You have entered a graduate program in particle physics and are learning about
the use of symmetry. You begin by repeating the analysis that led to the
prediction of the \(\Omega$$^-\) particle. Nine of the spin \(-\frac{3}{2}\)
baryons are four \(\Delta\) particles, each with mass 1232 \(MeV/c^2\),
strangeness 0, and charges \(+2e\), \(+e\), 0, and \(-e\); three \(\Sigma^*\)
particles, each with mass 1385 \(MeV/c^2\), strangeness -1, and charges \(+e\), 0,
and \(-e\); and two \(\Xi^*\) particles, each with mass 1530 \(MeV/c^2\),
strangeness -2, and charges 0 and \(-e\). (a) Place these particles on a plot of
\(S\) versus \(Q\). Deduce the \(Q\) and \(S\) values of the tenth spin \(-\frac{3}{2}\)
baryon, the \(\Omega^-\) particle, and place it on your diagram. Also label the
particles with their masses. The mass of the \(\Omega^-\) is 1672 \(MeV/c^2\); is
this value consistent with your diagram? (b) Deduce the three-quark
combinations (of \(u\), \(d\), and \(s\)) that make up each of these ten particles.
Redraw the plot of \(S\) versus \(Q\) from part (a) with each particle labeled by
its quark content. What regularities do you see?