Proton-PDF uncertainties in extracting nuclear PDFs from W± production in p+Pb collisions

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Proton-PDF uncertainties in extracting nuclear PDFs from W± production in p+Pb collisions © The Author(s) 2022 Published version Eskola, Kari J.; Paakkinen, Petja; Paukkunen, Hannu; Salgado, Carlos A. Eskola, K. J., Paakkinen, P., Paukkunen, H., & Salgado, C. A. (2022). Proton-PDF uncertainties in extracting nuclear PDFs from W± production in p+Pb collisions. European Physical Journal C, 82, Article 271. https://doi.org/10.1140/epjc/s10052-022-10179-2 2022 Eur. Phys. J. C (2022) 82:271 https://doi.org/10.1140/epjc/s10052-022-10179-2 Regular Article - Theoretical Physics Proton-PDF uncertainties in extracting nuclear PDFs from W± production in p+Pb collisions Kari J. Eskola1,2,a, Petja Paakkinen1,2,3,b, Hannu Paukkunen1,2,c, Carlos A. Salgado3,d 1Department of Physics, University of Jyvaskyla, P.O. Box 35, 40014 University of Jyvaskyla, Finland 2Helsinki Institute of Physics, University of Helsinki, P.O. Box 64, 00014 University of Helsinki, Finland 3Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de Santiago de Compostela, 15782 Galicia, Spain Received: 11 February 2022 / Accepted: 28 February 2022 © The Author(s) 2022 Abstract We discuss the recent CMS Collaboration measurement of W±boson production in p+Pb collisions at 8.16 TeV in terms of the constraining power on nuclear parton distribution functions (PDFs). The impact of the free-proton PDF uncertainties on the nuclear PDF extraction is quantified by using a theoretical covariance-matrix method and Hessian PDF reweighting. We discuss different ways to mitigate these theoretical uncertainties, including self-normalization, forward-to-backward ratios and nuclear-modification ratios. It is found that none of these methods offer perfect cancellation of the free-proton PDFs but, with the present data uncertainties, the residual free-proton-PDF dependence has, conveniently for the global analyses, little effect on the extraction of the nuclear modifications. Based on a simple estimate of obtainable statistics at the LHC Run 3, we argue that this will change in the near future and it becomes more important to propagate the proton-PDF uncertainties accordingly. Using the obtained information on the correlations of the free-proton uncertainties, we also identify a new charge asymmetry ratio, where the cancellation of the proton-PDF uncertainties is found to be extremely good. 1 Introduction The parton distribution functions (PDFs) of heavy nuclei, like their free-proton counterparts, are currently obtained most reliably from global analyses of experimental data. The bulk of these data comes from deep inelastic scattering (DIS) measurements which probe the nuclear structure directly and uniquely, with the precision limited at high enough scales ae-mail: [email protected] be-mail: [email protected] (corresponding author) ce-mail: [email protected] de-mail: [email protected] only by experimental uncertainties and perturbative accuracy. However, to constrain the full flavour dependence of the nuclear PDFs, it is necessary to additionally use proton– nucleus (p+A) processes, including fixed-target Drell–Yan (DY) dilepton as well as collider electroweak (EW) boson and (di)jet production data. For these processes, the collinearly factorized cross sections contain a convolution of the nuclear and free-proton PDFs, and as a consequence, the nuclear PDFs extracted from such data become inherently dependent on the assumed free-proton PDFs. One could then envisage two systematic approaches to treat the proton-PDF uncertainties in the nuclear-PDF analyses: First, one can try to reduce the proton-PDF uncertainties by using observables where one probes instead the nuclear modifications of the PDFs, which are then parametrized and fitted, and the “baseline” free-proton PDF dependence effectively drops out, as has been done systematically in the EKS– EPPS line of analyses [1–7], and by others [8–12]. This approach has been particularly attractive since much of the older DIS and DY data are in any case available only in terms of nuclear ratios. Or, second, one could allow using also absolute cross sections, taking into account all possible correlations with the free-proton PDFs, a program which has more recently been undertaken by the nNNPDF collaboration [13–15]. In some other instances the treatment of the baseline proton-PDF dependence have been less explicit [16–25], with the inherent assumption being that the free-proton uncertainties are in any case smaller than the nuclear-PDF ones and thus do not cause a significant bias in the fit. Until very recently, this has been a justifiable approximation. However, as the precision of data from the LHC p+Aprogram improves, it can become necessary to either propagate or mitigate the proton-PDF uncertainties in extracting the nuclear PDFs. 0123456789().: V,-vol 123 271 Page 2 of 14 Eur. Phys. J. C (2022) 82:271 One of the latest additions to the nuclear-PDF constraints is the CMS Collaboration measurement of W±boson production in LHC Run 2 p+Pb collisions at 8.16 TeV [26], with an eight-fold increase in the statistics compared to the Run 1 data taking at 5.02 TeV [27]. These data have been already included in nuclear-PDF analyses, where they have been seen to give constraints either specifically on the gluon and seaquark PDFs [25], strangeness [22], or on the flavour separation in more general [14]. The level at which the proton-PDF uncertainties are treated in these analyses varies, with Refs. [22,25] taking the proton-PDFs as fixed, ignoring their uncertainties, and Ref. [14] propagating the proton-PDF uncertainties in the analysis, but not discussing their importance in the fit. The role of the proton-PDF uncertainties in W±production in p+Pb collisions has been considered previously in Ref. [28]. In this paper, we elaborate their significance in the context of the aforementioned CMS Collaboration measurement [26] (Sect. 2) and different ratios constructed from the data (Sect. 4). We also extend the analysis of the importance of the proton-PDF uncertainties in nuclear-modification fitting by the tools of theoretical covariance matrix (Sect. 3) and Hessian PDF reweighting (Sect. 5). 2 Proton-PDF uncertainties in W±production in p+Pb collisions It is conventional to write the PDFs fA iof a nucleus with Z protons and Nneutrons in terms of bound-nucleon PDFs at momentum fraction xand scale Q2as fA i(x,Q2)=Zfp/A i(x,Q2)+Nfn/A i(x,Q2), (1) taking the bound-neutron PDFs fn/A ito be related to the bound-proton ones fp/A iby the isospin symmetry: un/A(x,Q2)=dp/A(x,Q2), dn/A(x,Q2)=up/A(x,Q2), ¯un/A(x,Q2)=¯ dp/A(x,Q2), ¯ dn/A(x,Q2)=¯up/A(x,Q2), (2) and fn/A i=fp/A ifor other flavours. This can be seen as an effective prescription, where the bound-nucleon PDFs should be understood as carrying information on the parton content of the “average” nucleon, used only to simplify the treatment of isospin dependence. In the region where x≤1, one can further write fp/A i(x,Q2)=Rp/A i(x,Q2)fp i(x,Q2), (3) where the nuclear modification factors Rp/A i, given the freeproton PDFs fp i, now encode all the information on the partonic structure of nuclei. It should be noted that the above steps can be taken without any loss of generality. In the end, if all correlations between the proton and nuclear PDFs are correctly taken into account, it should not matter whether one parametrizes the absolute nuclear PDFs or the nuclear modifications. One is simply mapping a set of unknown functions fA ito the same number of functions Rp/A i. In practice, however, simplifying assumptions are used in the nuclear-PDF analyses. We will use here the nuclear modifications from the EPPS16 analysis [6], and for full consistency, we use the CT14 NLO free-proton PDFs [29] but also validate the robustness of the results by comparing to the CT18 NLO PDFs [30]. This assumes that Rp/A i depend only on the nuclear mass number A=Z+N(i.e. that there is no non-trivial isospin