Mathematical Principles of Theoretical Physics

5.2. QUARK MODEL 285

Table 5.7

mesons I I 3 Q S Jπ m(MeV)

pn π+ 1 1 1 0 0 − ∼ 140

−pn π− 1 -1 -1 0 0 − ∼ 140

nn√−pp

2 π

0 1 0 0 0 0 − ∼ 140

pΛ K+^1212110 − ∼ 495

nΛ K^012 −^12010 − ∼ 495

Λn K

(^01)
2
1
2 0 -1^0

− ∼ 495

Λp K−^12 −^12 -1 -1 0 − ∼ 495

nn+p√p− 2 ΛΛ

6 η^00000

− ∼ 548

nn+p√p+ΛΛ

3 η

′ 0 0 0 0 0 − ∼ 548

In view of the irreducible representations ofSU( 3 ):

3 ⊗ 3 = 8 ⊕ 1 ,

3 ⊗ 3 ⊗ 3 = 10 ⊕ 8 ⊕ 8 ⊕ 1 ,

it is natural to guess the relations (5.2.18) or (5.2.19), i.e.

(5.2.66)

mesons= 3 ⊗ 3 = 8 ⊕ 1 ,

baryons= 3 ⊗ 3 ⊗ 3 = 10 ⊕ 8 ⊕ 8 ⊕ 1.

According to the Physical Explanation5.5, we infer immediately, from (5.2.66), that the
hadrons in the Eightfold Way are composed of three fundamental particles, denoted by

(5.2.67) q 1 , q 2 , q 3

and mesons consist of a fundamental particle and an antiparticle. In a nutshell, baryons
consist of three fundamental particles:

(5.2.68)

mesons=qiqj for 1≤i,j≤ 3 ,

baryons=qiqjqk for 1≤i,j,k≤ 3.

Physicists Gell-Mann and Zweig did this work independentlyin 1964, and presented the
celebrated Quark Model. The three fundamental particles (5.2.67) were termed the up, down
and strange quarks by Gell-Mann, denoted by

q 1 =u, q 2 =d, q 3 =s.

Zweig called these particles the aces.
By using the three quarks(u,d,s)to replace the sakataons(p,n,Λ), we can perfectly
explain all hadrons described by the Eightfold Way, i.e. thehadrons given by (5.2.1)-(5.2.4).
An important step to establish the quark model is to determine the quantum numbers of
quarks, which are introduced in the following several procedures: