So, I am not asking is the square of four-momentum of a particle an invariant to Lorentz trasnformations, but rather,is it invariant in dynamic situations? It seems to me that this also has to hold. So, is four-momentum squared same before and after collision, not the total, but for one particle in that collision?
As long as the particle has neither
- Changed kind (as happens in, for example, in charged-current weak scattering)
- Gotten excited (which can happen to atoms, nuclei, and hadrons; though in some cases this would be written as a change of type as in a proton turning into a Delta, for instance)
then the mass is the same.
But ... in a lot of ways what I wrote is a tautology. If a particle stays the same then it stays the same.