All the lepton masses from G, pi, e

[Hans de Vries]

Dear friends over at Physics Forums:

I have for more than two years been researching the possibility that
baryons may in fact be non-Abelian magnetic sources.

The result of this research are now formally and rigorously presented in
a paper at:

http://home.nycap.rr.com/jry/Papers/Baryon%20Paper.pdf

Thank you for presenting your work over here Jay. It's becoming more
interesting with each new paper.

I think we do share the viewpoint that much of the interesting and
important physics is the result of the special terms, like the spin term in
the Dirac equation for the fermions, and the extra non-Abelian term for
the Bosons. My feeling is that the universe would be just a dull homo-
geneous "soup" without these two terms.


Regards, Hans.

 

[Hans de Vries]

I never did expect numerical coincidences with hadron masses but
the pion shows a second one, again magnetic anomaly related.
Half the mass ratio of the (charged) two quark pion and the electron
is roughly equal to alpha. Now if we divide this value by 2pi then we
find a coincidence with the Muon's magnetic anomaly:

$$\mbox{\Huge \frac{1}{\pi} \frac{m_e}{\ \ m_{\pi\pm}}\ =\ 0.001165407 (3) }$$

We already had an Electron magnetic anomaly coincidence:

$$\mbox{\Huge \left(\ \frac{m_{\pi^\pm}}{m_{\pi^0}} - 1\ \right)^2\ =\ 0.00115821 (26) }$$

This adds to the list of Magnetic Anomaly related numerical coincidences:


0.001165920________ Muon Magnetic Analomy
0.001165407__(3)___ Electron / Pion mass ratio /pi.
0.001165892________ Electrom / Pion mass ratio +Vpi-Vmu

0.001159652________ Electron Magnetic Analomy
0.001159567________ Mass independent Magnetic Analomy
0.001158692__(27)__ Muon / Z boson mass ratio.
0.00115821___(26)__ Pion mass delta square.

0.0000063532_______ Muon Mag.Anomaly mass dependence
0.0000063558_(20)__ Electron / W mass ratio (2007)


The third value (+Vpi-Vmu) is closer. What we have done here
is to add the one-loop QED vacuum polarization difference for
a particle of muon mass and a particle of pion mass. A lepton
with the mass of a pion should have a higher magnetic anomaly
as the muon by this amount plus another 4-8% or so for higher
order VP loops which I can't calculate.

The "Mass independent Magnetic Anomaly" is the same for
all leptons. It is calculated here by subtracting the one loop VP
contribution from the electrons magnetic anomaly.

The third numerical coincidence was found earlier and relates
the electron / W mass ratio with the mass dependent part
of the Muon's magnetic anomaly. It was calculated by taking
the difference between the Muon and Electrons magnetic
anomalies and adding the Electrons largest (one-loop) VP term.Regards, Hans

 One loop vacuum polarization:

 m1/m2 = 273.132044975751,   one-loop vpol = 0.000006389950   (pion +/-)
 m1/m2 = 206.768283800000,   one-loop vpol = 0.000005904060   (muon)
 m1/m2 = 1.00000000000000,   one-loop vpol = 0.000000084641   (electron)

 

[Jay R. Yablon]

Thank you for presenting your work over here Jay. It's becoming more
interesting with each new paper.

I think we do share the viewpoint that much of the interesting and
important physics is the result of the special terms, like the spin term in
the Dirac equation for the fermions, and the extra non-Abelian term for
the Bosons. My feeling is that the universe would be just a dull homo-
geneous "soup" without these two terms.

Regards, Hans.

Hi Hans, thank you for your encouragement.

As you know I mentioned over on SPF, the most important results in the baryon paper have to do with confinment. I have extracted and consolidated these results and posted them at the link below.

http://home.nycap.rr.com/jry/Papers/Confinement%20Paper.pdf

This paper may provide an exact analytical solution to the problem of QCD confinement and the existence of short-range meson mediators of the nuclear interactions.

As you noted over on SPF, this paper evolves from the non-vanshing three form P = dF which would be the Yang-Mills (non-abelian) "magnetic current."

