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.

7 years ago

OK, so now I've spotted some more accurate low mass quark masses ... all six masses fit with

ReplyDelete$(u,c,t): \mu=22420, x=1.760, \theta = 2/27$

$(d,s,b): \mu=575, x=1.714, \theta = 1/6$

Probably already mentioned, but it is worthwhile to stress this: If we consider that 1 + x cos represents a perturbation of a diagonal identity matrix with a traceless matrix, then it could be interesting to ask for the value, in Mev, of the (multiple of) identity matrix. For charged leptons, it is sqrt(313.84 MeV).

ReplyDeleteIt puzzles me, because I thing that the value of the absolute mass of the muon is the important parameter, being as it is so near of the pion. But if you don't buy this argument, them 313.84 MeV is a nice quantity, specially if you have three of these. It predicts the mass of the proton (or perhaps the eta'...). We call it the "current quark" mass