Partial Wave Analysis of J/ψ Decays into ρρπ
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1 Commun. Theor. Phys. (Beijing, China) 48 (2007) pp c International Academic Publishers Vol. 48, No. 2, August 5, 2007 Partial Wave Analysis of J/ψ Decays into ρρπ ZHANG Zhen-Xia, and WU Ning 2 Department of Physics, Peking University, Beijing 0087, China 2 Institute of High Energy Physics, the Chinese Academy of Sciences, Beijing 00049, China (Received August 28, 2005) Abstract The possible place to search for exotic states in J/ψ hadronic decays is in J/ψ ρρπ. Because of the symmetry of identical particle and the symmetry of isospin, the physical analysis on this channel is quite complicated. In this paper, the method to use the partial wave analysis based on covariant helicity amplitude analysis to study the invariant mass spectrum of ρπ and to find the evidence of exotic states in ρπ spectrum is discussed. The decay amplitude for the decay sequence J/ψ ρx, X ρπ is given first. Then we discuss how to realize the identical particle symmetry and the isospin symmetry in the decay amplitude, which is the key point in the analysis of this channel. Then the total decay amplitude of this channel including all decay components is given. After that, how to identify the exotic states in the ρπ spectrum is discussed. What is discussed in this paper is the theoretical basis on experimentally searching for exotic states at BEPC/BES. PACS numbers:.80.et, k, 3.25.Gv Key words: covariant helicity-coupling amplitudes, exotic state Introduction Quantum chromodynamics (QCD) predicts a spectrum of mesons beyond the q q bound states of the conventional quark model. The spin (J), parity (P), and charge conjugation (C) quantum numbers of a q q system are: J = L + S, L is the relative angular momentum between the quarks and S is the total quark spin, P = ( ) L+, and C = ( ) L+S. The J PC combinations: 0 +, +,2 +,... are not allowed and are called exotic states. Some phenomenological models predict the existence of hybrid mesons, glueballs and four-quark states, and this new states can have exotic J PC quantum numbers. The observation of them and measurement of their properties provide experimental input necessary for an understanding of interaction of quark and gluon. Searching for these new stats of matter is a hot topic in the study of hadron spectroscopy. QCD predicts the existence of non-q q meson statesgluon ball (gg, ggg) hybrid state (q qg), and four-quark state (qq q q), etc. Greater progresses will be achieved theoretically and experimentally for looking for and affirming these non-q q meson states. The E852 collaboration reported a signal for + state decaying into ρπ with a mass of around.6 GeV. [] A + state at.4 GeV with a decay into ηπ was also reported by E852, [2] Grystal barrel, [3] VES, and KEK. The VES experiment has studied the effective πη and πη scattering and found a rather small enhancement in the πη channel at.4 GeV, but a much more pronounced signal in the πη channel at.5.6 GeV. [4] A π (600) is also consistent with the analysis of the BNL-E852 results. In addition, evidence of existence of + exotic state is also shown in charge exchange ρ 0 π + photo-production experiment. [5] At the same time, theorists have also done much work to study the rationality of the existence of exotic states and their structures according to some model-dependent calculations, indicating the mass range of them. One interpretation thinks of exotic state J pc I G = + as hybrid meson ( q qg) of constituent quark model. They study the interpretation of exotic states π (400) and π (600), decaying respectively into ηπ, η π, f π, and ρπ. The lowest exotic state + may be built in two ways: l g = (gluon-exited) corresponding to the angular momentum between one gluon and the (q q) system, while l q q = (quarks exited) corresponds to the angular momentum between q and q. They indicate that π (400) is unlikely to be hybrid state, but the resonance state π (600) is considered as hybrid meson in quark-exited mode. [6] Others think it is possible for the states seen experimentally to be really four-quark QQ QQ states. [7] QCD sum rule method including zero mode contribution predicts the mass of exotic state, decaying into ηπ and ρπ, is around.4.5 GeV, which is compatible with experimental candidates π (370) and π (440). [8] From lattice gauge theory, the mass of the light + hybrid is around 2 GeV. [7] There are also calculations of hybrid meson at large N c that predict towers of hybrid meson which, within expected /N c corrections, naturally accommodate the + experimental hybrid candidates (π (400), The project supported in part by National Natural Science Foundation of China under Grant Nos , , , and zxzhang@ihep.ac.cn
2 34 ZHANG Zhen-Xia and WU Ning Vol. 48 π (600)). [9] According to an estimation of the production rate of the various particle species based simply on counting powers of the electromagnetic and strong coupling constants, one obtains the hierarchies: Γ(J/ψ MH) > Γ(J/ψ MM ) Γ(J/ψ MG). Here M stands for ordinary q q meson, H for hybrid and G for gluon. Note that hybrid states, if they exist, can be produced with a fairly large production rate in J/ψ decay. [0] BESII which has got 58 M J/ψ events provides a good opportunity to search for exotic states at BES. The calculation of perturbation QCD theory still predict, the hadron decay process of J/ψ is propitious to study the production of hybrid. Moreover, many model calculations indicate that the mass of exotic hybrid state is less than 2.2 GeV. [] Therefore, it is suitable to look for J pc = + exotic states at BEPC/BES, for BES has large enough J/ψ events data sample. [2] So our purpose is to investigate how to study J/ψ ρρπ decay process by covariant helicity coupling amplitude method of partial wave analysis and present the theoretical formulas used for doing with experiment data. Then we give the definition of likelihood and introduce the method of study. 2 Total Covariant Helicity Coupling Amplitude The exotic states in ρπ spectrum have spin-parity quantum number: J PC = + and the recoil particle has spin-parity quantum number, [3] so we choose J/ψ ρρπ decay process to research exotic states in spectrum of ρπ which recoils ρ. 2. Helicity Coupling Amplitude of Single Decay Vertex First of all, let us simply introduce the amplitude M sa λ b λ c of single decay vertex of decay process, a b + c. Details on the analysis method used in this work can be found in Refs. [4] [7]. The spin-parity J p of mother particle a is J ηj and those of daughter particles b and c are respectively S ηs and σ ησ, and respective helicities are M, λ, ν, respectively. The p, q, k, and φ(δ), ω(λ), ε( ν) are respectively the four-momentum and wave function of particles a, b, c. The decay amplitude Mλ J b λ c of decay vertex of a b + c is Mλν(θ,φ,M) J DMδ(φ,θ,0)F J λν J, F J λν = JMλν M JM. Here θ and