Monday, December 5, 2011

Back to Koide IV

Today Alejandro Rivero writes about the new Koide triplets, where the two charged quark triplets are interwoven to create a sequence of six quarks, including the special $(b,c,s)$ triplet.

This brings my attention to Koide's 2000 paper on Fourier operations in mixing. Here Koide perturbs the basic circulant parameters to fit masses for charged leptons and charged quark triplets. In this case, he observes that one can use approximate phases

$\cos^2 \theta_{l} = \frac{11}{24}$, $\cos^2 \theta_{d} = \frac{9}{24}$, $\cos^2 \theta_{u} = \frac{7}{24}$

But this looks familiar, doesn't it? The numerators sum to the tetractys total of $27$. The $24$ is the number of leptons and quarks without the three mirror neutrinos. The complementarity $27 = 11 + 16$ appears often in M Theory. The paper also discusses fixing CKM parameters using the fitted rest masses. For example, $\sqrt{m_t / m_u}$ gives a small parameter $V_{\textrm{ub}}$, close to $0.0035$.


  1. I have to mention here Krolikowski's prediction of the top quark mass; it has a "1, 4, 24" numerology, arising from Krolikowski's model of the generations, as due to three copies of the Dirac equation, differing by the number of "algebraic partons". I'm sure that sounds familiar...

  2. Thanks, Mitchell.

    Note also that if we take $2/3$ of the sum of the three $\cos^2$ values above ($= 9/8$) we get $(\Sigma \cos \theta_i)^2$ such that the sum is $\sqrt{3}/2$. That is, we get the cosine for the $\pi /6$. Thus the Koide rule relates $27$ and $24$ via their ratio, $9/8$.

    And don't forget that there are three times as many quarks as leptons, due to color.

  3. By which I mean that $3/4$ of the total $24$ are quarks, where $3/4$ is $\cos^2 \pi/6$.

  4. So I see that Krolikowski used $29 = 24 + 4 + 1$, whereas $27$ is far more elegant, in the modern octonionic setting.

    Now I am wondering about the $7$, $9$, $11$. They are obtained in many ways. For instance, $3 + 2 + 6Q$ works, where $Q$ is positive charge, $3$ is generation number and $2$ perhaps picks out a neutrino pair.

  5. Taking another look at Foot's Koide paper, his geometric description can be lifted to Hermitian matrix space. For a given Koide mass matrix M, the angle between it and the identity matrix is $\pi/4$ given by:


    Any automorphism of the Hermitian matrix algebra (as a D=3(dim(K)+1) dimensional vector space) will preserve this relationship, since it's merely built from the trace. This is expected as such automorphisms are isometries of the corresponding projective space.

  6. All right, good. As usual, I am thinking that the full quark lepton complementarity (with all masses and mixings) comes down to a special 3D (ie. number of circulant parameters) set of polytopes and index simplices. Half of the parameters we have are already obviously there in the Euclidean geometry.


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