A Generalization of the Brodsky-Lepage Formalism

We present an approach that generalizes in a natural way the perturbative QCD formalism developed by Brodsky and Lepage for the study of exclusive hadronic processes to the case of $L\neq 0$ mesons. As an application of our approach we consider here the pr

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aINFNCA-TH9709hep-ph/9706349June1997AGENERALIZATIONOFTHEBRODSKY-LEPAGEFORMALISM L.HOURA-YAOU,P.KESSLER,J.PARISILaboratoiredePhysiqueCorpusculaire,Coll`egedeFrance11,PlaceMarcelinBerthelot,F-75231ParisCedex05,FranceF.MURGIAIstitutoNazionalediFisicaNucleare,SezionediCagliariCasellaPostale170,I-09042Monserrato(CA),ItalyJ.HANSSONDepartmentofPhysics,Lule aUniversityofTechnology,S-95187Lule a,SwedenWepresentanapproachthatgeneralizesinanaturalwaytheperturbativeQCDformalismdevelopedbyBrodskyandLepageforthestudyofexclusivehadronicprocessestothecaseofL=0mesons.Asanapplicationofourapproachweconsiderheretheproductionofmesonpairs,involvingtensorandpseudotensormesons,inphoton-photoncollisions.Inthelastyearsalotoftheoreticalworkhasbeendevotedtothestudyofhadronicexclusiveprocessesathightransfermomentum,usingfactorizationtechniquesandperturbativeQCD(PQCD)1.WhilethePQCDformalismseemstobequitesuccessful,itappliesonlytothecaseofhadronswithinternalorbitalangularmomentumL=0.Recently2wehaveproposedanapproachwhichgeneralizesthePQCDformalismtothecaseoftheproductionand/ordecayofmesonswitha(qq¯)valenceFockstateandwithanyvalueofthe

orbitalangularmomentumL.

Inthiscontributionwegiveashort,qualitativepresentationofourap-proach,anddiscussitsapplicationtothecaseofmesonpairproductioninphoton-photoncollisions.PredictionsfortheproductionofpseudoscalarandvectormesonpairshavebeenalreadygivenbyBrodskyandLepage3.Itistheninterestingtoextendthisstudytotheproductionofothermesonpairs,involvingtensorandpseudotensormesons.

Letus rstrecallhowthehelicityamplitudesforagenericexclusivehadronicprocessathightransfermomentumareevaluatedinperturbative

We present an approach that generalizes in a natural way the perturbative QCD formalism developed by Brodsky and Lepage for the study of exclusive hadronic processes to the case of $L\neq 0$ mesons. As an application of our approach we consider here the pr

ingfactorizationtechniques,thoseamplitudesareexpressedasaconvolutionamongdi erentcontributions:i)Ahard-scatteringamplitudein-volvingthevalencepartonsofallparticipatinghadrons,assumedtobe,insideeachhadron,inacollinearcon guration;ii)Soft,nonperturbativedistributionamplitudes(DA)for:a) ndingthe(collinear)valencepartonsintheincominghadrons;b)the nalpartonicstatetoformtheobservedoutgoinghadrons.

Noticethat:i)OnlyleadingFockstatesareconsidered(i.e.,|qq¯ formesons,|qqq forbaryons);ii)Valencepartonmassesareneglected(this,to-getherwithcollinearcon guration,forcestheorbitalangularmomentumLofeachhadrontobevanishing).

Inourapproach2weextendthePQCDformalismtothecaseofL=0mesonsbycombiningitwithabound-statemodelofweaklyboundvalencequarksforthemesonsinvolved.Ageneralprescriptioncanthusbegivenwhichrelatesthehelicityamplitudesfortheoverallreaction(involvingtheobservedmesons)tothepartonicamplitudes(involvingthecollinearvalencequarksofthosemesons).ThisprescriptionimpliestheusualconvolutionofthepartonicamplitudewiththemesonicDA’s,asdiscussedbefore.Inaddition,itinvolvesalsoaconvolutionwiththemomentum-spacewavefunctionofthevalenceqq¯pairineachmeson,giveninthecorrespondingmesonrest-frame.ALorentzboostconnectstherestframeofeachmesonwiththecenterofmassframeoftheinitialparticles;weassumethatthemesons,asseeninthisoverallframe,areextreme-relativistic(thisentailssomesimpli cationsinactualcalculations).Anotherbasicingredientofourapproachistheassumptionthat,foreachmeson,themomentum-spacewavefunctionofthevalenceqq¯pairissharplypeakedaroundzero,sothataseriesexpansioninpowersoftherelativeqq¯momentumcanbemadeandonlytheleadingtermshavetobeconsidered,in rstapproximation.

Thefollowingequationshowstheresultingrelationshipbetweentheoverallandthepartonichelicityamplitudesinthecaseofinteresthere,thatismesonpairproductioninphoton-photoncollisions,γγ′→QQ′:

MLSJΛ,L′S′J′Λ′

λγλγ′(E,Θ)=fL fL′ ′CLSJL′S′J′

×β,lim1β′→02dL 0ΛΛC0Λ′Λ′d(cosθ′)

0,0(θ)

We present an approach that generalizes in a natural way the perturbative QCD formalism developed by Brodsky and Lepage for the study of exclusive hadronic processes to the case of $L\neq 0$ mesons. As an application of our approach we consider here the pr

Q′);fListhenormalizationconstantrelatedtotheQmesonbound-statewavefunction;β=2|k|/MQ,θ,refertotherelativeqq¯momentum,k,insidemesonQ,initsrestframe;ΦN(x)istheQmesondistributionamplitude;xisthefractionofQmesonlongitudinalmomentumcarriedbyquarkq; nally,d-matricesandClebsch-Gordancoe cientscomefromopportunelycombiningtheqq¯spinandrelativeangularmomentumstatesto ttheQmesonquantumnumbers.NoticealsothatinthepartonicamplitudeTthehelicitiesofeachqq¯pairhavealreadybeencombinedtogivetherequiredspinandhelicityforthecorrespondingmeson.

