Toponium Discovered

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

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

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

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

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

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

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

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

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

The introduction explains that:

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

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

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

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

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

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

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

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

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

. . . 

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

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

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

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

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

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

An Electroweak Centric Model For Standard Model Mass Generation

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

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

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

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

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

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

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

The introduction of the paper is as follows:

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

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

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

The paper's conclusion states:

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

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

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

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

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

Two Papers Related To Quark Masses

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

Background

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

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

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

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

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

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

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

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

Why does this matter?

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

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

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

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

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

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

Charge parity violation and the quark masses

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

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

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

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

Bonus content

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

A New Strong Force Coupling Constant Determination

The Particle Data Group value for the strong force coupling constant is 0.1180 ± 0.0009. This new determination, based upon earlier runs of LHC dijet data and lower energy HERA data, is consistent with the PDG value at the 0.1 sigma level. 

The strong force coupling constant is pervasively important in almost all high energy physics calculations, but it known much less precisely (with just one part per 131 parts precision) than most other Standard Model or fundamental physical constants. So, pinning this down more precisely is always big deal.

The beta function that describes how the strong force coupling constant runs with energy scale is an exact theoretical prediction of the Standard Model, with no experimental uncertainties. The conference presentation's confirmation that the strong force coupling constant runs with energy scale just as predicted in the Standard Model, over four orders of magnitude of energy scale, is arguably an even more important confirmation of the Standard Model, because there are fewer experimental confirmations of this in the literature.

In this talk we present a determination of the strong coupling constant αs and its energy-scale dependence based on a next-to-next-to-leading order (NNLO) QCD analysis of dijet production. 

Using the invariant mass of the dijet system to probe αs at different scales, we extract a value of αs(mZ) = 0.1178 ± 0.0022 from LHC dijet data. 

The combination of various LHC datasets significantly extends the precision and scale reach of the analysis, enabling the first determination of αs up to 7 TeV. By incorporating dijet cross sections from HERA, we further probe αs at smaller scales, covering a kinematic range of more than three orders of magnitude. Our results are in excellent agreement with QCD predictions based on the renormalization group equation, providing a stringent test of the running of the strong coupling across a wide energy range.

João Pires, "Precision determination of αs from Dijet Cross Sections in the Multi-TeV Range" arXiv:2507.01670 (July 2, 2025) (Contribution to the 2025 QCD session of the 59th Rencontres de Moriond).

Hadron Molecules v. True Composite States

Any time that you have four to six quarks in some sort of bound state, the question that is presented is whether it is a true tetra-, penta-, or hexaquark, or whether it is a hadron molecule with the same valence quarks.

In a true tetra-, penta-, or hexaquark, the valence quarks are bound direct to each other by gluons. In a hadron molecule, mesons or baryons are bound to each other either by something analogous to the nuclear binding force (carried by mesons in atomic nuclei, mostly, but not entirely, pions) or electromagnetically (as in an ordinary molecule made up of atoms bound electromagnetically).

A new preprint looks at four pentaquark states with a valence charm quark and finds that all of them are hadron molecules rather than true pentaquarks.

Muon g-2 White Paper Updated

Next week, on June 2, 2025, the final round of experimental results for muon g-2 will be announced. Ahead of that there is an update of the Muon g-2 White paper that got the Standard Model predicted value for muon g-2 badly wrong. The revised version acknowledges this mistake and remarks that the revised prediction is spot on with the experimental value of muon g-2. 

The revised state of the art Standard Model prediction will still be about four times less precise than the experimentally measured value after June 3, 2025, however. The predicted value's uncertainty is greater than the experimentally measured uncertainty almost entirely due to the uncertainties in the QCD (quantum chromodynamics a.ka. strong force) calculation of the leading order hadronic vacuum polarization contribution to muon g-2. 

These uncertainties are hard to reduce, since the values of the fundamental physical constants relevant to the calculation, like the value of the strong force coupling constant's value and the light quark masses, have uncertainties of the same magnitude as the total HVP calculation.

The consistency of the experimental value of muon g-2 and the value for it predicted in the Standard Model, is a broad global high precision measurement of the consistency of all parts of the low to medium energy scale Standard Model of Particle Physics with the real world.

