S.D. Ellis, 1 J. Huston, 2 and M. Tönnesmann 3

Transcription

1 BUILDING BETTER CONE JET ALGORITHMS S.D. Ellis, 1 J. Huston, 2 an M. Tönnesmann 3 1 Seattle, WA USA 2 E. Lansing, MI USA 3 Föhringer Ring 6, München, Germany (Date: August 16, 2003) The search for physics beyon the Stanar Moel is being greatly enhance by improve theoretical tools an ieas at the same time that vast amounts of new high energy ata are becoming availiable. To make the most of this situation it is essential that we simultaneously improve our phenomenological tools, such as jet algorithms, to more reliably brige the gap between theory an experiment. We present recent results on the evelopment of better cone jet algorithms. I. INTRODUCTION A common facet of essentially all physics stuies at current an future accelerators is the important role of haronic jets in the characteriation of the nal states. These jets are intene to serve as surrogates for the unerlying partons (quarks an gluons) in terms of which, along with leptons, high energy scattering events are most easily analye. The explicit connection between the sprays of nal state particles observe in the etectors an the unerlying partons is typically supplie by a jet algorithm. These algorithms are employe to map nal states, both in QCD perturbation theory (i.e., in terms of a relatively small number of energetic partons) an in the ata (i.e., in terms of the observe harons an/or calorimeter towers), onto jets. Thus, in principle, we can connect the observe nal states, in all of their complexity, with the short-istance perturbative nal states, which are easier to interpret an to analye theoretically. For example, one woul like to be able to use jets to ientify an reconstruct the presence of W s, top quarks an Higgs bosons from their haronic ecays into quarks (an gluons). Given the large challenge inherent in such a proceure, it is no surprise that an important component of the preparations[1] for Run II at the Tevatron, an for future ata taking at the LHC, has been the stuy of ways in which to improve jet algorithms. It is this topic that we will aress in the following iscussion. In orer to unerstan how jet algorithms work it is useful to have in min a simple (essentially classical) spacetime picture of how har-scattering events evolve. Think of time ero as agge by the basic har-scattering process that, via the exchange of large transverse momentum, transforms 2 energetic but longituinally moving (initial state) partons into 2, or more, energetic ( nal state) partons with substantial transverse momentum. This picture serves to characterie the short istance ( 1 fermi) con guration accurately escribe by perturbative QCD. At next-to-leaing orer (NLO) in perturbative QCD there are at most 3 such large transverse momentum nal state partons, with at most 2 in a single jet. As the nal state evolves in time an space there is further (softer an largely collinear) raiation or showering. At istance scales of orer a fermi or larger (non-perturbative) con nement physics or haroniation is relevant. The partons with nonero QCD charge organie themselves into color-singlet harons. To accurately perform the task of connecting xe orer perturbative nal states to the experimentally observe nal states the jet algorithms must be robust uner the impact of both higher orer perturbative an non-perturbative physics. In the context of a theoretical calculation the jet algorithm must be insensitive to the corners of phase space where the perturbation theory is potentially ivergent, e.g., insensitive to emission of soft an/or collinear gluons. Likewise the resulting jets shoul not epen on the etails of the long istance process by which the partons associate themselves into color singlet harons. This inclues both the inherently nonperturbative haroniation process an the essentially perturbative, but all orers showering process. All of these processes ten to smear out the local structure of the low orer perturbative escription. On the experimental sie we esire insensitivity to the e ects introuce by the etectors themselves. We also nee to limit the sie of the require numerical analysis in orer to ensure timely analysis with nite computer resources. The past implementation of jet algorithms has invariably involve compromises on at least some of these issues, as we will iscuss. The (optimistic) quantitative goal is a precision of orer 1% in the comparison between theory an experiment. In this note we will ignore the uncertainties ue to higher orer perturbative corrections an uncertainties in the parton istributions functions an focus on the issues that arise from the algorithms themselves.

