Furong Xu （许甫荣）

Many-body calculations

with realistic and phenomenological nuclear forces

Outline

I. Nuclear forces

II. N3LO (LQCD): MBPT, BHF, GSM (resonance + continuum)

III. SM based on Gogny force

SKLNPT ， School of Physics, Peking University

JCNP2015, Osaka, Nov. 7-12, 2015

In nuclear structure calculations (starting from realistic forces ） ,

there are three key problems

1. Nuclear forces

2. Renormalization (softening) of nuclear forces

3. Many-body methods

In 1935, Yukawa discovered nuclear interaction by exchange meson,

predicted π meson (Nobel prize in 1949)

2

( )4

m cr

g eV r

r

p

p

-

=-h

If the meson mass , it becomes electromagnetic interaction, exchanging photons

0m®

1( )V r

r:

Nuclear force has a finite range, mass range Electromagnetic force has an infinite range!

0m=

Nuclear force is not a fundamental interaction, but an effective force!

Its nature has not been well known.

From Machleidt 4

Most general two-body potential under those symmetries (Okubo and Marshak, Ann. Phys. 4, 166 (1958))

with

NNV central

tensorquadratic spin-orbit

spin-orbit

another tensor

Symmetries ：1. parity

2. spin

3. isospin

1. Meson-exchange potential

2. QCD-based Chiral effective filed

theory (Chiral EFT)

Van der Waals force

+-

+-

The effective interaction between

neutral atoms: the residual force

of electromagnetic interaction

outside atom.

Nuclear force

Residual force of the QCD strong interaction outside the nucleon

What is the nature of nuclear force?

Quarks and gluons are confined into colorless hadrons

Analogy

Weinberg (1990’s)

Chiral EFT=nucleons+pions (symmetries: spin, isospin, parity, chiral symmetry broken spontaneously)

At low energy, the effective degrees of freedom are nucleon and pion, rather than quark and gluon!

QCD=quarks + gluons (symmetries: spin, isospin, parity, chiral symmetry broken spontaneously)

8

2N forces 3N forces 4N forces

Leading Order

Next-to-Next-to Leading Order

Next-to-Next-to-Next-to Leading Order

Next-to Leading Order

Chiral EFT

Power counting ：Q (π mass ） soft scale;

Λ (heavier mesons), hard scale

/Q L

-9-

1) Start from realistic nuclear force !

Reproduce experimental NN scattering phase shifts .

2) A “good enough” theoretical method solves the many-body problem of nucleus

What is ab-initio calculations?

The calculation is from the beginning of physics principles

No Core Shell Model (NCSM)

Coupled Cluster (CC)

Many-Body Perturbation Theory (MBPT)

Greens Function

Bruckner-Hartree-Fock

Gamow Shell model

Lattice Nuclear Chiral EFT, . . .

Realistic nuclear forces:

Ab-initio many-body methods

Chiral EFT (N3LO), CD Bonn, AV18 …

Vlow k , OLS, SRG, UCOM, G-Matrix, …

Renormalization process to soften nuclear force to speed up convergence

Ab initio calculation usually contains three steps

Our recent calculations:

Starting with N3LO (LQCD) plus SRG or Vlow k

1. Many-body perturbation theory

2. Brueckner Hartee Fock

3. Gamow Shell Model (for weakly bound nuclei,

to describe resonance and continuum)

4. Shell Model based on Gogny force

NCSM S.K. Bogner et al., arXiv0708.3754v2 (2007) Our MBPT calculations

4He

N3LO+SRG without 3NF

16O

Our MBPT: N3LO+SRG without 3NF

Our MBPT calculations with N3LO+SRG: convergence in radius

BHF calculations with N3LO

LQCD force provided by Sinya Aoki and Takashi Inoue,

We renormalize it using V low-k

Our MBPT calculations based on LQCD

Eexpt = -28.3 MeV

Eexpt = -127.6 MeV Eexpt = -342.0 MeV

LQCD + MBPT

Ab-initio calculations for the resonance states of weakly bound nuclei: GSM using N3LO

0.1 1

0.120.13

0.1

GSM USDB Exp.

MBPT for symmetric nuclear matter

L. Coraggio et al., PRC 89, 044321 (2014)

The importance of 3NF

NCSM with chiral 2N and 3N forces

By P. Navratil et al.

2N (N3LO) only

2N (N3LO)

+3N (N2LO)

SM calculations with Gogny force

in-medium (density-dependent) three-body force

3NF

TBMEs: Gogny vs. USDB

18O 40Ca

Gogny-force SM calculations for sd-shell nuclei

In the existing Gogny force,

taking: χ0=1 and α=1/3

Core (16O) binding energy and

single-particle energies can be

calculated by the model itself.

Density iteration

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.05

0.1

0.15

0.2

0.25

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.05

0.1

0.15

0.2

0.25

H.O. density distribution Iterated density distribution

r(fm)r(fm)

Single-particle energies in the spherical H.O. basis :

Core energy (i.e., 16O for sd shell):

Close-shell kinetic energy:

Close-shell interaction energy:

Center-of-mass energy:

Binding Energy calculations:

is calculated by diagonalizing all configurations within in valence shell space.

The two-body Coulomb energy:

Gogny-force SM calculations for spectroscopy: any excited states

Binding energy calculations

psd-shell calculations with much larger model space

Automatically smooth cutoff2 21 2|k | /4i ke m- -v v

Advantages of ab-initio calculations:

i) To understand the nature of nuclear forces;

Summary

• Nuclear force is still a big issue in nuclear physics

• Many-body methods need to be developed

ii) To understand many-body correlations;

Our recent works:

Starting from N3LO (LQCD)

i) Ab-initio MBPT and BHF

ii) Ab-initio GSM to describe resonance and continuum

iii) Shell model with Gogny force

An-initio nuclear structure at PKU

Furong Xu

Zhonghao Sun Baishan Hu Weiguang Jiang Qiang Wu Mr. Sijie Dai

http://www.phy.pku.edu.cn/~frxu

Thank you for your attention

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JCNP2015, Osaka, Nov. 7-12, 2015