All the lepton masses from G, pi, e

[enotstrebor]

Plus c and h, of course.

The idea is to collect here in only a thread all the approximations voiced out during the summer. .

For a single $$4\pi$$ relationship for the electron generations you might want to see Apeiron 16(4) 475-484 (2009).

The $$\mu$$ (n=1)to e (n=0) mass ratio $$m_{\mathrm{e_1}}/m_{\mathrm{e_0}}$$ is $$\sqrt{2}(4\pi\varrho_1)^{(3-1)}$$ where $$\varrho_1=.96220481$$
while the $$\tau$$ (n=2) to $$\mu$$ (n=1) mass ratio $$m_{\mathrm{e_2}}/m_{\mathrm{e_1}}$$ is $$\sqrt{2}(4\pi\varrho_2)^{(3-2)}$$ where $$\varrho_2=.94635968$$.

Thus the first and second (n=1,2) generation mass ratio ($$m_{\mathrm{e}(n)}/m_{\mathrm{e}(n-1)}$$) form is $$\sqrt{2}(4\pi\varrho_n)^{3-n}$$.

As the W, proton and electron masses are a function of $$(4\pi\varrho)^{3}$$, i.e

$$m_x = M_{\mathrm{sp}} ~(2S ~(4\pi\varrho)^3/\varsigma )^{(S ~C ~M)}$$

where$$M_{\mathrm{sp}}=\sqrt{m_p \cdot m_e}$$, $$\varrho=0.9599737853$$, S is the spin quantum number (1/2,1), C is the charge quantum number (+1,-1) and M is the matter quantum number (matter= +1 antimatter= -1),

You can get the masses of the W, proton, electron and its generations using the single formula for particle x (x=W,p,e,mu,tau)

$$m_x = M_{\mathrm{sp}(n)} ~(2S~(4\pi\varrho)^3/\varsigma)^{(S ~C ~M)}$$,

where $$M_{\mathrm{sp}(n)} = M_{\mathrm{sp}} ~S^{-n/2}(4\pi\varrho_n)^{(6S n - S n(n+1)))}$$ and $$\varrho_n = 1 - log(1 + 64.7564~n/S)/(112S)$$ are used, and generation n is $$\{0,1,2\}$$.

 

[legna777]

$$m_{e}=\frac{V_{H}}{\sqrt{2}\cdot\Bigl(e^{12}+\frac{m_{Z}}{m_{e}}\Bigr)}$$


$$V_{H}=\; vacuum\; Higgs\; \; "mass"=\frac{246,2205691\cdot E^{9}\cdot1,602176487\cdot E^{-19}\: Gev}{c^{2}}\medskip{}$$


$$m_{Z}=mass\; Z\; boson=91,1876\; Gev$$

$$e^{\Bigl(12+\frac{\cos(2\pi/5)}{10}\Bigr)=\frac{m_{Z}+m_{W}}{2m_{e}}\;\; m_{e}=\: electr\acute{o}n\; mass\; m_{W}=bos\acute{o}n\: W=80,398\: Gev}\medskip{}$$


$$\[<br /> \frac{V_{H}}{\sqrt{2}}+\frac{V_{H}}{\pi^{4}}=\sum_{q=1}^{6}m_{q}-\sum_{l=1}^{6}m_{l}\]<br /> \medskip{}$$

$$\[<br /> \sum_{q=1}^{6}m_{q}=sum\; over\; all\; six\; quarks\;\;\]<br /> \medskip{}$$

$$\[<br /> \sum_{l=1}^{6}m_{l}=m_{e}+m_{\mu}+m_{\tau}+\nu_{e}+\nu_{\mu}+\nu_{\tau}=sum\; leptons\; masses\]<br />$$


$$\[<br /> \frac{V_{H}}{\sqrt{2}}-\frac{V_{H}}{\pi^{4}}=m_{Z}+m_{w}\]<br /> \medskip{}$$

$$\[<br /> bos\acute{o}n\; Higgs\; mass=\frac{\sqrt{2}\cdot V_{H}}{e}=m_{H}\backsimeq128\; Gev\]<br /> \medskip{}$$

$$e^{\bigl(12+(\sin^{2}eff^{lep}(\phi_{W}))^{-1}/10\bigr)}=\frac{m_{H}}{m_{e}}$$

$$<br /> \sin eff^{lep}(\phi_{W})=\sqrt{0,2315...}=\ln(\varphi)\medskip{}<br />$$

