HYP2006 Mainz Quark-model baryon-baryon interactions and their applications to few-body systems Y. Fujiwara （ Kyoto) Y. Suzuki （ Niigata ） C. Nakamoto (Suzuka) M. Kohno （ Kyushu Dental ） K. Miyagawa （ Okayama) M. Kohno （ Kyushu Dental ） K. Miyagawa （ Okayama) 1. Introduction 2. B 8 B 8 interactions fss2 and FSS: spin-flavor SU 6 symmetry 3. B 8  interactions by quark-model G-matrix 4. Some applications 4.1.  N interaction and 3  H Faddeev calculation 4.2 effective  potential and 9  Be Faddeev calculation 4.3.  s. p. potential and  ,  (3N) potentials 4.4.  N total cross sections and  potential 5. Summary

HYP2006 Mainz B 8 B 8 interactions by fss2 A natural and accurate description of NN, YN, YY interactions in terms of ( 3 q)-( 3 q) RGM Short-range repulsion and LS by quarks Medium-attraction and long-rang tensor by S, PS and V FSS meson exchange potentials (fss2) (Cf. FSS without V) Model Hamiltonian + (U ij Conf +U ij FB +∑ β U ij Sβ +∑ β U ij PSβ + ∑ β U ij Vβ ) 6 i<j ∑ 6 i =1 ∑ H = (m i +p i 2 /2m i ) +  (3 q )  (3 q )| E-H| A {  (3 q )  (3 q )  ( r )}  =0 Phys. Rev. C64 (2001) Phys. Rev. C65 (2002) Phys. Rev. C54 (1996) 2180 QMPACK homepage QMPACK homepage PPNP in press Oka – Yazaki (1980)  Arndt : SAID Nijmegen : NN-OnLine

Lippmann-Schwinger (LS) RGM Solve [  - H 0 - V RGM (  ) ]  =0 with V RGM (  )=V D +G+  K in the mom. representation (  = E - E int ) Born kernel  q f |V RGM (  ) |q i   T-matrix, G-matrix 3-cluster Faddeev formalism using 3-cluster Faddeev formalism using V RGM (  ) P.T.P. 103 (2000) 755 1) non-local 2) energy-dependent 3) Pauli-forbidden states in  N -  N (I=1/2),  -  N -  (I=0),  -  (I=1/2) 1 S 0 : i.e. SU 3 (11) s P.T.P. 107 (2002) 745; 993 self-consistency equation for   : Ku=u

HYP2006 Mainz B 8  interaction by quark-model G-matrix G (p, p’; K, , k F ) G (k’, q’; q 1, q’) V (k, q) V (p f, p i ) Wigner transform V W (R, q) : Wigner transform U(R)=V W (R,  (h 2 /2  )(E-U(R)) Transcendental equation Schrödinger equation Lippmann - Schwinger equation exact E B,  (E) E B W,  W (E) k’=p’- p, q’=(p+p’)/2 k=p f - p i, q=(p f +p i )/2  - cluster folding B8B8B8B8  : “ (0s) 4 ” =0.257 fm -2 incident q 1 relative q’ in total c. m. k F =1.35 fm -1 q 1 =q for direct and knock-on k=k’

HYP2006 Mainz n  RGM by G-matrix of fss2 q 1 =0 q’=  3/5 k F k F =1.35 fm -1 exp “constant K, , k F ” S 1/2 P 3/2 P 1/2 n  sactt. phase shift

B 8 B 8 systems classified in the SU 3 states with (,  ) [ ‐ (11) a +(30)] [(11) a +(30)] (03) [(11) s +3(22)] [3(11) s ‐（ 22 ） ] (22)    ‐3‐3 ― (11) a [ ‐ (11) a + (30)+(03)] [(30) ‐ (03)] ― [2(11) a + (30)+(03)] ― (11) s + (22)+ (00) (11) s ‐ (22)+ (00) (11) s + (22) ｰ (11) s + (22) (11) s － (22) － (00) ― (22)        (30) ― (22)   [ ‐ (11) a +(03)] [(11) a +(03)] (30) [(11) s +3(22)] [3(11) s ‐ (22)] (22)    ‐1‐1 (03) ― (22) NN(0) NN(1) 3 E, 1 O ( P =antisymmetric) 1 E, 3 O ( P =symmetric)B 8 B 8 (I)S (11) s complete Pauli forbidden (30) almost forbidden (  =2/9) ‐2‐2 0 ‐4‐4

