Aspects of High Scale Leptogenesis with Low-Energy Leptonic CP Violation

Using the density matrix equations (DME) for high scale leptogenesis based on the type I seesaw mechanism, in which the CP violation (CPV) is provided by the low-energy Dirac or/and Majorana phases of the neutrino mixing (PMNS) matrix, we investigate the 1-to-2 and the 2-to-3 flavour regime transitions, where the 1, 2 and 3 leptogenesis flavour regimes in the generation of the baryon asymmetry of the Universe $\eta_B$ are described by the Boltzmann equations. Concentrating on the 1-to-2 flavour transition we determine the general conditions under which $\eta_B$ goes through zero and changes sign in the transition. Analysing in detail the behaviour of $\eta_B$ in the transition in the case of two heavy Majorana neutrinos $N_{1,2}$ with hierarchical masses, $M_1 \ll M_2$, we find, in particular, that i) the Boltzmann equations in many cases fail to describe correctly the generation of $\eta_B$ in the 1, 2 and 3 flavour regimes, ii) the 2-flavour regime can persist above (below) $\sim 10^{12}$ GeV ($\sim 10^9$ GeV), iii) the flavour effects in leptogenesis persist beyond the typically considered maximal for these effects leptogenesis scale of $10^{12}$ GeV. We further determine the minimal scale $M_{1\text{min}}$ at which we can have successful leptogenesis when the CPV is provided only by the Dirac or Majorana phases of the PMNS matrix as well as the ranges of scales and values of the phases for having successful leptogenesis. We show, in particular, that when the CPV is due to the Dirac phase $\delta$, there is a direct relation between the sign of $\sin \delta$ and the sign of $\eta_B$ in the regions of viable leptogenesis in the case of normal hierarchical light neutrino mass spectrum; for the inverted hierarchical spectrum the same result holds for $M_1 \lesssim 10^{13}$ GeV. The considered scenarios of leptogenesis are testable and falsifiable in low-energy neutrino experiments.


Introduction
In spite of the fact that the leptogenesis idea of the origin of the matter-antimatter, or baryon, asymmetry of the Universe is 35 years old [1,2], the leptogenesis scenario of the asymmetry generation continues to be actively investigated (see, e.g., the recent review article [3] which includes also an extended list of references). A very attractive version of leptogenesis is that based on type I seesaw mechanism of neutrino mass generation [4][5][6][7][8], which also corresponds to the original scenario proposed in [1]. The type I seesaw mechanism provides a natural explanation of the smallness of neutrino masses and via leptogenesis establishes a link between the existence and smallness of neutrino masses and the existence of the baryon asymmetry. A basic ingredient of the seesaw scenario are the singlet RH neutrinos ν lR (singlet RH neutrino fields ν lR (x)), by which the Standard Model (SM) can be extended without modifying its fundamental properties. Such an extension with two RH neutrinos is the minimal set-up in which leptogenesis can take place, satisfying the three Sakharov conditions [9] for a dynamical generation of the baryon asymmetry. In leptogenesis the requirement of lepton charge nonconservation is satisfied, as is well known, due to a Majorana mass term of the RH neutrinos ν lR and the Yukawa coupling L Y (x) of ν lR with the Standard Model lepton and Higgs doublets, ψ lL (x) and Φ(x), while the requisite C-and CP-symmetry violations are ensured by the ν lR Majorana mass term and/or the Yukawa coupling L Y (x). Both the ν lR Majorana mass term and L Y (x) respect the SU (2) L × U (1) Y W symmetry of the SM. In the diagonal mass basis of the RH neutrinos ν lR and the charged leptons l ± , l = e, µ, τ , L Y (x) and the Majorana mass term are given by: where Y li is the matrix of neutrino Yukawa couplings (in the chosen basis) and N i (N i (x)) is the heavy Majorana neutrino (field) possessing a mass M i > 0.
In the present article we revisit the high (GUT) scale flavoured leptogenesis scenario, which is realised for masses of the heavy Majorana neutrinos in the range M i ∼ (10 9 − 10 14 ) GeV, i.e., for M i by a few to several orders of magnitude smaller than the unification scale of electroweak and strong interactions, M GU T ∼ = 2 × 10 16 GeV. The values of the masses M i , i.e., their scale and spectrum, set the scale of leptogenesis. In this scenario, the out-of-equilibrium decays N j → l + + Φ (−) and N j → l − + Φ (+) of the heavy Majorana neutrinos N j , caused by the CP non-conserving neutrino Yukawa couplings in Eq. (1), proceed with different rates, producing CP violating (CPV) asymmetries in the flavour lepton charges L l , l = e, µ, τ , and in the integral lepton charge L = L e + L µ + L τ , of the Universe. The lepton asymmetries thus generated are converted into a baryon asymmetry by (B−L)-conserving, but (B+L)-violating, sphaleron processes which exist in the SM and are effective at temperatures T ∼ (132 − 10 12 ) GeV.
For heavy Majorana neutrinos N 1,2,3 with hierarchical mass spectrum, M 1 M 2 M 3 , there exists, as is well known, a lower bound on the mass of the lightest N 1 for which the matter-antimatter asymmetry can be generated in leptogenesis: M 1 ∼ > 10 9 GeV [10]. Furthermore, when flavour effects in leptogenesis [11][12][13] (see also [14][15][16]) are taken into account, leptogenesis was shown to be possible at scales compatible with the quoted lower bound with the requisite CP violation provided exclusively by the Dirac and/or Majorana phases in the Pontecorvo, Maki, Nakagawa, Sakata (PMNS) neutrino (lepton) mixing matrix U PMNS [17][18][19][20][21][22][23][24]. Rather detailed studies performed in 2018 in [25,26] have shown that in the case of spectrum of masses of the heavy Majorana neutrinos with mild hierarchy, M 2 ∼ 3M 1 , M 3 ∼ 3M 2 , the "flavoured" leptogenesis can successfully generate the observed baryon asymmetry at scales as low as ∼ 10 6 GeV, i.e., for M 1 ∼ > 10 6 GeV, and that in this case as well the Dirac or Majorana CPV phases present in the PMNS matrix can be the unique source of the required CP violation. Moreover, in [27] the same conclusion concerning the Dirac or Majorana phases being the source of CP violation in leptogenesis, was shown to hold in the case of the so-called "Neutrino Option" seesaw scenario [28] in which the term quadratic in the Higgs field in the Higgs potential, responsible for the breaking of the SM electroweak symmetry, is generated radiatively at one loop by the neutrino Yukawa coupling in Eq. (6).
The studies performed in [25,26] were based on the density matrix equations (DME) for leptogenesis [29][30][31], in which, in particular, the flavour decoherence effects associated with the charged lepton Yukawa couplings are accounted for in the various regimes they go through continuously (from negligible to non-negligible but non-thermalised to fully thermalised) in the expanding and cooling Universe. In contrast, the Boltzmann equations (BE) describe flavour effects in leptogenesis only for either negligible or fully thermalised charged lepton Yukawas. As a consequence, the DME approach to leptogenesis opened up the possibility to investigate quantitatively the behaviour of the baryon asymmetry η B in transitions between the different flavour regimes in leptogenesis. The analysis done in [26] revealed that in the case of heavy Majorana neutrinos with hierarchical mass spectrum, M 1 M 2 M 3 , the baryon asymmetry η B can go through zero, changing sign at certain scale M 1 in the transition between the unflavoured (or single flavoured) and the two flavoured leptogenesis regimes, associated with the τ -Yukawa coupling. Moreover, when the CP violation is provided in leptogenesis by the low-energy CPV phases present in the PMNS matrix, the width of the transition was shown to become extremely large and to lead to the existence of a "plateau" in the baryon asymmetry dependence on the scale M 1 .
In the present article we continue to investigate the transitions between the different flavour regimes in high scale leptogenesis based on type I seesaw mechanism with hierarchical heavy Majorana neutrinos, M 1 M 2 M 3 , began in [26]. We consider the case in which the CP violation (CPV) in leptogenesis is provided by the low-energy Dirac or/and Majorana phases of the PMNS neutrino mixing matrix. Using the density matrix equations (DME) to describe high scale leptogenesis, we investigate in detail the 1-to-2 and the 2-to-3 flavour regime transitions, where the 1, 2 and 3 leptogenesis flavour regimes in the generation of the baryon asymmetry η B are described by the Boltzmann equations. Concentrating on the 1-to-2 flavour transition we determine the general conditions under which η B goes through zero and changes sign in the transition and |η B | reaches a plateau as M 1 increases. We analyse further in detail the behaviour of η B in the transition under these conditions in the case of two heavy Majorana neutrinos N 1,2 with hierarchical masses, M 1 M 2 , and identify, in particular, cases in which the baryon asymmetry exhibits a "non-standard" behaviour in the transition. We determine the minimal scale M 1min as well as the corresponding ranges of M 1 and of the Dirac and Majorana CPV phases for which we can have successful leptogenesis when the requisite CP violation in leptogenesis is provided only either by the Dirac or by the Majorana phases of the PMNS matrix.
The paper is organised as follows. In Section 2 we summarise the existing data on the neutrino masses and mixing that we use in our analysis, the basics of the type I seesaw scenario, and introduce the 1, 2 and 3 flavour Boltzmann equations as well as the density matrix equations that are employed in our study, elucidating the role of the charged lepton Yukawa couplings. In Section 3 we investigate the baryon asymmetry (η B ) sign change in the transi-tion between the one and two flavour regimes in the case of three heavy Majorana neutrinos N 1,2,3 with hierarchical masses, M 1 M 2 M 3 , M 1 10 9 GeV, determine the general conditions under which the sign change takes place and then perform a detailed analysis of the transitions between the different flavour regimes. To make the discussion as transparent as possible we investigate in Section 4 the behaviour of the baryon asymmetry under the general conditions under which the sign change of the baryon asymmetry η B in the 1-to-2 flavour regimes is possible in the case of decoupled N 3 , in which the number of parameters is significantly smaller than in the general case with three heavy Majorana neutrinos. The thorough analysis is performed in this Section with the CP violation necessary for the generation of the baryon asymmetry provided by the low-energy Dirac or/and Majorana phases present in the PMNS neutrino mixing matrix. We conclude in Section 5 with a summary of our results.

