Deuterium scattering experiments in CTEQ global QCD analyses: a comparative investigation
SSMU-HEP-20-08JLAB-THY-21-3313
Deuterium scattering experiments in CTEQ global QCD analyses:a comparative investigation
A. Accardi ( a,b ) , T. J. Hobbs ( c,b,d, ∗ ) , X. Jing ( c,b ) , and P. M. Nadolsky ( c ) ( a ) Hampton University, Hampton, VA 23668, U.S.A. ( b ) Jefferson Lab, Newport News, VA 23606, U.S.A. ( c ) Department of Physics, Southern Methodist University, Dallas, TX 75275-0181, U.S.A. ( d ) Department of Physics, Illinois Institute of Technology, Chicago, IL 60616, U.S.A. ∗ (Dated: February 3, 2021)Experimental measurements in deep-inelastic scattering and lepton-pair production on deuteriumtargets play an important role in the flavor separation of u and d (anti)quarks in global QCDanalyses of the parton distribution functions (PDFs) of the nucleon. We investigate the impactof theoretical corrections accounting for the light-nuclear structure of the deuteron upon the fitted u, d -quark, gluon, and other PDFs in the CJ15 and CT18 families of next-to-leading order CTEQglobal analyses. The investigation is done using the L sensitivity statistical method, which providesa common metric to quantify the strength of experimental constraints on various PDFs and ratiosof PDFs in the two distinct fitting frameworks. Using the L sensitivity and other approaches,we examine the compatibility of deuteron data sets with other fitted experiments under variedimplementations of the deuteron corrections. We find that freely-fitted deuteron corrections modifythe PDF uncertainty at large momentum fractions and will be relevant for future PDFs affectingelectroweak precision measurements. Contents
I. Introduction
II. Low-energy QCD effects
III. Methods L sensitivity technique 8 IV. Comparison of deuteron data impact in the CJ15 and CT18 fits d/u
PDF ratio 12C. Impact on the valence PDFs in the LHC EW precision region 13D. Impact of deuteron corrections on the gluon PDF 15E. Valence-sector PDF pulls: the d/u ratio 16F. PDF pulls in the gluon and light-quark sea sectors 19
V. Conclusion Acknowledgments A. Comparison procedure and technical details References ∗ Electronic address: ∗ corresponding author: [email protected], a r X i v : . [ h e p - ph ] F e b I. INTRODUCTIONA. Light-parton structure of the nucleon in electroweak precision measurements
Electroweak precision tests of the Standard Model (SM) at hadron colliders are nontrivially sensitive to the partonflavor composition of initial-state hadrons. For spin-independent inclusive observables at the CERN Large HadronCollider (LHC) – or, indeed, at any high-enough energy facility such as the Relativistic Heavy Ion Collider, theJefferson Lab CEBAF accelerator, or the future Electron-Ion Collider – this flavor composition is typically specifiedby helicity-averaged parton distribution functions (PDFs) of the proton. The PDFs, f ( x, Q ), have long been ofstrong interest from the perspective of both fundamental Quantum Chromo-Dynamics (QCD) as well as particlephenomenology, given that they quantify the probability of resolving a quark or gluon constituent of flavor f carryinga fraction x of the proton’s longitudinal momentum in a scattering process with a squared energy scale Q > ∼ .For this reason, the PDFs play a central role in predicting cross sections for pp collisions at the LHC, and, in particular,their accuracy influences the ability of LHC measurements or other high-energy data to constrain the SM parameters,including in the electroweak sector.Due to the challenge of reducing their uncertainties and empirically distinguishing among their parton flavors,PDFs have historically been determined most robustly through “global QCD fits” [1–4], now increasingly performedat the next-to-next-to-leading order (NNLO) accuracy in α s , and drawing upon large collections of experimentalmeasurements sensitive to QCD and different underlying PDF combinations. In spite of the growing number of LHCmeasurements, deeply-inelastic scattering (DIS) experiments involving fixed hadronic or nuclear targets at BCDMS,NMC, SLAC, and JLab continue to provide key information to disentangle the PDFs in recent global QCD analysessuch as CJ15 [5], ABMP16 [6], CT18 [7], NNPDF3.1 [8], and MSHT20 [9]. The fixed-target experiments complementanalogous DIS collisions at ep collider HERA by extending the momentum fraction coverage to larger x valuesand adding measurements on deuterium targets. In fact, such experiments provide the leading constraints on the(anti)quark PDFs at low scales Q and large momentum fractions x > ∼ .
05, as well as on the gluon PDF by observingscaling violations over the same kinematic region [10–12].In precision tests of the electroweak sector, the substantial PDF dependence of the involved theoretical calculationsaffects experimental determinations of SM parameters, such as the weak-mixing angle θ W extracted from the A FB forward-backward asymmetry measured in the production of Z bosons during Runs 1 and 2 of the LHC. Fig. 1illustrates typical Hessian correlations [13–15] of PDFs (right) and PDF combinations (left) with the sin θ W valuesextracted from 8 TeV A FB measurements at the LHC. Here, the correlations are computed using the preliminary(unpublished) ATLAS Run-1 data [16] on sin θ W extracted with individual PDF eigenvector sets of the CT14 NNLOensemble [17].In the left subfigure, we see that the values of the extracted sin θ W are strongly correlated with the valencecombinations of light-quark PDFs at Q = 81 .
45 GeV, d val ( x, Q ) ≡ d ( x, Q ) − ¯ d ( x, Q ) at x ≈ . − .
05 and u val d val ( s + s - )/( u - + d - ) d / u u - / d - - - - - C o rr e l a t i on i nde x Correlation of sin θ w ( A FB @ LHC TeV ) and CT14 NNLO PDFs at Q = GeV d - u - g udscb10 - - - - C o rr e l a t i on i nde x Correlation of sin θ w ( A FB @ LHC TeV ) and CT14 NNLO PDFs at Q = GeV
FIG. 1: Hessian correlations [13–15] for the values of sin θ W extracted from Z boson production at the LHC 8 TeV.Left: correlations with valence PDFs and PDF ratios at Q =81.45 GeV, plotted as a function of x for CT14 NNLOPDFs. Right: the same, for correlations with PDFs of individual flavors. HERAI + IIBCDMS dNMC rat.CCFR F3E866 rat.E866ppZyCDF2LHCb7ZWrapATL7ZWLHCb8WZTotal12.0 12.5 13.0 - u val ( ) ( Δ χ ) e x p t. CT18Z NNLO
TotalHERAI + IIBCDMS pBCDMS dNMC rat.CCFR F3E605E866ppCMS7Masy2LHCb7ZWrapATL7ZWCMS8WasyLHCb8WZCMS8 jets6.8 6.9 7.0 7.1 - - d val ( ) ( Δ χ ) e x p t. CT18Z NNLO
FIG. 2: Lagrange Multiplier scans on d val ( x = 0 . , Q = 85 GeV) (left) and u val ( x = 0 . , Q = 85 GeV) (right),showing the changes in the χ for all data sets and most sensitive experimental data sets in the CT18Z NNLOglobal QCD analysis [7]. u val ( x, Q ) ≡ u ( x, Q ) − ¯ u ( x, Q ) at x ≈ . − .
1. In addition, significant correlations with the extracted sin θ W existat higher x > ∼ . d/u , and again for d val . Remarkably, the correlations are weakerwith the PDFs of individual parton flavors (shown at right) than with the valence combinations. An anti-correlationwith the d -quark (green dashed line) at x ∼ .
3, affecting A FB at smaller x via the valence-quark sum rule, is evidentin this case, though not exceptionally strong.The sizable correlations between fitted PDFs and sin θ W in Fig. 1 are consistent with the significant PDF uncer-tainties on these and similar BSM-sensitive quantities, including the W boson mass, M W , and Higgs cross section, σ H . For this reason, the realization of next-generation precision in the determination of these electroweak parametersis critically dependent on the reduction of their associated PDF uncertainties, including the high- x uncertainty of the d -quark and gluon ( g ) PDFs, as well as of d val and d/u .We might therefore ask where the experimental constraints on the relevant PDF combinations for LHC electroweakprecision tests arise from. While direct measurements at the LHC will supply increasing information on the PDFsaffecting A FB [18] and other observables [19], recent CTEQ studies [5, 7, 11, 12, 20] find that deep-inelastic scatteringexperiments on nuclear targets will continue to provide strong constraints on the down-quark PDFs in the nucleonin the near future. In fact, in a global QCD analysis, experimental measurements made solely on proton targets areat present insufficient for full separation of parton distributions for d , s , g , and anti-quark flavors. Assuming parton-level charge symmetry, d p ( x, Q ) ≈ u n ( x, Q ), between the PDFs in the proton p and the neutron n , and correcting forlow-energy nuclear effects [10], one can then use scattering processes on the deuteron or heavier nuclei to constrainthe down-type PDFs in the proton.We illustrate the importance of fixed target data in the determination of the weak mixing angle with the help ofLagrange Multiple (LM) scans [21] in Fig. 2, in which we examine the dependence of the figure-of-merit function χ in the CT18Z NNLO analysis [7] on the values of the valence u val (left) and d val (right) quarks at x = 0 .
