Supersymmetric $SO(10)$-inspired leptogenesis and a new $N_2$-dominated scenario

We study the supersymmetric extension of $SO(10)$-inspired thermal leptogenesis showing the constraints on neutrino parameters and on the reheat temperature $T_{\rm RH}$ that derive from the condition of successful leptogenesis from next-to-lightest right handed (RH) neutrinos ($N_2$) decays and the more stringent ones when independence of the initial conditions (strong thermal leptogenesis) is superimposed. In the latter case, the increase of the lightest right-handed neutrino ($N_1$) decay parameters helps the wash-out of a pre-existing asymmetry and constraints relax compared to the non-supersymmetric case. We find significant changes especially in the case of large $\tan\beta$ values $(\gtrsim 15)$. In particular, for normal ordering, the atmospheric mixing angle can now be also maximal. The lightest (ordinary) neutrino mass is still constrained within the range $10 \lesssim m_1/{\rm meV} \lesssim 30$ (corresponding to $75\lesssim \sum_i m_i/{\rm meV} \lesssim 120$). Inverted ordering is still disfavoured, but an allowed region satisfying strong thermal leptogenesis opens up at large $\tan\beta$ values. We also study in detail the lower bound on $T_{\rm RH}$ finding $T_{\rm RH}\gtrsim 1 \times 10^{10}\,{\rm GeV}$ independently of the initial $N_2$ abundance. Finally, we propose a new $N_2$-dominated scenario where the $N_1$ mass is lower than the sphaleron freeze-out temperature. In this case there is no $N_1$ wash-out and we find $T_{\rm RH} \gtrsim 1\times 10^{9}\,{\rm GeV}$. These results indicate that $SO(10)$-inspired thermal leptogenesis can be made compatible with the upper bound from the gravitino problem, an important result in light of the role often played by supersymmetry in the quest of a realistic model of fermion masses.


Introduction
There is no evidence so far of new physics at the electroweak scale or below, in particular not of the kind that would be required in order to address the problem of the matter-antimatter asymmetry of the Universe within the Standard Model. 1 On the other hand, the lightness of neutrino masses, within a minimal type I seesaw mechanism [1], would point to the existence of a very high energy scale intriguingly close to the grand-unified scale. This encourages the idea that the cosmological matter-antimatter asymmetry might have been generated in the early Universe well above the electro-weak energy scale. Traditional high energy leptogenesis [4] scenarios based on the minimal type I seesaw mechanism naturally realise this interpretation of the current phenomenological picture. However, testing these scenarios is challenging, relying on the possibility to find the way to over-constrain the large seesaw parameter space imposing successful leptogenesis within a definite model of new physics embedding the type I seesaw mechanism.
A traditional, and somehow paradigmatic, example of models able to embed the type I seesaw mechanism realising leptogenesis is given by SO(10)-inspired models [5,6]. In these models the fermion mass matrices, including the RH neutrino Majorana mass matrix, are not independent of each other but linked by relations that reduce the number of independent parameters establishing connections, for example between the quark and the lepton sector. In particular, the Dirac neutrino masses are typically not too different from the up quark masses. Moreover the mismatch between the flavour basis, where the charged lepton mass matrix is diagonal and the Yukawa basis, where the neutrino Dirac mass matrix is diagonal, can be described by a unitary matrix acting on the left-handed neutrino fields with mixing angles comparable to those of the CKM matrix in the quark sector.
SO(10)-inspired relations are realised not only within traditional SO(10) models [7] but, mentioning some recent examples, also within models combining grand-unification with discrete flavour symmetries [8] or with extra dimensions [9]. Barring fine tuned cancellations in the seesaw formula, the resulting RH neutrino mass spectrum would be highly hierarchical with the RH neutrino masses proportional to the squares of the upquark masses with typical values (M 1 , M 2 , M 3 ) ∼ (10 5 , 10 11 , 10 15 ) GeV. In this case the final asymmetry has to be necessarily dominantly produced by the N 2 decays, since the 1 The recent diphoton excess reported by the ATLAS and CMS collaborations [2], if confirmed, might or might not have direct relevance for baryogenesis. It might have if the excess is explained for example by a new scalar as predicted in the NMSSM that would be able to re-open electroweak baryogenesis viability [3]. Or, more indirectly, the excess could be associated to a new resonance signalling the existence of new strong dynamics that might originate within a grand-unified theory embedding leptogenesis. contributions both from the N 1 and from the heaviest RH neutrinos (N 3 ) decays are too small to explain the observed value: an N 2 -dominated scenario of leptogenesis is therefore naturally realised [10]. It is interesting that this scenario necessarily requires the existence of at least three RH neutrino species in order for the N 2 CP asymmetries to get a sizeable contribution from the interference between tree level N 2 decays and one loop graphs with the exchange of virtual N 3 's. Therefore, there is an intriguing convergence between the SO(10) prediction for the existence of three RH neutrino species and the requirements of N 2 -dominated leptogenesis.
A challenging crucial aspect of this scenario is the necessity for the asymmetry produced by the N 2 decays to survive the N 1 wash-out. Flavour effects [11,12,13] greatly enhance the region in the space of parameters where the N 1 wash-out is negligible since this acts separately on the three charged lepton flavours [14,15]. In this way it has been shown that flavour effects indeed rescue SO(10)-inspired models with strong hierarchical RH neutrino spectrum [16]. Interestingly, imposing successful SO(10)-inspired leptogenesis one obtains constraints on low energy neutrino parameters that can be testable [17]. These have been also derived and explained analytically in the approximation where the mismatch between the flavour and the Yukawa basis is neglected [18]. In particular the lightest left-handed (LH) neutrino mass is constrained within the range 1 meV m 1 300 meV. The upper bound 2 has now been tested by latest cosmological results that place an upper bound on the sum of the neutrino masses i m i 0.23 eV [21], translating into m 1 70 meV. 3 Another interesting constraint is placed on the atmosperic mixing angle. This has to be necessarily in the second octant in the case of inverted ordered (IO) neutrino masses. More stringent constraints on the low energy neutrino parameters can be obtained superimposing additional conditions. An interesting possibility is to impose the so called strong thermal condition, the requirement that the asymmetry is independent of the initial conditions. This is indeed nicely realised within SO(10)-inspired models [22] and results into a 'strong thermal SO(10)-inspired solution' characterized by normally ordered (NO) neutrino masses, lightest neutrino mass in the range 10 meV m 1 30 meV, atmospheric mixing angle in the first octant and Dirac phase δ ∼ −45 • , in very nice agreement with current best fit results from neutrino oscillation experiments global analyses [23].
