Double-Beta Decay, CP Violation, and Nuclear Structure

Transcription

1 Double-Beta Decay, CP Violation, and Nuclear Structure J. Engel University of North Carolina March 10, 2014 n p W e! " x n W p e

2 Double-Beta Decay

3 Neutrinos: What We Know Come in three flavors, none of which have definite mass. ν e ν 1 ν µ = U ν ν 2 = mass eigenstates m ν τ ν i 1 ev 3 U contains three mixing angles (and a few phases).

4 Neutrinos: What We Know Come in three flavors, none of which have definite mass. ν e ν 1 ν µ = U ν ν 2 = mass eigenstates m ν τ ν i 1 ev 3 U contains three mixing angles (and a few phases). From oscillation experiments:

5 Neutrinos: What We Know Come in three flavors, none of which have definite mass. ν e ν 1 ν µ = U ν ν 2 = mass eigenstates m ν τ ν i 1 ev 3 U contains three mixing angles (and a few phases). From oscillation experiments: Solar-ν s: m 2 sol ev 2 θ sol 34

6 Neutrinos: What We Know Come in three flavors, none of which have definite mass. ν e ν 1 ν µ = U ν ν 2 = mass eigenstates m ν τ ν i 1 ev 3 U contains three mixing angles (and a few phases). From oscillation experiments: Solar-ν s: m 2 sol ev 2 θ sol 34 Atmospheric-ν s: m 2 atm ev 2 θ atm 45

7 Neutrinos: What We Know Come in three flavors, none of which have definite mass. ν e ν 1 ν µ = U ν ν 2 = mass eigenstates m ν τ ν i 1 ev 3 U contains three mixing angles (and a few phases). From oscillation experiments: Solar-ν s: m 2 sol ev 2 θ sol 34 Atmospheric-ν s: m 2 atm ev 2 θ atm 45 Reactor ν s: θ reac 9

8 What We Still Don t Know Hierarchy : normal or inverted?

9 What We Still Don t Know Hierarchy : normal or inverted? Overall mass scale =?

10 What We Still Don t Know Hierarchy : normal or inverted? Overall mass scale =? Neutrinos their own antiparticles (Majorana fermions)?

11 Neutrinoless Double-Beta Decay

12 Neutrinoless Double-Beta Decay If energetics are right (ordinary beta decay forbidden)..... Z+1, N-1 Z, N Z+2, N-2

13 Neutrinoless Double-Beta Decay If energetics are right (ordinary beta decay forbidden)..... Z+1, N-1 and neutrinos are their own antiparticles... Z, N Z+2, N-2

14 Neutrinoless Double-Beta Decay If energetics are right (ordinary beta decay forbidden)..... Z+1, N-1 and neutrinos are their own antiparticles... Z, N Z+2, N-2 can observe two neutrons turning into protons, emitting two electrons and nothing else. n p W! " x e n W p e

15 Neutrinoless Double-Beta Decay If energetics are right (ordinary beta decay forbidden)..... Z+1, N-1 and neutrinos are their own antiparticles... Z, N Z+2, N-2 can observe two neutrons turning into protons, emitting two electrons and nothing else. Different from already observed two-neutrino process. n n W W p p e! "! " e

16 Usefulness of Double-Beta Decay n p W e! " x n W p e Rate proportional to square of effective neutrino mass m eff i m i U 2 ei

17 Usefulness of Double-Beta Decay n p W e! " x n W p e Rate proportional to square of effective neutrino mass m eff i m i U 2 ei If lightest neutrino is light:

18 Usefulness of Double-Beta Decay n p W e! " x n W p e Rate proportional to square of effective neutrino mass m eff i m i U 2 ei If lightest neutrino is light: m eff m 2 sol sin2 θ sol (normal)

19 Usefulness of Double-Beta Decay n p W e! " x n W p e Rate proportional to square of effective neutrino mass m eff i m i U 2 ei If lightest neutrino is light: m eff m 2 sol sin2 θ sol m eff m 2 atm cos 2θ sol (normal) (inverted)

20 Usefulness of Double-Beta Decay n p W e! " x n W p e Rate proportional to square of effective neutrino mass m eff i m i U 2 ei!! If lightest neutrino is light: m eff m 2 sol sin2 θ sol m eff m 2 atm cos 2θ sol (normal) (inverted)

