物理点格子 QCD によるストレンジネス S=-1 セクタのバリオン間力
根村英克 1,for HAL QCD Collaboration1

大阪大学核物理研究センター

arXiv:1810.04046 [hep­lat]Baryon force in strangeness S=-1 sector fromphysical point lattice QCD
H. Nemura1,for HAL QCD Collaboration
S. Aoki2, T. Aoyama2, T. Doi3, T. M. Doi3, F. Etminan4,
S. Gongyo3, T. Hatsuda3, Y. Ikeda1, T. Inoue5,
T. Iritani3, N. Ishii1, D. Kawai2, T. Miyamoto2,
K. Murano1, and K. Sasaki2,Osaka University,2Kyoto University, 3RIKEN, 4University of Birjand,5Nihon University
arXiv:1810.04046 [hep­lat]1Outline Introduction
Importance of LN-SN tensor force for

hypernuclei
Brief introduction of HAL QCD methodEffective block algorithm for various baryonbaryon channels, CPC207,91(2016)[1510.00903]
 Preliminary results of LN-SN potentials at nearly
physical point; update from [1702.00734]
LN-SN(I=1/2), central and tensor potentials
SN(I=3/2), central and tensor potentials
Phase shifts of SN(I=3/2) scattering SummaryPlan of researchQCD

Baryon interaction

J-PARC,JLab, GSI, MAMI, ...
YN scattering,
hypernuclei

Structure and reaction of(hyper)nuclei
Equation of State (EoS)of nuclear matter
Neutron star andsupernova

A =3A =4A =5pn pnn  ,ppn ppnn What is realistic picture of hypernuclei?
 B(total)=B( He)+BΛ(Λ He)45

 A conventional picture:B(total)= B(4He)+BΛ(Λ5He)= 28+3 MeV.

 A (probably realistic) picture:B(total)= (B(4He)−∆Ec)+(BΛ(Λ5He)+∆Ec)= ??+?? MeV.

Comparison between d=p+n and core+YSD3pAV8G3RS3npn

TD STPhaseshif(MeV)
(MeV)
8.57
11.31
( なるか?)

10.84

L=0

L=2α Λα' Σ

VNN(central) VNN(tensor)
(MeV)
(MeV)
−4.46
−16.64−7.29−11.46

5.64

TY-cΛ TY-cΣ+∆Hc

VYN( のこり )

2VΛN-ΣN(tensor)

Λ He

5

9.11

3.88+4.68−0.86−19.51

4 *

Λ H

5.30

2.43+2.02

0.01−10.6747.12

2.94+2.16

−5.05

−9.22

Λ H

VNN(LS)
(MeV)
−1.02
0.00

HN, Akaishi, Suzuki, PRL89, 142504 (2002).

Rearrangement effect of Λ5HeHN, Akaishi, Suzuki, PRL89, 142504 (2002).

 Phase shif Carlson
 Nogga ( なるか ?)2A −1A −1p2i
 N N
N Y H =∑ m ic −T C M  ∑ vij  ∑ viY = H core  H Y −core ,2 mii=1i ji=1

AA −12i2∑ A −1pi

A −1pi=1
H core= ∑− ∑ vijN N =T coreV N N .
2 A −1 m N i ji=1 2 m N

What is realistic picture of hypernuclei?
 B(total)=B( He)+BΛ(Λ He)45

 A conventional picture:B(total)= B(4He)+BΛ(Λ5He)= 28+3 MeV.

 A (probably realistic) picture:B(total)= (B(4He)−∆Ec)+(BΛ(Λ5He)+∆Ec)= 24+7 MeV.

