Schiff Moments J. Engel October 23, 2014 One Way Things Get EDMs - - PowerPoint PPT Presentation
Schiff Moments J. Engel October 23, 2014 One Way Things Get EDMs Starting at fundamental level and working up: N g Underlying fundamental theory generates three T -violating NN vertices: ? New physics Then neutron gets EDM,
October 23, 2014
Starting at fundamental level and working up: Underlying fundamental theory generates three T-violating πNN vertices: Then neutron gets EDM, e.g., from chiral-PT diagrams like this:
New physics
Nucleus can get one from nucleon EDM or T-violating NN interaction:
π ¯ g γ
VPT ∝
g0τ1 · τ2 − ¯ g1 2 (τz
1 + τz 1) + ¯
g2 (3τz
1τz 2 − τ1 · τ2)
−¯ g1 2 (τz
1 − τz 2) (σ1 + σ2)
mπ|r1 − r2| + contact term Finally, atom gets one from nucleus. Electronic shielding makes relevant nuclear object the “Schiff moment” S ≈
p r2 pzp + . . ..
Job of nuclear theory: calculate dependence of S on the ¯ g’s (and on the contact term and nucleon EDM).
Theorem (Schiff)
The nuclear dipole moment causes the atomic electrons to rearrange themselves so that they develop a dipole moment
electrons and a point nucleus the electrons’ dipole moment exactly cancels the nuclear moment, so that the net atomic dipole moment vanishes.
Consider atom with non-relativistic constituents (with dipole moments dk) held together by electrostatic forces. The atom has a “bare” edm d ≡
k
dk and a Hamiltonian H =
p2
k
2mk +
V ( rk) −
Ek = H0 +
k(1/ek)
dk · ∇V ( rk) = H0 + i
(1/ek)
pk, H0
dipole perturbation
The perturbing Hamiltonian Hd = i
(1/ek)
pk, H0
|˜ = |0 +
|m m| Hd |0 E0 − Em = |0 +
|m m| i
k(1/ek)
dk · pk |0 (E0 − Em) E0 − Em =
(1/ek) dk · pk
The induced dipole moment d′ is
= ˜ 0|
ej rj |˜ = 0|
k(1/ek)
dk · pk
j ej
rj
k(1/ek)
dk · pk
= i 0|
rj,
k(1/ek)
dk · pk
= − 0|
= −
= − d So the net EDM is zero!
The nucleus has finite size. Shielding is not complete, and nuclear T violation can still induce atomic EDM DA. Post-screening nucleus-electron interaction proportional to Schiff moment: S ≡
ep
p − 5
3R2
ch
If, as you’d expect, S ≈ R2
Nuc DNuc, then DA is down from
DNuc by O
Nuc/R2 A
Fortunately the large nuclear charge and relativistic wave functions offset this factor by 10Z 2 ≈ 105. Overall suppression of DA is only about 10−3.
The nucleus has finite size. Shielding is not complete, and nuclear T violation can still induce atomic EDM DA. Post-screening nucleus-electron interaction proportional to Schiff moment: S ≡
ep
p − 5
3R2
ch
If, as you’d expect, S ≈ R2
Nuc DNuc, then DA is down from
DNuc by O
Nuc/R2 A
Fortunately the large nuclear charge and relativistic wave functions offset this factor by 10Z 2 ≈ 105. Overall suppression of DA is only about 10−3.
S largest for large Z , so experiments are in heavy nuclei but Ab initio methods are making rapid progress, but Interaction (from chiral EFT) has problems beyond A = 50. Many-body methods not yet ready to tackle soft nuclei such as 199Hg, or even those with rigid deformation such as 225Ra. so for the near future must rely on nuclear density-functional theory: Mean-field theory with phenomenological “density-dependent interactions” (Skyrme, Gogny, or successors) plus corrections, e.g.: projection of deformed wave functions onto states with good particle number, angular momentum inclusion of small-amplitude zero-point motion (RPA) mixing of mean fields with different character (GCM) . . .
HFB, 2-D lattice, SLy4 + volume pairing
Ref: Artur Blazkiewicz, Vanderbilt, Ph.D. thesis (2005)
β
Nuclear ground state deformations (2-D HFB)
Ref: Dobaczewski, Stoitsov & Nazarewicz (2004) arXiv:nucl-th/0404077
Need to calculate S =
0| S |m m| VPT |0 E0 − Em + c.c. where H = Hstrong + VPT . Hstrong represented either by Skyrme density functional or by simpler effective interaction, treated on top of separate mean field. VPT either included nonperturbatively or via explicit sum
Nucleus either forced artificially to be spherical or allowed to deform.
