Transcript Document
Application of coupled-channel
Complex Scaling Method to the KbarN-πY system
A. Doté (KEK Theory center / IPNS/ J-PARC branch)
T. Inoue (Nihon univ.)
T. Myo (Osaka Inst. Tech. univ.)
1.
Introduction
2.
coupled-channel Complex Scaling Method
•
•
For resonance states
For scattering problem
Set up for the calculation of KbarN-πY system
3.
•
•
“KSW-type potential”
Kinematics / Non-rela. Approximation A. Dote, T. Inoue, T. Myo,
Nucl. Phys. A912, 66-101 (2013)
4.
Results
•
•
5.
I=0 channel: Scattering amplitude / Resonance pole
I=1 channel: Scattering amplitude
Summary and future plans
International workshop “Inelastic Reactions in Light Nuclei”,
09.Oct.’13 @ IIAS, The Hebrew univ., Jerusalem, Israel
1. Introduction
Kaonic nuclei = Nuclear system with Anti-kaon “Kbar”
Attractive KbarN interaction!
Excited hyperon Λ(1405) = a quasi-bound state of K- and proton
Difficult to explain by naive 3-quark model
Consistent with “repulsive nature” indicated by
KbarN scattering length and 1s level shift of kaonic hydron atom
KbarN threshold = 1435MeV
T. Hyodo and D. Jido,
Prog. Part. Nucl. Phys. 67, 55 (2012)
u
ud
s
u
d
ubar
s
Kaonic nuclei = Nuclear system with Anti-kaon “Kbar”
Attractive KbarN interaction!
Excited hyperon Λ(1405) = a quasi-bound state of K- and proton
Difficult to explain by naive 3-quark model
Consistent with “repulsive nature” indicated by
KbarN scattering length and 1s level shift of kaonic hydron atom
Anti-kaon can be deeply bound and/or form dense state?
Antisymmetrized Molecular Dynamics with a phenomenological KbarN potential
Anti-kaon bound in light nuclei with ~ 100MeV binding energy
Dense matter (average density = 2～4 ρ0)
Relativistic Mean Field applied to medium to heavy nuclei with anti-kaons
• AMD study
A. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, PLB 590 (2004) 51; PRC 70 (2004) 044313.
• RMF study
D. Gazda, E. Friedman, A. Gal, J. Mares, PRC 76 (2007) 055204; PRC 77 (2008) 045206;
A. Cieply, E. Friedman, A. Gal, D. Gazda, J. Mares, PRC 84 (2011) 045206.
T. Muto, T. Maruyama, T. Tatsumi, PRC 79 (2009) 035207.
Kaonic nuclei = Nuclear system with Anti-kaon “Kbar”
Attractive KbarN interaction!
Excited hyperon Λ(1405) = a quasi-bound state of K- and proton
Difficult to explain by naive 3-quark model
Consistent with “repulsive nature” indicated by
KbarN scattering length and 1s level shift of kaonic hydron atom
Anti-kaon can be deeply bound and/or form dense state?
Antisymmetrized Molecular Dynamics with a phenomenological KbarN potential
Anti-kaon bound in light nuclei with ~ 100MeV binding energy
Dense matter (average density = 2～4 ρ0)
Relativistic Mean Field applied to medium to heavy nuclei with anti-kaons
Prototype of kaonic nuclei = “K-pp”
Variational calculation with Gaussian base
Av18 NN potential
Chiral SU(3)-based KbarN potential
A. D., T. Hyodo, W. Weise,
PRC79, 014003 (2009)
B. E. ~ 20MeV, Γ = 40 ~70MeV
pp distance ~ 2.1 fm
Theoretical studies of K-pp
“K-pp” = Prototype of Kbar nuclei
(KbarNN, Jp=1/2-, T=1/2)
•Doté, Hyodo, Weise
PRC79, 014003(2009)
Variational with a chiral SU(3)-based KbarN potential
•Akaishi, Yamazaki
ATMS
PRC76, 045201(2007)
with a phenomenological
•Ikeda, Sato
KbarN potential
PRC76, 035203(2007)
Faddeev with a chiral SU(3)-derived KbarN potential
•Shevchenko, Gal, Mares
PRC76, 044004(2007)
Faddeev with a phenomenological KbarN potential
•Barnea, Gal, Liverts
PLB712, 132(2012)
Hyperspherical harmonics with a chiral SU(3)-based KbarN potential
•Wycech, Green
Variational with a phenomenological
•Arai, Yasui, Oka/ Uchino, Hyodo, Oka
Λ* nuclei model
•Nishikawa, Kondo
PRC79, 014001(2009)
KbarN
potential (with p-wave)
PTP119, 103(2008)
/PTPS 186, 240(2010)
PRC77, 055202(2008)
Skyrme model
All studies predict that K-pp can be bound!
Theoretical studies of K-pp
“K-pp” = Prototype of Kbar nuclei
(KbarNN, Jp=1/2-, T=1/2)
•Doté, Hyodo, Weise
PRC79, 014003(2009)
Variational with a chiral SU(3)-based KbarN potential
•Akaishi, Yamazaki
ATMS
PRC76, 045201(2007)
with a phenomenological
•Ikeda, Sato
KbarN potential
PRC76, 035203(2007)
Faddeev with a chiral SU(3)-derived KbarN potential
•Shevchenko, Gal, Mares
PRC76, 044004(2007)
Faddeev with a phenomenological KbarN potential
•Barnea, Gal, Liverts
PLB712, 132(2012)
Hyperspherical harmonics with a chiral SU(3)-based KbarN potential
•Wycech, Green
Variational with a phenomenological
•Arai, Yasui, Oka/ Uchino, Hyodo, Oka
Λ* nuclei model
•Nishikawa, Kondo
PRC79, 014001(2009)
KbarN
potential (with p-wave)
PTP119, 103(2008)
/PTPS 186, 240(2010)
PRC77, 055202(2008)
Skyrme model
All studies predict that K-pp can be bound!
Typical results of theoretical studies of K-pp
Width (KbarNN→πYN) [MeV]
0
20
40
60
80
100
120
140
0
Barnea, Gal, Liverts [5]
(HH, Chiral SU(3))
Doté, Hyodo, Weise [1]
(Variational, Chiral SU(3))
-20
-40
Akaishi, Yamazaki [2]
(Variational, Phenomenological)
Shevchenko, Gal, Mares [3]
(Faddeev, Phenomenological)
-60
-80
Ikeda, Sato [4]
(Faddeev, Chiral SU(3))
-100 MeV
-100
-120
Exp. : FINUDA [6]
if K-pp bound state.
-140
[1] PRC79, 014003 (2009)
[2] PRC76, 045201 (2007)
[3] PRC76, 044004 (2007)
[4] PRC76, 035203 (2007)
[5] PLB94, 712 (2012)
[6] PRL94, 212303 (2005)
[7] PRL104, 132502 (2010)
Exp. : DISTO [7]
if K-pp bound state.
Using S-wave KbarN potential
constrained by experimental data.
… KbarN scattering data,
Kaonic hydrogen atom data,
“Λ(1405)” etc.
Typical results of theoretical studies of K-pp
Width (KbarNN→πYN) [MeV]
0 threshold
20
40
Kbar+N+N
60
80
100
120
140
0
Barnea, Gal, Liverts [5]
(HH, Chiral SU(3))
Doté, Hyodo, Weise [1]
(Variational, Chiral SU(3))
-20
-40
Akaishi, Yamazaki [2]
(Variational, Phenomenological)
Shevchenko, Gal, Mares [3]
(Faddeev, Phenomenological)
-60
-80
Ikeda, Sato [4]
(Faddeev, Chiral SU(3))
-100
π+Σ+N-120threshold
Exp. : DISTO [7]
if K-pp bound state.
Exp. : FINUDA [6]
if K-pp bound state.
From theoretical viewpoint,
Using S-wave K N potential
constrained by experimental data.
bar
… K N scattering
data,
K pp exists between K -N-N and π-Σ-N
thresholds!
Kaonic hydrogen atom data,
-140
[1] PRC79, 014003 (2009)
[2] PRC76, 045201 (2007)
[3] PRC76, 044004 (2007)
[4] PRC76, 035203 (2007)
bar
[5] PLB94, 712 (2012)
[6] PRL94, 212303 (2005)
[7] PRL104, 132502 (2010)
bar
“Λ(1405)” etc.
Key points to study kaonic nuclei
Kbar + N + N
Coupling of KbarN and πY
“Kbar N N”
Bound below KbarN threshold,
but a resonant state above πY threshold.
π +Σ+N
Their nature?
“coupled-channel Complex Scaling Method”
1. Consider a coupled-channel problem
2. Treat resonant states adequately.
Similar to the bound-state calculation
3. Obtain the wave function to help the analysis of the state.
4. Confirmed that CSM works well on many-body systems.
KbarN (-πY) interaction??
“Chiral SU(3)-based potential”
… Anti-kaon = Nambu-Goldstone boson
2. Complex Scaling Method
Λ(1405) = a building block of kaonic nuclei
KProton
• For Resonance states
• For Scattering problem
Λ(1405) with c.c. Complex Scaling Method
Kbar + N
1435
Λ(1405)
π +Σ
1332
[MeV]
π
(Jπ=0-, T=1)
Kbar
(Jπ=0-, T=1/2)
L=0
L=0
N
(Jπ=1/2+, T=1/2)
Σ
(Jπ=1/2+, T=1)
KbarN-πΣ coupled system with s-wave and isospin-0 state
Complex Scaling Method for Resonance
Complex rotation of coordinate (Complex scaling)
U :
i
r re ,
k ke
H U H U
1
,
i
E H H
U
By Complex scaling, …
• Resonant state
R
exp ik R r exp i i r
Divergent
κ, γ : real, >0
U R
exp i i
re i
exp r sin cos
0
exp ir cos
tan
Damping under some condition
sin
Complex Scaling Method for Resonance
Complex rotation of coordinate (Complex scaling)
U :
i
r re ,
k ke
H U H U
1
,
i
E H H
U
By Complex scaling,
Resonance wave function: divergent function ⇒ damping function
Boundary condition is the same as that for a bound state.
Resonance energy (pole position) doesn’t change.
ABC theorem
“The energy of bound and resonant states is independent of scaling angle θ.”
† J. Aguilar and J. M. Combes, Commun. Math. Phys. 22 (1971),269.
E. Balslev and J. M. Combes, Commun. Math. Phys. 22 (1971),280
Diagonalize Hθ with Gaussian base,
we can obtain resonant states, in the same way as bound states!
Complex Scaling Method for Resonance
Test calculation with a phenomenological potential
Akaishi-Yamazaki potential†
• Local Gaussian form
• Energy independent
V
AY ,I 0
(I 0)
V
KbarN
(I 0)
V
436
412
exp r
KbarN
[MeV]
2
0.66 fm
πΣ
412
M eV
0
B. E. (KbarN) = 28.2 MeV
Γ = 40.0 MeV
… Nominal position of Λ(1405)
†Y.
πΣ
Akaishi and T. Yamazaki, PRC65, 044005 (2002)
=30 deg.
pS
continuum
KbarN continuum
2. Complex Scaling Method
Calculation of KbarN scattering amplitude
• For Resonance states
• For Scattering problem
Calc. of scattering amplitude with CSM
With help of the CSM, all problems for bound, resonant
and scattering states can be treated with Gaussian base!
Calc. of scattering amplitude with CSM
A. T. Kruppa, R. Suzuki and K. Katō, PRC 75, 044602 (2007)
1. Separate incoming wave
j l kr
+
• Unknown
• Non square-integrable
hl
x exp i kr
lπ
2
2. Complex scaling r → reiθ
square-integrable for 0 < θ < π
Cauchy theorem
dz j kz V z l , k
SC
z
0
Expanding with square-integrable basis function
(ex: Gaussian basis)
iθ
r escattering
3. Calculate
amplitude with help of Cauchy theorem
0
r
Born term is OK!
Scattered part
is unknown.
Scattered wave function along reiθ
is known!
By Cauchy theorem, we can obtain as
Calc. of scattering amplitude with CSM
A. T. Kruppa, R. Suzuki and K. Katō, PRC 75, 044602 (2007)
Scattering problem can be solved
as bound-state problem by matrix calculation!
Equation to be solved:
can be expanded with Gaussian basis
square-integrable for 0 < θ < π
Linear equation to be solved with matrix calculation!
3. Set up
for the calculation of
KbarN-πY system
• “KSW-type potential” … Chiral SU(3)-based
• Kinematics
• Non-rela. approximation of KSW-type potential
“KSW-type potential” … chiral-SU(3) based
Effective Chiral Lagrangian
• Delta-function type (→ Yukawa / separable type)
• Up to order q2
Pseudopotential
N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)
KbarN
“KSW-type potential”
(I 0)
(I )
(I )
V ij
r
C ij
8f
2
i
j
M iM
s i
j
(I )
g ij
r
j
• Local Gaussian form in r-space
(I )
g ij
r
1
p
3/2
a
exp r a i j
(I )
(I )3
ij
2
C ij
3
πΣ
3
2
4
a(I)ij : range parameter [fm]
→ Easy to handle in many-body calculation with Gaussian base
• Weinberg-Tomozawa term
→ Relative strength between channels determined by the SU(3) algebra
• Energy dependence
← Chiral SU(3) theory
Constrained by KbarN scattering length (Martin’s value)
aKN(I=0) = -1.70+i0.67fm, aKN(I=1) = 0.37+i0.60fm
A.D.Martin, NPB179, 33(1979)
Kinematics
1. Non-relativistic
2
p
mc M c
2c
2
H
NR
cK
bar
N ,p Y
c
E Mc
c
c
V
K SW
2
kc
2M c
2
mc
M cmc
2
kc
2mc
Reduced mass
M c mc
k c 2 c E M c m c
1/ 2
2. Semi-relativistic
H
SR
cK
bar
N ,p Y
2
2
m
p
c
V
K SW
2
2
Mc p c
E c c
Mc
2
2
kc
2
mc
2
2
c
c
c c
c c
Reduced energy
2
2
k c c m c
1/ 2
2
kc
Non-rela. approximation of KSW potential
- Original
(I )
V
(I )
ij
r
C ij
2
8 fp
• Normalized Gaussian
(I )
i
j
M iM
j
s i
j
g
(I )
ij
g ij
r
→ (NRv1) Non-rela. approx. version 1
i
r
1
p
3/2
d
(I ) 3
ij
exp r d ij
(I )
2
• Range parameter
for coupling potential
(I )
d ij
d ii d jj
(I )
(I )
2
(I )
V
(I )
ij
r
C ij
2
8 fp
i
j
M iM
1
j
j
s
g
(I )
ij
r
Comparison of the flux factor
for differential cross section
between non-rela. and rela.
→ (NRv2) Non-rela. approx. version 2
mi i , M i i 1
@ non-rela. limit (small p2)
(I )
V
(I )
ij
r
C ij
2
8 fp
i
j
1
m m j
(I )
g ij
r
Kinematics
Semi-rela.
Potential
- Original
(I )
V
(I )
ij
r
C ij
i j
2
8 fp
M iM
j
s i
j
(I )
g ij
r
(NRv1) Non-rela. approx. ver. 1
(I )
Non-rela.
V
(I )
ij
r
C ij
2
8 fp
i
j
1
M iM
E T ot
j
j
(I )
g ij
r
(NRv2) Non-rela. approx. ver. 2
(I )
V
(I )
ij
r
C ij
2
8 fp
i
j
1
m m j
(I )
g ij
r
4. Result
Using Chiral SU(3) potential “KSW-type”
… r-space, Gaussian form, Energy-dependent
• I=0 channel
• I=0 KbarN scattering length
Range parameters of KSW-type potential
• Scattering amplitude
• Resonance pole … wave function and size
I=0 KbarN scattering length
Range parameters of KSW-type potential
V
(I 0)
ij
r V
(I 0)
ij
s
1
p
3/2
Range parameters:
fπ=110 MeV
d ij
3
exp r d ij
2
Pion decay constant “fπ”
as a parameter
fπ = 90 ~ 120 MeV
d K bar N , K bar N
d ij
Kinematics
KSW-type pot.
Range
parameters [fm]
dKN,KN
dπΣ,πΣ
KbarN scatt.
length [fm]
Re
Re aI=0KN
Im aI=0KN
Im
d K ba r N ,p S
d p S ,p S
d K bar N ,p S d K bar N , K bar N d p S ,p S
Non-rela.
NR-v1
NR-v2
Semi-rela.
SR-A
SR-B
0.44
0.61
0.44
0.64
0.50
0.71
0.37
0.35
-1.701
0.681
-1.700
0.681
-1.700
0.681
-1.696
0.681
Found sets of range parameters (dKN,KN, dπΣ,πΣ)
to reproduce the Martin’s value.
Two sets found in semi-rela. case. (SR-A, SR-B)
2
Martin
-1.70
0.68
Scattering amplitude
… Non-rela. / KSW-type NRv1
(I 0)
V
(I 0)
ij
r
C ij
2
8 fp
i
j
πΣ
1
s
M iM
j
j
g ij r
KbarN
Re
fπ=110 MeV
πΣ
KbarN
Im
KbarN → KbarN
πΣ → πΣ
Resonance structure
at 1413 MeV
Resonance structure
at 1405 MeV
Resonance structure appears in KbarN and πΣ channels.
Scattering amplitude
… Non-rela. / KSW-type NRv2
(I 0)
V
(I 0)
ij
r
C ij
2
8 fp
i
j
πΣ
1
m m j
g ij r
KbarN
Re
fπ=110 MeV
πΣ
KbarN
Im
KbarN → KbarN
πΣ → πΣ
Resonance structure
at 1416 MeV
Essentially same as NRv1 case.
Resonance structure
at 1408 MeV
Scattering amplitude
… Semi-rela. / KSW-type SR-A
(I 0)
V
(I 0)
ij
r
C ij
2
8 fp
i j
πΣ
M iM
j
s i
j
g ij r
KbarN
Re
fπ=110 MeV
πΣ
KbarN
Im
KbarN → KbarN
πΣ → πΣ
Resonance structure
at 1410 MeV
Resonance structure
at 1398 MeV
Qualitatively similar to non-rela. amplitudes.
Scattering amplitude
… Semi-rela. / KSW-type SR-B
(I 0)
(I 0)
V ij
r
C ij
2
8 fp
i
j
πΣ
(Another set of SR)
M iM
j
s i
j
g ij r
KbarN
Re
fπ=110 MeV
Im
KbarN → KbarN
πΣ
KbarN
???
πΣ → πΣ
Resonance structure
at 1419 MeV
Resonance structure
at 1421 MeV
Very different behavior of πΣ amplitude from other cases.
Pole position of the resonance
90
100
110
120
NRv1
M
1419.8
1417.3
1416.6
1416.9
NRv2
M
1419.9
1418.0
1417.8
1418.3
-Γ/2
-26.0
-23.1
-19.5
-16.7
-Γ/2
-23.1
-19.8
-16.6
-14.0
SR-A
M
1423.6
1421.5
1419.5
1417.5
SR-B
M
1419.0
1419.6
1420.0
1418.9
-Γ/2
-26.4
-26.3
-25.0
-22.9
-Γ/2
-14.4
-13.2
-12.8
-11.7
M [MeV]
[MeV]
- -ΓΓ/ /22[MeV]
fpi
fπ=90 - 120MeV
120
120
120
90
90
120
90
90
Complex energy plane
Pole position of the resonance
90
100
110
120
NRv1
M
1419.8
1417.3
1416.6
1416.9
NRv2
M
1419.9
1418.0
1417.8
1418.3
-Γ/2
-26.0
-23.1
-19.5
-16.7
-Γ/2
-23.1
-19.8
-16.6
-14.0
SR-A
M
1423.6
1421.5
1419.5
1417.5
-Γ/2
-26.4
-26.3
-25.0
-22.9
SR-B
M
1419.0
1419.6
1420.0
1418.9
-Γ/2
-14.4
-13.2
-12.8
-11.7
M [MeV]
[MeV]
- -ΓΓ/ /22[MeV]
fpi
fπ=90 - 120MeV
120
Non-rela.
120
120
90
90
120
90
(M, Γ/2)
90
= (1418.2 ± 1.6, 21.4 ± 4.7) … NRv1
(1418.9 ± 1.1, 18.6 ± 4.6) … NRv2
Pole position of the resonance
90
100
110
120
NRv1
M
1419.8
1417.3
1416.6
1416.9
NRv2
M
1419.9
1418.0
1417.8
1418.3
-Γ/2
-26.0
-23.1
-19.5
-16.7
-Γ/2
-23.1
-19.8
-16.6
-14.0
SR-A
M
1423.6
1421.5
1419.5
1417.5
SR-B
M
1419.0
1419.6
1420.0
1418.9
-Γ/2
-26.4
-26.3
-25.0
-22.9
-Γ/2
-14.4
-13.2
-12.8
-11.7
M [MeV]
[MeV]
- -ΓΓ/ /22[MeV]
fpi
fπ=90 - 120MeV
120
120
120
90
(M, Γ/2)
90= (1420.5 ± 3, 24.5 ± 2) … SR-A
120
Semi-rela.
90
90
Pole position of the resonance
90
100
110
120
NRv1
M
1419.8
1417.3
1416.6
1416.9
NRv2
M
1419.9
1418.0
1417.8
1418.3
-Γ/2
-26.0
-23.1
-19.5
-16.7
-Γ/2
-23.1
-19.8
-16.6
-14.0
SR-A
M
1423.6
1421.5
1419.5
1417.5
-Γ/2
-26.4
-26.3
-25.0
-22.9
SR-B
M
1419.0
1419.6
1420.0
1418.9
M [MeV]
[MeV]
- -ΓΓ/ /22[MeV]
fpi
fπ=90 - 120MeV
Different behavior
from other cases
120
120
120
90
Semi-rela.
(Another set)
120
90
90
(M, Γ/2) 90
= (1419.5 ± 0.5, 13.0 ± 1.4) … SR-B
-Γ/2
-14.4
-13.2
-12.8
-11.7
“Wave function” of the resonance pole
Non-rela. (NRv2)
fπ=110 MeV
θ=30°
Semi rela. (SR-A)
πΣ component also localized
due to Complex scaling.
? Somehow small.
Mean distance between meson and baryon
2
r MB
fm
B
m
NR: ～ 1.3 - 0.3 i [fm]
SR: ～ 1.2 - 0.5 i [fm]
cf) 1.9 fm @ M = 1423 MeV†
(B = 12 MeV)
- Difference of binding energy?
- Different definition of pole?
Gamow state or bound state
† A. D., T. Hyodo, W. Weise,
PRC79, 014003 (2009)
4. Result
Using Chiral SU(3) potential (KSW-type)
… r-space, Gaussian form, Energy-dependent
• I=1 channel
• Range parameters of the KSW-type potential
• Scattering amplitude
I=1 channel … KbarN - πΣ - πΛ
Potential … Weinberg-Tomozawa term only
( I 1)
V
( I 1)
ij
r
C ij
2
8 fp
i
j flux factor
I 1
I 1
a K bar N f K bar N
1
p
3/2
d ij
s K bar N thr .
3
exp r d ij
2
Data : aI=1KbarN = 0.37 + i 0.60 fm
(A. D. Martin)†
Range parameters:
d K bar N , K bar N
d ij
d K bar N ,p S
d p S ,p S
d K bar N ,p
d p S ,p
d p ,p
† A. D. Martin, Nucl. Phys. B 179, 33 (1981)
I=1 channel … KbarN - πΣ - πΛ
Potential … Weinberg-Tomozawa term only
( I 1)
V
( I 1)
ij
r
C ij
2
8 fp
i
j flux factor
I 1
I 1
a K bar N f K bar N
1
p
3/2
d ij
s K bar N thr .
3
exp r d ij
2
Data : aI=1KbarN = 0.37 + i 0.60 fm
(A. D. Martin)†
Range parameters:
d K bar N , K bar N
d ij
d K bar N ,p S
d p S ,p S
d K bar N ,p
d p S ,p
d p ,p
×
×
d K bar N ,p S d K bar N , K bar N d p S ,p S
2
SU(3) C.G. coefficients
for I=1 channel
C
( I 1)
ij
1
1
2
3 2
0
0
I=1 channel … KbarN - πΣ - πΛ
Potential … Weinberg-Tomozawa term only
( I 1)
V
( I 1)
ij
r
C ij
2
8 fp
i
j flux factor
I 1
I 1
a K bar N f K bar N
1
p
3/2
d ij
s K bar N thr .
3
exp r d ij
2
Data : aI=1KbarN = 0.37 + i 0.60 fm
(A. D. Martin)†
Range parameters:
d K bar N , K ba r N
d ij
d K b a r N ,p S
d p S ,p S
d K b a r N ,p
d p S ,p
d p ,p
×
×
d K bar N ,p S d K bar N , K bar N d p S ,p S
2
SU(3) C.G. coefficients
for I=1 channel
C
( I 1)
ij
1
1
2
3 2
0
0
I=1 channel … KbarN - πΣ - πΛ
Potential … Weinberg-Tomozawa term only
( I 1)
V
( I 1)
ij
r
C ij
2
8 fp
i
j flux factor
I 1
I 1
a K bar N f K bar N
1
p
3/2
d ij
s K bar N thr .
3
exp r d ij
2
Data : aI=1KbarN = 0.37 + i 0.60 fm
(A. D. Martin)†
“Isospin symmetric choice”
Range parameters:
I 0
d bar bar
K
N ,K
N
d ij
I 0
d K ba r N ,p S
I 0
d p S ,p S
d K ba r N ,p
d p S ,p
d p ,p
×
×
d K bar N ,p S d K bar N , K bar N d p S ,p S
2
Ref.) chiral unitary model
SU(3) C.G. coefficients
for I=1 channel
C
( I 1)
ij
1
1
2
3 2
0
0
I=1 channel … KbarN - πΣ - πΛ
• “Iso-symmetric choice” : {dKN,KN, dπΣ,πΣ} fixed to I=0 channel ones
• Search dKN,πΣ to reproduce Re or Im aI=1KbarN of Matin’s value.
A. D. Martin : aI=1KbarN = 0.37 + i 0.60 fm
(I=0 one)
(Search)
(fit)
(fit)
(fit)
When Re aI=1KbarN is reproduced, Im aI=1KbarN deviates largely from Martin’s value.
Difficult to reproduce Re aI=1KbarN Martin’s value within our model.
(fit)
I=1 channel … KbarN - πΣ - πΛ
fπ=110 MeV
Search dKN,πΣ to reproduce Im aI=1KbarN of Matin’s value.
NRv2 (c): aI=1KbarN = 0.657 + i 0.599 fm
KbarN
πΣ
πΛ
πΣ
πΛ
SR-A (c): aI=1KbarN = 0.659 + i 0.600 fm
KbarN
I=1 channel … KbarN - πΣ - πΛ
• Only dKN,KN fixed to I=0 channel ones
In a study with separable potential, the cutoff parameter for KbarN is not so different
between I=0 and 1 channels.†
• Search {dπΣ,πΣ, dKN,πΣ} to reproduce Re and Im aI=1KbarN of Matin’s value.
† Y. Ikeda and T. Sato, PRC 76, 035203 (2007)
(I=0 one)
(Search)
(fit)
(fit)
(fit)
(fit)
When the only dKN,KN is fixed iso-symmetrically, we can find a set of {dπΣ,πΣ, dKN,πΣ}
to reproduce simultaneously Re and Im of Martin’s value
I=1 channel … KbarN - πΣ - πΛ
fπ=110 MeV
Search {dπΣ,πΣ, dKN,πΣ} to reproduce Re and Im aI=1KbarN of Matin’s value.
NRv2 (a): aI=1KbarN = 0.376 + i 0.606 fm
πΣ
KbarN
πΛ
A narrow resonance exists
at a few MeV below πΣ threshold
SR-A (a): aI=1KbarN = 0.375 + i 0.605 fm
KbarN
πΣ
πΣ repulsive ???
πΛ
5. Summary
and
Future plan
5. Summary
KbarN-πY system is essential for the study of Kbar nuclear system
which is expected to be an exotic nuclear system with strangeness.
Scattering and resonant states of KbarN-πY system is studied with
a coupled-channel Complex Scaling Method using a chiral SU(3) potential
• A Chiral SU(3) potential “KSW-type” … r-space, Gaussian form, energy dependence
• Calculated scattering amplitude with help of CSM
Scattering states as well as resonant and bound states are treated with Gaussian base.
Non-rela. / Semi-rela. kinematics and two types of non-rela. approximation
of KSW-type potential are tried.
Determined by the KbarN scattering length obtained by Martin’s analysis of old data.
fπ dependence (fπ = 90 ～ 120MeV)
Found two sets of range parameters in SR case.
5. Summary and future plans
I=0 channel (KbarN-πΣ)
• Pole position
(M, Γ/2)
NRv1 : (1418.2 ± 1.6, 21.4 ± 4.7)
NRv2 : (1418.9 ± 1.1, 18.6 ± 4.6)
SR-A : (1420.5 ± 0.5, 24.5 ± 2)
SR-B : (1419 ± 1, 13.0 ± 2)
• “Size” of the I=0 pole state
NR: ～ 1.3 – 0.3i fm
SR: ～ 1.2 – 0.5i fm
Another self-consistent solution
NR: (~1360, 40~90), SR: (1350~1390, 30~100) … Lower pole of double pole?
I=1 channel (KbarN-πΣ-πΛ)
• Difficult to reproduce Re aI=1KbarN of Martin’s value within our model,
in case of “iso-symmetric choice” of range parameter.
Future plans
Three-body system (KbarNN-πYN); Updated data of K-p scattering length by SHIDDARTA
Thank you very much!
A. D. is thankful to Prof. Katō for his advice on
the scattering-amplitude calculation in CSM,
and to Dr. Hyodo for useful discussion.
Backup slides
Rough estimation of charge radius
Distance
K-
p
CM
rK C M
rp C M
M
D istance
N
mK M
mK
mK M
2
N
D istance
rc
r p C M rK C M
2
N
Pole position [MeV]
<rc2> [fm2]
NRv2
1417.8 – i 16.6
-0.49 + i 0.24
SR-A
1419.5 – i 25.0
-0.37 + i 0.37
1426.11 – i 16.54
-0.131 + i 0.303
(Sekihara et al)†
2
† Calculated from electric form factor with chiral unitary model
T. Sekihara, T. Hyodo and D. Jido, PLB 669, 133 (2008)
Charge radius
Charge radius derived from electric form factor
T. Sekihara, T. Hyodo and D. Jido, PLB 669, 133 (2008)
Rough estimation
Chiral (HW-HNJH): B～ 12 MeV, Distance = 1.86 fm → <r2c> = -1.07 fm2
Kbar nuclei (Nuclei with anti-koan) = Exotic system !?
I=0 KbarN potential … very attractive
Deeply bound (Total B.E. ～100MeV)
Highly dense state formed in a nucleus
Interesting structures that we have never seen in normal nuclei…
3He
K- ppn
K- ppp
Antisymmetrized Molecular Dynamics
method with
a phenomenological KbarN potential
3 x 3 fm2
0.0
0.15
0.0
2.0 0.0
A. Dote, H. Horiuchi, Y. Akaishi and
T. Yamazaki, PRC70, 044313 (2004)
2.0
[fm-3]
Relate to various interesting physics such as …
Restoration of chiral symmetry in dense matter
Interesting structure
Neutron star
Excited hyperon Λ（１４０５）
s
u
d
u
ud
T. Hyodo and D. Jido,
Prog. Part. Nucl. Phys. 67, 55 (2012)
ubar
s
I=0 Proton-K- bound state
with 30MeV binding energy
Not 3 quark state,
← can’t be explained with
a simple quark model…
But rather a molecular state
Today, K-pp has been focused in theor. and exp. studies!
As the first trial of our study with ccCSM, consider Λ(1405).
KProton
=
Λ(1405)
Building block of
Kaonic nuclei
3HeK-,
K
P
P
Prototype of
Kaonic nuclei
pppK-,
4HeK-, pppnK-,
…
8BeK-,
…
Complicated
nuclear system
with K-
Asymptotic behavior of the wave function
before/after complex scaling
• Bound state
B
exp B r
Damping
κB : real, >0
exp B re
exp under
r cos some
exp i condition
r sin
Damping
U B
i
B
B
0
p 2
• Resonant state
R
exp ik R r
Diverging
exp i i r
κ, γ : real, >0
exp i i
U R
re i
exp i under
i r cos
i sin
Damping
some
condition
exp r sin cos exp ir cos sin
0
tan
• Continuum state
k
ex p ikr
Oscillating
k : real, >0
U k
exp i ke
i
Oscillating
exp ikr
re
i
Complex Scaling Method for Resonance
Great success in nuclear physics … in particular, unstable nuclei.
Well applied to usual calculation of bound states… diagonalize with Gaussian base
(Gaussian Expansion Method, Correlated Gaussian)
Tiny modification of matrix elements
2
T e
2 i
d
2
2 m dr
2
,
V
V 0 exp r a e
i
2
Possible to study many-body systems … up to 5-body system (4He+n+n+n+n)
“Cluster-Orbital Shell Model”
T. Myo, R. Ando, K. Kato, PLB691, 150 (2010)
Calc. of scattering amplitude with CSM
A. T. Kruppa, R. Suzuki and K. Katō, PRC 75, 044602 (2007)
Point 1
Separate the incoming wave j l kr !
+
• Unknown
• Non square-integrable
hl
x
lπ
exp i kr
2
Point 2
Complex scaling
r → r eiΘ
square-integrable for 0 < θ < π
Expanding with square-integrable basis function
(ex: Gaussian basis)
Calc. of scattering amplitude with CSM
A. T. Kruppa, R. Suzuki and K. Katō, PRC 75, 044602 (2007)
•Calculation of scattering amplitude
Point 3
SC
We don’t have l , k r
which is a solution along r,
Cauchy theorem
but we have θ r
which is a solution along reiθ.
SC ,
l ,k
dz j kz V z
SC
l ,k
z
0
r eiθ
0
r
fl
SC
expressed with
Gaussian base
k is independent of θ.
Test calculation with AY potential
Akaishi-Yamazaki potential†
• Phenomenological potential
• Local Gaussian form
• Energy independent
Resonance pole
πΣ
KbarN
[MeV]
=30 deg.
V
AY ,I 0
(I 0)
V
KbarN
(I 0)
V
436
412
exp r
2
0.66 fm
πΣ
412
M eV
0
pS
continuum
KbarN continuum
B. E. (KbarN) = 28.2 MeV
Γ = 40.0 MeV
†Y.
Akaishi and T. Yamazaki, PRC65, 044005 (2002)
… Nominal position of Λ(1405)
Complex Scaling Method for Resonance
Study of unstable nuclei … Resonance of 8He
Bound state
“Cluster-Orbital Shell Model”
Resonance
Continuum
on 2θ line
T. Myo, R. Ando, K. Kato, PLB691, 150 (2010)
Test calculation with AY potential
Unitarity violation
of S-matrix
EKbarN [MeV]
Phase shift sum
Checked by
Continuum Level Density method
(R. Suzuki, A. T. Kruppa, B. G. Giraud,
and K. Katō, PTP119, 949(2008))
δKbarN + δπΣ
[deg.]
| |Det S| -1 |
EKbarN [MeV]
Test calculation with AY potential
KbarN (I=0) scattering amplitude
I 0
fKN KN
fm
Y. Akaishi and T. Yamazaki,
PRC65, 044005 (2002)
Scattering length
= -1.76 + i 0.46 fm
EKbarN [MeV]
Scattering length
(Scatt. amp. @ EKbarN=0)
= -1.77 + i 0.47 fm
Scattering amplitude
… Non-rela. / KSW-type SR
(I 0)
V
(I 0)
ij
r
C ij
2
8 fp
i j
πΣ
M iM
j
s i
j
KbarN → KbarN
Kinematics-potential
mismatched case
g ij r
πΣ
KbarN
Re
fπ=90 MeV
KbarN
Im
πΣ → πΣ
Singular behavior at πΣ threshold
… A virtual state exists.
