Statistical double Λ hypernuclear formation from Ξ− absorption at rest in light nuclei


 We investigate double $\Lambda$ hyperfragment formation from the statistical decay of double $\Lambda$ compound nuclei produced in the $\Xi^-$ absorption at rest in the light nuclei $^{12}\mathrm{C}$, $^{14}\mathrm{N}$, and $^{16}\mathrm{O}$. We examine the target and the $\Lambda\Lambda$ bond energy dependence of the double $\Lambda$ hyperfragment formation probabilities, especially of those double hypernuclei observed in experiments. For the $^{12}\mathrm{C}$ ($^{14}\mathrm{N}$) target, the formation probabilities of $^{\,\;\;6}_{\Lambda\Lambda}\mathrm{He}$ and $^{\;10}_{\Lambda\Lambda}\mathrm{Be}$ ($^{\;13}_{\Lambda\Lambda}\mathrm{B}$) are found to be reasonably large as they are observed in the KEK-E373 (KEK-E176) experiment. By comparison, for the $^{16}\mathrm{O}$ target, the formation probability of $^{\;11}_{\Lambda\Lambda}\mathrm{Be}$ is calculated to be small with $\Delta B_{\Lambda\Lambda}$ consistent with the Nagara event. We also evaluate the formation probability of ${}^{\,\;\;5}_{\Lambda\Lambda}\mathrm{H}$ from a $\Xi^-$–${}^{6}\mathrm{He}$ bound state, ${}^{7}_{\Xi}\mathrm{H}$.


Introduction
Formation of double hypernuclei (D HN) from − absorption in nuclei is of importance for several reasons. − absorption at rest in nuclei is the most efficient way to produce D HN, and uniquely identified D HN [1][2][3][4][5][6] provide strong constraints on the interaction [7,8]. The strength and density dependence of the interaction are the keys to solving the hyperon puzzle in neutron star physics. Until now, four D HN formation events have been uniquely identified, and more will be found in the J-PARC-E07 experiment, where 10 4 − absorption events in nuclei are expected to be observed. Let us comment on these points in order.
The baryon-baryon interaction has been one of the central subjects in nuclear physics. Compared with nucleon-nucleon interactions, hyperon-nucleon scattering data are much more scarce and single hypernuclear data are also used to constrain the N interaction. For the interaction, there are theoretical predictions in the meson exchange model [9], the quark cluster model [10], and lattice QCD calculations [11]. Experimentally, by comparison, it is not possible to perform scattering experiments, and hence the binding energies of D HN [1][2][3][4][5][6] and the correlation function data from high-energy nuclear collisions [12][13][14][15][16][17][18][19][20][21][22][23] have been utilized to experimentally constrain the interaction. While the correlation function technique has recently been applied to investigate several hadron-hadron interactions [12][13][14][15], we need further theoretical and experimental studies to constrain the interactions precisely [15][16][17][18][19][20][21][22][23]. At present, the strongest constraint on the interaction is provided by the bond energy of the D HN, 6 He, observed in the Nagara event [4]. The bond energy represents the strength of the interaction, and is defined as B ≡ S ( A Z) − 2S ( A−1 Z), where S and S are the separation (binding) energies of and . The bond energy of 6 He is found to be B ( 6 He) = 0.67 ± 0.12 MeV [5]. This bond energy can be fitted by a interaction with the low-energy scattering parameters of (a 0 , r eff ) = (−0.44 fm, 10.1 fm) [7], where a 0 and r eff are the scattering length and the effective range, respectively.
The interaction also plays a crucial role in neutron star physics. With most of the well-known attractive N two-body interactions, hyperon mixing is calculated to take place in neutron star matter at (2)(3)(4)ρ 0 and the equation of state (EOS) is softened [24,25]. Consequently, it is hard to support two-solar-mass neutron stars [26][27][28]. In order to solve this problem, known as the hyperon puzzle, several mechanisms have been proposed so far. One of the natural ways is to introduce repulsive three-baryon interactions [29][30][31][32][33][34]. For example, it is possible to support massive neutron stars by introducing repulsive three-body contact couplings in relativistic mean field models [34]. It should be noted that the three-baryon repulsion also needs to operate among N and , otherwise matter becomes more stable than nuclear matter with hyperon mixing at high densities. The N three-baryon interaction will cause effective density-dependent and N interactions in nuclear matter. If we can observe and uniquely identify many D HN in a wide mass region, it would be possible to deduce the density dependence of the interaction and the underlying N three-baryon interaction.
