Nuclear uncertainties in the determination of proton PDFs
Abstract
We show how theoretical uncertainties due to nuclear effects may be incorporated into global fits of proton parton distribution functions (PDFs) that include deepinelastic scattering and Drell–Yan data on nuclear targets. We specifically consider the CHORUS, NuTeV and E605 data included in the NNPDF3.1 fit, which used Pb, Fe and Cu targets, respectively. We show that the additional uncertainty in the proton PDFs due to nuclear effects is small, as expected, and in particular that the effect on the \(\bar{d}/\bar{u}\) ratio, the total strangeness \(s+\bar{s}\), and the strange valence distribution \(s\bar{s}\) is negligible.
1 Introduction
Modern sets of parton distribution functions (PDFs) [1] are currently determined for the proton from a global quantum chromodynamics (QCD) analysis of hardscattering measurements [2]. A variety of hadronic observables are used, including some from processes that do not (exclusively) involve protons in the initial state, such as deepinelastic scattering (DIS) and Drell–Yan (DY) in experiments with deuterium or heavy nuclear fixed targets. These experiments complement the protononly ones, providing important sensitivity to light PDF flavour separation [3], and are therefore included in most contemporary global QCD analyses.
The inclusion of nuclear data necessitates accounting for differences between the PDFs for free nucleons and those for partons contained within nuclei. In the past a variety of different approaches have been adopted: nuclear corrections can be ignored on the basis that they are small [3], included according to various nuclear models [4, 5, 6], or determined in a fit to the data [7]. Whatever approach is adopted, nuclear effects will necessarily increase the uncertainty in the proton PDFs, since nuclear corrections are not known very precisely. So far the size of this uncertainty has also been regarded as small [8, 9]. However, in recent years the inclusion of increasingly precise LHC measurements in global PDF fits has reduced PDF uncertainties to the level of a few percent [3]. Furthermore, nuclear effects have been claimed to alter the shape of the PDFs, especially at large values of the momentum fraction x [10], albeit without an estimate of the corresponding theoretical uncertainty. Given this, it is becoming increasingly desirable to provide PDF sets that include such an uncertainty.
In this paper we will show how this may be achieved by performing global fits which include nuclear uncertainties, in the framework of the NNPDF3.1 global analysis [3]. We focus on the DIS and DY datasets with heavy nuclear targets (Pb, Fe and Cu). We estimate the theoretical uncertainty due to neglecting the corresponding nuclear corrections, we include it in a fit along with the experimental uncertainty, and we assess its impact on the resulting PDFs. A similar exercise for DIS and DY datasets with deuterium targets will be carried out in a separate analysis.
Our study is accomplished within the formalism of Ref. [11], that was developed to include a broad class of theoretical uncertainties in a PDF fit. The method consists of adding to the experimental covariance matrix a theoretical covariance matrix, estimated in the space of the data according to the theoretical uncertainties associated with the theoretical predictions. In practice this means that the theoretical uncertainties are treated in much the same way as experimental systematics. Here we will estimate the theoretical uncertainties associated with nuclear effects. This will be done empirically, by directly comparing theoretical predictions computed using nuclear PDFs (nPDFs) to those computed using proton PDFs. This comparison will allow us to construct the covariance matrix associated with the nuclear effects, and thus incorporate these effects in a global proton PDF fit. The fitting methodology will otherwise be the same as in the NNPDF3.1 analysis, allowing for a direct comparison of the results.
The paper is organised as follows. In Sect. 2, we summarise the changes in methodology required to include theoretical uncertainties in a global fit. We then describe the nuclear dataset included in this analysis, emphasising to which PDF flavours it is most sensitive. We next provide two alternative prescriptions to estimate the theoretical covariance matrix, and discuss their implementation. In Sect. 3, we present the impact of nuclear data and theoretical corrections in a global fit of PDFs. We compare variants of the NNPDF3.1 determination obtained by removing the nuclear datasets completely, or by retaining them but accounting for nuclear uncertainties, and nuclear corrections. We study the fit quality and the stability of the PDFs. Given that the nuclear dataset is mostly sensitive to sea quark PDFs, in Sect. 4 we assess how the \(\bar{d}\)\(\bar{u}\) asymmetry, and the strangeness content of the proton, including the asymmetry between s and \(\bar{s}\) PDFs, are affected by nuclear uncertainties. We provide our conclusions and an outlook in Sect. 5.
2 Theoretical uncertainties due to nuclear corrections
In this section we describe how the NNPDF methodology can be adapted to incorporate theoretical uncertainties through a theoretical covariance matrix, as proposed in Ref. [11]. We then focus on the DIS and DY datasets in the NNPDF3.1 global dataset (described in detail in Sect. 2 of Ref. [3]) that involve nuclear targets other than deuterium. We first summarise the type of measured observables, their kinematic coverage, and their sensitivity to the underlying PDFs. We then show how the size of the nuclear effects may be estimated empirically by comparing the different theoretical predictions made with proton and nuclear PDFs. We finally offer two alternative prescriptions to estimate the theoretical covariance matrix for the nuclear uncertainties: the first of which simply provides a conservative estimate of the overall uncertainty; the second of which also applies a correction for nuclear effects, aiming to reduce the overall uncertainty.
2.1 Theoretical uncertainties in PDF fits
The datasets involving nuclear targets other than deuterium in NNPDF3.1. The kinematic range covered in each variable is given after cuts are applied (see the text for details)
Experiment  Obs.  References  \(N_\mathrm{dat}\)  \(\mathrm{Kin}_1\)  \(\mathrm{Kin}_2\) [GeV] 

CHORUS  \(\sigma ^{\nu }\)  [15]  607 (416)  \(0.045\le x\le 0.65\)  \(1.9\le Q\le 9.8\) 
\(\sigma ^{\bar{\nu }}\)  [15]  607 (416)  \(0.045\le x\le 0.65\)  \(1.9\le Q\le 9.8\)  
NuTeV  \(\sigma ^{\nu ,c}\)  45 (39)  \(0.02\le x\le 0.33\)  \(2.0\le Q \le 10.8\)  
\(\sigma ^{\bar{\nu },c}\)  45 (37)  \(0.02\le x\le 0.21\)  \(1.9\le Q \le \ \, 8.3\)  
E605  \(\sigma _\mathrm{DY}^p\)  [18]  119 (85)  \(0.2\le y_{\ell \ell }\le 2.9\)  \(7.1\le M_{\ell \ell }\le 10.9\) 
In this way we obtain a PDF replica \(f^{(k)}\) for each data replica \(D^{(k)}\). Since the distribution of the PDF replicas is a representation of the distribution of the data replicas, the ensemble of PDF replicas \(\{f^{(k)}\}\) gives us a representation of the PDFs and their correlated uncertainties.
We can incorporate theory uncertainties into the NNPDF methodology by supplementing the experimental covariance matrix \(C_{ij}\) with a theoretical covariance matrix \(S_{ij}\), estimated using the theoretical uncertainties associated with the theoretical predictions \(T_i[f]\). The experimental and theoretical uncertainties are by their nature independent. In Ref. [11] it was shown that if we assume that both experimental and theoretical uncertainties are independent and Gaussian, the two covariance matrices can simply be added; the combined covariance matrix \(C_{ij}+S_{ij}\) then gives the total uncertainty in the extraction of PDFs from the experimental data.
Note that since the predictions \(T_i[f]\) depend on the PDFs, this means the nuisance parameters \(\Delta _i^{(n)}\) and the theoretical covariance matrix \(S_{ij}\) will also depend implicitly on the PDF, albeit weakly. This can be dealt with in precisely the same way as in the \(t_0\) method [14]: \(S_{ij}\) is computed with an initial (central) PDF, which is then iterated to consistency. In practice the iterations can be performed simultaneously.
In this paper we will show how this procedure works by estimating the theoretical uncertainties due specifically to nuclear effects related to the use of data from scattering off heavy nuclear targets.
2.2 The nuclear dataset
The NNPDF3.1 dataset involving heavy nuclei consists of inclusive chargedcurrent DIS cross sections from CHORUS [15], DIS dimuon cross sections from NuTeV [16, 17], and DY dimuon cross sections from E605 [18]. In the case of CHORUS and NuTeV, neutrino and antineutrino beams are scattered off a lead (\(^{208}_{\ 82}\)Pb) and an iron (\(^{56}_{26}\)Fe) target, respectively; while in the case of E605 a proton beam is scattered off a copper (\(^{64}_{32}\)Cu) target. Henceforth we refer to the combined measurements from CHORUS, NuTeV and E605 as the nuclear dataset. In NNPDF3.1 there are additional DIS and DY datasets involving scattering from deuterium: nuclear corrections to these datasets will be considered in a future analysis. In NNPDF we do not use the CDHSW neutrinoDIS data [19], taken with an iron target.
An overview of the nuclear dataset is presented in Table 1, where we indicate, for each dataset: the observable, the corresponding reference, the number of data points before and after kinematic cuts, and the kinematic range covered in the relevant variables after cuts. Kinematic cuts match the nexttonexttoleading order (NNLO) NNPDF3.1 baseline fit: for DIS we require \(Q^2\ge 3.5\) GeV\(^2\) and \(W^2\ge 12.5\) GeV\(^2\), where \(Q^2\) and \(W^2\) are the energy transfer and the invariant mass of the final state in the DIS process, respectively; for DY, we require \(\tau \le 0.080\) and \(y_{\ell \ell }/y_\mathrm{max}\le 0.663\), where \(\tau =M_{\ell \ell }^2/s\) and \(y_\mathrm{max}=\frac{1}{2}\ln \tau \), with \(y_{\ell \ell }\) and \(M_{\ell \ell }\) the rapidity and the invariant mass of the dimuon pair, respectively, and \(\sqrt{s}\) the centreofmass energy of the DY process.
The sensitivity of the measured observables to different PDF flavours can be quantified by the correlation coefficient \(\rho \) (defined in Eq. (1) in Ref. [20]) between the PDFs in a given set and the theoretical predictions corresponding to the measured data points. Large values of \(\rho \) indicate that the sensitivity of the PDFs to the data is most significant. The correlation coefficient \(\rho \) is displayed in Fig. 2, from top to bottom, for the \(u_V\) and \(d_V\) PDFs from CHORUS, for the s and \(\bar{s}\) PDFs from NuTeV, and for the \(\bar{u}\) and \(\bar{d}\) PDFs from E605. Each point corresponds to a different datum in the experiments enumerated in Table 1: PDFs are taken from the NNDPF3.1 NNLO parton set, and are evaluated at a scale equal to either the momentum transfer \(Q^2\) (for DIS) or the centerofmass energy s (for DY) of that point. For DY, the value of x is computed from hadronic variables using LO kinematics. As anticipated, the correlation between the PDF flavours and the observables displayed in Fig. 2 is sizeable, in particular: between \(u_V\) (\(d_V\)) PDFs and the neutrino (antineutrino) charged current DIS cross sections from CHORUS in the range \(0.1\le x\le 0.7\) (\(0.2\le x \le 0.5\)); between s (\(\bar{s}\)) PDFs and the antineutrino dimuon DIS cross sections from NuTeV along all the measured range, \(0.02\le x\le 0.32\) (\(0.02\le x\le 0.21\)); and between \(\bar{u}\) (\(\bar{d}\)) PDFs and the dimuon DY cross sections from E605 in the range \(0.3\le x\le 0.7\). Correlations between the measured observables and other PDFs, not displayed in Fig. 2, are relatively small. We therefore expect that including theoretical uncertainties due to nuclear corrections will mainly affect the valencesea PDF flavour separation in the kinematic region outlined above.
2.3 Determining correlated nuclear uncertainties
In Sect. 2.1, we explained how, if we want to include theoretical uncertainties in a PDF fit, we first need to estimate the theoretical covariance matrix \(S_{ij}\) in the space of the data, using Eq. (5). In this section, we illustrate how we might achieve this for the nuclear uncertainties affecting the three datasets described in Sect. 2.2. First we provide two alternative definitions for the theoretical covariance matrix associated with nuclear uncertainties, then we describe how we can implement them in practice, and finally we discuss the results.
2.3.1 Definition
Various models of nuclear effects on PDFs exist in the literature (for a review, see, e.g., Ref. [21]). They are however based on a range of different assumptions, which often limit their validity. In our opinion, a better ansatz for nuclear effects, over all the kinematic range covered by the measurements in Table 1, is provided by global fits of nPDFs, since they are primarily driven by the data. The \(N_\mathrm{nuis}\) models in Eq. (5) can then be identified with different members of a nPDF set. The free proton PDF can instead be taken from a global set of proton PDFs: it should in any case be iterated to consistency at the end of the fitting procedure, as explained at the end of Sect. 2.1. The practical way in which nPDF members are constructed, the proton PDF is chosen, and the corresponding observables are computed is discussed in Sect. 2.3.2 below.
The two definitions are in principle different. In Eq. (6), the contribution of the nuclear data to the global fit is deweighted by an extra uncertainty, which encompasses both the difference between the proton and nuclear PDFs, and the uncertainty in the nPDFs. In Eq. (8) the theory is corrected by a shift. Here the uncertainty, and thus the deweighting, is correspondingly smaller, arising only from the uncertainty in the nPDFs. In principle, if the uncertainty in the nPDFs is correctly estimated, and smaller than the shift, the second definition should give more precise results. However if the shift is small, or unreliably estimated, the first definition will be better, and should result in a lower \(\chi ^2\) for the nuclear data, albeit with slightly larger PDF uncertainties.
2.3.2 Implementation
The goal of our exercise is to estimate the overall level of theoretical uncertainty associated to nuclear effects. Inconsistency (following from somewhat inconsistent parametrisations) should be part of that, therefore, instead of relying on a single nPDF determination in Eqs. (6–8), we find it useful to utilise a combination of different nPDF sets. Such a combination can be realised in a statistically sound way by following the methodology developed in Ref. [1], which consists in taking the unweighted average of the nPDF sets. The simplest way of realing it is to generate equal numbers of Monte Carlo replicas from each input nPDF set, and then merge them together in a single Monte Carlo ensemble. The appropriate normalisation in Eq. (5) is therefore \(\mathcal{N} =\frac{1}{N_\mathrm{nuis}}\), since each replica is equally probable; each nPDF member in Eqs. (6–8) is a replica in the Monte Carlo ensemble; and \(f_N=\langle f_N^{(n)}\rangle \) is the zeroth replica in the same Monte Carlo ensemble. The combination method of Ref. [1] has proven to be adequate when results are compatible or differences are understood, as is the case with nPDFs.
The Monte Carlo ensemble utilised to compute Eqs. (6–8) is determined as follows. We consider recent nPDF sets available in the literature, namely DSSZ12 [22], nCTEQ15 [23] and EPPS16 [24]. These are determined at nexttoleading order (NLO) from a global analysis of measurements in DIS, DY and protonnucleus (pN) collisions. A compilation of the data included in these sets is given in Table 2. A detailed description may be found in Refs. [22, 23, 24], and a critical comparison is documented, e.g., in Refs. [25, 26]. As one can see, all three determinations include a significant amount of experimental information, so should collectively provide a reasonable representation of nuclear modifications.
The nuclear datasets used in NNPDF3.1, and hence in the fits performed in this analysis, also enter some of the nPDFs selected above. This is the case of NuTeV measurements (also included in DSSZ12) and of CHORUS measurements (also included in DSSZ12 and EPPS16), see Tables 1 and 2. This does not lead to a double counting of these data because we only use the nPDFs as a model to establish an additional correlated source of uncertainty in the determination of the proton PDFs. This then leads to an increase in overall uncertainties, since the nuclear datasets are deweighted, while double counting would give a decrease in uncertainties.
The DIS, DY and pp data entering the nPDF sets used in this analysis
Observable  Experiment  References  \(N_\mathrm{dat}\) (DSSZ12)  \(N_\mathrm{dat}\) (nCTEQ15)  \(N_\mathrm{dat}\) (EPPS16) 

\(F_2^\mathrm{D}/F_2^\mathrm{D}\)  NMC  [27]    201   
\(F_2^\mathrm{He}/F_2^\mathrm{D}\)  NMC  [28]  17  12  16 
E139  [29]  18  3  21  
HERMES  [30]    17    
\(F_2^\mathrm{Li}/F_2^\mathrm{D}\)  NMC  [31]  17  11  15 
\(F_2^\mathrm{Li}/F_2^\mathrm{D}\) \(Q^2\) dep  NMC  [31]      153 
