1.1. Phase Space Volume

International Journal of Mo dern <a href="/tags/Physics/" rel="tag">Physics</a> A� f cWorld Scienti�c Publishing CompanyJETS AND KINEMATICS IN HADRONIC COLLISIONSANDREW R� BADENPhysics Department� University of Maryland�Col lege Park� MD �� USAReceived September �� Revised Decemb er �� Analysis of colliding�b eam hadron exp eriments �p � p � X and p � p � X � often dep end up on observation of �jets� � a highly collimated spray of particles suchas � �K�e��and� s forming clusters of energy dep osition in calorimeters� In this pap er we outline howto de�ne jets from such clusters and discuss the meaning of jet quantities such as transverse energy and mass� Data from the D� exp eriment are used to illustrate these concepts� In addition� we review the motivations for using certain optimal co ordinates for describing energy�<a href="/tags/Momentum/" rel="tag">momentum</a> ��vectors� and deriveinteresting relationships among the kinematic variables��� Kinematics of Hadron Collider EventsTo motivate some of the kinematic quantities traditionally used to describ e p p p exp eriments� one must �rst recognize that the center�of�mass energy � s � of the scattering in these exp eriments centers is not �xed� This is a direct consequence of� the proton substructure � where the scattering constituents are quarks and gluons that reside inside the nucleons� with each parton carrying a fraction x of the total nucleon energy� Scattering of partons of di�erent energy results in a center�of� momentum that do es not necessarily coincide with the lab oratory frame� A lackof knowledge of the center�of�mass energy guides the choice of kinematic variables��� Phase SpaceVolumeWe start with the de�nition of a relativistically invariant phase <a href="/tags/Space_(mathematics)/" rel="tag">space</a> volume d�� for an ob ject of momentum p and energy E �For a �xed particle mass� d� can b e written as �see �� dp dp dp d p x y z� � �� d� �E EDe�ning z as the pp collision axis� we see that of the ��momentum co ordinates p �p �p � and E � only the �rst two are invariant with resp ect to a Lorentz transfor� x y z mation from the lab frame along the �z direction� by virtue of their direction b eing normal to that of the b o ost� In addition� we see that even though the co ordinates p and � are not invariant with resp ect to b o osts along �z�invariance will b e preserved in the pro duct p cos � sin � �whichisp � xAchoice of momentum co ordinates which more explicitly re�ect the Lorentz invariance of the phase space elementis�p �y��m�� where p is the momentumT T transverse to �z direction� � is the azimuthal angle ab out �z� m is the mass� and y is the �rapidity� along the �z direction� de�ned as �� Jets and Kinematics in Hadronic Col lisions� � E � p �� cos � z y � � ln ln ��� E � p � � � � cos � z or� equivalently�� cos � � tanh y� �� where � � p�E � and � is� as ab ove� the p olar pro duction angle relativetothe�z axis� For a b o ost along thez � direction with some � � � � a transformation from b the lab frame to the new frame will change the rapidity as follows� y � y � y b where y �ln�� b b bBecause cos � � � for b o osts along �z� it follows that� �tanhy � b bThe advantage of using the ab ove set of co ordinates is clear� the e�ect of a b o ost along the z �axis changes only y � and that by only an additive constant�To calculate d� in these co ordinates� we �rst di�erentiate Eq� ��� ��y �y �E dy � dp � z�p �E �p z z� � p p E z z� � dp z E � p E � p E z z which simpli�es to dp z dy � �EThe comp onents dp and dp are related to the elements dp and d� via a trans� x y T formation from cartesian to p olar co ordinates in � dimensions�� d�� dp dp � p dp d� � dp x y T TT�We can now write the volume element d� as� dp dy d�� d� �T�Thus particles pro duced in pro cesses that have a matrix elementthatvaries slowly with rapidity�y � should b e distributed uniformly in y � This is the origin of the�rapidity plateau� for single�particle pro duction in hadronic collisions�Our chosen set of kinematic co ordinates is well suited to the analysis of physics issues� and is used extensively to describ e the results of measurements in high energy physics� In the following sub�sections� wedevelop formulae and relationships among p � �� y �andm and other quantities of interest�T�� Transverse Energy �E � and Momentum �p �T TJets and Kinematics in Hadronic Col lisions �One can de�ne the �transverse energy� E of a particle as its energy in the restT frame where its p � �� Because this quantityisinvariant to boosts along �z�we can z write in general� E � p � p � m � p � m � E � p ��T x y T zFrom Eq� �� we can solve for p � yielding� z p � E tanh y �� zCombining the ab ovetwo equations gives� E � E � E tanh yT� E �� tanh y �� E � cosh y or equiv alentlyE � E cosh y� ��TNote that the p is always de�ned in terms of the ��momentum p� asT p � p sin �� TWe wish to stress that these two relativistically invariantquantities �E and p �T T are not de�ned in the same way �that is� E � E sin � �� More on this b elow�TCombining Eq� �� and �� gives p � E sinh y �� z T�� Invariant MassThe invariant mass of two particles is de�ned as� �M � �p � p ��p � p ��� �� � ���E E � p � p � �� m � m� � �Using the �p ��y and m�variables� and applying equations from the pro ceedingT sections� we calculateE E � E cosh y E cosh y �� T � � T and� � p � p � p p � p p � p p� x� x� y � y � z � z �� p cos � p cos � � p sin � p sin � � p pT � � T T � � T z � z � p p cos �� p pT � T z � z � p p cos �� E sinh y E sinh yT � T T � � T � E E �� cos �� sinh y sinh y ��T � T T � T � where � � p �E � � sin � cosh y �and�� T T