New parton distribution functions from a global analysis of quantum chromodynamics
Sayipjamal Dulat, Tie Jiun Hou, Jun Gao, Marco Guzzi, Joey Huston, Pavel Nadolsky, Jon Pumplin, Carl Schmidt, Daniel Stump, C. P. Yuan
aa r X i v : . [ h e p - ph ] M a r New parton distribution functions from a global analysis of quantum chromodynamics
Sayipjamal Dulat,
1, 2, ∗ Tie-Jiun Hou, † Jun Gao, ‡ Marco Guzzi, § Joey Huston, ¶ PavelNadolsky, ∗∗ Jon Pumplin, †† Carl Schmidt, ‡‡ Daniel Stump, §§ and C.–P. Yuan ¶¶ School of Physics Science and Technology, Xinjiang University,Urumqi, Xinjiang 830046 China Department of Physics and Astronomy, Michigan State University,East Lansing, MI 48824 U.S.A. Department of Physics, Southern Methodist University,Dallas, TX 75275-0181, U.S.A. High Energy Physics Division, Argonne National Laboratory,Argonne, Illinois 60439, U.S.A. School of Physics & Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom
We present new parton distribution functions (PDFs) at next-to-next-to-leading order (NNLO)from the CTEQ-TEA global analysis of quantum chromodynamics. These differ from previous CTPDFs in several respects, including the use of data from LHC experiments, and the new DØ chargedlepton rapidity asymmetry data, as well as the use of a more flexible parametrization of PDFs that, inparticular, allows a better fit to different combinations of quark flavors. Predictions for importantLHC processes, especially Higgs boson production at 13 TeV, are presented. These CT14 PDFsinclude a central set and error sets in the Hessian representation. For completeness, we also presentthe CT14 PDFs determined at the leading order (LO) and the next-to-leading order (NLO) in QCD.Besides these general-purpose PDF sets, we provide a series of (N)NLO sets with various α s valuesand additional sets in general-mass variable flavor number (GM-VFN) schemes, to deal with heavypartons, with up to 3, 4, and 6 active flavors. PACS numbers: 12.15.Ji, 12.38 Cy, 13.85.QkKeywords: parton distribution functions; large hadron collider; Higgs boson ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected] ∗∗ Electronic address: [email protected] †† Electronic address: [email protected] ‡‡ Electronic address: [email protected] §§ Electronic address: [email protected] ¶¶ Electronic address: [email protected]
Contents
I. Introduction II. Setup of the analysis
III. Overview of CT14 PDFs as functions of x and Q IV. Comparisons with hadronic experiments
W/Z boson production 252. Influence of W boson charge asymmetry measurements at the LHC 273. Production of Drell-Yan pairs at ATLAS 30D. W ± charge asymmetry from the DØ experiment at the Tevatron 31E. Constraints on strangeness PDF from CCFR, NuTeV, and LHC experiments 32F. The CMS W + c production measurement 36 V. Impact on Higgs boson and t ¯ t cross sections at the LHC t ¯ t production cross section at the LHC 43 VI. Discussion and Conclusion Acknowledgments Appendix: Parametrizations in CT14 References I. INTRODUCTION
Run-1 at the Large Hadron Collider (LHC) was a great success, culminating in the discovery of the Higgs boson [1, 2].No physics beyond the standard model was discovered in this run, however Run-2, with a larger center-of-mass energyand integrated luminosity, will allow for an increased discovery potential for new physics. Precision measurementsof the Higgs boson and of various electroweak observables will be performed with extraordinary accuracy in newkinematic regimes in Run 2. Run-1 achievements, such as the combined ATLAS/CMS measurement of the Higgsboson mass with 0.2% accuracy [3], will soon be superseded. For both precision measurements and for discovery ofpossible new physics, it is important to have the proper tools for the calculation of the relevant cross sections. Thesetools include both matrix element determinations at higher orders in perturbative QCD and electroweak theory,and precision parton distribution functions (PDFs). The need for precision PDFs was driven home by the recentcalculation of the inclusive cross section for gluon-gluon fusion to a Higgs boson at NNNLO [4]. As this tour-de-forcecalculation has significantly reduced the scale dependence of the Higgs cross section, the PDF and α s uncertaintiesbecome the dominant remaining theoretical uncertainty (as of the last PDF4LHC recommendation).The CT10 parton distribution functions were published at next-to-leading order (NLO) in 2010 [5], followed bythe CT10 next-to-next-to leading order (NNLO) parton distribution functions in 2013 [6]. These PDF ensembleswere determined using diverse experimental data from fixed-target experiments, HERA and the Tevatron collider,but without data from the LHC. In this paper, we present a next generation of PDFs, designated as CT14. TheCT14 PDFs include data from the LHC for the first time, as well as updated data from the Tevatron and fromHERA experiments. Various CT14 PDF sets have been produced at the leading order (LO), NLO and NNLO andare available from LHAPDF [7].The CTEQ-TEA philosophy has always been to determine PDFs from data on inclusive, high-momentum transferprocesses, for which perturbative QCD is expected to be reliable. For example, in the case of deep inelastic leptonscattering, we only use data with Q >
W > . χ scheme [9–11] and is essential for obtaining correct predictions for LHC electroweak cross sections [12, 13].We make two exceptions to this rule, by including measurements for charged-current DIS and inclusive jet productionat NLO only. In both cases, the complete NNLO contributions are not yet available, but it can be argued based onour studies that the expected effect of missing NNLO effects is small relatively to current experimental errors (cf.Sec. II). For both types of processes, the NLO predictions have undergone various benchmarking tests. A numericalerror was discovered and corrected in the implementation of the SACOT- χ scheme for charged-current DIS, resultingin relatively small changes from CT10 (within the PDF uncertainties).As in the CT10 global analysis, we use a charm pole mass of 1.3 GeV, which was shown to be consistent with theCT10 data in Ref. [6]. The PDFs for u , d , s (anti-)quarks and the gluon are parametrized at an initial scale of 1.295GeV, and the charm quark PDF is turned on with zero intrinsic charm as the scale Q reaches the charm pole mass.The new LHC measurements of W/Z cross sections directly probe the flavor separation of u and d (anti-)quarks inan x -range around 0 .
01 that was not directly assessed by the previously available experiments. We also include anupdated measurement of electron charge asymmetry from the DØ collaboration [14], which probes the d quark PDFat x > .
1. To better estimate variations in relevant PDF combinations, such as d ( x, Q ) /u ( x, Q ) and ¯ d ( x, Q ) / ¯ u ( x, Q ),we increased the number of free PDF parameters to 28, compared to 25 in CT10 NNLO. As another importantmodification, CT14 employs a novel flexible parametrization for the PDFs, based on the use of Bernstein polynomials(reviewed in the Appendix). The shape of the Bernstein polynomials is such that a single polynomial is dominant ineach given x range, reducing undesirable correlations among the PDF parameters that sometimes occurred in CT10.In the asymptotic limits of x → x →
1, the new parametrization forms allow for the possibility of arbitraryconstant ratios of d/u or ¯ d/ ¯ u , in contrast to the more constrained behavior assumed in CT10.The PDF error sets of the CT14 ensemble are obtained using two techniques, the Hessian method [15] and Monte-Carlo sampling [16]. Lagrange multiplier studies [17] have also been used to verify the Hessian uncertainties, especiallyin regions not well constrained by data. This applies at NNLO and NLO; no error sets are provided at LO due to thedifficulty in defining meaningful uncertainties at that order.A central value of α s ( M Z ) of 0.118 has been assumed in the global fits at NLO and NNLO, but PDF sets atalternative values of α s ( m Z ) are also provided. CT14 prefers α s ( M Z ) = 0 . +0 . − . at NNLO (0 . ± .
005 at NLO)at 90 % confidence level (C.L.). These uncertainties from the global QCD fits are larger than those of the data fromLEP and other experiments included into the world average [19]. Thus, the central PDF sets are obtained using thevalue of 0 . α s ( M Z ) value of 0.130, and the other with a 2-loop α s ( M Z ) value of 0.118.The flavor composition of CT14 PDFs has changed somewhat compared to CT10 due to the inclusion of newLHC and Tevatron data sets, to the use of modified parametrization forms, and to the numerical modificationsdiscussed above. The new PDFs are largely compatible with CT10 within the estimated PDF uncertainty. The CT14NNLO PDFs have a softer strange quark distribution at low x and a somewhat softer gluon at high x , comparedto CT10 NNLO. The d/u ratio has decreased at high x in comparison to CT10, as a consequence of replacing the2008 DØ electron charge asymmetry (0 .
75 fb − [22]) measurement by the new 9 . − data set [14]. The d/u ratioapproaches a constant value in the x → d val and u val behaveas (1 − x ) a at x → a (reflecting expectations from spectator counting rules), but allowingfor independent normalizations. The ¯ d/ ¯ u ratio has also changed as a consequence of the new data and the newparametrization form.The organization of the paper is as follows. In Sec. II, we list the data sets used in the CT14 fit and discuss furtheraspects of the global fits for the central CT14 PDFs and for the error sets. In Sec. III, we show various aspects of theresultant CT14 PDFs and make comparisons to CT10 PDFs. In Sec. IV, we show comparisons of NNLO predictionsusing the CT14 PDFs to some of the data sets used in the global fits. Specifically, we compare to experimentalmeasurements of jet, W and Z , W + c cross sections. In Sec. V, we discuss NNLO predictions using the CT14 PDFsfor Higgs boson production via the gluon-gluon fusion channel and for top quark and anti-quark pair production. Ourconclusion is given in Sec. VI. II. SETUP OF THE ANALYSISA. Overview of the global fit
The goal of the CT14 global analysis is to provide a new generation of PDFs intended for widespread use in high-energy experiments. As we generate new PDF sets, we include newly available experimental data sets and theoreticalcalculations, and redesign the functional forms of PDFs if new data or new theoretical calculations favor it. Allchanges — data, theory, and parametrization — contribute to the differences between the old and new generations ofPDFs in ways that are correlated and frequently cannot be separated. The most important, but not the only, criterionfor the selection of PDFs is the minimization of the log-likelihood χ that quantifies agreement of theory and data.In addition, we make some ”prior assumptions” about the forms of the PDFs. A PDF set that violates them may berejected even if it lowers χ . For example, we assume that the PDFs are smoothly varying functions of x , withoutabrupt variations or short-wavelength oscillations. This is consistent with the experimental data and sufficient formaking new predictions. No PDF can be negative at the input scale Q , to preclude negative cross sections in thepredictions. Flavor-dependent ratios or cross section asymmetries must also take physical values, which limits therange of allowed parametrizations in extreme kinematical regions with poor experimental constraints. For example,in the CT14 parametrization we restricted the functional forms of the u and d PDFs so that d ( x, Q ) /u ( x, Q ) wouldremain finite and nonzero at x →
1, cf. the Appendix. We now review every input of the CT14 PDF analysis in turn,starting with the selection of the new experiments.
B. Selection of experiments
The experimental data sets that are included in the CT14 global analysis are listed in Tables I (lepton scattering) andII (production of inclusive lepton pairs and jets). There are a total of 2947 data points included from 33 experiments,producing χ value of 3252 for the best fit (with χ /N pt = 1 . χ in Tables I andII that the data and theory are in reasonable agreement for most experiments. The variable S n in the last columnis an “effective Gaussian variable”, first introduced in the Sec. 5 of Ref. [5] and defined for the current analysis inRefs. [6, 23]. The effective Gaussian variable quantifies compatibility of any given data set with a particular PDFfit in a way that is independent of the number of points N pt,n in the data set. It maps the χ n values of individualexperiments, whose probability distributions depend on N pt,n in each experiment (and thus, are not identical), onto S n values that obey a cumulative probability distribution shared by all experiments, independently of N pt,n . Valuesof S n between -1 and +1 correspond to a good fit to the n -th experiment (at the 68% C.L.). Large positive values( &
2) correspond to a poor fit, while large negative values ( . −
2) are fit unusually well.The goodness-of-fit for CT14 NNLO is comparable to that of our earlier PDFs, but the more flexible parametriza-tions did result in improved agreement with some data sets. For example, by adding additional parameters to the { u, u } and (cid:8) d, d (cid:9) parton distributions, somewhat better agreement was obtained for the BCDMS and NMC data atlow values of Q . The quality of the fit can be also evaluated based on the distribution of S n values, which follows astandard normal distribution (of width 1) in an ideal fit. As in the previous fits, the actual S n distribution (cf. thesolid curve in Fig. 1) is somewhat wider than the standard normal one (the dashed curve), indicating the presence ID N pt,n χ n χ n /N pt,n S n
101 BCDMS F p [24] 337 384 1.14 1.74102 BCDMS F d [25] 250 294 1.18 1.89104 NMC F d /F p [26] 123 133 1.08 0.68106 NMC σ pred [26] 201 372 1.85 6.89108 CDHSW F p [27] 85 72 0.85 -0.99109 CDHSW F p [27] 96 80 0.83 -1.18110 CCFR F p [28] 69 70 1.02 0.15111 CCFR xF p [29] 86 31 0.36 -5.73124 NuTeV νµµ SIDIS [30] 38 24 0.62 -1.83125 NuTeV ¯ νµµ
SIDIS [30] 33 39 1.18 0.78126 CCFR νµµ
SIDIS [31] 40 29 0.72 -1.32127 CCFR ¯ νµµ
SIDIS [31] 38 20 0.53 -2.46145 H1 σ br [32] 10 6.8 0.68 -0.67147 Combined HERA charm production [33] 47 59 1.26 1.22159 HERA1 Combined NC and CC DIS [34] 579 591 1.02 0.37169 H1 F L [35] 9 17 1.92 1.7TABLE I: Experimental data sets employed in the CT14 analysis. These are the lepton deep-inelastic scattering experiments. N pt,n , χ n are the number of points and the value of χ for the n -th experiment at the global minimum. S n is the effectiveGaussian parameter [5, 6, 23] quantifying agreement with each experiment. of disagreements, or tensions, between some of the included experiments. The tensions have been examined before[5, 53–55] and originate largely from experimental issues, almost independent of the perturbative QCD order or PDFparametrization form. A more detailed discussion of the level of agreement between data and theory will be providedin Sec. IV. Mean = = - - - S n N u m be r o f e x pe r i m en t s FIG. 1: Best-fit S n values of 33 experiments in the CT14 analysis. ID N pt,n χ n χ n /N pt,n S n
201 E605 Drell-Yan process [37] 119 116 0.98 -0.15203 E866 Drell-Yan process, σ pd / (2 σ pp ) [38] 15 13 0.87 -0.25204 E866 Drell-Yan process, Q d σ pp / ( dQdx F ) [39] 184 252 1.37 3.19225 CDF Run-1 electron A ch , p Tℓ >
25 GeV [40] 11 8.9 0.81 -0.32227 CDF Run-2 electron A ch , p Tℓ >
25 GeV [41] 11 14 1.24 0.67234 DØ Run-2 muon A ch , p Tℓ >
20 GeV [42] 9 8.3 0.92 -0.02240 LHCb 7 TeV 35 pb − W/Z dσ/dy ℓ [43] 14 9.9 0.71 -0.73241 LHCb 7 TeV 35 pb − A ch , p Tℓ >
20 GeV [43] 5 5.3 1.06 0.30260 DØ Run-2 Z rapidity [44] 28 17 0.59 -1.71261 CDF Run-2 Z rapidity [45] 29 48 1.64 2.13266 CMS 7 TeV 4 . − , muon A ch , p Tℓ >
35 GeV [46] 11 12.1 1.10 0.37267 CMS 7 TeV 840 pb − , electron A ch , p Tℓ >
35 GeV [47] 11 10.1 0.92 -0.06268 ATLAS 7 TeV 35 pb − W/Z cross sec., A ch [48] 41 51 1.25 1.11281 DØ Run-2 9 . − electron A ch , p Tℓ >
25 GeV [14] 13 35 2.67 3.11504 CDF Run-2 inclusive jet production [49] 72 105 1.45 2.45514 DØ Run-2 inclusive jet production [50] 110 120 1.09 0.67535 ATLAS 7 TeV 35 pb − incl. jet production [51] 90 50 0.55 -3.59538 CMS 7 TeV 5 fb − incl. jet production [52] 133 177 1.33 2.51TABLE II: Same as Table I, showing experimental data sets on Drell-Yan processes and inclusive jet production.
1. Experimental data from the LHC
Much of these data have also been used in previous CT analyses, such as the one that produced the CT10 NNLOPDFs. As mentioned, no LHC data were used in the CT10 fits. Nonetheless, the CT10 PDFs have been in goodagreement with LHC measurements so far.As the quantity of the LHC data has increased, the time has come to include the most germane LHC measurementsinto CT fits. The LHC has measured a variety of standard model cross sections, yet not all of them are suitable fordetermination of PDFs according to the CT method. For that, we need to select measurements that are experimentallyand theoretically clean and are compatible with the global set of non-LHC hadronic experiments.In the CT14 study, we select a few such LHC data sets at √ s = 7 TeV, focusing on the measurements that providenovel information to complement the non-LHC data. From vector boson production processes, we selected W/Z crosssections and the charged lepton asymmetry measurement from ATLAS [48], the charged lepton asymmetry in theelectron [47] and muon decay channels [46] from CMS, and the
W/Z lepton rapidity distributions and charged leptonasymmetry from LHCb [43]. The ATLAS and CMS measurements primarily impose constraints on the light quarkand antiquark PDFs at x & .
01. The LHCb data sets, while statistically limited, impose minor constraints on ¯ u and d PDFs at x = 0 . − . u ,¯ u , d , and ¯ d . In the absence of relevant experimental constraints in the pre-CT14 fits, the PDF parametrizations werechosen so as to enforce ¯ u/ ¯ d → u/d → x → x a is required to be thesame in all light-quark PDFs in the x → u v , d v , and ¯ u/ ¯ d at x →
0, with the spread constrained by the newly included LHC data.From the other LHC measurements, we now include single-inclusive jet production at ATLAS [51] and CMS [52].These data sets provide complementary information to Tevatron inclusive jet production cross sections from CDFRun-2 [49] and DØ Run-2 [50] that are also included. The purpose of jet production cross sections is primarily toconstrain the gluon PDF g ( x, Q ). While the uncertainties from the LHC jet cross sections are still quite large, theyprobe the gluon PDF across a much wider range of x than the Tevatron jet cross sections.One way to gauge the sensitivity of a specific data point to some PDF f ( x, Q ) at a given x and Q is to computea correlation cosine between the theoretical prediction for this point and f ( x, Q ) [13, 15, 56]. In the case of CT10NNLO, the sensitivity of the LHC charge asymmetry data sets to the valence PDF combinations at x = 0 . − . x that had been observed suggested the possibility that CT10 light-quark parametrizations were not sufficiently flexiblein the x region probed by the LHC charge asymmetry.Since CT14 has adopted more flexible parametrizations for the affected quark flavors, the above correlations with u v , d v , and d/u at small x are now somewhat relaxed, as illustrated by the newly computed correlations betweenCT14 NNLO and CMS A ch data in Fig. 2. Each line shows cos φ between f ( x, Q ) and the NNLO prediction for oneof the bins of the data. When the PDF uncertainty receives a large contribution from f ( x, Q ), cos φ comes out to beclose to ±
1, say, | cos φ | > .
7. With the new parametrization form, the CMS charge asymmetry is reasonably, butnot exceptionally, correlated with both ¯ d/ ¯ u and d/u at x ∼ .
01 corresponding to central-rapidity production of weakbosons at √ s = 7 TeV (indicated by a vertical dashed line in the figure). The correlation with u v and d v is smallerthan in CT10.For the ATLAS [51], CMS [52], CDF [49] and DØ [50] inclusive jet data sets, the correlation cosine, cos φ , for gluonPDF is plotted in Fig. 3 using NLO QCD theory to evaluate the theoretical cross section. Again, the lines correspondto individual p T j bins of the data. We observe that the CDF and DØ jet cross sections are highly correlated withthe gluon PDF g ( x, Q ) at x & .
05, and anticorrelated at small x as a consequence of the momentum sum rule. TheATLAS and CMS jet cross sections are highly correlated with g ( x, Q ) in a much wider range, x > . u ( x, Q ) in the Fig. 4, is at mostmoderate. The ATLAS and CMS jet data therefore have the potential to reduce the gluon uncertainty, but significantreduction will require the data from Run 2.
2. High-luminosity lepton charge asymmetry from the Tevatron
Forward-backward asymmetry ( A ch ) distributions of charged leptons from inclusive weak boson production at theTevatron are uniquely sensitive to the average slope of the ratio d ( x, Q ) /u ( x, Q ) at large x , of order 0.1 and above.In the CT14 analysis, we include several data sets of A ch measured at √ s = 1.8 and 1.96 TeV by the CDF andDØ Collaborations. The CDF Run-1 data set on A ch [40, 60], which was instrumental in resolving conflictinginformation on the large- x behavior of u ( x, Q ) and d ( x, Q ) from contemporary fixed-target DIS experiments [61–64],is supplemented by the CDF Run-2 data set at 170 pb − [41]. A ch data at √ s = 1 .
96 TeV from DØ in the electron u v (x,Q)Correlation between CMS W asym. and CT14 PDFs at 100 GeV-1-0.5 0 0.5 1 c o s φ – d(x,Q)/ – u(x,Q)-1-0.5 0 0.5 1 x d(x,Q)/u(x,Q)-1-0.5 0 0.5 1 -4 -3 -2 -1 FIG. 2: The correlation cosine cos φ [13] between the PDF f ( x, Q = 100 GeV) at the specified x value on the horizontal axisand NNLO predictions for muon CMS charge asymmetry [46]. c o s φ xCT14 NNLO g(x,100 GeV) CDF inclusive jet R=0.7, |y| < 2.1 -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 10 -4 -3 -2 -1 c o s φ xCT14 NNLO g(x,100 GeV) D0 inclusive jet R=0.7, |y| < 2.4 -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 10 -4 -3 -2 -1 c o s φ xCT14 NNLO g(x,100 GeV) ATLAS single inc. jet R=0.6, |y| < 4.4 -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 10 -4 -3 -2 -1 c o s φ xCT14 NNLO g(x,100 GeV) CMS single inc. jet R=0.7, |y| < 2.5 -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 10 -4 -3 -2 -1 FIG. 3: The correlation cosine cos φ [13] between the g-PDF at the specified x value on the horizontal axis and NLO predictionsfor the CDF [49] (upper left panel), DØ [50] (upper right panel), ATLAS [51] (lower left panel) and CMS [52] (lower rightpanel) inclusive jet cross sections at Q = 100 GeV. c o s φ xCT14 NNLO u(x,100 GeV) CDF inclusive jet R=0.7, |y| < 2.1 -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 10 -4 -3 -2 -1 c o s φ xCT14 NNLO u(x,100 GeV) D0 inclusive jet R=0.7, |y| < 2.4 -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 10 -4 -3 -2 -1 c o s φ xCT14 NNLO u(x,100 GeV) ATLAS single inc. jet R=0.6, |y| < 4.4 -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 10 -4 -3 -2 -1 c o s φ xCT14 NNLO u(x,100 GeV) CMS single inc. jet R=0.7, |y| < 2.5 -1-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1 10 -4 -3 -2 -1 FIG. 4: The correlation cosine cos φ [13] between the u -PDF at the specified x value on the horizontal axis and NLO predictionsfor the CDF [49] (upper left panel), DØ [50] (upper right panel), ATLAS [51] (lower left panel) and CMS [52] (lower rightpanel) inclusive jet cross sections at Q = 100 GeV. . − and 0 . − , are also included. In all A ch data sets, we includesubsamples with the cuts on the transverse momentum p T ℓ of the final-state lepton specified in Table II.The electron data set (9 . − ) from DØ that we now include replaces the 0 .
75 fb − counterpart set [22], firstincluded in CT10. This replacement has an important impact on the determination of the large- x quark PDFs; thus,these new A ch data sets are perhaps the most challenging and valuable among all that were added in CT14.The DØ A ch data have small experimental errors, and hence push the limits of the available theoretical calculations.Relatively small differences in the average slope (with respect to x ) of the d/u ratio in the probed region can producelarge variations in χ n for the Tevatron charge asymmetry [61–63]. By varying the minimal selection cuts on p T ℓ ofthe lepton, it is possible to probe subtle features of the large- x PDFs. For that, understanding of the transverse mo-mentum dependence in both experiment and theory is necessary, which demands evaluation of transverse momentumresummation effects.When the first Tevatron Run-2 A ch data sets were implemented in CT fits, significant tensions were discoveredbetween the electron and muon channels, and even between different p T ℓ bins within one decay channel. The tensionsprompted a detailed study in the CT10 analysis [5]. The study found that various p T ℓ bins of the electron and muonasymmetries from DØ disagree with DIS experiments and among themselves.In light of these unresolved tensions, we published a CT10 PDF ensemble at NLO, which did not include theDØ Run-2 A ch data and yielded a d/u ratio that was close to that ratio in CTEQ6.6 NLO. An alternative CT10WNLO ensemble was also constructed. It included four p T ℓ bins of that data and predicted a harder d/u behavior at x →
1. When constructing the counterpart CT10 NNLO PDFs in [6], we took an in-between path and included onlythe two most inclusive p T ℓ bins, one from the electron [22], and one from the muon [42] samples. This choice stillresulted in a larger d/u asymptotic value in CT10 NNLO than in CTEQ6.6.The new A ch data for 9 . − in the electron channel is more compatible with the other global fit in the data thatwe included. Therefore, CT14 includes the DØ A ch measurement in the muon channel with p T ℓ >
20 GeV [42] andin the electron channel with p T ℓ >
25 GeV [14]. The replacement does not affect the general behavior of the PDFs,except that the CT14 d/u ratio at high x follows the trends of CTEQ6.6 NLO and CT10 NLO, rather than of CT10WNLO and CT10 NNLO.
3. New HERA data
CT14 includes a combined HERA-1 data set of reduced cross sections for semi-inclusive DIS production of opencharm [33], and measurements of the longitudinal structure function F L ( x, Q ) in neutral-current DIS [35]. The formerreplaces independent data sets of charm structure functions and reduced cross sections from H1 and ZEUS [65–68].Using the combined HERA charm data set, we obtain a slightly smaller uncertainty on the gluon at x < .
01 and betterconstraints on charm mass than with independent sets [69]. The latter HERA data set, on F L , is not independentfrom the combined HERA set on inclusive DIS [34], but has only nine data points and does not significantly changethe global χ . Its utility is primarily to prevent unphysical solutions for the gluon PDF at small x at the stage of thePDF error analysis.3
4. Other LHC results
One class of LHC data that could potentially play a large role [13] in the determination of the gluon distribution,especially at high x , is the differential distributions of t ¯ t production, now available from ATLAS [59] and CMS [58, 70].However, these data are not included into our fit, as the differential NNLO t ¯ t cross section predictions for the LHCare not yet complete and the total cross section measurements lack statistical power. [71]. In addition, constraints onthe PDFs from t ¯ t cross sections are mutually correlated with the values of QCD coupling and top quark mass. NLOelectroweak corrections, playing an important role [72, 73] for these data, are still unavailable for some t ¯ t kinematicdistributions. Once these calculations are completed, they will be incorporated in future versions of CT PDFs. Fornow, we simply show predictions from CT14 for the t ¯ t distributions using the approximate NNLO calculations inSection V. C. Summary of theoretical calculations
1. QCD cross sections
The CT14 global analysis prioritizes the selection of published data for which NNLO predictions are available,and theoretical uncertainties of various kinds are well understood. Theoretical calculations for neutral-current DISare based on the NNLO implementation [8] of the S-ACOT- χ factorization scheme [9–11] with massive quarks. Forinclusive distributions in the low-mass Drell-Yan process, NNLO predictions are obtained with the program VRAP[75, 76]. Predictions for W/Z production and weak boson charge asymmetries with p T ℓ cuts are obtained with theNNLL-(approx. NNLO) program ResBos [77–80], as in the previous analyses.As already mentioned in the introduction, two exceptions from this general rule concern charged-current DIS andcollider jet production. Both have unique sensitivities to crucial PDF combinations, but are still known only to NLO.The CCFR and NuTeV data on inclusive and semi-inclusive charge-current DIS are indispensable for constraining thestrangeness PDF; single-inclusive jet production at the Tevatron and now at the LHC are essential for constrainingthe gluon distribution. Yet, in both categories, the experimental uncertainties are fairly large and arguably diminishthe impact of missing NNLO effects. Given the importance of these measurements, our approach is then to includethese data in our NNLO global PDF fits, but evaluate their matrix elements at NLO.According to this choice, we do not rely on the use of threshold resummmation techniques [81, 81, 82] to approximatethe NNLO corrections in jet production. Nor do we remove the LHC jet data due to the kinematic limitations of suchresummation techniques [83]. A large effort was invested in the CT10 and CT14 analyses to estimate the possibilityof biases in the NNLO PDFs due to using NLO cross sections for jet production [92, 93]. The sensitivity of the centralPDFs and their uncertainty to plausible NNLO corrections was estimated with a variation of Cacciari-Houdeau’smethod [84], by introducing additional correlated systematic errors in jet production associated with the residualdependence on QCD scales and a potential missing contribution of a typical magnitude expected from an NNLOcorrection. These exercises produced two conclusions. First, the scale variation in the NLO jet cross section isreduced if the central renormalization and factorization scales are set equal to the transverse momentum p T of theindividual jet in the data bin. This choice is adopted both for the LHC and Tevatron jet cross sections. In the4recently completed partial NNLO calculation for jets produced via gg scattering [85, 86], this scale choice leads to anNNLO/NLO K-factor that is both smaller than for the alternative scale equal to the leading jet’s p T , and is relativelyconstant over the range of the LHC jet measurements [87]. Second, the plausible effect of the residual QCD scaledependence at NLO can be estimated as a correlated uncertainty in the CT10 NNLO fit. Currently it has marginaleffect on the central PDF fits and the PDF uncertainty.The CT14 analysis computes NLO cross sections for inclusive jet production with the help of FastNLO [88] and
ApplGrid [89] interfaces to
NLOJET++ [90, 91] . A series of benchmarking exercises that we had completed [92, 93]verified that the fast interfaces are in good agreement among themselves and with an independent NLO calculationin the program MEKS [92]. Both ATLAS and CMS have measured the inclusive jet cross sections for two jet sizes.We use the larger of the two sizes (0.6 for ATLAS and 0.7 for CMS) to further reduce the importance of NNLOcorrections.