dependence in them) and that they are uncorrelated to fp i.1As mentioned in Sect. 1,the latter is true as a first approximation due to the use of appropriate ratio observables in EPPS16, but we will discuss later in this article the validity of this assumption in the presence of increasingly precise electroweak data. The advantage of this framework is that we can study the relative importance of the nuclear-modification and freeproton-PDF uncertainties in any observable of interest. Here, we study these in the context of W±production in p+Pb collisions at 8.16 TeV, as measured by the CMS Collaboration in the muon decay channel [26]. The lepton-rapidity differential cross sections, with a cut on lepton transverse momentum pμ T>25 GeV, is presented in Fig. 1. The theoretical next-to-leading order (NLO) perturbative QCD predictions are obtained with MCFM [31], and the PDF uncertainties from EPPS16 and CT14 are calculated with the conventional asymmetric prescription at the 90% confidence level. As can be seen from the figure, the baseline CT14 free-proton PDF errors (shown as yellow boxes) contribute significantly to the total theoretical uncertainty budget (light blue boxes) and can even exceed those from the EPPS16 nuclear modifications (blue hatching) in some bins. The smallness of nuclearmodification uncertainties in the negative (backward) rapidities originates from the good neutral and charged-current DIS constraints at the probed values of x. Going to positive (forward) rapidities, we enter the less-constrained small-xregion and the nuclear-modification uncertainties begin to grow. As was shown already in Ref. [26], the agreement between the CMS measurement and the NLO predictions from EPPS16 ×CT14 is excellent. The goodness of fit for this data set is given by χ2 C=(D−fnorm.T)TC−1(D−fnorm.T) +fnorm.−1 σnorm.2 ,(4) 1Note that fp/A iand fA iare still correlated to fp ithrough Eqs. (3) and (1). 123 Eur. Phys. J. C (2022) 82:271 Page 3 of 14 271 Fig. 1 Lepton-rapidity differential W±production cross sections in p+Pb collisions at 8.16 TeV with a breakdown of the theoretical uncertainties (EPPS16×CT14, light-blue boxes) into those from free-proton PDFs (CT14 NLO, yellow boxes) and from the nuclear modifications (EPPS16, blue hatching). The data from the CMS measurement [26] are presented with black markers, scaled with the optimal normalization factor explained in the text where Dand Tare vectors of dimension Ndata containing the data and theory values, and we have extracted the normalization uncertainty σnorm.=3.5% from the data covariance matrix C, thus avoiding the D’Agostini bias [32]. By doing so, the optimal normalization factor fnorm.can be solved analytically, and in the figures we multiply the data with a factor 1/fnorm.=1+σ2 norm.TTC−1T 1+σ2 norm.DTC−1T.(5) Taking Tas the central prediction from EPPS16×CT14, we then have 1/fnorm.=0.986 and χ2 C/Ndata =1.12, confirming the visibly good data-to-theory agreement. When fitting to these data, we therefore do not expect the central nuclear PDFs to change much from the EPPS16 results, but as the experimental uncertainties are much smaller than the nuclear-modification uncertainties especially at forward rapidities, we can expect a significant reduction in the latter. The large baseline free-proton uncertainties can however affect the obtainable constraints and we need to find a way to either quantify or mitigate the impact. 