Abstract
We explore connections between two common methods for quantifying the uncertainty in parton distribution functions (PDFs), based on the Hessian error matrix and Monte-Carlo sampling. CT14 parton distributions in the Hessian representation are converted into Monte-Carlo replicas by a numerical method that reproduces important properties of CT14 Hessian PDFs: the asymmetry of CT14 uncertainties and positivity of individual parton distributions. The ensembles of CT14 Monte-Carlo replicas constructed this way at NNLO and NLO are suitable for various collider applications, such as cross section reweighting. Master formulas for computation of asymmetric standard deviations in the Monte-Carlo representation are derived. A correction is proposed to address a bias in asymmetric uncertainties introduced by the Taylor series approximation. A numerical program is made available for conversion of Hessian PDFs into Monte-Carlo replicas according to normal, log-normal, and Watt-Thorne sampling procedures.
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S. Dulat et al., New parton distribution functions from a global analysis of quantum chromodynamics, Phys. Rev. D 93 (2016) 033006 [arXiv:1506.07443] [INSPIRE].
L.A. Harland-Lang, A.D. Martin, P. Motylinski and R.S. Thorne, Parton distributions in the LHC era: MMHT 2014 PDFs, Eur. Phys. J. C 75 (2015) 204 [arXiv:1412.3989] [INSPIRE].
NNPDF collaboration, R.D. Ball et al., Parton distributions for the LHC Run II, JHEP 04 (2015) 040 [arXiv:1410.8849] [INSPIRE].
ZEUS, H1 collaboration, H. Abramowicz et al., Combination of measurements of inclusive deep inelastic e ± p scattering cross sections and QCD analysis of HERA data, Eur. Phys. J. C 75 (2015) 580 [arXiv:1506.06042] [INSPIRE].
S. Alekhin, J. Blumlein and S. Moch, The ABM parton distributions tuned to LHC data, Phys. Rev. D 89 (2014) 054028 [arXiv:1310.3059] [INSPIRE].
A. Accardi, L.T. Brady, W. Melnitchouk, J.F. Owens and N. Sato, Constraints on large-x parton distributions from new weak boson production and deep-inelastic scattering data, Phys. Rev. D 93 (2016) 114017 [arXiv:1602.03154] [INSPIRE].
J. Pumplin et al., Uncertainties of predictions from parton distribution functions. 2. The Hessian method, Phys. Rev. D 65 (2001) 014013 [hep-ph/0101032] [INSPIRE].
W.T. Giele and S. Keller, Implications of hadron collider observables on parton distribution function uncertainties, Phys. Rev. D 58 (1998) 094023 [hep-ph/9803393] [INSPIRE].
W.T. Giele, S.A. Keller and D.A. Kosower, Parton distribution function uncertainties, hep-ph/0104052 [INSPIRE].
D. Stump et al., Uncertainties of predictions from parton distribution functions. 1. The Lagrange multiplier method, Phys. Rev. D 65 (2001) 014012 [hep-ph/0101051] [INSPIRE].
A.M. Cooper-Sarkar, Uncertainties on parton distribution functions from the ZEUS NLO QCD fit to data on deep inelastic scattering, J. Phys. G 28 (2002) 2669 [hep-ph/0205153] [INSPIRE].
J. Pumplin, D.R. Stump, J. Huston, H.L. Lai, P.M. Nadolsky and W.K. Tung, New generation of parton distributions with uncertainties from global QCD analysis, JHEP 07 (2002) 012 [hep-ph/0201195] [INSPIRE].
P.M. Nadolsky and Z. Sullivan, PDF uncertainties in WH production at Tevatron, eConf C 010630 (2001) P510 [hep-ph/0110378] [INSPIRE].
S. Forte, L. Garrido, J.I. Latorre and A. Piccione, Neural network parametrization of deep inelastic structure functions, JHEP 05 (2002) 062 [hep-ph/0204232] [INSPIRE].
NNPDF collaboration, L. Del Debbio et al., Neural network determination of parton distributions: the nonsinglet case, JHEP 03 (2007) 039 [hep-ph/0701127] [INSPIRE].
NNPDF collaboration, R.D. Ball et al., A determination of parton distributions with faithful uncertainty estimation, Nucl. Phys. B 809 (2009) 1 [Erratum ibid. B 816 (2009) 293] [arXiv:0808.1231] [INSPIRE].
R.D. Ball et al., Parton distributions with LHC data, Nucl. Phys. B 867 (2013) 244 [arXiv:1207.1303] [INSPIRE].
G. Watt and R.S. Thorne, Study of Monte Carlo approach to experimental uncertainty propagation with MSTW 2008 PDFs, JHEP 08 (2012) 052 [arXiv:1205.4024] [INSPIRE].
J. Gao and P. Nadolsky, A meta-analysis of parton distribution functions, JHEP 07 (2014) 035 [arXiv:1401.0013] [INSPIRE].
S. Carrazza, S. Forte, Z. Kassabov, J.I. Latorre and J. Rojo, An unbiased Hessian representation for Monte Carlo PDFs, Eur. Phys. J. C 75 (2015) 369 [arXiv:1505.06736] [INSPIRE].
S. Carrazza, S. Forte, Z. Kassabov and J. Rojo, Specialized minimal PDFs for optimized LHC calculations, Eur. Phys. J. C 76 (2016) 205 [arXiv:1602.00005] [INSPIRE].
NNPDF collaboration, R.D. Ball et al., Reweighting NNPDFs: the W lepton asymmetry, Nucl. Phys. B 849 (2011) 112 [Erratum ibid. B 854 (2012) 926] [arXiv:1012.0836] [INSPIRE].
R.D. Ball et al., Reweighting and unweighting of parton distributions and the LHC W lepton asymmetry data, Nucl. Phys. B 855 (2012) 608 [arXiv:1108.1758] [INSPIRE].
S. Carrazza, J.I. Latorre, J. Rojo and G. Watt, A compression algorithm for the combination of PDF sets, Eur. Phys. J. C 75 (2015) 474 [arXiv:1504.06469] [INSPIRE].
J. Butterworth et al., PDF4LHC recommendations for LHC Run II, J. Phys. G 43 (2016) 023001 [arXiv:1510.03865] [INSPIRE].
H.-L. Lai et al., New parton distributions for collider physics, Phys. Rev. D 82 (2010) 074024 [arXiv:1007.2241] [INSPIRE].
J. Gao et al., CT10 next-to-next-to-leading order global analysis of QCD, Phys. Rev. D 89 (2014) 033009 [arXiv:1302.6246] [INSPIRE].
A. Belyaev, J. Pumplin, W.-K. Tung and C.P. Yuan, Uncertainties of the inclusive Higgs production cross section at the Tevatron and the LHC, JHEP 01 (2006) 069 [hep-ph/0508222] [INSPIRE].
S. Dulat et al., Higgs boson cross section from CTEQ-TEA global analysis, Phys. Rev. D 89 (2014) 113002 [arXiv:1310.7601] [INSPIRE].
C. Schmidt et al., Uncertainties on Higgs and ttbar predictions at the LHC from CTEQ-TEA Global Analysis, PoS(DIS2014)146.
J. Pumplin, J. Huston, H.L. Lai, P.M. Nadolsky, W.-K. Tung and C.P. Yuan, Collider inclusive jet data and the gluon distribution, Phys. Rev. D 80 (2009) 014019 [arXiv:0904.2424] [INSPIRE].
J.F. Kenney and E.S. Keeping, Mathematics of statistics, 3rd edition, Van Nostrand, New York U.S.A. (1954), see chapters 4.7-4.9.
http://http://hep.pa.msu.edu/cteq/public/ct14/MC/.
