tens of MeV + NNN ab initio Intro:  define fundaments my model is „standing on” sp mean-field ( or nuclear DFT )  beyond mean-field ( projection after variation ) 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 structural effects  SD bands in 56 Ni ISB corrections to superallowed beta decay isospin impurities in ground-states of e-e nuclei Results

Skyrme-force-inspired local energy density functional (without pairing) LS  | v(1,2) |  average Skyrme interaction (in fact a functional!) over the Slater determinant local energy density functional Deformation (q) Total energy (a.u.) Symmetry-conserving configurtion Symmetry-breaking configurations SV is the only Skyrme interaction Beiner et al. NPA238, 29 (1975)

Euler angles in space or/and isospace gauge angle rotated Slater determinants are equivalent solutions where Beyond mean-field  multi-reference density functional theory

Find self-consistent HF solution (including Coulomb)  deformed Slater determinant |HF>: in order to create good isospin „basis”: Apply the isospin projector: Engelbrecht & Lemmer, PRL24, (1970) 607 Diagonalize total Hamiltonian in „good isospin basis” | ,T,T z >  takes physical isospin mixing  C = 1 - |a T=T z | 2 AR n=1  C = 1 - |b T=|T z | | 2 BR See: Caurier, Poves & Zucker, PL 96B, (1980) 11; 15

(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.Satuła, J.Dobaczewski, W.Nazarewicz, M.Rafalski, PRL103 (2009)

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

Excitation energy of the T=1 doorway state in N=Z nuclei 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 y = – x R= doorway state energy [MeV]  C [%] 100 Sn SkO SIII MSk1 SkP SLy5 SLy4 SkO’ SLy SkP SkM* SkXc  l=0,  n r =1   N=2

aligned configuration  anti-aligned configuration  or  or   T=0 Isospin projection T=1 T=0   Mean-field   four-fold degeneracy of the sp levels Spontaneous isospin mixing in N=Z nuclei in other but isoscalar configs  yet another strong motivation for isospin projection 

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

Exp. band 1 Exp. band 2 Th. band 1 Th. band 2 Angular momentum Excitation energy [MeV] Hartree-Fock Isospin-projection  C [%] band band 2 56 Ni  ph ph T=0 T=1 centroid   W.Satuła, J.Dobaczewski, W.Nazarewicz, M.Rafalski, PRC81 (2010)

SVD SVD eigenvalues (diagonal matrix ) Isospin-projection is non-singular: singularity (if any) at  is inherited by  1 + |N-Z| -  + |N-Z|+2k k is a multiplicity of zero singular values  > 3 in the worst case W.Satuła, J.Dobaczewski, W.Nazarewicz, M.Rafalski, PRC81 (2010) ij  =   i * O ij  j ~

 C [%] 2K isospin isospin & angular momentum 0.586(2)% 42 Sc – isospin projection from [K,-K] configurations with K=1/2,…,7/2 Isospin and angular-momentum projected DFT is ill-defined except for the hamiltonian-driven functionals

|OVERLAP|  T [rad] only IP IP+AMP  inverse of the overlap matrix space & isospin rotated sp state HF sp state T ij  =   i * O ij  j

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 10 cases measured with accuracy ft ~0.1% 3 cases measured with accuracy ft ~0.3%

~2.4% Marciano & Sirlin, PRL96, , (2006) nucleus-independent ~1.5% 0.3% - 2.0% e  NS-independent The 13 precisely known transitions, after including theoretical corrections, are used to NS-dependent Towner & Hardy Phys. Rev. C77, (2008) Towner, NPA540, 478 (1992) PLB333, 13 (1994) e  courtesy of J.Hardy

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 (4) (6) < |V ud | 2 +|V us | 2 +|V ub | 2 =0.9996(7) |V ud | = test of three generation quark Standard Model of electroweak interactions

Towner & Hardy Phys. Rev. C77, (2008) Liang & Giai & Meng Phys. Rev. C79, (2009) spherical RPA Coulomb exchange treated in the Slater approxiamtion  C =  C2 +  C1 shell model mean field Miller & Schwenk Phys. Rev. C78 (2008) ;C80 (2009) radial mismatch of the wave functions configuration mixing

Isobaric symmetry violation in o-o N=Z nuclei ground state is beyond mean-field! T=0  T=0 T=1  Mean-field can differentiate between  and  only through time-odd polarizations! aligned configurations     anti-aligned configurations or  or    CORE T z =-/+1 J=0 +,T=1  +/- (N-Z=-/+2) (N-Z=0) T z =0

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) <T~1,T z =+/-1,I=0| |I=0,T~1,T z =0>  +/- 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)

V ud = (26) V ud = (23) Ft=3071.4(8)+0.85(85) Ft=3070.4(9) |V ud | 2 +|V us | 2 +|V ub | 2 = = (61) W.Satuła, J.Dobaczewski, W.Nazarewicz, M.Rafalski, PRL106 (2011)

|V ud | superallowed  -decay  + -decay -decay T=1/2 mirror  -transitions H&T’08 Liang et al.

T z  T z +1 T z = -1 T z = 0  C [%] A N cutoff =10 N cutoff =12 A=18 A=38 A=58

[Isospin projection, unlike the angular-momentum and particle-number projections, is practically non-singular !!!] 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 0 +  0 + beta decay:  encomaps 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? Pairing & other (shape vibrations) correlations can be „realtively easily” incorporated into the scheme by combining projection(s) with GCM

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