We present an improved determination of the strange quark and antiquark parton distribution functions of the proton by means of a global QCD analysis that takes into account a comprehensive set of strangeness-sensitive measurements: charm-tagged cross sections for fixed-target neutrino–nucleus deep-inelastic scattering, and cross sections for inclusive gauge-boson production and W-boson production in association with light jets or charm quarks at hadron colliders. Our analysis is accurate to next-to-next-to-leading order in perturbative QCD where available, and specifically includes charm-quark mass corrections to neutrino–nucleus structure functions. We find that a good overall description of the input dataset can be achieved and that a strangeness moderately suppressed in comparison to the rest of the light sea quarks is strongly favored by the global analysis.
publisher-imprint-nameSpringervolume-issue-count12issue-article-count74issue-toc-levels0issue-pricelist-year2020issue-copyright-holderThe Author(s)issue-copyright-year2020article-contains-esmNoarticle-numbering-styleContentOnlyarticle-registration-date-year2020article-registration-date-month12article-registration-date-day11article-toc-levels0toc-levels0volume-typeRegularjournal-productNonStandardArchiveJournalnumbering-styleContentOnlyarticle-grants-typeOpenChoicemetadata-grantOpenAccessabstract-grantOpenAccessbodypdf-grantOpenAccessbodyhtml-grantOpenAccessbibliography-grantOpenAccessesm-grantOpenAccessonline-firstfalsepdf-file-referenceBodyRef/PDF/10052_2020_Article_8749.pdfpdf-typeTypesettarget-typeOnlinePDFissue-typeRegulararticle-typeOriginalPaperjournal-subject-primaryPhysicsjournal-subject-secondaryElementary Particles, Quantum Field Theoryjournal-subject-secondaryNuclear Physics, Heavy Ions, Hadronsjournal-subject-secondaryQuantum Field Theories, String Theoryjournal-subject-secondaryMeasurement Science and Instrumentationjournal-subject-secondaryAstronomy, Astrophysics and Cosmologyjournal-subject-secondaryNuclear Energyjournal-subject-collectionPhysics and Astronomyopen-accesstrueIntroduction
An accurate determination of the strange quark and antiquark parton distribution functions (PDFs) of the proton [1–3] is key to carrying out precision phenomenology at current and future colliders, specifically for measuring fundamental parameters of the standard model (SM) such as the mass of the W boson [4], the Weinberg angle [5], and electroweak parameters in general [6]. Because of the limited experimental information available, however, the strange quark and antiquark PDFs remain much more uncertain than the up and down sea quark PDFs.
The strange quark and antiquark PDFs have been determined from neutrino–nucleus deep-inelastic scattering (DIS) for a long time, specifically from measurements of dimuon cross sections, whereby the secondary muon originates from the decay of a charmed meson, νμ+N→μ+c+X\documentclass[12pt]{minimal}
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\begin{document}$$\nu _{\mu }+N\rightarrow \mu +c+X$$\end{document} with c→D→μ+X\documentclass[12pt]{minimal}
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\begin{document}$$c\rightarrow D\rightarrow \mu +X$$\end{document} [7–10]. When interpreted in terms of the ratio between strange and non-strange sea quark PDFs, Rs≡(s+s¯)/(u¯+d¯)\documentclass[12pt]{minimal}
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\begin{document}$$R_s\equiv (s+{\bar{s}})/({\bar{u}}+{\bar{d}})$$\end{document}, these measurements favor values around Rs≲0.5\documentclass[12pt]{minimal}
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\begin{document}$$R_s\lesssim 0.5$$\end{document} when PDFs are evaluated at values of the momentum fraction x=0.023\documentclass[12pt]{minimal}
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\begin{document}$$x=0.023$$\end{document} and scale Q=1.6\documentclass[12pt]{minimal}
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\begin{document}$$Q=1.6$$\end{document} GeV. Therefore, it came as a surprise when a QCD analysis of the W- and Z-boson rapidity distributions measured by the ATLAS experiment in proton–proton collisions [11], later corroborated by an analysis based on an increased integrated luminosity [12], suggested instead a ratio closer to Rs≃1\documentclass[12pt]{minimal}
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\begin{document}$$R_s\simeq ~1$$\end{document}. Complementary information on the strange quark and antiquark PDFs is provided by W-boson production in association with light jets [13] and charm quarks [14], the latter process being dominated by the partonic scattering g+s→W+c\documentclass[12pt]{minimal}
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\begin{document}$$g+s \rightarrow W+c$$\end{document}. Measurements of these processes were performed by the ATLAS [15, 16] and CMS [17, 18] experiments recently. Although ATLAS and CMS W\documentclass[12pt]{minimal}
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\begin{document}$$c$$\end{document} measurements turned out to be consistent at the parton level, different interpretations in terms of Rs\documentclass[12pt]{minimal}
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\begin{document}$$R_s$$\end{document} were claimed [16, 17].
This state of affairs has motivated studies of the proton strangeness within the CT, MMHT, and NNPDF global fits, with overall consistent findings. The NNPDF3.1 analysis [19] found that, whereas the ATLAS W, Z dataset [12] does indeed favour a larger total strangeness, its χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} remains non-optimal when fitted together with the neutrino dimuon data. The recent CT18 global analysis [20] also presented fits with and without the ATLAS measurement of [12], with the resulting PDFs differing by more than one-sigma both for the gluon and for the total strangeness. An update of the global PDF analysis from the MMHT collaboration [21], which for the first time accounted for the NNLO massive corrections to the neutrino dimuon cross sections within a PDF fit, also revealed an enhanced strangeness driven by the ATLAS W, Z dataset. The resulting PDFs were, however, consistent within uncertainties with the corresponding fit once this dataset was excluded. Additional dedicated studies of the strange quark and antiquark PDFs have been presented [22–26], however, these focused on a restricted set of processes or datasets, or were based on theoretical and methodological assumptions that can potentially bias the results.
