Decrypting $SO(10)$-inspired leptogenesis

Encouraged by the recent results from neutrino oscillation experiments, we perform an analytical study of $SO(10)$-inspired models and leptogenesis with hierarchical right-handed (RH) neutrino spectrum. Under the approximation of negligible misalignment between the neutrino Yukawa basis and the charged lepton basis, we find an analytical expression for the final asymmetry directly in terms of the low energy neutrino parameters that fully reproduces previous numerical results. This expression also shows that is possible to identify an effective leptogenesis phase for these models. When we also impose the wash-out of a large pre-existing asymmetry $N^{\rm p,i}_{B-L}$, the strong thermal (ST) condition, we derive analytically all those constraints on the low energy neutrino parameters that characterise the {\rm ST}-$SO(10)$-inspired leptogenesis solution, confirming previous numerical results. In particular we show why, though neutrino masses have to be necessarily normally ordered, the solution implies an analytical lower bound on the effective neutrino-less double beta decay neutrino mass, $m_{ee} \gtrsim 8\,{\rm meV}$, for $N^{\rm p,i}_{B-L}=10^{-3}$, testable with next generation experiments. This, in combination with an upper bound on the atmospheric mixing angle, necessarily in the first octant, forces the lightest neutrino mass within a narrow range $m_1 \simeq (10$-$30)\,{\rm meV}$ (corresponding to $\sum_i \, m_i \simeq (75$-$125)\,{\rm meV}$). We also show why the solution could correctly predict a non-vanishing reactor neutrino mixing angle and requires the Dirac phase to be in the fourth quadrant, implying $\sin\delta $ (and $J_{CP}$) negative as hinted by current global analyses. Many of the analytical results presented (expressions for the orthogonal matrix, RH neutrino mixing matrix, masses and phases) can have applications beyond leptogenesis.


Introduction
There is no observational evidence of primordial antimatter in our observable Universe that, therefore, is matter-antimatter asymmetric. The matter-antimatter asymmetry can be expressed in terms of the baryon-to-photon number ratio, currently very precisely and accurately measured by the Planck satellite from CMB temperature anisotropies, finding [1] η CMB B = (6.1 ± 0.1) × 10 −10 .
This value is too high to be explained within the Standard Model (SM) and, therefore, it can be regarded as an evidence of new physics. Leptogenesis [2] provides an attractive explanation since it is based on a minimal extension of the SM, the see-saw mechanism [3], able to address neutrino masses and mixing, soundly established by neutrino oscillation experiments. In this way leptogenesis realises a very interesting connection between two, apparently unrelated, fundamental physical observations: the cosmological matterantimatter asymmetry and neutrino masses and mixing. Recent important experimental findings support a traditional thermal picture and encourage further investigation in this direction. The confirmation of the BEH mechanism with the discovery of the Higgs boson at the LHC, an important ingredient of the seesaw mechanism, certainly corroborates the overall picture of leptogenesis. In addition, the non-discovery of new physics at LHC so far, quite strongly constrains models of baryogenesis at the TeV (or lower) energy scale. Moreover if the recent claim of a Bmode polarization signal in the CMB will be (even partially) confirmed (see [4] for critical analyses of BICEP2 results), it would support high inflationary scale V 1/4 ∼ 10 16 GeV [5]. In this way high values of the reheat temperature, though not necessarily implied, would be not only possible but even quite natural. This phenomenological picture is then well compatible with the original idea of a high energy scale thermal leptogenesis scenario, much above the TeV scale [2].
Values of T RH greater than those required by successful leptogenesis, T RH 10 9 GeV [6, 7], would be not only possible but even, as already mentioned, quite natural if the BICEP2 signal, even just a small fraction of it, will be confirmed as primordial. In any case it is legitimate to conceive that other mechanisms of baryogenesis might have generated an asymmetry much larger than the observed one prior to the onset of leptogenesis, one of the three main working assumptions in our study. For example, considering that we will work within the context of SO(10)-inspired conditions (another of our main working assumptions), the decays of GUT gauge bosons might have acted as the source of such a large pre-existing asymmetry [8]. It is then quite legitimate to impose, in addition to the successful leptogenesis condition requiring the right-handed (RH) neutrinos decays as the source of the genuine contribution from leptogenesis explaining the observed asymmetry, that the RH neutrino inverse processes at the same time wash-out a pre-existing (too) large asymmetry generated from an external mechanism: we will refer to this condition as the ST (equilibrium) condition. 1 Within a minimal type I extension of the SM and assuming a hierarchical RH neutrino mass spectrum, there is only one mass pattern able to realise successful ST leptogenesis, solving the problem of the initial conditions: the tauon N 2 -dominated scenario [9]. 2 In this scenario the lightest RH neutrino (N 1 ) mass M 1 10 9 GeV. This implies that the N 1 wash-out stage occurs in a three-flavoured regime [11,12] but also that the N 1 -decays cannot produce a sizeable contribution to the final asymmetry [6,7]. On the other hand the next-to-lightest RH neutrino (N 2 ) mass falls in the range 10 12 GeV M 2 10 9 GeV so that the asymmetry production from N 2 -decays occurs in the two-flavoured regime [11,12] and can be potentially sizeable. In this way a pre-existing tauon asymmetry can be washed-out by the N 2 -inverse processes while at the same time the N 2 -decays can re-generate an asymmetry in the same tauon flavour. This tauon component would then explain the observed asymmetry if it can escape the N 1 wash-out. This, on the other hand, has necessarily to suppress the electron and muon pre-existing asymmetries since they cannot be fully washed-out by the N 2 -inverse processes. This is because these can only wash-out the 2 -flavour parallel component but not the orthogonal one, where 2 is the charged lepton flavour produced by N 2 -decays [11,13].
It is quite interesting and non-trivial that so called SO(10)-inspired models [14,15,16] can indeed realise this seemingly contrived chain of conditions for successful ST leptogenesis [17]. This happens for a subset of the whole set of the solutions that realise successful SO(10)-inspired leptogenesis, [18,19]. This subset defines the ST SO(10)-inspired solution' [17].
Interestingly, this solution implies a predictive set of constraints on low energy neutrino parameters that can be almost completely 3 tested by (data-taking or planned) low energy 1 This name is justified by the fact that, as we will discuss, because of flavour effects the washout of a pre-existing asymmetry is possible only locking all possible ways out in flavour space, i.e. imposing thermal equilibrium in all flavours, something that can be indeed regarded as a 'strong' thermal equilibrium condition. 2 This conclusion holds for hierarchical RH neutrino spectra, where wash-out and asymmetry production from each RH neutrino species occurs sequentially. It has been claimed that quasi-degenerate RH neutrino spectra might provide an alternative way to realise ST leptogenesis [9]. This claim seems to be supported by recent calculations within specific models [10]. 3 We will comment in the final discussion why 'almost'.
neutrino experiments (including cosmological observations). One of the most striking features is that the value of the absolute neutrino mass scale is quite constrained within a narrow window, such that the lightest neutrino mass m 1 20 meV (while m i 100 meV). The Majorana phases are also constrained about particular values. In this way cancellations in the effective neutrino mass are very mild and m ee ∼ 0.8 m 1 15 meV. In addition parameters tested by neutrino oscillation experiments are also quite definitely predicted: neutrino masses have to be NO; the atmospheric mixing angle has to lie in the first octant and the Dirac phase in the fourth quadrant (δ ∼ −π/4) so that sign(J CP ) = −sign(η B ) < 0. It is also interesting that the solution has correctly predicted a nonvanishing value of the reactor mixing angle (θ 13 2 • -3 • ), as now robustly established by different experiments [20]. 4 This set of predictive constraints has so far been derived numerically, from scatter plots with points randomly generated [17]. A partial analytical insight was found for SO(10)inspired solutions (no strong thermal condition) in the hierarchical limit [18] , for m 1 10 meV, and, therefore, it does not apply to the strong thermal SO(10)-inspired solution that corresponds to semi-hierarchical (NO) neutrino masses. It has been recently shown analytically [22] that in general strong thermal leptogenesis requires, barring fine tuned models, m 1 10 meV for NO. This is in complete agreement with the analogous lower bound found numerically in the case of the strong thermal SO(10)-inspired leptogenesis solution.
Encouraged by the recent experimental results, in addition to the above mentioned discovery of a non-vanishing reactor mixing angle also hints for negative values of J CP [23,24] and, to a weaker extent, for a possible deviation of the atmospheric mixing angle from the maximal value [23,24] (see however [25]) in this paper we perform a systematic analytical derivation of the constraints coming from the strong thermal SO(10)-inspired solution confirming the numerical results. The derivation is made in the approximation V L I, where V L is the matrix describing the misalignment between the neutrino Yukawa basis and the charged lepton flavour basis but we also discuss the implication of small deviations at the level of the CKM quark mixing matrix.
The paper is organised as follows. In Section 2 we discuss the general set of conditions that lead to the solution. In Section 3 we discuss the current experimental results on neutrino parameters. In Section 4, imposing the SO(10)-inspired conditions, we derive compact analytic expressions for the RH neutrino mass spectrum and for the CP asymmetries improving and extending previous results [15,16,18]. In particular we are able to determine explicitly all the three phases associated to the eigenvalues of the Majorana mass matrix, an important step to write correctly the CP asymmetries. We also give an analytical expression for the orthogonal matrix that can be useful for model building considerations. In Section 5 we find an expression for the final asymmetry within SO(10)-inspired models in terms of the low energy neutrino parameters showing that it perfectly reproduces the numerical results for V L = I: this represents one of the main results of our work. We then impose the successful leptogenesis condition and we find, for NO, a lower bound and an upper bound on m 1 , an upper bound on the atmospheric mixing angle and different constraints on the low energy phases. In particular we show how SO(10)-inspired leptogenesis constraints are not symmetric under a change of the sign of J CP , given by sin δ. In Section 6 we finally impose the ST condition and, still for NO, we show how the lightest neutrino mass is constrained within a narrow range and how a lower bound on the reactor mixing angle holds. We also show how this condition necessarily selects values of δ in the fourth quadrant. In Section 7 we discuss IO neutrino masses showing how in this case the ST condition cannot be satisfied and, therefore, this strictly implies NO neutrino masses. In Section 8 we make some final remarks on the different approximations and assumptions behind the results and on the testability of the solution in next years. In Section 9 we summarise our results.

Set of conditions: the general picture
The ST SO(10)-inspired leptogenesis solution is obtained imposing the following set of conditions on the see-saw parameter space: (i) SO(10)-inspired conditions on the neutrino Dirac mass matrix; (ii) successful leptogenesis; (iii) strong thermal leptogenesis.
Let us briefly discuss these conditions in general, showing how (i) and (ii)+(iii) both independently select the N 2 -dominated scenario and how (iii) specifies that the asymmetry has to be necessarily tauon dominated [9].

SO(10)-inspired conditions
In the minimal see-saw mechanism the SM Lagrangian is augmented introducing RH neutrinos with Yukawa couplings h and a Majorana mass term M . In the (flavour) basis, where both charged leptons and Majorana mass matrices are diagonal, the leptonic mass terms after spontaneous symmetry breaking, can be written as (α = e, µ, τ and i = 1, 2, 3) where D m ≡ diag(m e , m µ , m τ ) and The neutrino Dirac mass matrix m D in the flavour basis can then be written in the 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. If we parametrize the three eigenvalues in the Dirac mass matrix in terms of the up quark masses, writing we define SO(10)-inspired models those respecting the following conditions 5 : With the condition i) we mean that the values of the three mixing angles in V L , that we indicate with θ L 12 , θ L 13 , θ L 23 in the usual PDG parametrisation, are not larger than the corresponding mixing angles in the CKM matrix and in particular θ L 12 θ CKM 12 ≡ θ C 13 • . In the see-saw limit, for M m D , the spectrum of neutrino mass eigenstates splits into a very heavy set, N i N iR + N c iR , 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 see-saw formula This is diagonalised by a unitary matrix U , 5 These conditions can be also satisfied beyond traditional SO(10)-models, see examples discussed in [26], the 5D-SO(10) model [27], the 'tetra-model' [28] or the 'A to Z' model [29]. Vice-versa not all SO(10)-models necessarily respect these conditions. For example they could give rise to a type II see-saw [30] contribution for the neutrino masses (e.g. [31]) and to alternative leptogenesis scenarios than those considered here.
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 When the current experimental information from neutrino oscillation experiments on the leptonic mixing matrix and on the neutrino masses is taken into account, the RH neutrino mass spectrum, barring special crossing level solutions [16], turns out to be highly hierarchical with approximately M 1 : implying M 1 10 9 GeV, 10 9 GeV M 2 10 12 GeV and M 3 10 12 GeV. 6 In this way the lightest RH neutrino is too light to contribute significantly to the final asymmetry when successful leptogenesis is imposed. The heaviest RH neutrino also gives vanishing or in any case negligible contribution, since either it is not thermalised at all or, even if it is thermalised, its total CP asymmetry is strongly suppressed. In this situation the only RH neutrino species that can give a sizeable asymmetry able to explain the observed one is N 2 and in this way, the SO(10)-inspired conditions naturally realise the N 2 -dominated scenario [33]. This is crucially possible thanks to flavour effects [11,12] that enlarge the regions where the lightest RH neutrino wash-out is negligible and the N 2 -asymmetry can survive [34,35].
In addition to a strong hierarchy of the mass spectrum, in the approximation V L I, the N 2 -flavoured CP asymmetries are also strongly hierarchical with ε 2τ ε 2µ ε 2e and this results into a tauon dominated final asymmetry [18]. Relaxing the approximation V L I this conclusion partially changes since a muon-dominated type solution becomes also possible. This, however, exists only for large values of the lightest neutrino mass m 1 0.01 eV so that it is fair to say that SO(10)-inspired conditions typically imply a tauon N 2 -dominated scenario. In the next Section we will discuss this result in detail.

Successful ST leptogenesis
Let us now shortly discuss the two conditions imposed by leptogenesis, successful and ST leptogenesis conditions. It is convenient to discuss them separately, in a way to highlight more clearly the different steps leading to the tauon N 2 -dominated scenario as the only way to realise successful ST leptogenesis. For illustrative purposes it is convenient to start from the ST condition.

ST leptogenesis
Since we are assuming a hierarchical RH neutrino mass spectrum, with M 3 3 M 2 and M 2 3 M 1 , the decays and wash-out processes associated to each RH neutrino species occur sequentially, with no overlap [36]. In this situation a pre-existing asymmetry, while temperature drops down in the expanding very early Universe, undergoes a multiple stage wash-out, involving sequentially each RH neutrino species, starting from the N 3 -washout stage (if N 3 is thermalised) and ending with the N 1 -washout stage.
However, since each RH neutrino species is able to wash-out only a component of the asymmetry in a flavour space direction i ∝ (|m Dei | 2 , |m Dµi | 2 , |m Dτ i )| 2 ) [11] and since, in general the three flavour directions ( 1 , 2 , 3 ) do not form an orthonormal basis 7 , for simple geometric reasons the only possibility to fully wash-out a generic pre-existing asymmetry is that (at least) the N 1 wash-out stage occurs in the three flavoured regime [13]. In this case the most straightforward way to have ST leptogenesis would be to impose strong N 1 wash-out in each flavour. This condition can be expressed as K 1α 1, having defined, for any α = e, µ, τ , the flavour decay parameters where Γ iα = Γ(N i → φ † l α ) andΓ iα = Γ(N i → φl α ) are the zero temperature limit of the flavoured decay rates into α leptons and anti-leptons in the three-flavoured regime and m 1.1 × 10 −3 eV is the equilibrium neutrino mass. This simple set of conditions is, however, too restrictive to allow also successful leptogenesis, since the strong lightest RH neutrino washout would act strongly not only on all components of the pre-existing asymmetry but also on the leptogenesis contribution from N 2 -decays. 8

Successful leptogenesis
The only way to realise successful ST leptogenesis is that the wash-out of the pre-existing asymmetry occurs in two separate steps. In a first step, imposing K 2τ 1, the tauon component of the pre-existing asymmetry is N 2 washed-out in the two-flavoured regime, implying 10 9 GeV M 2 10 12 GeV. In the second step the N 1 -washout stage has 7 On the other hand if they would form an orhtonormal basis, there would be no interference among the RH neutrinos and successful leptogenesis would be simply impossible since all CP asymmetries would vanish [37]. 8 On the other hand this set up might be useful to have the wash-out of a pre-existing B −L asymmetry in some baryogenesis model, e.g. electroweak baryogenesis, occurring at some energy scale lower than the N 1 wash-out.
still to occur in the three flavoured regime (M 1 10 9 GeV) in a way that, imposing K 1e , K 1µ 1, it can eventually suppress also the electron and muon components of the pre-existing asymmetries. This time, however, the tauon asymmetry generated by the N 2 -decays at the end of the N 2 wash-out stage, the genuine contribution from leptogenesis, can be sufficiently large to explain the observed asymmetry if at the same time the N 1 wash-out in the tauon flavour is weak, i.e. if K 1τ 1. In this way the successful ST leptogenesis necessarily leads to a tauon N 2 -dominated scenario, where the final asymmetry is dominated by the tauon flavour component produced by the out-of-equilibrium N 2 -decays.
It is quite interesting, and highly non-trivial, that SO(10)-inspired models naturally realise this kind of leptogenesis scenario and can, therefore, also potentially realise successful ST leptogenesis. However, as we have seen, successful ST leptogenesis also requires the additional conditions and it is then to be verified whether these can be met within SO(10)-inspired models. Summarising, we have a situation where both SO(10)-inspired and successful strong thermal leptogenesis might be simultaneously realised for an interesting (intersection) region in the space of see-saw parameters. If this region exists, then it is clearly very interesting to understand what are the corresponding constrains onto the low energy neutrino parameters. In the following discussion, our main objective will be to show analytically not only that such region exists, confirming the numerical results [17], but also that it indeed implies definite constraints on the low energy neutrino parameters if the pre-existing B − L asymmetry is sufficiently large. We will derive these constraints in the approximation V L I. Finally we will compare the analytical results with the numerical results obtained in [17] and also discuss how the constraints get slightly relaxed going beyond the approximation V L I.

Low energy neutrino data
As we will see the final asymmetry from SO(10)-inspired leptogenesis depends in such a non trivial way on low energy neutrino parameters that the successful leptogenesis conditions produces strong constrains on them, in a way that any experimental information on one parameter usually produces constraints also on the other parameters. The main goal is to be able to place constraints that can be tested in future experiments. In this Section we briefly review the current experimental information on low energy neutrino parameters that we will employ for the derivation of the constraints.
Neutrino oscillation experiments measure two mass squared differences, ∆m 2 atm and ∆m 2 sol . Neutrino masses can then be either NO, with m 2 2 − m 2 1 ≡ ∆m 2 sol and m 2 3 − m 2 2 = ∆m 2 atm , or IO, with m 2 3 − m 2 2 = ∆m 2 sol and m 2 2 − m 2 1 = ∆m 2 atm . For example, in a recent global analysis [24] it is found m atm ≡ m (s ij ≡ sin θ ij , c ij ≡ cos θ ij ) while for IO, within our convention for labelling light neutrino masses and adopting the usual definition for the thee mixing angles θ ij , the columns of the leptonic mixing matrix have to be permuted in a way that (12) 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 [23] for NO and IO respectively, It is interesting that 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, while all values [−π, +π] are still allowed at 3 σ.
4 From SO(10)-inspired conditions to RH neutrino masses and CP flavoured asymmetries In this section we show how the SO(10)-inspired conditions imply, in the hierarchical case, a N 2 -dominated RH neutrino mass spectrum [15]. Only for particular conditions on the low energy neutrino parameters, there exist crossing level solutions, in whose vicinity two or even all three RH neutrino masses are quasi-degenerate [16]. We derive compact analytic expressions both for the RH neutrino masses and for their CP asymmetries and compare them with the numerical results for some selected examples.

RH neutrino masses
Inserting the bi-unitary parameterisation eq. (3) into the diagonalised seesaw formula eq. (7) one obtains an expression for the (symmetric) inverse Majorana mass matrix in the Yukawa basis, This can be also easily inverted obtaining for the Majorana mass matrix in the Yukawa basis, From these expressions, either from M −1 or from M , one can derive the RH neutrino mass spectrum and the RH neutrino mixing matrix U R , as a function of the 9 low energy neutrino parameters in m ν (6 mixing parameters in U and 3 light neutrino masses m i ), the 6 parameters in the unitary matrix V L and the 3 Dirac neutrino masses m Di . This can be done diagonalising the hermitian matrix is a generic diagonal unitary matrix, also diagonalises M and M −1 . However, going back to the (Takagi) diagonalisation M = U R D M U † R and given a U R , one can unambiguously fix [19] If one is not in the vicinity of crossing level solutions, where at least two RH neutrino masses become degenerate, the RH neutrino mass spectrum is strongly hierarchical and analytical expressions can be easily found [15,16]. Here we adopt a slightly different procedure that yields simplified and more compact expressions in terms both of the light neutrino mass and of the inverse light neutrino mass entries. If we start from the eq. (16) for M , in the approximation V L I, we can write Considering that from the definition of U (cf. eq. (6)) one easily finds the eq. (19) can be also written more compactly as This equation shows that M i3 /M 33 = M 3i /M 33 ∝ m Di /m D3 and, therefore, in first approximation the LH side is in a block diagonal form and, neglecting terms At the same time the phase Φ 3 is also specified and one simply has The same procedure can be adopted for M −1 , rewriting the eq. (15) in the approximation V L I and imposing the Takagi diagonalization This time the RH side is approximately in a block-diagonal form with can be written as 1/M 1 |m νee |/m 2 D11 and, therefore, Also in this case the procedure allows to specify the phase Φ 1 , Finally, from the approximate expressions eq. (22) for M 3 and eq. (25) for M 1 , one can also easily find an approximate expression for M 2 . From the see-saw formula eq. (7) one has In this way we can write and for the phase It is easy to see from the above general expressions, that in the hierarchical limit, m 1 m sol (remember that we are assuming NO), the RH neutrino masses tend to the following simple expressions [15,16] where the last ones are obtained within the (rough) approximation s 13 = 0. In Fig. 1 we compare the found approximated analytic expressions for the RH neutrino masses (cf. eqs. (22), (25) and (28)) with the numerical solutions for the simple four sets of parameters yielding level crossings for special values of m 1 as discussed in [16] (note that for simplicity θ 13 = 0 and θ 23 = π/4). For the up quark masses at the leptogenesis scale, 10  values of m 1 for NO. The first two cases realise two different types of tauon N 2 -dominated leptogenesis solutions so called type A and type B [18,19] that we will fully describe analytically in Section 5. The third case is a strong thermal SO(10)-inspired solution [18,19] (realised for m 1 10 meV and N p,i B−L = 0.001). As we will discuss in Section 6, this can only emerge within type A solutions. As one can see this time there are no level crossings and the analytic solutions perfectly track the numerical ones for any value of m 1 . These results show explicitly how we can safely adopt the analytic solutions in our following discussion, though it should be made clear that the comparison is made for V L = I and, therefore, at this stage we are not testing the validity of the approximation V L I that will instead be discussed in 8.1.
It is also possible to find an analytical approximate expression for the RH neutrino mixing matrix U R . From the discussion above it should be clear that U R is of the form U R = I + ξ, where ξ ii = 0 and the ξ i =j leading terms are suppressed ∝ m Di /m Dj with equivalent to the expression in [16] but where we identified neutrino mass matrices combinations with entries of the inverse neutrino mass matrix, something that will prove very useful when we will extract the constraints on the low energy neutrino parameters. Details on the derivation can be found in the Appendix. It should be noticed that in this way we defined the phases Φ i in a way that Φ R 0, so that the eq. (29) for Φ 2 specialises into It can be also useful to calculate the orthogonal matrix Ω within SO(10)-inspired models. Starting from the orthogonal parameterisation for the neutrino Dirac mass matrix in the charged lepton basis [39], where Ω Ω T = I, and comparing with the bi-unitary parameterisation eq. (3), one finds straightforwardly an expression for the orthogonal parameterisation [18] M , that in term of the entries can be written as From the eq. (33) one can then find this approximate expression for Ω (see Appendix), that in the limit m 1 → 0 correctly reduces to the two RH neutrino limit form [40] Ω As we said in the introduction and clearly shown in the examples of Fig. 2, barring crossing level (resonant) solutions, the SO(10)-inspired RH neutrino mass spectrum naturally realises the N 2 -dominated scenario. This is because N 1 is too light to produce a sizeable asymmetry. At the same time, since M 3 10 12 GeV and the N 3 total asymmetry is strongly suppressed as ε 3 ∝ (M 2 /M 3 ) 2 , the N 3 -decays contribution to the final asymmetry is also negligible. In this way the only possibility to reproduce the final asymmetry relies on the N 2 -production that occurs in the two-flavoured regime since M 2 ∼ 10 10-11 GeV.
In the N 2 -dominated scenario the contribution to the asymmetry from leptogenesis can be calculated as the sum of the three (charged lepton) flavoured asymmetries ∆ α ≡ B/3 − L α . Normalising the abundance N X of a generic quantity X in a way that in the ultra-relativistic thermal equilibrium the abundance of RH neutrinos N eq N i (T M 1 ) = 1, the final asymmetry produced by the decays of the (N 2 ) RH neutrinos can then be written, in terms of the CP asymmetries ε 2α and the efficiency factors κ(K 2α ) at the production, as [34,35,41,37] N lep,f where K 2τ ⊥ 2 ≡ K 2e + K 2µ and ε 2τ ⊥ 2 ≡ ε 2e + ε 2µ . The baryon-to-photon number ratio can then be simply calculated as η B 0.01 N lep,f B−L . This is true assuming that any contribution from external sources can be neglected, a point that we will address in detail when we will discuss the strong thermal leptogenesis condition.

Flavoured CP asymmetries
It is now interesting to calculate the N 2 (flavoured) CP asymmetries within SO(10)inspired models and in particular to see how these are linked to the low energy neutrino parameters. Defining them as these can be calculated from [42] where we introduced Since in our case M 3 M 2 , we can use the approximation ξ(M 2 3 /M 2 2 ) 1 and neglect the second term ∝ J α 23 . Moreover, making use of the biunitary parameterisation (cf. eq. (3)) and the approximation V L I, one arrives to the following approximated expression for the flavoured CP asymmetries in SO(10)-inspired models, Using the approximated expression eq. (33) for U R and the relations (4), one finds the following hierarchical pattern for the ε 2α 's: As one can see, while ε 2µ is suppressed about four orders of magnitude (∼ m 2 c /m 2 t ) compared to ε 2τ , the electronic CP asymmetry is suppressed even about seven orders of magnitude compared to ε 2µ . For this reason the electron contribution to the final asymmetry is always completely negligible. 11 This is well shown in Fig. 3 where, for the same four sets of parameters of Fig. 2, the flavoured (and total) CP asymmetries are plotted versus m 1 , comparing the numerical result (solid lines) with the analytic

Successful leptogenesis condition
In this Section we finally impose the successful leptogenesis condition finding some first interesting constraints on the low energy parameters. In this respect we extend the results found in the hierarchical (LH) neutrino masses limit [18,19] to arbitrary values of m 1 . The final asymmetry should be calculated using the eq. (38). However, as we have seen, in the approximation V L I the tauon CP asymmetry is by far the dominant one the inclusion of the wash-out at the production cannot change the τ -dominance as a contribution to the final B − L asymmetry. However, it should be stressed that this result holds using the V L I approximation, while relaxing this approximation, a muon-dominated solution also appears for m 1 10 meV [19]. In any case the ST condition selects the tauon-dominated solution anyway and for this reason we can neglect the muon-dominated solution in our discussion. In this way the expression for the final asymmetry eq. (38) greatly simplifies into 11 It is curious to notice that since the contribution to ε 2α from the interference with N 3 is so suppressed, actually it becomes comparable to the term coming from the interference with N 1 that we are neglecting in the eq. (40).
Using the explicit expressions eqs. (33) and (36) for the U R and Ω matrices respectively, we are now able to express the final B − L asymmetry in SO(10)-inspired models in terms of the α i and the low energy neutrino parameters.