The most efficient reaction to form D HN is − absorption at rest in nuclei. D HN formation proceeds in the following steps. First, − particles are produced in the (K − , K + ) reactions on nuclei or protons [1][2][3][4][5][6] orp-nucleus collisions [35]. The produced − particle is absorbed in a nucleus, and converted to two particles via the − p → reaction in the nucleus. If two s are trapped in the nucleus in the pre-equilibrium stage, a double compound nucleus (D C) is formed [36]. The compound nucleus de-excites by emitting nucleons, s, αs, and other clusters. When these two particles occasionally stay in the same fragment with an excitation energy below the particle emission threshold, a D HN is formed. When the sequential weak decay of the D HN is observed, one can identify that an S = −2 nucleus is formed. The production and detection of D HN in heavy-ion collisions [37] are also expected, while there is no clear evidence of D HN formation in these reactions yet.
Until now, several experiments have been performed to find D HN, and four of them have been uniquely identified from − absorption in nuclei, as summarized in Table 1 [1][2][3][4][5]: 10 Be from the − + 12 C reaction at CERN [1,2], 13 B from − + 14 N in the KEK-E176 experiment [3], 6 He from − + 12 C (Nagara event) [4], and 10 Be * from − + 12 C (Demachi-Yanagi event) [5] in the KEK-E373 experiment. Another report on 6 He [38] was questioned [39] and found to be not consistent with the Nagara event [4,5]. It should be noted that the identifications of the above four D HN rely on consistency between the events. For example, B values of 10 Be in Refs. [1,2,5] can be made consistent by assuming the channel of 10 Be → 9 Be * + p + π − in the weak decay in the event of Refs. [1,2] and the formation of the excited state 10 Be * in the Demachi-Yanagi event [5]. In order to further observe D HN, the J-PARC-E07 experiment has been carried out. While the analysis is still ongoing, a new D HN formation event was discovered recently, Be from − + 16 O (Mino event) [6]. We summarize the uniquely identified D HN events in Table 1. We also show the candidate fragmentation reactions in the Mino event.
Since further events are expected to be observed from the J-PARC-E07 and future experiments, it would be possible to perform statistical analysis of fragment formation events from − absorption  [1][2][3][4][5]. We also show the possible formation channels in the Mino event, a new event observed in the J-PARC-E07 experiment [6]. at rest in light nuclei. Statistical decay of D C was studied by using a canonical fragmentation model [40], a sequential binary statistical decay model [41], and a microcanonical fragmentation model [42]. We also note that the − absorption reaction in 12 C was analyzed by using the direct reaction model in Ref. [43]. In Ref. [41], the − absorption reaction in 12 C was analyzed in a combined framework of a transport model and a statistical decay model of hypernuclei. The formation probability of the D C ( 13 B * ) was evaluated to be around P D C 30% in the pre-equilibrium stage by using the antisymmetrized molecular dynamics (AMD) transport model calculation [41], the sum of branching ratios to form D HN from D C in the statistical decay was found to be around P tot Br = 60%, and then the total D HN formation probability was found to be around P tot D HN = P D C × P tot Br 18% after the statistical decay. This analysis was performed before the discovery of the Nagara event [4], and a strongly attractive interaction was adopted, B ( 13 B) = 4.9 MeV, as suggested by the KEK-E176 experiment [3]. After the Nagara event, the KEK-E176 event was reinterpreted and the bond energy is now considered to be B ( 13 B) = 0.6 ± 0.8 MeV. With these updated, less attractive, interactions, it would be possible to predict the D HN formation probabilities in a more reliable manner.
It should be noted that the formation probability of D C (P D C ) depends on the definition of compound nucleus formation, and in practice it depends on the transport model adopted in describing the dynamical stage. For example, the D C formation probability in stopped − absorption in 12 C is calculated to be smaller, P D C (AMD-QL) = 16%, in AMD with additional quantum fluctuations (AMD-QL) [41]. In addition to the difference in P D C , emission of nucleons and light clusters in the dynamical stage would modify the mass dependence of the formation probability of D C. In Sect. 3.1 we compare the statistical decay model results from D C ( 13 B * ) and AMD-QL results with the statistical decay in Ref. [41].
