Analysis of (π±,K ＋ ) and (K － ,K ＋ ) spectra in DWIA

HYP06, Friday, Oct. 13, 2006, Mainz, Germany

H. Maekawa, K. Tsubakihara, A. Ohnishi

Division of Physics, Graduate School of Science,

Hokkaido University, Sapporo 060-0810, Japan

1. Introduction and our purpose

2. Model (DWIA with Green function method and Local optimal Fermi averaging)

3. Results(Λ 、 Σ 、 Ξ Quasi-Free spectra)

4. Summary

Do we understand hypernuclear Quasi-Free spectrum ?

Previous DWIA calculation of (K,π), (π,K) and (K,K) reactions Bound state region

Successful expression of the hypernuclear production spectra QF(continuum) region

It is not possible to reproduce QF spectrum well though there are a lot of attempts.(S.W.Hong et al. 1999, M.T.Lopez-Arias 1995)

Auerbach et al., Annals of Physics 148(1983)381.

Traditional Fermi averaging

Recent analysis of hypernuclear Quasi-Free spectrum

Theoretical Cal.

Distorted wave impulse approximation:

T. Harada, Y. Hirabayashi, Nucl. Phys. A 744 (2004) 323.

Semi Classical Distorted Wave model:

M. Kohno, Y. Fujiwara, M. Kawai et al., PTP112 (2004)895.

Cascade model:

Y. Nara, A. Ohnishi, T. Harada and A. Engel, NPA614(1997)433.

T. Harada, Y. Hirabayashi, Nucl. Phys. A 744 (2004) 323

The key in this problem→Fermi averaging with on-shell classical kinematics of t-matrix(Harada and Hirabayashi)

Purpose of our study

・ In optimal Fermi averaging, the t-matrix is averaged under the on-shell kinematics in the free space(no potential effects)

・ We would like to include potential effects with the on-shell condition into the Fermi averaging procedure.

・ To confirm the validity our extension of Fermi averaging with potential effects, we attempt to calculate Λ, Σ and Ξ hypernuclear spectrum on several targets with our modification.

・ In optimal Fermi averaging, the t-matrix is averaged under the on-shell kinematics in the free space(no potential effects)

・ We would like to include potential effects with the on-shell condition into the Fermi averaging procedure.

・ To confirm the validity our extension of Fermi averaging with potential effects, we attempt to calculate Λ, Σ and Ξ hypernuclear spectrum on several targets with our modification.

Λ

ΣRepulsivRepulsivee

-30MeV-30MeV-50MeV-50MeV -50MeV-50MeV

nucleon

nucleon

Model: Green function method by Morimatsu and Yazaki Ref) O.Morimatsu and K.Yazaki, Nucl. Phys. A483(1988)493.S.Tadokoro,Y.Akaishi,H.Kobayashi. Phys.Rev.C51(1995)2656.M.T.Lopez-Arias, Nucl. Phys. A582(1995)440.

)(),';()('Im

1)( ''

* rrrrrr fEGfddES

)()(2

ESd

d

ddE

d

Average

ele

YKnKK

π

σβ

σ

Elementary cross section

Kinematical factor

Meson distorted waves

if NK )()()()( )()*( rrrr

Include the hyperon potential in Green function

iHUTEG

CoreYY

1

KK

MK

MK

K

K

Y

K

pE

pE

p

pp

E

E

cos

1

)()()( )()*( rrr qr iK e

z

zKNN dzzdzz ')'(')'(exp)( ρσρσπr

Strength function

Distortion factor

Double differential cross section

“Green function”

Strength function

Local Optimal Fermi Averaging of t-matrix (LOFAt)

π

Ｎ

Ｋ

Ｙ

)()( 2*2 rmrE NNN p )(2)( 22* rVmmrm NNNN

)()( 2*2 rmrE YYY p )(2)( 22* rVmmrm YYYY

)()( rEErEE YKN

))()(()(

))()(()(),(),,(

4321)4(

4321)4(

rpprpppd

rpprppptstdrt

NN

NN

δρ

δρω

p

pq

We’d like to include the potential effects in the production points.

Local Optimal Fermi Averaging of t-matrix (LOFALocal Optimal Fermi Averaging of t-matrix (LOFAt)t)

Energy conservation equation

“potential”

→Include the potential effects into Fermi-averaging

sΛ

pΛ

dΛ

Λ hypernuclear production spectra on 28Si target

Local Optimal Fermi Averaging27Si+Λ

V0=-28[MeV],VLS=2[MeV],W0=-0.5[MeV],R=r0(A-1)1/3,r0=1.080+0.395A-2/3[fm]

Ref. 　 D. J. Millener,et.al. PRC38(1988)2700

Woods-Saxon parameters

Σ hypernucler production spectrum on 28Si target

-50MeV

0MeV

+90MeV

-10MeV

+10MeV

・ Σ Quasi-Free analysis(Noumi et al., Harada and Hirabayashi, Kohno et al.): Σ -nucleus pot.:Repulsive (Woods-Saxon),V=+30MeV ～ +90MeV

With potential effect

W0Σ= 20MeV

⇒QF spectrum can be reproduced by small repulsive potential.⇒QF spectrum can be reproduced by small repulsive potential.

-30MeV

+50MeV

Σ hypernucler production spectrum on 28Si target

We consider the two type potentials derived from the Σ atomic data.

1.Batty density dependent potential

2.SCL-RMF model by Tsubakihara, Maekawa, Ohnishi(talk in previous session)

Batty-DD

SCL-RMF1

SCL-RMF2

Is the Quasi-Free data reproduced ??Is the Quasi-Free data reproduced ??

