Past exam of the gauge field theory course of the University of Cambridge / 2013 / Paper 1 / 1 / ii / Solution

In the stronger limit $m_H\ll m_W$, both intermediate W bosons must be virtual. Their heavy propagators suppress the four-fermion decay amplitude, while the small available phase space suppresses its rate further. At energies small compared with $m_W$, integrating out the two vectors gives a local interaction proportional to $G_{F}H{J}_{\mu }^{+}{J}^{-\mu }/v$. Its dimension-seven coefficient implies a rate of order $G_F^3{m}_H^7$ when all four final fermions can be treated as massless, with channel-dependent constants. This is a low-energy scaling statement, not the on-shell longitudinal-width formula. Other accessible Higgs boson decay channels, such as a pair of fermions or loop-induced two-photon decay, must then be compared using their actual masses and couplings.