Photoproduction of Vector Meson φ off Deuteron in QCD Inspired Model

Transcription

1 Commun. Theor. Phys. (Beijing, China) 52 (2009) pp c Chinese Physical Society and IOP Publishing Ltd Vol. 52, No. 2, August 15, 2009 Photoproduction of Vector Meson φ off Deuteron in QCD Inspired Model MENG Cheng-Ju, 1 PAN Ji-Huan, 1 PENG Jin-Song, 1 MA Wei-Xing, 2 and YUAN Tong-Quan 1 1 Department of Physics and Electronic Engineering, Hechi College, Yizhou , China 2 Institute of High Energy Physics, the Chinese Academy of Sciences, Beijing , China (Received August 4, 2008) Abstract Based on the quark-gluon contents of nucleon and strongly believing that the force mediators, Pomeron and its counterpart in the conventional approach of Regge theory, for high energy diffractive process would be the tensor glueball and Odderon respectively, we discuss photo-production of vector meson φ off the deuteron at energy less than 3 GeV in the QCD inspired model in which the quark gluon degrees of freedom and glueball, Odderon exchange are taken into account. A calculation is performed for γ + D φ + D, and the theoretical predictions of the differential cross section dσ γd /dt, are presented and compared with available experimental data. Our QCD inspired model reproduces data quite well in the whole range of the experimental measurements up to t = 0.4 GeV. Our results can be used to extract γn φn data, which cannot be measured in experiment. PACS numbers: Rd, Cy, Lg Key words: photoproduction of vector meson, QCD generalized vector meson dominance model, QCD 1 Introduction The vector meson photoproduction provides an interesting laboratory for studying the interplay between perturbative and nonperturbative QCD. A microscopic understanding of vector meson photoproduction remains an elusive goal of hadronic physics. The investigation of φ meson photoproduction at the energies, E γ = 1.6 GeV plays an important role in understanding the nonperturbative Pomeron exchange dynamics and the nature of the φn interaction. It was expected that in the diffractive region the dominant contribution comes from the Pomeron exchange, since processes associated with conventional meson (quark) exchange are suppressed by the Okubo Zweig Iizuka (OZI) rule. [1 7] An example of such a suppressed process is the pseudoscalar π and η meson exchange, which, as a rule, was considered as a small contribution to the dominant Pomeron exchange channel. The Pomeron with vacuum number couples to single constituent quarks as a charge conjugation number C = +1 isoscalar photon and has been regarded as the tensor glueball [8] with mass of 2.23 GeV, decay width Γ ξ 100 MeV and quantum numbers I G J P,G = 0 +, In this regarding, the Odderon exchange contribution, which corresponds to the crossing odd amplitude, should be considered in any theoretical prediction of observable as a counterpart of the tensor glueball exchange. Recent experimental data [9] strongly support the point of view that one has to introduce some exotic channels to modify the conventional Pomeron exchange model in order to fit the new data. In this paper, we mainly study the quark gluon degrees of freedom in φ meson photo-production off the deuteron. The coherent φ photoproduction off the deuteron in the diffraction region seems to be very useful for such a study. First of all, the isovector, hadrons are described in terms of quarks, anti-quarks, and gluons. As is wellknown, the isovector π meson exchange amplitude is eliminated in the case of the isoscalar target deuteron. Therefore, the appearance of the bump like structure in the energy dependence of the differential cross section of the reaction γd φd of vector meson would favor a modification of the conventional Pomeron exchange amplitude. The next step is an analysis of spin observable, in particular, the properties of the decay φ K + K with