\(F_2^\mathrm{Be}/F_2^\mathrm{D}\)  E139  [29]  17  3  20 
\(F_2^\mathrm{C}/F_2^\mathrm{D}\)  E665  [32]    3   
E139  [29]  7  2  7  
EMC  [33]  9  9    
NMC  17  24  31  
\(F_2^\mathrm{C}/F_2^\mathrm{D}\) \(Q^2\) dep  NMC  [28]  191    165 
\(F_2^\mathrm{N}/F_2^\mathrm{D}\)  HERMES  [30]    19   
BCDMS  [34]    9    
\(F_2^\mathrm{Al}/F_2^\mathrm{D}\)  E139  [29]  17  3  20 
\(F_2^\mathrm{Ca}/F_2^\mathrm{D}\)  NMC  [28]  16  12  15 
E665  [32]    3    
E139  [29]  7  2  7  
\(F_2^\mathrm{Fe}/F_2^\mathrm{D}\)  E049  [35]    2   
E139  [29]  23  6  26  
BCDMS    16    
\(F_2^\mathrm{Cu}/F_2^\mathrm{D}\)  EMC  19  27  19  
\(F_2^\mathrm{Kr}/F_2^\mathrm{D}\)  HERMES  [30]    12   
\(F_2^\mathrm{Ag}/F_2^\mathrm{D}\)  E139  [29]  7  2  7 
\(F_2^\mathrm{Sn}/F_2^\mathrm{D}\)  EMC  [33]  8  8   
\(F_2^\mathrm{Xe}/F_2^\mathrm{D}\)  E665  [32]    2   
\(F_2^\mathrm{Au}/F_2^\mathrm{D}\)  E139  [29]  18  3  21 
\(F_2^\mathrm{Pb}/F_2^\mathrm{D}\)  E665  [32]    3   
\(F_2^\mathrm{C}/F_2^\mathrm{Li}\)  NMC  [28]  24  7  20 
\(F_2^\mathrm{Ca}/F_2^\mathrm{Li}\)  NMC  [28]  24  7  20 
\(F_2^\mathrm{Be}/F_2^\mathrm{C}\)  NMC  [38]  15  14  15 
\(F_2^\mathrm{Al}/F_2^\mathrm{C}\)  NMC  [38]  15  14  15 
\(F_2^\mathrm{Ca}/F_2^\mathrm{C}\)  NMC  [28]  39  21  15 
\(F_2^\mathrm{Fe}/F_2^\mathrm{C}\)  NMC  [38]  15  14  15 
\(F_2^\mathrm{Sn}/F_2^\mathrm{C}\)  NMC  [39]  15    15 
\(F_2^\mathrm{Sn}/F_2^\mathrm{C}\) \(Q^2\) dep  NMC  [39]  145  111  144 
\(F_2^\mathrm{Pb}/F_2^\mathrm{C}\)  NMC  [38]  15  14  15 
\(F_2^\mathrm{\nu Fe}\)  NuTeV  [40]  75     
CDHSW  [19]  120      
\(F_3^\mathrm{\nu Fe}\)  NuTeV  [40]  75     
CDHSW  [19]  133      
\(F_2^\mathrm{\nu Pb}\)  CHORUS  [15]  63    412 
\(F_3^\mathrm{\nu Pb}\)  CHORUS  [15]  63    412 
\(\sigma _\mathrm{DY}^\mathrm{pC}/\sigma _\mathrm{DY}^\mathrm{pD}\)  E772  [41]  9  9  9 
\(\sigma _\mathrm{DY}^\mathrm{pCa}/\sigma _\mathrm{DY}^\mathrm{pD}\)  E772  [41]  9  9  9 
\(\sigma _\mathrm{DY}^\mathrm{pFe}/\sigma _\mathrm{DY}^\mathrm{pD}\)  E772  [41]  9  9  9 
\(\sigma _\mathrm{DY}^\mathrm{pW}/\sigma _\mathrm{DY}^\mathrm{pD}\)  E772  [41]  9  9  9 
\(\sigma _\mathrm{DY}^\mathrm{pFe}/\sigma _\mathrm{DY}^\mathrm{pBe}\)  E886  [42]  28  28  28 
\(\sigma _\mathrm{DY}^\mathrm{pW}/\sigma _\mathrm{DY}^\mathrm{pBe}\)  E886  [42]  28  28  28 
\(d\sigma _{\pi ^0}^\mathrm{dAu}/d\sigma _{\pi ^0}^{pp}\)  PHENIX  [43]  21  20  20 
STAR  [44]    12    
\(d\sigma _{\pi ^}^\mathrm{pW}/d\sigma _{\pi ^}^{pD}\)  NA10  [45]      10 
\(d\sigma _{\pi ^+}^\mathrm{pW}/d\sigma _{\pi ^}^{pW}\)  E615  [46]      11 
\(d\sigma _{\pi ^}^\mathrm{pPt}/d\sigma _{\pi ^}^{pH}\)  NA3  [47]      7 
\(W^\mathrm{\pm }\) pPb  CMS  [48]      10 
Z pPb  CMS  [49]      6 
ATLAS  [50]      7  
dijet pPb  CMS  [51]      7 
1579  1627  1811 
The three nPDF sets selected above are each delivered as Hessian sets, corresponding to 90% confidence levels (CLs) for nCTEQ15 and EPPS16. These CLs were determined by requiring a tolerance \(T=\sqrt{\Delta \chi ^2}\), with \(\Delta \chi ^2=35\), and \(\Delta \chi ^2=52\), for the two nPDF sets, respectively. Excursions of the individual eigenvector directions resulting in a \(\Delta \chi ^2\) up to 30 units, which correspond to an increase in \(\chi ^2\) of about \(2\%\), are tolerated in DSSZ12. We assume that this represents the \(68\%\) CL of the fit [52]. To generate corresponding Monte Carlo sets, we utilise the Thorne–Watt algorithm [53] implemented in the public code of Ref. [54], and rescale all uncertainties to 68% CLs. We then generate 300 replicas for each nPDF set. The size of the Monte Carlo ensemble is chosen to reproduce the central value and the uncertainty of the original Hessian sets with an accuracy of few percent. Such an accuracy is smaller than the spread of the nPDF sets, and therefore adequate for our purpose. We assume that all the three nPDF Monte Carlo sets are equally likely representations of the same underlying probability distribution, and we combine them by choosing equal numbers of replicas from each set. In the case of CHORUS and NuTeV, both lead and iron nPDFs are available from DSSZ12, nCTEQ15, and EPPS16, therefore the total number of replicas in the combined set is \(N_\mathrm{nuis}=900\); in the case of E605, copper nPDFs are only available from nCTEQ15 and EPPS16, therefore for E605 \(N_\mathrm{nuis}=600\). In principle, the large number of Monte Carlo replicas (and nuisance parameters) can be reduced by means of a suitable compression algorithm [55] without a significant statistical loss. A Monte Carlo ensemble of approximately the typical size of the starting Hessian sets could thus be obtained. However, we do not find it necessary to do this in the current analysis.
