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 Sci Advisor P: 1,135 These classical distance ratio's, $R_f$ for fermions and $R_b$ for bosons are realy full of surprices!! The whole list for sofar: 1) The ratio $R_b/R_f$ of the two is equal to the $m_W/m_Z$ ratio! (to within 0.063% or sigma 1.2) $$\cos \theta_W \ \ = \ \ \frac{m_W}{m_Z} \ \ = \ \ \frac{R_b}{R_f} \ \ = \ \ \frac{2 \beta_b/\alpha}{2 \beta_f/\alpha} \ \ = \ \ \frac{\beta_b}{\beta_f}$$ where $\beta_f, \beta_b$ 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% $$R_f^{-1} \ \ =\ \ \frac{m_e}{m_{\mu}}$$ which improves to 0.0000073% if we add a correction term involving the third lepton like this: $$R_f^{-1} \ \ = \ \ \frac{m_e}{m_\mu} \ (1 \ + \ \ \frac{4}{3} \ \frac{m_e}{m_\tau} \ )$$ 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: $$\beta_b R_f^{-1/2}}\ \ =\ \ \frac{m_{\mu}}{m_{\tau}}$$ which is accurate to within 0.090%. This becomes 0.000040% when we add the following correction term with the other lepton: $$\beta_b R_f^{-1/2}}\ \ =\ \ \frac{m_{\mu}}{m_{\tau}} \ (1 \ + \ \ \frac{3}{16} \ \frac{m_e}{m_\mu} \ )$$ 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 $\sqrt{ s(s+1}\ \hbar$" One gets a general solution for the 'classical velocity' of spin s: $$\beta_s \ \ \ \ \ \ = \ \ \ \ \sqrt{\sqrt{\ \ s(s+1) \ \ + \ \ ( \frac{1}{2} \ s(s+1) \ )^2 } \ \ - \ \ \frac{1}{2} \ s(s+1) }$$ 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 $\sin^2 \theta_W$ form as: $$\sin^2 \theta_W \ \ \ \ = \ \ \ \ 1 \ - \ \frac{\beta^2_f }{\beta^2_b} \ \ \ \ = \ \ \ \ 0.22310132230086634541466926662604$$ $$\sin^2 \theta_W \ \ \ \ = \ \ \ \ 1 \ - \ \frac { \sqrt{\ \ \frac{1}{2}(\frac{1}{2}+1) \ \ + \ \ ( \frac{1}{2} \ \ \frac{1}{2}(\frac{1}{2}+1) \ )^2 } \ \ - \ \ \frac{1}{2} \ \ \frac{1}{2}(\frac{1}{2}+1) } {\sqrt{\ \ 1(1+1) \ \ + \ \ ( \frac{1}{2} \ \ 1(1+1) \ )^2 } \ \ - \ \ \frac{1}{2} \ \ 1(1+1) }$$ The usual electro-weak parameters g1 and g2 would become: $$g_1^2 \ \ = \ \ e^2 \frac{\beta_b^2}{\beta_f^2} \ \ \ \ \ \ \ \ \ g_2^2 \ \ = \ \ \frac{e^2}{1-\frac{\beta_f^2}{\beta_b^2}}$$ where $e^2 = \alpha$ Regards, Hans