Coupled-channel effects in the pp ->ppK+K- reaction

The cross sections for the pp ->ppK+K- reaction were measured at three beam energies 2.65, 2.70, and 2.83 GeV at the COSY-ANKE facility. The shape of the K+K- spectrum at low invariant masses largely reflects the importance of K\bar{K} final state interactions. It is shown that these data can be understood in terms of an elastic K+K- rescattering plus a contribution coming from the production of a K0\bar{K}0 pair followed by a charge-exchange rescattering. Though the data are not yet sufficient to establish the size of the cusp at the K0\bar{K}0 threshold, the low mass behaviour suggests that isospin-zero production is dominant.

The COSY-ANKE collaboration has recently published data on the differential and total cross sections for the pp → ppK + K − reaction at three beam energies T p = 2.65, 2.70, and 2.83 GeV, which correspond to excess energies of ε = 50.6, 66.6, and 108.0 MeV, respectively [1]. The K + K − invariant mass M inv (KK) spectra show a strong signal for the production and decay of the φ meson. This sits upon an apparently non-resonant background. However, the distributions in the Kp and Kpp invariant masses prove that this background is strongly distorted by a K − p final state interaction (fsi ) [2]. After taking this fsi into account, as well as the one between the outgoing protons, most of the distributions are well described by Monte Carlo simulations. Nevertheless, as seen in the K + K − invariant mass spectrum at 2.65 GeV shown in Fig. 1, the simulation underestimates the experimental points for M inv (KK) < 995 MeV/c 2 . Of itself, this could be dismissed as a fluctuation, though it is important to realise that the ANKE spectrometer has a very good acceptance in this region [3]. Furthermore, a similar phenomenon was observed for the same mass region in the 2.70 and 2.83 GeV ANKE data, as well as in those obtained by the DISTO collaboration at a slightly higher energy [4]. Moreover, an analogous effect was also noted for the pn → dK + K − reaction, where the experimental systematics are rather different [5].
The simulation of the pp → ppK + K − spectrum of Ref. [1], shown in Fig. 1 for the 2.65 GeV data, includes only the final state interactions in the K − p and pp systems. To investigate the low K + K − mass region in finer detail, we have taken these results and divided them by the simulation. Although the resulting error bars are rather large, the ratios at all three energies are mutually consistent and Fig. 2 shows the weighted averages of the points at the three energies.
An enhancement at low K + K − masses is, of course, to be expected from an (1) to these data, with the parameters being given in Table 1. The dot-dashed curve is the best fit when the elastic rescattering is neglected and the dashed when the charge-exchange term is omitted.
elastic K + K − fsi, which was not included in the simulation of the ANKE data presented in Ref. [1]. However, the effect seems in all cases to be most prominent between the K + K − and K 0K 0 thresholds. It is therefore natural to speculate that it is also influenced by virtual K 0K 0 production and its subsequent conversion into K + K − through a charge-exchange fsi. If the swave K + K − ⇋ K 0K 0 coupling is strong, this would generate an observable cusp at the K 0K 0 threshold. Such phenomena can significantly distort spectra as seen, for example, in the case of the Λp invariant mass distribution from K − d → Λpπ − at the ΣN threshold [6].
Cusp effects can be described most economically within the K-matrix formalism and this, as well as the associated phenomena, has been discussed extensively by Dalitz and coworkers [7,8]. There are three basic simplifications that are justified in the treatment of a problem such as this, where the statistics are low. The first is that the elements of the K-matrix are taken to be constant, independent of energy, in the small region from the K + K − to a little above the K 0K 0 threshold. Secondly, we assume that isospin invariance is only broken by the mass difference between the charged and neutral kaons. Finally, since we have only a limited understanding of the KK dynamics, the distortions are taken to just first order, in which case the resulting formulae have very transparent interpretations. Figure 3 illustrates the three types of contribution that are to be considered. In all cases the large blob represents the pp → ppKK production distorted by the final state interactions in the pp andKp systems, as described in the Diagrammatic representation of (i) direct production of K + K − pairs, where the large blob includes the fsi effects in the pp and K − p systems considered in Ref. [1]. The elastic K + K − fsi is illustrated in (ii) and the production of virtual K 0K 0 pairs followed by a charge exchange fsi in (iii).
simulation presented in Ref. [1]. In addition to direct production of K + K − pairs shown in (i ), these may be distorted by an elastic rescattering shown in (ii ). The third term of panel (iii ) describes the possibility of the production of K 0K 0 pairs that are converted into K + K − through a charge exchange fsi.
Although the diagrams of Fig. 3 are easy to visualise in the charge basis, the evaluation is somewhat simpler in the isospin basis because the KK scattering lengths are normally quoted in this way. Let B 0 and B 1 be the bare pp → ppKK amplitudes for producing s-wave KK pairs in isospin-0 and 1 states, respectively. These amplitudes, which already include the fsi in the K − p and pp channels [1], are then distorted through a fsi corresponding to elastic scattering. This leads to enhancement factors of the form 1/(1 − ikA I ), where k is the momentum in the K + K − system and A I is the s-wave scattering length in each of the two isospin channels. The charge-exchange fsi of Fig. 3(iii ) depends upon the K 0K 0 → K + K − scattering length, which is proportional to the difference between A 0 and A 1 , and on the momentum q in the K 0K 0 system. In total therefore, the enhancement factor has a momentum dependence of the form where the 1 2 are isospin factors. In this way we have extended the K − p and pp fsi factorisation hypothesis of Ref. [1] to include also the KK system.
