MICROSCOPIC CALCULATIONS OF ISOSPIN IMPURITIES AND ISOSPINSYMMETRYBREAKING CORRECTIONS

MICROSCOPIC CALCULATIONS OF ISOSPIN IMPURITIES AND ISOSPIN-SYMMETRY-BREAKING CORRECTIONS USING ISOSPIN AND ANGULAR-MOMENTUM PROJECTD DFT Wojciech Satuła in collaboration with J. Dobaczewski, W. Nazarewicz & M. Rafalski 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 56 Ni superallowed beta decay symmetry energy – new opportunities of study Summary ab initio + NNN +. . tens of Me. V

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 2 ro. A 1/3 ~ 2 A 1/3 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 SV 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 isospin symmetry: VLS = V (in reality approximate) nn pp np parity… Mean-field solutions (Slater determinants) break (spontaneously) these symmetries Total energy (a. u. ) Symmetry-conserving configuration Symmetry-breaking configurations Elongation (q)

Restoration of broken symmetry Beyond mean-field multi-reference density functional theory Euler angles rotated Slater determinants are equivalent solutions gauge angle where

Isospin symmetry restoration There are two sources of the isospin symmetry breaking: Engelbrecht & Lemmer, - unphysical, unphysical caused solely by the HF approximation PRL 24, (1970) 607 - physical, physical caused mostly by Coulomb interaction (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 96 B, (1980) 11; 15 Apply the isospin projector: in order to create good isospin „basis”: Calculate the projected energy and the Coulomb mixing Before Rediagonalization: BR BR a. C = 1 - |b. T=|Tz||2

Diagonalize total Hamiltonian in „good isospin basis” |a, T, Tz> takes physical isospin mixing AR n=1 2 a. C = 1 - |a. T=T | z Isospin breaking: isoscalar, isovector & isotensor Isospin invariant

Numerical results: (I) Isospin impurities in ground states of e-e nuclei W. Satuła, J. Dobaczewski, W. Nazarewicz, M. Rafalski, PRL 103 (2009) 012502 Ca isotopes: BR SLy 4 AR 0. 4 Here the HF is solved without Coulomb |HF; e. MF=0>. a. C [%] 0. 2 Here the HF is solved with Coulomb |HF; e. MF=e>. e. MF = 0 0 1 0. 8 0. 01 0. 6 40 0. 4 44 48 52 56 60 e. MF = 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 [Me. V] a. C [%] (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 N=Z nuclei AR BR BR SLy 4 BR AR HF tries to reduce the isospin mixing by: Da. C ~30% in order to minimize the total energy Projection increases the ground state energy (the Coulomb and symmetry energies are repulsive) Rediagonalization (GCM) AR 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

Bohr, Damgard & Mottelson hydrodynamical estimate DE ~ 169/A 1/3 Me. V 35 mean values 30 Sliv & Khartionov PL 16 (1965) 176 Dl=0, Dnr=1 DN=2 DE ~ 2 hw ~ 82/A 1/3 Me. V 25 20 SIII SLy 4 Sk. P 20 based on perturbation theory 40 60 A 80 100 a. C [%] E(T=1)-EHF [Me. V] Position of the T=1 doorway state in N=Z nuclei 7 6 5 4 Sk. P MSk 1 SLy 5 SLy. Sk. P Sk. M* SLy 4 Sk. O’ Sk. Xc 100 Sn SIII Sk. O y = 24. 193 – 0. 54926 x R= 0. 91273 31. 5 32. 0 32. 5 33. 0 33. 5 34. 0 34. 5 doorway state energy [Me. V]

1 Isospin symmetry violation in superdeformed bands in 56 Ni f 5/2 p 3/2 f 7/2 4 p-4 h Nilsson [321]1/2 neutrons protons [303]7/2 space-spin symmetric 2 g 9/2 f 5/2 p 3/2 f 7/2 D. Rudolph et al. PRL 82, 3763 (1999) pp-h neutrons protons two isospin asymmetric degenerate solutions

Isospin projection Mean-field T=1 d. ET Excitation energy [Me. V] pph 20 centroid a. C [%] nph d. ET T=0 8 6 4 2 band 1 Isospin-projection Hartree-Fock 16 56 Ni 12 Exp. band 1 Exp. band 2 Th. band 1 Th. band 2 8 4 5 10 15 Angular momentum W. Satuła, J. Dobaczewski, W. Nazarewicz, M. Rafalski, PRC 81 (2010) 054310

Primary motivation of the project isospin corrections for superallowed beta decay Tz=-/+1 (N-Z=-/+2) J=0+, T=1 Qb t+/- t 1/2 BR d 5/2 8 8 2 2 J=0+, T=1 Tz=0 (N-Z=0) Experiment: Fermi beta decay: p 1/2 p 3/2 s 1/2 n p 14 O n p 14 N Hartree-Fock f statistical rate function f (Z, Qb) t partial half-life f (t 1/2, BR) GV vector (Fermi) coupling constant <t+/-> Fermi (vector) matrix element |<t+/->|2=2(1 -d. C)

Experiment world data survey’ 08 T&H, PRC 77, 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, PRL 96 032002 (2006)

