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.