Spin-flavor SU 6 symmetry 1. Quark-model Hamiltonian is approximately SU 3 scalar (assumption) ・ no confinement contribution (assumption) ・ Fermi-Breit int. … quark-mass dependence only ・ EMEP … automatic SU 3 relations for coupling constants phenomenologyCf. OBEP: exp data  g, f,  … (integrated) phenomenology Cf. OBEP: exp data  g, f,  … (integrated) 2.  -on plays an important role through SU 3 relations and FSB m 3. effect of the flavor symm. breaking (FSB) by m s >m ud, B, M masses Characteristics of SU 3 channels 1 S, 3 P ( P -symmetric) 3 S, 1 P ( P -antisymmetric) pp (22) attractive pp np (03) strongly attractive np  N(I=1/2) (11) s strongly repulsive  N(I=1/2)  N(I=3/2) (30) strongly repulsive  N(I=3/2) (00) strongly attractive H-particle channel H-particle channel  N(I=0) (11) a weakly attractive  N(I=0) “only this part is ambiguous” “only this part is ambiguous”

(22) S=0 S= ‐ 2 S= ‐ 3 S= ‐ 4 S= ‐ 1 1S01S01S01S0 1 S 0 phase shifts for B 8 B 8 interactions with the pure (22) state (fss2)

(03) (30) (11) a NN (03) central only (no  tensor) NN  (3/2)  N (0)  (0)  N (3/2) 3S13S13S13S1 fss2 3 S 1 phase shifts (30) : Pauli repulsion (11)a : weakly attractive

 + p differential cross sections and  + p,  p asymmetries a(  ) a exp =0.44±0.2 at p  =800±200 MeV/c Kadowaki et al. (KEK-PS E452) Euro. Phys. J. A15 (2002) 295 Ahnet al.(KEK-PS E251, E289) Ahn et al. (KEK-PS E251, E289) NP A648(1999)263, A761(2005) MeV/c  p lab  750 MeV/c Kurosawa et al. (KEK-PS E452B) KEK preprint (2006) reported by K. Nakai  + p elastic  p elastic +p+p

HYP2006 Mainz  N interaction by fss2 fss2 FSS from  3 He Faddeev  P-wave  N is weakly attractive N -  Ncoupling : 3 S D 1 by one-  tensor  N -  N coupling : 3 S D 1 by one-  tensor 1 P P 1 by FB LS ( － ) 1 P P 1 by FB LS ( － ) Backward/Forward ratio

HYP2006 Mainz 3  H (hypertriton) u d u u d d p n u d s Λ(∑ 0 ) ~2 fm ~5 fm “ ｄｅｕｔｅ ｒｏｎ ”  d = 2.22 MeV B Λ =130 ±50 keV  N on-shell properties are directly reflected fss2 289 keV 0.80 FSS 878 keV 1.36  NN = – = － 1.66  ｄ |= – = － 2.22 (MeV) 150 channel calculation P  (%) 1 S 0 / 3 S 1 relative strength close to NSC89 exp’t Phys. Rev. C70, (2004)  NN-  NN CC Faddeev

HYP2006 Mainz  N １ Ｓ 0 ａｎｄ ３ Ｓ １ effective range parameters  N １ Ｓ 0 ａｎｄ ３ Ｓ １ effective range parameters modela s (fm) r s (fm) a ｔ (fm) r t (fm) B  (keV)P  (%) FSS － － fss2 － － NSC89 － － “fss2” － － “fss2”: m  c 2 = 936 MeV  1,000 MeV Effect of the higher partial waves is large – 90 – 60 keV – vs. 20 – 30 keV in NSC89 favorable for  4 H (1 + ) B Λ exp =130 ±50 keV B  (keV)fss2“fss2” 6 ch (S) 15 ch (SD) 102 ch (J  4) 150 ch (J  6)

 effective local potentials by G-matrix B 8 B 8 interaction ND effective potentials quark-model  N-  N E B (exact) － － 3.62 MeV － － 3.18 MeV E B exp =3.12  0.02 MeV ‐‐ Cf. U  (0) ＝ ‐ 46 (FSS), ‐ 48 (fss2) MeV in symmetric matter

