Production of doubly strange hypernuclei via {\Xi}- doorways in the 16O(K-, K+) reaction at 1.8 GeV/c

We examine theoretically production of doubly strange hypernuclei, 16 {\Xi}-C and 16 {\Lambda}{\Lambda}C, in doublecharge exchange 16O(K-, K+) reactions using a distorted-wave impulse approximation. The inclusive K+ spectrum at the incident momentum pK- = 1.8 GeV/c and scattering angle {\theta}lab = 0^{\circ} is estimated in a one-step mechanism, K-p \to K+{\Xi}- via {\Xi}- doorways caused by a {\Xi}-p-{\Lambda}{\Lambda} coupling. The calculated spectrum in the {\Xi}- bound region indicates that the integrated cross sections are on the order of 7-12 nb/sr for significant 1- excited states with 14C(0+, 2+) \otimes s{\Lambda}p{\Lambda} configurations in 16 {\Lambda}{\Lambda}C via the doorway states of the spin-stretched 15N(1/2-, 3/2-) \otimes s{\Xi}- in 16 {\Xi}-C due to a high momentum transfer q{\Xi}- \approx 400 MeV/c. The {\Xi}- admixture probabilities of these states are on the order of 5-9%. However, populations of the 0+ ground state with 14C(0+) \otimes s2{\Lambda} and the 2+ excited state with 14C(2+) \otimes s2 {\Lambda} are very small. The sensitivity of the spectrum on the {\Xi}N-{\Lambda}{\Lambda} coupling strength enables us to extract the nature of {\Xi}N-{\Lambda}{\Lambda} dynamics in nuclei, and the nuclear (K-, K+) reaction can extend our knowledge of the S = -2 world.


I. INTRODUCTION
It is important to understand properties of Ξ hypernuclei whose states are regarded as "doorways" to access multi-strangeness systems as well as a two-body ΞN-ΛΛ system, and it is a significant step to extend study of strange nuclear matter in hadron physics and astrophysics [1]. Because the Ξ hyperon in nuclei has to undergo a strong ΞN → ΛΛ decay, widths of Ξ hypernuclear states give us a clue to a mechanism of Ξ absorption processes in nuclei. A pioneer study of Ξ hypernuclei by Dover and Gal [2] has found that a Ξnucleus potential has a well depth of 24 ± 4 MeV in the real part on the analysis of old emulsion data. However, our knowledge of these Ξ-nucleus systems is very limited due to the lack of the experimental data [3]. Indeed, the missing-mass spectra of a double-charge exchange (DCX) reaction (K − , K + ) on a 12 C target have suggested the Ξ well depth of [14][15][16]5]. Several authors [6]  The (K − , K + ) reaction is one of the most promising ways of studying doubly strange systems such as Ξ − hypernuclei for the forthcoming J-PARC experiments [3]. One expects that these experiments will confirm the existence of Ξ hypernuclei and establish properties of the Ξ-nucleus potential, e.g., binding energies and widths. This reaction can also populate a ΛΛ hypernucleus through a conventional DCX two-step mechanism as K − p → π 0 Λ followed by π 0 p → K + Λ [7][8][9], as shown in Fig. 1(a). Such an inclusive K − spectrum in the ΛΛ bound region is rather clean with much less background experimentally. Early theoretical predictions for two-step 16 O(K − , K + ) reactions at the incident momentum p K − = 1.1 GeV/c and scattering angle θ lab = 0 • [7,8] have indicated small cross sections for the ΛΛ states, for example, ∼0.1 nb/sr for the 0 + (s 2 Λ ) ground state and ∼2 nb/sr for the 2 + (s 2 Λ ) excited state in 16 ΛΛ C when we took 0.61 mb/sr and 0.32 mb/sr as the laboratory cross sections at 0 • for K − p → π 0 Λ and π 0 p → K + Λ, respectively.
It should be noticed that another exotic production of ΛΛ hypernuclei in the (K − , K + ) reactions is a one-step mechanism, K − p → K + Ξ − via Ξ − doorways caused by a Ξ − p → ΛΛ transition, as shown in Fig. 1(b). The ΞN-ΛΛ coupling induces the Ξ − admixture and the in the ΛΛ-nuclear states [10][11][12][13][14], and its coupling strength is also related to widths of Ξ-hypernuclear states [15,16]. For a viewpoint of S = −2 studies, it is very important to extract quantitative information concerning the ΞN-ΛΛ coupling from spectroscopy of the Ξ and ΛΛ hypernuclei [17,18].
