 Solution

Below $m_H=2m_W$, the on-shell two-body decay channel is kinematically closed. An imaginary $p^{\ast }$ 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, $H\to W{W}^{\ast }\to Wf{\bar{f}}^{\prime }$, 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.