# 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.

3. May 15, 2017

### mitchell porter

A few times in this thread (e.g. #129, #132), @arivero mentions that in a first approximation to his "waterfall" mass ansatz, the mass of the bottom quark comes out as exactly 12 times the constituent quark mass. Earlier this month, in a speculative new framework for hadron masses and systematics, I ran across the claim that the mass of the rho meson is sqrt(6) times the constituent quark mass (see footnote 6, bottom of page 6). This seems to be derivable within the framework of a so-called "gauged quark-level linear sigma model". The original linear sigma model is a famous 1960 creation of Gell-Mann and Levy, with nucleon, pion, and sigma meson fields. The quark-level linear sigma model employs quark fields in the place of the nucleon fields, and the gauged version of this adds vector mesons as well. This is, more or less, the framework I was citing in #134.

What I want to observe here, is that this first approximation to the bottom quark mass, as 12 times the constituent quark mass, could be expressed as 2 . sqrt(6) . sqrt(6) times the constituent quark mass. Furthermore, all these quantities (in the extended LSM framework) come from couplings to QCD condensates, where the value of the coupling is an exact quantity determined nonperturbatively. Also, the double appearance of one factor, is reminiscent of how Koide's yukawaon models work. So maybe this is a hint regarding how to realize the waterfall.

Last edited: May 15, 2017