Problem 55
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?
Problem 58
Suppose that positron-electron annihilations occur on the line 3 cm from the center of the line connecting two detectors. Will the resultant photons be counted as having arrived at these detectors simultaneously? (a) No, because the time difference between their arrivals is 100 ms; (b) no, because the time difference is 200 ms; (c) yes, because the time difference is 0.1 ns; (d) yes, because the time difference is 0.2 ns.
