Calculating Quantum Numbers in Two Decays

In part (a), the conservation laws can be examined by adding the quantum numbers of the decay products and comparing them with the parent particle. In part (b), the same procedure can reveal if a conservation law is broken or not.

Before the decay, the ${\text{Ξ}}^{-}$ has strangeness $S=-2$. After the decay, the total strangeness is –1 for the ${\text{Λ}}^0$, plus 0 for the ${\pi }^{-}$. Thus, total strangeness has gone from –2 to –1 or a change of +1. Baryon number for the ${\text{Ξ}}^{-}$ is $B=\text{+}1$ before the decay, and after the decay the ${\text{Λ}}^0$ has $B=\text{+}1$ and the ${\pi }^{-}$ has $B=0$ so that the total baryon number remains +1. Charge is –1 before the decay, and the total charge after is also $0-1=-1$. Lepton numbers for all the particles are zero, and so lepton numbers are conserved.

The ${\text{Ξ}}^{-}$ decay is caused by the weak interaction, since strangeness changes, and it is consistent with the relatively long $1\text{.}\text{64}\times {\text{10}}^{-\text{10}}\text{-s}$ lifetime of the ${\text{Ξ}}^{-}$.

The decay $K^{+}\to {\mu }^{+}+{\nu }_{\mu }$ is allowed if charge, baryon number, mass-energy, and lepton numbers are conserved. Strangeness can change due to the weak interaction. Charge is conserved as $s\to d$. Baryon number is conserved, since all particles have $B=0$. Mass-energy is conserved in the sense that the $K^{+}$ has a greater mass than the products, so that the decay can be spontaneous. Lepton family numbers are conserved at 0 for the electron and tau family for all particles. The muon family number is $L_{\mu }=0$ before and $L_{\mu }=-1+1=0$ after. Strangeness changes from +1 before to 0 + 0 after, for an allowed change of 1. The decay is allowed by all these measures.

This decay is not only allowed by our reckoning, it is, in fact, the primary decay mode of the $K^{+}$ meson and is caused by the weak force, consistent with the long $1\text{.}\text{24}\times {\text{10}}^{-8}\text{-s}$ lifetime.