Recently it has been shown that flavour coupling [11,24,25,12] can help to open new 2 Notice that this is more relaxed compared to the upper bound holding in the N 1 dominated scenario where m 1 0.1 eV [19,20,15]. 3 Future cosmological observations should be able to constraint m 1 10 meV at 95% C.L. and in this case they would test most of the window allowed by SO(10)-inspired leptogenesis.
solutions [26] and these can be crucial to realise successful leptogenesis within specific models. An explicit example has been recently obtained in [27] within a specific realistic grand unified model, the 'A to Z model' [8], obtaining quite definite predictions on the atmospheric mixing angle (θ 23 ∼ 52 • ), Dirac phase (δ ∼ 20 • ) and on the ordering (NO). Alternatively, at the expense of very highly fine tuned seesaw cancellations, in the vicinity of a crossing level solutions one can have a departure from a very highly hierarchical pattern [6] in a way that M 1 can be uplifted and its CP asymmetry strongly enhanced.
Recently this kind of solution has been realised within a realistic fit of quark and neutrino parameters within SO(10) models. In this case the uplift of M 1 is also accompanied by a simultaneous decrease of M 3 so that a compact spectrum is obtained [28] and this can also lead to successful leptogenesis [29]. An unpleasant feature of these solutions, in addition to the very high fine tuning, is that, because of the uplift of M 1 , they predict NO and too small values for the neutrinoless double beta decay effective neutrino mass m ee to be measured [18]. Supersymmetric extensions of SO(10)-models are important since they offer a traditional way to address naturalness. At the same time they help improving the goodness of fits of lepton and quark parameters [30,28]. Recently [28] good fits of the fermion parameters have been obtained within SO(10) models with hierarchical RH neutrino masses and interestingly IO light neutrino masses, leading to values of m ee well in the reach of next generation neutrinoless double beta decay experiments. However, supersymmetry is typically implemented as a local symmetry leading to supergravity and in this case one has to worry whether successful thermal leptogenesis can be achieved with values of T RH compatible with the upper bound from the solution of the gravitino problem [31]. A quite conservative model independent upper bound, T RH 10 10 GeV, comes from preventing Dark Matter over abundance, where the Dark Matter particle can be either the neutralino or the gravitino itself or some other hidden sector lighter particle depending whether the gravitino is or it is not the lightest supersymmetric particle. 4 In this paper we extend the study of SO(10)-inspired leptogenesis to the supersymmetric case, showing how the constraints derived in the non-supersymmetric case change, with a particular focus on the lower bound on T RH . We find that in a traditional scenario, where the lightest RH neutrino wash-out has to be taken into account, this can be as low 4 It should be noticed, however, that different ways to circumvent even this upper bound have been proposed. For example thanks to entropy production diluting Dark Matter abundance [32] or in models with mixed axion/axino Dark Matter [33] or yet another way to evade completely this upper bound is that the gravitino is heavier than ∼ 10 7 GeV in a way that its life-time is so short to decay before neutralino dark matter freeze-out [34]. as ∼ 10 10 GeV or even below admitting some fine tuning in the seesaw parameters and an initial thermal N 2 abundance. These results indicate that, in those supersymmetric scenarios where the gravitino is heavier than ∼ 30 TeV and decays prior to the onset of BBN, SO(10)-inspired thermal leptogenesis can be indeed reconciled with the gravitino problem. Similar analysis, though for more specific choices of the parameters, has been also done in [35], finding a much more stringent lower bound T RH 5×10 11 GeV and concluding that thermal SO(10)-inspired leptogenesis is incompatible with the upper bound from the gravitino problem thus motivating a non-thermal scenario. We will comment on this difference between our results and those of [35].
We also propose a new scenario where the lightest RH neutrino mass is comparable or below the sphaleron freeze-out temperature T out sph ∼ 100 GeV [36] in a way that the lightest RH neutrino wash-out occurs too late to wash-out the baryon asymmetry. In this case we show that values of T RH as low as ∼ 10 9 GeV are possible. Therefore, our results indicate that supersymmetric SO(10)-inspired thermal leptogenesis can be reconciled with the gravitino problem and is certainly not ruled out model independently.
The paper is organised in the following way. In Section 2 we show how the calculation of the asymmetry can be extended to a supersymmetric N 2 -dominated scenario. In Section 3 we study SO(10)-inspired (supersymmetric) leptogenesis deriving the constraints on the low energy neutrino parameters and comparing them with those obtained in the nonsupersymmetric case in [17,22,18]. In Section 4 we discuss in detail the lower bound on T RH showing that values as low as 1 × 10 10 GeV are possible. In Section 5 we discuss a new N 2 -dominated scenario where the lightest RH neutrino mass is lower than the sphaleron freeze out temperature, so that the N 1 wash-out is absent. We recalculate the lower bound on T RH in this scenario obtaining T RH 1 × 10 9 GeV, enlarging even more the region of compatibility with the gravitino problem. In Section 6 we draw some final remarks and conclude.
2 Calculation of the asymmetry within supersymmetric N 2 -dominated leptogenesis In this section we extend the calculation of the asymmetry in the N 2 -dominated scenario, as rising from SO(10)-inspired conditions, to a supersymmetric framework.
