Non-perturbative uncertainties on the transverse momentum distribution of electroweak bosons and on the determination of the W boson mass at the LHC
NNon-perturbative uncertainties on the transverse momentum distribution ofelectroweak bosons and on the determination of the W boson mass at the LHC Giuseppe Bozzi
1, 2, ∗ and Andrea Signori † Dipartimento di Fisica, Universit`a di Pavia, via Bassi 6, I-27100 Pavia, Italy INFN, Sezione di Pavia, via Bassi 6, I-27100 Pavia, Italy Physics Division, Argonne National Laboratory9700 S. Cass Avenue, Lemont, IL 60439 USA (Dated: Monday 7 th January, 2019)In this contribution we present an overview of recent results concerning the impact of a possi-ble flavour dependence of the intrinsic quark transverse momentum on electroweak observables. Inparticular, we focus on the q T spectrum of electroweak gauge bosons produced in proton-protoncollisions at the LHC and on the direct determination of the W boson mass. We show that theseeffects are comparable in size to other non-perturbative effects commonly included in phenomeno-logical analyses, and should thus be included in precise theoretical predictions for present and futurehadron colliders. Contents
I. Introduction II. Formalism III. Effects on the q T spectrum of the W IV. Impact on the determination of the W boson mass V. Outlook and future developments A. Conventions for nonperturbative parameters Acknowledgments References ∗ [email protected] - ORCID: 0000-0002-2908-6077 † [email protected] - ORCID: 0000-0001-6640-9659 a r X i v : . [ h e p - ph ] J a n I. INTRODUCTION
Electroweak precision observables are interesting benchmarks to test the limits of the Standard Model and todiscriminate between different scenarios for new physics. The mass of the W boson, m W , is an example of such anobservable.The Standard Model prediction for the W boson mass from the global fit of the electroweak parameters ( m W =80 . ± .
008 GeV) [1] has a very small uncertainty that represents a natural target for the precision of theexperimental measurements of m W at hadron colliders.Direct measurements of m W at hadronic colliders have been performed at the Tevatron p ¯ p collider with the D0 [2]and CDF [3] experiments, and at the LHC pp collider with the ATLAS [4] experiment, with a total uncertainty of23 MeV, 19 MeV and 19 MeV, respectively. The current world average, based on these measurements and theones performed at LEP, is m W = 80 . ± .
012 GeV [5]. Fig. 1 presents an overview of these measurementscompared to the electroweak global fit. The CPT theorem [6, 7] implies that the mass and lifetime of a particleand its anti-particle are the same. The
ATLAS measurement of the W + and W − mass difference yields: m W + − m W − = − ±
28 MeV [4]. The experimental determinations are based on a template-fit procedure applied to
Page 44 of 61 Eur. Phys. J. C (2018) 78 :110
Table 13
Results of the m W + − m W − measurements in the electronand muon decay channels, and of the combination. The table showsthe statistical uncertainties; the experimental uncertainties, divided intomuon-, electron-, recoil- and background-uncertainties; and the mod- elling uncertainties, separately for QCD modelling including scale vari-ations, parton shower and angular coefficients, electroweak corrections,and PDFs. All uncertainties are given in MeVChannel m W + − m W − [MeV] Stat. Unc. Muon Unc. Elec. Unc. Recoil Unc. Bckg. Unc. QCD Unc. EW Unc. PDF Unc. Total Unc. W → e ν − . W → µ ν − . − . [MeV] W m ALEPHDELPHIL3OPALCDFD0 + ATLAS W − ATLAS W ± ATLAS W
ATLAS
MeasurementStat. UncertaintyFull Uncertainty
Fig. 28
The measured value of m W is compared to other publishedresults, including measurements from the LEP experiments ALEPH,DELPHI, L3 and OPAL [25–28], and from the Tevatron collider exper-iments CDF and D0 [22,23]. The vertical bands show the statisticaland total uncertainties of the ATLAS measurement, and the horizontalbands and lines show the statistical and total uncertainties of the otherpublished results. Measured values of m W for positively and negativelycharged W bosons are also shown In this process, uncertainties that are anti-correlatedbetween W + and W − and largely cancel for the m W measure-ment become dominant when measuring m W + − m W − . On thephysics-modelling side, the fixed-order PDF uncertainty andthe parton shower PDF uncertainty give the largest contribu-tions, while other sources of uncertainty only weakly dependon charge and tend to cancel. Among the sources of uncer-tainty related to lepton calibration, the track sagitta correc-tion dominates in the muon channel, whereas several residualuncertainties contribute in the electron channel. Most lep-ton and recoil calibration uncertainties tend to cancel. Back-ground systematic uncertainties contribute as the Z and mul-tijet background fractions differ in the W + and W − channels.The dominant statistical uncertainties arise from the size ofthe data and Monte Carlo signal samples, and of the controlsamples used to derive the multijet background.The m W + − m W − measurement results are shown inTable 13 for the electron and muon decay channels, and forthe combination. The electron channel measurement com-bines six categories ( p ℓ T and m T fits in three | η ℓ | bins), while [MeV] W m LEP Comb.
