in collaboration with J. Dobaczewski, W. Nazarewicz & M. Rafalski
Intro:  define fundaments my model is „standing on”
sp mean-field (or nuclear DFT)  beyond mean-field (projection after variation)
Symmetry (isospin) violation and restoration:
 unphysical symmetry violation  isospin projection
 Coulomb rediagonalization (explicit symmetry violation)
Results
isospin impurities in ground-states of e-e nuclei
structural effects  SD bands in 56Ni
ISB corrections to superallowed beta decay
Summary
ab initio
+ NNN + ....
tens of MeV
Skyrme-force-inspired local energy density functional
(without pairing)
Y | v(1,2) | Y
average Skyrme interaction
(in fact a functional!) over
the Slater determinant
local energy density functional
SV is the only Skyrme interaction
Beiner et al. NPA238, 29 (1975)
Skyrme (nuclear) interaction conserves:
 rotational (spherical) symmetry
LS
LS
LS

Total energy
(a.u.)
Symmetry-conserving
configurtion
Symmetry-breaking
configurations
Deformation (q)
Beyond mean-field  multi-reference density functional theory
rotated Slater
determinants
are equivalent
solutions
Euler angles
gauge angle
in space or/and isospace
where
There are two sources of the isospin symmetry breaking:
-
, caused solely by the HF approximation
, caused mostly by Coulomb interaction
Engelbrecht & Lemmer,
PRL24, (1970) 607
(also, but to much lesser extent, by the strong force isospin non-invariance)
Find self-consistent HF solution (including Coulomb)  deformed
Slater determinant |HF>:
See: Caurier, Poves & Zucker,
PL 96B, (1980) 11; 15
Apply the isospin projector:
aCBR = 1 - |bT=|Tz||2
in order to create good isospin
„basis”:
Diagonalize total Hamiltonian in
„good isospin basis” |a,T,Tz>
 takes physical isospin mixing
AR
n=1 2
aC = 1 - |aT=T
|
z
(I) Isospin impurities in ground states of e-e nuclei
W.Satuła, J.Dobaczewski, W.Nazarewicz, M.Rafalski, PRL103 (2009) 012502
Ca isotopes:
BR
SLy4
AR
0.4
Here the HF is solved
without Coulomb
|HF;eMF=0>.
aC [%]
0.2
Here the HF is solved
with Coulomb
|HF;eMF=e>.
eMF = 0
0
1.0
1
0.1
0.8
0.01
0.6
40
0.4
44
48
52 56
60
eMF = e
0.2
In both cases rediagonalization
is performed for the total
Hamiltonian including
Coulomb
0
40
44
56
48
52
Mass number A
60
E-EHF [MeV]
aC [%]
(II) Isospin mixing & energy in the ground states of
e-e N=Z nuclei:
6
5
4
3
2
1
0
1.0
0.8
0.6
0.4
0.2
0
HF tries to reduce the
isospin mixing by:
DaC ~30%
in order to minimize
the total energy
N=Z nuclei
SLy4
BR
AR
Projection increases the
ground state energy
(the Coulomb and symmetry
energies are repulsive)
Rediagonalization (GCM)
20 28 36 44 52 60 68 76 84 92 100
A
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
Bohr, Damgard & Mottelson
hydrodynamical estimate
DE ~ 169/A1/3 MeV
35
mean
values
30
Sliv & Khartionov PL16 (1965) 176
Dl=0, Dnr=1  DN=2
DE ~ 2hw ~ 82/A1/3 MeV
25
20
SIII
SLy4
SkP
20
based on perturbation theory
40
60
A
80
100
aC [%]
E(T=1)-EHF [MeV]
Excitation energy of the T=1 doorway state in N=Z nuclei
7
6
5
4
SkP SLy5
MSk1
100Sn
SLy SkP
SkM*
SLy4
SkO’ SkXc
SIII
SkO
y = 24.193 – 0.54926x R= 0.91273
31.5 32.0 32.5 33.0 33.5 34.0 34.5
doorway state energy [MeV]
Spontaneous isospin mixing in N=Z nuclei in
