Schiff Moments. J. Engel. May 9, 2017

Transcription

1 Schiff Moments J. Engel May 9, 2017

2 Connection Between EDMs and T Violation Consider non-degenerate ground state g.s. : J, M. Symmetry under rotations R y (π) for vector operator like d i e i r i implies: g.s. : J, M d z g.s. : J, M = g.s. : J, M d z g.s. : J, M. R 1 R R 1 R

3 Connection Between EDMs and T Violation Consider non-degenerate ground state g.s. : J, M. Symmetry under rotations R y (π) for vector operator like d i e i r i implies: g.s. : J, M d z g.s. : J, M = g.s. : J, M d z g.s. : J, M. T takes M to M, like R y (π). But d is odd under R y (π) and even under T, so for T conserved g.s. : J, M d z g.s. : J, M = + g.s. : J, M d z g.s. : J, M. T 1 T T 1 T

4 Connection Between EDMs and T Violation Consider non-degenerate ground state g.s. : J, M. Symmetry under rotations R y (π) for vector operator like d i e i r i implies: g.s. : J, M d z g.s. : J, M = g.s. : J, M d z g.s. : J, M. T takes M to M, like R y (π). But d is odd under R y (π) and even under T, so for T conserved g.s. : J, M d z g.s. : J, M = + g.s. : J, M d z g.s. : J, M. Together with the first equation, this implies d z = 0. If T is violated, argument fails because T takes g : JM to states with J, M, but different energy.

5 One Way Things Get EDMs Starting at fundamental level and working up: Underlying fundamental theory generates three T-violating πnn vertices in chiral PT: N? ḡ π New physics γ Then neutron gets EDM from chiral-pt diagrams like this: π ḡ g n p n

6 How Diamagnetic Atoms Get EDMs γ Nucleus gets one from nucleon EDM and T-violating NN interaction: ḡ π {[ V PT g 0 τ 1 τ 2 g ] 1 2 (τz 1 + τz 1 ) + g 2 (3τz 1 τz 2 τ 1 τ 2 ) (σ 1 σ 2 ) g 1 2 (τz 1 τz 2 ) (σ 1 + σ 2 ) + contact term } ( 1 2 ) exp ( m π r 1 r 2 ) m π r 1 r 2 Finally, atom gets one from nucleus. Electronic shielding makes relevant nuclear object the Schiff moment S p r2 pz p Job of nuclear theory: calculate dependence of S on the g i (and on the contact term and nucleon EDM).

7 How Does Shielding Work? Theorem (Schiff) The nuclear dipole moment causes the atomic electrons to rearrange themselves so that they develop a dipole moment opposite that of the nucleus. In the limit of nonrelativistic electrons and a point nucleus the electrons dipole moment exactly cancels the nuclear moment, so that the net atomic dipole moment vanishes.

8 How Does Shielding Work? Proof Consider atom with non-relativistic constituents (with dipole moments d k ) held together by electrostatic forces. The atom has a bare edm d d k and a Hamiltonian k H = k p 2 k 2m k + k V( r k ) k d k E k = H 0 + k (1/e k ) d k V( r k ) = H 0 + i k [ ] (1/e k ) d k p k, H 0 K.E. + Coulomb dipole perturbation

9 How Does Shielding Work? The perturbing Hamiltonian H d = i k [ ] (1/e k ) d k p k, H 0 shifts the ground state 0 to 0 = 0 + m = 0 + m ( = 1 + i k m m H d 0 E 0 E m m m i k (1/e k) d k p k 0 (E 0 E m ) E 0 E m ) (1/e k ) d k p k 0

10 How Does Shielding Work? The induced dipole moment d is d = 0 e j r j 0 j ( = 0 1 i ) (1/e k ) d k p k e j r j ( k j 1 + i ) (1/e k ) d k p k 0 k = i 0 e j r j, (1/e k ) d k p k 0 j k = 0 k d k 0 = k d k = d So the net EDM is zero!

