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As we know, the $uds$ transforms in fundamental representations of $SU(3)$. It has the antifundamental partner. According to representation theory, $$ \mathbf{3} \otimes \mathbf{\bar{3}}= \mathbf{8} \oplus \mathbf{1} $$ where the $\mathbf{8} $ describes the mesons.

My question is if I am not taking $\mathbf{\bar{3}}$ to do tensor product but uses $$ \mathbf{3} \otimes \mathbf{3}= \mathbf{6} \oplus \mathbf{3} $$ are there any physical fields transforming in $\mathbf{6}$? Any applications of it in particle physics?

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  • $\begingroup$ The symmetric square of the natural rep would amount a combination of two quarks. IIRC there are no such elementary particles. A meson needs a quark and an antiquark, (anti)baryons three of either. $\endgroup$
    – Jyrki Lahtonen
    Commented Aug 13, 2020 at 17:02
  • $\begingroup$ For QCD, gluons are made by one color (rgb) and one anti color (\bar{rgb}). Is there any theory in QCD describing the state with two color? Such as rr? $\endgroup$
    – user39511
    Commented Aug 13, 2020 at 17:07
  • $\begingroup$ Search google for "color sextet". There are dozens of fanciful hypotheticals, none of them experimentally validated. $\endgroup$
    – Cosmas Zachos
    Commented Aug 13, 2020 at 20:05

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