M Theory Lesson 259
Complex magic matrices also multiply to yield new magic matrices. If the row and column sums of two magic matrices are $e^{i \theta_1}$ and $e^{i \theta_2}$, the row and column sum of their product will be $e^{i (\theta_1 + \theta_2)}$.
Carl's parameterization of the CKM matrix $V$ results in a row sum phase with $\theta = -0.27308859$, close to a 23rd root of unity. In other words, the row sum is the complex number $0.96294248 - 0.26970686 i$. The $n$th power of such a complex matrix will have a row sum with $n$ times the angle, $n \theta$.
Observe that the number 0.96294248 is very close to $26/27$. This corresponds to the fact that $2 - 2 \times 0.9629 = 8/9$, which is the real part of a factor in a product form for the CKM matrix. That is, let $V = AB$. Now assume that the row sums for $A = A_1 + i A_2$ and $B = B_1 + i B_2$ are such that $A_1 + A_2 = B_1 + B_2 = 1$, where these numbers may be complex. Then it follows that the real part of $A_1$ equals $8/9$. We should probably check to see how an exact figure of $8/9$ compares with experiment.
Carl's parameterization of the CKM matrix $V$ results in a row sum phase with $\theta = -0.27308859$, close to a 23rd root of unity. In other words, the row sum is the complex number $0.96294248 - 0.26970686 i$. The $n$th power of such a complex matrix will have a row sum with $n$ times the angle, $n \theta$.
Observe that the number 0.96294248 is very close to $26/27$. This corresponds to the fact that $2 - 2 \times 0.9629 = 8/9$, which is the real part of a factor in a product form for the CKM matrix. That is, let $V = AB$. Now assume that the row sums for $A = A_1 + i A_2$ and $B = B_1 + i B_2$ are such that $A_1 + A_2 = B_1 + B_2 = 1$, where these numbers may be complex. Then it follows that the real part of $A_1$ equals $8/9$. We should probably check to see how an exact figure of $8/9$ compares with experiment.
posted by Kea | 2:30 PM
20 Comments:
Sigh. Sorry. It works better with Re(A1) = 28/27 so that Re(A2) = -1/27. Gee, I need a whiteboard.
Remote Kea
When you say that that angle is "close to 26/27" you really mean it. Around 25.9994/27. Hmmm.
Carl, it should be precise enough to use as a constraint on your 6 parameters, reducing them to 4.
Remote Kea
Since it's a restriction of phase, shouldn't that be a loss of one degree of freedom rather than 2?
No, I don't think so, because the real and imaginary parts count separately in the 1circ/2circ decompositions.
Remote Kea
Okay, I'm catching up with you. I had some trouble figuring out what you were doing because the phase isn't listed on the page you linked to, and when I ran my software to repeat that calcualtion, it gave a magic matrix with cos(theta) = 0.973314444 not 26/27.
But now I hit the "Fourier Transform" button and I see that you are not working on the CKM matrix, but instead on its Fourier Transform, and sure enough, the top left corner of the Fourier transform (i.e. the 1x1 block, not the 2x2 block) is a complex phase with cosine = 26/27 to high accuracy.
However, my program only does the Fourier transform of the total thing, not the FT of the separate 1-circ and 2-circ parts, so I have to rewrite some code to catch up with you there.
Er, Carl, I was looking at your Victory post, which is NOT the Fourier transform, no? Heh, that sounds pretty cool then, that is if the 26/27 appears in both versions of the CKM.
Remote Kea
No, hang on a minute, it's your gamma parameter from the linked post, so it probably is the Fourier transform.
But you do use gamma in the row sums for the actual CKM, so it does appear in both matrices. Good.
Remote Kea
Okay, I'm at least 50% convinced you've got the key. The FT are all very close to integers if you square the (real) entries and multiply by 729. This sounds like numerology but it is not as the accuracy is incredibly high.
The errors in the 1x1 block are
0.000020 + i 0.000073
The 2x2 block has only two complex entries and so only a total of 4 errors. The diagonal errors are:
0.00037 + i 0.0061
and the off diagonal errors are:
0.0000066 + i 0.0000068
The one high error, the imaginary contribution to the diagonal entry comes from a value of
i 0.006123343
to put this in sqrt(n)/27 form, you multiply by 27 and square it. This gives 0.027, which, expressed as an integer, is closest to 0. However, it is also fairly close to 1/27 so it could be a third order effect.
For the other values, the integers show up rather well. The real part of the 2x2 block diagonal is 0.978500428, and on multiplying this by 27 and squaring, one obtains 697.9906, which is close to the integer 698 = 729 - 31.
The other things one will be replacing by integers are 26.789 ~= 27 and and 4.087 ~= 4 for the off diagonal 2x2 block, and of course 675.97 ~= 676 = 26^2 and 53.028 ~= 53 for the 1x1 block.
