Covariant Helicity Amplitude Analysis for J/ψ→γPP

Commun. Theor. Pbys. (Beijing, China) 35 (2001) pp 547 552International Academic PublishersVol. 35. No.乐May 15, 2001Covariant Helicity Amplitude Analysis for J/ψ- +→γPP *WU Ning' and RUAN Tu-Nan12.3'Division 1, Institutc of High Energy Physics, P.O. Box 918- 1, Beiing 100039, China2Department of Modern Physics, University of Science and Technology of China, Hefei 230026. China3CCAST (World Laborutory), B.O. Box 8730, Beijing 10080 China(Received April 26, 2000; Revised July 3, 2000)Abstract Covariant helicity amplitude analysis for the process of J/ψ→γPP is discussed. Starting froun the S-matrix elemnents of decay process, we deduce the formulae of helicity coupling amoplitudes for lwo-body dexay procss.These formulae are used to analyze intermediate resonance states in tbe proccss ofJ/u decay to γππ, γKK. rm' ctc.PACS numbers: 11.80.Et, 11.80.CrKey words: helicity analysis, partial wave analysis1 Introductionthe particle a.入and入c are helicities of two daughter par-It is generally believed that quantum chromodynamics ticles band C, J and M are spin and magnetic quantum(QCD) which has SU(3)。symmetry is the most prospec-numbers along Z-axis of mother particle u. The normnal-tive theory for strong interactions. QCD predicts the exis- ization for two-particle diret product state istence of gluon ald bound states of glon, such as glueballand hybrid states. The most direct method to prove thevalidity of QCD is to search for the evidence of the ex-= (2r)°2Ep2Fg8(% - D1)(p - R:8zxx, (3a)istence of glueball and hybrid states in experiments. Ac-and for the sake of simplicity, the noralization for initialcording to theoretical calculation, glueballs will be abun)-state wavefunction is selected asdantly produced in J/ψ radiative decays. So, it is anideal place for us to search for the evidence of the exis-(P&J"' M'lpaIM) = 8*(p% - ywyjs.Aar. (3b)tence of glueball in J/4 radiative decays. Many J/ψ ra-Because S matrix is transation-invariant, we getdiative decay channels belong to the type of γPP, whereP stands for a pseudoscalar meson, such as J/ψ decays(popoxbxclSlpuJM)to γππ, yKK, rmm, etc. Sormne important glueball= (ET-8(p-Pp-p.)(mnhr!SIpaJM>， (4)candidates or states which contain large ghon content,2such as fo(1500), fJ(1710) and ξ(2230), appear in thesc :where Q is the volume of infinite four dimensionalchannels. In this paper, we give out the covariant helic- Minkowski space. For the final state two pariclc system,ity amplitude analysis for J/ψ→γPP which is provedwe can separate the center of mass motion from the rela-to be a powerful tool in experimental data analysis. We tive motion17,8!have used this method in BES physics analyses and have4VS,already obtained some meaningful results.|瓯苑AX〉= (2π)0的AtX)P),(52 S-Matrix for Two-Body Decay Processwhere s is the invariant Imass of the final state two-particleFirst, let us dscuss the fllowing two-body decay pro- system, |P) stands for the planc wave of center of mass mo-cesstion. The renormalizations of the states !P) and |0λc) .a→b+c.(1respectively areThe S-matrix element for this process isl1- 6](P'|P)=δ(P'-P),(6)(BcλbXdSlpaJ M)，(2(bφ'的出的φ的》= 8(cos0' - cos0)where所:司are space momenta of two final state parti-中国煤化工6u20 ()cles, Pa is the four-momentum of the initial state particle.P ist!; systemThe state |paJM) is the direct product state of the planeMYHCNMH Gwarvefunction and the angular mornentun wavefunction ofPμ= (P6 + Pc)n.(8)"The project supported by Natonal Natural Scicncc Foundation of China and Postdoctoral Science Foundation of China48WU Ning and RUAN Th-NanVol. 35Iu the center of mass system, the space momentum of par- spin-0 particle is a scalar, the spin wavefunction of a spin-ticleb is p and the space momentum of particlecis -p 1 particle is a four-vector, and the spin wavefunction of aThe magnitude of momentum p is denoted as卧Ac spin-n particle is an nth-order tensor. So, we always setcording to Eq. (5), the S matrix element will be changed the spin wavefunction of a spin-0 particle to 1. In canoni-intocal rest frame, the spin-1 wavefunction is 1siually selectedas:pcAbλelSpaJM)(2π)7, A4√58*(P - pa)(P0OM\bx!SIpJM). (9)0°(土)=干方(0,1,+i,0).(16)nV同φ°(0)= (0.0,0,1).(17)The staste |POPA) is a reducible state. We can writeThe spin wavefunction for higher spin can be constructedit into another formfrom spin-1 wavefunction by using Clebsch-Gordan coef-Peasd)= E/2i+1Dma(φ0)Pjmλt)， (10)ficient. Such as, the spin-2 wavefunction is4nm'φ=0(m)= E (1m1 Im2m)*(m)p°(m2)，(18) .!Pjm)\)= 12i+1m1m24πwhere (1m1 1m2|2m) is the Clebsch-Gordon cofficient.And the spin-(n + 1) wavefunction issinθdθdφDm\(φ0)IPθpxbX)，(11)(1a2:-an+(m)= S ; (nm lm2(n+ 1)m)where λ = λ- λg.“ The renormalization of the statePjmλX) isx φ1a2--an (m1)$*n+(m2).(19)(P'j'm' A%X[PjmXa)It can be strictly proved that the spin-n wavefunction is= 8'(P' - P);yom6x28xx. (12) a symmetrie and trceles lenso, and it is orthogonal toThe set of all states |PjmAh) is a complete set, theythe mormentum of the particlesatisfy the following completeness relation∞9Q_~an(m)=φ2a1:an(m) =..=∞Qn42" (m), (20)(21)| dPE 2 |Pjmxoa)(PjmXsx.|=1. (13)Pa; φ1ax_an(m) =0.(22)jm入。Using all these relations, we can rewtite the S-matrix as These states are orthonormal and conplete states(Bpe\b\[|SlpaJM),ia.an (m)°ng_an(m')= (-1)"dmm'，(23)=(2m1PV4√5I8(p D1)10()似(01) Soo o(m)hx .p.(m) = P!9.，.. (24)wherewhere..a0...... is a projetion operator for nth-norder symmetric and traceless operator. The above spin成= (PJMNsS'(P)lpoJM). (15)wavefunctions are only for rest particles. After making aRelation (14) is the S-matrix element for two-body deLorentz transformation, we can obtain the spin wavefunc-cay process. In the above relation, Fx。is called thetions for moving particles.helicity coupling amplitude. The part of S- matrix whichThe total spin angular mormentum is constructed fromdepends on angular argument is completely contained in two spin angular momenta of final state particle system,:he D-function. The belicity coupling arnplitude Fxe isit isindependent of angular argument, it only depends on the φlmaqagmnmn+*(P,m)= 2 (numrnem2/nm)mass of center of mass system, the relative momentum ofmym2two final state particles and their helicities.x w12."an (Pb, m)ent+1an+e-"no+nc (pe, m2), (25)3 Helicity Coupling A mplitude中国煤化ithe lotal spin quanHelicity coupling amplitude Fx。can be constructed tumand c have diferentfrom spin wavefunction of all three particles and relative momeTHCNMHGisnotacompletely>rbital angular moment lm wavfunction of two final state symmetric and traceless tensor. Therefore, this total spinparticles.5- -8. It is known that the spin wavefunction of a angular momentum wavefunction cannot be obtained fromNo 5Covariant Helicity Amplitude Analysis for J/v→γPP549projection operator for completely symmetric and trace lar momentum wavefunction, .less tensor. According to quantum mechanics, the projec-(n+1)ana. an+(m)= 2 (nm1 Im2(n+ 1)rn)tion operator istnjm2p(n)(pbpe)x tna)aam(m(m)l)ao+(m2).(34)F a1a2.