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Nuclear Physics A 914 (2013) 85–90

www.elsevier.com/locate/nuclphysa

Hypernuclear ΛΛ production by (K−,K+) reactionsand 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, Japanc 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

We theoretically study production of a ΛΛ hypernucleus 16ΛΛC in a (K−,K+) reaction on an 16O target

at pK− = 1.8 GeV/c, within a distorted-wave impulse approximation. The calculated spectrum providespromising peaks of the ΛΛ bound and excited states of 16

ΛΛC in a one-step mechanism K−p → K+Ξ− via

Ξ− doorways caused by a Ξ−p → ΛΛ coupling, rather than in a two-step mechanism as K−p → π0Λ

followed by π0p → K+Λ. The cross sections and Ξ− admixture probabilities are discussed in terms ofthe ΞN–ΛΛ coupling in the nucleus.© 2013 Elsevier B.V. All rights reserved.

Keywords: Hypernuclei; DWIA; ΞN–ΛΛ coupling

1. Introduction

The double-charge exchange (DCX) reaction (K−,K+) is one of the most promising waysof studying doubly strange systems as Ξ− hypernuclei [1] for the forthcoming J-PARC experi-ments [2]. One expects that these experiments will confirm the existence of Ξ hypernuclei andestablish 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: harada@isc.osakac.ac.jp (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

86 T. Harada et al. / Nuclear Physics A 914 (2013) 85–90

Fig. 1. Diagrams for DCX nuclear (K−,K+) reactions (a) in a two-step mechanism, K−p → π0Λ followed byπ0p → K+Λ, and (b) in a one-step mechanism, K−p → K+Ξ− via Ξ− doorways caused by the Ξ−p–ΛΛ cou-pling. (c) Calculated inclusive Ξ spectra in the 16O(K−,K+) reaction at 1.8 GeV/c (0◦), with a detector resolutionof 1.5 MeV FWHM: VΞ = −24 or −14 MeV is used. These spectra correspond to the ΛΛ–Ξ ones by the one-stepmechanism without the ΛΛ–Ξ coupling potential.

ΛΛ hypernucleus through a conventional two-step mechanism as K−p → π0Λ followed byπ0p → K+Λ, as shown in Fig. 1(a). Theoretical predictions for two-step 16O(K−,K+) reac-tions at the incident momentum pK− = 1.1 GeV/c and scattering angle θlab = 0◦ [6,7] haveindicated that the cross sections for the ΛΛ states in 16

ΛΛC are on the order of 0.1–1 nb/sr. Fur-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 ΞN–ΛΛ coupling gives decays of Ξ -hypernuclearstates, and also induces the Ξ− admixture and the additional energy shift to �BΛΛ in the ΛΛ

states where the ΛΛ energy shift �BΛΛ ≡ BΛΛ( AΛΛZ) − 2BΛ(A−1

ΛZ) naively comes from theΛΛ interaction. Therefore, it is very important to extract quantitative information concerning theΞN–ΛΛ coupling from spectroscopy of the Ξ and ΛΛ hypernuclei [8,10,11].

In this article, we theoretically study production of a ΛΛ hypernucleus 16ΛΛC in the DCX

(K−,K+) reaction on an 16O target at pK− = 1.8 GeV/c and θlab = 0◦ within a distorted-waveimpulse 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 → π0Λ followedby π0p → K+Λ in the nuclear (K−,K+) reaction within the eikonal approximation [6,7], andone-step processes of K−p → K+Ξ− via Ξ− doorways caused by a ΞN–ΛΛ coupling in thereaction.

