Scaling properties of inclusive W± production at hadron colliders

This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Author(s): Title: Year: Version: Please cite the original version: All material supplied via JYX is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Scaling properties of inclusive W± production at hadron colliders Arleo, François; Chapon, Émilien; Paukkunen, Hannu Arleo, F., Chapon, É., & Paukkunen, H. (2016). Scaling properties of inclusive W± production at hadron colliders. European Physical Journal C, 76(4), Article 214. https://doi.org/10.1140/epjc/s10052-016-4049-1 2016 Eur. Phys. J. C (2016) 76:214 DOI 10.1140/epjc/s10052-016-4049-1 Regular Article - Theoretical Physics Scaling properties of inclusive W±production at hadron colliders François Arleo1,a, Émilien Chapon1,b, Hannu Paukkunen2,3,4,c 1Laboratoire Leprince-Ringuet, École polytechnique, CNRS/IN2P3, Université Paris-Saclay, 91128 Palaiseau, France 2Department of Physics, University of Jyvaskyla, P.O. Box 35, 40014 University of Jyvaskyla, Finland 3Helsinki Institute of Physics, P.O. Box 64, 00014 University of Helsinki, Finland 4Departamento de Física de Partículas and IGFAE, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Galicia, Spain Received: 7 December 2015 / Accepted: 30 March 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract We consider the hadroproduction of W gauge bosons in their leptonic decay mode. Starting from the leading-order expressions, we show that by defining a suitable scaling variable the centre-of-mass dependence of the cross sections at the LHC energies can be essentially described by a simple power law. The scaling exponent is directly linked to the small-xbehaviour of parton distribution functions (PDF) which, at the high virtualities involved in W production, is largely dictated by QCD evolution equations. This entails a particularly simple scaling law for the lepton chargeasymmetryandalsopredictsthatmeasurementsindifferent collision systems (p–p, p–p, p–Pb Pb–Pb) are straightforwardly related. The expectations are compared with the existing data and a very good overall agreement is observed. It is shown that the PDF uncertainty in certain cross-section ratios between nearby centre-of-mass energies can be significantly reduced by taking the ratios at fixed value of the scaling variable instead of fixed rapidity. 1 Introduction The production of W gauge bosons in hadronic collisions is a process which is sensitive to practically all aspects of Standard Model, from electro-weak couplings to QCD dynamics and the non-perturbative parton content of the hadrons [1]. One of the most precisely measured observables at hadron colliders is the rapidity (y) dependence of the lepton charge asymmetry, C, C(y)≡dσ+/dy−dσ−/dy dσ+/dy+dσ−/dy,(1) ae-mail: [email protected] be-mail: [email protected] ce-mail: [email protected] where the charged lepton (=e,μ) originates from the leptonic decay of the W boson. This observable is a useful probe of proton parton distribution functions (PDFs), in particular, to disentangle the flavour dependence [2–4] which is not well constrained by the deep inelastic scattering.1Today, the charge asymmetry has been studied in detail by the CDF [5,6]andD0[7–10]experimentsinp–¯pcollisionsattheTevatron as well as the ATLAS [11,12], CMS [13,14], and LHCb [15,16] experiments in p–p collisions at the LHC. While the broad features of the experimental data