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Two doublets and a triplet

  1. Jun 3, 2012 #1
    Updated: I was reading a paper discussing a Higgs triplet under SU(2)L. In addition to the usual Higgs doublet. Both are written as:

    [tex] D = \begin{pmatrix} x \\ y \end{pmatrix} [/tex]
    [tex] T = \begin{pmatrix} X \\ Y \\ z \end{pmatrix} [/tex]

    Now, at some point, the author says the lagrangian contains a term like,

    [tex]D D T^{\dagger} [/tex]

    [tex] \sim c \begin{pmatrix} x \\ y \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \begin{pmatrix} X & Y & z \end{pmatrix} [/tex]

    In the paper, c is a factor that doesn't carry any group index.

    I have a naive question:

    How can: (2 ×1) (2 ×1) (1×3) be an allowed term? Also, this reminded me of something I heard and didn't understand, which is that two doublets can produce a triplet. But I don't see how.

    Please note that I encounter group theory very little in my studies, hence I find myself constantly confused about some of its techniques.
     
    Last edited: Jun 3, 2012
  2. jcsd
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