Question 14.2: State whether each of the following processes can occur. If ...
State whether each of the following processes can occur. If the reaction cannot occur or if it can only occur by the weak interaction, state which conservation law is violated.
(a) π^{-} + p →\Lambda +\overline{\Sigma } ^{0}
(b) p + p →\Sigma ^{+}+n+K^{0}+\pi ^{+}+\pi ^{0}
(c) μ^{-}→e^{-}+v_{e}+v_{ μ}
(d) \Lambda →p+e^{-}+\overline{v}_{e}
(e) \Xi^- →\Sigma ^{0}+K^{-}
(a) This reaction does not occur because baryon number is not conserved. The proton (p) and lambda (\Lambda ) are baryons having baryon number equal to +1, while the \overline{\Sigma } ^{0} is the antiparticle of a baryon and has baryon number equal to −1. Thus, the total baryon number of the initial state is +1, while the total baryon number of the final state is 0.
(b) This process is allowed. Since the baryon number of p, \Sigma ^{+}, and n are all equal to +1, the total baryon number of the initial and final states are equal to +2. Since the strangeness quantum number S is equal to −1 for the \Sigma ^{+} and +1 for K^{0}, the total strangeness is equal to zero for both the initial and final states. No leptons are involved in this reaction.
(c) This process is not allowed because electron lepton number L_{e} is not conserved. The total electron lepton number of the initial state is zero; however, the electron lepton number of both e^{−} and ν_{e} are equal to +1, and, hence, the total electron number of the final state is equal to +2.
(d) Since \Lambda has a strangeness quantum number S equal to −1 and all the particles in the final state have S = 0, strangeness is not conserved in this reaction, and the reaction can only occur by the weak interaction.
(e) While the lepton numbers, baryon number, and strangeness are all conserved for this process, the process will not occur because energy is not conserved. According to the data given in Tables 14.4 and 14.3, the rest energies of \Xi ^{-}, \Sigma ^{0}, and K^{−} are 1321.3 MeV, 1192.6 MeV, and 493.7 MeV, respectively. Hence, the energy of the initial state is 1321.7 MeV, while the minimum energy of the final state is 1686.3 MeV.
| TABLE 14.4 Properties of a few of the lightest baryons. Data is that given by the Particle Data Group in 2008. |
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| Name | Symbol | Composition | Mc^{2} (MeV) | Decay Mode | Lifetime/Width |
| Nucleon | p | uud | 938.3 | > 10^{31}yr | |
| n | udd | 939.6 | n\rightarrow p+e^{-}+\overline{v_{e}} | 885.7 s | |
| Lambda | \Lambda | uds | 1115.7 | \Lambda \rightarrow p+\pi ^{-} | 2.63 × 10^{-10}s |
| Sigma | \Sigma ^{+} | uus | 1189.4 | \Sigma ^{+}\rightarrow p+\pi ^{0} | 8.02 × 10^{-11}s |
| \Sigma ^{0} | uds | 1192.6 | \Sigma ^{0}\rightarrow \Lambda +γ | 7.4 × 10^{-20}s | |
| \Sigma ^{-} | dds | 1197.5 | \Sigma ^{-}\rightarrow n+\pi ^{-} | 1.48 × 10^{-10}s | |
| Delta | \Delta ^{++} | uuu | 1232 | \Delta ^{++}\rightarrow p+\pi ^{+} | 118 MeV |
| \Delta ^{+} | uud | 1232 | \Delta ^{+}\rightarrow n+\pi ^{+} | 118 MeV | |
| \Delta ^{0} | udd | 1232 | \Delta ^{0}\rightarrow p+\pi ^{-} | 118 MeV | |
| \Delta ^{-} | ddd | 1232 | \Delta ^{-}\rightarrow n+\pi ^{-} | 118 MeV | |
| Xi | \Xi ^{0} | uss | 1314.9 | \Xi ^{0}\rightarrow \Lambda +\pi ^{0} | 2.90 × 10^{-10}s |
| \Xi ^{-} | dss | 1321.7 | \Xi ^{-}\rightarrow \Lambda +\pi ^{-} | 1.639 × 10^{-10}s | |
| Omega | \Omega ^{-} | sss | 1672.5 | \Omega ^{-}\rightarrow \Lambda +K^{-} | 8.21 × 10^{-11}s |
| TABLE 14.3 Properties of a few of the lightest mesons. Data is that given by the Particle Data Group in 2008. |
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| Name | Symbol | Composition | Mc² (MeV) | Decay Mode | Lifetime/Width |
| pion | π^{−} | d \overline{u} | 139.6 | π^{−}→ μ^{−} + \overline{ν_{μ}} | 2.603 × 10^{−8} s |
| π^{+} | u\overline{d} | 139.6 | π^{+} → μ^{+} + ν_{μ} | 2.603 × 10^{−8} s | |
| π^{0} | u\overline{u},d\overline{d} | 135.0 | π^{0}→ 2γ | 8.4 × 10^{−17} s | |
| kaon | K^{+} | u\overline{s} | 493.7 | K^{+} → μ^{+} + ν_{μ} | 1.238 × 10^{−8} s |
| K^{−} | s\overline{u} | 493.7 | K^{−} → μ^{−}+ \overline{ν_{μ}} | 1.238 × 10^{−8} s | |
| \overline{K}^{0} | d\overline{s} | 497.6 | |||
| K^{0} | \overline{d}s | 497.6 | |||
| eta | η | u\overline{u},d\overline{d},s\overline{s} | 547.8 | η → 2γ | 1.30 keV |
| rho | ρ^{+} | u\overline{d} | 775.5 | ρ^{+}→ π^{+} + π^{0} | 149.4 MeV |
| ρ^{−} | d\overline{u} | 775.5 | ρ^{−} → π^{-} + π^{0} | 149.4 MeV | |
| ρ^{0} | u\overline{u} , \overline{d} | 775.5 | ρ0 → 2π^{0} | 149.4 MeV | |
| omega | ω | u\overline{u},d\overline{d} | 782.7 | ω → π^{+} + π^{−} | 8.49 MeV |