are well captured by fixed-order perturbative QCD calculations [17,18], the simultaneous reproduction of the D0 data in bins of different kinematic cuts is known to pose difficulties [19–22]. The first measurements of W production in p–Pb collisions have recently appeared [23–25] and various observables seem to favour the use of EPS09 nuclear PDFs (nPDFs) [26] instead of a naive superposition of free nucleon PDFs (similar conclusion can be expected in the case of other sets of nPDFs [27–29]). In addition, these measurements may also help to probe, for the first time, the flavour dependence of nuclear modifications in quark densities [23]. The production of W bosons in heavy-ion collisions is also of paramount importance. Measurements by ATLAS [30] and CMS [31] in Pb–Pb collisions have revealed that the production rate approximately scales with the number of binary nucleonnucleon collisions. This is in sharp contrast to hadronic observables (high-transverse momentum hadrons [32–34] and jets [35–37]) which are strongly suppressed as compared to p–p collisions. Thus, the leptons from W decays are valuable “messengers” from the initial state of heavyion collisions and could also be used to constrain the nPDFs [38–40]. In this paper, our main focus is on the centre-of-mass energy (√s) systematics of the production cross sections 1Unless a deuterium target, complicated by possible nuclear corrections, is used. 123 214 Page 2 of 10 Eur. Phys. J. C (2016) 76:214 dσ±/dyin hadronic collisions and the consequent scaling properties of the lepton charge asymmetry. First, in Sect. 2, we show how the scaling laws for absolute cross sections and charge asymmetries emerge from the relatively simple leading-order expressions. In Sect. 3we then contrast these expectations against next-to-leading order (NLO) computations. Section 4presents comparisons with the existing world data from LHC and Tevatron experiments as well as demonstrates how PDF uncertainties in some ratios of W cross sections can be suppressed by carefully choosing the rapidity binning. Finally, we summarise our main findings in Sect. 5. 2 Derivation of the scaling properties 2.1 Absolute cross sections We consider the inclusive production of W bosons in highenergy collisions of two hadrons, H1and H2,followedbythe decay of W to a charged lepton and a neutrino, H1+H2→W−+X→−+¯ν+X, H1+H2→W++X→++ν+X. At leading order, the production cross section double differential in the charged lepton rapidity yand transverse momentum pTreads [41,42], d2σ±(s) dydpT=πpT 24s2αem sin2θW2 i,j δeqi+eqj,±1|Vij|2 ×1 xmin 2 dx2x2−pT √se−y−1(x1x2)−1 x1x2s−M2 W2+M2 WΓ2 W ×ˆ t+ˆu±ˆ t∓ˆu2qH1 i(x1,Q2)qH2 j(x2,Q2) +ˆ t+ˆu∓ˆ t±ˆu2qH1 j(x1,Q2)qH2 i(x2,Q2),(2) where the symbols αem,θW, and Vij refer to the finestructure constant, the weak-mixing angle, and the elements of the Cabibbo–Kobayashi–Maskawa matrix, respectively. The mass and width of the W boson are denoted by MW and ΓW. The lower limit of the x2integral is given by xmin 2=(pTe−y)/(√s−pTey)and the momentum argument x1=(x2pTey)/(x2√s−pTe−y). The Mandelstam variables ˆ tand ˆuare ˆ t=− √sp Tx1e−y,ˆu=− √sp Tx2ey.(3) The PDFs are denoted by qHk i(x,Q2)(with Q2=O(M2 W)) and the sum runs over all flavours i,jsuch that the electric charges eqiof the quarks sum up to ±1. Since the total width of the W boson is much smaller than its mass, ΓWMW, we can make use of a delta-function identity /(x2+2)→ πδ(x),as→0,toperform theremainingintegralin Eq.