Friday, April 30, 2010

M Theory Lesson 320

Consider the triple product of three such scaled $R_2$ factors, where $a$, $b$ and $c$ are real. By design, the matrix $M$ is the sum of a real $1$-circulant and an imaginary $2$-circulant. It follows that it is a doubly magic matrix, determined by a $2 \times 2$ block. Consider the block $B$ of the norm squares of $M$ given by After renormalising, this is fitted to the canonical $2 \times 2$ block of the CKM matrix (norm squares): The full CKM matrix follows from the magic properties. I used Maxima for this, and a row sum parameter for $|M|^2$ of $192.9986$, which turns out to be close to the natural normalisation of $M$ given by
$(a^2 + 1)(b^2 + 1)(c^2 + 1)$

3 comments:

  1. Of course, this does use 4 parameters to fix the small entries of the CKM.

    If you look at the complex matrix M more closely, you can see that the scaled values (by roughly sqrt(192)) of a,ab,c are really three real entries from the non squared CKM matrix.

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  2. That would be the Up row of the CKM matrix. That is, the Up row generates all the other rows.

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  3. OK, so the best result for the 4 parameter method (ie. with a row sum shift $k$ used to adjust the small values from the 2x2 block) is:

    $a = 0.2308985616$
    $b = 13.103671706$
    $c = -0.065260610$
    $k = 181.932270916$

    resulting in (all within exp. errors):

    $0.974400$, $0.224793$, $0.003584$;
    $0.224651$, $0.973556$, $0.041476$;
    $0.008737$, $0.040703$, $0.999133$.

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