33 MeV ± Tevatron Comb.
16 MeV ± LEP+Tevatron
15 MeV ± ATLAS
19 MeV ± Electroweak Fit ± W mStat. UncertaintyFull Uncertainty ATLAS
Fig. 29
The present measurement of m W is compared to the SM pre-diction from the global electroweak fit [16] updated using recent mea-surements of the top-quark and Higgs-boson masses, m t = . ± .
70 GeV [122] and m H = . ± .
24 GeV [123], and to the com-bined values of m W measured at LEP [124] and at the Tevatron col-lider [24] the muon channel has four | η ℓ | bins and eight categories intotal. The fully combined result is m W + − m W − = − . ± . ( stat. ) ± . ( exp. syst. ) ± . ( mod. syst. ) MeV = − . ± . , where the first uncertainty is statistical, the second corre-sponds to the experimental systematic uncertainty, and thethird to the physics-modelling systematic uncertainty.
12 Discussion and conclusions
This paper reports a measurement of the W -boson mass withthe ATLAS detector, obtained through template fits to thekinematic properties of decay leptons in the electron andmuon decay channels. The measurement is based on proton–proton collision data recorded in 2011 at a centre-of-massenergy of √ s = − . The measurement relies FIG. 1: Overview of the measurements of the W boson mass. The indirect determination via the electroweak fit sets theprecision for the measurements via direct determinations. Figure from Ref. [4]. differential distributions of the W decay products: in particular, the transverse momentum of the final lepton, p (cid:96) T ,the transverse momentum of the neutrino p ν T (only at the Tevatron), and the transverse mass m T of the lepton pair(where m T = (cid:112) p (cid:96) T p ν T (1 − cos( φ (cid:96) − φ ν )), with φ (cid:96),ν being the azimuthal angles of the lepton and the neutrino,respectively). The transverse momentum of the lepton pair, though not directly used in the template-fit procedure,is relevant for reweighing purposes (see, for instance, Sec. 6 of Ref. [4]).At leading order the W boson is produced with zero transverse momentum ( q W T ), but perturbative and non-perturbative corrections give rise to non-vanishing values of q W T . While perturbative and flavour-independentnon-perturbative corrections have received much attention and reached a high level of accuracy (see, for instance,Ref. [8, 9] and references therein), a possible flavour dependence of the intrinsic transverse momentum ( k T ) of theinitial state partons has been less investigated.In Fig. 2 we examine the decomposition in flavour channels of the cross section for Z and W ± productiondifferential with respect to q V T , V = Z, W ± . A non-trivial interplay among the different flavours and the gluonis observed. The role of the gluon becomes increasingly important at larger values of the transverse momentum.In the region of the peak, instead, the dominant channels involve combinations of u val , d val , ¯ u and ¯ d (where a = a val + a sea and ¯ a = a sea ). For this reason, we consider important to study the impact of flavour-dependenteffects on the production of electroweak bosons and on the determination of m W .In this contribution we give an overview of selected studies related to flavour-dependent effects, focusing inparticular on the results obtained in Ref. [11, 12], showing that they can be non-negligible compared to othersources of theoretical uncertainty and should thus be included in precision physics programs at hadron colliders. FIG. 2: From top to bottom: the decomposition in flavour channels of the cross section for Z , W + , W − productiondifferential with respect to the transverse momentum of the produced electroweak boson q V T , V = Z, W + , W − . The crosssection is calculated by means of CuTe [10] at LHC √ s = 8 TeV. The non-perturbative correction is implemented as aflavour-independent Gaussian smearing, governed by the parameter Λ NP (see Ref. [10] and App. A). The channels add toone. II. FORMALISM
In processes with a hard scale Q and a measured transverse momentum q T , for instance the mass and thetransverse momentum of an electroweak boson produced in hadronic collisions, we can distinguish three regions:a small q T region ( q T (cid:28) Q ), where large logarithms of q T /Q have to be properly resummed; a large q T region( q T (cid:38) Q ), where fixed-order perturbation theory provides reliable results; and an intermediate region, where aproper matching procedure between all-order resummed and fixed-order contributions is necessary. For a concisediscussion (and for additional relevant references) on the development of the different frameworks available to resumthe logs of q T /Q and on their matching to fixed-order perturbative