Extension of Λ NN → NNN Weak Interactions and the Two-Nucleon Induced Hypernuclear Nonmesonic Decays: Roles of Strange-Meson and Δ ₃₃-Induced (ρρ)-Pair Exchanges Open Access

Abstract

1. Introduction

Two-nucleon induced nonmesonic decay is one of the unique phenomena of hypernuclei and, in fact, the experimental data by the KEK E462/E508 [1,2] and FINUDA [3,4] groups showed a strong and clear impact. The understanding of these weak decay mechanisms has continued to be an interesting theoretical subject in recent years [5–7]. Alberico et al. initiated the idea that a virtual pion emitted from |$\Lambda$| could be absorbed by a pair of nucleons as |$np$|⁠, producing a three-body decay [8]. Bauer et al. developed the formalism of evaluating the |$\Lambda$| self-energy diagrams corresponding to two-nucleon induced decay in the lowest order in the nuclear matter in which the |$\Lambda$| hyperon interacts in the weak process with two-particle–two-hole nucleon excitations connected to the ground-state correlation of the core-nucleus [9–11]. Then the theory was applied to finite hypernuclei using the local density approximation.

For such hypernuclear weak decays, we presented a new type of treatment by taking account of the possible importance of the three-body weak interactions due to the meson-pair exchange (MPE) in the previous paper [12] (referred to as Paper I hereafter).¹ In Paper I, the three-body |$\Lambda NN \rightarrow NNN$| weak interactions are constructed in the MPE model, such as |$(\pi \sigma )$|-, |$(\pi \omega )$|-, |$(\sigma \sigma )$|-, and Fujita–Miyazawa |$(\pi \pi )$|-pair exchanges. Since it is very difficult to calculate two-nucleon induced decay rates with three-body interactions for finite hypernuclei, we adopt the following approach. First of all, the effective two-body |$\Lambda N \rightarrow NN$| potentials are deduced from the three-body |$\Lambda NN \rightarrow NNN$| interactions by integrating the “third” nucleon coordinates with use of the Loiseau–Nogami–Ross (LNR) approximation [13] in the nuclear matter. This procedure is considered to be rather standard. Then, by making use of these effective two-body potentials, the nonmesonic decay rates are calculated with the microscopic hypernuclear wave functions in the usual way. The resultant theoretical estimates turned out to be well comparable to the experimental decay rates measured in several hypernuclei in the s- and p-shell regions.

In order to complete the above-mentioned framework, we extend the three-body |$\Lambda NN \rightarrow NNN$| interactions to include new exchange diagrams of strange mesons such as K, |$K^{*}$|⁠, and |$\kappa$| in the model of Paper I. The Feynman diagrams are depicted in Fig. 1 with the details being explained in the next paragraph. The extension also includes a diagram of |$\Delta _{33}$|-isobar induced |$(\rho \rho )$|-pair-meson exchange, as shown in Fig. 2, which is new but parallel to the |$(\pi \pi )$|-pair exchange of the Fujita–Miyazawa model. The primary aim of this paper consists of these extensions, deduction of the corresponding effective two-body weak potentials, and new presentation of their configuration space expressions of |$V^{(\text {eff.2B})}_{(K \pi )\kappa }$|⁠, |$V^{(\text {eff.2B})}_{(K \pi )K^{*}}$|⁠, and |$V^{(\text {eff.2B})}_{D33(\rho \rho )}$|⁠.

Fig. 1.

Feynman diagrams of |$(K \pi )\kappa$| and |$(K \pi )K^{*}$| exchanges for the three-body |$\Lambda NN \rightarrow NNN$| interactions. For each exchange, the diagrams (a1)–(a2) of |$V_{12;3}$| and (b1)–(b2) of |$V_{13;2}$| are shown.The open circle |$(\circ )$| represents a weak vertex.

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Fig. 2.

Feynman diagrams of |$\Delta _{33}$|-induced |$(\rho \rho )$|-pair-exchange three-body |$\Lambda NN \rightarrow NNN$| interaction. (a) shows the diagram of |$V_{12;3}$| and (b) shows that of |$V_{13;2}$|⁠. The open circle |$(\circ )$| represents a weak vertex and the solid circle |$(\bullet )$| represents a |$\Delta _{33}$|-induced NN|$(\rho \rho )$| vertex.

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As the second aim of this paper, we explain why and how to include the strange-meson exchanges and the other |$(\rho \rho )$|-pair exchange within the model of Paper I. In hypernuclear nonmesonic weak decays, it is well known that the K-meson exchange gives rise to characteristic cancellation of the strong |$\pi$|-meson-exchange contribution in the tensor potentials of the |$^{3}S_1 \rightarrow {^{3}D_1}$| channel, leading to a substantial reduction in the total decay rate. On the other hand, the parity-violating vector potentials of the |$^{3}S_1 \rightarrow {^{3}P_1}$| transition channel in those exchanges work additively, thus enhancing the |$\Gamma _n /\Gamma _p$| ratio in favor of the experimental data [14–16]. Similarly, certain roles of |$K^{*}$| and |$\kappa$| exchanges have also been recognized theoretically by several authors [14–16]. In view of these important factors, however, the |$(K \pi )$|-meson pair directly coupled to the |$\Lambda$|–N (Lambda–nucleon) line could not be treated in Paper I. This is because the Nijmegen ESC model cannot determine the strong coupling constant |$g^{}_{\Lambda N(K \pi )}$| due to the lack of availability of experimental data for |$\Lambda N$| scattering at present. To overcome this drawback, here we introduce new diagrams for constructing |$\Lambda NN \rightarrow NNN$| interactions and devise to include the strange-meson exchanges. Figure 1 shows two types of diagrams for |$(K \pi )\kappa$|- and |$(K \pi )K^{*}$|-exchange |$\Lambda NN \rightarrow NNN$| interactions, respectively. In the former, there are two possible |$(K \pi )\kappa$|-exchange processes, as depicted by Figs. 1(a1) and (a2). In Fig. 1(a1), |$\kappa$| emitted from an N-nucleon (say |$N_3$|⁠) in a weak process dissociates into K and |$\pi$| in the intermediate state, and then K and |$\pi$| are absorbed by |$\Lambda$| and another N-nucleon (⁠|$N_2$|⁠), respectively. In Fig. 1(a2), on the other hand, K emitted from a nucleon (⁠|$N_3)$| in a weak process and |$\pi$| emitted from another nucleon (⁠|$N_2)$| are coupled to produce |$\kappa$| in the intermediate state, and then |$\kappa$| is absorbed by |$\Lambda$|⁠. Correspondingly, similar diagrams can be drawn for the |$(K \pi )K^{*}$| exchange in which |$\kappa$| is simply replaced with |$K^{*}$|⁠, so that both are expressed by (a1) and (a2) collectively. As another extension of the model of Paper I, Fig. 2 is introduced, showing the process of |$\Delta _{33}$|-isobar induced |$(\rho \rho )$|-pair-exchange three-body interaction in which the |$NN(\rho \rho )$| vertex is constructed based on the |$N\Delta \rho$| coupling interaction. The diagrams are quite similar in type to those of the Fujita–Miyazawa |$(\pi \pi )$|-pair-exchange process except that the |$(\pi \pi )$|-pair is replaced by a |$(\rho \rho )$| one. Note that the |$\Delta _{33}$|-isobar induced |$(\pi \rho )$|-pair exchange is not considered here, because the three-body |$\Lambda NN \rightarrow NNN$| interaction due to this exchange does not induce the finite effective two-body |$\Lambda N \rightarrow NN$| potential under the present approximation method.