unpolarized and polarized photon beams. The incoherent φ photoproduction in the γd φpn reaction allows one to extract observable of the γn φn reaction, which can be used for a simultaneous analysis of photoproduction off the neutron and proton targets in order to get additional and independent evidence of a manifestation of possible exotic channels. Schematically, the coherent and incoherent φ meson photoproduction processes are exhibited in Fig. 1 The internal dashed lines in Figs. 1(b) and 1(d) correspond to diagonal (m = φ) and nondiagonal (m = π, ρ, ω,...) transitions, respectively. In this paper, we study the φ meson photoproduction at the energies of E γ < 3 GeV in forward photoproduction angles with momentum transfer t 0.4 GeV 2, where the single scattering processes are dominant. The coherent φ meson photoproduction at higher values of t is controlled by the double scattering processes, which can provide important information about the cross section of the φn scattering. [10] However, this interesting topic is beyond the scope of our present analysis, now we just focus on the extremely forward φ meson photoproduction, where some hints of an anomaly in the differential cross section of γp φp reaction was found. [11] Supported by National Natural Science Foundation of China under Grant Nos and

2 296 MENG Cheng-Ju, PAN Ji-Huan, PENG Jin-Song, MA Wei-Xing, and YUAN Tong-Quan Vol. 52 Some theoretical estimation for the coherent vector meson photoproduction from the deuteron have been given in Ref. [12]. The first experimental data on the γd φd reaction were reported recently in Refs. [13,14]. stands for vector mesons such as φ, ω, ρ, J/ψ, and υ. Our numerical calculations and the corresponding theoretical results are given in Sec. 4. Finally, summary and concluding remarks stemmed from our present research work are reserved for Sec Generalized Vector Meson Dominance Model and QCD Inspired Model Our calculations of the γd φd cross section are performed in the QCD inspired model and in the QCD generalized vector meson dominance model. Therefore, we should briefly introduce the two models at first in this section. Fig. 1 Diagrammatic representation of coherent [(a) and (b)] and incoherent [(c) and (d)] φ meson photoproduction in the γd reaction with [(a) and (c)], and [(b) and (d)] being single and double scattering term, respectively. Differing from the other theoretical studies, which base on the conventional models with Pomeron exchange [15] or two perturbative gluons exchange, [16] we investigate the γd φd in our proposed QCD inspired model and the QCD generalized vector meson dominance model. Our paper consists of 5 parts. In Sec. 2, we briefly introduce the QCD generalized vector meson dominance model and the QCD inspired model. In Sec. 3, we present mechanism of the vector meson φ photo-production, γd φd, in the QCD inspired model. In particular, we set up a relationship between the differential cross sections of γn V N and γd V D reactions where V 2.1 QCD Generalized Vector Meson Dominance Model Gribov [17] was the first one to observe that a photon (even virtual photon) fluctuates into a hadron system with life time (coherence length) τ = l c = 1 mx B, x B = Q2 s, (1) where Q 2 is the photon virtuality (momentum), and m is the mass of the target nucleon. This life time is much larger at high energy than the size of the target and therefore, we can consider the photon-hadron interaction as a process, which proceeds in two stages. (i) Transition γ into hadrons (a pair of quarkantiquark), which is not affected by the target hadron, therefore, looks similar to electro-positron annihilation; (ii) Hadron-target interaction, which can be treated as standard hadron-hadron interaction, for example, in the Pomeron (Reggeon) exchange approach (Glueball and/or Odderon exchange in QCD). The two steps are shown in Fig. 2. Fig. 2 A sketch representation of the generalized vector meson dominance model in QCD. V stands for the vector meson Υ(b b). For φ(s s) production, V = φ(s s). These two separate stages of the photon-hadron interaction allow us to use a dispersion relation with respect to the masses M and M to describe the photon-hadron interaction. Since the coherence length l c = 1/mx B R N, the target size. For Υ production off the nucleon, the threshold energy is E γ = 8.2 GeV, and because of the large mass of the charmed quark (m b = 1.5 GeV), the b b fluctua-

