I’ve been busy. There is a lot I’d like to say here, but I’ve been writing the actual science papers. Can’t keep up with myself, let alone everything else. I am prompted to write here now because of a small rant by Maury Goodman in the neutrino newsletter he occasionally sends out. It resonated with me.

First, some context. Neutrinos are particles of the Standard Model of particle physics. They come in three families with corresponding leptons: the electron (νe), muon (νμ), and tau (ντ) neutrinos. Neutrinos only interact through the weak nuclear force, feeling neither the strong force nor electromagnetism. This makes them “ghostly” particles. Their immunity to these forces means they have such a low cross-section for interacting with other matter that they mostly don’t. Zillions are created every second by the nuclear reactions in the sun, and the vast majority of them breeze right through the Earth as if it were no more than a pane of glass. Their existence was first inferred indirectly from the apparent failure of some nuclear decays to conserve energy – the sum of the products seemed less than that initially present because the neutrinos were running off with mass-energy without telling anyone about it by interacting with detectors of the time.

Clever people did devise ways to detect neutrinos, if only at the rate of one in a zillion. Neutrinos are the template for WIMP dark matter, which is imagined to be some particle from beyond the Standard Model that is more massive than neutrinos but similarly interact only through the weak force. That’s how laboratory experiments search for them.

While a great deal of effort has been invested in searching for WIMPs, so far the most interesting new physics is in the neutrinos themselves. They move at practically the speed of light, and for a long time it was believed that like photons, they were pure energy with zero rest mass. Indeed, I’m old enough to have been taught that neutrinos must have zero mass; it would screw everything up if they didn’t. This attitude is summed up by an anecdote about the late, great author of the Standard Model, Steven Weinberg:

A colleague at UT once asked Weinberg if there was neutrino mass in the Standard Model. He told her “not in my Standard Model.”

Steven Weinberg, as related by Maury Goodman

As I’ve related before, In 1984 I heard a talk by Hans Bethe in which he made the case for neutrino dark matter. I was flabbergasted – I had just learned neutrinos couldn’t possibly have mass! But, as he pointed out, there were a lot of them, so it wouldn’t take much – a tiny mass each, well below the experimental limits that existed at the time – and that would suffice to make all the dark matter. So, getting over the theoretical impossibility of this hypothesis, I reckoned that if it turned out that neutrinos did indeed have mass, then surely that would be the solution to the dark matter problem.

Wrong and wrong. Neutrinos do have mass, but not enough to explain the missing mass problem. At least not that of the whole universe, as the modern estimate is that they might have a mass density that is somewhat shy of that of ordinary baryons (see below). They are too lightweight to stick to individual galaxies, which they would boil right out of: even with lots of cold dark matter, there isn’t enough mass to gravitationally bind these relativistic particles. It seems unlikely, but it is at least conceivable that initially fast-moving but heavy neutrinos might by now have slowed down enough to stick to and make up part of some massive clusters of galaxies. While interesting, that is a very far cry from being the dark matter.

We know neutrinos have mass because they have been observed to transition between flavors as they traverse space. This can only happen if there are different quantum states for them to transition between. They can’t all just be the same zero-mass photon-like entity, at least two of them need to have some mass to make for split quantum levels so there is something to oscillate between.

Here’s where it gets really weird. Neutrino mass states do not correspond uniquely to neutrino flavors. We’re used to thinking of particles as having a mass: a proton weighs 0.938272 GeV; a neutron 0.939565 GeV. (The neutron being only 0.1% heavier than the proton is itself pretty weird; this comes up again later in the context of neutrinos if I remember to bring it up.) No, there are three separate mass states, each of which are fractional probabilistic combinations of the three neutrino flavors. This sounds completely insane, so let’s turn to an illustration:

Neutrino mass states, from Adrián-Martínez et al (2016). There are two possible mass hierarchies for neutrinos, the so-called “normal” (left) and “inverted” (right) hierarchies. There are three mass states – the different bars – that are cleverly named ν1, ν2, and, you guessed it, ν3. The separation between these states is measured from oscillations in solar neutrinos (sol) or atmospheric neutrinos (atm) spawned by cosmic rays. The mass states do not correspond uniquely to neutrino flavors (νe, νμ, and ντ); instead, each mass state is made up of a combination of the three flavors as illustrated by the colored portions of the bars.

So we have three flavors of neutrino, νe, νμ, and ντ, that mix and match to make up the three mass eigenstates, ν1, ν2, and ν3. We would like to know what the masses, m1, m2, and m3, of the mass eignestates are. We don’t. All that we glean from the solar and atmospheric oscillation data is that there is a transition between these states with a corresponding squared mass difference (e.g., Δm2sol = m22-m12). These are now well measured by astronomical standards, with Δm2sol = 0.000075 eV2 and Δm2atm = 0.0025 eV2 depending a little bit on which hierarchy is correct.

OK, so now we guess. If the hierarchy is normal and m1 = 0, then m2 = √Δm2sol = 0.0087 eV and m3 = √(Δm2atm+m22) = 0.0507 eV. The first eigenstate mass need not be zero, though I’ve often heard it argued that it should be that or close to it, as the “natural” scale is m ~ √Δm2. So maybe we have something like m1 = 0.01 eV and m2 = 0.013 eV in sorta the same ballpark.

Maybe, but I am underwhelmed by the naturalness of this argument. If we apply this reasoning to the proton and neutron (Ha! I remembered!), then the mass of the proton should be of order 1 MeV not 1 GeV. That’d be interesting because the proton, neutron, and electron would all have a mass within a factor of two of each other (the electron mass is 0.511 MeV). That almost sounds natural. It’d also make for some very different atomic physics, as we’d now have hydrogen atoms that are quasi-binary systems rather than a lightweight electron orbiting a heavy proton. That might make for an interesting universe, but it wouldn’t be the one we live in.

One very useful result of assuming m1 = 0 is that it provides a hard lower limit on the sum of the neutrino masses: ∑mi = m1 + m2 + m3 > 0.059 eV. Here the hierarchy matters, with the lower limit becoming about 0.1 eV in the inverted hierarchy. So we know neutrinos weigh at least that much, maybe more.

There are of course efforts to measure the neutrino mass directly. There is a giant experiment called Katrin dedicated to this. It is challenging to measure a mass this close to zero, so all we have so far are upper limits. The first measurement from Katrin placed the 90% confidence limit < 1.1 eV. That’s about a factor of 20 larger than the lower limit, so in there somewhere.

Katrin on the move.

There is a famous result in cosmology concerning the sum of neutrino masses. Particles have a relic abundance that follows from thermodynamics. The cosmic microwave background is the thermal relic of photons. So too there should be a thermal relic of cosmic neutrinos with slightly lower temperature than the photon field. One can work out the relic abundance, so if one knows their mass, then their cosmic mass density is

Ωνh2 = ∑mi/(93.5 eV)

where h is the Hubble constant in units of 100 km/s/Mpc (e.g., equation 9.31 in my edition of Peacock’s text Cosmological Physics). For the cosmologists’ favorite (but not obviously correct) h=0.67, the lower limit on the neutrino mass translates to a mass density Ων > 0.0014, rather less than the corresponding baryon density, Ωb = 0.049. The experimental upper limit from Katrin yields Ων < 0.026, still a factor of two less than the baryons but in the same ballpark. These are nowhere near the ΩCDM ~ 0.25 needed for cosmic dark matter.

Nevertheless, the neutrino mass potentially plays an important role in structure formation. Where cold dark matter (CDM) clumps easily to facilitate the formation of structure, neutrinos retard the process. They start out relativistic in the early universe, becoming non-relativistic (slow moving) at some redshift that depends on their mass. Early on, the represent a fast-moving component of gravitating mass that counteracts the slow moving CDM. The nascent clumps formed by CDM can capture baryons (this is how galaxies are thought to form), but they are not even speed bumps to the relativistic neutrinos. If the latter have too large a mass, they pull lumps apart rather then help them grow larger. The higher the neutrino mass, the more damage they do. This in turn impacts the shape of the power spectrum by imprinting a free-streaming scale.

