Parton Distribution Functions at the LHC

Transcription

1 Parton Distribution Functions at the LHC Robert Thorne February 5th, 203 University College London IPPP Research Associate With contributions from Alan Martin, James Stirling, Graeme Watt and Arnold Mathijssen and Ben Watt UCL February 203

2 UCL February 203

3 UCL February 203 2

4 Obtaining PDF sets General procedure. Start parton evolution at low scale Q 2 0 GeV 2. In principle different partons to consider. u, ū, d, d, s, s, c, c, b, b, g m c, m b Λ QCD so heavy parton distributions determined perturbatively. Leaves 7 independent combinations, or 6 if we assume s = s (just started not to). u V = u ū, d V = d d, sea = 2 (ū + d + s), s + s d ū, g. Input partons parameterised as, e.g. MSTW, xf(x, Q 2 0) = ( x) η ( + ɛx γx)x δ. Evolve partons upwards using LO, NLO (or increasingly NNLO) DGLAP equations. df i (x,q 2,α s (Q 2 )) d ln Q 2 = j P ij(x, α s (Q 2 )) f j (x, Q 2, α s (Q 2 )) UCL February 203 3

5 Fit data above 2GeV 2. Need many types for full determination. - Lepton-proton collider HERA (DIS) small-x quarks (best below x 0.05). Also gluons from evolution (same x), and now F L (x, Q 2 ). Also, jets moderate-x gluon.charged current data some limited info on flavour separation. Heavy flavour structure functions gluon and charm, bottom distributions and masses. - Fixed target DIS higher x 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 neutrinoantineutrino comparison asymmetry. Only for x > Drell-Yan production of dileptons quark-antiquark annihilation (E605, E866) high-x sea quarks. Deuterium target ū/ d asymmetry. - High-p T jets at colliders (Tevatron) high-x gluon distribution x > W and Z production at colliders (Tevatron/LHC) different quark contributions to DIS. UCL February 203 4

6 This procedure is generally successful and is part of a large-scale, ongoing project. Results in partons of the form shown. MSTW 2008 NLO PDFs (68% C.L.) ) 2.2 ) 2.2 xf(x,q 2 Q 2 = 0 GeV xf(x,q Q = 0 GeV g/0 g/ u 0.6 b,b u 0.4 d 0.4 c,c d 0.2 c,c s,s u d 0.2 s,s d u x x Various choices of PDF MSTW, CTEQ, NNPDF, AB(K)M, HERA, Jimenez-Delgado et al etc.. All LHC cross-sections rely on our understanding of these partons. UCL February 203 5

7 Leads to very accurate and precise predictions. Comparison of MSTW prediction for Z rapidity distribution with data. Z/γ* rapidity shape distribution from D Run II /σ dσ/dy 0.3 MRST2006 NNLO PDFs LO Vrap NLO Vrap 0.2 NNLO Vrap y UCL February 203 6

8 Parton Fits and Uncertainties. Two main approaches. Parton parameterization and Hessian (Error Matrix) approach first used by H and ZEUS, and extended by CTEQ. χ 2 χ 2 min χ2 = i,j H ij(a i a (0) i )(a j a (0) j ) The Hessian matrix H is related to the covariance matrix of the parameters by C ij (a) = χ 2 (H ) ij. We can then use the standard formula for linear error propagation. ( F ) 2 = χ 2 i,j F a i (H) ij This is the most common approach. F a j, UCL February 203 7

9 Can find and rescale eigenvectors of H leading to diagonal form χ 2 = i z2 i Implemented by CTEQ, then MRST/MSTW, HERAPDF. Uncertainty on physical quantity then given by ( F ) 2 = i ( (+) F (S i ) F (S ( ) i ) ) 2, where S (+) i direction. and S ( ) i are PDF error sets displaced along eigenvector Must choose correct χ 2 given complication of errors in full fit and sometimes conflicting data sets. UCL February 203 8

10 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. Where r p (k) are random numbers following Gaussian distribution. Hence, include information about measurements and errors in distribution of O art,(k) i,p. Fit to the data replicas obtaining PDF replicas q (net)(k) i al.) (follows Giele et Mean µ O and deviation σ O of observable O then given by µ O = N rep Nrep O[q (net)(k) i ], σ 2 O = N rep Nrep (O[q (net)(k) i ] µ O ) 2. 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. UCL February 203 9

11 Showed equivalence of replica approach to Hessian study with eigenvectors for given χ 2. However, can generate random PDF sets directly from parameters and variation from eigenvectors. Ratio to MSTW 2008 NLO Up valence distribution at Q = 0 GeV -4 0 MSTW 2008 NLO (68% C.L.) Random params. (40) Random params. (40) Random PDFs (000) x Ratio to MSTW 2008 NLO Down valence distribution at Q = 0 GeV -4 0 MSTW 2008 NLO (68% C.L.) Random params. (40) Random params. (40) Random PDFs (000) x a i (S k )=a 0 i + n j= e ( ) ij ±t ± j Rjk (k =,..., N pdf ). Or from eigenvectors directly (see LHCb study and De Lorenzi thesis). Far quicker. F (S k )=F (S 0 ) + [ j F (S ± j ) F (S 0) ] R jk Can use in reweighting studies as NNPDF, i.e. weight random set by χ 2 of fit to new data set(s) to get new distribution. Ratio to MSTW 2008 NLO Ratio to MSTW 2008 NLO Up antiquark distribution at Q = 0 GeV MSTW 2008 NLO (68% C.L.) Random params. (40) Random params. (40) Random PDFs (000) Strange quark distribution at Q = 0 GeV MSTW 2008 NLO (68% C.L.) Random params. (40) Random params. (40) Random PDFs (000) - 0 x x Ratio to MSTW 2008 NLO Ratio to MSTW 2008 NLO Down antiquark distribution at Q = 0 GeV MSTW 2008 NLO (68% C.L.) Random params. (40) Random params. (40) Random PDFs (000) Gluon distribution at Q = 0 GeV MSTW 2008 NLO (68% C.L.) Random params. (40) Random params. (40) Random PDFs (000) x x UCL February 203 0

12 Speed of convergence of prediction for Z cross section. Ratio to best-fit prediction NLO Z cross section at the LHC ( s = 7 TeV) MSTW 2008 NLO PDFs (68% C.L.) Average and s.d. over N predictions pdf Same random numbers for each value of N pdf Ratio to best-fit prediction NLO Z cross section at the LHC ( s = 7 TeV) MSTW 2008 NLO PDFs (68% C.L.) Average and s.d. over N predictions pdf Different random numbers for each value of N pdf Number of random predictions, N 2 pdf Number of random predictions, N 2 pdf 3 Left, add a new random set to existing ones sequentially. increasing numbers of independent random sets. Right, Very good with 40 sets. Excellent with 00 sets. UCL February 203

13 Speed of convergence of prediction for W + /W cross section ratio. Ratio to best-fit prediction NLO W /W cross-section ratio at the LHC ( MSTW 2008 NLO PDFs (68% C.L.) s = 7 TeV) Average and s.d. over N predictions pdf Same random numbers for each value of N pdf Ratio to best-fit prediction NLO W /W cross-section ratio at the LHC ( MSTW 2008 NLO PDFs (68% C.L.) Average and s.d. over N predictions pdf Different random numbers for each value of N pdf s = 7 TeV) Number of random predictions, N 2 pdf Number of random predictions, N 2 pdf 3 Left, add a new random set to existing ones sequentially. increasing numbers of independent random sets. Right, Very good with 40 sets. Excellent with 00 sets. UCL February 203 2

14 Speed of convergence of prediction for t t cross section. Ratio to best-fit prediction NLO tt cross section at the LHC ( MSTW 2008 NLO PDFs (68% C.L.) Average and s.d. over N predictions pdf Number of random predictions, N 2 s = 7 TeV) Same random numbers for each value of N pdf pdf 3 Ratio to best-fit prediction NLO tt cross section at the LHC ( MSTW 2008 NLO PDFs (68% C.L.) Average and s.d. over N predictions pdf Number of random predictions, N 2 s = 7 TeV) Different random numbers for each value of N pdf pdf 3 Left, add a new random set to existing ones sequentially. increasing numbers of independent random sets. Right, Very good with 40 sets. Excellent with 00 sets. UCL February 203 3

15 Speed of convergence of prediction for H cross section. Ratio to best-fit prediction NLO gg H at the LHC ( s = 7 TeV) for M = 20 GeV H MSTW 2008 NLO PDFs (68% C.L.) Average and s.d. over N predictions pdf Same random numbers for each value of N pdf Number of random predictions, N 2 pdf 3 Ratio to best-fit prediction NLO gg H at the LHC ( s = 7 TeV) for M = 20 GeV H MSTW 2008 NLO PDFs (68% C.L.) Average and s.d. over N predictions pdf Different random numbers for each value of N pdf Number of random predictions, N 2 pdf 3 Left, add a new random set to existing ones sequentially. increasing numbers of independent random sets. Right, Very good with 40 sets. Excellent with 00 sets. UCL February 203 4

16 - Can combine PDF sets, e.g. comparison to PDF4LHC prescription. ) (nb) - l + l 0 B(Z 0 Z σ NNLO Z l l at the LHC ( s = 7 TeV) Solid lines: envelope and midpoint. Dashed lines: statistical combination. 4 MSTW08 CT0 NNPDF2.3 α 2 S (M ) = 0.7 α 2 ) = 0.80 Z S (M α 2 ) = 0 Z S (M Z 3 Open markers: usual best-fit and 68% C.L. Hessian uncertainty. Closed markers: average and s.d. over random predictions. G. Watt (November 202) / σ W + W σ R ± NNLO W /W ratio at the LHC ( s = 7 TeV) Solid lines: envelope and midpoint. Dashed lines: statistical combination..4 MSTW08 CT0 NNPDF2.3 α 2 S (M ) = 0.7 α 2 ) = 0.80 Z S (M α 2 ) = 0 Z S (M Z.4 Open markers: usual best-fit and 68% C.L. Hessian uncertainty. Closed markers: average and s.d. over random predictions. G. Watt (November 202) (pb) tt σ NNLO+NNLL tt cross sections at the LHC ( Solid lines: envelope and midpoint. Dashed lines: statistical combination. pole Top++ (v.4), µ = µ = m t = 73.8 GeV R F MSTW08 α 2 S (M ) = 0.7 Z CT0 α 2 S (M ) = 0.80 Z NNPDF2.3 α 2 S (M ) = 0 Z s = 7 TeV) Open markers: usual best-fit and 68% C.L. Hessian uncertainty. Closed markers: average and s.d. over random predictions. G. Watt (November 202) (pb) σ H NNLO gg H at the LHC ( s = 8 TeV) for M = 26 GeV H Solid lines: envelope and midpoint. Dashed lines: statistical combination. ggh@nnlo (v.4.), µ = µ = M / 2 R F H 9 MSTW08 CT0 NNPDF2.3 α 2 S (M ) = 0.7 α 2 ) = 0.80 Z S (M α 2 ) = 0 Z S (M Z 8.5 Open markers: usual best-fit and 68% C.L. Hessian uncertainty. Closed markers: average and s.d. over random predictions. Smaller uncertainty and shifted central value if disagreement between individual predictions. (Plots by G. Watt at G. Watt (November 202) UCL February 203 5

17 Comparison to LHC data. (arxiv:2.25, A.D. Martin et. al.) Start with ATLAS jets. Comparison clearly fine. UCL February 203 6