If I were to put the paper in one sentence it would be:

Baryons are non-Abelian (Yang-Mills) magnetic monopoles.

And if permitted a second sentence:

These Yang Mills monopoles exhibit all the properties of a baryon, including three fermions which via exclusion we connect with quarks, confinement of these quarks and their mediating gluons such that there is never any net flux of gluons or individual quarks across any closed two-dimensional surface of integration through the baryon current density, and emission of mesons which have short range such that these mesons, which account for interactions between baryons, are the only entities for which there is a net flux through the integration surface.

I appreciate any comments which the Physics Forums readers may have.

Jay.
_____________________________
Jay R. Yablon
Email: jyablon@nycap.rr.com
Web site: http://home.nycap.rr.com/jry/FermionMass.htm
sci.physics.foundations co-moderator

 

[arivero]

0.001165407__(3)___ Electron / Pion mass ratio /pi.

Well, that is rare, as every you find ! I would expect the quotient between charged and neutral pion to be related somehow to the fine structure constant, and then this natural connection would propagate to the rest of findings in the thread. But again, too much exactitude.

In a related thems, did I tell that I was interested on the history of the calculation of g-2? It seems that a guy, who helped Kino****a to set up the software, is now in the dark side of accidental numeric coincidences.

I have been for the last four months working in a BOINC supercomputing platform, Zivis. I wonder how much time could take to repeat K. et al calculations; I guess that programming time should be higher than computing time.

 

[Jay R. Yablon]

The so-called Yang- Mills "mass gap"

Dear Physics Forum friends:

I have posted an updated draft of my confinement paper to:

http://home.nycap.rr.com/jry/Papers/Confinement%20Paper.pdf

Last evening, I uncovered and added to the paper, a line of calculation
which explicitly and formally gives a non-zero rest mass to the meson
mediators of QCD without resort to the Higgs Mechanism, naturally
eliminates the propagator poles and therefore allows one to have "real"
rather than "virtual" particles without ad hoc tricks, formally renders
QCD a short-range interaction, and solves the Yang-Mills mass gap
problem. This is in addition to the confinement solution which has
already been posted for several days now.

Interested in feedback.

Jay.
_____________________________
Jay R. Yablon
Email: jyablon@nycap.rr.com
Web site: http://home.nycap.rr.com/jry/FermionMass.htm
sci.physics.foundations co-moderator

 

[arivero]

Gerald Rosen

I believe we have not included G. Rosen in the references for the topic of this thread.

His http://prola.aps.org/abstract/PRD/v4/i2/p275_1 , for instance, gives one of these formulae with exponential relationship between Planck and electron masses. Other papers look for masses of electron and quark, value of the fine structure constant, etc. http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=find+a+rosen%2C+g&FORMAT=WWW&SEQUENCE=

 

[arivero]

Then we use http://wwwusr.obspm.fr/~nottale/ukmachar.htm to get the mass of the electron

$$\ln (M_P/m_e) = \alpha^{-1} \sin^2 \theta_W$$

Note that this remark is mostly an elaborate rewritting of the argument about the self energy of the electron. For instance in a modern text such as Polchiski's string textbook, it appears as
$$\delta m \approx \alpha m \ln {1 \over m l}$$
without any reference except "known since the 1930s", and then it is explained that $l$ should to be expected to ultimate cutoff, thus the length of Planck, and that $\delta m \over m$ should be expected to be of order O(1).

Thus the formula can be rewritten as
$$\alpha^{-1} {\delta m_e \over m_e} \approx \ln {m_P \over m_e}$$

And Nottale's statement translates, in standard knowledge, as telling that the electromagnetic contribution to the mass of the electron is about 3/8.

While most of the "rediscoverers" of this relationship are outsiders, "out of the loop" people, in the case of Nottale it is sort of astonishing that he forgets to tell that he is recovering a "known" relationship and not a new one (he does not have, at all, new arguments for the 3/8 adjustment neither).

