Forexampleifa080GeVkaonbeamisincidentonaprotontargetthecenterofmassenergyis1699GeVandthecenterofmassmomentumofeitherparticleis0442GeVItisalsousefultonotethat1lab1lablabCITATIONSEidelmanetal ID: 514006
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38.Kinematics38.KINEMATICSRevisedJanuary2000byJ.D.Jackson(LBNL).Throughoutthissectionunitsareusedinwhich=1.Thefollowingconversionsareuseful:=197.3MeVfm,(=0.3894(GeV)mb.38.1.LorentztransformationsTheenergyand3-momentumofaparticleofmassforma4-vectorwhosesquare.Thevelocityoftheparticleis.Theenergyandmomentum()viewedfromaframemovingwithvelocityaregivenby=(1and)arethecomponentsofperpendicular(parallel)to.Other4-vectors,suchasthespace-timecoordinatesofevents,ofcoursetransforminthesameway.Thescalarproductoftwo4-momentaisinvariant(frameindependent).38.2.Center-of-massenergyandmomentumInthecollisionoftwoparticlesofmassesandthetotalcenter-of-massenergycanbeexpressedintheLorentz-invariantformistheanglebetweentheparticles.Intheframewhereoneparticle(ofmassisatrest(labframe),1labThevelocityofthecenter-of-massinthelabframeislab1lablab1laband1labThec.m.momentaofparticles1and2areofmagnitudelab Forexample,ifa0.80GeV/kaonbeamisincidentonaprotontarget,thecenterofmassenergyis1.699GeVandthecenterofmassmomentumofeitherparticleis0.442GeV/Itisalsousefultonotethat1lab1lablabCITATION:S.Eidelmanetal.,PhysicsLettersB592,1(2004)availableonthePDGWWWpages(URL:)January4,200512:11 38.Kinematics38.3.Lorentz-invariantamplitudesThematrixelementsforascatteringordecayprocessarewrittenintermsofaninvariantamplitude.Asanexample,the-matrixfor22scatteringisrelated Thestatenormalizationissuchthat=(238.4.ParticledecaysThepartialdecayrateofaparticleofmassintobodiesinitsrestframeisgivenintermsoftheLorentz-invariantmatrixelement ;:::;pisanelementof-bodyphasespacegivenby;:::;p Thisphasespacecanbegeneratedrecursively,viz.;:::;p;:::;pq;p;:::;p.Thisformisparticularlyusefulinthecasewhereaparticledecaysintoanotherparticlethatsubsequentlydecays.Survivalprobability:Ifaparticleofmasshasmeanproperlifetime(=1�)andhasmomentum(),thentheprobabilitythatitlivesforatimegreaterbeforedecayingisgivenbyandtheprobabilitythatittravelsadistanceorgreaterisJanuary4,200512:11 38.Kinematics Figure38.1:Denitionsofvariablesfortwo-bodydecays.Two-bodydecaysIntherestframeofaparticleofmass,decayinginto2particleslabeled1and2, 2M;(38:15)jp1j=jp2j=M2�(m1+m2)2M2�(m1�m2)21=2 and 2jMj2jp1j (cos)isthesolidangleofparticle1.Three-bodydecays Figure38.2:Denitionsofvariablesforthree-bodydecays.Deningand,thenand,whereistheenergyofparticle3intherestframeof.Inthatframe,themomentaofthethreedecayparticleslieinaplane.Therelativeorientationofthesethreemomentaisxediftheirenergiesareknown.ThemomentacanthereforebespeciedinspacebygivingthreeEulerangles( ;; )thatspecifytheorientationofthenalsystemrelativetotheinitialparticle[1].Then )51 d d(cosd :January4,200512:11 38.KinematicsAlternatively )51 where()isthemomentumofparticle1intherestframeof1and2,and istheangleofparticle3intherestframeofthedecayingparticle.andaregivenby and [ComparewithEq.(38Ifthedecayingparticleisascalarorweaverageoveritsspinstates,thenintegrationovertheanglesinEq.(3818)gives )31 8M jMj21dE2=1 )31 32M3 ThisisthestandardformfortheDalitzplot.Dalitzplot:Foragivenvalueof,therangeofisdeterminedbyitsvalueswhenisparallelorantiparalleltomax E22�m22�q E23�m232;:22a)(m2)=(E2+E3)2�q E22�m22+q Hereandaretheenergiesofparticles2and3intherestframe.ThescatterplotinandiscalledaDalitzplot.If isconstant,theallowedregionoftheplotwillbeuniformlypopulatedwithevents[seeEq.(3821)].Anonuniformityintheplotgivesimmediateinformation.Forexample,inthecaseofK,bandsappearwhenre ectingtheappearanceofthedecaychainKJanuary4,200512:11 38.Kinematics 012345 10 m12 (GeV2)m23 (GeV2) (m1+m2)2(M-m3)2 (M-m1)2(m2+m3)2 (m23)min Figure38.3:Dalitzplotforathree-bodynalstate.Inthisexample,thestate at3GeV.Four-momentumconservationrestrictseventstotheshadedregion.Kinematiclimits:Inathree-bodydecaythemaximumof,[givenbyEq.(3820)],isachievedwhen,particles1and2havethesamevectorvelocityintherestframeofthedecayingparticle.If,inaddition,thenmaxmaxmaxMultibodydecays:Theaboveresultsmaybegeneralizedtonalstatescontaininganynumberofparticlesbycombiningsomeoftheparticlesinto\e ectiveparticles"andtreatingthenalstatesas2or3\e ectiveparticle"states.Thus,ifijk:::,thenijk::: ijk:::andijk:::maybeusedinplaceofintherelationsinSec.38.4.3or38.4.3.1above. Figure38.4:Denitionsofvariablesforproductionofan-bodynalstate.January4,200512:11 38.Kinematics38.5.CrosssectionsThedi erentialcrosssectionisgivenby 4q ;:::;p[SeeEq.