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)]

r

eg rmNN

4

2

attra

cti

on

Repuls

ive

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, 054012 (1999). P.Pennanen et al, NPPS83, 200 (2000), A.M.Green et al., PRD61, 014014 (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 functionIshii, 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:0907.1915 [hep-lat]

Method, which utilizes BS wave function

)(

)()()( 0

x

xHErV

Sch

rod

ing

er

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 = En

(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 En) satisfy this equation simultaneously.

inNNyNxNyx ,|)()(|0)(

)(),()()( 30 yyxUydxHE EE

kr

klkrAr l )(2/sin

)(

)(xE

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.

}),({)()()(),( 212 rVSLrVSrVrVxV DLSTC

)(

)()()( 0

x

xHErV

E

EC

)()()()(),( 212

OSLrVSrVrVxV LSTC

)(}),({)()()(),( 3212

OrVSLrVSrVrVxV DLSTC

)(),(),( yxxVyxU

Example (1S0): Only VC(r) survives for 1S0 channel:

)( 2

O

)()()()( 0 xrVxHE ECE

Central Potential

Central potential (leading order) by quenched QCD

Repulsive core: 500 - 600 MeV Attraction: ~ 30 MeV

Ishii, Aoki, Hatsuda, PRL99,022001(2007).

Qualitative features of the nuclear force are reproduced.

)(

)()()( 0

x

xHErV

E

EC

Quark mass dependence

1S0

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 ?

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 examinedin order to see its validity in SRC momentum region (300 – 600 MeV).

)(}),({)()()(),( 212 yxrVSLrVSrVrVyxU DLSTC

)(),()()( 30 yyxUydxHE EE

kr

klkrAr l )(2/sin

)(

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 ?

MeV46CM E

Potential at is constructed by anti-periodic BC0E

fm4.4L

)(}),({)()()(),( 212 yxrVSLrVSrVrVyxU DLSTC

How good is the derivative expansion ? (cont'd)

RESULT

[K.Murano@Lattice2009]

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 ECM=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 ECM=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 ECM to such a good accuracy.)

E=44.5 MeV (APBC)

E=45 MeV (APBC)

phase shifts from potentials

[K.Murano@Lattice2009]

Tensor Potential

d-wave BS wave function

BS wave function for JP=1+ (I=0) consists of two components:

s-wave (L=0) and d-wave (L=2)

Og

S rgrPr )(24

1)]([)( 1)(

　

)()()]([)( )()( rrrQr SD

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

)ˆ()ˆ(6

2

)ˆ(6

2)ˆ(

)()(

)()(

1,20,2

0,21,2

)()(

)()(

rYrY

rYrY

rr

rrDD

DD

d-wave “spinor harmonics”

Angular dependence Multi-valuedAlmost Single-valued

is dominated by d-wave.

JP=1+, M=0

)(D

NOTE:(0,1) [blue] E-representation(0,0) [purple] T2-representation

Difference of these two violation of SO(3)

devid

e it

by Y

lm a

nd

by C

G

facto

r

Tensor force (cont'd)

Derivative expansion up to local terms

Schrodinger eq for JP=1+(I=0)

Solve them for VC(r) and VT(r) point by point

)()()()( 120 rErSrVrVH TC

}),({)()()(),( 212 rVSLrVSrVrVxV DLSTC

)()()()()(

)()()()()(

012

012

rQHErQSrVrQrV

rPHErPSrVrPrV

TC

TC

)(

)()(

)(

)(

)()(

)()(0

12

12

rQ

rPHE

rV

rV

rQSrQ

rPSrP

T

C

(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.

1cos35

4

1),( 2

0,2

Y unobtainable points:

nnn ,,

Tensor potential (quark mass dependence)

Tensor potential is enhancedin the light quark mass region

Energy dependence of the tensor force

)(13 rVD T )(1

3 rVS C

Energy dependence is weak for JP=1+(deuteron channel)

Small energy dependence implies These potentials are valid in the energy region ECM = 0 – 46 MeV.( relative momentum region 0 – 240 MeV)

MeV46E

E ～ 46 MeV

E ～ 0 MeV

2+1 flavor QCD

2+1 flavor PACS-CS gauge configurationPACS-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 323×64 lattice

O(a) improved Wilson quark (clover) action with a non-perturbatively improved coefficient cSW=1.715

1/a=2.17 GeV (a ～ 0.091 fm). L=32a ～ 2.91 fm

PACS-CS

κud=0.13700κs =0.13640

Mpi=701 MeV L=2.9 fm

κud=0.13727κs =0.13640

Mpi=570 MeV L=2.9 fm

κud=0.13754κs =0.13640

Mpi=411 MeV L=2.9 fm

κud=0.13754κs =0.13660

Mpi=384 MeV L=2.9 fm

κud=0.13770κs =0.13640

Mpi=296 MeV L=2.9 fm

κud=0.13781κs =0.13640

Mpi=156 MeV L=2.9 fm

The followings are already publicly available through ILDG/JLDG

PACS-CS Coll. is currently generating2+1 flavor gauge config's at physical pointwith 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 results

quenched 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_B1900Kud01370000Ks01364000C1715] L ～ 1.8fm [RC20x40_B1900Kud013700Ks013640C1715]

central(1

S0)_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.

)(

)(1)(

2

C x

x

mErV

N

(for 1S0)

fm9.0

fm9.0

O

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 Ci(x)

The higher the dimension of Oi(0) the lower the singularity of Ci(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

i

ii OxCNxN )0()()0()(

i

ii inNNOxCinNNNxN ,)0(0)(,)0()(0

0),;()()(22)(

gxCgdgdg iOONN ii

)2/()2(2 0)0()0(

))log(()(

iONNiO rrrCdd

i

operator local:)0(

functionnumber -csingular :)(

i

i

O

xC

YrDDx YX ))log(()(

[S.Aoki@lattice2009]

2

1))log(()()(

r

r

D

DrV

Y

YX

YC

110 ))log(()( YrCCrVT

βY = -12/29 (spin singlet)βY = -4/29 (spin triplet)

(Preliminary)

[Finite at the origin]

1/r2 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 spacingsto 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.

[T.Inoue@Lattice2009]

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 dependenceAPBC relative momentum ～ 370 MeV (∈SRC momentum region)

To respect the chiral symmetry, we have to wait a few(?) years.

( prel = 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 %