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An answer about former question about QCD interactions

  1. Aug 26, 2015 #1
    Hi all,

    In former threads, namely:

    "Color factors of color -- octet scalars",
    and

    "The double line notation and the adjoint representation"

    I were asking about the difference between the interaction vertices among:

    * three gluons (GGG), as in SM,
    * three colored octet scalars, call ## S = S^A T^A ##, where ## T^A ## are the ## SU_C(3)## generators, and A=1,..,8,
    * GSS.


    Let me summarize the answer here:

    In SM, the interactions between three gluons comes from the Lagrangian term;
    $$ \mathcal{L} = F_{\mu \nu} F^{\mu\nu}, $$
    where ## F_{\mu \nu} ## is the YM field strength, given by
    $$ F_{\mu\nu} = \partial_{[\mu} A_{\nu]} + i g [ A_\mu, A_\nu], $$
    So that GGG vertex comes from ## ig \text{Tr}(\partial_\mu A_\nu [ A^\mu, A^\nu]) \sim \text{Tr}(T_A [T_B,T_C]) \sim g f_{ABC} . ##


    The interactions between three coloured octet fields (i.e. in the adjoint representation as gluons), is given by the interaction term:
    \begin{equation*}
    \begin{split}
    \textrm{Tr} (S^A S^B S^C) &= \text{Tr} (T^A T^B T^C) S^A S^B S^C
    \\& = \frac{1}{4} (d^{ABC} + i f^{ABC} ) S^A S^B S^C \sim d^{ABC}~~ S^A S^B S^C .
    \end{split}
    \end{equation*}
    The term includes ## f^{ABC} ## has vanished because ## f^{ABC} ## is a totally symmetric tensor times a symmetric product.


    The interactions between gluons and octet scalars come from the covariant derivative:
    $$ \mathcal{L}_S = D^\mu S^\dagger D_\mu S, $$
    \begin{equation*}
    \begin{split}
    ( D_\mu S) ^A & = \partial_\mu S^A - i g A_{\mu B} (T_B)^{AC} S^C
    \\ &= \partial_\mu S^A + g A_{\mu B} f_{ABC} S^C.
    \end{split}
    \end{equation*}
    So that ## G_A S_B S_C ## vertex ## \sim f_{ABC} ##.
    Where in the adjoint representation ## (T_{adj}^a) = -i f^{abc}##


    A useful reference for that is of course:

    An Introduction To Quantum Field Theory (Frontiers in Physics) (Michael E. Peskin, Dan V. Schroeder), Ch:15,

    Hopefully that's useful for you and thanks for the advisors who helped me, fzero & samalkaiat

    :)
    S


    [/PLAIN] [Broken]
     
    Last edited by a moderator: May 7, 2017
  2. jcsd
  3. Aug 31, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
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