The Higgs-top-$Z$ mass coincidence relation after NNLO matching

Abstract

The relation $M_H^2\simeq M_ZM_t$, previously proposed as a non-trivial Higgs mass coincidence, is reconsidered with present electroweak inputs and with a scheme-consistent matching analysis. With the 2025 PDG values for $M_Z$, $M_W$ and $M_H$, and the ATLAS-CMS direct top-mass combination, the pole-level ratio is $ρ_{Zt}=M_ZM_t/M_H^2=1.00362\pm0.00261$. Thus an exact pole-level geometric relation predicts either $M_H=125.426\pm0.120\,\mathrm{GeV}$ or $M_t=171.898\pm0.302\,\mathrm{GeV}$, which is still a $1.4σ$ test rather than an exclusion. By contrast, the companion arithmetic relation gives $ρ_{Wt}=(M_W+M_t)/(2M_H)=1.00994\pm0.00159$ and is not a viable exact mass sum rule. We then evaluate the complete NNLO weak-scale $\overline{\mathrm{MS}}$ matching formulae at $μ=M_t$. In the standard convention one obtains $\widehatρ_{Zt}(M_t)=\sqrt{g_2^2+g_Y^2}\,y_t/(4\sqrt2λ)=0.96714\pm0.00361$. Consequently, the exact running-coupling boundary condition $λ=g_Zy_t/(4\sqrt2)$ at the top scale would predict $M_H=123.19\pm0.20\,\mathrm{GeV}$, or equivalently $M_t=177.81\pm0.50\,\mathrm{GeV}$ when $M_H$ is held fixed. This is incompatible with the measured point. A possible symmetry explanation must therefore act on pole-level threshold quantities, or provide a finite matching factor $κ_{\rm th}=1.0340\pm0.0039$ at the electroweak scale. We formulate this requirement as a target for custodial/top-Higgs or triality-like symmetry extensions.