diabnb998.com

Eur.
Phys.
J.
C54,73–87(2008)THEEUROPEANPHYSICALJOURNALCDOI10.
1140/epjc/s10052-007-0515-0RegularArticle–ExperimentalPhysicsExperimentaldisagreementswithextendedunitarityD.
V.
BuggaQueenMary,UniversityofLondon,MileEndRd.
,LondonE14NS,UKReceived:12December2007/Publishedonline:19January2008Springer-Verlag/Societ`aItalianadiFisica2008Abstract.
Inproductionprocesses,e.
g.
J/Ψ→ωππorpp→3π,theσandf0(980)overlapinthesamepar-tialwave.
Theconjectureofextendedunitarity(EU)statesthattheππpairshouldhavethesamephasevariationasππelasticscattering.
ThisisanextensionofWatson'stheorembeyonditsoriginalderivation,whichstatedonlythatthes-dependenceofasingleresonanceshouldbeuniversal.
ThepredictionofEUisthatthedeepdipobservedinππelasticscatteringcloseto1GeVshouldalsoappearinproductiondata.
Foursetsofdatadisagreewiththisprediction.
Allrequiredierentrelativemagnitudesofσandf0(980).
Thatbeingso,afreshconjectureistorewritethe2-bodyunitarityrelationforproductionintermsofobservedmagnitudes.
ThisleadstoapredictiondierenttoEU.
CentralproductiondatafromtheAFSexperimenttnaturallytothishypothesis.
PACS.
13.
25.
-k;13.
25.
Gv;13.
75.
Lb1IntroductionInitssimplestform,theideaofextendedunitarity(EU)statesthattheππpairinasinglepartialwaveshouldhavethesamephasevariationwithsinallreactionsasinelas-ticscattering.
ThisideaoriginatesfromAitchison[1]andhasbeenadoptedinvariousguisesbymanyauthors.
HisargumentswillbepresentedindetailinSect.
2,soastoex-posetheassumptionsandconsequences.
Atthetimetheideawasintroduced,itwasareasonableconjecture;nowmoderndataallowittobecheckedaccurately,butdisagreewithit.
ManyexperimentalgroupshavemadeextensivetstoproductiondatausingaK-matrixapproachbasedonEU.
Thesetsareexcellent;nocriticismisintendedoftheirquality.
Experimentalistshavefoundempiricallytheneces-saryfreedomtogetgoodtstodata.
However,oncloseinspection,thisfreedomisinconsistentwithstrictEU.
Thiswholetopichasbeenthesubjectofextensivedis-cussionwithmanyauthors.
Thereisabewilderingjungleofclaimsandcounter-claims.
Myobjectiveistocutapaththroughthistangleandexposewhereproblemslie;thismakesthepresentationpedanticinplaces.
Aitchison'sessentialpointisthatallprocessesshouldbedescribedbyauniversaldenominator[1iρ(s)K(s)],whereKisthesameasforelasticscattering;ρisLorentzinvariantphasespace.
TheassumptionwhichisbeingmadeisthatWatson'stheorem[2]appliestothecoherentsumofallcomponentsintheJP=0+partialwave.
ThisisastepbeyondWatson'sderivation,whichreferredonlyae-mail:D.
Bugg@rl.
ac.
uktoasingleeigenstate;Watsondidnotconsideroverlappingresonances.
Inππelasticscattering,thef0(980)issuperimposedonaslowlyrisingamplitudeassociatedwiththeσpole.
Cern–Munichdata[3]showthatthephasesofthesetwocompo-nentsadd.
BelowtheKKthreshold,bothσandf0(980)T-matricesT=eiδsinδareconnedtotheunitaritycircleifweneglectthetinyinelasticityduetoππ→γγ.
Unitar-itymaybesatisedbymultiplyingS-matricesS=e2iδofσandf0(980),assuggestedbyDalitzandTuan[4].
Thiststhedatawithinerrorsof3.
5.
Figure1showstheArganddiagramfortheI=0ππS-wavefrommyrecentre-analysisofthese(andother)data[5].
FromBESdataonJ/Ψ→φπ+π,thef0(980)hasafull-widthathalf-maximumof34±8MeV[6].
Thecom-binedphaseshiftrisesrapidlyfrom90at0.
88GeVto270near1.
1GeV.
Thereisadeepdipinthecrosssectionwherethecombinedphasegoesrapidlythrough180.
Thecru-cialpointofEUisthatthisfeatureshouldbecommontoproductionprocesses.
BESdataonJ/Ψ→φππ[6]immediatelyrequireamodicationoftherudimentaryformofEU.
TheππmassspectruminthesedataisreproducedinFig.
2.
Thereisadominantf0(980)contributionandasmallinterferingσcontribution;thisisverydierenttoelasticscattering.
L¨ahdeandMeissner[7]modifytheconjectureofEUtoap-plyseparatelytostrangeandnon-strangecomponents,i.
e.
tothescalarformfactorsforππandKK.
Thedipintheelasticcrosssectionat989MeVisaverydelicatefeature.
If,foranyreason,relativemagnitudesofσandf0change,thezeroat989MeVcandisappearquickly;hereandelsewhere,f0willdenotef0(980)unlessthereis74D.
V.
Bugg:ExperimentaldisagreementswithextendedunitarityFig.
1.
ArgandplotoftheππI=0S-waveinelasticscatter-ing;massesaremarkedinGeVFig.
2.
TheππmassprojectionforBESdataonJ/Ψ→φπ+π:theupperhistogramshowsthecurrentttoexperi-mentalpoints[5];thelowerhistogramshowsthettedσcom-ponentconfusionwithotherf0's.
Ifthephaseoff0changeswithrespecttotheσ,themassatwhichthedipappearswilllikewisechange.
Theinterferenceregionbetweenσandf0(980)isanidealplacetocheckextendedunitarity.
Twoconsiderationswillplayacriticalrole:unitarityandanalyticity.
Unitarityisoftenquotedandplaysanes-sentialroleinsettingupthecurrentK-matrixformalismwhichtreatsbothelasticscatteringandproductiononthesamebasis.