--ang-ne Bi 2--ngtnc=2 (u-ang+n(P,m),) .n. . (P,m). (26)In the canonical rest frame, the relative momentum ismalong the Z-axis, s0 the orbital angular monentun isIfn=nb+neandp=p= 0,this projectionorthogonal to the Z-axis. It means that the orbitaloperator will become the projection operator for (n +angular mormenturm in canonical rest frarne should bene )th-order symmetric and traceless tensor. The nor~t°n2”an(m = 0). The orbital angular mornentum wave-malization of the spin angular momentum wavefunctionfunction can also be obtained by using the projection op-erator for completely symmetric and traceless tensor. In(nB-n_a-*n+ne(P:m) isthis way, it is defined by(pn)a1a-.n = na)-mn-n-n-5-.3.3.-.r. (35)=(-1)"btneδmm' .(27)where ra is the relative Inomnentum four-vector andGenerally speaking, this spin wavefunction is not orthog-pl(n)n-0.uhB. is the projection operator for sym-onal to the four-momentum P = Pbu +Peu, but its first metric traceles tensor. Two methods give the sane ro-R indices are orthogonal to four momentum Pb, and the sults, becauselast ne indices are orthogonal to four. -momentun Pe,r1(n)ana2-*n(m= 0) = (e)a2an(36)p"lax-an+n.(P,m) =0,(28) where r is the magnitude of relative momentum.The helicity coupling anplitude is a scalar which canpζp+1中r--ano+on (P,m)= 0.be constructed from spin wavefunction. orbital angularBesides, the first nb indices are traceless and the last ne momentum wavefunction, energy- momnent urn four-vectorindices are tracelessof initial and final state particles, netric tensor and Levi-Civita tensor. Now, we will use the technique of covariancega(q)a.amp+m.(P,m)=0,(30)analysis to construct the helicity coupling amplitude forgan+tians+2QU)(P,m)=0. (31) the two body decays which will be used in J/4→yPP.We will use the fllwing otations to represent a definiteThe orbital angular mormentum wavefunction in mO-decay processmentum space is also a tensor. Generally speaking, if theJ→s+σ;1.1η7s70 .(37)orbital angular momentum quantum number is l, the cor-responding wavefunction is represented by an Ith-order where J,s and σ represent the spins of initial and finalteasor. For S-wave, its l equals zero. Its orbital angu- state particles, nηs, 7。o and no reprsent parity of the cor-lar momentum wavefunction in cordinale space is homa reponding paricles. Suppse that party is consered ingeneous. The orbital angular momentum wavefunctionthe decay process, then the helicity coupling amplitudein mormentum space is obtained after a Fourier transfor-will have the following symmetrymation, it is independent of relative mornentum F ForF呢= n7.no(-1)-5-0F2-.(38)P-wave,l=1. Its wavefunction isa four-vector. In mo and orbital angular momentum quantum nunber l mustmentum space, its wavefunction is similar to spin-1 wave satisfy the following relationfunction. They are(-1'nn.n。= +1.(39)()0(+)=干-一(0,1, ti;,),(32)The helicity coupling anplitudes which will be used in()0(0) = (0,0,0,1).(33)the analyses of the process J/v→γPP are respectivelydiscuse中国煤化工The orbital angular momentum wavefunction with higher0)jTYHCNMH Gquantum noumbers can be constructed from l = 1 wave-Theause of parity confunction, such as the l = n+ 1 orbital angular momentun servation, the orbital angular momentum quanturn num.wavefunction can be constructed from l = n orbital angu- ber l must be 0 or 2; and F& = F、Becase the550WU Ning and RUAN Th-NanVoL. 35helicities of photon can only be土1, there is only onc in-three independent heticit coupling aruplitudes, they aredependent helicit coupling amplitude, itis所Denote FA. FH and贴Denote the tolal spin wavefunction asthe polarization four-vectors for J/ψ and photonasφand ()(J = 3,4,5), which is constructed from spir-l and。