View source for Electron electric dipole moment

{{short description|Quantity relating an electron's potential energy to electric field strength}}

The '''electron electric dipole moment''' {{math|''d''<sub>e</sub>}} 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 [[Spin magnetic moment|magnetic moment]] (spin).<ref>{{Cite journal |doi=10.1103/PhysRevLett.109.193003 |pmid=23215379 |title=Limit on the electron electric dipole moment using paramagnetic ferroelectric Eu0.5Ba0.5TiO3 |journal=Physical Review Letters |volume=109 |issue=19 |page=193003 |year=2012 |last1=Eckel |first1=S. |last2=Sushkov |first2=A.O. |last3=Lamoreaux |first3=S.K. |arxiv=1208.4420|bibcode=2012PhRvL.109s3003E |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 {{nowrap|10<sup>−38</sup> ''e''⋅cm}},<ref name=Pospelov>{{Cite journal |last1=Pospelov |first1=M. |last2=Ritz |first2=A. |doi=10.1016/j.aop.2005.04.002 |title=Electric dipole moments as probes of new physics |journal=Annals of Physics |volume=318 |issue=1 |pages=119–169 |year=2005 |arxiv=hep-ph/0504231 |bibcode=2005AnPhy.318..119P|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 (physics)|parity invariance]] and [[T-symmetry|time reversal invariance]].<ref>{{cite book |last1=Khriplovich |first1=I.B. |last2=Lamoreaux |first2=S.K. |year=1997 |title=CP violation without strangeness: Electric dipole moments of particles, atoms, and molecules |publisher=[[Springer-Verlag]]}}</ref><ref>P. R. Bunker and P. Jensen (2005), ''Fundamentals of Molecular Symmetry'' (CRC Press) {{ISBN|0-7503-0941-5}}[https://www.routledge.com/Fundamentals-of-Molecular-Symmetry/Bunker-Jensen/p/book/9780750309417] Chapter 15</ref>

==Implications for Standard Model and extensions==
In the Standard Model, the electron EDM arises from the [[CP violation|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 particle|virtual]] quarks are created, interact with the electron, and then are annihilated.<ref name=Pospelov/>{{efn|More precisely, a non-zero EDM does not arise until the level of four-loop [[Feynman diagram]]s and higher.<ref name=Pospelov/>}}

If neutrinos are [[Majorana particle]]s, a larger EDM (around {{val|e=-33|u=''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 model]]s predict {{abs|'''d'''<sub>e</sub>}} that ranges from 10<sup>−27</sup> to 10<sup>−29</sup>&nbsp;''e''⋅cm.{{Citation needed|date=October 2008}} Some [[supersymmetric]] models predict that {{nowrap|{{abs|'''d'''<sub>e</sub>}} > 10<sup>−26</sup>&nbsp;''e''⋅cm}}<ref>{{cite journal |last1=Arnowitt |first1=R. |last2=Dutta |first2=B. |last3=Santoso |first3=Y. |year=2001 |title=Supersymmetric phases, the electron electric dipole moment and the muon magnetic moment |journal=[[Physical Review D]] |volume=64 |issue=11 |page=113010 |doi=10.1103/PhysRevD.64.113010 |arxiv=hep-ph/0106089 |bibcode=2001PhRvD..64k3010A|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">{{Cite web |title=Ultracold Atomic Physics Group |url=https://web2.ph.utexas.edu/~coldatom/EDM.html |department=Physics |website=U. Texas |access-date=2015-11-13 |df=dmy-all}}</ref> would place further limits on which theory takes precedence.

[[File:Electron EDM limits.png|thumb|Historical record of electron electric dipole moment measurements in leptonic systems.]]

