Adjoint fermions at large-$N_c$ on the lattice

Lattice simulations of Yang-Mills theories coupled with $N_f$ flavours of fermions in the adjoint representation provide a way to probe the non-perturbative regime of a plethora of different physical scenarios, such as Supersymmetric Yang-Mills theory to BSM models. We are interested in the large-$N_c$ limit of these theories, for which standard lattice techniques are limited by high-computational costs. Our approach makes use of the well-established twisted volume reduction, which allows one to simulate these theories on a $1^4$ lattice. In this talk, we are going to present a detailed study of the scale setting of these theories, performed with Wilson flow techniques, but endowed with a procedure that allowed us to reduce finite-volume (finite $N_c$) systematic effects. We will apply this procedure to our configurations for $N_f=0,\frac{1}{2},1,2$ and analyze the dependence of the scale on the coupling, the adjoint fermion mass and the number of flavours. For the cases in which enough couplings are available we compared the resulting $\beta$-function with the predictions of perturbation theory, finding very good agreement.