containing massless gauge bosons is straightforward.

Now suppose the classical Lagrangian of this gauge theory contains the following quadratic parts[^3] (summation over repeated indices is implicitly assumed): $$\mL_{quad}\supset\frac{1}{2}V_{ij}(\partial_\mu G_i)(\partial^\mu G_j)
+F_{pi}Z_p^\mu(\partial_\mu G_i)-\frac{1}{2}(\bM_G^2)_{ij}G_i G_j
+\frac{1}{2}(\bM_V^2)_{pq}Z_{p\mu}Z_q^\mu
\label{eq:quad}$$ Here $V$ is a real invertible $n_S\times n_S$ symmetric matrix, $F$ is a real $n_M\times n_S$ matrix, $\bM_G^2$ is a $n_S\times n_S$ symmetric matrix the rank of which does not exceed $n_E\equiv n_S-n_M$ [^4], and $\bM_V^2$ is a real $n_M\times n_M$ symmetric matrix which has $n_M$ positive eigenvalues. The elements of the four matrices $V,F,\bM_G^2,\bM_V^2$ depend only on the model parameters, not on field variables. For convenience let us define $$\tilde{G}_p=F_{pi}G_i,\quad p=1,2,...,n_M$$ Then the vector-scalar two-point transition term (the second term on the right hand side of Eq. ) is simply $Z_p^\mu\partial_\mu\tilde{G}_p$.

To carry out perturbation theory, it is preferable to eliminate the vector-scalar two-point transitions, make the scalar kinetic terms canonically normalized and at the same time diagonalize the scalar and vector mass terms. We will see that the procedure involved actually goes hand in hand with the quantization of the theory. Also, the tight structure of the gauge theory greatly facilitates the diagonalization process.

In gauge field theories, the vector-scalar two-point transitions are 