Given its phenomenological relevance for precision physics at the LHC, a global reinterpretation of all of the strangeness-sensitive measurements within an accurate theoretical and methodological framework appears to be therefore timely and compelling. This paper fulfills this purpose: we present an improved determination of the strange quark and antiquark PDFs, accurate to next-to-next-to-leading order (NNLO) in perturbative QCD where available, by expanding the NNPDF3.1 analysis [19] in two respects. First, we take into account several new pieces of experimental information which are relevant in constraining the strange quark and antiquark PDFs: charm-tagged to inclusive cross section ratios measured by the NOMAD experiment [10] in fixed-target neutrino–nucleus DIS; and an extended set of cross sections for inclusive gauge-boson production and W-boson production in association with light jets or charm quarks measured by the ATLAS [12, 15, 16] and CMS [17, 18] experiments in proton–proton collisions. Second, we improve the theoretical description of dimuon neutrino DIS structure functions, by implementing NNLO charm-quark mass corrections, and of W\documentclass[12pt]{minimal}
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\begin{document}$$c$$\end{document} production data, by including a theoretical uncertainty that accounts for the unknown NNLO QCD corrections; we also explicitly enforce the positivity of the F2c\documentclass[12pt]{minimal}
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\begin{document}$$F_2^c$$\end{document} structure function.
The outline of this paper is as follows. In Sect. 2 we discuss the experimental data and the theoretical details used in this analysis, along with the PDF fits performed. In Sect. 3, we present the results of these fits, we assess their quality, and we use them to understand how the datasets and the theoretical framework affect the PDFs, in particular in relationship with the strangeness content of the proton. We finally provide a summary of our work in Sect. 4.
Analysis settings
In this section we present the experimental datasets used as input to our analysis, we then discuss the details of the corresponding theoretical computations, and we finally explain which PDF fits we perform to study their impact on the proton strangeness.
Experimental data
The bulk of the dataset included in our analysis corresponds to the one used in [27], which is in turn a variant of the dataset used in the NNPDF3.1 NNLO analysis [19]. It contains in particular measurements of the dimuon neutrino–nucleus DIS cross sections from the NuTeV experiment [9], and of inclusive gauge-boson production in proton–(anti)proton collisions from several Tevatron and LHC experiments [12, 28–31]. These measurements represented the most constraining source of experimental information on the strange quark and antiquark PDFs in the NNPDF3.1 analysis.
We supplement this dataset with a number of new measurements. Concerning neutrino–nucleus DIS, we include measurements of the ratio of dimuon to inclusive charged-current cross sections, Rμμ(ω)=σμμ(ω)/σCC(ω)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {R}}_{\mu \mu }(\omega )=\sigma _{\mu \mu }(\omega )/\sigma _{\mathrm{CC}}(\omega )$$\end{document}, from the NOMAD experiment [10], see Sect. 2.2 for details. The data is presented for three kinematic variables ω\documentclass[12pt]{minimal}
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\begin{document}$$E_\nu $$\end{document}, the only variable which is directly measured by the experiment among the three. We will nevertheless verify that similar results can be obtained for instance with the s^\documentclass[12pt]{minimal}
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\begin{document}$$\sqrt{{\hat{s}}}$$\end{document}-dependent dataset. The kinematic sensitivity of the NOMAD measurements is roughly 0.03≲x≲0.7\documentclass[12pt]{minimal}
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\begin{document}$$0.03 \lesssim x \lesssim 0.7$$\end{document}, as illustrated by the coverage of the x-dependent dataset.
Concerning proton–proton collisions, we augment the inclusive gauge-boson production measurement from the ATLAS experiment at a center-of-mass energy (c.m.e.) of 7 TeV [12] with the off-peak and forward rapidity bins (not included in NNPDF3.1) for a total of ndat=61\documentclass[12pt]{minimal}
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\begin{document}$$n_{\mathrm{dat}}=61$$\end{document} data points. Furthermore, we include the ndat=37\documentclass[12pt]{minimal}
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\begin{document}$$n_{\mathrm{dat}}=37$$\end{document} data points corresponding to the ATLAS (at a c.m.e. of 7 TeV) [16] and CMS (at a c.m.e. of 7 TeV and 13 TeV) [17, 18] W\documentclass[12pt]{minimal}
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\begin{document}$$c$$\end{document} measurements; for ATLAS, we consider the charm-jet dataset, which is amenable to fixed-order calculations (instead of the D-meson dataset). Finally we take into account the ndat=32\documentclass[12pt]{minimal}
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\begin{document}$$n_{\mathrm{dat}}=32$$\end{document} data points corresponding to the ATLAS W+jets measurement (at a c.m.e. of 8 TeV) differential in the transverse momentum of the W boson [15]. Overall, these LHC datasets are sensitive to the proton strangeness in the region 10-3≲x≲0.1\documentclass[12pt]{minimal}
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\begin{document}$$n_{\mathrm{dat}}=4096$$\end{document} data points; experimental correlations within each dataset are available for all of the new measurements considered here and are therefore included in our analysis.
Theoretical calculations
The measurements outlined in the previous section correspond to hadronic observables already considered in [19], except for the ratio Rμμ\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {R}}_{\mu \mu }$$\end{document} measured by the NOMAD experiment, and for the production of W bosons in association with light jets measured by the ATLAS experiment. Likewise, the theoretical settings adopted in the present analysis closely follow those described in the NNLO analysis of [19, 27] (whereby, in particular, the charm PDF is fitted), except for some improvements. In this section we discuss in turn the new NOMAD observable and the theoretical details unique to the present analysis.
