Schiff Moments - Physics Department at UMass AmherstThe nuclear dipole moment causes the atomic...

Schiff MomentsJ. Engel

October 23, 2014

One Way Things Get EDMsStarting at fundamental level and working up:

Underlying fundamental theorygenerates three T -violatingπNN vertices:

Then neutron gets EDM, e.g.,from chiral-PT diagrams like this:

N

?π

g

n p n

π−

γ

g g

New physics

How Diamagnetic Atoms Get EDMsNucleus can get one from nucleonEDM or T -violating NN interaction: π

g

γ

VPT ∝[g0τ1 · τ2 − g12 (τz1 + τz1 ) + g2 (3τz1τz2 − τ1 · τ2)] (σ1 − σ2)− g12 (τz1 − τz2 ) (σ1 + σ2) · (∇1 −∇2) exp (−mπ |r1 − r2|)

mπ |r1 − r2|+ contact termFinally, atom gets one from nucleus. Electronic shielding makesrelevant nuclear object the “Schiff moment” 〈S〉 ≈ 〈∑p r2pzp + . . .〉.

Job of nuclear theory: calculate dependence of 〈S〉 onthe g’s (and on the contact term and nucleon EDM).

How Does Shielding Work?Theorem (Schiff)The nuclear dipole moment causes the atomic electrons torearrange themselves so that they develop a dipole momentopposite that of the nucleus. In the limit of nonrelativisticelectrons and a point nucleus the electrons’ dipole momentexactly cancels the nuclear moment, so that the net atomicdipole moment vanishes.

How Does Shielding Work?ProofConsider atom with non-relativistic constituents (with dipolemoments ~dk ) held together by electrostatic forces. The atomhas a “bare” edm ~d ≡∑k~dk and a Hamiltonian

H = ∑k

p2k2mk +∑

kV (~rk ) −∑

k

~dk · ~Ek

= H0 +∑k (1/ek )~dk · ~∇V (~rk )= H0 + i∑

k(1/ek ) [~dk · ~pk , H0]

K.E. + Coulomb dipole perturbation

How Does Shielding Work?The perturbing HamiltonianHd = i

∑k

(1/ek ) [~dk · ~pk , H0]shifts the ground state |0〉 to|0〉 = |0〉+∑

m

|m〉〈m|Hd |0〉E0 − Em

= |0〉+∑m

|m〉〈m| i∑k (1/ek )~dk · ~pk |0〉 (E0 − Em)

E0 − Em= (1 + i∑

k(1/ek )~dk · ~pk) |0〉

How Does Shielding Work?The induced dipole moment ~d′ is~d′ = 〈0|∑

jej~rj |0〉

= 〈0|(1− i∑k (1/ek )~dk · ~pk) (∑j ej~rj

)×(1 + i∑k (1/ek )~dk · ~pk) |0〉

= i 〈0| [∑j ej~rj ,∑k (1/ek )~dk · ~pk] |0〉

= − 〈0|∑k

~dk |0〉 = −∑k

~dk

= − ~d

So the net EDM is zero!

Recovering from ShieldingThe nucleus has finite size. Shielding is not complete, andnuclear T violation can still induce atomic EDM DA.Post-screening nucleus-electron interaction proportional toSchiff moment:〈S〉 ≡

⟨∑pep(r2p − 53〈R2ch〉

)zp

⟩+ . . .If, as you’d expect, 〈S〉 ≈ R2Nuc 〈DNuc〉, then DA is down from〈DNuc〉 by

O(R2Nuc/R2

A

)≈ 10−8 .

Fortunately the large nuclear charge and relativistic wavefunctions offset this factor by 10Z 2 ≈ 105.Overall suppression of DA is only about 10−3.

Can other T -odd moments play a significant role?

Theory for Heavy Nuclei〈S〉 largest for large Z , so experiments are in heavy nucleibutAb initio methods are making rapid progress, butInteraction (from chiral EFT) has problems beyond A = 50.Many-body methods not yet ready to tackle soft nucleisuch as 199Hg, or even those with rigid deformation suchas 225Ra. sofor the near future must rely on nuclear density-functionaltheory: Mean-field theory with phenomenological“density-dependent interactions” (Skyrme, Gogny, orsuccessors) plus corrections, e.g.:projection of deformed wave functions onto states withgood particle number, angular momentuminclusion of small-amplitude zero-point motion (RPA)mixing of mean fields with different character (GCM). . .

