MeVs and GeVs II

I was thinking again, about whether there is some way to explain the Higgs mass as a multiple of the nucleon mass. Not only is the fact that 5³=125 surely a red herring, 125 or 126 wouldn't even be the dimensionless quantity of interest. The Higgs is about 126 GeV, but the nucleon mass is somewhat less than 1 GeV...

And then I remembered that other number beloved of physics numerologists, 137. Specifically, I vaguely recalled that there is some instability for a nucleus with atomic number of 137, precisely because the fine structure constant is about 1/137. I was reminded of the recently discovered fact that the value of the Higgs mass (along with the specific value of the top quark mass, and a few other parameters) places the standard model vacuum on the brink of instability.

It is tempting to suppose that some unknown physics has forced the Higgs to a critical value. In the previous post I speculated that "the Higgs field could be a QCD meson condensate weighed down by virtual nucleons". Could it be that the density of these virtual nucleons is bounded by an analogue of this 137-instability?! A crackpot idea, yes; but the first thing is to check what the actual ratio of the masses is.

Let us say that the Higgs mass is 125-126 GeV. The nucleon mass is about 939 MeV. This gives a ratio between 133 and 134. To my mind, this is close enough to 137 that one should persist a while longer with the idea. So what is the mechanism that destabilizes element 137 - which is jocularly known as "feynmanium", because Feynman was the one who noted the instability?

It turns out that the problem emerges first in the Bohr model - the innermost electron would orbit the nucleus faster than light ... and then in a more sophisticated version when using the Dirac equation - a ground-state instability ... and that even more sophisticated analyses push the problem out to atomic number 173, or entirely abolish it. The fact that the "137 instability" appears in different formalisms is mildly encouraging, since it suggests a phenomenon at work that might still exist, even in a wildly different theoretical context.

The next step was to see whether anyone else has had thoughts along these lines. The numerology mentioned by "fzero" in the previous post is getting there, but it's a little back-to-front: it uses the running of the fine-structure constant, to reach a scale where it is approximately 1/125. But as I have already mentioned, 1 GeV is only a ballpark number; 939 MeV is the objectively interesting quantity, and that suggests that we should go with the low-energy value, ~ 1/137.

A search for "feynmanium higgs" turned up a blog comment by "Juan F." which is halfway there. Feynmanium is mentioned, but Juan F. is still using something more like fzero's relation, with the number 126 mooted as significant because it is a "magic number" in nuclear physics.

String theory and Mulla Sadra

Mulla Sadra is a notable Iranian philosopher, among whose concepts is one that is translated into English as "substantial motion". I do not have a scholar's precise understanding of the notion, but my rough understanding is that the previous notion of change in an entity was that its "accidents" might change but its essence would remain the same, whereas Sadra proposed that it could change in its being. Thus, rather than change just being motion through space, it could also take the form of change "within" a substance.

This reminds me of an elementary duality which appears early in the development of string theory. The string is a one-dimensional entity moving through an n-dimensional space. But it may also be conceived as a one-dimensional space in itself, with a number of fields existing within it. The values of these fields within the string, correspond to the position vectors of the points on the string, in the first picture. Thus this duality exchanges motion of the string through space, for substantial motion within the string. The second perspective is arguably the basis of perturbative string theory, which employs conformal field theory to describe the dynamics of fields within the string (worldsheet fields) corresponding to motion within a particular geometric background.

Proton mass and top quark mass

"In this picture, we discover that M_p ∝ m_t^2/27: despite the negligible population of virtual top-antitop pairs in the nucleon, the top-quark mass influences the proton mass."

This is not some wild-eyed vixra prophet of numerology speaking, but Chris Quigg, who has had a long and very respectable career, working especially on hadrons.

Koide-ologists ought to take a keen interest in this claim, for several reasons.

First, the "Brannen phases" for the triples u-c-t, d-s-b, e-mu-tau, are all multiples of 2/27.

