Special relativity with particles

[tskuzzy]

Homework Statement

Consider the event $p + \gamma \to p + \pi$, where $p$ is a cosmic ray of a proton, $\gamma$ is a microwave background photon, and $\pi$ is a generated meson. What is the minimum energy of the proton for such an event to happen? Proton has a rest mass of 1 GeV/c2, π particle has a rest mass of 100 MeV/c2, and microwave background photon has an energy of 2.5*10^-4 eV.

Homework Equations

Conservation of energy and momentum equations.
$$ E = mc^2 $$

The Attempt at a Solution

From conservation of energy, we get the equation:
$$
\begin{align*}
E_i &= E_f \\
E_{p_i} + E_{\gamma} &= E_{p_f} + E_{\pi} \\
E_{p_i} + 2.5 \times 10^{-4} &= \gamma_{p_f}(1 \times 10^6) + \gamma_{\pi}(1 \times 10^5)
\end{align*}
$$

In order to minimize the energy of the initial proton, is it reasonable to simply set the gammas on the right side equal to 1 (i.e. let them be at rest)?

 

[vela]

Nope, that would violate the conservation of momentum.

Try solving the problem in the center-of-mass frame. In that frame, the resulting proton and pion will be at rest. Then transform the results back to the lab frame.

 

[tskuzzy]

Nope, that would violate the conservation of momentum.

Try solving the problem in the center-of-mass frame. In that frame, the resulting proton and pion will be at rest. Then transform the results back to the lab frame.

Thanks for the speedy response!

Could you explain why the resulting proton and pion would be at rest in the COM frame? Wouldn't that imply that the two are traveling together with the same velocity? Why couldn't they fly off in different directions wrt the COM?

 

[vela]

Because you're looking for the minimum energy. Some of the energy goes into creating the pion. Any extra ends up as the kinetic energy of the resulting particles, so to find the minimum, you want the kinetic energy to be as small as possible.