PKU-CUSTIPEN 2015 Dirac Brueckner Hartree Fock and beyond Herbert Müther Institute of Theoretical Physics

2 | © 2015 Universität Tübingen Realistic NN Interaction: Fit 2N Data Local and Nonlocal NN Interactions

3 © 2015 Universität Tübingen Potential uncorrel wave correl wave r Energy of nuclear matter in HF approx. Correlations: more attractive larger

4 | © 2015 Universität Tübingen Does Vlowk solve our problem?  requires a density dependent CT or 3N force, to obtain saturation  this phenomenological CT dominates No !

Def:particle propagation hole propagation Fourier Transformation: Spectral function S h (k,  ): probability for the removal of particle with momentum k and energy  n(k) = Single-particle Greens function:

infinite matter Solve Dyson equation Calculate Greens function Evaluate self-energy  (k,  ) self-consistency required: one – and two-body Greens functions

Bonn 2005, 7 Examples for Spectral functions S(k,w) QPGF: quasiparticle approx. in evaluating  SCGF: self-consistent Spectral function and momentum distribution: with T. Frick

Explore momentum distribution in (e,e‘)p ? Nukleons with large momenta only at large „missing energies“  <<  F large excitation energy in residual nucleus Theory: LDA, Exp: Rohe, Sick et al. We need: Better description of spectral function at low energies

SCGF and saturation in Nuclear Matter Success: detailed information on energy and momentum distribution of nucleons Symmetry conserving approach (Hughenholtz – van Hove) But: saturation density and binding energy to large Results close to BHF (with rearrangement terms): RBHF

10 | © 2015 Universität Tübingen Renormalized BHF Bethe-Goldstone eq. Total energy Single-part. energy with

11© 2015 Universität Tübingen Dirac Effects and Saturation in Nuclear Matter DBHF: Selfenergy of Nucleons shows strong scalar and vector components Enhancement of small Dirac component in NM Less attractive Meson Exchange Interaction

12 © 2015 Universität Tübingen Three roads to DBHF in Finite Nuclei  Analyze the density- and momentum dependence of the nucleon self-energy in nuclear matter in terms of an effectiv meson exchange..(  )  Effective coupling constants g ..(  )   Use these coupling constants in mean field calculation of Finite Nuclei  Determine the density- momentum- and energy dependence of the relativistic self-energy in nuclear matter (  ), (  ), (  )  Dirac components of self-energy  s (  k),  0 (  k),  i (  k)   Use these Dirac components in a Local Density Approx.  Simulate relativistic effects in terms of 3N interaction  Perform BHF calculation of nuclear matter and fit strength of 3N interaction   Perform BHF calculation of Finite Nuclei with 3N force Explicit treatment of Dirac effects Explicit treatment of Dirac effects Explicit treatment of Correlations Explicit treatment of Correlations

13 © 2015 Universität Tübingen Effective Field Theory: Determine Parameter in NM with Eric v.Dalen

14 © 2015 Universität Tübingen Effective Field Theory: Saturation in Nuclei  Bulk properties of Nuclei can be described in a very reasonable way  Results are Model-Dependent  Bulk properties of Nuclei can be described in a very reasonable way  Results are Model-Dependent

Fingerprints for relativistic effects in nuclei Dirac equation: 2 coupled diff. equations for f(r) and g(r),the small and large components Schrödinger kind eq: 1 diff. equation of second order for  (r) with Schrödinger equivalen potential U contains energy dependent central part spin orbit term

16 © 2015 Universität Tübingen Effective Field Theory: Spin Orbit Example around 16 O Note: l s splitting larger for l =1 than l =2 !! Relativistic effect does not enhance the spin orbit at the surface, i.e. small densities

17 © 2015 Universität Tübingen Effective Field Theory: Optical Model with Ruirui Xu & Zhongyu Ma with Ruirui Xu & Zhongyu Ma Example: Elastic Scattering n 27 Al

18 © 2015 Universität Tübingen 3N Force in Nuclear Matter Almost standard BHF withCDBONN, but:  no angle-average Pauli Operator  details of single-particle spectrum  effects of rearrangement terms Almost standard BHF withCDBONN, but:  no angle-average Pauli Operator  details of single-particle spectrum  effects of rearrangement terms Simulate relativistic effects in terms of 3N force:  simple local form (part of Urbana 3N)  adjust 1 parameter Simulate relativistic effects in terms of 3N force:  simple local form (part of Urbana 3N)  adjust 1 parameter with A.H. Lippok

19 © 2015 Universität Tübingen 3N Force in Finite Nuclei: 16 O Problem: Choice of single- particle spectrum in BG eq. Here: With Pauli Operator Q and C adjusted to describe calculated spectrum Optimal C ??? Problem: Choice of single- particle spectrum in BG eq. Here: With Pauli Operator Q and C adjusted to describe calculated spectrum Optimal C ??? Conclusion: We are able to describe bulk properties of Nuclei (Energy and Radius), but  Choice of H 0 ?  3N has no effect on spin-orbit splitting Conclusion: We are able to describe bulk properties of Nuclei (Energy and Radius), but  Choice of H 0 ?  3N has no effect on spin-orbit splitting

20 | © 2015 Universität Tübingen Conclusions  We are able to describe bulk properties of nuclear systems in terms of realistic NN interactions, including correlation and Dirac effects  Open problems in description of finite nuclei:  consistent single-particle spectrum  explicit treatment of relativistic and correlation effects  Effects on bulk can be simulated in terms of many-nucleon forces  Explore characteric features beyond bulk properties as  spectral function, nucleon knock-out  optical model potential  spin-orbit splitting

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