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.

The SM Expectation For Higgs Boson Pair Production

A new study makes a state of the art prediction of the Higgs boson pair production rate from gluon fusion in the Standard Model. 

Some day when Higgs boson pair production experiments are about 1000 times more precise than they are today, this prediction can be compared to the experimental data, which is one way to determine is the Higgs boson self-coupling is consistent with the Standard Model prediction or if it instead has a value more consistent with a beyond the Standard Model value. 

Gluon fusion is one of the main mechanisms by which Higgs bosons and Higgs boson pairs are created, and combined with Standard Model predictions for the other possible mechanisms, can be compared to the actual experimentally observed rates of Higgs boson pair production at particle collider experiments.

Despite the lengths of many authors go to in order to make the calculation that considers all sorts of higher order corrections, however, the uncertainties are still large. 

But the experimental measurements currently aren't any better. They show that the actual rate of Higgs boson pair production is merely less than 2.4 times the Standard Model expectation (i.e. less than about 87.3 fb) with a 95% confidence interval. Higgs boson pair production rates are 0.06% of the overall Higgs boson production. In the Standard Model, Higgs boson pair production predominantly (90%) comes from the gluon fusion mechanism that the new study calculates considering all feasible to calculate factors.

Total Higgs boson production at 13 TeV is about 55.6 pb (+6% -8% uncertainties at one sigma) of which 48.4 (87% of the total) comes from gluon fusion with the remaining 7.2 pb coming from six other main production mechanisms. Higgs boson pair production at 13 TeV using a gluon fusion rate of 33 fb is 36.36 fb, of which 3.36 fb come from five other main non-gluon fusion production mechanisms. And, 1 picobarn (pb) = 1,000 femtobarns (fb).

This study (see below) concludes that double Higgs boson pair production at 13 TeV from gluon fusion is actually 30.4 fb (but subject to a roughly + 10% -23% uncertainty, so its is consistent with the earlier less exhaustively calculated result quoted in the Particle Data Group review below the fold which has roughly the same uncertainty on a percentage basis; the new result has a central value which is about 8% smaller than the old one). A ± 0.2 GeV change in the Higgs boson mass from 125.0 GeV shifts the predicted value by only about + 0.3% (if it is lighter) - 0.4% (if it is heavier), so the gluon fusion Higgs boson pair production rate isn't very sensitive to tweaks to the Higgs boson mass within the current range of uncertainty, but is probably a little bit less than 30.4 fb.

The paper and its abstract are as follows:

In this contribution, the higher-order QCD and electroweak corrections to Standard Model Higgs boson pair production via the gluon-fusion mechanism, gg→hh, are summarized and the different sources of theoretical uncertainty are assessed. The discussion includes finite top quark mass effects, matching to parton showers, approximate NNLO and N3LO QCD corrections, NLO electroweak effects, and uncertainties associated with the top quark mass scheme and perturbative scale choices. In addition, we provide an updated state-of-the-art recommendation for the inclusive gluon-fusion Higgs boson pair production cross section and the corresponding Higgs boson pair invariant-mass distribution.

Ajjath A H, et al., "Higgs Boson Pair Production via Gluon Fusion: Higher-Order Corrections and Theoretical Uncertainties" arXiv:2609.04868 (September 4, 2026) (Contribution to CERN Report 5 approved by LHC Higgs Working Group, Working Group 4 Report number LHCHWG-2026-010).

It concludes that:

Notably, while this prediction is sensitive to the Higgs boson mass, it is not sensitive enough to meaningfully distinguish Higgs boson masses experimentally because the differences due to the Higgs boson mass are smaller than the uncertainty in the prediction.

The conclusion explains:

This report has summarized the current status of precision predictions for Standard Model Higgs boson pair production via gluon fusion. The discussion brings together NLO QCD calculations with full top quark mass dependence, approximate NNLO QCD predictions, N3LO QCD corrections and soft-gluon resummation, NLO electroweak corrections, and details the main sources of theoretical uncertainty entering the theoretical prediction. 

The final recommendations provide state-of-the-art SM reference predictions for phenomenological studies and LHC analyses. They combine higher-order QCD (exact NLO, approximate NNLO and N3LO + N3LL) and EW (NLO) corrections, together with a full uncertainty budget. The combined inclusive cross sections, including the dominant uncertainty associated with the top-quark mass scheme, are collected in Table 12, while their dependence on the Higgs-boson mass is given in Table 13. Additionally we provide differential distributions in m(hh) (Section 7.5), along with corresponding K-factors from the higher-order calculations. These numbers should be used as the definitive predictions of this report, superseding the intermediate results shown in the preceding sections where different input parameters or PDF choices are used. It is worth noting that while the present work does not reduce the overall uncertainty with respect to the previous recommendation, its central prediction includes N3LO+N3LL QCD corrections in the HTL, NLO electroweak effects and updated PDF sets, and should therefore provide a more accurate reference value. 

Further improvements in the SM prediction will come from reducing uncertainties associated with finite top quark mass effects and mass-scheme choice, extending fully differential predictions with consistently combined higher-order QCD and electroweak effects, and updating the recommendations as parton distributions and input parameters evolve.

Background from the Particle Data Group (with somewhat icky formatting) appears below the fold. 

Handedness

 

From here.

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.

New Combined Standard Model Constant Measurements

A new study tries to extract several Standard Model Constant measurements from the same data set and obtains results generally consistent with prior efforts to measure the same constants, but with greater uncertainty than the state of the art measurements of these quantities.

The Latest Global Electroweak Fits Of Standard Model Physical Constants

A global electroweak fit combines experimentally measured values of Standard Model physical constants with the theoretical relationships between those constants in the electroweak sector of the Standard Model to determine where, within the range of uncertainties in the experimental measurements the true value of those physical constants is most likely to be. Basically, it uses theory to eke out a bit more precision in our determination of these constants than the measurements make possible in isolation.

The fact that it is possible with experimentally measured value of Standard Model physical constants without serious tensions (which it is) also provides a global test of the consistency of the Standard Model with reality.

The latest paper using up to date experimental data to make a global electroweak fit of Standard Model physical constants can be found here. The discussion of how the input values are chosen (basically, an educated best summary of the data to date) in the paper is also noteworthy.

The global fit of the Z boson mass is 91.1882 ± 0.0019 GeV and the global fit of the Z boson width is 2.4945 ± 0.0006 GeV.

The global fit of the W boson mass is 80.3584 ± 0.0048 GeV and the global fit of the W boson width is 2.090 ± 0.001 GeV.

The Higgs boson mass is 125.13 ± 0.11 GeV. The Standard Model expectation for the Higgs boson width is 4.10 ± 0.06 MeV; a complete global electroweak fit of the data produces 3.78 + 0.30 − 0.27 MeV, which is consistent with the Standard Model expectation. The couplings of the Higgs boson in an electroweak global fit are within roughly 1% ± 1% of the Standard Model expectation. 

The global fit of the charm quark pole mass (in the MS scheme) is 1.273 ± 0.003 GeV.

The global fit of the bottom quark pole mass (in the MS scheme) is 4.183 ± 0.004 GeV.

The global fit of the top quark pole mass is 172.67 ± 0.56 GeV.

The global fit of the strong force coupling constant at the Z boson squared energy scale is 0.1179 ± 0.0009.

The global fit of the effective leptonic weak mixing angle is sin^2(theta) = 0.23149 ± 0.00005.

The Modest Excess Higgs Boson Production Explained

The Standard Model is stochastic (i.e. probabilistic) and not deterministic. It doesn't say, if you do X then Y will happen. It says, if you do X, Y with happen Z percent of the time.

