Glueballs

A first installment on the non-vanilla hadron blogging project that I mentioned in a previous post notes a new review article on the topic. 

Glueballs are always bosons, which means that they can blend with other bosons with the same quantum numbers. As the article below explains:

If glueballs are easily studied in the quenched approximation, they have proved remarkably elusive in practice. This is because, in the real world, quarks are light and dynamical. Because the lowest-lying glueballs carry J^PC = 0++, 2++ and 0−+, all of which are quantum numbers that ordinary isoscalar q¯q mesons also carry, nothing forbids mixing. As a result, any physical resonance in these channels is a superposition of the possible bare resonances[.]

This is one of several reasons that it is hard to precisely predict the mass of glueball resonances even though, naively, it should be simpler than other hadron mass calculations because the only experimentally measured physical constant that enters into the calculation at leading order is the strong force coupling constant. Gluons have no electromagnetic charge, don't decay via the weak force, are themselves massless (although their energy gives rise to an emergent mass for glueballs), and don't need to take into account quark masses at leading order.

The theoretical calculations put essentially all of the potential glueball states in a narrow mass range of about 1.3-5.0 GeV, which is a mass range that is also crowded with a background of all sorts of more conventional hadron resonances, which further complicates the process of determining whether a resonance has a significant glueball component.

But despite these challenges, some experimentally observed resonances have been identified with a probable high scalar glueball or pseudoscalar glueball content, validating a key prediction of quantum chromodynamics (QCD). There are a couple of other possible glueball types that are harder to match to experimentally observed resonances.

Glueballs are colour-singlet bound states built from gluons alone. They are an unavoidable consequence of the non-Abelian structure of Quantum Chromodynamics (QCD), and, in the pure Yang--Mills limit, they are the only physical excitations of the theory. The present article reviews what is known about them. 

After establishing which spin, parity and charge-conjugation quantum numbers two- and three-gluon states can carry, the pure-gauge spectrum is surveyed. Each of the main theoretical approaches and their respective conclusions are briefly presented. They include lattice QCD, constituent-gluon and Coulomb-gauge models, functional Dyson--Schwinger and Bethe--Salpeter equations, holographic models, and QCD sum rules. Particular attention is paid to the scale-setting ambiguity that limits how precisely a quenched glueball mass can be converted into physical units. 

The discussion then turns to full QCD. After discussing the meaning of a glueball assignment in this context, unquenching effects and questions related to glueball--qq¯ mixing are addressed. The large-N(c) counting that underpins the mixing picture, as well as mass-matrix and effective-Lagrangian treatments of mixing are presented. The selection rules and dynamical mechanisms that shape glueball decays are also discussed. 

The gluon-rich production mechanisms used experimentally are finally reviewed. The candidates are assessed sector by sector: f(0)(1370)/f(0)(1500)/f(0)(1710) and the competing interpretations of the scalar sector, η(1405)/η(1475) and X(2370) in the pseudoscalar sector, the crowded tensor region, and the essentially unexplored C=−1 sector and its connection to the Odderon. 

Outlooks on the programmes that could help in validating some of these candidates or identifying others are reviewed as a conclusion.

Cyrille Chevalier, Vincent Mathieu, "Glueballs: hadrons without quarks" arXiv:2609.16790 (September 15, 2026) (Submission to Encyclopedia of Nuclear Physics (Elsevier)).

Hadron Physics To Do

One of my long-standing to dos for a blog post, which keeps getting put off because it is a pretty big project, is to survey the current state of the literature regarding hadron and/or hadron molecule resonances that aren't simple pseudo-scalar valance quark-antiquark, and simple three valance quark baryons, with u, d, s, c, and b valance quarks.

These include scalar mesons, axial-vector mesons, tetraquarks, pentaquarks, hexaquarks (if any), quarkonia, toponium, glueballs, mixed/blended meson resonances, glueball-quark hybrids, hadron molecules, excited hadron resonances, and other XYZ resonances. 

There are also "leptonic atoms" which substitute positively charged leptons for protons in an atomic nucleus and are bound by quantum electrodynamics (i.e. by electromagnetism) rather than by the strong force, that probably belong in the same discussion (and generally have a mass of less than 4 GeV).

As a prelude, the big bottom line is that there is not a global solution, really, even to any large group of unclassified resonances. Each resonance has to be figured out on its own. It is sometimes quite an epic effort to discriminate between plausible explanations of their structure. 

But there is also no BSM physics. QCD can explain it, but you have to be open to more involved hadron and hadron molecule structures than the vanilla mesons and baryons display. Thus, we are slowly and painfully, but inexorably, reaching a point where essentially all resonances have a Standard Model explanation.

Also, except for toponium, this highly sophisticated analysis and classification of hadron resonances, while it requires lots of data points, doesn't require the extreme high energies of the 13-14 TeV LHC (Large Hadron Collider). 

Generally speaking, all hadron resonances are somewhere between 135 MeV (the lightest pion) and about 30 GeV (a hypothetical six b quark hexaquark), and the lower middle part of this range is very crowded with all sorts of resonances. This is comfortably below the energy scale of even a W or Z or Higgs boson, and is also below the energy scale of a top quark-antitop quark pair. 

Maybe a post just spelling out the possibilities would be a good prelude to a post putting forth the leading theories about which resonances are most likely matches to which possibilities.

Hadronic B Decay Anomalies

The anomaly of the day is an anomaly in a certain kind of B meson decay. I'm very skeptical and think it will go away and is probably due to poor modeling of the Standard Model prediction, but I'll note its existence in this post for further analysis.

The decays B→PP, where the pseudoscalar P is a π or K, have been studied under the assumption of flavour SU(3) symmetry [SU(3)F]. The global fit shows a 3.6σ discrepancy with the Standard Model (SM). 

Separate fits for ΔS=0 and ΔS=1 decays find parameter sets that differ by a factor of 10, suggesting 1000% SU(3)F breaking, significantly larger than the ∼ 30% breaking expected in the SM. This study has been extended to include final states with η and η′ mesons. The resulting global fit, once again under the assumption of SU(3)F symmetry, is worse, with a 4.1σ deviation from the SM. When theoretical constraints |C˜/T˜| = 0.2 or A˜ = 0 are imposed, the fits worsen, with the discrepancy approaching 5σ. These results hint at new-physics contributions to these decays.

Marianne Bouchard, David London, "Anomalies in Hadronic B Decays" arXiv:2608.11298 (August 11, 2026) (Contribution to the Proceedings of the XVI International Conference on Beauty, Charm, Hyperons in Hadronic Interactions (BEACH 2026), 7-12 June 2026, Firenze, Italy).

A Possible Glueball Resonance

A glueball is a strong force bound system without any valence quarks that binds gluons, the carrier bosons of the strong force instead of quarks. A well-established resonance seen in experiments at discovery level significance since 2011, whose internal structure is unclear, is shown in a new preprint to be consistent with a nearly pure pseudoscalar glueball state (i.e. a spin-0, electromagnetically neutral boson with odd parity and no valence quarks).

The properties of glueballs (which depend primarily on a single experimentally measured physical constant, the strong force coupling constant), have been calculated from the very early days of quantum chromodynamics (QCD), which is the Standard Model theory of the strong force. And, there are only a modest number of theoretically possible glueballs. This resonance is on the low end of, but consistent with, the mass predicted for a pseudoscalar glueball.

But distinguishing a glueball resonance from a non-glueball resonance is difficult, and because glueballs are always bosons and share quantum numbers with bosons that include valence quarks, they have a natural tendency to blend with similar bosons making a mere glueball component in a resonance rather than a pure glueball, something that is probably common in reality.

There have only been a few resonances that have been convincingly interpreted as a near pure glueball, and this is one of them. But the fact that this is an analysis of a lone author which has not yet been published in a peer reviewed journal bodes caution in accepting this conclusion as definitive. But to the extent that this analysis holds up, it confirms an important qualitative prediction of QCD (i.e. the existence of glueballs with various quantum numbers that have their predicted masses).

In this work, we take the X(2370) with J^PC = 0−+ as a glueball consists of three valence gluons, and construct a six-quark current based on rigorous current-field duality to obtain the glueball-quark Lagrangian. Then we perform Fierz transformation to bosonize the quark current into a series of three pseudoscalar mesons. At last, we obtain ratios among the partial decay widths of the glueball to three pseudoscalar mesons in a model-independent way, which are compatible with the experimental data from the BESIII Collaboration and support assigning the X(2370) as a glueball.

Zhi-Gang Wang, "Analysis of the X(2370) as a glueball based on rigorous current-field duality" arXiv:2608.03362 (August 4, 2026).

In other physics news, the Higgs boson continues to be consistent with Standard Model expectations in newly observed ways.

A Notable Coincidence Related To The Proton Mass And Charge Radius

There is a functional relationship between the mass of the proton and the charge radius of the proton that is consistent with experimental measurements of those quantities, that doesn't have an obvious cause.

The simple proton mass and charge radius relationship

From @dandb at Physics Stack Exchange on May 5, 2016 (ten years ago). This can also be stated another way:

The charge radius of the proton is almost exactly four times the reduced Compton wavelength of the proton.

The reduced Compton wavelength is a natural representation of mass on the quantum scale and is used in equations that pertain to inertial mass, such as the Klein–Gordon and Schrödinger equations.

Equations that pertain to the wavelengths of photons interacting with mass use the non-reduced Compton wavelength. A particle of mass m has a rest energy of E = mc^2. The Compton wavelength for this particle is the wavelength of a photon of the same energy.

The reduced Planck's constant, h-bar, is Planck's constant divided by 2π. So, this relationship could also be stated as r = 2h/πmc, for Planck's constant h, the proton charge radius r, and the proton mass m.

This relationship is consistent with experimental measurements made to 0.05% precision

The uncertainty in the "predicted" value of the charge radius of the proton from this relationship, which is 0.84124 to five significant digits, is negligible, because the speed of light (c) and the reduced Planck's constant (h-bar) are quantities used to define SI units of measurement which are thus known "exactly" in terms of SI units of measurement, and the mass of the proton is known to the exquisite precision of about one part per hundred billion. See the Particle Data Group table of physical constants.

The Particle Data Group world average value is currently 0.8409(4) fm, i.e. a one sigma range of 0.8405 to 0.8413 This is a relative uncertainty of 0.048% (i.e. about one part per two thousand).

The PDG value is also consistent with a February 11, 2026 measurement of the charge radius of the proton with a relative uncertainty of 0.18% published in the prestigious peer reviewed journal Nature found it to be rp = 0.8406(15) fm, i.e. a one sigma range of 0.8392 to 0.8421 fm. 

So, the conjectured relationship is consistent with the experimentally measured value of the charge radius of the proton. 

At the time that this Physics Stack Exchange post was written, there was a discrepancy between the electron measurement of the proton charge radius and the muon measurement of the proton charge radius, but that has since been resolved. The muon measurement was found to be correct, and the electron measurement was found to have been incorrect due to experimental measurement errors not fully reflected in the stated uncertainty of the measurement.

This "prediction" is also notable because it is a testable hypothesis. As measurements of the proton charge radius grow more precise, we can find out if the experimentally measured value continues to be consistent with this prediction.

For example, if this hypothesis is merely numerology with no deeper meaning, it would be highly likely that it would grow less consistent with the experimental measurement if the experimental measurement's precision were increased by a factor of ten. And, in fact, experiments to do that are on the agenda of the physics community.

Analysis

What makes this relationship surprising?

Since the charge radius of the proton and the mass of the proton are both, in principle, derived quantities in the Standard Model, that this isn't actually a "coincidence" so much as it is a simple relationship arising from Standard Model physics whose source isn't trivially obvious.

The reason that it isn't trivially obvious is that the calculation of the mass and charge radius of the proton in the Standard Model are primarily functions at leading order of (1) the QCD coupling constant (which describes the strength of the "strong force") evaluated with non-perturbative QCD, (2) the mass of the up quark, (3) the mass of the down quark, and (4) the electromagnetic coupling constant. Yet, none of these experimentally measured physical constants have a functional relationship to Planck's constant or the speed of light.

There are comparatively minor contributions to these quantities that tweak their value beyond the leading order values from the masses of the other quarks (especially the strange quark), the weak force coupling constant, the W boson mass, and the CKM matrix elements (especially the  two elements of the nine elements in the matrix involving up-down quark transitions and up-strange quark transitions).

The reason that this relationship is surprising is that there is no known functional relationship between the reduced Planck's constant or the speed of light, and the other experimentally measured determinants of the proton mass and the proton charge radius (such as the Standard Model coupling constants, the quark masses, and the CKM matrix elements).

Three possible explanations

The stack exchange thread linked above contains some speculations as to why this is true, some more credible than others, but they are only speculations. For example, Michell Porter notes that:

Via P.R. Silva (eqn 6), I have run across a heuristic model of the nucleon in which M = 4/R (in natural units). Here R is the radius of the bag in the "bag model". See Xiangdong Ji, "Mass of the hadron", slide 20. I have not found where this argument originates, but a remark in a 1994 paper by Ji (see paragraph beginning "In the chiral limit...", on the final page) hints at it.

One possibility, which is to some extent the default one, is that this numerical coincidence of these two values has no deep meaning or connection and doesn't point to anything. In other words, this relationship just happens to hold for one hadron out of hundreds, for one of a large set of possible combinations of other physical constants that have no actually physical relationship to each other.

Another reason that this could be true is that the contributions of the experimentally measured constants cancel out in the combination of the proton mass and the proton charge radius, since the same experimentally measured constants enter into both calculations.

If true, this would suggest that should be a way of calculating the proton charge radius from first principles that more transparently and obviously reveals this cancelation.

This would be very interesting, would provide us to a deeper understand of the Standard Model and hadron physics. 

It would also suggest that this relationship ought be to generalizable in some way to the relationship between hadron mass and hadron charge radius for many hadrons (hadrons are composite particles made up of quark and/or gluons bound by the strong force of the Standard Model).

A calculation in this form would also have practical use, because the first principles Standard Model calculation of the proton mass has less than one part per thousand precision (vastly less than the precision of the experimentally measured value). And, in general, this would provide a quick and easy way to calculate hadron charge radii (which are no more precise than first principles calculations of hadron masses using current methods, see also here) which could then be compared to experimental measurements of hadron charge radii.

A third possibility, which would be even more grand, is that the values of the physical constants of the Standard Model that go into calculating the mass and charge radius of the proton actually have some deep functional connection to Planck's constant and the speed of light that has not previously been recognized or hypothesized.

Surfaceology

A new technique called "surfaceology" (described in the linked Quanta magazine article) provides a profoundly more efficient method than the path integrals implied by Feynman diagrams to calculate the probability of Standard Model interactions. 

It is also useful in doing calculations in "double copy" approaches to quantum gravity, in which on does a calculation in QCD and "squares" it, to get an answer for a parallel problem in quantum gravity. 

Surfaceology flows from the same line of reasoning as the amplituhedron of theoretical physics superstar Nima Arkani-Hamed (which only works for supersymmetry theories) and was devised by a junior member of his research group, Carolina Figueiredo, in 2022, with a pair of preprints (here and here) first published in September of 2023. But, it works for real Standard Model particles and not just for simplified theoretical physics models.

Further developments in the winter of 2023-2024 described outcomes that were considered with many calculations in Feynman diagram calculations that eventually revealed that these outcomes were effectively impossible called "hidden zeros." Figueiredo and Arkani-Hamed, along with Qu Cao, Jin Dong, and Song He, posted theses findings in a series of preprints.

More efficient calculations that this method facilitates could turn many particle physics and quantum gravity problems that were theoretically possible to calculate, but as a practical matter, impossible to numerically work out, into practically solvable problems, and can very difficult calculations vastly easier to solve.

Hat tip to 4Gravitons.

(opens a new tab

Theoretical X17 Considerations And Related Conjectures

Could the X17 resonance, if it is even real, be an electromagnetically bound light quark-light antiquark meson?

This explanation is much more attractive than a new fundamental particle, as it wouldn't involve beyond the Standard Model physics, and would instead involve a low energy electromagnetically bound up-antiup or down-antidown pair of quarks.

It has to be electromagnetically bound, rather than strong force bound, because a neutral light quark-antiquark pair bound by the strong force, i.e. a neutral pion, has a mass of about 135 MeV, mostly due to the binding energy of the gluons confining them in a hadron. 

This said, this theory has a big problem. 

Why aren't the light quarks confined in a QCD bound hadronic state? 

The only times quarks are not in QCD bound hadronic states that have so far been observed are shortly after top quarks form (because they almost always decay before they can hadronize, although we just learned in 2025 that in rare cases a top anti-top quark pair can form toponium in a QCD bound state the persists very briefly) and in quark-gluon plasma at temperatures corresponding to about 1-2 GeV (i.e. 11-23 trillion Kelvin).

The invariant mass spectrum of e+e− pairs produced in high-energy Pb-emulsion collisions at 160 A GeV at CERN SPS exhibits a complex structure of many resonances resting on top of a broad enhancement at invariant masses below 50 MeV, with the prominent resonance at 19 ±1 MeV providing independent support for the hypothetical X17 particle. 

We show that this complex structure may be coherently described as signatures for the neutral color-singlet qq¯ quark matter in both its deconfined and confined phases. That is, the broad enhancement may arise from thermal annihilation of QED(U(1))-deconfined quarks and antiquarks into e+e− pairs at the phase transition temperature Tc(QED), theoretically estimated to be 4.75 ± 1.2 MeV from the transitional equilibrium condition. The observed 3±1 and 7±1 MeV resonances may correspond to the QED(U(1))-deconfined dd¯ and uu¯ Coulomb bound states near their quark rest masses, respectively, whereas the observed 19 ± 1 MeV resonance may correspond to the QED(U(1))-confined isoscalar QED meson. 

The approximate agreement between the theoretical and the experimental spectrum suggests that both QED(U(1))-confined and QED(U(1))-deconfined neutral color-singlet qq¯ quark matter may have been produced in these high-energy Pb-emulsion collisions. We propose future experiments to confirm or refute these findings.

Cheuk-Yin Wong, "Possible Evidence for Neutral Color-Singlet qq¯ Quark Matter from High-Energy Pb-Emulsion Collisions" arXiv:2604.23473 (April 25, 2026) (21 pages).

Some conjectures

What would work without breaking the rules of the Standard Model, however, is if the 3 and 7 MeVs were light quark-antiquark pairs that were produced and immediately annihilated before  they could hadronize, and if the 19 MeV resonance was an electromagnetically bound positron-electron state (i.e. positronium). Positronium has a ground state mass of 1.022 MeV  (twice the 0.511 MeV mass of an electron or positron), however, with excited states varying in mass by single digit eV amounts per state, which wouldn't generate a single resonance at 17-19 MeV. 

Another possibility is that the observed 3 ± 1 MeV resonances may correspond to the QED(U(1))-deconfined uu¯ Coulomb bound state near its quark rest masses, that the 7 ± 1 MeV resonances correspond to the QED(U(1))-deconfined dd¯ Coulomb bound state and also to uu¯uu¯ Coulomb bound state near their respective quark rest masses, and that the observed 19 ± 1 MeV resonance may correspond to the QED(U(1))-deconfined dd¯dd¯ Coulomb bound state.

The light quark masses, according to the Particle Data Group (admittedly at the 1-2 GeV energy scale and not the low single digit to tens of MeVs energy scale) is as follows:

The rest mass of four d-quarks is 18.8 MeV, which is right where the resonance is observed.

In this hypothesis, these resonances fail to hadronize because the e+e− pairs that produced one or two light quark-antiquark pairs didn't have enough mass-energy to form a 135 MeV neutral pion, so they instead formed one or two deconfined quark-antiquark pairs that quickly annihilate again because the system had enough energy to create the quarks, but not enough energy to create the bound system of quarks and gluons necessary to form a pion. This has the virtue, again, of not requiring any BSM fundamental particles or new forces.

A four quark solution requires angular momentum that wouldn't normally be present in a simple e+e− pair, but if there were two e+e− pairs in close proximity, both with only modest kinetic energy, which is plausible in the context of the complex overall environment of the high-energy Pb-emulsion collisions generating the data here, or the interactions of the full fledged multi-nucleon atoms present in other contexts where there are claimed sightings of the X17 resonance, a coincidence of two low energy e+e− pairs would be expected with some calculable frequency.

This explanation would still be ground breaking, as it would represent a third circumstance, previously unknown and not predicted, where quarks are (briefly) deconfined. But it would be far less radical than most of the alternative explanations.

Does a(0) Evolve Over Time?

The radical acceleration relation (RAR) which is implied by MOND but isn't necessary caused by MOND, holds true for all low-z observations (i.e. nearby galaxies). But this study concludes that while the RAR still holds in intermediate age galaxies (i.e. those that are farther away), that Milgrom's constant a(0) for these galaxies has a numerical value that is a factor of two greater than what it is for low-z galaxies.

The radial acceleration relation (RAR) is a tight empirical correlation between the observed radial acceleration (a_tot) and the baryonic radial acceleration (a_bar) measured across galaxy radii: these two accelerations start to deviate significantly from each other below a characteristic acceleration scale, a0. So far, observational studies of the RAR have predominantly focused on galaxies in the local Universe, leaving its evolution with cosmic time largely unexplored. 

Using high signal-to-noise data from the MUSE Hubble Ultra Deep Field survey, we investigate the RAR with a sample of 79 star-forming galaxies (complete above M* >10^8.8 Msun) at intermediate redshifts (0.33 < z <1.44). We estimate the observed intrinsic acceleration and the baryonic acceleration from a disk-halo decomposition that incorporates stellar, gas, and dark matter components, with corrections for pressure support, using 3D forward modelling. 

We find a RAR in our intermediate-z sample offset from the local relation, with a higher characteristic acceleration scale, a0(z~1) = 2.38+/-0.1* 10^-10 m/s^2, and a larger intrinsic scatter (~0.17 dex). Dividing the sample into redshift bins and refitting the RAR in each bin, we find a characteristic acceleration scale that systematically increases with z. Parametrizing the z-dependence as a0(z)= a0(0) + a1 * z, we obtain a1 = 1.59 +/- 0.1 * 10^-10 m/s^2, providing evidence for a z-evolution. 

We find similar results using various dark matter halo profiles as well as the Modified Newtonian Dynamics framework in our 3D forward modelling. Our results show that the RAR persists at intermediate redshift, with statistically significant redshift evolution of the characteristic acceleration, pointing to a possible evolution of the baryon-missing mass connection over cosmic time.

B. I. Ciocan, N. F. Bouché, J. Fensch, D. Krajnović, J. Freundlich, H. Desmond, B. Famaey, R. Techi, "MUSE-DARK III: The evolution of the radial acceleration relation at intermediate redshifts" arXiv:2604.22613 (April 24, 2026) (Accepted in A&A).

For reference z=0.33 is about 3.7 to 3.8 billion years ago, z=1 is about 7.7 to 8 billion years ago, and z=1.44 is about 9 to 10 billion years ago. The universe is about 13.8 billion years old. A variation of 0.17 dex is about ± 48%. The intrinsic scatter in the recent time SPARC galaxy sample is about ± 8% (0.034 dex), which is about is small as possible given the precision of the astronomy instrumentation involved. Milgrom's constant is about a(0) ≈ 1.2 × 10^−10 m/s^2.

Ciocan (2026), above, and the cluster data, both point to something very like MOND, except that a(0) evolves under certain circumstances to higher values. 

Missing baryonic matter (i.e. matter made up of ordinary atoms) is, at least, a partial explanation and one that could evolve other time. Indeed, it should evolve over time, because over time more baryonic matter ends up in stars, which are easy for astronomers to see, rather than interstellar gas and dust, which are hard for astronomers to see (and hence often called "missing" when it isn't seen and couldn't be seen even if it was there with current instrumentation). Still, missing baryonic matter may not be the entire explanation, because the magnitude of the change in a(0) may not be big enough, and changes in the naively measured value of Milgrom's constant shouldn't be very uniform since some galaxies are forming starts more actively than others (although this may be reflected in the greater dispersion of Milgrom's constant measurements in older samples).

Deur (who bibliography is linked in the sidebar) argues that the missing piece for cluster scale phenomena is the geometry of the mass distributions, by an appealing analogy to similar phenomena in QCD (which is attractive theoretically because in many respects gravity behaves like QCD squared). (QCD stands for quantum chromodynamics which is the Standard Model theory of the strong force that holds hadrons together and indirectly through hadron mediated forces accounts for the nuclear binding force that binds atomic nuclei together.)

Stacy McGaugh at Triton Station has another post about MOND v. dark matter particles (DM) and why the evidence favors something like MOND but the sociology of astrophysics favors dark matter particles.

The search for a final explanation of dark matter phenomena continues, and while toy-model MOND isn't the final solution, it does a remarkably good job over a very wide range of masses. McGaugh is surely right that the final solution looks a lot more like MOND than it does like most DM models, because for DM to describe the universe we see, we need a theory that explains how DM particles consistently form in a way entirely predicted by baryonic mass distribution, which contrary to protests that it has, it hasn't.

Even if a(0) changes over time, it provides a vastly smaller degree of freedom in how galaxy dynamics can vary than DM, especially if the variation is systemic between galaxies and galaxy clusters, or between galaxies over billions of years of time, and not just random.

Calculating Light Meson Masses From First Principles In QCD

How good are current Standard Model calculations at predicting the experimental values of the light meson masses?

A new paper that makes that attempt for most light mesons under 1.5 GeVs of mass (except scalar mesons). And, physicists are finally starting to do a pretty good job of describing the meson mass spectrum which has been an elusive target for decades, even for axial vector mesons, which had long been challenging.

As explained in the introduction:

In the present work we employ the procedure described above to compute the masses of relatively light mesons, namely mesonic states no heavier than about 1.5 GeV. Specifically, for mesons composed of u and ¯d quarks, we compute the masses of π±, ρ(770), b1(1235), a1(1260), π±(1300), and ρ±(1450). For the strange sector, we calculate the masses of the states K±, K∗(890), K1A, K1B, and K±(1460). 

In general, the computed masses are in good agreement with the experimental values. In fact, our findings represent a definite improvement over the results obtained within the standard rainbow-ladder truncation [84], where the masses of axial-vector mesons and radially excited states tend to deviate considerably from the observed values.

Notably, this omits the f(0)(500) scalar meson a.k.a. the sigma meson and seven other true scalar mesons with masses under 1.5 Gev. The other omitted scalar mesons are the f(0)(980), f(2)(1270), f(1)(1285), f(0)(1370), f(1)(1420), f(2)(1430) and f(0)(1500). This may be because their internal structures are less well understood.

The actual procedure used is too technical to discuss at this blog, which is aimed at an education layman readership.

The money chart is as follows:

With the exception of spin-1 kaons (where the relationship is inverted for some reason), the experimental values (in red) tend to be at the very high end of the theoretically predicted values using their methods (in blue), and their predictions, in turn, tend to be more massive than those made using a previous "rainbow ladder" truncation method (in green).

The predictions (and measurements) of excited state light meson masses are much less precise than the predictions (and measurements) of ground state light meson masses.

The Proton Spin Puzzle

The total spin of the hadrons can be determined trivially by simply adding up the 1/2 spins of its valence quarks, with possible plus and minus values for each one. Each combination of plus or minus 1/2 spins adds up to a total spin, and each possible sum of spins for the valence quarks equals the possible total spins of hadrons with those valence quarks. Minimal values for a set of valence quarks are more stable, so protons and neutrons having a minimal possible combination of spins (i.e. they have spin 1/2 equal to 1/2 + 1/2 -1/2) since it is stable. All non-minimal spin sums are unstable hadrons

Surprisingly, however, this simple formula doesn't reflect the actual spin of the full array of valence quarks, sea quarks, and gluons that add up to spin-1/2 in an actual proton. Reality gets to the same result, but in a much more complicated way.

A new PhD dissertation (250 pages) exhaustively examines this puzzle and uses a novel method to try to solve it with a formula (i.e. analytically) rather than with a numerical approximation, extrapolating down to the 3 color, 6 flavor reality, from more complex models with larger numbers of colors and flavors.

The proton spin puzzle denotes the challenge of describing the proton's spin in terms of the angular momenta of the quarks and gluons which comprise it. These quarks and gluons carry a fraction x of the proton's momentum. Contributions from small-x quarks and gluons, which only possess a little of the proton's momentum, are difficult to measure, since this requires very high energy experiments. Furthermore, early theoretical work in the 1990s predicted substantial contributions to the proton spin from these small-x particles. We need theoretical control over this corner of phase space in order to resolve the spin puzzle.

In this dissertation, we build upon an existing framework for studying spin at small-x. Previously, several sets of small-x evolution equations were derived in this formalism -- one in the large-N(c) limit and one in the large-N(c) & N(f) limit. Here N(c) and N(f) are the numbers of quark colors and flavors [ed. there are three colors, three anti-colors, and six flavors in the Standard Model]. These equations were numerically solved but no analytic solutions had been found. In this dissertation we detail the construction of such analytic solutions, first in the large-N(c) limit and then in the large-N(c) & N(f) limit, after deriving an important correction to the existing large-N(c) & N(f) equations due to the contributions of quark-to-gluon transition operators.

From the solutions constructed here, we can predict the behavior of the quark and gluon helicity distributions at asymptotically small-x (and large-N(c) or large- N(c) & N(f)), both as a general power law and further as explicit analytic expressions in the asymptotic limit. Our solutions also allow us to predict all four polarized DGLAP anomalous dimensions in the same limits, yielding expressions exact to all orders in the strong coupling. The expansions of our predictions agree completely with the full extent of existing finite-order calculations, to three loops.

Jeremy Borden, "Searching for the Proton's Missing Spin: Small-x Helicity Evolution Equations and Their Analytic Solutions" arXiv:2603.20906 (March 21, 2026).

The dissertation's introduction does a good job of laying out the puzzle:

A relatively naive — but in some ways still very successful — model takes the proton to be made of three quarks (the general class of particles we call baryons are described in this way as bound states of three quarks). The quarks in this model are nonrelativistic and, like the proton, are spin-1/2 fermions. In such a model, it is easy to intuitively understand the proton’s spin — that is, its intrinsic angular momentum. Two of the constituent quarks have their spins pointed in the direction aligned with the proton’s spin, while the third constituent quark’s spin is in the opposite direction, as visualized in Fig. 1.1. 

In this model 100% of the proton’s spin is accounted for by the spin of the quarks. Perhaps unsurprisingly, more sophisticated models of the proton were also developed (see e.g. the bag model of [10]). But even these sophisticated models which tried to accommodate more complicated phenomena like special relativity and confinement still predicted that a substantial quantity of the proton’s spin must be carried by the quark spins, typically somewhere on the order of 60%. 

Figure 1.1: Naive quark model of the proton P with two quark q spins aligned and one anti-aligned relative to the proton spin. 

Then in the late 1980s, the European Muon Collaboration (EMC) utilized polarized muon-proton scattering experiments to measure the net amount of the proton’s spin carried by the quark spins. The shocking result was a measured value of around 6% [11,12]. Even allowing for the maximal experimental uncertainties, this was an irreconcilable difference compared to theoretical predictions. Thus began the proton spin puzzle. The majority of the proton’s spin could not be accounted for by the theoretical models of the time. 

The good news, however, is that today we have many powerful tools at our disposal to better understand the rich internal structure of the proton, chief among them Quantum Chromodynamics (QCD). Beginning in the 1970s, QCD began to emerge as the presumptive theoretical description of the strong force — the force that binds protons and neutrons together in atomic nuclei, and as would come to be understood, the force that governs the complicated internal structures of the proton and neutron themselves, along with a host of other strongly-bound particles. 

QCD is a non-abelian SU(Nc) gauge theory which describes the fundamental degrees of freedom of the strong force as quarks and gluons. The quarks of QCD are spin-1/2 fermions with fractional electric charges, although they are not exactly the same as the ‘constituent’ quarks shown in Fig. 1.1. There are six flavors of quarks in the Standard Model, varying in their masses and electric charges. In addition to electric charge, the quarks are also charged under the strong force. This color charge comes in Nc = 3 varieties called red, green, and blue (and the antiparticles of the quarks, the antiquarks, can carry anti-red, anti-green, or anti-blue color charge). The quarks form a color triplet and transform under the fundamental representation of SU(Nc). Meanwhile gluons — the strong-force-carriers — are spin-1 bosons that also carry a net color charge, a combination of color and anti-color (the color octet), and transform under the adjoint representation of SU(Nc). Notably the fact that gluons are charged under the strong force means they can interact with other gluons. This is a crucial difference from abelian theories like quantum electrodynamics [13] where the force-carrying particles (photons) do not self-interact. 

Among the consequences of the gluonic self-interactions in QCD is asymptotic freedom [14,15], a remarkable property that tells us the particles of QCD interact very weakly at short distances (or large momentum transfer). This has critical implications for perturbative QCD calculations. The strong coupling — the physical parameter that controls the strength of the force — becomes relatively small at these short distance scales, and so we have a small dimensionless parameter in which we can make a reliable perturbative expansion. This perturbative regime of QCD is the backdrop for this entire dissertation and so the applicability of perturbation theory is critical here. Note however, that the running of the strong coupling — that is, how the coupling changes with momentum scale — also has important implications in the low-momentum/long distance regime. Whereas at high momentum scales the coupling is smaller and we can employ perturbation theory, at low momentum scales the coupling becomes very strong and perturbative methods break down. 

This also hints at the perplexing notion of confinement [16], whereby free color charges cannot be isolated. They are always confined in color neutral combinations. A complete theoretical understanding of confinement is still lacking. 

A particularly useful model of the the proton (and other hadrons) at high energy is Feynman’s parton model [17], where the proton is taken to be a system of point-like particles called partons. Particularly effective in understanding the results of deep inelastic scattering (DIS) of electrons and protons1 at SLAC [18], the model treats the proton (in a frame where the proton is moving ultrarelativistically) as a collection of these co-moving point-like partons which do not interact with each other. 