2.3.3 Discussion
The nuclear corrections to the observables themselves are shown in Figs. 4, 5, and 6, where, for each data point i (after kinematic cuts) in the CHORUS, NuTeV and E605 datasets, we display the observables computed with nuclear PDFs, normalised to the expectation value with the proton PDF, \(T_i^N[f_N]/\langle T_i^N[f_p]\rangle \). Results are shown for the DSSZ12, EPPS16 and nCTEQ15 sets separately. The central value of the same ratio obtained from the Monte Carlo combination of all the three nPDF sets is also shown. Looking at Figs. 4, 5, and 6, we observe that the shape and size of the ratios between observables closely follow that of the ratios between PDFs in the shaded region of Fig. 3. For CHORUS, all the nPDFs modify the nuclear observable in a similar way, with a slight enhancement at smaller values of x, and a suppression at larger values of x. Overall, the uncertainty in the nuclear observable is comparable for the three sets. All sets are mutually consistent within uncertainties. For NuTeV, the nuclear corrections are markedly inconsistent for DSSZ12 and nCTEQ15, with the former being basically flat around one, while the latter deviates very significantly from one in some bins. Both nPDF sets are included within the much larger uncertainties of the EPPS16 set, which has now the largest stated uncertainty, especially in the case of antineutrino beams. This is likely a consequence of the fact that the strange quark nPDF was fitted independently from the other quark nPDFs in EPPS16, while it was related to lighter quark nPDFs in DSS12 and nCTEQ15 [26]. For E605, the nuclear correction gives a mild reduction in most of the measured kinematic range. The nCTEQ15 nPDF set appears to be more precise than the EPPS16 set, and they are reasonably consistent with each other. No DSSZ12 set is available for Cu. On average the size of the nuclear correction shift, as quantified by the ratio \(T_i^N[f_N]/T_i^N[f_p]\), is of order \(10\%\) for CHORUS, \(20\%\) for NuTeV, and \(5\%\) for E605.
3 Impact of theoretical corrections in a global PDF fit
In this section, we discuss the impact of the theoretical uncertainties due to nuclear corrections, as computed in the previous section, in a global fit of proton PDFs. We first summarise the experimental and theoretical settings of the fits, then we present the results.
3.1 Fit settings
The PDF sets discussed in this section are based on a variant of the NNLO NNPDF3.1 global analysis [3]. In particular the experimental input, and related kinematic cuts, are exactly the same as in the NNPDF3.1 NNLO fit. On top of the nuclear measurements presented in Sect. 2.2, the dataset is made up of: fixedtarget [15, 16, 17, 27, 60, 61, 62, 63] and collider [64] DIS inclusive structure functions; charm and botton cross sections from HERA [65]; fixedtarget DY cross sections [66, 67, 68]; gauge boson and inclusive jet production cross sections from the Tevatron [69, 70, 71, 72, 73]; and electroweak boson production, inclusive jet, Z \(p_T\), total and differential toppair cross sections from ATLAS [74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88], CMS [89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100] and LHCb [101, 102, 103, 104, 105]. The theoretical input is also the same as in NNPDF3.1: the strong running coupling at the Zboson mass is fixed to \(\alpha _s(m_Z)=0.118\), consistent with the PDG average [106]; heavyquark mass effects are included using the FONLL C generalmass scheme [107, 108], with pole masses \(m_c=1.51\) GeV for charm and \(m_b=4.92\) GeV for bottom, consistent with the Higgs cross section working group recommendation [109]; the charm PDF is fitted in the same way as the other light quark PDFs [110]; and the initial parametrisation scale is chosen just above the value of the charm mass, \(Q_0=1.65\) GeV. All fits are performed at NNLO in pure QCD, and result in ensembles of \(N_\mathrm{rep}=100\) replicas.
In comparison to NNPDF3.1, we have made small improvements in the computation of the CHORUS and NuTeV observables. In the case of CHORUS, cross sections were computed in NNPDF3.1 following the original implementation of Ref. [12], where the target was assumed to be isoscalar, and the data were supplemented with a systematic uncertainty to account for their actual nonisoscalarity. We now remove this uncertainty, and we compute the cross sections taking into account the nonisoscalarity of the target, as explained in Sect. 2.3.2. This increases the \(\chi ^2\) per data point a little (from 1.11 to 1.25). In the case of NuTeV, we update the value of the branching ratio of charmed hadrons into muons. This value, which is used to reconstruct charm production cross sections from the neutrino dimuon production cross sections measured by NuTeV, was set equal to 0.099 in NNPDF3.1, following the original analysis of Ref. [8]. Previously, the uncertainty on the branching ratio was not taken into account. We now utilise the current PDG result, \(0.086\pm 0.005\) [106], and include its uncertainty as an additional fully correlated systematic uncertainty. This reduces the \(\chi ^2\) per data point a little (from 0.82 to 0.66).

a Baseline fit, based on the theoretical and experimental inputs described above, and without any inclusion of theoretical nuclear uncertainties;

a “No Nuclear” fit, NoNuc, equal to the Baseline, but without the datasets that utilise nuclear targets, i.e. without CHORUS, NuTeV and E605;

a “Nuclear Uncertainties” fit, NucUnc, equal to the Baseline, but with the inclusion of theoretical nuclear uncertainties applied to CHORUS, NuTeV and E605, according to Eqs. (3)–(4), the theory covariance matrix being computed using (5) with the prescription Eq. (6) for the nuisance paremeters;

a “Nuclear Corrections” fit, NucCor, equal to the Baseline, but with the inclusion of theoretical nuclear uncertainties applied to CHORUS, NuTeV and E605, according to Eqs. (3)–(4), with the nuclear correction \(\delta T_i^N\), Eq. (8), added to the theoretical prediction \(T_i[f]\) used in Eq. (4), and the theory covariance matrix computed using Eqs. (5) and (7) for the nuisance parmeters.