18 Being more quantitative Using APPLGrid or FastNLO (B. Watt). MSTW fit very good, see χ 2 per point above. Always close to, and sometimes the best of any PDF set, particularly for R = 0.6. Not a huge variation in PDFs across groups though. Can see effect of data on the gluon by using the reweighting technique, R = 0.6 (R = 0.4 similar). Clearly little pull with present data. Ratio to MSTW Central Value Before Reweighting (N pdf =000) After Reweighting (N eff =774) g(x) at q 2 =0000 (GeV) x UCL February 203 7

19 Comparison of MSTW2008 to total W, Z excellent. CMS - 36 pb at s = 7 TeV σ B ( W ) 87 ± exp. ± th. + σ B ( W ) 82 ± exp. ± th. - lumi. uncertainty: ± 4% σ B ( W ) 93 ± 0.00 exp. ± th. σ B ( Z ).002 ± 0.00 exp. ± th. 8 ± 0.00 exp. ± 0.05 R W/Z th. 90 ± 0.0 exp. ± R +/- th Ratio (CMS/Theory) UCL February 203 8

20 Comparisons between different sets for Inclusive predictions. ν) (nb) - l NNLO W and W cross sections at the LHC ( s = 7 TeV) NNLO W and Z cross sections at the LHC ( s = 7 TeV) CMS, L = 36 pb - ATLAS, L = pb - + R(W /W ) = ) (nb) - l + l B(W σ W uncertainties PDF+α S 68% C.L. PDF MSTW08 CT0 NNPDF2.3 nolhc NNPDF2.3 HERAPDF.5 ABM JR G. Watt (November 202) σ + B(W l ν) (nb) W CMS, L = 36 pb - ATLAS, L = pb R(W/Z) = B(Z 0 σ Z uncertainties PDF+α S 68% C.L. PDF MSTW08 CT0 NNPDF2.3 nolhc NNPDF2.3 HERAPDF.5 ABM JR09 G. Watt (November 202) ± ± σ W ± B(W l ν) (nb) (Plots by G. Watt at UCL February 203 9

21 (pb) tt σ NNLO+NNLL tt cross sections at the LHC ( % C.L. PDF MSTW08 CT0 NNPDF2.3 nolhc NNPDF2.3 HERAPDF.5 ABM JR09 LHC combined, - L = fb (prel., Sept. 202) s = 7 TeV) G. Watt (November 202) (pb) σ H NNLO gg H at the LHC ( % C.L. PDF MSTW08 CT0 NNPDF2.3 nolhc NNPDF2.3 HERAPDF.5 ABM JR09 s = 8 TeV) for M = 26 GeV H G. Watt (November 202) 20 pole Top++ (v.4), µ = µ = m t = 73.8 GeV R F PDG α S (M ) Z 8 PDG ggh@nnlo (v.4.), µ = µ = M / 2 R F H α S (M ) Z Note variety of values of α S (MZ 2 ) used by different groups. UCL February

22 Correlates with PDF luminosity plots Ratio to MSTW 2008 (68% C.L.) NNLO Σ q (qq) luminosity at LHC ( MW MSTW 2008 CT0 NNPDF2.3 nolhc NNPDF2.3 MZ s = 8 TeV) s (GeV) G. Watt (November 202) Ratio to MSTW 2008 (68% C.L.) NNLO Σ q (qq) luminosity at LHC ( MW MSTW 2008 HERAPDF.5 ABM JR09 MZ s = 8 TeV) s (GeV) G. Watt (November 202) UCL February 203 2

23 Ratio to MSTW 2008 (68% C.L.) NNLO gg luminosity at LHC ( s = 8 TeV) MSTW 2008 CT0 NNPDF2.3 nolhc NNPDF2.3 M Higgs 2m top s (GeV) G. Watt (November 202) Ratio to MSTW 2008 (68% C.L.) NNLO gg luminosity at LHC ( s = 8 TeV) MSTW 2008 HERAPDF.5 ABM JR09 M Higgs 2m top s (GeV) G. Watt (November 202) Some pretty clear difference between some of the groups. More later. UCL February

24 MSTW also pretty good for inclusive distributions. Except some problems with asymmetry. Comparison gives χ 2 /N pts = 60/30 [pb] Z dσ/d y l Theory/Data [pb] dσ/d η l Theory/Data ATLAS - L dt = pb Data 200 ( s = 7 TeV) MSTW08 HERAPDF.5 ABKM09 JR Z l l Uncorr. uncertainty Total uncertainty y Z ATLAS - L dt = pb Data 200 ( s = 7 TeV) MSTW08 HERAPDF.5 ABKM09 JR η l - - W l ν Uncorr. uncertainty Total uncertainty l [pb] dσ/d η Theory/Data A l ATLAS - L dt = pb Data 200 ( s = 7 TeV) MSTW08 HERAPDF.5 ABKM09 JR W l νl Uncorr. uncertainty Total uncertainty η l Data 200 ( s = 7 TeV) MSTW08 HERAPDF.5 ABKM09 JR09 - L dt = pb ATLAS Stat. uncertainty Total uncertainty η l UCL February

25 MSTW pretty good for ATLAS W, Z distributions, except some problems with asymmetry. Electron Charge Asymmetry CMS 840 pb at s = 7 TeV p (e) > 35 GeV T W eν Electron Pseudorapidity η - MCFM: CT0 HERAPDF.5 MSTW2008NLO NNPDF2.2 (NLO) theory bands: 68% CL Similar for CMS data (will return to this later), though depends on p T cut. Generally very good for LHCb. A µ p µ T > 20 GeV/c LHCb, Datastat Datatot s = 7 TeV MSTW08 ABKM09 JR09 NNPDF2 HERA5 CTEQ6M (NLO) µ η UCL February

