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ON OBSERVING TOP QUARK PRODUCTION AT THE TEVATRON

arXiv:hep-ph/9205246v2 30 May 1992TUFTS-TH-92-G01hep-ph/9205246May 15, 1992ON OBSERVING TOP QUARK PRODUCTION AT THE TEVATRONGary R. Goldstein and K. SliwaDepartment of PhysicsTufts UniversityMedford, Massachusetts 02155 USAR.H. DalitzDepartment of Theoretical PhysicsUniversity of Oxford1 Keble Road, Oxford OX1 3NP UKABSTRACTA technique for separating top quark production from Standard Model background eventsis introduced.

It is applicable to the channel in which one top quark decays semi-leptonicallyand its anti-quark decays hadronically into three jets, or vice versa. The method is shown todiscriminate dramatically between Monte Carlo generated events with and without simulated topquarks of mass around 120 GeV and above.

The simulations were performed with CDF detectorcharacteristics incorporated, showing that the method is applicable to existing data.

With the discoveries of the tau lepton and then of the bottom quark, it became clear thatthe completion of a third generation in the Standard Model required a sixth quark flavor, nownamed ”top”. The successes of this model are now so numerous and so detailed that few doubtthe existence of the top quark[1], yet its discovery has remained elusive.

The only firm knowledgethat we have about its mass is that it exceeds 89 GeV[2], a bound deduced from the empiricaldi-lepton inclusive total cross-section corresponding to top- antitop production followed by decayswhich include a lepton. Some theoretical arguments suggest that its mass may not be larger thanabout 200 GeV, and probably less.

Unless its mass is much greater than this, top -antitop pairsshould be produced by the Tevatron collider, but formidible backgrounds have made its detectionquite difficult.Owing to its uncommonly large mass, top decay is expected to have some unique features[3,4].The most important of these are that decay through the (V-A) weak interaction, to a b-quarkand a real W boson will dominate, and that,if the top quark mass mt is above about 120 GeV,this weak decay will occur more rapidly than hadronization[3].Recently, two of us [3] developed a scheme for analysing top-antitop pair production in proton-antiproton annihilation through their leptonic decay modes. The two-step decay t→bW+, W+→lν, led us to construct a paraboloid surface of possible top three-momenta, based on the momentaof the b-jet and the lepton.

Each possible top energy corresponds to a circular cross-section of it;each possible top mass to an elliptic section of it.A t¯t pair is produced by the interaction of a parton of the proton with an antiparton of theantiproton. It is known empirically that the partons have quite limited transverse momentum.Hence the simplest parton model requires that the net transverse momentum of the producedpair be zero; however, to leading order in QCD, some transverse momentum can arise from gluonbremsstrahlung or the emission of hard gluons.

The result[3] is that a real t¯t pair corresponds topoints on the transverse plane projection of their two ellipses, such that there is near cancellationof their transverse momenta. This criterion has been used for the analysis of the CDF t¯t candidateevent [5], with the conclusion that the mass mt must be about 131 GeV[6], if this identificationof the event is correct.

Although this event was very striking, and well-fitted in this way, thisanalysis was not enough to prove that this event was an example of t¯t pair production.The W also decays hadronically, through the two quark channel, more probably than leptonically(by a factor of about 6 to 1 for the number of q−¯q combinations versus any single lepton channel).Hence a larger signal for top pair production is the event configurationp + ¯p →t + ¯t + X(1)t →¯l + ν + (b −jet)¯t →(q −jet) + (¯q −jet) + (¯b −jet)in which there is a single very energetic lepton (from the W emitted by one of the top quarks)accompanied by 4 energetic jets (3 from the decay of the other top quark and one accompanyingthe lepton).For high mass top quarks the cross section for this channel, (1), is expected to stand outover the anticipated background[7].which is approximated by the production of a W bosonalong with multiple jets originating from hard gluon bremsstrahlung and quark antiquark pairproduction. When felicitous variables are chosen, the separation can be quite pronounced, andthat is the major result of this paper.