The NOMAD ratio
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\begin{document}$$E_\nu $$\end{document}-dependent dataset, which we include by default, we haveσi(Eν)=∫x01dxx∫Qmin2Qmax2(x)dQ2d2σidxdQ2(x,Q2,Eν),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \sigma _i(E_\nu ) = \int _{x_0}^1\frac{\mathrm{{d}}x}{x} \int _{Q^2_{\mathrm{min}}}^{Q^2_{\mathrm{max}}(x)}\mathrm{{d}}Q^2 \frac{\mathrm{{d}}^2\sigma _i}{\mathrm{{d}}x\mathrm{{d}}Q^2}(x,Q^2,E_\nu )\,, \end{aligned}$$\end{document}where Qmax2(x)=2mpEνx\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&\frac{\mathrm{{d}}^2\sigma _i}{\mathrm{{d}}x\mathrm{{d}}Q^2}(x,Q^2,E_\nu ) = \frac{G_F^2M_W^2}{4\pi }\frac{1}{(Q^2+M_W^2)^2} \nonumber \\&\qquad \times \left[ \left( Y_+ - \frac{2m_p^2x^2y^2}{Q^2}\right) F_2^i(x,Q^2) - y^2 F_L^i(x,Q^2) \right. \nonumber \\&\qquad \left. + Y_-xF_3^i \right] K^i\,. \end{aligned}$$\end{document}The kinematic factors Y±=1±(1-y)2\documentclass[12pt]{minimal}
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\begin{document}$$b=6.7\pm 1.8$$\end{document}. The corresponding uncertainty is included in the experimental covariance matrix of the measurement.
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\begin{document}$$i=\mathrm{CC}$$\end{document}) structure functions Fpi\documentclass[12pt]{minimal}
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\begin{document}$$p=2,L,3$$\end{document}) entering Eq. (2.2) are evaluated with APFEL [32]. We benchmarked our results against those obtained from an independent computation based on [33]. After the correction of a bug in APFEL, which affected the computation of the large-x DIS coefficient functions at next-to-leading (NLO) order, the relative difference between the two is found to be of the order of permille, apart from the lowest Eν\documentclass[12pt]{minimal}
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Theoretical improvements
In comparison to the earlier NNPDF analyses [19, 27], here we introduce several theoretical improvements, which are summarized in turn below.
NNLO massive corrections in neutrino DIS We incorporate the recently computed NNLO charm-quark massive corrections [33, 34] in the description of the NuTeV and NOMAD measurements. We do so by multiplying the NLO theoretical prediction in the FONLL general-mass variable flavor number scheme [35, 36] by a K-factor defined as the ratio between the NNLO result in the fixed-flavor number (FFN) scheme with and without the charm-mass correction in the matrix elements (ME); NNLO PDFs are used in both cases. The resulting K-factor,KNNLO≡σFFN(NNLOPDFs,NNLOME)σFFN(NNLOPDFs,NLOME),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} K_{\mathrm{NNLO}} \equiv \frac{\sigma _{\mathrm{FFN}}(\mathrm{NNLO~PDFs,~NNLO~ME})}{\sigma _{\mathrm{FFN}}(\mathrm{NNLO~PDFs,~NLO~ME)}} \, , \end{aligned}$$\end{document}is such that the prediction for the NuTeV dimuon cross sections becomesd2σμμdxdQ2|FONLL(NNLOME)=KNNLO×d2σμμdxdQ2|FONLL(NLOME),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&\frac{\mathrm{{d}}^2\sigma _{\mu \mu }}{\mathrm{{d}}x\mathrm{{d}}Q^2}\Bigg |_{\mathrm{FONLL\,(NNLO\,ME)}} \nonumber \\&\quad = K_{\mathrm{NNLO}} \times \frac{\mathrm{{d}}^2\sigma _{\mu \mu }}{\mathrm{{d}}x\mathrm{{d}}Q^2}\Bigg |_{\mathrm{FONLL\,(NLO\,ME)}} \, , \end{aligned}$$\end{document}and an analogous expression holds for the NOMAD observables.
This approach provides a good approximation of the exact result, because theoretical predictions in the FFN and FONLL schemes are very close for the NuTeV and NOMAD kinematics. This fact was demonstrated in [36] in the case of NuTeV data; we nevertheless checked that it remains true with the independent computation of [33, 34], and that it also applies to the NOMAD measurements. To this purpose, we computed the relative difference between the FONLL-A and FFN scheme predictions for the NuTeV and NOMAD datasets based on structure functions accurate to O(αs)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {O}}(\alpha _s)$$\end{document}. We found that differences were less than 1% in the entire kinematic range for NuTeV, and of about 1.5% irrespective of the value of Eν\documentclass[12pt]{minimal}
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\begin{document}$$E_{\nu }$$\end{document} for NOMAD. These differences are well below the experimental and the PDF uncertainties.1 We therefore conclude that using a NNLO K-factor determined in the FFN scheme in fits that otherwise use the FONLL scheme throughout is unlikely to affect the final results. Given the current implementation of the FONLL scheme in the DIS observables [35] the matching between the NNLO massless and massive calculations would require non-trivial modifications of the code of [33], e.g. to extract the collinear logarithms, with little practical advantage.
The K-factors are in general smaller than unity, and thus enhance the (anti-)strange quark PDF when accounted for in the fit. This fact is consistent with what was already observed in [21], and is further illustrated in the right panel of Fig. 1, where we display the charm production cross section, Eq. (2.1), with i=μμ\documentclass[12pt]{minimal}
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\begin{document}$$n_f=3$$\end{document}) at different perturbative orders using the NNPDF3.1 NNLO PDF set (consistently with nf=3\documentclass[12pt]{minimal}
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\begin{document}$$n_f=3$$\end{document}). The inset displays the ratio to the leading order (LO) calculation. Higher-order corrections clearly suppress the cross section, in particular as Eν\documentclass[12pt]{minimal}
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NNLO corrections in collider gauge-boson production Theoretical predictions for inclusive W- and Z-boson production and for W-boson production in association with charm quarks or light jets are evaluated at NLO using MCFM+APPLgrid [37, 38], and are supplemented with NNLO QCD K-factors. These are evaluated with FEWZ [39] for inclusive gauge-boson production, and with the Njetty\documentclass[12pt]{minimal}
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\begin{document}$$_{\mathrm{jetty}}$$\end{document} program [40, 41] for W-boson production with light jets. In the first case, a fixed factorization and renormalization scale is used, equal to the mass of the gauge boson; in the second case, a dynamic scale is used, where the factorization and renormalization scales are defined as μF=μR=mW2+pT,j2\documentclass[12pt]{minimal}
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\begin{document}$$W + c$$\end{document} production have been presented only very recently [42], in this case we accompany the data with an additional correlated uncertainty, estimated from the 9-point scale variations of the NLO calculation [43, 44].
A list of the PDF fits presented in this work; see the text for details
Nuclear corrections in neutrino DIS Neutrino-DIS measurements from the NuTeV and NOMAD experiments are subject to nuclear corrections, because they both utilize an iron (Fe) target. In this analysis, however, we do not include such corrections because they are expected to be subdominant in comparison to other sources of uncertainties. For NuTeV, they were found to be moderate in a global fit based on the same methodology used here [45]; for NOMAD, they are known to approximately cancel out in the ratio Rμμ\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {R}}_{\mu \mu }$$\end{document}. To verify this last statement, we recomputed the NOMAD ratio Rμμ\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {R}}_{\mu \mu }$$\end{document} with the recently presented nNNPDF2.0 NLO Fe nuclear PDF set [46], and compared the result with the predictions obtained with the NLO free proton PDF set consistently determined in [46]. The full set of correlations between Fe and proton PDFs were therefore appropriately taken into account. The relative difference between the two computations (with and without nuclear PDF corrections) turned out to range between 3%, in the lowest Eν\documentclass[12pt]{minimal}
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\begin{document}$$E_\nu $$\end{document} bin, and a fraction of percent, in the bins at the highest Eν\documentclass[12pt]{minimal}
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\begin{document}$$E_\nu $$\end{document}. These differences are smaller than both the data and the PDF uncertainties, therefore ignoring nuclear PDF uncertainties is a well-justified approximation. We note that nuclear non-isoscalarity effects are treated as in [45]. In the future, it might be interesting to extend the present analysis in a way that systematically accounts for nuclear PDF uncertainties.
Positivity of cross sections We enforce the positivity of the charm structure function F2c\documentclass[12pt]{minimal}
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\begin{document}$$F_2^c$$\end{document} with a procedure similar to that described in [47] for light quarks. This additional positivity constraint is required to prevent the fitted charm PDF becoming unphysically negative once the new datasets are included in the fit.
PDF sets
We assess the impact of the datasets and of the theoretical choices outlined in Sects. 2.1–2.2 on PDFs by performing the series of fits summarized in Table 1. All of them are accurate to NNLO in perturbative QCD (where available), and are based on the NNPDF methodology; see [47] and the references therein for a comprehensive description.
Values of χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} per data point for the strangeness-sensitive datasets discussed in this work obtained from the str_base, str_prior, str, str_s_hat, and str_pch fits; see Table 1. Values in square brackets are for datasets not included in the corresponding fit
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\begin{document}$$F_2^c$$\end{document}, and with the removal of the 2010 and 2011 ATLAS W, Z inclusive measurements of [11, 12]. We will present a comparison of this baseline fit with the NNPDF3.1 PDF set of [19] in Sect. 3.3. This fit is then supplemented with all the new LHC data, including the ATLAS W, Z measurements from [11, 12], to obtain the second fit (str_prior), for which we generate Nrep=850\documentclass[12pt]{minimal}
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\begin{document}$$N_{\mathrm{rep}}=850$$\end{document} Monte Carlo replicas. The exclusion of the ATLAS W, Z measurements of [11, 12] from the str_base fit allows us to quantify the effect of the full LHC strangeness-sensitive dataset by comparing this fit with the str_prior one. This second fit, str_prior, is finally further supplemented with the NOMAD data, specifically the set that depends on Eν\documentclass[12pt]{minimal}
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\begin{document}$$E_\nu $$\end{document}, to determine the third fit (str). Bayesian reweighting and unweighting [48, 49] are used in this last step, because they allow one to evaluate the two-dimensional integral in Eq. (2.1) only once, a task that would otherwise be computationally very intensive in a fit. After reweighting, one ends up with Neff=105\documentclass[12pt]{minimal}
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\begin{document}$$N_{\mathrm{rep}}=100$$\end{document} replicas. Second, in order to assess the impact of parametrizing the charm PDF, we performed the str_prior_pch and str_pch fits. These fits are equivalent to the str_prior and str fits, except for the fact that the charm PDF is generated perturbatively off the gluon and the light-quark PDFs. In this case, we produced only Nrep=500\documentclass[12pt]{minimal}
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Results
In this section we present the main results of our analysis. First, we discuss the quality of the fits that we performed; then, we compare the data to our theoretical predictions; next, we present the PDFs that we determine; and finally, we revisit the strangeness content of the proton in the light of these. We conclude by focusing on the impact of the NOMAD dataset and of the implications that the treatment of the charm PDF has on our results.
