Isospin Violation & Nuclear Decays

Transcription

1 Isospin Violation & Nuclear Decays University of Jyväskylä & University of Warsaw with Wojtek Satuła, Witek Nazarewicz, Maciek Konieczka, Tomek Werner, and Maciek Zalewski Fundamental Symmetry Tests with Rare Isotopes October 23-25, 214 UMass Amherst 1/29

2 Outline 1. DFT we use 2. Isospin projection & Coulomb rediagonalization 3. Superallowed beta decay 4. No-core configuration interaction model 5. Conclusions 2/29

3 DFT we use 3/29

4 How the nuclear EDF is built? Energy Density Functional (EDF) Energy Density Direct Exchange 4/29

5 Skyrme interaction - specific (quasilocal) realization of the nuclear effective interaction: lim d a a 1(11) parameters LO NLO density dependence spin-orbit relative momenta spin exchange Y v(1,2) Y Slater determinants (s.p. HF states are equivalent to the Kohn-Sham states) 5/29

6 Skyrme (hadronic) interaction conserves such symmetries like: parity advantages: builts in correlations into single Slater determinant disadvantages: symmetry must be restored to compare theory to data Restoration beyond mean-field multi-reference density functional theory 6/29

7 Isospin projection & Coulomb rediagonalization 7/29

8 There are two sources of the isospin symmetry breaking: -, caused solely by the HF approximation -, caused mostly by Coulomb interaction (also, but to much lesser extent, by the strong force isospin non-invariance) Engelbrecht & Lemmer, PRL24, (197) 67 Find self-consistent HF solution (including Coulomb) deformed Slater determinant HF>: Apply the isospin projector: See: Caurier, Poves & Zucker, PL 96B, (198) 11; 15 in order to create good isospin basis : Diagonalize total Hamiltonian in good isospin basis a,t,t z > takes physical isospin mixing See: Satuła et al., PRL 13, 1252 (29) AR n=1 a C = 1 - a T=T z 2 8/29

9 E-E HF [MeV] a C [%] N=Z nuclei E. Farnea et al. PLB551, 56 (23) A. Corsi et al. PRC84, 4134 (211) SLy 4 BR AR A 1 This is not a single Slater determinat There are no constraints on mixing coefficients HF reduces the isospin mixing by: Da C ~3% 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 9/29

10 E-E HF [MeV] a C [%] N=Z nuclei E. Farnea et al. PLB551, 56 (23) A. Corsi et al. PRC84, 4134 (211) SLy 4 BR AR A 1 This is not a single Slater determinat There are no constraints on mixing coefficients HF reduces the isospin mixing by: Da C ~3% 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 1/29

11 Collectivity beyond mean field, ground-state correlations, shape coexistence, symmetry restoration, projection on good quantum numbers, configuration interaction, generator coordinate method, multi-reference DFT, etc. True for interaction 11/29

12 OVERLAP only IP b T [rad] p 12/29

13 Superallowed beta decay 13/29

14 adopted from J.Hardy s, ENAM 8 presentation 1 cases measured with accuracy ft ~.1% 3 cases measured with accuracy ft ~.3% test of the CVC hypothesis (Conserved Vector Current) 1.5%.3% - 1.5% ~2.4% weak eigenstates CKM Cabibbo-Kobayashi -Maskawa mass eigenstates Towner & Hardy Phys. Rev. C77, 2551 (28) V ud = test of unitarity of the CKM matrix V ud 2 + V us 2 + V ub 2 =.9997(6).949(4).57(4) <.1 14/29

15 T z =-/+1 (N-Z=-/+2) J= +,T=1 <T +/- > 2 =2(1-d C ) J= +,T=1 T z = (N-Z=) n n p p aligned configurations or n p CORE n p n n p p anti-aligned configurations or n p CORE n p n p T= Mean-field can differentiate between n p and n p only through time-odd polarizations! n p T=1 T= T=1 state is not representable within 15/29 np separable mean-field theory!

