Charge operator applied to matrix multiplets

[lalo_u]

In the context of SM ($SU(3)_C\otimes SU(2)_L\otimes U(1)_Y$) the charge operator is $Q_{SM} = T_3 + \frac{Y}{2}\mathbb{I}_2$ and gives us the fermions charges. Here $T_3=\frac{1}{2}\sigma_3$ is the third $SU(2)$ generator.
For example, assuming $Y=-1$ for the left lepton doublet, $Q_{SM}\Psi=\begin{pmatrix}0&0\\0&-1\end{pmatrix}\begin{pmatrix}\nu_L \\ e_L\end{pmatrix}=\begin{pmatrix}0 \nu_L \\ -1 e_L\end{pmatrix}$ and the charge components are obtained.
This process works fine for the scalar doublet too, $Q_{SM}\begin{pmatrix}\phi^+\\\phi^0\end{pmatrix}=\begin{pmatrix}+1 \phi^+\\0 \phi^0\end{pmatrix}$, with $Y=+1$.
On the other hand, the same calculation can be applied for $SU(3)_C\otimes SU(3)_L\otimes U(1)_X$ extension in which the charge operator is $Q_{331} = T_3-\sqrt{3}T_8+X\mathbb{I}_3$ where $T_3=\frac{1}{2}\lambda_3$ and $T_8=\frac{1}{2}\lambda_8$ are the $SU(3)$ diagonal generators.
For example, $Q_{331}\Psi_1=\begin{pmatrix}0&0&0\\0&-1&0\\0&0&1\end{pmatrix}\begin{pmatrix}\nu_L \\ e_L \\e_L^c\end{pmatrix}=\begin{pmatrix}0 \nu_L \\ -1 e_L \\ +1 e_L^c\end{pmatrix}$ and the charge components are obtained again with $X=0$.
In the same way as before, we can do this with the scalar multiplets. For example with one of the triplets called $\eta$, $Q_{331}\eta=\begin{pmatrix}1&0&0\\0&0&0\\0&0&2\end{pmatrix}\begin{pmatrix}\eta^+ \\ \eta^0 \\ \eta^{++}\end{pmatrix}=\begin{pmatrix}+1 \eta^+ \\ 0 \eta^0 \\ +2 \eta^{++}\end{pmatrix}$ and the charge components are obtained again with $X=+1$.
The question is, ¿what about the matrix scalar multiplets?, for example a 331 sextet is defined as $S=\begin{pmatrix} \sigma_1^0&h_2^-&h_1^+ \\ h_2^- & H_1^{--} & \sigma_2^0 \\ h_1^+ & \sigma_2^0 & H_2^{++}\end{pmatrix},$ with hypercharge $X=0$.
When you try to do the same process, but with $Q_{331}^{\dagger}SQ_{331}$ for being $S$ a matrix and not a vector, i can't obtain the charges. Here $Q_{331}$ is the same as for $\Psi_1$.
¿Any idea?

 

[AndyW]

Thank you for your post. It seems like you have a good understanding of the charge operator in the context of the Standard Model and its extensions. However, when dealing with matrix scalar multiplets, the charge operator may not be as straightforward.

In general, the charge operator is defined as the generator of gauge transformations. In the case of the Standard Model, the charge operator is a linear combination of the generators of the gauge group, as you have shown for both $SU(3)_C\otimes SU(2)_L\otimes U(1)_Y$ and $SU(3)_C\otimes SU(3)_L\otimes U(1)_X$. However, when dealing with matrix representations, the charge operator may not be a simple linear combination of the generators.

In the case of the 331 sextet, the charge operator is still defined as the generator of gauge transformations, but it may not be a linear combination of the generators of the gauge group. This is because the sextet is a matrix representation of the gauge group, and the generators of the gauge group may not commute with each other when acting on the sextet. This means that the charge operator cannot be written as a simple linear combination of the generators.

To determine the charge components of the sextet, you will have to use the explicit form of the charge operator for the 331 gauge group. This will involve using the commutation relations between the generators of the gauge group and the sextet representation, and may require some algebraic manipulation. I suggest consulting a textbook or research paper on the 331 gauge group for more information on how to calculate the charge components for the sextet.

In summary, when dealing with matrix representations, the charge operator may not be a simple linear combination of the generators of the gauge group. It will require some additional calculations and algebraic manipulation, and may not have a simple and intuitive form like in the case of vector representations. I hope this helps.