Neutrino Masses and Neutrino (Lepton) Mixing
Throughout the present study we employ the reference 3-neutrino mixing scheme (see, e.g., [32]): where ν αL (x), α = e, µ, τ , is the left-handed (LH) flavour neutrino field (which enters into the expression of the weak interaction Lagrangian), ν jL (x), j = 1, 2, 3, is the LH component of the field of a light neutrino ν j with mass m j , and U is the 3 × 3 unitary Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino (lepton) mixing matrix. We consider the case of light massive neutrinos ν j being Majorana particles and will use in our analysis the standard parametrisation of the PMNS matrix [32] in this case: Here c ij ≡ cos θ ij , s ij ≡ sin θ ij , the angles θ ij = [0, π/2], δ = [0, 2π) is the Dirac CP violation (CPV) phase, and α 21 and α 31 are the two Majorana CPV phases [33], α 21(31) = [0, 4π] 1 . The Dirac and Majorana phases can be sources of low-energy leptonic CP violation. In the case of CP invariance, we have δ = 0, π and α 21(31) = k 2(3)1 π, k 2(3)1 = 0, 1, 2, 3, 4. In what concerns the light neutrinos masses m 1,2,3 , we use the "standard" convention of numbering the neutrino mass eigenstates in which ∆m 2 21 ≡ m 2 2 − m 2 1 > 0 and ∆m 2 31(32) ≡ m 2 3 −m 2 1 (2) are associated, together respectively with the angles θ 12 and θ 23 , with the observed flavour conversion of solar (electron) neutrinos ν e and the dominant oscillations of atmospheric muon neutrinos and antineutrinos, ν µ andν µ , while the angle θ 13 , together with ∆m 2 31(32) , is associated with the reactorν e oscillations observed in the Daya Bay, RENO and Double Chooz experiments [32]. The enormous amount of neutrino oscillation data accumulated over 1 Within the type I seesaw mechanism of neutrino mass generation we will consider the mass-eigenstate neutrinos to be Majorana fermions and it proves convenient to work with this extended range of possible values of the two Majorana phases α21,31 [22] (see further).  Table 1: Best-fit values and 1σ allowed ranges of the neutrino mixing angles θ 12 , θ 13 , θ 23 , and of the ∆m 2 ≡ ∆m 2 21 and ∆m 2 atm ≡ ∆m 2 31 (∆m 2 atm ≡ ∆m 2 32 ) in the case of NO (IO) light neutrino mass spectrum, obtained in [36]. We quote also the best-fit value and 1σ allowed ranges of the Dirac CPV phase δ from [36]. However, these data on δ are not used in our analyses. many years of research (see, e.g., [34]) made it possible to determine ∆m 2 21 , sin 2 θ 12 , |∆m 2 31 | (|∆m 2 32 |), sin 2 θ 23 and sin 2 θ 13 with remarkably high precision (see, e.g., [35,36]). We report in Table 1 the best-fit values and 1σ ranges of the three neutrino mixing (or PMNS) angles and the two neutrino mass squared differences obtained from the global neutrino oscillation data analysis in [36]. In the numerical analyses we will perform, we will use the best-fit values of the three neutrino mixing angles and of the two neutrino mass squared differences quoted in Table 1.
The existing neutrino data, as is well known, do not allow to determine the sign of ∆m 2 31 (32) and the two values of sgn(∆m 2 31(32) ) correspond to two possible types of light neutrino mass spectrum -with normal ordering (NO) and inverted ordering (IO), which is reflected in Table  1. In the adopted convention, the two spectra read: • Normal Ordering (NO): m 1 < m 2 < m 3 , ∆m 2 31 ≡ ∆m 2 atm > 0; • Inverted Ordering (IO): m 3 < m 1 < m 2 , ∆m 2 32 ≡ ∆m 2 atm < 0.
Depending on the value of the lightest neutrinos mass, the neutrino mass spectrum can also be: • Normal Hierarchical (NH): 0 m 1 m 2 < m 3 , with m 2 ∆m 2 21 and m 3 ∆m 2 31 ; • Inverted Hierarchical (IH): 0 m 3 m 1 < m 2 , with m 1 |∆m 2 32 | − ∆m 2 21 and m 2 |∆m 2 32 |; All considered spectra are compatible with the existing data on the light neutrino masses [32]. The IO spectrum is disfavoured at approximately 2.7σ C.L. with respect to the NO spectrum by the global neutrino oscillation data [36]. In our further analyses we will mostly be interested in the NH and IH spectra. A few comments relevant for our further discussion are in order. As it follows from  [37,38] that the Dirac phase δ ∼ 3π/2, no other experimental information on the Dirac and Majorana CPV phases in the PMNS matrix is available at present. The values of δ obtained in the global analyses [35,36] have relatively large uncertainties. In view of this we will treat both the Dirac phase δ and the Majorana phases α 21 and α 31 as free parameters in our study. We recall that with θ 13 ∼ = 0.15, the Dirac phase δ can generate CP violating effects in neutrino oscillations [33,39,40], i.e., a difference between the probabilities of the ν α → ν β andν α →ν β oscillations, α = β = e, µ, τ . The magnitude of CP violation in ν α → ν β andν α →ν β oscillations (α = β) is determined by [41] the rephasing invariant which 2 in the standard parametrisation of the PMNS matrix has the form: cos θ 13 sin 2θ 12 sin 2θ 23 sin 2θ 13 sin δ .
If the hints that δ has a value close to 3π/2 are confirmed by future more precise data one would have J CP ∼ = − 0.03, implying that the CP violating effects in neutrino oscillations would be relatively large and observable in currently running and/or future neutrino oscillation experiments (T2K, NOνA, T2HK, DUNE, see, e.g., [32,34]).
Our interest in the Dirac and Majorana CPV phases present in the neutrino mixing matrix is stimulated also by the intriguing possibility that the Dirac phase and/or the Majorana phases in the PMNS matrix U can provide the CP violation necessary for the generation of the observed baryon asymmetry of the Universe [17,18]. In the present article we continue to explore this intriguing and very appealing possibility.
Finally we comment briefly on the current limits on the absolute scale of light neutrino masses (or equivalently on the lightest neutrino mass). Using the existing best lower bounds on the (ββ) 0ν -decay half-lives of 136 Xe [47] and 76 Ge [48] one can obtain the following "conservative" upper limit on the light Majorana neutrino masses, which is in the range of the QD spectrum [49]: m 1,2,3 0.58 eV.
The most stringent upper limit on the light neutrino masses, which does not depend on the nature of massive neutrinos, was obtained in KATRIN experiment by measuring the spectrum of electrons near the end point in 3 H β-decay [50,51]: m 1,2,3 < 0.8 eV (90% C.L.).
The Cosmic Microwave Background (CMB) data of the WMAP and PLANCK experiments, combined with supernovae and other cosmological and astrophysical data can be used to obtain information in the form of an upper limit on the sum of neutrino masses. Depending on the model complexity and the input data used one typically finds [52] (see also [35]): j m j < (0.11 − 0.54) eV (95% CL).

The Seesaw Mechanism
The leptogenesis we are going to discuss in the present article is based on the type I seesaw mechanism of neutrino mass generation [4][5][6][7][8]. This rather simple mechanism is realised, as is well known, by extending the Standard Model (SM) with n ≥ 2 right-handed (RH) neutrinos ν lR ( RH neutrino fields ν lR (x)), that are singlets under SU (2) L × U (1) Y W , possess a Majorana mass term and couple through a Yuakawa-type interaction to the SM lepton and Higgs doublets, (ψ αL (x)) T = (ν T αL (x) T αL (x)), with α = e, µ, τ , and (Φ(x)) T = (Φ (+) (x) Φ (0) (x)). The minimal type I seesaw scheme in which leptogenesis can be realised is with n = 2 RH neutrinos. In this scenario the lightest neutrino -ν 1 (ν 3 ) for NO (IO) neutrino mass spectrum -is massless at tree and one-loop level. We will consider leptogenesis with both n = 3 and n = 2 RH neutrinos.
Without loss of generality, we work in the basis in which i) the Majorana mass matrix of RH neutrinos, M , is diagonal and positive, M = diag(M 1 , M 2 , M 3 ) with M i > 0, and ii) the charged lepton Yukawa couplings are flavour diagonal. In the chosen basis, the neutrino Yukawa and the RH neutrino Majorana mass terms are given by: where Y αi is the matrix of the neutrino Yukawa coupling and ) T and C being the charge-conjugation matrix. The fields N 1,2,3 (x) correspond to Majorana neutrinos N 1,2,3 with masses M 1,2,3 > 0 which in high scale leptogenesis can have values M 1,2,3 ∼ (10 6 − 10 14 ) GeV, and so we will refer to N 1,2,3 further on as "heavy Majorana neutrinos" or just "heavy neutrinos". After the spontaneous breaking of the electroweak symmetry, the neutral component of the Higgs doublet acquires a non-vanishing vacuum expectation value (VEV) v = 246 GeV, generating a neutrino Dirac mass term which, together with the N 1,2,3 mass term, can be cast in the form: where α, β = e, µ, τ (i, j = 1, 2, 3) and ν c βR (x) ≡ C(ν βL (x)) T . This mass term can be diagonalised by means of the Takagi transformation, which, at leading order in the seesaw expansion parameter |vY αi |/M i 1, leads to the well known expression for the tree-level light neutrino mass matrix m tree ν : In the version of the high scale leptogenesis with heavy Majorana neutrino masses which are not hierarchical and have relatively low values, M 1 ∼ 10 6 GeV, M 2 ∼ = 3M 1 , M 3 ∼ = 3M 2 , the 1-loop radiative correction to the light neutrino mass matrix can be non-negligible [25,26] 3 . We will be interested in high scale leptogenesis with hierarchical masses of the heavy Majorana neutrinos, M 1 M 2 M 3 , in which the CP violation is provided exclusively by the Dirac and/or Majorana CPV phases present in the PMNS matrix. In this case successful leptogenesis is possible for M 1 10 10 GeV [18,26]. Under these conditions the 1-loop contribution to the light neutrino mass matrix m ν , as our numerical study has shown, is sub-leading and amounts to ∼(10% -20%) effects. Nevertheless, we have included it in our calculations. This contribution is given by [54][55][56] (see also, e.g., [57]): where R is a complex orthogonal matrix. In the present study we adopt the following parametrisation of the R-matrix: where c j ≡ cos(x j + iy j ) and s j ≡ sin(x j + iy j ), x j and y j being free real parameters (j = 1, 2, 3).