16 I= +,T=1,T z =-1 I= +,T=1, T z = 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=) (and perform Coulomb rediagonalization) Project on good isospin (T=1) and angular momentum (I=) (and perform Coulomb rediagonalization) <T~1,T ~ z =+/-1,I= T +/- I=,T~1,T~ z => 2 =2(1-d C ) 16/29

17 d C [%] d C [%] T z =-1 T z = Ft=371.4(8)+.85(85) V ud =.97418(26) 1..5 PRL16, (211) Ft=37.4(9) V ud =.97444(23) T z = T z =1 V ud 2 + V us 2 + V ub 2 = =1.31(61) PRC86, (212) Ft=373.6(12) V ud =.97397(27) V ud 2 + V us 2 + V ub 2 = =.99935(67) A 17/29

18 V ud Tests of the weak-interaction flavor-mixing sector of the Standard Model of elementary particles (a) (b) (c) (d) superallowed + + b-decay p-decay n-decay mirror T=1/2 nuclei (a) (b) (c) (d) superallowed + + b-decay p-decay n-decay mirror T=1/2 nuclei Vud 2 + V us 2 + V ub 2 (a) I.S. Towner and J. C. Hardy, Phys. Rev. C 77, 2551(28). Ft=371.4(8)+.85(85); V ud =.97418(26) H. Liang, N. V. Giai, and J. Meng, (b) Phys. Rev. C 79,64316 (29). (c,d) W. Satuła, J. Dobaczewski, W. Nazarewicz, M. Rafalski, Phys. Rev. C 86, 54314(212). Ft=37.4(9); V ud =.97374(23) SHZ2 Ft=373.6(12); V ud =.97397(27) SV 18/29

19 d C [%] DFT SM+WS DFT results from: W. Satuła, J. Dobaczewski, W. Nazarewicz, and M. Rafalski, Phys. Rev. C86, 54314(212). SM+WS results from: N. Severijns, M. Tandecki, T. Phalet, and I. S. Towner, Phys. Rev. C 78, 5551 (28) A 19/29

20 Basis-size dependence: ~5% 2/29

21 Configuration dependence: SV: Functional dependence: Ft=373.6(12) V ud =.97397(27) V ud 2 + V us 2 + V ub 2 = =.99935(67) SHZ2: Ft=375.(12) V ud =.97374(27) V ud 2 + V us 2 + V ub 2 = =.9989(67) a sym =42.2MeV!!! DE [MeV] d C [%] x j p i l s A=34 SV j n DE HF l j l p s s y y i y i j n x 34 Ar 34 Cl 34 Cl 34 S DE I=,T=1 (TO) DE IV Relative orientation of shape and current x j n j p 21/29

22 No-core configuration interaction model 22/29

23 MEAN-FIELD compute n self-consistent Slater determinants corresponding to low-lying p-h excitations j 1 j 2 j 3 j n PROJECTION non-orthogonal set of K- and T-mixed states { I> (1) } k1 { I> (2) } k2 { I> (3) } k3.. { I> (n) } kn STATE MIXING Hill-Wheeler equation : Hu=ENu E i I i > 23/29

24 DE (MeV) W. Satuła, et al., Acta Phys. Pol. B45, 167 (214) Cl ( + ) I= + I=1 + I=2 + (2 + ) (2 + ) (2 + ) (2 + ) I=3 + theory exp W.Satula, J.Dobaczewski, M.Konieczka, W.Nazarewicz, Acta Phys. Polonica B45, 167 (214) 24/29

25 32 S I= Cl I=1 + W. Satuła, et al., Acta Phys. Pol. B45, 167 (214) (kev) 4622, 4636 (kev) T 1 4 T theory 72keV experiment theory kev experiment Experiment: δ C 5.3(9)% SM+WS calculations: δ C 4.6(5)%. D. Melconian et al., Phys. Rev. Lett. 17, (211). 25/29

26 W.Satuła, J.Dobaczewski & M.Konieczka, arxiv: No-core configuration-interaction formalism based on the isospin and angular momentum projected DFT K.G. Leach et al. PRC88, 3136 (213) NCCI 26/29

27 W.Satuła, J.Dobaczewski & M.Konieczka, arxiv: DE [MeV] Static approach gives: d C =8.9% 3. EXP d C =1.5% I= +, T= d C =1.7% Ca K 4 Slaters 3 Slaters 27/29

