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Your terminology is hard to comprehend. My understanding regarding your question goes as follow. What we know is group $U(1)$ isomorphic to $SO(2)$-rotation in a 2D plane. On the other hand the Lie algebra of $SU(2)$ is same as of $SO(3)$. Which means $SU(2)$ is isomorphic (locally they have same Lie algebra) to $SO(3)$. One can write \begin{equation} ...


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Yes, of course, the "diagonal $U(1)$ symmetry inside $SU(2)$" just refers to the group of matrices that are diagonal. The $2\times 2$ matrices that are diagonal are ${\rm diag}(a,b)$. Their belonging to $SU(2)$ means that $|a|^2=|b|^2=1$ – from the unitarity – and $ab=1$, from the special condition (unit determinant). So the diagonal $SU(2)$ matrices are ...


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I gather your unfortunate edit, which further confused things, to the point of intractability, came from verbatim reproduction of eqn (3.23) of the Duff et al paper on Stochastic Local Operations and Classical Communication. One of about a dozen problems with your malformed question is that you used the wrong coset space to be described by the Cartan pair ...


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Basically the reason is that the classical conformal symmetry no longer holds at the quantum level due to the presence of the trace anomaly. More precisely, the tracelessness of the quantum stress-energy tensor is incompatible with the normal ordering needed to define it. By cohomological reasons, the trace of the stress-energy tensor, although ...


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Apologies for evincing magisterial cluelessness about what your diagrams represent and what you'd want to achieve, but I'd array the standard facts on tetraquarks avoiding Young diagrams, although they are self evident in the Dynkin labelling, which I also give, next to the tensor labelling. They may be useful to what you appear to be after--but I can't ...



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