Re: A neutral kaon at rest decays, what is the momentum of the pions?

Dear Mike,

This is a great question. You're right about the mass of the kaon being
about 498 MeV/c2, and one of the two kinds of neutral kaons, the k short,
does decay to pi+pi- some 69% of the time. (We won't go in to CP violation,
the difference between K short and K long, etc.) However, according to my
July 2004 Particle Physics Booklet (published by the Particle Data Group at
Lawrence Berkeley Labs), the mass of the pi+- is 139 MeV/c2.

Now that we've got our numbers squared away, we can look at the problem.
We've really got two conservation laws we have to consider, namely,
conservation of energy and conservation of momentum. The way to tackle this
problem is to think of the state before and after the kaon decays. Before
the decay we have a total momentum of zero (the particle is at rest), and
we have a total energy of 498 MeV (since E = mc2 and the m = 498 MeV/c2).
Therefore, in the final state we must have a total momentum of zero and a
total energy of 498 MeV.

After the decay, we have two particles with mass 139 MeV/c2 for a total of
278 MeV of energy that's tied up in mass. That means we need to have (498 -
278 =) 220 MeV energy that we've got to use. Since the initial particle was
at rest, total momentum of the pions has to be zero, too. So, they're
either at rest themselves, or they're moving in opposite directions. Since
they have the same mass, they have to be moving with the same speed, too.
That means the momentum vectors for the two pions have the same magnitude
but are in opposite directions. And, since we've got 220 MeV of energy
still to consider, we know that they've got non-zero velocities and thus
non-zero momentum. 

Remember that E = |p|c, where |p| is the magnitude of the momentum of the
system. Since we've got two particles, E = |p1|c + |p2|c where p1 and p2
are the momenta of the two pions. That then gives us |p1| + |p2| = E/c. But
remember that we said that, since the masses of the two pions were equal,
the magnitudes of the momenta were the same, |p1| = |p2| and thus 2|p1| =
E/c or |p1| = E/2c. Therefore, the momentum of each pion is |p1| = 220
MeV/2c = 110 MeV/c.

Hope this helps!