In this Letter, we study theoretically production of a doubly strange hypernucleus in the DCX (K − , K + ) reaction on an 16 O target at p K − = 1.8 GeV/c and θ lab = 0 • within a distorted-wave impulse approximation (DWIA). Thus we focus on the ΛΛ-Ξ spectrum for 16 ΛΛ C and 16 Ξ − C in the Ξ − bound region considering the one-step mechanism, K − p → K + Ξ − via Ξ − doorways caused by the ΞN-ΛΛ coupling in the nuclear (K − , K + ) reaction, rather than the two-step mechanism as K − p → π 0 Λ followed by π 0 p → K + Λ [7,8]. These different mechanisms are well separated kinematically. The forward cross section for the K − p → K + Ξ − elementary process is at its maximum at p K − = 1.8-1.9 GeV/c, whereas the K − p → π 0 Λ reaction at p K − = 1.1 GeV/c leads to the maximal cross section for the π 0 p → K + Λ process. The present study is the first attempt to evaluate a production spectrum of the ΛΛ-Ξ hypernucleus via the ΞN-ΛΛ coupling from the inclusive (K − , K + ) reaction, and to extract the Ξ − admixture probability in the ΛΛ hypernucleus from the spectrum. We also discuss a contribution of the two-step processes in the (K − , K + ) reactions within the eikonal approximation.

II. CALCULATIONS
Let us consider the DCX (K − , K + ) reaction on the 16 O target at 1.8 GeV/c within a DWIA and examine the production cross sections and wave functions of the doubly strange hypernucleus. To fully describe the one-step process via Ξ − doorways, as shown in Fig. 1 we perform nuclear ΛΛ-Ξ coupled-channel calculations [13,14], which are assumed to effectively represent the coupling nature in omitting other ΛΣ and ΣΣ channels for simplicity.
Here we employ a multichannel coupled wave function of the ΛΛ-Ξ nuclear state for a total spin J B within a weak coupling basis. It is written as where r Λ 1 (r p ) denotes the relative coordinate between the 14 C core-nucleus and the Λ (proton), and r Λ 2 (r Ξ ) denotes the relative coordinate between the center of mass of the 14 C-Λ ( 15 N) subsystem and the Λ (Ξ − ). Thus ϕ (Λ) jp describe the relative wave functions of shell model states (that occupy j 1,2 , j 3 and j p orbits) for the Λ, Ξ − and proton, respectively; Φ J ( 14 C) is a wave function of the 14 C corenucleus state, and A is the anti-symmetrized operator for nucleons. The energy difference be- and Λ hyperons, respectively. We take the 15 N core-nucleus states with J π = 1/2 − (g.s.) and 3/2 − (6.32 MeV), and the 14 C core-nucleus states with J π = 0 + (g.s.) and 2 + (7.01 MeV) that are given in (0p −1 [7,8]. Because we assume only natural-parity π = (−1) J B states via Ξ − doorways that are selectively formed by non-spin-flip processes in the forward K − p → K + Ξ − reaction, we consider a spin S = 0, ΛΛ pair in the hypernucleus. If the ΛΛ component is dominant in a bound state, we can identify it as a state of the ΛΛ hypernucleus 16 ΛΛ C, in which the Ξ − admixture probability can be estimated by under the normalization of j 1 j 2 ϕ (Λ) After we set up the 15 Λ C and 15 N configurations in our model space with Eq. (1), we calculate the wave functions of ϕ (Λ) j 2 (r Λ 2 ) and ϕ (Ξ − ) j 3 (r Ξ ) taking into account their channel coupling. Thus, the complete Green's function G(ω) [19] describes all information concerning ( 15 Λ C ⊗ Λ) + ( 15 N ⊗ Ξ − ) coupled-channel dynamics, as a function of the energy transfer ω. It is numerically obtained as a solution of the N-channels radial coupled equations with a hyperon-nucleus potential U [20,21], which is written in an