Final asymmetry in terms of the low energy neutrino parameters
Let us start from the derivation of an expression of ε 2τ . First of all we can specialise the eq. (43) to the case α = τ , obtaining Using then the expressions found for U R and M 2 , we arrive to where we have introduced the effective SO(10)-inspired leptogenesis phase α L (in the approximation V L = I) and where from the eqs. (23) and (34) one has so that we can write for the effective leptogenesis phase Let us now calculate the efficiency factor at the production κ(K 2τ ). First of all, from the expression eq. (8), one can easily find, for V L I, a general expression for the K iα 's, From this one we can then obtain a specific expression for where in the second approximation we made use of the eqs. (33) and (28).
From the general expression eq. (51) we can also write an expression for K 1τ describing the exponential suppression of the lightest RH neutrino wash-out (cf. eq. (45)) From this one we can then obtain an explicit expression in terms of the mixing angles and low energy phases that will prove useful, (54) We can then finally put together all the results finding, from the eq. (45), an expression in terms of the low energy neutrino parameters that can be , It is interesting to notice that: • The asymmetry does not depend on α 1 and on α 3 [18]. This is an important point since the only left non-observable parameter is α 2 on which however one can place a lower bound and, within SO(10)-inspired models cannot be in any case too large.
• The effective neutrinoless double beta decay mass m ee ≡ |m νee | plays a direct role and it can be noticed that successful leptogenesis implies the existence of a lower bound.
• The successful leptogenesis condition links together all low energy neutrino parameters constraining them to lie on a hypersurface described by the only left theoretical parameter α 2 .
In Fig. 4 we have plotted η B vs. m 1 for the same three sets of parameters of Figs. 2 and 3 comparing the numerical results (blue solid lines) with the analytical results (black dashed lines) obtained from the eq. (55). As one can see the analytical results perfectly match the numerical ones.
We also made a more general comparison between the constraints derived from the analytical expression eq. (55) and the numerical constraints (for V L = I). In Fig. 5 we show, with orange points, the results of a scatter plot for V L = I imposing successful SO(10)-inspired leptogenesis for α 2 = 5. The asymmetry is calculated from the eq. (45) where RH neutrino masses and mixing matrix U R are calculated numerically. The mixing angles are uniformly random generated within the same ranges adopted in [18], with the only exception of θ 23 that is allowed to be slightly lower, as adopted in [17]. The results confirm those obtained in [18,19], simply here a much higher (about thousand times) amount of points has been obtained and the constraints are much sharper. We have then produced corresponding scatter plots using directly the analytical expression for the final asymmetry eq. (55). The results are shown in Fig. 6 and as one can see they perfectly reproduce the numerical results shown in Fig. 5 (orange points). We can then conclude that the eq. (55) provides a very precise analytical way to calculate the final asymmetry in SO(10)-inspired models (barring crossing level resonant solutions) in the approximation V L I and can be regarded as one of the main results of our investigation. Indeed it can reliably be applied in all models where SO(10)-inspired conditions hold in order impose the successful leptogenesis condition using directly predictions on low energy neutrino data (the only additional parameter that has to be calculated is α 2 ).
Having done this important cross check, we can now safely proceed further deriving analytical constraints on the low energy neutrino parameters imposing successful leptogenesis fully trusting our eq. (55). Because of the intricate dependence of the eq. (55) on the low energy neutrino parameters, we need to understand the behaviour at low and

Lower bound on m 1
Let us now calculate the final asymmetry in the limit m 1 → 0 showing that this tends to vanish and, therefore, that successful SO(10)-inspired leptogenesis implies a lower bound on the absolute neutrino mass scale [19]. It is convenient to start from K 1τ . In the limit m 1 /m sol → 0 the eq. (53) simplifies into We can now consider the low m 1 limit of ε 2τ , obtaining from the eq. (47), where the asymptotic limit for the effective leptogenesis phase is given by α L 2 (ρ − σ). Notice that we have retained the term ∝ s 13 in m ee since, as we have seen, it has to be non-vanishing. The expression is maximised for σ − δ n π with n integer and clearly for sin α L = 1, finding (61) Finally the the asymptotic limit for K 2τ for m 1 /m sol 1 is given by K 2τ c 2 23 m atm /m 25 [18]. This shows that in the low m 1 limit the wash-out at the production is strong and in this case one can use a simple approximation for the efficiency factor [7], κ(K 2τ ) 0.5/K 1.2 2τ 0.01. Combining together the results found for the three terms, one finds that in the low m 1 limit the baryon-to-photon number ratio is maximised by where the last inequality has been obtained for the values of the mixing angles within the ranges eq. (56) that minimise the lower bound. The result is in very good agreement with the results of the scatter plots shown in Fig. 5 (orange points) and Fig. 6 and confirms, in more detail, the value obtained in [18]. Finally it should be noticed that the three conditions for maximal asymmetry on the three low energy phases, 2 σ − δ m π, σ − δ n π and sin[2 (ρ − σ)] 1, with n, m integers, imply that in the low m 1 limit one has σ = (2 n − m) π and δ = 2 π n, results that are confirmed by the results, at low m 1 , of the scatter plot shown in the two panels in Fig. 5 (and Fig. 6 as well) for σ and δ vs. m 1 (orange points). One also finds ρ = π/4 + q π,with q integer. It can be seen however that at small m 1 the value of ρ is actually ρ 0.35 π + q π. The reason for this shift is understood from the more complete expression eq. (54) for K 1τ . For ρ = π/2 the term m 1 e 2iπρ = −m 1 adds to the term −m 2 in a way that K 1τ 1 for slightly lower values of s 23 . However, because of the strong dependence ε 2τ ∝ s −4 23 , a shift of ρ towards π/2 maximises the asymmetry even though the phase α L is not maximal. We will be back soon on the value of θ 23 that cannot be too large.

Upper bound on m 1
Together with a lower bound on m 1 , there is also an upper bound on m 1 . We can work in the quasi-degenerate neutrino limit m 1 m 2 m 3 and then check whether the upper bound does indeed fall in the quasi-degenerate regime.
Let us first calculate separately the quasi-degenerate limit of K 1τ , K 2τ and ε 2τ , the three relevant quantities determining the final asymmetry. For K 1τ from the eq. (54) one can immediately see that if ρ = n π, with n integer, then This expression shows that for m 1 0.1 eV in any case one has K 1τ 4, where the maximum is saturated for σ = 2π m and δ = π/2 + k π, so in any case it cannot be too large and in any case it can be always made vanishing.
Let us now calculate the asymptotic limit for ε 2τ for ρ n π. We can neglect all sub-dominant terms ∝ s 2 13 containing δ in a way that the dependence on δ cancels out.
First of all notice that m ee ≡ |m νee | → m 1 an asymptotic limit that is clearly visible in the panels of Fig. 3 and 4. At the same time one has, In this way putting all terms together one finds from the eq. (47) the following asymptotic limit for ε 2τ where the asymptotic limit of α L is given by α L → −4 σ. Finally we can calculate the asymptotic limit of K 2τ from the eq. (52), finding Putting all together in the eq. (45) for N leo,f B−L , using the approximation κ(K 2τ ) (1 + 2 K 1. where, remember, we assumed ρ = n π.
Here we can notice that the asymptotic limit depends mainly on the value of σ, since in any case ρ n π in order to have K 1τ 1 and here the residual dependence on δ can be neglected. Guessing that the value 2σ 1 in order to minimise K 2τ implies that η B , using simply sin α L 1, is maximised by where we defined x ≡ s 2 23 c 2 23 |e −2iσ − 1| 2 and maximized on it finding x = 2.5 1.2 (m /m 1 ), that indeed implies σ 1 as guessed. Imposing the successful leptogenesis bound one then straightforwardly finds the upper bound very well reproducing the result from the scatter plots (Fig. 5, Fig. 6 and Fig. 7) even though it does not fall in a full quasi-degenerate regime. This upper bound is shown in Fig. 7 (dot-dashed line). In conclusion the upper bound on m 1 is mainly due to an interplay between minimising simultaneously both K 1τ and K 2τ while maximising the CP asymmetry. In the top-left panel of Fig. 8 a scatter plot of K 2τ vs. m 1 (orange points) confirms how for m 1 10 meV the value of K 2τ becomes smaller and smaller for growing m 1 in order to minimise the wash-out at the production that suppresses the asymmetry ∝ m /m 1 . The upper bound on m 1 is saturated for an analytical minimum value of K 2τ 2.5 well in agreement with the numerical result.

