Σ-admixture in neutron-rich Li Λ hypernuclei in a microscopic shell-model calculation

We systematically investigate the structures of the Λ hypernuclei ΛLi with the mass number A = 7– 10 in shell-model calculations considering the ΛN-ΣN coupling in the first-order perturbation method. We find that the calculated Σ-mixing probabilities and energy shifts due to the ΛN-ΣN coupling increase with the neutron number. The Fermi-type and Gamow-Teller-type couplings, which are related to the β-transition properties of the nuclear core state, coherently contribute to the energy shift in neutron-rich hypernuclei.


Introduction
One of the most important subjects in strangeness nuclear physics is a study of neutron-rich Λ hypernuclei [1].It is expected that a Λ hyperon plays a glue-like role in neutronrich nuclei, together with a strong ΛN-ΣN coupling [2,3], which might induce a Σ-mixing in nuclei.The knowledge of the behavior of hyperons in a neutron-excess environment will significantly affect our understanding of neutron stars, because adding hyperons softens the Equation of State [4].The purpose of our study is to theoretically clarify the structure of neutron-rich Λ hypernuclei and contribution of the ΛN-ΣN coupling by a shell model, which has successfully been applied for description of the neutronexcess nuclei [5][6][7].
The overbinding problem in 5  Λ He, which is regarded as the underbinding problem in 4  Λ H and 4 Λ He, was known in the calculation of energy levels of s-shell hypernuclei [8].The spin-spin component of the effective ΛN interaction contributes to the splitting of the 0 + and 1 + doublet states of 4  Λ H and 4 Λ He but cannot fully explain this splitting.Recently, Akaishi and his collaborators [2,3] suggested that a coherent ΛN-ΣN coupling, which is not included in effective ΛN interactions, can resolve the underbinding problem.Because the coherent ΛN-ΣN coupling is not so large in N ≈ Z p-shell hypernuclei [9], the Σ-mixing probabilities and the energy shifts is not significant.However, the authors [10] found first an enhancement of the Σ-mixing probabilities and the energy shifts in the neutron-rich hypernucleus 10  Λ Li in the shell-model calculation, where Σnuclear states are included in the model space by the firstorder perturbation.Thus it is expected that an investigation of the ΛN-ΣN coupling in neutron-rich hypernuclei gives a new knowledge of behavior of hyperons.a e-mail: aumeya@riken.jpRecently, Saha and his collaborators have performed the first successful measurement of a neutron-rich Λ hypernucleus 10  Λ Li by the double-charge exchange reaction (π − , K + ) on a 10 B target at p π = 1.2 GeV/c [11].The theoretical analysis of this reaction [12] suggests that a onestep process, π − p → K + Σ − via Σ − doorways due to the Σ − p ↔ Λn coupling is dominant rather than two-step processes.This means the importance of the Σ-mixing in the Λ hypernuclei.More experiments for productions of neutronrich hypernuclei by the (π − , K + ) reactions are planned at J-PARC [13] and analyses of these reactions provide to examine precisely a wave function involving the Σ-mixing in neutron-rich Λ hypernuclei as well as mechanisms of these reactions.
In this report, we investigate the structure of Λ Li hypernuclei with the mass number A = 7-10 by focusing on the Σ-mixing probabilities and the energy shifts in shell-model calculations including ΛN-ΣN coupling.Also, we discuss the ΛN-ΣN coupling strengths in neutron-rich Λ hypernuclei in terms of the β-transition properties of the nuclear core state.

Shell model for Λ hypernuclei
A shell-model Hamiltonian is given as where H Λ and H Σ are the Hamiltonians in the Λ and Σ configuration spaces, respectively.V ΛΣ and its Hermitian conjugate V ΣΛ denote the two-body ΛN-ΣN coupling interaction, ΛN ↔ ΣN.We treat the ΛN-ΣN coupling interactions, V ΛΣ and V ΣΛ , as perturbation because a Σ hyperon has a larger mass than a Λ hyperon by 80 MeV.When taking into account up to the first-order terms, we obtain the EPJ Web of Conferences ν-th eigenstate with the isospin T and the angular momentum J of the Hamiltonian H as where the Λ-nuclear and Σ-nuclear eigenstates, |ψ Λ µ ; T J⟩ and |ψ Σ µ ′ ; T J⟩, are obtained by solving the eigenvalue equations, respectively, and the coefficients are given by The Σ-mixing probability P Σ;ν and the eigenenergy E ν for the |( A Λ Z)νT J⟩ eigenstate are given by in the first-order perturbation, where E Λ ν = E Λ µ=ν is given in Eq. ( 4) and is the energy shift by the ΛN-ΣN coupling.

Shell-model setup
In the present shell-model calculations, we construct wave functions in the following model space: Basis states are described on the isospin base.The nucleon part of the wave function consists of the inert 4 He core and valence nucleons in the p-shell (0p 3/2 and 0p 1/2 ) orbits.The hyperon part consists of a hyperon (Λ or Σ) in the 0s 1/2 orbit.For the effective NN interaction, we adopt the Cohen-Kurath (8-16) 2BME [14] that is obtained by a fit to 35 energy data of A = 8-16 nuclei by using two single-particle energies and fifteen two-body matrix elements as parameters.The effective Y N interaction is written as where V, ∆, S + , S − and T are radial integrals [15][16][17].s N and s Y are spin operators for nucleons and the hyperon, respectively.ℓ N is the angular momentum operator for nucleons and is proportional to the relative ℓ for states with the 0s 1/2 hyperon.The tensor operator S 12 is defined by with σ = 2s and r = (r We use the effective ΛN interaction given in Ref. [9] and the effective ΛN-ΣN and ΣN interactions [9,18] based on the NSC97e,f potentials [19].The values of radial integrals for these effective Y N interactions are listed in Ref. [10].
3 Numerical results and discussion

Σ-mixing probabilities and energy shifts
We perform numerical calculations for the Λ Li hypernuclei and evaluate the Σ-mixing probabilities and the energy shifts which are obtained by Eqs. ( 7)- (9).In Fig. 1, we show the schematic energy levels for the Λ-nuclear and Σ-nuclear ground states of 10 Y Li.Here we assume that the difference between Λ and Σ threshold energies is E( 9 Li g.s.+Σ) − E( 9 Li g.s.+Λ) = 80 MeV, (12) and then the energy of the Σ-nuclear ground state |ψ Σ g.s.⟩ is calculated to be measured from that of the Λ-nuclear ground state |ψ Λ g.s.⟩.In Table 1, we show the calculated Σ-mixing probabilities and energy shifts and find that these values are the order of 0.1 % and 0.1 MeV, respectively, and increase with the neutron number (or isospin).The Σ-mixing probability is about 0.34 % and the energy shift is about 0.28 MeV 19 th International IUPAP Conference on Few-Body Problems in Physics for the neutron-rich 10  Λ Li ground state, which are about 3 times larger than those for 7  Λ Li.In order to check our shellmodel calculation, we also compared our numerical results for the Z = N Λ hypernuclei to Millener's work [9].We obtain the energy shifts, e.g., ∆ϵ = 0.085 and 0.073 MeV for the ground states of 7  Λ Li and 11 Λ B, which are comparable to the Millener's results of 0.078 and 0.066 MeV, respectively.Therefore, we conclude that our calculations agree with Millener's results.

ΛN-Σ N coupling strengths
We examine an enhancement of the Σ-mixing probability and the energy shift in 10  Λ Li isotope by the configuration mixing in the Σ-nuclear states, which couple to the Λnuclear states by the ΛN-ΣN coupling.≈ 80 MeV considerably contribute to the Σ-mixing.These contributions are coherently enhanced by the configuration mixing which is caused by the ΣN interaction.It is shown that the nature of the Σ-nuclear states plays an important role in the ΛN-ΣN coupling.
It is worth discussing microscopically one mechanism of ΛN-ΣN coupling in Λ hypernuclei.When a Λ-nuclear state in Λ Li converts to a Σ − -nuclear state through the ΛN-ΣN coupling interaction, the Li core state changes into the Be core state.In other words, the β − -transition, Li → Be, occurs in the core-nuclear state.We stress that the ΛN-ΣN coupling strengths |D µ ′ | 2 are extremely sensitive to the strengths of β-transitions between the core-nuclear states.The two-body ΛN-ΣN coupling interaction V ΣΛ can be approximately rewritten as where in the particle representation: VΣΛ and ∆ ΣΛ are radial integrals of central potentials.In the isospin representation, these terms can be rewritten as where t N is the isospin operator for a nucleon and ϕ ΣΛ is the operator that changes a Λ hyperon into a Σ hyperon, Because PSfrag replacements PSfrag replacements V GT ΣΛ are regarded as the Fermi-type and Gamow-Tellertype coupling interactions.The calculated strength distributions of these couplings,  = 0.098 % for the Gamow-Teller type.In order to see the potentiality of the neutron-rich nuclei clearly, we consider the reason why the energy shift in 10  Λ Li is about 3 times larger than that in 7 Λ Li.The enhancement of the Σ-mixing probabilities in neutron-rich Λ hypernuclei is mainly due to the Fermi-type coupling interaction V F ΣΛ , which might correspond to the coherent ΛN-ΣN coupling suggested in Refs.[2,3].When we assume the weak-coupling limit for the ΛN interaction and the approximation in which all the energy differences E Σ µ ′ − E Λ ν between Λ-nuclear and Σ-nuclear states are replaced by a constant value ∆E, we can write a sum over the Fermi-type coupling strengths as Thus the Fermi-type coupling strengths play an important role in the Σ-mixing probability of neutron-rich 10 Λ Li eigenstates because these states have T = 3  2 .On the other hand, in the case of 7  Λ Li with T = 0, the Fermi-type coupling strengths vanish and then the Σ-mixing probability is generated by the Gamow-Teller-type coupling strengths.
In the Gamow-Teller transitions for ordinary nuclei, the model independent formula

Summary
We have investigated the structure of the Λ Li hypernuclei in shell-model calculations including ΛN-ΣN coupling in perturbation theory.We found that the Σ-mixing probabilities and the energy shifts are the order of 0.1 % and 0.1 MeV, respectively, and they increase with the neutron number (or isospin).The reasons why the Σ-mixing probabilities are enhanced are summarized as follows: (i) The Σ-nuclear excited states can be strongly coupled with the Λ-nuclear ground state with the help of the ΣN interaction.(ii) The strong ΛN-ΣN coupling is coherently enhanced by the Fermi-type and Gamow-Teller-type coupling components.(iii) The Fermi-type coupling becomes more effective in a neutron-rich environment, increasing as T (T + 1).
t N and σ N t N denote the Fermi and Gamow-Teller β-transition operators for a nucleon, respectively, V F ΣΛ and

Table 1 .
The calculated Σ-mixing probabilities P Σ and energy shifts ∆ϵ for ground states of Λ Li isotopes.