Short-range nuclear force in lattice QCD Noriyoshi Ishii (Univ. of Tokyo) for HALQCD Collaboration S.Aoki (Univ. of Tsukuba), T.Doi (Univ. of Tsukuba), T.Hatsuda (Univ. of Tokyo), Y.Ikeda (Univ. of Tokyo), T.Inoue (Univ. of Tsukuba), K.Murano (Univ. of Tokyo), H.Nemura (Tohoku Univ.) K.Sasaki (Univ. of Tsukuba)

Introduction

Nuclear Force  long distance (r > 2 fm) OPEP [H.Yukawa(1935)] (One Pion Exchange)  medium distance (1 fm < r < 2fm) multi-pion, ρ, ω, "σ" attraction  bound nuclei  short distance (r < 1 fm) repulsive core [R.Jastrow(1950)] attraction Repulsive core The short distance property is difficult. Internal structure of nucleon may play a key role.  Studies based on QCD are desirable.

Lattice QCD approach to hadron potentials  Method, which utilizes static quarks D.G.Richards et al., PRD42, 3191 (1990). A.Mihaly et la., PRD55, 3077 (1997). C.Stewart et al., PRD57, 5581 (1998). C.Michael et al., PRD60, (1999). P.Pennanen et al, NPPS83, 200 (2000), A.M.Green et al., PRD61, (2000). H.R Fiebig, NPPS106, 344 (2002); 109A, 207 (2002). T.T.Takahashi et al, ACP842,246(2006), T.Doi et al., ACP842,246(2006) W.Detmold et al.,PRD76,114503(2007)  Method, which utilizes Bethe-Salpeter wave function Ishii, Aoki, Hatsuda, PRL99,022011(2007). Nemura, Ishii, Aoki, Hatsuda, PLB673,136(2009). Aoki, Hatsuda, Ishii, CSD1,015009(2008).  Strong coupling limit (NEW) Ph. de Forcrand and M.Fromm, arXiv: [hep-lat]

Method, which utilizes BS wave function Schrodinger eq. Ishii,Aoki,Hatsuda, PRL99,022001(’07). Advantages:  An extension to Luscher's finite volume method for the scattering phase shift: CP-PACS Coll., PRD71,094504(2005). (Bethe-Salpeter wave function  scattering phase shift)  It opens a way to lattice QCD construction of realistic NN potential. lattice QCD wave function nuclear force

Plan of the talk  General Strategy to construct potentials on the lattice  How good is the derivative expansion  Tensor force  2+1 flavor QCD results  Activities and possibilities to study SRC on the lattce (NOW and FUTURE)  Finite volume artifact is small  Operator product expansion  Flavor SU(3) structure of inter-baryon potential  Summary

Comments: (1)Exact phase shifts at E = E n (2)As number of BS wave functions increases, the potential becomes more and more faithful to the phase shifts. (3)U(x,y) does NOT depend on energy E. (4)U(x,y) is most generally non-local. General Strategy  Bethe-Salpeter (BS) wave function (equal time)  An amplitude to find 3 quark at x and another 3 quark at y  desirable asymptotic behavior as r  large.  Definition of (E-independent non-local) potential U(x,y) is defined by demanding (at multiple energies E n ) satisfy this equation simultaneously.

General Strategy: Derivative expansion We obtain the non-local potential U(x,y) step by step  Derivative expansion of the non-local potential  Leading Order: Use BS wave function of the lowest-lying state(s) to obtain:  Next to Leading Order: Include BS wave function of excited states to obtain terms:  Repeat this procedure to obtain higher derivative terms. Example ( 1 S 0 ): Only V C (r) survives for 1 S 0 channel:

Central Potential

Central potential (leading order) by quenched QCD Repulsive core: MeV Attraction: ~ 30 MeV Ishii, Aoki, Hatsuda, PRL99,022001(2007). Qualitative features of the nuclear force are reproduced.

Quark mass dependence 1S01S0 In the light quark mass region, The repulsive core grows rapidly. Attraction gets stronger. Aoki,Hatsuda,Ishii, CSD1,015009(2008).

How good is the derivative expansion ?

 The non-local potential U(x,y) is faithful to scattering data in wide range of energy region (by construction)  Def. by effective Schroedinger eq.  Desirable asymptotic behavior at large separation  The local potential at E～0 (obtained at leading order derivative expansion) may not be faithful to scattering data at different energy.  Validity of the derivative expansion has to be examined in order to see its validity in SRC momentum region (300 – 600 MeV).

Strategy:  Calculate two local potentials at different energies  Difference  size of higher order effects (= non-locality) The following two are closely related to each other Energy dependence  Non-locality of the potential How good is the derivative expansion ? Potential at is constructed by anti-periodic BC

How good is the derivative expansion ? (cont'd) RESULT phase shifts from potentials E=44.5 MeV (APBC)  Small discrepancy at short distance. (really small)  With a fine tuning of E for APBC, two phase shifts agree  Derivative expansion works. Potentials at the leading order serves as a good starting point for the E-independent non-local potential.  The local potential is valid in the energy region E CM =0-46 MeV. (  relative momentum region 0 – 240 MeV)

How good is the derivative expansion ? (cont'd) phase shifts from potentials  Small discrepancy at short distance. (really small)  With a fine tuning of E for APBC, two phase shifts agree  Derivative expansion works. Potentials at the leading order serves as a good starting point for the E-independent non-local potential.  The local potential is valid in the energy region E CM =0-46 MeV. (  relative momentum region 0 – 240 MeV)  The phase shift is quite sensitive to the precise value of E. (At the moment, it is not easy to calculate E CM to such a good accuracy.) E=44.5 MeV (APBC) E=45 MeV (APBC) phase shifts from potentials

Tensor Potential

d-wave BS wave function BS wave function for J P =1 + (I=0) consists of two components: s-wave (L=0) and d-wave (L=2) On the lattice, we decompose (1)s-wave (2)d-wave Up to possible contamination from l ≧ 4 due to the cubic group, s-wave and d-wave can be separated on the lattice.

d-wave BS wave function d-wave “spinor harmonics” Angular dependence  Multi-valued Almost Single-valued  is dominated by d-wave. J P =1 +, M=0 NOTE: (0,1) [blue]  E-representation (0,0) [purple]  T 2 -representation Difference of these two  violation of SO(3) devide it by Y lm and by CG factor

Tensor force (cont'd)  Derivative expansion up to local terms  Schrodinger eq for J P =1 + (I=0)  Solve them for V C (r) and V T (r) point by point (s-wave) (d-wave)

Tensor potential (cont'd) Qualitative features are reproduced. Data at very short distance may be afflicted with discretization artifact. A spike at r = 0.5 fm is due to zero of the spherical harmonics. Ishii,Aoki,Hatsuda,PoS(LAT2008)155. unobtainable points:

Tensor potential (quark mass dependence) Tensor potential is enhanced in the light quark mass region

Energy dependence of the tensor force Energy dependence is weak for J P =1 + (deuteron channel) Small energy dependence implies  These potentials are valid in the energy region E CM = 0 – 46 MeV. (  relative momentum region 0 – 240 MeV) E ～ 46 MeV E ～ 0 MeV

2+1 flavor QCD

2+1 flavor PACS-CS gauge configuration PACS-CS coll. is generating 2+1 flavor gauge configurations in significantly light quark mass region on a large spatial volume  2+1 flavor full QCD [PACSCS Coll. PRD79(2009)034503].  Iwasaki gauge action at β=1.90 on 32 3 ×64 lattice  O(a) improved Wilson quark (clover) action with a non-perturbatively improved coefficient c SW =1.715  1/a=2.17 GeV (a ～ fm). L=32a ～ 2.91 fm PACS-CS κ ud = κ s = Mpi=701 MeVL=2.9 fm κ ud = κ s = Mpi=570 MeVL=2.9 fm κ ud = κ s = Mpi=411 MeVL=2.9 fm κ ud = κ s = Mpi=384 MeVL=2.9 fm κ ud = κ s = Mpi=296 MeVL=2.9 fm κ ud = κ s = Mpi=156 MeVL=2.9 fm The followings are already publicly available through ILDG/JLDG PACS-CS Coll. is currently generating 2+1 flavor gauge config's at physical point with L ～ 6 fm.

NN potentials Comparing to the quenched ones, (1)Repulsive core and tensor force becomes significantly stronger. (Reasons are under investigation) (2)Attractions at medium distance are similar in magnitude. 2+1 flavor resultsquenched results

NN potentials (quark mass dependence) In the light quark mass region,  Repulsive core grows.  Attraction becomes stronger.  Tensor force gets enhanced.

NN (phase shift from potentials) They have reasonable shapes.

NN (phase shift from potentials) We have no deuteron so far. They have reasonable shapes. The strength is much weaker.  Use of physical quark mass is important.

Activities and possibilities to study SRC by lattice QCD (NOW and FUTURE)

Finite size artifact is weak at short distance !  Finite size artifact on the potential is weak at short distance ! (Example ） L ～ 3 fm [RC32x64_B1900Kud Ks C1715] L ～ 1.8fm [RC20x40_B1900Kud013700Ks013640C1715] central( 1 S 0 ) _ tensor force  Central force has to shift by E=24～29MeV. However, due to the missing asymptotic region, zero adjustment does not work. (Central force has uncertainty in zero adjustment due to the finite size artifact.)  Tensor force is free from such uncertainty.  Multivaluedness of the central force is due to the finite size artifact.  Finite size artifacts of short distance part of these potentials are weak !  SRC may be studied by using small volume lattice QCD. (for 1 S 0 )

Short distance limit by Operator Product Expansion  Operator Product Expansion(OPE)  OPE controls the behavior of BS wave function in the short distance limit  Renormalization group equation for C i (x) The higher the dimension of O i (0)  the lower the singularity of C i (x)  Short distance is dominated by op. with smallest dimensions (6 quark operators)_.  Anomalous dimensions of 6 quark operators are classified by chirality  Operators with "zero" anomalous dimension always exists. Others are all "negative"  Extension to the tensor force leads us to 09] β Y = -12/29 (spin singlet) β Y = -4/29 (spin triplet) (Preliminar y) [Finite at the origin] 1/r 2 with a log correction.

Short distance limit by Operator Product Expansion (cont'd)  Good agreement with lattice QCD data is not achieved yet.  Possible explanations: (1) Validity of OPE is restricted to very short distance (?) (2) Lattice discretization artifacts (?) (3) Chiral symmetry may play an important role (?) The chiral symmetry is not respected by our lattice QCD data so far.  Future plan:  Identification of the discretization artifacts. Calculations with several different lattice spacings to extrapolate them to continuum limit.  Calculations which respect chiral symmetry. Overlap gauge configurations provided by JLQCD Coll. Spatial volume is not large. But it is possible to study SRC. (Finite volume artifact is weak at short distance.)

Flavor SU(3) structure of inter-baryon potential  Flavor SU(3) structure is expected to provides the physics insights into inter-baryon potential.  The activity is currently going on for central force in flavor SU(3) limit:  Aims: (1) Qualitative understanding of the flavor SU(3) structure of inter- baryon potentials. (2) Physical origin of the repulsive core (2.1) vector meson exchange (2.2) constituent quark picture (2.3) others  Work in progress now (Central force)  Flavor SU(3) structure of tensor force (at short distance) can be also studied. which may provide physics insights into SRC. 09]

Summary  General strategy (NN potentials from BS wave functions)  These potentials are faithful to the phase shift data (by construction)  Derivative expansion works. (  Energy dependence is weak) The local potential obtained at leading order at E ～ 0 MeV is valid (at least) below 46 MeV.  Numerical results  Qualitative features are reproduced.  Significant quark mass dependence  Importance of physical quark mass  Full QCD (with 2+1 flavor PACS-CS gauge config's) The repulsive core and tensor force are enhanced  Importance of full QCD.  Finite volume artifact is weak at short distance  SRC may be studied by small volume lattice QCD  Short distance limit by OPE (Operator Product Expansion) Good agreement is not achieved yet. (Chiral quark, Discretization artifact, etc.)  Now and Future  Flavor SU(3) structure to obtain physics insights (work in progress)  Possibility to perform direct lattice QCD studies in SRC region  2+1 flavor PACS-CS config's on L ～ 2.9 fm with physical quark mass PBC and APBC to examine the energy dependence APBC  relative momentum ～ 370 MeV ( ∈ SRC momentum region)  To respect the chiral symmetry, we have to wait a few(?) years. ( p rel = 0 – 240 MeV )

End

Backup Slides

Explosion at r > 1.5 fm is due to small contamination of excited state dominant (ground state) contamination (excited state) + ～ 1 %