26 Asymmetry used (G. Watt, R.T.) in reweighting (JHEP 208 (202) 052), and moves u V d V up near x = where parameterisation perhaps underestimates uncertainty. (ATLAS left, CMS p T > 25GeV right). Lepton charge asymmetry l ν W ± l ± ν at the LHC ( s = 7 TeV) with p > 20 GeV, E T > 25 GeV, M T T ATLAS (l = e+µ), L = pb MSTW08 NLO PDFs (68% C.L.) Hessian, χ 2 / = 2.7 = 000, N pdf = 264, N eff χ 2 / = 2.90 χ 2 / =.27 > 40 GeV Lepton charge asymmetry ± ± W l ν at the LHC ( s = 7 TeV) with p > 25 GeV CMS (l = e), L = 36 pb CMS (l = µ), L = 36 pb - - MSTW08 NLO PDFs (68% C.L.) Hessian, χ 2 /2 =.74 = 000, N pdf = 485, N eff l T χ 2 /2 = 2.0 χ 2 /2 = Lepton pseudorapidity, η Ratio to MSTW 2008 NLO u v - d v distribution at Q = 0 GeV MSTW08 (68% C.L.) MSTW08 (N = 000) pdf MSTW08 (N = 264) eff 0 - l x Lepton pseudorapidity, η Ratio to MSTW 2008 NLO u v - d v distribution at Q = 0 GeV MSTW08 (68% C.L.) MSTW08 (N = 000) pdf MSTW08 (N = 485) eff 0 - l x UCL February

27 Investigation of Parameterisation Issues In the light of Monte Carlo studies investigate parameterisation dependence, initially concentrating on valence quarks. Decide to use Chebyshev polynomials (looked at other possibilities) xf(x, Q 2 0) = A( x) η x δ ( + n a n T n (y)) i.e. keep high and low x limits. Choose y = 2 x. T i y x 2 x T i y x 2 x T i y x 2 x 0.25 T i y x cos Π x x Same choice as in Pumplin study. Slightly different to Glazov, Moch and Radescu. UCL February

28 It is an open question how many parameters are required to accurately fit a simple shape such as that for xu V (x, Q 2 ) or xd V (x, Q 2 ). Investigate by generating pseudo-data for various types of parton distribution using a function with very many parameters, then fitting with smaller numbers. Fairly similar conclusion independent of type of parameterisation. UCL February

29 Example, fit to pseudo-data for valence quark generated between x = 0.0 and x = 0.7. For precise match to pseudo-data need 4 polynomials. Fractional deviation Fractional deviation x x Look at χ 2 distribution with increasing terms in polynomial Very good fits with 4 parameters. More tends to give over-fitting and peculiarities outside of range of x fit. UCL February

30 Fits to same data as MSTW2008 with 2 extra parameters for valence quarks params. Also with 2 extra parameters for valence quarks and sea params. No extra parameters in the gluon found to be necessary. χ 2 = 29, MSTW2008CP pdfs. Only real change in smallx u V, and sea at low Q ratio of u V (x,q 2 =GeV 2 ) MSTW MSTWCPv MSTWCP x 0 -. ratio of d V (x,q 2 =GeV 2 ) MSTW MSTWCPv MSTWCP x 0 - ratio of sea(x,q 2 =GeV 2 ) MSTW MSTWCPv MSTWCP x ratio of u V (x,q 2 =0000GeV 2 ) MSTW MSTWCPv MSTWCP x 0 -. ratio of d V (x,q 2 =0000GeV 2 ) MSTW MSTWCPv MSTWCP x 0 - ratio of sea(x,q 2 =0000GeV 2 ) MSTW MSTWCPv MSTWCP x 0 -. ratio of g(x,q 2 =5GeV 2 ). ratio of g(x,q 2 =0000GeV 2 ) MSTW MSTWCPv MSTWCP x 0 - MSTW MSTWCPv MSTWCP x 0 - UCL February

31 Also do the same fits with a freedom in the deuterium corrections more stable than previous MSTW studies. MSTW2008CPdeut PDFs ratio of u V (x,q 2 =GeV 2 ) MSTW MSTWCP MSTWCPdeut x ratio of u V (x,q 2 =0000GeV 2 ) MSTW MSTWCP MSTWCPdeut x 0 - ratio of d V (x,q 2 =GeV 2 ) ratio of d V (x,q 2 =0000GeV 2 ) MSTW MSTWCP MSTWCPdeut 0.8 MSTW MSTWCP MSTWCPdeut x x 0 - Now also get variation in d V (x) for higher x due to deuterium correction (seen before) and x 0.03 due to parameterization and corrections. Fit to ATLAS W, Z rapidity data at NLO improves to 49/30 for MSTWCP and 46/30 for MSTWCPdeut. UCL February

32 Shown is change in central value and uncertainty for u V (x) d V (x) at Q 2 = 0, 000 GeV 2. Biggest effect at lower x than probed at the LHC (yet) x(u V -d V )(x,q 2 =0000GeV 2 ) MSTW MSTWCP MSTWCPdeut Uncertainty sets have 23 eigenvectors (20 in MSTW2008). Main effect in uncertainty an increase in d V (x, Q 2 ) due to deuterium correction uncertainties, and minor valence uncertainty increase from extra parameter x MSTW MSTWCP MSTWCPdeut percentage difference at Q 2 =0000GeV x 0 - UCL February 203 3

33 Increases lepton asymmetry, but very preferentially for high p T cut. (Curves made here with LO calculations) LHC 7 TeV MSTW2008 NLO Most of the effect already obtained for parameterisation extension, but some from deuterium study. A(y l ) lepton asymmetry, variable p Tl (min) MSTW2008 MSTW2008CP MSTW2008CPdeut y l UCL February

