What is new with Koide sum rules?

1. Jan 24, 2017

mitchell porter

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 $Z (1 + b_f X)^{-1} Z$, mutiplied by a mass scale $m_{0f}$, where
$$Z = \frac 1 {\sqrt{m_e +m_μ + m_τ}} \begin{pmatrix} \sqrt{m_e} & 0 & 0 \\ 0 & \sqrt{m_μ} & 0 \\ 0 & 0 & \sqrt{m_τ} \end{pmatrix}$$
$$X = \frac 1 3 \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix}$$
and $b_f$ is a free parameter. f indicates the family, e, u, or d, and $b_f$ and $m_{0f}$ take different values according to the family.

For the e family, $b_e$ is just zero, so the matrix is $Z^2$, and it just gives the (e,μ,τ) masses by construction. For the d family, $b_d$ is a random-looking number. But for the u family, we have
$$b_u = -1.011$$
$$\frac {m_{0u}} {m_{0e}} = 3.121$$
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

arivero

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.