photoproduction based on chiral unitary model
Recent CLAS data for the invariant mass distributions (line-shapes) in the reaction are theoretically investigated. The line-shapes have peaks associated with the excitation. Our model consists of gauge invariant photo-production mechanisms, and the chiral unitary model that gives the rescattering amplitudes where is contained. It is found that, while the line-shape data in the region are successfully reproduced by our model for all the charge states, the production mechanism is not so simple that we need to introduce parameters associated with short-range dynamics to fit the data. Our detailed analysis suggests that the nonresonant background contribution is not negligible, and its sizable effect shifts the peak position by several MeV. We also analyze the data using a Breit-Wigner amplitudes instead of those from the chiral unitary model. We find that the fitted Breit-Wigner parameters are closer to the higher pole position for of the chiral unitary model. This work sets a starting point for a fuller analysis in which line-shape as well as angular distribution data are simultaneously analyzed for extracting pole(s).
pacs:13.30.Eg, 13.60.Le, 13.60.Rj, 14.20.Gk 14.20.Jn,
The pole structure of the resonance is a key issue to understand the nature of and the interaction. Because decays exclusively into the channel with by the strong interaction, a signal associated with is expected to be observed in the invariant mass distributions (to be referred to as “line-shape”) of certain production reactions.
In old bubble chamber experiments, bumps associated with the excitation have been observed in the line-shapes of hadron induced reactions, such as Thomas:1973uh (), Hemingway:1984pz () and Braun:1977wd (). The observed bumps in the first two experiments are consistent with the resonance at 1405 MeV, while the reaction with a deuteron target found the resonance at 1420 MeV. Recently, the line-shape for the energies was also measured in hadronic reactions, such as with 514-750 MeV/c kaon momenta by Crystal Ball Collaboration Prakhov:2004an (), and with 3.65 GeV/c proton beam at COSY-Jülich Zychor:2007gf ().
Although there have been several data for the spectrum from the hadron beam experiments as mentioned above, the quality of the data is still not sufficient for extracting pole(s). The situation has been changed by recent photon-beam experiments. The first photoproduction of the resonance was observed at SPring 8 by LEPS collaboration in the reaction with the photon energy of 1.5-2.4 GeV Ahn:2003mv (); Niiyama:2008rt (). In this experiment, the and line-shapes were measured, and they were found to be different from each other, owing to the interference between the resonant and non-resonant contributions. A high statistics, wide angle coverage experiment for the reaction was performed at Jefferson Laboratory by the CLAS collaboration, for center-of-mass energies GeV Moriya:2013eb (); Moriya:2013hwg (). In this experiment, all the three charge states of the channels were simultaneously observed in the scattering for the first time, and the differential cross sections were measured for the line-shape and for the angular distribution. This is the cleanest data that cover the kinematics of excitation, which encourages theorists to seriously work on extracting the pole(s) from data for the first time. Very recently the spectral shape of has been also observed in electroproduction in the range of Lu:2013nza ().
The coupled-channel approach based on the chiral effective theory (chiral unitary model) suggests that the resonance is composed of two poles located between the and thresholds Hyodo:2011ur () and these states have different masses, widths and couplings to the and channels. One pole is located at MeV with a dominant coupling to , while the other is sitting at MeV with a strong coupling to Jido:2003cb (). These two states are generated dynamically by the attractive interaction in the and channels with Hyodo:2007jq (). Because the resonance is composed of two states which have different weight to couple with and , the spectral shape of the line-shape in the region depends on how is produced, as pointed out in Ref. Jido:2003cb (). Reference Jido:2003cb () predicts that the resonance in the channel has a peak at 1420 MeV with a narrower width because the higher pole strongly couples to the channel. The study of Ref. Jido:2009jf () showed that, in the reaction, is dominantly produced by , and the line-shape has a peak at 1420 MeV as seen in the old bubble chamber experiment Braun:1977wd ().
It is important to confirm the two-pole structure by analyzing the new CLAS data, and if so, it will be interesting to see how the two-pole structure plays a role in the line-shape. In order to extract the resonance pole(s) from the production data, we develop a model that consists of production mechanism followed by the final state interaction (FSI); is excited in the FSI. Through a careful analysis of the data, we can pin down the production mechanism as well as the scattering amplitude responsible for the FSI. Then the pole(s) will be extracted from the scattering amplitude. Such an analysis of the new CLAS data has been done in Refs. roca1 (); roca2 () using a simple production mechanism.