Second, the centrality of the top is consistent with Rivero-ist ideas such as the waterfall (in which the quark masses can be understood as arising from a chain of Koide triplets starting with t-b-c) and the sbootstrap (in which all the SM fermions are superpartners of quark-(anti)quark composites of the 5 "light" quarks, i.e. everything but the top) and even "84"-ism.

Third, it legitimizes, in a new way, the appearance of QCD scales as "Brannen mass scales" for e-mu-tau and s-c-b.

A dark identity

Back in March 2013, the Planck collaboration announced a new estimate for the fraction of today's universe that consists of dark energy; about 68%. As chronicled here, someone soon noticed that this number is approximately 1-1/π. A few months later, it was noticed that this number is also approximately ln(2). If we combine these two observations, we discover what I shall call the "dark identity"

ln(2) + 1/π ≈ 1

The terms on the left-hand side correspond to "dark energy" and "everything else", respectively.

This little gem takes crackpot physics numerology to a new level, because even just as mathematics, it's anomalous; but it also encodes the universe.

One mission of this blog is to look at such relationships and to not immediately dismiss them. But so far, it has been a matter of asking whether particular numerical relationships between quantities from physics could be something other than coincidence.

Here, we first need to ask this of the strictly numerical aspect of the anomaly. Could this near-relationship between π, the number 1, and the natural logarithm of 2, conceivably be something other than a coincidence?

For this to be so, I think it would have to derive from some simple and significant mathematical object. It needs to be simple because numerically, the coincidence is not supernaturally close.

Suppose we learned, for example, that this formula approximates an exact identity concerning the discriminants of an obscure type of number field. This might be unimpressive if there are thousands of equally obscure number fields. But it would be impressive if it was a very special number field, which e.g. played a role in some important construction.

Obviously one should look for mathematical topics where both π and the natural logarithm of 2 play a role.

So much for pure math. What about the physics? Right away a problem presents itself: 68% is the density fraction of dark energy now; but this number changes as the universe expands. It is not impossible that this number means something fundamental; but it will require some unusual assumptions about physics for this to be so, and one must take care, in constructing such unusual models, that one does not violate the more conventional assumptions that the Planck team used to derive the number in the first place.

Similarly, one has to suppose that the hypothesized number field (or whatever) which is mathematically responsible for the dark identity, also plays a role in the mathematics of the physics of the dark sector. This is another constraint on model building; potentially a very stringent constraint, if no mathematical explanation of the identity is forthcoming.

Having spelled out some of the reasons why this may be a barren line of inquiry, now let me say why it's still interesting to think about. Mostly it's interesting because it offers an unexpected perspective on the coincidence problem of dark energy - why the amount of dark energy is about the same as the amount of everything else, rather than being different by orders of magnitude. If you're completely in the dark about how to solve a problem, then even a crazy idea with delirious logic can be a good starting point; in trying to make it work, you may discover some other idea that is similar, but better.

Also, this is a reminder of how one wants physics to work - a harmony in which deep mathematical facts are responsible for deep physical facts.

Finally, let me note that I may not have ever noticed this "dark identity", were it not for L. Edgar Otto's musings on approximations of π, which are apparently part of a larger poetic metaphysics which also includes "dark fluid dynamics".

Cosmon field in AdS space?

In the previous post, I mentioned Marni Sheppeard's theory of mass, as arising from an interaction of ordinary and mirror matter. She also recently posted a sketch of her cosmological ideas. They are sort of mysterious, but they may make a little more sense if you've seen Christof Wetterich's recent work on "A universe without expansion", which almost seems to be saying that whether the universe is expanding is a point of view - you can change coordinate systems, and also rescale certain fields, and the universe might be static, or even shrinking, depending on what picture you have adopted. Sheppeard's universe seems to be static, but baryonic mass and mirror mass change over time, and this gives the appearance of expansion. Or something like that; this post isn't about her cosmology, but I mention it because it's in the same territory as what follows. (An Iranian review of nonstandard cosmologies may also be of interest.)