The power spectrum is a key measurement fit by ΛCDM. Indeed, it is arguably its crowning glory. The power spectrum is well fit by ΛCDM assuming zero neutrino mass. If Ων gets too big, it becomes a serious problem.

Consequently, cosmological observations place an indirect limit on the neutrino mass. There are a number of important assumptions that go into this limit, not all of which I am inclined to grant – most especially, the existence of CDM. But that makes it an important test, as the experimentally measured neutrino mass (whenever that happens) better not exceed the cosmological limit. If it does, that falsifies the cosmic structure formation theory based on cold dark matter.

The cosmological limit on neutrino mass obtained assuming ΛCDM structure formation is persistently an order of magnitude tighter than the experimental upper limit. For example, the Dark Energy Survey obtains ∑mi < 0.13 eV at 95% confidence. This is similar to other previous results, and only a factor of two more than the lower limit from neutrino oscillations. The window of allowed space is getting rather narrow. Indeed, it is already close to ruling out the inverted hierarchy for which ∑mi > 0.1 eV – or the assumptions on which the cosmological limit is made.

This brings us finally to Dr. Goodman’s rant, which I quote directly:

In the normal (inverted) mass order, s=m1+m2+m3 > 59 (100) meV. If as DES says, s < 130 meV, degenerate solutions are impossible. But DES “…model(s) massive neutrinos as three degenerate species of equal mass.” It’s been 34 years since we suspected neutrino masses were different and 23 years since that was accepted. Why don’t cosmology “measurements” of neutrino parameters do it right?

Maury Goodman

Here, s = ∑mi and of course 1 eV = 1000 meV. Degenerate solutions are those in which m1=m2=m3. When the absolute mass scale is large – say the neutrino mass were a huge (for it) 100 eV, then the sub-eV splittings between the mass levels illustrated above would be negligible and it would be fair to treat “massive neutrinos as three degenerate species of equal mass.” This is no longer the case when the implied upper limit on the mass is small; there is a clear difference between m1 and m2 and m3.

So why don’t cosmologists do this right? Why do they persist in pretending that m1=m2=m3?

Far be it from me to cut those guys slack, but I suspect there are two answers. One, it probably doesn’t matter (much), and two, habit. By habit, I mean that the tools used to compute the power spectrum were written at a time when degenerate species of equal mass was a perfectly safe assumption. Indeed, in those days, neutrinos were thought not to matter much at all to cosmological structure formation, so their inclusion was admirably forward looking – or, I suspect, a nerdy indulgence: “neutrinos probably don’t matter but I know how to code for them so I’ll do it by making the simplifying assumption that m1=m2=m3.”

So how much does it matter? I don’t know without editing & running the code (e.g, CAMB or CMBEASY), which would be a great project for a grad student if it hasn’t already been done. Nevertheless, the difference between neutrino mass states and the degenerate assumption is presumably small for small differences in mass. To get an idea that is human-friendly, let’s think about the redshift at which neutrinos become non-relativistic. OK, maybe that doesn’t sound too friendly, but it is less likely to make your eyes cross than a discussion of power spectra Fourier transforms and free-streaming wave numbers.

Neutrinos are very lightweight, so start out as relativistic particles in the early universe (high redshift z). As the universe expands it cools, and the neutrinos slow down. At some point, they transition from behaving like a photon field to a non-relativistic gas of particles. This happens at

1+znr ≈ 1987 mν/(1 eV)

(eq. 4 of Agarwal & Feldman 2012; they also discuss the free-streaming scale and power spectra for those of you who want to get into it). For a 0.5 eV neutrino that is comfortably acceptable to the current experimental upper limit, znr = 992. This is right around recombination, and would mess everything up bigly – hence the cosmological limit being much stricter. For a degenerate neutrino of 0.13 eV, znr = 257. So one way to think about the cosmological limit is that we need to delay the impact of neutrinos on the power spectrum for at least this long in order to maintain the good fit to the data.

How late can the impact of neutrinos be delayed? For the minimum masses m1 = 0, m2 = 0.0087, m3 = 0.0507 eV, zero mass neutrinos always remain relativistic, but z2 = 16 and z3 = 100. These redshifts are readily distinguishable, so maybe Dr. Goodman has a valid point. Well, he definitely has a valid point, but these redshifts aren’t probed by the currently available data, so cosmologists probably figure it is OK to stick to degenerate neutrino masses for now.