03 and Q = 85 GeV. We plot the change in χ , as compared to the value in the best fit, for all data sets (labeled as “Total”in the figure) and for individual experiments with the highest sensitivity to this PDF combination. The curves forthe experimental data sets are labeled according to the convention in Table I. The LM scans show that a small groupof DIS experiments – NMC ratio of d and p DIS cross sections [22], inclusive HERA I+II DIS [23], BCDMS p and d reduced cross sections [24, 25], CCFR F structure function [26] – contribute the largest variations in ∆ χ when thevalence PDFs are varied, together with several lepton pair production experiments by ATLAS, LHCb, E605, and E866.In the case of d val in the right Fig. 2, the BCDMS measurements on p and d show somewhat different preferences,with the deuteron data clearly preferring a higher d val (0 . ,
85 GeV). In these and other cases, the deuteron DISmeasurements, with their large numbers of precise data points, have large statistical significance in the global fit. Asthis LM scan also illustrates, the proton and deuteron data sets in the CT18 fit sometimes prefer somewhat differentvalues of d val ( x, Q ), which in turn may hamper the efforts for reducing the PDF uncertainty in the relevant EWprecision measurements.In this article, we will employ a statistical technique called the L sensitivity [12] that can be viewed as a fastapproximation to the LM scans that are usually very computing-extensive. With this technique, we will surveyagreement between the constraints on the PDFs from the deuteron and other data sets in a wide range of x andfor various treatments of deuteron corrections. While we have investigated sensitivities for various PDF flavors, ourpresentation will focus on the sensitivities to u , d , and g PDFs that are most affected.Apart from its significance for the LHC and SM phenomenology, the physics of large- x quark PDFs is interestingin its own right. Nonperturbative QCD approaches [1, 27, 28] and lattice QCD [29, 30] provide increasingly completepredictions for the flavor composition of unpolarized protons at x →
1, which in turn inform one on the role of colorconfinement in the binding of the valence quarks. These predictions can then be confronted with phenomenologicaldeterminations of Mellin moments and PDFs at large x . For example, Ref. [31] observes that the proton (BCDMS F p , E866 pp DY, HERAI+II ep DIS) and deuteron (BCDMS F d , NMC p/d ratio) experiments in the CT18 NNLOanalysis have somewhat different preferences for the effective exponents β controlling the (1 − x ) β falloff of the valenceup and down quark PDFs at x →
1. In turn, these differences impact comparisons of phenomenological PDFs againstlarge- x predictions from quark counting rules [32–34] and other nonperturbative approaches [27, 28]. The analysismethods utilized in this paper can shed light on these issues. B. The role of nuclear-medium effects
Extracting parton-level information from nuclear data sets involving the deuteron or heavier targets requires anunderstanding of the effects of the nuclear environment [4, 10]. The trivial dependence on the nuclear atomic number A and charge Z is normally implemented by constructing a nuclear PDF as a linear combination of the PDFs onfree protons and neutrons, as reviewed, e.g., in Sec. 5. A of Ref. [3]. On top of this trivial ( A, Z ) dependence, low-energy interactions in the nuclear medium may modify the quark and gluon distributions relatively to those in freenucleons. The nuclear corrections that account for these deviations can be computed with increasing sophisticationand connection to the formal theory describing low-energy dynamics. One can, for example, utilize phenomenological,data-driven ratios to convert nuclear-target cross sections to free-nucleon ones [7, 35]; parametrize and fit the nucleardeformation of the PDFs either inside the deuteron in a nucleon PDF fit [9, 36] or in a heavy nucleus in a nuclearPDF fit [37–42]; or, finally, adopt a dynamical model of the low-energy nucleon-nucleon interactions and calculatethe hard cross-section as a double convolution of parton distributions inside the nucleons and nucleon wave functionsinside the nuclear targets [5, 43, 44]. The resulting extractions of the nucleon PDFs from nuclear data then have adependence on the assumed nuclear corrections.The specific methods used typically differ when analyzing light nuclear targets ( e.g. , the deuteron) or heavy nuclei( e.g. , Fe). Here, we concentrate on and summarize the techniques utilized in the CT18 and CJ15 fits which form thestarting point for the study presented in this paper, and then briefly mention other approaches:1.
Deuteron corrections . A dynamic deuteron correction in the CJ15 next-to-leading order (NLO) PDF fit [5]was applied to any process involving interaction with a deuterium target, including both DIS and Drell-Yanexperiments, as detailed at greater length in Sec. II A. This correction allows the CJ15 NLO fit to include thefixed-target DIS from SLAC and JLab at the largest x not accessed by other groups. The correction can beunderstood as arising from several dynamical effects, such as the relativistic Fermi motion of bound nucleons,binding corrections, and nucleon off-shellness effects. In practice these mechanisms are taken into accountvia convolutions of free-hadron cross sections with nuclear smearing functions calculated starting from boundnucleon wave-functions. Nuclear correction mostly affects the intermediate and large regions of x .The CJ15 analysis also applies a phenomenological parametrization for the off-shell deformation of the scatterednucleon’s structure function (in short, “off-shell corrections”) with parameters fitted to data to increase themodel flexibility. Care is taken that the valence quark number inside the nucleon is not modified; since theoff-shell function is flavor independent and has no significant dependence on Q , it must then change the sign inthe interval of [0 , { x, Q } region where the deuteron corrections are small compared to the precision of data,as is done, e.g., in the CT18 analysis [7].2. Heavy-nucleus effects . Nucleon PDF fits may include DIS experiments performed on heavy nuclear targets,such as CCFR [45] and NuTeV [46], involving Fe, and CHORUS [47], with Pb. It has been known empiricallyfor some time that the structure functions of these heavier nuclear targets exhibit x - and A -dependent deviationsfrom the structure function of the physical deuteron, owing to a variety of physical processes characterizing thenuclear medium [48–51], including the heavy-nucleus analogue of the EMC and Fermi-motion effects discussedfor the deuteron at high x , and nuclear (anti-)shadowing phenomena at lower x .To address these effects, the CT group corrects DIS cross sections on iron and copper to the corresponding crosssections on deuterium using a phenomenological parametrization of the nuclear-to-deuteron cross section ratiosbased on results in [49] (see also [52], Fig. 2a), which depends on x but not on Q in the fitted region. To includethe heavy-nuclear data in the MSHT20 [9] and earlier MMHT fits, a nuclear correction factor, R f , [36] havingthe form f A ( x, Q ) = R f ( x, Q , A ) f ( x, Q ), where f A ( x, Q ) is defined to be the PDF of a proton bound in anucleus of mass number A , was determined. This was assessed using the de Florian et al. nuclear PDF (nPDF)of Ref. [39], then weighted by a 3-parameter modification factor as in Ref. [36], which is actively refitted alongwith the PDF-associated parameters. NNPDF [53] determines the uncertainty due to heavy-nuclear effects usinga similar statistical procedure as for the deuteron.As can be seen from this summary, global analyses vary in their treatments of the nuclear effects, but with thefrequent conclusion that the resulting differences are marginal in comparison to contemporary PDF uncertainties.Indeed, the experiments in question are fitted reasonably well, while the higher-order QCD and parametrizationuncertainties on the PDFs are comparable for the most part to the nuclear ones at NLO or even NNLO. In thepresent study, however, we identify several areas where the deuteron corrections will play a prominent role in thenear future, as the field advances toward higher accuracy in the determination of nucleon PDFs. We compare,in particular, the effects of deuteron corrections in two independent PDF global fits by the CTEQ-JLab (CJ) andCTEQ-TEA groups (CT) which differ in their phenomenological focus, data selection and, crucially, the treatment ofscattering process in nuclear targets. We find that the deuteron effects will have pronounced consequencesfor both the fitted PDFs in the large- x region and the correlations among the PDFs and quantitiesderived from them in an extended x range. More specifically, this paper will elucidate constraints on the d -quark, gluon, and other PDFs arising in the CT18 andCJ15 global fits. We will accomplish this by analyzing the L sensitivity to various PDFs [12], a simple informativefigure of merit that allows us to look inside the CJ and CT fits and understand the constraints from the fittedexperiments on various parton flavors in an expansive region of x and Q . C. Paper organization
After this introduction, the remainder of the article is as follows. In Sec. II, we briefly present the deuteron-structurecorrections (Sec. II A) with which this investigation is primarily concerned, as well as power-suppressed QCD effects,also relevant to fits involving nuclear data and/or at lower Q and W ( in Sec. II B). In Sec. III, we summarize theessential features of the CT and CJ fitting frameworks relevant for this study, and the special modifications madein each fit to allow direct comparisons of the two analyses. In Sec. III C, we review the L sensitivity method. InSec. IV, we apply this method to analyze the impact of deuteron-structure corrections on the fit results, and examinethe patterns of PDF pulls obtained in the several iterations of CT/CJ under different assumed treatments of thedeuteron corrections. As representative cases, we will concentrate on the d/u ratio in Secs. IV B and IV E, andon the gluon in Secs. IV D and IV F. In Section IV C we also explore the implications of our analysis to precisionmeasurements of the weak mixing angle. The conclusions in Sec. V are followed by a technical appendix describing anumerical procedure to reconcile the error analyses in the CJ and CT approaches. II. LOW-ENERGY QCD EFFECTSA. Deuteron-structure effects
The critical low-energy effect considered in this study, which arises due to MeV-scale dynamics characterizing nuclearbound states, is the modification of the parton-level substructure of nucleons embedded in the nuclear medium — inparticular, those effects arising in the most weakly bound nuclear system, the two-body deuteron.In the CJ framework, these corrections are treated as nuclear wave-function effects, and the deuteron partondistributions f d are calculated as a convolution of the bound nucleon’s parton distributions, (cid:101) f N , with a suitablenucleonic “smearing function,” S N/d : f d ( x, Q ) = (cid:90) dzz (cid:90) dp N S N/d ( z, p N ) (cid:101) f N ( x/z, p N , Q ) . (1)Here, z represents the momentum fraction of the (isoscalar) nucleon within the deuteron, defined as z ≡ ( M d /M N )( p N · q/p d · q ); p d,N are the deuteron and nucleon four-momenta; and M d,N are their respective on-shell masses. This rep-resentation is grounded on the so-called Weak Binding Approximation (WBA) to the calculation of nuclear structurefunctions [44, 54], where the S N/d smearing function is calculable based on an assumed nuclear potential; as in Ref. [5],we assume the AV18 potential. Since p N is generically off-shell for a bound nucleon, but typically by only a smallamount, one can further expand the bound-nucleon PDF, (cid:101) f N , in powers of its off-shellness, ω = ( p N − M N ) /M N , as (cid:101) f q/N ( y, p N , Q ) = f N ( y, Q ) + p N − M N M N δf ( y, Q ) + O ( ω ) . (2)The first term, corresponding to p N = M N , gives the PDF of the free, on-shell nucleon. In the second term, the O ( ω )coefficient (also known as “off-shell function”) can be phenomenologically parametrized and determined in a global fitfrom the interplay of data involving deuterium targets and information involving free-nucleon-based observables like W boson production at the Tevatron or LHC. Like in Ref. [5], we assume the flavor-independent 3-parameter shapefunction δf ( x ) = C ( x − x )( x − x )(1 + x − x ) , (3)with x fixed by requiring the off-shell PDFs to satisfy the quark-number sum rule. Further technical details and adiscussion of the fit results can be found in Ref. [5].Section IV considers three main scenarios for implementing the deuteron corrections (d.c.) discussed above:(i) an uncorrected scenario for which no nuclear effects are included for the deuteron;(ii) a fixed scenario in which the nuclear wave-function effects (on- and off-shell) are frozen to the AV18-informedchoice of Ref. [5], and the off-shellness correction, δf N ( x ), is set to the CJ15 central fit; and(iii) a free scenario particular to CJ, in which the parameters in Eq. (3) for the off-shell nucleon are allowed to vary.The dynamical deuteron corrections we have discussed above are natively implemented in the CJ framework, andthe off-shell parameters can be simultaneously fitted with the PDFs. So far, however, the CT code only supportsdeuteron corrections given in the form of analytic interpolations, such as the one obtained from the correction in [55].To implement the fixed CJ15 deuteron correction in the CT framework and render it more directly comparable to CJwith respect to its treatment of deuteron target data, we instead multiply the experimental DIS deuteron structurefunction by the F N /F D nucleon-to-deuteron ratio plotted in Fig. 3, thus transforming it into an isoscalar combinationof neutron and proton structure functions. This can then be directly compared to uncorrected theoretical calculationsof the isoscalar deuteron DIS structure function. On this logic, the CT and CJ fits with a fixed CJ15 correction areplaced on similar theoretical footing regarding the implementation of the deuteron effects, with the main differencebeing whether the correction is imposed within the theoretical structure function calculation or in the F d experimentaldata — a fact which is immaterial for the sake of evaluating the χ -function and allows us to compare the impact ofthe same fixed correction on the CJ and CT frameworks.The size and x dependence of the deuteron corrections, as quantified by the isoscalar nucleon-to-deuteron structurefunction ratio F N /F d , are shown in the left panel of Fig. 3 for several representative choices of Q . One immediatelynotices that deuteron corrections depend on the DIS scale and, at large x , increase with Q toward a fixed point inthe Q → ∞ Bjorken limit; as such, deuteron corrections become effectively scale independent for Q > ∼
50 GeV .For each plotted value of Q , the figure also indicates the maximum x values below which data are accepted in theCJ and CT fits according to their W > .
