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ISOSPIN MIXING PHENOMENA IN THE VICINITY OF N=Z LINE

ISOSPIN MIXING PHENOMENA IN THE VICINITY OF N=Z LINE. Wojciech Satuła. in collaboration with J. Dobaczewski , W. Nazarewicz, M. Rafalski & M. Borucki. Intro : effective low-energy theory for medium mass and heavy nuclei 

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ISOSPIN MIXING PHENOMENA IN THE VICINITY OF N=Z LINE

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  1. ISOSPIN MIXING PHENOMENA IN THE VICINITY OF N=Z LINE WojciechSatuła incollaborationwith J. Dobaczewski, W. Nazarewicz, M. Rafalski & M. Borucki Intro: effectivelow-energytheory for medium mass and heavy nuclei mean-field(ornuclear DFT)  beyondmean-field (projection) Symmetry (isospin) violation and restoration: • unphysicalsymmetryviolation  isospinprojection • Coulomb rediagonalization(explicitsymmetryviolation) isospinimpuritiesinground-states of e-e nuclei structuraleffects SD bandsin56Ni superallowed beta decay symmetry energy – newopportunitiesof studyingwiththeisospinprojection Summary ab initio + NNN + .... tens of MeV

  2. Effectivetheories for low-energy(low-resolution) nuclearphysics (I): Low-resolution separation of scaleswhichis a cornerstone of alleffectivetheories

  3. Thenucleareffectivetheory There exist an „infinite” number of equivalent realizations of effective theories is based on a simple and very intuitive assumption that low-energy nuclear theory is independent on high-energy dynamics ultraviolet cut-off Fourier regularization Coulomb Long-range part of the NN interaction (must be treated exactly!!!) hierarchy of scales: 2roA1/3 2A1/3 ~ local correcting potential ro ~ 10 denotes an arbitrary Dirac-delta model where przykład Gogny interaction

  4. Skyrme interaction - specific (local) realization of the nuclear effective interaction: lim da a 0 spin exchange relative momenta LO NLO 10(11) SV density dependence parameters spin-orbit Skyrme-force-inspiredlocal energy densityfunctional Y | v(1,2) | Y Slater determinant (s.p. HF states are equivalent to the Kohn-Sham states) local energy density functional

  5. Symmetry-conserving configuration Total energy (a.u.) Symmetry-breaking configurations Elongation (q) Skyrme (nuclear) interactionconservessuchsymmetrieslike:  rotational (spherical) symmetry  isospinsymmetry: Vnn= Vpp= Vnp(in reality approximate)  parity… LS LS LS Mean-fieldsolutions (Slaterdeterminants) break (spontaneously) thesesymmetries

  6. Restoration of brokensymmetry Euler angles gauge angle Beyond mean-fieldmulti-referencedensityfunctionaltheory rotated Slater determinants are equivalent solutions where

  7. Applytheisospinprojector: in order to creategoodisospin „basis”: BR 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 Calculatetheprojected energy and the Coulomb mixing BeforeRediagonalization:

  8. n=1 AR aC= 1 - |aT=Tz|2 Diagonalizetotal Hamiltonian in „goodisospinbasis” |a,T,Tz>  takesphysicalisospinmixing Isospin breaking: isoscalar, isovector & isotensor Isospin invariant

  9. 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

  10. 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

  11. Isospinmixingin 80Zr from GDR gamma decaystudies 80Zr SLy7 MSk1 SkP communicated by Franco Camera atthe Zakopane’10 meeting SLy5 4.5 SLy4 SkO’ SkM* SkXc aC (AR) aC [%] 4.0 aC SIII (BR) SkM* 3.5 MSk1 SLy4 SkXc SkO’ SIII SkP SLy7 3.0 SLy5 (AR) ET=1 [MeV] 29.5 30.0 30.5 31.0 31.5 doorway state energy

  12. 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)

  13. 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

  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: d5/2 8 8 p1/2 p3/2 2 2 f statisticalratefunctionf (Z,Qb) s1/2 n t partialhalf-lifef (t1/2,BR) n p p GVvector (Fermi) couplingconstant 14N 14O <t+/-> Fermi (vector) matrix element Hartree-Fock |<t+/->|2=2(1-dC)

  15. Experimentworld data survey’08 PRC77, 025501 (2008) 10 casesmeasuredwithaccuracyft ~0.1% 3 casesmeasuredwithaccuracyft ~0.3% nucleus-independent ~1.5% 0.3% - 1.5% ~2.4% Marciano & Sirlin, PRL96 032002 (2006)

  16. Whatcan we learn out of it? From a single transiton we candetermineexperimentally: GV2(1+DR)  GV=const. From many transitions we can:  test of the CVC hypothesis (ConservedVectorCurrent)  exoticdecays Test for presence of a ScalarCurrent

  17. 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

  18. Model dependence Hardy &Towner Phys. Rev. C77, 025501 (2008) dC=dC1+dC2 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

  19. 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!

