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PF Gold
P: 2,884
 Quote by taarik Refering to first version, it ia again amazing that the mass of the neutral pion is deteremined very precisely in terms of the three leptons. Now neutral pion has no relation with the three leptons. it does not decay into any of these particles. On the other hand the charged pion decays into electron and muon. Hence charged pion mass should have been related to lepton masses.
I agree, it is misterious. Furthermore, forgetting the issue of integer multiples and the squaring of masses, the formula is very reminiscent of charged pion decay, you know, these $\prop m_\mu^2 ( m_\pi_+^2 - m_\mu^2)$ from textbooks.

To put more intrigue, the mass difference between eta and the average of pion and muon (say, diff=427.2 MeV) also fits roughly in the obvious permuted formulae:
$$\sqrt {m_\mu} \sqrt {m_\tau - m_e} \approx \sqrt {m_\tau} \sqrt {m_\mu - m_e} \approx \sqrt {m_\tau \pm m_e} \sqrt {m_\mu \pm m_e} \approx \sqrt {m_\mu} \sqrt {m_\tau} = 433.27 MeV$$

EDITED: a purpose of the above formulas is to consider the limit $m_\mu \approx m_\tau$ where the former formula cancels and the two first ones in the above become the same. Also, the same cancellation and similarity happens in the other limit $m_e \to 0$. Simultaneous limit conflicts with Koide's.