Combined Analysis of \eta Meson Hadro- and Photo-production off Nucleons

Combined Analysis of Meson Hadro- and Photo-production off Nucleons

K. Nakayama nakayama@uga.edu Department of Physics and Astronomy, University of Georgia, Athens, GA 30602, USA Institut für Kernphysik (Theorie), Forschungszentrum Jülich, 52425 Jülich, Germany    Yongseok Oh yohphy@knu.ac.kr Department of Physics, Kyungpook National University, Daegu 702-701, Korea    H. Haberzettl helmut.haberzettl@gwu.edu Center for Nuclear Studies, Department of Physics, The George Washington University, Washington, DC 20052, USA
July 13, 2019
Abstract

The -meson production in photon- and hadron-induced reactions, namely, , , , and , are investigated in a combined analysis in order to learn about the relevant production mechanisms and the possible role of nucleon resonances in these reactions. We consider the nucleonic, mesonic, and nucleon resonance currents constructed within an effective Lagrangian approach and compare the results with the available data for cross sections and spin asymmetries for these reactions. We found that the reaction could be described well with the inclusion of the well-established , , , and resonances, in addition to the mesonic current. Consideration of other well-established resonances in the same mass region, including the spin-5/2 resonances, and , does not further improve the results qualitatively. For the reaction , the resonance is found to be important for reproducing the structure observed in the differential cross section data. Our model also improves the description of the reaction to a large extent compared to the earlier results by Nakayama et al. [Phys. Rev. C 68, 045201 (2003)]. For this reaction, we address two cases where either the or the dominates. Further improvement in the description of these reactions and the difficulty to uniquely determine the nucleon resonance parameters in the present type of analysis are discussed.

Eta meson production, Photon-nucleon scattering, Pion-nucleon scattering, Nucleon-nucleon scattering, Nucleon resonances
pacs:
25.20.Lj, 13.60.Le, 13.75.Gx, 14.20.Gk

I Introduction

One of the primary motivations for studying the production of mesons off nucleons is to investigate the structure and the properties of nucleon resonances and, in the case of heavy-meson productions, to learn about hadron dynamics at short range. In particular, a clear understanding of the production mechanisms of mesons heavier than the pion still requires further theoretical and experimental investigation. Apart from pion production, the majority of theoretical investigations of meson-production processes are performed within phenomenological meson-exchange approaches. Such an approach forces one to correlate as many independent processes as possible within a single model if one wishes to extract meaningful physics information. Indeed, this is the basic motivation behind the coupled-channels approaches.

In this paper, we present the result of our investigation of -meson production in both photon- and hadron-induced reactions, which include

(1)

More specifically, we perform a combined analysis of the reactions, , , , and based on an effective Lagrangian approach.

The amount of data available for -meson production is now considerable. In particular, in photoproduction processes off protons, a new generation of high-precision data is now available, not only for total and differential cross sections in a wide range of energies starting from threshold photo-xsc-data (); photo-xsc-data1 () but also for beam and target asymmetries photo-spin-data (). Taken in combination, these data sets offer better opportunities for investigating the properties of nucleon resonances. In particular, much more detailed studies than in the past are possible now for resonances that may perhaps couple strongly to , but only weakly to . In this respect, the recent data on the quasi-free process KCGG07 () have attracted much interest in -production processes in connection to the possible existence of a narrow (crypto-exotic) baryon resonance with a mass near  GeV, which is still under debate exotic (). (See also Refs. 6; 7; 8.)

There are a large number of theoretical investigations of photoproduction, mostly from the early 1990s to the present photo-theory () and, especially, off protons. Most of them focus on the role of nucleon resonances and the extraction of the corresponding resonance parameters, but a variety of issues have also been addressed, such as the coupling constant TBK94 (), the anomaly BWW00 (), and the extended chiral symmetry FMU06 (). Among the more recent calculations that analyze the recent high-precision data are those of Refs. 13; 14; 15. The -MAID approach MAID () is an isobar model that includes, in addition to the -channel vector-meson exchange or its Reggeized version, a set of well-established spin-1/2, -3/2, and -5/2 resonances. This model has been applied to the analyses of data for photoproduction as well as for electroproduction of the meson. The Bonn-Gatchina-Giessen group ASBK05 () has developed a model which has been employed in a combined partial wave analysis of the photoproduction data with , , , and final states. This model considers fourteen nucleon resonances and seven Delta resonances for achieving a reasonable overall agreement with the whole database considered in their analysis.111The resonances do not couple to the and the channels because of isospin conservation. He, Saghai, and Li have analyzed the -photoproduction reaction in a chiral constituent quark model CQM () by considering all of the one- to four-star-rated resonances listed in the Review of Particle Data Group (PDG) PDG06 ().

In hadronic reactions, a noticeable amount of data for has been accumulated NNeta-xsc-data (); COSY-TOF-02 (); COSY11 (); CELSIUS-WASA (); NNeta-spin-data (). Here, we have and invariant mass distributions and the analyzing power near threshold, in addition to differential and total cross sections. The process is particularly relevant for studying the role of the final-state interaction (FSI). Most of the existing calculations (see, e.g., Refs. 22, 23 and references therein; see also Ref. 24) take into account the effects of the FSI in one way or another, which is well-known to influence the energy dependence of the cross section near threshold. Calculations that include the FSI to lowest order BGHH03 () reproduce the bulk of the energy dependence exhibited by the data. However, they are not sufficient to reproduce the invariant mass distribution measured by the COSY-TOF COSY-TOF-02 () and COSY-11 Collaborations COSY11 (). Thus, in order to explain the observed invariant mass distribution, the importance of the three-body nature of the final state (in the -wave) has been emphasized FA03 () or an extra energy dependence in the basic production amplitude has been suggested Deloff03 (). In spite of this, another possibility has been offered, which is based on the higher partial wave (-wave) contribution NHHS03 (). We observe that what is actually required to reproduce the measured invariant mass distribution is an extra dependence, where denotes the relative momentum of the final subsystem. Obviously, this can be achieved either by an -wave or by a -wave contribution. Note that the -wave () can also yield a flat proton angular distribution, as observed in the corresponding data. The model of Ref. 28, however, underpredicts the measured total cross near threshold to a large extent. One of the objectives of the present study is to resolve this discrepancy. In any case, as pointed out in Ref. 28, the measurements of the spin-correlation functions should help settle the question of the -wave versus the -wave contributions in a model-independent way.

