A New Phenomenological Relationship Of Neutrino Masses

A new phenomenological relationship of the neutrino mass differences is interesting, and close matches to simple formulas like this one are rare.

In the standard three-flavor paradigm, two of the three neutrino mass-squared differences are independent. The existence of an additional dependence between these oscillation parameters was recently predicted on empirical grounds, taking the form

The JUNO Collaboration has now reported the most precise measurements to date of both Δm(21)^2 and Δm(31)^2. We demonstrate that this relation is in excellent agreement with the experimental data, which yield 1.4143 +0.0036 −0.0038 corresponding to a relative precision of 0.27%. Remarkably, the central value differs by less than 0.03 standard deviations from the prediction (2‾√≈1.4142). This provides strong experimental support for the novel constraint and thereby suggests an underlying symmetry governing the neutrino mass spectrum, rather than purely independent parameters.

I. Alikhanov, "Novel dependence between neutrino mass splittings strongly supported by initial JUNO results" arXiv:2610.06738 (October 5, 2026). This updates:

I. Alikhanov, "Another relation among the neutrino mass-squared differences?", arXiv:2601.18781 (January 26, 2026).

There are more comparisons along this line from a different author, Vernon Barger, at https://arxiv.org/html/2603.00810v2.

What Is Fundamental In Physics?

This is a question and my answer from the Physics Forums (officially written at the graduate school level, but dumbed down since the question indicates that the person asking isn't actually that sophisticated and knowledgable. The question is in a block quote, but my answer is not, for ease of reading.

Wiktorius I said:

TL;DR: Does physics establish that mass, particles, and fields are fundamental, or could they be emergent properties of a deeper physical structure?

Are mass, particles, and fields fundamental physical entities, or could they be observable consequences of a deeper physical structure? I am particularly interested in what experimental evidence allows us to distinguish between these possibilities.They aren't necessarily fundamental, but we have no real idea what kind of deeper physical structure could cause what we see and there are no experiments or observations that are likely to take place in the next few decades that can tell us.

Generally, going as deep as we possibly can, we think of particles in the context of an effective field theory as local excitations of fields.

But, most popular accounts, because it is more intuitive to understand, talk in terms of particles, with fermions corresponding to what we think of as matter in a colloquial sense, and bosons corresponding to "force fields" for the fundamental forces of the Standard Model.

Generally speaking, the mass of particles (both fundamental and composite) is not understood as fundamental. Massive Standard Model fundamental particles like electrons, muons, tau leptons, and quarks and W bosons and Z bosons and the Higgs boson (with the probable exception of the neutrinos whose mechanism for acquiring mass isn't well understood at this point) get their mass from their interactions with the Higgs field. Massive composite particles (like protons and neutrons) get their mass from a combination of the masses of the fundamental Standard Model particles that they are made up of (to a small extent) and from the energy of the gluon fields that bind them together (to a great extent). But the sometimes elusive concept of mass-energy is something that is fundamental and conserved.

There is no real consensus over whether gravity is something that arises from the exchange of massless spin-2 gravitons (which are bosons) in a way analogous to the way that electromagnetism arises from the exchange of massless spin-1 photons, or whether gravity is a property of space-time itself which is either classical and smooth, or as in the quantum gravity approach of loop quantum gravity, in which case space-time itself is discrete at some ultra-fine scale.

There is no consensus on this because we don't have any observational tools sufficiently powerful to distinguish between the hypotheses, and because while the math behind classical general relativity is hard, we haven't cracked the math behind quantum gravity theories of various kinds outside some very specific isolated cases in theories that don't hold true for all circumstances and have internal mathematical issues with them that make quantum gravity theories in the general case intractable.

We don't have any well accepted theories that explain why the couple dozen plus experimentally measured physical constants of the Standard Model plus general relativity have the values that they do, although electroweak theory does establish that a few of the Standard Model physical constants are not actually mathematically independent of each other, so the number of independent degrees of freedom in the experimentally measured physical constants of the Standard Model is a few less than the number of experimentally measured physical constants (e.g. the electromagnetic coupling constant, weak force coupling constant, W boson mass, and Z boson mass are related to each other by a formula which has fewer than four independent degrees of freedom).

There are some theories out there that go part of the way, providing a road map for possible ways to explain more of the experimentally measured physical constants of the Standard Model and general relativity. But while these are bit more than speculative, none of them are established in a settled way that can secure a scientific consensus or anything close to that. Moreover, as impressive as our scientific methods are, imprecision in measuring some of these experimentally measured physical constants limits the degree to which we can distinguish one explanation for their values from another. Many key physical constants are measured only to percent or part per thousand levels.

But the failure of scientists so far to find any meaningful deviations from the Standard Model in particle accelerator experiments and other terrestrial tests of it (like ultra-precise measurements of the anomalous magnetic moment of the muon) profoundly constrains any alternatives that would predict anything experimentally different from what the Standard Model predicts.

I can't recall ever meeting a physicist who thinks that there isn't some deeper theory that could explain why those physical constants have the values that they do. But most physicists, having seen the parade of physicists before them who have failed to find that deeper theory, are content to not go looking for it very seriously.

As a practical matter, even if you found the deeper theory that explained why most of those physical constants have the values that they do, it wouldn't necessary change much for most physicists, because we've already measured those physical constants to high precision in most of the cases where precise values are important to precise calculations. Having the deeper theory might allow you to assign more precise values to the physical constants whose values are hardest to measure, which would be a helpful thing on the margins as a practical matter and very personally satisfying, but that might be just about it.

Of course, if we knew why those physical constants had the values that they do, we would probably also rule out all sorts of beyond the Standard Model theories and models in a way that would be more satisfying than the way that we do now. But it wouldn't have many very practical consequences.