As you are aware, but others may not be, 729 = 27^2 is a very important natural value for the denominators. It is the 6th power of 1/3, where 1/3 is the transition probability from color to color. This shows up repeatedly in my crap, for example, see the neutron proton mass splitting in post #305 and Fermi coupling constant in post #308. I'll publish the full CKM matrix later tonight.
The next step is to compute the inverse Fourier transform of the integerized matrix, and then see how close it is to the CKM.
And congratulations!
Doesn't quite get it, but it's pretty close. The errors show up in the (1,3) and (3,1) entries, that is, the CKM entries that are closest to zero.
The approximation based on the sqrt(n)/27 with a zero entry for the imaginary part of the 2x2 block, gives equal values for the CKM matrix in the (1,3) and (3,1) positions. Both work out to be 0.006229382 in absolute magnitude. They should be
(1,3): 0.00359(16) and
(3,1): 0.00874(37).
Thus the values that get stuck in these two entries work out to be the average of the measured values. I'm pretty sure that putting a nonzero value in there would dial the thing in, but I'm not sure of a justification.
Maybe the best justification is just "1st order approximation" and all that. By the way, it could also be that the MNS matrix is not exactly tribimaximal.
I should add that the other 3x3-2 = 7 CKM matrix entries are all compatible with experimental measurement and most are close to the centers. By the way, the values for the integers n_ij are:
(676 53),( 0 0 ),( 0 0 ),
( 0 0 ),(698 0 ),( 27 4),
( 0 0 ),( 27 4),(698 0).
To fix it, I need to change that (698 0) to (698 0.273), the resulting values in the (1,3) and (3,1) corners are better but not within the quoted error bars. Funny thing is that they are more compatible with the values given in 0706.3588 which analyzes the CKM matrix from the point of view of unitarity. Of course I am also doing a purely unitary calculation.
By the way, leaving that (698 0) entry makes the whole CKM matrix exactly symmetric.
There's another thing I should note, and that is that there are only 4 degrees of freedom in the CKM parameterization, but the Fourier transform involved 6 real variables. Consequently, I think it's unlikely that the integers given result in a CKM matrix that is unitary.
Okay, I see what you're saying about theta, alpha, beta, and gamma. The 26/27 is a statement about gamma.
So I went and did the approximation thingy on these angles. The result is that the angle alpha is approximately zero. This makes the CKM matrix symmetric. The squares of the cosines of theta, alpha, beta, and gamma, times 729, are:
(698, 0, 96, 676). This gives CKM matrix values of:
[.97415, .22582, .00647]
[.22582, .97333, .04032]
[.00647, .04032, .99916]
These are guaranteed to be consistent with unitary, but now three of them are excluded by experiment, the corners, and the 04032 in the rightmost column.
The thing about these angles, is that my choice of parameterization may be somewhat arbitrary as there are products of trig functions in there. (See the first page of the Victory post.)
OK, all good. Sorry, it's late here now, so I'll get back to this later. I'm glad somebody has better tools than my lousy calculator!
Remote Kea
And it's late here. My recent suspicions have been that the important numbers are not the amplitudes but instead their squares.
This amounts to treating the 1-circ and 2-circ parts as two separate probability matrices. The square roots just allow them to be combined in such a way that their squared magnitudes are the correct total probabilities.
So I took the squares of the real and imaginary components and summed them up. Their ratio is 18 plus 1/729. That is, the real values are larger.
OK, Carl, this is not just numerology. First note that the row sum of the cubed root of V (the CKM matrix) has a real part given by cos(1/3 cos^(-1)Re(Vsum) ) where Vsum = 26/27 is the row sum (of the 1x1 piece at least) for the Fouriered CKM. This number is given by
sqrt(723/729)
which happens to be precisely the complement of the small 0.064 values in the small amplitude spots that make up the difference from 1 for the 2x2 piece of the Fouriered CKM. That is, 0.064 = sqrt(3/729) and two lots of three add to the 723 to give 729. The 697/729 gives the 0.9778 and the remaining 29/729 gives the 0.19945.
Remote Kea
If we consider the adjusted Fourier CKM (ie. with the 0.064 non-zero entries included as a correction from perfect 1x1/2x2 blockness) then what do we get for V? Plenty of number crunching to do here.
Unfortunately, tomorrow I will be working hard at a local vineyard, but I suppose that's nicer than waitressing, or building gib board ceilings and lining a room with insulation, which I did last week.
Remote Kea
To clarify: there is a real unitary magic matrix within which the 2x2 piece of the Fourier CKM sits. In place of zeroes there are numbers sqrt(3/729) and in the 1x1 space is the real part of the row sum for the cubed root of the CKM! This matrix is symmetric.
This matrix (x 729) is therefore specified by the array
(723)(3)(3),
(3)(697)(29)
(3)(29)(697)
Remote Kea
Observe now that 697 + 26 = 723 and that 29 = 26 + 3. The whole CKM therefore hinges on the numbers 26 (= 27 - 1) and 3.
Now that's cool.
Remote Kea
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