3 No. 2 Photoproduction of Vector Meson φ off Deuteron in QCD Inspired Model 297 tion of the photon travels over a short coherence length. l c = 2 (Eγ /4m 2 b ) = 0.36 fm. The large mass of the bottom quark also imposes a. small transverse size γ = 1/mb = 0.13 fm on this fluctuation. The minimum value allowed for the momentum transfer is large, t min 1.7 GeV 2 at threshold, and t min 0.6 GeV 2 at E γ = 10 GeV. This bottom and antibottom quark pair production near threshold implies a small impact parameter, b 1/ t = 0.2 fm. All five quarks (the two heavy bottom quarks in the probe and three light quarks in the target proton) must be in the small interaction volume. As a consequence, all the quarks must be involved in the reaction mechanism. For nucleon target, this implies that three-gluon exchange may dominate two-gluon and one-gluon exchange and opens the way for the study of correlations between valence quarks in the proton target. The formation length, l F, over which the b b pair evolves into a Υ after its interaction with a nucleon is given by 2 E Υ l F 0.22E γ. (2) m Υ m Υ 2m b At near threshold l F is about 1 fm, closer to the size of the proton, 0.86 fm, than to the size of the nucleus. Based on the above idea we can write a general formula for the total cross section of the photon-hadron interaction, σ(γn) = α em Γ(M 2 )dm 2 3π Q 2 + M 2 σ(m2, M 2, s) Γ(M 2 )dm 2 Q 2 + M 2, (3) where Γ 2 (M 2 ) = R(M 2 ) and σ(m 2, M 2, s) is proportional to the imaginary part of the forward amplitudes for V +p V +p, where V and V represent the vector meson states with masses M and M. For the case of the diagonal transition (M = M ), the σ(m 2, M 2, s) = σ(m 2, s), which is the total cross section for V +p V +p interaction. Experimentally, it is known that a diagonal coupling of the Pomeron (glueball in QCD) is stronger than an offdiagonal transition. Therefore, in leading approximation we can neglect the off-diagonal transition and substitute σ(m 2, M 2, s) = σ(m 2, s)m 2 δ(m 2 M 2 ) (4) into Eq. (3). Then, the resulting photon-hardron cross section can be written as the following σ(γn) = α em R(M 2 )M 2 dm 2 3π (Q 2 + M 2 ) 2 σ(m 2, s), (5) where R(M 2 ) is defined as the ratio R(M 2 ) = σ(e+ e all-hadrons) σ(e + e µ + µ ). (6) As we mentioned before, M 2 is the mass squared of the hadronic system, Γ 2 (M 2 ) = R(M 2 ) and σ(m 2, s) is the cross section for the hadronic system to scatter off the nucleon target. Experimentally, R(M 2 ) can be described in a two component picture: the contribution of resonances such as ρ, ω, φ, J/Ψ, Ψ and so on, and the contributions from quarks, which give a more or less constant term changes abruptly with every new open quark-multiquark channel. R(M 2 ) = 3Σ q e 2 q, where e q is the charge of the quark and the summation is done over all active quarks terms. If we take into account only the contribution of the ρ- meson, ω-meson, φ, and J/Ψ resonances in R(M 2 ), we obtain the so-called vector dominance model (VDM), which gives for the total cross section the following formulism σ tot (γn) = α em 3π σ tot(ρ + p) R(M 2 = Mv) 2 ν { M 2 } 2 v, (7) Q 2 + Mv 2 where M V is the mass of vector meson, Q 2 is the value of the photon virtuality and R(M 2 = MV 2 ) is the value of the R in the mass of the meson, which can be rewritten