# Presentation: 941147

## Lattice QCD calculation of nuclear force

**Noriyoshi Ishii (Univ. Tsukuba)**

_**Contents:**_
- Backgrounds
- Main contents
- Bethe-Salpeter wave function
- (effective) central force
- Quark mass dependence of the central force
- Tensor force
- Energy dependence of nuclear force.--- toward more realistic nuclear force ---
- 2+1 flavor QCD results of nuclear force by using PACSCS gauge configuration
- Summary

- _In collaboration with_
- H. Nemura (RIKEN)K. Murano (Univ. Tsukuba)S. Aoki      (Univ. Tsukuba)T. Hatsuda (Univ. Tokyo)for PACS-CS Collaboration

**Notes:**
KPS meeting
23 October, 2008
25+5 minuites talk

## Background

** ****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 attraction in this region is responsible for bound nuclei.
** ****short distance**** ****(r < 1fm)**    strong repulsive core [R.Jastrow (1951)]  It is important in many phenomena.     However, its physical origin is still an open problem.

_**Background**_

**NN potential****=****nuclear force**

- _Realistic nuclear force_

- Reid93 is fromV.G.J.Stoks et al., PRC**49**, 2950 (1994).
- AV16 is fromR.B.Wiringa et al., PRC**51**, 38 (1995).

**Repusive core**

- attraction

- Two nucleon may overlap at such a short distance
- Repulsive core should reflect the quark/gluon structure of nucleon
- Lattice QCD first principle study of nuclear force is desired.

## Lattice QCD approaches to NN potentials

_**Lattice QCD approaches to NN potentials**_

_**Method, which utilizes the static quark**_
_**Method, which uses Bethe-Salpeter wave function **_(**We use this**)

- Bethe-Salpeter (BS) wave function for NNis generated by lattice QCD
- NN potentials are reconstructed from the wave function by using Schroedinger equation.

**Schrodinger eq.**

- N.Ishii, S.Aoki, T.Hatsuda, Phys.Rev.Lett.99,022001(’07).

**extension**

- S.Aoki et al.(CP-PACS Collab.),Phys. Rev. D**71**,094504(2005).

_**Advantage:**_**NN potential is so constructed as to reproduce (BS) wave function.**It can also reproduce **the phase shift**, which is embeded in the wave function This method will lead us to the lattice QCD construction of realistic nuclear force.

- Schematically,

## Lattice QCD parameters

**Standard Wilson gauge action**.
**β= 6/g****2 ****= 5.7 (******** gauge coupling)**
**a ****~****0.14 fm**
- 324 lattice (**L****~****4.4**** ****fm**)
- 1000-2000 gauge configs
**Standard Wilson quark action**
**κ=0.1665  (******** quark mass) is mainly used.**
**m****π**** ****~****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.

_**Lattice QCD parameters**_

## Wave function

-  Quantum mechanical NN wave function is an approximate concept in QCD.
-  The closest concept is provided by **equal-time**** ****Bethe-Salpeter(BS) wave function**** **
-  This calculates an amplitude to find three quarks at x and another three quarks at y.
-  desirable asympototic behavior at |x-y|→large
-  It satisfies (so called) **the effective Schroedinger equation**.   (1) U(r,r’) on r.h.s. plays the role of the interaction kernel.  (2) U(r,r’) is most generally non-local  (3) It is possible to define U(r,r’) so that it does not depend on the total energy E.

- 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, arXiv:0805.2462[hep-ph].

## How to obtain BS wave function

_**How to obtain BS wave function**_

- In the large t region,contributions from excited states are suppressed exponentially.We are left with the contribution from the _**lowest-lying state**_.

- BS wave function is obtained from NN amplitude in the large (imaginary) time region.

- A state consisting of p and n, whose relative momentum is nearly zero(due to the finiteness of the box).

## (Effective) central force

-  The derivative expansion to the non-local interaction U(r,r’)    after imposing constraints from various symmetries
** ****J****P ****= 0****+**** case (****1****S****0 ****)**     Schrodinger equation becomes
** ****J****P ****= 1****+**** case (****3****S****1 ****-****3****D****1****)**                                                                                 **As the first step**, we neglect VT(r) and VLS(r) , which leads to   ”**effective central force**” = ”a central force, which can reproduce the exact 3S1 W.F.”

## Lattice QCD result of NN wave function

- Shrink at short distance indicates an existence of repulsion at short distance.

- r < 0.7 fm region is fully calculated.
- r > 0.7 fm region is partially calculated.

## (Effective) central force

- Repulsive core: 500 - 600 MeV

- Attractive pocket: about 30 MeV

- Both of these are smaller than we expect.This is answered by the quark mass dependence.

- The following diagram is contained.
- This leads to the one pion exchangeat the large spatial separation.