other but isoscalar configs
 yet another strong motivation for isospin projection 
Mean-field
four-fold degeneracy of the sp levels
n
n
n p
p
or
p
n p
aligned configuration
n
n
n p
p
or
p
n p
anti-aligned configuration
Isospin projection
T=1
n p
T=0
n p
T=0
1
Isospin symmetry violation in
superdeformed bands in 56Ni
f5/2
p3/2
f7/2
4p-4h
Nilsson
[321]1/2
neutrons protons
[303]7/2
space-spin symmetric
2
g9/2
f5/2
p3/2
f7/2
D. Rudolph et al. PRL82, 3763 (1999)
pp-h
neutrons protons
two isospin asymmetric
degenerate solutions
T=1
dET
Excitation energy [MeV]
pph
20
centroid
aC [%]
nph
dET
T=0
8
6
4
2
band 2
band 1
Isospin-projection
Hartree-Fock
16
56Ni
12
Exp. band 1
Exp. band 2
Th. band 1
Th. band 2
8
4
5
10
15
Angular momentum
5
10
15
Angular momentum
W.Satuła, J.Dobaczewski, W.Nazarewicz, M.Rafalski, PRC81 (2010) 054310
Isospin-projection is non-singular:
W.Satuła, J.Dobaczewski, W.Nazarewicz, M.Rafalski, PRC81 (2010) 054310
SVD eigenvalues
(diagonal matrix)
SVD
singularity
(if any) at b=p
is inherited by r
r~ =S yi* Oij-1 jj
ij
1 + |N-Z| - h + |N-Z|+2k
in the worst case
k is a multiplicity of
zero singular values
h> 3
42Sc
– isospin projection from [K,-K]
configurations with K=1/2,…,7/2
isospin &
angular momentum
40
aC [%]
isospin
30
20
10
0.586(2)%
0
1
3
2K
5
7
Isospin and angular-momentum projected DFT is ill-defined
except for the hamiltonian-driven functionals
|OVERLAP|
1
0.1
only IP
0.01
0.001
IP+AMP
0.0001
0.0 0.5
1.0
1.5
2.0
r =S yi* Oij jj
T
-1
2.5
3.0
bT [rad]
p
ij
HF sp state
inverse of the
overlap matrix
space & isospin rotated
sp state
Tz=-/+1
(N-Z=-/+2)
J=0+,T=1
t+/J=0+,T=1
Tz=0 (N-Z=0)
d5/2
8
8
|<t+/->|2=2(1-dC)
p1/2
p3/2
s1/2
2
p
f  statistical rate function f (Z,Qb)
2
n
p
Hartree-Fock
n
t  partial half-life f (t1/2,BR)
GV vector (Fermi) coupling constant
<t+/-> Fermi (vector) matrix element
10 cases measured with accuracy ft ~0.1%
3 cases measured with accuracy ft ~0.3%
NS-independent
g e
NS-dependent
n
nucleus-independent
g e
~1.5%
0.3% - 2.0%
Towner,
NPA540, 478 (1992)
PLB333, 13 (1994)
~2.4%
Marciano & Sirlin,
PRL96, 032002, (2006)
The 13 precisely known transitions, after including
theoretical corrections, are used to
n
Towner & Hardy
courtesy of J.Hardy
Phys. Rev. C77, 025501 (2008)
With the CVC being verified and knowing Gm (muon decay)
one can determine
weak eigenstates
CKM
Cabibbo-Kobayashi-Maskawa
mass eigenstates
|Vud| = 0.97425 + 0.00023
 test unitarity of the CKM matrix
|Vud|2+|Vus|2+|Vub|2=0.9996(7)
0.9491(4)
0.0504(6)
<0.0001
test of three generation quark Standard Model of
electroweak interactions
Towner & Hardy
Liang & Giai & Meng
Phys. Rev. C79, 064316 (2009)
spherical RPA
Coulomb exchange treated in the
Slater approxiamtion
Phys. Rev. C77, 025501 (2008)
dC=dC2+dC1
mean
field
radial mismatch of
the wave functions
Miller & Schwenk
Phys. Rev. C78 (2008) 035501;C80 (2009) 064319
shell
model
configuration
mixing
J=0+,T=1
Tz=-/+1
(N-Z=-/+2)
t+/-
Isobaric symmetry violation
in o-o N=Z nuclei
J=0+,T=1
n
n
p
p
n
n
CORE
CORE
aligned configurations
n p or n p
n p
T=0
Mean-field can differentiate between
n p and n p
only through time-odd polarizations!