2. Figure-of-merit function
In accord with the general procedure summarized in Ref. [6], the most probable solutions for CT14 PDFs are foundby a minimization of the function χ global = N exp X n =1 χ n + χ th . (1)This function sums contributions χ n from N exp fitted experiments and includes a contribution χ th specifying theoreti-cal conditions (“Lagrange Multiplier constraints”) imposed on some PDF parameters. In turn, the χ n are constructedas in Eq. (14) of [6] and account for both uncorrelated and correlated experimental errors. Section 3 of that paperincludes a detailed review of the statistical procedure that we continue to follow. Instead of repeating that review,we shall briefly remind the reader about the usage of the tolerance and quasi-Gaussian S variables when constructingthe error PDFs.The minimum of the χ global function is found iteratively by the method of steepest descent using the program MINUIT .The boundaries of the 90% C.L. region around the minimum of χ global , and the eigenvector PDF sets quantifying theassociated uncertainty, are found by iterative diagonalization of the Hessian matrix [15, 17]. The 90% C.L. boundaryin CT14 and CT10 analyses is determined according to two tiers of criteria, based on the increase in the global χ global summed over all experiments, and on the agreement with individual experimental data sets [5, 6, 23]. The first typeof condition demands that the global χ does not increase above the best-fit value by more than ∆ χ = T , wherethe 90% C.L. region corresponds to T ≈
10. The second condition introduces a penalty term P , called Tier-2 penalty,in χ when establishing the confidence region, which quickly grows when the fit ceases to agree with any specificexperiment within the 90% C.L. for that experiment. The effective function χ ff = χ global + P is scanned along eacheigenvector direction until χ eff increases above the tolerance bound, or rapid χ eff growth due to the penalty P istriggered.The penalty term is constructed as P = N exp X n =1 ( S n ) k θ ( S n ) (2)5from the equivalent Gaussian variables S n that obey an approximate standard normal distribution independently ofthe number of data points N pt,n in the experiment. Every S n is a monotonically increasing function of the respective χ n given in [18, 23]. The power k = 16 is chosen so that ( S n ) k sharply increases from zero when S n approaches 1.3,the value corresponding to the 90% C.L. cutoff. The implementation of S n is fully documented in the appendix ofRef. [23].
3. Correlated systematic errors
In many of the data sets included in the CT14 analysis, the reported correlated systematic errors from experimentalsources dominate over the statistical errors. Care must therefore be taken in the treatment of these systematic errorsto avoid artificial biases in the best-fit outcomes, such as the bias described by D’Agostini in [94, 95].Our procedure for handling the systematic errors is reviewed in Secs. 3C and 6D of [6]; see also a related discussionin the appendices of [21] and [93]. The correlated errors for a given experiment, and effective shifts in the theoryor data that they cause, are estimated in a linearized approximation by including a contribution in the figure-of-merit function χ proportional to the correlation matrix. A practical implementation of this approach runs into adilemma of distinguishing between the additive and multiplicative correlated errors, which are often not separatedin the experimental publications, but must follow different prescriptions to prevent the bias. It is the matrix β i,α of relative correlated errors that is typically published; the absolute correlated errors must be reconstructed from β i,α by following the prescription for either the additive or multiplicative type.In inclusive jet production, the choice between the additive and multiplicative treatments modifies the large- x behavior of the gluon PDF. This has been studied in the CT10 NNLO analysis, cf. Sec. 6D of [6]. In general, thedominant sources of systematic error, especially at the Tevatron and LHC, should be treated as multiplicative ratherthan additive; that is, by assuming that the relative systematic error corresponds to a fixed fraction of the theoreticalvalue, and not of the central data value. The final CT14 PDFs were derived under this assumption, by treating thesystematic errors as multiplicative in all experiments. ∗ Of course, this is just one option on the table: alternativecandidate fits of the CT14 family were also performed, by treating some correlated errors as additive. They producedthe PDFs that generally lie within the quoted uncertainty ranges, as in the previous exercise documented in [6].
III. OVERVIEW OF CT14 PDFS AS FUNCTIONS OF x AND Q Figure 5 shows an overview of the CT14 parton distribution functions, for Q = 2 and 100 GeV. The function xf ( x, Q ) is plotted versus x , for flavors u, u, d, d, s = s , and g . We assume s ( x, Q ) = ¯ s ( x, Q ), since their difference isconsistent with zero and has large uncertainty [96]. The plots show the central fit to the global data listed in Tables Iand II, corresponding to the lowest total χ for our choice of PDF parametrizations.The relative changes between the CT10 NNLO and CT14 NNLO ensembles are best visualized by comparing theirPDF uncertainties. Fig. 6 compares the PDF error bands at 90% confidence level for the key flavors, with each band ∗ According to terminology adopted in Refs. [6, 93], CT14 implements the correlated errors according to the “extended T ” prescriptionfor all experiments, i.e., by normalizing the relative correlated errors by the current theoretical value in each iteration of the fit. g H x,Q L(cid:144) - bar d - bars - bar0.001 0.003 0.01 0.03 0.1 0.3 10.0.20.40.60.8 x x f H x , Q L a t Q = G e V CT14 NNLO g H x,Q L(cid:144) - bard - bars - bar0.001 0.003 0.01 0.03 0.1 0.3 10.0.20.40.60.8 x x f H x , Q L a t Q = G e V CT14 NNLO
FIG. 5: The CT14 parton distribution functions at Q = 2 GeV and Q = 100 GeV for u, u, d, d, s = s , and g . normalized to the respective best-fit CT14 NNLO PDF. The blue solid and red dashed error bands are obtained forCT14 and CT10 NNLO PDFs at Q = 100 GeV, respectively.Focusing first on the u and d flavors in the upper four subfigures, we observe that the u and ¯ u PDFs have mildlyincreased in CT14 at x < − , while the d and ¯ d PDFs have become slightly smaller. These changes can beattributed to a more flexible parametrization form adopted in CT14, which modifies the SU (2) flavor composition ofthe first-generation PDFs at the smallest x values in the fit.The CT14 d -quark PDF has increased by 5% at x ≈ .
05, after the ATLAS and CMS
W/Z production data sets at7 TeV were included. At x & .
1, the update of the DØ charge asymmetry data set in the electron channel, reviewedin Sec. II B 2, has reduced the magnitude of the d quark PDFs by a large amount, and has moderately increased the u ( x, Q ) distribution.The ¯ u ( x, Q ) and ¯ d ( x, Q ) distributions are both slightly larger at x = 0 . − . x = 0 . − .
5, where there are only very weak constraints on the sea-quark PDFs, the new parametrization form ofCT14 results in smaller values of ¯ u ( x, Q ) and larger values ¯ d ( x, Q ), as compared to CT10, although for the most partwithin the combined PDF uncertainties of the two ensembles.The central strangeness PDF s ( x, Q ) in the third row of Fig. 6 has decreased for 0 . < x < .
15, but withinthe limits of the CT10 uncertainty, as a consequence of the more flexible parametrization, the corrected calculationfor massive quarks in charged-current DIS, and the inclusion of the LHC data. The extrapolation of s ( x, Q ) below x = 0 .
01, where no data directly constrain it, also lies somewhat lower than before; its uncertainty remains large andcompatible with that in CT10. At large x , above about 0.2, the strange quark PDF is essentially unconstrained inCT14, just as in CT10.The central gluon PDF (last frame of Fig. 6) has increased in CT14 by 1-2% at x ≈ .
05 and has been somewhatmodified at x > . R a t i o t o r e f e r en c e f i t C T NN L O xu(x,Q) Q = 100 GeV, 90% c.l.CT14NNLOCT10NNLO/CT14NNLO0.80.91.01.11.2 10 -4 -3 -2 -1 R a t i o t o r e f e r en c e f i t C T NN L O xd(x,Q) Q = 100 GeV, 90% c.l.CT14NNLOCT10NNLO/CT14NNLO0.80.91.01.11.2 10 -4 -3 -2 -1 R a t i o t o r e f e r en c e f i t C T NN L O x – u(x,Q) Q = 100 GeV, 90% c.l.CT14NNLOCT10NNLO/CT14NNLO0.50.60.70.80.91.01.11.21.31.41.5 10 -4 -3 -2 -1 R a t i o t o r e f e r en c e f i t C T NN L O x – d(x,Q) Q = 100 GeV, 90% c.l.CT14NNLOCT10NNLO/CT14NNLO0.50.60.70.80.91.01.11.21.31.41.5 10 -4 -3 -2 -1 R a t i o t o r e f e r en c e f i t C T NN L O xs(x,Q) Q = 100 GeV, 90% c.l.CT14NNLOCT10NNLO/CT14NNLO0.00.51.01.52.0 10 -4 -3 -2 -1 R a t i o t o r e f e r en c e f i t C T NN L O xg(x,Q) Q =100 GeV, 90% c.l.CT14NNLOCT10NNLO/CT14NNLO0.60.81.01.21.4 10 -4 -3 -2 -1 FIG. 6: A comparison of 90% C.L. PDF uncertainties from CT14 NNLO (solid blue) and CT10 NNLO (red dashed) error sets.Both error bands are normalized to the respective central CT14 NNLO PDFs. and by the other factors discussed above. For x between 0.1 to 0.5, the gluon PDF has increased in CT14 as comparedto CT10.Let us now review the ratios of various PDFs, starting with the ratio d/u shown in Fig. 7. The changes in d/u in CT14 NNLO, as compared to CT10 NNLO, can be summarized as a reduction of the central ratio at x > . . − DØ charge asymmetry data, and an increased uncertainty at x < .
05 allowed by the newparametrization form. At x > .
2, the central CT14 NNLO ratio is lower than that of CT10 NNLO, while theirrelative PDF uncertainties remain about the same. This can be better seen from a direct comparison of the relativePDF uncertainties (normalized to their respective central PDFs) in the third inset. The collider charge asymmetry8 d ( x , Q ) / u ( x , Q ) xCJ12 NLOQ =10 GeV, 90% c.l.CT14 NNLOCT10 NNLO0.00.20.40.60.81.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 R a t i o t o r e f e r en c e f i t C T NN L O xd/u at Q = 2 GeV, 90% c.l.CT14NNLOCT10NNLO/CT14NNLO0.00.51.01.52.0 10 -4 -3 -2 -1 R e l a ti v e P D F e rr o r s xd(x,Q)/u(x,Q) at Q =2 GeV, 90% c.l.CT14NNLOCT10NNLO0.00.51.01.52.0 10 -4 -3 -2 -1 FIG. 7: A comparison of 90% C.L. uncertainties on the ratio d ( x, Q ) /u ( x, Q ) for CT14 NNLO (solid blue) and CT10 NNLO(dashed red), and CJ12 NLO (green lines) error ensembles. – d ( x , Q ) / – u ( x , Q ) xQ =2 GeV, 90% c.l.CT14NNLOCT10NNLO 0 0.5 1 1.5 2 2.5 10 -4 -3 -2 -1 ( s + – s ) / ( – u + – d ) xQ =2 GeV, 90% c.l.CT14NNLOCT10NNLO 0 0.5 1 1.5 2 10 -4 -3 -2 -1 FIG. 8: A comparison of 90% C.L. uncertainties on the ratios ¯ d ( x, Q ) / ¯ u ( x, Q ) and ( s ( x, Q ) + ¯ s ( x, Q )) / (cid:0) ¯ u ( x, Q ) + ¯ d ( x, Q ) (cid:1) , forCT14 NNLO (solid blue) and CT10 NNLO (red dashed) error ensembles. d/u at x up to about 0.4. At even higher x , outside of the experimental reach, the behavior of theCT14 PDFs reflects the parametrization form, which now allows d/u to approach any constant value at x → x , the CTEQ-JLab analysis (CJ12) [98] has independently determined the ratio d/u at NLO, byincluding the fixed-target DIS data at lower W and higher x that is excluded by a selection cut W > . d/u at NNLO lies for the most part between the CJmin and CJmax predictions atNLO that demarcate the CJ12 uncertainty, cf. the first inset of Fig. 7. We see that the CT14 predictions on d/u at x > .
1, which were derived from high-energy measurements that are not affected by nuclear effects, fall withinthe CJ12 uncertainty range obtained from low-energy DIS with an estimate of various effects beyond leading-twistperturbative QCD. The ratio should be stable to inclusion of NNLO effects; thus, the two ensembles predict a similartrend for collider observables sensitive to d/u .Turning now to the ratios of sea quark PDFs in Fig. 8, we observe that the uncertainty on ¯ d ( x, Q ) / ¯ u ( x, Q ) in theleft inset has also increased at small x in CT14 NNLO. At x > .
1, we assume that both ¯ u ( x, Q ) and ¯ d ( x, Q ) areproportional to (1 − x ) a with the same power a ; the ratio ¯ d ( x, Q ) / ¯ u ( x, Q ) can thus approach a constant value thatcomes out to be close to 1 in the central fit, while the parametrization in CT10 forced it to vanish. The uncertaintyon ¯ d/ ¯ u has also increased across most of the x range.The overall reduction in the strangeness PDF at x > .
01 leads to a smaller ratio of the strange-to-nonstrange seaquark PDFs, ( s ( x, Q ) + ¯ s ( x, Q )) / (cid:0) ¯ u ( x, Q ) + ¯ d ( x, Q ) (cid:1) , presented in the right inset of Fig. 8. At x < .
01, this ratio isdetermined entirely by parametrization form and was found in CT10 to be consistent with the exact SU (3) symmetryof PDF flavors, ( s ( x, Q ) + ¯ s ( x, Q )) / (cid:0) ¯ u ( x, Q ) + ¯ d ( x, Q ) (cid:1) → x →
0, albeit with a large uncertainty. The SU (3)-symmetric asymptotic solution at x → x = 10 − . The uncertainty of strangeness hasincreased at such small x and now allows ( s ( x, Q ) + ¯ s ( x, Q )) / (cid:0) ¯ u ( x, Q ) + ¯ d ( x, Q ) (cid:1) between 0.35 and 2.5 at x = 10 − . IV. COMPARISONS WITH HADRONIC EXPERIMENTSA. Electroweak total cross sections at the LHC
Measurements of total cross sections for production of massive electroweak particles at hadron colliders providecornerstone benchmark tests of the Standard Model. These relatively simple observables can be both measured withhigh precision and predicted in NNLO QCD theory with small uncertainties. In this subsection, we collect NNLOtheory predictions based on CT14 and CT10 NNLO PDFs for inclusive W and Z boson production, top-quark pairproduction, Higgs-boson production (through gluon-gluon fusion), at the LHC with center-of-mass energies of 8 and13 TeV. These theoretical predictions can be compared to the corresponding experimental measurements. We alsoexamine correlations between PDF uncertainties of the total cross sections in the context of the Hessian formalism,following the approach summarized in Ref. [13]. PDF-driven correlations reveal relations between PDF uncertaintiesof QCD observables through their shared PDF parameters.The masses of the top quark and Higgs boson are set to m polet = 173 . m H = 125 GeV, respectively, inthis work. The W and Z inclusive cross sections (multiplied by branching ratios for the decay into one charged lepton0 óóõõ % c.l. regions at LHC 8 TeVCT10 NNLOCT14 NNLO Σ W + H in nb L Σ W - H i nnb L óóõõ % c.l. regions at LHC 13 TeVCT10 NNLOCT14 NNLO Σ W + H in nb L Σ W - H i nnb L FIG. 9: The CT14 and CT10 NNLO 90% C.L. error ellipses for the W − and W + cross sections, at the LHC 8 and 13 TeV. óóõõ % c.l. regions at LHC 8 TeVCT10 NNLOCT14 NNLO Σ W + (cid:144) - H in nb L Σ z H i nnb L óóõõ % c.l. regions at LHC 13 TeVCT10 NNLOCT14 NNLO Σ W + (cid:144) - H in nb L Σ z H i nnb L FIG. 10: CT14 and CT10 NNLO 90% C.L. error ellipses for Z and W ± cross sections, at the LHC 8 and 13 TeV. flavor), are calculated by using the Vrap v0.9 program [75, 76] at NNLO in QCD, with the renormalization andfactorization ( µ R and µ F ) scales set equal to the invariant mass of the vector boson. The total inclusive top-quarkpair cross sections are calculated with the help of the program Top++ v2.0 [99, 100] at NNLO+NNLL accuracy, withQCD scales set to the mass of the top quark. The Higgs boson cross sections via gluon-gluon fusion are calculated atNNLO in QCD by using the iHixs v1.3 program [101], in the heavy-quark effective theory (HQET) with finite topquark mass correction, and with the QCD scales set equal to the invariant mass of the Higgs boson.Figs. 9 – 12 show central predictions and 90% C.L. regions for ( W + , W − ), ( Z , W ± ), ( t ¯ t , Z ) and ( t ¯ t , ggH ) pairs ofinclusive cross sections at the LHC 8 and 13 TeV. In each figure, two elliptical confidence regions are shown, obtainedwith either CT14 or CT10 NNLO PDFs. These can be used to read off PDF uncertainties and correlations for eachpair of cross sections. For example, Figs. 9 and 10 indicate that the PDF induced uncertainties, at the 90% C.L.,1 óóõõ % c.l. regions at LHC 8 TeVCT10 NNLOCT14 NNLO Σ Z H in nb L Σ tt H i nnb L óóõõ % c.l. regions at LHC 13 TeVCT10 NNLOCT14 NNLO Σ Z H in nb L Σ tt H i nnb L FIG. 11: CT14 and CT10 NNLO 90% C.L. error ellipses for t ¯ t and Z cross sections, at the LHC 8 and 13 TeV. óó õõ % c.l. regions at LHC 8 TeVCT10 NNLOCT14 NNLO Σ ggh H in pb L Σ tt H i nnb L óó õõ % c.l. regions at LHC 13 TeVCT10 NNLOCT14 NNLO Σ ggh H in pb L Σ tt H i nnb L FIG. 12: CT14 and CT10 NNLO 90% C.L. error ellipses for t ¯ t and ggH cross sections, at the LHC 8 and 13 TeV. are about 3.9%, 3.7%, and 3.7% for W + , W − , and Z boson production at the LHC 13 TeV, respectively, with CT14NNLO PDFs. As compared to the results using CT10 NNLO PDFs, the ratio of the total inclusive cross sections of W + to W − productions at the LHC 13 TeV is smaller by about one percent when using CT14 NNLO PDFs whichalso provide a slightly larger error (by about half percent) in that ratio. Specifically, the CT14 NNLO predictionsof that ratio at the 68% C.L. are 1 . +1 . − at LHC 8 TeV, and 1 . +1% − at LHC 13 TeV, respectively. The centralpredictions at 8 TeV are in agreement with the recent CMS measurements [102]. They also show that the electroweakgauge boson cross sections are highly correlated with each other; in fact, much of the uncertainty is driven in thiscase by the small- x gluon [13].In Fig. 11, we observe a moderate anti-correlation between the top-quark pair and the Z boson production crosssections. This is a consequence of the proton momentum sum rule mediated by the gluon PDF [13]. In Fig. 12, theHiggs boson cross section through gluon-gluon fusion does not have a pronounced correlation or anti-correlation with2the top-quark cross section, because they are dominated by the gluon PDF in different x regions. The Higgs bosonand t ¯ t cross section predictions are further examined in Section V. As a result of the changes in PDFs from CT10to CT14, both the calculated Higgs boson and top-quark pair production cross sections have increased slightly, whilethe electroweak gauge boson cross sections have decreased. However, the changes of the central predictions are withinthe error ellipses of either CT14 or CT10. B. LHC and Tevatron inclusive jet cross sections
We now turn to the comparisons of CT14 PDFs with new LHC cross sections on inclusive jet production. Weargued in Section II that PDF uncertainty of inclusive jet production at the LHC is strongly correlated with the gluonPDF in a wider range of x than in the counterpart measurements at the Tevatron. The true potential of LHC jets forconstraining the gluon PDF also depends on experimental uncertainties, which we can now explore for the first timeusing the CMS and ATLAS data on inclusive jet cross sections at 7 TeV.We first note that, in the context of our analysis, the single-inclusive jet measurements at the LHC are found tobe in reasonable consistency with the other global data, including Tevatron Run-2 single-inclusive jet cross sectionsmeasured by the CDF and DØ collaborations. The values of χ for the four jet experiments (ID=504, 514, 535, and538) are listed at the end of Table II. We obtain very good fits ( χ /N pt =1.09 and 0.55) to the DØ and ATLAS jetdata sets, and moderately worse fits ( χ /N pt =1.45 and 1.33) to the CDF and CMS data sets. The description ofthe Tevatron jet data sets has been examined as a part of the CT10 NNLO study [6], where it was pointed out thatthe χ for the CDF Run-2 measurement tends to be increased by random, rather than systematic, fluctuations of thedata. In regards to describing the Tevatron jet data, the CT14 NNLO PDFs follow similar trends as CT10 NNLO.
1. CMS single-inclusive jet cross sections
Figure 13 shows a comparison between the measurements for the CMS inclusive jet data at 7 TeV [52] and NLOtheory prediction [88, 103, 104] utilizing CT14 NNLO PDFs. We discussed earlier in the paper that the missingNNLO contributions to the hard-scattering cross section can be anticipated to be small under our QCD scale choices,compared to the experimental uncertainty.The CMS data, with 5 fb − of integrated luminosity, employ the anti- k T jet algorithm [105] with jet radius R = 0 . p T of the jet, with a total of133 data points. The theoretical prediction based on the CT14 NNLO PDFs reproduces the behavior of experimentalcross sections across thirteen orders of magnitude.Fig. 14 provides a more detailed look at these distributions, by plotting the shifted central data values divided bythe theory. The data are shifted by optimal amounts based on the treatment of the systematic errors as nuisanceparameters, cf. Ref. [6]. The error bars for the shifted data include only uncorrelated errors, i.e. statistical anduncorrelated systematic errors added in quadrature. Here we notice moderate differences (up to a few tens of percentof the central prediction) between theory and shifted data, which elevate χ for this data set by about 2.5 standarddeviations for the central CT14 PDF set, or less for the error PDF sets.Although they are not statistically significant, the origin of these mild discrepancies can be further explored by3 d σ / d P T dy [ pb / G e V ] P T [GeV]CMS, √ s=7 TeV, L=5.0 [fb] -1 , R=0.7 |y| < )0.5 < |y| < )1.0 < |y| < )1.5 < |y| < )2.0 < |y| < ) -6 -2 FIG. 13: Comparison of data and theory for the CMS 7 TeV inclusive jet production, for CT14 NNLO PDFs. Measurements of d σ/dp T dy for 5 rapidity bins are plotted as functions of jet p T . The points are data with total experimental errors, obtainedby adding the statistical and systematic errors in quadrature. The bands are theoretical calculations with 68% C.L. PDFuncertainties. studying the correlated shifts allowed by the systematic uncertainties. In our implementation of systematic errors[6], each correlated uncertainty α is associated with a normally distributed random nuisance parameter λ α . When λ α = 0, it may effectively shift the central value of the data point i in the fit by β i,α λ α = σ i,α X i λ α , where σ i,α is the published fractional 1- σ uncertainty of data point i due to systematic error α . X i is the cross sectionvalue that normalizes the fractional systematic uncertainty [6], set equal to the theoretical value T i in the procedureof the current analysis.Each λ α is adjusted to optimize the agreement between theory and data. Fig. 15 shows a histogram of the best-fit λ α for the 19 sources of the systematic errors published by CMS [52]. In an ideal situation, the optimized { λ α : α = 1 ... } would be normally distributed with a mean value of 0 and standard deviation of 1. The actual distribution of the λ α values in Fig. 15 appears to be somewhat narrower than the standard normal one. This and relatively high χ /N pt = 1 .
33 may indicate that either uncorrelated systematic uncertainties are underestimated, or higher-ordertheoretical calculations are needed to describe the data.
2. ATLAS single-inclusive jet cross sections
Equivalent comparisons for the ATLAS 7 TeV inclusive jet production with 37 pb − of integrated luminosity [51]are presented in Figs. 16 – 18. In this case, we compare to data in 7 bins of rapidity for the anti- k T jet algorithm [105]with jet radius R = 0 .
6. The agreement is excellent in all figures, not the least because both statistical and systematicerrors are still large in this early data set. Among 119 sources of experimental errors that were identified, many havelittle impact on the best fit. The resulting distribution of the nuisance parameters in Fig. 18 at the best fit is muchnarrower than the ideal Gaussian distribution, indicating that most of the correlated sources need not deviate from4 s h i f t e d d a t a / T h e o r y P T [GeV]CMS, √ s=7 TeV, L=5.0 [fb] -1 , R=0.7 |y| < s h i f t e d d a t a / T h e o r y P T [GeV]CMS, √ s=7 TeV, L=5.0 [fb] -1 , R=0.7 0.5 < |y| < s h i f t e d d a t a / T h e o r y P T [GeV]CMS, √ s=7 TeV, L=5.0 [fb] -1 , R=0.7 1.0 < |y| < s h i f t e d d a t a / T h e o r y P T [GeV]CMS, √ s=7 TeV, L=5.0 [fb] -1 , R=0.7 1.5 < |y| < s h i f t e d d a t a / T h e o r y P T [GeV]CMS, √ s=7 TeV, L=5.0 [fb] -1 , R=0.7 2.0 < |y| < FIG. 14: Same as Fig. 13, shown as the ratio of shifted data for CMS 7 TeV divided by theory. The error bars correspond tototal uncorrelated errors. The shaded region shows the 68% C.L. PDF uncertainties. their nominal values when the PDFs are fitted.To summarize, Figs. 13-18 demonstrate that CT14 PDFs agree with both sets of CMS and ATLAS single-inclusivejet cross sections. The ATLAS collaboration also measured inclusive jet production at center of mass energy √ s =2.76 TeV and published ratios between the 2.76 and 7 TeV measurements in Ref. [57]. These two measurementsare well described by the theory prediction using CT14, with a χ /N pt ≈
1. Furthermore, the ATLAS collaborationpublished the inclusive jet measurements using another choice of jet radius of 0.4 [51]. Both ATLAS and CMScollaborations measured cross sections for dijet production [51, 52] based on the same data sample of the single-inclusivejet measurements. These measurements are not included in the CT14 global analysis because of the correlations5 B i n c oun t Best-fit nuisance parameters λ α CMS inclusive jet production at 7 TeV 0 1 2 3 4 5-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0
FIG. 15: Histogram of optimized nuisance parameters λ α for the sources of correlated systematic errors of the CMS 7 TeVinclusive jet production. The curve is the standard normal distribution expected in the ideal case. d σ / d P T dy [ pb / G e V ] P T [GeV]ATLAS, √ s=7 TeV, L=37 [pb] -1 , R=0.6 |y| ≤ )0.3 ≤ |y| < )0.8 ≤ |y| < )1.2 ≤ |y| < )2.1 ≤ |y| < )2.8 ≤ |y| < -3 )3.6 ≤ |y| < -6 ) -9 -6 -3
100 1000
FIG. 16: Comparison of data on d σ/dp T dy and NLO theory for the ATLAS 7 TeV inclusive jet production, using CT14NNLO PDFs. between the two (di-jet and single-inclusive jet) data sets. However, it has been verified that the CT14 analysis givesa good description for all these data sets as well. C. Differential cross sections for lepton pair production at the LHC
1. Charged lepton pseudorapidity distributions in
W/Z boson production
Differential cross sections for production of massive vector bosons set important constraints on the flavor compositionof the proton, notably on the u and d quarks, anti-quarks and their ratios. Figure 19 compares CT14 NNLO theoreticalpredictions with pseudorapidity ( | η ℓ | ) distributions of charged leptons from inclusive W ± and Z production and decayin the 2010 ATLAS 7 TeV data sample with 33-36 pb − of integrated luminosity [48]. Theoretical predictions arecomputed using the program ResBos . The black data points represent the unshifted central values of the data. The6 s h i f t e d d a t a / T h e o r y P T [GeV]ATLAS, √ s=7 TeV, L=37 [pb] -1 , R=0.6 |y| < s h i f t e d d a t a / T h e o r y P T [GeV]ATLAS, √ s=7 TeV, L=37 [pb] -1 , R=0.6 0.3 ≤ |y| < s h i f t e d d a t a / T h e o r y P T [GeV]ATLAS, √ s=7 TeV, L=37 [pb] -1 , R=0.6 0.8 ≤ |y| < s h i f t e d d a t a / T h e o r y P T [GeV]ATLAS, √ s=7 TeV, L=37 [pb] -1 , R=0.6 1.2 ≤ |y| < s h i f t e d d a t a / T h e o r y P T [GeV]ATLAS, √ s=7 TeV, L=37 [pb] -1 , R=0.6 2.1 ≤ |y| < s h i f t e d d a t a / T h e o r y P T [GeV]ATLAS, √ s=7 TeV, L=37 [pb] -1 , R=0.6 2.8 ≤ |y| < s h i f t e d d a t a / T h e o r y P T [GeV]ATLAS, √ s=7 TeV, L=37 [pb] -1 , R=0.6 3.6 ≤ |y| < FIG. 17: Same as Fig. 16, shown as the ratio of shifted data for ATLAS 7 TeV divided by the NLO theory. The error barscorrespond to total uncorrelated errors. The shaded region shows the 68% C.L. PDF uncertainties. B i n c oun t Best-fit nuisance parameters λ α ATLAS inclusive jet production at 7 TeV, R=0.6 0 5 10 15 20 25 30 35 40 45 50-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0
FIG. 18: Histogram of optimized nuisance parameters λ α for the sources of correlated systematic errors of the ATLAS 7 TeVinclusive jet production. d σ / d | η l | ( pp → W - → l - - ν + X ) [ pb ] | η l |ATLAS, √ s=7 TeV, 33-36 [pb] -1 DATA 2010CT14, 68% C.L. 300 340 380 420 460 500 0 0.5 1 1.5 2 2.5 d σ / d | y Z | [ pb ] |y Z |ATLAS, √ s=7 TeV, 33-36 [pb] -1 DATA 2010CT14, 68% C.L. 40 60 80 100 120 140 160 0 0.5 1 1.5 2 2.5 3 3.5 d σ / d | η l e p | ( pp → W + → l + ν l + X ) [ pb ] | η l |ATLAS, √ s=7 TeV, 33-36 [pb] -1 DATA 2010CT14, 68% C.L. 520 560 600 640 680 720 0 0.5 1 1.5 2 2.5
FIG. 19: Comparison between the 2010 ATLAS measurements [48] of the W ± charged-lepton pseudorapidity and Z bosonrapidity distributions at √ s = 7 TeV, and the ResBos theory using CT14 NNLO PDFs. error bars indicate the total (statistical+systematic) experimental error. The blue band is the CT14 PDF uncertaintyevaluated at the 68% C.L. These three measurements share correlated systematic errors. From the figures, we seethat the data are described well by theory over the entire rapidity range, even in the absence of correlated systematicshifts. The PDF uncertainties are similar in their size to those of the experimental measurements and, overall, thetheory predictions are within one standard deviation of the data.