Foraproductionreaction,Aitchisonconjec-turesaunitarityrelationfortheproductionamplitudeF:ImF=FTel.
(1)HedeneshisFtobeproportionaltoT(p)/ρ,whereT(p)referstoproduction:Fρ=αT(p).
(2)Dividingbothsidesof(1)byα,ImT(p)=T(p)Tel.
(3)ItisoddthatT(p)ontheright-handsideismultipliedbyTel,unlessT(p)=Tel.
However,experimentwillrequiredif-ferentcontributionstoT(p)fromσandf0.
Considernextanalyticity.
Dispersionrelationsconnectmagnitudesandphases.
Iftherelativemagnitudesoff0andσchangefromthoseofelasticscatteringbecauseofmatrixelements,theirrelativephasesmustalsochange.
Conversely,analyticitypredictsthatifthephasevariationwithsoftheamplitudeisuniversal,asEUdemands,soisthevariationwithsofthemagnitude(uptoaconstantscalingfactor);forthesimplestsituationwhereonlyreso-nancesarepresent,therelativemagnitudesofσ,f0(980)andanyfurtherf0mustbealmostthesameinproductionaselasticscattering.
Thisisapointwhichhasalmostal-waysbeenignored.
Theword'almost'representsacaveat:theremayinadditionbeapolynomialisswhichcanbedierentbe-tweenelasticscatteringandproduction.
Itturnsoutthatonecanplausiblylimitdeviationsofrelativemagnitudeswithin12%.
ThisquestionisdiscussedinSect.
2.
1.
Experi-mentrequireslargerdeviationsthanthisinthefoursetsofdatadiscussedhere.
ThisimpliesphasesmustchangefromthosepredictedbyEU.
Experimentalistshavecorrectlyal-lowedforthisbyusingcomplexcouplingconstantsintheisobarmodel.
Section3comparesthepredictionofEUwith3setsofdata.
TherstconcernsBESdataforJ/Ψ→ωπ+π[8].
Thenon-strangecomponentsofσandf0dominatebothelasticscatteringandJ/Ψ→ωππ.
FromAitchison'salge-braandthatofL¨ahdeandMeissner,itfollowsthatthef0amplitudeshouldhavealmostthesamemagnitudeastheσamplitude,aswellasthesamephaseaselasticscatter-ing.
Thispredictioniscontradictedbythedata,wherenof0(980)isvisibleandattothedataplacesalowlimitonit.
ThenexttwosetsareCrystalBarreldataforpp→3π0,whereσandf0(980)areclearlyvisible[9].
Onesetisforannihilationinliquidhydrogenandtheotherforgaseoushydrogen.
Annihilationfromthe3P1initialstateis13%inliquidand48%ingas,allowingaclearseparationofampli-tudesforproductionofσandf0from1S0and3P1.
Resultsforbothareinconsistentwiththedeepdipofelasticscat-teringpredictedbyEU.
Section4concernsdatafromtheAFSexperimentoncentralproduction:pp→ppπ+π[10].
Hereoneexpectstheprotonsinthenalstatetoactasspectators.
However,EUstillfailsconspicuouslytotthedata.
Thisimportantresultleadstoarevisedformoftheunitarityrelation,asfollows.
Figure3sketchestheusualdiagrammaticapproachtotheunitarityrelation.
Itmaybederivedbycuttingthedia-gramdownthemiddle,alongthedashedline.
Fora2-bodysystemofππ,KK,ηη,etc.
theresultingrelationiswellknown:ImTel=TelTel.
(4)Theapplicationof2-bodyunitarityassumesthatthepionsinteractonlywithoneanother,notwithanyspectator.
Inmostsetsofdatatherearelargesignalswherepionsdoin-D.
V.
Bugg:Experimentaldisagreementswithextendedunitarity75Fig.
3.
Unitaritydiagramforππ→ππteractwiththespectator.
ForJ/Ψ→ωππ,asanexample,theb1(1235)πchannelaccountsfor40%ofevents[8].
Someofitmaybegeneratedbypionsfromdecaysofσorf0rescatteringfromthespectator;thisisaso-calledtrianglegraph.
Aitchisonhimselfremarksthatthiscandistorttheunitarityrelation.
ThisprovidesonereasonwhyEUmayfailfortherstthreesetsofdata;itdoesnotexplainthefourth,wheresomefurthereectisrequired.
Therearefundamentaldierencesbetweenelasticscat-teringandproduction.
InJ/Ψ→ωππ,forexample,matrixelementsJ/Ψ|ωσandJ/Ψ|ωf0dictatethemagnitudesoftheseamplitudes;anyvaluesarepossible.
Thisdiersfromelasticscattering,whereσandf0magnitudesarexedpurelybytheircouplingconstantsgπtoππ.
Equa-tion(3)isanasymmetricrelation,allowingσandf0tobeproducedwithdierentmagnitudes,butrequiringthattheyrescatteraninelasticscattering.
Amorelogicalalter-nativeisthesymmetricrelationImT(p)=T(p)T(p),(5)henceImF=FT(p).
ThisrelationtsAFSdataforcen-tralproductionnaturally,whereasEUdoesnot.
Ifthef0isabsentfromproductiondata,(5)reducestotheobviousrelationImTσ=|Tσ|2;Aitchison'sformoftherelation,takenwithanalyticitydoesnotallowthef0tobeabsent,asweshallseeinSect.
2.
Section5suggestsanewwayoftting2-bodydata.
Section6thensummarisesconclusions.
2ThehypothesisofextendedunitarityInatwo-bodyprocess,thescatteringofapairofpionstonalstatesππ,KK,ηη,4πandγγmustsatisfyunitarity.
TheT-matrixforthesecoupledchannelsmaybewrittenintermsofarealK-matrixasTel=ρK(1iρK)1.
(6)ItisnormalisedheresothatTππ=(ηe2iδ1)/2i.
BelowtheinelasticthresholdρK=tanδ.
(7)TheT-matricesusedherewillincludecouplingstoallchannels.
