The Eightfold Way — Question 5

Question 5

The missing corner of a decuplet predicts three additive numbers

A spin-3/2 baryon decuplet has rows of isospin 3/2, 1, 1/2 and 0 at Y=1,0,-1,-2. All states have B=1. Suppose only the last row is unobserved.

1. Count the states and infer S and Q of the missing member.

2. Distinguish the missing state total isospin from its spatial spin and give each multiplicity.

Question 5 - Solution

The missing corner of a decuplet predicts three additive numbers

(a) Method and calculation

$$\boxed{N=4+3+2+1=10}.$$$$\boxed{S=Y-B=-3,\quad Q/e=0-2/2=-1}.$$

(b) Interpretation and further calculation

$$\boxed{I=0:\ 2I+1=1\ \text{flavor state}}.$$$$\boxed{J=3/2:\ 2J+1=4\ \text{spin projections}}.$$

Physical check

The predicted corner has the quantum numbers of the Omega minus. Four spin projections do not create four separate weights in this diagram.