One of the many things that the Standard Model predicts is the Higgs boson production rate, as a probability distribution of the rate at which Higgs bosons are produced in given circumstances. The calculation is in the form of an infinite series of terms with leading order, next to leading order, next to next to leading order, etc. terms.

You haven't read much about the physics of Higgs boson production at this blog because its a lot less simple and intuitive than Higgs boson decays, which are much more straightforward and rely on simpler, less complicated processes and rules. This makes Higgs boson production harder to write good blog posts about than Higgs boson decays. Also, the experimental anomalies compared to Standard Model predictions for Higgs boson production have been less striking, with more uncertainty and not very striking discrepancies, even though the discrepancies in Higgs boson production rates have been quite persistent.

In practice, scientists calculate the Standard Model prediction for the Higgs production rate with as many terms as are practically feasible for them to calculate, and then they try to estimate the uncertainty arising from the omitted terms as best they can.

Usually, each slight incremental improvement in the accuracy of the calculation takes disproportionately more work to calculate than the amount of work that was necessary to make the previous improvement of that magnitude. 

But, now and then, scientists unexpectedly find a previous omitted term from their calculations that is really important, although figuring out which terms will be especially fruitful to include is still at a more art than science level right now. Research programs like the amplituhedron approach and related developments from it are trying to bring more science to that search, but we aren't quite there yet.

Experiments since 2012, when the Higgs boson was first discovered, have shown that Higgs production usually exceeds the rate calculated by the best available Standard Model prediction calculations, although either not by a statistically significant amount, or with only a mild statistical tension with the best available predicted value for the Standard Model Higgs boson production rate.

Initially, some scientists though that this could be because the Higgs boson was detected sooner than it would have been otherwise because of a statistical fluke of higher than expected Higgs production. At first, that was a plausible proposal.

But it has been 14 years now, so it probably wasn't that, because the slight bias towards higher the expected Higgs boson production rates hasn't completely gone away, as the sample size of Higgs bosons detected has surged and reduced statistical uncertainties (but not always systemic uncertainties in the measurements of the Higgs boson production rates). 

Of course, like every anomaly in high energy particle physics, some theorists have, instead, tried to explain this persistent, not very large anomaly, with beyond the Standard Model physics.

But, a new paper now explains most or all of what has been going on. It turns out that the Higgs boson that physicists have observed is behaving more like than Standard Model Higgs boson to higher precision than ever, once again.

The new paper recalculates the Standard Model predicted Higgs boson production rate and determines that some next to leading order terms contributing to the predicted Higgs boson production rate were more important than had been expected. It turns out that these omitted terms can led to up to 10% more Higgs bosons being produced than would have been predicted without them in some circumstances.

Including the omitted terms explains most or all of the excess of experimentally observed Higgs boson production over the old calculation of the SM predicted value. This also, by the way, tends to imply that the uncertainties in the old experimental measurements were probably overestimated, which is a common reality in electroweak physics (as opposed to QCD or astronomy where uncertainties are often underestimated).

This new discovery feels like a reprise of the comparisons between the experimentally measured values of muon g-2 and state of the art calculations of the Standard Model prediction. In both cases, the gap has been mostly bridged by improving the quality of the calculations of the Standard Model predictions with an immense amount of hard calculation work, rather than by improving experimental accuracy or discovery new beyond the Standard Model physics. And, like the muon g-2 discrepancies, the part of the Higgs boson production calculation that has impaired the accuracy of the Standard Model prediction has mostly been the very hard to calculate strong force/hadronic/quark based part of what is primarily an extremely precise electroweak calculation.

The new paper and its abstract are as follows:

We present the mixed QCD-electroweak corrections to Higgs boson pair production in the quark-antiquark channel. 

The virtual amplitudes are computed fully analytically using the method of differential equations. We determine the integration constants by matching our expressions to the large mass expansion limit of the canonical integrals. We implement the results in the POWHEG-BOX framework for phenomenological studies. 

The corrections are found to have a significant impact on the shapes of differential cross sections, reaching up to +10% for the invariant mass distribution of the Higgs boson pair near the production threshold. This channel has not been considered before in calculations of the next-to-leading order electroweak corrections to Higgs boson pair production.

Marco Bonetti, Gudrun Heinrich, Philipp Rendler, William J. Torres Bobadilla, "Electroweak corrections to Higgs boson pair production: The quark channel" arXiv:2606.25928 (June 24, 2026) (contribution to the proceedings of Loops and Legs in Quantum Field Theories 2026, Bayreuth, Germany).

The new paper above is a physics conference summary of a more detailed paper on the same topic released in January of this year.

Standard Model Muon g-2 Calculation Closely Matches Experimental Data

The most accurate ever calculation of the Standard Model predicted value of muon g-2 matches the world average experimentally measured value to 0.7 sigma (with the prediction and the experimental measurement having a precision of 310 and 124 parts per billion, respectively).

The new theoretically calculated value for muon g-2 is: 

aμ = (116,592,052 ± 36) × 10−11.

The most precise available experimental measurement is as follows:

Fermilab (2025): (116,592,070.5 ± 14.8) × 10−11.

The difference is (18.5 ± 38.9) × 10−11

The relative experimental result has an uncertainty of 0.127 ppm. The new calculation of the Standard Model expected value has a relative uncertainty of 0.31 ppm.

The error weighted experimental world average, which has a relative uncertainty of 0.124 ppm is: 

(116,592,071.5 ± 14.5) × 10−11. 

This final result is recapped in an exhaustive final muon g-2 experimental data report at arXiv:2606.17323.

The difference between the world average and the new SM prediction calculation is 

(28.5 ± 38.8) × 10−11, which is 0.7 sigma (which is still closer than than one sigma expected by a random distribution of uncertainties if the results are identical).

This global test of the Standard Model (which implicates all three of its forces) at low energies passes with flying colors.

For 50 years, the standard model of particle physics has been very successful in describing subatomic phenomena. In the past quarter of a century, this was challenged by a mismatch between its predictions and precision measurements of the anomalous magnetic moment of the muon, a(μ). This disagreement was eventually reconciled, first through a determination in an ab initio lattice calculation of the most uncertain theoretical contribution, the leading-order hadronic vacuum polarization (LO-HVP), a(μ)^(LO-HVP) and subsequently by experimental results and updates of the reference standard-model predictions using lattice results for a(μ)^(LO-HVP).

Here we present a new calculation for this crucial quantity, obtaining 

. This reduces the uncertainty by a factor of 1.6 compared with our earlier computation. We use a hybrid approach that includes a small, long-distance contribution from experiments in a low-energy regime in which they all agree. Our approach combines the strengths of experimental and lattice data in different energy ranges, achieving better precision than with either alone. Our lattice quantum chromodynamics (QCD) simulations are performed on finer lattices . . . allowing for an even more accurate continuum extrapolation. 

Combined with the calculations of the other standard-model contributions . . . our result leads to a prediction that differs from the recent measurement of a(μ) by only 0.5 standard deviations. This provides a notable validation of the standard model to 11 digits.

A. Boccaletti, et al., "Hybrid calculation of hadronic vacuum polarization in muon g − 2 to 0.48%." 653 (8814) Nature 373 (April 22, 2026) (open access) DOI: 10.1038/s41586-026-10449-z

While the hadronic part of the calculation accounts for a fairly modest part of the total value, it is the source of almost all of the uncertainty in the calculation:

Cold Dark Matter Still Doesn't Work

The missing local baryon problem

Stacy McGaugh at Triton Station explores one of the many bits of empirical evidence, which he calls the missing local baryon problem, that really convincingly disfavors any kind of cold dark matter paradigm.

Basically, he utilizes a proof by contradiction. 

He assumes a standard cold dark matter model, analyzes the data on the share of the mass of galaxies and galaxy clusters that is made up of ordinary baryonic matter (which is about 15.7% in the cold dark matter paradigm), in line for the percentage for the whole universe in that paradigm. Then, he shows how the proportion of baryonic matter gets systemically lower in a very predictable manner as the absolute amount of baryonic matter in a galaxy falls.

The problem is that in the cold dark matter paradigm, galaxy clusters form as galaxies cluster together, and larger galaxies form from the merger of smaller galaxies. But this leaves open the question of how the proportion of baryonic matter in a pair of merged galaxies that form a larger galaxy can be systemically and precisely greater in a merged larger galaxy than it was in any of the smaller galaxies whose merger formed it.

Keep in mind that Standard Model physics demonstrates that in all but ultra-extreme circumstances (which haven't existed since the first few seconds after the Big Bang, at most) the total number of baryons in any system (less the total number of anti-baryons in any system) is constant (which has been experimentally confirmed to extreme precision), and that baryons profoundly outnumber anti-baryons in the universe (on the order of 10^10 to one), so there is no plausible physical mechanism by which new baryons are being created in galaxy mergers.

Indeed, even the proportion of the baryonic mass of the universe of each kind of atomic element, something that can only occur in nuclear fission and nuclear fusion reactions that happen mostly in mature stars, has changes only incrementally from the proportions of those atoms predicted to have been present fifteen minutes after the Big Bang, and even then, in amounts and by mechanisms mostly associated with the nuclear physics of stars, that are reasonably well understood. This strongly reinforces the idea that the new baryons aren't being created in galaxy mergers.

So, the shortfall of baryons in a dark matter particle paradigm, that is present in every system smaller than a galaxy cluster, would have to come from the intergalactic medium (IGM) of cold interstellar gas between galaxies and the circumgalactic medium (CGM) of cold interstellar gas in the dark matter halos of galaxies.

Fig. 1 of McGaugh et al. (2026): Conceptual elements of a galaxy: the stars (yellow/blue) and atomic gas (green) of NGC 6946 (Spitzer 3.6µ and 21 cm data: F. Walter et al. 2008) are shown embedded in an extended dark matter halo (black). The dark matter density decreases continuously with radius so the halo has no hard edge, but for convenience we adopt the common convention that the radius r200 marks the boundary of the dark matter halo and the dividing line between the circumgalactic medium (CGM) and the intergalactic medium (IGM; orange). The stars and atomic gas illustrated here appear within r < 20 kpc while r(200) ≈ 220 kpc (not shown to scale).

One kpc (i.e. kiloparsec) equals 32,600 light years.

But while this is the only possible solution to the local missing baryon problem in essentially all galaxies (but especially the smaller ones) in the dark matter particle paradigm, there is basically no way to make this work.

Therefore, cold dark matter models are inconsistent with what we observe.

CDM predicts excessive dwarf galaxy masses

Another example demonstrates that in the Local Group that includes the Andromeda galaxy and the Milky Way, one of its minor galaxies should have more mass than its two biggest galaxies and even more mass than the Local Group as a whole, which is contrary to the kinetic dynamics of the system as a whole and contrary to the conservation of matter. As McGaugh explains:

One signature of this misfit is the occurrence of very large V(200) for dwarf galaxies with small V(f). Taken literally, this would mean that some of the smallest dwarf galaxies reside in dark matter halos that outweigh those of giants like the Milky Way. This seems absurd, and it is. For example, by this approach, the dwarf galaxy NGC 3109 residing just outside the Local Group outweighs the Local Group and both its giants, Andromeda and the Milky Way, put together. But it is pretty clear from the local velocity field that the entire Local Group is not orbiting this little dwarf.

Real galaxies rarely have NFW halo distributions 

In dark matter particle paradigms, inferred dark matter halos have a "pseudo-isothermal" distribution, while collisionless cold dark matter must theoretically have, as an inexorable consequence of a very simple statistical mechanics style calculation that applies to dark matter particle with these very simple properties, what is called an NFW distribution, which is a very poor fit to the vast majority of galaxies.

Figure 2 from McGaugh et al. (2026): The observed flat velocity V(f) as it relates to the fitted V(200) for pseudo-isothermal (left panel) and NFW (right panel) halos (Li et al. 2020). Filled points have formal uncertainties < 20% in V(200); open points are less accurate fits. The solid line shows V(f) = V(200). The gray line in the right panel shows Equation (2a) of Katz et al. (2019), which corresponds roughly to f(v) ≈ 1.4.

V(f) is the rotational velocity of a galaxy at about a 65,000 light year radius, V(200) is the velocity of a galaxy at about 715,000 light year radius, and f(v) is equal to V(200)/V(f). 

The bottom line is that pseudo-isothermal dark matter halo distributions are a decent fit to what is observed with f(v) approximately equal to 1 and little scatter in the data (and scatter mostly associated with data points that have high uncertainties), while an NFW dark matter halo distribution has f(v) approximately equal to 1.4 with a great deal of scatter in the data.

This is a problem for the dark matter paradigm because coming up with a dark matter candidate with properties the naturally form pseudo-isothermal halos (for a candidate that isn't excluded by other data) is a challenging enterprise. Indeed, pseudo-isothermal dark matter halo density distributions aren't even theoretically stable.

CDM predicts the wrong slope for the Tully-Fischer scaling law

In a cold dark matter paradigm, the baryonic Tully-Fischer relationship (which roughly speaking related galaxy size to the speed of its flat rotation) has a slope of four when the observed relationship has a slope of three. 

When your power law exponent is a power of three rather than a predicted power of four, you have a seriously flawed functional form for your model.

Gravity based solutions compared

Toy-model MOND has challenges of its own (especially in galaxy clusters, although the intra-cluster medium of cold interstellar gas that was recently estimated makes the discrepancy smaller), but it is much more descriptive of the data, and predictive, than the cold dark matter paradigm. It even fits clusters reasonably well also with a tweak to just one of its parameters, rather than to the model as a whole. 

Deur chalks up the different gravitational behavior of galaxy clusters and galaxies to the different geometries of the mass distributions involved.

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

The Standard Model Still Works (Again)

The LHCb experiment at the Large Hadron Collider (LHC) has made a statistically significant observation (although not an absolutely certain discovery) a rare decay of a particular kind of positively charged bottom quark meson (to a positively charged pion and an electron-positron pair, which is an example of what is called a semi-leptonic decay because it is a mix of a hadron, the pion, and leptons like electrons and positrons) with a frequency of one decay per 40 million decays of this kind of meson (a kind of meson which, itself, doesn't make up a large share of mesons produced at LHCb). 

This just happens to be statistically consistent with the frequency of this kind of decay of this kind of meson that the Standard Model predicts of B(B+→ π+ℓ+ℓ−) = (2.04 ± 0.21) × 10^−8, which is about one per 50 million decays. The same decay, but with muons, was first seen in 2012 at a branching fraction of one per 55 million decays that was also statistically consistent with the Standard Model expectation (which is the same for electrons and for muons due to lepton universality).

The first evidence for the decay B+→π+e+e− is reported using proton-proton collision data recorded by the LHCb experiment at centre-of-mass energies of 7, 8 and 13 TeV, corresponding to an integrated luminosity of 9 fb^−1. 

A signal excess with a significance of 3.2σ is observed and the branching fraction is measured to be B(B+→ π+e+e−) (2.4+0.9−0.8+0.4−0.2) × 10^−8, where the first set of uncertainties is statistical and the second is systematic. The result is consistent with the Standard Model expectation.

LHCb collaboration, "First evidence of the decay B+→π+e+e−" arXiv:2604.26784 (April 29, 2026).