$$\varphi=\frac{1+\sqrt{5}}{2}$$

$$\[<br /> \frac{\sin\theta_{C}}{\sqrt{4\pi\alpha}}\Bigl(\frac{V_{H}}{\pi^{4}}\Bigr)=\sum_{l=1}^{6}m_{l}=m_{e}+m_{\mu}+m_{\tau}+\nu_{e}+\nu_{\mu}+\nu_{\tau}\]<br /> \medskip{}$$

$$\theta_{C}=Cabibbo\; angle=13,04^{\circ}\medskip{}$$

$$\alpha=fine\; structure\; constant=(137,035999084...)^{-1}$$

"En este punto aun tenemos 4 bosones gauge $$(Wi\textgreek{m}(x) y B\textgreek{m}(x))$$ y 4 escalares $$\xi\overrightarrow{(x)}$$ y h(x)), todos ellos sin masa, lo que equivale a 12 grados de libertad[/color] (Conviene notar que un bosón vectorial de masa nula posee dos grados de libertad, mientras que un bosón vectorial masivo adquiere un nuevo grado de libertad debido a la posibilidad de tener polarización longitudinal: 12 = 4[bosones vectoriales sin masa] × 2 + 4[escalares sin masa]). P. W. Higgs fue el primero en darse cuenta de que el teorema de Goldstone no es aplicable a teorías gauge, o al menos puede ser soslayado mediante una conveniente selección de la representación. Así, basta con escoger una transformación:"

http://es.wikipedia.org/wiki/Mecanismo_de_Higgs


http://arxiv.org/PS_cache/hep-ph/pdf/0001/0001283v1.pdf

A Finely-Predicted Higgs Boson Mass from A Finely-Tuned Weak Scale

Lawrence J. Hall, Yasunori Nomura
(Submitted on 13 Oct 2009 (v1), last revised 19 Oct 2009 (this version, v2))

Abstract: If supersymmetry is broken directly to the Standard Model at energies not very far from the unified scale, the Higgs boson mass lies in the range 128-141 GeV. The end points of this range are tightly determined. Theories with the Higgs boson dominantly in a single supermultiplet predict a mass at the upper edge, (141 \pm 2) GeV, with the uncertainty dominated by the experimental errors on the top quark mass and the QCD coupling. This edge prediction is remarkably insensitive to the supersymmetry breaking scale and to supersymmetric threshold corrections so that, in a wide class of theories, the theoretical uncertainties are at the level of \pm 0.4 GeV. A reduction in the uncertainties from the top quark mass and QCD coupling to the level of \pm 0.3 GeV may be possible at future colliders, increasing the accuracy of the confrontation with theory from 1.4% to 0.4%. Verification of this prediction would provide strong evidence for supersymmetry, broken at a very high scale of ~ 10^{14 \pm 2} GeV, and also for a Higgs boson that is elementary up to this high scale, implying fine-tuning of the Higgs mass parameter by ~ 20-28 orders of magnitude. Currently, the only known explanation for such fine-tuning is the multiverse.

http://arxiv.org/abs/0910.2235

 

[legna777]

$$\[<br /> \theta_{C}=Cabibbo\; angle\]<br /> \medskip{}$$

$$<br /> \[<br /> \tan\theta_{c}=\sin^{2}eff^{lep}(\phi_{W})=0,2315...=\ln^{2}(\varphi)\]<br /> <br /> <br /> <br /> <br />$$


http://pdg.lbl.gov/2009/reviews/rpp2009-rev-vud-vus.pdf

 

[legna777]

"Standard" parameters


A "standard" parameterization of the CKM matrix uses three Euler angles ($\theta_{12}\,,\,\theta_{13}\,,\,\theta_{23}$)and one CP-violating phase $(\delta_{13})$
Couplings between quark generation i and j vanish if $\theta_{ij}=0$ .Cosines and sines of the angles are denoted $c_{ij}$ and $c_{ij}$ $s_{ij}$ respectively. $\theta_{12}$ is de Cabibbo angle.

