Study of the near threshold reaction
in view of the final state interaction
††thanks: Presented at Excited QCD 2010
Measurements of the reaction, performed near the kinematical threshold with the experiment COSY-11 at the Cooler Synchrotron COSY, reveal a significant discrepancy between obtained excitation function and theoretical expectations neglecting interactions of kaons. In order to deepen our knowledge about the low energy dynamics of the system we investigated population of events for the reaction as a function of the invariant masses of two particle subsystems. Based for the first time on the low-energy invariant mass distributions and the generalized Dalitz plot analysis, we estimated the scattering length for the interaction.
13.75.Lb, 13.75.Jz, 25.40.Ep, 14.40.Aq
The basic motivation for investigation of the reaction near the kinematical threshold at COSY was an attempt to understand the nature of the scalar resonances (980) and (980). In addition to the standard interpretation as states , these particles were also proposed to be tetraquarks , molecules [3, 4], hybrid /meson-meson systems  or even quark-less gluonic hadrons . With regard to the formation of the molecule the strength of the interaction becomes a crucial quantity, and it can be probed for example in the near threshold reaction. First measurements of this reaction were conducted at cooler synchrotron COSY by the COSY-11 collaboration [7, 8]. A precise determination of the collision energy, in the order of fractions of MeV, permitted us to deal with the rapid growth of cross sections  and thus to take advantage of the threshold kinematics like full phase space coverage achievable with dipole magnetic spectrometer being rather limited in geometrical acceptance. These experiments revealed, however, that the total cross section at threshold is by more than seven orders of magnitude smaller than the total proton-proton production cross section making the study difficult due to low statistics. A possible influence from the or on the pair production appeared to be too weak to be distinguished from the direct production of these mesons on the basis of the COSY-11 data . However, the combined systematic collection of data obtained by the collaborations COSY-11 [7, 8, 10], ANKE  and DISTO  reveal a significant signal in the shape of the excitation function which may be a manifestation of the interaction among particles in the final state.
2 Total cross sections for the reaction
Results of all the measurements are presented in Fig. 1 together with curves representing three different theoretical expectations normalized to the DISTO data point at Q = 114 MeV . The dashed curve represents the energy dependence from four-body phase space when we assume that there is no interaction between particles in the final state. These calculations differ by two orders of magnitude form data at Q = 10 MeV and by a factor of about five at Q = 28 MeV.
Inclusion of the –FSI (dashed-dotted line in Fig. 1), using parametrization known from the three body final state  with the four body phase space, is closer to the experimental data, but does not fully account for the difference . The enhancement may be due to the influence of and interaction which was neglected in the calculations. Indeed, the inclusion of the –FSI (solid line) reproduces the experimental data for excess energies down to Q = 28 MeV. These calculations of the cross section were accomplished under the assumption that the overall enhancement factor, originating from final state interaction in the system, can be factorised into enhancements in the and two subsystems :
where , and stand for the relative momenta of the particles in the first subsystem, second subsystem and subsystem, respectively. Factors describing the enhancement originating from the –FSI are parametrized using the scattering length approximation, with the scattering length amounting to fm . However the inclusion of the and final state interaction fail to describe the data very close to threshold (see Fig. 1). This indicates that in this energy region the influence of the interaction is significant and cannot be neglected111 In this calculations also the interaction was neglected. It is repulsive and weak and hence it can be interpreted as an additional attraction in the system . . Therefore we decided to perform more detailed analysis of the COSY-11 data at excess energies of Q = 10 MeV and 28 MeV including studies of both the differential cross section distributions  and the strength of the final state interaction between the and .
3 Analysis of the final state interaction
The final state interaction may manifest itself even stronger in the distributions of the differential cross sections than in the shape of the excitation function . Thus, we have performed an analysis of the generalized Dalitz plots [15, 16] for the low energy data at Q = 10 MeV (27 events) and Q = 28 MeV (30 events), in spite of the quite low statistics available.
Complementary to previous derivations [17, 18, 19, 20] here we estimate the scattering length directly from the low energy differential mass distributions of and pp pairs from the system produced at threshold. The raw data (represented by black points in Figs. 2(a) and 2(b)) were first binned and then for each bin corrected for the acceptance and detection efficiency of the COSY-11 facility . The resulting Goldhaber plots are presented together with the raw distributions in Figs. 2(a) and 2(b). In order to estimate the strength of the interaction, the derived cross sections were compared to results of simulations generated with various parameters of the interaction taking into account strong final state interaction in the and subsystems. To describe the experimental data in terms of final state interactions between i) the two protons, ii) the and protons and iii) the and , the enhancement factor was introduced such that Eq. 1 changes to:
As for the case of the –FSI, the was calculated in the scattering length approximation:
where is the effective scattering length and stands for the relative momentum of the kaons in their rest frame. Using this parametrization we compared the experimental event distributions to the results of Monte Carlo simulations treating the scattering length as an unknown parameter, which has to be determined. In order to estimate the real and imaginary part of we constructed the Poisson likelihood statistic derived from the maximum likelihood method [22, 23]. Data collected at both excess energies have been analysed simultaneously . The best fit to the experimental data corresponds to fm and fm. The final state interaction enhancement factor in the scattering length approximation is symmetrical with respect to the sign of , therefore only its absolute value can be determined.
The analysis of the reaction measured by COSY-11 collaboration at excess energy Q = 10 MeV and Q = 28 MeV has been extended to the determination of the differential cross sections in view of the final state interaction. The extracted scattering length amounts to:
= 0.5 fm
= 3 3 fm.
Due to the low statistics the uncertainties are rather large. In this analysis we cannot distinguish between the isospin I = 0 and I = 1 states of the system. However, as pointed out in , the production with I = 0 is dominant in the reaction independent of the exact values of the scattering lengths.
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