5. A pi0 meson whose rest mass is 135 MeV/c^2 is moving with a kinetic energy of 1 GeV.
It decays in flight into two photons whose paths are along the direction of motion of the
meson. Find the energies of the two photons.
[Ans: 4 and 1131 MeV.]
The Attempt at a Solution
Working in the CM frame (the rest frame of the pion) and conserving energy in that frame:
mc^2 = 2E*, where E* are the energies of the photons in the CM frame
I then thought I could use a lorentz transformation Ea=gamma(E* + betaE*) where Ea is the energy of one photon in lab frame and gamma and beta are the gamma and beta for the pion moving in lab frame, with gamma = (energy of pion)/mc^2 and beta = (pc)/E
But first i was checking something, and i cannot get the simple conservation of energy to work with the answers given. So in the lab frame, i thought the energy of the pion must equal the sum of the energies of the photons, given as 1135? And the energy of the pion
= sqrt[(pc)^2 + m^2c^4]
i thought this
=sqrt[1000^2 + 135^2]
as the kinetic energy is given as 1GeV=1000MeV and the mass is given in units of MeV/c^2. Clearly I'm misunderstanding something, but what?!? Will my method work once this misunderstanding is resolved (and is there a better method or different method just for future reference)?