All the lepton masses from G, pi, e

  • Thread starter arivero
  • Start date

Multiple poll: Check all you agree.

  • Logarithms of lepton mass quotients should be pursued.

    Votes: 21 26.3%
  • Alpha calculation from serial expansion should be pursued

    Votes: 19 23.8%
  • We should look for more empirical relationships

    Votes: 24 30.0%
  • Pythagorean triples approach should be pursued.

    Votes: 21 26.3%
  • Quotients from distance radiuses should be investigated

    Votes: 16 20.0%
  • The estimate of magnetic anomalous moment should be investigated.

    Votes: 24 30.0%
  • The estimate of Weinberg angle should be investigated.

    Votes: 18 22.5%
  • Jay R. Yabon theory should be investigate.

    Votes: 15 18.8%
  • I support the efforts in this thread.

    Votes: 43 53.8%
  • I think the effort in this thread is not worthwhile.

    Votes: 29 36.3%

  • Total voters
    80
  • #26
marcus
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Hans de Vries said:
I found it a week or two before posting. tried to see if it
could be further simplified but couldn't resist to post it
anymore...

well I am glad you posted it here
where we can see and discuss some

I suppose the next thing is two things:

1.write email to some physicists
(I would think immediately to write Lee Smolin because
it could have some connection with his CNS scheme for
iteratively generating the fundamental dimensionless constants)

2. prepare an article to post on http://arxiv.org

Alejandro knows about arXiv, has some experience.
Once it is on arXiv then it will always come up when people do
keyword searches. Then it may be of use to someone who
finds it by accident and is building a theory to explain alpha.

I think anyone here would be glad to help. It is quite interesting
that a coincidence (let us call it that) should go out 10 decimal places
 
  • #27
arivero
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marcus said:
2. prepare an article to post on http://arxiv.org
Alejandro knows about arXiv, has some experience. Once it is on arXiv then it will always come up when people do keyword searches. Then it may be of use to someone who finds it by accident and is building a theory to explain alpha.

Yep, I know from the ArXiV :frown: , and I can see it is not easy to fit there, as the goal of the preprint distributions is to keep researchers up-to-date about advancements in *their* area. It is clearly not a hep-th/ , as it does not have a theory under, and probably not a math-ph/ . It could be focused as sort of "state-of-art memotecnics" or something in a line able to fit as physics/ (think about the American Journal of Physics kind of articles).
 
  • #28
Hans de Vries
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I found an interesting one:

The idea is to see if Nature's limitation to three gene-
rations might have some relation with the identity:

[tex]3^2 + 4^2 = 5^2[/tex]

For the three consecutive integers 3,4 and 5, leading
to only 3 states rather than a whole series. We write
for the three lepton masses:
.
.

[tex]3^2 \ ln(m_e) \ \ \ + \ \ \ (4^2+n)\ ln(m_\tau) \ \ \ = \ \ \ (5^2+n)\ ln(m_\mu) [/tex]
.
.


We then look at the value of n. We do find [itex] n \approx 1.00086[/itex]
using the following Codata values for the mass ratio’s:

__3477.48 ____ (57)__ tau/electron
___206.7682838 (54)__ muon/electron
____16.8183 ___(27)__ tau/muon


The least well known value, [tex]ln(m_\tau)[/tex] has to be
exact to within 0.005% to get a result so close to
1.0000. This is less than the current experimental
uncertainty which means that the exact value of
n=1.0000 is still within the experimental uncertainty.

Regards, Hans
.
.
.

PS. Please be aware that the chance for coincidences
might be bigger than you think!


PPS: The formula is invariant under the transformation
[tex]\{ m_e, m_\mu, m_\tau \} \rightarrow \{ xm^y_e, \ xm^y_\mu, \ xm^y_\tau \}[/tex]
where x and y
are arbitrary numbers.
 
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  • #29
arivero
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Actually there are two "pythagorean sequences", the other one is (-1,0,1), having as square the 1,0,1 sequence. Intriguingly, in your combination there is a twist, because the -1 is not going along with the 3, but with the 4.