After getting the additional effective two-body |$\Lambda N \rightarrow NN$| potentials in the configuration space, we give theoretical estimates for two-nucleon induced nonmesonic weak decays of several hypernuclei and discuss roles of various meson-exchange processes consistently within the proposed framework. This is the third aim of this paper.

In Section 2, the coupling Hamiltonians are given. In Section 3, the three-body |$\Lambda NN \rightarrow NNN$| interactions and the effective two-body |$\Lambda N \rightarrow NN$| potentials are derived. Numerical calculations and discussions for the results are given in Section 4. Erroneous overestimates found in Paper I are corrected in this section. A summary is given in Section 5. In Appendix  A, the effective two-body |$\Lambda N \rightarrow NN$| potentials in the configuration space are presented.

2. Hamiltonians

The present paper introduces the three-body |$\Lambda NN \rightarrow NNN$| weak interactions due to |$(K \pi )\kappa$|-, |$(K \pi )K^{*}$|-, and |$\Delta _{33}$|-isobar induced |$(\rho \rho )$|-pair meson exchanges. The necessary interaction Hamiltonians for the strong vertices in Figs. 1 and 2 are the following.

Strong:

where |$\boldsymbol{\rho }^{\mu }$| represents a |$\rho$|-meson field of isovector and Lorentz four-vector |$(\rho ^0, \boldsymbol{\rho })$|⁠, |$\boldsymbol{T}$| is a transition operator connecting isospin 3/2 and 1/2 states, and |$\phi ^{K} = \big (^{\phi ^{K^+}} _{\phi ^{K^0}}\big )$|⁠, |$\phi ^{\kappa } = \big (^{\phi ^{\kappa ^+}} _{\phi ^{\kappa ^0}}\big )$|⁠, and |$\phi ^{K^{*}} = \big (^{\phi ^{{K^{*}}^+}} _{\phi ^{{K^{*}}^0}}\big )$| are used.

There are four types of |${\cal H}_{NN(\rho \rho )_{33}}$|⁠, which signifies the Hamiltonian that couples a |$\Delta _{33}$|-isobar induced |$(\rho \rho )$|-pair to the N–N particle line. Here we list two of them, |${\cal H}^{(1)}_{NN(\rho \rho )_{33}}$| (⁠|$\boldsymbol{\rho }_1 \boldsymbol{\rho }_2$|-related term) and |${\cal H}^{(2)}_{NN(\rho \rho )_{33}}$| (⁠|${\rho }^{0}_1 {\rho }^{0}_2$|-related term). The others, which have either |${\rho }^{0}_1 \boldsymbol{\rho }_2$|-related or |$\boldsymbol{\rho }_1 {\rho }^{0}_2$|-related terms, are not relevant to the final two-body |$\Lambda N - NN$| potentials:

In Eqs. (9)–(10), |$\boldsymbol{k}_1 = - i \boldsymbol{\nabla }_1$| represents an incoming |$\rho$|-meson momentum and |$\boldsymbol{k}_2 = + i \boldsymbol{\nabla }_2$| an outgoing |$\rho$| one, and |$\boldsymbol{q} = (\boldsymbol{p} + \boldsymbol{p}^{\prime } )$|/2.

For interaction Hamiltonians at weak vertices in Figs. 1 and 2 (marked by open circles), we employ the following.

Weak:

The weak Hamiltonian for |$NN(\rho \rho )_{33}$| coupling is not listed since it is not known at present. The isospurion |$(^0_1)$| appearing in Eqs. (11)–(14) is used to ensure that the weak Hamiltonian should follow the |$\Delta I = 1/2$| rule.

3. Three-body |$\Lambda NN \rightarrow NNN$| interactions and effective two-body |$\Lambda N \rightarrow NN$| potentials

3.1. Three-body |$\Lambda NN \rightarrow NNN$| interactions due to the strange mesons and the |$\Delta _{33}$|-isobar induced |$(\rho \rho )$|-pair meson exchanges

Figure 1 shows the diagrams of the |$(K \pi )\kappa$|- and |$(K \pi )K^{*}$|-exchange three-body |$\Lambda NN \rightarrow NNN$| weak interactions. It is noticed that there is no diagram that contains the weak vertex in the |$\Lambda$|–N line. Three-body |$\Lambda NN \rightarrow NNN$| interactions are calculated based on the Fig. 1 diagrams in momentum space. The momenta |$(\boldsymbol{p}_{1\Lambda }, \boldsymbol{p}_2 , \boldsymbol{p}_3 )$| are assigned for the initial |$(\Lambda _1, N_2, N_3)$| particles and |$(\boldsymbol{p}^{\prime }_1, \boldsymbol{p}^{\prime }_2 , \boldsymbol{p}^{\prime }_3 )$| are assigned for the final |$(N^{\prime }_1, N^{\prime }_2, N^{\prime }_3)$| ones. We evaluate first |$( V^{\rm (a1)}_{12;3})_{(K \pi )\kappa } \, (\boldsymbol{p}^{\prime }_1, \boldsymbol{p}^{\prime }_2 , \boldsymbol{p}^{\prime }_3 ; \boldsymbol{p}_{1 \Lambda } , \boldsymbol{p}_2, \boldsymbol{p}_3)$| and |$( V^{\rm (a2)}_{12;3})_{(K \pi )\kappa } \, (\boldsymbol{p}^{\prime }_1, \boldsymbol{p}^{\prime }_2 , \boldsymbol{p}^{\prime }_3 ; \boldsymbol{p}_{1 \Lambda } , \boldsymbol{p}_2, \boldsymbol{p}_3)$| for Figs. 1(a1) and (a2) of the |$(K \pi )\kappa$| diagram, respectively. Similarly, |$( V^{\rm (b1)}_{13;2})_{(K \pi )\kappa } \, (\boldsymbol{p}^{\prime }_1, \boldsymbol{p}^{\prime }_2 , \boldsymbol{p}^{\prime }_3 ; \boldsymbol{p}_{1 \Lambda } , \boldsymbol{p}_2, \boldsymbol{p}_3)$| and |$( V^{\rm (b2)}_{13;2})_{(K \pi )\kappa } \, (\boldsymbol{p}^{\prime }_1, \boldsymbol{p}^{\prime }_2 , \boldsymbol{p}^{\prime }_3 ; \boldsymbol{p}_{1 \Lambda } , \boldsymbol{p}_2, \boldsymbol{p}_3)$| are calculated for Figs. 1(b1) and (b2) of |$(K \pi )\kappa$| diagram, respectively. The complete three-body interaction of the |$(K \pi )\kappa$| exchange is obtained by the sum of these four types of meson-exchange interactions as

Introducing a new set of variables,

the three-body interaction |$V^{\rm W3B}_ {(K \pi )\kappa }$| in Eq. (15) is expressed in the momentum space as

Quite analogously to |$V^{\rm W3B}_ {(K \pi )\kappa }$|⁠, the |$(K \pi )K^{*}$|-exchange three-body interaction |$V^{\rm W3B}_ {(K \pi )K^{*}}$| is derived from the |$(K \pi )K^{*}$| diagrams in Fig. 1. It is written in the momentum space as

The three-body |$\Lambda NN \rightarrow NNN$| interaction due to the |$\Delta _{33}$|-isobar induced |$(\rho \rho )$|-pair exchange is calculated based on the Figs. 2(a) and (b) diagrams in the momentum space. We often use the abbreviation |$\Delta _{33}$|-induced |$(\rho \rho )$| or |$D33(\rho \rho )$| for the |$\Delta _{33}$|-isobar induced |$(\rho \rho )$|-pair exchange in this paper. The three-body |$D33(\rho \rho )$|-exchange interaction is written as

In the momenta defined in Eq. (16), |$V^{\rm W3B}_{D33(\rho \rho )}$| is expressed as

The configuration space interaction is given by the Fourier transform of the interaction |$V^{\rm W3B} \, (\boldsymbol{k}_1, \boldsymbol{k}_2 , \boldsymbol{k}_3 ; \boldsymbol{q}_1 ,\boldsymbol{q}_2, \boldsymbol{q}_3)$| in the momentum space. If the interaction depends only on the |$\boldsymbol{k}_{i}$| variables, one obtains

If the interaction has |$\boldsymbol{q}_i$|-dependence in |$V^{\rm W3B} \, (\boldsymbol{k}_1, \boldsymbol{k}_2 , \boldsymbol{k}_3 ; \boldsymbol{q}_1 ,\boldsymbol{q}_2, \boldsymbol{q}_3)$|⁠, we need to handle the interaction in an approximation in which the nonlocal operator with |$\boldsymbol{q}_i$|-dependence is replaced by the operator with no |$\boldsymbol{q}_i$|-dependence.

3.2. Deduction of effective two-body |$\Lambda N \rightarrow NN$| potentials

In order to obtain the effective two-body potentials from the three-body interactions, further requirements are imposed on the momenta of the three- and/or two-particle systems that are described in Paper I. Then the following relations hold in the two-body CM frame as

Thus one sees

For the |$\boldsymbol{q}_i$|-dependent operator, we note the following. |$V^{\rm W3B}_{(K \pi )K^{*}}$| has a |$(\boldsymbol{\sigma }_3 \cdot \boldsymbol{q}_3)$| term, but the term can be omitted because it vanishes in the LNR approximation, which is described later as |$Tr_{\lbrace \rm spin-3\rbrace } \, (\boldsymbol{\sigma }_3 \cdot \boldsymbol{q}_3) \, = 0$|⁠. |$V^{\rm W3B}_{D33(\rho \rho )}$| has terms like |$\boldsymbol{q}^{2}_3$|⁠, |$(\boldsymbol{k}_1 \cdot \boldsymbol{q}_1)^{2}$|⁠, |$(\boldsymbol{\sigma }_1 \cdot \boldsymbol{q}_1)$|⁠, and |$(\boldsymbol{\sigma }_1 \cdot \boldsymbol{q}_1) (\boldsymbol{k}_1 \cdot \boldsymbol{q}_1)$|⁠. The operator |$\boldsymbol{q}^{2}_3$| for the nucleon-“3” is replaced by the average |$\lt \boldsymbol{q}^{2}_3 \gt $| in nuclear matter and |$\lt \boldsymbol{q}^{2}_3 \gt \, = \, \lt \boldsymbol{p}^{2}_3 \gt \, = \frac{3}{5} k^{2}_F$|⁠, where |$k_F = 278.0$| MeV/c is the Fermi momentum. Other terms are neglected simply for their nonlocal nature. Under these prescriptions for the |$\boldsymbol{q}_i$|-dependent operators in |$V^{\rm W3B}$| interactions, we have the interaction form as |$V^{\rm W3B} \, (\boldsymbol{k}_1, - \boldsymbol{k}_1 , 0 \, )$|⁠.

Now we can obtain the effective two-body |$\Lambda N \rightarrow NN$| potential from |$V^{W3B} \, ( \boldsymbol{x}_1, \boldsymbol{x}_2, \boldsymbol{x}_3)$| of Eq. (21) by integrating the nucleon-“3” coordinate with adopting the LNR approximation [13] as

In Eq. (24), the |$Tr$| symbol signifies trace over the spin and isospin operators of nucleon-“3” as |$Tr = Tr_{\lbrace \rm spin-3\rbrace }Tr_{\lbrace \rm isospin-3\rbrace }$| and |$\rho ^{}_{\rm NM} = 2 k^{3}_F/3{\pi }^3$| is a nuclear matter density.

The form factor is introduced for each meson propagator in the three-body interaction |$V^{\rm W3B} \, (\boldsymbol{k} , - \boldsymbol{k} , 0 )$| such that

and we take a Gaussian form

where |$\varLambda _{m}$| is a cutoff mass for the meson m. For |$V^{\rm W3B}$| in the case of |$\Delta _{33}$|-induced |$(\rho \rho )$| exchange, we take

and |$\varLambda _{D3\rho } = \varLambda _{\rho }$| for the |$\rho$| meson.

3.3. Configuration space effective two-body |$\Lambda N \rightarrow NN$| potentials

The effective two-body |$\Lambda N \rightarrow NN$| potentials in the configuration space that contain the strange-meson exchange and the |$\Delta _{33}$|-induced |$(\rho \rho )$|-pair exchange are listed in Appendix  A. The spin–orbit-like part of the potentials is neglected in the configuration space for simplicity. We write |$\boldsymbol{r} \, = \boldsymbol{x}_1 - \boldsymbol{x}_2$| in the expressions.

4. Calculations of nonmesonic decay rates and roles of new effective |$\Lambda N \rightarrow NN$| potentials

4.1. FULL potential model and calculation method

In the model of Paper I, four kinds of effective two-body potentials are adopted, and we call the contents “MOD-I” for convenience:

In this paper we add three kinds of new potentials arising from strange-meson exchanges such as |$(K \pi )\kappa$| and |$(K \pi )K^{*}$| and the |$\Delta _{33}$|-induced |$(\rho \rho )$|-pair exchange to exhaust possible diagrams within the MPE model. Here we remark especially that the new diagrams for the |$(K \pi )\kappa$|- and |$(K \pi )K^{*}$|-exchange potentials depicted in Fig. 1 are substitutes for the |$(K \pi )$|-pair exchange one. Therefore we call the total sum the “FULL” potential model for simplicity:

The coupling constants of the Hamiltonians are given in Refs. [12,16] with the addition of the following constants: |$g^{}_{K^{*} K \pi } \, (= - \frac{1}{2} \, g^{}_{\rho \pi \pi } \, ) = 2.95$| [16,17] and |$f^{}_{N \Delta \rho } = 7.799$| [18]. The cutoff masses |$\varLambda _{m}$| for the meson m in the form factors are found in Refs. [12,16]. As we know that the calculated nonmesonic decay rate is highly dependent on the coupling constants, we choose two typical different sets of potentials used in Paper I with the same notation:

where E16 denotes the coupling constants adopted in Refs. [19,20] and E19 denotes those in Refs. [21,22]. Thus, we have two potential models (MOD-I and FULL) and two sets of typical coupling constants listed above, and therefore we will discuss the results of four kinds of calculations: MOD-I (E16), MOD-I (E19), FULL (E16), and FULL (E19).