In the following we will show that for events satisfyingcertain kinematic criteria, a probabilistically weighted mass distribution function, to be referredto as the accumulated probability distribution, can be defined that will distinguish t¯t events fromStandard Model background.In an event like (1) the 4-momenta of the three jets associated with the hadronic decay willdetermine a total 4-momentum for the hadronically decaying top; both the momentum ⃗pt andthe mass mt* will be determined. The lepton and remaining b-jet 4-momenta, along with the

mass mt* just determined, define an ellipse of possible 3-momenta ⃗pt′ for the other top quark, asalready shown[3]. What is the relation between ⃗pt and the ellipse for the presumed mass?The hypothesis that the production mechanism involved limited transverse momentum leadsto the requirement that the resulting transverse momentum projection (⃗pt)T +(⃗pt′)T be near zero.Then the negative of the transverse vector -(⃗pt)T will be near points on the ellipse, as illustratedin Fig.1.

This near coincidence severely restricts the kinematics of top production events. Thecombination of the existence of an ellipse for mt* and a near cancellation of transverse momentaconsitutes our kinematic fitting criteria.

A kinematic configuration satisfying those criteria willconstitute a ”match” to the criteria.The probabilistic weighting for any particular ”matched” kinematic configuration for process(1) depends on several factors. The (normalized) leptonic decay probability depends on the topmass and the lepton and b-jet 4-momenta through the relationPl(b, l) = 2(2b · l −M 2W )(m2t −m2b + M 2W −2b · l)(m2t −m2b)2 + M 2W (m2t + M 2b ) −2M 4W(2)where b and l are the b-jet and lepton 4-momenta.

The hadronic decay probability can be writtenonly when the flavors of the three quarks or their jet fragmentation is known, a circumstancethat will be difficult to sort out experimentally (hence we will simply take 1 for this factor in thecalculations that follow). The total transverse momentum will be distributed normally about zerowith a width on the order of 10 % of the produced mass[8].

For subsequent calculations we willtake a step function of width 0.1mt for transverse momentum, for simplicity.The production mechanism favored for any particular kinematic match depends on the Bjorkenx values for the initiating partons, that is the structure functions F1(x1) and F2(x2), and the partonsubprocess center-of-mass energy and momentum transfer through the cross section. The relativeprobability for producing the kinematical configuration is then given byPx1,x2 =Pi=qq,gg Fi(x1)Fi(x2) dσdˆt (ˆs, ˆt)iPi=qq,ggdσdˆt (ˆs, ˆt)i(3)where the relevant variables are obtained from the energies and momenta of the top quarks inthe p¯p lab frame,x1,2 = (Et −Et′ ± (ptL + pt′L))/2P;ˆs = x1x2s;ˆt = m2t −x1p(s)(Et −ptL)with P the proton momentum, s the square of the p¯p center-of-mass energy, and ptL the toplongitudinal momentum.

The a priori probability for a top quark production through this channel(1) with a given kinematical configuration will be the product of the above probability factorsfor the variables obtained, along with whatever probabilities are necessary to account for thesystematic errors involved in real measurements of jet 4-momenta.What real p¯p collider events would be candidates for this channel? Ideally, for a perfectlymeasured event containing a high transverse energy lepton, considerable missing transverse energy(for a neutrino) and at least four jets with clearly identified flavor, the task of checking whetheror not the top production hypothesis is likely to be correct is straightforward.

The likelihood forthat event, with a specific identification of jet flavors, would be achieved by simply evaluating thea priori probability for the given kinematic configuration and applying Bayesian statistics (thiswas feasible with the di-lepton event, where there were few combinations of jets possible[3,6]).For real data, however, there are two complications that could have considerable impact: thesizeable uncertainties in jet energy measurements and the lack of flavor identification for the jets.

The latter shortcoming leads to the necessity for considering all possible jet combinations in agiven event. Will this necessity wash out any ability to discriminate top events from a ubiquitousbackground?