The consistency which exists strongly disfavors the discovery of any beyond the Standard Model physics at a next generation particle collider (even though there one could cherry pick potential modifications of the Standard Model that haven't already been ruled out by other high energy physics data, that could have no impact on muon g-2, or would have an impact that is too negligible to discern).

This summary chart appears in the introduction to the paper:

A chart from the conclusion shows how the old White Paper Standard Model prediction for muon g-2 and the new one differ.

We present the current Standard Model (SM) prediction for the muon anomalous magnetic moment, aμ, updating the first White Paper (WP20) [1]. 

The pure QED and electroweak contributions have been further consolidated, while hadronic contributions continue to be responsible for the bulk of the uncertainty of the SM prediction. Significant progress has been achieved in the hadronic light-by-light scattering contribution using both the data-driven dispersive approach as well as lattice-QCD calculations, leading to a reduction of the uncertainty by almost a factor of two. 

The most important development since WP20 is the change in the estimate of the leading-order hadronic-vacuum-polarization (LO HVP) contribution. A new measurement of the e+e−→π+π− cross section by CMD-3 has increased the tensions among data-driven dispersive evaluations of the LO HVP contribution to a level that makes it impossible to combine the results in a meaningful way. At the same time, the attainable precision of lattice-QCD calculations has increased substantially and allows for a consolidated lattice-QCD average of the LO HVP contribution with a precision of about 0.9%. 

Adopting the latter in this update has resulted in a major upward shift of the total SM prediction, which now reads a(SM)(μ) = 116592033(62) × 10^−11 (530 ppb). When compared against the current experimental average based on the E821 experiment and runs 1-3 of E989 at Fermilab, one finds a(exp)(μ)−a(SM)(μ) = 26(66) × 10^−11, which implies that there is no tension between the SM and experiment at the current level of precision. The final precision of E989 is expected to be around 140 ppb, which is the target of future efforts by the Theory Initiative. The resolution of the tensions among data-driven dispersive evaluations of the LO HVP contribution will be a key element in this endeavor.

R. Aliberti, et al., "The anomalous magnetic moment of the muon in the Standard Model: an update" arXiv:2505.21476 (May 27, 2025) (188 pages).

The conclusion explains that:

By comparing the uncertainties of Eq. (9.5) and Eq. (9.4) it is apparent that the precision of the SM prediction must be improved by at least a factor of two to match the precision of the current experimental average, which will soon be augmented by the imminent release of the result based on the final statistics of the E989 experiment at Fermilab. We expect progress on both data-driven and lattice methods applied to the hadronic contributions in the next few years. Resolving the tensions in the data-driven estimations of the HVP contribution is particularly important, and additional experimental results combined with further scrutiny of theory input such as from event generators should provide a path towards this goal. Further progress in the calculation of isospin-breaking corrections, from both data-driven and lattice-QCD methods, should enable a robust SM prediction from τ data as well. For lattice-QCD calculations of HVP continuing efforts by the world-wide lattice community are expected to yield further significant improvements in precision and, hopefully, even better consolidation thanks to a diversity of methods. The future focus will be, in particular, on more precise evaluations of isospin-breaking effects and the noisy contributions at long distances. 

The role of aµ as a sensitive probe of the SM continues to evolve. We stress that, even though a consistent picture has emerged regarding lattice calculations of HVP, the case for a continued assessment of the situation remains very strong in view of the observed tensions among data-driven evaluations. New and existing data on e+e− hadronic cross sections from the main collaborations in the field, as well as new measurements of hadronic τ decays that will be performed at Belle II, will be crucial not only for resolving the situation but also for pushing the precision of the SM prediction for aµ to that of the direct measurement. This must be complemented by new experimental efforts with completely different systematics, such as the MUonE experiment, aimed at measuring the LO HVP contribution, as well as an independent direct measurement of aµ, which is the goal of the E34 experiment at J-PARC. The interplay of all these approaches, various experimental techniques and theoretical methods, may yield profound insights in the future, both regarding improved precision in the SM prediction and the potential role of physics beyond the SM. Finally, the subtleties in the evaluation of the SM prediction for aµ will also become relevant for the anomalous magnetic moment of the electron, once the experimental tensions in the determination of the fine-structure constant are resolved.