Comparison between the theoretical predictions, obtained from the str_prior and str (left) or str_s_hat (right) fits, and the experimental data for the Eν\documentclass[12pt]{minimal}
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Comparison between theoretical predictions and experimental data for the neutrino (left) and antineutrino (right) charm dimuon cross sections measured by the NuTeV experiment [9]. Data and theory are normalized to the central value of the former; data points are sorted by their ID values, roughly corresponding to increasing x and Q2\documentclass[12pt]{minimal}
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Fit quality
In Table 2 we summarize the values of χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} of the str_prior_pch fit is not reported, because it is not particularly more informative than the one of the str_pch fit, which includes the complete dataset. In all cases, the value of χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} per data point correspond to the definition given in Eqs. (3.2)–(3.3) in [50]. The values in square brackets are for datasets not included in the corresponding fit.
We first assess the general consistency of the new experimental data, by comparing the values of χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} of the first three fits. The description of the new datasets – which, in particular, is not optimal for the ATLAS W, Z dataset in the str_base fit and for the NOMAD dataset in the str_base and str_prior fits – markedly improves as soon as they are included in subsequent fits. The largest effect is witnessed by the NOMAD dataset, whose χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} per data point decreases from about 9 in the str_base and str_prior fits to about 0.6 in the str fit. The value of χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} for all of the other datasets is in general not affected upon the addition of the NOMAD dataset in the str fit, except for the NuTeV dataset, whose χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} is further improved in comparison to the str_prior fit. We therefore conclude that the global dataset is overall consistent and satisfactorily described in the final str fit.
We then assess the consistency of alternative NOMAD datasets by comparing χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} of the str and str_s_hat fits. We recall that they include, respectively, the Eν\documentclass[12pt]{minimal}
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\begin{document}$$\sqrt{{\hat{s}}}$$\end{document}-dependent distributions. A very similar fit quality is achieved in the two cases, not only for the NOMAD dataset, but also for all of the other datasets: the differences in the values of χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} between the two fits are smaller than statistical fluctuations. This fact suggests that the alternative NOMAD datasets are consistent between them and with the rest of the dataset. This conclusion is in line with the observation that a similar number of effective replicas is obtained by reweighting the str_prior fit with either dataset; see the discussion in Sect. 2.3.
We finally assess the effect of parametrizing the charm PDF (or not) by comparing the values of χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} of the str and str_pch fits. We recall that the two fits contain exactly the same datasets, however, in the former the charm PDF is parametrized on the same footing as the other light-quark PDFs, while in the latter it is generated perturbatively off the light quarks and the gluon. The fitted charm fit (str) achieves a better description of the strangeness-sensitive datasets, and of the global dataset overall, than the perturbative charm fit (str_pch). We note in particular the χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} values of the ATLAS W, Z and of the total datasets, which increase, respectively, from 1.67 to 1.80 and from 1.17 to 1.20 when comparing the str and the str_pch fits. We therefore confirm previous studies indicating that fitting the charm PDFs improves the description of the experimental data within a global PDF analysis.
Comparison with experimental data
We now compare the strangeness-sensitive datasets included in our analysis with the corresponding theoretical predictions. Our aim is to assess the impact of the various datasets. To this purpose, we compare the fits obtained without and with a specific dataset included.
We first focus on the neutrino-DIS datasets. In Fig. 2 we display the comparison for the Eν\documentclass[12pt]{minimal}
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\begin{document}$$\sqrt{{\hat{s}}}$$\end{document}-dependent NOMAD measurements. We compare the theoretical predictions obtained from the str_prior fit and, respectively, either from the str or the str_s_hat fits. The insets display the ratio to the central value of each measured data point. In the two cases, we observe a consistent picture: the theoretical prediction obtained from the str_prior fit overshoots the data points by about 20% (10%) for the Eν\documentclass[12pt]{minimal}
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\begin{document}$$\sqrt{{\hat{s}}}$$\end{document}-dependent) dataset; after reweighting, the theoretical prediction nicely describes the data points with an uncertainty consistently reduced by up to a factor of 4. We explicitly checked that the same reduction of the uncertainty occurs also in the case of perturbative charm fits without (str_prior_pch) and with (str_pch) the Eν\documentclass[12pt]{minimal}
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\begin{document}$$E_\nu $$\end{document}-dependent NOMAD dataset included. In this case, however, the underlying PDFs and the strangeness ratio Rs\documentclass[12pt]{minimal}
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\begin{document}$$R_s$$\end{document} vary in comparison to fitted charm fits, as discussed in Sect. 3.4.
In Fig. 3 we display the data/theory comparison for the charm dimuon cross sections from the NuTeV measurement of [9] (for both neutrino and antineutrino beams). In this case, predictions are determined from the str_base and str PDF input sets, and are normalized to the central value of the data points. These are sorted by their ID value, roughly corresponding to increasing x and Q2\documentclass[12pt]{minimal}
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\begin{document}$$Q^2$$\end{document} values (for fixed pseudo-rapidity bins y) when the plot is read from left to right. A fair agreement between data and theory is observed, as expected from the pattern of the χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} values reported in Table 2. The inclusion of the NOMAD data in the str fit suppresses the theoretical expectation for the NuTeV neutrino cross sections (but not for the antineutrino ones, for which no analogue observable is measured by NOMAD); uncertainties are reduced by up to a factor of 2 (again, more markedly for the neutrino data points than for the antineutrino ones). Both the shift in the central value and the reduction of the uncertainty remain smaller than the comparatively large experimental uncertainty.
We now turn to the hadron collider data. In Fig. 4 we display: the W+c\documentclass[12pt]{minimal}
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\begin{document}$$W^-$$\end{document}) of [17, 18] (respectively at a c.m.e. of 7 TeV and 13 TeV); and the Z dilepton rapidity distributions from the ATLAS measurement of [12] at a c.m.e. of 7 TeV (for both the central and the forward selection cuts). The insets display the ratio of the theory to the central value of the experimental measurement. As in Fig. 3, theoretical predictions are evaluated with the str_base and str PDF sets.
Comparison between theoretical predictions and experimental data for some of strangeness-sensitive proton collider measurements used in this work. Top: the W+c\documentclass[12pt]{minimal}
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\begin{document}$$W^-$$\end{document}) corresponding to the CMS measurements at 7 TeV [17] and 13 TeV [18]. Bottom: the Z dilepton rapidity distributions from the ATLAS measurement of [12]. The insets display the theory to data ratio. Theoretical predictions are evaluated with the str_base and str fits
A fair agreement between data and theory is found in all cases, as expected from the pattern of χ2\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2$$\end{document} values reported in Table 2. However, we clearly see that the size of the PDF uncertainty relative to the size of the data uncertainty depend on the dataset. Concerning the ATLAS and CMS W+c\documentclass[12pt]{minimal}
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\begin{document}$$W+ c$$\end{document} measurements, experimental uncertainties span the range between 10 and 20%, and are consistently larger than PDF uncertainties. We note that the PDF uncertainties in the theory predictions are markedly reduced in the str fit in comparison to str_base fit, as highlighted by the ratios in the insets. Concerning the ATLAS Z distribution, the total experimental uncertainty is much smaller than the W counterpart, around 2% for the central rapidity bin, and in the central region it is comparable to the PDF uncertainty. We therefore expect this measurement to be one of the most constraining among all of the LHC measurements considered in this work. Interestingly, once the NOMAD dataset is included in the fit, the central value of the theoretical prediction approaches the central value of the ATLAS data, and PDF uncertainties are slightly reduced. A similar trend can be observed for the forward selection data. This behavior is a further sign of the good overall compatibility of all of the datasets, and in particular of neutrino DIS and LHC gauge-boson production measurements.
Parton distributions
We now turn to study the impact of the theoretical assumptions and of the new datasets considered in this analysis on the PDFs. We first present a comparison between the str_base and the NNPDF3.1 parton sets, and then a comparison among the str_base, str_prior and str PDF sets. In the latter case, because the new datasets are expected to mainly affect the strange quark and antiquark distributions, we will focus on the total and valence strange distributions, first, and on the other PDFs, then.
Comparison with NNPDF3.1
Our baseline fit str_base differs from NNPDF3.1 [19] in several respects. As explained in Sect. 2.2, these include: the treatment of inclusive jet production from ATLAS and CMS with NNLO K-factors, see [27]; an updated treatment of non-isoscalarity effects in neutrino-DIS data, see [45]; the inclusion of the NNLO massive corrections to the NuTeV structure functions; the new F2c\documentclass[12pt]{minimal}
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\begin{document}$$F_2^c$$\end{document} positivity constraint; and the correction of the APFEL bug found in the benchmark reported in Fig. 1, which affected the large-x implementation of the NLO coefficient functions. Furthermore, following the motivation presented in Sect. 2.3, the ATLAS W, Z rapidity distributions from [11, 12] that were part of NNPDF3.1 are excluded from str_base.
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Comparison between the NNPDF3.1 NNLO fit [19] and the baseline fit used in this work, str_base. From top to bottom and left to right we show the up (valence and sea), down (valence and sea), strange (valence and total), gluon, and charm distributions at a scale Q=100\documentclass[12pt]{minimal}
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A comparison of the total (top) and valence (bottom) strange distributions, s+\documentclass[12pt]{minimal}
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\begin{document}$$s^+$$\end{document} is normalized to the central value of the str_base fit. The insets display the corresponding relative (δs+/s+\documentclass[12pt]{minimal}
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Total and valence strange distributions
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\begin{document}$$W + c$$\end{document} and the W + jets datasets are not part of any of these PDF sets. We also emphasize that, apart from the more extensive dataset, our analysis differs from all of the other PDF determinations shown in Fig. 6 in that the charm-quark PDF is fitted on the same footing as the other light-quark PDFs [53]. This feature was demonstrated to improve the description of DIS and LHC datasets, and in particular to partially relieve tensions between the NuTeV and the ATLAS W, Z datasets [19]. The insets in Fig. 6 display the relative and absolute PDF uncertainties for the total (δs+/s+\documentclass[12pt]{minimal}
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A comparison among the str_base, str_prior and str fits reveals that the impact of the data is consistent for the total and valence strange distributions. The inclusion of the LHC datasets in the str_prior fit does not alter the central value of the PDFs in a significant way, while it narrows the PDF uncertainty across most of the x range. The inclusion of the NOMAD dataset in the str fit is associated to a larger effect: the central value of both s+\documentclass[12pt]{minimal}
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A comparison among the str fit and other recent parton sets reveals differences in the shape of the central value of the s+\documentclass[12pt]{minimal}
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Light-quark, charm, and gluon PDFs
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Comparison between the fits presented in this work. From top to bottom and left to right we show the up (valence and sea), down (valence and sea), gluon and charm distributions resulting from the str_base, str_prior and str fits at Q=100\documentclass[12pt]{minimal}
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From these comparisons, we observe that the new datasets have a little impact on the gluon PDF, both on central values and on uncertainties, as expected. A bigger effect is observed instead on the quark PDFs. For light quarks and antiquarks, the electroweak LHC datasets constrain the distributions at low to mid values of x, x≲0.1\documentclass[12pt]{minimal}
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\begin{document}$$0.01\lesssim x \sim 0.1$$\end{document} by a few percent, and to make all the light valence and sea quark PDFs more precise: overall, uncertainties are reduced by up to a factor 2 in the same region for the str fit. For the charm PDF, the central value is suppressed in the str fit; uncertainties are reduced by up to a factor 2 for x≃0.05\documentclass[12pt]{minimal}
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\begin{document}$$x\simeq 0.05$$\end{document}. This effect is almost entirely due to the NOMAD data, which is indirectly sensitive to the charm PDF through its interplay with the sc¯\documentclass[12pt]{minimal}
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\begin{document}$${\bar{s}}c$$\end{document} contributions to W-boson production.
