M Theory Lesson 300

Last time we considered the relation between ${\Omega }_B$, the baryonic mass fraction, and two phases appearing in Carl Brannen's mass matrices. A more natural choice involves the angles

$\phi =\frac{1}{9\pi }$
$\alpha =\frac{1}{24}$,

where $\alpha$ comes from the phase $\pi /12$. The difference $(\alpha -\phi )$ appears in mass relations for neutrinos, charged leptons and hadrons. We now have

$24\alpha -27\phi ={\Omega }_B$

where $24\alpha =1$, the total of baryonic and dark matter components. Both the numbers $24$ and $27$ have a multitude of interesting number theoretic properties which M theorists enjoy. For example, $27$ is the dimension of the octonionic exceptional Jordan algebra, which in terms of $3\times 3$ Hermitian matrices has a $24$ dimensional component coming from three octonion elements. The number $24$ decomposes into $2^3\cdot 3$, and in Brannen's path integrals the $3$ indexes particle generation number, and $2$ stands for the fermionic spin quantum number.