First of all we assume a minimal type I seesaw extension of the MSSM introducing three RH neutrinos N iR with Yukawa couplings h and Majorana mass M . In the flavour basis, where the charged lepton and the Majorana mass matrices are both diagonal, the masses and Yukawa couplings relevant for leptogenesis are given by the following terms in the superpotential [37,38] (α = e, µ, τ ) where and is the totally anti-symmetric tensor. After spontaneous symmetry breaking the two neutral Higgs field vev's generate the Dirac masses for the charged leptons and for the neutrinos, respectively where v is the SM Higgs vev. The Dirac mass matrix in the flavour basis can be expressed through the singular value decomposition (or bi-unitary parameterisation) as where D m D ≡ diag(m D1 , m D2 , m D3 ) is the neutrino Dirac mass matrix in the Yukawa basis and V L and U R are the unitary matrices acting respectively on the LH and RH neutrino fields in the transformation from the flavour basis to the Yukawa basis.
In the seesaw limit, for M m D , the spectrum of neutrino mass eigenstates splits into a very heavy set with masses almost coinciding with the Majorana masses M i and into a light set ν i ν iL + ν c iL , with a symmetric mass matrix m ν given by the seesaw formula This is diagonalised by a unitary matrix U , where D m ≡ diag(m 1 , m 2 , m 3 ) with m 1 ≤ m 2 ≤ m 3 , corresponding to the PMNS leptonic mixing matrix, in a way that we can write Assuming SO(10)-inspired conditions, i) where m q i are the three up quark masses, m u , m c and m t for i = 1, 2, 3 respectively, the RH neutrino masses are approximated by the following simple analytical expressions [6,18,27] where m ν ≡ V L m ν V T L is the light neutrino mass matrix in the Yukawa basis. These expressions show that under SO(10)-inspired conditions, barring fine tuned conditions on ( m ν ) 11 and ( m −1 ν ) 33 5 , the RH neutrino masses are highly hierarchical and in particular M 1 10 9 GeV and M 2 10 9 GeV, in a way that the N 2 -dominated scenario is realised, where the asymmetry is necessarily produced by the N 2 's.
A general calculation of the asymmetry valid for any mass regime should proceed within a density matrix formalism [11,13,39,40]. However, except for some transition regimes, the mass of the N 2 producing the asymmetry, M 2 , falls within so called fully flavoured regimes where the density matrix equation simplifies into Boltzmann equations [40] and in this case the final asymmetry can be calculated using simple approximate analytic expressions.
We will neglect flavour coupling effects [11,12,25,26], that can in some cases produce dominant contributions to the final asymmetry [26,27] and have been studied in detail in the supersymmetric case in [41], but we will comment in the conclusions on the impact they can have on our results. We will also not pursue here the case of soft leptogenesis, offering a way to lower the scale of leptogenesis circumventing the gravitino problem [42].
It is important to notice that, within a supersymmetric framework, the N 2 -production of the asymmetry for a fixed mass M 2 can occur in different fully flavoured regimes depending on the value of tan β since charged lepton interaction rates involving leptons are ∝ (1 + tan 2 β) [12]. On the other hand since, because of our working assumptions, one has M 1 10 9 GeV, the lightest RH neutrino produced asymmetry is always negligible and the N 1 wash-out occurs always in the three-flavoured regime independently of the value of tan β. We can then distinguish three fully flavoured regimes for the calculation of the asymmetry: • In the unflavoured 6 regime, for M 2 5 × 10 11 GeV (1 + tan 2 β), the final B − L asymmetry can be calculated using 5 Recently it has been noticed that such a fine tuning can be precisely quantified in terms of the orthogonal matrix [27]. 6 Here we refer to an 'unflavoured' regime rather than to a 'one-flavoured' regime, as sometimes it is done, since we refer only to the number of charged lepton flavours.
• In the two-(fully) flavoured regime, for 5 × 10 11 GeV (1 + tan 2 β) M 2 5 × 10 8 GeV (1 + tan 2 β), the final B − L asymmetry can be calculated using where we indicated with τ ⊥ 2 the electron plus muon component of the quantum flavour states produced by the As we discussed, in the transition regimes, about M 2 ∼ 5 × 10 11 GeV (1 + tan 2 β) and M 2 ∼ 5 × 10 8 GeV (1 + tan 2 β), the asymmetry should be calculated using density matrix equations. We will describe these transition regimes switching abruptly from one fully flavoured regime to another at the two given values of M 2 . We will also comment on the impact of this 'step approximation'. The total and flavoured decay parameters, K i and K iα respectively, can be still written as but the equilibrium neutrino mass is now given by [43,35] having taken into account that: i) RH neutrinos and sneutrinos have a doubled number of decay channels compared to the SM case that simply doubles the rates and ii) the number of ultra-relativistic degrees of freedom g M SSM = 915/4. This implies that the (total and flavoured) decay parameters are ∼ √ 2 larger than in the SM. 7 This will clearly tend to enhance the wash-out both at the production, depending on the K 2α 's, and from the lightest RH neutrinos, depending on the K 1α 's. The wash-out at the production is described by the efficiency factor κ(K 2α ) that for an initial thermal N 2 abundance can be calculated as [20,44] For an initial vanishing N 2 abundance this is the sum of a negative and a positive contribution [20], that are approximated by the following expressions [44] and where and p 0 2α = K 2α /K 2 is the tree level probability that the lepton quantum state produced by a N 2 -decay is measured as an α flavour eigenstate. If the asymmetry is produced in the strong wash-out regime, the two expressions converge to the same asymptotic limit and there is no dependence on the initial N 2 abundance.
The other important modification to be taken into account, compared to the nonsupersymmetric case, is that now there are also more interference terms contributing to the CP asymmetries and one obtains [37] where we defined [45], (model dipendent) supersymmetric models where M 1 is low enough that some supersymmetric degrees of freedom associate to heavier particles get suppressed in between. In Section 5 we will consider a scenario with M 1 100 GeV but in that case we will point out that the N 1 wash-out at all can be neglected so in any case this point has no relevance.
