格子格子 QCDQCD によるによる S=-2
S=-2 バリオン間相互作用のバリオン間相互作用のクォーク質量依存性の研究
クォーク質量依存性の研究
Kenji Sasaki (CCS, University of Tsukuba)for HAL QCD collaboration

HAL (Hadrons to Atomic nuclei from Lattice) QCD Collaboration
S. AokiB. Charron(Univ. of Tsukuba) (Univ. of Tokyo)

T. Doi(RIKEN)F. Etminan
T. Hatsuda(Univ. of Tsukuba) (RIKEN)

Y. Ikeda
T. Inoue(Tokyo Inst. Tech.) (Nihon Univ.)

N. Ishii
K. Murano(Univ. of Tsukuba) (RIKEN)

M. Yamada(Univ. of Tsukuba)
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration

H. Nemura(Univ. of Tsukuba)Introduction
Introduction

Introduction
Introduction
Baryon-baryon
Baryon-baryon interactions
interactions areare crucial
crucial forfor nuclear
nuclear andand astrophysics
astrophysics
BBBB phase
phase shift
shift

For NN, large amount of scattering data
For YN and YY, experimental data are scarce.
Meson
Meson exchange
exchange modelmodelBB
BB interaction
interaction (potential)(potential)

Described by hadron dof
with phenomenological repul. coreΝΝ ΛπΛσΝΝ ΛΛ
Repulsive core is generatedby ω meson exchange.
Nucleus
Quark
Quark cluster
cluster model
model
Effective meson ex+ quark anti-symmetrization
u dsuudHypernucleus

Quark Pauli effects
are taking into accountKenji Sasaki (University of Tsukuba) for HAL QCD collaboration

Introduction
Introduction
Study
Study thethe hyperon-nucleon
hyperon-nucleon (YN)
(YN) andand hyperon-hyperon
hyperon-hyperon (YY)(YY) interactions
interactions
Strangeness brought the deeper understanding of BB interaction.

NNNN sector
sectorThree
Three flavor
flavor (u,d,s)(u,d,s) worldworldnp

Σ- Λ Σ0 Σ+Ξ-Ξ0n×p

Σ- Λ Σ0 Σ+Ξ-

Ξ0Flavor symmetric=

18S

27Flavor anti-symmetricH-dibaryon
H-dibaryon state
state isis expected
expected
WideWide variety
variety ofof BBBB interaction
interaction

8A

10

10SU(3)
SU(3) breaking
breaking278s1ΣΣΝΞ, ΛΣ
ΛΛ, ΝΞ, ΣΣKenji Sasaki (University of Tsukuba) for HAL QCD collaboration

I=1 states
8s, 27 mixing
I=0 states
1, 8s, 27 mixing

Introduction
Introduction
S=-2
S=-2 BBBB interaction
interaction

Direct scattering experiment is very difficult.
All irreducible representations are involved.
H-dibaryon
H-dibaryon

Flavor singlet object (6q / tightly bound)u dsusdNo Pauli blocking was proved by quark cluster model.
Strong color magnetic (attractive) interaction
Recent Lattice QCD studies
HAL QCD: SU(3) limit
BE = 26MeV mπ = 470MeV
NPLQCD: SU(3) breaking
BE = 13MeV mπ = 390MeV

Conclusions of the “NAGARA Event”
K.Nakazawa and KEK-E176 & E373 collaborators

Λ−Ν attraction
Λ−Λ weak attractionmH ≥ 2mΛ - 6.9MeV

Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration

HAL
HALQCD
QCD strategy
strategyQCD
QCD toto hadronic
hadronic interactions
interactions
QCD
QCD Lagrangian
Lagrangian

1 a aμνL QCD= q̄ (i γμ D −m) q+ F μ ν F4μLattice QCD simulationu ds〈O( ̄q , q ,U )〉uud=∫ eHAL
HALQCD
QCDmethod
method−S E

Lattice
Lattice QCD
QCD simulation
simulation cancan connect
connect
thethe fundamental
fundamental QCD
QCD withwith nuclear
nuclear physics
physics

O dU d ̄q dq

TheThe potential
potential throughthrough ourour method
methodreproducereproduce toto thethe phase
phase shift
shift byby QCD
QCD

NBS wave function

BB interaction (potential)

N. Ishii, S. Aoki and T. Hatsuda, Phys. Rev. Lett. 99 (2007) 022001
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration

Nambu-Bethe-Salpeter
Nambu-Bethe-Salpeter wave
wave function
function
Definition
Definition :: equal
equal time
time NBS
NBS w.f.
w.f.
Ψ ν (E ,t−t 0 ,⃗r ) = ∑ 〈0∣B i (t , ⃗x + ⃗r )B j (t ,⃗x )∣E , ν ,t 0 〉abcT

B=ϵ (q a C γ 5 qb )q c⃗x

The ket stands for the eigenstate of the complete set of observables
E : Total energy of systemν : other observables which needs to form the complete setLocal composite interpolating operators
pα = ϵc c c (C γ5 )d d δd α u(ξ1 ) d(ξ 2 )u(ξ 3 )1 23123Σα0 = − ϵc c c (C γ 5 )d d δd α
1 23123Λ α = − ϵc c c (C γ 5 )d d δd α12 3123√√

1[ d(ξ1)s(ξ2)u(ξ3 )+u(ξ1) s(ξ2)d (ξ3 )]21[ d(ξ1)s(ξ2)u(ξ3 )+s(ξ1)u(ξ2)d (ξ3 )−2u(ξ1) d(ξ2) s(ξ3)]
6

NBS wave function has a same asymptotic form with quantum mechanics.
(NBS wave function is characterized from phase shift)Ψ(t −t 0, ⃗r )≃ Asin( pr +δ( E))
prKenji Sasaki (University of Tsukuba) for HAL QCD collaboration

N. Ishii et al. [HAL QCD Collab.], Phys. Lett. B 712 (2012) 437

SchrSchröödinger
dinger equation
equation
Define the energy-independent potential in Schrödinger equation(most general form)(2)

k−H 0 Ψ( ⃗x)=∫ U ( ⃗x , ⃗y )Ψ ( ⃗y)d 3 y
2μRecent development : Time dependent method.
We replace ψ to R defined below−Et2A Ψα (⃗x , E)epα∂t Rα (⃗x , E) ≡ ∂t∝−Rα (⃗x , E)
−m t −m t
2μe eα(AB)Performing the derivative expansion for the interaction kernel()− ∂ −H 0 R( ⃗x )=∫ U ( ⃗x , ⃗y) R( ⃗y )d y∂t
3Taking the leading order of derivative expansion of non-local potential

U ( ⃗x , ⃗y) ≃ V 0 ( ⃗x )δ( ⃗x −⃗y) + V 1 (⃗x , ∇ )δ ( ⃗x − ⃗y ) ⋯Finally local potential was obtained as∂t R( ⃗r ) 1 ∇ 2 R(⃗x )V ( ⃗x) = −+R( ⃗v ) 2μ R( ⃗x )Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration

S.Aoki [HAL QCD collab.] Proc. Jpn. Acad., Ser. B, 87 509

Coupled
Coupled channel
channel Schr
Schröödinger
dinger equation
equation
Preparation
Preparation forfor thethe NBS
NBS wave
wave function
functionΨα (E ,t ,⃗r ) = ∑ 〈0∣(B1 B 2 )α (t ,⃗r )∣E 〉⃗xΨ (E ,t , ⃗r ) = ∑ 〈0∣(B1 B 2 ) (t , ⃗x )∣E 〉ββ

Two-channel coupling case
TheThe same
same “in”“in” statestate⃗x

InsideInside thethe interaction
interaction range
range
In theleading
orderof velocity
expansionof non-local potential,
Assuming
thatpotentials
are energy
independent,Factorization
Factorization ofof interaction
interaction kernel
kernelCoupled channel Schrödinger equation.

()

22pα
∇
ααααβ+ψ (⃗x , E )=V α (⃗x ) ψ (⃗x , E )+ V β (⃗x ) ψ (⃗x , E )2μ α 2μ αµα : reduced masspα : asymptotic momentum.

Asymptotic momentum are replaced by the time-derivative of R.
B1 B 2RI((t , ⃗r )=∑⃗x 〈 0∣B1 (t , ⃗x+ ⃗r ) B2 (t , ⃗x ) ̄I (0)∣0〉 e(m1+m 2)tV αα (⃗r )βV α (⃗r ) x−1()(22∇α∇∂(− ) R I1 (⃗r , E) (− ∂ ) RβI2 (⃗r , E)
−1αV β (⃗r ) x2μα ∂t2μβ ∂tR αI1 (⃗r , E) RβI1 (⃗r , E)
=αββ22R I2 (⃗r , E) R I2 (⃗r , E)
∇α∇βV β (⃗r )∂∂(− ) R (⃗r , E) (− ) R (⃗r , E)
2μα ∂t I12μβ ∂t I2))

x=exp (−(m α + mα )t )11Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration2

exp (−(mβ + mβ ) t)
2

Results
Results

Numerical
Numerical setup
setup
2+1 flavor gauge configurations by PACS-CS collaboration.
RG improved gauge action & O(a) improved clover quark action
β = 1.90, a-1 = 2.176 [GeV], 323x64 lattice, L = 2.902 [fm].
κs = 0.13640 is fixed, κud = 0.13700, 0.13727 and 0.13754 are chosen.
Flat wall source is considered to produce S-wave B-B state.
The KEK computer system A resources are used.
In unitof MeV

πK

Esb
Esb 11

Esb
Esb 22701±1570±2

411±2

789±1
0.89

713±2
0.80

635±20.65Λ

1585±51644±51411±121504±101215±121351± 8Σ

1660±41531±111400±10Ξ

1710±51610± 91503± 7mπ/mΚ
NEsb
Esb 11

Esb
Esb 33

Esb
Esb 22

Esb
Esb 333300

3200

2702MeV
2718MeV
2800MeVΛΛ : 3288MeVΝΞ : 3295MeV3000ΣΣ : 3320MeV
3100

3008MeV
3021MeV
3062MeV

2900

2800

2700u,d quark masses lighter

SU(3) breaking effects becomes larger

Kenji Sasaki (University of Tsukuba) for HAL QCD collaborationΛΛ,(I=0) S
S00 channelchannelΛΛ, ΝΞ,ΝΞ, ΣΣΣΣ (I=0)
11

Esb1
Esb1::mmππ==701
701MeVMeV
Esb2
Esb2::mmππ==570
570MeVMeV
Esb3
Esb3::mmππ==411
411MeVMeV

Diagonal elements

shallow attractive pocket
Off-diagonal elements

Deeper attractive pocket

Strongly repulsive

All channels have repulsive coreRelatively weaker than the others
In this channel, our group found the “H-dibaryon” in the SU(3) limit.
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration

Comparison
Comparison ofpotential matricesof potentialmatrices
Transformation of potentials
from the particle basis to the SU(3) irreducible representation (IR) basis.
SU(3) Clebsh-Gordan coefficients    VV N  V V1
∣1 〉∣  〉
NNN
=,V  U t V8∣ 8 〉 U ∣N  〉 U V   VV 27∣27 〉∣  〉V  V N V  
In the SU(3) irreducible representation basis,the potential matrix should be diagonal in the SU(3) symmetric configuration.

Off-diagonal part of the potential matrix in the SU(3) IR basis
would be an effectual measure of the SU(3) breaking effect.
We will see how the SU(3) symmetry of potential will be broken
by changing the u,d quark masses lighter.
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration111,1, 88ss,, 27
27 (I=0)(I=0) S
S00 channel
channel

Esb1Esb1::mmππ==701
701MeVMeV
Esb2
Esb2::mmππ==570
570MeVMeV
Esb3
Esb3::mmππ==411
411MeVMeV

Diagonal elements

Strongly attractive
H-dibaryon channel

Pauli blocking effect

Off-diagonal elements

Mixture of singlet and octet
Is relatively larger than the others

27 plet does not mix so much to the other representations

Kenji Sasaki (University of Tsukuba) for HAL QCD collaborationΛΛand ΝΞphase shifts
shiftsΛΛ andΝΞ phase
Esb1
Esb1::mmππ==701
701MeVMeV

Esb2
Esb2::mmππ==570
570MeVMeV

Esb3
Esb3::mmππ==411
411MeV
MeV

Preliminary!

Esb1:
Bound H-dibaryon
Esb2:
H-dibaryon is near the ΛΛ threshold
Esb3:
The H-dibaryon resonance energy is close to ΝΞ threshold..
We can see the clear resonance shape in ΛΛ phase shifts for Esb2 and 3.
The “binding energy” of H-dibaryon from ΝΞ threshold becomes smaller
as decreasing of quark masses.
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration

Summary
Summary andand outlook
outlook
We have investigated the S=-2 BB system from lattice QCD.
In order to deal with a variety of interactions, we extend our method
to the coupled channel formalism.
Potentials are derived from NBS wave functions calculated withPACS-CS configurations
Quark mass dependence of potentials can be seen not in long range
region but in shot distances as an enhancement of repulsive core.
Small mixture between different SU(3) IRs can be seen as
the flavor SU(3) breaking effect.
SU(3) breaking effects are still smalleven in mπ/mΚ=0.65 situation
but it would be change drasticallyat physical situation mπ/mΚ=0.28.
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration