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PDF’s for the LHC: LHC cross section benchmarking and error prescriptions

PDF’s for the LHC: LHC cross section benchmarking and error prescriptions. J. Huston Michigan State University MCTP Spring Symposium on Higgs Boson Physics. First, a brief introduction.

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PDF’s for the LHC: LHC cross section benchmarking and error prescriptions

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  1. PDF’s for the LHC:LHC cross section benchmarking and error prescriptions J. Huston Michigan State University MCTP Spring Symposium on Higgs Boson Physics

  2. First, a brief introduction • The calculation of PDF uncertainties for LHC cross sections is becoming more topical, as LHC exp cross sections themselves are more topical • The LHC experiments have gone/are going through exercises tabulating important cross sections and their uncertainties (Jianming has for Higgs) • In many cases, the estimates of cross sections and uncertainties from the PDF groups (such as CTEQ, MSTW, NNPDF…) are closer than many people thought • A discussion, started at Les Houches, was formalized within the PDF4LHC working group to perform some benchmarking tests to understand the commonalities and differences between the predictions and uncertainties of the different PDF groups • First comparisons in meeting (Mar 26) of PDF4LHC see for example, A. Vicini’s talk at the Jan. 29, 2010 PDF4LHC meeting

  3. PDF errors • So now, seemingly, we have more consistency (at least in some cases) in the size of PDF errors • NB: MSTW2008 uses a dynamic tolerance for their error determination rather than a fixed Dc2 of 50 • CTEQ6.6 still uses Dc2 of 100; dynamic in CT10 • The eigenvector sets (or NNPDF equivalent) represent the PDF uncertainty due to the experimental errors in the datasets used in the global fitting process To estimate the error on an observable X(a),from the experimental errors, we use the Master Formula where Xi+ and Xi- are the values for the observable X when traversing a distance corresponding to the tolerance T(=sqrt(Dc2)) along the ith direction

  4. as uncertainties • Another uncertainty is that due to the variation in the value of as • MSTW has recently tried to better quantify the uncertainty due to the variation of as, by performing global fits over a finer range, taking into account correlations between the values of as and the PDF errors • Procedure is a bit complex • …more recent studies by CTEQ and NNPDF

  5. as(mZ) and uncertainty: a complication • Different values of as and of its uncertainty are used • CTEQ and NNPDF use the world average (actually 0.118 for CTEQ and 0.119 for NNPDF), where MSTW2008 uses 0.120, as determined from their best fit • Latest world average (from Siggi Bethke->PDG) • as (mZ) = 0.1184 +/- 0.0007 • What does the error represent? • Siggi said that only one of the results included in his world average was outside this range • suppose we say that +/-0.002 is a reasonable estimate of the uncertainty G. Watt Mar 26 PDF4LHC meeting

  6. as(mZ) and uncertainty • Could it be possible for all global PDF groups to use the world average value of as in their fits, plus a prescribed 90% range for its uncertainty (if not 0.002, then perhaps another acceptable value)? • After that, world peace • For the moment, we try determining uncertainties from as over a range of +/- 0.002 from the central value for each PDF group; we also calculate cross sections with a common value of as=0.119 for comparison purposes

  7. My recommendation to PDF4LHC/Higgs working group • Cross sections should be calculated with MSTW2008, CTEQ6.6 and NNPDF • Upper range of prediction should be given by upper limit of error prediction using prescription for combining as uncertainty with error PDFs • in quadrature for CTEQ6.6 and NNPDF • using eigenvector sets for different values of as for MSTW2008 • (my suggestion) as standard, use 90%CL limits • Ditto for lower limit • So for a Higgs mass of 120 GeV at 14 TeV, the gg cross section lower limit would be defined by the CTEQ6.6 lower limit (PDF+as error) and the upper limit defined by the MSTW2008 upper limit (PDF+as error) • with the difference between the central values primarily due to as • To fully understand similarities/differences of cross sections/uncertainties conduct a benchmarking exercise, to which all groups are invited to participate

  8. PDF Benchmarking Exercise 2010 • Benchmark processes, all to be calculated (i) at NLO (in MSbar scheme) (ii) in 5-flavour quark schemes (definition of scheme to be specified) (iii) at 7 TeV [ and 14 TeV] LHC (iv) for central value predictions and +-68%cl [and +- 90%cl] pdf uncertainties (v) and with +- as uncertainties (vi) repeat with as(mZ)=0.119 (prescription for combining with pdf errors to be specified) • Using (where processes available) MCFM 5.7 • gzipped version prepared by John Campbell using the specified parameters and exact input files for each process (and the new CTEQ6.6 as series)->thanks John! • sent out on first week of March (and still available to any interested parties) • statistics ok for total cross section comparisons

