Visualizing the Mass and Width Spectrum of Unstable Particles

Particles

N.L.Harshman

Formerly:DepartmentofPhysicsandAstronomy

RiceUniversityHouston,TX

Currently:DepartmentofComputerScience,AudioTechnologyandPhysics

AmericanUniversityWashington,DC

February2,2008

Abstract

Severalgraphicalrepresentationsofthemassandwidthspectrumofunstablesubatomicparticlesarepresented.Suchplotsareusefultoolsforintroducingstudentstotheparticlezooandprovidestudentsanalternatewaytoorganizeconceptuallywhatcanseemlikeanoverwhelmingamountofdata.Inparticular,suchgraphshighlightphenomenologicalfeaturesofunstableparticlescharacter-isticofdifferentenergyandtimescales.

1Introduction

Whenconfrontingthevastarrayofsubatomicparticlesdetectedandcreatedinparticlephysicsexperiments,newstudents(andoldresearchers)maybecomeoverwhelmedwiththe

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volumeofdescriptiveinformation.Thepurposeofthisnoteistopresentavisualtooltosupplementotherschemesusedtoorganizethediversityofunstableparticles.Theideaissimple:whatdoesaplotofwidthversusmassofsubatomicparticleslooklikeandwhatinformationdoesitconveyaboutthekinematicsanddynamicsoftheseparticles?Ihopethatinstructorsofparticlephysicsmayfinditusefulwhenintroducingnewvisitorstotheparticlezooandthatpractitionersmayfindanalternateperspectivestimulating.

Themostusefulfeatureofthesegraphsisnotthattheysummarizeinformationaboutthespectrumorstructureofunstableparticles,butthattheyhighlightthedistinctionsbetweenthephenomenologicalsignaturesofunstableparticlesatdifferentenergyandtimescales.Thesegraphsprovideanaturalwayofgroupingtogetherparticles,notbytheirphysicalcontent,butbyhowtheyareproduced,howtheydecay,andhowtheircharacteristicparametersaremeasured.Thisalternatetaxonomydistinguishesthreeclassesofunstableparticles:decayingstates,whichdecayviatheweakinteractionandwhoseexponentialdecayratecanbemeasured,resonancestates,whichdecayprimarilythroughthestronginteractionandaremeasuredasfeaturesinthecrosssection,andamiddleclassof“platypus”states,whoseinstabilityparametersrequiremoresubtlemeasurementstodetermine.Also,thesegraphscanbeusedtocallattentiontohowthelimitationsandprejudicesofpastexperimentsaffectwhatisknownaboutthemassandwidthspectrumofparticles.

2WhichUnstableParticles?

Figure1depictsthemassMandwidthΓof139unstableparticles,withmassplottedlogarithmicallyonthehorizontalaxisandwidthplottedlogarithmicallyvertically.Theshapeofeachplottedpointindicatesthetypeofparticleitis(gaugeboson,e.g.)andforthehadronsthestyle(black,gray,orhollow)indicatessomeinformationaboutthequarkcontent.

Thedataforthe139unstableparticlescomefromthe2002editionofTheReviewofPar-ticlePhysics[1],andinparticularthelistofwell-known,reasonablywell-measuredunstableparticlesfromthe“SummaryTablesofParticleProperties”therein.Noteveryparticleinthe“SummaryTable”hasbeenincluded;onlythoseparticlesfoundinthefile[2]thattheParticleDataGrouptabulates,ofthemassandwidthdata,foruseinMonteCarloeventgeneratorsanddetectorsimulators.ForunstableparticleswhoselifetimesτarequotedintheReview[1]andnottheirwidths,thewidthvaluesintheMonteCarlofilearefoundusingtheWeisskopf-WignerrelationΓ=󰀁/τ(morewillbesaidaboutthislater).

ThelistofparticlesfromtheMonteCarlofile[2]hasbeenmodifiedandappliedinthefollowingwayinFig.1andsubsequentfigures:

1.Thestableparticles,theproton,electron,photonandneutrinos,areexcluded.

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2.Thenearly-stableneutronisneglectedforreasonsofscale.