The nonmesonic decay rates of hypernuclei are calculated by using the shell-model wave functions for the initial hypernucleus and the final residual nucleus along with the outgoing two-nucleon scattering wave. The initial-state correlation for the |$\Lambda N$| relative S state is taken into account by solving the Bethe–Goldstone equation with use of the Nijmegen model-D |$\Lambda N$| interaction [23,24]. The final two-nucleon scattering states are solved with the Nijmegen model-D |$NN$| interaction [23,24]. Then the initial state |$\phi _{n \ell = 0}(r, b_{r}) \, {\cal Y}_{ \ell = 0, S, {\cal J} }$| is replaced as

where |$f_{S} (r)$| is the correlation function for the spin-S state and |$f_{2} (r)$| is the induced D-wave “|${^3D_1}$|” for the initial |$^3S_{1}$| state. The partial wave |$i^{{\ell }_{0} } j^{}_{ {\ell }_{0}} (k, r) \, {\cal Y}_{ {\ell }_{ 0}, S^{\prime }_{2}, {\cal J}}$| for the outgoing wave is replaced as

Here the function |$\psi _{ {\ell }_{0} S^{\prime }_{2} } (k, r)$| denotes the scattering wave. The functions |$\chi _{2} (k,r)$| and |$\chi _{0} (k,r)$| are the induced D-wave |$(^3D_1)$| for the final |$^3S_1$| state and the induced S-wave |$(^3S_1)$| for the final |$^3D_1$| state, respectively, due to the |$NN$| tensor interaction.

The calculations are made for the proton-induced decay rate |$\Gamma ^{(\text {eff.2B})}_{p} (\Lambda p \rightarrow np)$| and neutron-induced one |$\Gamma ^{(\text {eff.2B})}_{n} (\Lambda n \rightarrow nn)$| by the effective two-body |$\Lambda N \rightarrow NN$| potentials. Noting that there exist decay rates based on the pure two-body |$\Lambda N \rightarrow NN$| potentials, we regard the sum

as giving the “estimate” of the two-nucleon induced nonmesonic decay rate. This is because the effective two-body |$\Lambda N \rightarrow NN$| potentials originate from the three-body |$\Lambda NN \rightarrow NNN$| interactions and have a different nature from the pure two-body |$\Lambda N \rightarrow NN$| potentials. We therefore treat those two types of potentials as not interfering with each other. The decay rates calculated in the effective two-body potentials give the decay rates for the two-nucleon induced nonmesonic weak decay within the present approximation.

4.2. Corrections of nonmesonic decay rates calculated by MOD-I in Paper I

Before going on to discuss the results of the FULL framework of calculations, we have to correct the erroneous overestimates found in Paper I where the computations were done within the model MOD-I for the hypernuclei |$^5_\Lambda {\rm He}$|⁠, |$^{11}_\Lambda {\rm B}$|⁠, and |$^{12}_\Lambda {\rm C}$|⁠. The error was that in the decay channel-3 (main: |$^{3}S_1 \rightarrow \, ^{3}S_1$|⁠) the transition from |$^{3}S_1$| to the induced |$^{3}D_1$| due to the final-state |$NN$| interaction was not treated properly for some meson-pair exchanges except the |$(\sigma \sigma )$|-pair one. Table 1 lists the corrected values of the decay rates |$\Gamma ^{(\text {eff.2B})}_{\rm nm}$| obtained within the MOD-I framework together with reference values appearing in Paper I. In Table 1 those decay rates are shown for two cases of coupling constant |$g_{(\sigma \sigma )} / 4\pi = -0.300$| and |$-0.325$| in |$V_{(\sigma \sigma )}$|⁠, and for the |$V{\rm E19}$| potential set. The corrected decay rates for these hypernuclei turned out to be a little smaller, respectively, than the estimates reported in Paper I.

4.3. Results of the FULL model and roles of various effective two-body |$\Lambda N \rightarrow NN$| potentials

Here we discuss the results for the FULL model calculations for two-nucleon induced nonmesonic decay rates in order to make clear the roles of the adopted two-body potentials, especially the strange-meson-exchange and |$\Delta _{33}$|-induced |$(\rho \rho )$|-pair-exchange potentials. In the following we often leave out the superfix symbol “(eff.2B)” for the effective two-body potential, e.g. |$V_{(\pi \sigma )}$| for |$V^{(\text {eff.2B})}_{(\pi \sigma )}$|⁠.

Table 2 lists the divided contributions from various combinations of effective two-body potentials to the nonmesonic decay rate |$\Gamma ^{(\text {eff.2B})}_{\rm nm}$| for the |$^{12}_\Lambda {\rm C}$| hypernucleus. Two cases of potential sets |$V{\rm E16}$| and |$V{\rm E19}$| are considered and the coupling constant |$g_{(\sigma \sigma )}/4\pi = -0.325$| is adopted here for |$V_{(\sigma \sigma )}$|⁠. First of all we note the gross features of the effective two-body potentials: The potential set |$V{\rm E16} = \lbrace V_{(\pi \sigma )} + V_{(\pi \omega )}\, \rbrace _{(cc.{\rm E16})}$| has weak central, strong tensor, and weak parity-violating vector potential, while the set |$V{\rm E19} = \lbrace V_{(\pi \sigma )} + V_{(\pi \omega )}\, \rbrace _{(cc.{\rm E19})}$| has weak central, very strong tensor, and moderately strong vector potential. |$V_{(\sigma \sigma )}$| has strong central, no tensor, and very weak vector potential.

For the strange-meson-exchange potentials introduced here, we emphasize that the potential set |$V_{(K \pi )\kappa } + V_{(K \pi )K^{*}}$| has weak central, moderately strong tensor, and weak vector potential. We find that the sign of the central potential for the |$^3S_1 \rightarrow {^3S_1}$| channel (the tensor one for |$^3S_1 \rightarrow {^3D_1}$|⁠, and the vector one for |$^3S_1 \rightarrow {^1P_1}$|⁠) of |$V_{(K \pi )\kappa } + V_{(K1 )K^{*}}$| is almost opposite (completely opposite, and completely opposite) to those of |$V_{(\pi \sigma )} + V_{(\pi \omega )}$| of |$V{\rm E16}$| and of |$V{\rm E19}$|⁠.

|$V_{\rm FM(\pi \pi )}$| is known to have very weak central, strong tensor, and weak vector potential. On the other hand, |$V_{D33(\rho \rho )}$| is shown to have weak central, very weak tensor, and very weak vector one, mainly because of the large mass of the |$\rho$| meson. It is also remarked that the sign of the tensor potential of |$V_{D33(\rho \rho )}$| for the |$^3S_1 \rightarrow {^3D_1}$| channel is opposite to that of |$V_{\rm FM(\pi \pi )}$|⁠.

Table 2 summarizes the contributions to the decay rate |$\Gamma ^{(\text {eff.2B})}_{\rm nm}$| from the various combinations of effective potentials indicated in the first column. The decay rates under the heading of |$\lbrace V{\rm E16}\rbrace$| (⁠|$\lbrace V{\rm E19}\rbrace$|⁠) correspond to the theoretical estimates obtained by using the set of coupling constants E16 (E19), respectively. It is interesting to discuss these estimates in some more detail. First we discuss the decay rates in the third column heading of |$\lbrace V{\rm E19}\rbrace$|⁠. The potential set |$V{\rm E19} = \lbrace V_{(\pi \sigma )} + V_{(\pi \omega )} \rbrace _{(cc.{\rm E19})}$| gives an extraordinary large decay rate as |$0.272 \, \Gamma _{\Lambda }$|⁠, which is attributed to the special features of the two-body potentials VE19, i.e. weak central and very strong tensor nature. The potential |$V_{(\sigma \sigma )}$| has strong central force so that it alone gives a large decay rate |$0.158\, \Gamma _{\Lambda }$|⁠.