2. Influence of W boson charge asymmetry measurements at the LHC Another handy observable for determining the parton distribution functions is the charge asymmetry for W + and W − bosons produced in pp or p ¯ p collisions. This process has been measured both at the Tevatron and at the LHC.As the asymmetry involves a ratio of the cross sections, many experimental systematic errors cancel, leading to veryprecise results. Without these collider data, the main information about the difference between the light flavors, d, d and u, u , would come from the BCDMS and NMC experiments, which are measurements of muon deep-inelasticscattering on proton and deuteron targets. Under the assumption of charge symmetry between the nucleons, thedifference of the proton and deuteron cross sections distinguishes between the u and d PDFs in a nucleon. However,the deuteron measurements are subject to nuclear binding corrections, which have been estimated by introducingnuclear models [97, 98, 106], but are not calculable from first principles. In contrast, the W ± charge asymmetrydata from the Tevatron and LHC colliders directly provide information about the difference between d and u flavors,8 A l | η l |ATLAS, √ s=7 TeV, 33-36 [pb] -1 DATA 2010CT14, 68% C.L.0.120.160.200.240.280.32 0 0.5 1 1.5 2 2.5
FIG. 20: W ± charge asymmetry as a function of lepton pseudorapidity measured by the ATLAS Collaboration, compared tothe 68% C.L. CT14 NNLO uncertainty band. The kinematic requirements are p Tℓ >
20 GeV, p Tν ℓ >
25 GeV and M ℓν ℓ T > M uon c h a r g e a s y mm e t r y | η µ |CMS, √ s=7 TeV, L=4.7 [fb] -1 P T >
25 GeVCT14, 68% C.L.0.100.150.200.250.30 0 0.5 1 1.5 2 2.5 M uon c h a r g e a s y mm e t r y | η µ |CMS, √ s=7 TeV, L=4.7 [fb] -1 P T >
35 GeVCT14, 68% C.L.0.100.150.200.250.30 0 0.5 1 1.5 2 2.5 E l ec t r on c h a r g e a s y mm e t r y | η e |CMS, √ s=7 TeV, L=840 [pb] -1 P T >
35 GeVCT14, 68% C.L.0.080.120.160.200.24 0 0.5 1 1.5 2 2.5
FIG. 21: Charge asymmetry of decay muons and electrons from W ± production measured by the CMS experiment. The datavalues have p Tℓ >
25 or 35 GeV for the muon data and p Tℓ >
35 GeV for the electron data. The vertical error bars on the datapoints indicate total (statistical and systematic) uncertainties. The curve shows the CT14 theoretical calculation; the shadedregion is the PDF uncertainty at 68% C.L. without the need for nuclear corrections. By including the ATLAS and CMS charge asymmetry data, we are able toobtain, for the first time, direct experimental constraints on the differences of the quark and antiquark PDFs for u and d flavors at x ≈ .
02 typical for the 7 TeV kinematics.Figure 20 shows a comparison of data and theory, for the lepton charge asymmetry of inclusive W ± production,9 A µ η µ LHCb, √ s=7 TeV, 35 [pb] -1 P T >
20 GeVCT14, 68% C.L.-0.8-0.6-0.4-0.20.00.20.4 2 2.5 3 3.5 4 4.5
FIG. 22: Charge asymmetry of decay muons from W ± production measured by the LHCb experiment. from the ATLAS experiment at the LHC 7 TeV [48]. These asymmetry data are correlated with the W/Z rapiditymeasurements discussed in the previous subsection; all four
W/Z data sets are included in the CT14 global analysisusing a shared correlation matrix from the ATLAS publication [48]. The measurement was carried out with severalkinematic cuts. The lepton transverse momentum was required to be greater than 20 GeV, the missing transverseenergy to be greater than 25 GeV, and the lepton-neutrino transverse mass to be greater than 40 GeV. The shadedregion is the PDF uncertainty of CT14 NNLO at 68% C.L. Again the points with error bars represent the unshifteddata with the experimental errors added in quadrature. The data fluctuate around the CT14 predictions and aredescribed well by the CT14 error band.Figure 21 presents a similar comparison of the unshifted data and CT14 NNLO theory for the charge asymmetry ofdecay muons [46] and electrons [47] from inclusive W ± production from the CMS experiment at the LHC 7 TeV. Theasymmetry for muons is measured with 4.7 fb − of integrated luminosity, with p T ℓ >
25 and 35 GeV; the asymmetryfor electrons is measured with 840 pb − and p T ℓ >
35 GeV. Here we note that the CMS measurement does not applya missing E T cut to A ch , contrary to the counterpart ATLAS A ch measurement. Theory predictions are the samefor both the muon and electron channels with the same cuts. The muon and electron data are consistent with oneanother, but the muon data have smaller statistical and systematic uncertainties, as is apparent in Fig. 21. All threesubsets of CMS A ch agree with predictions using CT14; their χ is further improved by optimizing the correlatedshifts. The electron data and the muon data with the p T ℓ cut of 35 GeV are included in the CT14 global analysis.The muon data with a p T ℓ cut of 25 GeV are not included in the CT14 analysis, but nevertheless are well described.In the LHCb measurement of the charged lepton asymmetry at 7 TeV [43], the muons are required to have atransverse momentum greater than 20 GeV. The corresponding comparison of the CT14 NNLO predictions to theLHCb A ch data is shown in Fig. 22. The LHCb case is especially interesting, as the LHCb acceptance for chargedleptons extends beyond the rapidity range measured by ATLAS and CMS. Thus, the LHCb results are sensitive tothe u and d quark PDFs at larger x values than at the ATLAS or CMS. Good agreement between data and theoryis again observed.0 d σ / d m ee [ pb / G e V ] m ee [GeV]ATLAS, √ s=7 TeV, L=4.9 [fb] -1 pT > 25 GeV, | η |<2.5CT14, 68% C.L.10 -6 -5 -4 -3 -2 -1
100 1000 s h i f t e d d a t a / T h e o r y m ee [GeV]ATLAS, √ s=7 TeV, L=4.9 [fb] -1 FIG. 23: Invariant mass distributions of Drell-Yan pairs in the high-mass region by ATLAS 7 TeV [108], with superimposedNNLO predictions based on CT14 NNLO PDFs. The left subfigure shows the differential cross sections as a function of thedilepton mass m ee . The right subfigure shows the ratio of ATLAS shifted data to CT14 theory predictions. d σ / d m ll [ pb / G e V ] m ll [GeV]ATLAS, √ s=7 TeV, L=1.6 [fb] -1 pT > 12 and 15 GeV, | η |<2.4CT14, 68% C.L. 0.5 1 1.5 2 2.5 3 3.5 20 25 30 35 40 45 50 55 60 65 s h i f t e d d a t a / T h e o r y m ll [GeV]ATLAS, √ s=7 TeV, L=1.6 [fb] -1 d σ / d m ll [ pb / G e V ] m ll [GeV]ATLAS, √ s=7 TeV, L=35[pb] -1 pT > 6 GeV, | η |<2.4CT14, 68% C.L.-5 0 5 10 15 20 25 30 10 15 20 25 30 35 40 45 50 s h i f t e d d a t a / T h e o r y m ll [GeV]ATLAS, √ s=7 TeV, L=35 [pb] -1 FIG. 24: Same as in Fig. 23, for ATLAS 7 TeV differential distributions of Drell-Yan pairs in the low-mass and extendedlow-mass regions [109].
3. Production of Drell-Yan pairs at ATLAS
In Figs. 23 and 24, we compare CT14 NNLO predictions to ATLAS 7 TeV measurements of differential cross sectionsfor production of high-mass [108] and low-mass [109] Drell-Yan pairs, plotted as a function of dilepton invariant mass m ℓℓ . The experimental cross sections correspond to the “electroweak Born level”, unfolded from the raw data bycorrecting for electroweak final-state radiation. The high-mass data sample corresponds to 116 < m ℓℓ < < m ℓℓ <
66 GeV for L = 1 . − in the upper row, as well as to the muon sample at 12 < m ℓℓ <
66 GeV for L = 35 pb − in the lower row. Fiducialacceptance cuts on the decay leptons are specified inside the figures. Correlated experimental uncertainties areincluded in the comparison.On the theory side, the cross sections are calculated at NNLO in QCD with ApplGrid interface [89] to FEWZ[110–113], and including photon-scattering contributions. Experimental uncertainties in these cross sections tend tobe larger than the PDF uncertainties, as illustrated by the figures, hence we only compare these data to the CT14predictions a posteriori , without actually including them in the CT14 fit.It can be observed in the figures that CT14 NNLO PDFs agree well with the high-mass and low-mass data samplesboth in terms of the cross sections (in the left subfigures) and ratios of the shifted data to theoretical predictions(right subfigures). The PDF uncertainty bands, indicated by light-blue color, approximate the average behavior ofthe experimental data without systematic discrepancies. D. W ± charge asymmetry from the DØ experiment at the Tevatron We reviewed above that, historically, measurements of W ± charge asymmetry at the Tevatron have been importantin the CTEQ-TEA global analysis. For example, the CTEQ6 PDFs (circa 2002) and CT10 PDFs (circa 2010-2012)included the W ± asymmetry data from the CDF and DØ experiments to supplement the constraints on u and d quarkPDFs at x > . slope in x of the PDFs for u and d flavors.A new W ± charge asymmetry measurement from the DØ experiment at the Tevatron has recently been published,using the full integrated luminosity (9 . − ) from Run-2 [14]. The experimental uncertainties, both statistical andsystematic, are smaller than in the previous A ch measurement [22]. Figure 25 compares the DØ Run 2 data andvarious theoretical predictions at NNLO for both the latest (left) and the previous DØ data set (right). We show theunshifted data with the total experimental errors as error bars, and the 68% C.L. PDF uncertainties as the shadedregions. As an alternative representation, Figure 26 shows the differences between theory and shifted data, where theerror bars represent the uncorrelated experimental errors. From the two figures, we conclude that it is difficult to fitboth data sets well, given the smallness of the systematic shifts associated with A ch . While the 9 . − electron dataset is in better agreement with the global data, including the DØ muon [42] and CDF [41] A ch measurements, thebest-fit χ /N pt for the 9 . − sample remains relatively high (about 2) and is sensitive to detailed implementationof NNLO corrections. In-depth studies on the DØ asymmetry data will be presented in a forthcoming paper. Whenthe high-luminosity DØ A ch measurement was substituted for the low-luminosity one, we observed reduction in the d/u ratio at x > . W/Z differential cross sections and asymmetries lead to importantchanges in the quark sector PDFs, as documented in Sec. III. At x . .
02, we obtain more realistic error bands forthe u , ¯ u , d , ¯ d PDFs upon including the ATLAS and CMS data sets. At x > .
1, the high-luminosity DØ chargeasymmetry and other compatible experiments predict a softer behavior of d ( x, Q ) /u ( x, Q ) than in CT10W.2 E l ec t r on c h a r g e a s y mm e t r y | η e | D0, L=9.7 fb -1 CT14 NNLOCT14 NNLOCT10 NNLOCTEQ6.6 NLOE eT >
25 GeV, E
T,miss >
25 GeV-0.8-0.6-0.4-0.20.00.2 0 0.5 1 1.5 2 2.5 3 E l ec t r on c h a r g e a s y mm e t r y | η e | D0, L=0.75 fb -1 CT10 NNLOCT10 NNLOCT14 NNLOCTEQ6.6 NLOE eT >
25 GeV, E
T,miss >
25 GeV-0.8-0.6-0.4-0.20.00.2 0 0.5 1 1.5 2 2.5 3
FIG. 25: Charge asymmetry of decay electrons from W ± production measured by the DØ experiment in Run-2 at theTevatron with high (left) and low (right) luminosities, compared to several generations of CTEQ-TEA PDFs. T h e o r y - S h i f t e d D a t a | η e |D0, L=9.7 fb -1 CT14 NNLOCT14 NNLOCT10 NNLOCTEQ6.6 NLO-0.10-0.050.000.05 0 0.5 1 1.5 2 2.5 3 T h e o r y - S h i f t e d D a t a | η e |D0, L=0.75 fb -1 CT10 NNLOCT10 NNLOCT14 NNLOCTEQ6.6 NLO-0.2-0.10.00.10.2 0 0.5 1 1.5 2 2.5 3
FIG. 26: Same as Fig. 25, plotted as the difference between theory and shifted data for A ch from DØ Run-2 (9 . − ). E. Constraints on strangeness PDF from CCFR, NuTeV, and LHC experiments
Let us now turn to the strangeness PDF s ( x, Q ), which has become smaller at x > .