φ are the polar angular and azimuthal angle of particle b in the center of mass frame of mother particle a and δ = λ ν. Parity conservation in the decay leads to the relationship F J λν = η J η s η σ ( ) J s σ F J λ ν. One may write an explicitly covariant expression (Lorentz scalar) for the helicity-coupling amplitudes F J λν = a g a A a (λν), A a (λν) = [p n,r l,ω(λ),ε( ν),φ (δ)]. The variables α stands for the set {l,s}. l is the quantum number of relative angular momentum of particles b and c and r = q k is the relative four-momentum between particles b and c, while g a is the coefficient determined. It is to be noted that n = for s + σ + l J odd and n = 0 otherwise. So we can find the covariant helicity coupling amplitude Fλν J which has no relation with the Euler angular but is included in D-function can be written out. Up to now, the vertex amplitude of single decay process can be obtained. 2.2 Amplitude of Sequential Decay Process Now we calculate the total covariant helicity coupling amplitude of this sequential decay process J/ψ ρ 0 ρ + π (the final states π + π π + π π 0 ): J/ψ ρ 0 + X, X ρ + + π. The decay mechanism is shown in Fig.. J/ψ ρ0 X ρ + π π + π π + π 0 Fig. The decay structure of J/ψ ρ 0 + X, X ρ + + π. Here the resonance X has definite J P. In this sequential decay process, there are four decay vertexes and three intermediate resonances, which are described by Breit Wigner function, m x Γ X BW X (s X,m X,Γ X ) = m 2, X + s X + im X Γ X s X is the invariant mass square of resonance X. m X and Γ X are the mass and width of resonance X respectively. One can obtain the total amplitude of this
3 No. 2 Partial Wave Analysis of J/ψ Decays into ρρπ 35 sequential decay process by multiplying the amplitude of decay vertexes and Breit Wigner function of intermediate resonance. For one definite resonance X i with definite J P, the total covariant helicity coupling amplitude is A(X i,ρ 0,ρ +,π +,π,π+ 2,π0,π 2 ) = MJ J/ψ λ ρ 0λ X (θ 2,φ 2,M) BW(s ρ 0,m ρ 0,Γ ρ 0) M Jρ 00 (θ,φ,λ ρ 0) BW(s X,m X,Γ X ) M JX λ ρ0 (θ 3,φ 3,λ X ) BW(s ρ +,m ρ +,Γ ρ +) M Jρ 00 (θ 4,φ 4,λ ρ +), θ and φ, θ 2 and φ 2, θ 3 and φ 3, θ 4 and φ 4 are respectively the polar angle and azimuthal angle of π + in the center mass system of ρ 0, X, the center mass system of J/ψ, ρ + in the center mass system of X, and π + in the center mass system of ρ +. In the calculation of total amplitude, we can add intermediate resonance decaying into ρπ, which exists in the J/ψ ρρπ decay process. For J pc = 0 + resonance, there are π(300), π(800), and a (260), a (640) are for ++. For J pc = + exotic states we include π (400), π (600) and π (2267). In addition, there are a 2 (320) and a 2 (700) for 2 ++, π 2 (670) and π 2 (200) for 2 + and a 4 (2040) for Now let us begin to calculate the helicity coupling amplitudes which will be used in the partial wave analysis of this decay channel. (i) +0, η J η s η σ =. Because of parity conservation, must be and Fλ0 = F λ0. So there is only one independent helicity coupling amplitude F0, F 0 = m r, m is the mass of mother particle, and other helicity coupling amplitudes which are not written here vanish. (ii) 0 + 0, η J η s η σ =. Because of parity conservation, must be. There is only one independent helicity coupling amplitude F00, 0 F 0 00 = γ s r, γ s is rapidity of the first daughter particle. (iii) +, η J η s η σ =. The possible spin angular momentum quantum numbers are 0,, and 2. Because of parity