3 Theoretical covariance matrix One way to quantify the impact of a certain theoretical source of uncertainty is to use the method of theoretical covariance matrix [33]. For the free-proton uncertainties, taken from the CT14 PDFs, this matrix is given by SCT14 ij = kTi[Sk,+ CT14]−Ti[Sk,− CT14] 2×1.645 ×Tj[Sk,+ CT14]−Tj[Sk,− CT14] 2×1.645 ,(6) where the sum goes over the CT14 parameter eigendirections kand Ti[Sk,± CT14]are the corresponding predictions for the ith data point with positive and negative parameter variations in 10 20 30 40 10 20 30 40 j i Cij /D iDj 10 20 30 40 10 20 30 40 j i SCT14 ij /D iDj −2 −1 0 1 2 ·10−3 Fig. 2 The experimental (excluding overall normalization uncertainty) and theoretical free-proton-PDF covariance matrices for the p+Pb W± measurement at 8.16 TeV. Indices i,jfollow the same ordering as the data points in Fig. 1, with the indices 1 through 24 corresponding to the W−production and 25 through 48 to W+ that eigendirection (all calculated with the central EPPS16 nuclear modifications). The factors 1.645 in the denominators are used to scale the nominally 90% confidence-level uncertainties of CT14 to a 68% (one standard deviation) level in order not to overestimate their impact with respect to the experimental uncertainties. The CT14 theoretical covariance matrix is presented in Fig. 2with a comparison to the experimental covariance matrix. To ease the interpretation, we have excluded the large fully correlated luminosity component from the experimental covariance matrix, as in Eq. (4), and divided each matrix element with the product of the corresponding data values. We see that the proton-PDF uncertainties are comparable or larger than the correlated non-luminosity experimental uncertainties but still mostly smaller than the combined statistical and non-luminosity systematical uncertainties in the diagonal elements. Clearly, the free-proton PDFs contribute a non-negligible uncertainty component to the nuclear-modification fitting. With the theoretical CT14 proton-PDF uncertainties taken into account, the figure of merit for the nuclear-modification d.o.f.s takes the form [33] 123 271 Page 4 of 14 Eur. Phys. J. C (2022) 82:271 Fig. 3 As Fig. 1, but now for the self-normalized cross sections 10 20 30 40 10 20 30 40 j i Cnorm. ij /D norm. iDnorm. j 10 20 30 40 10 20 30 40 j i SCT14,norm. ij /D norm. iDnorm. j −2 −1 0 1 2 ·10−3 Fig. 4 As Fig. 2, but now for the self-normalized cross sections. Indices i,jfollow the same ordering as the data points in Fig. 3, with the indices 1 through 24 corresponding to the W−production and 25 through 48 to W+ χ2 C+SCT14 =(D−fnorm.T)T(C+SCT14)−1(D−fnorm.T) +fnorm.−1 σnorm.2 ,(7) and we find χ2 C+SCT14 /Ndata =0.85 for EPPS16. Comparing this to the value χ2 C/Ndata =1.12 for EPPS16×CT14, we see that the chosen proton PDFs can indeed have a significant impact on the level of agreement with the data. Interestingly, the proton-PDF uncertainties are strongly positively correlated, behaving almost like an additional normalization uncertainty. It is exactly this positive correlation (and the positive correlation with the corresponding proton– proton cross section) which makes the uncertainty reduction with the ratios discussed in Sect, 4possible. We note also that the optimal data normalization that we find for EPPS16 ×CT14 from Eq. (5)is1/fnorm.=0.986, well within the 3.5% normalization uncertainty. This should be compared to the value of 0.960 in the nCTEQ15WZ fit for these data [22]. Since the proton PDFs contribute significantly to the normalization of the predictions, we can speculate whether the larger than 1 ×σnorm.normalization shift in the nCTEQ15WZ analysis originates from the used freeproton baseline. This possibility is also corroborated by the fact that when testing the robustness of the results presented here by changing the free-proton PDFs to CT18 NLO, the main effect was a change in the normalization, with the optimal data-scaling factor 1/fnorm.changing to a value 0.997. The CT18 uncertainties were also observed to be slightly less correlated across different rapidities, but the uncertainties were found to be almost the same, and this had no impact on our conclusions. 4 Reducing proton-PDF uncertainties As we have shown that the free-proton uncertainties are important in describing the p+Pb W±data, it makes sense to explore ways to reduce these uncertainties. Since the covariance matrix of the CMS measurement is available to us, we can propagate the data uncertainties to any desired observable keeping also track of the correlations by using Cnew =JCJ T,(8) where Jis the Jacobian of the transformation. We note that also perturbative higher-order corrections can (partially) cancel in many of the considered ratios, which supports their use in nuclear-PDF analyses, but the importance of missing higher orders is left outside the scope of this article. 