2 2 II. CONE JET HISTORY We begin by reviewing cone jet algorithms, which have forme the basis of jet stuies at haron colliers. First employe at the at CERN[2], the iea of the cone jet was explicitly spelle out in the Snowmass Algorithm[3], which was evelope by a collaboration of theorists an experimentalists. The Snowmass Algorithm le to the jet algorithms use by the CDF an DØ collaborations uring Run I at the Tevatron, although neither Collaboration followe the Snowmass prescription precisely. The intuitive picture is that jets shoul be compose of harons or partons that are, in some sense, nearby each other. The funamental cone jet iea is that nearness be e ne in a simple geometric fashion: jets are compose of harons or partons whose 3-momenta lie within a cone e ne by a circle in the angular variables ( ), where = ln(cot 2) is the pseuorapiity an is the aimuthal angle. This iea of being nearby in angle can be contraste with an algorithm base on being nearby in transverse momentum as illustrate by the so-calle Algorithm[4] that has been wiely use at + an colliers, an is now being stuie at the Tevatron[5]. Intuitively we also expect the jets to be aligne with the most energetic particles in the nal state an corresponingly we expect to be able to ientify a unique set of jets on an event-by-event basis. This uniqueness expectation is realie in the Snowmass Algorithm by ientifying jets with stable cones. A stable cone has the property that the geometric center of the cone coincies with the weighte centroi of the particles in the cone. We imagine e ning a nal state in terms of a list of nal state partons (theory) or harons an/or calorimeter towers (experiment) each labele by an inex, a irection ( ) an a scalar transverse energy (assuming that we start with massless 4-vectors), = sin = j! j. Accoring to the Snowmass Algorithm the constituents of a jet ( ) of cone raius an the corresponing jet properties are e ne by the following set of equations 2 : ( ) 2 + ( ) 2 2 = X = X (1) 2 2 = X 2 The rst line speci es that the jet consists of all partons, harons or towers that lie within the istance of the jet cone center, ( ), while the secon line speci es that the weighte centroi of this list of particles coincies with the geometric jet cone center (i.e., the secon line is the stability conition). The thir line e nes the Snowmass total scalar for the jet. Typically the value = 0 7 has been employe as it leas to fairly stable results both theoretically an experimentally. It is important to recognie that jet algorithms involve two istinct steps. The rst step is to ientify the constituents of the jet, i.e., the list of calorimeter towers or harons or partons that comprise the stable cone that is the jet. The secon step involves constructing the kinematic properties that will characterie the jet, i.e., etermine into which bin the jet will be place. In the Snowmass Algorithm the weighte variables e ne in Eq. 1 are use both to ientify an bin the jet, i.e., in terms of an. The simplicity an intuitive appeal of the cone algorithm justify its wie sprea usage at haron colliers an it has prove to be a very useful tool. However, its practical implementation has involve several compromises that now constitute limits to its precision, as we will attempt to explain. In a theoretical calculation[6] the Snowmass Algorithm can be applie quite literally by integrating only over the portions of multi-parton phase space corresponing to parton con gurations that satisfy the stability conitions in Eq. 1. In the experimental case it was imagine that the Snowmass Algorithm woul be applie to each event via an iterative process to search the entire nal state, i.e., the entire particle/tower list mentione above, for sets of nal state particles/towers that satisfy the stability constraint. In practice[1] the experimental implementation of the cone algorithm has been more complicate, involving at least 3 etaile steps without irect analogues in the theoretical implementation. First, experimentalists have employe various short cuts to minimie the (computer) search time. In particular, Run I algorithms mae use of both sees an pre-clustering. The iterative search for stable cones was initiate only at locations provie by preclusters assemble from contiguous see towers, i.e., calorimeter towers with eposite energy exceeing a pree ne limit (typically ~1 GeV). Preclusters are constraine to exten no more than 2 in either or. Starting only at such a seee location in ( ), a list is constructe of the particles (towers) within a istance of the see. Then the weighte centroi for the particles in the list is foun (calculate as in Eq. 1). If the calculate centroi is consistent with the initial cone center, a stable cone has been ienti e. If not, the calculate centroi is use as the center of a new cone with a new list of particles insie an the calculation of the centroi is repeate. This process is iterate (as envisione in the Snowmass Algorithm), with the cone

3 center migrating with each repetition, until a stable cone is ienti e or until the cone centroi has migrate out of the ucial volume of the etector. Note, however, that since the search starts only at the sees, this process may not ientify all possible stable cones an thus may not n all jets. Of particular concern are the nal states that arise from two energetic partons of nearly equal energy that are separate in angle by nearly 2. This con guration will be ienti e as a single jet in the theoretical perturbative analysis with the center of the jet approximately mi-way between the partons. The smeare version of this con guration in the actual (or simulate) nal state, on the other han, will likely exhibit a local minimum of energy (a salepoint) at the mipoint between the partons (assuming the angular smearing ue to showering an haroniation is not large compare to ), with sees only at the location of the partons. As iscusse in more etail below, the experimental iterative search initiate (i.e., seee) only at the location of the two partons will likely n two jets centere on the partons, rather than the higher energy solution containing the remnants of both partons foun by the theoretical analysis. When all of the stable cones in an event have been ienti e, there will typically be some overlap between cones. This situation raises the next issue that has no obvious, nontrivial analogue in the perturbative calculation (at least at low orer) an that was not envisione in the Snowmass Algorithm. It is aresse with a splitting/merging routine in the experimental jet algorithms. In practice the splitting/merging routine involves the e nition of a parameter, typically with values in the range such that, if the overlapping transverse energy fraction (the transverse energy in the overlap region ivie by the smaller of the total energies in the two overlapping cones) is greater than, the two cones are merge to make a single jet. If this constraint is not met, the calorimeter towers/harons in the overlap region are iniviually assigne to the cone whose center is closer. This situation yiels 2 nal jets. As the initial seeing/preclustering process an the nal merging/splitting process were not foreseen in the Snowmass Algorithm, it is perhaps not surprising that the etaile speci cation of these steps were somewhat i erent for CDF an DØ in Run I (e.g., CDF use = 0 75, while DØ use = 0 5). Note that the trivial application of the merging routine to the NLO perturbative analysis explains why the theoretical algorithm is e ne to always keep the jet containing two partons out to a separation of 2, whenever such a solution exists. In such a con guration the 2 possible jets that are compose 1 parton each exhibit 100% overlap with the 2-parton jet. Thus by the usual merging routines, the single parton jets shoul always be merge with the 2-parton jet. The nal point that istinguishes the theoretical algorithm from the experimental algorithms is speci c to the JetClu algorithm employe by the CDF Collaboration. The JetClu algorithm exhibits a not wiely ocumente an troubling feature calle ratcheting. All see towers initially inclue in a see cone (the rst cone aroun a see) remain associate with that cone (i.e., on its list) even as the center of the cone migrates uring the search for a stable solution to Eq. 1. This remains true, by construction, even if the center moves further than R away. This ratcheting feature leas to several results not mimicke in the theoretical algorithm, or in other experiments. It leas to nal stable cones that are often not cone-shape (although this issue alreay arises from splitting/merging). Ratcheting also implies that all of the initial see towers are inclue in (at least one of) the nal jets. More importantly it means that cones ten to become stable after only a short migration, but in a fashion epenent on the etails of the showering/haroniation process (an the initial pre-clustering process). This property argues against using JetClu for precision comparisons with theory. The essential challenge in the use of jet algorithms is to unerstan the quantitative implications of the i erences between the algorithms as applie by i erent experiments an as applie in theoretical calculations. This is the only way to control the uncertainties. It is this goal that is the primary concern of this paper. Clearly i erences that lea to substantial uncertainties ( 1%) shoul be eliminate, or well enough unerstoo that they can be correcte for. Consier rst the better known issues raise above. As suggeste earlier the use of sees in the experimental algorithms means that certain con gurations kept by the theoretical algorithm are likely to be misse by the experimental one. In particular the symmetric con guration of two energetic partons nearly 2 apart can satisfy the strict application of Eq. 1 to form a single jet in a perturbative calculation, with no energy at the center of the jet. Yet the same con guration, incluing the e ects of showering an haroniation, will likely lea to two well separate sees (with only limite in between) an thus two jets in the experimental algorithm, which may not be subsequently merge. When this point was recognie, the theoretical parameter [7] was introuce in the perturbative calculation to mimic this feature of the experimental analysis. With this parameter inclue two partons with angular separation q = ( ) 2 + ( ) 2 2 (2) are not allowe to comprise a single jet, even if they satisfy Eq. 1. Comparison of the theory to ata[7] suggeste that the value = 1 3 provie the best escription of the ata at NLO. In completely inepenent analyses[8] the CDF an DØ Collaborations also foun that two initially inepenent jets (selecte, for example, from i erent events) ha to place within ' 1 3 of each other in a simulate event in orer to be ienti e as a single jet by the experimental algorithms. Thus the theoretical parameter is presumably 3

