tens of MeV

tens of MeV+ NNN + ....

3D angular momentum and isospin restored calculationswith the Skyrme EDF

Wojciech Satuła

ab initio

Intro: define fundaments my model is „standing on” sp mean-field (or nuclear DFT) beyond mean-field (projection after variation)

Summary

Symmetry (isospin) violation and restoration: unphysical symmetry violation isospin projection Coulomb rediagonalization (explicit symmetry violation)

in collaboration with J. Dobaczewski, W. Nazarewicz & M. Rafalski

structural effects SD bands in 56Ni ISB corrections to superallowed beta decay

isospin impurities in ground-states of e-e nucleiResults

Skyrme-force-inspired local energy density functional(without pairing)

Skyrme (nuclear) interaction conserves: rotational (spherical) symmetry isospin symmetry: Vnn = Vpp = Vnp (in reality approximate)

LS LS LS

Mean-field solutions (Slater determinants) break (spontaneously) these symmetries

Y | v(1,2) | Yaverage Skyrme interaction (in fact a functional!) over

the Slater determinant

local energy density functional

Deformation (q)

Tota

l ene

rgy

(a.u

.)

Symmetry-conserving

configurtion

Symmetry-breaking

configurations

SV is the only Skyrme interactionBeiner et al. NPA238, 29 (1975)

Euler anglesin space or/and isospace

gauge angle

Restoration of broken symmetry

rotated Slater determinantsare equivalent

solutions

where

Beyond mean-field multi-reference density functional theory

There are two sources of the isospin symmetry breaking:- unphysical, caused solely by the HF approximation- physical, 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>:

in order to create good isospin„basis”:

Apply the isospin projector:

Isospin symmetry restoration

Engelbrecht & Lemmer, PRL24, (1970) 607

Diagonalize total Hamiltonian in„good isospin basis” |a,T,Tz> takes physical isospin mixing

aC = 1 - |aT=Tz

|2AR n=1

aC = 1 - |bT=|Tz||2

BR

See: Caurier, Poves & Zucker, PL 96B, (1980) 11; 15

0

0.2

0.4

0.6

0.8

1.0

aC

[%]

40 44 48 52 56 60Mass number A

0.01

0.1

1

44 48 52 5640 60

0

0.2

0.4 BRARSLy4

Ca isotopes:

eMF = 0

eMF = e

Numerical results:(I) Isospin impurities in ground states of e-e nuclei

Here the HF is solved without Coulomb |HF;eMF=0>.

Here the HF is solved with Coulomb |HF;eMF=e>.

In both cases rediagonalizationis performed for the total Hamiltonian including Coulomb

W.Satuła, J.Dobaczewski, W.Nazarewicz, M.Rafalski, PRL103 (2009) 012502

0123456

00.20.40.60.81.0

20 28 36 44 52 60 68 76 84 92A

ARBR

SLy4

aC [%

]E

-EH

F [M

eV]

N=Z nuclei

100

This is not a single Slater determinatThere are no constraints on mixing coefficients

AR

AR

BR

BR

(II) Isospin mixing & energy in the ground states of e-e N=Z nuclei:

~30%DaC

HF tries to reduce the isospin mixing by:

in order to minimize the total energy

Projection increases the ground state energy(the Coulomb and symmetryenergies are repulsive)

Rediagonalization (GCM)

lowers the ground state energy but only slightlybelow the HF

Excitation energy of the T=1 doorway state in N=Z nuclei

20

25

30

35

20 40 60 80 100A

SIII SLy4 SkP

E(T

=1)-

EH

F [M

eV]

meanvalues

Sliv & Khartionov PL16 (1965) 176

based on perturbation theoryDE ~ 2hw ~ 82/A1/3 MeV

Bohr, Damgard & Mottelsonhydrodynamical estimate

DE ~ 169/A1/3 MeV

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]

4567

aC [%

] 100Sn

SkO

SIIIMSk1

SkP SLy5

SLy4SkO’

