Realistic calculations of Kbar-N-N, Kbar-N-N-N, and Kbar-Kbar-N-N quasibound states

Binding energies and widths of three-body KbarNN, and of four-body KbarNNN and KbarKbarNN nuclear quasibound states are calculated in the hyperspherical basis, using realistic NN potentials and subthreshold energy dependent chiral KbarN interactions. Results of previous K^-pp calculations are reproduced and an upper bound is placed on the binding energy of a K^-d quasibound state. A self consistent handling of energy dependence is found to restrain binding, keeping the calculated four-body ground-state binding energies to relatively low values of about 30 MeV. The lightest strangeness -2 particle-stable Kbar nuclear cluster is most probably KbarKbarNN. The calculated Kbar N ->pi Y conversion widths range from approximately 30 MeV for the KbarNNN ground state to approximately 80 MeV for the KbarKbarNN ground state.


Introduction
Unitarized coupled-channel chiral dynamics in the strangeness S = −1 sector, constrained by fitting to K − p low-energy and threshold data, gives rise to a (KN) I=0 s-wave quasibound state (QBS), as detailed in recent works [1,2]. The relationship of this QBS to the observed Λ(1405) resonance, which was predicted long ago by Dalitz and Tuan [3] within a phenomenological study ofKN − πΣ coupled channels, has been recently reviewed by Hyodo and Jido [4]. With that strong (KN) I=0 interaction,K mesons are expected to bind to nuclear clusters beginning with the (KNN) I=1/2 J π = 0 − QBS, loosely termed K − pp. While several few-body calculations confirmed that K − pp is bound, as reviewed in Ref. [5], we here focus on those calculations using chiral interaction models in which the strong subthreshold energy dependence of the inputKN interactions, essential inK nuclear few-body calculations, is under sound theoretical control. Such calculations yield binding energies in the range B(K − pp) ∼ 10 − 20 MeV [6,7], in contrast to values of 100 MeV or more obtained upon relegating peaks observed in final-state Λp invariant-mass spectra from FINUDA [8] and DISTO [9] to the QBS decay K − pp → Λp. To reinforce this discrepancy we note that none of the other published K − pp calculations based onKN phenomenology [10,11,12,13] managed to get as large K − pp binding energy as 100 MeV.
Given this unsettled state of affairs for K − pp, it is desirable to provide chiral model predictions for heavierK nuclear clusters starting with fourbody systems and, in particular, to study the onset of binding for S = −2 clusters. 1 A good candidate isKKNN which of all four-bodyK nuclear clusters has the largest number ofKN bonds (four out of six). Furthermore, for the I = 0, J π = 0 + lowest energy QBS, and limiting the nuclear isospin to I N = 1 corresponding to the dominant s-wave NN configuration, this QBS has the most advantageous IK N = 0, 1 composition of V In this Letter we present fully four-body nonrelativistic calculations of theK nuclear clustersKNNN andKKNN in the hyperspherical basis. Realistic NN interactions and effective subthresholdKN interactions derived within a chiral model [15] are used. The energy dependence of the subthresholdKN interactions is treated self consistently, extending a procedure suggested and practised in Refs. [16,17,18]. This provides a robust mechanism to restrain the calculated binding energies ofK nuclear clusters. Our calculations in the three-body sector reproduce the K − pp calculations of Doté et al. [6] and provide an upper bound on the binding energy of a K − d J π = 1 − QBS. In the four-body sector we find binding energies close to 30 MeV, in strong disagreement with predictions of over 100 MeV made in phenomenological, non-chiral models forKNNN [19] andKKNN [20,21].

Input and Methodology
In this section we (i) briefly review the hyperspherical basis in which K-nuclear cluster wavefunctions are expanded and in which calculations of ground-state energies are done, (ii) specify the two-body NN,KN,KK input interactions, and (iii) discuss the choice ofKN subthreshold energy to be used self consistently in the binding energy calculations.

Hyperspherical basis
The hyperspherical-harmonics (HH) formalism is used here similarly to its application in light nuclei [22] and recently in four-quark clusters [23]. In the present case, the N-body wavefunction (N = 3, 4) consists of a sum over products of isospin, spin and spatial components, antisymmetrized with respect to nucleons and symmetrized with respect toK mesons. Focusing on the spatial components, translationally invariant basis functions are constructed in terms of one hyper-radial coordinate ρ and a set of 3N −4 angular coordinates [Ω N ], substituting for N − 1 Jacobi vectors. The spatial basis functions are of the form where R [K] ([Ω N ]) are the HH functions in the angular coordinates Ω N expressible in terms of spherical harmonics and Jacobi polynomials. Here, the symbol [K] stands for a set of angular-momentum quantum numbers, including those ofL 2 ,L z andK 2 , whereK is the total grand angular momentum which reduces to the total orbital angular momentum for N = 2. The HH functions Y [K] are harmonic polynomials of degree K.

