# Fundamental invariants of the electroweak sector?

In a previous question, I asked what the matrix representation of the electroweak fields is, and I was told they are identical to the Faraday tensors, but come in a set of three ($$W_i, i\in \{1,2,3\}$$), associated to the $$SU(2)$$ symmetry and another one ($$B$$), associated to the $$U(1)$$ symmetry for a total of four.

I also know that the expressions $$||\mathbf{E}||^2-||\mathbf{B}||^2$$ and $$\mathbf{E}\cdot \mathbf{B}$$ are the two fundamental invariants of the electromagnetism. My question is what are the invariants of the $$SU(2)$$ symmetry and the $$U(1)$$ symmetry. Based on the similarity, my guess is the $$SU(2)$$ invariants are 3 orthogonal elements of the same form as the invariants of the electromagnetism:

$$(||\mathbf{A}_1||^2-||\mathbf{D}_1||^2)\sigma_x\\ (||\mathbf{A}_2||^2-||\mathbf{D}_2||^2)\sigma_y\\ (||\mathbf{A}_3||^2-||\mathbf{D}_3||^2)\sigma_z$$

That can be squared as follows:

$$(||\mathbf{A}_1||^2-||\mathbf{D}_1||^2)^2+(||\mathbf{A}_2||^2-||\mathbf{D}_2||^2)^2+(||\mathbf{A}_3||^2-||\mathbf{D}_3||^2)^2$$

and the $$U(1)$$ symmetry is of the same form as that of electromagnetism:

$$||\mathbf{F}||^2-||\mathbf{G}||^2$$

Would be nice to know $$SU(3)$$ invariants as wells.

• You are using maximally opaque and disruptive language for this. In the covariant formulation, you are talking about $F_{\mu\nu} F^{\mu\nu}$ and $\epsilon^{\mu\nu\kappa\lambda}F_{\mu\nu} F_{\kappa\lambda}$. The close analog expression for the YM field strength also holds. What, exactly, is holding you up? Jul 14 '20 at 13:04