== 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 factor (quantum field theory)|form factors]] <math>F_i(q^2)</math> appearing in the matrix element<ref name=Nowakowski>{{Cite journal |last1=Nowakowski |first1=M. |last2=Paschos |first2=E.A. |last3=Rodriguez |first3=J.M. |doi=10.1088/0143-0807/26/4/001 |title=All electromagnetic form factors |journal=European Journal of Physics |volume=26 |issue=4 |pages=545–560 |year=2005 |arxiv=physics/0402058|bibcode=2005EJPh...26..545N |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 [[Electron magnetic moment|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 electron]]s 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 line]]s. The sensitivity to <math>\mathbf d_{\rm e}</math> scales approximately with the [[nuclear charge]] cubed.<ref>{{cite arXiv |last1=Alarcon |first1=Ricardo |last2=Alexander |first2=Jim |last3=Anastassopoulos |first3=Vassilis |last4=Aoki |first4=Takatoshi |last5=Baartman |first5=Rick |last6=Baeßler |first6=Stefan |last7=Bartoszek |first7=Larry |last8=Beck |first8=Douglas H. |last9=Bedeschi |first9=Franco |last10=Berger |first10=Robert |last11=Berz |first11=Martin |last12=Bethlem |first12=Hendrick L. |last13=Bhattacharya |first13=Tanmoy |last14=Blaskiewicz |first14=Michael |last15=Blum |first15=Thomas |date=2022-04-04 |title=Electric dipole moments and the search for new physics |class=hep-ph |eprint=2203.08103 }}</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 {{nowrap|{{abs|'''d'''<sub>e</sub>}} < {{val|0.11|e=-28|u=''e''⋅cm}}}}.<ref>{{cite web |url=https://pdg.lbl.gov/2020/tables/rpp2020-sum-leptons.pdf |title=Electron listing |department=Particle Data Group |website=Lawrence Berkeley Laboratory |year=2020}}</ref> Here is a list of some electron EDM experiments after 2000 with published results:
{| class="wikitable"
|+List of Electron EDM Experiments
!Year
!Location
!Principal Investigators
!Method
!Species
!Experimental upper limit on {{abs|'''d'''<sub>e</sub>}}
|-
| 2002
| [[University of California, Berkeley]]
| [[Eugene D. Commins|Eugene Commins]], [[David DeMille]]
| [[Atomic beam]]
| [[Thallium|Tl]]
| {{val|1.6|u=''e''⋅cm|e=-27}}<ref>{{Cite journal |last1=Regan |first1=B.C. |last2=Commins |first2=Eugene D. |last3=Schmidt |first3=Christian J.|last4=DeMille |first4=David |date=2002-02-01 |title=New Limit on the Electron Electric Dipole Moment |journal=Physical Review Letters |volume=88 |issue=7 |page=071805 |doi=10.1103/PhysRevLett.88.071805 |pmid=11863886 |bibcode=2002PhRvL..88g1805R |s2cid=32396668 |df=dmy-all|url=https://digital.library.unt.edu/ark:/67531/metadc741308/ }}</ref>
|-
| 2011
| [[Imperial College London]]
| [[Edward Hinds]], Ben Sauer
| [[Molecular beam]]
| [[Ytterbium|Yb]][[Monofluoride|F]]
| {{val|1.1|e=-27|u=''e''⋅cm}}<ref>{{cite journal |last1=Hudson |first1=J.J. |last2=Kara |first2=D.M. |last3=Smallman |first3=I.J. |last4=Sauer |first4=B.E. |last5=Tarbutt |first5=M.R. |last6=Hinds |first6=E.A. |year=2011 |title=Improved measurement of the shape of the electron |journal=[[Nature (journal)|Nature]] |volume=473 |issue=7348 |pages=493–496 |bibcode=2011Natur.473..493H |doi=10.1038/nature10104 |pmid=21614077 |hdl=10044/1/19405|s2cid=205224996 |url=http://spiral.imperial.ac.uk/bitstream/10044/1/19405/2/Nature_473_7348_2011.pdf |hdl-access=free }}</ref>
|-
| 2014
| [[Harvard University|Harvard]]-[[Yale University|Yale]]<br/>(ACME I experiment)
| [[David DeMille]], [[John Doyle (academic)|John Doyle]], [[Gerald Gabrielse]]
| [[Molecular beam]]
| [[Thorium|Th]][[Monoxide|O]]
| {{val|8.7|e=-29|u=''e''⋅cm}}<ref>{{cite journal |author=The ACME Collaboration |date=January 2014 |title=Order of Magnitude Smaller Limit on the Electric Dipole Moment of the Electron |url=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 |journal=[[Science (journal)|Science]] |volume=343 |issue=6168 |pages=269–272 |arxiv=1310.7534 |bibcode=2014Sci...343..269B |doi=10.1126/science.1248213 |pmid=24356114 |s2cid=564518 |access-date=2014-06-24 |archive-url=https://web.archive.org/web/20150427085647/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 |archive-date=2015-04-27 |url-status=dead }}</ref>
|-
| 2017
| [[JILA]]
| [[Eric Allin Cornell|Eric Cornell]], [[Jun Ye]]
| [[Ion trap]]
| [[Hafnium|Hf]][[Fluoride|F]]+
| {{val|1.3|e=-28|u=''e''⋅cm}}<ref>{{Cite journal |last1=Cairncross |first1=William B. |last2=Gresh |first2=Daniel N. |last3=Grau |first3=Matt |last4=Cossel |first4=Kevin C. |last5=Roussy |first5=Tanya S. |last6=Ni |first6=Yiqi |last7=Zhou |first7=Yan |last8=Ye |first8=Jun |last9=Cornell |first9=Eric A. |date=2017-10-09 |title=Precision Measurement of the Electron's Electric Dipole Moment Using Trapped Molecular Ions |journal=Physical Review Letters |volume=119 |issue=15 |pages=153001 |doi=10.1103/PhysRevLett.119.153001 |pmid=29077451 |arxiv=1704.07928 |bibcode=2017PhRvL.119o3001C |s2cid=44043558 }}</ref>
|-
| 2018
| [[Harvard University|Harvard]]-[[Yale University|Yale]]<br/>(ACME II experiment)
| [[David DeMille]], [[John Doyle (academic)|John Doyle]], [[Gerald Gabrielse]]
| [[Molecular beam]]
| [[Thorium|Th]][[Monoxide|O]]
| {{val|1.1|e=-29|u=''e''⋅cm}}<ref>{{cite journal |author=The ACME Collaboration |date=October 2018 |title=Improved Limit on the Electric Dipole Moment of the Electron |journal=[[Nature (journal)|Nature]] |volume=562 |issue=7727 |pages=355–360 |doi=10.1038/s41586-018-0599-8|pmid=30333583 |bibcode=2018Natur.562..355A |s2cid=52985540 |url=https://authors.library.caltech.edu/89472/3/41586_2018_599_MOESM1_ESM.pdf }}</ref>
|-
| 2022
| [[JILA]]
| [[Eric Allin Cornell|Eric Cornell]], [[Jun Ye]]
| [[Ion trap]]
| [[Hafnium|Hf]][[Fluoride|F]]+
| {{val|4.1|e=-30|u=''e''⋅cm}}<ref>{{Cite journal |last1=Roussy |first1=Tanya S. |last2=Caldwell |first2=Luke |last3=Wright |first3=Trevor |last4=Cairncross |first4=William B. |last5=Shagam |first5=Yuval |last6=Ng |first6=Kia Boon |last7=Schlossberger |first7=Noah |last8=Park |first8=Sun Yool |last9=Wang |first9=Anzhou |last10=Ye |first10=Jun |last11=Cornell|first11=Eric A. |date=2023 |title=An improved bound on the electron's electric dipole moment |journal=Science |volume=381 |issue=6653 |pages=46–50 |doi=10.1126/science.adg4084 |arxiv=2212.11841}}</ref> <ref>{{Citation  |last1=Roussy |first1=Tanya S. |last2=Caldwell |first2=Luke |last3=Wright |first3=Trevor |last4=Cairncross |first4=William B. |last5=Shagam |first5=Yuval |last6=Ng |first6=Kia Boon |last7=Schlossberger |first7=Noah |last8=Park |first8=Sun Yool |last9=Wang |first9=Anzhou |last10=Ye |first10=Jun |last11=Cornell|first11=Eric A. |date=2023-07-06 |title=A new bound on the electron's electric dipole moment |journal=Science  |volume=381 |issue=6653 |pages=46–50 |doi=10.1126/science.adg4084|url=https://www.science.org/doi/10.1126/science.adg4084|arxiv=2212.11841 }}</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>{{Cite web|url=http://doylegroup.harvard.edu/edm/index.html|title = ACME Electron EDM}}</ref><ref>{{Cite journal |last1=Ang |first1=D. G. |last2=Meisenhelder |first2=C. |last3=Panda |first3=C. D. |last4=Wu |first4=X. |last5=DeMille |first5=D. |last6=Doyle |first6=J. M. |last7=Gabrielse |first7=G. |date=2022-08-15 |title=Measurement of the $H^3\Delta_1$ radiative lifetime in ThO |url=https://link.aps.org/doi/10.1103/PhysRevA.106.022808 |journal=Physical Review A |volume=106 |issue=2 |pages=022808 |doi=10.1103/PhysRevA.106.022808|arxiv=2204.05904 }}</ref>

