2 -2 t t -1 -1 -2 -2 t t -1 -1 10 10 10 10 10 10 10 10 10 10 10 10 M (GeV) M (GeV) H H s s / s / s s s / s / s MSTW, NNPDF and CTEQ are converging somewhat. Birmingham – February 2012 76

Σ Σ (q q ) luminosity at LHC ( s = 7 TeV) (q q ) luminosity at LHC ( s = 7 TeV) q q 1.2 1.2 1.2 1.2 Ratio to MSTW 2008 NLO (68% C.L.) Ratio to MSTW 2008 NLO (68% C.L.) Ratio to MSTW 2008 NLO (68% C.L.) Ratio to MSTW 2008 NLO (68% C.L.) MSTW08 MSTW08 NLO 1.15 1.15 1.15 1.15 CTEQ6.6 CTEQ6.6 NNPDF2.0 CT10 1.1 1.1 1.1 1.1 HERAPDF1.0 NNPDF2.1 1.05 1.05 1.05 1.05 G. Watt (March 2011) 1 1 1 1 0.95 0.95 0.95 0.95 0.9 0.9 0.9 0.9 0.85 0.85 0.85 0.85 W Z W Z 0.8 0.8 0.8 0.8 -3 -3 -3 -3 -2 -2 -1 -1 -2 -2 -1 -1 10 10 10 10 10 10 10 10 10 10 10 10 s s / s / s s s / s / s Same for quark-antiquark luminosities. Birmingham – February 2012 77

Σ Σ (q q ) luminosity at LHC ( s = 7 TeV) (q q ) luminosity at LHC ( s = 7 TeV) q q 1.2 1.2 1.2 1.2 Ratio to MSTW 2008 NLO (68% C.L.) Ratio to MSTW 2008 NLO (68% C.L.) Ratio to MSTW 2008 NNLO (68% C.L.) Ratio to MSTW 2008 NNLO (68% C.L.) MSTW08 NNLO MSTW08 NLO 1.15 1.15 1.15 1.15 HERAPDF1.0 HERAPDF1.0 HERAPDF1.5 HERAPDF1.5 ABKM09 1.1 1.1 1.1 1.1 ABKM09 JR09 GJR08 NNPDF2.1 G. Watt (September 2011) G. Watt (September 2011) 1.05 1.05 1.05 1.05 1 1 1 1 0.95 0.95 0.95 0.95 0.9 0.9 0.9 0.9 0.85 0.85 0.85 0.85 W Z W Z 0.8 0.8 0.8 0.8 -3 -3 -2 -2 -1 -1 -3 -3 -2 -2 -1 -1 10 10 10 10 10 10 10 10 10 10 10 10 s s / s / s s s / s / s Not all luminosity differences the same at NLO as at NNLO, e.g. HERAPDF q ¯ q . Birmingham – February 2012 78

The PDFs are related to the issue of the use and uncertainty of α S ( M 2 Z ) . There is a significant systematic change in value from fit as one goes from NLO to NNLO. Seen in (most) other extractions. Also highlighted in stability of predictions. Consider percentage change from NLO to NNLO in MSTW08 predictions for best fit α S compared to fixed α S ( M 2 Z ) = 0 . 119 . σ W ( Z ) 7TeV σ W ( Z ) 14TeV σ H 7TeV σ H 7TeV MSTW08 best fit α S 3.0 2.6 25 24 MSTW08 α S = 0 . 119 5.3 5.0 32 30 α S ( M 2 Z ) is not a physical quantity. In (nearly) all PDF related quantities (and many others) shows tendency to decrease from order to order. Noticeable if one has fit at NNLO. Any settling on, or near common α S ( M 2 Z ) has to take this into account. Birmingham – February 2012 79

