χ cj Decays into B B in Quark-Pair Creation Model

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1 Commun. Theor. Phys. (Beijing, China) 41 (4) pp c International Academic Publishers Vol. 41, No., February 15, 4 χ cj Decays into B B in Quark-Pair Creation Model PING Rong-Gang 1, and JIANG Huan-Qing 1,,,4 1 CCAST (World Laboratory), P.O. Box 87, Beijing 18, China Institute of High Energy Physics, the Chinese Academy of Sciences, P.O. Box 918(4), Beijing 19, China Institute of Theoretical Physics, the Chinese Academy of Sciences, P.O. Box 75, Beijing 18, China 4 Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou 7, China (Received June 1, ; Revised July, ) Abstract A quark pair creation model is introduced to study the χ cj exclusive decays into baryon-antibaryon pairs. The decay widths for processes χ cj B B (J =, ; B = Λ, Σ, Ξ ) are evaluated phenomenologically with an explicit inclusion of the properties for outgoing baryons described by wave functions in the naive quark model. The results show that states χ cj (J =, ) decay into Λ Λ pair with a larger branching ratio than into p p pair. PACS numbers: 1.5.Gv, 1.9.Jh, 14..Jn Key words: χ cj, baryon, quark-pair creation model 1 Introduction Exclusive decays of charmonium into hadrons have attracted continuous interests of both theoretical and experimental experts. 1,] The earliest perturbative QCD treatment on exclusive decays of charmonium with S-wave into the light hadrons was carried out by Lepage and Brodsky. ] Their calculation was based on the assumption that the annihilation of the heavy quark and antiquark is a short distance process, which can be calculated in perturbative theory due to the asymptotic freedom of QCD. Due to the large Q value involved in these processes, the mass of the quarks emitted from a gluon can be ignored. Thus it leads to the famous QCD helicity selection rule, 4] which states that a spin- particle cannot decay into two fermions with opposite helicities. Then it immediately forbids the decay χ c p p. However, the experimental value for this channel does not vanish. The massless quarks approximation seems to fail for these exclusive processes. The mass correction to this forbidden charmonium decay has been considered by Anselmino et al. 5] They found that, assigning to the quarks a constituent rather than a current mass, one obtains nonzero value for this process. Other corrections from the nonrelativistic approximation are also supposed to be essential for explaining this forbidden decay. Recently, the contributions from higher order Fock states have also been considered as a most important scheme to explain these exclusive decays by Wong, 6] however, his numerical results showed that the decay widths for χ c1, χ c states to p p are all smaller than to other octet baryon-antibaryon pairs. Recently, a new measurement from BES collaboration shows that the branching ratios of χ cj Λ Λ (J =, 1, ) are almost over two times larger than χ cj p p (J =, 1, ). 7] In this paper, a quark pair creation model is presented for one to evaluate phenomenologically the branching ratios for decays χ cj (J =, ) B B (B = Λ, Σ, Ξ ) with an explicit inclusion of the baryonic properties in the simple quark model. We will show that, in this model, the forbidden decay χ c B B is well reproduced, but the decay χ c1 B B is forbidden. Model Description and Formalism In this model, the states χ cj (J =, 1, ) are simply assumed as bound states of c c quarks in p-wave states, namely, one only considers the lowest Fock order for χ cj states. The exclusive decay processes χ cj B B are assumed via two steps as illustrated in Fig. 1. At first step, the c c quarks annihilate into two gluons under the restriction of the conservation of J P C quantum number. For simplicity, the p-wave states are only described in the level of hadrons, and the contribution from the c c quark dynamics, as well as from the bound state, is all parameterized into an overall constant, which only associates with their quantum numbers. Phenomenologically, this constant can be determined by the measurement of a specific decay width, e.g. the processes χ cj p p. Since we will evaluate the relative decay widths for the processes χ cj B B (B = Λ, Σ, Ξ ) over the processes χ cj p p, this treatment would be reasonable at least in the limit of the nonrelativistic approximation, because it directly leads to that the contributions from the bound states to the transitional amplitudes for the processes χ cj gg The project supported in part by National Natural Science Foundation of China under Grant Nos. 1555, 155 and the CAS Knowledge Innovation Project No. KJCX-SW-NO Mailing address

