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tens of MeV + NNN +.... 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 structural effects SD bands in 56 Ni superallowed beta decay isospin impurities in ground-states of e-e nuclei symmetry energy – new opportunities of study
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Effective theories for low-energy (low-resolution) nuclear physics (I): Low-resolution separation of scales which is a cornerstone of all effective theories
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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
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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
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Elongation (q) Total energy (a.u.) Symmetry-conserving configuration Symmetry-breaking configurations LS
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Euler angles gauge angle rotated Slater determinants are equivalent solutions where Beyond mean-field multi-reference density functional theory
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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
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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
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(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) 012502
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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
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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
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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
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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.Satuła, J.Dobaczewski, W.Nazarewicz, M.Rafalski, PRC81 (2010) 054310
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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
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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% T&H, PRC77, 025501 (2008)
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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 see J.Hardy, ENAM’08 presentation
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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 0.9490(4) 0.0507(4) <0.0001 |V ud | 2 +|V us | 2 +|V ub | 2 =0.9997(6) |V ud | = 0.97418 + 0.00026 test of three generation quark Standard Model of electroweak interactions Towner & Hardy Phys. Rev. C77, 025501 (2008)
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Hardy &Towner Phys. Rev. C77, 025501 (2008) Liang & Giai & Meng Phys. Rev. C79, 064316 (2009) spherical RPA Coulomb exchange treated in the Slater approxiamtion C = C1 + C2 shell model mean field Miller & Schwenk Phys. Rev. C78 (2008) 035501;C80 (2009) 064319 radial mismatch of the wave functions configuration mixing
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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
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0 10 20 30 40 1357 C [%] 2K isospin isospin & angular momentum 0.586(2)% 42 Sc – isospin projection from [K,-K] configurations with K=1/2,…,7/2 -7/2 7/2 -5/2 -3/2 -1/2 1/2 3/2 5/2 f 7/2
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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 T
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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)
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V ud =0.97418(26) V ud =0.97444(23) Ft=3071.4(8)+0.85(85) Ft=3070.4(9) |V ud | 2 +|V us | 2 +|V ub | 2 = =1.00031(61)
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0.970 0.971 0.972 0.973 0.974 0.975 0.976 |V ud | superallowed -decay + -decay -decay T=1/2 mirror -transitions H&T’08 Liang et al.
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0510152025303540 CC (SV) CC (EXP) C [%] Z of daughter 0 0.5 1.0 1.5 2.0 2.5 Confidence level test based on the CVC hypothesis T&H PRC82, 065501 (2010) C = 1+ NS - Ft ft(1+ R ) ‚ (EXP) Minimize RMS deviation between the caluclated and experimental C with respect to Ft 2 /n d =5.2 for Ft = 3070.0s 75% contribution to the 2 comes from A=62
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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
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[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 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? How to include pairing into the scheme?
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-1.5 -0.5 0.0 0.5 202530354045 A Fig43:110427 E exp - E th [MeV] Nolen-Schiffer anomaly in mirror symmetric nuclei V pp -V nn SLy4 (HF) SV (HF) SV (PROJ)
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