4.1 Self-normalized cross sections Since the free-proton PDF uncertainties were found to be strongly correlated, almost normalization-like, a viable option to reduce them is by self-normalizing the cross sections dσW±,norm. pPb /dημ=1 σW± pPb dσW± pPb /dημ,(9) where σW± pPb =1.93 −2.86 dημdσW± pPb /dημ(10) are the fiducial integrated cross sections of each W±charge. We perform here the normalization for each charge separately, but it would be also possible to do this by dividing with the charge-summed integrated cross section. The obtained normalized cross sections are presented in Fig. 3 and the experimental Cnorm.(from Eq. (8)) and theoretical SCT14,norm.(calculated directly from the CT14 error sets) covariance matrices in Fig. 4. We observe a very good, but not perfect cancellation of the free-proton uncertainties, with 123 Eur. Phys. J. C (2022) 82:271 Page 5 of 14 271 Fig. 5 As Fig. 1, but now for the forward-to-backward ratios the remaining CT14 uncertainties being largest at the large negative rapidities, ημ<−1.93. In addition to cancelling the normalization uncertainty, the self-normalization changes the correlation pattern of the remaining statistical and systematical experimental uncertainties, which become mostly anticorrelated across different rapidity bins. Importantly, even the originally uncorrelated (statistical) uncertainties become correlated in the self-normalized cross sections. Another important thing to notice here is that neither the experimental nor the theoretical covariance matrix is invertible, with det Cnorm.= det SCT14,norm.=0. This simply follows from the fact that a self-normalized set of data forms an overdetermined system: given all but one data point, the last one can be solved from the requirement that the data integrate to one. Another way to see this is to notice that the self-normalization is not a bijection, with an immediate consequence that det J=0. For this property, one should fit to the self-normalized data by leaving one point out. Due to the fully correlated nature of the normalized data, it does not matter which data point is left out, manifesting the loss of information in the normalization.2 4.2 Forward-to-backward ratios The forward-to-backward ratios RW± FB =dσW± pPb /dημ|ημ dσW± pPb /dημ|−ημ (11) have been considered earlier in Ref. [28], where it was realised that they do not yield as good a cancellation of the free-proton uncertainties as e.g. the same ratios for Z-boson or dijet [36] production. Since the experimental acceptance is not symmetric with respect to the p+Pb center-of-mass 2We note, however, that e.g. in the case of the CMS measurement of self-normalized dijet cross sections in p+p and p+Pb collisions at 5.02 TeV [34], which we have studied in Ref. [35], the data correlations were not published and it is less clear how to treat the data statistically accurately in a fit. Without knowing the correlations, it would matter which data point was left out. frame, one has to drop part of the data points, in this case those for ημ<−1.93. Furthermore, by taking the ratio, the number of data points is still halved, leading to a significant loss of information. The remaining ten data points for each charge are shown in Fig. 5, where we see that the proton- PDF cancellation is good, but starts to worsen towards larger rapidities. This can be understood by taking the large-rapidity limit ημ0, where we can take the large momentum-fraction x1to probe only valence quarks and the small momentumfraction x2then probes the sea quarks. At this limit, neglecting the Cabibbo suppressed quark-mixing effects and denoting x1,2:=x1,2|ημ=x2,1|−ημ, we can approximate at leading order RW− FB ημ0 ≈ x1large x2small ZRp/A ¯u(x2)+N¯ dp(x2) ¯up(x2)Rp/A ¯ d(x2) ZRp/A dV(x1)+Nup V(x1) dp