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3D angular momentum and isospin restored calculations w ith the Skyrme EDF. Wojciech Satuła. in collaboration with J. Dobaczewski , W. Nazarewicz & M. Rafalski. Intro :  define fundaments my model is „standing on”

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  1. 3D angularmomentum and isospinrestoredcalculations with the Skyrme EDF WojciechSatuła incollaborationwith J. Dobaczewski, W. Nazarewicz & M. Rafalski Intro:  definefundaments my model is „standing on” spmean-field (ornuclear DFT)  beyondmean-field (projectionaftervariation) Symmetry (isospin) violation and restoration: • unphysicalsymmetryviolation  isospinprojection • Coulomb rediagonalization(explicitsymmetryviolation) isospinimpuritiesinground-states of e-e nuclei Results structuraleffects SD bandsin56Ni ISB corrections to superallowed beta decay Summary ab initio + NNN + .... tens of MeV

  2. Skyrme-force-inspiredlocal energy densityfunctional (withoutpairing) Y | v(1,2) | Y averageSkyrmeinteraction (in fact a functional!) over the Slater determinant localenergydensityfunctional SV is the onlySkyrmeinteraction Beiner et al. NPA238, 29 (1975) Skyrme (nuclear) interactionconserves:  rotational (spherical) symmetry  isospinsymmetry: Vnn= Vpp= Vnp(in reality approximate) LS LS LS Mean-fieldsolutions (Slaterdeterminants) break (spontaneously) thesesymmetries Symmetry-conserving configurtion Total energy (a.u.) Symmetry-breaking configurations Deformation(q)

  3. Restoration of brokensymmetry Eulerangles in spaceor/and isospace gauge angle Beyond mean-fieldmulti-referencedensityfunctionaltheory rotated Slater determinants are equivalent solutions where

  4. Applytheisospinprojector: in order to creategoodisospin „basis”: AR aC= 1 - |aT=Tz|2 BR aC= 1 - |bT=|Tz||2 Isospinsymmetryrestoration • Therearetwosources of theisospinsymmetrybreaking: • unphysical, causedsolely by the HF approximation • physical, caused mostly by Coulomb interaction • (also, but to much lesserextent, by the strong force isospin non-invariance) Engelbrecht & Lemmer, PRL24, (1970) 607 Findself-consistent HF solution (including Coulomb)  deformed Slater determinant |HF>: See: Caurier, Poves & Zucker, PL 96B, (1980) 11; 15 Diagonalizetotal Hamiltonian in „goodisospinbasis” |a,T,Tz>  takesphysicalisospinmixing n=1

  5. eMF = 0 eMF = e Ca isotopes: 0.4 BR SLy4 AR 0.2 0 aC [%] 1 1.0 0.1 0.8 0.01 0.6 60 40 44 48 52 56 0.4 0.2 0 56 40 48 44 52 60 Mass number A Numericalresults: • Isospinimpuritiesingroundstates of e-e nuclei W.Satuła, J.Dobaczewski, W.Nazarewicz, M.Rafalski, PRL103 (2009) 012502 Herethe HF issolved without Coulomb |HF;eMF=0>. Herethe HF issolved with Coulomb |HF;eMF=e>. In bothcasesrediagonalization isperformed for thetotal Hamiltonian including Coulomb

  6. DaC ~30% SLy4 BR AR aC [%] 6 N=Z nuclei 5 4 3 1.0 2 0.8 1 E-EHF [MeV] 0.6 0 0.4 0.2 0 20 28 36 44 52 60 68 76 84 92 100 A (II) Isospinmixing & energy inthegroundstates of e-e N=Znuclei: HF tries to reduce the isospin mixing by: AR in order to minimize the total energy BR Projectionincreasesthe ground state energy (the Coulomb and symmetry energiesarerepulsive) BR Rediagonalization (GCM) AR lowerstheground state energy but onlyslightly belowthe HF This is not a single Slater determinat There are no constraints on mixing coefficients

  7. SIII SLy4 SkP 100Sn 35 7 30 aC [%] SkP 6 SLy5 SkP SLy MSk1 SkM* SLy4 5 SIII SkXc SkO’ 25 4 SkO DE ~ 2hw ~ 82/A1/3 MeV 20 y = 24.193 – 0.54926x R= 0.91273 doorway state energy [MeV] 31.5 32.0 32.5 33.0 33.5 34.0 34.5 20 40 60 80 100 Excitationenergyof the T=1 doorwaystate in N=Z nuclei Bohr, Damgard & Mottelson hydrodynamical estimate DE ~ 169/A1/3 MeV mean values E(T=1)-EHF [MeV] Sliv & Khartionov PL16 (1965) 176 Dl=0, Dnr=1  DN=2 based on perturbation theory A

  8. n n p p n n p p n p n p n p T=0 Spontaneousisospinmixing in N=Z nuclei in other but isoscalarconfigs  yetanotherstrongmotivation for isospinprojection four-fold degeneracy of the sp levels Mean-field n p or or n p aligned configuration anti-aligned configuration n p T=1 Isospin projection T=0

  9. Nilsson f5/2 p3/2 [321]1/2 [303]7/2 f7/2 protons neutrons 2 g9/2 pp-h f5/2 p3/2 f7/2 protons neutrons two isospin asymmetric degenerate solutions 1 Isospin symmetry violation in superdeformed bands in 56Ni 4p-4h space-spin symmetric D. Rudolph et al. PRL82, 3763 (1999)

  10. T=1 nph dET centroid pph band 2 dET T=0 Isospin-projection 20 16 8 6 56Ni 4 12 2 Exp. band 1 Exp. band 2 Th. band 1 Th. band 2 8 5 10 15 4 Isospin projection Mean-field aC [%] band 1 Hartree-Fock Excitation energy [MeV] 5 10 15 Angular momentum Angular momentum W.Satuła, J.Dobaczewski, W.Nazarewicz, M.Rafalski, PRC81 (2010) 054310

  11. -1 ij Isospin-projectionis non-singular: SVD eigenvalues (diagonal matrix) W.Satuła, J.Dobaczewski, W.Nazarewicz, M.Rafalski, PRC81 (2010) 054310 SVD singularity (ifany) atb=p isinherited by r r =Syi*Oijjj ~ in the worstcase h> 3 1 + |N-Z| - h + |N-Z|+2k kis a multiplicity of zero singularvalues

  12. 40 30 20 10 0 1 3 5 7 42Sc – isospinprojection from [K,-K] configurations with K=1/2,…,7/2 isospin & angularmomentum isospin aC [%] 0.586(2)% 2K Isospin and angular-momentumprojected DFT isill-defined except for the hamiltonian-drivenfunctionals thereis no alternative but Skyrme SV

  13. 1 0.1 0.01 0.001 -1 0.0001 ij 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Why we have to to useSkyrme-V? onlyIP |OVERLAP| IP+AMP p bT [rad] r =Syi*Oijjj T HF sp state space & isospinrotated sp state inverse of the overlap matrix

  14. Primarymotivation of theproject isospincorrections for superallowed beta decay Tz=-/+1 J=0+,T=1 (N-Z=-/+2) t1/2 t+/- Qb J=0+,T=1 (N-Z=0) BR Tz=0 Experiment: Fermi beta decay: 8 8 d5/2 |<t+/->|2=2(1-dC) p1/2 p3/2 2 2 f statisticalratefunctionf (Z,Qb) s1/2 t partialhalf-lifef (t1/2,BR) n n p p GVvector (Fermi) couplingconstant 14N 14O <t+/-> Fermi (vector) matrix element 10 casesmeasuredwithaccuracyft ~0.1% Hartree-Fock 3 casesmeasuredwithaccuracyft ~0.3%

  15. NS-independent g nucleus-independent e ~1.5% 0.3% - 2.0% n n Marciano & Sirlin, PRL96, 032002, (2006) ~2.4% NS-dependent Towner, NPA540, 478 (1992) PLB333, 13 (1994) g e The 13 preciselyknowntransitions, afterincluding theoreticalcorrections, areused to test the CVC hypothesis Towner & Hardy Phys. Rev. C77, 025501 (2008) courtesy of J.Hardy

  16. Withthe CVC beingverified and knowingGm(muondecay) one candetermine mass eigenstates CKM Cabibbo-Kobayashi-Maskawa weakeigenstates |Vud| = 0.97425 + 0.00023  test unitarity of the CKM matrix |Vud|2+|Vus|2+|Vub|2=0.9996(7) 0.9491(4) test of threegenerationquark Standard Model of electroweakinteractions 0.0504(6) <0.0001

  17. „Hidden” model dependence Towner & Hardy Phys. Rev. C77, 025501 (2008) dC=dC2+dC1 Liang & Giai & Meng Phys. Rev. C79, 064316 (2009) shell model mean field spherical RPA Coulomb exchangetreatedinthe Slaterapproxiamtion radialmismatch of thewavefunctions configuration mixing Miller & Schwenk Phys. Rev. C78 (2008) 035501;C80 (2009) 064319

