Three Little-Known Equalities in Food For TheorDev

[arivero]

Three little known equalities:

From Smirnov:
$$\theta_{\mbox{sun}} + \theta_{\mbox{cabibbo}} = {\pi \over 4}$$

From De Vries:
$$\ln {m_\tau \over m_\mu} = \pi - {1 \over \pi}$$

From Smirnov again:
$$\sqrt {m_\mu \over m_\tau} \sim \sin \theta_{\mbox{Cab.} }$$

Can anyone predict them?

de Vries has a secondary formula for the electron-muon relationship, namely
ln(mu/me) / (2pi-3/pi) = 1.000627.

In principle one could recast them in terms of hyperbolic cosines and sines, for instance 1pi-1/pi= 2 sinh(ln(pi)) but it does not clarify the situation.

See also:
https://www.physicsforums.com/showthread.php?t=36624
http://arxiv.org/abs/hep-ph/0405088
http://www.chip-architect.com/news/2004_07_27_The_Electron.html

 

[arivero]

For instance, taking logarithms we have

$$\ln (\sin \theta_C) \sim - \sinh (\ln \pi)$$