f^a)^{1/2}$ is the length of the complex vector $f^a$. Next, we introduce vectors that are orthogonal to $f^a_U$, $f_D^a$, $f^{\prime\,a}_U$, and $f^{\prime\,a}_D$, respectively, g\_U\^a&&f\_U\^[b]{} \^[ba]{},g\_D\^af\_D\^[b]{} \^[ba]{},\
g\_U\^[a]{}&&f\_U\^[b]{} \^[ba]{}, g\_D\^[a]{}f\_D\^[b]{} \^[ba]{}. These are pseudo-vectors with respect to U(2) Higgs basis transformations,[^8] \[gtransform\] g\^a\_[U,D]{}([det]{} U)\^[-1]{}U\_[ab]{} g\^b\_[U,D]{},g\^[a]{}\_[U,D]{}([det]{} U)\^[-1]{}U\_[ab]{} g\^[b]{}\_[U,D]{}, due to the appearance of the complex phase, ${\rm det}~U$.

We now define U(2)-invariant, hypercharge-one Higgs fields as follows: H\_U& &f\_U\^[a]{}\_a,H’\_Uf\_U\^[a]{}\_a,\[invH1\]\
H\_D& & f\_D\^[a]{}\_a,H’\_Df\_D\^[a]{}\_a.\[invH2\] One can also define a corresponding set of hypercharge $-1$ fields, e.g., \[invH3\] H\_D\^i\^[ij]{}H\_D\^[j]{},H\_D\^[i]{}\^[ij]{}H\_D\^[j]{}. It is also convenient to define U(2) pseudo-invariant Higgs fields (denoted by calligraphic fonts), \_U& &g\_U\^[a]{}\_a,\^\_Ug\_U\^[a]{}\_a,\[invH4\]\
\_D& & g\_D\^[a]{}\_a,\^\_Dg\_D\^[a]{}\_a.\[invH5\] It then follows that: \_a&=& f\_U\^a H\_U +g\_U\^a \_U= f\_D\^a \_D +g\_D\^a \_D \[phia1\]\
&=& f\_U\^[a]{} H’\_U +g\_U\^[a]{} ’\_U= f\_D\^[a]{} ’\_D +g\_D\^[a]{} ’\_D,\[phia\] after using $\hat f_{U,D}^a \hat f_{U,D}^{a\,*}=
\hat g_{U,D}^a \hat g_{U,D}^{a\,*}=1$ and $\hat g_{U,D}^a \hat f_{U,D}^{a\,*}=\hat f_{U,D}^a \hat g_{U,D}^{a\,*}=0$.

There is some motivation for this proliferation of Higgs field definitions. In particular, as we show later in , the choices of \[four\] {\_U,H\_U}, {\^\_U,H\^\_U}, {\_D,\_D},[and]{} {\^\_D,\^\_D}, correspond to four different basis choices for the hypercharge-one Higgs doublet fields.

One can express the gaugino-higgsino-Higgs interaction Lagrangian in a manifestly U(2)-invariant form. For example, using the definitions 