∅ ∅ D Run II inclusive jet data (cone, R = 0.7) D Run II dijet data (cone, R = 0.7) σ µ σ µ (Ratio w.r.t. NLO with = p using MSTW08 NNLO PDFs) (Ratio w.r.t. NLO with = p using MSTW08 NNLO PDFs) T T 1.4 1.4 1.4 1.4 T T T T = p = p = p = p 1.3 1.3 1.3 1.3 JET JET µ µ µ µ 0.0 < |y | < 0.4 0.4 < |y | < 0.8 0.0 < |y| < 0.4 0.4 < |y| < 0.8 with with with with 1.2 1.2 1.2 1.2 max max 1.1 1.1 1.1 1.1 σ σ σ σ Ratio w.r.t NLO Ratio w.r.t NLO Ratio w.r.t NLO Ratio w.r.t NLO 1 1 1 1 0.9 0.9 0.9 0.9 µ µ × µ µ × = = 0.5 p = = 0.5 p R F R F 0.8 T 0.8 0.8 T 0.8 µ µ × µ µ × σ σ = = 1.0 p = = 1.0 p Solid lines: NLO Solid lines: NLO R F R F T T 0.7 µ µ × 0.7 σ 0.7 µ µ × 0.7 ≡ = = 2.0 p Dashed lines: NLO + 2-loop threshold = = 2.0 p p (p +p )/2 R F R F T T T T1 T2 0.6 0.6 0.6 0.6 0.2 0.3 0.4 0.5 1 0.2 0.3 0.4 0.5 1 2 2 10 10 JET JET p (GeV) p (GeV) M (TeV) M (TeV) T T JJ JJ 1.4 1.4 1.4 1.4 T T T T = p = p = p = p 1.3 1.3 1.3 1.3 JET JET µ µ µ µ 0.8 < |y | < 1.2 1.2 < |y | < 1.6 0.8 < |y| < 1.2 1.2 < |y| < 1.6 with with with with 1.2 1.2 1.2 1.2 max max 1.1 1.1 1.1 1.1 σ σ σ σ Ratio w.r.t NLO Ratio w.r.t NLO Ratio w.r.t NLO Ratio w.r.t NLO 1 1 1 1 0.9 0.9 0.9 0.9 0.8 0.8 0.8 0.8 0.7 0.7 0.7 0.7 0.6 0.6 0.6 0.6 2 2 0.2 0.3 0.4 0.5 1 0.2 0.3 0.4 0.5 1 10 10 JET JET p (GeV) p (GeV) M (TeV) M (TeV) T T JJ JJ 1.4 1.4 1.4 1.4 T T T T = p = p = p = p 1.3 1.3 1.3 1.3 JET JET µ µ µ µ 1.6 < |y | < 2.0 2.0 < |y | < 2.4 1.6 < |y| < 2.0 2.0 < |y| < 2.4 with with with with 1.2 1.2 1.2 1.2 max max 1.1 1.1 1.1 1.1 σ σ σ σ Ratio w.r.t NLO Ratio w.r.t NLO Ratio w.r.t NLO Ratio w.r.t NLO 1 1 1 1 0.9 0.9 0.9 0.9 0.8 0.8 0.8 0.8 0.7 0.7 0.7 0.7 0.6 0.6 0.6 0.6 0.2 0.3 0.4 0.5 1 0.2 0.3 0.4 0.5 1 2 2 10 10 JET JET p (GeV) p (GeV) M (TeV) M (TeV) T T JJ JJ Shape of corrections as function of p T at NLO and also at approx. NNLO in inclusive case. Problem at highest p T and rapidity even for inclusive jets. Birmingham – February 2012 80

NLO PDF (with NLO ˆ σ ) µ = p T / 2 µ = p T µ = 2 p T MSTW08 1.45 (0.89) 1.08 (0.20) 1.05 (1.22) CTEQ6.6 1.62 (1.15) 1.56 (0.59) 1.61 (1.35) CT10 1.39 (0.88) 1.26 (0.37) 1.32 (1.29) NNPDF2.1 1.41 (0.87) 1.29 (0.20) 1.22 (0.96) HERAPDF1.0 1.73 (0.27) 1.84 (0.74) 1.83 (2.79) HERAPDF1.5 1.78 (0.29) 1.87 (0.75) 1.84 (2.81) ABKM09 1.39 (0.35) 1.43 (1.07) 1.63 (3.66) GJR08 1.90 (1.46) 1.34 (0.45) 1.03 (0.51) NNLO PDF (with NLO+2-loop ˆ σ ) µ = p T / 2 µ = p T µ = 2 p T MRST06 3.19 (5.00) 1.77 (3.22) 1.25 (1.50) MSTW08 1.95 (0.90) 1.23 (0.44) 1.08 (0.35) HERAPDF1.0, α S ( M 2 Z ) = 0 . 1145 2.11 (0.37) 1.68 (0.35) 1.41 (0.63) HERAPDF1.0, α S ( M 2 2.28 (0.95) 1.50 (0.40) 1.17 (0.21) Z ) = 0 . 1176 ABKM09 1.68 (0.79) 1.55 (1.21) 1.63 (2.04) JR09 1.84 (0.47) 1.61 (0.36) 1.58 (0.50) Table 3: Values of χ 2 /N pts . for the DØ Run II inclusive jet data using a cone jet algorithm with N pts . = 110 and N corr . = 23 , for different PDF sets and different scale choices. At most a 1- σ shift in normalisation is allowed. Birmingham – February 2012 81

