We analyze whether and how the neutron resonance mode in unconventional
superconductors is affected by higher order corrections in the coupling between
spin excitations and fermionic quasiparticles and find that in general such
corrections cannot be ignored. In particular, we find that in two spatial
dimensions (d=2) the corrections are of same order as the leading, weak
coupling contributions demonstrating that the neutron resonance mode in
unconventional superconductors is a strong coupling phenomenon. The origin of
this behavior lies in the quantum-critical nature of the low energy spin
dynamics in the superconducting state and the feedback of the resonance mode
onto the fermionic excitations. While quantum critical fluctuations occur in
any dimensionality smaller than the upper critical dimension d_{uc}=3, they can
be analyzed in a controlled fashion by means of the \epsilon-expansion
(\epsilon =3-d), such that the leading corrections to the resonance mode
position are small. Regardless of the strong coupling nature of the resonance
mode we show that it emerges only if the phase of the superconducting gap
function varies on the Fermi surface, making it a powerful tool to investigate
the microscopic structure of the pair condensate.Comment: 26 pages, 8 figure
