The primary decay of the Ξ^- is Ξ^- → Λ^0 + π^-. The two decay products in turn decay. Find the first decay listed in Table 14.4 for both the Λ^0 and π^- and write the original decay and two subsequent decays in terms of their quark composition. Strategy We look up the two subsequent decays and

Question

The primary decay of the [latex]\Xi^{-} ext {is } \Xi^{-} ightarrow \Lambda^{0}+\pi^{-}[/latex]. The two decay products in turn decay. Find the first decay listed in Table 14.4 for both the [latex]\Lambda^{0} ext { and } \pi^{-}[/latex] and write the original decay and two subsequent decays in terms of their quark composition.

Strategy We look up the two subsequent decays and find [latex]\Lambda^{0} ightarrow p+\pi^{-} ext {and } \pi^{-} ightarrow \mu^{+}+ u_{\mu}[/latex]. We use Table 14.6 to find the quark composition. The quark structures of the antiparticles are just the corresponding antiquarks of the particles.

Accepted Answer

We list first the reaction and then the quark composition. Keep in mind that these decays all occur by the weak interaction in which quark transformations can occur. We have not discussed this, but it is because these decays occur by exchange of the [latex]W ^{\pm}[/latex] gauge bosons.

[latex]\Xi^{-} ightarrow \Lambda^{0}+\pi^{-} \quad d s s ightarrow u d s+\bar{u} d[/latex]

A s quark is changed to a d quark, and an [latex]\bar{u} u[/latex] pair is created.

[latex]\Lambda^{0} ightarrow p+\pi^{-} \quad u d s ightarrow u u d+\bar{u} d[/latex]

Again a s quark is changed to a d quark, and an [latex]\bar{u} u[/latex] pair is created.

[latex]\pi^{-} ightarrow \mu^{+}+ u_{\mu} \quad \bar{u} d ightarrow ext { no quarks }[/latex]

The d quark can change to an u quark and annihilate the existing [latex]\bar{u}[/latex] quark.

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)

Table 14.6 Quark Composition of Selected Hadrons
	Quark
Particle	Composition
Mesons
[latex]\pi^{+}[/latex]	[latex]u \bar{d}[/latex]
[latex]\pi^{-}[/latex]	[latex] \bar{u} d[/latex]
[latex]K^{+}[/latex]	[latex]u \bar{s}[/latex]
[latex]K^{0}[/latex]	[latex]d \bar{s}[/latex]
[latex]D^{+}[/latex]	[latex]c \bar{d}[/latex]
[latex]D^{0}[/latex]	[latex]c \bar{u}[/latex]
Baryons
Particle	uud
n	udd
[latex]\Lambda[/latex]	uds
[latex]\Sigma^{+}[/latex]	uus
[latex]\Sigma^{0}[/latex]	uds
[latex]\Xi^{0}[/latex]	uss
[latex]\Xi^{-}[/latex]	dss
[latex]\Omega^{-}[/latex]	sss
[latex]\Lambda_{C}^{+}[/latex]	udc

Question 14.9: The primary decay of the Ξ^- is Ξ^- → Λ^0 + π^-. The two dec...

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