The redshifts z2 = 16 and z3 = 100 are coincident with other important events in cosmic history, cosmic dawn and the dark ages, so it is worth considering the potential impact of neutrinos on the power spectra predicted for 21 cm absorption at those redshifts. There are experiments working to detect this, but measurement of the power spectrum is still a ways off. I am not aware of any theoretical consideration of this topic, so let’s consult an expert. Thanks to Avi Loeb for pointing out these (and a lot more!) references on short notice: Pritchard & Pierpaoli (2008), Villaescusa-Navarro et al. (2015), Obuljen et al. (2018). That’s a lot to process, and more than I’m willing to digest on the fly. But it looks like at least some cosmologists are grappling with the issue Dr. Goodman raises.

Any way we slice it, it looks like there are things still to learn. The direct laboratory measurement of the neutrino mass is not guaranteed to be less than the upper limit from cosmology. It would be surprising, but that would make matters a lot more interesting.

31 thoughts on “The neutrino mass hierarchy and cosmological limits on their mass

    1. Not according to some, which is why I included Weinberg’s quote. But I’ve heard it said that this was something he simply left out, and having neutrino mass is perfectly compatible with the Standard Model. I’m don’t know, but I wouldn’t think there would be so much discussion of the Majorana see-saw mechanism if it were that simple.

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    1. What Indranil says is all true, though I can imagine other scenarios. Indeed, a neutrino of modest mass (say, 0.5 eV) would help explain the CMB, the residual mass discrepancy in clusters, and help tame structure formation in MOND, which if anything is a tad too fast: http://astroweb.case.edu/ssm/mond/LSSinMOND.html
      More generally though, I suspect these are indications of some deeper but as-yet unrecognized entity that we can approximate using a massive neutrino (or a dark matter particle, or dark energy, etc., each in its place).

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  1. Neutrinos could be crucial to reconciling MOND with cosmology, as described in this previous post on TritonStation:

    tritonstation.com/2020/10/23/big-trouble-in-a-deep-void/amp/

    The title links to the paper MNRAS, 499, 2845, which is open access. The nuHDM cosmological model advocated there also explains El Gordo, which is incompatible with the standard cosmological model (LCDM) at over 5 sigma significance:

    https://darkmattercrisis.wordpress.com/2021/01/16/54-the-interacting-galaxy-cluster-el-gordo-a-massive-blow-to-%ce%bbcdm-cosmology/

    The relevant paper is MNRAS, 500, 5249.

    Including an extra dark matter component in MOND may seem a bit contrived, but we recently did a fairly sensitive test of this idea with regards to a problem it was not designed to solve, namely the tidal stability of dwarf galaxies in a galaxy cluster, in this case Fornax:

    https://darkmattercrisis.wordpress.com/

    So neutrinos could play quite an important role, and I think could be the dark matter needed to explain cosmological, but not galactic, observations. At least with the exception of galaxies at the centre of a cluster, or like in the example above where the cluster gravity is important to the problem in addition to the galaxy’s own self-gravity.

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  2. There are several papers that consider the full range of allowed neutrino masses for the normal or inverted hierarchy; two of the most recent are:
    https://arxiv.org/abs/2003.03354
    https://arxiv.org/abs/2006.09395

    A slightly older review that gives an overview of some of the physics is:
    https://ui.adsabs.harvard.edu/abs/2014NJPh…16f5002L/abstract

    Boltzmann codes are capable of handling all of this, although one has to be a bit careful about the assumptions made there (which are generally slightly different for the different codes). See, e.g., section 2 here:

    Click to access CAMB.pdf


    versus lines 107-114 here:
    https://github.com/brinckmann/montepython_public/blob/3.4/input/base2018TTTEEE.param

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  3. “We know that neutrinos have mass because they have been observed to transition between flavours as they traverse space.”

    I find this a difficult statement to understand because it sounds too much like “We know that dark matter exists because…” I think it would be more accurate to say that we infer that neutrinos have mass, based on our current theoretical understanding. It is not accurate to say that they have been observed to transition between states: what has been observed is that they exist in one state when they are produced, and they exist in another state when they are annihilated. Nobody has any idea what happens in between, and it is in principle unknowable.

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    1. If you want to call it an inference, sure. But I do not consider it to be in the same category of inferring dark matter, for which we assume that the equations that lead to its inference are immutable. For neutrinos, we understand how they’re produced and how many should arrive from a given source, and that number differs but other flavors arrive. So something transitioned. Calling it that presupposes little more than a quantum framework… that they do this violated then-existing theory.

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      1. I think Robert’s point is worth pondering. Using the standard model of particle physics to support your argument is a bit dodgy when neutrinos are not really supposed to have mass in this framework. Neutrinos could be shown to have mass using an experiment like KATRIN, but this has not yet happened. I certainly can’t imagine how they can all be massless, and would definitely assign an extremely low likelihood to this, much less than say the Universe having no dominant dark matter component. But it is important to be open minded about these things. Certainly if neutrinos turned out to all be massless, it would ruin much more of physics than if it turns out the matter in the Universe is mostly baryons.

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        1. Yes, it would be even more interesting if somehow neutrinos transitioned between flavors without having mass. Even having mass is already a shock and surprise, so of course we should be open to that possibility… IF it is a possibility.

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          1. Yes, I think that really is the point. IS it possible that such a transition could happen without mass? I believe it is, and I have put forward some suggestions along these lines in some of my arXiv papers. They have not been refereed, so it is probably premature to discuss this here. All I would say at this point is that in order to discuss neutrino mass, one really needs to have a theory for how the neutrino interacts with a (presumably quantised) gravitational field. Without such a theory, anyone’s guess is as good as anyone else’s, I suggest.

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            1. In the hopes of not giving an obvious answer: Yes, it is possible in the presence of matter where the neutrinos can interact with their corresponding charged lepton. This is commonly called MSW effect. Indeed, this has been used to infer that m_2>m_1 (this conclusion is not possible from vacuum oscillations alone since only the squared mass differences appear in the transition probability formulae). In the sun the electron density is sufficiently high to significantly influence the transition rates of the highly energetic neutrinos (e.g. B8 or Be7 neutrinos) which is then noticeable as an even higher rate of muon neutrinos in our detectors (or rather a greater lack of electron neutrinos) than explained by pure vacuum oscillations. In contrast the neutrinos with lower kinetic energies (e.g. pp neutrinos) don’t interact as much with the matter (read electrons) in the sun and consequently have a much higher survival probability when they arrive on Earth. See for example https://arxiv.org/abs/0808.2868v1
              In principle it is possible that all neutrino oscillations are just a result of interactions with matter while the neutrinos themselves are massless. However, in my understanding the various energy and oscillation length ranges over which neutrino oscillations have been measured, rule this possibility out as the necessary parameters wouldn’t match.

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        2. I don’t want to belabour the point, but there is also a problem in even defining the term “mass” – as the whole MOND debate illustrates quite clearly. So I am quite willing to admit that there may be very reasonable definitions of mass in which neutrinos do indeed have masses in the .1eV range, as measured on Earth. But I do find it hard to accept that such a definition of mass can be universal.

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      1. If that comment is addressed to me, I can assure you that I have read a great deal about the solar neutrino problem, and thought hard about it for ten years. My comments are not based on ignorance, but on a thorough analysis of the experimental and theoretical evidence.

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        1. Hi Robert, Yes I was replying to you. Sorry for my misunderstanding. I guess I picture neutrino oscillations as similar to neutral K- mesons. (Kinda a standard grad level physics problem. See G. Baym “Lectures on Quantum Mechanics” for instance.) When you think about it deeply QM is just weird. I mean forget neutrino’s, and take correlated and/or entangled photons. All we know is what happens at the source and then the detector. What’s going on in between is (almost by definition) unknowable. In some ways I find entangled photons stranger than neutrino mass, but that is probably just due to my ignorance about neutrino physics.