However,itsimpliesthepresentationofessen-tialpointstoreducetheformalisminitiallytoasingleππchannel.
Thissimplicationissucienttoexposethebasicissues,andcanbegeneralisedlatertoincludeinelasticity.
TheapproachofAitchison[1]willnowbeoutlined.
Iamgratefultohimforclarifyingthealgebrainmoredetailthanistobefoundintheoriginalpublication.
SupposeS-matricesmultiply,i.
e.
phasesadd.
LetKAandKBbeK-matricesforσandf0respectively.
Theelementaryex-pressionfortan(δA+δB)thengivesaK-matrixforelasticscatteringKel=KA+KB1ρ2KAKB,(8)fromwhichitfollowsthattheT-matrixforelasticscatter-ingisTel=(KA+KB)ρ(1iρKA)(1iρKB).
(9)AitchisonnowconjecturesthatanamplitudeFforpro-ducingatwo-bodychannelpresentinKelmaybewrittenintermsofavectorP,withF=(1iρKel)1P,(10)P=αAKA+αBKB1ρ2KAKB,(11)whereαAandαBareconstantsforproductioncouplings.
Withthisansatz,therelationImF=FTel,(12)knownasextendedunitarity,isautomaticallysatised.
Itisaconsequenceof(10)thatFhasthesamephaseasTel.
Substituting(8)and(11)in(10)gives,inthisone-channelcaseF=αAKA+αBKB(1iρKA)(1iρKB),(13)=αA[TA(1+iTB)+βTB(1+iTA)]/ρ,(14)whereβ=αB/αA.
From(13),thephaseofFisindeedδA+δB,asimposedby(10).
Equation(14)willplaythedecisiveroleincomparisonswithexperiment.
In(14),TA(1+iTB)=expi(δA+δB)sinδAcosδBandTB(1+iTA)=expi(δA+δB)cosδAsinδB.
At989MeV,δf=90andδσ=92.
Sobothtermsareveryclosetozero,regardlessofthevaluesofαAandαB.
Thispredictsthatproductiondatashouldhavethesamedeepdipatthisen-ergyaselasticscattering.
Thereisafurtherpoint.
Inthesecondterm,(1+iTA)0overasizablemassrange.
Intherstterm,TBshouldbeconspicuous,sinceithasarapidphasevariationandthesamepeakmagnitudeasTA,whichisitselfclearlyvisibleinallsetsofdataconsideredhere.
However,thedataallrequirethemagnitudeofthef0(980)tobesmallerthanpredicted.
Thekeypointisthatthefactor1/(1iρKB)of(13)leadsdirectlytothefactor(1+iTB)inthersttermof(14).
Therstandthirdsetsofdatawillrequireβof(14)tobesmall.
Intheelasticregion,thersttermbecomesFiαA1+ieiδBsinδB.
76D.
V.
Bugg:ExperimentaldisagreementswithextendedunitarityThebizarreconclusionofEUisthatthephaseofthef0(980)ispresenteventhoughαB0.
Thisisinconsistentwithanalyticity.
ItwillbeshowninSect.
3thatexperimentdisagreeswithEUevenwithouttheconstraintofanalytic-ity.
Howeverthisadditionalconstraintmakesconclusionsmoredenitive.
2.
1AnalyticityForpurelyelasticscattering,theOmn`esrelation[11]reads(includingafactorρ(s)inN(s)):Tel(s)=N(s)/D(s),(15)D(s)=eiδ(s)exps4m2ππPdss4m2πδ(s)ss,(16)wherePdenotestheprincipalvalueintegral.
Weshallnotactuallyneedtoevaluate(16).
Itplaysonlyaconceptualroleandthisneedsconsiderableexplanation.
ThebasicpointisthatD(s)containsbothrealandimaginaryparts,soδ(s)determinesboth.
ForelasticscatteringN(s)isreal.
Itarisesfromtheleft-handcut,i.
e.
mesonexchangesbe-tweenthetwopions.
Withinelasticity,correspondingrelationsmaybewrit-tenina2-channelform.
Thenδ(s)isreplacedbyφ(s),theangleTmakestotherealaxiswhenmeasuredfromtheoriginoftheArganddiagram,seeFig.
12ofSect.
4.
1.
IfEUisvalid,theproductionamplitudemaybewrit-tenF=X(s)/D(s).
InprincipleX(s)couldbeanything,dependingonproductiondynamics.
However,wehavequitepreciseexperimentalinformationaboutit.
Anex-tremeviewisthatαAandαBof(13)canbearbitraryandcomplex.
Howeveriftheyarecomplexthisleadsdi-rectlytoaconictwithEU.
Equation(14)containstwopartsTA(1+iTB)andTB(1+iTA).
SubstitutingBreit–WignerformulaeforTAandTB,thersttermbecomesg2A(M2Bs)/DA(s)DB(s).
Thisisrealbuthasaspecics-dependenceinthenumeratoraswellasinthedenomina-tor.
IfαAorαBbecomescomplex,thenumeratorbecomescomplex.
ThisthenintroducesaphasevariationseparatefromD(s).
The"prediction"ofEUisdistortedbythisex-traphase.
OnlyifX(s)isrealdoesEUsurviveinitsstrictform.
ManyexperimentalgroupshaveusedtheP-vectorap-proachusingcomplexcouplingcoecients,withoutrealis-ingthatthisdestroystheuniversalityofthephasevaria-tionwiths.
Thisiswhatexperimentdemands,sotheyhavedonetherightthing.
Buttheuseoftheuniversaldenomi-nator[1iρK(el)]isnolongerlogicallycorrect.
OnemightaswelltdirectlyintermsofcomplexcouplingconstantsandindividualT-matricesforeachresonance.
2.
2FormfactorsThereisafurtherfundamentalpoint.
Forelasticscatter-ing,N(el)isuniquelyrelatedtoImD(s)bybothunitarityandanalyticity.
Atrstsightitappearsthatanalyticityre-latesX(s)inthesamewaytoD(s)inproductionreactions,withtheresultX(s)=αN(el),whereαisaconstant.