(3811).]Intherestframeof(lab), 1lab;(38whileinthecenter-of-massframe 1cm Two-bodyreactions Figure38.5:Denitionsofvariablesforatwo-bodynalstate.Twoparticlesofmomentaandandmassesandscattertoparticlesofmomentaandandmassesand;theLorentz-invariantMandelstamvariablesaredenedbyandtheysatisfyThetwo-bodycrosssectionmaybewrittenas dt=1 1 1cmJanuary4,200512:11 38.KinematicsInthecenter-of-massframe1cm3cm1cm3cm1cm3cmsin1cm3cmsinistheanglebetweenparticle1and3.Thelimitingvalues=0)and)for22scatteringare 2p 1cm3cmIntheliteraturethenotationmax)for)issometimesused,whichshouldbediscouragedsince.Thecenter-of-massenergiesandmomentaoftheincomingparticlesare1cm 2p 2cm 2p For3cmand4cm,changeand.Then and1cm1lab p Herethesubscriptlabreferstotheframewhereparticle2isatrest.[ForotherrelationsseeEqs.(38.2){(38.4).]Inclusivereactions:Choosesomedirection(usuallythebeamdirection)forthe-axis;thentheenergyandmomentumofaparticlecanbewrittenasy;psinhisthetransversemassandtherapidityisdenedby 2E+pz =ln =tanh Underaboostinthe-directiontoaframewithvelocitytanh.HencetheshapeoftherapiditydistributiondN=dyisinvariant.Theinvariantcrosssectionmayalsoberewritten d3p=d3 d dyp dydJanuary4,200512:11 38.KinematicsThesecondformisobtainedusingtheidentitydy=dp,andthethirdformrepresentstheaverageoverFeynman'svariableisgivenby max maxInthec.m.frame, p sinh p andmax=ln( For,therapidity[Eq.(3837)]maybeexpandedtoobtain 2)+ sin2)+lntan(wherecos.Thepseudorapiditydenedbythesecondlineisapproximatelyequaltotherapidityforand,andinanycasecanbemeasuredwhenthemassandmomentumoftheparticleisunknown.Fromthedenitiononecanobtaintheidentitiessinh=cotsintanh=cosPartialwaves:TheamplitudeinthecenterofmassforelasticscatteringofspinlessparticlesmaybeexpandedinLegendrepolynomialsk; +1)(cosisthec.m.momentum,isthec.m.scatteringangle,1,andisthephaseshiftofthepartialwave.Forpurelyelasticscattering,=1.Thedi erentialcrosssectionis k;Theopticaltheoremstatesthat andthecrosssectioninthepartialwaveisthereforebounded: +1)+1) January4,200512:11 38.Kinematics 1/21/2 al d Figure38.6:Argandplotshowingapartial-waveamplitudeasafunctionofenergy.Theamplitudeleavestheunitarycirclewhereinelasticitysetsin(Theevolutionwithenergyofapartial-waveamplitudecanbedisplayedasatrajectoryinanArgandplot,asshowninFig.38.6.TheusualLorentz-invariantmatrixelement(seeSec.38.3above)fortheelasticprocessisrelatedtok;)by k; lab=0)andarethecenter-of-massenergysquaredandmomentumtransfersquared,respectively(seeSec.38.4.1).Resonances:TheBreit-Wigner(nonrelativistic)formforanelasticamplitudewitharesonanceatc.m.energy,elasticwidth�,andtotalwidth� isthec.m.energy.AsshowninFig.38.7,intheabsenceofbackgroundtheelasticamplitudetracesacounterclockwisecirclewithcenter2andradiuswheretheelasticity.TheamplitudehasapoleatThespin-averagedBreit-Wignercrosssectionforaspin-resonanceproducedinthecollisionofparticlesofspinand+1) +1)(2+1) k2BB�2tot isthec.m.momentum,isthec.m.energy,andandarethebranchingfractionsoftheresonanceintotheentranceandexitchannels.The2factorsarethemultiplicitiesoftheincidentspinstates,andarereplacedby2forphotons.Thisexpressionisvalidonlyforanisolatedstate.Ifthewidthisnotsmall,�cannotbetreatedasaconstantindependentof.Therearemanyotherformsfor,allofwhichareequivalenttotheonegivenhereinthenarrow-widthcase.Someoftheseformsmaybemoreappropriateiftheresonanceisbroad.January4,200512:11 38.Kinematics 1/21/2 el/2 Figure38.7:Argandplotforaresonance.TherelativisticBreit-WignerformcorrespondingtoEq.(3850)is: Abetterformincorporatestheknownkinematicdependences,replacing ),where�)isthewidththeresonanceparticlewouldhaveifitsmasswere ,andcorrespondingly )where�)isthepartialwidthintheincidentchannelforamass s:a`=�p s�(s) s�m2+ip Fortheboson,allthedecaysaretoparticleswhosemassesaresmallenoughtobeignored,soondimensionalgrounds� ,where�denesthewidthofthe,and�)isconstant.Afulltreatmentofthelineshaperequiresconsiderationofdynamics,notjustkinematics.ForthethisisdonebycalculatingtheradiativecorrectionsintheStandardModel.1.See,forexample,J.J.Sakurai,ModernQuantumMechnaics,Addison-Wesley(1985),p.172,orD.M.BrinkandG.R.Satchler,AngularMomentum,2nded.,OxfordUniversityPress(1968),p.20.January4,200512:11