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.

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