3.TheMonteCarlofileincludessomeparticlesforwhichonlyanupperboundofthewidthhasbeenmeasured.Theyhavebeenexcluded.Examplesincludesomelight

∗±

unflavoredmesonresonanceslikethef0(980),othermesonresonancessuchastheDsandχb0(1P),andafewbaryonresonancesliketheΣc(2520)+.4.Thetopquark,nottrulyanindependentparticleliketheothersinthelist,isnotincluded.5.Thesymbolplottedforaparticlealsorepresentsitsantiparticle,exceptfortheneutral

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K-mesons.Forthese,themasseigenstatesKSandKLareplottedinsteadoftheflavor

00¯.eigenstatesKandK6.Asinglesymbolisplottedforalldifferentcharge-speciesofabaryonunlessdifferent

massesfordifferentchargeshavebeenmeasured.Forexample,eachpointrepresentinga∆baryonrepresentsallfourchargespecies{++,+,0,−}correspondingtoquarkcontents{uuu,uud,udd,ddd}.Thenwhatunstableparticlesareincluded?TheweakgaugebosonsWandZareatthehighenergyextremeandthemuonisatthelowenergyextreme.Theotherunstablelepton,thetau,isinthemiddle,alongwithahostofhadronsmadeupoffiveoutofthesixquarks:up,down,strange,charmandbottom.Whilethegaugebosonsandleptonsaretoourbestknowledgestructureless;thehadronsarecomposite.Subsequentreferencestoparticlesreferjusttothissetofwell-established,well-measuredunstableparticles,andthereforeshouldnotbetakentorefertoallpossibleparticlesthathaveorhavenotbeenobservedortheorized.LookingatFigure1,itmaybetemptingtoaskifthereisafunctionaldependenceofthewidthonthemass.InprinciplethewidthsofunstableparticlesarecalculablefromthemassesofthequarksandleptonsandotherStandardModelparameters,althoughinpracticesuchcalculationsaredifficultorimpossible,especiallyforhadrons.Forthepurposesofthisarticle,thewidthandmassareconsideredindependentphenomenologicalparameterstobedeterminedfromexperiment.

Figure1doesmakeapparentthegeneraltrendofincreasingwidthwithincreasingmass,whichisexplainedbyphasespaceeffects.Thedecayrateisroughlyproportionaltothephasespaceofthedecayproductsandthemoremassivetheunstableparticle,themoredecaychannelsareavailable.

TobetterelucidatethepropertiesofthescattereddistributioninFigure1,severalotherpartitionsorsectionsofthedataareincludedbelow.Figure2plotsthemassesofthe139unstableparticlesinorderofincreasingmass,i.e.inrankorderfromtheleastmassiveto

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themostmassive.Figure3issimilar,exceptitplotstheminrankorderofincreasingwidth.Figure4plotstheunitlessratioofwidthtomassinincreasingorder.Thesedifferentgraphsgivefurthercluesaboutthestructureofthemassandwidthspectrumandhowtoidentifyphenomenologically-similargroupsofunstableparticles.

3MassSpectrum

AstrikingfeatureofFigure1isthatwhilethewidthsrunoverarangeof18ordersofmag-nitude,themassesareconstrainedwithinthreedecades,withmostofthembetween1GeVand10GeV.Thisfactsaysmoreaboutthetypesofexperimentsthathavebeenperformedthanaboutthe“essentialnature”ofthemassspectrum.Farmoreparticlesearcheshavefocussedonthisenergyrangeforavarietyofhistoricalandpracticalreasons.

AnotherperspectiveonthemassspectrumcanbegainedfromFigure2.Qualitatively,gapsintheplotindicatehowquarkcompositionaffects(andeffects)themassspectrumofhadrons.Forexample,atthelowend,twelveofthefirstthirteenpointsaremesonsconsistingofthethreelightquarks:up,downandstrange.Thenon-mesonamongthethirteenisthemuon,thelightestunstableparticle,whichhistoricallywasmistakenforameson.Thetwelvemesonshavemasseslessthanthelightest(undepicted)baryons,theprotonandneutronatabout940MeV.Thereisagapbetweenthefirstthreepoints,whichrepresentthemuon,π0andπ±,andthenextset,whichincludethelightestmesonscomposedofstrangequarks,

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theK±,KS,KLandη.

