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CRC110 Workshop on Nuclear Dynamics and Threshold Phenomena

CRC110 Workshop on Nuclear Dynamics and Threshold Phenomena Ruhr- Universität Bochum, April 5-7, 2017. Relativistic Brueckner-Hartree-Fock Theory for finite nucleus. Jie MENG （孟 杰） School of Physics ， Peking University （北京大学物理学院）. Outline. Introduction

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CRC110 Workshop on Nuclear Dynamics and Threshold Phenomena

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  1. CRC110 Workshop on Nuclear Dynamics and Threshold Phenomena Ruhr-Universität Bochum, April 5-7, 2017 Relativistic Brueckner-Hartree-Fock Theory for finite nucleus Jie MENG （孟 杰） School of Physics，Peking University（北京大学物理学院）

  2. Outline • Introduction • Full Relativistic Brueckner-Hartree-Fock • Calculation for 4He, 16O, 40Ca, and 48Ca • Summary & Perspectives

  3. Nuclear Energy Density Functional Nuclear Energy Density Functionals: the many-body problem is mapped onto a one-body problem without explicitly involving inter-nucleon interactions! Kohn-Sham Density Functional Theory For any interacting system, there exists a local single-particle potential h(r),such that the exact ground-state density of the interacting system can be reproduced by non-interacting particles moving in this local potential. The practical usefulness of the Kohn-Sham scheme depends entirely on whether Accurate Energy Density Functionalcan be found!

  4. Density functional theory in nuclei Nuclear DFT has been introduced by effective Hamiltonians: by Vautherin and Brink (1972) using the Skyrme model as a vehicle Based on the philosophy of Bethe, Goldstone, and Brueckner one has a density dependent interaction in the nuclear interior G(ρ) At present, the ansatz for E(ρ) is phenomenological: • Skyrme: non-relativistic, zero range • Gogny: non-relativistic, finite range (Gaussian) • CDFT:Covariant density functional theory

  5. Why Covariant? P. Ring Physica Scripta, T150, 014035 (2012) • Spin-orbit automatically included • Lorentz covariance restricts parameters • Pseudo-spin Symmetry • Connection to QCD: big V/S ~ ±400 MeV • Consistent treatment of time-odd fields • Relativistic saturation mechanism • … Pseudospin symmetry Hecht & Adler NPA137(1969)129 Arima, Harvey & Shimizu PLB 30(1969)517 Ginocchio PRL 78, 436 Brockmann & Machleidt, PRC42, 1965 (1990)

  6. The relativistic mean field (RMF) theory, based on either the finite-range meson-exchange or point-coupling interactions, has received much attention due to its successful description of numerous nuclear phenomena in stable as well as exotic nuclei International Review of Nuclear Physics (Vol 10) Relativistic Density Functional for Nuclear Structure World Scientific(Singapore (2016) B. Serot, J.D. Walecka, Adv. Nucl. Phys. 16 (1986) 1. P.-G. Reinhard, Rep. Prog. Phys. 52 (1989) 439 P. Ring, Prog. Part. Nucl. Phys. 37 (1996) 193. D. Vretenar, A. Afanasjev, G. Lalazissis, P. Ring, Phys. Rep. 409 (2005) 101. J. Meng, H. Toki, S.G. Zhou, S.Q. Zhang, W.H. Long, L.S. Geng, Prog. Part. Nucl. Phys. 57 (2006) 470.

