The augmented field theory proposed by Klauder is shown to be equivalent (in its Euclidean formulation) to the conventional field theory with the term $log\left|\phi \mathrm{}\right(x\left)\right|$ added to the Lagrangian. Gradient-free models (arising by omitting the kinematical term of the Lagrangian) with the Lagrangian of the form $\mathcal{p}\left(\phi \mathrm{}\right(x\left)\right)+log\left|\phi \mathrm{}\right(x\left)\right|$ (where $\mathcal{p}$ is a polynomial) are studied (in the Euclidean lattice limit formulation). Their generating functionals are explicitly calculated. It is shown that these models (nonrenormalizable in the perturbative approach) are renormalizable in nonperturbative theory. The problem of the limit with "vanishing interaction constant" is considered. Its solution depends heavily on clarifying the notion of the "vanishing interaction constant." When introducing the "physical" mass and interaction constants as parameters of generating functionals, this limit gives the usual free theory even for the models studied by Klauder.

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