tens of MeV

tens of MeV+ NNN + ....

MICROSCOPIC CALCULATIONS OF ISOSPIN IMPURITIES AND ISOSPIN-SYMMETRY-BREAKING CORRECTIONS USING ISOSPIN AND ANGULAR-MOMENTUM PROJECTD DFT

Wojciech Satuła

ab initio

Intro: effective low-energy theory for medium mass and heavy nuclei mean-field (or nuclear DFT) beyond mean-field (projection)

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 superallowed beta decay

isospin impurities in ground-states of e-e nuclei

symmetry energy – new opportunities of study

Effective theories for low-energy (low-resolution) nuclear physics (I):

Low-resolution separation of scales which isa cornerstone of all effective theories

Fourierlocal

correctingpotential

hierarchy of scales:2roA1/3

ro~ 2A1/3

is based on a simple and very intuitive assumption that low-energy

nuclear theory is independent on high-energy dynamics

~ 10

The nuclear effective theory

Long-range part of the NN interaction(must be treated exactly!!!)

where

regularizationCoulomb

ultravioletcut-off

denotes an arbitrary Dirac-delta model

Gogny interaction

przykład

There exist an „infinite” number

of equivalent realizationsof effective theories

lim daa 0

Skyrme interaction - specific (local) realization of the nuclear effective interaction:

spin-orbitdensity dependence10(11)

parameters

Y | v(1,2) | YSlater determinant

(s.p. HF states are equivalent to the Kohn-Sham states)

Skyrme-force-inspired local energy density functional

local energy density functional

relative momenta spin exchange

LO

NLO

SV

Elongation (q)

Tota

l ene

rgy

(a.u

.)

Symmetry-conserving

configuration

Symmetry-breaking

configurations

Skyrme (nuclear) interaction conserves such symmetries like: rotational (spherical) symmetry isospin symmetry: Vnn = Vpp = Vnp (in reality approximate) parity…

LS LS LS

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

Euler angles 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>:

Calculate the projected energy andthe Coulomb mixingBefore Rediagonalization:

BR

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

BR

in order to create good isospin„basis”:

Apply the isospin projector:

Isospin symmetry restoration

Engelbrecht & Lemmer, PRL24, (1970) 607

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

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

Isospin invariant

Isospin breaking: isoscalar, isovector & isotensor

aC = 1 - |aT=Tz

|2AR n=1

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

Position 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 estimateDE ~ 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

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

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

Experiment world data survey’08

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

nucleus-independent

~2.4%Marciano & Sirlin, PRL96 032002 (2006)

~1.5% 0.3% - 1.5%

T&H, PRC77, 025501 (2008)

What can we learn out of it?From a single transiton we can determine experimentally:

GV2(1+DR) GV=const.

From many transitions we can: test of the CVC hypothesis

(Conserved Vector Current)

exotic decays Test for presence of a Scalar Currentse

e J.H

ardy

, ENA

M’0

8 pr

esen

tatio

n

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

0.9490(4) 0.0507(4) <0.0001

|Vud|2+|Vus|2+|Vub|2=0.9997(6)

|Vud| = 0.97418 + 0.00026

test of three generation quark Standard Model of electroweak interactions

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

Hardy &TownerPhys. Rev. C77, 025501 (2008)

Model dependence

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

spherical RPACoulomb exchange treated in the

Slater approxiamtion

dC=dC1+dC2shell

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

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

-7/2

7/2

-5/2-3/2-1/2 1/2 3/2 5/2

f7/2

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

pr =S yi

* Oij jjij

-1

inverse of theoverlap matrix

space & isospin rotatedsp state

HF sp state

T

Singularities force us to use interaction-driven functional SV

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=0.97444(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)

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 hypothesisT&H 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

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

Summary and outlook

[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 beta decay: encomapsses 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?

How to include pairing into the scheme?

-1.5

-1.0

-0.5

0.0

0.5

20 25 30 35 40 45A

Fig43:110427DE

exp-DE

th [

MeV

]

Nolen-Schiffer anomaly in mirror symmetric nucleiVpp-Vnn

SLy4 (HF)SV (HF)SV (PROJ)

Mirror-symmetric nuclei (preliminary)