Most data for the more basic (two-body) reaction have been obtained in the 1960s through 1970s; they are rather scarce and less accurate piNetaN-data () than the data for the other two reactions mentioned above. Recently, the Crystal Ball Collaboration has measured the differential and the total cross sections of this reaction near threshold CB05 (). Theoretically, this reaction has been studied mostly in conjunction with other reactions in a combined analysis BT90 () or in coupled-channels approaches IOV01 (); PM02c (); GHHS03 (); CSZ06a (); MSL06 () in order to constrain some of the model parameters. Recently, Zhong et al. ZZHS07 () extended the chiral constituent quark model for meson photoproduction CQM () to this reaction. Also, has been studied within a heavy-baryon chiral perturbation theory Krippa01 (). Arndt et al. ABMS05 () investigated the role of on the scattering length within a coupled-channels analysis of this reaction and elastic scattering.

In the present work, we consider the three reactions mentioned above in the following manner: The photoproduction reaction is calculated by considering the -, - and -channel Feynman diagrams plus a generalized contact term NH04-NH05 (), which ensures gauge invariance of the total amplitude, in addition to accounting for the final-state interaction effects. (See Ref. 42 for details.) The reaction is calculated in the tree-level approximation, including the -, -, and -channels. To the extent that this reaction is dominated by the excitation of the resonance at least for energies close to the threshold, this should be a reasonable approximation if we confine ourselves to energies not too far from the threshold. The process is calculated in the distorted-wave Born approximation (DWBA), where both the FSI and the initial-state interaction (ISI) are taken into account explicitly NSL02 (). The FSI is known to be responsible for the dominant energy dependence observed in the total cross section (apart from the dependence due to the phase space) arising from the very strong interaction in the -wave states at very low energies NSHHS98 (). As for the basic meson-production amplitude, our model includes the nucleonic, mesonic, and nucleon resonance currents, which are derived from the relevant effective Lagrangians.

Ultimately, our goal is to perform a more complete model calculation in which the relevant FSIs are taken into account explicitly. However, before being able to undertake a complex calculation that couples many channels, we need to learn some of the basic features of meson production (in particular, those of -meson production) within a simplified model where these basic features may be revealed and analyzed in a much easier manner. In this regard, one of the major purposes of the present investigation is to show that consideration of the hadronic reactions , in conjunction with more basic two-body reactions, would greatly help in the study of nucleon resonances, especially, in imposing much stricter constraints on the extracted resonance-nucleon-meson () coupling strength involving a meson other than the pion. In fact, currently, our knowledge of the branching ratios of the majority of the known resonances is very limited PDG06 ().

This paper is organized as follows. In the next section, we briefly describe our model for the reactions listed in Eq. (1). The results of the corresponding model calculations are presented and discussed in Sec. III. Section IV contains a summary and a conclusion. Some details of the present model are given in the Appendix.

Ii Reaction Mechanisms

In the present work, the -meson production processes are treated within a relativistic meson-exchange approach, whose dynamical contents are summarized by the Feynman diagrams displayed in Figs. 1, 2, and 3 for the reactions , , and , respectively. We employ phenomenological form factors at hadronic vertices to account for the structures of the corresponding hadrons.

Figure 1: Feynman diagrams contributing to . Time proceeds from right to left. The wavy, solid, and dashed lines represent the photon, the nucleon, and the meson, respectively. The intermediate baryon states are denoted by N and R for the nucleon and the nucleon resonances, respectively. The intermediate mesons in the -channel include the and the . The external legs are labeled by the four-momenta of the respective particles, and the labels s, u, and t of the hadronic vertices correspond to the off-shell Mandelstam variables of the respective intermediate particles. The top-right diagram is the generalized contact current.

For the photoproduction process, the total amplitude in the present work is given by the Feynman diagrams displayed in Fig. 1. In these Feynman diagrams, the three diagrams in the lower part are transverse individually while the three diagrams in the upper part are not. Gauge invariance of the total amplitude is ensured by the generalized contact current given in Refs. 41 and 42, which follows the general formalism of Refs. 44; 45; 46. This contact term provides a rough phenomenological description of the FSI and is not treated explicitly here HNK06 (). The details of the present approach are fully described in Ref. 41, where -meson production in photon- and hadron-induced reactions was investigated, and they will not be repeated here. One new feature in the present work, however, is the inclusion of spin-5/2 resonance contributions. For our discussion, we refer the nucleonic current (NUC) to the diagrams shown in the top line of Fig. 1. The meson-exchange current (MEC) and the resonance current contributions correspond to the leftmost diagram and the two diagrams on the right of the bottom line of Fig. 1, respectively.

Figure 2: Feynman diagrams contributing to . The notation is the same as in Fig. 1.

As mentioned in Sec. I, the reaction is calculated within the tree-level approximation. To the extent that it is dominated by the resonance contribution, this is a reasonable approximation, at least near threshold. The total amplitude for this reaction is, therefore, given by the Feynman diagrams displayed in Fig. 2. Here, the nucleonic current (NUC) corresponds to the first two diagrams on the top line while the meson-exchange current (MEC) and the resonance current contributions correspond, respectively, to the rightmost diagram on the top line and the two diagrams on the bottom line in Fig. 2.