  9. Cross Sections • W+, W-, and Z total cross sections and rapidity distributions total cross section ratios W+/W- and (W+ + W-)/Z, rapidity distributions at y = -4,-3,...,+4 and also the W asymmetry: A_W(y) = (dW+/dy - dW-/dy)/(dW+/dy + dW-/dy) using the following parameters taken from PDG 2009 • MZ=91.188 GeV • MW=80.398 GeV • zero width approximation • GF=0.116637 X 10-5 GeV-2 • other EW couplings derived using tree level relations • BR(Z-->ll) = 0.03366 • BR(W-->lnu) = 0.1080 • CKM mixing parameters from eq.(11.27) of PDG2009 CKM review 0.97419 0.2257 0.00359 V_CKM = 0.2256 0.97334 0.0415 0.00874 0.0407 0.999133 • scales: mR = mF = MZ or MW

  10. Cross Sections 2. gg->H total cross sections at NLO • MH = 120, 180 and 240 GeV • zero Higgs width approximation, no BR • top loop only, with mtop = 171.3 GeV in sigma_0 • scales: mR = mF = MH 3. ttbar total cross section at NLO • mtop = 171.3 GeV • zero top width approximation, no BR • scales: mR = mF = mtop

  11. For CTEQ: as series • Take CTEQ6.6 as base, and vary as(mZ) +/-0.002 (in 0.001 steps) around central value of 0.118 • Blue is the PDF uncertainty from eigenvectors; green is the uncertainty in the gluon from varying as • We have found that change in gluon due to as error (+/-0.002 range) is typically smaller than PDF uncertainty with a small correlation with PDF uncertainty over this range • as shown for gluon distribution on right • PDF error and as error can be added in quadrature • expected because of small correlation • in recent CTEQ paper, it has been proven this is correct regardless of correlation, within quadratic approximation to c2 distribution arXiv:1004.4624; PDFs available from LHAPDF So the CTEQ prescription for calculating the total uncertainty (PDF+as) involves the use of the 45 CTEQ6.6 PDFs and the two extreme as error PDF’s (0.116 and 0.120)

  12. Higgs cross sections and uncertainties • Linear dependence of Higgs cross section at NLO with as can be observed • as and gluon distribution are anti-correlated in this range, but the Higgs cross section has a large K-factor (NLO/LO), so as dependence from the higher order contribution • Predictions are from new CTEQ paper; not exactly the same setup as for MCFM benchmark predictions, so numerical values aren’t identical, but linear as dependence is clear

  13. Some results from the benchmarking • …from G. Watt’s presentation at PDF4LHC meeting on March 26 • See also S. Glazov’s summary in the March 31 MC4LHC workshop at CERN • CTEQ/MSTW predictions for W cross section/uncertainty in very good agreement • small impact from different as value • similar uncertainty bands • NNPDF prediction low because of use of ZM-VFNS • HERAPDF1.0 a bit high because of use of combined HERA dataset

  14. W/Z ratio • Good agreement among the PDF groups

  15. Some results from the benchmarking • …from G. Watt’s presentation at PDF4LHC meeting on March 26 • Similar gluon-gluon luminosity uncertainty bands, as noted before • Cross sections fall into two groups, outside 68% CL error bands • But, slide everyone’s prediction along the as curve to 0.119 (for example) and predictions agree reasonably well • within 68% CL PDF errors

  16. tT production

  17. Two other Higgs masses • Of course, for gg->Higgs, we know the NNLO corrections have to be taken into account for precision comparisons • This requires NNLO PDFs, but since most of the NNLO correction is from the ME and not the PDFs, one can calculate an approximate NNLO prediction from those sets not yet at NNLO • CTEQ NNLO PDFs currently in development

  18. New from CTEQ-TEA (Tung et al)->CT10 PDFs • Combined HERA-1 data • CDF and D0 Run-2 inclusive jet data • Tevatron Run 2 Z rapidity from CDF and D0 • W electron asymmetry from CDFII and D0II (D0 muon asymmetry) (in CT10W) • Other data sets same as CTEQ6.6 • All data weights set to unity (except for CT10W) • Tension observed between D0 II electron asymmetry data and NMC/BCDMS data • Tension between D0 II electron and muon asymmetry data • Experimental normalizations are treated on same footing as other correlated systematic errors • More flexible parametrizations: 26 free parameters (26 eigenvector directions) • Dynamic tolerance: look for 90% CL along each eigenvector direction • within the limits of the quadratic approximation, can scale between 68% and 90% CL with naïve scaling factor • Two series of PDF’s are introduced • CT10: no Run 2 W asymmetry • CT10W: Run 2 W asymmetry with an extra weight

