All the lepton masses from G, pi, e

[arivero]

Let me add, Ganzfeld, that I must apologize by not noticing your equations during the two years we have run this thread. In my discharge let me explain that I am 200km away from the nearest deposit of Nature, and our electronic access does not include access to the historic repository.

 

[physmike]

Physmike, The hardest thing after finding a valid numerical coincidence
is to put it in the context of real physics. Please don't throw in all these
things like pyramids, tonal systems, and other stuff. It's much better to
refrain from all this wired stuff here if there isn't at least some kind of link
to accepted mainstream physics since this is after all a physics website.

Hans, Conner's work is partially based on "accepted mainstream physics", specifically including the work of Kepler, Helmoltz, Einstein, Schrodinger, and Bohm. He also uses admittedly "off the beaten track" sources of Pythagoras, Tesla, Russell, Cathie, and others. His books could use some good editing work too. I reference him because I think he is "on to something" in bringing together these different sources toward an explanation of the physics involved here. Though his phi-nesting spirals, fibonacci mirrors, and a few other ideas are original, unique, and brilliant. And I agree, it is hard to put a "valid numerical coincidence" in the "context of real physics"; the new concepts introduced here, relate to the understanding of this.
Judgments should respect the possiblity of physical reality, and regard the width of our educational conditioning.

In constructing his Quadrispiral, he used a transposition technique from electrical engineering which gives the result that, in particular, some harmonic values are increased by powers of ten, most notably, 10^4.
The harmonics of the Pythagorean Table produce a factor of 10 as well.
And finally, the harmonic system itself, addresses this too, as noted before.


Some general references to show partially related physics:

On the Sensations of Tone , Hermann Helmholtz

Fundamentals of Musical Acoustics, Arthur H. Benade

Theoretical Acoustics, Philip M. Morse, K. Uno Ingard

Music, Physics and Engineering, Harry F. Olson

On the mathematical structure of Tonal Harmony, Segre, Gavriel
"Some little step forward is made in the analysis of the mathematical structure of Tonal Harmony, a task begun by Galilei, Euler and the Lagrange" http://arxiv.org/abs/math/0402204

The Geometry of Musical Chords
Dmitri Tymoczko, Princeton University
"Musical chords have a non-Euclidean geometry..."
http://music.princeton.edu/~dmitri/voiceleading.pdf

G. Mazzola. The Topos of Music. Geometric Logic of Concepts...

The Geometry of White’s Dimensional-Shift Operator
Douglass A. White, Observer Physics www.dpedtech.com/Geo.pdf


Physics and pyramids ... some history,

Mystery of 'chirping' pyramid decoded, Philip Ball
http://www.nature.com/physics/highlights/7020-2.html

http://www.physicstoday.org/pt/vol-57/iss-9/p29b.shtml[/URL]

l1. L. W. Alvarez et al., Science 167, 832 (1970); see also L. Alvarez, Adventures in Exp. Phys. 1, 157 (1972).
2. Tesla Foundation, Unfolding Pyramids' Secrets Using Modern Physics, film, narrated by L. W. Alvarez and B. C. Maglich, directed by Victoria Vesna (1988).

 

[arivero]

I expect at least you will notice you are mad for common standards, physmike. It seems we can not help you here (I *really* wished to be able to help you), you can not help us, and our interaction does not help physics as a whole pursuit.

Worst, your random launching of unvalorated data-links murks any geometrical or algebraic truth which could be extracted from the proportion law you are interested in (the infamous a/b = b / (a+b) )

 

[physmike]

Sorry arivero, Perhaps you made an unconsidered connection with selected, non-random, references posted for general review by Hans (concerned about standards of physics, and should not have been needed), with a proportion that I did not talk about?

Thanks for really wanting to help, ...you did. (Guess it does not show)

All the best, physmike

 

[CarlB]

-they do not depend on a measuring unit (Meters, GeV, kilograms, etc)

Amusingly enough, my version of the Koide coincidence, got accused of depending on units by a professional physicist:

The paper on Lepton masses is interesting, but there is not really any physics, just a coincidence of numbers. Even that may not be as remarkable as it seems, since you predict 3 charged Lepton masses using 4 variables: \mu, \nu , \delta and the units of mass.
http://groups.yahoo.com/group/QM_from_GR/message/1056

I consider this another piece of evidence in favor of the thesis that nowadays, physicists will faithfully attempt to read out of the ordinary papers only to the extent that they trust the author. Instead of reading the paper, they will instead scan it for the first "error" they can find, and then move on to more interesting things.

Carl

 

[Ganzfeld]

Insofar as the essence of any piece of numerology is along the lines of 'surely this is beyond chance ?', perhaps the most useful thing would be to be able to determine just exactly what the probability is for any result. To be able to say ' the odds against chance are 1 in a million ' would at least be impressive.

Of course, it's not exactly clear just exactly how one would do so. It would require some sort of 'a priori' statement of just exactly what sort of results one was looking for...and that's not something that exists after the event. But I do think some sort of 'contrivance factor' could be incorporated, whereby all results with a similar degree of simplicity or complexity get weighed accordingly.

Surely the best response to any numerology, rather than saying there's no physics so it can't be physics, is to show that any result is just exactly what one would expect by chance.

 

[arivero]

I consider this another piece of evidence in favor of the thesis that nowadays, physicists will faithfully attempt to read out of the ordinary papers only to the extent that they trust the author. Instead of reading the paper, they will instead scan it for the first "error" they can find, and then move on to more interesting things.

Yes, and this is the goal of the introductory section of the paper: to get the trusting of the reader/referee. Old papers were a lot shorter, and I think it was because they did not rely on this need of introduction.

 

[arivero]

2) An history of numerological approaches to the fine structure constant is contained in
H. Kragh, "Magic Number: A Partial History of Fine-Structure Constant", Arch Hist Exact Sci 57 (2003) p 395-431

A related paper by the same author is "The fine structure constant before quantum mechanics". There, a subsection "Dimensionless speculations" tells about early numerology and conjectures.

 

[Kea]

Alpha

Now might be a good time to resurrect the de Vries truncated formula

e^{2} \textrm{exp}(\frac{\pi^{2}}{4}) = e + e^{3} + \frac{e^{5}}{2 \pi}

where e is electron charge.

 

[Hans de Vries]

Now might be a good time to resurrect the de Vries truncated formula

e^{2} \textrm{exp}(\frac{\pi^{2}}{4}) = e + e^{3} + \frac{e^{5}}{2 \pi}

where e is electron charge.

Hi, Kea.

If anything it would be the truncated version to pursue now. It leads to
a value of:

1/137.035 999 528 369 Interesting now would be a new direct measurement of alpha maybe from
a photon recoil experiment. The newly indirect measurement derived from
the new record setting precision electron's magnetic anomaly experiment
is:

1/137.035999710 (96)

http://hussle.harvard.edu/~gabrielse/gabrielse/papers/2006/NewFineStructureConstant.pdf

So we are off with about 1.9 sigma here. While the paper mentions many
direct measurements of alpha, it remains unclear to me why they ignore
the NIST/CODATA value which is currently the best direct measurement:

1/137.035999110(460)

http://physics.nist.gov/cgi-bin/cuu/Value?alphinv|search_for=fine+structure

But OK, We get our value for alpha by starting of with an nice analytical
'bare' vale for e (= sqrt(alpha)) of:

e^{-\pi^2/4}

and then use a Charge Renormalization Factor given by:

\Gamma\ =\ 1+\alpha+\frac{\alpha^2}{2\pi}

To get the renormalized alpha of 1/137.035 999 528 369

The last term brings the result from:
1/137.038339943 to:
1/137.035999528
So it improves the result by a factor 12800 ...Pfff, I've spend so much time to explain the truncated series from
geometrical physics... I've countless numbers of scratch pages with
potentials, propagators, Green functions, Fourier transforms.

Well it's at least useful for something I guess.Regards, Hans.

 

[Hans de Vries]

The electrons magnetic anomaly, being the result of a (very) complex
series, is:

0.00115965218085 (76).

Now could there be a direct analytical calculation of this series?
An interesting starting point seems to be:

\frac{e^{-e-e^{-1}}}{(2\pi)^2}\ =\ 0.0011570109\ \approx \ \frac{\alpha}{2\pi} + ...

That's in the right range and it becomes more interesting if we look at the
error in the ratio between the two:

g/g' = 1.0022828 = 1.00114076²

So in the error we again see our target value, which may be a sign that
our expression could be a simplified version of a more complex analytical
formula.Regards, Hans.

 

[Hans de Vries]

Mass relations between the vector bosons and the leptons.If we interpret the frequency m\ c^2/h of the leptons as a precession,
then there must also be another frequency: The frequency at which it spins.

(Simply like: Spinning top frequency versus the frequency at which it
precesses. The harder we try to tilt the top, the faster it precesses.)

The first which comes to mind is the magnetic anomaly. This is the ratio
between the orbit frequency of a (lepton) in a magnetic field and the
frequency at which its spin direction precesses.

Now, we already found the following on this thread:


0.00115965__ = electron magnetic anomaly
0.00115869__ = muon / Z mass ratio


It gives a lepton/vector_boson mass ratio. Now, can a charge-less
particle precess in a EM field? What about light by light scattering via
charged virtual particles (vacuum polarization).

The following we also found earlier on this thread:


0.0000063522 = muon g vacuum polarization terms.
0.0000063537 = electron / W mass ratio.


This gives another lepton/vector_boson mass ratio. The numbers fit very
well. Remarkable is that the properties of the muon and Z explain the
electron/W mass ratio, while in the first case it was the other way around:
The properties of the electron and W explained the muon/Z mass ratio...And then the tau. where does this leave the tau-lepton? If this is also a
precession/spin ratio, then it would need a significantly stronger coupling
because its mass (=frequency) is 16.8183 times higher as that of the muon.

Well, we naively use the (photon) diagrams of the magnetic anomaly
in first order to try to see how large this coupling should be. (even though
the only particles which have similar propagators are the gluons)

\mbox{Anomaly}\ \ \ =\ \ \frac{\alpha_?}{2\pi} + ...

This leads us to a new numerical coincident:


0.1224 ________ = required coupling constant.
0.1216 (0.0017) = the coupling constant \alpha_s(m_Z)


So, the coupling constant required to give the tau / Z mass ratio,
assuming massles propagators, leads us to the strong coupling
constant at mZ energies...

Now, "who ordered" the s of strong here? Well, at least the use of
massless (gluon) propagator diagrams fits ... :^)Regards, Hans.PS: See also http://arxiv.org/abs/hep-ph/0503104
(and http://arxiv.org/abs/hep-ph/0604035 for \alpha_s(m_Z))

 

[Severian]

I have sort of missed this entire thread, and I am not going to read through 11 pages, so I apologise if this point has already been made.

<br /> \alpha^{-1/2}+ (1+{\alpha \over 2 \pi }) \alpha^{1/2}=e^{\pi^2 \over 4}<br />

If this equation is fundamental (and I agree that it is rather a huge coincidence if it is not) then the obvious question is: Why should there be a relation for the asymptotic low energy value of alpha? Why not alpha(Mz) or alpha(MGUT)? The latter would seem to make more sense to me, but clearly it won't work with the formula.

In other words, usually we think of the most fundamental physics existing at high energies, but this is a low energy equation.

Incidentally, I have seen a similar but slightly different form:

\alpha = \Gamma^2 e^{-\pi^2/2}

with
\Gamma = 1+\frac{\alpha}{(2\pi)^0} \left(1+\frac{\alpha}{(2\pi)^1} \left(1+\frac{\alpha}{(2\pi)^2} \left(1+ ... \right. \right. \right.

This is not quite as nice, but also gives the right low energy alpha. Both of these equations can't be fundamental(?), so there has to be at least one coincidence.

 

[Hans de Vries]

If this equation is fundamental (and I agree that it is rather a huge coincidence if it is not) then the obvious question is: Why should there be a relation for the asymptotic low energy value of alpha? Why not alpha(Mz) or alpha(MGUT)? The latter would seem to make more sense to me, but clearly it won't work with the formula.

In other words, usually we think of the most fundamental physics existing at high energies, but this is a low energy equation.

Hi, Severian.

I would expect such a formula to be much more complicated and
depending on all what's out there in the vacuum, including the things
which are still hiding out there.