As for the reaction, the total amplitude is calculated within the DWBA:

(2)

where denote the ISI (FSI), stands for the corresponding propagator, and denotes the basic production amplitude displayed in Fig. 3 and is constructed from the interaction Lagrangians given in Appendix. Further details of the present approach to this reaction, including all the values of the coupling constants and the cutoff parameters of the corresponding form factors that enter in the definition of the basic production amplitude , can be found in Ref. 22. Also, we use the interaction based on the Paris potential LLRV80 (), which includes the Coulomb interaction as well NADS00 (). The nucleonic, resonance, and meson-exchange contributions correspond, respectively, to the first, second, and third lines of the Feynman diagrams on the right-hand side in Fig. 3.

Figure 3: Basic production amplitude for . The full amplitude, including the ISI and FSI contributions, is given by Eq. (2). As in Fig. 1, N and R denote the intermediate nucleon and resonances, respectively, and M incorporates all exchanges of mesons , , , , , and (former ) for the nucleon graphs and , , , and for the resonance graphs. External legs are labeled by the four-momenta of the respective particles as in Fig. 1. Diagrams with are understood although not displayed here.

In the Appendix, we present all the hadronic and electromagnetic interaction Lagrangians and propagators necessary for computing the diagrams displayed in Figs. 1, 2, and 3 within the present approach. The phenomenological form factors used in this model are also given in the Appendix. The free parameters of our model — the resonance parameters, the coupling constant, and the cutoff parameter at the electromagnetic vector-meson exchange vertex — are fixed so as to reproduce the available data in a global fitting procedure of the three reaction processes listed in Eq. (1).

Iii Results and Discussion

In this section, we present and discuss the results of our model calculation. The basic strategy of our approach is, in principle, the same as that of Ref. 41; namely, we start with the nucleon plus meson-exchange currents and add resonance contributions one by one as needed in the fitting procedure until achieving a reasonable description of the available experimental data for the reactions listed in Eq. (1). Apart from the dominant resonance, we allow for other well-established resonances of spin-, -, and - in this model. We confirm the earlier finding photo-theory () that, in photoproduction, in addition to spin- resonances, at least spin-3/2 resonances — in particular, resonances — are needed in order to obtain a reasonable description of the data.

Following Ref. 41, for each resonance, we take into account only the branching ratios , , and corresponding to the respective hadronic decay channels , , and . The latter accounts effectively for all the other open decay channels. Note that the branching ratios and are related to the corresponding and coupling constants in the interaction Lagrangians, and, as such, they are not free parameters of the model. The same is true for the branching ratio , associated with the radiative decay channel, which is related to the corresponding coupling constants. Then, in view of the constraint given by Eq. (38), the branching ratio is not a free parameter either. Here, we emphasize that, in contrast to Ref. 41 — where some assumptions were made concerning the values of the branching ratios — no such assumptions are enforced in the present work because the simultaneous consideration of the reaction processes listed in Eq. (1) allows us, in principle, to extract the and coupling constants separately.

The coupling constants of the () interaction Lagrangians are required in the calculation of the reaction. Therefore, in principle, the hadronic branching ratios involving vector mesons (such as the and the ) should also be taken into account. However, since we have restricted ourselves to nucleon resonances for which the decay channels are either closed or nearly closed, we have set the associated branching ratios to zero ().222Strictly speaking, even these resonances can have non-vanishing branching ratios to channels due to their large widths. Obviously, in a more refined calculation, this condition should be relaxed. In a more complete (coupled-channels) dynamical model approach, the branching ratios and total widths will be generated by the model via the dressing mechanism of the corresponding vertices and resonance masses.

The results shown here are not necessarily the best fits achievable within the present approach. Rather, they are sample fits that illustrate different dynamical features that may be obtained in this type of analysis. The resonance parameters are obtained by global fitting to the available data for the reactions mentioned above.

For this end, we consider four models. Although these four models contain the same nucleonic and mesonic currents described in Figs. 1, 2, and 3, they include different nucleon resonances and different resonance parameters. In model (A), we consider only the , , , and resonance currents. Model (B) includes the same resonances as in model (A), but the parameters of those resonances are different from those of model (A). The parameters are obtained by using different starting values for the search in parameter space during the global fit procedure. The implication of the differences between these two models will be discussed later. In addition to the resonances considered in model (A) and (B), model (C) includes the resonance. Finally, we consider model (D), which takes into account the contributions from , , and in addition to the resonances considered in model (C). In the following, we present and discuss the results for each reaction.

Nucleonic current:
(, ) (, 0.0)
Mesonic current:
(MeV)
current: PDG PDG
(MeV) [] []
(, ) (, ) (, )
(, ) (, ) (, )
() (, ) (, )
() (, ) (, )
(MeV) [] []
(%) [] []
(%) [] []
(%) [] []
(%) [] []
current: PDG PDG
(MeV) 1520 [] 1700 []
() (, ) (, )
() (, , 0.0) (, , 0.0)
() (, , 0.0) (, , 0.0)
(MeV) [] []
(%) [] []
(%) [] []
(%) [] []
(%) [] []
Table 1: Parameters of model (A) fitted to the reactions listed in Eq. (1). (See the Appendix for an explanation of the parameters.) Values in boldface are not fitted. The branching ratios and () are not free parameters, but are extracted from the corresponding coupling constants, except for which is obtained from Eq. (38). The values in square brackets are the range estimates quoted in PDG PDG06 (). The data set for , , and was used in the fit.
Figure 4: (Color online) Results for the reaction in model (A), i.e., with the parameters of Table 1. (a) Total cross section as a function of the total energy of the system . The line styles identified here apply to all four parts of this figure. (b) angular distribution in the center-of-mass frame. (c) Beam asymmetry and (d) target asymmetry in the center-of-mass frame. The numbers in (b, c, and d) are the incident photon laboratory energies in MeV. In (c and d), only the total results are shown. The data are from Refs. 1 and 3.

iii.1

We first consider the model in which only the , , , and resonance currents are considered in addition to the nucleonic and the mesonic currents (cf. Fig. 1). We find that this comprises the minimal set of resonances that are required to achieve a reasonable description of the reaction processes listed in Eq. (1). The resulting fitted parameters, which are obtained by fitting to the experimental data shown in Fig. 4 (however, not taking into account the total cross sections for ) are given in Table 1. Note that for photoproduction, in addition to the electromagnetic couplings, only the hadronic vertices involving the meson are required for the present calculation. In the reaction, only the hadronic vertices involving and are needed while all the hadronic vertices given in the Appendix are required for the calculation of the reaction.

