terms in Eq.  yield mass matrices for the charged and neutral gauge bosons. The photon remains massless and is given by the exact expression $$A_{\mu} = \frac{e}{g} W_{0 \mu}^3 + \frac{e}{\tilde g} W_{1 \mu}^3
  + \frac{e}{g^{\prime}} B_{\mu},$$ where $e$ is the electromagnetic coupling. Normalizing the photon eigenvector, we get the relation between the coupling constants: $$\frac{1}{e^2}=\frac{1}{g^2}+\frac{1}{\tilde{g}^2}+\frac{1}{g'^2}.$$ This invites us to conveniently parametrize the gauge couplings in terms of $e$ by

$$g = \frac{e}{\sin\theta \cos\phi} = \frac{g_0}{\cos\phi}, \qquad \qquad
  \tilde g = \frac{e}{\sin\theta \sin\phi} = \frac{g_0}{\sin\phi}, 
  \qquad \qquad
   g^{\prime} = \frac{e}{\cos\theta}.
\label{eqn:gauge couplings}$$

We will take $\tilde g \gg g$, which implies that $\sin\phi \equiv x$ is a small parameter. The result of the diagonalization of the gauge boson matrices perturbatively in $x$ is summarized in Appendix A.

Counting the number of degrees of freedom, we see that there are six scalar degrees of freedom on the technicolor side ($\Sigma_{01},
\Sigma_{12}$) and four on the topcolor side ($\Phi$). Six of these will be eaten to form the longitudinal components of the $W^{\pm}$, $Z^0$, $W^{\prime \pm}$, and $Z^{\prime 0}$. This leaves one isospin triplet of scalars and the top-Higgs $H_t$ as physical states in the spectrum. While the interactions in Eq.  are 