In the hierarchical RH neutrino mass limit one has ξ(x) → 1 and moreover terms ∝ I α 21 ξ(M 2 1 /M 2 2 ), J α 21 , J α 23 are strongly suppressed in the N 2 -dominated scenario so that the ε 2α 's can be approximated simply by Compared to the SM case, for a given set of values of the seesaw parameters, the CP asymmetries are double. Finally the baryon-to-photon number ratio can be calculated from the final B − L asymmetry produced by the RH neutrinos (or sneutrinos), as having taken into account a factor 2 from the sum of the asymmetry generated by RH neutrinos and sneutrinos, the sphaleron conversion coefficient a M SSM sph = 8/23 [46] and that the number of photons at recombination is given by (N rec γ ) M SSM = 4 g M SSM /(3 g rec ) 78 since g M SSM = 915/4, g rec 3.91 and one has to consider that in the portion of co-moving volume containing one RH neutrino in ultra-relativistic equilibrium there are 4/3 photons.
In the non supersymmetric case, and in the approximation V L I, the solutions are tauon dominated [16] and, as shown in [18], the asymmetry is well described by a full analytical expression. We can extend this analytical expression, for the tauon contribution to the final asymmetry, to the supersymmetric case with the simple modifications we discussed 8 , obtaining 8 In addition we are correcting a typo that we found in [18] where instead of the term We will have of course to check whether the tau dominance, holding for V L = I in the non-supersymmetric case, still holds in the supersymmetric case. Finally, we also want to give the expression for the relic value of a pre-existing asymmetry and the condition for its wash-out (strong thermal leptogenesis condition) that we will superimpose to the successful leptogenesis condition, extending the results found in the non-supersymmetric case [22].
If the production occurs in the unflavoured regime, for M 2 5×10 11 GeV (1+tan 2 β), then it is impossible to realise successful strong thermal leptogenesis since the N 2 washout cannot suppress completely the pre-existing asymmetry in any of the three (charged lepton) flavours. The pre-existing asymmetry can be only washed-out by the lightest RH neutrinos in all three flavours [47] but in this way it also suppresses the N 2 produced asymmetry and one cannot attain successful leptogenesis. On the other hand if the N 2 production occurs in the two fully-flavoured regime, for 5×10 11 GeV (1+tan 2 β) M 2 5 × 10 8 GeV (1 + tan 2 β), then the relic value of the pre-existing B − L asymmetry is given by where In this case imposing K 2τ , K 1µ , K 1e 1 and K 1τ 1 one can wash-out the pre-existing asymmetry but not the tauonic component of the N 2 produced asymmetry [48]. This is the case holding in the non-supersymmetric case or in the supersymmetric case for small tan β values. On the other hand for sufficiently large tan β values, such that M 2 5×10 8 GeV (1+tan 2 β), the production occurs in the thee-flavoured regime and in this case the relic value of the final flavoured asymmetries are simply modified by the replacement K 1µ → K 1µ +K 2µ and K 1e → K 1e +K 2e in the exponentials. In this way the conditions for the wash-out of the pre-existing asymmetry are now less stringent since one has to impose K 2τ , K 1µ + K 2µ , K 1e + K 2e 1, so that one can also have K 2µ 1 and K 1µ 1, washingout the pre-existing asymmetry and having a final muon (instead of tauon) dominated asymmetry, a new situation compared to the non-supersymmetric case.

Constraints on the low energy neutrino parameters
In the SO(10)-inspired scenario of leptogenesis that we described, the asymmetry formally depends on the nine parameters in the low energy neutrino mass matrix, on the six parameters in the matrix V L and on the three α i . As we discussed the 3 RH neutrino masses M i and the RH neutrino mixing matrix U R can be expressed in terms of these parameters. However, since the finally asymmetry is dominated by the N 2 contribution, the dependence on α 1 and α 3 cancels out (this can be seen analytically in the eq. (23) for V L = I but the result remains true for a generic V L ) and this is crucial to understand why one gets constraints on the low energy neutrino parameters.
We have numerically calculated the final asymmetry and imposed the condition of successful leptogenesis in the SO(10)-inspired case producing scatter plots in the space of parameters for (α 1 , α 2 , α 3 ) = (1, 5, 1). As in [16,17,18], for the up quark masses at the leptogenesis scale we adopted the values m u = 1 MeV, m c = 400 MeV and m t = 100 GeV [49]. We verified that indeed constraints do not depend on α 1 and α 3 but only on α 2 as in the non-supersymmetric case [17]. 9 The value α 2 = 5 can be considered a close-tomaximum value in a way that the constraints obtained for this value have to be regarded close to the most conservative ones. Moreover this value has been used as a benchmark value both in the non-supersymmetric case [16,17] and also in [35], allowing us a useful comparison among the results.
We have to distinguish 'small tan β values' for which the production, as in the nonsupersymmetric case, occurs in the two-flavoured regime, from 'large tan β values', for which the production occurs in the three-flavoured regime. Since for successful SO(10)inspired leptogenesis one typically has M 2 10 11 GeV and since the transition from the two to the three flavoured regime occurs for M 2 5 × 10 8 GeV(1 + tan 2 β), one can say that for tan β 15 the production occurs mainly in the three flavoured regime, while for tan β 15 it occurs mainly in the two-flavoured regime. We made the calculation for two extreme values , tan β = 5 and tan β = 50. In the first case the production occurs almost entirely in the two flavoured regime, except for very special points, while in the second case the production occurs mostly in the three flavoured regime.
We also performed the scatter plots both for NO and for IO neutrino masses so that in total we have four cases to consider.
In addition to successful leptogenesis, we also show the results when the condition of strong thermal leptogenesis, such that a large pre-existing asymmetry is washed-out, is superimposed. As in the non-supersymmetric case [22], this singles out a sub set of the solutions out of those satisfying successful leptogenesis. In the scatter plots we highlight these sub sets in blue (light blue for V L = I and dark blue for I ≤ V L ≤ V CKM ).