3.2 Fit quality and parton distributions
The values of the \(\chi ^2\) per data point for the various fits described in the text. Data sets that utilise a nuclear target other than a deuteron are highlighted in boldface
Experiment  \(N_\mathrm{dat}\)  Baseline  NoNuc  NucUnc  NucCor 

NMC  325  1.31  1.33  1.31  1.29 
SLAC  67  0.79  0.93  0.72  0.73 
BCDMS  581  1.23  1.17  1.20  1.21 
CHORUS \(\nu \)  416  1.29  –  0.97  1.04 
CHORUS \(\bar{\nu }\)  416  1.20  –  0.78  0.83 
NuTeV dimuon \(\nu \)  39  0.41  –  0.31  0.40 
NuTeV dimuon \(\bar{\nu }\)  37  0.90  –  0.62  0.83 
HERA I+II incl.  1145  1.15  1.15  1.15  1.15 
HERA \(\sigma _c^\mathrm{NC}\)  37  1.40  1.52  1.46  1.44 
HERA \(F_2^b\)  29  1.11  1.10  1.10  1.11 
E866 \(\sigma ^d_\mathrm{DY}/\sigma ^p_\mathrm{DY}\)  15  0.47  0.33  0.44  0.44 
E886 \(\sigma ^p\)  89  1.35  1.22  1.69  1.66 
E605 \(\sigma ^p\)  85  1.18  –  0.85  0.89 
CDF Z rap  29  1.41  1.29  1.39  1.41 
CDF Run II \(k_t\) jets  76  0.88  0.82  0.89  0.92 
D0 Z rap  28  0.60  0.57  0.59  0.59 
D0 W asy  17  2.11  2.06  2.10  2.09 
ATLAS total  360  1.08  1.04  1.04  1.05 
ATLAS W, Z 7 TeV 2010  30  0.93  0.93  0.92  0.93 
ATLAS HM DY 7 TeV  5  1.68  1.60  1.57  1.54 
ATLAS lowmass DY 7 TeV  6  0.91  0.87  0.88  0.89 
ATLAS W, Z 7 TeV 2011  34  1.97  1.78  1.87  1.94 
ATLAS jets  180  1.00  0.97  1.00  1.00 
ATLAS Z \(p_T\) 8 TeV  92  0.93  0.95  0.95  0.92 
ATLAS \(t\bar{t}\)  13  1.32  1.22  1.24  1.21 
CMS total  409  1.07  1.07  1.07  1.07 
CMS W asy  22  1.23  1.41  1.30  1.31 
CMS Drell–Yan 2D 2011  110  1.30  1.33  1.31  1.29 
CMS W rap 8 TeV  22  0.94  0.88  0.95  0.96 
CMS jets  214  0.93  0.89  0.90  0.91 
CMS Z \(p_T\) 8 TeV \((p_T^{ll},y_{ll})\)  28  1.31  1.38  1.35  1.35 
CMS \(t\bar{t}\)  13  0.77  0.73  0.76  0.75 
LHCb total  85  1.46  1.27  1.32  1.37 
LHCb Z  26  1.25  1.25  1.26  1.25 
LHCb \(W,Z\rightarrow \mu \)  59  1.55  1.28  1.35  1.42 
Total  4285  1.177  1.144  1.073  1.086 
Inspection of Table 3 reveals that the global fit quality improves either if nuclear data are removed, or if they are retained with the supplemental theoretical uncertainty. The lowest global \(\chi ^2\) is obtained when the theoretical covariance matrix is included, with the deweighted implementation NucUnc leading to a slightly lower value than the corrected implementation NucCor. In all cases, the improvent is mostly driven by the fact that the \(\chi ^2\) for all the Tevatron and LHC hadron collider experiments decreases. This suggests that there might be some tension between nuclear and hadron collider data in the global fit.
Interestingly, however, the relatively poor \(\chi ^2\) of the ATLAS W, Z 7 TeV 2011 dataset, which is sensitive to the strange PDF, only improves a little if the NuTeV dataset, which is also sensitive to the strange PDF, is either removed from the fit or supplemented with the theoretical uncertainty. This suggests that the poor \(\chi ^2\) of the ATLAS W, Z 7 TeV 2011 dataset does not arise from tension with the NuTeV dataset. The two datasets are indeed sensitive to different kinematic regions, and there is little interplay between the two, as we will further demonstrate in Sect. 4.2.
The fit quality of the DIS and fixedtarget DY data is stable across the fits. Fluctuations in the values of the corresponding \(\chi ^2\) are small, except for a slight worsening in the \(\chi ^2\) of the HERA charm cross sections and of the fixedtarget proton DY cross section. Concerning nuclear datasets, their \(\chi ^2\) always decreases in the NucUnc and NucCor fits in comparison to the Baseline fit, with the size of the decrease being slightly larger in the NucUnc fit than in the NucCor fit. This suggests that when the shift is used as a nuclear correction, such a correction is reasonably reliable, in the sense that its uncertainty is not substantially underestimated. However, the NucUnc implementation is more conservative than the NucCor one, and leads to a better fit.
We now make the most important differences in the PDF central values and uncertainties among the four fits explicit. In Fig. 10 we compare the light quark and antiquark PDFs between the Baseline and the NoNuc fits. In Fig. 11 we compare the sea quark PDFs between the Baseline and either the NucUnc or the NucCor fits. In both figures, results are normalised to the Baseline fit. In Fig. 12 we compare the absolute PDF uncertainty for the light quark and antiquark flavours from the four fits. All results are displayed at \(Q=10\) GeV.
Inspecting Figs. 10, 11 and 12 we may draw a number of conclusions. Firstly, the data taken on nuclear targets add a significant amount of information to the global fit. All light quark and antiquark PDFs are affected. The effect on the u and d PDFs, which are expected to be already very constrained by proton and deuteron data, consists of a slight distortion of the corresponding central values, which however remain always included in the onesigma uncertainty of the comparing fit. More importantly, uncertainties are reduced by up to a factor of one third in the region \(x\gtrsim 0.1\). The effect on the sea quark PDFs is more pronounced. Concerning central values, the nuclear data suppresses \(\bar{u}\) and \(\bar{d}\) PDFs below \(x\sim 0.1\), and enhances them above \(x\sim 0.1\). It also suppresses s and \(\bar{s}\) above \(x\sim 0.1\). Uncertainties can be reduced down to two thirds (for \(\bar{u}\) and \(\bar{d}\)) and to one quarter (for s and \(\bar{s}\)) of the value obtained without the nuclear data. All these effects emphasise the constraining power of the nuclear data in a global fit, as already noted in Sect. 4.11 of Ref. [3].
Second, the inclusion of theoretical uncertainties in the fit mostly affects sea quark PDFs. Central values are generally contained within the onesigma uncertainty of the Baseline fit, that includes the nuclear data but does not include any theoretical uncertainty, irrespective of the PDF flavour and of the prescription used to estimate the theoretical covariance matrix. The nature of the change in the central value is similar in both the NucUnc and the NucCor fits. For \(\bar{u}\) and \(\bar{d}\) PDFs, we observe an enhancement in the region \(0.2\simeq x \simeq 0.3\) followed by a strong suppression for \(x\gtrsim 0.3\). For s and \(\bar{s}\) PDFs, we observe: a slight suppression at \(x\lesssim 0.10.2\); an enhancement at \(0.10.2\lesssim x \lesssim 0.40.5\); and a strong suppression at \(x\gtrsim 0.40.5\). Uncertainties are always increased when the theoretical covariance matrix is included in the fit, irrespective of the way it is estimated, for all PDF flavours. Such an increase is only marginally more apparent in the NucUnc fit, as a consequence of this being the more conservative estimate of the nuclear uncertainties.
