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What is new with Koide sum rules?

  1. Jan 24, 2017 #141
    Koide and Nishiura's latest (it came out today) contains new numerology.

    In their model, each fermion family e (e,μ,τ), u (u,c,t), and d (d,s,b) gets its masses as eigenvalues of a matrix [itex]Z (1 + b_f X)^{-1} Z[/itex], mutiplied by a mass scale [itex]m_{0f}[/itex], where
    [tex]Z = \frac 1 {\sqrt{m_e +m_μ + m_τ}}
    \sqrt{m_e} & 0 & 0 \\
    0 & \sqrt{m_μ} & 0 \\
    0 & 0 & \sqrt{m_τ}
    [tex]X = \frac 1 3
    1 & 1 & 1 \\
    1 & 1 & 1 \\
    1 & 1 & 1
    and [itex]b_f[/itex] is a free parameter. f indicates the family, e, u, or d, and [itex]b_f[/itex] and [itex]m_{0f}[/itex] take different values according to the family.

    For the e family, [itex]b_e[/itex] is just zero, so the matrix is [itex]Z^2[/itex], and it just gives the (e,μ,τ) masses by construction. For the d family, [itex]b_d[/itex] is a random-looking number. But for the u family, we have
    [tex]b_u = -1.011[/tex]
    [tex]\frac {m_{0u}} {m_{0e}} = 3.121[/tex]
    i.e. very close to the integer values, -1 and 3.

    All that is from section 4.1 (page 12). The model itself is a seesaw as displayed in section 2.1 (page 5 forward). There is no explanation for the values of those numbers.
  2. Jan 24, 2017 #142


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    It looks very much as the discarded corrections for the lepton sector.

    For b=-1 the matrix is singular, so perhaps it is just a signal of how massive the top quark is.
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