Hans de Vries said:
PS. Please be aware that the chance for coincidences
might be bigger than you think!
I can see your conflict; your first set of formulae and this third one have different assignations for the quotient of logarithms,
[tex]{\ln m_\mu/m_\tau \over \ln m_e/m_\mu}={\pi^2-1 \over 2\pi^2-3}; \,
{\ln m_\mu/m_\tau \over \ln m_e/m_\mu}={3^2\over 4^2+1}[/tex]
so surely at least one if them is a random coincidence.
(EDITED: Or, perhaps, there is some happy circunstance where an expression happens during some series expansion of the another.
(EDITED again: in fact, using the expansion [tex]\pi^2/6=\zeta (2)=\sum_n 1/n^2[/tex], its first term is [tex]10/18[/tex], ie [tex]9+1 \over 17+1[/tex] so both expresions are no so far away: the second formula can be seen, at least, as the first term of the expansion of [tex]{\pi^2-1-1/2 \over 2\pi^2-3-1/2}[/tex]. Good enough if one takes into account that both HdV formulae are approximate guesses of an hypothetical exact formula.
Said this, both relations have an intense geometrical flavour. We are exploring two avenues of discretization of mechanics which could, in the long run, provide some justification: either to parametrise the ambiguity in taking two derivatives of the position to get Newton law, or to parametrise the ambiguity in the two sequencial derivatives of Lie Bracket.
 
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  • #30
arivero
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spinors?

Perhaps it is physics after all; I have just revised (ah, google!) a point that I should have noticed instantaneously: that the Pythagorean condition appears naturally in the definition of Cayley spinors. A guy called Andrzej Trautman has worked out the diophantine case SL(2,Z). If interested in this line, check http://math.ucr.edu/home/baez/week196.html and references therein.
 
  • #31
Hans de Vries
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Alejandro,

I'm trying to go back to more physics but first this. The two
lepton ratio formula's may also be combined into the following:

[tex]\ln {m_\tau \over m_\mu} = {\pi-{1 \over \pi}}[/tex]_______ [tex]\ln {m_\mu \over m_e} = {4^2+1 \over 3^2} (\pi-{1 \over \pi})[/tex]_______[tex]\ln {m_\tau \over m_e} = {5^2+1 \over 3^2} (\pi-{1 \over \pi})[/tex]


Now if and only if the term [itex]\pi-1/\pi[/itex] was exact then all three
formulae would be within experimental range. Now it isn't but the
[itex]\pi-1/\pi[/itex] term is the only thing that I could connect to some real
physics up to now. It's inspired on the way how you rewrote:


[tex]\ln {m_\tau \over m_\mu} = {\pi-{1 \over \pi}}\ \ \ \ \ \ \equiv \ \ \ \ \ \ \ln {m_\tau \over m_\mu} = 2\sinh(\ln \pi)\ \ \ \ \ \ \equiv \ \ \ \ \ \ {m_\tau \over m_\mu} = |\exp( \sinh(\ln\pi) )|^2[/tex]


the sinh() gives us something in the space-time domain or in
momentum space if we consider it to be a boost like in:

[tex]\sinh \xi=\frac{ v/c}{\sqrt{1-v^2/c^2}}[/tex]____[tex]\cosh \xi=\frac{1}{\sqrt{1-v^2/c^2}}[/tex]
with:

[tex]\tanh \xi=v/c[/tex] ____[tex]\exp \xi=Doppler Ratio[/tex]

The Doppler Ratio now becomes [itex]\pi[/itex] interestingly enough (for
blueshift) and [itex]1/\pi[/itex] (for redshift) corresponding with a speed
v/c (of rotation?) The term [itex]\pi-1/\pi[/itex] could for instance
correspond with the imbalance in momentum change when
absorbing a photon from the back and a photon from the front.

The term [itex]|\exp( )|^2[/itex] may possibly be associated with going from
phase space (defined in x and ct) to probability space. if the
masses were to be defined as "mass density probabilities"
(mass * wave function) then the imaginary part of the
argument would define phase while the real part would
lead to the mass ratio at each point of the wave function.

Another, although numerological, reason to become interested
in this approach is that the most exact equations I got up to
now are found in the "boost domain", that is:

[tex]\ln {m_\mu \over m_e} \ = \ 2\sinh(a) * 1.000000093 , [/tex]____[tex]\ln {m_\tau \over m_\mu} \ = \ 2\sinh(b) * 1.0000047[/tex]

with:

[tex]a = 1 +\sqrt{1/2} \ \ \ and \ \ \ ab^2 = \sqrt 5[/tex]

I don't know if a and b are really a pair but at least there is a
simple relationship.

That's the best I can do for sofar in the hope to get some
physical meaning out of it.

