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 π πNN vertices: ? New physics γ Then neutron gets EDM, e.g., π − g ¯ g from chiral-PT diagrams like this: n p n

How Diamagnetic Atoms Get EDMs γ Nucleus can get one from nucleon g ¯ EDM or T -violating NN interaction: π �� � g 0 τ 1 · τ 2 − ¯ g 1 2 ( τ z 1 + τ z g 2 (3 τ z 1 τ z V PT ∝ ¯ 1 ) + ¯ 2 − τ 1 · τ 2 ) ( σ 1 − σ 2 ) � − ¯ g 1 · ( ∇ 1 − ∇ 2 ) exp ( −m π | r 1 − r 2 | ) 2 ( τ z 1 − τ z 2 ) ( σ 1 + σ 2 ) m π | r 1 − r 2 | + contact term Finally, atom gets one from nucleus. Electronic shielding makes relevant nuclear object the “Schiff moment” �S� ≈ � � p r 2 p z p + . . . � . Job of nuclear theory: calculate dependence of � S � on the ¯ g’s (and on the contact term and nucleon EDM).

How Does Shielding Work? Theorem (Schiff) T he 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.

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 ≡ � k � d k and a Hamiltonian p 2 � � � k d k · � � H = + V ( � r k ) − E k 2 m k k k k + � k (1 /e k ) � d k · � = H 0 ∇ V ( � r k ) � � � � = H 0 + i (1 /e k ) d k · � p k , H 0 k dipole perturbation K.E. + Coulomb

How Does Shielding Work? The perturbing Hamiltonian � � � � H d = i (1 /e k ) d k · � p k , H 0 k shifts the ground state | 0 � to | m � � m | H d | 0 � � | ˜ 0 � = | 0 � + E 0 − E m m | m � � m | i � k (1 /e k ) � d k · � p k | 0 � ( E 0 − E m ) � = | 0 � + E 0 − E m m � � � (1 /e k ) � d k · � = 1 + i p k | 0 � k

How Does Shielding Work? d ′ is The induced dipole moment � � � � ˜ r j | ˜ d ′ = 0 | e j � 0 � j � � �� � 1 − i � k (1 /e k ) � d k · � = � 0 | p k j e j � r j � � 1 + i � k (1 /e k ) � × d k · � p k | 0 � �� � r j , � k (1 /e k ) � = i � 0 | j e j � d k · � p k | 0 � � � � � − � 0 | d k | 0 � − d k = = k k − � = d So the net EDM is zero!

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: �� � � � p − 5 r 2 3 � R 2 � S � ≡ e p ch � z p + . . . p If, as you’d expect, � S � ≈ R 2 Nuc � D Nuc � , then D A is down from � D Nuc � by � � ≈ 10 − 8 . R 2 Nuc /R 2 O A Fortunately the large nuclear charge and relativistic wave functions offset this factor by 10 Z 2 ≈ 10 5 . Overall suppression of D A is only about 10 − 3 .

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: �� � � � p − 5 r 2 3 � R 2 � S � ≡ e p ch � z p + . . . p Can other T -odd moments play a significant role? If, as you’d expect, � S � ≈ R 2 Nuc � D Nuc � , then D A is down from � D Nuc � by � � ≈ 10 − 8 . R 2 Nuc /R 2 O A Fortunately the large nuclear charge and relativistic wave functions offset this factor by 10 Z 2 ≈ 10 5 . Overall suppression of D A is only about 10 − 3 .

Theory for Heavy Nuclei � 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 199 Hg, or even those with rigid deformation such as 225 Ra. 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) . . .

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 � ��� ����������� ��������������������������������������� �������� ����������������������������� ��

Applied Everywhere Nuclear ground state deformations (2-D HFB) Ref: Dobaczewski, Stoitsov & Nazarewicz (2004) arXiv:nucl-th/0404077 �������� ����������������������������� ���

Varieties of Recent Schiff-Moment Calculations Need to calculate � 0 | S | m � � m | V PT | 0 � � � S � = + c.c. E 0 − E m m where H = H s t rong + V PT . H s trong 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 explicit sum over intermediate states. Nucleus either forced artificially to be spherical or allowed to deform.

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. g 2 (e fm 3 ) � S � Hg ≡ a 0 g ¯ g 0 + a 1 g ¯ g 1 + a 2 g ¯ a 0 a 1 a 2 SkM ⋆ 0 . 009 0 . 070 0 . 022 SkP 0 . 002 0 . 065 0 . 011 0 . 010 0 . 057 0 . 025 SIII 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. . .

Deformation and Angular-Momentum Restoration If deformed state | Ψ K � has good intr. J z = K , average over angles gives: � | J, M � = 2 J + 1 D J∗ MK (Ω) R (Ω) | Ψ K � d Ω 8 π 2 Matrix elements (with more detailed notation): � � � d Ω d Ω ′ × (some D-functions) �J, M| S m |J ′ , M ′ � ∝ j × � Ψ K | R − 1 (Ω ′ ) S n R (Ω) | Ψ K � rigid defm. − − − − − − → (Geometric factor) × � Ψ K | S z | Ψ K � Ω ≈ Ω ′ � �� � � S � intr. For expectation value in J = 1 2 state: � � S � intr. spherical nucleus � S � = � S z � J = 1 2 = ⇒ 2 ,M = 1 1 3 � S � intr. rigidly deformed nucleus Exact answer somewhere in between.

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 limit of current technology 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 4 δ ρ p (arb.) Oscillating PT -odd 2 density distribution 0 5 indicates delicate -2 4 3 Schiff moment. -4 z (fm) 2 1 0 1 2 3 0 4 5 r ⊥ (fm) 1 Has 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 . . .

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

What to Do About Discrepancy 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.

Schiff Moment with Octupole Deformation Here we treat always V PT as explicit perturbation: � 0 | S | m � � m | V PT | 0 � � � S � = + c.c. E 0 − E m m 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: → � 0 | S | ¯ 0 � � ¯ 0 | V PT | 0 � � S � intr . � V PT � intr . � S � − + c.c. ∝ E 0 − E ¯ E 0 − E ¯ 0 0