SLySkPSkM*

SkXc

Dl=0, Dnr=1 DN=2

aligned configuration

p pnn

anti-aligned configurationn p or n porn p n p

T=0

Isospin projection

T=1

T=0n p n p

Mean-field

nn p p

four-fold degeneracy of the sp levels

Spontaneous isospin mixing in N=Z nuclei in other but isoscalar configs

yet another strong motivation for isospin projection

D. Rudolph et al. PRL82, 3763 (1999)

f7/2

f5/2p3/2

neutrons protons

4p-4h

[303]7/2

[321]1/2

Nilsson

1

space-spin symmetric

2

f7/2

f5/2p3/2

neutrons protons

g9/2 pp-h

two isospin asymmetricdegenerate solutions

Isospin symmetry violation insuperdeformed bands in 56Ni

4

8

12

16

20

5 10 15 5 10 15

Exp. band 1Exp. band 2Th. band 1Th. band 2

Angular momentum Angular momentum

Exc

itatio

n en

ergy

[MeV

] Hartree-Fock Isospin-projection

aC [%

]

band 12468 band 2

56Ni

Mean-field

pph

nph

T=0

T=1centroiddET

dET

Isospin projection

W.Satuła, J.Dobaczewski, W.Nazarewicz, M.Rafalski, PRC81 (2010) 054310

SVD

SVD eigenvalues(diagonal matrix)

Isospin-projection is non-singular:

singularity (if any) at b=p

is inherited by r

1 + |N-Z| - h + |N-Z|+2kk is a multiplicity of zero singular values

h > 3in the worst case

W.Satuła, J.Dobaczewski, W.Nazarewicz, M.Rafalski, PRC81 (2010) 054310

ij

-1r =S yi* Oij jj

~

0

10

20

30

40

1 3 5 7

aC

[%]

2K

isospin

isospin & angular momentum

0.586(2)%

42Sc – isospin projection from [K,-K] configurations with K=1/2,…,7/2

Isospin and angular-momentum projected DFT is ill-defined except for the hamiltonian-driven functionals

there is no alternative but Skyrme SV

0.0001

0.001

0.01

0.1

1

0.0 0.5 1.0 1.5 2.0 2.5 3.0

|OV

ER

LA

P|

bT [rad]

only IP

IP+AMP

p

inverse of theoverlap matrix

space & isospin rotatedsp state

HF sp state

Why we have to to use Skyrme-V?

T

ij

-1r =S yi* Oij jj

Primary motivation of the project isospin corrections

for superallowed beta decay

s1/2

p3/2

p1/2

p2

8

n p2

8

n

d5/2

14O 14NHartree-Fock

Experiment:Fermi beta decay:

f statistical rate function f (Z,Qb)t partial half-life f (t1/2,BR)

GV vector (Fermi) coupling constant <t+/-> Fermi (vector) matrix element

|<t+/->|2=2(1-dC)

Tz=-/+1 J=0+,T=1

J=0+,T=1t+/-

BR

(N-Z=-/+2)

(N-Z=0)Tz=0Qb

t1/2

10 cases measured with accuracy ft ~0.1% 3 cases measured with accuracy ft ~0.3%

~2.4%Marciano & Sirlin,

PRL96, 032002, (2006)

nucleus-independent

~1.5% 0.3% - 2.0%

eg

n

NS-independent

The 13 precisely known transitions, after including theoretical corrections, are used to

NS-dependent

test the CVC hypothesis

Towner & HardyPhys. Rev. C77, 025501 (2008)

Towner, NPA540, 478 (1992)PLB333, 13 (1994)

eg

n

courtesy of J.Hardy

one can determine

mass eigenstates

CKMCabibbo-Kobayashi-Maskawaweak eigenstates

With the CVC being verified and knowing Gm (muon decay)

test unitarity of the CKM matrix

0.9491(4) 0.0504(6) <0.0001

|Vud|2+|Vus|2+|Vub|2=0.9996(7)

|Vud| = 0.97425 + 0.00023

test of three generation quark Standard Model of electroweak interactions

Towner & HardyPhys. Rev. C77, 025501 (2008)

„Hidden” model dependence

Liang & Giai & MengPhys. Rev. C79, 064316 (2009)

spherical RPACoulomb exchange treated in the

Slater approxiamtion

dC=dC2+dC1shell

modelmeanfield

Miller & SchwenkPhys. Rev. C78 (2008) 035501;C80 (2009) 064319

radial mismatch of the wave functions

configuration mixing

Isobaric symmetry violation in o-o N=Z nuclei

ground stateis beyond mean-field!

