2006.10.13 HYP2006 Mainz

Quark-model baryon-baryon interactionsQuark-model baryon-baryon interactionsand their applicationsand their applications

to few-body systemsto few-body systems

Y. Fujiwara Y. Fujiwara （（ Kyoto) Y. Suzuki Kyoto) Y. Suzuki （（ NiigataNiigata ） ） C. Nakamoto C. Nakamoto (Suzuka)(Suzuka)

　 　 M. KohnoM. Kohno （（ Kyushu DentalKyushu Dental ） ） K. MiyagawaK. Miyagawa （（ Okayama)Okayama)1. Introduction2. B8 B8 interactions fss2 and FSS: spin-flavor SU6 symmetry3. B8 interactions by quark-model G-matrix4. Some applications 4.1. N interaction and 3H Faddeev calculation 4.2 effective potential and 9Be Faddeev calculation 4.3. s. p. potential and , (3N) potentials 4.4. N total cross sections and potential5. Summary

2006.10.13 HYP2006 Mainz

B8B8 interactions by fss2A natural and accurate description of NN, YN, YY interactions in terms of (3q)-(3q) RGM• Short-range repulsion and LS by quarks• Medium-attraction and long-rang tensor by S, PS and V meson exchange potentials (fss2) (Cf. FSSFSS without V)

Model HamiltonianModel Hamiltonian

+ (UijConf+Uij

FB+∑βUijSβ

+∑βUijPSβ + ∑βUij

Vβ)

6i<j

∑

6i=1∑H = 　　 (mi+pi

2/2mi)+

(3q)(3q)|E-H|{(3q)(3q)(r)}=0

Phys. Rev. C64 (2001) 054001Phys. Rev. C64 (2001) 054001Phys. Rev. C65 (2002) 014001Phys. Rev. C65 (2002) 014001

Phys. Rev. C54 (1996) 2180Phys. Rev. C54 (1996) 2180

QMPACK homepageQMPACK homepage http://qmpack.homelinux.com/~qmpack/index.php

PPNP in pressPPNP in press

Oka – Yazaki (1980) Oka – Yazaki (1980)

Arndt : SAID Arndt : SAID Nijmegen : NN-OnLineNijmegen : NN-OnLine

Lippmann-Schwinger (LS) RGMLippmann-Schwinger (LS) RGM Solve [ - H0 - VRGM() ] =0 with VRGM()=VD+G+ K in the mom. representation ( = E - Eint ) Born kernel qf|VRGM() |qi T-matrix, G-matrix

3-cluster Faddeev formalism using 3-cluster Faddeev formalism using VRGM()

P.T.P. 103P.T.P. 103(2000) 755(2000) 755

1) non-local 2) energy-dependent3) Pauli-forbidden states in N - N (I=1/2), - N

- (I=0), - (I=1/2) 1S0 : i.e. SU3 (11)s

P.T.P. P.T.P. 107107(2002) (2002) 745; 745; 993993

0

| |( ) ( , )(

0)G E T E h

u P u

RGM| ( ) |P h V P self-consistencyequation for

:: Ku=u Ku=u

2006.10.13 HYP2006 Mainz

B8 interaction by quark-model G-matrix

G (p, p’; K, , kF)

G (k’, q’; q1, q’)

V (k, q)

V (pf , pi)

VW (R, q) : Wigner transformWigner transform

U(R)=VW(R, (h2/2)(E-U(R))Transcendental equationTranscendental equation

Schrödinger equationSchrödinger equation Lippmann - Schwinger equationLippmann - Schwinger equation

exact exact E EB B , , ((EE))EEBB

WW , , WW((EE))

k’=p’- p , q’=(p+p’)/2k’=p’- p , q’=(p+p’)/2

k=pk=pf f - p- pii , q= , q=((ppff+p+pii))/2/2

- cluster folding- cluster folding

BB88

: “: “(0s)(0s)44””=0.257 fm-2

incident incident qq11

relative relative q’q’

in total c. m.in total c. m.

kF=1.35 fm-1

qq11=q=q for direct and knock-onfor direct and knock-onk=k’

