View Poll Results: Multiple poll: Check all you agree. Logarithms of lepton mass quotients should be pursued. 24 27.91% Alpha calculation from serial expansion should be pursued 22 25.58% We should look for more empirical relationships 26 30.23% Pythagorean triples approach should be pursued. 21 24.42% Quotients from distance radiuses should be investigated 16 18.60% The estimate of magnetic anomalous moment should be investigated. 26 30.23% The estimate of Weinberg angle should be investigated. 21 24.42% Jay R. Yabon theory should be investigate. 16 18.60% I support the efforts in this thread. 47 54.65% I think the effort in this thread is not worthwhile. 30 34.88% Multiple Choice Poll. Voters: 86. You may not vote on this poll

# All the lepton masses from G, pi, e

by arivero
Tags: koide formula, lepton masses
HW Helper
P: 1,204
 Quote by granpa from post #342: it allows 4 generations if you count m=0
If you'll look carefully at the formula, you will find that putting 4 into the equation gives you the same masses as if you'd put 1. This is because trig functions repeat every 2 pi.

I'm preparing a submission for Phys Math Central on the subject of Koide's mass formulas for the hadrons. I'm supposed to have something ready by January 10th. The latest cut is here:

Carl
 P: 2,258 I was using this formula: a mass of zero can be added to numerator and denominator without changing anything.
PF Patron
P: 2,873
 Quote by CarlB I I'm preparing a submission for Phys Math Central on the subject of Koide's mass formulas for the hadrons. I'm supposed to have something ready by January 10th. The latest cut is here: http://www.brannenworks.com/koidehadrons.pdf Carl
Hi Carl,

I would use some more group notation. Instead of your 6x6=36, I'd vote by 6 x bar6 = sum of irreps.

Second, besides to the "extraordinary" case you put, I'd include more of the "Ordinary" case, meaning the original argument of Koide about his formula for composite particles, along the lines of the original PhysRev.
P: 2,258
there are obviously an infinite number of formulas relating the masses of the electron muon and tauon. the beauty of the koide formula is its simplicity and elegance. especially the fact that Q=2/3 which is exactly halfway between the upper and lower limits of 1 and 1/3.

as wikipedia puts it:
 Not only is this result odd in that three apparently random numbers should give a simple fraction, but also that Q is exactly halfway between the two extremes of 1/3 and 1.
the formula involving cos given above seems to me to lack this elegance. it should be obvious that anyone can always find a formula involving cos that will pass through any 3 points quite easily.
HW Helper
P: 1,204
 Quote by arivero Hi Carl, I would use some more group notation. Instead of your 6x6=36, I'd vote by 6 x bar6 = sum of irreps.
Hmmm. The only place I've got 36 in the thing right now is the calculation for the square of a set of MUBs. In terms of color, this would be
$$(3+\bar{3})\times (3+\bar{3})$$
i.e. (R+G+B+/R+/G+/B)^2,
and I guess the symmetry would be, (hems and haws whilst trying to recall Georgi), uh, let's see, maybe
$$(3+\bar{3})\times (3+\bar{3}) = 8+\bar{8}+6+\bar{6}+3+\bar{3}+1+\bar{1}$$?

Yes, as usual, your comment is brilliant and I will incorporate it.

In a certain sense, what I'm doing is taking the
$$3\times \bar{3} = 8+1$$
definition of gluons and claiming that I can compute in qubits with it using 3x3 matrices. [If you can read this, then PhysicsForums let me edit out a silly comment about 6+3-bar.]

Because both can be faithfully represented by 3x3 complex matrix addition and multiplication. From the symmetry point of view, it's a matter of selecting a certain ratio of the coupling constants.

 Second, besides to the "extraordinary" case you put, I'd include more of the "Ordinary" case, meaning the original argument of Koide about his formula for composite particles, along the lines of the original PhysRev.
You mean the 1982 article I suppose. I guess I'll have to hike over to the university and read the article. Somehow I don't think I've ever downloaded it. One of the things I'm trying to do is to avoid talking about the preon model. There are two reasons for this. First, it doesn't apply to hadrons, and second, it gets into hot water with Coleman Mandula or (worse) spin statistics.

