Multifractal worldsheet

The opinion is spreading that the real discovery of the LHC was that the Higgs boson mass is special. The most impressive prediction was Shaposhnikov and Wetterich 2006, which got the right value from the assumption that gravity is asymptotically safe.

This creates cognitive dissonance for anyone who appreciates the string-theoretic model of quantum gravity. Asymptotic safety isn't even consistent with the holographic principle, is it?

Well, asymptotic safety is one of several heterodox approaches to quantum gravity in which the dimension of spacetime seems to change from 4 to 2 at the smallest scales. Sabine Hossenfelder lists a few others and says, "It is difficult to visualize what is happening with the dimensionality of space if it goes down continuously, rather than in discrete steps".

Fractals can have non-integer dimensionality. But they are typically embedded in a larger space. Meanwhile, in string theory, one has a 2d worldsheet embedded in a "target space" that usually has more than two dimensions. So what if the world sheet embeds in the target space as a multifractal surface that is 4d on large scales but 2d on small scales?

A formula for α

On page 4 of "Naturally Speaking" by G.F. Giudice, after a short list of numerological formulas for the fine-structure constant α, one finds a formula for α according to physical orthodoxy, i.e. grand unification. I reproduce it here for the edification of passing numerologists: 

α = { α_s sin²θ_W (b₁−b₃)+3/5 cos²θ_W (b₃−b₂) } / (b₁−b₂) + higher-order terms.

"Here, the fine-structure constant α, the strong coupling constant α_s and the weak mixing angle θ_W are evaluated at the same renormalization scale and b_1,2,3 are the gauge β-function coefficients. Higher-order terms cannot be neglected to achieve a prediction that matches the experimental accuracy."

Higgs, top, 750 GeV II

The idea that the 750 GeV anomaly might be a top-antitop bound state has been taken to a new level by Froggatt & Nielsen, who have sketched a whole phenomenology for their particle. The reasoning is "crude" (their word), but still on a much higher plane than any mere numerology of masses.

So things may be about to get serious there. Meanwhile, I want to enumerate a few relationships which are still just numerology, but have the potential to be part of a genuine theoretical synthesis.

m_H ~ m_t/√2
H_vev ~ 2 m_H
m₃₇₅ ~ 3 m_H
m₇₅₀ ~ 6 m_H

m_H is the Higgs boson mass, m_t is the top quark mass, H_vev is the Higgs field vev. m₇₅₀ is the mass of the 750 GeV particle. m₃₇₅ is the mass of a 375 GeV particle that Lubos may have found in the data. 

The picture I get is that the Higgs field is a top quark condensate, the 750 is a sort of loose bound state of 6 Higgs bosons (that is a "1S" toponium when analyzed at the level of quarks), and the 375 is like the 750 but with only half of the available top states occupied.

Higgs, top, 750 GeV

It is a long-standing idea that the Higgs might be a top-antitop bound state. (I have proposed a bootstrap version of this idea.)

My credence for that idea has just gone way up, now that I have discovered another long-standing proposal, that there might be a light bound state of 6 tops and 6 anti-tops. The number 6 appears because the top quark has two spin states and three color states, so this is the maximum number of tops in the same wavefunction that is allowed by the Pauli exclusion principle.

I had already wondered if the LHC bump at 750 GeV was somehow 6 Higgs bosons bound by top loops, since 750 GeV = 6 x 125 GeV, the Higgs mass. But if the Higgs is already a top-antitop bound state...

Proton charge radius

A new user at Physics Stack Exchange, "dandb", has made an observation which I express as follows:

"The charge radius of the proton (in muonic hydrogen) is almost exactly four times the reduced Compton wavelength of the proton."

Susy II

In some ways, the MSSM seems promising as a framework for tHWZ numerology. The reason may be seen at the end of section 3.4 in Stephen Martin's primer: susy determines the couplings of the scalar potential, in terms of coupling constants elsewhere in the theory. If I am interpreting that passage correctly, the quadratic coupling will be set by the top yukawa, and the quartic by the gauge couplings.

However, the MSSM has a lot of annoying particles like gauginos and sfermions which get in the way. Here last decade's ideas about split susy are useful. In particular, in section 2.3.3 of "Predictive Landscapes..." we read about a framework midway between split and supersplit susy, in which only the Higgsino is light. That sounds worth exploring.

H, Z, susy

I finally noticed that the Higgs mass parameter μ, 89 GeV, is very close to the Z boson rest mass, 91 GeV (and the width of the Z is a few GeV).

In the standard model, these quantities should be independent. But in the MSSM, the Z boson is the upper bound on the tree-level mass of the Higgs.

I am too tired to develop an interpretation. But tomorrow is another day.

t, H, W, Z in 2016

Recently I have been puzzling again, over the Dharwadker-Khachatryan sum rule

m_H = m_W + 1/2 m_Z

The problem being that it works quite well; but theory tends to favor relations among the squares of these masses (e.g. the "Veltman condition").

The primary purpose of this post is just to observe that you can get such a relation by squaring both sides of the D-K equation.

You do also get a term m_W m_Z. Perhaps it could result from a geometric mean, as in Torrente-Lujan.

Another simple thing that I want to observe, is that you might obtain something like D-K, by taking the square root of a Veltman-like sum rule. In other words, it could be approximately true, not by chance and not because it is directly implied by a fundamental theory, but as an algebraic side-effect of the truly fundamental relationship.

(The same applies to the Lopez-Castro - Pestieau - Garces-Doz sum rule, previously discussed here, which does involve masses squared, and therefore even more closely resembles Veltman's condition.)

P.S. Dharwadker also has a numerology for the ratio baryonic matter : dark matter : dark energy, which he deduces to be 1:5:18.