and $g$ is the weak interaction coupling constant. The relevant CKM factor has been set equal to 1.

The squared, spin and colour averaged matrix element for the $\mathcal{O} \left(\alpha _s \right)$ real-emission correction to the decay $t \rightarrow W b$ is $$\begin{gathered}
|\mathcal{M}_R|^2 = g^2 g_s^2 C_F \left\{ - \frac{|\mathcal{M}_B|^2} {g^2} \left(\frac{p_b}{p_b.p_g} - \frac{p_t} {p_t.p_g} \right)^2 + \right. \\ \left. \left(\frac{p_g.p_t}{p_b.p_g} + \frac{p_b.p_g}{p_t.p_g} \right) \left(1 + \frac{m_t^2}{2m_w^2} + \frac{m_b^2}{2m_w^2} \right) - \frac{1}{m_w^2} \left(m_t^2+m_b^2 \right)\right\},\end{gathered}$$ where $g_s$ is the strong coupling constant, $C_F= \frac{4}{3}$ and $p_t$, $p_b$, $p_W$ and $p_g$ are the four-momenta of the top quark, bottom quark, $W$ boson and gluon. In general, the orientation of the decay products in the three-body final state is such that the emitting parton absorbs the transverse recoil coming from the emission of the gluon and the spectator particle continues to lie along the negative $z$-direction. When the radiation originates from the top quark, the bottom quark effectively acts as the emitting particle so that we remain in the rest frame of the top quark. Therefore, for emission from both the top and the bottom quarks, the momenta of the decay products are p\_W = (E\_W,0,0,- ), p\_b = (E\_b, -p\_T ( 