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 parts. Flavor non-universal effects are completely captured by examining three options for the top mixing: fully composite qL3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {q}_L^3 $$\\end{document}, equal compositeness, and fully composite tR. In the most phenomenologically viable case of a fully composite tR, we show that the strongest bound on the natural parameter space comes from next-to-leading log running of the 4-top operator Ott=t¯RγμtRt¯RγμtR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{O}}_{tt}=\\left({\\overline{t}}_R{\\gamma}_{\\mu }{t}_R\\right)\\left({\\overline{t}}_R{\\gamma}^{\\mu }{t}_R\\right) $$\\end{document} 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∗ ≳ 25 TeV, and test the naturalness of the electroweak scale at the ≲ 10−4 level.
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