Type A solution (m 1 m sol )
We now describe what happens for values of m 1 between the lower and the upper bound. From this point of view, as we will see, the value of m sol 10 meV represents a kind of border between two different solutions, the τ A and the τ B solutions, thought the border is not sharp and the two solutions overlap somehow around m 1 10 meV. This distinction will be useful when we will discuss the ST solution in the next Section. Let us start from values m 1 m sol .

Upper bound on θ 23
An important feature of SO(10)-inspired leptogenesis is that, for NO as we are considering, it places an upper bound on θ 23 . In the case of low values of m 1 m sol , from the eq. (57), imposing K 1τ 1 and taking into account the dominant term ∝ m 1 e 2 i ρ in K 1τ , and approximating ρ π/2, one finds the upper bound where the maximum value on the right-hand side is obtained clearly in the limit m 1 /m sol → 0. In Fig. 7 we show the results of a specific scatter plot obtained starting from the analytic eq.

Sign of the asymmetry and low energy phases
Here we want to show how the sign of the asymmetry influences the values of the phases. Looking at the K iα one could indeed think that constraints on ρ and σ should exhibit a π/2-periodicity and constraints on δ a π periodicity. This is because these quantities are defined in absolute values and that an overall change of sign of the argument leaves them unchanged. However it can be seen from the Figs. 5 and 6 that actually σ and ρ constraints have a π periodicity and constraints on δ have a π periodicity. This is an effect of the sign of the asymmetry that is clearly given by For the τ A solution, for m 1 m sol , one has α L 2 (ρ − σ) and, therefore, one has that ρ − s π/4 would maximise the asymmetry. We have already discussed how in the limit of lowest m 1 the value of ρ 0.35 π while indeed the value of σ 0.1 π. This is because values ρ = π/2 would maximise the amplitude of the CP asymmetry and at the same time minimise K 1τ . Values ρ 0.35 π are, therefore, a compromise that maximise the total final asymmetry. At the same time the value 2σ − δ 0 in order to minimise K 1τ , while σ − δ 0 in order to maximise m ee .
When m 1 /m sol increases, the values of the phases can be understood from the eq. (54) for K 1τ . The first term in the numerator ∝ m 1 e 2 i ρ becomes non-negligible. Since ρ is slightly different from π/2, this term has some non-vanishing imaginary part that has also to be cancelled in order not to have K 1τ 1. Since, as we have seen, at low m 1 necessarily s 13 = 0, this is cancelled by the term ∝ m 3 e i (2σ−δ) with 2σ − δ < 0. For increasing values of m 1 , the value of ρ has necessarily to tend to ρ = n π in order to have a cancellation of m 1 e 2 i ρ − m 2 . In this case there are two possibilities: either ρ > π/2 and in this case 2σ − δ > 0 or ρ < π/2 and in this case 2σ − δ < 0. The latter is clearly the dominant one, since, as we said, at lowest values of m 1 one has necessarily ρ 0.35π < π/2. For this case it is possible at the same time to have maximal phase and small value of K 1τ . It is important to realise that the dominance of this case is driven by the positive sign of the asymmetry.
In order to clearly see these results from the scatter plots in a clear way, we have performed new scatter plots finally constraining the reactor mixing angle in the current 3σ experimental range (cf. eq, (13)). The results are shown in Fig. 9. In the first (top-left) panel we show the ρ vs. m 1 scatter plot. One can first of all see that because of the much more restricted θ 13 range, many points disappeared compared to the corresponding plot in Fig. 5 and the behaviour is much cleaner. At the lower bound m 1 1 meV one can see how indeed ρ 0.35 π. For increasing values of m 1 there are two branches for ρ: in a first 'high' brunch the value of ρ increases to π and in a second 'low' branch it decreases to 0, where the two branches actually merge because of the π periodicity. It is clearly noticeable how the low branch clearly dominates, since it corresponds to values of ρ that produce the correct sign of the asymmetry and to maximal leptogenesis phase (α L ) values already at minimum m 1 -values, while the high-ρ branch is suppressed since it corresponds to non-maximal α L values.
In the second (top-right) panel we show the 2σ − δ scatter plot. This clearly shows how the 'low-ρ' branch corresponds to (dominant) 2σ − δ values below 2 nπ, while the 'high-ρ' branch corresponds to (sub-dominant) 2σ − δ values greater than 2 nπ. The next step is to understand what are the corresponding values of σ. In the third panel of Fig. 9 we show the σ vs. m 1 scatter plot. As it could be expected by the fact that α L → 2(ρ − σ) for m 1 m sol , the (subdominant) high ρ values branch corresponds to high sub-dominant σ values branch (σ n π) while the dominant low ρ values branch corresponds to a low (dominant) σ values branch.
Finally, combining these results on σ with the results shown on 2σ − δ, we can deduce the behaviour of δ. For the dominant low-ρ values branch, corresponding to a low-σ values branch and values of 2σ − δ 2 n π one concludes that δ shifts toward negative values. Vice-versa the sub-dominant high ρ values branch, corresponding to 2σ − δ > 2n π values and high σ > m π values, one has positive δ values. The results are shown in the last (bottom-right) panel of Fig. 9. One can see the clear dominance of values of δ in the fourth quadrant. This conclusion is strengthened even more by a scatter plot of δ vs. θ 23 showing Figure 9: Scatter plots of points satisfying successful leptogenesis generated using the analytical expression (55) for the final asymmetry in different planes. The mixing angle θ 13 values are uniformly randomly generated within the 3 σ allowed experimental range in eq. (13). Panels should be compared with the corresponding ones in Fig. 6, in particular the last one for δ vs. θ 23 .
that actually positive values of δ are even more constrained if one imposes the current 3σ lower bound θ 23 38 • . At this level this result should be mainly be meant as a proof that within SO(10)-inspired leptogenesis the sign of the asymmetry yields asymmetric constraints between positive and negative sin δ values. However, it would be certainly desirable to verify how these constraints relax going beyond the V L = I approximation since this could provide quite an effective way to test SO(10)-inspired leptogenesis with future experimental results.

