A fundamental question about charge and speed of a particle Hi everybody and happy 2019. In my teaching sessions sometimes someone asks questions i cannot truly answer (although i have many arguments on it) and here's one that really puzzles me:

A massless (real) particle moves at $c$; now, since electric charge is always
  associated with the presence of mass, if you move at $c$ you cannot
  carry any electric charge, nevertheless you can still carry color charge, why?

I think that a more fundamental question for this would be to ask why if you are massless you actually must move at speed $c$ plus to understand the nature of the e.m. interaction and strong force one. Any help to give my student an answer that is not a labirynth of theorical burning hell?
The point here is not if charged massless particle can or cannot exist but if in principle this apply to all kind of charges (e.g. elecrtic, color, etc.) that is to say if the property of a filed of self-interaction is somehow necessarely linked to the property of having no mass (or alternatively spin 0)
 A: 
...if you move at $c$ you cannot carry any electric charge, nevertheless you can still carry color charge...

This seems to be assuming that there is some kind of massless particle that carries color charge. As far as I know, that is not correct. Unlike photons, gluons do not occur as isolated particles traveling at the speed of light, as explained here:
How do we know that gluons travel at the speed of light?
Many popular (and, confusingly, many technical) presentations do use the massless-particle language to describe gluons, but it's important to realize that what they're really describing is a computational procedure, not a physical prediction, as described in this post:
Why do nearby charges increase the probability of virtual particles?
In quantum chromodynamics (QCD), it is true that the gluon field doesn't have any associated mass term in the definition of the model. However, even if we also set the quark mass terms to zero in the definition of the model (the up- and down-quark masses are nearly zero anyway), what the model predicts is a spectrum of color-neutral particles: baryons, mesons, and color-neutral glueballs. For a simplified-but-defensible graphic depiction of each of these three types of object, I recommend figure 1.5 in


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*"Hadronic Flux Tube in Dual Superconducting Vacuum of QCD," http://user.numazu-ct.ac.jp/~koma/thesis/D-thesis.pdf.


It's a technical paper, but the figure is simple and still manages to convey some legitimate intuition about the structure of color-neutral objects in QCD. Here's the idea: The electromagnetic field does not carry electric charge, so it does not interact directly with itself. It only interacts with charge particles (which all have non-zero mass). In contrast, the gluon field does carry color charge, so it does interact directly with itself. The interaction is attractive, so the gluon field tends to gather itself into a rope-like structure called a flux tube that connects the quarks to each other. This is related to why quarks are confined into color-neutral baryons and mesons: the energy in the rope is (roughly) proportional to its length (and the proportionality constant is called the string tension), so the force doesn't get any weaker with distance. And here's the important part: What does the gluon field do if there are no quarks? Well, it still gathers itself into a rope-like flux tube, but this time the rope doesn't have any ends: its more like a circle, a flux loop. This is a (somewhat legitimate) picture of a glueball. A glueball is color-neutral and has a non-zero mass. 
The bottom line is that, as far as I know, there are no massless particles that carry color charge, with the caveats explained in the first post cited above. 
(By the way, some candidates for real-life glueballs have been noted in particle-accelerator experiments, but as far as I know, none of them have been unambiguously confirmed. Here's a recent example of a paper analyzing one such candidate: https://arxiv.org/abs/1810.08067)