Another remark nobody does is that the argument can be reverted to predict the GUT-Planck-String scale from the low energy scale. Thus one could to try to predict muon or electron masses from the electroweak scale, then use the cutoff argument to predict the GUT scale, and the to use the seesaw to predict the neutrino scale.

EDITED: Nottale presentation can be found in http://luth2.obspm.fr/~luthier/nottale/ukmachar.htm Also, it is argued in http://luth2.obspm.fr/~luthier/nottale/arDNB.pdf that better results are met if one uses in a very peculiar way the fine structure constant running value instead of the infrared one. Indeed using alpha at m_e, one really "predicts" (except the justification of the 3/8) the value of Newton Constant. In any case, I am ashamed 1) about our inability to note that it was just the renormalisation of electron mass as usual, and 2) that Nottale hides or ignores this fact, after being in contact with QED theory during decades.

 

[arivero]

For numerical "coincidences", hep-ex/0412028 remembers that "the infamous Florida 2000 presidential election recount with the official result of 2,913,321 Republican vs. 2,913,144 Democratic votes, with the ratio equal to 1.000061".

 

[Hans de Vries]

Once in a while one runs into a nice numerical coincident like this one
relating alpha to a Yukawa potential:

$$\mbox{\Huge \frac{\alpha}{2\pi}\ =\ \frac{e^{-2\pi r}}{2\pi r}}$$

Where $r$ is the quotient of two radii:

$$r\ =\ \frac{r_c}{r_l}$$

in which rc is the Compton radius of the electron, muon or tau lepton and
rl is the corresponding cut-off radius for which the electrostatic energy
is equal to the magnetostatic energy of the classical electron, muon or
tau lepton. It's the magnetic moment which is responsible for the magneto-
static energy. One gets the following relations:

$$\alpha =1/137.05268 -\ \quad \mbox{for the electron}$$
$$\alpha =1/137.04743 -\ \quad \mbox{for the muon}$$
$$\alpha =1/137.03796 -\ \quad \mbox{for the tau lepton}$$
$$\alpha =1/137.03599971\ \quad \mbox{experimental}$$

The differences stem from the differences of the magnetic moment analomies. Regards, Hans==================================================

PS: For the EM fields the following expressions were used:

$$\textsf{E}_x\ = \frac{q}{4\pi\epsilon_o}\ \frac{x}{r^3}, \quad \textsf{E}_y\ = \frac{q}{4\pi\epsilon_o}\ \frac{y}{r^3}, \quad \textsf{E}_z\ = \frac{q}{4\pi\epsilon_o}\ \frac{z}{r^3}$$

$$\textsf{B}_x\ = \frac{\mu_o\mu_e}{2\pi}\left(\frac{3z}{r^5}x \right), \quad \textsf{B}_y\ = \frac{\mu_o\mu_e}{2\pi}\left(\frac{3z}{r^5}y \right), \quad \textsf{B}_z\ = \frac{\mu_o\mu_e}{2\pi}\left(\frac{3z}{r^5}z-\frac{1}{r^3} \right)$$

And for the total energy:$$\mbox{Energy:} \qquad E\ =\ \frac{1}{2}\left(\epsilon_o\textsf{E}^2 + \frac{1}{\mu_o}\textsf{B}^2\right)$$

$$E\ =\ \int \left\{ \frac{q^2}{32\pi^2\epsilon_o r^4}\ +\ \frac{\mu_o\mu_e^2}{8\pi^2}\left(\frac{3z^2}{r^8}+\frac{1}{r^6} \right) \right\} dx^3\ =\ \frac{q^2}{8\pi\epsilon_o r_o}\ + \ \frac{\mu_o\mu_e^2}{4\pi}\ \frac{3}{r_o^3}$$

For an electron:

$$\frac{q^2}{8\pi\epsilon_o} = 1.1535385\ 10^{-28},\quad \frac{3\mu_o\mu_e^2}{4\pi}\ =\ 2.5862051\ 10^{-53}$$

and ro is the cut-off radius.

