boson fields to their mass eigenstates before adding the gauge-fixing terms Eq. . This offers great convenience for the diagonalization of scalar mass matrix afterwards.

On the other hand, if the gauge-fixing terms in Eq.  are added when $Z_p^\mu$’s are not mass eigenstate fields, although this way of gauge-fixing is also legitimate, it would cause further inconveniences. Firstly, after rotation to gauge boson mass eigenstates, the term $-\frac{1}{2\xi^p}(\partial_\mu Z_p^\mu)^2$ will induce kinetic mixing between gauge bosons in a general $R_\xi$ gauge, spoiling the diagonalization of gauge boson kinetic terms. Secondly, from Eq.  it is obvious that now the $\tilde{G}_p$’s are not orthogonal to each other. Therefore the diagonalization of scalar mass terms would not be straightforward. Due to the above considerations in the following we adopt the procedure in which gauge-fixing terms Eq.  are added after rotating gauge boson fields to their mass eigenstates.

Suppose the gauge boson mass matrix $\bM_V^2$ can be diagonalized as follows $$R\bM_V^2 R^{-1}=\bM_{DV}^2\equiv\text{diag}\{\mu_1^2,\mu_2^2,...,\mu_{n_M}^2\}$$ Here $R$ is a $n_M\times n_M$ orthogonal matrix, and $\mu_1^2,\mu_2^2,...,\mu_{n_M}^2$ are positive. Let us define $$G_p^m\equiv\frac{R_{pq}}{\mu_p}\tilde{G}_q=\frac{(RF)_{pi}}{\mu_p}G_i,\quad p=1,2,...,n_M\,\text{(no summation over $p$)}$$ (superscript $m$ denotes canonically-normalized mass eigenstates). Now we can check with the help of Eq. (no summation over $p,q$) $$\begin{aligned}
\langle G_p^m|G_q^m\rangle=\frac{1}{\mu_p\mu_q}(R\bM_V^2 