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.