CERN-TH-2019-139, DESY 19-169, PSI-PR-19-21

Parton Density Uncertainties and the Determination of Electroweak Parameters atHadron Colliders

Emanuele Bagnaschi1, ∗ and Alessandro Vicini2, †

1Paul Scherrer Institut, CH-5232 Villigen, Switzerland2TH Department, CERN 1 Esplanade des Particules, Geneva 23, CH-1211, Switzerland and

Dipartimento di Fisica “Aldo Pontremoli”, University of Milanoand INFN Sezione di Milano, Via Celoria 16, 20133 Milano, Italy

(Dated: February 8, 2021)

We discuss the determination of electroweak parameters from hadron collider observables, focusingon the W boson mass measurement. We revise the procedures adopted in the literature to includein the experimental analysis the uncertainty due to our imperfect knowledge of the proton structure.We show how the treatment of the proton parton density functions (PDFs) uncertainty as a sourceof systematic error, leads to the automatic inclusion in the fit of the bin-bin correlation of thekinematic distributions with respect to PDF variations. In the case of the determination of MW

from the charged lepton transverse momentum distribution, we observe that the inclusion of thiscorrelation factor yields a strong reduction of the PDF uncertainty, given a sufficiently good controlover all the other error sources. This improvement depends on a systematic accounting of thefeatures of the QCD-based PDF model, and it is achieved relying only on the information availablein current PDF sets. While a realistic quantitative estimate requires to take into account the detailsof the experimental systematics, we argue that, in perspective, the proton PDF uncertainty will notbe a bottleneck for precision measurements.

INTRODUCTION

The values of the W boson mass mW and of the sinusof the leptonic effective weak mixing angle sin2 θ`eff arevery precise predictions in the electroweak (EW) sectorof the Standard Model (SM) and allow stringent testsat the level of the quantum corrections. The measure-ments of these two parameters at the Tevatron and atLHC indicate [1–6] the imperfect knowledge of the pro-ton structure as one of the main sources of systematicuncertainty of theoretical origin. The latter affects thecomputation of the templates used in the fit of the kine-matic distributions and eventually the determination ofthe EW parameters.

The proton collinear parton distribution functions(PDFs) suffer from different uncertainties of experimen-tal as well as theoretical origin. The impact of the er-ror of the data from which the PDFs are extracted isrepresented by sets of functions, Hessian eigenvectors orMonte Carlo replicas, that span in a statistically signifi-cant way the functional space of all possible parameter-isations. The propagation of the experimental error inthe prediction of any observable is achieved by simplyrepeating the evaluation of the latter several times, withall the available members of the PDF set; mean and stan-dard deviation are eventually computed, the latter beingthe propagation of the experimental error to the observ-able under study. All the members of a PDF set sharesome common theoretical features, like the fact that theyall obey the perturbative QCD (PQCD) evolution equa-tions and sum rules, and are thus correlated with eachother. While this correlation is automatically includedin the propagation of the experimental PDF error to the

prediction of any observable, the determination of a pa-rameter extracted from the simultaneous fit of severalobservables requires a careful discussion.

In the case of the W boson mass determination, therole of the PDFs has been discussed in the articles pre-senting the experimental results, for those PDF sets usedin the analyses, while a more general comparison of differ-ent parameterisations has been presented in Refs. [7, 8];in all cases the common outcome is that an uncertainty∆PDFmW at the 10 (20) MeV level is expected in thelepton-pair transverse mass (lepton transverse momen-tum) case, with the precise value depending on severaldetails of the analyses and on which parameterisationsare included in the study. All these studies consideredthe fit of a kinematic distribution, by combining the in-formation of different bins weighed by their statisticaland systematic errors, but neglecting any bin-bin corre-lation with respect to PDF variations. In Ref. [9] thedependence of the uncertainty on the rapidity range in-cluded in the acceptance region was exploited to quantifythe benefit given by an mW measurement at LHCb to thefinal combination of all the available results, in terms ofa reduced PDF uncertainty. The possibility of a sys-tematic extraction of very precise information about theDrell-Yan parton-parton luminosities has been studied inRefs. [10–14], aiming at a better modelling of the initialstate and to a consequent reduction of the PDF error onmW . The impact of measurements at different collidersand energies has been scrutinised in Ref. [15].

We plan to revise the propagation of the PDF uncer-tainties of experimental origin, in the determination of aparameter obtained via the fit of a kinematic observable.

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BIN-BIN CORRELATION AND TEMPLATE FITDEFINITIONS

At hadron colliders mW is determined in the charged-current (CC) Drell-Yan (DY) process, from the measure-ment of observables such as the charged-lepton transversemomentum dσ/dp`⊥ and the lepton-pair transverse massdistributions, for which the Jacobian peak enhances thesensitivity to the position of the pole of the W propa-gator. The finite rapidity detector acceptance and otherkinematical constraints induce a sensitivity of the shapeof these observables, defined in the transverse plane, tothe initial state proton collinear PDFs.

The PDF uncertainty, represented by a set of repli-cas with Nrep members, affects the normalisation butalso the shape of the observables. Different bins of thesame distribution are correlated with respect to a PDFreplica variation, as it can be seen in Fig. 1, because ofkinematic constraints and due to the theoretical frame-work shared by all the replicas. In Fig. 1 the sudden and

25 1000.0001 0.1

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plT [GeV]

x1

x1

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

ρij

FIG. 1. Correlation with respect to PDF replica variationsof the differential cross sections with respect to the chargedlepton transverse momentum and to the partonic x1 variable.Results obtained with normalised distributions. We show theself correlation of dσ/dx1 (lower right), of dσ/dp`⊥ (upper left)and the cross correlation of the two distributions. Fluctua-tions due to finite MC statistics are visible as a stripe-likepattern in the plots.

strong change of sign of the dσ/dp`⊥ self-correlation isquite evident across the Jacobian peak at p`⊥ ∼ 40 GeV;the self-correlation of the dσ/dx1 distribution also signalsthe existence of two partonic x ranges, below and abovex ∼ 4 · 10−3. The cross correlations thus establish a linkbetween the parton-parton luminosities, i.e. the source ofthe PDF uncertainty, and the dσ/dp`⊥ distribution, fromwhich mW is determined, with a non-trivial underlyingcorrelation pattern.

The determination of a Lagrangian parameter froma kinematic distribution via a template fit requires thechoice of a Lagrangian density (in our case the SM one)

and of a tool that simulates the observables computed inthat model in a well defined setup. The simulation toolis fully specified by the choice of a proton PDF param-eterisation, while the parameter, e.g. mW , is left free tovary when comparing to the experimental data. In thisconstruction the PDF replicas represent a one parameterfamily of models to analyse the data.

The equivalence of the replicas in the proton descrip-tion represents a source of theoretical systematic error,when we try to determine mW from the fit of a kine-matic distribution. We account for this systematics inthe following χ2 definition:

χ2k =

∑i∈bins

((T0,k)i − (Dexp)i −

∑r∈R αr(Sr,k)i

)2σ2i

+∑r∈R

α2r

(1)where, in the bin i of the distribution, we have the fol-lowing quantities: T0,k is our fitting model based on theaverage replica 0 of the PDF set R and it has been com-puted with the k-th W -boson mass hypothesis mW,k;Dexp is the experimental value and σ2

i is its statisticalerror; the differences Sr,k = Tr,k − T0,k are computed foreach member r of the PDF set and are treated as nui-sances with fit parameters αr. The quadratic penaltyfactor

∑r∈R α

2r corresponds to having assumed a Gaus-

sian penalty for the replicas with respect to the centralreplica of the set. Since the templates are in generalaffected by statistical Monte Carlo and experimental er-rors, we should take that into account by consideringσ2i = (σexp,stat

i )2 + (σMCi )2.

By repeating the minimisation of χ2k, with respect to

the αr, for different values of mW,k, the minimum of thesequence labelled by k selects the preferred mW valueand the ∆χ2 = 1, 4, 9 rules identify the 1,2,3 standarddeviations intervals due to the PDF uncertainty. For agiven mW,k, the minimum of the χ2 expression in Eq. 1can be written [16] with the bin-bin covariance matrixcomputed with respect to PDF variations and includingthe statistical and systematic error contributions [17].

χ2k,min=

∑(r,s)∈bins

(T0,k −Dexp)r(C−1

)rs

(T0,k −Dexp)s (2)

C = ΣPDF + Σstat + ΣMC + Σexp,syst (3)

(ΣPDF )rs=

〈(T − 〈T 〉PDF )r(T − 〈T 〉PDF )s〉PDF (4)

〈O〉PDF ≡1

Ncov

Ncov∑l=1

O(l) (5)

where Σstat is a diagonal matrix with the statistical vari-ances on each bin of the distribution, estimated for agiven integrated luminosity L, ΣMC is the diagonal ma-trix of the squared Monte Carlo error of the templatesand Ncov is the number of PDF replicas used to com-pute the PDF covariance matrix [18]. We introduce inthe full covariance matrix an additional term Σexp,syst

3

to account for experimental systematics, although theirfaithful description depends on the details of each exper-iment. In Eq. 3 we approximate Σexp,syst by using thedetector model of the Compact Muon Solenoid (CMS)presented in Ref. [19]. We stress that in this note all thereplicas are treated as equivalent, i.e. we do not antici-pate the impact that future measurements may have inreducing the PDF uncertainty. The approach that weare proposing to include the PDF uncertainty on an EWparameter has to be compared with what has been usedin the past, e.g. in Refs. [7, 8], where the analysis reliedon the minimisation of a χ2 defined as

χ2k,r,no−cov =

∑i∈bins

(T0,k −Dr)2i /σ

2i (6)

treating the contributions of different bins as indepen-dent and weighing them with their statistical error; thetemplates were generated with the central PDF replica 0,for different mass hypotheses k; the distributions, com-puted with Nrep different replicas, were treated as inde-pendent pseudodata and the minimisation was repeatedseparately for each of them; the resulting Nrep preferredmW,r values were eventually analysed computing meanvalue and standard deviation and ignoring the associatedvalues of χ2

k,r,no−cov; the standard deviation was taken as

the estimate of the PDF uncertainty. A similar χ2 def-inition, including only diagonal contributions, has beenused up to now by the experimental collaborations atTevatron and LHC.

NUMERICAL RESULTS

We perform all the simulations using the CC-DY eventgenerator provided in the POWHEG-BOX [20, 21], show-ered with Pythia8.2 [22], setting

√S = 13 TeV. We

restrict ourselves to W+ production without hinderingthe generality of our arguments. We apply the accep-tance cut |ηl| < 2.5. We use for our analysis the PDF setNNPDF30 nlo as 0118 1000 [23], featuring Nrep = 1000replicas.

In Eq. 2, the templates are computed using the replica0 of the PDF set, scanning mW with a 1 MeV spacingin the interval mW ∈ [80.035, 80.735] GeV. We let thedistribution computed with the central replica 0 of thePDF set, and with a fixed mW,0 = 80.385 GeV value,play the role of the experimental data Dexp; this choicedoes not spoil the validity of the method and of the con-clusion and offers a sanity check on the fit results. Thecovariance matrix is evaluated with the Nrep replicas. Wechecked that the dependence of the covariance matrix onmW , in the interesting range of ±20 MeV around thecentral value of 80.385 GeV; is small and therefore weneglected it in the numerical analysis. The statistical er-ror on the pseudodata is estimated assuming 2 differentluminosities, 1, and 300 fb−1.

30 32 34 36 38pmin⊥ [GeV]

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0.020

Bozzi et al.

ATLAS 7 TeV

0.000

0.005

0.010

0.015

0.020

0.025

±∆mW

[GeV

]

FIG. 2. PDF error estimated by computing the standard de-viation of the mW values corresponding to the minima of thefit of 200 replicas onto the template pseudodata representedby the central replica.

Since the value of the PDF uncertainty affectingthe mW determination is sensitive to the fit window[pmin⊥ , pmax

⊥ ], we perform a scan in the two values

pmin,max⊥ and plot, for each point in this plane, the un-

certainty value corresponding to the half-width of the∆χ2 = 1 interval.

To present a comparison with the previous approaches,we perform an analysis using the prescription of Eq. 6,using 200 replicas, this time with a fixed mW,0 = 80.385GeV value, as distinct pseudodata distributions; we gen-erate the templates with the replica 0. In Fig. 2 we showthe analysis of distributions normalised to the cross sec-tion integrated in the fitting interval. The results, con-sistent with those presented in Ref. [8] and labeled in allthe Figures by ”Bozzi et al., show a weak sensitivity tothe upper limit of the fit window, but a clear dependenceon its lower limit.

In Figs. 3, 4, we present the results based on Eq. 2,in the case of normalised distributions, assuming an ex-perimental integrated luminosity Lint respectively equalto 1, and 300 fb−1, and no template Monte Carlo error.Fig. 5 also corresponds to 300 fb−1, but we now includea Monte Carlo error extrapolated to a statistics of 1010

events [24]. The statistical error is dominant in Fig. 3,while it becomes negligible at high luminosity, puttingin evidence a strong reduction of the PDF uncertainty,down to the O(1 MeV) level. The Monte Carlo error ofthe templates has a visible impact, as shown in Fig. 5,and would become negligible in a sample with 200 bilionsevents. We remark the weak sensitivity of the results tothe fit window. We eventually show in Fig. 6 the resultscorresponding to 300 fb−1 and a systematic error on themuon momentum reconstruction simulated via the modelof Ref. [19]. The covariance matrix used in Fig. 5 is addedto the one coming from the detector effects, estimated us-ing 100 toys. The negative impact in the description of

4

30 32 34 36 38 40pmin⊥ [GeV]

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80pm

ax

⊥[G

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Lint = 1 fb−10.014

0.015

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ATLAS 7 TeV

0.000

0.005

0.010

0.015

0.020

0.025

±∆mW| ∆χ

2=

1[G

eV]

FIG. 3. PDF error as a function of the fit window expressedby its minimum and maximum p`⊥ values. Error estimatedfrom a fit of shape distributions and using a covariance matrixobtained by summing the PDF one with a statistical (diago-nal) error on the pseudodata corresponding to 1 fb−1.

30 32 34 36 38 40pmin⊥ [GeV]

40

50

60

70

80

pmax

⊥[G

eV]

Lint = 300 fb−1 0.002

0.003

0.0050.006

0.0080.010

Bozzi et al.

ATLAS 7 TeV

0.000

0.005

0.010

0.015

0.020

0.025

±∆mW| ∆χ

2=

1[G

eV]

FIG. 4. Same as in Fig. 3, but assuming Lint = 300 fb−1.

30 32 34 36 38 40pmin⊥ [GeV]

40

50

60

70

80

pmax

⊥[G

eV]

Lint = 300 fb−1 0.005

0.006

0.0080.010

0.02

0

Bozzi et al.

ATLAS 7 TeV

0.000

0.005

0.010

0.015

0.020

0.025

±∆mW| ∆χ

2=

1[G

eV]

FIG. 5. Same as in Fig. 4, but including also a Monte Carloerror on the templates corresponding to 1010 events.

30 32 34 36 38 40pmin⊥ [GeV]

40

50

60

70

80

pmax

⊥[G

eV]

Lint = 300 fb−10.002

0.003

0.00

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0.00

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0.00

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0

0.02

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Bozzi et al.

ATLAS 7 TeV

0.000

0.005

0.010

0.015

0.020

0.025

±∆mW| ∆χ

2=

1[G

eV]

FIG. 6. Same as in Fig. 4, but including also detector effectsmodelled according to Ref. [19].

the peak region balances the improved control on the tailsof the distribution, increasing the size of the total error.We have checked that a reduction by a factor of 10 of thegaussian smearing of the lepton momentum, would leadto uncertainties close to the ones shown in Fig. 4. Othersources of theoretical systematics, such as perturbativeQCD or parton shower uncertainties, could become oneof the limiting factor for the mW determination, and willbe considered in a future publication.

We observe that this approach strongly reduces the im-pact of PDF uncertainties because of the specific struc-ture of the bin-bin PDF covariance matrix ΣPDF of thep`⊥ distribution, with the presence of quite distinct blocksformed by the bins below and above the Jacobian peak[25]. The eigenvalues spectrum of ΣPDF covers morethan 7 orders of magnitude between the largest and thesmallest elements in absolute value. The broad rangeof the eigenvalues induces a very narrow shape of theχ2 distribution as a function of mW , implying a strongpenalty factor for all the templates that do not perfectlyoverlay their peak position with the one of the data. Thepenalty applies to the differences in the tails of the p`⊥distribution, while, at the same time, an excellent sen-sitivity to mW , at the 1 MeV level given by the tem-plates granularity, is preserved, as we explicitly verifiedas a sanity check of the approach. The important roleplayed by ΣPDF is partially smeared by the interplay be-tween PDF and statistical and systematic errors. SinceC = ΣPDF + Σstat + ΣMC + Σexp,syst, at low luminosi-ties or low template accuracy the statistical error has anon-trivial interplay with the PDF error, yielding largeruncertainties than the values obtained for each class oferrors alone; at high-luminosities, with highly-accuratetemplates, instead we approach the limit C ' ΣPDF andthe corresponding strong uncertainty reduction. Simi-lar comments apply to the inclusion of the experimentalsystematic errors.

5

The PDF uncertainty band of the p`⊥ distributionis given by a combination of perturbative and non-perturbative effects, which can not be analytically sepa-rated; although the pQCD elements (Dokshitzer-Gribov-Lipatov-Altarelli-Parisi equations, QCD sum rules) in theproton description can not be qualified as uncertaintysources, they nevertheless enter in the generation of theuncertainty band, because of their entanglement with thedata. The covariance matrix allows the effective encod-ing of a substantial piece of information of pQCD origin,which should not be qualified as uncertainty, and includesit in the fit. The description of the proton in terms of aQCD-inspired model and the representation of the uncer-tainty via Monte Carlo replicas are thus the two elementsallowing the PDF uncertainty reduction. The discussionof the PDF sets representing the associated uncertaintyvia Hessian eigenvectors will be presented in a future pub-lication.

In conclusion, we have studied the theoretical system-atic error due to the PDF uncertainty, focusing on thedetermination of the W boson mass from the DY dσ/dp`⊥distribution. We included this systematics in the χ2 thatwe use in the data fitting, achieving the automatic in-clusion of the bin-bin correlation with respect to PDFvariations. We observe a drastic reduction of the PDFuncertainty on mW , which we explain as a consequenceof the strong kinematic correlation, of pQCD origin, ofthe bins above and below the Jacobian peak of the dis-tribution. The interplay of the PDF with the statisticaland experimental systematic errors yields non trivial re-sults, when the statistics is limited and systematics notfully understood. We consider this approach promisingin view of the reduction of one of the bottlenecks limitingso far the high-precision determination of mW at hadroncolliders. The formulation of Eq. 2 is well suited for adirect and efficient inclusion of the PDF uncertainty inthe analysis of the experimental data. The use of thisinformation should not be limited to the fit of mW , butit should also be part of the determination of any La-grangian parameter derived in the analysis of LHC ob-servables. The inclusion of further observables sensitiveto the QCD model in a global fit, such as pZ⊥, and ofthe corresponding cross-correlations, might provide ad-ditional benefits to the mW determination. However toproperly assess the impact of such a procedure, a detailedstudy beyond the scope of this article is needed.

ACKNOWLEDGEMENTS

E.B. is supported by the Paul Scherrer Institut. Wethank L. Bianchini, S. Camarda, E. Manca, M. Mangano,G. Rolandi and L. Silvestrini for useful discussions. Wealso thank the DESY IT department for making us avail-able the computational resources of the BIRD/NAF2cluster, which have been extensively used for this work.

A.V. is supported by the European Research Council un-der the European Unions Horizon 2020 research and in-novation Programme (grant agreement number 740006).

∗ emanuele.bagnaschi@psi.ch† alessandro.vicini@mi.infn.it

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[24] Our simulations were performed with only 15 millions

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