Λ(1405) and “Kpp” - J-PARC Theory top...
Transcript of Λ(1405) and “Kpp” - J-PARC Theory top...
Λ(1405) and “K-pp”
on a coupled-channel scheme
1. Introduction
2. Method
• Complex Scaling Method (CSM)
• Chiral SU(3)-based KbarN interaction
3. Results
• Λ(1405) as a KbarN-πΣ system with coupled-channel CSM
• “K-pp” as a KbarNN-πYN system with ccCSM+Feshbach method
4. Further analysis of “K-pp”
5. Summary and future plan
Akinobu Doté (KEK Theory Center, IPNS / J-PARC branch)
Takashi Inoue (Nihon university)
Takayuki Myo (Osaka Institute of Technology)
「ストレンジネス核物理の発展方向」03. Aug. ’15 @ KEK Tokai campus (Tokai 1st building), Tokai., Ibaraki, Japan
1. Introduction
Phenomenologically derived I=0 KbarN potential … Very attractive
• Deeply bound (Total B.E. ~100MeV)
• Quasi-stable (πΣ decay mode closed)
• Highly dense state … anti-kaon attracts nucleons
Excited hyperon Λ(1405) = quasi-bound state of I=0 KbarN
Y. Akaishi and T. Yamazaki, PRC65, 044005 (2002)
Kaonic nuclei = Exotic system !?
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)
Restoration of chiral symmetry in dense matter
Interesting structure, Neutron star, …
Relate to various physics:
“Prototype of
kaonic nuclei”
K-
Proton
Λ(1405) “Building block of kaonic nuclei”
… used to determine KbarN interaction
K-P P
K-pp
Most essential kaonic nucleus
3HeK-, pppK-, 4HeK-, pppnK-,
…, 8BeK-,…
Nuclear many-body system with K-
Kaonic nuclei
at J-PARC
P PK-
Prototype system = K- pp
Experiments of K-pp searchFINUDA DISTO
J-PARC E15 SPring8/LEPSJ-PARC E27
• p + p → p + Λ + K+ at Tp=2.85 GeV
• K- pp???B. E. = 103 ±3 ±5 MeVΓ = 118 ±8 ±10 MeV
PRL104, 132502 (2010)
• Stopped K- on Li, C, Al, V targets
• K- pp???B. E.= 115 MeVΓ = 67 MeV
PRL 94, 212303 (2005)
• 3He(K-, n) X at PK = 1 GeV/c
• Attraction in K-pp subthreshold regionarXiv:1408.5637 [nucl-ex],
to appear in PTEP
• d(γ, π+K+) X at Eγ = 1.5 – 2.4 GeV
• No evidence of K-pp bound statePLB 728, 616 (2014)
• d(π+, K+) X at Pπ= 1.69 GeV/c
• ΣN cusp, Y* shiftPTEP 101D03 (2014)
• K-pp???
B. E. ~ 95 MeV
Γ ~ 162 MeV
PTEP 021D01 (2015)
Typical results of theoretical studies of K-pp
DHW AY BGL IS SGM
B(K-pp) 20±3 47 16 9 ~ 16 50 ~ 70
Width Γ 40 ~ 70 61 41 34 ~ 46 90 ~ 110
Method Variational
(Gauss)
Variational
(Gauss)
Variational
(HH)
Faddeev-AGS Faddeev-AGS
Potential Chiral
(E-dep.)
Pheno. Chiral
(E-dep.)
Chiral
(E-dep.)
Pheno.
Kinematics Non-rel. Non-rel. Non-rel. Non-rel. Non-rel.
Dependent on the type of KbarN interaction
Chiral SU(3)-based potential (Energy dependent)
-->> Shallow binding: B(K-pp) ~ 20 MeV
Phenomenological potential (Energy independent)
-->> Deep binding: B(K-pp) = 40~70 MeV
B(K-pp) < 100 MeV
B.E. of “K-pp”(=KbarNN-πYN with Jπ=0-, I=1/2) is less than 100 MeV.