 

[arivero]

This last comment reminders me about Born calculation of the Lamb shift,
in the train cominb back from Shelter Island. He had no serious argument to fix the cut-off in the renormalized equation, so he simply choose the compton length of the electron because it is the cutoff of particle creation.

 

[Walter L. Wagne]

Evidence for a moving Magnetic Monopole in 1975

See:

"Evidence for Detection of a Moving Magnetic Monopole", Price et al., Physical Review Letters, August 25, 1975, Volume 35, Number 8.

This was the last of a series of balloon flights, launched in 1973, but not analyzed by myself until 1975, due to higher priority cosmic rays analysis then ongoing.

The suggestion that the anomalous track could have been caused by a doubly fractionating normal nucleus is untenable. One would have expected to have seen billions of similar tracks, not quite as closely matched to the expected track of a magnetic monopole, first. No such similar events were ever detected.

For further information, contact the administrator who can email me, as I do not regularly post at this forum. Or check www.sciforums.com where I do regularly post, and PM me.

Whether the Large Hadron Collider [LHC] will create a magnetic monopole is highly debatable. It might also create miniature black holes, or strangelets.

 

[taarik]

Mass quantization in terms of pion and muon mass difference

Please see my paper about mass quantization @ arxiv. hep-ph/0702140

 

[arivero]

Please see my paper about mass quantization @ arxiv. hep-ph/0702140

Indeed your paper, your reference list
http://www.slac.stanford.edu/spires/find/hep/wwwrefs?key=7074930
and the list of citations of McGregor
http://www.slac.stanford.edu/spires/find/hep/www?c=NUCIA,A58,159
are interesting for the topics of this thread. A problem of quantisation of M instead of quantisation of M^2 is that it has some scent of classical group theory, thus one needs to see how many of the relationships are already explained in the quark model and check the extant cases.

 

[taarik]

Mass quantization

Thiis paper has been accepted for publication in Modern Physics Letters A.
The important thing is that while the charged pions and muon are related in sense via the decay of former into latter, the neutral pion and muon are not related in any sense. Yet there mass difference serves as a basic mass unit for both leptons and hadrons.

 

[arivero]

Thiis paper has been accepted for publication in Modern Physics Letters A.
The important thing is that while the charged pions and muon are related in sense via the decay of former into latter, the neutral pion and muon are not related in any sense. Yet there mass difference serves as a basic mass unit for both leptons and hadrons.

very good news.

the question, to me, is not why muon and pion have different mass, but why have they got almost the same mass. A conjecture is SUSY.

 

[taarik]

Here is another surprise
the t lepton mass can be obtained by taking 57 jumps of 29.318 MeV from the muon mass i.e t mass= 57x 29.318. Now this 57 number also helps us to include the electron mass as 57 times electron mass= 29.127 very close to 29.318. This leads us to thing that like in Nambu's and many other cases the basic unit appears from the electron mass.
Now this also means the pion -muon= 57x electron , tau - muon =57x 29.318 =57x57x electron. which in turn leads to tau- pion =56x 29.318 =56 x 57 x electron.
Hence the lightest hadron i.e pion and lightest unstable lepton i.e muon , two leptons muon and tau , lightest hadron and heaviest lepton i.e tau are all related through electron mass.

 

[arivero]

Now this also means the pion -muon= 57x electron , tau - muon =57x 29.318 =57x57x electron. which in turn leads to tau- pion =56x 29.318 =56 x 57 x electron..

I prefer to write then

$${ m_\pi - m_\mu \over m_e}= \sqrt {m_\tau - m_\mu \over m_e}$$

It should be nice to have a mathematical (group theoretical) argument for 57.

EDIT: It is a bit puzzling that if we fix the mass of tau, mu and electron to the experimental values, the above formula "predicts" 134.88 MeV, to be compared with the mass of the neutral pion (134.976 MeV). Naively one could expect the result to be more related with the mass of the charged pion, which is 4.6 MeV above.

EDIT2. Perhaps Krolikowski has some argument for 58/2. Also, Ramanna (eg pg 16 of nucl-th/9706063)

 

[arivero]

I prefer to write then...

