Parton correlations in same-sign W pair production via double parton scattering at the LHC
Federico Alberto Ceccopieri, Matteo Rinaldi, Sergio Scopetta
aa r X i v : . [ h e p - ph ] F e b Parton correlations in same-sign W pair production via double parton scattering atthe LHC Federico Alberto Ceccopieri,
1, 2
Matteo Rinaldi, and Sergio Scopetta Dipartimento di Fisica e Geologia, Universit`a degli Studi di Perugia and Istituto Nazionale di Fisica Nucleare,Sezione di Perugia, via A. Pascoli, I - 06123 Perugia, Italy IFPA, Universit´e de Li`ege, B4000, Li`ege, Belgium ∗ Departament de Fisica Te`orica, Universitat de Val`encia and Institut de Fisica Corpuscular,Consejo Superior de Investigaciones Cient´ıficas, 46100 Burjassot (Val`encia), Spain (Dated: September 10, 2018)Same-sign W boson pairs production is a promising channel to look for signatures of double partoninteractions at the LHC. The corresponding cross section has been calculated by using double partondistribution functions, encoding two parton correlations, evaluated in a Light-Front quark model.The obtained result is in line with previous estimates which make use of an external parameter, theso called effective cross section, not necessary in our approach. The possibility to observe for thefirst time two-parton correlations, in the next LHC runs, has been established. PACS numbers:Keywords:
It is known since a long time that a proper descriptionof final states in hadronic collisions requires the inclusionof processes where more than one pair of partons partic-ipate in a single hadronic collision, the so-called multiplepartonic interactions (MPI) [1, 2]. Due to LHC opera-tion, the wide subject of MPI is having in these yearsa renewed interest [3]. At low transverse momenta, MPIenhance particle production and affect particle multiplic-ities and energy flows. The effect of MPI is present also inhard scattering processes. In this letter, we are interestedin double parton scattering (DPS), in which parton pairsfrom two hadrons interact between each other, and bothcollisions are hard enough to apply perturbative tech-niques. While these processes need to be well controlledsince they could represent a background to New Physicssearches, the main focus of this work is the sensitivity ofDPS to relevant features of the non-perturbative nucleonstructure, not accessible otherwise. In particular, DPScross section depends on non-perturbative quantities, theso-called double parton distribution functions (dPDFs).The latter represent the number density of parton pairswith longitudinal fractional momenta x , x , at a relativetransverse distance ~b ⊥ . If extracted from data, dPDFswould offer for the first time the opportunity to investi-gate two-parton correlations, as noticed long time ago [4].Since dPDFs are two-body distributions, this informa-tion is different and complementary to the one encodedin one-body distributions, such as ordinary and general-ized parton distributions [5]. The present letter aims atestablishing to what extent this novel information can beaccessed in the next runs of LHC, looking at a specificfinal state, namely, the production of a pair of W bosons ∗ Electronic address: [email protected] with the same-sign ( ssW W ). In fact, this channel hasbeen found to be promising for DPS observation [6–8],since single parton scattering (SPS) at tree-level startscontributing to higher order in the strong coupling [9].For such reasons, diboson production via DPS has beentheoretically investigated in detail [10–13].Let us define now the quantities we are going to cal-culate. If final states A and B are produced in a DPSprocess, the corresponding cross section can be sketchedas [1] dσ ABDP S = m X i,j,k,l Z d~b ⊥ D ij ( x , x ; ~b ⊥ ) × D kl ( x , x ; ~b ⊥ ) d ˆ σ Aik d ˆ σ Bjl , (1)where m = 1 if A and B are identical and m = 2 other-wise, i, j, k, l = { q, ¯ q, g } are the parton species contribut-ing to the final states A ( B ). In Eq. (1) and in the fol-lowing, dσ is used for the cross section, differential in therelevant variables. The functions D ij in Eq. (1) are thedPDFs which depend additionally on factorization scales µ A ( B ) , D ij ( x , x ; ~b ⊥ , µ A , µ B ). To date, dPDFs arevery poorly known, so that it has been useful to describeDPS cross section independently of the dPDFs concept,using the approximation: dσ ABDP S ≃ m dσ ASP S dσ BSP S σ eff , (2)where dσ ASP S is the SPS cross section with final state A : dσ ASP S = X i,k f i ( x , µ A ) f k ( x , µ A ) d ˆ σ Aik ( x , x , µ A ) . (3)In Eq. (3) f i ( j ) are parton distribution functions (PDFs)and an analogous expression holds for the final state B .The physical meaning of Eq. (2) is that, once the pro-cess A has occured with cross section σ ASP S , the ratio σ BSP S /σ eff represents the probability of process B to oc-cur. A constant value of σ eff has been assumed in allthe experimental analyses performed so far, so that thetechnical implementation of Eq. (2) is rather easy. Inthis way, different collaborations have extracted valuesof σ eff which are consistent within errors, irrespectiveof center-of-mass energy of the hadronic collisions and ofthe final state considered. A comprehensive compilationof experimental results on σ eff is reported in Ref. [14],where the latest DPS measurement in the four jet finalstate is presented.To understand the approximation leading to Eq. (2)from Eq. (1), let us write dPDFs in the latter in a fullyfactorized form: D ij ( x , x , µ A , µ B ,~b ⊥ ) = f i ( x , µ A ) f j ( x , µ B ) T ( ~b ⊥ ) , (4)where the function T ( ~b ⊥ ), describes the probability tohave two partons at a transverse distance ~b ⊥ . Then, in-serting Eq. (4) into Eq. (1), one obtains Eq. σ eff , (2),as follows σ − eff = Z d~b ⊥ [ T ( ~b ⊥ )] , (5)with T ( ~b ⊥ ) controlling the double parton interactionrate. It is clear that, as a consequence of the approxima-tion (4), σ eff does not show any dependence on partonfractional momenta, hard scales or parton species.Actually, if factorized expressions are not used, σ eff depends on longitudinal momenta. Since dPDFs are ba-sically unknown, and only sum rules relating them toPDFs are available [15, 16], model calculations, de-veloped at low energy, but able to reproduce relevantfeatures of nucleon parton structure, can be useful andhave been proposed. In such model calculations, fac-torized structures, Eq. (4), do not arise, and σ eff de-pends non-trivially on longitudinal momenta. In par-ticular, this was found in a Light-Front (LF) Poincar´ecovariant constituent quark model (CQM), reproducingthe sum rules of dPDFs [17, 18], as well as in a holo-graphic approach [19]. In this letter we will evaluate DPScross sections, using different models of dPDFs, to estab-lish wether forthcoming LHC data will exhibit (for theconsidered final state) such features, not yet seen in thepresent uncertain experimental scenario.Let us now summarise our calculation. We first con-sider the SPS W ± production and subsequent decay intomuon at center-of-mass energy √ s : pp → W ± ( → µ ± ν ( − ) µ ) X , (6)indicating with σ ± the corresponding cross sections.Defining quarks according to their charge, i.e. D = d, s, b and U = u, c, t , we consider the following partonic sub-processes U ( p a ) ¯ D ( p b ) → µ + ( p µ ) ν µ ( p ν ) , (7) D ( p a ) ¯ U ( p b ) → µ − ( p µ )¯ ν µ ( p ν ) , (8) where particle four-momenta are indicated in parenthe-sis. Differential cross sections are calculated in terms ofthe muon transverse momentum p T = | ~p T | and pseudo-rapidity η µ , defined in the hadronic center-of-mass frame.The partonic Lorentz invariants ˆ u and ˆ t , in terms of thesevariables, readˆ t = ( p a − p µ ) = − x a √ sp T e − η µ , ˆ u = ( p b − p µ ) = − x b √ sp T e η µ , (9)from which parton fractional momenta can be calculatedas x a = e η µ M W √ s ( A ± B ) , x b = e − η µ M W √ s ( A ∓ B ) , (10)with A = M W / (2 p T ), B = √ A − M W the W -boson mass. The unobserved neutrino causes anunder-determination of the W -rapidity and, in turns, thetwofold ambiguity in Eq. (10). Cross sections are eval-uated in the narrow width approximation, i.e. at fixedˆ s = ( p a + p b ) = M W , and read d σ pp → W + ( → µ + ν ) X dηdp T = G F s Γ W V U ¯ D √ A − × h f U ( x a , µ F ) f ¯ D ( x b , µ F )ˆ t + f ¯ D ( x a , µ F ) f U ( x b , µ F )ˆ u i , where G F is the Fermi constant, Γ W the W boson decaywidth and V ij the CKM matrix elements whose values aretaken from Ref. [20]. The σ − cross section is obtainedexchanging U ↔ D and ˆ t ↔ ˆ u in Eq. (11). The PDFsappearing in Eq. (11) are evaluated at a factorizationscale µ F = M W and therefore PDFs from CQM calcula-tions, related to low momentum scales, need to be prop-erly evolved. The evolution is performed at LO by usingDGLAP equations. We adopt a variable flavour numberscheme and parameters as in LO version of MSTW08 dis-tribution [21]. In particular heavy quark masses are setto m c = 1 . m b = 4 .
