1
$\begingroup$

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

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

and

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

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

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

$\endgroup$
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.