= Solution
Below $m_H=2m_W$, the on-shell two-body <decay channel> is kinematically closed. An imaginary $p^*$ in the preceding formula is therefore a sign that this formula no longer applies. Slightly below threshold, the physical process can instead contain an on-shell <W boson> and an off-shell one, $H\to WW^*\to Wf\bar f'$, followed by decay of the remaining <W boson>. Its rate requires the virtual-vector <propagator>, its allowed fermionic decay channels and the corresponding multiparticle <Lorentz-invariant phase space>. The nonzero <decay width> of the <W boson> smooths the idealized threshold. A polarization-resolved rate for two on-shell longitudinal vectors must not be analytically continued and treated as that multiparticle rate.
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