1. Kilka słów o strukturze jądra atomowego
dla matematyków
Wojciech Satuła
From ab initio toward „rigorous” effective theory for light nuclei
Principles of low-energy nuclear physics  effective theories for
medium-mass and heavy systems
Nuclear DFT  coupling constants & fitting strategies
 single-particle fingerprints of tensor interaction
Extensions of the nuclear DF up to N3LO
Multi-reference DFT  beyond mean-field theories
 isospin (and angular momentum) projection
Summary
ab initio
+ NNN
tens of MeV

2. Paramters (~40) are fitted to two-nucleon data
Argonne V18 NN potential
short range is phenomenological
long range – pion exchange
Paramters (~40) are fitted to two-nucleon data
Triton binding: th: 7.62MeV exp: 8.48MeV

3. A=4-10 GFMC calculations using Argonne V18 NN potential
and Illinois-2 NNN interaction
Pion three-body sector of Urbana 3-body potential
plus phenomenological short-range 3-body

4. From „infinite basis” ab initio towards finite basis
„rigorous” effective theory
Strategy: adopt a Hamiltonian and a basis, compute matrix elements and diagonalize N =0 =1 =2 =4 =3 =5
In ab initio many-body
theory H acts in „infinite”
Hilbert space
Select HO Slater determinat
basis and retain all A-body
determinants below an oscillator
Ecutoff (Nmax)
finite subspace
(P-space)
Heff
„P – space”

5. H : E , K Expansion in (Q/L)n VNN Repulsive core in VNN
Realistic local NN interaction H : E 1 , 2 3 K d
eff P
model space
dimension
Properties of Heff for A-nucleon system
A-body operator even if H is 2- or 3-body
For P  1 Heff  H
Expansion in (Q/L)n
VNN
Repulsive core in VNN
cannot be accommodated
in this truncated HO basis
needs regularization
Vlow-k, PT
and further renormalization
to the finite basis
(Lee-Suzuki-Okamoto)

6. Modern Mean-Field Theory º Energy Density Functional
Effective theories for low-energy nuclear physics in
heavy(ier) nuclei:
Hohenberg-Kohn-Sham
Modern Mean-Field Theory º Energy Density Functional j,
r, t, J, T, s, F,

7. The nuclear effective theory
is based on a simple and very intuitive assumption that low-energy
nuclear theory is independent on high-energy dynamics
ultraviolet
cut-off
regularization
Fourier
Coulomb
Long-range part of the NN interaction
(must be treated exactly!!!)
hierarchy of scales:
2roA1/3 ~
2A1/3 ro
correcting
potential
local ~ 10
There exist an „infinite” number
of equivalent realizations
of effective theories
where
denotes an arbitrary Dirac-delta model
przykład
Gogny interaction

8. Spin-force inspired local energy density functional
Skyrme interaction - specific (local) realization of the
nuclear effective interaction:
lim da
a 0 LO
NLO
10(11)
density dependence
parameters
spin-orbit
relative momenta
spin exchange
Spin-force inspired local energy density functional
Y | v(1,2) | Y
Slater determinant
(s.p. HF states are equivalent to the Kohn-Sham states)
local energy density functional

9. ZOO– 20 parameters are fitted to: Skyrme-inspired functional
is a second order expansion
in densities and
currents:
tensor
spin-orbit
density rg
dependent CC
20 parameters are fitted to:
Symmetric NM:
- saturation density ( ~0.16fm-3)
- energy per nucleon ( MeV)
- incompresibility modulus ( MeV) +
- isoscalar effective mass (0.8)
Asymmetric NM:
isovector effective mass
(GDR sum-rule enhancement)
- symmetry energy ( 30 2MeV) +
neutron-matter EOS
(Wiringa, Friedmann-Pandharipande)
Finite, double-magic nuclei
[masses, radii, rarely sp levels]:
surface properties
ZOO–

10. How many parameters are really needed?
How many parameters can be constrained by fitting global properties?
SLy4
(original)
Can we learn more from the sp
properties?
linear (re)fit to masses
Bertsch, Sabbey, and Uusnakki
Phys. Rev. C71, (2005)
De(f5/2-f7/2) [MeV] 5 6 7 8
40Ca
48Ca
56Ni a) b)
neutrons
protons
SkO av
as + asym + s-o

11. Fitting strategies of the tensorial coupling constants
- the idea - C1 J C0 1 3 5 7
0.7
0.8
0.9
40Ca
-40
-30
-20
-10
56Ni
f7/2-f5/2
p3/2-p1/2
f7/2-d3/2 2 4 6 8
-80
-60
from binding energies
48Ca
f7/2-p3/2
Single-particle energies [MeV]
1) Fit of the isoscalar SO strength
j<
j> F
40Ca
2) Fit of the isoscalar tensor strength:
j> F
j<
56Ni
3) Fit of the isovector tensor strength
or, more precisely, C1J/C1 D J
48Ca
48Ni or 78Ni are needed
in order to fix SO-tensor sector
f7/2
f5/2
splittings around
SkPT T0=-39(*5);T1=-62(*-1.5);SO*0.8

12. SLy4T Single-particle fingerprints of tensor interaction
spin-orbit splittings
Spin-orbit splittings [MeV]
SLy4T T0=-45;T1=-60; SO*0.65 n 1h 1i
f7/2-f5/2
g9/2-g7/2 1 3 5 7
16O
40Ca
48Ca
56Ni
90Zr
132Sn
208Pb p
SLy4T
M.Zalewski, J.Dobaczewski, WS, T.Werner, PRC77, (2008)

13. f7/2 f5/2 p3/2 4p-4h Nilsson DE [MeV] SkO SkOTX SkOT’ DEtensor [MeV]
neutrons
protons
4p-4h
[303]7/2
[321]1/2
Nilsson
Rudolph et al. PRL82, 3763 (1999) 2 4 6 8 10 12
0.1
0.2
0.3
0.4
DE [MeV]
tensor
spin-orbit
deformacja b2
SkO
SkOTX
SkOT’ -6 -5 -4 -3
DEtensor [MeV]
0.1
0.2
0.3
0.4 b2

14. Singular value decomposition
Fits of s.p. energies
EXP: M.N. Schwierz, I. Wiedenhover,
and A. Volya, arXiv:
Singular value decomposition
M. Kortelainen et al., Phys. Rev. C77, (2008) =

15. Possible extensions: N2LO, N3LO – higher order derivatives
explicit reconstruction of the NDF
N2LO, N3LO – higher order derivatives
Mass dependent coupling constants
New higher-order physics-motivated terms: ~t( r)2 D

16. Beyond mean-field Total energy (a.u.) Elongation (q)
multi-reference density functional theory
Spontaneous Symmetry Breaking (SSB)
Elongation (q)
Total energy (a.u.)
Symmetry-conserving
configuration
Symmetry-breaking
configurations

17. Restoration of broken symmetry
Euler angles
gauge angle
rotated Slater
determinants
are equivalent
solutions
where

18. Determination of Vud matrix element of the CKM matrix
Motivation:
Determination of Vud matrix element of the CKM matrix
from superallowed beta decay
 test of unitarity  test of three generation quark
Standard Model of electroweak interactions
J=0+,T=1
N-Z=-2
N-Z=0 T+
Tz=-1
vector (Fermi) cc
Tz=0
d5/2
nucleus-independent 8 8
p1/2
p3/2 2 2
s1/2 p n p n

19. Isospin symmetry breaking and restoration – general principles
There are two sources of the isospin symmetry breaking:
unphysical, related to the HF approximation itself
physical, caused mostly by Coulomb interaction
(also, but to much lesser extent, by the strong force isospin non-invariance)
Engelbrecht & Lemmer,
PRL24, (1970) 607
Solve SHF (including Coulomb) to get isospin symmetry broken
(deformed) solution |HF>:
See: Caurier, Poves & Zucker,
PL 96B, (1980) 11; 15
in order to create good-isospin „basis”:
Apply the isospin projection operator: BR
Compute projected (PAV) energy
and Coulomb mixing before
rediagonalization:
aC = 1 - |bT=|Tz||2 BR

20. aC = 1 - |aT=Tz|2 Rediagonalize the Hamiltonian in
the good-isospin „basis” |a,T,Tz>
in order to remove spurious
isospin-mixing:
aC = 1 - |aT=Tz|2 AR
n=1
Isospin breaking: isoscalar, isovector & isotensor
Isospin invariant

21. Few numerical results:
Isospin mixing in ground states of e-e nuclei
0.2
0.4
0.6
0.8
1.0
aC [%] 40 44 48 52 56 60
Mass number A
0.01
0.1 1 BR AR
SLy4
Ca isotopes:
eMF = 0
eMF = e
Here the HF is solved without
Coulomb |HF;eMF=0>.
Here the HF is solved with
Coulomb |HF;eMF=e>.
In both cases rediagonalization
is performed for the total
Hamiltonian including
Coulomb

22. aC [%] (II) Isospin mixing & energy in the ground states of
e-e N=Z nuclei:
HF tries to reduce the
isospin mixing by: 1 2 3 4 5 6
0.2
0.4
0.6
0.8
1.0 20 28 36 44 52 60 68 76 84 92 A AR BR
SLy4
aC [%]
E-EHF [MeV]
N=Z nuclei
100
~30%
DaC
in order to minimize
the total energy
Projection increases the
ground state energy
(the Coulomb and the symmetry
energies are repulsive)
Rediagonalization (GCM)
lowers the ground state
energy but only slightly
below the HF
This is not a single Slater determinat
There are no constraints on mixing coefficients

23. Position of the T=1 doorway state in N=Z nuclei
Bohr, Damgard & Mottelson
hydrodynamical estimate
DE ~ 169/A1/3 MeV 20 25 30 35
mean
values
E(T=1)-EHF [MeV]
Sliv & Khartionov PL16 (1965) 176
Dl=0, Dnr=1  DN=2
DE ~ 2hw ~ 82/A1/3 MeV
SIII
SLy4
SkP
based on perturbation theory
31.5
32.0
32.5
33.0
33.5
34.0
34.5
y = – x R=
doorway state energy [MeV] 4 5 6 7
aC [%]
100Sn
SkO
SIII
MSk1
SkP
SLy5
SLy4
SkO’
SLy
SkM*
SkXc 20 40 60 80
100 A

24. Isobaric symmetry breaking in odd-odd N=Z nuclei
Let’s consider N=Z o-o nucleus disregarding, for a sake of simplicity,
time-odd polarization and Coulomb (isospin breaking) effects
4-fold
degeneracy n p n p
CORE
CORE
aligned configurations
anti-aligned configurations n p n p n p or or n p
T=0
After applying „naive” isospin projection we get:
T=1
ground state
is beyond mean-field! n p
Mean-field can differentiate between
and
only through time-odd polarizations!

25. no time-odd polarizations included
-66
-65
-64
-63
-62
-61
-60
E [MeV]
Hartree-Fock
10C
10B
Isospin projection &
Coulomb rediagonalization
4275
2098
T=0
T=1
exp: 1908
Isospin projection &
Coulomb rediagonalization
T=1
exp: 6424
T=0
42Sc
42Ca
7784
907
Hartree-Fock
42Ca
42Sc
-360
-358
-356
-354
E [MeV]

26. Qb values in super-allowed transitions
time-even
time-odd 4
Qb – Qb [MeV] th
exp 3
isospin projected
isospin projected 2
0,2%
4,1%
0,9%
2,5%
29,9%
10,1% 1
15,1%
21,7%
1,5%
3,7%
0,9%
26,3%
0,8%
7,9%
Hartree-Fock -1
Hartree-Fock 10 20 30 40 50 60 10 20 30 40 50 60
Atomic number
Atomic number
time-odd
T=1,Tz=-1
T=1,Tz=0
T=1,Tz=1
e-e
o-o
Isospin symmetry violation due to time-odd fields in the intrinsic system
Isobaric analogue
states:

27. AMP+IP projection from the „anti-aligned” Slater determinant
(very preliminary tests – no Coulomb rediagonalization)
10C
very preliminary
(qualitative)
-65
-64
-63
-62
-61
-60
-59
J=0+
J=0+,T=1
J=0+,T=1
J=0+
Energy [MeV]
very preliminary
(qualitative)
10B
J=1+,T=0
J=1+
J=3+,T=0
Hartree
Fock
AMP
+IP
AMP

28. |OVERLAP| p r =S yi* Oij jj only AMP IP+AMP bT [rad] 0.0001 0.001 0.01
0.1 1
0.0
0.5
1.0
1.5
2.0
2.5
3.0
|OVERLAP|
bT [rad]
only AMP
IP+AMP p
r =S yi* Oij jj ij -1
original sp state
space-isospin rotated
sp state
inverse of the
overlap matrix

29. Isospin symmetry violation in superdeformed bands in 56Ni1
Isospin symmetry violation in
superdeformed bands in 56Ni
4p-4h
f7/2
f5/2
p3/2
neutrons
protons
[303]7/2
[321]1/2
Nilsson
space-spin symmetric 2
f7/2
f5/2
p3/2
neutrons
protons
g9/2
pp-h
two isospin asymmetric
degenerate solutions
D. Rudolph et al. PRL82, 3763 (1999)

30. Mean-field versus isospin-projected mean-field
interpretation
pph
nph
T=0
T=1
centroid
dET 2 4 6 8
band 2
aC [%]
band 1
Hartree-Fock
Isospin-projection 4 8 12 16 20
56Ni
Excitation energy [MeV]
Exp. band 1
Exp. band 2
Th. band 1
Th. band 2 5 10 15 5 10 15
Angular momentum
Angular momentum

31. SUMMARY
(verbal)

32. Local Density Functional Theory for Superfluid Fermionic Systems:
The Unitary Gas
Aurel Bulgac, Phys. Rev. A 76, (2007)
running coupling constant
in order to renormalize....
ultraviolet
divergence in
pairing tensor
ab initio
calculations by:
Chang & Bertsch
Phys. Rev. A76,
von Stecher, Greene & Blume,
E-print: v1

33. From two-body, zero-range tensor interaction towards the EDF:
mean-field
averaging