Nuclear structure calculations with realistic nuclear forces Furong Xu Outline I. Core Gamow Shell Model (CGSM) with CD Bonn (resonance + continuum) II. Ab-initio MBPT with N3LO (LQCD) ECT* workshop “Towards consistent approaches for nuclear structure and reactions”, June 6-10, 2016, Trento, Italy

Yukawa’s nuclear force by π meson exchange (long range) Nuclear force is not a fundamental interaction, but an effective force! 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!

Symmetries： parity spin isospin Meson-exchange potential QCD-based Chiral effective field theory (Chiral EFT) Symmetries： parity spin isospin

What is the nature of nuclear force? Analogy 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 Quarks and gluons are confined into colorless hadrons

QCD=quarks + gluons (symmetries: spin, isospin, parity, chiral symmetry broken spontaneously) Weinberg At low energy, the effective degrees of freedom are nucleons and pions , rather than quark and gluon! Chiral EFT=nucleons+pions (symmetries: spin, isospin, parity, chiral symmetry broken spontaneously) Machleidt Hamiltonian可能包含的对称性：I, T, parity? CP symmetry? In Chiral EFT （手征有效场理论），由于pion的质量不为零，所以手征对称性是破缺的,但是具有CP对称性？正像QCD中，由于u和d夸克的质量不为零，所以QCD强相互作用的手征对称性是破缺的，但具有CP对称性（CP=手征×宇称），弱相互作用又破坏CP对称性。 理论方法要好。 第二项公式的第一项是质心系动能，但（p_i-p_j)不是相对动量，如果两个物体质量相同，则相对动量是(p_i-p_j)/2，具体推导见笔记。

Chiral EFT 2N forces 3N forces 4N forces Leading Order Next-to Leading Order Power counting ： Next-to-Next-to Leading Order Next-to-Next-to-Next-to Leading Order

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

Our recent calculations: Starting with N3LO (also LQCD) or CD-Bonn using SRG or Vlow k I. Core Gamow Shell Model (for weakly bound nuclei: resonance and continuum) II. Hartree-Fock + Many-body perturbation theory (MBPT)

I. Core Gamow Shell Model P Q Core

The radial wave function T. Berggren, Nucl. Phys. A109 (1968) 265 Single-particle Berggren basis in complex-k plane, describing bound, resonance and scattering on equal footing. The radial wave function Outgoing solution at large distance Orthogonality and Completeness Discretized

R.J. Liotta et al., PLB 367, 1 (1996)… Used Berggren basis to describe α-particle resonances and decays in nuclei with Woods-Saxon potential, and later two-particle resonance (Betan et al., PRL 89, 042601 (2002) For many-body systems

4He core Michel, Nazarewicz, Ploszajczak, Rotureau et al., 2003-- Michel, Nazarewicz, Płoszajczak, Vertse, Phys. G: Nucl. Part. Phys. 36 (2009) 013101 4He core

Hagen, Hjorth-Jensen et al., 2004-2012 Core Gamow shell model with realistic nuclear forces Papadimitriou, Rotureau, Michel, Płoszajczak, Barrett, Phys. Rev. C 88, 044318 (2013) Ab initio no-core Gamow shell model calculations Tritium [`trɪtɪəm], hydrogen

Realistic nuclear forces Gamow shell model calculations Our CGSM with Vlow k CD-Bonn Realistic nuclear forces Gamow shell model calculations 𝑽 𝒍𝒐𝒘 𝒌 or SRG Bare forces: Strong repulsion，slow convergence To remove hard core, but still keep good descriptions of NN scattering phase shifts Q

We have calculated sd-shell oxygen isotopes with CD-Bonn CGSM Berggren s. p. states Continuum discretized, and test stability with increasing the number of Legendre points for d3/2 orbit in 20O

No 3-body force Test convergence against Vlow k cutoff Λ in 20O Test Q-box starting-energy dependence for sd shell (20O) No 3-body force

CD-Bonn CGSM, compared with conventional H.O. SM (Λ=2.6 fm-1)

II. Our ab-initio calculations with MBPT ; , First we did HF calculations (in HO basis); The HF state is chosen as a reference state. In the HF basis, we make MBPT corrections up to 3rd order using j-j coupling:

Perturbation (MBPT) For the ground state: Rayleigh-Schrodinger method resolvent [re,solvent] Rayleigh-Schrodinger method

HF energy φ0就是HF wave function, 给出HF energy, i.e., E(0)+E(1) HF

E(0)+E(1) E(2) E(3) Anti-Symmetrized Goldstone (ASG) diagram expansion HF energy E(3) Vertex [və:’teks] 2p4h =

ASG diagrams for wave functions ψ(1) ψ(2)

HF 2nd order 2nd order

4He NCSM S.K. Bogner et al., arXiv0708.3754v2 (2007) Our MBPT calculations N3LO+SRG without 3NF 4He

Our MBPT: N3LO+SRG without 3NF

Our MBPT calculations with N3LO+SRG: convergence in radius

R. Roth et al. (2006) PRC 73, 044312 AV18, UCOM, corrections to 3rd order in energy, 2nd order in radius

Point-proton rms radius HF-MBPT calculations for 4He with N3LO-SRG, Nshell=13, hΩ=35 MeV Binding energy rp(NCSM)=1.418 fm with Nmax=10 Point-proton rms radius

Point-proton rms radius HF-MBPT calculations for 16O with N3LO-SRG, Nshell=13, hΩ=35 MeV Binding energy 3NF important! Point-proton rms radius

LQCD was provided by Aoki and Inoue We renormalize it using V low-k LQCD MBPT calculations LQCD was provided by Aoki and Inoue We renormalize it using V low-k Preliminary Eexpt = -28.3 MeV

LQCD + MBPT Preliminary Eexpt = -127.6 MeV Eexpt = -342.0 MeV

…… III. Summary Advantages of ab-initio calculations: Starting with realistic nuclear forces and renormalizations I. CGSM with CD Bonn + Vlow k for weakly bound nuclei (resonance & continuum), Oxygen isotopes II. MBPT with N3LO (LQCD) + SRG for close-shell nuclei Advantages of ab-initio calculations: i) To understand the nature of nuclear forces; ii) To understand many-body correlations; ……

Collaborators: Zhonghao Sun (Peking University) Baishan Hu (Peking University) Qiang Wu (Peking University) Sijie Dai (Peking University) Yuanzhuo Ma (Peking University) J.P. Vary (Iowa State University)

Thank you for your attention ECT* workshop “Towards consistent approaches for nuclear structure and reactions” June 6-10, 2016, Trento, Italy 36