Kilka słów o strukturze jądra atomowego dla matematyków Wojciech Satuła From ab initio toward „rigorous” effective theory for light nuclei Principles of low-energy nuclear physics  effective theories for medium-mass and heavy systems Nuclear DFT  coupling constants & fitting strategies  single-particle fingerprints of tensor interaction Extensions of the nuclear DF up to N3LO Multi-reference DFT  beyond mean-field theories  isospin (and angular momentum) projection Summary ab initio + NNN + .... tens of MeV

Paramters (~40) are fitted to two-nucleon data Argonne V18 NN potential short range is phenomenological long range – pion exchange Paramters (~40) are fitted to two-nucleon data Triton binding: th: 7.62MeV exp: 8.48MeV

A=4-10 GFMC calculations using Argonne V18 NN potential and Illinois-2 NNN interaction Pion three-body sector of Urbana 3-body potential plus phenomenological short-range 3-body

From „infinite basis” ab initio towards finite basis „rigorous” effective theory Strategy: adopt a Hamiltonian and a basis, compute matrix elements and diagonalize N =0 =1 =2 =4 =3 =5 In ab initio many-body theory H acts in „infinite” Hilbert space Select HO Slater determinat basis and retain all A-body determinants below an oscillator Ecutoff (Nmax) finite subspace (P-space) Heff „P – space”

H : E , K Expansion in (Q/L)n VNN Repulsive core in VNN Realistic local NN interaction H : E 1 , 2 3 K d ¥ eff P model space dimension Properties of Heff for A-nucleon system A-body operator even if H is 2- or 3-body For P  1 Heff  H Expansion in (Q/L)n VNN Repulsive core in VNN cannot be accommodated in this truncated HO basis   needs regularization Vlow-k, PT and further renormalization to the finite basis (Lee-Suzuki-Okamoto)  

Modern Mean-Field Theory º Energy Density Functional Effective theories for low-energy nuclear physics in heavy(ier) nuclei: Hohenberg-Kohn-Sham Modern Mean-Field Theory º Energy Density Functional ® j, r, t, J, « T, s, F,

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

Spin-force inspired local energy density functional Skyrme interaction - specific (local) realization of the nuclear effective interaction: lim da a 0 LO NLO 10(11) density dependence parameters spin-orbit relative momenta spin exchange Spin-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

ZOO– 20 parameters are fitted to: Skyrme-inspired functional is a second order expansion in densities and currents: tensor spin-orbit density rg dependent CC 20 parameters are fitted to: Symmetric NM: - saturation density ( ~0.16fm-3) - energy per nucleon (-16 0.2MeV) - incompresibility modulus (210 20MeV) + - isoscalar effective mass (0.8) Asymmetric NM: isovector effective mass (GDR sum-rule enhancement) - symmetry energy ( 30 2MeV) + neutron-matter EOS (Wiringa, Friedmann-Pandharipande) Finite, double-magic nuclei [masses, radii, rarely sp levels]: surface properties ZOO–

How many parameters are really needed? How many parameters can be constrained by fitting global properties? SLy4 (original) Can we learn more from the sp properties? linear (re)fit to masses Bertsch, Sabbey, and Uusnakki Phys. Rev. C71, 054311 (2005) De(f5/2-f7/2) [MeV] 5 6 7 8 40Ca 48Ca 56Ni a) b) neutrons protons SkO av as + asym + s-o

Fitting strategies of the tensorial coupling constants - the idea - C1 J C0 1 3 5 7 0.7 0.8 0.9 40Ca -40 -30 -20 -10 56Ni f7/2-f5/2 p3/2-p1/2 f7/2-d3/2 2 4 6 8 -80 -60 from binding energies 48Ca f7/2-p3/2 Single-particle energies [MeV] 1) Fit of the isoscalar SO strength j< j> F 40Ca 2) Fit of the isoscalar tensor strength: j> F j< 56Ni 3) Fit of the isovector tensor strength or, more precisely, C1J/C1 D J 48Ca 48Ni or 78Ni are needed in order to fix SO-tensor sector f7/2 f5/2 splittings around SkPT T0=-39(*5);T1=-62(*-1.5);SO*0.8

SLy4T Single-particle fingerprints of tensor interaction spin-orbit splittings Spin-orbit splittings [MeV] SLy4T T0=-45;T1=-60; SO*0.65 n 1h 1i f7/2-f5/2 g9/2-g7/2 1 3 5 7 16O 40Ca 48Ca 56Ni 90Zr 132Sn 208Pb p SLy4T M.Zalewski, J.Dobaczewski, WS, T.Werner, PRC77, 024316 (2008)

f7/2 f5/2 p3/2 4p-4h Nilsson DE [MeV] SkO SkOTX SkOT’ DEtensor [MeV] neutrons protons 4p-4h [303]7/2 [321]1/2 Nilsson Rudolph et al. PRL82, 3763 (1999) 2 4 6 8 10 12 0.1 0.2 0.3 0.4 DE [MeV] tensor spin-orbit deformacja b2 SkO SkOTX SkOT’ -6 -5 -4 -3 DEtensor [MeV] 0.1 0.2 0.3 0.4 b2

Singular value decomposition Fits of s.p. energies EXP: M.N. Schwierz, I. Wiedenhover, and A. Volya, arXiv:0709.3525 Singular value decomposition M. Kortelainen et al., Phys. Rev. C77, 064307 (2008) =