There are a variety of reasons why the most naively obvious alternative to the list of Standard Model particles being fundamental, which would be that they are composite particles of a smaller number of more fundamental particles (generically called preons), is unlikely to be an explanation. The reasons are a bit technical (e.g. the binding energy necessary for this to be the case is too great to be consistent with the hypothesis), but the consensus is that this is pretty much a dead end.

String theory, at first, seemed to be another path to a more fundamental explanation of the laws of nature than the Standard Model plus general relativity (which aren't themselves compatible with each other which implies that one or the other or both aren't truly fundamental) but this research agenda in the many decades since it was first introduced has turned out to be a quagmire that even many of its leading theorists don't claim is an accurate description of the world we actually live in, even though the things we learned along the way investigating it may have value.

So, while our core theories, the Standard Model and general relativity, probably aren't the most fundamental and complete explanations of the laws of nature that are possible, we really have no solid leads on alternatives.

Perhaps the most promising area where we might gain a better understanding is from astronomy where ill-understood concepts summarized as dark matter and dark energy might be better understood with more observational astronomy evidence. But we are far from reaching a consensus understanding of them, and are less confident about the baseline explanation of them called Lambda CDM (for cold dark matter) theory now than we have been since Lambda CDM was formulated.

The prospects for something more fundamental didn't always seem so bleak. In the late 1980s and early 1990s, a group of theories called "supersymmetry" and their gravitational extension, "supergravity" looked very promising. It would have explained many of the fundamental constants in an overall framework (although not all of them), it would have provided a path to unifying the Standard Model coupling constants at higher energies into a single force, it would have provided a candidate for dark matter particles that was sufficient to fit the astronomy and particle physics data available at the time, there was no evidence that the cosmological constant wasn't constant, it would have provided a bridge between the Standard Model and string theory, and a path to a workable theory of quantum gravity.

But bit by bit, that fell apart. Supersymmetry predicted particles with well defined properties, but when particle collider scientists looked for those predicted particles they weren't were they were assumed to be. Better data showed that the Standard Model coupling constants weren't on a path to converging at high energies. Better astronomy observations found that dark matter phenomena behaved differently than it would have if the supersymmetry candidates were what explained it. Better astronomy observations found that the cosmological constant might not be constant after all. The discovery that neutrinos had mass complicated the situation and the failure to detect neutrinoless beta decay complicated one of the most elegant explanations of neutrino mass that might fit better into the supersymmetry/string theory framework. The math necessary for string theory and quantum gravity turned out to be much more problematic than early researchers in these fields had hoped or expected at a very fundamental level. Strong experimentally confirmed anomalies that would deviate from the Standard Model to prove that supersymmetry was correct, instead failed to materialize. The experimental anomalies that seemed to potentially support supersymmetry went away as theoretical predictions and experimental measurements grew more precise.

Supersymmetry isn't definitively ruled out yet, but any version of it that is discovered won't solve the problems that a large share of the physics community hoped and believed that it would in the late 1980s and early 1990s, before they had enough experimental data and mathematical understanding to realize that their beautiful solution to all of the unsolved problems in fundamental physics with a deeper theory wasn't going to hold together.

A generation or two later, we have bits and pieces of ideas - MOND, entropic gravity, axion-like ultralight dark matter particles, Koide's rule and extrapolations of it, multiple Higgs doublet theories, see saw theories of neutrino mass, and more. There are many more partial theories to explain one little thing that surfaced as an anomaly or possibility for a while that were later ruled out. But we have no real solid agenda for developing a deeper theory.

Who knows?

Maybe we're just a genius or three away from ending a period of non-consensus and confusion about a way forward and finding the deeper theories we need to explain the outstanding unsolved problems of fundamental physics. But I'm not that genius and there aren't a lot of good candidates among living physicists today for people who could be that genius.

POST SCRIPT (not in the original post):

It is notable that MOND and Koide's rule, two of the most successful enduring beyond the Standard Model theories, albeit both phenomenological only, both date to the early 1980s, before supersymmetry and string theory had become dominant trends in fundamental physics.

Neutrino physics

Neutrino mass was first discovered in 1998. This finding was replicated and confirmed in 2001. In 2015 Nobel prizes were awarded for the discovery that neutrinos are massive. Experimental determination of some of the parameters of neutrino oscillation (such as the CP violating phase) is still very crude and the absolute neutrino masses are still also not tightly established.

Supersymmetry and string theory advocates tend to strongly favor Majorana mass for neutrinos. The absolute masses of the neutrinos are still not known, only their mass differences. But there are strong hints from cosmology and from the magnitude of the mass differences, that neutrino masses are so small that we wouldn't expect to have observed definitive neutrinoless beta decay signals that would be proof of Majorana mass in neutrinos yet even if that was the source of neutrino mass. I personally find the hypothesis that neutrinos are their own antiparticles to be highly unlikely given that the main reason that they were hypothesized in the first place was to provide particle-antiparticle balance that conserves lepton number. There is a good chance that the bounds on absolute neutrino mass and the non-discovery of neutrinoless double beta decay will exclude a vanilla version of the Majorana mass hypothesis for neutrinos in my lifetime.

This said, I'm also not very impressed with see saw models, in which neutrinos have a Dirac mass arising from oscillations with one to three otherwise undetectable right handed neutrinos that are much more massive than their Standard Model active neutrino counterparts. 

The most plausible alternatives that I imagine, with only thin support from academic research, are that neutrinos either acquire their mass through a combination of their self-interactions via the weak force and their interactions with W bosons more generally, or acquire their mass through a second neutral scalar boson analogous to the Higgs boson.

More generally, I suspect that the role of the W boson in imparting mass to fundamental particles is underrated relative to the rule of the Higgs boson in doing so.

