Study On Deeply Bound Kaonic Nuclear States

Recently, many theoretical and experiment efforts have been devoted to investigation of deeply-bound kaonic nuclei. In order to seek possible narrow discrete nuclear bound states, Akaishi and Yamazaki have investigated theoretically several few-body systems and found deeply bound nuclear states with very small widths. They considered the Lamda(1405) state as the bound state of the kaon and proton and its width is caused by coupling to theπΣchannel. The binding energy of K ? ppn( K ? ppnn) they obtained is 108 (86) MeV and the width is 20 (34) MeV. A few years later, in search for quasi-bound states in the K ? ppsystem a three-body K ? NN?πΣNcoupled-channel Faddeev calculation by Shevchenko et al. yields a quasibound state with B K~55-70 MeV, andΓ~90-110 MeV. On the other hand, the strange tribaryons S 1(3140)with T =0 and S 0(3115) with T =1were observed by T. Suzuki and M.Iwasaki et al. in the interaction of stopped K ?mesons with 4 He. Another indication of K ? ppbound state was reported by the FINUDA Collaboration, the binding energy (width) is 115(67) MeV.But the phenomenological K N interaction Akaishi and Yamazaki given is simple, It contains only one range parameter (0.66 fm) and VπΣ,πΣis set equal to zero. It is known that the latter is very important for the coupled channel consideration. To make phenomenological K N interaction more flexible, we (the first stage) have looked for a new phenomenological K N interaction ( VπΣ,πΣ≠0) with two range parameters. We considered the Lamda (1405) state as the bound state of the kaon and proton, too. The new I =0interaction produces a resonance stateΛ(1405) with EΛ=1406.5-i25 MeV, and gives a scattering length of a I=0=-1.76+i0.6fm which can be compared to Martin's empirical value a I=0=-(1.70±0.07) + i(0.68±0.04)fm. The relative weight of g I=0with respect to g I=1is 1:5 for K ? pnn, 1:1 for K ? ppn, 3:1 for K ? pp, 1:3 for K ? pn. Within the framework of the Bruechner-Hartree-Fock we have studied these kaonic nuclear states, the binding energy of K ? ppn( K ? ppnn) we obtained is 124.1 MeV (94.1 MeV) and the width is 11.8 MeV (25.8 MeV). Our results show that these kaonic nuclear states are more bound and stable. On the other hand, in order to investigate the influence of VπΣ,πΣ, we have considered the interaction of model E with VπΣ,πΣ=0. We find Model E is similar to the model of Akaishi and Yamazaki, but B Kof Model E is bigger and the widthΓis narrow comparing other models'. So VπΣ,πΣis very important for the coupled channel consideration and can not discard easily!However, the results of Akaishi and Yamazaki were criticized by Oset et al. , who agued that the model of Akaishi and Yamazaki is unrealistic. Within the SU(3) chiral unitary model, they found that there are not just one Lamda (1405) but two states, and the shape obtained in experiments comes from a superposition of the two resonances with different weights for different reactions. Recently, Magas et al. have provided firm evidence of the two poles structure of the Lamda(1405). Recently, M.Iwasaki et al. have performed a couple of experiments, KEK PS-E549/570, for the detailed study of the strange tribaryon S 0(3115)obtained in KEK PS-E471. In contrast to the previous proton spectrum, no narrow peak structure were found, the strange tribaryon S 0(3115) was an experimental artifact. CorrelatedΛdpairs emitted after the absorption of negative kaons at rest K ? A→ΛdA′in light nuclei 6Li and 12 C were performed with the FINUDA spectrometer at DAΦNE(LNF), and it was found that MΛd =3251±6 MeV from theΛdinvariant mass spectrum of 6Li. The corresponding binding energy is BKppn?=58±6 MeV, and K?ppnΓ=37 MeV. A similar experiment was performed by Suzuki et al. They found that MΛd≤3220 MeV.So in second stage, we construct a quantitative K Ninteraction phenomenologic- ally using free K Nscattering data, the KpX data of kaonic hydrogen, and the binding energy and width of the two resonances. For I =0K N interaction, we construct two models which reproduce two resonances: one appears around 1420 MeV, which has a width of 40 MeV and couples mostly to K N; While the other one appears around 1392 MeV, has a width about 130 MeV and and couples mostly toπΣ. Furthermore, in the second resonance which is always seen from the invariant mass of its only strong decay channel, the strength of theπΣ→πΣis about 4/3 times larger than that of K N→KNwhich is in good agreement with the result provided by the chiral Lagrangians. The two models both gave a scattering length of a I=0=-1.90+ i0.88fm. If the relative weight of the first resonance with respect to the second one is 7:5, that is 7Λ(1420)/12+5Λ(1392)/12, we find that the superposition of the two resonances is in good agreement with the Lamda(1405) obtained in experiments with EΛ=1406.5-i25 MeV. We also investigate nuclear K ?bound states within the framework of Bruechner-Hartree-Fock theory. The binding energy of K ? ppn(T =0) is 93 MeV and the width is 13 MeV. The binding energy we obtained is smaller than Akaishi and Yamazaki's about 15 Mev. A similar situation holds for K ? ppnn(T =1/2), the binding energy BKppnn ?=72 MeV and the width is about 25 MeV. Our results show that these nuclear K ?states are deeply bound. The binding energy (width) we obtained is 3 MeV (56 MeV) for K ? pnn(T =1), and 23 MeV (62 MeV) for K ? pp(T =1/2)。On the other hand, instead of the resonance stateΛ(1405) the first resonance is considered as an isospin I =0bound state of K N. For example, A.Dote et al. have investigated the K ? pp system with chiral SU(3) effective interaction, they found that the system is weakly bound, with a binding energy BKpp =(19±3) MeV and a decay width K ?pp→ΣNΓπ~40-70 MeV. In order to investigate these strange systems in detail, we have also studied two models, only considering the first resonance or the second one as a bound K Nstate. For the former, the binding energy (width) we obtained is 80 MeV (33 MeV) for K ? ppn(T =0) and 22 MeV (56 MeV) for K ? pp. For the latter, the binding energy for K ? ppn(T =0) is 102 MeV and the width is 12 MeV. Thus, if we consider only one of the two resonances as a bound K Nstate, the results show that the strange tribaryons system is deeply bound, too.Akaishi and Yamazaki have investigated theoretically several few-body systems and found the deeply bound K ?nuclear states with very small widths considering the Lamda(1405) state as the bound state of the kaon and proton. While E.Oset et al. found that there are not just one Lamda (1405) but two states, and they predicted that the binding energy of nuclear K states is smaller, but the width are larger. However, our results show that, contrary to Oset's expectation, these nuclear K states are deeply bound.