LetusstressthatEq.(1)reducesexactlytotheusualPQCDresultwhenL=0.

ThepartonichelicityamplitudeTinEq.(1)canbeevaluatedatlead-ingorderinperturbativeQCD.ThenormalizationconstantsfL,fL′maybe xedbyevaluatingwithinthesametheoreticalapproachthetwo-photonde-caywidthsΓ(Q→γγ)andcomparingthemwiththecorrespondingavail-ableexperimentalresults.InordertocheckthedependenceofourresultsonthemesonDA’sappearinginEq.(1)wehaveconsideredseveralindicativechoices:i)Forthepion,theasymptotic(ASY)DA,ΦASYN,π(x)=6x(1 x),

2andtheChernyak-Zhitnitsky(CZ)DA,ΦCZN,π(x)=30x(1 x)(2x 1);ii)

Fortensorandpseudotensormesonstheso-callednonrelativistic(NR)DA,LΦNRN(x)=δ(x 1/2),andageneralizedasymptotic(GASY)DA,ΦN(x)∝xL+1(1 x)L+1,whichisalsorequiredtogetridofpossibledivergencesin=122thedenominatorofthepartonicamplitudes(e.g.,ΦLN(x)=30x(1 x)for=233tensormesons,ΦLN(x)=140x(1 x)forpseudotensormesons).

Letusnowpresentsomenumericalresultsfortheproductionof:i)Tensormesonpairs;ii)Pseudotensormesonpairs;iii)Pairsmadebyapionandapseudotensormeson.AmorecompletediscussioncanbefoundinRef.2;herewelimitourselvestopresentsomeindicativeexamples.

InFig.1thescalingdi erentialcrosssectionE8dσ/dt,asafunctionofcos2Θispresented(Θisthescatteringangleinthetwo-photonc.m.frame).Resultsaregivenforanumberoftensormesonpairs(usingtheGASYDA).Forcomparison,analogousresultsforpionpairsarealsopresented,usingtheasymptoticDA.

Tables1and2show,forallthemesonpairsconsidered,theintegratedcrosssections(overthepTofthemesonsinthelab.frame,startingfromagivenminimumpT)fortheoverallprocesse e+→e e+QQ′,intwokinematicalconditions,correspondingroughlytothoseofLEP2andofapossibleB-factory.Again,severalchoicesforthemesonDA’sareconsidered.

FromtheresultsofFig.1andTables1,2(seealsoRef.2)onecanmakethefollowingconclusions:i)Theresultsobtaineddonotdependstronglyonthe

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We present an approach that generalizes in a natural way the perturbative QCD formalism developed by Brodsky and Lepage for the study of exclusive hadronic processes to the case of $L\neq 0$ mesons. As an application of our approach we consider here the pr

Fig.1:Di erentialcrosssectionE8[dσ/dt]innb×GeV6,asafunctionof

cos2Θ,fortheprocessγγ→QQ′involvingtheproductionoftensor-mesonpairs;thegeneralizedasymptotic(GASY)DAwasusedfortensormesons;

forcomparison,analogouscurvesforγγ→π+π andγγ→π0π0,usingtheasymptoticDAarealsoshown.

choiceofthedistributionamplitudefortensorandpseudotensormesons;ontheaverage,theGASYDAleadstoresultsslightly(byafactorof2or3)higherthanthenonrelativisticone;ii)Asexpected,thechargedchannelsgiverisetosigni cantlyhigheryieldsthantheneutralones,andarethenthefavoritecandidatesforexperimentalsearches;iii)Theproductionofthechargedmesonpairshereconsideredmaybecomemeasurablewithhigh-energye e+collidersofthenextgeneration,withintegratedluminositiesofabout1040cm 2.Acknowledgments

ThisworkhasbeenpartiallysupportedbytheEUprogram“HumanCapitalandMobility”undercontractCHRX-CT94-0450.

References

1.See,e.g.,S.J.BrodskyandG.P.Lepage,inPerturbativeQuantumChro-

modynamics,ed.A.H.Mueller(WorldScienti c,Singapore,1989).

2.L.Houra-Yaou,P.Kessler,J.Parisi,F.MurgiaandJ.Hansson,preprint

hep-ph/9611337,Z.Phys.C,inpress,1997.

3.S.J.BrodskyandG.P.Lepage,Phys.Rev.D24,1808(1981).

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We present an approach that generalizes in a natural way the perturbative QCD formalism developed by Brodsky and Lepage for the study of exclusive hadronic processes to the case of $L\neq 0$ mesons. As an application of our approach we consider here the pr

Table1:Integratedcrosssections(in10 40cm2)fortheprocesse e+→√e e+QQ′,at

NRGASY

s=10GeV,pT>1GeV.

QQ′σ(e e+→e e+QQ′)[10 40cm2]

f2f2

0a02a21.42.3

1.5

0.04

20.1

10.0

41.6

12.0

3.0

341.1

122.01.43.62.30.0668.61.672.424.58.8536.7193.2f2a02′′f2f2a 20π2 π20πCZπ200πASYπ2+ πCZπ2

+ πASYπ2a+20π2+π20

5

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