$\phi$ photoprodution with coupled-channel effects

We study phi photoproduction with various hadronic rescattering contributions included, in addition to the Pomeron and pseudoscalar meson-exchange diagrams. We find that the hadronic box diagrams can explain the recent experimental data in the vicinity of the threshold. In particular, the bump-like structure at the photon energy E_gamma \approx 2.3$ GeV is well explained by the K Lambda(1520) rescattering amplitude in the intermediate state, which is the dominant contribution among other hadronic contributions. We also find that the hadronic box diagrams are consistent with the observed spin-density matrix elements near the threshold region.


I. INTRODUCTION
The φ(1020) meson is distinguished from other vector mesons, since it contains mainly strange quarks. Because of its dominant strange quark content, its decays to lighter mesons and coupling to the nucleon are known to be suppressed by the Okubo-Zweig-Iizuka (OZI) rule. In fact, the strange vector form factors of the nucleon, which is implicitly related to the φ meson via the vector-meson dominance, is reported to be rather small [1]. This large ss content of the φ meson makes the meson-exchange picture unfavorable in describing photoproduction of the φ meson. Thus, the Pomeron [2,3] is believed to be the main contribution to φ photoproduction, since it explains the slow rise of the differential cross sections of φ photoproduction as the energy increases. However, while it is true in the higher energy region, a recent measurement reported by the LEPS collaboration [4] shows a bump-like structure around the photon energy E γ ≈ 2.3 GeV. It seems that the Pomeron alone cannot account for this bump-like structure and requires that one should consider other production mechanism of φ photoproduction near the threshold energy. Moreover, a recent measurement of the spin-density matrix elements near the threshold region [5] implies that hadronic degrees of freedom play essential role in the vicinity of the threshold.
So far, the theoretical understanding of the production mechanism for the φ photoproduction can be summarized as follows: • General energy-dependence of the cross sections is mainly explained by Pomeron exchange that can be taken as either a scalar meson or a vector meson with charge conjugation C = +1. While the Pomeron explains the increase of the differential cross section dσ/dt in the forward direction, it cannot describe the behavior of dσ/dt near the threshold.
• The exchange of neutral pseudoscalar mesons (π 0 , η) provides a certain contribution to dσ/dt near the threshold but it is not enough to explain the threshold behavior of dσ/dt [6]. Moreover, π 0 and η exchanges wrongly predict the spin-density observables and, in particular, ρ 1 1−1 matrix element (see Appendix for its definition).
• Usual vector meson-exchanges such as ρ and ω are forbidden due to their negative charge conjugations (C = −1). Otherwise, the charge conjugation symmetry will be broken.
• Vector meson-exchanges with exotic quantum number such as I(J P C ) = 1(1 −+ ) are allowed but those vector mesons are not much known experimentally. Moreover, as for the experimental data from the deuteron target, exchange of isoscalar mesons is more plausible. On the other hand, there is no experimental evidence for isoscalar hybrid-exotic mesons [7].
• The contribution of scalar mesons such as σ and f 0 are negligibly small for dσ/dt [6].
Understanding this present theoretical and experimental situation in φ photoproduction, Ozaki et al. [8] proposed a coupled-channel effects based on the K-matrix formalism. They considered the γN → KΛ * (1520) and KΛ * → φN kernels [9] in the coupled-channel formalism in addition to γN → φN and φN → φN . It is a very plausible idea, since the threshold energy for the KΛ * is quite close to that for the bump-like structure (E γ ≈ 2.3 GeV), the Λ * (1520) resonance may influence φ photoproduction. Moreover, the γp → KΛ * (1520) reaction can be regarded as a subreaction for the γp → KKp process together with the γp → φp one in Ref. [9]. In addition, a possible nucleon resonance (J P = 1/2 − ) with large ss content was also taken into account. Interestingly, the coupled-channel effects were shown to be not enough to explain the bump-like structure E γ ≈ 2.3 GeV. On the other hand, the bump-like structure was described by their possible N * resonance and was interpreted as a destructive interference arising from the N * resonance [10,11].
