from a unique Lagrangian term reflecting the coupling of the top-Higgs to the top quark: $$\mathcal{L}_{top}=-\lambda_{t}\bar{\psi}_{L0}\,\Phi\, t_{R}+h.c.\label{eq:top quark mass L}$$ Details of the fermion masses and mass eigenstates are given in Appendix A.

Physical top-pions and their couplings {#sec:couplings}
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The next step towards understanding top-pion phenomenology is to identify the combination of degrees of freedom which make up the physical (uneaten) top-pions. While the top-Higgs $H_t$ remains a mass eigenstate, the pions $\pi_0^a$, $\pi_1^a$ and $\pi_t^a$ mix. We can identify the physical top-pions as the linear combination of states that cannot be gauged away. We do this by isolating the Goldstone boson states that participate in interactions of the form $V_{\mu} \partial^{\mu} \phi$ in the Lagrangian.

We start by expanding the nonlinear sigma fields to first order in $\pi/F$, $$\begin{aligned}
   \Sigma_{01} &=& 1 + \frac{2i\pi_{0}^{a}\tau^{a}}{F} 
   + \mathcal{O}\left(\frac{\pi^2}{F^2}\right),
   \label{eqn:sigma01} \\
   \Sigma_{12} &=& 1 + \frac{2i\pi_{1}^{a}\tau^{a}}{F}
   + \mathcal{O}\left(\frac{\pi^2}{F^2}\right).
\label{eqn:sigma12} \end{aligned}$$ Plugging this in Eq. [^1], we can read off the various interaction terms. The complete details are given in Appendix B. Here, we concentrate on the gauge-Goldstone mixing terms: $$\mathcal{L}_{\textrm{mixing}} =
   \frac{g}{2} W_0^{a\mu} \partial_{\mu} \left[F\pi_0^a + f\pi_t^a \right] 
   + \frac{\tilde{g}}{2} W_1^{a\mu} \partial_{\mu} 
   \left[F \pi_1^a - F \pi_0^a \right]
   - \frac{g^{\prime}}{2} B_2^{\mu} 