Standard Model decompositions of larger group representations?

In summary: The multiplets in ##(18.15)## aren't a subgroups of ##SU(3)\times SU(2)\times U(1)##?No, they're not.
  • #1
Anchovy
99
2
When reading about GUTs you often come across the 'Standard Model decomposition' of the representations of a given gauge group. ie. you get the Standard Model gauge quantum numbers arranged between some brackets. For example, here are a few SM decompositions of the SU(5) representations [itex]\textbf{5}, \textbf{10}, \textbf{15}[/itex] and [itex]\textbf{24}[/itex].

SU(5)_SM_decompositions.png


So, for instance this is telling us that the representation [itex]\textbf{5}[/itex] will contain fields that are either

([itex]SU(3)_{C}[/itex] triplet, [itex]SU(2)_{L}[/itex] singlet, hypercharge [itex]\tfrac{1}{2}Y = -\tfrac{1}{3}[/itex]) for the (3, 1, [itex]-\tfrac{1}{3}[/itex]),

or

([itex]SU(3)_{C}[/itex] singlet, [itex]SU(2)_{L}[/itex] doublet, hypercharge [itex]\tfrac{1}{2}Y = \tfrac{1}{2}[/itex]) for the (1, 2, [itex]\tfrac{1}{2}[/itex]).

That's straightforward enough. However, I can't seem to find anything online explaining how these have been determined. I can find plenty about how you might go about constructing the [itex]\textbf{10}, \textbf{15}[/itex] and [itex]\textbf{24}[/itex] starting from combinations of the fundamental [itex]\textbf{5}[/itex] by the 'Young's Tableaux' method, but nothing about starting with one of these SU(5) representations and breaking them down. Can anyone explain or link to an explanation?
 
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  • #2
Try chapter 18 of "Lie algebras in particle physics" by Georgi.
 
  • #3
Andrea M. said:
Try chapter 18 of "Lie algebras in particle physics" by Georgi.

OK, found a pdf version, will check it out very soon.
 
  • #4
I'm wondering something about what it says here. Specifically I am trying to understand the motivation for the part that's in the red box. I have also highlighted earlier parts in green boxes that I suspect might be related, but I don't quite understand.

SU(5)_SM_decompositions__Georgi_book.png


So it wants to get the SM decomposition of the [itex]\textbf{5}[/itex] by choosing from constituents of the equation (18.13) that combine to form a 5-dimensional subset. Fair enough. What I don't understand is the [itex]SU(2) \times U(1)[/itex] part. Why must the [itex]\textbf{5}[/itex] incorporate that specifically? Why not require, say, [itex]SU(3) \times SU(2) \times U(1)[/itex]?
 
  • #5
Anchovy said:
Why must the 5\textbf{5} incorporate that specifically? Why not require, say, SU(3)×SU(2)×U(1)SU(3) \times SU(2) \times U(1)?
The multiplets in ##(18.15)## aren't a subgroups of ##SU(3)\times SU(2)\times U(1)##?
 

What is the Standard Model decomposition of larger group representations?

The Standard Model decomposition of larger group representations is a mathematical framework that describes the fundamental particles and interactions in the universe. It breaks down larger group representations into smaller, more manageable representations that can be used to understand the behavior of particles.

Why is the Standard Model decomposition important?

The Standard Model decomposition is important because it provides a unified description of all known particles and their interactions. It has been successfully tested and validated through numerous experiments and is considered the most accurate and comprehensive model of particle physics.

What are the limitations of the Standard Model decomposition?

While the Standard Model decomposition is an incredibly successful theory, it does have limitations. For example, it does not account for gravity and dark matter, and it does not provide a complete understanding of the origins of particle masses. It is also difficult to incorporate into theories of quantum gravity.

How is the Standard Model decomposition used in research?

The Standard Model decomposition is used in research to make predictions about the behavior of particles and their interactions. It is also used to design experiments and analyze data in order to test the validity of the model. Researchers also use the Standard Model decomposition to search for new particles and interactions beyond what is predicted by the model.

What are some current challenges in the study of Standard Model decompositions of larger group representations?

One current challenge in the study of Standard Model decompositions is the search for new physics beyond the Standard Model. While the model has been incredibly successful, there are still unanswered questions and potential limitations that researchers are actively trying to address. Additionally, incorporating the Standard Model into a more comprehensive theory that includes gravity is a major challenge in current research.

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