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          1. Yes, I also picture neutrino oscillations as similar to neutral K-mesons. That is another set of experiments that I have thought long and hard about. Again, I find the accepted explanation hard to swallow, and have been searching for a better one. I don’t want to discuss my personal theories here, but I have put forward some suggestions in one or two papers on the arXiv this year.

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            1. No worries, QM is weird, no doubt.
              You can do photon entanglement on a table top…
              blows my mind.

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  4. Off topic, I note that you are cited in Deur’s most recent paper. Alexandre Deur, “Effect of gravitational field self-interaction on large structure formation” arxiv.org: 2108.04649 (July 9, 2021) (Accepted for publication in Phys. Lett. B) DOI: 10.1016/j.physletb.2021.136510 His work is summarized at http://dispatchesfromturtleisland.blogspot.com/p/deurs-work-on-gravity-and-related.html provides a very promising way to secure MOND phenomenology from a different way of operationalizing classical general relativity in weak fields in large, complex system like galaxies, than is conventionally applied, while curing some of its struggles at the galactic cluster and cosmology scale and providing an explanation for dark energy phenomena without a cosmological constant.

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  5. Lab resolution of the normal v. inverted hierarchy was predicted in a preprint today to be resolvable experimentally at five sigma with combined results by 2025 (2 to get all of the experiments collecting data and 2 to collect it) if the answer is a normal hierarchy and second quadrant theta23 with values close to those already measured, and by 2029 in a worst case scenario, with an inverted hierarchy and significant flaws in current PMNS parameter measurements. If that doesn’t work, single detector resolution by 2031. https://arxiv.org/abs/2108.06293

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  6. The latest cosmology bound claimed is 87 meV, which would rule out an inverted hierarchy at 95% C.I. https://arxiv.org/abs/2106.15267 (the abstract rounds that to 0.09 eV).

    The Katrin bound is somewhat worse than you suggest relative to the cosmology bound and minimum neutrino mass sum. The tightest directly measured bound on the mass of the lightest neutrino mass eigenstate is that it is less than 800 meV/c^2 with 90% confidence (this limit from the KATRIN experiment is expected to ultimately be reduced to 200 meV/c^2 with 90% confidence when the experiment has run its course and collected all of the data it plans to collect) which is 2459 meV for the sum of the three masses that cosmology is looking at. https://arxiv.org/abs/2105.08533

    I think that one of the reasons to use the sum of three masses in cosmology is to reflect the averaged results of neutrino oscillation, although I say that with less than absolute certainty.

    I think there is good reason to place a non-zero floor on the lightest neutrino mass because zero mass particles are qualitatively different from massive ones, even if the mass is negligible.

    I also recommend Gustavo F. S. Alves, Enrico Bertuzzo, Gabriel M. Salla, “An on-shell perspective on neutrino oscillations and non-standard interactions” arXiv (March 30, 2021) which explains in the body text that “we have obtained the PMNS matrix without having to ever talk about mass diagonalization and mismatches between flavor and mass basis.” It is somewhat comforting to recognize that there is a mathematically equivalent description of neutrino oscillation and mass that doesn’t require the blend of mass eigenstate and weak eigenstate description.

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    1. Very interesting. Di Valentino et al appear to phrase everything in terms of the sum of neutrino masses without addressing Dr. Goodman’s question, but it is a very impressive combination of data that on its face appears to rule out the inverted hierarchy – provided we believe everything else, of course. The window is so ridiculously tight that we might as well set the sum to 0.074 +/- 0.015 eV and derive the individual masses from the normal hierarchy.

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      1. Been there, done that: mv3 = 64.7 ± 7.7 meV; mv2 = 15.2 ± 6.7 meV; mv1= 6 ± 6 meV (with mv1 neutrino mass uncertainty correlated 100% with most of the uncertainty in the other values).