The model MOD-I (E19) adopted in Paper I, which consists of the combinations |$V{\rm E19} + V_{(\sigma \sigma )} + V_{\rm FM(\pi \pi )}$|⁠, yields the decay rate |$0.188 \, \Gamma _{\Lambda }$|⁠. This number is understandable as follows. Owing to the final-state |$NN$| tensor correlations in particular, the transition amplitude of the strong central potential, with a positive sign dominantly of |$V_{(\sigma \sigma )}$|⁠, and that of the strong tensor potential, with a negative sign of |$V{\rm E19} + V_{\rm FM(\pi \pi )}$|⁠, work destructively. As a result the decay rate in channel-3 of the |$(^3S_1)_{\Lambda p} \rightarrow (^3S_1)_{np}$| transition is much reduced, but, in contrast, the two factors work additively to produce the large decay rate in channel-4 of the |$(^3S_1)_{\Lambda p} \rightarrow (^3D_1)_{np}$| transition. Actually, the transition amplitudes of channel-3 and channel-4 are expressed as

Further, it is noted that the tensor potential with a positive sign of |$V_{\rm FM(\pi \pi )}$| works to weaken the tensor potential of |$V{\rm E19}$|⁠. These effects as a whole produce the decay rate of the model MOD-I (E19).

Now we remark that the set of strange-meson-exchange potentials, |$V_{(K \pi )\kappa } + V_{(K \pi )K^{*}}$|⁠, gives a small decay rates as |$0.022 \, \Gamma _{\Lambda }$| because of the weak central, moderately strong tensor, and weak vector force characters. It is worth noting, however, that the combination of potentials |$V{\rm E19} + V_{(K \pi )\kappa } + V_{(K \pi )K^{*}}$| gives rise to a much reduced decay rate |$0.150 \, \Gamma _{\Lambda }$| in comparison with |$0.272 \, \Gamma _{\Lambda }$| by the set |$V{\rm E19}$| alone. The obtained results clearly demonstrate special roles of the strange-meson-exchange potentials |$V_{(K \pi )\kappa } + V_{(K \pi )K^{*}}$|⁠. In fact, we found the reason that specifically the tensor and vector force parts of |$V_{(K \pi )\kappa } + V_{(K \pi )K^{*}}$| work in opposite signs to those of |$V{\rm E19}$|⁠, so yielding the remarkable reduction of the decay rate. The reduction factor amounts to |$\Gamma ^{(\text {eff.2B})}_{\rm nm}\, \lbrace V{\rm E19} + V_{(K \pi )\kappa } + V_{(K \pi )K^{*}} \rbrace / \Gamma ^{(\text {eff.2B})}_{\rm nm}\, \lbrace V{\rm E19} \rbrace$||$\approx 1/1.84$|⁠, and this value can be roughly estimated by the ratio of the tensor force for the two potential sets at a certain typical interaction radius R, because the central forces of those potentials are quite small. Since the ratio |$\lbrace V{\rm E19} + V_{(K \pi )\kappa } + V_{(K \pi )K^{*}} \rbrace ^{T} / \lbrace V{\rm E19} \rbrace ^{T}$| gives a measure of the transition amplitude ratio, the decay rate ratio is given by the square of the amplitude ratio. Then we see that

which may be compared with |$1/1.84$|⁠. It is noted that the deviation from the value |$1/1.84$| can be accounted for by another reduction effect of the parity-violating vector forces of |$V{\rm E19} + V_{(K \pi )\kappa } + V_{(K \pi )K^{*}}$| compared to |$V{\rm E19}$|⁠. In fact, the square of the vector force ratio in channel-5 of the |$(^3S_1)_{\Lambda p} \rightarrow (^1P_1)_{np}$| transition is |$| \, \lbrace V{\rm E19} + V_{(K \pi )\kappa } + V_{(K \pi )K^{*}} \rbrace ^{V} / \lbrace V{\rm E19} \rbrace ^{V} \, |^{2} = 1/2.24$| at |$R = 0.75$| fm.  Then the reduction effects for both the tensor and vector forces in |$V{\rm E19} + V_{(K \pi )\kappa } + V_{(K \pi )K^{*}}$| compared to |$V{\rm E19}$| explain the decay rate ratio |$\Gamma ^{(\text {eff.2B})}_{\rm nm}\, \lbrace V{\rm E19} + V_{(K \pi )\kappa } + V_{(K \pi )K^{*}} \rbrace / \Gamma ^{(\text {eff.2B})}_{\rm nm}\, \lbrace V{\rm E19} \rbrace$|⁠.

The contribution of |$V_{D33(\rho \rho )}$| to the decay rate is very limited and the sign of the tensor force of |$V_{D33(\rho \rho )}$| is negative and thus opposite to that of |$V_{\rm FM(\pi \pi )}$| in the |$^3S_1 \rightarrow {^3D_1}$| channel, which is known by comparing the decay rates in cases of using the set |$V_{\rm FM(\pi \pi )} + V_{D33(\rho \rho )}$| and |$V_{\rm FM(\pi \pi )}$| alone.

The present model FULL (E19) = MOD-I (E19)|$+V_{(K \pi )\kappa } + V_{(K \pi )K^{*}} + V_{D33(\rho \rho )}$| is characterized by strong central, moderately strong tensor, and moderately strong parity-violating vector potential parts. The decay rate |$\Gamma ^{(\text {eff.2B})}_{\rm nm}$| by the model FULL (E19) turns out to be |$0.139 \, \Gamma _{\Lambda }$|⁠, which is small compared with the value |$0.188 \, \Gamma _{\Lambda }$| calculated by MOD-I (E19). The reduction of the decay rate by the FULL (E19) model from that by MOD-I (E19) originates mostly due to the weaker tensor force in FULL (E19) than in MOD-I (E19). This is the important consequence of introducing the strange-meson and |$\Delta _{33}$|-induced |$(\rho \rho )$|-pair exchanges in the three-body |$\Lambda NN \rightarrow NNN$| interactions and therefore in the effective two-body |$\Lambda N \rightarrow NN$| potentials. The obtained nonmesonic decay rate of |$0.139 \, \Gamma _{\Lambda }$| for the |$^{12}_{\Lambda } {\rm C}$| hypernucleus may be small, but it is within the error bars of |$\Gamma ^{\rm exp}_{2N}$| listed in Table 2.

Next we move on to discuss the decay rates in the second column heading of {|$V{\rm E16}$|}. The potential set |$V{\rm E16} = \lbrace V_{(\pi \sigma )} + V_{(\pi \omega )} \rbrace _{(cc.{\rm E16})}$| gives a smaller decay rate, which amounts to 1/3 of |$0.272 \, \Gamma _{\Lambda }$| for |$V{\rm E19}$|⁠. Since both potential sets show tensor-force dominance, the above ratio can be accounted for almost by the square of the tensor-force ratio in the |$^3S_1 \rightarrow {^3D_1}$| transition as |$\big | \, \lbrace V{\rm E16}\rbrace ^{T}/ \lbrace V{\rm E19}\rbrace ^{T} \, \big |^{2} \simeq 1/3$| at |$R = 0.75$| fm.

The potentials of MOD-I (E16) including |$V{\rm E16}$| have strong central (positive sign) and moderately strong tensor-force (negative sign) character. The calculated decay rate can be understood from the mutual interference between the two types of forces due to the strong |$NN$| final-state tensor correlations as shown in Eqs. (33)–(34), which are that the central force and tensor one work destructively on the decay rate in the |$(^3S_1)_{\Lambda p} \rightarrow (^3S_1)_{np}$| channel, while both forces work additively on the decay rate in the |$(^3S_1)_{\Lambda p} \rightarrow (^3D_1)_{np}$| channel.

