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Hans de Vries
Hans de Vries is offline
#44
Nov26-04, 02:34 AM
Sci Advisor
P: 1,136
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