(2). We find d2σ±(s) dydpT≈π2 24sαem sin2θW21 MWΓW ×pT 1−4p2 T/M2 W i,j|Vij|2δeqi+eqj,±1 ×1∓1−4p2 T/M2 W2 qH1 i(x+ 1)qH2 j(x+ 2) +1±1−4p2 T/M2 W2 qH1 i(x− 1)qH2 j(x− 2) +1±1−4p2 T/M2 W2 qH1 j(x+ 1)qH2 i(x+ 2) +1∓1−4p2 T/M2 W2 qH1 j(x− 1)qH2 i(x− 2), (4) where the momentum arguments of the PDFs are x± 1≡M2 Wey 2pT√s1∓1−4p2 T/M2 W,(5) x± 2≡M2 We−y 2pT√s1±1−4p2 T/M2 W.(6) Let us first consider a situation with2y0, that is, x± 2< x± 1. In terms of a dimensionless variable ξ1(which coincides with x± 1when pT→MW/2), ξ1≡MW √sey,(7) the momentum fractions in Eq. (6) become x± 1≡MW 2pT ξ11∓1−4p2 T/M2 W,(8) x± 2≡M3 W 2pTsξ11±1−4p2 T/M2 W.(9) At sufficiently small x, the sea-quark densities at high Q2∼ M2 Wshould be reasonably well approximated by a power law [43] xqi(x,Q2)≈xqi(x,Q2)≈Nix−α(Q2),(10) where the exponent α(Q2)>0 and the normalisations Ni should both be almost flavour independent. Such a behaviour (though not exactly a power law [44]) is expected by considering the small-xand large Q2limit (the so-called double logarithmic approximation [45]) of Dokshitzer–Gribov– Lipatov–Altarelli–Parisi parton evolution equations [46–52] and it is also consistent with the observations in deep inelastic scattering [53] with the Q2dependence of the exponent 2For simplicity, y0(y0) should be understood as ey1 (ey1) in the remainder of the paper. 123 Eur. Phys. J. C (2016) 76:214 Page 3 of 10 214 α(Q2)being roughly logarithmic. However, in the following, the “running” of α(Q2)does not directly show up since we will always set Q2=M2 W. For brevity, we will denote α≡α(Q2=M2 W)from now on. By using the approximation Eq. (10)inEq.(4) and trading the rapidity variable y with ξ1, we find d2σ±(s,ξ 1) dpTdξ1≈sα×f±(ξ1,pT,H1,H2), y0,(11) where f±(ξ, pT,H1,H2)is a function that does not depend explicitly on sor y. Since the expression of Eq. (4) is peaked at pT≈MW/2andthe pTdependenceofthe probedmomentumfractions inEq.(6) isnotparticularly fierce,the xinterval spanned by integration over pTwith a typical kinematic cut pT20 GeV remains sufficiently narrow such that approximation of Eq. (10) stays valid. Under these conditions, the scaling law in Eq. (11) holds also for pT-integrated cross sections, dσ±(s,ξ 1) dξ1≈sα×F±(ξ1,H1,H2), y0,(12) where F±(ξ1,H1,H2)≡dpTf±(ξ1,pT,H1)θ(pT− pmin T). In the backward direction with y0, the appropriate scaling variable is ξ2≡MW √se−y,(13) such that dσ±(s,ξ 2) dξ2≈sα×G±(ξ2,H1,H2), y0,(14) where G±(ξ2,H1,H2)is a function that does not depend explicitly on sor y.IfH 1=H2, then F±(ξ1,H1,H2)= G±(ξ2,H1,H2). Here, we emphasise the fact that at fixed ξ1(ξ2)thex region at which the PDFs of hadron H1(H2) is sampled becomes approximately independent of √s;seeEq.(8). Going to forward (backward) direction pushes this region to large xwhere the parameterisation dependence of PDFs may be large. As a consequence, one could hope that the PDF uncertainties on cross-section ratios between two different values of √swould better cancel out if performed at fixed ξ1,2than at fixed rapidity (as has been done e.g. by LHCb collaboration [16]).Atsmallx, the probed xregions will be different for two different √s;seeEq.(9), but at large Q2 the xdependence is almost purely dictated by the DGLAP evolution (in our scaling law approximated by a power law) and less prone to PDF uncertainties. We will come back to this later on in Sect. 4.2. 