In the present work, we want to scrutinize in detail the nontrivial hadronic contributions arising from hadronic box diagrams in addition to Pomeron and pseudoscalar meson exchanges. Extending the idea of Ref. [8], we consider seven possible box diagrams with intermdiate ρN , ωN , σN , πN , KΛ(1116), K * Λ(1116), and KΛ(1520) states. However, it is quite complicated to compute these box diagrams explicitly, so that we use the Landau-Cutkosky rule [12,13], which yields the imaginary part of the box diagrams by its discontinuity across the branch cut. Though their real part may contribute to the transition amplitude, we will show that the imaginary part already illuminates the coupledchannel effects on the production mechanism of γp → φp near the threshold. The parameters such as the coupling constants and cut-off masses of the form factors will be fixed by describing the corresponding processes and by using experimental and empirical data. Yet unknown parameters are varied as compared to the present experimental data. In addition, we tune the strength of the Pomeron amplitude near the threshold region, where the hadronic contribution seems more significant. It is a legitimate procedure, since the Pomeron gets more important as the energy increases. Thus, we determine the threshold parameter in such a way that the Pomeron exchange becomes effective in the higher energy region. We did not consider any N * resonance, since we do not have much information on them above the φN threshold [7]. We will show that the coupled-channel effects are indeed essential in explaining the recent LEPS data, which is the different conclusion from Ref. [8].
The present paper is organized as follows. In Section II, we explain the basic formalism. We show how to compute the box diagrams mentioned above. In Section III, we present the numerical results such as the energy dependence of the forward cross sections, the angular distributions, and the spin observables. We also discuss how the KΛ * (1520) channel can explain the bump-like structure together with the Pomeron exchange tuned. We discuss in detail the spin-density matrix elements for φ-photoproduction. The final Section is devoted to summary and outlook. In the Appendix, we present the definition of the spin-density matrix elements for reference.

II. GENERAL FORMALISM
In the present work, we will employ the effective Lagrangian approach in addition to the Pomeron-exchange. In  (1520), among which the last one was already considered in Ref. [8]. From now on, we will simply define the ρN box diagram as that with intermediate ρ and N states, and so on. We also define the 4-momenta of the incoming photon, outgoing φ, the initial (target) proton and the final (recoil) proton as k 1 and k 2 , p 1 and p 2 , respectively. In the center of mass (CM) frame, these variables are written as k 1 = (k, k), k 2 = (E φ , p), , and E p = m 2 p + |p| 2 , respectively.

A. Pomeron exchange
The amplitude of the Pomeron-exchange [14][15][16] is given by where φ and γ are the polarization vectors of the φ meson and photon. M µν is where the transition operator Γ µν is defined as withp = (p 1 + p 2 )/2. Note that the Pomeron amplitude preserves gauge invariance k ν 1 M µν = 0. The corresponding invariant amplitude M (s, t) in Eq.(2) is written as where s = (k 1 + p 1 ) 2 and t = (k 1 − k 2 ) 2 . F N (t) is the isoscalar form factor of the nucleon, whereas F φ (t) is the form factor for the photon-φ meson-Pomeron vertex. They are parameterized, respectively, as The Pomeron trajectory α p (p) = 1.08 + 0.25 t in Eq.(4) is determined from hadron elastic scattering in the highenergy region. The prefactor C p in Eq.(4) governs the overall strength of the amplitude and s th determines the starting energy at which the Pomeron-exchange comes into play. We will discuss the determination of these two parameters in Section III.
B. π-and η-exchanges To calculate pseudoscalar meson (ϕ = π 0 , η) exchange in the t channel, we introduce the following effective Lagrangians: where φ ν , A β , and N denote the φ vector meson, photon, and nucleon fields, respectively. m φ and M N stand for the φ meson and nucleon masses respectively. e represents the electric charge. The t-channel amplitude then takes the following form: where r is the four momentum of an exchanged pseudoscalar meson. We introduce the monopole-type form factors for each vertex F ϕN N (t) and F φγϕ defined as The coupling constants and the cutoff masses for the pseudoscalar-exchange, we follow Ref. [6]: g πN N = 13.26, g ηN N = 3.527 for the πN N and ηN N coupling constants, respectively. The cutoff masses are taken to be Λ πN N = 0.7 GeV and Λ ηN N = 1 GeV. Though these values are somewhat different from the phenomenological nucleon-nucleon potentials [17,18], the effects of the pseudoscalar meson-exchanges on φ photoproduction are rather small. Thus, we will take the values given above typically used in φ photoproduction. Those of the coupling constants for the φγϕ vertices are determined by using the radiative decays of the φ meson to π and η. Using the data from the Particle Data Group (PDG) [7], one can find g φγπ = −0.141 amd g φγη = −0.707. The negative signs of these coupling constants were determined by the phase conventions in SU(3) symmetry as well as by π photoproduction [6]. We choose the cut-off masses for the φγπ and φγη form factors as follows: Λ φγπ = 0.77 GeV and Λ φγη = 0.9 GeV, respectively.
