There is a certain amount of confusion on the relationship between Einstein's Theory of Relativity and the recent experimental results that seem to point towards neutrinos that are faster than light by an amount of about 7 km/s. So let me try to clarify things by answering to the following question:

If neutrinos travel faster than light by 7 km/s, do we need to modify Relativity?

The answer is a clear-cut "Yes"; let me explain why.

Lorentz invariance, which is embodied in the theory of Relativity, has the unescapable consequence that there exists a precise relationship between a free particle's energy E, its momentum p and its mass m:

c being the speed of light. From this equation one immediately obtains the velocity v as:

Then, from m2 >0 follows v<c. This holds for all known particles, with the possible exception of neutrinos for which measurements of the involved physical quantities are much harder than, say, for electrons.

Is it possible to admit v>c in a Lorentz-invariant framework?

In principle yes, by assuming a negative squared mass (m2<0) and hence an imaginary mass. In this case the particle is called a "tachyon" and its velocity is greater than that of light. Tachyons give me a headache, since they lead to mathematically inconsistent theories and to problems with causality. But even forgetting these (major) problems, from a phenomenological point of view OPERA neutrinos cannot be tachyons. In order to understand this, we need a quantitative argument.

If we suppose neutrinos are tachyons, then in order to comply with the value published  by OPERA:

which corresponds to the above mentioned 7 km/s, neutrinos must have an imaginary mass with modulus |m c2 | ~ 0.2 GeV. But such a huge scale is totally incompatible with 80 years of neutrino physics. Just to mention one (of many) constraints, the corresponding value for neutrinos coming from the 1987A Supernova would be of the order of 10-6 GeV: five orders of magnitudes smaller than the value required by OPERA.

The "tachyonic way" for OPERA neutrinos is dead, and one has to abandon Lorentz invariance, which is at the heart of Einstein's special relativity. So be it.

If one gives up Lorentz invariance, then the usual relationship between energy and momentum can be broken and one can, in principle, be compatible both with OPERA and with SN1987A. However, then come the real problems as I have signaled in my article:

Unfortunately, it is in italian; I plan to write an updated version in English and to post it in Science 2.0.