NLO PDF (with NLO ˆ σ ) µ = p T / 2 µ = p T µ = 2 p T MSTW08 1.40 ( + 1.05 ) 1.08 ( − 0.55) 0.85 ( − 2.25 ) CTEQ6.6 1.52 ( − 1.61 ) 1.25 ( − 2.88 ) 1.01 ( − 4.02 ) CT10 1.39 ( − 0.66) 1.11 ( − 2.02 ) 0.90 ( − 3.35 ) NNPDF2.1 1.41 ( + 0.37) 1.23 ( − 1.22 ) 0.95 ( − 2.67 ) HERAPDF1.0 1.55 ( − 2.16 ) 1.38 ( − 3.51 ) 1.07 ( − 4.52 ) HERAPDF1.5 1.63 ( − 1.98 ) 1.45 ( − 3.35 ) 1.12 ( − 4.40 ) ABKM09 1.25 ( − 1.90 ) 1.04 ( − 3.20 ) 0.89 ( − 4.44 ) GJR08 1.72 ( + 2.14 ) 1.34 ( + 0.53) 0.98 ( − 1.05 ) NNLO PDF (with NLO+2 + loop ˆ σ ) µ = p T / 2 µ = p T µ = 2 p T MRST06 2.92 ( + 2.66 ) 1.70 ( + 1.31 ) 1.25 ( + 0.44) MSTW08 1.87 ( + 1.34 ) 1.23 ( + 0.09) 1.08 ( − 0.87) HERAPDF1.0, α S ( M 2 Z ) = 0 . 1145 2.11 ( − 0.82) 1.52 ( − 2.03 ) 1.14 ( − 2.61 ) HERAPDF1.0, α S ( M 2 2.28 ( + 0.94) 1.50 ( − 0.49) 1.11 ( − 1.23 ) Z ) = 0 . 1176 ABKM09 1.48 ( − 2.33 ) 1.13 ( − 3.35 ) 1.02 ( − 4.03 ) JR09 1.84 ( + 0.63) 1.61 ( − 0.60) 1.50 ( − 1.35 ) Table 4: Values of χ 2 /N pts . for the DØ Run II inclusive jet data using a cone jet algorithm with N pts . = 110 and N corr . = 23 , for different PDF sets and different scale choices. No restriction is imposed on the shift in normalisation and the optimal value of “ − r lumi . ” is shown in brackets. Birmingham – February 2012 82

Comparison of the raw comparison to CDF inclusive jet data using the k T and cone algorithms. CDF Run II inclusive jet data (k , D = 0.7) CDF Run II inclusive jet data (Midpoint) T (data points before systematic shifts, show total errors) (data points before systematic shifts, show total errors) 3 3 4 4 Data / Theory Data / Theory Data / Theory Data / Theory 3.5 3.5 JET JET JET JET 2.5 2.5 0.0 < |y | < 0.1 0.1 < |y | < 0.7 0.0 < |y | < 0.1 0.1 < |y | < 0.7 3 3 2 2 2.5 2.5 2 2 1.5 1.5 1.5 1.5 1 1 1 1 0.5 0.5 0.5 0.5 2 2 2 2 10 10 10 10 JET JET JET JET p (GeV) p (GeV) p (GeV) p (GeV) T T T T 3 3 4 4 Data / Theory Data / Theory Data / Theory Data / Theory 3.5 3.5 JET JET JET JET 2.5 2.5 0.7 < |y | < 1.1 1.1 < |y | < 1.6 0.7 < |y | < 1.1 1.1 < |y | < 1.6 3 3 2 2 2.5 2.5 2 2 1.5 1.5 1.5 1.5 1 1 1 1 0.5 0.5 0.5 0.5 2 2 2 2 10 10 10 10 JET JET JET JET p (GeV) p (GeV) p (GeV) p (GeV) T T T T 3 4 Data / Theory Data / Theory µ µ × µ µ × JET JET = = 1.0 p = = 1.0 p 3.5 JET JET 2.5 R F R F 1.6 < |y | < 2.1 1.6 < |y | < 2.1 T T 3 2 2.5 NNLO PDFs, 76 data points NNLO PDFs, 72 data points 2 1.5 2 2 χ χ MSTW08, = 34 MSTW08, = 31 1.5 2 2 1 χ χ ABKM09, = 68 ABKM09, = 51 1 0.5 0.5 Inner error bars: only uncorrelated Inner error bars: only uncorrelated 2 2 10 10 JET JET p (GeV) p (GeV) Outer error bars: total (add in quadrature) Outer error bars: total (add in quadrature) T T Data/theory the same shape for both. Good compatibility. Verified by fits. Birmingham – February 2012 83