In this article we discuss the formation of D HN from the statistical decay of D C formed via the − absorption at rest in nuclei. Specifically, we concentrate on the target nuclei 12 C, 14 N, and 16 O. These are the main light components of emulsion, and some D HN have been reported to be formed. We mainly discuss the formation probabilities 6 He and 10 Be from − + 12 C, 13 B from − + 14 N, and 11 Be from − + 16 O. We also discuss the dependence of the formation probabilities on the bond energy, and the decay of the hypernucleus formed in the 7 Li(K − , K + ) reaction. The paper is organized as follows. In Sect. 2 we provide a general idea of statistical decay of compound hypernuclei. In Sect. 3, we evaluate D HN formation probabilities from the absorption reaction at rest in 12 C, 14

Statistical decay of double compound nuclei and hypernuclear binding energies
In this paper, the statistical decay of D C is calculated by using a sequential binary statistical decay model (SDM) [44]. In the SDM, an excited nucleus 1 is assumed to decay into nuclei 2 and 3 with a decay rate of where E i and J i denote the excitation energy and the angular momentum of the ith nucleus. We adopt the back-shifted Fermi gas model to evaluate the level density ρ i (E i , J i ) [44,45]. Since we are interested in decays of compound nuclei in equilibrium, the density of narrow excited levels needs to be considered and the level density is reduced at excitation energies above the threshold for chargeneutral particle emission and above the Coulomb barrier for charged particle emission [45]. The factor T L = (L c − L) is the transmission coefficient of the partial wave L in the fusion reaction 2 + 3 → 1 with the incident energy corresponding to the excitation energies and the Q-value. We assume strong absorption in the inverse fusion process 2 + 3 → 1, and take the form T L = (L c − L) with L c being the maximum orbital angular momentum of the fusion. The statistical decays are assumed to be binary and to proceed until all the fragments are in their ground states. When combined with the transport models describing the pre-equilibrium stage, the SDM has been found to work well for intermediate energy heavy-ion collisions [46][47][48] and for hypernuclear formation reactions [41,49]. We would also like to mention here that the combined framework of the transport model and the SDM has been successfully applied to the light-ion induced reaction, p + 12 C at 45 MeV, where the excitation energy of the compound nucleus is similar to the − absorption reaction at rest in nuclei and essentially the same SDM program is used [41]. The essential inputs of the SDM are the binding energies of nuclei that would be formed during the sequential decay processes. While the binding energies of normal nuclei are well known, information on single and double hypernuclear binding energies is limited. We have constructed the mass table based on the following assumptions. First, all existing normal nuclei plus and are assumed to form single and double hypernuclei, respectively. Normal nuclei include those whose ground states are resonance and unstable to particle emission. We also consider dineutron (nn) and 2 He (pp) as resonance nuclei, and their energies are assumed to be 100 keV above the threshold, in order to mimic three-body decays. Second, for single hypernuclei we adopt the separation energy if measured. We use the separation energy data summarized by Bando et al. (BMZ) [50] and by Hashimoto and Tamura (HT) [51]. We add 0.5 MeV to the separation energies measured in (π + , K + ) experiments [51] to take account of the recalibration as shown in Ref. [52]. If not measured, we adopt the separation energy S , parameterized as a function of the mass number and fitted to the observed separation energies, where

Double hypernuclear formation from − absorption in nuclei
We now discuss D HN formation from − absorption reactions at rest in 12 C, 14 N, and 16 O. We assume that − is absorbed from the 3D atomic orbit, so the − binding energy is B = μα 2 Z 2 /2n 2 = 0.126, 0.174, and 0.230 MeV for 12 C, 14 Table 2.
The open channels including D HN from the − absorption reactions are shown in Fig. 2. Many channels, including two-, three-, and four-body, can be populated in the final state. We have performed the statistical decay model calculation from the D C until all the fragments are de-excited to the ground state. In order to handle the variety of decay paths, the decay channel, excitation energies, and angular momenta of daughter fragments are chosen with a Monte Carlo method. Table 2. Relevant energies in the stopped − absorption reaction at rest in 12 C, 14 N, and 16 O, showing the target binding energies (B T ), the proton separation energies (S p ), and the energy from the emission threshold (E th ). We also show the case of the 6 He target, a core nucleus of 7 H, which may be formed in the 7 Li(K − , K + ) reaction. 12 Table 3 summarizes the formation probabilities of D HN in the SDM calculation from the D C, 13 B * ( − + 12 C), 15 C * ( − + 14 N), and 17 N * ( − + 16 O). The total D HN formation probability in the SDM is calculated to be P tot Br = (25-80)%. If we assume that the D C formation probability is common in the three targets and set to be P D C = 30% [41], the total D HN formation probability is P tot D HN = P D C × P tot Br = (7.5-24)%, which is larger than the lower limit of the 2 trapping probability (double and twin hypernuclear formation), 4.8% for light nuclei, evaluated by the KEK-E176 collaboration [53,54]. By comparison, the calculated D HN formation probability is larger than the 2 trapping probability captured by light nuclei, 5.0 ± 1.7% [55]. The small 2 trapping probability data may suggest small bond energies as in model B.

3.1.
− absorption in 12 C First, we discuss the − absorption in 12 C, where 6 He [4] and 10 Be [1,2,5] were observed. In Fig. 3, we show the D HN mass distribution obtained in the SDM calculation of the D C, 13 B * , assumed to be formed in the − absorption in 12 C. We find that D HN with A = 11 is most frequently formed; P Br = (18.9-27.1)% for 11 Be and P Br = (4.3-8.7)% for 11 B. The next most frequently formed D HN is 10 Be with a probability of P Br = (5.5-18.7)%. While the B model dependence is not significant for heavier (A ≥ 11) hypernuclei, small B in models B and C suppresses the formation probabilities of lighter hypernuclei. For example, the formation probability of 6 He is calculated to be P Br = 4.3% in model A, while it becomes P Br = 1.6% and 1.3% in models B and C, respectively.
In the SDM, the formation probability strongly depends on the Q-value and the number of fragments in the final state, as shown in Fig. 4. Since the statistical binary decay favors decays into the excited state having large level densities, two-body decays with large Q-values do not necessarily exhaust the probabilities. The formation and decay of 6 He in the Nagara event [4] have been uniquely 6/17 PTEP 2020, 063D01 A. Ohnishi et al. Table 3. Formation probabilities of D HN in the statistical decay model from the D C, 13 B * ( − + 12 C), 15  identified to take place as − + 12 C → 6 He + α + t, 6 He In the formation of 10 Be [1,2], the following sequence was found not to be inconsistent with the Nagara event [56,57]: − + 12 C → 10 Be + d + n, 10 Be → 9 Be * + p + π − .  4)) in the SDM is calculated to be P Br = 2.6%, 1.0%, and 0.9% (P Br = 8.8%, 3.5%, and 5.0%) in models A, B, and C, respectively. The results are summarized in Table 4.    We shall now evaluate the expected number of events in experiments. In the KEK-E373 experiment, about 10 3 events of − absorption in emulsion nuclei were analyzed [5]. About half of the − absorption events are those with light nuclei, 12 C, 14 N, and 16 O; it would therefore be reasonable to assume that about 160 events of − absorption in 12 C were analyzed. The formation probability of D C, 13 B * , is about P D C 30% in AMD [41]. Thus the number of decays in a given channel may 8 be given as N ev = 160 × P D C × P Br , with P Br being the branching ratio (event probability) in the statistical decay. For the fragmentation channel in Eq. (3) (for the formation of 6 He), the estimated number of events in the E373 experiment is N ev = 1.2, 0.5, and 0.4 (N ev = 2.1, 0.8, and 0.6) in models A, B, and C, respectively. Since these numbers are close to unity, it is not unreasonable that the KEK-E373 experiment observed one event in the channel of Eq. (3). Since the B models B and C are more realistic, we can judge that the KEK-E373 experiment was reasonably lucky provided that the detection efficiency of 6 He is high enough. As for the fragmentation channel in Eq. (4) (for the formation of 10 Be), the estimated number of events in the E373 experiment is N ev = 4.2, 1.7, and 2.4 (N ev = 14.1, 6.0, and 8.2) in models A, B, and C, respectively. The observation of 10 Be in the E373 experiment was reasonable, as long as the detection efficiency of the sequential weak decay is not small. Now let us discuss the transport model dependence. In Fig. 3, we also show the D HN formation probabilities obtained by using AMD with additional fluctuations (AMD-QL) with the statistical decay effects [41] normalized by P D C = 30%. We show the results with strong fluctuations at g 0 = 0.5, with g 0 being the fluctuation strength parameter, and a strongly attractive interaction similar to model A was adopted. Additional fluctuations promote nucleon and emissions in the dynamical stage, then the total D C formation probability is found to be around half, P D C (AMD-QL) = 16%, compared with the AMD results, and the total D HN formation probability is also reduced. By comparison, lighter D C with smaller excitation energies are found to be formed in AMD-QL, and the formation probabilities of some D HN are enhanced. For example, 6 He is formed at P D HN ( 6 He) = 2.0%, which corresponds to P Br ( 6 He) 6.7% and is larger than the SDM calculation results. When normalized by P D C = 30%, the differences of the results in AMD-QL with statistical decays from the SDM results using model A are within a factor of two. If we assume similar differences in the SDM with models B and C, which are more realistic, the expected numbers of events of 10 Be and 6 He in KEK-E373 become closer to unity with additional quantum fluctuation effects in the dynamical stage. Thus these differences are not negligible, but the conclusion in the previous paragraph, that the observations of 10 Be and 6 He in KEK-E373 are reasonable, does not change.

3.2.
− absorption in 14 N Next, we proceed to discuss the − absorption in 14 N, where 13 B was observed [3]. Figure 5 shows the D HN mass distribution in the SDM calculation of 15 1-3.6)%) are also frequently formed. Since the proton separation energy is smaller and the initial 15 C * energy is larger for 14 N, the B model dependence is smaller than other target nuclei.
As shown in Fig. 6, this decay channel is found to have the largest event probability, P Br = 17.9%, 10.4%, and 14.6% (P Br = 18.5%, 11.2%, and 15.5%) for the event probability of this channel 9/17 PTEP 2020, 063D01 A. Ohnishi et al.  (formation probability of 13 B) in models A, B, and C, respectively. In the KEK-E176 experiment, the number of events was estimated to be 77.6 ± 5.1 +0.0 −12.2 for − absorption at rest in emulsion [54], and 31.1 ± 4.8 for absorption events on light nuclei [53]. Then we may roughly estimate the number of − absorption events for 14 N at around 10. With the D C formation probability of around P D C 30%, the expected number of events in the fragmentation channel in Eq. (5) (formation of 13 B) is N ev = 0.5, 0.3, and 0.4 (N ev = 0.6, 0.3, and 0.5) in models A, B, and C, respectively. We can judge that the KEK-E176 experiment was also reasonably lucky provided that the detection efficiency of the sequential weak decay of 13 B is high.

− absorption in 16 O
We now discuss the − absorption in 16 O, where Be was reported to be observed [6]. In Fig. 7, we show the D HN mass distribution in the SDM calculation of 17 N * from − + 16 O. The most frequently formed D HN is 14 C, which is a bound state of 12 C + + , and the formation probability in the SDM is P Br = 59.3%, 38.5%, and 52.8% in models A, B, and C, respectively. The next most frequently formed one is 15 C, which is formed via two-nucleon evaporation from 17 N * . We also find that formation of D HN with A = 7 and 8 is strongly suppressed in B models B and C.
The J-PARC E07 experiment is expected to detect 100 D HN formation events among 10 4 stopped − events. As of November 2018, 920 − absorption events were analyzed [58], and 8 and 6(+2) double and twin hypernuclear formation events were detected. Among them, there is one event (the Mino event) where the D HN formed was identified to be Be [6]. The event was interpreted as 10/17 PTEP 2020, 063D01 A. Ohnishi et al.