In addition to the Pomeron-and pseudoscalar meson-exchanges, we include the seven different box diagrams: ρN , ωN , σN , πN , KΛ(1116), K * Λ(1116), and KΛ(1520). Since the KΛ(1520) box diagram is the most significant one among several possibie box diagram in describing φ photoproduction, we first discuss the K + Λ(1520) one and then deal with all other box diagrams in the next subsection. In The γN → K + Λ(1520) process was investigated within an effective Lagrangian method in Ref. [9] of which the results were in good agreement with the experimental data. Thus, we will take the formalism developed in Ref. [9] so that we may take into account the KΛ(1116) coupled-channel effects more realistically.
The effective Lagrangians for γN → K + Λ(1520) are written as where K and Λ * µ denote the K meson and Λ(1520) fields. For Λ(1520), we utilize the Rarita-Schwinger formalism. M K is the kaon mass. The KN Λ * coupling constant is taken from Ref. [9], since we use the amplitude derived in it. The φKK coupling constant can be determined from the experimental data for the decay width Γ φ→KK . On the other hand, g φN N is not much known experimentally. Recent experiments measuring the strange vector form factors imply that the strange quark gives almost no contribution to the nucleon electromagnetic (EM) form factors [1]. One can deduce from this experimental fact that the φN N coupling constant should be very small. In Ref. [20], the φN N was estimated by using a microscopic hadronic model with πρ continuum: g φN N = ±0.25 and κ φ = 0.2, which are compatible with the recent data for the strange vector form factors. Thus, we will take these values in the present work. However, note that the s-channel contribution with the φN N vertex is almost negligible. In Table I, the relevant strong coupling constants and anomalous magnetic moments are listed. Based on the effective Lagrangians given in Eq.(9), we can write down the amplitude for the K + Λ * (1520) box diagram. It contains both real and imaginary parts. The real part is divergent, which is also the case for other box diagrams and the rigorous calculation is rather involved. Thus we consider that the real part can be taken into account effectively by the reenormalization of various coupling constants, and calculate only the imaginary part explicitly. The reasoning behind is similar to the concept of K-matrix formalism for the S-matrix. Physically, the imaginary part corresponds to rescattering and is obtained by the Landau-Cutkosky rule, Ref. [12,13].
Having computed the Lorentz-invariant phase space volume factors, we obtain the imaginary part of the amplitude as where r is the magnitude of the K + momentum. This imaginary part of the amplitude is schematically drawn in Fig. 2. The shaded ellipse in the left-hand side represents the invariant amplitude for γp → K + Λ * , which is basically the same as that of Ref. [9] except for different form factors as will be explained later. It consists of three different types of the Feynman diagrams as shown below the left dashed arrow. On the other hand, the right ellipse stands for the K + Λ * → φp process that contains the diagrams below the right arrow, generically. Note that we use a similar method as in Ref. [8] but we choose the different form factors and parameters. The corresponding invariant amplitudes M L (γp → K + Λ * ) and M R (K + Λ * → φp) with the form factors are defined as follows: where M L,s (M R,s ), M L,t (M R,t ), and M L,c (M R,c ) represent the s-channel, the t-channel, and the contact-term contributions to the γp → K + Λ * (K + Λ * → φp) process, respectively: We introduce the form factors F R (s, t) and F L (s, t) for M R and M L , respectively, in particular, in a gauge-invariant manner for the γp → K + Λ * rescattering: where the cut-off masses Λ i and powers n i are fitted to the experimental data for the γp → K + Λ * and γp → φp, which are listed in Table II. In Fig. 3, we draw the numerical result of the total cross section for γp → K + Λ * in comparison with the experimental data taken from Ref. [22]. It is in good agreement with the data.

D. All other box diagrams
In the same manner as done for the K + Λ * box diagram, we consider the six intermediate box diagrams as shown in Fig.4, i.e. the ρN , ωN , σN , πN , KΛ(1116), and K * Λ(1116) box diagrams. ρ photoproduction has been studied theoretically [11,[23][24][25] in which the contributions of the t-channel π-and σ-exchanges were considered and σexchange was found to be the dominant one, since it selects the isovector part of the EM current. Thus, we take into account the ρp box diagram with the σ-and π-exchanges in the t-channel, as shown in the first diagram of Fig. 4. We  will show later in Fig. 5 that indeed the σ-exchange describes qualitatively well the γp → ρp reaction. In Ref. [23] ω photoproduction was also discussed within the same framework. In contrast to the γp → ρp reaction, the π-exchange appeared to be dominant, since it picks up the isoscalar part of the EM current. Correspondingly, we consider the ωp box contribution as in the second diagram of Fig. 4, where ω is produced by the one pion exchange. The σp and πp box diagrams are obtained by reversing the ρp and ωp box diagrams. The γp → KΛ(1116) and γp → K * Λ(1116) reactions were measured by several experimental collaborations [26][27][28][29][30][31] and were investigated theoretically [32][33][34][35][36].
