Electron electric dipole moment
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:
- <math>U = \mathbf d_{\rm e} \cdot \mathbf E.</math>
The electron's electric dipole moment (EDM) must be collinear with the direction of the electron's magnetic moment (spin).<ref>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): 193003. arXiv:1208.4420. Bibcode:2012PhRvL.109s3003E. doi:10.1103/PhysRevLett.109.193003. PMID 23215379. S2CID 35411253.</ref> 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,<ref name=Pospelov>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.</ref> 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.<ref>Khriplovich, I.B.; Lamoreaux, S.K. (1997). CP violation without strangeness: Electric dipole moments of particles, atoms, and molecules. Springer-Verlag.</ref><ref>P. R. Bunker and P. Jensen (2005), Fundamentals of Molecular Symmetry (CRC Press) ISBN 0-7503-0941-5[1] Chapter 15</ref>
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.<ref name=Pospelov/><ref group="lower-alpha"></ref>
If neutrinos are Majorana particles, a larger EDM (around 10−33 e⋅cm) is possible in the Standard Model.<ref name=Pospelov/>
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] Some supersymmetric models predict that |de| > 10−26 e⋅cm<ref>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. S2CID 17341766.</ref> 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,<ref name="web2.ph.utexas.edu">"Ultracold Atomic Physics Group". Physics. U. Texas. Retrieved 13 November 2015.</ref> 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
- <math>
\mathbf d_{\rm e} = \int ({\mathbf r} - {\mathbf r}_0) \rho({\mathbf r}) d^3 {\mathbf r} </math> depends on the point <math>{\mathbf r}_0 </math> about which the moment of the charge distribution <math> \rho({\mathbf r}) </math> is taken. If we were to choose <math>{\mathbf r}_0 </math> to be the center of charge, then <math>\mathbf d_{\rm e}</math> would be identically zero. A more interesting choice would be to take <math>{\mathbf r}_0 </math> 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 <math>F_i(q^2)</math> appearing in the matrix element<ref name=Nowakowski>Nowakowski, M.; Paschos, E.A.; Rodriguez, J.M. (2005). "All electromagnetic form factors". European Journal of Physics. 26 (4): 545–560. arXiv:physics/0402058. Bibcode:2005EJPh...26..545N. doi:10.1088/0143-0807/26/4/001. S2CID 119097762.</ref>
- <math>
\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)
</math> of the electromagnetic current operator between two on-shell states with Lorentz invariant phase space normalization in which
- <math> \langle p_f \vert p_i \rangle= 2E (2\pi)^3 \delta^3({\bf p}_f-{\bf p_i}).</math>
Here <math>u(p_i)</math> and <math>\bar u(p_f)</math> are 4-spinor solutions of the Dirac equation normalized so that <math>\bar u u=2m_e </math>, and <math>q^\mu=p^\mu_f-p^\mu_i</math> is the momentum transfer from the current to the electron. The <math>q^2=0</math> form factor <math>F_1(0) = Q</math> is the electron's charge, <math>\mu = \tfrac{F_1(0)\ +\ F_2(0)}{2m_{\rm e}}</math> is its static magnetic dipole moment, and <math>\tfrac{-F_3(0)}{2m_{\rm e}}</math> provides the formal definition of the electron's electric dipole moment. The remaining form factor <math>F_4(q^2)</math> 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 <math>U = \mathbf d_{\rm e} \cdot \mathbf E</math> as a slight shift of spectral lines. The sensitivity to <math>\mathbf d_{\rm e}</math> scales approximately with the nuclear charge cubed.<ref>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].</ref> 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 |de| < 0.11×10−28 e⋅cm.<ref>"Electron listing" (PDF). Particle Data Group. Lawrence Berkeley Laboratory. 2020.</ref> Here is a list of some electron EDM experiments after 2000 with published results:
| 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<ref>Regan, B.C.; Commins, Eugene D.; Schmidt, Christian J.; DeMille, David (1 February 2002). "New Limit on the Electron Electric Dipole Moment". Physical Review Letters. 88 (7): 071805. Bibcode:2002PhRvL..88g1805R. doi:10.1103/PhysRevLett.88.071805. PMID 11863886. S2CID 32396668.</ref> |
| 2011 | Imperial College London | Edward Hinds, Ben Sauer | Molecular beam | YbF | 1.1×10−27 e⋅cm<ref>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.</ref> |
| 2014 | Harvard-Yale (ACME I experiment) |
David DeMille, John Doyle, Gerald Gabrielse | Molecular beam | ThO | 8.7×10−29 e⋅cm<ref>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. S2CID 564518. Archived from the original (PDF) on 2015-04-27. Retrieved 2014-06-24.</ref> |
| 2017 | JILA | Eric Cornell, Jun Ye | Ion trap | HfF+ | 1.3×10−28 e⋅cm<ref>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.</ref> |
| 2018 | Harvard-Yale (ACME II experiment) |
David DeMille, John Doyle, Gerald Gabrielse | Molecular beam | ThO | 1.1×10−29 e⋅cm<ref>The ACME Collaboration (October 2018). "Improved Limit on the Electric Dipole Moment of the Electron" (PDF). Nature. 562 (7727): 355–360. Bibcode:2018Natur.562..355A. doi:10.1038/s41586-018-0599-8. PMID 30333583. S2CID 52985540.</ref> |
| 2022 | JILA | Eric Cornell, Jun Ye | Ion trap | HfF+ | 4.1×10−30 e⋅cm<ref>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. (2023). "An improved bound on the electron's electric dipole moment". Science. 381 (6653): 46–50. arXiv:2212.11841. doi:10.1126/science.adg4084.</ref> <ref>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. (2023-07-06), "A new bound on the electron's electric dipole moment", Science, 381 (6653): 46–50, arXiv:2212.11841, doi:10.1126/science.adg4084</ref> |
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.<ref>"ACME Electron EDM".</ref><ref>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): 022808. arXiv:2204.05904. doi:10.1103/PhysRevA.106.022808.</ref>
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<ref>Aggarwal, Parul; Bethlem, Hendrick L.; Borschevsky, Anastasia; Denis, Malika; Esajas, Kevin; Haase, Pi A.B.; Hao, Yongliang; Hoekstra, Steven; Jungmann, Klaus; Meijknecht, Thomas B.; Mooij, Maarten C.; Timmermans, Rob G.E.; Ubachs, Wim; Willmann, Lorenz; Zapara, Artem (2018). "Measuring the electric dipole moment of the electron in BaF". The European Physical Journal D. 72 (11). arXiv:1804.10012. doi:10.1140/epjd/e2018-90192-9. S2CID 96439955.</ref>
- John Doyle (Harvard University), Nicholas Hutzler (California Institute of Technology), and Timothy Steimle (Arizona State University): YbOH molecular trap<ref>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.</ref>
- EDMcubed collaboration, Amar Vutha (University of Toronto), Eric Hessels (York University): oriented polar molecules in an inert gas matrix<ref>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. S2CID 3349485.</ref><ref>"EDMcubed". www.yorku.ca. Retrieved 2023-10-31.</ref>
- David Weiss (Pennsylvania State University): Cs and Rb atoms trapped inside an optical lattice<ref>"Search for the Electron EDM Using Cs and Rb in Optical Lattice Traps". Penn State. Retrieved 2022-09-09.</ref>
- TRIUMF: Fountain of laser cooled Fr<ref>"Report Summary | TRIUMF : Canada's National Laboratory for Particle and Nuclear Physics". mis.triumf.ca. Retrieved 2022-09-09.</ref>
- EDMMA collaboration: Cs in an inert gas matrix<ref>"Moment dipolaire électrique des électrons à l'aide de Cs en matrice cryogénique - LAC". www.lac.universite-paris-saclay.fr. Retrieved 2022-09-09.</ref>
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