Atthehighendofthemassspectrum,exceptforthegaugebosons,thepointsaredominatedbyhadronscontainingheavyquarks.Thelightesthadroncontainingacharmquark,theD0,iseighty-fifthonthelistwithamassof1865MeVandthelightesthadronscontainingabottomquarkaretheB-mesonsB±andB0withranksof127and128andmassesof5279MeV.Similartothestrangequarkmassthresholdjumpseenatthelowerenergies,herethereareslightjumpsat117,thelightestparticlewithtwocharmquarksηc,at127,thelightestparticlewithonebottomquarkB±,andat132,thelightestparticlewithtwobottomquarksΥ(1S).

Thefactthatabovenumber109allpointsrepresenthadronscontainingcharmorbottomquarksdoesnotmeanthatunstableparticlescontainingonlylightquarksarenotfoundinthismassrange.Itonlyindicatesthatsuchresonanceshavenotbeenthefocusofexperi-mentalsearchesinthatenergyrange.Also,atsuchhighenergiesproductionoflightmesonsissocopiousthatwide,light-mesonresonancesgetlostinthebackground.

Finally,thetrueheavy-weights,thegaugebosons,withmassesnear100GeV,standaloneatthefarsidesofFigure1andFigure2.Theirisolationisagainanartifactofexperimentalparticlephysicshistory,notofsomefundamentalnatureofthemassspectrumofparticles.

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Tomakeprecisemeasurementsoftheseparticlesentailedbuildingexperimentswithfarmoreenergyandatafargreatercost.TobetterexplorethephysicsatthisscaleandtosearchforpotentiallyheavierparticlessuchastheHiggsbosonwillrequirethenextgenerationofparticleaccelerators.

4WidthSpectrum

Unlikethemassspectrum,thewidthspectrum(i.e.,theplotofthewidthsofthe139unstableparticlesinincreasingorder)depictedinFigure3isnotasdependentontheenergyscalesofexperimentsthathavebeenperformed.Figure3spansmanydecadesofthewidth,andeventhemostmassiveparticlescanhaveverysmallorverylargewidths.Forexample,theΛbbaryon,withmassandwidth(5.62GeV,5.36×10−13GeV),hasamassrankof130outof139,butawidthrankof15.Itisconsideredunlikelythatwewilldiscoveranymorestableorextremelylong-livedsubatomicparticles,sofuturediscoverieswilllikelyeitherfitintothisgraphor,perhapsfornewultra-massiveparticlesliketheHiggsboson,beappendedtotheend,beyondthegaugebosons.Thereforetheshapeofthisgraphisunlikelytochangemuchasunstableparticlesareadded,whereasnewparticlediscoverieswillprobablysmoothoutthehighenergy,quark-massthresholdgapsinthemassspectrumFigure2.

LookingatFigure3,itseemsnaturaltoroughlybreakthegraphintothreeparts:particleswithwidthsΓ<10−8MeV,10−5MeV<Γ<10MeV,andΓ>10MeV.ThisdivisionbecomesevenmoresensiblewhenFigure4,whichliststheparticlesinorderofthewidth-to-massratioΓ/M,isconsidered.Thefirst40particlesinbothFigures3and4havenearlythesameorder;onlyafterthatisthereasubstantialreshuffling.InFigure4aheuristicdivisionbetweennearlythesamethreeclassesofparticlescouldbemadeatΓ/M<10−10,10−8<Γ/M<10−2.5andΓ/M>10−2.5.

4.1FirstClass:DecayingStates

Theinterestingthingabouttheseclassesisthattheyhavephysicalsignificance.InFigure1,thisfirstclassofparticlesisthearcofleptons,mesonsandbaryonsalongthelowerpartofthegraph.Fromthepointofviewoffundamentalinteractions,whattheseparticleshaveincommonisthattheydecayviatheweakinteraction,andconsequentlytheyarelong-lived.Long-livedisofcoursearelativeterm,butwidthsΓ<10−8MeVcorrespondtolifetimesτ>10−14s.Intermsofphenomenology,thatmeansforparticlesinthisfirstclasstheexponentialdecayratecanbemeasureddirectly.Thismeasurementinvolvesfindingthedistancetraveledbetweentheproductionlocationandthedecayvertex.Thisinformation,combinedwithkinematicinformationofmass,momentum,energyand/orspeed,canbe

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usedtofindthetime-of-flightintherestframeoftheparticle.Ahistogramofallthetime-of-flightsforacertainparticletypecanbefittoanexponentialtogetthelifetime.Longlifemakesthisfirstclassseemlikethemost“particle-like”ofalltheunstableparticles.Inmanycalculationstheycanbeapproximatedasstablebecausetheyarestablewithrespecttothemuchshortertimescalesdictatedbythestrongandelectromagneticinteractions.Theirwidth/massratiosΓ/Maresosmallthatanymassuncertaintycanbeneglectedinkinematiccalculations.

