Low energy kaon-hyperon interaction

In this work we study the low energy kaon-hyperon interaction considering effective chiral Lagrangians that include kaons, $\sigma$ mesons, hyperons and the corresponding resonances. We calculate the scattering amplitudes, and then the total cross sections, angular distributions, polarizations and the $S$ and $P$ phase shifts.


I. INTRODUCTION
Until today a subject that is very interesting and remains not very well studied is the low energy hyperon interactions. Despite the fact that experimental data for many hyperon processes are not available (as for example the KΛ and KΩ interactions) and that by the theoretical side they are not fully described, this kind of interaction is a fundamental element for several physical systems of interest.
In the study of the hypernuclei structure [1]- [4], the knowledge of the nucleon-hyperon and hyperon-hyperon interactions is an essential aspect. In order to understand these interactions, and to determine the potentials of interest, an accurate understanding of the meson-hyperon interactions is needed.
Another system where hyperon interactions are required is in the study of the hyperon stars. After the proposal of the hypothesis where hyperons could be produced inside neutron stars at high densities, many models have been proposed, as for example in [5]- [8], and the effect of the presence of hyperons in the equations of state, and consequently in the determination of the star masses have been studied. The indeterminations in the nucleon-hyperon and hyperon-hyperon potentials cause difficulties in the understanding of these stars.
In high energy physics this kind of interaction is very important also. When studying hyperon polarization, produced in proton-nucleus and nucleus-nucleus collisions [9]- [18], in [19]- [21] the final interactions of the hyperons and antihyperons with the produced pions is a central ingredient in order to explain the final polarizations. As it has been shown, the effect of the hyperon interactions with the surrounding hot medium, composed predominantly of pions, is very important. The observed differences between the polarizations of hyperons and antihyperons are very difficult to be explained in another way. The effect of the final kaon-hyperon interactions has not been considered yet, and it may cause corrections in the final polarization. For this reason, this work is very * magwwo@gmail.com † barros.celso@ufsc.br important and this effect must be investigated. Recent results from RHIC [22] and even the hyperons produced in the LHC may be studied in a similar form, and in order to obtain accurate results these interactions must be considered. For these reasons, this work will be devoted to the study of the kaon-hyperon (KY ) and antikaon-hyperon (KY ) interactions. This work may be considered as a continuation of the study proposed in [23]- [25], where the pion-hyperon interactions have been described with a model based in effective chiral Lagrangians where the exchange of mesons and baryons has been taken into account. In this model [23], the resonances dominate many channels of the reactions as it may be seen in the results. This behavior may be considered as an experimental feature, fact that is similar to what happens in the low energy pion-nucleon interactions, where the isospin 3/2 and spin 3/2 channel is dominated by the ∆ ++ resonance. Comparison with the data from the HyperCP experiment [26], [27] shows a very good accord with the results obtained for the πΛ scattering in [23]. So, the work that will be shown in this paper is based on the ideas presented in this model. This paper will present the following content: in Sec. II, the basic formalism will be shown, in Sec. III, the kaon-lambda (KΛ) interactions will be studied, in Sec. IV, the antikaon-lambda (KΛ), and in Sec. V, the kaonsigma will be shown. In Sec. VI we will present the antikaon-sigma interactions (KΣ) and finally, the discussions and conclusions in Sec. VII. In the Appendix, some expressions of interest will be presented.