Comparison of the raw comparison to D0 inclusive jet data using the cone algorithms and D0 dijet data. ∅ ∅ D Run II inclusive jet data (cone, R = 0.7) D Run II dijet data (cone, R = 0.7) (data points before systematic shifts, show total errors) (data points before systematic shifts, show total errors) 2 2 1.8 1.8 Data / Theory Data / Theory Data / Theory Data / Theory 1.8 1.8 1.6 1.6 JET JET 1.6 1.6 0.0 < |y | < 0.4 0.4 < |y | < 0.8 0.0 < |y| < 0.4 0.4 < |y| < 0.8 1.4 1.4 max max 1.4 1.4 1.2 1.2 1.2 1.2 1 1 1 1 0.8 0.8 µ µ 0.8 0.8 = = p NNLO PDFs, 110 data points NNLO PDFs, 71 data points 0.6 0.6 R F T χ µ µ × χ 2 2 JET 0.6 0.6 MSTW08, = 48 MSTW08, = 23 ≡ = = 1.0 p 0.4 0.4 p (p +p )/2 χ χ 2 ABKM09, 2 = 133 R F 0.4 ABKM09, = 137 0.4 0.2 0.2 T T T1 T2 0 0 0.2 0.2 0.2 0.3 0.4 0.5 1 0.2 0.3 0.4 0.5 1 2 2 10 10 JET JET p (GeV) p (GeV) M (TeV) M (TeV) T T JJ JJ 2 2 1.8 1.8 Data / Theory Data / Theory Data / Theory Data / Theory 1.8 1.8 1.6 1.6 JET JET 1.6 1.6 0.8 < |y | < 1.2 1.2 < |y | < 1.6 0.8 < |y| < 1.2 1.2 < |y| < 1.6 1.4 1.4 max max 1.4 1.4 1.2 1.2 1.2 1.2 1 1 1 1 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.4 0.2 0.2 0 0 0.2 0.2 2 2 0.2 0.3 0.4 0.5 1 0.2 0.3 0.4 0.5 1 10 10 JET JET p (GeV) p (GeV) M (TeV) M (TeV) T T JJ JJ 2 2 1.8 1.8 Data / Theory Data / Theory Data / Theory Data / Theory 1.8 1.8 1.6 1.6 JET JET 1.6 1.6 1.6 < |y | < 2.0 2.0 < |y | < 2.4 1.6 < |y| < 2.0 2.0 < |y| < 2.4 1.4 1.4 max max 1.4 1.4 1.2 1.2 1.2 1.2 1 1 1 1 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 Inner error bars: only uncorrelated Inner error bars: only uncorrelated 0.4 0.4 0.4 0.4 Outer error bars: total (add in quadrature) Outer error bars: total (add in quadrature) 0.2 0.2 0 0 0.2 0.2 0.2 0.3 0.4 0.5 1 0.2 0.3 0.4 0.5 1 2 2 10 10 JET JET p (GeV) p (GeV) M (TeV) M (TeV) T T JJ JJ Not such good compatibility as for the two CDF sets. Birmingham – February 2012 84

Three-jet cross-sections Recent results from D0 ( arXiv 1104.1986 ) on three jets cross sections. All the same work already done. Birmingham – February 2012 85

Broadly similar results. Birmingham – February 2012 86

Sometimes the reason for cross section differences is unexpected. Warsinsky at recent Higgs- LHC working group meeting. m b values bring CTEQ and MSTW together but exaggerate NNPDF difference. Couplings have assumed common mass value. Birmingham – February 2012 87