In this work, we focus on the photoproduction of in , and investigate the new CLAS data for the line-shape Moriya:2013eb (). The first study of the reaction was done in Ref. nacher (), in which a simple diagram was considered for the production mechanism and the is described by the chiral unitary approach. Related calculations were also done in Refs. nacher2 (); borasoy (). Although the calculation of Ref. nacher () was to get a rough estimate of the cross sections, in the advent of the fairly precise data, it is necessary to develop a quantitative model to extract the properties from the data. In this work, we extend and refine the model of Ref. nacher () by considering more production mechanisms that are gauge invariant at the tree-level. We consider relevant meson-exchange mechanisms, and contact terms that simulate short-range mechanisms. We explain details of the model, and successfully fit the CLAS data with it. Then we discuss a role played by each mechanism, effects of non-resonant contributions, and a possibility of a single-pole solution of . By doing so, we set a starting point for a full analysis in which we simultaneously analyze the data for line-shape Moriya:2013eb () and the angular distribution Moriya:2013hwg () to study the pole structure of . Such a full analysis is left to a future work. We expect the angular distribution data are an important information to pin down the production mechanism.
The rest of this paper is organized as follows: We give a detailed description of our model in Sec. II. Then we show numerical results and discuss them in Sec. III, followed by a summary in Sec. IV. Expressions for Lagrangians and photo-production operators, and also model parameters are collected in Appendices.
ii.1 Kinematics and cross section formula
First we define kinematical variables. We consider the reaction in which the variables in the parentheses are four-momenta for the particles in the total center-of-mass system. The differential cross section for the reaction is derived following a standard procedure, and given as
where , , and are the masses of the proton, , and , respectively, and the Källen function is denoted by . The symbol is the squared total energy of the system, and is related to the four-momenta by , while the invariant mass of the subsystem is . The kinematical variables with asterisk stand for the quantities in the center-of-mass system. The summation of spin and polarization states in initial and final particles are indicated by ; the average factor, 1/4, for the initial states is already included in the factor of the formula. All information about the dynamics is encoded into the reaction amplitude in Eq. (1), and is discussed in detail in the next subsection. The line-shape of the spectrum is obtained by integrating Eq. (1) over the angular part of and , and given as
ii.2 Photo-production mechanism
As stated in the introduction, we describe the reaction by a set of tree-level mechanisms for ( : a set of meson and baryon) followed by rescattering. We use an index to specify , , respectively. Thus the reaction amplitude introduced in Eq. (1), , is given by
where is a tree-level photo-production mechanism. In the next paragraph, we specify the tree mechanisms that go into our calculation. The summation of runs over all of the tree-level photoproduction mechanisms included in our calculation. Contribution from the rescattering is denoted by . The rescattering amplitude is calculated with a partial wave expansion with respect to the relative motion of , and and partial waves are considered; and are the total and orbital angular momenta for . The partial wave amplitude is given, with the on-shell factorization, by
where and are partial wave amplitudes of and , respectively, and are calculated with the on-shell momenta of relevant particles. More details about the partial wave expansion, including the relation between and , are given in Appendix A. For the scattering amplitudes , we use those from the chiral unitary model given in Ref. ORB () for wave, and in Ref. JOR () for wave. The wave contains as double poles, while the wave does not include any resonance and provide a smooth background. It is turned out that the contribution from the wave rescattering is small. We use the meson-baryon Green function, , calculated with the dimensional regularization as follows:
where and are the masses of a baryon and a meson , respectively, and we use the values listed in the Particle Data Group pdg () for the masses. The relative on-shell momentum of corresponding to is denoted by . The symbol is the subtraction constant for the regularization scale , and we set MeV for all channels. The subtraction constants can depend on a channel as well as a production mechanism ; we will come back to this point at the end of this section.
We consider gauge-invariant tree-level photo-production mechanisms () as follows: minimal substitution to the lowest order chiral meson-baryon interaction such as the Weinberg-Tomozawa terms (Fig. 1) and the Born terms (Fig. 2); vector-meson exchange mechanisms (Fig. 3).
Thus, for specifying each mechanism , we use the label for each figure in Figs. 1-3 so that 1(a), 1(b), .., 3(k). These photo-production mechanisms are expanded in terms of , and and terms are considered in our calculation. Explicit expressions for , as well as our model Lagrangians from which are derived, are shown in Appendix B and C. Coupling constants contained in of Figs. 1-3 are fixed either by data (other than ) if possible, or by SU(3) relation if poorly constrained by data. More details about the couplings are given in Appendix B.
