Minimal seesaw mechanism

In the framework of the seesaw mechanism, and adopting a typical form for the Dirac neutrino mass matrix, we discuss the impact of minimal forms of the Majorana neutrino mass matrix. These matrices contain four or three texture zeros and only two parameters, a scale factor and a hierarchy parameter. Some forms are not compatible with large lepton mixing and are ruled out. Moreover, a normal mass hierarchy for neutrinos is predicted.


I. INTRODUCTION
There is now a strong evidence for neutrino oscillations, especially through the SuperKamiokande, K2K, and SNO, KamLAND experiments [1]. Neutrino oscillations are naturally accounted for if neutrinos have small masses, so that leptons can mix in a similar way as quarks do [2]. Moreover, small neutrino masses can be achieved by means of the seesaw mechanism [3]. In this framework, the effective (Majorana) mass matrix of neutrinos M L is related to the Dirac neutrino mass matrix M ν and the heavy Majorana neutrino mass matrix M R by the relation As a matter of fact, the seesaw formula (1) is valid at the high M R scale, and therefore one should determine both M ν and M L at that scale, in order to find a consistent model. The effective matrix M L is partially described at the low scale through the analysis of several experiments [4]. On the other hand, the Dirac matrix M ν is based on theoretical hints. Both have to be renormalized to the M R scale. Then the problem is to find models for M ν and M R which reproduce the phenomenological forms of M L according to the master relation (1). Such a problem has been addressed in many papers (see, for instance, the review [5]). In the present article we consider a structure for M ν which is usually adopted for charged fermion mass matrices, and minimal models for M R . We select minimal forms which are compatible with phenomenology.
In section II we discuss the effective neutrino mass matrix. In section III we describe the Dirac and Majorana mass matrices of our minimal framework. Then, in section IV, the seesaw formula is applied and the resulting neutrino mass matrix is compared to phenomenology. A brief discussion is finally proposed. hierarchy m 1 < m 2 ≪ m 3 , and the inverse hierarchy m 1 ≃ m 2 ≫ m 3 . For the normal hierarchy the dominant elements are given by with m 3 ≃ ∆m 2 32 , and for the inverse hierarchy they are given by with m 1,2 ≃ ∆m 2 32 . Both contain a democratic µτ block, due to near maximal U µ3 . The difference stands in the element ee, which is suppressed in (6) but dominant in (7). Now, according to Ref. [7], the general structure of M L is not changed by renormalization. Therefore, we can take matrices (6) and (7) as simple forms at the high scale, the zero elements meaning suppressed with respect to dominant elements. They correspond to distinct predictions for the double beta decay parameter M ee , since for the normal hierarchy we get M ee ∼ ∆m 2 21 , while for the inverse hierarchy we have M ee ∼ ∆m 2 32 .

III. DIRAC AND MAJORANA MASS MATRICES
In order to apply the seesaw formula, we need the expression of the Dirac and Majorana mass matrices. We take a typical form for the three mass matrices of charged fermions [8]: for down and up quarks, and for charged leptons. Then, since M e ∼ M d , a natural choice is also M ν ∼ M u . In fact, we have [9] where λ = 0.22 is the Cabibbo parameter. The renormalization of quark mass matrices does not affect their expression in terms of powers of λ [10]. Therefore, for the Dirac neutrino mass matrix we take with a ≪ b ∼ c ≪ 1. As order in λ we have a ∼ λ 6 , c ∼ λ 4 . Expressions (8) and (9) lead to small quark mixings, while lepton mixings U e2 and U µ3 are large. The matrix M R should produce, through the seesaw formula, large lepton mixings [11]. For this Majorana mass matrix we consider minimal forms. These include matrices with four texture zeros: and matrices with three texture zeros, that is the diagonal form and the Zee-like form [12] M Parameters A, B, C can take values 1 and r < 1. Therefore, in such matrices there is a scale factor, related to m R , and possibly one hierarchy parameter r.

IV. SEESAW MECHANISM
In this section we calculate the effective neutrino mass matrix by means of the seesaw formula (1) on mass matrices discussed in the previous section. Then we look for structure of the kind (6) and (7). We exclude possible cancellations during our analysis.

Matrix (12) leads to
Condition M µτ ∼ M τ τ gives A/B ∼ ac. Then A = r ∼ ac and B = 1. Condition M µµ ∼ M τ τ is satisfied as a consequence. See also Ref. [13] for a discussion on this structure. Matrix (13) leads to Here condition M µτ ∼ M τ τ is satisfied while M µµ ∼ M τ τ requires A/B ∼ a 2 /c. Hence A = r ∼ a 2 /c and B = 1. Matrix (14) leads to so that M µτ ∼ M τ τ gives B/A c 2 /a. Then A = 1 and B = r c 2 /a. The condition M µµ ∼ M τ τ is valid as a consequence.
The normal hierarchy is achieved in all cases. However, note the three different values for the scale m R , that is m 2 t /m 3 , cm 2 t /m 3 , am 2 t /m 3 , respectively.

B. Diagonal form
In this case the effective neutrino mass matrix is given by Here condition M µτ ∼ M τ τ gives B/C c 2 . Then C = 1 and B = r c 2 . Both A = r and A = 1 are consistent with M µµ ∼ M τ τ . The normal hierarchy is obtained. The scale m R is given by m 2 t /m 3 . Large lepton mixing can indeed be obtained even by means of small mixing in M ν and zero mixing in M R (see Ref. [14]). In particular, for A ≃ B ≃ c 2 we get with k = a/c. This form of M L has already been proposed several times [15]. Moreover, the same form is realized in (19) for B ≃ c 2 /a, but with the scale m R suppressed by the factor a with respect to (20).

C. Zee-like form
In this case, apart from an overall scale m 2 t /2m R , we get the following approximate effective matrix (22) In order to have a useful µτ block, the leading terms must be those with AC in the denominator. Then the normal hierarchy is achieved for A = C = r c and B = 1. Here the scale m R is about m 2 t /m 3 .

V. DISCUSSION
We have studied the seesaw mechanism assuming simple forms of the fermion mass matrices and in particular minimal forms for the heavy neutrino mass matrix, which contain four or three texture zeros, a scale factor and a hierarchy parameter. Our minimal framework allows only the normal hierarchy for light neutrinos and not the inverse hierarchy. Large lepton mixing is achieved by tuning the hierarchy parameter r in the heavy neutrino mass matrix.
There is another possible mass spectrum for neutrinos, the degenerate spectrum, m 1 ≃ m 2 ≃ m 3 , which gives the dominant elements M L ∼ diag(1, 1, 1)m 1,2,3 or One can easily check that such forms are not reproduced in our minimal framework. However, both the inverse hierarchy and the degenerate spectrum can be achieved in some nonminimal models [16]. Indeed, generally it is quite hard to yield degeneracy in M L from hierarchy in M ν by means of M R in the seesaw formula.
The present framework could also be embedded in a unified SO(10) model. In fact, the relations M e ∼ M d and M ν ∼ M u can be the result of a quark-lepton symmetry, and the high M R scale can as well be related to the unification or intermediate breaking scale of the supersymmetric or nonsupersymmetric model, respectively [17]. Then, matrix (13) and especially matrix (14) correspond to the nonsupersymmetric model, while matrices (12), (15) and possibly (16) correspond to the supersymmetric model.
We thank F. Buccella for discussions.