J.M. Campbell, J.W. Huston and W.J. Stirling, Hard interactions of quarks and gluons: a primer for LHC physics, Rept. Prog. Phys. 70 (2007) 89 [hep-ph/0611148] [INSPIRE].
T. Carli et al., A posteriori inclusion of parton density functions in NLO QCD final-state calculations at hadron colliders: the APPLGRID project, Eur. Phys. J. C 66 (2010) 503 [arXiv:0911.2985] [INSPIRE].
J. Alwall et al., The automated computation of tree-level and next-to-leading order differential cross sections and their matching to parton shower simulations, JHEP 07 (2014) 079 [arXiv:1405.0301] [INSPIRE].
V. Bertone, R. Frederix, S. Frixione, J. Rojo and M. Sutton, aMCfast: automation of fast NLO computations for PDF fits, JHEP 08 (2014) 166 [arXiv:1406.7693] [INSPIRE].
J.R. Andersen et al., Les Houches 2015: Physics at TeV colliders standard model working group report, arXiv:1605.04692 [INSPIRE].
ATLAS collaboration, Measurement of inclusive jet and dijet production in pp collisions at \( \sqrt{s}=7 \) TeV using the ATLAS detector, Phys. Rev. D 86 (2012) 014022 [arXiv:1112.6297] [INSPIRE].
Z. Nagy, Three jet cross-sections in hadron hadron collisions at next-to-leading order, Phys. Rev. Lett. 88 (2002) 122003 [hep-ph/0110315] [INSPIRE].
Z. Nagy, Next-to-leading order calculation of three jet observables in hadron hadron collision, Phys. Rev. D 68 (2003) 094002 [hep-ph/0307268] [INSPIRE].
CMS collaboration, Measurement of the electron charge asymmetry in inclusive W production in pp collisions at \( \sqrt{s}=7 \) TeV, Phys. Rev. Lett. 109 (2012) 111806 [arXiv:1206.2598] [INSPIRE].
CMS collaboration, Measurement of the muon charge asymmetry in inclusive pp → W + X production at \( \sqrt{s}=7 \) TeV and an improved determination of light parton distribution functions, Phys. Rev. D 90 (2014) 032004 [arXiv:1312.6283] [INSPIRE].
S. Alekhin et al., HERAFitter, Eur. Phys. J. C 75 (2015) 304 [arXiv:1410.4412] [INSPIRE].
ZEUS, H1 collaboration, F.D. Aaron et al., Combined measurement and QCD analysis of the inclusive e ± p scattering cross sections at HERA, JHEP 01 (2010) 109 [arXiv:0911.0884] [INSPIRE].
R. Thorne, Summary of structure functions and PDFs working group, in the summary talk of the XXIV Workshop on Deep-Inelastic Scattering (DIS) and Related Subjects (DIS’2016), Aprile 11–15, Hamburg, Germany (2016).
NNPDF collaboration, R.D. Ball et al., A determination of the charm content of the proton, Eur. Phys. J. C 76 (2016) 647 [arXiv:1605.06515] [INSPIRE].
J. Pumplin, Experimental consistency in parton distribution fitting, Phys. Rev. D 81 (2010) 074010 [arXiv:0909.0268] [INSPIRE].
http://http://hep.pa.msu.edu/cteq/public/.
S. Dulat et al., Intrinsic charm parton distribution functions from CTEQ-TEA global analysis, Phys. Rev. D 89 (2014) 073004 [arXiv:1309.0025] [INSPIRE].
T. Lewis, A simple improved-accuracy normal approximation for χ 2, Austral. J. Statist. 30A (1988) 160.
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Hou, TJ., Gao, J., Huston, J. et al. Reconstruction of Monte Carlo replicas from Hessian parton distributions. J. High Energ. Phys. 2017, 99 (2017). https://doi.org/10.1007/JHEP03(2017)099
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DOI: https://doi.org/10.1007/JHEP03(2017)099