Explain the unusually short lifetime of the Σ^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^-10s

Question

Explain the unusually short lifetime of the [latex]\Sigma^{0}[/latex] 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 [latex]10^{-10} s[/latex]except for the [latex]\Sigma^{0}[/latex], which has a lifetime of [latex]7 imes 10^{-20} s[/latex]. We suspect that something is different about the [latex]\Sigma^{0}[/latex] because of its short lifetime.

Accepted Answer

Note that the decay is [latex]\Sigma^{0} ightarrow \Lambda+\gamma[/latex], which has S = 1 on the left side and [latex]S=-1+0=-1[/latex] on the right side. It is an allowed transition. The [latex]\Sigma^{0}[/latex] 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 [latex]\Sigma^{+} ext {and } \Sigma^{-}[/latex] decay to nucleons, which violates strangeness [latex](|\Delta S|=1) . ext { The } \Xi^{0}[/latex] and [latex]\Xi^{-} ext {both decay to } \Lambda[/latex]. which violates strangeness because the left side has S = -2 whereas the right side has S = -1. Similarly, the [latex]\Omega^{-} ext {decays to either } \Xi^{0} ext { or } \Lambda[/latex]; both violate strangeness because [latex]S=-3 ext { for } \Omega^{-}[/latex]. These decay times are all on the order of [latex]10^{-10}[/latex] s, which is characteristic of the weak interaction.

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

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

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