Low energy process $e^{+}e^{-} \rightarrow K^{+} K^{-}$ in the extended Nambu-Jona-Lasinio model

In the extended Nambu-Jona-Lasinio model, the process of production of charged kaon pair on colliding electron-positron beams in the energy interval 1 - 1.7 GeV is described. In studying of this process, we take into account the contact terms when the pairs $K^{+}$ $K^{-}$ are generated by an intermediate photon, and processes with intermediate vector mesons $\rho$, $\omega$ and $\phi$ in the ground and in the first radially excited states. The results obtained here are compared with experimental data recently received in Novosibirsk and in Stanford.


Introduction
the form factor f k 2 should be equated to zero. After the introduction of the first radially excited meson states, nondiagonal terms appear in the NJL model in the free Lagrangian. These terms correspond to exchanges between the meson states with and without form factors. The free Lagrangian is diagonalized by introducting mixing angles [25,27,28,29]. After diagonalizing the free Lagrangian, we can describe various meson interactions in both the ground and first radially excited states without introducing any arbitrary parameters. In particular for the description of e + e − → K + K − at energy 1-1.7 GeV intermediate ρ, ω, φ, ρ(1450), ω(1420), φ(1680) vector states are considered.

Effective quark-meson Lagrangian
In the extended NJL model, the quark-meson interaction Lagrangian for pseudoscalar K ± , vector ρ, ω, φ mesons in the ground and first radially excited states takes the form: where q andq are the u-, d-and s-constituent quark fields with masses m u = m d = 280 MeV, m s = 420 MeV [28,29,32], K ± , ρ, ω and φ are the pseudoscalar and vector mesons, the excited states are marked with prime, f k 2 is the form factor, θ a and θ 0 a are the mixing angles for the mesons in the ground and excited states [25,28,29]. The slope parameters and mixing angles are The matrices The coupling constants: Z K is the factor corresponding to the K − K 1 transitions, M K 1 = 1272 MeV [33] is the mass of the axial-vector K 1 meson, and the integral I 2 has the following form: GeV is the cut-off parameter [25,27]. All these parameters were defined earlier and are standard for the extended NJL model. 3 The amplitude of the process e + e − → K + K − The diagrams of the process e + e − → K + K − are shown in Figs.1,2.
The process e + e − → K + K − contains contributions of following amplitudes: where s = (p(e − ) + p(e + )) 2 , l µ =ēγ µ e is the lepton current. Unfortunately, the NJL model can not describe a relative phase between different states. Thus, we take a phase from e + e − annihilation experiments [1,2] (e iπ factor in the φ mesons). The contribution of the contact diagram is where γ = 1. The sum of ρ and ρ meson contributions reads  The contribution of the diagrams with the intermediate ω, ω meson is The contribution of the diagrams with the intermediate φ, φ meson is MeV are the masses of the intermediate vector mesons [33]. Here, instead of the constant decay width, we used Γ(s) like in [6]: where β(s, M K ) = 1 − 4M K 2 /s. The numerical coefficients C a are obtained from the quark loops in the transitions of the photon into the intermediate vector mesons: where m 1 and m 2 are the masses of the u-quarks or the s-quarks depending on the quark structure of the intermediate vector meson. The integrals are obtained from the quark triangles, a( k 2 ), b( k 2 ) and c( k 2 ) are the coefficients defined in (5).

Numerical estimations
The cross section of the process e + e − → K + K − as a function of energy is shown in Figs 3 and 4.  The results are in satisfactory agreement with the experimental data of SND, CMD-3 [4,5] and the BaBar Collaboration [6] in the energy range 1 -1.6 GeV (see fig.3,4). It is also interesting to compare our results with the results obtained in other fenomenological models [10,34], in particular, the vector-meson-dominance model, which is based on the chiral perturbation theory mentioned in the introduction. These results together with our results are summarized in Table 1. In the table, the absolute values of the numerators of the Breit-Wigner propagators of the form factors are given. In reference [10] in table 4 a comparison of four possible different variants are considered. We selected third one for comparison, which is based on comparisons with [34]. In the NJL model, these numerators in the amplitude correspond to the following expression: where the coefficient C V , as shown in (15), is obtained from the quark loops in the transitions of the photon into the intermediate vector mesons and the numerical coefficients are a ρ = a ρ = 1/2, a ω = a ω = 1/6, a φ = a φ = 1/3.

Conclusion
The satisfactory agreement with the experimental data in the energy interval 1-1.6 GeV has been obtained in the extended NJL model for the e + e − → K + K − process. At the same time, in the region >1.6 GeV, the intermediate vector mesons ρ(1700), ω(1650) significantly affect the final results. However, they are not taken into account in our version of the extended NJL model. So in this region we can not claim to satisfactory estimates with experiment. Once again, we emphasize that our results are obtained without any arbitrary parameters. The absolute values of the numerators of the Breit-Wigner propagators of the form factors that are qualitatively consistent with the results obtained in other phenomenological models are obtained.