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In Y.Grossman and Y.Nir "The Standard Model" book in chapter 4 (non abelian symmetrys) they present the law of whom we can have a triplet and singlet out of 2 doublets name them $\phi_a$ and $\phi_b$, under $SU(2)$ symmetry

the law is $2 * 2 = 1 + 3$ and to produce a triplet under $SU(2)$ out of 2 doublets there is another formula: $$(\phi_a\phi_b)_{ij}=(\phi_a)_i(\phi_b)_j-\frac{\delta_{ij}}{2}(\phi_a)_k(\phi_b)_k$$

How do I write it explicity this 3 dimensional vector if for instance $\phi_a=\binom{A}{B}$ and $\phi_b=\binom{C}{D}$, the indices made me very confused.

Daniel Vainshtein
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In a product of two doublets $(A\ B)$ and $(C\ D)$, the components of the triplet are proportional to $AC$, $AD+BC$, and $BD$ (omitting normalization factors). The singlet is $AD-BC$.

Another way to describe this is that the triplet is the symmetric part of the product of the two doublets, and the singlet is the antisymmetric part.

Chiral Anomaly
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