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Lattice QCD calculation of Nuclear Forces. ★ Plan of the talk: Introduction Formalism Lattice QCD results NEXT STAGE: Subtleties of our potential (at short distance) and Inverse scattering Summary. Noriyoshi ISHII (Univ. of Tsukuba) in collaboration with

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lattice qcd calculation of nuclear forces

Lattice QCD calculation of Nuclear Forces

  • ★ Plan of the talk:
  • Introduction
  • Formalism
  • Lattice QCD results
  • NEXT STAGE: Subtleties of our potential (at short distance) and Inverse scattering
  • Summary

Noriyoshi ISHII (Univ. of Tsukuba)

in collaboration with

Sinya AOKI (Univ. of Tsukuba)Tetsuo HATSUDA (Univ. of Tokyo)Hidekatsu NEMURA (RIKEN)


For detail, see N.Ishii, S.Aoki, T.Hatsuda, Phys.Rev.Lett.99,022001(2007).


The nuclear force

Reid93 is fromV.G.J.Stoks et al., PRC49, 2950 (1994).

AV16 is fromR.B.Wiringa et al., PRC51, 38 (1995).

CAS A remnant

  • Nuclear force isthe most important building block in nuclear physics.
  • long distance (r > 2fm) OPEP(one pion exchange) [H.Yukawa (1935)]
  • medium distance(1fm < r < 2fm)multi pion and heavier meson exchanges(“σ”, ρ, ω,..) The attractive pocket is responsible for bound nuclei.
  • short distance (r < 1fm) Strong repulsive core [R.Jastrow (1951)] The repulsive coreplays an important role for(a) stability of nuclei (b) super nova explosion of type II (c) maximum mass of neutron starsPhysical origin of the repulsive core has not yet been understood.(1) vector meson exchange model (2) constituent quark model Pauli forbidden states and/or color magnetic interaction (3) etc.Since this region is expected to reflectthe internal quark & gluon structure of the nucleon,one has desired the QCD understanding of the nuclear force for a long time.


nn potentials from lattice qcd
NN potentials from lattice QCD


  • Methods employing static quarks
  • Method employing NN wave function (We use this.)We extended the method recently proposed by CP-PACS collab for ππ scattering length inCP-PACS collab., S. Aoki et al., PRD71,094504(2005)

cf) T.T.Takahashiet al, AIP Conf. Proc.842,249 (’06). D.Arndt et al., Nulc.Phys.A726,339(’03).

  • Sketch of the method:
  • NN wave function is constructed by using lattice QCD.
  • The NN potential is constructed from the wave functionby demanding that the wave function should satisfy the Schrodinger equation.

An advantage:Since it is obtained from wave function, the result can be faithful to the NN scattering data.

cf) N.Ishii et al., PRL99,022001(2007).

the formalism
The Formalism

reduced mass

  • Schrodinger-like eq. for NN system.
    • For derivation, see C.-J.D.Lin et al., NPB619,467 (2001). S.Aoki et al., CP-PACS Collab., PRD71,094504(2005). S.Aoki, T.Hatsuda, N.Ishii in preparation.
    • The interaction kernel U(r,r’) is non-local.
  • Various symmetries restrict possible forms of U(r,r’). Derivative expansion at low energy leads us to⇒ NN Schrodinger equation

These three interactions play the most important role in the conventional nuclear physics VC(r) central “force” VT(r) tensor “force” VLS(r) LS “force”

construction of nn potentials
Construction of NN potentials

★ Schrodinger equation:

★ 1S0 channel ⇒only VC(r) survives

★ 3S1 channel ⇒ VT(r) and VLS(r) survive.

(deuteron channel)

Effective central force, which takes into account the effect of VT(r) and VLS(r).

nn wave function
NN wave function
  • In QCD, the quantum mechanical NN wave function is only an approximate concept.
  • The closest concept is provided byequal-timeBethe-Salpeter(BS) wave function
  • It is a probability amplitude to find three quarks at x and another three quarks at y.
  • Asymptotic behavior at large r ≡ |x-y|,
  • BS wave function for NN is obtained from the nucleon4 point amplitude in large t region.


BS wave function for the ground state is obtained from the large t region.

lattice qcd parameters
Lattice QCD parameters
  • Quenched QCD is used.
  • Standard Wilson gauge action.
    • β= 6/g2 = 5.7
    • a ~0.14 fm(from ρmass in the chiral limit)
    • 324 lattice (L~4.4fm)
    • 1000-2000 gauge configs(3000 sweeps for thermalization. The gauge config is separated by 200 sweeps)
  • Standard Wilson quark action
    • κ=0.1665 (⇔ quark mass)
    • mπ ~0.53 GeV, mρ~0.88 GeV, mN~1.34 GeV(Monte Carlo calculation becomes the harder for the lighter quark mass.)
    • Dirichlet BC along temporal directionWall source on the time-slice t = t0 = 5NN wave function is measured on the time-slice t-t0= 6
  • Blue Gene/L at KEK has been usedfor the Monte Carlo calculations.

cf) M.Fukugita et al., Phys. Rev. D52, 3003 (1995).

lattice qcd result for nn wave function
Lattice QCD result for NN wave function
  • ★ 3D plot of wave function
  • For r > 0.7 fm, calculation is performed only in the neighborhood of the coordinate axis to save the computational time. To obtain all the data, the calculation becomes 60 times as tough.

★ shrink at short distance suggests an existence of repulsion.

lattice qcd result of nn potential
Lattice QCD result of NN potential

The effective central force for 3S1,which can effectively take into account thecoupling of 3D1 via VT(r) (and VLS(r)) .

★ Our calculation includes the following diagram, which leads to OPEP at long distance.

★ quenched artifacts appear in the iso-scalar channel. Cf) S.R.Beane et al., PLB535,177(’02).

  • Medium range attraction: 30 MeV
  • heavy mπ makes it weaker..⇒heavy virtual pion cannot propagate long distance. For lighter pion mass, the attraction is enhanced.
  • The effective central force in 3S1 tends to be stronger than in the central force in 1S0. (This is good for deuteron)

Repulsive core: 500 - 600MeV

constituent quark model suggestsit is enhanced in the light quark mass region

tensor force
Tensor force

Previous treatment:

Recently, we have proceeded one step further.We obtained (effective) tensor force separately.

★Schrodinger equation (JP=1+):

Projection op’s



tensor force1
Tensor force

The wave function

  • Nconf=1000
  • time-slice: t-t0=6
  • mπ=0.53 GeV, mρ=0.88 GeV, mN=1.34 GeV
  • The wave function for 3D1 is smaller by a factor of ~10 than W.F. for 3S1.


  • At short distance, due to the centrifugal barrier, experimental information of tensor force has uncertainty.
  • This shape of tensor force is expected from cancellation of contributions from π and ρ
  • VCeff(r) tends to be a bit more attractive than VC(r) (as is expected).
  • It is straightforward to extend the procedure to LS-force by providing a Schrodinger eq. in another channel.
quark mass dependence
quark mass dependence

(1) mπ=379MeV: Nconf=2034[28 exceptional configurations have been removed](2) mπ=529MeV: Nconf=2000(3) mπ=731MeV: Nconf=1000


Volume integral of potential

  • The repulsive core at short distance grows rapidlyin the light quark mass region.
  • The medium range attraction is enhanced.
    • less significant in magnitude
    • range of attraction tends to become wider
  • Monte Carlo calculation in the light quark mass region is important.

attractive part

repulsive part


our potential gives correct scattering length by construction
Our potential gives “correct” scattering length by construction

Scattering length calculated on our latticeusing Luescher’s method.

mq dependence of a0from one boson exchange potential

From Y.Kuramashi, PTPS122(1996)153. see also M.Fukugita et al., PRD52, 3003(1995).

