Possible methods for the determination of the $P$-parity of the $Theta^+$-pentaquark in NN-

NN-collisions.

MichailP.Rekalo

NationalScienceCentre-KharkovInstituteofPhysicsandTechnology,

Akademicheskaya1,61108Kharkov,Ukraine

EgleTomasi-Gustafsson

DAPNIA/SPhN,CEA/Saclay,91191Gif-sur-YvetteCedex,France

(Dated:February1,2008)

WepresenttwopossibilitiestodeterminetheP-parityofthepentaquarkΘ+,inamodelindepen-dentway,viathemeasurementofpolarizationobservablesinp+p→Θ++Σ+,orn+p→Θ++Λ0,inthenearthresholdregion.Besidesthemeasurementofthespincorrelationcoefficient,Axx=Ayy,(incollisionsoftransversallypolarizednucleons),thecoefficientDxxofpolarizationtransferfromtheinitialprotontothefinalΣ+(Λ0)hyperonisalsounambiguouslyrelatedtotheΘ+parity.

PACSnumbers:13.75.Cs,21.10.Hw,13.88.+e,14.20.Jn

arXiv:hep-ph/0401162v1 22 Jan 2004In1999,N.K.PakandM.Rekaloproposed[1]twonewmethodsforthedeterminationoftheP-parityoftheK-meson,throughthemeasurementofdifferentpolarizationobservablesinK-mesonproductioninprotonprotoncollisionsnearthreshold,p+p→K++Y0+p(Y0=ΛorΣ-hyperon).OnemethodisbasedonthemeasurementofthesignofthespincorrelationcoefficientAyy,forcollisionsoftransversallypolarizedprotons.Thesecondoneisbasedonthemeasurementofthepolarizationtransfercoefficientfromtheinitialprotontotheproducedhyperon,Dnn.Bothmethodsapplyinthresholdconditions,whereallfinalparticlesareinS-state.Note,inthisrespect,thattheDISTOcollaborationshowedthefeasibilityofthesecondmethod,bymeasuringtheDnncoefficientatprotonmomentumof3.67GeV/c,whichshowedthat”Dnnislargeandnegative(≃−0.4)overmostofthekinematicregion”[2].Itwasmentionedin[1]thatanonzerovalueofDnn,inthethresholdregion,canbeconsideredastheexperimentalconfirmationofthepseudoscalarnatureoftheK+meson.

ItisstraightforwardtoadaptthesemethodstothedeterminationoftheP-parityoftheΘ+-hyperon,whichispresentlyobjectofanintensivetheoreticaldiscussion.

ThesimplestreactionsofNN-collisions,whichcanbeconsideredforthisaimarethefollowing:

p+n→Λ0+Θ+,p+p→Σ++Θ+,

p+p→π++Λ0+Θ+,

(1)(2)(3)

inthresholdconditions(S-waveproduction).ItisimportanttonotethattheΛ0ortheΣ+hyperons,producedinthesereactions,areselfanalyzingparticles,thereforethepolarizationtransfermethod-withmeasurementofthepolarizationtransfercoefficientDnnseemsmorepreferable,fromtheexperimentalpointofview.

Thefollowinganalysisofreactions(1-3)isbasedongeneralsymmetrypropertiesofthestronginteraction,suchastheP-invariance,theconservationofthetotalangularmomentum,thePauliprinciple,forthepp-system,andthegeneralizedPauliprincipleforthenpsystem,whichholdsattheleveloftheisotopicinvarianceofthestronginteraction.

Thesesymmetryproperties,beingappliedtoS-waveproductionintheprocesses(1-3)allowtoestablishthespinstructureofthecorrespondingmatrixelements,forbothcasesoftheconsideredP-parity.

Weconsiderhere,forsimplicity,thecaseofspin1/2Θ+hyperon,butthisformalismcanbeextendedtoanyΘ+spin.

Firstly,letusestablishthegeneralspinstructureofdoublepolarizationobservablesfortheprocesses(1-3)atthreshold.

󰀑1andP󰀑2ofthecollidingnucleonsThedependenceofthecrosssection,totalordifferential,onthepolarizationsP

canbewrittenas:

ˆ󰀑󰀑ˆ󰀑󰀑1·P󰀑2+A2󰀑󰀑1,P󰀑2)=σ0(1+A1PkPσ(P1·kP2),

(4)

ˆ

whereσ0isthecrosssectionforthecollisionofunpolarizednucleons,󰀑kistheunitvectoralongthethreemomentumofthecollidingnucleons,inthereactionCMsystem.TherealcoefficientsA1andA2,whicharedifferentfordifferent

ˆ

reactions,dependontheparityoftheΘ+hyperon.Takingthez-axisalong󰀑k,onecanfindthefollowingexpression

2

forthespincorrelationcoefficientsAab,intermsofA1andA2:

Axx=Ayy=A1,Azz=A1+A2.