The currently best known values for the standard parameters are:


$$\theta_{12}=13.04\pm0.05^{\circ}$$
$$\theta_{13}=0.201\pm0.011^{\circ}$$
$$\theta_{23}=2.38\pm0.06^{\circ}$$
$$\delta_{13}=1.20\pm0.08^{\circ}<br />$$


$$\[<br /> \begin{bmatrix}c_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i\delta_{13}}\\<br /> -s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta_{13}} & c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta_{13}} & s_{23}c_{13}\\<br /> s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta_{13}} & -c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta_{13}} & c_{23}c_{13}\end{bmatrix}\]<br />$$

http://en.wikipedia.org/wiki/Cabibbo–Kobayashi–Maskawa_matrix

$$\[<br /> \frac{\alpha}{\sin(\phi_{W})}=\frac{\theta_{13}}{\theta_{12}}\]$$


$$\[<br /> \theta_{12}+\theta_{13}+\theta_{23}-\delta/2=\frac{2\pi}{24}\; rad.\]<br />$$

$$\[<br /> 2\theta_{12}+\theta_{13}+\theta_{23}\backsimeq\phi_{W}\]$$

 

[legna777]

Alexis Monnerot-Dumaine: The Fibonacci Fractal

http://alexis.monnerot-dumaine.neuf.fr/articles/fibonacci fractal.pdf

$$\[<br /> \frac{\sin eff^{lep}(\phi_{W})}{\frac{m_{W}}{m_{Z}}}=\frac{D_{H}}{3}=\frac{\ln\varphi}{\ln(1+\sqrt{2})}\]$$

Hausdor Dimension=3=Dh

FRACTAL GEOMETRY IN QUANTUM MECHANICS, FIELD THEORY AND
SPIN SYSTEMS

H. Kröger

Physics Reports 322 (2000) 81-181

[Link Deleted]

It´s free, not my work. Great work

In geometry, an icosahedron (Greek: εικοσάεδρον, from eikosi twenty + hedron seat; pronounced /ˌaɪkɵsəˈhiːdrən/ or /aɪˌkɒsəˈhiːdrən/; plural: -drons, -dra /-drə/) is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of five Platonic solids.

It has five trianglular faces meeting at each vertex. It can be represented by its vertex figure as 3.3.3.3.3 or 35, and also by Schläfli symbol {3,5}. It is the dual of the dodecahedron, [/color]which is represented by {5,3}, having three pentagonal faces around each vertex.

http://en.wikipedia.org/wiki/Icosahedron

$$<br /> \[<br /> -\frac{\sin\theta_{C}}{\sqrt{4\pi\alpha}}\backsimeq\cos(138,186965^{\circ})=\cos(\widehat{\Omega})_{d}icoSHD=\]<br />$$cosine icosaedral dihedral angle

Symmetry :dodecahedral manifold

$$\[<br /> \frac{2\pi}{12+\varphi^{-1/2}}=\widehat{\phi_{W}}\; rad\]<br />$$

$$\[<br /> \frac{2\cdot m_{Z}}{m_{e}}+\alpha_{s}^{-1}(m_{Z})\cdot\varphi^{2}=\exp(12+\varphi^{-1/2})\]<br /> \medskip{}$$

$$\[<br /> \exp(\varphi^{5}+1)+\alpha_{s}^{-1}(m_{Z})\cdot\varphi^{2}=\frac{m_{Z}}{m_{e}}\]<br /> \medskip{}$$

$$\[<br /> -\sin\Bigl(\frac{2\pi}{\varphi^{2}}\Bigr)-\cos\Bigl(\frac{2\pi}{\varphi^{2}}\Bigr)=\frac{\alpha(q=0)}{\alpha_{s}(m_{Z})}=\frac{(137,035999084)^{-1}}{0,1178}\]<br />$$

 

[Vanadium 50]

First, the paper link you posted to is not "free". It is "stolen". I have since deleted it.

Second, Kroeger doesn't mention icosahedra - or polyhedra at all - in his paper.

Third, if this thread has taught us anything, it's that one can always toss a simple equation together to get some "interesting" value. By itself, this means nothing. This is why physicists call this "numerology".

Fourth, the ratio of two fundamental constants at two different scales cannot possibly be physical - as they are at two different scales.

Fifth, even if one takes everything at face value, the agreement is not impressive. Plugging in alpha_s at the Z, one gets a value for alpha at q=0 of 1/(137.4 ± 2.3). While that covers the correct value, the agreement is not so impressive when the error bars are added.

Finally, I'd like to remind everyone in this thread about the PF rules for overly speculative posts.

 

[arivero]

Yes, I am sorry that while this thread was active people was more aware of how touchy the issue is. We could close it perhaps.