34 Lepton charge asymmetry ± ± W l ν at the LHC ( s = 7 TeV) with p - CMS (l = e), L = 840 pb NLO PDFs (with NLO K-factors) l T > 35 GeV MSTW08 (68% C.L.), χ 2 / = 5.34 MSTW08CP, χ 2 / =.53 MSTW08CPdeut, χ 2 / = Lepton pseudorapidity, η Prediction for p T > 35GeV CMS asymmetry data. Note no change to data fit, just parameterisation and some from deuterium corrections. Main deuterium effect absence of shadowing used in default fit. l UCL February

35 Lepton charge asymmetry ± W l ± ν at the LHC ( s = 7 TeV) with p - ATLAS (l = e+µ), L = pb l T ν > 20 GeV, E T NLO PDFs (with NLO K-factors) > 25 GeV, M T MSTW08 (68% C.L.), χ 2 / = 2.7 MSTW08CP, χ 2 / =.38 > 40 GeV 0.2 MSTW08CPdeut, χ 2 / = Lepton pseudorapidity, η Prediction for ATLAS asymmetry data. Note no change to data fit, just parameterisation and some from deuterium corrections. l UCL February

36 Big change in high-p T cut asymmetry, but this is very specifically sensitive to u V (x, Q 2 ) d V (x, Q 2 ). What about other quantities? Other PDFs changed little. α S free but tiny change. Expect little variation. Seen clearly on luminosity plots (G. Watt). A not totally insignificant change in the high-x gluon luminosity for the MSTWCPdeut set. Due to softer d V Ratio to MSTW 2008 (68% C.L.) Ratio to MSTW 2008 (68% C.L.) Ratio to MSTW 2008 (68% C.L.) NLO Σ q (qq) luminosity at LHC ( MW MSTW08 MSTW08CP MSTW08CPdeut MZ s = 8 TeV) NLO gg luminosity at LHC ( MSTW08 MSTW08CP MSTW08CPdeut M Higgs 2m top s = 8 TeV) NLO GG luminosity at LHC ( MSTW08 MSTW08CP s (GeV) s s (GeV) s = 8 TeV), G g + 4/9 Σ (q+q) q MSTW08CPdeut UCL February (GeV)

37 The percentage change of various cross sections due to the modifications of the MSTW2008 PDFs. In order to demonstrate the small changes in the cross sections, we also show, in the final column, the symmetrized PDF + α S (M 2 Z ) percentage uncertainties for MSTW2008 PDFs. UCL February

38 Can see variation of cross sections with energy % change in total (NLO) LHC cross sections from MSTW2008 to MSTW2008CPdeut W + δσ/σ (%) gg H(25) Z 0 W _ t t s (TeV) UCL February

39 Choices for Heavy Flavours in DIS. (Extension of work in by RT in Phys.Rev. D86 (202) ) Near threshold Q 2 m 2 H state. massive quarks not partons. Created in final Described using Fixed Flavour Number Scheme (FFNS). F (x, Q 2 ) = C F F,n f k (Q 2 /m 2 H) f n f k (Q2 ) Does not sum αs n lnn Q 2 /m 2 H terms in perturbative expansion. Usually achieved by definition of heavy flavour parton distributions and solution of evolution equations. Additional problem FFNS known up to NLO (Laenen et al.), but are not fully known at NNLO αs 3 F,3 CF 2,Hi unknown. Approximations based on some or all of threshold, low-x and high-q 2 limits can be derived, see Kawamura, et al.,, and are sometimes used in fits, e.g. ABM and MSTW (at low Q 2 ). Generally not large except at threshold and very low x. UCL February

40 Variable Flavour - at high scales Q 2 m 2 H heavy quarks behave like massless partons. Sum ln(q 2 /m 2 H ) terms via evolution. Zero Mass Variable Flavour Number Scheme (ZM-VFNS). Ignores O(m 2 H /Q2 ) corrections. F (x, Q 2 ) = C ZM,n f j f n f j (Q 2 ). Partons in different number regions related to each other perturbatively. f n f + j (Q 2 ) = A jk (Q 2 /m 2 H) f n f k (Q2 ), Perturbative matrix elements A jk (Q 2 /m 2 H ) (Buza et al.) containing ln(q 2 /m 2 H ) terms relate f n f i (Q 2 ) and f n f + i (Q 2 ) correct evolution for both. Want a General-Mass Variable Flavour Number Scheme (VFNS) taking one from the two well-defined limits of Q 2 m 2 H and Q2 m 2 H. UCL February

41 The GM-VFNS can be defined by demanding equivalence of the n f light flavour and n f + light flavour descriptions at all orders above transition point n f n f + F (x, Q 2 )=C F F,n f k (Q 2 /m 2 H ) f n f k (Q2 )=C V F,n f + j (Q 2 /m 2 H ) f n f + j (Q 2 ) C V F,n f + j (Q 2 /m 2 H ) A jk(q 2 /m 2 H ) f n f k (Q2 ). Hence, the VFNS coefficient functions satisfy C F F,n f k (Q 2 /m 2 H ) = CV F,n f + j (Q 2 /m 2 H ) A jk(q 2 /m 2 H ), which at O(α S ) gives C F F,n f,() 2,Hg ( Q2 m 2 H ) = C V F,n f +,(0) 2,HH ( Q2 m 2 H ) Pqg 0 ln(q 2 /m 2 H )+CV F,n f +,() 2,Hg ( Q2 ), m 2 H The VFNS coefficient functions tend to massless limits as Q 2 /m 2 H. However, C V F j (Q 2 /m 2 H ) only uniquely defined in this limit. Can swap O(m 2 H /Q2 ) terms between C V F,0 2,HH (Q2 /m 2 F, H ) and CV 2,g (Q 2 /m 2 H ), i.e. get scheme variations. UCL February