I think I should be ready to release a first cut in a few days. It still doesn't have "results" and the abstract is obsolete. But I've redone the results section to be make an argument that hangs together better.
 HW Helper Sci Advisor P: 1,204 Okay, I've got this thing past the first cut. It is time to ask for advice and for people to point out obvious problems etc. http://www.brannenworks.com/koidehadrons.pdf
 HW Helper Sci Advisor P: 1,204 The most recent arXiv included a hep-th article that I thought was pretty cute. Symmetries of Nonrelativistic Phase Space and the Structure of Quark-Lepton Generation Piotr Zenczykowski. According to the Hamiltonian formalism, nonrelativistic phase space may be considered as an arena of physics, with momentum and position treated as independent variables. Invariance of x^2+p^2 constitutes then a natural generalization of ordinary rotational invariance. We consider Dirac-like linearization of this form, with position and momentum satisfying standard commutation relations. This leads to the identification of a quantum-level structure from which some phase space properties might emerge. Genuine rotations and reflections in phase space are tied to the existence of new quantum numbers, unrelated to ordinary 3D space. Their properties allow their identification with the internal quantum numbers characterising the structure of a single quark-lepton generation in the Standard Model. In particular, the algebraic structure of the Harari-Shupe preon model of fundamental particles is reproduced exactly and without invoking any subparticles. Analysis of the Clifford algebra of nonrelativistic phase space singles out an element which might be associated with the concept of lepton mass. This element is transformed into a corresponding element for a single coloured quark, leading to a generalization of the concept of mass and a different starting point for the discussion of quark unobservability. http://arxiv.org/abs/0901.2896
PF Patron
P: 2,873
 Quote by Taunus Let Me be the mass of the electron, Mp be the mass of the proton, and Mn be the mass of the neutron. Then observe that Mn/Me - Mp/Me is approximately equal to ln(4*pi) = 2.5310...
Of course, it is really the quotient between the electromagnetic isospin breaking, (Mn-Mp), and the mass of the electron.

 ellipsoid of volume 4pi(4pi-1/pi)(4pi-2/pi) = 1836.15... Which is approximately equal to Mp/Me (a very old result)
Hmm not sure how old is this "ellipsoid". A really old approximation to Mp/Me as 6 pi^5 has been discussed in the middle of this thread.

 I hesitate to put in measured values since Mp/Me has changed several times. At one time the ellipsoid volume was within one standard deviation of the accepted CODATA value, but the accepted value changed. I simply rest my case with the mathematical formulae.
A pseudoconvention in this thread is to put both the exact quotient between left and right sides of the equality AND the one-sigma error of this quotient. In this way it is clear how much we can expect the fit to change.

A tradition in this thread is to put both the exact quotient between left and right sides of the equality AND the one-sigma error of this quotient. In this way it is clear how much we can expect the fit to change.

In any case, according current pdg data:

mn − mp = 1.2933317 ± 0.0000005 MeV
Me= 0.510998910 ± 0.000000013 MeV

thus
(mn-mp)/me=2.530987199 ± .000001 MeV
and
(mn-mp)/me :: ln(4*pi) = 0.999 985 36 ± 0.000 000 4

As Hans says, the important point is that the agreement is of 99.998 5 %. The one-sigma adjustment is not important because the explerimental precision is a lot better. But this thread stresses that reaching the empirical bar of 99.99% in a simple formula in not easy.
 HW Helper Sci Advisor P: 1,204 Walter Smilga has rewritten his justification of Wyler's approximation of the fine structure constant in a nice clean article: Probing the mathematical nature of the photon field Walter Smilga The mathematical content of the interaction term of quantum electrodynamics is examined under the following assumption: It is presumed that the apparent degrees-of-freedom of the photon field reflect the kinematical degrees-of-freedom of the two-particle state space of massive fermions, rather than independent degrees-of-freedom of the photon field. This assumption is verified by reproducing the numerical value of the fine-structure constant. http://arxiv.org/abs/0901.4917 Of course I see everything through the lens of what I'm doing. So I see the above as evidence that in understanding the color force, one should do something similar with gluons. My recent paper analyzes the mesons in terms of what happens to the quarks, pretty much ignoring the gluons. And it also finds that things turn out simpler than expected.
P: 22
http://foros.astroseti.org/viewtopic...=asc&start=180

