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Abstract

We propose a self-consistency equation for the $\beta$-function for theories
with a large number of flavours, $N$ , that exploits all the available
information in the critical exponent, $\omega$, truncated at a fixed order in
$1/N$. We show that singularities appearing in critical exponents do not
necessarily imply singularities in the $\beta$-function. We apply our method to
(non-)abelian gauge theory, where $\omega$ features a negative singularity. The
singularities in the $\beta$-function and in the fermion mass anomalous
dimension are simultaneously removed providing no hint for a UV fixed point in
the large-$N$ limit.