= Solution
The <Standard Model> is a <chiral gauge theory>: left-chiral quarks and leptons form <SU(2)> weak doublets, while the corresponding right-chiral charged fields are weak singlets. The covariant kinetic term contains $g_W\bar Q_L\gamma^\mu t^aQ_LW_\mu^a$. With $W^\pm=(W^1\mp iW^2)/\sqrt2$, its off-diagonal generators give the <weak charged current>
$$
\mathcal L_{\rm cc}\supset\frac{g_W}{\sqrt2}\bar b_L\gamma^\mu V_{tb}^*t_LW_\mu^-
+\mathrm{h.c.}
=\frac{g_W}{2\sqrt2}\bar b\gamma^\mu(1-\gamma^5)V_{tb}^*tW_\mu^-+\mathrm{h.c.}
$$
An overall interaction sign depends on the convention for the <covariant derivative> and has no effect on this decay rate. The <chiral projector> $P_L=(1-\gamma^5)/2$ explains the vector-minus-axial combination. The <Cabibbo-Kobayashi-Maskawa matrix> comes from the different left-handed rotations that diagonalize up- and down-type <fermion mass matrices>. Set $V_{tb}=1$ and $m_b=0$ as stipulated.
Let $P_{\mu\nu}=-g_{\mu\nu}+p_{3\mu}p_{3\nu}/m_W^2$, the <massive vector polarization sum>. Sum over final spins and average over the two equally populated initial <top quark> spins. The <fermion spin sum> and the <Dirac adjoint> of the vertex give
$$
\overline{|\mathcal M|^2}
=\frac{g_W^2}{16}P_{\mu\nu}\operatorname{tr}
\left[\not p_2\gamma^\mu(1-\gamma^5)(\not p_1+m_t)\gamma^\nu(1-\gamma^5)\right].
$$
The $m_t$ term vanishes because $(1-\gamma^5)\gamma^\nu(1-\gamma^5)=0$. Anticommuting $\gamma^5$ through the remaining <gamma matrices>, using $(1-\gamma^5)^2=2(1-\gamma^5)$, then using cyclicity of the <matrix trace> and the symmetry of $P_{\mu\nu}$ yields
$$
\boxed{\overline{|\mathcal M|^2}
=\frac{g_W^2}{8}P_{\mu\nu}\operatorname{tr}
\left[(1-\gamma^5)\not p_1\gamma^\mu\not p_2\gamma^\nu\right].}
$$
The extra factor two from the chiral algebra compensates the initial spin-average factor. The term with $\gamma^5$ is antisymmetric in $\mu,\nu$ and vanishes upon contraction with $P_{\mu\nu}$. The remaining <gamma-matrix trace> is
$$
4\left[p_1^\mu p_2^\nu+p_1^\nu p_2^\mu-(p_1\cdot p_2)g^{\mu\nu}\right].
$$
Its contraction gives
$$
\overline{|\mathcal M|^2}
=\frac{g_W^2}{2}(p_1\cdot p_2)
+\frac{g_W^2}{m_W^2}(p_1\cdot p_3)(p_2\cdot p_3).
$$
Here $p_1=p_2+p_3$, $p_1^2=m_t^2$, $p_2^2=0$ and $p_3^2=m_W^2$, so
$$
p_1\cdot p_2=p_2\cdot p_3=\frac{m_t^2-m_W^2}{2},\qquad
p_1\cdot p_3=\frac{m_t^2+m_W^2}{2}.
$$
Therefore
$$
\overline{|\mathcal M|^2}
=\frac{g_W^2}{4m_W^2}(m_t^2-m_W^2)(m_t^2+2m_W^2).
$$
The <two-body Lorentz-invariant phase space> has rest-frame <momentum> $p^*=(m_t^2-m_W^2)/(2m_t)$. Substitution into $\Gamma=p^*\overline{|\mathcal M|^2}/(8\pi m_t^2)$ gives the <two-body top-quark decay width>
$$
\boxed{\Gamma(t\to bW^+)
=\frac{g_W^2m_t^3}{64\pi m_W^2}
\left(1-\frac{m_W^2}{m_t^2}\right)^2
\left(1+2\frac{m_W^2}{m_t^2}\right).}
$$
The squared threshold factor combines the vanishing final <momentum> with the corresponding kinematic factor in the squared <scattering amplitude>. The longitudinal <W boson> <polarization> contributes the high-mass enhancement proportional to $m_t^3/m_W^2$. The result is at tree level, for an on-shell <W boson> and $m_t>m_W$; finite bottom <mass> and <radiative corrections> change it. Restoring quark mixing multiplies the width by $|V_{tb}|^2$.
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