Nuclear Schiff moment · From the experiment this implies for the Schiff moment ∣S Hg ∣ 0.75×10−11efm3 ∣dp∣ 3.8×10 −24 ecm, ∣d n∣ 4.0×10 −25 ecm. Existing upper

Nuclear Schiff moment

V.F. Dmitriev,Budker Institute of Nuclear Physics,

Novosibirsk, Russia

R.A. Sen'kov, I.B. Khriplovich, V.V. Flambaum

Schiff theorem

(0) 0=E

extE

The energy of a neutral atom with a point like nucleus in an external electric field does not depend on a nuclear dipole moment.

This is true for electron dipole moment as well.

intE

Schiff moment operator (P- T-violating nuclear forces)

( ) ( )2

0 2a

a a b ext a exta a b a

H e V em

φ≠

= − + − − ⋅ + ⋅∑ ∑ ∑pr r r d E rΕ

0[ , ] 0e H⟨ ⟩ =P

( ) 0a a exta

e ZeEφ⟨ ∇ ⟩ − =∑ rr r

In first order in a dipole moment and an external field we can add to the Hamiltonian the following expression:

( )a a exta

dd E

Zφ− ⋅ ∇ + ⋅∑ r

rrr r

One should remember that in this equation ( )φ r

should be understood as a mean value over nuclear ground state,since

( ) ( )nuc nuc nucd dφ φ⟨ ⋅∇ ⟩ ≠ ⟨ ⟩ ⋅ ⟨∇ ⟩r rr rr r

We obtain ( )0' a a exta

dH H d E

Zφ= − ⋅ ∇ + ⋅∑ r

rrr r

( ) ( )2

'2

aa a b a ext

a a b a

H e V em

φ≠

= − + − + ⋅∑ ∑ ∑pr r r rΕ%

( ) ( ) ( )3 3' '' '| ' | | ' |

c cdd r d r

Ze

ρ ρφ ⋅ ∇= +

− −∫ ∫r r

rr r r r

r r%where

Expansion of ( )cρ r

( ) ( ) ( ) ( ) ( )

( )

1

2

1...

6

i i i j i jc p p p p

p p p

i j k i j kp p p

p

e Ze er er r

er r r

ρ δ δ δ δ

δ

= − ≈ − ∇ + ∇ ∇

− ∇ ∇ ∇ +

∑ ∑ ∑

∑

r r r r r r

r

1 1( ) ( )

5 5i j k k i j k i j

ij jk ik ij jk ikδ δ δ δ δ δ∇ ∇ ∇ − ∇ + ∇ + ∇ ∆ + ∇ + ∇ + ∇ ∆

Dipole component

( )4 i iSπ δ∇ r where

21 1 110 6 15

iji i i jp c p

p p p

QS e r r e r r e r

Z

⟨ ⟩= ⟨ ⟩ − ⟨ ⟩⟨ ⟩ − ⟨ ⟩∑ ∑ ∑

Si=1

10∑ p er p

2 r i−53∑ p er i ⟨r c

2 ⟩−23

⟨Qij ⟩

Ze∑ p er p

j

The form of the above equation for the schiff moment is unique.The coefficient 5/3 is fixed by  condition of  absence of the ghostdipole mode which corresponds to displacement of the nucleusas a whole.

Under  small displacement the nuclear isoscalardensity transforms as 0 ra=0 r a

∂0

∂ r

The transition density is: tr r =∂0

∂ r

The absence of the ghost mode means that the integralof the isoscalar part of the Schiff moment operatorwith this transition density must yield zero.

∫d3 r S0i r ∂0

∂ ri=∫d3 r 1

10r2 r i−

53

r i ⟨r 02 ⟩−

23

Qij

Zr j∂0

∂r i=0

P and T violating NN­interaction

W rr 1−rr 2=−g

8m p

[ g0 r1⋅r2g2 r1⋅r2−3132

3 r1− r2g113 r1−2

3 r2]⋅r∇ 1

exp −mr12

r12

.

Opposite parity contribution to nuclear mean field

U dir rr =gm

2

m p

r⋅rn3∫0

∞r ' 2 dr ' br , r ' [ g0−2g2 pr ' −n r ' g1p r ' nr ' ]

Correction to a single particle orbit wave function r

⟨ ∣S∣ ⟩=⟨∣S∣ ⟩⟨∣S∣ ⟩

Z−NA

   Core polarizationResponse of the core particles to the strong residual interactionwith the valence nucleon creates an additional contribution to theSchiff moment.

⟨ '∣%S∣⟩=⟨ '∣S0∣ ⟩∑ ' ⟨ ∣%S∣ ' ⟩

n−n '

− '⟨ ' '∣FW∣⟩

∣⟩

⟨ ∣%S∣ ⟩=⟨∣%S∣ ⟩⟨∣%S∣ ⟩⟨∣ S∣ ⟩

Elimination of the ghost mode contribution

S r∣= ar 2−0

2br∣=0O

2

Eex2

J nr =∮n S r∣d 0≪∣∣≪E ex

b r∣=0 =J−1r 0

2=J 3r / J1r

The proton component of the Schiff moment. Solid curve is therenormalized operator after subtraction of the ghost mode, dashed curveis the bare operator.

The neutron component of the renormalized Schiff moment.

Results for Mercury nucleus.

S=−0.0004g g0−0.055gg10.009g g2[ e⋅ fm3 ]

For atomic EDM we have the upper bound d Hg2.1×10−28 e⋅cm

Combining our result with calculations of the atomic EDM(Flambaum et al.) we obtain g10.5×10−11 .

The standard model estimates is g1≈10−17

S=−0.007gg0−0.071gg10.018gg2

J. H. de Jesus and J. Engel (2005)

0.057 ÷ 0.090

Nucleon dipole moments contribution

( ) ( ) ( )3 3' '' '| ' | | ' |

c cdd r d r

Ze

ρ ρφ ⋅ ∇= +

− −∫ ∫r r

rr r r r

r r%

crr =∑ p

err−rr p−∑a

rda⋅r∇ rr−rra

rd=∑arda

Repeating the expansion over nucleon coordinates and separating thedipole component we obtain for the Schiff moment

S i=16∑a da

i r a2−⟨r 2 ⟩c

115∑a da

j [3r ai r a

j−ij ra2−⟨Qij ⟩

Ze]

⟨ '∣%S∣⟩=⟨ '∣S0∣ ⟩∑ ' ⟨ ∣%S∣ ' ⟩

n−n '

− '

⟨ ' '∣F∣⟩

Polarization effects for theneutron component of the Schiffmoment.

Radial dependence of the protoncomponent of the Schiff momentinduced by core polarization. Dashedcurve is the tensor component.

Results for Hg

S=s p d psn d n s p=0.20±0.02 fm2 sn=1.895±0.035fm2

From the atomic calculations, cited above, wehave

d=−2.8×10−17S [e fm3 ]From the experiment this implies for the Schiff moment

∣SHg∣ 0.75×10−11e fm3

∣dp∣ 3.8×10−24 e cm , ∣dn∣ 4.0×10−25 e cm.Existing upper limit for the neutron dipole momentis: dn 0.63×10−25 e cm

For the proton dipole moment the existing upperlimit was:

dp −4±6×10−23 e cm

Theoretical uncertainty

The value +- 0.02 cited above does not reflects the real accuracy of the theory.It came from the differences in adopted values of the residual interactionconstants. The uncertainty in calculations of the core polarization using RPAwith effective forces can be estimated from the following considerations. UsingRPA with effective forces we can fit different nuclear moments in one nucleus.Then, in neighbor nuclei the calculated moments will differ from the data. Thisdifference can be regarded as an uncertainty in our theory. In our experiencethis difference is of the order of 20% on average, reaching sometimes the valueof 30%. To be safe we can adopt a conservative 30% uncertainty in calculationsof . It gives sp

sp=0.2±0.06 fm2

For the proton dipole moment we obtain ∣dp∣ 5.4×10−24 e cm

Relativistic correctionsRelativistic corrections appear in higher order terms of charge densityexpansion. The first correction comes from

( )1

120i j k l m i j k l mp p p p p

p

r r r r r δ− ∇ ∇ ∇ ∇ ∇∑ r

The atomic matrix element for the correction contains higher order derivatives of the electron wave functions at r= 0.It is convenient to expand the product of radial wave functions near r= 0.For Dirac wave functions

( ) ( ) ( )1

( ) ks p s p k

k

f r f r g r g r b r=

+ = ∑With this expansion all the corrections can be summed in a new P- and T- odd so called LOCAL DIPOLE MOMENT

1 1

1,3,5...

1 4

( 1)( 4) 3k k

kk

ke b r r

k k+ +

=

+ = ⟨ ⟩ − ⟨ ⟩⟨ ⟩ + + ∑L r r

The first term is just the Schiff moment and the other terms arethe relativistic corrections. The coefficients bk can be calculated analyticallyfor a uniformly charged sphere. For the first correction we have

2

3 1 1/ 22

3 ( ) for atomic transition

5 N

Zb b s p

R

α= − −

2

3 1 3/ 22

9 ( ) for atomic transition

20 N

Zb b s p

R

α= − −

Results for the first correction4 41 7

'28 3

r r = ⟨ ⟩ − ⟨ ⟩⟨ ⟩

L r r

Conclusions

Calculations of nuclear Schiff moments combined withatomic structure calculations and data from atomicexperiments can bring new information about suchfundamental quantities like nucleon dipole moments.Effects of nuclear polarization are important and should betaken into consideration in calculations of the Schiffmoment. Relativistic corrections are sizable for heavy atoms andshould not be ignored either.