The cusp structure arises because q changes from being purely real above the K 0K 0 threshold to purely imaginary below this point. The strength of the effect clearly depends upon A 0 − A 1 , but the shape of the signal also depends upon the interference with the direct K + K − production amplitude.
The KK scattering lengths are significant due to the presence of the (a 0 , f 0 ) scalar resonances [9], but there is a large uncertainty in their numerical values, which reflects the uncertainty in the positions and widths of these states. A useful summary of the different estimates before 2004 is to be found in Ref. [10]. It is generally agreed that the isospin I = 1 scattering length has a small real part [11,12,13,14] and we take A 1 = (0.1 + i0.7) fm. Values of the isoscalar scattering length can be extracted from many fits to data [14,15,16,17] but that deduced by the BES collaboration, A 0 = (−0.45 + i1.63) fm [18] seems to be the most reliable.
There is an even bigger uncertainty in the (complex) ratio of the production amplitudes, which is completely unknown a priori. We have therefore fitted the points shown in Fig. 2 with Eq. (1) so as to determine the values of the unknown magnitude C and phase φ C within this approach. This has been done separately for the cases where (i ) all the terms in Eq. (1) are retained, (ii ) for purely elastic fsi, when the q[A 1 − A 0 ] terms are neglected, and (iii ) purely charge-exchange fsi, when the kA I terms are discarded. The corresponding results are presented in Table 1 and the associated curves in Fig. 2. Table 1 The fit results for the magnitude and phase of the ratio of the I = 1 and I = 0 amplitudes of Eq. (2). The data of Fig. 2  Although all three sets of curves in Fig. 2 reproduce the data in an acceptable way, it has to be stressed that the two final state interactions must be bought as a single package. The effects of neglecting either the elastic or charge-exchange fsi have only been considered in order to indicate the separate influences of the two types of contribution.
The full fit of Fig. 2 demonstrates a cusp effect at the K 0K 0 threshold but only as a sharp discontinuity in the slope of the cross section ratio. This is qualitatively similar to the case where the elastic rescattering is neglected. On the other hand, the scenario where only elastic rescattering is retained gives (statistically) an equally good description of the data. This shows a smooth increase down to the K + K − threshold and no anomalous behaviour at the K 0K 0 threshold. The nature of the solutions also differs in that the full one requires mainly I = 0 production, whereas the contributions from the two isospins would be rather similar if the channel coupling were disregarded.
The KK elastic and charge-exchange fsi both enhance the cross section at low masses and, since this region represents a larger fraction of the total spectrum at low excess energies, it is clear that these will also affect the energy dependence of the total production cross section [19]. This is indeed the case, as shown by Fig. 4, where the extra contributions from the fsi allow the low and high energy data to be described simultaneously.  Fig. 4. Total cross section for the pp → ppK + K − reaction as a function of the excess energy. The experimental data are taken from Refs. [1] (closed circles), [4] (open circle), [2] (closed squares), [20] (open triangle), and [21] (open square). The dotdashed curve is that of four-body phase space normalised on the 108 MeV point. The dashed curve corresponds to the fit which includes final state interactions between the K − and the protons and between the two protons themselves, as described in Ref. [1]. The further inclusion of the fsi between the kaon pair leads to the solid curve.
The KK final state interaction approach used here does not rely on knowing the basic production mechanism. The analysis of the pp → ppK + K − data given in Ref. [1] suggests that the main terms driving the reaction might be linked to Y * excitation and decay. However, even if one assumed that the major contribution to the cross section came from a combination of a 0 and f 0 production, where these resonances are described by Flatté shapes [22], this would lead to a similar structure to that of the present work, though only in the vicinity of the KK thresholds. Thus the observation of cusps or smooth enhancements at low K + K − invariant mass should not be taken as evidence that the underlying production mechanism is necessarily driven by the formation of these scalar resonances.
On the other hand, if (a 0 , f 0 ) production were indeed dominant, then one could put the kaon mass difference directly into the Flatté descriptions of these resonances [23,24]. However, after fitting the pp → ppK + K − data in terms of a 0 and f 0 production amplitudes, such a procedure would give results that differed little from ours in the near-threshold region, provided that the corresponding values of the scattering lengths were used.
On the basis of the parameters quoted in Table 1, it is seen that the best fit is achieved with a production of I = 0 KK pairs in the near-threshold region that is about three times stronger than for I = 1. This sensitivity originates mainly from the very different I = 0/I = 1 scattering lengths, which is a general feature of the various analyses [10,11,12,13,14,15,16,17,18]. This suggests that the production of I = 0 pairs is dominant in the pp → ppK + K − reaction, independent of the exact values of the scattering lengths.
In summary, on the basis of the existing knowledge of the low energy KK interaction, we would expect there to be a cusp-like structure in the K + K − invariant mass spectrum from the pp → ppK + K − reaction. The details, however, are unclear because of the uncertainties in the relative amplitudes for I = 0 or I = 1 production of KK pairs, as well as in the KK scattering lengths. As seen in Fig. 2, the data themselves would be consistent with either a cusp or simply a strong but smooth low mass enhancement.
Although the energy dependence of the total cross section is better reproduced when the KK rescattering is included, this is mainly a reflection of it enhancing the cross section at low K + K − masses. A similar improvement is found if the charge-exchange fsi is neglected in the fitting process. To establish the actual nature of the behaviour in the cusp region, better data are needed and it is hoped that this might be achieved by working at lower excess energy [25].
Extensive discussions with Dr. C. Hanhart have been particularly valuable when carrying out the work reported here. We are grateful to correspondence on this subject with Professors D.V. Bugg, A. Gal, and B.S. Zou. Support from other members of the ANKE collaboration is also much appreciated.