More recently, I was reminded of a problem I have with Wetterich's concept - doesn't it require baryonic mass to vary with the VEV of his "cosmon field", in a way which is not consonant with the complexity of how mass is actually produced in QCD? As of this date, the comments beneath my question say that Wetterich's idea is just about trivial rescaling of the metric, but I don't think that's correct, I think its viability depends on there being a certain mechanism of mass generation as well. I note that Wetterich has tried to produce a Higgs-like effective theory of QCD, and perhaps he was motivated by this very problem.

But here is today's idea. Let's suppose that Wetterich's scenario works after all. Could it even be applied within anti-de-Sitter space, so that the cosmic geometry is really AdS, but the evolution of the mass-generating cosmon field produces the appearance of universal expansion? (I owe this concept to a conversation with A. Hattawi about Wetterich's ideas, and a conversation with T. Seletskaia about applications of AdS to cosmology.)

I could also go further and mention A. Rivero's latest speculation, about dark energy as the slight positive excess remaining from a near-cancellation of AdS negative curvature, and the vacuum energy arising from a Higgs VEV. (Also see posts here, 1 2.) And if you want the perspective of people who know what they're talking about, see e.g. Polchinski and Silverstein, page 5, for the three different contributions to the vacuum energy of a typical Freund-Rubin compactification (AdS curvature, curvature of KK-like compact dimensions, energy densities of stringy fluxes).

MeVs and GeVs

It began with a discussion of Koide relations. "fzero" said he thought it was all numerology, just like the fact that 1/alpha, at the Higgs mass scale, is approximately equal to the Higgs mass in units of GeV.

Initially I concurred that the latter fact, at least, must be a coincidence. But then I noticed that 1 GeV is rather close to the nucleon mass (939 MeV). So I decided to think the impossible for a while. Could it actually mean something, that the Higgs boson weighs about the same as 125 protons? In fact it is somewhat more than that, but 125 would serve as a placeholder in my deliberations.

Later I recalled that the VEV of the Higgs field is about twice the Higgs boson mass. This was more promising. One of the mysteries of Koide-ology is the appearance of quantities from QCD, as the mass scales of the e-mu-tau and b-c-s triples. In the standard model, the masses of those particles are obtained as (yukawa coupling) x (Higgs VEV).

The standard model, even without a Higgs field, still has a Higgs mechanism, thanks to the quark-antiquark condensate. But the VEV is measured in MeVs rather than GeVs. I began to develop the notion of such a condensate, being somehow weighed down with proton-antiproton pairs - 125 of them...

I came up with a silly visualization based on the idea of a pentagon "cubed". A pentagon is made of five line segments, a square is a line segment times a line segment, a cube is a line segment cubed. It should be possible to multiply three pentagons in a certain sense, to produce a six-dimensional object made up of 125 cubes, the cubes consisting of every possible product of an edge from each pentagon.

In the 1990s, Witten discovered a notion of baryons as branes with strings hanging off them (attached by just one end to the brane); the strings are the quarks. So here you should imagine a torus in each cube of the pentagon-cubed - the torus represents a virtual proton-antiproton loop - and then up- and down-flavored quark-strings hanging off the tori.

Finally you should suppose that this construct exists at every point in space - perhaps in extra dimensions surrounding our brane-world - and that the virtual up and down quarks form the meson condensate of the Higgsless standard model.

And that was as far as I got. So imagine my surprise when the next day, I saw Marni Sheppeard blogging about how to get Koide triple mass scales from "three pentagons". The coincidence was not only uncanny, but also somewhat unwelcome, since that part of her theory looks messy and complicated to me.

About a week after that, I was reading the latest version (number 5) of her opus on scattering. Sheppeard's ideas defy summary, but let's say that in her theory, standard model fermions are braids (that are actually morphisms in a category), the dark sector is made of mirror braids, and rest mass comes from a cohomological composition of braids and mirror braids. ("Cohomology" is little more than a word to me, but I believe that taking the product of a vector and a 1-form is the algebraic prototype here.)

