Solution (source code)

= Solution

For fixed $(P,Q)\ne(0,0)$ the effective <Fermi interaction> predicts $d\sigma/d\Omega\propto G_F^2s$. The angular odd term integrates to zero, while $\int d\Omega(1+\cos^2\theta)=16\pi/3$, so
$$
\boxed{\sigma_{\mathrm{eff}}(s)=\frac{G_F^2s}{24\pi}(P^2+Q^2)^2.}
$$
The corresponding fixed-spin <scattering amplitudes> grow as $G_Fs$, eventually conflicting with <partial-wave unitarity>. This is a failure of extrapolating the <effective field theory>, whose contact approximation requires $s\ll M_W^2$, not a failure of the <Standard Model>.

Restoring the virtual <W boson> <Feynman propagator> replaces its constant low-energy denominator by $M_W^2/(M_W^2-s)$, away from the resonance. The massless external <Dirac equations> give zero contraction of each current with the exchanged momentum, so the longitudinal numerator does not contribute. Its squared modulus multiplies the effective cross-section. For $s\gg M_W^2$ the resulting tree-level rate decreases as $1/s$ rather than growing as $s$. Near the pole one must include the <W boson> <decay width>. These statements assume the same massless conserved external currents; the contact expression alone is not valid there.