Mathematical Principles of Theoretical Physics

5.2. QUARK MODEL 283

Then we can compute that

m= 1 for

m= 15 for

m= 4 for

m= 20 for and

Hence we deduce that

(5.2.61)

4 ⊗ 4 = 1 ⊕ 15 ,

4 ⊗ 4 ⊗ 4 = 4 ⊕ 20 ⊕ 20 ⊕ 20.

5.2.5 Sakata model of hadrons

In 1950’s, many hadrons were discovered, leading to many attempts to investigate the deeper
hadron structure. Based on the irreducible representation(5.2.60) ofSU( 3 ), i.e..

(5.2.62) 3 ⊗ 3 = 8 ⊕ 1 ,

Sakata presented a model for hadron structure, called the Sakata model. This was an early
precursor to the quark model, and also resulted in the physical implications of irreducible
representations as stated by Physical Explanation5.5.
Sakata model proposed three particles

(5.2.63) p, n, Λ

as the fundamental particles for all strong interacting particles. In his scheme, Sakata used the
three particles in (5.2.63) as a basis ofSU( 3 ), such that each hadron consists of a fundamental
particle and an antiparticle as

(5.2.64) hadron=SiSj for 1≤i,j≤ 3 ,

where(S 1 ,S 2 ,S 3 ) = (p,n,Λ)are called the sakataons.
In (5.2.62), the left-hand side represents the pairsSiSj, and the right-hand side represents
the following eight mesons:

(5.2.65) π+,π−,π^0 ,K+,K−,K^0 ,K^0 ,η.

It is a coincidence that the eight particles just constitutean eight multiple state of hadrons,
and can be illustrated by the Eightfold Way; see Figure5.3and Figure5.6.