= Solution
Count <strangeness> before drawing a <weak interaction> diagram. An <Omega baryon> has $S=-3$, a <Lambda baryon> has $S=-1$, a negatively charged <kaon> has $S=-1$, and a <pion> has $S=0$. Thus the two <nonleptonic baryon decays> require
$$
\Delta S(\Lambda^0K^-)=+1,\qquad \Delta S(\Lambda^0\pi^-)=+2.
$$
The <strong interaction> cannot supply either net change of <strangeness>.
\Image[/past-exam-of-the-particle-and-nuclear-physics-course-of-the-university-of-cambridge/2019-omega-weak-diagrams.png]
{title=One-W and two-W quark diagrams for the two Omega decay channels}
In the upper <Feynman diagram>, one initial <strange quark> undergoes the <weak charged current> transition
$$
s\longrightarrow u+W^-,\qquad W^-\longrightarrow d+\bar u.
$$
The remaining two <strange quarks> are <spectator quarks>. Combine the outgoing $u,d$ and one spectator $s$ into the <Lambda baryon>, and the remaining $s$ with $\bar u$ into the <kaon>. This uses one <W boson> propagator and two weak <interaction vertices>. Its <Cabibbo-Kobayashi-Maskawa matrix> factor has magnitude $|V_{us}V_{ud}|\simeq\sin\theta_C\cos\theta_C$.
For the lower <Feynman diagram>, let the upper initial $s$ first emit a $W^-$ and become a virtual $u$, then emit a $W^+$ and become a $d$. The $W^+$ is absorbed by a second initial $s$, turning it into a $u$; the third initial $s$ remains a <spectator quark>. The emitted $W^-$ produces $d\bar u$. The complete <quark flavour> flow is therefore
$$
\begin{aligned}
s_1&\longrightarrow u^*+W^-,&u^*&\longrightarrow d+W^+,\\
s_2+W^+&\longrightarrow u,&W^-&\longrightarrow d+\bar u,\\
s_3&\longrightarrow s.
\end{aligned}
$$
The final $d,u,s$ form the <Lambda baryon> and the other $d\bar u$ form the <pion>. <Electric charge conservation> holds at every vertex. Internal lines are virtual; this is a contribution to one <decay amplitude>, not a sequence of freely propagating on-shell constituent-quark decays.
This is a possible lowest-weak-order diagram for a <second-order strangeness-changing weak decay>: it has two <W boson> propagators and four weak <interaction vertices>. Its <Cabibbo-Kobayashi-Maskawa matrix> factor has magnitude $|V_{us}|^2|V_{ud}|^2$ in the displayed light-quark topology. Other contributions can exist, but a single charged-current insertion cannot turn two net strange quarks into nonstrange ones. No extra <loop Feynman diagram> is compulsory: the displayed two-W tree already realizes the required <quark flavour> change.
At low <energies>, one-W exchange produces one power of the <Fermi constant> in an effective <decay amplitude>; the second process requires an additional weak insertion. Hadronic <matrix elements> and propagators supply the accompanying dimensions, so a numerical ratio cannot be obtained by simply squaring a dimensionful constant. The robust comparison is
$$
\boxed{\Gamma(\Omega^-\to\Lambda^0\pi^-)\ll\Gamma(\Omega^-\to\Lambda^0K^-).}
$$
The dominant reason is the extra weak order, with additional <Cabibbo suppression> in this representative topology. It is not a comparison between a strong decay and a weak decay.
Back to article page