For the low energy neutrino parameters we adopted the same values and ranges as in [18]. In particular for the solar neutrino mass scale m sol ≡ m 2 2 − m 2 1 = 0.0087 eV and for the atmospheric neutrino mass scale m atm ≡ m 2 3 − m 2 1 = 0.0495 eV, the best fit values found in a recent global analysis [23]. When these values are combined with the upper bound on the sum of the neutrino masses from the Planck satellite, i m i < 0.23 eV (95% C.L.) [21], one obtains an upper bound on the lightest neutrino mass The mixing angles, respectively the reactor, the solar and the atmospheric ones, are now measured with the following best fit values and 1σ (3σ) ranges [50] for NO and IO respectively, Current experimental data also start to put constraints on the Dirac phase and the following best fit values and 1σ errors are found for NO and IO respectively, δ/π = −0.61 +0.38 −0.27 and δ/π = −0.69 +0.29 −0.33 , though all values [−π, +π] are still allowed at 3 σ. They do not yet favour one of the two orderings over the other.

Normal ordering
Let us first present the results for NO neutrino masses. As mentioned, we also discuss separately the results for 'low tan β' values and for 'high tan β values'.  orange points are the subset for V L = I. These results for tan β = 5 are similar to those obtained in the non-supersymmetric case and they are well explained and understood [16,17,18]. Even though they have been obtained for initial thermal N 2 abundance, they are actually very marginally dependent of the initial N 2 abundance since for the tauon-dominated solutions (K 1τ 1) one has K 2τ 1 and for the muon-dominated solutions (K 1µ ) one has K 2τ ⊥ 2 1 (except for very few points with K 2τ ⊥ 2 1). As we will discuss, there are also some electron dominated solutions that entirely depend on the initial N 2 abundance, since K 2τ ⊥ 2 1, but for low tan β values they are marginal and do not influence the constraints on the low energy neutrino parameters. They correspond to the sparse point at K 1τ 1 and in the range 1 meV m 1 10 meV. The low density indicates that these solutions are marginal and require some fine tuning to realise weak wash-out at the production, i.e. K 2τ ⊥ 2 1, and to enhance the CP asymmetry ε 2α . Coming back to the leading tauon dominated solutions, notice that in principle since the washout is stronger compared to the SM case, because of the smaller value of m M SSM compared to m SM (cf. eq. (12)), one could think that it should be more difficult to realise the condition K 1τ 1. However, from the analytical expression given in [18] for V L = I, extended to the supersymmetric case with the simple replacement m SM → m M SSM , explicitly one can see that the slightly lower value of m M SSM plays just a marginal role since the condition K 1τ 1 produces conditions on the phases marginally dependent on m M SSM . Actually the increase of the asymmetry of a factor ∼ √ 2 at the production, due to the doubled CP asymmetry only partly compensated by a stronger wash-out, enlarges the allowed region in the plane θ 23 vs. m 1 at values m 1 50 meV, the so called τ B solution and, more generally, the allowed range of m 1 gets slightly wider (for example the upper bound relaxes from 0.06 eV to 0.1 eV).
The blue points in Fig. 1 are the subset satisfying the strong thermal condition (dark blue for I ≤ V L ≤ V CKM , light blue for V L = I) for an initial pre-existing asymmetry N p,i B−L = 10 −3 . Also in this case we can compare the results with the non-supersymmetric case. This time there is one significant difference since in the supersymmetric case the strong thermal region is more extended and in particular it allows higher values of the atmospheric mixing angle. Indeed while in the non-supersymmetric case one has a quite stringent upper bound on the atmospheric mixing angle θ 23 43 • , in the supersymmetric case this now gets relaxed to θ 23 46 • , a relaxation that might be relevant in view of the next expexted results from long baseline experiments.
This relaxation is quite well explained analyticaly for V L = I extending to the supersymmetric case the discussion in [18]. The upper bound on θ 23 indeed originates from the requirement K 1e 1 from the strong thermal condition. This condition first translates into a lower bound on the 0νββ effective neutrino mass m ee 8 meV and then into one on the lightest neutrino mass m 1 1.3 m ee 10 meV [22]. Since in the supersymmetric case all K iα are ∼ √ 2 larger, this requirement is now more easily satisfied and one has m ee 6 meV giving m 1 7 meV, well explaining the constraints in the plane m ee vs. m 1 (see bottom left panel in Fig. 1), and this in turn implies indeed θ 23 46 • .
In Fig. 2 we show 6 panels, for integer values of α 2 from one to six, of the RH neutrino masses M i and of the minimum requested value of T RH (we will discuss this in detail separately in Section 4). In these panels we have highlighted the flavour that dominates the asymmetry associating a different colour to each flavour (blue for tauon, green for muon, red for electron). The points are calculated in the case I ≤ V L ≤ V CKM and again for initial thermal N 2 abundance. As one can see, in addition to muon (green points) and tauon (blue points) flavour dominated solutions, also electron flavour dominated solutions are present. At low α 2 values (α 2 = 1, 2) these are even the only solutions for m 1 20 meV. For V L = I the electron flavour asymmetries are many order of magnitude suppressed compared to the muonic and even more compared to the tauonic [18] but when V L = I this sharp flavour dominance does not hold [17]. In the non-supersymmetric case we have also found electron-flavour dominated solutions but in a very marginal way. This means that these solutions realise successful leptogenesis only for very special conditions in the non-supersymmetric case and the maximum possible asymmetry is just very slightly above the observed value. In the supersymmetric case, since the CP asymmetries double and the wash-out at the production is only ∼ √ 2 stronger, the B − L asymmetry at the production is ∼ √ 2 higher and this helps the marginal electron-dominated solutions to be realised for a slightly wider region in parameter space in any case without really opening up new allowed regions in the low energy neutrino parameters. At the same time it is important to stress that since these solutions are realised for K 2τ ⊥ 2 1, they are strongly dependent on the initial N 2 abundance and, in particular, they completely disappear for initial vanishing N 2 abundance.
