Magic Neutrino Mass Matrix and the Bjorken-Harrison-Scott Parameterization

Observed neutrino mixing can be described by a tribimaximal MNS matrix. The resulting neutrino mass matrix in the basis of a diagonal charged lepton mass matrix is both 2-3 symmetric and magic. By a magic matrix, I mean one whose row sums and column sums are all identical. I study what happens if 2-3 symmetry is broken but the magic symmetry is kept intact. In that case, the mixing matrix is parameterized by a single complex parameter $U_{e3}$, in a form discussed recently by Bjorken, Harrison, and Scott.

We shall not discuss such models, but shall persue possible deviations through a symmetry structure of the neutrino mass matrix.
The HPS mixing possesses two remarkable properties. In the basis where the charged lepton mass matrix is diagonal, it gives rise to a neutrino mass matrix that is 2-3 symmetric, so that when we interchange the second and the third rows, and simultaneously the second and the third columns, the mass matrix remains the same. See eq. (2) below. Such a 2-3 symmetry has been extensively studied [4]. It leads to maximal atmospheric mixing with sin 2 θ 23 = 1, and a vanishing reactor angle with sin 2 θ 13 = 0. The mixing matrix is real, so no Dirac phase is present, but the three eigen-masses are unknown and could be complex, thus admitting arbitrary Majorana phases and neutrino masses.
As we shall show, the HPS neutrino mass matrix is also magic, in the sense that the sum of each column and the sum of each row are all identical. I call such a matrix magic because it reminds me of magic squares, though the latter also have identical diagonal sums.
It is this magic property of the mass matrix that we want to investigate, with or without 2-3 symmetry. For specific examples, see, for example, Ref. [5]. It turns out that the mixing matrix for magic mass matrices can be parameterized in the way proposed recently by Bjorken, Harrison, and Scott (BHS) [6], with arbitrary Majorana phases and neutrino masses.

II. HPS MIXING MATRIX
The HPS form of the MNS [7] mixing matrix is If m i are the neutrino masses, possibility complex to absorb the two Majorana phases, and if m = diag(m 1 , m 2 , m 3 ), then the neutrino mass matrix may be written It is 2-3 symmetric. It is also magic, because the row sums and column sums are each equal to m 2 . Conversely, we shall show in the next section that a magic and 2-3 symmetric matrix gives rise to the mixing structure (1), so that a magic 2-3 symmetry for the mass matrix is synonymous to a HPS mixing structure.

III. MAGIC MATRIX
An n × n matrix A will be called magic if the row sums and the column sums are all equal to a common number α: For example, every permutation matrix of n objects has a single 1 in every row and every column, and 0 elsewhere, so it is a magic matrix with α = 1.
It is easy to see that magic matrices are closed under addition, multiplication, inverse, and scalar multiplication. In other words, if c is a number, and A, B are magic matrices with common sums α, β, then A + B, AB, A −1 , and cA are all magic, with common sums α + β, αβ, α −1 , and cα. Such closure properties are also true for 2-3 symmetric matrices.
Eq. (3) tells us that A has an eigenvector v, with eigenvalue α. Namely, if all the n components v i of the vector v are equal, then This eigenvector is normalized if we choose v i = 1/ √ n.
We specialize now to 3 × 3 matrix M that is magic. To satisfy (3), it must have the general form This is the case with the HPS mass matrix in (2).
Our task is to find the mixing matrix U for a symmetric magic mass matrix M, i.e., a unitary matrix U so that U T MU becomes a diagonal matrix m = diag(m 1 , m 2 , m 3 ). Since U is unitary, we can write this relation as MU = U * m, so that if u 1 , u 2 , u 3 are the three column vectors of U. Since the normalized eigenvector v in (4) is real, it satisfies (6) and is one of the three column vectors u i . By comparing with the HPS mixing matrix (1), we see that v = u 2 . In other words, the second neutrino mass eigenstate ν 2 always mixes democratically with all three neutrino flavor states, thus a magic mass matrix leads to a trimaximal mixing for ν 2 . This is the basic assumption involved in the Bjorken-Harrison-Scott (BHS) parameterization [6] of the mixing matrix. The rest of the matrix elements in U is determined by unitarity and allowed phase choices. The result, as given by BHS, is where U e3 is complex so it contains the Dirac phase, allowing CP violation, and C = 1 − 3|U e3 | 2 /2.

Note that
for j = 1 and 3, because these two columns must be orthogonal to the second column.
If M is 2-3 symmetric as well, then U e3 = 0 [4], so U = U HP S as claimed.
Conversely, in terms of the neutrino masses m i (possibly complex after absorbing the Majorana phases), the mass matrix M = UmU T can be calculated from (7). Such a matrix is necessarily magic on account of (8): IV. CONCLUSION In conclusion, present experimental data are consistent with the HPS mixing of eq. (1), which exhibits a trimaximal mixing for ν 2 and a bimaximal mixing for ν 3 . In the basis where the charged-lepton mass matrix is diagonal, the neutrino mass matrix M possesses both magic symmetry and 2-3 symmetry. If magic symmetry is broken but 2-3 symmetry is kept, then [4] the bimaximal structure of ν 3 is retained, but the trimaximal nature of ν 2 is broken, thus keeping atmospheric mixing maximal and the reactor angle zero, but the solar mixing is left as a free parameter. If 2-3 symmetry is broken but magic symmetry is kept, then it retains the trimaximal structure of ν 2 while breaking the bimaximal nature of ν 3 . The mixing matrix in this case is parameterized by Bjorken, Harrison, and Scott [6], with a single complex parameter U e3 , thus allowing CP violation. There are also two relations between the mixing parameters, arising from the trimaximal structure U e2 = U µ2 = U τ 2 = 1/ √ 3. In terms of the usual Chau-Keung parameterization of the mixing angles [8], one relation is sin θ 12 cos θ 13 = 1/ √ 3, and the other involves the Dirac phase angle.
If both the 2-3 and the magic symmetries are violated, then we are back to the general case with a full blown Chau-Yeung parameterization.
This research is support by the Engineering and Natural Science Research Council of Canada.