2006.10.13 HYP2006 Mainz

nn RGM by RGM by GG-matrix of fss2-matrix of fss2 qq11==00

q’=q’=3/53/5 k kFF

kkFF==1.351.35 fmfm-1-1

expexp

“constant K , , kF”

SS1/21/2PP3/23/2

PP1/21/2

nn sactt. phase shift sactt. phase shift

B8B8 systems classified in the SU3 states with (, )

[‐(11)a+(30)]

[(11)a+(30)]

(03)

　　　 [(11)s+3(22)]

　　　 [3(11)s‐ （ 22 ） ] 　(22)

‐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

(03)

―

―

(22)

NN(0)

NN(1)

3E, 1O (P =antisymmetric)1E, 3O (P =symmetric)B8B8(I)S

10

1

10

1

10

1

10

1

2

1

2

1

2

1

2

1

2

1

3

1

6

1

5

1

5

1

5

3

5

3

5

3

5

2

5

2

302

9

10

322

1

2

1

102

1

8

3

(11)s complete Pauli forbidden (30) almost forbidden (=2/9)

‐2

0

‐4

Spin-flavor Spin-flavor SUSU66 symmetry symmetry

1. Quark-model Hamiltonian is approximately SU3 scalar ・ no confinement contribution (assumption)(assumption)　　・ Fermi-Breit int. … quark-mass dependence only　 ・ EMEP … automatic SU3 relations for coupling constants phenomenologyphenomenology CfCf. OBEP: exp data . OBEP: exp data gg, , ff, , … (integrated) … (integrated)2. -on plays an important role through SU3 relations and FSB3. effect of the flavor symm. breaking (FSB) by ms>mmud , B, M masses

Characteristics of SUCharacteristics of SU33 channels channels

1S, 3P (P-symmetric) 3S, 1P (P-antisymmetric)

(22) attractive pppp (03) strongly attractive np np

(11)s strongly repulsive NN((II=1/2)=1/2) (30) strongly repulsive NN((II=3/2)=3/2)

(00) strongly attractive HH-particle channel-particle channel

(11)a weakly attractive NN((II=0)=0) ““only this part is ambiguous”only this part is ambiguous”

(22)(22)

S=0S=0

S=‐2S=‐2

S=‐3S=‐3

S=‐4S=‐4S=‐1S=‐1

11SS00

1S0 phase shifts for B8B8 interactions with the pure (22) state (fss2)

(03)

(30)

(11)a

NN (03) central only(no tensor)

NNNN

(3/2)(3/2)

N N (0)(0) (0)(0)

N N (3/2)(3/2)

33SS11

fss2

3S1 phase shifts

(30) : Pauli repulsion

(11)a : weaklyattractive

+p differential cross sectionsand +p, p asymmetries a()

aexp=0.44±0.2at p=800±200 MeV/c

Kadowaki Kadowaki et al.et al. (KEK-PS E452)(KEK-PS E452)Euro. Phys. J. A15 (2002) 295Euro. Phys. J. A15 (2002) 295

AhnAhn et al.et al. (KEK-PS E251, E289)(KEK-PS E251, E289)NP A648(1999)263, A761(2005)41NP A648(1999)263, A761(2005)41350 MeV/c plab 750 MeV/c

Kurosawa Kurosawa et al. et al. (KEK-PS E452B)(KEK-PS E452B)KEK preprint 2005-104 (2006)KEK preprint 2005-104 (2006)

reported byreported byK. NakaiK. Nakai

+p elastic

p elastic

+p

2006.10.13 HYP2006 Mainz

NN interaction by fss2 interaction by fss2

fss2 FSSfrom 3He Faddeev

P-waveN is weakly attractive

N N -- NN coupling :coupling : 33SS1 1 ++ 33DD11 by one-by one- tensor tensor 11PP11 ++ 33PP11 by FB by FB LS LS (( －－ ))

Backward/Forward ratio

2006.10.13 HYP2006 Mainz

33H (hypertriton)H (hypertriton)

ud

u

ud

d

pp

nn

ud

sΛ(∑Λ(∑00 ) )

