8th International Conference on Quarks and Nuclear Physics (QNP218) Nov. 13 17, 218 Hypernuclear photoproduction spectra calculated with multi-configuration wave functions Atsushi UMEYA (Nippon Inst. of Tech.) Toshio MOTOBA (Osaka E-C Univ., YITP) Kazunori ITONAGA (Gifu Univ.) 1
Recent (e, e K + ) reaction experiments done at the Jefferson Lab c: " ' 1:) " Fig. 7. dσ/dω K per.3 MeV [nb/sr] 8 6 4 2-1 1 (MeV I PRC93, 34314 (216) The predicted excitation E Λ [MeV] function for the 5 1 JLab E5-115 1 B(e,e K + ) 1 Be Λ #1 #2 (Fit I) #3 a #4-1 -8-6 -4-2 2 4 6 -B Λ [MeV] Shell-model prediction T. Motoba et al., PTPS117, 123 (1994) Core nucleus calculated with standard p-shell model Λ in s-orbit Recent experimental result T. Gogami et al., This experiment has confirmed the major peaks (#1, #2, #3, #4) predicted in DWIA by emplying the Λ particle in s-orbit coupled with the nuclear core states confined within the p-shell configuration. However, it is interesting to observe extra strengths at E Λ = MeV excitation (a). The extension of the model space is necessary and interesting challenge in view of the present hypernuclear spectroscopy. 2
Extension of the model space in the shell model ( 1 Be case) Λ Model space for 9 Be core (A) standard model space J core (s) 4 (p) 5 (p-h) (B) extended model space J + core (s) 3 (p) 6 (s) 4 (p) 4 (sd) 1 (1p-1h) Standard model space for 1 Λ Be (I) J core s Λ 1 Λ Be(J ) (II) J core p Λ 1 Λ Be(J+ ) Extension (1) 1p-1h (1ħω) core excitation is taken into account (a) J core s Λ 1 Λ Be(J ) (b) J core p Λ 1 Λ Be(J+ ) (c) J + core s Λ 1 Λ Be(J+ ) (d) J + core p Λ 1 Λ Be(J ) Extension (2) Configrations mixed by ΛN interaction J core s Λ J + core p Λ 1 Λ Be(J ) J core p Λ J + core s Λ 1 Λ Be(J+ ) 3
Configration mixing in 1 Be unnatural parity states Λ 9 Be (J core) + 1 Λ(p) ΛBe (J ) 9 Be (J + core) 1 ω 9 Be (J core) + Λ(s) 9 Be (Jcore) Λ(p) 1 ΛBe (J + ) ΛBe (J + ) 1 Mixing 9 Be (J core) 1 ω 9 Be (Jcore) 1 Λ(s) ΛBe (J ) In the standard shell model, only natural-parity nuclaer-core states (J core ) are taken into account. Λ particle is in the s orbit in 1 Λ Be(J ). In 1 Λ Be(J+ ), the energy difference between Λ(s) and Λ(p) is 1ħω, and the energy difference between 9 Be(J core ) and 9 Be(J + core ) is 1ħω. By ΛN interaction, natural-parity nuclaer-core configurations and unnatural-parity nuclaer-core configurations can be mixed. 4
Extended model space for target nucleus 1 B 1 ΛBe (J ) (s) 4 (p) 5 (s Λ ) 1 (s) 3 (p) 6 (p Λ ) 1 (s) 4 (p) 4 (sd) 1 (p Λ ) 1 1 B (J + ) (s) 4 (p) 6 (s) 3 (p) 6 (sd) 1 (s) 4 (p) 5 (fp) 1 (s) 2 (p) 8 (s) 4 (p) 4 (sd) 2 1 ΛBe (J + ) (s) 4 (p) 5 (p Λ ) 1 (s) 3 (p) 6 (s Λ ) 1 (s) 4 (p) 4 (sd) 1 (s Λ ) 1 extended model space for the target nucleus standard model space extended model space for hypernuclei Extension of model space for target nucleus 1 B up to 2p-2h (2ħω) allows the 1 Be production through various configurations. Λ 5
ΛN interaction and Λ single-particle energy NΛ V NΛ Nijmegen NSC97e Th. A. Rijken, V. G. J. Stoks, Y. Yamamoto, PRC59, 21 (1999) ε Λ s and ε Λ p are determined to reproduce the #1 (2 ) and #6 ( ) peaks in 12 Λ B production cross-section. ε Λ s and ε Λ p are applied to 1 Λ Be. JLab Hall C, E5-115 L. Tang et al., PRC9, 3432 (214) Theoretical calculation T. Motoba et al., PTPS185, 224 (21) e K ) 12 B experimental spectrum (top) taken from JLab E5-115 experiment [9 6
Results : Energy levels of 9 Be and 1 Λ Be Energy (MeV) 12 11 1 9 8 7 6 5 4 3 2 1 7/2 9/2 + 7/2 3/2 3/2 + 1/2 5/2+ 1/2 + 3/2 9 Be (exp.) 9 1 Be (cal.) ΛBe (cal.) 1 4 1 2 3 5 + 2 1 4 3 4 + 1 + 1 + 2 + 2 + 1 + 3 2 3 4 2 1 1 2 3 1 2 2 + 4 + 5 + dominant configurations blue J ; 9 Be(J core) Λ(s) green J + ; 9 Be(J + core) Λ(p) 2 + 1 + 2 + + 1+ magenta J + ; 9 Be(J core) Λ(p) red J + ; 9 Be(J + core) Λ(s) 4 + 2 1 + 3 1+ + + 1 + 2 + 2 + 1 + 4 3 1 2 1 7
Results : Energy levels of 1 Be (comparison with JLab experiments) Λ Energy (MeV) 12 11 1 9 8 7 6 5 4 3 2 1 7/2 9/2 + 7/2 3/2 3/2 + 1/2 5/2+ 1/2 + 3/2 1 4 1 2 3 5 4 3 + 2 1 4 3 4 + 1 + 1 + 2 + 4 + 2 1 3 1.83 + 1+ + + 1 + 1 2 + 1 + 2 1 2 + 4 + 5 + 28.34 + 3 2 2 + 1 New bump + 3 4 2 1 2 + 1 + 1 2 3 1 2 2 + + 1+ 6.26 2.78. 9 Be (exp.) 9 1 Be (cal.) Λ Be (cal.) 1 Λ Be (exp.) T. Gogami et al., PRC93, 34314 (216) 8
Results : Spectroscopic factors of the pickup reaction, 1 B 9 Be Energy (MeV) 11 1 9 8 7 6 5 4 3 2 1 9/2 + 5/2 7/2 3/2 7/2 3/2 + 1/2 5/2 + 1/2 + 3/2.668.958 1. 1.299 7/2 9/2 + 7/2 3/2 3/2 + 1/2 5/2+ 1/2 + 3/2.942 1.2 1. 1.355 1 2 C 2 S rel (Exp.) 1 2 C 2 S rel (Cal.) 9
Results : Cross sections of the 1 B (γ, K + ) 1 Be reaction (1) Λ 5 Eγ= L 1.5 GeV,θγ= L 7 Cross Section d 2 σ/dωde (nb/sr/mev) 4 3 2 1 1 B (γ, K + ) 1 Λ Be 1 2 2 3 3 4 1+3+ 2 + 3 4 2 + 4 + 12 1 8 6 4 2 2 4 6 8 1 Hypernuclear Energy E Λ (MeV) QF 1
Results : Cross sections of the 1 B (γ, K + ) 1 Be reaction (2) Λ dσ/dω K per.3 MeV [nb/sr] 8 6 4 2 5 1 JLab E5-115 1 B(e,e K + ) 1 Be Λ #1 #2 (Fit I) #3 a E Λ [MeV] #4 T. Gogami et al., PRC93, 34314 (216) Cross Section d 2 σ/dωde (nb/sr/mev) 5 4 3 2 1-1 -8-6 -4-2 2 4 6 -B Λ [MeV] Eγ= L 1.5 GeV,θγ= L 7 1 B (γ, K + ) 1 Λ Be 1 2 2 3 3 4 1+3+ 2 + 3 4 2 + 4 + 12 1 8 6 4 2 2 4 6 8 1 Hypernuclear Energy E Λ (MeV) QF Our new calculation reproduces the four major peaks (#1, #2, #3, #4). Our new calculation explains the new bump (a) as a sum of cross sections of some J + states. 11
Results : Cross sections of the 1 B (γ, K + ) 1 Be reaction (3) Λ Eγ = 1.5 GeV EXP = T. Gogami et al, PRC93 (216) 9 Be (Ji) Λ 1 Be (Jk) CAL θ = 7 deg EXP Fit I Ji Ei (exp) Ei (cal) Jk Ex BΛ dσ/dω exp Ex BΛ dσ/dω C2S C2S [MeV] [MeV] [nb/sr] peak [MeV] [MeV] [nb/sr] 3/2.. 1. 8.6 9.69 1.(rel) 1.(rel) 2.165 8.435 12.8 21.62 #1. 8.55±.7 17.±.5 2.429 2.644 2 2.712 5.888 11.654.958 1.2 3 2.86 5.74 9.391 7/2 6.38 6.189 3 6.183 2.417 7.625.668.942 4 6.37 2.23 13.55 21.5 #2 2.78±.11 5.76±.9 16.5±.5 21.13 #3 6.26±.16 2.28±.14 1.5±.3 2 + (3) 7.87.793 4.495 1 + (3) 7.935.665 4.968 (2) 8.712.112 6.15 2 + (4) 8.828.228 1.431 2 + (5) 9.2.42 9.893 (3) 9.59.459 2.434 9.46 19.91 (29.37) #a 8.34±.41.2±.4 23.2±.7 7/2 11.283 1.241 3 1.15 1.55 3.913 1.299 1.355 4 1.455 1.855 17.985 1 + (5) 1.828 2.228 4.598 4 + (3) 11.318 2.718 11.185 (5) 11.543 2.943 13.759 21.9 #4 1.83±.1 2.28±.7 17.2±.5 29.54 (51.44) 12
Results : Configrations of J + states corresponding to the new bump J π n( B Λ [MeV]) [J π core] j Λ [J π core] j Λ [J π core] j Λ XS [nb/sr] 2 + 3 (.739) 4.49 1 + 3 (.665) 4.97 2 + 4 (.228) 1.43 2 + 5 (.42) 9.89 2 (.112) 6.15 3 (.459) 2.43 [5/2 + 2 ]sλ 1/2 87.5% [5/2 + 2 ]sλ 1/2 11.3% [5/2 + 2 ]sλ 1/2 31.6% [5/2 + 2 ]sλ 1/2 67.5% [3/2 1 ](p 3/2 p 1/2 )Λ 82.5% [3/2 1 ](p 3/2 p 1/2 )Λ 79.5% [3/2 1 ](p 3/2 p 1/2 )Λ 9.4% [3/2 1 ](p 3/2 p 1/2 )Λ 7.9% [3/2 1 ]pλ 3/2 55.4% [3/2 1 ]pλ 3/2 27.1% [ 1 ](p 3/2 p 1/2 )Λ 15.8% [ 1 ]pλ 3/2 17.9% [ 1 ](p 3/2 p 1/2 )Λ 2.4% [ 1 ](p 3/2 p 1/2 )Λ 1.8% [ 1 ](p 3/2 p 1/2 )Λ 9.7% [ 1 ](p 3/2 p 1/2 )Λ 2.7% 13
Results : Cross sections of the 1 B (K, π ) 1 Λ B reaction Cross Section d 2 σ/dωde (µb/sr/mev) Cross Section d 2 σ/dωde (nb/sr/mev) 12 1 8 6 4 2 1 B (K,π ) 1 Λ B 2 3 3 3 p L K =.8 GeV/c,θL γ= 2 12 1 8 6 4 2 2 4 6 8 1 p n Hypernuclear Energy Ep Λ (MeV) Λ 5 Eγ= L 1.5 GeV,θγ= L 7 α 4 3 2 1 1 B (γ, K + ) 1 Λ Be α 1 2 2 3 3 4 3 4 2 + 4 + 1+3+ 9 Be 2 + 9 12 1 8 6 4 2 2 4 6 8 1 Λ Hypernuclear Energy Eα Λ (MeV) α α Λ Be α [p 1 p Λ ] QF p In the (K, π ) reaction, the large peak at E Λ = 4.4 MeV is a p-substitutional state via the p N 3/2 pλ 3/2, which is strongly excited by recoilless reaction. The small peak at E Λ = MeV corresponds to the new bump and is explained as a mixture of s Λ and p Λ states. The large peak at E Λ = 4.4 MeV in 1 Λ Be corresponds to the [p 1 p Λ ] state in 9 Λ Be (9 Be analog state). The small peak at E Λ = MeV in 1 Λ Be corresponds to the [p 1 p Λ // ] state in 9 Λ Be. [p 1 p Λ // ] 14
[p 1 p Λ ] and [p 1 p Λ // ] states of 9 Λ Be 9Be (K, π ) 9 Λ Be T. Motoba et al., PTPS81, 42 (1985) R. Bertini et al. (H-S-S Collaboration), NPA368, 365 (1981) 15
Summary We have calculated the cross sections in 1 ΛBe productions by using the extended shell model to describe the unnatural-parity nuclear core. dσ/dω K per.3 MeV [nb/sr] 8 6 4 2 5 1 JLab E5-115 1 B(e,e K + ) 1 Be Λ #1 #2 (Fit I) #3 a E Λ [MeV] #4-1 -8-6 -4-2 2 4 6 -B Λ [MeV] Cross Section d 2 σ/dωde (nb/sr/mev) 5 4 3 2 1 1 B (γ, K + ) 1 Λ Be 1 2 2 3 3 4 1+3+ 2 + E L γ= 1.5 GeV,θ L γ= 7 3 4 2 + 4 + 12 1 8 6 4 2 2 4 6 8 1 Hypernuclear Energy E Λ (MeV) Our new calculation explains the new bump in the JLab experimental results as a sum of cross sections of some J + states. These states have a large mixture of unnatural- and natural-parity nuclear-core states. The new bump in 1 Λ Be corresponds to the [p 1 p Λ // ] state in 9 Λ Be. 16 QF