21 Usefulness of Double-Beta Decay n p W e! " x n W p e Rate proportional to square of effective neutrino mass m eff i m i U 2 ei!! new expts. If lightest neutrino is light: m eff m 2 sol sin2 θ sol m eff m 2 atm cos 2θ sol (normal) (inverted)

22 Usefulness of Double-Beta Decay n p W e! " x n W p e Rate proportional But rate also to square dependsofon a nuclear matrix element! effective neutrino mass m eff i m i U 2 ei!! new expts. If lightest neutrino is light: m eff m 2 sol sin2 θ sol m eff m 2 atm cos 2θ sol (normal) (inverted)

23 one, Elliott, and Engel: Double beta decay, Majorana neutrinos, and Other Mechanisms p n p n If neutrinoless decay 41 occurs then _ ν s are Majorana, no matter what: (ν) R e ββ(0ν) e ν L ere are no W R s in the ator is roughly propor- FIG. 3. Majorana propagator resulting from MeV is a typical Schechter and Valle, but light neutrinos may not drive the decay: en, instead of Eq. 41, 42 be approximately equal Mohapatra 1999; Cir- Exchange of heavy right-handed neutrino in left-right symmetric model. 43 W or more zero-mass bosons, called Majorons, be emitted along with the two electrons in d decay 0, Chikashige et al., 1981; G Roncadelli, 1981; Georgi et al., Appar ever, it is difficult for such a process to have a amplitude. Second, even if some exotic lepto violating physics exists and light neutrino e not responsible for the decay, the occurrence still implies that neutrinos are Majorana par nonzero mass Schechter and Valle, Th that any diagram contributing to the decay W

24 one, Elliott, and Engel: Double beta decay, Majorana neutrinos, and Other Mechanisms p n p n If neutrinoless decay 41 occurs then _ (ν) ν s are Majorana, no matter what: R e ν ββ(0ν) e L ere are no W R s in the ator is roughly propor- FIG. 3. Majorana propagator resulting from MeV is a typical Schechter and Valle, but light neutrinos may not drive the decay: en, instead of Eq. 41, Exchange of heavy right-handed neutrino or more zero-mass bosons, called Majorons, in left-right symmetric be emitted model. along with the two electrons in d 42 decay 0, Chikashige et al., 1981; G Amplitude of exotic mechanism: Roncadelli, 1981; Georgi et al., Appar be approximately Z heavy ( ) 4 equal ever, ( it) is difficult for such a process to have a 0ν WL meff amplitude. Second, even if some exotic lepto Mohapatra Z light 1999; Cir- W 0ν R q violating 2 m N q MeV 2 physics exists and light neutrino e 1 if mnot R responsible 1 TeV andform the eff decay, mthe 2 occurrence atm still implies that neutrinos are Majorana par So exotic processes 43 cannonzero occur with mass roughly Schechter the same and Valle, rate as1982. Th light-ν exchange. But we that don t anyknow diagram if any contributing exotics exist, toso the... decay W W

25 Focus on Light-ν Exchange Hadronic Part Integral over times produces a factor n f J µ L ( x) n n J ν L ( y) i q 0 (E n + q 0 + E e2 E i ) + ( x, µ y, ν), with q 0 the virtual-neutrino energy and the J the weak current.

26 Focus on Light-ν Exchange Hadronic Part Integral over times produces a factor n f J µ L ( x) n n J ν L ( y) i q 0 (E n + q 0 + E e2 E i ) + ( x, µ y, ν), with q 0 the virtual-neutrino energy and the J the weak current. In impulse approximation: ( May not be adequate. p J µ (x) p = e iqx u(p) g V (q 2 )γ µ g A (q 2 )γ 5 γ µ ) ig M (q 2 ) σµν q ν + g P (q 2 )γ 5 q µ u(p ). 2m p

27 Focus on Light-ν Exchange Hadronic Part Integral over times produces a factor n f J µ L ( x) n n J ν L ( y) i q 0 (E n + q 0 + E e2 E i ) + ( x, µ y, ν), with q 0 the virtual-neutrino energy and the J the weak current. In impulse approximation: ( May not be adequate. p J µ (x) p = e iqx u(p) g V (q 2 )γ µ g A (q 2 )γ 5 γ µ ) ig M (q 2 ) σµν q ν + g P (q 2 )γ 5 q µ u(p ). 2m p q 0 typically about inverse inter-nucleon distance, 100 MeV, so denominator can be taken constant and sum done in closure.