2.2 Charge asymmetries Since the √sdependence in Eqs. (12) and (14) is completely in the common prefactor sα, it follows that the lepton chargeasymmetry equation (1) should obey a particularly simple scaling law, CH1,H2 (s,ξ 1)≈F(ξ1,H1,H2), y0,(15) CH1,H2 (s,ξ 2)≈G(ξ2,H1,H2), y0, where F(ξ, H1,H2)≡F+(ξ, H1,H2)−F−(ξ, H1,H2) F+(ξ, H1,H2)+F−(ξ, H1,H2),(16) and similarly for G. In other words, at fixed ξ1or ξ2,the charge asymmetry should be approximately independent of the centre-of-mass energy. In fact, here one can allow the exponent αto depend also on √sand ξ1,2and it is only required that the PDFs are locally well approximated by a power law in the relevant region at small-x. Another, and also a bit surprising feature of the charge asymmetry is that at sufficiently large |y|it effectively dependsonlyon thenucleonthatisprobedat large x.Thisfollows from the facts that when |y|is sufficiently large, either udor du partonic process eventually dominates, and that the light-sea-quark distributions are expected to be approximately SU(2) symmetric at small x, u(x,Q2)≈u(x,Q2)≈d(x,Q2)≈d(x,Q2), x1, (17) and thus symmetric with respect to charge conjugation and isospin rotation. For example, one would expect that Cp,p (s,ξ 1)≈Cp,p (s,ξ 1)at large ξ1. In the case of nuclei the nPDFs fA i(x,Q2)are built from the free nucleon PDFs fproton i(x,Q2)and nuclear modification factors Rproton,A iby (see e.g. [26]) fA i(x,Q2)=Zfproton,A i(x,Q2)+Nfneutron,A i(x,Q2),(18) where fproton,A i(x,Q2)=Rproton,A ifproton i(x,Q2), (19) fneutron,A i(x,Q2)=fproton,A i,u↔d(x,Q2). (20) At small-xone expects modest shadowing (Rproton,A i<1), which, however, should not significantly alter the scaling exponent α(particularly at high Q2∼M2 Winvolved here) and, to a good approximation, the effect of shadowing is just a slight overall downward normalisation in the absolute cross sections which should largely disappear in the case of charge asymmetry. In other words, we can encapsulate the scaling law for lepton charge asymmetry as 123 214 Page 4 of 10 Eur. Phys. J. C (2016) 76:214 -5 -4 -3 -2 -1 0 1 2 3 4 5 10-3 10-2 10-1 1 y,-y √s=13TeV √s=7TeV √s=5.02 TeV √s=1.96 TeV ξ1,2 Fig. 1 Relation of rapidity yand scaling variables ξ1,2for a few values of √s CH1,H2 (s,ξ 1)≈F(ξ1,H1), y0, CH1,H2 (s,ξ 2)≈G(ξ2,H2), y0,(21) independentlyofthenature ofhadron(nucleon,anti-nucleon, nucleus) probed at small x. 3 Scaling vs. NLO calculation Most of our plots in the rest of the paper will use the scaling variables ξ1,2, which are related to the rapidity yand centre-of-mass energy √svia Eqs. (7) and (13). To ease the interpretation in what follows, this dependence is illustrated in Fig. 1. According to Eq. (10), the scaling exponent αin Eq. (12) should reflect the small-xbehaviour of quark distributions and it can be straightforwardly extracted from cross sections at two different centre-of-mass energies. To verify this correspondence and the consistency of our derivation, we have computed the full NLO cross sections at √s=7,8,13 TeV for p–p collisions using MCFM Monte-Carlo code [54] and CT10NLO PDFs [22]. From these cross sections, we have evaluated the effective scaling exponent αeff by αeff(ξ) =log σ±(s,ξ)/dξ σ±(s,ξ)/dξlog−1s s,(22) taking √s=7 TeV and √s=8,13 TeV. The outcome is plotted in the upper panel of Fig. 2. To first approximation, towardlargeξ1,2theeffectivescalingexponentisαeff ≈0.35 and independent of the lepton charge. In more detail, the scaling exponent is not exactly constant but some