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

* [[University of Groningen]]: [[Barium fluoride|BaF]] [[molecular beam]]<ref>{{Cite journal |arxiv=1804.10012 |doi=10.1140/epjd/e2018-90192-9 |title=Measuring the electric dipole moment of the electron in BaF |journal=The European Physical Journal D |volume=72 |issue=11 |year=2018 |last1=Aggarwal |first1=Parul |last2=Bethlem |first2=Hendrick L. |last3=Borschevsky |first3=Anastasia |last4=Denis |first4=Malika|last5=Esajas |first5=Kevin |last6=Haase |first6=Pi A.B. |last7=Hao |first7=Yongliang |last8=Hoekstra |first8=Steven |last9=Jungmann |first9 = Klaus |last10=Meijknecht |first10=Thomas B. |last11=Mooij |first11=Maarten C. |last12=Timmermans |first12=Rob G.E. |last13=Ubachs |first13=Wim |last14=Willmann |first14=Lorenz |last15=Zapara |first15=Artem|s2cid=96439955 }}</ref>
* [[John Doyle (physicist)|John Doyle]] ([[Harvard University]]), [[Nicholas Hutzler]] ([[California Institute of Technology]]), and [[Timothy Steimle]] ([[Arizona State University]]): YbOH [[Magnetic trap (atoms)|molecular trap]]<ref>{{Cite journal |last1=Kozyryev |first1=Ivan |last2=Hutzler |first2=Nicholas R. |date=2017-09-28 |title=Precision Measurement of Time-Reversal Symmetry Violation with Laser-Cooled Polyatomic Molecules |journal=Physical Review Letters |volume=119 |issue=13 |pages=133002 |doi=10.1103/PhysRevLett.119.133002 |pmid=29341669 |arxiv=1705.11020 |bibcode=2017PhRvL.119m3002K |s2cid=33254969 }}</ref>
* [[EDMcubed collaboration]], [[Amar Vutha]] ([[University of Toronto]]), [[Eric Hessels]] ([[York University]]): oriented polar molecules in an inert gas matrix<ref>{{Cite journal |last1=Vutha |first1=A.C. |last2=Horbatsch |first2=M. |last3=Hessels |first3=E.A. |date=2018-01-05 |title=Oriented polar molecules in a solid inert-gas matrix: A proposed method for measuring the electric dipole moment of the electron |journal=Atoms |language=en |volume=6 |issue=1 |pages=3 |arxiv=1710.08785 |bibcode=2018Atoms...6....3V |doi=10.3390/atoms6010003 |s2cid=3349485 |doi-access=free }}</ref><ref>{{Cite web |title=EDMcubed |url=https://www.yorku.ca/edmcubed/ |access-date=2023-10-31 |website=www.yorku.ca}}</ref>
* [[David Weiss (physicist)|David Weiss]] ([[Pennsylvania State University]]): [[Caesium|Cs]] and [[Rubidium|Rb]] atoms trapped inside an [[optical lattice]]<ref>{{Cite web |title=Search for the Electron EDM Using Cs and Rb in Optical Lattice Traps |url=https://pennstate.pure.elsevier.com/en/projects/search-for-the-electron-edm-using-cs-and-rb-in-optical-lattice-tr |access-date=2022-09-09 |website=Penn State |language=en}}</ref>
* [[TRIUMF]]: Fountain of [[Laser cooling|laser cooled]] [[Francium|Fr]]<ref>{{Cite web |title=Report Summary {{!}} TRIUMF : Canada's National Laboratory for Particle and Nuclear Physics |url=https://mis.triumf.ca/science/experiment/view/S1324LOI |access-date=2022-09-09 |website=mis.triumf.ca}}</ref>
* [[EDMMA collaboration]]: Cs in an inert gas matrix<ref>{{Cite web |title=Moment dipolaire électrique des électrons à l'aide de Cs en matrice cryogénique - LAC |url=http://www.lac.universite-paris-saclay.fr/?emploi=moment-dipolaire-electrique-des-electrons-a-laide-de-cs-en-matrice-cryogenique |access-date=2022-09-09 |website=www.lac.universite-paris-saclay.fr}}</ref>

==See also==
* [[Neutron electric dipole moment]]
* [[Electron magnetic moment]]
* [[Anomalous electric dipole moment]]
* [[Anomalous magnetic dipole moment]]
* [[Electric dipole spin resonance]]
*{{slink|Parity (physics)|Parity violation}}
* [[CP violation]]
* [[Charge conjugation]] 
* [[T-symmetry]]

== Footnotes ==
{{notelist|1}}

==References==
{{reflist|25em}}

{{DEFAULTSORT:Electron Electric Dipole Moment}}
[[Category:Electric dipole moment]]
[[Category:Electromagnetism]]
[[Category:Particle physics]]