1. 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

2. 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!

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

4. 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

5. 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。

6. 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

7. 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

8. 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)

9. I. Core Gamow Shell ModelP Q
Core

10. 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

11. 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, (2002)
For many-body systems

12. 4He core Michel, Nazarewicz, Ploszajczak, Rotureau et al., 2003--
Michel, Nazarewicz, Płoszajczak, Vertse,
Phys. G: Nucl. Part. Phys. 36 (2009)
4He core

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

14. 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

15. 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

16. 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

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

20. 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:

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

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

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

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

25. HF
2nd order
2nd order

26. 4He NCSM S.K. Bogner et al., arXiv0708.3754v2 (2007)
Our MBPT calculations
N3LO+SRG without 3NF
4He

27. Our MBPT: N3LO+SRG without 3NF

28. Our MBPT calculations with N3LO+SRG: convergence in radius

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

30. 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

31. 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

32. 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 = MeV

33. LQCD + MBPT
Preliminary
Eexpt = MeV
Eexpt = MeV

34. …… 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; ……

35. 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)

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