Texture and cofactor zeros of the neutrino mass matrix

We study Majorana neutrino mass matrices that have two texture zeros, or two cofactor zeros, or one texture zero and one cofactor zero. The two texture/cofactor zero conditions give four constraints, which in conjunction with the five measured oscillation parameters completely determine the nine independent real parameters of the neutrino mass matrix. We also study the implications that future measurements of neutrinoless double beta decay and the Dirac CP phase will have on these cases.


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2 Two texture zeros The condition for a vanishing element M αβ = M * αβ = 0 is Since there are two such constraints that depend linearly on the masses, the masses are related by where c j are complex numbers that are quartic in the matrix elements of U . Then, with δm 2 = m 2 2 − m 2 1 and ∆m 2 = |m 2 3 − 1 2 (m 2 1 + m 2 2 )|, we get two equations that relate m 1 to the oscillation parameters and the Dirac phase δ, where the plus and minus signs correspond to the normal hierarchy (NH) and the inverted hierarchy (IH), respectively. (For the NH the lightest mass is m 1 , and for the IH the lightest mass is m 3 = m 2 1 + 1 2 δm 2 − ∆m 2 .) For a fixed set of oscillation parameters each of these two equations give m 1 as a function of δ, and the intersections of the curves give the allowed values of m 1 and δ. We use the data from the latest global fit of ref. [12] to find the 2σ allowed regions for the lightest mass and δ that satisfy eqs. (2.3) and (2.4). Note that if we replace δ by −δ, the two constraints from the eqs. (2.3) and (2.4) will be the same since the magnitude of c i does not depend on the sign of δ, but because the latest global fit has a preference for negative values of δ [12], the allowed regions for 0 ≤ δ ≤ 180 • are a little larger than for 180 • ≤ δ ≤ 360 • .
For two texture zeros in the mass matrix, there are 6! 2!4! = 15 different cases to consider. If two off-diagonal entries vanish, the mass matrices are block diagonal and have one neutrino decoupled from the others, which is inconsistent with the data. Therefore, we only need to consider 12 cases that can be divided into three categories: 1. One zero on diagonal, off-diagonal zero sharing column and row. The six possibilities of this type, X 1 , X 2 , X 3 , X 4 , X 5 , and X 6 , are displayed in table 1. Using the unitarity of U and the fact that the cofactors of U ij are equal to U * ij , e.g., U e1 U µ2 − U e2 U µ1 = U * τ 3 , we obtain the simplified expressions for c 1 , c 2 , and c 3 provided in table 1. From a numerical analysis, we find that at the 2σ level, only X 1 , X 2 and X 5 are allowed for the normal hierarchy and X 5 and X 6 are allowed for the inverted hierarchy. The allowed regions for X 1 and X 2 for the normal hierarchy are shown in figures 1 and 2. The allowed regions for X 5 for the normal and inverted hierarchy are shown in figures 3 and 4, respectively. For the best-fit values of the measured oscillation parameters, X 2 NH and X 5 IH are not allowed, and the best-fit points for X 1 NH and X 5 NH are shown in figures 1 and 3 respectively. Both hierarchies for X 5 have nearly maximal JHEP09(2014)013 CP violation, i.e., δ close to 90 • or 270 • , and a lower bound on the lightest mass of about 30 meV. For X 5 NH and X 5 IH, the upper bound on the lightest mass is about 290 meV and 250 meV, respectively. For comparison, the 95% C.L. limit from cosmology is m i < 660 meV [13]. The allowed region for X 6 IH is very similar to that for X 5 IH.
2. One zero on diagonal, off-diagonal zero not sharing column and row. The three possibilities of this type, Y 1 , Y 2 and Y 3 , and the corresponding c i 's are displayed in table 2. At the 2σ level, Y 1 and Y 2 are allowed for the inverted hierarchy, and their allowed regions are very similar to that for X 5 IH; Y 1 is also allowed for the normal hierarchy and the allowed region is very similar to that for X 5 NH; Y 3 is excluded at 2σ. All the allowed cases have nearly maximal CP violation, and a lower bound on the lightest mass of about 30 meV, similar to X 5 NH and X 5 IH; see figures 3 and 4.     Table 2. The expressions for c 1 , c 2 and c 3 for Class Y. The symbol × denotes a nonzero matrix element.
the inverted hierarchy is allowed at the 2σ level, and the allowed regions are shown in figure 5. Z 1 for the normal hierarchy is excluded at 2σ for m 1 < 0.3 eV, which is consistent with the result of ref. [14].
Although the allowed regions for the seven acceptable textures of ref. [1] have been further restricted by the determination of θ 13 , all seven textures remain allowed. Further restrictions on the Dirac CP phase δ [4,5] can also be placed by the latest global fit [12] for each case. Table 3. The expressions for c 1 , c 2 and c 3 for Class Z. The symbol × denotes a nonzero matrix element.

