+ NNN +.... tens of MeV ab initio Intro: effective low-energy
theory for medium mass and heavy nuclei mean-field ( or nuclear DFT
) beyond mean-field ( projection ) Summary Symmetry (isospin)
violation and restoration: unphysical symmetry violation isospin
projection Coulomb rediagonalization (explicit symmetry violation)
in collaboration with J. Dobaczewski, W. Nazarewicz, M. Rafalski
& M. Borucki structural effects SD bands in 56 Ni superallowed
beta decay isospin impurities in ground-states of e-e nuclei
symmetry energy new opportunities of studying with the isospin
projection
Slide 2
Effective theories for low-energy (low-resolution) nuclear
physics (I): Low-resolution separation of scales which is a
cornerstone of all effective theories
Slide 3
Fourier local correcting potential hierarchy of scales: 2r o A
1/3 roro ~ 2A 1/3 is based on a simple and very intuitive
assumption that low-energy nuclear theory is independent on
high-energy dynamics ~ 10 The nuclear effective theory Long-range
part of the NN interaction (must be treated exactly!!!) where
regularization Coulomb ultraviolet cut-off denotes an arbitrary
Dirac-delta model Gogny interaction przykad There exist an infinite
number of equivalent realizations of effective theories
Slide 4
lim a a 0 Skyrme interaction - specific (local) realization of
the nuclear effective interaction: spin-orbit density dependence
10(11) parameters | v(1,2) | Slater determinant (s.p. HF states are
equivalent to the Kohn-Sham states) Skyrme-force-inspired local
energy density functional local energy density functional relative
momenta spin exchange LO NLO
Slide 5
Elongation (q) Total energy (a.u.) Symmetry-conserving
configuration Symmetry-breaking configurations LS
Slide 6
Euler angles gauge angle rotated Slater determinants are
equivalent solutions where Beyond mean-field multi-reference
density functional theory
Slide 7
Find self-consistent HF solution (including Coulomb) deformed
Slater determinant |HF>: BR C = 1 - |b T=|T z | | 2 BR in order
to create good isospin basis: Apply the isospin projector:
Engelbrecht & Lemmer, PRL24, (1970) 607 See: Caurier, Poves
& Zucker, PL 96B, (1980) 11; 15
Slide 8
Diagonalize total Hamiltonian in good isospin basis | ,T,T z
> takes physical isospin mixing Isospin invariant Isospin
breaking: isoscalar, isovector & isotensor C = 1 - |a T=T z | 2
AR n=1
Slide 9
(I)Isospin impurities in ground states of e-e nuclei Here the
HF is solved without Coulomb |HF;e MF =0>. Here the HF is solved
with Coulomb |HF;e MF =e>. In both cases rediagonalization is
performed for the total Hamiltonian including Coulomb W.Satua,
J.Dobaczewski, W.Nazarewicz, M.Rafalski, PRL103 (2009) 012502
Slide 10
0 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 1.0 20283644526068768492 A AR
BR SLy4 C [%] E-E HF [MeV] N=Z nuclei 100 This is not a single
Slater determinat There are no constraints on mixing coefficients
(II) Isospin mixing & energy in the ground states of e-e N=Z
nuclei: ~30% C HF tries to reduce the isospin mixing by: in order
to minimize the total energy Projection increases the ground state
energy ( the Coulomb and symmetry energies are repulsive)
Rediagonalization (GCM) lowers the ground state energy but only
slightly below the HF
Slide 11
3.0 3.5 4.0 4.5 29.530.030.531.031.5 C [%] CC CC (AR) (BR) 80
Zr E T=1 [MeV] (AR) MSk1 SkO SkP SLy5 SLy4 SLy7 SkM* SkXc SkP SLy5
SLy4 SkM* SLy7 SkXc SIII communicated by Franco Camera at the
Zakopane10 meeting doorway state energy
Slide 12
D. Rudolph et al. PRL82, 3763 (1999) f 7/2 f 5/2 p 3/2 neutrons
protons 4p-4h [303]7/2 [321]1/2 Nilsson 1 space-spin symmetric 2 f
7/2 f 5/2 p 3/2 neutrons protons g 9/2 p-h two isospin asymmetric
degenerate solutions Isospin symmetry violation in superdeformed
bands in 56 Ni
Slide 13
4 8 12 16 20 51015 51015 Exp. band 1 Exp. band 2 Th. band 1 Th.
band 2 Angular momentum Excitation energy [MeV] Hartree-Fock
Isospin-projection C [%] band 1 2 4 6 8 band 2 56 Ni ph ph T=0 T=1
centroid W.Satua, J.Dobaczewski, W.Nazarewicz, M.Rafalski, PRC81
(2010) 054310
Slide 14
s 1/2 p 3/2 p 1/2 p 2 8 np 2 8 n d 5/2 Hartree-Fock f
statistical rate function f (Z,Q ) t partial half-life f (t 1/2,BR)
G V vector (Fermi) coupling constant Fermi (vector) matrix element
| | 2 =2(1- C ) T z =-/+1 J=0 +,T=1 +/- (N-Z=-/+2) (N-Z=0) T z
=0
Slide 15
10 cases measured with accuracy ft ~0.1% 3 cases measured with
accuracy ft ~0.3% nucleus-independent ~2.4% Marciano & Sirlin,
PRL96 032002 (2006) ~1.5% 0.3% - 1.5% PRC77, 025501 (2008)
Slide 16
From a single transiton we can determine experimentally: G V 2
(1+ R ) G V =const. From many transitions we can: test of the CVC
hypothesis (Conserved Vector Current) exotic decays Test for
presence of a Scalar Current
Slide 17
one can determine mass eigenstates CKM
Cabibbo-Kobayashi-Maskawa weak eigenstates With the CVC being
verified and knowing G (muon decay) test unitarity of the CKM
matrix 0.9491(4) 0.0504(6)
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) ( and perform Coulomb rediagonalization) +/- H&T C
=0.330% L&G&M C =0.181% ~ ~ Project on good isospin (T=1)
and angular momentum (I=0) ( and perform Coulomb
rediagonalization)
Slide 22
0 0.5 1.0 1.5 2.0 3040506070 C [%] T z =0 T z =1 A 0 0.2 0.4
0.6 0.8 1.0 1.2 10152025303540 C [%] T z =-1 T z =0 A V ud
=0,97418(26) V ud =0,97466 Ft=3071.4(8)+0.85(85) Ft=3069.2(8)
Slide 23
0 2 4 6 1020304050 a sym [MeV] SV SLy4 L SkM L * SLy4 A (N=Z)
T=0 T=1 E sym = a sym T(T+1) 1 2 a sym a sym =32.0MeV a sym
=32.8MeV SLy4: In infinite nuclear matter we have: SV: a sym
=30.0MeV SkM*: a sym = e F + a int m m* SLy4: 14.4MeV SV: 1.4MeV
SkM*: 14.4MeV
Slide 24
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 ) Superallowed beta decay: encomapsses
extremely rich physics: CVC, V ud, unitarity of the CKM matrix,
scalar currents connecting nuclear and particle physics there is
still something to do in c business 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 !!!
Slide 25
0.0001 0.001 0.01 0.1 1 0.00.51.01.52.02.53.0 |OVERLAP| T [rad]
only IP IP+AMP = i * O ij j ij inverse of the overlap matrix space
& isospin rotated sp state HF sp state
Slide 26
0 1 2 3 4 0102030405060 Q Q [MeV] th exp Atomic number 0,2%
0,8% 0,9% 1,5% 2,5% 3,7% 4,1% time-even Hartree-Fock isospin
projected 0102030405060 Atomic number 0,9% 7,9% 15,1% 10,1% 26,3%
29,9% 21,7% time-odd Q values in super-allowed transitions time-odd
T=1,T z =-1 T=1,T z =0 T=1,T z =1 e-e o-o Isospin symmetry
violation due to time-odd fields in the intrinsic system Isobaric
analogue states: isospin projected Hartree-Fock
0.20 0.25 0.30 0.35 681012 Number of shells C [%] Towner &
Hardy 2008 Liang et al. (NL3)
Slide 29
aligned configurations anti-aligned configurations Isobaric
symmetry breaking in odd-odd N=Z nuclei Lets consider N=Z o-o
nucleus disregarding, for a sake of simplicity, time-odd
polarization and Coulomb (isospin breaking) effects or or T=0 After
applying naive isospin projection we get: T=0 T=1 ground state is
beyond mean-field! Mean-field can differentiate between and only
through time-odd polarizations! 4-fold degeneracy CORE
Slide 30
Position of the T=1 doorway state in N=Z nuclei 20 25 30 35
20406080100 A SIII SLy4 SkP E(T=1)-E HF [MeV] mean values Sliv
& Khartionov PL16 (1965) 176 based on perturbation theory E ~
2h ~ 82/A 1/3 MeV Bohr, Damgard & Mottelson hydrodynamical
estimate E ~ 169/A 1/3 MeV 31.532.032.533.033.534.034.5 y = 24.193
0.54926x R= 0.91273 doorway state energy [MeV] 4 5 6 7 C [%] 100 Sn
SkO SIII MSk1 SkP SLy5 SLy4 SkO SLy SkP SkM* SkXc l=0, n r =1
N=2