The combination of the potential |$V{\rm E16} + V_{(K \pi )\kappa } + V_{(K \pi )K^{*}}$| produces a decay rate of |$0.030 \, \Gamma _{\Lambda }$| which is 1/5 of the value calculated by |$V{\rm E19} + V_{(K \pi )\kappa } + V_{(K \pi )K^{*}}$|⁠. This ratio is roughly explained by the square of the tensor-force ratio as |$\big | \, \lbrace V{\rm E16} + V_{(K \pi )\kappa } + V_{(K \pi )K^{*}} \rbrace ^{T} / \lbrace V{\rm E19} + V_{(K \pi )\kappa } + V_{(K \pi )K^{*}} \rbrace ^{T} \, \big |^{2} \simeq \, 1/4.3$| at |$R = 0.75$| fm.

Finally, the model FULL (E16) = MOD-I (E16) |$+ V_{(K \pi )\kappa } + V_{(K \pi )K^{*}} + V_{D33(\rho \rho )}$| gives a decay rate of |$0.171 \, \Gamma _{\Lambda }$| whose value exceeds the decay rate by the model FULL (E19) including |$V{\rm E19}$|⁠. This is because the two models FULL (E16) and FULL (E19) have different features, especially for the tensor-force part for the |$^3S_1 \rightarrow {^3D_1}$| transition; i.e. FULL (E19) has a moderately strong one with a negative sign at |$0.5 \lt R \lt \, 2.0$| fm, while FULL (E16) has an effectively weak tensor with a sign change from positive to negative at |$R \approx 0,75$| fm. Then in FULL (E16) the central-force dominance is almost realized and the destructive and additive interferences due to the final-state correlations in channel-3 and channel-4, respectively, are small. However, in FULL (E19) the tensor force is nonnegligible and the central force is strong so that the destructive and additive interferences in channel-3 and channel-4, respectively, are effective in finally producing a quite small decay rate. This is the reason for the different decay rates |$\Gamma ^{(\text {eff.2B})}_{\rm nm}$| for the two models FULL (E16) and FULL (E19).

Table 3 shows the calculated two-nucleon induced decay rates of |$^5_\Lambda {\rm He}$|⁠, |$^{11}_\Lambda {\rm B}$|⁠, and |$^{12}_\Lambda {\rm C}$| hypernuclei in the models FULL (E16) and FULL (E19) for two coupling constants |$g_{(\sigma \sigma )}/4\pi = -0.300$| and |$g_{(\sigma \sigma )}/4\pi = -0.325$| of |$V_{(\sigma \sigma )}$|⁠. The experimental data for the two-nucleon induced decay rate |$\Gamma ^{\rm exp}_{2N}$| and the total nonmesonic one |$\Gamma ^{\rm exp}_{\rm NM}$| are listed for comparison. The calculated decay rates vary depending on the models FULL (E16) and FULL (E19), and the coupling constant |$g_{(\sigma \sigma )}/4\pi$|⁠. We should say that the pair-meson coupling constants, E16 and E19, in the ESC model still have ambiguities and |$g_{(\sigma \sigma )}/4\pi$| is not well constrained. Considering these situations, although the calculated decay rates come within the experimental error bars of |$\Gamma ^{\rm exp}_{2N}$|⁠, more studies are necessary, including other possibilities for the decay mechanism of two-nucleon induced nonmesonic decays.

5. Summary

In the preceding paper we introduced three-body |$\Lambda NN \rightarrow NNN$| weak interactions and proposed a framework to deduce the effective two-body |$\Lambda N \rightarrow NN$| potentials to be used for estimates of nonmesonic decay rates of hypernuclei. There we examined the contributions from four kinds of deduced potentials of |$V^{(\text {eff.2B})}_{(\pi \sigma )}$|⁠, |$V^{(\text {eff.2B})}_{(\pi \omega )}$|⁠, |$V^{(\text {eff.2B})}_{(\sigma \sigma )}$|⁠, and |$V^{(\text {eff.2B})}_{\rm FM(\pi \pi )}$|⁠, respectively. In order to complete the framework, we have extended the treatment to include new diagrams arising from the strange-meson exchanges and the |$\Delta _{33}$|-isobar induced |$(\rho \rho )$|-pair exchange. Thus we present new forms of |$V^{(\text {eff.2B})}_{(K \pi )\kappa }$|⁠, |$V^{(\text {eff.2B})}_{(K \pi )K^{*}}$|⁠, and |$V^{(\text {eff.2B})}_{D33(\rho \rho )}$| potentials. By adding these potentials to the preceding four, we have investigated in full the FULL model outputs of the present framework with particular attention to the roles of various types of the effective |$\Lambda N \rightarrow NN$| potentials in hypernuclear nonmesonic decays.

According to the present investigation wthin the FULL potential model, the effective two-body |$\Lambda N \rightarrow NN$| potentials deduced from the three-body |$\Lambda NN \rightarrow NNN$| weak interactions can be classified into the following major groups on the basis of their dominant characters:

• The effective two-body potentials deduced from three-body weak interactions involving |$(\pi \sigma ) + (\pi \omega )$|-pair exchanges lead to a tensor-force-dominant character similar to the “|$\pi$|-exchange” potential in the single-meson-exchange model.

• The effective two-body potentials deduced from three-body weak interactions involving |$(K \pi )\kappa + (K \pi )K^{*}$| exchanges lead to a tensor-force-dominant character similar to the “K-exchange” potential.

• The effective two-body potential deduced from three-body weak interactions involving |$(\sigma \sigma )$|-pair exchange is revealed to have central-force dominance and no tensor term similar to the “|$\sigma$|-exchange” potential.

• The effective two-body potentials deduced from three-body weak interactions involving |$\Delta _{33}$|-induced pair-meson exchanges are found to have rather strong tensor force due to |${\rm FM(\pi \pi )} + D33(\rho \rho )$| exchanges.

Here we note that these results are not trivial ones, but are remarkable findings realized through the effective two-body potential representations.

We found that the new potential set |$V_{(K \pi )\kappa } + V_{(K \pi )K^{*}}$| has weak central (negative sign in |$^3S_1 \rightarrow {^3S_1}$|⁠), moderately strong tensors (positive sign in |$^3S_1 \rightarrow {^3D_1}$|⁠), and weak parity-violating vector (positive sign in |$^3S_1 \rightarrow {^1P_1}$|⁠) forces, and also that their signs are completely opposite to those of |$V_{(\pi \sigma )} + V_{(\pi \omega )}$|⁠, respectively. Moreover, it is interesting to remark that even the moderately strong tensor forces of |$V_{(K \pi )\kappa } + V_{(K \pi )K^{*}}$| are sufficient to diminish the effect of the strong tensor ones of |$V_{(\pi \sigma )} + V_{(\pi \omega )}$| because of their opposite signs. As a result, such interplay tends finally to reduce the two-nucleon induced nonmesonic decay rates in the FULL (E19) potential model in comparison with the decay rate predicted in the limited MOD-I (E19) model in Paper I. We emphasize that this is a special and notable role played by the strange-meson-exchange |$V_{(K \pi )\kappa } + V_{(K \pi )K^{*}}$| potentials that are introduced in this paper. Moreover we found that |$V_{D33(\rho \rho )}$| has a tendency to work in the opposite direction with respect to |$V_{\rm FM(\pi \pi )}$| but with a limited role in the estimate of two-nucleon induced decay rates.