42 Difference between FFNS and GM-VFNS NLO NNLO F c 2(x,Q 2 ) MSTW08 MSTW08FFNS F c 2(x,Q 2 ) MSTW08 MSTW08FFNS 0 x= Q 2 0 x= Q F c 2(x,Q 2 ) MSTW08 MSTW08FFNS F c 2(x,Q 2 ) MSTW08 MSTW08FFNS x= Q 2 0 x= Q MSTW08 MSTW08FFNS 0.05 MSTW08 MSTW08FFNS F c 2(x,Q 2 ) F c 2(x,Q 2 ) 0 x= Q 2 0 x= Q 2 As seen at higher Q 2 charm structure function for FFNS nearly always lower than any GM-VFNS. NNLO uses O(αS 2 ) coefficient functions for F2(x, c Q 2 ). Approx. O(αS 3 ) raises slightly F 2(x, c Q 2 ) at higher x and lowers it at small x (except very small x at low Q 2 ). UCL February 203 4

43 evolved/fixed-order µ=00 GeV c (2) (N f =5) b (2) (N f =5) Σ (2) (N f =5) G (2) (N f =5) x Results for F2(x, c Q 2 ) in GM-VFNS compared to those for FFNS similar to results for PDFs by Alekhin et al. in Phys.Rev. D8 (200) comparing NNLO evolution to the fixed order result up to O(αS 2 ). Details depend on PDF set and α S (MZ 2 ) value used. UCL February

44 Can lead to over 4% changes in the total F 2 (x, Q 2 ) if the same input PDFs are used in two schemes..04 F 2 (x,q 2 )/MSTW08 NNLO MSTW08 MSTW08FFNS x= At higher x mainly due to F c 2(x, Q 2 ). At lower x there is a large contribution from light quarks evolving slightly more slowly in FFNS. At much higher x difference dies away. Charm component becomes very small and light quark evolution not much different. (Light quarks slightly bigger at the highest x.) F 2 (x,q 2 )/MSTW F 2 (x,q 2 )/MSTW Q x= Q x= Q UCL February

45 This is certainly important given the precision and range of the data on F 2 (x, Q 2 ) included in PDF fits. 2 MSTW 2008 σ ~ (x,q 2 ) HERA (F 2 (x,q 2 ) FixedTarget ) + c x= (c=0.35) x= (c=0.4) x=0.003 (c=0.3) H ZEUS NMC BCDMS SLAC Mainly from HERA, but NMC data also relevant..5 x= (c=0.25) x=0.008 (c=0.2) x=0.03 (c=0.5) x=0.05 (c=0.) 0.5 x=0.8 (c=0.05) x=0.35 (c=0.0) NNLO NLO LO Q 2 (GeV 2 ) 0 3 UCL February

46 Performed a series of NLO fits using the FFNS scheme and NNLO with up to O(αS 2 ) heavy flavour coefficient functions.(approximations to the O(αS 3 ) expressions change results very little). Fit to only DIS and Drell-Yan data but also effectively fit to Tevatron Drell- Yan or Tevatron jet data, if necessary, in 5-flavour scheme as FFNS calculations do not exist. Fits to DIS and Drell-Yan data usually at least a few tens of units worse than MSTW08 to same data (even without refitting MSTW08 to restricted data sets). Often slightly better for F c 2(x, Q 2 ), but flatter in Q 2 for x 0.0 for inclusive structure function. As well as (usually) a worse fit to DIS and Drell-Yan data only, in FFNS the fit quality for the DIS and low-energy Drell Yan data deteriorates by in general 50 units when all jet data is included as opposed to < 0 units when using a GM-VFNS. UCL February

47 PDFs evolved up to Q 2 = 0, 000GeV 2 (using variable flavour evolution for consistent comparison) different in form to MSTW08... FFNS/2008 at NLO for g(x,q 2 ).05 5 MSTW08NLO FFNSDIS-0.87 FFNS+DY-0.85 FFNSjet FFNSjetZ FFNS/2008 at NNLO for g(x,q 2 ).05 5 MSTW08NNLO-0.7 FFNSDIS-0.44 FFNS+DY-0.52 FFNSjet-0.8 FFNSjetZ FFNS/2008 at NLO for u(x,q 2 ).05 5 MSTW08NLO FFNSDIS FFNS+DY FFNSjet FFNSjetZ FFNS/2008 at NNLO for u(x,q 2 ).05 5 MSTW08NNLO FFNSDIS FFNS+DY FFNSjet FFNSjetZ In contrast in standard MSTW2008 fit PDFs usually within uncertainties if Tevatron jet data left out. Main effect loss of tight constraint on α S (MZ 2). UCL February

48 GMVFNSa/2008 at NLO for g(x,q 2 ) GMVFNSa/2008 at NLO for u(x,q 2 ) MSTW08 GMVFNS GMVFNS2 GMVFNS3 GMVFNSopt GMVFNS4 GMVFNS5 GMVFNS6 ZMVFNS MSTW08 GMVFNS GMVFNS2 GMVFNS3 GMVFNSopt GMVFNS4 GMVFNS5 GMVFNS6 ZMVFNS GMVFNSa/2008 at NNLO for g(x,q 2 ) GMVFNSa/2008 at NNLO for u(x,q 2 )..05 MSTW08NNLO GMVFNS GMVFNS2 GMVFNS3 5 GMVFNS4 GMVFNS5 GMVFNS6 GMVFNSopt MSTW08NNLO GMVFNS GMVFNS2 GMVFNS3 5 GMVFNS4 GMVFNS5 GMVFNS6 GMVFNSopt Using FFNS leads to much larger changes than any choice of GM- VFNS mainly due to fitting high-q 2 DIS data. UCL February

49 Low Q 2 Higher Twist. Potentially large corrections at low Q 2 and particularly low W 2. Usual MSTW cuts for Q 2 cut - 2GeV 2 W 2 cut - 5GeV 2 Have tried raising Q 2 cut to 5GeV 2 and 0GeV 2 and W 2 to 20GeV 2. Not much effect on PDFs or α S. Can also lower Wcut 2 to 5GeV 2 and try parameterising higher twist contributions by ( Fi HT (x, Q 2 ) = Fi LT (x, Q 2 ) + D ) i(x) Q 2 where i spans bins of x from x = 0.8 down to x = Previously no evidence for much higher twist except at low W 2. UCL February

