Coupled
Coupled channel
channel approach
approach to
to hyperon-nucleon
hyperon-nucleon
interaction
interaction from
from Lattice
Lattice QCD
QCD
Kenji Sasaki (CCS, University of Tsukuba)
for HAL QCD Collaboration
HAL (Hadrons to Atomic nuclei from Lattice) QCD Collaboration
S. Aoki
(YITP)
B. Charron
(Univ. of Tokyo)
T. Doi
(RIKEN)
F. Etminan
(Birjand U.)
T. Hatsuda
(RIKEN)
Y. Ikeda
(RIKEN)
T. Inoue
(Nihon Univ.)
N. Ishii
(RCNP)
K. Murano
(RCNP)
H. Nemura
(Univ. of Tsukuba)
M. Yamada
(Univ. of Tsukuba)
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration
Introduction
Introduction
Introduction
Introduction
BB interactions are inputs for nuclear structure, astrophysical phenomena
Once we obtain a “proper” nuclear potential,
we apply them to the structure of (hyper-) nucleus
and neutron star calculation.
Y dof
u ds
uu
d
How do we obtain the BB force?
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration
Introduction
Introduction
Traditional way to research the BB interaction / potential
BB
BB interaction
interaction (potential)
(potential)
BB
BB phase
phase shift
shift
NN
NN interaction
interaction
Large amount of scattering data
to determine theoretical parameters
YN
YN // YY
YY interaction
interaction
More strangeness, more difficult to access experimentally.
Experimental data are scarce.
Why do we consider strangeness?
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration
Introduction
Introduction
Strangeness brought us deeper understanding of BB interaction.
Three flavor (u, d, s) world
n p
0
+
Λ
Σ
Σ
Σ
8 x 8 = 1 + 8S + 10 + 8A + 10 + 10*
Ξ- Ξ0
Flavor symmetric
Spin singlet
Two flavor (u, d) world
2x2= 1+3
Flavor anti-symmetric
Spin triplet
Spin singlet
Flavor symmetric
H-dibaryon
H-dibaryon state
state is
is expected
expected
Spin triplet
8A
p
Isospin (Iz)
Wide variety
1
Flavor anti-symmetric
n
8S
Pauli
Pauli forbidden
forbidden state
state
27
Short
Short range
range repulsion
repulsion
10
Almost
Almost forbidden
forbidden state
state
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration
10
Introduction
Introduction
Approaches
Approaches to
to baryon-baryon
baryon-baryon interactions
interactions
Meson
Meson exchange
exchange model
model
Quark
Quark cluster
cluster model
model
Described by hadron dof
Ν
π,ρ
ω
Effective meson ex
+ quark anti-symmetrization
Ν
Ν
Ν
Repulsion is generated
by ω meson exchange and
phenomenological repul. core
u ds
Effective
Effective Field
Field theory
theory
Systematic calc. respecting with
symmetry of QCD
uu
d
Quark Pauli effects
Color magnetic int.
are taking into account
First
First principle
principle calculation
calculation
Direct derivation of nuclear force
from QCD
Short range int.
parametrized by
contact term
Particularly
difficult problem...
Derivation of BB interaction from QCD
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration
HAL
HALQCD
QCD method
method
QCD
QCD to
to hadronic
hadronic interactions
interactions
Start
Start with
with the
the fundamental
fundamental theory,QCD,
theory,QCD, to
to obtain
obtain aa “proper”
“proper” interaction
interaction
μ
L QCD= q
̄ (i γμ D −m) q+
1 a aμν
Fμ ν F
4
BB
BB scattering
scattering phase
phase shift
shift
Lattice QCD simulation
u ds
Lüscher's
Lüscher's finite
finite volume
volume method
method
uu
d
M. Lüscher, NPB354(1991)531
Guaranteed to be the same result
T.Kurth et al JHEP 1312 (2013) 015
NBS
NBS wave
wave function
function
BB
BB interaction
interaction (potential)
(potential)
HAL
HALQCD
QCDmethod
method
Ishii, Aoki, Hatsuda, PRL99 (2007) 022001
Widely applicable!
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration
Aoki, Hatsuda, Ishii, PTP123, 89 (2010).
Nambu-Bethe-Salpeter
Nambu-Bethe-Salpeter wave
wave function
function
Definition
Definition :: equal
equal time
time NBS
NBS w.f.
w.f.
−E(t−t 0 )
Ψ( E ,⃗r )e
= ∑ 〈0∣B i (t ,⃗x +⃗r )B j (t ,⃗x )∣E , ν , t 0 〉
E : Total energy of system
⃗x
Local
Local composite
composite interpolating
interpolating operators
operators
0
abc
T
B=ϵ (q a C γ 5 qb )q c
−
p = u d u n = u d d Ξ = s u s Ξ = sd s
1
Λ=
[ d s u+s ud −2u d s ]
6
1
Σ+ = −u s u Σ0 = −
[ d s u+u s d ] Σ− = −d s d
2
√
√
−E t
Four
Four point
point correlator
correlator F B B ( ⃗r , t)=〈0 ⌈T [ B1 ( ⃗r ,t )B 2 (0, t)( B̄2 B̄1 )t ] ⌉0〉=∑ n An Ψ( E n , ⃗r )e
n
1
2
0
2
2
It satisfies the Helmholtz eq. in asymptotic region : ( p + ∇ ) Ψ (E , ⃗r )=0
( p2 + ∇ 2 ) Ψ (E , ⃗r )=K ( E ,⃗r )
In interaction region :
NBS wave function has a same asymptotic form with quantum mechanics.
(NBS wave function is characterized from phase shift)
Ψ(E , ⃗r )≃ A
sin ( pr + δ(E ))
pr
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration
N. Ishii et al Phys. Lett. B712(2012)437
Time-dependent
Time-dependent method
method
Let's start with the normalized four-point correlator.
R
B 1 B2
I
(t , ⃗r )=F B B ( ⃗r ,t )e
1
Each wave functions satisfy
Schrödinger eq. with proper energy
(m1 +m2)t
2
−(E 0−m1−m2 )t
= A0 Ψ(⃗r , E 0) e
(
−(E 1−m1−m2)t
+ A1 Ψ(⃗r , E 1)e
)
2
p0 ∇ 2
3
+
Ψ (⃗r , E 0 )=∫ U (⃗r ,⃗r ' ) Ψ (⃗r ' , E 0 )d r '
2μ 2μ
(
)
2
p1 ∇ 2
3
+
Ψ (⃗r , E1 )=∫ U (⃗r ,⃗r ' ) Ψ(⃗r ' , E 1 )d r '
2μ 2μ
2
pn
E n −m 1−m 2≈
2μ
(
)
B B
∇2 B B
3
∂
− +
R I (t ,⃗r )=∫ U (⃗r ,⃗r ' ) R I (t , ⃗r )d r '
∂ t 2μ
1
2
1
2
A single state saturation is not required!!
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration
+⋯
Treatment
Treatment of
of non-local
non-local potential
potential
(
)
B B
∇2 B B
3
∂
− +
R I (t ,⃗r )=∫ U (⃗r ,⃗r ' ) R I (t , ⃗r )d r '
∂ t 2μ
1
2
1
2
General nonlocal poten.al is inconvenient.
Derivative expansion of U to treat its non-locality
2
U (⃗r , ⃗r ' ) = V C (r )+ S 12 V T ( r )+ ⃗L⋅S⃗s V LS (r )+ ⃗
L⋅S⃗a V ALS (r)+O (∇ )
Leading
Leading order
order
Next
Next to
to leading
leading order
order
Negative parity sector
K. Murano et al Phys.Lett. B735 (2014) 19
by N. Ishii
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 for
for the
the NBS
NBS wave
wave function
function
α
−Et
β
−Et
Ψ ( E ,⃗r )e
= ∑ 〈0∣(B1 B2 ) (t ,⃗r )∣E〉
In asymptotic region : ( p2 + ∇ 2 ) Ψ (E , ⃗r )=0
= ∑ 〈0∣(B1 B2 ) (t , ⃗r )∣E〉
In interaction region : ( p2 + ∇ 2 ) Ψ (E , ⃗r )=K ( E , ⃗r )
α
⃗x
Ψ (E ,⃗r )e
Two-channel coupling case
β
⃗x
Inside
Inside the
the interaction
interaction range
range
In the
leading
order
of derivative
expansion
of non-local potential, U ( ⃗r , ⃗r ' )=V ( ⃗r )δ( ⃗r ' −⃗r )
Assuming
that
potentials
are energy
independent,
Coupled channel Schrödinger equation.