## Quark mass dependence of the central force:  

- (1) mπ=380MeV: Nconf=2034    [28 exceptional configurations     have  been removed](2) mπ=529MeV: Nconf=2000(3) mπ=731MeV: Nconf=1000

**1****S****0**

- Strong quark mass dependence is found.
- In the light quark mass region,
-  the repulsive core grows rapidly.
-  attractive pocket is enhanced mildly.
- The lattice QCD calculation at light quark mass region is quite important.

## Net interaction is attraction

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

**★ ****comments:**(1) **attractive scattering length**     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.

**m****phys**

**Scattering length by Luescher’s formula**

## Tensor force (JP=1+ channel)

_**NN potentials**_

- It is possible to take into account 3S1-3D1 coupling.
- By neglecting LS force, we calculate **central **and **tensor force** separatelyfrom coupled Schrodinger eq of s-wave and d-wave.

_**Comments:**_
- Below is obtained as a result of cancellation   between the contributions from pi and rho
-  VCeff(r) gets attraction    due to the 2nd order perturbation effects of VT(r). However, the difference is weaker than expected.  tensor force is weaker than expected.  this may be due to the heavy quark mass.

_**BS wave function**_

## d-wave BS wave function

- plot of

- plot of                  after removing Ylm and Clebsch-Gordan factors

- BS wave function for d-waveshould be proportional to the “spinor harmonics”

- Multi-valuedness is due to the angular dependence.

- Almost Single-valued function is obtained.
-  ψ(D) is dominated by d-wave.

- JP=1+, M=0

## Quark mass dependence of tensor force

- A strong quark mass dependence is found.Tensor force grows in the light quark mass region.

- (1) mπ=380MeV: Nconf=2020    [28 exceptional configurations     have  been removed](2) mπ=529MeV: Nconf=1947(3) mπ=731MeV: Nconf=1000

-  It is important to perform a lattice QCD calculation      directly in the light quark mass region.

## The hyperon potentials

**J-PARC**

-  It serves as an important input for the hyper nuclear physics.
-  It provides a necessary information on the hyperon matter    in the neutron star.
-  Only a limited information is so far available.    (No accelerator facility generates direct hyperon beam.)
-  It is directly connected to the J-PARC project.

_**NΞ potential (I=1)**_

-  The main target at J-PARC DAY-1 experiment.
-  Experimental information is limited.
- H.Nemura et al., arXiv:0806.1094[nucl-th].

_**NΛ potential **_

**PRELIMINARY**

- H.Nemura’talk at LATTICE2008
- Work in progress

**Notes:**
pion_mass.jack 3.6688618486007929e-01 4.0869070225408091e-04 0.509879600406672 0.000567977374331237 
kaon_mass.jack 4.3443364123829398e-01 3.4968238897158636e-04 0.603753590455509 0.000485970647344162 
rho_mass.jack 6.1824867705485886e-01 1.6619799602629827e-03 0.859210298498719 0.00230973449803231 
kaon_vector_mass.jack 6.4894341481036288e-01 1.1407409372187259e-03 0.901868270554365 0.00158534324059773 
proton_mass.jack 9.3332236963789572e-01 3.1792315444770054e-03 1.29708355484431 0.00441833292282847 
xizero_mass.jack 1.0181830004485026e+00 2.6650399752485331e-03 1.41501850664543 0.00370373585521013 
lambda_mass.jack 9.7173193779306777e-01 2.6606988122818819e-03 1.35046320245961 0.00369770272959767 
sigmaplus_mass.jack 9.8694679962034737e-01 2.6612605836607230e-03 1.37160803698559 0.0036984834506441 
lattice scale a=0.141987517056546 fm hbarc=197.326968 MeV fm energy scale (=hbarc/a) = 1389.74870531341 MeV 

## More realistic nuclear force 

-  The asympototic behavior at large distance:
-  Our potential is so constructed as to reproduce the used BS wave functions simultaneously.
- It can simultaneously reproduce the phase shiftat all these four energies.

_**Single **__**energy independent **__**potential**_,which can simultaneously reproduce BS wave functions at various energies.**Potential, which is faithful to the NN scattering.****(******** ****Scattering phase shift at these energies becoms faithful to experiments.****)**

- This topic is interesting.Through the inverse scattering theory, it is related to
-  Effect of possible distortion of nucleon at short distance. ( Uncertainty originating from a particular choice of      the interpolating field of nucleon)
-  Orthogonality of Bethe-Salpeter Wave function.
-  Non-locality of the potential
-  Energy dependence of the potential

## Energy dependence of the potential (The first step toward this direction) 

_**Energy dependence of the potential **__(The first step toward this direction)__** **_

-  Our potential so far is constrcted    from a single BS wave function at E=E0~0.     Strictly speaking,    the validity is limited only to the scattering length.    (~ phase shift at E~0)
-  Validity at other E can be examined   by constructing the potential                            from BS wave function at E=E1≠0.
-  If the shape does not change,    validity of the potential is               extended to an energy region [E0, E1].

_**Potential at E=E**__**1**__**~**__**50 MeV (CM frame)**_
-  spatial **anti-periodic BC** on quark fields
-  nucleon fields also satisfy the **anti-periodic BC** (nucleon fields consist of odd number of quark fields)
-  Spatial momentum of nucleon is **discretized** as
-  **Even the lowest energy states has to have****    ****the minimum momentum**  (about 240 MeV for L ~ 4.4 fm) E1~50 MeV (CM frame)

##  Energy dependence of the potential (cont’d)

- The result so far indicates    that there is an energy dependence at short distance.
-  The data is quite noisy.    (4000 gauge config are used to obtain this result)    We need more studies for final conclusion.
-  _**OUR PLAN:**_    Energy dependent potential                 will be converted to    energy independent but non-local potential by using derivative expansion.

- Enlarged figure

**PRELIMINARY**

**PRELIMINARY**

## Full QCD result of NN potential by using 2+1 flavor PACS-CS gauge configurations

_**Full QCD result of NN potential **__**by using 2+1 flavor PACS-CS gauge configurations**_

- PACS-CS collaboration is generating 2+1 flavor gauge configurations in _significantly light quark mass region_ on a _large spatial volume_
-  2+1 flavor full QCD    S.Aoki et al., PACSCS Collab., arXiv:0807.1661[hep-lat]
-   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

- We show preliminary results of nuclear force obtained by using some of the gauge config’s.
-  κud=0.13700, κs=0.13640 (mpi~702MeV,  L=2.9 fm)
-  κud=0.13727, κs=0.13640 (mpi~560 MeV, L=2.9 fm)
-  κud=0.13770, κs=0.13640 (mpi~296 MeV, L=2.9 fm)

**PACS-CS**

## Nuclear forces (mpi~702 MeV) from 2+1 flavor full QCD

_**Comments:**_
-  mpi~700 MeV
-  results from time-slice t=10.     where ground state saturation is expected.
-  Comparing to the quenched case,    a remarkable difference is found in
-  the strength of the repulsive core
-  the strength of the tensor force
-  We are currently increasing the statistics.

**PRELIMINARY**

**PRELIMINARY**

- Quenched VC(r) for comparison.

## Nuclear forces (mpi~570 MeV) from 2+1 flavor full QCD

_**Comments:**_
-  mpi~570 MeV
-  results from time-slice t=10.     where ground state saturation is expected.
-  Comparing to mpi~702 MeV,    a slight enhancement is found in
-  the strength of the repulsive core
-  the strength of the tensor force
-  We are currently increasing the statistics.

**PRELIMINARY**

**PRELIMINARY**

## Nuclear forces (mpi~296 MeV) from 2+1 flavor full QCD

_**Comments:**_
-  mpi~296 MeV
-  results from time-slice t=6
-  Statistics has to be improved significantly.

- Tensor force is not available yet.(Error bar is too large !)

**PRELIMINARY**

## Full QCD at physical quark mass point

_**Full QCD at physical quark mass point**_

-  Full QCD calculation at the physical quark mass point play the key role in providing a more realistic nuclear force.
-  **This is now within our reach** owing to **Progress of lattice QCD Monte Carlo calculation** & **recent high performance super computers** _One of the groups, who made a significant progress in this direction_
-   from Y.Kuramashi’s talk(PACS-CS) at LATTICE 2008

- They are generating 2+1 flavor gauge configs
- at physical point on L~6fm lattice.

## Summary

-  We have reported the lattice QCD approach to the nuclear force.
-  We have used a new method,                           where the potential is constructed from BS wave function generated by lattice QCD.
-  The resulting (effective) central force posses a repulsive core of~600 MeV at short distance     and an attractive pocket of ~30MeV at medium distance.
-  Strong quark mass dependence is found:    the repulsive core grows rapidly and the attraction is enhanced mildly in the light quark mass region.     It is important to perform the Monte Carlo calculation in the light quark mass region directly.
-  The preliminary result on the tensor force.
-  The data so far indicates that there is an energy dependence at short distance.    For for final conclusion, we need more study. (Data is quite noisy)    According to our plan,    energy dependent potential   will be converted to   energy independent but non-local potential.
-  Preliminary full QCD result by using 2+1 flavor gauge config. generated by PACS-CS Collaboration.    Compared to the quenched calculation,    significantly strong repulsive core is obtained   and   strength of tensor force is enhanced.
-  _Future plans:_
-  Origin of the repulsive core.
-  LS force. Potentials for P-wave. Three body force.
-  deuteron, form factor.
-  Collaboration with specialists of the quantum few body system.
-  Full QCD calculations at the physical quark mass point in a large spatial volume.

## END

## Backup slides

## General form of NN potential 

- where
- ★ If we keep the terms up to O(p), we are left with the convensional form of the potential in nuclear physics:

_**General form of NN potential**_

- ★ By imposing following constraints:
- Probability (Hermiticity):
- Energy-momentum conservation:
- Galilei invariance:
- Spatial rotation:
- Spatial reflection:
- Time reversal:
- Quantum statistics:
- Isospin invariance: