In colaboration with: Jacek Dobaczewski, Pawe Bczyk, Maciek - PowerPoint PPT Presentation

In colaboration with: Jacek Dobaczewski, Paweł Bączyk, Maciek Konieczka, Koichi Sato, Takashi Nakatsukasa Isospin symmetry studies using SR-DFT and MR-DFT-rooted approaches: - new developments: DFT-rooted NCCI involving angular-momentum and isospin projections pn-mixed SR functionals charge-dependent functionals - physics highlights nuclear structure and beta decays strong-force isospin symmetry breaking effects (TDE/MDE)

Third Law of Progress in Theoretical Physics by Weinberg: “You may use any degrees of freedom you like to describe a physical system, but if you use the wrong ones, you’ll be sorry!”

Effective or low-energy (low-resolution) theory explores separation of scales. Its formulation requires: in coordinate space: � define R to separate short- and long-distance physics or, in momentum space: � define Λ (1/ R ) to separate low and high momenta replace (complicated and, in nuclear physics, unknown) short distance (or high momentum) physics by a LCP (local correcting potential) (there is a lot of freedom how this is done concerning both the scale and form but physics is (should be!) independent on the scheme!!!) emergence of 3NF due to finite resolution from Hammer et al. RMP 85, 197 (2013)

Nuclear effective theory for EDF (nuclear DFT) is based on the same simple and very intuitive assumption that low-energy nuclear theory is independent on high-energy dynamics ultraviolet cut-off Λ regularization Fourier Coulomb Long-range part of the NN interaction (must be treated exactly!!!) hierarchy of scales: 2r o A 1/3 2A 1/3 ~ correcting potential r o local ~ 10 denotes an arbitrary Dirac-delta model where There exist an „infinite” number Gaussian regulator of equivalent realizations J. Dobaczewski, K. Bennaceur, of effective theories F. Raimondi, J. Phys. G 39, 125103 (2012)

J. Dobaczewski, K. Bennaceur, F. Raimondi, J. Phys. G 39, 125103 (2012)

Skyrme interaction - specific (local) realization of the lim δ a nuclear effective interaction: a 0 LO NLO 10(11) density dependence parameters spin-orbit spin exchange relative momenta direct term exchange term

Skyrme (hadronic) interaction conserves such symmetries like: � LS LS LS � � particle numer, parity… advantages: builts in correlations into single Slater determinant disadvantages: symmetry must be restored to compare theory to data NCCI MR-DFT SR-DFT

There are two sources of the isospin symmetry breaking: - , caused solely by the HF approximation Engelbrecht & Lemmer, PRL24, (1970) 607 - , 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, Apply the isospin projector: PL 96B, (1980) 11; 15 in order to create good isospin „basis”: Diagonalize total Hamiltonian in „good isospin basis” | α ,T,T z > � takes physical isospin mixing AR n=1 = 1 - | a T=Tz | 2 α C

HF tries to reduce the isospin mixing by: 6 A. Corsi et al. PRC84, 041304 (2011) ∆α C N=Z nuclei ~30% 5 α C [%] in order to minimize 4 E. Farnea et al. the total energy 3 PLB551, 56 (2003) 2 1 SLy4 Projection increases the 0 ground state energy BR AR 1.0 ( the Coulomb and symmetry E-E HF [MeV] energies are repulsive) 0.8 0.6 0.4 Rediagonalization (GCM) 0.2 0 lowers the ground state 20 28 36 44 52 60 68 76 84 92 100 energy but only slightly A below the HF This is not a single Slater determinat There are no constraints on mixing coefficients

10 cases measured with accuracy ft ~0.1% 3 cases measured with accuracy ft ~0.3% adopted from J.Hardy’s, ENAM’08 presentation � test of the CVC hypothesis (Conserved Vector Current) 1.5% 0.3% ~2.4% - 1.5% Towner & Hardy Phys. Rev. C77, 025501 (2008) |V ud | = 0.97418 + 0.00026 - � test of unitarity of the CKM matrix |V ud | 2 +|V us | 2 +|V ub | 2 =0.9997(6) CKM weak mass 0.9490(4) 0.0507(4) <0.0001 eigenstates Cabibbo-Kobayashi eigenstates -Maskawa

0.976 1.0025 |V ud | 2 +|V us | 2 +|V ub | 2 0.975 1.0000 0.974 (a) (a) |V ud | (c) (d) (c)(d) 0.973 0.9975 (b) ν -decay ν -decay 0.972 (b) 0.9950 0.971 superallowed 0 + � 0 + superallowed 0 + � 0 + 0.9925 0.970 β -decay β -decay mirror T=1/2 mirror T=1/2 π -decay nuclei π -decay nuclei (a) I.S. Towner and J. C. Hardy, Phys. Rev. C 77 , 025501(2008). (b) H. Liang, N. V. Giai, and J. Meng, [%] 0.5 Phys. Rev. C 79 ,064316 (2009). W. Satuła, J. Dobaczewski, (c,d) W. Nazarewicz, M. Rafalski (HT) 0 Phys. Rev. C 86 , 054314 (2012) δ C - δ C (SV) For the NCCI study see: W.Satuła, P.Bączyk, J.Dobaczewski -0.5 & M.Konieczka, Phys. Rev. C94, 024306 (2016) O. Naviliat-Cuncic and N. Severijns, Eur. Phys. J. A 42 , 327 (2009); 10 20 30 40 50 60 70 A Phys. Rev. Lett. 102 , 142302 (2009).