Higgs physics

The Higgs boson was discovered in 2012, although it was predicted about forty years earlier and efforts continue at the Large Hadron Collider (LHC), and in reanalysis of Tevatron collider data, to confirm that its properties are actually fully consistent with the Standard Model Higgs boson predicted in the early 1970s (so far no experimental data disproves this hypothesis and the extent to which it could deviate from that hypothesis is heavily constrained by the data that we do have). The measured mass of the Higgs boson according to the Particle Data Group, which is 125.11  ± 0.11 GeV/c², significantly constrains that parameter space of many supersymmetry (a.k.a. SUSY) theories.

One largely back burner issue is that a global electroweak fit would favor a Higgs boson mass of 90 + 21 -18 GeV/c², which is consistent at the two sigma level with the measured Higgs boson mass, but has a best fit value 1.7 sigma below the measured value which is still a bit puzzling.

There are exclusions of additional Higgs bosons (neutral, charged) proposed by some beyond the Standard Model theories, which the Particle Data Group also underplays, but which are indeed not quite as insurmountable as the exclusions for supersymmetric particles.

Supersymmetry

While supersymmetry papers are still being published, this line of beyond the Standard Model physics inquiry has greatly fallen out of favor from its heyday. A large share of physicists expected it to be discovered at the LHC but that didn't happen.

The Particle Data Group has a full page of supersymmetry exclusions from experiments, which in my humble opinion is too conservative. But it is a difficult set of experimental data points to present because many of the exclusions of model dependent. The three linked review articles on that page, however, are helpful.

The main reasons that supersymmetric particles are disfavored as dark matter candidates (called WIMPs) now are that: 

(1) direct dark matter detection experiments (such as Lux-Zepplin which detected a single first candidate for a direct dark matter detection signal earlier this month mostly like with a mass of about 1 TeV/c² which is six times the top quark mass and four times the mass of a uranium atom) imply a far smaller cross-section of interaction that the cross-section of interaction predicted by supersymmetry theories;

(2) galaxy dynamics (particularly the core-cusp problem) disfavor a dark matter candidate in the 1 TeV/c² or more mass range (even though plenty of preprints hypothesizing that the LZ candidate is a WIMP are out there) and really favor dark matter particle candidates in the 10 keV/c² or less mass range; and

(3) particle collider bounds on the minimum mass of supersymmetric particles that could plausibly be dark matter candidates are generally in the hundreds of GeV/c² or more range.

The consistency of the experimentally measured value of the anomalous magnetic moment of the muon (muon g-2) and the Standard Model predicted value of that quantity, strongly disfavors the existence of supersymmetry below very high energies. 

Indo-Aryan v. Indo-Iranian Religion (Speculative)

The Vedic religion of South Asia, which ultimately gave rise to modern Hinduism, can be understood as a mash-up of the proto-Indo-European pantheon, which also influenced Nordic/Germanic and Greco-Roman polytheism, with an indigenous Harappan religious substrate that had a somewhat more animistic and a somewhat less anthropomorphic character. It is a process not unlike the process seen in Europe.

In contrast, the Indo-Iranian religion of present day Iran, which appears to have emerged contemporaneously with the Vedic religion from Indo-European contributors very similar to their Indo-European counterparts, was the progenitor of Zoroastrianism, which marks a dramatic departure from proto-Indo-European polytheism, replacing it with what is typically simplified to being described as dualism rather than a polytheistic pantheon.

The open question is, what made Indo-Iranian religion depart so markedly from other Indo-European religious landscapes.

We have a fairly rich trove of written sources, both Jewish and non-Jewish, that sheds light on how Jewish monotheism emerged from Eastern Mediterranean Semitic polytheism to henotheism (in which there are multiple gods but one is pre-eminent and the center of worship) to true monotheism, with some influence from a brief and failed flirtation in Egypt with a monotheistic or henotheistic religion that was heralded by a pharaoh, then abandoned not longer after that by another one.

But in West Asia, we have pretty much only the Avesta, the core and original Zoroastrian scripture, itself to guide us.

One possibility is that the Indo-Iranians had Zoroaster who just happened to be a truly remarkable individual that no other contemporaneous culture had, to make the leap in a way that secured large numbers of followers somehow. Maybe Zoroaster charmed and won over the most powerful kings and princes of West Asia who imposed his religion upon their people. Maybe Zoroastrianism secured a mass grass roots following somehow and was co-opted by West Asia's leaders.

Alternatively, perhaps Zoroastrianism is the legacy of a pre-existing religious tradition and worldview of West Asia that had a powerful influence on the conquering Indo-Iranians, which would certainly not be unprecedented. The Hittites, for example, heavily borrowed from and adopted the Hattic religion of their predecessors, keeping the Hattic language alive as a liturgical language long after it had ceased to be a living liturgical language. The linguistically Semitic Akkadians did something similar with the Sumerian religion. The fact that the language of the Avesta is written in an Indo-Iranian dialect that is a close sister to Sanskrit, somewhat disfavors this hypothesis, but it could be that the fact that Sumerian and Hattic were written languages, while the pre-Indo-European languages of South Asia and West Asia were not, could also explain this fact.

In any of these scenarios, however, there is another point. What about West Asia in the Bronze Age caused Zoroastrianism to fill a spiritual need better filled by dualism than polytheism, that was absent from contemporaneous polytheistic cultures which came under Indo-European rule?

Religions tend to imagine the world of the divine that is parallel to the leadership structure of their own societies in their formative eras. 

Animistic religions are associated with hunter-gather societies with weak political structures of fragmentary local bands. Polytheistic religions tended to emerge in chiefdomships where a chief ruled in concern with subordinate but independently powerful other leaders in a sort of council. Egypt's flirtation with monotheism and Jewish monotheism emerged in solitary monarchies with a single clear king or queen. Chinese conceptions of the spiritual world mirrored the vast bureaucracies of its own quick emerging empire.

Was the political situation in West Asia such that the dualism of Zoroastrianism better mirrored its political reality than the chiefdom style of rule of other polytheistic early Bronze Age societies?