28 Final Form of Nuclear Part M 0ν = M GT 0ν g2 V ga 2 M0ν F +... with M GT 0ν = f a,b H(r ab, E) σ a σ b τ + a τ + b i +... M F 0ν = f a,b H(r ab, E)τ + a τ + b i +... H(r, E) 2R πr 0 sin qr dq q + E (E i + E f )/2

29 Final Form of Nuclear Part M 0ν = M GT 0ν g2 V ga 2 M0ν F +... with M GT 0ν = f a,b H(r ab, E) σ a σ b τ + a τ + b i +... M F 0ν = f a,b H(r ab, E)τ + a τ + b i +... H(r, E) 2R πr 0 sin qr dq q + E (E i + E f )/2 Corrections ( forbidden terms, weak form factors) 30%.

30 Calculations of Matrix Elements Nuclear-Structure Theory In light nuclei, theory has made transition from art to science. In medium-mass nuclei nuclei, it s now somewhere between the two. Is it enough of a science yet to get accurate double-beta matrix elements?

31 Recent Level of Agreement From P. Vogel, 2010 proton-neutron (pn) QRPA Shell Model Interacting Boson Model Projected HFB Generator Coordinates Same level of agreement in Not so great. Results of recent calculations, references and comments on request

32 Recent Level of Agreement From P. Vogel, 2010 proton-neutron (pn) QRPA Shell Model Interacting Boson Model Projected HFB Generator Coordinates Same level of agreement in Not so great. Results of recent calculations, references and comments on request I m working to improve all these methods; will focus on shell model here.

33 All Approaches Start with Mean Field/Potential In shell model oscillator potential is added to Hamiltonian (and then subtracted later).

34 The Nuclear Shell Model Once mean field is obtained, build in correlations: QRPA Shell Model protons neutrons

35 The Nuclear Shell Model Once mean field is obtained, build in correlations: QRPA Shell Model protons neutrons Shell Model: Small single-particle space in simple spherical mean field; arbitrarily complex correlations within the space.

36 Corrected Shell Model Partition of Full Hilbert Space P P ˆP H ˆP Q ˆP H ˆQ P = valence space Q = the rest Task: Find unitary transformation to make H block-diagonal in P and Q, with H eff in P reproducing d most important eigenvalues. Q ˆQH ˆP ˆQH ˆQ Shell model done here

37 Corrected Shell Model Partition of Full Hilbert Space P P H eff Q P = valence space Q = the rest Task: Find unitary transformation to make H block-diagonal in P and Q, with H eff in P reproducing d most important eigenvalues. Q H eff-q Shell model done here

38 Corrected Shell Model Partition of Full Hilbert Space P P H eff Q P = valence space Q = the rest Task: Find unitary transformation to make H block-diagonal in P and Q, with H eff in P reproducing d most important eigenvalues. Q H eff-q For transition operator ˆM, must apply same transformation to get ˆM eff. Shell model done here

39 Corrected Shell Model Partition of Full Hilbert Space P P H eff Q P = valence space Q = the rest Task: Find unitary transformation to make H block-diagonal in P and Q, with H eff in P reproducing d most important eigenvalues. Q H eff-q For transition operator ˆM, must apply same transformation to get ˆM eff. This is as difficult as solving full problem. But the idea is that N-body effective operators may not be important for N > 2 or 3. Shell model done here

40 Corrected Shell Model Partition of Full Hilbert Space P Q P = valence space Q = the rest P H eff Task: Find unitary transformation to make H block-diagonal in P Version of this (plus phenomenology) and usedq, towith get Hshell-model eff in P reproducing interactions, but not the decay operator. Bare d most operator important generally eigenvalues. used. Q H eff-q For transition operator ˆM, must apply same transformation to get ˆM eff. This is as difficult as solving full problem. But the idea is that N-body effective operators may not be important for N > 2 or 3. Shell model done here

41 Peturbation-Theory Approach Q-Box (Interaction) c a d ˆQ = b c a d V low-k + b c a d b +... c d c d + + a b a b c d c d a b a b

42 Peturbation-Theory Approach Q-Box (Interaction) a c b d ˆQ = a c b d V low-k + a c b d a c b d + a c b d + a b d c + a c b d +... X-Box (Decay Operator) a c b d ˆX = a c b d M + a c b d + a c b d + a b d c + a b d c + a c b d + a c b d +...

