Electron electric dipole moment

The electron electric dipole moment d_e is an intrinsic property of an electron such that the potential energy is linearly related to the strength of the electric field:

${\displaystyle U=\mathbf {d} _{\rm {e}}\cdot \mathbf {E} .}$

The electron's electric dipole moment (EDM) must be collinear with the direction of the electron's magnetic moment (spin).^[1] Within the Standard Model of elementary particle physics, such a dipole is predicted to be non-zero but very small, at most 10⁻³⁸ e⋅cm,^[2] where e stands for the elementary charge. The discovery of a substantially larger electron electric dipole moment would imply a violation of both parity invariance and time reversal invariance.^[3][4]

Implications for Standard Model and extensions

In the Standard Model, the electron EDM arises from the CP-violating components of the CKM matrix. The moment is very small because the CP violation involves quarks, not electrons directly, so it can only arise by quantum processes where virtual quarks are created, interact with the electron, and then are annihilated.^[2][a]

If neutrinos are Majorana particles, a larger EDM (around 10⁻³³ e⋅cm) is possible in the Standard Model.^[2]

Many extensions to the Standard Model have been proposed in the past two decades. These extensions generally predict larger values for the electron EDM. For instance, the various technicolor models predict |d_e| that ranges from 10⁻²⁷ to 10⁻²⁹ e⋅cm.^{[citation needed]} Some supersymmetric models predict that |d_e| > 10⁻²⁶ e⋅cm^[5] but some other parameter choices or other supersymmetric models lead to smaller predicted values. The present experimental limit therefore eliminates some of these technicolor/supersymmetric theories, but not all. Further improvements, or a positive result,^[6] would place further limits on which theory takes precedence.

Formal definition

As the electron has a net charge, the definition of its electric dipole moment is ambiguous in that

${\displaystyle \mathbf {d} _{\rm {e}}=\int ({\mathbf {r} }-{\mathbf {r} }_{0})\rho ({\mathbf {r} })d^{3}{\mathbf {r} }}$ 

depends on the point ${\displaystyle {\mathbf {r} }_{0}}$  about which the moment of the charge distribution ${\displaystyle \rho ({\mathbf {r} })}$  is taken. If we were to choose ${\displaystyle {\mathbf {r} }_{0}}$  to be the center of charge, then ${\displaystyle \mathbf {d} _{\rm {e}}}$  would be identically zero. A more interesting choice would be to take ${\displaystyle {\mathbf {r} }_{0}}$  as the electron's center of mass evaluated in the frame in which the electron is at rest.

Classical notions such as the center of charge and mass are, however, hard to make precise for a quantum elementary particle. In practice the definition used by experimentalists comes from the form factors ${\displaystyle F_{i}(q^{2})}$  appearing in the matrix element^[7]

${\displaystyle \langle p_{f}|j^{\mu }|p_{i}\rangle ={\bar {u}}(p_{f})\left\{F_{1}(q^{2})\gamma ^{\mu }+{\frac {i\sigma ^{\mu \nu }}{2m_{\rm {e}}}}q_{\nu }F_{2}(q^{2})+i\epsilon ^{\mu \nu \rho \sigma }\sigma _{\rho \sigma }q_{\nu }F_{3}(q^{2})+{\frac {1}{2m_{\rm {e}}}}\left(q^{\mu }-{\frac {q^{2}}{2m_{e}}}\gamma ^{\mu }\right)\gamma _{5}F_{4}(q^{2})\right\}u(p_{i})}$ 

of the electromagnetic current operator between two on-shell states with Lorentz invariant phase space normalization in which

${\displaystyle \langle p_{f}\vert p_{i}\rangle =2E(2\pi )^{3}\delta ^{3}({\bf {p}}_{f}-{\bf {p_{i}}}).}$ 

Here ${\displaystyle u(p_{i})}$  and ${\displaystyle {\bar {u}}(p_{f})}$  are 4-spinor solutions of the Dirac equation normalized so that ${\displaystyle {\bar {u}}u=2m_{e}}$ , and ${\displaystyle q^{\mu }=p_{f}^{\mu }-p_{i}^{\mu }}$  is the momentum transfer from the current to the electron. The ${\displaystyle q^{2}=0}$  form factor ${\displaystyle F_{1}(0)=Q}$  is the electron's charge, ${\displaystyle \mu ={\tfrac {F_{1}(0)\ +\ F_{2}(0)}{2m_{\rm {e}}}}}$  is its static magnetic dipole moment, and ${\displaystyle {\tfrac {-F_{3}(0)}{2m_{\rm {e}}}}}$  provides the formal definition of the electron's electric dipole moment. The remaining form factor ${\displaystyle F_{4}(q^{2})}$  would, if nonzero, be the anapole moment.

Experimental measurements

Electron EDMs are usually not measured on free electrons, but instead on bound, unpaired valence electrons inside atoms and molecules. In these, one can observe the effect of ${\displaystyle U=\mathbf {d} _{\rm {e}}\cdot \mathbf {E} }$  as a slight shift of spectral lines. The sensitivity to ${\displaystyle \mathbf {d} _{\rm {e}}}$  scales approximately with the nuclear charge cubed.^[8] For this reason, electron EDM searches almost always are conducted on systems involving heavy elements.