Tz=0 (N-Z=0)
p
p
anti-aligned configurations
n p or n p
n p
T=1
T=0
ground state
is beyond mean-field!
ground state
in N-Z=+/-2 (e-e) nucleus
antialigned state
in N=Z (o-o) nucleus
Project on good isospin
(T=1) and angular
momentum (I=0)
Project on good isospin
(T=1) and angular
momentum (I=0)
rediagonalization)
rediagonalization)
(and perform Coulomb
(and perform Coulomb
<T~1,T
~ z=0>
~ z=+/-1,I=0| t+/- |I=0,T~1,T
H&T  dC=0.330%
L&G&M  dC=0.181%
1.4
1.2
Tz= 1
Tz=0
Ft=3071.4(8)+0.85(85)
Vud=0.97418(26)
dC [%]
1.0
0.8
0.6
Ft=3070.4(9)
Vud=0.97447(23)
0.4
0.2
0
10 14 18 22 26 30 34 38 42 A
|Vud|2+|Vus|2+|Vub|2=
=1.00031(61)
2.0
Tz=0
Tz=1
dC [%]
1.5
1.0
0.5
0
26
34
42
50
58
66
74 A
W.Satuła, J.Dobaczewski,
W.Nazarewicz, M.Rafalski,
PRL106 (2011) 132502
0.976
0.975
H&T’08
|Vud|
0.974
0.973
0.972
0.971
Liang et al.
n-decay
superallowed b-decay
0.970
p+-decay
T=1/2 mirror
b-transitions
Confidence level test based on the CVC hypothesis
Towner & Hardy, PRC82, 065501 (2010)
2.5
(EXP)
dC(SV)
dC [%]
2.0
dC = 1+dNS -
dC
(EXP)
‚
ft(1+dR)
Minimize RMS deviation
between the caluclated
and experimental dC with
respect to Ft
1.5
1.0
0.5
0 0
Ft
5
10 15
c2/nd=5.2
for Ft = 3070.0s
75% contribution to the
2
20 25 30 35 40 c comes from A=62
Z of daughter
Ncutoff=12
Ncutoff=10
2.5
Tz  Tz+1
dC [%]
2.0
A=58
A=38
1.5
1.0
A=18
0.5
0
0
Tz = -1
Tz = 0
10
20
30
40
A
50
60
70
80
Elementary excitations in binary systems may differ
from simple particle-hole (quasi-particle) exciatations
especially when interaction among particles posseses
additional symmetry (like the isospin symmetry in nuclei)
Projection techniques seem to be necessary to account
for those excitations - how to construct non-singular EDFs?
[Isospin projection, unlike the angular-momentum and particle-number
projections, is practically non-singular !!!]
Superallowed 0+0+beta decay:
 encomaps extremely rich physics: CVC, Vud,
unitarity of the CKM matrix, scalar currents…
connecting nuclear and particle physics
 … there is still something to do
in dc business …
Pairing & other (shape vibrations) correlations can be
„realtively easily” incorporated into the scheme by
combining projection(s) with GCM
T=1
n p
T=0 a’sym
SLy4
SLy4L
SkML*
SV
6
a’sym [MeV]
1 a’ T(T+1)
E’sym = 2
sym
4
In infinite nuclear
matter we have:
SLy4:
asym=32.0MeV
SV:
asym=32.8MeV
SkM*:
asym=30.0MeV
2
asym=
0
10
20
30
40
A (N=Z)
50
m
m*
eF + aint
SLy4: 14.4MeV
SV:
1.4MeV
SkM*: 14.4MeV