  7. s w r Meson-exchange version of CDFT CDFT：Relativistic quantum many-body theory based on DFT and effective field theory for strong interaction Strong force: Meson-exchange of the nuclear force (Jp T)=(0+0) (Jp T)=(1-0) (Jp T)=(1-1) Sigma-meson: attractive scalar field Omega-meson: Short-range repulsive Rho-meson: Isovector field Electromagnetic force: The photon

  8. Point-coupling version of CDFT Isoscalar-scalar Elementary building blocks Isoscalar-vector Isovector-scalar Energy Density Functional Densities and currents Isovector-vector

  9. Covariant density functional Meson Exchange Point Coupling Nonlinear parameterizations: Nonlinear parameterizations: NL3, NLSH, TM1, TM2, PK1, … PC-LA, PC-F1, PC-PK1 … Density dependent parameterizations: Density dependent parameterizations: TW99, DD-ME1, DD-ME2, PKDD, … DD-PC1, … 2

  10. Covariant density functional PC-PK1 Zhao, Li, Yao, Meng, PRC 82, 054319 (2010)

  11. CDFT for Nuclear matter Saturation properties:

  12. Kτ = Kτ,v + Kτ,s A-1/3 Data from from Umesh Garg, also H. Sagawa et al., Phys. Rev. C 76, 034327 (2007)

  13. CDFT: Meson-exchange/Point Coupling Very successful in nuclear physics Ring PPNP1996, Vretenar Phys.Rep.2005, Meng PPNP2006 • Spin-orbit splitting • Pseudo-spin symmetry • Nuclear saturation properties • Exotic nuclei • Excellent reproduction of nuclear properties • …… Meng, Peng, Zhang, Zhao, Front. Phys., 8(2013) 55–79 13

  14. The relativistic Hartree-Fock theory Bouyssy PRC 1987 Bernardos PRC 1993 Marcos JPG 2004 Bürvenich PRC 2002 • In addition of the RMF advantages, there are • Pion contribution included • the contributions due to the exchange (Fock) terms • the pseudo-vector π-meson. • Nuclear effective mass • Fully self-consistent description for spin-isospin excitation • …… Long PLB2006, Long PRC2007, Liang PRL2008, Liang PRC2009 14

  15. DDRHF(B): pion and Exchange term Long, Giai, Meng, Physics letters B640 (2006) 150 Long, Ring. Giai, Meng, PHYSICAL REVIEW C81 (2012) 024308 Long, Ring. Giai, Bertulani, Meng, PHYSICAL REVIEWC 81(2010) 031302 • RHFB equation • Single particle Hamiltonian: Kinetic energy: Local potentials: Non-local Potentials: • Pairing Force: Gogny D1S • Dirac Woods-Saxon Basis:solve the integro-differential RHFB equation 15

  16. Charge-exchange excitation modes RH + RPA • No contribution from isoscalar mesons (σ,ω), because exchange terms are missing. • π-meson is dominant in this resonance. • zero-range pionic counter-term g’ has to be refitted to reproduce the data. RHF + RPA • Isoscalar mesons (σ,ω) play an essential role via the exchange terms. • While, π-meson plays a minor role. • g’ = 1/3 is kept for self-consistency. Liang, Giai, Meng, PRL 101, 122502 (2008)

  17. Lcalized form of Fock terms The fine structure of spin-dipole excitations in O-16 is reproduced quite well in a fully self-consistent RPA calculation based on the RHF theory • A localized form of Fock terms is proposed with considerable simplicity as compared to the conventional Fock terms. • Based on this localized RHF theory, the spin-dipole excitation in Zr-90 is well reproduced with a RPA calculation. Liang, Giai, Meng PRL 101, 122502 (2008) Liang, Zhao, Meng Phys. Rev. C 85, 064302 (2012) Liang, Zhao, Ring, Roca-Maza, Meng Phys. Rev. C 86, 021302 (2012)

  18. ab initio calculation Relativistic Brueckner-Hartree-Fock Theory for finite nucleus

  19. ab initio calculation in nuclear physics • ab initio----- “from the beginning” • without additional assumptions • without additional parameters • ab initio in nuclear physics • with realistic nucleon-nucleon interaction • with some few-body methods and many-body methods, such as Monte Carlo method, shell model and energy density functional theory • ab initio in nuclear matter • Variational method Akmal PRC1998 • Green’s function method Dickhoff PPNP2004 • Chiral Perturbation theory Kaiser NPA2002 • Brueckner-Hartree-Fock (BHF) theory Baldo RPP2012 • Relativistic BHF (RBHF) theory Brockmann PRC1990 • ........... BHF 19

  20. 2020/1/3 ab initio calculation in nuclear physics ab initio calculation for light nuclei • Gaussian Expansion Method Hiyama PPNP2003 • Green Function Monte Carlo Method Pieper PRC2004 • Lattice Chiral Effective Field Theory Lee PPNP2009 • No-Core Shell Model Barrett PPNP2012 • …… ab initio calculation for heavier nuclei • Coupled Channel method Hagen PRL2009 • BHF theory Hjorth-Jensen Phys.Rep.1995 With HJ potential Dawson Ann.Phys.1962 With Reid potentialMachleidt NPA1975 With Bonn potentials Muether PRC1990 16O in BHF method in Bonn potential

  21. ab initio calculation for nucleus Relativistic Brueckner Hartree-Fock: nuclear matter • Nuclear matter Anastasio PRep 1978 Brockmann PLB 1984 ter Haar PRep. 1987 • Defining an effective medium dependent meson-exchange interaction based upon the nuclear matter G matrix Brockmann PRC1990 Brockmann PRL 1992 Fritz PRL 1993 ab initio attempt for finite nucleus: extracted interaction from the ab initio calculation for nuclear matter • Density-dependent relativistic mean field theory Brockmann PRL1992 • Density-dependent relativistic Hartree-Fock theory Fritz PRL1993

  22. FullRelativistic Brueckner–Hartree–Fock • Ab initio calculations, the description of finite nuclei with a bare nucleon-nucleon (NN) interaction, form a central problem of theoretical nuclear physics. • Nonrelativistic potentials systematically failed to reproduce the saturation properties of infinite nuclear matter in ab initio calculations. It has been concluded that the bare three-body force plays an essential role. • Relativistic Brueckner–Hartree–Fock theory, based on RHF calculations with the 𝐺-matrix, can reproduce empirical saturation data without three-body forces. • Inspired by the success of the RBHF theory in nuclear matter, it is a natural extension to study finite nuclei in the same framework. Shi-Hang Shen, Jin-Niu Hu,Hao-Zhao Liang, Jie Meng, Peter Ring, Shuang-Quan Zhang, Relativistic Brueckner–Hartree–Fock Theory for Finite Nuclei . Chin. Phys. Lett. 33 (2016) 102103

  23. T-Matrix and G-Matrix Lippmann-Schwinger Equation Bethe-Goldstone Equation Lippmann Phys. Rev 1950 Brueckner Phys. Rev 1955 • E is the starting energy • Q is the Pauli operator • G-matrix is for many-body problem • V is the realistic NN interaction • E is the incident energy • T-matrix is for two-body scattering The corresponding EOS in HF The corresponding EOS in HF

  24. Bethe-Goldstone Equation K. A. Brueckner, C. A. Levinson, and H. M. Mahmoud, Phys. Rev. 95, 217 (1954) • Sum up ladder diagrams to infinite order • Bethe-Goldstone equation • Wis the starting energy, its value depends on the position of G-matrix in the diagram. • εm , εnare the RHF single particle energies. • Q is the Pauli operator which forbids the states being scattered below Fermi surface.

  25. Relativistic Brueckner-Hartree-Fock Theory • Relativistic Hartree-Fock equation in complete basis, • for details, e.g. • where D are the expansion coefficients: • RBHF total energy • together with exchange term, W. Long, N. Van Giai, and J. Meng, PLB 640, 150 (2006) m’ n’ RHF + + + . . . a b a m n b a m n b RBHF = a b

  26. Iteration of RBHF Solution • Initial single-particle basis trial for RBHF final solution • Bethe-Goldstone equation • Single-particle potential • RHF iteration If converged , RBHF iteration finishes. • Basis transformation Go back to step 2. Solving with matrix inversion method M. Haftel and F. Tabakin, NPA 158, 1 (1970) = =

  27. Single-Particle Potential in Nuclear Matter • Bethe-Brandow-Petschek (BBP) theorem: if the s.p. potential of hole states (k <= kF) defined in the following way, certain higher order diagrams will be canceled: • For s.p. potential of particle states (k > kF), there are several choices: • Continuous choice : same as hole states. • Gap choice: U = 0 for k > kF. • In between: ε’ = energy of hole states. H. Bethe, B. Brandow, and A. Petschek, Phys. Rev. 129, 225 (1963) ... ... R. Rajaraman and H. Bethe, Rev. Mod. Phys. 39, 745 (1967)

  28. Single-Particle Potential in Finite Nuclei • Single-particle potential in finite nuclei : • In this work, ε’ is chosen in the range of energy of occupied states. K. Davies, et al., Phys. Rev. 177, 1510 (1969)

  29. Center-of-Mass Motion • Intrinsic Hamiltonian • with • Projection after variation (PAV) • Projection before variation (PBV) Ref: P. Ring and P. Schuck, The Nuclear Many-Body Problem, Springer, 1980

  30. RHF Check • Motivation: by repeating previous RHF calculation, we can check the correctness of • Two-body matrix elements. • RHF equation solving. • Numerical details: • Object: 16O. • Interaction: Bouyssy (a,b,c,e) A. Bouyssy, et al., PRC 36, 380 (1987). • Box size: R = 10 fm (step size 0.05 fm). • Basis space: lcut = 3; εDcut = -1800 MeV, εFcut = 300 MeV. • Convergence condition: max| Uij(new) – Uij (old) |< 10-4 fm-1. • Center-of-mass motion: No.

  31. RHF Check • Binding energy per nucleon -E/A (MeV), charge radius rc (fm), and proton 1p spin-orbit splitting ΔLS (MeV) with our code (Shen), in comparison with Bouyssy A. Bouyssy, et al., PRC 36, 380 (1987) and results from Long’s DDRHF code W. Long, N. Van Giai, and J. Meng, PLB 640, 150 (2006). RH: σ + ω without Coulomb exchange RHF: σ + ω RHF: σ + ω + π with Coulomb exchange • Correctness of RHF part is confirmed. RHF: σ + ω + π + ρ

  32. Numerical Details for RBHF Calculation • Object: 4He, 16O, 40Ca, 48Ca. • Interaction: Bonn A potential. R. Machleidt, Adv. Nucl. Phys. 19, 189 (1989) • Cut-offs for basis space: lcut = 20, εcut = 1100 MeV, εDcut = -1700 MeV. • Basis: fixed Dirac Woods-Saxon basis and self-consistent RHF basis. • Coulomb exchange term: relativistic local density approximation. • Center-of-mass correction: included in variation. as in density functional SLy6 E. Chabanat, et al., NPA 635, 231 (1998). • Box size: R = 7 fm. S. Zhou, J. Meng, and P. Ring, PRC 68, 034323 (2003) H. Gu, et al., PRC 87, 041301 (2013)

  33. Convergence to Energy Cut-Off • Total energy and charge radius of 16O as a function of energy cut-off. Rough estimation of the dimension of |ab> : (20 × 2 × 20)2 = 640,000 (symmetry) ≈ 60,000 lsn Storage: 256 GB CPU time: 224 h Intel(R) Xeon(R) CPU E5-4627 v2 @ 3.30GHz • Satisfactory convergence is achieved near εcut = 1.1 GeV.

  34. Center-of-mass Motion • Intrinsic Hamiltonian • Include -Hcm in RBHF iteration (PBV) or c.m. correction after RBHF iteration (PAV). • Total energy E and energy of c.m. motion Ecm at each iteration step. Δ ≈ 9 MeV Δ ≈ 0.7 MeV • Behavior of Ecm from PBV is similar to PAV. • Energy E from PBV is similar to PAV after the first RHF iteration (G1), but very different from the second RHF iteration (G2).

  35. Center-of-mass Motion • Single-particle spectrum for PAV and PBV for 1st RHF iteration (G1), and the final RHF iteration (G4). • As PBV includes -Hcm in RHF equation, the sp energies are lower than that of PAV, especially those high lying states. • Different s.p. spectrum give different G-matrix. • PAV is a good approximation for PBV in RHF calculation, but not in RBHF calculation. • In RBHF calculation, PBV method should be used.

  36. Self-Consistent Basis for RBHF • Total energy of RBHF calculation at each iteration step with different initial DWS basis • Self-consistency is very important for RBHF calculation.

  37. Different Choices for Single-Particle Potential • Uncertainty in the definition of s.p. potential for particle states (a, b > kF), • We will investigate the following two situations: • Gap choice: U = 0. • Similar as continuous choice but with ε’ = energy of occupied states, e.g. επ1p1/2 or εν1s1/2 . • Results • Choice ε’ = επ1p1/2 will be used.

  38. Different Choices for Single-Particle Potential • Energy spectrum and charge density distribution of RBHF calculation for 16O with different s.p. potentials for particle states. • There is a “gap” between particle states and hole states with gap choice. • Choice with ε’ below Fermi surface give similar results. • Central density given by gap choice is smaller than the other choices.

  39. RBHF for 16O • Total energies and charge radii of 16O obtained with RBHF, RBHF with EDA, and BHF. For EDA and BHF: H. Müther, R. Brockmann, and R. Machleidt, PRC42, 1981 (1990). • RBHF improves the description over EDA or BHF. • Binding energy and charge radius given by Bonn potentials are smaller than experimental data.

  40. RBHF for 16O • Single-particle spectrums for Bonn A, B, C with RBHF, in comparison with experimental dataL. Coraggio, et al., PRC68, 034320 (2003). • Bonn A gives the best description within RBHF framework. • The gap at Fermi surface may be due to lack of higher order configurations • E. Litvinovaand P. Ring, PRC73, 044328 (2006).

  41. RBHF for 16O • Charge density distributions for Bonn A, B, C with RBHF, in comparison with experiment H. De Vries, C. De Jager, and C. De Vries, At. Data Nucl. Data Tables36, 495 (1987). • Central density distributions given by Bonn potentials are larger than data. • Density distribution gets more concentrated in the center: Bonn C  B  A.

  42. Comparison for 16O • Energy, charge radius, matter radius, and π1p spin-orbit splitting in 16O with Bonn A/B/C in RBHF; Bonn A innonrelativistic renormalized BHFH. Müther, R. Brockmann, and R. Machleidt, PRC42, 1981 (1990); Vlow−k derived from Argonne v18 in BHFB. Hu, et al., arXiv:1609.02675; N3LO inno core shell model (NCSM)R. Roth, et al., PRL 107, 072501 (2011) and coupled-cluster theory (CC) G. Hagen, et al., PRC 80, 021306 (2009). • Relativistic effect is important to improve description. • The results of RBHF are similar to other state-of-the-art ab initio results.

  43. Comparison for 4He • Energy, charge radius and proton radius of 4He. Results of Bonn A in RBHF with PBV and PAV, comparing with CD-Bonn inFaddeev-Yakubovsky(FY) equations A. Nogga, H. Kamada, and W. Glöckle, PRL85, 944 (2000 N4LO in FY S. Binder, et al., PRC 93, 044002 (2016); N2LO innuclear lattice effective field theory (NLEFT) T. Lähde, et al., PLB 732, 110 (2014); N3LO in NCSMP. Navrátil, Few Body Syst. 41, 117 (2007); Vlow−k derived from Argonne v18 in BHFB. Hu, et al., arXiv:1609.02675. • Treatment of center-of-mass motion need to be careful for light system. • Radii given by ab initio calculations are generally larger than data.

  44. RBHF for 40Ca and 48Ca • Energies, charge radii, matter radius, and π1d spin-orbit splittings of 40Ca and 48Ca. Results of Bonn A in RBHF, comparing with N3LO in CC G. Hagen, et al., PRC82, 034330 (2010); Vlow−k derived from Argonne v18 innon-relativistic BHFB. Hu, et al., arXiv:1609.02675, NCSMR. Roth, et al., PRL99, 092501 (2007); and CC G. Hagen, et al., PRC76, 044305 (2007). Storage: 1100 GB CPU time: 1720 h Storage: 1800 GB CPU time: 4900 h For RBHF • Results for 40Ca and 48Ca given by RBHF are similar as for 16O. • Radius given by ab initio calculations are generally larger than experiment.

  45. RBHF for 40Ca and 48Ca • Charge density distribution of 40Ca and 48Ca calculated by RBHF theory with Bonn A potential, in comparison with experimental dataH. De Vries, C. De Jager, and C. De Vries, At. Data Nucl. Data Tables36, 495 (1987). • Central density given by RBHF theory is higher than experimental value, while the relation between 40Ca and 48Ca is similar as experiment.

  46. RBHF for Neutron Skin of 48Ca • Neutron skin of 48Ca by RBHF theory with Bonn A potential, in comparison with ab initio CC calculation G. Hagen, et al., Nat. Phys.12, 186 (2015)., other theoretical analysis (quasielastic charge-exchange reactions P. Danielewicz, P. Singh, and J. Lee, NPA958, 147 (2017), Coulomb energies J. Bonnard, S. Lenzi, and A. Zuker, PRL116, 212501 (2016)), and experimentJ. Birkhan, et al., arXiv:1611.07072, X. Roca-Maza, et al., PRC92, 064304 (2015). • Neutron skin of 48Ca given by RBHF with Bonn A potential agrees with other ab initio calculation and experiment.

  47. RBHF for 40Ca and 48Ca • Single-particle spectrum and localized 1s1/2 single-particle potential of 40Ca (left) and 48Ca (right) in RBHF with Bonn A interaction, in comparison with experimental dataL. Coraggio, et al., PRC68, 034320 (2003), A. Bouyssy, et al. PRC36, 380 (1987). • RBHF theory is very promising to describe the heavy nuclei.

  48. Summary and Perspectives • Summary • Full relativistic Brueckner-Hartree-Fock equations have been solved self-consistently for the first time for finite nuclei. • 16O is taken as an example to show the convergence • 16O, 40Ca and 48Ca have been studied with Bonn A interaction. The resulting binding energies and charge radii are reasonably good. The spin-orbit splittings, and the neutron skin of 48Ca are well reproduced.

  49. Summary and Perspectives • Future Work • To parallelize the RBHF code and calculate on supercomputer for heavy nuclei. • To learn from the G-matrix and provide information to nuclear energy density functional and other studies. • To investigate with relativistic LQCD force or relativistic chiral force Ren et al. arXiv:1611.08475. • To investigate the three-body force in RBHF framework. • To improve the convergence, e.g. by using V-low k or SRG technique. • To investigate the effect of higher orders in the hole-line expansion beyond RBHF. • ... Thank you for your attention!

  50. Ground state properties CDFT, implemented with self-consistency and taking into account various correlations by spontaneously broken symmetries, provide an excellent description for the ground-state properties including total energy and other physical observables as the expectation values of local one-body operators Even for exotic nuclei with extreme neutron or proton numbers, where novel phenomena such as halos may appear. • Meng, Toki, Zhou, Zhang, Long, Geng, Prog. Part. Nucl. Phys. 57 (2006) 470 • Meng and Zhou, J. Phys. G: Nucl. Part. Phys. 42 (2015) 093101

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