(aπΣ, re)=(61,-6.3) fm satisfies -aπΣ /2 < re†.
† Y. Ikeda et al, PTP 125, 1205 (2011)
fπ=110 MeV
I=0 KbarN scattering length
Range parameters of KSW-type potential
Data : aI=0KbarN = -1.70 + i 0.68 fm (A. D. Martin)
Kinematics
KSW-type pot.
Range
parameters [fm]
dKN,KN
dπΣ,πΣ
KbarN scatt.
length [fm]
Re
Re aI=0KN
Im aI=0KN
Im
Non-rela.
NR-v1
NR-v2
Semi-rela.
SR-A
SR-B
0.44
0.61
0.44
0.64
0.50
0.71
0.37
0.35
-1.701
0.681
-1.700
0.681
-1.700
0.681
-1.696
0.681
Found sets of range parameters (dKN,KN, dπΣ,πΣ)
to reproduce the Martin’s value.
Two sets found in semi-rela. case. (SR-A, SR-B)
Martin
-1.70
0.68
Resonant structure
in the scattering amplitude
Energy @ Re fKbarN → KbarN = 0
fπ=90 - 120MeV
KSW - Semi rela.
KbarN → KbarN
fπ = 90MeV
Energy @ Re fπΣ → πΣ = 0
Resonant structure
in the scattering amplitude
Energy @ Re fKbarN → KbarN = 0
fπ=90 - 120MeV
Energy @ Re fπΣ → πΣ = 0
Strange behavior?
SR-B
SR-A
“Wave function” of the resonance pole
Non-rela. (NRv2)
fπ=110 MeV
θ=30°
Semi rela. (SR-A)
Semi rela. (SR-B)
πΣ component also localized
due to Complex scaling.
* The wave functions shown above are multiplied by
a phase factor so that the KbarN wfn. becomes real at r=0.
“Size” of the resonance pole state
fπ=90 - 120MeV
Mean distance between meson and baryon
2
r MB
fm
M
Re
B
Im
? Somehow small.
cf) 1.9 fm @ M = 1423 MeV†
(B = 12 MeV)
- Difference of binding energy?
- Different definition of pole?
Gamow state or bound state
† A. D., T. Hyodo, W. Weise,
PRC79, 014003 (2009)
NR: ～ 1.3 - 0.3 i [fm]
SR: ～ 1.2 - 0.5 i [fm]
Strange nuclear physics
Hypernuclei…
d
u
s
Hyperon
baryon = qqq
Nucleus
Strangeness is introduced
through baryons.
Strange nuclear physics
Strangeness is introduced
through mesons …
ubar
s
K- meson
(anti-kaon, Kbar)
meson = qqbar
Nucleus
Kaonic nuclei !
Key points to study kaonic nuclei
Kbar + N + N
KbarN couples with πY.
“Kbar N N”
Bound state for KbarN channel
but a resonant state for πY channel
π +Σ+N
Their nature?
“coupled-channel Complex Scaling Method”
1. Consider a coupled-channel problem
2. Treat resonant states adequately.
Similar to the bound-state calculation
3. Obtain the wave function to help the analysis of the state.
KbarN (-πY) interaction??
“Chiral SU(3)-based potential”
Excited hyperon Λ（１４０５） p + K
Energy [MeV]
1435
1405
-
Λ(1405)
1325
Σ+π
1250
Λ+π
1190
Σ
1115
Λ
939
p,n
s
u
d
Mysterious state; Λ(1405)
Quark model prediction … calculated as 3-quark state
Λ(1405) can’t be well reproduced
as a 3-quark state!
calculated Λ(1405)
observed Λ(1405)
q
qq
N. Isgar and G. Karl, Phys. Rev. D18, 4187 (1978)
３．KSW非相対論版の考察
(I 0)
(I 0)
V ij
r
C ij
2
8 fp
i
j
M iM
j
s i
j
赤字のファクターは、微分断面積で適切な
相対論的flux factorを得るために導入されている。
g r
KSWの（１４）式、Born近似での微分断面積の表式 （相対論的な場合）
ki: meson momentum、 ωi: reduced energy
ki
kj
i j
1
2p
d r exp i k j k i r V ij r
3
2
2
３．KSW非相対論版の考察
非相対論的な場合のBorn近似での微分断面積
d ij
di
vi
2
f ij
vj
ki i
2
f ij
kj j
fij : 散乱振幅
vi : 速度
f ij
1
2p
d ij
di
d
3
r exp i k j k i r V ij r
ki i
kj j
ki
kj
kij : 運動量
μi : 換算質量
j
1
2p
1
2p
1
2
4
d
3
r exp i k j k i r V ij r
d
3
2
r exp i k j k i r V ij r
2
３．KSW非相対論版の考察
相対論・非相対論でのBorn近似微分断面積の表式の比較
相対論的 ：
(KSW)
d ij
di
ki
kj
j
1
2p
d r exp i k j k i r V ij r
3
2
2
は１ に取っ て いる 。
非相対論的 ：
d ij
di
ki
kj
j
1
2p
1
2
4
d r exp i k j k i r V ij r
3
微分断面積の表式は、相対論的なものと非相対論的なもので、
明らかに対応している。
非相対論的な表式の換算質量（μi）が、
相対論的表式では換算エネルギー（ωi）に対応。
2
３．KSW非相対論版の考察
推測
KSWのpseudo-potential中のファクター
M iM
j
s i j
が微分断面積の表式とつじつまが合うように導入されているのならば、
このポテンシャルの非相対論版を考える際には、
前ページの対応関係からこのファクター中の換算エネルギーを
換算質量に置き換えればいいのでは？
つまり
(I 0)
(I 0)
V ij
r
C ij
2
8 fp
i
j
1
M iM
E Tot
j
j
g r
カイラル理論を尊重するので、メソンのエネルギー ωi は残しておく。
（続きは次ページ）
• 1 / √sはどうするか？そのまま、系の全エネルギー（ETot）として残しておく？？
Comparison with other studies
of I=0 channel
Comparison with other studies of I=0 KbarN-πΣ
Kaiser-Siegel- Hyodo-Weise
Weise
Ikeda-Hyodo- Dote-InoueJido-Kamano- Myo
Sato-Yazaki
Base
Charge
Isospin
Isospin
Isopin
Terms
All terms up to q2
WT term
WT term
WT term
Regularization
Yukawa
/ Separable form
Dimensional
regularization
Dimensional
regularization
/ Separable form
Gaussian form
Low energy
scattering data,
Low energy
scattering data,
Branching ratio at
KbarN threshold
Branching ratio at
KbarN threshold
KbarN scattering
length
/ Λ(1405) pole
KbarN scattering
length
NPA 594, 325 (1995)
PRC 77, 035204 (2008)
PTP 125, 1205 (2011)
NPA 912, 66 (2013)
Constraint
Ref.
Comparison with other studies of I=0 KbarN-πΣ
Different energy dependence of interaction kernel
IHJKSY, HW:
Not proportional to meson energy
… including relativistic q2 correction
and non-static effect of baryons
DIM:
Proportional to just meson energy
Comparison with other studies of I=0 KbarN-πΣ
IHJKSY
HW
Comparison with other studies of I=0 KbarN-πΣ
Use the same interaction kernel as that of IHJKSY and HW
in our ansatz (Flux factor and Gaussian form factor)
Suppressed the enhancement near the πΣ threshold,
but still different from the amplitudes of IHJKSY.
Different interaction kernel
Use of flux factor
Different regularization scheme (Gaussian form factor vs Dimensional reg.)
* Gaussian form factor enhances the amplitude far below the threshold as
|E| exp [c|E|] with E → -∞. (c>0)
1. Introduction
Neutron star in universe
Radius ～ 10km
Mass ～ Msun
Maybe …
High dense at core ～ > 2 ρ0
Involving strangeness
Kbar nuclear system on the Earth
… light nucleus with K- mesons
Involving strangeness via K- meson
• Self-bound
Kbar-nuclear system
1. Introduction
Neutron star in universe
Radius ～ 10km
Mass ～ Msun
Maybe …
High dense at core ～ > 2 ρ0
Involving strangeness
Kbar nuclear system on the Earth
3He
3HeK-
4×4 fm2
… light nucleus with K- mesons
Involving strangeness via K- meson
Strong attraction by K- meson
⇒ Dense nuclear system
Density [fm-3]
0.00
0.14
Density [fm-3]
0.0
0.75
1.5
A phenomenological KbarN potential†
⇒ B(K) ～ 100MeV, 2～4ρ0
† A. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, PLB 590 (2004) 51; PRC 70 (2004) 044313.
Experimental studies of K-pp
• FINUDA collaboration (DAΦNE, Frascati)
• K- absorption at rest on various nuclei (6Li, 7Li, 12C, 27Al, 51V)
• Invariant-mass method
p
p
K-p
If it is K-pp, …
Total
binding energy = 115
Λ
Decay width
= 67
Strong correlation between
emitted p and Λ (back-to-back)
6 3
5 4
14 2
11 3
MeV
MeV
Invariant mass of p and Λ
PRL 94, 212303 (2005)
Experimental studies of K-pp
• Re-analysis of KEK-PS E549
- K- stopped on 4He target
- Λp invariant mass
Strong Λp back-to-back correlation is confirmed.
Unknown strength is there
in the same energy region as FINUDA.
T. Suzuki et al (KEK-PS E549 collaboration),
arXiv:0711.4943v1[nucl-ex]
• DISTO collaboration
- p + p -> K+ + Λ + p @ 2.85GeV
- Λp invariant mass
- Comparison with simulation data
K- pp???
B. E.= 103 ±3 ±5 MeV
Γ = 118 ±8 ±10 MeV
T. Yamazaki et al. (DISTIO collaboration), PRL104, 132502 (2010)
as of 2011
Experiments of kaonic nuclei at J-PARC
E15: A search for deeply bound kaonic nuclear states
by 3He(inflight K-, n) reaction
(M. Iwasaki (RIKEN), T. Nagae (Kyoto))
E17: Precision spectroscopy of kaonic 3He atom
3d→2p X-rays
(R. Hayano (Tokyo), H. Outa (Riken))
E27: Search for a nuclear Kbar bound state K-pp in the d(π+, K+) reaction
(T. Nagae (Kyoto))
E31: Spectroscopic study of hyperon resonances below KN threshold
via the (K-, n) reaction on deuteron
(H. Noumi (Osaka))
as of 2011
Experiments of kaonic nuclei at J-PARC
E15: A search for deeply bound kaonic nuclear states
at K1.8BR beam line
by 3He(inflight K-, n) reaction
(M. Iwasaki (RIKEN), T. Nagae (Kyoto))
E17: Precision spectroscopy of kaonic 3He atom
3d→2p X-rays
(R. Hayano (Tokyo), H. Outa (Riken))
1.8GeV/c
E27: Search
for a nuclear Kbar bound state K-pp in the d(π+, K+) reaction
(T. Nagae (Kyoto))
E31: Spectroscopic study of hyperon resonances below KN threshold
via the (K-, n) reaction on deuteron
(H. Noumi (Osaka))
Dr. Fujioka’s talk (KEK workshop, 7-9. Aug. 08)
as of 2011
Experiments of kaonic nuclei at J-PARC
Invariant
E15: A search for deeply bound kaonic nuclear
states mass
at K1.8BR beam line
by 3He(inflight K-, n) reaction
spectroscopy
(M. Iwasaki (RIKEN), T. Nagae (Kyoto))
E17: Precision spectroscopy of kaonic 3He atom
3d→2p X-rays
(R. Hayano (Tokyo), H. Outa (Riken))
1.8GeV/c
E27: Search
for a nuclear Kbar bound state K-pp in the d(π+, K+) reaction
Missing
mass
(T. Nagae (Kyoto))
spectroscopy
E31: Spectroscopic study of hyperon resonances below KN threshold
via the (K-, n) reaction on deuteron
All emitted
particles will be measured.
(H. Noumi (Osaka))
「完全実験」
Dr. Fujioka’s talk (KEK workshop, 7-9. Aug. 08)
as of 2011
Experiments of kaonic nuclei at J-PARC
π+
K+
E15: A search for deeply bound kaonic nuclear states
Missing
Use
lots
of
pion
3
by He(inflight K , n) reaction
(M. Iwasaki (RIKEN), T. Nagae (Kyoto))
n
3He atom
d spectroscopy of kaonic
Λ(1405)
E17: Precision
p
3d→2p X-rays
(R. Hayano (Tokyo), H. Outa (Riken))
mass
Kp
p
E27: Search for a nuclear Kbar bound state K-pp in the d(π+, K+) reaction
(T. Nagae (Kyoto))
Preceding E15,
E31: Spectroscopic study of hyperon resonances below KN threshold
-, n) reaction on
via
the (K
deuteron
E27
experiment
was
performed in June.
(H. Noumi (Osaka))
Analysis is now undergoing!
as of 2011
Experiments of kaonic nuclei at J-PARC
E15: A search for deeply bound kaonic nuclear states
by 3He(inflight K-, n) reaction
K-pp
(M. Iwasaki (RIKEN), T. Nagae (Kyoto))
E17: Precision spectroscopy of kaonic 3He atom
3d→2p X-rays
KbarN interaction
(R. Hayano (Tokyo), H. Outa (Riken))
E27: Search for a nuclear Kbar bound state K-pp in the d(π+, K+) reaction
(T. Nagae (Kyoto))
K-pp
E31: Spectroscopic study of hyperon resonances below KN threshold
via the (K-, n) reaction on deuteron
(H. Noumi (Osaka))
Λ(1405)
These days, K-pp has been focused
in both of theoretical and experimental studies!
KProton
=
Λ(1405)
3HeK-,
K
P
P
Most essential
kaonic nucleus!
pppK-,
4HeK-, pppnK-,
…
8BeK-,
…
Complicated
nuclear system
with K-
Coupled Channel Chiral Dynamics
(Chiral Unitary model)
We are thinking about KbarN interaction.
S=-1 meson-baryon system is constrained by
Chiral SU(3) dynamics !
The leading couplings between
Low mass pseudo-scalar meson octet
(Nambu-Goldston bosons)
and
Baryon octet
are determined by
Spontaneous breaking of
SU(3)×SU(3) Chiral symmetry
Coupled Channel Chiral Dynamics
(Chiral Unitary model)
Chiral low-energy theorem tells us …
Weinberg-Tomozawa term derived from Chiral SU(3) effective Lagrangian
f : Psuedo-scalar meson decay constant
Energy and mass of baryon
in channel i
For I=0 channel,
KbarN
and
πΣ
ηΛ
KΞ
Remark
• There are no free parameters
as for coupling.
• There is an attractive interaction
in πΣ-πΣ channel,
while AY potential doesn’t have it.
Coupled Channel Chiral Dynamics
(Chiral Unitary model)
1. Weinberg-Tomozawa term derived from Chiral SU(3) effective Lagrangian:
2. Using the WT-term as a building block,
3. Solve coupled channel Bethe-Salpeter equation.
=
+…+
…
T-matrix of coupled channel scattering
• K-p scattering length
• Threshold branching ratio
• Total cross section of K-p scattering
+…
Coupled Channel Chiral Dynamics
(Chiral Unitary model)
• Threshold branching ratio
T. Hyodo, S. I. Nam, D. Jido, and A. Hosaka,
Phys. Rev. C68, 018201 (2003)
• Total cross section of K-p scattering
Coupled Channel Chiral Dynamics
(Chiral Unitary model)
1. Weinberg-Tomozawa term derived from Chiral SU(3) effective Lagrangian:
2. Using the WT-term as a building block,
3. Solve coupled channel Bethe-Salpeter equation.
=
+…+
…
+…
T-matrix of coupled channel scattering
• K-p scattering length
• Threshold branching ratio
• Total cross section of K-p scattering
Λ(1405) is dynamically generated as meson-baryon system.
Coupled Channel Chiral Dynamics
(Chiral Unitary model)
I=0 πΣ mass distribution
Dynamical generation of Λ(1405) !
T. Hyodo, S. I. Nam, D. Jido, and A. Hosaka,
Phys. Rev. C68, 018201 (2003)
Remark:
Calculated with
πΣ-πΣ scattering
amplitude.
Chiral SU(3) potential (KSW)
N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)
Original: δ-function type
(I )
(I )
V ij
r
C ij
8f
2
i j
M iM
j
s i
j
Energy dependence is determined by
Chiral low energy theorem.
← Kaon: Nambu-Goldstone boson
r
Present: Normalized Gaussian type
(I )
(I )
V ij
r
C ij
8f
2
M iM
i j
j
s i
(I )
g ij
r
r
(I )
g ij
j
1
p
3/2
a
exp r a i j
(I )
(I )3
ij
2
a: range parameter [fm]
Mi , mi : Baryon, Meson mass in channel i
Ei : Baryon energy, ωi : Meson energy
Ei
s
2
mi M i
2
2 s
2
, i
s
Flavor SU(3) symmetry
2
mi M i
2
2
KbarN
2 s
(I 0)
Reduced energy: i
i Ei
i Ei
Energy dependence of Vij is
controlled by CM energy √s.
C ij
3
πΣ
3
2
4
Chiral SU(3) potential (KSW)
N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)
Energy dependence
( I 0)
V ij
r 0, s
@ a 0.5 fm , f p 100 M eV
KbarN-πΣ
√s [MeV]
KbarN-KbarN
πΣ-πΣ
πΣ threshold
KbarN threshold
Calculational procedure
Chiral SU(3) potential = Energy-dependent potential
Self consistency for the energy!
Assume the values of the CM energy √s.
s Assum ed
H T V
MB
s
M
Perform the Complex Scaling method.
Then, find a pole of resonance or bound state.
Check
s Calculated
s C alculated
If Yes
s Assumed
If No
Finished !
4. Experiments
related to
bar
K nuclear physics
Ｋ原子核に関係する実験
Ｋ原子 (Kaonic atom)
• Kaonic 4He atom, 2pレベルシフト (3d→2p X線測定) @ KEK (E570)
S. Okada et. al., Phys. Lett. B653, 387 (2007)
• Kaonic 3He atom, 2pレベルシフト (3d→2p X線測定)
@ J-PARC (E17, DAY-1)
• Kaonic hydrogen atom, 1sレベルシフト @ DEAR group, DAΦNE,
G. Beer et al., Phys. Rev. Lett. 94, 212302 (2005)
Frascati National Labor
• Kaonic hydrogen, deuterium @ SIDDHARTA group
M. Bazzi et al., Phys. Lett. B704, 113 (2011)
Λ(1405)
πΣ invariant mass測定 γ + p → K+ + Λ(1405), Λ(1405) → π Σ
• LEPS / SPring-8 J. K. Ahn, Nucl. Phys. A835, 329 (2010)
K. Moriya and R. Schumacher, Nucl. Phys. A835, 325 (2010)
• CLAS / JLab
π-Σ+, π0Σ0, π+Σ- が全て押さえられた
Ｋ原子核に関係する実験
Ｋ原子 (Kaonic atom)
• Kaonic 4He atom, 2pレベルシフト (3d→2p X線測定) @ KEK (E570)
S. Okada et. al., Phys. Lett. B653, 387 (2007)
• Kaonic 3He atom, 2pレベルシフト (3d→2p X線測定)
@ J-PARC (E17, DAY-1)
• Kaonic hydrogen atom, 1sレベルシフト @ DEAR group, DAΦNE,
G. Beer et al., Phys. Rev. Lett. 94, 212302 (2005)
Frascati National Labor
• Kaonic hydrogen, deuterium @ SIDDHARTA group
M. Bazzi et al., Phys. Lett. B704, 113 (2011)
Λ(1405)
πΣ invariant mass測定 γ + p → K+ + Λ(1405), Λ(1405) → π Σ
• LEPS / SPring-8 J. K. Ahn, Nucl. Phys. A835, 329 (2010)
K. Moriya and R. Schumacher, Nucl. Phys. A835, 325 (2010)
• CLAS / JLab
π-Σ+, π0Σ0, π+Σ- が全て押さえられた
DEAR exp. for kaonic hydrogen atom
G. Beer et al., Phys. Rev. Lett. 94, 212302 (200
Kaonic hydrogen atom, 1sのレベルシフト
@ DEAR Collaboration, DAΦNE, Frascati National Laboratories
cf) KEK exp.
M.Iwasaki et al., Phys. Rev. Lett. 78, 3067 (1997)
シフトの符号は同じだが、ＫＥＫの前回の実験（ＫｐＸ）と重ならない
KEK exp.
DEAR
Coupled channel chiral dynamics
(Chiral unitary model) で
DEARの結果を合わすのには苦労する。
かろうじてギリギリ合わせられる程度。。。
B. Borasoy et al., Phys. Rev. Lett. 94, 213401 (2005)
SHIDDARTA exp. for kaonic hydrogen atom
M. Bazzi et al., Phys. Lett. B704, 113 (201
Kaonic hydrogen atom, 1sのレベルシフト
@ SHIDDARTA Collaboration, DAΦNE, Frascati National Laboratories
K-p散乱長が精密に決定
理論計算にとって重要な
インプットに強い拘束条件
KbarN subthresholdでの
散乱振幅の振る舞い、
Λ(1405)のポールの位置、
が制限される。
KEK実験（ＫｐＸ）とコンシステントな結果
Y. Ikeda, T. Hyodo and W. Weise,
Phys. Lett. B706, 63 (2011)
Ｋ原子核に関係する実験
Ｋ原子 (Kaonic atom)
• Kaonic 4He atom, 2pレベルシフト (3d→2p X線測定) @ KEK (E570)
S. Okada et. al., Phys. Lett. B653, 387 (2007)
• Kaonic 3He atom, 2pレベルシフト (3d→2p X線測定)
@ J-PARC (E17, DAY-1)
• Kaonic hydrogen atom, 1sレベルシフト @ DEAR group, DAΦNE,
G. Beer et al., Phys. Rev. Lett. 94, 212302 (2005)
Frascati National Labor
• Kaonic hydrogen, deuterium @ SIDDHARTA group
M. Bazzi et al., Phys. Lett. B704, 113 (2011)
Λ(1405)
πΣ invariant mass測定 γ + p → K+ + Λ(1405), Λ(1405) → π Σ
• LEPS / SPring-8 J. K. Ahn, Nucl. Phys. A835, 329 (2010)
K. Moriya and R. Schumacher, Nucl. Phys. A835, 325 (2010)
• CLAS / JLab
π-Σ+, π0Σ0, π+Σ- が全て押さえられた
KEK E570 for kaonic 4He atom
S. Okada et. al., Phys. Lett. B653, 387 (2007)
Kaonic 4He atom, 2pのレベルシフト
3d→2p X線測定
@ KEK, E570
“Kaonic helium puzzle”
理論の予言がほぼ0eVに対して、
過去の実験ではシフトは平均-43 eV
S. Hirenzaki et al.,
Phys. Rev. C61, 055205 (2000)
シフトは 0 eV とconsisitent
パズルは解けた！
J-PARC for kaonic 3He atom
Kaonic 3He atom, 2pのレベルシフト
3d→2p X線測定
Kaonic 4He atom, 2pのレベルシフト
…ほぼ 0 eV と確定
@ J-PARC, E17 DAY-1
S. Okada et. al., Phys. Lett. B653, 387 (2007)
赤石氏の計算
Y. Akaishi, Proceedings of EXA’05,
Austrian Academy of Sciences press, Vienna, 2005, p.45
＋
KbarN potentialの強度が
二つの領域に絞られた。
※特定領域研究「ストレンジネスで探るクォーク多体系」研究会2007 での岡田氏のスライドより引用
さらに3Heでシフトが測定されることで
KbarN potentialの強度に絞りを
掛けることが期待できる。
Ｋ原子核に関係する実験
Ｋ原子 (Kaonic atom)
• Kaonic 4He atom, 2pレベルシフト (3d→2p X線測定) @ KEK (E570)
S. Okada et. al., Phys. Lett. B653, 387 (2007)
• Kaonic 3He atom, 2pレベルシフト (3d→2p X線測定)
@ J-PARC (E17, DAY-1)
• Kaonic hydrogen atom, 1sレベルシフト @ DEAR group, DAΦNE,
G. Beer et al., Phys. Rev. Lett. 94, 212302 (2005)
Frascati National Labor
• Kaonic hydrogen, deuterium @ SIDDHARTA group
M. Bazzi et al., Phys. Lett. B704, 113 (2011)
Λ(1405)
πΣ invariant mass測定 γ + p → K+ + Λ(1405), Λ(1405) → π Σ
• LEPS / SPring-8 J. K. Ahn, Nucl. Phys. A835, 329 (2010)
K. Moriya and R. Schumacher, Nucl. Phys. A835, 325 (2010)
• CLAS / JLab
π-Σ+, π0Σ0, π+Σ- が全て押さえられた
Λ(1405) - πΣ invariant mass 測定 •LEPS / Spring-8
• CLAS / Jefferson Laboratory
J. K. Ahn, Nucl. Phys. A835, 329 (2010)
K. Moriya and R. Schumacher, Nucl. Phys. A835, 325 (2010)
p (γ, K+ π) Σ at Eγ = 1.5-2.4GeV
Charged πΣを測定
γ + p → K+ + Λ(1405), Λ(1405) → π Σ
三つの異なる電荷状態が抑えられた
理論
ピークの順番が理論
(chiral unitary)と違う？
Highest peak
実験： Σ+ π理論： Σ- π+
What is the object observed experimentally?
• DISTO collaboration
A bound state of K-pp,
or another object such as πΣN ???
Only what we can say from only this spectrum is that
“There is some object with B=2, S=-1, charge=+1”…
Self-consistent solutions for complex energy
Chiral potential … Energy-dependent potential
(I 0)
V
(I 0)
ij
r
C ij
2
8 fp
i
j
M iM
j
s i
j
ga r
Self-consistency for complex energy should be considered!
H Z in K N p S I 0 Z out K N p S I 0
Z E i 2
Self-consistent solution …
: complex energy
Z in Z out
Ex) KSW ; fπ = 100 MeV, a= 0.47 fm ( θ=35°)
Self-consistency
achieved after a
few times iteration.
A. D., T. Inoue and T. Myo,
Proc. of BARYONS’ 10,
AIP conf. proc. 1388, 549 (2010)
Scattering amplitude
• fπ = 90 MeV
Scattering amplitude
… KSW org. – Non. rela.
πΣ
fπ=90 MeV
(I 0)
V
(I 0)
ij
r
KbarN
C ij
2
8 fp
Im
Re
KbarN → KbarN
KbarN → πΣ
πΣ → πΣ
• Phase shift (πΣ)
δ（πΣ）
πΣ→ KbarN
i
j
M iM
j
s i
j
g ij r
Scattering amplitude
… KSW NRv1 – Non. rela.
πΣ
fπ=90 MeV
(I 0)
V
(I 0)
ij
r
C ij
2
8 fp
KbarN
Re
KbarN → KbarN
KbarN → πΣ
Im
πΣ → πΣ
• Phase shift (πΣ)
δ（πΣ）
πΣ→ KbarN
i
j
1
M iM
E T ot
j
j
g ij r
Scattering amplitude
… KSW NRv2 – Non. rela.
πΣ
fπ=90 MeV
(I 0)
V
(I 0)
ij
r
C ij
8 fp
KbarN
Re
KbarN → KbarN
KbarN → πΣ
Im
πΣ → πΣ
• Phase shift (πΣ)
δ（πΣ）
πΣ→ KbarN
2
i
j
1
m m j
g ij r
Scattering amplitude
… KSW org. – Semi rela.
πΣ
fπ=90 MeV
(I 0)
V
(I 0)
ij
r
KbarN
C ij
2
8 fp
Im
Re
KbarN → KbarN
KbarN → πΣ
πΣ → πΣ
• Phase shift (πΣ)
δ（πΣ）
πΣ→ KbarN
i
j
M iM
j
s i
j
g ij r
Complex Scaling Method
Succeeded in nuclear physics
… Especially used in the study of unstable nuclei.
Resonant state of 6He
n
4He
S. Aoyama, T. Myo, K. Kato and K. Ikeda,
Prog. Theor. Phys. 116, 1 (2006)
n
Resonance
Continuum
Result
• Scattering amplitude
• Position of resonant structure appeared in
the scattering amplitude
• “Wave function” of the resonance pole
• “Size” of the resonance pole state
• θ dependence of “Size” and “Wave function”
• Comparison with Hyodo-Weise’s result
Scattering amplitude
• Potential: KSW original
• Kinematics: Non. rela.
Scattering amplitude
… KSW org. – Non. rela.
fπ=90 MeV
Im
Re
KbarN → KbarN
KbarN → πΣ
πΣ → πΣ
• Phase shift (πΣ)
δ（πΣ）
πΣ→ KbarN
Scattering amplitude
… KSW org. – Non. rela.
fπ=100 MeV
Im
Re
KbarN → KbarN
KbarN → πΣ
πΣ → πΣ
• Phase shift (πΣ)
δ（πΣ）
πΣ→ KbarN
Scattering amplitude
… KSW org. – Non. rela.
fπ=110 MeV
Re
Im
KbarN → KbarN
KbarN → πΣ
πΣ → πΣ
• Phase shift (πΣ)
δ（πΣ）
πΣ→ KbarN
Scattering amplitude
… KSW org. – Non. rela.
Re
KbarN → KbarN
KbarN → πΣ
fπ=120 MeV
Im
πΣ → πΣ
• Phase shift (πΣ)
δ（πΣ）
πΣ→ KbarN
Scattering amplitude
• Potential: KSW NRv1
• Kinematics: Non. rela.
Scattering amplitude
… KSW NRv1 – Non. rela.
Re
KbarN → KbarN
KbarN → πΣ
fπ=90 MeV
Im
πΣ → πΣ
• Phase shift (πΣ)
δ（πΣ）
πΣ→ KbarN
Scattering amplitude
… KSW NRv1 – Non. rela.
Re
KbarN → KbarN
KbarN → πΣ
fπ=100 MeV
Im
πΣ → πΣ
• Phase shift (πΣ)
δ（πΣ）
πΣ→ KbarN
Scattering amplitude
… KSW NRv1 – Non. rela.
fπ=110 MeV
Im
Re
KbarN → KbarN
KbarN → πΣ
πΣ → πΣ
• Phase shift (πΣ)
δ（πΣ）
πΣ→ KbarN
Scattering amplitude
… KSW NRv1 – Non. rela.
fπ=120 MeV
Im
Re
KbarN → KbarN
KbarN → πΣ
πΣ → πΣ
• Phase shift (πΣ)
δ（πΣ）
πΣ→ KbarN
Scattering amplitude
• Potential: KSW NRv2
• Kinematics: Non. rela.
Scattering amplitude
… KSW NRv2 – Non. rela.
Re
KbarN → KbarN
KbarN → πΣ
fπ=90 MeV
Im
πΣ → πΣ
• Phase shift (πΣ)
δ（πΣ）
πΣ→ KbarN
Scattering amplitude
… KSW NRv2 – Non. rela.
fπ=100 MeV
Im
Re
KbarN → KbarN
KbarN → πΣ
πΣ → πΣ
• Phase shift (πΣ)
δ（πΣ）
πΣ→ KbarN
Scattering amplitude
… KSW NRv2 – Non. rela.
fπ=110 MeV
Im
Re
KbarN → KbarN
KbarN → πΣ
πΣ → πΣ
• Phase shift (πΣ)
δ（πΣ）
πΣ→ KbarN
Scattering amplitude
… KSW NRv2 – Non. rela.
fπ=120 MeV
Im
Re
KbarN → KbarN
KbarN → πΣ
πΣ → πΣ
• Phase shift (πΣ)
δ（πΣ）
πΣ→ KbarN
Scattering amplitude
• Potential: KSW original
• Kinematics: Semi rela.
Scattering amplitude
… KSW org. – Semi rela.
fπ=90 MeV
Im
Re
KbarN → KbarN
KbarN → πΣ
πΣ → πΣ
• Phase shift (πΣ)
δ（πΣ）
πΣ→ KbarN
Scattering amplitude
… KSW org. – Semi rela.
fπ=100 MeV
Im
Re
KbarN → KbarN
KbarN → πΣ
πΣ → πΣ
• Phase shift (πΣ)
δ（πΣ）
πΣ→ KbarN
Scattering amplitude
… KSW org. – Semi rela.
fπ=110 MeV
Im
Re
KbarN → KbarN
KbarN → πΣ
πΣ → πΣ
• Phase shift (πΣ)
δ（πΣ）
πΣ→ KbarN
Scattering amplitude
… KSW org. – Semi rela.
fπ=120 MeV
Im
Re
KbarN → KbarN
KbarN → πΣ
πΣ → πΣ
• Phase shift (πΣ)
δ（πΣ）
πΣ→ KbarN
Position of
resonant structure
appeared in
the scattering amplitude
Position of resonant structure
appeared in the scattering amplitude
Kinematics
KSW potential
fπ=90 MeV
Semi-rela.
Non-rela.
Non-rela. v1 Non-rela. v2 Original
Original
0.593
0.541
0.576
0.725
0.574
0.751
0.487
0.457
-1.701
0.679
-1.700
0.677
-1.700
0.687
-1.703
0.677
Resonance position [MeV]
E
f11=0
bar
bar
K N→K N
E(KbarN)
1402.5
-32.5
1408.4
-26.6
1411.1
-23.9
1417.0
-18.0
f22=0
E
E(KbarN)
1354.6
-80.4
1388.8
-46.2
1393.0
-42.0
1422.0
-12.9
δ(πΣ)=0
E
E(KbarN)
1354.6
-80.4
1388.8
-46.2
1393.0
-42.0
1422.1
-12.9
Range
parameter [fm]
a(KbarN)
a(πΣ)
KbarN Scatt.
leng. [fm]
Re
Im
πΣ → πΣ
Position of resonant structure
appeared in the scattering amplitude
Kinematics
KSW potential
fπ=100 MeV
Semi-rela.
Non-rela.
Non-rela. v1 Non-rela. v2 Original
Original
0.517
0.477
0.503
0.665
0.501
0.695
0.421
0.395
-1.698
0.676
-1.702
0.680
-1.702
0.683
-1.703
0.673
Resonance position [MeV]
E
f11=0
bar
bar
K N→K N
E(KbarN)
1404.8
-30.2
1411.2
-23.8
1413.6
-21.4
1418.0
-17.0
f22=0
E
E(KbarN)
1384.7
-50.3
1399.3
-35.7
1402.6
-32.4
1421.5
-13.5
δ(πΣ)=0
E
E(KbarN)
1384.7
-50.3
1399.3
-35.7
1402.6
-32.4
1421.5
-13.5
Range
parameter [fm]
a(KbarN)
a(πΣ)
KbarN Scatt.
leng. [fm]
Re
Im
πΣ → πΣ
Position of resonant structure
appeared in the scattering amplitude
Kinematics
KSW potential
fπ=110 MeV
Semi-rela.
Non-rela.
Non-rela. v1 Non-rela. v2 Original
Original
0.453
0.419
0.440
0.605
0.438
0.636
0.369
0.348
-1.703
0.683
-1.701
0.681
-1.700
0.681
-1.696
0.681
Resonance position [MeV]
E
f11=0
bar
bar
K N→K N
E(KbarN)
1407.2
-27.8
1413.4
-21.6
1415.5
-19.5
1418.9
-16.1
f22=0
E
E(KbarN)
1394.7
-40.3
1405.3
-29.7
1408.1
-26.9
1421.4
-13.6
δ(πΣ)=0
E
E(KbarN)
1394.7
-40.3
1405.3
-29.7
1408.1
-26.9
1421.4
-13.6
Range
parameter [fm]
a(KbarN)
a(πΣ)
KbarN Scatt.
leng. [fm]
Re
Im
πΣ → πΣ
Position of resonant structure
appeared in the scattering amplitude
Kinematics
KSW potential
fπ=120 MeV
Semi-rela.
Non-rela.
Non-rela. v1 Non-rela. v2 Original
Original
0.398
0.368
0.386
0.549
0.384
0.581
0.327
0.310
-1.698
0.682
-1.699
0.674
-1.700
0.669
-1.690
0.669
Resonance position [MeV]
E
f11=0
bar
bar
K N→K N
E(KbarN)
1408.6
-26.4
1415.0
-20.0
1417.0
-18.0
1419.3
-15.7
f22=0
E
E(KbarN)
1399.7
-35.3
1409.1
-25.9
1411.7
-23.3
1420.9
-14.1
δ(πΣ)=0
E
E(KbarN)
1399.7
-35.3
1409.1
-25.9
1411.7
-23.3
1420.9
-14.1
Range
parameter [fm]
a(KbarN)
a(πΣ)
KbarN Scatt.
leng. [fm]
Re
Im
πΣ → πΣ
Position of resonant structure
appeared in the scattering amplitude
Resonance position
in KbarN → KbarN
Resonance position
in πΣ → πΣ
“Wave function”
of
the resonance pole
“Wave function” of the resonance pole
Org. – Non.rela.
Org. – Semi rela.
NRv1 – Non.rela.
NRv2 – Non.rela.
fπ=90 MeV
θ=30°
“Wave function” of the resonance pole
Org. – Non.rela.
Org. – Semi rela.
NRv1 – Non.rela.
NRv2 – Non.rela.
fπ=100 MeV
θ=30°
“Wave function” of the resonance pole
Org. – Non.rela.
Org. – Semi rela.
NRv1 – Non.rela.
NRv2 – Non.rela.
fπ=110 MeV
θ=30°
“Wave function” of the resonance pole
Org. – Non.rela.
Org. – Semi rela.
NRv1 – Non.rela.
NRv2 – Non.rela.
fπ=120 MeV
θ=30°
“Size” of
the resonance
pole state
“Size” of the resonance pole state
Kinematics
KSW potential
Range
parameter [fm]
a(KbarN)
a(πΣ)
KbarN Scatt.
leng. [fm]
Pole position
[MeV]
fπ=90 MeV
θ=30°
Non-rela.
Semi-rela.
Non-rela. v1 Non-rela. v2 Original
Original
0.593
0.541
0.576
0.725
0.574
0.751
0.487
0.457
Re
Im
-1.701
0.679
-1.700
0.677
-1.700
0.687
-1.703
0.677
E
B(KbarN)
Γ/2
1423.3
11.7
28.5
1419.8
15.2
26.0
1419.9
15.1
23.1
1419.0
16.0
14.4
1.19
-0.45
1.20
-0.35
1.25
-0.35
1.21
-0.49
Meson-Baryon distance[fm]
KbarN ch.
Re
Im
πΣ ch.
Re
Im
0.29
0.21
0.33
0.24
0.34
0.23
0.12
-0.09
KbarN+πΣ
Re
Im
1.19
-0.39
1.20
-0.28
1.25
-0.29
1.21
-0.49
“Size” of the resonance pole state
Kinematics
KSW potential
Range
parameter [fm]
a(KbarN)
a(πΣ)
KbarN Scatt.
leng. [fm]
Pole position
[MeV]
fπ=100 MeV
θ=30°
Non-rela.
Semi-rela.
Non-rela. v1 Non-rela. v2 Original
Original
0.517
0.477
0.503
0.665
0.501
0.695
0.421
0.395
Re
Im
-1.698
0.676
-1.702
0.680
-1.702
0.683
-1.703
0.673
E
B(KbarN)
Γ/2
1421.7
13.3
28.0
1417.3
17.7
23.1
1418.0
17.0
19.8
1419.6
15.4
13.2
1.20
-0.45
1.27
-0.34
1.32
-0.34
1.20
-0.49
Meson-Baryon distance[fm]
KbarN ch.
Re
Im
πΣ ch
Re
Im
0.34
0.16
0.39
0.15
0.38
0.13
0.11
-0.08
KbarN+πΣ
Re
Im
1.22
-0.40
1.31
-0.29
1.36
-0.30
1.21
-0.49
“Size” of the resonance pole state
Kinematics
KSW potential
Range
parameter [fm]
a(KbarN)
a(πΣ)
KbarN Scatt.
leng. [fm]
Pole position
[MeV]
fπ=110 MeV
θ=30°
Non-rela.
Semi-rela.
Non-rela. v1 Non-rela. v2 Original
Original
0.453
0.419
0.440
0.605
0.438
0.636
0.369
0.348
Re
Im
-1.703
0.683
-1.701
0.681
-1.700
0.681
-1.696
0.681
E
B(KbarN)
Γ/2
1420.5
14.5
26.6
1416.6
18.4
19.5
1417.8
17.2
16.6
1420.0
15.0
12.8
1.20
-0.48
1.31
-0.37
1.37
-0.37
1.18
-0.49
Meson-Baryon distance[fm]
KbarN ch.
Re
Im
πΣ ch
Re
Im
0.36
0.11
0.39
0.05
0.37
0.04
0.11
-0.06
KbarN+πΣ
Re
Im
1.23
-0.43
1.36
-0.34
1.42
-0.34
1.18
-0.49
“Size” of the resonance pole state
Kinematics
KSW potential
Range
parameter [fm]
a(KbarN)
a(πΣ)
KbarN Scatt.
leng. [fm]
Pole position
[MeV]
fπ=120 MeV
θ=30°
Non-rela.
Semi-rela.
Non-rela. v1 Non-rela. v2 Original
Original
0.398
0.368
0.386
0.549
0.384
0.581
0.327
0.310
Re
Im
-1.698
0.682
-1.699
0.674
-1.700
0.669
-1.690
0.669
E
B(KbarN)
Γ/2
1419.5
15.5
25.6
1416.9
18.1
16.7
1418.3
16.7
14.0
1418.9
16.1
11.7
1.18
-0.49
1.33
-0.39
1.40
-0.39
1.15
-0.42
Meson-Baryon distance[fm]
KbarN ch.
Re
Im
πΣ ch
Re
Im
0.37
0.06
0.36
-0.01
0.34
-0.02
0.14
-0.04
KbarN+πΣ
Re
Im
1.22
-0.45
1.38
-0.38
1.44
-0.38
1.16
-0.42
“Size” of the resonance pole state
2
Mean distance between meson and baryon
r MB
fm
Re
Im
θ=30°
Pole position of the resonance
90
100
110
120
Org-NR
M
1423.3
1421.7
1420.5
1419.5
NRv1-NR
M
1419.8
1417.3
1416.6
1416.9
-Γ/2
-28.5
-28.0
-26.6
-25.6
-Γ/2
-26.0
-23.1
-19.5
-16.7
NRv2-NR
M
1419.9
1418.0
1417.8
1418.3
-Γ/2
-23.1
-19.8
-16.6
-14.0
Org-SR
M
1419.0
1419.6
1420.0
1418.9
-Γ/2
-14.4
-13.2
-12.8
-11.7
M [MeV]
- Γ / 2 [MeV]
fpi
120
120
120 110
110
100
90
110
100
100
90
120
90 110
100
90
θ dependence of
“Size” and
“Wave function”
θ dependence of “Size”
θ
15
20
25
30
35
1416.8
18.5
1419.2
22.9
1419.8
23.1
1419.9
23.1
1419.9
23.1
1419.8
23.1
Meson-Baryon distance[fm]
KbarN ch. (Re) KbarN
0.95
(Im) KbarN
-0.43
1.24
-0.41
1.25
-0.35
1.25
-0.35
1.25
-0.35
1.24
-0.35
12.36
4.27
3.61
4.81
0.71
1.17
0.35
0.33
0.34
0.23
0.34
0.23
KbarN+πΣ(Re) KbarN+πΣ
12.38
(Im) KbarN+πΣ4.23
3.61
4.67
0.87
0.44
1.23
-0.26
1.25
-0.29
1.25
-0.29
πΣ ch.
(Re) πΣ
(Im) πΣ
[fm]
Pole position
E
[MeV]
Γ/2
10
NRv2-NR
fπ=90 MeV
θ [deg]
θ dependence of “Size”
θ
15
20
25
30
35
1416.7
14.5
1418.9
14.5
1419.0
14.4
1419.0
14.4
1419.0
14.4
1418.9
14.4
Meson-Baryon distance[fm]
KbarN ch. (Re) KbarN
0.98
(Im) KbarN
-0.69
1.21
-0.50
1.21
-0.49
1.21
-0.49
1.21
-0.49
1.21
-0.49
8.62
7.05
0.73
2.33
0.13
0.51
0.06
-0.11
0.12
-0.09
0.12
-0.09
KbarN+πΣ(Re) KbarN+πΣ8.60
(Im) KbarN+πΣ6.99
0.54
2.00
1.10
-0.48
1.21
-0.49
1.21
-0.49
1.22
-0.49
πΣ ch.
(Re) πΣ
(Im) πΣ
[fm]
Pole position
E
[MeV]
Γ/2
10
Org-SR
fπ=90 MeV
θ [deg]
θ dependence of “Wave function”
NRv2-NR
fπ=90 MeV
KbarN
πΣ
πΣ resonant state
KbarN bound state
r [fm]
r [fm]
θ =10 deg.
θ dependence of “Wave function”
NRv2-NR
fπ=90 MeV
KbarN
πΣ
πΣ resonant state
KbarN bound state
r [fm]
r [fm]
θ =15 deg.
θ dependence of “Wave function”
NRv2-NR
fπ=90 MeV
KbarN
πΣ
πΣ resonant state
KbarN bound state
r [fm]
r [fm]
θ =20 deg.
θ dependence of “Wave function”
NRv2-NR
fπ=90 MeV
KbarN
πΣ
πΣ resonant state
KbarN bound state
r [fm]
r [fm]
θ =25 deg.
θ dependence of “Wave function”
NRv2-NR
fπ=90 MeV
KbarN
πΣ
πΣ resonant state
KbarN bound state
r [fm]
r [fm]
θ =30 deg.
θ dependence of “Wave function”
NRv2-NR
fπ=90 MeV
KbarN
πΣ
πΣ resonant state
KbarN bound state
r [fm]
r [fm]
θ =35 deg.
Comparison with
Hyodo-Weise’s result
Comparison with Hyodo-Weise’s result
Scattering amplitudes
DIM
KbarN → KbarN
HW: PRC 77, 035204 (2008)
πΣ → πΣ
HW
Fig. 4 I=0 KbarN/πΣ scattering amplitude in two channel model
Comparison with Hyodo-Weise’s result
Pole position
HW: PRC 77, 035204 (2008)
HW
(Upper pole in
two channel model)
DIM
z = 1419 – 14.4i MeV
Fig. 5 Pole positions of I=0 scattering amplitude
c.c. CSM for resonant states
Schrödinger equation to be solved
Phenomenological potential
= Energy independent potential
Y. Akaishi and T. Yamazaki,
PRC 52 (2002) 044005
V
Chiral SU(3) potential
= Energy dependent potential
N. Kaiser, P. B. Siegel and W. Weise,
NPA 594, 325 (1995)
Wave function expanded with Gaussian base
: complex parameters to be determined
c.c. CSM for resonant states
Schrödinger equation to be solved
Complex scaling of coordinate
Phenomenological potential
= Energy independent potential
Y. Akaishi and T. Yamazaki,
PRC 52 (2002) 044005
V
Chiral SU(3) potential
= Energy dependent potential
N. Kaiser, P. B. Siegel and W. Weise,
NPA 594, 325 (1995)
ABC theorem
Wave function expanded with Gaussian base
The energy of bound and resonant states is
independent of scaling
angle θ.
: complex parameters to be determined
Complex-rotate
J. Aguilar and J. M. Combes, Commun. Math. Phys. 22 (1971),269
E. Balslev and J. M. Combes, Commun. Math. Phys. 22 (1971),280
, then diagonalize
with Gaussian base.
Chiral SU(3) potential (KSW)
N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)
Original: δ-function type
( I 0)
(I 0)
V ij
r
C ij
8f
2
i j
M iM
j
s i
j
Energy dependence is determined by
Chiral low energy theorem.
← Kaon: Nambu-Goldstone boson
r
Present: Normalized Gaussian type
( I 0)
(I 0)
V ij
r
C ij
8f
2
i
j
M iM
j
s i
j
g r
g r
1
r a 2
exp
3/2 3
p a
a: range parameter [fm]
Mi , mi : Baryon, Meson mass in channel i
Ei : Baryon energy, ωi : Meson energy
Ei
s
2
mi M i
2
2 s
2
, i
s
Flavor SU(3) symmetry
2
mi M i
2
2
KbarN
2 s
(I 0)
Reduced energy: i
i Ei
i Ei
Energy dependence of Vij is
controlled by CM energy √s.
C ij
3
πΣ
3
2
4
Chiral SU(3) potential (KSW)
N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)
Energy dependence
( I 0)
V ij
r 0, s
@ a 0.5 fm , f p 100 M eV
KbarN-πΣ
√s [MeV]
KbarN-KbarN
πΣ-πΣ
πΣ threshold
KbarN threshold
Calculational procedure
Chiral SU(3) potential = Energy-dependent potential
Self consistency for the energy!
Assume the values of the CM energy √s.
s Assum ed
H T V
MB
s
M
Perform the Complex Scaling method.
Then, find a pole of resonance or bound state.
Check
s Calculated
s C alculated
If Yes
s Assumed
If No
Finished !
やったこと
1. Chiral SU(3) potentialで、r-space ガウス型を仮定
( I 0)
(I 0)
V ij
r
C ij
2
8 fp
i
j
M iM
j
s i
j
g r
g r
1
r a 2
exp
3/2 3
p a
N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)
レンジパラメータ a、相互作用の強さ fπ の決定が必要
2. A.D. Martinの解析による I=0 KbarN散乱帳 aKbarN(I=0)を再現するような (a, fπ)を探す。
a K barN 1.70 0.07 i 0.68 0.04 fm
(I 0)
散乱振幅の計算は、Kruppa流に複素スケーリングを使って計算
A. T. Kruppa, R. Suzuki and K. Kato, PRC 75, 044602 (2007)
3. 見つけた(a, fπ)に対して、これまでの結合チャネル複素スケーリングで、
Λ(1405)に相当するKbarN-πΣ系の共鳴エネルギーを求める。
Chiral SU(3) potentialはエネルギー依存性をもつが、
複素エネルギーに対してself consistencyを課して計算。
結果のまとめ
1. この形のポテンシャルではMartinの散乱長の実部・虚部を同時に
合わせることはできない。fπ=90～120MeVの範囲では。
fπ=75.5MeVというカイラルリミットより小さい値を使えば、一応再現できる。
2. 散乱長の実部・虚部、一方だけなら再現できるポテンシャルが存在。
各々の場合で、複素エネルギーに対してself-consistentなKbarN-πΣ系の
共鳴エネルギーが求まる。
散乱長実部を尊重した場合： (EKbarN, Γ/2) = (-19, 27) ～ (-5,8) MeV
虚部を尊重した場合: (EKbarN, Γ/2) = (-32～-24, 28～66) MeV
※ fπ=75.5で無理やり散乱長を再現した場合: (EKbarN, Γ/2) = (-23, 34) MeV
3. 散乱長の再現に関わらず、このポテンシャルではPDGに載っている
Λ（1405）のエネルギー及び幅 (EKbarN, Γ/2) = (-29±4, 25±1) MeV は
再現できない。
Mysterious state; Λ(1405)
Quark model prediction … calculated as 3-quark state
Λ(1405) can’t be well reproduced
as a 3-quark state!
calculated Λ(1405)
observed Λ(1405)
q
qq
N. Isgar and G. Karl, Phys. Rev. D18, 4187 (1978)
Λ(1405) with c.c. Complex Scaling Method
Kbar + N
1435
Λ(1405)
B. E. (KbarN) = 27 MeV
Γ (πΣ)
～ 50 MeV
Jπ = 1/2I = 0
π +Σ
1332
[MeV]
π
(Jπ=0-, T=1)
Kbar
(Jπ=0-, T=1/2)
L=0
L=0
N
(Jπ=1/2+, T=1/2)
Σ
(Jπ=1/2+, T=1)
KbarN-πΣ coupled system with s-wave and isospin-0 state
Λ(1405) with c.c. Complex Scaling Method
Schrödinger equation to be solved
Complex scaling of coordinate
Phenomenological potential
= Energy independent potential
Y. Akaishi and T. Yamazaki,
PRC 52 (2002) 044005
V
Chiral SU(3) potential
= Energy dependent potential
N. Kaiser, P. B. Siegel and W. Weise,
NPA 594, 325 (1995)
ABC theorem
Wave function expanded with Gaussian base
The energy of bound and resonant states is
independent of scaling
angle θ.
: complex parameters to be determined
Complex-rotate
J. Aguilar and J. M. Combes, Commun. Math. Phys. 22 (1971),269
E. Balslev and J. M. Combes, Commun. Math. Phys. 22 (1971),280
, then diagonalize
with Gaussian base.
Calc. of scattering amplitude with CSM
“Kruppa method”
A. T. Kruppa, R. Suzuki and K. Katō,
PRC 75, 044602 (2007)
•Radial Schroedinger eq.
V(r) : short-range potential
•Asymptotic behavior of wave func.
()
j l kr , h l
kr : Ricatti-Bessel, Hankel func.
Point 1
Search the solution in the form
Separate the incoming wave j l kr !
•Equation to be solved
•Asymptotic behavior of scattering part of wave func.
l , k r is not square-integrable
SC
hl
x
lp
exp i kr
2
Calc. of scattering amplitude with CSM
“Kruppa method”
A. T. Kruppa, R. Suzuki and K. Katō,
PRC 75, 044602 (2007)
•Complex scaling
i
r re ,
r
r e i 2 r e i
Equation to be solved
Point 2
square-integrable for 0 < θ < π
•Expanding with square-integrable basis function (ex: Gaussian basis)
i r : square integrable function
Linear equation:
Calc. of scattering amplitude with CSM
“Kruppa method”
A. T. Kruppa, R. Suzuki and K. Katō,
PRC 75, 044602 (2007)
•Calculation of scattering amplitude
Point 3
Cauchy theorem
dz j kz V z
r eiθ
0
SC
We don’t have l , k r
which is a solution along r,
r
SC
l ,k
z
0
but we have r
which is a solution along reiθ.
S C ,
l ,k
2. Application of CSM
to Λ(1405)
• Coupled-channel Complex Scaling Method (ccCSM)
• Energy-independent KbarN potential
Phenomenological potential (AY)
Y. Akaishi and T. Yamazaki, PRC 52 (2002) 044005
Energy-independent potential
KbarN
V
AY , I 0
436
412
πΣ
412
exp r
0
0.66 fm
2
1. free KbarN scattering data
2. 1s level shift of kaonic hydrogen atom
3. Binding energy and width of Λ(1405)
= K- + proton
Remark !
The result that I show hereafter is not new,
because the same calculation was done
by Akaishi-san, when he made AY potential.
M eV
Λ(1405) with c.c. Complex Scaling Method
trajectory
• # Gauss base (n) = 30
• Max range (b) = 10 [fm]
E
= 0 deg.
[MeV]
Λ(1405) with c.c. Complex Scaling Method
trajectory
• # Gauss base (n) = 30
• Max range (b) = 10 [fm]
E
[MeV]
= 5 deg.
Λ(1405) with c.c. Complex Scaling Method
trajectory
• # Gauss base (n) = 30
• Max range (b) = 10 [fm]
E
[MeV]
=10 deg.
Λ(1405) with c.c. Complex Scaling Method
trajectory
• # Gauss base (n) = 30
• Max range (b) = 10 [fm]
E
[MeV]
=15 deg.
Λ(1405) with c.c. Complex Scaling Method
trajectory
• # Gauss base (n) = 30
• Max range (b) = 10 [fm]
E
[MeV]
=20 deg.
Λ(1405) with c.c. Complex Scaling Method
trajectory
• # Gauss base (n) = 30
• Max range (b) = 10 [fm]
E
[MeV]
=25 deg.
Λ(1405) with c.c. Complex Scaling Method
trajectory
• # Gauss base (n) = 30
• Max range (b) = 10 [fm]
E
[MeV]
=30 deg.
Λ(1405) with c.c. Complex Scaling Method
trajectory
• # Gauss base (n) = 30
• Max range (b) = 10 [fm]
E
[MeV]
=35 deg.
Λ(1405) with c.c. Complex Scaling Method
trajectory
• # Gauss base (n) = 30
• Max range (b) = 10 [fm]
E
[MeV]
=40 deg.
Λ(1405) with c.c. Complex Scaling Method
trajectory
pS
KbarN
Resonance!
(E, Γ/2) = (75.8, 20.0)
E
[MeV]
Measured from KbarN thr.,
B. E. (KbarN) = 28.2 MeV
Γ = 40.0 MeV
… (1405) ! KbarN continuum
pS
continuum
=30 deg.
3. ccCSM with an
energy-dependent potential
for Λ(1405)
Result
Range parameter (a) and
pion-decay constant fπ are ambiguous
in this model.
Various combinations (a,fπ) are tried.
fπ = 95 ～ 105 MeV
Self-consistency for complex energy
s C alculated R e E
s C alculated E
com plex en erg y
Search for such a solution that both of real and imaginary parts of energy
are identical to assumed ones.
(B.E., Γ)Calculated = (B.E., Γ)Assumed
More reasonable?
Pole search of T-matrix is done on complex-energy plane.
T Z
V Z
V Z G0 Z T Z
Z: complex energy
KSW
fπ = 100 MeV
Self consistency for complex energy
a=0.47, θ=35°
obtained by the self-consistency
for the real energy
KbarN
-B
[MeV]
1 step
-Γ/2
[MeV]
KSW
fπ = 100 MeV
Self consistency for complex energy
a=0.47, θ=35°
KbarN
-B
[MeV]
-Γ/2
2 steps
[MeV]
KSW
fπ = 100 MeV
Self consistency for complex energy
a=0.47, θ=35°
KbarN
-B
[MeV]
-Γ/2
3 steps
[MeV]
KSW
fπ = 100 MeV
Self consistency for complex energy
a=0.47, θ=35°
KbarN
-B
[MeV]
-Γ/2
4 steps
[MeV]
KSW
fπ = 100 MeV
Self consistency for complex energy
a=0.47, θ=35°
KbarN
-B
[MeV]
Self consistent!
-Γ/2
5 steps
[MeV]
Self consistency for complex energy
KSW
fπ = 100 MeV
Assumed
Calc.
Assumed
Calc.
Self consistency for complex energy
-B [MeV]
KbarN
a=0.60
a=0.50
a=0.47
Repulsively
shifted!
a=0.45
S.C. for real energy
S.C. for complex energy
KSW
fπ = 100 MeV
Mean distance between Kbar (π) and N (Σ)
Kbar (π)
Distance
N (Σ)
Chiral (HW-HNJH): B～ 12 MeV, Distance = 1.86 fm
Λ（１４０５）、
他の計算との比較
Rough estimation of charge radius
Distance
KrK C M
rp C M
p
CM
M
D istance
N
mK M
mK
mK M
2
N
D istance
rc
r p C M rK C M
2
2
N
Chiral (HW-HNJH): B～ 12 MeV, Distance = 1.86 fm → <r2c> = -1.07 fm2
Charge radius
Charge radius derived from electric form factor
T. Sekihara, T. Hyodo and D. Jido, PLB 669, 133 (2008)
Rough estimation
Chiral (HW-HNJH): B～ 12 MeV, Distance = 1.86 fm → <r2c> = -1.07 fm2