Hypernuclear ΛΛ production by (K−,K+) reactions and the ΛΛ–Ξ mixing in hypernuclei

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Nuclear Physics A 914 (2013) 85–90 www.elsevier.com/locate/nuclphysa

Hypernuclear ΛΛ production by (K − , K +) reactions and the ΛΛ–Ξ mixing in hypernuclei Toru Harada a,b,∗ , Yoshiharu Hirabayashi c , Atsushi Umeya d a Research Center for Physics and Mathematics, Osaka Electro-Communication University, Neyagawa, Osaka,

572-8530, Japan b J-PARC Branch, KEK Theory Center, Institute of Particle and Nuclear Studies, KEK, Tokai, Ibaraki, 319-1106, Japan c Information Initiative Center, Hokkaido University, Sapporo, 060-0811, Japan d Faculty of Engineering, Nippon Institute of Technology, Saitama, 345-8501, Japan

Received 1 December 2012; received in revised form 10 January 2013; accepted 17 January 2013 Available online 23 January 2013

Abstract 16 C in a (K − , K + ) reaction on an 16 O target We theoretically study production of a ΛΛ hypernucleus ΛΛ at pK − = 1.8 GeV/c, within a distorted-wave impulse approximation. The calculated spectrum provides 16 C in a one-step mechanism K − p → K + Ξ − via promising peaks of the ΛΛ bound and excited states of ΛΛ − − Ξ doorways caused by a Ξ p → ΛΛ coupling, rather than in a two-step mechanism as K − p → π 0 Λ followed by π 0 p → K + Λ. The cross sections and Ξ − admixture probabilities are discussed in terms of the Ξ N–ΛΛ coupling in the nucleus. © 2013 Elsevier B.V. All rights reserved.

Keywords: Hypernuclei; DWIA; Ξ N –ΛΛ coupling

1. Introduction The double-charge exchange (DCX) reaction (K − , K + ) is one of the most promising ways of studying doubly strange systems as Ξ − hypernuclei [1] for the forthcoming J-PARC experiments [2]. One expects that these experiments will confirm the existence of Ξ hypernuclei and establish properties of the Ξ -nucleus potential [3–5]. Such DCX reactions can also populate a * Corresponding author at: Research Center for Physics and Mathematics, Osaka Electro-Communication University, Neyagawa, Osaka, 572-8530, Japan. E-mail address: [email protected] (T. Harada).

0375-9474/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nuclphysa.2013.01.020

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Fig. 1. Diagrams for DCX nuclear (K − , K + ) reactions (a) in a two-step mechanism, K − p → π 0 Λ followed by π 0 p → K + Λ, and (b) in a one-step mechanism, K − p → K + Ξ − via Ξ − doorways caused by the Ξ − p–ΛΛ coupling. (c) Calculated inclusive Ξ spectra in the 16 O(K − , K + ) reaction at 1.8 GeV/c (0◦ ), with a detector resolution of 1.5 MeV FWHM: VΞ = −24 or −14 MeV is used. These spectra correspond to the ΛΛ–Ξ ones by the one-step mechanism without the ΛΛ–Ξ coupling potential.

ΛΛ hypernucleus through a conventional two-step mechanism as K − p → π 0 Λ followed by π 0 p → K + Λ, as shown in Fig. 1(a). Theoretical predictions for two-step 16 O(K − , K + ) reactions at the incident momentum pK − = 1.1 GeV/c and scattering angle θlab = 0◦ [6,7] have 16 C are on the order of 0.1–1 nb/sr. Furindicated that the cross sections for the ΛΛ states in ΛΛ thermore, 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 [8,9], as shown in Fig. 1(b). The Ξ N –ΛΛ coupling gives decays of Ξ -hypernuclear states, and also induces the Ξ − admixture and the additional energy shift to BΛΛ in the ΛΛ A Z) − 2B (A−1 Z) naively comes from the states where the ΛΛ energy shift BΛΛ ≡ BΛΛ (ΛΛ Λ Λ ΛΛ interaction. Therefore, it is very important to extract quantitative information concerning the Ξ N –ΛΛ coupling from spectroscopy of the Ξ and ΛΛ hypernuclei [8,10,11]. 16 C in the DCX In this article, we theoretically study production of a ΛΛ hypernucleus ΛΛ − + 16 ◦ (K , K ) reaction on an O target at pK − = 1.8 GeV/c and θlab = 0 within a distorted-wave impulse approximation (DWIA), in order to extract the Ξ − admixture probability in the ΛΛ hypernucleus from the spectrum [12]. We discuss two-step processes of K − p → π 0 Λ followed by π 0 p → K + Λ in the nuclear (K − , K + ) reaction within the eikonal approximation [6,7], and one-step processes of K − p → K + Ξ − via Ξ − doorways caused by a Ξ N –ΛΛ coupling in the reaction. 2. Two-step mechanism, K − p → π 0 Λ followed by π 0 p → K + Λ In the (K − , K + ) reaction, ΛΛ hypernuclear states can be populated by the two-step mechanism, K − p → π 0 Λ followed by π 0 p → K + Λ [6,7], as shown in Fig. 1(a). Following the procedure by Dover [6,13], a crude estimate is obtained for the contribution of the two-step processes in the eikonal approximation using a harmonic oscillator model. The cross section at scattering angle θlab = 0◦ for quasielastic ΛΛ production on an 16 O target at pK − = 1.8 GeV/c is given [13] as