The Baryon Asymmetry of the Universe and Flavoured Leptogenesis
The baryon asymmetry of the Universe (BAU) can be parametrised by the baryon-to-photon ratio where n B , nB and n γ are the number densities of baryons, anti-baryons and photons, respectively. Alternatively, it can be expressed in terms of the baryonic density parameter where m p is the proton mass, ρ c is the critical density of the Universe and h is the Hubble expansion rate of the Universe (H) per unit of 100 (km/s)/Mpc (the numerical values of the constants are taken from [59]). The present BAU has been determined independently from the estimates of the Big Bang Nucleosynthesis (BBN) and, with higher precision, from the measurements of the Cosmic Microwave Background (CMB) anisotropies made by the Planck observatory. The following results have been reported (at 68% C.L.) in [60,61] 4 : From both we obtain the best-fit value of that is going to be our reference value in the further analyses. The generation of a matter-antimatter asymmetry in the expanding Universe can naturally be accomplished within the type I seesaw framework through thermal leptogenesis. Provided the Yukawa couplings in Eq. (6) are CP violating, the out-of-equilibrium decays of the heavy Majorana neutrinos N i to leptons and Higgs doublets in the early Universe generate CPV asymmetries in the individual lepton flavour charges L α , as well as in the total lepton charge L. The so generated lepton asymmetry is then translated into an asymmetry in the baryon charge B by the SM (B + L)-violating, but (B − L)-conserving, sphaleron processes, which are effective at temperatures T ∼ = 132 − 10 12 GeV.
In this work we will concentrate on the case in which the heavy neutrinos N 1,2,3 have hierarchical masses, namely M 1 M 2 M 3 . In this case generically only the CPV decays of N 1 contribute to the generation of the CPV lepton asymmetry. We shall report in this section the relevant equations for one decaying heavy neutrino that will be used in our analysis.
The charged lepton final states in the decays of the heavy neutrino N i , N i → Φ + ψ i and N i → Φ − ψ i , are a superposition of the charged lepton flavour states, namely, with the coefficients C iα and C iα given by We are interested in the decays of N 1 , N 1 → Φ + ψ 1 and N 1 → Φ − ψ 1 , so index i should be replaced with 1 in Eqs. (20) - (22). If it were not for the SM charged lepton Yukawa interactions, the quantum states |ψ 1 and |ψ 1 would be coherent superpositions of the charged lepton flavour states. However, when these interactions are in thermal equilibrium, i.e., their rates are larger than the expansion rate of the Universe, given the difference between the charged lepton Yukawa couplings, h e , h µ , h τ , the flavour states become distinguishable and each flavour state experiences a different time-evolution -actually, it is enough for the SM τ -and µ-Yukawa interactions to be in equilibrium for the three lepton flavours to be distinguishable. If the SM charged lepton Yukawa interactions are faster than the process of the heavy neutrino decay into (anti)leptons, then the coherence in |ψ 1 (|ψ 1 ) is efficiently destroyed [30] (see, e.g., also [62]) -in this sense these are decoherence effects. The relevant processes are the interchanges between the LH leptons with their respective RH components and vice verse through scattering processes involving the Higgs doublet. By means of the optical theorem, the rates of these processes involving the tauon and the muon, Γ τ and Γ µ , are given by the imaginary part of the τ , µ thermal self-energy and read [25,31] (see also, e.g., [63] and references therein): Γ τ, µ ∼ = 8 × 10 −3 h 2 τ, µ T . The comparison of Γ τ and Γ µ with the Hubble expansion rate H gives 5 : where M P ∼ = 1.22 × 10 19 GeV is the Planck mass. At T 10 12 GeV, the rates of the τ -and µ-Yukawa interactions are much smaller than the expansion rate of the Universe as Γ τ, µ 1. As a consequence, the flavour states are indistinguishable and the (anti)leptons produced via the N 1 's decay are always found in the coherent superposition defined in Eq. (20) ( (21)). This is the unflavoured or single-flavour regime. For M 10 12 GeV, leptogenesis proceeds in the unflavoured regime for its entire duration and is usually studied within the single-flavour approximation, under which the µ-and τ -decoherence effects are neglected. Correspondingly, this scenario is typically dubbed unflavoured or single-flavoured leptogenesis. In the singleflavour approximation, the time-evolution of the number densities of N 1 and B − L charge can be described by the set of semi-classical single-flavoured Boltzmann equations (1BE1F): where z ≡ M 1 /T . The quantities N N 1 and N B−L are respectively the number of heavy neutrinos N 1 and B − L asymmetry in a comoving volume. In the present work the comoving volume is normalised as in [25][26][27] so that it contains one photon at z = 0, i.e., N eq N 1 (0) = 3/4. This normalisation within the Boltzmann statistics is equivalent to using N eq N 1 (z) = 3 8 z 2 K 2 (z), where K n (z), n = 1 , 2 , ..., is the modified n th Bessel function of the second kind.
The decay parameter D 1 is given by: where κ 1 is defined as the ratio between the total decay rate of N 1 at zero temperature, Γ , and the Hubble expansion rate H at z = 1. It proves convenient to write κ 1 in the following form: 5 The τ -and µ-Yukawa couplings are given by hτ = √ 2mτ /v ∼ = 1.02×10 −2 and hµ = √ 2mµ/v ∼ = 6.08×10 −4 , where mτ and mµ are the τ ± and µ ± masses, respectively, and v = 246 GeV. Given the smallness of the e-Yukawa coupling he = √ 2me/v ∼ = 2.94 × 10 −6 , me being the e ∓ mass, the e-Yukawa interactions come into thermal equilibrium only at T 10 5 GeV, being therefore ineffective at the temperatures of interest in the present work.
g * = 106.75 being the number of relativistic degrees of freedom at z = 1. The wash-out parameter W 1 reads: where N eq is the equilibrium number density of leptons at z = 0, which, within the adopted normalisation, is given by N eq = N eq N 1 (0) = 3/4 6 . Finally, the CPV-asymmetry parameter (1) is given by [65][66][67] We note that for large x, ξ(x) = 1 + O(1/x), so that, in the hierarchical limit Since the Yukawas enter in (1) only through the product Y † Y , there is no dependence on the PMNS matrix. There is therefore no contribution to (1) from the CPV Dirac and Majorana phases in the PMNS matrix.
As the mass scale of leptogenesis is lowered to M 1 ∼ 10 12 GeV, the single-flavour approximation becomes inaccurate since the SM τ -Yukawa interactions enter in equilibrium during the generation of the lepton asymmetry, i.e. Γ τ /Hz ∼ 1. This is a transition regime, which we will refer to as 1-to-2 flavour transition, where the τ -decoherence effects cannot be neglected. Moreover, as was noticed in [26], when the requisite CP violation in leptogenesis is provided exclusively by the Dirac and/or Majorana CPV phases of the PMNS matrix, the 1-to-2 flavour transition proceeds with an unusual behaviour of the baryon asymmetry η B , which extends into the region of the unflavoured regime at M > 10 12 GeV. This unusual behaviour will be investigated in detail in our work. Here it suffices to mention that due to CP violating quantum decoherence effects caused by the SM τ -Yukawa interactions, in which CP violation is provided by the low-energy leptonic CPV phases, the generation of BAU in the single-flavour approximation as described by Eqs. (25) and (26) fails and that the observed BAU can still be generated at M 1 > 10 12 GeV even if (1) = 0 and the τ -Yukawa interactions are not in full thermal equilibrium.
For 10 9 T /GeV 10 12 , the τ -Yukawa interactions are in thermal equilibrium while that of µ are not, namely Γ τ /H 1 while Γ µ /H 1 . Correspondingly, the τ -(anti)lepton state becomes distinguishable from the other flavour states and the coherence in |ψ 1 (|ψ 1 ) gets eventually destroyed. As a consequence, the CPV asymmetry in L τ evolves differently with respect to the asymmetry in the sum of L e and L µ charges, L τ ⊥ ≡ L e+µ ≡ L e + L µ . This corresponds to the two-flavour regime of leptogenesis or two-flavoured leptogenesis.
For 10 9 M 1 /GeV 10 12 , the τ -Yukawa (µ-Yukawa) interactions enter in thermal equilibrium at z 1 (z 1) and leptogenesis can be studied within the two-flavour approximation under which only the µ-decoherence effects are neglected. If in addition the τ -Yukawa interactions are assumed to be infinitely (="sufficiently") fast during the whole period of leptogenesis, the two-flavoured Boltzmann equations (1BE2F) can be used to describe the time-evolution of the CPV asymmetries in the L τ and L τ ⊥ charges and of BAU. The set of 1BE2F equations in the two-flavour approximation reads: where p 1τ = |C 1τ | 2 and p 1τ ⊥ = |C 1e | 2 +|C 1µ | 2 = 1−p 1τ , while N τ τ and N τ ⊥ τ ⊥ are respectively the values of the asymmetries in the charges 1 3 B − L τ and 2 3 B − L τ ⊥ in a comoving volume, so that N B−L = N τ τ + N τ ⊥ τ ⊥ . The expressions for the relevant CPV lepton asymmetries µµ will be given below. As the mass scale is lowered to M 1 ∼ 10 9 GeV, leptogenesis approaches the 2-to-3 flavour transition, where the µ-decoherence effects cannot be neglected since the µ-Yukawa interactions enter in equilibrium, Γ µ /Hz ∼ 1. Therefore, the two-flavour approximation ceases to be accurate. Actually, as we are going to show in the present study, there are choices of the parameters for which the 1BE2F equations are never accurate and cannot be used for the description of leptogenesis in the whole range 10 9 M 1 /GeV 10 12 . In addition, in certain regions of the parameter space, the scale below which the 1BE2F set of equations starts to be valid can be significantly lower than ∼ 10 12 GeV.
At T 10 9 GeV, also the µ-Yukawa interactions are in thermal equilibrium, i.e. Γ µ /H 1. This is the three-flavour regime: all the flavours are distinguishable, the coherent superposition in |ψ 1 (|ψ 1 ) is fully destroyed and the CPV lepton asymmetries in each of the charges L α (α = e, µ, τ ) evolve separately. At M 1 10 9 GeV, both the µ-and τ -Yukawa interactions enter in equilibrium at z 1 corresponding to the three-flavoured leptogenesis scenario. If the µ-and τ -Yukawa interactions are assumed to be infinitely (≡"sufficiently") fast, then leptogenesis can be described by the three-flavoured Boltzmann equations (1BE3F), namely: where p 1α = |C 1α | 2 , while N αα is the value of the asymmetry in the charge 1 3 B − L α in a comoving volume, so that N B−L = α N αα , with α = e, µ, τ .
To obtain a better description of the physics of leptogenesis, the decoherence effects should always be included in the calculations. As already shown in, e.g., [29][30][31], the density matrix equations (DMEs) provide an accurate tool to study thermal leptogenesis accounting for quantum decoherence processes, especially when these are neither infinitely fast nor their effects negligible. The DMEs describe the time-evolution of the entries of the density matrix, which, in the three-flavour basis, is given by with α, β = e, µ, τ . The diagonal entries N αα are the already defined number densities for the 1 3 B − L α asymmetry, so that N B−L = Tr(N ) = α N αα . The off-diagonal elements N αβ describe the degree of coherence between the flavour states. The DMEs in the three-flavour basis explicitly read [29][30][31]: where I τ and I µ are 3 × 3 matrices such that (I τ ) αβ = δ ατ δ βτ and (I µ ) αβ = δ αµ δ βµ , and are projection matrices which generalise the notion of the projection probability. They appear in the anti-commutator structure, which explicitly reads: 9 The need for a double flavour index will later be clarified as in the quantum treatment also the off-diagonal terms are relevant.
In the numerical analyses that follow, we will use the ULYSSES Python package [69] to solve the sets of equations that we have introduced in the present section. The code computes, in particular, N B−L = N ee +N µµ +N τ τ , which is then converted into the baryon asymmetry of the Universe η B expressed in terms of the baryon-to-photon ratio using the following relation: where 28/79 is the SM sphaleron conversion coefficient and the 1/27 factor comes from the dilution of the baryon asymmetry due to the change of the photon density between leptogenesis and recombination [64].