through the ratio of the electron-positron decay width to the total width of the resonance. [18] Since Gribov s work, the interaction between the photon and hadronic matter has been remarkably well described by use of the vector meson dominance model. The model assumes that the hadronic components of the vacuum polarization of the photon consist exclusively of the known vector mesons (ρ, ω, φ, J/Ψ, Υ). This is certainly an approximation, but in the region around the vector meson masses, it appears to be a very good approximation. As vector mesons are believed to be bound states of quark-antiquark pair, for example, ρ + = u d, ρ = dū, ρ 0 = 1 (d d uū), 2 ω = 1 2 (d d + uū), φ = s s, J Ψ = c c, Υ = b b. (8) It is tempting to try to establish a connection between the old language of VMD and the standard model. In the standard model, quarks, being charged, couple to the photon and so the strong sector contribution to the photon propagator arises, in a manner analogous to the electron-positron loops in QED. According to our above discussions, we may understand why we use the mechanisms given in Fig. 3 for φ meson photo-production. The diagrams in Fig. 3 contain fully dressed quark propagators, glueball and Odderon propagators and their coupling vertices with quarks and antiquark, the photon-quark vertex as well as the coupling vertex of quark-antiquark into vector meson φ, which is solution of the Bethe Salpeter equations. In QCD language of VMD, photon fluctuates into a quark-antiquark pair and then the photon-hadron scattering process is described by a quark-antiquark pair-hadron scattering process. We call this theory as QCD generalized vector meson dominance theory. Of course, it is a model theory but a

4 298 MENG Cheng-Ju, PAN Ji-Huan, PENG Jin-Song, MA Wei-Xing, and YUAN Tong-Quan Vol. 52 good approximation to describe vector meson photo- and electro-production and has been successfully verified by fitting experimental data. 2.2 QCD Inspired Model We use the QCD inspired model to study φ photoproduction. In order to understand our proposed QCD inspired model, we would like, at first, to briefly remind readers of Quantum Chromodynamics (QCD), the fundamental theory of strong interaction, which was developed in the seventies of 20 century, along the general lines of QED Rationale of QCD QCD, as it is normally known in high energy physics, is the quantum field theory that describes the strong interactions. It is the SU C (3) gauge theory of the current Standard Model for elementary particles and forces, SU C (3) SU(2) U(1), which encompasses the strong, electromagnetic and weak interactions. The symmetry group of QCD, with its eight conserved charges, is referred to as color SU C (3) group. As is characteristic of quantum field theories, each field may be described in terms of quantum waves or particles. Because it is a gauge field theory, the fields that carry the forces of QCD transform as vectors under the Lorentz group. Corresponding to these vector fields are the particles called gluons, which carry an intrinsic angular momentum, or spin of one in units of. The strong interactions are understood as the cumulative effects of gluons, interacting among themselves and with the quarks, the spin 1/2 particles of the Dirac quark fields. There are six quark fields of varying masses in QCD. Of these, three are called light quarks, and other three are called heavy quarks. The light quarks are the up (u), down (d), and strange (s), while the heavy quarks are the charm (c), bottom (b) and top (t). Their electric charge are famously e f = 2/3e (f = u, c, t) and e f = 1/3e (f = d, s, b), with e the positron charge. The gluons interact with each quark field in an identical fashion, and the relatively light masses of three of the quarks provide the theory with a number of approximate global symmetries that profoundly