There is a lot of experimental research activity into whether the electron has an electric dipole moment. The electron, however, has a net charge, and so its dipole moment $$ {\bf \mu}= \int ({\bf r}- {\bf r}_0)\rho({\bf r}) \,d^3r $$ depends on the chosen origin ${\bf r}_0$. Indeed, if one takes moments about the center of charge, then - by definition - the electric dipole moment is zero.
Now I know that what is really meant by the experimentalists is that their "electric dipole moment" corresponds to adding to the Dirac Lagrangian a term propertional to $$ \frac 12 \bar \psi \sigma_{\mu \nu}\psi\, ^*F^{\mu\nu}, $$ where $^*F^{\mu\nu} = \frac 12 \epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$ is the dual Maxwell field. So I have two questions:
a) What point ${\bf r}_0$ does this correspond to? I'd guess that it it is something like the center of energy of the electron's wavepacket measured in its rest frame. Is there a way to see this?
b) If my guess in (a) is correct what would happen if the electron were massless? There is then no rest frame, and the center of energy is frame-dependent. I imagine therefore that the electric dipole moment would have to be zero. Is this correct? Certainly $\bar \psi \sigma_{\mu \nu}\psi$ is zero for a purely left or right helicity particle obeying a Weyl equation as $\gamma_0 [\gamma_\mu,\gamma_\nu]$ is off-diagonal in the helicity basis
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