It is our expectation that the kinematic restrictions favoring small net transversemomentum for decay products and requiring one jet pair to have invariant mass near the W mass,will enhance the probability for correct jet combinations.Each event in any sample of real Tevatron data, or simulated data as we will consider here,has a set of ”measured” lepton and jet 4-momenta. The first step in putting the event throughthe fitting procedure is to choose a particular combination of lepton and jets, with a set ofenergy and rapidity cuts that assure the clean determination of the topology of the events.

Acombination of four jets is tentatively assigned: one to the semi-leptonic decaying top and three tothe hadronically decaying partner. The three jets, with 4-momenta p1, p2, p3, form the tentativetop or anti-top momentum ⃗t=⃗p1+⃗p2+⃗p3, with top ”mass” mt*2=t2.The lepton, say electron, and the tentative b-jet 4-momenta, pe and pb, define a continuumof ellipses for different top masses.

The minimum allowed value of the square of the mass is(M 2W + 2pe · pb)(m2b + 2pe · pb)/2pe · pbso mt*2 must be greater than this for a match to be possible. If mt* can define one of the ellipses,the next step is to determine the total transverse momentum of the t, ⃗tT , with a point chosenon the transverse projection of the ellipse.

To choose such a point the ellipse is parameterizedby an uniform angular variable, the angles are discretized in 5o units, to be specific, and eachcorresponding point on the ellipse will be in consideration. Choose one point on the ellipse, asillustrated in Fig.1.

Graphically the vector -⃗tT should lie within 0.1mt* of the given point on thetransverse ellipse. If that condition is satisfied the kinematic configuration is a match.

Next moveto another angular point on the ellipse. For every point satisfying this transverse momentumcondition, a match is defined.

Usually then, there will be an arc of matches on the ellipse for agiven mt*. Each match is assigned an a priori probability according to its overall kinematics.

Theprobabilities for all these matches are summed to make one entry in the probability distributionfor this particular t vector, mass and ellipse.Two of the jets must be from W decay for real top decay, so the invariant mass of at least onepair should lie in the region of the W mass. However, the spread in pair mass values estimatedfrom Monte Carlo simulations of W decays in CDF gives a broad distribution, from 40 to 150 GeV,because of uncertainties in determining jet energies.

If the measured pair mass can be broughtto the W mass while staying within the expected measurement uncertainties, the pair will be acandidate for a W decay. Fixing the pair mass to MW provides a strong constraint, as we willsee.How are the uncertainties in real measurements to be accounted for in this fitting scheme?The measured value of a jet energy represents one of a continuum of possible ”true” values, withfrequency of occurence for any particular energy given by an empirical probability distribution.Jet measurement distributions have been studied carefully by the CDF group with a resultingalgorithm for evaluating the width, σ, of a roughly gaussian distribution of energies[10].

(Theuncertainty in the lepton measurement is quite small compared to the jets and will be ignoredhere.) So to account for the distribution in a single jet 4-momentum, the σ for that energy isdetermined.

A fixed number, say N, of discrete weighted values are chosen for the energy varyingover E(measured)± a constant number of σ’s.Since we are selecting relatively high energyjets, and directional measurements are better than energy measurements at CDF, we assume thevariation in 3-momentum will be in magnitude rather than direction and will follow the energyvariation (jet mass being ignored).There are four jets in the particular combination we are considering at this point.In theparameterization just described, each jet’s 4-momentum (and gaussian measurement weight)would vary over N values as their four energies vary independently over a hypercubic (N)4 lattice.Thus there would be (N)4 kinematic configurations for this combination of jets. Because of the

constraint that one pair must have an invariant mass equal to MW or 80.6 GeV, the variationof the pair’s energies can not be independent; an independent variation of each jet’s energy inthe pair could force the invariant mass farther from the W mass, and make the correct kinematicreconstruction more problematic. The independent variation (over several standard deviations) ofthe two energies, say E1 and E2, would fill a rectangular area in the plane spanned by the twovariable energy values.

But each pair of energies in that plane corresponds to a different invariantmass for the pair. Forcing the two energies to give the W mass (E1 ·E2 is proportional to the pairmass) thereby constrains the energies to lie on a section of an hyperbola.

The uniform divisionof that hyperbola into N equal ”angles” (in the parameterization through hyperbolic functions)provides the proper analog to the single jet variation. Associated with each pair of ”corrected”energies on the hyperbola is the product of the gaussian probability values for the displacement tothe corrected value of each energy from its measured value.

That product of gaussian probabilitieswill be biggest for a pair that has its measured invariant mass nearest to the W mass.With this W mass constraint, and its parameterization, there are (N)3 values over which theparticular combination of jets’ kinematic configuration can vary. Each kinematic configuration issubmitted to the same fitting procedure, matches are obtained and the a priori probability forthat mass mt* obtained.

When all the (N)3 configurations are processed a probability distributionversus top mass for that jet combination is obtained. It is a measure of the probability that thatcombination fits the top hypothesis for each mass.For a given event, which one of the combinations is the ”correct” one?In the simulatedevents, to be discussed, that is known, of course, but for real data either a criterion must be set forselecting a subset or all combinations should be treated equally.

For different combinations of jets,including the ”real” decay fragments, the correctly assigned jet combination would be expected tohave the highest integrated probability. That suggests a procedure in which all jet combinationsthat pass the constrained fitting criteria be accumulated in the probability for that event.

Whilewrong choices will contribute to this accumulated distribution, the correct choice usually shoulddominate. Furthermore, the relative probability for each combination is a meaningful quantity,since each kinematic configuration is treated uniformly with the same parameterization.So the distribution with mass of the sum of the probabilities for the different combinationsin an event is a distribution of probabilistically weighted ”events”.

How are other events to betreated? Since there is no way to determine which combination is correct, there is no distinctionbetween a combination formed from one event or any other.

This leads to adopting a procedure inwhich all combinations of jets from all events are treated uniformly. Any kinematic configurationwith a non-vanishing probabilistic weight at a given mass should be counted in the distributionat that mass.

Hence all the individual weighted combinations should be added together to formthe ”accumulated” probability distribution as a function of the hypothetical top quark mass. Thisaccumulated probability distribution is just the sum of the probability distributions from eachevent.

This distribution will provide the means to separate ”top” events from non-top events.To test the fitting procedure advocated here, we have used several sets of simulated data,belonging to two categories. Simulations of top pair production events, with different levels ofcomplications due to real detectors, were generated and will be referred to as TTbar sets.

Thesimplest of these, a ”toy model”, is a parton level scheme in which the top pair is producedback-to-back in the center -of-mass of the incoming quark-antiquark or gluon pair. The energiesand angles of the t¯t are chosen at random as are the directions of their decay products (throughthe physical W) in their respective rest frames, all subject to correct kinematic constraints.

Amore complex set will also be used for TTbar, below.A second category consists of Standard Model Monte Carlo simulations that produce jetsand a W that decays leptonically. These are the W+3jet and W+4jet sample.

Both categoriesare generated by starting with the relevent tree level Feynman diagrams. For t¯t production theq¯q annihilation or gluon fusion amplitudes are relevent.For competing non-top mechanisms,processes in which quarks or gluons produce a W boson and four hard gluons, or two gluonsand a light quark pair, are initiators[7].

Then the outgoing partons must fragment into hadron

jets through some standard iteration scheme[9]. Finally the measurement of the kinematics ofthose jets and leptons depends on the detection system and the various jet finding procedures andfragmentation codes.

We have used the CDF detector simulation code QFL, which incorporatesthe efficiencies and peculiarities of that particular system as understood in the 1988-1989 run.To begin with we show several probability distributions in Fig.2 for different jet combinationsin a single t¯t simulated toy event for mt=140 GeV. The probabilities are added together to obtaina probability distribution for that event, as we proposed above.

Note that the peak near 140GeV is due to the ”correct” combination and dominates over the contributions from ”wrong”combinations, as anticipated.When 100 tightly constrained toy events are randomly generated and put through the fit-ting procedure the resulting accumulated probability distribution, Fig.3a, is sharply peaked, withsmaller, insignificant structures arising from wrong jet combinations, as expected. The broadeningof the peak is from the gaussian probabilities for the CDF indeterminancy of jet energies.To simulate the mismeasurements arising from the detection efficiencies as well as smearingthrough fragmentation and soft gluon bremsstrahlung, we randomly alter the values of the jets’4-momenta using the same CDF determination of the standard deviation in energy determinationand the corresponding inverse gaussian or error function to reproduce the gaussian distributionof energy values.

The likelihood distribution for these ”smeared” events, Fig.3b, remains peakedat the fixed mass of 140 GeV, although there is some broadening and the wrong combinationsare relatively more significant. Nevertheless, it remains quite clear that the identification of thetop peak, as well as the mass determination, are quite striking in the accumulated distributionfunction.

To be more realistic, cuts were applied: i) three jets originating from the ”hadronic” topdecay were required each to have high transverse momentum; ii) all four jets were required to havetheir pseudo-rapidities limited; iii) the missing transverse energy in the entire event was requiredto be large. These cuts are motivated by Monte Carlo studies and the general characteristics ofexisting p¯p detectors, with CDF as an example.To include the detector effects a full simulation of the CDF detector has to be performed inthe environment of jets of hadrons.

A distribution for a sample of ISAJET+QFL events[9], withthe cuts indicated above, was generated with mt=130 GeV, and is shown in Figure 4. Again, thedistribution exhibits a sharp, somewhat displaced peak.What of the background for the lepton and jets ?

Simulation of the expected Standard Modelcontribution to the relevant final states has been constructed through the mechanism of partonsproducing a W boson along with hard gluons and/or quark-anti-quark pairs[7]. Applying the CDFdetector constraints to pseudo-rapidity and transverse energy of the parton level processes resultedin a prediction that the background would be no more than roughly 50% of the signal[7].

Butwill that be sufficient to muddy the clean determination of the toy model by enhancing particularregions of the three jet mass ? To test that possibility we took a large sample of the W + 3jets and W + 4 jets generated[11] with VECBOS Monte Carlo program which implements theQCD calculation of Berends, et.al.[7].

A sample of W+4jet events corresponds to an integratedluminosity of 112 pb−1 and that of W+3jet events to 128 pb−1. The events were subjected tothe same analysis as t¯t events.

The resulting accumulated probability distributions as a functionof mt and normalized to the luminosity of 4 pb−1 (integrated luminosity of 1988-1989 CDFrun) are shown in Figure 5. The likelihood distribution for t¯t ISAJET+QFL Monte Carlo events,normalized to the same luminosity, has been superimposed on the W+jets’ graphs.There isvery little background left above mt=120 GeV.

The completely different forms of the probabilitydistributions for TTbar simulations versus W+jets demonstrates that the accumulated probabilitydistribution function for real data will effectively distinguish t¯t production from Standard Modelbackground.To summarize our results, first note that the geometric method offers many advantages overa multi- dimensional, multiply constrained fit of kinematic variables. The parameterization for asingle ellipse is one-dimensional, uniform and continuous.

So the corresponding linear phase space

density, which is also fairly uniform, is well represented by a uniform grid (set by dividing the ellipseinto equal angular steps). The uniform parameterization is the same for all ellipses, all top massesand all events.

Hence the probability assignment for a given mass in a given event is commensuratewith the probability assignment for any other mass in any other event. The events can be combinedto form an accumulated distribution with the relative probability of each event being a meaningfulquantity.

The continuity of the ellipses provides for the unambiguous determination of kinematicalconfigurations, without the two-fold ambiguity in solving algebraic constraint equations that canlead to spurious solutions in handling large samples. The single parameter ellipse is the geometricrealization of the continuum of solutions to constraint equations.From a practical point of view, the ellipse provides an efficient, systematic and unambiguousrealization of the kinematic fitting procedure.