All these effects lead us to conclude that theoretical uncertainties related to nuclear data are generally small in comparison to the experimental uncertainty of a typical global fit, in that deviations do not usually exceed onesigma. Nevertheless, slight distortions in the central values and increases in the uncertainty bands (especially for s and \(\bar{s}\)) become appreciable when theoretical uncertainties are taken into account. The systematic inclusion of nuclear uncertainties in a global fit is thus advantageous whenever conservative predictions of sea quark PDFs are required.
4 Impact on phenomenology
As discussed in the previous section, the most sizeable impact of theoretical uncertainties is on the light sea quark PDFs, which display slightly distorted central values, and appreciably inflated uncertainties in comparison to the Baseline fit. In this section, we study the implications of these effects on the sea quark asymmetry, and on the strangeness fraction of the proton, including a possible asymmetry between s and \(\bar{s}\) PDFs.
4.1 The sea quark asymmetry
A sizeable asymmetry in the antiup and antidown quark sea was observed long ago in DY first by the NA51 [112] and then by the NuSea/E866 experiments [113]. Perturbatively, the number of antiup and antidown quarks in the proton is expected to be very nearly the same, because they originate primarily from the splitting of gluons into a quarkantiquark pair, and because their masses are very small in comparison to the confinement scale. The observed asymmetry must therefore be explained by some nonperturbative mechanism, which has been formulated in terms of various models over the years [114].
Inspection of Fig. 13 makes it apparent that the effect of nuclear data on the \(\bar{d}/\bar{u}\) ratio is significant, in particular in the region \(0.03\lesssim x \lesssim 0.3\). In this region, the central value of the Baseline fit is enhanced by around two sigma with respect to the NoNuc fit. The corresponding uncertainty bands do not show any significant difference in size, but they barely overlap. The two fits differ by approximately \(\sqrt{2}\) sigma. The inclusion of the nuclear uncertainty, even at its most conservative, makes little difference to the \(\bar{d}/\bar{u}\) ratio: the central value and the uncertainty of the NucUnc result are almost unchanged in comparison to the Baseline, and the ratio remains significantly larger than the NoNuc result.
4.2 The strange content of the proton revisited
Baseline  NoNuc  NucUnc  NucCor  Ref. [83]  

\(R_s(x=0.023,Q=1.38\text { GeV})\)  \(+\,0.69 \pm 0.15\)  \(+\,0.68 \pm 0.14\)  \(+\,0.65 \pm 0.14\)  \(+\,0.64 \pm 0.15\)  \(+\,1.13 \pm 0.11\) 
\(R_s(x=0.023,Q=91.2\text { GeV})\)  \(+\,0.81 \pm 0.05\)  \(+\,0.79 \pm 0.07\)  \(+\,0.79 \pm 0.06\)  \(+\,0.77 \pm 0.06\)  – 
\(K_s(1.38\text { GeV})\)  \(+\,0.63 \pm 0.09\)  \(+\,0.97 \pm 0.18\)  \(+\,0.63 \pm 0.09\)  \(+\,0.61 \pm 0.10\)  – 
\(K_s(91.2\text { GeV})\)  \(+\,0.80 \pm 0.05\)  \(+\,0.98 \pm 0.09\)  \(+\,0.80 \pm 0.05\)  \(+\,0.79 \pm 0.06\)  – 
Inspection of Table 4 and Fig. 14 makes it apparent that the effect of nuclear data on \(R_s\) at low values of x, namely at \(x=0.023\), is negligible. The central values and the uncertainties of \(R_s\) are remarkably stable across the four fits. This is unsurprising, as \(x=0.023\) is at the lower edge of the kinematic region covered by the nuclear data, see Fig. 1. At larger values of x, instead, particularly in the range \(0.03\lesssim x\lesssim 0.2\), the nuclear data affects \(R_s\) quite significantly, as is apparent from comparison of the NoNuc and Baseline fits. In the Baseline (which includes the nuclear data), the uncertainty on \(R_s\) is reduced by a factor of two without any apparent distortion of the central value. Likewise, the effect of nuclear uncertainties is mostly apparent in a similar x range. If one compares the NucUnc and the Baseline fits, an increase of the uncertainty on \(R_s\) by up to one third can be seen. However, this effect remains moderate, and is mostly washed out when it is integrated over the full range of x. The value of \(K_s\) is indeed almost unchanged by the inclusion of nuclear uncertainties in the Baseline fit, irrespective of whether they are implemented as the NucUnc or the NucCor fit, especially at high values of Q.
We therefore conclude that the inclusion of nuclear uncertainties does nothing to reconcile the residual tension between ATLAS and NuTeV data, the reason being that they probe the strangeness in kinematic regions of x and Q that barely overlap. Further evidence of the limited interplay between ATLAS and NuTeV data is provided by the \(\chi ^2\) of the former, which remains poor for all the four fits considered in this analysis, see Table 3. Achieving a better description of the ATLAS data or an improved determination of the strange content of the proton might require the inclusion of QCD corrections beyond NNLO and/or of electroweak corrections, or the analysis of other processes sensitive to s and \(\bar{s}\) PDFs, such as kaon production in semiinclusive DIS. All this remains beyond the scope of this work.
Finally, we investigate the effect of the nuclear data on the strange valence distribution \(xs^(x,Q)=x[s(x,Q)\bar{s}(x,Q)]\), which we display as a function of x at two representative values of Q in Fig. 15. Results are shown for each of the four fits performed in this analysis. From Fig. 15, we see once again the constraining power of the nuclear data. By comparing the Baseline and the NoNuc fits, it is apparent that the strange valence distribution is almost unconstrained when the nuclear data is removed from the fit, with large uncertainty and completely unstable shape.