  • The following two are formally equivalent in the limit L→∞.
  • scattering length obtained with our potential by using the standard scattering theory.
  • scattering length from Luescher’s finite volume method.

Idea of proof:Luescher’s finite volume method uses the information on the long distance part of BS wave function. Our potential is so constructed to generate exactly the same BS wave function at the energy of the input BS wave function.

Luescher’s method ⇒ a=0.123(39) fmOur potential ⇒ a=0.066(22) fm(The discrepancy is due to the finite size effect)

★ comments:(1) net interaction is attraction.Born approx. formula: Attrractive part can hide the repulsive core inside. (2) The empirical values: a(1S0)~20fm, a(3S1)~-5 fm our value is considerably small. This is due to heavy pion mass.

next stage

Subtleties of our potential (at short distance)

★ Operator dependence at short distance:

Short distance part of BS wave function depends on a particular choice of interpolating fieldunlike its long distance behavior, where the universal behavior is seen


★ Orthogonality of BS wave functions:

Most probably, the orthogonality of equal-time BS wave functions will fail, i.e., ⇒ non-existence of single Hermitian Hamiotonian. ⇒ Our potential is either energy dependent or (energy independent but) non-Hermitian.

We propose to avoid these subtleties by using a knowledge of the inverse scattering theory.

The inverse scattering theory guranteesthe existence of the uniqueenergy-independentlocal potential for phase shifts at all Efor fixed l.

  • Advantages:
  • The result does not depend on a particular choice of nucleon operator. It is constructed from the scattering phase shift, which is free from the operator dependence problem.
  • There are many ways to define non-local potentials. The local potential is unique. ← Inverse scattering
  • The resulting potential is Hermitian by construction.
  • Local potentials are simpler to be used in practice than non-local ones.
outline next stage
Outline (NEXT STAGE)

We may consider as a matrixwith respect to index (x’, Ei).

Instead of using the inverse scattering directly on the lattice, we go the following way:

  • Construction of non-local potential UNL(x,x’):
  • Deform the short distance part of BS wave function without affecting the long distance part
  • Search for such Λ(x,x’) that can lead to a local potential VL(x) through the relationExistence and uniqueness of such VL(x) are supported by the inverse scattering theory.

★For practical lattice calculation, it is difficult to obtain all BS wave function.⇒ derivative expansion to take into account the non-locality of UNL(x,x’) term by term.In this context, our potential at PREVIOUS STAGE serves as 0-th step of this procedure.

  • NN potential is calcuclated with lattice QCD.We extended the method, which was recently proposed by CP-PACS collaboration in studying pion-pion scattering length.
    • All the qualitative properties of nuclear force have been reproduced
      • Repulsive core(~600 MeV)at short distance (r < 0.5fm).
      • Attraction (~30 MeV) at medium distance (0.5 fm < r < 1.2 fm ). (The attraction is weak due to the heavy pion (mπ~530 MeV).)
    • Results of quark mass dependence suggest that, in the light quark mass region,
      • Repulsive core at short distance is enhanced.
      • Medium range attraction is enhanced.
    • We have presented our preliminary result on the tensor force.
  • NEXT STAGE: To avoid the subtleties of operator dependence at short distance, we proposed to resort to the inverse scattering theory. The inverse scattering theory guarantees the unique existence of the local energy-independent Hermitian potential in an operator independent way.
  • Future plans:
    • Physical origin of the repulsive core
    • Hyperon potential (Sigma N, Lambda N, Lambda Lambda, Xi N, etc.) is going on. [Next Speaker]
    • LS force VLS(r) and tensor force VT(r), etc are going on.
    • Performing the plan in NEXT STAGE.
    • unquenched QCD, light (physical) quark mass, large spatial volume,finer discretization, chiral quark actions, ...