Given $SU(2)$ doublet fermions $\Psi^1$ and $\Psi^2$ and a $SU(2)$ triplet Higgs $H$, how does the correct Yukawa term look like in tensor notation?

Schematically, we have

$$ 2 \otimes 2 \otimes 3 \stackrel{!}{=} 1$$


$$ \Psi^1 \otimes \Psi^2 \otimes H $$

A first guess could be $\Psi_i^1 \Psi_j^2 H_{ij}$, but we have $ 2\otimes 2 = 3 \oplus 1$ and therefore the naive tensor product $\Psi_i \Psi_j = M_{ij}$ is not an element of the $3$-dimensional representation, but belongs to the reducible $ 3 \oplus 1$. This means in abstract notation

$$ \Psi_i^1 \Psi_j^2 H_{ij} \hat = (3 \oplus 1) \otimes 3 \neq 1 $$

Therefore, we must first remove the singlet $1$ from the product $2 \otimes 2$ before we multiply the $\Psi_1 \otimes \Psi_2$ term with the triplet Higgs. In other words we must make sure our product $\Psi_1 \otimes \Psi_2$ yields an element of the $3$, because the only chance to get a singlet is by multiplying the triplet with another triplet $3 \otimes 3 = 1 \oplus ...$.

In tensor notation this last step, as soon as we have formed a $3$ from the two doublets, is ordinary matrix multiplication and taking the trace.

Two guess:

  • Is there some additional tensor $\eta$ involved that makes sure we get an element of the $3$ from two doublets, i.e.

$$ 2\otimes 2= 3 \hat = \Psi^1_i \Psi^2_j \eta_{ijkl} ?$$

  • Or do we simply need to remove the trace

$$ 3 = \Psi^1_i \Psi^2_j - \frac{1}{2} Tr(\Psi^1_i \Psi^2_j) ? $$


The first thing to notice is that composing two spin 1/2 particles (doublets) symmetrically yields spin 1 (a triplet) and anti symmetrically, spin 0, a singlet. So to suppress the singlet component in the Clebsch -Gordan reduction, you just symmetrize. What you are seeking has already been written down in 1964, in the celebrated σ-model of Gell-Mann and Levy, in the context of isospin, for one isodoublet, namely $g\overline{\psi}(\sigma +i\vec{\tau}\cdot\vec{\pi} \gamma_5)\psi$, an SU(2) invariant.

You see that the scalar term is antisymmetric, by fermion interchange, but the pseudoscalar one is symmetric, so isotriplet, dotted onto the isotriplet of π s. So all you need is $i\overline{\Psi}^1 \vec{\tau}\cdot\vec{H} \gamma_5~\Psi^2$. But since I don't know the context of your construction, I cannot seek better γ-matrix implementations suitable for your application: the important thing is that they should result in a symmetric fermion bilinear.


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