conservation, the orbital angular momentum quantum number must be 2 and Fλν = F λ ν. So there are three F 0 = 2 AW (3g (γ s + γ σ ) + g 2 ( 4γ s + 2γ σ )r 2 g 3 (4γ s + 2γ σ )r 2 ), F = γ s r, F 0 = γ s r. (iv) 2 + 0, η J η s η σ =. Because of parity conservation, must be and 3 and Fλ0 = F λ0. So there are two F 0 = γ s (g r 0.4g 2 r 3 ), F 00 = g r γ 2 s + 0.2g 2 r 3 ( + 2γ 2 s). (v) , η J η s η σ =. Because of parity conservation, can be 0, 2, and 4 and Fλ0 = F λ0. So there are three F 2 00 = 3 g (2γ 2 s +.0) + 9 g 2(4γ 2 s.0)r g 3(2γ 2 s +.0)r 4, F 2 0 = 20 γ s(20g + 35g 2 r 2 48g 3 r 4 ), F 2 20 = g 3 g 2 r g 3 r 4. (vi) , η J η s η σ =. Because of parity conservation, can be and 3, and Fλ0 = F λ0. So there are two F 2 0 = m γ s (0.5g r + 0.4g 2 r 3 ), F 2 20 = m(g r 0.2g 2 r 3 ). (vii) 4 2+0, η J η s η σ =. Because of parity conservation, can be 4 and 5, and Fλ0 4 = F λ0 4. So there are two F 4 20 = m( 3g + g 2 r 2 r 3 ), F 4 0 = m γ s (3g + 2g 2 r 2 r 3 ). (viii) 2 +0, η J η s η σ =. Because of parity conservation, can be 2 and Fλ0 2 = F λ0 2. So there is only one independent helicity coupling amplitude: F 2 0 = m r 2. (ix) 2+, η J η s η σ =. The possible spin angular momentum quantum numbers are, 2, and 3. Because of parity conservation, the orbital angular momentum quantum numbers can be and 3, and Fλν = F λ ν. So there are four F 2 = AC 2 g + AC 22 g 2 + AC 23 g 3 + AC 24 g 4, F = AC g + AC 2 g 2 + AC 3 g 3 + AC 4 g 4, F 0 = AC 0 g + AC 02 g 2 + AC 03 g 3 + AC 04 g 4, F 0 = AC 0 g + AC 02 g 2 + AC 03 g 3 + AC 04 g 4,
4 36 ZHANG Zhen-Xia and WU Ning Vol. 48 AC 2 = 0 r m( γ s γ σ + 3β 2 β 3 γ s γ σ ), AC 22 = 6 r m(.0 γ s γ σ + β 2 β 3 γ s γ σ ), AC 23 = 30 r3 m(.0 2γ s γ σ + 3γ s γ σ ), AC 24 = 37.5 r3 m(.0 + γ s γ σ + 3γ s γ σ ), and and AC = AC 2 = AC 4 = 0, AC 3 = r3 m γ s, and AC 0 = AB r m( γ s γ σ 45β 2 β 3 γ s γ σ ), AC 02 = AB r m( γ s γ σ 25β 2 β 3 γ s γ σ ), AC 03 = AB r 3 m(5 0γ s γ σ + 5γ s γ σ ), AC 04 = AB r 3 m(6 + 6γ s γ σ + 48γ s γ σ ), AB = , AC 0 = AB 2 r m( 35 5γ s γ σ + 5β 2 β 3 γ s γ σ ), AC 02 = AB 2 r m(25 25γ s γ σ + 25β 2 β 3 γ s γ σ ), AC 03 = AB 2 r 3 m( 5 0γ s γ σ 5β 2 β 3 γ s γ σ ), AC 04 = AB 2 r 3 m(8 + 8γ s γ σ + 24β 2 β 3 γ s γ σ ), For β 2 and β 3 we have β 2 = AB 2 = γ 2 s γ s, β 3 = γ 2 σ γ σ, γ s and γ σ are respectively the rapidity of the first and second daughter particles. The other helicity coupling amplitudes can be found in Ref. [3]. 2.3 Four Kinds of Decay Modes of J/ψ ρρπ It is known that ρ and X are all iso-vector particles. When the final states are π + π π + π π 0, there are four kinds of decay modes: J/ψ ρ 0 + X 0, X 0 ρ + + π, ρ 0 π + + π, ρ + π + + π 0 ; J/ψ ρ 0 + X 0, X 0 ρ + π +, ρ 0 π + + π, ρ π + π 0 ; J/ψ ρ + + X, X ρ 0 + π, ρ + π + + π 0, ρ 0 π + + π ; J/ψ ρ + X +, X + ρ 0 + π + ρ π + π 0, ρ 0 π + + π. Here we must pay attention to one thing that J/ψ ρ 0 ρ 0 π 0 decay mode is forbidden, for it violates charge conjugate symmetry, though its final state of decay is also π + π π + π π 0. All of the amplitudes of these four decay processes will be added together as the total amplitude of the whole decay process of J/ψ ρρπ. 2.4 Identical Particle Exchange Symmetry It is known that π + π + or π π are identical particles, and according to quantum mechanism, the decay amplitude should have particle exchange symmetry. This requires to consider the symmetry of identical particle exchange by adding all the amplitudes of every kinds of decay mode. So in each kind of decay process described above, the amplitude should include four amplitudes from identical particle exchange. In all, the total amplitude of the whole decay process of J/ψ ρρπ includes 6 sub-amplitudes, which come from four different decay processes and identical particle exchange symmetry. 2.5 Total Amplitude of Decay Channel J/ψ ρρπ The four kinds of decay modes of J/ψ ρρπ and identical particle exchange symmetry should be considered in the total decay amplitudes. For two J/ψ ρ 0 ρ + π decay