The Baryon Asymmetry Sign Change
We first consider two-flavoured leptogenesis in the case of three heavy Majorana neutrinos N 1,2,3 with hierarchical masses, M 1 M 2 M 3 , M 1 10 9 GeV. In this case generically only the CPV decays of the lighter Majorana neutrino N 1 contribute to the generation of CPV lepton asymmetry which is converted into a baryon asymmetry by the sphaleron effects. The density matrix equations (DMEs) describing the evolution of the number of N 1 in a comoving volume, N N 1 , and of the CPV asymmetries in the lepton charges L τ and L τ ⊥ = L e+µ = L e +L µ in the two-flavoured leptogenesis have the following form in the case of interest: The B − L asymmetry is given by N B−L = N τ τ + N τ ⊥ τ ⊥ . We find that (see Appendix A): where z 0 corresponding to the beginning of leptogenesis, which we set to z 0 = 10 −3 in all our numerical calculations, and The term N 1BE1F B−L is the solution to the single-flavoured Boltzmann equations (1BE1F) and vanishes if the CP violation in leptogenesis is due only to the physical Dirac and/or Majorana CPV phases in the PMNS matrix since in that case [18] As was shown in [26], N decoh B−L can be the only source of CPV lepton asymmetry if the CP violation in leptogenesis is provided exclusively by the physical CPV phases in the PMNS matrix. In the discussion that follows we focus on this case.
The factor Λ τ ≡ Γ τ /Hz = const./M 1 and can be taken out of the integration in (56). In the high scale regime (M 10 12 GeV) we can work in the limit of Λ τ → 0 and neglect all the terms of order O(Λ 2 τ ) in the lepton asymmetry. Since in our case N decoh B−L is the only source of lepton asymmetry, at M 10 12 GeV we have . Solving Eq. (52) at zero order in Λ τ 10 with the integrating factor method we find: Inserting this result in Eq. (56) we get: where and we have used the relation 2 for a derivation of this relation). To leading order in Λ τ the final asymmetry reads: τ τ and Since (  mass scale M 1 . Thus, when the CP violation is provided by the CPV phases of the PMNS matrix, at M 1 > 10 12 GeV there should exist an interval of values of M 1 in which the baryon asymmetry η B is constant, i.e., does not change with M 1 . Indeed, the numerical solutions of the DMEs show the existence of a plateau at values of M 1 > 10 12 GeV [26], as is illustrated in Fig. 1, right panels. We note that if the CP violation in leptogenesis is due to the Casas-Ibarra matrix and thus (1) = 0, N 1BE1F B−L ∝ M 1 eventually starts to dominate over N decoh B−L as M 1 increases, recovering the single-flavour approximation as is clearly seen in Fig. 1, left panels.
As We discuss in the next subsections the circumstances under which the sign change of η B can take place in the cases of strong and weak wash-out regimes, for which κ 1 1 and κ 1 1, respectively. We concentrate on the physically interesting possibility of the requisite CP violation provided only by the Dirac and/or Majorana CPV phases of the PMNS matrix, which leads also to the existence of a "plateau" at M 1 > 10 12 GeV where to a good approximation η B does not depend on M 1 .

Strong Wash-Out Regime
In the strong wash-out regime the solution to the 1BE2F does not depend on the initial conditions since any initially generated asymmetry is erased by the strong wash-out processes. A sufficiently accurate analytic expression of the solution to the 1BE2F, valid in the strong wash-out regime, is given by (see Appendix B.1): Since p 1τ (1 − p 1τ ) > 0, a difference in sign between the solution N 1BE2F B−L (z f ) given above and the solution of Eq. (60) occurs when I 2 (κ 1 ; z f ) is negative. We show in Fig. 2 the behaviour of I 2 (κ 1 ; z f ) for z f = 1000 computed numerically with the ULYSSES Python package [69]   from Eq. (61) 12 in the cases of vanishing initial abundance (VIA) of N 1 , N N 1 (z 0 ) = 0, and thermal initial abundance (TIA) of N 1 , N N 1 (z 0 ) = N eq N 1 (z 0 ). As follows from the behaviour of I 2 (κ 1 ; z f ) shown in Fig. 2, a sign change of η B at M 1 ∼ 10 12 GeV in the strong wash-out regime always happens for VIA, but never for TIA.

Weak Wash-Out Regime
In the weak wash-out regime we need to consider separately the cases of the two different initial conditions -VIA (N N 1 (z 0 ) = 0) and TIA (N N 1 (z 0 ) = N eq N 1 (z 0 )).

Vanishing Initial Abundance
In the VIA case the asymmetry of interest in the two-flavoured leptogenesis reads (see Appendix B.2.1): where to get the last equation we have used the fact that (1) Using this condition also in Eq. (60) we find: It is then clear from the comparison of the last two equations that I 2 (κ 1 ; z f ) needs to be positive in order for a sign change of η B to occur at M 1 ∼ 10 12 GeV. Since, as shown in Fig.  2, I 2 (κ 1 ; z f ) is always negative if N N 1 (z 0 ) = 0, the transition at M ∼ 10 12 GeV in the weak wash-out regime in the VIA case always takes place without a sign change.

Thermal Initial Abundance
In the case of TIA, for which N N 1 (z 0 ) = N eq N 1 (z 0 ), and CP violation due only to the CPV phases in the PMNS matrix, the asymmetry of interest in the two-flavoured leptogenesis is described by the following analytic expression (see Appendix B.2.2): where The comparison of (65) with Eq. (64) tells us that no sign change of η B should occur in the 1-to-2 flavour transition also in this case, given the fact that I 2 (κ 1 ; z f ) is always positive for TIA.

Transitions Between Different Flavour Regimes: Detailed Analysis
The discussion and the results obtained in the preceding subsections led to the important conclusion that we should expect a sign change of the baryon asymmetry at the transition between the single-and two-flavoured leptogenesis in the case of VIA and strong wash-out regime of baryon asymmetry generation. However, certain important points could not be addressed within the approach used in the discussion leading to this conclusion. For example, the intermediate cases in which the asymmetries in different flavours N τ τ and N τ ⊥ τ ⊥ are generated in different wash-out regimes -strong and weak -could not and have not been considered. The analysis performed by us also does not allow to determine the mass scale M 10 at which η B = 0 and the transition between the two different flavour regimes considered takes place. Clearly, having different wash-out regimes for the different flavour asymmetries and having a value of M 10 which differs significantly from ∼ 10 12 GeV, might be possible, in principle, for choices of the parameters, namely the R-matrix angles x 1 +iy 1 , x 2 +iy 2 , x 3 +iy 3 , the PMNS phases δ, α 21 , α 31 and the mass of the lightest neutrino m 1 , which differ from the choices considered by us. To address, in particular, the aforementioned points, we use an alternative approach to the problem of interest. We start from the following equation for N B−L in the case of (1) = 0 (a detailed derivation of this equation is provided in Appendix A), which is valid as long as M 1 10 9 GeV where Λ µ can be safely neglected: The functions W 1 (z) and λ(z) are given in Eqs. (30) and (56), respectively. Within this apparently simple equation for N B−L , we have encoded all the decoherence effects on the system in the term W 1 λ, which in particular contains both a source and a wash-out term, as will be clarified later on. The term W 1 N B−L is the usual wash-out term which tends to cancel any initially generated asymmetry.
Taking into account the expression for N τ τ ⊥ given in Eq. (116) of Appendix A, the function λ(z) can be cast in the form: The first term in this expression for λ(z) is the only source of B − L asymmetry, while the second is an integrated wash-out term. In the limit of Λ τ → 0, i.e., for M 10 12 GeV, the first term scales as Λ τ , while the second term scales as Λ 2 τ and can be neglected. We note also that the integrated wash-out term can be suppressed by a small value of p 1τ p 1τ ⊥ as well. Given that the source term is proportional to Λ τ , also the B − L asymmetry will be proportional to it at any z: whereÑ B−L can only depend, apart from z, on κ 1 , Λ τ and p 1τ (through the product p 1τ p 1τ ⊥ ). This means that the function λ(z) can be written as: with It follows from Eqs. (67) -(71) that in the VIA case of interest we have: The last equation, combined with Eq. (69), cannot be used to compute the final asymmetry because inside S a dependence onÑ B−L is "hidden". However, it is clear from the last expression and Eqs. (69) and (70) that, given the signs of (1) τ τ and of (p 1τ ⊥ −p 1τ ) = (1−2 p 1τ ), the sign of the asymmetry N B−L depends on the sign evolution of S. We therefore analyze the behaviour of the function S to better understand the sign change at the 1-to-2 flavour transition. We construct the function S by first solving numerically the full set of DMEs with the ULYSSES Python package [69] and then we compute explicitly S using the definition of λ(z) in Eq. (56) together with Eqs. (70) and (115) 13 .
We consider the case of heavy Majorana neutrinos having a vanishing initial abundance (VIA), i.e., N N 1 (z 0 ) = 0. At the beginning of leptogenesis at z > z 0 , but z relatively close to z 0 , both the term involving N N 1 (z ) and the integrated wash-out term in Eq. (71) are much smaller than the term involving N eq N 1 (z ) 14 , so that S starts its evolution with a negative sign. As z increases, S receives contributions from both terms in Eq. (71). At values of z > z eq , where z eq corresponds to the time of evolution at which N N 1 = N eq N 1 , we have, as our numerical analysis shows, N N 1 (z) − N eq N 1 (z) > 0. As z increases, the source term in Eq. (71) goes through zero and becomes positive. Let us callz Λ the value of z at which S = 0, so that at z <z Λ (z >z Λ ) we have S < 0 (S > 0).
It should be clear from Eq. (72) that for z <z Λ ,Ñ B−L < 0 and therefore also the second (integrated wash-out) term in Eq. (71) is positive. However, it is significantly smaller than the absolute value of the negative source term involving N eq N 1 . At z =z Λ , this negative term is compensated by the sum of the source term involving N eq N 1 (z ) and the integrated wash-out term. At z >z Λ , S is positive and remains so as z increases.
In the TIA case, as our numerical analysis shows,z Λ does not exist since, in particular, for z > z 0 and correspondingly the function S has a positive sign for the entire period of leptogenesis. This explains why no sign change can be present in the TIA case, as proven in the previous section. In what follows we focus our discussion on the VIA case only.
If in the VIA case the B − L asymmetry is frozen at z f <z Λ , then, as we have discussed, To highlight this behaviour we focus for a moment on the strong wash-out regime. Suppose that κ 1 1, so that there exist two moments z in and z out for which W 1 (z in < z < z out ) 1. Then, for z in < z < z out to a good approximation we have dN B−L /dz ∼ = 0 15 (see, e.g., [70]), and following the same steps as in Appendix B, from Eq. (67) we get: After z out the asymmetry gets frozen so that: Hence, as follows from Eq. (70), the sign of the final asymmetry reads: At z out =z Λ we have S = 0 and therefore N B−L (∞) = 0. Analytic expression for both z in and z out are given in [64]: In the weak wash-out regime the analysis is more complicated as the asymmetry may freeze at z f = z out and we do not have any analytic expression for this case. However, on the basis of the numerical analysis we did, we expect leptogenesis to end at z f smaller than a few tens.
We note thatz Λ depends only on p 1τ , κ 1 and Λ τ , i.e.z Λ =z Λ (p 1τ , κ 1 , Λ τ ). If we neglect the weak dependence on the mass scale M 1 of κ 1 , which comes from the loop contribution to the light neutrino masses [25], the only dependence ofz Λ on M 1 is inside Λ τ ∝ 1/M 1 . Therefore we have 16z Λ z Λ (p 1τ , κ 1 , M 1 ). In addition, in the limit of Λ τ → 0, i.e., at M 1 10 12 GeV, the integrated wash-out term -the second term in Eq. (71) -can be neglected so that the dependence of S, and therefore ofz Λ , on p 1τ drops off, i.e.,z Λ z Λ (κ 1 , M 1 ). In terms of the Casas-Ibarra parametrisation this means thatz Λ does not depend on the PMNS phases. As the mass scale M 1 decreases, the integrated wash-out term becomes non-negligible activating a dependence on the PMNS phases through the product p 1τ p 1τ ⊥ .
In the general case of three heavy Majorana neutrinos having non-generate but also nonhierarchical masses (e.g., M 3 = 3M 2 , M 2 = 3M 1 ), the discussion is rather complicated due to the large number of parameters present in the Casas-Ibarra parametrisation. To make the discussion as transparent as possible we consider the case of decoupled N 3 , in which the number of parameters is significantly smaller than in the general case.

The Case of Decoupled N 3
In the case of decoupled heavy Majorana neutrino N 3 (M 1 M 2 M 3 ), the lightest neutrino, as is well known, is massless at tree and one loop level, i.e., m 1 ∼ = 0 (m 3 ∼ = 0), and the light neutrino mass spectrum is normal (inverted) hierarchical, denoted as NH (IH). The set of parameters relevant for our discussion includes: the masses of the two heavy Majorana neutrinos M 1 and M 2 ; the three CPV phases δ, α 21 , α 31 of the PMNS matrix; the real and imaginary parts x and y of the complex angle of the Casas-Ibarra orthogonal R-matrix. The R-matrix for the NH and IH light neutrino mass spectra of interest has the form: with θ = x + iy. Both R-matrices in Eqs. (77) and (78) have det(R) = 1. In the literature, the factor ϕ = ±1 is sometimes included in the definition of R to allow for the both cases det(R) = ±1. We choose instead to work with matrices in Eqs. (77) and (78) but extend the range of the Majorana phases α 21(31) from [0, 2π] to [0, 4π], which effectively accounts for the case of det(R) = − 1 [22], so that the same full set of R and Yukawa matrices is considered.
In what follows we will analyse the case of hierarchical mass spectrum of the two heavy Majorana neutrinos, M 1 M 2 . In the subsequent numerical analysis we use M 2 = 10M 1 .

CP Violation from Low-Energy CPV Phases of the PMNS Matrix
We are interested in the scenario of leptogenesis in which the CP violation is due exclusively to the low-energy CPV phases present in the PMNS matrix. Correspondingly, we should avoid contributions to CP violation in leptogenesis associated with the R-matrix. To satisfy this  Figure 3: The decay parameter κ 1 versus x for NH (blue) and IH (orange) light neutrino mass spectra with real R-matrix, i.e., y = 0. As the figure shows, κ 1 1, meaning that leptogenesis occurs always in the strong wash-out regime. The figure illustrates also the periodicity of π in the κ 1 dependence on x. The top panel illustrates the small oscillations of the IH curve. requirement we can [18] either set i) y = 0 and x = 0, which corresponds to a real R-matrix; or 2) x = kπ, k = 0, 1, 2, and y = 0, so that in the case of NH (IH) spectrum the product R 12 R 13 (R 11 R 12 ) of the R-matrix elements (see Eqs. (77) and (78)), which enters into the expression for the CPV asymmetry (1) τ τ , is purely imaginary. We first report the expressions for κ 1 , p 1τ and (1) τ τ in the NH case for y = 0 (real R 12 R 13 ), relevant for our further analysis: where m * and f (M 1,2 ) are defined in Eqs. (29) and (11). The corresponding expressions for the IH spectrum can formally be obtained from those given above by changing m 2(3) → m 1 (2) and U τ 2(τ 3) → U τ 1(τ 2) .
In the NH (IH) case, only the Majorana CPV phase difference (phase) α 21 − α 31 (α 21 ) is physically relevant 17 . We note also that the dependence on δ is always suppressed by sin θ 13 . Thus, for the NH (IH) neutrino mass spectrumz Λ is predominantly a function of the Majorana phase α 23 (α 21 ), of the real part of the R-matrix angle x and of the mass scale M 1 , exhibiting also subleading dependence on δ.
For y = 0 and x = 0, i.e., for real elements of the R-matrix, the final baryon asymmetry η B is always suppressed in the IH case with respect to that in the NH case [18]. This is a consequence of the fact that the CPV asymmetry (1) τ τ in the NH and IH cases, (NH) and (IH) , are proportional respectively to m 3 − m 2 and m 2 − m 1 (see Eq. (81) and the subsequent discussion), m 3,2 ≡ m 3,2 (NH) and m 2,1 ≡ m 2,1 (IH) being the corresponding light neutrino masses of the two spectra (see section 2.1), so that As a consequence, it is impossible to have a successful leptogenesis for IH neutrino mass spectrum with CP violation provided only by the CPV phases in the PMNS matrix and real R-matrix for M 1 10 13 GeV. The suppression can be avoided in the considered scenario if the product R 11 R 12 of the R-matrix elements is purely imaginary [18], i.e., if x = kπ, k = 0, 1, 2, and y = 0. Under the conditions x = kπ and y = 0, the expressions for κ 1 , p 1τ and (1) τ τ in 17 We will call the Majorana phase difference α23 simply "Majorana phase" and will denote it as α23 in what follows.  Figure 4: The decay parameter κ 1 as a function of y for NH (blue) and IH (orange) light neutrino mass spectra in the case of purely imaginary product R 12 R 13 (R 11 R 12 ), i.e., x = 0, π, ... . As the figure shows, κ 1 1, meaning that leptogenesis always takes place in the strong wash-out regime.
the IH case take the form: The corresponding expressions for the NH spectrum can formally be obtained from those given above by changing m 1(2) → m 2(3) and U τ 1(τ 2) → U τ 2(τ 3) . The suppression of η B in the IH case is now avoided due to the presence of the factor (m 1 + m 2 ) in the CPV-asymmetry (1) τ τ . For x = kπ, k = 0, 1, 2, and y = 0, the strong wash-out condition κ 1 1 is always satisfied, as is shown in Fig. 4. Thus, also in this case we can rely on Eq. (75) to discuss the the sign change of η B at the 1-to-2 flavour transition.

CP Violation due to the Dirac Phase
We consider in this subsection the scenario of leptogenesis with decoupled N 3 and CP violation due only to the Dirac phase δ. To this end the Majorana phase α 23 (α 21 ) is set in the NH (IH) case to the following CP conserving values: i) ±2nπ (2nπ), n = 0, 1, 2, when x = 0 and y = 0; ii) ±(2n + 1)π ((2n + 1)π), n = 0, 1, when x = kπ, k = 0, 1, 2, and y = 0. With the choices of the values of α 23 (α 21 ) made we avoid the situation in which one of the sources of CP violation in leptogenesis is the interplay of CP conserving Majorana phases and R-matrix elements [18] that could be generated by the first term in the r.h.s. of Eq. (85) (Eq. (87)).

The Case of Real
τ τ )sgn(p 1τ ⊥ − p 1τ ). Alternatively, if thez Λ curve lies below the z out one, then sgn(N B−L (∞)) = sgn( (1) τ τ )sgn(p 1τ ⊥ − p 1τ ). The intersection points correspond to a vanishing η B and mark the 1-to-2 flavour transition. In the bottom-left panel the generated baryon asymmetry η B (with the correct sign) at different mass scales M 1 is also depicted. In the right panel we show the behaviour with x of the relevant quantities: (1) τ τ (normalised to its absolute maximal value), p 1τ , p 1τ p 1τ ⊥ and p 1τ ⊥ − p 1τ .
For the considered choice of the parameters, depending on x we can have different scenarios. At x 10 • and x 60 • , the 10 13, 14 GeV curves lie above the z out line while the 10 12, 11 ,10 ,9 GeV curves lie below (with the 10 12 GeV curve lying near the z out line). We can conclude that, for x 10 • and x 60 • , the 1-to-2 flavour transition occurs at values of M 1 slightly larger than 10 12 GeV and with a sign change. According to the bottom-left panel, in the indicated ranges of x we can have successful leptogenesis for values of x ≈ 150 • at M 10 13 GeV. At x min 22.2 • the magenta curve in the right panel of Fig. 5, corresponding to p 1τ p 1τ ⊥ , reaches an absolute minimum for the chosen value of α 23 = 0. We then note that in the range 10 • x 60 • , the transition occurs at M 10 12 GeV, and the more the range of x is squeezed around x min , the lower is the mass scale of the transition. At x min no sign change occurs at the transition, given that all thez Λ curves obtained for M 1 > 10 9 GeV lie above the z out line. As is shown in the bottom-left panel of Fig. 5, at x min the final baryon asymmetry η B is positive at any mass scale and reaches the observed value at 10 11 M 1 /GeV 10 12 .
In Fig. 6 we plot η B as a function of M 1 , calculated using the density matrix equations For x = 150 • corresponding to the top-left panel in Fig. 6, successful leptogenesis takes place at M 1 4.0 × 10 12 GeV and the 1-to-2 flavour transition happens around M 1 10 12 GeV, as one would have expected from the considerations made after Eq. (23). This panel shows an example of what we will call "standard" scenario of a 1-to-2 flavour transition at ∼ 10 12 GeV under the assumption made about the source of CP violation as well as the strong wash-out regime of η B generation. We note also that the 2-to-3 flavour transition happens at M 1 ∼ 10 9 GeV with no sign change, as the DME solution (solid blue line) interpolates smoothly between the 1BE2F (green) line and the 1BE3F (red) one.
The remaining three panels of Fig. 6 represent examples of "non-standard" scenarios of the transition of interest, as the 1-to-2 flavour transition happens at a mass scale which decreases from 10 12 GeV as x decreases approaching x min , the value of x at which the sign change does not occur and, as we will discuss, (1) τ τ has an absolute maximum (bottom-right panel). The scenarios in these panels are "non-standard" for the following reasons. Firstly, the product p 1τ p 1τ ⊥ p 1τ 2.5 ×  integrated wash-out term in λ is additionally strongly suppressed, allowing the plateau due the τ τ ⊥ -decoherence contribution to extend below 10 12 GeV. Secondly, since for x = 30 the CPV asymmetry in the τ ⊥ -flavour is generated in the strong wash-out regime, while the CPV asymmetry in the τ -flavour is produced in the weak wash-out regime. This scenario could not and was not considered in Sec. 3 18 . Finally, Fig. 6 shows that the two-flavour approximation in the range 10 9 M 1 /GeV 10 12 based on 1BE2F is not always accurate.
For the case considered in the bottom-right panel of the figure, the DME solution for the asymmetry η B is enhanced by a factor of ∼ 10 with respect to the asymmetry obtained by solving the Boltzmann equations in the two-flavour approximation. This leads, in particular, to successful leptogenesis at M 1 2.5 × 10 11 GeV. Most remarkably, in the case shown in the bottom-left panel, the asymmetry η B predicted by the DMEs (blue solid curve) at M 1 4.8 × 10 11 GeV has the correct sign allowing for successful leptogenesis, while the 1BE2F solution (green curve) gives η B < 0 in the indicated range of M 1 and thus non-viable leptogenesis.
We note also that, as the top-right and bottom-left panels in Fig. 6 show, at the 2-to-3 flavour transitions at M 1 ∼ = 10 9 GeV, the baryon asymmetry η B changes sign going through zero, in contrast to the behaviour of η B shown in the top-left and bottom-right panels. For the chosen values of the CPV phases of the PMNS matrix the presence of this zero in η B , as Fig. 6 indicates, depends on the value of x. For x = 22.2 • (bottom-right panel), the 2-to-3 flavour transition takes place with η B not going through zero but only through a relatively shallow minimum at M 1 ∼ = 10 10 GeV.

