## Tuesday, May 4, 2010

### M Theory Lesson 324

In the $(a,b,c)$ parameterisation, with $a = -0.231$ and $b = 24$, the value of $c$ was very small. Now a zero $r$ in an $R_2$ factor gives a weak identity, since the factor only rephases and switches rows or columns. In other words, setting $c = 0$ in a triple $R_2$ product results in a two factor product, like the tribimaximal neutrino mixing matrix.

Since $c$ is small, setting it to zero does not substantially alter the matrix entries. In fact, we observe that the parameter values $(a,b,c) = (-0.231,24,0)$ maintain all experimental CKM values except for the smallest entry, which is now forced to zero. Thus the $(ub)$ term may be considered as a correction to a tribimaximal form of the CKM matrix.

Since people are so unhappy with parameters, I will do my best to reduce the number even further in future.

1. This means that we can also view the neutrino matrix as a triple product with an $r = 0$ factor. Similarly, the charged lepton case is covered by the triple $(0,-1,+1)$.

2. The tribi approximation to CKM is:

0, 0.04163054, 0.99913307;
0.97434190, 0.22487786, 0.00936991;
0.22507298, 0.97349722, 0.04056238.

3. A natural prediction then is that the $(ut)$ factor may remain above $0.009$, which is a little outside current error estimates.

4. I still don't understand the physical concept behind the algebra. If I expand a matrix of flavor-changing amplitudes into a product of two or three matrices, it sounds like I'm saying that it's really a two- or three-step process, and that these matrix factors provide the amplitudes for the intermediate processes. Is that part of the picture?

5. Mitchell, maybe it helps to look at it that way, but I don't myself, because for me the ontology of quantum gravity isn't about quantum states running around all independently in some fixed space. In fact, one could interpret the 3 factors as 3 time directions (each one labeled by one of those Z boson factors). I don't think we will understand it well until the cosmological elements are clearer.

For now, think of each 2d factor as selecting a pair of generations independently of the third, and the triple product as the canonical cyclic way to do the mixing of all generations. For instance, if you write down the $24$ factor you will see that it makes the (cs)(bt) block look like the bottom quark numbers, and $(ud)$ becomes 1.

6. Another naive summary from me...

So, we have three mixing matrices, CKM, MNS (for neutrinos), and "Koide" (the "identity mixing matrix" for electron/muon/tauon). They can all be written in this form. For MNS and Koide the c-parameter is zero, for CKM it is nonzero but small.

I can't find the (a,b) parameters for MNS anywhere, though maybe equation 14 in viXra:0907.0011 is what I'm looking for.

7. Off-topic: we had arxiv, we had vixra, now we have snarxiv!

8. Marni, while your mind is focussed on patterns in matrices, I'm linking here to a table of semi-empirical lepton and quark mass prediction formulae. I don't know if this is helpful for your and Carl's analysis at all or not. If definitely unhelpful, just delete this comment. It might be interesting if it could be generated by some kind of matrix multiplication scheme?

9. Snarxiv! Wow, cool! Thanks, Nigel, I will take a look. Mitchell, for the MNS it is $(1, \sqrt{2})$.

11. Another note: when we take the TBM type parameters $(24, 1/ \sqrt{24} ,0)$ we get the correct up row and bottom column:

0.0, 0.0416305, 0.9991330;
0.979796, 0.199827, 0.008326;
0.199999, 0.978946, 0.040789.

The $0.2$ in the bad block comes from $\sqrt{1/25}$. So we can also look at the CKM as a $2 \times 2$ correction to this matrix.

12. Note also that Carl's original CKM circulants have the same $0.009$ problem. I was reluctant to consider the possibility that this measurement might actually be wrong, but Carl convinced me that experimenters are capable of mistakes.