II. THE METHOD
In order to study the KY and KY interactions, we will use a model proposed with the purpose of studying the low energy pion-hyperon interactions [23]- [25], that is based in an analogy with models successfully used to describe the πN interactions considering chiral effective Lagrangians. These interactions are very well studied, as for example in [28]- [31], both theoretically, where many models have been proposed, and experimentally, with a large amount of experimental data available. A basic characteristic of this system is the dominance of reso-nances in the scattering amplitudes at low energies. The ∆ ++ , for example, dominates the cross section of the π + p scattering at low energies. As this particle has spin 3/2 and isospin 3/2, it may be introduced in the theory by considering a Lagrangian in the form of eq. (2). In the study of the pion-hyperon interactions [23], a similar behavior has been observed, so we expect that in the KY interactions it also occurs.
In this section we will present the basic formalism that will be used to study the kaon-hyperon interactions (that is the same one worked out in the study the pion-hyperon interactions [23]) and how the observables may be obtained. In the method that will be followed in this work, some important characteristics of the interacting particles will be implemented, the spin, the isospin, and the masses of each one of them. These characteristics determine which Lagrangian have to be used in order to build the model.
For example, in [28], the Lagrangians considered to study the πN scattering are given by where N , ∆, φ, ρ are the nucleon, delta, pion, and rho fields with masses m, m ∆ , m π , and m ρ , respectively, µ P and µ n are the proton and neutron magnetic moments [36], M and τ are the isospin recombination matrices, and Z is a parameter representing the possibility of the off-shell-∆ having spin 1/2. The parameters g, g ∆ and g 0 are the coupling constants. In [23] similar Lagrangians have been used to study the pion-hyperon interactions, and in this work the same procedure will be adopted. So, in the following sections these Lagrangians will be adapted to the kaon-hyperon systems.
Calculating the diagrams, considering the interactions described by the Lagrangians above for an arbitrary process, the scattering amplitudes may be written in the form that is a sum over all the I isospin states where P I is a projection operator,the indices α and β are relative to the initial and final isospin states of the π, and T I is an amplitude for a given isospin state that may be written as where u( p) is a spinor representing the initial baryon, incoming with four-momentum p µ . The final baryon has a spinor u( p ), four-momentum p µ , and k µ and k µ are the incoming and outgoing meson four-momenta. The amplitudes A I and B I are calculated from the diagrams. So, if these amplitudes are determined, the T I amplitudes may be obtained and then we will be able to compute the observables of interest. The scattering matrix for an isospin state is given by the expression which may be decomposed into the spin-non-flip and spin-flip amplitudes f I (k, θ) and g I (k, θ), defined in terms of the momentum k = | k| and x = cos θ, θ the scattering angle, as wheren is a vector normal to the scattering plane, and may be expanded in terms of the partial-wave amplitudes a l± with These amplitudes may be calculated using the Legendre polynomials orthogonality relations where E is the baryon energy in the center-of-mass frame and √ s is given by a Mandelstam variable (see the Appendix). At low energies the S (l =0) and P (l =1) waves dominate the scattering amplitudes, and for higher values of l the amplitudes are much smaller (almost negligible), so they may be considered as small corrections.
Calculating the amplitudes at the tree-level, the results obtained will be real, and then violate the unitarity of the S matrix. As it is usually done, we may reinterpret these results as elements of the K reaction matrix [23]- [25] and then obtain unitarized amplitudes The differential cross sections may be calculated using the previous results and integrating this expression over the solid angle we obtain the total cross sections The phase shifts are given by and finally, the polarization , An important task to achieve is to determine the coupling constant for each resonance that will be considered. We will adopt the same procedure considered in [23], comparing the amplitude obtained in the calculations with the relativistic Breit-Wigner expression, that is determined in terms of experimental quantities where Γ 0 is the width, k 0 = | k 0 | is the momentum at the peak of the resonance in the center-of-mass system, m r is its mass and J the angular momentum, considering the data from [36]. We will consider the coupling constant that better fits this experession in each case.
In the following sections we will apply this formalism in the study of the reactions of interest.

III. KAON-LAMBDA INTERACTION
Since the Λ hyperon has isospin 0, the scattering amplitude for the KΛ interaction will have the form with the variables defined in section II. Comparing this expression with (5), we have a simple result, P βα 1/2 = 1, as the kaon has isospin 1/2, and just one isospin amplitude.
In FIG. 1 we show the diagrams and the particles considered to formulate the KΛ interaction. The particles considered for each diagram are shown in Tab. I.  For the calculation of the contribution of particles with spin-1/2 (N and Ξ) in the intermediate state (FIG. 1a  and c ), the Lagrangian of interaction is (considering the necessary adaptations from eq. (1)) where φ represents the kaon field, B the intermediate baryon field, with mass m B , and Λ, the hyperon field, with mass m Λ . Calculating the Feynman diagrams and comparing with eq. (20) we find the amplitudes for the N (spin-1/2) particles contribution and for the Ξ (spin-1/2) hyperon in the crossed diagram (FIG. 1c) the contribution is where u is a Mandelstam variable, defined in the appendix and g ΛKN (Ξ) are the coupling constants.
In a similar way, we adapted the interaction Lagrangian (2) for the exchange of spin-3/2 resonances, shown in FIG. 1b and Calculating the amplitude for the exchange of a spin-3/2 N * (FIG. 1b) we have For the spin-3/2 Ξ * resonance (FIG. 1d), the amplitudes are where where t, q N * , E N * are defined in the appendix and m K , m N * are the kaon and the N * masses respectively. For A andB we change the N * parameters, inserting the Ξ * ones in eqs. (29) and (30). g ΛKN * (Ξ * ) are the coupling constants.
The parameters considered in the KΛ interaction are shown in Tab. II, the masses are taken from [36].  The coupling constants g ΛKN and g ΛKΞ are determined using SU (3) [37,38] and the ones of the Λ with the resonances, by using the Breit-Wigner expresion eq. (19), as described above, in the same way it has been done in [23].
In FIG. 2, we show our results for the total elastic cross section and the phase shifts as functions of the kaon momentum k, defined in the center-of-mass frame. Figure  FIG. 3 shows the angular distributions and the polarizations as functions of x = cos θ and k.
Observing the figure we can note that the resonances, and in special the N (1650) contribution, dominate the total cross section when k ∼ 150 MeV, as it was expected. At higher energies, the other resonances also have an important effect. The polarization oscilates for k < 150 MeV, but as the momentum increases, it becomes negative.

