Electron electric dipole moment

Short description: Quantity relating an electron's potential energy to electric field strength

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

U=−𝐝e⋅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, 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]

## Contents

* 1 Implications for Standard Model and extensions
* 2 Formal definition
* 3 Experimental measurements
* 3.1 Future proposed experiments
* 4 See also
* 5 Footnotes
* 6 References

## 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][lower-alpha 1]

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 |de| that ranges from 10−27 to 10−29 e⋅cm.[5] Some supersymmetric models predict that |de| > 10−26 e⋅cm[6] 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,[7] would place further limits on which theory takes precedence.[8]

Measurements of the electron's electric dipole moment.

## Formal definition

As the electron has a net charge, the definition of its electric dipole moment is ambiguous in that 𝐝e=∫(𝐫−𝐫0)ρ(𝐫)d3𝐫 depends on the point 𝐫0 about which the moment of the charge distribution ρ(𝐫) is taken. If we were to choose 𝐫0 to be the center of charge, then 𝐝e would be identically zero. A more interesting choice would be to take 𝐫0 as the electron's center of mass evaluated in the frame in which the electron is at rest.[9] 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 Fi(q2) appearing in the matrix element[10] ⟨pf|jμ|pi⟩=u¯(pf){F1(q2)γμ+iσμν2meqνF2(q2)+iϵμνρσσρσqνF3(q2)+12me(qμ−q22meγμ)γ5F4(q2)}u(pi) of the electromagnetic current operator between two on-shell states with Lorentz invariant phase space normalization in which ⟨pf|pi⟩=2E(2π)3δ3(𝐩f−𝐩i). Here, u(pi) and u¯(pf) are 4-spinor solutions of the Dirac equation normalized so that u¯u=2me, and qμ=pfμ−piμ is the momentum transfer from the current to the electron. The q2=0 form factor F1(0)=Q is the electron's charge, μ=F1(0) + F2(0)2me is its static magnetic dipole moment, and −F3(0)2me provides the formal definition of the electron's electric dipole moment. The remaining form factor F4(q2) would, if nonzero, be the anapole moment.[3]

## 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 U=−𝐝e⋅𝐄 as a slight shift of spectral lines. The sensitivity to 𝐝e scales approximately with the nuclear charge cubed.[11] For this reason, electron EDM searches almost always are conducted on systems involving heavy elements.[6]

To date, no experiment has found a non-zero electron EDM. As of 2020 the Particle Data Group publishes its value as |de| < 0.11×10−28 e⋅cm. 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[12]
2011  | Imperial College London | Edward Hinds, Ben Sauer  | Molecular beam | YbF | 1.1×10−27 e⋅cm[13]
2014  | Harvard-Yale
(ACME I experiment)  | David DeMille, John Doyle, Gerald Gabrielse | Molecular beam | ThO | 8.7×10−29 e⋅cm[14]
2017  | JILA | Eric Cornell, Jun Ye | Ion trap | HfF\+  | 1.3×10−28 e⋅cm[15]
2018  | Harvard-Yale
(ACME II experiment)  | David DeMille, John Doyle, Gerald Gabrielse | Molecular beam | ThO | 1.1×10−29 e⋅cm[16]
2022  | JILA | Eric Cornell, Jun Ye | Ion trap | HfF\+  | 4.1×10−30 e⋅cm[17][18]

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.[19][20]

### 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[21]
* John Doyle (Harvard University), Nicholas Hutzler (California Institute of Technology), and Timothy Steimle (Arizona State University): YbOH molecular trap[22]
* EDMcubed collaboration, Amar Vutha (University of Toronto), Eric Hessels (York University): oriented polar molecules in an inert gas matrix[23][24]
* David Weiss (Pennsylvania State University): Cs and Rb atoms trapped inside an optical lattice[25]
* TRIUMF: Fountain of laser cooled Fr[26]
* EDMMA collaboration: Cs in an inert gas matrix[27]
* Laboratori Nazionali di Legnaro DOCET : BaF in a parahydrogen matrix [28]

## See also

* Neutron electric dipole moment
* Electron magnetic moment
* Nuclear electric dipole 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

