A relationship between the Higgs boson, top quark, and Z boson masses

Maybe a coincidence, maybe meaningful. The only relation that fits, using pole masses, holds at 1.4 sigma, but tends to predict either a rather high Higgs boson mass, or a rather low top quark mass.

I have little doubt that there are deeper functional relationships between the fundamental constants of the Standard Model (or at least some of them) than are contained within the Standard Model (what I call "within the Standard Model new physics" as opposed to "beyond the Standard Model new physics"). And, even if this particular relationship is not actually true, it is close enough that it is fruitful to ask, if there is some deeper source for these experimentally measured physical constant values, what kind of relationship would produce a close coincidence like this one.

For example, I wonder if an approximation of this relationship is favored in some way by the LP & C relationship that the square of the Higgs vev is equal to the sum of the squares of the fundamental SM particle masses, or by the approximate, but not exact, equality between the sum of the squares of the fundamental fermion masses and the sum of the squares of the fundamental boson masses.

Indeed, the paper notes that: 

After the Higgs discovery the numerical observation M(H)^2 ≃ M(Z)*M(t) (1) was proposed as a possible electroweak mass coincidence involving the heaviest spin-0, spin-1/2 and spin-1 representatives of the Standard Model (SM) spectrum.

(The citation for this sentence is to a paper by the author of the current paper: E. Torrente-Lujan, "The Higgs mass coincidence problem: why is the Higgs mass M2 H = MZMt?", Eur. Phys. J. C 74 (2014) 2744, arXiv:1209.0474.)

This suggests that there might be a fuller relationship that involves addition masses on the Standard Model spectrum beyond the heaviest ones that might provide correcting terms bringing the relationship to a more exact one.

I also seem to recall that there theoretically expected mass of the W boson in the Standard Model is a function of the Z boson mass, the top quark mass, and the Higgs boson mass, based upon electroweak unification in some way, but have never seen that relationship spelled out in detail. I have only seen the abbreviated leading order relationship between the W boson mass and Z boson mass that is related to the electromagnetic force and weak force coupling constants. 

The paper doesn't evaluate the "arithmetic relation" with the theoretically predicted W boson mass of 80.357 ± 0.006 GeV, but because that value is lower than the PDG value of 80.3692 ± 0.0133 GeV, it should be a somewhat better fit with the theoretically predicted W boson mass. As it turns out, this difference doesn't matter much, and neither does the difference between the Higgs boson mass that they use of 125.2 ± 0.11 GeV and the lower value of the Higgs boson mass of 125.09 GeV that is sometimes used. Indeed, the greater precision of the theoretically estimated W boson mass may increase the statistical significance of the discrepancy somewhat.

The relation M(H)^2 ≃ M(Z)*M(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(Z)*M(t)/M(H)^2 = 1.00362 ± 0.00261. Thus an exact pole-level geometric relation predicts either M(H) = 125.426 ± 0.120 GeV or M(t) = 171.898 ± 0.302 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±0.00159 and is not a viable exact mass sum rule. 
We then evaluate the complete NNLO weak-scale MS bar matching formulae at μ=M(t). In the standard convention one obtains ρˆ(Zt(M(t)) = √(g(2)^2+g(Y)^2) * y(t)/(4√2λ) = 0.96714±0.00361. Consequently, the exact running-coupling boundary condition λ =  g(Z)y(t)/(4√2) at the top scale would predict M(H) = 123.19 ± 0.20 GeV, or equivalently M(t) = 177.81 ± 0.50GeV 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(th) = 1.0340 ± 0.0039 at the electroweak scale. We formulate this requirement as a target for custodial/top-Higgs or triality-like symmetry extensions.
E. Torrente-Lujan, "The Higgs-top-Z mass coincidence relation after NNLO matching" arXiv:2605.21721 (May 20, 2026) (Report number: FISPAC-TH/3145-26, UQBAR-TH/26-97234).

Another key chart from the paper is this one:

The Lambda predictions are very far from the mark, as are the "arithmetic relation" estimates. The other relationship still has some tension with the experimental results but isn't ruled out.

