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Stability and leptogenesis in left-right symmetric seesaw models

In left-right symmetric seesaw models an eight-fold degeneracy among the right-handed neutrino mass matrices is known to exist. We use stability and viability of leptogenesis as criteria in order to discriminate among the degenerate solutions and to partially lift the eight-fold degeneracy.


Introduction
It has become an established fact that neutrinos are massive, although very light. The Standard Model (SM) of particle physics is insufficient in explaining in a natural way why the neutrinos are so much lighter than the charged leptons. One attractive and elegant solution to this problem is provided by the seesaw mechanism [2], where the small masses of the neutrinos are induced by the presence of heavy particles, such as right-handed Majorana neutrinos or Higgs triplets. The seesaw mechanism has in addition the attractive feature of a built-in mechanism for generating the baryon asymmetry of the Universe (BAU) through the baryogenesis via leptogenesis mechanism [3], whereby the decays of the right-handed neutrinos produce a lepton asymmetry, which is then partly converted into a baryon asymmetry by sphaleron processes.
Recently, it was demonstrated [4,5] that in a certain class of left-right symmetric seesaw models, the seesaw mass formula can be inverted to yield eight (in the three-flavor case) possible mass matrices for the right-handed Majorana neutrinos, for a given light neutrino mass matrix and Dirac Yukawa couplings as input. Since the right-handed neutrino sector is not directly accessible by laboratory experiments, it is important to find ways of discriminating among the eight seesaw solutions. This was the objective of Refs. [1,6], where leptogenesis and stability of the solutions were used as selection criteria and it was shown that it is possible to partially lift the degeneracy in the righthanded neutrino sector.

The left-right symmetric seesaw model
We will consider the left-right symmetric seesaw model with gauge group G = SU (2) L ×SU (2) R ×U (1) B−L [7,8]. The SM field content is extended to include three heavy right-handed Majorana neutrinos N i and the Higgs sector contains an SU (2) L × SU (2) R bidoublet φ as well as SU (2) L,R triplets ∆ L,R . When the neutral components of the Higgs fields acquire vacuum expectation values (VEVs) we obtain the type I+II seesaw mass formula where v, v L , and v R denote the VEVs of the neutral components of φ and ∆ L,R , y denotes the Dirac Yukawa coupling matrix and f the triplet Yukawa coupling matrix. In general, the triplet Yukawa couplings for ∆ L,R could be different, but we will restrict to the case where there is a discrete left-right symmetry such that f L = f R ≡ f and y is symmetric. This case has much fewer parameters and is more predictive. We will assume that the Dirac Yukawa coupling matrix y equals the up-type quark Yukawa coupling matrix y u , which is motivated by grand unified theories (GUTs). Many of the results depend mostly on the fact that y has hierarchical eigenvalues. The free parameters will then be the absolute neutrino mass scale m 0 , the ratio of the triplet VEVs, and the light neutrino mass hierarchy. The seesaw formula Eq. (1) can now be inverted to yield 2 N solutions (for N generations) for f , and hence 2 N possible right-handed Majorana mass matrices, which are given by M R = f v R . In the one-generation case, the two solutions admit a simple analytical form Analytical solutions exist also in the three-flavor case and we label the eight solutions as '± ± ±' where '−'('+') denotes that the corresponding mass eigenvalue is type The labeling starts with the largest eigenvalue in the small v R /v L regime. Thus, there will be two solutions which are either pure type I or type II dominated for large v R /v L and six mixed solutions.
In Fig. 1 we show the eigenvalues and mixing for the solution '+++' for m 0 = 0.1 eV and inverted light neutrino mass hierarchy. As a measure of mixing, we have introduced the parameters u i which are related to the off-diagonal elements of the unitary matrix U which diagonalize f as: In the next section, we will consider ways of selecting among the eight solutions.
3 Discriminating among the degenerate solutions

Stability
First we will introduce a notion of naturalness, or stability, in order to select among the degenerate solutions. We pose the question if, for a given light neutrino mass matrix, the triplet Yukawa coupling matrix f has to be very special or fine-tuned in the sense that a marginally different f would give very different lowenergy neutrino phenomenology. We consider such a situation unnatural. In order to quantify the stability of the solutions we introduce the following stability measure The matrices f and m ν are determined by the real coefficients f k and m l according to where T k , k ∈ [1, 6], form a normalized basis of complex symmetric 3 × 3 matrices. This stability measure can be shown to be basis independent [1]. Fig. 2 shows the stability measure Q for the eight solutions, for m 0 = 0.1 eV and an inverted light neutrino mass hierarchy. First we note that all solutions are unstable for small v R /v L , which is due to the fact that in this regime there must be a very precise cancellation between the type I and type II contributions to the light neutrino mass matrix in Eq. (1). In addition, most of the solutions also become unstable for large v R /v L . This happens when there is a large spread in the eigenvalues, which imply suppressed mixing and fine-tuning between f and y (for a detailed discussion see Ref. [1]). The most stable solution is the purely type II dominated '+ + +' solution. If one allows for a fine-tuning at the percent level, Q 10 3 , then the two solution '± + +', for v R /v L 10 18 and the solutions '± − +' with v R /v L ≃ 10 20 are favored. The stability results do not change qualitatively, when changing m 0 or the light neutrino mass hierarchy. Additional CP violating phases only change the results marginally.