4 also proviing some level of simulation of the splitting/merging proceure. One coul also imagine explicitly incluing sees in the perturbative calculations at higher orers, e.g., requiring the presence of a soft gluon between the two partons just mentione. However, this explicit see e nition introuces a highly unesirable (logarithmic) epenence on the see cut (the minimum require to be treate as a see cell)[9]. The kinematic con gurations corresponing to this scenario are suggeste in Fig. 1, where the two energetic partons 4 FIG. 1: Two partons in two cones or in one cone with a (soft) see present. coul be in either 2 cones (LHS) or a single cone seee by a soft gluon (RHS). Of course, the most esirable approach is ree ne the experimental algorithms in orer to remove the nee for the parameter as will be escribe in more etail below. Concerning the issue of ratcheting in the JetClu algorithm, we note that there is no reasonable way to simulate the role of ratcheting in the theoretical calculations, since its role epens in etail on the level of seconary (soft) raiation present in the event. As suggeste above, this point shoul be interprete as a strong argument against the continue use of JetClu. For reliable comparisons of theory an experiment it is essential that we eliminate such i erences between the theoretical an experimental algorithms. III. RUN II IMPROVEMENTS With the avent of a new, higher energy an higher luminosity, Run II at the Tevatron, theorists an experimentalists have been o ere another opportunity to construct algorithms whose application to perturbation theory an to ata analysis will be more in parallel. This will in turn allow greater precision in comparing theory an experiment. In particular the Run II Workshop Proceeings[1] recommen that the CDF an DØ Collaborations employ, as much as possible, ientical jet algorithms, incluing the use of 4-vector kinematics (4-vector instea of scalar summation an = 0 5 ln[( + ) ( )] instea of = ln(cot 2)) an a common splitting/merging proceure. The Workshop Proceeings also provie algorithms evelope speci - cally to aress the issue of sees, in particular the Mipoint Algorithm an the Seeless Algorithm. With the Mipoint Algorithm one procees initially as in the original algorithms, i.e., starting with sees. Using the stable cones thus ienti e, one explicitly tests for the possibility of a new stable cone centere ( in or ) between two stable cones alreay foun by placing a new see miway between all pairs of stable cones separate by less than 2. The Mipoint algorithm is inclue in the analysis iscusse below. The central innovation of the Seeless Algorithm is to place an initial trial cone at every point on a ranomly locate but regular lattice in ( ). For maximum coverage the lattice shoul be approximately as ne-graine as the etector. It is not so much that this algorithm lacks sees, but rather that the algorithm puts see cones everywhere of interest. The Seeless Algorithm can be streamline by imposing the constraint that a given trial cone is remove from the analysis if the center of the cone migrates outsie of its original lattice cell uring the iteration process. The streamline version still samples every lattice cell for stable cone locations, but is less computationally intensive. Our experience with the streamline version of this algorithm suggests that there can be problems ning stable cones with centers locate very close to the cell bounaries of the lattice. This technical i culty is easily aresse by enlarging the istance that a trial cone must migrate before being iscare. For example, if this istance is 60% of the lattice cell with instea of the efault value of 50%, the problem essentially isappears with only a tiny impact on the require time for analysis. While this algorithm is less familiar than the Run I algorithms an apparently is more computer intensive than

