Here is a press report http://phys.org/news/2012-10-physicists-special-relativity.html#jCp on paper discussing new way to treat special relativity permitting faster than light particles (tachyons). The momentum of an infinite-velocity particle is in this picture a finite and nonzero consnt depending only on the particle-type. Apparently then you can get conserved real momentum and energy and these kinds of particle could only travel faster than light, never below. I have not yet read the paper itself (behind paywall). It seems to me, though, that such tachyons could interact with exactly-lightspeed=massless particles like photons & gravitons. (Also, e.g, W-bosons have zero mass until they gain effective mass from the Higgs mechanism.) Those in turn could interact with normal particles. Thus effectively tachyons & normals could interact. That would permit transmission of information from normal matter into the past of normal matter, creating paradoxes. If such paradoxes are considered forbidden, we'd presumably still need to conclude that tachyons cannot exist.