Pion Photoproduction Homework: Proton Energy Calculation

[Arcturus7]

I have been given the following problem on a homework sheet;

"The CMB has an (almost) uniform temperature of (almost) 3K. If we take the energy of a photon to be E=3kT, what minimum energy must a proton have in order to interact with a CMB photon in order to produce neutral Pi Mesons?"

Homework Equations

I've come at this problem using nothing but the definition of 4-Momenta and the Invariant Mass, plus the given equation for the photon energy.

The Attempt at a Solution

Immediately we can calculate the photon energy to be 7.76 x 10^-4eV.

Initially, in the Center of Momentum frame, we have that the sum of the momenta is 0. This leaves us with;

W=E_Proton+E_Photon

And then after the interaction, since the resultant particles will be at rest (minimum energy is specified in the question) we can conclude that the invariant mass is just the sum of the individual masses;

W=M_Proton+M_Pion

By equating the two we conclude that;

E_Proton=M_Proton+M_Pion-E_Photon=1.073GeV

I've heard a lot of classmates say their answers were very different (some many many orders of magnitude larger). I can't really see a flaw in my argument, but that being said, it seems fairly simplistic. What have I missed?

 

[mfb]

You mix energies in different reference frames. In the center of momentum frame, the photon energy won't be 0.776 meV.

The correct answer is many, many orders of magnitude larger.