In various places, she remarks that maybe the mirror partners of SM fermions are dark baryons. That sounds crazy, I thought... then I realized it is not so far removed from the notions that I was just describing. There is even such a thing as baryonic cohomology.

So where do things stand?

I find it very hard to believe that the number 125 has any deep meaning here. Common sense says it just served to inspire a visual picture, which in turn only matters as a gateway to a more abstract idea, that the Higgs field could be a QCD meson condensate weighed down by virtual nucleons. That, I believe, has the potential to explain the coexistence of Koide numerology and the SM Higgs mechanism.

But it's interesting to note the multiple points of contact with Sheppeard's work. They represent one of the more exotic directions one could take the idea, alongside a more conservative field-theoretic approach.

Weak-interaction bootstrap

In two research notes from the mid-2000s, Alejandro Rivero reported that the Z boson decay width lies on the same curve (proportional to mass cubed) as the pion and several other mesons, and that the width is minimized for a value of the Weinberg angle which is realized at the GUT scale in grand unified theories.

It is unclear to me whether this is unusual. The width is not a fundamental property, and it could be that these observations can be completely explained in terms of the SM, or a GUT, respectively; I would have to check.

But if it is unusual... and if it is not to be dismissed as a coincidence... then it seems it might need a "bootstrap" explanation. The bootstrap philosophy, also known as S-matrix theory and as nuclear democracy, was an idea of the 1960s which sought to explain the behavior of hadrons, not through reductionism, but through an algebraic holism of "Regge trajectories" and scattering dualities. (In the mainstream histories, the bootstrap is regarded as having been superseded by QCD and the standard model, but as Ron Maimon has explained in recent years, the bootstrap also gave rise to string theory.)

This holistic approach, though hard to penetrate, seems appropriate for explaining relations between quantities that aren't fundamental. But the main problem is that the bootstrap was just about the strong force; there seems no opportunity for weak and electromagnetic forces to enter the synthesis.

So here I would like to unearth an unpublished preprint from 1975, “Instability of Collective Strong-Interaction Phenomena in Hadron Production as a Possible Origin of the Weak and Electromagnetic Interactions” by Richard C. Arnold, simply to quote its opening remarks:

Recent attempts, in the context of local field theories, to unify all interactions (strong, electromagnetic, and weak) have led to serious consideration of the possibility that all these interactions become indistinguishable at sufficiently small distances, or large momenta. If this were true, then methods applicable to strong interactions such as the self-consistent S-matrix approach ("bootstrap") should be equally well relevant for the other interactions, leading to the expectation that symmetries combining all interactions would be found, as in the strong-interaction dynamics alone. This phenomenon cannot be seen in a low energy ("old") bootstrap theory, since the weak and electromagnetic forces are negligible compared to the strong at low energies. However, a self-consistent S-matrix theory which relies on high-energy, high-multiplicity intermediate states should make manifest such an interplay between classes of interactions. 

I cannot judge the merits of Arnold's particular idea, of producing leptons and the electroweak interactions from "t-channel Regge poles". But if the peculiar properties of Z boson decay mentioned above are real and not a coincidence and not already explained by standard theory, here is a place to start looking...

t, H, W, Z: Weinberg, Veltman

1) I have so far failed to note the completely orthodox relationship between m_W, m_Z, and the Higgs VEV v, that exists in the standard model, as set out e.g. in equations 1.4 and 1.14 in this thesis. It is:

m_W = 1/2 g v

m_Z = 1/2 sqrt(g'²+g²) v

where g is the SU(2)_L coupling, and g' is the U(1)_Y coupling.

2) A paper today informs us of a more arcane t,H,W,Z sum rule, due to Veltman and motivated by naturalness:

m_H² + 2m_W² + m_Z² - 4m_t² = 0

"which is satisfied for a value of the Higgs mass m_H ~ 314 GeV in flagrant conflict with experimental data."

