# Kinematics of Decay of Ultra-Relativistic Particle

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.

• Compare to the alleged process $\gamma \to e^+ + e^-$ in free space (about which we have quite a few questions). – dmckee --- ex-moderator kitten Apr 15 '18 at 18:31
• Is it massive as $m>>M$? – Oktay Doğangün Apr 15 '18 at 18:35
• So because the energy of an ultra-relativistic particle is almost entirely momentum we can approximate its energy as just being momentum and then it's impossible to conserve energy and linear momentum at the same time as for photon decay? – John Doe Apr 15 '18 at 18:54