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## Q. 14.7

Explain the unusually short lifetime of the $\Sigma^{0}$ relative to the other hyperons (baryons with nonzero strangeness numbers) shown in Table 14.4.

Strategy We should be able to explain the lifetimes by examining the conservation laws. All the lifetimes for the hy perons are on the order of $10^{-10} s$except for the $\Sigma^{0}$, which has a lifetime of $7 \times 10^{-20} s$. We suspect that something is different about the $\Sigma^{0}$ because of its short lifetime.

## Verified Solution

Note that the decay is $\Sigma^{0} \rightarrow \Lambda+\gamma$, which has S = 1 on the left side and $S=-1+0=-1$ on the right side. It is an allowed transition. The $\Sigma^{0}$ is able to decay by the strong interaction to another strange particle with the same value of strangeness.

All the other hyperons in Table 14.4, however, have a decay that violates strangeness. Both $\Sigma^{+} \text {and } \Sigma^{-}$ decay to nucleons, which violates strangeness $(|\Delta S|=1) . \text { The } \Xi^{0}$ and $\Xi^{-} \text {both decay to } \Lambda$. which violates strangeness because the left side has S = -2 whereas the right side has S = -1. Similarly, the $\Omega^{-} \text {decays to either } \Xi^{0} \text { or } \Lambda$; both violate strangeness because $S=-3 \text { for } \Omega^{-}$. These decay times are all on the order of $10^{-10}$ s, which is characteristic of the weak interaction.

The decay of the $\Omega^{-} \text {into the } \Lambda$ is particularly interesting. At first glance it appears to violate strangeness by $|\Delta S|=2 \text {, but note that the reaction is actually } \Omega^{-} \rightarrow \Lambda+ K ^{-}$, so $S=-3 \text { on the left side, and } S=-1-1=-2$ on the right side; thus, $\Delta S=1 \text {, because both } \Lambda \text { and } K ^{-} \text {have } S=-1$.

 Table 14.4 The Hadrons Baryon Strangeness Charm Particle Anti- Mass Mean Main Decay Number Number Number Name Symbol particle $\left( MeV / c^{2}\right)$ Lifetime (s) Modes Spin B S C Mesons Pion $\pi^{-}$ $\pi^{+}$ 140 $2.6 \times 10^{-8}$ $\mu^{+} \nu_{\mu}$ 0 0 0 0 $\pi^{+}$ Self 135 $8.4 \times 10^{-17}$ $2 \gamma$ 0 0 0 0 Kaon $K ^{+}$ $K ^{-}$ 494 $1.2 \times 10^{-8}$ $\mu^{+} \nu_{\mu}, \pi^{+} \pi^{0}$ 0 0 1 0 $K _{S}^{0}$ $\overline{K _{S}^{0}}$ 498 $8.9 \times 10^{-11}$ $\pi^{+} \pi^{-}, 2 \pi^{0}$ 0 0 1 0 $K _{L}^{0}$ $\overline{K _{L}^{0}}$ 498 $5.1\times 10^{-8}$ $\pi^{\pm} e^{\mp} \nu_{e}, 3 \pi^{0}$, 0 0 1 0 $\pi^{\pm} \mu^{\mp} \nu_{\mu}$, $\pi^{+} \pi^{-} \pi^{0}$ Eta $\eta^{0}$ Self 548 $5\times 10^{-19}$ $2 \gamma, 3 \pi^{0}$, 0 0 0 0 $\pi^{+} \pi^{-} \pi^{0}$ Charmed $D ^{+}$ $D ^{-}$ 1870 $1.0 \times 10^{-12}$ $e^{+}, K^{\pm}, K^{0}$, 0 0 0 1 D’s $\overline{{K}^{0}}+\text { anything }$ $D ^{0}$ $\overline{ D^{0} }$ 1865 $4.1 \times 10^{-13}$ $\text { Same as } D ^{+}$ 0 0 0 1 $D _{S}^{+}$ $\overline{ D^{-}_{S} }$ 1968 $5.0 \times 10^{-13}$ Various 0 0 1 1 Bottom B’s $B^{+}$ $B^{-}$ 5279 $1.6 \times 10^{-12}$ Various 0 0 0 0 $B^{0}$ $\overline{ B }^{0}$ 5279 $1.5 \times 10^{-12}$ Various 0 0 0 0 J/Psi $J / \psi$ Self 3097 $7.1 \times 10^{-21}$ Various 0 0 0 0 Upsilon $\Upsilon (1 S)$ Self 9460 $1.2 \times 10^{-20}$ Various 0 0 0 0 Baryons Proton p $\bar{p}$ 938.3 Stable (?) $\frac{1}{2}$ 1 0 0 Neutron n $\bar{n}$ 939.6 886 $p e^{-}\bar{\nu}_{e}$ $\frac{1}{2}$ 1 0 0 Lambda $\Lambda$ $\bar{\Lambda}$ 1116 $2.6 \times 10^{-10}$ $p \pi^{-}, n \pi^{0}$ $\frac{1}{2}$ 1 -1 0 Sigmas $\Sigma^{+}$ $\bar{\Sigma}^{-}$ 1189 $8.0 \times 10^{-11}$ $p \pi^{0}, n \pi^{+}$ $\frac{1}{2}$ 1 -1 0 $\Sigma^{0}$ $\bar{\Sigma}^{0}$ 1193 $7.4 \times 10^{-20}$ $\Lambda \gamma$ $\frac{1}{2}$ 1 -1 0 $\Sigma^{-}$ $\bar{\Sigma}^{+}$ 1197 $1.5 \times 10^{-10}$ $n \pi^{-}$ $\frac{1}{2}$ 1 -1 0 Xi $\Xi^{0}$ $\bar{\Xi}^{0}$ 1315 $2.9 \times 10^{-10}$ $\Lambda \pi^{0}$ $\frac{1}{2}$ 1 -2 0 $\Xi^{-}$ $\Xi^{+}$ 1322 $1.6 \times 10^{-10}$ $\Lambda \pi^{-}$ $\frac{1}{2}$ 1 -2 0 Omega $\Omega^{-}$ $\Omega^{+}$ 1672 $0.82 \times 10^{-10}$ $\Lambda K ^{-}, {\Xi}^{0} \pi^{-}$ $\frac{1}{2}$ 1 -3 0 Charmed $\Lambda_{C}^{+}$ $\bar{\Lambda}_{C}^{-}$ 2286 $2.0 \times 10^{-13}$ Various $\frac{1}{2}$ 1 0 1 lambda Review of Particle Physics, K. Nakamura et al. (Particle Data Group), Journal of Physics G37, 075021 (2010)