In Table 1, the parameter values in boldface are fixed and are not allowed to vary during the fitting procedure. The pure pseudovector coupling choice () at the vertex was motivated by the massless chiral limit of the meson. (See the Appendix for the definition of .)333We will also consider the pseudoscalar coupling choice (). See the discussions in subsection III.2. Also, the most general form of the vertex for spin-3/2 and -5/2 resonances involves three independent coupling constants, as exhibited in Eqs. (15) and (17). At present, however, information on the corresponding coupling constants is extremely scarce, especially, on defined in Eqs. (15) and (17). In the present work, therefore, we allow only two structures at those vertices by setting the coupling constant to zero.

In this work, we use the resonance masses from the centroid values quoted in PDG PDG06 (). The exceptions to this are the masses of the and the resonances. They were allowed to vary in the fitting process in order to reproduce accurately the position of the large cross-section peak exhibited by the photoproduction data near the threshold. The quantities in the square brackets in Table 1 are the range estimates quoted in PDG PDG06 () and are given here for an easy comparison with the fitted values extracted in the present model. As one can see, some parameter values are considerably outside the range quoted in PDG. However, we note that, although various parameters are highly correlated to each other, the data used in the present study are not sufficient to uniquely constrain the model parameters. As a result, different parameter sets may provide fits of comparable quality; some of them are discussed in this paper.

Figure 4 displays the results for photoproduction observables corresponding to the parameter set of Table 1. The total cross section shown in Fig. 4(a) provides the line styles used in all four parts. We start with the discussion of the angular distribution shown in Fig. 4(b). As one can see, the flat angular distribution near the threshold is dominated by the resonance. When the photon incident energy (in the laboratory frame) is larger than about  GeV, both the resonance and the mesonic currents become relevant for reproducing the shape of the measured differential cross sections. As the energy increases, the shape becomes more and more forward-peaked, which is a well-known feature of the -channel mesonic current contribution. In the energy region of  GeV, some details of the measured angular distribution are still not well explained, which indicates that our model should be improved by using a more refined and quantitative calculation. We leave such an investigation to a future work. The nucleonic current contribution is negligible because of the very small coupling constant resulting from the fit, which is preferred by the small angular distribution measured at backward angles and higher energies. The nucleonic current contributes mostly at backward angles and at high energies through the -channel diagram (cf. Fig. 1). We will come back to this issue later in subsection III.2.

Shown in Fig. 4(a) are the total cross sections for photoproduction obtained with the resonance parameters given in Table 1. Here, we mention that the total cross section data for photoproduction were not included in the global fitting process. As one can see, the large peak that rises sharply from threshold is due to the dominating and resonances. Although both the resonance and the mesonic currents are small, their contribution beyond GeV cannot be ignored because of the interference with the large current contribution, a feature that has already been pointed out in earlier works photo-theory (). Around GeV, all the resonance and mesonic currents become comparable to each other, but at higher energies, the mesonic current yields the largest contribution, and the resonance contribution becomes negligible, as expected from the relative low mass. The structure shown by the data at  GeV seems to suggest possible contributions from other higher mass resonances. Analyses of this structure and other details are left to a future work.

Figure 4 also shows the results for beam and target asymmetries. We find that, by and large, the beam asymmetry () and the target asymmetry () are described reasonably well.444Quite recently, the GRAAL Collaboration KPBJ08 () re-analyzed the beam asymmetry in , revealing a sharp structure at GeV for and suggesting the presence of a narrow resonance. The most visible room for improvement exists for the latter, in particular, at lower energies, where the data show a dependence, while the model yields nearly flat and vanishing results. This is a feature common to all the parameter sets and not just to the particular parameter set of Table 1. The difficulty in reproducing the target asymmetry is a known feature from earlier works. (See, in particular, the work of Tiator et al. photo-theory ().) Perhaps, the difficulty in reproducing the target asymmetry may be understood if we write the quantities in terms of the four amplitudes a la the CGLN decomposition CGLN (); NL04-NL05 ()

(3)

We then have

(4)

where the expression for the recoil polarization is also given. The above results reveal that, unlike the beam asymmetry, the target asymmetry involves the imaginary part of the product of amplitudes . This means that this observable is more sensitive to the effects of the final state interaction, a feature that is not accounted for explicitly in the present type three-level calculations. Furthermore, it is also clear that the same difficulty should be present in describing the recoil polarization. In fact, this seems to be the case, as reported in Ref. 52.

We anticipate here that, in contrast to low energies, the target asymmetry becomes sensitive to the dynamical content of the model as the energy increases (cf. the results in Figs. 4, 5, 6, and 7). Therefore, we expect that this observable at higher energies will be useful in imposing extra constraints on the model parameters.

Nucleonic current:
(, ) (, 0.0)
Mesonic current:
(MeV)
current: PDG PDG
(MeV) [] []
(, ) (, ) (, )
(, ) (, ) (, )
() (, ) (, )
() (, ) (, )
(MeV) [] []
(%) [] []
(%) [] []
(%) [] []
(%) [] []
current: PDG PDG
(MeV) 1520 [] 1700 []
() (, ) (, )
() (, , 0.0) (, , 0.0)
() (, , 0.0) (, , 0.0)
(MeV) [] []
(%) [] []
(%) [] []
(%) [] []
(%) [] []
Table 2: Another set of fitted parameters [model (B)]. See the caption of Table 1 for details.
Figure 5: (Color online) Same as Fig. 4 but for model (B), i.e., with the fitted parameter set of Table 2.