Calculated decay rates of the two-nucleon induced nonmesonic decays of |$^5_\Lambda {\rm He}$|⁠, |$^{11}_\Lambda {\rm B}$|⁠, and |$^{12}_\Lambda {\rm C}$| are presented, showing that the two kinds of theoretical estimates using the models of FULL (E16) and FULL (E19) are slightly different from each other depending on their coupling constants, although the predicted decay rates are well comparable with the experimental values. Finally we point out that the pair-meson coupling constants of the ESC model [19–22] and |$g_{(\sigma \sigma )}$| still have some ambiguities, and so extensive studies on the weak decay mechanism of two-nucleon induced decays are necessary.

Acknowledgments

One of the authors, K.I., would like to thank T. Matsuda of University of Miyazaki for his interest in and warm support for this work. T.M. has been supported in part by JSPS KAKENHI Grant Number JP20K03950.

Appendix A. Configuration space effective two-body |$\Lambda N \rightarrow NN$| potentials of strange-meson and |$\Delta _{33}$|-induced |$(\rho \rho )$|-pair exchange

1. |$(K \pi )\kappa$| exchange

   $$\begin{eqnarray} V^{(\text {eff.2B})}_{(K \pi )\kappa } ( r ) & = & \frac{4\pi \rho ^{}_{\rm NM}}{m^{3}_{\pi }} \frac{G^{}_{\rm F} m^{2}_{\pi } g^{}_{K \pi \kappa }}{4\pi } \frac{g^{}_{\Lambda NK} g^{}_{NN\pi }}{4\pi } \left( \frac{m^{2}_{\pi }}{m^{2}_{\kappa }} \right) \, \Big [ \frac{1}{2} C^{pc}_{\kappa } + D^{pc}_{\kappa } \, \Big ] \\ && \times \left\lbrace (\boldsymbol{\sigma }_1 \cdot \boldsymbol{\sigma }_2) (\boldsymbol{\tau }_1 \cdot \boldsymbol{\tau }_2) \frac{1}{3} \, \Big [ \frac{m^{2}_{\pi }}{m^{2}_{K} - m^{2}_{\pi } } \frac{m^{2}_{\pi }}{4\bar{M} M_{N}} \phi ^{1}_{C}\left(m_{\pi }, \sqrt{2} \varLambda _{\pi }, r\right) \, \right. \\ && - \frac{m_{\pi } m_{K}}{m^{2}_{K} - m^{2}_{\pi } } \frac{m^{2}_{K}}{4\bar{M} M_{N}} \phi ^{1}_{C}(m_{K}, \sqrt{2} \varLambda _{K}, r) \, \Big ] \\ && + \, S_{12} \, (\boldsymbol{\tau }_1 \cdot \boldsymbol{\tau }_2) \, \Big [ \frac{m^{2}_{\pi }}{m^{2}_{K} - m^{2}_{\pi } } \frac{m^{2}_{\pi }}{4\bar{M} M_{N}} \phi ^{0}_{T}(m_{\pi }, \sqrt{2} \varLambda _{\pi }, r) \, \\ && \left. - \frac{m_{\pi } m_{K}}{m^{2}_{K} - m^{2}_{\pi } } \frac{m^{2}_{K}}{4\bar{M} M_{N}} \phi ^{0}_{T}(m_{K}, \sqrt{2} \varLambda _{K}, r) \, \Big ] \, \right\rbrace \\ && + \frac{4\pi \rho ^{}_{\rm NM}}{m^{3}_{\pi }} \frac{G^{}_{\rm F} m^{2}_{\pi } g^{}_{K \pi \kappa }}{4\pi } \frac{g_{\Lambda N\kappa } g^{}_{NN\pi }}{4\pi } \left( \frac{m^{2}_{\pi }}{m^{2}_{K}} \right) \, \Big [ \frac{1}{2} C^{pv}_{K} + D^{pv}_{K} \, \Big ] \\ && \times \left\lbrace i \, (\boldsymbol{\sigma }_2 \cdot \hat{\boldsymbol{r}} ) (\boldsymbol{\tau }_1 \cdot \boldsymbol{\tau }_2) \, \frac{m^{2}_{\pi }}{m^{2}_{\kappa } - m^{2}_{\pi } } \, \Big [ \frac{m_{\pi }}{2M_{N}} \phi ^{0}_{V}(m_{\pi }, \sqrt{2} \varLambda _{\pi }, r) \, \right. \\ && - \frac{m^{3}_{\pi }}{2M_N \bar{M}^{2}} \frac{1}{8} \, \Big \lbrace \phi ^{0}_{V}(m_{\pi }, \sqrt{2} \varLambda _{\pi }, r) \, \\ && - \frac{1}{2\sqrt{\pi } } \Big ( \frac{\sqrt{2} \varLambda _{\pi }}{m_{\pi } } \Big )^{4} \, \frac{\sqrt{2} \varLambda _{\pi } r}{2} \, e^{ - \Big ( \frac{\sqrt{2} \varLambda _{\pi } r}{2} \Big )^{2} } \Big \rbrace \, \Big ] \, \\ && - \, i \, (\boldsymbol{\sigma }_2 \cdot \hat{\boldsymbol{r}} ) (\boldsymbol{\tau }_1 \cdot \boldsymbol{\tau }_2) \, \frac{m_{\pi } m_{\kappa } }{m^{2}_{\kappa } - m^{2}_{\pi } } \, \Big [ \frac{m_{\kappa }}{2M_{N}} \phi ^{0}_{V}(m_{\kappa }, \sqrt{2} \varLambda _{\kappa }, r) \, \\ && - \frac{m^{3}_{\kappa }}{2M_N \bar{M}^{2}} \frac{1}{8} \, \Big \lbrace \phi ^{0}_{V}(m_{\kappa }, \sqrt{2} \varLambda _{\kappa }, r) \, \\ && \left. - \frac{1}{2\sqrt{\pi } } \Big ( \frac{\sqrt{2} \varLambda _{\kappa }}{m_{\kappa } } \Big )^{4} \, \frac{\sqrt{2} \varLambda _{\kappa } r}{2} \, e^{ - \Big ( \frac{\sqrt{2} \varLambda _{\kappa } r}{2} \Big )^{2} } \Big \rbrace \, \Big ] \, \right\rbrace \, . \end{eqnarray}$$

