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結合チャネル複素スケーリング法による“ K-pp” の研究
1. Introduction
2. Fully coupled-channel Complex Scaling Method
for the essential kaonic nucleus “K-pp”
3. Result Phenomenological KbarN potential
Chiral SU(3)-based KbarN potential
4. Discussion
5. Summary and future prospects
Akinobu Doté(KEK Theory Center, IPNS / J-PARC branch)
KEK理論センター研究会 『ハドロン・原子核物理の理論研究最前線 2017』, 22. Nov., ’17 @ KEK (Tsukuba campus), Japan
Takashi Inoue (Nihon univ.)
Takayuki Myo (Osaka Inst. Tech.)
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Kaonic nuclei = Nuclear system with Kbar mesons
Nucleus
qbar
s
Kbar meson(K0bar, K-)
Jπ=0-, I=1/2
Involve strange quarks (Strangeness).
Real mesons are involved
as a constituent of the system.
Expect “Dense and Cold” state Y. Akaishi, T. Yamazaki, PRC65, 044005 (2002)
A. Dote, H. Horiuchi, Y. Akaishi, T. Yamazaki, PRC70, 044313 (2004)
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K-
Proton
KbarN two-body system
Excited hyperon Λ(1405) = K- proton quasi-bound state
• Low energy scattering data, 1s level shift of kaonic hydrogen atom
• Hard to describe Λ(1405) with Quark Model as a 3-quark state
• Success of Chiral Unitary Model with a meson-baryon picture
Excited hyperon Λ(1405)
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K-
Proton
KbarN two-body system
Excited hyperon Λ(1405) = K- proton quasi-bound state
• Low energy scattering data, 1s level shift of kaonic hydrogen atom
• Hard to describe Λ(1405) with Quark Model as a 3-quark state
• Success of Chiral Unitary Model with a meson-baryon picture
Doorway to dense matter†
→ Chiral symmetry restoration in dense matter
Interesting structure†
Neutron star
† A. D., H. Horiuchi, Y. Akaishi and T. Yamazaki, PRC70, 044313 (2004)
3HeK-, pppK-, 4HeK-, pppnK-,
…, 8BeK-,…
Kaonic nuclei
Strongly attractive KbarN potential
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K-
ProtonKbarN two-body system = Λ(1405)
Kaonic nuclei
= Nuclear many-body system with antikaons
P PK-
Prototype system = K- pp
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P PK-
Prototype system = K- pp
K-pp bound state???
Studied with various methods,
since K-pp is a three-body system
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Clear evidence of K-pp bound state
J-PARC E15; Exclusive exp. 3He(K-, Λp)nmissing
K-pp bound state!T. Sekihara, E. Oset, A. Ramos,
PTEP 2016, 123D03
Theoretical study
J-PARC E15 (2nd run)
Yamaga’s talk
(JPS symposium 2017)
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P PK-
2. Fully coupled-channel
Complex Scaling Method
for
the essential kaonic nucleus “K-pp”
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Resonant state of KbarN-πΣ coupled-channel system
•Hyperon resonance Λ(1405)
According to early studies ...
