Associated strangeness production in the and reactions
The total and differential cross sections for associated strangeness production in the and reactions have been studied in a unified approach using an effective Lagrangian model. It is assumed that both the and final states originate from the decay of the resonance which was formed in the production chain . The available experimental data are well reproduced, especially the ratio of the two total cross sections, which is much less sensitive to the particular model of the entrance channel. The significant coupling of the resonance to is further evidence for large components in the quark wave function of the resonance.
pacs:13.75.-n.; 14.20.Gk.; 13.30.Eg.
The isobar has proved to be a controversial resonance for many years. In the simple three-quark constituent model, the odd parity should be the lowest spatially excited nucleon state, with one quark in a wave. However, the even parity has in fact a much lower mass, despite requiring two units of excitation energy. This is the long-standing mass inversion problem of the nucleon spectrum.
The resonance couples strongly to the channel pdg2008 but a large coupling has also been deduced garciaplb582 ; liuprl96 ; gengprc79 through the analysis of BES data on decays BES and COSY data on the reaction near threshold COSY11 . Analyses Mosel ; Saghai of recent SAPHIR ELSA and CLAS CLAS data also indicate a large coupling of the to .
In a chiral unitary coupled channel model it is found that the resonance is dynamically generated, with its mass, width, and branching ratios in fair agreement with experiment garciaplb582 ; osetprc65 ; kaisernpa612 ; nievesprd64 ; doeringepja43 . This approach shows that the couplings of the resonance to the , and channels could be large compared to that for . Data on the dugger and reactions caoxuprc78 suggest also a coupling of the isobar to . In addition, there is some evidence for a coupling from the and doringprc78 ; xieprc77 as well as the caoxuprc80 reactions.
The mass inversion problem could be understood if there were a significant components in the wave function zoureview ; zhangan and this would also provide a natural explanation of its large couplings to the strangeness , and channels. It would furthermore lead to an improvement in the description of the helicity amplitudes in photoproduction anepja39 . We wish to argue in this paper that a hidden strangeness component in the might play a much wider role in associated strangeness production in medium energy nuclear reactions.
The can be considered as the strangeness counterpart of the and its structure is possibly even more controversial. In quark model calculations, it is described as a wave baryon isgurprd18 but it can also be explained as a molecule dalitz or pentaquark state inouenpa790 . On the other hand, within unitary chiral theory kaisernpa612 ; garciaplb582 ; chiral , two overlapping states are dynamically generated and in this approach the shape of any observed spectrum might depend upon the production process. In a recent study of the reaction zychorplb660 the resonance was clearly identified through its decay and no obvious mass shift was found. However, this result is inconclusive in the sense that the data could also be well described in the two-resonance scenario gengepja34 . For simplicity we shall here work within the single framework with parameters as reported in the PDG review pdg2008 .
In parallel with the measurement, Maeda et al. also extracted differential and total cross sections for kaon pair production in the reaction maedaprc77 . These results show clear evidence for the excitation and decay of the meson sitting on a smooth background, whose shape resembles phase space. It has been suggested WIL09 that the could be important for the non- kaon pair production through the reaction. This would, of course, only be relevant for the isospin contribution but this is likely to dominate the low mass region because of the presence of the . It is therefore the purpose of the present paper to analyze simultaneously the available data on and production at a beam energy of 2.83 GeV zychorplb660 ; maedaprc77 within a unified phenomenological model, where the isobar acts as a doorway state for both production processes.
The foundation of the model is the assumption that there are large components in the quark wave function of the isobar and that these induce a significant coupling. This in turn allows the possibility that the production of the in proton-proton and collisions could be dominated by the excitation and decay of the resonance below the threshold. Within this picture, we calculate the and reactions using an effective Lagrangian approach. We show that the pion-induced data are indeed compatible with the large coupling resulting from the components in the . The resulting theoretical estimates of the and differential and total cross sections describe well the available COSY experimental data zychorplb660 ; maedaprc77 . In particular, the ratio of these two cross sections, where many of the theoretical uncertainties cancel, is reproduced within the total theoretical and experimental uncertainties.
Ii Formalism and ingredients
We study the and reactions in an effective Lagrangian approach on the assumption that the production of the pair is dominantly through the excitation and decay of the sub-threshold resonance. It is generally assumed that the production of mesons in nucleon-nucleon collisions near threshold passes mainly through the , which has a very strong coupling to . However, there is far from unanimity in the modelling of these processes within a meson-exchange picture, with different groups considering , , , and exchanges to be important xieprc77 ; models . Fortunately, the estimation of the cross section in our model is only sensitive to the production rate of the and single pion exchange is sufficient for this purpose. By neglecting and exchanges, we can present a unified picture of pion- and proton-induced production processes, though our results are more general than this would suggest.
The basic Feynman diagrams for the -channel exchanges in reaction and the channel diagram for are depicted in Figs. 1 and 2, respectively. For the reaction, only diagrams in Figs. 1(a) and 1(b) need to be considered, while for the reaction, the exchange terms 1(c) and 1(d) have also to be included.
We employ the commonly used interaction Lagrangian for the vertex,
with an off-shell form factor taken from the Bonn potential model mach
To evaluate the invariant amplitudes corresponding to the diagrams of Figs. 1 and 2, we also need to know the interaction Lagrangians involving the and resonances. In Ref. zouprc03 , a Lorentz-covariant orbital-spin (-) scheme for couplings was studied in detail. Within this approach, the , , and effective couplings become,
To minimize the number of free parameters, a similar dipole form factor to that of Eq. (2) will be used for the vertex, with the same value of the cut-off parameter.
The , and coupling constants are determined from the partial decay widths of these two resonances pdg2008 . With the effective interaction Lagrangians of Eqs. (3), the coupling constants are related to the partial decay widths by
and correspondingly for the decay in terms of the coupling constant. Here is the triangle function,
Although the mass differences do not allow one to obtain directly similar results for the vertex, the requisite information can be extracted from data, provided that this reaction is dominated by the -channel pole of Fig 2. The corresponding invariant amplitude becomes:
where and are the baryon spin projections.
The propagator is written in a Breit-Wigner form liang :
where is the energy-dependent total width. Keeping only the dominant and decay channels pdg2008 , this can be decomposed as
where GeV/, GeV/, and the two-body phase space factors, , are
and is the threshold value for the decay channel.
A similar representation is adopted for the propagator and form factor, with the same value of the cut-off parameter GeV/. Because the resonance lies slightly below the threshold, the only nominally allowed decay channel is . Nevertheless, ever since the pioneering work of Dalitz and Tuan dalitz it has been known that there is also a strong coupling to . The ensemble of low energy data on and related channels has been described in terms of a separable potential model gal . In contrast to the unitary chiral approach gengepja34 , the separable model produces only a single pole and from this we can investigate its effects above the threshold. These can be parametrized in terms of an energy dependent partial width
where GeV/, GeV/ and the two-body phase space factors are given in Eq. (10). By using
the width equation (11) leads to a coupling constant at the threshold.
We now evaluate the total cross section as a function of the center-of-mass energy. The value of the coupling constant leads to the predictions that are compared with experimental data pipdata in Fig. 3. Although the agreement is reasonable, it must be stressed that the predictions are not very sensitive to the mass of the , provided it lies well below the threshold. As can be judged from the figure, a very similar shape would be obtained if one used for example the second resonance . However, it has been shown xieprc77 that a large component in the resonance is not consistent with its smaller coupling to than . It should also be noticed that any possible contributions from - and -channel exchanges have also been neglected. The value of this coupling constant is given along with others in Table. 1.
The full invariant amplitude for the reaction is composed of four parts, corresponding to the diagrams shown in Fig. 1;
To take account of the antisymmetry of the protons in the initial and final states, factors and are introduced. It is important to note that only and should be considered for the reaction.
Each amplitude can be derived straightforwardly with the effective couplings given. We give as an example the form of the amplitude:
where and represent the spin projections and four-momenta of the two initial and two final protons, respectively. The and are the four-momenta of intermediate and resonances, while is the four-momentum of the final meson. The pion propagator is
The final-state-interaction(FSI) between the two emerging protons in the wave in the case is taken into account using the Jost function formalism gill , with
The normalization is chosen such that the differential cross section is
with the flux factor
The factor before the -function in Eq. (17) results from having two final identical protons and must be omitted for the final state.
