Application of coupled-channel Complex Scaling Method to the K bar N -πY system
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Transcript of Application of coupled-channel Complex Scaling Method to the K bar N -πY system
Application of coupled-channel Complex Scaling Method to the KbarN-πY system
1. Introduction
2. coupled-channel Complex Scaling Method• For resonance states• For scattering problem
3. Set up for the calculation of KbarN-πY system• “KSW-type potential”• Kinematics / Non-rela. Approximation
4. Results• I=0 channel: Scattering amplitude / Resonance pole• I=1 channel: Scattering amplitude
5. Summary and future plans
A. Doté (KEK Theory center / IPNS/ J-PARC branch)T. Inoue (Nihon univ.)
T. Myo (Osaka Inst. Tech. univ.)
International workshop “Inelastic Reactions in Light Nuclei”, 09.Oct.’13 @ IIAS, The Hebrew univ., Jerusalem, Israel
A. Dote, T. Inoue, T. Myo,Nucl. Phys. A912, 66-101 (2013)
1. Introduction
Kaonic nuclei = Nuclear system with Anti-kaon “Kbar”
Excited hyperon Λ(1405) = a quasi-bound state of K- and protonAttractive KbarN interaction!
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
ubar
suu d
su d
T. Hyodo and D. Jido, Prog. Part. Nucl. Phys. 67, 55 (2012)
KbarN threshold = 1435MeV
Kaonic nuclei = Nuclear system with Anti-kaon “Kbar”
Excited hyperon Λ(1405) = a quasi-bound state of K- and protonAttractive KbarN interaction!
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
Antisymmetrized Molecular Dynamics with a phenomenological KbarN potential
Anti-kaon can be deeply bound and/or form dense state?
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 studyA. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, PLB 590 (2004) 51; PRC 70 (2004) 044313.
• RMF studyD. 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”
Excited hyperon Λ(1405) = a quasi-bound state of K- and protonAttractive KbarN interaction!
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
Antisymmetrized Molecular Dynamics with a phenomenological KbarN potential
Anti-kaon can be deeply bound and/or form dense state?
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
B. E. ~ 20MeV, Γ = 40 ~70MeVpp distance ~ 2.1 fm
A. D., T. Hyodo, W. Weise,
PRC79, 014003 (2009)
Theoretical studies of K-pp
• Doté, Hyodo, Weise PRC79, 014003(2009)
Variational with a chiral SU(3)-based KbarN potential• Akaishi, Yamazaki PRC76, 045201(2007)
ATMS with a phenomenological KbarN potential• Ikeda, Sato 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 PRC79, 014001(2009)
Variational with a phenomenological KbarN potential (with p-wave)
• Arai, Yasui, Oka/ Uchino, Hyodo, Oka PTP119, 103(2008)
Λ* nuclei model /PTPS 186, 240(2010)• Nishikawa, Kondo PRC77, 055202(2008)
Skyrme model
“K-pp” = Prototype of Kbar nuclei(KbarNN, Jp=1/2-, T=1/2)
All studies predict that K-pp can be bound!
Theoretical studies of K-pp
• Doté, Hyodo, Weise PRC79, 014003(2009)
Variational with a chiral SU(3)-based KbarN potential• Akaishi, Yamazaki PRC76, 045201(2007)
ATMS with a phenomenological KbarN potential• Ikeda, Sato 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 PRC79, 014001(2009)
Variational with a phenomenological KbarN potential (with p-wave)
• Arai, Yasui, Oka/ Uchino, Hyodo, Oka PTP119, 103(2008)
Λ* nuclei model /PTPS 186, 240(2010)• Nishikawa, Kondo PRC77, 055202(2008)
Skyrme model
“K-pp” = Prototype of Kbar nuclei(KbarNN, Jp=1/2-, T=1/2)
All studies predict that K-pp can be bound!
50 60 70 80 90 100 110 120 130
-140
-120
-100
-80
-60
-40
-20
0
Width (KbarNN→πYN) [MeV]
- B.E
. [M
eV]
Doté, Hyodo, Weise [1](Variational, Chiral SU(3))
Akaishi, Yamazaki [2](Variational, Phenomenological)
Exp. : FINUDA [6]if K-pp bound state.
Exp. : DISTO [7]if K-pp bound state.
Using S-wave KbarN potentialconstrained by experimental data. … KbarN scattering data, Kaonic hydrogen atom data, “Λ(1405)” etc.
Ikeda, Sato [4](Faddeev, Chiral SU(3))
Shevchenko, Gal, Mares [3](Faddeev, Phenomenological)
[1] PRC79, 014003 (2009) [5] PLB94, 712 (2012)[2] PRC76, 045201 (2007)[3] PRC76, 044004 (2007)[4] PRC76, 035203 (2007)
[6] PRL94, 212303 (2005)[7] PRL104, 132502 (2010)
Typical results of theoretical studies of K-pp
Barnea, Gal, Liverts [5](HH, Chiral SU(3))
-100 MeV
50 60 70 80 90 100 110 120 130
-140
-120
-100
-80
-60
-40
-20
0
Width (KbarNN→πYN) [MeV]
- B.E
. [M
eV]
Doté, Hyodo, Weise [1](Variational, Chiral SU(3))
Akaishi, Yamazaki [2](Variational, Phenomenological)
Exp. : FINUDA [6]if K-pp bound state.
Exp. : DISTO [7]if K-pp bound state.
Using S-wave KbarN potentialconstrained by experimental data. … KbarN scattering data, Kaonic hydrogen atom data, “Λ(1405)” etc.
Ikeda, Sato [4](Faddeev, Chiral SU(3))
Shevchenko, Gal, Mares [3](Faddeev, Phenomenological)
[1] PRC79, 014003 (2009) [5] PLB94, 712 (2012)[2] PRC76, 045201 (2007)[3] PRC76, 044004 (2007)[4] PRC76, 035203 (2007)
[6] PRL94, 212303 (2005)[7] PRL104, 132502 (2010)
Typical results of theoretical studies of K-pp
Barnea, Gal, Liverts [5](HH, Chiral SU(3))
Kbar+N+N threshold
π+Σ+N threshold
From theoretical viewpoint, K-pp exists between Kbar-N-N and π-Σ-N thresholds!
Key points to study kaonic nuclei
Similar to the bound-state calculation
1. Consider a coupled-channel problem
Kbar + N + N
“Kbar N N”
π + Σ + N
2. Treat resonant states adequately.
Coupling of KbarN and πY
Bound below KbarN threshold, but a resonant state above πY threshold.
Their nature?
3. Obtain the wave function to help the analysis of the state.
“coupled-channel Complex Scaling Method”
KbarN (-πY) interaction??
4. Confirmed that CSM works well on many-body systems.
“Chiral SU(3)-based potential”… Anti-kaon = Nambu-Goldstone boson
2. Complex Scaling Method
Λ(1405) = a building block of kaonic nuclei
K-
Proton
• For Resonance states
• For Scattering problem
KbarN-πΣ coupled system with s-wave and isospin-0 state
Λ(1405) with c.c. Complex Scaling Method
Kbar
(Jπ=0-, T=1/2)
N (Jπ=1/2+, T=1/2)
L=0
π(Jπ=0-, T=1)
Σ(Jπ=1/2+, T=1)
L=0
Kbar + N
Λ(1405)
π + Σ
1435
1332 [MeV]
Complex Scaling Method for Resonance
By Complex scaling, …
Complex rotation of coordinate (Complex scaling) : ,i iU e e r r k k
1 ,H U HU U
E H H
• Resonant state
exp expR Rik r i i r
κ, γ : real, >0Divergent
Damping under some condition
exp
exp sin cos exp cos sin
0 tan
iRU i i re
r ir
Complex Scaling Method for ResonanceComplex rotation of coordinate (Complex scaling)
: ,i iU e e r r k k
1 ,H U HU U
E H H
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
πΣ KbarN
pS continuum KbarN continuum
[MeV]
=30 deg.
B. E. (KbarN) = 28.2 MeVΓ = 40.0 MeV… Nominal position of Λ(1405)
Akaishi-Yamazaki potential†
• Local Gaussian form• Energy independent
2( 0), 0 exp 0.66 fmI
AY IV V r
KbarN πΣ
( 0) 436 412MeV
412 0IV
†Y. Akaishi and T. Yamazaki, PRC65, 044005 (2002)
Test calculation with a phenomenological potential
2. Complex Scaling Method
• For Resonance states
• For Scattering problem
Calculation of KbarN scattering amplitude
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 CSMA. T. Kruppa, R. Suzuki and K. Katō, PRC 75, 044602 (2007)
1. Separate incoming wave
• Unknown• Non square-integrable
exp+
llπh x i kr -2
lj kr
square-integrable for 0 < θ < π
2. Complex scaling r → reiθ
3. Calculate scattering amplitude with help of Cauchy theorem
Born term is OK!
Scattered part is unknown.
Scattered wave function along reiθ is known!By Cauchy theorem, we can obtain as
Expanding with square-integrable basis function (ex: Gaussian basis)
r0
r eiθ
, 0SCl kdz j kz V z z
Cauchy theorem
Calc. of scattering amplitude with CSMA. T. Kruppa, R. Suzuki and K. Katō, PRC 75, 044602 (2007)
square-integrable for 0 < θ < π
can be expanded with Gaussian basis
Scattering problem can be solved as bound-state problem by matrix calculation!
Equation to be solved:
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
N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)
Effective Chiral Lagrangian
Pseudopotential • Delta-function type (→ Yukawa / separable type)• Up to order q2
• Local Gaussian form in r-space
• Weinberg-Tomozawa term
• Energy dependence
“KSW-type potential”
(( )
( )2
)
8ij
ij
Ii jI
i ijj
Ij
i
C M Mr
f sg rV
2( )
3/ 2 ( )3( )1 expI
ij Ii
Ii
jjg r r a
ap
a(I)ij : range parameter [fm]
→ Easy to handle in many-body calculation with Gaussian base
→ Relative strength between channels determined by the SU(3) algebra
← Chiral SU(3) theory
( 0)332
4ij
IC
KbarN πΣ
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)
Kinematics1. Non-relativistic
2. Semi-relativistic
2 2 2 2
1/ 2
2 2
2
c cc c
c c
c cc
c c
c c c c
k kE M mM m
M mM m
k E M m
2 2 2 2 2 2
1/ 22 2
c c c c c c
c cc
c c
c c c
E M k m k
k m
2 22 2
,bar
SR c cc K N Y
KSW
H m M c c
V
p
p p
2
, 2bar
NR c cc K N Y c
KSW
H m M c c
V
p
p
Reduced mass
Reduced energy
Non-rela. approximation of KSW potential- Original
( )
( ) ( )28
IijI I
ij ii j
i jj ij
M Ms
CV r g r
fp
2( )( )3/ 2 ( ) 3
1 exp IIij ijI
ij
g r r ddp
( ) ( ) ( ) 2I I Iij ii jjd d d
• Normalized Gaussian
• Range parameter for coupling potential
→ (NRv1) Non-rela. approx. version 1
( )
( ) ( )2
18
IijI I
ij ii j
jj
ij
Mr
sg
fMC
V rp
Comparison of the flux factor for differential cross sectionbetween non-rela. and rela.
i
→ (NRv2) Non-rela. approx. version 2
( )
( ) ( )2
18
IijI I
ij i j ijjm m
CV r g r
fp
, 1i i i im M @ non-rela. limit (small p2)
Kinematics
- Original
(NRv1) Non-rela. approx. ver. 1
(NRv2) Non-rela. approx. ver. 2
( )
( ) ( )28
IijI I
ij ii j
i jj ij
M Ms
CV r g r
fp
( )
( ) ( )28
1IijI I
ij i j iji j
Tot j
CV r g r
MEf
M
p
( )
( ) ( )2
18
IijI I
ij i j ijjm m
CV r g r
fp
Non-rela.
Semi-rela.
Potential
4. Result
• I=0 channel• I=0 KbarN scattering length
Range parameters of KSW-type potential• Scattering amplitude• Resonance pole … wave function and size
Using Chiral SU(3) potential “KSW-type”… r-space, Gaussian form, Energy-dependent
I=0 KbarN scattering lengthRange parameters of KSW-type potential
2( 0) ( 0)33/ 2
1 expI Iij ij
ijijdV r V s r
dp
,
,
,babar bar rK Nij
K N K Nd d
dd
p
p pS
S
S
Range parameters: ,, ,
2bar bar barK N K N K Nd d dp pp S SS
Pion decay constant “fπ” as a parameter
fπ = 90 ~ 120 MeV
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)
Kinematics MartinKSW- type pot. NR- v1 NR- v2 SR- A SR- B
Range 0.44 0.44 0.50 0.37parameters [fm] 0.61 0.64 0.71 0.35
KbarN scatt. Re - 1.701 - 1.700 - 1.700 - 1.696 - 1.70length [fm] Im 0.681 0.681 0.681 0.681 0.68
Non- rela. Semi- rela.
dKN,KN
dπΣ,πΣ
Re aI=0KN
Im aI=0KN
fπ=110 MeV
Scattering amplitude … Non-rela. / KSW-type NRv1
fπ=110 MeV
( 0)
( 0)2
18
Iij i jI
ijj
i j ij
C M MV r g r
f sp
πΣ KbarN πΣ KbarN
Resonance structure appears in KbarN and πΣ channels.
ReIm
KbarN → KbarN πΣ → πΣ
Resonance structure at 1413 MeV
Resonance structure at 1405 MeV
Scattering amplitude … Non-rela. / KSW-type NRv2
fπ=110 MeV
πΣ KbarN πΣ KbarN
( 0)
( 0)2
18
IijI
ij i j ijj
CV r g r
f m mp
Essentially same as NRv1 case.
ReIm
KbarN → KbarN πΣ → πΣ
Resonance structure at 1416 MeV
Resonance structure at 1408 MeV
Scattering amplitude … Semi-rela. / KSW-type SR-A
fπ=110 MeV
πΣ KbarN πΣ KbarN
Qualitatively similar to non-rela. amplitudes.
( 0)
( 0)28
Iij i jI
ij i j iji j
C M MV r g r
f sp
ReIm
KbarN → KbarN πΣ → πΣ
Resonance structure at 1410 MeV
Resonance structure at 1398 MeV
Scattering amplitude … Semi-rela. / KSW-type SR-B
fπ=110 MeV
πΣ KbarN πΣ KbarN
Very different behavior of πΣ amplitude from other cases.
( 0)
( 0)28
Iij i jI
ij i j iji j
C M MV r g r
f sp
ReIm
KbarN → KbarN πΣ → πΣ
(Another set of SR)
???
Resonance structure at 1419 MeV
Resonance structure at 1421 MeV
Pole position of the resonance fπ=90 - 120MeV
fpi NRv1 NRv2 SR- A SR- BM - Γ / 2 M - Γ / 2 M - Γ / 2 M - Γ / 2
90 1419.8 - 26.0 1419.9 - 23.1 1423.6 - 26.4 1419.0 - 14.4100 1417.3 - 23.1 1418.0 - 19.8 1421.5 - 26.3 1419.6 - 13.2110 1416.6 - 19.5 1417.8 - 16.6 1419.5 - 25.0 1420.0 - 12.8120 1416.9 - 16.7 1418.3 - 14.0 1417.5 - 22.9 1418.9 - 11.7
- Γ /
2 [M
eV]
90
120
90
120
90
120
M [MeV]
120
90
- Γ /
2 [M
eV]
Complex energy plane
Pole position of the resonance
- Γ /
2 [M
eV]
fπ=90 - 120MeV
fpi NRv1 NRv2 SR- A SR- BM - Γ / 2 M - Γ / 2 M - Γ / 2 M - Γ / 2
90 1419.8 - 26.0 1419.9 - 23.1 1423.6 - 26.4 1419.0 - 14.4100 1417.3 - 23.1 1418.0 - 19.8 1421.5 - 26.3 1419.6 - 13.2110 1416.6 - 19.5 1417.8 - 16.6 1419.5 - 25.0 1420.0 - 12.8120 1416.9 - 16.7 1418.3 - 14.0 1417.5 - 22.9 1418.9 - 11.7
90
120
90
120
90
120
120
90
M [MeV]
Non-rela.
(M, Γ/2) = (1418.2 ± 1.6, 21.4 ± 4.7) … NRv1 (1418.9 ± 1.1, 18.6 ± 4.6) … NRv2
- Γ /
2 [M
eV]
Pole position of the resonance
- Γ /
2 [M
eV]
fπ=90 - 120MeV
fpi NRv1 NRv2 SR- A SR- BM - Γ / 2 M - Γ / 2 M - Γ / 2 M - Γ / 2
90 1419.8 - 26.0 1419.9 - 23.1 1423.6 - 26.4 1419.0 - 14.4100 1417.3 - 23.1 1418.0 - 19.8 1421.5 - 26.3 1419.6 - 13.2110 1416.6 - 19.5 1417.8 - 16.6 1419.5 - 25.0 1420.0 - 12.8120 1416.9 - 16.7 1418.3 - 14.0 1417.5 - 22.9 1418.9 - 11.7
90
120
90
120
90
120
M [MeV]
120
90
Semi-rela.
(M, Γ/2) = (1420.5 ± 3, 24.5 ± 2) … SR-A
- Γ /
2 [M
eV]
Pole position of the resonance
- Γ /
2 [M
eV]
fπ=90 - 120MeV
fpi NRv1 NRv2 SR- A SR- BM - Γ / 2 M - Γ / 2 M - Γ / 2 M - Γ / 2
90 1419.8 - 26.0 1419.9 - 23.1 1423.6 - 26.4 1419.0 - 14.4100 1417.3 - 23.1 1418.0 - 19.8 1421.5 - 26.3 1419.6 - 13.2110 1416.6 - 19.5 1417.8 - 16.6 1419.5 - 25.0 1420.0 - 12.8120 1416.9 - 16.7 1418.3 - 14.0 1417.5 - 22.9 1418.9 - 11.7
90
120
90
120
90
120
120
90
M [MeV]
Semi-rela.(Another set)
(M, Γ/2) = (1419.5 ± 0.5, 13.0 ± 1.4) … SR-B
Different behavior from other cases
- Γ /
2 [M
eV]
“Wave function” of the resonance pole fπ=110 MeVθ=30°
Non-rela. (NRv2)
πΣ component also localizeddue to Complex scaling.