abbreviated notation as where G (0) (ω) is a free Green's function. In our calculations, for example, we deal with N = 28 for the J π B = 1 − state. The nuclear optical potentials U Y (Y = Ξ or Λ) can be written as where f is the Woods-Saxon (WS) form, f (r, R, a) = [1 + exp ((r − R)/a)] −1 , and g is the Coulomb potential with the nuclear finite size R C = 1.25A 1/3 = 3.15 fm [22]. The spreading imaginary potential in Eq. (5), Im U Y , expresses complicated excited states via Ξ − N → ΛΛ conversion processes in 16 Ξ − C or 16 ΛΛ C above the 15 ΛΛ C+n threshold at 8.2 MeV, as a function of the excitation energy E ex measured from an energy of the 16 ΛΛ C ground state, as often used in nuclear optical models. Since we have no criterion for a choice of W Ξ or W (D) Ξ in the limited experimental data, we adjust appropriately the strength parameter of W Ξ in the WS-type to give widths of Ξ − quasibound states in recent calculations [5,6,23]. In 14 C-ΛΛ channels, we should use a 15 Λ C-Λ potential, which can be constructed in folded potential models [24]: where . U CΛ and V ΛΛ denote an optical potential for 14 C-Λ as given in Eq. (5) and a ΛΛ residual interaction, respectively. Here we neglected V ΛΛ for simplicity. The real part of U CΛ leads to B Λ = 12.2 MeV for the (0s) Λ state and B Λ = 1.6 MeV for the (0p) Λ state in 15 Λ C [25], and its imaginary part exhibits a flux loss of the wave functions through the core excitations of for nucleon were obtained in Ref. [26] because the well depth of the imaginary potential for Λ is by a factor of 4 weaker than that for nucleon in g-matrix calculations [27].
The ΛΛ-Ξ coupling potential U X in off-diagonal parts of U is the most interesting object in this calculation [10][11][12][13][14][15][16]. It can be obtained by a two-body ΞN-ΛΛ potential v ΞN,ΛΛ (r ′ , r) in a real potential for simplicity, where v 0 ΞN,ΛΛ is the strength parameter that should be connected with volume integral v ΞN,ΛΛ (r)dr = v 0 ΞN,ΛΛ [13,14,16]. Thus the matrix elements can be easily estimated by use of Racah algebra [29]: where ] j is a spin-orbit function and C J B LSK (J ′ J ′′ ) is a purely geometrical factor [29]; F J ′ J ′′ LSK (r) is the nuclear form factor including a recoupling coefficient of U(Jj 1 J ′′ K; J ′ j p ) [16], a parentage coefficient for proton removal from 15 N(1/2 − , 3/2 − ) [30] and the center-ofmass correction of a factor A/(A − 1) [31]. The factor 1/2 comes from the procedure handling a transition between pΞ − and ΛΛ states in the nucleus.
The inclusive K + double-differential laboratory cross section of the ΛΛ-Ξ production in the nuclear (K − , K + ) reaction can be written within the DWIA [32,33] using the Green's function method [19]. In the one-step mechanism, K − p → K + Ξ − via Ξ − doorways, it is given [21] as for the target with a spin J A and its z-component M z , where [J A ] = 2J A +1, and a kinematical factor β [34] that expresses the translation from the two-body K − -p laboratory system to the K − -16 O laboratory system [2]. The production amplitude F α Ξ is where f K − p→K + Ξ − is a Fermi-averaged amplitude for the K − p → K + Ξ − reaction in nuclear medium [2], and χ where α denotes the complete set of eigenstates for the system. It should be recognized that the ΛΛ-Ξ coupled-channel Green's function with the spreading potential provides an advantage of estimating contributions from sources both as ΛΛ components in Ξ − -nucleus eigenstates [16] and as Ξ − p → ΛΛ quasi-scattering processes in the nucleus [15].
Because the momentum transfer is very high in the nuclear (K − , K + ) reaction at 1.8 GeV/c, i.e., q Ξ − ≃ 360-430 MeV/c, the distorted waves for outgoing K + and incoming K − in Eq. (9) are calculated with the help of the eikonal approximation [32,35]. As the distortion parameters, we use total cross sections of σ K − N = 28.9 mb for K − N scattering and σ K + N = 19.4 mb for K + N scattering [6], and α K − N = α K + N = 0. We take 35 µb/sr as the laboratory cross section of dσ/dΩ =ᾱ|f K − p→K + Ξ − | 2 including the kinematical factor α [5,9]. For the target nucleus 16 O with J π A = 0 + , we assume the wave functions for the proton hole-states in the relative coordinate, which are calculated with central (WS-type) and spin-orbit potentials [22], by fitting to the charge rms radius of 2.72 fm [36]. For the energies (widths) for proton-hole states, we input 12.1 (0.0), 18.4 (2.5) and 36 (10) MeV for 0p −1 1/2 , 0p −1 3/2 and 0s −1 1/2 states, respectively. Three parameters, V Ξ , W Ξ and v 0 ΞN,ΛΛ , are very important for calculating the inclusive spectra with the one-step mechanism. These parameters are strongly connected each other for the shape of the spectrum and its magnitude, as well as for the Ξ − binding energies and widths of the Ξ − states. Several authors [10,14,16] investigated the effects of the ΞN-ΛΛ coupling in light nuclei evaluating the volume integrals for k F -dependent ΞN-ΛΛ effective interactions based on Nijmegen potentials [28], in which these values are strongly model dependent; for example, 250.9, 370.2, 501.5, 582.1 and 873.9 MeV·fm 3 for NHC-D, NSC97e, NSC04a, NHC-F and NSC04d potentials (k F = 1.0 fm −1 ), respectively [14,28].
The Ξ − p → ΛΛ conversion cross section of (vσ) Ξ − p→ΛΛ ≃ 7.9 mb also yields to be about 544 MeV·fm 3 [16]. To see the dependence of the spectrum on the ΞN-ΛΛ coupling strength, here, we choose typical values of v 0 ΞN,ΛΛ = 250 and 500 MeV, which approximate the volume integrals of NHC-D and NSC04a, respectively. We take the spreading potential of Im U Ξ to be W Ξ ≃ −3 MeV at the 15 N + Ξ − threshold [5,6,14,18]. It should be noticed that this spreading potential expresses nuclear core breakup processes caused by the Ξ − p → ΛΛ conversion in the 15 N nucleus, and its effect cannot be involved in U X .

III. RESULTS AND DISCUSSION
Now let us discuss the inclusive spectrum in the 16 O(K − , K + ) reaction at 1.8 GeV/c (0 • ) in order to examine the dependence of the spectrum on the parameters of V Ξ and v 0 ΞN,ΛΛ . We consider contributions of the ΛΛ-Ξ nuclear bound and resonance states to the Ξ − p → ΛΛ conversion processes in the Ξ − bound region.
In Fig. 2(a), we show the calculated spectra in the Ξ − bound region without the ΛΛ-Ξ coupling potential when we use V Ξ = −24 MeV or −14 MeV with the Coulomb potential.
The calculated spectra are in agreement with the spectra obtained by previous works [6]. In the case of V Ξ = −24 MeV, we find that a broad peak of the MeV with a sizable width of Γ = 3.5 MeV, and a clear peak of the Integrated cross sections indicate dσ(0 • )/dΩ ≃ 28 nb/sr for the 1 − state and 77 nb/sr for the 2 + state in 16 Ξ − C. In the case of V Ξ = −14 MeV, which is favored in recent calculations [6,13,14,18], we have the cross sections indicate dσ(0 • )/dΩ ≃ 6 nb/sr for the 1 − state and 9 nb/sr for the 2 + state.
Note that the Ξ − p → ΛΛ conversion processes that can be described by the absorption potential Im U Ξ , must appear above the 15 ΛΛ C + n decay threshold at ω = 360.4 MeV (which corresponds to B ΛΛ = 16.7 MeV). We confirm that no clear signal of the Ξ − bound state is measured if V Ξ is sallow such as −V Ξ ≤ 14 MeV and/or W Ξ is sizably absorptive (−W Ξ ≥ 3 MeV at the 15 N + Ξ − threshold) in U Ξ . Nevertheless, the production of these Ξ − states as well as Ξ − states coupled to a 15 N(3/2 − ) nucleus is essential in this model because these states act as doorways when we consider the ΛΛ states formed in the one-step mechanism.