43 Perturbative Effective Decay Operator Evaluated ββ version of these (which are for Hamiltonian).2 ε f V low-k ε i

44 Results Bare Effective 82 Se Ge

45 Results Bare Effective 82 Se Ge Can we really believe the results? Convergence is an issue, but a deeper one may be effect of many-body induced operators.

46 Nonperturbative Version 1. Use nearly-exact framework, e.g.coupled-clusters theory: to solve the closed shell + 2 (and closed-shell + 3) problems, 2. Map lowest eigenstates onto valence shell, determine effective Hamiltonian and decay operator (up to three-body). 3. Start adding nucleons to shell and use effective interaction and operator to calculate stuff, e.g. matrix element for 76 Ge. Already done this for two-body effective Hamiltonian, (with done with Jannsen, Hagen, Navratil, Signoracci).

47 Coupled Clusters in sd Shell: Oxygen 3 Energy [MeV] /2 + 9/2 + 3/2 + 5/2 + 7/2 + 1/2 + 9/ O 3/2 + 9/2 + 5/2 + 5/2 + 3/2 + 5/2 + 7/2 + 9/2 + 1/2 + 3/2 + * 9/2 + 5/2 + * 3/2 + 5/2 + * 7/2 + * 9/2 + * 1/2 + * O * 4 + * 2 + * 5/2 + 9/2 + 5/2 + 7/2 + 3/2 + 1/ O 9/2 + 7/2 + 7/2 +? 1/2 + 5/2 + 3/2 + 9/2 + 5/2 + 7/2 + 3/2 + 1/ Energy [MeV] /2 + 3/ O 3/2 + 5/ /2 + * 5/2 + * /2 + 7/2 + 3/2 + 5/2 + 3/2 + 5/2 + 1/ O 0 + 3/2 + 5/2 + 1/ * 5/2 + 9/2 + 7/2 + 3/2 + 5/2 + 3/2 + 5/2 + 1/2 + 5/ O 5/2 +? /2 + * CCEI Exp. USD CCEI Exp. USD CCEI Exp. USD

48 Part II: EDMs and CP Violation

49 The T Operator in QM is Different.

50 The T Operator in QM is Different. Not linear: T [x, p]t 1 = [x, p] so i is odd under T. Has no eigenstates in the conventional sense: T a = a T (α a ) = α T a = α a α a for α complex. Typical physical states J, M not even close to eigenstates of T.

51 The T Operator in QM is Different. Not linear: T [x, p]t 1 = [x, p] so i is odd under T. Has no eigenstates in the conventional sense: T a = a T (α a ) = α T a = α a α a for α complex. Typical physical states J, M not even close to eigenstates of T. As a result, T violation doesn t show up as mixing of states with opposite T."

52 Is T Violated in the Real World? Yup! Violation is seen in decay of K-mesons (direct) and B-mesons (through CP violation). And we strongly believe that T ( CP ) violation played an important role in the early universe, causing excess of matter over antimatter.

53 What is the Source of T -Violation? K and B phenomena almost certainly due to a phase in the 3 3 CKM matrix, which converts (d, s, b) to weak eigenstates" that couple to W and Z.

54 What is the Source of T -Violation? K and B phenomena almost certainly due to a phase in the 3 3 CKM matrix, which converts (d, s, b) to weak eigenstates" that couple to W and Z. But this violation is too weak to cause baryogenesis, which must arise outside the standard model, e.g. through supersymmetry heavy neutrinos Higgs sector...

55 What is the Source of T -Violation? K and B phenomena almost certainly due to a phase in the 3 3 CKM matrix, which converts (d, s, b) to weak eigenstates" that couple to W and Z. But this violation is too weak to cause baryogenesis, which must arise outside the standard model, e.g. through supersymmetry heavy neutrinos Higgs sector... To confuse things more, there s the strong CP problem.

56 What is the Source of T -Violation? K and B phenomena almost certainly due to a phase in the 3 3 CKM matrix, which converts (d, s, b) to weak eigenstates" that couple to W and Z. But this violation is too weak to cause baryogenesis, which must arise outside the standard model, e.g. through supersymmetry heavy neutrinos Higgs sector... To confuse things more, there s the strong CP problem. We need to see T -violation outside mesonic systems to understand its sources. EDM s are not sensitive to CKM T violation, but are to other sources. They re already putting extreme pressure on supersymmetry.

57 Connection Between EDMs and T Violation

58 EDMs Sensitive to New Physics In standard model only one phase. Diagrams cancel to high order, e.g.: i sinδ f W γ f f f f f + i sin δ +... f W γ f

59 EDMs Sensitive to New Physics In standard model only one phase. Diagrams cancel to high order, e.g.: i sinδ f W γ f f f f f + i sin δ +... SUSY has many phases. Low-order diagrams uncanceled, e.g.: f γ ~ f W γ f f e iθ f

60 EDMs Sensitive to New Physics In standard model only one phase. Diagrams cancel to high order, e.g.: i sinδ f W γ f f f f f + i sin δ +... SUSY has many phases. Low-order diagrams uncanceled, e.g.: f γ ~ f W γ f f e iθ f Thus, EDMs are insensitive to standard-model CP sensitive to extra-standard-model CP. but

61 One Way Things Get EDMs Starting at fundamental level and working up: Underlying fundamental theory generates three T -violating πn N vertices: N? ḡ π New physics

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

63 How Diamagnetic Atoms Get EDMs Nucleus can get one from nucleon EDM or T -violating NN interaction: ḡ π γ {[ V P T ḡ 0 τ 1 τ 2 ḡ1 ] 2 (τ 1 z + τ1 z ) + ḡ 2 (3τ1 z τ2 z τ 1 τ 2 ) (σ 1 σ 2 ) } ḡ1 2 (τ 1 z τ2 z ) (σ 1 + σ 2 ) ( 1 2 ) exp ( m π r 1 r 2 ) m π r 1 r 2 + contact term

64 How Diamagnetic Atoms Get EDMs Nucleus can get one from nucleon EDM or T -violating NN interaction: ḡ π γ {[ V P T ḡ 0 τ 1 τ 2 ḡ1 ] 2 (τ 1 z + τ1 z ) + ḡ 2 (3τ1 z τ2 z τ 1 τ 2 ) (σ 1 σ 2 ) } ḡ1 2 (τ 1 z τ2 z ) (σ 1 + σ 2 ) ( 1 2 ) exp ( m π r 1 r 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 r2 pz p +....

65 How Diamagnetic Atoms Get EDMs Nucleus can get one from nucleon EDM or T -violating NN interaction: ḡ π γ {[ V P T ḡ 0 τ 1 τ 2 ḡ1 ] 2 (τ 1 z + τ1 z ) + ḡ 2 (3τ1 z τ2 z τ 1 τ 2 ) (σ 1 σ 2 ) } ḡ1 2 (τ 1 z τ2 z ) (σ 1 + σ 2 ) ( 1 2 ) exp ( m π r 1 r 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 r2 pz p Job of nuclear theory: calculate dependence of S on the ḡ s (and on the contact term and nucleon EDM).

66 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.

67 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 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 ) dk p k, H 0 K.E. + Coulomb dipole perturbation

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

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

70 All is Not Lost, Though... 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 p e p ( r 2 p 5 3 R2 ch ) z p +...

71 All is Not Lost, Though... 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 p e p ( r 2 p 5 3 R2 ch ) z 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) 10 8, Ughh! Fortunately the large nuclear charge and relativistic wave functions offset this factor by 10Z Overall suppression of D A is only about 10 3.

72 Theory for Heavy Nuclei S Z 2, so experiments are in heavy nuclei.

73 Theory for Heavy Nuclei S Z 2, so experiments are in heavy nuclei. Paradigm: Density functional Theory Höhenberg-Kohn-Sham: Can get exact density from Hartree calculation with effective interaction (density functional).

74 Theory for Heavy Nuclei S Z 2, so experiments are in heavy nuclei. Paradigm: Density functional Theory Höhenberg-Kohn-Sham: Can get exact density from Hartree calculation with effective interaction (density functional). Nuclear version: Mean-field theory with density-dependent interactions (named after Skyrme) built from delta functions and derivatives of delta functions, plus corrections, e.g.: projection of deformed wave functions onto states with good angular momentum mixing of several mean fields... Density functional still largely phenomenological.