Iii Numerical results and discussion
The predictions for the variation of the total cross section with excess energy , calculated using a Monte Carlo multiparticle phase-space integration program, are shown in Fig. 4. Although the general shape of the experimental data is described, nevertheless the results very close to threshold are underestimated. This may be due to the neglect of a final state interaction dzyubaplb668 , which might be associated with the influence of the and scalar resonances maedaprc77 .
The predicted invariant mass spectrum for the reaction at GeV ( MeV) is compared in Fig. 5 to the experimental data from the ANKE group maedaprc77 . The theoretical model reproduces well the shape of the data, being much more peaked to lower invariant masses than the four-body phase-space distribution, which is also shown. As already indicated in Fig. 4, the predicted 100 nb coincides with the experimental value of nb, where the first error is statistical and the second systematic maedaprc77 .
The corresponding results for the invariant mass distribution for the reaction at the same beam energy, but excess energy MeV, are shown in Fig. 6 together with the ANKE data zychorplb660 . Although the statistics are low, the shape of the spectrum is described correctly, with a rather asymmetric peak that is strongly influenced by the opening of the threshold, that is by the energy dependence of the width parametrized by Eq. (11). On the other hand, the overall normalization of the prediction is too high, giving a cross section of 4.0 b compared to an experimental value of b zychorplb660 . The predicted normalization could, of course, be reduced by considering the initial state interaction but that would then lower also the value for the channel.
Many effects cancel out in the estimation of the ratio of the to total cross sections. These include initial state distortions and most of the parameters connected with the . Combining the two experimental results one finds that, at a proton beam energy of 2.83 GeV,
where only non- events have been considered. This is to be compared with a value of obtained within the framework of the present model. The theoretical uncertainties are hard to quantify because they reside to a large extent in the modelling of the low energy system gal , which is based upon a limited experimental data set. In addition there are possibly small contributions from -wave pairs or, for the higher masses, also some -wave contributions. In view of the large experimental and theoretical uncertainties, the good agreement for the ratio is very satisfactory.
Iv Summary and Conclusions
The total and differential cross sections for associated strangeness production in the and reactions have been studied in a unified approach using an effective Lagrangian model. The basic assumptions are that both the and systems come from the decay of the resonance. This state itself results from the excitation of the isobar, for which there is strong evidence for the importance of hidden strangeness components. Although only pion exchange has been kept in the reaction, our predictions are sensitive to the production rate and pion exchange provides a reasonable description of this. Within the model, the energy dependence of the total cross section is well reproduced, as are the characteristic and invariant mass distributions.
Of particular interest is the ratio of the and total cross sections because in the estimation of many unknowns drop out. Apart from initial state distortion, which has been completely neglected in our work, the details of the doorway state are largely irrelevant provided that this state lies well below the threshold. Thus the very satisfactory prediction for would remain the same if one assumed that the processes were driven for example by the isobar. On the other hand, it is the absolute value of either cross section that depends upon the hypothesis and it is the reasonable description here that gives further weight to the idea of large components in this isobar.
The link between and production could be established through the use of much low energy data, which led to the phenomenological separable potential description of the coupled systems gal . Although this particular model gives rise to a single pole it is merely a parametrization of measured scattering data and we cannot rule out the possibility that similar results would be obtained if one used a chiral unitary description which requires two poles gengepja34 .
The production of resonances, such as the scalars pdg2008 , can clearly not contribute to the reaction. Consequently, even if the model presented here is only qualitatively correct it would suggest that non- production in is driven dominantly through the excitation of -hyperon pairs rather than non-strange mesonic resonances.
Further experimental data are needed and some should be available soon on the reaction at the slightly higher energy of 3.5 GeV from the HADES collaboration FAB2010 . It would, however, be highly desirable to have data on kaon pair production at a similar energy in order to provide an independent check on the value of and hence on the approach presented here.
Acknowledgements.We wish to thank Xu Cao, A. Gal, M. Hartmann, Feng-Kun Guo, J. Nieves, N. Shevchenko, and Bing-Song Zou for useful discussions, and the CAS Theoretical Physics Center for Science Facilities for support and hospitality during the initiation of this work. This research was partially supported by Ministerio de Educación “Estancias de movilidad de profesores e investigadores extranjeros en centros españoles”, Contract No. SB2009-0116.
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