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:

[math]\displaystyle{ 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).^[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][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 |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.

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]\displaystyle{ \mathbf d_{\rm e} = \int ({\mathbf r} - {\mathbf r}_0) \rho({\mathbf r}) d^3 {\mathbf r} }[/math]

depends on the point [math]\displaystyle{ {\mathbf r}_0 }[/math] about which the moment of the charge distribution [math]\displaystyle{ \rho({\mathbf r}) }[/math] is taken. If we were to choose [math]\displaystyle{ {\mathbf r}_0 }[/math] to be the center of charge, then [math]\displaystyle{ \mathbf d_{\rm e} }[/math] would be identically zero. A more interesting choice would be to take [math]\displaystyle{ {\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]\displaystyle{ F_i(q^2) }[/math] appearing in the matrix element^[7]

[math]\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) }[/math]

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

[math]\displaystyle{ \langle p_f \vert p_i \rangle= 2E (2\pi)^3 \delta^3({\bf p}_f-{\bf p_i}). }[/math]

Here [math]\displaystyle{ u(p_i) }[/math] and [math]\displaystyle{ \bar u(p_f) }[/math] are 4-spinor solutions of the Dirac equation normalized so that [math]\displaystyle{ \bar u u=2m_e }[/math], and [math]\displaystyle{ q^\mu=p^\mu_f-p^\mu_i }[/math] is the momentum transfer from the current to the electron. The [math]\displaystyle{ q^2=0 }[/math] form factor [math]\displaystyle{ F_1(0) = Q }[/math] is the electron's charge, [math]\displaystyle{ \mu = \tfrac{F_1(0)\ +\ F_2(0)}{2m_{\rm e}} }[/math] is its static magnetic dipole moment, and [math]\displaystyle{ \tfrac{-F_3(0)}{2m_{\rm e}} }[/math] provides the formal definition of the electron's electric dipole moment. The remaining form factor [math]\displaystyle{ 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]\displaystyle{ U = \mathbf d_{\rm e} \cdot \mathbf E }[/math] as a slight shift of spectral lines. The sensitivity to [math]\displaystyle{ \mathbf d_{\rm e} }[/math] 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:

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][18]

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^[19]
• John Doyle (Harvard University), Nicholas Hutzler (California Institute of Technology), and Timothy Steimle (Arizona State University): YbOH molecular trap^[20]
• EDMcubed collaboration, Amar Vutha (University of Toronto), Eric Hessels (York University): oriented polar molecules in an inert gas matrix^[21][22]
• David Weiss (Pennsylvania State University): Cs and Rb atoms trapped inside an optical lattice^[23]
• TRIUMF: Fountain of laser cooled Fr^[24]
• EDMMA collaboration: Cs in an inert gas matrix^[25]

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

References

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