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Eta meson production in nucleon-nucleon collisions within an effective Lagrangian model

We investigate the pp -->pp eta and pn -->pn eta reactions within an effective Lagrangian model for laboratory kinetic energies ranging from very close to eta meson production threshold to about 10 GeV. Production amplitudes include contributions from the mechanism of excitation, propagation and decay of N*(1535), N*(1650), and N*(1710) baryonic resonances. The initial interaction between two incoming nucleons is modeled by the exchange of pi, rho, omega and sigma mesons where the vertex parameters are taken to be the same as those used in the previous applications of this model. Parameters of the resonance vertices also have been taken from our earlier studies wherever applicable. Calculations have been done for total as well as differential eta production cross sections. To describe the data for energies closer to the production threshold final state interactions among the out-going particles have been included by means of a generalized Watson-Migdal method. Terms corresponding to the excitation of N*(1535) resonance and pion exchange process dominate the cross sections. With a single set of vertex parameters our model is able to describe the available data well on total cross sections for beam energies ranging from close to threshold to upto 10 GeV.


I. INTRODUCTION
The low energy behavior of the quantum chromodynamics (QCD) is not accessible to the perturbative approaches; the lattice gauge theory [1] is the ideal tool for this purpose.
Despite enormous computational power necessary for the numerical realization, lattice QCD calculations have started, very recently, to describe masses and other constants of the baryonic ground as well as excited states [2]. Experimentally, the determination of baryonic resonance properties proceeds indirectly by exciting the nucleon with the help of a hadronic or electromagnetic probe and performing measurements for their decay products (mesons and nucleons). The reliable extraction of nucleon resonance properties from such experiments is a major issue of the hadron physics.
In recent years, important advances have been made in the experimental investigation of meson production reactions in nucleon-nucleon (NN) collisions, particularly at beam energies very close to respective production thresholds [3,4,5,6,7,8,9,10,11,12,13,14,15]. Low incident energies also provide the opportunity to investigate the meson-nucleon interactions through these reactions since in this energy regime the final state interactions among the outgoing particles affect strongly the meson production cross sections.
The η meson, which is the next lightest nonstrange member in the meson mass spectrum, has been a subject of considerable interest. It has been thought of as a probe for the ss component in the nucleon wave function [16]. There is also interest in measuring the rare decays of η which could provide a new rigorous test of the standard model [17] or even of the physics beyond this. The nucleon resonance N * (1535) [S 11 (1535)] with spin 1 2 , isospin 1 2 , and odd parity, has a remarkably large ηN branching ratio. It lies very close to the threshold of the NN → NNη reaction and contributes to the amplitude of this reaction even at the threshold. Therefore, the study of η-meson production in NN collisions at the near threshold beam energies provides the unique opportunity to investigate the properties of N * (1535) which have been the subject of some debate recently (see, e.g., [18]). The attractive nature of the η-nucleon interaction may lead to the formation of bound (quasibound) η-nucleus states (see, e.g., [19,20,21,22,23]). This subject has been a topic of intense discussion at a recent work shop [24].
Production of η meson in heavy ion collisions also is of great interest. Due to the high threshold of the elementary production reaction, η mesons in such collisions are produced only by very energetic nucleons and reflect therefore, the tails of the nucleon momentum distributions as they arise in a high density and high temperature phase of the collision [25].
The elementary NNη production cross sections are a crucial ingredient in the transport model studies of the η-meson production in the nucleus-nucleus collisions.
Since lattice QCD calculations are still far from being amenable to the low and intermediate energy scattering and reaction processes, the effective methods are mostly used for the description of the dynamics of the meson production reactions in hadronic collisions. These approaches introduce the baryonic resonance states explicitly in their framework and try to extract their properties by comparing the theoretical results with experimental observables.