05 in CT14 compared toour previous analyses, CT10 and CTEQ6.6. Although the CT14 central s ( x, Q ) lies within the error bands of eitherearlier PDF set, it is important to verify that it is consistent with the four fixed-target measurements that are knownto be sensitive to s ( x, Q ): namely, measurements of dimuon production in neutrino and antineutrino collisions withiron targets, from the CCFR [31] and NuTeV [30] collaborations (ID=124-127).Predictions using previous CTEQ PDFs were in agreement with these four experiments. In Table I for CT14,the four corresponding χ values are also good. Supporting evidence comes from the point-by-point comparisons inFigs. 27 and 28, between the theoretical cross sections for CT14 NNLO PDFs and the dimuon data from the NuTeVexperiment in neutrino and antineutrino scattering. The analogous comparisons for the CCFR experiment are inFigs. 29 and 30. Given the size of the measurement errors and of the PDF uncertainty, it is clear that CT14 centralpredictions provide a good description of the dimuon cross sections. Also, our estimate for the uncertainty of thestrange PDF looks reasonable: it is comparable to the measurement errors for these cross sections, which are knownto be sensitive mostly to the strange quark PDF.3 NuTeV Neutrino, d σ ( ν µ N -> µ + µ - X)/dxdy [pb/(GeV)] E ν =88.29 GeV, y=0.3240.000.100.200.30 0 0.1 0.2 0.3 E ν =88.29 GeV, y=0.5580.000.100.200.30 0 0.1 0.2 0.3 E ν =88.29 GeV, y=0.7710.000.100.200.30 0 0.1 0.2 0.3E ν =174.29 GeV, y=0.3240.000.100.200.30 0 0.1 0.2 0.3 E ν =174.29 GeV, y=0.5580.000.100.200.30 0 0.1 0.2 0.3 E ν =174.29 GeV, y=0.7710.000.100.200.30 0 0.1 0.2 0.3E ν =247 GeV, y=0.3240.000.100.200.30 0 0.1 0.2 0.3 E ν =247 GeV, y=0.5580.000.100.200.30 0 0.1 0.2 0.3 E ν =247 GeV, y=0.7710.000.100.200.30 0 0.1 0.2 0.3 FIG. 27: Comparison of data and theory for the NuTeV measurements of dimuon production in neutrino-iron collisions. Thedata are expressed in the form of d σ/dxdy and shown as a function of x for a certain y and neutrino energy. NuTeV Anti-Neutrino, d σ ( – ν µ N -> µ + µ - X)/dxdy [pb/(GeV)] E – ν =77.88 GeV, y=0.3490.000.100.200.30 0 0.1 0.2 0.3 E – ν =77.88 GeV, y=0.5790.000.100.200.30 0 0.1 0.2 0.3 E – ν =77.88 GeV, y=0.7760.000.100.200.30 0 0.1 0.2 0.3E – ν =143.74 GeV, y=0.3490.000.100.200.30 0 0.1 0.2 0.3 E – ν =143.74 GeV, y=0.5790.000.100.200.30 0 0.1 0.2 0.3 E – ν =143.74 GeV, y=0.7760.000.100.200.30 0 0.1 0.2 0.3E – ν =226.79 GeV, y=0.3490.000.100.200.30 0 0.1 0.2 0.3 E – ν =226.79 GeV, y=0.5790.000.100.200.30 0 0.1 0.2 0.3 E – ν =226.79 GeV, y=0.7760.000.100.200.30 0 0.1 0.2 0.3 FIG. 28: Same as Fig. 27, for the NuTeV measurements of dimuon production in antineutrino-iron collisions. CCFR Neutrino, d σ ( ν µ N -> µ + µ - X)/dxdy [pb/(GeV)] E ν =109.46 GeV, y=0.320.000.100.200.30 0 0.1 0.2 0.3 E ν =109.46 GeV, y=0.570.000.100.200.30 0 0.1 0.2 0.3 E ν =109.46 GeV, y=0.7950.000.100.200.30 0 0.1 0.2 0.3E ν =209.9 GeV, y=0.320.000.100.200.30 0 0.1 0.2 0.3 E ν =209.9 GeV, y=0.570.000.100.200.30 0 0.1 0.2 0.3 E ν =209.9 GeV, y=0.7950.000.100.200.30 0 0.1 0.2 0.3E ν =332.7 GeV, y=0.320.000.100.200.30 0 0.1 0.2 0.3 E ν =332.7 GeV, y=0.570.000.100.200.30 0 0.1 0.2 0.3 E ν =332.7 GeV, y=0.7950.000.100.200.30 0 0.1 0.2 0.3 FIG. 29: Same as Fig. 27, for the CCFR measurements of dimuon production in neutrino-iron collisions.
CCFR Anti-Neutrino, d σ ( – ν µ N -> µ + µ - X)/dxdy [pb/(GeV)] E – ν =87.4 GeV, y=0.3550.000.100.200.300.400.50 0 0.1 0.2 0.3 E – ν =87.4 GeV, y=0.5960.000.100.200.300.400.50 0 0.1 0.2 0.3 E – ν =87.4 GeV, y=0.8020.000.100.200.300.400.50 0 0.1 0.2 0.3E – ν =160.5 GeV, y=0.3550.000.100.200.300.400.50 0 0.1 0.2 0.3 E – ν =160.5 GeV, y=0.5960.000.100.200.300.400.50 0 0.1 0.2 0.3 E – ν =160.5 GeV, y=0.8020.000.100.200.300.400.50 0 0.1 0.2 0.3E – ν =265.8 GeV, y=0.3550.000.100.200.300.400.50 0 0.1 0.2 0.3 E – ν =265.8 GeV, y=0.5960.000.100.200.300.400.50 0 0.1 0.2 0.3 E – ν =265.8 GeV, y=0.8020.000.100.200.300.400.50 0 0.1 0.2 0.3 FIG. 30: Same as Fig. 27, for the CCFR measurements of dimuon production in antineutrino-iron collisions.
W/Z data, and moreflexible parameterizations for all PDF flavors.The ATLAS and CMS experimental collaborations have recently published studies on the strangeness content ofthe proton and have come to somewhat discrepant conclusions. On the ATLAS side, two papers were published, onein 2012 [116], and one in 2014 [114]. In the 2012 study, the inclusive DIS and inclusive W ± and Z boson productionmeasurements [48] were employed to determine the strangeness fraction of the proton for one value of ( x, Q ). In the2014 study, the ATLAS 7 TeV W + c -jet, W + D ( ⋆ ) [114], and inclusive W ± /Z cross sections were used. These twoanalyses determined the ratio ( r s ) of strange to down-sea quark PDF, r s ≡ . s + s ) d at x = 0 . , Q = 1 . . (3)They find r s = 1 . +0 . − . , ATLAS (2012) ,r s = 0 . +0 . − . ATLAS (2014) , (4)which imply a rather large strangeness density.In 2014, the CMS collaboration [46] determined the ratios R s ≡ ( s + s ) u + d , at x = 0 . , Q = 1 . κ s ( Q ) = R x (cid:2) s ( x, Q ) + s ( x, Q ) (cid:3) dx R x (cid:2) u ( x, Q ) + d ( x, Q ) (cid:3) dx , (6)by using inclusive DIS, the charge asymmetry of decay muons from W ± production [46], and W + charm productiondifferential cross sections [115] at 7 TeV. They obtain R s = 0 . +0 . − . ,κ s ( Q = 20 GeV ) = 0 . +0 . − . CMS (2014) . (7)Notice that ATLAS and CMS use two different definitions, r s and R s , for the strangeness fraction, which are supposedto coincide at the initial scale Q = 1 . u ( x, Q ) = ¯ d ( x, Q ). † For comparison, at the factorization scale Q = 1 . x = 0 . r s CT14 NNLO = ¯ s ( x, Q )¯ d ( x, Q ) = 0 . ± . ,r s CT10 NNLO = ¯ s ( x, Q )¯ d ( x, Q ) = 0 . ± . . (8) † Both ATLAS and CMS studies are performed in the HERAFitter framework [107] and assume SU(2)-symmetric sea quark PDFparametrizations at the initial scale Q = 1.4 GeV. r s ratio is smaller for CT14 than for CT10.The NOMAD Collaboration has also completed a study of the strange quark PDF, relying on ν + F e → µ + + µ − + X measurements [117] at lower energies than NuTeV and CCFR. They find that the strangeness suppression factor is κ s (20 GeV ) = 0 . ± . , (9)also yielding a smaller strangeness density than the ATLAS result. In another recent study by S. Alekhin andcollaborators [118], the strange quark distribution and the ratios r s and κ s were determined in a QCD analysisincluding the NuTeV, CCFR, NOMAD and CHORUS measurements. The study uses the fixed-flavor-number (FFN)scheme for the heavy-flavor treatment. Their main result is κ s (20 GeV ) = 0 . ± . κ s CT14 NNLO = 0 . ± . ,κ s CT10 NNLO = 0 . ± . . (10)The CT14 calculation is consistent with the NOMAD central value. However, the CT14 PDF uncertainty is consider-ably larger than the uncertainty quoted in the NOMAD paper, partly because of a different convention for the PDFuncertainty. F. The CMS W + c production measurement Another experimental measurement that has direct access to the strange quark distribution is the associated pro-duction of W boson and charm quark at the LHC. Such a measurement was reported by the CMS collaboration for √ s = 7 TeV and a total integrated luminosity of 5 fb − [115]. Cross sections and ratios of cross sections with theobserved W + and W − bosons were measured differentially with respect to the absolute value of the pseudorapidity ofthe charged lepton from the W boson decay. As the theoretical cross section is not yet known at NNLO, the data werenot directly included in the global fit, but are compared here to NLO calculations based on MCFM 6.0 [119], assuminga non-zero charm quark mass, and excluding contributions from gluon splitting into a c ¯ c pair. The renormalizationand factorization scales are set to the virtuality of the W boson. The transverse momentum of the charged lepton isrequired to be at least 25 GeV. The theoretical calculation applies the same kinematical cuts as in the experimentalanalysis, but at the parton level.The left panel of Fig. 31 shows the pseudorapidity distribution of the decay charged lepton from W boson decayin W ± + c production at 7 TeV. The format of the figures is the same as in the previous comparisons. The totalexperimental errors in the figures are reasonably close to the 68% C.L. PDF uncertainties. With further experimentaland theoretical improvements, the process may contribute to the reduction of the PDF uncertainty.The right panel shows the ratio of charged lepton rapidity distributions in W + + ¯ c and W − + c production, whichprovides a handle on the strangeness asymmetry, s − ¯ s . The CT14 parametrization allows for no intrinsic s -asymmetryat the initial scale Q . (At higher scales, a tiny asymmetry is generated by 3-loop DGLAP evolution.) Our predictionreproduces the average trend of the data, however, the experimental errors are larger than the PDF uncertainties.7 CT14unshifted dataCMS 7 TeV, cross section of W ± + c p T , l >
25 GeV È Η l È d Σ H W ± + c L (cid:144) d È Η l È @ pb D CT14unshifted dataCMS 7 TeV, ratio of W ± + c p T , l >
25 GeV È Η l È Σ H W + + c L (cid:144) Σ H W - + c L FIG. 31: Comparison of the CT14 predictions to W ± + c differential cross sections (left) and to the ratio of W + + ¯ c to W − + c cross sections (right) from the CMS measurement at 7 TeV. PDF Correlations of d Σ H W ± + c)/d| | Η with CT14 PDFs at Q=100 GeVs quarkgluon - - - - - - x c o s Φ PDF Correlations of Σ H W + + c L(cid:144) Σ H W - + c L with CT14 PDFs at Q=100 GeVs quarkd quark - - - - - - x c o s Φ FIG. 32: Correlation cosines between the PDFs of select flavors, the W ± + c cross section, and the W + /W − cross section ratio,as a function of x in the PDF. The specific x ranges that are probed by CMS W ± + c cross sections can be identified by plotting correlationcosines [13] between the PDFs of various flavors, the W ± + c cross section, or cross section ratios. Fig. 32 shows suchcorrelation cosines for the s quark, gluon, and d quark PDFs at the factorization scale of 100 GeV. Lines in darkercolors correspond to bins with larger rapidities. In the case of the differential cross section, the PDF correlations aremost significant for the strange quark distributions at x = 0 . − .
1, as indicated by their strong correlations withcos φ ∼
1. The gluon does not play a significant role, due to its relatively smaller uncertainty in the same x region. Inthe case of the cross section ratio, the correlation with the strangeness is still dominant. But also, at large rapidity,the d quark contribution to the W − cross sections is mildly anti-correlated, indicating that the ratio has marginal8 Correlation betweens(x,Q) at Q=100 GeV and Σ H W + (cid:144) - L(cid:144) Σ H Z L CT10 NNLO, LHC 7 TeVCT10 NNLO, LHC 13 TeVCT14 NNLO, LHC 7 TeVCT14 NNLO, LHC 13 TeV - - - - - - x c o s Φ FIG. 33: Correlation cosines between the ratio of total cross sections for W ± production and Z boson production, and s quarkPDF from CT14 and CT10 NNLO sets, for LHC 7 and 13 TeV. sensitivity to d ( x, Q ) at x around 0 . W ± and Z boson production [13]. The correlation cosine between σ ( W ± ) /σ ( Z ) and s ( x, Q ) can be viewedin Fig. 33. As expected, we observe strong anti-correlation in a certain x range at all LHC center-of-mass energies.Compared to CT10, the x region of the strongest sensitivity shifts to higher x , and the x dependence gets flatter inCT14. V. IMPACT ON HIGGS BOSON AND t ¯ t CROSS SECTIONS AT THE LHC
Gluon fusion provides the largest cross section for production of a Higgs boson. It was the most important processfor the discovery of the Higgs boson in 2012, and it continues to be essential for detailed studies of Higgs bosonproperties. A great deal of benchmarking of hard-cross sections and PDFs for the gg initial state was carried outboth before and after the discovery [20, 92, 93, 121–123]. This was motivated in part by the fact that the PDFuncertainty for the gg initial state was comparable to the renormalization and factorization scale uncertainties in thetheoretical cross section at NNLO for producing a Higgs boson through gluon fusion. The recent calculation of thegluon fusion process at NNNLO [4] has reduced the scale uncertainty in the hard cross section still further, makingthe PDF uncertainty even more critical.Similarly, production of a t ¯ t final state is crucial to many analyses at the LHC, as both a standard model signal andas a background to new physics. By far the dominant subprocess for t ¯ t production at the LHC is gg → t ¯ t , making t ¯ t production an important benchmark for understanding the gg PDF luminosity [13], especially with the currentcalculation of the t ¯ t total inclusive cross section now available at NNLO [99, 100].Using CT10 PDFs, we have recently performed detailed analyses of the predictions for gg → H and t ¯ t cross sections,as well as their uncertainties from both the PDFs and the strong coupling α s [36, 124]. In this section we update9these studies and review CT14 predictions for gg → H and t ¯ t total and differential cross sections. A. Higgs boson from gluon fusion at the LHC
We begin with an analysis of the PDF and α s uncertainties for gg → H . For this, we have utilized the NNLOcode iHixs 1.3 [101], choosing the Higgs boson mass to be M H = 125 GeV, and with both the renormalization andfactorization scales fixed at µ = M H . Here, we have included the finite top quark mass correction (about 7%) to thefixed-order NNLO result obtained using the HQET (with infinite top quark mass approximation).To calculate the 90% C.L. PDF and α s uncertainties of an arbitrary cross section X according to the most con-ventional (Hessian) method [15], we provide error PDFs (56 in the case of CT14) to probe independent combinationsof the PDF parameters for the central α s ( M Z ) = 0 . α s ( M Z ) = 0 .