W. Satuła, J. Dobaczewski, W. Nazarewicz, M. Rafalski Phys. Rev. C 86, 054314(2012). 1.5 See also the NCCI study: pDFT M. Konieczka, P. Bączyk, W. Satuła, Phys. Rev. C 93 , 042501(R) (2016). SM+WS SM+WS results from: 1.0 N. Severijns, M. Tandecki, δ C [%] T. Phalet, and I. S. Towner, Phys. Rev. C 78 , 055501 (2008). 0.5 0 10 15 20 25 30 35 40 45 50 A

W.Satuła, P.Bączyk, J.Dobaczewski & M.Konieczka, Phys. Rev. C94, 024306 (2016)

For details see: W.Satuła, P.Bączyk, J.Dobaczewski & M.Konieczka, Phys. Rev. C94, 024306 (2016) -20 -25 2 + 0 + 3 + -30 1 + Energy [MeV] -35 1 + 3 + -40 1 + 2 + -45 -50 -55 -60

No-core configuration-interaction formalism based on the isospin and angular momentum projected DFT W.Satuła, J.Dobaczewski & M.Konieczka, arXiv:1408.4982; 62 Zn, I=0 + states below 5MeV JPS Conf. Proc. 6, 020015 (2015) π |312 5/2> -1 π 2 SM SM SV mix EXP Excitation energy of 0 + states [MeV] EXP π |310 1/2> 5 (GXPF1) (MSDI3) (6 Slaters) (new) (old) ν |312 3/2> -1 ν 2 ν |321 1/2> 4 π |312 5/2> -2 ππ 1 π |312 3/2> 2 ν 1 ν |312 3/2> -1 3 ν |310 1/2> W.S., J. Dobaczewski, M. Konieczka 2 arXiv:1408.4982 (2014) π |312 5/2> -1 π 1 JPS Conf. Proc. 6, 020015 (2015) π |312 3/2> K.G. Leach et al. PRC88, 031306 (2013) 1 I=0 + HF before 0 + ground state mixing 0

Gamow-Teller and Fermi matrix elements in T=1/2 sd- and ft- mirrors. The NCCI study Proof-of-principle calculation: 6 He(0 + ) 6 Li(1 + ) 2.5 |M GT | Knecht et al. PRL108, 122502 (2012) 2.0 NCCI in 6 Li NCCI in 6 He 6 He is fixed 6 Li is fixed T=1/2 mirrors: (E EXP -E TH )/E EXP (%) 1.5 1.0 0.5 0 -0.5 -1.0 T z =-1/2 -1.5 T z = 1/2 20 30 40 50 A

SM 4 NCCI |g A M GT | (TH) 3 2 B. A. Brown and B. H. Wildenthal, Atomic Data and Nuclear 1 Data Tables 33 , 347 (1985). G. Martinez-Pinedo et al. , 20 30 40 50 Phys. Rev. C 53 , R2602 (1996). A 5 T. Sekine et al. , Nucl. Phys. A 467 , (1987). quenching q~25%!!! 4 The NCCI takes into account |g A M GT | 3 (TH) a core and its polarization Completely different model 2 spaces Different treatment of 1 SM correlations NCCI Different interactions 0 0 1 2 3 4 5 (EXP) |M GT |

Menendez et al. PRL107, 062501 (2011) β − decays of 14 C and 22;24 O Ekstrom et al. PRL 113, 262504 (2014) q 2 ~0.84-0.92 (from Ikeda sum rule) See also: Klos et al. PRC89, 029901 (2013) q~0.9 Engel et al. PRC89, 064308 (2013)

K. Sato, J. Dobaczewski, T. Nakatsukasa, and W. Satuła, Phys. Rev. C88 (2013), 061301 λ x λ z normalized: theory (red curve) shifted by 3.2MeV p-n mixed separable separable solution solution solution + |n> |p> -

Class II corrects for TDE Class III corrects for MDE

0.40 EXP GFMC 0.35 SV T CD 0.30 a A,T,I (MeV) SkM CD SLy4 CD 0.25 0.20 0.15 (2) 0.10 0.05 0 10,1,0 12,1,1 8,1,2 A,T,I

3.0 GFMC 2.5 CD SV T a A,T,I (MeV) 2.0 1.5 (1) 1.0 V CD 0.5 0 10,1,0 12,1,2 8,1,2 A,T,I

(from Maciek Konieczka) Gamow-Teller matrix elements for 20 Na 20 Ne Beta decay to 20 Ne; 6 SD in NCCI 2 + 2 + Isospin symmetry breaking effects in GT decays of T=1 nuclei 2 + 2 + 0 + VERY PRELIMINARY RESULTS !!! NSM fom Brown, Wildenthal, Atomic and Nuclear Data Tables 33 (1985)