エキゾチックハドロン系の精密科学

根村英克
筑波大学数理物質科学研究科計算科学研究センター

Plan of research
QCD

Baryon interaction

J-PARC
hyperon-nucleon (YN)
scattering

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

A =3

A =4

A =5pn pnn  ,ppn ppnn Comparison between d=p+n and core+YS

D

3p

AV8G3RS

3np

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

10.84

5.64

TY-cΛ TY-cΣ+∆HcnL=0

L=2α Λα' Σ

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

2VΛN-ΣN(tensor)

5Λ He

9.11

3.88+4.68−0.86−19.51

Λ H

4 *

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

Introduction:
 Tensor ΛN-ΣN force plays a key role for thelight hypernuclei: An example: pnΛ + NNΣ three-body system.

2905 (1995).

Miyagawa, et al., PRC 51,Tensor
ΛN-ΣN

FY calculation with and w/o 3NF
 Three nucleon force does not change the BΛ

so much.

 A. Nogga, et al., PRL88, 172501 (2002).

ハイペロンポテンシャルは、 NN と切り離して決めることはできない。

Lattice QCD calculation

Baryon-baryon potentials from lattice QCD
H. Nemura1,for HAL QCD Collaboration
S. Aoki2, B. Charron3, T. Doi4, F. Etminan1,T. Hatsuda4, Y. Ikeda4, T. Inoue5, N. Ishii1,K. Murano2, K. Sasaki1, and M. Yamada1,

Center for Computational Science, University of Tsukuba, Japan2
Yukawa Institute for Theoretical Physics, Kyoto University, Japan3
Department of Physics, University of Tokyo, Japan4Theoretical Research Division, Nishina Center RIKEN, Japan
5
College of Bioresouce Science, Nihon University, Japan
6Strangeness Nuclear Physics, Nishina Center RIKEN, Japan1

Outline Introduction
 Formulation --- potential (central + tensor)
 Numerical results:
NΛ force ( VC + VT )N (I=3/2) force ( VC + VT )

 Recent work on lattice QCD
 Stochastic variational calculation of 4He withusing a lattice potential
 Summary and outlook

Introduction:
 Study of hyperon-nucleon (YN) and hyperon-hyperon (YY) interactions is one of theimportant subjects in the nuclear physics.
 Structure of the neutron-star core, Hyperon mixing, softning of EOS, inevitable strong

repulsive force, H-dibaryon problem, To be, or not to be, The project at J-PARC:
 Explore the multistrange world, However, the phenomenological description of YN

and YY interactions has large uncertainties,
which is in sharp contrast to the nice description of phenomenological NN potential.

Introduction:
 Study of hyperon-nucleon (YN) and hyperon-

hyperon (YY) interactions is one of theimportant subjects in the nuclear physics.
 Structure of the neutron-star core, Hyperon mixing, softning of EOS, inevitable strong

repulsive force, H-dibaryon problem, To be, or not to be, The project at J-PARC:
 Explore the multistrange world, However, the phenomenological description of YN

and YY interactions has large uncertainties,
which is in sharp contrast to the nice description of phenomenological NN potential.

Formulation
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−S  q,q,U=∫ dU det D  U e

− SU  U O  q ,q , U O  D −1  U N

1−1=lim ∑ O  D  U i N  ∞ N i =1p〈pt p

 t 0 〉Formulation
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

−S  q,q,U

=∫ dU det D  U e

− SU  U O  q ,q , U O  D −1  U N

1−1=lim ∑ O  D  U i N  ∞ N i =1p〈pt p t 0 〉Formulationi) basic procedure:
asymptotic region--> phase shiftii) advanced (HAL's) procedure: interacting region--> potential

Formulation
An example ofi) basic procedure:
Luscher’s formula
asymptotic region(or temporal correlation)--> scattering energy--> phase shift

2kE=2

2122k cot 0  k =
Z 00 1 ; k L / 2 =  O  k a0 L1123
Z 00 1 ;q =∑  n2− q2 sℜ s2 4 n∈ Z
3Luscher, NPB354, 531 (1991).
Aoki, et al., PRD71, 094504 (2005).

Formulation
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 〉

HAL formulationii) advanced procedure:
make better use of the latticeoutput ! (wave function)interacting region--> potential

Ishii, Aoki, Hatsuda,
PRL99, 022001 (2007);
ibid., arXiv:0805.2462[hep-ph].

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)....

HAL formulationii) advanced procedure:
make better use of the latticeoutput ! (wave function)interacting region--> potential

Ishii, Aoki, Hatsuda,
PRL99, 022001 (2007);
ibid., arXiv:0805.2462[hep-ph].