Thisrequiresβ=1:ifthephaseoftheππamplitudeisuniver-sal,relativemagnitudesofσandf0mustalsobeuniversal.
Thereishoweveracaveat.
Amorefundamentalformoftheunitarityrelation(12)isthatthediscontinuityofFacrosstheelasticbranchcutis2iFT.
ThenFmaybemul-tipliedbyapolynomialX(s),providingitdoesnothaveadiscontinuityalongthereals-axis.
Afewexampleswillhopefullyclarifyideas.
Firstly,aformfactorinsisonesuchexample,arisingfromthesizesofparticles,i.
e.
frommatrixelements.
Secondly,inφ→γf0,theE1transitionhasanintensityproportionaltothecubeofthephotonmomen-tum;thisinatesthelowersideofthef0(980).
Thirdly,in3P1pp→π0σ,thereisanL=1centrifugalbarrierfortheproductionprocess.
Fourthly,insomespecialcases,matrixelementsmaygothroughzeroasafunctionofs.
TakingX(s)tobereal,letuswriteingeneralF=X(s)N(el)/D(s).
(17)Afeatureofallproductiondataconsideredhereisastronglow-massππpeakduetotheσpole.
ThispeakisnotpresentinelasticscatteringbecauseofanAdlerzerointheelasticscatteringamplitudeatsA0.
41m2π,justbelowtheππthreshold.
Theelasticamplituderisesapproximatelylinearlywithsandthereisnolowmasspeak.
Theoriginofthedierencehasbeenknowntotheo-ristsforatleast20years.
Au,MorganandPennington[12]pointedoutthatthedierencebetweenelasticscatter-ingandcentralproductiondatacanbeaccomodatedbyusingthesameBreit–Wignerdenominatorforboth,butreplacingthenumeratorN(el)bysomethingclosetoacon-stant.
ThispolynomialisallowedbecausesAisoutsidethephysicalregion.
DatarequireX(s)N(el)1,henceX(s)1/N(el)1/(ssA).
Moreexactly,X(s)=1/[(ssA)(1+bs)expsM2A/A(18)fortheparametrisationoftheσamplitudein[13].
Inprac-tice,quadaticandcubictermsinsareverysmallandundertightcontrolfromtstodataupto1.
8GeV.
ForJ/Ψ→ωππ,theσpoleisvisiblebyeyeinFig.
4abelow.
Thephaseoftheσamplitudeinthisreactionisexperimentallythesameasinelasticscatteringwithin3.
5[14].
ValuesofN(prodn)=X(s)N(el)canbede-termineddirectlyfromthedata.
Thesameistrueoftheκ[15],whichlikewisehasanAdlerzerointhenumeratorforelasticscattering,butnotforproduction.
Inbothcases,N(prodn)isconsistentwithinerrorswithaconstant;theAdlerzerointhenumeratorofelasticscatteringhasdis-appeared.
Onecantryttingtheσandκpolesinproduc-tiondatawiththeconventionalformfactorN(prodn)=exp(k2R2/6),wherekismomentumintheproductionchannel.
Forboth,R2optimisesatslightlynegativevalues,whichareunphysical.
Fortheσ,R2104,becausethef0(1370)contributionisfartoolarge.
Evenifthef0(1370)isttedfreelyinmagnitude,χ2=137for37points.
Thet,shownbythechaincurveofFig.
10bisparticularlybadclosetotheKKthreshold,wherethedipofelasticscatteringispredictedbyEU.
ThettoKKdataisalsopoor,withaχ2of18for5points.
ThedottedcurveofFig.
10bshowstheeectofttingfreelyanadditionalcontributionfromtheI=2S-waves:χ2=59.
6.
Thisfailstocurethepoort.
ItmakesalmostnodierencewhethertheI=2amplitudeisdividedbyafactor(ssA)liketheσamplitude.
ThebasicdicultyisthattheslowlyvaryingI=2amplitudecannotcuretherapidstructureduetof0(980).
Furthermore,therelativemagnitudesofthettedI=2amplitudeandtheEUam-plitudeis0.
46,whereasitisonly0.
18forelasticscatteringat1GeV.
SuchalargeI=2amplitudeisimplausible.
ObviouslytheproblemwithEUisthatthemagnitudeofthef0amplitudeneedstobesmallerthanforelasticscattering.
Fromanalyticity,thisalsorequiresadierenceinphase.
Thatisalsoclearfromtheabsenceofthepre-dicteddipat989MeVinthedata.
Ifthephaseofthef0(980)amplitudeisconstrainedtotheEUvalue,butrelativemagnitudesofσandf0aresetfree,χ2=144,whichisbad.
Mostofχ2comesfrompointsimmediatelyaroundtheKKthreshold,showingthatthedatarejectalsothephasevariationofEU.
4.
1ProposedmodicationtoEUAtthispoint,onecouldarguethatthemechanismoftheproductionreactionisunknownandmightgenerateaphaseforf0(980)dierenttothatoftheσ.
Thisar-gumentisnotspecic,thoughtheisobarmodelcantthedatawell.
However,theusualargumentforadierentphaseforf0andσismultiplescatteringofthepionswithspectatorparticles.
InAFSdata,thereisanemptyrapiditygapisolatingthecentralregion.
RememberalsothatCern–Munichphaseshiftsarederivedintherstinstancefromdataonπp→ππpathighmomentumandsmallmomen-tumtransfer,asimilarconguration.
TheconjectureofEUwillnowbereplacedwithanal-ternativeansatz.
Thetreatmentofproductiondataneedstobeabletocopewiththecasewhereoneresonanceamplitudeiszero.
EUdoesnot,sinceauniversalphaseequaltoelasticscatteringrequiresaproductionamplitudeT(p)∝1/Dσ(s)Df0(s).
ThecorrectproductionamplitudeshouldreducetoTσwhenthef0isabsent.
Asmallf0am-plitudeshouldproduceasmallperturbationtoTσ.