Lattice QCD calculationpnMulti-hadron on latticei) basic procedure:
asymptotic region--> phase shiftii) HAL's procedure:interacting region--> potential

Multi-hadron on lattice
Lattice QCD simulation
1 a a a a
L =− G   Gq   i ∂ − g t A   q − m q q4〈 O  q ,q , U 〉=∫ dU d q dq e

−S  q,q,U

=∫ dU det D  U e

− SU  U O  q ,q , U O  D −1  U N1−1=lim ∑ O  D  U i N  ∞ N i =1p〈p

t p

 t 0 〉Multi-hadron on latticei) basic procedure:
asymptotic region(or temporal correlation)--> scattering energy--> phase shift

2kE=2

2
122k cot 0  k =Z 00 1 ; k L / 2 =  O  k a0 L
1123Z 00 1 ;q =∑  n2− q2 sℜ s2 4 n∈ Z
3Luscher, NPB354, 531 (1991).
Aoki, et al., PRD71, 094504 (2005).

Multi-hadron on lattice
Lattice QCD simulation
1 a a a a
L =− G   Gq   i ∂ − g t A   q − m q q
4〈 O  q ,q , U 〉=∫ dU d q dq e

F−S  q,q,U

=∫ dU det D  U e

 JM  =lim 1 N O  D∑NNi〈− SU  U O  q ,q , U O  D −1  U r ,t −t 0 ∞p

−1 U i =1
r ,t p

Calculate the scattering state,

 t 0 〉Multi-hadron on latticeii) HAL’s procedure:
make better use of the latticeoutput ! (wave function)interacting region--> potential

Ishii, Aoki, Hatsuda,
PRL99, 022001 (2007);
ibid., PTP123, 89 (2010).

NOTE:> Potential is not a direct experimental observable.
> Potential is a useful tool to give (and to reproduce)
the physical quantities. (e.g., phase shift)....

Multi-hadron on latticeii) HAL’s procedure:
make better use of the latticeoutput ! (wave function)interacting region--> potential

Ishii, Aoki, Hatsuda,
PRL99, 022001 (2007);
ibid., PTP123, 89 (2010).

=>> Phase shift> Nuclear many-body problems

The potential is obtained at moderately large
imaginary time; no single state saturation isrequired.
RECIPE:
Compute the 4pt correlator
Take into account the threshold energy differences for coupled-channel system

elastic
inelastic
Obtain the potential by using the appropriate equation(s); For spin-singlet,
For spin-triplet, the “tensor force” becomes active
Where
In the lowest few orders, we haveAn improved recipe for NY potential: cf. Ishii (HAL QCD), PLB712 (2012) 437.

 Take account of not only the spatial correla-tion but also the temporal correlation interms of the R-correlator:
123 '−∇ R t , r ∫ d r U  r , r '  R  t , r '=− ∂ R  t, r 2
∂t k2

U r ,r '=V N Y r ,∇ r−r '
 A general expression of the potential:

V N Y =V 0  rV   r  N ⋅Y 
V T  r S 12V LS  r L⋅S  2V ALS  r  L⋅S− O ∇ 2

R  t , r Determination ofbaryon-baryon potentials
at nearly physical point

Effective block algorithm forvarious baryon-baryon correlators
HN, CPC207,91(2016), arXiv:1510.00903(hep­lat)

Numerical cost (# of iterative operations) in this algorithm
22222221  N c  N c N  N c N   N c N  N c N =370In an intermediate step:B N c ! N   × N u ! N d ! N s ! ×2

N  N  − B0=3456

In a naïve approach:2B

 N c ! N   × N u ! N d ! N s ! =3,981,312

Generalization to the various baryon-baryon channels
strangeness S=0 to -4 systems

Make better use of the computing resources!
HN, CPC 207, 91(2016) [arXiv:1510.00903[hep-lat]],(See also arXiv:1604.08346)

Benchmark (from NN to XiXi channels)

Benchmark (from NN to XiXi channels)
arXiv:1510.00903(hep­lat)=2 2  ' 2  '2 x 16  y 16  z 16  c 52  t −t0 

Almost physical point lattice QCD calculation
using NF=2+1 clover fermion + Iwasaki gauge action
 APE-Stout smearing (=0.1, n=6)
 Non-perturbatively O(a) improved Wilson Cloveraction at β=1.82 on 963 × 96 lattice 1/a = 2.3 GeV (a = 0.085 fm)
 Volume: 964 → (8fm)4
 m=145MeV, mK=525MeV
stout DDHMC(ud) and UVPHMC(s) with preconditioning K.-I.Ishikawa, et al., PoS LAT2015, 075;

arXiv:1511.09222 [hep-lat].

