Furong Xu （许甫荣） Many-body correlations in ab-initio methods Outline I. Nuclear forces, Renormalizations (induced correlations) II. N 3 LO (LQCD) MBPT, BHF, GSM (resonance + continuum) III. Gogny-Force Shell Model with a density-dependent 3NF correlation SINAP-CUSTIPEN workshop, Shanghai, Dec , 2015 Students involved: Z.H. Sun, B.S. Hu, W.G. Jiang, Q. Wu, S.J. Dai

Yukawa’s nuclear force by π meson exchange (long range) If meson mass, nuclear force may have a similar form to electromagnetic interaction by exchanging photons Nuclear force has a finite range, mass range Electromagnetic force has an infinite range! Nuclear force is not a fundamental interaction, but an effective force!

From Machleidt 3 Most general two-body potential under those symmetries (Okubo and Marshak, Ann. Phys. 4, 166 (1958)) with central tensor quadratic 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)

7 2N forces 3N forces4N 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

What is ab-initio calculations? 3. Ab-initio methods to treat many-body problems 1. Starting from realistic nuclear forces 2. Renormalization (softening) to speed up convergence

Correlations in ab-initio many-body methods  Realistic nuclear forces: Strong short-range correlations (high momentum, repulsion) Long-range correlations (low momentum, attarction)  Renormalizations (softening) : deal with short-range correlations, which also induces many-body correlations. Do correlated transform which contains irreducible contributions to all particles An effective interaction is A-body interaction NCSM Renormalization is also a process from uncorrelated wave functions to correlated wave functions

Many-body calculations (ab initio) NCSM, CC, GF, Lattice EFT… Long-range correlations can be treated by Hartree Fock, but HF is not good enough to contain all long-range correlations, because HF takes only one Slater determinant. can be directly used for nuclear structure calculations (no need of parameter refitting, no need of additional parameters) BHF reorders HF single-particle levels by iterations, including a self-consistent interplay between the G-matrix interaction and single-particle states. G matrix scatters into all particle intermediate states, which brings more correlations. We need to go beyond, e.g., one can use MBPT to treat further long-range correlations which are lost in HF. A renormalized nuclear force: should reproduce experimental NN phase shifts at the same level as the original bare force;

Our recent calculations: Starting with N 3 LO (also LQCD) plus SRG or V low k or Brueckner 1. Many-body perturbation theory beyond HF 2. Brueckner Hartee Fock 3. Gamow Shell Model (for weakly bound nuclei: resonance and continuum) 4. Gogny-Force Shell Model

Anti-Symmetrized Goldstone (ASG) diagram expansion In the HF basis, MBPT corrections up to 3 rd order MBPT (Many-Body Perturbation Theory ) calculations

NCSM S.K. Bogner et al., arXiv v2 (2007) Our MBPT calculations 4 He N 3 LO+SRG without 3NF

16 O Our MBPT: N 3 LO+SRG without 3NF

Our MBPT calculations with N 3 LO+SRG: convergence in radius

R. Roth et al. (2006) PRC 73, AV18, UCOM, corrections to 3 rd order in energy, 2 nd order in radius

N 3 LO +MBPT 4 He 16 O

BHF calculations with N 3 LO

LQCD was provided by Aoki and Inoue We renormalize it using V low-k LQCD MBPT calculations E expt = MeV

E expt = MeVE expt = MeV LQCD + MBPT

N 3 LO V low-k Folded Diagrams+Q-Box GSM Resonance and continuum

MBPT for symmetric nuclear matter L. Coraggio et al., PRC 89, (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: short range and long range, and 3NF in-medium (density-dependent) three-body force 3NF

Automatically smooth cutoff Why choose Gogny force?

Gogny-force SM calculations for sd-shell nuclei In the existing Gogny force, taking: χ 0 =1 and α=1/3 Core ( 16 O) binding energy and single-particle energies can be calculated by the model itself.

TBMEs: Gogny vs. USDB 18 O 40 Ca

Density iteration H.O. density distribution Iterated density distribution r(fm) Single-particle energies in the spherical H.O. basis :

Core energy (i.e., 16 O 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

Extend model space to psd-shell calculations, without need of Gogny parameter refitting.

Advantages of ab-initio calculations: i) To understand the nature of nuclear forces; Summary Nuclear forces, renormalizations and induced correlations Ab-initio many-body calculations N 3 LO and LQCD + SRG or V low k or G-matrix to do MBPT, BHF, GSM, Gogny SM ii) To understand many-body correlations; Our recent works: Starting from N 3 LO (and also LQCD) i)Ab-initio MBPT and BHF ii)Realistic core GSM to describe resonance and continuum iii)Core shell model with Gogny force

An-initio nuclear structure at PKU Furong Xu Zhonghao Sun Baishan HuWeiguang Jiang Qiang Wu Sijie Dai

Thank you for your attention 36 SINAP-CUSTIPEN workshop, Shanghai Dec , 2015