  We study and discuss the extension of the rotating-wave
spin$\unicode{x2013}$boson model, together with more general models describing
a system$\unicode{x2013}$field coupling with a similar rotating-wave structure,
to interactions mediated by possibly singular (non-normalizable) form factors
satisfying a weaker growth constraint. To this purpose, a construction of
annihilation and creation operators as continuous maps on a scale of Fock
spaces, together with a rigorous renormalization procedure, is employed.
