Impact of Higgs precision measurements at the LHC and FCC-ee on the spectrum of composite Higgs models

We investigate the minimal composite Higgs model based on the symmetry-breaking pattern SU(4)→ Sp(4), SU ( 4 ) → Sp ( 4 ) , where electroweak symmetry breaking is governed by the vacuum alignment angle θ . θ . Through the leading-order relations κ V=cos θ κ V = cos θ and m_η =mₕ/sin θ , m η = m h / sin θ , precision measurements of Higgs couplings constrain the vacuum structure and, within this leading-order realization, the singlet pseudo-Nambu–Goldstone boson mass. Using the current ATLAS Run-2 measurement of κ V, κ V , we construct a Bayesian posterior for θ , θ , including the Jacobian from the transformation κ V→ θ κ V → θ (assuming a flat prior in κ V κ V ). We cross-check these results with an independent frequentist Δ χ ² Δ χ 2 analysis; the two approaches yield consistent bounds on θ θ . The same framework is applied to the projected sensitivities of the High-Luminosity LHC and FCC-ee. The current data imply a lower bound of approximately 440 GeV on the singlet mass at the 95% credibility level, conservative under the flat prior in κ V κ V with boundary truncation. The HL-LHC improves this to about 600 GeV, while FCC-ee reaches beyond 2 TeV as an indicative projected sensitivity. While the constraint on θ θ is comparatively model-independent, these m_η m η bounds hold within the specific leading-order potential considered here. Our results show that future Higgs precision measurements will probe the vacuum alignment of the minimal $${SU}(4)/{{Sp