The main objective of this paper is to investigate the η meson production in NN collisions in the framework of an effective Lagrangian Model (ELM) which has been used earlier rather successfully to describe the pion [33,34], associate kaon [35], and dilepton [36] production data in such collisions. The motivation here is to see as to how far can one explain the recently measured data on total and differential cross sections of pp → ppη [11,12,13,14,15] and the pn → pnη [10] reactions within this model using same sets of entrance channel and resonance channel parameters (for those resonances that appeared in earlier applications [35,36]).
Within the ELM, initial interaction between two incoming nucleons is modeled by an effective Lagrangian which is based on the exchange of π, ρ, ω and σ mesons. The coupling constants at the nucleon-nucleon-meson vertices are determined by directly fitting the T matrices of the NN scattering in the relevant energy region [37]. The effective Lagrangian uses the pseudovector (PV) coupling for the nucleon-nucleon-pion vertex, as it is consistent with the chiral symmetry requirement of the QCD [38,39,40] and also it leads to negligible contributions from the negative energy states ("pair suppression phenomena") [41]. The η meson production proceeds via excitation of N * (1535), N * (1650) and N * (1710  [29,30,32]. In applications of the ELM to describe the near threshold meson production reactions in NN collisions [34,35], the FSI effects were included within the Watson-Midgal theory [42] which is based on the assumption that the FSI effects are strong in relation to the production process and that they occur attractively between only one particular pair of out-going particles. In Ref. [35], this method was used somehow arbitrarily for all the 3 outgoing pairs of particles. However, Watson's method as such is not applicable, in strict sense, for those cases where the attraction between outgoing particles is not so pronounced or where interaction between more than one pair is to be included in calculations. In this paper, we employ a generalized Watson method in which three-body states are treated by splitting the total interaction into pairwise net interactions which leads to a series decomposition of the net scattering among all the particles in terms of separate total scattering between pairs of particles (see, e.g. [43]). However, the three-body interactions are neglected. In view of the arguments presented in Refs. [32,44] in favor of using the three-body scattering theory to describe the NNη process, it would be interesting to see to what extent this generalized method is able to explain the NNη production data.
In the next section, we present a brief description of the ELM where we describe the main ingredients of the theory and give all the input parameters used in our calculations. The generalized Watson method of FSI effects is also described in a subsection here. The results of calculations and their discussions are presented in section III. Summary and conclusions of our work are given in section IV.
At each interaction vertex, the following form factor is introduced where q i and m i are the four momentum and mass of the ith exchanged meson and λ i is the corresponding cut-off parameter. The latter governs the range of suppression of the contributions of high momenta which is done via the form factor. Since NN scattering cross sections decrease gradually with the beam energy (beyond certain value), and since we fit the elastic T matrix directly, the coupling constants are expected to be energy dependent.
Therefore, while keeping the cut-offs λ i [in Eq. (7)] energy independent, we take energy dependent meson-nucleon coupling constants of the following form where s is the square of the total CM energy. The parameters g 0 , ℓ, and λ were determined by fitting to the T matrix of the relevant proton-proton and proton-neutron scattering data at the beam energies in the range of 800 MeV to 4.00 GeV [37]. This procedure also fixes signs  . We would like to remark that the same parameters were also used to describe the initial NN interaction in the calculations reported in Refs. [33,35,36]. This ensures that the elastic NN elastic scattering channel remains the same in the description of various inelastic processes within this model.
The main criterion for choosing the meson exchanges as discussed above is to describe the NN scattering in the relevant beam energy region. We have left out the η-meson exchange in our description of the NN interaction since due to the psuedoscalar (PS) nature of its coupling, the contribution of the exchange terms of this particle (having a mass much larger than that of the psuedoscalar pion) is expected to be very small. Furthermore, as confirmed by several studies (see, e.g., [48,49,50,51]), the coupling constant for the NNη vertex is rather small.