Regards. Hans
 
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  • #32
arivero
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Yep, I intended a connection with relativity when casting sinh here, but I am still far to understand how it goes. There is some dinamical analogy between the tetrad of elementary particles, [tex]\nu, e, u, d[/tex] and the tetrad of relativistic coordinates [tex]t, \rho, \theta, \phi[/tex]. On one side quarks are unobservable, so they feel very much an angular coordinate -which lacks of metric scale-. On another neutrinos are sort of special, so they feel as the trivial time coordinate. But we are very far from getting real meaning of this analogy; my own research program is in the backburner fire already for more than five years (see hep-th/9804169 and "cited by" there) with no remarkable milestones.

Hans, are you using some symbolic algebra program to search for your equations? I ask because even if the results are not publishable as a physics article, it could then to be a very valid article for computational journals.
 
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  • #33
arivero
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arivero said:
TL attributes the second one to Lubos, but I have read it attributed to Feynman himself, just as an example of the kind we are discussing, random relationships. It could be interesting to know the origin of the first one to alpha, as it is a variant of HdV idea.

Let me to update on this. While Lubos Motl seems to have proposed the 6 pi^5 equation in an independent way, some previous postings in the net give it as folklorical knowledge, sometimes related to a couple of footnote-quoted papers by Armand Wiler:
Wyler,A., 'On the Conformal Groups in the Theory of Relativity and their Unitary Representations', Arch. Rat. Mech. and Anal.,31:35-50, 1968
Wyler,A., L'espace symetrique du groupe des equations de Maxwell' C. R. Acad. Sc. Paris,269:743 745
Wyler,A., 'Les groupes des potentiels de Coulomb et de ¥ukava', C..R. Acad. Sc. Paris,271:186 188
According F. D. T. Smith, this work of Wyler was noted in the Physics Today Aug and Sept 1971 (vol 24, pg 17-19 according M. Ibison).

Regretly old comptes rendues are not in the internet so I can not verify these papers. The Physics Today comment attracted some discussion in the Physical Review Letters,
http://prola.aps.org/abstract/PRL/v27/i22/p1545_1
http://prola.aps.org/abstract/PRL/v28/i7/p462_1
http://prola.aps.org/abstract/PRD/v15/i12/p3727_1
And well, there they refer to a "Wyler equation for alpha",
[tex]\alpha=(9/8 \pi^4)(\pi^5/2^4 5!)^{1/4}=1/137.03608[/tex]
not to an equation for proton mass. So we are still in doubt about the priority of the proton/electron quotient.

PS: please do not believe the Journal-Ref of Tony Smith, he is always exchanging trickeries with the arxiv (and please do not comment about it in this thread!!!). But Tony is always a good starting point to remember exotic, sometimes forgotten, research.
 
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  • #34
Kea
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Trautman

arivero said:
Perhaps it is physics after all; I have just revised (ah, google!) a point that I should have noticed instantaneously: that the Pythagorean condition appears naturally in the definition of Cayley spinors. A guy called Andrzej Trautman has worked out the diophantine case SL(2,Z). If interested in this line, check http://math.ucr.edu/home/baez/week196.html and references therein.

By the way, Trautman (I guess it's the same guy) considered
applying categories to gravity a long time ago. He presented
an article on it at Dirac's 70th birthday party.

The HdV work is cool. I'm tired of arguing with a string
phenomenologist that I know, who believes that it's in
principle impossible to calculate (from a fundamental theory)
the mass ratios.

Cheers
Kea
:smile:
 
  • #35
arivero
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I found finally the Mp/Me=6 pi^5 estimate. It appears in the Physical Review,

Friedrich Lenz "The Ratio of Proton and Electron Masses" Phys. Rev. 82, 554 (1951)
so neither Motl nor Feynman.

Bayley and Ferguson adscribe it to http://www.stat.org.vt.edu/holtzman/IJGood/CV_IJG.pdf [Broken], "On the masses of Proton, Neutron and Hyperons", Journal of the Royal Naval Science Service, v 12, p 82-83 (1957), an article which is not included in the standard biblography of this author. But it is true that Good worked on this topic and published some comments about it. The bibliography gives
Some numerology concerning the elementary particles or things, JRNSS 15 (1960) 213
The Scientists Speculates: An Anthology of Partly-Baked Ideas (London, 1962)
The proton and neutron masses and a conjecture for the gravitational constant. Phys Lett A 33, 6 (1970) p 383-384

As it happens, Gustavo R. Gonzalez-Martin, who wrote me earlier this year (and I ignored him), gives all the correct references. I believe he uses a volume quotient very in the spirit of Wyler, so it is not strange he has done the right research of earlier work.

