CKM as a Magic Unitary Matrix

Since writing the MNS as a magic unitary matrix, of course I’ve been working on writing the CKM matrix the same way. This involved learning a lot more about 3×3 unitary and 3×3 magic matrices, and writing a Java program to do the heavy lifting.

The first thing one must do to deal with magic unitary matrices is to define a parameterization of these matrices. A full parameterization of all unitary matrices requires 9 real variables. Five of these define the arbitrary complex phases that can be applied to any row or column (yes there are 3 rows and 3 columns, but one of them is redundant). The remaining 4 variables are usually written in the Wolfenstein parameterization. In this parameterization, three variables are mixing angles and the fourth variable defines the CP violation.

If one is given a magic unitary matrix, the effect of multiplying any row or column by a complex phase would be to destroy the magic. Consequently, putting a unitary matrix into magic form (if this can be done) amounts to choosing a set of unique phases. Therefore, the largest number of free parameters we can expect to need to define magic unitary matrices is the 4 used in the Wolfenstein parameterization (which also amounts to choosing a unique set of phases). In fact, I’ve found a parameterization of the unitary magic matrices on 4 real parameters so one supposes that any unitary matrix can be put into a (more or less unique) magic form.

First, let’s write some reduced notation. We will abbreviate the 1-circ and 2-circ matrices as follows:
Abbreviated circulant matrices
Note that that “E” and “F” are reversed in the above 2-circ definition. This is to make the Fourier transforms easier to deal with. In our abbreviated notation, we are to solve:
1-circulant + 2-circulant matrix as unitary

Sums of 1-circulants and 2-circulants
In solving the unitary magic matrix problem, the first thing to note is that the product of a 1-circ and a 2-circ is a 2-circ, and a product of two matrices of the same type gives a 1-circ result. The second thing to note is that the “dagger” operation preserves circularity. The right hand side of the above unitarity equation is a 1-circulant, so we can break the above equation into two equations, based on the circularity of the right and left hand sides:
Unitary magic equation split into circulant equations
In the above, I’ve abbreviated the circulant matrices even a little farther.

One can write out the above matrix products in terms of the A, B, C, D, E, and F. From the top equation one finds:
AA* + BB* + CC* + DD* + EE* + FF* = 1,
AB* + BC* + CA* + DE* + EF* + FD* = 0,
AC* + BA* + CB* + DF* + ED* + FE* = 0.
The top equation just says that the sum of the squared magnitudes of the six complex numbers must be 1. The other two equations are more difficult in that they mix terms, but they are identical equations, that is, one is the complex conjugate of the other. So we only need to consider one of them.

Eigenvectors and Eigenvalues
The two equations that mix terms can be written simply if we think of the (ABC) and (DEF) and kets and their complex conjugates as bras. Then the 2nd equation above can be written as:
Cross terms of 1-circulant part of magic unitary problem
This is in the form of a matrix element as used in quantum mechanics. In the above, the matrix is a permutation matrix, one that we have called “J” in our previous posts. It is useful to rewrite the (A B C) and (D E F) vectors to a basis of eigenvectors of the J matrix. The matrix is a 1-circulant matrix, such matrices have three eigenvectors:
(1 1 1) with eigenvalue 1
(1 w w*) with eigenvalue w
(1 w* w) with eigenvalue w*,
where w and w* are the complex cube roots of unity.

So to make the lower equation simple, we rewrite into the basis for the permutation matrix (as Kea says, this is a discrete Fourier transform, always a good thing to do):
(A B C) = A_1 (1 1 1) + A_2 (1 w w*) + A_3 (1 w* w),
(D E F) = D_1 (1 1 1) + D_2 (1 w w*) + D_3 (1 w* w).
Note that since our eigenvectors are unnormalized, we will end up with a factor of 3. But since the difficult equations we’re trying to solve are all crapola = 0, we can factor the sqrt(3) out, and we’ll just ignore this detail. And as far as a matrix element goes, the top equation is even simpler, it is the matrix element for the unit matrix. Accordingly, after Fourier transform, it will be unchanged; the squared magnitudes of the Fourier coefficients must still sum to 1.

The permutation matrix leaves the eigenvectors unchanged except for multiplication by the eigenvalues. Accordingly, the 2nd equation, where before the A, B, and C were mixed, become unmixed:
Fourier transformed cross terms for 1-circulant piece of magic unitary eqn
The above equation does not involve the phases of the parameters A_n and D_n. So the only source of complex phase are the w and w*. Accordingly, we can interpret the above as a statement about the magnitudes of the parameters.

On the complex plane, 1, w and w* are unit vectors separated by 120 degrees. For three real multiples of these constants to sum to 0, as in the above equation, amounts to a requirement that the sides of an equilateral triangle all be equal. Therefore we have that:
Results of equilateral triangle argument on 2nd 1-circ equation
Since the top 1-circ equation defined the sum of the squared magnitudes of the Fourier coefficients, we can parameterize the magnitudes of the Fourier coefficients with 3 real parameters as follows:
Parameterization of the magnitudes of the Fourier coefficients

The 2-circulant Part
This completely solves the parameterization problem for the 1-circ part of the problem. Now we move on to the 2-circ part, the part that mixes the (A B C) terms with the (D E F) terms:
The 2-circulant portion of the magic unitary matrix problem
As before, we have three independent complex equations, corresponding to the three complex parts of a 2-circulant matrix. After moving the terms around a little, we find:
(AD* + DA*) + (BF* + FB*) + (CE* + EC*) = 0,
(AF* + FA*) + (BE* + EB*) + (CD* + DC*) = 0,
(AE* + EA*) + (BD* + DB*) + (CF* + FC*) = 0.
The terms in parenthese are formulas for twice the real part of the various products beginning with AD*. Therefore we can rewrite these more tersely as:
Re( AD* + BF* + CE*) = 0,
Re( AF* + BE* + CD*) = 0,
Re( AE* + BD* + CF*) = 0.

The above three equations can also be written out in matrix form. As before, we find a set of 3 equations that look like matrix calculations for a quantum mechanics problem. But this time, the operator turns out to be the odd permutation matrices, the matrices we’ve previously referred to as “R”, “G”, and “B”. For example, the top equation becomes:
Example of 2-circulant magic unitary equation written in permutation operator form
The middle equation becomes the matrix element of G, and the bottom equation becomes the matrix element of B.

R, G, and B are odd permutation operators; they swap two elements of the vectors. So the (1 1 1) Fourier basis vector, which has all its elements equal, is left unchanged by these operators. The other basis vectors, (1 w w*) and (1 w* w) are swapped by R, G, and B, and, in addition, the vectors are multiplied by 1, w or w*:
R (1 w w*) = 1 (1 w* w),
R (1 w* w) = 1 (1 w w*),
G (1 w w*) = w*(1 w* w),
G (1 w* w) = w (1 w* w),
B (1 w w*) = w (1 w* w),
B (1 w* w) = w*(1 w w*).

This is not quite as convenient and simple as if they were eigenvectors of R, G, and B, but it is good enough. We end up with three Fourier transformed equations. As before, these all must be imaginary:
D_1* A_1 + 1 D_2* A_3 + 1 D_3* A_2,
D_1* A_1 + w D_2* A_3 + w* D_3* A_2,
D_1* A_1 + w* D_2* A_3 + w D_3*A_2.
Adding all three together, and noting 1+w+w* = 0, we find that D_1* A_1 is imaginary.

To get information on the other Fourier coefficients, take the sum and differences between the second and third equations. Since w+w* = -1, we have that
D_2*A_3 + D_3*A_2
is imaginary. And since w-w* is purely imaginary, we find that
D_2*A_3 – D_3*A_2
must be real.

As before, we treat D_2*A_3 and D_3*A_2 as vectors in the complex plane. Their sum is imaginary and their difference is real. For their sum to be imaginary, their real parts must be equal in magnitude and opposite in sign. And for their difference to be real, their imaginary parts must be equal. Therefore, we can write:
D_2* A_3 = R exp( i k)
D_3* A_2 = R exp( pi – i k).

Taking the magnitude of the above two equations, and remembering our parameterization of the magnitudes, we find that
Restriction on magnitude parameterization
and therefore, \theta_2 = \theta_3 if we allow for the magnitudes to be positive or negative (to account for the two solutions in x to tan(y) = x).

When one writes a magic matrix as the sum of a 1-circulant and a 2-circulant, one always has a symmetry (or redundancy) in that the sum is left unchanged by the transformation:
A_n = A_n + alpha,
D_n = D_n – alpha
for alpha any complex constant. To eliminate this redundancy, we can take alpha = D_1 and therefore assume D_1 = 0. This means we must have \theta_1 = 0 , and we have eliminated another one of the theta parameters; we have fully parameterized the magnitudes of A_n and D_n.

It remains to parameterize the phases. Since the magnitudes for D_2 and D_3 are the same, as are the magnitudes of A_2 and A_3, the cross terms given become pure phase equations. One ends up with three phase angle parameters. A complete parameterization is:
Parametric solution to magic unitary matrices
where \theta, \alpha, \beta, \gamma are the parameters. Note, in taking the above back through the inverse Fourier transform, one will have to deal with the factor of 3 that was left off earlier.

In the physics literature, the usual parameterization of a 3×3 unitary matrix is the Wolfenstein. The objective of the Wolfenstein parameterization is to make the matrix as real as possible. There are three Wolfenstein parameters that do not cause the matrix to become complex, while the 4th parameter gives the unavoidable complexity of a (CP violating) matrix. It is interesting that the parameterization I’ve found also splits its four parameters into three (\alpha, \beta, \gamma ) and a 1 (\theta ), but instead the three parameters are complex phases and the one parameter defines magnitudes which are real.

The CKM Matrix
Once one has a parameterization of the magic unitary 3×3 matrices, one is naturally drawn to see what the CKM matrix looks like in magic form. To do the calculation, I naturally wrote a java computer program. The program also checked my mathematics and corrected various errors. So in honor of the Java programming language, I’ll reproduce the actual screen shot (click to size it larger):

Magic CKM matrix

Magic CKM matrix

The magic unitary version of the CKM matrix is highlighted with the red bars.

While working all this out, I also found a few other things that I’m not sure has been mentioned before:

a) Every magic matrix can be written as the sum of a 1-circulant and a 2-circulant, and of course every sum of a 1-circulant and a 2-circulant is magic.

b) Every magic unitary 3×3 matrix has a complex phase as the sum of its rows and columns. To see this, note that the sum over a row or column is always A+B+C+D+E+F. When one converts to the Fourier transform, only the A_1 and D_1 contribute to this sum as the other terms appear multiplied by 1, w and w*, which sum to zero. Of course one can choose that phase to be 0, as the above paramaterization does.

c) And as noted, it appears that every unitary 3×3 matrix can be put into magic form, though so far I do not have a proof of this, only the fact that the number of parameters matches that of the Wolfenstein parameterization.


Filed under physics

21 responses to “CKM as a Magic Unitary Matrix

  1. Could the Chau-Keung parameterization be useful to you?
    It is as the product of 3 matrices

    c1 s1 0
    -s1 c1 0
    0 0 1

    c2 0 s2
    0 1 0
    -s2 0 c2

    1 0 0
    0 c3 s3
    0 -s3 c3

    where c1 and s1 mean cos and sin of angle 1 etc
    angle 1 mixed generations 1 and 2
    angle 3 mixes generations 2 and 3
    angle 2 mixes generations 1 and 3

    The complex phase is put in by
    multiplying s2 by exp( – I theta )
    multiplying -s2 by exp( – I theta )

    All three matrices have determinant 1


    1 0 0
    0 c3 s3
    0 -s3 c3

  2. Sorry for typos

    Should have multiplied -s2 by

    exp( + I theta )

    and I spelled net wrong for my web site

    I am trying to do this by iPhone
    with fat fingers


  3. carlbrannen

    Tony, the theory Kea and I are pursuing is that the generation structure of the elementary particles needs to be represented by circulant matrices and consequently the MNS and CKM matrices are most simple when written in circulant form. I don’t see how the C-K parameterization helps in this as it is not circulant.

    As you can see, I’ve got the CKM data in circulant form. The problem is that from here, there are still three degrees of freedom in how you write the matrix.

    You can move a constant times the democratic matrix (whose entries are all 1/3) between the 1-circulant and 2-circulant forms while keeping the same sum. And you can multiply by a complex phase. So with those two things, you have to specify 3 more real parameters.

    So our theory is that if you choose those three parameters wisely, you will end up with a simple form for the CKM matrix. I just haven’t found that simple form yet.

    And don’t worry, I’ll fix the .nev

  4. The connection I had in mind was

    Your A1 = CK c1
    Your D1 = CK s1

    Your A2 = CK c2
    Your D2 = CK s2

    Your A3 = CK c3
    Your D3 = CK s3

    This gives a specific although subtle correspondence between
    your matrix structures
    CK physical interpretations of the 3 angles as mixers of each of the 3 pairs of generations in 3-generation physics


  5. carlbrannen


    I don’t see any connection. I’m using an inherently complex parameterization while the Chau-Keung, like the other parameteizations, is designed to put all the complexity into a single parameter.

