Pion-pion, pion-kaon, and kaon-kaon interactions in the one-meson-exchange model

The SU ( 3 ) -symmetric one-meson-exchange mechanisms are used in order to describe the ππ , π K , and KK interactions at low energies ( √ s < 1 . 5 GeV). In the model, both s - and t -channel exchange diagrams are considered in ππ and π K scatterings, in which the coupling constants and cut-off masses in form factors are determined to reproduce experimental phase shifts with J < 3 . The form factors are examined both in monopole and Gaussian types. At the same time as giving quantitative predictions for KK interaction with J = 0 ∼ 3 , the investigation of the η channel effects, the ηη channel in ππ scattering, and the η K channel in π K scattering are also interesting discussion points in this paper. The scattering lengths of all states in ππ , π K , and KK interactions and the pole positions of the S -matrix are shown.


Introduction
The strong forces acting between mesons at low energies ( √ s < 1.5 GeV), especially the pionpion (ππ) and pion-kaon (π K ) forces, have been the subject of investigation for some decades, experimentally as well as theoretically. In many theoretical models, such as the SU (3) symmetric meson-exchange models [1] and the SU (3) and SU (2) chiral perturbation models [2,3], hadronhadron interactions have been treated for two decades. Oller and Oset [2] used a non-perturbative coupled-channel approach to deal with the meson-meson interaction in the strangeness S = 0 sector at energies below √ s = 1.2 GeV and found the poles for the f o (980) and a o (980) resonances. In 1999, a non-perturbative method, which combined constraints from chiral symmetry breaking and coupled-channel unitarity to describe the meson-meson interaction up to about 1.2 GeV in perturbation theory, was proposed by Oller, Oset, and Pelaez [3].
The purpose of our work is to construct a unified hadron-hadron potential model that is appropriate for all baryon-baryon (BB) [4], meson-baryon (MB) [5], and meson-meson (MM) interactions. This work is the first step towards extending our model to include meson-meson interactions. In this step, we do not explicitly try to obtain a model consistent with the BB and MB potential model.
In this paper, we not only construct the ππ and π K interactions but also give a prediction of the K K (KK ) interaction based on the SU (3)-symmetric one-meson-exchange mechanisms. Concretely, by using common parameters (coupling constants and form factors), we determine the coupled-channel ππ-KK -ηη and π K -ηK potentials in order to reproduce low-energy ( √ s < 1.5 GeV) ππ and π K scatterings and quantitatively predict the KK interaction. In Sect. 2, we give the formalism of the model. The vector, scalar, and tensor meson-exchange potentials are determined based on the SU (3) symmetric interaction Lagrangians. The method, which is used to calculate the potentials, is described in detail in Sect. 2.2. The potentials are given in the momentum space. The coupled-channel Lippmann-Schwinger (LS) equations are solved with relativistic kinematics in order to obtain the T -matrix. The S-matrix is then acquired from the onshell T -matrix. Solutions of the LS equations are obtained by a kind of numerical matrix inversion method that was developed by Haftel and Tabakin [6]. In s-channel exchange diagrams, we introduce bare masses for exchanged mesons and perform renormalization calculations to determine the poles of the S-matrix that correspond to physical resonance poles. In order to discuss the η channel effects, we consider the potential model without η channels. We find that the η channel effects are not small, but these effects can be fairly well compensated by the readjustment of coupling constants and form factors mainly in s-channel exchange diagrams.
In Sect. 3, numerical results are compared with experimental analysis and quantitative predictions for the KK interaction are given. In Sect. 4, we mainly discuss the effects of η channels in ππ-KK -ηη scattering. The last section is devoted to summarizing this paper and giving future prospects.

Formalism of the model
In meson-meson potentials, we consider two kinds of one-meson-exchange diagrams, i.e., the t-and s-channel exchange diagrams that are shown in Fig. 1. We do not need to treat the u-channel exchange, because it is already taken into account as the t-channel exchange. For example, the K * exchange in the π K interaction, which provides an exchange force, is regarded as a t-channel We use ρ, K * , ω, φ, f 2 , , and κ as exchanged mesons. We emphasize that both the t-and s-channel ρ(K * ) exchanges have to be taken into account for the ππ(π K ) channel to obtain a quantitative description of the experimental phase shifts. The scalar-isoscalar (1400), the isoscalar tensor (spintwo) meson f 2 (1270), and the strange scalar meson κ(1430) in s-channel exchange are necessary but their t-channel contributions are neglected because their large mass bring too short range an interaction. Both the t-and s-channels are used to construct the π K interaction with ρ, K * , and κ exchanged mesons. The KK interaction acts only on t-channel exchange diagrams with ρ, ω, and φ meson exchange. When extending the model for investigating the effect of ηη and ηπ channels in ππ scattering and ηK in π K scattering, and K * contribute to the coupling.

