More Neutrino Oscillation Physical Constant Measurements

Almost all of the experimental data favors a normal mass ordering for neutrinos over an inverted mass ordering, but given the limitations of current experiments, the preference is almost always a weak one.

The Particle Data Group value for delta m(32) squared in normal ordering is as follows:

Taking the square root, the PDG value is a gap of 49.5 meV.

A new paper's results are consistent with the world average. The paper, its abstract, and the chart below from its supplementary materials are as follows:

This Letter reports measurements of muon-neutrino disappearance and electron-neutrino appearance and the corresponding antineutrino processes between the two NOvA detectors in the NuMI neutrino beam. These measurements use a dataset with double the neutrino mode beam exposure that was previously analyzed, along with improved simulation and analysis techniques. 

A joint fit to these samples in the three-flavor paradigm results in the most precise single-experiment constraint on the atmospheric neutrino mass-splitting, Δm^2(32) = 2.431 +0.036 −0.034 (−2.479 +0.036 −0.036) × 10^−3 ~eV^2 if the mass ordering is Normal (Inverted). In both orderings, a region close to maximal mixing with sin^2(θ23) = 0.55 +0.06 −0.02 is preferred. 

The NOvA data show a mild preference for the Normal mass ordering with a Bayes factor of 2.4 (corresponding to 70% of the posterior probability), indicating that the Normal ordering is 2.4 times more probable than the Inverted ordering. When incorporating a 2D Δm^2(32) --sin^2(2*θ13) constraint based on Daya Bay data, this preference strengthens to a Bayes factor of 6.6 (87%).

NOvA Collaboration, "Precision measurement of neutrino oscillation parameters with 10 years of data from the NOvA experiment" arXiv:2509.04361 (September 4, 2025).

Toponium Discovered

Toponium is a hadron which is the bound state of a valance top quark and a valance top anti-quark. Oversimplified presentations often state that top quarks don't form hadrons, because they decay to bottom quarks extremely rapidly after they are created, leaving no time to form a hadron. And, the vast majority of the time, this is true. But, the lifetime of a top quark is only an average lifetime. Sometimes it decays faster and sometimes it decays slower. In the highly improbable case that a top quark and a top anti-quark are created at the same time and both last much longer than the average lifetime before decaying, they can form a hadron which is called toponium, and it is fairly elementary to determine how likely this is to happen at a given energy scale.

In the paper below, the CMS collaboration at the Large Hadron Collider (LHC) claims to have discovered a resonance which appears to be ground state toponium, which has a highly distinctive signature in collider, because toponium is profoundly more massive (at more than 344 GeV) than any other meson. The background that has to be distinguished from the signal is therefore pretty modest.

Another paper, whose preprint was released today, in the course of considering the possibility of a hadron which is a baryon with three top quarks (a profoundly difficult to form hadron since three top quarks or three antitop quarks need to be formed within about 3 x 10^-25 seconds in essentially the same place), asserts that the ATLAS collaboration at the LHC has also discovered a toponium resonance, although the citation in the preprint does not include any arXiv or journal reference. This citation is to: 

ATLAS Collaboration, “Observation of a cross-section enhancement near the t¯t production threshold in √s =13 TeV pp collisions with the ATLAS detector.” 

Presumably the authors have received advance word of this paper and plan to update the reference in their own paper when it is released.

This paper slightly overstates what the papers actually claim (which is that the resonance is consistent with toponium, but not that it definitely is toponium), but only modestly so.

Discovering this vanishingly rare and incredibly short lived meson, which is the heaviest possible meson (and has a mass about 68% greater than a uranium-235 atom confined to a space on the order of 100 times smaller than a proton in radius) is a remarkable accomplishment in and of itself, and also with more detections, could make it possible to measure the top quark mass to a precision of about ten times as great as current measurements (i.e. ± 0.3 GeV now v. ± 0.03 GeV with this improvement).