Note finally a funny coincidence: both De Vries and F. Lenz got his names as a famous heritage, of older scientists. But note that while HdV seems to exist in the network today, F. Lenz did not published anything in the Phys Rev except for that three-lines letter (hmm, it seems there is a contemporary F. Lenz publishing in the Z. Naturforsch.)
 
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  • #36
Hans de Vries
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Classical Distance Ratio Formula

.
.

This one is the most “physical” yet. It’s also the simplest
and most accurate expression involving lepton mass ratios.
(fully within experimental range and exact to 0.0000073%)
.
.
.
[tex]\frac{d_{crad}}{d_{spin}} \ \ = \ \ \frac{m_e}{m_\mu} \ (1 \ + \ \ a \ \frac{m_e}{m_\tau} \ ) , \ \ \ \ \ \ \ \ \ \ \ \ \ where \ \ a = \frac{1}{\frac
{1}{2}(1+\frac{1}{2})} = \frac{4}{3}[/tex]
.
.

It relates the ratio of two “classical distances” with the
lepton mass ratio’s. Since distances scale (inversely)
proportional to mass we may expect the distance ratio
to be directly proportional to the lepton masses as well.
So far so good.


The distance ratio is dependent only on the fine structure
constant and consequently can be calculated with a very
high precision:

1/206.68905011 (69) _ dist. ratio from fine structure const.
1/206.689035 (19) ___ dist. ratio calculated with mass ratios

1/206.7682838 (54) __ electron/muon mass ratio
1/3477.48 (57)_______ electron/tau mass ratio

The electron/muon ratio already equals the distance
ratio to within 0.038% A simple correction term using
the electron/tau ratio then brings it to 0.0000073%



The distance ratio is the same for all spin ½ particles.
The classical distances are:


dcrad:___ Classical (Electron) Radius
=====================================
A lepton and anti-lepton spaced at this distance,
(which is inversely proportional to their mass),
have a (negative) potential energy which is equal
to their rest mass energy.

dspin:___Classical (Electron) Spin ½ Orbital
=====================================
A lepton and anti-lepton spaced at this distance,
(which is inversely proportional to their mass),
and orbiting at the frequency corresponding to their
rest mass, have an angular momentum of a spin ½
particle: [itex]\sqrt{(\frac{1}{2}(1+\frac{1}{2}))} \ \hbar[/itex]


It turns out that the velocity at which the leptons
would have to orbit here is equal for all spin ½ particles.
It’s a dimension less constant when written as v/c and
the solution of the following equation:

[tex]\frac{v^2/c^2}{\sqrt{1-v^2/c^2}} \ \ = \ \ \sqrt{\frac{1}{2}(1+\frac{1}{2})}[/tex]

Where the right hand term corresponds to the angular
momentum. When solved it gives:

[tex]\frac{v}{c} \ \ = \ \ v_{spin\frac{1}{2}} \ \ = \ \ 0.754141435281767 \ \ = \ \ \sqrt{\frac{1}{8}\sqrt{57} - \frac{3}{8}}[/tex]

The relation between the distance ratio and the fine
structure constant is:

[tex] \frac{d_{crad}}{d_{spin}} \ \ = \ \ \frac{1}{2}\alpha / v_{spin\frac{1}{2}} [/tex]

So that we can write exact to within 0.0000073%

[tex]\alpha \ \ = \ \ 2 \ v_{spin\frac{1}{2}} \ \ \frac{m_e}{m_\mu} \ (1 \ + \ \frac{1}{\frac{1}{2}(1+\frac{1}{2})} \ \frac{m_e}{m_\tau} \ ) \ \ [/tex]


The following CoData 2002 values were used so you
can test it yourself. (Don’t forget to apply a factor of
[itex]\frac{1}{\sqrt{1-v^2/c^2}} [/itex] to the mass when calculating the angular
momentum)


fine structure const 0.07297352568 ____ 0.00000000024
Planck constant ____ 6.626 0693e-34 ___ 0.0000011e-34 J s
speed of light _____ 299792458 ________ (exact) m/s
electron mass ______ 9.1093826e-31 ____ 0.0000016e-31 kg
speed of light _____ 299792458 ________ (exact) m/s
electric constant __ 8.854187817e-12 __ (exact) F/m
electric charge ____ 1.60217653e-19 ___ 0.00000014e-19 C
clas_electron_rad __ 2.817940285e-15 __ 0.000000028e-15 m




These "classical distances" are loaded with theoretical problems.
The classical electron radius has been used to try to cut of
the electric field close to the electron in the hope to avoid
the infinity problem of the [itex]1/r^2[/itex] field: Doesn’t work.

