added much later: It turned out that the OPERA results were flawed.
I leave the text below and the comments as testimony of my confusion about all that when the surprising observation seemed somehow possible.
I wonder if neutrinos should travel slightly faster than photons according to standard physics, i.e. quantum field theory as we know it (*).
The reason would be the Scharnhorst effect.
If c denotes the 'bare' speed of light then real photons should actually travel a tiny bit slower than c, due to the interaction with virtual particles in a 'real vacuum': c(physical) < c. The bulk of the effect comes from interactions with virtual electrons and positrons (see e.g. this paper for more details).
But neutrinos interact only weakly with those and therefore the velocity of neutrinos should be closer to c; In other words neutrinos would be slightly faster than photons.
I am not sure if the OPERA experiment detected anything real, but if the results are indeed confirmed one should take a second look at the Scharnhorst effect imho; This time considering the difference between neutrinos and photons instead of looking at photons between Casimir plates.
with credit to A.M., a friend of mine and yet another quant interested in physics, who reminded me of this effect, which was discussed already in the 1990s. But if this is all b.s. the embarrassment is of course fully mine.
added later: A very simplified calculation shows that the Scharnhorst effect could have the right order of magnitude!
According to the Scharnhorst paper, equ. 10, the increase of the speed of photons between Casimir plates is to first order approximately
1 + 0.01*alpha2/(mL)4.
As we move the plates closer and reduce L (in a thought experiment) we eliminate more of the interaction with the virtual particles and the speed of photons gets closer to the 'bare' speed of light c.
However, we cannot reduce L below 1/m (in this approximation) and thus the Scharnhorst formula
gives a maximum correction
c = c(bare) = c(physical)*( 1 + 0.01*alpha2 ).
If we assume that neutrinos travel at a speed close to the 'bare' speed c, then they would travel faster than photons in the 'real vacuum' by a factor of approximately 1 + 10-6.
This is pretty much what OPERA measured.
(*) added even later: Heather Logan thinks this explanation cannot work and I changed the sentence to reflect that this is not 'standard' opinion; After all she is a physics professor and my last lesson in QED was about 20 years ago. On the other hand I don't see where the argument goes wrong. Obviously I appreciate any input.
added one day later: After some more thinking and reading I can now better formulate this idea as follows:
i) In QED the Ward identities ensure that after quantization of the electromagnetic field the longitudinal polarization of the photon vanishes.
ii) This implies (by comparison with classical fields?) that the photon mass remains zero and the velocity of photons is the light speed c.
iii) On the other hand, the Scharnhorst effect suggests that photons travel at a 'dressed' speed d and one can increase (in principle) d between Casimir plates (see all the above).
iv) The standard interpretation is that d = c to keep QED simple, but one has then to explain away the acausality from photons travelling at increased speed between Casimir plates.
v) In light of the OPERA experiment, the proposal is to assume d < c and (because the Ward identities hold) for all possible experiments the longitudinal polarization of the physical photon still vanishes, so that it would behave like a massless particle, although it travels slightly slower than the 'true' speed of light c. (By the way, I am not the first thinking along such lines.)
vi) The neutrino would travel very close to the 'true' speed of light c and thus slightly faster than photons. The estimate above suggests that the effect could be in the right ballpark,
see the comments for more