Supposethe2-bodyππamplitudeiswrittenasFρ1=αT(p)=α1+β2TA+βTBe2iΨ(s),(19)whereαandβarereal.
Thisallowsfreedominβandin-cludesaphaseΨ(s)whichbecomesthesameasforelasticscatteringwhenβ→1.
ItisnecessarytochooseasAthestatewiththelargeramplitudeonresonance,sothatβ≤1.
Ifonecouldcreatethisππsystemin'freespace',theappro-priate2-bodyunitarityrelationbelowtheKKthresholdwouldbeImT(p)=|T(p)|2.
(20)Analternativewayofformulatingthebasicphysics(withthesameoutcome)isintermsoftheSchwinger–Dysonequation.
InsteadoftheconventionalrelationTprod=Vprod+VprodGTel,(21)myconjectureistoreplacethiswithTprod=Vprod+VprodGTprod.
(22)HereVisthe'potential'generatingthenalstateandGisthepropagator.
Forthecaseofpurelyelasticscattering,aclosedformforΨ(s)of(19)maybederivedbysubstitutingTAandTBintheform(e2iδ1)/2iinto(20).
Aftersimplecancella-tionsbetweenleft-andright-handsides,sin(2Ψ+δB2δA)=βsinδB,(23)or2Ψ=2δAδB+sin1(βsinδB).
(24)Ifβ=1,thisisadierentrelationfrompurelyelasticscattering.
84D.
V.
Bugg:ExperimentaldisagreementswithextendedunitarityTheimprovementinthetisdramatic.
ImmediatelyanexcellentttopointsbelowtheKKthresholdisobtainedwithχ2=28.
4for29pointsand24degreesoffreedom.
Thetermsin1(βsinδB)in(24)diersfromδBby54.
Thisisjustwhatisneededtoproducetheinterferencebetweenσandf0observedintheisobarmodelt.
AbovetheKKthreshold,itistemptingtosatisfyuni-taritybyintroducingaK-matrix.
However,theK-matrixdependsontheassumptionthatthe2-bodysystemiscon-nedtotheunitarycircle,butthatisnolongerthecaseina3-bodysituation.
ThetmaybeextendedabovetheinelasticthresholdbywritingππandKKamplitudesasFππ=αTσ11+γTσ21+βTf11+Tf21e2iΨ/ρ1=αTσ111+γgσ2rgσ1+Tf11β+gf2rgf1e2iΨρ1,(25)FKK=αTσ12+γTσ22+βTf12+Tf22e2iΨ/ρ2=αTσ11gσ2gσ11r+γgσ2gσ1+Tf11gf2gf1β+gf2gf1e2iΨρ1.
(26)Equations(25)and(26)exposetheexplicitdependenceofT12ontheratior=ρ2/ρ1;theexperimentalgroupdi-videsoutthephasespaceρ2andρ1intheππandKKchannels.
AsexplainedinSect.
2.
1,allThavethenumer-atorofelasticscatteringreplacedbyaconstant.
MyT12isdenedsoastocontainafactor√ρ1ρ2andT22isdenedtocontainafactorρ2.
Withthesedenitions,theunitarityrelationsbecomeImT11=|T11|2+|T12|2,(27)ImT12=T11T12+T12T22,(28)ImT22=|T22|2+T21T12.
(29)Equations(25)and(26)satisfytheserelationsbycon-struction,exceptthatΨandΨneedtobeconstrainedtoobey(27)–(29).
AbovetheKKthreshold,thisisdoneusingfreelyttedΨandΨforeverydatapointandintro-ducingintoχ2apenaltyfunctionwhichapplies(27)–(29)with3%errors;inpractisetheseconstraintsareeasilysat-isedanddiscrepanciesattheendofthetarebelowthe1%level;thisiswellbelowexperimentalerrors.
InfacttheKKdataarenotveryprecise,leavinglargeexibilityinΨabovetheKKthreshold.
Inotherwords,thedataareeasytotabovetheKKthreshold,buthighlydenitivebelowit.
Adetailisthatg2needstoincludeformfactorsbothbe-lowandabovetheKKthreshold,suchthatitfallsquiterapidlyonbothsidesofthethreshold;formulaearegivenin[13].
Thebesttwithσandf0(980)alonehasχ2=46.
4for37pointsand32degreesoffreedom.
Thetisgoodupto1.
1GeV,butisinadequateforππdatanear1.
3GeV.
Thismaybecuredstraightforwardlybyaddingasmallf0(1370)component.
Atechnicaldetailisthatthef0(1370)amplitudeismultipliedbyafactorexp(2iΨ)and(24)isiterated;thecontributionoff0(1370)toT12andT22isneg-ligible.
Theresultingt,shownonFig.
11a,hasβ=0.
59±0.
06andthef0(1370)amplitudeis0.
18timesthatoftheσamplitudeat1.
3GeV.
Theχ2forππdatais28.
7for29de-greesoffreedom.
TheOmegacollaborationreportsasig-nicantcontributionfromf0(1370)totheirdataoncentralproductionofπ+π[30].
Theirttedmassandwidthagreecloselywiththeline-shapettedtof0(1370)in[5].
Figure11bshowsthettoAFSK+Kdata.
AdetailhereisthattheK+KdataofAFSarescaledupbyafac-tor4/3toallowforisospinClebsch–Gordancoecientsinπ+πandK+Ksystems.
TheKKdataconstraintheco-ecientsofT12andT22amplitudes.
ThelowestAFSKKpointhassmallacceptancewhichmayhavesignicantsys-tematicuncertainty[31].
Figure12showstheArganddiagramforthettedam-plitude.
Thisillustratestheformof(20).
Thereisageomet-ricalrelationbetweentheimaginarypartoftheamplitudeanditsmodulussquared,butitisadierentrelationtoEU.
Thef0(980)amplitudeissmallerthanthatoftheσandtheirrelativephasesaredierenttoEU.
ThesameisFig.
11.
FitstoAFSdata:ausingtherevisedformof2-bodyunitarity,bttoKKdataFig.
12.
TheArganddiagramoftheamplitudettingAFSdata;massesaremarkedinGeVD.