Type B solution (m 1 m sol )
Because of the upper bound m 1 m atm , we can approximate m 1 m 2 and m 3 m atm . In this case the general expression for K 1τ eq. (54) can be written as, (74) Here there are two possibilities to minimise K 1τ . In the case s 13 → 0, then very easily one can have ρ = n π and this immediately produces K 1τ = 0, showing that it is quite easy to find a way for the tauon asymmetry to escape the lightest RH neutrino wash-out. On the other hand for the measured values s 13 0.15 a non-vanishing value of the first term in the numerator is necessary in order to cancel the second term. The exact value of ρ depends on the value of m 1 . The value of δ is in this case able to cancel the imaginary part of e i2ρ but at the same time has to be such to keep σ − δ ν π in order to maximise the value of m ee . Moreover since α L −4 σ and has to be negative, this also leads to negative values of δ and favours positive values of ρ. This is confirmed by the first panel Fig. 9 we show a scatter plot of ρ vs. m 1 for θ 13 in the 3σ range eq. (13). It can be seen how this time, compared to the analogous plot of Fig. 5 where 0 ≤ θ 13 ≤ 11.54 • , ρ = n π only when m 1 saturates its upper bound.
We can maximise K 1τ taking in both cases ρ = n π even for m 1 m sol m atm and taking σ − δ = m π in order to maximise m ee , and in this case one finds showing that indeed the lightest RH neutrino wash-out can be avoided in any case.
Let us now consider the CP asymmetry ε 2τ and the wash-out at the production described by K 2τ . In the expressions one can still approximate m ee m 1 , as for the quasi-degenerate case. However, this time one has where we have approximated σ n π. These two expressions produce a dependence η B ∝ s −4 23 that strongly suppresses the asymmetry for increasing s 23 and produces a tight upper bound on θ 23 . On the other hand, however, now one also has η B ∝ m 1 /m 3 and this makes in a way that the upper bound gets relaxed at higher m 1 reaching a maximum toward m 1 0.035 meV. This is because for higher m 1 the term |(m −1 ν ) µτ | ∝ 1 − m 1 /m 3 gets suppressed.
Finally one also has Combining together all results, one finds an implicit form for the upper bound of s 23 vs. m 1 . In Fig. 7 we have plotted with the dot-dashed line the result. As one can see it somehow overestimates the allowed region but this is a consequence of the crude approximations used for the phases. In any case these results well explain the existence of an upper bound on θ 23 also for values m 1 m sol and how this gets relaxed for increasing values of m 1 upto a peak value that is reached for m 1 35 meV. It should be noticed how the region for the τ A and for the τ B solutions overlap to some extent for m 1 10 meV. This is not contradictory since they are realised for different values of the phases, in particular in the case of the τ A solution the phase ρ π/2 for m 1 → 0, while for the τ B solution one has ρ π for m 1 = m max 1 . Around m 1 10 meV the two solutions meet but the τ B has higher values of ρ resulting in lower values of K 2τ .

ST leptogenesis condition
In this section we finally over-impose the ST condition in addition to the SO(10)-inspired conditions deriving all the features of the ST-SO(10)-inspired solution [17]. All SO(10)inspired solutions, for V L = I, are already tauon-type solutions satisfying K 1τ 1. Therefore, we need to impose the conditions eqs. (9).
For pre-existing initial asymmetries N p,i ∆α in the different flavours α = e, µ, τ , one has to require [22] in order for these components to give a negligible contribution to the final asymmetry, where because of geometric factors in general a total pre-existing asymmetry N p,i B−L corresponds to values of the electron and muonic pre-existing asymmetries at the lightest RH neutrino wash-out of about N p,i ∆α N p,i B−L /6 [22].

The τ B solutions cannot realise ST leptogenesis
First of all it is easy to show that the τ B solutions, characterised by m 1 10 meV and ρ n π cannot satisfy the ST condition.
If one goes back to the expression (77) for K 2τ in the τ B case, one can immediately check that for m 1 m sol one has K 2τ 13, that would be still sufficient to wash-out a pre-existing asymmetry as large as ∼ 10 −2 . There is also no problem for τ B solutions to wash-out a pre-existing electronic asymmetric. Starting from the general expression eq. (8) for the K iα one can easily see that in the τ B case, m 1 m sol and ρ n π, one has K 1e m ee /m m 1 /m , that is sufficient to wash-out electronic pre-existing asymmetries as large as 10 −3 for m 1 10 meV and even larger if m 1 increases (as we will see soon this is indeed the origin of the lower bound on m 1 ). The real intrinsic problem for τ B solutions is the wash-out of a pre-existing muon component, since one can easily see that confirming in a general analytical way the numerical examples shown in [18,19,17]. Therefore, we conclude that τ B -type solutions cannot realise ST leptogenesis because they cannot wash-out a large pre-existing muon asymmetry. We can then now focus on τ A solutions in the following discussion.
Lower bounds on m ee and on m 1 As we said in the τ B case one finds easily from the general expression eq. (8) K 1e = m ee /m , interestingly showing how in SO(10)-inspired models a not too low neutrino-less double beta decay effective neutrino mass is required for the wash-out of the pre-existing electronic asymmetry. Indeed, from the eq. (78) we can immediately place the lower bound that is quite interesting since it predicts that, despite neutrino masses are normally ordered, next generation 00ββ experiments should find a signal. This lower bound translates into a lower bound on m 1 . From the explicit general expression for m ee , = |m 1 c 2 12 c 2 13 e 2 i ρ + m 2 s 2 12 c 2 13 + m 3 s 2 13 e 2 i (σ−δ) | (82) where we approximated m 1 m 2 and neglected the term ∝ m 3 s 2 13 . Considering that for the τ A type solutions one has 2ρ ±π/2, one arrives to m ee /m 1 c 4 12 + s 4 12 0.75, in very good agreement with the numerical results. From this result combined with the lower bound eq. (80), then one obtains the lower bound

Lower bound on θ 13
From the general expression eq. (8) for the K iα one obtains the following expression for K 1µ for ρ ±π/2 and m 1 m 2 , It is easy to see that for s 2 13 = 0 ,the condition K 1µ 10 (for N p B/3−Lµ = 10 −3 ) would imply m 1 30 meV that is clearly incompatible with the upper bound m 1 16 meV just obtained. However, for s 2 13 0.1, corresponding to θ 13 5 • , one can satisfy K 1µ 10 and m 1 16 meV. This is an interesting feature of the ST solution, since it is a prediction that is now verified by the experimental results.
It should be noticed that this lower bound strengthen the lower bound eq. (58) derived from the condition K 1τ 1.

Dirac phase in the fourth quadrant
As we discussed in 5.4.2, for τ A solutions the Dirac phase is driven toward negative values by its link with the phase σ in K 1τ , that requires, for non-vanishing θ 13 , 2σ − δ 0. In addition there is a subdominant solution for δ π When the ST condition is imposed this conclusion is strengthened even more by the more stringent lower bound on τ 13 . The last panel of Fig. 9, where we plotted δ vs. θ 23 for the experimental allowed 3σ range for θ 13 , clearly shows this situation. It also shows how δ is basically constrained in the fourth quadrant for θ 23 in the first octant and θ 23 38 • . We have just seen how the ST condition necessarily requires θ 23 in the first octant as a result of the lower bound on m 1 . Combining this result with the 3σ lower bound θ 23 38 • , we conclude that the only way for the ST condition to be satisfied for such high values of θ 23 is to have δ in the fourth quadrant (−π/2 δ 0). This is an interesting result in light of the experimental hint for sin δ < 0. The ST condition more definitely requires also cos δ > 0. From the Fig. 5 panel showing δ vs. θ 23 , it can be seen how, for N p B−L = 10 −3 , the highest value of θ 23 are obtained for δ −60 • and is given by θ 23 40 • (light blue points for V L = I).

