Seesaw model in SO(10) with an upper limit on right-handed neutrino masses
In the framework of gauge unification and the seesaw mechanism, we show that the upper bound on the mass of the heaviest right-handed neutrino GeV, given by the Pati-Salam intermediate scale of spontaneous symmetry breaking, constrains the observables related to the left-handed light neutrino mass matrix. We assume such an upper limit on the masses of right-handed neutrinos and, as a first approximation, a Cabibbo form for the matrix that diagonalizes the Dirac neutrino matrix . Using the inverse seesaw formula, we show that our hypotheses imply a triangular relation in the complex plane of the light neutrino masses with the Majorana phases. We obtain normal hierarchy with an absolute scale for the light neutrino spectrum. Two regions are allowed for the lightest neutrino mass and for the Majorana phases, implying predictions for the neutrino mass measured in Tritium decay and for the double beta decay effective mass .
The present status of neutrino oscillations, conceived many years ago by Pontecorvo , provides the following approximate values for the square mass differences and the mixing angles of the PMNS matrix [2, 3] :
The following experimental limits constrain the effective mass matrix of the left-handed neutrinos :
These upper limits are respectively obtained from the high energy spectrum of the electron in nuclear beta decay, from the upper limit on the rate in neutrinoless double beta decay (for Majorana neutrinos) and from cosmology.
The lower limits on and are respectively obtained from the bounds
An upper limit has also been found for the component of the along the third mass eigenstate, supposedly the heaviest, i.e. the one that is not involved in solar neutrino oscillations :
It is generally recognized that unified gauge theories  provide a very natural framework for the seesaw model , accounting naturally for the fact that left-handed neutrinos have masses several orders of magnitude smaller than the charged fundamental fermions. Indeed, the 16 representation of contains, besides the 10 and of , a singlet that can get a large mass, unrelated to the electroweak symmetry breaking scale.
Moreover, in the most appealing gauge unified model, the one with Pati-Salam  intermediate symmetry, is broken around GeV [7, 8, 9], providing the scale for right-handed neutrino masses by the vacuum expectation value (VEV) of the 126 representation.
In one expects a spectrum for the eigenvalues of the Dirac neutrino mass matrix that is similar to the masses of the quarks with charge , apart from some scale factor due to the different scale dependence of quark and leptons masses.
It is also very reasonable to assume that the matrix appearing in the biunitary trasformation that diagonalizes the Dirac neutrino mass matrix has the same structure as the Cabibbo-Kobayashi-Maskawa matrix , namely a hierarchical structure, the mixing angle between the first two generations being larger than the other angles. This statement is stricly correct within the simplifying hypothesis of assuming that the Higgs bosons providing the Dirac masses and mixing belong to representations.
2 The inverse seesaw
In this paper we intend to deduce the consequences of two main hypothesis :
(i) We assume an upper limit for the right-handed neutrino masses.
(ii) Within the gauge unification scheme, the Dirac mass matrix (eigenvalues and mixing) has the same structure as the up quark mass matrix (eigenvalues and mixing).
More quantitatively, we shall assume for the eigenvalues of the Dirac neutrino mass matrix the same values than in , namely :
Moreover we shall take for a matrix that, to begin with, has the Cabibbo form with only different from zero
which was a very instructive approximation . The rest of the angles are considered as perturbations relatively to the simple ansatz (12) and, as shown in , even the quantitative features of the light left-handed neutrino spectrum are correctly described.
Let us consider the inverse seesaw formula :
Diagonalizing the neutrino Dirac mass matrix by
one gets the relation
where the matrix is defined by  :
Moreover, within , with the electroweak Higgs boson belonging to the 10 and 126 representations, and no component along the 120 representation, the mass matrices are symmetric. As a consequence, the unitary matrices and that diagonalize Dirac neutrino matrix (14) are related :
and the matrix (15) becomes
The Cabibbo limit (12) taken by us would be a good approximation of in the limit of quark-lepton symmetry, with only components along the representations for the electroweak Higgs, where should be equal to .
The neutrino mass matrix is diagonalized by the PMNS unitary neutrino mixing matrix, which reads :
in the approximation that we will consider here for the angle (10)
In writing (19) we have taken the maximal mixing angle for atmospheric neutrino oscillation and () and the angles and are the Majorana phases. We use in (19) the notation of Davidson et al.  for the Majorana phases, that have the ranges . In the PDG Tables  they are defined as and , with .
Then, the left-handed neutrino light mass matrix reads
are the light neutrino masses, real positive parameters, since the Majorana phases have been factorized out, as it should.
For the inverse of the matrix (21) we will have :
Therefore, being symmetric and unitary, the matrix is also symmetric.