The spin angular momentum is much to high to fit into any
guess of the electron size. The angular orbit here is much
larger then electron radius! The electric force is more then
40 times higher then the centrifugal force to allow such an
orbit, et-cetera.

However, as a side remark, I've been looking at not so very
different scenarios were it seems that it may be possible to
bring the spin back into classical EM field theory without the
use of any new physics.


Regards, Hans
 
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  • #37
arivero
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Hans de Vries said:
However, as a side remark, I've been looking at not so very
different scenarios were it seems that it may be possible to
bring the spin back into classical EM field theory without the
use of any new physics.
Well, for sure spin is a non-classical parameter. It is measured as integer or half-integer multiples of the Planck Constant. So If you have a classical theory with spin, you have a classical theory predicting Planck Constant. And my, that's really new physics.

I am giving a glance to Gonzalez-Martin estimates, as well as Wyler's. At different decades, they postulated the use of quotient of volumes as a mean to determine mass ratios. It does not seem as accurate as this thread work.
 
  • #38
arivero
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a bit more bibliography

1) Hans, the thing that worries me about your last try is the dissappearance of the logarithmic expressions. It seems to be thus radically different.

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
According copyright, private emailing of the paper, not massive, is allowed. So if someone wants to have a read of it, please tell me at my standard email address, arivero@... (where ...=unizar.es)
 
  • #39
Hans de Vries
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arivero said:
1) Hans, the thing that worries me about your last try is the dissappearance of the logarithmic expressions. It seems to be thus radically different.

We have another situation now. We relate the mass ratio's
with two "classical physical quantities" In fact these quantities
are about the two most elementary you can think of.

The 1st relates the rest mass to the electric field energy.

The 2nd relates the rest mass to the spin 1/2 ang. momentum.

It's remarkable that the ratio between these two equals the
electron/muon mass ratio to within 0.038%

It's even more remarkable that a simple correction term using
the other (electron/tau) mass ratio produces the right result
to within 0.0000073%! And all within experimental accuracy:


___________[tex]\frac{d_{crad}}{d_{spin}} \ \ = \ \ \frac{m_e}{m_\mu} \ (1 \ + \ \ \frac{4}{3} \ \frac{m_e}{m_\tau} \ )[/tex]


The only coefficient (4/3) is equal to the inverse square of
the spin 1/2 angular momentum value:

___________________[tex]\frac{1}{\frac{1}{2}(1+\frac{1}{2})} = \frac{4}{3}[/tex]

Further, it's remarkable (and good news) that the ratio is
directly proportional to the mass ratio's. Both quantities are
distances and distances are supposed to behave that way.

For instance the classical electron radius is 206.7682838 times
larger than the classical muon radius, exactly the same ratio as
their masses have (but reversed).

These classical radii have another long forgotten property:
If something would rotate around this radius with a frequency
corresponding to the rest mass frequency then the speed
comes out to be c times alpha. This is true for the electron,
muon and tau.

The classical radii can be found in the Codata list. The other
value however has no public status. One has to solve a few
equations to get it. I called it a "classical value" because
probably nobody with a background in QFT or just QM in general
would ever bother to calculate it!

It simply looks which orbit a particle, with a certain rest mass
and a rotation frequency corresponding with that rest mass,
needs to have in order to give it the spin angular momentum
of a spin 1/2 particle.


Regards, Hans
 
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  • #40
arivero
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Renormalization group running of the coupling constant seems to inspire some of the formulae for alpha. While the real coupling is sensitive to the particle content and it should break down at the GUT scale, these theories prefer to work with some naive exponential coupling from the Planck scale, where they impose a simple value of alpha at these scale, say 2pi or 1 or infinity. So Nottale formula,
[tex]
M_P/m_e = e^{\frac38 \alpha^{-1}}
[/tex]
has alpha going down from infinity to 1/137.41.

By aumenting the complexity of the formula, the "prediction" can be more precise. Recently an anonymous nicknamed "Quantoken" has suggested
[tex](M_P/m_e)^2=2 \pi \alpha^{-1} e^{\frac23 \alpha^{-1}} [/tex]
which gives alpha a value near 2 pi at the planck scale, and a "prediction" of 1/137.07

I don't what to do of these formulae. My current opinion, as I said above, is that they work because they are kind of approximations to the running constant formulae. I had some curiosity because the factors 1/3 and 3/8 in the exponents differ a 1/24, so I wondered if another formula with such factor could be proposed. But to compare both formulae is not useful here, because the exponentials are nearly parallel.


