2007.6.6 inpc07

BB88 and and BB88(3(3NN)) potentials derived from thepotentials derived from theSUSU6 6 quark-model baryon-baryon interactionquark-model baryon-baryon interaction

Y. Fujiwara Y. Fujiwara （（ Kyoto) M. KohnoKyoto) M. Kohno （（ Kyushu DentalKyushu Dental ） ） Y. Suzuki Y. Suzuki （（ NiiNiigatagata ））

1. Motivation and background2. New folding method3. n RGM

4. kF dependence of potentials and s force5. Comparison of and (3N) potentials6. and (3NN) potentials7. Summary

2007.6.6 inpc07

MotivationMotivation Derive reliable Derive reliable BB88, , BB88(3(3NN) potentials) potentials based on the based on the GG-matrix -matrix

calculations of the quark-model calculations of the quark-model BB88 BB88 interactions. interactions.

BackgroundBackground• Quark-model Quark-model BB88BB88 int. fss2, FSS : int. fss2, FSS : QMPACK homepageQMPACK homepage http://qmpack.homelinux.com/~qmpack/index.php• GG-matrix and 3-cluster Faddeev calculations of the -matrix and 3-cluster Faddeev calculations of the ss-shell -shell and and p-p-shellshell hypernuclei : hypernuclei : Prog. Part. Nucl. Phys. 58 (2007) Prog. Part. Nucl. Phys. 58 (2007) 439439• Much interest to Much interest to -hypernuclei (J-PARC day-1 exp’t)-hypernuclei (J-PARC day-1 exp’t)

ImprovementImprovement •　　 BB88BB88 BB88, , BB88(3(3NN), … a new folding method), … a new folding method Cf.Cf. optical potential … optical potential … SS-wave, -wave, PP-wave local potentials-wave local potentials parameterizations : off-shell property is assumedparameterizations : off-shell property is assumed ！！　　• Correct treatment of the c.m. motion in light nuclei Correct treatment of the c.m. motion in light nuclei • Optimal treatment of the Optimal treatment of the GG-matrix parameters-matrix parameters

　　 starting energy starting energy , c.m. mom. , c.m. mom. KK, Fermi mom. , Fermi mom. kkF F

• Various exchange kernels in Various exchange kernels in nn RGM RGMOnly restrictionOnly restriction , (3, (3NN) clusters: simple (0) clusters: simple (0ss))44, (0, (0ss))33 h.o. wave functions with h.o. wave functions with

　　　　 =0.257 fm=0.257 fm-2-2, , =0.18 fm=0.18 fm-2-2 　　 (reproduce the rm(reproduce the rm

s radii)s radii) Fujiwara, Kohno and Suzuki, Nucl Phys. A784 (2007) 161Fujiwara, Kohno and Suzuki, Nucl Phys. A784 (2007) 161

2007.6.6 inpc07

B8 interaction by quark-model G-matrix

G (pp, pp’; K, , kF)

G (kk’, qq’; q1, q’)

V (kk, qq)

V (ppf , ppi)

VW (RR, qq) : Wigner transformWigner transform

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

Schrödinger equationSchrödinger equation Lippmann-Schwinger equationLippmann-Schwinger equation

EEB 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=1.20 fm-1

qq11=q=q for direct and knock-onfor direct and knock-onkk=kk’

(())

2007.6.6 inpc07

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

q’=q’=3/53/5 k kFF

kkFF==1.201.20 fmfm-1-1

expexp

“constant K, , kF”

SS1/21/2PP3/23/2

PP1/21/2

nn scatt. phase shifts scatt. phase shifts

2007.6.6 inpc07

kF dependence of central potential

EB (MeV)－－ 3.623.62－－ 4.544.54－－ 5.475.47－ 3.18－ 3.90－ 4.81exp (MeV)exp (MeV)－－ 3.12 3.12 0.020.02