With the meson-exchange production mechanisms and the subtraction constants ( in Eq. (5)) taken as the same as those in the chiral unitary amplitudes, we cannot reproduce the line-shape data for the reaction from the CLAS Moriya:2013eb (). Therefore, it is inevitable to introduce adjustable degrees of freedom to fit the data. Thus all of the meson-exchange mechanisms are multiplied by a common form factor of the following form:
where and are respectively the momenta of and in the center-of-mass frame of . The cutoff will be used to fit the data. In addition, we also consider phenomenological contact terms that can simulate mechanisms not explicitly considered, such as, in particular, and excitation mechanisms. We take couplings for the contact terms -dependent ( : total energy of the system), and will be determined by fitting the data Moriya:2013eb (). We consider three types of contact terms that are gauge-invariant at the tree-level, and are couple to and states of different charges, and thus we have 15 complex couplings at each . Expressions for the contact terms are presented in Eqs. (99)-(101) in Appendix. Also, for the mechanism index , we write , as in Eqs. (99)-(101). The form factor of Eq. (6) is not applied to the contact terms.
The subtraction constants included in Eq. (5) are also adjusted to fit the data, thereby changing the interference pattern between different production mechanisms. As already stated, can depend on the production mechanism 1(a), 1(b), .., 3(k), . However, some of ’s should have the same value for . Also, we do not want to have too many free parameters from the subtraction constants, because it will complicate fitting the data. Thus we classify the production mechanisms into several groups, and each group has its own real subtraction constant. In grouping, we try to classify important mechanisms into different groups so that we have effective freedom in fitting. In TABLE 1, we show the classification of the mechanisms into 11 groups labeled by A,B,…,K.
The subtraction constant for each group is denoted by where refers to one of the groups, A,B,…,K. Then will be used to fit the data Moriya:2013eb (). It is noted that we do not adjust the subtraction constants in the chiral unitary amplitudes in the fit. The subtraction constants we adjusted are all for the first loop of the rescattering, and for the renormalization of the production mechanism.
For the number of data points to be fitted, we now have rather many free parameters most of which are from contact terms. We find that this amount of degrees of freedom is necessary to obtain a reasonable fit to the data. This situation can be understood, considering that we do not explicitly consider short-range mechanisms (baryon resonances, coupled-channel effects) that will play a substantial role here. Because it will be a very difficult task to identify and/or fix each of the short-range mechanisms, we develop the production model in a practical manner as discussed above. Of course, our method could bring a model-dependence of pole(s) extracted from the data. The model-dependence of pole(s) must be assessed by analyzing the data with different form factors and/or contact terms. This will be a future work.
iii.1 Fitting data
Before presenting our results, we comment on the calculated quantity to be fitted to the line-shape data from CLAS Moriya:2013eb (). In the data analysis done by CLAS in Ref. Moriya:2013eb (), enhanced events due to the peak in the invariant mass spectrum has been subtracted. In our model, a mechanism of Fig. 3(i) without rescattering can create the peak. Thus we fit the data with the following modified differential cross section:
where contains all of the meson-exchange mechanisms and contact terms followed by the rescattering as discussed in Sec. II.2, and is calculated using Eq. (2). Meanwhile, the second term contains only the tree-level mechanism of Fig. 3(i). The subtraction in Eq. (7) is done at the cross section level, and the interference between the mechanism of Fig. 3(i) and others is kept to be consistent with the analysis of Ref. Moriya:2013eb ().
We present all numerical values for the fitting parameters (the cutoff from the form factor, the subtraction constants, and the complex couplings from the contact terms) in Appendix D.
iii.2 Line-shape results
Our results, after the fit, are presented in Fig. 4 where the CLAS data are also shown for comparison.
We fitted the data at , 2.1, 2.2 and 2.3 GeV. As seen in the figure, our model fits the data very well for all three different charge states of .
It is interesting to break down the line-shapes into contributions from different mechanisms, as shown in Fig. 5.