V.
Bugg:Experimentaldisagreementswithextendedunitarity85trueofFig.
5,theArganddiagramttingBESdata;therethef0(980)contributionisverysmall.
Letusnowreturnto(20)and(24)andreviewtheirgen-eralfeatures.
Thelasttermof(24),sin1(βsinδB)→δBasβ→1andδB→90.
Furthermoreitapproachesthislimitnon-linearlyasβ→1.
Theinterestingpointisthattwo-bodyelasticscatteringemergesasalimitingcaseofamoregeneralrelation.
FurthermoreithaspathologicalpropertiesastherelativemagnitudeofAandBcrosses1(or1).
IfBbecomeslargerthanA,itisnecessarytoin-terchangetherolesofAandB.
Asβapproaches1frombelow,Ψ→δAandonerecoversthestandardresultofelas-ticscattering.
Furtherpathologicalcasesariseifβ→1(impossibleinthe2-bodyelasticcase).
Elasticscatteringisinfactaveryspecialcase.
SoEUcanfailverybadlyasβdepartsfrom1.
Asβdropsfrom1,thephaseΨmeasuredfromtheori-ginoftheArganddiagramsoonchangesbyonlyamodestamountoverthef0.
Inthiscase,theisobarmodelbecomesanexcellentapproximation:thef0hasitsusualdepen-denceonsthroughitsphaseshiftδandthef0amplitudeaddsvectoriallytotheσ;theisobarmodelcanthentaconstantphasetobothquitesuccessfullyifthetermsin1(βsinδ)issmall.
Insummary,themodiedformofEUsuggestedin(20)and(24)givesamuchbettertthanEU.
ThetofMorganandPennington[29]usingEUrequiresanadditionalthird-sheetpoleatM=978i28MeV.
Thisadditionalpolecan-notbeaccomodatedbyBESφππdata,whichrequireonlyasecond-sheetpoleat998i17MeVandabroadthird-sheetpoleat851i418MeV.
Thesetwopoleshaveanat-uralexplanation.
Ifthecouplingoff0(980)toKKisal-lowedtodecreasegraduallytozero,leavingotherparam-etersunchanged,thetwopolescoalescetowardsthesamepolepositionM=968i82MeV;itistheeectofρ(KK)whichmovesthesecond-sheetpoletotheKKthresholdandmovesthethird-sheetpoleaway.
Thereisnopoleinthismassrangefromtheσamplitude.
Asecondremarkisthatallofσ,κ,a0(980)andf0(980)maybereproducedasanonetbythemodelofRuppandvanBeveren,wheremesonscoupletoaquarkloop[32].
Inthismodel,noadditionalpolelikethatofMorganandPenningtonappearsinthemassrangeclosetotheKKthreshold.
ThesetworesultssuggestthattheadditionalpoleisaconsequenceoftheconstraintofEU.
Therelation(20)isanewconjecture.
Arethereforsee-ablesnagsTheσandf0maymix,andthismixingcouldbedierentinelasticscatteringandproduction.
Thismix-ingwouldaltertheapparentwidthoff0(980)andcouldinduceanadditionalphasechangerelativetoσ.
Presentlythereisnoindicationofanyneedformixing.
Suchmix-ingwillbeabsentifσandf0(980)havestrictlyorthogonalwavefunctions,asisplausibleformembersofthesamenonet.
OnFig.
2,theσisdenitelyvisibleinφππdata.
Itwouldnotbesurprisingifφππandωππchannelsl-teroutorthogonalcombinationsofσandf0.
FromFig.
2,onecanestimatetherelativeintensitiesofσandf0.
Itisnecessarytoallowforthemassresolution,sincethef0am-plitudefallsextremelyrapidlyfromitspeakat989MeV,particularlyabovetheKKthreshold.
Doingthis,thein-tensityofσis4%off0atthepeak,i.
e.
20%inamplitde.
Thisismarginallyhigherthanthef0signalttedtoωππbutwithintheerror,supportingtheideaoforthogonalamplitudes.
5HowtotelasticdataabovetheKKthresholdManyauthorsusetheK-matrixtosatisfyunitarityfor2-bodyscattering,e.
g.
thecoupledchannelsππ,KK,ηη,ηηand4π.
ThepopularapproachistoaddK-matricesofallresonancesappearinginonepartialwave.
However,ifresonancesoverlap,asσandf0(980)do,theK-matrixpolesoccuratmasseswherecombinedphaseshiftshappentogothrough90,270,etc.
,i.
e.
at750and1200MeV.
Firstly,anexpansionintermsofthesepolesisproblemati-calforf0(980)unlessotherfactorsorhighpowersofsareincluded.
Secondly,therelationbetweenK-matrixpolesandT-matrixpolesisobscure.
AnyoneT-matrixpoleisbuiltupfromallK-matrixpoles;theconverseisobvious.
TheprescriptionthatS-matricesmultiplybelowthein-elasticthresholddoesnotappearnaturally,buthastobeenforcedbyttingdata.
AnattractivealternativecanbeconstructedfollowingthespiritofAitchison'sapproach(for2-bodyscattering).
Supposeonecombinestworesonancesaccordingtothepre-scriptionKij(total)=(KA+KB)ij10.
5ρiρj(KAKB+KBKA)ij.
(30)BelowtheKKthreshold,thisautomaticallygivesthere-sultthatphaseshiftsadd.
(Furtherresonancesmaybecombinedbyiteratingthisprescription).
Anicefeatureof(30)isthatonecanwriteKij=gigjM2s,(31)usingthesamemassMastheusualBreit–Wignerde-nominator.
Asecondattractivefeatureof(31)isthattheamplitudecontinuesnaturallythroughtheKKthreshold,becauseofthefactorρiρjinthedenominator.
Myownapproachinseveralpapers,[5,19]hasbeenclosetothis.
AlldiagonalelementsofS-matricesaremul-tiplied,asproposedhere.
Magnitudesofo-diagonalelem-entsoftheS-matrixneedtobecalculatedfromunitarityrelations.