calculations, we refer the reader to Ref. [13–17].In the Transverse-Momentum-Dependent (TMD) factorisation framework [18], the unpolarized TMD PartonDistribution Function (TMD PDF) for a parton with flavour a , carrying a fraction x of longitudinal momentum ata certain scale Q , can be written in b T -space (where b T is the variable Fourier-conjugated to the partonic transversemomentum k T ) as : (cid:101) f a ( x, b T ; Q ) = (cid:88) i = q, ¯ q,g (cid:0) C a/i ⊗ f i (cid:1) ( x, b T , µ b ) e S ( µ b ,Q ) e g K ( b T , λ ) ln( Q /Q ) (cid:101) f a NP ( b T , λ (cid:48) ) , (1)where µ b is the b T -dependent scale at which the collinear parton distribution functions are computed and Q isa hadronic mass scale. Eq. (1) is a generic schematic implementation of the perturbative and non-perturbativecomponents of a renormalized TMD PDF. Depending on the chosen perturbative accuracy, S includes the UV-anomalous dimension of the TMD PDF and the Collins-Soper kernel. Also, in principle the TMD PDF dependson two kinds of renormalization scales, related to the renormalization of UV and light-cone divergences. Here wespecify their initial and final values as µ b and Q respectively. Moreover, the perturbative scales can be chosen inposition or momentum space [10, 19–23]. For the implementation of all these details, we refer the reader to thedescription of the public codes that we are going to discuss.The C coefficients in Eq. (1), also called Wilson coefficients for the TMD distribution, are calculable in pertur-bation theory and are presently known at order α s in the unpolarized case [16, 24]. They are convoluted with thecorresponding collinear parton distribution functions f i according to (cid:0) C a/i ⊗ f i (cid:1) ( x, b T , µ b ) = (cid:90) x duu C a/i (cid:16) xu , b T , α s (cid:0) µ b (cid:1)(cid:17) f i ( u ; µ b ) , (2)The perturbative part of the evolution, the S factor in Eq. (1), can be written as: S ( µ b , Q ) = (cid:90) Q µ b dµ µ γ F [ α s ( µ ) , Q /µ ] − K ( b T ; µ b ) log Q µ b . (3)It involves, in principle, the UV-anomalous dimension γ F and the Collins-Soper kernel K , which can be decomposedas: γ F [ α s ( µ ) , Q /µ ] = − (cid:20) ∞ (cid:88) k =1 A k (cid:18) α s ( µ )4 π (cid:19) k (cid:21) ln (cid:18) Q µ (cid:19) + ∞ (cid:88) k =1 B k (cid:18) α s ( µ )4 π (cid:19) k , K ( b T , µ b ) = ∞ (cid:88) k =1 d k (cid:18) α s ( µ )4 π (cid:19) k . (4)The A k and B k coefficients are known up to NNNLL (at least, their numerical value) and the integration ofthe Sudakov exponent in Eq. (4) can be done analytically up to NNNLL (for the complete expressions see, e.g.,Ref. [25–27]). The perturbative coefficients of the kernel K are also known analytically up to NNNLL.A well known problem in the implementation of the QCD evolution of transverse-momentum-dependent distribu-tions (TMDs) is the divergent behaviour at large b T caused by the QCD Landau pole. Two common prescriptionsto deal with this divergence consist in replacing b T with a variable that saturates at a certain b T max , as suggestedby the CSS formalism [18, 28], or perform the b T integration on the complex plane in such a way that the Lan-dau pole is never reached [29]. On the other hand, also the small b T region needs to be regularized, in order toeliminate unjustified contributions from the evolution of TMDs in the intermediate and large q T regions and torecover the expression for the cross section in collinear factorisation upon integration over q T . Several prescriptionsexist [15, 22, 25, 30, 31] also in this case.Two intrinsically non-perturbative factors are introduced in Eq. (1) in order to account for the large b T behavior.The first one is named g K ( b T ; λ ) in the TMD/CSS literature [18]. It embodies the flavour-independent non-perturbative part of the evolution. The second one, (cid:101) f a NP ( b T ; λ (cid:48) ), accounts for a kinematic- and flavour-(in)dependentintrinsic transverse momentum of the parton with flavour a . The λ and λ (cid:48) are (vectors of) nonperturbativeparameters that can be fit to data. The λ (cid:48) parameters are related to the quantity (cid:104) k T (cid:105) a . For example, in case ofa simple Gaussian functional form, e − λ (cid:48) b T , we have λ (cid:48) = (cid:104) k T (cid:105) a /
4. For both the nonperturbative factors g K and (cid:101) f a NP , several implementations have been discussed, see, e.g., Ref. [22, 32] and references therein. In particular, akinematic- and flavour-dependent Gaussian parametrisation has been proposed in Ref. [21, 33].The studies that we discuss make use of three different computational tools: CuTe [10],
DyqT [19] and
DYRes [20].
