A critical look at $\beta$-function singularities at large $N$

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.