When a proton and an antiproton annihilate, the annihilation products are usually pions. (a) Suppose three pions are produced. What combination(s) of \(\pi^{+}, \pi^{-},\) and \(\pi^{0}\) are possible? (b) Suppose five pions are produced. What combination(s) of \(\pi^{+}, \pi^{-},\) and \(\pi^{0}\) are possible? (c) What is the maximum number of pions that could be produced if the kinetic energies of the proton and antiproton are negligibly small? The mass of a charged pion is \(0.140 \mathrm{GeV} / c^{2}\) and the mass of a neutral pion is \(0.135 \mathrm{GeV} / c^{2}.\)

: First, we'll analyze the case when three pions are produced. To conserve the charge, the total charge of the three pions must be zero, since the proton and antiproton have positive and negative charges, respectively. The possible combinations are: 1. One \(\pi^{+}\), one \(\pi^{-}\), and one \(\pi^{0}\) 2. Two \(\pi^{0}\) and one \(\pi^{0}\) So there are two possible combinations of pions for three-pion production.

: Next, we'll analyze the case when five pions are produced. To conserve the charge, the total charge of the five pions must still be zero. The possible combinations are: 1. Two \(\pi^{+}\), two \(\pi^{-}\), and one \(\pi^{0}\) 2. One \(\pi^{+}\), one \(\pi^{-}\), and three \(\pi^{0}\) So there are two possible combinations of pions for five-pion production.

: Now we'll find the maximum number of pions that can be produced, assuming the kinetic energies of the proton and antiproton are negligibly small. The masses of the proton and antiproton are both approximately \(0.938 \mathrm{GeV}/c^2\). The total rest mass energy of the proton-antiproton system is \(2 \times 0.938 \mathrm{GeV} = 1.876 \mathrm{GeV}\). The maximum number of pions is produced when the total mass of the produced pions is minimized, which occurs when the pions produced are all neutral since the mass of a neutral pion is less than that of charged pions (\(0.135 \mathrm{GeV}/c^2\) vs \(0.140 \mathrm{GeV}/c^2\)). Let \(n\) be the maximum number of pions produced. Then, the total mass of the pions is \(n \times 0.135 \mathrm{GeV}/c^{2}\). To find the maximum number of pions, we'll compare the total mass of the proton-antiproton system and the total mass of the neutral pions and use integer values for \(n\): \(n \times 0.135 \mathrm{GeV}/c^{2} \leq 1.876 \mathrm{GeV}\) Solving for \(n\) and rounding down to the nearest integer value, we find: \(n \approx \frac{1.876 \mathrm{GeV}}{0.135 \mathrm{GeV}/c^{2}} \approx 13.9\) Rounding down to the nearest integer value, we get: \(n \approx 13\) So the maximum number of pions that can be produced, assuming negligibly small kinetic energies of the proton and antiproton, is 13.