Quote by legna
 Quote by franc Entonces Legna, (perdona mi ignorancia) ¿qué significa la cifra resultante de 522? saludos
Existen, como sabes, 5 grupos excepcionales ( a veces los llamo especiales ) de Lie y son:

$$$E8,E7,E6,F4,G2$$$

$$$dim(E8)=248$$$

$$$dim(E7)=133$$$

$$$dim(E6)=78$$$

$$$dim(F4)=52$$$

$$$dim(G2)=14$$$

En el grupo excepcional $$$E8$$$ están contenidos los otros 4 grupos excepcionales de Lie.
Este grupo $$$E8$$$ tiene 240 raíces no nulas, número que es el producto
de los 5 primeros términos de la sucesión de Fibonacci.

$$$240=1\times2\times3\times5\times8$$$

$$$dim(E8)+dim(E7)+dim(E6)+dim(F4)+dim(G2)=\Omega$$$

$$$\Omega+(8-5-3-2-1)=522$$$

Saludos
 Quote by legna La tabla de caracteres del grupo $$$E8$$$ se compone de una matriz de $$$453060\times453060$$$ http://gaussianos.com/category/noticias/page/3/ $$$\frac{\pi^{5}}{120}$$$ es el factor de volumen de una hiperesfera de 10 dimensiones http://es.wikipedia.org/wiki/3-esfera Momento magnético anómalo del electrón Límite del error experimental máximo: $$$1+\frac{\Omega+(\frac{\pi^{5}}{120})^{-1}}{453060}=1+\frac{\alpha}{2\pi}+\frac{2\alpha^{2}}{3}(\frac{\alpha}{2 \pi})-\frac{4}{3}(\frac{\alpha}{2\pi})^{2}$$$ http://en.wikipedia.org/wiki/Anomalo..._dipole_moment Saludos
 P: 22
P: 22
Hola

Quote by legna
Quote by legna
$$$(\frac{m_{e}}{m_{e}}+\frac{m_{\mu}}{m_{e}}+\frac{m_{\tau}}{m_{e}})\appr oxeq(2^{7}+\frac{1}{\sqrt{2^{7}}})^{-1}(\frac{m_{W-}}{m_{e}}+\frac{m_{W-}}{m_{e}}+\frac{m_{W-}}{m_{e}})$$$

$$$(\frac{m_{e}}{m_{e}}+\frac{m_{\mu}}{m_{e}}+\frac{m_{\tau}}{m_{e}})\appr oxeq(\alpha^{-1}(M_{Z})-\sin(\frac{2\pi}{6}))^{-1}(\frac{m_{W-}}{m_{e}}+\frac{m_{W-}}{m_{e}}+\frac{m_{W-}}{m_{e}})$$$

$$$m_{e}=$$$ masa del electrón

$$$m_{\mu}=$$$ masa del muón

$$$m_{\tau}=$$$ masa del tauón

$$$m_{W-}=$$$ masa del bosón W con carga negativa -1= 80,398 Gev

$$$\alpha^{-1}(M_{Z})=128,95$$$ o valor de la "constante" estructura fina a la escala de unificación electrodebil o masa del bosón Z

 muon-electron mass ratio
http://physics.nist.gov/cgi-bin/cuu/...ch_for=atomnuc!

 tau-electron mass ratio
http://physics.nist.gov/cgi-bin/cuu/...ch_for=atomnuc!

Saludos
$$$[\frac{M_{Z}}{m_{e}}-(\frac{m_{e}}{m_{e}}+\frac{m_{\mu}}{m_{e}}+\frac{m_{\tau}}{m_{e}})]=174764=\sum_{p=2}^{137}p^{2}$$$

