Neutrino masses: hierarchy without hierarchy

A large hierarchy of the Dirac masses can result in a small hierarchy for the low energy masses of the active neutrinos. This can happen even if the Majorana masses of right-handed neutrinos are all equal. A realistic description of the observed neutrino masses and mixing can be obtained starting from a large hierarchy in the Dirac masses. A large mixing for solar neutrinos results from the neutrino sector. The small value of the MNS matrix element Ue3 is a natural consequence of the scheme. The masses of the two lighter neutrinos are related to the solar neutrino mixing angle: μ1/μ2 = tan 2 θ⊙. 1 Is there any mass hierarchy for active neutrinos? Let us start with the remark that the active neutrinos are exactly these particles which experimentalists are studying. They couple to W and Z bosons. There are three of them known as νe, νμ and ντ , and they have very small masses which are reflected in mass scales governing neutrino oscillations. In the oscillations of the solar and atmospheric neutrinos only differences[1, 2, 3] ∆m⊙ = |μ 2 2 − μ 2 1| ∼ 5.0× 10 −5eV 2 (1) ∆m@ = |μ 2 3 − μ 2 2| ∼ 2.5× 10 −3eV 2 (2) can be measured. The ratio of these two mass scales ρ−1 exp = ∆m 2 @/∆m 2 ⊙ ∼ 50 (3) seems to provide a clear answer to the question asked in the title of this section. Apparently yes. There is a hierarchy. However the correct answer may be more subtle. Let us compare what Nature tells us through eq.(3) with expectations based on a theory. The best theory of neutrino masses we know is the see-saw mechanism [4]. It explains why the masses of the active neutrinos are much smaller than the masses of all other fundamental fermions, i.e. charged leptons and quarks. The see-saw mechanism implies that the masses of the active neutrinos are composite low energy objects derived from more fundamental mass parameters. These more fundamental masses are the Dirac masses describing couplings between left-handed and right-handed neutrinos. and the Majorana masses of the right-handed neutrinos. The right-handed neutrinos are singlets of the standard model SU3 × SU2 × U1 local gauge symmetry, so their Majorana masses are not forbidden by gauge invariance. Majorana masses are not allowed for particles with non-zero electric charge. So, the masses of charged leptons and quarks are all of the Dirac type and they all exhibit a clear hierarchy me ≪ mμ ≪ mτ mu ≪ mc ≪ mt md ≪ ms ≪ mb (4) If we assume that this hierarchical structure is a common feature of all fundamental fermions, the Dirac masses of neutrinos should be also hierarchical, i.e. m1 ≪ m2 ≪ m3 (5) We still have to say something about the Majorana masses of the right-handed neutrinos. The most natural thing is to assume that they are all equal. So let us assume that there are three right-handed neutrinos and their Majorana masses are equal to M: MR = M1 (6) 1 Then the following sequence can be derived for the masses of three active neutrinos: μ1 = m1 M , μ2 = m2 M , μ3 = m3 M . (7) If m3/m2 ∼ mt/mc ∼ 10 is assumed, as suggested by many grand unified models, the ratio ρ−1 th ≈ μ3 μ2 = ( m3 m2 )4 ∼ 10 (8) is obtained. When viewed from this perspective the hierarchy exhibited in eq.(3) can be called a moderate one at best. It is much more appropriate in fact to consider this small hierarchy as a small perturbation of the situation without hierarchy. 2 Reducing hierarchy Are we then forced to abandon the assumed hierarchy (5) of the Dirac masses or the nice and economic postulate (6) of equal Majorana masses? Let us repeat the standard derivation of the mass formula for the active neutrinos. Our guiding principle is to reduce the resulting hierarchy as much as possible. The Dirac masses of neutrinos are described by a 3× 3 matrix N = URm UL (9) with m = diag (m1, m2, m3) (10) As an unitary matrix UL cannot affect the resulting mass spectrum, we assume UL = 1 (11) for simplicity. We may be led to reconsidering this when discussing the lepton mixing matrix. The mass spectrum of the active neutrinos is given by a dimension five operator N . This operator is obtained as a low energy approximation of a term resulting from the underlying renormalizable theory in the next-to-leading order. The result is N = NTM−1 R N = 1 M m T U RURm (ν) (12) As the Majorana mass M in (12) is huge the resulting masses of the active neutrinos are small. The spectrum is extremely sensitive to the form of a symmetric unitary matrix R = U RUR (13) 2 so, the matrix UR plays a very important role in low energy physics and its structure is imprinted in the masses of the active neutrinos. Unfortunately this mass spectrum is the only piece of information on UR accessible at our low energies. So we have to guess some form of R and hope that the results obtained may to some extend justify our cavalier attempt. R = 1 is not acceptable because this would lead us directly to the disastrous spectrum (7). Let us follow our guiding principle and try to reduce the hierarchy of the resulting spectrum as much as possible. Certainly