Possible extensions: N2LO, N3LO – higher order derivatives explicit reconstruction of the NDF N2LO, N3LO – higher order derivatives Mass dependent coupling constants New higher-order physics-motivated terms: ~t( r)2 D

Beyond mean-field Total energy (a.u.) Elongation (q) multi-reference density functional theory Spontaneous Symmetry Breaking (SSB) Elongation (q) Total energy (a.u.) Symmetry-conserving configuration Symmetry-breaking configurations

Restoration of broken symmetry Euler angles gauge angle rotated Slater determinants are equivalent solutions where

Determination of Vud matrix element of the CKM matrix Motivation: Determination of Vud matrix element of the CKM matrix from superallowed beta decay  test of unitarity  test of three generation quark Standard Model of electroweak interactions J=0+,T=1 N-Z=-2 N-Z=0 T+ Tz=-1 vector (Fermi) cc Tz=0 d5/2 nucleus-independent 8 8 p1/2 p3/2 2 2 s1/2 p n p n

Isospin symmetry breaking and restoration – general principles There are two sources of the isospin symmetry breaking: unphysical, related to the HF approximation itself physical, caused mostly by Coulomb interaction (also, but to much lesser extent, by the strong force isospin non-invariance) Engelbrecht & Lemmer, PRL24, (1970) 607 Solve SHF (including Coulomb) to get isospin symmetry broken (deformed) solution |HF>: See: Caurier, Poves & Zucker, PL 96B, (1980) 11; 15 in order to create good-isospin „basis”: Apply the isospin projection operator: BR Compute projected (PAV) energy and Coulomb mixing before rediagonalization: aC = 1 - |bT=|Tz||2 BR

aC = 1 - |aT=Tz|2 Rediagonalize the Hamiltonian in the good-isospin „basis” |a,T,Tz> in order to remove spurious isospin-mixing: aC = 1 - |aT=Tz|2 AR n=1 Isospin breaking: isoscalar, isovector & isotensor Isospin invariant

Few numerical results: Isospin mixing in ground states of e-e nuclei 0.2 0.4 0.6 0.8 1.0 aC [%] 40 44 48 52 56 60 Mass number A 0.01 0.1 1 BR AR SLy4 Ca isotopes: eMF = 0 eMF = e Here the HF is solved without Coulomb |HF;eMF=0>. Here the HF is solved with Coulomb |HF;eMF=e>. In both cases rediagonalization is performed for the total Hamiltonian including Coulomb

aC [%] (II) Isospin mixing & energy in the ground states of e-e N=Z nuclei: HF tries to reduce the isospin mixing by: 1 2 3 4 5 6 0.2 0.4 0.6 0.8 1.0 20 28 36 44 52 60 68 76 84 92 A AR BR SLy4 aC [%] E-EHF [MeV] N=Z nuclei 100 ~30% DaC in order to minimize the total energy Projection increases the ground state energy (the Coulomb and the symmetry energies are repulsive) Rediagonalization (GCM) 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

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

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 n p n p CORE CORE aligned configurations anti-aligned configurations n p n p n p or or n p T=0 After applying „naive” isospin projection we get: T=1 ground state is beyond mean-field! n p Mean-field can differentiate between and only through time-odd polarizations!

no time-odd polarizations included -66 -65 -64 -63 -62 -61 -60 E [MeV] Hartree-Fock 10C 10B Isospin projection & Coulomb rediagonalization 4275 2098 T=0 T=1 exp: 1908 Isospin projection & Coulomb rediagonalization T=1 exp: 6424 T=0 42Sc 42Ca 7784 907 Hartree-Fock 42Ca 42Sc -360 -358 -356 -354 E [MeV]

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

AMP+IP projection from the „anti-aligned” Slater determinant (very preliminary tests – no Coulomb rediagonalization) 10C very preliminary (qualitative) -65 -64 -63 -62 -61 -60 -59 J=0+ J=0+,T=1 J=0+,T=1 J=0+ Energy [MeV] very preliminary (qualitative) 10B J=1+,T=0 J=1+ J=3+,T=0 Hartree Fock AMP +IP AMP

|OVERLAP| p r =S yi* Oij jj only AMP IP+AMP bT [rad] 0.0001 0.001 0.01 0.1 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 |OVERLAP| bT [rad] only AMP IP+AMP p r =S yi* Oij jj ij -1 original sp state space-isospin rotated sp state inverse of the overlap matrix

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

Mean-field versus isospin-projected mean-field interpretation pph nph T=0 T=1 centroid dET 2 4 6 8 band 2 aC [%] band 1 Hartree-Fock Isospin-projection 4 8 12 16 20 56Ni Excitation energy [MeV] Exp. band 1 Exp. band 2 Th. band 1 Th. band 2 5 10 15 5 10 15 Angular momentum Angular momentum

SUMMARY (verbal)

Local Density Functional Theory for Superfluid Fermionic Systems: The Unitary Gas Aurel Bulgac, Phys. Rev. A 76, 040502 (2007) running coupling constant in order to renormalize.... ultraviolet divergence in pairing tensor ab initio calculations by: Chang & Bertsch Phys. Rev. A76, 021603 von Stecher, Greene & Blume, E-print:0705.0671v1

From two-body, zero-range tensor interaction towards the EDF: mean-field averaging