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## Q. 14.9

The primary decay of the $\Xi^{-} \text {is } \Xi^{-} \rightarrow \Lambda^{0}+\pi^{-}$. The two decay products in turn decay. Find the first decay listed in Table 14.4 for both the $\Lambda^{0} \text { and } \pi^{-}$ 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 $\Lambda^{0} \rightarrow p+\pi^{-} \text {and } \pi^{-} \rightarrow \mu^{+}+\nu_{\mu}$. We use Table 14.6 to find the quark composition. The quark structures of the antiparticles are just the corresponding antiquarks of the particles.

## Verified Solution

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 $W ^{\pm}$ gauge bosons.

$\Xi^{-} \rightarrow \Lambda^{0}+\pi^{-} \quad d s s \rightarrow u d s+\bar{u} d$

A s quark is changed to a d quark, and an $\bar{u} u$ pair is created.

$\Lambda^{0} \rightarrow p+\pi^{-} \quad u d s \rightarrow u u d+\bar{u} d$

Again a s quark is changed to a d quark, and an $\bar{u} u$ pair is created.

$\pi^{-} \rightarrow \mu^{+}+\nu_{\mu} \quad \bar{u} d \rightarrow \text { no quarks }$

The d quark can change to an u quark and annihilate the existing $\bar{u}$ quark.

 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)

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