5 the Mipoint Algorithm, we still avocate its use. It is inherently more similar in structure to the theoretical algorithm. 5 IV. NEW ISSUES The rest of this paper is evote primarily to an analysis of the Mipoint algorithm, especially as compare to the JetClu algorithm of CDF an the theoretical algorithm, with an eye towars ientifying issues that coul limit the precision of the comparison with theory. In particular, we have foun that there exist previously unrecognie nal state con gurations that are likely to be misse in the ata, compare to the theoretical result. We escribe the application of representative jet algorithms to ata sets that were generate with the HERWIG 6.1 Monte Carlo[10] an then processe through a CDF etector simulation. We inclue results from the JetClu algorithm both with an without the ratcheting feature escribe above. Following the recommenation of the Run II Workshop, we use 4-vector kinematics for the Mipoint Algorithm an place the cone at the mipoint in ( ), where is the true rapiity. In the JetClu Algorithm the value = 0 75 was use (as in the Run I analyses), while for the Mipoint an Seeless Algorithms the value = 0 5 was use, as suggeste in the Workshop Proceeings[1]. Finally, for completeness, we also inclue in our analysis a sample Algorithm. Starting with a sample of 250,000 events, which were generate with HERWIG 6.1 an processe through a CDF etector simulation an which were require to have at least 1 initial parton with 200 GeV, we applie the various algorithms to n jets with = 0 7 in the central region (j j 1). We then ienti e the corresponing jets from each algorithm by ning jet centers i ering by 0 1. The plots in Fig. 2 inicate the average i erence in for these jets as a function of the jet. From these results we can raw several conclusions. First, the Algorithm ienti es jets with values similar to those foun by JetClu, ning slightly more energetic jets at small an somewhat less energetic jets at large. We will not iscuss this algorithm further here except to note that DØ has applie it in a stuy of Run I ata[12] an in that analysis the Algorithm jets seems to exhibit somewhat larger than expecte from NLO perturbation theory. The cone algorithms, incluing the JetClu Algorithm without ratcheting, which is labele JetCluNR, ientify jets with systematically smaller values (by approximately 0.5% to 1 %) than those ienti e by the JetClu Algorithm (with ratcheting). Due to the rapi fallo of the jet cross section with the variable, this systematic isplacement in results in a corresponing systematic reuction of approximately 5% in the jet cross section at a given value. This sort of systematic variation falls well within our e nition of an important e ect. We believe that this systematic shortfall can be unerstoo as resulting from the smearing e ects of perturbative showering an non-perturbative haroniation, which have not previously been consiere. V. A SIMPLE MODEL To provie insight into the issues raise by Fig. 2 we now iscuss a simple, but informative analytic picture. It will serve to illustrate the impact of showering an haroniation on the operation of jet algorithms. We consier the scalar potential (! ) e ne as a function of the 2-imensional variable! = ( ) by the integral over the transverse energy istribution of either the partons or the harons/calorimeter towers in the nal state with the inicate weight function, Z (! ) = 1 2 = 1 X 2 2 ³ 2 (!! 2 ) ³ 2 (!! 2 ) (! ) (3) ³ 2 (!! 2 ) ³ 2 (!! 2 ) The secon expression arises from replacing the continuous energy istribution (! ) with a iscrete set, = 1 to of elta functions, representing the contributions of either a iscrete con guration of partons or a set of calorimeter towers (an harons). Each parton irection or the location of the center of each calorimeter tower is e ne in, by the 2-D vector! = ( ), while the parton/calorimeter cell has a transverse energy (or ) content given by. This function is clearly relate to the energy in a cone of sie containing the towers whose centers lie within a circle of raius aroun the point! as e ne by the function. More importantly it carries information about the locations of stable cones. The points of equality between the weighte centroi an the geometric center of the cone (i.e., the stable cones) correspon precisely to the

6 6 E T [GeV] JetClu E T KtClus JetClu - E T vs. E T (matching conition: R < 0.1) E T [GeV] JetClu E T MiPoint - E T JetClu vs. E T (matching conition: R < 0.1) E T [GeV] E T [GeV] E T [GeV] JetClu E T Seeless - E T JetClu vs. E T (matching conition: R < 0.1) E T [GeV] JetClu E T JetCluNR - E T JetClu vs. E T (matching conition: R < 0.1) E [GeV] T E [GeV] T FIG. 2: Comparison of in matche jets ienti e by various jet algorithms. minima of, as in familiar mechanics problems. The corresponing force or graient of this function has the form (note that the elta function arising from the erivative of the theta function cannot contribute as it is multiplie by a factor equal to its argument)! (! ) =! r (! ) = X (!! ) ³ 2 (!! 2 ) (4) This expression vanishes at points where the weighte centroi coincies with the geometric center, i.e., at points of stability (an at maxima of, points of extreme instability). One can think of this force as riving the ow of cone centers uring the iteration process escribe earlier. The corresponing expression for the energy in the cone centere at! is (! ) = X ³ 2 (!! 2 ) (5) Since the primary unerlying issue is uner what conitions two energetic partons are associate with the same nal jet, we can evelop our unerstaning of the above equations by consiering a scenario (containing all of the interesting e ects) involving 2 partons separate by j! 2! 1 j =. It is su cient to specify the

7 7 a) 2.0 b) E C (r), V(r), 1.5 = 0 = = E C (r) -0.1 V(r) r FIG. 3: 2-Parton istribution: a) transverse energy istribution; b) istributions ( ) an ( ) in the perturbative limit of no smearing. energies of the 2 partons simply by their ratio, = Now we can stuy what sorts of 2 parton con gurations yiel stable cones in the 2-D phase space speci e by 0 1, 0 2 (beyon 2 the 2 partons are surely in i erent cones). As a speci c example consier the case 1 = 0, 2 = = 1 0 an = 0 6 with = 0 7 (a typical experimental value). The unerlying (partonic) energy istribution is illustrate in Fig. 3a, representing a elta function at = 0 (with scale weight 1) an another at = 1 0 (with scale weight 0.7). This simple istribution leas to the functions ( ) an ( ) inicate in Fig. 3b, where the 1-D variable is the istance along the irection between the partons, =! (! 2! 1 ). In going from the true energy istribution to the cone istribution ( ) the energy is e ectively smeare over a range given by, each region having sharp bounaries. In the potential ( ) the istribution is further shape by the quaratic factor 2 ( ) 2. We see that ( ) exhibits 3 local minima corresponing to the expecte stable cones aroun the two original partons ( 1 = 0 an 2 = 1), plus a thir stable cone in the mile ( 3 = (1 + ) = in the current case). This mile cone inclues the contributions from both partons as inicate by the magnitue (1.6) of the mile peak in the function ( ). Note further that the mile cone is foun at a location where there is initially no energy in Fig. 3a, an thus no sees. One naively expects that such a con guration is not ienti e as a stable cone by the experimental implementations of the cone algorithm that use sees simply because they o not look for it. Note also that, since both partons are entirely within the center cone(j 1 3 j j 2 3 j ), the overlap fractions are unity an the usual merging/splitting routine will lea to a single jet containing all of the initial energy (1 + ). This is precisely how this con guration was treate in the NLO perturbative analysis of the Snowmass Algorithm[6] (i.e., only the leaing jet, the mile cone, was kept).

8 8 a) R sep = 2.0 cone jet LCR cones jet cones jets b) R sep = 1.3 cone jet 3 LCR cones jet cones jets FIG. 4: Perturbation theory cone/jet structure: a) = 2; b) = 1 3. The subscripts, LCR, refer to whether the cone/jet is foun at the position of the Left parton, Right parton or at a intermeiate Central position. The cross inicates the con guration explore in subsequent gures. Similar reasoning leas to Fig. 4a, which inicates the various 2 parton con gurations foun by the perturbative stable cone algorithm. In the 2-D omain of ( ) there are three istinct regions. For an all values of the potential ( ) has just a single minimum in the central region, as inicate by the C subscript ( = (1 + ) ). Thus this region yiels a single stable cone an a single jet containing both partons. In the triangular region (1 + ) the potential function has three minima (as above) an one ns 3 stable cones at the locations of the two partons ( L an R, = 1 = 0, = 2 = ) an at an intermeiate central location inicate by the C subscript ( = (1 + ) ). As in the speci c example escribe above, both partons lie within this central cone (j j =, j j = (1 ) (1 + ) (1 ) ) an by the usual merging rules (the overlap is 100%) all three cones are merge to a single central jet (at ), again with all of the energy. For (1 + ) the potential exhibits two well separate minima at = 0 an = with no overlap. Thus in this region we n 2 stable cones an 2 jets, each containing one parton, of scale energies 1 an. Thus, except in the far right region of the graph, the 2 partons are (in perturbation theory) always merge to form a single jet. In particular, the parton con guration of Fig. 3a, which is inicate by the cross symbol in Fig. 4a, yiels 3 stable cones an a single nal jet encompassing both partons. As suggeste above we expect that the use of sees in experimental algorithms will play a role in the mile, 3 LCR, region, but not elsewhere. While the potential ( ) in perturbation theory has three minima in this region, we may not n all three of the corresponing stable cones if we only look for stable cones near the sees. Looking back at Fig. 3a we expect to have sees near 1 an 2 but presumably not in the central region between the two partons where there is no energy. Trial cones near 1 will migrate to the L stable

9 position uner iteration while a similar ow will occur for sees near 2 to the R stable cone. With no sees in between there may be no trial cones owing to the C stable cone. The a hoc parameter was introuce to provie a (crue) simulation of this situation (see further iscussion below) in the NLO calculations[7]. The parameter is e ne such that stable cones containing 2 partons are not allowe for partons separate by. As a result cones are no longer merge in this kinematic region. Thus we are moeling the e ect of sees by the replacement 3 LCR cones! cones in this region. In the present language this situation is illustrate in Fig. 4b corresponing to = 1 3, = This speci c value for was chosen[7] to yiel reasonable agreement with the Run I ata. The conversion of much the 3 cones! 1 jet region ( 0 91, ( ) ) to 2 cones! 2 jets has the impact of lowering the average of the leaing jet an hence the jet cross section at a xe. Parton con gurations that naively prouce jets with energy characterie by 1 + now correspon to jets of maximum energy 1. However, the large, large part of phase space is far from the collinear an soft singularities of the perturbation theory an oes not make a major contribution to the cross section. Thus the resulting reuction in the jet cross section in going from = 2 0 to = 1 3 is only approximately 5.2% at = 100 GeV an 3.7% at = 400 GeV. This level of precision was not essential for Run I, but is relevant to the present iscussion. This magnitue of change in the jet cross section correspons approximately to a 1% ecrease in the average jet consistent with the level of mismatch exhibite in Fig. 2. Note that with this value of the speci c parton con guration in Fig. 3a, which is inicate by the cross symbol in Fig. 4b, will now yiel 2 jets (an not 1 merge jet) in the theoretical calculation. As mentione earlier, it has been recommene that this issue with sees in the 3 LCR region be aresse in Run II via the Mipoint an Seeless Algorithms. The expectation was that both woul succee in ientifying the missing stable cone at. However, as inicate in Fig. 2, neither of these two algorithms reprouces the results of JetClu. Further, they both ientify jets that are similar to JetClu without ratcheting. Thus we expect that there is more to this story. As suggeste earlier, a major i erence between the perturbative level, with a small number of partons, an the experimental level of multiple harons is the smearing that results from perturbative showering an nonperturbative haroniation. For the present iscussion the primary impact is that the starting energy istribution will be smeare out in the variable!. We can simulate this e ect in our simple moel using Gaussian smearing, ³ i.e., we replace the elta functions in Eq. 3 with 2-D Gaussians of with (/ (!! ) 2 2 ). (Since this correspons to smearing in an angular variable, we woul expect to be a ecreasing function of, i.e., more energetic jets are narrower. We also note that this naive picture oes not inclue the expecte color coherence in the proucts of the showering/haroniation process, nor its stochastic nature.) The rst impact of the simulate smearing is that some of the energy initially associate with the partons now lies outsie of the cones centere on the partons. This e ect, typically referre to as splashout in the literature, is (exponentially) small in this moel for. Here we will focus on less well known but phenomenologically more relevant e ects of the smearing. The istributions corresponing to Fig. 3b with = 1 0 an = 0 6, but now with = 0 10 (instea of = 0), are exhibite in Fig. 5a. With the initial energy istribution smeare by, the potential ( ) is now even more smeare an, in fact, we see that the mile stable cone (the minimum in the mile of Fig. 3b) has been washe out by the increase smearing. Thus the cone algorithm applie to ata (where such smearing is present) may not n the mile cone that is present in perturbation theory, not only ue to the use of sees but also ue to this new variety of smearing correction, which reners this cone unstable. Note that this conclusion obtains even though there is a clear maximum in ( ) at the location where the mile stable cone use to be. Since, as a result of this smearing correction, the mile cone is not stable, this problem is not aresse by either the Mipoint Algorithm or the Seeless Algorithm. Both algorithms look in the correct place, but they look for stable cones, which may not present in real ata. This point is presumably part of the explanation for why both of these algorithms isagree with the JetClu results in Fig. 2. Our stuies also suggest a further impact of the smearing of showering/haroniation that was previously unappreciate. This new e ect is illustrate in Fig. 5b, which shows ( ), still for = 0 6 an = 1 0 but now for = With the increase smearing the secon stable cone, corresponing to the secon parton, has now also been washe out, i.e., the right han local minimum has also isappeare. While the e ect of the smearing is intuitively reasonable, i.e., the istributions become increasingly smoother, the resulting loss of stable cones was previously unappreciate. To help make this scenario more visual Figs. 6 to 8 exhibit the 2-D structure inherent in Eq. 3 in terms of contour plots of the potential (! ). The arker shaes inicate lower values of potential, while the white ots inicate the location of the initial partons. In each gure the white circle(s) inicates the location of the nal jet(s). The smooth evolution of the potential as the smearing increases is clear. With no smearing symmetrical minima aroun each parton are apparent along with the eeper minimum in the center. This gure also makes the point that, even when the central minimum is present, it may serve as an attractor for only a small area in the ( ) plane. With some smearing as in Fig. 10 the central minimum has evolve into just a lobe on the left-han minimum an we n just two nal jets. 9

