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Dynamical coupled-channel model of kaon-hyperon interactions

The pi N -->KY and KY -->KY reactions are studied using a dynamical coupled-channel model of meson-baryon interactions at energies where the baryon resonances are strongly excited. The channels included are: pi N, K \Lambda, and K\Sigma. The resonances considered are: N^* [S_{11}(1650), P_{11}(1710), P_{13}(1720),D_{13}(1700)]; \Delta^* [S_{31}(1900), P_{31}(1910), P_{33}(1920)]; \Lambda ^* [S_{01}(1670), P_{01}(1810)] \Sigma^* [P_{11}(1660), D_{13}(1670)]; and K^*(892). The basic non-resonant \pi N -->KY and KY -->KY transition potentials are derived from effective Lagrangians using a unitary transformation method. The dynamical coupled-channel equations are simplified by parametrizing the pi N -->pi N amplitudes in terms of empirical pi N partial-wave amplitudes and a phenomenological off-shell function. Two models have been constructed. Model A is built by fixing all coupling constants and resonance parameters using SU(3) symmetry, the Particle Data Group values, and results from a constituent quark model. Model B is obtained by allowing most of the parameters to vary around the values of model A in fitting the data. Good fits to the available data for pi^- p to K^0 \Lambda, K^0 \Sigma^0 have been achieved. The investigated kinematics region in the center-of-mass frame goes from threshold to 2.5 GeV. The constructed models can be imbedded into associated dynamical coupled-channel studies of kaon photo- and electro-production reactions.