Small- x Theory Reason for this instability – at each order in α S each splitting function and coefficient function obtains an extra power of ln(1 /x ) (some accidental zeros in P gg ), i.e. s ( Q 2 ) ln m − 1 (1 /x ) . P ij ( x, α s ( Q 2 )) , C P i ( x, α s ( Q 2 )) ∼ α m BFKL equation for high-energy limit � 1 � ∞ dq 2 dx ′ f ( k 2 , x ) = f I ( Q 2 q 2 K ( q 2 , k 2 ) f ( q 2 , x ) , 0 )+ x ′ ¯ α S x 0 where f ( k 2 , x ) is the unintegrated gluon distribution � Q 2 g ( x, Q 2 ) = ( dk 2 /k 2 ) f ( x, k 2 ) , and K ( q 2 , k 2 ) is a 0 calculated kernel known to NLO. Physical structure functions obtained from σ ( Q 2 , x ) = ( dk 2 /k 2 ) h ( k 2 /Q 2 ) f ( k 2 , x ) � where h ( k 2 /Q 2 ) is a calculable impact factor. The global fits usually assume that this is unimportant in practice, and proceed regardless. Fits work well at small x , but could improve. Birmingham – February 2012 88

Good recent progress in incorporating ln(1 /x ) resummation Altarelli-Ball- Forte, Ciafaloni-Colferai-Salam-Stasto and White-RT. Include running coupling effects and variety (depending on group) of other corrections 2 80 Q 2 =100GeV 2 By 2008 very similar results coming 1 xg(x) from the competing procedures, 60 NLL+ 0 NLL(2)+ despite some differences in technique. 40 NLO+ -1 Full set of coefficient functions still 20 -2 Q 2 =1GeV 2 to come in some cases, but splitting 0 -3 functions comparable. -5 10 -4 10 -3 10 -2 10 -5 10 -4 10 -3 10 -2 10 -1 -1 10 1 10 1 x x Note, in all cases NLO corrections lead to dip in functions below fixed order values until slower growth (running coupling effect) at very small x . Birmingham – February 2012 89

A fit to data with NLO plus NLO resummation, with heavy quarks included (White,RT) performed. 4 H1 ZEUS NMC x=5 × 10 -4 3.5 x=6.32 × 10 -4 2 x=8 × 10 -4 3 x=1.3 × 10 -3 80 Q 2 =100GeV 2 1 x=1.61 × 10 -3 2.5 xg(x) x=2 × 10 -3 60 NLL+ F 2p (x,Q 2 ) + 0.25(9-i) 0 NLL(2)+ 2 x=3.2 × 10 -3 40 NLO+ -1 x=5 × 10 -3 1.5 20 x=8 × 10 -3 -2 Q 2 =1GeV 2 1 0 -3 0.5 NLL+ -5 10 -4 10 -3 10 -2 10 -5 10 -4 10 -3 10 -2 10 NLL(2)+ -1 -1 10 1 10 1 NLO+ x x 0 2 3 1 10 10 Q 2 (GeV 2 ) 10 → moderate improvement in fit to HERA data within global fit, and change in extracted gluon (more like quarks at low Q 2 ). Together with indications from Drell Yan resummation calculations (Marzani, Ball) few percent effect quite possible. Birmingham – February 2012 90

PDF Correlations The PDF uncertainty analysis may be extended to define a correlation between the uncertainties of two variables, say X ( � a ) and Y ( � a ) . The correlations were calculated using the MCFM NLO program (versions 5.8 and 6.0) with a common set of input files for all groups. Each group did their own calculations. � � For the groups using a Hessian approach the correlations were calculated using N � ∆ X ∆ Y 1 � � � � X (+) − X ( − ) Y (+) − Y ( − ) � cos ϕ = ∆ X ∆ Y = i i i i 4∆ X ∆ Y i =1 � N � � = 1 � 2 � � � X (+) − X ( − ) � � � � ∆ X = ∆ X � � � i i 2 i =1 or some similar variation. This included the most up-to-date published sets for each group, i.e. , ABKM09, CT10, GJR 08, MSTW08. The basic results for CT10 and MSTW08 are PDF only, whereas ABKM09 and GJR08 include α S as a parameter in the error matrix. Birmingham – February 2012 91

Due to the specific error calculation prescription for HERAPDF1.5 which includes parameterization and model errors, the correlations can not be calculated in exactly the same way. An alternative way is to use a formula for uncertainty propagation in which correlations can be expressed via relative errors of compounds and their combination: � X � Y σ 2 σ ( X ) + = 2 + 2 cos ϕ, σ ( Y ) where σ ( O ) is the PDF error of observable O calculated using the HERAPDF prescription. The correlations for the NNPDF prescription are calculated using � XY � rep − � X � rep � Y � rep ρ ( X, Y ) = σ X σ Y where the averages are performed over the N rep = 100 replicas of the NNPDF2.1 set. The averaging was done and diagrams made by J. Rojo. Birmingham – February 2012 92

Full study involves range of backgrounds shown by Huston at PDF4LHC- July 2011. Will be found at https://twiki.cern.ch/twiki/bin/view/LHCPhysics/PDFCorrelations Birmingham – February 2012 93

And similar for signals. However, too detailed for concise presentation when averaging/comparing, and precision much higher than spread between groups. Full list also not vital since W production is very similar to Z production, both depending on partons (quarks in this case) at very similar hard scales and x values. Similarly for WW and ZZ , and the subprocesses gg → WW(ZZ) and gg → H for M H = 200 GeV. Birmingham – February 2012 94

The up-to-date PDF4LHC average (CT10, MSTW08, NNPDF2.1) for the correlations between all signal processes with other signal and background processes for Higgs production considered here. The processes have been classified in correlation classes of width 0 . 2 . 120 GeV 160 GeV ggH VBF WH ttH ggH VBF WH ttH ggH 1 -0.6 -0.2 -0.2 ggH 1 -0.6 -0.4 0.2 VBF -0.6 1 0.6 -0.4 VBF -0.6 1 0.6 -0.2 WH -0.2 0.6 1 -0.2 WH -0.4 0.6 1 0 tt H -0.2 -0.4 -0.2 1 ttH 0.2 -0.2 0 1 W -0.2 0.6 0.8 -0.6 W -0.4 0.4 0.6 -0.4 -0.4 0.8 1 -0.2 -0.4 0.6 0.8 -0.2 WW WW WZ -0.2 0.4 0.8 -0.4 WZ -0.4 0.4 0.8 -0.2 W γ 0 0.6 0.8 -0.6 W γ -0.4 0.6 0.6 -0.6 Wbb -0.2 0.6 1 -0.2 Wbb -0.2 0.6 0.8 -0.2 tt 0.2 -0.4 -0.4 1 tt 0.4 -0.4 -0.2 0.8 tb -0.4 0.6 1 -0.2 tb -0.4 0.6 1 0 t( → b)q 0.4 0 0 0 t( → b)q 0.6 0 0 0 Birmingham – February 2012 95

200 GeV 300 GeV ggH VBF WH ttH ggH VBF WH ttH ggH 1 -0.6 -0.4 0.4 ggH 1 -0.4 -0.2 0.6 VBF -0.6 1 0.6 -0.2 VBF -0.4 1 0.4 -0.2 WH -0.4 0.6 1 0 WH -0.2 0.4 1 0.2 0.4 -0.2 0 1 0.6 -0.2 0.2 1 ttH ttH W -0.6 0.4 0.6 -0.4 W -0.6 0.4 0.4 -0.6 WW -0.4 0.6 0.8 -0.2 WW -0.4 0.6 0.8 -0.2 WZ -0.4 0.4 0.8 -0.2 WZ -0.6 0.4 0.6 -0.4 W γ -0.4 0.4 0.6 -0.6 W γ -0.6 0.4 0.4 -0.6 -0.2 0.6 0.8 -0.2 -0.2 0.4 0.8 -0.2 Wbb Wbb tt 0.6 -0.4 -0.2 0.8 tt 1 -0.4 0 0.8 tb -0.4 0.6 0.8 0 tb -0.4 0.4 0.8 -0.2 t( → b)q 0.6 -0.2 0 0 t( → b)q 0.4 -0.2 0 -0.2 Birmingham – February 2012 96