Interaction Lagrangians
For vertices in meson-exchange diagrams, we assume the following types of interaction Lagrangians L ppm (m = s, v, t): In the above equations, φ s , φ p , φ μ v , and φ μν t are the field operators for scalar, pseudoscalar, vector, and tensor mesons, respectively. By treating the interaction Hamiltonian H I = − Ld 3 x in timeordered perturbation theory, we construct the potentials from scalar-, vector-, and tensor-exchange diagrams.
For all interaction Lagrangians, we assume the derivative type of coupling. This type of coupling seems to be favorable from the viewpoint of chiral symmetry and the low-energy theorem. By comparing with the interaction Lagrangian L 2 and L 4 in the chiral perturbation model [3], we find that one-vector-exchange potentials give a physical interpretation for the main part of the L 2 contributions while the scalar-and tensor-exchange potentials, which provide short-range interaction, only pick up some essential contributions from copious L 4 contributions.

One-meson exchange potentials
In our calculations, all kinematical variables are defined in the center of mass system of two interacting mesons and employ relativistic kinematics. In practice, using the notation of Fig. 1, we define: 1 2 , ω 1 = m 1 2 + p 1 2 , The above equations are applied for t-channel exchange. For s-channel exchange, the equations are almost the same but: For vector exchange, using the relations, we always find where μ ( p α , λ α ) is the polarization vector for a vector meson with momentum p α and helicity λ α . By using the interaction Lagrangian L ppv given in Eq. (1), we obtained the potential for t-channel 3 vector meson exchange given by: The form factors F (t) will be given in Sect. 2.4. We need an additional factor 1 2 k with k = 0 when both the initial and final states consist of different particles: k = 1 when one of the states consists of identical particles, k = 2 when both are identical.
For the potential by s-channel vector meson exchange, we obtain: where p μ α = (m α , 0), ω p = ω 1 + ω 2 , ω p = ω 1 + ω 2 , m 0 α is bare mass, and m α is physical mass. The s-channel potential for gradient coupling is given as follows: In order to calculate the potential by the s-channel tensor meson ( f 2 exchange), we need the polarization tensor of the spin-two meson: where Then, the potential for the tensor meson is given by: where P 2 (cos θ) is the Legendre polynomial.
For the tensor meson f 2 and scalar meson 1 , we assume them to be flavor SU (3) singlet mesons (T 1 = f 2 and S 1 = ). In the above expressions, we assume symmetric (D-type) coupling for scalar and tensor mesons and anti-symmetric (F-type) for vector mesons. The singlet-octet mixing angles θ S , θ V , and θ P are introduced by: The values θ P , θ V , θ S are determined by the SU (6) quark models, θ P = 24 o by the Gell-Mann-Okubo mass formula, θ P = 35.3 o by the ideal mixing between ω and φ. By using the SU (3) coupling constants and the mixing angles θ P , θ V , we determined the coupling constants g abc , which correspond to g 1 and g 2 in Eqs. (5, 7, and 9).

Form factors
Form factors play a very important role in meson-meson scattering. In this paper, we attempt two types of form factors: monopole [1] and Gaussian type [4,5]. Meson-exchange potentials contain form factors that express the effects of short-range physics beyond the one-meson-exchange mechanism, e.g., many-meson exchange or the intrinsic structure of hadrons. Therefore, we treat the form factors as phenomenological quantities that are determined by fitting to the experimental data. For t-channel exchange, the form factor of monopole type can be written as follows: where m α and q α are the mass and 3-momentum of the exchanged mesons in the t-channel, respectively. For s-channel exchange, we take two kinds of monopole form factors: where ω p is the total energy of the incoming (outgoing) state. The fourth-order monopole type is necessary for the gradient coupling of the scalar meson as well as the tensor meson to obtain sufficient convergence in the high-momentum region.
The form factors of Gaussian type for t-and s-channel exchange are defined respectively by: We should note the difference between the two kinds of form factors. In t-channel exchange, for the momentum transfer p α ∼ = 0, which corresponds to forward scattering. In such a situation, short-range physics does not work and the latter may be reasonable. On the other hand, in s-channel exchange, F . But, in this case, the "momentum transfer" |p 1 | = |p 2 | is not small in general. The short-range properties of mesons may be responsible. Therefore, Gaussian form factors seem to be more reasonable than monopole ones.