2 No. χ cj Decays into B B in Quark-Pair Creation Model 5 are proportional to the derivatives of their radial wave functions at the origin. Thereafter, the two gluons are materialized into two quark-antiquark pairs. Due to the quark-gluon coupling, another quark pair is allowed by OZI rule to be created from QCD vacuum with the quantum number J P C = ++. Generally, the quark pairs could be allowed to generate anywhere from the QCD vacuum in momentum space with any color and any flavor, however, H I = i,j,α,β,s,s i,j,α,β,s,s d k k only those whose color-flavor wave function and spatial wave function overlap with those of outgoing baryons can make a contribution to the final decay width. Following the usual procedure, the Hamiltonian for the created quark pair can be defined in the modified P model 8] in terms of quark and anti-quark creation operators b and d, g Iū( k, s)v( k, s )]b α,i ( ks)d β,j ( k, s )δ αβ Ĉ I d k k ˆQ(k, s; k, s )b α,i ( ks)d β,j ( k, s )δ αβ Ĉ I, (1) where α(β) and i(j) are the flavor and color indexes of the created quarks (anti-quarks), and u(k, s) and v(k, s ) are free Dirac spinors for quarks and antiquarks, respectively. They are normalized as ū(k, s)u(k, s ) = v(k, s)v(k, s ) = mδ ss. Ĉ I = δ ij is the color operator for q q, and g I is the strength of the decay interaction. In the non-relativistic limit g I can be related to γ, the strength of the conventional P model, by g I = m q γ. 8] Similarly to a common assumption in the conventional quark-pair creation model, the probability of finding a strange quark-pair is assumed to be lower than the light quark-pairs. Hence, it is reasonable to assume that the phenomenological parameters g I, describing the strength of a quark pair created, are equal either for light or strange quark pairs in this model. Fig. 1 Feynman diagram to the lowest order α s for the decay χ cj B B. In the c.m. system of χ cj, the outgoing baryons move to opposite directions with a high momentum. So one would imagine that the constituent quarks move along the outgoing direction with a quite larger longitudinal momentum than in transverse directions. For simplicity, if one neglects the transverse momentum of the outgoing (anti-) quarks, and further assumes that the outgoing quarks and antiquarks have the same momentum but move oppositely, one has p i = x ip, p i = m i + p i, q i = x iq, q i = m i + q i, () where P and Q are the momentum vectors of the two outgoing baryons, and the quark helicity is defined as the projection of the quark s spin along the covariant fourvector, ɛ(p ) = m B P kp P m B where k = (1, ), and P = ] k, () M χ 4m B/, and we have p ɛ(p) =, (4) ɛ µ (p)ɛ µ (p) = 1. (5) Under those definitions, the matrix involved in those processes can be simplified by projecting the Dirac spinors onto the spin configurations, thus one gets ū (p i )γ u v (q i ) = ū (p i )γ u v (q i ) = m + p i m + p i q i x i P p i + m ɛ ()µ, (6) i ū (p i )γ u v (q i ) = p i + + p m(m i m + p i q i )ɛ (+) µ, (7) ū (p i )γ u v (q i ) = p i + + p m(m i m + p i q i )ɛ () µ, (8) ū (p i )v (q i ) = ū (x i P )v (x i Q) = P (m i + q i ) p i + m i, (9) ū (p i )v (q i ) = ū (p i )v (q i ) =, (1) where the vector ɛ (λi) is identical to the projection of the χ cj spin along the vector ɛ = ( P Q)/ P with a value of the helicity λ i. In the above equations, the normalized condition ūu = vv = m has been used. From these identities, it is worth while to note that only quark pairs

3 54 PING Rong-Gang and JIANG Huan-Qing Vol. 41 created with the opposite helicities can make contributions to the nonzero transitional amplitudes. The amplitudes for decays χ cj B B are expressed as M J = B B T P dx1 P J χ cj = dx dx ( ) p 1 p p δ x i 1 i=1 φ B (x 1, x, x )φ B(x 1, x, x )A J (x 1, x, x ), (11) where φ B or φ B is the radially baryonic wavefunction φ B (x 1, x ) = d p ρ d p λ φ B ( p ρ, p λ ), and A J is the transitional amplitudes for decays χ cj B B. For χ c B B process, we have A (x 1, x, x ) = c g µν Ô µ (x 1, λ 1 )Ôν (x, λ 1 ) ˆQ(x, λ ) (p 1 + q 1 ) (p + q ) + (1 ) + ( ), (1) where Ô(x i, λ i ) and ˆQ(x i, λ i ) are operators defined as Ô µ (x i, λ i ) = ū(p i, s i )γ µ v(q i, s i ), ˆQ(x i, λ i ) = g I ū(p i, s i )v(q i, s i ), (1) where the value λ i is the sum of the spin projections for the quark and antiquark along the vector ɛ = ( P Q )/ P. Since the spins of the created quarks are only limited to be oppositely parallel, there is only one configuration of the baryonic helicity, i.e. B B T χ c or B B T χ c to make contributions to the amplitudes A. In the context of the naive quark model, one attempts to take into account the properties of baryons by explicitly including an asymmetry wave function of the bound state. For example, the flavor-spin wave function of the proton may be explicitly constructed in the representation of the SU(6) group if one ignores the mass difference between the u, d light quarks. The spin and flavor wave functions of the proton and antiproton are takes as Ψ p SF = Ψ p SF = 1 (χ ρ φ ρ + χ λ φ λ ), (14) where χ ρ (φ ρ ) and χ λ (φ λ ) are the mixed-symmetry pair spin(isospin) wave functions. In the case of hyperons, one would like to use a basis that makes explicit SU() F symmetry breaking under an exchange of unequal mass quarks. The flavor wave functions for strange baryons are taken as φ Λ = 1 (ud du)s, φ Σ = 1 (ud + du)s, φ Ξ = ssd, (15) and the construction of the spin wave functions χ proceeds exactly with an analogous scheme to build the flavor wave functions. On projecting the transitional amplitudes onto those flavor-spin wavefunctions, one gets A (χ c p p) = A (χ c Λ Λ) = C (4p 1 p ) i=1, C (4p 1 p ) i=1, C (4p 1 p ) A (χ c Σ Σ, Ξ + Ξ ) = i=1, C (4p 1 p ) i=1, (m i + p i m + p i q i x i P ) P (m + q) p + m (m i + p i m + p i q i ) (m i + p i m + p i q i x i P ) P (m + q) ] p + m + (1 )] + ( )], (16) P (m + q) ] p + m + (1 )] + ( )] (m i + p i m + p i q i ) P (m + q) p + m, (17) C (4p 1 p ) + i=1, C (4p 1 p ) i=1, { C + (4p 1 p ) (m i + p i m + p i q i x i P ) (m i + p i m + p i q i ) P (m + q) p + m i=1, (m i + p i m + p i q i ) P (m + q) ] p + m + (1 )] + ( )] P (m + q) } p + m + {(1 )}. (18) For the processes χ c1 B B, A 1 (χ c1 B B) = i ɛ αβµν K α ɛ (λk)β A µν 1, (19) Mχ where K is the four-momentum vector of χ c1, and ɛ αβµν is Levi Civita tensor, and ɛ (λ k) is identical to the polarization vector of the χ c1 with a value of helicity λ k. Note that A µν ɛ (λi)µ ɛ (λj)ν, where ɛ (λi)µ and ɛ (λj)ν are polarization vectors corresponding to the two gluons which couple to the two quark-antiquark pairs with Lorentz indexes µ and ν, respectively. For fulfilling the requirement

4 No. χ cj Decays into B B in Quark-Pair Creation Model 55 of Levi Civita tensor to combine the polarization vectors to be asymmetric, there is only one kind of configuration, i.e., ɛ (+) ɛ () ɛ () and its arrangements to make contributions to the non-zero transitional amplitudes. Then upon projecting it onto the baryonic spin wavefunctions, one immediately gets Γ(χ c1 B B) =. () As for χ c B B decays, the calculation is quite lengthy since there are more configurations of the outgoing baryonic helicity to make contributes to the transitional amplitudes. One has C Φ µν A = (m)ôµ (x 1, λ 1 )Ôν (x, λ ) ˆQ(x ], λ ) (q 1 + q 1 ) (q + q ) + (1 )] + ( )], (1) where C is an overall constant and Φ µν (m) is the covariant spin wave function of χ c with the helicity value m. It can be built out of the polarization vector Φ(m) by the relation Φ µν (m) = 1m 1, 1m m Φ µ (m 1 )Φ ν (m ). () m 1,m The full decay widths for the processes χ cj B B are evaluated directly from the following expression, dγ(χ cj B B) = 1 dω π M J (s z, s z) P B Mχ, () cj s z,s z where P B is the momentum vector for the outgoing baryon. Numerical Results and Discussions The experimental data on exclusive decays χ cj B B are available only for the channel χ cj p p (J =, 1, ) by far. 7] For cancellation of the overall constant dependence, one would like to evaluate the decay width ratio Γ(χ cj B B)/Γ(χ cj p p). In the calculation, the spatial wavefunctions for baryons are assumed to be well described by a simple harmonic-oscillator eigenfunction in their center-of-mass (c.m.) system, i.e. φ B ( k ρ, k λ ) = 1 (πβ) / e( k λ + k ρ )/β, (4) where β = mω is the harmonic-oscillator parameter and kρ, k λ are defined as kρ = 1 6 ( p 1 + p p ), kλ = 1 ( p 1 p ), (5) where for simplicity, one assumes that the radial wavefunction both in its c.m. system and the laboratory system are the same, namely, the relativistic effects of the outgoing baryon are ignored. In the calculation, there are four parameters to be determined, i.e. the overall constant C J, the light quark mass m u,d, the strange quark mass m s, and the harmonicoscillator parameter β. The constant C J is associated with the channel χ cj p p. In most quark model calculations, the value β is chosen within the range of.6 GeV. GeV. The strange quark mass is always chosen within the range.4 GeV.5 GeV. In the study of the baryonic spectroscopy in a relativized quark model, if those parameters are taken as m u = m d =. GeV, m s =.4 GeV, β =.16 GeV, the spectroscopies of baryons, as well as their excited states are well reproduced. 