The authors go on to speculate that perhaps Veltman's condition might be satisfied at high scales instead. I will just note two things: one has to wonder whether some of the t,H,W,Z formulae I have chronicled here - especially those which actually work but are otherwise mysterious - might be produced by a Veltman-like argument; and Veltman's wrong prediction is rather close to the real Higgs boson mass, times 2.5.

t, H, W, Z: noncommutative edition

In May, I blogged about a Higgs-VEV-squared sum rule due to Lopez Castro and Pestieau (LC & P) - and which, I would like to repeat, was closely anticipated by A. Garces Doz, in a pseudonymous comment at Lubos Motl's blog, at a time when the Higgs mass was not yet known. (The comment seems to have been lost from Lubos's blog, perhaps when the old "JS-Kit" comment system was shut down, but fortunately I made a copy here.)

In their paper (second link above, page 2), LC&P express their equation in terms of standard model couplings. Rewritten slightly, it is:

2 λ + 1/4 g² + 1/4 (g²+g'²) + 1/2 Σ y_f² = 1

This may seem less enlightening than the original.

But consider this discussion between Urs Schreiber and Jacques Distler, dating from 2006, regarding the mysteries of the Chamseddine-Connes-Marcolli noncommutative standard model. From his personal notes, Urs reproduces the equation

g₃² = g₂² = 3 λ = 1/4 Σ g_Y²

I do not actually see how to obtain this from the CCM paper. Perhaps it's implied by something in section 5.4. It can at least be verified that there are odd formulae in which squares of yukawas appear, e.g. equation 5.25 - and that may be "enough". As I have previously noted, the LC&P sum rule is still true if you only use the top yukawa, i.e. if you replace the fourth term on the left-hand side with just "y_t²".

The important observation here is that these noncommutative models have a tendency to produce, in Jacques Distler's words, "relations among the coupling constants over and above those guaranteed by gauge invariance and renormalizability"; and that these can include squares of couplings, such as appear in the rewritten version of the LC&P sum rule with which I began this post. So perhaps these "noncommutative" or "spectral" models have at least a fighting chance of explaining it.

P.S.: While I'm here, I'll also observe that it would be interesting to see whether the derivation of the Higgs mass via asymptotic safety, can be extended or modified to also produce a Higgs VEV that is roughly twice the mass. Out of all the t, H, W, Z numerology that I've collected so far, that ought to be the simplest relation to add to the A.S. scenario.

t, H, W, Z, part 3

Malcolm Mac Gregor proposes that

(1) ... m_t = m_W + m_Z

Emilio Torrente-Lujan suggests (page 5) that custodial symmetry could produce the relation

(2) ... m_H ~ (m_W + m_t)/2

Together these would imply the Dharwadker-Khachatryan sum rule (page 56)

(3) ... m_H = m_W + m_Z/2

Higgsed by the vacuum, II

Continuing the previous post... What Alejandro Rivero actually said to me was,

"Yesterday I went to a popularisation talk of De Rujula, and I was thinking on the mismatch between vacuum energy due to higgs and vacuum energy seen in astrophysics. It sounds as the problem with the string energy scale, QCD string vs "gravity" string."

This remark helped to suggest a crackpot interpretation of the preceding Higgs-VEV numerology, as a sign that the higgsing in the Standard Model is done by the total vacuum energy, which mysteriously conforms to the new ansatz that "the zero-point energy of a quantum field is equal to the mass of one quantum of the field".

Now I want to take things a little further... One of the "movements" spun off from string theory is that of "large extra dimensions" (also sometimes called "universal extra dimensions"). In such models, the Planck scale can come all the way down to the TeV scale. Under such circumstances, the QCD string could well become a fundamental string. I don't remember the LED model-builders ever talking about this possibility, but it certainly dovetails with Alejandro's other ideas about the "sBootstrap".

So here is the new twist. As he observes, the astrophysical dark energy, which is often interpreted as vacuum energy, is orders of magnitude away from the Higgs field vacuum energy. But what if... we assume the new crackpot ansatz, that "higgsing is done by the total vacuum energy", and that the cosmologically relevant vacuum energy has been diluted by a large-extra-dimensions mechanism, perhaps similar to Randall-Sundrum? Or even by a non-geometric, noncommutative analogue of LED...

