Lattice
Lattice QCD
QCD studies
studies ofof strangenessstrangeness
SS == -2-2 baryon-baryon
baryon-baryon interactions
interactions
Kenji Sasaki (University of Tsukuba)for HAL QCD collaboration

HAL (Hadrons to Atomic nuclei from Lattice) QCD Collaboration
S. AokiT. Doi(Univ. of Tsukuba) (Univ. of Tokyo)
T. Inoue(Nihon Univ.)

T. Hatsuda(RIKEN)

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

Y. Ikeda(Tokyo Inst. Tech.)
H. Nemura(Tohoku Univ.)

Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration

IntroductionIntroduction
Strangeness in nuclei opened the new frontier of nuclear physics.
Experimental
Experimental side
side
Exploration of the multi-strangeness hadronicsystems is planned at J-PARC
Generalized BB interaction
Hypernuclear structure
Search of exotic hadrons
and so on

J-PARC

Theoretical
Theoretical side
side
Study
Study thethe hyperon-nucleon
hyperon-nucleon (YN)
(YN) andand hyperon-hyperon
hyperon-hyperon (YY)(YY) interactions
interactions
One of the most important subject in the (hyper-) nuclear physics
The phenomenological description of them haslarge uncertainties due to the shortage of experimental data.
Lattice QCD simulation can produce BB potential
directly from QCD complementary to an experiment.
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration

Introduction
Introduction
ThisThis workwork ::Baryon-baryon
Baryon-baryon interactions
interactions inin strangenessstrangeness S
S == -2-2 systemsystemThe first step towards the multi-strangeness world.
Structures of double-Λ hypernuclei and Ξ-hypernuclei
The SU(3) breaking effects in the BB interaction.
“H-dibaryon“ at physical point .
Information
Information ofof ΛΛΛΛ interaction
interaction andand H-dibaryon
H-dibaryon fromfrom experimentexperiment
Conclusions of the “NAGARA Event” (The double- hypernuclear event)
NAGARA event
Lower limit of “H“ mass : mH ≥ 2mΛ - 6.9MeV.
The Λ−Λ interaction is weakly attractive.

K.Nakazawa and KEK-E176 & E373 collaborators

Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration

SU(3)
SU(3) Classification
Classification ofof B-B
B-B statesstatesWithin S-wave total anti-symmetric states are constructed by combination of spin and flavor.
8 x 8 = 1 + 8S + 27 + 8A + 10 + 10*Flavor symmetric Flavor anti-symmetric
Spin singlet
Spin triplet

Hyper-charge (Strangeness)

Y=2(S=0)pΣ++Σ+p : flavor symmetric

Y=1(S=-1)pΣ+-Σ+p : flavor anti-symmetric
Y=0(S=-2)

Isospin (Iz)

There are 10 flavor combinations

ΛΛ, pΞ−, nΞ0, Ξ−p, Ξ0n, Σ+Σ−, Σ0Σ0, Σ−Σ+, ΛΣ0, Σ0Λ
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration

Classification
Classification ofof B-B
B-B statesstates withwith S=-2
S=-2
Flavor-Symmetric : spin singlet

I=2 state
27
I=1 states
8s

S=-2 BB states

1I=0 states

Flavor-Anti-symmetric : spin triplet
1010*
S=-2 BB states

8aKenji Sasaki (University of Tsukuba) for HAL QCD collaboration

I=1 states

I=0 state

Classification
Classification ofof B-B
B-B statesstates withwith S=-2
S=-2
Flavor-Symmetric : spin singletΣΣΛΣ
I=1 statesΝΞ 8s, 27 mixing

278s
1SU(3) IRSU(3) breaking

ΣΣ
ΝΞ
ΛΛ

I=0 states
1, 8s, 27 mixing

Flavor-Anti-symmetric : spin triplet
10
10*8aΣΣΛΣ I=1 statesΝΞ 8a, 10, 10* mixing

SU(3) breaking

ΝΞKenji Sasaki (University of Tsukuba) for HAL QCD collaboration

Channel
Channel coupling
coupling
EnergyEnergy levels
levels ofof baryon-baryon
baryon-baryon systemsystem inin thethe real
real worldworldΣΣ(Ι=0) ΣΣ(Ι=1) ΣΣ(Ι=2)

2400

NΞ(Ι=0)

2300

2200

N∆(I=1)NΣ(Ι=1/2)

small energy differenceNΞ(Ι=1)NΣ(Ι=3/2)

2100

NΛ(Ι=1/2)

2000

1900

ΛΛ(Ι=0)ΛΣ(Ι=1)The effects of the neighboring states
can not be ignored.

NN(I=0) NN(I=1)

1800

S=0

S=-1

S=-2

We have to extend our method to the coupled channel formalism.

Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration

HAL
HAL QCDQCD strategy
strategy
Calculate Bethe-Salpeter (BS) wave function on any gauge configuration.
 t−t 0 ,x  = ∑ 〈0∣Bt , x y B t ,x ∣BBt 0 〉yDefine the non-relativistic Schrödinger equation (general form)
2∇3E−  x =∫ U  x −y  y d y2

Performing the derivative expansion for the interaction kernelU  x − y  ≃ V 0  x  x − y   V 1  x , ∇  x−y ⋯The potential is given as1 ∇ 2  x V  x  = E −2    x This technique is widely applicable for hadronic systems

Extention to the YN and YY systems
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration

See details in S.Aoki et al arXiv:1106.2281 [hep-lat]

Coupled
Coupled channel
channel Schr
Schröödinger
dinger equation
equation
Using four-point correlator W with an optimized source such as,−Et

W α (⃗x , E )= A Ψ α (⃗x , E )eThe coupled channel Schrödinger equation can be rewritten as(2)pαα−H 0 W α ( ⃗x , E)=V αα (⃗x )W α ( ⃗x , E)+V αβ ( ⃗x )W β ( ⃗x , E)+V α γ (⃗x )W γ ( ⃗x , E)
2μ αDefine( )2W (⃗x , E)
pRα (⃗x , E) ≡ α∝ exp (−(E −M α )t )≃exp − α t
C α (t )2μ αTaking time derivative of R,

Product of single baryon correlators

2pα∂t Rα (⃗x , E) = −Rα (⃗x , E)
2 μαThus the potential matrix can be obtained as( )(VVVW Λ Λ ( ⃗x , E 0 ) W N Ξ ( ⃗x , E 0 ) W Σ Σ ( ⃗x , E 0 )(⃗x )(⃗x ) = W Λ Λ (⃗x , E 1 ) W N Ξ (⃗x , E 1 ) W Σ Σ (⃗x , E 1 )W Λ Λ ( ⃗x , E 2 ) W N Ξ ( ⃗x , E 2) W Σ Σ ( ⃗x , E 2 )
(⃗x )

ΛΛΛΛ
ΛΛNΞ
ΛΛ
ΣΣ−1)(α)−C Λ Λ ∂ RΛ Λ ( ⃗x , E 0 )−H 0 W Λ Λ ( ⃗x , E 0 )α−C Λ Λ ∂ R Λ Λ (⃗x , E 1 )−H 0 W Λ Λ (⃗x , E 1 )−C Λ Λ ∂ RΛ Λ ( ⃗x , E 2 )−H α0 W Λ Λ ( ⃗x , E 2 )Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration

Numerical
Numerical setup
setup
2+1 flavor gauge configurations by CP-PACS/JLQCD collaboration.
RG improved gauge action & O(a) improved clover quark action
β = 1.83, a-1 = 1.632 [GeV], a = 0.1209 [fm]163x32 lattice, L = 1.934 [fm].
κud = 0.13825, κs = 0.13710 was chosen (named Set3).
800 / 800 configurations are used.
Flat wall source is considered to produce S-wave B-B state.
16 shifted sources every 2 time-slices are considered
to enhance the S/N ratio.
The USQCD computer resources are used.
Cluster at FNAL
We acknowledge the USQCD for providing of computer resources.

Set3π

Kmπ/mΚ

N

Λ

Σ

Ξ

661±1

768±1

0.860

1482±31557±31576±31640±3
In unit of MeV

Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration

Isospin
Isospin combinations
combinations ofof BB
BB operator
operatorΛΛ, pΞ−, nΞ0, Ξ−p, Ξ0n, Σ+Σ−, Σ0Σ0, Σ−Σ+, ΛΣ0, Σ0Λ
I=0 operatorsFlavor symmetric








N=

1
1−0p − n
22

 =

1  −
1 0 01 −   −     333

I=1 operators





N=

1
1−0p  n
22

 =

1  −
1 − 
  −
 
22



Flavor anti-symmetric

I=2 operators
 =





1  −
4 0 0
1 −      
666
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration

ListsLists ofof channels
channels
I=0 statesSpin

BB channels

1S0ΛΛ3--S1

I=1 statesNΞ
NΞ

Strong attraction(H-dibaryon)
SU(3) representation

ΣΣ18s

27

--

8a--

--Attraction

Spin

BB channels

1S03S1NΞ
NΞ

--

ΣΣ

Strong repulsion
SU(3) representation

ΛΣΛΣ

--

8s

27

8a1010*I=2 states

RepulsionSpin

BB channels

SU(3) representation

1

ΣΣ

--

S0

Similar toThe NN potential

--

3

S1
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration

27

T. Inoue

Baryon-baryon
Baryon-baryon potential
potential inin thethe flavorflavor SU(3)
SU(3) limit.
limit.

Accessible by non-strange sector

Strong flavor dependence turns out with irreducible rep.
Various interaction are seen by extending to SU(3).
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration

11ΣΣ
(I=2)
S00 channelchannelΣΣ (I=2) SSetSet 33 :: mmππ== 661
661

3000
200
2000

Direct correspondence to the 27plet
in SU(3) irreducible representationVΣ Σ − Σ ΣV[MeV]

100

Similar behavior to the NN potential

01000-10000.5

11.5

000.5

r[fm]

11.5

Short range repulsion and mid-range attraction

33NΞ
(I=0)
NΞ (I=0) S
S11 channel
channel
3000
200
2000

Direct correspondence to 8aplet.

VΝ Ξ − Ν ΞV[MeV]

100Repulsive core is not so high01000
-100

00.5

11.5

More attractive than 27 plet potential

000.5

r[fm]

11.5

Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration

11N
Ξ,
ΛΣ
(I=1)
NΞ, ΛΣ (I=1) S
S00 channel
channel3000

Set
Set 33 :: mmππ== 661
661

3000
200

2000

200VΝ Ξ − Ν Ξ
2000

100

V[MeV]V[MeV]

10001000VΛ Σ − Λ Σ

01000

-100

00.5

1-100

1.5

000.5

11.5

000.5

r[fm]

1

1.5

000.5

r[fm]

1

1.58s-27 mixing channel.
100

Origin of strong repulsion

V[MeV]-1000
0

No attractive pocket in diagonal elements-100-2000

VΝ Ξ − Λ Σ-200
0-3000

00.5
0.5

r[fm]

1
1

1.5
1.5

Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration

33N
Ξ,
ΣΣ,
ΛΣ
(I=1)
NΞ, ΣΣ, ΛΣ (I=1) SS11 channelchannel
3000

3000

2000200VΝ Ξ − Ν Ξ
2000

01000

2000

01000-10000.5

1-10000.5

10.5

r[fm]

11.5-1000.5

r[fm]

11.5

10.5

r[fm]

1

Relatively small transition potential 200
2000-100-20000.50.5r[fm]110-100-2000

VΝ Ξ − Σ Σ

1.5-3000VΝ Ξ − Λ Σ0001000

-200

1.5

1.5

100

V[MeV]V[MeV]

-1000

01.5

VΣ Σ − Λ Σ

100-1000
V[MeV]

0.5

3000

100

0000-2000

0000100

1.500VΣΛΣ−ΛΣΣ−ΣΣ1000

1.50-3000

200

VΣ Σ − Σ Σ

100

V[MeV]

100

V[MeV]

3000

V[MeV]

200SetSet 33 :: mmππ== 661
661

0.50.5r[fm]11-100

1.5

00.5

11.50

1.500.5r[fm]Coupling of ΛΣ state to the other states are quite small.
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration

11.5

11
ΛΛ,
ΝΞ,
ΣΣ
(I=0)
S00 channel
channelΛΛ, ΝΞ, ΣΣ (I=0) S
3000

3000

3000

2000

200

VΛ Λ − Λ Λ
2000

0

2000

100

V[MeV]V[MeV]

100200VΝ Ξ − Ν Ξ

0-100

00.5

1-100

1.5

00.5

1-100

1.5

0.5

r[fm]

10

1.50

r[fm]101.5

3000
2002000-100-200000.5

r[fm]

111.5

11.50-100-2000-1001.5

r[fm]

-1000

0-2000.5

1.51001000VΛ Λ − Ν Ξ0VΛ Λ − Σ Σ

100

V[MeV]00.5

V[MeV]V[MeV]-1000

10

Relatively small

100-3000

0.5

0.5

0000

Totally repulsive

Most attractive
0010001000

1000

VΣ Σ − Σ Σ

100

V[MeV]

200SetSet 33 :: mmππ== 661
661

00.51VΝ Ξ − Σ Σ

1.5-2000000.5r[fm]11.5-3000

0In this channel, our group found the “H-dibaryon” in the SU(3) limit.

0.5
0.5r[fm]

11

1.5
1.5

T. Inoue [HAL QCD coll.]PRL106(2011)162002.

Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration

ListsLists ofof channels
channels
I=0 operators

Spin dependence of potential

Spin

BB channels

1S0

ΛΛ

3--

S1NΞ
NΞ

SU(3) representation

ΣΣ

18s

27--

8a--

--

I=1 operators

Isospin dependence of potential

Spin

BB channels

1S03S1NΞ
NΞ

--

ΣΣ

SU(3) representation

ΛΣΛΣ

--8s

278a1010*

I=2 operators
Spin

BB channels

SU(3) representation1ΣΣ--S0--

3S1
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration

27

Spin
Spin dependence
dependence ofof NNΞ,potentials
Ξ, ΛΣΛΣ potentials

Set
Set 33 :: mmππ== 661
661

6000

Ν Ξ (I=0)1S0
Ν Ξ (I=0)3S1150

V[MeV]

100

4000

0-100
000.5

0.5

11Λ Σ (I=1)1S0
Λ Σ (I=1)3S1150

V[MeV]

1000.511.5

1.51.5

r[fm]

6000

4000-50000050

2000-502000

100

400050Ν Ξ (I=1)1S0Ν Ξ (I=1)3S1

150

V[MeV]

200

6000

5000.51r[fm]

1.5

Spin triplet potentials are more attractive
than the spin singlet potentials

02000

-50
000

0.5

0.5

11r[fm]

1.5

The tensor potential is not separated yet
In spin triplet channel.

1.5

Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration

Isospin
Isospin dependence
dependence ofof NNΞ,potentials
Ξ, ΣΣΣΣ potentials

SetSet 33 :: mmππ== 6616616000

Ν Ξ (I=0)1S0Ν Ξ (I=1)1S0150

V[MeV]

100

4000

02000

8000

0.50.511300

Σ Σ (I=0)1S0
Σ Σ (I=2)1S0200

V[MeV]

6000

150
100

4000

00-50
0000.50.5

11r[fm]1.5

1.5In ΝΞ potentials the I=0 potentials are
more attractive than the I=1 potentials.
The short range behavior of potentials
are strongly depend on the choice of state.

50

2000

-50

1.5

1.5

r[fm]

250

000050

2000

-50
00100

400050Ν Ξ (I=0)3S1
Ν Ξ (I=1)3S1150

V[MeV]

6000

0.5

0.5

11r[fm]1.5

1.5

Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration

Summaries
Summaries andand outlooks
outlooks
We have investigated the S=-2 BB interactions from lattice QCD.
They are complement to experiments.
In order to deal with a variety of interactions, we extend our method
to the coupled channel formalism.
Asymptotic momentum can be determined from time derivativeof R-correlator.
The source optimization is not necessary in our formalism
by employing the time derivative treatment of the energy part.
We have found the strong state dependence of ΝΞ potential
especially at the short range region.
Realistic
potentials with lighter quark masses and large volume.
Toward the physical point !
Look for the physical “H-dibaryon”.
Coupled channel technique is powerful and widely applicable
We will tackle to reveal all baryon-baryon interactions
with S=0, -1, … -6 below the pion production threshold.
By the world's fastest “K computer”
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration