There was unexpected progress, posted at Physics Stack Exchange, on two problems that were low on my list.

First, numerology of the charge radius. See my 2017 update: I ran across a model of the nucleon in which the radius is 4 natural units, divided by the mass. That doesn't explain why the radius comes out a little different for muonic hydrogen compared to electronic hydrogen; but it can explain why dandb's ratio is approximately 4 in both cases.

Second, mystery of the Z0 decay width - that it lies on the same curve as a number of mesons. It's one of @arivero's minor observations, and not one that I spent any time on. I was just going through the motions of investigating it, when to my surprise, something turned up.

# theory

not endorsed by snarxiv

## Tuesday, March 7, 2017

## Friday, February 10, 2017

### tHWZ - latest formulation

During a discussion at PF, I found the following interesting way to think of these quantities:

m

m

H

m

The last one may look a little odd, but it allows us to approximate sin

The impetus was a comment by @arivero in which he pointed out that a tHWZ mass estimate due to Hans de Vries implies

(m

Now in many GUTs, at the GUT scale, we have that

m

So it's as if (m

We could even speculate that my set of four approximations above is an infrared fixed point. (The approximations are not exact, but one could think of these as valid at tree level.)

Unfortunately I don't see how any of this makes sense in terms of Hans de Vries's original physical hypothesis.

Anyway, I find that the LC&P formula also works neatly using the four approximations. And I would remark again that m

m

_{H}~ √2 m_{Z}m

_{t}~ 2 m_{Z}H

_{vev}~ 2 √2 m_{Z}m

_{W}~ √7 / 3 m_{Z}The last one may look a little odd, but it allows us to approximate sin

^{2}of the Weinberg angle as 2/9.The impetus was a comment by @arivero in which he pointed out that a tHWZ mass estimate due to Hans de Vries implies

(m

_{W}^{2}- m_{H}^{2}) / (m_{Z}^{2}- m_{t}^{2}) = 3/8Now in many GUTs, at the GUT scale, we have that

m

_{W}^{2}/ m_{Z}^{2}= 3/8So it's as if (m

_{W}^{2}- m_{H}^{2}) / (m_{Z}^{2}- m_{t}^{2}) is almost invariant under renormalization group flow, with m_{H}= m_{t}= 0 at the GUT scale.We could even speculate that my set of four approximations above is an infrared fixed point. (The approximations are not exact, but one could think of these as valid at tree level.)

Unfortunately I don't see how any of this makes sense in terms of Hans de Vries's original physical hypothesis.

Anyway, I find that the LC&P formula also works neatly using the four approximations. And I would remark again that m

_{Z}is very close to the standard model's μ parameter.## Wednesday, August 24, 2016

### 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?

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?

## Saturday, July 9, 2016

### 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

"Here, the fine-structure constant α, the strong coupling constant α

α = { α

_{s}sin^{2}θ_{W}(b_{1}−b_{3})+3/5 cos^{2}θ_{W}(b_{3}−b_{2}) } / (b_{1}−b_{2}) + 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."## Saturday, May 14, 2016

### 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

m

H

m

m

m

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.

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}/√2H

_{vev}~ 2 m_{H}m

_{375}~ 3 m_{H}m

_{750}~ 6 m_{H}m

_{H}is the Higgs boson mass, m_{t}is the top quark mass, H_{vev}is the Higgs field vev. m_{750}is the mass of the 750 GeV particle. m_{375}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.

## Wednesday, May 11, 2016

### 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...

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...

## Friday, May 6, 2016

### 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."

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

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