(5)

󰀑Yoftheproducedhyperon,ΛorΣ+,onthepolarizationP󰀑oftheinitialnucleonThedependenceofthepolarizationP

(beamortarget)canbewrittenas:

ˆˆ󰀑󰀑+p2󰀑󰀑Y=p1Pk󰀑k·P,P

(6)

wherep1,2arerealcoefficients,whichdependontheΘ+parity,sothatforthenon-zeropolarizationtransfercoeffi-cients,Dabonecanwrite:

Dxx=Dyy=p1,Dzz=p1+p2.

(7)

Letuscalculatethesecoefficientsforthereactions(1-3),ittermsofS-wavepartialamplitudes,consideringboth

valuesoftheΘ+parity.n+p→Λ+Θ+

3

p+p→Σ++Θ+

Thespinstructureofthecorrespondingmatrixelementscanbewrittenasfollows:

M(−)=f(−)I

󰀁

I,ifP(Θ+)=−1

M

(+)

=

(+)f1󰀑σ

ˆ󰀁ˆ󰀁(+)󰀑󰀑·kI+if2(󰀑σ×k)mσm,ifP(Θ+)=+1,

(20)

i.e.similartothereactionp+p→Σ++Θ+,butwithoppositeparity.So,forp+p→π++Λ+Θ+,thenecessary

polarizationobservablescanbedescribedbyEqs.(17)and(18),takingcaretointerchangetheP-parities.

Thisanalysisshowsthatallthereactions(1-3)arewelladaptedtothedeterminationofP(Θ+).Inallthesereactionstwopolarizationobservables,namelyAyyandDyyaresensitivetothisparity.Thesignatureoftheparityof(Θ+)isthesignoftheasymmetryAyy,oraavalueofthetransferpolarizationtensor,differentfromzero.Thesestatementsaremodelindependent.

Letusbrieflydiscusstheexpectedvaluesofthepolarizationobservables,inthereactions(1-3),whichdependontwoamplitudes,f1andf2.Forthisaim,wewilltake,asanexample,amodelofK-mesonexchange,whichhasbeen

4

FIG.1:K-exchangeforthereactionp+p→Σ++Θ+.

appliedtotheΛandΣ0productioninpp-collisions[5],andreproducesquitewellthesignandtheabsolutevalueof

󰀑+K++p.ConsideringthecontributionofbothdiagramsinFig.1,onecanfind:Dnnin󰀑p+p→Λ

f1

(−)

(Σ)=−f2(Σ)

(−)

(21)

Thisrelationdoesnotdependonmanydetailsofthereactionmechanism,suchasthevaluesofthetwocoupling

constants,gpΘKandgpΣK,onthewidthofΘ+andontheformofthephenomenologicalformfactors,whichhastobetakenintoaccountinsuchconsiderations.Therelation(21)allowstopredict:

(−)(−)Ayy(Σ)=+1/3,Dyy(Σ)=−2/3.

(−)

(22)

Letusnotethat|Dyy(Σ)|isdifferentfromzeroandlarge.Thiswillmakeeasiertodiscriminatethevalueofthe

P-parity.

(+)

Thesamerelationholdsalsofortheamplitudesf1,2(Λ)fortheprocessn+p→Λ+Θ+(forK-exchange),withcorrespondingpredictionsforpolarizationeffects.

Note,however,thatfinalstateΣΘ-interaction,whichisdifferent,generally,insingletandtripletstates,canaffecttherelation(21).Thereareargumentswhichshowthattheseeffectscannotbelarge[4].InanycaseweconsideredhereK-exchangeonlyforillustrativepurposes,foraquickestimationofpolarizationphenomena,withoutanyclaimthatthisisarealisticmodelforthesereactions[6]:themainresultofthispaperdoesnotdependonmodelconsiderations.

Theexperimentalstudyofallthreereactions(1-3)willgiveanonambiguoussignatureoftheΘ+parity.

[1][2][3][4][5][6]N.K.PakandM.P.Rekalo,Phys.Rev.D59,077501(1999).F.Balestraetal.,Phys.Rev.Lett.83,1534(1999).K.Hagiwaraetal.,Phys.Rev.D66,010001(2002).

A.W.Thomas,K.HicksandA.Hosaka,arXiv:hep-ph/0312083;C.Hanhartetal.,arXiv:hep-ph/0312236.P.Moskal,M.Wolke,A.KhoukazandW.Oelert,Prog.Part.Nucl.Phys.49,1(2002).S.I.Nam,A.HosakaandH.C.Kim,arXiv:hep-ph/0401074.