influence the manner in which QCD manifests itself in the Standard Model. These quark and gluon fields and their corresponding particles are enumerated with complete confidence by the community of high energy physicists. Yet none of these particles has ever been observed in isolation, as one might observe a photon or an electron. Rather, all known strongly interacting particles are colorless: most are mesons, combinations with the quantum numbers of a quark q and an antiquark q, or baryons with the quantum numbers of combinations of three quarks qqq. This feature of QCD is called confinement. The very existence of confinement requires new ways of thinking about field theory, and only with these was the discovery and development of QCD possible. In one word, the main degrees of freedom in QCD are quarks and gluons. The interaction between quarks and gluons are mediated by exchange of gluons. Unfortunately, gluons interact with each other as they carry the color charge that gives rise to the strong interactions its non-abeilian behavior, which makes us to face the so-called non-perturbative difficulty QCD Inspired Model Since QCD is non-abeilian theory and no one can solve the non-perturbative problem of QCD, one has to make some modeling assumptions to treat the non-perturbative difficulty in any theoretical calculations of observable. The QCD inspired model proposed by us previously is one of the phenomenological models. By the term of the QCD-inspired model, we mean that the scattering amplitude of high energy diffractive processes, such as vector meson photo- and electro-production off the proton, is expressed as a sum of contributions from quark-quark, gluon-gluon, quark-gluon interactions to crossing even amplitude F + (s, t) and the QCD Odderon terms to crossing odd amplitude F (s, t). The total scattering amplitude F(s, t) = F + (s, t) + F (s, t), which can be given by eikonal phase transition function χ where F + (s, t) is given by χ even and F (s, t) by χ odd. The charge conjugation even eikonal function χ even can be written [19] as χ even (s, b) = χ qq (s, b) + χ gg (s, b) + χ qg (s, b). (9) The individual term in Eq. (9) factorize into a product of two functions of σ(s) and W(b, µ). That is, χ ij (s, b) = iσ ij (s)w(b, µ ij ), where ij stands for qq, gg and qg. If one uses the dipole form of the nucleon electromagnetic form factor, W ij (b, µ ij ) can be expressed W ii (b, µ ii ) = [µ ii b] 3 K 3 (µ ii b)/8 with K 3 being a modified Bessel function. For i j, µ ij = µ ii µ jj, where i=q, g, quark or gluon. The charge conjugation odd eikonal function χ odd reads χ odd (s, b) = σ odd (s)w(b, µ odd ). (10) In the QCD inspired model, the mediators of the force are tensor glueball and Odderon. This is very different from the conventional treatment in hadronic level of high energy scattering in which the quark and gluon degrees of freedom have not been taken into account. 3 φ Photo-Productions off Deuteron in QCD Inspired Model We study coherent process of γd φd in this section. The four-momentum of the incoming photon γ, outgoing vector meson φ, and the initial and the final deuteron are denoted as k γ, q φ, P D, and P D, respectively. The standard Mandelstam variables are defined t D = (P D P D ) 2 = (k γ q ϕ ) 2, S W 2 D = (P D + k γ ) 2. (11) The space component of the momentum transfer to deuteron in the laboratory system is q 2 = t D (1

5 No. 2 Photoproduction of Vector Meson φ off Deuteron in QCD Inspired Model 299 t D /4M 2 D ), where M D is the deuteron mass. The mechanism of the process γd φd is depicted by Fig. 3, where Fig. 3(a) represents the general process in hadronic level with waving line stands for incoming real photon, dashed line denotes the produced vector meson φ, and the dark circle presents interaction. The deuteron consists of proton and neutron and they are represented by solid line in this figure. Fig. 3(b) describes more detailed mechanism of the process γd φd in our QCD inspired model and generalized vector meson dominance model, where the incoming photon fluctuates into a quark-antiquark pair according to the generalized vector meson dominance model and then the quark-antiquark pair interacts with target deuteron through exchange of tensor Glueball and Odderon instead of the usual Pomeron exchange, which is an imaginary object and its nature and origin are still unknown so far. Therefore, the process under our study proceeds through quark gluon degrees of freedom in QCD. In particular, the photon deuteron interaction is replaced by quark deuteron strong interaction. This is very important difference between our study and the others. Of course, Fig. 3(b) represents many different Feynman diagrams, which contribute to the γd φd cross section. The sub-process diagrams contained in Fig. 3(b) and the detailed calculations of their contributions to cross section can be found in our previous publications in Ref. [20], where the mechanism and calculations of the elementary process γ + P φ + P has been presented and performed numerically. Now we extend the previous study of γp φp reaction to γd φd process. 3.1 Differential Cross Section of γd φd Production In this subsection, we closely follow the treatment of Ref. [21]. We consider the φ meson photo-production off the deuteron at forward angles with t 0.4 GeV 2, where the dominant contribution comes from the single scattering process, shown in Fig. 1(a). In such a case, we can use a non-relativistic framework for the deuteron form factor based on utilizing the realistic NN interaction. In our analysis, we use the deuteron wave function calculated with the Paris potential [22] designed just for describing nuclear processes at high momentum transfer. Thus, it describes fairly well the deuteron electromagnetic form factor with momentum transfer up to t 0.9 GeV 2. [22] The total vector meson photo-production amplitude in the reaction γd V D is given by T D M f M i ;λ ν λ γ = 2 αβ M f λ ν, β T s βα,λ ν λ γ M i λ γ, α, (12) where M i, M f, λ γ, and λ V stand for the deuteron-spin projections of the initial and final states, helicities of the incoming photon and the outgoing vector meson φ, respectively. Tβα;λ s ν λ γ is the amplitude of the vector meson photoproduction from the isoscalar nucleon, T s βα;λ ν λ γ 1 2 (T p + T n ) βα;λν λ γ. (13) The indices α and β in Eqs. (12) and (13) refer to all quantum numbers of states before and after the collision. T P,n are the elementary photoproduction amplitude. The π exchange terms are canceled in the total amplitude, since T n π = T p π. Using the standard decomposition of the deuteron state in terms of S(U 0 ) and d(u 2 ) components of the deuteron wave functions, one can rewrite Eq. (12) in the explicit form [21] TM D f M i λ ν λ γ (t) = 2 4π λ ˆL ˆλ i L Y λµ(ˆq) C 1M 1 2 m mc1m 1 2 m mc1m i 1MLM L C 1M i 1ML M L CLM L L M L λµ CL0 L 0λ0 R LL λ(q 2 )T s m 1 m 1 λ νλ γ (t), (14) where ĵ = 2j + 1, and the radial integral R LL λ are expressed by R LL λ(q 2 ) = dru L (r)u L (r)j λ (qr/2). (15) Fig. 3 Schematic representation of the γd φd reaction in the QCD inspired model and the QCD generalized vector meson dominance model. (a) general schematic representation of γd φd in hadronic level; (b) mechanism of γd φd in QCD inspired model and the QCD generalized vector meson dominance model. For a qualitative analysis of the unpolarized differential cross section at small momentum transfer with θˆq 0, keeping only the spin/helicity conserving terms with natural T N and unnatural T U parity exchange in the total amplitude, one gets T N,U mm ;λ ν λ γ (t) = ( 1 2mλ γ ) δ mm δ λν λ γ T N,U 0 (t). (16)

