Relativity of particle physics

1. The problem statement, all variables and given/known data

(i) The [itex]\pi_{0}[/itex] meson predominantly decays into two photons.

(a) Write down an expression for the energy of each photon in the [itex]\pi_{0}[/itex] rest frame in terms of the [itex]\pi_{0}[/itex] mass [itex]m_{\pi}[/itex]. [2 marks]

(b) One of the photons is produced at spherical polar angle [itex]\theta[/itex] with respect to the positive z axis in the [itex]\pi_{0}[/itex] rest frame. What is the corresponding spherical polar angle of the other photon? [2 marks]


2. Relevant equations

Energy of a particle in its rest frame = mc².


3. The attempt at a solution

In the rest frame of the meson, the meson is at rest. So, its momentum is zero. By the law of conservation of momentum, therefore, the two photons have a total momentum equal to zero. So, the photons must be travelling in opposite directions with the same magnitude of momentum.

(a) The photons have the same magnitude of momentum. So, each must have half the rest energy of the meson. So, energy of each photon = [itex]\frac{m_{\pi}}{2}c^{2}[/itex].

(b) The photons must be travelling in opposite directions. So, the corresponding spherical polar angle of the other photon is [itex]\pi - \theta[/itex].

I would greatly appreciate if you could point out any mistakes and make some comments on my solutions.

 

If [itex]\pi-\theta[\itex] is not the answer, I can't think of any other answers. What could it possiby be?

Alos, I am not sure how I could write the solutions in a more formal way. I would be grateful if you could take a bit of your time to show the steps.