Flavour Chains at the TeV Edge

Abstract

We study chains of vector-like fermions that generate the Yukawa hierarchies as products of small mixings, with the Higgs coupled at a single link. Minimal Flavour Protection allows these flavour messengers to live at the TeV scale, while Higgs naturalness offers a reason to place them there: in cosmological solutions based on Higgs criticality, their large multiplicity lowers the critical point of the Higgs potential to the TeV scale. Crucially, the same Higgs interactions induce tree-level shifts in the $Z$ and $W$ couplings of light fermions. Existing LEP and SLD measurements already bound $M_F/|y_F|\gtrsim 3$ TeV, while a future Tera-$Z$ factory such as FCC-ee could probe up to $100$ TeV. It would test the parameter space most relevant to Higgs naturalness, reaching scenarios with Higgs-mass tuning at the level of $10^{-4}$ or worse in minimal realisations. Longer chains partially alleviate this tuning but do not eliminate it, making a Tera-$Z$ factory a decisive test of the framework.