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
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
cf) T.T.Takahashiet al, AIP Conf. Proc.842,249 (’06). D.Arndt et al., Nulc.Phys.A726,339(’03).
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).
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”
★ Schrodinger equation:
★ 1S0 channel ⇒only VC(r) survives
★ 3S1 channel ⇒ VT(r) and VLS(r) survive.
Effective central force, which takes into account the effect of VT(r) and VLS(r).
BS wave function for the ground state is obtained from the large t region.
cf) M.Fukugita et al., Phys. Rev. D52, 3003 (1995).
★ shrink at short distance suggests an existence of repulsion.
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).
Repulsive core: 500 - 600MeV
constituent quark model suggestsit is enhanced in the light quark mass region
Recently, we have proceeded one step further.We obtained (effective) tensor force separately.
★Schrodinger equation (JP=1+):
The wave function
(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
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).
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.
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.
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:
★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.