Ranges of M 1 and δ for Viable Leptogenesis
The case illustrated in the bottom-right panel of Fig. 6 is interesting for the following additional reasons. As our scan of the relevant parameter space shows, it is the case in which successful leptogenesis with two hierarchical in mass heavy Majorana neutrinos and CP violation provided by the Dirac phases δ takes place for the minimal for the considered scenario value of M 1min ∼ = 2.5 × 10 11 GeV. The value of x = x min = 22.2 • maximises the CPV asymmetry (1) τ τ . Indeed, (1) τ τ depends on x through the factor which has an absolute maximum at 19 x min = 22.2 • : f (x min ) ∼ = 1.00. The chosen value of δ also maximises | (1) τ τ |. As M 1 increases from the value M 1 ∼ = 2.5 × 10 11 GeV, η B also increases from η B = 6.1 × 10 −10 and, as Fig. 6 bottom-right panel shows, for x = x min and δ = 3π/2 reaches a plateau at M 1 = 2.7 × 10 12 GeV, where η B = 1.60 × 10 −9 and 20 is larger than the observed value of η B by the factor C P 1 ∼ = 2.62. For the value of M 1 2.7 × 10 12 GeV of the 18 Moreover, since p1τ κ1 10 −2 , the analytic approximation used in Sec. 3 in the weak wash-out regime is not sufficiently accurate [64]. 19 To be more precise, f (x) has an absolute maximum at x max = 0.5 arccos((m3 − m2)/(m3 + m2)) = 22.5 • , where we have made use of m3 = ∆m 2 31 , m2 = ∆m 2 21 and the best-fit values of ∆m 2 31 and ∆m 2 21 given in Table 1. However, as can be easily checked, f (x max ) − f (x min ) ∼ = 10 −4 . 20 As M1 increases beyond 2.7 × 10 12 GeV, ηB continues to grow very slowly due to the dependence of plateau, we have η B ∝ (− (1) τ τ ) (see Eq. (93)). Thus, η B will be compatible with the observed value of BAU for smaller value of (− (1) τ τ ) > 0, i.e., for smaller (− f (x) sin δ) > 0. The plateau value of η B corresponds to x = x min and δ = 3π/2 for which (− f (x min sin(3π/2)) = 1. Thus, fixing x = x min we get the minimal value of (− sin δ) > 0 for which we can have successful leptogenesis at M 1 2.7 × 10 12 GeV: The derived condition on δ is a necessary condition for successful leptogenesis within the considered scenario 21 .
As M 1 decreases from 2.7 × 10 12 GeV to 2.5 × 10 11 GeV, η B decreases from the value at the plateau to the observed value and correspondingly, the interval of values of δ for which one can have viable leptogenesis also decreases shrinking to the point δ = 3π/2 at M 1 = 2.5 × 10 11 . Clearly, there exists a correlation between the value of δ and the scale M 1 of viable leptogenesis. It follows from the preceding discussion also that in the considered scenario of CP violation provided by the Dirac CPV phase δ of the PMNS matrix, it is possible to reproduce the observed value of BAU for values of M 1 spanning at least three orders of magnitude, i.e., for 2.5 × 10 11 M 1 /GeV 10 14 .
In the case we have considered with α 23 = 0 and x = x min = 22.2 • , we can have successful leptogenesis, as Eq. (96) shows, for δ lying in the interval π < δ < 2π, where sin δ < 0. So, the sign of sin δ is anticorrelated with the sign of the observed η B . This result holds also for the alternative possible values of α 23 = ±2π and all possible value of x, for which we can have viable leptogenesis. In other words, in the case under study there are no values of δ from the interval 0 < δ < π where sin δ > 0, for which it is possible to reproduce the observed value of BAU.
Indeed, we note first that there is a periodicity of π in the dependence on x, and of 4π in the dependence on α 23 , of all relevant quantities on which the predicted sign of η B depends: (1) τ τ , k 1 and p 1τ (see Eqs. (79) -(88)). Therefore one gets the same results for x and x − π.
In the example with α 23 = 0 and x = 22.2 • we have considered, we get the same result for x = − 157.8 • (or equivalently x = 202.2 • ). If we set α 23 = 2π, the results will be the same for α 23 = − 2π. Therefore the only possibility to have viable leptogenesis with sin δ > 0 is when α 23 = 2π. The quantities (U * τ 2 U τ 3 ) sin 2x ∝ − β α sin 2x and (U * τ 2 U τ 3 ) sin 2x ∝ β α sin δ sin 2x, on which respectively p 1τ and (1) τ τ depend, change sign when α 23 is changed from 0 to 2π: β α = 1 (−1) for α 23 = 0 (2π). In addition (U * τ 2 U τ 3 ) and p 1τ exhibit a subleading dependence on δ via terms proportional to sin θ 13 cos δ. Thus, in what concerns the present discussion, changing the sign of sin δ has negligible effect on p 1τ . We recall that for α 23 = 0, p 1τ has a minimum at x min = 22.2 • where p 1τ 1, so that we have (1 − 2p 1τ ) > 0 for the quantity on which, in particular, the sign of η B depends. The change of the sign of (U * τ 2 U τ 3 ) sin 2x leads to a significant change of the value of p 1τ , leading for x min = 22.2 • to (1 − 2p 1τ ) < 0 and thus to non-viable leptogenesis for sin δ > 0. 21 In [18] in the same scenario the following condition for successful leptogeneis was obtained using the 1BE2F and assuming that the two-flavoured leptogenesis regime does not extend beyond M1 = 5 × 10 11 GeV: | sin θ13 sin δ| 0.090. In the same article the minimal scale of viable leptogenesis was found to be M1min ∼ = 2.2 × 10 11 GeV, to be compared with M1min ∼ = 2.5 × 10 11 GeV found by us. The lower limit on (− sin δ) we have obtained in Eq. (96) implies | sin θ13 sin δ| 0.057, where we have used the best fit value of θ13 from Table 1. It is clear that our results based on the DME, in particular, extend the ranges of δ and M1, for which we can have successful leptogenesis, derived in [18]. τ τ can only be compensated simultaneously by changing x to π − x, i.e., by a change of the sign of sin 2x. This implies that in addition to the solutions we have found for α 23 = 0 for certain ranges of x (e.g., for 0 < x < π/2) and of δ in the interval π < δ < 2π, for α 23 = 2π we will have successful leptogenesis in the range of π − x (e.g., for π/2 < x < π) and for δ in the same interval. Thus, in the case of α 23 = 2π, a value of δ from the interval 0 < δ < π with sin 2x < 0 (sin 2x > 0) leads either to a wrong sign of η B due to the interplay of the signs of p 1τ and (1) τ τ , or else to a value of η B which is smaller than the observed one.
The conclusions of the preceding discussions are confirmed by the numerical scan of the parameters δ and x in the case of α 23 = 0 and 2π and several fixed values of M 1 , the results of which are shown in Fig. 7. Thus, in the considered scenario there is a direct and unique relation between the sign of sin δ and the sign of the baryon asymmetry of the Universe. If the measurement of δ in the low-energy neutrino oscillation experiments will show that δ lies in the interval [0, π], the considered leptogenesis scenario will be ruled out. If, however, δ will be found to lie in the lower half-plane, π < δ < 2π, this will not only lend support for the discussed scenario, but also will allow to obtain constraints on the leptogenesis scale.
Given that for x = 0, y = 0 and CP violation due only to the Dirac phase δ leptogenesis is unsuccessful at any mass scale in the IH case (see Eq. (89) and the discussion related to it) we do not consider this case.

Purely imaginary
We discuss next the leptogenesis scenario in which CP violation is still provided by the Dirac phase only, but now x = k π, k = 0, 1, 2, and y = 0 so that the product R 12 R 13 (R 11 R 12 ) is purely imaginary in the NH (IH) case and the suppression of the CPV asymmetry shown in Eq. (89) is avoided.

NH Spectrum
The analysis is similar to that performed in the preceding subsection. We report below the results on the ranges of δ and M 1 for which one can have successful leptogenesis. For the minimal value of M 1 we get M 1 = 1.7 × 10 11 GeV, which is obtained for δ = π/2, y = 26 • (y = − 26 • ) and α 23 = π or − 3π (3π or −π). This case is illustrated by Fig. 8. The value of y maximises the factor in the expression for (1) τ τ , and thus maximises (1) τ τ with respect to y. For α 23 = π or (− 3π) the value of δ = π/2 maximises (1) τ τ which is proportional to sin δ. At the plateau which begins at M 1 ∼ = 2.1 × 10 12 GeV we have η B ∼ = C P 2 6.1 × 10 −10 with C P 2 ∼ = 3.9. Correspondingly, at  successful leptogenesis is possible for values of M 1 1.7 × 10 11 GeV, which span at least three orders of magnitude.
It follows from the preceding discussion that for α 23 = π or (− 3π), one can have successful leptogenesis for value of δ from the interval 0 < δ < π where sin δ > 0. Performing an analysis similar to that in the preceding subsection, we find that also in this case there is a direct relation between the sign of sin δ and the sign of the observed BAU in the sense that for the values of the parameters in the considered case, no region with viable leptogenesis exists for δ from the interval π < δ < 2π, where sin δ < 0. Changing the value of α 23 from π to 3π, for example, one finds that the viable regions of values of y and δ from the interval 0 < δ < π, for which it is possible to reproduce the observed value of BAU, shift to the regions corresponding to (− y) with δ remaining in the same interval 0 < δ < π. This is confirmed by the numerical scan of the y − δ parameter space for α 23 = π and 3π, the results of which are shown in Fig.  9.
Obviously, the discussed leptogenesis scenario will be ruled out if δ determined in neutrino oscillation experiments is found definitely to lie in the interval [π, 2π], while if δ is found to be in the upper half-plane, 0 < δ < π, the scenario will be proven viable and it will be possible to obtain also constraints on the leptogenesis scale of the scenario.