=>> Phase shift> Nuclear many-body problems

Numerical results

Full QCD calculations by using
NF=2+1 PACS-CS gauge configurations: S. Aoki, et al., (PACS-CS Collaboration),

PRD79, 034503 (2009), arXiv:0807.1661 [hep-lat].
 Iwasaki gauge action at β=1.90 on 323 × 64 lattice O(a) improved Wilson quark action
 1/a = 2.17 GeV (a = 0.0907 fm)

ΛN potentialVC(ΛN; 1S0)
− ∂ R  t , r 2ℏ 2 ∇ R  t , r ∂tV  r =R t , r
2  R  t , r 1S0, t-t0=8, 3pt
・ {27}+{8s}・ Similar to NN (1S0)・ Sizable contribution from time-derivative part

VC(ΛN; 3S1-3D1)
V r ∂∂t∇2・ {10*}+{8a}・ Sizable attractive contribution from time-derivative partVT(ΛN; 3S1-3D1)
V r ∂∂t∇2・ Weaker tensor force than NN
・ Small contribution from time-derivative partΣN(I=3/2) potentialVC(ΣN(I=3/2); 1S0)
− ∂ R  t , r 2ℏ 2 ∇ R  t , r ∂tV  r =R  t , r 2  R  t , r 

1S0, t-t0=8, 3pt
・ {27}・ Similar to NN (1S0) (as well as Lambda-N (1S0))・ Sizable contribution from time-derivative part

VC(ΣN(I=3/2); 3S1-3D1)
V r ∂∂t∇

2・ {10}・ Repulsive potential (consistent with quark model)
・ sizable repulsive contribution from time-derivative partVT(ΣN(I=3/2); 3S1-3D1)
V r ∂∂t∇2・ Weak tensor force
・ Small contribution from time-derivative partScattering phase shifts

Proton-Lambda scattering (preliminary)Parametrized
potential

Phase shift(Hyper-)Nuclear
few-body problem

Stochastic variational calculation of 4He
with using a lattice potential

 For NN potential, we use Inoue-san’s SU(3)potential at the lightest quark mass(m_ps =469 MeV), which has been reported to have a
4N bound state (about 5.1MeV) within atensor-included effective central potential;
NPA881, 28-43 (2011).Stochastic variational calculation of 4He
with using a lattice potential

 For NN potential, we use Inoue-san’s SU(3)potential at the lightest quark mass(m_ps =469 MeV), which has been reported to have a
4N bound state (about 5.1MeV) within atensor-included effective central potential;
NPA881, 28-43 (2011).

Stochastic variational calculation of 4He
with using a lattice potential

 For NN potential, we use Inoue-san’s SU(3)potential at the lightest quark mass(m_ps =469 MeV), which has been reported to have a4N bound state (about 5.1MeV) within atensor-included effective central potential;
NPA881, 28-43 (2011).

Summary

Summary(1) Lattice QCD calculation for hyperon potentials toward thephysical point calculation.
Lambda-N, Sigma-N: central, tensor(2) (hyper-)nuclear few-body calculation of stochastic
variational method(3) recent misc work
( 萌芽的研究プロジェクトに関連していそうなその他の報告 )萌芽的研究プロジェクト
分野５の計算科学技術推進体制構築における萌芽的研究プロジェクト
支援は将来の主要な研究開発課題になるべきプロジェクトを開拓することを目的として行っているものです。アイデアの豊富な若手研究者に自由な発想で研究する機会を与え、分野全体で新しい研究を育てることを目指しています。このため研究支援チームの皆さんには、ユーザからのアルゴリズム・コーディング等の支援要請への対応・共通コード作成と同時に、新しい発想に基づく萌芽的研究課題に取り組むことを推奨してきました。萌芽的研究プロジェクトの一覧は以下の通りです。

Recent work

(1) Porting the C++ program to Bridge++, which can calculate
the four-point correlation function of Lambda-Nucleon system.
The C++ program also has been used to study other baryon-baryon
potential for student.
(2) Improve the computational performance by implementing thehybrid parallel program with MPI and OpenMP(3) Generalize the target system to various baryon-baryon
Channels (e.g., 52 channels would be required to study thecomplete set of baryon-baryon potentials on 2+1 QCD calculation)(4) In this approach, the number of iterations to obtain thefour-point correlation function is remarkably smaller than thenumbers given in the unified contraction algorithm[2][1] H.N. Pos(LAT2013)426;(LAT2008)156;(LAT2009)152;(LAT2011)167.
[2] Doi and Endres, Comput. Phys. Commun. 184, 117 (2013).