10 a) = 0.1 E C (r), V(r), b) = 5 E C (r), V(r), E C (r) E C (r) V(r) V(r) r r FIG. 5: The istributions ( ) an ( ) with =1 0 an =0 6 for smearing with = a) 0.1; b) 5. FIG. 6: Contour plot of the potential ( ) in 2-D for =0 6 an =1 in the perturbative limit ( =0). With further smearing as in Fig. 11 the lobe extens to the right-han parton an only one minimum an one jet remain. This same, large smearing, situation is exhibite in the case of Monte Carlo ata by the lego plot in Fig. 9 inicating the jets foun by the Mipoint Algorithm in a speci c Monte Carlo event. The Mipoint Algorithm oes not ientify the energetic towers (shae in black) to the right of the (more lightly shae) energetic ienti e central jet as either part of that jet or as a separate jet, i.e., these obviously relevant towers are not

11 11 FIG. 7: Contour plot of the potential ( ) in 2-D with =0 6 an =1 for =0 1. FIG. 8: Contour plot of the potential ( ) in 2-D with =0 6 an =1 for =0 25. foun to be in a stable cone. The iteration of any cone containing these towers invariably migrates to the nearby higher towers. We can further escribe the impact of these smearing e ects by looking again at the stable cone content in the ( ) plane as we i in Fig. 4. The resulting structure is inicate in Figs. 10 an 11. The special case ( = 1 0, = 0 6) that we have alreay iscusse is again marke in the gures by the cross. Comparing to the perturbative result in Fig. 4b (the three regions of the previous graph with = 1 3 are inicate by the ashe-otte line as a reference) we note several features. The small separation region,, remains unchange. There is still a single stable cone intermeiate in position between the 2 partons, which will yiel a single central jet with both partons in the jet. The bulk of the region for has change from yieling 3 stable cones to yieling just a single central stable cone. This will yiel a single central jet with both partons inclue, This conclusion is presumably inepenent of the see issue as trial cones seee near 1 an 2 will now ow to the central stable cone an not be stoppe by the now absent LR stable cones as was expecte in the previous iscussion without smearing. Thus the jet structure in this region is largely unchange from the unsmeare case perturbative incluing the parameter. On the ne etails, e.g., the speci c location of the bounary, epens on the value of. Note that this e ect of smearing in this region is consistent with our use of the parameter (i.e., we assume that the central jet was ienti e in this region even if sees were use). The bulk of the region of larger separation,, now yiels two stable cones, each one containing a single parton an locate essentially at that parton, as inicate by the LR subscript. The example, = 1, = 0 6, stuie above falls in this region for the smaller smearing. For the smaller smearing case of Fig. 10 there is still a iscernible region where 3 stable cones are ienti e. Depening on the role of sees this may or may not yiel a central jet. Base on the iscussion above, we expect the use of sees in the experimental algorithms to mean that in most of this region only two jets, L an R for each parton, are foun an there is no central combine jet (as suggeste by the bounary). There is also a small region where only the left-han an central cones are still stable, yieling essentially the original central jet with both partons. Even with sees invoke, a trial cone seee by the right-han parton

12 E T FIG. 9: Result of applying the Mipoint Algorithm to a speci c Monte Carlo event in the CDF etector. shoul ow to this central stable cone. Both of these regions (2 LC an 3 LCR ) have essentially isappeare with the larger smearing case of Fig. 11. The other new con guration is a single stable cone near the location of the left-han parton, with both the central an right-han cones washe out by the smearing. The ashe bounary between the an 1 L regions is meant to inicate that this is a soft transition with the location of the single stable cone moving smoothly leftwar as increases. As iscusse earlier, our special case (the cross) falls into this region for the larger smearing (recall that there is only the L stable cone in Fig. 8). We expect that an e ective smearing intermeiate between = 0 1 an = 0 25 is the most realistic value for this parameter in the sense of most closely matching the e ects of real showering an haroniation for the purposes of the present iscussion. Thus Figs. 10 an 11 suggest that the a hoc parameter = 1 3 a ors a reasonable escription of how the cone algorithm works in practice an o ering at least a partial explanation of why it prouce perturbative results similar to the ata. Base on the current analysis, however, its role is more to simulate the e ects of smearing an less the e ect of sees. The e cacy of the parameter is perhaps best highlighte by reinterpreting the stable cones of Figs. 10 an 11 in terms of ienti e jets, assuming sees act as escribe above (3 LCR!, 2 LC! ), as illustrate in Fig. 12. The point is to notice the similarity between Fig. 12 an the perturbative result, with, in Fig. 4. This similarity is mae even stronger when we note that, in terms of the leaing jet, 1 L is essentially equivalent to at least for small values where the R jet has much lower energy. It is worthwhile mentioning that merging plays essentially no role here. In the region the two cones are too well separate to be merge, even if = 0 5. This will not be the case in the next section. VI. RATCHETING Since our goal is to unerstan the etaile i erences between theoretical an experimental applications of cone jet algorithms, we must say something about the role of the previously mentione ratcheting e ect in the CDF JETCLU jet algorithm. This feature of the algorithm ensures that as the trial cone migrates away from the initial see position, any calorimeter tower inclue in the initial trial cone stays with trial cone ( once in a cone, always in a cone ), which is therefore no longer cone shape in the ( ) plane. Thus with ratcheting turne on the history of a migrating trial cone matters, an e ect with no analogue in the perturbative calculations. Ratcheting means that the lists of towers to be summe over in Eqs. 1 an 3 grow with migration an a tower no longer nees to be within of the cone center to be inclue. In the current analysis we can e ne the integral (the potential) over the smeare pro les to take this ratcheting into account, with two istinct potentials epening on whether we are seee (by assumption) by the left-most

13 13 = 0.10 (R sep = 1.3) 3 LCR 2 LC 1 L 1 L FIG. 10: Stable cone structure foun by the cone jet algorithm with unerlying partons incluing gaussian smearing with with =0 1. The ashe-otte lines inicate the unsmeare, perturbative result with =1 3. = 5 (R sep = 1.3) 2 LC 3 LCR 1 L FIG. 11: Stable cone structure foun by the cone jet algorithm with unerlying partons incluing gaussian smearing with with =0 25. The ashe-otte lines inicate the unsmeare, perturbative result with =1 3.

14 14 a) b) = 0.10 (R sep = 1.3) = 5 (R sep = 1.3) 1 L 1 L FIG. 12: Ienti e jets structure in( ) plane incluing smearing an sees: a) =0 1, b) =0 25. or right-most parton. The stable cones are then the sum of the stable cones foun from both potentials. In general, the central stable cones ienti e by the two potentials (if present) are nearly coincient. On the other han, the right-han potential will never exhibit a left-han stable cone. Due to ratcheting, the trial cone cannot migrate that far. Likewise the left-han potential will never exhibit a right-han stable cone. With enough smearing present the left-han potential (base on the larger energy left-han parton) will ten to have only a left-han stable cone with its central stable cone washe out by the smearing. In contrast the right-han potential (base on the lower energy right-han parton) will still exhibit a central stable cone even when the right-han stable cone is lost ue to smearing. The analogues of the stable cone structures above, but now with ratcheting simulate as just escribe, are exhibite in Figs. 13 an 14. We see from these gures that the impact of ratcheting is to greatly expan the 2 LC an 3 LCR regions at the expense of the 1 L an regions. As suggeste above, the e ect of ratcheting is to keep the central stable cone even when smearing is present. Of particular importance is the appearance of a large region below the line = ( ) that is now of type 2 LC, instea of 1 L (or even as it was in the perturbative calculation), where the central stable cone arises from the ratchete cone seee by the right-han parton. Thus this central stable cone, ue to ratcheting, will inclue the energy from the right-han parton. Further these two stable cones are close enough to be merge into a single jet, 2 LC (stable cones)! (jet). This implies that in this region, where the cross section is large, we will ientify a merge jet (encompassing both partons) using the experimental algorithm with sees an ratcheting on (realistic) ata with smearing. Note that this is just the region corresponing to Fig. 9 where the i erence between 2 LC with ratcheting an 1 L without ratcheting explains why the ark towers in the LEGO were not assigne to any jet. This point is further illustrate in Fig. 15, which shows the ienti e jet structure an is the analogue of Fig. 12. As before we have assume that the impact of sees is to ensure that we miss the central cone in the con guration 3 LCR, i.e., 3 LCR!. These gures are to be compare to each other an the perturbative result in Fig. 4. Again we see that the parameter provies a reasonable simulation of see an smearing e ects. However, in the region below the line = ( ) mentione above, the result with ratcheting is i erent in that this region now has a leaing C type jet (with both partons) while the perturbative result an the non-ratchete result has leaing L type jets (with just the left-han parton). Since this merging of the two partons has the e ect of increasing the of the primary

