in collaboration with J. Dobaczewski, W. Nazarewicz, M. Rafalski & M. Borucki
Intro: effective low-energy theory for medium mass and heavy nuclei 
mean-field (or nuclear DFT)  beyond mean-field (projection)
Symmetry (isospin) violation and restoration:
 unphysical symmetry violation  isospin projection
 Coulomb rediagonalization (explicit symmetry violation)
isospin impurities in ground-states of e-e nuclei
structural effects  SD bands in 56Ni
superallowed beta decay
symmetry energy – new opportunities of studying with the isospin projection
Summary
ab initio
+ NNN + ....
tens of MeV
Effective theories for low-energy (low-resolution)
nuclear physics (I):
Low-resolution  separation of scales which is
a cornerstone of all effective theories
The nuclear effective theory
is based on a simple and very intuitive assumption that low-energy
nuclear theory is independent on high-energy dynamics
hierarchy of scales:
Long-range part of the NN interaction
(must be treated exactly!!!)
where
local
correcting
potential
2roA1/3
~ 2A1/3
ro
~ 10
denotes an arbitrary Dirac-delta model
przykład
Gogny interaction
Fourier
Coulomb
regularization
ultraviolet
cut-off
There exist an „infinite” number
of equivalent realizations
of effective theories
Skyrme interaction - specific (local) realization of the
lim da
nuclear effective interaction:
a
0
LO
NLO
10(11)
parameters
density dependence
spin-orbit
relative momenta
spin exchange
Skyrme-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
Skyrme (nuclear) interaction conserves such symmetries like:
 rotational (spherical) symmetry
LS
LS
LS

 parity…
Total energy
(a.u.)
Symmetry-conserving
configuration
Symmetry-breaking
configurations
Elongation (q)
Beyond mean-field  multi-reference density functional theory
Euler angles
rotated Slater
determinants
are equivalent
solutions
gauge angle
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:
in order to create good isospin
„basis”:
Calculate the projected energy and
the Coulomb mixing
:
BR
BR
aC = 1 - |bT=|Tz||2
Diagonalize total Hamiltonian in
„good isospin basis” |a,T,Tz>
 takes physical isospin mixing
AR
n=1 2
aC = 1 - |aT=T
|
z
Isospin breaking: isoscalar, isovector & isotensor
Isospin invariant
(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
29.5
30.5
SIII
SkXc
31.0 31.5
ET=1 [MeV]
(AR)
SIII
SkM*
SLy7
SkXc
SkM*
SkP
SLy5
SLy4
SkO’
30.0
SLy7
3.0
aC
a(BR)
C
SkP
SLy5
SLy4
3.5
(AR)
SkO’
4.0
MSk1
aC [%]
4.5
MSk1
80Zr
doorway state energy
communicated by
Franco Camera
at the Zakopane’10
meeting
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
Tz=-/+1
(N-Z=-/+2)
J=0+,T=1
t+/J=0+,T=1
Tz=0 (N-Z=0)
d5/2
8
8
2
2
p1/2
p3/2
s1/2
p
n
n
p
f  statistical rate function f (Z,Qb)
t  partial half-life f (t1/2,BR)
GV vector (Fermi) coupling constant
Hartree-Fock
<t+/-> Fermi (vector) matrix element
|<t+/->|2=2(1-dC)
PRC77, 025501 (2008)
10 cases measured with accuracy ft ~0.1%
3 cases measured with accuracy ft ~0.3%
nucleus-independent
~1.5%
0.3% - 1.5%
~2.4%
Marciano & Sirlin, PRL96 032002 (2006)
From a single transiton we can determine experimentally:
GV2(1+DR)  GV=const.
From many transitions we can:
 test of the CVC hypothesis
(Conserved Vector Current)
 exotic decays
Test for presence of a Scalar Current

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
Hardy &Towner
Liang & Giai & Meng
Phys. Rev. C79, 064316 (2009)
spherical RPA
Coulomb exchange treated in the
Slater approxiamtion
Phys. Rev. C77, 025501 (2008)
dC=dC1+dC2
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!
isospin &
angular momentum
isospin
30
20
(
(
10
0
(
1
0.586(2)%
(
aC [%]
40
3
2K
5
7
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%
1.2
1.0 Tz=-1  Tz=0
Ft=3071.4(8)+0.85(85)
Vud=0,97418(26)
dC [%]
0.8
0.6
Ft=3069.2(8)
Vud=0,97466
0.4
0.2
0
2.0
15
20
25
30
35
40 A
Tz=0  Tz=1
dC [%]
1.5
10
1.0
0.5
0
30
40
50
60
70
A
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
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?
Isopin projection, unlike angular-momentum and
particle-number projections, is practically non-singular !!!
Superallowed beta decay:
 encomapsses 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 …
|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
-1
2.5
3.0
bT [rad]
p
ij
HF sp state
inverse of the
overlap matrix
space & isospin rotated
sp state
Qb values in super-allowed transitions
time-odd
time-even
th
exp
Qb – Qb [MeV]
4
3
0,2%
0,9%
1
2,5%
1,5%
4,1%
10,1%
-1
20 30 40 50
Atomic number
29,9%
26,3%
21,7%
7,9%
Hartree-Fock
Hartree-Fock
10
15,1%
0,9%
3,7%
0,8%
0
0
isospin projected
isospin projected
2
60 0
10
20 30 40 50
Atomic number
60
Isospin symmetry violation due to time-odd fields in the intrinsic system
time-odd
Isobaric analogue
states:
T=1,Tz=1
T=1,Tz=-1
T=1,Tz=0
e-e
o-o
e-e
aC [%]
1.1
1.0
0.9
0.8
0.7
40Ca
SkO’
MSk1
SkXc
SkO
18
aC [%]
SkM*
1.1 40Ca
1.0
0.9
0.8
0.7SIII
SkP
SLy
SLy
SIII
7
100Sn
6
SkO’
5
18
19 20 21 22
asym(rNM/2) [MeV]
SkP MSk1
SkM*
SkXc
SLy
SLy5
SkO’
SkO
29
30
31
32
asym(rNM) [MeV]
100Sn
6 SIII
5
4
SkP
MSk1
SkXc
SLy
SIII
SkO
4
7
SkM*
19 20 21 22
asym(rNM/2) [MeV]
SkP
MSk1
SkM*
SkXc
SLy SLy5
SkO’
SkO
29
30
31
32
asym(rNM) [MeV]
0.35
Towner & Hardy 2008
dC [%]
0.30
0.25
0.20
Liang et al. (NL3)
6
8
10
Number of shells
12
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
n
p
p
n
n
CORE
CORE
aligned configurations
n p or n p
p
p
anti-aligned configurations
n p or n p
After applying „naive” isospin projection we get:
T=1
n p
T=0
Mean-field can differentiate between
n p and n p
only through time-odd polarizations!
n p
T=0
ground state
is beyond mean-field!
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]
Position 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]