Basically, the conclusion calls for scientists to get to the bottom of why the experiments that were used as a basis for the first White Paper prediction were wrong, and hopes against all reasonable expectations that the process of doing that will reveal new physics.

The paper's claim that the uncertainty in the Standard Model prediction can be cut dramatically "in the next few years" is pretty much wishful thinking.

This paper doesn't address in detail how completely this result ruled out new physics, but further papers by unaffiliated scientists will no doubt do just that not long after the new experimental results are released next week. 

Will A Next Generation Collider See A Sphaleron?

Overview

A new pair of preprints, whose abstracts and citations are set forth below, examine the sphaleron energy threshold with newly updated experimental values of Standard Model physical constants and rigorous calculations, and its other properties. This tells experimentalists very specifically where to look and what to look for when trying to observe a Standard Model sphaleron interaction.

The sphaleron interaction is the only time in the Standard Model of Particle Physics that baryon number and lepton number are not simultaneously conserved. 

But it requires extremely high collider energies to form in detectable numers (an order of magnitude greater collider energy than the nominal energy levels is required because the energy of the interaction must be confined so compactly, so a collider energy of something on the order of 91 TeV is needed to confidently assert that they have been discovered).

Not Yet Observed Standard Model Phenomena

The two biggest predictions of the Standard Model of Particle Physics that haven't been observed yet are the sphaleron and the failure to definitively observe pure glueballs (a strong force bound hadron with no quarks) and certain other hadrons predicted to exist in Standard Model quantum chromodynamics (QCD). 

Standard Model Hadron Predictions Not Yet Seen

Current experiments have more than enough energy to produce glueballs, which are predicted to have masses on the order of 0.5 GeV to 3 GeV in their ground states (while the LHC can create energies up to 14,000 GeV), at some experimentally observed mesons have been provisionally identified as likely glueballs. 

But identifying them definitively against other possible explanations of glueball-like resonances is challenging, since they are electromagnetically neutral, color charge neutral, don't interact via the weak force at the tree-level, and are bosons with integer spins shared by mesons in overlapping mass ranges. Glueballs have a natural tendency to blend into mesons with the same quantum numbers, resulting in mixed hadron resonances.

A similar issue applies to some of the heaviest hadrons predicted by the Standard Model but not yet definitively identified with resonances observed at sufficient statistical significance. But the most massive of these have ground states with masses of 20 GeV or less, and a few new ones are identified every year these days. Observing the last few is mostly just a matter of time.

Similarly, the project of identifying the underlying structure of hadron resonances other than simple pseudo-scalar valence quark-antiquark mesons and simple three valence quark baryons, is progressing one hard won resonance identification at a time with no sweeping explanations for large groups of resonances whose structures do not have a consensus explanation. 

There are even hints of exceeding improbable, short lived, and rare ultrahigh energy toponium (i.e. a meson made up of a top quark and anti-top quark with a mass on the order of 340 to 350 GeV), which would also be enhanced at a next generation higher energy particle collider, with a 73 TeV collider energy with a very large number of collisions, being a key threshold to detect this highly improbable resonance. It is rare not just because it takes high energies, but also because of the high risk that the valence top quark and valence anti-top quark necessary to form one would decay or annihilate with each other, before the quarks could hadronize into the most massive theoretically possible simple quark-antiquark meson. Toponium is almost guaranteed to be seen at a next generation collider along the lines of the LHC but more powerful.

Sphalerons

A sphaleron, in contrast, is basically the only major Standard Model prediction that is not yet confirmed, with an energy scale about 730 times that of a Higgs boson, that requires a bigger accelerator to confirm or rule out, even though the nominal sphaleron energy of 9.1 TeV is less than the 14 TeV peak energy of the LHC.

Analysis

To be clear, this study does not alter the long standing conclusion that sphaleron interactions cannot explain the baryon asymmetry of the universe (i.e. the extreme excess of matter over anti-matter in quarks, which is also true of charged leptons), although it does tweak estimates of what percentage of baryon asymmetry this interaction can explain with Standard Model physics.

I put even odds on whether sphalerons actually exist or not. A mathematically consistent modification of the Standard Model that would make baryon number and lepton number conserved symmetries of the Standard Model, which would make sphalerons impossible, would have almost no important phenomenological consequences for anything other than (i) baryogenesis and leptogenesis in the first few seconds after the Big Bang, which we don't really understand yet anyway and which could be explained without any baryon number and lepton number violations in other ways, such as a mirror universe model, and (ii) the presence or absence of sphaleron decay signatures in ultrahigh energy collider experiments. 

Non-detection of sphalerons would also disfavor a wide variety of grand unified theories (GUTs) and Theories of Everything (TOEs), which usually permit violations of baryon number and lepton number, which makes phenomena like proton decay, which is forbidden in the Standard Model of Particle Physics, rare but possible.

But, detecting sphalerons as predicted would also be a triumph for the Standard Model taken to the extremes of its domain of applicability.

The Standard Model prediction is that there would be a desert of new physics (within or beyond the Standard Model) at energies above those where a sphaleron are observed.

The Papers

The electroweak sphaleron is a static, unstable solution of the Standard Model classical field equations, representing the energy barrier between topologically distinct vacua. 

In this work, we present a comprehensive updated analysis of the sphaleron using current Standard Model parameters with the physical Higgs boson mass of m(H)=125.1 GeV and m(W)=80.4 GeV, rather than the m(H)=m(W) approximation common in earlier studies. The study includes: (i) a complete derivation of the SU(2)×U(1) electroweak Lagrangian and field equations without gauge fixing constraints, (ii) high-precision numerical solutions for the static sphaleron configuration yielding a sphaleron energy E(sph)≃9.1 TeV, (iii) an analysis of the minimum energy path in field space connecting the sphaleron to the vacuum (a 1D potential barrier as a function of Chern-Simons number), and (iv) calculation of the sphaleron single unstable mode with negative eigenvalue ω^2=−2.7m(W)^2, providing analytical fits for its eigenfunction. 

We find that using the measured Higgs mass modifies the unstable mode frequency, with important implications for baryon number violation rates in both early universe cosmology and potential high-energy collider signatures. These results provide essential input for accurate lattice simulations of sphaleron transitions and precision calculations of baryon number violation processes.

Konstantin T. Matchev, Sarunas Verner, "The Electroweak Sphaleron Revisited: I. Static Solutions, Energy Barrier, and Unstable Modes" arXiv:2505.05607 (May 8, 2025).

We present a comprehensive analysis of electroweak sphaleron decay dynamics, employing both analytical techniques and high-resolution numerical simulations. 

Using a spherically symmetric ansatz, we reformulate the system as a (1+1)-dimensional problem and analyze its stability properties with current Standard Model parameters (m(H)=125.1 GeV, m(W)=80.4 GeV). We identify precisely one unstable mode with eigenvalue ω^2 ≃ −2.7m(W)^2 and numerically evolve the full non-linear field equations under various initial conditions. Through spectral decomposition, we quantify the particle production resulting from the sphaleron decay. 

Our results demonstrate that the decay process is dominated by transverse gauge bosons, which constitute approximately 80% of the total energy and multiplicity, while Higgs bosons account for only 7-8%. On average, the sphaleron decays into 49 W bosons and 4 Higgs bosons. The particle spectra consistently peak at momenta k ∼ 1−1.5m(W), reflecting the characteristic size of the sphaleron. 

Remarkably, these properties remain robust across different decay scenarios, suggesting that the fundamental structure of the sphaleron, rather than specific triggering mechanisms, determines the decay outcomes. These findings provide distinctive experimental signatures of non-perturbative topological transitions in the electroweak theory, with significant implications for baryon number violation in the early universe and potentially for high-energy collider physics.

Konstantin T. Matchev, Sarunas Verner, "The Electroweak Sphaleron Revisited: II. Study of Decay Dynamics" arXiv:2505.05608 (May 8, 2025).

How Precise Are Astronomy And High Energy Physics?

* It is worth recalling that even the best telescopes often aren't very precise.

For example, the state of the art precision with which we can measure the distance from Earth to the reasonably close M87 galaxy is about ± 2-3%.

* By comparison, in high energy physics, scientists have recently detected a rare form of decay predicted by the Standard Model of Particle Physics at that frequency, from a three valence quark particle with two valence up quarks and one valence strange quark that is about 27% more massive than a proton, known as a sigma plus baryon, which accounts for just one in 100 million decays of this kind of particle, which is a tiny effect (although admittedly, the precision of that measurement is just ± 16%). 

