Quote Of The Day

The sociology in the simulation community seems to be to assert complete success in explaining everything at all times until the next batch of simulations completes running, then point out all the improvements. Everything is explained all the time, only more so as time goes on.

- From Stacy McGaugh at Triton Station on June 18, 2026.

A Theoretically Innovative MOG Theory

Canadian physicist John Moffat's MOG modified gravity theory is a long standing tensor, vector, scalar modification of General Relativity (GR). As the link explains:

Scalar–tensor–vector gravity theory, also known as MOdified Gravity (MOG), is based on an action principle and postulates the existence of a vector field, while elevating the three constants of the theory to scalar fields. In the weak-field approximation, STVG produces a Yukawa-like modification of the gravitational force due to a point source. Intuitively, this result can be described as follows: far from a source gravity is stronger than the Newtonian prediction, but at shorter distances, it is counteracted by a repulsive fifth force due to the vector field.

STVG has been used successfully to explain galaxy rotation curves, the mass profiles of galaxy clusters, gravitational lensing in the Bullet Cluster, and cosmological observations without the need for dark matter. On a smaller scale, in the Solar System, STVG predicts no observable deviation from general relativity. The theory may also offer an explanation for the origin of inertia.

Yukawa forces are forces carried by massive mediator bosons (in contrast to the massless mediator boson of electromagnetism, the photon, which has an infinite range as a result), whose range is a function of the mediator mass. 

The most familiar example of a Yukawa force is the nuclear binding force (sometimes called the residual strong force) that holds photons and neutrons in atomic nuclei together, which is mediated by like composite mesons, especially pions (neutral pions have a mass of about 135 MeV, while charged ones have a mass of about 140 MeV) that have an effective range on the order of femtometers, which is similar to the size of an atomic nucleus.

In contrast, GR without a cosmological constant (including Deur's approach to explaining dark matter phenomena as gravitional) is a tensor theory, and GR with a cosmological constant is a tensor-scalar theory. Newtonian gravity is a scalar theory. Several of the main relativistic generalizations of MOND are also tensor, vector, scalar theories.

MOG, while not the subject of as much scholarship as MOND (Israeli physicist Mordehai Milgrom's 1983 non-relativistic toy model modification of Newtonian gravity that does a good job of replicating dark matter phenomena is almost near equilibrium systems of galaxy size or smaller), MOG is still one of the older modified gravity theories, has received considerable investigation from scientists other than its inventor, is relativistic, is more easily generalized to cosmology scale problems, and unlike MOND, models galaxy cluster phenomena often attributed to dark matter more successfully, at the cost of being somewhat less intuitive to understand.

Moffat's latest short paper formulates his MOG theory in a manner, that while essentially identical to the original, is easier to apply to cosmology scale questions.

We develop a Stueckelberg gauge-invariant formulation of modified gravity (MOG). 

The massive vector field is made gauge-invariant by introducing a compensating scalar field, without requiring a Higgs field, spontaneous symmetry breaking, or a vacuum expectation value to fix the effective Newtonian gravitational coupling. This separates the gauge-invariant origin of the vector mass from the cosmological evolution of the gravitational coupling. 

The formulation preserves the finite-range vector interaction of MOG, while allowing the effective gravitational coupling to be treated as an independent scalar or scale-dependent quantity. This distinction is important for cosmological tests, since early-universe constraints and late-time large-scale gravitational phenomena need not be tied to a symmetry-breaking vacuum. The Stueckelberg formulation provides a gauge-invariant framework for comparing MOG with nucleosynthesis, cosmic microwave background, large-scale structure, lensing, and distance data.

John W. Moffat, "Stueckelberg Gauge Invariant Formulation of MOG" arXiv:2606.26427 (June 4, 2026).

Another new MOG paper constrains the value of one of that theory's physical constants (to a value inconsistent with the range in the previous literature on the topic):

The scalar-tensor-vector-gravity (STVG), a prototype of modified gravity developed by Moffat, can correctly explain galaxy rotation curves, cluster dynamics, Bullet Cluster phenomena and cosmological data without invoking the observationally elusive general relativistic (GR) dark matter. Further, recent observations of neutron star masses are shown to defy some GR predictions, whereas STVG turns out to be more consistent with those observations. These successes indicate that STVG could be a potential candidate for a new theory of gravity. 

However, an important question concerns the possible range of values of the STVG dimensionless parameter α imposed by various physical scenarios. In the literature, the range 0.03 < α < 2.47 corresponding to different central source masses has been suggested. We show here that the α can be considerably constrained into the range 0 < α < 10^−5 assuming that the updated GPS fluctuation does not exceed the α-dependent correction to the terrestrial Sagnac delay.

R. Kh. Karimov, R. N. Izmailov, K. K. Nandi, "Terrestrial Sagnac delay in scalar-tensor-vector-gravity" arXiv:2606.27033 (June 25, 2026).

A footnote on f(R) gravity

Probably the other modified gravity theory with significant scholarship from multiple astrophysicists that is most often used to explain dark matter phenomena gravitationally is f(R) gravity (the image below is from this link), which like GR with a cosmological constant, and unlike MOG or some relativistic generalizations of MOND, is a tensor-scalar theory. The way f(R) gravity modifies GR is not with an extra vector field, but with a higher order derivative term. The standard Ricci scalar R in the Einstein-Hilbert action is replaced by a general function of R (e.g., R + (alpha)*R^2). Mathematically and dynamically, this higher-order derivative theory is exactly equivalent to standard General Relativity coupled to a single, dynamical scalar field (known as the scalaron), rather than only having the static scalar dark energy field that is equivalent to the cosmological constant.

Like MOG, f(R) gravity has a Yukawa correction to the gravitational potential, which (at least in part, it also has a time and scale dependent gravitational constant) is how it can explain some or all dark matter phenomena without dark matter particles.

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.