Of interest for our discussion will be the consideration of the matrix that enters in r.h.s. of the expression (15) :
and in the limit (11) one has roughly
The expression (25) found for follows from the assumption (12) for the matrix . Let us notice that in all generality it will also depend on the square of the mixing angle between the third generation and the other two lighter ones, that is assumed to be small.
Let us first remark that a rather conservative upper limit on the mass of the heaviest right-handed neutrino of the order
Since and , according to (1) and (3) are monotonically increasing functions of , one has
and for the Majorana mass matrix of right-handed neutrinos one has :
3 Imposing an upper bound on the heaviest eigenvalue
Let us stress that large cancellations are required in (26) if we impose to the masses of the right-handed neutrinos the more stringent limit
i.e. the scale of spontaneous symmetry breaking in the unified gauge theory with Pati-Salam intermediate symmetry.
The trivial bound
would be effective to constrain to be smaller than GeV only in a region of unrealistically large neutrinos masses. In fact should be smaller about two orders of magnitude than , taking into account the upper limit on (7).
More precisely, only the third term in the r.h.s. of (33) would give rise at least, by assuming the largest value for consistent with the square masses differences fixed by the oscillations (3) and the rather conservative cosmological limit (7) on the sum of their masses, eV, to a mass around GeV, two orders of magnitude larger than the value expected in the ordinary unified model with Pati-Salam intermediate symmetry.
Therefore, we underline again that one needs a strong cancellation between the three terms in (25), which have moduli related by the square mass differences implied by neutrino oscillations.
Notice the very important point that this is already a hint for large relative Majorana phases. In this respect, it is interesting to look for the implications for the neutrinoless double beta decay effective mass (6) :
Owing to (25) can be exactly expressed in terms of and by the formula
4 A triangle in the complex plane of light neutrino masses and Majorana phases
Let us now examine carefully the consequences of the condition (38). This cancellation condition defines a triangle in the complex plane :
that we have drawn in Fig 1.
|Fig. 1. The triangle in the complex plane (39). The sides are given in terms of the inverses of|
|the light neutrino masses and the angles as functions of the Majorana phases and|
|of the light neutrino mixing matrix (19).|
Eqns. (1) and (3) give and in terms of and (39) may be satisfied if one has the inequality
which is violated for eV or in the range ( eV, eV).
We thus get two regions where the triangular relation holds :
On the boundaries of both regions one has but these two quantities can be reasonably large in their interior.
We plot in Figures 2 and 3 the dependence of the Majorana phases and for both regions I and II as functions of (in eV units).
|Fig. 2. The Majorana phases (red) and (blue) in Region I as function of in eV units|
|Fig. 3. The Majorana phases (red) and (blue) in Region II as function of in eV units|
As we see in Fig. 2, in Region I the phase decreases from to at and gets back to at , while increases from to (notice that we take in the range () and in the range (), but of course (39) holds also with the opposite choice). As we will see below, the fact that is rather close to for Region I will imply a strong cancellation in the effective mass of neutrinoless double beta decay.
In Region II both Majorana phases can get large values for moderate values of , where the sides of the triangle (39) are not very different.
Notice now the important remark that for both regions one must have normal hierarchy. The reason is the following. From eqn. (39) one gets the relation
which, for the inverted hierarchy (), would be about -1.
For the normal hierarchy (), eqn. (43) becomes and to have (eqn. (2)) one needs
As we have seen above, when vanishes, one has, from (37)
so that, once is fixed, the three quantities in (38) are also fixed, with and given by (1) and (3).
In the present scheme we have therefore for the appealing expression (45), which implies a negative interference between the two terms in (35) for small , and a positive one when approaches the cosmological bound.
On the other hand, the mass can be obtained from
4.1 A further discussion on the constraint GeV
Besides the main constraint (39), some words of caution are necessary to prevent a mass for the heavier right-handed neutrino to be not larger than GeV. We have also to check that
because multiplies the product of the two highest eigenvalues of the Dirac matrix, as we can see in (24), . It depends on the mixing angle and is given by
At the boundary of the allowed regions, we can tune the value of in order that holds, as following :
implying , at respectively, where , as we have seen above. In the first region, as soon as in the complex plane forms a large angle with , the cancellation between the two terms in (48) is impossible and, when they are just orthogonal, the coefficient of the term proportional to is at least of order , giving rise to two right-handed neutrinos around GeV and a lowest state around GeV.
In order to avoid a too small value for the mass of the lightest right-handed neutrino, a necessary condition is that is smaller than . This can be obtained by relaxing the condition . However, one gets anyway a too small mass for the lightest right-handed neutrino because of the range allowed for
would be equal to in the limit of vanishing . So, near the boundaries of Region I one gets the choice made recently  of a compact neutrino spectrum, as it is also the case with a large value of near . The other values of consistent with eqn. (37) imply a value higher than GeV for the two heaviest right-handed neutrinos, and a small value for the lightest one.