Hmm, funnier:

$$( m_{\pi_0} - m_\mu )= \sqrt {m_e} \sqrt {m_\tau - m_\mu}$$LHS and RHS still agree within a 0.3 %. No bad.

The above comments still apply. On other hand, if I recall correctly, the question about why the mass of the charged pion is higher, and not lower, than the neutral one was a touchy issue decades ago, and it required very high level theoretists to explain it.

 

[taarik]

Refering to first version, it ia again amazing that the mass of the neutral pion is deteremined very precisely in terms of the three leptons. Now neutral pion has no relation with the three leptons. it does not decay into any of these particles. On the other hand the charged pion decays into electron and muon. Hence charged pion mass should have been related to lepton masses.

 

[arivero]

Refering to first version, it ia again amazing that the mass of the neutral pion is deteremined very precisely in terms of the three leptons. Now neutral pion has no relation with the three leptons. it does not decay into any of these particles. On the other hand the charged pion decays into electron and muon. Hence charged pion mass should have been related to lepton masses.

I agree, it is misterious. Furthermore, forgetting the issue of integer multiples and the squaring of masses, the formula is very reminiscent of charged pion decay, you know, these $\prop m_\mu^2 ( m_\pi_+^2 - m_\mu^2)$ from textbooks.

To put more intrigue, the mass difference between eta and the average of pion and muon (say, diff=427.2 MeV) also fits roughly in the obvious permuted formulae:
$$\sqrt {m_\mu} \sqrt {m_\tau - m_e} \approx \sqrt {m_\tau} \sqrt {m_\mu - m_e} \approx \sqrt {m_\tau \pm m_e} \sqrt {m_\mu \pm m_e} \approx \sqrt {m_\mu} \sqrt {m_\tau} = 433.27 MeV$$

EDITED: a purpose of the above formulas is to consider the limit $m_\mu \approx m_\tau$ where the former formula cancels and the two first ones in the above become the same. Also, the same cancellation and similarity happens in the other limit $m_e \to 0$. Simultaneous limit conflicts with Koide's.

 

[arivero]

My thinking during the last two years was:
initially there is a symmetry where neutrinos have the same mass than neutral mesons and charged leptons have the same mass than charged mesons. Note the count of degrees of freedom. Of course one could also expect the dirac mass of neutrino and charged lepton to coincide.
Then seesaw moves the mass of neutrinos out of reach
and mixing, including CKM, and/or other unknown mechanism alter the mass eigenvalues of the mesons.
The mechanism could be related to a mismatch between isospin in mesons and leptons. Namely, third generation mesons do not exist except bB.

 

[arivero]

To put more intrigue, the mass difference between eta and the average of pion and muon (say, diff=427.2 MeV) also fits roughly in the obvious permuted formulae:
$$\sqrt {m_\mu} \sqrt {m_\tau - m_e} \approx \sqrt {m_\tau} \sqrt {m_\mu - m_e} \approx \sqrt {m_\tau \pm m_e} \sqrt {m_\mu \pm m_e} \approx \sqrt {m_\mu} \sqrt {m_\tau} = 433.27 MeV$$

Hmm, Gell-Mann Okubo value for unmixed $\eta_8$ is
569.32 GeV, so $$\eta_8 - \pi^0 = 434.34 MeV$$

 

[arivero]

Hmm, Gell-Mann Okubo value for unmixed $\eta_8$ is
569.32 GeV, so $$\eta_8 - \pi^0 = 434.34 MeV$$

For amateurs, it could be worthwhile to explain what the Gell-Mann Okubo is, or refer to a textbook. I like Donoghue Golowich Holstein, "Dynamics of the Standard Model". The formula appears in chapter VII, expression (1.6b). You get the formula from the following set of equations:
$$m^2_\pi=B_0 (m_u + m_d)$$
$$m^2_{K^0}=B_0 (m_s + m_d)$$
$$m^2_{K^\pm}=B_0 (m_s + m_u)$$
$$m^2_{\eta_8}=\frac 13 B_0 (4 m_s + m_u + m_d )$$
and so on.

