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Topic 8 of 8

SU(3) and the Standard Model

$SU(3)$ governs two of the most important symmetries in particle physics: the exact color symmetry of QCD (Quantum Chromodynamics), and the approximate flavor symmetry of the light quarks that led Gell-Mann to predict the $\Omega^-$ baryon before its discovery. With 8 generators, it produces 8 gluons and a rich multiplet structure that organizes all known hadrons.

Key Concepts

SU(3)

SU(3) = \{U \in M_{3\times3}(\mathbb{C}) : U^\dagger U = I,\, \det U = 1\}

. Dimension 8, rank 2. The fundamental representation has dimension 3.

Gell-Mann Matrices

The 8 generators

\lambda^a

(

a=1,\ldots,8

): traceless, Hermitian

3\times3

matrices. Normalized so

\text{Tr}(\lambda^a\lambda^b)=2\delta^{ab}

. The generators are

T^a = \lambda^a/2

Color Charge (QCD)

Quarks carry one of three colors (red, green, blue) — they transform in the fundamental representation

\mathbf{3}

SU(3)_c

. Gluons are in the adjoint

\mathbf{8}

. Only color singlets are observable.

Flavor SU(3)

Approximate symmetry among up, down, strange quarks (

m_u\approx m_d\approx m_s

at low energy). Predicts the hadron octet and decuplet. Led to prediction of the

\Omega^-

baryon.

Color Confinement

Only

SU(3)_c

singlets can exist as free particles.

\mathbf{3}\otimes\bar{\mathbf{3}} = \mathbf{1}\oplus\mathbf{8}

(mesons),

\mathbf{3}\otimes\mathbf{3}\otimes\mathbf{3} = \mathbf{1}\oplus\ldots

(baryons). The singlet

\mathbf{1}

is the physical hadron.

Key Equations

SU(3) Representation Decompositions

\mathbf{3}\otimes\bar{\mathbf{3}} = \mathbf{1}\oplus\mathbf{8}, \qquad \mathbf{3}\otimes\mathbf{3}\otimes\mathbf{3} = \mathbf{1}\oplus\mathbf{8}\oplus\mathbf{8}\oplus\mathbf{10}

Mesons = quark + antiquark; baryons = three quarks. The singlet is the observable hadron.

Standard Model Gauge Group

G_{\text{SM}} = SU(3)_c \times SU(2)_L \times U(1)_Y

Dimensions 8+3+1=12 gauge bosons: 8 gluons,

W^\pm, Z

, and

\gamma

Gell-Mann–Nishijima Formula

Q = T^3 + \frac{Y}{2}

Electric charge from weak isospin

T^3

and hypercharge

Y

Worked Example

Counting Gluons from $SU(3)$

Problem

Gauge bosons of a gauge theory with symmetry group $G$ correspond to generators of $G$ . How many gluons does QCD have, and why?

Solution

QCD has gauge group $SU(3)_c$ . The number of gauge bosons equals $\dim SU(3)$ .

\dim SU(n) = n^2 - 1 \implies \dim SU(3) = 9 - 1 = 8

Equivalently: gluons transform in the adjoint representation of $SU(3)$ , which has dimension 8.

The 8 gluons correspond to the 8 Gell-Mann matrices $\lambda^1,\ldots,\lambda^8$ . They carry color charge and interact with each other — unlike the electrically neutral photon.

Answer There are 8 gluons.

Practice

Exercises

7 problems

1 of 7

How many generators does $SU(3)$ have? (Use $\dim SU(n) = n^2 - 1$ .)

Your answer

2 of 7

In the decomposition $\mathbf{3}\otimes\bar{\mathbf{3}} = \mathbf{1}\oplus\mathbf{8}$ , what is the dimension of the octet?

Your answer

3 of 7

The Casimir eigenvalue of the fundamental representation of $SU(3)$ is $C_2 = 4/3$ . Enter this as a decimal to 3 decimal places.

Unlock Exercise 3