a) In the center-of-momentum frame, the final particles will have total momentum of zero (as do the initial particles). In order to determine the minimum
energy needed, we must consider the situation where the final products are at
rest. This means that none of the initial energy will be “wasted” in giving the
final particles any kinetic energy.. Therefore, the initial kinetic energy, “Kinitial ”
is determined from the Conservation of Mass-Energy:
Ei = Ef
Kinitial + mπ + mp = mπ + mn + mπ
Rearranging terms leads to:
Kinitial = mπ + mn - mp
= 139.6 - 939.6 - 938.2
= 141.0 MeV
b) The quark content of the initial state is:
ud + uud
which yields, after “canceling” quarks with anti-quarks:
udd
The final state is:
ud + udd + ud
which yields, after “canceling” quarks with anti-quarks:
ddu
which is the same as the initial state.