IV. ANTIKAON-LAMBDA INTERACTION
The KΛ interations may be studied exactly in same way as it has been done in the last section for the the KΛ interactions. Now we have the contributions presented in FIG. 4, where the Lagrangians take into account the N , Ξ, Λ and φ (representing the antikaon) fields The parameters considered are given before, m K = m K , and for the crossed diagrams in FIG. 4c and d

V. KAON-SIGMA INTERACTION
In this case the interacting particles have isospin 1/2 and 1 (K and Σ respectively). So, we have two possible total isospin states, 1/2 and 3/2, which allow also the exchange of ∆ particles.
The scattering amplitude has the general form and the considered projection operators are P βα where Q is the M matrix for ∆ (I = 3/2) or τ matrix for the N * and Ξ * (I = 1/2).
The resulting amplitudes for nucleons in the intermediate state (FIG. 7a) are and for the Ξ exchange in the diagram 7d (55) Figure 7b gives and for the crossed diagram shown in FIG. 7e, where a Ξ * exchange is taken into account, where the expressions forÂ,B,Â ,B , a 0 , b 0 , c 0 , d 0 , c z and d z are the same ones presented in Sec. III, but replacing the Λ hyperon for the Σ hyperon.
Thus, to calculate the observables for each reaction we use (44) and (45), resulting in the amplitudes  To determine the coupling constants g ΣKN , g ΣKΞ and the ones with resonances we take into account the same arguments presented in Sec. III.
Using the isospin formalism for the elastic and the charge exchange scattering, we can determine the amplitudes for the reactions (that we name C i , for simplicity) and with these amplitudes we can calculate all observables of interest. The total elastic cross sections and the phase shifts as functions of the kaon momentum are shown in FIG. 8. Figures 9 and 10 show the angular distributions and the polarizations.

VI. ANTIKAON-SIGMA INTERACTION
In this case, we will proceed in the same way as we have done in the last section for the KΣ interaction. The diagrams to be considered are shown in FIG. 11.
In this case we have the following reactions For diagram 11d the resonance to be considered is N * (1900). Using the parameters given in Tab. IV we have obtained the results for the KΣ scattering shown in Figures 12, 13 and 14.   son, we believe in the formulation of the proposed model at low energies (k < 0.4 GeV). In [23] a similar behavior has been observed, and the preditions of the model, when compared with the HyperCP data, showed to be very accurate.
For the KΛ scattering, at the Ω hyperon mass  considering that for other similar decays, no CP violation has been observed [26], [27]).
In the study of high energy hyperon polarization, produced in proton-nucleus and in heavy ion collisions, if we consider the polarizations obtained in the final-state interactions, the processes studied in this work may have some effect in the final polarization of the Λ, Λ, Σ and Σ hyperons produced in these reactions. In special, in some reactions of Fig. 14 considerable polarization may be observed, and some signs of this fact probably may be observed in the Σ and in the Σ polarizations. Probably this effect is smaller than the one obtained when considering the πY interactions [19]- [21], but as these in-teractions (KY ) provide polarizations of different signs, it is possible to obtain some differences in the final results. And certainly, a more accurate final result will be obtained.
These reactions are also important in the determination of nucleon-hyperon and hyperon-hyperon potentials, as they are subprocesses of these interactions, and as it has been discussed before, these interactions have a fundamental importance in the structure of the hypernuclei and in the hyperon stars.
It must be pointed that the study of the Ξ hyperons and other related reactions have been left for future works.
So, as it has been shown, the study presented in this work is very important for many physical systems of interest, and for this reason, must be continuated and improved in future works.

VIII. ACKNOWLEDGMENTS
We would like to thank CAPES for the financial support.

IX. APPENDIX
Considering a process where p and p are the initial and final hyperon four-momenta, k and k are the initial and final meson four-momenta, the Mandelstam variables are given by s = (p+k) 2 = (p +k ) 2 = m 2 +m 2 K +2Ek 0 −2 k. p , (80) u = (p −k) 2 = (p−k ) 2 = m 2 +m 2 K −2Ek 0 −2 k . p , (81) In the center-of-mass frame, the energies will be defined as and the total momentum is null We also define the variable where θ is the scattering angle. Other variables of interest are where m, m r and m K are the hyperon mass, the resonance mass and the kaon mass, respectively. For the energy and the 3-momentum of the intermediary particles we also have the relations where E B * and q B * are the energy and the momentum of intermediary baryon B * in the center-of-mass frame, respectively.