Near threshold kaon-kaon interaction in the reactions e+ e- -->K+ K- gamma and e+ e- -->K0 K0bar gamma

Strong interactions between pairs of the K+ K- and K0 K0bar mesons can be studied in radiative decays of phi(1020) mesons. We present a theoretical model of the reactions e+ e- -->phi -->K+ K- gamma and e+ e- -->phi -->K0 K0bar gamma. The K+ K- and K0 K0bar effective mass dependence of the differential cross sections is derived. The total cross sections and the branching fractions for the two radiative phi decays are calculated.


Description of the theoretical model
The kaon-kaon strong interaction near threshold is largely unknown. Also the parameters of the scalar resonances f 0 (980) and a 0 (980) are still imprecise. The φ(1020) meson decays into π + π − γ, π 0 π 0 γ and π 0 ηγ have been measured, for the φ transition into K 0K0 γ only the upper limit of the branching fraction has been obtained in Ref. [1] but there are no data for the φ → K + K − γ process.
In this paper we outline a general theoretical model of the e + e − reactions leading to final states with two pseudoscalar mesons and a photon. At the beginning we derive the amplitude A(m) for the e + e − → K + K − γ process. It is a sum of the four amplitudes corresponding to diagrams (a), (b), (c) and (d) in Fig. 1: Figure 1. Diagrams for the reaction e + e − → K + K − γ with final-state K + K − interaction. The K + K − elastic amplitude is labelled by T and R denotes the difference of the K + K − amplitudes T in Eq. (4).
One can show that the amplitude , is the inverse of the kaon propagator, m K is the charged kaon mass, the four-vectork = (0,k) with the unit three-vectork = k/|k|. In the above expressions q is the photon four-momentum, p = p e + + p e − is the φ meson four-momentum, ν is the photon polarization four-vector and J µ is defined as where e is the electron charge, s = (p 2 is the Mandelstam variable, v and u are the e + and e − bispinors, respectively, γ µ are the Dirac matrices and F K (s) is the kaon electromagnetic form factor. The K + K − elastic scattering amplitude is given by where m 2 = (k 1 + k 2 ) 2 is the square of the K + K − effective mass andT (m) is the KK scattering operator. The on-shell K + K − amplitude can be expressed as . The four-momenta of kaons in the K + K − center-of-mass frame are: K is the kaon momentum in the final-state. We can assume that T (k) is related to T K + K − (m) as follows: where g(k), as a real function of the modulus of the kaon three-momentum k ≡ |k|, takes into account the off-shell character of T (k). From Eqs. (6-7) one infers that g(k f ) = 1.
Under a dominance of the pole at m = 2E k ≡ 2 k 2 + m 2 K , the amplitude A(m) can be written in the following form: where the integral I(m) reads In Eq. (9) p 0 = m + ω, where ω is the photon energy in the K + K − center-of-mass frame, m φ is the φ meson mass andq = q/|q| is the unit vector indicating the photon direction.
In the model of Close, Isgur and Kumano [2] the momentum distribution of the interacting kaons has been expressed by the function with the parameter µ = 141 MeV. This function is normalized to unity at k = 0, however the function g(k) in Eq. (7) has to be normalized to 1 at k = k f , so the function g(k) corresponding to φ(k) should be defined as In Ref. [2] kaons are treated as extended objects forming a quasi-bound state. If the K + K − system is point-like, like in Refs. [3] or [4], then the function g(k) ≡ 1. Let us note that both models can be treated as special cases of our approach. Separable kaon-kaon potentials can be used to calculate the kaon-kaon amplitudes needed in practical application of the present model. Then the function g(k) takes the following form: where β is a range parameter. In Ref. [5] for the isospin zero KK amplitude the value of β close to 1.5 GeV has been obtained. In order to get an integral convergence at large k in Eq. (9) we use an additional cut-off parameter k max =1 GeV.

Numerical results
In Fig. 2 we show the K + K − effective mass distributions at the e + e − energy equal to m φ . On the left panel one observes some dependence on the form of the function g(k). A common feature is a presence of the maximum of the differential cross section situated only a few MeV above the K + K − threshold. On the right panel we see a comparison of our model (solid line) with two other models, named the "no-structure" model [6] and the "kaon-loop" model, and described in Refs. [3] and [4]. The parameters of the latter models have been taken by us from experimental analysis of the data of the reaction φ → π + π − γ [7] and then used in calculation of the results shown as dashed and dotted lines.  Figure 2. Dependence of the differential cross-section for the reaction e + e − → K + K − γ on the K + K − effective mass m. Left panel: the solid line corresponds to the case of the function g(k) (Eq. 12) with the parameter β ≈ 1.5 GeV and the cut-off k max = 1 GeV, the dotted line -to g(k) ≡ φ(k) given by Eq. (10) and the dashed curve -to g(k) from Eq. (11) with µ = 141 MeV; right panel: the dashed line is calculated for the no-structure model (Ref. [6]), the dotted one for the kaon-loop model of Ref. [4] with parameters obtained in Ref. [7] and the solid line is the same as in the left panel but with a different vertical scale.
The K 0K0 differential cross sections are presented in the left panel of Fig. 3. These cross sections are considerably lower than the K + K − cross sections seen in Fig. 2. This is due to a smaller K 0K0 phase space and to smaller absolute values of the transition amplitude T (K + K − → K 0K0 ) which replaces in this case the elastic K + K − amplitude in Eq. (8).
By integration of the K + K − and K 0K0 effective mass distributions, shown as solid lines in left panels of Figs. 2 and 3, one can calculate the total cross sections which are equal to 1.85 pb and 0.17 pb, respectively. The corresponding branching fractions are 4.5·10 −7 and 4.0·10 −8 . In the right panel of Fig. 3 we have plotted the contours of the branching fraction for the φ → K 0K0 γ decay as a function of the a 0 (980) resonance position. We see that it is possible to generate lower values of the branching fraction by a moderate change of not well known resonance mass and width.

Conclusions
The above theoretical results for the reactions with charged and neutral kaon pairs indicate that the measurements of the e + e − → K + K − γ process can provide a valuable information about the pole positions of the a 0 (980) and f 0 (980) resonances. A coupled channel analysis of the radiative φ transitions into different pairs of mesons in the final state is possible after a relevant generalization of the present model. This work has been supported by the Polish National Science Centre (grant no 2013/11/B/ST2/04245).