9] In this calculation, one would like to take those parameters as the preferable value since the dispersion relation p = m + p has been used. Selecting those parameters, the main numerical results are shown in Table 1, where the uncertainty of central values are corresponding to the change of the strange quark mass within.7 GeV.47 GeV. The decay width for the process χ cj Ξ + Ξ (J =, ) is the smallest among those decays since it contains two strange quarks and a lower phase space factor. The calculated value for χ c Λ + Λ is comparable with the measured one, while for the process χ c Λ + Λ it is smaller than the measured one. This might be improved by considering to cancel the assumptions in this model on the properties of the bound states χ cj and the transverse momentum of the outgoing (anti)quarks and Lorentz contraction of baryonic wave function, etc. Table 1 The ratios of the decay widths for Γ(χ cj B B)/Γ(χ cj p p) (J =, ). The central values are corresponding to the choice of parameters m u = m d =. GeV, m s =.4 GeV, β =.16 GeV, and the uncertainty of the central values is related to the change of the strange quark mass within.7 GeV.47 GeV, and the measured values are taken from Ref. 7]. Γ(χ cj B B) Γ(χ cj p p) Γ(χ c Λ Λ) Γ(χ c p p) Γ(χ c Σ Σ ) Γ(χ c p p) Γ(χ c Ξ Ξ+ ) Γ(χ c p p) Γ(χ c Λ Λ) Γ(χ c p p) Γ(χ c Σ Σ ) Γ(χ c p p) Γ(χ c Ξ Ξ+ ) Γ(χ c p p) Calculated values Measured values 1.91 ±..14 ± ±.1.7 ± How does the choice of the uds basis for hyperons affect the full decay widths? One would like to imagine that if one takes the limit of the SU() of the hyperons,

5 56 PING Rong-Gang and JIANG Huan-Qing Vol. 41 i.e., m s = m u,d, in addition to the difference of the phase space among those channels, the full decay widths of those processes should be the same. This is really true if the operators describing those decay processes do not change the configuration of the quark s spins. As is well known, if the operator ˆT does not change configurations of the quarks spins or isospins, it does not matter if we choose an asymmetric or symmetric spin configurations for s 1 s in the expression s 1 s ˆT A ( s 1 s ), where A is an asymmetric operator. However, the operators describing the quark currents, e.g., ū s (p)γ µ v s (q), do change the configurations of the quark spins as shown in Eqs. (6) (8). Hence in this calculation, the decay widths for processes χ cj B B (J =, ) are by no means equal in the limit of the SU() choice. The following numerical results are able to serve as an evidence for this argument. If the parameters are taken as m s = m u,d =. GeV, m Λ = m Σ = m Ξ = m p =.98 GeV, the ratio of the decay widths are Γ (p p) : Γ (Λ Λ) : Γ (Σ Σ) : Γ (Ξ + Ξ ) = 1 : 1. :.8 :.8 for χ c B B, and Γ (p p) : Γ (Λ Λ) : Γ (Σ Σ) : Γ (Ξ + Ξ ) = 1 : 1.1 :.94 :.94 for χ c B B. To conclude, a quark pair creation model is presented for one to study the χ cj B B exclusive decays phenomenologically with an explicit inclusion of the baryonic properties. In this simple model, the forbidden decays, i.e. χ c B B (B = Λ, Σ, Ξ ), by helicity selection rule, are reproduced, while the decays χ c1 B B (B = Λ, Σ, Ξ ) are forbidden. The results show that the decay widths for χ cj (J =, ) into Λ Λ are all greater than that into p p pair. Acknowledgments One of the authors would like to thank B.S. ZOU and H.Q. ZHOU for useful discussions and comments. References 1] L. Kopke and N. Wermes, Phys. Rep. 174 (1989) 67. ] V.A. Novirov, et al., Phys. Rep. C41 (1978) 1. ] G.P. Lepage and S.J. Brodsky, Phys. Rev. D (198) ] S.J. Brodsky and G.P. Lepage, Phys. Rev. D4 (1981) ] M. Anselmino, F. Caruso, and F. Murgia, Phys. Rev. D4 (199) 18; M. Anselmino and F. Murgia, Phys. Rev. D47 (199) 977; D5 (1994) 1. 6] S.M.H. Wong, Eur. Phys. J. C14 () 64. 7] BES collaboration, Phys. Rev. D67 () ] E.S. Ackleh, T. Barnes, and E.S. Swanson, Phys. Rev. D54 (1996) ] S. Capstick and N. Isgur, Phys. Rev. D4 (1986) 89.