EDITED: Probably Nigel's http://nigelcook0.tripod.com/ [Broken] falls also in the family here described. The key is to notice that any reference to Newton constant is, at the same time, a reference to the Planck mass scale.
 
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  • #41
Hans de Vries
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Magnetic Anomalies and Mass Ratios


I found this numerical relation intriguing as well:
Taken into account that there are only a handful
of mass-ratios to play with.


[tex]1 \ \ + \ \ \frac{m_\mu}{m_Z} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \ \ 1.001158692 (27) [/tex]

[tex]1 \ \ + \ \ \frac{m_\mu}{m_Z} \ \ + \ \ \frac{m_e}{m_W} \ \ = \ \ 1.001165046 (30) [/tex]


We recognize the magnetic moments of the leptons:


1.001159652187 ____ Electron’s Magnetic Moment
1.001158692(27) ___ 1 + mμ/mZ

1.0011659160 ______ Muon’s Magnetic Moment
1.001165046(30) ___ 1 + mμ/mZ + me/mW


The small term me/mW doesn’t do too bad either in
bridging the difference between the electron's and
muon’s (normalized) magnetic moments:


0.000006263813 ____ Difference of Magnetic Moments
0.000006353(3) ____ me/mW

The weak contributions are very small in the normal
anomaly calculations. For the electron they are
simply neglected. For the muon they are smaller than
the hadronic vacuum polarization terms.

At this scale one would expect to see nothing but QED.


Regards, Hans


PS: The CODATA mass values used:

W mass = 80.425___(38) GeV
Z mass = 91.1876__(21) GeV
e mass = 0.51099892(4) MeV
μ mass = 105.658369(9) MeV
 
  • #42
arivero
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This seems to be a case for "add 1 effect", as I.J. Good, above quoted, explains. The approximation becomes more impressive by counting from 1.00XXX... that from 0.00XXX...

Still, it could be pretty standard physics. The corrections to the magnetic moment are no very far away from formulae of this kind, with some powers of the quotient of masses etc. I'll check my QED.


EDITED: a recent review of muon magnetic moment calculations:
http://xxx.unizar.es/abs/hep-ph/0411168
 
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  • #43
arivero
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Hans de Vries said:
The weak contributions are very small in the normal
anomaly calculations. For the electron they are
simply neglected. For the muon they are smaller than
the hadronic vacuum polarization terms.
At this scale one would expect to see nothing but QED.

Yep you are right here, we only expect some corrections depending on alpha. Perhaps it is coincidence, perhaps a fine tunning mechanism for a whole summation of diagrams.

By the way, while checking this I have got a clue of where Nottale could have got his 3/8 coefficient from. While he claims to relate it to angular prediction from GUT, a older apparition of 3/8 appears associated to the infrared-catastrophic term of the traditional calculation for the anomalous moment. For instance Sakurai' formula 4.462 contains
[tex]{\alpha \over 3 \pi}{q^2 \over m_e^2} (\ln {m_e\over \lambda_{IR}} -\frac38)[/tex]
So perhaps some argument about the control of the infrared catastrophe up to planck mass scale could be retorted somehow to justify Nottale's formula.

Obviusly this 3/8 term was very apparent in all the old books when doing this calculation. I had seen it also in formulae 10.3 of Aitchison, who in turn refers to Bjorken Drell sect 8.6 and pg 172--6 for the infrared problem.
 
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  • #44
Hans de Vries
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Surprices!

These classical distance ratio's, [itex]R_f[/itex] for fermions and [itex]R_b[/itex]
for bosons are realy full of surprices!! The whole list for
sofar:

1)

The ratio [itex]R_b/R_f[/itex] of the two is equal to the [itex]m_W/m_Z[/itex] ratio!
(to within 0.063% or sigma 1.2)


[tex]\cos \theta_W \ \ = \ \ \frac{m_W}{m_Z} \ \ = \ \ \frac{R_b}{R_f} \ \ = \ \ \frac{2 \beta_b/\alpha}{2 \beta_f/\alpha} \ \ = \ \ \frac{\beta_b}{\beta_f}[/tex]

where [itex]\beta_f, \beta_b[/itex] are the classical spin 1/2 and spin 1 velocites.
This would mean that the electro-weak mixing angle would be a
numerical constant! with a physical geometric origin rather than
an arbitrary symmetry breaking parameter.