The most recent theoretical predictions for the branching fraction B(Σ+ → pµ+µ−) lie within the range [1.2, 7.8] × 10^−8. The experimentally measured value is [0.81, 1.25] × 10^−8 derived from 237 ± 16 observed decays of sigma plus baryons to a proton, a muon, and an anti-muon.

This is the smallest baryon decay branching fraction ever definitively observed, and implicitly rules out all manner of even very slight deviations from the Standard Model of Particle Physics at the ten parts per billion level in the processes that are involved in this decay.

This branching fraction measurement took a data set of about 24 billion sigma plus baryon decays, which themselves are created in only a small fraction of the many trillions of Large Hadron Collider (LHC) particle collision events that scientists at CERN have observed to date.

* As another example, another recent paper makes a first principles calculation using lattice QCD of the absolute frequency of a certain kind of particle decay (without reference to the frequency of other possible decays of that particle), using standard world averages for the physical constants involved, and compares this prediction to two experimental measurements of the frequency of that particle decay and their average value, producing the following results:

The predicted value has been calculated to a ± 6.4% precision. The experimental measurements and their average have a ± 2.4% to 6.4% precision. 

The uncertainties in the world average measured values of the two key physical constants used in making the prediction are about 0.6% and 2% respectively, which put a floor on how precise the calculation of the predicted value in the Standard Model of Particle Physics could be. The balance of the ± 6.4% uncertainty in the predicted frequency flows from the method used to approximately calculate the true value of predicted quantity using the equations of the Standard Model of Particle Physics with lattice QCD, because these calculations can't be done exactly with current mathematical methods.

The difference between the predicted frequency and each of the two experimentally measured frequency (as well as their average value) are within one standard deviation of each other, as are the experimentally measured frequencies with each other. This is, of course, what scientific theories and experiments are suppose to do.

* In contrast, calculations made using only the electromagnetic part of the Standard Model of Particle Physics (quantum electrodynamics or QED for short) and the weak force, without implicating quantum chromodynamics (QCD), which is the physics of the strong force, and without implicating neutrino oscillations, are many orders of magnitude more precise.

* While it is not precisely on topic in this post, another article today looks at the subtle differences between a single bound state of more than three valence quarks into a composite particle known as a hadron, and a "hadron molecule" that is made up of two composite particles with two or three valence quarks each, bound to each other in a manner analogous to the way that protons and neutrons in an atomic nucleus are bound to each other, at a theoretical level.

* Finally, a new paper examines a way of approximating QCD calculations for hadrons by assigning a rest mass of 450 MeV/c^2 to gluons (which is about half the mass of a proton), rather than following the Standard Model assumption that gluons are massless, and gets some promising results. 

This is, however, simply a calculation trick. We know this because pions, which are composed of light quarks bound by gluons, have a mass of less than 450 MeV.

A Muon g-2 Update

A new preprint recaps developments in establishing the Standard Model prediction for muon g-2 (my own latest recap, with more detail, is in a November 12, 2024 post and shows that the state of the art SM prediction is just 0.2 sigma from the experimental result). 

A 2020 "white paper" using a "data driven" model, showing a five sigma discrepancy between theory and experiment was badly wrong, because the data is relied upon was subtly flawed, or understated the uncertainty of its results.

New Lattice QCD calculations which corroborate each other, and some new experimental data to contribute to a data driven model, show that, in fact, it is extremely likely that there is no statistically significant difference between ultra-precise experimental measurements of muon g-2 and the best available calculation of the Standard Model predicted value of it.

[T]he final result for 𝑎𝜇 from the Muon g-2 experiment is expected in Spring 2025, preceded by a new Theory White Paper. 

A further measurement of 𝑎𝜇 is planned at J-PARC@KEK using a very different technique with a compact magnetic ring and low momentum 𝜇+. Data-taking should start in 2028 with 2 years of running needed to reach a result with ∼ 2°ø the uncertainty of Muon g-2.

A High Precision Estimate Of The Strong Force Coupling Constant

A new pre-print uses novel methods to make a high precision determination of the strong force coupling constant, at greater precision than all previous determinations combined, which is:

αs(mZ) = 0.11873(56)

The relative uncertainty is one part per 212.

This is important because all strong force (i.e. QCD) calculations rely upon this experimentally determined physical constant.

The Particle Data Group value is 0.1180(9) which is a relative uncertainty of one part per 131 using an error weighted average of all strong force coupling constant measurements.

The Lightest Neutron Star Ever? Or Something Else?

A new preprint argues that a newly observed object that looks lot like a neutron star, but is less massive than should have been possible theoretically, might be an exotic star.

But, since the observed mass, of 0.77 + 0.2 -0.17 solar masses, is still within two sigma of the theoretical minimum mass of a neutron star, which is 1.17 solar masses, I don't take the conclusion that it could be an exotic object (made up of color flavor locked quark matter), very seriously.

In other news, I have a dim opinion of any paper whose abstract begins:

The gauge singlet right-handed neutrinos are one of the essential fields in neutrino mass models that explain tiny masses of active neutrinos.

If you feel the need to create right handed neutrinos (with masses different from any of the three Standard Model active neutrinos) to explain anything, your model is probably wrong because you are too lazy to find a solution that doesn't need them, and there is no positive experimental evidence that they exist. This possibility has been a perennial source for a steady stream of dead end theoretical speculation for at least a decade or two. This paper is the work of dim bulbs in the physics community. Try harder until you come up with something better.

To be clear, I'm not saying that I'm a professional physicist coming up with something better myself. But you don't have to be a genius composer yourself to appreciate the difference between Mozart and a mediocre music theory student.

Proton And Neutron Structure

"Nucleons" are protons and neutrons. 

A new paper determines that their mass is distributed over a radius of consistent with one femtometer (a 10^-15 meter), and is spread over a larger area than their electromagnetic charges which are found in their valence quarks. The proton charge radius is 0.842 femtometers. This is because the electromagnetically neutral gluons which bind these quarks, whose energy is the source of most of the mass of protons and neutrons, are spread out more than the electromagnetically charged quarks in a proton or neutron. 

Being closely connected to the origin of the nucleon mass, the gravitational form factors of the nucleon have attracted significant attention in recent years. We present the first model-independent precise determinations of the gravitational form factors of the pion and nucleon at the physical pion mass, using a data-driven dispersive approach. 

The so-called ``last unknown global property'' of the nucleon, the D-term, is determined to be −(3.38+0.26−0.32). The root mean square radius of the mass distribution inside the nucleon is determined to be 0.97+0.02−0.03 fm. Notably, this value is larger than the proton charge radius, suggesting a modern structural view of the nucleon where gluons, responsible for most of the nucleon mass, are distributed over a larger spatial region than quarks, which dominate the charge distribution. We also predict the nucleon angular momentum and mechanical radii, providing further insights into the intricate internal structure of the nucleon.

Xiong-Hui Cao, Feng-Kun Guo, Qu-Zhi Li, De-Liang Yao, "Precise Determination of Nucleon Gravitational Form Factors" arXiv:2411.13398 (November 20, 2024).

Muon g-2 HLbL Developments

Background (mostly, but not entirely, from an October 27, 2022 post and a July 18, 2024 post)

The combined result of the experimental measurements of muon g-2 (all of the numbers that follow are in the conventional -2 and divided by two form times 10^-11) is:

116,592,059 ± 22 

This compares to the leading Standard Model predictions of: 

116,592,019 ± 38 (which is a relative error of 370 parts per billion). This is from A. Boccaletti et al., "High precision calculation of the hadronic vacuum polarisation contribution to the muon anomaly." arXiv:2407.10913 (July 15, 2024).

The gap is only 40 ± 44.9, with the Standard Model prediction still a bit lower than the experimental value.

The QED + EW predicted value is:

116,584,872.53 ± 1.1

About 90% of the combined uncertainty in this  QED + EW value is from the EW component (there may be an error in the standard QED prediction but it is so small that it is immaterial for these purposes).

The difference, which is the experimentally implied hadronic component value (HVP plus HLbL), is:

7186.47 ± 22.02

This has a plus or minus two sigma range of:

7,142.32 to 7,230.51

The hadronic QCD component is the sum of two parts: the hadronic vacuum polarization (HVP) and the hadronic light by light (HLbL) components.

In the Theory Initiative analysis the QCD amount is 6937(44) which is broken out as HVP = 6845(40), which is a 0.6% relative error and HLbL = 98(18), which is a 20% relative error.