Looking at, say, the calculations of the magnetic anomaly of the muon,
then it's always the low energy limit of alpha which is used, and the
vacuum polarization comes in from explicit terms defined in mass relations
like A2(mμ/me), A4(mτ/me). After that you get all the hadronic terms
and electroweak terms.

So a running alpha (which includes all these vacuum polarization terms)
is dependent on a complex function of all kinds of SM parameters like
the lepton mass ratios for the QED only part and getting much worse
for the hadronic contributions.

The point is that, on this thread we're looking for simple numerical
coincidences which just might have a physical origin. So, the shorter
the expression is, the better. Complex things generally lead to complex
expressions and we try to avoid them.

Incidentally, I have seen a similar but slightly different form:

\alpha = \Gamma^2 e^{-\pi^2/2}

with
\Gamma = 1+\frac{\alpha}{(2\pi)^0} \left(1+\frac{\alpha}{(2\pi)^1} \left(1+\frac{\alpha}{(2\pi)^2} \left(1+ ... \right. \right. \right.

This is not quite as nice, but also gives the right low energy alpha. Both of these equations can't be fundamental(?), so there has to be at least one coincidence.

Both formula's are from here The second one was an attempt to
extend the first one into a series. The first expression is just the first 3
terms of the series.Regards, Hans

 

[Severian]

Looking at, say, the calculations of the magnetic anomaly of the muon,
then it's always the low energy limit of alpha which is used, and the
vacuum polarization comes in from explicit terms defined in mass relations
like A2(mμ/me), A4(mτ/me). After that you get all the hadronic terms
and electroweak terms.

But there is a good reason that the alpha used in the magnetic moment of the muon (or the electron for that matter) is 1/137. It is a low energy observable. So it is really not the same thing as an equation that is supposed to derive alpha itself.

So a running alpha (which includes all these vacuum polarization terms)
is dependent on a complex function of all kinds of SM parameters like
the lepton mass ratios for the QED only part and getting much worse
for the hadronic contributions.

My theoretical prejudice would be that the number at high energies would have a simple form, while that at low energies was simply obtained by running to the low scale. The low energy value would then be the one which contains all the messy corrections. So I would be happier if you could you could reproduce an alpha at the GUT scale (or perhaps the Planck scale) which would provide the low energy value after running.

The point is that, on this thread we're looking for simple numerical
coincidences which just might have a physical origin. So, the shorter
the expression is, the better. Complex things generally lead to complex
expressions and we try to avoid them.

Fair enough. The coincidence I find most intriguing is the match between the Higgs vev and the top mass. Or why, to current experimental accuracy, is the top Yukawa exactly 1?

I see now that the two equations I quoted are the same (with one truncated). Silly me! Though I saw it on a different site (I will have a look for the link).

 

[arivero]

My theoretical prejudice would be that the number at high energies would have a simple form, while that at low energies was simply obtained by running to the low scale.

...


Fair enough. The coincidence I find most intriguing is the match between the Higgs vev and the top mass. Or why, to current experimental accuracy, is the top Yukawa exactly 1?

Yes, in a way this thread defies the current prjudice by suggesting that it is possible to find relationships in the asymptotic low energy limit. This could be explained in some ways:

- For mass relationships, it could be that the high energy GUT masses are zero, thus the mass is generated radiatively at low energy. This has been tryed a lot of time in the seventies to get the electron mass out of the muon one, but abandoned after (or because of) the third generation.

- For coupling constants, some eigenvalue in the renormalisation group flow could be reached when running to low energy, sort of universality. Adler did some speculative tries on this sense.

- For relationships between coupling constants, it could be related to the symmetry breaking mechanism. Meaning, that the symmetry breaking mechanism triggers at a energy scale with the coupling constant meeting some algebraic relationships. I have no idea of a mechanism of such kind.

Of course, we could also consider that our prejudice about high energy GUT is just that, a prejudice. For instance, Koide's relationship, related to mass quotients across the three generations, seems hard to be married with a radiative generation principle.

 

[arivero]

So, the coupling constant required to give the tau / Z mass ratio,
assuming massles propagators, leads us to the strong coupling
constant at mZ energies...

Do you mean alpha/2pi = mtau / mZ approx? Hmm the only thing I am afraid is about the experimental value of the strong coupling, a fuzzy bussiness.