50 .. PDF/MSTW08 at NLO for g(x,q 2 ).05 5 MSTW08NLO MSTW08HT-0.89 FFNSDIS-0.87 FFNSHT-0.88 PDF/MSTW08 at NNLO for g(x,q 2 ).05 5 MSTW08NNLO-0.7 MSTW08HT-0.75 FFNSDIS-0.44 FFNSHT PDF/MSTW08 at NLO for u(x,q 2 ).05 5 MSTW08NLO MSTW08HT FFNSDIS FFNSHT PDF/MSTW08 at NNLO for u(x,q 2 ).05 5 MSTW08NNLO MSTW08HT FFNSDIS FFNSHT Now more evidence for positive contribution also at very low x. Leads to lower input quarks, more gluon for evolution. Largely washes out quickly with Q 2. Similar effect using FFNS as for GM-VFNS. UCL February

51 .. PDF/MSTW08 at NLO for g(x,q 2 ).05 5 MSTW08NLO FFNS+DY-0.85 FFNS+DYHT*-0.65 PDF/MSTW08 at NLO for g(x,q 2 ).05 5 MSTW08NNLO-0.7 FFNS+DY-0.52 FFNS+DYHT* PDF/MSTW2008 at NLO for u(x,q 2 ) MSTW08NLO FFNS+DY FFNS+DYHT* PDF/MSTW2008 at NNLO for u(x,q 2 ) MSTW08NNLO FFNS+DY FFNS+DYHT* Restricting higher twist from lowest x value and omitting nuclear target data (except dimuon for strangeness) tends to keep values of α S lower by UCL February

52 Explains some PDF differences? MSTW FFNS ratios and ABKM ratios... PDF/MSTW08 at NLO for g(x,q 2 ).05 5 MSTW08NLO ABKM PDF/MSTW08 at NNLO for g(x,q 2 ).05 5 MSTW08NNLO-0.7 ABKM PDF/MSTW08 at NLO for u(x,q 2 ).05 5 MSTW08NLO ABKM09 PDF/MSTW08 at NNLO for u(x,q 2 ).05 5 MSTW08NNLO ABKM General trend is very similar to fits on previous page. UCL February 203 5

53 Conclusions There is a reasonable amount of variation in comparisons to data/predictions at the LHC using different PDF sets. The MSTWCP and MSTWCPdeut PDFs improve dramatically the pre(post)diction for lepton asymmetries from W bosons at the LHC. This is due to a very localised effect in small-x valence quark differences. There is extremely little change in parton luminosities or predictions for a wide variety of other cross sections. Lowering W 2 cut and allowing for higher twist terms in structure functions makes very little change to MSTW PDFs except at extremely high x Performing fits using an FFNS leads to worse fits to DIS and low energy Drell-Yan data than GM-VFNS, and much more tension with jet data. Light quarks in variable flavour number scheme are automatically larger in most regions for FFNS than for GM-VFNS. The gluon is smaller at high x and larger at small x in FFNS, and α S (MZ 2 ) smaller. Seems to be the likely explanation of some major differences in PDFs and LHC predictions. (Similar conclusions in NNPDF studies.) UCL February

54 Back-up UCL February

55 Calculate χ 2 /N = 60/30 for ATLAS W, Z data again at NLO using APPLGrid. Not best, but fairly close to any other set except CT0 which is best. Again look at eigenvectors. Fit improves markedly in one direction with eigenvector 9, gluon, which alters common shape and normalisation, and 4 and 8 which alter d V and u V, i.e. affect asymmetry. Not much variation in strange normalisation. MSTW Eigenvectors for WZ Fit Χ 2 per point Deviation From Central Value Eigenvector Number UCL February

56 Similar but slightly smaller effect on u V d V than asymmetry alone. Can also see the effect on the gluon. Slight raise near x = 0.0 preferred. Improves overall shape of rapidity distribution. After reweighting χ 2 = 48/ Before Reweighting (N pdf =000) After Reweighting (N eff =90) g(x) at q 2 =0000 (GeV) 2. Ratio to MSTW Central Value x UCL February

57 Ratio to MSTW 2008 NLO (90% C.L.) Gluon distribution at Q No Tevatron jets Fit only Run I jets Fit CDFII(kT) + D II jets JET Same with µ = µ = p /2 R F T 4 = 0 2 GeV 0.6 Fit CDFII(Midpoint) + D II jets Fit Run I + CDFII(kT) + D II jets In contrast in MSTW2008 fit central gluon hardly changed if Tevatron jet data left out, and only slight further rearrangement of quark flavours if Drell-Yan data left out. Main effect loss of tight constraint on α S (MZ 2 ). Much the same at NNLO. Similar results from various other groups. x UCL February

58 Given previous relationship between Tevatron asymmetry and deuterium corrections where partial success was noted revisit with extended parameterisation. Default for MSTW some shadowing for x < 0.0. Previously big improvement in fit, but unusual corrections. Now improvement again but much more stable, and sensible for deuterium corrections. (No shadowing favoured though.) correction factor MSTWCP - 4 deuteron parameters MSTWCP - 3 deuteron parameters MSTWCP - 2 deuteron parameters UCL February

