Mathematical Principles of Theoretical Physics

286 CHAPTER 5. ELEMENTARY PARTICLES

1) The spins of quarks have to beJ=^12 , as required by

(5.2.69)

spins of mesons: J= 0 (↑↓), J= 1 (⇈), J=− 1 (),

spins of baryons: J=

1

2

(⇈↓), J=−

1

2

(↑),

J=

3

2

(↑↑↑), J=−

3

2

(↓↓↓).

Namely, the unique choice to ensure (5.2.69) is thatJ=^12 for quarks.

2) The success of(p,n,Λ)for describing mesons suggests that for the strange numberS

and the isospin(I,I 3 ),uandp,dandn,sandΛshould be the same respectively. Hence

we have

u:(S,I,I 3 ) = ( 0 ,

1

2

,

1

2

),

d:(S,I,I 3 ) = ( 0 ,

1

2

,−

1

2

),

s:(S,I,I 3 ) = (− 1 , 0 , 0 ).

3) Since the baryon numbers of all baryons areB=1, by the constituents (5.2.68) of

baryons, it is natural that

u:B=

1

3

, d:B=

1

3

, s:B=

1

3

.

4) For all hadrons, the following formula, well known as the Gell-Mann-Nishijima rela-

tion, holds true:

(5.2.70) Q=I 3 +

B

2

+

S

2

.

This relation should also be valid for quarks. Hence, we deduce from (5.2.70) that the

electric charges of quarks are

u:Q=

2

3

, d:Q=−

1

3

, S:Q=−

1

3

.

The data derived in 1)-4) above are collected in Table5.3. According to the quantum
numbers of hadrons and quarks, we can determine the quark constituents of all hadrons.
For example, foruudits quantum numbers are derived from those ofuanddas follows

(5.2.71) uud:B= 1 ,Q= 1 ,S= 0 ,J=

1

2

,I=

1

2

,I 3 =

1

2

,

which dictate thatuudis the proton:
uud=p.

The quark constituents of the main hadrons are listed in Subsection5.1.1.