As seen in the figures, different mechanisms gives significant contributions that interfere with each other. We find that the contributions from the gauged Weinberg-Tomozawa terms (Fig. 1) are rather small. In fact, a diagram such as Fig. 1(e) gives a contribution comparable to those from the gauged Born mechanisms (Fig. 2). However, as a result of a destructive interference, the net contribution from the gauged Weinberg-Tomozawa terms is rather small. This destructive interference is not necessarily a result of the gauge invariance. Actually, a dominant term in Fig. 1(e) is gauge invariant itself. Rather, fitting the data have fixed relevant subtraction constants so that the diagrams in Fig. 1 with the rescattering cancel out each other. We find relatively large contributions from mechanisms of Fig. 1(e), 2(b), 2(e), 2(i), 2(l), 3(b), 3(f), that have two propagators rather than three in the other mechanisms; the propagators tend to suppress the contributions of the mechanisms. Meanwhile, the contact terms, which simulate short-range dynamics, also give a large contribution to bring the theoretical calculation into agreement with the data. As seen in TABLE 4, the contact terms have a rather strong coupling to the channels as a result of the fit. One may find in TABLE 4 that the -dependence of the contact couplings is rather irregular, and is not well under control. However, we note that the contact terms can have a resonant behavior. Also, in Fig. 6, we show the -dependence of the most important contact couplings, and for and . From the figure, it is hard to judge if the behavior of the couplings is out of control. As will be discussed later, however, we would be able to put them under better control if we fit not only the line-shape data but also other observables such as angular distributions. This will be a future work. Finally, we mention that coupled-channels effects are mostly from the and channels.
The difference in the line-shape between different charge states observed in Fig. 4 is a result of the interference between different isospin states. The has three isospin states (), and they are separately shown in Fig. 7.
A dominant contribution is from the state as expected due to the peak. The higher mass pole at MeV, that creates the prominent bump in the line-shapes, seems to play more important role than the lower mass pole. This is because the production mechanisms in our model generate more strongly than , and the final state interaction induces . As shown in the previous study Jido:2003cb (), the higher mass pole couples to the channel more strongly than the lower mass pole does. The state gives a smaller contribution, still plays an important role to generate the charge dependence of the line-shapes. As the energy increases, the contribution is larger. The state contribution is even more smaller, but still unnegligible. To see this point, we show in Fig. 7 the line-shape multiplied by 3. The difference between this and the line-shape is the effect of the interference between the and states. We can see that the interference with the state even changes slightly the peak position of the line-shape.
The different peak positions for differently charged states seen in Fig. 4 can be also understood as a result of the interference between a resonant and a background parts. To see this, it is useful to decompose the amplitude into the resonant (second term of Eq. (3)) and background (first term of Eq. (3)) parts. Each of the contributions to at GeV is shown in Fig. 8.
Interestingly, the background terms give smooth and significant contributions. Although the peak structures are due to the resonant contributions, the background can shift the positions of the peaks, particularly the peak of the line-shape. After all, the resonant contributions give the peaks at almost the same position, GeV, for all the differently charged states. Thus it seems that one of the poles at MeV plays a dominant role in the line-shapes. We will look into this observation in the next subsection.
iii.3 Single Breit-Wigner model
So far, the excitation of is described by the chiral unitary model, and has the double-pole structure. However, as seen above, only the higher mass pole seems to give the dominant contribution. Thus, it is interesting to see if a single Breit-Wigner can simulate the photo-induced excitation. For this purpose, we use a model in which the rescattering amplitude, in Eq. (4), is given by, instead of the chiral unitary amplitude, a single Breit-Wigner function in and isospin zero partial wave; the other rescattering partial waves amplitudes are set to zero. Here we assume that the rescattering amplitude couples to only and channels. Thus we have
where and are Breit-Wigner mass and width, respectively. A symbol is a complex coupling strength, and will be fitted to the data, along with the Breit-Wigner mass, width and other fitting parameters. Because of being isospin zero, have three independent complex values that we denote , , and : is for ; is for ; is for . We have defined the index at the beginning of Sec. II.2. Our Breit-Wigner form of Eq. (8) is more relaxed than the conventional one in which the coupling strengths are related to the width by , where the summation is taken over both channels and their particles’ phase-space. In this way, we can simulate non-resonant effects that are not considered explicitly in the rescattering amplitude. The single Breit-Wigner model is fitted to the data and is shown with the data in Fig. 9. The fitted parameters are presented in tables in Appendix D.