116 and 0.120. Using the error sets, the combined PDF+ α s uncertainty on X is estimated by addingthe PDF and α s uncertainties in quadrature [125]. The quadrature-based combination is exact if χ has a quadraticdependence, and X has a linear dependence on the PDF fitting parameters in the vicinity of the best fit. To accountfor some mild nonlinearities, asymmetric errors are allowed in the positive and negative directions of each eigenvectorin the fitting parameter space.Another method for estimating the PDF and α s uncertainties on X introduces Lagrange multipliers (LM) [17]. Itdoes not rely on any assumptions about the functional dependence of X on the PDF parameters. Instead, the PDFsare refitted a number of times, while fixing X to take some user-selected value in each fit. Then the uncertainty in X can be estimated by looking at how χ in the series of fits varies depending on the input value of X . The downside ofthe LM method is that it requires to repeat the PDF fit many times in order to calculate the uncertainty of each givenobservable. It is clearly impractical for general-purpose experimental analyses; however, it can be straightforwardlyperformed for a few selected observables. As a side benefit, the LM method also provides an easy way to see whichexperimental data sets in the PDF global analysis have the most impact on the PDF dependence of X . Thus, inthis section we will perform both the LM and Hessian analyses of the uncertainties for the Higgs boson and t ¯ t crosssections at the LHC.We first do these calculations while keeping the strong coupling fixed at its central value of α s ( M Z ) = 0 . χ in χ as a function of the tentative crosssection σ H for Higgs boson production via gluon fusion in pp collisions at energies √ s = 8 and 13 TeV. ∆ χ = 0corresponds to the best-fit PDFs to the CT14 experimental data set, so that the minimum of the approximatelyparabolic curves is at our best-fit prediction for σ H . Non-zero ∆ χ are obtained with an extra constraint thatenforces σ H to take the values on the horizontal axis that deviate from the best-fit ones. We have plotted the changesof both the simple χ (solid) and the χ +Tier-2 penalty (dashed), in order to see the effects of requiring that noparticular data set is too badly fit in the global analysis. (As defined in the Appendix of Ref. [23] and in [5], theTier-2 penalty makes use of the variable S n , which gives a measure of the goodness-of-fit for each individual dataset. A large S n means that the experiment is not consistent with the theory.) We see that the two curves are almostidentical over much of the range plotted, only beginning to diverge when σ H is far from the best-fit value, and one ormore experimental data sets can no longer be satisfactorily fit.0 ∆ χ σ H [pb]CT14 NNLO 8 TeV, α s = 0.118 ∆χ = 100.0 ∆χ = 36.96 χ χ +Tier-2 0 20 40 60 80 100 120 18 18.5 19 19.5 ∆ χ σ H [pb]CT14 NNLO 13 TeV, α s = 0.118 ∆χ = 100.0 ∆χ = 36.96 χ χ +Tier-2 0 20 40 60 80 100 120 40.5 41 41.5 42 42.5 43 43.5 44 44.5 45 FIG. 34: Dependence of the increase in χ in the constrained CT14 fit on the expected cross section σ H at the LHC 8 and13 TeV, for α s ( M Z ) = 0 . We can estimate asymmetric errors ( δσ H ) ± at the 90% C.L. by allowing a tolerance ∆ χ = T , with T of about 10.Given the nearly parabolic nature of these plots, we see that the 68% C.L. errors can be consistently defined using arange corresponding to ∆ χ = ( T / . . The 90% C.L. and 68% C.L. tolerance values are indicated by the upperand lower horizontal lines, respectively, in each of the plots. Finally, the red dots are the upper and lower 90% C.L.limits from the Hessian method analysis. They agree quite well with the LM analysis using the χ +Tier-2 penalty atboth 8 and 13 TeV. The effect of the Tier-2 penalty is modest, the deviations from the parabolic behavior are small.We next perform a LM scan by allowing both the σ H cross section and α s ( M Z ) to vary as “fitting parameters”,and by including the world-average constraints on α s ( M Z ) directly into the χ function. (Details can be obtained inRef. [36].) We examine χ as a function of ( α s ( M Z ) , σ H ) and trace out contours of constant χ +Tier-2 penalty inthe ( α s , σ H ) plane in Fig. 35, for √ s = 8 and 13 TeV.A contour here is the locus of points in the ( α s , σ H ) plane along which the constrained value of χ +Tier-2 isconstant. We see from Fig. 35 that the values of σ H and α s ( M Z ) are strongly correlated, as expected, since the gg fusion cross section is proportional to α s ( M Z ) . Larger values of α s ( M Z ) correspond to larger values of σ H for thesame goodness-of-fit to the global data, even though there is a partially compensating decrease of the gg luminosity.The effect of the Tier-2 penalty is very small, being most noticeable for values of α s around its global average of 0.118,which results in a squeezing of the ellipses in that region.Table III recapitulates the results from Figs. 34 and 35 by listing the central values of σ H , the PDF uncertainties,and combined PDF + α s uncertainties as obtained by the Hessian and LM methods. Here, the PDF + α s uncertaintyat 68% C.L. is obtained from the result at 90% C.L. by a scaling factor of 1 / . gg PDF luminosities for CT14, MMHT2014 [126] and NNPDF3.0 [83] PDFs at 13 TeV are shown in Fig. 36.The parton luminosity is defined as in Ref. [127]. All central values and uncertainty bands agree very well amongthe three global PDFs, in the x range sensitive to Higgs production. In Table IV, we compare the predictions for σ H from CT14 with those from MMHT2014, NNPDF3.0, and CT10. Compared to CT10, predicted σ H values for1 CT14 NNLO 8 TeV σ H [ pb ] ∆ χ +T2110100908070605040302010 0.116 0.117 0.118 0.119 0.12 α s σ H [ pb ] ∆ χ +T2110100908070605040302010 0.116 0.117 0.118 0.119 0.12 α s
40 41 42 43 44 45
FIG. 35: Contour plots of ∆ χ ( α s ( M Z ) , σ H ) plus Tier-2 penalty in the ( α s ( M Z ) , σ H ) plane, for σ H at the LHC 8 and 13 TeV. gg → H (pb), PDF unc., α s = 0 .
118 8 TeV 13 TeV68% C.L. (Hessian) 18 . . − .
3% 42 . . − . . − .
3% +2 . − . gg → H (pb), PDF+ α s unc. 8 TeV 13 TeV68% C.L. (Hessian) 18 . . − .
0% 42 . . − . . − .
9% +3 . − . σ H ( gg → H ) computed by the Hessian and LM methods, with Tier-2 penalty included. The 68%C.L. errors are given as percentages of the central values. The PDF-only uncertainties are for α s ( M Z ) = 0 . CT14 NNLO have increased by 1-1.5%. Along with the changes also present in the updated PDFs from the two otherPDF groups, the modest increase in the CT14 gluon brings σ H from the global PDF groups into a remarkably goodagreement. The projected spread due to the latest NNLO PDFs in the total cross section σ H at 13 TeV will be aboutthe same in magnitude as the scale uncertainty in its NNNLO prediction.Besides providing an estimate of the PDF uncertainty, the LM analysis allows us to identify the experimental datasets that are most sensitive to variations of σ H . In the LM scan of σ H , we monitor the changes of the equivalentGaussian variable S n for each included experimental data set. In the plots of S n values vs. σ H , of the type presented CT14 MMHT2014 NNPDF3.0 CT108 TeV 18 . +2 . − . . +1 . − . . +1 . − . . +1 . − .
13 TeV 42 . +2 . − . . +1 . − . . +1 . − . . +1 . − . TABLE IV: Higgs boson production cross sections (in pb) for the gluon fusion channel at the LHC, at 8 and 13 TeV center-of-mass energies, obtained using the CT14, MMHT2014, NNPDF3.0, and CT10 PDFs, with a common value of α s ( M Z ) = 0 . CT14 NNLOMMHT'14 NNLONNPDF3.0 NNLO10 50 100 500 10000.80.91.01.11.21.3 M X , GeV R a t i o t o C T NN L O Gluon - gluon luminosity, s = % c.l. CT14 NNLOMMHT'14 NNLONNPDF3.0 NNLO10 50 100 500 10000.80.91.01.11.21.3 M X , GeV R a t i o t o C T NN L O Gluon - gluon luminosity, s =
13 TeV, 68 % c.l. FIG. 36: The gg PDF luminosities for CT14, MMHT2014 and NNPDF3.0 PDFs at the LHC with √ s = 8 and 13 TeV, with α s ( M Z ) = 0 . in Fig. 37, we select the experiments whose S n (closely related to χ n ) depends strongly on σ H . Such experimentstypically impose the tightest constraints on σ H , when their S n quickly grows with σ H .We see that, although the CMS 7 TeV inclusive jet data (538) is relatively poorly fit by CT14 NNLO, it is alsonot very sensitive to the expected Higgs cross section. The data sets most relevant to the Higgs cross section are theHERA inclusive data set (159) at both larger and smaller values of σ H , as well as combined charm production crosssections from HERA (147); DØ Run 2 inclusive jet (514); and CCFR F p (110) at larger σ H . At small σ H , the mostsensitive data set is BCDMS F d (102), with some sensitivity also from E605 Drell-Yan (201) and LHCb 7 TeV chargeasymmetry (241). Sensitivity of σ H to CCFR dimuon data observed with CT10 [36] is no longer present.3 S n σ H [pb]CT14 NNLO 8 TeV α s =0.118 102110147159201241514538-1 0 1 2 3 4 18 18.5 19 19.5 S n σ H [pb]CT14 NNLO 13 TeV α s =0.118 FIG. 37: The equivalent Gaussian variable S n versus σ H at the LHC with √ s = 8 and 13 TeV. B. t ¯ t production cross section at the LHC Next, we consider theoretical predictions and their uncertainties for the total inclusive cross section for t ¯ t productionat the LHC, and also present some differential cross sections.In the t ¯ t case, the comparison between the Hessian and Lagrange multiplier methods for finding uncertainties isvery similar to that found for the Higgs cross section. Therefore, we just present our final estimates for the totalinclusive cross section from the Top++ code [100], given in Table V. Recent experimental measurements of the totalinclusive cross section for top-quark pair production at the LHC are given in Table VI, together with ATLAS andCMS combined determinations at √ s = 7 and 8 TeV. pp → t ¯ t (pb), PDF unc., α s = 0 .
118 7 TeV 8 TeV 13 TeV68% C.L. (Hessian) 177 + 4 . − .
7% 253 + 3 . − .
5% 823 + 2 . − . . − .
6% +2 . − . pp → t ¯ t (pb), PDF+ α s . − .
6% +5 . − .
4% +3 . − . . − .
7% +3 . − . For comparison, predictions and PDF-only errors using CT10NNLO PDFs give σ t ¯ t = 246 +4 . − . pb at 8 TeV and σ t ¯ t = 806 +2 . − . pb at 13 TeV at 68% C.L. Here we find that the Hessian and the LM methods are in very goodagreement in CT14 at √ s = 13 TeV, and agree slightly worse at √ s = 8 TeV. Measurements of t ¯ t pair production canpotentially constrain the gluon PDF at large x , if correlations between the gluon, α s ( M Z ) and the top-quark massare accounted for. Given the current experimental precision of t ¯ t measurements, the impact of such data in a globalPDF fit is expected to be moderate; related exploratory studies can be found in Refs. [74, 120].In Figs. 38, 39 and 40, the normalized top-quark transverse momentum p T and rapidity y distributions at approx-imate NNLO ( O ( α s )) are compared to the CMS [58] and ATLAS [59] measurements, at a center of mass energy4 σ exp t ¯ t (pb) 7 TeV (dilepton channel) 8 TeV (lep+jets)ATLAS [128],[129] 177 ± (stat) ± (syst) ± (lumi.) ± (stat)+22 − ± (lumi.) ± (beam) CMS [130],[131] 161 . ± . (stat)+5 . − . ± . (lumi.) . ± . (stat)+29 − ± (lumi.) . ± . (stat) ± . (syst) ± . (lumi.) . ± . (stat) + 5 . (syst) ± . (lumi.) TABLE VI: Measurements of total inclusive cross sections for top-quark pair production at LHC center-of-mass energies of 7,8, and 13 TeV, for an assumed top-quark mass of 172.5 GeV. √ s = 7 TeV. The yellow bands represent the CT14 PDF uncertainty evaluated at the 68%C.L. with the program DiffTop [74] based on QCD threshold expansions beyond the leading logarithmic approximation, for one-particleinclusive kinematics. The value of the top-quark mass here is m t = 173 . p T distribution and the momentum fraction x carriedby the gluon is shown, in four different p T bins at the LHC √ s = 8 and 13 TeV. The cosine correlation at √ s =7 TeV exhibits identical features to that of √ s = 8 TeV. It is therefore omitted. A strong correlation between the p T distribution and large x -gluon ( x ≈ .
1) is observed for both LHC energies, although the cosines exhibit differentpatterns of x dependence. Finally, in Fig. 42 we present the absolute, rather than normalized, differential p T and y distributions for top-quark production, together with the relative PDF uncertainties, at the LHC with √ s = 7, 8 and13 TeV. VI. DISCUSSION AND CONCLUSION
In this paper, we have presented CT14, the next generation of NNLO (as well as LO and NLO) parton distributionsfrom a global analysis by the CTEQ-TEA group. With rapid improvements in LHC measurements, the focus of theglobal analysis has shifted toward providing accurate predictions in the wide range of x and Q covered by the LHCdata. This development requires a long-term multi-prong effort in theoretical, experimental, and statistical areas.In the current study, we have added enhancements that open the door for long-term developments in CT14 method-ology geared toward the goals of LHC physics. This is the first CT analysis that includes measurements of inclusiveproduction of vector bosons [43, 46–48] and jets [51, 52] from the LHC at 7 and 8 TeV as input for the fits. Wealso include new data on charm production from DIS at HERA [33] and precise measurements of the electron chargeasymmetry from DØ at 9 . − [14]. These measurements allow us to probe new combinations of quark flavors thatwere not resolved by the previous data sets. As most of these measurements contain substantial correlated systematicuncertainties, we have implemented these correlated errors and have examined their impact on the PDFs.On the theory side, we have introduced a more flexible parametrization to better capture variations in the PDFdependence. A series of benchmark tests of NNLO cross sections, carried out in the run-up for the CT14 fit for all keyfitted processes, has resulted in better agreement with most experiments and brought accuracy of most predictions tothe truly NNLO level. We examined the PDF errors for the important LHC processes and have tested the consistencyof the Hessian and Lagrange Multiplier approaches. Compared to CT10, the new inputs and theoretical advancementsresulted in a softer d/u ratio at large x , a lower strangeness PDF at x > .
01, a slight increase in the large- x gluon5 p tT @ GeV D (cid:144) Σ d Σ (cid:144) dp T @ G e V - D m t = DiffTop approx NNLO è CMS data 5.0 @ fb - D CT14 68 % CL p tT @ GeV D da t a (cid:144) t heo r y LHC 7 TeV, m t = H central L , CT14NNLO è CMS data (cid:144)
DiffTop approx NNLO
PDF unc. 68 % CL FIG. 38: Normalized final-state top-quark p T differential distribution at CMS 7 TeV. (of order 1%), and wider uncertainty bands on d/u , ¯ d/ ¯ u , and q − ¯ q combinations at x of order 0 .
001 (probed by LHC
W/Z production). Despite these changes in central predictions, the CT14 NNLO PDFs remain consistent with CT10NNLO within the respective error bands.Some implications of CT14 predictions for phenomenological observables were reviewed in Sections IV and V.Compared to calculations with CT10 NNLO, the gg → H total cross section has increased slightly in CT14: by 1.6%at the LHC 8 TeV and by 1.1 % at 13 TeV. The t ¯ t production cross sections have also increased in CT14 by 2.7% at 8TeV and by 1.4% at 13 TeV. The W and Z cross sections, while still consistent with CT10, have slightly changed asa result of reduced strangeness. Common ratios of strangeness and non-strangeness PDFs for CT14 NNLO, shown inEqs. (8) and (10), are consistent with the independent ATLAS, CMS, and NOMAD determinations within the PDFuncertainties.The final CT14 PDFs are presented in the form of 1 central and 56 Hessian eigenvector sets at NLO and NNLO.The 90% C.L. PDF uncertainties for physical observables can be estimated from these sets using the symmetric [21]6 - - y t (cid:144) Σ d Σ (cid:144) d y t m t = % CLDiffTop approx NNLO
PDF unc. è CMS data 5.0 @ fb - D - - y t da t a (cid:144) t heo r y LHC 7 TeV, m t = H central L , CT14NNLO 68 % CL è CMS data (cid:144)
DiffTop approx NNLO
PDF unc.
