1. Gogny’s pairing forces in covariant density functional theory
ISTANBUL-06
Bruyères-le-Châtel, Dec. 10, 2015
Technical University Munich
Excellence Cluster “Universe”
Peking University
Peter Ring
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

2. Dogmata in nuclear physics:
A quantum field theory has to be renormalizable
Forces have to be „phase-shift equivalent“
Nuclei are non-relativistic systems
DFT has to be derived from a Hamiltonian
DFT has to be local …
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

3. Dogmata in nuclear physics:
A quantum field theory has to be renormalizable
Forces have to be „phase-shift equivalent“
Nuclei are non-relativistic systems
DFT has to be derived from a Hamiltonian
DFT has to be local …
16:58
First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

4. Dogmata in nuclear physics:
A quantum field theory has to be renormalizable
Forces have to be „phase-shift equivalent“
Nuclei are non-relativistic systems
DFT has to be derived from a Hamiltonian
DFT has to be local …
Egido-poles !
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

5. Dogmata in nuclear physics:
A quantum field theory has to be renormalizable
Forces have to be „phase-shift equivalent“
Nuclei are relativistic systems
DFT has to be derived from a Hamiltonian
DFT has to be local …
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

6. First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel
My conclusions:
Nuclei are relativistic systems (Covariant DFT)
The concept of one Hamiltonian
for mean field and pairing does not work in relativistic systems
Pairing is a non-relativistic property
We use Gogny-pairing together with a relativistic DFT
There is a separable pairing force equivalent to Gogny
Finite range is very important for pairing
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

7. Density functional theory in nuclei
● The nuclear chart is two-dimensional: isospin (p/n)
● Magic numbers need large spin-orbit: spin (↑↓)
● Most of the nuclei have open shells: pairing (u,v)
● Nuclei are relativistic systems: components (f,g)
We deal with spinors of dimension 16
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

8. First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel
Covariant DFT is based on the Walecka model
Dürr and Teller, Phys.Rev 101 (1956)
Walecka, Phys.Rev. C83 (1974)
Boguta and Bodmer, Nucl.Phys. A292 (1977)
The nuclear fields are obtained by coupling
the nucleons through the exchange of effective mesons
through an effective Lagrangian.
(J,T)=(0+,0)
(J,T)=(1-,0)
(J,T)=(1-,1)
sigma-meson:
attractive scalar field
omega-meson:
short-range repulsive
rho-meson:
isovector field
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

9. First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel
Covariant DFT is based on the Walecka model
Dürr and Teller, Phys.Rev 101 (1956)
Walecka, Phys.Rev. C83 (1974)
Boguta and Bodmer, Nucl.Phys. A292 (1977)
This model has only 4 parameters
(J,T)=(0+,0)
(J,T)=(1-,0)
(J,T)=(1-,1) gσ mσ gρ gω
We need in addition density dependence: gm(ρ)
and pairing
We do not need: t3 with strong repulsion
spin-orbit
time-odd terms
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

10. Relativistic Kohn-Sham equations:
S(r) ≈ -400 MeV
V(r) ≈ 350 MeV
scalar potential:
vector potential:
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

11. First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel
scalar potential:
S(r) ≈ -400 MeV
V(r) ≈ 350 MeV
vector potential:
Fermi sea
Dirac sea
2m* ≈ 1100 MeV
V-S ≈ 750 MeV
V+S ≈ 50 MeV
2m ≈ 1900 MeV
continuum
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

12. First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel
Why covariant ?
In medium QCD-sum rules
relate the scalar condensate and the quark density
to the scalar and vector self-energies of the nucleon in the medium:
1) In QCD we have very large scalar and vector fields
Cohen, Furnstahl, Griegel, PRC 67 (1992) 961
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

13. First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel
Why covariant ?
2) Similar fields in the Walecka model:
Walecka, Ann.Phys. (1974)
The EoS in the σω-model depends on
two parameters Gσ and Gω. They are
determined by the density ρ0 and
the binding energy E/A at saturation
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

14. First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel
Why covariant ?
2) Similar fields in the Walecka model: S = -400 MeV , V = +350 MeV
Walecka, Ann.Phys. (1974)
The EoS in the σω-model depends on
two parameters Gσ and Gω. They are
determined by the density ρ0 and
the binding energy E/A at saturation
16:58
First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

15. First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel
Why covariant ?
2) Similar fields in the Walecka model: S = -400 MeV , V = +350 MeV
Walecka, Ann.Phys. (1974)
The EoS in the σω-model depends on
two parameters Gσ and Gω. They are
determined by the density ρ0 and
the binding energy E/A at saturation
This gives the proper spin-orbit splitting in finite nuclei
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

16. First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel
Why covariant ?
Large fields V ≈ 350 MeV, S ≈ – 400 MeV
Large spin-orbit splittings in nuclei (a factor -40)
Success of relativistic Brueckner calculations
Success of intermediate energy proton scattering
Relativistic saturation mechanism
Consistent treatment of time-odd fields
Natural explanation of pseudospin symmetry s
Connection to underlying theories ?
Use as many symmetries as possible in phenomenology
Ch. Fuchs
No three-body forces
Coester-line
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

17. First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel
Why covariant ?
Large fields V ≈ 350 MeV, S ≈ – 400 MeV
Large spin-orbit splittings in nuclei
Success of relativistic Brueckner calculations
A relativistic saturation mechanism:
No parameter t3
The σ-field is the origin of attraction. Its source is the scalar density:
In the non-rel. case, Hartree with Yukawa forces would lead to collapse
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

18. First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel
Why covariant ?
Large fields V ≈ 350 MeV, S ≈ – 400 MeV
Large spin-orbit splittings in nuclei
Success of relativistic Brueckner calculations
A relativistic saturation mechanism:
Consistent description of time-odd fields: + -
nuclear magnetism
scalar potential
vector potential (time-like)
vector potential (space-like)
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

19. Moments of inertia in rotating nuclei:
Skyrme
RMF NL1
Dobaczewski, Dudek, PRC (1995)
Afanasjev, P.R. PRC (1996)
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

20. First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel
Why covariant ?
1) Relativistic kinematic is not important in nucl. structure
2) Large fields V ≈ 350 MeV , S ≈ – 400 MeV
3) Large spin-orbit splittings in nuclei
4) Success of relativistic Brueckner calculations
5) Relativistic saturation mechanism
6) Consistent treatment of time-odd fields
7) Natural explanation of pseudospin symmetry
9) Success of intermediate energy proton scattering
10) Use as many symmetries as possible in phenomenology
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

21. Pairing in a relativistic quantum field theory:
H. Kucharek et al, ZPA 1991
Leads to
Relativistic Hartree-(Fock)-Bogoliubov Theory
normal density anomaleous density (pairing tensor)
where:
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

22. First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel
neglect retardation
π,δ,η = +
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

23. Pairing in nuclear matter:
RMF+BCS
Gap equation:
e.g. ω-meson:
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

24. The pairing gap at the Fermi surface
maximal pairing at the Fermi surface:
potential ΔF(MeV) kF(fm-1)
Bonn A
Bonn B
Bonn C
Gogny D1S
free NN-forces, which reproduce the phase shift in the
1S0 channel give pairing similar to the Gogny force
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

25. Contributions of the various mesons in the Bonn-potential to pairing:
M. Serra, A. Rummel, P. R., PRC 65 (2002)
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

26. First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel
σ-ω model
All relativistic forces, overestimate nuclear pairing by a factor 3,
because of the very large cut off in momentum space
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

27. Pairing matrix elements in the σ-ω ´model
vpp(k,p=0.8)
mσ = 520 MeV
mσ = 390 MeV
k[fm-1]
H. Kucharek, P. R., Z. Phys. A 1991
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

28. Relativistic Hartree-Bogoliubov
therefore we neglect
total
scalar
vector time-like
vector spacelike
M. Serra, P. R., PRC 65 (2002)
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

29. Hybride model: RMF + Gogny pairing
Gonzales-Llarena, Egido et al,
PLB 379, 13 (1996)
pairing in
superdeformed
bands:
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

30. Separable ansatz fitted to Gogny force
perfect agreement for nuclear matter
P(k) Gaussian form
P(k) obtained by mapping
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

31. In finite nuclei: comparison with Gogny: 16:58
First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

32. dynamic results for the new force:
comparison with Gogny:
comparison with experiment:
Y. Tian, Z.Y. Ma, P.R. (2007)
A. Ansari PLB 623, 37 (2005)
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

33. Comparison between Gogny- and δ-pairing:
244Pb
Tian, Ma, P.R. PLB 676, 44 (2009)
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

34. Influence of pairing on fission barriers:
240Pu
Shell effects are washed out
with increasing pairing
Lalazissis et al PLB 689, 72 (2010)
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

35. Influence of the pairing window on fission barriers:
Barrier height as a function of the ground state gap
For Gogny pairing and for δ-pairing with various
Pairing windows.
Lalazissis et al PLB 689, 72 (2010)
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

36. Influence of the pairing window on fission barriers:
For each pairing window the strength is adjusted
such that the ground state gap ∆gs stays the same
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

37. First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel
Summary and outlook:
Nuclei are relativistic systems: Covariant DFT
No Hamiltonian common for pairing and bulk properties
Pairing is a non-relativistic phenomenon
RHB with Gogny-pairing is very successful
A simple separable form for Gogny-pairing
Applications for fission barriers:
strong dependence on pairing correlations
strong dependence on the pairing window for δ-pairing
Finite range pairing increases predictive power
Next problem:
implement separable pairing into Skyrme calculations
finite range pairing + zero range mean field
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

38. First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel
M. Serra† (TUM, Munich, Tokyo Univ.)
H. Kucharak (TUM, Munich)
J. Koenig (TUM, Munich)
J. L. Egido (UAM, Madrid)
L. Robledo (UAM, Madrid)
A.V. Afanasjev (Mississippi State)
Y. Tian (CIAE, Beijing)
Z.Y. Ma (CIAE, Beijing)
S. Karatzikos (Thessaloniki)
G.A. Lalazissis (Thessaloniki)
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

39. large gap caused by the repulsive part of the force:
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

40. First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

41. Contributions of the various mesons in the Bonn-potential to pairing:
M. Serra, A. Rummel, P. R., PRC 65 (2002)
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

42. Wave functions of the Cooper pair in r-space:
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

43. Wave functions of the Cooper pair in momentum space:
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

44. Influence of the repulsive core in Bonn-pot.:
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

45. Coherence length: M. Serra, A. Rummel, P. R., PRC 65 (2002) 014304
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

46. First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel
Density functional:
Density functional
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

47. First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel
normal deformed bands in the rare earth region
A.V. Afanasjev et al., PRC 62, (2000)
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

48. Relativistic Kohn-Sham equations:
V0(r)
Vi (r)
time-like
space-like
S(r)
scalar potential:
vector potential:
The space-like parts V break time reversal symmetry
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

49. First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel
Why covariant ?
2) Similar fields in the Walecka model: S = -400 MeV , V = +350 MeV
The EoS in the σω-model depends on
two parameters Gσ and Gω. They are
determined by the density ρ0 and
the binding energy E/A at saturation
This gives the proper spin-orbit splitting
σω - model
J.D. Walecka, Ann.Phys. (NY) 83, (1974) 491
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

50. Pairing in a relativistic quantum field theory:
Lagrangian with fermions and mesons :
quantization leads to a Hamiltonian
2 Greens functions:
2 equations of motion:
elimination of meson fields:
relativistic 2-body interaction:
neglect retardation
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

51. Relativistic Hartree-Bogoliubov (RHB) theory:
Gorkov factorization:
direct term exchange term pairing term
Relativistic Hartree-Bogoliubov equations:
quasiparticle energy
pairing field
Dirac hamiltonian
quasiparticle
wave function
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel
H. Kucharek et al, ZPA 1991

52. First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

53. First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel
effective pairing forces:
nucl. matter:
seniority force, constant G
zero range; δ-force
pairing part of Gogny D1S
Gonzales-Llarena et al, PLB 379, 13 (1996)
Gogny equivalent separable force:
Tian, Ma, P.R. PLB 676, 44 (2009)
finite nuclei:
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

54. Simplified pairing energy functional:
Y. Tian, Z.Y. Ma, P.R.
Gogny:
separable:
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

55. First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel
Content
Covariant density functional theory
Pairing in a relativistic quantum field theory
BCS-theory with meson-exchange forces
Rel. Hartree-Bogoliubov (RHB) with Gogny pairing
Applications with zero- and finite-range pairing forces
A separable version of Gogny‘s pairing force
Fission barriers with zero and finite range pairing
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel

56. Density functional Theory (DFT)
Motivation:
proton number Z
Density functional Theory (DFT)
Shell model
Coupled cluster
Ab initio
neutron number N
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First Gogny Conference, Dec. 8-11, 2015, Bruyéres le Châtel