⇒ “coupled-channel
Complex Scaling Method”
Resonant state of
KbarNN-πΣN-πΛN coupled channel
three-body system
•Prototype system of kaonic nuclei “K-pp”
Doté, Hyodo, Weise, PRC79, 014003(2009). Akaishi, Yamazaki, PRC76, 045201(2007)
Ikeda, Sato, PRC76, 035203(2007). Shevchenko, Gal, Mares, PRC76, 044004(2007)
Barnea, Gal, Liverts, PLB712, 132(2012)
Resonance & Channel coupling
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Complex Scaling Method… Powerful tool for resonance study of many-body system
S. Aoyama, T. Myo, K. Kato, K. Ikeda, PTP116, 1 (2006)
T. Myo, Y. Kikuchi, H. Masui, K. Kato, PPNP79, 1 (2014)
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Hamiltonian
1,2
3,MB MB MB MB
i
V V i
... symmetric for baryon’s site (i=1,2)
Glöckle, Miyagawa, Few-body Systems 30, 241 (2001)
1
2
3
Baryon
Baryon
Meson
1
62 285 285
0 0 MeV
0
bar
I
K N
V
Y. Akaishi and T. Yamazaki,
PRC 65, 044005 (2002)
R. B. Wiringa, V. G. J. Stoks, and R. Schiavilla, PRC 51, 38 (1995) NN potential = Av18 potential
KbarN-πY potential = Akaishi-Yamazaki potential
2
exp 0.66 fmI
IV V r
0 436 412MeV
0
bar
I
K N
V
Phenomenological energy-independent potential
Ignore YN and πN potentials
( ) ( )
,bar
NN MB MB
K N
H M T V V
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Wave function
Baryon-Baryon are antisymmetrized on space, spin and isospin as well as label (flavor).Glöckle, Miyagawa, Few-body Systems 30, 241 (2001)
Spatial part = Correlated Gaussian function
including 3 types of Jacobi coordinates
projected onto a parity eigenstate of B1B2,
M3
B2
B1
x1(3)
x2(3)
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P PK-
3. Result
Phenomenological KbarN potential(Energy-independent)
“K-pp” =
KbarNN – πΣN – πΛN (Jπ = 0-, T=1/2)
A. Dote, T. Inoue, T. Myo, PRC 95, 062201(R) (2017)
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Just diagonalize
the complex-scaled Hamiltonian matrix!
No channel elimination!!
KbarNN
- KbarNN
θKbarNN
- πΣN
KbarNN
- πΛN
πΣN
- πΣN
πΣN
- πΛN
πΛN
- πΛN
*
* *
Hij
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Complex eigenvalue distributionNN: Av18 pot.
KbarN-πY: Akaishi-Yamazaki pot.
(Pheno., E-indep.)
Scaling angle θ=30°Dimension = 6400
Λ*N cont.
Λ* pole
BKN = 28 MeV, ΓπΣ /2 = 20 MeV
The “K-pp” pole of AY potential :
BKNN = 51 MeV, ΓπYN /2 = 16 MeV
πΛN πΣN KbarNN
KbarNN cont.
πΛN cont. πΣN cont.
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Property of the “K-pp” pole
Norm 1.004 - i 0.286 -0.002 + i 0.276 -0.002 + i 0.010
BB distance [fm]
1.86 + i 0.14 1.50 + i 0.48 1.10 + i 0.23
KbarN
(I=0) 0.726 - i 0.213
(I=1) 0.278 - i 0.073
πΣ
(I=0) -0.009 + i 0.261
(I=1) 0.007 + i 0.015
πΛ
(I=0) ---
(I=1) -0.002 + i 0.010
MB distance [fm]
(I=0) 1.40 - i 0.01
(I=1) 1.95 + i 0.14
(I=0) 0.52 + i 1.31
(I=1) 0.80 + i 1.22
(I=0) ---
(I=1) 0.74 + i 0.74
NN: Av18 pot.
KbarN-πY: Akaishi-Yamazaki pot.
(Pheno., E-indep.)
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Property of the “K-pp” pole
Norm 1.004 - i 0.286 -0.002 + i 0.276 -0.002 + i 0.010
BB distance [fm]
1.86 + i 0.14 1.50 + i 0.48 1.10 + i 0.23
KbarN
(I=0) 0.726 - i 0.213
(I=1) 0.278 - i 0.073
πΣ
(I=0) -0.009 + i 0.261
(I=1) 0.007 + i 0.015
πΛ
(I=0) ---
(I=1) -0.002 + i 0.010
MB distance [fm]
(I=0) 1.40 - i 0.01
(I=1) 1.95 + i 0.14
(I=0) 0.52 + i 1.31
(I=1) 0.80 + i 1.22
(I=0) ---
(I=1) 0.74 + i 0.74
NN distance = 1.9 fm
⇒1.6 ρ0
NN: Av18 pot.
KbarN-πY: Akaishi-Yamazaki pot.
(Pheno., E-indep.)
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Property of the “K-pp” pole
Norm 1.004 - i 0.286 -0.002 + i 0.276 -0.002 + i 0.010
BB distance [fm]
1.86 + i 0.14 1.50 + i 0.48 1.10 + i 0.23
KbarN
(I=0) 0.726 - i 0.213
(I=1) 0.278 - i 0.073
πΣ
(I=0) -0.009 + i 0.261
(I=1) 0.007 + i 0.015
πΛ
(I=0) ---
(I=1) -0.002 + i 0.010
MB distance [fm]
(I=0) 1.40 - i 0.01
(I=1) 1.95 + i 0.14
(I=0) 0.52 + i 1.31
(I=1) 0.80 + i 1.22
(I=0) ---
(I=1) 0.74 + i 0.74
cf) Λ(1405) ~ KbarN (I=0)
... 1.25 fm
NN: Av18 pot.
KbarN-πY: Akaishi-Yamazaki pot.
(Pheno., E-indep.)
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P PK-
3. Result
Chiral SU(3)-based KbarN potential(Energy-dependent)
“K-pp” =
KbarNN – πΣN – πΛN (Jπ = 0-, T=1/2)
A. Dote, T. Inoue, T. Myo, arXiv: 1710.07589
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Chiral SU(3)-based KbarN potential
Coupled-channel chiral dynamics
(Chiral Unitary model)N. Kaiser, P.B. Siegel, W. Weise, NPA 594 (1995) 325
E. Oset, A. Ramos, NPA 635 (1998) 99
Weinberg-Tomozawa term
of effective chiral Lagrangian
Based on Chiral SU(3) theory
→ Energy dependence
• Anti-kaon, Pion = Nambu-Goldstone boson
... governed by chiral dynamics
Constrained by KbarN scattering length
• Old data: aKN(I=0) = -1.70 + i0.67 fm, aKN(I=1) = 0.37 + i0.60 fm A. D. Martin, NPB179, 33(1979)
• SIDDHARTA K-p data with a coupled-channel chiral dynamics:
aK-p = -0.65 + i0.81 fm, aK-n = 0.57 + i0.72 fm M. Bazzi et al., NPA 881, 88 (2012)
Y. Ikeda, T. Hyodo, W. Weise, NPA 881, 98 (2012)
( 0,1)
( 0,1)
2
1
8
I
ijI
ij ii j j
i j
CV r g r
f m m
3
2
/ 2 3ex
1p
ijij
ij
g rd
r d
Non-relativistic potential
: Gaussian form
ωi: meson energy
A. Dote, T. Inoue, T. Myo, NPA 912, 66 (2013)
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How to deal with E-dep. potential?
( 0,1)
( 0,1)
2
1
8
I
ijI
ij ii j j
i j
CV r g r
f m m
Chiral SU(3)-based potential is an energy-dependent potential.
How to treat energy-dependent potentials in many-body system?
Especially, in the KbarNN-πYN coupled-channel problem case???
Generalize earlier studies of the KbarNN single-channel problem
with E-dep. effective potential:A. D., T. Hyodo, W. Weise, PRC79, 014003 (2009)
A. D., T. Inoue, T. Myo, PTEP 2015, 043D02 (2015)
“Self-consistency for Meson-Baryon energy”
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Self-consistency for KbarN energy
[KbarNN single-channel case]
barK NN NNB K H H
1. Kaon’s binding energy:
: Hamiltonian of 2N
2
N K
KN N K
N K
M m B KE M
M m B K
2. Define a KbarN-bond energy in two ways:
Estimation the two-body energy in the three-body system
: Field picture
: Particle picture
barK NNH : Hamiltonian of 2N+Kbar
NNH
A. D., T. Hyodo, W. Weise,
PRC79, 014003 (2009)
An interacting KbarN pair
carries
... 100% of B(K)
... 50% of B(K)
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Self-consistency for MB energy
[KbarNN-πYN coupled-channel case]
1 2 1 2 MMB B B BB M H H m
1. Meson’s binding energy:
: Hamiltonian of 2B
( )
2
B M
B M
B M
M m B ME MB M
M m B M
2. Define a MB-bond energy in two ways:
Estimation the two-body energy in the three-body system
A. D., T. Hyodo, W. Weise,
PRC79, 014003 (2009)
: Field picture
: Particle picture
1 2MB BH : Hamiltonian of 2B+M
1 2B BH
Generalized version of
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Realization of self-consistency
Indicator Δ
[MeV]
- Re E(MB)In [MeV]
Chiral SU(3) pot.
