Mathematical Principles of Theoretical Physics

5.2. QUARK MODEL 269

S= 0 n p

S=− 1

S=− 2

Σ

−

Σ

0

Σ+

Λ

I 3

Ξ

− Ξ

0

Q=− 1 Q= 0 Q= 1

Y = 1

Y= 0

Y =− 1

-1 −^1

2 0

1

2 1

I 3

Figure 5.2: Baryon octet

The second group of hadrons consists of the eight lightest mesons:

(5.2.2) π+,π^0 ,π−,K+,K^0 ,K−,K
0
,η.

In the same fashion as the baryon octet (5.2.1), this group (5.2.2) fits into a hexagonal array
withπ^0 andηat the center; see Figure5.3.
The third group of hadrons consists of ten baryons:

(5.2.3) ∆++,∆+,∆−,∆^0 ,Σ∗+,Σ∗−,Σ∗^0 ,Ξ∗−,Ξ∗^0 ,Ω−,

which are fited into a triangular array as in Figure5.4.
The fourth group of hadrons is given by the eight messons:

(5.2.4) ρ+,ρ−,ρ^0 ,K∗+,K∗−,K∗^0 ,K
∗ 0
,ω,

which are arranged as in Figure5.5.

Remark 5.3.The Eightfold Way based on the patterns given by Figures5.2-5.5provides a
beautiful classification for hadrons. However, the most important point is that the Eightfold
Way and the irreducible representation ofSU(N)also provide key clues to discover the quark
model for the hadron structure.

5.2.2 Irreducible representations ofSU(N)

To better understand the process from the Eightfold Way to the quark model, it is necessary
to know the irreducible representation ofSU(N)and its connection with particle physics. We
proceed in a few steps as follows.