Skyrme DFT

"#"$#%&! '()*+,!-.+,/0*+,1!'/23+,.4)5! F!

Zr-102: normal density and pairing density

HFB, 2-D lattice, SLy4 + volume pairing Ref: Artur Blazkiewicz, Vanderbilt, Ph.D. thesis (2005)

G=HI!β"JKLM&N76! +OKI!β"

JKLM&N7"J8L!1!PNQN!G@/2R!+5!/)N1!9AS;N!T+<N!U!J"&&$L!

Applied EverywhereNuclear ground state deformations (2-D HFB)

Ref: Dobaczewski, Stoitsov & Nazarewicz (2004) arXiv:nucl-th/0404077

"#"$#%&! %V!'()*+,!-.+,/0*+,1!'/23+,.4)5!

Varieties of Recent Schiff-Moment CalculationsNeed to calculate

〈S〉 =∑m

〈0|S |m〉〈m|VPT |0〉E0 − Em + c.c.

where H = Hstrong + VPT .Hstrong represented either by Skyrme density functional orby simpler effective interaction, treated on top of separatemean field.VPT either included nonperturbatively or via explicit sumover intermediate states.Nucleus either forced artificially to be spherical orallowed to deform.

199Hg via Explicit RPA in Spherical Mean Field1. Skyrme HFB (mean-field theory with pairing) in 198Hg.2. Polarization of core by last neutron and action of VPT ,treated as explicit corrections in quasiparticle RPA, whichsums over intermediate states.〈S〉Hg ≡ a0 gg0 + a1 gg1 + a2 gg2 (e fm3)

a0 a1 a2SkM? 0.009 0.070 0.022SkP 0.002 0.065 0.011SIII 0.010 0.057 0.025SLy4 0.003 0.090 0.013SkO′ 0.010 0.074 0.018Dmitriev & Senkov RPA 0.0004 0.055 0.009Range of variation here doesn’t look too bad. But thesecalculations are not the end of the story. . .

Deformation and Angular-Momentum RestorationIf deformed state |ΨK 〉 has good intr. Jz = K , average over anglesgives:|J,M〉 = 2J + 18π2

∫DJ∗MK (Ω)R(Ω) |ΨK 〉 dΩ

Matrix elements (with more detailed notation):〈J,M|Sm |J ′,M ′〉 ∝

∫ ∫ ∑jdΩdΩ′ × (some D-functions)

× 〈ΨK |R−1(Ω′)Sn R(Ω) |ΨK 〉

rigid defm.−−−−−−→Ω≈Ω′ (Geometric factor)× 〈ΨK |Sz |ΨK 〉︸︷︷︸〈S〉intr.For expectation value in J = 12 state:

〈S〉 = 〈Sz〉J= 12 ,M= 12 =⇒ 〈S〉intr. spherical nucleus13 〈S〉intr. rigidly deformed nucleus

Exact answer somewhere in between.

Deformed Mean-Field Calculation Directly in 199HgDeformation actually small and soft — perhaps worst casescenario for mean-field. But in heavy odd nuclei, that’s thelimit of current technology1. VPT included nonperturbativelyand calculation done in one step. Includes more physics thanRPA (deformation), plus economy of approach. Otherwiseshould be more or less equivalent.

0 1 2 3 4 5r⊥ (fm) 0 1

2 3

4 5

z (fm)-4

-2

0

2

4

6

δ ρ p

(ar

b.) Oscillating PT -odddensity distributionindicates delicateSchiff moment.

1Has some “issues”: doen’t get ground-state spin correct, limited for now to axially-symmetric minima, which are sometimes a little unstable, true minimum probably notaxially symmetric . . .

Results of “Direct” CalculationLike before, use a number of Skyrme functionals:

Egs β Eexc. a0 a1 a2SLy4 HF -1561.42 -0.13 0.97 0.013 -0.006 0.022SIII HF -1562.63 -0.11 0 0.012 0.005 0.016SV HF -1556.43 -0.11 0.68 0.009 -0.0001 0.016SLy4 HFB -1560.21 -0.10 0.83 0.013 -0.006 0.024SkM* HFB -1564.03 0 0.82 0.041 -0.027 0.069Fav. RPA QRPA — — — 0.010 0.074 0.018

Hmm. . .

