The purpose of mitigating all potential effects that can detrimentally influence the sensitivity of present and future gravitational wave detectors has triggered specific research and development worldwide. One of the many issues to be solved is the noise induced by the electrostatic charge forming on test mass mirrors. At LIGO, a mitigation method has been proposed, studied, and successfully applied. This method requires long mirror's exposures to a relatively high pressure of ${\mathrm{N}}_{2}$ ions flux. It is difficult, if not impossible, to apply this method, the way it is now, when mirrors are at cryogenic temperatures, since a significantly thick condensed gas layer will develop on the mirror surface severely affecting its performance. Hence, we suggest a new method to neutralize test masses electrostatic charge that could be performed in ultra high vacuum (UHV) and could easily be applied to cryogenic mirrors. We suggest the use of selected energy electrons (between 10 to 100 eV) which, at very low doses, can impinge on the surface mirror. The energy of the incident beam can be tuned to neutralize positive and negative charges on the mirror's dielectric surface or part of it. Here, we experimentally prove that this is successful in the case of Si and ${\mathrm{SiO}}_{2}$, two prototypical materials for mirror surfaces. The method is briefly presented in its basic principles, and a number of further studies are identified in view of developing the appropriate enabling technology.