Description of scattering
We start with the partial wave expansion of the quasi-potential V i i J ( p p|z).
where θ is the angle between p and p . The T -matrix can be obtained from the Lippmann-Schwinger (LS) equation below: By solving the LS equation, we determine the T -and S-matrices. We will solve the integral equation. Firstly, we solve the equation given by (P is ignored for complex z) We define ρ  The S-matrix is obtained from the on-shell T -matrix by using the equation From the S-matrix we can calculate the scattering lengths and pole positions of the meson-meson interaction. The S-matrix element in coupled-channel scattering is parameterized by where η(z) is called inelasticity and δ i j are phase shifts. In this paper, we only discuss these quantities for the lowest channels (the ππ, π K , and KK channels). The potentials from the s-channel exchange diagrams have a pole at √ s = m 0 (bare mass of the exchanged meson). Such singular potentials should be treated by the renormalization procedure. Details of this procedure are given in Refs. [7] and [8].

Results
In this section, we show the quantitative results of phase shifts δ I J , in which I is the isospin and J is the total angular momentum of the ππ or π K system. In addition, we also propose quantitative predictions for the KK interaction.

Parameter values in the model
We use 18 free parameters to fit the meson-meson data. They are 4 coupling constants (g), 9 cut-off masses ( ) and 5 bare masses (m 0 ). There are two separate sets of parameters, as required to fit the data for each kind of form factor. All the parameters determined so as to reproduce ππ and π K scattering are given in Tables 1 and 2. All coupling constants in the vector meson-exchange process are interrelated to the ππρ coupling by SU (3) symmetry. Others involving the scalar ( , κ) and tensor ( f 2 ) mesons are independently adjusted to the data.

ππ scattering
In Fig. 2a, we show the isospin I = 1 P-wave ππ phase shift, δ 1 1 . In this partial wave, the ρ meson exchange makes a very important contribution in both the s-and t-channel exchange mechanisms. 7   The s-channel ρ meson exchange almost dominates the strong attraction of phase shift δ 1 1 . The agreement of the theoretical results with the experimental data indicates that both the position and the width of the ρ meson are described very well.
In fact, the resonance of I = J = 1 is at m = 765 MeV with a width of 146 MeV for the monopole form factor case (see Table 5). For the Gaussian form factor case, the resonance is at m = 772 MeV with a width of 138 MeV.
In Fig. 2b, we show the results of the I = 0, J = 2 phase shifts, δ 0 2 , which agree rather well with the experimental data. By including s-channel f 2 meson exchange in the potential, we get much more attractive phase shifts and a physical f 2 pole in ππ scattering. The I = 0, J = 2 channel is influenced by resonance in the s-channel-the f 2 (1270) meson. In fact, we have a resonance at m = 1253 MeV with a width of 164 MeV for the monopole form factor case (see Table 5). For the Gaussian case, the resonance is at m = 1265 MeV with a width of 168 MeV, nearer to the f 2 (1270) than the monopole case.
The I = 2, J = 0 and I = J = 2 states describe a simple one-channel of ππ scattering with only ρ meson exchange. Phase shifts δ 2 0 (see Fig. 3a) and δ 2 2 (see Fig. 3b) are all repulsive. In fact, the repulsive nature of these phase shifts definitely comes only from t-channel ρ meson exchange. To obtain the I = 2, J = 0 phase shifts, we use the coupling constant and the cut-off mass of the ππρ vertex in the t-channel. Our calculated I = J = 2 ππ phase shifts are fairly large in comparison with the experimental data.  The most interesting result of ππ interaction is the phase shift of the scalar-isoscalar channel, δ 0 0 (see Fig. 4). In this partial wave, KK channels play an essential role in reproducing the experimental data. In fact, the resonance at around √ s = 980 is interpreted as a quasi-bound state of the KK system, which is produced by strongly attractive t-channel vector meson (ρ, ω, φ) exchange contributions. In the S-wave with I = 0, the phase shift is very difficult to reconcile with the experimental data. The derivative coupling increases strongly with the pion momentum above 1 GeV and gives a sharp rise to a potential that difficultly changes the phase shifts below 1 GeV. When we adjust carefully the 1 meson with the coupling constant, the cut-off mass, and the bare mass, good agreement between theory and experiment is obtained throughout the whole energy range. The bare mass and the coupling constant of the 1 meson have been chosen to reproduce the phase shift in the high-energy region. As a result, both the strong direct KK interaction and the 1 meson are necessary to obtain agreement with the experiment. We have a resonance at m = 981 MeV with a width of 16.8 MeV for the monopole form factor case (see Table 5). For the Gaussian case, the resonance shifts by tens of MeV. The elasticity η 0 0 which is shown in Fig. 5  Refs. [19][20][21] for δ 1 2 0 and from Refs. [20] and [22] for δ 1 2 1 . bars in the experimental data for η 0 0 are quite large. Within the error bars, the agreement of η 0 0 with the experimental data is good.