rspectivly. The helicity coupling amplitudes forl=0 spim-4 wavefunctions. There are five covariant amplitudesaudl=2are Ao = and A2 = (wi(2)φ*) respectively.The total helicity coupling amplitude is the liner combi-Al= φ°∪s0 t(2)p0，A2 = φ°*ul3)，{(413-P0nation of themA3= φ°*y68) (4)3rpo， A4 = φ"°*y(5) [(4)87p0,FR = goAo() + 92A2().As=∞i6)(9)100In canouical rest framne, it isThe total helicity coupling amplitude is the linear combi-F()=go - 92r2/3,where 9o is a scalar.F% = grAl(2v)+ 92A2(2) + 93A3(2w)(i)1→2+1, nJn.no =+1+ g4A(Ax) + 9gsAs(2).The second particle is photon. Because of parity con-In canonical rest frame, they areservation, the orbital angular momentum quantum num-berlmustbe0,2or4,andFl=FIx-r.Thereare陆= 690万0:(- 13475g1r2 - 26950g17%2three independent helicity coupling amplitudes, they areFh, F and B. Denote the total spin wavefunction as+ 3465g2r4 + 13860g273+4 + 8919g3r4u)(J = 1.2 3), which is constructed from spin-1 and+ 3564g372-4 + 1584gqr4 + 6336g4)3+4spin-2 wavefunctions. ψ的) is a third-order tensor. There- 400g5+58 - 2880g9593-8 - 640gs72r),are five covariant amplitudesF-0S;7.(- 1925gr2 - 38509gn2+2A1 = φ°°ψga PA2= φ(y82.i(2)所,10395√7As=2(2)为，As= (*(:(21(3)gr,+ 495g2r* + 198092r-4 + 396g4r*As=φ°{出,(3)00 .+ 1580g43r4 + 480gr4 + 640gsγ2+°),The total belicity coupling aumplitude is the linear combi-.=.y。(-2695g.r2 - 5390gr3+26930V70 .natiou of them+ 693g2r4 + 2772g272+4 - 297g3r4F = glAl(\2) + g2A2(A山) + g3A3(A山)- 188937ζr4 + 792g4r4 + 3168g473r4+ gaA4(\v) + gsAs(Xw).. 200gsr5 - 1490236 - 3209gs25+%),where 91, 92, g3, g4 and 95 are scalars.Fh=--t 3(8g93(iv)0→0+0,刀J7。7o =+1The orbital angular momentum quantum number is 0.gs(3 + 4及)r*，525There is only one indepeudent helicity coupling amplitnde,h=;25V2二后2。(-315g1 + 105g2r2itisF'= FoO =g, where g is a scalar.(v)2→0+ 0, ηJηono = +1.+ 70gsr2 + 48gsr*),The orbital aungular munuentun quantum number is 2.F品=o(- 73591 + 245g2r2 - 175g>r2There is only one independent helicity coupling amplitude.1050√6The only one covariant amplitnde is A = φt2.+ 280gsr2 - 72gsr4 - 96gs3),The helicity coupling ampltude is proportional tcwbere 91, 92: 93;: 94 and 95 are scalars.the above covariant armplitnde. In the rest frame, it is中国煤化工(ii)1→4+1. ηJηn7o = +1vilMHCNMHGThe orbital angular momentum quantum runber is 4.ber l must be2, 4or 6, and FE = FLx- v . There are There is ouly one independent helicity coupling amplitude.No.5Covariant Helicity Amplitnde Analysis for J/ψ， γPP551The only one covariant amplitude is A = φ376°i38 isThe helicity coupling amplitude is proportional to thef(x.a)=do /o(46)dφ,above covariant amplitude. In the rest frame, it is FJ =Foo= gr4, where g is a scalar.where x represents a set of quantities W hich are me:asuredby experiments, a represents somne unknown parameters4 Sequential Decaywhich are needed to be determined. σ is the total crossThere are three fual state particles in the process ofsectionJjv→?PP. Suppose that this radiative decay is realizedthrough two body sequential decaysσ=W(面)g d师，(47)dφJ/中→V+Xwhere W(刺) is the integration weight. The totil crossψP+ P.(40) section can be determined by Monte Carlo integrationThe helicity coupling amplitudes which describe the above12 (IAo+ A2+ A4+- + BG). (48)process is generally written asNme =A, = cMSux BWx(s,m,I)M8X，(41) where Nmc is the total number of Monte Carlo events. Itwhere c is a constant, Sx is the spin of the resonance,is required that these Monte Carlo events are obtainedM2Ix is the decay amplitude for the processJ/ψ→VX,through real detector simulation and they have passed allMδ5 is the decay amplitude for the process X→PP,cut conditions which are used to obtain the data sarnpleBWx(s, m,I) is the Breit-Wigner propagator for the res-of this process.onance X,Maximum likeliood function is given by the adjointprobability density for all data. It isBWx(s,m,I)=--mTe!