The ratio Rs\documentclass[12pt]{minimal}
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\begin{document}$$R_s$$\end{document}, Eq. (3.1), as a function of x at Q=10\documentclass[12pt]{minimal}
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\begin{document}$$Q=10$$\end{document} GeV. The PDF used are from the str_base, str_prior and str fits (left), and from the str fit and from recent parton sets (right) see text for details. The insets display the corresponding relative uncertainty δRs/Rs\documentclass[12pt]{minimal}
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\begin{document}$$\delta R_s/R_s$$\end{document}
Same as Fig. 8, comparing fits to the Eν\documentclass[12pt]{minimal}
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\begin{document}$$E_\nu $$\end{document}-dependent (str) or to the s^\documentclass[12pt]{minimal}
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\begin{document}$$\sqrt{{\hat{s}}}$$\end{document}-dependent (str_s_hat) NOMAD dataset (left), and fits with fitted (str) or perturbative (str_pch) charm (right)
The strange content of the proton revisited
We finally revisit the strange content of the proton in the light of our results. To this purpose, we consider the strange fraction of proton quark sea and the corresponding ratio of momentum fraction, respectively, defined asRs(x,Q2)=s(x,Q2)+s¯(x,Q2)u¯(x,Q2)+d¯(x,Q2),Ks(Q2)=∫01dxxs(x,Q2)+s¯(x,Q2)∫01dxxu¯(x,Q2)+d¯(x,Q2).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&R_s(x,Q^2) = \frac{s(x,Q^2)+{\bar{s}}(x,Q^2)}{{\bar{u}}(x,Q^2)+{\bar{d}}(x,Q^2)} \,, \qquad \nonumber \\&K_s(Q^2) = \frac{\int _0^1 \mathrm{{d}}x\, x \left[ s(x,Q^2)+{\bar{s}}(x,Q^2)\right] }{\int _0^1 \mathrm{{d}}x\, x \left[ {\bar{u}}(x,Q^2)+{\bar{d}}(x,Q^2) \right] }\,. \end{aligned}$$\end{document}We first consider the ratio Rs\documentclass[12pt]{minimal}
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\begin{document}$$R_s$$\end{document}. In the left panel of Fig. 8 we display it for the str_base, str_prior and str fits at a scale Q=10\documentclass[12pt]{minimal}
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\begin{document}$$Q=10$$\end{document} GeV as a function of x. The inset displays the associated relative PDF uncertainty δRs/Rs\documentclass[12pt]{minimal}
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\begin{document}$$\delta R_s/R_s$$\end{document}. The impact of the new datasets is clearly visible. Concerning the central value, collider datasets do not alter its expectation (the results obtained from the str_base and str_prior fits are almost identical); the NOMAD dataset, instead, prefers a more suppressed strange sea for x≳0.1\documentclass[12pt]{minimal}
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\begin{document}$$x\gtrsim 0.1$$\end{document}. Concerning uncertainties, collider datasets lead to a reduction of the relative uncertainty on Rs\documentclass[12pt]{minimal}
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\begin{document}$$R_s$$\end{document} of about 4% for x≲0.1\documentclass[12pt]{minimal}
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\begin{document}$$x\lesssim 0.1$$\end{document}; the NOMAD dataset, instead, reduces it by about a factor of 2 for x≳0.1\documentclass[12pt]{minimal}
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\begin{document}$$x\gtrsim 0.1$$\end{document}. Overall, the impact of the new datasets depends on x, and is mostly significant for x=0.2\documentclass[12pt]{minimal}
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\begin{document}$$x\gtrsim 0.3$$\end{document} no experimental constraints are available, hence the PDF uncertainty blows up.
The right panel of Fig. 8 compares the ratio Rs\documentclass[12pt]{minimal}
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\begin{document}$$Q=10$$\end{document} GeV as a function of x, as obtained from the str fit and from the CT18/CT18A [20], MMHT14 [51], and ABMP16 [52] fits. The inset displays the relative PDF uncertainty δRs/Rs\documentclass[12pt]{minimal}
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\begin{document}$$\delta R_s/R_s$$\end{document}. Our str determination agrees with the CT18A and ABMP16 results within uncertainties in the data region. However, it overshoots the CT18 and MMHT14 results. Note that the very small PDF uncertainties of the ABMP16 result should be realistically rescaled by a tolerance factor T=χ2>1\documentclass[12pt]{minimal}
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\begin{document}$$T=\chi ^2>1$$\end{document} [1], which is, however, not accounted for in their analysis. With this caveat, our results for s+\documentclass[12pt]{minimal}
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\begin{document}$$R_s$$\end{document} are also the most precise, in particular around x∼0.1\documentclass[12pt]{minimal}
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\begin{document}$$x\sim 0.1$$\end{document}, thanks to the wider dataset (and specifically of NOMAD) employed to constrain the strange quark and antiquark PDFs.