5 No. 2 Partial Wave Analysis of J/ψ Decays into ρρπ 37 channel, there are two amplitudes and each one is made from four partial amplitudes caused by identical particle exchange: A(X i,ρ 0,ρ + ) = 2 A(X i,ρ 0,ρ +,π +,π,π+ 2,π0,π 2 ) + 2 A(X i,ρ 0,ρ +,π + 2,π,π+,π0,π 2 ) + 2 A(X i,ρ 0,ρ +,π +,π 2,π+ 2,π0,π ) + 2 A(X i,ρ 0,ρ +,π + 2,π 2,π+,π0,π ), A(X i,ρ +,ρ 0 ) = 2 A(X i,ρ +,ρ 0,π +,π0,π + 2,π,π 2 ) + 2 A(X i,ρ +,ρ 0,π + 2,π0,π +,π,π 2 ) + 2 A(X i,ρ +,ρ 0,π +,π0,π + 2,π 2,π ) + 2 A(X i,ρ +,ρ 0,π + 2,π0,π +,π 2,π ). And for two J/ψ ρ 0 ρ π decay channel, there are two amplitudes and each one is made from four partial amplitudes: A(X i,ρ 0,ρ ) = 2 A(X i,ρ 0,ρ +,π +,π,π 2,π0,π + 2 ) + 2 A(X i,ρ 0,ρ,π + 2,π,π 2,π0,π + ) + 2 A(X i,ρ 0,ρ,π +,π 2,π,π0,π + 2 ) + 2 A(X i,ρ 0,ρ,π + 2,π 2,π,π0,π + ), A(X i,ρ,ρ 0 ) = 2 A(X i,ρ +,ρ 0,π,π0,π +,π 2,π+ 2 ) + 2 A(X i,ρ,ρ 0,π 2,π0,π +,π,π+ 2 ) + 2 A(X i,ρ,ρ 0,π,π0,π + 2,π 2,π+ ) + 2 A(X i,ρ,ρ 0,π 2,π0,π + 2,π,π+ ). Adding four kinds of decay channels, the total amplitude of resonance x i is represented as A(X i ) = 2 A(X i,ρ 0,ρ + ) + 2 A(X i,ρ +,ρ 0 ) + 2 A(X i,ρ 0,ρ ) + 2 A(X i,ρ,ρ 0 ). Then A(J P ) is the total amplitude of all resonance whose spin-parity is J P. If N resonances X,X 2,X 3,...,X N are added into the decay process, A(J P ) = A(X ) + A(X 2 ) + A(X 3 ) + + A(X N ). 3 Total Cross Section There are two ways to perform PWA analysis on this channel. One way is to treat ρ 0 as an intermediate resonance which is described by Breit Wigner function, the cross section which describes the whole decay process J/ψ ρρπ is dσ d φ = M J/ψ A 2 + BG, A = A(0 ) + A( + ) + A( ) + A(2 + ) + A(2 ) + A(4 + ), and BG is the background term, which has no interference with resonances. M J/ψ is the helicity value of J/ψ and equals only and under the circumstance of BES experiment. Another way is to treat ρ 0 as a stable particle. In this case, we set the Breit Wigner function of ρ 0 to in the formula of amplitude A(X i ) and the other part is the same. After ρ 0 is treated as a stable particle, the π + and π which are decayed from ρ 0 is definite. Then one does not have to consider the symmetry of identical particle exchange and there are only four amplitudes coming from four kinds of decay processes, dσ dφ = M J/ψ A 2 + BG = M J/ψ A(0 ) + A( + ) + A( ) + A(2 + ) BG = M J/ψ A(X ) + A(X 2 ) + A(X 3 ) + + A(X N ) 2 + BG, the item A (X i ) A(X i ) denotes the contribution of resonance X i, and the item A (X i ) A(X j ) (i j) denotes the contribution of interference between resonance X i and X j. 4 Fitting by Likelihood Function Method 4. Definition of Likelihood Function The probability density function f(x j,α) that describe the data sample is f(x j,α) = dσ dφ σ, x j are 4-momenta of final state particles that are measured by experiment, α is a set of unknown parameters which just need to be determined in fitting and σ is the total cross section of this process, dσ σ = dφ dφ.