(fπ=110 MeV, Martin)
Field picture
Indicator of self-consistency
Δ=|E(MB)Cal – E(MB)In|
Self-consistent solution :
BKNN = 23.5 MeV, ΓπYN /2 = 9.1 MeV
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Result
Martin constraint SIDDHARTA constraint
Field: (B, Γ/2) = (19-36, 8-14)
Particle: (B, Γ/2) = (30-47, 12-14)
Field: (B, Γ/2) = (14-28, 8-15)
Particle: (B, Γ/2) = (22-38, 13-18)
(-BKNN, -Γ/2) [MeV]
120
100
90
(-BKNN, -Γ/2) [MeV]
fπ=120 MeV
90100
120
fπ=120 MeV
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P PK-
4. Discussion
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1. Dense matter or not?
P K-
P PK-
Chiral SU(3) potential
(E-dep.)A. Dote, T. Inoue and T. Myo,
NPA 912, 66 (2013)
B. E. (Λ*) ~ 15 MeV
⇒ Λ* = Λ(1420)
B. E. (“K-pp”) = 14-38 MeV
NN distance = 2.2 fm
⇒ ~ ρ0 (=0.17 fm-3)
Pheno. potential
(E-indep.)Y. Akaishi and T. Yamazaki,
PRC 65, 044005 (2002)
B. E. (Λ*) = 28 MeV
⇒ Λ* = Λ(1405)
B. E. (“K-pp”) = 51 MeV
NN distance = 1.9 fm
⇒ ~ 1.6 ρ0
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2. Comparison of Theory and Experiment
0 [MeV]
KbarNN thr.
-50 [MeV]
J-PARC E15
(1st run)
B ~ 16 MeV
Γ ~ 110 MeV
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2. Comparison of Theory and Experiment
0 [MeV]
KbarNN thr.
-50 [MeV]
J-PARC E15
(1st run)
B ~ 16 MeV
Γ ~ 110 MeV
J-PARC E15
(2nd run)
Full ccCSM calculation
Phenomenological pot. : BK-pp = 51 MeV(Martin)
Chiral SU(3)-based pot. : BK-pp = 14-28 MeV(SIDDHARTA, Field)
Chiral SU(3) pot.Pheno. pot.
Yamaga’s talk
(JPS symposium
2017)
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5. Summary
and
Future prospects
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“K-pp” studied with Fully coupled-channel Complex Scaling Method
• Chiral SU(3)-based KbarN potential constrained with the latest KbarN data (SIDDAHRTA)
K-pp (Jπ=0-, T=1/2) … (B, Γ/2) = (14--28, 8--15) MeV (Field picture)
(22--38, 13--18) MeV (Particle picture)
• Phenomenological KbarN potential (Akaishi-Yamazaki potential; Energy-independent)
K-pp (Jπ=0-, T=1/2) …. (B, Γ/2) = (51, 16) MeV
Self-consistency for meson-baryon energy is considered.
• At the moment, J-PARC E15 (2nd run) seems consistent with the result of K-pp (Jπ=0-, T=1/2) calculated with a phenomenological potential ???
• NN mean distance of “K-pp” system = 2.2 fm (Chiral pot.), 1.9 fm (AY pot.)
If KbarN potential is so attractive as Akaishi-Yamazaki potential, kaonic nuclei could be a gateway to dense matter.
Among kaonic nuclei, “K-pp” is the most essential system:
All theoretical studies predict B(K-pp) < 100 MeV.
“K-pp” = Resonance state of KbarNN-πYN coupled system
Kaonic nuclei are expected to have exotic nature, such as formation of
“Dense and cold” state, due to the strongly attractive KbarN interaction.
5. Summary
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Semi-relativistic treatment... Pion mass is small.
Lower pole of K-pp?… E-dep. Chiral SU(3) pot. may have the double pole structure.
Reaction spectrum... Direct comparison with experimental data.
It can be calculated using the Green function obtained
with ccCSM.
(“Morimatsu-Yazaki Green function method”)
...
5. Future prospects
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References:
• A. Dote, T. Inoue, T. Myo, PRC 95, 062201(R) (2017)• A. Dote, T. Inoue, T. Myo, arXiv: 1710.07589
Thank you very much!