variation is visible which reflects the fact that the PDFs do not follow a pure power law, especially when xis not very small 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 10-2 10-1 1 eff p-p, +,√s=7TeV,√s=8TeV p-p, −,√s=7TeV,√s=8TeV p-p, +,√s=7TeV,√s=13TeV p-p, −,√s=7TeV,√s=13TeV p± T>25 GeV CT10NLO ξ1,2 10-2 10-1 1 10 10-5 10-4 10-3 10-2 10-1 u d s c 0.122 ×x−0.35 CT10NLO Q2=M2 W x xf(x) Fig. 2 Scaling exponent extracted from NLO calculations (upper panel) and its comparison with CT10NLO PDFs (lower panel) (at small ξ1,2). The scaling exponent for −tends to have more slope and to be somewhat larger than that of +especially at small ξ1,2, which corresponds to midrapidity. This can be explained by the slightly steeper slope of the udistribution in comparison to ddistribution (see the lower panel of Fig. 2) and the fact that −production tends to be sensitive to somewhat larger values of xin the small-xside. The latter follows from the factors (1±1−4p2 T/M2 W)2that multiply PDFs in Eq. (4). These, in turn, originate from the parity non-conserving W couplings to quarks and leptons. The lower panel in Fig. 2compares the extracted exponent αeff ≈0.35 to the CT10NLO sea-quark PDFs. Evidently, there is a good correspondence between the scaling exponent αand the behaviour of the small-xquark PDFs. We can conclude that despite the complex higher-order QCD calculations, the centre-of-mass dependence of the cross sections beingdiscussedcanbeessentiallycapturedbyasimplepower law. 123 Eur. Phys. J. C (2016) 76:214 Page 5 of 10 214 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 10-2 10-1 1 1.96TeV p-p 1.96TeV p-Pb 5.02TeV p-p 7TeV p-p 8TeV p-p 13TeV Lepton charge asymmetry p-p p± T>25 GeV CT10NLO y>0 ξ1 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 10-2 10-1 1 p-Pb 5.02TeV Pb-Pb 2.76TeV Lepton charge asymmetry p± T>25 GeV CT10NLO y<0 ξ2 Fig. 3 Lepton charge asymmetry in p–¯p(√s=1.96 TeV), p–p(√s= 1.96,7,8 TeV), p–Pb (√s=5.02 TeV) and Pb–Pb (√s=2.76 TeV) collisions, for y>0(upper panel)andy<0(lower panel) Let us now discuss Eq. (21) and whether the nature of the hadronic projectile or nucleus probed at small xreally disappears as conjectured. To this end we have computed the leptonchargeasymmetry(again,atNLOaccuracy)invarious collision systems at centre-of-mass energies that correspond to existing Tevatron and LHC data. The results are shown in Fig. 3.Aty0, the curves corresponding to p–p, p–Pb and p–p tend to unite, whereas in the opposite direction, y0, p–Pb and Pb–Pb become approximately the same. Thus, as far as theoretical NLO expectations are concerned, the scaling law of Eq. (21) turns out to be a very good approximation, though not perfect. The largest deviations in Fig. 3are seen in the case of p–p at the Tevatron energy, √s=1.96 TeV. There, the probed values of xfor p are not small enough and especially the assumption of charge-conjugation symmetric quark distributions, Eq. (17), is not particularly accurate until almost the end of phase space (e.g. ξ1=1 corresponds to x2≈M2 W/s≈0.002). The p–p curve at the same centre-ofmass energy unites with the rest already at lower ξ1. At small fixed value of ξ, the lepton charge asymmetry in p–p collisions tends to decrease toward increasing centreof-mass energies. This can be interpreted in terms of the slightly different scaling exponent for +and −production (see Fig. 2). Denoting the scaling exponent for ±production by α±, and the