    In looking for the Cheu-Keung parametrization, I found a reference which also discusses magic matrices in the lepton mixing. It even uses the same notation! See: hep-ph/0606220.

  6. Thanks for the reference to 0606220 by Lam
    Lam says
    “… If both the 2-3 and the magic symmetries are violated, then we are back to the general case with a full-blown Chau-Keung parameterization …”.

    The Chau-Keung paper is Phys Rev Lett 53 (1984) 1802.


  7. In 0606220 Lam says
    “… If … magic symmetry is kept …
    In terms of the usual Chau-Keung parameterization …
    sin(theta12) cos(theta13) = 1 / sqrt(3) …”

    I think that experimentally roughly
    sin(theta12) = 0.22
    sin(theta13) = 0.0046

    so that
    cos(theta13) = about 1

    and the product s12 c13 = 0.22

    which is not close to 1/sqrt3

    Does that mean that the quark K-M 3×3 matrix cannot be magic consistently with experimental observations ?


  8. carlbrannen


    I must not have made it very clear, but the whole point of this blog post is that I managed to write the CKM matrix in magic form. Furthermore, I’m convinced (but do not yet have a proof) that all unitary matrices can be written in magic form. The form is given in the computer output at the bottom.

    By the way, it turns out that there’s a maybe 0.001% typo in the bottom right CKM absolute value that was used as an input to the program and I should fix it. If you want to start cranking on the numbers, ask me for newer data which is a little more accurate.

    The things I’m pursuing right now to write the CKM in a simple form is looking at how one can combine unitary matrices, and their transposes and inverses, together in such a way as to produce another unitary matrix.

    I’m thinking maybe the discrete Fourier decomposition shown in this post will be helpful for that.

  9. So is it fair to say given experimental results, that
    Lam shows that a Kobayashi-Maskawa 3×3 matrix with 3 angles and 1 phase cannot be magic
    you show that by introducing more phases you can make it magic?


  10. carlbrannen

    “So is it fair to say given experimental results, that
    Lam shows that a Kobayashi-Maskawa 3×3 matrix with 3 angles and 1 phase cannot be magic”

    No, as far as I know, EVERY unitary matrix can be put into magic form. One does this by changing the phases.

    With these mixing matrices, the phases of a unitary matrix are considered arbitrary as they can be individually adjusted according to the row or column. The reason for this is that measurements do not measure the phase but only the absolute value of the amplitude.

    To change the phase convention, one individually multiplies rows or columns by various complex phases. 3×3 matrices have 3 rows and 3 columns, so there are a total of 6 ways you can modify the phase convention this way, but if you multiply all the rows by a phase and also all the columns by its inverse, then the net result is no change to the matrix. Hence there are only 5 independent phases (real parameters) for a unitary matrix.

    To convert a unitary matrix into magic form, you choose these 5 phases so that the matrix becomes magic. I do not have a proof that this can always be done, but I’m pretty sure it can. If I was more of a mathematician and less of an engineer / phycisist, I’d either already know a proof or at least I’d care enough to find one.

    The reason I believe it can be done is from counting the independent real parameters used to define the magic unitary matrices (4), and comparing this to the independent real parameters used to define a unitary matrix having made the more conventional phase convention (also 4).

    My computer program gave the magic unitary matrix for the CKM matrix accurate to 3 or 4 places. It’s in the blog post, the last picture. For example, the top left element of the matrix is:
    0.949850840 + 0.216154559i.

    The middle left element of the matrix is:
    0.042330473 – 0.221757068i.
    To compare with the usual CKM matrix, you can take the squared magnitude of this complex number and it will be equal to the square of the experimental number which is 0.225766.
    That is:
    0.042330473^2 + 0.221757068^2
    = 0.225766^2.
    In other words, the matrix does give the experimentally measured CKM values. The error is shown in the botton right portion of the computer screen, the worst is 0.007 for the bottom right entry, but that’s because I put incorrect data into the input.

    To see that the matrix is unitary, take the Hermitian dot products between rows and verify that you get 1 for the same row and 0 for different ones. My program automatically does this (to check my computer programming) and it got a result of e^( – 71.48 ) for the error, which is zero to within the double precision arithmetic I used.

    To see that it is magic, add up a row or column, you will see that they all add to 1.

    Yes, it’s the CKM matrix and it is indisputably both unitary and magic.

  11. You make the matrix magic by fixing the 5 independent quark phases
    instead of incorporating them into the quark fields like most physicists do.
    What are the physical consequences of your quark field phase fixing?