4.2ThirdClass:Resonances

Incontrast,particlesinthethirdclasshavewidthsΓ>10MeV,correspondingtoparticleswithlifetimesτ<10−22,andtoawidth-to-massratioΓ/Mgreaterthanapartinathou-sand.Becauseofthismassuncertainty,theycanbeproducedinexperimentswithenergiessubstantiallylowerorhigherthantheircentral(quoted)massvalue.Thisclassofparti-clesdecaysprimarilythroughstronginteractionsandthevastlifetimedifferenceseparatingthemfromthefirstclassisattributabletothestrengthofthestronginteractionandthemassivenessoftheweakgaugebosons.

Itmayseemunjustifiedtoconsidertheseshort-livedunstableparticlesasthesamekindofobjectaslong-livedunstableparticles,sovastarethedifferencesintheirinstabilitypa-rameters.Thedistinctionbetweenthesetwoclassesissometimescodifiedinthelanguageofparticlephysics:thelong-livedparticlesarecalleddecayingstatesandtheshort-livedarecalledresonances.Thisdistinctionarisesbecauseofthedifferentwaysthatdecayingstatesandresonancesareobservedexperimentally.

Particleresonancesaredetectedasrapidvariations(usuallypeaks)inthecrosssection.Asthecenter-of-massenergyofacollisionisscannedoversomerange,theremayappearanenhancementoftheelasticcrosssectionorthecrosssectionintoaparticularsetofinelasticchannels.Afterextractingthebackgroundandaccountingforuncertaintiesintheprepara-tionanddetectionapparatusesandothereffects(suchasradiativecorrections),theresonantcrosssectionσRasafunctionofcenter-of-massenergy(orcenter-of-massenergysquareds)canbeextracted.Thisprocesscanbecomemorecomplicatediftherearemultipleresonancesinthesameenergyregion,interferingresonances,orbackground-resonanceinterference.Neglectingthesecomplications,theresonancecrosssectionσR(orresonancelineshapeorlinewidth)canbefittoatheoreticalfunctionandthemassandwidthextracted.Typically,thefunctionusedistheBreit-WignerorLorentzianfunction,whichcanbeparameterizedintermsofmassandwidthas

󰀁󰀁1

σ(s)∝󰀁󰀁

orbyseveralotherparameterizations.ThisfunctionalformfortheresonancelineshapecanbederivedbyassociatingtheresonancetoapoleinthescatteringS-matrix[3,4].Alter-natefunctionsexist;forexample,inperturbativequantumfieldtheory,theon-massshellrenormalizationschemeleadstoadifferentdefinitionformassandwidth(foradiscussionofthisasappliedtotheZ-boson,see[5]).Forsomeresonances,suchasthe∆baryons,theReview[1]citesvaluesformassandwidthcorrespondingtoboththeBreit-Wignerandperturbationtheorydefinitions.

Tomeasuretheresonancecrosssectionaccuratelyandtoextractavalueformassandwidth,whicheverfunctionandparameterizationareused,requiresexperimentalenergyreso-lutionpreciseenoughtotraceoutthelineshape.Practically,thismeanstheenergyresolutionshouldbesmallerthantheratioΓ/Mforthatparticularresonance.Asaresult,theline-shapesofparticlesinthefirstclass,withΓ/M<10−10,cannotbemeasuredinthisway.Measurementsofwidthandlifetimearephysicallyverydistinctprocessesthatapplytophenomenaatverydifferentenergyscales.