IH Spectrum
We analyse in somewhat greater detail the case of IH spectrum. We show in Fig. 10  at which η B = 0, we have η B < 0 (η B > 0). In the case illustrated in Fig. 10, we have M 10 10 12 GeV. When y changes from (−100 • ) to (−46.5 • ), M 10 decreases from 6.0 × 10 10 GeV to 2.4 × 10 10 GeV. Most importantly, the minimal value of M 1 at which one can have successful leptogenesis also decreases from M 1 = 1.6 × 10 11 GeV to M 1 = 6.2 × 10 10 , with both values being 10 12 GeV. Further, the DME solution for η B shown in the left (right) panel of Fig. 10 is at M 1 ≤ 10 12 GeV larger than (similar in magnitude to) |η B | found with 1BE2F, except in a narrow region around M 10 . Still, η B obtained from the 1BE2F equations shown in both panels, in contrast to that derived from DME ones, has a wrong sign, i.e., predicts η B < 0 and thus nonviable leptogenesis. The value of |η B | obtained with 1BE3F is in both cases, as the panels show, significantly smaller than those found with DME and 1BE2F. Moreover, the 2-to-3 flavour transition described by the DME solution takes place at M 1 10 8 GeV, with the µ-Yukawa interaction having the effect of enhancing the DME solution for |η B | in the interval 10 8 M 1 /GeV 10 10 . Both these features are in the region of values of M 1 for which the calculated |η B | is significantly smaller that the observed η B . However, they might be relevant in a leptogenesis scenario with three heavy Majorana neutrinos with non-hierarchical masses, e.g., with In what concerns the range of δ and M 1 for which we have successful leptogenesis, we find that (see Fig. 11): i) the minimal value of M 1 is M 1 ∼ = 4.6 × 10 10 GeV and corresponds to the values of α 21 = π (3π), y = − 73 • (y = + 73 • ) and δ = 211 • ; ii) the plateau of values of η B is present at M 1 1.2 × 10 12 GeV; iii) at the plateau η B ∼ = C P 3 6.1 × 10 −10 with C P 3 ∼ = 6.1. Correspondingly, at M 1 1.2 × 10 12 GeV successful leptogenesis is possible for where we have used the fact that the plateau value of η B corresponds to δ = 211 • . We note that for the chosen values of α 21 = π, y = − 73 • , τ τ is proportional to sin δ and thus at δ = 211 • | (1) τ τ | is smaller by the factor 0.515 than for δ = 3π/2. However, due to the fact that, as can be shown, the value of p 1τ at δ = 211 • is smaller approximately by a factor of 6 than that at δ = 3π/2, the minimal M 1 at which we can have successful leptogenesis is also smaller than the one for 3π/2 which reads M 1 ∼ = 10 11 GeV. At the same time, η B at the plateau for δ = 3π/2 is by a factor of approximately 1.9 times larger than the plateau value of η B for δ = 211 • and reads: η B = 7.3 × 10 −9 .
As we have seen, in the discussed case of Dirac CP violation and IH neutrino mass spectrum, successful leptogenesis is possible for values of δ from the interval π < δ < 2π where sin δ < 0. Performing a scan over y and δ for the possible values of α 21 = π and 3π with x = kπ, k = 0, 1, 2, we find that for M 1 10 13 GeV one can have a successful leptogenesis also for values of δ from the interval 0 < δ < π, and a small range of values of y from the interval 0 < y < 50 • . The appearance of this second region is related to the slow increase of (1) τ τ and thus of η B with M 1 due to the factor f −1 (M 2 )/M 2 . The results of the scan are presented graphically in Fig. 12. Thus, in this case we have a direct relation between the sign of sin δ and the sign of the baryon asymmetry of the Universe only for M 1 < 10 13 GeV.

CP Violation due to the Majorana Phases
We investigate next the considered leptogenesis scenario with two heavy Majorana neutrinos with hierarchical masses in which the CP violation is provided only by the Majorana phases of the PMNS matrix. Thus, the Dirac phase and the R-matrix elements are chosen not to contribute to the CP violation necessary for the generation of BAU. We note that in the case of CP violation due to the Majorana phases, the additional CP violation due to the Dirac phase has sub-leading effects in leptogenesis as a consequence of the suppression by the factor sin θ 13 . However, in certain cases these effects are non-negligible.

NH Spectrum
For real R-matrix it is impossible to have successful leptogenesis for IH light neutrino mass spectrum and CP violation originating from CPV phases of the PMNS matrix, as we have already discussed, so we consider only the case of NH spectrum. We show in Fig. 13 Fig. 13 correspond respectively to what we dubbed "standard" and "non-standard" behaviour of the baryon asymmetry η B . The salient features of the behaviour of η B in the two cases are analogous to those discussed in detail in the two preceding subsections and we are not going to comment on them further.
We show in the bottom-left (right) panel of Fig. 13 the dependence of η B on M 1 for real (purely imaginary) R 12 R 13 and δ = π in the case in which it is possible to have successful leptogenesis with CP violation provided by the Majorana phase α 23 for the minimal in the considered scenario value of M 1 . The other relevant parameters have the following values in the left (right) panels: x = 20 • , y = 0 (x = kπ, k = 0, 1, 2, y = 19 • ) and α 23 = 102 • (80 • ). We will discuss in some detail in what follows first the case of x = 20 • , y = 0 and α23 = 102 • , extending the discussion after that to the whole plane 0 < x < 180 • .
It follows from the preceding discussion that if we denote by M 10 ) extending into η B plateau region when 220 • < α 23 < 360 • (500 • < α 23 < 720 • ). Moreover, the results for 90 • < x < 180 • can be obtained from those derived for 0 < x < 90 • by making the simultaneous change x → π − x and α 23 → α 23 ± 2π and taking into account that the results are invariant with respect to the change α 23 → α 23 ± 4π.
These qualitative conclusions are essentially confirmed by a thorough numerical analysis, the results of which are shown graphically in Fig. 14. We next summarise briefly these results giving the ranges of α 23 and M 1 of viable leptogenesis in the four intervals of values of α 23 identified earlier for 5 • < x < 90 • . We will do it for the representative value of x = 20.5 • at which some of the ranges of interest are maximal, commenting first the results for the specific values of α 23 at which viable leptogenesis occurs for the minimal for the case value of M 1 . Viable leptogenesis in the scenario of interest is possible only for purely imaginary product R 11 R 12 of elements of the R-matrix (x = kπ, k = 0, 1, 2, y = 0). In Fig. 15 we show the modulus of η B versus M 1 for δ = π (3π/2) and values of the other parameters for which successful leptogenesis takes place for the minimal for the considered scenario value of M 1 : In both cases illustrated in Fig. 15, η B exhibits a "non-standard" behaviour as a function of M 1 going through zero in the 1-to-2 flavour transition at M 1 ∼ 10 10 GeV 10 12 GeV. In the left (right) panel the minimal value of M 1 at which the calculated η B matches the observed value of η B is M 1 = 4.3 × 10 10 (5.4 × 10 10 ) GeV. The plateau value of η B ∼ = 3.9 × 10 −9 (5.6 × 10 −9 ) and is reached at M 1 ∼ = 1.1 × 10 12 (1.8 × 10 12 ) GeV. Performing a detailed numerical analysis we have determined the regions of viable leptogenesis in the space of parameters in the considered scenario with IH spectrum, CP violation provided by α 21 , x = kπ, k = 0, 1, 2, y = 0 (purely imaginary R 11 R 12 ) and δ = π. The values of y were varied in the interval [− 180 • , +180 • ]. For y = 0 we have, obviously, η B = 0. Due to symmetries of the quantities involved in the generation of η B , the results for 0 ≤ y ≤ 180 • can formally be obtained from those derived for − 180 • ≤ y ≤ 0 by making the change y → − y and α 21 → α 21 ± 2π and using the invariance with respect to α 21 → α 21 ± 4π. The results of this analysis are shown graphically in Fig. 16. A few comments are in order.
As we have already pointed out, the minimal value of M 1 for successful leptogenesis is found to take place at y = −76 • and α 21 = 164 • . For y = −76 • and α 21 = 164 • , the plateau value of η B given earlier is larger than the observed value of η B by the factor C M 5 6.39. We can determine then the range of α 21 for viable leptogenesis the case of y = −76 • and δ = π There is an additional relatively small region of values of α 21 and M 1 around α 21 ∼ 500 • , for which it is possible to reproduce the observed value of BAU. It is clearly seen in Fig.  16. In this case the minimal mass scale takes place at α 21 493 • , y − 76 • and reads M 1 5.9 × 10 12 GeV. At the plateau we have η B C M 6 6.1 × 10 −10 , with C M 6 1.83. The range of α 21 of viable leptogenesis can be determined using the equation: Solving the preceding equation we get for the range of α 21 : The corresponding range of the leptogenesis scale is 5.9 × 10 12 M 1 /GeV 10 14 .
As in the previous scenarios discussed by us, also in the scenario considered in the present subsection there is a correlation between the low-energy phase -in this case α 21 -responsible for the CP violation in leptogenesis and the scale of leptogenesis M 1 . Clearly, obtaining constraints on α 21 in low-energy experiments can rule out the considered scenario or constrain the leptogenesis scale of the scenario.