75 GeV and the one-looprunning coupling is fixed at Z -boson mass scale to be α ( n f =5) s ( M Z ) = 0 . f d ( x, Q ) = 1 / f u ( x, Q ) , (12)at the hadronic scale Q , where three valence quarkscarry all proton momentum. Since this scale is gen-erally located in the infrared regime, PDFs evolutionand corresponding cross sections are very sensitive toits choice. defined. Therefore, in the present paper, Q is fixed requiring that σ + and σ − , calculated by us-ing evolved LF PDFs, match the corresponding predic-tions obtained with the DYNNLO code [22] at LO by using
MSTW08
PDFs [21]. For both simulations we set √ s = 13TeV and define the muon fiducial phase space in SPS tobe p µT >
20 GeV and | η µ | < .
4. As shown in Fig. 1,considering the cross section summed over the W bosoncharge, this procedure locates the central value of the σ − σ + σ + + σ − Q [GeV ] σ [ nb ] FIG. 1:
W production cross sections as predicted by LF PDFsas a function of Q , compared to DYNNLO predictions (straightlines) at LO by using LO
MSTW08 parton distributions in thefiducial region indicated in the text. initial scale at Q = 0 .
26 GeV (where α s ( Q ) = 1 . Q , a simultaneous de-scription of σ + and σ − can not be achieved, a fact whichis ascribed to the model assumption for PDFs in Eq. (12)and it is an example of typical drawback of PDFs CQMcalculations. In order to take into account this deficiency,we assign a theoretical error to Q , allowing it to vary inthe range 0 . < Q < .
28 GeV , where the limits arefixed requiring that cross sections obtained via LF modelreproduce σ + and σ − predicted by DYNNLO (straight linesin Fig. 1). Having fixed Q in SPS processes and beingdPDFs obtained within the same LF model adopted forPDFs, we can use the same Q range for dPDFs. In thisway the estimate of DPS cross sections does not requireadditional parameters. Double PDFs in the LF modelare defined at Q as [17] f du = f ud = f uu ( x , x , Q ,~b ⊥ ) . (13)At this scale, when integrated over ~b ⊥ , dPDFs satisfynumber and momentum sum rules [15]. Their perturba-tive QCD evolution is presently known only at leadinglogarithmic accuracy [23, 24], however the presence ofthe so-called inhomogeneous term in the evolution equa-tions is still under investigation [3, 16, 25]. In the presentpaper dPDFs are evolved with the same scheme and pa-rameters used for PDFs but using homogeneous evolu-tion equations valid at fixed values of ~b ⊥ [3, 26]. The pp , √ s = 13 TeV p leadingT,µ >
20 GeV, p subleadingT,µ >
10 GeV | p leadingT,µ | + | p subleadingT,µ | >
45 GeV | η µ | < .
420 GeV < M inv <
75 GeV or M inv >
105 GeVTABLE I:
Fiducial DPS phase space used in the analysis.