Mean distance between meson and baryon
2MBr fm
B
mNR: ~ 1.3 - 0.3 i [fm]SR: ~ 1.2 - 0.5 i [fm]
? 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)
4. Result
• I=1 channel
• Range parameters of the KSW-type potential• Scattering amplitude
Using Chiral SU(3) potential (KSW-type)… r-space, Gaussian form, Energy-dependent
I=1 channel … KbarN - πΣ - πΛ
Data : aI=1KbarN = 0.37 + i 0.60 fm
(A. D. Martin)† 1 1
.barbar barI I
K N thrK N K Na f s
, , ,
, ,
,
bar bar bar barK N K N K N K N
ij
d d d
d d dd
p p
p p p p
p p
S
S S S
Range parameters:
† A. D. Martin, Nucl. Phys. B 179, 33 (1981)
( 1)
2( 1)32 3/ 2
1 exp8
IijI
ij i jij
ij
CV r flux factor r
fd
dp
p
Potential … Weinberg-Tomozawa term only
I=1 channel … KbarN - πΣ - πΛ
1 1.barbar bar
I IK N thrK N K N
a f s
, , ,
, ,
,
bar bar bar barK N K N K N K N
ij
d d d
d d dd
p p
p p p p
p p
S
S S S
Range parameters:
( 1)
1 1 3 200
2IijC
SU(3) C.G. coefficients for I=1 channel
××
( 1)
2( 1)32 3/ 2
1 exp8
IijI
ij i jij
ij
CV r flux factor r
fd
dp
p
Data : aI=1KbarN = 0.37 + i 0.60 fm
(A. D. Martin)†
,, ,2bar bar barK N K N K N
d d dp pp S SS
Potential … Weinberg-Tomozawa term only
I=1 channel … KbarN - πΣ - πΛ
1 1.barbar bar
I IK N thrK N K N
a f s
, ,
,
,
,
,
bbar ba bar ar rK NK N K N
j
K N
i
d
d dd
d d
dpp
p p
p p
p p
S
S
S S
Range parameters:
( 1)
1 1 3 200
2IijC
SU(3) C.G. coefficients for I=1 channel
××
,, ,2bar bar barK N K N K N
d d dp pp S SS
( 1)
2( 1)32 3/ 2
1 exp8
IijI
ij i jij
ij
CV r flux factor r
fd
dp
p
Data : aI=1KbarN = 0.37 + i 0.60 fm
(A. D. Martin)†
Potential … Weinberg-Tomozawa term only
I=1 channel … KbarN - πΣ - πΛ
1 1.barbar bar
I IK N thrK N K N
a f s
,0 0
,
,
,
,0
,
babar bar ba rr K NI IK N K N K N
Iij
d
d d
dd
dd p p
p
p
p
p
p
p
S
S S S
Range parameters:
( 1)
1 1 3 200
2IijC
SU(3) C.G. coefficients for I=1 channel
××
“Isospin symmetric choice”
Ref.) chiral unitary model
( 1)
2( 1)32 3/ 2
1 exp8
IijI
ij i jij
ij
CV r flux factor r
fd
dp
p
Data : aI=1KbarN = 0.37 + i 0.60 fm
(A. D. Martin)†
,, ,2bar bar barK N K N K N
d d dp pp S SS
Potential … Weinberg-Tomozawa term only
I=1 channel … KbarN - πΣ - πΛ
A. D. Martin : aI=1KbarN = 0.37 + i 0.60 fm
• “Iso-symmetric choice” : {dKN,KN, dπΣ,πΣ} fixed to I=0 channel ones• Search dKN,πΣ to reproduce Re or Im aI=1
KbarN of Matin’s value.
Difficult to reproduce Re aI=1KbarN Martin’s value within our model.
When Re aI=1KbarN is reproduced, Im aI=1
KbarN deviates largely from Martin’s value.
(fit)(fit)
(fit)(fit)
(I=0 one)
(Search)
I=1 channel … KbarN - πΣ - πΛ
NRv2 (c): aI=1KbarN = 0.657 + i 0.599 fm
fπ=110 MeV
Search dKN,πΣ to reproduce Im aI=1KbarN of Matin’s value.
KbarN πΣ πΛ
KbarN πΣ πΛ
SR-A (c): aI=1KbarN = 0.659 + i 0.600 fm
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.
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
(fit)(fit)(fit)
(fit)
† Y. Ikeda and T. Sato, PRC 76, 035203 (2007)
(I=0 one)
(Search)
I=1 channel … KbarN - πΣ - πΛ
NRv2 (a): aI=1KbarN = 0.376 + i 0.606 fm
fπ=110 MeV
Search {dπΣ,πΣ, dKN,πΣ} to reproduce Re and Im aI=1KbarN of Matin’s value.
SR-A (a): aI=1KbarN = 0.375 + i 0.605 fm
KbarN πΣ πΛ
KbarN πΣ πΛ
A narrow resonance exists at a few MeV below πΣ threshold
πΣ repulsive ???
5. Summary and
Future plan
5. Summary
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.
KbarN-πY system is essential for the study of Kbar nuclear system which is expected to be an exotic nuclear system with strangeness.
5. Summary and future plans
Future plansThree-body system (KbarNN-πYN); Updated data of K-p scattering length by SHIDDARTA
• Pole position • “Size” of the I=0 pole stateNR: ~ 1.3 – 0.3i fmSR: ~ 1.2 – 0.5i fm
(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)
I=0 channel (KbarN-πΣ)
I=1 channel (KbarN-πΣ-πΛ)• Difficult to reproduce Re aI=1
KbarN of Martin’s value within our model, in case of “iso-symmetric choice” of range parameter.
Another self-consistent solution NR: (~1360, 40~90), SR: (1350~1390, 30~100) … Lower pole of double pole?
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 radiusK-
p
Distance
CM
NK CM
K N
Kp CM
K N
Mr Distancem M
mr Distancem M
2 2 2c p CM K CM
r r r
† Calculated from electric form factor with chiral unitary model T. Sekihara, T. Hyodo and D. Jido, PLB 669, 133 (2008)
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
(Sekihara et al)† 1426.11 – i 16.54 -0.131 + i 0.303
Charge radius
Chiral (HW-HNJH): B ~ 12 MeV, Distance = 1.86 fm → <r2c> = -1.07 fm2
Charge radius derived from electric form factor
T. Sekihara, T. Hyodo and D. Jido, PLB 669, 133 (2008)
Rough estimation
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 nucleusInteresting structures that we have never seen in normal nuclei…
3He
0.0 0.15
K- ppn
0.0 2.0
K- ppp
3 x 3 fm2
0.0 2.0 [fm-3]
Antisymmetrized Molecular Dynamics method with a phenomenological KbarN potential
A. Dote, H. Horiuchi, Y. Akaishi and T. Yamazaki, PRC70, 044313 (2004)
Relate to various interesting physics such as … Restoration of chiral symmetry in dense matter Interesting structure Neutron star
Excited hyperon Λ (1405)
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
ubar
suu d
su d
T. Hyodo and D. Jido, Prog. Part. Nucl. Phys. 67, 55 (2012)
Today, K-pp has been focused in theor. and exp. studies!As the first trial of our study with ccCSM, consider Λ(1405).
K-P P
Prototype of Kaonic nuclei
3HeK-, pppK-, 4HeK-, pppnK-, …8BeK-,…
Complicated nuclear system with K-
K-
Proton
Λ(1405)
=
Building block of Kaonic nuclei
Asymptotic behavior of the wave function before/after complex scaling
exp
exp cos exp sin
0 2
iB B
B B
U re
r i r
p
exp
exp cos sin
exp sin cos exp cos sin
0 tan
iRU i i re
i i r i
r ir
exp
exp
i ikU i ke re
ikr
• Bound state
• Resonant state
• Continuum state
expB Br
exp
expR Rik r
i i r
expk ikr
κB : real, >0
κ, γ : real, >0
k : real, >0
Damping Damping under some condition
Oscillating Oscillating
Diverging Damping under some condition
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 22 2 2
02 , exp2
i idT V V r am dr
e e
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 CSMA. T. Kruppa, R. Suzuki and K. Katō, PRC 75, 044602 (2007)
Complex scaling
r → r eiΘ
• Unknown• Non square-integrable
exp+
llπh x i kr -2
Separate the incoming wave ! lj krPoint 1
Point 2
square-integrable for 0 < θ < π
Expanding with square-integrable basis function (ex: Gaussian basis)
• Calculation of scattering amplitude
We don’t have which is a solution along r,
but we have which is a solution along reiθ.
,SCl k r
,,SCl k r θ
r0
r eiθ
, 0SCl kdz j kz V z z
Cauchy theoremPoint 3
Calc. of scattering amplitude with CSM
is independent of θ. SClf k
expressed with Gaussian base
A. T. Kruppa, R. Suzuki and K. Katō, PRC 75, 044602 (2007)
Test calculation with AY potentialResonance pole
πΣ KbarN
pS continuum KbarN continuum
[MeV]
=30 deg.
B. E. (KbarN) = 28.2 MeVΓ = 40.0 MeV
… Nominal position of Λ(1405)
Akaishi-Yamazaki potential†
• Phenomenological potential• Local Gaussian form• Energy independent
2( 0), 0 exp 0.66 fmI
AY IV V r
KbarN πΣ
( 0) 436 412MeV
412 0IV
†Y. Akaishi and T. Yamazaki, PRC65, 044005 (2002)
Complex Scaling Method for ResonanceStudy of unstable nuclei … Resonance of 8He
“Cluster-Orbital Shell Model”
T. Myo, R. Ando, K. Kato, PLB691, 150 (2010)
Bound state
Resonance
Continuumon 2θ line
Test calculation with AY potentialUnitarity violation of S-matrix
Phase shift sum
| |Det S| -1 |
EKbarN [MeV]
EKbarN [MeV]
Checked by Continuum Level Density method (R. Suzuki, A. T. Kruppa, B. G. Giraud, and K. Katō, PTP119, 949(2008))
δKbarN + δπΣ [deg.]