2. Method
• Complex Scaling Method
• Feshbach projection on
coupled-channel Complex Scaling Method
“ccCSM+Feshbach method”
A. D., T. Inoue, T. Myo,
PTEP 2015, 043D02 (2015)
Resonant state
Coupled-channel
system
⇒ “coupled-channel
Complex Scaling Method”
• Λ(1405) = Resonant state & KbarN coupled with πΣ
• “K-pp” … Resonant state of
KbarNN-πYN coupled-channel systemDoté, 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)
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)
Complex rotation (Complex scaling) of coordinate
Resonance wave function → L2 integrable : ,i iU e e r r k k
Diagonalize Hθ = U(θ) H U-1(θ) with Gaussian base,
Continuum state appears on 2θ line. Resonance pole is off from 2θ line, and independent of θ. (ABC theorem)
tan-1 [Im E / Re E] = -2θ
Chiral SU(3) potential with a Gaussian form
Weinberg-Tomozawa term
of effective chiral Lagrangian
Gaussian form in r-space
Semi-rela. / Non-rela.
Based on Chiral SU(3) theory
→ Energy dependence
Constrained by KbarN scattering length
aKN(I=0) = -1.70+i0.67fm, aKN(I=1) = 0.37+i0.60fm A. D. Martin, NPB179, 33(1979)
( 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
A non-relativistic potential (NRv2c)
: Gaussian form
ωi: meson energy
• Anti-kaon = Nambu-Goldstone boson
⇒ Chiral SU(3)-based KbarN potential
A. D., T. Inoue, T. Myo, Nucl. Phys. A 912, 66 (2013)
3. Hyperon resonance Λ(1405)
on ccCSM
Excited hyperon Λ(1405)
I=0 N-Kbar bound state
Not 3 quark state,
← A naive quark model fails.N. Isgar and G. Karl, Phys. Rev. D18, 4187 (1978)
But rather a molecular stateT. Hyodo and D. Jido,
Prog. Part. Nucl. Phys. 67, 55 (2012)
qbar
sqq q
q <<
KbarN-πY coupled system with s-wave and isospin-0 state
Λ* on c.c. Complex Scaling Method
Schrödinger equation to be solved
… I=0 KbarN-πΣ coupled potentialV
: complex parameters to be determined
Wave function expanded with Gaussian base
Poles of I=0 KbarN-πΣ system found by ccCSM
KbarN potential:
a chiral SU(3) potential
(NRv2, fπ=110)
πΣ continuum
KbarN continuum
A. D., T. Inoue, T. Myo,
Nucl. Phys. A 912, 66 (2013)
θ=30°
Λ*
Lower pole
πΣ KbarN
A. D., T. Myo,
Nucl. Phys. A 930, 86 (2014)
“Complex-range
Gaussian basis”
-Γ/2
[Me
V]
M [MeV]
Double-pole structure of Λ(1405)
θ=40°
Higher pole
D. Jido, J.A. Oller, E. Oset, A. Ramos, U.-G. Meißner, Nucl. Phys. A 725 (2003) 181
ccCSM wfnc. of double pole
Norm (KbarN)
1.115+0.098i
Norm (πΣ)
-0.115-0.098i
Norm (KbarN)
0.097+0.154i
Norm (πΣ)
0.903-0.154i
KbarN dominant
πΣ dominant
Higher pole
Lower pole
A. D., T. Myo,
Nucl. Phys. A 930, 86 (2014)
Double-pole structure of Λ(1405)
1435
1331
KbarN
πΣ
fπ = 90 MeV 100 MeV 110 MeV 120 MeV
1349
Γ=134
1371
Γ=221
1395
Γ=275
1425
Γ=327
1420
Γ=46
1418
Γ=40
1418
Γ=33
1418Γ=28
Lower pole does exist in our potential.
It depends strongly on a parameter fπ.
A. D., T. Myo,
Nucl. Phys. A 930, 86 (2014)
P PK-
3. Three-body “K-pp” resonance
on ccCSM+Feshbach projection
“K-pp” =
KbarNN – πΣN – πΛN (Jπ = 0-, T=1/2)
ccCSM+Feshbach method• Λ(1405) = two-body system of KbarN-πΣ
→ Explicitly treat coupled-channel problem
• “K-pp” = three-body system of KbarNN-πYN
… High computational cost
For economical treatment of “K-pp”, we construct an effective KbarN
single-channel potential by means of Feshbach projection on CSM.
Formalism of ccCSM + Feshbach method
Schrödinger eq.
in model space “P” and out of model space “Q”
Eff
PPP PT U E E
Eff
P P PQ Q QPU E v V G E V
Schrödinger eq. in P-space :
Effective potential for P-space
P P PQ P P
QP Q Q Q Q
T v VE
V T v
Elimination of channels by Feshbash method
1
Q
G EE H
Q-space Green function:
1 1
Q n n
nQQ n
G EE H E
n
: expanded with Gaussian base.
Express the GQ(E) with Gaussian base using ECR
1
Q
Eff
P P PQ QPU E v V U UG VE
Q
G E
Extended Closure Relation in Complex Scaling Method
QQ n n nH 1n n
BC R
1n n
n
Diagonalize HθQQ with Gaussian base,
Well approximated
T. Myo, A. Ohnishi and K. Kato, PTP 99, 801 (1998)
R. Suzuki, T. Myo and K. Kato, PTP 113, 1273 (2005)
1
QQ QQH U H U
Applying this tequnique
to the two-body KbarN-πY system,
Effective single-channel KbarN potential
is constructed.
Using the UEffKN in “K-pp” three-body calculation,
the KbarNN-πYN coupled-channel problem is reduced
to the KbarNN single-channel problem.
VKN-KN VKN-πY
VπY-KN VπY-πY
UEffKN
“K-pp” … KbarNN - πΣN - πΛN (Jπ=0-, T=1/2) “K-pp” … KbarNN - πΣN - πΛN (Jπ=0-, T=1/2)
Apply ccCSM + Feshbach method to K-pp
• Schrödinger eq. for KbarNN channel :
; 0,1
' ; 0,1
barV K N Y I
V Y Y I
( 0,1)bar
Eff
K N IU E
For the two-body system, P = KbarN, Q = πY
( ,1) ( ,1) (3) (3) ( ,1) (3) (3)
2 2 1 1/ 2" " , , 0KNN KNN KNN
a a a NN Ta
K pp C G G S K NN
1 1x x x x
( ,2) ( ,2) (3) (3) ( ,2) (3) (3)
2 2 0 1/ 2, , 0KNN KNN KNN
a a a NNT
a
C G G S K NN
1 1x x x x
Ch. 1: KbarNN, NN:1E
Ch. 2: KbarNN, NN:1O
• Trial wave function
(3)
( , ) (3) (3) ( , ) (3) (3) ( , )
2 2 (3)
2
, exp ,KNN i KNN i KNN i
a a aG N A
1
1 1
xx x x x
x
• Basis function = Correlated Gaussian
…including 3-types Jacobi-coordinates
Feshbach + ccCSM
(1,2
)bar barbar bar bi
arNNK NN K NN K NN
Eff
K N I K Ni
T EV EU
Self-consistency for complex KbarN energy
Kbar
NN
E(KN)Cal
( )
( )bari
Eff
K N II nE KNU
When E(KN)In= E(KN)Cal,
a self-consistent solution is obtained.
Effective KbarN potential has energy dependence…
• E(KN)In : assumed in the KbarN potential
• E(KN)Cal : calculated with the obtained K-pp
Self-consistency for complex KbarN energy
NNB K H H 1. Kaon’s binding energy: NNH : Hamiltonian of two nucleons
2
N K
KN N
N K
M m B KE M
M m B K
2. Define a KbarN-bond energy in two ways
How to determine the two-body energy in the three-body system? A. D., T. Hyodo, W. Weise,
PRC79, 014003 (2009)
: Field picture
: Particle picture
Particle pictureField picture
Self-consistent results
fπ=90~120MeV
NN pot. : Av18 (Central)
KbarN pot. : NRv2c potential
(fπ=90 - 120MeV)
fπ = 90
100
fπ = 90
100
110
120
Particle picture
(B, Γ/2) = (25~30, 15~32)
120110
100
fπ = 90×Unstable for scaling angle θ!
Field picture
(B, Γ/2) = (21~32, 9~16)
NN correlation density
Re ρNN
Im ρNN
NN repulsive core
NN pot. : Av18 (Central)
KbarN pot. : NRv2c potential
fπ=110, Particle pict.
Correlation density in Complex Scaling Method
,XNNN x x ,, NNNN x r x
,i
XN XN e r r 3 23 ,i ide e
R x R
N
N
Kbar
NN distance = 2.1 - i 0.3 fm
~ Mean distance of 2N in nuclear matter at normal density!
Energy composition of “K-pp” NN pot. : Av18 (Central)
KbarN pot. : NRv2c potential
(fπ=110MeV)
barKbar bar bar
NNNK NN K NN K N
EffV VH T V
Field pict. Particle pict.
TKNN 163.1 174.6
VNN -15.3 -15.9
VKN -153.3 -169.8
VKNEff -20.1 -16.1
-B.E. -25.6 -27.3
-5.5 -11.2
P PK-
4. Further analysis of “K-pp”
• SIDDHARTA constraint for K-p scattering length
• Another way of KbarN energy self-consistency
K-pp with SIDDHARTA data
M. Bazzi et al. (SIDDHARTA collaboration),
NPA 881, 88 (2012)
K-p scattering length (with improved Deser-Truman formula)
Y. Ikeda, T. Hyodo and W. Weise, NPA 881, 98 (2012)
Precise measurement of 1s level shift of kaonic hydrogen
Strong constraint for the KbarN interaction!
U. -G. Meissner, U. Raha and A. Rusetsky, Eur. Phys. J. C 35, 349 (2004)
(Prediction)
“K-pp” with Martine value NN pot. : Av18 (Central)
KbarN pot. : NRv2c potential
(fπ=90 - 120MeV)
fπ = 100
110120
fπ = 90
100
110
120
aKN(I=0) = -1.7 + i0.68 fm
aKN(I=1) = (0.37) + i0.60 fm
“K-pp” with SIDDHARTA value NN pot. : Av18 (Central)
KbarN pot. : NRv2a-IHW pot.
(fπ=90 - 120MeV)
fπ = 100
110120
fπ = 90
100
110
120
Particle picture
(B, Γ/2) = (16~19, 14~25)
Field picture
(B, Γ/2) = (15~22, 10~18)
aKN(I=0) = -1.97 + i1.05 fm
aKN(I=1) = 0.57 + i0.73 fm
“K-pp” with SIDDHARTA value NN pot. : Av18 (Central)
KbarN pot. : NRv2a-IHW pot.
(fπ=90 - 120MeV)
E(KN) = -B(K)
E(KN) = -B(K)/2
E(KN) = -B(K)/2 - ΔBarnea, Gal, Livertz, PLB 712, 132 (2012)
Averaged KbarN energy in many-body system
fπ = 100
110120
fπ = 90
100
110
120
aKN(I=0) = -1.97 + i1.05 fm
aKN(I=1) = 0.57 + i0.73 fm
Summary of “K-pp”
NRv2c NRv2a-IHW DHW-HNJHfp=110 fp=110
aKN (I=0) -1.70 + i 0.68 -1.97 + i 1.04 -1.54 + i 1.14aKN (I=1) 0.68 + i 0.60 0.57 + i 0.73 0.39 + i 0.53
Λ * (High) -17.2 - i 16.6 -8.1 - i 13.3 -11.5 - i 21.9Λ * (Low) -39.8 - i 137.9 -45.1 - i 138.8
K-ppField -25.6 - i 11.6 -18.2 - i 12.3 -16.9 - i 23.5Particle -27.3 - i 18.9 -17.8 - i 17.2 -20.8 - i 29.2
†
† Dote, Hyodo, Weise, PRC 79, 014003 (2009)
Imaginary part is treated perturbatively.
5. Summary and future plans
5. SummaryWe have investigated
an excited hyperon Λ(1405) as a resonant state of KbarN-πΣ coupled system and
a prototype of Kbar nuclei “K-pp” as that of KbarNN-πYN coupled system,
based on a coupled-channel Complex Scaling Method (ccCSM)
using a chiral SU(3)-based KbarN-πY potential.
Λ(1405)
• Double pole structure is confirmed in our chiral SU(3) potential with a Gaussian form.
• Higher pole appears at 1420 MeV with a half decay witdth of ~20 MeV stably,
while Lower pole position strongly depends on a parameter of the potential.
• Main component of Higher pole is the KbarN, while that of Lower pole is πΣ.
“K-pp” with ccCSM+Feshbach projection
• πY’s are eliminated to reduce the coupled-channel problem to a single KbarNN channel
problem with Feshbach projection, which is well realized with Extended Closure Relation.
• With a constraint of Martine KbarN scattering length,
B(K-pp) = 20 ~ 30 MeV and Γ/2 = 10 ~ 30 MeV. NN distance ~ 2.2 fm → Normal density
• With a constraint of SIDDAHRTA K-p scattering length
and a prediction of K-n one by a chiral unitary model,
B(K-pp) = 15 ~ 22 MeV and Γ/2 = 10 ~ 25 MeV.
… Shallower binding compared with Martine’s case.
Cats in KEK
Thank you for your attention!
References:
1. A. D., T. Inoue, T. Myo,
NPA 912, 66 (2013)
2. A. D., T. Myo, NPA 930, 86 (2014)
3. A. D., T. Inoue, T. Myo,
PTEP 2015, 043D02 (2015)
5. Future plans
Full-coupled channel calculation of K-pp
… Deailed study for the double pole structure of K-pp
Application to resonances of other hadronic systems