(
)
p 2α ∇ 2
+
Ψ α (⃗r , E)=V αα (⃗r )Ψ α (⃗r , E)+V αβ (⃗r ) Ψ β (⃗r , E)
2μ α 2μ α
µα : reduced mass
pα : asymptotic momentum.
Asymptotic momentum are replaced by the time-derivative of R.
B1 B 2
RI
(
(t , ⃗r )=∑⃗x 〈 0∣B1 (t , ⃗x+ ⃗r ) B2 (t , ⃗x ) ̄I (0)∣0〉 e
(m1+m 2)t
V αα (⃗r )
β
V α (⃗r ) x
−1
(
)
2
2
∇
α
∇
∂
(
− ) R I1 (⃗r , t) (
− ∂ ) RβI2 (⃗r , t)
−1
α
V β (⃗r ) x
2μα ∂t
2μβ ∂ t
R αI1 (⃗r ,t) RβI1 (⃗r ,t )
=
α
β
β
2
2
R I2 (⃗r ,t ) R I2 (⃗r , t)
∇
α
∇
β
V β (⃗r )
∂
∂
(
− ) R (⃗r , t) (
− ) R (⃗r , t)
2μα ∂t I1
2μβ ∂ t I2
exp (−(m α + mα )t )
)
(
)
x=
1
exp (−(mβ + mβ ) t)
1
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration
2
2
Numerical
Numerical results
results
m
mππ== 469MeV
469MeV
3
S1--3D1
1
S0
B-B
B-B potentials
potentials in
in SU(3)
SU(3) limit
limit
Two-flavors
Three-flavors
Quark Pauli principle can be seen at around short distances
No repulsive core in flavor singlet state
Strongest repulsion in flavor 8s state
Possibility of bound H-dibaryon.
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration
Looking
Looking for
for H-dibaryon
H-dibaryon in
in SU(3)
SU(3) limit
limit
Potential is getting more attractive
as decreasing quark masses
6q bound state is found in this mass range
with SU(3) symmetry.
What happens at the physical point?
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration
Recent
Recent status
status for
for H-particle
H-particle
Conclusions of the “NAGARA Event”
Lattice QCD simulations
K.Nakazawa and KEK-E176 & E373 collaborators
PRL87(2001)212502
Λ−Ν attraction
Λ−Λ weak attraction
mH ≥ 2mΛ - 6.9MeV
PRL106,162001
PRL106,162002
Conclusions of C(K ,K ΛΛ) reaction
12
-
+
C.J.Yoon and KEK-PS E522 collaborators
PRC75(2007)022201(R)
the enhancement
was found below 30 MeV
At the physical point
Fate of H-dibaryon??
Beyond SU(3) simulation is necessary
toward the physical point !!
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration
Numerical
Numerical setup
setup
2+1 flavor gauge configurations by PACS-CS collaboration.
Iwasaki gauge action & O(a) improved Wilson quark action
a = 0.08995(40) [fm], a−1 = 2.194(10) GeV, determined with mΩ
323x48 lattice, L = 2.8784 [fm], T = 4.3176 [fm].
Kud = 0.1373316, Ks = 0.1367526
402 confs x 48 sources.
Flat wall source is considered to produce S-wave B-B state.
Mass [MeV]
π
K
mπ/mΚ
N
509±1
612±1
0.83
Λ
1319±6
1383±5
Σ
1404±6
Ξ
1456±4
33MeV
21MeV
12MeV
10MeV
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration
S=-2
S=-2 sector
sector
11
ΛΛ,
ΝΞ,
ΣΣ
(I=0)
S00 channel
channel
ΛΛ, ΝΞ, ΣΣ (I=0) S
Diagonal elements
Off-diagonal elements
All diagonal element have a repulsive core.
ΣΣ−ΣΣ potential is strongly repulsive.
Off-diagonal potentials are relatively strong.
except for ΛΛ−ΝΞ transition
Existence of H-dibaryon can be seen
in ΛΛ phase-shift as a sharp resonance.
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration
m
mππ==510
510MeV
MeV
Comparison
Comparison of
potential matrices
of potential
matrices
Transformation of potentials
from the particle basis to the SU(3) irreducible representation (irrep) basis.
SU(3) Clebsh-Gordan coefficients
U
(
ΛΛ
ΛΛ
ΛΛ
V
V N Ξ V ΣΣ
NΞ
NΞ
NΞ
V ΛΛ V
V ΣΣ
V ΣΛΣΛ V ΣNΣ Ξ V Σ Σ
) (
U →
t
V1
V8
V 27
)
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) irrep basis
would be an effectual measure of the SU(3) breaking effect.
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration
11
1,
8,
27
irreps
(I=0)
1, 8, 27 irreps (I=0) S
S00 channel
channel
Diagonal elements
m
mππ==510
510MeV
MeV
Off-diagonal elements
Strongly attractive potential in SU(3) singlet state is still alive.
To generate the H-dibaryon state.
Strong repulsion in SU(3) octet state reflects the Pauli blocking effect.
Mixing between different irreps are still small.
Singlet-octet mixing potential is relatively larger than the others.
27plet does not mix to the other irreps.
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration
S=-1
S=-1 sector
sector
S0 --- ΛΝ−ΣΝ coupled channel
0
11
I=1/2
I=1/2
I=3/2
I=3/2
I=1/2
8s
27
ΛΝ
-√1/10
√9/10
ΣΝ
√9/10
√1/10
8s dominant state
LN interaction is a bit repulsive than NN
Due to the effect of 8s potential
Pure 27 state
S1-33D1 --- ΛΝ−ΣΝ coupled channel
1
1
33
I=1/2
I=1/2
I=3/2
I=3/2
I=1/2
8a
10*
ΛΝ
-√1/2
√1/2
ΣΝ
√1/2
√1/2
Similar potential strength between LN and SN channel
Pure 10 state
Summary
Summary and
and outlook
outlook
We have investigated hyperon interactions from lattice QCD.
In order to deal with a variety of interactions, we extend our method
to the coupled channel formalism.
Octet-Octet potentials are derived from NBS wave functions
calculated with PACS-CS configurations
S=-2 system is focused.
We found H-dibaryon as a resonance state
SU(3) breaking effects are still small
even in mπ/mΚ=0.83 situation
but it would be change drastically
at physical situation mπ/mΚ=0.28.
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration
Summary
Summary and
and outlook
outlook
Decuplet
Decupletsector
sector
Three
Threebody
bodyforce
force
Extension to YNN force
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration
S=-4
S=-4 sector
sector
Symmetry
Symmetry of
of BB
BB system
system
Similarity
Similarity and/or
and/or dissimilarity
dissimilarity to
to NN
NN system
system
Flavor component
p=[ u d ] u=̄s u
n=[ u d ] d =̄s d
n p
Σ- Λ Σ0 Σ+
Ξ- Ξ0
Octet baryon
Iso-doublet
0
Ξ =[ s u ] s= ̄d s
Ξ =[ s d ] s=̄u s
u ss
Conjugate flavor structure
Iso-triplet state
pp
=
pn+ np=
nn
=
̄s u ̄s u
̄s u ̄s d + ̄s d ̄s u
̄s d ̄s d
0
0
ΞΞ
=
Ξ0 Ξ+ ΞΞ 0=
ΞΞ
=
ss
d
̄
d s d̄ s
d̄ s u
̄ s+ u
̄ s d̄ s
u
̄ su
̄s
Iso-singlet state
pn−np= ̄s u ̄s d −̄s d ̄s u
Ξ0 Ξ−Ξ Ξ0= ̄d s ̄u s− ̄u s d̄ s
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration
ΞΞ channel
m
mππ==510
510MeV
MeV
Iso-singlet
Iso-singlet channel
channel
Potential of 10 irrep is expected to be repulsive due to the quark Pauli effect.
Iso-triplet
Iso-triplet channel
channel
EFT calculation found that the bound ΞΞ state in 1S0 channel [1].
Meson exchange model calculations: bound.[2] or unbound [3]
Bound ΞΞ state was found by Lattice QCD simulation at m=389MeV [4]
1). J. Haidenbauer Nucl.Phys.A881(2012)44
2). M. Yamaguchi PTP105(2001)627
3). Y. Fujiwara PPNP 58(2007)439
4). S.R. Beane PRD85(2012)054511
Search for the bound ΞΞ state is interesting at the physical point!
Kenji Sasaki (University of Tsukuba) for HAL QCD collaboration