are canonically normalized in their kinetic part, the vector-scalar two-point transitions in a gauge theory has the form [@Weinberg:1996kr] $$i\sum_{nm\alpha}\partial_\mu\phi'_n t_{nm}^\alpha A_{\alpha}^{\mu}v_m
\label{eq:cvs}$$ Here $\phi'_n$ is the shifted scalar field with zero vacuum expectation value, $v_m$ is the vacuum expectation value of the original scalar fields. $t^\alpha$ denotes the generator matrix with $\alpha$ being the adjoint index and $A_\alpha^\mu$ is the corresponding gauge field. On the other hand, the elements of the gauge boson mass matrix are [@Weinberg:1996kr] $$\mu_{\alpha\beta}^2=-\sum_{nml}t_{nm}^\alpha t_{nl}^\beta v_m v_l
\label{eq:cvv}$$ Compare Eq.  and Eq.  it is easy to find for our case the useful property $$\langle\tilde{G}_p|\tilde{G}_q\rangle=(\bM_V^2)_{pq},\forall p,q=1,2,...,n_M
\label{eq:gip}$$ A nonlinearly-realized scalar sector does not introduce additional difficulty in arriving at Eq. , because compared to the linearly-realized case, the relevant differences begin from quadratic terms in the field expansion and do not affect Eq.  and Eq. .

Eq.  suggests that if the gauge bosons are already in their mass eigenstates, then the related Goldstone boson vectors must be orthogonal to each other, which is exactly what we desire. Physically this implies that massive gauge bosons eat their corresponding Goldstone bosons along the directions dictated by their mass eigenstates. Therefore it would be desirable we rotate the gauge 