Σ － 27Al:UΣ WΣ

Σ hypernucler production spectrum on 28Si target

Batty’s DD

SCL-RMF1

Derived from Σ － X-ray data potential

⇒QF spectrum can be reproduced well using density dependent potentials derived from atomic data

(rather than the case of simple Woods-Saxon type potentials)

⇒QF spectrum can be reproduced well using density dependent potentials derived from atomic data

(rather than the case of simple Woods-Saxon type potentials)

SCL-RMF2

⇒Σ-nucleus potential is…

Structure of Attractive pocket and Repulsive coreAttractive pocket and Repulsive core is favored.

⇒Σ-nucleus potential is…

Structure of Attractive pocket and Repulsive coreAttractive pocket and Repulsive core is favored.

P. Khaustov et al., Phys. Rev. C61(2000) 054603-1.

Reasonable agreement between the data and theory is achieved by assuming a Ξ-nucleus potential well depth V0 of about 14 MeV within the Woods-Saxon prescription (DWIA calculation).

12C(K － ,K+)

PK=1.80GeV/c

Study of Ξ-nucleus potential by (K － ,K+) reaction

Theoretical curve:

DWIADWIA(Tadokoro et al,PRC51(1995)2656.)

INCINC(Y. Nara et al.,NPA614(1997)433.)

Ξ － hypernuclear production spectra on several targets

Woods-Saxon Potential: VWoods-Saxon Potential: V00ΞΞ=-=-

15MeV15MeV

Exp.Data:E17Exp.Data:E1766

Calculation in Green function method

Q.F.

Q.F.

Q.F.

Q.F.

Ξ － hypernuclear production spectra on 12C target

sΞ

pΞ

11B+Ξ-

Quasi-Free

Det. Res. :2MeVWoods-Saxon Potential

V0Ξ=-15MeV

W0Ξ= 1MeV

p3/2-1 s1/2

-1

s1/2-

1

p3/2-1

Ξ － hypernuclear production spectra on Al target

Det. Res. :2MeV

sΞ

pΞ

dΞ

12(deg.)

6(deg.)0(deg.)

Woods-Saxon Potential

V0Ξ=-15MeV

W0Ξ= 1MeV

26Mg+Ξ －

Ξ － hypernuclear production spectra on Ni target

Det. Res. :2MeV

Woods-Saxon Potential

V0Ξ=-15MeV

W0Ξ= 1MeV

57Co+Ξ －

Quasi-Free

Summary

DWIA with Quantum mechanical treatment of QF region(Green function method) Fermi averaging (In ordinary DWIA) On-shell classical kinematics (Optimal Fermi average, SCDW,INC) Potential effects at reaction points (Local optimal Fermi average; Ours)

are found to explain various hyperon production QF spectrum.

We propose the “Local optimal Fermi averaging” of t-matrix To include the potential effects into optimal Fermi averaging

We calculate the hypernucler Quasi-Free spectrum. Λ:With V0 ～－ 30 ＭｅＶ

Both of QF and Bound state spectrum are reproduced very well We confirm the validity of our extension of F.A.

Σ:QF spectra is compatible with atomic data Batty’s DD pot.,SCL-RMF→works well

Ξ: With V0 ～－ 15 ＭｅＶ , QF spectra on various targets are reproduced.

Ξ － hypernuclear production spectra on 12C target

Quasi-Free

11B+Ξ-Exp.Data:E17Exp.Data:E1766

p3/2-1

s1/2-1

Woods-Saxon Potential

V0Ξ=-15MeV

W0Ξ= 1MeV

Σ hypernucler production spectrum on 28Si target

Batty’s DD

Σ hypernucler production spectrum on 28Si target

-50MeV 0MeV

+90MeV

Batty’s DD-10MeV

+10MeV

・ Σ atomic data analysis(Batty et al., Mares et al.):Σ-nucleus pot.:Repulsive core + attractive pocket

・ Σ Quasi-Free analysis(Noumi et al., Harada and Hirabayashi, Kohno et al.):Σ-nucleus pot.:Repulsive (Woods-Saxon),V=+30MeV ～ +90MeV

Optimal Fermi averaging

W0Σ=

20MeV

Recent analysis of hypernuclear Quasi-Free spectrum

Experimental side:

Theoretical side:

Noumi et al.(E438)

Harada and Hirabayashi(DWIA) Kohno et al.(SCDW)

T. Harada, Y. Hirabayashi, Nucl. Phys. A 744 (2004) 323M. Kohno, Y. Fujiwara, M. Kawai et al., PTP112 (2004)895

Σnucleus potential

～ 90MeV

The key in this problem→Fermi averaging of t-matrix

Several Fermi averagings for t-matrix in DWIA

Previous procedure

Auerbach et al.

Annals of Physics 148(1983)381.

Recent extension of Fermi averaging(Optimal Fermi averaging)

T. Harada, Y. Hirabayashi, Nucl. Phys. A 744 (2004) 323

Do we understand hypernuclear Quasi-Free spectrum ?

Previous DWIA calculation of (K,π), (π,K) and (K,K) reactions Bound state region

Successful expression of the hypernuclear production spectra QF(continuum) region

It is not possible to reproduce QF well though there are a lot of attempts.

S.W.Hong et al. 1999 M.T.Lopez-Arias 1995

Λ hypernuclear production spectra on 51V

sΛ

pΛ

dΛ

fΛ

Ξ － hypernuclear production spectra(Bound region)

Det. Res. :2MeV

Woods-Saxon potential

V0Ξ=-15MeV

26Mg+Ξ －

0.5MeV1MeV

3MeV5MeV

sΞ

pΞ

dΞ

Ξ － hypernuclear production spectra(Bound region)

Det. Res. :2MeV

Woods-Saxon potential

V0Ξ=-15MeV59Co+Ξ －

0.5MeV1MeV

3MeV5MeV

sΞ

pΞ

dΞ

fΞ