  20. 40 30 20 isospin & angularmomentum 10 isospin 0 1 3 5 7 42Sc – isospinmixinginnKpKantialignedconfigurations for K=1/2,3/2,5/2, and 7/2 aC [%] ( ( 0.586(2)% ( ( 2K

  21. 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) ~5h ~50000h 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)

  22. H&T: 1.2 Ft=3071.4(8)+0.85(85) Tz=-1  Tz=0 1.0 Vud=0,97418(26) our (no A=38): 0.8 Ft=3069.2(8) dC [%] 0.6 Vud=0,97466 0.4 0.2 0 A 10 15 20 25 30 35 40 2.0 Tz=0  Tz=1 1.5 dC [%] 1.0 our 0.5 0 A 30 40 50 60 70

  23. „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)

  24. Summary and outlook Elementaryexcitationsinbinary systems maydiffer fromsimpleparticle-hole (quasi-particle) exciatations especiallywheninteractionamongparticlespossesesadditional symmetry (liketheisospinsymmetryinnuclei) Projectiontechniquesseem to be necessary to account for thoseexcitations - how to constructnon-singularEDFs? Isopinprojection, unlikeangular-momentum and particle-numberprojections, ispracticallynon-singular !!! Superallowed beta decay:  encomapssesextremelyrichphysics: CVC, Vud, unitarity of the CKM matrix, scalarcurrents… connectingnuclear and particlephysics  … thereisstillsomething to do indc business …

  25. 1 0.1 onlyIP |OVERLAP| 0.01 0.001 IP+AMP r =Syi*Oijjj -1 0.0001 p ij 0.0 0.5 1.0 1.5 2.0 2.5 3.0 bT [rad] HF sp state space & isospinrotated sp state inverse of the overlap matrix

  26. time-even time-odd Qb – Qb [MeV] exp th Isospin symmetry violation due to time-odd fields in the intrinsic system time-odd Isobaric analogue states: T=1,Tz=1 T=1,Tz=-1 T=1,Tz=0 0 0 10 10 20 20 30 30 40 40 50 50 60 60 o-o e-e e-e Qb values in super-allowed transitions 4 3 isospin projected isospin projected 2 0,2% 4,1% 29,9% 0,9% 2,5% 10,1% 1 21,7% 15,1% 0,9% 1,5% 3,7% 26,3% 0,8% 0 7,9% Hartree-Fock -1 Hartree-Fock Atomic number Atomic number

  27. 100Sn 100Sn 18 18 19 19 20 20 21 21 22 22 asym(rNM/2) [MeV] asym(rNM/2) [MeV] 1.1 1.1 7 7 1.0 1.0 40Ca 40Ca 6 6 0.9 0.9 5 5 0.8 0.8 4 4 aC [%] 0.7 0.7 29 29 30 30 31 31 32 32 asym(rNM) [MeV] asym(rNM) [MeV] SkP MSk1 SkM* MSk1 SkM* SkP SkXc SkO’ SkO’ aC [%] SLy SkXc SLy SIII SkO SLy SkO SIII MSk1 SLy SLy5 MSk1 SkP SkP SkM* SLy SkM* SLy5 SIII SkO’ SkXc SkXc SkO’ SkO SkO SIII

  28. 0.35 0.30 0.25 0.20 6 8 10 12 Towner & Hardy 2008 dC [%] Liang et al. (NL3) Number of shells

  29. n n n n p p p p n n n p p p n p n p n n p p After applying „naive” isospin projection we get: T=1 T=0 T=0 Mean-field can differentiate between ground state is beyond mean-field! only through time-odd polarizations! and Isobaric symmetry breaking in odd-odd N=Z nuclei Let’s consider N=Z o-o nucleus disregarding, for a sake of simplicity, time-odd polarization and Coulomb (isospin breaking) effects 4-fold degeneracy CORE CORE anti-aligned configurations aligned configurations n p or or

  30. Position of the T=1 doorway state in N=Z nuclei 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 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

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