The Eightfold Way — Question 2

Question 2

A shared neutral weight does not identify a unique baryon

The Sigma triplet has I=1, B=1, S=-1. The Lambda has I=0 with the same B and S. Use $Q/e=I_3+(B+S)/2$.

1. Find all Sigma charges and the Lambda charge.

2. Explain why a point at $I_3=Y=0$ contains two distinct octet states.

Question 2 - Solution

A shared neutral weight does not identify a unique baryon

(a) Method and calculation

$$\boxed{Y=1-1=0,\quad I_3(\Sigma)=-1,0,+1}.$$$$\boxed{Q(\Sigma)=-e,0,+e;\quad Q(\Lambda)=0}.$$

(b) Interpretation and further calculation

$$I^2|\Sigma^0\rangle=1(1+1)|\Sigma^0\rangle,\quad I^2|\Lambda\rangle=0.$$$$\boxed{(I_3,Y)=(0,0)\text{ has multiplicity }2}.$$

Physical check

Here I is dimensionless internal isospin. The distinct eigenvalues of its squared operator distinguish the neutral states.