2. Two-step mechanism, K−p → π0Λ followed by π0p → K+Λ

In the (K−,K+) reaction, ΛΛ hypernuclear states can be populated by the two-step mech-anism, K−p → π0Λ followed by π0p → K+Λ [6,7], as shown in Fig. 1(a). Following theprocedure by Dover [6,13], a crude estimate is obtained for the contribution of the two-stepprocesses in the eikonal approximation using a harmonic oscillator model. The cross section atscattering angle θlab = 0◦ for quasielastic ΛΛ production on an 16O target at pK− = 1.8 GeV/c

is given [13] as

T. Harada et al. / Nuclear Physics A 914 (2013) 85–90 87

Fig. 2. (a) Calculated inclusive ΛΛ–Ξ spectra by the one-step mechanism in the 16O(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 obtainedby v0

ΞN,ΛΛ= 250 and 500 MeV. (b) Comparison between the calculated cross sections of 16

ΛΛC and the E885 data as

a function of the ΛΛ excitation energy Eex, where the data show the 90% upper limit of production of 12ΛΛBe in the

12C(K−,K+) reaction at 1.8 GeV/c [18].

∑f

(dσ

(2)f

dΩL

)0◦

≈ 2πξ

p2π

⟨1

r2

⟩(α

dσ

dΩL

)K−p→π0Λ

0◦

(α

dσ

dΩL

)π0p→K+Λ

0◦N

pp

eff , (1)

where ξ = 0.022–0.019 mb−1 is the factor integrated over angle θ(π0)lab for K−p → π0Λ with

−θ(K+)lab for π0p → 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/r2〉 � 0.028 mb−1

is the mean inverse-square radial separation of the proton pair. Npp

eff � 1 is the effective num-ber of proton pairs including the nuclear distortion effects [6]. The elementary laboratorycross section (α dσ/dΩL)0◦ is estimated to be 1.57–1.26 mb/sr for K−p → π0Λ or 0.070–0.067 mb/sr for π0p → K+Λ depending on the nuclear medium corrections. Therefore, we have∑

f (dσ(2)f /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 expectthat the production probabilities for the ΛΛ bound states do not exceed 1% in the quasielasticΛΛ production, so that the cross section of the ΛΛ bound states in the two-step mechanism maybe 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-stepprocesses via Ξ− doorways, as shown in Fig. 1(b). For the 16O(K−,K+) reaction, we employ amulti-channel wave function coupled with |14C ⊗ Λ ⊗ Λ〉 and |15N ⊗ Ξ−〉 for ΛΛ–Ξ nuclearstates. We take the 15N core-nucleus states with Jπ = 1/2− (g.s.) and 3/2− (6.32 MeV), and the14C core-nucleus states with Jπ = 0+ (g.s.) and 2+ (7.01 MeV) that are given in (0p−1

1/20p−11/2)0+ ,

(0p−1 0p−1 )2+ and (0p−1 0p−1 )0+,2+ configurations on 16Og.s. [6,7]. If we identify it as a state

3/2 1/2 3/2 3/2

88 T. Harada et al. / Nuclear Physics A 914 (2013) 85–90

of the ΛΛ hypernucleus 16ΛΛC, we can obtain the Ξ− admixture probabilities in the nucleus by

calculating PΞ− = 〈15N ⊗ Ξ−|15N ⊗ 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-channelGreen’s function [12,15], which was extended from the Green’s function method [16]. We obtainthe complete Green’s function G by solving numerically a coupled-channel equation with thehyperon–nucleus potential U :

G = G(0) + G(0)UG, G =(

GΛ GX

GX GΞ

), U =

(UΛ UX

UX UΞ

), (2)

where G(0) is a free Green’s function, UY is the Woods–Saxon potential between the 14C (15N)core-nucleus and the hyperon for Y = Λ (Ξ−), and the spreading imaginary potential, ImUY ,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 obtainedwith a zero-range ΞN–ΛΛ potential in the 1S0T = 0 state, vΞN,ΛΛ(r ′, r) = v0

ΞN,ΛΛδ(r ′ − r)

for simplicity, where v0ΞN,ΛΛ is a strength parameter that should be connected with volume

integral∫

vΞN,ΛΛ(r) dr = v0ΞN,ΛΛ [8,9,14,17].

Now we discuss the inclusive spectrum in the 16O(K−,K+) reaction at 1.8 GeV/c (0◦) bythe one-step mechanism, in order to examine the dependence of the spectrum on the strengthparameters of VΞ and v0

ΞN,ΛΛ. In Fig. 1(c), we show the calculated spectra in the Ξ− boundregion without the ΛΛ–Ξ coupling potential when we use VΞ = −24 MeV or −14 MeV withthe attractive Coulomb potential for Ξ−. We confirm that no clear signal of the Ξ− bound stateis seen if VΞ is shallow such as −VΞ � 14 MeV and/or WΞ is sizably absorptive (−WΞ �3 MeV at the 15N + Ξ− threshold) in UΞ . In the case of VΞ = −14 MeV [5,9,14], we have the[15N(1/2−)⊗sΞ−]1− state at BΞ− = 6.8 MeV with Γ = 3.8 MeV and the [15N(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 =−14 MeV when the ΛΛ–Ξ coupling potential is switched on. For v0

ΞN,ΛΛ = 500 MeV [17], we

find that the significant peaks of the 1− excited states with 14C(0+) ⊗ sΛpΛ at ω = 362.1 MeV(BΛΛ = 15.1 MeV) and 14C(2+) ⊗ sΛpΛ at ω = 368.5 MeV (BΛΛ = 8.7 MeV), and smallpeaks of the 2+ excited states with 14C(0+) ⊗ p2

Λ at ω = 373.8 MeV (BΛΛ = 3.4 MeV) and14C(2+) ⊗ p2

Λ at ω = 380.4 MeV (BΛΛ = −3.2 MeV). We have an invisible peak of the 0+ground state with 14C(0+) ⊗ s2

Λ at ω = 352.3 MeV (BΛΛ = 24.9 MeV). The shape of thesespectra is quite sensitive to the value of v0

ΞN,ΛΛ. Therefore, the ΛΛ–Ξ coupling potential plays

an important role in making a production of the ΛΛ states via Ξ− doorways below the 15N + 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[15N(1/2−,3/2−) ⊗ sΞ−]1− → [14C(0+,2+) ⊗ sΛpΛ]1− and [15N(1/2−,3/2−) ⊗ pΞ−]2+ →[14C(0+,2+) ⊗ p2

Λ]2+ transitions. The integrated cross section at θlab = 0◦ for the 1− excitedstate with 14C(0+) ⊗ sΛpΛ (14C(2+) ⊗ sΛpΛ) is on the order of 7 nb/sr (12 nb/sr), and theΞ− 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 thetwo-step mechanism as obtained in Refs. [6,7]. Consequently, we believe that the one-stepmechanism acts in a dominant process of the (K−,K+) reaction at 1.8 GeV/c (0◦) when

T. Harada et al. / Nuclear Physics A 914 (2013) 85–90 89

v0ΞN,ΛΛ = 400–600 MeV. This implies that the (K−,K+) spectrum provides valuable infor-

mation concerning ΞN–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

excited states of the 12ΛΛBe hypernucleus in the 12C(K−,K+) reaction at 1.8 GeV/c from the

BNL-E885 experiment. In Fig. 2(b), we display the calculated result of integrated cross sectionsfor the 0+ ground state and the 1− excited states of 16

ΛΛC as a function of the ΛΛ excitationenergy Eex, in comparison with the upper limit of the E885 data. Here we took account of recentupdated data of ΛΛ binding energies [19] and Bcal

ΛΛ = 20.2 MeV in shell-model predictions [20].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 16O(K−,K+)

reaction at 1.8 GeV/c within DWIA calculations using the coupled-channel Green’s function.The calculated spectrum for the 16

ΛΛC hypernucleus in the one-step mechanism K−p → K+Ξ−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 → π0Λ followed by π0p → K+Λ. Wehave shown that the Ξ− admixture in the ΛΛ hypernucleus plays an essential role in producingthe ΛΛ states in the (K−,K+) reaction. The sensitivity to the potential parameters indicatesthat the nuclear (K−,K+) reactions have a high ability for the theoretical analysis of precisewave functions in the ΛΛ hypernuclei. New information on ΛΛ–Ξ dynamics in nuclei fromthe (K−,K+) data at J-PARC facilities will bring the S = −2 world development in nuclearphysics [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).

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