~2 fm

~5 fm~5 fm

“ ｄｅｕｔｅｒｏｎ”

dd == 2.22 MeV2.22 MeVBB=130 ±50 keV=130 ±50 keV

N on-shell properties are directly reflected

fss2 289 keV 0.80 FSS 878 keV 1.36 NNNN = 19.37 – 21.03 = = 19.37 – 21.03 = －－ 1.661.66

ｄｄ |= 17.50 – 19.72 = |= 17.50 – 19.72 = －－ 2.22 2.22 (MeV)(MeV)

150 channel calculation150 channel calculation

PP (%)(%)

1S0 / 3S1 relative strength

close to NSC89

exp’texp’t

Phys. Rev. C70, 024001 (2004)Phys. Rev. C70, 024001 (2004) NNNN--NNNN CC Faddeev CC Faddeev

2006.10.13 HYP2006 Mainz

N N １１ＳＳ 00 　ａｎｄａｎｄ　３３ＳＳ１１ effective range effective range parametersparameters 　

model as (fm) rs (fm) a ｔ (fm)

rt (fm) B(keV) P(%)

FSS － 5.41 2.26 －1.03

4.20 878 1.36

fss2 － 2.59 2.83 －1.60

3.01 289 0.80

NSC89 － 2.59 2.90 －1.38

3.17 143 0.5

““fss2”fss2” － 2.15 3.05 －1.80

2.87 145 0.53

““fss2”: fss2”: mm cc2 2 = 936 MeV = 936 MeV 1,000 MeV 1,000 MeV

Effect of the higher partial waves is large

90 –– 60 keV vs. 20 –– 30 keV in NSC89

favorable for 4H (1+)

BBΛΛexpexp=130 ±50 keV=130 ±50 keV

B (keV) fss2 ““fss2”fss2”

6 ch (S)15 ch (SD)102 ch (J4)150 ch (J6)

137198288289

44 85145145

effective local potentialseffective local potentials by by GG-matrix -matrix BB88BB88 interaction interaction

ND

effective potentialsquark-model N-N

EB (exact)

－－ 3.62 MeV－－ 3.18 MeV

EBexp=3.120.02

MeV

Cf. U(0) ＝‐‐ 46 (FSS), ‐‐48 (fss2) MeVin symmetric matter

2006.10.13 HYP2006 Mainz

(0)

(3.04 MeV)

0+

2+

-3.120.02 MeV3067(3) keV

3024(3) keV

-6.620.04 MeV

3026 keV

92 keV

1/2+

5/2+3/2+

8Be

+5He

9Be calc.

Eexp(3/2+ - 5/2+) = 43 43 5 5 keV Akikawa, Tamura Akikawa, Tamura et al. et al. (BNL E930)(BNL E930)Phys. Rev. Let. 88, 082501 (2002)Phys. Rev. Let. 88, 082501 (2002)

198 keV (fss2 quark+), 137 keV (FSS) : 3 5 times too large

2 Faddeev for 99BeBe

+ + RGM kernel (MN3R)effective pot. (SB u=0.98)exp’t

2828 keV

Phys. Rev. C70, 024002, 0407002 (2004)Phys. Rev. C70, 024002, 0407002 (2004)

s s splitting bysplitting by N LS N LS Born kernelBorn kernel

2006.10.13 HYP2006 Mainz

ss splitting of 99BeBe byby 22 Faddeev using Faddeev using

quark-model quark-model GG-matrix -matrix LSLS Born kernel Born kernel

0.5 0 0.70 0 N Born

kF (fm-1) 1.07 1.20 1.35 -

G-matrixS (MeV fm5)

fss2 (cont) ‐10.5 ‐10.6 ‐10.7 -10.9FSS (cont) -1.9 -2.9 -3.6 -

7.8Faddeev

E (keV)

fss2 (cont) 188 194 198 198FSS (cont) 7 34 59 137

E E expexp (keV)(keV) 43 43 5 5FSS (cont) reproduces E exp at kF=1.25 fm-1 !PP-wave -wave NN--NN coupling by coupling by LSLS(-)(-) is important. is important.

S-meson LS in fss2 is not favorable.