Interactions
For the NN interaction we used the Argonne AV4' potential [24] derived from the full AV18 potential by suppressing the spin-orbit and tensor interactions and readjusting the central spin and isospin dependent interactions. The AV4' potential provides an excellent approximation in s-shell nuclei to AV18. Its accuracy inK nuclear cluster calculations has been confirmed here by comparing our results for K − pp using AV4' with those of Ref. [6] using AV18.
ForKh interactions, where the hadron h is a nucleon orK meson, following Refs. [14,15] we have used a generic finite-range potential with b = 0.47 fm, where the superscript I denotes the isospin of theKh pair and √ s is the Mandelstam variable reducing to the total energy in the twobody c.m. system. ForKK, owing to Bose-Einstein statistics forK mesons, it is safe to assume that V = 313 MeV was obtained in Ref. [14] by fitting to the chiral leading-order Tomozawa-Weinberg s-wave scattering length. In the absence of nearby thresholds of coupled channels, no significant energy dependence is anticipated for this weakly repulsiveKK interaction.
TheKN interaction is an effective interaction based on chiral SU(3) meson-baryon coupled-channel dynamics with low-energy constants fitted to near-threshold K − p scattering and reaction data plus threshold branching ratios [15]. Its HNJH version [25] used here reproduces, a-posteriori, within error bars the K − p scattering length determined from the recent SID-DHARTA measurement of the 1s level shift and width of kaonic hydrogen [26]. The energy-dependent complex potential strengths V KN ( √ s) become gradually weaker for subthreshold arguments √ s 1420 MeV, a property shown below to be crucial in restraining the calculated binding energies ofK nuclear clusters. The absorptive imaginary parts Im V (I) KN ( √ s) that originate fromKN → πY conversion also become weaker, but much faster, practically vanishing at the πΣ threshold.

Energy dependence
The issue of energy dependence in near-thresholdKN interactions deserves discussion. For a singleK meson bound together with A nucleons we define an averageKN Mandelstam variable √ s av by approximating it near threshold, where B is the total binding energy of the system and B K = −E K . Note that all the terms on the r.h.s. following AE th are negative definite, so that √ s av ≈ √ s th + δ √ s with δ √ s < 0. Hence, the relevant two-body energy argument of VK N resides in the subthreshold region, forming a continuous distribution. The state of the art in non-FaddeevK nuclear calculations is to replace this distribution by an expectation value taken in the calculated QBS [6,16,17,18]. Transforming squares of momenta in (4) to kinetic energies, the following expression is derived: , T K is the kaon kinetic energy operator in the total c.m. frame and T N :N is the pairwise NN kinetic energy operator in the NN pair c.m. system. Eq. (5) refines the prescription δ √ s = −ηB K , with η = 1, 1/2, used in the two types of K − pp variational calculations in Ref. [6]. In the limit A ≫ 1, it agrees with the nuclear-matter expression given in Ref. [16] for use in kaonic atoms andK nuclear quasibound states.
A similar procedure is applied to theKKNN system by summing up the four pairwiseKN √ s contributions and expanding about √ s th : where T K:K is the pairwiseKK kinetic energy operator in theKK pair c.m. system. Eqs. (5) and (6)

