Hypercharge of the complex scalar doublet

Question

I often see the complex scalar doublet $Φ_A$, $A=1,2$ with the opposite hypercharge arising in the Yukawa couplings as $\tildeΦ_A = iτ{_2}_{AB}Φ_B^*$ where $τ_r$ $(r=1,2,3)$ denote isospin pauli matrices. How can i change the sign of the hypercharge without pauli matrices? Somewhere I saw such notation $\tildeΦ_A = ε_{AB}Φ_B^*$, is it a two-index Levi-Civita symbol or just a unitary antisymmetric tensor?

Answer 1

The hypercharge deals with both members of the doublet identically, and complex conjugation suffices to reverse its charge. In the subspace of Pauli matrices, it looks like the identity matrix! See this question.

Trying to avoid a counterproductive conversation on SU(5) technology orthogonal to the spirit of your stated question, here is the standard picture.

In SU(5), there are two independent renormalizable Yukawa interactions given by $$ λ_t (\mathbf{ 10 ~~ 10 ~~5}_H) + λ (\mathbf { 10 ~~\bar 5 ~~\bar 5}_H), $$ containing the SM colorless interactions $$ λ_t (\mathbf{ Q ~~u^c ~~ \tilde\Phi}) + λ (\mathbf{Q~~ d^c ~~\Phi }), $$ the first one with the v.e.v. upstairs, and the second with it downstairs.

Recall, if you thought the Yukawa with the conjugate rep of SU(2) flip is cockeyed, the SU(5) assignment of the SU(2) singlets $u^c$ in the 10 and the $d^c$ in the $\mathbf { \bar 5}$ more than makes up for it. I strongly suspect the facile flip you are envisioning would be out of place here.