What can we learn out of it? see J. Hardy, ENAM’ 08 presentation From a single transiton we can determine experimentally: GV 2(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 mass eigenstates test unitarity of the CKM matrix |Vud|2+|Vus|2+|Vub|2=0. 9997(6) 0. 9490(4) 0. 0507(4) <0. 0001 test of three generation quark Standard Model of electroweak interactions Towner & Hardy |Vud| = 0. 97418 + 0. 00026 Phys. Rev. C 77, 025501 (2008) Cabibbo-Kobayashi-Maskawa

Model dependence Hardy &Towner Liang & Giai & Meng Phys. Rev. C 79, 064316 (2009) spherical RPA Coulomb exchange treated in the Slater approxiamtion Phys. Rev. C 77, 025501 (2008) d. C=d. C 1+d. C 2 mean field radial mismatch of the wave functions Miller & Schwenk Phys. Rev. C 78 (2008) 035501; C 80 (2009) 064319 shell model configuration mixing

J=0+, T=1 Tz=-/+1 (N-Z=-/+2) Qb t+/- t 1/2 Isobaric symmetry violation in o-o N=Z nuclei J=0+, T=1 BR Tz=0 (N-Z=0) MEAN FIELD n n p p n n CORE aligned configurations n p or n p T=0 Mean-field can differentiate between n p and n p only through time-odd polarizations! p anti-aligned configurations n p or n p ISOSPIN PROJECTION n p p n p T=1 T=0 ground state is beyond mean-field!

42 Sc – isospin projection from [K, -K] configurations with K=1/2, …, 7/2 -5/2 -3/2 -1/2 3/2 5/2 7/2 f 7/2 isospin & angular momentum 40 a. C [%] isospin 30 20 10 0. 586(2)% 0 1 3 2 K 5 7

|OVERLAP| 1 0. 1 only IP 0. 01 0. 0001 IP+AMP 0. 0 0. 5 1. 0 1. 5 2. 0 r =S yi* Oij jj T -1 2. 5 3. 0 b. T [rad] p ij HF sp state inverse of the overlap matrix space & isospin rotated sp state Singularities force us to use interaction-driven functional SV

Hartree-Fock ground state in N-Z=+/-2 (e-e) nucleus Project on good isospin (T=1) and angular momentum (I=0) (and perform Coulomb CPU antialigned state in N=Z (o-o) nucleus ~ few h ~ few years Project on good isospin (T=1) and angular momentum (I=0) (and perform Coulomb rediagonalization) <T~1, T ~ z=0> ~ z=+/-1, I=0| T+/- |I=0, T~1, T 14 O 14 N H&T d. C=0. 330% L&G&M d. C=0. 181% our: d. C=0. 303% (Skyrme-V; N=12)

1. 4 1. 2 Tz = 1 H&T: Ft=3071. 4(8)+0. 85(85) Vud=0. 97418(26) our (no A=38): Ft=3070. 4(9) Vud=0. 97444(23) Tz=0 d. C [%] 1. 0 0. 8 0. 6 0. 4 0. 2 0 2. 0 10 14 18 22 26 30 34 38 42 A Tz=0 Tz=1 d. C [%] 1. 5 1. 0 0. 5 0 26 34 42 50 58 66 74 A |Vud|2+|Vus|2+|Vub|2= =1. 00031(61)

0. 976 0. 975 |Vud| 0. 974 0. 973 0. 972 0. 971 H&T’ 08 our model Liang et al. n-decay superallowed b-decay 0. 970 p+-decay T=1/2 mirror b-transitions

Confidence level test based on the CVC hypothesis T&H PRC 82, 065501 (2010) 2. 5 (EXP) d. C(SV) d. C [%] 2. 0 d. C (EXP) ‚ ft(1+d. R) Minimize RMS deviation between the caluclated and experimental d. C with respect to Ft 1. 5 1. 0 0. 5 0 0 d. C = 1+d. NS - Ft 5 10 15 20 25 30 35 Z of daughter c 2/nd=5. 2 for Ft = 3070. 0 s 75% contribution to the 2 40 c comes from A=62

„NEW OPPORTUNITIES” IN STUDIES OF THE SYMMETRY ENERGY: T=1 n p 1 a’ T(T+1) E’ = a’ sym T=0 2 sym SLy 4 L Sk. ML* SV a’sym [Me. V] 6 4 In infinite nuclear matter we have: SLy 4: asym=32. 0 Me. V SV: asym=32. 8 Me. V 2 Sk. M*: asym=30. 0 Me. V asym= 0 10 20 30 40 A (N=Z) 50 m m* e. F + aint SLy 4: 14. 4 Me. V SV: 1. 4 Me. V Sk. M*: 14. 4 Me. V

Summary and outlook 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? [Isospin projection, unlike the angular-momentum and particle-number projections, is practically non-singular !!!] 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 dc business … How to include pairing into the scheme?

Mirror-symmetric nuclei (preliminary) Nolen-Schiffer anomaly in mirror symmetric nuclei Vpp-Vnn DEexp-DEth [Me. V] Fig 43: 110427 0. 5 0. 0 -0. 5 -1. 0 SLy 4 (HF) SV (PROJ) -1. 5 20 25 30 A 35 40 45