T=0n pT=0

T=1n p

Mean-field can differentiate between n p and n p

only through time-odd polarizations!

aligned configurationsn p

nn p p

n panti-aligned configurations

or n por n p

nn p pCORE CORE

Tz=-/+1 J=0+,T=1

J=0+,T=1t+/-

BR

(N-Z=-/+2)

(N-Z=0)Tz=0Qb

t1/2

ISOSPIN PROJECTION

MEAN FIELD

Hartree-Fock

ground statein N-Z=+/-2 (e-e) nucleus

antialigned statein N=Z (o-o) nucleus

Project on good isospin (T=1) and angular momentum (I=0)

(and perform Coulomb rediagonalization)

<T~1,Tz=+/-1,I=0| |I=0,T~1,Tz=0>t+/-

CPU~ few h

~ few years

14O 14NH&T dC=0.330%

L&G&M dC=0.181%our: dC=0.303% (Skyrme-V; N=12)

~ ~

Project on good isospin (T=1) and angular momentum (I=0)

(and perform Coulomb rediagonalization)

10 14 18 22 30 34 42

d C [%

]

00.20.40.60.81.01.21.4

A

Tz= 1 Tz=0

26 42 50 66 74 A0

0.5

1.0

1.5

2.0

d C [%

]

Tz=0 Tz=1

26 38

34 58

Vud=0.97418(26)

Vud=0.97447(23)

Ft=3071.4(8)+0.85(85)

Ft=3070.4(9)our (no A=38):

H&T:

|Vud|2+|Vus|2+|Vub|2==1.00031(61)

W.Satuła, J.Dobaczewski, W.Nazarewicz, M.Rafalski,

PRL106 (2011) 132502

0.970

0.971

0.972

0.973

0.974

0.975

0.976

|Vud

|

superallowed b-decay

p+-decay

n-decay

T=1/2 mirrorb-transitions

H&T’08

Liang et al.

ourmodel

0 5 10 15 20 25 30 35 40

dC(SV)

dC(EXP)

d C [

%]

Z of daughter

0

0.5

1.0

1.5

2.0

2.5

Confidence level test based on the CVC hypothesisTowner & Hardy, PRC82, 065501 (2010)

dC = 1+dNS - Ftft(1+dR)

‚(EXP)

Minimize RMS deviationbetween the caluclated and experimental dC withrespect to Ft

c2/nd=5.2for Ft = 3070.0s

75% contribution to thec2 comes from A=62

0

0.5

1.0

1.5

2.0

0 10 20 30 40 50 60 70 80

2.5Tz Tz+1

Tz = -1Tz = 0

d C [

%]

A

Ncutoff=10 Ncutoff=12SUMMARY OF THE CALCULATIONS

A=18

A=38

A=58

Summary

[Isospin projection, unlike the angular-momentum and particle-number projections, is practically non-singular !!!]

Elementary excitations in binary systems may differfrom simple particle-hole (quasi-particle) exciatationsespecially when interaction among particles posseses additional symmetry (like the isospin symmetry in nuclei)

Superallowed 0+0+beta decay: encomaps extremely rich physics: CVC, Vud, unitarity of the CKM matrix, scalar currents… connecting nuclear and particle physics … there is still something to do in dc business …

Projection techniques seem to be necessary to account for those excitations - how to construct non-singular EDFs?

Pairing & other (shape vibrations) correlations can be„realtively easily” incorporated into the scheme by combining projection(s) with GCM

0

2

4

6

10 20 30 40 50

a’sy

m [

MeV

]

SV

SLy4LSkML*

SLy4

A (N=Z)

„NEW OPPORTUNITIES” IN STUDIES OF THE SYMMETRY ENERGY:

T=0

T=1n p

E’sym = a’symT(T+1)

12a’sym

asym=32.0MeV

asym=32.8MeV

SLy4:

In infinite nuclear matter we have:

SV:

asym=30.0MeVSkM*:

asym= eF + aintmm*

SLy4: 14.4MeV SV: 1.4MeV SkM*: 14.4MeV