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

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Discussion Overview

The discussion centers on the representation of baryons and mesons within the framework of SU(3) symmetry, specifically addressing the D(1,1) representation and its implications for understanding the baryon octet. Participants explore the theoretical underpinnings and the decomposition of representations relevant to the quark model.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions the application of the D(1,1) representation of SU(3) to both baryons and mesons, noting that D(1,1) corresponds to one quark and one antiquark, which seems to conflict with the three-quark nature of baryons.
  • Another participant explains that baryons are constructed from three-quark states, and provides the decomposition of the direct product of three fundamental representations, resulting in a decuplet, two octets, and a singlet.
  • A participant expresses a desire for an alternative method to construct the baryon octet without needing additional resources, indicating a preference for clarity in the representation without further literature.
  • One participant shares a link to a detailed review by the Particle Data Group, suggesting it as a resource for understanding the quark model and its representations.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the construction of the baryon octet from the D(1,1) representation, and multiple viewpoints regarding the representation of baryons and mesons remain evident.

Contextual Notes

The discussion highlights the complexity of SU(3) representations and their applications, with some assumptions about the nature of quark states and their combinations remaining implicit. There is also a noted lack of access to certain resources, which may limit the depth of exploration for some participants.

Who May Find This Useful

This discussion may be useful for those interested in particle physics, specifically in understanding the quark model and SU(3) symmetry, as well as for individuals exploring the theoretical frameworks behind baryon and meson classifications.

joneall
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TL;DR
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.
 
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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.)
 

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