When colliding with the electron, the system of partons interacts with the electron probe incoherently. Feynman was agnostic about what particles these partons might be but the interpretation that emerged, and the one we still use today, is that they are the quarks and gluons of QCD. The parton framework serves as a powerful tool for understanding how the properties of the proton emerge from the properties of the intrinsic QCD degrees of freedom. But note that we are not limited to the three quark model like that in Fig. 1.1. Instead we can have many partons and we will often label them with the Bjorken-x variable, which corresponds to the longitudinal momentum fraction of a given parton relative to the parent proton. Intuitively, a parton could have as little as zero longitudinal momentum (x = 0) and as much as the full momentum of the proton (x = 1) and so 0 < x < 1. 

The modern picture of the proton’s structure that has emerged holds that there are indeed three quarks that live at relatively large x (that is, close to x = 1) — these are the valence quarks. But we also have a rich sea of quarks and antiquarks at smaller values of x. These sea quarks can fluctuate in number, as particle-antiparticle pairs are created or annihilated, and the interactions among the sea quarks are mediated by gluons, which can themselves split into more gluons or recombine with each other. The interior structure of the proton is thus much less trivial than the naive diagram in Fig. 1.1, but could instead look (for illustrative purposes only) more like the representation in Fig. 1.2.

Figure 1.2: A more complicated but realistic illustration of the proton’s structure. In addition to the three valence quarks (the large spheres), we now have a sea of quarks and antiquarks (the smaller colorful spheres) along with many gluons (the corkscrew lines). The spins of the particles are not represented here, but each quark and gluon can contribute its spin — and also its orbital angular momentum — to the proton’s spin. 

To describe the spin of the proton, we can make the following decomposition: 

Sq +Lq +SG+LG = 1/2. (1.1) 

This is the Jaffe-Manohar sum rule [19]. Eq. (1.1) says that we can break the spin of the proton, which is 1/2 in units of ℏ, into the spins S and orbital angular momenta (OAM) L of the quarks q and gluons G.

Predicting Heavy Hadron Masses

This paper makes mass predictions for a huge number of three and five valence quark hadrons (in both ground states and excited states) made by both traditional methods from the literature and AI models, producing multiple estimates by different methods for each hadron considered. It is mostly a pattern recognition exercise, rather than a set of calculations from QCD first principles. It predicts several hundred composite particle masses.

This is easier for baryons (i.e. half-integer spin fermions) than for mesons (i.e. integer spin bosons) because baryons have far fewer quirky exceptions to general rules that flow, in part, from different mesons blending into each other, which is something that baryons don't do.

One observation is that these several hundred heavy baryons (in the broad sense of half integer spin hadrons, rather than the narrow sense of three valence quark hadrons) fill a pretty narrow range of masses, with the lightest having a mass of about 1.5 GeV, the heaviest having a mass of 11.4 GeV, and most of the predicted masses bunching up in the middle, with more than 4 GeV and less than 10 GeV. The lightest pentaquarks are a bit over 4 GeV.

Given that there are only a handful of possible quantum numbers for each hadron, the experimental task of distinguishing one heavy baryon from another would be challenging, with many possibilities near any given mass. 

While experimental mass measurement of heavy baryons typically have uncertainties of a few MeV, the uncertainties in the theoretical mass predictions are much greater. The theoretical uncertainties of the predictions range from about 100 to 2000 MeV, with most in the range of about 450 to 1200 MeV. The differences between theoretical mass predictions methods for the same hadron also frequently exceed the combined claimed uncertainties in the predictions, however, so the uncertainties are probably underestimated.

Since it is easy to make predictions if they are vague enough, which makes it easy for the predictions to be consistent with the experimentally observed values, the significance of these models shouldn't be exaggerated. They are making very ballpark estimates based upon very general considerations. 

But because it is so comprehensive, this is still somewhat useful in winnowing down candidates for a particular observed resonance with a particular observed mass from several hundred possibilities to perhaps a few dozen likely candidates of similar mass, which can be narrowed down further with measurements of the resonances spin, charge, and other quantum numbers to perhaps a dozen or fewer candidates.

In this article, we use two different methods for studying the mass spectra of fully-heavy baryons and pentaquarks. 

In the first section, we use state-of-the-art machine learning methods, such as deep neural networks and the Particle Transformer model architecture, to predict baryon masses directly from their quantum numbers, based on experimental information on hadrons from the Particle Data Group (PDG). We use this data-driven approach for the case of fully heavy baryons, and a large number of exotic pentaquark states, going much beyond the well-known P+c(4380) and $ P_c^+(4457) candidates. Subsequently, we extend the Gürsey-Radicati mass formula to incorporate the contributions of charm and bottom quarks, enabling analytical calculations for both ground and radially excited states of baryons and pentaquarks. 

The results obtained from both approaches demonstrate strong agreement with experimental data where available and make predictions for a number of unobserved states, including higher radial excitations. By addressing the question through both data-driven prediction and analytical modeling in different frameworks, this study offers complementary insights into the mass spectrum of conventional and exotic hadrons, guiding future experimental searches.

S. Rostami, A. R. Olamaei, M. Malekhosseini, K. Azizi, "Comprehensive Mass Predictions: From Triply Heavy Baryons to Pentaquarks" arXiv:2603.11259 (March 11, 2026).

From Quarks To Chemistry

Protons, neutrons, and hundreds of other much less stable hadrons  (i.e. systems of quarks and/or gluons bound by the strong force) are understood quite well with the Standard Model of Particle Physics, although there are challenges in understanding scalar mesons, axial vector mesons, and hadrons with four or more quarks, as well as in distinguishing true hadrons with four or more quarks from "hadron molecules", and predicting the full spectrum of hadrons from first principles.

