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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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slide1

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

slide2

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)

slide3

Restoration of brokensymmetry

Eulerangles

in spaceor/and isospace

gauge angle

Beyond mean-fieldmulti-referencedensityfunctionaltheory

rotated Slater

determinants

are equivalent

solutions

where

slide4

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

slide5

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

slide6

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

slide7

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

slide8

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

slide9

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)

slide10

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

slide11

-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

slide12

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

slide13

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

slide14

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%

slide15

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

slide16

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

slide17

„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

slide18

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!

slide19

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)

slide20

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

slide21

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

slide22

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

slide23

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

slide24

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

slide25

„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)