DPS cross section, Eq. (1), in the ssW W channel reads d σ pp → µ ± µ ± X dη dp T, dη dp T, = X i,k,j,l Z d ~b ⊥ × D ij ( x , x ,~b ⊥ , M W ) D kl ( x , x ,~b ⊥ , M W ) × d σ pp → µ ± Xik dη dp T, d σ pp → µ ± Xjl dη dp T, I ( η i , p T,i ) . (14)The function I ( η i , p T,i ) in Eq. (14) implements the kine-matical cuts reported in Tab. (I) which we mutuatefrom the 8 TeV analysis of Ref. [27]. In Eq. (14) weare neglecting the supposed small contributions comingfrom longitudinally polarized dPDFs [13]. Eq. (14) willbe evaluated with three different models of dPDFs de-scribed in the following in order of increasing complex-ity. In the simplest one, called MSTW, dPDFs are pa-rameterized as products of
MSWT08
PDFs according toEq. (4). In the second one, the so-called GS09model [15], the factorized form Eq. (4), properly cor-rected to fulfill dPDFs sum rules, is assumed only at amomentum scale Q . Such initial conditions are evolvedwith dPDFs evolution equations with the inhomogenousterm included [23, 24]. Therefore, with respect to modelMSTW, GS09 takes into account additional perturbativecorrelations [4, 15, 26, 28]. The DPS cross section basedon MSTW and GS09 models can be evaluated only as-suming a constant σ eff in Eq. (2). In the present work,we will use, as a reference value, ¯ σ eff = 17 . ± . W -boson plus dijet final state, the latter being theclosest to the one considered here. The only availableinformation in this channel is a lower limit, σ eff > . L =19.7 fb − at √ s = 8 TeV [27].In the last model [17], called QM, dPDFs have beenevaluated within the LF framework, generalizing the ap-proach of Ref. [31] for the calculation of PDFs. Asa result, fully correlated dPDFs are obtained [17]. Insuch a model, longitudinal and transverse correlationsare generated among valence quarks and propagated byperturbative evolution to sea quarks and gluons dPDFs.The use of this model in the present analysis is particu-larly relevant. First of all, within this model, the DPScross section can be calculated using Eq. (1), withoutany assumption on σ eff , at variance with MSTW and dPDFs σ ++ + σ −− [fb]MSTW 0 . +0 . − . ( δµ F ) +0 . − . ( δ ¯ σ eff )GS09 0 . +0 . − . ( δµ F ) +0 . − . ( δ ¯ σ eff )QM 0 . +0 . − . ( δµ F ) +0 . − . ( δQ )TABLE II: Model predictions for W -charge summed crosssections in fiducial region in Tab. (I). dPDFs σ ++ [fb] σ −− [fb] σ ++ /σ −− GS09 0.54 0.28 1.9QM 0.53 0.16 3.4GS09/QM 1.01 1.78 -TABLE III:
Ratio of cross sections for same sign muons pro-duction in fiducial region.
GS09. Moreover, the simultaneous use of single and dou-ble PDFs obtained from the same LF dynamics, allowsone to investigate the role of parton correlations on po-tentially sensitive observables. Theoretical systematicerrors are associated to our predictions as follows. Un-certainties related to missing higher order corrections,denoted by δµ F , are estimated for all models, varying µ F in the range 0 . M W < µ F < . M W ; the ones due Q -fixing, denoted by δQ , are given by varying this pa-rameter in the range 0 . < Q < .
28 GeV . A furthererror, δ ¯ σ eff , is assigned to MSTW and GS09 predictions,due to ¯ σ eff uncertainty. In Tab. II we report DPS crosssections, integrated in the fiducial volume, evaluated us-ing the above models. Predictions based on MSTW andGS09 are close, while QM one is smaller by around 15%,although they are all consistent within errors.For all the considered models, cross sections rise as µ F increases, an effect induced by the sea quark growth at h x i ∼ − (typical of this process). We have estimatedthat, if the integrated luminosity L is greater than 300fb − , the central values of the QM and GS09 predictionscan be discriminated. It is worth noting that, if the mea-surement were performed also in the eµ ( eµ + ee ) channel,the number of signal events would increase by a factorthree (four).In Tab. III, predictions of models GS09 and QM fordefault values of parameters of charged ssW W cross sec-tions (indicated by σ −− and σ ++ ) integrated in the fidu-cial volume, are compared. While agreement betweenmodel predictions is found for σ ++ , a rather smaller σ −− is obtained in model QM, due to the assumption inEq.(10). The ratio σ −− and σ ++ is therefore a suitableobservable to investigate the flavor structure of dPDFs.In order to analyze correlations encoded in dPDFs, weconsider the differential cross section in the variable η · η which, neglecting the boost generated by W -decay into δQ ⊕ δµ F QM δ ¯ σ eff ⊕ δµ F GS09 η · η N u m b e r o f e v e n t s FIG. 2:
Number of expected events with L − = 300 fb − asa function of product of muons rapidities. leptons, can be approximated via Eqs. (10) as η · η ≃
14 ln x x ln x x , (15)where fractional momenta are subject to the invariantmass constraint x x = x x = M W /s . The result, con-verted into per-bin number of events assuming an inte-grated luminosity L = 300 fb − , is presented in Fig. 2.The maximun is located at η · η ∼
0, where annihi-lating partons equally share the momentum fractions, x ∼ M W / √ s , in at least one scattering. At large andpositive (negative) values of η · η , muons are producedin the same (opposite) emisphere and the fast drop of thecross section is associated to the fall off of dPDFs as one( η · η ≪
0) or both ( η · η ≫
0) partons in the sameproton approach the large x limit. We note that pre-dictions based on GS09 and QM models show a rathersimilar shape and are compatible within their sizeableerrors. To deal with such large uncertainties, differentialcross sections, normalized to the total ones (Tab. II), maybe considered. In this way, the predictions of MSTW andGS09 models do not depend any more on the choice of σ eff and the related error cancels. Moreover, for modelQM, we verified that the scale variations δµ F and δQ ,acting basically on normalizations, almost cancel in theratio. A shape comparison can then be used to discrimi-nate among models and their factorized structure. In thepresent analysis, however, we prefer to discuss the effectsof correlations on a more familiar quantity, σ eff .To this aim we use the LF approach for both PDFs anddPDFs to evaluate SPS and DPS differential cross sec-tions, Eqs. (11) and (14), respectively, integrated in binsof η · η . With these ingredients we obtain, through Eq.(2), a prediction for σ eff intrinsic to the LF model, calledhereafter e σ eff . If a corresponding procedure is performedon cross sections integrated in the fiducial volume, one e σ eff h e σ eff i η · η e σ e ff [ m b ] FIG. 3: e σ eff and h e σ eff i as a function of product of muonrapidities. The error band represents scale variations addedin quadrature. obtains the constant value h e σ eff i = 21 . +0 . − . ( δQ ) +0 . − . ( δµ F ) mb . (16)This value is compatible, within errors, with ¯ σ eff exper-imentally determined. Both e σ eff and h e σ eff i are shownin Fig. 3 and, being ratios, are both stable against µ F and Q variations. The departure of e σ eff from a con- stant value is a measure of two parton correlations in theproton. These are primarily correlations in longitudinalmomenta but, as shown using the fully correlated modelQM, they are related to the ones in transverse space inan irreducible way [32]. We have estimated that thisdeparture could be appreciated with an integrated lumi-nosity L of around 1000 fb − , at 68 % confidence level,reachable in the planned LHC runs. Our conclusion isthat the extraction of this observable in bins of η · η is a convenient strategy to look for parton correlations.Summarising, we have calculated ssW W cross sectionsin a LF model for dPDFs, carefully estimating the corre-sponding uncertainties. Our predictions, completely in-trinsic to the approach, are in line with those obtainedby other approaches which make use of the external pa-rameter σ eff . This indicates that the model is able tocatch the transverse structure of the DPS process. Fur-thermore, we have established that, in this specific finalstate, transverse and longitudinal correlations, embodiedin dPDFs, could be observed in the next LHC runs.This work is supported in part through the project“Hadron Physics at the LHC: looking for signatures ofmultiple parton interactions and quark gluon plasma for-mation (Gossip project)”, funded by the “Fondo ricercadi base di Ateneo” of the Perugia University. Thiswork is also supported in part by Mineco under contractFPA2013-47443-C2-1-P and SEV-2014-0398. We warmlythank Livio Fan`o, Marco Traini and Vicente Vento formany useful discussions. [1] N. Paver and D. Treleani, Nuovo Cim. A , 215 (1982).[2] T. Sjostrand and M. Van Zijl, Nuovo Cim. A , 215(1982).[3] M. Diehl, D. Ostermeier and A. Schafer JHEP , 089(2012); M. Diehl and A. Schafer, Phys. Lett. B , 389(2011).[4] G. Calucci and D. Treleani, Phys. Rev.
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