 

[legna777]

First, the paper link you posted to is not "free". It is "stolen". I have since deleted it.

Second, Kroeger doesn't mention icosahedra - or polyhedra at all - in his paper.

Third, if this thread has taught us anything, it's that one can always toss a simple equation together to get some "interesting" value. By itself, this means nothing. This is why physicists call this "numerology".

Fourth, the ratio of two fundamental constants at two different scales cannot possibly be physical - as they are at two different scales.

Fifth, even if one takes everything at face value, the agreement is not impressive. Plugging in alpha_s at the Z, one gets a value for alpha at q=0 of 1/(137.4 ± 2.3). While that covers the correct value, the agreement is not so impressive when the error bars are added.

Finally, I'd like to remind everyone in this thread about the PF rules for overly speculative posts.

First, the paper is not "stolen": is public

Second: I do not say that the paper mentions nothing around icosahedra. I´ts is my opinion

Third: The fine structure constant QED at transfer momentun 0 ( or mass =0 )is:

(137,035999084... )-1

The
effective coupling equals the fine-structure constant a at
the Thomson limit(q2=
* 0)and is expected to increase
logarithmically as jq2
j increases at large jq2

PHYSICAL REVIEW LETTERS Volume 81 number 12 21 September 1998

"Measurement of the Running of Effective QED Coupling at Large Momentum Transfer
in the Space like Region"

The link is: PUBLIC, not stolen

http://dspace.lib.niigata-u.ac.jp:8080/dspace/bitstream/10191/1710/1/p2428_1.pdf





Four: Alpha_s is the coupling strong

Fifth: the ratio of two fundamental dimensionless constants at two different scales can possibly be physical

 

[Vanadium 50]

First, the paper is not "stolen": is public

No, it's not. It's in Physics Reports, which means that Elsevier owns the copyright. The fact that you have to download it from a file-sharing site rather than a scientific site should have been a big hint.

Second: I do not say that the paper mentions nothing around icosahedra. I´ts is my opinion

I'd like to remind everyone in this thread about the PF rules for overly speculative posts. .

 

[legna777]

No, it's not. It's in Physics Reports, which means that Elsevier owns the copyright. The fact that you have to download it from a file-sharing site rather than a scientific site should have been a big hint.



I'd like to remind everyone in this thread about the PF rules for overly speculative posts. .

Your are in true:

The fact that you have to download it from a file-sharing site rather than a scientific site should have been a big hint.

correctely deleted the link, I am sorry around this fact

 

[CarlB]

With journals, you used to get a pile of paper "reprints" to mail around to people. Now some journals instead give you an electronic copy of the paper. You're allowed to put this on your own website but it still isn't quite a public copy in that no one else is allowed to do this. I've got a paper to be published this month in IJMPD and here's what they say:

You may post the postprint on your personal website or your institution's repository, provided it is accompanied by the following acknowledgment:
Electronic version of an article published as [Journal, Volume, Issue, Year, Pages] [Article DOI] © [copyright World Scientific Publishing Company] [Journal URL]
http://www.worldscinet.com/authors/authorrights.shtml

So when you see a copy of a scientific paper on the web it's not necessarily stolen. If it's on a file sharing website that's a bad sign, but there are also papers that have been released by the journals.

 

[Hans de Vries]

I'd say that this last remark reveals all. Either legna is mocking of the rest of the thread (which is reasonable, but perhaps and argument more to close the thread after all these years) or a part of legna is mocking of other part of self. On other hand, it is true that the relationships are, if not under 1%, at least within 3%. But it is clear that they fail on the side of simplicity, when compared to the rest of the thread.

To make people realize that one can come up with hundreds of thousands
of formulas with an equal or lower error you could introduce the following
posting requirements:

1) Every numerical coincidence posted should include the error percentage.
2) The posts should contain the "prediction power" of the "formulas"

The "prediction power" can for instance be defined by the simple formula:

$$\mbox{PP} ~~=~~ \frac{10^{-\frac43 S}}{\mbox{error}}$$

where S is the number of symbols like 1,2,3,+,-,*/,sin,log...

The value of 1/PP is then the amount of similar complex formulas which give
an equal or better prediction, most likely to be a very big number...


Regards, Hans

 

[jtbell]

This thread is closed pending a moderation decision.

Update: it has been decided to leave this thread closed.