The latest LO-HVP calculation component is 7141 ± 33 (a relative error of just 0.46%).

As of November 1, 2024, it was clear that the Theory Initiative calculation of the Standard Model value of the HVP contribution to muon g-2 (which differs from 5.1 sigma from the experimental value) was the flawed one:

Fermilab/HPQCD/MILC lattice QCD results from 2019 strongly favour the CMD-3 cross-section data for e+e−→π+π− over a combination of earlier experimental results for this channel. Further, the resulting total LOHVP contribution obtained is consistent with the result obtained by BMW/DMZ, and supports the scenario in which there is no significant discrepancy between the experimental value for aμ and that expected in the Standard Model.

Similarly, the introduction to a new paper that is the motivation for this post notes that:

estimates based on τ data-driven approaches or lattice QCD calculations significantly reduce the tension between theoretical and experimental values to 2.0σ and 1.5σ, respectively (less than one σ in [44]). The latest CMD-3 measurement of σ(e+e− → π+π−) also points in this direction.

Reference [44] cited in the block quote above is the current state of the art calculation cited above.

The HLbL Contribution

The Theory Initiative HLbL calculation

None of the refinements of the muon g-2 HVP contribution discussed above tweak the Theory Initiative value of the Hadronic Light by Light (HLbL) contribution of 92 ± 18, even though it has the highest relative error of any of the components of the muon g-2 calculation of nearly 20%, because the HLbL is only 1.3% of the total hadronic contribution and still has only half the uncertainty of the HVP contribution.

But, progress has been made on the HLbL component as well, which is now getting more attention as the experimental result's increased precision and the progress on the HVP contribution makes it relevant.

The Chao (April 2021) HLbL Calculation 

On the day that the first new muon g-2 experimental results from Fermilab were released a "new calculation of the hadronic light by light contribution to the muon g-2 calculation was also released on arXiv." This wasn't part of the BMW calculation and increased the HLbL contribution from 92 ± 18 to 106.8 ± 14.7. That paper stated:

We compute the hadronic light-by-light scattering contribution to the muon g−2 from the up, down, and strange-quark sector directly using lattice QCD. Our calculation features evaluations of all possible Wick-contractions of the relevant hadronic four-point function and incorporates several different pion masses, volumes, and lattice-spacings. We obtain a value of aHlblμ = 106.8(14.7) × 10^−11 (adding statistical and systematic errors in quadrature), which is consistent with current phenomenological estimates and a previous lattice determination. It now appears conclusive that the hadronic light-by-light contribution cannot explain the current tension between theory and experiment for the muon g−2.

En-Hung Chao, et al., "Hadronic light-by-light contribution to (g−2)μ from lattice QCD: a complete calculation" arXiv:2104.02632 (April 6, 2021) (the failure of this pre-print to be published, three and a half years later, however, is somewhat concerning, as there is no obvious flaw in the paper from the eyes of an educated layman).

This would increase the Standard Model prediction's value and lower the uncertainty to:

116,592,033.8 ± 36

This would reduce the gap between this combined theoretical prediction and the world average experimental value to 25.2 ± 42.2 (just 0.6 sigma).

The Zimmerman (October 2024) HLbL Calculation

The most recent total HLbL calculation, from October 2024 reached value of 125.5 ± 11.6, which would reduce the HLbL relative uncertainty to 9% (cutting it in more than half from the Theory Initiative value). This would make the state of the art combined prediction of muon g-2:

116,592,052.5 ± 35

The gap between this combined state of the art calculations of the Standard Model value of muon g-2, and world average experimental value for muon g-2, would be 6.5 ± 41.3 (less than 0.2 sigma).