1. ↑ Eckel, S.; Sushkov, A.O.; Lamoreaux, S.K. (2012). "Limit on the electron electric dipole moment using paramagnetic ferroelectric Eu0.5Ba0.5TiO3". Physical Review Letters 109 (19). doi:10.1103/PhysRevLett.109.193003. PMID 23215379. Bibcode: 2012PhRvL.109s3003E.
2. ↑ 2.0 2.1 2.2 2.3 Pospelov, M.; Ritz, A. (2005). "Electric dipole moments as probes of new physics". Annals of Physics 318 (1): 119–169. doi:10.1016/j.aop.2005.04.002. Bibcode: 2005AnPhy.318..119P.
3. ↑ 3.0 3.1 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
5. ↑ Roussy, Tanya S.; Caldwell, Luke; Wright, Trevor; Cairncross, William B.; Shagam, Yuval; Ng, Kia Boon; Schlossberger, Noah; Park, Sun Yool et al. (2023-07-07). "An improved bound on the electron's electric dipole moment" (in en). Science 381 (6653): 46–50. doi:10.1126/science.adg4084. ISSN 0036-8075. PMID 37410848. Bibcode: 2023Sci...381...46R. https://www.science.org/doi/10.1126/science.adg4084.
6. ↑ 6.0 6.1 Arnowitt, R.; Dutta, B.; Santoso, Y. (2001). "Supersymmetric phases, the electron electric dipole moment and the muon magnetic moment". Physical Review D 64 (11). doi:10.1103/PhysRevD.64.113010. Bibcode: 2001PhRvD..64k3010A.
7. ↑ "Ultracold Atomic Physics Group". https://web2.ph.utexas.edu/~coldatom/EDM.html.
8. ↑ Andreev, V.; Ang, D. G.; DeMille, D.; Doyle, J. M.; Gabrielse, G.; Haefner, J.; Hutzler, N. R.; Lasner, Z. et al. (October 2018). "Improved limit on the electric dipole moment of the electron" (in en). Nature 562 (7727): 355–360. doi:10.1038/s41586-018-0599-8. ISSN 1476-4687. PMID 30333583. Bibcode: 2018Natur.562..355A. https://www.nature.com/articles/s41586-018-0599-8.
9. ↑ Verma, Mohit; Jayich, Andrew M.; Vutha, Amar C. (2020-10-08). "Electron Electric Dipole Moment Searches Using Clock Transitions in Ultracold Molecules" (in en). Physical Review Letters 125 (15). doi:10.1103/PhysRevLett.125.153201. ISSN 0031-9007. PMID 33095600. Bibcode: 2020PhRvL.125o3201V. https://link.aps.org/doi/10.1103/PhysRevLett.125.153201.
10. ↑ Nowakowski, M.; Paschos, E.A.; Rodriguez, J.M. (2005). "All electromagnetic form factors". European Journal of Physics 26 (4): 545–560. doi:10.1088/0143-0807/26/4/001. Bibcode: 2005EJPh...26..545N.
11. ↑ Alarcon, Ricardo; Alexander, Jim; Anastassopoulos, Vassilis; Aoki, Takatoshi; Baartman, Rick; Baeßler, Stefan; Bartoszek, Larry; Beck, Douglas H.; Bedeschi, Franco; Berger, Robert; Berz, Martin; Bethlem, Hendrick L.; Bhattacharya, Tanmoy; Blaskiewicz, Michael; Blum, Thomas (2022-04-04). "Electric dipole moments and the search for new physics". arXiv:2203.08103 [hep-ph].
12. ↑ 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). doi:10.1103/PhysRevLett.88.071805. PMID 11863886. Bibcode: 2002PhRvL..88g1805R. https://cloudfront.escholarship.org/dist/prd/content/qt33q46241/qt33q46241.pdf.
13. ↑ 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. doi:10.1038/nature10104. PMID 21614077. Bibcode: 2011Natur.473..493H. http://spiral.imperial.ac.uk/bitstream/10044/1/19405/2/Nature_473_7348_2011.pdf.
14. ↑ The ACME Collaboration (January 2014). "Order of Magnitude Smaller Limit on the Electric Dipole Moment of the Electron". Science 343 (6168): 269–272. doi:10.1126/science.1248213. PMID 24356114. Bibcode: 2014Sci...343..269B. http://laserstorm.harvard.edu/wiki/images/1/1a/Order_of_Magnitude_Smaller_Limit_on_the_Electric_Dipole_Moment_of_the_Electron._-_Baron_et_al._-_2014.pdf. Retrieved 2014-06-24.