What to Do About DiscrepancyAuthors of these papers need to revisit/recheck/interpolatebetween their results. (This will be done, at least to someextent.)Improve treatment further:Variation after projectionTriaxial deformation

Ultimate goal: mixing of many mean fields, aka “generatorcoordinates”Still a ways off because of difficulties marrying generatorcoordinates to density functionals.

Schiff Moment with Octupole DeformationHere we treat always VPT as explicitperturbation:〈S〉 =∑

m

〈0|S |m〉〈m|VPT |0〉E0 − Em + c.c.

where |0〉 is unperturbed ground state. Calculated 225Ra densityGround state has nearly-degenerate partner |0〉 with sameopposite parity and same intrinsic structure, so:〈S〉 −→ 〈0|S |0〉〈0|VPT |0〉E0 − E0 + c.c. ∝ 〈S〉intr. 〈VPT 〉intr.

E0 − E0

Why is this? See next slide.〈S〉 is large because 〈S〉intr. is collective and E0 − E0 is small.

Schiff Moment with Octupole DeformationHere we treat always VPT as explicitperturbation:〈S〉 =∑

m

〈0|S |m〉〈m|VPT |0〉E0 − Em + c.c.

where |0〉 is unperturbed ground state. Calculated 225Ra densityGround state has nearly-degenerate partner |0〉 with sameopposite parity and same intrinsic structure, so:〈S〉 −→ 〈0|S |0〉〈0|VPT |0〉E0 − E0 + c.c. ∝ 〈S〉intr. 〈VPT 〉intr.

E0 − E0Why is this? See next slide.

〈S〉 is large because 〈S〉intr. is collective and E0 − E0 is small.

A Little on Parity DoubletsWhen intrinsic state | 〉 is asymmetric, it breaks parity.In the same way we get good J , we average over orientationsto get states with good parity:|±〉 = 1√2( | 〉 ± | 〉

)These are nearly degenerate if deformation is rigid. So with|0〉 = |+〉 and |0〉 = |−〉, we get

〈S〉 ≈ 〈0|Sz |0〉〈0|VPT |0〉E0 − E0 + c.c.And in the rigid-deformation limit〈0|O|0〉 ∝ 〈 |O| 〉= 〈O〉intr.

again like angular momentum.

Spectrum of 225Ra350

9/2+ 321

300-

7l24 Ia27 g& -- -4

250 - 243 (13/2+)L (7/2+) 236

(~~2~)~

3/2_----_ 5L2+

200

i

5l2+ 379

150- 312 + 149

720 5f2--

x2+ fli K=3!2 bands

tOo-- 912i 'O"

712-A

50 i/2- 55 3f2i

42

5i2t25 312- 3l

O- l/2+- 0

K I TIP bands i

Fig. 5. Proposed grcxxping of the low-lying states OF 2zSRa into rotation& bands. T’ke two members of tke f? = $- band have been reported in a study of the ‘%?r decay 2oj; they are not observed in the

present study.

of the favored K * = z* band. (We have chosen to show in fig. 4 the M 1 multipolarity for the 134 keV y so that this apparent con%& in the data will not be overlooked by the reader.)

Definitive I” assignments for the remaining levels above 236 keV are difficult to make fram the available data, although the y-ray multipolarities and o-transition hindrance factors provide at least some insight. Again, the low value (23) of the hindrance factor of the rw-transition to the 394.7 keV Ievel is quite interesting, but no definite conclusion can be drawn regarding the I” assignment of this fevei.

Parity doublet|0〉|0〉

225Ra ResultsHartree-Fock calculation with our favorite interaction SkO’gives

〈S〉Ra = −1.5 gg0 + 6.0 gg1 − 4.0 gg2 (e fm3)Larger by over 100 than in 199Hg!

Variation a factor of 2 or 3. But, as you’ll see, we should beable to do better!

Current “Assessment” of UncertaintiesJudgment in review article from last year (based on spread inreasonable calculations):

Nucl. Best value Rangea0 a1 a2 a0 a1 a2

199Hg 0.01 ±0.02 0.02 0.005 – 0.05 -0.03 – +0.09 0.01 – 0.06129Xe -0.008 -0.006 -0.009 -0.005 – -0.05 -0.003 – -0.05 -0.005 – -0.1225Ra -1.5 6.0 -4.0 -1 – -6 4 — 24 -3 – -15

Uncertainties pretty large, particularly for a1 in 199Hg (rangeincludes zero). How can we reduce them?

Reducing Uncertainty: HgImproving many-body theory to handle soft deformation,though probably necessary, is tough. But can also try tooptimize density functional.

0 6 12 18 24 30 36 42Energy (MeV)

0

6

12

18

24

30

36

10−

3 S

tren

gth

(fm

6 /MeV

)

SkPSkO’SIII

EX2

EX1

Isoscalar dipole operatorcontains r2z just like Schiffoperator. Can see how wellfunctionals reproduce measureddistributions, e.g. in 208Pb.

More on Reducing Uncertainty in Hg

VPT probes spin density;functional should have goodspin response. Can adjustrelevant terms in, e.g. SkO’,to Gamow-Teller resonanceenergies and strengths.

More generally, examine correlations between Schiff momentand lots of other observables.

Reducing Uncertainty: RaImportant new developments here.

ex

p: 0

.94

(3)

225Ra

DN=0.9→0.6

224Ra 0.2

0.3

0.4

0.8 0.9 1.0 1.1 1.2 1.3 1.4

SKM*SKO'

SLy4

SKXc

SIIIUNEDF0

Proton octupole moment (10 fm)3

0.2

0.3

0.4

HFBCS

Sc

hif

f m

om

en

t [(

10 f

m)3

]

SKM*

SKO'SLy4

UNEDF0SKXc

SIII SKM*

SKO'

SLy4

SKXc

SIIIUNEDF0

0.29

0.3

0.31

0.32

0.33

0.34

0.35

0.9 0.94 0.98 1.02

Sch

iff m

omen

t in

225 R

a (1

0 fm

)3

Proton octupole moment in 224Ra (10 fm)3

SkO’

Experiment

Label is ∆N 0.60

0.65

0.70

0.75

0.80

0.85

0.90-30

-20

-10

0

10

20

30

40

Yu

kaw

a en

erg

ies

[keV

]

-10

0

10

2.8 3 3.2 3.4 3.6

Octupole moment Q30

[(10 fm)3]

SIII

SkXc

SkO’

SLy4

Landau

δ (δ (δ (δ (Time Odd)

g0

g1

g2

SkM*

225Ra

HF

〈S〉intr. correlated with octupolemoment, which will be extractedfrom measured E3 transitions.

Coulomb Excitation

5

Projectile (Z1,A1)

Target (Z2,A2)

b$

v

Sommerfeld parameter:

!"

#$%&'()*+,+-

./

0/

1/

2/

3!

"!

4!

!0

!0

!0

!0

!0

!0

!0

!0!0

!0!0

!"

!"

!"

!"

!"

1

“Safe” Coulex:

Reduced matrix elements: < 0+||E3||3 >224RaGaffney et al., NatureTransitions in 225Ra to bemeasured soon?

More on Reducing Uncertainty in RaWhat about matrix element of VPT ?In one-body approximation

VPT ≈ ~σ · ~∇ρ .

The closest simple one body operator isOAC = ~σ · ~r .

Q: Can we measure 〈0|OAC |O〉 or something like it?Doesn’t occur in electron scattering.Occurs in weak neutral current, but with large corrections frommeson exchange.p,p’ reactions? Something else? Input would be really useful.

Finally: A Little on Contact Term in VPTFirst approximation: simply use the interaction as given.Something like this has been done with two-body weakcurrents in shell-model calculations of double-beta decay.

Ultimately need to renormalize the contact interaction toaccount for the omission of high-energy states in many-bodycalculation. Renormalization scheme that preserves localnature of interaction will be useful, should be feasible.

The FutureCalculations have become sophisticated, but we still have a lotof work to do.In tne near future, that work must be in nuclear DFT.

In Hg, a GCM calculation eventually needed.And need correlation analysis, good proxies for Schiffdistributions (e.g. isoscalar dipole distribution), VPTdistribution.In ocutpole-deformed nuclei, improved techniquesprobably won’t change things drastically.But again, need correlation analysis. Have good proxy for〈S〉int., need one for 〈VPT 〉int..

THE ENDThanks for your kind attention.

Schiff Moments - Physics Department at UMass AmherstThe nuclear dipole moment causes the atomic...

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