Now, "who ordered" the s of strong here? Well, at least the use of
massless (gluon) propagator diagrams fits ... :^)

Actually we have already an uninvited aparision of alpha_s in the relationship between Z0 decay and Pi0 decay (somewhere in the middle ot the long long thread). It is hidden because we speak of the "pion decay constant", but this pion decay constant is actually a sum of radiative corrections coming from the strong force.

 

[arivero]

I see. In other words you are telling that the relationship between tau and muon mass is as the one between electromagnetic and strong coupling constants. Gsponer did me a related observation time ago, that if we drove electron mass to zero, Koide relationship should imply a quotient between tau and muon, and this quotient was about the same magnitude that the nuclear strong force (the pion-mediated force between nucleons). Now, electron mass to zero with a fixed muon mass should be equivalent to electromagnetic coupling going to zero. Hmm.

 

[Hans de Vries]

Actually we have already an uninvited aparision of alpha_s .

One might blame it on the infamous ninth gluon... :^) , the white/anti white one,
sometimes speculated to be the photon. ( which would leave it with the
wrong coupling constant)Regards, Hans

 

[arivero]

Just adding this one to the thread for reference

http://es.arxiv.org/abs/hep-ph/0609131
G. Lopez Castro, J. Pestieau
The unit of electric charge and the mass hierarchy of heavy particles

Some more:
Jean Pestieau http://es.arxiv.org/abs/hep-ph/0105301
G. Lopez Castro, J. Pestieau http://arxiv.org/abs/hep-ph/9804272
Jean Pestieau* and Probir Roy http://prola.aps.org/abstract/PRL/v23/i6/p349_1

 

[arivero]

For example...the combined mass of the particles of the meson octet is 3.14006 times the proton mass. Pretty close to pi. Amazingly, the combined mass of the particles in the baryon octet is around 9.8 times the proton mass. Close to pi squared.

Let me to evaluate these quantities because, as we know, these multiplets at at the heart of SU(3) flavour mass formulae.

The meson octect is \pi^0 \pi^+ \pi^- K^0 \bar K^0 K^+ K^- \eta_8. The latter mixes with the singlet to produce \eta and \eta' which are the actually measured masses, and we can use \eta if we think that the mix is very small. But we could also consider the full non irreducible nonet or the extended 16-plet with the charmed particles.

The barion octect is p n \Lambda \Sigma^0 \Sigma^+ \Sigma^- \Xi^- \Xi^0. No mixing issue here, but Sigma^0 is a very fascinating piece of physics on itself, particularly its decay mechanism. As for adding charm, we could, going then to a 20-plet.

figures are drawn in http://pdg.lbl.gov/2006/reviews/quarkmodrpp.pdf

Now the meson sum is

134.9766+139.57018+139.57018+547.51+497.648+497.648+493.677+493.677
=2944.277

2944.27696/938.27203=3.1379, which is 0.12% off from pi, thus qualifies for the thread. It fails pi for about 3.5 MeV, so the error bars can not be blamed, perhaps the mixing can.

938.27203+939.56536+1115.683+1192.642+1189.37+1197.449+1321.31+1314.83=9209.12139=
9.815 M_p, while pi square is about 9.87. So off by 0.55% this time but no mixing to be blamed here.

 

[Demystifier]

Numerology

I do not know if somebody already said that, but I suspect that it is possible to write a computer program that will find a "numerical coincidence" (up to a reasonably specified accuracy) between any two (or more) specified numbers. Essentially, such a program tries various algebraic relations and combinations of small integer numbers and constants such as "pi" and "e", until it finds a "good" one.

 

[arivero]

I do not know if somebody already said that, but I suspect that it is possible to write a computer program that will find a "numerical coincidence" (up to a reasonably specified accuracy) between any two (or more) specified numbers. Essentially, such a program tries various algebraic relations and combinations of small integer numbers and constants such as "pi" and "e", until it finds a "good" one.