6 300 MENG Cheng-Ju, PAN Ji-Huan, PENG Jin-Song, MA Wei-Xing, and YUAN Tong-Quan Vol. 52 Here, T N,U 0 (t) is the spin-independent part of the amplitudes. Using Eq. (14) with Eq. (16), we get the following result for the natural and unnatural parity-exchange parts of the total amplitude: T DN M f M i ;λ ν λ γ = 2δ Mi M f δ λγ λ ν (δ ±1Mi S N 1 + δ 0Mi S N 0 )T N 0, T DU M f M i ;λ ν λ γ = 2M i λ γ δ Mi M f δ λν λ γ δ ±1Mi S U 1 T U 0. (17) with The form factors S N,U i read [21] S N 1 = F C 2F Q, S N 0 = F C + 2 2F Q, S U 1 = F M (18) F C = R R 220, R Q = R R 220, F M = R R R 202. (19) Taking into account the cancellation of the unnatural parity π exchange contribution and neglecting weak η meson exchange, one can express the differential cross section of the γd φd reaction by the cross section of the φ photo-production from the isoscalar nucleon N as dσ γd 4Z(t) dσγn, (20) dt dt where t = t D and Z(t) is the structure factor Z(t) = F 2 C(t) + F 2 Q(t). (21) The dependence of Z(t) and F C,Q on t D is rather symbolic. In fact, these factors depend on the spatial part of the four-momentum transfer in the laboratory system q. The relation between t D and q 2 is given by t D = 2M D ( q 2 + MD 2 M D) with M D being deuteron mass. The structure factor Z(t D ) as a function of t D is shown in Fig. 4. Fig. 4 Dependence of the structure factor Z(t) Z(t D ) = FC(t 2 D ) + FQ(t 2 D ) on t t D. The structure factor Z(t) is given by numerical calculations according to Eqs. (15,21). The structure factor Z(t D ) is also related to the wellknown structure function of electron elastic scattering on the deuteron e + D e + D as A(t) Z(t)G 2 D(t), (22) with G D (t) = [1 t/0.71] 2, (23) which is so-called the dipole electromagnetic form factor of the proton. Equation (20) allows one to extract the cross section of the γn reaction from the measured cross section of the γd reaction as dσ γn dt 1 dσγd [4Z(t)]. (24) dt Equation (23) makes it possible that we can extract the cross section of γn φn photoproduction with the goal of a subsequent combined analysis of γp and γn reactions to seek for a possible manifestation of exotic channels such as tensor Glueball and Odderon exchange channels. 3.2 φ Photoproductions off Deuteron in QCD Inspired Model Assuming F(s, t) is scattering amplitude in general, then the total cross sections σ tot (s), ratio of the real part to the imaginary part of the forward scattering amplitude F(s, t = 0), ρ(s), and the nuclear slope parameter function β(s) are normalized in such a way that [23] σ tot (s) = 1 Im F(s, t = 0), (25) s Re F(s, t = 0) ρ(s) = Im F(s, t = 0), (26) β(s) = d [ dσ(s, t) ] ln dt dt, (27) t=0 with dσ(s, t)/dt, in Eq. (27), being the differential cross sections of the process under study, which is determined by the reaction amplitude F(s, t) and normalized such that dσ(s, t) dt = 1 16πs 2 F(s, t) 2. (28) Equations (20) and (25) (28) are the fundamental formulae of our present study on the diffractive φ photoproduction off the deuteron at energies of E γ = GeV. 4 Numerical Calculations and Theoretical Predictions for γd φd In order to reproduce the available experimental data measured at LEPS [24] and CLAS [25] collaborations, we use Eq. (20) to make our predictions. First, we calculate differential cross section dσ γn /dt in our proposed QCD inspired model as what we have done exactly for the process of γn φn in Ref. [19]. The structure factor Z(t D ) defined by Eq. (19) are given by numerical calculations with the paris potential. The resulting Z(t D ) is plotted in Fig. 4. Our predictions γd φd for are shown in Fig. 5 together with data, where the solid curve stands for our theoretical results while the dark points and deltas with

