Kinematics of Decay of Ultra-Relativistic Particle

Question

An ultra-relativistic particle with $pc>>Mc^2$ is supposed to decay into two massive particles of mass $m$. I've been told that this is impossible and have tried the following to reach that conclusion:

The four momentum of the ultra-relativistic particle in its rest frame is $p^{\mu} = (M,0)$ and the decay products will have identical energy and momenta so writing the four momentum of the second particle in terms of the first and the parent particle: $p_2^{\mu} = p^{\mu} -p_1^{\mu}$ we can take magnitudes and find: $-m^2+m^2 = -M^2 +2ME$ where $E$ is the energy of the decay products. So $E =M/2$ and the magnitude of the daughter's four momenta are: $|p| = \frac{\sqrt{M^4 -4M^2 m^2}}{2M}$. I don't see how to use the ultra-relativistic condition to reach a contradiction here.

Answer 1

The “ultra-relativistic” part is a statement about motion relative to an observer, e.g. you. The decaying particle simply doesn’t care about that: it’s decaying or not in its own rest frame.

So if it can decay, it eventually will.

The only thing “ultra-relativistic” brings to the discussion is how long you observe it to take.

Answer 2

"I've been told that this is impossible..." Who told you that? The particle will decay, but it's lifetime will be extended by a factor of E/M. this means that it will probably leave your laboratory before it decays. Your not seeing it decay is different than "impossible".