Experimentally, the second candidate ( 11 Be formation) is considered to be the most probable, and the 11 Be formation probability is reasonably large (P Br = 1.1%) in the SDM with model A, as shown in Table 3. By comparison, 11 Be formation is less probable, P Br = 0.2% and 0.3% in models B and C, respectively. By assuming the number of − absorption events in 16 O to be around 150 and a D C formation probability of P D C 30%, the expected numbers for the second candidate are N ev = 0.14, 0.04, and 0.06 for models A, B, and C, respectively. In the SDM, the third candidate ( 12 Be * formation) is more probable: the expected numbers are N ev = 0.25, 0.15, and 0.17 for models A, B, and C, respectively. The probabilities are summarized in Fig. 8 and Table 5.

Double hypernuclear formation from − nucleus
In addition to − absorption in nuclei, conversion of nuclei would also make D HN. (K − , K + ) reactions on nuclear targets populate certain hypernuclear states, which become doorway states to D HN. In particular, the 7 Li(K − , K + ) reaction may produce the neutron-rich hypernucleus 7 H = − + 6 He, if it exists. Kumagai-Fuse and Akaishi proposed that the branching ratio of 7 H conversion to form 5 H is surprisingly large, at around P Br = 90% [59]. The first reason for this 11/17 PTEP 2020, 063D01 A. Ohnishi et al.  large branching ratio is the limited decay channels, 7 H → 5 H + n + n, 4 H + + n + n, 4 H * + + n + n, 3 H + + + n + n, (9) if the separation energy of in 7 H is around 2 MeV or more. In the left panel of Fig. 9 we show the decay channels from 7 H. There are two channels, 3 H + + 3n and 2 H + 2 + 3n, 1.7 and 1.6 MeV below the − + 6 He threshold, respectively. If the binding energy of in 7 H is larger than 1.7 MeV, 7 H does not decay to these channels. The second reason for the large branching ratio is in the small Q-values. Since a large part of the released energy in p − → (∼28.6 MeV) is exhausted in breaking the α cluster in 7 H, the threshold energy of − + 6 He is only 5.1 MeV above the emitting threshold ( 5 H + + threshold). This energy difference is much smaller than in the case of − absorption in 12 C (12.7 MeV), 14 N (21.1 MeV), and 16 O (16.5 MeV). Furthermore, the binding energy of − suppresses the Q-values. The Q-values in the channels shown in Eq. (9) are estimated as ∼11, 7, 6, and 5 MeV, respectively [59]. These Q-values are small and the corresponding "temperatures" are small. Hence, three-body decays would be favored over four-or five-body decays. As a result, the decay process will be dominated by the 5 H + 2n channel having the largest Q-value and three bodies in the final state. In Ref. [59], approximate but explicit calculations were performed for the decay process of 7 H, and a large branching ratio decaying to 5 H is obtained. Based on this estimate of a high branching ratio to form 5 H, Fujioka et al. proposed a new experiment at J-PARC (P75) [60]. Since the branching ratio is crucial to determining the feasibility of the experiment, it is valuable to evaluate the branching ratio in different approaches.
We shall now discuss the branching ratio of the 7 H decay to 5 H + 2n in the SDM described in Sect. 2 and used in Sect. 3; B model C is applied as an example. There are several differences from Ref. [59]. First, the matrix elements are evaluated statistically as given in Eq.  states in the bound as well as unbound energy regions are also taken into account. Second, several other channels are included. As discussed in Sect. 2, nuclei with ground states being resonances are included. These nuclei include 5 H, 5 H, 6 H, and 6 H. By including decay channels containing these nuclei, 7 H * can decay in sequential binary decay chains, for example 7 H * → 6 H + n → 5 H + n + n and 7 H * → 6 H * + → 5 H + + n → 4 H + + n + n. The third point is the binding energy differences. We adopt the measured and fitted S for single hypernuclei, and model C is used for B . The threshold energies of decay channels in the present treatment are shown in the right panel of Fig. 9. Figure 10 shows the branching ratios of 7 H → 5 H + n + n (P Br ( 5 H)) and 7 H → 4 H + + n + n (P Br ( 4 H)) obtained in the SDM calculation using B model C as functions of the binding energy of − in 7 H (solid curves). The − binding energy in 15 C was evaluated from the twin hypernuclear formation in the Kiso event, − + 14 N → 15 C → 10 Be ( * ) + 5 He, as B = 3.87 ± 0.21 MeV or 1.03 ± 0.18 MeV [57,61]. These two values are for decays to the ground state and the excited state of 10 Be. The binding energy would be smaller in 7 H, so the binding energy region of 0 ≤ B ≤ 4 MeV would be enough.