Perhaps, for example, the Eastern Iranian region to which the Avesta refers was ruled by a strong unified monarchy, rather than a mere chiefdomship, of the conquering Indo-Iranians, but faced a strong insurgency, perhaps from the pre-Indo-Iranian king who was deposed (perhaps associated with the Jiroft culture) or from an arguably more legitimate heir to the throne who was passed over and was seeking to regain the rule of his dynasty, at the time that Zoroaster formulates Zoroastrianism.

The fact that West Asia was an early place of metallurgy mastery and the Copper Age first and then the Bronze Age is suggestive of the possibility that its political culture may indeed have in some way differed materially from that of its Bronze Age peers, so this isn't an implausible hypothesis.

More Thoughts On Red River

The first twelve episode cours of the anime Red River a.k.a. Anatolia story, based upon the long hit manga series of the same name finished yesterday, and has prompted me to research the actual history of the period depicted in this historical fiction piece (another twelve episodes are contracted for the fourth quarter of this year).

One of the points highlighted by the series that often gets overlooked in academic research is that: (1) the Hittites, the Mittani kingdom, a Mesopotamian regime, and the Egyptian 18th dynasty, were all contemporaneous, and (2) that these kingdoms meaningfully interacted with each other, exchanging correspondence, diplomats, and brides with each other, and engaging in military conflicts with the intermediate Hittite empire.

At least two royal Egyptian brides were from the Mittani Kingdom in roughly the 14th century BCE when Red River is set.

This is especially notable because, while the everyday commoners in the Mittani Kingdom were Hurrian language speakers (a language associated with the highlands of the Zargos Mountains and to a lesser extent the eastern Anatolian highlands where the possibly related Hattic language was spoken), the Mittani elites in the 1300s BCE spoke a Sanskrit derived Indo-Aryan language (that gave rise, most famously, to the Hindi language of northern India) and worshiped Indo-Aryan gods in their pantheon that became part of the Hindu religion rather than the Avestan language and religion that was starting to emerge in what is now Iran at around this time. (A cultural affiliation that neither the anime or the manga make much of.)

The Indo-Aryan linguistic and religious link to western Iran and connection the Anatolia and Egypt, disappears, however, when the Mittani kingdom falls, for historical reasons that like its brief appearance in the Mittani kingdom have since been lost and are currently unknown.

The Mittani kingdom featured in Red River as a major military adversary of the Hitties, therefore, provides a concrete human, person to person bridge between the culture of South Asian Indo-Europeans and late Bronze Age ancient Egypt: two cultural worlds usually thought of as unrelated and independent of each other despite being contemporaneous.

Actual history is unclear over whether the famous ancient Egyptian Queen Nefertiti was one of those Mittani brides or not, as the manga and anime assert, because it is a legitimate historical possibility and makes for a good story. But the historicity of the Mittani kingdom providing two royal brides to Egypt in the historical era in which Red River is set is not seriously questioned, nor is the Red River story point that at least one of those marriages was driven to a significant extent by a Mittani expectation of receiving large amounts of gold from Egypt in exchange (although, historically, the Egypt failed to deliver the gold).

A Single Possible Direct Dark Matter Detection

The Lux-Zepplin direct dark matter experiment has detected a single event that could be a dark matter particle (see also here). But given the immense amount of searching which drives up look elsewhere effects and the myriad other possible explanations for a single outlier data point, it is not a definitive dark matter detection yet.

From my posts on this at the link:

With this detector does that mean that the dark matter candidate would interact with the weak nuclear force?

Or some novel fifth force that has a cross-section of interaction much weaker (by factors of millions or billions or so) than the SM weak force.

The DM cross-section of interaction of atomic nuclei (and hence the weak force charge of DM particles) would have to be profoundly weaker than that of neutrinos if it is a weak force interaction, which would be surprising since every SM particle with weak force interactions has the same weak force charge.

What would even be a standard for credible detection here? 3 events? 10? When would they start calling Stockholm?

Not sure, does the paper say what statistical significance this event has?

The paper estimates the global statistical significance at 2.6 sigma, with a local statistical significance of up to 3.4 sigma. But the number of events probably isn't sufficient to determine the significance. It also depends, in part, about the details of each event and how far those details are from the expected background events on the chart in the OP. 

But given the amount of searching that has been done with multiple direct dark matter detection experiments that are all roughly similar to each other, the significance after the look elsewhere effect should be much lower than the local statistical significance.

You'd probably need a local significance of something like 10 sigma to get a global significance that meets the 5 sigma discovery threshold. Also, it isn't just 5 sigma, you also need a theoretical framework to attach the result to and replication, to be a true discovery.

So, you'd need (1) to do a lot of analysis with outside peer reviewers to rule out extremely faint backgrounds that weren't considered in the original analysis or other possible non-DM sources of this outlier data point, and (2) you'd need another experiment in addition to LZ to see it.

But, this makes funding direct dark matter detection experiments similar to LZ for the purpose of replicating this result and tuned to the parameters space where this was seen a no brainer.

Another issue is that even if the event is some non-SM particle, it doesn't necessarily follow that it is a significant component of DM.

Direct DM experiments are based upon the assumptions that the total DM mass flux and DM particle momentum can be pretty well determined from Milky Way dynamics, so you are looking at a parameter space in which DM particle mass and DM particle cross-section of interaction with nucleons trade off against each other for any given result.

This outlier data point, if it is real, points to a fairly high DM particle mass (some arXiv phenomenology papers are speculating in the 1 TeV order of magnitude). But anything much above 10 keV of mass presents real problems as a major component of DM since inferred DM distributions which are "cored" rather than "cuspy" suggest that you need much smaller DM masses to reduce the core-cusp problem, and plausible self-interaction strengths of heavy DM particles still don't solve that problem in simulations.

So, even if it is DM, it might be a type of DM particle that makes up, for example, only 1% of DM, as a DM analog to something like carbon atoms in interstellar space, while the predominant component of particle DM, as a DM analog of something like hydrogen atoms, might be too light for LZ to detect significantly due to neutrino backgrounds.

Analysis of this result in other preprints include:
https://arxiv.org/abs/2609.01475
https://arxiv.org/abs/2609.01504
https://arxiv.org/abs/2609.01592
https://arxiv.org/abs/2609.02608
https://arxiv.org/abs/2609.02775
https://arxiv.org/abs/2609.02868
https://arxiv.org/abs/2609.02823
https://arxiv.org/abs/2609.02807

If there are multiple kinds of DM particles and this is only a rare and heavy member of that set, this implies that the cross-section of interaction can be much higher (and thus, much closer to the neutrino-nucleon cross section of interaction). This is because the cross-section of interaction calculations assume that there is only one kind of DM particle, so that the actual events recorded comes from interactions with 100% of the DM flux through LZ. But if this assumption is wrong and only, for example, 1% of DM particles are massive enough for LZ to detect, then the actual cross-section of interaction implied by a given number of events is 100 times greater in that example.

If DM particles of this mass are extremely rare (the DM analog to uranium or lead, perhaps, making up only one in a million or billion DM particles) within the universe of DM particles, then perhaps the cross-section of interaction could be equivalent to the strength of the weak force interaction of SM particles.

The effective lower bound of DM particle mass that LZ can detect is about 0.2-0.5 GeV, and the signal to noise ratio starts to degrade meaningfully for DM particle masses below 10 GeV. And, there are, to repeat, strong suggestions from the inferred shape of DM distributions, that the predominant share of DM particles (assuming that they exist) should be about 10 keV or less, which is about 20,000 times less massive than DM particles that can give rise to events detected by LZ which are distinguishable from background events, and about 1,000,000 times less massive than DM particles that LZ can detect with maximum efficiency.

My suspicions

I think that this result is either a fluke in the background events (which at 2.6 sigma global significance, a global significance that I suspect is actually overstated, is entirely plausible), or a methodological error.

Even if it is, however, a genuine BSM particle, the one observed seems unlikely to be a DM particle, and particularly unlikely to be a Higgsino, which is what many of the linked papers suggest. A Higgsino is fairly tightly constrained in supersymmetry theories to have properties that this particle is unlikely to have, and might even be possible to rule out with further analysis of this particular data point.

Further, supersymmetry theories are simply not credible as theoretical frameworks in the broader sense for a variety of reasons, even though a Higgsino mass of 1.1 TeV was predicted in 2012 as the mass of a Higgsino that was the sole component of dark matter according to Hall, Lawrence J.; Nomura, Yasunori (2012). "Spread Supersymmetry". Journal of High Energy Physics. 2012: 82. arXiv:1111.4519 doi:10.1007/JHEP01(2012)082

The non-detection of any hint of a Higgsino at the LHC also casts doubt on this hypothesis, although the formal exclusions from the LHC (which is always a bit dicey because it depends on the Higgsino model used and the mass splitting between it and certain other supersymmetric particles) only go up to about 1.025 TeV.

There are also lots of strong reasons from astronomy, as I just scratched the surface of above, to think that either gravity (modified, non-perturbative, otherwise) or a fifth force, rather than dark matter particles make more sense, and that heavy dark matter particles (1 TeV or more particles that are at the fringe of what LZ and other direct detection experiments can be sensitive to), at least as a primary source of dark matter, are among the least observationally favored dark matter particle hypotheses.

This doesn't inherently rule out the possibility of a new fundamental particle, and the spectrum of composite bound quark and gluon structures, except toponium (which has a very distinct signature and set of conditions in which it can be formed) pretty much top out in the low tens of GeV, far below the 1 TeV scale. Toponium is 344-347 GeV, which is still below the TeV scale. The most massive observed atom, Oganesson (element 118), specifically its isotope Oganesson-294, has a mass of only about 274 GeV, which is still well below 1 TeV. A mass of 1 TeV would require a rather large and complex molecule (not hadron molecule, but a normal molecule made up of ordinary chemical elements), so an event like this, if being properly interpreted is not a good fit for any known fundamental particle, any known or possible hadron, any known or plausible future atom or atomic element. So, if there is a real detection of a 1 TeV mass particle, it is very much beyond the Standard Model and new physics. But even if it is real, that doesn't mean that it is an important contributor to dark matter. It could be a BSM particle with nothing meaningful to do with dark matter phenomena.

Indeed, the lack of well-motivated candidates for a 1 TeV particle with a very low cross-section of interaction with nucleons (at least no larger than the weak force coupling), makes this extraordinary claim require extraordinary proof and compels a very hard look for other explanations, especially given its only modest statistical significance so far.

More Physics Quick Hits

A relationship between the Higgs boson, top quark, and Z boson masses

Maybe a coincidence, maybe meaningful. The only relation that fits, using pole masses, holds at 1.4 sigma, but tends to predict either a rather high Higgs boson mass, or a rather low top quark mass.

I have little doubt that there are deeper functional relationships between the fundamental constants of the Standard Model (or at least some of them) than are contained within the Standard Model (what I call "within the Standard Model new physics" as opposed to "beyond the Standard Model new physics"). And, even if this particular relationship is not actually true, it is close enough that it is fruitful to ask, if there is some deeper source for these experimentally measured physical constant values, what kind of relationship would produce a close coincidence like this one.

For example, I wonder if an approximation of this relationship is favored in some way by the LP & C relationship that the square of the Higgs vev is equal to the sum of the squares of the fundamental SM particle masses, or by the approximate, but not exact, equality between the sum of the squares of the fundamental fermion masses and the sum of the squares of the fundamental boson masses.

Indeed, the paper notes that: 

After the Higgs discovery the numerical observation M(H)^2 ≃ M(Z)*M(t) (1) was proposed as a possible electroweak mass coincidence involving the heaviest spin-0, spin-1/2 and spin-1 representatives of the Standard Model (SM) spectrum.

(The citation for this sentence is to a paper by the author of the current paper: E. Torrente-Lujan, "The Higgs mass coincidence problem: why is the Higgs mass M2 H = MZMt?", Eur. Phys. J. C 74 (2014) 2744, arXiv:1209.0474.)

This suggests that there might be a fuller relationship that involves addition masses on the Standard Model spectrum beyond the heaviest ones that might provide correcting terms bringing the relationship to a more exact one.

I also seem to recall that there theoretically expected mass of the W boson in the Standard Model is a function of the Z boson mass, the top quark mass, and the Higgs boson mass, based upon electroweak unification in some way, but have never seen that relationship spelled out in detail. I have only seen the abbreviated leading order relationship between the W boson mass and Z boson mass that is related to the electromagnetic force and weak force coupling constants. 

The paper doesn't evaluate the "arithmetic relation" with the theoretically predicted W boson mass of 80.357 ± 0.006 GeV, but because that value is lower than the PDG value of 80.3692 ± 0.0133 GeV, it should be a somewhat better fit with the theoretically predicted W boson mass. As it turns out, this difference doesn't matter much, and neither does the difference between the Higgs boson mass that they use of 125.2 ± 0.11 GeV and the lower value of the Higgs boson mass of 125.09 GeV that is sometimes used. Indeed, the greater precision of the theoretically estimated W boson mass may increase the statistical significance of the discrepancy somewhat.

The relation M(H)^2 ≃ M(Z)*M(t), previously proposed as a non-trivial Higgs mass coincidence, is reconsidered with present electroweak inputs and with a scheme-consistent matching analysis. With the 2025 PDG values for M(Z), M(W) and M(H), and the ATLAS-CMS direct top-mass combination, the pole-level ratio is ρ(Zt)=M(Z)*M(t)/M(H)^2 = 1.00362 ± 0.00261. Thus an exact pole-level geometric relation predicts either M(H) = 125.426 ± 0.120 GeV or M(t) = 171.898 ± 0.302 GeV, which is still a 1.4σ test rather than an exclusion. 

By contrast, the companion arithmetic relation gives ρ(Wt) = (M(W)+M(t))/(2M(H))=1.00994±0.00159 and is not a viable exact mass sum rule. 

We then evaluate the complete NNLO weak-scale MS bar matching formulae at μ=M(t). In the standard convention one obtains ρˆ(Zt(M(t)) = √(g(2)^2+g(Y)^2) * y(t)/(4√2λ) = 0.96714±0.00361. Consequently, the exact running-coupling boundary condition λ =  g(Z)y(t)/(4√2) at the top scale would predict M(H) = 123.19 ± 0.20 GeV, or equivalently M(t) = 177.81 ± 0.50GeV when M(H) is held fixed. This is incompatible with the measured point. 

A possible symmetry explanation must therefore act on pole-level threshold quantities, or provide a finite matching factor κ(th) = 1.0340 ± 0.0039 at the electroweak scale. We formulate this requirement as a target for custodial/top-Higgs or triality-like symmetry extensions.

E. Torrente-Lujan, "The Higgs-top-Z mass coincidence relation after NNLO matching" arXiv:2605.21721 (May 20, 2026) (Report number: FISPAC-TH/3145-26, UQBAR-TH/26-97234).

Another key chart from the paper is this one:

The Lambda predictions are very far from the mark, as are the "arithmetic relation" estimates. The other relationship still has some tension with the experimental results but isn't ruled out.

Bounds on new neutrino physics

The constraints on BSM physics continue to narrow. In the Standard Model, the neutrino magnetic moment is predicted to be far below the threshold of current experimental detection, and the neutrino has an exactly zero electromagnetic charge. These results are consistent with those predictions and thus constrain BSM neutrino physics to very slight deviations from the SM predictions. Non-standard interactions of neutrinos are likewise constrained materially.

CODATA 2022 gives the value:

     ${\displaystyle {\sin }^2{\theta }_{\mathsf{\text{w}}}=1-{\left(\frac{m_{\mathsf{\text{W}}}}{m_{\mathsf{\text{Z}}}}\right)}^2=0.22305(23).}$

So, the Weinberg angle measurement from the CONUS collaboration, while consistent with the world average measurement at the two sigma level, is too imprecise by two orders of magnitude to add meaningfully to our knowledge of that physical constant compared to the value obtained from simply plugging in the world average measured values of the W boson mass and the Z boson mass (or from calculating it in electroweak unification theory from the electromagnetic coupling constant and the weak force coupling constant so e^2/g^2 = sin^2 theta(W).

Its detections with pion-decay-at-rest, solar and recently with reactor antineutrinos by the CONUS collaboration render coherent elastic neutrino-nucleus scattering (CEνNS) an established tool for investigations within and beyond the Standard Model (SM). The CONUS experiment located at the nuclear power plants in Brokdorf (Germany) and Leibstadt (Switzerland) operates Germanium semiconductor detectors in a compact shield at close distance to the reactor core. An observation with 3.7σ significance is reported at the Leibstadt site, showing good agreement with its SM prediction.

Physics investigations performed with the last datasets collected at the Brokdorf reactor and with the first data obtained at the Leibstadt site are summarized. By using the experimental analysis framework, the presented results contain the full systematics that underlie the experiment. 

Previously determined limits with neutrino-electron scattering on the neutrino magnetic moment and a neutrino millicharge are improved to μ(ν) < 5.18⋅10^−11μB and q(ν) < 1.76⋅10^−12e0 (90% C.L). Further, the scale of new physics related to NSIs is improved to ΛNSI = 145 GeV and limits on the coupling of light new mediators are lowered down to 4⋅10−7 (90% C.L.) with the new data. Finally, the determination of the Weinberg angle with CEνNS and reactor antineutrinos yields sin(θ(W))^2  = 0.28 +0.03 −0.04 at a momentum transfer of ∼10 MeV.

N. Ackermann, et al., "New constraints on physics within and beyond the standard model from the latest CONUS datasets" arXiv:2605.22815 (May 21m 2026).

The abstract is unclear about what there is 3.7 sigma evidence of, but the body text clarifies that: "With data collected there since 2023, a successful CEνNS detection was reported with 3.7σ significance."

The body text explains some of the theoretical motivation for the paper:

On theory side CEνNS has become an interesting tool for investigations within and beyond the standard model (BSM) because of its flavor-blind and, in principle, threshold-free properties [21–27]. Within the SM it enables measurements of the Weinberg angle sin^2 θ(W) at the MeV scale with neutrinos and probe modifications of the involved couplings, i.e. via radiative correction [28–31] or investigation of the nuclear form factor when deviating from full coherence, i.e. with higher neutrino energies from πDAR sources [32–36]. BSM searches can be performed by testing for new neutrino interactions, for example in the context of heavy new physics via non-standard neutrino interactions (NSIs)[37–43] or new light mediators [44–53]. Neutrino (electromagnetic) properties [26, 27, 54–56] or emerging new particles may be probed as well [57–59]. Furthermore, future applications in the context of multi-messenger astronomy [60–62] or nuclear safeguarding seem promising [63, 64]. Experimental upscaling in the near future will allow such investigations via precision CEνNS measurements.

Ancient Tooth Proteins Tell Tales

An analysis of six ancient Homo erectus tooth proteins from three locations in China, a Denisovan tooth protein, and some modern human and animal tooth proteins reveal some notable insights. Bernard's Blog has the story. Proteins in teeth are easier to recover than ancient DNA and can serve as a proxy for it in cases like these.

Interestingly, it implies that a tooth enamel protein found in 1% of humans in or near the former Denisovan range, and Denisovans have a tooth enamel protein also found in Chinese Homo erectus, probably due to introgression from Homo erectus to Denisovans and then from Denisovans to modern humans, in the view of the researchers.

John Hawk has a discussion of the paper, however, that interprets the data differently:

All six teeth share two derived amino acid changes, both in the sequence of an enamel matrix protein known as ameloblastin, or AMBN. One of these hasn’t before been seen in hominins: a change from alanine to glycine at position 253 of the sequence, or A253G. The other is 20 positions downstream, swapping in valine for the ancestral methionine, M273V. The DNA mutation encoding this change is shared by both the Denisova 3 and Denisova 25 genomes. The M273V amino acid change itself is in the Harbin and Penghu 1 dental proteomes—part of why they align with Denisovans. A number of genomes from modern people also share this change, possibly from Denisovan ancestors.

The hypothesis presented by Fu and coworkers is that Homo erectus was the source of M273V, and its presence in Denisovans is a result of introgression. In support of this idea of introgression, they note earlier research on the Denisova 3 and Denisova 25 genomes that suggests a contribution from a “superarchaic” source population. Many—including me—have speculated that this superarchaic ancestry came from H. erectus. Fu’s team may have just proved it.

But I don’t think these teeth are Homo erectus.

Their estimated ages, all around 400,000 years ago, are prime Denisovan time. Fu and coworkers find that all the teeth share a derived link with later Denisovan genomes. For me, the most likely hypothesis is that these teeth come from a population within the Denisovan branch of humanity.

This situation is basically the same as the Sima de los Huesos fossils. Those remains are around 430,000 years old. Those fossils look like Neanderthals in some subtle ways, but until DNA was recovered from them, many researchers considered them to be part of a different group, often called Homo heidelbergensis. DNA revised both the timeline and their identification.

It may seem heretical, but I think protein data may be about to do the same for fossils from East Asia.

Physics Quick Hits

Lots of interesting papers today. Little time to write, so only minimal commentary for now.

The reactor antineutrino anomaly still isn't real.

The Reactor Antineutrino Anomaly refers to the deficit observed between the average event rate measured in reactor antineutrino experiments with respect to the theoretical prediction. This anomaly was first identified in 2011 (2.5σ) as a consequence of the Huber-Muller reactor antineutrino flux calculation. It was thought to be resolved in 2021 as a result of new reactor antineutrino flux calculations, with a reduction to about 1σ. In this work, we examine the latest reactor antineutrino flux calculation published in 2023 by a French research group. This work represents the first summation model to include a comprehensive uncertainty budget. The result indicates a revival of the Reactor Antineutrino Anomaly at the level of 2.2σ. We also consider the usual simplest explanation of the Reactor Antineutrino Anomaly by active-sterile neutrino oscillations. We present the constraints on the oscillation parameters and we derive a tension of 3.8σ with the results of gallium source experiments (Gallium Anomaly) taking into account also the solar neutrino and KATRIN bounds, that of the combined short-baseline reactor spectral ratio measurements, and that of the Daya Bay search for a sub-eV sterile neutrino. Since the tension may be due to underestimated systematic uncertainties and the main tension is between the gallium data and the other data, we finally present the results of a global analysis with enlarged gallium uncertainties, which reduce the global tension to 1.3σ.

C. Giunti, Y.F. Li, R.P. Zhang, "Revival of the Reactor Antineutrino Anomaly" arXiv:2605.10353 (May 11, 2026).

Intriguing.

