Decay of the Particle The lambda (Λ) particle (Table 26.4) is an uncharged, heavy baryon that can decay via the reactions Λ → p^+ + π^- and Λ → n + π^0. Examine both these reactions from the standpoint of conservation laws.

Question

Decay of the Particle

The lambda (Λ) particle (Table 26.4) is an uncharged, heavy baryon that can decay via the reactions [latex]\Lambda ightarrow p^{+}+\pi^{-} ext {and } \Lambda ightarrow n+\pi^{0}[/latex]. Examine both these reactions from the standpoint of conservation laws.

TABLE 26.4 Table of Hadrons
Particle name	Symbol	Anti- particle	Rest energy (MeV)	Mean lifetime (s)	Main decay modes	Spin	Baryon number B	Strangeness number S	Charm number C
Mesons
Pion	[latex]\pi^{-}\ \pi^{0}[/latex]	[latex]\pi^{+}[/latex] Self	140 135	[latex]2.6 imes 10^{-8}\8.4 imes 10^{-17}[/latex]	[latex]\mu^{+} v_{\mu}\ 2 \gamma[/latex]	0 0	0 0	0 0	0 0
Kaon	[latex]K ^{+}\K _{S}^{0}\K _{L}^{0}[/latex]	[latex]K ^{-}\\overline{ K }_{S}^{0}\ \overline{ K }_{L}^{0}[/latex]	494 498 498	[latex]1.2 imes 10^{-8}\9.0 imes 10^{-11}\ 5.1 imes 10^{-8}[/latex]	[latex]\mu^{+} u_{\mu}, \pi^{+} \pi^{0}\\pi^{+} \pi^{-}, 2 \pi^{0}\\pi^{\pm} e^{\mp} u_{e}, 3 \pi^{0}\\pi^{\pm} \mu^{\mp} u_{\mu}[/latex][latex]\pi^{+} \pi^{-} \pi^{0}[/latex]	0 0 0	0 0 0	1 1 1	0 0 0
Eta	[latex]\eta^{0}[/latex]	Self	548	[latex]5 imes 10^{-19}[/latex]	[latex]2 \gamma, 3 \pi^{0}\\pi^{+} \pi^{-} \pi^{0}[/latex]	0	0	0	0
Charmed D's	[latex]D ^{+}\D ^{0}\D _{S}^{+}[/latex]	[latex]D ^{-} \ \overline{ D }^{0}\ \overline{ D }_{S}^{-}[/latex]	1870 1865 1968	[latex]1.0 imes 10^{-12}\ 4.1 imes 10^{-13}\ 5.0 imes 10^{-13}[/latex]	[latex]e^{+}, K ^{\pm}, K ^{0}[/latex] [latex]\overline{ K }^{0}+ ext { anything }\ ext { Same as } D ^{+}[/latex]	0 0 0	0 0 0	0 0 1	1 1 1
Bottom B's	[latex]B ^{+} \ B ^{0}[/latex]	[latex]B ^{-}\ \overline{ B }^{0}[/latex]	5280 5280	[latex]1.6 imes 10^{-12}\ 1.5 imes 10^{-12}[/latex]	Various	0 0	0 0	0 0	0 0
J/Psi	[latex]J / \psi[/latex]	Self	3097	[latex]10^{-20}[/latex]	Various	0	0	0	0
Upsilon	[latex]\Upsilon( ext { IS })[/latex]	Self	9460	[latex]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
Lambda	Λ	[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	0	0
Sigmas	[latex]\Sigma^{+} \ \Sigma^{0} \ \Sigma^{-}[/latex]	[latex]\bar{\Sigma}^{-}\ \bar{\Sigma}^{0}\ \bar{\Sigma}^{+}[/latex]	1189 1193 1197	[latex]8.0 imes 10^{-11}\ 7.4 imes 10^{-20}\ 1.5 imes 10^{-10}[/latex]	[latex]p \pi^{0}, n \pi^{+}\\Lambda \gamma \ n \pi^{-}[/latex]	[latex]\frac{1}{2}\ \frac{1}{2}\ \frac{1}{2}[/latex]	1 1 1	-1 -1 -1	0 0 0
Xi	[latex]\Xi^{0}\ \Xi^{-}[/latex]	[latex]\bar{\Xi}^{0}\\Xi^{+}[/latex]	1315 1321	[latex]2.9 imes 10^{-10} \ 1.6 imes 10^{-10}[/latex]	[latex]\Lambda \pi^{0}\ \Lambda \pi^{-}[/latex]	[latex]\frac{1}{2}\ \frac{1}{2}[/latex]	1 1	-2 -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{3}{2}[/latex]	1	-3	0
Charmed lambda	[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

Accepted Answer

It's best to consider the conservation laws one at a time. For energy conservation, note that a lambda particle's mass is larger than the total mass of a proton and pion or neutron and pion. Therefore, a lambda particle has enough rest energy for these decays. Charge is conserved in both decays, because the net charge of the proton and [latex]\pi^{-} ext {is } e-e=0 [/latex]. Similarly, the neutron and neutral pion are both uncharged. Baryon number is conserved in each decay, because the lambda has baryon number 1, and so do the proton and neutron. The pions are mesons, with baryon number zero. Thus the net baryon number is 1 on each side of both reactions. None of the particles involved carry charm, so there's no issue with conservation of charm. The lambda particle has strangeness -1, but none of the decay products do. Thus, strangeness is not conserved. This indicates that the weak force is involved in this decay, because strangeness need not be conserved in weak-force reactions.

