Into the conformal window: multi-representation gauge theories.

We investigate the conformal window of four-dimensional gauge theories containing fermionic matter fields in multiple representations. Of particularly relevant examples are the ultra-violet complete models with fermions in two distinct representations considered in the context of composite Higgs and top-partial compositeness. We first discuss various analytical approaches to unveil the lower edge of the conformal window and their extension to the multiple matter representations. In particular, we argue that scheme-independent series expansions for the anomalous dimension of a fermion bilinear $\gamma_{\bar{\psi}\psi}$ at an infrared fixed point, combined with the conjectured critical condition $\gamma_{\bar{\psi}\psi} = 1$ or equivalently $\gamma_{\bar{\psi}\psi} (2-\gamma_{\bar{\psi}\psi})=1$, can be used to determine the boundary of conformal phase transition on fully physical grounds. In illustrative cases of $SU(2)$ and $SU(3)$ theories with $N_f$ fundamental flavors, we access our results by comparing to other analytical and lattice results.