Decay of electron? Have we detected any decays of electrons to an electron neutrino and $W$-boson in Fermilab or in CERN? Are neutrinos the only possible stable leptons inside an electroweak field?
 A: 
Have we detected any decays of electrons to an electron neutrino and W- boson in Fermilab or in CERN?Are neutrinos the only possible stable leptons inside an electroweak field?

This cannot happen because energy and charge  conservation are absolute laws. Particles can decay to other particles when the sum of the masses of the decay products is smaller than the mass of the original particle, (and also various quantum number consrvations laws are obeyed).  See the table

The electron is a charged lepton, leptons can decay to other leptons, but not the electron since the other two charged leptons are heavier.

A: A very interesting question. If we ignore charge conservation law, then electron could decay into neutrino and photon :
$$ e^− \to ν_e + \gamma $$
Current estimates gives that life time of electron $\gt 10^{26} \,\text{years}$. Feynman diagram of such electron decay :

It is calculated that energy needed to break electron into neutrino + photon is on the order $\approx 10^{22} \,\text{GeV} $. So it's by $1'000$ times greater than Plank energy ! 
As far as I know CERN achieved biggest record of it's collision energies $\approx 10^6 \text{GeV}$ at 25 Nov 2015. 
No need to say that this is only very very tiny amount of energy required for actually breaking the electron. Probably for this event to occur we need a particle accelerator with diameter comparable to our Sun planetary system size or even greater. Which would cost some gazillions of USA yearly budget. So as for now technologically we will not be able to test if charge conservation law holds or not for electron. 