difference by Δ≡α−−α+,toafirst approximation, CH1,H2 (s,ξ)=CH1,H2 (s,ξ) +Δ 21−CH1,H2 (s,ξ) 2log s s+O(Δ2). (23) Since Δ>0, we have a condition CH1,H2 (s,ξ)<CH1,H2 (s,ξ), if s>s,(24) which explains the decreasing trend of lepton charge asymmetries in p–p collisions toward higher centre-of-mass energies at fixed, small ξ. 4 Data and predictions 4.1 Comparison with existing data The currently most accurate experimental measurements for inclusive W production from Tevatron and LHC experiments are summarised in Table 1. A direct comparison of various measurementsiscomplicatedbythekinematiccutsforlepton pT, missing transverse energy / ET, and transverse mass mT of the neutrino–lepton system, which vary among the experiments and have to be accounted for. Here, we have chosen to “correct” the data to pT>25 GeV (the default cut in CMS measurements)by MCFM evaluatingtheobservablesfirstwith the true cuts shown in Table 1, then with pT>25 GeV and takingtheratio(absolute crosssections)ordifference(charge asymmetry). We stress that if the kinematic cuts were the same in all experiments, this step would be unnecessary. The available absolute cross sections are compared in Fig. 4.The p–p and p–Pb data are plotted together at forward rapidity (left-hand panels) and Pb–Pb and p–Pb data together at backward rapidity (right-hand panels). In these plots, the data has been scaled by a factor (s/GeV2)−α, where a constant value α=0.4 has been used for the scaling exponent as a compromise between the expected exponent at small and large ξ;see Fig.2. Keepingin mind the “non-constantness” of thescaling exponent and that at forward (backward) direction the p–Pb (Pb–Pb) data are presumably affected by small-xshadowing in comparison to p–p (p–Pb), an exact match with p–p (p–Pb) is not expected. Nevertheless, there is clearly a rough correspondence between the data from different collision systems and different √s. 123 214 Page 6 of 10 Eur. Phys. J. C (2016) 76:214 Table 1 The experimental data sets Experiment System √s(TeV) Kinematic cuts Refs. D0 p–p1.96 pT>25 GeV, / ET>25 GeV [10] ATLAS Pb–Pb 2.76 pT>25 GeV, / ET>25 GeV, mT>40 GeV [30] CMS p–Pb 5.02 pT>25 GeV [23] ALICE p–Pb 5.02 pT>10 GeV [24] CMS p–p 7 pT>25 GeV [13] ATLAS p–p 7 pT>20 GeV, / ET>25 GeV, mT>40 GeV [55] LHCb p–p 7 pT>20 GeV [15] LHCb p–p 8 pT>20 GeV [16] CMS p–p 8 pT>25 GeV [14] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 10-2 10-1 1 ATLAS, p-p, 7TeV LHCb, p-p, 7TeV LHCb, p-p, 8TeV CMS, p-p, 8TeV CMS, p-Pb, 5.02TeV ALICE, p-Pb, 5.02TeV CT10NLO, p-Pb, 5.02TeV ξ1 s GeV2−0.40 ×ξdσ + dξ [pb] p+ T>25 GeV y>0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 10-2 10-1 1 ATLAS, Pb-Pb, 2.76TeV CMS, p-Pb, 5.02TeV ALICE, p-Pb, 5.02TeV CT10NLO, p-Pb, 5.02TeV ξ2 s GeV2−0.40 ×ξdσ + dξ [pb] p+ T>25 GeV y<0 0.0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 10-2 10-1 1 ATLAS, p-p, 7TeV LHCb, p-p, 7TeV LHCb, p-p, 8TeV CMS, p-p, 8TeV CMS, p-Pb, 5.02TeV ALICE, p-Pb, 5.02TeV CT10NLO, p-Pb, 5.02TeV ξ1 s GeV2−0.40 ×ξdσ − dξ [pb] p− T>25 GeV y>0 0.0 0.1 0.2 0.3 0.4 0.5 10-2 10-1 1 ATLAS, Pb-Pb, 2.76TeV CMS, p-Pb, 5.02TeV ALICE, p-Pb, 5.02TeV CT10NLO, p-Pb, 5.02TeV ξ2 s GeV2−0.40 ×ξdσ − dξ [pb] p− T>25 GeV y<0 Fig. 4 Absolute spectra of charged leptons (upper panels for +,lower panels for −) in p–p (√s=7,8 TeV) and p–Pb (√s=5.02 TeV) collisions for y>0(left-hand panels),andinPb–Pb( √s=2.76 TeV) and p–Pb (√s=5.02 TeV) collisions for y<0. The data has been