In conclusion for low tan β values the low energy neutrino constraints are only slightly more relaxed than in the non-supersymmetric case and in particular, as we have seen, the strong thermal condition is satisfied for slightly lower m 1 values.   Let us see now what happens when tan β is large enough that the production occurs in the three-flavoured regime. There is no explicit dependence of the asymmetry on tan β, the dependence is all encoded in the values of M 2 marking the transitions between two different flavoured regimes. The results will be the same for all tan β values large enough to lead to a production in the three flavoured regime for all allowed values of M 2 . Since the condition for the three-flavoured regime is M 2 5 × 10 8 GeV (1 + tan 2 β) and since one expects M 2 10 10 GeV, some solutions occurring in the three-flavoured regime are expected to appear for tan β 5. On the other hand since there are no solutions for M 2 3 × 10 12 GeV, for tan β 80 all solutions fall in the three flavoured regime. We choose for definiteness tan β = 50. This is sufficiently large that basically all solutions fall in the three flavoured regime so that constraints on low energy neutrino data are saturated increasing tan β. The results are shown in Fig. 3. The panels, the colour codes and all benchmark values are the same as in Fig. 1, so that there can be a straightforward comparison with the results obtained for tan β = 5. Looking at the yellow (and orange) points, those satisfying only the successful leptogenesis condition, one can notice that the constraints are even more relaxed than in the previous case for tan β = 5 compared to the non-supersymmetric case. There is still a lower bound on the lightest neutrino mass Figure 4: Scatter plots as in Fig. 2 but for tan β = 50. In the top central panel for α 2 = 2, the dark (light) red points are solutions for |Ω ij | 2 > 3 (|Ω ij | 2 < 3) able to lower T RH below 10 10 GeV. m 1 1 meV that is just very slightly relaxed compared to non-supersymmetric case (the CP asymmetry doubles but the value of K 2τ determining the wash-out at the production gets ∼ √ 2 higher and the two effects almost cancel out). This is interesting because one can conclude that the lower bound on m 1 is quite a stable and general feature of SO(10)inspired models that, therefore, predict some deviation from the hierarchical limit though this might well be below current experimental sensitivity. Indeed in the most optimistic case cosmological observations should place a 2σ upper bound m 1 10 meV [51,52].
From Fig. 3 it should be also noticed how the region satisfying K 1e 1 now greatly enlarges compared to the large tan β case. Indeed, if one looks at the panels in Fig. 4, showing again (as in Fig. 2 but now for tan β = 50) what flavour dominates the final asymmetry, one can notice how this time there are plenty of electron dominated solutions, in the range for 2 meV m 1 10 meV, as anticipated. This region was very marginal, almost absent, in the non-supersymmetric case and it was still quite marginal also for tan β = 5, as discussed. However now, for tan β = 50, it becomes quite significant and as we will discuss in the next section, it allows a relaxation of the lower bound on T RH below 10 10 GeV for 1 α 2 2. We should however stress again that these electron dominated solutions occur in the weak wash-out regime at the production (K 2e 1) and, therefore, they strongly depend on the initial N 2 abundance. They exist for initial thermal N 2 abundance but for vanishing initial N 2 abundance they completely disappear (i.e. they do not realise successful leptogenesis). The reason why they are obtained much easier at large tan β values compared to low tan β values is because now the condition of weak wash-out at the production is more relaxed, K 2e 1 instead of K 2τ ⊥ 2 ≡ K 2e + K 2µ 1. If we again consider the subset of points satisfying also the strong thermal condition (dark and light blue points), we can see that, as in the low tan β case, the region is now much more extended, even more than before. This happens because of the effect explained at the end of Section 2: one can now have K 1µ 1 and at the same time wash-out the pre-existing asymmetry in all flavours imposing K 1e , K 2µ , K 2τ 1. This opens up a new (muon dominated) region at large values of m 1 0.05 eV, though notice that this is now largely excluded by the upper bound Eq. (27).

Inverted ordering
Let us now discuss the IO case distinguishing, as we did for NO, small tan β values ( 15) from large tan β values ( 15). Figure 5: Scatter plots as in Fig. 1 but for IO and tan β = 5.

Small tan β values
For small tan β values the situation is, as for NO, similar to the non-supersymmetric case though the allowed regions are slightly more relaxed. In Fig. 5 we show the results again for tan β = 5 and one can see in particular that: • there is a lower bound m 1 10 meV corresponding to m i 130 meV that will be in a close future tested by the cosmological observations; • this time, differently from the non-supersymmetric case, there is no lower bound on the atmospheric mixing angle, though values in the first octant require higher values of the absolute neutrino mass scale on the verge of being excluded by the cosmological observations.
We can therefore conclude again, as in the non-supersymmetric case, that the IO case is disfavoured compared to the NO case.
In Fig. 6 we show again, with the same colour code as in Fig. 2 and 4 for NO, the solutions for various values of α 2 indicating the flavour that dominates the final asymmetry. This time one can see that, even for initial thermal N 2 -abundance, there are no electron dominated solutions. The reason is simply that in the IO case one has K 1e = Figure 6: Scatter plots as in Fig. 2 but for IO and tan β = 5. m ee /m M SSM 70 [18] and, therefore, the electron asymmetry is completely washed-out by the lightest RH neutrinos inverse processes.

Large tan β values
For large tan β values and imposing successful leptogenesis condition, the situation is qualitatively similar to the case of small tan β as one can see from Fig. 7 (orange and yellow points) but simply the allowed regions slightly further enlarge. For example now one has m 1 7 meV. In the panels of Fig. 8 we show the dominant flavour and one can see that, for the same reason, there are no electron dominated solutions (no red points). The real difference is that now there is a large amount of solutions satisfying the strong thermal leptogenesis condition. The reason is that for large tan β the fact that K 1µ tends not to be too large (see central bottom panel in Fig. 7), is not a problem, since the condition for the wash-out of the pre-existing asymmetry now requires K 1µ + K 2µ 1 and it can be more easily satisfied even for low K 1µ values. We can conclude that in all cases supersymmetry helps realising the strong thermal condition.

Lower bound on T RH
Thermal leptogenesis requires the initial temperature of the radiation dominated regime, T RH within inflation, to be sufficiently high for the RH neutrinos to be thermally produced before their interactions with the thermal bath, in particular their inverse decays, go outof-equilibrium.