7 No. 2 Photoproduction of Vector Meson φ off Deuteron in QCD Inspired Model 301 error bars denote the existing experimental data. One can see that our calculations describe rather well the data. Fig. 5 Differential cross section of the γd φd reaction as a function of momentum transfer (t = t D ), with LEPS [15] and CLAS [16] data. 5 Summary and Concluding Remarks Based on the QCD generalized vector meson dominance model, we study vector meson φ photoproduction off deuteron at small momentum transfers in the QCD inspired model. The QCD inspired model takes quark gluon degrees of freedom into account and assumes mediators of exchange force as the tensor glueball and Odderon, which consist of two Reggeinized gluons and three gluons, respectively, in a bound state with colorless. For this purpose, we use our previous numerical calculations of the elementary γp φp reaction as an input for present calculations. We obtain a quite reasonable agreement between our model predictions and the existing experimental data in diffractive region. To summarize, we can conclude that the existing data on γd φd support our QCD inspired model. The very important point stemmed from our current investigation is that we can extract the cross section of γn V n (V = φ, ω, ρ, J/ψ, Υ) from the γd cross section, that is, our QCD inspired model and results can be used for the extraction of the information on γn φn photoproduction from the observable of γd data. The measurement of the process γn φn photoproduction is impossible in experiment since neutron is unstable. It could not be a target for measurement. The contributions from the individual terms in the reaction γd φd to its cross section, and theoretical prediction on the γn φn photoproduction will appear soon in our coming papers. References [1] T. Nakano and H. Toki, in Proceedings of the International Workshop on Exiting Physics and New Accelerator Facilities, Spring-8, Hyogo, 1997, World Scientific, Singapore (1998) p. 48. [2] M.A. Pichowsky and T.-S.H. Lee, Phys. Rev. D 56 (1997) [3] A.I. Titov, Y. Oh, S.N. Yang, and T. Morii, Phys. Rev. C 58 (1998) [4] Q. Zhao, Z. Li, and C. Bennhold, Phys. Lett. B 436 (1998) 42; Phys. Rev. C 58 (1998) [5] R.A. Williams, Phys. Rev. C 57 (1998)223. [6] J.M. Laget, Phys. Lett. B 489 (2000) 313. [7] A.I. Titov and T.S.H. Lee, Phys. Rev. C 67 (2003) [8] Zhou Li-Juan and Ma Wei-Xing, Commun. Theor. Phys. 45 (2006) 1085; Zhou Li-Juan, He Xiao-Rong, and Ma Wei-Xing, Commun. Theor. Phys. 44 (2005) 509; Ma Wei-Xing, Zhou Li-Juan, and L.C. Liu, Chin. Phys. 26 (2002) 309. [9] T. Mibe, et al., [LEPS Collaboration], Phys. Rev. Lett. C 95 (2005) [10] L.L. Frankfurt, J. Mutzbauer, W. Koepf, G. Piller, M. Sargsian, and M. I. Strikman, Nucl. Phys. A 622 (1997) 511; L. Frankfurt, G. Piller, M. Sargsian, and M. Strikman, Eur. Phys. J. A 2 (1998) 301; T.C. Rogers, M.M. Sargsian, and M.I. Strikman, Phys. Rev. C 73 (2006) [11] C.H.J. Besch, et al., Nucl. Phys. B 70 (1974) 257; E. Anciant, et al., [CLAS Collaboration], Phys. Rev. Lett. 85 (2000) [12] A.I. Titov, M. Fujiwara, and T.S.-H. Lee, Phys. Rev. C 66 (2002) (R). [13] W.C. Chang, et al., [LEPS Collaboration], arxiv:nucl. ex/ [14] T. Mibe, et al., [CLAS Collaboration], arxiv:nucl. ex/ [15] A. Donnachie and P.V. Landshoff, Phys. Lett. B 185 (1987) 403. [16] J.M. Laget, Phys. Lett. B 489 (2000) 313; M.G. Ryskin, Z. Phys. C 57 (1993) 89; J.R. Cudell and I. Royen, Phys. Lett. B 397 (1997) 317. [17] V.N. Gribov, Sov. Phys. JETP 30 (1970) 709. [18] J.J. Sakurai and D. Schildknecht, Phys. Lett. B 40 (1972) 121. [19] Lu Juan, Ma Wei-Xing, and He Xiao-Rong, Commun. Theor. Phys. 45 (2006) 819. [20] Pan Ji-Huan, Ma Wei-Xing, Gu Yun-Ting, and Luo Liang-Zi, Commun. Theor. Phys. 52 (2009) 108. [21] A.I. Titov and B. Kampfer, Phys. Rev. C 76 (2007) [22] M. Lacombe, B. Loiseau, R. Vinh Mau, J. Cote, P. Pires, and R. de Tourreil, Phys. Lett. B 101 (1981) 139; M. Lacombe, B. Loiseau, J.M. Richard, R. Vinh Mau, J. Conte, P. Pires, and R. de Tourreil, Phys. Rev. C 21 (1980) 861. [23] Zhou Li-Juan, Liu Bao-Rong, and Ma Wei-Xing, Commun. Theor. Phys. 48 (2007) 3, 519.