CuTe implements the SCET formalism, where the resummation is performed in terms of factorisation formulae thatinvolve Soft Collinear Effective Theory operators and matching coefficients. It gives the transverse momentumspectrum of on-shell electroweak bosons at NLO ( O ( α s )) accuracy in the C Wilson coefficients and at NNLL in theSudakov exponent . DyqT and
DYRes are based on Ref. [19, 20] and perform soft gluon resummation in b T -space. The first computes the q T spectrum of an electroweak boson produced in hadronic collisions. The second also provides the full kinematicsof the vector boson and of its decay products, allowing for the application of arbitrary cuts on the final-statekinematical variables and giving differential distributions in form of bin histograms. The accuracy of both codes isup to NNLL in the resummed part, and up to NLO ( O ( α s )) at large q T .A simple Gaussian parametrisation of the non-perturbative effects is present in these codes, as in most of thecomputational tools used to analyse the electroweak observables relevant for the determination of the W bosonmass. A single non-perturbative parameter, g NP , usually encodes both the (flavour-independent) effect of g K andthe distribution in the (potentially flavour-dependent) intrinsic transverse momentum : e − g NP b T ≡ e g K ( b T ; λ ) ln( Q /Q ) (cid:101) f a NP ( b T ; λ (cid:48) ) (cid:101) f a (cid:48) NP ( b T ; λ (cid:48) ) . (5)The values of the non-perturbative parameters used in fitting the W boson mass are usually obtained through fitson Z production data [34], for which the relevant partonic channels are of the type q i ¯ q i , and then used to predict W ± production, despite the process being sensitive to different partonic channels, q i ¯ q j ( i (cid:54) = j ). This procedureessentially neglects any possible flavour dependence of the intrinsic partonic transverse momentum.In order to introduce the flavour dependence, one can simply decompose g NP in the LHS of Eq. (5) into thesum g aNP + g a (cid:48) NP , where the flavour indices span the range a, a (cid:48) = u v , u s , d v , d s , s, c, b, g (the subscripts referringto the valence and sea components, respectively), additionally disentangling the non-perturbative contributionto the evolution and the intrinsic transverse momentum distribution. Thus, for each parton with flavour a , thenonperturbative contribution (cid:101) f a NP and g K in Eq. (1), (5) are included in the corresponding term in the flavoursum of the TMD factorisation formula. More details regarding the non-perturbative parameters in the codes underconsideration have been collected in App. A. III. EFFECTS ON THE q T SPECTRUM OF THE W The impact of a flavour-dependent intrinsic (cid:104) k T (cid:105) on the q T spectrum of the electroweak bosons has been firststudied in Ref. [11] and here we partly summarize the findings therein. Part of the analysis is devoted to the shiftsinduced in the position of the peak for the distribution in q V T , V = W + , W − and Z . Flavour-independent (f.i.) andflavour-dependent (f.d.) variations of the average intrinsic transverse momentum squared are considered, togetherwith the uncertainties associated to other non-perturbative factors, such as the collinear PDFs, the renormalisationscale, and the value of the strong coupling constant. As justified in Sec. I and Fig. 2, it is assumed that the intrinsictransverse-momentum depends on five flavours only: u v , d v , u s , d s , s , where s collectively refers to the strange, charmand bottom quarks and to the gluon.The numerical results are obtained by means of a modified (i.e. flavour-dependent) version of CuTe [10]. Namely,the non-perturbative parameter 2Λ NP (see App. A), which corrects the whole cross section at large b T , is splitinto a sum of two flavour-dependent non-perturbative contributions, Λ i,j , such that Λ i + Λ j = 2Λ NP . Thisdecomposition reabsorbs the non-perturbative contribution to QCD radiation into Λ i,j . The flavour dependence ofΛ i,j is compatible with the ratios fitted in Ref. [33]. The goal is to combine flavour