Asuming isospin, up and down have the same mass, and thus you can get a combination of neutral kaon, pion and eta8.

If works well with the neutral particles; it is not only that it does not account for isospin; the idea does not account for EM interactions neither. Old timers extract an extra EM relation via "Dashen's theorem", but I think to remember there was some work of Witten or some other genious about this kind of corrections.

EDITED: Indeed we could use the above expressions to reformulate our equations in terms of the mass $m_s$ and $\hat m \equiv m_u = m_d$, with SU(3) flavour breaking to global SU(2) isospin x U(1) as it happened in the papers of 1960s on global symmetries.

$$m^2_\pi = (m_\mu + \sqrt { m_e (m_\tau-m_\mu)})^2 = B_0 \hat m$$
$$m^2_{\eta_8} = (m_\pi + \sqrt { m_\mu (m_\tau-m_e)})^2 = \frac 23 B_0 (2 m_s + \hat m)$$
Here you can see also one of the themes which were debatable in the sixties: the use of mass square instead of plain mass. For instance, it is because of it that our resulting equations
do not allow to cancel $B_0$ out.

 

[arivero]

Here is another surprise
the t lepton mass can be obtained by taking 57 jumps of 29.318 MeV from the muon mass i.e t mass= 57x 29.318. Now this 57 number...

http://adsabs.harvard.edu/abs/1962RSPSA.268...57D

Dirac gets 53 times the electron mass for the muon, in a paper that has been lately recalled by the guys working on strings and branes.

 

[arivero]

A pretty amazing coincident isn't it? The relation:

$$\mbox{\Huge \frac{m_{\pi^\pm}}{m_{\pi^0}}\ =\ 1+ \left(\frac{m_\mu}{m_Z}\right)^\frac{1}{2}\ =\ 1.0340344(55)}$$

following from the previous post is as exact as:

1 : 1.0000067 (42)

About this one, it can have interesting implications: if the mass of the muon goes to zero, assume so it happens with the masses of up and down, then (global) isospin symmetry is restored. On the contrary, if Z goes to cero (and W) but the quark masses are different, the restored symmetry is only gauge G-W-S isospin and the charged pion decays into the neutral one. What is amazing in this argument is that if Z goes to infinity the two pions can not tunnel one into another, but from the point of view of the electroweak scale the masses of quarks are negligible, thus global isospin is restored again.

The traditional current algebra formula for the pion mass(^2) difference puts it in terms of the fine structure constant and the pion decay constant, $e^2 / F_\pi^2$ times some other factors.

Entering the octet, we are touching deep problems of the elders. There is a short work of witten http://www.slac.stanford.edu/spires/find/hep/www?j=PRLTA,51,2351 about how the mass of the charged pion must always be higher than the neutral pion, even if only to avoid tachions in the limit of zero pion mass. Also, the mixing between [itex]\eta_8[\itex] and [itex]\eta_0[\itex] to give [itex]\eta[\itex] and [itex]\eta'[\itex] was the U(1) headache, addressed by t'Hoft, http://www.slac.stanford.edu/spires/find/hep/www?j=NUPHA,B159,213 and http://www.slac.stanford.edu/spires/find/hep/www?j=NUPHA,B156,269 independently, and according Okubo still unclear. I have found even some recent work in the context of strings: http://www.slac.stanford.edu/spires/find/hep/wwwrefs?key=5864437

 

[arivero]

http://arxiv.org/abs/0710.2429 (g-2)_mu status and prospects

 

[Count Iblis]

You could try to use the LLL algorithm to find formulae.

 

[arivero]

You could try to use the LLL algorithm to find formulae.

It is an interesting idea. Actually, the team of http://crd.lbl.gov/~dhbailey/dhbpapers/ have tried in the past to input the standard model masses etc in some of their algorithms; but I had no idea about their new work using LLL.

 

[Count Iblis]

Actually, the LLL algorithm is a bit older than Bailey's PSLQ algorithm. The LLL is a little less efficient, but that's only a problem if you have thousands of significant digits of some number and try to find a formula in terms of known constants for it.