While we consider all the relevant diagrams for the KΛ * (1520) box contribution because of its significance, we will take into account only the K-exchange diagrams in the t-channel for the KΛ and K * Λ box diagrams, since these two box diagrams turn out to have tiny effects on φ photoproduction. The relevant effective Lagrangians for these box diagrams are given as follows: where the coupling constants and the cut-off masses are listed in Table III. The invariant amplitudes for these box where the subscripts 1, · · · 6 correspond to the box diagrams appearing in Fig. 4 in order. The other subscripts L and R denote the γp → M B and M B → φp parts, respectively. In Figs. 5 and 6 we draw the results of the total cross sections for the γp → ρp and γp → ωp reactions, respectively. The results are qualitatively in agreement with the experimental data.

III. NUMERICAL RESULTS AND DISCUSSION
We are now in a position to discuss the numerical results for φ photoproduction. We start with the differential cross section at the forward angle dσ/dt(θ = 0) as a function of the photon energy E γ in the laboratory frame. The parameters are determined in the following manner. Since the the Pomeron-exchange in the low-energy region is not much understood, we fit the parameter for the overall strength C p and that for the threshold s th in Eq.(4) in such a way that the Pomeron-exchange reproduces the high energy behavior of the differential cross section: C p = 8 GeV −1 and s th = 3.83 GeV 2 . On the other hand, We fix the cut-off parameters for the KΛ * (1520) box diagrams to describe the E γ dependence of dσ/dt in lower energy region, in particular, to explain the well-known bum-like structure around E γ ≈ 2.3 GeV. The parameters of all other hadronic diagrams are taken from existing references as explained in the previous section. Figure 7 illustrates various contributions to dσ/dt(θ = 0) as a function of the photon energy E γ from the Pomeron-  FIG. 6. Total cross-section of the γp → ωp reaction. The solid curve depicts the present result obtained from the t-channel π-exchange diagram. The closed squares denote the experimental data from Ref. [45] whereas the open circles represent those from Ref. [43]. exchange, the t-channel π-and η exchanges, and seven box diagrams. The solid curve with symbol P draws the contribution of the the Pomeron-exchange to dσ/dt. As expected, it governs E γ dependence in the higher energy region (E γ ≥ 3GeV). Note, however, that the Pomeron does not contribute to dσ/dt below around E γ = 2.3 GeV in the present work. The π-and η-exchanges provide a certain amount of effects on the differential cross section (solid curve with symbol T ). The contribution of the π-and η-exchanges start to increase from the threshold energy and then it decreases very slowly when it reaches approximately 3 GeV. Thus, the effects of the π-and η-exchanges are quite important in the lower E γ energy region up to 3 GeV, where the Pomeron-exchange overtakes the π-and η-exchanges.
Except for the KΛ * (1520) box diagram, all other box contributions turn out to be negligibly small. However, the KΛ * (1520) box diagram plays an essential role in describing the experimental data for dσ/dt in the lower E γ region, in particular, in explaining the bump-like structure near 2.3 GeV. This is very different from the conclusion of Ref. [8], where the KΛ * (1520) seems to be suppressed in the K-matrix formalism. The reason lies in the fact that we have introduced different form factors for the γp → KΛ * and KΛ * → φp reactions. In general, form factors are given as functions of two Mandelstam variables for the box diagrams, i.e. F (s, t), since we have two off-shell particles in the s-channel and other two off-shell particles in the t-channel. However, it is very difficult to preserve the gauge invariance in the presence of the form factors. Thus, we have introduced a type of overall form factors to keep the   gauge invariance in the γp → KΛ * part, as written in Eq. (13). To keep the consistency, we also have included a similar type of the form factors in the KΛ * → φp part. With these form factors considered, we find that the KΛ * box diagram is indeed enhanced as shown in Fig. 7 in comparision with Ref. [8]. The contribution of the KΛ * box diagram increases sharply up to E γ ≈ 2 GeV and then falls off linearly. The result of the KΛ * box diagram indicates that the off-shell effects, which arise from the form factors and the rescattering equation, may come into play.
Considering the fact that the K * Λ threshold energy (E th ≈ 2 GeV) is very close to that of φ photoproduction (E th ≈ 1.96 GeV), one might ask why the contribution of the K * Λ is suppressed. While the KΛ * (1520) channel (E th ≈ 2 GeV) is directly related to φp, since both are the subreactions of the γp → KKp process, the γp → K * Λ reaction is distinguished from those two reactions, because the K * Λ channel is related to γp → πKΛ reaction. Thus, one can qualitatively understand why the contribution of the K * Λ box diagrams is suppressed.
In Fig. 8, the differential cross section as a function of the scattering angle is depicted at E γ = 2 GeV. Since the Pomeron-exchange is suppressed at this photon energy because of s th = 2.3 GeV, we can examine each hadronic contribution to the differential cross section more in detail. Figure 8 clearly shows that the KΛ(1520) box diagram is the most dominant one among the hadronic contributions. Adding all the effects of the box diagrams, we find that the box contributions almost describe the θ dependence. Together with the π-and η-exchanges, the result of the differential cross section is in good agreement with the experimental data [4,44].
The differential cross section as a function the scattering angle are drawn in Fig. 9. The left and right panels correspond to the photon energies E γ = 3 and 3.7 GeV, respectively. As expected, the hadronic contribution is dominant over the Pomeron-exchange at the lower photon energy, while at E γ = 3.7 GeV, the Pomeron governs the γp → φp process. Interestingly, the effects of the box diagrams, in particular, the KΛ * (1520) one, turn out to be larger than those of the π-and η-exchanges, whereas the box diagrams seem to be suppressed at higher photon energies. It implies that the KΛ * (1520) box diagram influences φ photoproduction only in the vicinity of the threshold energy. Figure 10 depicts the results of the differential cross section as a function of t + |t| min with eight different photon energies, where |t| min is the minimum 4-momentum transfer from the incident photon to the φ meson. The results are in good agreement with the experimental data taken from the measurement of the LEPS collaboration [4].
It is of great importance to examine the angular distribution of the φ → K + K − decay in the φ rest frame or in the Gottfried-Jackson (GJ) frame, since it makes the helicity amplitudes accessible to experimental investigation [46,47]. The detailed formalism for the angular distribution of the φ meson decay can be found in Refs. [6,47]. The decay angular distribution of φ photoproduction was measured at SAPHIR/ELSA [48] but the range of the photon energy is too wide. On the other hand, the LEPS collaboration measured the decay angular distribution at forward angles   (−0.2 < t + |t| min ) in two different energy regions: 1.97 < E γ < 2.17 GeV and 2.17 < E γ < 2.37 GeV [4], which are related to the energy around the local maximum of the cross section and that above the local maximum, respectively. Therefore, we have computed the decay angular distributions at two photon energies, i.e. E γ = 2.07 GeV and E γ = 2.27 GeV, which correspond to the center values of the given ranges of E γ in the LEPS experiment. The one-dimensional decay angular distributions W (cos θ), W (φ − Φ), W (φ) are presented in Fig. 11, which are expressed respectivley as where θ and φ denote the polar and azimuthal angles of the decay particle K + in the GJ frame. Φ represents the azimuthal angle of the photon polarization in the center-of-mass frame. P γ stands for the degree of polarization of the photon beam. ρ 1 1−1 , ∆ 1−1 , and ρ are defined as The expressions for the spin-density matrix elements ρ α λλ with the helicities λ and λ for the φ meson can be found in Appendix A .
The panel (a) of Fig. 11 draws the one-dimensional decay polar-angle distributions W (cos θ). As pointed out by Refs. [4,5], the decay distribution behaves approximately as ∼ (3/4) sin 2 Ψ, which indicates that the helicityconserving processes are dominant as shown in Eq. (16). It means that t-exchange particles with unnatural parity at the tree level do not contribute to W (cos θ). As will be discussed later, ρ 0 00 from the π-and η-exchanges, which  Fig. 11 shows the results of W (φ − Φ), which are in good agreement with the LEPS data, whereas the panel (c) depicts those of W (φ), W (φ + Φ), and W (Φ), respectively, which deviate from the data. In fact, the data show somewhat irregular behavior which does not seem to be easily reproduced.  As shown in Fig. 11, the decay angular distributions shed light on the production mechanism of the φ meson, since they make it possible to get access experimentally to the spin-density matrix elements, or the helicity amplitudes of φ photoproduction. It has important physical implications, because even though some diagrams seem to contribute negligibly to the cross sections, they might have definite effects on the decay angular distributions. In Table IV, The contributions of each box diagram to the various spin-density matrix elements at E γ = 2 GeV are listed. As expected, the π-and η-exchanges contribute only to ρ 1 1−1 . The hadronic box diagrams mainly contribute to ρ 0 00 and ρ 1 1−1 and are almost negligible to other components. Interestingly, the ρp box diagram is the dominant one for ρ 0 00 , even though it provides much smaller effects on the differential cross section than the KΛ * (1520) one.
