There is a related question at What defines the mass of elementary particle? but my question is more fundamental.
Why do we treat particle masses as exact values? If a particle is indeed an excitation/resonance of a field, then like any excitation/resonance, should that resonance not have a finite "width"?
I ask because it is not clear to me whether we defined the mass to be exact, a priori, or if we assume as much because variation would be difficult to handle, or if our measurements are not accurate enough to resolve any variation about some mean mass for any given particle.
Is there a fundamental reason why we treat, say, all electrons as having the same mass? I recall the assumption that electrons are identical particles and I understand that measurements support this, but can any of our observations actually test for variation in mass between any two electrons?