top decays, $t \to \Pi_t^+ b$. The Tevatron experiments have searched for this process in the context of two-Higgs-doublet models and set upper bounds of about 10–20% on the branching fraction of $t \to H^+ b$, with $H^+$ decaying to $\tau \nu$ or $c \bar s$ [@Aaltonen:2009ke; @Abazov:2009zh] - we can use this to set a lower bound on the top-pion mass. In our model, below the $t \bar b$ threshold $\Pi_t^+$ decays via its mixing with $\pi_0^+$ to lighter SM fermions, with couplings controlled by the fermions’ SM Yukawa couplings. The branching fraction of $\Pi_t^+$ to $\tau\nu$ is therefore about 70%, with the remainder of decays to $c \bar s$. The Tevatron studies can then be applied directly to the top-pion. The relevant limit is BR($t \to \Pi_t^+ b) \lesssim 0.2$ based on D0 data [@Abazov:2009zh].

In our model, the branching fraction of $t \to \Pi_t^+ b$ is controlled by the top-pion mass, the pion mixing angle $\omega$, and the coupling $\lambda_t$: $$\Gamma(t \to \Pi_t^+b) = \frac{G_F m_t}{8 \sqrt{2} \pi}
    \left[m_t^2 R^2 + \mathcal{O}(m_b^2)\right]
    \left[1 - \frac{M_{\Pi_t^+}^2}{m_t^2} \right]^2,
\label{eqn:gamma_tb}$$ with $$R \equiv \cos\omega \frac{\lambda_t}{\lambda_t^{\rm SM}}
  = \cot\omega \left[ \frac{M_D^2 (\epsilon_L^2 + 1) - m_t^2}{M_D^2 - m_t^2}
    \right].$$ To evaluate 