500 GeV ggH VBF WH ttH ggH 1 -0.4 0 0.8 VBF -0.4 1 0.4 -0.2 WH 0 0.4 1 0 ttH 0.8 -0.2 0 1 -0.6 0.4 0.2 -0.6 W WW -0.4 0.6 0.6 -0.4 WZ -0.6 0.4 0.6 -0.4 W γ -0.6 0.4 0.2 -0.6 Wbb -0.4 0.4 0.6 -0.4 tt 1 -0.4 0 0.8 -0.4 0.4 0.8 -0.2 tb 0.2 -0.2 0 -0.2 t( → b)q Generally the results expected, i.e. gluon dominated processes correlated with each other as are quark dominated processes. Little correlation between the two. However, see that breakdown of correlation between gluon probed at different x values, e.g gg → H for M H = 120 GeV and tt since from momentum conservation gluon changes in one place (high x ) are compensated by those in another (low x ), and the crossing point is between 0 . 01 and 0 . 1 but varies slightly between groups. Birmingham – February 2012 97

The same for the correlations between background processes. W WW WZ W γ Wbb tt tb t( → b)q W 1 0.8 0.8 1 0.6 -0.6 0.6 -0.2 WW 0.8 1 0.8 0.8 0.8 -0.4 0.8 0 WZ 0.8 0.8 1 0.8 0.8 -0.4 0.8 0 W γ 1 0.8 0.8 1 0.6 -0.6 0.8 0 Wbb 0.6 0.8 0.8 0.6 1 -0.2 0.6 0 tt -0.6 -0.4 -0.4 -0.6 -0.2 1 -0.4 0.2 tb 0.6 0.8 0.8 0.8 0.6 -0.4 1 0.2 t( → b)q -0.2 0 0 0 0 0.2 0.2 1 Similar conclusions as for signals. What about variations between groups? Birmingham – February 2012 98

Correlation between the Vector Boson Fusion WH ttH NNPDF2.1 NNPDF2.1 NNPDF2.1 gluon fusion gg → H CT10 CT10 CT10 LHC HiggsXSWG 2011 LHC HiggsXSWG 2011 LHC HiggsXSWG 2011 MSTW08 MSTW08 MSTW08 PDF4LHC Average PDF4LHC Average PDF4LHC Average Correlation with ggF HERAPDF1.5 HERAPDF1.5 HERAPDF1.5 GJR08 GJR08 GJR08 process and the other ABKM09 ABKM09 ABKM09 1 1 1 0.5 0.5 0.5 0 0 0 signal and background -0.5 -0.5 -0.5 -1 -1 -1 120 160 200 300 500 120 160 200 300 500 120 160 200 300 500 processes as a function M H ( GeV ) M H ( GeV ) M H ( GeV ) W WW WZ of M H . NNPDF2.1 NNPDF2.1 NNPDF2.1 CT10 CT10 CT10 LHC HiggsXSWG 2011 LHC HiggsXSWG 2011 LHC HiggsXSWG 2011 MSTW08 MSTW08 MSTW08 PDF4LHC Average PDF4LHC Average PDF4LHC Average Correlation with ggF HERAPDF1.5 HERAPDF1.5 HERAPDF1.5 GJR08 GJR08 GJR08 ABKM09 ABKM09 ABKM09 1 1 1 0.5 0.5 0.5 0 0 0 -0.5 -0.5 -0.5 The class width of 0 . 2 -1 -1 -1 120 160 200 300 500 120 160 200 300 500 120 160 200 300 500 M H ( GeV ) M H ( GeV ) M H ( GeV ) is typical of the scatter W γ Wbb tt NNPDF2.1 NNPDF2.1 NNPDF2.1 of most deviations CT10 CT10 CT10 LHC HiggsXSWG 2011 LHC HiggsXSWG 2011 LHC HiggsXSWG 2011 MSTW08 MSTW08 MSTW08 PDF4LHC Average PDF4LHC Average PDF4LHC Average Correlation with ggF HERAPDF1.5 HERAPDF1.5 HERAPDF1.5 GJR08 GJR08 GJR08 between groups. ABKM09 ABKM09 ABKM09 1 1 1 0.5 0.5 0.5 0 0 0 -0.5 -0.5 -0.5 -1 -1 -1 120 160 200 300 500 120 160 200 300 500 120 160 200 300 500 M H ( GeV ) M H ( GeV ) M H ( GeV ) tb t(->b)q NNPDF2.1 NNPDF2.1 CT10 CT10 LHC HiggsXSWG 2011 LHC HiggsXSWG 2011 MSTW08 MSTW08 PDF4LHC Average PDF4LHC Average Correlation with ggF HERAPDF1.5 HERAPDF1.5 GJR08 GJR08 ABKM09 ABKM09 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 -1 120 160 200 300 500 120 160 200 300 500 M H ( GeV ) M H ( GeV ) Birmingham – February 2012 99