Higgsed by the vacuum?

The Higgs mechanism for generating mass can be realized in many different ways, not just the one employed in the standard model. Theories can have multiple Higgs fields, in string theory a wide variety of scalar quantities can serve as the Higgs, and so on.

Meanwhile, what to do with the vacuum energy or zero-point energy of quantum field theory is one of the subject's outstanding problems. The leading idea still seems to be that there are numerous positive and negative contributions to the cosmological constant, and they have to cancel for anthropic reasons, but otherwise there's no particular pattern to it. But in the literature you can find numerous unlikely ideas for why the vacuum energy is cancelled or cut off or doesn't gravitate.

I mention all this as a prelude to a discussion of the curious relation mentioned in the previous post. I should mention two further things: in the standard model, it's the Higgs VEV, not the Higgs boson mass, which determines the mass scale of the other particles; and, in the standard model, the Higgs VEV and the Higgs boson mass are independent quantities, separately determined by different parameters in the Higgs potential.

Conceivably, a specific model of the Higgs field could produce a relation between Higgs VEV and Higgs boson mass. But what are we to make of a formula which relates the magnitude of the Higgs VEV to every particle with mass? And furthermore, each such particle appears exactly once, and in the same way, in the formula.

So here is today's wacky concept: (1) The zero-point energy of a quantum field is equal to the mass of one quantum of the field; (2) The sum of the squares of these energies provides the v^2 term in the Higgs potential of the standard model. Thus, it is the vacuum energy (as described by this mysterious new ansatz) which does the Higgsing.

There is a circular or bootstrap aspect to this idea, since the masses appearing in (1) are themselves supposed to be generated by yukawa couplings to this "vacuum energy" whose magnitude is a function of those masses.

I am quite aware that, even given my initial remarks about the variety of realizations of the Higgs mechanism and the various desperate ideas meant to deal with the vacuum energy problem, this is a proposal that doesn't make much sense, according to the way these concepts are normally employed. But it's the only idea I have, to explain what that formula might mean and where it could come from.

I especially regard it as significant that each massive species appears once and once only. It somehow suggests that a property of each field as a whole, rather than a property of quanta of the field, is at work here.

t, H, W, Z again

(A continuation of "t, H, W, Z".)

A recent paper informs us that 

(m_H)² + (m_W)² + (m_Z)² + Σ_quarks(m_q)² + Σ_leptons(m_l)² = (HiggsVEV)²

... almost identical to the formula

(m_H)² + (m_W)² + (m_Z)² + (Σ_quarksm_q)² + (Σ_leptonsm_l)² = (HiggsVEV)²

which appeared as part of the very first exercise of physics numerology examined by this blog (in July 2011), due to A. Garcés Doz. (I cannot find the formula in any of his papers at vixra.org - search for "author=doz" for a full listing - but perhaps it's there too.)

The similarity may be attributed to the fact that the top mass, m_t, is so much greater than all the other fermions, that it hardly matters whether you add the masses and square them, or square the masses and then add them. Just the following formula still works quite well:

(m_H)² + (m_W)² + (m_Z)² + (m_t)² = (HiggsVEV)²

Curiously, when Garcés Doz posted his formula two years ago, he was using m_H = 119 GeV, a value which must have come from someone's supersymmetric model; but the value of the Higgs VEV is then off by a few GeV. It actually works much better using m_H = 125 GeV, the measured value; so he could have actually predicted the Higgs mass using this formula.

(Caution: I have been a little careless in checking these facts and calculations; will check them more closely, later.)

Cosmic numerology from Texas

Some guy from Texas thinks he can obtain the non-integer part of the cosmic parameter "N_eff" using the formula (4/7)(43/57)^4/3. "There's no crazy like Texas crazy"!

