# Why does D(1,1) representation of SU(3) give baryon octet?

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• joneall
In summary, the discussion is about the baryon octet and its relation to the D(1,1) representation of SU(3). The speaker references Greiner's book on QM and symmetries and explains that in the SU(3)-flavor model, the direct product of the three fundamental representations can be decomposed into irreducible parts. The speaker also mentions that the rules for this decomposition can be found in Sakurai and Lipkin's books, as well as in a detailed review by the Particle Data Group. The speaker then discusses the decomposition for mesons and mentions that there is a better way of constructing the baryon octet, but does not elaborate on it due to lack of access to English books
joneall
Gold Member
TL;DR Summary
D(1,1) means one quark and one antiquark, which corresponds perfectly to mesons. But how can it explain baryons?
The question may be ambiguous but it's really simple. One says that the baryon octet is the D(1,1) representation of SU(3), but then uses the same one for mesons. D(1,1) means one quark and one antiquark, which corresponds perfectly to mesons. But how can it explain baryons?

My information and notation comes from Greiner's excellent, though old, book on QM and symmetries.

Baryons are three-quark states. In the SU(3)-flavor model ("eightfold way") you decompose the direct product of the three fundamental representations into irreducible parts. The rules are nicely explained in Sakurai, Modern Quantum Mechanics as well as in Lipkin, Lie groups for pedestrians. There's also a very detailed review by the Particle Data Group:

https://pdg.lbl.gov/2021/reviews/rpp2020-rev-quark-model.pdf

You get ##3 \otimes 3 \otimes 3=10 \oplus 8 \oplus 8 \oplus 1##, i.e., a decuplet, two octets, and a singulet.

Mesons are quark-antiquark states. Here the decomposition is of the fundamental and the conjugate complex fundamental representation (which are not equivalent 3D irreducible representations for SU(3)): ##3 \otimes \bar{3}=8 \oplus 1## (octet and singulet).

Thanks, but that does not answer my question. Greiner uses 1 quark and 1 antiquark to construct the 10-d SU(3) octet, then uses it for mesons and baryons. Is there a better way of constructing the baryon octet? Preferably without my purchasing yet one more QM book. (I don't have access to a library of books in English.)

## 1. Why is the D(1,1) representation used for SU(3) baryon octet?

The D(1,1) representation is used for SU(3) baryon octet because it is the simplest representation that can accommodate three quarks, which are the building blocks of baryons. This representation also satisfies the symmetry requirements of SU(3), making it a suitable choice for describing baryon properties.

## 2. How does the D(1,1) representation relate to the baryon octet?

The D(1,1) representation is a mathematical representation of the baryon octet, which consists of eight baryons: the proton, neutron, lambda, sigma, xi, and omega particles. Each of these baryons is described by a combination of three quarks in the D(1,1) representation.

## 3. What is the significance of the baryon octet in particle physics?

The baryon octet is significant in particle physics because it represents the fundamental building blocks of matter. Baryons, which are made up of three quarks, are the building blocks of the visible universe. Understanding the properties and interactions of baryons is crucial in understanding the nature of matter.

## 4. How does the D(1,1) representation explain the properties of baryon octet particles?

The D(1,1) representation provides a framework for understanding the properties of baryon octet particles. The three quarks in the representation have specific charges and spin values, which determine the overall charge and spin of the baryon. The representation also allows for the prediction of other properties, such as mass and decay modes, based on the symmetry requirements of SU(3).

## 5. Are there other representations of SU(3) that can describe the baryon octet?

Yes, there are other representations of SU(3) that can describe the baryon octet, such as the D(1,2) and D(2,1) representations. However, the D(1,1) representation is the simplest and most commonly used representation for describing baryons. Other representations may be used to describe more complex baryon systems or to study specific properties of baryons.

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