π K scattering
Most of the parameters used in the π K interaction are determined from the previous investigation of the ππ interaction and the SU (3) symmetry relations. The t-channel exchange contributions are completely determined by the coupled-channel ππ-KK interaction, whereas additional pole contributions in the s-channel with κ meson exchange have to be adjusted separately.
The strong attractive phase shifts for the I = 1 2 P-wave π K interaction are shown in Fig. 6b. The agreement between the theoretical calculation and the experimental data is good in the low-energy region; however, it is a little large at higher energies (E > 1.0 GeV). The resonant state of the P-wave with isospin I = 1 2 , J = 1 is known as K * (892). In Table 5, we can see that the physical mass of vector meson exchange is m K * = 885 MeV with the width K * = 44 MeV for the monopole form factor and m K * = 895 MeV, K * = 40 MeV for the Gaussian form factor.
An excellent test for meson-exchange interaction is δ 3 2 0 and δ 3 2 1 . The interactions in these partial waves are completely determined by the parameters in the ππ-KK interaction. As we can see in

Prediction of KK scattering
By extending our vector meson exchange potentials without any additional parameters, we construct the KK interaction. The KK interaction is simply constructed by only t-channel with ρ, ω, and φ meson exchange. The properties of KK interaction are described by phase shifts in S-, P-, D-, and F-waves.
By investigating the contributions from ρ, ω, and φ meson exchanges, we can confirm the realizability of our predicted results. In Fig. 8a, the weakly attractive phase shift in the I = 0, J = 1 state is shown. This attractive δ 0 1 is quite reasonable because the contributions to the KK interaction are repulsive with ω and ϕ meson exchange, while with ρ meson exchange the contribution is attractive. Otherwise, the I = 1, L = 0 KK interaction is really strongly repulsive because of all the repulsive contributions from the ρ, ω, and ϕ meson exchanges. Using both types of form factors, we obtain almost the same results of phase shifts, as shown in Figs. 8 and 9. Unfortunately, we have not yet found any experimental data with which to compare these results.
From the contributions of meson exchange in KK interaction (see details in Table 3), we can see the important role of the ρ meson in the meson-exchange model. At the isospin I = 1 state, the vector meson ρ made both the KK and KK interactions repulsive. The ω and φ meson exchanges make the KK interaction repulsive and the KK interaction attractive. Therefore, we get a strongly repulsive 11    , L = J is the angular momentum. Experimental and theoretical scattering lengths for ππ and π K scattering. The experimental data are taken from Ref. [25].

Scattering lengths
In this section, we show the result for the scattering lengths of all states in ππ, π K , and KK scatterings. In our treatment, the experimental data for the scattering lengths were not used in determining the parameter values given in Table 1. All the experimental values were only reserved for checking the interaction model at very low energies. In Table 4, we show the results of scattering lengths a I J in comparison with the experimental data for all the channels of the ππ, π K , and KK interactions. For the ππ scattering in the I = 2, J = 0, 2 states, because of the repulsive nature of the phase shifts, the scattering lengths have negative values. The scattering lengths a 2 0 and a 2 2 are completely generated by the t-channel interaction. The scattering lengths of the I = 2, J = 0, 2 states are a little smaller than those of the experimental data. The influence of the meson on the scattering length a 0 0 is weaker than other t-channel interactions. However, in both the monopole and Gaussian form factor cases, a 0 0 is in very good agreement with the experimental value. In contrast, the influence of the ρ meson on a 1 1 is very strong: the calculated a 1 1 is nearly same as the experimental value in both the monopole and Gaussian types of form factors. In the case of a 0 2 , the contributions of tchannel interactions and of the f 2 meson are strong. The calculated scattering length a 0 2 is smaller than that of the experimental data. On the whole, the theoretical results of scattering lengths are in good agreement with the experimental data, but a 1 1 in ππ scattering, a

Poles and widths
In order to determine the physical mass and width of the ρ, f 2 , f 0 , κ, and K * resonances in the ππ, π K interactions, one has to search for the positions of poles in the physical amplitudes in the complex  Table 5.
For ππ and π K scatterings, there are five resonances that correspond to the physical mass of f 0 , f 2 , ρ, κ, and K * .