The branching ratio P Br ( 5 H) increases with increasing B , as expected, and takes values between P Br = 43.6% (B = 0 MeV) and P Br = 57.8% (B = 4 MeV) in the B region of interest. These values are smaller than those in Ref. [59], but still a large branching ratio of around P Br = 50% seems to be probable in the SDM.
Let us discuss the difference of the branching ratio in the present study and in Ref. [59]. We show the SDM results without dineutron emission by dotted lines in Fig. 10. Model C is used for B . In this case, 6  These values are slightly smaller than those with dineutron emission. This is because the direct threebody decay of 7 H * into 5 H + 2n simulated by dineutron emission is suppressed, while the effect of dineutron emission is not significant and hypernuclear formation is dominated by binary decay sequences such as 7 H * → 6 H ( * ) + n → 5 H + 2n and 7 H * → 6 H ( * ) + → 5 H + + n. Since the Q-values are small, it is reasonable to expect that two-body (binary) decays are expected to be favored over three-or four-body decays. We also show the branching ratios in the SDM with the B values and channels used in Ref. [59] by dashed lines in Fig. 10. Here we adopt 13/17 PTEP 2020, 063D01 A. Ohnishi et al.

B
= 1.92 MeV and ignore unstable hypernuclei with respect to particle decay, 5 H, 6 H, and 6 H, in the decay processes. The branching ratio to 5 H takes larger values than those with particle unstable nuclei, and takes values of P Br ( 5 H) = 46.7% (B = 0 MeV) and 67.8% (B = 4 MeV). These values are still smaller than those in Ref. [59]. Therefore, the larger branching ratio around P Br = 90% in Ref. [59] is found to be a result of an explicit evaluation of the transition width in a specific model treatment in addition to kinematical reasons such as the limited number of decay channels and ignoring particle unstable states during the decay processes. For a more serious estimate, we need calculations with updated − N and interactions, and experimental confirmation of the three-body decay width in the same theoretical treatment.
It should be noted that the formation of D C and its statistical decay may be a less reliable picture of hypernuclear formation for lighter nuclear targets. For example, while AMD+SDM roughly explains the 4 H formation probability from the K − absorption reaction at rest in 12 C and 16 O, the combined framework underestimates [49] the probability for 7 Li and 9 Be targets [62]. These light target nuclei, 7 Li and 9 Be, have a cluster structure in the ground states and easily dissociate to fragments after the conversion process of K − N → π . Then, D HN formation may take place without going through D C.

Summary and discussion
We have investigated double hypernuclear (D HN) formation from the double compound nuclei (D C), 13 B * , 15 C * , 17 N * , and 7 H * in the statistical decay model. The first three compound nuclei would be formed in − absorption at rest in light nuclear targets in emulsion, − + 12 C → 13 B * , − + 14 N → 15 C * , and − + 16 O → 17 N * , and the last one would be formed from -hypernuclei, 7 Li(K − , K + ) 7 H gs followed by 7 H gs → 7 H * . The SDM has been demonstrated to work well in describing fragment formation processes, especially when combined with the transport model calculation to populate the compound nuclei and the excitation energies. In the antisymmetrized molecular dynamics calculations of − absorption in 12 C, the D C are dominated by 13 B * and its formation probability is around P D C 30%. Assuming that this mechanism also applies to the − absorption in other target nuclei, we have evaluated the formation probabilities of D HN. We have also applied the same SDM to evaluate the branching ratio of the hypernucleus 7 H to form 5 H.
We have examined the target and the bond energy ( B ) dependence of the D HN formation probabilities, and the event probabilities are also examined for the channels in which D HN have been observed in experiments. The bond energy is given as B

In models A and B B
is assumed to be independent of the hypernuclei, and in the model C B is assumed to be a linear function of the D HN mass number. Models B and C are consistent with the Nagara event result.
In the − absorption in 12 C, 6 He and 10 Be are observed in the fragmentation channel of − + 12 C → 6 He + α + t and − + 12 C → 10 Be + d + n, respectively. In the SDM, these channels are found to have relatively large probabilities, and the expected numbers of events are consistent with unity within a factor of three. For a 14 N target, 13  This channel is found to have a small event probability of P Br = (0.2-0.3)% and the expected number of events is (0.04-0.06) in B models B and C. If the observation was not accidental, we may need other mechanisms than the statistical decay of D C.
The B model dependence of the D HN formation probabilities is not significant in the main decay channels, where a few nucleons are evaporated: two-nucleon emission for 12 C and 14 N targets and three-nucleon emission for 16 O, as shown in Figs. 3, 5, and 7. One-nucleon emission to excited levels is favored in each step of the statistical binary decay because of the large level densities in the daughter nuclei, ρ ∝ A −5/3 e 2 √ aE * , with a being the level density parameter, a A/8 MeV −1 [44,45]. Since the mass number dependence of B is assumed to be small, the B difference results in a shift of the excitation energies of the parent and daughter nuclei simultaneously and does not strongly affect the decay width of one-nucleon emission. By comparison, lighter D HN formation depends more strongly on B . Since the initial energy is fixed in − absorption in nuclei, the Q-value difference of around 4 MeV in emitting light D HN is significant in the total released energy of (10-30) MeV in the fragmentation. Transport model dependence is also discussed via comparison with the results in AMD with quantum fluctuation effects [41], and a factor of two difference may appear in the D HN formation probabilities.
We also discussed the branching ratio to form 5 H from a hypernucleus 7 H, which can be formed in the 7 Li(K − , K + ) reaction and is assumed to be converted to a D C, 7 H * . It was proposed that the branching ratio would be around P Br = 90% [59]. In SDM calculations, the branching ratio is found to be P Br = (40-60)% in a binding energy range of (0-4) MeV. This branching ratio is still large, but lower than that in Ref. [59]. The difference seems to come from the method used to evaluate the decay width: explicit few-body calculations in Ref. [59] and the statistical assumptions in the present work.
The theoretical framework adopted in the present work may be too simplified to describe realistic D HN formation processes, and we need improvements in the theoretical frameworks for more quantitative estimates of the formation probabilities of D HN from − absorption reactions at rest and hypernuclei. First, the initial D C formation probabilities need to be evaluated with updated B values consistent with the Nagara event and for 14 C, 14 N, and 16 O target nuclei in a consistent way. We have adopted a formation probability of D C in the initial pre-equilibrium stage of around P D C 30%, based on the AMD calculation of − absorption at rest in 12 C [41]. This probability is obtained with a strongly attractive interaction, which gives the bond energy B = 4.9 MeV in 13 B, and 14 N and 16 O targets were not considered. In heavier targets ( 14 N and 16 O), the trapping probabilities are found to be larger in the case of K − absorption at rest [49], and then similar D C formation probabilities or more are also expected, while this expectation may be too optimistic and is premature. Second, we need more care in discussing hypernuclear formation in reactions on light nuclear targets such as 7 Li and 9 Be. When the ground state of the target has a developed cluster structure, direct reaction processes are found to be more important than in reactions on C, N, and O targets [49]. This may also apply to the decay of light − nuclei such as 7 H, as discussed in Sect. 4. Thirdly, we may need other quantum effects or multifragmentation mechanisms to understand D HN formation consistently with twin hypernuclear formation, where two single hypernuclei are formed. The twin hypernuclear formation probability is known to be comparable to that of D HN, but the SDM predicts a much lower probability of twin hypernuclear formation. In Ref. [41], quantum fluctuation effects are considered and are found to promote twin hypernuclear formation, but the twin hypernuclear formation probability is still underestimated. One of the possible mechanisms for producing twin single hypernuclei efficiently is to go through the 15 resonance states of two single hypernuclei around the threshold energy of − and target nuclei [43]. This mechanism seems plausible, since resonance states tend to appear around the threshold, and there are several twin hypernuclear channels around the − target threshold.Another candidate mechanism is multifragmentation, simultaneous decay to three or more fragments. An excitation energy of E * = (30-50) MeV corresponds to a temperature of T √ E * /a ∼ 5 MeV, and multifragmentation may be relevant at this temperature. Actually, canonical and microcanonical statistical fragmentation models are applied to hypernuclear formation in Refs. [40,42]. Thus it is desirable to examine D HN formation probabilities in multifragmentation models.