Leptogenesis
The second selection criteria we use is viability of leptogenesis for a given solution. Thus, we ask whether a solution can reproduce the observed BAU, η B = (6.1 ± 0.2) × 10 −10 [9]. The baryon asymmetry can be parametrized as where η is the so-called efficiency factor that takes into account washout effects, the initial density of the righthanded neutrinos and the deviation from equilibrium in their decays and ǫ N1 denotes the CP-asymmetry parameter, which measures the CP-asymmetry in the decay of the lightest right-handed neutrino. In Eq. (6) we have assumed that the lightest right-handed neutrino is much lighter than the other heavy neutrinos, such that it gives the dominating contribution to the lepton asymmetry. In Ref. [10] we study leptogenesis in the same framework, but where instead the decay of the triplet ∆ L is the source of the lepton asymmetry. The CP-asymmetry parameter is defined as From a numerical solution of the Boltzmann equations, it is possible to derive the following expression for the efficiency parameter assuming a mass hierarchy in the right-handed neutrino sector. Here we have also introduced the effective neutrino massm The notation with a hat indicates that all matrices are evaluated in the basis where f is real and diagonal. The CP-asymmetry parameter receives contributions of type I (from the right-handed neutrinos) and type II (from the triplets). In the limit where the lightest right-handed neutrino is considerably lighter than the heavier neutrinos as well as the triplets, the CPasymmetry can be written as [11] Our numerical results indicate that the upper bound on the CP asymmetry found in Ref. [11] can be saturated. An important feature of the left-right symmetric model is the presence of additional physical Majorana phases, which can increase the CP asymmetry and improve the prospects for leptogenesis. As demonstrated in Ref. [1], this allows for leptogenesis even with only one right-handed neutrino. In the case of three generations there are more sources of CP violation, although mixing can increase the effective neutrino mass, thereby increasing the washout. Fig. 3 shows that baryon-to-photon ratio η B for the solution '+ + +', when a Majorana phase κ is attributed to the electron neutrino. We have chosen κ = π/4 (π/8). The mass of the lightest right-handed neutrino required to reproduce the observed BAU is m N1 1.4 × 10 9 GeV (m N1 2.5 × 10 8 GeV). We find that leptogenesis is possible for four out of the eight solutions. For the other solutions, leptogenesis is not possible, since for these cases the mass of the lightest right-handed neutrino never exceeds 10 6 GeV. It should be noted that the results depend on the assumption that y = y u . For a different choice of Yukawa couplings it is possible to relax the mass bounds. However, the choice y = y u , as discussed before, is motivated by GUTs. Our results complements the results of the leptogenesis analysis in Ref. [6], since the washout is less severe for our specific choice of parameters.

Summary
We have studied the left-right symmetric type I+II seesaw mechanism with a hierarchical Dirac Yukawa coupling matrix motivated by GUTs. It has been shown previously that it is possible to invert the seesaw mass formula to obtain eight possible triplet Yukawa coupling matrices for a light neutrino mass matrix and Dirac Yukawa coupling matrix as input. Our goal was to discriminate among these degenerate solutions using stability properties and viability of leptogenesis as criteria. As a measure of stability we have introduced the parameter Q (see Eq. (4)) which quantifies the amount of fine-tuning.
Our results are summarized in Tab. 1. The stability criterion disfavors the four solutions of the type '±±−' and favors the solutions '±−+' provided that v R /v L ≃ 10 20 . The remaining two solutions '± + +' are stable, for v R /v L 10 18 . In addition, we find that successful leptogenesis is possible for the four solution of the ± + + ± − + ± ± − Stability vR/vL > 10 18 vR/vL ≃ 10 20 disfavored Leptogenesis vR/vL > 10 18 vR/vL > 10 18 excluded Gravitinos vR/vL < 10 21 unconstrained unconstrained type '± ± +' if v R /v L 10 18 . This requires the existence of additional Majorana-type phases, not present in the pure type I seesaw framework. Additional constraints would also be present in a supersymmetrized version, coming from the potentially dangerous overproduction of gravitinos in the early universe. For our specific choice of Yukawa couplings, y = y u , only the solutions of the type '± + +' are affected by this constraint. This leads to the requirement v R /v L 10 21 . Thus, we have shown, that it is possible, within the chosen framework, to partially lift the eight-fold degeneracy among the right-handed neutrino mass matrices in the left-right symmetric seesaw model.