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

TL;DR

This paper reevaluates the Higgs-top-Z mass relation using NNLO matching and current electroweak data, finding it consistent within uncertainties but not as an exact sum rule, and explores symmetry explanations for the relation.

Contribution

It provides a scheme-consistent NNLO analysis of the Higgs-top-Z mass relation and investigates potential symmetry-based explanations for its near-coincidence.

Findings

The pole-level ratio $ ho_{Zt}$ is close to unity within uncertainties.

Exact geometric relation predicts $M_H$ or $M_t$ within 1.4σ of measured values.

Running-coupling analysis suggests a finite matching factor at the electroweak scale.

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 $\rho_{Zt}=M_ZM_t/M_H^2=1.00362\pm0.00261$. 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 $1.4\sigma$ test rather than an exclusion. By contrast, the companion arithmetic relation gives $\rho_{Wt}=(M_W+M_t)/(2M_H)=1.00994\pm0.00159$ and is not a viable exact mass sum rule. We then evaluate the complete NNLO weak-scale $\overline{\mathrm{MS}}$ matching formulae at $\mu=M_t$. In the standard convention one obtains $\widehat\rho_{Zt}(M_t)=\sqrt{g_2^2+g_Y^2}\,y_t/(4\sqrt2\lambda)=0.96714\pm0.00361$. Consequently, the exact running-coupling boundary condition $\lambda=g_Zy_t/(4\sqrt2)$ 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 $\kappa_{\rm th}=1.0340\pm0.0039$ at the electroweak scale. We formulate this requirement as a target for custodial/top-Higgs or triality-like symmetry extensions.