59 The GM-VFNS can be defined by demanding equivalence of the n f light flavour and n f + light flavour descriptions at all orders above transition point n f n f + F (x, Q 2 )=C F F,n f k (Q 2 /m 2 H ) f n f k (Q2 )=C V F,n f + j (Q 2 /m 2 H ) f n f + j (Q 2 ) Hence, the VFNS coefficient functions satisfy C F F,n f C V F,n f + j (Q 2 /m 2 H ) A jk(q 2 /m 2 H ) f n f k (Q2 ). k (Q 2 /m 2 H) = C V F,n f + j (Q 2 /m 2 H) A jk (Q 2 /m 2 H), which at O(α S ) gives C F F,n f,() 2,Hg ( Q2 m 2 H ) = C V F,n f +,(0) 2,HH ( Q2 m 2 H ) Pqg 0 ln(q 2 /m 2 H)+C V F,n f +,() 2,Hg ( Q2 m 2 ), H The VFNS coefficient functions tend to the m=0 limits as Q 2 /m 2 H. However, C V F j (Q 2 /m 2 H ) only uniquely defined in this limit. Can swap O(m 2 H /Q2 ) terms between C V F,0 2,HH (Q2 /m 2 F, H ) and CV2,g (Q 2 /m 2 H ). UCL February

60 The results for F 2 (x, Q 2 ) when refits are performed. As seen very little change when using GM-VFNS with no jets. Much more tension and worse fits for FFNS..04 F 2 (x,q 2 )/MSTW F 2 (x,q 2 )/MSTW08 NNLO MSTW08 MSTW08nojet FFNSDIS FFNSjetZ x= Q x= Q F 2 (x,q 2 )/MSTW08 x= Q UCL February

61 F c, Q 2 = 20 GeV F c, Q 2 = 00 GeV NNLO / NLO.3 NLO scale.2 NNLO approx..2 NNLO scale x ABM, m 2 = 2 GeV 2 c x Approximate O(αS 3 ) corrections to F 2(x, c Q 2 ) by Kawamura et al. Nucl.Phys. B864 (202) in Similar results for O(αS 3 ) approximation used by MSTW at low Q2 extended to higher Q 2. UCL February

62 partons/mstw2008 at NLO for g(x,q 2 ) partons/mstw2008 at NLO for u(x,q 2 ) Preliminary Q 2 =0,000GeV 2 MSTW08 NLO Q 2 =5GeV 2, W 2 =20GeV 2 Q 2 =0GeV 2, W 2 =20GeV Preliminary Q 2 =0,000GeV 2 MSTW08 NLO Q 2 =5GeV 2, W 2 =20GeV 2 Q 2 =0GeV 2, W 2 =20GeV partons/mstw2008 at NNLO for g(x,q 2 ) partons/mstw2008 at NNLO for u(x,q 2 ) Preliminary MSTW08 NNLO Q 2 =5GeV 2, W 2 =20GeV 2 Q 2 =0GeV 2, W 2 =20GeV Preliminary MSTW08 NNLO Q 2 =5GeV 2, W 2 =20GeV 2 Q 2 =5GeV 2, W 2 =20GeV Similar to effect of higher twist, particularly at NNLO. Remember lose data at lowest x. UCL February 203 6

63 .. PDF/MSTW2008 at NLO for g(x,q 2 ).05 5 MSTW08NLO FFNSHT*DIS-0.62 FFNS+DYHT*-0.65 FFNSjetHT*-9 FFNSjetZHT*-0.25 PDF/MSTW08 at NNLO for g(x,q 2 ).05 5 MSTW08NNLO-0.7 FFNSDISHT*-0.32 FFNS+DYHT*-0.36 FFNSjetHT*-0.55 FFNSjetZHT* PDF/MSTW08 at NLO for u(x,q 2 ).05 5 MSTW08NLO FFNSDISYHT* 2 L FFNS+DYHT* FFNSjetHT* FFNSjetZHT* PDF/MSTW08 at NNLO for u(x,q 2 ).05 5 MSTW08NNLO FFNSDISHT* FFNS+DYHT* FFNSjetHT* FFNSjetZHT* Restricting higher twist from lowest x value and omitting nuclear target data (except dimuon for strangeness). Same trends as for standard fits but slightly lower α S UCL February

64 Explains some PDF differences? MSTW FFNS ratios and ABKM ratios... PDF/MSTW08 at NLO for g(x,q 2 ).05 5 MSTW08NLO ABKM ABM-0.80 PDF/MSTW08 at NNLO for g(x,q 2 ).05 5 MSTW08NNLO-0.7 ABKM ABM PDF/MSTW08 at NLO for u(x,q 2 ).05 5 MSTW08NLO ABKM09 ABM PDF/MSTW08 at NNLO for u(x,q 2 ).05 5 MSTW08NNLO ABKM09 ABM Better to compare to ABKM09 as mass scheme and data fit are more similar. UCL February

65 xg(x,q 2 =0000GeV 2 ) xu(x,q 2 =0000GeV 2 ) x 0-5 percentage difference at Q 2 =0000GeV x 0-5 percentage difference at Q 2 =0000GeV x x 0 - Change in MSTW2008 NNLO PDFs when fitting HERA combined data. UCL February

66 Comparison to LHC data. Start with ATLAS jets. Use APPLGrid or FastNLO at NLO (Ben Watt) and correlated errors treated as in the formula, χ 2 = N pts. i= ( ˆDi T i σ uncorr. i ) 2 + N corr. k= r 2 k, where ˆD i D i N corr. k= r k σk,i corr. D i are the data points allowed to shift by the systematic errors in order to give the best fit, and σk,i corr. is a fractional uncertainty. Normalisation is treated as the other correlated uncertainties. MSTW fit very good (χ 2 per point below left ), though numbers lower for inclusive data. Always close to, if not best, particularly for R = 0.6. Not huge variation in PDFs though. Scale pt/2 pt 2pT Inclusive (R=0.4) Inclusive (R=0.6) Dijet (R=0.4) Dijet (R=0.6) r k < < r k < 2 2< r k < 3 3< r k < 4 Inclusive (R=0.4) Inclusive (R=0.6) Dijet (R=0.4) Dijet (R=0.6) UCL February