  19. CT10/CT10W predictions No big changes with respect to CTEQ6.6 See H-L Lai talk at Pheno2010

  20. LO PDFs • Workhorse for many predictions at the LHC are still LO PDFs • Many LO predictions at the LHC differ significantly from NLO predictions, not because of the matrix elements but because of the PDFs • W+ rapidity distribution is the poster child • the forward-backward peaking obtained at LO is an artifact • large x u quark distribution is higher at LO than NLO due to deficiencies in the LO matrix elements for DIS

  21. Modified LO PDFs • Try to make up for the deficiencies of LO PDFs by • relaxing the momentum sum rule • including NLO pseudo-data in the LO fit to guide the modified LO distributions • Results tend to be in better agreement with NLO predictions, both in magnitude and in shape • Some might say that the PDFs then have no predictive power, but this is true for any LO PDFs • See arXiv:0910.4183; PDFs available from LHAPDF

  22. gg->Higgs • Higgs K-factor is too large to absorb into PDFs (nor would you want to) • Shape is ok with LO PDF’s, improves a bit with the modified LO PDFs

  23. K-factor table with the modified LO PDFs mod LO PDF K-factors for LHC slightly less K-factors at Tevatron K-factors with NLO PDFs at LO are more often closer to unity Note K-factor for W < 1.0, since for this table the comparison is to CTEQ6.1 and not to CTEQ6.6, i.e. corrections to low x PDFs due to treatment of heavy quarks in CTEQ6.6 “built-in” to mod LO PDFs

  24. K-factor table • Some rules-of-thumb • NLO corrections are larger for processes in which there is a great deal of color annihilation • gg->Higgs • gg->gg • K(gg->tT) > K(qQ -> tT) • these gg initial states want to radiate like crazy (see Sudakovs) • NLO corrections decrease as more final-state legs are added • K(gg->Higgs + 2 jets) < K(gg->Higgs + 1 jet) < K(gg->Higgs) • unless can access new initial state gluon channel • Can we generalize for uncalculated HO processes? • What about effect of jet vetoes on K-factors? Signal processes compared to background. Of current interest. Casimir for biggest color representation final state can be in Simplistic rule Ci1 + Ci2 – Cf,max L. Dixon Casimir color factors for initial state (not the full story, but indicative)

  25. Aside: multi-parton final states • In addition to the considerations outlined on the previous slide, the K-factors for multi-parton final states will tend to decrease due to the different behavior of jet algorithms at LO and NLO • See arXiv:1001.2581 (or extra slides)

  26. Les Houches wishlist • …is being completed faster than I can refill it • See arXiv:1003.1241 • tT+2jets and VVbB finished • V+4 jets on the horizon

  27. Finally, PDF correlations • Consider a cross section X(a), a function of the Hessian eigenvectors • ith component of gradient of X is • Now take 2 cross sections X and Y • or one or both can be pdf’s • Consider the projection of gradients of X and Y onto a circle of radius 1 in the plane of the gradients in the parton parameter space • The circle maps onto an ellipse in the XY plane • The angle f between the gradients of X and Y is given by • The ellipse itself is given by • If two cross sections are very • correlated, then cosf~1 • …uncorrelated, then cosf~0 • …anti-correlated, then cosf~-1

  28. Correlations with Z, tT tT Z Define a correlation cosine between two quantities • If two cross sections are very • correlated, then cosf~1 • …uncorrelated, then cosf~0 • …anti-correlated, then cosf~-1

  29. Correlations with Z, tT Define a correlation cosine between two quantities • If two cross sections are very • correlated, then cosf~1 • …uncorrelated, then cosf~0 • …anti-correlated, then cosf~-1 • Note that correlation curves to Z • and to tT are mirror images of • each other • By knowing the pdf correlations, • can reduce the uncertainty for a • given cross section in ratio to • a benchmark cross section iff • cos f > 0;e.g. D(sW+/sZ)~1% • If cos f < 0, pdf uncertainty for • one cross section normalized to • a benchmark cross section is • larger • So, for gg->H(500 GeV); pdf • uncertainty is 4%; D(sH/sZ)~8% tT Z

