= Solution
Work in the <W boson> <rest frame>, taking the spin-quantization axis along its laboratory motion. Neglect charged-lepton and <neutrino> <masses>. The <weak charged current> produces a <Helicity>[negative-helicity] $\ell^-$ and a <Helicity>[positive-helicity] <antineutrino>. Their <momenta> are opposite, so both <spin> <Vectors> point opposite to the <charged lepton>'s direction $\hat{\mathbf n}$; the final total <Spin>[spin projection] along $\hat{\mathbf n}$ is $-1$.
Let $\lambda_W=+1,0,-1$ denote the parent <Spin>[spin projection] along $z$. In the given <probability amplitude>, use $j=1$ and the <rotation> element $d^1_{-1,\lambda_W}(\theta)$. The supplied spin-one <rotation> elements give
$$
\boxed{|\mathcal M_{+1}|^2=m_W^2g^2\frac{(1-\cos\theta)^2}{4},\qquad
|\mathcal M_0|^2=m_W^2g^2\frac{\sin^2\theta}{2},\qquad
|\mathcal M_{-1}|^2=m_W^2g^2\frac{(1+\cos\theta)^2}{4}.}
$$
The transverse states have <Spin>[spin projections] $\pm1$; the longitudinal state has projection zero, not total <spin> zero. The original diagram below distinguishes <momenta> from <spins> and shows the corresponding normalized angular profiles.
\Image[/past-exam-of-the-particle-physics-course-of-the-university-of-oxford/2018-w-polarization-and-angles.png]
{title=<W boson> <Spin>[spin projections], <lepton> <helicities> and decay <Angular distribution>[angular distributions]}
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