[Tr]{}\[S\^[j]{}\_8 S\_[8 j]{}\] + \_2 H\^[i]{} H\_j [Tr]{}\[S\^[j]{}\_8 S\_[8 i]{}\]\
&+&\
&+& \_6 [Tr]{}\[S\^[i]{}\_8 S\_[8 i]{} S\^[j]{}\_8 S\_[8 j]{}\] + \_7 [Tr]{}\[S\^[i]{}\_8 S\_[8 j]{} S\^[j]{}\_8 S\_[8 i]{}\]\
&+& \_8 [Tr]{}\[S\^[i]{}\_8 S\_[8 i]{}\] [Tr]{}\[S\^[j]{}\_8 S\_[8 j]{}\] + \_9 [Tr]{}\[S\^[i]{}\_8 S\_[8 j]{}\] [Tr]{}\[S\^[j]{}\_8 S\_[8 i]{}\]\
&+& \_[10]{} [Tr]{}\[S\_[8 i]{} S\_[8 j]{}\] [Tr]{}\[S\^[i]{}\_8 S\^[j]{}\_8\] + \_[11]{} [Tr]{}\[S\_[8 i]{} S\_[8 j]{}\] [Tr]{}\[S\^[j]{}\_8 S\^[i]{}\_8\]. In the potential the index is an $\rm SU(2)$ index and the trace is over the down flavour index. We again rotate the phase of $S_8$ (relative to $H$) so that the $\hat{\lambda}_3$ term is real, then the couplings and $\hat{\lambda}_{4,5}$ and the $\eta$’s are in general complex. In the above potential there are no linear terms in $S_8$ after $H$ gets its vacuum expectation value and so it is natural for it not to have a vev.

In the potential and the ${\hat Y}'s$ one can also insert arbitrary numbers of $g_D \, g_D^\dagger$ matrices between contractions of a down index. We work in the down basis so that $g_D = {\rm diag}(\sqrt{ 2}m_d/v,\sqrt{2}m_s/v,\sqrt{2}m_b/v)$. The interactions in the potential do not change flavour and are suppressed by $m_b^2/v^2$ so we neglect them.

Keeping just the leading term in, $\sqrt{2}m_t/v$, the 