\partial_{\mu} 
   \left[F \pi_1^3 + f \pi_t^3 \right].
   \label{eq:gauge-Goldstone}$$ Note that the pion combination in the third term can be written as a linear combination of those appearing in the first two terms: $$F \pi_1^3 + f \pi_t^3 = [F \pi_0^3 + f \pi_t^3] + [F \pi_1^3 - F \pi_0^3].$$ The two eaten triplets of pions span the linear combinations that appear in the first two terms of Eq. , leaving the third linear combination as the remaining physical top-pions, which we will denote $\Pi_t^a$: $$\Pi_t^a = - \sin\omega \left(\frac{\pi_0^a + \pi_1^a}{\sqrt{2}} \right)
   + \cos\omega \, \pi_t^a,$$ where we have normalized the state properly using the definitions of $F$ and $f$ in Eq. .

The physical top-pions can also be identified by expanding the top-Higgs potential given in Eq.  and collecting the mass terms. The masses of $H_t$ and $\Pi_t^a$ are given by, $$M^2_H = 2 v^2 (\kappa + 4 \lambda) \sin^2{\omega},
   \qquad \qquad M^2_{\Pi_t} = 2 v^2 \kappa \tan^2{\omega},
\label{eq:pimass}$$ while the other two linear combinations of pions are massless, as true Goldstone bosons should be. Equation (\[eq:Vhiggs\]) also contains trilinear couplings between $H_t$ and two top-pions; we find $$\begin{aligned}
  \lambda_{H_t \Pi_t^0 \Pi_t^0} = \lambda_{H_t \Pi_t^+ \Pi_t^-}
  &=& 2 v 