ThewidthandlifetimeofaparticlearerelatedtheoreticallybytheWeisskopf-Wignerformulaτ=󰀁/Γ,whichwasoriginallyproposedinthecontextofatomicelectronicline-shapes[6].Thisrelationisconsideredsostandardastoappeartobeadefinitionoridentityordirectconsequenceoftheuncertaintyprinciple,buttothebestknowledgeofthisau-thor,thisrelationshiphasneverbeenverifiedexperimentallyintheregimeofsubatomicphysics[7].TheWigner-WeisskopfrelationbetweenthewidthanddecayrateorinverselifetimeΓ/󰀁isderivedasanapproximationinnon-relativisticscatteringinmanytextbooks(forexample[3,11]),andcanbeprovenidenticallyinthenon-relativisticandrelativisticcaseforthemathematicalobjectcalledaGamowvector[12].

4.3MiddleClass:Troublemakers

Themiddleclassofparticlesonthechart,withwidthsroughlybetween10−5MeV<Γ<10MeVandwidth-to-massratios10−8<Γ/M<10−2.5,areawkwardlyplacedfromanexper-imentalpointofview.Theirlifetimesaresoshortthatdirectmeasurementofexponentialdecayisextremelydifficultorimpossible;theirwidthsaresonarrowthatfewexperimentshavetheenergyresolutiontoaccuratelytracetheirlineshape.Whilethemiddleclassiseas-ilydistinguishablefromthedecayingstatesinFigure1,theyblendintothebottomoftheresonancestates.Thereisnocleargapcharacteristicofthedifferenceinstrengthbetweenelectromagnetically-andstrongly-decayingstates.

Physically,whatdostatesinthismiddleclasshaveincommon?Someofthesestateshavesubstantialbranchingratiosintoelectromagneticdecaychannelsliketheπ0andΣ0anddonotdecayviathestronginteraction.Some,liketheD∗(2020)±arebarelyabovetheenergythresholdoffortheirprimarydecaychannelsandtheirdecaysaretherefore

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phase-spacesuppressed.Allthemesonsinthisclass(excepttheD∗(2020)±)areunflavored,i.e.composedofquarksandantiquarksofthesameflavor(orsuperpositionsofsame-flavorquark/antiquarkpairs)andarenotenergeticenoughtodecayintostrongly-favoredchannels.Neitherdecayingstatesnorresonancesbythedefinitionsdescribedabove,howarethewidthsofthesestatesmeasured?Below,theexperimentaldeterminationsofsomeoftheseparticles’widthsaredescribedtogiveaflavorforsomeoftheothertoolsatthedisposalofparticlephysicists.

•Withalifetimeof(8.4±0.6)×10−17s,theexponentialdecayrateoftheπ0isveryhardtomeasure.Theπ0decaysalmostexclusivelyintoapairofphotonsviatheelectromag-neticinteraction.AprecisionexperimentbyAtherton,etal.[13],fitanexponentialtothreedatapointsextractedfromthedistancebetweentheproductionandthedecayoftheπ0intotwophotons.Themeasurementagrees(withinerror)withanothermethodusedtomeasurewidthsandlifetimes:thePrimakoffeffect[14].Bombardingheavynucleiwithgammarays,aninteractionbetweentheincomingphotonandavirtualphotoncanproducetheπ0;thisprocessiscalledphoto-production.OnecanmeasurethecrosssectionforthisprocessandrelateittheoreticallyviathetransitionamplitudetothepartialwidthordecayrateΓ2γfortheprocessπ0→γγ.Fromthis,thetotalwidthisdeterminedbydividingthepartialwidthΓ2γbytheindependently-measurablebranchingratioB2γintothetwophotondecaychannel,i.e.,Γ=Γ2γ/B2γ.

•ThePrimakoffeffecthasalsobeenusedtomeasurethelifetimeoftheΣ0baryonas(7.4±0.7)×10−20s.TheΣ0decaysalmostexclusivelytoΛγ.

•Forthecc¯resonancesJ/ψandψ(2S)andtheb¯bresonancesΥ(1S),Υ(2S)andΥ(3S)thepartialwidthΓeeisextractedfromtheintegratedcrosssection,forexample,intheelasticprocesse+e−→J/ψ→e+e−.ThentheratiooftheelasticcrosssectiontothetotalcrosssectionindependentlyprovidesthebranchingratioBee.Thetotalwidth(asabove)isΓ=Γee/Bee.

•BoththeabovemethodandthePrimakoffeffecthavebeenappliedtotheηmesongivingconsistentresultsandavalueforthewidthofΓ=1.18±0.11keV.

±

•Finally,theD∗0(2010),withawidthofΓ=96±4±22keV,hasamassjustabovethethresholdforitsmaindecaychannelsD0π±andD±π0.Itswidthcannotbemeasureddirectly,butcanbeextractedfromfittingtosimulationsofthetheenergydistributionofdecayproducts[15].

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5SummaryandFurtherConsiderations

Themass-widthspectruminFigure1doesnotrevealasmuchinformationaboutunstableparticlesasHertzsprung-Russelldiagramsrevealaboutstellarcompositionandevolution[16].Thesegraphsdonotcontributetofindingaperturbative,renormalizable,eleganttheoryforpredictingmassesandwidthsofhadronsbasedonstandardmodelparameters.Nonethe-less,Figure1andsubsequentpartitionsofitdoshowsomestructuresthatcorrespondtosignificantphenomenologicalfeatures.Asaresult,Ithinkthesegraphsprovideanexcellenttool(oratleastastartingpoint)forinstructingstudentsofparticlephysicsaboutahostofphysicalphenomenaandexperimentalprocedures.Apartiallistofthesefeaturesorideaswouldinclude:

•Relativestrengthsofthefundamentalinteractions.

•Connectionofthetransitionamplitudeandscatteringmatrixtomeasurablequantities.•Experimentalmeasurementofexponentialdecay.•Resonanceproductionanddecay.•Crosssectionmeasurements.

•Phasespacedependenceofwidthanddecayrate.

Additionally,notjusttheviewingofthesegraphs,buttheproductionofthesekindsofplotswouldbeanexcellentexerciseforstudents,undergraduateorgraduate.Whilefamiliarizingthemselveswiththeparticlezoo,theycouldalsopracticeusinggraphinganddatamanagementsoftware.Asanexampleofotherkindsofplotspossible,Fig.5depictsthewidthversusmassofjustthemesonsonalinearscale.

Finally,visualizingthemassandwidthspectrumofunstableparticleswithsuchgraphsmakesthetaskofconceptuallyorganizingthephysicaldataofhundredsofunstablepar-ticlesalittleeasier.Itprovidesanalternatewaytogroupthemintophenomenologically-meaningfulfamilies,complimentingstandardorganizationschemesaccordingtoflavorand/orquarkcontent.Manyphysicistshavesomebiological,taxonomicalpartoftheirbrainstowhichIhopethesegraphsappeal.

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References

[1]K.Hagiwaraetal.,“TheReviewofParticlePhysics,”Phys.Rev.D66,0100011-974

(2002).[2]http://pdg.lbl.gov/rpp/mcdata/mass_width_02.mc

[3]Forexample,seeM.L.Goldberger,K.M.Watson,CollisionTheory(Wiley,NewYork,

1964),chap.8.[4]Seealso,ArnoBohm,“QuantumMechanics:FoundationsandApplications,”(Springer,

NewYork,1994),3rded.,chap.18.[5]ArnoR.BohmandN.L.Harshman,“OnthemassandwidthoftheZ-bosonandother

relativisticresonances,”Nucl.Phys.B581,91-115(2000).[6]V.WeisskopfandE.P.Wigner,“Berechnungdernat¨urlichenLinienbreiteaufGrund

derDiracsichenLichttheorie,”Z.f.Physik63,54-73(1930)and“Uberdienat¨urlichLinienbreiteinderStrahlungdesharmonischenOszillators,”65,18-27(1930).[7]TheWigner-Weisskopfrelationhasbeenverifiedsometimeagoforsomenuclearunsta-blestates[8,9]usingtheM¨ossbauereffecttoachievetherequiredenergyresolutionandmorerecentlyforoneatomicunstablestate[10]usingultra-coldatomsforthesamepurpose.[8]Forexample,seeH.Frauenfelder,TheM¨ossbauerEffect(W.A.Benjamin,NewYork,

1962),pp.50-51.[9]P.A.DeYong,P.L.Jolivette,andN.Rouze,“ExperimentalverificationoftheHeisenberg

uncertaintyprinciple—anadvancedundergraduatelaboratory,”Am.J.Phys.61(6),560-563(1993).[10]U.Volz,M.Majerus,H.Liebel,A.Schmitt,andH.Schmoranzer,“PrecisionLifetime