2)

Rf is equal to the muon-electron mass ratio to within 0.038%

[tex]R_f^{-1} \ \ =\ \ \frac{m_e}{m_{\mu}}[/tex]

which improves to 0.0000073% if we add a correction term
involving the third lepton like this:

[tex]R_f^{-1} \ \ = \ \ \frac{m_e}{m_\mu} \ (1 \ + \ \ \frac{4}{3} \ \frac{m_e}{m_\tau} \ )[/tex]

1/206.6890501 first expression
1/206.7682987 second expression
1/207.7682838 mass ratio

3)

We can do something similar for the tau-muon mass ratio:

[tex]\beta_b R_f^{-1/2}}\ \ =\ \ \frac{m_{\mu}}{m_{\tau}}[/tex]

which is accurate to within 0.090%. This becomes 0.000040%
when we add the following correction term with the other lepton:

[tex]\beta_b R_f^{-1/2}}\ \ =\ \ \frac{m_{\mu}}{m_{\tau}} \ (1 \ + \ \ \frac{3}{16} \ \frac{m_e}{m_\mu} \ )[/tex]

1/16.80305 first expression
1/16.81829 second expression
1/16.8183 mass ratio

----
----

We did see that the clasical velocity can be defined as:

“The velocity of a mass with spin s rotating on an orbit
with a frequency corresponding to its rest mass and an
angular momentum [itex]\sqrt{ s(s+1}\ \hbar[/itex]"


One gets a general solution for the 'classical velocity' of spin s:


[tex]\beta_s \ \ \ \ \ \ = \ \ \ \ \sqrt{\sqrt{\ \ s(s+1) \ \ + \ \ ( \frac{1}{2}
\ s(s+1) \ )^2 } \ \ - \ \ \frac{1}{2} \ s(s+1) } [/tex]

Which solutions are dimensionless constants, independent
of the mass of the particle. The values of the common spins
are given below:


spin 0.0: __ 0.00000000000000000000000000000000
spin 0.5: __ 0.75414143528176709788873548859945
spin 1.0: __ 0.85559967716735219296923576621118
spin 1.5: __ 0.90580479773844104117525862119228
spin 2.0: __ 0.93433577808377694874713811004304
spin inf: __ 1.00000000000000000000000000000000




Weinberg’s Electro-Weak mixing angle becomes a dimension-
less constant as well and is given in the [itex]\sin^2 \theta_W[/itex] form as:


[tex] \sin^2 \theta_W
\ \ \ \ = \ \ \ \ 1 \ - \ \frac{\beta^2_f }{\beta^2_b}
\ \ \ \ = \ \ \ \ 0.22310132230086634541466926662604[/tex]

[tex] \sin^2 \theta_W
\ \ \ \ = \ \ \ \ 1 \ - \ \frac
{ \sqrt{\ \ \frac{1}{2}(\frac{1}{2}+1) \ \ + \ \ ( \frac{1}{2} \ \
\frac{1}{2}(\frac{1}{2}+1) \ )^2 } \ \ - \ \ \frac{1}{2} \
\ \frac{1}{2}(\frac{1}{2}+1) }
{\sqrt{\ \ 1(1+1) \ \ + \ \ ( \frac{1}{2} \ \
1(1+1) \ )^2 } \ \ - \ \ \frac{1}{2} \
\ 1(1+1) } [/tex]

The usual electro-weak parameters g1 and g2 would become:

[tex]g_1^2 \ \ = \ \ e^2 \frac{\beta_b^2}{\beta_f^2} \ \ \ \ \ \ \ \ \ g_2^2 \ \ = \ \ \frac{e^2}{1-\frac{\beta_f^2}{\beta_b^2}}[/tex]

where [itex]e^2 = \alpha[/itex]


Regards, Hans
 
Last edited:
  • #45
Hans de Vries
Science Advisor
Gold Member
1,091
26
Hans de Vries said:
1)

The ratio [itex]R_b/R_f[/itex] of the two is equal to the [itex]m_W/m_Z[/itex] ratio!
(to within 0.063% or sigma 1.2)


[tex]\cos \theta_W \ \ = \ \ \frac{m_W}{m_Z} \ \ = \ \ \frac{R_b}{R_f} \ \ = \ \ \frac{2 \beta_b/\alpha}{2 \beta_f/\alpha} \ \ = \ \ \frac{\beta_b}{\beta_f}[/tex]

where [itex]\beta_f, \beta_b[/itex] are the classical spin 1/2 and spin 1 velocites.
This would mean that the electro-weak mixing angle would be a
numerical constant! with a physical geometric origin rather than
an arbitrary symmetry breaking parameter.

