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Standard Model decompositions of larger group representations?

  1. Sep 29, 2015 #1
    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?
     
  2. jcsd
  3. Sep 29, 2015 #2
    Try chapter 18 of "Lie algebras in particle physics" by Georgi.
     
  4. Sep 29, 2015 #3
    OK, found a pdf version, will check it out very soon.
     
  5. Sep 29, 2015 #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]?
     
  6. Sep 30, 2015 #5
    The multiplets in ##(18.15)## aren't a subgroups of ##SU(3)\times SU(2)\times U(1)##?
     
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