For your work, the LLL may be better, because with the LLL you can look for simultaneous relations, the PSLQ can't do that (at least not in general, you can work with complex numbers, quaternions,... but you'll reach a limit beyond which you can't go). If you have 5 numbers that are known to ten digits then you have 50 digits of information. Ten digits may not be enough to detect a relation, but 50 may be enough provided, of course, that the five numbers are given by formulae of the same form that are specified by the same constants.

See the Appendix of this article for details.

 

[arivero]

The significance of numerical coincidences in nature
Authors: Brandon Carter

http://arxiv.org/abs/0710.3543

 

[jal]

I found that your link gave a good and understandable explanation.
However,
p. 9
...They can be categorised as three coupling constants,
and three mass ratios, and their empirically determined numerical
values are approximately:12
gS .=4, e.=1/12, mN.=1/2× 10−10
The values of the coupling constants are rather more familiar in their
squared forms: thus we have the gravitational fine structure constant
… the ordinary (electromagnetic) fine structure constant, e2 .= 1/137, and …
----------
I need explanation with e.=1/12 and , e2 .= 1/137
I always thought that 12X12=144 not 137
jal

 

[arivero]

Well, I guess he takes the nearest simple fraction to sqrt(137), and he chooses 1/12. So it really compares 12 against 11.7, arguably not so bad.

A peculiarity of this article is that a couple ways ago I have been told that the relationship between mass of electron and pion (!) had circulated as a conjecture in the sixties. I though it came from McGregor, but it is formula (6) in pg 11 of this paper.

 

[jal]

"So it really compares 12 against 11.7, arguably not so bad."
If you find anything else that can help I'll put it in "How to build a universe"
jal

 

[CarlB]

I've had some interesting results in rewriting the Koide equation as a sort of "field energy" equation. The idea is to treat the square in the mass as coming from the energy of a field.

Field energies are quadratic, for instance, E&M field gives mass as m = (E^2 + B^2)/c^2, where I've left off some units. So begin with electromagnetism as a toy example.

Then the thing to notice is that E and B end up quantized at different amplitudes. Magnetic monopoles are much heavier than electrons, so assume that when you quantize E and B, the contribution of B dominates, giving you m = B^2/c^2.

From there, you assume that the angle I've called "delta" is 2/9exactly, and that the reason this doesn't exactly fit the electron, muon and tau masses is cause "E" does contribute slightly.

That converts the Koide formula from being a two parameter fit, with mu and delta, (the mass scale and the angle), to being a two parameter fit with a B scale and an E scale. To write the masses we have

$$m_n = |B|^2(1 + \sqrt{2}\cos(2/9+2n\pi/3))^2 + |E|^2(\sqrt{2}\sin(2/9+2n\pi/3))^2$$

where B and E are constants. The contribution to B is split into two parts, $$1 + \sqrt{2}\cos(2/9+2n\pi/3)$$, so we write $$B_v = B$$, $$B_s = \sqrt{2}\cos(2/9+2n\pi/3)$$, and $$E_s=\sqrt{2}\sin(2/9+2n\pi/3)$$. That is, the "v" field is a valence field that is shared between the electron, muon, and tau, and the "s" field is a sea field that distinguishes the three generations.

Then the mass equation is $$m = (B_v + B_s)^2 + E_s^2$$.

What's more interesting is that if you write down the vectors $$(B_v,B_s,E_s)$$ for the electron, muon, and tau, you get the tribimaximal mixing matrix (after scaling the B stuff and E stuff so that each vector has length 1).

Another way of saying this is that the vectors $$(B_v,B_s,E_s)$$ are orthogonal. Making them orthonormal defines the tribimaximal neutrino mixing matrix. (Except that when you see it in the literature, it is usually has two columns reversed so you should put the three contributions in the order $$(B_s,B_v,E_s)$$ instead.)

