Faster-than-$c$ photons As far as I know, according to quantum field theory, there are some photons that go faster than c, which is the speed of light in vacuum. 
However, there seems to be a paper and a corresponding experiment that show every photon obeys the speed limit of $c$. (http://physics.aps.org/synopsis-for/10.1103/PhysRevLett.106.243602)
So, my question is:


*

*Is this experiment accepted universally?

*Regardless of the acceptance of the experiment, if every single photon is shown to obey the speed limit of $c$, what does this mean for quantum field theory?
 A: 
As far as I know, according to quantum field theory, there are some photons that go faster than c, which is the speed of light in vacuum.

No, this is not correct. Photons always travel at $c$. This is an accepted fact throughout the physics community, and it is based on many different experiments. 

Regardless of the acceptance of the experiment, if every single photon is shown to obey the speed limit of c, what does this mean for quantum field theory?

It means that quantum field theory works - or rather, that it can work. The alternative, that photons might travel faster than $c$, would mean that Lorentz invariance fails, and since QFT relies on that invariance, it would mean that QFT would not be a generally valid theory.
A: you're terribly mistaken. But i understand the source of your confusion. It is more or less well known, and not from QFT per se, that light can be sped up faster than c IN SOME SENSE and I stress this carefully: ONLY IF YOU CONSIDER "PHASE VELOCITY" as the speed of this light, then it is correct that this can happen. HOWEVER, GROUP VELOCITY (A different concept) is the speed at which any information or physical "pulses" travel. According to relativity (and therefore, also according to QFT) group velocity can never violate the speed limit c. Now, take into account as you look up the definitions of phase velocity and group velocity, that the concept of wave packet is important and essential if you want to talk about group velocity and phase velocity. Also, the experiment you mention in that link says something new but not surprising: a single photon (would you consider it a wave packet?) obeys the speed limit... not much of a punch, anyway...Well, one more remark: there are actually some cases when the group velocity exceeds c but in these cases physical information travels at a lower speed called signal velocity and still obeys the speed limit.
A: Something like photons and even massive particles moving faster than the speed of light in vacuum ($c$) appears in Feynman diagrams. But these virtual particles don't transfer information and can't be directly detected. The experiment about which you are talking regarded real photons and does not contradict quantum mechanics. See more in Wikipedia sections Variable speed of light § Varying c in quantum theory and Propagator § Faster than light?.
It's hard to hide that the article Scharnhorst effect suggests that superluminal signaling based on that effect could be possible, but probably it's a misconception and Wikipedians just haven't found explanation clear enough to write more than "However, several authors (including Scharnhorst) argue that the Scharnhorst effect cannot be used to create causal paradoxes."
Problems appearing in other answers are rather unrelated examples of other apparent contradictions with the "law" that "nothing can travel faster than light".

*

*Cherenkov radiation is emitted by particles travelling faster than light in water or other medium.

*Some experiments have shown that in some media speed of light (even group velocity) can exceed $c$, but this is quite clearly explained on Wikipedia

The group velocity of a wave (e.g., a light beam) may also exceed $c$ in some circumstances. In such cases, which typically at the same time involve rapid attenuation of the intensity, the maximum of the envelope of a pulse may travel with a velocity above $c$. However, even this situation does not imply the propagation of signals with a velocity above $c$, even though one may be tempted to associate pulse maxima with signals. The latter association has been shown to be misleading, basically because the information on the arrival of a pulse can be obtained before the pulse maximum arrives. For example, if some mechanism allows the full transmission of the leading part of a pulse while strongly attenuating the pulse maximum and everything behind (distortion), the pulse maximum is effectively shifted forward in time, while the information on the pulse does not come faster than c without this effect.
Faster-than-light § Group velocities above c
  
  Correct rule is that no information can travel faster than light in vacuum (Actually, this is specific speed appearing in special relativity, and light is just the best known example of something travelling with this speed).

A: Special Relativity only requires that light travel at c in vacuum.  In any normal dielectric the speed of light will be less than c.  This is what gives rise to Cherenkov radiation.  https://en.wikipedia.org/wiki/Cherenkov_radiation
If you consider phase and group velocity, the issue is really only straightforward in homogeneous dielectrics.  Without going into a lot of details, there exist real media in which the permittivity is negative.  Near a resonance of the dispersion relation the phase and group velocity can be anything.  In metals, below the plasma frequency the permittivity is negative.  Finally, in random media quantities like wavevector and velocity are not well defined.  You have to look at average phase and group delay, etc.  But I suspect, just the first paragraph is what you're after.
