Is there any way in which a bound state could consist only of massless particles? If yes, would this "atom" of massless particles travel on a light-like trajectory, or would the interaction energy cause it to travel on a time-like trajectory?
Answer
John Rennie has answered the first part of the question. The second part was this:
If yes, would this "atom" of massless particles travel on a light-like trajectory, or would the interaction energy cause it to travel on a time-like trajectory?
The answer is that it would have a timelike world-line, and this is independent of any (probably uncertain) details of the system's dynamics or binding energy.
Mass is not additive. Mass is defined (in units with $c=1$) by $m^2=E^2-p^2$, where $E$ is the mass-energy and $p$ is the momentum. $(E,p)$ is the momentum four-vector, and the squared mass is its squared norm. For a massless particle, the momentum four-vector is lightlike. If four-vectors $p$ and $p'$ are both lightlike and future-directed, but not parallel, then $p+p'$ is timelike. Therefore a system of interacting, massless particles is guaranteed to have a nonzero mass.
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