We also expect to extract properties of the Ξ-nucleus potential such as V Ξ and W Ξ from the Ξ − continuum spectra in the (K − , K + ) reactions on nuclear targets, as already discussed for studies of the Σ − -nucleus potential in nuclear (π − , K + ) reactions [38,39]. In Fig. 2(b), we show the calculated spectra with the ΛΛ-Ξ coupling potential when V Ξ = −14 MeV. We recognize that the shape of these spectra is quite sensitive to the value of v 0 ΞN,ΛΛ , and it is obvious that no ΞN-ΛΛ coupling cannot describe the spectrum of the ΛΛ states below the 14 C + Λ + Λ threshold. The calculated spectrum for v 0 ΞN,ΛΛ = 500 MeV has a fine structure of the ΛΛ excited states in 16 ΛΛ C. We find that significant peaks of the 1 − excited states with 14 C(0 + ) ⊗ s Λ p Λ at ω = 362.
where the Ξ − admixture probabilities of these states amount to P Ξ − = 5.2% and 8.8%, respectively. It should be noticed that the cross sections are on the same order of magnitude as those for the 1 − and 2 + quasibound states that are located at B Ξ − = 6.8 MeV and 0.5 MeV, respectively, in the 16 Ξ − C hypernucleus. Therefore, such ΛΛ excited states below the 14 C + Λ + Λ threshold will be measured experimentally at the J-PARC facilities [3]. There is no production in the 2 + state with 14 C(2 + ) ⊗ s 2 Λ under the angular-momentum conservation in the 16 O(K − , K + ) reactions by the one-step mechanism. The contribution of these states to the ΛΛ spectrum in the one-step mechanism is completely different from that in the two-step mechanism as obtained in Refs. [7,8].
In the (K − , K + ) reaction, ΛΛ hypernuclear states can be also populated by the two-step mechanism, K − p → π 0 Λ followed by π 0 p → K + Λ [7][8][9], as shown in Fig. 1(a). Following the procedure by Dover [7,9], a crude estimate can be obtained for the contribution of this twostep processes in the eikonal approximation using a harmonic oscillator model. The cross section at 0 • for quasielastic ΛΛ production at p K − = 1.8 GeV/c in the two-step mechanism, which is summed over all final state, is given [9] as where ξ = 0.022-0.019 mb −1 is a constant nature of the angular distributions of the two elementary processes, p π ≃ 1.68 GeV/c is the intermediate pion momentum, and 1/r 2 ≃ 0.028 mb −1 is the mean inverse-square radial separation of the proton pair. N pp eff ≃ 1 is the effective number of proton pairs including the nuclear distortion effects [7]. The elementary laboratory cross section (αdσ/dΩ L ) 0 • is estimated to be 1.57-1.26 mb/sr for K − p → π 0 Λ and 0.070-0.067 mb/sr for π 0 p → K + Λ depending on the nuclear medium corrections. This which is half smaller than ∼0.14 µb/sr at 1.1 GeV/c. Considering a high momentum transfer q ≃ 400 MeV/c in the (K − , K + ) reactions, we expect that the production probability for the ΛΛ bound states does not exceed 1% in the quasielastic ΛΛ production, so that an estimate of the ΛΛ hypernucleus in the two-step mechanism may be on the order of 0.1-1 nb/sr. This cross section is smaller than the cross section for the ΛΛ 1 − states we mentioned above in the one-step mechanism. Consequently, we believe that the one-step mechanism acts in a dominant process in the (K − , K + ) reaction at 1.8 GeV/c (0 • ) when v 0 ΞN,ΛΛ = 400-600 MeV. This implies that the (K − , K + ) spectrum provides valuable information concerning ΞN-ΛΛ dynamics in the S = −2 systems such as ΛΛ and Ξ hypernuclei, which are often discussed in a full coupling scheme [40].

IV. SUMMARY AND CONCLUSION
We have examined theoretically production of doubly strange hypernuclei in the DCX 16 O(K − , K + ) reaction at 1.8 GeV/c within DWIA calculations using coupled-channel Green's functions. We have shown that the Ξ − admixture in the ΛΛ hypernuclei plays an essential role in producing the ΛΛ states in the (K − , K + ) reaction.