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I am trying to do a Higgs triplet extension of the standard model but I don't know how to find the corresponding vacuum expectation value for the Higgs triplet. I read a paper where they found the expectation value for doublet to be $$<\phi>_0 =\frac{1}{\sqrt{2}}\Big{(}\begin{matrix}0\\v\end{matrix}\Big{)}$$and the triplet expectation value is found to be $$<\Delta>_0\Big{(}\begin{matrix}0&0\\v_T& 0\end{matrix}\Big{)}$$

but I was expecting a 3x1 matrix for the VEV of the Higgs triplet, why is it a 2x2 matrix

Qmechanic
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The "triplet" representation is the adjoint representation of the $\mathfrak{su}(2)$-algebra, i.e. the representation of the algebra upon itself through the commutator. Although the algebra is three-dimensional, it is customary to identify it with the traceless Hermitian algebra of 2x2 matrices spanned by the Pauli matrices. It's still three-dimensional since the conditions of being Hermitian and traceless eliminate the additional degrees of freedom a 2x2 matrix has.

ACuriousMind
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