2006.10.13 HYP2006 Mainz

potentials potentials ((VVWWC C ((RR, 0)), 0)) by quark-model by quark-model

GG-matrix interaction-matrix interaction

I=3/2 I=3/2

I=1/2 I=1/2

total total

fss2FSS

The Pauli repulsion of N(I=3/2) 3S1 is very strong.

2006.10.13 HYP2006 Mainz

(3(3NN) potentials by quark-model ) potentials by quark-model GG-matrix -matrix interaction ( 0interaction ( 0++, T, T=1/2 channel)=1/2 channel)

3/ 2 1/ 2 1/ 2(3 ) ( 0, 1/ 2) (4 / 3) (3 / 2) (1/ 6)N s t sV S T V V VS = = = + +

EEBB(exact)(exact) ＝－＝－ 3.79 3.79 MeVMeV EEBB(exact)(exact) ＝－＝－ 5.70 5.70 MeVMeV

FSS fss2

consistent with 4He (0+) resonance

(3N): (0s)3

=0.22 fm=0.22 fm-2-2

qq11=0=0

（（－－ , , KK++)) inclusive spectra oninclusive spectra on 2828SiSiexp: Noumi exp: Noumi et al.et al. PRL 89, 072301 (2002) ; 90, 049902 (E) (2003) PRL 89, 072301 (2002) ; 90, 049902 (E) (2003) Saha Saha et al.et al. Phys. Rev. C70, 044613 (2004) Phys. Rev. C70, 044613 (2004)

poster sessionposter sessionby M. Kohnoby M. Kohno

Repulsive U (q) in symmetric nuclear matter is experimentally confirmed.

potentials potentials ((VVWWC C ((RR, 0)), 0)) by quark-model by quark-model

GG-matrix interaction-matrix interaction

I=1

I=0

I=1

totaltotal

I=0

Some attraction in the surface region.

FSS fss2

FSS fss2

- (in medium) = 30.7±6.7 mb (eikonal approx.)= 20.9±4.5 mb

+3.7 -3.6+2.5 -2.4-p /-n =1.1 at plab=550 MeV/c

+1.4+0.7 -0.7 -0.4

Tamagawa Tamagawa et al.et al. (BNL-E906)(BNL-E906) Nucl. Phys. A691 (2001) 234cNucl. Phys. A691 (2001) 234cYamamoto Yamamoto et al.et al. Prog. Theor. Phys. 106 (2001)363Prog. Theor. Phys. 106 (2001)363

Ahn Ahn et al.et al.Phys. Lett. BPhys. Lett. B633 (2006) 214633 (2006) 214

More experiments are needed.

2006.10.13 HYP2006 Mainz

ＳｕｍｍＳｕｍｍａｒｙａｒｙQuark-model description for the baryon-baryoninteraction is very successful to reproduce manyexperimental data. In particular, the extension of the (3q)-(3q) RGM study for the NN and YNinteractions to the strangeness S=-2, -3, -4 sectors has clarified characteristic features of the B8B8

interactions. The results seem to be reasonable if we consider1)1) spin-flavor spin-flavor SUSU66 symmetry symmetry2) weak π-on effect in the strangeness sector2) weak π-on effect in the strangeness sector3) effect of the flavor symmetry breaking3) effect of the flavor symmetry breaking 　　　We have analyzed B8, B8(3N) interactions based onthe 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 1S0 and 3S1 attraction (relative strength) ( 3H Faddeev calculation: 289 keV for fss2) ・ small s splitting in 9Be excited states (FSS)

・ N (I=1/2 1S0), N (I=3/2 3S1) repulsion repulsive s. p. and potentials … fss2, FSSS ＝‐ 2 interaction is not much attractive !

・ interaction |V|<|VN|<|VNN|

B 1 MeV (Nagara event 6He) … fss2 ・ N in-medium total cross section (fss2, FSS) … strong isospin dependence of s.p. potential

・ N (I=0 3S1): (11)a 0 or weakly attractive (fss2, FSS)

vs. ESC04(d): strongly attractive

Characteristics of Characteristics of fss2fss2 and and FSSFSS