The MNS matrix (in tribimaximal form, which is compatible with all experimental measurements) can be written in elegant form as a unitary matrix which is the sum of a real 1-circulant matrix and an imaginary 2-circulant matrix. See Doubly Magic Matrices and the MNS. Now I’ve got the CKM matrix in the same form. And here it is:
The cool thing about the above is that it only involves 6 real numbers. Three reals define the 1-circulant matrix, +0.973313178, -0.008576543, and +0.000466480, while three more reals define the 2-circulant, +0.225761835, +0.040012680, and -0.004273188.
I’ve marked the larger contributions with red to help you see the symmetry. The real matrix has each of its rows shifted one to the right (i.e. 1-circulant), while the imaginary matrix has them shifted two to the right (i.e. 2-circulant). Note that the above matrix is unitary, which you will have to verify by taking dot products of its rows with the complex conjugatges of its rows, and the same for the columns. (You should get 1s and 0s.)
The experimental values I’m using for the CKM matrix come from hep-ph/0706.3588) and are:
If you square the elements of a row or column of the above, and add them up, you get 1, but it is not unitary. To put it into unitary form, we supplement each of the above real numbers by multiplying by a complex phase. The matrix is unitary when the resulting rows and columns are orthonormal.
The experimental measurement has 9 real numbers. Multiplying them by complex phases so as to make the matrix unitary is going to give you 18 real numbers. And yet I’ve been able to do this, put the experimental numbers into unitary form, and ended up with only 6 real numbers, and those are used in an elegant and symmetrical manner. This is by far the most elegant version of the CKM matrix around. It is the number crunched result of an observation by Marni Sheppeard that the CKM matrix is approximately the sum of a 1-circulant and a 2-circulant matrix. Well, now we have it exactly that way. Victory! But more to come.
To get an experimental measurement from the new magic form, compute the magnitude of the complex number. For instance, to get the top right value of 0.0042982, compute
|+0.000466480 – 0.004273188 i| = sqrt(0.000466480^2 + 0.004273188^2)
Uh, I get 0.0042986, which is within rounding errors on the experimental measurement number.
More generally, the nine elements of the experimental CKM matrix can be found by taking one element from the set {+0.973313178, -0.008576543, +0.000466480} along with one element from the set {+0.225761835, +0.040012680, -0.004273188} and computing the RMS value.
So what does this mean?
It means I’m awfully close to writing the CKM matrix in a “tribimaximal” kind of form. What I am thinking about right now is unitary processes that convert between colors of three preons where two of the preons are identical and the third is not. In this, the 1-circulant values correspond to even permutations of color, while the 2-circulant numbers correspond to permutations of color that are odd. The 1-circulant amplitude +0.973313178 (“I”) corresponds to no color changes (which is even), while the 2-circulant amplitude +0.225761835 (“B”) corresponds to a swap of the colors of the identical particles. The other 2-circulant amplitudes, +0.040012680 (“R”), and -0.004273188 (“G”), correspond to swaps of the colors between the odd preon and one of the two identical preons. And the other two 1-circulant amplitudes, -0.008576543 (“J”), and +0.000466480 (“K”), are for the cyclic permutations of the colors. The letters I,J,K,R,G,B are the standard letters we use around here for the 6 permutations on 3 colors. Here we assume that Blue is the odd preon, and that the Red and Green preons are identical.
My intuition says that the +0.040012680 and -0.004273188 amplitudes are different, and the same for the -0.008576543, and +0.000466480 because of CP violation. That is, if these were equal (i.e. |R|=|G| and |J|=|K|), then there would be no CP violation.
The amplitude for a swap of two colors, 0.225761835 i, reminds me of the correction factor in the Koide formulas for the charged and neutral leptons. It’s apparently the amplitude for the basic preon interaction.
So what’s going on with the CKM matrix is telling us is that the two lowest generations of quarks, the u/c and d/s, are more similar to each other than the generation 3 quarks. And this suggests that the eigenvectors for the third generation is (1,1,1)/sqrt(3), while the eigenvectors for the two lower generations are (1,w,w*)/sqrt(3) and (1,w*,w)/sqrt(3) where w is the complex cubed root of unity. And this numbering is consistent with the generation numbers for the Koide formulas for the leptons.
Mass 
I wish I had a calculator, but its way down on my shopping list and I probably won’t get one until I have a postdoc somewhere. The Windows calculator is surprisingly bad. Anyway, let’s see if at least some phenomenologists find this fascinating. Shame on them if they don’t!
Marni, try my reverse polish notation, scientific java calculator. It’s designed so that you can copy numbers on and off of it with ease.
The large white space in the bottom is your “scrap paper”. You can write on it. If you press the “base” key, it prints the bottom of the stack to the bottom of the sheet of paper in the current base.
To put a number into the stack, just type it into the space, or copy and paste it there.
And it does Koide transformations. That is, you can convert 3 masses into sea, valence, and delta.
By the way, the new magic form for the CKM matrix has rows and columns that add up to the complex phase exp(i 15.1591642595 degrees) or exp(i 0.264577328) in radians. Sound familiar? Not to me. Yet.
Thanks, but I’m busy playing with Fourier operators!
Well, 0.264577328 is close to sqrt(pi * 9/2)^(-1). And one could use the first Riemann zero instead of 2/9.
Kea, I think that Fourier operators are probably the key.
Before we try and fit the 6 variables, which correspond to the I, J, K, R, G and B of the permutation group, we should do the reverse Fourier transform on them.
There should be some MUB theory in there. And I think that we will eventually be forced to model the quarks as linear superpositions of fairly messy combinations. And that should have something to do with why the MNS decomposition to real 1-circulant + imaginary 2-circulant is so balanced and simple, while the CKM decomposition is so much closer to 1.
Marni, there was another comment I wanted to add.
The cool thing about this decomposition is that there it is unique. It has no arbitrary complex phases. The usual parameterizations are constructed by making the CKM matrix as real as possible. What’s left is the CP violating part. However, there is a great deal of freedom in where you leave that CP violating part.
By contrast, this decomposition is unique. There are no choices. That means that those 6 numbers are not due to arbitrary choices of how to parameterize the matrix. They are fundamental constants of the CKM matrix and must have fundamental meanings.
Part of the reason I’ve been gung-ho on doing this is that I don’t believe in arbitrary complex phases. Density operator formalism eliminates the arbitrary complex phases of the usual state vector formalism. So if density operator formalism is better, the CKM matrix has to be written in a way such that arbitrary complex phases are eliminated. This is that way.
Agreed. Perhaps you could write a post summarising the precise values of the usual parameters for the CKM.
Kea, I think you’re jumping to the conclusion I’ve got the problem completely solved. No, I’m just convinced I’m on the road. I think those 6 numbers are approximations of a very simple formula that will involve the usual suspects; rational numbers, maybe some square roots.
But to make a guess, I need to understand it better. What I really want is a way of getting to the CKM matrix from the MNS matrix. Right now, I don’t have an explanation for why the MNS matrix is so simple in this form. I just think it’s a flashing neon sign saying “study me”.
It’s OK Carl. I realise we need to do more work.
Carl,
quite nice result! Are the numbers appearing in the rows of CKM rationals or approximations to algebraic numbers or reals? Have you tried to work U and D matrices which might have even simpler form?
The reason why I am asking this is that for years ago I constructed a CKM matrix by using as the basic mathematical input number theoretical universality stating that CKM as well as U and D make sense both as real and p-adic sense for a suitable algebraic extension of p-adics. See this
I also discussed a variational principle based on interpretation of the matrices U and D as describing the mixing of topologies with genus 0,1,2 (sphere, torus, …) for light-like wormhole throat associated with elementary fermion. The proposal was that U and D maximize their total entropy defined in terms of probabilities defined by the rows. The physical motivation was that parton orbits can be regarded as random light-like orbits so that entropy associated with the mixing by topology changing transitions for wormhole throat might be maximized. The variational principle predicts the probabilities as algebraic numbers and is consistent with the matrix found if I remember correctly.
Extrema usually have symmetries. For instance, entropy maximation would suggest that the entropies of different rows could be same in absence of constraints. Different mass squared values of quarks (integers when p-adic length scale is extracted out) however define a constraint. In any case, the composition of your CKM matrix might reflect a symmetry related to an extremum of entropy.
SU(3) in this case corresponds to a dynamical SU(3) assignable to genus. Fermions correspond to wormhole contacts with single light-like throat labelled by genus g and gauge bosons to pair of light-like wormhole throats of wormhole contact and labelled by (g1,g2). The prediction is that besides ordinary bosons which are SU(3) singlets also heavier variants which form SU(3) octet should exist and induce also neutral generation changing currents.
Thanks Matti,
The numbers are approximations consistent with the best experimental estimate of the CKM as a unitary matrix.
My feeling on them is that they have to do with the sum of a bunch of Feynman diagrams arranged according to how they change colors, and with coupling constants of something that has a phase factor of
times a real factor of something like 
.
The reason for the phase factor is because each time you swap a pair of colors you end up with a factor of i. The odd permutations have one swap and hence end up imaginary. The even permutations have zero or two swaps and hence end up real.
This defines the complex phase factor
. However, this constant is also what you get when you compute the Berry or Berry-Pancharatnam phase for a state that goes through three Pauli algebra MUBs. That is, this is the phase picked up in the product:
.
The sqrt(2)/3 comes from looking at the largest term other than the identity, which is close to 2/9.
I’d prefer 1/3 because that is color democracy; all colors treated equally. However, note that in the MNS matrix, the corresponding six constants are {sqrt(1/3),sqrt(1/6),0} and {i\sqrt(1/3),-i\sqrt(1/6),0}. This means that the bare preons do not have color democracy.
I think all this has to do with the MUBs of the Pauli algebra but I’m still working on a calculation.
Thank you for explaining.
It would be nice to try to understand what kind of forms for U and D could produce this kind of CKM matrix (U and D separately are not of course so fundamental in standard view as they are in mine).
Carl and Kea,
Beautiful work! This blog is living up to it’s title.
“This means that the bare preons do not have color democracy.”
Could this be related to whatever factor causes parity violation in beta decays? That ain’t democratic either.
Cheers,
Kris
Sorry to jump on something tangential on an impressive result, but to respond to Kea’s remark on a calculator and Carl’s clever RPN calculator with scratch space: if one uses Gedit on linux, one can use the multiple tabbed terminal plugin (http://fazibear.blogspot.com/2008/04/gedit-with-multi-terminal.html) and thus use any Computer Algebra System one would like…and then directly copy/paste the results in a TeX-nical note.
Thanks Angry, but no linux around here. Could someone please check the numbers at:
http://kea-monad.blogspot.com/2008/10/ckm-rules-iii_10.html