  18. n n n n p p p p n n n p p p n p n p n n p p T=0 Isobaricsymmetryviolation in o-o N=Znuclei Tz=-/+1 J=0+,T=1 (N-Z=-/+2) t1/2 t+/- Qb J=0+,T=1 (N-Z=0) BR Tz=0 MEAN FIELD CORE CORE anti-aligned configurations aligned configurations n p or or ISOSPIN PROJECTION T=1 T=0 Mean-fieldcandifferentiatebetween ground state isbeyondmean-field! and onlythroughtime-oddpolarizations!

  19. Hartree-Fock antialigned state inN=Z (o-o) nucleus ground state inN-Z=+/-2 (e-e) nucleus CPU Project on goodisospin (T=1) and angularmomentum (I=0) (and perform Coulomb rediagonalization) Project on goodisospin (T=1) and angularmomentum (I=0) (and perform Coulomb rediagonalization) ~ few h ~ fewyears t+/- |I=0,T~1,Tz=0> <T~1,Tz=+/-1,I=0| ~ ~ H&TdC=0.330% 14N L&G&MdC=0.181% 14O our:  dC=0.303% (Skyrme-V; N=12)

  20. H&T: 1.4 Ft=3071.4(8)+0.85(85) Tz= 1 Tz=0 1.2 Vud=0.97418(26) 1.0 our (no A=38): 0.8 Ft=3070.4(9) dC[%] 0.6 Vud=0.97447(23) 0.4 0.2 |Vud|2+|Vus|2+|Vub|2= =1.00031(61) 0 A 10 14 18 22 30 34 42 26 38 2.0 Tz=0 Tz=1 1.5 dC [%] 1.0 0.5 W.Satuła, J.Dobaczewski, W.Nazarewicz, M.Rafalski, PRL106 (2011) 132502 0 A 26 42 50 66 74 34 58

  21. 0.976 0.975 H&T’08 our model 0.974 |Vud| 0.973 Liang et al. n-decay 0.972 superallowedb-decay 0.971 0.970 T=1/2 mirror b-transitions p+-decay

  22. Confidencelevel test based on the CVC hypothesis 2.5 (EXP) Towner & Hardy, PRC82, 065501 (2010) 2.0 Ft dC = 1+dNS - (SV) dC ‚ ft(1+dR) (EXP) dC 1.5 MinimizeRMS deviation between the caluclated and experimentaldC with respect to Ft 1.0 dC [%] 0.5 c2/nd=5.2 for Ft = 3070.0s 75% contribution to the c2comes from A=62 0 Z of daughter 0 5 10 15 20 25 30 35 40

  23. 0 10 20 30 40 50 60 70 80 SUMMARY OF THE CALCULATIONS Ncutoff=12 Ncutoff=10 2.5 Tz Tz+1 A=58 2.0 A=38 1.5 dC [%] 1.0 A=18 0.5 Tz= -1 Tz= 0 0 A

  24. Summary Elementaryexcitationsinbinary systems maydiffer fromsimpleparticle-hole (quasi-particle) exciatations especiallywheninteractionamongparticlesposseses additionalsymmetry (like the isospinsymmetry in nuclei) Projectiontechniquesseem to be necessary to account for thoseexcitations - how to constructnon-singularEDFs? [Isospinprojection, unlike the angular-momentum and particle-number projections, ispractically non-singular !!!] Superallowed 0+0+beta decay:  encomapsextremelyrichphysics: CVC, Vud, unitarity of the CKM matrix, scalarcurrents… connectingnuclear and particlephysics  … thereisstillsomething to do indc business … Pairing & other (shapevibrations) correlationscan be „realtivelyeasily” incorporatedinto the scheme by combiningprojection(s) with GCM

  25. „NEW OPPORTUNITIES” IN STUDIES OF THE SYMMETRY ENERGY: n p a’sym E’sym = a’symT(T+1) In infinitenuclear matter we have: SLy4 SLy4L 1 6 SLy4: SkML* 2 asym=32.0MeV SV 4 SV: a’sym[MeV] asym=32.8MeV SkM*: 2 m asym=30.0MeV m* asym= eF + aint 0 T=1 T=0 SLy4: 14.4MeV SV: 1.4MeV SkM*: 14.4MeV 10 20 30 40 50 A (N=Z)

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