Results and Discussion
We now present the results of self-consistent three-body and four-body calculations ofK andKK nuclear clusters. The three-body calculations have been tested by comparing with similar calculations for K − pp [6].
For aK nuclear cluster with global quantum numbers I, L, S, J π , the potential and kinetic energy matrix elements were evaluated in the HH basis. The interactions specified in Section 2.2 conserve L and S, the latter is given by the nuclear spin S N . Since no L = 0 QBS are likely to become particle stable upon switching off Im VK N , we limit our considerations to L = 0, resulting in J = S = S N with parity ± for even/odd number ofK mesons, respectively. Although the total isospin I is conserved by these charge-independent interactions, the isospin dependence of VK N induces ∆I N = 1 nuclear charge-exchange transitions, so that the nuclear isospin I N need not generally be conserved. Suppressing Im VK N , the g.s. energy E g.s. was calculated in a model space spanned by HH basis functions with eigenvalues K ≤ K max . Self-consistent calculations were done for √ s from theKN threshold down to 80 MeV below, at which value the error incurred by the near-threshold approximation (4) is only 2.4 MeV. Self consistency in δ √ s was reached after typically five cycles. The convergence of binding energy calculations for particle-stable g.s. configurations is shown in Fig. 2 as a function of K max . With the exception of the (KNNN ) I=1 cluster, good convergence was reached for values of K max ≈ 30 − 40. The poorer convergence for (KNNN ) I=1 is apparently due to its proximity to the (KNN ) I=1/2 + N threshold. Asymptotic values of E g.s. were found by fitting the constants C and γ of the parametrization to values of E(K max ) calculated for sufficiently high values of K max . The accuracy reached is better than 0.1 MeV in the three-body calculations and about 0.2 MeV in the four-body calculations. The conversion width Γ was then evaluated through the expression where VK N sums over all pairwiseKN interactions. Since |Im VK N | ≪ |Re VK N |, this is a reasonable approximation for the width. The dependence of the calculated width Γ ofK nuclear clusters on the input δ √ s value used for the subthresholdKN energy is demonstrated in Fig. 3 for the samē K nuclear clusters depicted in Fig. 2. The width is seen almost invariably to decrease upon increasing −δ √ s, i.e. upon going deeper below threshold.
This is similar to the dependence of E g.s. on the input δ √ s, as displayed for (KKNN) I=0 in Fig. 1. It is worth noting that the calculated widths of the single-K nuclear systems are clustered roughly in a range of 30 − 40 MeV. Given a calculated width ΓK N = 43.6 MeV for the underlying (KN) I=0 QBS, a scale of Γ(singleK) approximately 40 MeV appears quite natural. In contrast, the width calculated for the double-K system (KKNN) I=0 is about twice larger, approximately 80 MeV.
In Table 1 we compare results of the present work for (KN) I=0 and (KNN) I=1/2 QBS with those by Doté et al. [6]. Our (KN) I=0 calculation reproduces that of Ref. [14] and agrees with that in Ref. [6] to within 0.1 MeV out of binding energy B ≈ 11.5 MeV and 0.2 MeV out of width Γ ≈ 43.7 MeV, a precision of better than 1%. We note that this Λ(1405)-like QBS is bound considerably weaker than a QBS required by construction to reproduce Λ(1405) nominally, with B Λ(1405) ≈ 27 MeV [19]. For a more complete discussion of this point we refer to [15]. ForKNN with I = 1/2 and J π = 0 − , loosely termed K − pp, we compare the present calculation with the type-I HNJH-versed DHW variational calculation [6] for which the implied effective δ √ s value is close to our self-consistent δ √ s value. From their type-I,II calculations one concludes that δB/ δ √ s ≈ 0.24, so that our binding energy value B should come out smaller by approximately 1 MeV than their listed type-I B. The remainder 0.2 MeV of the 1.2 MeV difference between rows 3 and 4 in the table is attributed to using slightly different NN interactions: AV4' here, AV18 in Ref. [6]. Rows 5 and 6 of the table demonstrate the effect of limiting the model space to I N = 1, compatible with the dominant s-wave NN configuration. This results in a decrease of the calculated binding energy by 4.8 ± 0.1 MeV. The 1 MeV difference between rows 5 and 6 is consistent with the estimate made above for δB/ δ √ s , with no room within NN s waves for any marked difference arising from the difference between using AV4' (BGL) and AV18 (DHW). Finally, the differences of order 10 −15% between the two width calculations, and between the two r.m.s. distance calculations, reflect the sensitivity of these entities to details of the three-body wavefunction, particularly through the effective δ √ s value used.
We have also searched for aKNN QBS with I = 1/2 and J π = 1 − , loosely termed K − d. The possibility of a QBS with these quantum numbers has hardly been discussed in the literature, apparently since it was realized from the very beginning [27] that K − d is less exposed than K − pp, by a ratio close to 1 ÷ 3, to the strongly attractive V (0) KN interaction. We are not aware of any genuine three-body calculation for K − d. 2 Our calculations did not produce any I = 1/2, J π = 1 − QBS below the (KN) I=0 + N threshold, i.e. with total binding energy exceeding 11 MeV. Whether or not such a QBS exists above the (KN) I=0 + N threshold is an open question which cannot be resolved within the present HH calculations that normally converge at the lowest energy state for given quantum numbers.
In Table 2 we present new results forKNNN andKKNN QBS. The first two rows concern theKNNN system essentially based on the I N = 1/2 mirror nuclei 3 H and 3 He which are bound by 8.99 MeV in this calculation. TheK nuclear interaction splits the two resultant I = 0, 1KNNN QBS such that the I = 0 QBS is the lower of the two. The 11 MeV isospin splitting is small compared to the approximately 30 MeV conversion width of each of these states. We note that the I = 0 QBS is bound weakly compared to the tight binding over 100 MeV predicted for it by Akaishi and Yamazaki [19]. Its spatial dimensions, with interparticle distances all exceeding 2 fm, also do not indicate a very tight structure. The imposition of self consistency in the binding energy calculation is responsible for the relatively low value B(KNNN) I=0 = 29.3 MeV, compared to a considerably higher value B(KNNN)  (5), see Table 1. The last two rows of Table 2 report on the S = −2 (KKNN) I=0 QBS which has been highlighted as a possible gateway to kaon condensation in self-bound systems, given its large binding energy over 100 MeV predicted by Yamazaki et al. [20]. Our calculated value B = 32.1 MeV is comparable with that for the S = −1 (KNNN) I=0 QBS, and is a factor of two larger than for the lowestKNN QBS with I = 1/2 and J π = 0 − . Note, however, that (KKNN) I=0 is bound by less than 10 MeV with respect to the threshold for decay to a pair of (KN) I=0 Λ(1405)-like QBS. This apparent relatively weak binding of (KKNN) I=0 is owing to the restraining effect of handling self consistently the energy dependentKN interaction. Finally, the last row of the table shows what happens when the repulsive VKK is switched off. The effect is mild, increasing B by only 4 MeV. Nevertheless, inspection of the r.m.s. distances in (KKNN) I=0 reveals a more compact structure than (KNNN) I=0 , which is also reflected by the large value of Γ(KKNN) I=0 .

Conclusion
In conclusion, we have performed calculations of three-bodyKNN and four-bodyKNNN andKKNN QBS systems. Using practically identical interactions to those used in the K − pp chiral model calculations by Doté et al. [6], we were able to test our calculations for this QBS against theirs. Given the low binding energy B(K − pp) ≈ 16 MeV and sizable conversion width Γ conv (K − pp) ≈ 40 MeV, it might be difficult to identify such a near-threshold QBS unambiguously in ongoing experimental searches. This situation gets further complicated by two additional factors: (i) the possible presence of a near-threshold K − d QBS in the same charge state as the one in which K − pp is searched on, and (ii) additional two-nucleon absorption widths ∆Γ abs accounting for the poorly understood non-pionic processesKNN → Y N. For K − pp we note the estimate ∆Γ abs (K − pp) 10 MeV [6]. Appreciable p-wave contributions to the K − pp width were also suggested in Ref. [6], but doubts have been recently expressed on the effectiveness of a p-waveKN interaction by testing its role in kaonic atoms [16]. Altogether, the widths of KNN QBS are likely to be dominated by their conversion widths.
For the four-body QBS systemsKNNN andKKNN we found relatively modest binding, of order 30 MeV in both, with conversion widths ranging from about 30 MeV for each of theKNNN QBS to about 80 MeV for the lowestKKNN QBS. These systems, although somewhat more compact than K − pp, are not as compact as suggested by Yamazaki et al. [19,20,21]. Their KN r.m.s. distances do not fall below that of the Λ(1405)-likeKN QBS, and their NN r.m.s. distances exceed that of nuclear matter (≈ 1.7 fm). For a conservative estimate of the absorption widths ∆Γ abs in these systems, we count the number of nucleons n available to join a givenKN correlated pair, one pair per eachK. This gives twice as large n for each of the fourbody systems (n = 2) with respect to K − pp (n = 1). Hence, neglecting three-nucleon absorption, ∆Γ abs (KNNN,KKNN) ∼ 20 MeV.
The energy dependence of the subthresholdKN effective interaction, constructed in Ref. [15] within a coupled channel chiral model, was found to be instrumental in restraining the binding of the four-bodyK nuclear clusters through the self-consistency requirement derived here for these light systems. A strongKN interaction operates to form tightly bound compact structures, necessarily accompanied by large kinetic energies. This leads by Eqs. (5) and (6) to substantial values of the energy shift δ √ s which give rise to weaker inputKN interactions, resulting in less binding as demonstrated in Fig. 1 forKKNN. However, dispersive contributions to the binding energy of QBS cannot be excluded. Recent fits to kaonic atoms [16,17] suggest that ∆B disp ∼ ∆Γ abs , so that these binding energies could reach values B(K − pp) ∼ 25 MeV and B(KNNN,KKNN) ∼ 50 MeV. For heavierKnuclear clusters where the nuclear density is closer to nuclear-matter density, a restraining mechanism similar to the one discussed here has been shown to be operative [18]. Other restraining, or saturation mechanisms are likely to be operative such as the increasedKK repulsion upon addingK mesons [30]. It is therefore quite unlikely that strange dense matter is realized throughK mesons as argued repeatedly by Yamazaki et al. [20,21].