B. Resonance-nucleon-meson vertex
As the η meson has zero isospin, only isospin-1 2 nucleon resonances are allowed. Below 2 GeV center of mass (c.m.) energy, N * (1535) has a prominent decay branching ratio of 40-60% into the Nη channel (see, e.g., the latest review of the particle data group [52]). On the other hand, N * (1650) and N * (1710) resonances have a small but non-negligible decay branching ratios of 3 -10% and 6 ± 1%, respectively, to this channel. In comparison to these, the branching ratio for the decay of N * (1520) resonance to Nη channel is negligibly small and we have not included it in out description. In several previous studies of the NN → NNη reaction, contributions from only N * (1535) resonance have been considered.
Since all the three resonances can couple to the meson-nucleon channels considered in the previous section, we require the effective Lagrangians for all the four resonance-nucleonmeson vertices corresponding to all the included resonances. At the spin-1 2 resonances -N − π (η) vertices, we have the choice of PS or PV couplings. The corresponding effective Lagrangians can be written as [35,48,53,54] where M = (m N * ± m N ), with upper sign for even parity and lower sign for odd parity resonance. The operators Γ, Γ µ , are given by, for resonances of even and odd parities, respectively. We have performed calculations with both of these couplings. The effective Lagrangians for the coupling of resonances to other mesons are, where operators Γ ′ and Γ µν are, for resonances of even and odd parities, respectively.
We assume that the off-shell dependence of the NN * vertices are determined solely by multiplying the vertex constants by form factors. Similar to Refs. [46,55], we use the following form factors for N * N-meson vertices The resonance couplings are determined from the experimentally observed branching ratios for the decay of the resonances to the corresponding channels. Since the resonances considered in this study have no known branching ratios for the decay into the Nω channel, we determine the coupling constants for the N * Nω vertices by the strict vector meson dominance (VMD) hypothesis [56], which is based essentially on the assumption that the coupling of photons to hadrons takes place through a vector meson. For details of these calculations we refer to [35].
The resonance properties and the values of various coupling constants are given in Table   II. Value of the cut-off parameter (λ N * i ) is taken to be 1.2 GeV for all the vertices, which is the same as that used in Refs. [36,55]. Fixing λ N * i to one value minimizes the number of free parameters.
It should however, be stresses that the branching ratios determine only the square of the corresponding coupling constants, thus their signs remain uncertain in this method.
Predictions from independent studies are used as a guide to fix these signs. We have followed here the results of Ref. [46] for this purpose. The propagators for various mesons and nucleon resonances in the calculation of the amplitudes have been taken to be the same as those discussed in [35,36].

C. Amplitudes and cross sections
After having established the effective Lagrangians, coupling constants and form of the propagators, the amplitudes for various diagrams associated with the NN → NNη reaction can be calculated in straight forward manner by following the well known Feynman rules.
The isospin part is treated separately. This gives rise to a constant factor for each graph, which is shown in Table III. It should be stressed here that signs of various amplitudes are fixed, by those of the effective Lagrangians, coupling constants and propagators as described above. These signs are not allowed to change anywhere in the calculations.
The general formula for the invariant cross section of the NN → NNη reaction is written where T f i represents the total amplitude, P i and P f the sum of all the momenta in the initial and final states, respectively, and p a the momenta of the three particles in the final state.
The corresponding cross sections in the laboratory or center of mass systems can be written from this equation by imposing the relevant conditions.

D. Final state interaction
For describing the data for the NN → NNη reaction at beam energies very close to the η production threshold, consideration of the final state interaction (FSI) among the three out going particles is important. We follow here an approximate scheme in line with the Watson-Migdal theory of FSI [42]. In this approach the energy dependence of the cross section due to FSI is separated from that of the primary production amplitude. This is based on the assumption that the reaction takes place over a small region of space, a condition fulfilled rather well in near threshold reactions involving heavy mesons. This method has been extensively applied to study the low momentum behavior of the pion [34,57,58], η meson [59,60,61], associated hyperon [35,62] and φ-meson [63] production in NN collisions.
The total amplitude is written as where T 0 (NN → NNη) is the primary production amplitude, while T f f describes the rescattering among the final particles which goes to unity in the limit of no FSI. The factorization of the total amplitude into those of the FSI and primary production (Eq. (22)), enables one to pursue the diagrammatic approach for the latter within an effective Lagrangian model and investigate the role of various meson exchanges and resonances in describing the reaction.
Watson's original method [42] was developed for those cases where the final state interaction is strong in relation to the production process and where it is confined only to one particular pair of particles (mostly among nucleons in case of nucleon-nucleon-meson final states). On the other hand, in certain cases it may be necessary to include FSI among all the three outgoing particles since even if the meson-baryon interactions are weak, they can still be influential through interference. In Ref. [35], the T matrix T f f was written (without providing any proof) as a coherent sum of the transition matrices describing the final state interaction among all three two-body subsystems of the final state nucleon-nucleon-meson system. We show here that this result can be obtained (in a slightly different form) by following the technique of multiple FSI as discussed in Ref. [43]. To that end, we first try to get a representation for the total amplitude T f i in terms of an expression similar to that given by Eq. (22) where the FSI amplitude T f f is appropriately constructed.
To introduce the treatment of the FSI for the three-particle system, we assume that the three-particle final states can be represented by additive potentials of the form U = With this assumption, the total amplitude T f i can be written as a iteration series in terms of the production amplitude T 0 (defined in Eq. (22)) and the three-body final state pair interaction amplitudes T ij ≡ T k (see Appendix A for details).
where the final state interaction transition matrices T k are as defined in Appendix A.
It may be noted that in this form, each particle lies on the mass shell and three momentum is conserved in the intermediate processes. Introducing the total and relative momenta, and evaluating the integral in the barycentric frame of 1 and 2, we obtain,  For further evaluation of the integral, we make a partial wave decomposition of the amplitude and for each partial wave we rewrite it as Now we make the assumption that the ratio within the square brackets in the integrand of Eq. (26) is constant upto a certain energy ξ c and zero thereafter. Extending this procedure to all the three interacting pairs, we get in the low energy and s-wave limit In Eq. (27) T f f is defined as where It is obvious that Eq. (28) allows interference among the final state scattering amplitudes.
We further note that apart from the factor c ij , this equation is similar to that used in Ref. [35].
The derivation of Eq. (27) where q η is the momentum in the ηN center of mass (c.m.) system. Seven sets of values of the parameters a, r 0 and s are given in Ref. [65]. We found that the best description of the data (within the realm of our overall input parameter sets given in Tables I and   II Ref. [66]. However, we note that in the theoretical description of the pp → ppη reaction as reported in Refs [27,67], the value of a R was similar to that used by us. A smaller a R is also consistent with that extracted in Ref. [54] in an effective Lagrangian model analysis of the meson-nucleon scattering. Furthermore, it was noted in Ref. [68] that within a three-body model, the shapes of the np → ηd cross sections can be explained over a wide energy range only with a a R around 0.42 fm. A smaller value of a R is also consistent with the Jülich model [69]. The FSI amplitude T N N has been calculated by following the Jost function method using the effective range expansion (ERE) of the NN phase shifts as discussed in Refs. [34,35,42].
In case of the proton-proton sub-state, the Coulomb modified effective range expansion has been used [70]. The effective range parameters for the NN channel have been taken to be the same as those used in Ref. [34].
It should however, be mentioned here that the use of on-shell forms to describe the FSI T matrices T if has been criticized by some authors. It has been argued in Refs. [71,72] that the absolute magnitudes of the cross sections obtained by such a procedure could be uncertain because of the off shell effects. Even the Jost function method has been shown [73] to produce inadequate results in an study where the scattering length parameters for the Λ − p final state has been extracted from the pp → pΛK + data. In the next section we have examined the role of the off shell effects in the NN sub-state in some more details.

III. RESULTS AND DISCUSSIONS
The major aim of this paper is to check the suitability of our model and the vertex parameters appearing therein to describe the η production cross sections over a wide range of beam energies. We have therefore, applied our approach to describe the total cross sections for the pp → ppη reaction for beam energies ranging from near threshold to upto 10 GeV, and for the pn → pnη reaction for beam energies from threshold to upto 1.6 GeV. These are the energy regimes in which experimental data are available for the two reactions. We have also used this method to describe one set of the exclusive data, namely the η angular distributions for the former reaction. Calculations have been performed by using both the PS and PV couplings for N * Nπ and N * Nη vertices. We note that the cross sections remain almost unchanged by switching from one type of coupling to another. In all the calculations shown below the coupling constants and cut off parameters for various vertices were the same as those discussed in section II.
A cleaner check of the vertex parameters used in calculating the amplitude T 0 (NN → NNη), is provided by the comparison of our calculations with the data for beam momenta above 3 GeV/c, since at these energies FSI effects are most likely to be unimportant. In

reaction.
We see that the ELM is able to describe both the energy dependence and the absolute magnitudes of the experimental cross sections for the η meson production in both pp and pn channels for excess energies > 15 MeV. However, it underpredicts the pp channel data for ǫ values below 15 MeV. Such a trend has also been seen in calculations presented in Refs. [29,30] where the η meson production in NN collisions has been investigated within a relativistic meson exchange model including the initial state interactions and FSI only in the NN sub-system. These authors have attributed the near threshold underestimation of the experimental total cross section to the non-inclusion of the ηp FSI in their model.
Since the ingredients of the primary production amplitude of our model have already Ref. [61] to investigate the effects of using off shell pp FSI on the near threshold η production cross sections. In Fig. 6, we have compared the results for the total cross sections for the pp → ppη reaction obtained by including FSI in pp substate calculated within Jost function technique and that obtained with the method described in Ref. [61] which includes off-shell effects. In the results presented in this figure, no ηp FSI has been considered. We note that the off shell effects in the pp FSI do increase the cross section for ǫ < 60 MeV. However, it is not enough to explain the underprediction of the experimental data by our theory at smaller energies. One needs to have a better understanding of the ηN scattering amplitude.
Further improvement may come by including the three-body terms in the expansion of the scattering amplitude given by Eq. (23). It has been shown in Ref. [32] that NNη FSI effects calculated within a three-body scattering theory leads to enhanced cross sections at very low values of ǫ.
It should be remarked here that the differences in the cross sections of pn → pnη and pp → ppη reactions are not only due to different isospin factors, but also due to differences in the FSI effects. The low energy scattering parameters between pp and pn cases are different; the latter involves also a triplet spin state together with the singlet one. A crucial difference between them is the Coulomb interaction. This is not included in the three-body model calculations of the pp → ppη reaction reported in Ref. [32]. Inclusion of this term is likely to reduce the cross section for beam energies very close to the threshold.

In Figs 7 and 8 we show the individual contributions of various nucleon resonances to
the total cross sections of the pp → ppη and pn → pnη reactions, respectively, at the near threshold beam energies. Similar to the situation at higher beam energies, the cross sections are dominated by the N * (1535) resonance excitation. Since N * (1535) is the lowest energy baryonic resonance having an appreciable branching ratio for the decay into the Nη channel, its dominance in this reaction even at beam energies near the η production threshold is to be expected. The contribution of N * (1650) resonance state is small and that of the N * (1710) resonance is even smaller at these lower beam energies. It should, however be noted that the resonance-resonance interference terms are not negligible. For the pp → ppη reaction, the total cross sections are smaller than those corresponding to the N * (1535) resonance alone.
For the pn → pnη reactions the difference between the two is not visible in Fig. 8.
We found that the inclusion of the amplitudes corresponding to the nucleon intermediate

states (the nucleon bremsstrahlung) made a negligible difference in the results reported in
Figs. 1-8 if the value of the coupling constant g N N η is taken below 3.0. With the largest value of g N N η used in the literature (6.14), the results were affected to the extent of only a few percent. This result is in agreement with that reported in Ref. [29]. It is obvious that due to a considerable amount of uncertainty in the value of g N N η (see, e.g. [48,75,76]), the nucleon excitation amplitudes are quite uncertain (see, e.g. [48,75,76]) and their inclusion makes an insignificant difference to the results reported above.
In Fig. 9, we show contributions of various meson exchanges to the pp → ppη reaction at near threshold beam energies. The dashed, dotted, dashed-dotted and dashed-double dotted curves represent the contributions of π, ρ, σ and ω meson exchanges, respectively. Their coherent sum is shown by the solid line. We see that the one pion exchange graphs make the largest contribution to the reaction in this energy regime. However, a striking feature of this figure is that despite a larger value for the g N * N ρ coupling used in our calculations, the contributions of ρ meson exchange is still much smaller than that of the pion exchange graphs. Hence in contrast to the results reported in Refs. [67,77] the ρ meson exchange terms do not dominate the total NNη production cross sections. To understand this difference we note that while in Refs. [67,77] γ 5 γ µ couplings have been used for the ρNN * vertex, we have taken a γ 5 σ µν coupling for the same which is an extension of the γNN * couplings (due to vector meson dominance reasons [48]). This is also compatible with the forms of the ρNN * couplings used in the literature [46,78,79]. Since the ρ meson exchange amplitudes calculated with the γ 5 σ µν couplings involve delicate cancellations among leading terms, contributions of this exchange diagrams to the η production cross sections are weakened. This is the main reason for differences between our results and those of Refs. [67,77].
In Ref [29], although the form of the ρNN * coupling is the same as ours, relatively lower ρ meson exchange cross sections result from the use of a very small value for the coupling constant g N * N ρ which is based on the lower limit of the branching ratio for the radiative decay of this resonance. Our value for this constant, on the other hand, is calculated from the branching ratio of the decay of this resonance to Nρ channel as quoted in Ref. [52].
We see that the ω meson exchange process contributes insignificantly to the NNη production but the σ meson exchange terms are relatively more important. Larger contributions from the latter has also been seen in other subthreshold reactions analyzed within our model. mentum mismatch involve in the meson production in NN collisions [80,81].
In fig. 10, we investigate the effects of using PS or PV couplings for the N * Nπ(η) vertices.
We notice hardly any difference in the cross sections calculated by the two types of couplings.
Similar results were also observed in Ref [46]. This result is not surprising since the two couplings are constructed in such a way that both are equivalent on the mass shell. Of course, they start having different energy behavior in the far off shell region where resonance contribution is anyway suppressed due to dominance of the corresponding propagator. It is only in the NNπ case that difference in the PS and PV couplings are obvious with a clear preference for the PV coupling in line with the chiral symmetry as discussed earlier.
After establishing the dynamical content of our model vis a vis the description of the total production cross sections, we now turn our attention to more exclusive data. In Fig. 11, it is difficult to conclude if they show a trend different from our calculations. It will be useful to have better quality data with less statistical errors in order to determine if other mechanisms which may show a trend different from ours, are also important. In any case, it is quite unlikely that the ρ meson exchange mechanism which might lead to a distribution different from that seen in our calculations [28] is a dominant mechanism as has already been discussed.

IV. SUMMARY AND CONCLUSIONS
In this paper we investigated the η meson production in proton-proton and protonneutron collisions for beam energies ranging from near threshold to about 10 GeV within an effective Lagrangian model which has been used previously to describe successfully the pion, associated kaon and dilepton production in NN collisions. The interaction between two To describe the data at the near threshold beam energies, the FSI effects among the outgoing particles are included by a generalized Watson-Migdal method which allows to have these effects in all the three two body sub-systems of the out-going channel. This method involves a parameter which has been determined from the constrained that in the limit of FSI in only NN sub-system the result of the usual Watson method is reproduced.
In this paper we presented the analysis of the data for total cross sections for the pp → ppη and pn → pnη reactions and for the η angular distributions in the former reaction. With the same set of vertex parameters, the model is able to provide a good description of the data for the pp → ppη reaction at higher as well as near threshold beam energies except for the excess energies below 15 MeV where our calculations underpredict the experimental data.
The experimental total cross sections of the pn → pnη reaction are also well described by our model. The data for the η angular distributions in the case of pp → ppη reaction are also well reproduced at two beam energies. Imprecise knowledge of the ηN scattering amplitudes and non-inclusion of the three-body effects are the most likely reasons for underestimation of the pp channel data by our model at very low beam energies.
In this study we have not studied the observables related to meson energy dependence e.g., final pp and ηp invariant mass distributions. There are some open theoretical issues concerning the explanation of the corresponding experimental data. While in Ref. [30], the inclusion of contributions of the non-s-wave states in the pp subsystem were found to be essential to explain these data, the three-body effects in the ppη system and not the contribution of the higher partial waves were shown to be crucial for this purpose by the authors of Ref. [32]. At this stage, our theory excludes both these effects. Extension of our model to include these mechanisms is vital before we can make some meaningful contribution towards settlement of this issue.
Within our model, one pion exchange processes make the largest contributions to cross sections in the entire energy regime. Despite our using a large coupling constant for the N * (1535)Nρ vertex, the cross sections of the ρ meson exchange process are still lower than those of the pion exchange mechanism. Therefore, ρ meson exchange being the dominant mechanism of N * (1535) resonance excitation [28] is not supported by our calculations. The individual contributions of the ω meson exchange diagrams are very small every where. On the other hand, the σ exchange terms make relatively larger contributions.
The excitation of the N * (1535) resonance dominates the NNη production at both higher as well as near threshold beam energies. The contributions of N * (1650) and N * (1710) are small in comparison. However, the interference among various resonance contributions is not negligible. Unlike the NNπ vertex where there is a clear preference for the PV coupling, the present reaction does not distinguish between PS and PV couplings at the N * Nη vertex involving spin-1/2 even or odd parity resonances. We point out that the mechanism of the ppη production via preferential excitation of the N * (1535) intermediate baryonic resonance state in one-pion-exchange process has received support recently from an experimental study [82] of the analyzing powers of thep + p → p + p + η reaction.
This work fixes the parameters of the effective Lagrangian model for most of the vertices involve in the eta meson production processes. An interesting further check of this model will be provided by the analysis of the eta photoproduction data on nucleons (see,e.g. [83,84]).
An exciting recent result is that the integrated cross section of the photoproduction of η meson on neutrons shows an additional maximum at center of mass energies around 1.66 GeV. This has recently been explained in terms of the excitation of the N * (1650) and N * (1650) resonance states [85]. Furthermore, the vertex parameters derived by us will also be useful in applications of effective Lagrangian method in describing the production of eta-mesic nuclei in proton and photon induced reactions (see, e.g. [86,87]).

V. ACKNOWLEDGMENTS
The author is thankful to Andrzej Deloff for several useful correspondences regarding the off shell pp FSI method presented in Ref. [ We give here some clarifications and steps leading to the derivation Eq. (23).
The total Hamiltonian of the three-particle system is written as H = H 0 + U where H 0 is the kinetic energy operator of the system and the interaction U is taken as U = U 23 + U 31 + U 12 ≡ U 1 + U 2 + U 3 assuming that the three-particle states interact by means of the additive pair interactions represented by U k . The Green's functions corresponding to H and H 0 are, respectively We shall also need the Hamiltonian describing the two particles interacting while the third one is free, namely, H k = H 0 + U k and the corresponding Green's functions The full three-body transition operator T satisfies the Lipmann-Schwinger equations Substituting (A8) into (A7) one gets after some manipulation, where χ i is the initial plane wave state. The final state interactions of interest are contained in φ (−) f . In applications of relevance to us, the second term of Eq. (A9) would vanish because we assume V can not create real mesons. Eq. (23) can be obtained by using an iteration similar to that given by Eq. (A6) in the remaining (first) term of Eq. (A9). The amplitude T 0 is defined in the same way as the first term of Eq. (A9) with a plane wave final state. The amplitude T k in the second term of Eq. (23) represents the matrix elements of the interaction V k between the plane wave and the scattering states of particle ij. We define V k for the partition k in the same way as U k .