  12. carlbrannen


    There’s no difference in how I’m fixing quark phases than how everyone else does it. I’m just choosing a different convention. In the usual sense, it’s no more physically significant than choosing one convention over another to write the CKM matrix. They’re all unitary matrices, and they all are compatible with experiment. For the C-K parameterization, one could obtain the same sort of thing by changing the order of the matrices in the product, for example. One would get a different parameterization, but not physically different.

    But it does seem to me that the MNS matrix having such a simple form this way could be significant. And that makes me wonder about the significance of writing a mixing matrix this way. The topic seems significant enough to need a blog post to describe at length.

  13. So are you saying that the conventional view
    that the 6 quark flavor field phases are real – 5 independent 6 real quark phases that get conventionally merged plus the 1 surviving KM phase
    amount to a choice of choice of zero complex phase – real axis – in your model
    you choose 5 (or 6?) fixed complex phases to put the KM matrix into magic form?

    Are you not introducing new phase degrees of freedom that should be given physical interpretation?


  14. carlbrannen


    “6 quark flavor field phases are real – 5 independent 6 real quark phases that get conventionally merged plus the 1 surviving KM phase”

    I think I understand the confusion. The standard parameterizations end up with 6 phases and 3 angles = 9 total real parameters. My way of doing it ends up with 8 phases and 1 angle also 9 real parameters.

    Either way uses 9 parameters to describe the CKM matrix. Both use 5 of these as the quark field phases which cannot be measured in experiment. The standard methods use three more as the theta_jk mixing angles. And their final parameter is the CP violation delta, which is a phase.

    In my parameterization, again 5 of these are quark field phases. Then I have 3 complex phases, alpha, beta, and gamma. And finally, the last parameter is the real angle theta.

    In either case there are 9 parameters to describe all possible unitary matrices. The actual values of the 5 quark field phases are different between the two cases just as the 5 quark phases would be different between two different conventional parameterizations such as the Wolfenstein versus the K-M, but the way they’re applied is the same (typically by premultiplying and post multiplying by diagonal matrices with phases down the diagonals).

    If you are given a unitary matrix to parameterize both ways, the matrix looks very different in the usual parameterizations than in mine. That’s due to how the two parameterizations use their remaining 4 parameters.

    In the conventional cases, these 4 parameters describe a matrix that is mostly real. Mine is mostly complex. But they’re equivalent under the right transformation of quark phases.

  15. So the difference is that

    Conventionally there are “… three … mixing angles …” with 3 real dimensions


    You have “… three complex phases …”

    I guess what confuses me is that it seems to me that your “three complex phases” seem to me to have 3 complex dimensions and so 6 real dimensions and so 3 more degrees of freedom than the conventional approach.


  16. More explicitly it seems to me that

    each of the 3 conventional.angles gives one real magnitude such as sin(alpha)


    each of your “three complex phases” seems to me to give both a magnitude and a phase angle.


  17. carlbrannen

    No, a complex phase is equivalent to one real degree of freedom.

    A real angle makes terms look like:
    (cos(theta), sin(theta) ).
    A complex phase looks like:
    cos(theta) +i sin(theta).

    Either way there’s only one real degree of freedom. These are the two ways of describing a circle. In 2-real dimensions as in (x,y), and in 1 complex dimension as in x+iy. But in either case, the circle is a 1-dimensional object described by a single real parameter. The complex phases don’t have a magnitude degree of freedom. Their magnitudes are all = 1. In both cases, the degree of freedom is an angle, but in the case when it shows up as cos + i sin, I like to call it a “phase” instead of an angle.

    I’ve got this stuff rewritten in a much more elegant fashion using idempotents and 2-d discrete Fourier transform. Very elegant. I’ll write it up tonight, God willing.

  18. In the conventional picture the sin(alpha) and cos(alpha) are not in orthogonal spaces like (sin(alpha),cos(alpha)) 2-dim space but are just two numbers on the same 1-dim real line
    which is why they are added that way in Lam’s paper when he discusses the magic properties

    So when you go to complex cos + i sin you really are introducing new degrees of freedom in order to get magic properties and the new degrees of freedom should have some physical interpretation


    Carl: You’re completely wrong on this and I give up trying to explain it to you. You’ll just have to figure it out for yourself.

  19. Pingback: Doubly Magic Matrices and the MNS « Mass

  20. Pingback: CKM as a magic unitary matrix II « Mass

  21. Pingback: A New Parameterization for 3×3 Unitary Matrices « Mass

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