5 Phenomenological implications for low-energy physics
In conclusion, the choice of a compact spectrum seems the most natural, but it is useful to describe the phenomenological consequences of the other scenarios. We shall write the phenomenological consequences for the quantities, for which there are the limits written in (5)-(7) for the two regions (41) and (42) in the triangle (39). For the sum of the moduli of the neutrino masses we find in Region I values slightly above the lower limit + eV, while in Region II the sum of the neutrino masses is at least eV, it grows almost linearly and saturates the bound at eV.
We get always a small value for , in the range eV in Region I, while in Region II the relevant range is eV. We have limited the evaluation in Region II to eV, according to the bound (7).
For , the neutrino mass intervening in the tritium decay, it is confined to the ranges eV for Region I and eV for Region II.
To summarize, we obtain the following numerical results :
With reasonable hypotheses in the framework of unified theories, and by imposing the simple assumption of an upper bound on the mass of the heaviest right-handed neutrino GeV, as suggested by a Pati-Salam intermediate scale of spontaneous symmetry breaking, one gets interesting predictions for the physical quantities related to the effective mass matrix of the light left-handed neutrinos, namely on the mass of the lightest neutrino and on the Majorana phases.
Using the inverse seesaw formula, we have shown that our hypothesis of an upper bound for the right handed neutrino masses implies a triangular relation in the complex plane of the light neutrino masses with the Majorana phases. In a straightforward way we thus have predicted, on the one hand, normal hierarchy for the light neutrinos and a lower limit and an exclusion region for the mass of the lightest left-handed neutrino , implying an absolute scale for the light neutrino spectrum.
The allowed regions for are the range eV and the lower bound eV. For small , one of the Majorana phases can be close to , and we get a strong cancellation in the effective mass of neutrinoless double beta decay, and for light neutrino masses near the cosmological bound we obtain a positive interference for this quantity. Within our scheme we obtain also an interesting formula for just in terms of the three light neutrino masses, that is valid in both domains allowed for .
It is a pleasure to aknowledge a very inspiring discussion with Prof. A. Abada.
-  B. Pontecorvo, Sov. Phys. JETP 6, 429 (1958); 7, 172 (1958).
-  Z. Maki, M. Nakagawa and S. Sakata, Progr. Theor. Phys. 28, 870 (1962).
-  S.M. Bilenky and B. Pontecorvo, Phys. Rept. 41, 225 (1978).
-  H. Georgi, in Particles and Fields, ed. C. Carlson (AIP, New York, 1975); H. Fritzsch and P. Minkowski, Ann. Phys. 93, 193 (1975).
-  P. Minkowski, Phys. Lett. B 67, 421 (1977); M. Gell-Mann, P. Ramond and R. Slansky in Supergravity, eds. P. von Nieuwenhuizen and D. Freedman (North-Holland, Amsterdam, 1979); T. Yanagida in Unified Theories and Baryon Number in the Universe, eds. O. Sawada and A. Sugamoto (KEK, Tsukuba, 1979); R. N. Mohapatra and G. Senjanovic, Phys. Rev. Lett. 44, 912 (1980).
-  J.C. Pati and A. Salam, Phys. Rev. D 10, 275 (1974).
-  F. Buccella, L. Cocco, A. Sciarrino and T. Tuzi, Nucl. Phys. B 274, 559 (1986).
-  Dae-Gyu Lee, R.N. Mohapatra, M.K. Parida and M. Rani, Phys. Rev. D 51, 229 (1995).
-  F. Acampora, G. Amelino-Camelia, F. Buccella, O. Pisanti, L. Rosa and T. Tuzi, Nuovo Cim. A 108, 375 (1995).
-  N. Cabibbo, Phys. Rev. Lett. 10, 531 (1963); M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973).
-  E.K. Akhmedov, M. Frigerio and A.Yu. Smirnov, JHEP 0309, 021 (2003).
-  F. Buccella, D. Falcone and L. Oliver, Phys. Rev. D 83, 093013 (2011).
-  F. Buccella, D. Falcone, Mod. Phys. Lett. A 19, 2993 (2004).
-  S. Davidson, E. Nardi and Y. Nir, 0802.2962 [hep-ph], Phys. Rept. 466, 105 (2008).
-  K. Nakamura et al., J. Phys. G : Nucl. Part. Phys. 37, 075021 (2010) (Review of Particle Properties, p. 179).
-  M. Abud and F. Buccella, Int. J. Mod. Phys. A 16, 609 (2001).