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\begin{document}$$Q=100$$\end{document} GeV (right). The PDF sets are from the str_base, str_prior, str fits and from other recent PDF analyses; see the text for details
The ratio Ks\documentclass[12pt]{minimal}
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\begin{document}$$Q=100$$\end{document} GeV (right). The PDF sets are the same as in Fig. 10
We explicitly assessed how the results for Rs\documentclass[12pt]{minimal}
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\begin{document}$$R_s$$\end{document} obtained with our optimal fit str depend on the specific choice of the NOMAD dataset and on the fact that the charm PDF is parametrized on the same footing as light-quark PDFs. In Fig. 9 we display the ratio Rs\documentclass[12pt]{minimal}
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\begin{document}$$Q=10$$\end{document} GeV: in the left panel we compare results obtained with the str and str_s_hat fits; in the right panel, we compare results obtained with the str and str_pch fits. In the first case, both the central value and the PDF uncertainties of Rs\documentclass[12pt]{minimal}
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\begin{document}$$R_s$$\end{document} are very similar. This fact confirms the independence of our results upon the choice of the NOMAD dataset included in the fit. In the second case, while PDF uncertainties turn out to be very similar in both the perturbative and the fitted charm fits, the former prefers a central value which is systematically larger than the one obtained from the latter. The size of the shift, however, is at most as large as one-sigma in units of the PDF uncertainties, in line with previous studies [19].
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\begin{document}$$Q=100$$\end{document} GeV. Figure 10 makes it clear the consistent effect of the new datasets included in our analysis. Considering the results for Rs\documentclass[12pt]{minimal}
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\begin{document}$$Q=1.6$$\end{document} GeV, the value of , Rs=0.69±0.22\documentclass[12pt]{minimal}
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\begin{document}$$R_s=0.69 \pm 0.22$$\end{document} in the str_base fit is made more precise by the LHC datasets, which reduce its uncertainty by about a factor 2, , while also increasing its central value, Rs=0.76±0.12\documentclass[12pt]{minimal}
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\begin{document}$$R_s=0.76\pm 0.12$$\end{document}; then the neutrino-DIS NOMAD dataset shifts this number towards a lower value by a half-sigma bringing in also a further moderate reduction of the uncertainty, Rs=0.71±0.10\documentclass[12pt]{minimal}
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\begin{document}$$R_s=0.71\pm 0.10$$\end{document}. We therefore conclude that the result Rs=1.13±0.11\documentclass[12pt]{minimal}
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\begin{document}$$R_s=1.13 \pm 0.11$$\end{document}, reported in [12] from an analysis of HERA and ATLAS W, Z data within the xFitter framework [54], is not compatible with ours, possibly because it is affected by a restricted dataset and/or methodological limitations. Similar remarks apply to the results for Rs\documentclass[12pt]{minimal}
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\begin{document}$$R_s$$\end{document}, and a larger reduction of the uncertainty due to the stronger effect of NOMAD data at larger x; see Fig. 8. Our results are compatible, within uncertainties, with those of the other PDF determinations, but they are generally more precise.
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\begin{document}$$R_s$$\end{document}, in particular, PDF uncertainties are reduced by a factor of 2 in the str fit with respect to the str_base fit. The values of Ks\documentclass[12pt]{minimal}
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\begin{document}$$K_s=1$$\end{document}) scenarios. As for Rs\documentclass[12pt]{minimal}
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Summary
By means of a state-of-the-art global analysis, which combines all the relevant experimental and theoretical inputs, we have achieved a precise determination of the strangeness content of the proton. We have demonstrated the compatibility of a wide range of strangeness-sensitive datasets; quantified their relative impact on the fit; compared our results to other recent global analyses; and assessed the robustness of our results with respect to various methodological choices. Our analysis demonstrates that the strange PDF can be precisely determined and that, after all, the proton is not too strange: the momentum fraction carried by strange quark and antiquark PDFs ranges between about 65% and 80% of the momentum fraction carried by the other light sea quarks in a wide energy range (1.6 GeV ≤Q≤100\documentclass[12pt]{minimal}
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\begin{document}$$\le Q\le 100$$\end{document} GeV). The present determination of the strangeness content of the proton is found to agree, within uncertainties, with the results of other recent global PDF analyses.
Pivotal to this result is the complementary between the LHC gauge-boson production data and of the charmed-tagged neutrino-DIS data, in particular from the NOMAD experiment. Our str PDF set, which combines all this information, is available in the LHAPDF format [55] together with its perturbative charm counterpart from
http://nnpdf.mi.infn.it/nnpdf3-1strangeness/
This analysis represents an important input for phenomenology, for instance to carry out improved determinations of fundamental parameters of the SM or to be used as baseline in the determination of nuclear PDFs, where strange distributions are not well known [46, 56, 57]. Our determination of the strange and antistrange quark PDFs could be further stress-tested with more exclusive processes, e.g., measurements of kaon production in semi-inclusive DIS (SIDIS). Studies of the strange PDFs based on SIDIS [58–60] notoriously prefer a suppressed strangeness, but are also subject to the potential bias coming from their sensitivity to the fragmentation of the strange quarks into kaons.
Acknowledgements
We are grateful to Jun Gao for providing us with the NNLO K-factors for the NuTeV and the NOMAD measurements, and for help into the benchmark of the corresponding NLO calculations. We thank Valerio Bertone for assistance with the usage of APFEL, Lucian Harland-Lang for discussions on the massive corrections to dimuon production, and Rabah Abdul-Khalek, Amanda Cooper-Sarkar, Stefano Forte, Katerina Lipka, Gavin Pownall and Cameron Voisey for comments on the manuscript. E.R.N. is supported by the European Commission through the Marie Skłodowska-Curie Action ParDHonS FFs.TMDs (Grant number 752748). J.R. is partially supported by the Netherlands Organization for Scientific Research (NWO). M.U. and S.I. are partially supported by the STFC grant ST/L000385/1 and by the Royal Society grant RGF/EA/180148. The work of M.U. is also funded by the Royal Society grant DH150088 and supported by the European Research Council under the European Union’s Horizon 2020 research and innovation Programme (Grant agreement n.950246).
Data Availability Statement
This manuscript has associated data in a data repository [Authors’ comment: This manuscript has its associated data available from the following url: http://nnpdf.mi.infn.it/nnpdf3-1strangeness/.]
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