T � Combining the ab ove results gives� Jets and Kinematics in Hadronic Col lisions M � m � m ��E E �cosh �y � � � cos �� t� t t� t� � where �y � y � y and cosh �y � cosh y cosh y � sinh y sinh y �We will discuss� � � the consequences of Eq� �� in the limit � � � in a subsequent section��� Pseudo�RapidityIn order to determine the rapidity of a particle� one needs b oth its cos � and velo city� � Since the velo city requires measuring b oth E and the mass m� this precludes using a single detector comp onent to measure rapidity�However� in the limit of� � � �or m � p �� Eq� �� b ecomes cos � �tanhy �which often pro vides anT excellentapproximation to use for the rapidity�We therefore de�ne the �pseudo� rapidity� � as the rapidity of a particle with zero mass �or � � �� y j � mUsing Eq� �� for y � and setting � � �� one obtains�� cos� � � ln � � ln tan � �� � � � cos � � and the inverse equation� which is often useful cos � � tanh �� Eliminating cos � from Eqs� �� and �� wehave the relationship tanh y � � tanh �� This last equation shows clearly that in the limit m � �or � � �� wehave � � y �The pseudo�rapidity � � de�ned to b e indep endentof� � is therefore another wayto measure the lo cation of a particle� equivalent to measuring its p olar angle � �In addition� we can use Eq� �� to derive the transformation d� � � sin �d� � �d� � cosh �� Since � is always � � and the function tanh�x� is monotonic� the pseudo�rapidity j� j of a particle of given p is larger than its true rapidity jy j� Figure � shows theT di�erence b etween � and y as a function of � for protons� for several choices of constant p � and Figure � shows a similar plot for pions� In each plot� a dashedT line is drawn at constant j� j�jy j �� corresp onding to the size of the calorimeter readout pad at D� � Rapidity and pseudo�rapidity are equal to within �� for pion p of � �� GeV�c� and proton p of � �� GeV�c� for all � � This means that oneT T can use � in place of y in kinematic calculations involving pions and protons in this range of p values�TIt is imp ortant to recall that� for �xed p � the velo city � is a function of pseudo�T p rapidity � �To see the explicit form� we start with � � p�E � E � p � m �useEq� �� for p � and Eq� �� to relate � to � � to getT pT p�� � m tanh � ET which gives p tanh �T q� tanh y �� m tanh � ETGiven the fact that j� j�jy j�wewould exp ect that for �xed p �any single�T particle sp ectrum that is indep endentofy �d� �dy is constant� would b ecome de� pleted in the central region when plotting as a function of � �Thistyp e of e�ect would b e esp ecially large for particles with low p and signi�cant mass� TJets and Kinematics in Hadronic Col lisions �Fig� �� j� j� jy j for protons at various p �TFig� �� j� j� jy j for pions at various p � T� Jets and Kinematics in Hadronic Col lisions d� dy dy d�� � � k � � d� dy d� d� where k is a constant� An analytic form for dy �d� can b e derived starting with Eq��� substituting for � using Eq� �� di�erentiating and rearranging terms to get the result that dy� � �� d� with � �� given by Eq� �� This result� although somewhat surprising� can b e understo o d in the following sense� at large y �and therefore large � �� the function tanh y converges to �� Hence� for large y � b oth the tanh y and tanh � converge to�� which means � � � �whichwould b e true for large p and �xed p �� In this case�T there is little if any functional dep endence of y on � �Atsmally �we can use the approximation tanh y � y and tanh � � � to get the relation y � �� whichgives dy �d� � � �We therefore exp ect dy �d� to b e constant for large y or � �� with all of the � dep endence coming at small � �Figure � shows a histogram of dN �d� for �� pions generated with constant p �� GeV�c and uniformly in dN �dy � The smo oth curve is the function � �� T that is dy �d� for pions with �xed p �� GeV�c�TFig� �� The histogram shows dy �d� for pions generated uniformly in rapidity y � with �xed p ��TGeV�c� The smo oth curve is the corresp onding function � �� �� Invariant Masses for � � �In the limit m � �� y � � and � � � � �� We can then write the invariantmassT of two massless particles �using Eq� �� as���E E �cosh �� cos �� MT � T �Jets and Kinematics in Hadronic Col lisions �This expression is often quite useful� as weshowbelow��� Transverse Mass and �Missing� Transverse EnergyNon�interacting particles such as neutrinos can b e detected only via an apparent imbalance of momentum in an event� In p p �or pp�interactions� where most of the longitudinal �parallel to the b eam � �z �momentum in the collision is lost down the b eam pip e� it is essentially imp ossible to measure momentum conservation along �z�However� given a detector with su�ciently large acceptance� measuring momentum balance in the plane transverse to the b eam direction is relatively straightforward�Thus� that amount of transverse momentum required to satisfy momentum con� servation �the �missing� transverse momentum� can b e in large part attributed to the presence of any unobserved non�interacting particles� esp ecially when the total momentum �and energy� in the detector is large �since for small total detected en� ergy� small �uctuations in the measured momentum along anygiven direction can give a small� statistically insigni�cant� missing transverse momentum�� Since the neutrino is massless �certainly on these scales�� this quantity �missing transverse momentum� is virtually equivalenttothe missing transverse energy �or E� � in theT event�For particles such as the W b oson� which decayinto lepton�neutrino pairs� the lack of a measurementofp for the neutrino precludes a direct