28 Conclusions 1. DFT properly accounts for the balance between the long-range Coulomb interaction and strong-force neutron-proton attraction. 2. Rediagonalization after the isospin projection properly accounts for the isospin mixing. 3. The DFT isospin-mixing corrections to superallowed beta decays, obtained in individual nuclei, are different than the SW+SM values. 4. Final V ud values are almost the same as those obtained in the SW+SM approach. 5. No-core configuration interaction model is a promising tool to study beyond-mean-field effects in N~Z nuclei. 28/29

29 Thank you 29/29

30 T z =-/+1 (N-Z=-/+2) J= +,T=1 <T +/- > 2 =2(1-d C ) J= +,T=1 T z = (N-Z=) n n p p aligned configurations or n p CORE n p n n p p anti-aligned configurations or n p CORE n p n+p n+p T=1 Proton-neutron mixed states isorotated by p/2 around the x axis describe T=1 states 3/29

31 K. Sato, et al., Phys. Rev. C88, 6131(R) (213) Triple energy differences (TED) 3.2 MeV 31/29

32 W. Satuła, et al., Acta Phys. Pol. B45, 167 (214) Triple energy differences (TED) 32/29

33 W.Satuła, J.Dobaczewski, W.Nazarewicz, M.Rafalski, PRC81 (21) 5431 MRDFT: H(r) GWT H(r ~ ) ~ ~ -1 r =S y * i O ij y j ij SVD SVD eigenvalues (diagonal matrix) Power [cos(b/2)] counting: singularity (if any) at b=p is inherited by a transition density r 1 + N-Z - h + N-Z +2k k is a multiplicity of zero singular values in the worst case i.e. for N=Z and k=1 h > 3 to get a singularity 33/29

34 spherical RPA Coulomb exchange treated in the Slater approximation mean field d C =d C1 +d C2 radial mismatch of the wave functions Miller & Schwenk Phys. Rev. C78 (28) 3551;C8 (29) shell model configuration mixing 34/29

35 d C [%] Confidence level test based on the CVC hypothesis T&H PRC82, 6551 (21) (SV) d C (EXP) d C (EXP) d C = 1+d NS - Ft ft(1+d R ) Minimize RMS deviation between the calculated and experimental d C with respect to Ft Z of daughter c 2 /n d =5.2 for Ft = 37.s 75% contribution to the c 2 comes from A=62 From: W. Satuła, J. Dobaczewski, W. Nazarewicz, M. Rafalski Phys. Rev. Lett. 16, (211). 35/29

36 ~2-3MeV d C [%] CI scenario 1.5 averages mixing 1. E X ~E Y ~E Z.5 T z =-1 T z = A T&H W.Satula, J.Dobaczewski, M.Konieczka, W.Nazarewicz, Acta Phys. Polonica B45, 167 (214 36/29

37 d C [%] E HF (MeV) Energy (MeV) Stability of the NCCI calculation for 62 Zn g.s. + p 1 + n 1 + n 2 + p 2 + pp 1 EXP -526 ~2keV normalized n 1 p 1 Z X ~ Y 37/29

38 Eigenvalue of the norm matrix Zn.1 62 Ga Eigenstate of the norm matrix 38/29

39 Eigenvalues of norm matrix 46 V 46 Ti 5 Mn 5 Cr MeV d C =.563% MeV d C =.477% MeV d C =.486% ALL INCLUDED MeV d C =1.345% MeV d C =1.66% ALL INCLUDED 39/29

40 Excitation energy [MeV] deuteron-like correlations? DE (MeV) SV SHZ2 nk pk 42 Sc 1/2 3/2 5/2 7/2 Mixing of states projected from the antialigned configurations: K Sc T= T= angular momentum 4/29

41 Excitation energy [MeV] Energy [MeV] Li 8 Li SPIN Li Li 41/29

42 DE (MeV) DE (MeV) bare Z=2 gap EXP p-h 2p-2h 4p-4h 6p-6h 2 spin 4 6 p-h spherical rotational correction Bare gaps in 4 Ca: SV: 11.53MeV SIII: 5.66MeV EXP: 7.2MeV 42/29