I. INTRODUCTION
Investigation of kaon-nucleon and nucleon-hyperon interactions with hadronic probes has a long history in strangeness physics. However, the interactions involving an additional relevant kaon-hyperon channel have received marginal attention, because of lack of data.
More recently, strangeness reactions are also receiving considerable attention in associated strangeness production with incident photon and electron beams. With the advent of facilities such as JLab, ELSA, Spring-8, and GRAAL, copious and high precision data on meson electromagnetic production on both nucleon and nuclear targets are becoming available.
Measurements of the strangeness associated production channels focus on the energy region of E lab γ ≤ 2.5 GeV, corresponding to the total center-of-mass energy W≤2.3 GeV, which cover the baryon resonances region. A result of our earlier work [1] on the γp → K + Λ reaction showed that multi-step processes, due to the coupling with the πN channel, can have as much as a 20% effect on the total cross-section. To investigate very recent strangeness production data, it is necessary to extend that work, which was limited to the KΛ channel, to include all of the KΣ channels: γp → KY , with K ≡ K + , K • and Y ≡ Λ, Σ • , Σ + . Accordingly, a dynamical coupled-channel investigation of these processes requires realistic models to describe πN → πN, KY , and KY → KY processes. The purpose of this paper is to report on our progress in this direction.
The importance of developing coupled-channel approaches to meson-baryon reactions is summarized as follows: • Such an approach is required for a proper extraction of fundamental resonance decay properties, which are ultimately to be predicted by basic quark dynamics. In short, information about baryon resonance properties can only be reliably extracted within the context of an appropriate reaction theory. The importance of this interplay between extraction of fundamental information and the need for a consistent reaction theory has been emphasized by Sato and Lee [2] in the pion sector. Here we extend their investigation to the kaon sector.
• Impressive amounts of high quality data from JLab [3], ELSA [4], LEPS [5], and GRAAL [6] offer us the opportunity to pin down the underlying reaction mechanism and to study the role and/or properties of intervening baryon resonances. Such an effort is a prerequisite for any attempt to search for missing resonances [7]. Combining models from a chiral constituent quark formalism [8,9] with a coupled-channel approach, as presented in this work, is expected to provide reliable insights into the elementary strangeness photo-and electro-production reactions.
In recent years, coupled-channel effects on meson-baryon reactions with strangeness production have been investigated using two approaches. Kaiser et al. [10] applied a coupled-channel approach with chiral SU(3) dynamics to investigate pion-and photon-induced meson production near the KY threshold. Although their recent results [11] include p-wave multipoles, and thus reproduce data slightly above the threshold region, their chiral SU(3) dynamics model can not provide the higher partial waves that are important in describing the data at higher energies. Similar approaches have also been taken in Refs. [12,13,14,15]. Given the relevant W range mentioned above, their simplified dynamics represents of course only a first step. Indeed, comparisons with the πN → KY data clearly show that SU(3) models of Refs. [10,15], even when p-waves interactions are included [11], greatly miss fitting the differential cross-sections: theoretical predictions produce slopes opposite to that required by the data.
The second coupled-channel approach used in the literature to investigate photon-and meson-induced reactions is based on using effective Lagrangians along with a K-matrix method, developed by the Giessen Group [16,17,18,19]. In the K-matrix approach, all intermediate states are put on-energy-shell and hence possibly important off-shell dynamical effects are not accounted for. The advantage of this K-matrix approach is its numerical simplicity in handling a large number of coupled channels. However, the extracted N * parameters may suffer from interpretation difficulties in terms of existing hadron models, as explicitly demonstrated in an investigation [2] of the ∆ resonance.
In this paper, we present a dynamical coupled-channel model in which the meson-baryon off-shell interactions are defined in terms of effective Lagrangians. This off-shell approach is achieved by directly extending existing dynamical models [2] for πN scattering and pion photoproduction, to include KY channels. The main feature of our approach is that the strong interaction matrix elements of πN → KY and KY → KY transition operators are derived from effective Lagrangians using the unitary transformation method of Ref. [2]. This derivation marks our major differences with chiral SU(3) coupled-channel models mentioned above [10,11,12,13,14,15] since it allows one to include all relevant higher partial waves and our approach is also applicable at energies way above threshold. The dynamical content of our approach is also clearly very different from the on-shell K-matrix coupled-channel models [16,17,18,19].
It is necessary to indicate more precisely, and within a more general theoretical framework, the differences between our and other approaches . Similar to the well-studied meson-exchange models of NN and πN scattering, we also start with relativistic quantum field theory. With a model Lagrangian, there are two approaches for deriving models of meson-baryon scattering. The most common approach [20] is to find an appropriate three-dimensional reduction of the ladder Bethe-Salpeter equation of the considered model Lagrangian. Meson-baryon interactions are then identified with the driving terms of the resulting three-dimensional scattering equation; such as the Blankenbecler-Sugar [21] or Gross [22] equations. A fairly extensive study of the three-dimensional reductions for πN scattering is given by Hung et al. [23]. Extending such reduction methods to derive coupledchannel equations with stable two-particle channels is straightforward. In fact, the K-matrix coupled-channel equations employed in Refs. [16,17] can be derived along this line if one further neglects that the principal-value parts of the meson-baryon propagators, which account for the off-shell dynamics. The scattering equations used in SU(3) chiral models of Refs. [10,11,12,13,15] can also be derived from the ladder Bethe-Salpeter equation using a procedure similar to a three-dimensional reduction, although this simplification is not spelled out explicitly by the authors. In Ref. [14], the Bethe-Salpeter equation is solved directly, but only for the simplified case that the interaction kernel is of separable form due to the use of contact interactions. The difficulties in solving the Bethe-Salpeter equation, even with the ladder approximation, are well documented [24].
Alternatively, one can construct models of meson-baryon interactions by deriving an effective Hamiltonian H ef f defined in a chosen channel-space from a specific model Lagrangian.
The scattering equation within the considered channel-space is then governed by standard scattering theory where α, β represent the relevant channels, S is the S-matrix and T is the scattering operator.
Here we have defined H ef f = H 0 + H I , with H 0 denoting the free Hamiltonian and H I defining the interactions between considered channels. This approach for πN scattering and pion photo-and electro-production reactions has been pursued by Sato and Lee [2]. They applied the unitary transformation method of Refs. [25,26] where v πN,πN is the non-resonant interaction and the ∆ excitation is described by the vertex interaction Γ πN ↔∆ . With the Hamiltonian Eq. (3), it is straightforward (as explained in Ref. [2]) to show that the solution of Eq.(2) can be cast into the following form: where the non-resonant scattering operator t πN,πN is defined by the non-resonant potential v πN,πN , where the πN propagator is defined by with E α (p) = p 2 + m 2 α . The resonant amplitudes (in the center-of-mass frame) is It is clear from the above equations that the resonant operator t R contains off-shell effects due to the non-resonant interaction v πN,πN . Such off-shell effects must be accounted for in with the propagator defined by where q = k−k ′ = p ′ −p is the three-momentum transfer. In this Hamiltonian formulation, all particles are on their mass-shell, but the energies are not conserved during the collisions Our goal is to construct models for describing all available data of differential crosssections and polarization observables of the πN → KY reactions in the total center-of-mass energy range of W≈1.3 to 2.3 GeV. These data [27,28,29,30,31,32] have been obtained a few decades ago with low intensity beams and therefore are not very extensive and not of high quality. Nevertheless, we will show that they give sufficient constraints on constructing models of KY interactions.
In section II, we present the dynamical coupled-channel equations and explain our strategy in solving these equations. The results are given in section III. Section IV is devoted to Summary and Conclusions.