Test calculation with AY potential
KbarN (I=0) scattering amplitude
Scattering length (Scatt. amp. @ EKbarN=0) = -1.77 + i 0.47 fm
Y. Akaishi and T. Yamazaki, PRC65, 044005 (2002)
Scattering length = -1.76 + i 0.46 fm
0IKN KNf fm
EKbarN [MeV]
Scattering amplitude … Non-rela. / KSW-type SR
fπ=90 MeV
πΣ KbarN πΣ KbarN
( 0)
( 0)28
Iij i jI
ij i j iji j
C M MV r g r
f sp
Kinematics-potential mismatched case
ReIm
KbarN → KbarN πΣ → πΣ
… A virtual state exists. (aπΣ, re)=(61,-6.3) fm satisfies -aπΣ /2 < re
†. † Y. Ikeda et al, PTP 125, 1205 (2011)
Singular behavior at πΣ threshold
I=0 KbarN scattering lengthRange parameters of KSW-type potential
fπ=110 MeV
Data : aI=0KbarN = -1.70 + i 0.68 fm (A. D. Martin)
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)
Kinematics MartinKSW- type pot. NR- v1 NR- v2 SR- A SR- B
Range 0.44 0.44 0.50 0.37parameters [fm] 0.61 0.64 0.71 0.35
KbarN scatt. Re - 1.701 - 1.700 - 1.700 - 1.696 - 1.70length [fm] Im 0.681 0.681 0.681 0.681 0.68
Non- rela. Semi- rela.
dKN,KN
dπΣ,πΣ
Re aI=0KN
Im aI=0KN
Resonant structure in the scattering amplitude
Energy @ Re fKbarN → KbarN = 0 Energy @ Re fπΣ → πΣ = 0
fπ=90 - 120MeV
KSW - Semi rela.KbarN → KbarNfπ = 90MeV
Resonant structure in the scattering amplitude
Energy @ Re fKbarN → KbarN = 0 Energy @ Re fπΣ → πΣ = 0
fπ=90 - 120MeV
Strange behavior?
SR-B
SR-A
“Wave function” of the resonance pole fπ=110 MeVθ=30°
πΣ component also localizeddue to Complex scaling.
* The wave functions shown above are multiplied by a phase factor so that the KbarN wfn. becomes real at r=0.
Non-rela. (NRv2) Semi rela. (SR-A)
Semi rela. (SR-B)
“Size” of the resonance pole state
2MBr fm
Mean distance between meson and baryon
Re
Im
fπ=90 - 120MeV
NR: ~ 1.3 - 0.3 i [fm]SR: ~ 1.2 - 0.5 i [fm]
B
M
? 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)
Strange nuclear physics
Nucleus
Hypernuclei…u
d sHyperon
baryon = qqq
Strangeness is introduced through baryons.
Strange nuclear physics
Nucleus
Kaonic nuclei !
ubar
sK- meson(anti-kaon, Kbar)
Strangeness is introduced through mesons …
meson = qqbar
Key points to study kaonic nuclei
Similar to the bound-state calculation
1. Consider a coupled-channel problem
Kbar + N + N
“Kbar N N”
π + Σ + N
2. Treat resonant states adequately.
KbarN couples with πY.
Bound state for KbarN channelbut a resonant state for πY channel
Their nature?
3. Obtain the wave function to help the analysis of the state.
“coupled-channel Complex Scaling Method”
KbarN (-πY) interaction??
“Chiral SU(3)-based potential”
Excited hyperon Λ (1405)
939 p,n
1115 Λ
1190 Σ
p + K-1435
Λ(1405)1405
1250 Λ + π
Σ + π1325
Ener
gy [M
eV]
su d
Mysterious state; Λ(1405)Quark model prediction … calculated as 3-quark state
N. Isgar and G. Karl, Phys. Rev. D18, 4187 (1978)
qq q
Λ(1405) can’t be well reproduced as a 3-quark state!
observed Λ(1405)calculated Λ(1405)
( 0)
( 0)28
ij
ij
i j
i j
II
i j
CV r r
fM
sg
M
p
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。
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
23
2
1 exp2
ii j j i ij
j
k d i Vk
p
r k k r r
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
31 exp2ij j i ijf d i Vp
r k k r r
2 2ij i i iij ij
i j j j
d v kf fd v k
fij : 散乱振幅vi : 速度
kij : 運動量μi : 換算質量
23
23
2 4
1 exp2
1 1 exp2
ij i ij i ij
i j j
ij j i ij
j
d k d i Vd k
k d i Vk
p
p
r k k r r
r k k r r
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
23
2 4
1 1 exp2
ij ij j i ij
i j
d k d i Vd k
p r k k r r
23
2
1 exp2
ij ij j i ij
i j
d k d i Vd k
p r k k r r相対論的 :
(KSW)
非相対論的 :
は1に取っている。
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。
3. KSW 非相対論版の考察
推 測 KSW の pseudo-potential 中のファクター
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?
i j
i j
M Ms
つまり
( 0)( 0)
2
18
ij
ij
Ii jI
ijTo
jt
C M MV r g r
Efp
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??
3. KSW 非相対論版の考察
Comparison with other studies of I=0 channel
Comparison with other studies of I=0 KbarN-πΣ
Kaiser-Siegel-Weise
Hyodo-Weise Ikeda-Hyodo-Jido-Kamano-Sato-Yazaki
Dote-Inoue-Myo
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 formGaussian form
Constraint
Low energy scattering data,
Branching ratio at
KbarN threshold
Low energy scattering data,
Branching ratio at KbarN threshold
KbarN scattering length
/ Λ(1405) pole
KbarN scattering length
Ref. NPA 594, 325 (1995) PRC 77, 035204 (2008) PTP 125, 1205 (2011) NPA 912, 66 (2013)
Comparison with other studies of I=0 KbarN-πΣ
IHJKSY, HW:
DIM:
Different energy dependence of interaction kernel
Proportional to just meson energy
Not proportional to meson energy… including relativistic q2 correction and non-static effect of baryons
Comparison with other studies of I=0 KbarN-πΣ
IHJKSY
HW
Comparison with other studies of I=0 KbarN-πΣUse 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. IntroductionNeutron star in universe
Radius ~ 10kmMass ~ Msun
High dense at core ~ > 2 ρ0
Maybe …
Involving strangeness
Kbar nuclear system on the Earth
• Self-bound Kbar-nuclear system
K-
… light nucleus with K- mesons Involving strangeness via K- meson
1. IntroductionNeutron star in universe
Radius ~ 10kmMass ~ Msun
High dense at core ~ > 2 ρ0
Maybe …
Involving strangeness
Kbar nuclear system on the Earth
Density [fm-3]0.00 0.14
3He
Density [fm-3]0.0 0.75 1.5
3HeK- 4×4 fm2… light nucleus with K- mesons
Involving strangeness via K- meson
Strong attraction by K- meson ⇒ Dense nuclear system
† A. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, PLB 590 (2004) 51; PRC 70 (2004) 044313.
A phenomenological KbarN potential† ⇒ B(K) ~ 100MeV, 2 ~ 4ρ0
• FINUDA collaboration (DAΦNE, Frascati)
• K- absorption at rest on various nuclei (6Li, 7Li, 12C, 27Al, 51V)• Invariant-mass method
Strong correlation between emitted p and Λ (back-to-back)
Invariant mass of p and Λ
K-pp
p
ΛIf it is K-pp, …Total binding energy = 115 MeVDecay width = 67 MeV
PRL 94, 212303 (2005)
6 35 4
14 211 3
Experimental studies of K-pp
• Re-analysis of KEK-PS E549
• DISTO collaboration
- p + p -> K+ + Λ + p @ 2.85GeV- Λp invariant mass- Comparison with simulation data
T. Yamazaki et al. (DISTIO collaboration), PRL104, 132502 (2010)
- K- stopped on 4He target- Λp invariant mass
T. Suzuki et al (KEK-PS E549 collaboration), arXiv:0711.4943v1[nucl-ex]
K- pp???B. E.= 103 ±3 ±5 MeVΓ = 118 ±8 ±10 MeV
Strong Λp back-to-back correlation is confirmed.Unknown strength is there in the same energy region as FINUDA.
Experimental studies of K-pp
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 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
Dr. Fujioka’s talk (KEK workshop, 7-9. Aug. 08)
at K1.8BR beam line
1.8GeV/c
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
Missing mass spectroscopy
at K1.8BR beam line
1.8GeV/c
Invariant mass spectroscopy
Dr. Fujioka’s talk (KEK workshop, 7-9. Aug. 08)
All emitted particles will be measured. 「完全実験」
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
d
π+ K+
Λ(1405)np p
pK-
Preceding E15, E27 experiment was performed in June. Analysis is now undergoing!
Use lots of pion Missing mass
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
KbarN interaction
Λ(1405)
K-pp
K-pp
These days, K-pp has been focused in both of theoretical and experimental studies!
K-
Proton
Λ(1405)
3HeK-, pppK-, 4HeK-, pppnK-, …8BeK-,…
Complicated nuclear system with K-
=
K-P P
Most essential kaonic nucleus!
Coupled Channel Chiral Dynamics (Chiral Unitary model)
We are thinking about KbarN interaction.S=-1 meson-baryon system is constrained by
The leading couplings between
Low mass pseudo-scalar meson octet (Nambu-Goldston bosons)
and Baryon octet
are determined by
Spontaneous breaking ofSU(3)×SU(3) Chiral symmetry
Chiral SU(3) dynamics !
Chiral low-energy theorem tells us …
KbarN πΣ ηΛ KΞ
For I=0 channel,
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.
Energy and mass of baryon in channel i
and
Weinberg-Tomozawa term derived from Chiral SU(3) effective Lagrangian
f : Psuedo-scalar meson decay constant
Coupled Channel Chiral Dynamics (Chiral Unitary model)
3. Solve coupled channel Bethe-Salpeter equation.
T-matrix of coupled channel scattering
+ … + + … …
=
2. Using the WT-term as a building block,
1. Weinberg-Tomozawa term derived from Chiral SU(3) effective Lagrangian:
• K-p scattering length • Threshold branching ratio• Total cross section of K-p scattering
Coupled Channel Chiral Dynamics (Chiral Unitary model)
• Threshold branching ratio • Total cross section of K-p scattering
T. Hyodo, S. I. Nam, D. Jido, and A. Hosaka, Phys. Rev. C68, 018201 (2003)
Coupled Channel Chiral Dynamics (Chiral Unitary model)
3. Solve coupled channel Bethe-Salpeter equation.
T-matrix of coupled channel scattering
+ … + + … …
=
2. Using the WT-term as a building block,
Λ(1405) is dynamically generated as meson-baryon system.