measurementofthe z invariant mass of the W� However� one can calculate an in variant transverse mass of these two particles �denoted by M �� de�ned as the invariant ��particle mass usingT only the momentum comp onents in the transverse plane �p and p � i�� From xi yi this de�nition� the transverse mass is de�ned as� �M � �p � p ��p � p ���T � �T TT � T �� where p ��E � p �� For two zero�mass particles� �E � p �wehaveTi Ti Ti TiTi� �� ��E E � p � p � MT � T T � T TE E �� cos �� T � T ���� E E sin �T � T � or� equivalently� p��� M � � sin� E E � ��T T � T �We can see that this formula is equivalent to Eq� �� with the condition � � � �� that is the transverse mass and the invariant mass are equivalent for particles which are pro duced at the same rapidity�� As such� M � M �ignoring theT � detector resolution and intrinsic width � �� As a rule� M will deviate from MW T � as �� gets large� that is as the particles decay more �forward�backward� in the lab frame� Distributions in M will therefore have a tail at low M �forward�backward��T T will p eak at M � and will fall to zero ab ove M in the same manner as M �� � �It is imp ortant to note that these formulae require that E b e de�ned using Eq�T as E sin � � They are equivalent only in the limit m�p � �� Of course� �� and notTEq� �� is true in general� As will b e shown b elow� in the small mass limit of � � �� the di�erence b etween E and p b ecomes insigni�cant�T TAn interesting relationship involving the invariant mass of � particles is M � M � M � M � m � m � m� � � � which in the limit of m � �becomes i� Jets and Kinematics in Hadronic Col lisions M � M � M � M � ��� � �This formula is interesting to keep in mind when searching for ��b o dy decays� sayA � a � b � c� If it is appropriate to ignore the three masses m � m �andm �then a b c you do not havetoknowwhich of the three b o dies is a� b�or c sp eci�cally in order to calculate the correct value for m �you only havetoknow that these are the �A b o dies of whichyou are interested�Lastly� since the mass of the W particle is well known �we can constrain the invariantmassofthe e� � pair� and solve for the longitudinal momentum of the neutrino� To do this� we can use Eq� ��M ��E E �cosh �� cos ��T � T WRewriting this expression� wegetMW cosh �� cos�� �E ET � T Solving for �� gives p r � � r ��� ln � ��� where r is the right�hand side of Eq� �� Because �� is the di�erence in pseu� dorapiditybetween the electron and the neutrino� there are two solutions to the problem� That is� there is no way of resolving the ambiguity of whether the neu� trino is at a lower or higher rapidity relative to the electron as seen from the fact that the hyp erb olic cosine cosh �� is even in �� Both solutions are p ossible� at least in principle��� Jets Quantum Chromo dynamics �QCD � is the currently accepted theory of the strong interaction b etween particles �partons� that carry �color�� These partons �quarks q and gluons g � app ear in detectors as �sprays�� or �jets�� of particles� In this section� we will discuss issues concerning the observation of jets and their origin in QCD��� De�nition of a JetThere are various ways to de�ne jets� For instance� most of the LEP�PEP�and�PETRA e e exp eriments de�ne jets using information just from tracking cham� bers � One of the less ambiguous ways to de�ne jets is motivated bywhatwe understand to b e the physics of fragmentation� namely that a jet is the result of the hadronization of a parent parton� A way to visualize hadronization is through wed virtual gluon emission by the parentparton�q � q � g and g � g � g �� follo by gluon splitting to q q pairs� forming the color singlet hadrons that we measure��For high precision studies of parton QCD fragmentation in e e collisions� this mo del is to o naive� However� in hadronic collisions� where the presence of the soft�underlying event� part of a high Q collision complicates the analysis� the naive mo del is adequate for describing how a parton turns into a jet�To see how this simple fragmentation mo del can b e related to the concept of a jet� we consider the transverse momentum �k � of �radiated� gluons relativetoT the jet axis� The distribution of k has b een found to b e exp onential in �k � withTT p� �� and DORIS energies � hk i� �� MeV�c at SPEAR s � ��GeV �� increasingT� to hk i��GeV �c at SppS energies� We de�ne � as the angle of the gluon withT resp ect to some reference in the plane p erp endicular to the momentum direction ofJets and Kinematics in Hadronic Col lisions �Fig� �� Radiated gluons make an angle � with resp ect to an arbitrary direction in the plane p erp endicular to the jet axis� the parent parton �see Figure �� Since the physics of gluon radiation is indep endent of this angle� we exp ect the probability distribution for � will b e uniform in � �The jet axis can b e de�ned by a cluster of calorimeter cells �less often bytracks in the� tracking chamb er� in space that minimize the total vector k of the jet remnants�TThis means requiring�X� k �� TiWe can use Eq� �� to form the jet cluster into an equivalent particle with����momentum p ��E� p�� that can b e used� for example� to calculate an invariant mass �for example W � jet � jet �� Let us sp ecialize to calorimetry� and assume that the jet will b e constructed as a cluster of cells �or towers for pro jective geometry calorimeters�� If one represents the energy E in each cell of a calorimeter i as a i��particle� with zero mass� one can then de�ne E � E n� where n� is the unit vector i i i i pointing from the interaction vertex to cell i� The jet is therefore de�ned in the rest frame of the detector as an ob ject