QG v v G

e

e= -H0

=k.e.+U(q1) +UN(q2) < 0< 0

= 00.700.50

centralcentral

2007.6.6 inpc07

kF dependence of s potential

FSS

fss2

no no SS-meson -meson LSLS

with with SS-meson -meson LSLS

onlyonly FB FB LSLS

1.351.35

1.071.07

UULS LS ((RR)) s s

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

fitted

2007.6.6 inpc07

ss splitting of 99BeBe byby 22 Faddeev using Faddeev using

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

0.50 0.70 0 N Born

kF (fm-1) 1.07 1.20 1.35 -G-matrix

S (MeV fm5)Scheerbaum factor

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

7.8Faddeev

E (keV)

fss2 (cont) 188 194 198 198FSS (cont) 77 3434 5959 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.((11PP1 1 - - 33PP11))

2007.6.6 inpc07

Zero-momentum Zero-momentum Wigner transform Wigner transform 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.

2007.5.30 Colloquium

(3(3NN) potentials by quark-model ) potentials by quark-model GG-matrix i-matrix interaction ( 0nteraction ( 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 ＝－＝－ 5.915.91 　　 MeVMeV EEBB ＝－＝－ 7.00 7.00 MeVMeV

fss2 fss2

consistent with 4He (0+) resonance

(3N): (0s)3

=0.18=0.18-0.22 fm-0.22 fm-2-2

=0.22 fm=0.22 fm-2-2=0.18 fm-2=0.18 fm-2

2007.5.30 Colloquium

B8 (3N) bound-state energies

fss2 (0s)3 =0.18 fm=0.18 fm-2-2 (0s)3 =0.22 fm=0.22 fm-2-2 exp

kF (fm) 1.07 1.20 1.20 1.35

(3N)0+ -2.11-2.11 -1.66 -2.06 -1.57 4H (He) (0+) -2-2

(3N)1+ -1.13-1.13 -0.78 -1.02 -0.66 4H (He) (1+) -1-1

(3N)0+ -8.43-8.43 -5.91 -7.00 -6.45 4He (0+)

EB= --4.64.6 MeV=7.9 MeV

unit: MeV

3H 4H 4He 5He

- 0.13- 0.99

- 2.04

- 1.24

- 2.39 - 3.12

1+ 1+

0+0+ExpExp

T. Nagae T. Nagae et al. et al. Phys. Rev. Lett. Phys. Rev. Lett. 80, 1605 (1998)80, 1605 (1998)

2007.6.6 inpc07

Zero-momentum Zero-momentum Wigner transform Wigner transform 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

(3(3NN) potentials are almost) potentials are almostrepulsive !repulsive !

2007.6.6 inpc07

(ESC04d is not supported)

SummarySummary

Quark-model baryon-baryon interaction can reproducemany experimental data of light s-shell hypernuclei1)1) nn RGM: reasonable RGM: reasonable LSLS splitting (fss2, FSS) splitting (fss2, FSS)2) 2) and and (3(3NN) bound states (fss2)) bound states (fss2)3) very3) very small s splitting insplitting in 9Be excited statesexcited states (FSS)(FSS)4) 4) NN ( (II=1/2 =1/2 11SS00), ), NN ( (II=3/2 =3/2 33SS11) repulsion) repulsion

repulsive s.p. and repulsive s.p. and potentials (fss2, FSS) potentials (fss2, FSS) 5) 5) potential is weakly attractive (fss2, FSS) potential is weakly attractive (fss2, FSS)

B8, B8(3N) potentials are derived from the GG-matrix interactions of fss2 and FSSkF=1.20 - 1.35 fm-1 =0.257 fm-2 () =0.18 fm-2 (3N)

Further analysis ofFurther analysis of BB88, , BB88(3(3NN)) interactionsinteractions by the recentEnergy-independent Energy-independent renormalized RGM kernelRGM kernel for the for the quark-modelquark-model BB88BB88 interactionsinteractions

Future problemsFuture problems(ESC04d is not supported)(ESC04d is not supported)