In 2012, a large amount of attention was given to the OPERA experiment's apparent sighting of faster than light neutrinos. This turned out to be erroneous due to a faulty cable, and similar experiments confirmed the same results. However, while this was occurring, a distinct point was made: some attempts to determine the mass of the electron neutrino(one of the three known neutrino types) found that the square of the mass was apparently negative, which would be consistent with an imaginary mass and thus electron neutrinos would be tachyons. While little attention was paid to at the time, a new paper by Robert Ehrlich looks again at this approach. Ehrlich points out that six different experimental results seem to yield an imaginary mass for the electron neutrino, and what is more, all the results are in close agreement, with an apparent square of the mass being close to -0.11 electron-volts squared.
There are at least two major difficulties with Ehrlich's suggestion, both of which were also issues for OPERA aside from any philosophical or metaconcerns like desire to preserve causality. First, it is difficult to reconcile with Ehrlich's suggestion is one of the same data points that apparently tripped up OPERA, that is the neutrinos from SN 1987A neutrinos. In the SN 1987A supernova (the first observed in 1987 hence the name), the supernova was close enough that we were actually able to detect the neutrinos from it. The neutrinos arrived about three hours before the light from the supernova. But that's not evidence for faster than light neutrinos, since one actually expects this to happen. In the standard way of viewing things, the neutrinos move very very close to the speed of light, but during a core-collapse supernova like SN 1987A, the neutrinos are produced in the core at the beginning of the process. They then flee the star without interacting with the matter, whereas the light produced in the core is slowed down by all the matter in the way, so the neutrinos get a few hours head start.
The problem for FTL neutrinos is that if the neutrions were even a tiny bit faster than the speed of light they should have arrived much much earlier. This is strong evidence against FTL neutrinos. In the paper in question, Ehrlich mentions SN 1987A in the context of testing his hypothesis in an alternate way using a supernova and the exact distribution of the neutrinos from one but doesn't discuss anywhere I can see the more basic issue of the neutrinos arriving at close to the same time as the light. It is conceivable that electron neutrinos are the only neutrinos which are tachyons, and if this is the case, then it seems like neutrino oscillation (the tendency for neutrinos to change types spontaneously) could account for part of what is going on here, but having only some types of neutrinos be tachyons would possibly lead to other problems.
Second, there's reason to believe that tachyons if they existed would emit Cherenkov-like radiation. Andrew Cohen and Sheldon Glashow wrote a paper showing that this would be a major issue in the context of OPERA. Ehrlich seems to claim in the new paper that this shouldn't be an issue in the context he is working in, but does not provide any reasoning. Hopefully someone who is more of an expert can comment on what is going on there.
This seems like potentially stronger evidence for tachyonic neutrinos than the OPERA experiment since this is the same result from a variety of different experiments all giving very close to the same results.