Protons and neutrons in atomic nuclei are not bound together primarily by the strong force itself. Instead, the nuclear binding force between protons and neutrons in an atomic nucleus is the sum of the forces arising from the exchange of several kinds of light mesons (primarily pions but also other light mesons including kaons).

We are not quite there in terms of using Standard Model physics to explain the physics and chemistry of atomic nuclei, although we are getting closer to achieving this vertical integration of subatomic and atomic scale phenomena, and we making great progress on this front. Part of the hold up is the challenge of explaining "parton distribution functions" (PDFs), a property of hadrons that, in principle, can be worked out from first principles with Standard Model physics. But until the past few years, PDFs have actually been determined almost entirely from brute force raw data collection and organization from particle accelerator data.

We also mostly understand the way electrons interact with atomic nuclei, which is almost entirely an electromagnetic quantum electrodynamics (QED) phenomena.

The next layer above understanding atoms, is chemistry, which pertains mostly to how atoms interact with each other, much of which ultimately flow from the physics of atomic nuceli.

We extend the QCD Parton Model analysis by employing a factorized nuclear structure model that explicitly accounts for both individual nucleons and correlated nucleon pairs. This novel framework establishes a paradigm that directly links the nuclear physics description of matter (in terms of protons and neutrons) to the particle physics schema (in terms of quarks and gluons). 

Our analysis of high-energy data from lepton Deep-Inelastic Scattering, Drell-Yan, and W/Z production simultaneously extracts the universal effective distribution of quarks and gluons inside correlated nucleon pairs, and their nucleus-specific fractions. 

The successful extraction of these universal distributions marks a significant advance in our understanding of nuclear structure, as it directly connects nucleon-level and parton-level quantities.

Fredrick Olness, "Bridging the Gap: Connecting Atomic Nuclei to Their Quantum Foundations" arXiv:2511.15659 (November 19) ("Talk presented at the 32nd International Workshop on Deep Inelastic Scattering and Related Subjects (DIS 2025), Capetown, South Africa, 24-28 March 2025. To appear in the proceedings").

Because Deur Is Awesome, Even At His Day Job

Alexandre Deur's side hustle, described in the sidebar link, is his work on a gravitational explanation for dark matter and dark energy phenomena, which would solve several of the greatest unsolved problems in physics.

His day job is as a QCD physicist at Jefferson Lab, a U.S. Department of Energy particle physics facility in Newport News, Virginia. There, his progress in determining the value of the least accurately known Standard Model model coupling constant, and confirming that its running with energy scale is consistent with the Standard Model, is also a good thing. 

Unsurprisingly, the research done by him and his colleagues confirms that the Standard Model's running of the strong force coupling constant of quantum chromodynamics determined experimentally confirms the Standard Model over a huge range of energy scales (from hundreds of MeVs to about 14,000,000 MeV).

The strong force coupling constant is usually quoted with values converted using the beta-function that describes its running with energy scale in the Standard Model to the Z-boson mass of 91.188 ± 0.002 GeV (according to the Particle Data Group, inverse error weighted world average measurement). Its world average value converted to that energy scale is 0.1180(9).

The numerical values shown below are in a normalized scale, so the numerical value doesn't match the familiar number.

We discuss how the Bjorken sum rule allows access to the QCD running coupling αs at any scale, including in the deep infrared IR domain. The Bjorken sum data from Jefferson Lab, together with the world data on αs reported by the Particle Data Group, allow us to determine the running of α(s)(Q) over five orders of magnitude in four-momentum Q. We present two possible future measurements of the running of α(s)(Q) using the Bjorken sum rule: the first at the EIC, covering the range 1.5 < Q < 8.7 GeV, and the second at Jefferson Lab at 22 GeV, covering the range 1.0 < Q < 4.7 GeV.

A. Deur, "The strong coupling from the IR to the UV extremes: Determination of α(s) and prospects from EIC and JLab at 22 GeV" arXiv:2510.19556 (October 22, 2025) (Contribution to the proceedings of the "QCD at the Extremes" workshop, Sept. 1-11 2025).

The paper above discusses how proposed low energy experiments at the electron-ion collider at the Brookhaven Lab on Long Island, New York (EIC) and JLab would greatly reduce uncertainties in the measurement of the strong force coupling constant measurement at low energies (i.e. below 5,000 MeV) as shown by the chart below.

Does Non-Perturbative QCD Have A Cosmological Constant Analog?

A new paper explores a potential parallel between non-perturbative quantum chromodynamics (the physics of the strong force that binds quarks into hadronic structures) and gravity. This isn't entirely surprising, as both are non-abelian gauge theories. And, it suggests that features like the cosmological constant may have a natural source in a non-abelian quantum gravity theory.

Einsteins gravity with a cosmological constant Λ in four dimensions can be reformulated as a λϕ^4 theory characterized solely by the dimensionless coupling λ∝G(N)Λ (G(N) being Newton's constant). The quantum triviality of this theory drives λ → 0, and a deviation from this behavior could be generated by matter couplings. Here, we study the significance of this conformal symmetry and its breaking in modeling non-perturbative QCD. The hadron spectra and correlation functions are studied holographically in an AdS(5) geometry with induced cosmological constants on four-dimensional hypersurface. 

Our analysis shows that the experimentally measured spectra of the ρ and a(1) mesons, including their excitations and decay constants, favour a non-vanishing induced cosmological constant in both hard-wall and soft-wall models. Although this behavior is not as sharp in the soft-wall model as in the hard-wall model, it remains consistent. Furthermore, we show that the correction to the Gell-Mann-Oakes-Renner relation has an inverse dependence on the induced cosmological constant, underscoring its significance in holographic descriptions of low-energy QCD.

Mathew Thomas Arun, Nabeel Thahirm, "On the role of cosmological constant in modeling hadrons" arXiv:2510.06380 (October 7, 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.