 NBS wf is measured using wall quark source withCoulomb gauge fixing, spatial PBD and temporal
DBC; #stat=207configs x 4rotation x Nsrc
(Nsrc=4 → 20 → 52 → 96 (2015FY+))LN-SN potentials
at nearly physical point
The methodology for coupled-channel V is based on:
Aoki, et al., Proc.Japan Acad. B87 (2011) 509.
Sasaki, et al., PTEP 2015 (2015) no.11, 113B01.
Ishii, et al., JPS meeting, March (2016).
Nemura, et al., [1702.00734].
#stat: (this/scheduled in FY2015+) 0.2V C  S 0

1.00

t −t 0 =5−12,t −t 0 =5−14

 N − N  I =1 /2 10.543333V C  S 1− D 1

V T  S 1− D 1

V C  3 S 1− 3 D 1

V T 3 S 1− 3 D 1  N  I =3 /2 V C  1 S 0

LN-SN potentials
at nearly physical point
The methodology for coupled-channel V is based on:Aoki, et al., Proc.Japan Acad. B87 (2011) 509.
Sasaki, et al., PTEP 2015 (2015) no.11, 113B01.
Ishii, et al., JPS meeting, March (2016).
Nemura, et al., [1702.00734].
#stat: (this/scheduled in FY2015+) 0.2

0.54

1.00
2.00

t −t 0 =5−12,t −t 0 =5−14

 N − N  I =1 /2 1V C  S 0

3333V C  S 1− D 1

V T  S 1− D 1

V C  3 S 1− 3 D 1

V T 3 S 1− 3 D 1  N  I =3 /2 V C  1 S 0

Effective mass plot of the single baryon’s correlation function

PRELIMINARY
Potentials obtained at t-t0 = 5 to 12 are shown;
For the largest statistics, t-t0 = 13 and 14 will also be shown.

Results ––– potential (cont.) The net interaction of the NΞ (I=1) is at-tractive.
Wavefunction↓
Energy(k2)E=−0.4(2)MeV(1S0)
E=−0.8(2)MeV(3S1)22ℏ ∇  r
V  r= E 2    rOka, Shimizu and Yazaki (1987)

Very preliminary result of LN potential at thephysical point
V C  1S 0 N N − NPRELIMINARY N − NN

Very preliminary result of LN potential at thephysical point
V C  3 S 1− 3 D 1N N − N

PRELIMINARY N − N

N

Very preliminary result of LN potential at thephysical point
V T 3 S 1− 3 D 1 N N − N

PRELIMINARY N − N

N

Very preliminary result of LN potential at thephysical point3 N  I =3 / 2 

3V C  S 1− D 1

PRELIMINARY1V C  S 033V T S 1− D 1 Very preliminary results of the SN(I=3/2)
phase shift at the physical point N  I =3 / 2 

PRELIMINARY
More or less qualitatively similar to(recent) phenomenological approaches:
Fujiwara, et al., PRC54(1996)2180,Arisaka, et al., PTP104(2000)995,
Haidenbauer et al., NPA915(2013)24.

Summary(I-1) Latest results of LN-SN potentials at nearly physical point.
(Lambda-N, Sigma-N: central, tensor)
Statistics approaching to 1.96 => 2.0 (=present/scheduled in 2015)
Increasing statistics, the result at large t converges to theresult at next-to-smaller t (with poor statistics).
Signals in spin-triplet are relatively going well smoothly.
The channels that the quark model predicts strongly repulsive
have relatively poor signals; Simultaneous calcs (NN to XiXi) is
the point we have to take into account for the comprehensive
perspective as well as energy-computing-resource efficiency.
(I-2) Effective hadron block algorithm for the various
baron-baryon interaction
Comput.Phys.Commun.207,91(2016) [arXiv:1510.00903(hep-lat)]
The algorithm will be applied to more wide range problems.
Future work:(II-1) Physical quantities including the binding energies of fewbody problem of light hypernuclei with the lattice YN (and NN)
potentials