II. DYNAMICAL COUPLED-CHANNEL EQUATIONS
In this work, we consider a coupled-channel formulation obtained by extending Eqs. (3)- (9) to include the KY channels. Specifically, we are interested in solving where α, β ≡ πN, KY . The non-resonant scattering operator is defined by where the propagators are defined by withΓ In momentum-space, the matrix element of Eq. (13) in the center-of-mass frame is (19) and the matrix element of the dressed vertex interaction Eq. (17) is The integrals in the above equations extend over the relative momentum p, the off-shell dynamics is hence included in determining the scattering amplitudes. The K-matrix coupledchannel equation limit used by others can be obtained from the above equations only if one

A. Non-resonant amplitudes
Our first task is to define the nonresonant potentials for solving the coupled-channel equation (19). In the KΛ threshold energy region, it is reasonable to derive the potentials involving the KY channel using effective Lagrangians with SU(3) symmetry. On the other hand, it is not clear how to derive πN potential v πN,πN for energies well above the πN threshold region. We thus circumvent deriving v πN,πN and instead use a phenomenological procedure to include its effect using empirical πN amplitudes [33]. Accordingly, the main outcome from our calculations are scattering operators for πN → KY and KY → KY transitions, which are also needed for dynamical coupled-channel studies of γN → KY reactions.
To proceed, we first derive from Eq. (13) the following equations: where the effective KY potential V KY,KY (E) is defined by with We see that the operators T KY,KY and T KY,πN can be obtained by solving Eqs. (21)- (23) using the matrix elements of v KY,KY , v KY,πN and t πN,πN . We will calculate v KY,KY , v KY,πN from effective Lagrangians. On the other hand, we will use a phenomenological procedure to set where p 0 is the on-shell momentum defined by is the empirical πN amplitudes taken from the dial-in program SAID [34], and we have introduced an off-shell function The matrix elements of v πN,KY and v KY,KY are calculated from effective Lagrangians by using the unitary transformation method of Ref. [2]. The effective Lagangians we consider are given in Appendix A. The resulting potentials are the following: where Ξ is a baryon with the strangeness S = −2 and isospin I = 1/2, and Ξ * its excited states; K * indicates possible strange vector mesons including K * (892) and K 1 (1270); ρ here stands for all possible vector mesons (ρ, ω, φ).
However, not every term in Eqs. (27) and (28) is computed in our calculation for a variety of reasons. We do not consider Ξ and Ξ * exchange terms, v Ξ E and v Ξ * E , because of their unknown coupling strength. The vector meson t-channel exchange terms, v ρ and K 1 (1270), are also not included because of their unknown couplings as well as the duality hypothesis [35]. Since on the one hand, our formalism can handle all N * resonances with spin≤ 3/2, in the s-channels, and on the other hand, contributions from higher spin N * s are found [36] to be negligible in the processes studied here, it should be a reasonable approximation to keep only the t-channel contributions from K * (892). Due to the above considerations, the potentials used in this work are as illustrated in Figs. (2) and (3). Their matrix elements can be calculated from the usual Feynman diagrams except that the propagators of intermediate particles are defined by the procedures illustrated in Eqs. (10)- (11). For Y * resonance exchange terms v Y * E , the width is included in the propagators using the following Breit-Wigner form: In appendix B, and in the next Section, we show how we determine the coupling constants associated with the resulting potentials using SU(3) symmetry and constituent quark models.
To solve the coupled-channel equations Eqs.

