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