Combining the statistical and systemic uncertainties, the total uncertainty is about 2.4 ± 0.9 x 10^-8, which a larger branching fraction (i.e. more events) actually slightly favored over a smaller one (i.e. fewer events), relative to the best fit value.

The deviation from the Standard Model expectation in the muon measurement was about 0.7 sigma (in the opposite direction of the deviation in the electron experiment, from the best fit value), while the deviation from the Standard Model expectation in the electron measurement was about 0.4 sigma. This suggests that the systemic uncertainty estimate in the Standard Model prediction and in the experiments was probably conservatively somewhat high.

This particular hadron decay isn't extremely significant (hadrons are either mesons like the B+ or baryons like the proton). But comparing the decay rate of a positively charged pion with a muon-antimuon pair to the decay rate of a positively charge pion with an electron-positron pair is a good test of "lepton universality" (i.e. the Standard Model rule that electrons, muons, and tau leptons have properties that are identical except for their masses). For several years there were experimental anomalies that made it appear that lepton universality was violated, but those anomalies were recently resolved in favor of the Standard Model prediction that lepton universality is not violated.

There are about a hundred plain vanilla mesons and baryons in the Standard Model like the B+ meson studied here, and some of the heavier ones have perhaps hundreds of decay modes with a predicted branching fraction of less than one decay per billion decays. So, the universe of Standard Model predicted meson decays to look for is somewhere on the order of 10,000.

The B+ meson has two "valance quarks" an up quark and an anti-b quark. It has a rest mass of 5279.26 ± 0.17 MeV/c^2 (about 5.6 times the mass of a proton and a little less massive than a Lithium-6 atom). It has total angular moment (a.k.a. "spin") of 0 and odd (i.e. negative) parity, which means that it is a "pseudo-scalar" meson. It is ephemeral, it has a mean lifetime of (1.638 ± 0.004) × 10^−12 seconds (i.e. a little more than a trillionth of a second). It has more than two dozen measured decay modes that happen in more than one in a million decays, and the vast majority of the time B+ mesons decay to particles that include some kind of charm quark hadron. It has hundreds of decay modes more probable than this one.

The Standard Model was devised in the early 1970s, the b quark was discovered at Fermilab in 1977. The full set of fundamental particles (except the Higgs boson, which was discovered at Fermilab in 2012 and the discovery that the neutrinos were massive), was in place in 1995, more than three decades ago. 

The Standard Model prediction for the frequency of this particular B+ meson decay was cited in connection with the first observation of the parallel muon decay in 2012 and in 2015, and derive from a 2008 paper (i.e. it was made more than 14 years before this decay was observed just as predicted).

'The Higgs boson and the neutrino masses don't (meaningfully) enter into the calculation of the branching fractions of the B+ meson, so the only thing that has changed in the Standard Model since 1995 that is relevant to this calculation is that the measurements of some of the fundamental physical constants involved in the calculation, especially the relevant CKM matrix elements (as noted at page 13 of the 2008 paper) have gotten more precise over that time. (The accuracy with which we know another non-fundamental physical constant, called the "form factor" of the B+ meson, which is too hard to calculate from first principles at this point, has also improved and is material to this calculation.)

The physical constant whose improved precision matters most in this context are the CKM matrix elements for the b quark to up quark transition probability in W boson interactions and the top quark to down quark transition probability in W boson interactions, which are low: about 0.14% and 0.007% respectively. 

The respective 3% and 2% uncertainties in the world average measurement of these physical constants are probably some of the leading sources of the roughly 10% uncertainty in the Standard Model prediction of the frequency of this B+ meson decay branching fraction. It is hard to say exactly how much of a share of the uncertainty in the predicted value is from this source, however, because while the respective papers linked above provide an error budget chart for the uncertainties in their experimental measurements, none of the papers provide an exact error budget chart for their Standard Model predictions for this decay frequency, probably because this was considered too elementary to publish. 

Computer processing capacity has also improved greatly since then, which makes these calculations much less cumbersome to actually make.