Two cofactor zeros
In ref. [10] it was shown that for matrices with two zero cofactors, the lightest mass can vanish only if θ 13 = 0. Since θ 13 is nonzero at the 7.7σ level [15], we assume there are no vanishing neutrino masses and the mass matrix is invertible.
The above equation is the same as eq. (2.1), except that the m i 's are replaced by their inverses. Hence, we can follow a procedure similar to that for the TT case to find the favored values of the lightest mass and δ. Since the cofactor matrix is also diagonalized by the mixing matrix V , it cannot be block diagonal, and only 12 different patterns need to be considered. It is possible to employ the notation for the TT case if the locations of the two zeros are the same in the cofactor matrix as in the mass matrix. Then all the c i 's are identical, and the only difference from the TT case is that eqs. (2.3) and (2.4) are replaced by As for the TT case, there are three categories: 1. One zero on diagonal, off-diagonal zero sharing column and row. An interesting fact is that the two cofactor zero cases in this class yield the same allowed regions as for the two texture zero cases in the same class [6][7][8][9][10]; the correspondence is listed in

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Two cofactor zeros X 1 X 2 X 3 X 4 X 5 X 6 Two texture zeros X 3 X 4 X 1 X 2 X 6 X 5 Table 4. The correspondence between the two cofactor zero cases and two texture zero cases for Class X. table 4. The reason for this is that the two cofactor zero conditions in this category imply either two texture zeros, or three cofactor zeros in a row or column. The latter possibility which gives a vanishing mass is excluded since θ 13 = 0. From table 4, we readily find the cases that are allowed at 2σ: X 3 , X 4 and X 6 for the normal hierarchy, and X 5 and X 6 for the inverted hierarchy. The allowed regions in the m 1 (m 3 )-δ plane are the same as those for the corresponding cases in the TT ansatz.
2. One zero on diagonal, off-diagonal zero not sharing column and row. There are three possibilities of this type: Y 1 , Y 2 and Y 3 ; see table 2. At the 2σ level, Y 1 and Y 2 are allowed for the inverted hierarchy, and their allowed regions are very similar to that for TT X 5 IH; Y 2 is also allowed for the normal hierarchy and the allowed region is very similar to that for TT X 5 NH; Y 3 is excluded at 2σ. All the allowed cases have nearly maximal CP violation, and a lower bound on the lightest mass of about 30 meV, similar to TT X 5 NH and and TT X 5 IH.
3. Two zeros on diagonal. There are three possibilities of this type: Z 1 , Z 2 and Z 3 ; see table 3. We find numerically that Z 1 is allowed at 2σ for the normal hierarchy only. The other cases are excluded at 2σ. The allowed regions for Z 1 for the normal hierarchy are shown in figure 6.

One texture zero and one cofactor zero
There are 36 possibilities with one texture zero and one cofactor zero, of which 21 are equivalent to a TT case [11]. So we only need to study the remaining 15 cases listed in table 5. The two constraints M αβ = 0 and C α ′ β ′ = 0 can be written as and where A i = U * αi U * βi , and B i = U α ′ i U β ′ i for i = 1, 2, 3. Solving these two equations, we get Case Conditions  where Taking the absolute values of the above equations, we can find the two mass ratios, σ = m 2 /m 1 and ρ = m 3 /m 1 . Then, A numerical study shows that at 2σ, only 2D, 3F and 4B are allowed for the normal hierarchy, and only 2A, 2D, 3A, 3F , 4B and 6C are allowed for the inverted hierarchy. The allowed regions for 2D and 3F for the normal hierarchy are shown in figures 7 and 8, and the allowed regions for 2A, 4B and 6C for the inverted hierarchy are shown in figures 9, 10 and 11, respectively. The allowed region for 3A IH is very similar to that for 2A IH. The allowed region for 4B NH is very similar to that for TT X 5 NH, and the allowed regions for 2D IH and 3F IH are very similar to that for TT X 5 IH. They have nearly maximal CP violation, and a lower bound on the lightest mass of about 30 meV. For 3F NH and 6C IH there are four best-fit points since there are four solutions to the one texture zero and one cofactor zero conditions. JHEP09(2014)013