Resonant terms
The calculation of the resonant terms t R πN,KY and t R KY,KY using Eqs. (16)-(18) requires bare form factors Γ N * ,πN and Γ N * ,πN from a hadron model. The number of the resonances we need to consider is rather large and such calculations are not very certain at the present time. To make progress, we postpone such that more fundamental approach and simply adopt the following Breit-Wigner form: with the total width We will only consider the known resonances and hence the above resonant amplitudes can be evaluated using the information provided by the Particle Data Group [37]. For poorly determined decay strengthsΓ N * ,πN andΓ N * ,KY , we use a SU(3) quark model [38,39] to fix them.

III. RESULTS AND DISCUSSION
In this Section we will first use the formalism developed in the previous sections to build models by fitting the existing differential cross-section and hyperon polarization asymmetry data for the following processes: in the center-of-mass energy region ranging from threshold to W ≈ 2.3 GeV. We then present our predictions based on this coupled-channel model for the following reactions: To our knowledge no empirical or theoretical information about the above KY → KY reactions is available, although it constitutes an important ingredient in strangeness physics, especially in dynamical coupled-channel studies of hyperon photoproduction reactions, as discussed in Introduction.
To proceed, we need to construct the driving terms v KY,πN , Figs To produce numerical results for observables, the first step is to select a set of resonances relevant to the reaction mechanism, to be included in the calculation. To keep the number of adjustable parameters reasonable, we need some guidance from independent investigations on the relevant reaction mechanism, or in other words, on the intervening resonances in different s-, u-, and t-channels. As mentioned in previous Sections, our final aim is to apply this formalism to study associated strangeness production using electromagnetic probes. We therefore consider resonant states that were found to be important in this realm (see e.g. Refs. [9,40,41]) (though our formalism allows us to introduce any nucleon and/or hyperon We will construct two models. The first model, henceforth called model A, is obtained by fitting the data with most of coupling constants fixed by combining SU(3)-symmetry, with central values reported in the Particle Data Group [37], and the predictions from constituent quark models [38]. In the second model, henceforth called model B, in fitting the data, we let the rather poorly determined coupling constants used in model A, vary within the ranges permitted by the estimated broken SU(3)-symmetry or by the uncertainties (δ) corresponding to the ranges reported in the PDG [37] . More precisely, those adjustable parameters are allowed to vary within ±2δ. Accordingly, the fixed and adjustable parameters within our models can be classified into three sets, as explained below.
Set I: The coupling constants πNN, πNN * , and KNY * channels can be found in the literature. They are determined from using either the SU(3) predictions or from the partial decay widths listed by the Particle Data Group [37]. Those coupling constants are listed in Table I and  and Y * → πY,KN decay amplitudes tabulated in Refs. [38,39], the resonance coupling constants, as defined by the effective Lagrangians given in Appendix A, can be determined straightforwardly. These coupling constants are listed in the 4 th column of Table II and are used, with no adjustments, in our construction of model A. In model B, they are treated as adjustable parameters, within ±2δ as explained above.
Set III: The set includes three categories, and were treated as free parameters in constructing both models A and B, as listed in Table III.
ii) The off-shell parameter for describing the propagator of the spin 3/2 resonances, as introduced in Ref. [41]. For simplicity, we assume this off-shell parameter, X in Table III, is the same for all three spin 3/2 resonances considered.
iii) The K * NY coupling constants for evaluating K * -exchange mechanism illustrated in Fig. (2c).
At this point, we wish to summarize the content of our models A and B, and discuss briefly the extracted free parameters. In the fitting procedure, to save computation time, we have used a data-base of about 500 points for differential cross-sections and polarization asymmetries in the whole energy range of interest here. However, the resulting fits are compared with the complete data-base, and a representative set of data are shown in the rest of this Section.