15 15 = 0.10 (Rsep = 1.3) 3 LCR 2 LC 1.2 FIG. 13: Stable cone structure foun by the cone jet algorithm with unerlying partons incluing gaussian smearing with with =0 1 an ratcheting. The ashe-otte lines inicate the unsmeare, perturbative result with =1 3. jet in the event, we expect the MiPoint Algorithm (without ratcheting) to yiel a smaller on average than JETCLU, just as we saw in Fig. 2. The current analysis also suggests a similar i erence from the NLO result with, i.e., the e ects escribe here suggest that the inclusive jet cross section e ne by JETCLU shoul be larger by 3-5% than that e ne by NLO perturbation theory with = 1 3. VII. POSSIBLE IMPROVEMENT In summary, we have foun that the impact of smearing ue to showering an haroniation is expecte to be much more important than simply the leaking of energy out of the cone (splashout). Certain stable cone con gurations, present at the perturbative level, can isappear from the analysis of real ata ue to the e ects of showering an haroniation. This situation leas to corrections to the nal jet yiels that are relevant to our goal of 1% precision in the mapping between perturbation theory an experiment. Compare to the perturbative analysis of the 2-parton con guration, both the central stable cone an the stable cone centere on the lower energy parton can be washe out by smearing. Further, this situation is not aresse by either the Mipoint Algorithm or the Seeless Algorithm, both of which assume that the central stable cone is present. One possibility for aressing the missing mile cone woul be to eliminate the stability requirement for the ae mipoint cone in the Mipoint Algorithm. However, if there is enough smearing to eliminate also the secon (lower energy) cone, even this scenario will not help, as we o not n two cones to put a thir cone between. Another approach, which we will explore here, is to attempt to reuce the impact of smearing. Recall that nothing preclues us from employing two i erent cone sies. We can use one uring the search for the stable cones an the secon uring the calculation of the jet properties. Since the e ective smearing in the potential (! ) that escribes the location of the stable cones is a combination of that ue to the physical e ects of showering an haroniation an that ue to the sie of the cone, using a smaller value of cone raius, e.g., 0 = 2, uring the search for stable cones will reuce the smearing e ects. We can still use = 0 7 in the jet construction phase. With this choice of 0 we n the stable cone structure illustrate in Fig. 16. We can unerstan this gure by noting that in rst approximation the bounaries in these graphs are function of. This when we ecrease by a factor of 2, the bounaries move to the left by a corresponing amount. The major change with this xe algorithm is that the bounary between 1 stable cone an 2 stable cones occurs at much smaller. There is also a new region labele 2 LR separate from the usual region by a soft bounary inicate by the ashe line (similar in spirit to the!1 L bounary), that is intene to inicate that in this region the 2 stable cones are actually both rather central, i.e., not at the parton locations.

16 16 = 5 (Rsep = 1.3) 2 LC 3 LCR 1.2 FIG. 14: Stable cone structure foun by the cone jet algorithm with unerlying partons incluing gaussian smearing with with =0 25 an ratcheting. The ashe-otte lines inicate the unsmeare, perturbative result with =1 3. a) b) = 0.10 (Rsep = 1.3) = 5 (Rsep = 1.3) FIG. 15: Ienti e jets structure in( ) plane incluing smearing, sees an ratcheting: a) =0 1, b) =0 25.

17 17 a) b) 2 'LR' R = R/2 = 0.35 = LCR 2 LC R = R/2 = 0.35 = 5 1 L 1 L FIG. 16: Stable cone structure foun using 0 = 2 for the two values of the smearing parameter: a) = 0 1, b) =0 25. This changes smoothly as increases. To convert these stable cones into jets we use their locations for cones of sie = 0 7 to etermine what energy is in the jet. Unlike the situations iscusse above, the merging of the cones to make a single jet is important for this analysis. Two stable cones of raius 0 can be consierably closer together than those of sie. Incluing the splitting/merging algorithm with = 0 5 we n the jet structure exhibite in Fig. 17. To interpret this gure it is important to note for, a jet of the type 1 L is essentially ientical to one labele in the sense that, once we have switche to the larger cone raius, essentially all of the energy from both partons is inclue in that jet. For practical purposes all of the single jet region correspons to jets with both partons. The essential change from the un xe case of Fig. 12 occurs in the region, ( ). Here the change from 1 L jets to jets is essential. The 1 L jets in the un xe case contain only the left-han parton, while the jets in the xe case inclue both partons. This leas to both higher energy jets, similar to those ienti e by JETCLU, an ensures that there are few instances of the situation illustrate in Fig. 9 where some energetic calorimeter towers are not assigne to any jet. The improve agreement between the JetClu results an those of the Mipoint Algorithm with the last x (using the smaller 0 value uring iscovering but still requiring cones to be stable) is inicate in Fig. 18. Clearly most of the i erences between the jets foun by the JetClu an Mipoint Algorithms are remove in the xe version of the latter. The small x suggeste for the Mipoint Algorithm can also be employe for the Seeless Algorithm but, like the Mipoint Algorithm, it will still miss the mile (now unstable) cone. To further illustrate the improvement o ere by this x an to con rm that the i erence between JETCLU with ratcheting an the MiPoint algorithm arises primarily from the, ( ) region of the ( ) plane we have performe the following exercise. Using the set of Monte Carlo events note earlier, we have ienti e the jets foun with JETCLU an the various versions of the MiPoint algorithm. Then we remove the calorimeter towers an particles containe in those jets from event an performe a secon pass search for jets with the same algorithms. The point is that clusters of towers that are potential jets but not ienti e as such in the rst pass, e.g., the cluster of towers in Fig. 9, may still be foun as jets in this secon pass when the stability of the corresponing cone is not compromise by the nearby ienti e jet. Then we can ask how the secon pass jets are relate to the nearest rst pass jets in terms of their location (as pairs) in the ( ) plane. A scatter plot of this result is shown in Fig. (I want to put Matthias s plot _vs plt_new_6