A model-building resource for cosmic numerologists

"Stationary dark energy" presents a model of energy transfer between dark energy, dark matter, and baryonic matter, in which the density fractions can asymptote to desired values: "Choosing the parameters we can ensure that the final state is, for instance, Ω_ϕ = 0.7 and Ω_m = 0.3. Once reached, these values will remain fixed forever." (ϕ is a dark-energy scalar.) So if your theory of everything requires that Ω_DE should be 2/3, or 1-1/π, or 9/4π, here's a phenomenological model, complete with equations, that you can build on.

2/3^n

The quantities 2/3, 2/9, 2/27 show up in the world of Koide numerology as angles (in radian units) appearing in mass formulas.

Last year, Marni Sheppeard had a paper in which she tried to derive Louise Riofrio's cosmological dark-sector numerology from the "2/9".

This year, the Planck satellite has given us some new estimates for the dark-sector density fractions. As already noted here, one quantity is close to 1 - 1/π, which could be the starting point for a new cosmic numerology.

However, it amused me to notice that, with much less precision, the density fractions for dark energy, dark matter, and baryonic matter, are not too far from that sequence, 2/3, 2/9, 2/27.

In conventional cosmology, all these density fractions evolve throughout the history of the universe, such that it shouldn't make much sense to focus on their values at a specific moment in cosmic history (like now) as a clue to anything deep.

However, it's also true that conventional cosmology has a notorious multiple coincidence problem in explaining why those density fractions are even of the same order of magnitude. (Some conventional papers trying to explain these coincidences: 1 2. A less conventional paper: 3.)

While it's very unlikely that the cosmic densities are actually such a direct echo of whatever it is that produces the Koide patterns... it might still be worth trying to make a model in which that is the case, because it would take us in new directions and make us think of new things.

So I'm recording here an idea about where to start in such an effort. It's simply the nuMSM of Asaka and Shaposhnikov, coupled to the cosmon field of Wetterich, which for him serves as both inflaton and dynamical dark energy. And in the quest for a deeper theory, one might start with Q6 symmetry.

What is the logic of this proposal? The angle 2/9, like Koide's original formula, applies to leptons. But here we want to see it show up as the cosmic density fraction for dark matter; and in the nuMSM, the dark matter is leptonic, a keV-mass sterile neutrino.

Meanwhile, Wetterich's cosmology is wacky enough that the universe might even be shrinking, rather than expanding; and yet (he says; I haven't verified) it can be transformed through a change of frame into a far more standard cosmology. It's the best opportunity I can see, to start with something that's at least semi-orthodox, but in which it might actually make sense to model the time evolution of the density fractions as "2/3^n + ϵ_n(t)", the second term being a time-dependent correction with - one hopes - a deeper explanation. 

Finally, the finite group Q6 shows up in an attempt to explain the nuMSM's odd neutrino mass spectrum. The hope would be that this could be joined up with an explanation of Koide relations, perhaps via the group S3.

Return of the dark lepton jets

A year ago I made a silly post, "Dark matter and powers of two", proposing a connection between Weniger's gamma-ray line, and the alleged "lepton jets" that had caused a brief blog-scandal (when Tommaso Dorigo opined that a theory paper by Arkani-Hamed and Weiner must have been written with advance knowledge of the lepton-jet claim; Tommaso later retracted his allegation).

Today I see two papers talking about multi-tau states in connection with the AMS-02 confirmation of an excess positron fraction at high energies, that might be due to dark matter annihilation: Cholis and Hooper, and Jin, Wu and Zhou... I suppose the serious physics issue is this: it would be interesting if the lepton jets were real, and if they had a role in producing the positron excess; and conversely, it would be interesting if the intermediate states producing the lepton jets (assuming them to be real!) were dark-matter states.

New numerology for dark energy

At Physics Forums, a poster called "zeroace" has pointed out that, using the new values from the Planck satellite, the dark energy density is very close to 1 - 1/π.

