Recently, Kea observed that the CKM matrix can be written as the sum of 1-circulant and 2-circulant matrices. The CKM matrix defines the relationship between flavor eigenstates and mass eigenstates for the quarks. Her observation naturally suggests one should look also at the MNS matrix, which defines the relationship between flavor and mass eigenstates for the leptons.
Current experimental measurements show that the MNS matrix is approximately tribimaximal:
The form of the matrix is chosen so that the rows and columns are orthonormal. This is the mathematical definition of unitary, it means that when the matrix is multiplied by its conjugate transpose, it will give the identity matrix. The physics definition of unitary is a little looser, it only requires that probabilities sum to 1.
For the case of a mixing matrix, probabilities are given by the squares of the absolute values of the entries. Conservation of probabilities means that the sum of the probabilities for any row or column is 1. Since in quantum mechanics, the phase of any quantum state is arbitrary, we can consider modifications to the tribimaximal matrix by multiplying any row or column by an arbitrary phase. The result will be a new matrix but its probabilities will be unchanged (and therefore consistent with experiment).
Now 1-circulant and 2-circulant 3×3 matrices are “magic” in the limited sense that the sum of a row or column are all equal (but not necessarily the sum of a diagonal). This is a requirement that is similar to the unitary requirement, but without taking the absolute value and squaring. Is it possible to modify the tribimaximal matrix so that it is also magic? Yes it is, and it turns out that there is only one way to do it, subject to the transformation . What’s more, the new matrix will have rows and columns that all add to 1, just like they sum in magnitude squared. The new MNS matrix is:
The reader will observe that the above matrix is magic in that each row or column sums to 6/6 = 1. In addition, taking the squared magnitude of the entries gives the probability matrix:
which is also magic, with sums of 1 for any row or column, and each probability is the same as that of the MNS matrix. The above is a very special matrix. Is it unique? Is there only one complex 3×3 matrix whose rows and columns sum to 1 and whose squared magnitude entries also have this property? I’ll look into this.
The 1-circulant and 2-circulant matrices are important for a bunch of reasons, not least of which is that their bases consist of the permutation on the group with 3 elements. Since we are doing physics here, and since we use R, G, and B for the three colors being permuted by Feynman diagrams corresponding to ways that a gluon type force can permute the colors of a state, we will use R, G, and B to designate the permutations that leave R, G, and B unchanged and swap the other two colors (these are the odd permutations on 3 elements). We will use I for the identity permutation that swaps nothing, and we will use J and K for the even permutations other than the identity.
The even permutations I,J,K pack together to fill a 3×3 1-circulant matrix, and the odd permutations R,G,B do the same, to fill a 3×3 2-circulant matrix:
We wish to write the new MNS matrix as a sum M = iI + jJ + kK + rR + gG + bB where i,j,k,r,g and b are complex numbers.
Since the R,G,B and I,J,K matrices all sum to the “democratic matrix” that has all entries equal to 1, our writing the MNS matrix in this form is not unique. We can always move a constant c between the (R,G,B) and the (I,J,K) matrices. That is, a split into 1-circulant and 2-circulant matrices defined by the constants (i,j,k,r,g,b) implies another split of form (i-c,j-c,k-c,r+c,g+c,b+c). Accordingly, we can assume that r+g+b = 0.
Then it is easy to find the values i,j,k. For example, the sum of the diagonal terms will be i+i+i + r+g+b = 3i + (r+g+b) = 3i. Similarly we obtain j and k. And then r,g, and b follow by subtraction. The resulting split into 1-circulant and 2-circulant matrices is:
As written, the 6 constants are written using a basis of 1, i, sqrt(2), and sqrt(2)i, all divided by 18. This acts like a vector space. The above can be simplified by choosing the constant c appropriately, however, this new result is fresh off the notebook, it is late and I will leave this post here. The reader can imagine that I will be checking beautiful result and looking for particularly simple versions of i,j,k,r,g,b while I should perhaps be attending to my nephews and nieces while visiting family this week.
But before I go, I should add that the fact that a matrix can be written as a sum of 1-circulant + 2-circulant seems to be equivalent to requiring that all the rows and columns of the matrix have the same value when dotted the vector (1,1,1). Geometrically, this means that all the rows and columns are an equal angle away from (1,1,1). Of course this is a familiar geometric interpretation of the Koide mass formula. Another way of expressing the same requirement is that the above new MNS matrix, when multiplied by the democratic matrix all of whose 3×3 = 9 elements are 1, will give the democratic matrix. In other words, the new MNS matrix is an eigenstate of the democratic matrix.