= Solution
At momentum transfers small compared with $M_W$, the two <weak charged current> vertices each carry $g/(2\sqrt2)$ when written with $(1-\gamma_5)$, and the <W boson propagator> reduces to a local factor proportional to $1/M_W^2$. Matching their product to the <Fermi interaction> gives, at tree level,
$$
\boxed{\frac{G_F}{\sqrt2}=\frac{g^2}{8M_W^2},\qquad G_F=\frac1{\sqrt2v^2}\quad(M_W=gv/2).}
$$
The <Fermi constant> is thus determined by the electroweak coupling and symmetry-breaking scale; the four-fermion description is a low-energy effective interaction.
Only the cross term $J_L^\dagger J_H$ can annihilate the pion and create the stated charged-lepton/antineutrino pair. Its leptonic <matrix element> is $\bar u_e(k)\gamma^\mu(1-\gamma_5)v_\nu(q)$. <Lorentz covariance> permits a vacuum-to-<spin>-zero current <matrix element> only proportional to $p^\mu$. At rest this has only a temporal component. The vacuum is parity-even and the pion is parity-odd, while $V_H^0$ is parity-even, so $\langle0|V_H^0|\pi\rangle$ equals its own negative and vanishes. <Lorentz covariance> then makes the full vector <matrix element> vanish. The temporal <axial current> is parity-odd and is allowed: this is the <vacuum-to-pseudoscalar current selection rule>.
The minus sign in $J_H=V_H-A_H$ cancels the interaction's minus sign, up to an irrelevant common S-matrix phase. Therefore
$$
\mathcal M=\frac{G_F}{\sqrt2}\bar u_e\gamma^\mu(1-\gamma_5)v_\nu\langle0|A_{H\mu}|\pi^-\rangle
=iG_FF_\pi\cos\theta_C\,p_\mu\bar u_e\gamma^\mu(1-\gamma_5)v_\nu.
$$
The <pion decay constant> convention here includes the given $\sqrt2$ in its current <matrix element>. Momentum conservation and the external <Dirac equations> give $p=k+q$, $\bar u_e\not k=m_e\bar u_e$, $\not qv_\nu=0$. Since $\not q(1-\gamma_5)=(1+\gamma_5)\not q$, the neutrino term vanishes and
$$
\boxed{\mathcal M=iG_FF_\pi\cos\theta_C\,m_e\bar u_e(1-\gamma_5)v_\nu.}
$$
This is <helicity suppression>: the amplitude vanishes in the zero charged-lepton-mass limit despite the larger available phase space.
For the <spin sum>, the Dirac adjoint of $1-\gamma_5$ is $1+\gamma_5$, and the completeness relations give
$$
\begin{aligned}
\sum_s|\mathcal M|^2
&=G_F^2F_\pi^2\cos^2\theta_C\,m_e^2
\operatorname{tr}[(\not k+m_e)(1-\gamma_5)\not q(1+\gamma_5)]\\
&=2G_F^2F_\pi^2\cos^2\theta_C\,m_e^2
\operatorname{tr}[(\not k+m_e)\not q(1+\gamma_5)]\\
&=8G_F^2F_\pi^2\cos^2\theta_C\,m_e^2\,k\cdot q
=4G_F^2F_\pi^2\cos^2\theta_C\,m_e^2(m_\pi^2-m_e^2).
\end{aligned}
$$
The chiral projectors automatically eliminate the unused neutrino helicity when the full massless completeness relation is inserted. There is no initial <spin> average for a <spin>-zero pion.
In the pion rest frame, let $r=|\boldsymbol k|=|\boldsymbol q|$, $E=\sqrt{r^2+m_e^2}$. After integrating the spatial delta function and angles, the <two-body Lorentz-invariant phase space> becomes
$$
\Phi_2=\frac1{4\pi}\int_0^\infty\frac{r^2}{Er}\,\delta(m_\pi-E-r)dr.
$$
The root is $r_*=(m_\pi^2-m_e^2)/(2m_\pi)$ and $E_*=(m_\pi^2+m_e^2)/(2m_\pi)$. The absolute derivative of the delta-function argument is $r_*/E_*+1=m_\pi/E_*$. Thus $\Phi_2=r_*/(4\pi m_\pi)$. Multiplying by the <spin sum> and the initial factor $1/(2m_\pi)$ yields the <leptonic pseudoscalar decay width>
$$
\boxed{\Gamma(\pi^-\to e^-\bar\nu_e)=\frac{G_F^2F_\pi^2\cos^2\theta_C}{4\pi}\,m_\pi m_e^2\left(1-\frac{m_e^2}{m_\pi^2}\right)^2.}
$$
If the decay constant were instead defined by a <matrix element> $if_\pi p^\mu$ without $\sqrt2$, then $f_\pi=\sqrt2F_\pi$ and the width coefficient would be $f_\pi^2/(8\pi)$; mixing these conventions would introduce a factor-of-two error.
The same derivation for a muon cancels the common hadronic and weak factors in the <pion electron-to-muon decay ratio>:
$$
\boxed{R_{e/\mu}^{(0)}=\frac{m_e^2}{m_\mu^2}\left(\frac{1-m_e^2/m_\pi^2}{1-m_\mu^2/m_\pi^2}\right)^2\approx1.28\times10^{-4}.}
$$
The numerical value uses approximate masses $0.511,105.66,139.57$ MeV. Agreement with the corresponding <Standard Model> prediction tests <lepton universality> and the chiral, helicity-suppressing structure of the <weak charged current>, rather than the poorly calculable hadronic decay constant. A difference in the electron and muon weak couplings would multiply the ratio by their squared coupling ratio. Precision agreement must be assessed after <radiative corrections>, with matching <photon>-inclusive experimental definitions; the displayed tree-level ratio is not itself the radiatively corrected precision prediction.
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