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:

${\displaystyle U=\mathbf {d} _{\rm {e}}\cdot \mathbf {E} .}$

The electron's

Formal definition

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^[7]

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

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

${\displaystyle \langle p_{f}\vert p_{i}\rangle =2E(2\pi )^{3}\delta ^{3}({\bf {p}}_{f}-{\bf {p_{i}}}).}$

Here ${\displaystyle u(p_{i})}$ and ${\displaystyle {\bar {u}}(p_{f})}$ are 4-spinor solutions of the Dirac equation normalized so that ${\displaystyle {\bar {u}}u=2m_{e}}$, 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, ${\displaystyle \mu ={\tfrac {F_{1}(0)\ +\ F_{2}(0)}{2m_{\rm {e}}}}}$ is its static magnetic dipole moment, and ${\displaystyle {\tfrac {-F_{3}(0)}{2m_{\rm {e}}}}}$ provides the formal definition of the electron's electric dipole moment. The remaining form factor ${\displaystyle F_{4}(q^{2})}$ would, if nonzero, be the

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 ${\displaystyle U=\mathbf {d} _{\rm {e}}\cdot \mathbf {E} }$ as a slight shift of spectral lines. The sensitivity to ${\displaystyle \mathbf {d} _{\rm {e}}}$ scales approximately with the

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