Another parameter set resulting from the global fit, employing the same set of nucleon resonances as in Table 1, is shown in Table 2. This set [model (B)] was obtained by using different starting values for the search in parameter space during the fit procedure. As a result, many parameter values of this set are quite different from those of Table 1 not only in magnitude but also in relative signs for some coupling constants, which provides an indication of the general reliability of such global fits. The corresponding observables are shown in Fig. 5. Here, although both the measured total and differential cross sections are reproduced with a comparable fit quality to the results in Fig. 4, the dynamical content is quite different. In particular, the resonance contribution dominates over the other currents in the energy region of  GeV (i.e.,  GeV). It is interesting to note that, searching for the pole positions of the -matrix in the complex plane, as well as performing Breit–Wigner parameterizations of resonances, the recent partial-wave analysis by Arndt et al. ABSW06 () of the elastic and charge-exchange processes, combined with the reaction , finds no resonance. Also, some of the coupled-channels dynamical models KHKS00 (); GHHS03 () find no necessity for this resonance to fit the phase shifts. In this respect, the present results corresponding to the parameter set of Table 1, where the contribution of the resonance is much smaller than that of Table 2, are more in line with these findings. It would be most interesting to see if the inclusion of the channel in those coupled-channels analyses mentioned above would require the resonance. Overall, the spin observables in Fig. 5 exhibit the same features as in Fig. 4, but for higher energies, the target asymmetry shows very different angular dependences for the two parameter sets, which points to the importance of spin asymmetries in reducing the ambiguities that otherwise would exist.

Nucleonic current:
(, ) (, 0.0)
Mesonic current:
(MeV)
current: PDG PDG
(MeV) [] []
(, ) (, ) (, )
(, ) (, ) (, )
(MeV) [] []
(%) [] []
(%) [] []
(%) [] []
(%) [] []
current: PDG PDG PDG
(MeV) 1520 [] 1700 [] 1720 []
() (, ) (, ) (, )
(MeV) [] [] []
(%) [] [] []
(%) [] [] []
(%) [] [] []
(%) [] [] []
Table 3: The parameter set for model (C), where the resonance is added to see whether it further improves the fit. No data were used for this fit.
Figure 6: (Color online) Same as Fig. 4, but for model (C), i.e., with the parameter set of Table 3.

Table 3 [model (C)] displays a parameter set including the resonance, in addition to those considered in the previous two sets. Here, we have considered only the and the reaction data in the fitting procedure. The corresponding results for the observables are shown in Fig. 6. As can be seen, the inclusion of the resonance does not improve significantly the description of the data for this photon-induced reaction. However, as we will see in the following subsection, this resonance considerably improves the fit quality of the hadronic reaction at higher energies.

Nucleonic current:
(, ) (, 0.0)
Mesonic current:
(MeV)
current: PDG PDG PDG
(MeV) [] [] 1710 []
( (, ) (, ) (, )
(MeV) [] [] []
(%) 0.26 [] 0.06 [] 0.01 []
(%) [] [] []
(%) [] [] []
(%) [] [] []
current: PDG PDG PDG
(MeV) 1720 [] 1520 [] 1700 []
(MeV) [] [] []
(%) 0.12 [] 0.10 [] 0.54 []
(%) [] [] []
(%) [] [] []
(%) [] [] []
current: PDG PDG
(MeV) 1675 [] 1680 []
(MeV) [] []
(%) 0.02 [] 0.25 []
(%) 45 [] 65 []
(%) [] []
(%) [] []
Table 4: Same as Table 1. The parameter set for model (D). Here, more resonances are added here to see whether they would further improve the fit. Here we have considered only the reaction in the fitting procedure.
Figure 7: (Color online) Same as Fig. 4, but for model (D), i.e., with the parameter set of Table 4.

Next, we extend our model by including all of the well-established resonances in the mass region of  MeV in order to verify whether these resonances can lead to a qualitatively superior description of the data compared to the previous cases, where a more limited set of resonances was considered. To this end, we concentrate only on the reaction. Table 4 shows the parameter set, where the and , as well as the , resonances are included, in addition to those considered in Table 3. Here, following Ref. 41, we treat the branching ratio as a free parameter to be fitted while the branching ratio is extracted from the product of the coupling constants in conjunction with the assumed branching ratio for the radiative decay. The resulting observables with the parameter set of Table 4 [model (D)] are shown in Fig. 7. We see that the overall fit quality does not change significantly from the fit qualities for the previous sets. Here, the resonance gives the largest contribution to the cross section in the energy region of  GeV, a feature similar to that already exhibited in Fig. 5.

Figure 8: (Color online) -meson angular distributions in the center-of-mass frame in at  MeV and  MeV. (a) The results corresponding to the parameter set of Table 4, with . (b,c) The results obtained with two other parameter sets (not given in this work) using the pseudovector () and the pseudoscalar () coupling choices, respectively, at the vertex. The resulting coupling constant values are and , respectively. The contributions from the other nucleon resonances are practically negligible at these energies and are not displayed. is the sum of spin- resonance contributions. The data are from Credé et al. photo-xsc-data ().
Figure 9: (Color online) (a) Beam asymmetry and (b) target asymmetry in at  MeV and  MeV as functions of -emission angle in the center-of-mass frame. The solid curves represent the results obtained with the parameter set of Table 4, which correspond to Fig. 8(a). The dashed and the dash-dotted curves are obtained with the two other parameter sets that correspond to Figs. 8(b) and (c), respectively.

