Exploring EDM enhancements in quadrupole-octupole interacting

boson model

Alexander Volya Florida State University

Dipole-moment and violation of P and T-symmetries

spin

spin

d dT-reversal

spin

spin

d dP-reversal

Observation of the dipole moment is an indication of parity and time-reversal violation

Limit on EDM in electron Experiment 10-27 e cmStandard model » 10-38 e cmPhysics beyond SM » 10-28 e cm

Why is this interesting?

• Sensitive test of CP violation in the standard model • Baryon asymmetry in the universe. • Physics beyond standard model.

system EDM limit SM

e (electron) 10-27 10-40

n (neutron) 3.0x10-26 10-32

225Ra 1.4x10-23 10-33

199Hg 7.4x10-30 10-33

T. E. Chupp, P. Fierlinger, M. J. Ramsey-Musolf, and J. T. Singh, Electric dipole moments of atoms, molecules, nuclei, and particles, Rev. Mod. Phys. 91, 015001 (2019).

Schiff Moment• The nuclear dipole moment causes

polarization of the electron cloud so that the total atomic dipole moment for point particles vanishes!

• For a finite size nucleus the screening is not complete. The net result is determined by the vector called the Schiff moment

R. A. Sen'kov, N. Auerbach, V. V. Flambaum, and V. G. Zelevinsky, Phys. Rev. A 77, 014101 (2008).

PT mixing NN-interaction

O. P. Sushkov, V. V. Flambaum, and I. B. Khriplovich, Feasibility of Investigation of P-Odd and T-Odd Nuclear-Forces in Atomic and Molecular Experiments, Z. Eksp. Theor. Fiz. 87, 1521-1540 (1984).

Parity conservation

PT mixing in a parity-doublet

Small parity violating interaction W

Perturbed ground state

Non-zero Schiff moment

Single-particle parity-doublet

No enhancement in close-lying states!

O. P. Sushkov, V. V. Flambaum, and I. B. Khriplovich, Feasibility of Investigation of P-Odd and T-Odd Nuclear-Forces in Atomic and Molecular Experiments, Z. Eksp. Theor. Fiz. 87, 1521-1540 (1984).

199Hg example

5.8 MeV

1/2+

1/2-

Many-body enhancements

• Proximity of levels of opposite parity • Many-body complexity

• Chaotic dynamics • Collective dynamics (parity doublets)

• Solid state or molecular mechanisms

Parity-doublet octupole shape

Strongly enhanced Schiff moment

Body-fixed frame:

Lab frame:

V. Spevak, N. Auerbach, and V. V. Flambaum, Enhanced T-odd, P-odd electromagnetic moments in reflection asymmetric nuclei, Phys. Rev. C 56, 1357--1369 (1997).

The path to a large Schiff moment

• 1. The quadrupole +octupole deformation leads to large intrinsic Schiff moments [1].

• 2. The parity doublets in the reflection asymmetric nuclei are very close in energy.

• 3. Parity and time reversal violating matrix element between the members of the doublet is relatively large.

• 4. Heavy nuclei - large Z.• Candidates 223Ra, 225Ra, 223Rn, 221Fr, 223FrQuestions:• Is enhancement possible in dynamical regions of soft

shape?

[1] V. Spevak, N. Auerbach, and V. V. Flambaum Phys. Rev. C 56, 1357 - 1369 (1997)

Soft octupole mode in a deformed nucleus

V. V. Flambaum and V. G. Zelevinsky, Enhancement of nuclear Schiff moments and time-reversal violation in atoms due to soft nuclear octupole vibrations, Phys. Rev. C 68, 035502 (2003).

Soft octupole

Soft dipole + octupole?

Soft dipole + soft octupole?

Vibration-rotation shape transition?

Coupling between phonons?

Interacting Boson Model (IBM)

Consistent Q model

F. Iachello and A. Arima, The Interacting Boson Model (Cambridge University Press, 1987).

R. Casten (ed.) Algebraic Approaches to Nuclear Structure, Interacting Boson and Fermion Models (Harwood Academic Publishers, Switzerland, 1993).

IBM shape triangle

prolateoblate

0+

2+

4+

IBM: Quadrupole moment of the 2+

0+

2+

4+

f-boson

Angular momentum

Schiff operator

Quadrupole-octupole coupling

3-

3-

2+

Coupling between modes, three-body force?

Mueller, et.al. PRC 73, (2006) 014316 V. Zelevinsky Phys. At. Nucl. 72, 1107 (2009)

Quadrupole-octupole collectivity

3-

3-

2+

Coupling between modes, three-body force?Cocks, et.al. PRL 78, 2920

Mueller, et.al. PRC 73, (2006) 014316 V. Zelevinsky Phys. At. Nucl. 72, 1107 (2009) B=0.7 MeV

Quadrupole-octupole coupling

Quadrupole-octupole coupling (revisited)

From Xe

Schiff moment

vibrational rotational

Schiff moment

Prolate to oblate transition

oblate prolate

Particle-core coupling

IBM s.p.

Particle-core coupling

Explicit PT-violation

Dipole moment of excited states

0+

2+

E-field

The “minimal” Shell Model Hamiltonian Two-level system with single-particle states of

opposite parity j+ and j-

Parity Mixing operator

Schiff vector

Dynamics• Single-particle gap: • Pairing: P-pair operator• Quadrupole interaction: Q2 -phonon• Octupole interaction: Q3 –phonon

V. Zelevinsky, A. Volya, and N. Auerbach, Phys. Rev. C 78, 014310 (2008).

Details

V. Zelevinsky, A. Volya, and N. Auerbach, Phys. Rev. C 78, 014310 (2008).

Spectra of the two-level model

Spectra of the two-level model

Spectra of the two-level model

Spectra of the two-level model

Degenerate levels level interchange symmetry

•Even particle system can be classified with the additional symmetry•Odd particle system: all states are degenerate parity doublets.

Degenerate levels level interchange symmetry

•Even particle system can be classified with the additional symmetry•Odd particle system: all states are degenerate parity doublets.

Two-level j=17/2 model Schiff and Weak Operators G=0.2 κ3=1

Left: N=6 system•Schiff operator <2|S|3>•Excitation energy of 2+ and 3- states Right: N=7 system•Energy of the g.s. doublet 17/2+ and 17/2-

•Schiff operator <17/2+|S|17/2->•Weak Matrix element <17/2+|W|17/2->•Enhancement product

Two-level j=17/2 model Schiff and Weak Operators G=0.2 κ3=1

Left: N=6 system•Schiff operator <2|S|3>•Excitation energy of 2+ and 3- states Right: N=7 system•Energy of the g.s. doublet 17/2+ and 17/2-

•Schiff operator <17/2+|S|17/2->•Weak Matrix element <17/2+|W|17/2->•Enhancement product

Support:• U.S. Department of Energy DE-SC0009883. • Binational Science Foundation USA-Israel • U.S. National Science Foundation

CollaboratorsNaftali AuerbachVladimir ZelevinskyCharles White

Acknowledgments