Using the best PDG numbers for the electron and muon masses to predict the tau mass, the equations for the charged lepton masses are (ignore the precision, I haven't had time to compute the ranges and fix everything up yet):

$$\begin{array}{rcl} m_n &=& 313.8561002547\;\textrm{MeV}\;(1 + \sqrt{2}\cos(2/9 + 2n\pi/3))^2\\ &&+4.6929703\;\textrm{eV}\; (\sqrt{2}\sin(2/9+2n\pi/3))^2 \end{array}$$

And the three vectors (which ignore the phase angle 2/9 because it is presumably canceled in the neutrinos) are:
$$\begin{array}{ccc} (1,& \sqrt{2},& 0)\\ (1,& -\sqrt{2}/2,& +\sqrt{3/2})\\ (1,& -\sqrt{2}/2,& -\sqrt{3/2}) \end{array}$$

In the above, note that the angle 2/9 has been removed as it is presumably canceled in the neutrinos, which also use a similar angle. And the scaling to B and E has been removed because in computing phases, one needs to normalize by particle number rather than energy.

After dividing by the lengths of the vectors, sqrt(3), and turning the three vectors into a matrix, one has:
$$\left(\begin{array}{ccc} \sqrt{1/3},& \sqrt{2/3},& 0\\ \sqrt{1/3},& -\sqrt{1/6},& +\sqrt{1/2}\\ \sqrt{1/3},& -\sqrt{1/6},& -\sqrt{1/2} \end{array}\right)$$


Carl

Koide paper giving Tribimaximal mixing matrix, see eqn (3.2):
http://arxiv.org/abs/hep-ph/0605074

 

[Hans de Vries]

After dividing by the lengths of the vectors, sqrt(3), and turning the three vectors into a matrix, one has:
$$\left(\begin{array}{ccc} \sqrt{1/3},& \sqrt{2/3},& 0\\ \sqrt{1/3},& -\sqrt{1/6},& +\sqrt{1/2}\\ \sqrt{1/3},& -\sqrt{1/6},& -\sqrt{1/2} \end{array}\right)$$

Carl

Koide paper giving Tribimaximal mixing matrix, see eqn (3.2):
http://arxiv.org/abs/hep-ph/0605074

Interesting, I noticed that one can write the above tribimaximal
matrix for neutrino mixing as the x,y,z coordinates of a tetrahedron
with sides of $\sqrt{1/2}$ with its top down and the origin in h/2,
thus:

$$\left(\begin{array}{ccc} z_1,& y_1,& x_1\\ z_2,& y_2,& x_2\\ z_3,& y_3,& x_3 \end{array}\right)\ =\ \left(\begin{array}{ccc} \sqrt{1/3},& \sqrt{2/3},& 0\\ \sqrt{1/3},& -\sqrt{1/6},& +\sqrt{1/2}\\ \sqrt{1/3},& -\sqrt{1/6},& -\sqrt{1/2} \end{array}\right)$$

The angle of 2/9 radians is then a simple rotation around the z-axis
to get your form of Koide's lepton mass formula:

$$\sqrt{m_n}\ =\ \sqrt{1/3} + \sqrt{2/3}\cos(2/9 + 2n\pi/3)$$I see that this "A4-symmetry" was already found here by Ma:
http://arxiv.org/PS_cache/hep-ph/pdf/0606/0606024v1.pdf

and that there is another group X24 which is larger which could
incorporate quarks here:
http://aps.arxiv.org/PS_cache/hep-ph/pdf/0701/0701034v3.pdf

On the other hand, Garrett Lisi uses a 3d quark matrix here:
http://arxiv.org/PS_cache/arxiv/pdf/0711/0711.0770v1.pdf
https://www.physicsforums.com/showthread.php?t=196498

which is the same except for some coordinate switching:

$$\left(\begin{array}{ccc} -\sqrt{1/3},& -\sqrt{1/3}, & -\sqrt{1/3} \\ -\sqrt{1/2},& +\sqrt{1/2}, & 0 \\ -\sqrt{1/6},&-\sqrt{1/6},& \sqrt{2/3} \end{array}\right)$$

see (2.4) in the paper and also page 18

Regards, Hans