consisting of a set of these zero mass particles� E �� one p er calorimeter cell� Note that this representation of �and therefore p i i a jet �using zero mass particles� do es not imply that the jet also has zero mass� or p � E � since the constituents �the cells� will eachhave an op ening angle b etween them� and this op ening angle will �generate� a jet mass� To see this explicitly� consider Eq� �� the invariant mass of two particles� If the particles have zero mass� we drop the terms m and m �set E � p and E � p � leaving� � � M ��E E �� cos � ��E E sin ��� � � where � is the angle b etween the two particles� We see clearly that if � ��the invariant mass is also zero��� Jets and Kinematics in Hadronic Col lisionsTo form a jet from these calorimeter cells� wemust apply Eq� �� relativeto the axis that de�nes the jet direction� The transformation of the cells to this axis involves an Euler rotation with � degrees of freedom� � and � � The rotation jet jet� � matrix M� de�ned by E � M E � is a function of the yet to b e determined Euler i i� angles� and is given by� �� sin � cos � � jet jet� A� cos � cos � � cos � sin � sin � jet jet jet jet jet sin � cos � sin � sin � cos � jet jet jet jet jetThe transformation equations which relate the cell energies in the frame of the jet�primed� to those in the frame of the detector �unprimed� are thereforeE � �E sin � � E cos � xi jet yi jet xi� E sin ��Ti i� �E cos � cos � � E cos � sin � � E sin � E xi jet jet yi jet jet zi jet yi� � cos � E cos �� sin� E jet Ti i jet ziE � E sin � cos � � E sin � sin � � E cos � xi jet jet yi jet jet zi jet zi� sin � E cos �� cos� E � �� jet Ti i jet zi where E � E sin � �and�� As a check of Eq� �� we canTi i i i i jet imagine a jet with a single tower which when rotated to the jet axis should retrieve E � E � � and E � E �Thus� setting �� remember that �� is the jet x y z di�erence in � between the jet and the tower�� the ab ove equations yield E �� x E � E sin � � E cos � � � �using tan � � E �E �� and E � E sin � � z jet T jet jet T z T jet y z sin � and E � E cos � �� Here wehave E cos � � E �using E � E jet z jet z jet jet T assumed that suchahyp othetical jet would b e massless since it is contained in a single tower� allowing E � p in the ab ove� This is discussed further b elow� jet jetThe momentum vectors of eachtower can now b e rotated into the frame of the jet� summed over the transverse comp onents E � k and E � k �and xi yi xi yi separately set to zero� as required by Eq� ��X X XE � � sin � � E �cos� � E �� jet xi jet yi xi i i i andX X� � E � � � cos � cos E xi jet jet yi i iX XE � sin � � E cos � sin � � yi jet zi jet jet i i� �� We therefore havetwo equations �� and �� and two unknowns �� and � �� jet jetSolving Eq� �� yields the following relation�P PE sin � ETi i yiP P� � �� tan � � jetE E cos � xi Ti iSolving Eq� �� givesJets and Kinematics in Hadronic Col lisions ��P P cos � � E � sin � � E jet xi jet yi i iP tan � � � �� jetE zi iEquations �� and �� describ e the co ordinates of the jet cluster in a waywhich is consistent with the de�ning Eq� ��The jet is a true physical ob ject� which mighthave originated through the show�� ering of some virtual parton� and it must therefore havea��vector �E � p �that jet jet corresp onds to an ob ject of some �nite mass� We can de�ne the angles of our jet in the lab oratory�asfollows� p yj et tan � � �� jet p xj et and p tj et�� tan � � jet p zj et or equivalently� using the relation that sinh � tan � �� p zj et sinh � � � �� jet p tj etEquating the numerators and denominators of Eqs� ��and �� we obtain the result� tow er sX p � E xj et xi i� tow er sXE sin � cos � � i i i i� tow er sXE cos � � �� Ti i i� and tow er sX p � E yj et yi i� tow er sXE sin � sin � � i i i i� tow er sX� E sin � � ��Ti i i�By comparing the numerators and denominators of Eqs� �� and �� weget�� Jets and Kinematics in Hadronic Col lisions towersX p � E zj et zi i� towersX� E cos � i i i� towersXE tanh � � i i i� towersX� E sinh � � ��Ti i i� where wehave again used the relationship sinh � tan � � ��andnotshown the details of the algebra� Wehave also used the fact that� for the towers� E � E sin �Ti i i rememb ering that in our approximation each calorimeter tower b e represented as a particle of zero mass�It should b e recognized that the co ordinate � was derived from Eq� �� jet and the co ordinate � from Eq� �� using the relationship � � � log tan �� jetConsequently� the true � do es not corresp ond to the energy weighted result � � jetP PE � � E �Ti i TiThe energy of the jet is given by an energy sum over all towers in the cluster� tow er sXE � �� E � i jet i�Wehave therefore established a self�consistent prescription for turning a �cluster� of energy towers into a physical jet� which is a particle with ��momentum in the detector frame given byX X X X�� p �� E � p �� E � E n� � �� i i i i i jet and with directional co ordinates � �or � �and� derived from the ab ove jet jet jet momentum comp onents� or as calculated using Eq� �� through ��It is worth investigating expressions for the jet ��vector that involve not just sums o ver towers at sp eci�c angles � and � � but rather sums that involve angles i i relative to the jet axis� �� and �� i jet i i jet iThe expression for p is straightforward� Weusethenumerator of Eq� ��TjetX X p � cos � � E �sin� � ETjet jet xi jet yi i i and substitute in E � E cos � and E � E sin � �Thisgives xi Ti i yi Ti iX X sin � E cos � � sin � � E p � cos � � i Ti i jet Ti Tjet jet i iXE cos �� Ti i iJets and Kinematics in Hadronic Col lisions �� The expression for E is not as straightforward� First� we form E � E � pTjet