Mathematical Principles of Theoretical Physics

5.2 Quark Model

Table 5.4: Baryons

Baryons J m Qe Qw Qs B I I 3 S Y τ

p 1 / 2 938.3 1 9 3 1 1 / 2 1 / 2 0 1 ∞

n 1 / 2 939.6 0 9 3 1 1 / 2 − 1 / 2 0 1 885.7

Λ 1 / 2 1115.7 0 9 3 1 0 0 -1 0 2. 7 × 10 −^10

Σ+ 1 / 2 1189.4 1 9 3 1 1 1 -1 0 8 × 10 −^11

Σ− 1 / 2 1197.5 -1 9 3 1 1 0 -1 0 1. 5 × 10 −^10

Σ^01 / 2 1192.6 0 9 3 1 1 -1 -1 0 7. 4 × 10 −^20

Ξ− 1 / 2 1321.3 -1 9 3 1 1 / 2 1 / 2 -2 -1 1. 6 × 10 −^10

Ξ^01 / 2 1314.8 0 9 3 1 1 / 2 − 1 / 2 -2 -1 2. 9 × 10 −^10

∆++ 3 / 2 1230 2 9 3 1 3 / 2 3 / 2 0 1 5. 6 × 10 −^24

∆+ 3 / 2 1231 1 9 3 1 3 / 2 1 / 2 0 1 5. 6 × 10 −^24

∆− 3 / 2 1234 -1 9 3 1 3 / 2 − 1 / 2 0 1 5. 6 × 10 −^24

∆^03 / 2 1232 0 9 3 1 3 / 2 − 3 / 2 0 1 5. 6 × 10 −^24

Σ∗+ 3 / 2 1383 1 9 3 1 1 1 -1 0 1. 8 × 10 −^23

Σ∗− 3 / 2 1387 -1 9 3 1 1 0 -1 0 1. 8 × 10 −^23

Σ∗^03 / 2 1384 0 9 3 1 1 -1 -1 0 1. 8 × 10 −^23

Ξ∗− 3 / 2 1535 -1 9 3 1 1 / 2 1 / 2 -2 -1 6. 9 × 10 −^23

Ξ∗^03 / 2 1532 0 9 3 1 1 / 2 − 1 / 2 -2 -1 6. 9 × 10 −^23

Ω− 3 / 2 1672 -1 9 3 1 0 0 -3 -2 8. 2 × 10 −^11

5.2 Quark Model

5.2.1 Eightfold way

We see from Tables5.4and5.5that there are many hadrons, and in fact their number arrives
at hundreds. The abundance of hadrons implies that there must be some rules to classify them
into groups based on certain common properties. The Eightfold Way can be regarded as a
successful classification for hadrons. In fact, the Eightfold Way, together with the irreducible
representations ofSU( 3 ), has played a navigation role for the introduction of the quark model.
The Eightfold Way was introduced by Gell-Mann and Ne’eman independently in 1961.
This scheme arranges the baryons and mesons into certain geometric patterns, according
to their charge, strangeness, hypercharge and isospinI 3. These geometric patterns include
hexagons and triangles, and all particles put in each diagram are considered as a class.
The first group of hadrons consists of the eight lightest baryons:

(5.2.1) n,p,Σ−,Σ^0 ,Σ+,Λ,Ξ−,Ξ^0 ,

which fit into a hexagonal array with two particlesΣ^0 andΛat the center; see Figure5.2.
The eight baryons (5.2.1) withJ= 1 /2 are known as the baryon octet. Note that particles
in Figure5.2range with charges lie along the downward-sloping diagonalline:Q=1 forp
andΣ+,Q=0 forn,Λ,Σ^0 andΞ^0 , andQ=−1 forΣ−andΞ−. Horizontal lines associate
particles of the same strangenessS=0 and hyper-chargeY=1 for the neutron and proton,
S=−1 andY=0 forΣ−,Σ^0 ,ΛandΣ+,S=−2 andY=−1 forΞ−andΞ^0. The third
componentI 3 of isospin indicates particles in column.