Although the quality of the fit is a little worse than the previous model with the chiral unitary amplitude (Fig. 4), still it is an acceptable level. Thus, the line-shape data for only do not rule out the possibility of single pole solution for . The fit gives MeV and MeV that are close to the middle of the two poles from the chiral unitary amplitude, but still closer to the higher mass pole than to the lower one. As seen in TABLE 5, the Breit-Wigner amplitude couples strongly (weakly) to the () channel, which is also similar to the character of the higher mass pole of in the chiral unitary model.
iii.4 angular distribution
So far, we fitted only the line-shape data for the reaction. We found that fitting only the line-shape data can lead to several solutions whose quality of the fit to the line-shape data are comparable. However, they can have very different angular distribution. Therefore, angular distribution data will be useful information to constrain the production mechanism. Recently the CLAS Collaboration reported data for the angular distributions Moriya:2013hwg (). In Figs. 4-8, we actually presented the line-shapes from the model that gives angular distributions relatively close to the new data of Ref. Moriya:2013hwg (). Here we use the same model to calculate the angular distributions, and show them in Fig. 10.
At GeV, our model captures overall trend of the data. However, for the reaction at GeV, there is a sharp rise in the data at while rather smooth behavior is found in the calculated counterpart. We actually tried fitting the angular distributions data, but this sharply rising behavior cannot be fitted with the current setup. It seems that we need to search for a mechanism that is responsible for this behavior. We leave such a more detailed analysis of the angular distribution to a future work.
We calculated the line-shapes of the photoproduction process . This was motivated by the recent CLAS collaboration’s report Moriya:2013eb () that found peaks due to excitation in the line-shapes. We employed the scattering amplitudes from the chiral unitary model to describe the final state interaction where the is excited. For the tree-level photo-production mechanisms, we introduced a gauge invariant model that consists of the gauged Weinberg-Tomozawa terms, the gauged Born terms and the vector-meson exchange terms. We also introduced freedom to fit the data such as contact terms modelling short-range dynamics like baryon resonances, and subtraction parameters that can be different from those determined in the chiral unitary model. These are necessary to reproduce the line-shape data from the CLAS. This implies that the mechanism for the photo-production is not so simple that the several important terms interfere. It is noted that we do not adjust any parameters (subtraction constants, couplings) of the chiral unitary amplitudes in the fit.
Our model reproduces the line-shape data quite well. Breaking down the calculated line-shape into contributions from each production mechanism, we found that the contribution from the gauged Weinberg-Tomozawa terms is not so important due to rather destructive interference between the terms. More important contributions are from the born and vector-meson exchange terms. In addition, the short range contact terms give large contributions to reproduce the line-shape data. We also decomposed the calculated line-shapes into the resonant and nonresonant parts for each charge state of the channels. We found that even though the resonant part dominates the spectra and generates the peak structure, the nonresonant background contribution is not so negligible and its sizable effect shifts the peak position. The direction of the shift is a consequence of complicated interference of many terms and it is hard to pin down the main mechanism for the shift. One can say that it is not the case that the shift is solely caused by an interference between the and components, because one can see the shift of the peak position also in the channel.
We also made a check of the amplitude obtained by the chiral unitary model. We refitted the line-shape data with a single Breit-Wigner amplitude for the amplitude instead of those from the chiral unitary model. The quality of the fit is fairly good, indicating that the line-shape data alone do not rule out a one-pole solution for . We found that the Breit-Wigner mass and width obtained by the fit are closer to the higher pole position of the in the chiral unitary model. Also the Breit-Wigner amplitude strongly couples to the channel, sharing the similar property of the higher mass pole. This implies that, in the model including the chiral unitary amplitude, the higher resonance pole plays a more important role for the photoproduction.
In future work, we will simultaneously analyze data for the line-shape and the angular distribution, and then extract pole(s). We presented the angular distribution from the current model obtained by fitting only the line-shape data. Although our model captures overall behavior of the data in many cases, there is also a qualitative difference that cannot be fixed by fitting with the current setup. Identifying a mechanism that can fill the difference will be a challenge in the future work. Also an important task is to address a model-dependence of the extracted pole(s) because we are using a rather phenomenological production mechanisms. This can be done by using contact terms and/or form factors of different forms.
Acknowledgements.We thank Reinhard Schumacher and Kei Moriya for useful discussions. SXN is the Yukawa Fellow and his work is supported in part by Yukawa Memorial Foundation, by the Yukawa InternationalãProgram for Quark-hadron Sciences (YIPQS), and by Grants-in-Aid for the global COE program âThe Next Generation of Physics, Spun from Universality and Emergenceâ from MEXT. This work was partially supported by the Grants-in-Aid for Scientific Research (No. 25400254 and No. 24105706).
