# Electron electric dipole moment

The electron electric dipole moment (EDM) de 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 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 existence of a non-zero electron electric dipole moment would imply a violation of both parity invariance and time reversal invariance.[3] 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] More precisely, a non-zero EDM does not arise until the level of four-loop Feynman diagrams and higher.[2] An additional, larger EDM (around 10−33 e·cm) is possible in the standard model if neutrinos are majorana particles.[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 de that ranges from 10−27 to 10−29 e·cm.[citation needed] Supersymmetric models predict that |de| > 10−26 e·cm.[4] The present experimental limit is therefore close to eliminating some of these theories. Further improvements, or a positive result,[5] would place further limits on which theory takes precedence.

## Formal Definition of the Electron EDM

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[6]

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

of the electromagnetic current operator between two on-shell states. Here ${\displaystyle u(p_{i})}$ and ${\displaystyle {\bar {u}}(p_{f})}$ are 4-spinor solution of the Dirac equation normalized so that ${\displaystyle {\bar {u}}u=2m_{\rm {e}}}$, 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, ${\displaystyle \mu =(F_{1}(0)+F_{2}(0))/2m_{\rm {e}}}$ is its static magnetic dipole moment, and ${\displaystyle -F_{3}(0)/2m_{\rm {e}}}$ provides the formal definion of the electron's electric dipole moment. The remaining form factor ${\displaystyle F_{4}(q^{2})}$ would, if non zero, be the anapole moment.

## Experimental Measurements of the Electron EDM

A non-zero electron EDM has not yet been found in all experiments to date. The Particle Data Group publishes its value as <0.87×10−28 e·cm.[7] Here is a list of electron EDM experiments starting in the 2000s which have published a result:

List of Electron EDM Experiments
Location Principal Investigators Method Species Experimental result Year
University of California, Berkeley Eugene Commins, David DeMille Atomic beam Tl |de| < 1.6×10−27 e·cm[8] 2002
Imperial College London Edward Hinds, Ben Sauer Molecular beam YbF |de| < 1.1×10−27 e·cm[9] 2011
Harvard-Yale David DeMille, John Doyle, Gerald Gabrielse Molecular beam ThO |de| < 8.7×10−29 e·cm[10] 2014
JILA Eric Cornell, Jun Ye Ion trap HfF+ |de| < 1.3×10−28 e·cm[11] 2017
Harvard-Yale David DeMille, John Doyle, Gerald Gabrielse Molecular beam ThO |de| < 1.1×10−29 e·cm[12] 2018

### Future proposed experiments

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

## References

1. ^ Thomas, Jessica. Synopsis: Particle Physics with Ferroelectrics.
2. ^ a b c d Pospelov, M.; Ritz, A. (2005). "Electric dipole moments as probes of new physics". Annals of Physics. 318: 119–169. arXiv:hep-ph/0504231. Bibcode:2005AnPhy.318..119P. doi:10.1016/j.aop.2005.04.002.
3. ^ Khriplovich, I. B.; Lamoreaux, S. K. (1997). CP Violation Without Strangeness: Electric Dipole Moments of Particles, Atoms, and Molecules. Springer-Verlag.
4. ^ Arnowitt, R.; Dutta, B.; Santoso, Y. (2001). "Supersymmetric phases, the electron electric dipole moment and the muon magnetic moment". Physical Review D. 64 (11): 113010. arXiv:hep-ph/0106089. Bibcode:2001PhRvD..64k3010A. doi:10.1103/PhysRevD.64.113010.
5. ^ a b "Ultracold Atomic Physics Group @ UT". web2.ph.utexas.edu. Retrieved 2015-11-13.
6. ^ Nowakowski, M.; Paschos, E. A.; Rodriguez, J. M. (2005). "All electromagnetic form factors". European Journal of Physics. 26: 545–560. arXiv:physics/0402058. doi:10.1088/0143-0807/26/4/001.
7. ^ http://pdg.lbl.gov/2018/listings/rpp2018-list-electron.pdf
8. ^ Regan, B. C.; Commins, Eugene D.; Schmidt, Christian J.; DeMille, David (2002-02-01). "New Limit on the Electron Electric Dipole Moment". Physical Review Letters. 88 (7): 071805. Bibcode:2002PhRvL..88g1805R. doi:10.1103/PhysRevLett.88.071805.
9. ^ 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". Nature. 473 (7348): 493–496. Bibcode:2011Natur.473..493H. doi:10.1038/nature10104.
10. ^ The ACME Collaboration (January 2014). "Order of Magnitude Smaller Limit on the Electric Dipole Moment of the Electron" (PDF). Science. 343 (6168): 269–272. arXiv:1310.7534. Bibcode:2014Sci...343..269B. doi:10.1126/science.1248213. PMID 24356114.
11. ^ 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.
12. ^ The ACME Collaboration (October 2018). "Improved Limit on the Electric Dipole Moment of the Electron". Nature. 562: 355–360. doi:10.1038/s41586-018-0599-8.
13. ^ "D. S. Weiss : Electron electric dipole moment search @ Penn State Physics". physwww.science.psu.edu. Retrieved 2016-03-13.
14. ^ https://arxiv.org/pdf/1804.10012.pdf
15. ^ 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.
16. ^ Vutha, A. C.; Horbatsch, M.; Hessels, E. A. (2018-01-05). "Oriented Polar Molecules in a Solid Inert-Gas Matrix: A Proposed Method for Measuring the Electric Dipole Moment of the Electron". Atoms. 6 (1): 3. arXiv:1710.08785. Bibcode:2018Atoms...6....3V. doi:10.3390/atoms6010003.