Strong thermal leptogenesis
It is quite straightforward to understand why the ST condition cannot be satisfied in the IO case. Even though the N 1 wash-out of the electron pre-existing component is strong since again one has K 1e = m ee /m m 2 /m 50, on the other hand the wash-out of the muonic component is very weak since one has This happens because, since m 2 m 3 , there is a cancellation of the two terms ∝ m 2 , m 3 (that are not suppressed by s 2 13 ). This is again, as for NO, a particular example of what happens more generally, beyond SO(10)-inspired models, for IO: the wash-out of a muonic pre-existing asymmetry is weakened by this cancellation and produces a lower bound on m 1 [22]. In the case of SO(10)-inspired models the cancellation is basically almost exact in a way that the lower bound becomes incompatible with the CMB upper bound and one can conclude that ST SO(10)-inspired leptogenesis is simply not viable for IO neutrino masses.

Theoretical approximations and uncertainties
The eq. (55) constraints all low energy neutrino parameters to lie on a hyper-surface in the low energy neutrino parameter space. However, there are different effects of different nature to be considered that make in a way that this hypersurface has actually a thickness: the experimental errors on the low energy neutrino parameters; the theoretical uncertainties in the calculation; accounting for deviations from V L = I introduces a dependence on the parameters of the V L that since are not measured lead to an intrinsic indetermination within SO(10)-inspired models. Of course within a specific model one is able to specify the V L and in this case one can expect and calculate specifically the deviation from the hypersurface described by the eq. (55).
Beyond the approximation V L I In Fig. 5 we have included the results of a scatter plot, for α = 5 and NO, of points respecting successful leptogenesis for I ≤ V L ≤ V CKM (yellow points), confirming once more previous results [19,22]. When these are compared with the results obtained for V L = I (the orange points), one can see that there are constraints that do not get strongly modified (e.g. the lower bound on m 1 ) and constraints that are more greatly modified (e.g. the upper bound on m 1 ). The most remarkable difference can be noticed in the panel θ 23 vs. m 1 where a complete new region at large values m 1 m sol appears and that is now largely excluded by the CMB upper bound on m 1 . This region is due to the appearance of a muon-type solution that is possible since when deviations of V L from unity are taken into account the strong hierarchy in the CP asymmetries (cf. eq. (44)) gets much milder and a muonic solution becomes possible.
Another clear difference is that for τ type B solution the upper bound on θ 23 is much more relaxed. On the other hand the constraints for the τ type A solution do not change dramatically, except for the well known effect that now there is no lower bound on θ 13 .
We have also compared the results obtained in the approximation V L = I (light blue points) and for I ≤ V L ≤ V CKM (dark blue points) in the case of ST. One can see how the constrains get moderately relaxed. The lower bound on θ 13 gets relaxed from θ 13 5 • to θ 13 2 • . The upper bound on θ 23 gets relaxed from θ 23 42 • to θ 23 43 • , probably the most important effect in light of the current experimental constraints on θ 23 that tend to favour θ 23 40 • at least at 2σ. One can leo see how the lower bound on m ee gets greatly relaxed.
In Fig. 4 we have plotted of η B vs. m 1 for three examples where V L = I comparing them to the three examples for V L = I. In all three cases the low energy neutrino parameters are unchanged. One can see how turning on angles in the V L can significantly enhance the asymmetry, though not dramatically (approximately up to a factor 2).

Theoretical uncertainties
A detailed discussion of the theoretical uncertainties can be found in [22]. Here we just remind that the main sources of corrections to our results should come by inclusion of flavour coupling and (in the case of the muonic solution) of phantom terms [41]; account of decoherence for M 2 10 11 GeV within a density matrix formalism [43,44,45,37,46]. Minor effects should come from the running of neutrino parameters [47] and a full relativistic calculation of the wash-out rates [48].

Summary
We have seen how SO(10)-inspired models motivate an interesting scenario of high energy scale leptogenesis with hierarchical RH neutrinos able to reproduce the correct asymmetry when the production from the decays of the next-to-lightest RH neutrinos is taken into account. This scenario implies constraints on the low energy neutrino parameters that will be partially testable with low energy neutrino experiments. In the approximation of negligible misalignment between the Yukawa basis and the charged lepton basis (V L = I), we found a very accurate analytical expression that links all low energy neutrino parameters in quite a non-trivial way. Constraints on each individual parameter depend on the experimental information on the other parameters and interesting predictions can gradually emerge with more experimental information. For example, we have seen how the discovery of a non-vanishing θ 13 seems to produce combined constraints on the Dirac phase, the atmospheric mixing angle and the absolute neutrino mass scale. This potential interesting feature should, however, be confirmed relaxing the approximation V L = I.
In addition, quite interestingly, for a subset of the successful leptogenesis solutions, SO(10)-inspired leptogenesis realise the ST condition, in a way that a large pre-existing asymmetry is efficiently washed-out and the final asymmetry is independent of the initial conditions. This produces very tight constraints on the low energy neutrino parameters characterising the SO(10)-inspired ST solution whose predictions will be (almost) completely tested during next years. For example, the discovery of a IO neutrino mass spectrum would certainly rule out the solution like also a value of the atmospheric angle in the second octant. Vice-versa a discovery of NO and atmospheric neutrino mixing angle in the first octant should should certainly be regarded as a strong support to the solution. In this respect it is interesting that some of the current global analyses [23,24] mildly support either IO and θ 23 in the second octant or NO and θ 23 in the first octant: in the first case the solution would be undoubtedly ruled out, while the second case should be regarded as a very successful test.
It is also interesting that the experimental value of the reactor mixing angle falls just within quite a narrow range of values allowed by the solution. Interestingly the solution, within the allowed current range for the atmospheric mixing angle, is also potentially able to explain the emerging hint for a negative value of sin δ, and therefore of J CP , with the additional prediction cos δ > 0, with δ ∼ −π/4. We managed to provide a full analytical description of these constraints. In particular we showed why non-vanishing values of the reactor mixing angle are required. Very importantly the ST leptogenesis condition also forces m 1 to lie within a narrow range about m 1 20 meV. This narrow range necessarily implies the atmospheric angle θ 23 in the first octant and this also indirectly forces δ to lie in the fourth quadrant.
The NOνA long baseline experiment [49] is currently taking data and, combined with the results from the other neutrino oscillation experiments, in particular T2K constraints on δ [50] and θ 13 determination [20], will allow in the next years to test in quite a significant way the predictions on the leptonic mixing matrix unknowns from the ST SO(10)-inspired solution. At the same time cosmological observations are starting to corner quasi-degenerate neutrino and might in a close future start to test the narrow window, 75 meV m i 125 meV, required by the solution corresponding to semi-hierarchical NO neutrino masses. In longer terms neutrinoless double beta decay experiments should also be able to test the range of values predicted by the solution for m ee centred about m ee 15 meV. In this way one would just miss a further experimental complementary constraint on the Majorana phases for a complete test of the solution (explaining why the solution can 'almost' completely be tested). It is then quite exciting that with SO(10)-inspired leptogenesis one has quite a well motivated opportunity to probe at the same time not only leptogenesis and the see-saw mechanism, but also the SO(10)-inspired conditions, sheding light on the model for the origin of neutrino masses and mixing.
Finally, from the second eq. in (A.1) and eq. (A.2), one can write leading to the eq. (33). In writing these equations we have made use of the fact that the entries of the inverse neutrino mass matrix are related to the entries of the neutrino mass matrix by (A.8) The orthogonal matrix entries are given by the eq. (35). Using the eq. (33) and that (m ν U ) ei = −(U D m ) ei one arrives to the eq (36).