Model A:
As described above, only parameters listed in Table III  of the data. The model A gives a reduced χ 2 of 3.28.

Model B:
As mentioned above, the parameters in Table I  In the following, we compare the results of models A and B with relevant data. For the π − p → K • Σ • reaction, the situation is different: the model B shows a significantly better agreement with the data up to W ≈ 2.1 GeV. At the two highest energies, models A and B produce comparable results and they both miss the bump around cos(θ) ≈ 0.3.
The main gross features of our results might be explained by the ingredients of the reaction mechanisms in our models. The K • Λ channel is dominated by the N * resonances.
In our models the included resonances are around M ≈ 1.7 GeV. To cure the above short comings, we probably need to include higher mass resonances, especially the P 13 (1900). This hypothesis is endorsed by the results reported in Ref. [18]. In the case of the K • Σ • channel, the ∆ resonances embodied in our models are around M ≈ 1.9 GeV and ensure a much better reproduction of the data.  Fig. (4).
We have noticed that by loosening the constraints on the adjustable parameters (±3δ instead of ±2δ), the model-data agreement gets improved for the K • Λ channel, especially at backward angles. However, we feel that the extracted values are less meaningful.
Interpretation of recent data from JLab [3] and ELSA [4] within a constituent quark model is in progress [9]. That work will allow us to determine the most pertinent resonances with respect to strangeness electromagnetic production. Afterwards, the present formalism will be used to imbed those resonances into planned future coupled-channel investigations of associated strangeness photo-and electro-production.

B. Polarization asymmetry for
The quality of the final state hyperon polarization asymmetry data, shown in Figs. 6 and 7 is clearly very poor. Nevertheless, as already noticed by the Giessen Group [18], the inclusion of those data in the fitting procedure has a significant effect on the extracted  Fig. (4).
Data are from Refs. [27,30]. Tables II and III. The main features of the polarized Λ asymmetries (Fig. (6)) are that they are large and positive up to W ≈ 1.8 GeV, and above they show nodal structures. The model B shows a better agreement with the data at lower energies. The short comings at higher energies could again be attributed to the lack of higher mass N * s in our models.

coupling constants reported in
The most noticeable differences between models A and B are in the shapes of the Σ− polarization asymmetry for W ≤ 1.8 GeV (Fig. 7). The higher mass ∆ resonances seem to play a less important role here than in the case of the differential cross-sections (Fig. 5).  Fig. (4).

C. Role of the nucleon resonances in the reactions
It is interesting to identify the role of nucleon resonances within our model B. To do so, we have turned off the nucleon resonances, one at a time, by putting the relevant couplings in Table II to zero, and have calculated the observable without any readjustment of the other parameters. The excitation functions, at three angles, for the cross-sections and the polarization asymmetries are depicted in Figures (8) and (9) for the reactions π − p → K • Λ and K • Σ • , respectively.
The most striking feature here is the angular dependence of the role played by each resonance.
In the K • Λ channel (Fig. 8, left column), the effects on the differential cross-sections due to the S 11 (1650) goes from highly dominant at forward angles to marginal at large backward angles.
For the P-wave resonances, P 11 (1710) and P 13 (1720), we observe strong effects at extreme angles, which also reveal large interference phenomena.
The D-Wave resonance, D 13 (1700), has a significant effect only below ≈ 1.7 GeV and show a sharp increase at intermediate and large angles. The possible role played by Dwave resonances has not been investigated in other recent works [11,18]. The polarized Λ asymmetries (Fig. 8, right column) are depicted in Fig. 9.
Here the S 11 (1650) resonance has a significant effect on both observables and at all angles.
The P 11 (1710) shows small contributions in the whole phase space for both observables, while the other P-wave with higher spin, P 13 (1720), produces significant effects at forward angles in the differential cross-section below W ≈ 2.0 GeV, and even more at large backward angles. The role played by the D-Wave resonance, D 13 (1700), in the differential cross-section increases with angle and becomes comparable to that of S 11 (1650) at large backward angles.
Finally, the polarization observable does not show any significant sensitivity to the P 13 (1720) and D 13 (1700) resonances.
Such a partial-wave decomposition has been also performed by the Giessen Group [18] on the total cross-section observables, leading also to small contributions from the P 11 resonances. Effects found there for the other three resonances are compatible with our findings. For the reaction π − p → K • Λ the two models give comparable results, and model B does slightly better at lower energies.
In the case of π − p → K • Σ • channel the situation is very different: model B gives a significantly better agreement with the data than does model A.
Both features reflect our comments about the differential cross-sections, showing that the method used to extracted total cross-section data from differential cross-section measurements is sufficiently reliable.