15. ↑ Cairncross, William B.; Gresh, Daniel N.; Grau, Matt; Cossel, Kevin C.; Roussy, Tanya S.; Ni, Yiqi; Zhou, Yan; Ye, Jun et al. (2017-10-09). "Precision Measurement of the Electron's Electric Dipole Moment Using Trapped Molecular Ions". Physical Review Letters 119 (15). doi:10.1103/PhysRevLett.119.153001. PMID 29077451. Bibcode: 2017PhRvL.119o3001C.
16. ↑ The ACME Collaboration (October 2018). "Improved Limit on the Electric Dipole Moment of the Electron". Nature 562 (7727): 355–360. doi:10.1038/s41586-018-0599-8. PMID 30333583. Bibcode: 2018Natur.562..355A. https://authors.library.caltech.edu/89472/3/41586_2018_599_MOESM1_ESM.pdf.
17. ↑ Roussy, Tanya S.; Caldwell, Luke; Wright, Trevor; Cairncross, William B.; Shagam, Yuval; Ng, Kia Boon; Schlossberger, Noah; Park, Sun Yool et al. (2023). "An improved bound on the electron's electric dipole moment". Science 381 (6653): 46–50. doi:10.1126/science.adg4084. PMID 37410848. Bibcode: 2023Sci...381...46R.
18. ↑ Roussy, Tanya S.; Caldwell, Luke; Wright, Trevor; Cairncross, William B.; Shagam, Yuval; Ng, Kia Boon; Schlossberger, Noah; Park, Sun Yool et al. (2023-07-06), "A new bound on the electron's electric dipole moment", Science 381 (6653): 46–50, doi:10.1126/science.adg4084, PMID 37410848, Bibcode: 2023Sci...381...46R, https://www.science.org/doi/10.1126/science.adg4084
19. ↑ "ACME Electron EDM". http://doylegroup.harvard.edu/edm/index.html.
20. ↑ Ang, D. G.; Meisenhelder, C.; Panda, C. D.; Wu, X.; DeMille, D.; Doyle, J. M.; Gabrielse, G. (2022-08-15). "Measurement of the $H^3\Delta_1$ radiative lifetime in ThO". Physical Review A 106 (2). doi:10.1103/PhysRevA.106.022808. https://link.aps.org/doi/10.1103/PhysRevA.106.022808.
21. ↑ Aggarwal, Parul; Bethlem, Hendrick L.; Borschevsky, Anastasia; Denis, Malika; Esajas, Kevin; Haase, Pi A.B.; Hao, Yongliang; Hoekstra, Steven et al. (2018). "Measuring the electric dipole moment of the electron in BaF". The European Physical Journal D 72 (11). doi:10.1140/epjd/e2018-90192-9.
22. ↑ 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). doi:10.1103/PhysRevLett.119.133002. PMID 29341669. Bibcode: 2017PhRvL.119m3002K.
23. ↑ 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" (in en). Atoms 6 (1): 3. doi:10.3390/atoms6010003. Bibcode: 2018Atoms...6....3V.
24. ↑ "EDMcubed". https://www.yorku.ca/edmcubed/.
25. ↑ "Search for the Electron EDM Using Cs and Rb in Optical Lattice Traps" (in en). https://pennstate.pure.elsevier.com/en/projects/search-for-the-electron-edm-using-cs-and-rb-in-optical-lattice-tr.
26. ↑ "Report Summary | TRIUMF: Canada's National Laboratory for Particle and Nuclear Physics". https://mis.triumf.ca/science/experiment/view/S1324LOI.
27. ↑ "Moment dipolaire électrique des électrons à l'aide de Cs en matrice cryogénique - LAC". http://www.lac.universite-paris-saclay.fr/?emploi=moment-dipolaire-electrique-des-electrons-a-laide-de-cs-en-matrice-cryogenique.
28. ↑ Lippi, Alessandro; Carugno, Giovanni; Calabrese, Roberto; Chiossi, Federico; Guarise, Marco; Makhdoom, Madiha M.; Messineo, Giuseppe; Pazzini, Jacopo (2026). "Neutral barium in solid neon: Optical spectroscopy and first excited state lifetime". The Journal of Chemical Physics 164 (14): 144303. doi:10.1063/5.0316792.

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