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Fig. 2. (a) Calculated inclusive ΛΛ–Ξ spectra by the one-step mechanism in the 16 O(K − , K + ) reaction at 1.8 GeV/c (0◦ ) with a detector resolution of 1.5 MeV FWHM, using VΞ = −14 MeV with the ΛΛ–Ξ coupling potential obtained 0 16 by vΞ N,ΛΛ = 250 and 500 MeV. (b) Comparison between the calculated cross sections of ΛΛ C and the E885 data as 12 Be in the a function of the ΛΛ excitation energy Eex , where the data show the 90% upper limit of production of ΛΛ 12 C(K − , K + ) reaction at 1.8 GeV/c [18].

 dσf(2)  f

dΩL

0◦

0  +    −  0 2πξ 1 dσ K p→π Λ dσ π p→K Λ pp ≈ 2 Neff , α α dΩL 0◦ dΩL 0◦ pπ r 2

(π 0 )

where ξ = 0.022–0.019 mb−1 is the factor integrated over angle θlab (K + )

(1)

for K − p → π 0 Λ with

−θlab for π 0 p → K + Λ to restore θlab = 0◦ in the angular distributions of the two elementary processes, pπ  1.68 GeV/c is the intermediate pion momentum, and 1/r 2  0.028 mb−1 pp is the mean inverse-square radial separation of the proton pair. Neff  1 is the effective number of proton pairs including the nuclear distortion effects [6]. The elementary laboratory cross section (α dσ/dΩL )0◦ is estimated to be 1.57–1.26 mb/sr for K − p → π 0 Λ or 0.070– 0.067 mb/sr for π 0 p → K + Λ depending on the nuclear medium corrections. Therefore, we have  (2) f (dσf /dΩL )0◦  0.06–0.04 µb/sr, 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 probabilities for the ΛΛ bound states do not exceed 1% in the quasielastic ΛΛ production, so that the cross section of the ΛΛ bound states in the two-step mechanism may be on the order of 0.6–0.4 nb/sr at θlab = 0◦ [6]. 3. One-step mechanism, K − p → K + Ξ − via Ξ − doorways caused by a Ξ − p → ΛΛ transition Let us consider nuclear ΛΛ–Ξ coupled-channel calculations [9,14] to fully describe one-step processes via Ξ − doorways, as shown in Fig. 1(b). For the 16 O(K − , K + ) reaction, we employ a multi-channel wave function coupled with |14 C ⊗ Λ ⊗ Λ and |15 N ⊗ Ξ − for ΛΛ–Ξ nuclear states. 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 0p −1 ) + , 1/2 1/2 0 −1 −1 −1 −1 (0p3/2 0p1/2 )2+ and (0p3/2 0p3/2 )0+ ,2+ configurations on 16 Og.s. [6,7]. If we identify it as a state

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16 C, we can obtain the Ξ − admixture probabilities in the nucleus by of the ΛΛ hypernucleus ΛΛ calculating PΞ − = 15 N ⊗ Ξ − |15 N ⊗ Ξ − . The inclusive K + double-differential laboratory cross section of the ΛΛ–Ξ production in the nuclear (K − , K + ) reaction can be calculated within the DWIA using the coupled-channel Green’s function [12,15], which was extended from the Green’s function method [16]. We obtain the complete Green’s function G by solving numerically a coupled-channel equation with the hyperon–nucleus potential U :     GΛ GX U Λ UX ,U= , (2) G = G(0) + G(0) U G, G = GX GΞ UX UΞ