Other Recent HLbL work

As the introduction to the new paper explains in a nice overview of the HLbL calculation:

The HVP data-driven computation is directly related to the experimental input from σ(e+e− → hadrons) data. HLbL in contrast, requires a decomposition in all possible intermediate states. Recently, a rigorous framework, based on the fundamental principles of unitarity, analyticity, crossing symmetry, and gauge invariance has been developed, providing a clear and precise methodology for defining and evaluating the various low energy contributions to HLbL scattering. The most significant among these are the pseudoscalar-pole (π(0), η and η′) contributions. Nevertheless, subleading pieces, such as the π± and K± box diagrams, along with quark loops, have also been reported, with the proton-box representing an intriguing follow-up calculation. Specifically, a preliminary result obtained from the Heavy Mass Expansion (HME) method —which does not consider the form factors contributions— for a mass of M ≡Mp=938 MeV, yields an approximate mean value of ap−box µ = 9.7 ×10−11. This result is comparable in magnitude to several of the previously discussed contributions, thereby motivating a more realistic and precise analysis that incorporates the main effects of the relevant form factors. In this work, we focus on the proton-box HLbL contribution. We apply the master formula and the perturbative quark loop scalar functions, . . . (which we verified independently), together with a complete analysis of different proton form factors descriptions, which are essential inputs for the numerical integration required in the calculations.

The new paper concludes the proton box contribution to HLbL which was preliminarily estimated at 9.7 is actually 0.182 ± 0.007, which is about 50 times smaller than the preliminary result and immaterial in the total, making the neutral and charged pion, the eta, the eta prime, charged kaon, and quark loop contributions as the primary components of the HLbL contribution to muon g-2.

Another new paper calculates the neutral pion contribution to HLbL which is the single largest component of the HLbL contribution, which accounts for more than half (almost two-thirds) of the HLbL contribution:

We develop a method to compute the pion transition form factors directly at arbitrary photon momenta and use it to determine the 

π0

-pole contribution to the hadronic light-by-light scattering in the anomalous magnetic moment of the muon. The calculation is performed using eight gauge ensembles generated with 2+1 flavor domain wall fermions, incorporating multiple pion masses, lattice spacings, and volumes. By introducing a pion structure function and performing a Gegenbauer expansion, we demonstrate that about 98% of the π0-pole contribution can be extracted in a model-independent manner, thereby ensuring that systematic effects are well controlled. After applying finite-volume corrections, as well as performing chiral and continuum extrapolations, we obtain the final result for the π0-pole contribution to the hadronic light-by-light scatterintg in the muon's anomalous magnetic moment, aπ0−poleμ=59.6(2.2)×10−11, and the π0 decay width, Γπ0→γγ=7.20(35)eV.

Tian Lin, et al., "Lattice QCD calculation of the π(0)-pole contribution to the hadronic light-by-light scattering in the anomalous magnetic moment of the muon" arXiv:2411.06349 (November 10, 2024).

The relative uncertainty in the neutral pion contribution is 3.7%, which is a much larger relative uncertainty than in the EM, weak force, or HVP components, but much smaller than the relative uncertainty in the HLbL calculation as a whole.

This neutral pion contribution calculaton is an incremental improvement in the precision of this estimate, compared to most other recent attempts, and produces in value in the same ballpark as previous attempts (i.e. it is statistically consistent with them):

This also implies that the uncertainty from the charged pion, the eta, the eta prime, charged kaon, and quark loop contributions to HLbL, while small in magnitude (about 29-45 * 10^-11 from all of them combined) have combined uncertainties on the order of 13-17 * 10^-11. This is on the order of 35-45% relative uncertainty, which is far more than any other part of the muon g-2 calculation. 

Future Prospects

As the uncertainty in the HVP calculation falls (and this calculation currently approaches the maximum relative precision possible in QCD), this becomes more material in the overall accuracy of the muon g-2 calculation, and the greater precision will be important as the precision of the experimental measurement continues to improve. QCD calculations definitely can get more precise than the HLbL calculations are today, and especially more precise than the HLbL calculations other than the neutral pion contribution. 

But, it will be quite challenging, and may require a major breakthrough in QCD calculations generally, to get the uncertainty in the muon g-2 calculation to below 33-34 * 10^-11, which would be only about a 3-6% improvement from the best available combination of calculations so far. Therefore, the experimental result will probably be more precise than the QCD calculation for the foreseeable future.

Still, the bottom line, which has been clear since the BMW calculation was published at the time of the first Fermilab muon g-2 measurement, is that there is no muon g-2 anomaly since the predicted value and the measured value are consistent at the 0.2 sigma level. 

This global test of beyond the Standard Model physics at relatively low energies reveals that the Standard Model physics is complete and accurate at sub-parts per million levels, at least at relatively low energies of on the order of low GeVs or less.