Rcently, the LEPS experiment measured the spin-density matrix elements for γp → φp [5] in the range of E γ = 1.6−2.4 GeV in which the Pomeron-exchange does not play any important role, in particular, in the present approach.  Thus, we can examine the hadronic contributions to each spin-density matrix elements. Figure 12 illustrates the various spin-density matrix elements, compared with the LEPS data. Since the experimental data are given in the finite range of E γ , we just take the three center values corresponding to the ranges, i.e. E γ = 1.87, 2.07, 2.27 GeV. The hadronic diagrams considered in the present work describe quantitatively Reρ 0 10 , ρ 0 1−1 and ρ 1 11 . However, the deviations are found in other spin-density matrix elements as t − |t| min increases.

IV. SUMMARY AND OUTLOOK
In the present work, we aimed at investigating the coupled-channel effects arising from the hadronic intermediate box diagrams to φ photoproduction near the threshold region in addition to the Pomeron-, π-, and η-exchanges. In particular, the bump-like structure near E γ ≈ 2.3 GeV, which was reported by the LEPS experiment [4], sheds light on the production mechanism of the φ meson in the vicinity of the threshold, since the Pomeron-exchange was shown to be not enough to explain this peculiar structure of φ photoproduction. Thus, we studied in detail the effects of the seven different box diagrams such as ρN , ωN , σN , πN , KΛ(1116), K * Λ(1116), and KΛ(1520). In order to take into account the rescattering terms, we employed the Landau-Cutkosky rule in dealing with these box diagrams.
Since it turned out that the KΛ * (1520) box diagram played a dominant role among hadronic contributions in the lower-energy region, we scrutinized its contribution to φ photoproduction. We introduced the form factors depending on both the s and t Mandelstam variables in such a way that the total cross section of the γp → KΛ * (1520) reaction was well reproduced. All other box diagrams were constructed by utilizing the previous theoretical works and by reproducing the corresponding experimental data when they were available. We examined each contribution carefully by computing the differential cross section of φ photoproduction. While the KΛ * box diagram was found to be the most dominant near the 2 GeV, all other box diagrams turned out to be very small. The results were in good agreement with the LEPS data including the bump-like structure. We also computed the differential cross section as a function of t + |t| min and found it to be in good agreement with the experimental data.
We investigated the contributions of hadronic box diagrams to the decay angular distributions. While the onedimensional angular distributions W (cos θ) and W (φ − Φ) were in good agreement with the experimental data, other three angular distributions seemed to deviate from the LEPS experimental data. We also examined the various spin-  density matrix elements, which were measured recently by the LEPS collaboration. We found that the hadronic box diagrams describe the experimental data for Reρ 0 10 , ρ 0 1−1 and ρ 1 11 were well reproduced. While the present results explain near t − |t| min ≈ 0 relatively well for other spin-density matrix elements, they deviated from the expeimental data as t − |t| min ≈ 0 increased.
As shown in the present work, the intermediate box diagrams, in particular, the KΛ * (1520) one, play crucial roles in explaining the cross sections of the γp → φp reaction in the vicinity of the threshold. Other box diagrams also provided certain effects on the part of the spin-density matrix elements. We have considered in this work only the imaginary part of the transition amplitudes of the box diagrams based on the Landau-Cutkosky rule. However, the results of the spin-density matrix elements already indicate that we should carry out a theoretical analysis of φ photoproduction more systematically and quantitatively. Thus, we need to investigate a full coupled-channel formalism and to solve rescattering equations with the real parts of the box diagrams fully taken into account. Another interesting and important problem is to extend our approach to the neutron target, since some of considered amplitudes are isospindependent. The corresponding works are under way.
where λ γ , λ i , and λ f represent the helicities for the photon and the initial and final nucleons, respectively, whereas λ and λ denote those for the φ meson. The normalization factor N is defined as N = |T λ f ,λ;λi,λγ | 2 . (A2)