   (A1)
2. |$(K \pi )K^{*}$| exchange

   $$\begin{eqnarray} V^{(\text {eff.2B})}_{(K \pi )K^{*}} ( r ) & = & \frac{4\pi \rho ^{}_{\rm NM}}{m^{3}_{\pi }} \frac{G^{}_{\rm F} m^{2}_{\pi } g^{}_{K^{*} K \pi }}{4\pi } \frac{g^{}_{\Lambda NK} g^{}_{NN\pi }}{4\pi } \Big ( \frac{m^{2}_{\pi }}{m^{2}_{K^{*}}} \Big ) \, 2M_{N} \frac{M_{N} - M_{\Lambda } }{2M_{N}} \\ && \times \Big [ \frac{1}{2} C^{pc,V}_{K^{*}} + D^{pc,V}_{K^{*}} \, \Big ] \\ && \times \left\lbrace (\boldsymbol{\sigma }_1 \cdot \boldsymbol{\sigma }_2) (\boldsymbol{\tau }_1 \cdot \boldsymbol{\tau }_2) \frac{1}{3} \, \Big [ \frac{m^{2}_{\pi }}{m^{2}_{K} - m^{2}_{\pi } } \frac{m^{2}_{\pi }}{4\bar{M} M_{N}} \phi ^{1}_{C}(m_{\pi }, \sqrt{2} \varLambda _{\pi }, r) \, \right. \\ && - \frac{m_{\pi } m_{K}}{m^{2}_{K} - m^{2}_{\pi } } \frac{m^{2}_{K}}{4\bar{M} M_{N}} \phi ^{1}_{C}(m_{K}, \sqrt{2} \varLambda _{K}, r) \, \Big ] \\ && \ + \, S_{12} \, (\boldsymbol{\tau }_1 \cdot \boldsymbol{\tau }_2) \, \Big [ \frac{m^{2}_{\pi }}{m^{2}_{K} - m^{2}_{\pi } } \frac{m^{2}_{\pi }}{4\bar{M} M_{N}} \phi ^{0}_{T}(m_{\pi }, \sqrt{2} \varLambda _{\pi }, r) \, \\ && \left. - \frac{m_{\pi } m_{K}}{m^{2}_{K} - m^{2}_{\pi } } \frac{m^{2}_{K}}{4\bar{M} M_{N}} \phi ^{0}_{T}(m_{K}, \sqrt{2} \varLambda _{K}, r) \, \Big ] \, \right\rbrace \, . \end{eqnarray}$$

   (A2)
3. |$\Delta _{33}$|-induced |$(\rho \rho )$|-pair exchange

   $$\begin{eqnarray} V^{(\text {eff.2B})}_{D33 (\rho \rho )} ( r ) & = & \frac{4\pi \rho ^{}_{\rm NM}}{m^{3}_{\pi }} \frac{G^{}_{\rm F} m^{2}_{\pi } g^{}_{NN\rho }}{4\pi } \frac{f^{2}_{N \Delta \rho } }{4\pi } \, m_{\pi } \, \Big ( \frac{m^{2}_{\pi }}{m^{2}_{\rho }} \Big ) \, \frac{m_{\rho } }{M_{\Delta } - M_{N} } \, \frac{8}{9} \\ && \times \left\lbrace (\boldsymbol{\tau }_1 \cdot \boldsymbol{\tau }_2) \, \Big [ \alpha _{\rho } \, \Big ( \frac{m^{2}_{\rho }}{4\bar{M} M_{N} } + \frac{m^{2}_{\rho }}{4{\bar{M}}^{2}_{\Delta N} } \, \Big ) \, \Big \lbrace \phi ^{1}_{C}(m_{\rho }, \frac{\varLambda _{D3\rho }}{\sqrt{2}}, \, r) \, \right. \\ && + \, \Big (1 + \frac{2m^{2}_{\rho }}{{\varLambda }^2_{D3\rho }} \Big ) \, \phi ^{0}_{C}(m_{\rho }, \frac{\varLambda _{D3\rho }}{\sqrt{2}}, \, r) \, - \frac{1}{4} \chi ^{0}_{C}(m_{\rho }, \frac{\varLambda _{D3\rho }}{\sqrt{2}}, \, r) \, \Big \rbrace \\ && - \, \alpha _{\rho } \frac{1}{{\bar{M}}^{2}_{\Delta N} } \, \frac{3}{5} k^{2}_{F} \, \Big \lbrace \Big (1 + \frac{2m^{2}_{\rho }}{{\varLambda }^2_{D3\rho }} \Big ) \, \phi ^{0}_{C}(m_{\rho }, \frac{\varLambda _{D3\rho }}{\sqrt{2}}, \, r) \, \\ && - \frac{1}{4} \chi ^{0}_{C}(m_{\rho }, \frac{\varLambda _{D3\rho }}{\sqrt{2}}, \, r) \, \Big \rbrace \, \Big ] \\ && + (\boldsymbol{\sigma }_1 \cdot \boldsymbol{\sigma }_2) (\boldsymbol{\tau }_1 \cdot \boldsymbol{\tau }_2) \, (\alpha _{\rho } + \beta _{\rho }) \, \Big ( 1 + \frac{f_{NN\rho } }{g_{NN\rho } } \, \Big ) \, \frac{2}{3} \, \frac{m^{2}_{\rho }}{4\bar{M} M_{N} } \\ && \times \, \Big [ \, \phi ^{1}_{C}(m_{\rho }, \frac{\varLambda _{D3\rho }}{\sqrt{2}}, \, r) + \Big (1 + \frac{2m^{2}_{\rho }}{{\varLambda }^2_{D3\rho }} \Big ) \phi ^{0}_{C}(m_{\rho }, \frac{\varLambda _{D3\rho }}{\sqrt{2}}, \, r) \, \\ & & - \frac{1}{4} \chi ^{0}_{C}(m_{\rho }, \frac{\varLambda _{D3\rho }}{\sqrt{2}}, \, r) \, \Big ] \\ & & - \, S_{12} \, (\boldsymbol{\tau }_1 \cdot \boldsymbol{\tau }_2) \, (\alpha _{\rho } + \beta _{\rho }) \, \Big ( 1 + \frac{f_{NN\rho } }{g_{NN\rho } } \, \Big ) \, \frac{m^{2}_{\rho }}{4\bar{M} M_{N} } \\ & & \times \, \Big [ \, \Big (1 + \frac{2m^{2}_{\rho }}{{\varLambda }^2_{D3\rho }} \Big ) \phi ^{0}_{T}(m_{\rho }, \frac{\varLambda _{D3\rho }}{\sqrt{2}}, \, r) \, - \frac{1}{6} \, (m_{\rho } r) \, \phi ^{0}_{V}(m_{\rho }, \frac{\varLambda _{D3\rho }}{\sqrt{2}}, \, r) \, \Big ] \\ & & + \, i \cdot i \, ( (\boldsymbol{\sigma }_1 \times \boldsymbol{\sigma }_2) \cdot \hat{\boldsymbol{r}} \, ) (\boldsymbol{\tau }_1 \cdot \boldsymbol{\tau }_2) \, \epsilon _{\rho } \, \Big ( 1 + \frac{f_{NN\rho } }{g_{NN\rho } } \, \Big ) \, \frac{m_{\rho }}{2M_{N} } \\ & & \left. \times \, \Big [ \, \Big (1 + \frac{2m^{2}_{\rho }}{{\varLambda }^2_{D3\rho }} \Big ) \phi ^{0}_{V}(m_{\rho }, \frac{\varLambda _{D3\rho }}{\sqrt{2}}, \, r) \, - \frac{1}{2} \, (m_{\rho } r) \, \phi ^{0}_{C}(m_{\rho }, \frac{\varLambda _{D3\rho }}{\sqrt{2}}, \, r) \, \Big ] \, \right\rbrace \, , \\ \end{eqnarray}$$

   (A3)

   where |$\bar{M}_{\Delta N} = (M_{\Delta } + M_{N})/2$|⁠.

In the above equations, the functions |$\phi ^{0}_{C}(m,\varLambda , r)$|⁠, |$\phi ^{1}_{C}(m,\varLambda , r)$|⁠, and |$\phi ^{0}_{T}(m,\varLambda , r)$| are expressed in Ref. [17], and |$\phi ^{0}_{V}(m,\varLambda , r)$| and |$\chi ^{0}_{C}(m,\varLambda , r)$| are defined in Ref. [12].

Footnotes

References