E. Total cross sections of KY → KY processes
Using the models we have constructed, one can predict the KY → KY amplitudes.
These amplitudes, although presently inaccessible experimentally, are needed for dynamical coupled-channel investigations of the electromagnetic production of hyperons. As an example, we show in Fig. 11 the predicted total cross sections for the We therefore expect that the more realistic model B will generate very different final state KY scattering effects on Kaon electromagnetic production reactions. Our investigations in this direction will be published elsewhere.

IV. SUMMARY AND CONCLUSIONS
Based on an extension of the dynamical model of Ref. [2], we have developed an approach to construct the coupled-channel models for describing the πN → KY and KY → KY  Fig. (4). Data are from Refs. [27,29,30,31].
to also investigate kaon electromagnetic production reactions. Undoubtedly, our objective is very limited compared to a more rigorous coupled-channel approach, which necessarily includes more channels, such as π∆, ρN, and ωN. However, our approach can be used to include additional nucleon and hyperon resonances with spin ≤ 3/2.
Given that no attempt is made to also fit the πN elastic scattering data, we solve the coupled-channel equations with a simplification that the πN → πN scattering t-matrix elements are parameterized in terms of the empirical πN partial-wave amplitudes and a phenomenological off-shell function. On the other hand, the basic non-resonant πN → KY and KY → KY transition potentials are derived rigorously from effective Lagrangians using a unitary transformation method.
We have constructed two models. The first one (model A) is built by assuming that all A in fitting the data. Good fits to the available differential cross section and spin observable data for π − p → K • Λ, K • Σ • have been achieved. The investigated kinematics region in the center-of-mass frame goes from threshold to 2.5 GeV.
The constructed models will facilitate coupled-channel studies of kaon photo-and electroproduction reactions. In particular, the predicted KY → KY amplitudes, which are inaccessible experimentally, are needed to predict coupled-channel effects, such as that due to the γN → KΛ → KΣ transition. Our effort in this direction will be published elsewhere. The effective Lagrangians used in this work are given in this appendix for reference.

Born term interaction
The 0 − meson-1 2 + baryon interactions are usually described using either pseudoscalar(PS) or pseudovector (PV) coupling, In this work, the pseudovector coupling is used for both π and K sectors. Using SU (3) symmetry as discussed later, we can express the interaction Lagrangians in each particle basis. For example, the Lagrangian for the (K + p Λ) vertex can be written as where the field operators are denoted by the particle's identity.

b. Numerical considerations
The central values of two main KYN couplings are determined using Eqs. (B3) and (B4), with α D = 0.644 (Ref. [47]), and g πN N = 14.11 (Ref. [42]). For those couplings, the allowed ranges in the fitting procedure are in line with the findings of a recent work [48] based on generalized Goldberger-Treiman relation combined with the Dashen-Weinstein sum rule, which puts the following uncertainties on the g KΛN and g KΣN couplings: ±36% and ±55%, respectively. We hence find: