Central ΛN Research Articles

ΛN and ΣN Interactions from Lattice QCD

We present our recent study on ΛN and ΣN (isospin I = 3/2) interactions by measuring Nambu–Bethe–Salpeter wave functions on the Lattice QCD. The lattice QCD calculation is performed by using the Nf = 2 + 1 gauge configurations generated by PACS-CS collaboration together with employing an improved method to obtain potentials in lattice QCD simulations. For the 1S0 channel, the central ΣN (I = 3/2) potential and the central ΛN (1S0) potential are found to be very similar. For the spin triplet (3S1−3D1) channels, the central ΛN(3S1−3D1) potential is attractive while the central ΣN(I = 3/2, 3S1−3D1) potentials is repulsive. Tensor potentials, on the other hand, are rather weak in both ΛN and ΣN(I = 3/2) systems.

Binding energies of hypernuclei and ΛN interaction

A central ΛN potential VΛN is found on the basis of an analysis of the binding energies of 1s-shell hypernuclei and Λp scattering. Within the exprerimental errors, this potential makes it possible to reproduce the binding energies of three-, four-, and five-particle ground and excited states of hypernuclei and the angular and energy dependences of the cross sections for Λp scattering. The accuracy and reliability of the calculation of three-, four-, and five-particle hypernuclear and the respective nuclear systems is ensured by determining both upper and lower bounds on the energy. Within the Λ plus core model, the potential VΛN is matched with binding energies of heavy hypernuclei.

Binding energies of hypernuclei and ΛN interaction

A central ΛN potential V ΛN that describes, in the limits of experimental errors, the binding energies of three-, four-, and five-particle ground and excited states of hypernuclei, as well as energy and angular dependence of Λp scattering cross sections, is found on the basis of a conjoint analysis of binding energies of 1s-shell hypernuclei and Λp scattering. The reliability and accuracy of three-, four-, and five-particle calculations of hypernuclei and related nuclear systems are assured by the estimation of, an upper, as well as a lower bound of energy. In the framework of the model Λ+core, V ΛN agrees with binding energies of heavy hypernuclei.

Hypernuclei and baryon-baryon interaction

A central ΛN potential containing spin and charge dependences is constructed on the basis of an analysis of fundamental hypernuclear data (Λp scattering, the binding energies of the ground and excited states of hypernuclei). The lightest hypernuclei ( Λ 3 H, Λ 4 H, Λ 4 He) are calculated as a system of three or four particles on the basis of the highly efficient variational method. Heavier hypernuclei are considered in terms of a twobody model (Λ + undeformed nucleon core) corrected by comparison with direct 3–5-particle calculations. By comparing the results of the calculations of single and double Λ hypernuclei and taking the considerations of meson theory into account, it is possible to estimate the binding energies of charmed and other (b- and t-quark) supernuclei.

Nuclear core dynamics and rearrangement energy for hypernuclei

The effect of the nuclear core dynamics on the binding energies of Λ hypernuclei is studied in the framework of variational correlated wave functions. In addition to the core energy and to a Λ single-particle contribution ϵ ν , we exhibit a rearrangement energy E R which in particular depends on the ΛN correlations F and on the difference δφ N between the core wave function and its exact ground-state value. E R is nonzero only if there are both ΛN correlations ( F ≠ 1) and if δφ N ≠ 0. The latter can result from core polarization by the Λ or from errors in the ground-state variational wave function of the core. E R varies linearly with δφ N for small δφ N . Detailed numerical results for central ΛN and NN potentials with repulsive cores confirm these general features. For a give core wave function ΛN correlations reduce the radius of the core nucleons relative to their c.m.; however E R leads to an expansion of the core rather than to a contraction with only ϵ ν . The core polarization eneegy is quite small ⋍ 0.4 MeV. We study the effects of errors in the variational core wave function by detuning this as a function of one of the variational parameters.

The magnetic moment of the hypertriton

We present a method for calculating the magnetic moment of Λ 3 H . Assuming a central ΛN interaction and a central np interaction including a tensor component, it is found that the contribution of the predominant spatially symmetric 2S state is decreased by the presence of a D-state component in the wavefunction.

Hypernuclear binding energies in the α-particle model

The binding energies BΛ for the hypernuclei with baryon number A = 4N + 1 up toΛ25Mg have been calculated by the variational method in the framework of Brink's α-particle model. The central ΛN potential is adjusted to the binding energy ofΛ5He. Due to their strong dependence on the nuclear density, the BΛ energies are very sensitive to the choice of the effective nuclear interaction and have reasonable values for large A with the potential B1 of Brink and Boeker. The comparison with the BΛ values obtained in the limiting shell model and the consideration of two different Λ-particle wave functions both indicate that the effect of nuclear clustering on BΛ is significant only forΛ9Be andΛ13C. The nuclear distortion due to the Λ-particle binding is also evaluated in this calculation.

Reaction-matrix calculations of the Λ-particle binding in nuclear matter for central ΛN potentials

Extensive results, obtained in the G-matrix approximation for a wide range of central ΛN Yukawa potentials both with hard cores (HC) and soft repulsive cores, are presented for the Λ well depth (binding energy of a Λ-particle in nuclear matter) and related quantities. The HC results, in particular, span a wide range of scattering lengths a, effective ranges r 0 and hard-core radii c, sufficient to include any potentials likely to be proposed. The s-state well depth D s is to a good approximation determined by a, r 0 and the s-state correlation volume (wound integral) κ s NM which characterizes the core in an effectively shape-independent way. The depth D s decreases with κ s NM for given a, r 0. A successful search was made for a “scattering” parameter κ s SC which also characterizes the core in a shape-independent way but which is determined only by (low-energy) s-wave scattering calculations; κ s SC is the (zero-energy) scattering correlation volume up to the separation distance d. The parameterization of D s in terms of a, r 0 and κ s SC is then effectively universal for local potentials, permitting one to obtain D s for any reasonable local potential by use only of s-wave scattering calculations. The p-wave depth D p is generally quite large with only a moderate dependence on c for given a and r 0 but a strong dependence on r 0. The p-wave correlation volume is not a useful shape-independent parameter for D p. The d-wave depth is quite small although not negligible. Limited results are obtained for the dependence on density. The implications of a comparison of the calculated and the phenomenological well depths for the ΛN interaction are discussed.

ΛN-ΣN coupling for ΛN scattering and for the Λ-particle binding in nuclear matter

The effect of coupling of the ΛN to the (virtual) ∑N channel on the Λ-particle binding in nuclear matter (the Λ-well depth) is studied with perturbation theory and especially in the reaction-matrix approach with use of the g-matrix approximation. The g-matrix is calculated self-consistently by use of the Kallio-Day version of the reference-spectrum method, the free kinetic energies being assumed for the unoccupied single-particle states. A large variety of interactions is considered including in particular, hard-core interactions with (static) one-pion exchange (OPE) and ϱp-meson exchange coupling potentials. The couplings have both central and tensor components. A meaningful measure of the suppression of the coupling in nuclear matter is obtained in perturbation theory by considering the effective nonlocal central potentials that represent the (s-state) effect of the coupling for nuclear matter and for ΛN scattering. For hard-core interactions we use one-channel central ΛN potentials with the same hard core and the same low-energy scattering characteristics as the coupled-channel interactions to define the “unsuppressed” well depths appropriate to the effect of the coupling for scattering. The suppression δD of the well depth arises from the effect of the exclusion principle and of the spectrum gaps (binding-energy effects) on the coupling. δD depends strongly on the details of the interaction and is relatively much larger for central than for tensor couplings and is less for shorter ranges of the same type of coupling potential. Furthermore, δD depends on the “unsuppressed” (i.e., intrinsic) strength of the coupling. For the OPE coupling the tensor component is dominant and the central components contribute only very slightly; reasonable Λ∑πn coupling constants can give large values of δD (as much as 15 MeV). For the ϱ-exchange coupling the central components for both spin states, S = 0 and 1, can give quite large δD (possibly as much as 5 MeV for S = 0) in spite of the short range. Interesting possibilities arise from possible destructive interference between the OPE and ϱ-exchange tensor couplings because the interference for nuclear matter differs from that for scattering, since the OPE coupling is more strongly suppressed than the ϱ-exchange one. The possibility of 15 MeV suppression implies that agreement of the calculated with the phenomenological depth (≈ 30 MeV) can be obtained for reasonable hard-core radii (≈ 0.4 fm) and for an appreciable p-state interaction of about half the strength of the s-state one.

Reaction-matrix calculation of the Λ-particle binding in nuclear matter

The two principal procedures for obtaining the phenomenological Λ well depth D from the experimental separation energies B Λ are discussed. A value D ∼ 30 ± 2 MeV, with an upper limit D ≲ 35 MeV, is consistent with the results obtained from both procedures. A reactionmatrix calculation of D is made for central ΛN potentials. This is exact in the g-matrix approximation which gives the term proportional to the density ρ in an expansion in powers of p. To calculate the g-matrix the reference-spectrum method is used in the Kallio-Day form. The free kinetic energies are used for the unoccupied states. The difference between the singleparticle energies of the bound and excited states of the Λ involves a gap which is identified with the well depth. This leads to a self-consistency condition for the determination of D. The nucleon-spectrum parameters are taken from the calculations of Sprung et al. The angle-average approximation for the exclusion-principle operator, appropriate to a Λ in nuclear matter, is examined and found to be excellent. For hard-core potentials, a perturbation expansion in the strength of the attractive tail is in general inadequate and, especially for potentials of short range, it is necessary to obtain the g-matrix exactly for the complete potentials. For given lowenergy scattering parameters, the presence of a sizable hard core very substantially reduces D, The p-state contribution to D can be quite large, especially for the phenomenological potentials with hard cores and with larger intrinsic ranges. Thus for Herndon and Tang's potentials which fit the binding energies of the A = 3 and 4 hypernuclei as well as the Λp scattering data and which have rather large hard cores, one can get close to the phenomenological value of D by assuming zero p-state interaction. With some p-state interaction — about half of that for the s-state — reasonable suppression of the ΛN-σN coupling in nuclear matter can give agreement with the phenomenological depth. The well depth is found to depend fairly linearly on ρ over a large range and the Λ effective mass can be as small as 0.65 M Λ .

Binding energy of a Λ particle in nuclear matter with tensor Λ N interaction

The tensor component in the modified Harndon and Tang ΛN potential (MHT) is found to increase the binding of a Λ particle in nuclear matter by about 5 MeV. A comparison with a pure central ΛN potential of similar low energy Λp scattering parameters as the MHT potential gives about 2 MeV for the tensor suppression effect.