A search for resonances in top quark pair (tt¯) production in final states with two charged leptons and multiple jets is presented, based on proton-proton collision data collected by the CMS experiment at the CERN LHC at s√ = 13 TeV, corresponding to 138 fb−1. The analysis explores the invariant mass of the tt¯ system and two angular observables that provide direct access to the correlation of top quark and antiquark spins. A significant excess of events is observed near the kinematic tt¯ threshold compared to the nonresonant production predicted by fixed-order perturbative quantum chromodynamics (pQCD). The observed enhancement is consistent with the production of a color-singlet pseudoscalar (1S[1]0) quasi-bound toponium state, as predicted by nonrelativistic quantum chromodynamics. Using a simplified model for 1S[1]0 toponium, the cross section of the excess above the pQCD prediction is measured to be 8.8 +1.2−1.4 pb.

CMS Collaboration, "Observation of a pseudoscalar excess at the top quark pair production threshold" arXiv:2503.22382v2 (March 28, 2025, published version from Rep. Prog. Phys. 88 (2025) 087801 released on August 23, 2025).

The introduction explains that:

The discovery of the top quark in 1995 at the Fermilab Tevatron collider was a major milestone in particle physics. Uniquely among quarks, the top quark’s lifetime is shorter than the hadronization timescale. This causes the spin of the top quark to be transferred directly to its decay products, enabling precise measurements of spin properties via angular distributions. While the individual polarizations of the top quark and antiquark (t and t) are small when produced via the strong interaction, their spins are correlated in the standard model (SM), which was experimentally confirmed at both the Tevatron and the LHC. 

Although tt pairs do not form stable bound states given the short lifetime of the top quark, calculations in nonrelativistic quantum chromodynamics (NRQCD) predict bound state enhancements at the tt threshold. Since this effect is present only when the tt pairs are in the color singlet configuration, the dominant contribution at the LHC is from the gluon-gluon initial state, leading to the production of the 1S[1]0 “toponium” quasi-bound state ηt. 

Contributions from other spin states are much smaller at the LHC; for instance, the 3P[1]0 state χt is suppressed by additional powers of the top quark velocity, which is nearly zero at the threshold. The color octet configuration, on the other hand, is suppressed below the tt threshold because of a repulsive interaction between the top quarks, and has a steeply rising cross section as a function of the tt invariant mass mt t above the threshold. The presence of such an ηt state would therefore manifest itself as an enhancement in the number of events near the production threshold with distinctive patterns in tt spin correlation observables caused by its pseudoscalar nature. However, due to the possibility of initial- and final-state radiation, the color configurations of the tt pairs are not necessarily the same as the partons in the initial state, making theoretical predictions of toponium production challenging. 

This Letter reports the observation of a threshold enhancement in tt production consistent with pseudoscalar toponium. The analyzed proton-proton (pp) collision data at √s = 13TeV were recorded by the CMS experiment at the CERN LHC in 2016–2018, corresponding to an integrated luminosity of 138fb−1. The analysis, whose tabulated results are provided in the HEPDatarecord, is conducted within the context of a search for neutral spin-0 bosons produced through gluon-gluon fusion and decaying to tt. Here, we focus on the threshold production of a composite CP-odd pseudoscalar ηt and a CP-even scalar χt as signal hypotheses, where CP refers to the charge-parity symmetry. These represent the simplest hypotheses that can explain the observation, since they arise naturally within NRQCD. However, the available experimental data does not exclude alternative explanations like additional pseudoscalar bosons, whose existence is predicted by several theoretical models beyond the SM. This possibility is explored in Ref. [21], the companion paper to this publication, where the same data is interpreted in terms of limits on additional scalar and pseudoscalar bosons over a large mass range. 

The analysis considers final states with two charged leptons (electrons and/or muons) and at least two jets, referred to as the ℓℓ channel. A similar analysis was previously performed by the CMS experiment using the data sample collected in 2016 and considering the ℓj channel (i.e., f inal states with one charged lepton and at least four jets) in addition to the ℓℓ channel. 

In that analysis, a moderate pseudoscalar-like deviation with a mass at the lowest investigated value of 400GeV was found. Compared to that superseded analysis, we consider only the ℓℓ channel here, but use more than three times the data, consider resonances with masses below the tt production threshold, and add a second angular observable that provides direct access to tt spin correlation. 

Similar searches have also been conducted by the ATLAS Collaboration using data at √ s =2 8 [23] and 13TeV [24]. The results presented in Ref. [24] use the data sample collected in 2015-2018 and combine the ℓℓ and ℓj channels, with the latter being predominant. The analysis in the ℓℓ channel differs from our approach in that it investigates the invariant mass mbbℓℓ of the bbℓ+ℓ− system rather than mtt and it utilizes an angular variable whose sensitivity to tt spin correlation is significantly diluted by kinematic effects. 

We have verified that incorporating these differences into our analysis would not result in a significant enhancement at the threshold. Consequently, the conclusions of Ref. [24] are not directly comparable to the ones reported in this paper, nor do they refute or confirm the findings reported herein. 

Moreover, our findings are consistent with enhancements at the threshold in previous tt differential cross section measurements reported by ATLAS and CMS. Similarly, the mild tension between the observed and expected measurement of spin correlation in the tt threshold region, which has been reported by both ATLAS and CMS as part of their studies of quantum entanglement, has been reproduced by this analysis. 

. . . 

[22] CMS Collaboration, “Search for heavy Higgs bosons decaying to a top quark pair in proton-proton collisions at √s = 13TeV”, JHEP 04 (2020) 171, doi:10.1007/JHEP04(2020)171, arXiv:1908.01115. 

[23] ATLAS Collaboration, “Search for heavy Higgs bosons A/H decaying to a top quark pair in pp collisions at √s = 8TeV with the ATLAS detector”, Phys. Rev. Lett. 119 (2017) 191803, doi:10.1103/PhysRevLett.119.191803, arXiv:1707.06025. 

[24] ATLAS Collaboration, “Search for heavy neutral Higgs bosons decaying into a top quark pair in 140fb−1 of proton-proton collision data at √ s =13TeV with the ATLAS detector”, JHEP 08 (2024) 013, doi:10.1007/JHEP08(2024)013, arXiv:2404.18986. 

The discovery is basically a side effect of LHC searches for a neutral heavy Higgs boson. Preprints with more analysis can be found here and here and here and here and here and here and here and here and here and here and here.

This is a substance that has a greater mass per volume than a neutron star or an atomic nucleus by a long shot (it is about 344 million times as dense). If you use a definition of density for a black hole of mass within the spatial volume of an event horizon, it even has more mass per volume than a stellar or greater mass black hole, although some primordial black holes, if they exist, would have a greater density.

The Schwarzchild radius of toponium is about 2.2 x 10^-29 meters, which is about 10^11 times shorter than the estimated radius of toponium, and about 10^14 times shorter than the size of a proton or neutron. So, there is no risk of the LHC or a future collider creating a primordial black hole when this hadron is formed.

An Electroweak Centric Model For Standard Model Mass Generation

The basic intuitive gist of the proposal of this paper is one that I've entertained myself, although I don't have the theoretical physics chops to spell it out at this level of formality and technical detail (and I'm really not qualified to evaluate the merits to this proposal at that level). I've seen one or two other papers (not recent ones) that take a similar approach.

The ratio of the electron mass to the lightest neutrino mass eigenstate is roughly the same as the ratio of the electromagnetic coupling constant to the weak force coupling constant, and both are masses are similar to what would be expected from the self-interactions of electrons and neutrinos via the electromagnetic and weak forces with themselves. Electrons interact via both of these forces, while neutrinos interact only via the weak force.

The down quark mass is about twice as much as the up quark mass, just as the absolute value of the down quark electromagnetic charge is twice the absolute value of the up quark electromagnetic charge. All quarks have the same magnitude of strong force color charge. And all of the fundamental fermions of the Standard Model have the same magnitude of weak force charge. Quarks interact via the strong force, the electromagnetic, and the weak force, so their self-interactions might be expected to be larger than for the electron which doesn't interact via the strong force.

Figuring out how this can work in concert with the three fundamental fermion generations is particularly challenging. I'm inclined to associate it with a W boson mediated dynamic process that sets the relative values of the Higgs Yukawas. This paper doesn't attempt to look beyond the first generation of fundamental fermions in implementing its model.

I'm not thrilled with the "leptoquark" component of this theory, but the fact that it gives rise to neutrino mass without either Majorana mass or a see-saw mechanism is very encouraging.

In the Standard Model of elementary particles the fermions are assumed to be intrinsically massless. Here we propose a new theoretical idea of fermion mass generation (other than by the Higgs mechanism) through the coupling with the vector gauge fields of the unified SU(2) ⊗ SU(4) gauge symmetry, especially with the Z boson of the weak interaction that affects all elementary fermions. The resulting small masses are suggested to be proportional to the self-energy of the Z field as described by a Yukawa potential. Thereby the electrically neutral neutrino just gets a tiny mass through its Z-field coupling. In contrast, the electrically charged electron and quarks can become more massive by the inertia induced through the Coulomb energy of the electrostatic fields surrounding them in their rest frames.

Eckart Marsch, Yasuhito Narita, "On the Lagrangian and fermion mass of the unified SU(2) ⊗ SU(4) gauge field theory" arXiv:2508.15332 (August 21, 2025) (13 pages).

The introduction of the paper is as follows:

According to the common wisdom of the Standard Model (SM) of elementary particle physics, the fermions are intrinsically massless, but they gain their masses via phase transition from the vacuum of the Higgs field. However, this notion introduces many free parameters (the Yukawa coupling constants) that are to be determined through measurements. These have been made at the LHC only for some members of the second and third family of heavy leptons and quarks, yet not for the important first family of fermions, of which the stable and long-lived hadrons form according to the gluon forces of quantum chromodynamics (QCD). 

Here we just consider the first fermion family of the SM and propose a new idea of the fermion mass generation. The key assumption is that their masses may be equal to the relevant gauge-field energy in the rest frames of these charged fermions carrying electroweak or strong charges. Their masses are suggested to originate from jointly breaking the chiral SU(2) symmetry combined with the hadronic isospin SU(4) symmetry, as described in the recent model by Marsch and Narita, following early ideas of Pati and Salam and their own work. Unlike in the SM, in their model both symmetries are considered as being unified to yield the SU(2) ⊗ SU(4) symmetry, which then is broken by the same procedures that are applied successfully in the electroweak sector of the SM. 

The outline of the paper is as follows. We briefly discuss the extended Dirac equation and its Lagrangian including the Higgs, gauge-field and fermion sectors. Especially, the covariant derivative is discussed and the various gauge-field interactions are described. Also the different charge operators (weak and strong) are presented. Then the CPT theorem is derived for the extended Dirac equation including the gauge field terms. The remainder of the paper addresses the idea of mass generation from gauge field energy in the fermion rest frame. Finally we present the conclusions.

The paper's conclusion states:

In this letter, we have considered a new intuitive idea of how the elementary fermions might acquire their finite empirical masses. We obtained diagonal mass matrices as Kronecker products within the framework of the unified gauge-field model of Marsch and Narita. The mass matrices still commute with the five Gamma matrices of the extended free Dirac equation without gauge fields. However, when including them the chiral SU(2) and the hadronic SU(4) symmetries both are broken by the mass term. Thus, the breaking of the initial unified SU(2) ⊗ SU(4) symmetry by the Higgs-like mechanism gives the fermions their different charges as well as specific masses. 

In the SM the initial common mass m is assumed to be zero, and then the Dirac spinor splits into two independent two-component Weyl spinors. But when the gauge fields are switched on, their self-energy gives inertia and thus mass to the fermions in their rest frame. The breaking of gauge symmetry yields the electromagnetic massless photon field E(µ) and the weak boson field Z(µ), which becomes very massive via the Higgs mechanism. It also induces inertia for all eight fermions, yet the resulting masses are rather small owing to the very small Compton wavelength of the Z boson. The neutrino and electron can acquire masses in this way, which yet differ by six orders of magnitude. The hadronic charge of the leptons is zero, and thus they decouple entirely from QCD. It is responsible by confinement through the gluons for the mass of the various resulting composite fermions, in particular for the proton mass. 

The masses of the light fermions are thus argued to originate physically from the major self-energy of the electrostatic field as well as from the minor self-energy of the Z-boson field, which is proportional to the Higgs vacuum that determines the Z-boson mass. It is clear, however, that the masses of heavy composite hadrons, in particular of the proton and neutron, involve dominant contributions from the energy of the binding gluon fields, as the QCD lattice simulations have clearly shown. 

In conclusion, the extended Dirac equation contains a physically well motivated mass term. It remedies the shortcoming of the SM that assumes massless fermions at the outset, whereas the empirical reality indicates that they are all massive. Therefore, the neutrino cannot be a Majorana particle, as it has often been suggested in the literature. This notion is in obvious contradiction to the observed neutrino oscillation, implying clearly finite masses. Chiral symmetry is broken in our theory, yet the parity remains intact. 

Finally, we like to mention the masses of the heavy gauge bosons involved in the above covariant derivative and related matrix. In the reference of the particle data group we find in units of MeV/c^2 the values: M(Z) = 91.2 and M(W) = 80.4. For the “leptoquark" boson V we obtain M(V) = 35.4. For the sum of these masses we find the following surprising results: M(V) + M(Z) = 126.6, which equals within less than a one-percent margin the measured mass of the Higgs boson, M(H) = 125.3. Also, M(W) + M(Z) = 171.6, which again equals within less than a one-percent margin the measured mass of the top quark, M(T) = 172.7. Whether this is just a fortuitous coincidence or indicates a physical connection has to remain open. 

Two Papers Related To Quark Masses

How much of a nucleon's mass is due to Higgs field sourced quark mass?

Background

To the nearest 0.1 MeV, the mass of the proton is 938.3 MeV (the experimentally measured value is 938.272 089 43 (29) MeV) and the mass of the neutron is 939.6 MeV (the experimentally measured value is 939.565 421 94 (48) MeV).

A proton has two valence up quarks and one valence down quark. A neutron has one valence up quark and two valence down quarks.

According to the Particle Data Group (relying on state of the art averages of Lattice QCD calculations that extra this from measurable masses of particles made up of quarks bound by gluons which are called hadrons) concludes that the average of the up quark mass and the down quark mass is measured to be 3.49 ± 0.4 MeV, the up quark mass is 2.16  ± 0.4 MeV, and the down quark mass is 4.70  ± 0.4 MeV. 

What would the masses of the proton and the neutron be, hypothetically, if the up quark and down quark has zero mass?

A new paper calculates that the proton mass would be 882.4 ± 2.5 MeV (about 94% of its measured value), while the mass of the neutron would be 883.7 MeV ± 2.5 MeV (about 94.1% of its measured value). Thus, this would reduce the total nucleon mass by 55.9 MeV ± 2.5 MeV, and ignoring the effect of the difference between the up quark and down quark masses (which has a roughly 3.5 MeV effect in the average massless quark estimate, according to the body text of the paper). 

These masses can be conceptualized as the combined pure gluon and electroweak field source mass of a proton or neutron in a minimum energy ground state.

Naively, one would think that the reduction from the measured value would be smaller, because the sum of three times the average of the up and down quark mass is about 10.5 MeV, and this figure is often cited on popular science discussions of the proton and neutron mass. 

But, massive quarks indirectly impact the strength of the gluon field between the three valence quarks of a nucleon, and this indirect effect has a magnitude of roughly 45.4 MeV.

Why does this matter?

Prior to this paper, there was a large gap between the values produced by different kinds of calculations of this amount, which the new paper reconciles.

Also, this is not entirely a hypothetical question, because it is part of, for example, how one calculates the mass of protons and neutrons at higher energy scales, and how one can reverse engineer the quark masses of the proton and neutron masses.

At higher momentum transfer scales (a.k.a. energy scales) the Higgs field is weaker and the quark masses get smaller, and eventually, extrapolated to high enough energy scales, the Higgs field goes to zero and the quarks really are massless. 

The strong force coupling constant also runs with energy scale, however, and also gets weaker at higher energies, although not at the same rate at the quark masses. 

There is also a modest electroweak contribution to the proton and neutron masses and the electromagnetic force (which predominates over the weak force component) gets stronger at higher energy scales, modestly mitigating the declining quark masses and strong force field strength.

So, in order to be able to make a Standard Model calculation of the expected mass of protons and neutrons at high energies, you need to be able to break these distinct sources of the proton and neutron masses into their respective components, because the different components run with energy scale in different ways.