iii.2 Coupling Constant

In the previous subsection, we have shown that the present calculations yield very small values of the coupling constant — compatible with zero — due to the smallness of the measured cross sections at backward angles where is large. As mentioned before, the angular distribution becomes very sensitive to at these kinematics through the -channel nucleonic current contribution. However, one must be cautious in drawing conclusions about the extracted value of from calculations based on approaches such as the present one. This is due to the fact that we cannot completely discard the possibility of a relatively large nucleonic current contribution interfering destructively with contributions from resonances in order to yield the angular distributions observed at those kinematics. This point is illustrated in Fig. 8, where the results for the angular distribution are shown, together with the data reported by Credé et al. (CB/ELSA Collaboration) photo-xsc-data (), at the two highest energies. Figure 8(a) shows the results corresponding to the parameter set of Table 4, where the nucleonic current contribution is practically zero555Note that the value of is in Table 4. and cannot be seen in the figure. The corresponding nucleon resonance contributions are also very small. The mesonic current dictates to a large extent the behavior of the angular distribution at forward angles. Figures 8(b) and (c) display the results corresponding to two additional parameter sets (not given here) with the same set of nucleon resonances as in Table 4. Overall, the two parameter sets yield a fit quality comparable to that of Fig. 7, but with very different resonance parameter values, which points to the ambiguity of the fit results if one relies solely on differential cross-section data. Shown in Fig. 8(b) is the result obtained by using pure pseudovector coupling () as in Fig. 7 while in Fig. 8(c), pure pseudoscalar coupling () is adopted. The corresponding coupling constants are and , respectively. As can be seen in Figs. 8(b) and (c), in these cases the nucleonic current contribution at backward angles is as large as that of the mesonic current at forward angles. However, the resonance contribution exhibits an angular dependence similar to that of the nucleonic current with a comparable magnitude and interferes destructively with the nucleonic current. The destructive interference is almost complete and results in very small cross sections at backward angles as observed in the data. Overall, everything else being very similar between Figs. 8(b) and (c), we find no real sensitivity of the differential cross sections at these energies as to whether pseudovector or pseudoscalar couplings are employed.

The situation changes when one considers spin observables. While there is no real difference at lower energies, at the high energies ( MeV) considered here, the beam asymmetry shown in Fig. 9(a) can distinguish clearly between the parameter sets corresponding to Fig. 8(a) on the one hand and Figs. 8(b) and (c) on the other. The marked differences are due to the marked differences in the values for the coupling constant , which is vanishingly small () for the set corresponding to Fig. 8(a) and much larger (and about the same, and , respectively) for Figs. 8(b) and (c). This finding shows that the beam asymmetry at higher energies can impose more stringent constraints, in particular, on the coupling constant. In any case, judging from the results in the present investigation, we expect the upper limit of the coupling constant to be not much larger than .

These parameter sets also lead to noticeable differences — albeit not quite as large — for the target asymmetry , as shown in Fig. 9(b). Of particular importance, however, is that this observable may distinguish between the use of the pseudoscalar or the pseudovector coupling at the vertex, which the results of Figs. 8(b) and (c) and the respective curves of Fig. 9(a) cannot do. Of course, one should keep in mind that the target asymmetry can be more sensitive to the effects of the FSI than the beam asymmetry does, as Eq.(4) indicates. Therefore, one should be cautious in drawing strong conclusions from the present results, which do not account for the FSI explicitly.

Before closing this subsection, we remark that, quite recently, the authors of Ref. 12 addressed the issue of chiral symmetry in -meson photoproduction through the pseudoscalar-pseudovector mixing parameter at the vertex by investigating this reaction close to the threshold. Our study reveals that one must be careful with such an investigation for the reasons mentioned above. In particular, if the coupling constant turns out to be very small, it will be very difficult to determine the value of the mixing parameter .

We also note that in Ref. 58 an alternative way of extracting the coupling is discussed, where, in contrast to the present work, one makes use of the cross section data at very low energies. There, the assumption is made that all the production mechanisms are known, except for the nucleonic current. The extracted value of is consistent with the present findings.

iii.3

In this subsection, we discuss the reaction with the parameter sets determined above. The results for the total and the differential cross sections for the reaction corresponding to the parameter sets of Tables 1 and 2 are displayed in Figs. 10(a) and (b) and in Figs. 10(c) and (d), respectively. The cross-section results show that the various dynamical contributions of the two sets are very similar. The total cross section is rather well reproduced up to  GeV, where it is dominated by the resonances, especially, by the resonance. Here, both the nucleonic and the mesonic currents yield very small contributions. However, we do not reproduce the total cross section at higher energies due to the absence of the higher-mass resonances in these parameter sets and the absence of the contribution via the coupled channel IOV01 (); GHHS03 () in this model. For differential cross sections, we again note that these parameter sets are unable to reproduce the structure exhibited by the data at higher energies.

As we have shown before, the inclusion of the does not improve the results for the reaction significantly. However, this resonance provides an important contribution to reproduce the structure exhibited by the differential cross section data in . This is illustrated in Fig. 11 corresponding to the parameter set of Table 3. In addition, the resonance also helps improve, to some extent, the fit quality for the total cross section at energies above  GeV, corroborating the finding of Ref. 34.

Figure 10: (Color online) Results for corresponding to the parameter set of Table 1. (a) Total cross section as a function of the total energy of the system . (b) angular distribution in the center-of-mass frame. Here, stands for contributions and contributions. (c and d) Same as (a) and (b), but with the parameter set of Table 2. The numbers in (b) and (d) denote the total center-of-mass energy in MeV. The labelings of the curves in (b and d) are the same to that of (a and c). The experimental data are from Ref. 29; 30.
Figure 11: (Color online) Same as Fig. 10 but with the parameter set shown in Table 3. Here, is the sum of contributions from , , and .
Figure 12: (Color online) Same as Fig. 11 but with another parameter set (not given here) that includes the resonance. Here, is the sum of contributions from , , and .

Figure 12 shows an alternative fit for the total and the differential cross sections by using the same set of resonances as those of Fig. 11 plus . The corresponding fit results for photoproduction are of comparable quality to those shown in Sec. III.1. As one can see from Fig. 12(a), the small bump near GeV in the spin-1/2 resonance contribution (dashed line) is caused by the resonance, which makes the bump in the total contribution (solid line) more pronounced compared to the result of Fig. 11(a). The resonance seems also to affect the differential cross section in the vicinity of MeV, improving the agreement with the data to some extent. It is interesting to note that the chiral constituent quark-model calculation of Ref. 37 shows a dominant contribution from the resonance at an energy around  MeV, in contrast to the present approach, where the dominant contribution arises from the spin-3/2 resonances. However, the authors of Ref. 37 have also found that the agreement with the measured differential cross sections in the  MeV energy range can be improved if the sign of their partial wave amplitude is reversed and if they employ a larger total decay width of MeV. The effect of the resonance is, then, very similar to that found in the present calculation. We note that the total decay width of the resonance in Fig. 12 is MeV.

The reaction has been also investigated in Ref. 38 within a coupled-channel approach. There, the role of the -wave resonances – in particular of the and – in the differential cross sections has been studied. The former resonance has not been considered in the present work.

iii.4

Figure 13: (Color online) Results for the reaction with the parameter set of Table 1. (a) Total cross section as functions of the excess energy in and collisions. (b) angular distribution in the overall center-of-mass frame. (c) Final proton angular distribution. (d) invariant-mass distribution. (e) invariant-mass distribution. (f) Analyzing power. In (f), only the total contributions are shown (solid curves); the dashed curves represent the correspond results of another parameter set (not given in this work) which yields practically the same results for other observables considered in this reaction. The labelings of the curves in (b, c, d, and e) are the same as in (a). The data are from Refs. 17; 18; 19; 21.
Figure 14: (Color online) Same as Fig. 13 but for the parameter set of Table 2.
Figure 15: (Color online) Predictions for the invariant-mass distributions for () and () in the reaction corresponding to the excess energy of  MeV for the parameter sets of Tables 1 (solid line) and 2 (dashed line). Data are from Ref. 20 measured at MeV. They have not been included in the fitting procedure.

In this subsection, we turn our attention to the reaction. Although the model calculation of Ref. 28, which is based on a strong -wave contribution, reproduces nicely the shape of the measured invariant mass distributions, it largely underestimates the total cross section data near the threshold. Here, we present new results based on a combined analysis of the reactions listed in Eq. (1).

Shown in Fig. 13 are the results for the reaction corresponding to the parameter set of Table 1. It can be seen that the present model reproduces rather reasonably all the considered data for this reaction, including the energy dependence of the total cross sections at lower energies. This is a considerable improvement over the results of Ref. 28. However, this model still underestimates the total cross section in collisions by a factor of when the excess energy is  MeV. We expect the inclusion of FSI and possibly the three-body effects to resolve this discrepancy once they are properly taken into account. Here, the major difference of our results from those of the previous calculation of Ref. 28 is that we have a much stronger spin-3/2 resonance contribution to the cross sections, especially at lower energies. This is due to the large coupling of the resonances to the and the vector mesons, resulting from the global fitting procedure (cf. Table 1). In contrast, the resonance contribution to the cross section is surprisingly small, especially at lower energies, which is due to the strong destructive interference among the exchanged mesons in the excitation of resonances. Note that in the present calculation the resonance coupling constants involving vector mesons, (), are basically fixed by the reaction while the coupling constants to pseudoscalar mesons, (), are fixed by the and the reactions to a large extent. We also found that the mesonic current yields a very small contribution to the cross sections. The nucleonic current contribution is also negligible because of the very small coupling constant that results from the fit to the photoproduction data, as discussed above.

It should be emphasized that it still remains to be verified whether the dominance of the resonances discussed above is indeed true. In fact, in spite of the present lack of information on the corresponding coupling constants, , the obtained values (cf. Table 1) may be too large to be realistic. For example, a rough estimate of these coupling constants from the PDG helicity amplitudes PDG06 (), in conjunction with vector-meson dominance, yields

(5)

and

(6)

These values are corrected to the normalization point, at , by writing

(7)

where is the effective hadronic coupling function, which includes the form factor given by Eq. (24). We emphasize in this context that these coupling constants cannot be determined uniquely in the present analysis. Indeed, as shown in Fig. 14, a scenario in which the resonance dominates over the resonance current can be achieved. Therefore, the consistency of these coupling constant values with other independent reaction processes should be examined in detail. For this purpose, vector-meson production processes, such as , and , are of particular interest VM ().

In Ref. 22, where the dominant -production mechanism is the excitation of the resonance via the pion-exchange, the analyzing power exhibits a zero at . In the present calculation, where the dominant production mechanism is the resonance excitation, the zero of is shifted toward backward angles at larger . As has been pointed out in Ref. 22, unlike the total and the differential cross sections, the analyzing power is very sensitive to the reaction dynamics. In fact, the dashed curves in Fig. 13 for represent the results of another parameter set (not given here) that yields practically the same results for the other observables considered here (see also the results shown in Fig. 14 below). Unfortunately, the data are not accurate enough to disentangle these different scenarios. More accurate data will, therefore, impose more stringent constraints so as to help distinguish different dynamics of production in collisions.

The results corresponding to the parameter set of Table 2 are shown in Fig. 14. They are of comparable fit quality to those in Fig. 13 overall. However, the dynamical content is quite different. In this case, the resonance dominates over the resonance contributions to the cross sections. Unlike the results shown in Fig. 13, there is no strong destructive interference among the exchanged meson contributions to excite the intermediate resonances. Of course, the smallness of the resonance contributions is directly correlated to the very small constants for the coupling of the resonances to vector mesons (cf. Table 2). The analyzing power for this parameter set at  MeV exhibits a qualitatively different behavior from that found in Fig. 13, again, the quality of the current experimental data does not allow any definite conclusion to be drawn.

It should be emphasized that the much larger difference in the dynamics in between the two parameter sets considered above as compared to those in and stems from the much richer interference effects in the former reaction. In particular, note that, for each meson exchanged (), there are two coupling vertices involved in the nucleon resonance currents (i.e., and , as shown in Fig. 3). This allows for an interference among all the exchange mesons involving the coupling constants, a feature that is absent in the other two reactions. Therefore, the meson-production reactions in collisions, in conjunction with other basic photon- and (two-body) hadron-induced reactions, should help in extracting the resonance parameters to a large extent. In particular, adding the vector meson () production channels, , and , to the list of reactions given in Eq. (1), should impose more stringent constraints on the resonance coupling constants.

Finally, Fig. 15 shows predictions using the parameter sets of Tables 1 and 2 for the invariant-mass distributions for and at the excess energy of  MeV. The results are compared with the recent CELSIUS-WASA data CELSIUS-WASA () measured at  MeV. Note that these data have not been included in the present fit. We see that, while the invariant mass distribution is well reproduced except in the small region, the invariant mass distribution shows big differences between the prediction of the present model and the measured data, indicating a deficiency in the model. Note that the present model does not account for the FSI. Further investigation is required to identify the origin of the discrepancies in both the and the invariant mass distributions.

Iv Summary

In the present work, a combined analysis of the reactions , , , and has been carried out within a relativistic meson-exchange model of hadronic interactions. Both the and the reactions have been treated in the tree-level approximation with the former reaction containing a generalized contact current that ensures gauge invariance of the reaction amplitude. The reaction has been treated in the DWBA approximation with the explicit treatment of the ISI and FSI. The free parameters of the model, especially the nucleon resonance parameters, are then fixed in a global fitting procedure.

Overall, the photoproduction data can be described reasonably well with the inclusion of the well-established , , , and resonances as the minimally required set of resonances to achieve a reasonable fit to the currently available data. The inclusion of additional well-known resonances in the same mass region [including the spin-5/2 and resonances] does not further improve the quality of the overall description of the data. The measured angular distributions at higher energies and backward angles are compatible with a vanishing coupling constant. However, in order to extract this coupling constant unambiguously within an approach of the type pursued here, one needs to go beyond the resonance region to avoid possible interference effects. On the other hand, as we have seen in the results of Fig. 9, the beam asymmetry can impose constraints on the coupling constant that are much more stringent than the differential cross sections because of the interference between the nucleonic and the resonance currents. One difficulty of the present calculation is to reproduce the dependence exhibited by the measured target asymmetry near threshold, and this shows that further investigations are necessary to understand better the production mechanism of this reaction.

Our model can also explain the available data on reasonably well, at least for energies not too far from the threshold. At higher energies the total cross section is underestimated due to the lack of the contribution via the coupled channel.

Our model also describes the reaction data rather well. The problem of the underestimation of the total cross section near threshold NHHS03 () has been cured to a large extent in the present approach. However, we emphasize that the scenario of strong coupling of the resonance to the channels () still remains to be verified, for the couplings cannot be fixed unambiguously in the present study. In this connection, vector meson production reactions, such as the , , , should be investigated. We also verified that the analyzing power is sensitive to the reaction dynamics. Unfortunately, however, the currently available data are not accurate enough to unambiguously distinguish different dynamical contributions.

As we have illustrated with some selected examples, the present approach is unable to determine a unique set of (resonance) parameter values. In fact, it was shown that different parameter sets that describe the data equally well lead to results that exhibit quite different reaction dynamics. This is especially true for the reaction, where we found quite different values of the coupling constants. As mentioned above, the inclusion of vector-meson production reactions into the combined analysis should help constrain those coupling constants. Consequently, our study reveals that one must be cautious in interpreting the resonance parameters extracted from these kinds of analyses, especially, if only a single reaction process is considered. It is clear that in order to extract more accurate information on the nucleon resonances, one must combine the investigation of hadron-induced meson production with the corresponding photon-induced reactions. To help in this, ample data sets are now available for photo- and electro-production processes from various accelerator facilities, including the Thomas Jefferson National Accelerator Facility, SPring-8, CB/ELSA, GRAAL, etc. By contrast, data for hadron-induced production processes are much more limited, and we clearly need more of them.

In this connection, we note that the progress in the study of meson-production processes in collisions, both experimentally and theoretically, has reached such a level that it allows us to address certain concrete physics issues, especially, when they are investigated in conjunction with other independent reactions. This has been illustrated in the present work for the specific case of production, where some information on nucleon resonances can be extracted. In particular, the consideration of meson-production processes in collisions, in conjunction with photon- and (two-body) hadron-induced reactions aimed at a resonance parameters extraction, should help impose more stringent constraints on these parameters. As pointed out in the last subsection, meson production in collisions exhibits much richer interference effects, a feature that is absent in more basic two-body reactions. Furthermore, the inclusion of this reaction in the resonance-parameter extraction is especially relevant because the existing data for meson production (other than for the pion) in two-body hadronic reactions are rather scarce and of relatively low accuracy. Currently, there exist only very limited efforts to improve or extend the corresponding database CB05 (). On the other hand, the available data on meson production in collisions are much more accurate; moreover, the corresponding database can be and is being expanded, especially at the COSY facility.

Acknowledgements.
This work was supported by the Forschungszentrum Jülich under FFE Grant No. 41445282 and by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No. 2010-0009381).

*

Appendix A

In this Appendix, we give the ingredients that define our models described in Sec. II. Throughout this paper, we use the notation and for the nucleon and the nucleon resonance fields, respectively; denotes the mass of the baryon (). We also use , , and to denote the scalar, pseudoscalar, and vector meson fields, respectively. The vector notation refers to the isospin space. For isovector mesons, , , and . The mass of the meson is denoted by . The photon field is denoted by . We define and .

We use the superscript in the Lagrangian densities () involving the nucleon resonance to denote the spin-parity of that resonance. Furthermore, for convenience, we define

(8)

a.1 Hadronic Interaction Lagrangians

The following interaction Lagrangian densities describe the hadronic vertices. The Lagrangians for meson-nucleon interactions are

(9)