HYP2006 Mainz (0) (3.04 MeV)  0.02 MeV 3067(3) keV 3024(3) keV  0.04 MeV 3026 keV 92 keV 1/2 + 5/2 + 3/2 + 8 Be  +  5 He  9 Be calc. 43  5  E exp (3/ /2 + ) = 43  5 keV Akikawa, Tamura et al. (BNL E930) Phys. Rev. Let. 88, (2002) 198 keV (fss2 quark+  ), 137 keV (FSS) : 3  5 times too large 9  Be 2  Faddeev for 9  Be  + + + +  RGM kernel (MN3R) effective  pot. (SB u=0.98) exp’t 2828 keV Phys. Rev. C70, , (2004) s splitting by  N LS Born kernel s splitting by  N LS Born kernel

HYP2006 Mainz s 9  Beby2  Faddeev using quark-model G-matrix  LS Born kernel s splitting of 9  Be by 2  Faddeev using quark-model G-matrix  LS Born kernel  0.5   0  0  N Born k F (fm -1 ) G-matrix S  ( MeV fm 5 ) fss2 (cont) ‐ 10.5 ‐ 10.6 ‐ FSS (cont) Faddeev  E (keV) fss2 (cont) FSS (cont)  E exp (keV) 43  5 FSS (cont) reproduces  E exp at k F =1.25 fm -1 ! P-wave  N-  N coupling by LS (-) is important. S-meson LS in fss2 is not favorable.

HYP2006 Mainz  potentials (V W C (R, 0)) by quark-model G-matrix interaction I=3/2 I=1/2 total fss2FSS The Pauli repulsion of  N(I=3/2) 3 S 1 is very strong.

HYP2006 Mainz  (3N) potentials by quark-model G-matrix interaction ( 0 +, T=1/2 channel) E B (exact) ＝－ 3.79 MeV E B (exact) ＝－ 5.70 MeV FSS fss2 consistent with 4  He (0 + ) resonance (3N): (0s) 3 =0.22 fm -2 =0.22 fm -2 q 1 =0

（  －, K + ) inclusive spectra on 28 Si exp: Noumi et al. PRL 89, (2002) ; 90, (E) (2003) Saha et al. Phys. Rev. C70, (2004) Saha et al. Phys. Rev. C70, (2004) poster session by M. Kohno Repulsive U  (q) in symmetric nuclear matter is experimentally confirmed.

 potentials (V W C (R, 0)) by quark-model G-matrix interaction I=1 I=0 I=1 total I=0 Some attraction in the surface region. FSS fss2

FSSfss2   -  (in medium) = 30.7±6.7 mb (eikonal approx.)= 20.9±4.5 mb   - p /   - n =1.1 at p lab =550 MeV/c Tamagawa et al. (BNL-E906) Nucl. Phys. A691 (2001) 234c Nucl. Phys. A691 (2001) 234c Yamamoto et al. Prog. Theor. Phys. 106 (2001)363 Prog. Theor. Phys. 106 (2001)363 Ahn et al. Phys. Lett. B 633 (2006) 214     More experiments are needed.

HYP2006 Mainz Ｓｕｍｍ ａｒｙ Quark-model description for the baryon-baryon interaction is very successful to reproduce many experimental data. In particular, the extension of the (3q)-(3q) RGM study for the NN and YN interactions to the strangeness S= - 2, - 3, - 4 sectors has clarified characteristic features of the B 8 B 8 interactions. The results seem to be reasonable if we consider 1) spin-flavor SU 6 symmetry 2) weak π-on effect in the strangeness sector 3) effect of the flavor symmetry breaking 3) effect of the flavor symmetry breaking We have analyzed B 8 , B 8 (3N) interactions based on the G-matrix calculations of fss2 and FSS.

S=0 ・ triton binding energy … fss2: +150 keV (3 body force?) S ＝ ‐ 1  p and  + p interactions are progressively known. ・  + p total and differential cross sections and polarization … fss2, FSS ・  N 1 S 0 and 3 S 1 attraction (relative strength) ( 3  H Faddeev calculation: 289 keV for fss2) ・ small s splitting in 9  Be excited states (FSS) ・  N (I=1/2 1 S 0 ),  N (I=3/2 3 S 1 ) repulsion  repulsive s. p. and  potentials … fss2, FSS S ＝ ‐ 2  interaction is not much attractive ! ・  interaction |V  |<|V  N |<|V NN |  B   1 MeV (Nagara event 6  He) … fss2 ・  N in-medium total cross section (fss2, FSS) … strong isospin dependence of  s.p. potential ・  N (I=0 3 S 1 ): (11) a  0 or weakly attractive (fss2, FSS) vs. ESC04(d): strongly attractive Characteristics of fss2 and FSS