scaled by (s/GeV2)−0.40 The data for lepton charge asymmetries Care compiled in Fig. 5. We note that some experimental uncertainties, luminosity above all, cancel in the measurement of the lepton charge asymmetries as compared to absolute cross sections. As previously, the data from p–p, p–p, and Pb–Pb collisions are plotted only in the direction where they are supposed 123 Eur. Phys. J. C (2016) 76:214 Page 7 of 10 214 -0.8 -0.6 -0.4 -0.2 0.0 0.2 10-2 10-1 1 D0 p- , 1.96TeV CMS p-p, 7TeV CMS p-p, 8TeV LHCb p-p, 7TeV LHCb p-p, 8TeV CMS p-Pb, 5.02TeV ALICE p-Pb, 5.02TeV CT10NLO p-Pb, 5.02TeV Lepton charge asymmetry p ξ1 p± T>25 GeV y>0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 10-2 10-1 1 CMS p-Pb, 5.02TeV ALICE p-Pb, 5.02TeV ATLAS Pb-Pb, 2.76TeV CT10NLO p-Pb, 5.02TeV Lepton charge asymmetry ξ2 p± T>25 GeV y<0 Fig. 5 Lepton charge asymmetry in p–¯p( √s=1.96 TeV), p–p(√s= 7,8 TeV), p–Pb (√s=5.02 TeV) and Pb–Pb (√s=2.76 TeV) collisions. The dotted curve is to guide the eye and corresponds to Cp,Pb at √s=5.02 TeV to merge with p–Pb data. To a very good approximation, the experimental data indeed line up to the same underlying curve which corresponds to the charge asymmetry in p–Pb collisions. Two CMS p–Pb data points at negative rapidities appear to lie below the NLO predictions and could potentially require additional nuclear modifications in PDFs (as also pointed out in Ref. [23]). However, the ATLAS Pb–Pb data shows no sign of such a disagreement with the theory at those values of rapidity indicating that there appears to be some tension between these two data sets and that the both data sets cannot be optimally reproduced with the same set of (nuclear) PDFs. We can also compress all the data into a single plot. This is done by choosing a certain reference centre-of-mass energy √sref (we take √sref =5.02 TeV) and plotting the data as a function of variable -0.8 -0.6 -0.4 -0.2 0.0 0.2 -4 -3 -2 -1 0 1 2 3 4 D0 p- , 1.96TeV CMS p-p, 7TeV CMS p-p, 8TeV LHCb p-p, 7TeV LHCb p-p, 8TeV CMS p-Pb, 5.02TeV ALICE p-Pb, 5.02TeV ATLAS Pb-Pb, 2.76TeV CT10NLO p-Pb, 5.02TeV Lepton charge asymmetry p yref(√sref =5.02 TeV) p± T>25 GeV Fig. 6 The world data on lepton charge asymmetry as a function of yref taking √sref =5.02 TeV yref ≡y±1 2log sref s,y≷0,(25) such that ξ1(y,√s)=ξ1(yref,√sref), y>0,(26) ξ2(y,√s)=ξ2(yref,√sref), y<0. Such a plot is shown in Fig. 6. In order to keep the plot readable Pb–Pb data is plotted only at y<0, and p–p, p–p data is plotted only at y>0. 4.2 Cross-section ratios In Sect. 2.1 we noted that the ratios of cross sections at two nearby √sat fixed values of scaling variable ξ1,2could become less prone to large-xPDF uncertainties in comparison to taking the ratios at fixed rapidity. To investigate this statement quantitatively, we have computed (p–p collisions, NLO precision) ratios R+ √s/√s(yref)=dσ+(√s)/dyref dσ+(√s)/dyref ≈√s √s2α ,(27) R− √s/√s(yref)=dσ−(√s)/dyref dσ−(√s)/dyref ≈√s √s2α ,(28) R√s/√s(yref)= R+ √s/√s(yref) R− √s/√s(yref)≈1,(29) where the predictions from scaling laws are also indicated. For comparison we evaluate the same ratios also at fixed rapidity (instead of fixed yref). We have used PDF4LHC15_30NLO setofPDFs[56] available from the LHAPDF library [57]. This is a hybrid set that combines [58] information from independent PDF fits (CT14 [59], 123 214 Page 8 of 10 Eur. Phys. J. C (2016) 76:214 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 