Now there is an interesting story behind this:

I was looking at this peculiar coincidence:

0.00115869 = muon / Z mass ratio
0.00115965 = electron magnetic anomaly
0.00000635 = electron / W mass ratio
0.00000626 = difference between muon and electron magnetic anomaly

Now whatever, what we can say is that the magnetic
anomaly is totally dominated by photon (spin 1) interactions
coming from the first order [itex]\alpha/2\pi[/tex] term while the difference
of the muon and electron anomaly is almost entirely vacuum
polarization interaction (spin 1/2), the result of virtual electrons
and muons. This causes an asymmetry that results in a different
anomaly for the muon and the electron which otherwise would
be the same.

when I saw this I immediately tried if Rf/Rb had any relation
with mW/mZ. To my big surprise it turned out that they were
equal!

Regards, Hans
 
  • #46
arivero
Gold Member
3,362
100
Hans de Vries said:
0.00115869 = muon / Z mass ratio
0.00115965 = electron magnetic anomaly
0.00000635 = electron / W mass ratio
0.00000626 = difference between muon and electron magnetic anomaly

Now whatever, what we can say is that the magnetic
anomaly is totally dominated by photon (spin 1) interactions
coming from the first order [itex]\alpha/2\pi[/tex] term while the difference
of the muon and electron anomaly is almost entirely vacuum
polarization interaction (spin 1/2), the result of virtual electrons
and muons.

I see. In the first case we are having neutral particles, and the Z ratio gets a role. In the second case we have a contribution from charged particles, and the W ratio gets a role. And Z is a neutral particle, and W is a charged particle. So at least we have a logical link.
 
  • #47
arivero
Gold Member
3,362
100
Some remarks from the historic literature: According H. Kragh (above quoted article), Sommerfeld definition of alpha is not very far from the quotient you are using for the rest of the electroweak thing. To be precise, the fine structure constant is defined as the quotient between wave K angular momentum, h/2pi and "limiting momentum", e^2/c. Or equivalently (H. Kragh, "the fine structure constant before quantum mechanics"), between the orbital velocity in wave K and the maximum velocity c.

(Let me recall here that the eigenvalues of electron velocity are +-c only).

Related to this we have the so-called "Sommerfeld puzzle". In the early time, before Dirac, he applied his quantization procedure to an orbiting electron plus relativistic mass correction, and well, he got the right formula for the fine structure of hidrogen. Without any reference to spin! When Dirac equation was known, time after, its only real news were to justify an arbitrary selection rule for level intensity.

Hans, I can not read Sommerfeld german, but I guess you can, so I'd suggest you to try to find his relativistic calculation. It is around 1915-1916, a couple articles in Annalen der Physik and Akad. der Wiss Muenchen Sitzungsberichte.
 
Last edited:
  • #48
259
0
arivero said:
Related to this we have the so-called "Sommerfeld puzzle". In the early time, before Dirac, he applied his quantization procedure to an orbiting electron plus relativistic mass correction, and well, he got the right formula for the fine structure of hidrogen. Without any reference to spin! When Dirac equation was known, time after, its only real news were to justify an arbitrary selection rule for level intensity
He may have got the right correction, but he did not get the total number of states right - he only got half because he didn't have spin. The Sommerfeld calculation was one of many false steps in the early days of QM, unlike the Dirac equation which actually made for progress.
 
  • #49
Nereid
Staff Emeritus
Science Advisor
Gold Member
3,392
3
This is all fascinating stuff PFers!

Are you any closer to making some specific, concrete predictions? Testable ones would be nice.
 
  • #50
arivero
Gold Member
3,362
100
zefram_c said:
He may have got the right correction, but he did not get the total number of states right - he only got half because he didn't have spin. The Sommerfeld calculation was one of many false steps in the early days of QM, unlike the Dirac equation which actually made for progress.
Actually, if my lecture notes are right, he got not half but double, and then he was forced to impose a selection rule to have it right. The selection rule being a change in one h unit of angular momentum, it is now clear, as you say, that it was related to a spin flip.

But again, please count that I can not read German so I am looking at third hand -or more. lecture notes.
 

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