FIG. 39: Normalized final-state top-quark rapidity distribution at CMS 7 TeV. or asymmetric [5, 56] master formulas by adding contributions from each pair of sets in quadrature. These PDFsare determined for the central QCD coupling of α s ( M Z ) = 0 . α s value. Forestimation of the combined PDF+ α s uncertainty, we provide two additional best-fit sets for α s ( M Z ) = 0 .
116 and0.120. The 90% C.L. variation due to α s ( M Z ) can be estimated as a half of the difference in predictions from the two α s sets. The PDF+ α s uncertainty, at 90% C.L., and including correlations, can also be determined by adding thePDF uncertainty and α s uncertainty in quadrature [125].At leading order, we provide two PDF sets, obtained assuming 1-loop evolution of α s and α s ( M Z ) = 0 . α s and α s ( M Z ) = 0 . α s ( M Z ) = 0 . − .
123 and additional sets in heavy-quark schemes with up to 3, 4, and 6 active flavors.Phenomenological applications of the CT14 series and the special CT14 PDFs (such as allowing for nonperturbativeintrinsic charm contribution) will be discussed in a follow-up study [134].Parametrizations for the CT14 PDF sets are distributed in a standalone form via the CTEQ-TEA website [136], or7 p tT @ GeV D (cid:144) Σ d Σ (cid:144) d P T @ G e V - D m t = DiffTop approx NNLO è ATLAS data 4.6 @ fb - D CT14 68 % CL p tT @ GeV D da t a (cid:144) t heo r y LHC 7 TeV, m t = H central L , CT14NNLO è ATLAS data (cid:144)
DiffTop approx NNLO
PDF unc. 68 % CL FIG. 40: Normalized final-state top-quark p T differential distribution at ATLAS 7 TeV. as a part of the LHAPDF6 library [7]. For backward compatibility with version 5.9.X of LHAPDF, our website alsoprovides CT14 grids in the LHAPDF5 format, as well as an update for the CTEQ-TEA module of the LHAPDF5library, which must be included during compilation to support calls of all eigenvector sets included with CT14 [137]. Acknowledgments
This work was supported by the U.S. DOE Early Career Research Award de-sc0003870; by the U.S. Departmentof Energy under Grant No. DE-FG02-96ER40969, de-sc0013681, and DE-AC02-06CH11357; by the U.S. NationalScience Foundation under Grant No. PHY-0855561 and PHY-1417326; by Lightner-Sams Foundation; and by theNational Natural Science Foundation of China under Grant No. 11165014 and 11465018.8 -1-0.5 0 0.5 1 0.0001 0.001 0.01 0.1 1 c o s ( φ ) x for the gluon(x)LHC 8 TeV, m top(pole) = 173.3 GeV, CT14NNLO PDFs1 < p T < 100 GeV100 < p T < 200 GeV200 < p T < 300 GeV300 < p T < 400 GeV -1-0.5 0 0.5 1 0.0001 0.001 0.01 0.1 1 c o s ( φ ) x for the gluon(x)LHC 13 TeV, m top(pole) = 173.3 GeV, CT14NNLO PDFs1 < p T < 100 GeV100 < p T < 200 GeV200 < p T < 300 GeV300 < p T < 400 GeV FIG. 41: The correlation cosine as a function of x -gluon for the top-quark p T distribution in t ¯ t production at the LHC at √ s = 8 and 13 TeV. d σ / dp T [ pb / G e V ] p T [GeV]DiffTop approx NNLO, m top(pole) = 173.3 GeV, CT14 NNLO 68% C.L.7 TeV8 TeV13 TeV 0 50 100 150 200 250 300−3 −2 −1 0 1 2 3 d σ / dy t [ pb ] y t DiffTop approx NNLO, m top(pole) = 173.3 GeV, CT14 NNLO 68% C.L.7 TeV8 TeV13 TeV
FIG. 42: Absolute differential p T and y distributions for the final-state top-quark in t ¯ t production at the LHC at √ s = 7, 8,and 13 TeV. Appendix: Parametrizations in CT14
Parton distribution functions are measured by parameterizing their x -dependence at a low scale Q . For each choiceof parameters, the PDFs are computed at higher scales by DGLAP evolution; and the parameters at Q are adjustedto optimize the fit to a wide variety of experimental data. Traditional parametrizations for each flavor are of the form x f a ( x, Q ) = x a (1 − x ) a P a ( x ) (11)where the x a behavior at x → − x ) a behavior at x → P a ( x ) is assumed to be slowly varying, because there is no reason to expect finestructure in it even at scales below Q , and evolution from those scales up to Q provides additional smoothing.In the previous CTEQ analyses, P a ( x ) in Eq. (11) for each flavor was chosen as an exponential of a polynomial in x or √ x ; e.g., P ( x ) = exp( a + a √ x + a x + a x ) (12)for u v ( x ) or d v ( x ) in CT10 [6]. The exponential form conveniently enforces the desired positive-definite behavior for9the PDFs, and it suppresses non-leading behavior in the limit x → √ x , which is similar to what wouldbe expected from a secondary Regge trajectory. However, this parametrization has two undesirable features. First,because the exponential function can vary rapidly, the power laws x a and (1 − x ) a , which formally control the x → x → x (say x . . x (say x & . a √ x ), exp( a x ), and exp( a x ) to each other causes the parameters a , a , a to be strongly correlated with each other in the fit. This correlation may destabilize the χ minimization andcompromise the Hessian approach to uncertainty analysis, since that approach is based on a quadratic dependence of χ on the fitting parameters, which is only guaranteed close to the minimum.We introduce a better style of parametrization in CT14. We begin by replacing P a ( x ) by a polynomial in √ x ,which avoids the rapid variations invited by an exponential form. Low-order polynomials have been used previouslyby many other groups; however, polynomials with higher powers were less widespread. We add them now to providemore flexibility in the parametrization. In particular, for the best-constrained flavor combination u v ( x ) ≡ u( x ) − ¯u( x )we use a fourth-order polynomial P u v = c + c y + c y + c y + c y , (13)where y = √ x . But rather than using the coefficients c i directly as fitting parameters, we re-express the polynomialas a linear combination of Bernstein polynomials : P u v = d p ( y ) + d p ( y ) + d p ( y ) + d p ( y ) + d p ( y ) , (14)where p ( y ) = (1 − y ) ,p ( y ) = 4 y (1 − y ) ,p ( y ) = 6 y (1 − y ) ,p ( y ) = 4 y (1 − y ) ,p ( y ) = y . (15)This re-expression does not change the functional form of P u v : it is still a completely general fourth-order polynomialin y = √ x . But the new coefficients d i are less correlated with each other than the old c i , because each Bernsteinpolynomial is strongly peaked at a different value of y . (The flexibility of the parametrization can be increased byusing higher order polynomials; the generalization of Eq. (15) to higher orders is obvious—the numerical factors arejust binomial coefficients.)In practice, we refine this procedure as follows. First, as a matter of convenience, we set d = 1 and supply inits place an overall constant factor, which is determined by the number sum rule R u v ( x ) dx = 2. We then set d = 1 + a / − x ) a behavior of u v ( x ) at large x by canceling the first subleadingpower of (1 − x ) in P u v : x u v ( x ) → const × (1 − x ) a × (cid:2) O ((1 − x ) ) (cid:3) for x → . (16)We use the same parametrization for d v ( x ) ≡ d( x ) − ¯d( x ), with the same parameter values a and a ; but, ofcourse, independent parameters for the coefficients of the Bernstein polynomials and the normalization, which is setby R d v ( x ) dx = 1.0Tying the valence a parameters together is motivated by Regge theory, and supported by the observation thatthe value of a obtained in the fit is not far from the value expected from Regge theory. (The a values for u , d , ¯ u ,and ¯ d are expected to be close to 0 from the Pomeron trajectory; but that leading behavior is expected to cancel inu v = u − ¯u and d v = d − ¯d, revealing the subleading vector meson Regge trajectory at a ≃ . a and a parameters for d v to be independent of those for u v wouldreduce χ ≈ x , where the fractional uncertainty isalready very large. The additional fractional uncertainty at small x generated by allowing different a powers is alsonot important, because that uncertainty only appears in the valence quantities u ( x ) − ¯ u ( x ) and d ( x ) − ¯ d ( x ); whilemost processes of interest are governed by the much larger u ( x ), d ( x ), ¯ u ( x ), ¯ d ( x ) themselves.In addition to theoretical arguments that the power laws a should be the same for u v and d v [135], χ tends to beinsensitive to the differences. A large portion of the data included in the global fit are from electron and muon DISon protons, which is more sensitive to u and ¯ u than to d and ¯d because of the squares of their electric charges. Hence,when similar parametrizations are used for P u v and P d v , the uncertainties of a (d v ) and a (d v ) are relatively large.Our assumption a (u v ) = a (d v ) forces u v ( x ) / d v ( x ) to approach a constant in the limit x →
1. It allows ourphenomenological findings to be relevant for the extensive discussions of what that constant might be [97, 98]. However,the experimental constraints at large x are fairly weak: we can find excellent fits over the range − . < a (d v ) − a (u v ) < . χ . Hence both u v ( x ) / d v ( x ) → v ( x ) / d v ( x ) → ∞ at x → a (u v ) = a (d v ) does not restrict the calculateduncertainty range materially in regions where it is not already very large.By way of comparison, if we use the CT10 NNLO [6] form (12) for u v and d v , we obtain a slightly better fit ( χ lower by 8) with an unreasonable a ≈ . a = 0 . v and d v that is equivalent toEq. (13) with c = c = 0, with the power-law parameters a and a allowed to differ between u v and d v . If we usethis MSTW parametrization for the valence quarks at our Q = 1 . χ increases by 64, even though the total number of fitting parameters is the same. This decline in the fitquality comes about because the freedom to have a (u v ) = a (d v ) and a (u v ) = a (d v ) is not actually very helpful,as noted above; so setting c = c = 0 does not leave an adequate number of free parameters.The more recent MMHT2014 [126] PDF fit uses full fourth-order polynomials for u v and d v . In our fit, however,we find that no significant improvement in χ would result from treating d ( u v ) and d ( d v ) as free parameters, ratherthan choosing them to cancel the first subleading behavior at x →
1, as we have done.Meanwhile the HERA PDF fits [34, 139, 140] use much more restricted forms, equivalent to c = c = c = 0 for u v and c = c = c = c = 0 for d v . Those forms are far too simple to describe our data set: using them in place ofour choice increases χ by more than 200.We made a case in previous work [141] to repackage polynomial parametrizations like (13) as linear combinationsof Chebyshev polynomials of argument 1 − √ x . This method has been adopted in the recent MMHT2014 fit [126].However, we now contend that repackaging based on a linear combination of Bernstein polynomials, as we do in CT14,1is much better. The full functional forms available in the fit are, of course, the same either way. But, because each ofthe Bernstein polynomials has a single peak, and the peaks occur at different values of x , the coefficients that multiplythose polynomials mainly control distinct physical regions, and are therefore somewhat independent of each other.In contrast, every Chebyshev polynomial of argument 1 − √ x has a maximum value ± x = 0 and x = 1,along with an equal maximum magnitude at some interior points. All Chebyshev polynomials are important over theentire range of x , so their coefficients are strongly correlated in the fit. This causes minor difficulties in finding thebest fit and major difficulties in using the Hessian method to estimate uncertainties based on orthogonal eigenvectors.Furthermore, using Bernstein polynomials makes it easy to enforce the desired positivity of the PDFs in the x → x → P g ( y ) = g [ e q ( y ) + e q ( y ) + q ( y )] (17)where q ( y ) = (1 − y ) ,q ( y ) = 2 y (1 − y ) ,q ( y ) = y . (18)However, in place of y = √ x , we use the mapping y = 1 − (1 − √ x ) = 2 √ x − x . (19)This mapping makes y = 1 − (1 − x ) / O ((1 − x ) ) and hence P g ( y ) → const + O (cid:0) (1 − x ) (cid:1) (20)in the limit x → − x ) at x →
1. We have 5free parameters to describe the gluon distribution, including g which governs the fraction of momentum carried bythe gluons. The best fit has a = 3 .
8, with the range 2 . < a < . χ .In contrast, CT10 NNLO [6] again used the form (12) for the gluon distribution, where a was frozen at an arbitraryvalue of 10 because χ was rather insensitive to it. That left the same number of free parameters as are used here,but didn’t allow anything to be learned about the behavior at very large x .If we use (12) for the gluon in our present fit, the resulting χ is nearly as good, but again this choice yields almostno information about the sixth parameter a : a range of ∆ χ = 1 includes − . < a <
12. The negative a part ofthat range corresponds to an integrably singular gluon probability density at x →
1, which is not actually forbiddentheoretically; but would be totally unexpected. This older parametrization would bring in unmotivated complexity inthe large-x region that is not indicated by any present data. To test that our parametrization has adequate flexibility,we made similar fits using somewhat higher order Bernstein polynomials, including up to a total of 10 more freeparameters. We calculated the uncertainty for the gg → H cross section at 8 TeV using the Lagrange Multipliermethod, and found very little variation in the range of the prediction. We also calculated the range of uncertainty in α s ( m Z ) obtained from our fits at 90% confidence (including our Tier 2 penalty). The extra freedom in parametrization2increased the uncertainty range only slightly: 0 . − .
121 using the CT14 parametrization; 0 . − .
123 using themore flexible one.The sea quark distributions ¯ d and ¯ u were parametrized using fourth-order polynomials in y with the same mapping y = 2 √ x − x that was used for the gluon. We assumed ¯ u ( x ) / ¯ d ( x ) → x →
0, which implies a (¯ u ) = a ( ¯ d ). Asthe strangeness content is constrained rather poorly, we used a minimal parametrization P s +¯ s = const, with a tiedto the common a of ¯ u and ¯ d . Even fewer experimental constraints apply to the strangeness asymmetry, so we haveassumed s ( x ) = ¯ s ( x ) in this analysis. Thus, we have just two parameters for strangeness in our Hessian method: a and normalization. In view of more upcoming data on measuring the asymmetry in the production cross sections of W + ¯ c and W + c from the LHC, we plan to include s ( x ) = ¯ s ( x ) in our next round of fits.In all, we have 8 parameters associated with the valence quarks, 5 parameters associated with the gluon, and13 parameters associated with sea quarks, for a total of 26 fitting parameters. Hence there are 52 eigenvector setsgenerated by the Hessian method that captures most of the PDF uncertainty.The Hessian method tends to underestimate the uncertainty for PDF variations that are poorly constrained, becausethe method is based on the assumption that χ is a quadratic function of the fitting parameters; and that assumptiontends to break down when the parameters can move a long way because of a lack of experimental constraints. Thiscan be seen, for example, for the case of the small- x gluon uncertainty, by a Lagrange Multiplier scan in which a seriesof fits are made with different values of the independent variable g ( x, Q ) at x = 0 . Q = Q .In order to include the wide variation of the gluon distribution that is allowed at small x , we therefore supplementthe Hessian sets with an additional pair of sets that were obtained using the Lagrange Multiplier method: one withenhanced gluon and one with suppressed gluon at small x , as was already done in CT10. In CT14, we also include anadditional pair of sets with enhanced or suppressed strangeness at small x ; although it is possible that treating a ( s )as a fitting parameter independent from a (¯ u ) = a ( ¯ d ) would have worked equally well.In summary, we have a total of 56 error sets: 2 ×
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