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Limit of vanishing regulator in the functional renormalization group

The non-perturbative functional renormalization group equation depends on the choice of a regulator function, whose main properties are a "coarse-graining scale" $k$ and an overall dimensionless amplitude $a$. In this paper we shall discuss the limit $a\to0$ with $k$ fixed. This limit is closely related to the pseudo-regulator that reproduces the beta functions of the $\overline{\text{MS}}\,$ scheme, that we studied in a previous paper. It is not suitable for precision calculations but it appears to be useful to eliminate the spurious breaking of symmetries by the regulator, both for nonlinear models and within the background field method.