Particles, Particles, Particles — Question 5

Question 5

Two detected photons reconstruct the mass of an unseen parent

Two photons have energies $E_1=.300$ GeV and $E_2=.400$ GeV, with opening angle $120^\circ$. Assume they are the only decay products of one parent.

1. Derive the two-photon invariant mass.

2. Evaluate the parent rest energy and compare it with the total observed photon energy.

Question 5 - Solution

Two detected photons reconstruct the mass of an unseen parent

(a) Method and calculation

$$M^2c^4=(E_1+E_2)^2-c^2|\mathbf p_1+\mathbf p_2|^2.$$$$\boxed{M^2c^4=2E_1E_2(1-\cos\theta)}\quad(p_ic=E_i).$$

(b) Interpretation and further calculation

$$\boxed{Mc^2=\sqrt{2(.300)(.400)(1+1/2)}=.600\,\mathrm{GeV}}.$$$$\boxed{E_{\mathrm{parent}}=.700\,\mathrm{GeV},\quad K_{\mathrm{parent}}=.100\,\mathrm{GeV}}.$$

Physical check

The lab energy exceeds the invariant rest energy because the parent was moving. Summing photon energies gives rest energy only in the parent rest frame.