p-p, fixed y p-p, fixed yref 0.99 0.995 1.0 1.005 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Relative uncert. (8TeV/7TeV)2×0.35 y,yref( √ sref =7TeV) dσW+(8TeV)/dσW+(7TeV) p± T>25 GeV PDF4LHC15 30 1.1 1.2 1.3 1.4 1.5 1.6 p-p, fixed y p-p, fixed yref 0.99 0.995 1.0 1.005 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Relative uncert. (8TeV/7TeV)2×0.35 y,yref( √ sref =7TeV) dσW−(8TeV)/dσW−(7TeV) p± T>25 GeV PDF4LHC15 30 0.95 1.0 1.05 1.1 1.15 1.2 1.25 1.3 p-p, fixed y p-p, fixed yref 0.99 0.995 1.0 1.005 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Double ratio Relative uncert. y,yref( √ sref =7TeV) p± T>25 GeV PDF4LHC15 30 √s=8TeV,√s=7TeV Fig. 7 Ratios of +(left)and−(middle) spectra computed at √s= 8 TeV and √s=7 TeV centre-of-mass energies. In red color are the results binned in lepton rapidity y,andingreen the results binned in yref taking √sref =7TeV.The dashed lines indicate the prediction of scaling law Eq. (12). The right-hand panel shows the double ratio of Eq. (27) 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 p-p, fixed y p-p, fixed yref 0.98 1.0 1.02 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Relative uncert. (13TeV/8TeV)2×0.35 y,yref(√sref =8TeV) dσW+(13TeV)/dσW+(8TeV) p± T>25 GeV PDF4LHC15 30 1.5 2.0 2.5 3.0 3.5 p-p, fixed y p-p, fixed yref 0.98 1.0 1.02 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Relative uncert. (13TeV/8TeV)2×0.35 y,yref(√sref =8TeV) dσW−(13TeV)/dσW−(8TeV) p± T>25 GeV PDF4LHC15 30 1.0 1.2 1.4 1.6 1.8 2.0 p-p, fixed y p-p, fixed yref 0.98 1.0 1.02 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Double ratioRelative uncert. y,yref(√sref =8TeV) p± T>25 GeV PDF4LHC15 30 √s= 13TeV,√s=8TeV Fig. 8 As Fig. 7but using √s=13 TeV and √s=8TeV MMHT14 [60], NNPDF3.0 [61]) thereby giving a better idea of the uncertainties than when sticking to a one particular PDF provider. The results are shown in Fig. 7(√s=8TeV,√s= 7 TeV), Fig. 8(√s=13 TeV, √s=8 TeV), and Fig. 9 (√s=14 TeV, √s=13 TeV). The histograms in red indicate the outcome when the ratios are taken at fixed rapidity intervals and the green ones correspond to making the ratios at fixed yref (equivalent to fixed ξ1). One can observe that in the case of W−production and the double ratio the PDF uncertainties indeed tend to cancel out better when the ratios are taken at fixed yref.ForW +production it appears that there is no definite advantage (in the sense that PDF uncertainties would decrease) in binning as a function of yref.We attribute this to the fact that in the case of W+, the integrand (in Eq. (4)) in xis broader for W+production than what it is for W−production and the PDF uncertainties do not cancel as effectively. The LHCb collaboration has recently reported [16] ratios similar to ones discussed here (though integrated over the rapidity interval 2 <y<4.5), and has observed some deviations between the measurements and NLO calculations. Our results suggest that by making the rapidity intervals equal in yref, the PDF uncertainties especially in the double ratio can be suppressed and the significance of the measurement thereby increased.3 5 Summary We havediscussedthescalingproperties of inclusivecharged leptons from decays of W bosons created in hadronic 3An even better precision could be attained by considering the ratios of total cross sections [62], which, however, are more difficult to measure for the finite acceptance of the experimental apparatuses. 123