For a specific solution this occurs at a temperature T lep (K 2α ) M 2 /z B (K 2α ), where α is the flavour in equilibrium that dominates the asymmetry, either τ or τ ⊥ 2 in the two fully flavoured regime, or α = e, µ or τ in the three fully flavoured regime. 10 At higher temperatures, in the strong wash-out regime (K 2α 1), the produced asymmetry is efficiently washed-out, while at lower temperatures, since the RH neutrino abundance is dropping exponentially, the produced asymmetry is negligible. In this way the asymmetry, in each flavour in equilibrium, is produced within quite a well defined range of temperatures between M 2 /[z B (K 2α ) − 2] and M 2 /[z B (K 2α ) + 2] [20]. Therefore, for a specific solution the reheat temperature has to be greater than T min . In the weak wash-out regime one cannot identify such a sharp interval of temperatures and moreover the process of production of the asymmetry depends on the initial N 2 abundance. In this case one can say that T RH M 2 for the final asymmetry to be equal to the asymptotic value at high temperatures. An expression that interpolates quite well T min RH (K 2α ) between the strong and weak wash-out regime is given then by [20] T min RH (K 2α ) This expression gives, for each solution with specified values of K 2α and M 2 , the minimum T RH . The lower bound on T RH can then be calculated minimising over all the found solutions, i.e. T min RH ≡ min[T min RH (K 2α )]. In the non-supersymmetric case it was obtained T min RH 1 × 10 10 GeV for α 2 = 1 [17], a lower bound that cannot by excluded by any experimental observation or theoretical argument. However, this result could somehow suggest that also in the supersymmetric case one can expect a similar or even more stringent lower bound because of the increased wash-out, leading in this case to a tension with the gravitino problem upper bound that, as discussed in the introduction, in a conservative way can be assumed to be T RH 10 10 GeV in order not to overproduce the gravitino abundance. 11 This potential tension was confirmed by a dedicated analysis made in the supersymmetric case [35]. Here it was obtained (for α 2 = 5) T RH 10 11 GeV, a result that would suggest that SO(10)-inspired thermal leptogenesis is incompatible with the upper bound from the gravitino problem unless, as discussed in the introduction, one assumes very specific supersymmetric models.
We plotted T min RH (K 2α ) for each point satisfying successful leptogenesis. The values of T min RH (K 2α ) are shown with grey points in all plots where also the RH neutrino masses are plotted. These plots are shown in Figs. 2, 4, 6, 8 for NO low tan β, NO high tan β, IO low tan β and IO high tan β respectively for six specific integer values of α 2 (from 1 to 6) since one can expect a non trivial dependence on α 2 . This is because for decreasing α 2 one has that M 2 decreases and this would go into the direction to lower T RH . On the other hand the final asymmetry decreases as ∝ α 2 2 so that there is also a lower bound on α 2 coming from successful leptogenesis. In these figures one can see indeed how the allowed range of values for T min RH (K 2α ) depends on α 2 . Finally, in Fig. 9 we summarised the results plotting the lower bound T min RH as a function of α 2 indicating, with the same colour code as in Figures 2, 4, 6 and 8, which flavour dominates the asymmetry for each value of α 2 . The results are shown both for initial thermal N 2 abundance (thin lines) and for vanishing initial N 2 abundance (thick lines). The main difference is that in the second case there are no electron-dominated solutions since these all have weak wash-out at the production (K 2e 1) and the asymmetry is strongly suppressed in the case of initial N 2 vanishing abundance. In the left (right) panels we show the results for low (high) values of tan β, in the top (bottom) panels the results for NO (IO). In the case of low tan β values (left panels) one can see how the results do not actually differ that much from those in the non-supersymmetric case [17]. There is actually even a ∼ √ 2 relaxation due to the fact that the asymmetry increases by a factor ∼ 2 because of the doubled CP asymmetry and the efficiency factor decreases of a factor ∼ √ 2 (the efficiency factor is approximately inversely proportional to the decay parameters that increase of a factor √ 2). However, in the right panels, for large tan β values, one can see how the red branch, corresponding to the electron flavour dominated solutions now, for α 2 = 1-2, allows T min RH (5-10) × 10 9 GeV, showing that it is possible to go even below 10 10 GeV. Notice however that these electron-flavour dominated solutions have two drawbacks. First they exist only for initial thermal N 2 abundance, a case that should justified within models where for example the RH neutrinos are produced by Z particles, heavier than the N 2 's, of a left-right symmetry after SO(10) breaking [53]. Moreover those solutions minimising T min RH below 10 10 GeV are characterised by large values of the squared modules of the orthogonal matrix entries (implying strongly fine tuned cancellations in the see-saw formula). This happens because the N 2 CP asymmetries are not upper bounded and they are enhanced when |Ω ij | 2 1. In the α 2 = 2 panel of Fig. 4 (bottom central panel) these fine tuned solutions correspond to the dark red points. One can see how they also correspond to uplifted values of M 1 and reduced values of M 2 (they are indeed in the vicinity of crossing level solution).
For these reasons these solutions should not be over emphasized, though they still represent a possibility that should not be disregarded. On the other hand the muon and the tauon-flavour dominated solutions, that give T min RH ∼ 10 10 GeV, are not fine tuned and correspond to typical SO(10)-inspired solutions. Moreover in the case of the tauon dominated solutions they are also genuinely strong washed solutions independent of the initial N 2 abundance.
Notice that the lower bound on T RH that we found for α 2 = 5, T RH 1.5 × 10 10 GeV, is more than one order of magnitude below the lower bound found in [35]. We cannot explain the origin of the discrepancy since details of the calculation of the final asymmetry (for example how the matrix U R is calculated) are not specified in [35]. We can only report that in [35] the tauon-dominated solutions that we find are absent and the result on the T RH lower bound mainly relies on the electron dominated solutions and therefore on the assumption of initial thermal N 2 abundance.
