Generalized Dalitz plot analysis of the near-threshold $\mathit{pp}\to \mathit{pp}{K}^{+}{K}^{-}$ reaction in view of the $K^{+}{K}^{-}$ final state interaction

The excitation function for the $\mathit{pp}\to \mathit{pp}{K}^{+}{K}^{-}$ reaction revealed a significant enhancement close to threshold that may plausibly be assigned to the influence of the ${\mathit{pK}}^{-}$ and $K^{+}{K}^{-}$ final-state interactions. In an improved reanalysis of COSY-$11$ data for the $\mathit{pp}\to \mathit{pp}{K}^{+}{K}^{-}$ reaction at excess energies of $Q=10$ and 28 MeV, including the proton-$K^{-}$ interaction, the enhancement is confirmed. Invariant mass distributions for the two- and three-particle subsystems allow us to test at low excess energies the ansatz and parameters for the description of the interaction in the $\mathit{pp}{K}^{+}{K}^{-}$ system as derived from the COSY-ANKE data. Finally, based for the first time on the low-energy $K^{+}{K}^{-}$ invariant mass distributions and the generalized Dalitz plot analysis, we estimate the scattering length for the $K^{+}{K}^{-}$ interaction to be $|\mathrm{Re}(a_{K^{+}{K}^{-}})|={0.5}_{-0.5}^{+4.0}$ fm and $\mathrm{Im}(a_{K^{+}{K}^{-}})=3.0\pm 3.0$ fm.

Figure 1

Excitation function for the $\mathit{pp}\to \mathit{pp}{K}^{+}{K}^{-}$ reaction. Triangle and circles represent the DISTO and ANKE measurements, respectively. The four points close to the threshold are results from the COSY-11 measurements. The curves are described in the text.

Figure 2

Schematic view of the COSY-11 detector with a typical event of the $\mathit{pp}\to \mathit{pp}{K}^{+}{K}^{-}$ reaction channel. For the description see text.

Figure 3

Differential cross sections for the $\mathit{pp}\to \mathit{pp}{K}^{+}{K}^{-}$ reaction at $Q=10$ MeV. Circles and dashed bars (in black) denote spectra where the acceptance was determined taking into account only the $\mathit{pp}$-FSI, and triangles with solid bars (in red) denote results where $\mathit{pp}$- and ${\mathit{pK}}^{-}$-FSI was taken into account in the acceptance calculations. They are hardly distinguishable. Vertical bars indicate statistical errors, whereas the horizontal bars stand for the invariant mass intervals for which the cross-section values were established.

Figure 4

Differential cross sections for the $\mathit{pp}\to \mathit{pp}{K}^{+}{K}^{-}$ reaction at $Q=28$ MeV. For the description see the caption to Fig. 3.

Figure 5

The distributions of the absolute values of ratios $R_{\mathit{pK}}$ and $R_{\mathit{pp}K}$ for data at $Q=10$ MeV [(a) and (b)] and $Q=28$ MeV [(c) and (d)]. Solid curves represent theoretical expectations calculated taking into account $\mathit{pp}$ and ${\mathit{pK}}^{-}$ final-state interaction. It should be stressed that the absolute values are not fitted but are the result of the model and used parameters [19].

Figure 6

Goldhaber plots for the $\mathit{pp}\to \mathit{pp}{K}^{+}{K}^{-}$ reaction. The solid lines of the triangles show the kinematically allowed boundaries. Raw data are shown in (a) and (b) as black points. The superimposed squares represent the same distributions but binned into intervals of $\Delta M=2.5$ MeV/$c^2(\Delta M=7\hspace{0.5em}\mathrm{MeV}/c^2$) widths for an excess energy of $Q=10$ MeV (28 MeV), respectively. The size of the square is proportional to the number of entries in a given interval. In (c) and (d) Monte Carlo results are presented. In the simulated distributions both the $\mathit{pp}$ and the ${\mathit{pK}}^{-}$-FSI are taken into account.

Figure 7

${\chi }^2\text{−}{\chi }_{\min }^2$ distribution as a function of $|\mathrm{Re}(a_{K^{+}{K}^{-}})|$ and $\mathrm{Im}(a_{K^{+}{K}^{-}})$. ${\chi }_{\min }^2$ denotes the absolute minimum with respect to parameters $\alpha$, $|\mathrm{Re}(a_{K^{+}{K}^{-}})|$, and $\mathrm{Im}(a_{K^{+}{K}^{-}})$.