Nonrelativistic one-hadron-exchange k−p interaction model

We have derived a nonrelativistic K−p interaction model as one-hadron-exchanges. The exchanged particles are the scalar meson σ, the vector mesons ω, ρ, the hyperons Λ, Σ, and the resonances Λ* (1600), Σ*(1385). Parameters in this model are determined by fitting to experimental data of spin-averaged differential cross section for kaon laboratory energy range of 51.27 MeV up to 900.64 MeV. The K-p scattering is calculated using a 3D technique without partial wave expansion.


Introduction
One of interesting topics in the field of nuclear and particle physics is how hadrons interact with each other. The purpose of this work is to derive an interaction model for Kp to be used in calculations of nonrelativistic processes. The interaction is formulated as one-hadron-exchange potential, since it is more practical in the application. The exchanged hadrons are the scalar meson σ, the vector mesons ω, ρ, the hyperons Λ, Σ, and the resonances Λ* (1600), Σ*(1385). The model is derived from Feynman diagrams describing Kp scattering by referring to the work in Reference [1]. Note, we also add the Σ* exchange from K + p model to the calculation [7]. To make it applicable to nonrelativistic calculations, we perform some reduction following Blankenbecler-Sugar reduction [2]. The parameter of the potential are determined by fitting to the data of the spin-averaged differential cross section for kaon laboratory energy range of 51.27 MeV up to 900.64 MeV.
As nowadays experiments can be done at higher energies, suitable theoretical calculation technique is needed. This technique does not use the standard partial wave expansion and is usually called a threedimensional (3D) technique. We implement a 3D technique described in Reference [3] to calculate Kp scattering.

Kp scattering
The potential model is derived from Feynman diagrams describing scattering and is modified using Blankenbecler-Sugar reduction, so it can be used in nonrelativistic process. Parameters of this model are obtained by fitting to the experiment data using MINUIT subroutine. The experiment data are the spin-averaged differential cross section for Kaon laboratory energy of 51.27 MeV to 900.64 MeV [4,5].  [3], we define the basis state as a combination of the free state , the spin state , and the total-isospin state of the system: with being the relative momentum, the spin projection in an arbitrary fixed z axis, the total isospin, and the z-component of the total isospin. The normalization of the basis state of Equation (1) is (2) and the completeness relation is as follows

Potential matrix elements
In the 3D basis state given in Equation (1), we define the potential matrix elements as (4) with is the potential matrix elements in the momentum space for the given total isospin. The general form of is with being spin-independent functions and is a spin-operator, where is the Pauli matrix. Substituting Equation (5) into Equation (4) we get the potential matrix elements as follows For ^ ^, we get ^ Because of the Kronecker delta, we can put as an overall factor in Equation (7). Thus, the potential matrix elements ^ depends on the azimuthal angle

T-Matrix Elements and Spin-Averaged Differential Cross Section
We define the T-matrix elements in the 3D basis state similarly as (12) The T-matrix elements defined in Equation (12)  The matrix elements satisfy the following integral equation with being defined as We obtain a symmetry relation for as Applying Equation (17), we obtain a symmetry relation for being similar to that for given in Equation (11). Thus, As usual in scattering calculations, the incident momentum is chosen as the z-axis, thus and  (18), we obtain the spin-averaged differential cross section as In Equation (20) The second diagram applies also to * and * exchange [7].
We construct the Kp interaction model as a one-hadron-exchange potential. The Feynman diagram of the interaction is shown in Figure. 1. Note that we also include the resonance of the hyperon ( * and *). After applying the Blankenbecler-Sugar reduction [2], we obtain the interaction model in operator form as with and being the momentum transfers or momenta of the particles being exchanged:

Results and discussion
We fit to the experiment data of the differential cross section for Kaon laboratory energy of 51.27 MeV to 900.64 MeV [4]- [5] to get the parameters used in this model. First, we give the initial values of the parameters that are to be fitted with the experiment data to obtain the new values of the parameters. The initial values of the parameters are taken from Reference [1], but for the nucleonnucleon-meson vertices we refer to Reference [6]. The unfitted parameters are shown in Table 1. Note that mass is not a constant, thus it also has to be fitted with the initial value of 720 MeV and 550 MeV for and , respectively. The indices 0 and 1 for the particle are the total isospin of the Kp system. The initial value for the parameters are shown in Table 2. The fitting process gives ⁄ value of 10.39 and new values of the parameters as shown in Table 3. New parameter value of and mass respectively 900 MeV and 700 MeV. The ⁄ value is obtained using Equation (35).
Parameters value obtained from the fitting process are to be used in the differential cross sections calculation. We choose 3 energy to represent the low energy, medium energy, and high energy respectively 60.79 MeV, 500.54 MeV, and 900.64 MeV. The differential cross sections for those 3 energy shown in Figure 2. Figure 2. shows that at higher energies the cross sections obtained from the calculations match better the experiment data.
To see the significance of the exchanged particles, we eliminate the contribution of each exchanged particle by setting the coupling constant equals zero ( and ). After that, we recompute the differential cross sections and compare it with the previous calculations. If those two calculations show a big differences, it means the contribution of the particle is large.   Figures 3 -4 it is clearly shown that the exchanged particles , , , , and have large contribution. On the contrary, shown in Figure 5 the exchanged particles * and * show little contributions. Furthermore, the particle has a significant contribution at forward angles in medium and high energy ranges, while in low energy range it contributes significantly at forward and backward angle. The particle has significant contribution at forward angle in the whole energy range. The particle contributes significantly at all angle in all energy ranges.
particle has a significant contribution at the backward angle in all energies, but contributes largely at all angle in low energy ranges. particle has a small contribution in medium and high energy ranges. While Figure 5 shows that * and * have a very small contribution in this model. This small contribution also shown in Table 3 that * and * have an almost-zero coupling constant value.

Conclusion
We have derived a Kp interaction model as one-hadron-exchanges for nonrelativistic calculations. The potential parameters are determined by means of fitting processes to experimental data of differential cross section for kaon's laboratory energy up to around 900 MeV. We obtain a ⁄ value of 10.39, which is a large number. The model, Thus still needs to be improved. We check the contributions from each particle exchange and found that * and * give a very small contribution.