dependent parameters in such a CuTe is labelled NNLL in the SCET language but NNLL (cid:48) in standard pQCD language. The accuracy of NNLL (cid:48) is considered lowerthan that of the full NNLL, in which Wilson coefficients are computed at NNLO. ResBos [34] is a counter-example, but it does not account for the flavour dependence of the intrinsic transverse momentum. way to respect the values of Λ NP fitted on the Z data, generating at the same time different values Λ i,j to be usedin the calculation of the differential cross section for W ± (we refer the reader to Ref. [11] for the precise values ofΛ i,j used in the study).The shifts (quantified in GeV) induced by different perturbative and non-perturbative contributions are summa-rized in Tab. I. The renormalisation scale is varied between 1 / µ c and 2 µ c , with µ c = q T + q (cid:63) , where q (cid:63) is a cutoffintroduced in the SCET formalism to avoid the Landau pole [10]. The scale in the hard part has not been varied.Regarding the impact of the collinear PDFs, the result shown in the table is the smallest interval which contains68% or 90% of peak positions, computed for every member of the NNPDF3.0 set [35]. The strong coupling is variedby ± .
003 from the central value of 0.118.
TABLE I: Summary of the shifts in GeV induced on the peak position in q T spectra of W ± /Z , generated by different effects.“f.i.” stands for flavour-independent, whereas “f.d.” for flavour-dependent. “Max W ± ” effect indicates the maximum shiftinduced on the peak position of the W ± q T spectrum by flavour-dependent variations of (cid:104) k T (cid:105) that keep the peak of the Z q T spectrum unchanged. For the values of the flavour-dependent non-perturbative parameters we refer the reader to Ref. [11]. W + W − Zµ R = µ c / , µ c +0 . − .
09 +0 . − .
06 +0 . − . .
03 +0 .
03 +0 .
04 +0 .
00 +0 . − . . − .
05 +0 . − .
02 +0 . − . α s = 0 . ± .
003 +0 . − .
12 +0 . − .
14 +0 . − . (cid:104) k T (cid:105) = 1 . , .
96 +0 . − .
16 +0 . − .
14 +0 . − . (cid:104) k T (cid:105) (max W + effect) +0 . − . ± (cid:104) k T (cid:105) (max W − effect) − .
03 +0 . ± The shift induced in the peak position from flavour-dependent (cid:104) k T (cid:105) is smaller than that induced by scale variation, α s variation and flavour-independent (cid:104) k T (cid:105) , but comparable in magnitude. It is also bigger than the uncertaintyfrom the PDF set, which is the only other uncertainty where the shifts are not almost perfectly correlated betweenthe three vector bosons. With flavour-dependent variations of (cid:104) k T (cid:105) , the peaks of the W + and W − distributions shiftin different directions. Since the (cid:104) k T (cid:105) parameters are selected under the constraint that the Z q T -distribution is leftunchanged (see Tab. I), the channels for W + and W − move in different directions. The anticorrelation of the shiftsbetween W + and W − is a peculiarity of the uncertainty generated by flavour-dependent variations of the intrinsic k T . This means that the uncertainty stemming from the non-perturbative hadron structure in the transverse planecan affect the determination of m W + and m W − in different ways. Indeed, this feature nicely emerges in the analysissummarized in Sec. IV.The analysis in Ref. [11] thus shows that the uncertainty on the peak position for W ± bosons arising from theflavour dependence of the intrinsic transverse momentum is not negligible with respect to the other sources oftheoretical uncertainties and comparable in magnitude with the uncertainties due to the collinear PDFs.We now analyse the ratios of the q T -differential cross section calculated with a flavour-independent set of non-perturbative parameters in (cid:101) f a NP ( b T ; λ (cid:48) ) over the same cross section calculated with flavour-dependent parameters.The results are presented in Fig. 3 for Z , W + , W − . The calculation has been performed by means of a flavour-dependent modification of DyqT , where the non-perturbative contributions in Eq. (1) have been coded as:exp {− g aNP } = exp {− [ g evo ln( Q /Q ) + g a ] b T } . (6)The values for g evo , Q , g a are taken from Ref. [22] and the flavour-dependence in g a is inspired to the flavour ratiosin Ref. [33]. The curves in Fig. 3 correspond to 50 sets of flavour-dependent non-perturbative parameters builtaccording to these criteria. The perturbative accuracy is NLL and the collinear PDF set used is NNPDF3.1 [36].As predicted by the TMD formalism, the effect induced by the non-perturbative corrections is more evident atlow q T . In particular, it is stronger for q T < q T = 10 GeV. The flavour dependence of theintrinsic transverse momentum can modify the shape of dσ/dq T by ∼ −
10% at very low transverse momentum.This observable affects the cross section differential with respect to the kinematics of the final state particles, namelythe distributions in p (cid:96) T , p ν T , m T , and thus has an impact also on the determination of the W boson mass. q T [ GeV ] [ d σ / dq T f l a v o r - i ndep . ]/[ d σ / dq T f l a v o r - dep . ] pp ⟶ ZLHC s = - independent set vs 50 flavor - dependent sets q T [ GeV ] [ d σ / dq T f l a v o r - i ndep . ]/[ d σ / dq T f l a v o r - dep . ] pp ⟶ W + LHC s = - independent set vs 50 flavor - dependent sets q T [ GeV ] [ d σ / dq T f l a v o r - i ndep . ]/[ d σ / dq T f l a v o r - dep . ] pp ⟶ W - LHC s = - independent set vs 50 flavor - dependent sets FIG. 3: In these figures the ratio dσ V dq T ( f.i. ) / dσ V dq T ( f.d. ) is plotted for the three different electroweak bosons ( V = Z , W + , W − respectively), with a single set of flavour-independent (f.i.) non-perturbative parameters in the transverse part of the TMDPDFs and 50 different flavour-dependent (f.d.) sets of the same parameters. The values of the non-perturbative parametershave been chosen from the results in Ref. [22, 33]. IV. IMPACT ON THE DETERMINATION OF THE W BOSON MASS
As previously mentioned, the measurements of m W at hadron colliders rely on a template-fit procedure performedon selected observables, i.e., the distributions in the transverse mass of the lepton pair and the lepton/neutrinotransverse momentum. Both CDF and D0 experiments at Tevatron use data from all the three observables. In the ATLAS case, however, the transverse momentum of the (anti)neutrino is used for consistency checks only, since it isaffected by larger uncertainties with respect to m T and p (cid:96) T .In this section we consider selected results concerning the estimate of the uncertainties of non-perturbative originon the determination of m W . In particular, we focus on shifts of the W ± mass induced by possible configurationsfor the flavour dependence of the intrinsic transverse momentum, and we will compare them with the correspondingshifts generated by the uncertainties in the collinear PDFs.In the template-fit procedure, several histograms are generated with a specific theoretical accuracy and descriptionof detector effects, letting the fit parameter(s) (only m W , in this case) vary in a range: the histogram best describingthe experimental data selects the measured value for m W . The details of the theoretical calculations used tocompute the templates (the choice of the scales, of the collinear PDFs, of the perturbative order, the resummationof logarithmically enhanced contributions, the nonperturbative effects, etc.) affect the result of the fit and definethe theoretical systematics.This procedure can also be used to estimate the effect of each single theoretical uncertainty, by generating sets ofpseudodata (with the same event generator used for the templates, but at a lower statistics) differing by the valueof the parameter(s) controlling that uncertainty [37, 38]. Fig. 4 contains a graphical illustration of the flowchart forthe template-fit procedure, specified to the comparison of one set of pseudo-data generated with flavour-dependentparameters with 30 templates generated with one set of flavour-independent parameters and 30 values of m W (80385 ±
