Weight Diagrams of SU(3) Representations

The Fundamental Representations: Quarks and Antiquarks

Let's begin with the simplest non-trivial representations, which serve as the basic building blocks for all others.

The Quark Triple: The (1,0) or 3 Representation

The representation with highest weight label $(k,l)=(1,0)$ is called the fundamental representation, denoted simply as 3. Physically, this three-dimensional representation describes the "flavor" states of the three lightest quarks: up (u), down (d), and strange (s).

Let's first see how to derive the weights directly. In a previous lesson, we considered the standard representation where SU(3) matrices act on vectors in ${\mathbb{C}}^3$. The following video segment calculates the weights for the standard basis vectors.

The video shows that the weights are $(1,0)$, $(0,1)$, and $(-1,-1)$ in the $(k,l)$ basis used there. Now, let's confirm this using the highest weight construction method from our last lesson. The procedure is detailed in the following text.

Interpreting the Diagram: From Abstract Weights to Physical Quantum Numbers

To connect this mathematical diagram to physics, we need to choose our axes appropriately. In the context of the strong interaction and flavor symmetry, the two commuting generators of the su(3) algebra are chosen to correspond to two conserved quantities:

1. Isospin (third component), $I_3$: This quantum number distinguishes the "up" and "down" type quarks. It is proportional to the generator ${\lambda }_3$.
2. Hypercharge, $Y$: This is related to strangeness and baryon number. It is proportional to the generator ${\lambda }_8$.

The following reading explains this choice of basis.

With this physical basis, the weight diagram for the 3 representation is universally plotted with $I_3$ on the horizontal axis and $Y$ on the vertical axis. The three weights of the representation correspond to the three light quarks.

The resulting diagram is an upright triangle, with the quark states having the following quantum numbers:

• Up (u): $(I_3,Y)=(+1/2,+1/3)$
• Down (d): $(I_3,Y)=(-1/2,+1/3)$
• Strange (s): $(I_3,Y)=(0,-2/3)$

The Antiquark Triple: The (0,1) or $\bar{3}$ Representation

The representation with highest weight label $(0,1)$ is the conjugate representation of 3, denoted $\bar{3}$ (3-bar). If the 3 describes quarks, the $\bar{3}$ describes their corresponding antiparticles: the antiquarks.

A conjugate representation $\bar{D}$ is related to the original representation $D$ by complex conjugation. For a Lie algebra representation, this has a simple and profound consequence for the weights.

Because the weights are negated, the weight diagram for the $\bar{3}$ is an inverted version of the diagram for the 3. Its highest weight is ${\mu }_2$, and its lowest weight is $-{\mu }_1$. The three states correspond to the anti-up ($\bar{u}$), anti-down ($\bar{d}$), and anti-strange ($\bar{s}$) quarks, with their quantum numbers being the negative of their particle counterparts.

• Anti-down ($\bar{d}$): $(I_3,Y)=(+1/2,-1/3)$
• Anti-up ($\bar{u}$): $(I_3,Y)=(-1/2,-1/3)$
• Anti-strange ($\bar{s}$): $(I_3,Y)=(0,+2/3)$

The diagrams for the 3 and $\bar{3}$ representations from the lecture notes LINK perfectly illustrate this relationship.

The Adjoint Representation: The Meson Octet or Gluon Octet

The representation with highest weight label $(1,1)$ is the adjoint representation, denoted 8. This eight-dimensional representation is special because its vector space is the Lie algebra su(3) itself. It describes either the eight gauge bosons of the strong force (the gluons) or the octet of light mesons, which are quark-antiquark bound states.

First, let's formally define the adjoint representation.

The weights of the adjoint representation are called roots, and the weight diagram is called the root diagram. The non-zero weights are precisely the six roots of the su(3) algebra, which we have seen before. The video below provides a detailed construction of this weight diagram.

The final diagram looks like this:

The fact that the central weight space has dimension 2 is a non-trivial result. It follows from the "shell-stripping" algorithm we learned in the last lesson (the outer hexagon has multiplicity 1; stripping it leaves the origin, whose multiplicity becomes $1+1=2$). A formal proof is also instructive.

Physically, this octet diagram famously classifies the light pseudoscalar mesons. The two states at the center with $(I_3,Y)=(0,0)$ are the ${\pi }^0$ meson and the $\eta$ meson. This octet structure arises naturally when we combine a quark and an antiquark, as we will see when we study tensor products.

Conclusion

In this lesson, we have constructed and interpreted the weight diagrams for the three most important low-dimensional representations of SU(3). We have seen how abstract mathematical structures, derived systematically from highest weight theory, map directly onto the fundamental particles and composites of flavor SU(3) symmetry.

Key Takeaways:

• The 3 representation, or $(1,0)$, has a triangular weight diagram and describes the u, d, and s quarks.
• The $\bar{3}$ representation, or $(0,1)$, is the conjugate of the 3. Its weight diagram is an inverted triangle, and it describes the corresponding antiquarks.
• The 8 representation, or $(1,1)$, is the adjoint representation. Its weight diagram is a hexagon with a two-dimensional center. It describes the gluon octet or the light meson octet.
• The axes of these diagrams are chosen to represent physical conserved quantities: the third component of isospin ($I_3$) and hypercharge ($Y$).

We have now established a solid foundation in the structure of su(3) irreps. In our upcoming lessons, we will build upon this by introducing more general and compact ways to label these representations, such as Dynkin labels, and then explore the crucial technique for combining them: the tensor product, which we will analyze using tools like Young Tableaux. This will allow us to understand how composite particles like mesons ($q\bar{q}$) and baryons ($qqq$) are formed.