25 GeV kinematic cuts, respectively. For CJ, which extends tothe low- Q values shown in the right panel of Fig. 3, it is imperative to correctly account for the Q dependence ofthe deuteron correction in order to avoid conflicts with the leading-twist logarithmic Q evolution that constrains thefitted gluon distribution in DIS experiments. For CT, with its larger W cut, the deuteron corrections are small andnearly scale independent, as seen in the left Fig. 3, except for the less precise BCDMS deuteron points with x > ∼ . B. Power-suppressed effects
Due to their less conservative kinematical restrictions on Q and W , the CJ global fits extend into a region for whichpower-suppressed corrections are non-negligible, as depicted in Fig. 3 (right). On the one hand, dynamical higher-twistcorrections of O (Λ /Q ) emerge because of the presence of multi-parton correlations within the soft portion of the x Q = 5 GeV = 10 GeV = 50 GeV = 100 GeV F N2 / F d2 CT W cuts
FIG. 3:
Left:
We plot the correction ratio, F N /F d , based on the central CJ15 prediction. This correction factorrepresents the K -factor applied to correct the physical deuteron structure function data fitted in CT to the isoscalardeuteron. We show the correction factor for a range of DIS scales, Q , and lines terminate at high x according tothe W cut of each fitting framework. The termini in x due to the CT W cut are indicated by vertical boxes. Right:
Kinematics of the DIS data included in the fits discussed in this paper. The HERA data were only taken on protontargets; the fixed target SLAC, JLab, BCDMS and NMC experiments include both proton and deuterium targetdata at approximately the same kinematics. The W = 12 .
25 GeV and W = 3 GeV cuts adopted, respectively,by the CT and CJ fits are shown by dashed and dot-dashed lines, respectively.factorized DIS process, for which the first subleading contribution to the twist expansion for unpolarized scatteringare matrix elements of twist-4 operators [56, 57]. As in CJ, these are often determined phenomenologically using formslike F ( x, Q ) = F LT2 ( x, Q ) (cid:2) C ( x ) (cid:14) Q (cid:3) , where F LT2 represents the leading-twist structure function, and a fittedcoefficient, C ( x ) = αx β (1 + γx ), parametrizes the power-suppressed twist-4 corrections. On the other hand, target-mass corrections of O ( M N /Q ) are due to the non-negligible mass, M N , of the struck nucleon, and are implementedvia the operator product expansion of Georgi and Politzer [58, 59] or related prescriptions, as extensively reviewed in[60, 61]. Both corrections are natively implemented in the CJ framework.In contrast, CT imposes more restrictive kinematical cuts in W , such that the standard CT data sets lie beyondthe region for which the finite ∼ /Q corrections are significant. In the past CT studies it mattered little whetherthe deuteron correction was included according to a specific model or not included at all (the default choice). Whilewe expect some interplay between the deuteron and power-suppressed effects, we do not systematically isolate thelatter and leave their investigation to future studies. III. METHODSA. Selections of experimental data sets
The CT18 and CJ15 global PDF fits each describe expansive data sets consisting of both high-energy measurementsas well as data down to the few-GeV region, especially for CJ in the latter case. Despite their somewhat differingphenomenological emphases — with CT generally aimed toward high-energy processes, and CJ toward the large- x region probed at facilities like JLab and SLAC — there are substantial overlaps with respect to the key experimentsthey include. In Table I, we provide a complete listing of the experiments included in each global analysis. Ofparticular importance for interpreting our results in Sec. IV, the leftmost column of Table I designates a process-based category label for each experiment, placing, e.g. , the BCDMS F d inclusive structure function data in Group4: DIS Deuterium. Our article will investigate the agreement and tensions between these groups of experiments withthe help of L sensitivities. In contrast, previous studies [11, 12, 31] employing the same technique focused primarilyon the individual experiments.While the CJ and CT analyses include a large number of the same measurements, Table I shows that the CJfits include additional DIS data at fixed-target energies from SLAC, HERMES, JLab, and NMC. They also includeTevatron measurements of charge asymmetries reconstructed to the level of W bosons and cross sections for photonplus leading jet production. The CT PDF fits more extensively cover collider observables. They include HERA heavy-quark production, an assortment of cross sections and cross section asymmetries in electroweak boson production atthe LHC, as well as LHC cross sections for inclusive jet and t ¯ t production. The CT fits include CCFR and NuTeVcross sections for both inclusive (Group ) and semi-inclusive (Group , for opposite-sign dimuon production)charge-current DIS on iron. The CT fits, however, implement only the most direct measurements of Tevatron andLHC charge asymmetries in W → (cid:96) lepton decay, presented as a function of the rapidity and transverse momentumof the charged lepton. They do not include the CDF and DØ boson-level charge asymmetries fitted by CJ, whichdirectly probe the large- x PDF ratios, while they also involve additional recursive unfolding of the data that utilizesa PDF-dependent calculation to reconstruct the weak boson’s rapidity.
B. Modifications in the fitting methodologies
For the study presented in this article, we modified some default settings of the CJ15 and CT18 fits, fully describedrespectively in Ref. [5] and [7], to place the two fitting frameworks on a common footing and isolate the impact ofvarious assumed treatments of the deuteron structure.1. We match perturbative orders between the two fits at NLO in α s . In practice, this means that in the CT fits,performed by default at NNLO, we instead compute the hard cross sections, perturbative PDF evolution, andrunning of α s at O ( α s ) accuracy to agree with the default NLO settings used in CJ.2. We perform supplementary fits by excluding some data sets that appear in one fit only. While both CJ and CTfits include Tevatron lepton charge asymmetry measurements presented as a function of the charged lepton’srapidity, the CJ fit also includes the fixed-target low W and Q DIS data from SLAC [69] and JLab [70],as well as the CDF [80] and DØ [81] W boson charge asymmetry with reconstructed weak boson kinematics.On the other hand, CT makes use of neutrino-initiated DIS data sets on heavy nuclear targets (both inclusiveand semi-inclusive DIS [SIDIS] di-muon production in ν -A scattering). In CT, data on heavy-nuclear targetsare fitted at the isoscalar level after being corrected in the fit using a phenomenological parametrization of the F A ( x, Q ) /F d ( x, Q ) ratio from Ref. [52]. To isolate the impact of these extra experiments, we performed CJfits without the W asymmetry and SLAC DIS data sets, and CT fits without the inclusive ν -A DIS data.3. As in the original CJ and CT publications, we estimate the final PDF uncertainties using the Hessian method [13],but in this paper we fix the tolerance to be T = 10 for both global analyses, in between the nominal T = 2 . T = 37 value (at the 68% probability level) used in the CT18 fits. Furthermore, we donot include the additional “Tier-2” tolerance contribution [7, 101] that is applied in the CT18 fits to preventthe error PDFs from running into strong disagreements with individual experiments, but content ourselves withthe “Tier-1” tolerance as defined in [102].Regarding the lattermost point, in the CJ analysis it is additionally necessary to implement a numerical prescriptionat the level of individual eigenvector directions of the diagonalized Hessian matrix to guarantee ∆ χ = T to theneeded accuracy and to ensure the validity of the analysis methods utilized in this article. These technical details arereviewed in the appendix. C. The L sensitivity technique In the next two sections, we will investigate the impact of the DIS deuteron data on the PDFs with the help of the“ L sensitivity” [12]. The method is easy to use and has already been applied to clarify the sensitivity of the globaldata sets to the CT18 NNLO PDFs [7], LHC parton luminosities [103, Sec. II.2], and effective exponents of the high- x PDF falloff [31]. Here we give a quick summary of the L technique and refer the interested reader to Ref. [12] foradditional details.The L sensitivity provides a fast approximation to the information contained in the LM scans typified by Fig. 2.It does so by quantifying the extent to which variations in the fitted PDFs drive the shifts in the log-likelihood χ E for each experiment E . For example, the L plots discussed below supply the change in an experiment’s value of χ given a defined increase in the value of a PDF of interest, as a function of x for fixed Q . That being the case,if two experiments are in relative tension with respect to the behavior of a given PDF — say, one favors a largerPDF value in a given x range, the other favors a smaller value — an increase in the PDF would result in a negative Group γ +jet DØ γ +jet 301-304 [62] Jets Tevatron CDF Run-2 inclusive jet production 504 204 [63]DØ Run-2 inclusive jet production 514 203 [64] DIS Proton HERA Run I+II 160 80,82,83,93-96 [23]BCDMS F p
101 3 [25]H1 σ br
145 [65]Combined HERA charm production 147 [66]H1 F L
169 [67]HERMES proton 17 [68]SLAC proton 5 [69]JLab proton 7 [70]NMC F
51 [71] DIS Deuterium BCDMS F d
102 4 [24]NMC F d / F p
104 53 [22]BoNuS F n /F d
55 [72]SLAC deuteron 6 [69]JLab deuteron 8 [70]HERMES deuteron 18 [68] WZ Tevatron CDF Run-1 lepton A ch , p Tl >
25 GeV 225 [73]CDF Run-2 electron A ch , p Tl >
25 GeV 227 128 [74]DØ Run-2 muon A ch , p Tl >
20 GeV 234 [75]DØ Run-2 Z rapidity 260 [76]CDF Run-2 Z rapidity 261 140 [77]DØ Run-2 9.7 fb − electron A ch , p Tl >
25 GeV 281 130 [78]DØ Z
141 [79]CDF W asymmetry 131 [80]DØ W asymmetry 132 [81]DØ lepton asymmetry 13 134 [82] WZ LHC LHCb 7 TeV 1.0 fb − W/Z forward rapidity cross sec. 245 [83]LHCb 8 TeV 2.0 fb − Z → e − e + forward rapidity cross sec. 246 [84]CMS 8 TeV 18.8 fb − muon charge asymmetry A ch
