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

Abstract We investigate the minimal composite Higgs model based on the symmetry-breaking pattern $${\textrm{SU}}(4)\rightarrow {\textrm{Sp}}(4),$$ SU ( 4 ) → Sp ( 4 ) , where electroweak symmetry breaking is governed by the vacuum alignment angle $$\theta .$$ θ . Through the leading-order relations $$\kappa _V=\cos \theta $$ κ V = cos θ and $$m_\eta =m_h/\sin \theta ,$$ 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 $$\kappa _V,$$ κ V , we construct a Bayesian posterior for $$\theta ,$$ θ , including the Jacobian from the transformation $$\kappa _V\rightarrow \theta $$ κ V → θ (assuming a flat prior in $$\kappa _V$$ κ V ). We cross-check these results with an independent frequentist $$\Delta \chi ^2$$ Δ χ 2 analysis; the two approaches yield consistent bounds on $$\theta $$ 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 $$\kappa _V$$ 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 $$\theta $$ θ is comparatively model-independent, these $$m_\eta $$ 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 $${\textrm{SU}}(4)/{\textrm{Sp