M Theory Lesson 254

Dave at Carl's blog has pointed out that

$\textrm{cos} (\theta) = \frac{2 \sqrt{2}}{\sqrt{9}}$

where $\theta$ is the sum of row (or column) entries in the circulant decomposition of the tribimaximal mixing matrix. In M Theory, we are used to coming across this number $2/9$. For example, it appears as a determinant,

$\frac{2}{9} = 1 - (\frac{1}{\sqrt{3}} + \frac{2}{3} i)(\frac{1}{\sqrt{3}} - \frac{2}{3} i)$

in the 2D component of the transform of the tribimaximal mixing matrix. The number

$\frac{1}{\sqrt{3}} + \frac{2}{3}$

also appears in the 2D component of $| R_3 R_2 |$, a MUB analogue of the (absolute value of) the tribimaximal matrix. That is, this number is a normalised form of $(1 + i \omega^2)$, where $\omega$ is the usual cubed root of unity.