Our result for the lower bound on the reheat temperature, T min RH 1 × 10 10 GeV, is approximately equal to the value that one needs in order to produce the Dark Matter gravitino abundance depending on the value of the gluino masses. This coincidence is similar to what happens in the case of traditional N 1 -dominated leptogenesis [54] so that one could intriguingly relate matter-antimatter asymmetry production in thermal leptogenesis to gravitino Dark Matter production. On the other hand the recent LHC results on the lower bound of gluino masses [2] make the upper bound on T RH more stringent, at the level of T RH 5 × 10 9 GeV within the pMSSM [55] and this seems to corner this intriguing scenario of thermal leptogenesis combined with gravitino Dark Matter . However, as already mentioned, for large values of the gravitino mass ( 30 TeV) the large T RH required by SUSY SO(10)-inspired thermal leptogenesis (T RH 10 10 GeV) can be reconciled with the gravitino problem. Of course within specific realistic models one should verify whether the lower bound T min RH ∼ 10 10 GeV can be indeed saturated. There is, however, still another possibility, never considered so far, that can allow a relaxation of T min RH even below 10 10 GeV for usual tauon-flavour solutions.

A new scenario of N 2 -dominated leptogenesis
It is usually assumed that the lightest RH neutrino mass M 1 T out sph 100 GeV, where T out sph is the sphaleron freeze-out temperature [56]. In this case the lightest RH neutrino wash-out has to be taken into account. However, if M 1 is below such a temperature, then the lightest RH neutrino wash-out acts only on the lepton asymmetry but not on the frozen baryon asymmetry produced earlier by N 2 out-of-equilibrium decays. 12 In this case the final asymmetry is given by the expressions eqs. (8), (9), (10) without the exponentials encoding the lightest RH neutrino wash-out since this is negligible.
We have then repeated the calculation of T min RH in this scenario and the results are shown in the four panels of Fig. 10 that correspond to the same cases of the panels in Fig. 9. This time the minimum is always realised by tauon dominated solutions with strong wash-out at the production where the final asymmetry is independent of the initial N 2 abundance. It can be seen how values of T RH as low as 10 9 GeV are possible. In this case the gravitino overabundance problem can be circumvented for a wider range of gravitino masses compared to the traditional scenario discussed in the previous sections.
From the expression eq. (7) for M 1 one can see how this scenario requires values α 1 0.1 (for hierarchical neutrinos, if m 1 10 meV one can have higher values). This would also imply somehow that also m D3 T out sph ∼ 100 GeV in order for the seesaw formula to be valid implying α 3 1. One can wonder whether this can be achieved in some realistic models. Interestingly in a recent study of realistic SO(10) models [28] one of the found best fit cases, a supersymmetric model with 10 H , 120 H ,1 26 H Higgs representations, is realised for M 1 1 TeV corresponding to α 1 0.3. Since this case also has a very small χ 2 min 0.6, one can wonder whether with some deviation from the best fit one could get M 1 T sph RH with still an acceptable value of χ 2 min . In any case this specific example seems to suggest that this scenario might be indeed realised within some realistic model. Notice that within this scenario we are not showing the low energy neutrino constraints since these simply evaporate. Indeed these constraints exist mainly because of the presence of the lightest RH neutrino wash-out, as stressed in previous papers [16,17,18]. It should also be made clear that though we are presenting this scenario in a supersymmetric framework, where it nicely allows T RH values below 10 10 GeV, it might be also realised and find applications within a non-supersymmetric framework. 12 More precisely the N 1 wash-out acts in an interval of temperatures T = [M 1 /z in , M 1 /z out ] with z in 2/ √ K 1α [20]. Therefore, more precisely one has to impose M 1 z in T out sph .  Fig. 9 with the difference that this time there is no distinction between thin and solid lines since there is no dependence on the initial N 2 abundance.

Conclusions
We extended the study of SO(10)-inspired leptogenesis, previously discussed in a nonsupersymmetric framework, to the supersymmetric case calculating the constraints on the low energy neutrino parameters and the lower bound on T RH that has a particular importance because of the tension with the upper bound from the gravitino problem. Our results show that, in the usual case, where the lightest RH neutrino mass is heavier than the sphaleron freeze-out temperature and N 1 wash-out is present, values of T RH as low as T RH 1 × 10 10 GeV are possible without any fine-tuning and for a final asymmetry independent of the initial N 2 -abundance. We have then proposed a novel scenario where M 1 is below the sphaleron freeze-out temperature so that the N 1 wash-out is absent. In this case without any fine-tuning reheat temperature values as low as T RH 1 × 10 9 GeV are allowed. In our calculation the main neglected effects that could produce some significant modifications are flavour coupling effects [26] arising from a redistribution of the asymmetry among quarks, right handed charged leptons and above all Higgs (and of course also among the supersymmetric particles [41]). This could open new way to circumvent the N 1 wash-out but in any case it should be clear that an account of these effect can at most relax the reheat temperature in the scenario where M 1 T out sph to the minimum value, T min RH 1 × 10 9 GeV found in the case where M 1 T out sph . We have also described the transition between different fully flavoured regimes for a changing value of M 2 with a step approximation, while a full description would require solution of density matrix equations [11,13,39,40]. Another important effect that we neglected and that might be important in the supersymmetric case for large tan β is the running of low energy neutrino parameters that might modify the constraints in this case [57]. However, this effect would not change our main results on the lower bound on T RH . In conclusion, we have shown the existence of a window for the viability of thermal SO(10)-inspired leptogenesis in the supersymmetric case without gravitino overabundance problem. Independently whether supersymmetry is found at the LHC, these results are interesting in connection with the current debate of identifying a realistic grand-unified model able also to realise successful leptogenesis, since supersymmetric extensions might more easily provide good fits of the parameters even if supersymmetry breaking occurs above the scale testable at colliders. With more experimental information on the neutrino mixing parameters coming in a close future, a particular successful model (or class of models) might emerge with interesting further phenomenological predictions (e.g proton life-time) and even more specific links between leptonic and quark sector and possibly with the identification of the DM candidate and new predictions at colliders. In this exciting search, leptogenesis might play a primary role, increasing the predictive power of the model and solving the matter-antimatter asymmetry cosmological puzzle.