MeasurementsonNaI3p2P1/2and3p2P3/2byBeam-Gas-LaserSpectroscopy,”Phys.Rev.Lett.76,2862-2865(1996);C.W.Oates,K.R.Vogel,andJ.L.Hall,“HighPrecisionLinewidthMeasurementofLaser-CooledAtoms:ResolutionoftheNa3p2P3/2LifetimeDiscrepancy,”Phys.Rev.Lett.76,2866-2869(1996).[11]Foramoreelementarydiscussionthanthatin[3]basedontime-dependentperturbation

theory,seeEugenMerzbacher,QuantumMechanics(Wiley,NewYork,1970),2nded.,chap.18,whichincludesthequotation“Toobtainnonreversingtransitionsanda

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progressivedepletionoftheinitialstateitisessentialthatthediscreteinitialstatebecoupledtoaverylargenumberofstateswithsimilarfrequencies.However,thefactremainsthattheexponentialdecaylaw,forwhichwehavesomuchempiricalsupportinradioactiveprocesses,isnotarigorousconsequenceofquantummechanicsbuttheresultofsomewhatdelicateapproximations,”pp.484-485.

[12]GamowwasthefirsttouseeigenfunctionsoftheHamiltonianwithcomplexenergy

foraheuristicdescriptionofunstablestatesinG.Gamow,“ZurQuantentheoriederAtomkernes,”Z.Phys.51,204-218(1928).Foramodernnon-relativistictreatment,seechap.21of[4];forarelativisticdiscussion,see[5];also,A.Bohm,N.L.HarshmanandH.Walther,“RelatingtheLorentzianandexponential:Fermi’sapproximation,theFouriertransform,andcausality,”Phys.Rev.A66,0121071-11(2002).[13]H.W.Athertonetal.,“Directmeasurementofthelifetimeoftheneutralpion,”Phys.

Lett.15B,81-84(1985).[14]H.Primakoff,“Photo-productionofneutralmesonsinnuclearelectricfieldsandthe

meanlifeoftheneutralmeson,”Phys.Rev.81,899(1951).[15]A.Anastassovetal.(CLEOCollaboration),“FirstmeasurementofΓ(D∗+)andpreci-sionmeasurementofmD∗+−mD0,”Phys.Rev.D65,0320031-11(2002).[16]M.Zeilik,S.A.Gregory,E.v.P.Smith,IntroductoryAstronomyandAstrophysics,

ThirdEdition,(HarcourtBraceJavanovich,FortWorth,1992),p.249-267.

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Fig.1.Log-logplotofMass/MeVversusWidth/MeV.Choiceof139unstableparticlesplotteddescribedintext.Key:hollowcircles—gaugebosons;blackstars—leptons;blacktriangles—lightunflavoredmesons;graytriangles—strangemesons;hollowtriangles—flavoredcharmedmesons(includingcharmed-strangemesons);blackdiamonds—unflavoredcharmed(cc¯)mesons;graydiamonds—flavoredbottommesons(includingbottom-strangeandbottom-charmedmesons);hollowdiamonds—unflavoredbottom(b¯b)mesons;blacksquares—Nand∆baryons;graysquares—strangebaryons(includingΛ,Σ,ΞandΩbaryons);hol-lowsquares—charmedandbottombaryons.

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Fig.2.Log-linearplotofMass/MeVversusparticlerankinorderofincreasingmass(outof139selectedparticles).KeyissameasinFig.1.

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Fig.3.Log-linearplotofWidth/MeVversusparticlerankinorderofincreasingwidth(outof139selectedparticles).KeyissameasinFig.1.

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Fig.4.Log-linearplotofwidth-to-massratioΓ/Mversusrankinorderofincreasingwidth-to-massratio(outof139selectedparticles).KeyissameasinFig.1.

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Fig.5.Linearplotofwidthversusmassofunstablemesons.Key:blackdiamonds—light,unflavoredmesons,blackstars—strangemesons;blacktriangles—charm-flavoredmesons(includingcharmed/strange-flavoredmesons);hollowtriangles—cc¯-mesons;blacksquares—bottom-flavoredmesons(includingbottom/strange-andbottom/charmed-flavoredmesons);hollowsquares—b¯b-mesons.

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