Charge parity violation and the quark masses

Another new paper argues that because the Standard Model has CP-violation, the masses of the up quarks must be related to the masses of the down quarks, giving rise to five independent degrees of freedom for quark masses rather than six.

A physically viable ansatz for quark mass matrices must satisfy certain constraints, like the constraint imposed by CP-violation. In this article we study a concrete example, by looking at some generic matrices with a nearly democratic texture, and the implications of the constraints imposed by CP-violation, specifically the Jarlskog invariant. This constraint reduces the number of parameters from six to five, implying that the six mass eigenvalues of the up-quarks and the down-quarks are interdependent, which in our approach is explicitly demonstrated.

A. Kleppe, "On CP-violation and quark masses: reducing the number of free parameters" arXiv:2508.11081 (August 14, 2025).

This relies on some assumptions, but the assumptions are quite general, and its basic conclusion is that up-type quark masses (i.e. the Higgs Yukawas of up-type quarks) can't arise from a mechanism independent of that for down-type quark masses (i.e. the Higgs Yukawas of down-type quarks), something that is the case, for example, in an extended Koide's rule approach.

Bonus content

Supersymmetry (a.ka. SUSY) is still a failure, when it comes to describing reality. It does not describe the world we live in.

A Technical But Interesting Paper On Fermion Mass Ratios

A new paper looks at fundamental fermion mass ratios from the perspective of something similar to an extended Koide's rule approach.

We revisit the "three generations" problem and the pattern of charged-fermion masses from the vantage of octonionic and Clifford algebra structures. Working with the exceptional Jordan algebra J3(OC) (right-handed flavor) and the symmetric cube of SU(3) (left-handed charge frame), we show that a single minimal ladder in the symmetric cube, together with the Dynkin Z2 swap (the A2 diagram flip), leads to closed-form expressions for the square-root mass ratios of all three charged families. The universal Jordan spectrum (q - delta, q, q + delta) with a theoretically derived delta squared = 3/8 fixes the endpoint contrasts; fixed Clebsch factors (2, 1, 1) ensure rung cancellation ("edge universality") so that adjacent ratios depend only on which edge is taken. The down ladder determines one step, its Dynkin reflection gives the lepton ladder, and choosing the other outward leg from the middle yields the up sector.

From the same inputs we obtain compact CKM "root-sum rules": with one 1-2 phase and a mild 2-3 cross-family normalization, the framework reproduces the Cabibbo angle and Vcb and provides leading predictions for Vub and Vtd/Vts. We perform apples-to-apples phenomenology (common scheme/scale) and find consistency with current determinations within quoted uncertainties. Conceptually, rank-1 idempotents (points of the octonionic projective plane), fixed symmetric-cube Clebsches, and the Dynkin swap together account for why electric charge is generation-blind while masses follow the observed hierarchies, and they furnish clear, falsifiable mass-ratio relations beyond the Standard Model.

Tejinder P. Singh, "Fermion mass ratios from the exceptional Jordan algebra" arXiv:2508.10131 (August 13, 2025) (90 pages).

Another interesting paper develops a relationship between the mixing ratios of the unitary triangle in the CKM matrix and the CP violating phase of that matrix. The abstract below deviates from my usual editing conventions to preserve the details of superscripts and subscripts in the notion without a lot of extra editing work that is prone to human error:

In this letter, we obtain a rephasing invariant formula for the CP phase in the Kobayashi--Maskawa parameterization δKM=arg[−VuddetVCKM/VusVubVcdVtd]. General perturbative expansion of the formula and observed value δKM≃π/2 reveal that the phase difference of the 1-2 mixings ei(ρd12−ρu12) is close to maximal for sufficiently small 1-3 quark mixings su,d13. Moreover, by combining this result with another formula for the CP phase δPDG in the PDG parameterization, we derived an exact sum rule δPDG+δKM=π−α+γ which relating the phases and the angles α,β,γ of the unitarity triangle.

Masaki J. S. Yang, "Rephasing Invariant Formula for the CP Phase in the Kobayashi-Maskawa Parametrization and the Exact Sum Rule with the Unitarity Triangle δPDG + δKM = π −α +γ" arXiv:2508.10249 (August 14, 2025) (6 pages).