Arcadian Functor

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Marni D. Sheppeard

Thursday, October 09, 2008

CKM Rules

In a timely manner, Carl Brannen has posted his latest analysis of the CKM matrix for quarks. In the spirit of simplicity, the experimental matrix is given as a sum of a 1-circulant and 2-circulant, using only 6 real numbers. The 1-circulant is the real part and the 2-circulant the imaginary part. It is very precise, forming a new set of predictions for post standard model physics.

The plethora of recent blogposts on CKM has provided us with a goldmine of interesting links. For example, one historical paper mentioned here is a result [1] by Cecilia Jarlskog, whom I had the pleasure of meeting at Neutrino08.

One considers two $3 \times 3$ mass matrices $M_1$ and $M_2$, for $(u,c,t)$ and $(d,s,b)$. If $U_1$ (resp. $U_2$) diagonalises $M_1$ (resp. $M_2$) then one expects the CKM matrix to be of the form $V = U_1 U_{2}^{\dagger}$. If $M_1$ and $M_2$ are pure circulants, the same Fourier operator will diagonalise both, leading to the result $V = I$. So, as Carl has shown, the asymmetry between the quarks is what leads to CP violation and the CKM values.

[1] C. Jarlskog, Phys. Rev. Lett. 55 (1985) 1039

4 Comments:

Blogger CarlBrannen said...

Marni, I'm pretty sure that they already knew that there was some asymmetry in the quarks.

But I love the new form for its simplicity.

It uses 6 real numbers, and it is unique (up to sign of i). But there should be a parameterization that uses 4 real numbers only. I guess I'll be looking for this.

The computer program I used to find the formula does have four degrees of freedom, but it produces a matrix set up so that all the sums give 1. I guess I can work backwards through the modifications I made to split the reals and imaginaries into 1-circ and 2-circ, and so figure out the parameterization using just four variables. I suspect it will be complicated.

I should have time to look at this in the next few days. My gut feeling is that there is a more natural parameterization that has something about the summation of non commutative Feynman diagrams of MUBs about it. If so, that will be wonderful.

October 09, 2008 10:08 AM  
Blogger Kea said...

Sorry, I guess the word asymmetry is a bit vague in this context, but of course you know what I mean. I am playing with Fourier operators to see if I can recover such a decomposition.

October 09, 2008 10:19 AM  
Blogger Matti Pitkanen said...

I already posted this to Carl's blog but it might be worthwhile to add it also here.

I had two questions.

*Are the numbers appearing in the rows of CKM rationals or approximations to algebraic numbers or reals?

*Could U and D matrices have even simpler form?

The reason for asking 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 the CKM matrix proposed by Carl might reflect a symmetry related to an extremum of entropy.

SU(3) in my 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.

October 09, 2008 4:45 PM  
Blogger Kea said...

Thanks for the comment, Matti. I feel we are coming closer to TGD these days. Although, as Carl points out, we only know the entries as approximations, I am certainly picturing them as algebraic numbers myself, since the hierarchy of allowed truth values only constructs algebraic numbers up to dimension omega. And a lousy dimension 4 or 5 should be enough for impressive accuracy, I think.

October 10, 2008 10:00 AM  

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