Appendix A Partial wave expansion
We summarize a partial wave expansion of an amplitude for the (: meson; : baryon) reaction in the center-of-mass system of , with respect to the relative motion of . We denote the amplitude by and its partial wave where and are total and orbital angular momenta of , respectively. Here we show equations for the partial waves relevant to this work; .
The relation between and for (, )=(1/2,0) is
where is the momentum for and . For (, )=(1/2,1), we have
and for (, )=(3/2,1),
where is an arbitrary vector. The operator () is a baryon spin transition operator from spin 1/2 to 3/2 (3/2 to 1/2), and it can be expressed by
With the partial wave amplitudes defined above, the original amplitude is written as
Appendix B Model Lagrangians
We present a set of Lagrangians from which we derive photo-production mechanisms graphically shown in Figs. 1-3. We follow the convention of Bjorken-Drell. We use symbols , , and to denote octet baryon, octet meson, nonet vector meson, and electromagnetic fields, respectively. Also, we use curly symbols to denote creation or annihilation operators. For example, is the annihilation operator contained in , and its normalization is .
b.1 Hadronic interactions
We work with the lowest order chiral Lagrangian for the octet pseudoscalar mesons () coupled to the octet baryons (), as given by 111When , , and are enclosed by the trace brackets, they are SU(3) matrix. Otherwise, they are understood to be one of particles contained in the SU(3) matrix elements. The same applies to the curly symbols for those fields.
where the symbol denotes the trace of SU(3) flavor matrices, is the baryon mass and
For the couplings and , we use , , and with MeV.
The meson and baryon fields in the SU(3) matrix form are
From Eq. (14), we will particularly use the interaction, as contained in the covariant derivative, given by
and also use the interaction, as in and terms, as
where and , and we have introduced the traces and defined by
We will also use a notation defined by
From here, we discuss interactions involving the nonet vector mesons. In the SU(3) matrix form, the vector meson nonet is given by
where the ideal mixing between the neutral vector mesons is assumed. With the matrix, the interactions we use are
where the coupling is related to the coupling by , and we use determined from the decay width. The vector part of the interactions are given by
We use the coupling constants , and from the universality assumption. The relative phase between the and interactions is also fixed by the universality. We also use , so that and . The interactions also contain the tensor coupling, as seen in the common expressions for the and interactions:
where is the proton mass. We use and , based on an average of and reaction models tensor (). The tensor couplings for the other vector mesons are fixed using the SU(3) relation for the magnetic coupling, and an explicit expression will be given later in Eq. (43). The interactions are given by
The interactions we use are based on the hidden local symmetry model HLS (), and are given by
where , and we use the convention, .
b.2 Electromagnetic interactions
The photon coupling to the baryonic current is given by
The symbol is the electric charge (in unit of ) of a baryon for , but zero otherwise. The anomalous magnetic moment is denoted by for which we use experimental values listed in the Particle Data Group pdg () 222 The magnetic moment for has not been measured, and we use a quark model prediction pdg (). We also use the quark model to fix the sign for . .
The photon coupling to the pseudoscalar meson current is given by
where is the quark charge matrix .
The electromagnetic interactions involving the vector mesons are due to the U(1) axial anomaly, and are given by
The couplings are determined by experimental decay widths pdg (), and the relative phases are fixed by the SU(3) relation. The numerical values for are given in TABLE 2. 333 Although we find GeV from data, the SU(3) predicts it to be zero, and we cannot find its phase. In this work, we set it to be zero.
b.3 Matrix elements and coupling constants
In this subsection, we evaluate matrix elements of the Lagrangians defined above, in order to introduce a coupling constant for a given set of incoming and outgoing particles. The coupling constants will be used to write down the photoproduction amplitudes in the next section. We will often use an index to specify a pair of meson and baryon, , respectively. Also we denote four-momenta for and as and , respectively.
The matrix element of the interaction defined in Eq. (18) is given by
where the energy for a particle is denoted by where is the mass of ; the values of are from Ref. pdg (). The couplings are tabulated in TABLE 1 of Ref. OR (). The matrix element of the interaction defined in Eq. (19) is given with a coupling constant as