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  7. If you want to get just a bit more speculative on what is “natural”, a century old notion about the rest mass of the electron is very close to the mass associated with the electromagnetic potential energy field that an electron generates if one assumes that rather than being point-like it has a radius which dimensional analysis naturally suggests, and make some clever choices about how to do the math. “The empirical electron mass fixes an electron radius (as expressed already more than 100 years ago by Lorentz (and Poincare)) by the formula e^2/r ≈ me*c^2: for r ≈ nuclear radius (= 2.8 · 10−15 m), the mass comes out to be ≈ 1/2 MeV.” http://www1.jinr.ru/Pepan/2011_v42/v-42-5/04_boya.pdf See also Georgi H., Glashow S. L. “Attempts to Calculate the Electron Mass” 7 Phys. Rev. D. 2467 (1973). The ratio of the electron to lightest neutrino mass scale is on the same order as what might be expected by the ratio of the electromagnetic force to the weak force.

    Also, it is worth observing that all Standard Model fundamental particles with known mass (except the lightest neutrino whose mass or lack thereof is unknown) interact with the W boson via the weak force, and that all Standard Model fundamental particles that have zero rest mass do not interact with the W boson. It would also be particularly odd for a neutrino whose oscillations experience CP violation to be massless since massless particles don’t experience time in their own reference frame.

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    1. You point out, correctly, that one cannot consider neutrino oscillations, and neutrino masses or lack of thereof, independently of CP-violation. They come as a package, and if we are to consider the possibility of zero neutrino mass, we must also consider the possibility that CP-violation is not quite what it seems. This may well be dismantling too much of the standard model for most people’s comfort, but it is worth remembering that CP-violation came as a nasty shock when first demonstrated experimentally in 1964. Coming on top of P-violation in 1957, it profoundly changed our perceptions of the quantum world. What I find extremely strange, however, is that these successive discoveries of strange anomalies never once seem to have suggested the possibility that the experiments had run up against the boundary between the standard model and quantum gravity. Instead, the model has been successively altered in ad hoc ways, on the (obviously false) assumption that gravity does not exist.

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  8. A very interesting topic, Professor McGaugh, is always a pleasure to read you.

    If I may go off-topic, new papers have come out about the galaxies without DM NGC 1052 DF2 and DF4:

    https://arxiv.org/abs/2104.03319 A Tip of the Red Giant Branch Distance of 22.1±1.2 Mpc to the Dark Matter Deficient Galaxy NGC1052-DF2 from 40 Orbits of Hubble Space Telescope Imaging

    https://iopscience.iop.org/article/10.3847/1538-4357/abc340/meta
    The Galaxy “Missing Dark Matter” NGC 1052-DF4 is Undergoing Tidal Disruption

    Do these new results change something about the points you made in previous posts about this topic?

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  9. Hi Stacy, very instructive addition to your blog as usual.
    Neutrino physics is obviously very important to the understanding of the universe, but the thing that strikes me is that all those mass estimates (through oscillation theory or cosmological constraints) are just way too model dependent to my taste: I won’t personally put too much weight on the conclusions they may or may not reach. Nothing can beat direct measurements IMHO and currently they do allow neutrino mass sum to be in the MeV range (the tau neutrino upper experimental mass limit lies around 20MeV…). An MeV range fundamental and neutral particle that could be of the neutrino family has the potential to explain MOND phenomenology if such heavy neutrino virtual pairs are gravitational dipoles (within the paradigm of antigravity for anti-matter) and dominate the Gravitational Quantum Vacuum for Gravitational polarization effects. Anyway, let’s keep an open mind on the very sensitive neutrino mass topic, as I suspect that there is much more going for these lil’ evasive dudes than meets the eyes (or physicist imagination in this case!). I’ll enclose a link on another interesting paper that discusses the possibility for so-called ‘heavy’ neutrinos: https://warwick.ac.uk/fac/sci/physics/staff/academic/boyd/warwick_week/neutrino_physics/lec_neutrinomass_writeup.pdf

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    1. While the cosmology bound on the sum of the neutrino masses is subject to all sorts of theoretical limitations, the bounds on the differences between the neutrino masses due to neutrino oscillation measurements are not.

      The Particle Data Group lists upper bounds on the electron neutrino mass as < 1.1 eV at 90% confidence, muon neutrino mass as less than 0.19 MeV at 90% confidence, and tau neutrino < 18.2 MeV at 95% confidence. But, even in an inverted hierarchy case, that maximum spread of 0.1 eV between the three masses. So, this places a combined limit of about 1.3 eV at 90% confidence on muon and tau neutrino masses.

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