  30. Summary • PDF4LHC/Les Houches benchmarking exercise has provided useful understanding of commonalities/differences among the different PDF groups • and hopefully we can continue towards more standardization • Benchmark cross sections will be made available in a reference document, with predictions from all participating PDF groups • see https://wiki.terascale.de/index.phptitle?=PDF4LHC_WIKI • …and recommendations on how to estimate PDF/as uncertainties for LHC cross sections where such uncertainties are critical/non-critical • gg->Higgs is one case for NNLO corrections are large, so NNLO cross sections are necessary • Available from only a few groups to date • Since most of the impact arises from the matrix element and not the change in PDF’s, I’ve suggested providing approximate NNLO predictions in cases where NNLO PDFs are not available • NNLO ME + NLO PDF • Mod LO PDFs, as series available now • CT10, CT10W soon

  31. EXTRA SLIDES

  32. Now consider W + 3 jets Consider a scale of mW for W + 1,2,3 jets. We see the K-factors for W + 1,2 jets in the table below, and recently the NLO corrections for W + 3 jets have been calculated, allowing us to estimate the K-factors for that process. Is the K-factor (at mW) at the LHC surprising?

  33. Is the K-factor (at mW) at the LHC surprising? The K-factors for W + jets (pT>30 GeV/c) fall near a straight line, as do the K-factors for the Tevatron. By definition, the K-factors for Higgs + jets fall on a straight line. Nothing special about mW; just a typical choice. The only way to know a cross section to NLO, say for W + 4 jets or Higgs + 3 jets, is to calculate it, but in lieu of the calculations, especially for observables that we have deemed important at Les Houches, can we make some rules of thumb? Related to this is: - understanding the reduced scale dependences/pdf uncertainties for cross section ratios we have been discussing -scale choices at LO for cross sections uncalculated at NLO

  34. Is the K-factor (at mW) at the LHC surprising? The K-factors for W + jets (pT>30 GeV/c) fall near a straight line, as do the K-factors for the Tevatron. By definition, the K-factors for Higgs + jets fall on a straight line. Nothing special about mW; just a typical choice. The only way to know a cross section to NLO, say for W + 4 jets or Higgs + 3 jets, is to calculate it, but in lieu of the calculations, especially for observables that we have deemed important at Les Houches, can we make rules of thumb? Related to this is: - understanding the reduced scale dependences/pdf uncertainties for the cross section ratios we have been discussing -scale choices at LO for cross sections calculated at NLO -scale choices at LO for cross sections uncalculated at NLO Will it be smaller still for W + 4 jets?

  35. Jet algorithms at LO/NLO • Remember at LO, 1 parton = 1 jet • By choosing a jet algorithm with size parameter D, we are requiring any two partons to be > D apart • The matrix elements have 1/DR poles, so larger D means smaller cross sections • it’s because of the poles that we have to make a DR cut • At NLO, there can be two (or more) partons in a jet and jets for the first time can have some structure • we don’t need a DR cut, since the virtual corrections cancel the collinear singularity from the gluon emission • but there are residual logs that can become important if D is too small • Increasing the size parameter D increases the phase space for including an extra gluon in the jet, and thus increases the cross section at NLO (in most cases) d z=pT2/pT1 For D=Rcone, Region I = kT jets, Region II (nominally) = cone jets; I say nominally because in data not all of Region II is included for cone jets not true for WbB, for example

  36. Is the K-factor (at mW) at the LHC surprising? The problem is not the NLO cross section; that is well-behaved. The problem is that the LO cross section sits ‘too-high’. The reason (one of them) for this is that we are ‘too-close’ to the collinear pole (R=0.4) leading to an enhancement of the LO cross section (double- enhancement if the gluon is soft (~20 GeV/c)). Note that at LO, the cross section increases with decreasing R; at NLO it decreases. The collinear dependence gets stronger as njet increases. The K-factors for W + 3 jets would be more normal (>1) if a larger cone size and/or a larger jet pT cutoff were used. But that’s a LO problem; the best approach is to use the appropriate jet sizes/jet pT’s for the analysis and understand the best scales to use at LO (matrix element + parton shower) to approximate the NLO calculation (as well as comparing directly to the NLO calculation). cone jet of 0.4 For 3 jets, the LO collinear singularity effects are even more pronounced. blue=NLO; red=LO x NLO pTjet 20 GeV pTjet =20 GeV LO 30 GeV =30 GeV =40 GeV 40 GeV x NB: here I have used CTEQ6.6 for both LO and NLO; CTEQ6L1 would shift LO curves up

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