To date, no experiment has found a non-zero electron EDM. As of 2020 the Particle Data Group publishes its value as |d_e| < 0.11×10⁻²⁸ e⋅cm.^[9] Here is a list of some electron EDM experiments after 2000 with published results:

List of Electron EDM Experiments

Year	Location	Principal Investigators	Method	Species	Experimental upper limit on |de|
2002	University of California, Berkeley	Eugene Commins, David DeMille	Atomic beam	Tl	1.6×10^−27 e⋅cm^[10]
2011	Imperial College London	Edward Hinds, Ben Sauer	Molecular beam	YbF	1.1×10^−27 e⋅cm^[11]
2014	Harvard-Yale (ACME I experiment)	David DeMille, John Doyle, Gerald Gabrielse	Molecular beam	ThO	8.7×10^−29 e⋅cm^[12]
2017	JILA	Eric Cornell, Jun Ye	Ion trap	HfF+	1.3×10^−28 e⋅cm^[13]
2018	Harvard-Yale (ACME II experiment)	David DeMille, John Doyle, Gerald Gabrielse	Molecular beam	ThO	1.1×10^−29 e⋅cm^[14]
2022	JILA	Eric Cornell, Jun Ye	Ion trap	HfF+	4.1×10^−30 e⋅cm^[15] ^[16]

The ACME collaboration is, as of 2020, developing a further version of the ACME experiment series. The latest experiment is called Advanced ACME or ACME III and it aims to improve the limit on electron EDM by one to two orders of magnitude.^[17]

Future proposed experiments

Besides the above groups, electron EDM experiments are being pursued or proposed by the following groups:

• University of Groningen: BaF molecular beam^[18]
• John Doyle (Harvard University), Nicholas Hutzler (California Institute of Technology), and Timothy Steimle (Arizona State University): YbOH molecular trap^[19]
• Amar Vutha (University of Toronto), Eric Hessels (York University): oriented polar molecules in an inert gas matrix^[20]
• Pennsylvania State University: Cs and Rb atoms trapped inside an optical lattice^[21]
• TRIUMF: Fountain of laser cooled Fr^[22]
• EDMMA collaboration: Cs in an inert gas matrix^[23]

See also

• Neutron electric dipole moment
• Electron magnetic moment
• Anomalous electric dipole moment
• Anomalous magnetic dipole moment
• Electric dipole spin resonance
• Parity (physics) § Parity violation
• CP violation
• Charge conjugation
• T-symmetry

Footnotes

1. More precisely, a non-zero EDM does not arise until the level of four-loop Feynman diagrams and higher.^[2]

References

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2. Pospelov, M.; Ritz, A. (2005). "Electric dipole moments as probes of new physics". Annals of Physics. 318 (1): 119–169. arXiv:hep-ph/0504231. Bibcode:2005AnPhy.318..119P. doi:10.1016/j.aop.2005.04.002. S2CID 13827759.
3. Khriplovich, I.B.; Lamoreaux, S.K. (1997). CP violation without strangeness: Electric dipole moments of particles, atoms, and molecules. Springer-Verlag.
4. P. R. Bunker and P. Jensen (2005), Fundamentals of Molecular Symmetry (CRC Press) ISBN 0-7503-0941-5[1] Chapter 15
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11. Hudson, J.J.; Kara, D.M.; Smallman, I.J.; Sauer, B.E.; Tarbutt, M.R.; Hinds, E.A. (2011). "Improved measurement of the shape of the electron" (PDF). Nature. 473 (7348): 493–496. Bibcode:2011Natur.473..493H. doi:10.1038/nature10104. hdl:10044/1/19405. PMID 21614077. S2CID 205224996.
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13. Cairncross, William B.; Gresh, Daniel N.; Grau, Matt; Cossel, Kevin C.; Roussy, Tanya S.; Ni, Yiqi; Zhou, Yan; Ye, Jun; Cornell, Eric A. (2017-10-09). "Precision Measurement of the Electron's Electric Dipole Moment Using Trapped Molecular Ions". Physical Review Letters. 119 (15): 153001. arXiv:1704.07928. Bibcode:2017PhRvL.119o3001C. doi:10.1103/PhysRevLett.119.153001. PMID 29077451. S2CID 44043558.
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15. Roussy, Tanya S.; Caldwell, Luke; Wright, Trevor; Cairncross, William B.; Shagam, Yuval; Ng, Kia Boon; Schlossberger, Noah; Park, Sun Yool; Wang, Anzhou; Ye, Jun; Cornell, Eric A. (2022-12-22). "A new bound on the electron's electric dipole moment". arXiv:2212.11841.
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19. Kozyryev, Ivan; Hutzler, Nicholas R. (2017-09-28). "Precision Measurement of Time-Reversal Symmetry Violation with Laser-Cooled Polyatomic Molecules". Physical Review Letters. 119 (13): 133002. arXiv:1705.11020. Bibcode:2017PhRvL.119m3002K. doi:10.1103/PhysRevLett.119.133002. PMID 29341669. S2CID 33254969.
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