When the nuclear data is included, similar effects are observed whether nuclear uncertainties are implemented conservatively or as a correction. Concerning the central values, in comparison to the Baseline fit the NucUnc and NucCor fits have a slightly suppressed valence distribution in the region \(x \lesssim 0.3\). Overall, nuclear effects do not alter the asymmetry between s and \(\bar{s}\) PDFs. Concerning the uncertainties themselves, both the NucUnc and the NucCor fits show an increased uncertainty in the strange valence distribution in the region \(x \lesssim 0.3\), which is a little more pronounced for the the NucCor fit than the NucUnc fit, contrary to what would be expected if the small nuclear correction obtained from the nPDFs were a genuine effect.
In conclusion, nuclear effects have negligible impact on the \(\bar{d}\bar{u}\) asymmetry, on the total strangeness, and on the \(s \bar{s}\) asymmetry. We do not observe any evidence in support of the use of nuclear corrections (NucCor): in the global proton fit, the fit quality to the nuclear datasets (and the overall fit quality) is always a little worse when nuclear corrections are implemented. This may be due to the slight inconsistencies between the different sets of nPDFs, visible in Fig. 3. The use of the more conservative nuclear uncertainties (NucUnc) in global proton PDF fits is thus the recommended option, at least until more reliable nPDFs become available.
5 Summary and outlook
In this paper we revisited the rôle of the nuclear data commonly used in a global determination of proton PDFs. Specifically, we considered: DIS data taken with Pb and Fe targets, from CHORUS and NuTeV experiments, respectively; and DY data taken with a Cu target, from the E605 experiment. We studied the fit quality and the stability of the proton PDFs obtained in the framework of the NNDPF3.1 global analysis, by comparing a series of determinations: one in which the nuclear dataset is removed from the fit; one in which it is included without any nuclear correction (the baseline fit); and two in which it is included with a theoretical uncertainty that takes into account nuclear effects.
The two determinations which included a theoretical uncertainty were realised by constructing a theoretical covariance matrix that was added to the experimental covariance matrix, both when generating data replicas and when fitting PDFs. These covariance matrices were constructed using a Monte Carlo ensemble of nuclear PDFs, determined from a wide set of measurements using nuclear data. In the first determination, the theoretical covariance matrix elements were constructed by finding the difference between each nuclear PDF replica and a central proton PDF, and then taking an average over replicas. This gives a conservative estimate of the nuclear uncertainty, increasing uncertainties overall, and deweighting the nuclear data in the fit. In the second determination, the theoretical covariance matrix elements were constructed by finding the difference between each nuclear PDF replica and the central nuclear PDF. Additionally, the theoretical prediction was shifted by the difference between the predictions made with central nuclear and proton PDFs. This correction procedure takes the nuclear effects and their uncertainties as determined by the nuclear fits at face value, but is less conservative than the first prescription.
We confirm that the nuclear dataset contains substantial information on the proton PDFs, even when the uncertainties due to nuclear effects are taken into account. In particular it provides an important constraint on the light sea quark PDFs, consistent with constraints from LHC data, as we explicitly demonstrated by inspecting the individual PDFs, the \(\bar{d}/\bar{u}\) ratio, the strangeness fractions \(R_s\) and \(K_s\), and the strange valence distribution \(s\bar{s}\). Therefore, it should not be dropped from current global PDF fits.
A conservative estimate of the additional theoretical uncertainty due to the use of a nuclear rather than a proton target in these measurements gives only small changes in the central values of the proton PDFs, with a slight increase in their overall uncertainties. In particular, nuclear effects are insufficient to explain any residual tension in the global fit between fixed target and ATLAS determinations of the strangeness content of the proton. This is largely because the corresponding measurements are sensitive to different kinematic regions that have a limited interplay. We nevertheless recommend that nuclear uncertainties are always included in future global fits, to eliminate any slight bias, and as a precaution against underestimation of uncertainties.
We should emphasise that all our results were determined from the most recent publically available nuclear PDFs. Despite the fact that these are obtained from a global analysis of experimental data taken in a wide variety of processes, including DIS, DY and pN collisions, some inconsistencies between the nPDF sets, in particular in the estimation of the uncertainties, were observed. This suggests that nPDFs are not yet sufficiently reliable to justify the use of a nuclear correction in the fit. Indeed, when we attempted to use them to give a nuclear correction to the predictions with proton PDFs, the fit quality was always a little worse than that obtained without nuclear corrections. Even so, the effect on the proton PDFs, in particular the \(\bar{d}/\bar{u}\) ratio, the total strangeness \(s+\bar{s}\), and the valence strange distribution \(s\bar{s}\) was negligible. Nevertheless, we recommend that when estimating nuclear effects using current nPDF sets, the more conservative approach is adopted, in which nuclear effects give an additional uncertainty, but not a correction.
Our work can be extended in various different directions. First, we intend to reconsider our results when newer more reliable nPDF sets become available. In this respect, a consistent determination based on the NNPDF methodology [117] would be particularly helpful, both because it would give a more reliable assessment of the uncertainties on nPDFs, and because a consistent treatment, determining proton PDFs, nPDFs and nuclear corrections iteratively, would then be possible.
Second, our analysis can be extended to deuterium data. In principle, this suffers from similar uncertainties as the nuclear data considered here, though they are expected to be much smaller. Theoretical uncertainties might be estimated by studying the spread between various model predictions for nuclear effects, or by attempting to determine them empirically from the data, in iterating them to consistency with the proton data.
Finally, the general method of accounting for theoretical uncertainties in a PDF fit by estimating a theoretical covariance matrix that is added to the experimental one in the definition of the \(\chi ^2\) can have many other applications [11]: missing higherorder uncertainties [118], highertwist uncertainties, and fragmentation function uncertainties in the analysis of semiinclusive data. In this last respect, the analysis of kaon production data might be useful to obtain more information on the strangeness content of the proton.
The PDF sets presented in this work are available in the LHAPDF format [119] from the authors upon request.
Notes
Acknowledgements
We thank R. Sassot for providing us with the nuclear PDF sets of Ref. [22]. We acknowledge useful discussions on the inclusion of theoretical uncertainties in PDF fits with our colleagues in the NNPDF Collaboration. The authors are supported by the UK Science and Technology Facility Council through Grant ST/P000630/1. E.R.N. is also supported by the European Commission through the Marie SkłodowskaCurie Action ParDHonS_FFs.TMDs (Grant number 752748).
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