1. Weinberg-Tomozawa term derived from Chiral SU(3) effective Lagrangian:
• K-p scattering length • Threshold branching ratio• Total cross section of K-p scattering
Coupled Channel Chiral Dynamics (Chiral Unitary model)
I=0 πΣ mass distribution
T. Hyodo, S. I. Nam, D. Jido, and A. Hosaka, Phys. Rev. C68, 018201 (2003)
Dynamical generation of Λ(1405) !
Remark:Calculated with πΣ-πΣ scattering amplitude.
Coupled Channel Chiral Dynamics (Chiral Unitary model)
Chiral SU(3) potential (KSW)N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)
a: range parameter [fm]Mi , mi : Baryon, Meson mass in channel iEi : Baryon energy, ωi : Meson energy
2 22 2 2 2
,2 2
i i i i
i i
s s M
sE
s
m M m
Reduced energy: i ii
i i
EE
( 0)332
4ij
IC
KbarN πΣ
Energy dependence of Vij is controlled by CM energy √s.
Flavor SU(3) symmetry
Original: δ-function type
Present: Normalized Gaussian type
(( )
( )2
)
8ij
ij
Ii jI
i ijj
Ij
i
C M Mr
f sg rV
2( )
3/ 2 ( )3( )1 expI
ij Ii
Ii
jjg r r a
ap
a: range parameter [fm]
Energy dependence is determined by Chiral low energy theorem. ← Kaon: Nambu-Goldstone boson
( )
( )28
ij
ij
Ii jI
i ji j
C M MV r
fr
s
Chiral SU(3) potential (KSW)Energy dependence
( 0) 0, @ 0.5 , 100ij
IV r s a fm f MeVp
√s [MeV]
KbarN-KbarN
πΣ-πΣ
KbarN-πΣ
N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)
KbarN thresholdπΣ threshold
Chiral SU(3) potential = Energy-dependent potential
Calculational procedure
Perform the Complex Scaling method.
Then, find a pole of resonance or bound state. Calculateds
Calculated Assumeds sCheck Finished !If Yes
Self consistency for the energy!
Assume the values of the CM energy √s.
MBH T V s M Assumeds
If No
4. Experiments related to Kbar nuclear
physics
K原子核に関係する実験
• LEPS / SPring-8
πΣ invariant mass 測定
K原子 (Kaonic atom)
Λ(1405)
• CLAS / JLab
• 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 hydrogen atom, 1s レベルシフト @ DEAR group, DAΦNE, G. Beer et al., Phys. Rev. Lett. 94, 212302 (2005) Frascati National Laboratories
• Kaonic hydrogen, deuterium @ SIDDHARTA group
• Kaonic 3He atom, 2p レベルシフト (3d→2p X 線測定 ) @ J-PARC (E17, DAY-1)
γ + p → K+ + Λ(1405), Λ(1405) → π Σ
K. Moriya and R. Schumacher, Nucl. Phys. A835, 325 (2010)J. K. Ahn, Nucl. Phys. A835, 329 (2010)
π-Σ+, π0Σ0, π+Σ- が全て押さえられた
M. Bazzi et al., Phys. Lett. B704, 113 (2011)
K原子核に関係する実験
• LEPS / SPring-8
πΣ invariant mass 測定
K原子 (Kaonic atom)
Λ(1405)
• CLAS / JLab
• 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 hydrogen atom, 1s レベルシフト @ DEAR group, DAΦNE, G. Beer et al., Phys. Rev. Lett. 94, 212302 (2005) Frascati National Laboratories
• Kaonic hydrogen, deuterium @ SIDDHARTA group
• Kaonic 3He atom, 2p レベルシフト (3d→2p X 線測定 ) @ J-PARC (E17, DAY-1)
γ + p → K+ + Λ(1405), Λ(1405) → π Σ
K. Moriya and R. Schumacher, Nucl. Phys. A835, 325 (2010)J. K. Ahn, Nucl. Phys. A835, 329 (2010)
π-Σ+, π0Σ0, π+Σ- が全て押さえられた
M. Bazzi et al., Phys. Lett. B704, 113 (2011)
DEAR exp. for kaonic hydrogen atom
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!
Kaonic hydrogen atom, 1s のレベルシフト @ DEAR Collaboration, DAΦNE, Frascati National Laboratories
G. Beer et al., Phys. Rev. Lett. 94, 212302 (2005)
cf) KEK exp. M.Iwasaki et al., Phys. Rev. Lett. 78, 3067 (1997)
B. Borasoy et al., Phys. Rev. Lett. 94, 213401 (2005)
KEK exp.
DEARCoupled 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。。。
SHIDDARTA exp. for kaonic hydrogen atom
KEK 実験(KpX)とコンシステントな結果
Kaonic hydrogen atom, 1s のレベルシフト @ SHIDDARTA Collaboration, DAΦNE, Frascati National Laboratories
M. Bazzi et al., Phys. Lett. B704, 113 (2011)
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。
Y. Ikeda, T. Hyodo and W. Weise, Phys. Lett. B706, 63 (2011)
K原子核に関係する実験
• LEPS / SPring-8
πΣ invariant mass 測定
K原子 (Kaonic atom)
Λ(1405)
• CLAS / JLab
• 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 hydrogen atom, 1s レベルシフト @ DEAR group, DAΦNE, G. Beer et al., Phys. Rev. Lett. 94, 212302 (2005) Frascati National Laboratories
• Kaonic hydrogen, deuterium @ SIDDHARTA group
• Kaonic 3He atom, 2p レベルシフト (3d→2p X 線測定 ) @ J-PARC (E17, DAY-1)
γ + p → K+ + Λ(1405), Λ(1405) → π Σ
K. Moriya and R. Schumacher, Nucl. Phys. A835, 325 (2010)J. K. Ahn, Nucl. Phys. A835, 329 (2010)
π-Σ+, π0Σ0, π+Σ- が全て押さえられた
M. Bazzi et al., Phys. Lett. B704, 113 (2011)
KEK E570 for kaonic 4He atom
シフトは 0 eV と consisitent
Kaonic 4He atom, 2p のレベルシフト 3d→2p X 線測定 @ KEK, E570
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
“Kaonic helium puzzle”
S. Okada et. al., Phys. Lett. B653, 387 (2007)
パズルは解けた!S. Hirenzaki et al., Phys. Rev. C61, 055205 (2000)
J-PARC for kaonic 3He atomKaonic 3He atom, 2p のレベルシフト 3d→2p X 線測定 @ J-PARC, E17 DAY-1
S. Okada et. al., Phys. Lett. B653, 387 (2007)
Kaonic 4He atom, 2p のレベルシフト … ほぼ 0 eV と確定
赤石氏の計算
KbarN potential の強度が二つの領域に絞られた。
+
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。
Y. Akaishi, Proceedings of EXA’05, Austrian Academy of Sciences press, Vienna, 2005, p.45
※ 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
K原子核に関係する実験
• LEPS / SPring-8
πΣ invariant mass 測定
K原子 (Kaonic atom)
Λ(1405)
• CLAS / JLab
• 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 hydrogen atom, 1s レベルシフト @ DEAR group, DAΦNE, G. Beer et al., Phys. Rev. Lett. 94, 212302 (2005) Frascati National Laboratories
• Kaonic hydrogen, deuterium @ SIDDHARTA group
• Kaonic 3He atom, 2p レベルシフト (3d→2p X 線測定 ) @ J-PARC (E17, DAY-1)
γ + p → K+ + Λ(1405), Λ(1405) → π Σ
K. Moriya and R. Schumacher, Nucl. Phys. A835, 325 (2010)J. K. Ahn, Nucl. Phys. A835, 329 (2010)
π-Σ+, π0Σ0, π+Σ- が全て押さえられた
M. Bazzi et al., Phys. Lett. B704, 113 (2011)
Λ(1405) - πΣ invariant mass 測定 -
三つの異なる電荷状態が抑えられた
• LEPS / Spring-8K. Moriya and R. Schumacher, Nucl. Phys. A835, 325 (2010)
• CLAS / Jefferson Laboratory
γ + p → K+ + Λ(1405), Λ(1405) → π Σ
ピークの順番が理論(chiral unitary) と違う?
Highest peak実験: Σ+ π-
理論: Σ- π+
理論
J. K. Ahn, Nucl. Phys. A835, 329 (2010)
p (γ, K+ π) Σ at Eγ = 1.5-2.4GeVCharged πΣ を測定
What is the object observed experimentally?
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”…
• DISTO collaboration
Self-consistent solutions for complex energy
0 0KN I Ki N In outZ ZH p p S S 2Z E i : complex energy
Self-consistent solution … in outZ Z
Ex) KSW ; fπ = 100 MeV, a= 0.47 fm ( θ=35°)
Self-consistency for complex energy should be considered!
Chiral potential … Energy-dependent potential
( 0)
( 0)28
Iij i jI
i ji j
j i a
C M MV r g r
f sp
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)
•fπ = 90 MeV
Scattering amplitude
Scattering amplitude … KSW org. – Non. rela.
• Phase shift (πΣ)
KbarN → KbarN
Re
Im
πΣ → πΣ
KbarN → πΣ
δ ( πΣ )
πΣ→ KbarN
fπ=90 MeV
( 0)
( 0)28
Iij i jI
ij i j iji j
C M MV r g r
f sp
πΣ KbarN
Scattering amplitude … KSW NRv1 – Non. rela.
fπ=90 MeV
• Phase shift (πΣ)
ReIm
KbarN → KbarN
KbarN → πΣ
πΣ→ KbarN
πΣ → πΣ
δ ( πΣ )
( 0)
( 0)2
18
Iij i jI
ijTot
jj
i ij
C M MV
Er g r
f p
πΣ KbarN
Scattering amplitude … KSW NRv2 – Non. rela.
fπ=90 MeV
• Phase shift (πΣ)
ReIm
KbarN → KbarN
KbarN → πΣ
πΣ→ KbarN
πΣ → πΣ
δ ( πΣ )
( 0)
( 0)2
18
IijI
ij i j ijj
CV r g r
f m mp
πΣ KbarN
Scattering amplitude … KSW org. – Semi rela.
fπ=90 MeV
• Phase shift (πΣ)
Re
Im
KbarN → KbarN
KbarN → πΣ
πΣ→ KbarN
πΣ → πΣ
δ ( πΣ )
( 0)
( 0)28
Iij i jI
ij i j iji j
C M MV r g r
f sp
πΣ KbarN
Succeeded in nuclear physics … Especially used in the study of unstable nuclei.
Resonant state of 6He
4He
n
n
S. Aoyama, T. Myo, K. Kato and K. Ikeda, Prog. Theor. Phys. 116, 1 (2006)
Continuum
Resonance
Complex Scaling Method
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
•Potential: KSW original
•Kinematics: Non. rela.
Scattering amplitude
Scattering amplitude … KSW org. – Non. rela.
fπ=90 MeV
• Phase shift (πΣ)
δ ( πΣ )
ReIm
KbarN → KbarN πΣ → πΣ
KbarN → πΣ
πΣ→ KbarN
δ ( πΣ )
Scattering amplitude … KSW org. – Non. rela.
fπ=100 MeV
• Phase shift (πΣ)
δ ( πΣ )
ReIm
KbarN → KbarN πΣ → πΣ
KbarN → πΣ
πΣ→ KbarN
δ ( πΣ )
Scattering amplitude … KSW org. – Non. rela.
fπ=110 MeV
• Phase shift (πΣ)
δ ( πΣ )
ReIm
KbarN → KbarN πΣ → πΣ
KbarN → πΣ
πΣ→ KbarN
δ ( πΣ )
Scattering amplitude … KSW org. – Non. rela.
fπ=120 MeV
• Phase shift (πΣ)
δ ( πΣ )
ReIm
KbarN → KbarN πΣ → πΣ
KbarN → πΣ
πΣ→ KbarN
δ ( πΣ )
•Potential: KSW NRv1
•Kinematics: Non. rela.
Scattering amplitude
Scattering amplitude … KSW NRv1 – Non. rela.
fπ=90 MeV
• Phase shift (πΣ)
δ ( πΣ )
ReIm
KbarN → KbarN πΣ → πΣ
KbarN → πΣ
πΣ→ KbarN
δ ( πΣ )
Scattering amplitude … KSW NRv1 – Non. rela.
• Phase shift (πΣ)
fπ=100 MeV
δ ( πΣ )
πΣ → πΣ
KbarN → πΣ
πΣ→ KbarN
δ ( πΣ )
Re Im
KbarN → KbarN
Scattering amplitude … KSW NRv1 – Non. rela.
• Phase shift (πΣ)
δ ( πΣ )
fπ=110 MeV
πΣ → πΣ
KbarN → πΣ
πΣ→ KbarN
δ ( πΣ )
ReIm
KbarN → KbarN
Scattering amplitude … KSW NRv1 – Non. rela.
• Phase shift (πΣ)
πΣ → πΣ
δ ( πΣ )
fπ=120 MeV
δ ( πΣ )
πΣ → πΣ
KbarN → πΣ
πΣ→ KbarN
δ ( πΣ )
Re
Im
KbarN → KbarN
•Potential: KSW NRv2
•Kinematics: Non. rela.
Scattering amplitude
Scattering amplitude … KSW NRv2 – Non. rela.
• Phase shift (πΣ)
πΣ → πΣ
δ ( πΣ )
fπ=90 MeV
δ ( πΣ )
Re Im
KbarN → KbarN πΣ → πΣ
KbarN → πΣ
πΣ→ KbarN
δ ( πΣ )
Scattering amplitude … KSW NRv2 – Non. rela.
• Phase shift (πΣ)
fπ=100 MeV
δ ( πΣ )
ReIm
KbarN → KbarN πΣ → πΣ
KbarN → πΣ
πΣ→ KbarN
δ ( πΣ )
Scattering amplitude … KSW NRv2 – Non. rela.
• Phase shift (πΣ)
πΣ → πΣ
δ ( πΣ )
fπ=110 MeV
δ ( πΣ )
Re
Im
KbarN → KbarN πΣ → πΣ
KbarN → πΣ
πΣ→ KbarN
δ ( πΣ )
Scattering amplitude … KSW NRv2 – Non. rela.
• Phase shift (πΣ)
πΣ → πΣ
δ ( πΣ )
fπ=120 MeV
δ ( πΣ )
πΣ → πΣ
KbarN → πΣ
πΣ→ KbarN
δ ( πΣ )
Re
Im
KbarN → KbarN
•Potential: KSW original
•Kinematics: Semi rela.
Scattering amplitude
Scattering amplitude … KSW org. – Semi rela.
• Phase shift (πΣ)
πΣ → πΣ
δ ( πΣ )
fπ=90 MeV
δ ( πΣ )
Re
Im
KbarN → KbarN πΣ → πΣ
KbarN → πΣ
πΣ→ KbarN
δ ( πΣ )
Scattering amplitude … KSW org. – Semi rela.
• Phase shift (πΣ)
πΣ → πΣ
δ ( πΣ )
fπ=100 MeV
δ ( πΣ )
Im
KbarN → KbarN πΣ → πΣ
KbarN → πΣ
πΣ→ KbarN
δ ( πΣ )
Re
Scattering amplitude … KSW org. – Semi rela.
• Phase shift (πΣ)
πΣ → πΣ
δ ( πΣ )
fπ=110 MeV
δ ( πΣ )
Im
KbarN → KbarN πΣ → πΣ
KbarN → πΣ
πΣ→ KbarN
δ ( πΣ )
Re
Scattering amplitude … KSW org. – Semi rela.
• Phase shift (πΣ)
πΣ → πΣ
δ ( πΣ )
fπ=120 MeV
δ ( πΣ )
Im
KbarN → KbarN πΣ → πΣ
KbarN → πΣ
πΣ→ KbarN
δ ( πΣ )
Re
Position of resonant structure
appeared in the scattering amplitude
Position of resonant structure appeared in the scattering amplitude
fπ=90 MeV
Kinematics Non- rela. Semi- rela.KSW potential Original Non- rela. v1 Non- rela. v2 Original
Range a(KbarN) 0.593 0.576 0.574 0.487parameter [fm] a(π Σ ) 0.541 0.725 0.751 0.457
KbarN Scatt. Re - 1.701 - 1.700 - 1.700 - 1.703leng. [fm] Im 0.679 0.677 0.687 0.677
Resonance position [MeV]f11=0 E 1402.5 1408.4 1411.1 1417.0
E(KbarN) - 32.5 - 26.6 - 23.9 - 18.0
f22=0 E 1354.6 1388.8 1393.0 1422.0E(KbarN) - 80.4 - 46.2 - 42.0 - 12.9
δ (π Σ )=0 E 1354.6 1388.8 1393.0 1422.1E(KbarN) - 80.4 - 46.2 - 42.0 - 12.9
KbarN → KbarN
πΣ → πΣ
Position of resonant structure appeared in the scattering amplitude
fπ=100 MeV
Kinematics Non- rela. Semi- rela.KSW potential Original Non- rela. v1 Non- rela. v2 Original
Range a(KbarN) 0.517 0.503 0.501 0.421parameter [fm] a(π Σ ) 0.477 0.665 0.695 0.395
KbarN Scatt. Re - 1.698 - 1.702 - 1.702 - 1.703leng. [fm] Im 0.676 0.680 0.683 0.673
Resonance position [MeV]f11=0 E 1404.8 1411.2 1413.6 1418.0
E(KbarN) - 30.2 - 23.8 - 21.4 - 17.0
f22=0 E 1384.7 1399.3 1402.6 1421.5E(KbarN) - 50.3 - 35.7 - 32.4 - 13.5
δ (π Σ )=0 E 1384.7 1399.3 1402.6 1421.5E(KbarN) - 50.3 - 35.7 - 32.4 - 13.5
KbarN → KbarN
πΣ → πΣ
Position of resonant structure appeared in the scattering amplitude
fπ=110 MeV
Kinematics Non- rela. Semi- rela.KSW potential Original Non- rela. v1 Non- rela. v2 Original
Range a(KbarN) 0.453 0.440 0.438 0.369parameter [fm] a(π Σ ) 0.419 0.605 0.636 0.348
KbarN Scatt. Re - 1.703 - 1.701 - 1.700 - 1.696leng. [fm] Im 0.683 0.681 0.681 0.681
Resonance position [MeV]f11=0 E 1407.2 1413.4 1415.5 1418.9
E(KbarN) - 27.8 - 21.6 - 19.5 - 16.1
f22=0 E 1394.7 1405.3 1408.1 1421.4E(KbarN) - 40.3 - 29.7 - 26.9 - 13.6
δ (π Σ )=0 E 1394.7 1405.3 1408.1 1421.4E(KbarN) - 40.3 - 29.7 - 26.9 - 13.6
KbarN → KbarN
πΣ → πΣ
Position of resonant structure appeared in the scattering amplitude
fπ=120 MeV
Kinematics Non- rela. Semi- rela.KSW potential Original Non- rela. v1 Non- rela. v2 Original
Range a(KbarN) 0.398 0.386 0.384 0.327parameter [fm] a(π Σ ) 0.368 0.549 0.581 0.310
KbarN Scatt. Re - 1.698 - 1.699 - 1.700 - 1.690leng. [fm] Im 0.682 0.674 0.669 0.669
Resonance position [MeV]f11=0 E 1408.6 1415.0 1417.0 1419.3
E(KbarN) - 26.4 - 20.0 - 18.0 - 15.7
f22=0 E 1399.7 1409.1 1411.7 1420.9E(KbarN) - 35.3 - 25.9 - 23.3 - 14.1
δ (π Σ )=0 E 1399.7 1409.1 1411.7 1420.9E(KbarN) - 35.3 - 25.9 - 23.3 - 14.1
KbarN → KbarN
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.
NRv1 – Non.rela. NRv2 – Non.rela.
Org. – Semi rela.
fπ=90 MeVθ=30°
“Wave function” of the resonance pole
Org. – Non.rela. Org. – Semi rela.
NRv1 – Non.rela. NRv2 – Non.rela.
fπ=100 MeVθ=30°
“Wave function” of the resonance pole
Org. – Non.rela.
NRv1 – Non.rela. NRv2 – Non.rela.
Org. – Semi rela.
fπ=110 MeVθ=30°
“Wave function” of the resonance pole
Org. – Non.rela.
NRv1 – Non.rela. NRv2 – Non.rela.
Org. – Semi rela.
fπ=120 MeVθ=30°
“Size” of the resonance
pole state
Kinematics Non- rela. Semi- rela.KSW potential Original Non- rela. v1 Non- rela. v2 Original
Range a(KbarN) 0.593 0.576 0.574 0.487parameter [fm] a(π Σ ) 0.541 0.725 0.751 0.457
KbarN Scatt. Re - 1.701 - 1.700 - 1.700 - 1.703leng. [fm] Im 0.679 0.677 0.687 0.677
Pole position E 1423.3 1419.8 1419.9 1419.0[MeV] B(KbarN) 11.7 15.2 15.1 16.0
Γ / 2 28.5 26.0 23.1 14.4
Meson- Baryon distance[fm]KbarN ch. Re 1.19 1.20 1.25 1.21
Im - 0.45 - 0.35 - 0.35 - 0.49
π Σ ch. Re 0.29 0.33 0.34 0.12Im 0.21 0.24 0.23 - 0.09
KbarN+π Σ Re 1.19 1.20 1.25 1.21Im - 0.39 - 0.28 - 0.29 - 0.49
“Size” of the resonance pole state fπ=90 MeVθ=30°
“Size” of the resonance pole state fπ=100 MeVθ=30°
Kinematics Non- rela. Semi- rela.KSW potential Original Non- rela. v1 Non- rela. v2 Original
Range a(KbarN) 0.517 0.503 0.501 0.421parameter [fm] a(π Σ ) 0.477 0.665 0.695 0.395
KbarN Scatt. Re - 1.698 - 1.702 - 1.702 - 1.703leng. [fm] Im 0.676 0.680 0.683 0.673
Pole position E 1421.7 1417.3 1418.0 1419.6[MeV] B(KbarN) 13.3 17.7 17.0 15.4
Γ / 2 28.0 23.1 19.8 13.2
Meson- Baryon distance[fm]KbarN ch. Re 1.20 1.27 1.32 1.20
Im - 0.45 - 0.34 - 0.34 - 0.49
π Σ ch Re 0.34 0.39 0.38 0.11Im 0.16 0.15 0.13 - 0.08
KbarN+π Σ Re 1.22 1.31 1.36 1.21Im - 0.40 - 0.29 - 0.30 - 0.49
“Size” of the resonance pole state fπ=110 MeVθ=30°
Kinematics Non- rela. Semi- rela.KSW potential Original Non- rela. v1 Non- rela. v2 Original
Range a(KbarN) 0.453 0.440 0.438 0.369parameter [fm] a(π Σ ) 0.419 0.605 0.636 0.348
KbarN Scatt. Re - 1.703 - 1.701 - 1.700 - 1.696leng. [fm] Im 0.683 0.681 0.681 0.681
Pole position E 1420.5 1416.6 1417.8 1420.0[MeV] B(KbarN) 14.5 18.4 17.2 15.0
Γ / 2 26.6 19.5 16.6 12.8
Meson- Baryon distance[fm]KbarN ch. Re 1.20 1.31 1.37 1.18
Im - 0.48 - 0.37 - 0.37 - 0.49
π Σ ch Re 0.36 0.39 0.37 0.11Im 0.11 0.05 0.04 - 0.06
KbarN+π Σ Re 1.23 1.36 1.42 1.18Im - 0.43 - 0.34 - 0.34 - 0.49
“Size” of the resonance pole state fπ=120 MeVθ=30°
Kinematics Non- rela. Semi- rela.KSW potential Original Non- rela. v1 Non- rela. v2 Original
Range a(KbarN) 0.398 0.386 0.384 0.327parameter [fm] a(π Σ ) 0.368 0.549 0.581 0.310
KbarN Scatt. Re - 1.698 - 1.699 - 1.700 - 1.690leng. [fm] Im 0.682 0.674 0.669 0.669
Pole position E 1419.5 1416.9 1418.3 1418.9[MeV] B(KbarN) 15.5 18.1 16.7 16.1
Γ / 2 25.6 16.7 14.0 11.7
Meson- Baryon distance[fm]KbarN ch. Re 1.18 1.33 1.40 1.15
Im - 0.49 - 0.39 - 0.39 - 0.42
π Σ ch Re 0.37 0.36 0.34 0.14Im 0.06 - 0.01 - 0.02 - 0.04
KbarN+π Σ Re 1.22 1.38 1.44 1.16Im - 0.45 - 0.38 - 0.38 - 0.42
“Size” of the resonance pole state
θ=30°
2MBr fmMean distance between meson and baryon
Re
Im
Pole position of the resonancefpi Org- NR NRv1- NR NRv2- NR Org- SR
M - Γ / 2 M - Γ / 2 M - Γ / 2 M - Γ / 290 1423.3 - 28.5 1419.8 - 26.0 1419.9 - 23.1 1419.0 - 14.4100 1421.7 - 28.0 1417.3 - 23.1 1418.0 - 19.8 1419.6 - 13.2110 1420.5 - 26.6 1416.6 - 19.5 1417.8 - 16.6 1420.0 - 12.8120 1419.5 - 25.6 1416.9 - 16.7 1418.3 - 14.0 1418.9 - 11.7
- Γ /
2 [M
eV]
M [MeV]
90100110
120 90
100
110
120
90100
110
12090
100110
120
θ dependence of “Size” and
“Wave function”
θ dependence of “Size” NRv2-NRfπ=90 MeV
θ [deg]
[fm]
θ 10 15 20 25 30 35
Pole positionE 1416.8 1419.2 1419.8 1419.9 1419.9 1419.8[MeV] Γ / 2 18.5 22.9 23.1 23.1 23.1 23.1
Meson- Baryon distance[fm]KbarN ch. (Re) KbarN 0.95 1.24 1.25 1.25 1.25 1.24
(Im) KbarN - 0.43 - 0.41 - 0.35 - 0.35 - 0.35 - 0.35
π Σ ch. (Re) π Σ 12.36 3.61 0.71 0.35 0.34 0.34(Im) π Σ 4.27 4.81 1.17 0.33 0.23 0.23
KbarN+π Σ(Re) KbarN+π Σ12.38 3.61 0.87 1.23 1.25 1.25(Im) KbarN+π Σ 4.23 4.67 0.44 - 0.26 - 0.29 - 0.29
θ dependence of “Size” Org-SRfπ=90 MeV
θ [deg]
[fm]
θ 10 15 20 25 30 35
Pole positionE 1416.7 1418.9 1419.0 1419.0 1419.0 1418.9[MeV] Γ / 2 14.5 14.5 14.4 14.4 14.4 14.4
Meson- Baryon distance[fm]KbarN ch. (Re) KbarN 0.98 1.21 1.21 1.21 1.21 1.21
(Im) KbarN - 0.69 - 0.50 - 0.49 - 0.49 - 0.49 - 0.49
π Σ ch. (Re) π Σ 8.62 0.73 0.13 0.06 0.12 0.12(Im) π Σ 7.05 2.33 0.51 - 0.11 - 0.09 - 0.09
KbarN+π Σ(Re) KbarN+π Σ 8.60 0.54 1.10 1.21 1.21 1.22(Im) KbarN+π Σ 6.99 2.00 - 0.48 - 0.49 - 0.49 - 0.49
θ dependence of “Wave function” NRv2-NRfπ=90 MeV
KbarN bound state πΣ resonant state
KbarN
πΣ
r [fm]
r [fm]
θ =10 deg.
θ dependence of “Wave function” NRv2-NRfπ=90 MeV
KbarN bound state πΣ resonant state
KbarN
πΣ
r [fm]
r [fm]
θ =15 deg.
θ dependence of “Wave function” NRv2-NRfπ=90 MeV
KbarN bound state πΣ resonant state
KbarN
πΣ
r [fm]
r [fm]
θ =20 deg.
θ dependence of “Wave function” NRv2-NRfπ=90 MeV
KbarN bound state πΣ resonant state
KbarN
πΣ
r [fm]
r [fm]
θ =25 deg.
θ dependence of “Wave function” NRv2-NRfπ=90 MeV
KbarN bound state πΣ resonant state
KbarN
πΣ
r [fm]
r [fm]
θ =30 deg.
θ dependence of “Wave function” NRv2-NRfπ=90 MeV
KbarN bound state πΣ resonant state
KbarN
πΣ
r [fm]
r [fm]
θ =35 deg.
Comparison with Hyodo-Weise’s result
Comparison with Hyodo-Weise’s result
KbarN → KbarN πΣ → πΣ
Scattering amplitudes
DIM
HW
HW: PRC 77, 035204 (2008)
Fig. 4 I=0 KbarN/πΣ scattering amplitude in two channel model
Comparison with Hyodo-Weise’s resultPole position HW: PRC 77, 035204 (2008)
DIM
HW
Fig. 5 Pole positions of I=0 scattering amplitude
(Upper pole in two channel model)
z = 1419 – 14.4i MeV
Schrödinger equation to be solved
: complex parameters to be determined
Wave function expanded with Gaussian base
c.c. CSM for resonant states
Phenomenological potentialY. Akaishi and T. Yamazaki,
PRC 52 (2002) 044005= Energy independent potential
Chiral SU(3) potential N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)= Energy dependent potential
V
Schrödinger equation to be solved
: complex parameters to be determined
Wave function expanded with Gaussian base
Complex-rotate , then diagonalize with Gaussian base.
c.c. CSM for resonant states
Phenomenological potentialY. Akaishi and T. Yamazaki,
PRC 52 (2002) 044005= Energy independent potential
Chiral SU(3) potential N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)= Energy dependent potential
V
Complex scaling of coordinate
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),269E. Balslev and J. M. Combes, Commun. Math. Phys. 22 (1971),280
Chiral SU(3) potential (KSW)N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)
Original: δ-function type
Present: Normalized Gaussian type
( 0)
( 0)28
ij
ij
Ii jI
i ji j
C Mg
MV r r
f s
2
3/ 2 3
1 expg r ra
ap
a: range parameter [fm]Mi , mi : Baryon, Meson mass in channel iEi : Baryon energy, ωi : Meson energy
2 22 2 2 2
,2 2
i i i i
i i
s s M
sE
s
m M m
Reduced energy: i ii
i i
EE
( 0)332
4ij
IC
KbarN πΣ
Energy dependence of Vij is controlled by CM energy √s.
( 0)
( 0)28
ij
ij
Ii jI
i ji j
C M Mr rV
f s
Energy dependence is determined by Chiral low energy theorem. ← Kaon: Nambu-Goldstone boson
Flavor SU(3) symmetry
Chiral SU(3) potential (KSW)Energy dependence
( 0) 0, @ 0.5 , 100ij
IV r s a fm f MeVp
√s [MeV]
KbarN-KbarN
πΣ-πΣ
KbarN-πΣ
N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)
KbarN thresholdπΣ threshold
Chiral SU(3) potential = Energy-dependent potential
Calculational procedure
Perform the Complex Scaling method.
Then, find a pole of resonance or bound state. Calculateds
Calculated Assumeds sCheck Finished !If Yes
Self consistency for the energy!
Assume the values of the CM energy √s.
MBH T V s M Assumeds
If No
やったこと1. Chiral SU(3) potential で、 r-space ガウス型を仮定
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。
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
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。
( 0)
( 0)28
ij
ij
Ii jI
i ji j
C M MV r
sg r
fp
2
3/ 2 3
1 expg r ra
ap
N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)
( 0) 1.70 0.07 0.68 0.04KbarN
Ia i fm
A. T. Kruppa, R. Suzuki and K. Kato, PRC 75, 044602 (2007)
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
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。
※ fπ=75.5 で無理やり散乱長を再現した場合 : (EKbarN, Γ/2) = (-23, 34) MeV
Mysterious state; Λ(1405)Quark model prediction … calculated as 3-quark state
N. Isgar and G. Karl, Phys. Rev. D18, 4187 (1978)
qq q
Λ(1405) can’t be well reproduced as a 3-quark state!
observed Λ(1405)calculated Λ(1405)
KbarN-πΣ coupled system with s-wave and isospin-0 state
Λ(1405) with c.c. Complex Scaling Method
Kbar + N
Λ(1405)
π + Σ
1435
1332 [MeV]
B. E. (KbarN) = 27 MeVΓ (πΣ) ~ 50 MeV
Jπ = 1/2-
I = 0
Kbar
(Jπ=0-, T=1/2)
N (Jπ=1/2+, T=1/2)
L=0
π(Jπ=0-, T=1)
Σ(Jπ=1/2+, T=1)
L=0
Schrödinger equation to be solved
: complex parameters to be determined
Wave function expanded with Gaussian base
Complex-rotate , then diagonalize with Gaussian base.
Λ(1405) with c.c. Complex Scaling Method
Phenomenological potentialY. Akaishi and T. Yamazaki,
PRC 52 (2002) 044005= Energy independent potential
Chiral SU(3) potential N. Kaiser, P. B. Siegel and W. Weise, NPA 594, 325 (1995)= Energy dependent potential
V
Complex scaling of coordinate
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),269E. Balslev and J. M. Combes, Commun. Math. Phys. 22 (1971),280
Calc. of scattering amplitude with CSM “Kruppa method” A. T. Kruppa, R. Suzuki and K. Katō,
PRC 75, 044602 (2007)
• Radial Schroedinger eq.
• Asymptotic behavior of wave func.
• Equation to be solved
V(r) : short-range potential
• Asymptotic behavior of scattering part of wave func.
exp2
llh x i kr p
( )
, llj kr h kr
: Ricatti-Bessel, Hankel func.
is not square-integrable ,SCl k r
Search the solution in the form
Separate the incoming wave ! lj kr
Point 1
Calc. of scattering amplitude with CSM “Kruppa method” A. T. Kruppa, R. Suzuki and K. Katō,
PRC 75, 044602 (2007)
• Complex scaling
Equation to be solved
2,i i ir r e r r e r e
• Expanding with square-integrable basis function (ex: Gaussian basis)
square-integrable for 0 < θ < π
Point 2
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
We don’t have which is a solution along r,
but we have which is a solution along reiθ.
,SCl k r
,,SCl k r
r0
r eiθ
, 0SCl kdz j kz V z z
Cauchy theoremPoint 3
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
2, 0
436 412exp 0.66 fm MeV
412 0AY IV r
KbarN πΣ
1. free KbarN scattering data2. 1s level shift of kaonic hydrogen atom3. Binding energy and width of Λ(1405)
= K- + proton
The result that I show hereafter is not new, because the same calculation was done by Akaishi-san, when he made AY potential.
Remark !
trajectory• # Gauss base (n) = 30• Max range (b) = 10 [fm]
E /
2
[MeV]
= 0 deg.
Λ(1405) with c.c. Complex Scaling Method
E /
2
[MeV]
= 5 deg.
trajectory• # Gauss base (n) = 30• Max range (b) = 10 [fm]
Λ(1405) with c.c. Complex Scaling Method
E /
2
[MeV]
=10 deg.
trajectory• # Gauss base (n) = 30• Max range (b) = 10 [fm]
Λ(1405) with c.c. Complex Scaling Method
E /
2
[MeV]
=15 deg.
trajectory• # Gauss base (n) = 30• Max range (b) = 10 [fm]
Λ(1405) with c.c. Complex Scaling Method
E /
2
[MeV]
=20 deg.
trajectory• # Gauss base (n) = 30• Max range (b) = 10 [fm]
Λ(1405) with c.c. Complex Scaling Method
E /
2
[MeV]
=25 deg.
trajectory• # Gauss base (n) = 30• Max range (b) = 10 [fm]
Λ(1405) with c.c. Complex Scaling Method
E /
2
[MeV]
=30 deg.
trajectory• # Gauss base (n) = 30• Max range (b) = 10 [fm]
Λ(1405) with c.c. Complex Scaling Method
E /
2
[MeV]
=35 deg.
trajectory• # Gauss base (n) = 30• Max range (b) = 10 [fm]
Λ(1405) with c.c. Complex Scaling Method
E /
2
[MeV]
=40 deg.
trajectory• # Gauss base (n) = 30• Max range (b) = 10 [fm]
Λ(1405) with c.c. Complex Scaling Method
E /
2
[MeV]
trajectory
=30 deg.
pS KbarN
pS continuum
KbarN continuum
Resonance!(E, Γ/2) = (75.8, 20.0)
Measured from KbarN thr.,
B. E. (KbarN) = 28.2 MeVΓ = 40.0 MeV… (1405) !
Λ(1405) with c.c. Complex Scaling Method
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
ReCalculateds E
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
Calculated complex en ys E erg
More reasonable?
0T V V G ZTZ Z Z Z
Pole search of T-matrix is done on complex-energy plane.
Z: complex energy
Self consistency for complex energy KSWfπ = 100 MeV
a=0.47, θ=35°
-B [MeV]
1 step-Γ/2 [MeV]
obtained by the self-consistency for the real energy
KbarN
Self consistency for complex energy KSWfπ = 100 MeV
a=0.47, θ=35°
-B [MeV]
2 steps-Γ/2 [MeV]
KbarN
Self consistency for complex energy KSWfπ = 100 MeV
a=0.47, θ=35°
-B [MeV]
3 steps-Γ/2 [MeV]
KbarN
Self consistency for complex energy KSWfπ = 100 MeV
a=0.47, θ=35°
-B [MeV]
4 steps-Γ/2 [MeV]
KbarN
Self consistency for complex energy KSWfπ = 100 MeV
a=0.47, θ=35°
5 steps-Γ/2 [MeV]
-B [MeV]
Self consistent!
KbarN
Self consistency for complex energy KSWfπ = 100 MeV
Assumed
Calc.
Assumed
Calc.
Self consistency for complex energy KSWfπ = 100 MeV
-B [MeV]
-Γ /
2 [M
eV]
a=0.60
a=0.50
a=0.47
a=0.45
S.C. for real energyS.C. for complex energy
KbarN
Repulsively shifted!
Mean distance between Kbar (π) and N (Σ)
Kbar (π)
N (Σ)
Distance
Chiral (HW-HNJH): B ~ 12 MeV, Distance = 1.86 fm
Λ (1405)、他の計算との比較
Rough estimation of charge radiusK-
p
Distance
Chiral (HW-HNJH): B ~ 12 MeV, Distance = 1.86 fm → <r2c> = -1.07 fm2
CM
NK CM
K N
Kp CM
K N
Mr Distancem M
mr Distancem M
2 2 2c p CM K CM
r r r
Charge radius
Chiral (HW-HNJH): B ~ 12 MeV, Distance = 1.86 fm → <r2c> = -1.07 fm2
Charge radius derived from electric form factor
T. Sekihara, T. Hyodo and D. Jido, PLB 669, 133 (2008)
Rough estimation