+ 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

Effective theories for low-energy (low-resolution) nuclear physics (I): Low-resolution  separation of scales which is a cornerstone of all effective theories

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 przykład There exist an „infinite” number of equivalent realizations of effective theories

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

Elongation (q) Total energy (a.u.) Symmetry-conserving configuration Symmetry-breaking configurations LS

Euler angles 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>: 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

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

(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

 C [%] CC CC (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 Zakopane’10 meeting doorway state energy

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)

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% nucleus-independent ~2.4% Marciano & Sirlin, PRL (2006) ~1.5% 0.3% - 1.5% PRC77, (2008)

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

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

Hardy &Towner Phys. Rev. C77, (2008) Liang & Giai & Meng Phys. Rev. C79, (2009) spherical RPA Coulomb exchange treated in the Slater approxiamtion  C =  C1 +  C2 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

 C [%] 2K isospin isospin & angular momentum 0.586(2)% ( ( ( (

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)

 C [%] T z =0  T z =1 A  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)

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

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 !!!

|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

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 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

a sym (  NM /2) [MeV] a sym (  NM ) [MeV]  C [%] a sym (  NM /2) [MeV] a sym (  NM ) [MeV]  C [%] 40 Ca 100 Sn SkO SkO’ SkM* MSk1 SkXc SIII SLy SkP SIII SkP MSk1 SkXc SkM* SLy5SLy SkO’ SkO SkO’ SkM* SIII SkO SkP SLy SkXc MSk1 SkP MSk1 SkXc SkM* SLy SkO’ SLy5

Number of shells  C [%] Towner & Hardy 2008 Liang et al. (NL3)

aligned configurations     anti-aligned configurations 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 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

Position 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