(42)δ-m2+imTNwnC= I f(x.a).(49)In the above relation, m and T are the mass and widthinIof the resonance X, Tel is the partial width of the reso-Then we defiuenance, s is the iuvariant mass of two final state particles.S=-InC.(50)The MSs。is used to represent the decay amplitude of thefollowing processIn our data analysis, the goal is to find the set of values;a→b+c,a, which minimize S.Sa°→S[+S?e(43) 5 Conclusion .It isIn the global helicity analysis, not ornly the informa-Mis。=FSx. Diea-小,(44) tion about the mass spectrun, but also the infornationwhere入,入and入c are helicities of the correspoudingabout the angular distribution are used. Because moreinformation is used, the results obtained by this methodparticles.In this process, there usually are several resonanccsare more reliable. This method is used in BES physicswhich bave different spins and parities. Besides, there areanalysis. It has the following three advantages:background events. The total differential cross section(i) Supposed that there are sorne peaks in the masswhich describes the whole process isspectrum which are not real resonances. If we used和= E |A0+A2+A++BG， (4)the conventional Breil- Wigner function t fit them,we cannot diferentiate them frorn the real reso-where A, represents the total amplitude of spin 8,重rep-nances. But if we use the global helicity analysisresents all angular arguments and BG represents the con-中国煤化Inclnding mass spec-tribution from background.CHCNMHG: we can diferentiateMaximum likelihood method is used to ft the wholeuem uymuass anu wiuwl Mmaul. Those peaks causedmass spectrum. The normalized probability density func-by statistical fuctuations have 1n0 structure in thetion which is used to describe the whole decay processmass and width scan.55WU Ning and RUAN TU-NanVol. 35(i) In the traditional Breit Wigner ftting, we usuallyAuctuations.need to divide the whole mass spectrum into sev-(ili) In the global helicity analysis, the interference be-eral bins. And the results sometimes depend on thetween varions resonances is cousidered in the fit-manner how we separate the mass spectrum. Butting. And because of destructive interference, a1n the global helicity analysis, we do global fttingresonancc sonnetimes shows a dip, rather than ato the whole spectrum. So, our results are independent of the manner how we separate the masspeak, in the nass spectrum. This phenomenon isspectrum and are not afected by small statisticalfound in our ftting [9.10|1References[7] WU Ning, "Helicity Analysis of Relativistic Particles"Ph. D. Thesis, University of Science and Technology of[1] M. Jacob and G.C. Wick, Ann. Phys. (NY) 7 (1959) 404.China (1997).[8] WU Ning, "Covariant Helicity Coupling Amplitude Anal[2] A. Mcerrell, Nuovo Cimento 34 (1964) 1298.ysis and Its Application to BES Plhysics Analysis' , Post-[3] AJ. Macfarlane, J. Math. Phys. 4 (1963) 490.Doctoral Research Rcport, Institute of High EnergyPhysics, The Chincse Academy of Sciences.[4] S.U. CHUNG, "Spin Formalism" , CERN Yellow Report,[9] BES Collaboration, "Experimental Study on J/ψ →No. CERN 71-8, Geneva, Switzerland (1971).rπ+ π一”, (in preparation).[5] S.U. CHUNG, Phys. Rev. D48 (1993) 1225.[10] BES Collaboration, "Experimental Study on J/ψ + .6] S.U. CHUNG, Phys. Rev. D57 (1998) 431.γK+ K~", (in preparation).中国煤化工MYHCNMHG