199Hg via Explicit RPA in Spherical Mean Field
treated as explicit corrections in quasiparticle RPA, which sums over intermediate states. SHg ≡ a0 g¯ g0 + a1 g¯ g1 + a2 g¯ g2 (e fm3) a0 a1 a2 SkM⋆ 0.009 0.070 0.022 SkP 0.002 0.065 0.011 SIII 0.010 0.057 0.025 SLy4 0.003 0.090 0.013 SkO′ 0.010 0.074 0.018 Dmitriev & Senkov RPA 0.0004 0.055 0.009 Range of variation here doesn’t look too bad. But these calculations are not the end of the story. . .
If deformed state |ΨK has good intr. Jz = K, average over angles gives: |J, M = 2J + 1 8π2
MK(Ω)R(Ω) |ΨK dΩ
Matrix elements (with more detailed notation): J, M| Sm |J′, M′ ∝
j
dΩ dΩ′ × (some D-functions) × ΨK| R−1(Ω′) Sn R(Ω) |ΨK
rigid defm.
− − − − − − →
Ω≈Ω′
(Geometric factor) × ΨK|Sz|ΨK
For expectation value in J = 1
2 state:
S = SzJ= 1
2 ,M= 1 2 =
⇒
spherical nucleus
1 3 Sintr.
rigidly deformed nucleus Exact answer somewhere in between.
Deformation actually small and soft — perhaps worst case scenario for mean-field. But in heavy odd nuclei, that’s the limit of current technology1. VPT included nonperturbatively and calculation done in one step. Includes more physics than RPA (deformation), plus economy of approach. Otherwise should be more or less equivalent.
1 2 3 4 5 r⊥ (fm) 1 2 3 4 5 z (fm)
2 4 6 δ ρp (arb.)
Oscillating PT-odd density distribution indicates delicate Schiff 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 not axially symmetric . . .
Like before, use a number of Skyrme functionals: Egs β Eexc. a0 a1 a2 SLy4 HF
0.97 0.013
0.022 SIII HF
0.012 0.005 0.016 SV HF
0.68 0.009
0.016 SLy4 HFB
0.83 0.013
0.024 SkM* HFB
0.82 0.041
0.069
QRPA — — — 0.010 0.074 0.018
Authors of these papers need to revisit/recheck/interpolate between their results. (This will be done, at least to some extent.) Improve treatment further:
Variation after projection Triaxial deformation
Ultimate goal: mixing of many mean fields, aka “generator coordinates” Still a ways off because of difficulties marrying generator coordinates to density functionals.
Here we treat always VPT as explicit perturbation: S =
0| S |m m| VPT |0 E0 − Em + c.c. where |0 is unperturbed ground state.
Calculated 225Ra density
Ground state has nearly-degenerate partner |¯ 0 with same
S − → 0| S |¯ 0 ¯ 0| VPT |0 E0 − E¯ + c.c. ∝
E0 − E¯
Here we treat always VPT as explicit perturbation: S =
0| S |m m| VPT |0 E0 − Em + c.c. where |0 is unperturbed ground state.
Calculated 225Ra density
Ground state has nearly-degenerate partner |¯ 0 with same
S − → 0| S |¯ 0 ¯ 0| VPT |0 E0 − E¯ + c.c. ∝
E0 − E¯
Why is this? See next slide.
Here we treat always VPT as explicit perturbation: S =
0| S |m m| VPT |0 E0 − Em + c.c. where |0 is unperturbed ground state.
Calculated 225Ra density
Ground state has nearly-degenerate partner |¯ 0 with same
S − → 0| S |¯ 0 ¯ 0| VPT |0 E0 − E¯ + c.c. ∝
E0 − E¯
Why is this? See next slide.
S is large because Sintr. is collective and E0 − E¯
0 is small.
When intrinsic state | is asymmetric, it breaks parity. In the same way we get good J, we average over orientations to get states with good parity: |± = 1 √ 2
± |
|0 = |+ and |¯ 0 = |−, we get S ≈ 0| Sz |¯ 0 ¯ 0| VPT |0 E0 − E¯ + c.c. And in the rigid-deformation limit 0|O|¯ 0 ∝ |O| = Ointr. again like angular momentum.
350 9/2+ 321 300-
7l24 Ia27 g&
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+-
K I TIP bands
i
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.
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 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA
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 |¯
225Ra Results
Hartree-Fock calculation with our favorite interaction SkO’ gives SRa = −1.5 g¯ g0 + 6.0 g¯ g1 − 4.0 g¯ g2 (e fm3) Larger by over 100 than in 199Hg! Variation a factor of 2 or 3. But, as you’ll see, we should be able to do better!
Judgment in review article from last year (based on spread in reasonable calculations):
Nucl. Best value Range a0 a1 a2 a0 a1 a2
199Hg
0.01 ±0.02 0.02 0.005 – 0.05
0.01 – 0.06
129Xe
225Ra
6.0
4 — 24
Uncertainties pretty large, particularly for a1 in 199Hg (range includes zero). How can we reduce them?
Improving many-body theory to handle soft deformation, though probably necessary, is tough. But can also try to
6 12 18 24 30 36 42
Energy (MeV)
6 12 18 24 30 36
10
−3 Strength (fm 6/MeV)
SkP SkO’ SIII EX2 EX1
Isoscalar dipole operator contains r2z just like Schiff
functionals reproduce measured distributions, e.g. in 208Pb.
VPT probes spin density; functional should have good spin response. Can adjust relevant terms in, e.g. SkO’, to Gamow-Teller resonance energies and strengths. More generally, examine correlations between Schiff moment and lots of other observables.
Important new developments here.
D →0.
moment, which will be extracted from measured E3 transitions.
!"
./ 0/ 1/ 2/ 3! "! 4!
!0 !0 !0 !0 !0 !0 !0 !0 !0 !0 !0 !" !" !" !" !"
Reduced matrix elements:
224Ra Gaffney et al., Nature
Transitions in 225Ra to be measured soon?
Important new developments here.
exp: 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 SIII UNEDF0
Proton octupole moment (10 fm)3 0.2 0.3 0.4 HF BCS Schiff moment [(10 fm)3]
SKM* SKO' SLy4 UNEDF0 SKXc SIII SKM* SKO' SLy4 SKXc SIII UNEDF0
moment, which will be extracted from measured E3 transitions.
!"
./ 0/ 1/ 2/ 3! "! 4!
!0 !0 !0 !0 !0 !0 !0 !0 !0 !0 !0 !" !" !" !" !"
Reduced matrix elements:
224Ra Gaffney et al., Nature
Transitions in 225Ra to be measured soon?
Important new developments here.
exp: 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 SIII UNEDF0
Proton octupole moment (10 fm)3 0.2 0.3 0.4 HF BCS Schiff moment [(10 fm)3]
SKM* SKO' SLy4 UNEDF0 SKXc SIII SKM* SKO' SLy4 SKXc SIII UNEDF0 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.9 0.94 0.98 1.02 Schiff moment in 225Ra (10 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
10 20 30 40 Yukawa energies [keV]
10 2.8 3 3.2 3.4 3.6 Octupole moment Q30 [(10 fm)3]
S I I I S k X c S k O ’ S L y 4
Landau δ ( δ ( δ ( δ (Time Odd) g0 g1 g2
S k M *
225Ra
HF
moment, which will be extracted from measured E3 transitions.
!"
./ 0/ 1/ 2/ 3! "! 4!
!0 !0 !0 !0 !0 !0 !0 !0 !0 !0 !0 !" !" !" !" !"
Reduced matrix elements:
224Ra Gaffney et al., Nature
Transitions in 225Ra to be measured soon?
What about matrix element of VPT ?
In one-body approximation VPT ≈ σ · ∇ρ . The closest simple one body operator is OAC = σ · r .
0| OAC |O or something like it? Doesn’t occur in electron scattering. Occurs in weak neutral current, but with large corrections from meson exchange. p,p’ reactions? Something else? Input would be really useful.
First approximation: simply use the interaction as given. Something like this has been done with two-body weak currents in shell-model calculations of double-beta decay. Ultimately need to renormalize the contact interaction to account for the omission of high-energy states in many-body
nature of interaction will be useful, should be feasible.
Calculations have become sophisticated, but we still have a lot
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 Schiff distributions (e.g. isoscalar dipole distribution), VPT distribution.
In ocutpole-deformed nuclei, improved techniques probably won’t change things drastically.
But again, need correlation analysis. Have good proxy for Sint., need one for VPT int..
Calculations have become sophisticated, but we still have a lot
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 Schiff distributions (e.g. isoscalar dipole distribution), VPT distribution.
In ocutpole-deformed nuclei, improved techniques probably won’t change things drastically.
But again, need correlation analysis. Have good proxy for Sint., need one for VPT int..
Thanks for your kind attention.