Discussion
There are 7 cases that are allowed at the 2σ level for the two texture zero ansatz, 7 cases that are allowed for the two cofactor zero ansatz, and 6 cases that are allowed for the one texture and one cofactor zero ansatz. Seven cases allow both hierarchies, so there are a total of 27 possible two-zero cases allowed at 2σ. However, there are many similarities among the allowed regions for these cases. In ref. [16] we noted that any case with a homogeneous relationship among elements of M with one mass hierarchy yields predictions for the oscillation parameters and phases similar to those given by a case with the same homogeneous relationship among cofactors of M with the opposite mass hierarchy. The only exceptions are when the lightest mass is small, of order 20 meV or less, or when the allowed ranges of the oscillation parameters differ significantly for the two mass hierarchies. The latter situation occurs for θ 23 , which is constrained at the 2σ level to be less than about 45.3 • for the NH but can have larger values for the IH.
A texture or cofactor zero is the simplest homogeneous relationship; therefore, CC cases can be dual to TT cases (and, of course, vice versa), and some TC cases can be dual to other such cases. 1 We can identify 8 cases where allowed regions are similar due to the dual-case argument, 6 cases where a case is allowed at 2σ but its dual case is not because the lightest mass is small, 5 cases where an IH case is allowed but its dual case is disfavored because θ 23 must be larger than 45.3 • . A complete listing of dual case relationships is given in table 6.
We note that the allowed regions for the CC Z 1 NH case ( figure 6) are similar to the allowed regions for its dual case, TT Z 1 IH (figure 5). The region for 90 • ≤ δ ≤ 180 • in figure 5 does not appear in figure 6 because θ 23 has values that are larger than 45.3 • for the inverted hierarchy for 90 • ≤ δ ≤ 180 • , while such values of θ 23 are disfavored for the normal hierarchy.
We can also use the effective Majorana mass for the neutrinoless double beta decay to differentiate two-zero cases. In table 7, we list the minimum and maximum values of |M ee | at the 2σ level for each case. Note that for TT X 1 and X 2 , and CC X 3 and X 4 , |M ee | is identically zero, and therefore they are not listed in the table. We also omit the cases of CC X 5 IH, X 6 NH and X 6 IH because they give the same phenomenology as the corresponding cases in the TT class, as given in table 4.
We find that there are four different types of cases phenomenologically:  Table 6. A listing of which allowed cases have dual cases that are also allowed, and which do not. The "Maybe" designation is for situations in which the dual case has a NH and θ 23 > 45.3 • ; the global analysis of ref. [12] suggests that for a NH, θ 23 < 45.3 • at 2σ. "Maybe" indicates that the exclusion of the dual case on this basis is not robust.
3F ), all of which have a minimum value for the lightest mass of about 30 meV. In all NH cases of this type, the maximum value for m 1 is about 290 meV and 35 meV < ∼ |M ee | < ∼ 290 meV; in all IH cases, the maximum value for m 3 is about 250 meV and 55 meV < ∼ |M ee | < ∼ 250 meV. Therefore it will be very difficult to distinguish these cases from each other.
3. Cases in which maximal CP violation is approached for larger values of the lightest mass. This group includes TT Z 1 IH and CC Z 1 NH (figures 5 and 6). In these cases a wide range of δ is possible, although maximal CP violation is not allowed.

4.
Cases that are a mixture of types 1 and 2. The TC cases 2D NH, 3F NH, 4B IH, and 6C IH allow values of the lightest mass less than 10 meV, and also have nearly maximal CP violation when the lightest mass is above about 30 meV (figures 7, 8, 10, and 11).