Proving a Uniform Energy Distribution of B in Decay A -> B + C

[ryanwilk]

Homework Statement

Consider the decay A -> B + C (where A is not at rest). In the rest frame of A, B is emitted in a random direction (all directions have equal probability) and I need to show that in the lab frame, the energy distribution of B is uniform.

(We assume that B has negligible mass)

Homework Equations

(Let c=1)

The Attempt at a Solution

So I started by writing down the 4-momenta of A and B in the rest frame of A (choosing the momentum of B to be along the x axis):

P_A' = (m_A,0,0,0)
P_B' = (E_B',E_B',0,0)

In the lab frame:

P_A = (E_A,(E_A²-m_A²)^1/2,0,0)
P_B = (E_B,E_B,0,0)

Lorentz boosting along the x-axis, I can determine the maximum and minimum energy that B can have:

$$E_B^{\mathrm{min,max}} = \frac{E_A}{2} \bigg(1-\frac{m_C^2}{m_A^2}\bigg) \bigg(1 \pm \sqrt{1-\frac{m_A^2}{E_A^2}}\bigg)$$

using:

$$\gamma = \frac{E_A}{m_A}\>,\gamma \beta = \frac{p_A}{m_A},\>E_A' = \frac{m_A^2-m_C^2}{2m_A}$$

I could also Lorentz boost in all other directions to get expressions for the energy. But I have no idea how to show that each of these energies in equally probable?

Any help would be appreciated.

Thanks!

 

[vela]

Try calculating E_A as a function of θ and show that the dE/dΩ is a constant.

 

[ryanwilk]

Try calculating E_A as a function of θ and show that the dE/dΩ is a constant.

Ok, so using 3D spherical polars, I get $$E_B(\theta') = \frac{E_A}{2} \bigg(1-\frac{m_C^2}{m_A^2}\bigg) \bigg(1 + \sqrt{1-\frac{m_A^2}{E_A^2}}\>\mathrm{cos}(\theta ') \bigg)$$

(I defined it so that there is max. energy at θ'=0 and min. at θ'=π)

Then dΩ = sin(θ')dθ'dφ' and so since E_B is independent of φ', dE/dΩ is a constant?

 

[vela]

Not quite. Nothing depends on φ', so you can integrate that out and deal only with θ', so you have dΩ' = 2π sin θ' dθ' where θ' goes from 0 to π. It's convenient to change variables to cos θ' so you have dΩ' = d(cos θ') where cos θ' runs from -1 to 1. (The minus sign goes into switching the order of the limits.) When you say that a decay is isotropic, that means that the distribution N is flat as a function of cos θ'. What you want to show is that dN/dE_B is constant, using the fact that$$\frac{dN}{dE_B} = \frac{dN}{d(\cos \theta')} \frac{d(\cos\theta')}{dE_B}$$(which is simply an application of the chain rule).