where G(0) is a free Green’s function, UY is the Woods–Saxon potential between the 14 C (15 N) core-nucleus and the hyperon for Y = Λ (Ξ − ), and the spreading imaginary potential, Im UY , can describe a loss of flux to complicated excited-states, as often used in nuclear optical models. UX in off-diagonal parts denotes the ΛΛ–Ξ coupling folding potential, which can be obtained 0 with a zero-range Ξ N –ΛΛ potential in the 1 S0 T = 0 state, vΞ N,ΛΛ (r , r) = vΞ N,ΛΛ δ(r − r) 0 for simplicity, where vΞ N,ΛΛ is a strength parameter that should be connected with volume  0 integral vΞ N,ΛΛ (r) dr = vΞ N,ΛΛ [8,9,14,17]. Now we discuss the inclusive spectrum in the 16 O(K − , K + ) reaction at 1.8 GeV/c (0◦ ) by the one-step mechanism, in order to examine the dependence of the spectrum on the strength 0 − parameters of VΞ and vΞ N,ΛΛ . In Fig. 1(c), we show the calculated spectra in the Ξ bound region without the ΛΛ–Ξ coupling potential when we use VΞ = −24 MeV or −14 MeV with the attractive Coulomb potential for Ξ − . We confirm that no clear signal of the Ξ − bound state is seen if VΞ is shallow such as −VΞ  14 MeV and/or WΞ is sizably absorptive (−WΞ  3 MeV at the 15 N + Ξ − threshold) in UΞ . In the case of VΞ = −14 MeV [5,9,14], we have the [15 N(1/2− ) ⊗ sΞ − ]1− state at BΞ − = 6.8 MeV with Γ = 3.8 MeV and the [15 N(1/2− ) ⊗ pΞ − ]2+ at BΞ − = 0.5 MeV with Γ = 1.1 MeV. In Fig. 2(a), we show the calculated spectra by the one-step mechanism in the case of VΞ = 0 −14 MeV when the ΛΛ–Ξ coupling potential is switched on. For vΞ N,ΛΛ = 500 MeV [17], we − 14 + find that the significant peaks of the 1 excited states with C(0 ) ⊗ sΛ pΛ at ω = 362.1 MeV (BΛΛ = 15.1 MeV) and 14 C(2+ ) ⊗ sΛ pΛ at ω = 368.5 MeV (BΛΛ = 8.7 MeV), and small 2 at ω = 373.8 MeV (B peaks of the 2+ excited states with 14 C(0+ ) ⊗ pΛ ΛΛ = 3.4 MeV) and 14 C(2+ ) ⊗ p 2 at ω = 380.4 MeV (B = −3.2 MeV). We have an invisible peak of the 0+ ΛΛ Λ 2 14 + ground state with C(0 ) ⊗ sΛ at ω = 352.3 MeV (BΛΛ = 24.9 MeV). The shape of these 0 spectra is quite sensitive to the value of vΞ N,ΛΛ . Therefore, the ΛΛ–Ξ coupling potential plays an important role in making a production of the ΛΛ states via Ξ − doorways below the 15 N + Ξ − threshold. This result also comes from the fact that the high momentum transfer qΞ −  400 MeV/c leads to a preferential population of the spin-stretched Ξ − doorways states followed by the [15 N(1/2− , 3/2− ) ⊗ sΞ − ]1− → [14 C(0+ , 2+ ) ⊗ sΛ pΛ ]1− and [15 N(1/2− , 3/2− ) ⊗ pΞ − ]2+ → 2 ] + transitions. The integrated cross section at θ ◦ − [14 C(0+ , 2+ ) ⊗ pΛ lab = 0 for the 1 excited 2 14 + 14 + state with C(0 ) ⊗ sΛ pΛ ( C(2 ) ⊗ sΛ pΛ ) is on the order of 7 nb/sr (12 nb/sr), and the Ξ − admixture probability is on the order of 5.2% (8.8%). The contribution of these ΛΛ 1− states to the ΛΛ spectrum in the one-step mechanism is completely different from that in the two-step mechanism as obtained in Refs. [6,7]. Consequently, we believe that the one-step mechanism acts in a dominant process of the (K − , K + ) reaction at 1.8 GeV/c (0◦ ) when

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0 − + vΞ N,ΛΛ = 400–600 MeV. This implies that the (K , K ) spectrum provides valuable information concerning Ξ N –ΛΛ dynamics in the S = −2 systems such as ΛΛ and Ξ hypernuclei. Yamamoto et al. [18] discussed the 90% upper limit of production of the ground state and 12 Be hypernucleus in the 12 C(K − , K + ) reaction at 1.8 GeV/c from the excited states of the ΛΛ BNL-E885 experiment. In Fig. 2(b), we display the calculated result of integrated cross sections 16 C as a function of the ΛΛ excitation for the 0+ ground state and the 1− excited states of ΛΛ energy Eex , in comparison with the upper limit of the E885 data. Here we took account of recent cal = 20.2 MeV in shell-model predictions [20]. updated data of ΛΛ binding energies [19] and BΛΛ It is shown that our calculated result is not contradictory to the E885 data.

4. Summary We have theoretically examined production of ΛΛ hypernuclei in the DCX 16 O(K − , K + ) reaction at 1.8 GeV/c within DWIA calculations using the coupled-channel Green’s function. 16 C hypernucleus in the one-step mechanism K − p → K + Ξ − The calculated spectrum for the ΛΛ − via Ξ doorways provides promising peaks of the ΛΛ bound and excited states in the reactions, rather than those in the two-step mechanism, K − p → π 0 Λ followed by π 0 p → K + Λ. We have shown that the Ξ − admixture in the ΛΛ hypernucleus plays an essential role in producing the ΛΛ states in the (K − , K + ) reaction. The sensitivity to the potential parameters indicates that the nuclear (K − , K + ) reactions have a high ability for the theoretical analysis of precise wave functions in the ΛΛ hypernuclei. New information on ΛΛ–Ξ dynamics in nuclei from the (K − , K + ) data at J-PARC facilities will bring the S = −2 world development in nuclear physics [2]. Acknowledgements This work was supported by Grants-in-Aid for Scientific Research on Priority Areas (Nos. 17070002 and 20028010) and for Scientific Research (C) (No. 22540294). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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