The charged-lepton Koide relation remains a striking empirical regularity in Standard-Model flavor data. We prove that for any positive mass set with Koide ratio Q0, the one-particle extension Q(m1,…,mN,x) has a unique global minimum Qmin=Q0/(1+Q0) at m∗=[(∑imi)/(∑imi‾‾‾√)]2. This exact kinematic result defines a unique extension benchmark. For the measured charged leptons it gives mℓ∗=1.25534(16)GeV and Qexp4,min=0.3999978(43); in the ideal Koide limit QKℓ=2/3, the corresponding minimum is exactly 2/5. In the effective-participant language Neff≡1/Q, the optimal one-particle extension increases Neff by one, while the equal-k multiplet extension increases it by k. The one-particle Neff profile is exactly Lorentzian in a dimensionless share-mismatch coordinate u, which we interpret kinematically rather than dynamically. Using charged-lepton pole masses with the PDG~2024 own-scale MS⎯⎯⎯⎯⎯⎯⎯⎯⎯ charm mass gives Q(e,μ,τ,c)=0.4000025(64), i.e. 11.7ppm above the measured-input benchmark and 6.2ppm above 2/5. This intentionally mixed-definition comparison is treated only as a phenomenological coincidence. To calibrate it within a stated benchmark class, we perform an exhaustive common-scale scan over non-neutrino Standard Model 2-body and 3-body seeds with one added mass. The charged-lepton-plus-charm continuation ranks 33/12,720 in the raw trial set, 24/2,640 after collapsing repeated scale realizations, and 6/756 within the fermion-only collapsed subset. We present the charm case as an empirically calibrated example of the theorem, not as a dynamical flavor model.

K. Hübner, "A minimization theorem for the Koide ratio and its Standard Model calibration" arXiv:2605.09651 (May 10, 2026).

So what?

Koide's charged-lepton relation suggests that (me‾‾‾√,mμ‾‾‾√,mτ‾‾‾√) is the natural family vector. We construct an effective compact-cycle model in which this vector is sampled from one real amplitude Z(ϕ) on an internal circle, while the masses are quadratic overlaps, ma∝|Z(2πa/3)|2. The amplitude is built from the two lowest antiperiodic modes on the circle; their symmetric square is periodic and gives the minimal three-harmonic family space e^iϕ,1,e^−iϕ. A reality condition together with the requirement that the amplitude comes from the square of one two-component spinor fixes the relative weights required by Koide's 45º geometry. The remaining orientation angle is fixed by matching one C3 family shift to transport on the full circle: integrating out the higher Fourier harmonics gives the Berry dressing that enters the determinant term and selects θℓ=−2/9. Using me and mμ as inputs, the model predicts mτ=1776.97MeV.

Kirill Shulga, "Charged-Lepton Koide Geometry from a Green-Dressed Compact Family Cycle" arXiv:2605.10245 (May 11, 2026).

Similar to another recent paper.

We show how, by exploiting the process of Coherent Elastic neutrino (v) Nucleus Scattering (CEvNS), neutrinos produced by nuclear reactor experiments appear to corroborate the evidence of the so-called X17 particle, which has been invoked to explain the ATOMKI anomaly. We base our analysis primarily on CONUS+ and Dresden-II data, which, when combined with CEvNS data from COHERENT and neutrino oscillation data from IceCube, single out a unique region of couplings to neutrinos and nuclei.

Johan Rathsman, Joakim Cederkäll, Yasar Hicyilmaz, Else Lytken, Stefano Moretti, "The X17 Existence Hinted at by Nuclear Reactor Neutrinos" arXiv:2605.10689 (May 11, 2026) (Short version of 2603.15246 using a different model for the X17).

Neutrinos do not have negative mass, so something isn't quite right in the model to estimate its masses from cosmology.

Recent baryon acoustic oscillation (BAO) distance measurements, when combined with Cosmic Microwave Background (CMB) observations in the ΛCDM framework, lead to a preference for negative neutrino masses. We investigate whether this neutrino mass anomaly can be alleviated by a class of astrophysically motivated reionization histories. Using a frequentist analysis, we find that some reionization histories can move the best-fit value of ∑mν to a positive value and bring ∑mν ≃ 0.06 eV into the 95% confidence interval. To separate the effect of the total optical depth from that of the details of the reionization history, we compare a high-τ history with a two-step tanh-like reionization history of the same τ. The resulting Δχ2(∑mν) profiles are nearly identical. This indicates that the effect is mainly driven by the total optical depth, while the details of the reionization history play only a minor role.

Yi Cheng Dai, Wei Liao, "Reionization History and Neutrino Mass" arXiv:2605.10116 (May 11, 2026).

As expected.

Modified Newtonian Dynamics (MOND) is a paradigm that can do away with dark matter at galaxy scales, but displays a residual missing mass discrepancy in galaxy clusters. Prompted by the updated JWST-based gravitational lens model of the Bullet Cluster, I confirm here that this cluster exhibits the same residual missing mass discrepancy as other clusters of similar mass in the MOND context. Moreover, this missing mass should be mostly collisionless, since it is centred on the galaxies of the Bullet Cluster.

Benoit Famaey, "On the residual missing mass of the Bullet Cluster" arXiv:2605.10022 (May 11, 2026).

Color me skeptical. It will take a closer look to poke holes in it, however. I suspect that while it may point out problems in toy-model MOND and some other models, this data could actually point the way towards a better modified gravity theory rather than towards dark matter particles which have myriad problems of their own that are ignored in this study.

Modified gravity theories such as Modified Newtonian Dynamics (MOND) and Scalar-Tensor-Vector Gravity (STVG) have been proposed as alternatives to dark matter, but decisive tests have been hindered by degeneracies between baryonic structure and gravitational laws. Here we break this degeneracy using independent, high-precision constraints: the Milky Way radial rotation curve, vertical phase-space spirals from Gaia, and a broken-exponential stellar disk. A joint reconstruction of the radial and vertical gravitational fields reveals a structural inconsistency in modified gravity -- no model can simultaneously reproduce both observations. Our results strongly disfavor MOND at >13σ and STVG at >4σ. In contrast, dark matter halo models naturally explain the observations, providing a self-consistent test of gravity on galactic scales.

Zheng-long Wang, Yue-Lin Sming Tsai, Lan Zhang, Yin Wu, Haining Li, Xiang-Xiang Xue, Hongsheng Zhao, Yi-Zhong Fan, "Milky Way Dynamics Favor Dark Matter over Modified Gravity Models" arXiv:2605.10857 (May 11, 2026).