249 [85]LHCb 8 TeV 2.0 fb − W/Z cross sec. 250 [86]ATLAS 8 TeV 20.3 fb − , Z p T cross sec. 253 [87]CMS 7 TeV 4.7 fb − muon A ch , p Tl >
35 GeV 266 [88]CMS 7 TeV 840 pb − electron A ch , p Tl >
35 GeV 267 [89]ATLAS 7 TeV 35 pb − W/Z cross sec., A ch
268 [90] Drell-Yan E605 201 [91]E866, σ pd /σ pp
203 [92]E866, σ pp
204 108 [93]E866, σ pd
110 [93] ν -A incl. DIS CDHSW F
108 [94]CDHSW xF
109 [94]CCFR F
110 [95]CCFR xF
111 [26] ttbar production CMS 8 TeV 19.7 fb − , t ¯ t norm. top p T and y cross sec. 573 [96]ATLAS 8 TeV 20.3 fb − , t ¯ t p tT , m t ¯ t abs. spectrum 580 [97] ν -A dimuon SIDIS NuTeV νµµ SIDIS 124 [45]NuTeV ν ¯ µµ SIDIS 125 [45]CCFR νµµ
SIDIS 126 [46]CCFR ν ¯ µµ SIDIS 127 [46] Jets LHC CMS 7 TeV 5 fb − , single incl. jet cross sec., R = 0.7 542 [98]ATLAS 7 TeV 4.5 fb − , single incl. jet cross sec., R = 0.6 544 [99]CMS 8 TeV 19.7 fb − , single incl. jet cross sec., R = 0.7 545 [100] TABLE I: A comprehensive listing of experiments included within the CT and CJ frameworks for this study,grouped according to the experimental process they represent.0change, ∆ χ E <
0, in the χ for the first experiment, and a positive shift, ∆ χ E >
0, for the second experiment.As shall be seen below, these competing pulls are visualized by the L method as opposing deviations from zero inthe negative/positive directions. In this fashion, the L sensitivity can encapsulate, on a flavor-by-flavor basis, thepatterns of pulls exerted on the PDFs (often called “PDF pulls” below for short) by various experiments or groups ofexperiments as a function of x and Q . In fact, the method is not limited to the influence of data sets on the PDFs,but can be extended to any observable that can be calculated starting from those.More formally, the L sensitivity yields an approximation of the shift ∆ χ E in the value of the log-likelihood forexperiment E caused by an upward 1 σ variation of the chosen PDF or PDF-dependent theoretical prediction. Weevaluate the L sensitivity in the Hessian approximation as S f,L ( E ) ≡ (cid:126) ∇ χ E · (cid:126) ∇ f | (cid:126) ∇ f | = ∆ χ E cos ϕ ( f, χ E ) , (4)where cos ϕ ( f, χ E ) represents the cosine of the correlation angle between a PDF of flavor f (or, indeed, any PDFderived quantity) and the experimental χ E , evaluated over the 2 N Hessian eigenvector sets for each N -dimensional fitexamined in this study. Recalling that both CT and CJ frameworks make use of the Hessian formalism in determiningPDF uncertainties, this correlation cosine can be computed as indicated in Ref. [11]:cos ϕ ( f, χ E ) = 14∆ f ∆ χ E N (cid:88) j =1 ( f + j − f − j )([ χ E ] + j − [ χ E ] − j ) , (5)where ∆ f = (cid:12)(cid:12)(cid:12) (cid:126) ∇ f (cid:12)(cid:12)(cid:12) = 12 (cid:118)(cid:117)(cid:117)(cid:116) N (cid:88) j =1 (cid:0) f + j − f − j (cid:1) , (6)and “ ± ” denote the PDF variations along the positive and negative direction of the j -th Hessian eigenvector. �� - � �� - � ���� ���� ���� ��� ��� ��� ��� - �� - ����� � Δ χ � � � � � �� � � � � � � � � �� ���� ����� ��� ��������� ( � ) � � = � ��� u - d - gdusc �� - � �� - � ���� ���� ���� ��� ��� ��� ��� - �� - ����� � Δ χ � � � � � �� � � � � � � � � �� ���� ����� DIS ��������� ( � ) � � = � ��� u - d - gdusc FIG. 4: A comparison of the PDF pulls of the deuterium DIS data computed according to the L method at 2 GeVfor the CT fixed d.c. (left) and
CJ fixed d.c. (right) fits; both cases are for the scenario with fixed deuteroncorrections.When L sensitivities are summed over all experiments, the resulting sum should be close to zero by construction,assuming deviations from a symmetric Gaussian probability distribution are negligible (see App. A for further dis-cussion). As an example, Fig. 4 shows the combined L sensitivities of the experiments in the DIS deuteron group( in Table I) to each parton flavor separately, calculated for the CT and CJ fixed d.c. fits, where the deuteroncorrections were fixed to the central value determined in the CJ15 analysis. These figures can be interpreted as thestatistical pulls at fixed Q = 2 GeV from this group of experiments on each PDF flavor, f ( x, Q ), at the x valuesspecified on the horizontal axis. One can observe quite large deviations from zero, with S f,L values nearly reaching ±
10 units in some regions of x . This non-negligible pull by the deuteron DIS experiments is ultimately offset bycontributions from other groups of experiments to obtain a zero result (within about one unit of χ ) when summing1over all of these. It is therefore interesting to investigate which experimental groups pull significantly against the DISdeuteron data sets, as we do below in Sec. IV E and IV F.Fig. 4 contains a substantial amount of information. For example, looking at the left panel for CT fixed d.c. , the negative S f,L for the d quark at x ≈ .
25 indicates that the deuteron DIS data prefer a higher d quark at x ≈ . d -quark PDF in the full fit. Similarly, the positive S f,L for the u quark at the same x indicates apreference for a relatively lower u -quark PDF. In totality, the deuteron DIS data prefer a higher d/u at x = 0 . − . d/u in this x interval is further confirmed inFig. 8 of Sec. IV E.)From the right panel of Fig. 4, we also read that the deuteron DIS data in the CJ fixed d.c. fit prefer an enhanced d/u over a slightly higher interval x = 0 . − .
7. Comparative analyses of this kind between different global QCD fitscan thus indicate which features of a fit are data-driven, as opposed to being determined by particular theoreticaland/or phenomenological choices.Regarding other flavors and x ranges, in the left panel of CT fixed d.c. we observe a significant preference ofthe deuteron DIS group for lower u - and d -quark PDFs at x = 0 . − .
1, in the region relevant for LHC W -bosonproduction. One also notices a preference for a larger gluon PDF in the interval, x = 0 . − .
1, relevant for Higgs-boson production at the LHC, and, at slightly lower x , for a larger perturbative charm-quark PDF, which is radiativelygenerated from the gluon.Finally, we remark that the Hessian sensitivity is most effective in identifying the top 5-10 experiments or groupsof experiments sensitive to variations in the chosen PDF, as has been verified by comparing rankings of the mostsensitive experiments obtained from Hessian sensitivities and more precise LM scans [11, 12]. However, especially forsubleading experiments, detailed rankings depend on the chosen definition of the sensitivity indicator and deviationsfrom the simple linear approximation that we assumed when using symmetric finite derivatives in S f,L ( E ) as inEqs. (5) and (6). IV. COMPARISON OF DEUTERON DATA IMPACT IN THE CJ15 AND CT18 FITS
In accordance with the preceding discussion in Sec. II and III, our analysis will be based on a series of fits namedaccording to the following convention:1. Fits without deuteron corrections:
CT no d.c. , CJ no d.c. ;2. Fits with the fixed
CJ15 correction:
CT fixed d.c. , CJ fixed d.c. ;3. A CJ fit in which the off-shellness correction is freely varied:
CJ free d.c. ;4. Fits with the fixed CJ15 deuteron correction and variations in the fitted data sets:
CT no nu-A (removinginclusive ν -A DIS experiments from CT), and CJ no-W slac (removing the CDF [131] and DØ [132] W bosonasymmetry data and SLAC DIS [proton and deuteron] sets from CJ).In each enumerated case, the CT and CJ fits are methodologically comparable to each other, but not identical. Thedifferences between the fits in the first two categories will highlight the nontrivial effects of deuteron corrections onboth the PDF central values and their L sensitivities. We then consider several variations of the fixed d.c. fits.Firstly, the comparison of the CJ sensitivity plots in categories 2 and 3 demonstrates that freeing the d.c. parametersin the CJ fit tangibly improves the agreement among all categories of experiments. Secondly, we try to make the CTand CJ fits comparable not only methodologically but also (partially) with respect to data selections. We do this incategory 4 by removing the indicated data sets from each fit. A. Overall agreement of theory and data
We first review the overall quality of these fits by referring to Table II, which lists values for the total χ per point( χ /N pt ) according to the experimental groups listed in Table I. The χ values assess the global agreement among thedata and must be considered together with other indicators of the goodness-of-fit, such as the L sensitivities whichquantify local compatibility among fitted experiments, as well as an assortment of other statistical tests reviewedin Ref. [3]. Table II shows the χ values for SLAC DIS, Tevatron W -boson asymmetry, and ν -A DIS data sets inseparate lines.The χ values in Table II indicate that the deuteron data agree globally with the published fits of both groups.Deuteron corrections are essential to the CJ analysis, which includes deuteron DIS data at the largest values of x from SLAC as well as the very sensitive DØ data on the reconstructed W -charge asymmetry [5]. Indeed, even if the2total χ /N pt may seem only marginally improved, the highlighted data sets clearly require deuteron corrections toreconcile the SLAC DIS deuteron data with the deuteron-independent DØ W -asymmetry data.In CT, the inclusion of deuteron corrections also improves the description of the DIS deuteron data (especially theBCDMS F d measurements), producing a 14-unit reduction in χ for the N pt = 373 points fitted in Group , withan additional 6-unit reduction once the inclusive ν -A data are removed. Mainly due to the absence of the SLAC DISdata in CT, this is a smaller relative reduction than that observed for CJ in the left columns of Table II.It is also interesting to note far more substantial shifts in χ within Table II among the other CT experimentalgroups: the introduction of the fixed deuteron correction in CT improves the χ of the DY data ( ) by more than 100units, while at the same time, increasing the χ of the LHC weak-boson production ( ) by an opposing ∆ χ changeof 80 units. The χ for group 5, “WZ Tevatron”, also increases by 16 units. The inclusive ν -A DIS data set ( ) isfitted well globally, with χ /N pt = 0 .
80 (0 .
83) in the
CT no d.c. ( CT fixed d.c. ) fit. In sum, the fixed deuteroncorrection improves the CT total χ by 18 units for 3670 data points. Since the ν -A DIS data are well-described,removing them from the CT fit actually increases the total χ /N pt , but also seems to release some tension with theWZ LHC data, whose χ value decreases by 32 units. While these look like modest changes, overall, we will see nextthat they do influence some PDF flavors and the local compatibility among select groups of experiments.The quality of the fits to other data sets, as measured by their χ /N pt , is comparable across the performed fits;nonetheless, deuteron corrections have nontrivial, indirect consequences on those also, as we discuss in the remainderof this section. Group γ +jet 62/56 61/56 60/56 – – – Jets Tevatron 37/182 36/182 36/182 225/182 225/182 229/182 DIS proton 3007/2548 2973/2548 2330/1848 1818/1523 1812/1523 1806/1523 DIS deuteron 1363/1389 1214/1389 704/671 401/373 387/373 381/373
SLAC deuteron (only) 507/582 376/582 – – – – WZ Tevatron 193/117 133/117 99/90 113/101 129/101 137/101
CDF W asym. (only) 14/13 17/13 – – – –DØ W asym. (only) 82/14 12/14 – – – –6 WZ LHC – – – 267/185 347/185 315/185 Drell-Yan 302/250 284/250 272/250 454/318 348/318 344/318 ν -A incl. DIS – – – 269/336 279/336 –9 t ¯ t production – – – 44/31 45/31 46/31 ν -A dimuon SIDIS – – – 103/149 104/149 103/149 Jets LHC – – – 594/483 595/483 598/483
TOTAL 4963/4542 4699/4542 3501/3097 4289/3670 4271/3670 3959/3334
TABLE II: For each fit, we report the total χ per point, χ /N pt , as well as its breakdown according to theexperiment categories listed in Table I. The data sets removed in the CJ no-W slac fit are singled out and emboldened in their respective categories. The ν -A inclusive data excluded in the CT no nu-A fit are also emboldened . B. Impact of deuteron corrections on the d/u
PDF ratio
As discussed in Sec. I A, experimental information involving deuterium targets, especially measurements of the DISstructure function of the deuteron (Group , “DIS Deuterium,” in Table I), has been pivotal for separating the d -quark content of the proton from other parton flavors. The leading impact of deuteron corrections thus primarilyinfluences the extracted d -PDF at high x , where the deuteron most prominently differs from a superposition of a freeproton and a free neutron. In contrast, the effect of the deuteron corrections on the u -type PDFs is comparativelymild, as these are most directly constrained by measurements of the proton’s structure function (Group , “DISProton”).To gauge the leading impact of deuteron corrections, in this subsection we now examine the x dependence of the d/u ratio within the CT/CJ frameworks, before proceeding in Sec. IV C to an examination of the indirect effects onthe lower- x d val PDF relevant for sin θ W and on the gluon PDF in Sec. IV D (the PDF pulls will be considered in3Secs. IV E and IV F).Fig. 5 illustrates the impact of the F d corrections, as introduced in Sec. II A, on d/u . The upper row shows the d/u ratio and its error band obtained in the discussed series of fits, normalized to the central value obtained in the fitswith no deuteron corrections. The lower row shows the unnormalized d/u ratios themselves, using a linear horizontalscale to better visualize the x > . x → d/u in both CT and CJ, especially at x > ∼ .
1, with evidence of a mild, few-percentenhancement of the fitted d/u ratio over the no d.c. fits for 0 . < ∼ x < ∼ . x > ∼ .
5, beyond which d/u is strongly affected bythe 2-body nucleon-nucleon corrections included in the F d calculation. For CJ, this suppression is larger than in theCT case, but compatible with the latter within the respective uncertainties of each fit. After discussing the absolutevalues of the d/u ratio below, we will return to this difference, arguing for its origin in the specific choice of large- xd -quark parametrization as well as the standard kinematical cuts implemented within each fitting framework.The qualitative x dependence of the deuteron-corrected fit of d/u in the top rows of Fig. 5 closely follows the F N /F d ratio plotted in left panel of Fig. 3, in which F N and F d represent the deuteron structure function computedusing the isoscalar and full nuclear predictions, respectively. Indeed, in the no d.c. fits, F d is effectively fitted as anisoscalar target, but in the CT fixed d.c. , for example, the F d data for the physical deuteron are corrected to F N ,which leads to a relative suppression of the fitted d/u PDF ratio for x > ∼ . fixed d.c. fits (red curves), the d.c. parameters are held constant at their values obtainedfrom the central CJ15 fit. If, on the other hand, the d.c. parameters in Eq. (3) are actively fitted with the PDFshape parameters, we obtain the same central PDFs but a narrower uncertainty band on d/u , as shown by the CJ free d.c. error band in the right panels of Fig. 5. This reduction in the PDF uncertainty of d/u , which atfirst sight is paradoxical because we have increased the number of fit parameters, is actually a consequence of thecorrelation between the treatment of F d structure function data and extracted d -PDF. More specifically, when thenucleon off-shellness parameters are freed, the variations in the d -quark parameters that were necessary to encompassthe F d data in the CJ fixed d.c. fit are partly absorbed by the parameters of Eq. (3). In other words, releasing theoff-shell parameters reduces tensions in parameter space, and ultimately also diminishes the overall experiment-by-experiment χ E variations in the PDF analysis, as we shall note again below in Sec. IV E and IV F. We have verifiedthat, over the same x range, the relative uncertainty in the determination of other flavors, such as, e.g. , the gluon,does correspondingly increase. This is an instance of the fact that, typically, the constraining power of a fit can beenhanced by a greater number of free parameters only in a limited sector of parameter space.Given the pattern of pulls displayed by the fits just discussed, it is clear that the different choices of data sets in theCT and CJ fits also affect the fitted d/u PDF ratio. For example, unlike CJ, the CT fit includes DIS data from inclusive ν -A collisions, multiplied by a phenomenological parametrization of the heavy-nuclear structure function relative tothe deuteron. The green bands in Fig. 5 (left) are for the CT no nu-A variant of the CT fit, and show that the removalof this data augments the shifts in the CT d/u ratio induced by including the fixed deuteron correction, which nowis comparable to the CJ result. The reason for this may simply lie in the lack of further deuteron-to-isoscalar protonplus neutron corrections, or may also be related to possible discrepancies between neutral- and charge-current DISdata or interactions [50, 104, 105].Conversely, the CT fits do not include the low- W SLAC data and the reconstructed W -boson asymmetry from theDØ experiment that are influential upon the large- x d -quark fit in CJ [5]. When these are removed from the CJ fitas well — see the green CJ no-W slac bands in the right column of Fig. 5 — we obtain an enlarged uncertainty thatincludes both the deuteron corrected and uncorrected bands. The uncertainty on d/u at x → CJ no-W slac fitis wider than in the
CT no nu-A fit in part in reflection of different parametrization forms and selection of experimentsbetween the CJ and CT analyses.
C. Impact on the valence PDFs in the LHC EW precision region
The effects of including deuteron structure corrections to F d , while most pronounced at x > .
1, have someconsequences for the PDFs in the low- x region as well. In the CT result shown in Fig. 5(left), modest enhancementsin d/u at about x ∼ − can be seen for the CT fixed d.c. fit. While the sea-quark PDFs are relatively unaffectedin this kinematic region, the valence component of the d -quark and, to a lesser extent, the u -quark PDFs in thissmall- x region are sensitive to the inclusion and theoretical treatment of both neutrino-nucleus and deuterium dataat large x . This sensitivity is mainly a consequence of corresponding valence sum rules.Fig. 6 illustrates this feature. In the left panel, the fixed d.c. CT fit (red dotted line) prefers a slightly lower d val at x ≈ . − .
13 than in the no d.c. fit, and a slightly higher one at x ≈ . − .
45, shown in this case at Q = 81 GeV ≈ M W . This deviation becomes still more pronounced if the neutrino-nucleus DIS data are removed4FIG. 5: Upper row: The PDF ratios d/u and their asymmetric error bands for T = 10 at scale Q = 2 GeV, asdetermined within the two fitting frameworks examined in this analysis, CT (left) and CJ (right). To visualize therelative differences between the fits, we normalize all d/u error bands to the ratio from the central no d.c. fit(without any assumed deuteron correction). The left panel shows the CT no d.c. , CT fixed d.c. , and
CT no nu-A error bands. The right panel shows the analogous
CJ no d.c. , CJ fixed d.c. , CJ no-W slac , and the
CJ freed.c. fits. The abscissas are scaled to highlight the impact of the deuteron corrections over the whole x range: atlarge x , where the impact is most pronounced, as well as the modest enhancement in the d/u ratio for x < ∼ .
01 in CTat left. Lower row: now showing the absolute d/u ratios on a linear x -axis scale to highlight the behavior at high x .5(leading to the green dot-dashed line). In the right panel, we observe for CJ a qualitatively similar trend and associated x dependence over slightly shifted x regions of 0 . − . . − . A FB , are sensitive to the valence d - and u -PDFs in an x -region about ∼ . i.e. , where Fig. 6indicates a dependence of these PDFs upon the treatment of the deuteron/heavy-nucleus data at higher- x values. D. Impact of deuteron corrections on the gluon PDF
The deuteron data sets can also impact the gluon density through Q -scaling violations encoded by DGLAPevolution; this is particularly true when these measurements cover a large range of the four-momentum squared, Q , of the exchanged boson. Similarly to the case for d val in Sec. I A, the Lagrange Multiplier scans [7] and PDFsensitivity techniques [11, 12] in the CT18 analysis collectively demonstrate that the gluon at x > . x region, and also with LHC and Tevatron inclusive jetproduction, which cover a wide x range but involve complex arrays of systematic effects which remain under activestudy.To address this point, in Fig. 7 we plot the error bands for the gluon PDF at Q = 2 GeV as determined in theseries of fits discussed at the beginning of this Section, again normalized to the central value obtained in the nod.c. fits. As seen in the left panel of Fig. 7, the gluon PDF in the CT fits exhibits a modest sensitivity to thechosen deuteron correction treatment, with a dependence that is somewhat moderated by the adopted W > . cut. Still, as with d/u , there is a qualitative tendency for the fixed deuteron correction to reduce the high- x gluon PDF, with this modification being enhanced by the exclusion of the inclusive ν -A DIS data. Like CT, the CJfits seen in Fig. 7 (right), which include the SLAC DIS data, similarly display a relative suppression of the gluonfor x > ∼ . T = 10 tolerance used to determine the uncertainty bands. For CJ, itwill be interesting to confirm this effect by fitting the full JLab 6 GeV inclusive data set [106], and, even more so,the JLab 12 GeV data which will augment the precision of the available DIS measurements over a wide Q range atlarge x , once available. (.) "O a: d val (x,Q) at 0=81. GeV, T = 10 1.3 - CT no d.c.-········ CT fixed d.c. --- •-· CT no nu A X d val (x,Q) at 0=81. GeV, T = 10 1.3 1.2 u "O a: 0.9 0.8 0.7 - CJ no d.c.......... CJ fixed d.c. ---. -· CJ no-W _slac--· CJ free d.c . X FIG. 6: The valence d -quark PDFs and their uncertainties, normalized to the central values obtained in the nod.c. fits. The fits and conventions are the same as for Fig. 5.6 E. Valence-sector PDF pulls: the d/u ratio
In Fig. 8, we plot the L sensitivities to the d/u PDF ratio of the experimental groups fitted in CT and CJ, whereinwe varied the implementation of the deuteron corrections. L sensitivities for fits exploring data-set variations areshown in Fig. 9, which we also discuss below.Starting with the no d.c. fits that either do not include (CT) or remove (CJ) deuteron corrections, we notice that,in both cases, the landscape of PDF pulls tends to be dominated by a few competing groups of experiments, whichdiffer between the two fitting frameworks. For CT in the left panel, these are the LHC W/Z production (Group ) and inclusive nuclear DIS (Group ), which possess the sharpest opposing pulls at x ∼ .
2, for example, in thedirection of either favoring or disfavoring a larger d/u ratio, respectively. At slightly higher x > . d/u behavior as x →
1, these are joinedby the DIS-deuterium ( ) and Drell-Yan ( ) groups of experiments. Turning to the CJ case, displayed in thetop-right panel, it is the DIS deuterium ( ) and gamma-jet ( ) groups that dominate the landscape of PDF pulls— and quite strikingly at large x — with lesser but also significant pulls from the Tevatron W/Z production data( ). The large- x pulls are expected, since the SLAC data are quite sensitive to nuclear dynamics in the deuterontarget, as already noted.Once fixed deuteron-structure corrections are introduced into the respective fixed d.c. fits, the relative patterns ofPDF pulls experience an intriguing series of shifts, which we display in the middle row of Fig. 8. For CJ, the deuteroncorrections largely resolve the huge tensions between the photon+jet and DIS deuteron data, because dynamicalnuclear effects in the latter are now included in the theoretical calculation of the deuteron DIS cross section withoutforcing the d -quark to deform to compensate for the missing nuclear effect. A residual tension between the DISdeuteron ( ) and W/Z
Tevatron data ( ) data is still visible at x ≈ .
5. While the large- x tensions are reduced,the small- x pulls visibly change in shape for several flavors.For CT, the introduction of the fixed deuteron correction detailed in Sec. II A instead preserves to a large extent thequalitative x dependence of the L pulls ( i.e. , the shapes) but softens their magnitude in a few cases, for example, forthe Drell-Yan ( ) and the small- x LHC
W/Z production ( ) data. Notably, the DIS-deuterium sensitivity shiftsto closely resemble that of the LHC W/Z data ( ) in favoring a larger d/u for x ∼ .
2. At the same time, the sizeof the competing pulls at high x > ∼ . ν -A data ( ) is enhanced, while theopposing pulls of other experiments are modestly reduced over the same range in x . This is especially clear for theCT Drell-Yan data ( ), which in the CT no d.c. fit had preferred a softer d/u ratio at low x < .
01 and an enlargedvalue of d/u over 0 . < ∼ x < ∼ .
2. In
CT fixed d.c. , these preferences mostly vanish. At the same time, outsideFIG. 7: Same as Fig. 6, but now for the gluon PDF.7 � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � ��� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� ���� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� ���� - � �� - � ���� ���� ���� ��� ��� ��� ��� - �� - �� - ������� � Δ χ � � � � � �� � � � � � � � � �� �� ����� � ( ��� )/ � ( ��� )( �� � ��� ) � Tevatron jets � DIS Proton � DIS Deuterium � WZ Tevatron � WZ LHC � Drell - Yan � ν A incl. DIS � ttbar production �� ν A μμ SIDIS �� LHC jets � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � ��� - � �� - � ���� ���� ���� ��� ��� ��� ��� - �� - �� - ������� � Δ χ � � � � � �� � � � � � � � � �� �� ����� � ( ��� )/ � ( ��� )( �� � ��� ) ������ up to ~ 30 down to ~ -22 γ + jet Tevatron jets
DIS P roton DIS D euterium WZ Teva tr on D rell -Y an � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � ��� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� ���� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� ���� - � �� - � ���� ���� ���� ��� ��� ��� ��� - �� - �� - ������� � Δ χ � � � � � �� � � � � � � � � �� ���� ����� � ( ��� )/ � ( ��� )( �� � ��� ) � Tevatron jets � DIS Proton � DIS Deuterium � WZ Tevatron � WZ LHC � Drell - Yan � ν A incl. DIS � ttbar production �� ν A μμ SIDIS �� LHC jets � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � ��� - � �� - � ���� �������� ��� ��� ��� ��� - �� - �� - ������� � Δ χ � � � � � �� � � � � � � � � �� ���� ����� � ( ��� )/ � ( ��� )( �� � ��� ) ������ γ + jet Tevatron jets
DIS
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Deuterium WZ Tevatron
Drell - Yan � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � ��� - � �� - � ���� ���� ���� ��� ��� ��� ��� - �� - �� - ������� � Δ χ � � � � � �� � � � � � � � � �� ���� ����� � ( ��� )/ � ( ��� )( �� � ��� ) ������ γ + jet Tevatron jets
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FIG. 8: The L sensitivities computed according to Eq. (4) for Q = 2 GeV, giving the pulls on the d/u PDF ratio ofthe process-dependent data sets fitted by CT (left) and CJ (right). Upper, middle, lower rows: results for the nod.c. , fixed d.c. , and free d.c. fits discussed at the beginning of Sec. IV.8this interval of very high x , the opposing pulls of the inclusive ν -A data (Group ) and both the DIS deuterium( ) and LHC W/Z ( ) experiments increase and sharpen for x > ∼ .
01. In fact, this is the same collection ofexperiments for which in Table II we observed increases (in the case of Groups and ) in their respective valuesof χ /N pt upon introducing deuteron corrections for F d . Both the χ values in Table II and the L analysis for CTtherefore indicate a noticeable rearrangement of the pulls of the inclusive neutrino-nucleus DIS data and select otherexperiments introduced by the correction to the deuteron DIS data. This rearrangement, being presently of similarorder with respect to other contributing effects, will require attention in the future.In the last row of Fig. 8 we present the L pulls of the CJ free d.c. fit in which the deuteron off-shellnessdegrees-of-freedom in Eq. (3) are released . Comparing the vertical extents of the peaks with those of the CJ fixedd.c. fit in the middle row, we see that freeing the offshell parameters moderates the PDF pulls over the whole x range, especially those at x > . ) and the WZ Tevatron CJ Drell-Yan ( ) data thatremained after including fixed deuteron corrections. This behavior can be generically understood as a consequence ofincreasing the number of free parameters, but is not guaranteed, for example, in the presence of incompatible datasets. The CJ free d.c. plot is thus an indication of a good level of consistency between the considered data sets,when the PDFs and deuteron corrections are fitted together.The results discussed so far suggest a nontrivial relationship between the treatment of the DIS deuterium dataand the description of other data sets in each fitting framework — a relationship that depends on the details of theimplementation of deuteron corrections. This observation marks one essential conclusion of this study:
The impactof deuteron-structure corrections in a global fit like CT and CJ cannot generally be expected to applyto deuterium data alone, but have secondary effects on the patterns of pulls of other data sets.
It is therefore interesting to study variations in the choices of experimental data sets in both fits, in particular,removing from each analysis those data sets that showed especially strong sensitivity to deuteron corrections orotherwise played a major role in the foregoing discussion.For CT, we remove the entire collection of inclusive ν -A measurements (Group ), and refit the with the fixeddeuteron corrections of Sec. II A; the resulting L sensitivity plot is displayed in the left panel of Fig. 9. Overall, themagnitude of the PDF pulls is slightly reduced, with the biggest change occurring for the DIS Deuterium Group ( ),that now is now more closely aligned with the pulls exerted by the DIS proton data ( ) throughout the plotted � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � ��� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� ���� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� ���� - � �� - � ���� ���� ���� ��� ��� ��� ��� - �� - �� - ������� � Δ χ � � � � � �� � � � � � � � � �� �� �� - �� � ( ��� )/ � ( ��� )( �� � ��� ) � Tevatron jets � DIS Proton � DIS Deuterium � WZ Tevatron � WZ LHC � Drell - Yan � ttbar production �� ν A μμ SIDIS �� LHC jets � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � ��� - � �� - � ���� ���� ���� ��� ��� ��� ��� - �� - �� - ������� � Δ χ � � � � � �� � � � � � � � � �� �� � - ����� � ( ��� )/ � ( ��� )( �� � ��� ) ������ γ + jet Tevatron jets
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FIG. 9: As in Fig. 8, we plot the PDF pulls on d/u at 2 GeV with fixed deuteron corrections present, but, in thiscase, removing select experiments which have shown significant competing pulls with respect to the DIS deuteriumsets. For CT (left panel), we remove the inclusive ν -A data (Group ), while for CJ, we remove the SLAC DISexperiments (part of Group ) and W -asymmetry information from the Tevatron (part of Group ). This fit correspond to the published CJ15 fit [5], except we are using here a T = 10 tolerance, and have adjusted the error sets accordingto the procedure detailed in Appendix A. x . When considered in parallel with Fig. 8, and in the light of the previous discussion of Fig. 8, the leftpanel of Fig. 9 suggests a connection between the pulls of the DIS deuteron and ν -A data in fits with and withoutdeuteron corrections. For both groups of experiments, the interplay between the theoretical description of deuteriumand heavy-nuclear data is relevant. To that end, investigating the systematic treatment of nuclear effectsfor light and heavy nuclei is a critical subject for future global analyses that aim to use such data forconstraining the nucleon PDFs to higher accuracy. A similar consideration arises for CJ. As we have discussed, the combination of W -boson charge asymmetries andSLAC DIS data is strongly constraining on the d/u ratio at large x , and the d.c. treatment influences also the PDFpulls at smaller x values (as seen see the right panels of Fig. 8). We therefore remove these data sets from the fitto obtain the CJ no-W slac fit shown in the right panel of Fig. 9. In this instance, the removal of the combined W and SLAC DIS data relieves the small x tensions seen in CJ fixed d.c. , for x < .
1. However, the large- x tensionbetween DIS deuteron and WZ Tevatron data (that now only include the W → (cid:96) decay lepton asymmetries) remainlargely intact, and can in fact also be seen in the CT panel on the left of Fig. 9. It remains to be seen whether this isof experimental origin, or due to an as yet incomplete treatment of nuclear corrections in the deuteron target. F. PDF pulls in the gluon and light-quark sea sectors
At first sight, it might seem reasonable to suppose that deuteron-structure corrections, being most sizable at high x and more immediately connected to extractions of the d -quark, would be relatively inconsequential for determinationsof the gluon PDFs. In actuality, constraining the gluon PDF through DIS data requires an adequate prediction ofthe scale dependence of both proton and deuteron DIS data sets, with the latter simultaneously sensitive to the ( Q dependent) deuteron corrections discussed in Sec. II A, especially at larger values of x . Moreover, the momentum sumrule requires that the changes in the total momentum fraction from the large- x and small- x quark and gluon PDFscompensate one another. The practical implementation of the deuteron correction can therefore impact g ( x, Q ) overa still broader range beyond high x .We therefore turn to an examination of the pulls on the gluon PDF in fits with and without deuteron corrections,presented in Fig. 10, before examining CT and CJ fits with the modified data sets in Fig. 11. Comparing the CJ nod.c. fit in the upper-right panel of Fig. 10 to the
CJ fixed d.c. fit in the middle-right panel, one sees that adding afixed deuteron correction clearly aligns the pulls of the DIS proton group ( ) and DIS deuteron group ( ) on thegluon. The x dependence of these pulls was effectively uncorrelated without the deuteron correction. In the presenceof the fixed correction, however, they are aligned and pronounced over the whole x range and are opposed mostly bythe strong pull of the W Z
Tevatron data (Group ). Furthermore, after the off-shell parameters in the CJ deuteroncorrection are freed (lowermost panel), the tension between Groups , , and is relieved, resulting in a moreconsistent data set, with weak pulls ( < ∼ d/u sector discussed in the previous subsection, and therefore seems to be a robust feature of fitting the deuteroncorrections simultaneously with the PDF parameters.In the two CT fits in the upper left and middle left panels of Fig. 10, a somewhat different pattern emerges.Inclusion of the deuteron correction in CT fixed d.c. does have the effect of partially aligning the pulls of the DISProton (Group ) and DIS deuteron (Group ) on the gluon, but this effect is restricted to a narrower interval, x ∈ [0 . , . W Z and fixed-target Drell-Yan production experiments (Groups and ). The weaker dependence of the gluon pulls in the CT fit on the deuteron correction compared to the CJ caseis most likely due to the absence of the SLAC DIS data from the former fit. One can therefore investigate the effectof the removal of the SLAC data on the CJ fit. Intriguingly, as shown in the right panel of Fig. 11, simultaneouslyremoving the SLAC DIS data and the Tevatron W -boson asymmetry data largely alleviates the competing gluonpulls, which are now smaller than those observed in the CT fit, especially from the LHC sets not included in CJ.Clearly, the gluon pulls in the CT fit are due to the data other than the large- x SLAC and W production data. Inparticular, we notice a strong preference for a harder gluon at x ≈ . ν -A DIS experiments (Group )both with and without the nuclear correction. In fact, the preference of the CDHSW ν -A DIS deuteron data for aharder gluon at large x had already been identified in the CT18 analysis [7], although the net effect of including thisdata set in the CT18 fit does not exceed the PDF uncertainty. However, as the left panel of Fig. 11 shows, removingthese data from the fit does not substantially alter the pulls of the remaining experiments shown in the CT plots ofFig. 10, which are led by the jet and W/Z measurements from the LHC.0 � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � ��� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� ���� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� ���� - � �� - � ���� ���� ���� ��� ��� ��� ��� - �� - �� - ������� � Δ χ � � � � � �� � � � � � � � � �� �� ����� � ( �� � ��� ) � Tevatron jets � DIS Proton � DIS Deuterium � WZ Tevatron � WZ LHC � Drell - Yan � ν A incl. DIS � ttbar production �� ν A μμ SIDIS �� LHC jets � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � ��� - � �� - � ���� ���� ���� ��� ��� ��� ��� - �� - �� - ������� � Δ χ � � � � � �� � � � � � � � � �� �� ����� � ( �� � ��� ) ������ γ + jet Tevatron jets
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Deuterium WZ Tevatron
Drell - Yan
FIG. 10: Analogous to Fig. 8, for the PDF pulls on g ( x, Q ) at Q = 2 GeV.1 � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � ��� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� ���� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� ���� - � �� - � ���� ���� ���� ��� ��� ��� ��� - �� - �� - ������� � Δ χ � � � � � �� � � � � � � � � �� �� �� - �� � ( �� � ��� ) � Tevatron jets � DIS Proton � DIS Deuterium � WZ Tevatron � WZ LHC � Drell - Yan � ttbar production �� ν A μμ SIDIS �� LHC jets � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � ��� - � �� - � ���� ���� ���� ��� ��� ��� ��� - �� - �� - ������� � Δ χ � � � � � �� � � � � � � � � �� �� � - ����� � ( �� � ��� ) ������ γ + jet Tevatron jets
DIS
Proton
DIS
Deuterium WZ Tevatron
Drell - Yan
FIG. 11: PDF pulls on g at 2 GeV after removing the inclusive ν -A experiments for CT (left) and W chargeasymmetry and SLAC DIS data from CJ (right), with deuteron corrections fixed. V. CONCLUSION
In this analysis, we have for the first time undertaken a comparative analysis of two global fitting frameworks,CTEQ-JLab (CJ) and CTEQ-TEA (CT), using the L sensitivity statistical metric developed in Refs. [11, 12]. Thismetric allowed our study to take advantage of the complementary strengths of the two frameworks: the extendedexperimental coverage and various theoretical developments implemented in the two approaches, as well as the flexiblePDF parametrizations available in CT and the unique capabilities of CJ in describing low-energy and nuclear dynamics.In doing so, we made a number of technical adjustments to each framework (discussed in detail in the appendix) inorder to reconcile the CT and CJ treatment of PDF uncertainties and thereby render them sufficiently similar to bemeaningfully juxtaposed.We have, in particular, concentrated on evaluating the impact on PDF determinations of nuclear corrections whichtake into account the two-baryon structure of the deuteron. In fact, as discussed in Sec. I A, DIS and Drell-Yanmeasurements on deuterium are very informative in providing flavor separation of d -type quarks from other partonflavors (under an assumption of nucleon charge symmetry). At the same time, the introduction of deuterium datainto proton PDF fits brings along its own uncertainties associated with nuclear and power-suppressed effects. Globalanalyses take diverse approaches in handling the deuteron and heavy-nuclear effects, from selection of the leastaffected experimental data [7–9], to including some fixed [7] or free [5, 9] nuclear corrections and performing Bayesianmarginalization [35, 53] with respect to the nuclear parameters. It is therefore important to understand the role ofthe deuteron corrections in a controlled setting, by isolating them from other factors that affect the existing PDFensembles at comparable levels. By examining the fitted PDFs and resulting PDF pulls of experimental data under several theoretical scenarios forthe treatment of deuteron corrections, in Sec. IV we have gathered a substantial number of results that clarify thesequestions. We reiterate here our overriding conclusions based on this investigation: • While the compilation of χ values in Sec. IV A indicates good global agreement of CJ and CT NLO theoreticalpredictions with deuteron data sets, the L sensitivity additionally provides insights about local compatibility offitted experiments in an x -dependent fashion. In the case of CJ, the model of deuteron dynamic effects is crucialfor the description of the informative low- Q DIS data from SLAC. The dependence on the deuteron correctionis reduced in the CT analysis with its more conservative cut on W . Still, including the CJ deuteron correctionreduces χ for the aggregated DIS-deuteron experiments by about ∼
14 units and also reduces the cumulative See examples in [7] for comparable variations in S f,L ( E ) caused by various assumptions. χ for vector-boson production data sets by several tens of units, with the modifications potentially comparableto the NNLO scale dependence in an analysis like CT. Another effect of the deuteron correction is to alleviatethe competing pulls of deuteron and some other experiments in the large- x region. Examining aspects of thisstudy in the context of an NNLO analysis will be a valuable activity for future work. • The impact of a fixed deuteron correction is particularly evident in the high- x distribution for the d -quark, or theassociated d/u PDF ratio. A number of commonalities exist between the CT and CJ analyses in the qualitativeeffect of this correction on the extracted high- x PDFs. The fixed deuteron correction of Sec. II A generally leadsto the suppression of the d/u ratio at x > . • Due to the influence of sum rules and nontrivial correlations among fitted PDFs of different parton flavors,deuteron corrections to DIS structure functions at large x can have important secondary effects on, e.g. , thegluon or sea-quark PDFs over a range of x , as well as the d val distribution at lower x ∼ .
03 of relevance toprecision studies in the electroweak sector. • In both fitting frameworks, the modifications caused by the deuteron-structure corrections are moderated bythe inclusion of some non-deuteron data sets; for CT, these are inclusive neutrino DIS data on heavy nucleartargets, while for CJ, a combination of high- x SLAC DIS data and reconstructed boson-level Tevatron chargeasymmetry requires special attention. Disentangling the interplay among these fitted experiments will requirea further study at NNLO accuracy, including additional investigation of the implementation of theoreticalcorrections for nuclear data sets (including both deuteron and heavier targets) and the treatment of
W/Z data.As the drive to realize next-generation accuracy in PDF analyses gains speed with preparations for the High-Luminosity LHC [107], Electron-Ion Collider [108], and Long-Baseline Neutrino Facility [109], we recommend consid-eration of deuteron corrections and broader nuclear effects in PDF analyses, as well as continued phenomenologicaland model-based studies [55, 110–112] of deuteron structure in parallel. Deuteron effects will become particularly un-avoidable with increasing PDF precision and in PDF-benchmarking studies, most obviously for the d -PDF at x > ∼ . x . Consid-eration of the parton-level violation of the charge symmetry in the deuteron [113] may become relevant as precisiongoals advance still further. As emphasized in Sec. I A and IV C, the achievement of ultimate precision in tests of theSM in the electroweak sector will partly depend upon the successful treatment of the issues described in this work. Acknowledgments
We appreciate insightful discussions with Aurore Courtoy, Lucian Harland-Lang, Robert Thorne, and our colleagueswithin the CTEQ-JLab and CTEQ-TEA collaborations, with special thanks to Jun Gao, Shujie Li, Wally Melnitchouk,and C.-P. Yuan. The work at SMU was partially supported by the U.S. Department of Energy under Grant No. DE-SC0010129. T. J. Hobbs acknowledges support from a JLab EIC Center Fellowship. The work of A. Accardi wassupported by the U.S. Department of Energy contract DE-AC05-06OR23177, under which Jefferson Science AssociatesLLC manages and operates Jefferson Lab. A. Accardi and and X. Jing were additionally supported by DOE contractDE-SC0008791.
Appendix A: Comparison procedure and technical details
To meaningfully compare two distinct global fits on a common footing, as done in this article, it has been necessaryto conciliate their methodologies, cf. Sec. III B. Part of this consisted in making a few technical adjustments to ensurethe mutual compatibility of the CJ and CT computations of the L sensitivity introduced in Sec. III C and employedin Sec. IV. In this Appendix, we provide more detail about these adjustments.The L sensitivity can be interpreted as a fast approximation, based on the Hessian error formalism [13], of the∆ χ E shifts contained in the LM scans shown in the panels of Fig. 2. For that reason, we expect the sum of L sensitivities over all fitted experiments E to vanish for each parton flavor f ; i.e. , L tot2 ,f ≡ (cid:88) E S f,L ( E ) ≈ ∀ f . (A1)This desired result — that the “total” L tot2 sensitivities approximately sum to zero for all flavors within a given fit— represents an ideal scenario in which all data sets demonstrate full mutual compatibility, and uncertainties aresmall enough to validate a quadratic approximation for the χ function around the central PDF parameters, (cid:126)a . In3 �� - � �� - � ��� ��� - ���� - ����������� �������� ���� ��� ��� L � ,f t o t �� ���� ����� original eigenvectors, � = � ��� u - d - gdusc � �� - � �� - � ���� ���� ���� ��� ��� ��� ��� - � - ���� � �� ���� ��� , adjusted eigenvectors � � = � ��� u - d - gdusc L � ,f t o t �� - � �� - � ���� ���� ���� ��� ��� ��� ��� - � - ���� � �� �� ���� , ���� � + �� � = � ��� u - d - gdusc L � ,f t o t �� - � �� - � ���� ���� ���� ��� ��� ��� ��� - � - ���� � �� no d.c., ���� �� � = � ��� u - d - gdusc L � ,f t o t FIG. 12: Total sensitivities, L tot2 , summed over all experiments as in Eq. A1. The top left panel shows L tot2 for the CJ free d.c.
PDF ensemble with the standard CJ15 computation of the Hessian eigenvectors. The top right panelillustrates the same result after adjusting each
CJ free d.c. eigenvector to have ∆ χ = T = 10 in both thepositive and negative direction, thereby correcting the departures from Gaussianity observed for poorly constrainedeigenvectors in the standard CJ Hessian analysis. In the bottom row, the left panel shows L tot2 for the CT nod.c.
PDF ensemble computed with the CT18 two-tier uncertainty constraints with T = 10. On the right, the samebut including only the Tier-1 uncertainty constraint that do not introduce a priori deviations from the Gaussianitycondition.practice, however, neither condition was perfectly realized when generating the Hessian eigenvector sets, causing L tot2 to deviate from zero for both the CJ15 and CT18 eigenvector ensembles. We illustrate the graphs of L tot2 obtainedwith the standard eigenvector computations, for tolerance T = 10, in the left panels of Fig. 12. In the upper figure,the total sensitivities in the standard CJ free d.c. show very pronounced deviations from zero. In the lower figure,we see milder but not entirely negligible deviations from zero in the counterpart default
CT no d.c. fit.We have identified the sources of this behavior and remedied this situation as follows. • CJ : The standard CJ PDF error sets are obtained by the Hessian analysis around the best-fit PDF parameters, (cid:126)a , with each eigenvector scaled by a given tolerance factor T to nominally produce an increase of T above theminimum in the χ function ( T = 1 .
646 in the CJ15 analysis [5], T = √
10 in this paper). For the computation ofthe sensitivities, the χ function is instead scanned along every eigenvector starting from the best-fit parameters, (cid:126)a , until parameters (cid:126)a i are found in the plus- and minus-directions such that∆ χ ( (cid:126)a i +1 ) = ∆ χ ( (cid:126)a i ) = T ∀ i = 1 , . . . , N par . (A2)This way, the error PDFs correspond exactly to a given likelihood L ∝ e − T / , and, by construction, L tot2 = 0except for numerical uncertainties, see Eq. (5). The total sensitivities after this adjustment of the eigenvectorexcursions are shown in the upper right panel of Fig. 12. All L tot2 values are now well within ± • CT : The published general-purpose CT fits [7, 17, 114] apply a a two-tier procedure to determine the publishederror bands. Excursions along each eigenvector are constrained both by the Tier-1 penalty imposed by the4increase of the global χ above T units, and by the dynamical Tier-2 penalty based on effective Gaussianvariables that ensure that no single-experiment χ E value increases above its best-fit value by more than itsuncertainty at the requested confidence level. The Tier-2 procedure may lead to unequal excursions in theplus- and minus-directions along a given eigenvector, therefore introducing a departure from the condition for L tot2 = 0. For this reason, we revert to the Tier-1 PDF error determination, which nominally satisfies the desiredcondition on L tot2 . The resulting total sensitivities are compared to the standard CT18 calculation in the lowerrow of Fig. 12. All total sensitivities with the Tier-1 error sets are within ± δτ E Expt ID Expt1 37.2555 160 HERA run I+II2 32.6485 204 E866 proton proton Drell-Yan process3 28.1273 545 CMS 8 TeV 19.7 fb − , single incl. jet cross sec., R = 0.74 24.7211 250 LHCb 8 TeV 2.0 fb − W/Z cross sec.5 24.2715 101 BCDMS F p − , single incl. jet cross sec., R = 0.77 19.1007 245 LHCb 7 TeV 1.0 fb − W/Z forward rapidity cross sec.8 18.9541 102 BCDMS F d − , single incl. jet cross sec., R = 0.6... ... ... ...39 0.446136 145 H1 σ br TABLE III: Impact of a given experiment on the difference between the Tier-1+2 and Tier-1 CT total sensitivities,quantified by the δτ E distance defined in Eq. (A4). The first 3 experiments are well separated from the remainingones, which are more closely spaced.The very bad CJ L tot2 obtained before the adjustment in the upper left panel of Fig. 12 was traced primarily to asubstantial deviation from Gaussianity observed along a small number of eigenvector directions. This can happen ina fit where certain linear combinations of parameters are poorly constrained by the data, especially when exploitingan extended parametrization such as in the CJ case, where additional parameters are included to allow for a non-vanishing d/u quark at large x , to describe off-shell deformation of bound nucleons in a deuteron target, and to fithigher-twist corrections to the standard twist-2 calculation of electron-nucleon DIS. In fact, we have verified thatthe very large values of the total sensitivity are primarily driven by the combined HERA data set, which accountsfor most of the calculated sensitivities across all parton flavors, with a secondary large contribution provided by theDØ W -asymmetry measurements.In the CT analysis, both the Tier-1+2 and Tier-1 total sensitivities in the lower row of Fig. 12 are of a natural order,being generally ≈
0, especially in the Tier-1 calculation at lower-right. This is aside from the high- x L tot2 values for the g - and c -PDFs, which are somewhat larger than in the CJ15 adjusted analysis, but still < ∼ L tot2 ≈ χ imbalance , τ Ei ≡ [ χ i ] E − [ χ i − ] E , i = 1 , . . . , N par . (A3)between the PDF error sets in each positive and negative eigenvector direction, see Eq. (5). We can therefore quantifythe impact of a given experiment on the observed differences between the total sensitivities for the Tier-1+2 andTier-1 CT analysis by calculating δτ E = |−→ τ E Tier − − −→ τ E Tier − | . (A4) As a technical observation, both Tier-1 and Tier-1+2 predictions in this case are computed assuming T = 10, without modifying theexcursions of the effective Gaussian parameters [114] in the Tier-2 penalty compared to the default CT18 setup. χ imbalances. However, our goal here is to identify the experiments with the largest impact on the reduction of thetotal sensitivity, rather than performing a detailed quantitative analysis and ranking of each experiment.We collect the largest values of the δτ E metric in Table III, where experiments have been ordered from the highestto lowest impact. Intriguingly, the most affected experiment in the CT case is also the HERA I+II combined dataset. We believe that the underlying reason(s) for this commonality with the CJ analysis are the high precision andlarge kinematic coverage of the HERA data, which are therefore sensitive to small variations in the PDFs and subjectto secondary constraints, such as those imposed on the gluon distribution by PDF sum rules. [1] P. Jimenez-Delgado, W. Melnitchouk, and J. F. Owens, J. Phys. G , 093102 (2013), arXiv:1306.6515 [hep-ph] .[2] J. Gao, L. Harland-Lang, and J. Rojo, Phys. Rept. , 1 (2018), arXiv:1709.04922 [hep-ph] .[3] K. Kovaˇr´ık, P. M. Nadolsky, and D. E. Soper, Rev. Mod. Phys. , 045003 (2020), arXiv:1905.06957 [hep-ph] .[4] J. J. Ethier and E. R. Nocera, Ann. Rev. Nucl. Part. Sci. , 43 (2020), arXiv:2001.07722 [hep-ph] .[5] A. Accardi, L. T. Brady, W. Melnitchouk, J. F. Owens, and N. Sato, Phys. Rev. D93 , 114017 (2016), arXiv:1602.03154[hep-ph] .[6] S. Alekhin, J. Bl¨umlein, S. Moch, and R. Placakyte, Phys. Rev.
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