Donde los corchetes en
$$$[\frac{M_{Z}}{m_{e}}-(\frac{m_{e}}{m_{e}}+\frac{m_{\mu}}{m_{e}}+\frac{m_{\tau}}{m_{e}})]$$ significa la función parte entera de lo que contiene Y $$\[ \sum_{p=2}^{137}p^{2}$$$ es el sumatorio del cuadrado de todos los números primos menores o iguales a 137

$$$M_{Z}=91,1876\; Gev$$$

$$$[(\frac{M_{W}}{m_{e}}){\cos\theta_{W}}]=\sum_{p=2}^{127}p^{2}=138834$$$

Recuérdese que para la inversa de la "constante" de acoplamiento electromagnética en la escala de unificación elctrodébil ( masa bosón Z ), 128,95; el número primo inmediatamente inferior a su parte entera [128,95] es el primo de Mersenne 127

$$$127=2^{7}-1$$$

Saludos
P: 22
Hola

 Quote by legna $$$N_{C}(E8)+(1+\cos\theta_{W})(1^{3}+2^{3}+3^{3}+5^{3}+8^{3})=(137\cdot\l n137)^{2}$$$ $$$1\times2\times3\times5\times8=240$$$ Cantidad de raíces no nulas del grupo E8. Los 5 primeros números de la sucesión de Fibonacci. $$$N_{C}(E8)=453060=$$$ Número de la matriz generadora de los caracteres del grupo E8 $$$(1^{3}+2^{3}+3^{3}+5^{3}+8^{3})=[\sqrt{N_{C}(E8)}]=[\sqrt{453060}]=673$$$
P: 22
Hola

 Quote by legna $$\frac{1}{\ln(\frac{\alpha^{-1}-21\varphi}{2})}=\frac{\pi^{4}}{384}$$ Donde Pi^4/384 es la densidad de enpaquetamiento de hiperesferas en dimensión 8. Grupo E8. http://mathworld.wolfram.com/HyperspherePacking.html Alpha= constante estructura fina= (137,035999084...)^-1 Phi= 1+SQR(5) /2 = número aureo= 1,618033989...
P: 22
Hola

 Quote by legna $$$(\frac{m_{PK}}{m_{e}})^{2}\cdot\varphi\cdot\cos(2\pi/10)\approxeq\alpha^{-21}$$$ $$$m_{PK}=$$$ masa de Planck= 2.176 44 x 10E-8 kg http://physics.nist.gov/cgi-bin/cuu/...r=universal_in! $$$m_{e}=$$$ masa del electrón= 9.109 382 15 x 10E-31 kg http://physics.nist.gov/cgi-bin/cuu/...ch_for=atomnuc! $$$\alpha^{-1}=137,035999084$$$ $$$\varphi=\frac{1+\sqrt{5}}{2}$$$= límite del cociente entre 2 términos consecutivos de la sucesión de Fibonacci. $$$21=1\times3\times7$$$= septimo término de la sucesión de Fibonacci 1, 2, 3, 5, 8, 13, 21,.... $$$[\alpha^{-1}]=137$$$ $$$[\alpha^{-1}]=137=2^{2}-1+2^{3}-1+2^{7}-1$$$ Saludos
P: 22
Hola

 Quote by legna $$$\ln((\frac{\ln\ln\varphi}{\sum_{p}^{137}1/P}+137)/21)\approxeq\sum_{p}^{137}1/p$$$ Donde: $$$\sum_{p}^{137}1/p=1,872603466...$$$ es el sumatorio del inverso de los números primos menores o iguales a 137 $$$\varphi=\frac{1+\sqrt{5}}{2}=1,618033989...$$$
P: 22
Hola

Quote by legna
$$$(\frac{M_{Z}}{m_{e}})(\frac{\alpha}{2\pi})=\frac{m_{\mu}}{m_{e}}a_{\tau }a_{e}$$$

$$$m_{Z}=91,1876$ \: Gev$$= 1.625566473 E-25 Kg

$$$m_{e}=9.10938215E-31\: Kg$$$

$$$\alpha=1/137,035999084$$$

$$$a_{\tau}$$$= momento magnético anómalo del tau=$$$\frac{g-2}{g}$$$=$$$1.00117721$$$

 The tau lepton anomalous magnetic moment
http://arxiv.org/abs/hep-ph/0702026

$$$a_{e}=\frac{g_{e}-2}{g_{e}}=1.00115965218$$$= momento magnético anómalo de electrón.

http://physics.nist.gov/cgi-bin/cuu/...ch_for=atomnuc!

$$$m_{\mu}=1.8835313E-28\: Kg$$$=masa del muón
P: 22
Hola

Quote by legna
 Quote by legna
Espacio de 10 dimensiones ( 9 espaciales y una temporal )