zT and use the relation cosh x � sinh x � � to get E � E � pTjet jet zj et � E �cosh � � sinh � � � p �cosh � � sinh � � jet jet jet zj et jet jet � �E cosh � � p sinh � � � �E sinh � � p cosh � � jet jet zj et jet jet jet zj et jetIf we add and subtract the quantity�E p sinh � cosh � � and then use Eqs� jet zj et jet jet�� and �� in the ab ove equation weget E � �E cosh � � p sinh � � ��E sinh � � p cosh � � jet jet zj et jet jet zj et jetTjetX X X X � � E cosh � � E sinh � � �� E sinh � � E cosh � � i jet zi jet i jet zi jetX X� � E cosh � cosh � � E sinh � sinh � � �Ti i jet Ti i jetX X� E cosh � sinh � � E sinh � cosh � �Ti i jet Ti i jetX X � � E cosh �� E sinh �� Ti i Ti iThis equation is not altogether transparent� However� consider the symmetry of the sinh and cosh functions � the former is o dd� the latter is even� If the jet is symmetric �in energy and physical extent� ab out the jet axis� then the sum overE sinh �� will vanish� leavingTi iXE � E cosh �� Tjet Ti iEqs� �� and �� will prove useful later in our exp osition��� Jet �or Clustering� AlgorithmThe �nal missing ingredient is an algorithm for determining which calorimeter cells to include in a cluster that is consistent with Eq� �� One suchscheme� usually� � � referred to as a ��xed�cone� algorithm used byUA� � CDF � and D� �works in the following way�� All towers ab ove a mo derate energy threshold serve as �seed towers� for initial�proto�clusters��� All immediately neighb oring towers within some cone cuto� R in �� space�a common range for the cone cuto� R is �� relative to the center of p �� are added to form a new cluster� The the proto�cluster �R � centroids �or �st moments� of the cluster in � and � are recalculated� These towers mayormay note b e sub ject to a small energy threshold�� If the jet centroid in �� changes by more than some small preset value� then all towers within the cone R are re�combined to form a new cluster� Again the centroids are calculated in � and �� This pro cedure is iterated until the centroid of the cluster remains stable�� After all clusters are found� some maycontain overlapping �same� calorimeter towers� If so� such clusters are merged� dep ending on the fraction of transverse energy present in the shared towers� The smaller E cluster is then subsumed� T�� Jets and Kinematics in Hadronic Col lisionsA threshold parameter controls the merging� Forthedatatobeshown in this rep ort� the merging parameter was set to �� of the energy shared by anytwo clusters will cause merging��Another algorithm� called the �nearest�neighb or� algorithm� was the primary clus�� tering algorithm used by UA�� and is describ ed elsewhere��� The Shape of a JetEquation �� can b e used to calculate the comp onent of the tower energy �or momentum� that is transverse to the jet direction� k � E � E �For those xi yiTi towers that are �close� to the jet axis� the small�angle limit of Eq� �� givesE � E ��Ti i xiE � sin � E � cos � E jet zi jet Ti yi� sin � E cos � � cos � E sin � jet i i jet i i� E sin �� i i� E �� i i� �E ��Ti i where wehave used �� sin �� from Eq� �� and the relation E � E sin � �Ti i iSquaring and adding the comp onents gives k � E �� Ti Ti i i or� equivalently� k � E R ��Ti Ti i where R is the distance b etween the jet and cell i in �� space de�ned by i � �� R i i iThus� the momentum of each �particle� in the �� plane is given by the pro duct of the E of the particle �in the lab frame� and the distance in �� space from the jetT axis� Since the k are exp ected to b e symmetrically distributed ab out the jet axisTi ve circular shap es in �� space� Of course� this �Figure �� this suggests that jets ha small angle limit is not a go o d approximation for �particles� �cells� that are far o� the jet axis�It is b est to de�ne the shap e of a jet using an energy weighting pro cedure� This reduces the dep endence of the shap e on low energy towers that are b oth far from the jet axis and have a negligible contribution to the total jet energy� The shap e of a jet is often characterized byanE �weighted second moment�variance� of theTi distance in the transverse plane� Since the �rst momentvanishes�X X�� k � E R ��Ti Ti i wehaveP P E �� E RTi Ti i i iP P� �� RE ETi Ti which is equivalentto� � � � �� �� RJets and Kinematics in Hadronic Col lisions �� when we de�ne � and � as��� P P E �� E ��Ti Ti i iP P� � �� �� E ETi TiReferring to data� for some sp eci�c R we �rst de�ne q p� � � � � �� �� RR using events from D� describ ed in App endix � �Sample �� This data sample was selected to minimize jet merging �� to �� a pro cess that has origin in higher�orderQCD e�ects that are distinct from those of jet fragmentation�Figure � shows the distribution of the second moment � for three jet coneR cuto�s of R �� and �� The p oints are D� data� The e�ect of the cone cuto� is clear� the larger the cone� the larger the �nd moment in b oth mean and width� The main source of the broadening is simply due to the presence of additional towers for the larger cone cuto�s� Figure � also shows the results of Gaussian �ts to the distributions� The �ts reveal that for unmerged jets� the distribution of jet widths is well describ ed by a Gaussian� with a central value that is a function of the cone cuto�� This is not to say that the energy �ow around the jet axis has aGaussian distribution � rather� this is only describing the �uctuations in � �the jetRRMS� around the average � �which is itself a function of the jet cone parameter�RTable � shows the results for the �ts� and Figure � shows that the central value ofTable �� Results of a Gaussian �t to distributions of � using the data from Sample ��RR � RMS�� R R�� �� �� the distribution in � �mean of the Gaussian� scales linearly with the cone cuto� �RThe small angle approximation used ab ove is clearly excellent� Figure � shows that even for jets with R �� ab out �� of the time� This shows thatR the ab ove approximationisvalid for all but p erhaps �� of the R �� jets �with largest RMS widths���� Jet �� Correlation MatrixDetector e�ects� gluon radiation� and the underlying event� all contribute to dis� torting the shap e of jets� Figure � shows a sketchofatypical elliptical �real� jet� with � de�ned as the smaller of the angles b etween the semi�ma jor axis and either the � or � axis� The angle � can b e calculated from the variance in � and ��as de�ned in Eq� �� and from the corresp onding correlation term � � de�ned as��PE �� Ti i iP� �� ��ETiThe shap e of the jet can b e characterized by the correlation term � and the���diagonal� terms � and � in the shap e matrix��� � �� ��� � ��� � Up on diagonalization� the two eigenvalues corresp onding to the semi�ma jor �� and semi�minor �� axes are ��� Jets and Kinematics in Hadronic Col lisionsFig� �� Jet probability distribution in � for single unmerged jets�RFig� �� Results of a Gaussian �t to distributions in � as a function of the cone cuto� R�TheR straight line is from a linear �t to the p oints yielding � �� R� RJets and Kinematics in Hadronic Col lisions ��ηα φFig� �� Jet ellipse� Semi�ma jor and �minor axes represent the square ro ot of the second moment in � �� and � �� �� �� Jets and Kinematics in Hadronic Col lisions s� �� � � � � ��� � � � � � ����� � It is clear that for a circular jet �� and � � � ��� �Wenow return to Eq� �� which de�ned the quantity � �the RM S width ofR the jet in �� space� This quantity is of course the square ro ot of the sum of the two diagonal terms of the shap e matrix� and not the ro ot of the sum of the squares of these two terms� The o��diagonal term � do es not a�ect � ��� R�� The E of a JetTA common way for exp erimenters to characterize the hard scattering energy scale�Q � for QCD collisions which pro duce jets is to use the transverse energy E ofT a pro duced jet� The measure of E which is most often used to compare data toTQCD theory is tow er sXE � E � ��T Ti i�It has the advantage that one can hop e to establish a corresp ondence b etween the energy observed in eachtower �exp erimen t� and the particles radiated by a jet �a hadronized parton�� There are� however� two other ways to de�ne E � one of whichT is more amenable for treating a jet as a collection of physical ob jects �particles�� and p ossibly more appropriate for calculating invariant masses involving jets� The�rst is�E � E sin � ��T jet jet where sin � is related to � through Eq� �� and E is calculated using Eq� jet jet jet�� The second way to de�ne E uses Eqs� �� and �� As long as there isT consistency in comparing theory to exp eriment� it is relatively arbitrary howone de�nes E �given that all internal variables are calculated consistently�� However�T b ecause of the di�erences b etween p and E � the de�nition in Eq� �� should notT T b e used in calculating the invariantmassgiven in Eq� �� For such cases� only the E de�ned in Eq� �� is appropriate�T To see this clearly�we use the relation E � p � m and write �E sin � � ��p � m � sin � � p � m sin ��TWe see that this de�nition gives a systematical ly lower value for the transverse energy E than the relativistically correct de�nition from Eq� �� The di�erence�T of course� is insigni�cant in the high energy limit �jet velo city � � �� and for jets that are pro duced at �� to the b eam axis �so�called �central jets��Wecanquantify the di�erence in the two de�nitions of E through the quantity�T q p � m sin �T E sin � p�� ET p � mT which measures the fractional �error� in using Eq� �� Expanding Eq� �� for small mass� and keeping only the lowest p owers of m �p � givesT m tanh �� �� �p TJets and Kinematics in Hadronic Col lisions ��This shows the explicit dep endence of the di�erence �� on p �for arbitrary valuesT of � �� For small � �tanh � � � and �rapidly approac hes zero� Coupled with the fact that m�E � for all but the smallest�E jets� we conclude that� typically�Eq�T T�� and Eq� �� agree at the few p ercentlevel for ��Figure � shows the distribution in �for the highest� E jet in ��jet events fromT our Sample � �App endix �� Figure � shows the integrated probability as a function of a lower cuto� on ��the p ercen tage of jets that would b e mismeasured� if �w ere greater than the value given on the abscissa�� The distributions clearly broaden with jet cone size� Because �is prop ortional to m � the broadening can b e attributed to the broadening of the mass of a jet due to the increase in the numberofjet towers with the cone size �including towers that might not �b elong� in the jet��Such broadening can also come ab out from the failure to merge twojetsinto one� where one of the two jets mighthave an energy b elowanE threshold� �SuchaT threshold is usually employed by the exp eriments to eliminate the numb er of jets found at very low energy��Fig� �� Distribution in � as de�ned using Eq� �� for jets of Sample �� required to have j� j��The E de�ned in Eq� �� also underestimates the true E of the jet �as de�nedT T using Eq� �� This can b e seen as follows� Supp ose there is a cluster with just � towers� then Eq� �� would yield � �E � E � � �p � p � E � p� z � z z � �E � E � � �E cos � � E cos � �� � � � E � E ��E E �� cos � cos � �� � T � T �� while equation �� would yield�� Jets and Kinematics in Hadronic Col lisionsR R� � dP �� where P �� is the distribution in Figure �� Jets were dP �� Fig� �� I �� � required to have j� j� ��X � E � � �E � E �Ti t� t i� �E sin � � E sin � �� � ��E E sin � sin � � � E � E� � T T ���If we form the di�erence of the two expressions� wegetX �E � p � � � E � � E E sin ��Ti � z i where �� We see that this di�erence is always positive� which means that� using Eq� �� underestimates the true E of a jet�TAgain� we form the quantityPETi i� � � � ��ET whichshows the fractional �error� in using Eq� �� for the transverse energy�as opp osed to Eqs� �� and �� for the �true� E � Figure �� shows the distributionT in �as so de�ned� As can b e seen in the �gure� the t wo di�erent de�nitions of transverse energy give similar results� Di�erences are not large� and� in fact� Eq��� gives a smaller �error� than one obtains when using Eq� ���� The Mass of a JetJets and Kinematics in Hadronic Col lisions ��Fig� �� Same as Figure �� E de�ned using Eq� ��TThe mass of a jet can b e calculated from its transverse momentum and energy usingEqs� �� and �� � p � E M jet jet jet � E � pTjet Tjet� �E � p ��E � p � ��Tjet Tjet Tjet Tjet Note that in the ab ove equation� we formed the mass by using either E � p or E � p � One should not confuse the mass of the jet as de�ned by E � p withT T T T what wehave previously describ ed as the �transverse mass�� In fact� the transverse mass is only de�ned when dealing with � �or more� ��vectors� A jet do es not have a transverse mass�In the limit when the jet energy gets large� the jet narrows � �� and i�� We expand Eq� �� for small �� and Eq� �� for small �� yielding i i iX�� i� � �� E �� pTi Tjet� andX�� iE � E �� Tjet Ti�Calculating the di�erence and sum b etween E and p �wehaveTjet TjetX� � � �� E � p � E ��Tjet Tjet Ti i i ��� Jets and Kinematics in Hadronic Col lisionsX� E � p � E �� Tjet Tjet Ti i i�X� � E ��Ti where wehave ignored the terms in �� and �� in the second equation� Inserting i i these relations into Eq� �� givesX X M � E E �� Ti Ti jet i iNowwe pro ceed to the small angle �thin jet� limit for large�E jets� As in theTP sections ab ove� we can use the approximation E � E � and combine withTjet TiEq� �� to obtain the relation for the invariant mass of the jet asM � � � E � �� jet R TjetThis equation con�rms what we knowintuitively ab out the invariant mass of a multiparticle <a href="/tags/System/" rel="tag">system</a>� namely that the mass of the system is �generated by� the angles b etween the �approximately zero mass� particles� ws the normalized distribution in mass �M � for jets that were Figure �� sho jet reconstructed with cone cuto�s of �� and �� The distributions broaden as the cuto� increases due to inclusion of more towers further away from the jet center�Fig� �� Mass distribution for jets of di�erent cone size�In Figure �� we plot the ratio � � E �M � onaneventbyevent basis� TheseR Tjet jet distributions show that the ratio is well within the few p ercent level of unity for all jet cones �note the vertical log scale used���� Jet MergingJets and Kinematics in Hadronic Col lisions ��Fig� �� The distribution in the ratio � � E �M for di�erent cone sizes� jetR TjetThe merging of two jets is usually re�ected in the value of � � and� of course� inR the size of the elements of the shap e matrix� To study merging� wehave used data from D�� paying particular attention to whether there is evidence of any merging or splitting of jets as controlled by the jet algorithm parameters �see section �� These data constitute our samples as describ ed in the app endices�Figure �� shows the normalized distributions in � for merged and not mergedR jets using our Sample �� for R �� and �� cone sizes� We see a marked in� crease in the width of the merged jets for all cone cuto�s� �This sample is somewhat biased by the requirement that� after merging� there must b e � and only � jets for all three de�ning cone sizes��Figure �� shows the normalized distributions in � using jets with cone cuto�sR of �� and �� for events from Sample � �App endix B�� This sample was chosen to maximize the probability of collecting merged jets �unlike Sample �� whichwas chosen to maximize the probability of collecting unmerged jets�� The crosses and the dots are� resp ectively� � sp ectra for the merged and unmerged jets resp ectivelyR of Figure �� It should b e understo o d that each jet with R �� in Sample� � is required to b e reduced to � jets of R ��Figure��shows thatR correlates well� on average� with whether or not a jet is the result of a merging�We can therefore use the distributions from Figure �� to construct a likeliho o d for a jet b eing from a merged or unmerged sample� Figure �� shows the ratio of the probability distributions for non�merged and merged jets versus � for cones of ��R�� and �� The p oints indicate the values of � for relativelikeliho o ds of � �equalR probability of b eing from either distribution�� these are listed in Table ��Finally�weinvestigate the shap e of jets via correlations in � �� with resp ect to� � merging� If jets were circular in �� space� wewould exp ect that the quantity�� p p� � � would b e Gaussian�distributed around �� If jets result from merging��� then we exp ect the shap e of the jet in �� space to b ecome elongated� forming an ellipse� and the quantity�� would deviate from �� We use the normalized quantity�� Jets and Kinematics in Hadronic Col lisionsFig� �� Normalized distributions in � for jets that were merged �b ottom� and those that haveR not b een merged �top�� for di�erent cone sizes�Fig� �� Normalized distributionsof � for single jets that are asso ciated with � jets found byR way of a smaller cone cuto� �see text��Jets and Kinematics in Hadronic Col lisions ��Fig� �� Relative likeliho o d for a jet b eing non�merged to a jet b eing merged as a function of � �R for three jet cones� Relativelikeliho o ds of � are indicated by the p oints on the curves�Table �� Value for � corresp onding to a a relative likeliho o d of unity for jets to b e from non�R merged or merged distributions�R � ��R�� �� �� �� Jets and Kinematics in Hadronic Col lisions� � ��� � ���� � ��� which� to �rst order� should b e indep endent of the cone size for symmetric jets�This is indeed observed in Figure �� where weshow � for jets from Sample ��� with cones �� and �� Figure �� shows the same distribution from Sample ��� ���Fig� �� The distribution in � � using unmerged jets with cone cuto�s of �� and��� ����� from Sample ��As demonstrated ab ove� these jets are broader than �pure� unmerged jets� howeverFigure �� also shows that the broadening of � can b e asso ciated with a broadeningR in either � or � �� ��� SummaryThe ab ove sections are intended to give a p edagogic description of jets� towards an understanding which is self consistent and illuminating� For the particle physicist� the next step is to learn more ab out the physics of jets� i�e� characteristics of jet pro duction� which in turn will allow understanding of the fundamental pro cesses underlying the physics of interest�App endix A � Events Used in Sample �A sample of events was used with the following requirements��i� Pathologies �Main Ring splash� cosmic rays� etc��were eliminated��ii� The p osition of the collision vertex was required b e within ��cm of the longitudinal center of the D� detector��iii� Eacheventhadtohave � and only � reconstructed jets for cones of �� and �� with none of the jets formed from merging of twoormorejetsoflowerJets and Kinematics in Hadronic Col lisions ��� ��� using merged jets with cone cuto�s of �� and Fig� �� The distribution in � ���� ����� from Sample �� transverse energy �by merging� we mean that there was no explicit merging by the jet cone algorithm as describ ed ab ove���iv� All � � electron� and <a href="/tags/Photon/" rel="tag">photon</a> candidates are rejected by requiring a minimum for cluster � and � � dep ending on the cone size� The cuto�s used were�� �� and �� for cones of �� and �� resp ectively�These requirements were chosen to maximize the probability of selecting events with only � jets at tree level� Nevertheless� we cannot exclude the presence of events with� or more jets� where jets were merged two at a time a priori due to their extreme closeness in �� space� Suchcontamination should b e small� In addition� there are�probably small� biases due to e�g� trigger thresholds �since the D� jet trigger starts with a single trigger tower threshold� thinner jets are more likely than fat jets to pass the trigger requirement� and primary vertex cuts �item ii ab ove is not a very restrictive cut�� However� the results presented here are intended to b e purely qualitative in nature�App endix B � Events selected for Sample �These events were chosen to have a sample of merged jets� Events were required to satisfy the same criteria � and � used to de�ne Sample �� but only events that had� jets reconstructed with a cone size of �� and � jets with a cone size of ��or�� were analyzed� guaranteeing the presence of twowell separated clusters of energy within a jet� Ab out �� of all events were classi�ed in this manner��� V� Barger and R� Phillips� Col lider Physics �Addison�Wesley� Reading� MA� ���� D� Perkins� Introduction to High Energy Physics� �rd Edition �Addison�Wesley� Reading� MA� �� pp� ���� Jets and Kinematics in Hadronic Col lisions�� H� B�ggild and T� Ferb el� �Inclusive Reactions�� Annual Review of Nuclear andParticle Science �� �� D� Collab oration� S� Abachi et al�� NIM A�� �� As of this submission� the most precise published value of the W mass is �� ��GeV �c from a combination of D� CDF� and UA� data� By adding the most recent LEP� data� the W mass changes to �� GeV �c � These re� sults were presented byD�Wo o d at the June �� Hadron Collider Physics con� ference at Stony Bro ok� Pro ceedings for this conference will b e available online at http��sbhep��physics�sunysb�edu � hadrons�pro ceedin gs�html��� D� Gross and F� Wilczek� Phys� Rev� Lett� �� �� H� Politzer� Phys� Rev� Lett� �� �� J� Huth and M� Mangano� �QCD Tests in Proton�Antiproton Collisions�� Ann� Rev�Nucl� Part� Sci� �� �� S� Bethke� in Proceedings of the XXIII International Conference on High EnergyPhysics� Berkeley� �� ed� S� Loken� �World Scienti�c� �� Volume I I� pp� ������ G� Hanson� in Proceedings XIII Rencontre de Moriond� �� ed� J� Tran ThanhVan� Vol� I I� p� ���� PLUTO Collab oration� Ch� Berger et al�� Phys� Lett� ��B �� �� G� Thompson� in Proceedings of the XXIII International Conference on High En� ergy Physics� Berkeley� �� ed� S� Loken� �World Scienti�c� �� Vol I I� p� ���� Herb ert Goldstein� <a href="/tags/Classical_mechanics/" rel="tag">CLASSICAL MECHANICS</a> ��nd Edition�� Addison�Wesley��� pp� �� The Euler transformation used here is given by equation �� with the condition � �� and � � � � ���� UA� Collab oration� Arnison� G� et al�� Phys� Lett� ��B� �� B� ���� and ��B� �� �� CDF Collab oration� F� Ab e et al�� Phys� Rev� Lett� �� and p erhaps most imp ortantly Phys� Rev� D�� �� D� Collab oration� S� Abachi et al�� Phys� Rev� D�� �� UA� Collab oration � M� Banner et al�� Phys� Lett� ��B� ��

1.1. Phase Space Volume

Details

Download

Copyright

We respect the copyrights and intellectual property rights of all users. All uploaded documents are either original works of the uploader or authorized works of the rightful owners.

Support