Increase in momentum under a boost for relativistic two body kinematics

I'm trying to understand the momentum distribution in a 1 to 2 body particle decay.

A statement which I'm trying to verify is that in situations with uneven masses for the daughter particles, the heavier particle will have higher momentum in real experiments which produce the parent particle with some initial momentum. Of course in the rest frame of the decay, there must be momentum balance, so this is equal to the statement that the heavier particle will gain more momentum in the boost from the rest frame of the parent to the lab frame.

You can work out the kinematics and see that the energy of a daughter $i$ is

$$E_i = \frac{M^2 + m_i^2 - m_j^2}{2M}$$

where $m_j$ is the mass of the other daughter. Of course the momentum must be $p_i = \sqrt{E_i^2 -m_i^2}$. I'm having a very hard time with the algebra to show that under a lorentz boost

$$(E_i, p_i) \rightarrow (\gamma E_i - \gamma \beta p_i, \gamma p_i - \gamma \beta E_i)$$

that $\left| p_i \right|>\left| p_j \right|$ if $m_i > m_j$. My strategy has been to try and show that $p'_{i} - p'_{j} \lt 1$, which boils down to showing that

$$\frac{\beta}{M}(m_i^2 - m_j^2) \lt p_i - p_j,$$

but I am finding this very challenging to show since $p_i$ and $p_j$ are quite nasty. Is there a more clever way to understand this, or does anyone know of a derivation of this result I could study?

• What stops the lab frame from being equal to the rest frame of the heavier daughter particle, in which it has zero momentum? If the statement is true, there must be some additional assumptions you're not telling us. – Mark A Nov 8 '16 at 6:54
• Hi @MarkA, thank you for your insightful comment. You are right, as stated this can not be the case. The proper statement is that when the difference in mass between the daughters and mother goes to zero, then the heavier particle will have more momentum. This is not too hard to show and I will update the question. – Bobak Hashemi Nov 9 '16 at 13:06
• I also should add that it is a statement should be about the velocities in the rest frame of the mother particle, that more directly addresses your comment – Bobak Hashemi Sep 15 '18 at 0:33