Non-universal probes of composite Higgs models: New bounds and prospects for FCC-ee

We study the leading loop-level phenomenology of composite Higgs models via the effective field theory of a strongly interacting light Higgs and top quark (SILH+TQ). We systematically analyze the renormalization group evolution (RGE) of tree-generated operators in the SILH+TQ scenario, finding large mixings of flavor non-universal operators into those affecting electroweak precision observables. We show that these model-independent RG contributions are more important than typical estimates for finite matching terms. Flavor non-universal effects are completely captured by examining three options for the top mixing: fully composite $q_L^3$, equal compositeness, and fully composite $t_R$. In the most phenomenologically viable case of a fully composite $t_R$, we show that the strongest bound on the natural parameter space comes from a 2-loop double-log contribution of the 4-top operator $O_{tt} = (\bar t_R \gamma_\mu t_R)(\bar t_R \gamma^\mu t_R)$ into the Peskin-Takeuchi $T$ parameter. In general, we find that this 2-loop effect allows existing electroweak precision data to give better constraints on 4-top operators than high-energy probes from top production at the LHC. Independent of the top mixing, we find that a future tera-$Z$ machine such as FCC-ee has the potential to probe composite Higgs models up to a scale of $m_* \gtrsim 25$ TeV, and test the naturalness of the electroweak scale at the $\lesssim 10^{-4}$ level.