Indeed this program has been written and it is quoted somewhere in the middle of the thread; even the output is available in a web page, sorted by decimal ordering. It was done by a couple of computer scientists and the motivation is to try to define some kind of complexity of an algebraic expression. Another researcher, I.J.Good, tryed to use the same approach to single out "low entropy" expressions. Such GIGO (measure garbage In to determine the Garbage out) methods usually trip on the electron/proton quotient

In some sense a problem with these programs is that they concentrate in algebra instead of geometry (nor to speak of dynamics). So 6 \pi^5 is reported as "less complex" than, for instance, e^\pi - {1 \over e^{\pi}}

Let me add that the study if the rings generated by the rationals plus some finite set of irrational numbers are a very prolific field of study in algebra. Still, its truncation to some n-digits decimal expansion is not very studied as far as I know; a friend of me, J. Clemente, tried time ago to work out the algebraic setting of IEEE "real" numbers and we did not found too much bibliography on it.

 

[Demystifier]

I am really happy to know that such a program has been made.
Thanks arivero.

 

[arivero]

Via an old comment of Baez in Woits blog a year ago, I note the http://www.nbi.dk/~predrag/papers/finitness.html about the perturbative expansion of QED. It its weak form, it claims that the growth of diagrams is not combinatorial. In its strong form, it seems to claim that the coefficients are of order unity, a point that Kino****a considers refuted after the 8th-order calculation.

Lovers of gossip and theoretists of science would like to check also the remarks
http://www.nbi.dk/~predrag/papers/g-2.html
http://www.nbi.dk/~predrag/papers/DFS_pris.ps.gz

I do not know what to do of his "social experience". I remember, as undergradute student, how excited I was about the articles of Cvitanovic on Chaos theory, and how a time later I was not so fond of them. But he got to bail out into another physics area, at least. On the other hand, 20 years later, Kino****a (could someone edit it our of the politically correct spelling rules of PF, please!) is the leader of the perturbative calculation effort, and it is because of him, perhaps, that the (g-2) is still an important test of the standard model.

And yes, the 8 order term seems nowadays very much as -sqrt(3). The 6th order term is now 1.181241..., it was 1.195(26) already in the 1974 paper; the 0.922(24) refers to a particular subset, see http://www.nbi.dk/~predrag/papers/PRD10-74-III.pdf

 

[arivero]

On sociology, it is sad to think that a "pet theory" had privated QED of one of its more dedicated calculators. Now consider if the observations of Hans about g-2 could attract some interest, or more probably to excite old memories of conflict.

 

[CarlB]

Did we include this one already?
http://federation.g3z.com/Physics/#MassCharge

I can't give you the name of the author as I don't know it. Mark Hopkins maybe?

 

[arivero]

Did we include this one already?
http://federation.g3z.com/Physics/#MassCharge

I can't give you the name of the author as I don't know it. Mark Hopkins maybe?

It refers to Yablon, and some of the mass formulae have been discussed in usenet news.

 

[Hans de Vries]

It refers to Yablon, and some of the mass formulae have been discussed in usenet news.

Good to see you posting!

We should compare the recently improved W mass with the results of the
thread. The new world average for the W mass is now 80.398 (25) GeV.
http://www.interactions.org/cms/?pid=1024834
The value for the Z mass is still 91.1876 (21) GeV as far as I know.
We had two numerical coincidences for the mW/mZ mass ratio on this thread:

0.881418559878 ___ from the spin half / spin one ratio
0.881373587019 ___ the value arcsinh(1)

Using the more precise value of Z we can get values for the W mass:

80.374  ( 2)    Derived W mass from spin half / spin one ratio
80.370  ( 2)    Derived W mass from arcsinh(1) 
80.376  (19)    Experimental W mass: from sW on page 8 of hep-ph/0604035 
80.398  (25)    Experimental W mass: New world average
80.425  (38)    Experimental W mass: Old world average

This is certainly an improvement for both the values as well as the sigmas
which are around 1 for both now. (mid value difference: 0.030% and 0.035%)

The last value was discussed here:
https://www.physicsforums.com/showpost.php?p=958122&postcount=202Regards, Hans

 

[arivero]

Good to see you posting!

Hi! Instead of a sabbatical I got increased workload, so I read the blogs and forums but I do not calculate

In fact I had not checked the new values. So, before, the 1 sigma low point was 80.387 and now it is 80.377 so the results are better than in 2004. No surprise, as the word average is to be calculated with similar patterns than hep-ph/0604035. This means that the coincidence is here to remain, unexplained or not. Any future deviations could be covered with radiative corrections, if it comes from a fundamental theory.