The measurement of the dipole moment relies on a simple idea: electric dipole moment gives rise to additional precession if one has parallel magnetic and electric fields. The additional electric field is now that associated with the molecule containing electrons plus strong molecular electric field in the direction of spin quantization axes. One puts the molecules containing the electrons into magnetic field and measure the precession of spins by detecting the photons produced in the process. The deviation of the precession frequency from its value in magnetic field only should allow to deduce the upper bound for the dipole moment.
Semiclassically the non-vanishing dipole moment means asymmetric charge distribution with respect to the spin quantization axis. The electric dipole coupling term for Dirac spinors comes to effective action from radiative corrections and has the same form as magnetic dipole coupling involving sigma matrices except that one has an additional γ5 matrix bringing in CP breaking. The standard model prediction is of order de≈ 10-40 e× me,: this is by a factor 10-5 smaller than Planck length!
The new upper bound is de ≈ .87 × 10-32 e×me and still much larger than standard model prediction. Standard SUSY predicts typically non-vanishing dipole moment for electron. The estimate for the electron dipole moment coming from SUSYs and is by dimensional considerations of form de= c ℏ e× me/16π2M2, where c is of order unity and M is the mass scale for the new physics. The Feynman diagram in question involves the decay of electron to virtual neutrino and virtual chargino and the coupling of the latter to photon before absoption.
This upper bound provides a strong restriction on "garden variety" SUSY models (involving no fine tuning to make dipole moment smaller) and the scale at which SUSY could show itself becomes of order 10 TeV at least so that hopes for detecting SUSY at LHC should be rather meager. One can of course do fine tuning. "Naturality" idea does not favor fine tunings but is not in fashion nowadays: the existing theoretical models do not simply allow such luxury. The huge differences between elementary particle mass scales and quite "too long" proton lifetime represent basic example about "non-naturality" in the GUT framework. For an outsider like me this strongly suggests that although Higgs exist, Higgs mechanism provides only a parametrization of particle masses - maybe the only possible theoretical description in quantum field theory framework treating particles as point like - and must be eventually replaced with a genuine theory. For instance, Lubos does not see this fine tuning is not seen as reason for worrying too much. Personally I however feel worried since my old-fashioned view is that theoretical physicists must be able to make predictions rather than only run away the nasty data by repeated updating of the models so that they become more and more complicated.