## Friday, July 23, 2010

### M Theory Lesson 341

For eigenvalues

$\lambda_{i} = 1 + x \textrm{cos} (\phi \pm \pi/12 + \omega_{i})$

the zero eigenvalue at

$\frac{1}{x} = \textrm{sin} (\phi \pm \pi/4)$

helps us search for Koide triplets with large gap scalings. For example, consider the two quark triplets $(u,c,t)$ and $(d,s,b)$. For a scale factor of $\mu$, an initial fit to these triplets (in MeV$/c^{2}$) is given by the parameters

$(u,c,t):$ $\mu = 22280$, $x = 1.782$, $\phi = -0.183$ (resp. $+ \pi/12$)
$(d,s,b):$ $\mu = 580$, $x = 1.718$, $\phi = 0.095$ (resp. $- \pi/12$)

where the angles $\phi$ are not far from the zero eigenvalue condition on $x$. Note that these fits are only rough, since there are large errors on the quark masses. Now including the $\pi /12$ factors, we observe that the two phases are close to $1/12$ and $-1/6$ respectively. In fact, if we adjust the $(u,c,t)$ $x$ parameter to $x = 1.774$, these exact phases provide all six quark masses to well within experimental error. That is, for a Koide matrix phase of $\theta$, and an arbitrary phase conjugation,

$(u,c,t):$ $\mu = 22280$, $x = 1.774$, $\theta = 1/12$
$(d,s,b):$ $\mu = 580$, $x = 1.718$, $\theta = 1/6$

These exact phases (in radians) are analogous to the $2/9$ phase for the charged leptons.

$(u,c,t): \mu=22420, x=1.760, \theta = 2/27$
$(d,s,b): \mu=575, x=1.714, \theta = 1/6$