Because there is a single parameter for each mass,the probability distribution will be smoothly varying with systematic variation of the measured jetenergies. There is no possibility of falling into local extrema in multi-dimensional phase space, asthere will be in the traditional constrained fit approach.Finally we note that the accumulated probability distribution as a function of mass is completelyconsistent with the requirements for an efficient discriminator.

The peaks for top events are wherethey should be - masses are well determined. The Standard Model background peaks at low mass,giving virtually no substantial contribution in the region above 120 GeV.

If real events containeda top contribution in the mass range above 120 GeV (as suggested in the dilepton event), thisprocedure would provide a dramatic demonstration of the existence of such a top quark.ACKNOWLEDGEMENTSTwo of us (G.R.G. and K.S.) acknowledge the U.S. Department of Energy for partial supportduring the course of this research.

REFERENCES1. For a recent review see G.L.Kane, ”Top Quark Physics”, University of Michigan preprint,UM-TH-91-32 (1991).2.

K. Sliwa (CDF Collaboration), in Zo Physics, Proc. 25th Rencontre de Moriond, Les Arcs,France, 1990, edited by J.Tran Thanh Van (Editions Frontieres, Gif-s,r-Yvette, 1990), p.459;F. Abe, et.al., Phys.Rev.

Letters 68, 447 (1992); P.Langacker and M.Luo, Phys.Rev. D44,817(1991).3.

R.H.Dalitz and Gary R. Goldstein, Phys.Rev. D45, 1531 (1992).4.

I.Bigi, Phys.Lett. B175, 233 (1986); I.Bigi, et.al., Phys.

Lett. B181, 157 (1986).5.

F. Abe et.al. (CDF Collaboration) Phys.Rev.Letters 64 (1990) 147; ”Results from HadronColliders”, Lee Pondrom, in Proceedings of XXV-th International Conference on High EnergyPhysics, Singapore 1990; ”Search for top quark at Fermilab Collider”, K. Sliwa, in Proceedingsof 4-th Heavy Flavour Conference, Orsay 1991.6.

R.H.Dalitz and Gary R. Goldstein, ”The analysis of top-antitop production and dileptondecay events and the top quark mass”, University of Oxford, Department of Physics - TheoreticalPhysics, preprint (1992).7. F.A.Berends, H.Kuijf, B.Tausk, W.T.Giele, Nucl.Phys.B357, 32 (1991).8.

F. Abe et.al. (CDF Collaboration), Phys.Rev.Lett.

64, 157 (1990); P. Bagnaia et.al. (UA2Collaboration), Phys.Lett.

144B, 283 (1984).9.ISAJET is described in F.Paige and S.Protopopescu, in ”Proceedings of the SummerStudy on the Physics of the SSC, Snowmass, Colorado, 1986, edited by R.Donaldson and J.Marx(DPF,American Physical Society, New York, 1986), p.320. QFL is described in C.Newman-Holmesand J.Freeman, in Proc.

of the Workshop on Detector Simulation for the SSC, Argonne, 1987,ed. L. Price (A.N.L.

Report no. ANL-HEP-CP-80-51),pp.190,285.10.

CDF Collaboration, F. Abe et al, Phys. Rev.

Lett. 68 (1992) 1104; we appreciate thehelp of Naor Wainer for providing us with the subroutine which incorporates all the informationon jet errors.11.

We appreciate the work of Jose Benlloch in generating the VECBOS samples.FIGURE CAPTIONS1. The vector and ellipse construction for a possible match of transverse momentum for asimulated top pair production event.2.

Probability distributions for three individual events, generated with Mt=140 GeV. Probabil-ities for ALL combinations for each single event are added together to form a single distributionfor a single event.3.

Accumulated probability distributions for a sample of 100 events generated with Mt=140GeV without (a) and with (b) smearing of the jets according to standard CDF parametrization ofthe error on the jet energy measurement.4. Accumulated probability distributon for a sample of 100 ISAJET+QFL events generated withMt=130 GeV.

The mass shift (from 130 GeV) is understood as an artifact of some incompatibilitiesin simulation.5. Accumulated probability distributions for VECBOS W+3jets and W+4jets samples.

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