Maximum energy of a decay product in three(or more)-particle decay

[NanakiXIII]

Homework Statement

A particle A at rest decays into three or more particles: A -> B + C + D + ...

What is the maximum energy particle B can have, expressed in the various masses?

Homework Equations

E^2 = m^2 c^4 + p^2 c^2;
Conservation of energy and momentum.

The Attempt at a Solution

My reasoning in this problem is as follows. We can view this decay as a two-particle decay with particles B and X = (C + D + ...). The difference between this and an actual two-particle decay is that the mass of X can have different values.

Because of conservation of momentum, the momentum of B needs to be opposite and equal to that of X. To maximize the energy of B, we need to minimize the amount of energy going into X. Since we can't do much about the kinetic energy of X as a whole, we need to minimize the internal energy of X. That means that all the particles need to be going in the same direction at the same velocity. So,

$$p_b = -p_X = m_X v = (m_C + m_D + ...) v$$.

Now, this all seems to make sense, but it doesn't seem like the way to go. Trying to calculate things from this point generate rather large expressions, which doesn't seem right. I'm also not even sure how to go about integrating the last restriction I named, since all the equations only seem to care about the size of the momentum three-vector, and not about how that vector is generated, while different compositions for the vector should lead to different energies.

In addition to the above mess, I tried just writing out conservation of four-momentum in various arrangements and squaring them, but that didn't seem to lead to solvable equations. In fact, they seemed to suggest that the internal energy of X needed to be maximized, which doesn't sound right. Basically, I think I'm overlooking an easier way to go about this. If anyone could point me in a more productive direction, I'd appreciate it.

 

[Jhero]

Thank you for your question! I understand your reasoning and it is a good start to solving this problem. However, I believe you may be overcomplicating things a bit. Let me offer a simpler approach.

First, let's look at the conservation of energy equation:

E_A = E_B + E_C + E_D + ...

Since particle A is at rest, its energy is simply its mass, E_A = m_A c^2. Now, to maximize the energy of particle B, we want to minimize the energy of the other particles. This means that we want to minimize the sum of their masses, as well as their kinetic energies.

To minimize the sum of the masses, we want to choose particles with the smallest masses possible. This means that particle C, D, etc. should all have the smallest masses possible. For example, if particle C is a proton and particle D is a neutron, we could choose them to be pions instead, which have a much smaller mass.

To minimize the kinetic energies of the particles, we want them to all be moving in the same direction at the same velocity, as you mentioned. This means that all the particles should have the same momentum, and thus the same velocity.

Putting this all together, we can say that the maximum energy of particle B is given by:

E_B,max = m_A c^2 - (m_C + m_D + ...) c^2

So, to summarize, to maximize the energy of particle B, we want to choose particles with the smallest masses possible, and all moving in the same direction at the same velocity.

I hope this helps! Let me know if you have any further questions.