V(x1)Rp/A uV(x1) (12) and RW+ FB ημ0 ≈ x1large x2small ZRp/A ¯ d(x2)+N¯up(x2) ¯ dp(x2)Rp/A ¯u(x2) ZRp/A uV(x1)+Ndp V(x1) up V(x1)Rp/A dV(x1) ,(13) where we have suppressed for simplicity the relevant phasespace integrations and the scale-dependence of the PDFs. We note that this approximation is not exact in the data region as there can still be sizeable (but subleading) contributions also from the ¯c+sand c+¯schannels [38]. In any case, we see that the forward-to-backward ratios depend on the freeproton PDFs through uV/dVand ¯u/¯ dratios, which determine the relative size of the contributions from the different nuclear modifications and give a non-cancelling contribution to the theoretical uncertainty. The resulting covariance matrices for the forward-to- backward ratios are presented in Fig. 6, where we observe a positive correlation of the proton-PDF uncertainties between same-charge bins, but an anticorrelation between different charges. Indeed, one could reduce the proton-PDF uncertainties further by taking the forward-to-backward ratio of 123 271 Page 6 of 14 Eur. Phys. J. C (2022) 82:271 5101520 5 10 15 20 j i CFB ij /D FB iDFB j 5101520 5 10 15 20 j i SCT14,FB ij /D FB iDFB j −2 −1 0 1 2 ·10−3 Fig. 6 As Fig. 2, but now for the forward-to-backward ratio. Indices i,jfollow the same ordering as the data points in Fig. 5, with the indices 1 through 10 corresponding to the W−production and 11 through 20 to W+ the differential cross section summed over the two charges, as considered in Ref. [26], but this leads to a further loss of information compared to taking the ratio separately for different charges, and the constraints for nuclear-PDF analyses are rather limited. 4.3 Nuclear-modification ratios We now study the possibility of using nuclear-modification ratios to cancel free-proton uncertainties. For the 8.16 TeV p+Pb data, no same-energy p+p reference is available, but one could construct “mixed-energy” ratios RW± pPb =dσW± pPb,8.16 TeV/dημ AdσW± pp,8.0TeV/dημ (14) with the p+p measurements at 8.0 TeV, where the probed xranges are almost the same between the two energies. We use here the measurements from the CMS Collaboration [37], shown in Fig. 7along with the predictions from the CT14 PDFs. Again, the 2.6% normalization uncertainty is taken into account in presenting the data. The optimal shift, 0.969, is slightly larger than what we found for p+Pb. The agreement in normalization could be again improved by using CT18 5101520 10 20 30 40 j i Jpp ij [pb−1] 5101520 5 10 15 20 j i Cpp ij /D pp iDpp j −2 −1 0 1 2 ·10−3 Fig. 8 The Jacobian matrix Jpp ij for propagating the p+p uncertainties into RpPb uncertainties (cf. Eq. (15)) and the experimental (excluding overall normalization uncertainty) covariance matrix Cpp ij for the p+p W±measurement at 8.0 TeV. Indices i,jin Cpp ij and the index jin Jpp ij follow the ordering of the data points in Fig. 7, with the indices 1 through 11 corresponding to the W−production and 12 through 22 to W+,and the index iin Jpp ij follows the ordering in Fig. 9, with the indices 1 through 22 corresponding to the W−production and 23 through 44 to W+ PDFs, to 0.996, but the size of the PDF uncertainties stays almost the same. The rapidity binning is the same in p+p and p+Pb measurements up to |ημ|<1.6. For larger rapidities, we associate the p+Pb bins with the most-overlapping one in p+p. Therefore, for the p+Pb bins with 1.6<|ημ|<1.8wetake the ratio with p+p bin 1.6<|ημ|<1.85 and the p+p bin 1.85 <|ημ|<2.1 is used in obtaining three of the RpPb bins: 1.8<η μ<1.93, −1.93 <η μ<−1.8 and −2.2<η μ< −1.93. Finally, for the p+Pb bin −2.4<η μ<−2.2, the p+p bin 2.1<|ημ|<2.4 is used. For ημ<−2.4 we run out of p+p bins and we discard the remaining two p+Pb data points for both W±charges. The loss of information is therefore slightly larger than in the self-normalized cross sections, but significantly smaller than in the forward-to-backward ratios. Since the correlations between the p+Pb and p+p measurements are not known, the covariance matrix for the ratio is calculated with CRpPb =JpPb CpPb (JpPb)T+Jpp Cpp (Jpp)T.