REFLECT Note that these are the only two likely decay modes for the lambda particle. Baryon conservation dictates that the lambda must decay into another baryon, which must have less mass than the lambda in order to conserve energy. This leaves only the proton and neutron. With the rest energy of the proton or neutron committed, the only mesons light enough to form from the leftover energy are pions.

Question 26.4: Decay of the Particle The lambda (Λ) particle (Table 26.4) i...

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TABLE 26.4 Table of Hadrons
Particle name	Symbol	Anti- particle	Rest energy (MeV)	Mean lifetime (s)	Main decay modes	Spin	Baryon number B	Strangeness number S	Charm number C
Mesons
Pion	$\pi^{-}\\ \pi^{0}$	$\pi^{+}$ Self	140 135	$2.6 \times 10^{-8}\\8.4 \times 10^{-17}$	$\mu^{+} v_{\mu}\\ 2 \gamma$	0 0	0 0	0 0	0 0
Kaon	$K ^{+}\\K _{S}^{0}\\K _{L}^{0}$	$K ^{-}\\\overline{ K }_{S}^{0}\\ \overline{ K }_{L}^{0}$	494 498 498	$1.2 \times 10^{-8}\\9.0 \times 10^{-11}\\ 5.1 \times 10^{-8}$	$\mu^{+} \nu_{\mu}, \pi^{+} \pi^{0}\\\pi^{+} \pi^{-}, 2 \pi^{0}\\\pi^{\pm} e^{\mp} \nu_{e}, 3 \pi^{0}\\\pi^{\pm} \mu^{\mp} \nu_{\mu}$ $\pi^{+} \pi^{-} \pi^{0}$	0 0 0	0 0 0	1 1 1	0 0 0
Eta	$\eta^{0}$	Self	548	$5 \times 10^{-19}$	$2 \gamma, 3 \pi^{0}\\\pi^{+} \pi^{-} \pi^{0}$	0	0	0	0
Charmed D’s	$D ^{+}\\D ^{0}\\D _{S}^{+}$	$D ^{-} \\ \overline{ D }^{0}\\ \overline{ D }_{S}^{-}$	1870 1865 1968	$1.0 \times 10^{-12}\\ 4.1 \times 10^{-13}\\ 5.0 \times 10^{-13}$	$e^{+}, K ^{\pm}, K ^{0}$ $\overline{ K }^{0}+\text { anything }\\\text { Same as } D ^{+}$	0 0 0	0 0 0	0 0 1	1 1 1
Bottom B’s	$B ^{+} \\ B ^{0}$	$B ^{-}\\ \overline{ B }^{0}$	5280 5280	$1.6 \times 10^{-12}\\ 1.5 \times 10^{-12}$	Various	0 0	0 0	0 0	0 0
J/Psi	$J / \psi$	Self	3097	$10^{-20}$	Various	0	0	0	0
Upsilon	$\Upsilon(\text { IS })$	Self	9460	$10^{-20}$	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
Lambda	Λ	$\bar{\Lambda}$	1116	$2.6 \times 10^{-10}$	$p \pi^{-}, n \pi^{0}$	$\frac{1}{2}$	1	0	0
Sigmas	$\Sigma^{+} \\ \Sigma^{0} \\ \Sigma^{-}$	$\bar{\Sigma}^{-}\\ \bar{\Sigma}^{0}\\ \bar{\Sigma}^{+}$	1189 1193 1197	$8.0 \times 10^{-11}\\ 7.4 \times 10^{-20}\\ 1.5 \times 10^{-10}$	$p \pi^{0}, n \pi^{+}\\\Lambda \gamma \\ n \pi^{-}$	$\frac{1}{2}\\ \frac{1}{2}\\ \frac{1}{2}$	1 1 1	-1 -1 -1	0 0 0
Xi	$\Xi^{0}\\ \Xi^{-}$	$\bar{\Xi}^{0}\\\Xi^{+}$	1315 1321	$2.9 \times 10^{-10} \\ 1.6 \times 10^{-10}$	$\Lambda \pi^{0}\\ \Lambda \pi^{-}$	$\frac{1}{2}\\ \frac{1}{2}$	1 1	-2 -2	0
Omega	$\Omega^{-}$	$\Omega^{+}$	1672	$0.82 \times 10^{-10}$	$\Lambda K ^{-}, \Xi^{0} \pi^{-}$	$\frac{3}{2}$	1	-3	0
Charmed lambda	$\Lambda_{C}^{-}$	$\bar{\Lambda}_{C}^{-}$	2286	$2.0 \times 10^{-13}$	Various	$\frac{1}{2}$	1	0	1