The η channel effects
In order to understand the effects of the η channels in the ππ and π K interactions, we investigate the ηη channel in ππ scattering and the ηK channel in π K scattering. When introducing the η channels in our model, we have to redetermine the parameters of the coupling constants of I = 0, J = 0, 2 for the ππ interaction and I = 1 2 , J = 0, 1 for the π K interaction as given in Table 1. In this section, we discuss the η channel effects in I = 0, J = 0, 2 for ππ scattering and I = 1 2 , J = 0, 1 for π K scattering. In I = J = 0 ππ scattering, we deal with the coupled-channel ππ-KK -ηη problem. Only 1 and K * meson exchange contribute to ηη channel-coupling. K * (t-channel) exchange does not cause direct ππ-ηη coupling. For this reason, the effect of the ηη channel on ππ scattering by K * exchange is very small. On the other hand, s-channel 1 exchange provides the coupling between all three channels. This effect is not small, as shown in Fig. 10. However, it can be compensated by readjustment of the bare mass of 1 and the coupling constant g ππ 1 .
When we add t-channel K * exchange in ππ-KK -ηη(πη), the phase shifts of the isospin I = 0(1) states are almost the same; therefore, the η channel effects on the K * exchange mechanism are small. The η channels with the 1 exchange mechanism have a large effect on the I = 0, J = 0 phase shift. As shown in Fig. 10, the δ 0 0 with η channel suddenly gets a large variation of shape in comparison with the experimental data. After fitting the coupling constants, cut-off parameters, and bare mass of the meson, we obtain the phase shifts in the low-energy region ( √ s < 900 MeV) to show a good agreement with the experimental data, while the phase shifts at √ s > 900 MeV are small in comparison with experimental data.
In π K scattering with I = 1 2 , J = 0, the η channel with κ meson exchange has a smaller effect than the I = 0, J = 0 state in ππ scattering. We make a small adjustment of the coupling-constant parameters in π K with the κ meson exchange reactions but the phase shifts δ   1 . The dashed lines are results with conventional monopole form factors. The solid lines are those with Gaussian form factors. The red lines represent the no-η-meson model, the green lines the model with the η meson, and the blue lines the models with the η meson in which the parameters are adjusted. The experimental phase shift analyses are taken from Refs. [19][20][21] for δ 1 2 0 and from Refs. [20] and [22] for δ 1 2 1 .
the π K -ηK model. For the purposes of confirming the effects of the η meson in the π K -ηK model in I = 1 2 , J = 1, we adjust the cut-off parameter and bare mass of the K * meson. After adjusting, we find that the phase shift δ 1 2 1 in Fig. 11b agrees with the experimental data. As a result, the η channel effects in ππ scattering are larger than those in π K scattering and fitting δ 0 0 is not easy to do. In the future, we will refine our model with η channels for δ 0 0 , δ

Summary
For the purposes of describing the ππ, π K , and KK interactions in a unified way, we constructed a potential model that assumes the one-meson-exchange model based on the interaction Lagrangian that satisfies the SU (3) symmetry. In order to determine the parameters of the potential, which include form factors and coupling constants, we used the phase shift data of the ππ and π K scatterings at low energies. We gave a prediction for the KK interaction by extending the ππ and π K scatterings. In the KK interaction, t-channel ρ, ω, and φ meson exchange plays an important role. In order to confirm resonance positions, we calculated the S-matrix in the complex energy plane. The scattering lengths of all states in ππ, π K , and KK scatterings were proposed. Moreover, the ππ-KK -ηη and π K -ηK interactions have been discussed to investigate the role of the η meson in the ππ and π K scatterings. In the future, we will treat the complete mesonmeson interaction (ππ-πK -πη-ηη and π K -ηK ) with the meson-exchanged model. In the πη-KK interactions, we will examine the I = 1, J = 0 states to show the phase shift and look for the pole of the a 0 meson. In the KK interaction, we will propose the phase shift δ 0 1 and search for the existence of the ϕ, ω poles.