I now this is a little late, but I might be able to answer. Look in this paper. As stated in the comments, non-zero mass is consequence of the Bargmann extension of the Galilean Lie algebra. This leads to the Bargmann superselection rule, which forbids the superposition of two particles with unequal masses. On page 4 of the paper I link, the authors state that
The mass appears as a central element of the extended Lie algebra... that is, it commutes with all the other elements, in particular with the Hamiltonian H. It is thus a conserved quantity. It is even a ''super-conserved" quantity, giving rise to a superselection rule ...This is due to the fact that the extension...is nontrivial so that a physically trivial transformation may result in a definite modification of the phase of the state vector, depending on the mass of the system. In order not to alter the physical properties of the system by such a transformation, we must forbid any superposition of states with different masses, thus obtaining Bargmanns superselection rule. It has the effect of breaking the Hilbert space into mutually incoherent eigenspaces of the mass operator M. The consequences of this superselection rule are rather far-reaching with respect to the possible types of particle reactions allowed by Galilean invariance. Suppose, for instance, that we treat a theory with only one kind of particle, with definite mass m. In this case, conservation of the mass implies conservation of the number of particles, thus forbidding any type of production process.
Hence, for one kind of particle with one kind of mass, the Bargmann superselection rule directly implies particle number conservation.