75 Nuclear Deformation

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

77 Deformation and Angular-Momentum Restoration If deformed state Ψ K has good intrinsic J z = K, average over angles gives: J, M = 2J + 1 8π 2 DMK(Ω)R(Ω) J Ψ K dω

78 Deformation and Angular-Momentum Restoration If deformed state Ψ K has good intrinsic J z = K, average over angles gives: J, M = 2J + 1 8π 2 DMK(Ω)R(Ω) J Ψ K dω 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. Ω Ω (Geometric factor) Ψ K S z Ψ K }{{} S intr.

79 Deformation and Angular-Momentum Restoration If deformed state Ψ K has good intrinsic J z = K, average over angles gives: J, M = 2J + 1 8π 2 DMK(Ω)R(Ω) J Ψ K dω 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. Ω Ω (Geometric factor) Ψ K S z Ψ K }{{} S intr. For expectation value in J = 1 2 state: { S S = S z J= 1 2,M= 1 = intr. spherical nucleus S intr. rigidly deformed nucleus Exact answer somewhere in between.

80 Schiff Moment with Octupole Deformation Here we treat always V P T as explicit perturbation: S = m 0 S m m V P T 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 P T 0 E 0 E 0 + c.c. S intr. V P T intr. E 0 E 0

81 Schiff Moment with Octupole Deformation Here we treat always V P T as explicit perturbation: S = m 0 S m m V P T 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 P T 0 E 0 E 0 + c.c. S intr. V P T intr. E 0 E 0 Why is this? See next slide.

82 Schiff Moment with Octupole Deformation Here we treat always V P T as explicit perturbation: S = m 0 S m m V P T 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 P T 0 E 0 E 0 + c.c. S intr. V P T intr. 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.

83 A Bit More Detail 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 ( ± )

84 A Bit More Detail 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 P T 0 E 0 E 0 + c.c.

85 A Bit More Detail 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 P T 0 E 0 E 0 + c.c. And in the rigid-deformation limit 0 O 0 O = O intr. like angular momentum.

86 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

87 225 Ra Results Hartree-Fock calculation with our favorite interaction SkO gives S Ra = 1.5 gḡ gḡ gḡ 2 (e fm 3 ) Larger by over 100 than in 199 Hg! Variation a factor of 2 or 3.

88 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 g 1 in 199 Hg (range includes zero). How can we reduce them?

89 Grounding the Calculations Important new developments here. S intr. correlated with octupole moment, which will be extracted from measured E3 transitions. Schiff moment [(10 fm) 3 ] HF BCS SKM* SLy4 SKO' UDF0 SKX c SIII SKM* SLy4 SKO' SIII SKX c UDF0 225 Ra N = Rn P = Ra 229 Pa SkO L 2/!0 1/!0 0/!0./!0!0!0!"!"!"!"!"!"!0!0!0!0 3! "!!0 4! Reduced mat 224 Ra Octupole moment Q 30 [(10 fm) 3 ] Gaffney et al., Nature

90 Grounding the Calculations Important new developments here. S intr. correlated with octupole moment, which will be extracted from measured E3 transitions. Schiff moment [(10 fm) 3 ] HF BCS SKM* SLy4 SKO' UDF0 SKX c SIII SKM* SLy4 SKO' SIII SKX c UDF0 225 Ra N = Rn P = Ra 229 Pa SkO L 2/!0 1/!0 0/!0./!0!0!0!"!"!"!"!"!"!0!0!0!0 3! "!!0 4! Reduced mat 224 Ra Octupole moment Q 30 [(10 fm) 3 ] Gaffney et al., Nature

91 Grounding the Calculations Important new developments here. S intr. correlated with octupole moment, which will be extracted from measured E3 transitions. Schiff moment [(10 fm) 3 ] HF BCS SKM* SLy4 SKO' UDF0 SKX c SIII SKM* SLy4 SKO' SIII SKX c UDF0 225 Ra N = Rn P = Ra 229 Pa SkO L 2/!0 1/!0 0/!0./!0!0!0!"!"!"!"!"!"!0!0!0!0 3! "!!0 4! Reduced mat 224 Ra Octupole moment Q 30 [(10 fm) 3 ] Gaffney et al., Nature Transitions in 225 Ra to be measured soon.

92 In Sum... Calculations have become sophisticated, but we still have a lot of work to do. Octupole-deformed nuclei more under control than 199 Hg.

93 In Sum... Calculations have become sophisticated, but we still have a lot of work to do. Octupole-deformed nuclei more under control than 199 Hg. THE END Thanks for your kind attention.