6 38 ZHANG Zhen-Xia and WU Ning Vol. 48 It can be given by Monte Carlo integration in the fitting: σ = dσ dφ dφ = N mc N mc i=( A 2 + BG), M J/ψ A = A(0 ) + A( + ) + A( ) + A(2 + ) + A(2 ) + A(4 + ), N mc is the number of Monte Carlo events after all selected conditions, and N mc is generated events number. M J/ψ represents different helicity of J/ψ. For the two kinds of methods whether ρ 0 is considered as intermediate resonance, there are two kinds of Monte Carlo samples to calculate the total cross section σ. The Monte Carlo samples are respectively generated by the following process. (i) if ρ 0 is considered as a resonance J/ψ π + π π + π π 0. (ii) if ρ 0 is considered as a final state particle and fixed in event selection, J/ψ ρ 0 π + π π 0, ρ 0 π + π. Maximum likelihood method is utilized in the fitting, and the likelihood function L is given by the joint probability density of all experimental data: L = N event j= f(x j,α). Our purpose is to look for the maximum of likelihood function by maximum likelihood method. Otherwise, all of our calculation tools can only get the minimum of one function, so it needs to transform finding the maximum of likelihood function to finding the minimum of another function. One can define N event S = ln L = f(x j,α). j= So in the data fitting, we can obtain the minimum of S function by adjusting the set of parameter α. 4.2 How to Fit S Function There are two groups of input parameters : the masses and widths of all resonances and the coefficients from covariant helicity coupling amplitude, of which the masses and widths of resonances are set to the PDG values at the beginning of fitting, the later coefficients just wait to be fitted by finding the minimum of S function. All of those coefficients waiting to be fitted are complex numbers except background coefficient. Having no interference with the resonances, background parameter waiting to be fitted is just a real number. 5 Summary In this paper we have given the formula of covariant helicity coupling amplitude of the hadronic decays J/ψ ρρπ. The identical particle symmetry and the isospin symmetry in the decay amplitude are very complicated but are the key points in the analysis of the channel. The paper provides the theoretical basis on experimentally searching for exotic states in this channel. Acknowledgments The authors acknowledge helpful discussions with H.Q. Zheng and X.Y. Shen. References [] G.S. Adams, et al., Phys. Rev. Lett. 8 (999) 576. [2] S.U. Chung, et al., hep-ex/ v2; Phys. Rev. D 60 (999) 9200; D.R. Thompson, et al., (E852 Collaboration), Phys. Rev. Lett. 79 (997) 630 and BNL Press Release [3] A. Abele, et al. (Crystal Barrel), Phys. Lett. B 423 (998) 75. [4] D.V. Amelin, et al. (VES), Proceedings of Hadron 200 A.I.P. Conf. Proc. 69, (2002) 43. [5] Andrei V. Afanasev and Adam P. Szczepaniak, hepph/ v (999). [6] F. Iddir and A.S. Safir, hep-ph/0002 v2 (200). [7] Craig McNeile, hep-lat/ v (999). [8] Z.F. Zhang and H.Y. Jin, hep-ph/ v2 (2005). [9] Silas R. Beane, hep-ph/ (200). [0] L. Köpke and N. Wermes, CERN-EP/88-93, Phys. Rep. 74 (989) 67. [] R.L. Jaffe and K. Johnson, Phys. Lett. B 60 (976) 20; T. Barnes, et al., Nucl. Phys. B 224 (983) 24; N. Isgur, et al., Phys. Rev. Lett. 54 (985) 869; N. Isgur and J. Paton, J. Phys. Rev. D 3 (985) 290; I.I. Balitsky, et al., Z. Phys. C 33 (986) 265; J.I. Latorre, et al., Z. Phys. C 34 (987) 347; N.A. Campbell, et al., Nucl. Phys. B 306 (988) 5; P. Lacock, et al., Phys. Lett. B 40 (997) 308. [2] Shen Qi-Xing and Yu Hong, High Energy Physics And Nuclear Physics 23 (999) 54. [3] Ning Wu, et al., Commun. Theor. Phys. (Beijing China) 35 (200) 547. [4] M. Jacob and G.C. Wick, Ann. Phys. (NY) 7 (959) 404; A. Mckerell, Nuovo Cimento 34 (964) 298; A.J. Macfarlane, J. Math. Phys. 4 (963) 490. [5] S.U. Chung, Phys. Rev. D 57 (998) 43. [6] Ning Wu, et al., Commun. Theor. Phys. (Beijing China) 37 (2002) 309. [7] Ning Wu, Helicity Analysis of Relativestic Particles, Ph.D. Thesis, University of Science and Technology of China (997).
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