YN Potentials of Strangeness S=-1
from Lattice QCD
H. Nemura1,for HAL QCD Collaboration
S. Aoki1, T. Doi2, T. Hatsuda3, Y. Ikeda4, T. Inoue5,
N. Ishii1, K. Murano6, and K. Sasaki1,

1

Center for Computational Science, University of Tsukuba, Japan
2
Center for Nuclear Study, University of Tokyo, Japan
3
Theoretical Research Division, Nishina Center RIKEN, Japan
4
Department of Physics, Tokyo Institute of Technology, Japan
5
College of Bioresouce Science, Nihon University, Japan
6
Strangeness Nuclear Physics, Nishina Center RIKEN, Japan

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

 Recent improvement for Vc and VT
 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 YNand YY interactions has large uncertainties,
which is in sharp contrast to the nice description of phenomenological NN potential.

The purposes of this work
 NYforces from lattice QCD

 Spin dependence Potential (central + tensor)
 Numerical calculation:
 Full lattice QCD by using N =2+1 PACS-CS
F

full QCD gauge configurations with thespatial lattice volume (2.86 fm)3
 We also use the N =2+1 gauge configuraF
tions by CP-PACS/JLQCD, with the spatial lattice volume (1.93 fm)3

Formulationi) basic procedure:
asymptotic region--> phase shiftii) advanced (HAL's) procedure: interacting region--> potential

Formulation
An example ofi) basic procedure:
Luscher’s formulaasymptotic 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
1123
Z 00 1 ;q =∑  n2− q2 sℜ s
2 4 n∈ Z
3Luscher, NPB354, 531 (1991).
Aoki, et al., PRD71, 094504 (2005).

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

2kE=
2

2122k cot 0  k =
Z 00 1 ; k L / 2 =  O  k a0
 L
1123
Z 00 1 ;q =∑  n2− q2 sℜ s
2 4 n∈ Z
3Luscher, NPB354, 531 (1991).
Aoki, et al., PRD71, 094504 (2005).

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

An improved recipe for lattice potential: cf. Ishii (HAL QCD, 2010); Talk tomorrow.

 Take account of the temporal correlation as

well as the spatial correlation of the NBS
amplitude in terms of the R-correlator:
C YN  t , r R  t , r =C Y t C Nt − t H

R  t  t , r =e
R  t , r =1− t H  R  t , r Time-dependent effective Schroedinger eq.:− ∂ R  t , r = H R  t , r ∂tAn improved recipe for NY potential: cf. Ishii (HAL QCD, 2010); Talk tomorrow.

 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 A recipe for NΛ potential:.

 The equal time BS wave function withangular momentum (J,M) on the lattice,J M r =∑ 〈 0∣ p  rx     x ∣ p  ;k ,J M 〉xp   x = abc  ua  x  C 5 d b  x  u c  x  ,   x =abc { d a C 5 sb  u c  sa C 5 ub  d c −2 u a C 5 d b  sc  } TheFJ M 

4-point NΛ correlator on the lattice,〈∣ x ,y ,t−t0= 0 p  x ,t  y ,t=∑ An

J Mp t0 

J M n

J M p∣〉

 t0 0− E n t− t0〈 0∣ p  x   y ∣E 〉 enwall source at t = t0

A recipe for NΛ potential: cf. Ishii (HAL QCD, 2010); Talk tomorrow.

 Calculate the 4-point NΛ correlator on the lattice,− E  t−t0 N   x − y e

∝ 〈 p  x ,t   y ,t   '0, t0  p ' 0, t0〉 Which has the physical meanings of, Create a NΛ state and making imaginary time

evolution, in order to have the lowest state of
the NΛ system.
 Take the R-correlator R (t-t , x −y ), which can0be understood as a wave function of the
non-relativistic quantum mechanics.
 Obtain the effective central potential from theeffective Schroedinger equation.

2ℏ−∇ 2 V  r  R  t , r =− ∂ R  t , r 2∂t➔− ∂ R  t , r 22ℏ ∇ R  t , r ∂tV  r =R  t , r 2  R  t , r A recipe .for NY potential: (contd.) For J = 1,  comprises S-wave and D-wave,

where,   =  S  +  D  S  = P    = ( 1/24 ) R  O R    D  = Q    = ( 1 − P )    Therefore, we have 2-component Schrödinger eq.

S-wave:P (T + VC + VT S12 )  = -∂/∂t P D-wave:

Q (T + VC + VT S12 )  = -∂/∂t Q  Obtain the V ( r ) and the V ( r ) simultaneously.
CTNumerical results:

8 ⊗8 =27 ⊕8 s ⊕1 ⊕10⊕10⊕8 a

⊗
=⊕⊕

1⊕⊕

⊕

> SU(3) limit{27 }{8 s }
{1 }

{10 }

{10 }

{8 a }

Inoue, et al. (HAL QCD), PTP124, 591 (2010).

8 ⊗8 =27 ⊕8 s ⊕1 ⊕10⊕10⊕8 a

⊗
=⊕
⊕

1⊕⊕

⊕

An exercise calculation by using
NF=2+1 CP-PACS+JLQCD gauge configurations: RC16x32_B1830Kud013710Ks013710C1761
 a=0.1209(fm) Wall source, point sink
 700 gauge configs.

1S0
ΛN{27}+{8s}ΣN(I=1/2) {27}+{8s}ΣN(I=3/2) {27}{27}

3S1-3D1{8a}+{10*}{8a}+{10*}
{10}

{10}

1S0
ΛN{27}+{8s}ΣN(I=1/2) {27}+{8s}ΣN(I=3/2) {27}{27}

3S1-3D1{8a}+{10*}{8a}+{10*}
{10}

1S0
ΛN{27}+{8s}ΣN(I=1/2) {27}+{8s}ΣN(I=3/2) {27}{8s}3S1-3D1{8a}+{10*}{8a}+{10*}
{10}

1S0
ΛN{27}+{8s}ΣN(I=1/2) {27}+{8s}ΣN(I=3/2) {27}3S1-3D1{8a}+{10*}{8a}+{10*}
{10}

{10*}

1S0
ΛN{27}+{8s}ΣN(I=1/2) {27}+{8s}ΣN(I=3/2) {27}3S1-3D1{8a}+{10*}{8a}+{10*}
{10}

{8a}

The main 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

VC(ΛN; 3S1-3D1)
V r ∂∂t∇2

VT(ΛN; 3S1-3D1)
V r ∂∂t∇2

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

VC(ΣN(I=3/2); 3S1-3D1)
V r ∂∂t∇2

VT(ΣN(I=3/2); 3S1-3D1)
V r ∂∂t∇2

Summary: The lattice QCD study for Lambda-Nucleon andSigma-nucleon(I=3/2) interactions.
 pΛ:
 Central, tensor. For full QCD
 Time-derivative terms enhance the attractive force.

 Qualitatively similar to well-known nuclear forces.
 Repulsive at short distance.
 Attractive well at medium to long distance.
 N(I=3/2):
 Central, tensor. For full QCD
 The 1S0 potential is similar to Lambda-N potential
 The 3S1 potential is repulsiveOutlook: Quark mass dependence.
 Scattering lengths.
spin-dependence.
Comparison with the hypernuclear data.
 Coupled-channel potential.

Proton-Lambda interaction (preliminary)Parametrized
potential
Phase shift