Bounds on new neutrino physics

The constraints on BSM physics continue to narrow. In the Standard Model, the neutrino magnetic moment is predicted to be far below the threshold of current experimental detection, and the neutrino has an exactly zero electromagnetic charge. These results are consistent with those predictions and thus constrain BSM neutrino physics to very slight deviations from the SM predictions. Non-standard interactions of neutrinos are likewise constrained materially.

CODATA 2022 gives the value:

     sin2⁡θw=1−( mW mZ)2=0.22305(23) .{\displaystyle \sin ^{2}\theta _{\textsf {w}}=1-\left({\frac {\ m_{\textsf {W}}\ }{m_{\textsf {Z}}}}\right)^{2}=0.22305(23)~.}

So, the Weinberg angle measurement from the CONUS collaboration, while consistent with the world average measurement at the two sigma level, is too imprecise by two orders of magnitude to add meaningfully to our knowledge of that physical constant compared to the value obtained from simply plugging in the world average measured values of the W boson mass and the Z boson mass (or from calculating it in electroweak unification theory from the electromagnetic coupling constant and the weak force coupling constant so e^2/g^2 = sin^2 theta(W).

Its detections with pion-decay-at-rest, solar and recently with reactor antineutrinos by the CONUS collaboration render coherent elastic neutrino-nucleus scattering (CEνNS) an established tool for investigations within and beyond the Standard Model (SM). The CONUS experiment located at the nuclear power plants in Brokdorf (Germany) and Leibstadt (Switzerland) operates Germanium semiconductor detectors in a compact shield at close distance to the reactor core. An observation with 3.7σ significance is reported at the Leibstadt site, showing good agreement with its SM prediction.
Physics investigations performed with the last datasets collected at the Brokdorf reactor and with the first data obtained at the Leibstadt site are summarized. By using the experimental analysis framework, the presented results contain the full systematics that underlie the experiment. 
Previously determined limits with neutrino-electron scattering on the neutrino magnetic moment and a neutrino millicharge are improved to μ(ν) < 5.18⋅10^−11μB and q(ν) < 1.76⋅10^−12e0 (90% C.L). Further, the scale of new physics related to NSIs is improved to ΛNSI = 145 GeV and limits on the coupling of light new mediators are lowered down to 4⋅10−7 (90% C.L.) with the new data. Finally, the determination of the Weinberg angle with CEνNS and reactor antineutrinos yields sin(θ(W))^2  = 0.28 +0.03 −0.04 at a momentum transfer of ∼10 MeV.
N. Ackermann, et al., "New constraints on physics within and beyond the standard model from the latest CONUS datasets" arXiv:2605.22815 (May 21m 2026).

The abstract is unclear about what there is 3.7 sigma evidence of, but the body text clarifies that: "With data collected there since 2023, a successful CEνNS detection was reported with 3.7σ significance."

The body text explains some of the theoretical motivation for the paper:
On theory side CEνNS has become an interesting tool for investigations within and beyond the standard model (BSM) because of its flavor-blind and, in principle, threshold-free properties [21–27]. Within the SM it enables measurements of the Weinberg angle sin^2 θ(W) at the MeV scale with neutrinos and probe modifications of the involved couplings, i.e. via radiative correction [28–31] or investigation of the nuclear form factor when deviating from full coherence, i.e. with higher neutrino energies from πDAR sources [32–36]. BSM searches can be performed by testing for new neutrino interactions, for example in the context of heavy new physics via non-standard neutrino interactions (NSIs)[37–43] or new light mediators [44–53]. Neutrino (electromagnetic) properties [26, 27, 54–56] or emerging new particles may be probed as well [57–59]. Furthermore, future applications in the context of multi-messenger astronomy [60–62] or nuclear safeguarding seem promising [63, 64]. Experimental upscaling in the near future will allow such investigations via precision CEνNS measurements.
Copyright Andrew Oh-Willeke (2012-2020)