15 MeV with steps of 1 MeV). This method has been also used to estimate the shift in m W induced bythe variation of the collinear PDF set in fitting the transverse mass [39, 40] and the lepton p T [40, 41] both atTevatron and at the LHC in the central rapidity region of the produced electroweak boson ( | η | < . | η | < . < η < . FIG. 4: Flowchart for a template-fit procedure to estimate shifts in m W induced by the flavour dependence of the intrinsicquark transverse momentum. In the transverse mass case, the total error (envelope) induced by three different PDF sets (CTEQ6.6 [43],MSTW2008 [44], NNPDF2.1 [45]) is less than 10 MeV both at the Tevatron and at the LHC [39]. The results areshown in the left plot of Fig. 5. The analysis has been performed at fixed-order NLO QCD ( O ( α s )), thus withoutall-order resummation, since the m T -shape is mildly sensitive to soft gluon emission from the initial state. The keyfactor in reducing the PDF uncertainty is the use of normalised differential distributions in the fitting procedure,in such a way to eliminate normalisation effects which are irrelevant for m W .A similar analysis applied to the lepton p T observable reveals a much larger error due to PDF variations (CT10 [46],MSTW2008CPdeut [44], MMHT2014 [47], NNPDF2.3 [48], NNPDF3.0 [35]), as shown in the right plot of Fig. 5.While the individual sets provide non-pessimistic estimates ( O (10 MeV)), the distance between the best predictionsof the various sets ranges between 8 and 15 MeV, and the total envelope ranges between 16 and 32 MeV (dependingon the collider, the energy and the final state) [39]. While soft gluon emission already provides a non-vanishingtransverse momentum, additional contributions may come from the intrinsic transverse momentum of the collidingpartons. The study of the impact of a possible flavour-dependent intrinsic k T on the determination of m W hasbeen first performed in Ref. [12], using the same template-fit procedure described above and sketched in Fig. 4,performed with modified versions of the DYqT [19] and
DYRes [20] codes. In this case, the pseudodata are built withthe Gaussian widths g a associated to the different flavours in Eqs. (6). m W ( G e V ) NLO-QCD, normalized transverse mass distributionTEV LHC7W + LHC7W - LHC14W + LHC14W - CTEQ6.6MSTW2008NNPDF2.1Nominal value M W ( G e V ) LHC8 W + LHC8 W − LHC13 W + LHC13 W − TEV W + NNPDF2.3NNPDF3.0CT10MSTW2008CPdeutMMHT2014
FIG. 5: Shifts induced on m W by the choice of different PDF sets, obtained through a template-fit performed on thetransverse mass m T (left) and the lepton p T (right) observables (left figure from Ref. [39], right figure from Ref. [41]). In order to estimate the impact of the flavour dependence, it is necessary to first identify the “ Z -equivalent”sets of parameters, i.e., those sets in agreement with the Z transverse momentum distribution measured at hadroncolliders. To this extent: • a single flavour-independent ( i.e. , using a version of Eq. (6) without a -dependence) q T -spectrum for the Z boson is produced based on the parameters presented in Ref. [22]; • each bin of this flavour-independent spectrum is assigned an uncertainty equal to the one quoted by the CDF and
ATLAS experiments; • several flavour-dependent sets for g a in Eq. (6) are generated randomly within a variation range consistentwith the information obtained in previous TMD fits (in particular, taking into account the estimate for theflavour-independent contribution to the non-perturbative part of the evolution obtained in Ref. [22]); • a flavour-dependent set is defined “ Z -equivalent” if the associated q T spectrum for the Z has a ∆ χ ≤ CDF and
ATLAS who pass this filter are treated as the pseudodata of the template-fitprocedure, while the flavour-independent one is used for the generation of the templates at high statistics. Thenumber of events corresponds to 135M for the pseudodata and 750M for the templates. Only 9 sets out of the 30ones which are “ Z -equivalent” both with respect to CDF and
ATLAS uncertainties have been investigated. The valuesof the flavour-dependent parameters for each set are given in Tab. II. A summary of the shifts obtained throughthis procedure is given in Tab. III.
Set u v d v u s d s s g aNP parameter in Eq. 6 for the flavours a = u v , d v , u s , d s , s = c = b = g . Units are GeV . The statistical uncertainty of the template-fit procedure has been estimated by considering statistically equivalentthose templates for which ∆ χ = χ − χ min ≤
1. Overall, the quoted statistical uncertainty on the results in Tab. IIIis ± . W ± transverse momentum, the correspondingshifts are compatible with zero considering the statistical uncertainty of the template-fit procedure. On the contrary,0 ∆ m W + ∆ m W − Set m T p (cid:96) T m T p (cid:96) T m W ± (in MeV) induced by the corresponding sets of flavour-dependent intrinsic transverse momentaoutlined in Tab. II (Statistical uncertainty: 2.5 MeV). in the p (cid:96) T case the shifts can be incompatible with statistical fluctuations and are comparable to the ones inducedby collinear PDFs, with an envelope of 15 MeV in the case of W + production and 11 MeV for W − production. Wealso notice a hint of a possible anti-correlation between the shifts in the W + and W − cases, as it was also noticedin Sec. III.Along this line, we also stress that ATLAS measured m W + − m W − = − ±
28 MeV [4]. From Tab. III, we can inferthat part of the discrepancy between the mass of the W + and the W − can be artificially induced by not consideringthe flavour structure in transverse momentum. For example, the sets 1 and 2 in Tab. II feature δm W − > δm W + (induced by p (cid:96) T ). This implies that for templates built with sets 1 and 2, instead of flavour-independent values, thedifference between the two masses would be reduced. An opposite result would be obtained if building templateswith flavour-dependent sets for which δm W − < δm W + (e.g. sets 3 and 5, for the p (cid:96) T case). V. OUTLOOK AND FUTURE DEVELOPMENTS
The selected results presented in this contribution point out that the impact of a possible flavour dependence of theintrinsic partonic transverse momentum should not be neglected, even in the kinematic region where nonperturbativeeffects are expected to be small [49–51], such as for electroweak boson production at the LHC.This kind of uncertainty directly affects the electroweak observables relevant for the measurement of m W : thetransverse momentum distribution for the W and the decay lepton, and the transverse mass distribution of thelepton pair. The numerical results presented in Sec. III and Sec. IV indicate that flavour-dependent effects arecomparable in size to other uncertainties of (non-)perturbative origin (for example, the choice of collinear PDFset). Thus, a flavour-blind analysis is not a sufficiently accurate option for a program of precision electroweakmeasurements at the LHC and at future colliders.Moreover, in hadron colliders at a lower energy such as RHIC and a possible fixed-target experiment at the LHC,the non-perturbative effects can play an even more significant role (due to the larger x -values probed) and affectthe study of polarised TMDs [52] and the structure of the light sea quarks [53].A detailed knowledge of TMD distributions is thus important, not only for nucleon tomography beyond thecollinear picture [54–62], but also to constrain fundamental parameters of the Standard Model, thus providing adirect connection between hadron physics and the high-energy phenomenology.In light of these results, we call for improved investigations of the impact of nonperturbative effects linked tothe hadron structure at hadron colliders and for the inclusion of these effects in the event generators employed inexperimental and theoretical investigations of high-energy physics. Appendix A: Conventions for nonperturbative parameters
For convenience, we collect in this Appendix the naive translation of the nonperturbative parameters used in thenumerical codes cited in the text. In the conventions of [21, 33], the nonperturbative parameters appear as:d σ ∝ exp (cid:18) − (cid:0) (cid:104) k T (cid:105) q + (cid:104) k T (cid:105) q (cid:1) b T (cid:19) . (A1)1In CuTe [10] there is a single nonperturbative parameter entering the cross section:d σ ∝ exp (cid:0) − NP b T (cid:1) . (A2)The same happens in DyqT [19] and DYRes [20], in terms of the nonperturbative parameter g NP :d σ ∝ exp (cid:0) − g NP b T (cid:1) . (A3)We obtain the parameter employed in CuTe asΛ NP = (cid:114)
18 ( (cid:104) k T (cid:105) q + (cid:104) k T (cid:105) q ) , Λ NP = (cid:112) g NP / . (A4)and similarly for the DYqT parameter: g NP = 14 (cid:0) (cid:104) k T (cid:105) q + (cid:104) k T (cid:105) q (cid:1) ,g NP = 2Λ NP . (A5)Here we report the most important values, based on Eq. (A5): Λ NP g NP (cid:104) k T (cid:105) ( q = q )CuTe default 0.60 0.72 1.44DYqT conservative estimate 0.77 1.2 2.40 Acknowledgments
We thank Alessandro Bacchetta and Marco Radici for the fruitful collaboration on this topic, Alessandro Vicinifor many suggestions and discussions, Chao Shi for carefully reading the manuscript and Piet Mulders and MathiasRitzmann for contributing to the investigations summarized in this article. AS acknowledges support from theU.S. Department of Energy, Office of Science, Office of Nuclear Physics, contract no. DE-AC02-06CH11357. GBacknowledges support from the European Research Council (ERC) under the European Union’s Horizon 2020research and innovation program (grant agreement No. 647981, 3DSPIN). [1]
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