The HWZ sum rule needs a Sumino mechanism

By "the HWZ sum rule", I mean the Dharwadker-Khachatryan formula that predicted the Higgs mass. (Perhaps we should call it the Dharwadker-Khachatryan sum rule.)

By a "Sumino mechanism", I mean something like Yukinari Sumino's mechanism which allows the Koide relation to be exact within experimental limits, despite the running of the masses.

This thought was prompted by Lykken's talk, mentioned in the previous post, in which a dark matter scalar controls the running of the Higgs mass. Perhaps some version of this model can also protect the DK formula from quantum corrections.

Scalar trinity

Michio Kaku is being criticized on a number of physics blogs for saying (on TV) that the "God particle" is the missing piece in the "Big Bang theory". While it would be apt for the particle dubbed God to be the first cause, there's no particular reason to think that the scalar responsible for inflation (the inflaton) is the same as the scalar responsible for mass (the Higgs boson).

Meanwhile, Joseph Lykken has described a model (see slide 26 forwards) in which the dark matter is yet another scalar - this possibility is sometimes called a "darkon" - which also ultimately serves to generate the Higgs mass in a finetuned way. So that would make three scalars doing fundamental things: the Higgs, the inflaton, and the darkon. Someone should tell the Pontifical Academy of Sciences about this.

t, b, tau

Of the three generational triples, t-b-tau works best (Andrew had to fudge the first generation a little, and the second is completely off); which shouldn't be a surprise, since t-b-c is already known to work well, and the masses of tau and charm are relatively close. It wouldn't surprise me if t-b-tau is already in the Koide-inspired literature (it is, incidentally, common to speak of t-b-tau mass unification in GUTs, but the idea there is that their masses are the same at the GUT scale).

Andrew's paradigm doesn't especially favor consideration of generational triples (with all the non-neutrino fermions of a generation) beyond the first generation. But a more conventional approach might hope to produce such triples through a type of "horizontal symmetry" (that skips the electrically neutral neutrinos). Indeed, one could reasonably also dub a generational triple a "horizontal triple", and a family triple could be a "vertical triple".

Andrew Oh-Willeke's paradigm

Andrew has a new post up describing his new way to think about Koide relations.

I'd say it's characterized by two ideas. First, the Higgs boson is somehow bundled with the W and Z bosons, and so mass generation is regarded as intimately linked with the weak force. Second, Koide relations arise from a sort of equilibrium of weak-force transitions between the fermions in question.

There's a lot to be said about the compatibility (or incompatibility) of these ideas with more commonplace theoretical notions, like seesaw or Froggatt-Nielsen, and with standard notions of the weak force, the Higgs mechanism, QFT, etc; and also much to be said regarding how these ideas could be given sufficient specificity to become the basis of a mathematical theory. Much to think about!

e, u, d

Andrew Oh-Willeke suggests a new Koide triple: electron, up, down.

I consider it rather unlikely (and it depends on the up mass being zero, an old idea which features in Rivero's Koide waterfall but which is out of fashion, presumably because it is empirically disfavored). But I'll say this for Andrew's triple, that it does have a little logic; those are all the first-generation fermions except for the electron-neutrino, and the neutrino masses differ from the others by orders of magnitude.

In the Koide lore of fringe physics, we have what I call "family triples" like the original electron/muon/tauon, and "sequential triples" like top/bottom/charm; I suppose this is a "generational triple".

Andrew now needs to look at the corresponding neutrinoless triples for the other two generations. If they work well, that's really interesting; if they don't, it doesn't necessarily kill the idea; there may just be other influences dominating the masses of the second and third generations, in that case.

Although I have much more confidence in the reality of the sequential triples appearing in Alejandro's waterfall for the quark masses, it may actually be easier to build models in which Andrew's generational triple is something real (i.e. has a cause, such as a new symmetry). The sequential triples alternate between up-type and down-type quarks in a way which makes them the most difficult to accommodate.

In general this also seems a positive development simply because it rounds out the picture, regarding possible generalizations of Koide's relation to all the fermions.