Summary and Conclusions
We have considered the generation of the baryon asymmetry of the Universe η B in the high (GUT) scale leptogenesis based on type I seesaw mechanism of neutrino mass generation. Using the density matrix equations (DME) for high scale leptogenesis in which the CP violation is provided by the low-energy Dirac or/and Majorana phases of the neutrino mixing (PMNS) matrix, we have investigated the 1-to-2 and the 2-to-3 flavour regime transitions, where the 1, 2 and 3 flavour regimes are described by the Boltzmann equations. Concentrating on the 1-to-2 flavour transitions in leptogenesis with three heavy Majorana neutrinos N 1,2,3 with hierarchical mass spectrum, M 1 M 2 M 3 , we have determined the general conditions under which the baryon asymmetry η B goes through zero and changes sign in the transition. We have shown, in particular, that the asymmetry η B goes through zero changing its sign when leptogenesis proceeds in the strong wash-out regime with zero initial abundance of the heavy Majorana neutrinos.
In order to make the discussion of all the salient features of the transitions between the different flavour regimes of interest as transparent as possible, we have investigated further the case of decoupled heaviest Majorana neutrinos N 3 , in which the number of parameters in leptogenesis is significantly smaller than in the general case of three heavy Majorana neutrinos. In particular, the complex orthogonal matrix R, which makes part of the Casas-Ibarra parametrisation of the neutrino Yukawa couplings we have employed in the analysis, depends only on one complex angle θ CI = x + i y, where x and y are real parameters. With only two heavy Majorana neutrinos (N 1,2 ) active in the seesaw mechanism, the light neutrino mass spectrum can only be either normal hierarchical (NH) with m 1 ∼ = 0, or inverted hierarchical (IH), with m 3 ∼ = 0. Furthermore, in the case of interest of CP violation in leptogenesis provided only by the low-energy CPV phases of the PMNS matrix, one can avoid the contributions to the CP violation associated with the R-matrix only if the angle θ CI is such that for NH (IH) spectrum sin 2θ CI = 0 is real (purely imaginary) [18], sin 2θ CI = sin 2x (sin 2θ CI = ±i sinh 2y).
Analysing in detail the behaviour of η B in the transition in the case of two heavy Majorana neutrinos N 1,2 with hierarchical masses, M 1 M 2 , allowed us to gain a better understanding of the transitions but also to discover new unexpected features of the transitions. We have found, in particular, that: i) the Boltzmann equations in many cases fail to describe correctly the generation of the baryon asymmetry η B in the 1, 2 and 3 flavour regimes, in particular, underestimating η B by a factor ∼ 10 in certain cases; ii) depending on the values of the relevant parameters, the transitions between the different flavour regimes can be "non-standard" as, e.g., the 1-to-2 and the 2-to-3 flavour transitions can take place at the same M 1 , with η B going through a relatively shallow minimum at the transition value of M 1 ; iii) the two-flavour regime can persist above 5 × 10 11 − 10 12 GeV (below ∼ 10 9 GeV), and iv) the flavour effects in leptogenesis persist beyond what is usually thought to be the maximum leptogenesis scale for these effects of ∼ 10 12 GeV, with the requisite CP violation provided by the Dirac or/and Majorana phases present in the low-energy PMNS neutrino mixing matrix.
At M 1 ∼ 10 12 GeV, |η B | reaches a "plateau" where it remains practically constant as M 1 increases and flavour effects are fully operative. We further have determined the minimal scale M 1min at which we can have successful leptogenesis when the CP violation is provided only by the Dirac (δ) or Majorana (α 23 ≡ α 21 − α 31 or α 21 ) phases of the PMNS matrix as well as the ranges of the scales and the values of the phases for having successful leptogenesis. In the case of Dirac phase CP violation we found that M 1min ∼ = 2.5 (1.7) × 10 11 GeV for NH light neutrino mass spectrum and δ lying in the interval π < δ < 2π (0 < δ < π) for real (purely imaginary) Casas-Ibarra parameter sin 2θ CI (Figs. (7) and (9)). As M 1 increases from M 1min to M 1 ∼ = 2.7 (2.1) × 10 12 GeV, at which η B reaches the plateau value, the range of interest of δ increases from the point δ = 3π/2 (π/2) to 202.4 • δ 337.6 • (14.6 • δ 166.4 • ) and remains practically the same up to M 1 ∼ 10 14 GeV. We get similar results for M 1min and the ranges of δ and M 1 in the case of IH light neutrino mass spectrum and purely imaginary Casas-Ibarra parameter sin 2θ CI , with M 1min ∼ = 4.6 × 10 10 GeV for δ = 3π/2 and 185 • δ 355 • for 1.2 × 10 12 GeV M 1 10 14 GeV (Fig. (12)). We found also that in the case of NH spectrum there is a direct relation between the sign of sin δ and the sign of the baryon asymmetry of the Universe in the regions of viable leptogenesis; for IH spectrum such a relation holds for M 1min ∼ = 4.6 × 10 10 GeV M 1 10 13 GeV.
We have investigated also the generation of η B when the CP violation is provided solely by the Majorana phases α 21 /2 and α 31 /2 of the PMNS matrix. In the considered scenario with two heavy Majorana neutrinos only the phase α 23 ≡ α 21 − α 31 (α 21 ) is physically relevant for NH (IH) light neutrino mass spectrum. We have performed a thorough analysis and have determined the ranges of values of the Majorana phase α 23 (α 21 ) and the related ranges of the scale M 1 , for which we can successful leptogenesis with CP violation provided exclusively by α 23 (α 21 ) in the case of NH (IH) spectrum, real (purely imaginary) Casas-Ibarra factor sin 2θ CI = sin 2x = 0 (sin 2θ CI = ±i sinh 2y = 0) and CP conserving value of δ = π (the results of these analyses are presented graphically in Figs. (13), (14), (15) and (16)). Our results show, in particular, that there exist relatively large regions of the relevant spaces of parameters where it is possible to reproduce the observed value of BAU as well as that in these regions the values of the respective Majorana phases providing the requisite leptogenesis CP violation are strongly correlated with the value of leptogenesis scale M 1 .
We note that in [71], the RGE-corrections to the neutrino Yukawa matrices are included in the calculations of the CPV lepton asymmetry through a modified Casas-Ibarra parametrisation. Without these corrections, within the formalism employed in [71], which is based on the Boltzmann equations, all lepton flavour dependence (contained in the PMNS matrix) cancels at leptogenesis scales M LG 10 12 GeV from the total CPV asymmetries. Once the RGE-corrections are included, this cancellation no longer occurs and a corrective term approximately proportional to the square of the τ -Yukawa coupling is added to the total CPV asymmetry. The authors of [71] then show that this correction is sufficient in some regions of the parameter space at M LG 10 12 GeV for successful leptogenesis from purely low-scale CP violation due to the phases in the PMNS matrix in the absence of the usual flavour effects. However, this correction is subdominant to flavour effects discussed in this work, typically being by a factor of ∼ 10 to ∼ 100 smaller in the regions of the parameter space of the scenar-ios we have considered, where leptogenesis successfully generates the observed BAU. Thus, the mechanism of generation of BAU considered in [71] is subdominant to the mechanism discussed in this work.
It follows from the results obtained in the present article that, in particular, viable leptogenesis based on type I seesaw mechanism with two hierarchical in mass heavy Majorana neutrinos and CP violation provided by the physical low-energy Dirac or/and Majorana phases, present in the PMNS neutrino mixing matrix, is possible for rather wide ranges of values of the CP violating phases and of the scale of leptogenesis. The scenarios of leptogenesis investigated by us are obviously falsifiable in low-energy experiments on the nature -Dirac or Majorana -of massive neutrinos. As far as the nature of massive neutrinos is not known or if the massive neutrinos are proven to be Majorana particles, the cases of leptogenesis we have considered are still testable and falsifiable in low-energy experiments on CP violation in neutrino oscillations, on determination of the type of spectrum neutrino masses obey and of the absolute neutrino mass scale. The data from these experiments can severely constrain the corresponding leptogenesis parameter spaces and even rule out some of, if not all, the cases studied in detail by us. We are looking forward to these experimental data that can provide crucial tests of the leptogenesis scenarios discussed in the present paper.
An important relation that derives from Eqs. (107) and (108) and that is going to be used further is with α, β = e, µ, τ, τ ⊥ . We now concentrate again on the hierarchical case for which only the decay of the heavy neutrino N 1 is relevant for leptogenesis (i.e. i = 1). Firstly, we sum the equations for N ee and N µµ , which result from taking α = β = e, µ in Eqs. (44) respectively, and get an equation for N τ ⊥ τ ⊥ = N ee + N µµ : The second line of the above equation is actually p 1τ ⊥ N τ ⊥ τ ⊥ . This can be shown by considering the equations for p 1µ N ee , p 1e N µµ and 2 C 1e C * 1µ N ee , from which we can write: 2 C 1e C * 1µ N µe = p 1µ N ee + p 1e N µµ , which leads to p 1e N ee + p 1µ N µµ + 2 C 1e C * 1µ N µe = p 1τ ⊥ N τ ⊥ τ ⊥ .
We then define and N τ ⊥ τ = N * τ τ ⊥ , so that the equations for N τ τ and N τ ⊥ τ ⊥ can be recast in the forms given in Eqs. (50) and (51). By using the relation C iτ ⊥ (i) τ µ (with i = 1 in our case), which follows from Eqs. (107) and (109), combined with all the previous relations, we get the equation for N τ τ ⊥ as in Eq. (52) and the DMEs in the two-flavour basis are recovered.
The formal expression of N τ τ ⊥ can be obtained by solving Eq. (52) with the integrating factor method, which leads to where the initial asymmetry was assumed to be zero, namely N τ τ ⊥ (z 0 ) = 0. Notice that the above expression contains a term with N B−L which cannot be ignored in general, if not, e.g., in the limit of Γ τ /Hz → 0 (see Sec. 3).
To get the resulting equation for N B−L = N τ τ + N τ ⊥ τ ⊥ we first notice that: Then we write the equation for p 1τ ⊥ N τ τ + p 1τ N τ ⊥ τ ⊥ , that is: Then, given that we get Since all the asymmetries are assumed to be zero at z 0 , the above relation converts to with λ(z) defined as in Eq. (56). We note that λ(z) = 0 in the single-flavour approximation, namely for Γ τ /Hz = 0. Finally, by summing Eqs. (50) and (51) and using the previous relations we get an equation for N B−L that reads: In the case of λ(z) = 0, the above equation corresponds to the Boltzmann equation for the B − L asymmetry in the single-flavour approximation given in Eq. (26). Moreover, when (1) = 0, as in the case of CP violation solely provided by the PMNS phases, Eq. (122) reduces to Eq. (67). The formal solution to Eq. (122) reads: where, as usual, we have assumed vanishing initial asymmetry N B−L (z 0 ) = 0.

B Approximated Solutions to the BEs in Various Regimes
In this appendix we illustrate the passages that lead to the analytical approximations to the Boltzmann Equations (BEs) in various regimes. Useful references with similar calculations are [64,72]. The BEs are: where α = τ, τ ⊥ or e, µ, τ in the two-or three-flavour basis, respectively. The single-flavour BEs can be recovered by formally substituting N αα with N B−L and setting p 1α = 1 in (125). The strength of the decays and inverse decays is quantified by κ 1 p 1α . When κ 1 p 1α 1, the flavour α is said to be in the strong wash-out regime. Conversly, if κ 1 p 1α 1, the flavour α is in the weak wash-out regime. The formal solution to the BEs can be found by means of the integrating factor method: We define z eq as the time at which N N 1 (z eq ) = N eq N 1 (z eq ), that corresponds to a maximum for N N 1 (z). Indeed, from Eq. (124) it follows that, at z eq , dN N 1 /dz = 0 and d 2 N N 1 /dz 2 = D 1 dN eq N 1 /dz < 0. The number of RH neutrinos at z eq can be computed using some analytical approximations such as in [64], of which we employ the same result: For z < z eq , we can assume N eq N 1 N N 1 . Then, from Eq. (127) and using Eq. (30), we find that the asymmetry up to z eq reads: N (κ 1 ) 2N eq For 1 < z ≤ z D , the equilibrium number density is exponentially dropped so that N eq N 1 (z) N N 1 (z) and we have Then the asymmetry up to z D is roughly zero. At z z D the heavy neutrinos start to decay effectively and their abundance for z z D is exponentially damped: gives approximately the mass scale of the 1-to-2 flavour transition. In terms of M 10 the above condition reads: For the values of the parameters used to obtain the bottom-left (right) panel of Fig. 6, for example, we get 4p 1τ p 1τ ⊥ ∼ = 0.1 (6 × 10 −3 ), and correspondingly the mass scale of the transition is found at M 10 ∼ = 10 11 GeV (6 × 10 9 GeV), in agreement with the figure. We find, however, that the above approximation can underestimate M 10 by a factor O(1.5 − 2.5) when M 10 ≈ 10 12 GeV. The approximation is more accurate when M 10 10 12 GeV, as in the "non-standard" scenarios discussed in the present paper.