11 Recovering from Shielding The nucleus has finite size. Shielding is not complete, and nuclear T violation can still induce atomic EDM D A. Post-screening nucleus-electron interaction proportional to Schiff moment: ( S e p r 2 p 5 ) 3 R2 ch z p +... p If, as you d expect, S R 2 Nuc D Nuc, then D A is down from D Nuc by O ( R 2 Nuc /R2 A) Fortunately, the large nuclear charge and relativistic wave functions offset this factor by 10Z Overall suppression of D A is only about 10 3.

12 Theory for Heavy Nuclei S largest for large Z, so experiments are in heavy nuclei. Ab initio methods are making rapid progress, but Interaction (from chiral EFT) has problems beyond A = 50. Many-body methods not quite ready to tackle soft nuclei such as 199 Hg, or even those with rigid deformation such as 225 Ra. so for now we 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)...

13 Nuclear Deformation

14 Skyrme DFT Zr-102: normal density and pairing density HFB, 2-D lattice, SLy4 + volume pairing Ref: Artur Blazkiewicz, Vanderbilt, Ph.D. thesis (2005) β β

15 Applied Everywhere Nuclear ground state deformations (2-D HFB) Ref: Dobaczewski, Stoitsov & Nazarewicz (2004) arxiv:nucl-th/

16 Varieties of Recent Schiff-Moment Calculations Need to calculate S m 0 S m m V PT 0 E 0 E m + c.c. where H = H strong + V PT. H strong represented either by Skyrme density functional or by simpler effective interaction, treated on top of separate mean field. V PT either included nonperturbatively or via the explicit sum over intermediate states above. Nucleus either forced artificially to be spherical or allowed to deform.

17 199 Hg via Explicit RPA in Spherical Mean Field 1. Skyrme HFB (mean-field theory with pairing) in 198 Hg. 2. Polarization of core by last neutron and action of V PT, treated as explicit corrections in quasiparticle RPA, which sums over intermediate states. S Hg a 0 gg 0 + a 1 gg 1 + a 2 gg 2 (e fm 3 ) a 0 a 1 a 2 SkM SkP SIII SLy SkO Dmitriev & Senkov RPA Range of variation here doesn t look too bad. But these calculations are not the end of the story...

18 Deformation and Angular-Momentum Restoration If deformed state Ψ K has good intr. J z = K, one averages over angles to get: J, M = 2J + 1 8π 2 dω D J MK (Ω)R(Ω) Ψ K Matrix elements (with more detailed notation): J, M S m J, M dω dω (some D-functions) n Ψ K R 1 (Ω ) S n R(Ω) Ψ K rigid defm. (Geometric factor) Ψ K S z Ψ K Ω Ω }{{} For expectation value in J = 1 2 state: { S = S z J= 1 2,M= 1 2 = S intr. Exact answer somewhere in between. S intr. spherical nucleus 1 3 S intr. rigidly deformed nucleus

19 Deformed Mean-Field Calculation Directly in 199 Hg Deformation actually small and soft perhaps worst case scenario for mean-field. But in heavy odd nuclei, that s the best that has been done 1. V PT 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. 6 δ ρ p (arb.) r (fm) z (fm) Oscillating PT-odd density distribution indicates delicate Schiff moment. 1 Has some issues : doen t get ground-state spin correct, limited for now to axiallysymmetric minima, which are sometimes a little unstable, true minimum probably not axially symmetric...

20 Results of Direct Calculation Like before, use a number of Skyrme functionals: E gs β E exc. a 0 a 1 a 2 SLy4 HF SIII HF SV HF SLy4 HFB SkM* HFB Fav. RPA QRPA Hmm...

21 What to Do About Discrepancy Revisit/recheck existing calculations. 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.

22 Schiff Moment with Octupole Deformation Here we treat always V PT as explicit perturbation: S = m 0 S m m V PT 0 E 0 E m + c.c. where 0 is unperturbed ground state. Calculated 225 Ra density Ground state has nearly-degenerate partner 0 with same opposite parity and same intrinsic structure, so: S 0 S 0 0 V PT 0 + c.c. E 0 E 0

23 Schiff Moment with Octupole Deformation Here we treat always V PT as explicit perturbation: S = m 0 S m m V PT 0 E 0 E m + c.c. where 0 is unperturbed ground state. Calculated 225 Ra density Ground state has nearly-degenerate partner 0 with same opposite parity and same intrinsic structure, so: S 0 S 0 0 V PT 0 + c.c. S intr. V PT intr. E 0 E 0 E 0 E 0 Why is this? See next slide.

24 Schiff Moment with Octupole Deformation Here we treat always V PT as explicit perturbation: S = m 0 S m m V PT 0 E 0 E m + c.c. where 0 is unperturbed ground state. Calculated 225 Ra density Ground state has nearly-degenerate partner 0 with same opposite parity and same intrinsic structure, so: S 0 S 0 0 V PT 0 + c.c. S intr. V PT intr. E 0 E 0 E 0 E 0 Why is this? See next slide. S is large because S intr. is collective and E 0 E 0 is small.

25 A Little on Parity Doublets 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 ( ± ) These are nearly degenerate if deformation is rigid. So with 0 = + and 0 =, we get S 0 S z 0 0 V PT 0 E 0 E 0 + c.c. And in the rigid-deformation limit 0 O 0 O = O intr. again like angular momentum.

26 Spectrum of 225 Ra 350 9/ g& (13/2+)L (7/2+) 236 3/2_----_ 5L2+ (~~2~)~ 7l24 Ia l i x2+ fli 5f K=3!2 bands too-- 912i 'O" 712-A i/2-55 3f2i 0 5i2t l O- l/ Parity doublet K I TIP bands i Fig. 5. Proposed grcxxping of the low-lying states OF 2zSRa into rotation& bands. T ke two members of

27 225 Ra Results Hartree-Fock calculation with our favorite interaction SkO gives S Ra = 1.5 gg gg gg 2 (e fm 3 ) Larger by over 100 than in 199 Hg! Variation a factor of 2 or 3. But, as you ll see, we should be able to do better!

28 Current Assessment of Uncertainties Judgment in recent review article (based on spread in reasonable calculations): Nucl. Best value Range a 0 a 1 a 2 a 0 a 1 a Hg 0.01 ± Xe Ra Uncertainties pretty large, particularly for a 1 in 199 Hg (range includes zero). How can we reduce them?

29 Reducing Uncertainty: Hg Improving many-body theory to handle soft deformation, though probably necessary, is tough. But can also try to optimize density functional Strength (fm 6 /MeV) E X1 E X2 SkP SkO SIII Isoscalar dipole operator contains r 2 z just like Schiff operator. Can see how well functionals reproduce measured distributions, e.g. in 208 Pb Energy (MeV)

30 More on Reducing Uncertainty in Hg V PT 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.

31 Reducing Uncertainty: Ra Important new developments here. S intr. correlated with octupole moment, which will be extracted from measured E3 transitions. 2/ 3! 1/ "! 4! 0/./ Reduced 224 Ramatrix #$%&'()*+,+- Gaffney et al., Nature Transitions in 225 Ra to be measured soon?

32 Reducing Uncertainty: Ra Schiff moment [(10 fm) 3 ] exp: 0.94(3) Important new developments here SKM* HF SLy4 BCS SKO' UNEDF0 SKX c SIII SKM* SLy4 SKO' SIII UNEDF0 225 Ra SKX c SKX c SKO' SIII D N = SLy4 UNEDF0 SKM* 224 Ra Proton octupole moment (10 fm) 3 S intr. correlated with octupole moment, which will be extracted from measured E3 transitions. 2/ 1/ 0/./ #$%&'()*+,+- Gaffney et al., Nature Transitions in 225 Ra to be measured soon? 3! "! 4! Reduced 224 Ramatrix

33 Reducing Uncertainty: Ra Schiff moment [(10 fm) 3 ] exp: 0.94(3) Important new developments here. Schiff moment in 225 Ra (10 fm) SKM* HF SLy4 BCS SKO' UNEDF0 SKX c SIII SKM* Label SLy4is N SKO' 0.60 Experiment SIII UNEDF Ra SKX c 0.70 SKO' 0.90 D N = SLy SKM* SkO UNEDF SIII Ra SKX Proton c octupole moment in 224 Ra (10 fm) Proton octupole moment (10 fm) 3 S intr. correlated with octupole moment, which will be extracted from measured E3 transitions. 2/ 1/ 0/./ #$%&'()*+,+- Gaffney et al., Nature Transitions in 225 Ra to be measured soon? 3! "! 4! Reduced 224 Ramatrix

34 Reducing Uncertainty: Ra Schiff moment [(10 fm) 3 ] exp: 0.94(3) Important new developments here. Schiff moment in 225 Ra (10 fm) 3 Yukawa energies [kev] SKM* HF SLy4 Landau BCS HFSKO' UNEDF0 SKX c SIII SKM* g1 Label SLy4is N SKO' 0.60 Experiment SIII UNEDF0 225 Ra g0 Ra SKX c 0.70 g D N = SKO' 0.90 SKM* SLy4 SkO δ (Time Odd) UNEDF SIII Ra SKX Proton c octupole moment in 224 Ra (10 fm) 3 SIII SkXc SkO SLy4 SkM* Proton Octupole octupole moment moment Q [(10 (10 fm) fm) 3 ] 3 30 S intr. correlated with octupole moment, which will be extracted from measured E3 transitions. 2/ 1/ 0/./ #$%&'()*+,+- Gaffney et al., Nature Transitions in 225 Ra to be measured soon? 3! "! 4! Reduced 224 Ramatrix

35 Reducing Uncertainty: Ra Schiff moment [(10 fm) 3 ] exp: 0.94(3) Important new developments here. Schiff moment in 225 Ra (10 fm) 3 Yukawa energies [kev] Schiff moment [(10 fm) 3 ] SKM* 40 HF SLy4 SKM* Landau 0.4 BCS HF HFSKO' SLy BCS UNEDF0 SKO' 0.35 SKX c UDF0 20 SIII SKX SKM* 0.3 g1 c SIII Label SLy4is SKM* SKO' SLy4 N SIII UNEDF0 225 SKO' Ra g0 Experiment UDF0 Ra 0.2 SKX c SIII Ra 0.32 SKX -10 c 0.75 g2 D N 0.80 = P = SKO' 0.90 SKM* SLy4 SkO δ (Time Odd) Pa 0.3 UNEDF SIII 0.98 Ra Ra SKX Proton c octupole moment in 224 Ra (10 fm) 223 Rn SkO 3 L Proton Octupole 2.0octupole moment 2.5moment Q [(10 3.0(10 fm) fm) 3 ] Octupole moment Q 30 [(10 fm) 3 ] SIII SkXc N = SkO SLy4 SkM* S intr. correlated with octupole moment, which will be extracted from measured E3 transitions. 2/ 1/ 0/./ #$%&'()*+,+- Gaffney et al., Nature Transitions in 225 Ra to be measured soon? 3! "! 4! Reduced 224 Ramatrix

36 More on Reducing Uncertainty in Ra What about matrix element of V PT? In one-body approximation V PT σ ρ. The closest simple one body operator is O AC = σ r. Q: Can we measure 0 O AC O or something like it? Doesn t occur in electron scattering, but does occurs in weak neutral current. Neutrino scattering on Ra?

37 The Future Calculations have become sophisticated, but we still have a lot of work to do. In the near future, that work involve nuclear DFT. In Hg, a GCM calculation eventually needed. And need correlation analysis, good proxies for Schiff distributions (e.g. isoscalar dipole distribution), V PT distribution. In ocutpole-deformed nuclei, improved techniques probably won t change things drastically. But again, need correlation analysis. Have good proxy for S int., need one for V PT int..

38 The Future Calculations have become sophisticated, but we still have a lot of work to do. In the near future, that work involve nuclear DFT. In Hg, a GCM calculation eventually needed. And need correlation analysis, good proxies for Schiff distributions (e.g. isoscalar dipole distribution), V PT distribution. In ocutpole-deformed nuclei, improved techniques probably won t change things drastically. But again, need correlation analysis. Have good proxy for S int., need one for V PT int.. THE END. Thanks for your kind attention.