Forexample,fora3-channelsystem:|S12|2=(1/2)1+|S33|2|S11|2|S22|2.
(32)Thephaseoftheseo-diagonalelementshasbeenttedempirically,whereas(30)wouldpredictthesephases.
Thisapproachsuccessfullytselasticdata,ππ→KKandηηwithoneproviso:agoodtrequiresinclusionofmixingbetweenσ,f0(1370)andf0(1500)[5],usingtheformulaeofAnisovich,AnisovichandSarantsev[33];theseformu-laearethemodernequivalentoftheBreit–Rabiequationofmolecularspectroscopy,generalisedtoincluderesonancewidths.
86D.
V.
Bugg:Experimentaldisagreementswithextendedunitarity6ConclusionsCrystalBarreldatahavef0(980)andσcomponentswithrelativemagnitudesseriouslydierenttothosepredictedbyEUplusanalyticity.
Furthermore,theyaredierentin3S1and3P1annihilation.
TheBESdataforJ/Ψ→ωπ+πdonotreproducethedeepdipofelasticscatteringat989MeV.
AFSdatalikewisedonotcontainthesamedipatthismass.
Alltheseresultsareinconictwith(14),whichisadirectconsequenceofEU.
ThisshowsunambiguouslythattheremustbeamajorawinthehypothesisofEU.
Experimentalistshavedealtwiththisproblembyusingcomplexcouplingconstantsforeachresonance.
However,asemphasisedinSect.
2.
1,theimaginarypartofthecoup-lingconstantintroducesintothenumeratoroftheam-plitudeans-dependentphasevariationwhichalterstheuniversalphasecomingfromthedenominator[1iρK(s)].
Thisdestroystheoriginalideaofauniversalphase.
OnemightaswelltdirectlyintermsoftheT-matrixofeachindividualresonance,alongthelinesoutlinedinSect.
5.
TheformoftheK-matrixsuggestedtherewouldeliminatedierencesbetweenK-matrixandT-matrixpoles,makinginterpretationofresultsmoredirect.
NotallexperimentalistsadopttheP-vectorapproach.
SometdirectlyintermsofindividualT-matricesforeachresonance,includingsequentialdecaysfromoneresonancetoadaughterwithdierentcomplexcouplingconstantsforeachdecaymode.
AscoliandWyld[34]andSchultandWyld[35]consideramultiplescatteringseriesofthetypeR→(12)3→1(23),etc.
,whereRisa3-bodyresonanceandbracketsindicateresonancesintwo-bodysub-systems;thisisaunitarityeectofadierentformtothatconsid-eredhere.
InviewofthefailureofEUinthe4casesconsideredhere,eachnewsetofdatashouldbeinspectedonitsmerits.
LetusexaminewaysoftryingtosaveEU.
Firstly,itispossiblethatunspeciedbackgroundscanbeaddedtotheP-vectorsoastoside-steptheconict.
However,analyt-icityindependentlylimitsrelativemagnitudesoff0(980)andσwithin12%.
Experimentaldeterminationsofσandκphasesin[14]and[15]constrainphaseswithin±5.
Theprobabilitythatunspeciedbackgroundscanevade(14)tothisaccuracyinfoursetsofdataisvanishinglysmall.
Furthermore,ifsuchbackgroundsareintroduced,EUlosesanypredictivepower.
Secondly,CrystalBarreldataandAFSdatacannotbettedwithEUwhetherornottheI=2S-waveamplitudeisincluded.
Sothisisnotasatisfactoryescaperoute.
Athirdlikelypossibility,applicabletotherstthreesetsofdata,isthatpionsfromsigmaand/orf0(980)rescatterfromthespectatorparticle,leadingtoabreak-downofEU.
Aitchisonhimselfpointedthisout.
Today,weknownthatsuchgraphshavemagnitudestypically25%ofthoseoftheparentprocessesbeforetherescattering.
Thisissucienttointroducelargephasechangesinsomecases,butnotall.
Forallofthesethreesetsofdata,theisobarmodelpro-videsanexcellentt.
TheproductionamplitudeisthenwrittenF=αANA(el)/DA+αBNB(el)/DBwithcomplexαAandαB.
Inthisform,novestigeremainsofthecon-straintthatS-matricesmustmultiplyasinelasticscat-tering.
RelativemagnitudesofαAandαBcanarisefrommatrixelementscouplingtheinitialstatetoeachreson-ance.
ForJ/Ψ→ωππand3P1pp→3π0,thef0componentissosmallthatonecannottellwhetherthephasealonefol-lowsEUornot.
However,for1S0pp→3π0,thef0signalislargeenoughtoruleoutthispossibility.
ThefourthpointisthatonewouldstillexpectEUtoworkforAFSdata,butitdoesnot.
AnexcellenttmaybeobtainedbyreplacingtheunitarityrelationImT(p)=T(p)TelbythenewrelationImT(p)=|T(p)|2.
(33)ThiscorrespondstotherelationImF=FT(p),(34)ratherthanthecommonlyusedformImF=FTel.
Equations(20)and(24)havepathologicalbehaviourinthevicinityofthe2-bodyelasticlimitβ=1.
OnecannowseethebasicproblemofEU.
Itattemptstoimposeonthe3-bodysystemaveryspecialbehaviourwhichisnarrowlyrestrictedto2-bodyscattering.
Awayfromthespecialcaseβ=1,theisobarmodelworkssuccessfully.
Therearetwopointsaboutthenewunitarityrelations.
Firstly,itwasarguedinSect.
2.
1thatauniversalphaseinthedenominatoroftheamplitudealsorequires,viaan-alyticity,almostuniversalmagnitudes;theword'almost'coversthepossibilitythattheremaybeslowlyvaryingformfactorsorcentrifugalbarrierfactorsinproductionre-actionswithoutcorrespondingchangestoD(s).
Ifrelativemagnitudesofresonancesdierbylargeamountsbetween2-bodyscatteringandproduction,theirrelativephasesmustalsochange.
Secondly,thenewunitarityrelation(33)succeedsquantitativelyinaccountingfortheobservedrelativephasebetweenσandf0(980)incentralproduction.
Thatisanon-trivialresult.
ThettoAFSdatathenrequiresonlytwopolesinthevicinityoff0(980),inagreementwiththeBESline-shape(asdoestheisobarmodel).
EUrequiresanextrapoleforwhichthereisnoobviousexplanation.
TheformofthisnewunitarityrelationisillustratedbytheArganddiagramsofFigs.
5and12.
Inboth,thef0(980)amplitudeissmallorfairlysmallandsoissin1(βsinδB)of(19).
Asaresult,thephaseΨmeasuredfromtheoriginoftheArganddiagramchangesratherlittleoverthef0.
Thisisanextrasourceofphasesappearingintheisobarmodel,andhasnotbeenappreciatedbefore.
However,onethenneedstoaskwhetherthisnewrela-tioncanbeuseduniversallyintheisobarmodel.
Doesit,forexample,correctlydescribetherelativephasesofσandf0in1S0→3π0dataandinJ/Ψ→φππTheanswerisno.
Forthesetworeactions,theagreementbetweendataand(20)and(24)improvessubstantiallyoverEU.
How-ever,therearestilldiscrepancieswiththenewunitarityrelationof20–30,whichisstillsignicant.
Itseemslikelythatrescatteringofpionsfromthespectatorintroducessomeadditionalphases.
Thenewunitarityrelationneedstobetestedelsewhere.
ApossibletestinggroundisinKloedataonφ→γπ0π0,D.
V.
Bugg:Experimentaldisagreementswithextendedunitarity87wherebothσandf0maycontribute,butthedecayiselec-tromagnetic;thesmallamplitudefromφ→ρπ0introducesaperturbationofonly4%inamplitudeandonlyinwelldenedpartsoftheDalitzplot.
Theremedywhichsucceedswellinttingnearlyalldataistheisobarmodel,wherebothmagnitudesandphasesofresonancesarebothttedfreely.
Itneedstobeemphasisedthatexperimentalgroupshaveadoptedtheexibilityneededtotexistingdata,sotheirresultsareessentiallysoundandarenotinquestion.
Thehypothe-sisofEUhasmostlybeenadoptedbytheoristsformakingpredictions.
Thosepredictionsnowneedtobeviewedwithsuspicion.
Althoughthescalarformfactoriswelldeter-minedforelasticscattering,itisdangeroustoassumethatthisformfactorisuniversalandcanpredictproductionprocesses.
Acknowledgements.
IamgreatlyindebtedtoI.
J.
R.
Aitchisonforextensiveandilluminatingcomments.
IamalsogratefultoProf.
G.
RuppandProf.
E.
vanBeverenforextensivediscus-sions.
References1.
I.
J.
R.
Aitchison,Nucl.
Phys.
A189,417(1972)2.
K.
M.
Watson,Phys.
Rev.
88,1163(1952)3.
B.
Hyamsetal.
,Nucl.
Phys.
B64,134(1973)4.
R.
H.
Dalitz,S.
Tuan,Ann.
Phys.
(NewYork)10,397(1960)5.
D.
V.
Bugg,Eur.
Phys.
J.
C52,55(2007)6.
BESCollaboration,M.
Ablikimetal.
,Phys.
Lett.
B607,243(2005)7.
T.
A.
L¨ahde,U.
G.
Meissner,Phys.
Rev.
D74,034021(2006)8.
BESCollaboration,M.
Ablikimetal.
,Phys.
Lett.
B598,149(2004)9.
A.
Abeleetal.
,Nucl.
Phys.
A609,562(1996)10.
T.
Akessenetal.
,Nucl.
Phys.
B264,154(1986)11.
R.
Omn`es,NuovoCim.
8,316(1958)12.
K.
L.
Au,D.
Morgan,M.
R.
Pennington,Phys.
Rev.
D35,1633(1987)13.
D.
V.
Bugg,J.
Phys.
G34,151(2007)14.
D.
V.
Bugg,Eur.
Phys.
J.
C37,433(2004)15.
D.
V.
Bugg,Phys.
Lett.
B632,471(2006)16.
W.
Ochs,PhDthesis(UniversityofMunich,1974)17.
S.
Pislaketal.
,Phys.
Rev.
D57,072004(2003)18.
I.
Caprini,G.
Colangelo,H.
Leutwyler,Phys.
Rev.
Lett.
96,132001(2006)19.
D.
V.
Bugg,Eur.
Phys.
J.
C47,45(2006)20.
V.
Baruetal.
,Eur.
Phys.
J.
A23,523(2005)21.
ParticleDataGroup,J.
Phys.
G33,1(2006)22.
A.
Aloisioetal.
,Phys.
Lett.
B537,21(2002)23.
BESCollaboration,M.
Ablikimetal.
,Phys.
Lett.
B603,138(2004)24.
I.
Aitchison,privatecommunication25.
M.
G.
Bowleretal.
,Nucl.
Phys.
B97,227(1975)26.
I.
J.
R.
Aitchison,M.
G.
Bowler,J.
Phys.
G3,1503(1977)27.
J.
L.
Basdevant,E.
L.
Berger,Phys.
Rev.
D16,657(1977)28.
G.
Reifenrother,E.
Klempt,Nucl.
Phys.
A503,886(1989)29.
D.
Morgan,M.
R.
Pennington,Phys.
Rev.
D48,1185(1993)30.
D.
Barberisetal.
,Phys.
Lett.
B462,462(1999)31.
A.
A.
Carter,privatecommunication32.
E.
vanBeveren,D.
V.
Bugg,F.
Kleefeld,G.
Rupp,Phys.
Lett.
B641,265(2006)33.
A.
V.
Anisovich,V.
V.
Anisovich,A.
V.
Sarantsev,Z.
Phys.
A359,173(1997)34.
G.
Ascoli,H.
W.
Wyld,Phys.
Rev.
D12,43(1975)35.
R.
L.
Schult,H.
W.
Wyld,Phys.
Rev.
D16,62(1977)