In isolation, this experimental confirmation of the Standard Model prediction could be just a lucky fluke, although a quite remarkable one, even on its own. But together with thousands of other measured hadron branching decay fractions, the Standard Model is really unstoppable. 

Experimental result anomalies where there are deviations from the Standard Model prediction are few, far between, modest in statistical significance, and usually go away quickly for closer inspections and more experiments and analysis. Experiments testing the Standard Model in contexts other than hadron decay branching fractions that involve completely different kinds of calculations are just as consistently correct. It is an extremely robustly tested theory.

Even if there are beyond the Standard Model physics gaps that are missing from the Standard Model, it is very close to the truth. The open parameter space for deviations from it are very small.

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 Latest News In Top Quark Physics

The latest indirect measurement of the top quark pole mass is surprisingly precise (exceeding the precision of the world average in a single measurement) despite the method used, which has historically had large error bars. The Particle Data Group world averages are as follows:

This will probably drag up the world average a little bit, to about 172.7 GeV.

We present an indirect determination of the top-quark pole mass mt within a global analysis of parton distribution functions (PDFs), based on the public NNPDF framework. 

We consider a wide range of measurements, including both single- and double-differential observables, computed at NNLO QCD accuracy with EW corrections, and analyse their individual as well as combined impact on the joint (α(s),m(t)) parameter space, while accounting for PDF evolution up to approximate N3LO QCD accuracy with QED corrections. We account for missing higher order QCD uncertainties by default. 

Unique to our analysis are the inclusion of, first, toponium contributions around the tt¯ threshold, second, state-of-the-art constraints on αs from the lattice, and finally, a detailed sensitivity study of the various ATLAS and CMS differential cross-section measurements at 8 and 13 TeV. We demonstrate explicitly how a combined determination requires the refitting of the PDFs in order to correctly correlate uncertainties. 

We find mt = 172.80 ± 0.26 GeV at approximate N3LO QCD including NLO QED, EW and toponium corrections.

Richard D. Ball, Jaco ter Hoeve, Roy Stegeman, "A Determination of the Top Mass from a Global PDF Analysis" arXiv:2603.28865 (March 30, 2026).

Another new paper on top quark physics (with an abstract devoid of much of an interesting description of the paper) confirms that: 

(1) the experimentally measured top quark-antitop quark pair production rates are consistent with the Standard Model expectation, 

(2) toponium has been discovered by both the ATLAS and CMS experiments at the Large Hadron Collider (LHC), and 

(3) the Higgs field Yukawa of the top quark is experimentally confirmed to be not more than 2.1 times the Standard Model expectation (the coupling should be proportionate to the top quark's pole mass in the Standard Model).

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.

Nailing Neutrino-Nucleus Interaction Rates

An experimental test of how frequently neutrinos interact with atomic nuclei has a best fit value of the Standard Model expectation and an uncertainty of less than ± 15%, which is impressive given how slight the interaction is, fit 10^22 trials yielding just 124 observed interactions.

The COHERENT collaboration reports the most precise measurement of the coherent elastic neutrino-nucleus scattering cross section to date. This measurement was performed with COHERENT's germanium detector array, Ge-Mini, at the Spallation Neutron Source at Oak Ridge National Laboratory. 

A cumulative exposure of 4.68 × 10^22 protons on target yielded a total number of observed counts of 124 + 14 −12 and a flux-averaged cross section of 1.00 ± 0.10 (statistical) ± 0.10 (systematic) relative to the standard-model expectation of 5.9 × 10^−39 cm^2. 

The well-understood energy and timing distributions of the neutrino source allow for independent measurements of muon- and electron-neutrino scattering rates. This information is used to improve constraints on non-standard neutrino interactions mediated by heavy particles.

M. Adhikari, et al., "Measurement of coherent elastic neutrino nucleus scattering on germanium by COHERENT" arXiv:2603.17951 (March 18, 2026).

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).