(15) Fig. 7 Lepton-rapidity differential W±production cross sections in p+p collisions at 8.0 TeV with theoretical uncertainties from the free-proton PDFs (CT14 NLO, yellow boxes). The data from the CMS Collaboration measurement [37] are presented with black markers, scaled with the optimal normalization factor 00.511.522.5 400 500 600 |ηµ| dσpp/dηµ[pb] 00.511.522.5 600 700 800 |ηµ| dσpp/dηµ[pb] W− →μ−¯νµpµ T>25 GeV W+→μ+νµpµ T>25 GeV CT14 CMS data ×0.969 123 Eur. Phys. J. C (2022) 82:271 Page 7 of 14 271 Fig. 9 As Fig. 1, but now for the nuclear-modification ratio with the p+p reference taken from Ref. [37] 10 20 30 40 10 20 30 40 j i CRpPb ij /D RpPb iDRpPb j 10 20 30 40 10 20 30 40 j i SCT14,RpPb ij /D RpPb iDRpPb j −2 −1 0 1 2 ·10−3 Fig. 10 As Fig. 2, but now for the nuclear-modification ratio with the p+p reference taken from Ref. [37]. Indices i,jfollowthesameordering as the data points in Fig. 9, with the indices 1 through 22 corresponding to the W−production and 23 through 44 to W+ This is a conservative estimate: in a direct experimental analysis some of the systematic uncertainties could be cancelled in the ratio. The Jacobian Jpp and the covariance matrix Cpp for the p+p data are presented in Fig. 8, visualising also how each p+p point contributes to multiple RpPb bins. The correlations arising from this are then taken correctly into account in Eq. (15). The obtained mixed-energy nuclear-modification ratios and the corresponding experimental and theoretical covariance matrices are presented in Figs. 9and 10 , respectively. The data are again well described by the EPPS16×CT14 predictions, but due to the larger optimal downward normalization shift in p+p compared to p+Pb, we find the optimal shift for the nuclear-modification ratio to be 1/fnorm.=1.032, still within the combined normalization uncertainty of 4.36%. Compared to the previously discussed ratios, we can expect a more “local” cancellation of the proton-PDF dependence. However, since we use different collision energies for p+p and p+Pb and the rapidity binning does not exactly match outside mid-rapidity, the probed xregions in the ratio can be slightly different, which can make the proton-PDF cancellation less than perfect.3We observe still a very good cancella- 3In a direct experimental measurement of the ratio, one could consider binning the data in a shifted rapidity variable yref as in Ref. [39]to minimize the effect of using different energies. tion, comparable or better than in the previous ratios, and the CT14 uncertainties in Fig. 10 are now clearly smaller than the diagonal elements of the experimental covariance matrix in all bins (note that we have also here omitted the overall experimental normalization uncertainty from the presentation of the matrix, in accordance with Eq. (4)). Consequently, the free-proton PDFs have smaller impact on the agreement with data, as we find χ2 C/Ndata =0.77 with the pure experimental uncertainties and χ2 C+SCT14 /Ndata =0.75 after taking the CT14 uncertainties into account. The cancellation somewhat deteriorates towards larger rapidities. In the far-backward region, the momentumfraction from the nuclear side x2is large and we can approximate, at leading order and neglecting the small shifts in the momentum fractions due to the different energies, RW− pPb ημ0 ≈ x2large Z ARp/Pb dV(x2)+N A up V(x2) dp V(x2)Rp/Pb uV(x2)(16) and RW+ pPb ημ0 ≈ x2large Z ARp/Pb uV(x2)+N A dp V(x2) up V(x2)Rp/Pb dV(x2), (17) where we see that the proton uV/dVratio again sets the limit to how well the proton-PDF uncertainties are cancelled. Note that in Eq. (16) we have the ratio uV/dV, leading to an enhancement at the probed backward rapidities, whereas in Eq. (17) we have its reciprocal, leading to a suppression, even in absence of nuclear modifications. In the far-forward region the nuclear momentum-fraction x2is small and we have RW− pPb ημ0 ≈ x2small Z ARp/Pb ¯u(x2)+N A ¯ dp(x2) ¯up(x2)Rp/Pb ¯ d(x2)(18) and RW+ pPb ημ0 ≈ x2small Z ARp/Pb ¯ d(x2)+N A ¯up(x2) ¯ dp(x2)Rp/Pb ¯u(x2). (19) 123 271 Page 8 of 14 Eur. Phys. J. C (2022) 82:271 Fig. 11 The W±production charge asymmetry, with a breakdown of theory uncertainties as in Fig. 1 At this limit, we see from Fig. 9that the ratios of both charges approach the value 0.9, reflecting the fact that at these large scales, one is probing an almost flavour symmetric quark sea generated through g→q¯qsplittings, and therefore also the probed nuclear modifications are almost the same. 4.4 Charge asymmetries As discussed already in Ref. [28], the traditional charge asymmetry ApPb =dσW+ pPb /dημ−dσW− pPb /dημ dσW+ pPb /dημ+dσW− pPb /dημ (20) is very sensitive to the free-proton uncertainties. As shown in Fig. 11, the excellent data-to-theory agreement continues to be valid also in this observable, but as now the free-proton and nuclear-modification uncertainties are of the same order, one is probing a non-trivial combination of the two, and the usefulness for nuclear-PDF fits is rather limited. In particular, at large positive rapidities this observable probes mostly the uV−dVasymmetry in proton [40], with the nuclear uncertainties having a strong cancellation. It is, however, possible to construct asymmetries with more direct sensitivity to the nuclear modifications. In Ref. [28], a charge ratio of forward–backward differences ˜ ApPb =dσW+ pPb /dημ|ημ−dσW+ pPb /dημ|−ημ dσW− pPb /dημ|ημ−dσW− pPb /dημ|−ημ ,(21) shown in Fig. 12 (left), was proposed, motivated by the finding that for cross sections differential in the W±boson rapidity, the proton-PDF uncertainties cancel extremely well in this quantity. Here, with the experimentally measurable lepton rapidity, we find the cancellation to be slightly worse, but it still gives far better access to the nuclear modifications than the traditional charge asymmetry. In particular, the measured data differ significantly from the predictions with free-proton PDFs taking into account the isospin effects only, i.e. neglecting the nuclear modifications in the bound-nucleon PDFs. Close to midrapidity this observable is experimentally problematic since the denominator approaches zero, and we find with the linear error propagation the statistics to be insufficient for any constraints at |ημ|<0.4. We can now use our knowledge of the proton-PDF correlations to our advantage. As can be seen from Figs. 4and 9, after taking away the overall normalization-like contribution, there is an anticorrelation in the proton-PDF uncertainties between the same-charge forward and backward cross sections. This anticorrelation is reflected also in the imperfect free-proton-PDF cancellation in the forward-to-backward ratios. Conversely, and quite unexpectedly, there appears to be a positive correlation between the forward production of one charge and the backward production of the other and an anticorrelation between the two charges at the same rapidity. Based on the approximation in Eqs. (16) through (19), this appears to be possible only if the ratio ¯up(x2)/ ¯ dp(x2)at small x2and up V(x1)/dp V(x1)at large x1are positively correlated. We find this to be true at the EW scale for CT14 (and also for CT18, MSHT20 [41] and NNPDF4.0 [42]) as long as x2<0.03, independently of x1. With this information in mind, we construct here a new forward-to-backward ratio of rapidity-mirrored charge difference A∗ pPb =dσW+ pPb /dημ|ημ−dσW− pPb /dημ|−ημ dσW+ pPb /dημ|−ημ−dσW− pPb /dημ|ημ ,(22) shown in Fig. 12 (right). The free-proton-PDF uncertainties in this observable are negligible and it avoids the problem of vanishing denominator that appeared in Eq. (21)astheW+ cross section is always larger than the W−one. Within the approximation used in Eqs. (12) and (13), it can be written in the large-rapidity limit as A∗ pPb ημ0 ≈ x1large x2small ZRp/A ¯ d(x2)−¯up(x2) ¯ dp(x2) dp V(x1) up V(x1)Rp/A dV(x1)+N¯up(x2) ¯ dp(x2)Rp/A ¯u(x2)−Rp/A uV(x1) ZRp/A uV(x1)−¯up(x2) ¯ dp(x2) dp V(x1) up V(x1)Rp/A ¯u(x2)+Ndp V(x1) up V(x1)Rp/A dV(x1)−Rp/A ¯ d(x2).(23) 123