Leptogenesis from Low Energy $CP$ Violation

We revisit the possibility of producing the observed baryon asymmetry of the Universe via thermal leptogenesis, where $CP$ violation comes exclusively from the low-energy phases of the neutrino mixing matrix. We demonstrate the viability of thermal flavoured leptogenesis across seven orders of magnitude $\left(10^{6}<T \text{ (GeV)}<10^{13}\right)$, using modern numerical machinery, where the lower bound can be reached only if flavour effects are taken into account and its value depends on the allowed degree of cancellation between the tree-level and radiative contributions to the light neutrino masses. At very high scales $\left( T \gg 10^{12} \text{ GeV} \right)$, we clarify that thermal leptogenesis is sensitive to the low-energy phases, in contradiction with what is usually claimed in the literature. In particular we demonstrate that Majorana-phase leptogenesis is in general viable while Dirac-phase leptogenesis requires some level of fine-tuning.


Introduction
The Standard Model (SM) cannot explain the observed baryon asymmetry of the Universe (BAU) in spite of qualitatively satisfying the Sakharov conditions. 1 Similarly, physics beyond the SM is often invoked to explain the existence of the non-zero neutrino masses [2]. Leptogenesis is a mechanism by which some lepton-number-violating theories, which may also explain the origin of neutrino masses, produce a lepton asymmetry which is subsequently converted into a baryon asymmetry through the non-perturbative (B +L)-violating but (B − L)-conserving sphaleron processes of the SM [3,4]. A minimal implementation of leptogenesis occurs in the type I seesaw framework in which a number of heavy Majorana neutrinos are added to the SM [5][6][7][8]. The decay of these heavy Majorana neutrinos in leptons and Higgs bosons is both lepton-number-and CP -violating and occurs out of thermal equilibrium, thereby satisfying the Sakharov conditions and potentially producing the observed baryon asymmetry [9] (see also, e.g., [10][11][12][13] and articles quoted therein). CP violation is fundamental to the creation of the matter-antimatter asymmetry. In thermal leptogenesis, the decays of the heavy Majorana neutrinos are CP -asymmetric and this results from both CP -violating low-scale measurable phases and high-scale immeasurable ones. In the original conception of leptogenesis, the flavour-dependent interactions due to charged lepton Yukawa couplings between the leptons and the early Universe plasma were disregarded. If leptogenesis occurs at high scales, where the temperature T 10 12 GeV, then this approximation is ordinarily justified and a basis may be chosen in which essentially only one flavour of lepton ever appears in the theoretical description. Consequently, it was expected that the low-energy CP -violating phases contained in the neutrino mixing matrix play no physical role in the production of the lepton and therefore baryon asymmetry [11,12].
Under certain ad hoc assumptions the high-scale CP -violating phases can be related to the CP -violating phases in the Pontecorvo-Maki Nakagawa-Sakata (PMNS) matrix (see, e.g., [14]) which then participate in the production of the lepton asymmetry -a possibility that was investigated in [15][16][17][18][19]. If leptogenesis occurs at temperatures somewhat below 10 12 GeV 10 9 GeV , the Yukawa interactions of the tau charged lepton (of the muon) come into thermal equilibrium, causing decoherence between this and the remaining flavour components of the charged lepton state [20][21][22][23][24] such that two (three) lepton flavour states must be separately considered and the CP -violating phases of the PMNS matrix have physical significance. Historically, the possibility that the CP violation in leptogenesis may be strictly due to Dirac and/or Majorana phases of the PMNS matrix was first apparent in this regime [25][26][27][28][29][30][31][32] (for a review see, e.g., [33]).
There have been other works which have investigated the impact of low energy phases on the BAU. Indeed, CP conservation at the high-scale and CP violation at the low-scale in the context of leptogenesis can be theoretically motivated by minimal flavour violation [34,35], flavour symmetries [36][37][38] or a generalised CP symmetry [39][40][41]. Beyond the type I seesaw mechanism, there have been other studies which connect the Dirac phase, δ, with the BAU using an extended Higgs sector [42].
The primary aims of this work are twofold: • We revisit the possibility of producing the observed BAU where the only source of CP violation comes from the phases of the PMNS matrix and calculate the lepton asymmetry generated across seven orders of magnitude (10 6 < T (GeV) < 10 13 ) using our modern numerical machinery which properly incorporates radiative corrections to the light neutrino masses.
• We clarify a misunderstanding regarding the high-scale regime: as discussed above, it was initially thought that at very high scales, T ∼ 10 12 GeV, the low-energy CP phases could not produce the observed BAU. This came from the observation that in the one-flavoured regime, and barring any relation between the high-scale and low-energy CP -violating phases, the CP asymmetry is vanishing (1) = 0 for CP violation solely coming from low energy phases. This led to the conclusion that in this case no lepton asymmetry could be generated. However, the individual flavour contributions to the CP asymmetries are not zero and, even in this scenario, where flavour effects are very weak, the washouts of lepton asymmetries are flavour dependent. We show that these flavour effects may be sufficient to produce the observed baryon asymmetry. We discuss this analytically and demonstrate the viability of leptogenesis in this scenario numerically. This implies the CP violation present in the low energy phases can, in principle, generate the observed matter antimatter asymmetry between 10 6 T (GeV) 10 13 .
The remainder of this work is structured as follows: in Section 2 we present the relevant theoretical framework beginning with Section 2.1 where we briefly review neutrino masses, mixing and radiative corrections to the light neutrino mass matrix in the type I seesaw. In Section 2.2 we present the C and CP properties of the light and heavy Majorana neutrinos of the type I seesaw. This will be crucial in understanding the textures of the neutrino Yukawa matrices which contain CP -violating low energy phases and CP -conserving high energy phases. Subsequently, in Section 2.3 we discuss the allowed structures of the Yukawa matrix when there is a high-scale CP symmetry. The kinetic equations and effects of flavour are discussed in Section 2.4. In Section 3 we revisit the link between leptonic CP violation at low energies and successful leptogenesis in the two-flavour regime (or more precisely when 10 9 ≤ T (GeV) ≤ 10 12 ). We bring a more sophisticated set of numerical tools to more convincingly answer the question than in earlier work on the subject [25]. In Section 4 we demonstrate that successful leptogenesis with only low energy CP violation is possible at much lower scales than previously considered T ∼ 10 6 GeV , if one allows for fine-tuning in the light neutrino masses. In Section 5 we establish that even if leptogenesis occurs at very high scales (T 10 12 GeV), then it is still possible for successful leptogenesis to result from purely low energy CP violation. This latter case is in stark contrast with the conclusions of the current literature.   Table 1: Best fit and 1σ ranges from a global fit to neutrino data [44].

Neutrino Masses and Mixing
Neutrino oscillation experiments have provided compelling evidence that neutrinos have small but non-zero masses and mix (for a review see, e.g., [14]). The mass and flavour states of neutrinos are misaligned, with this misalignment described by the PMNS matrix U : where α ∈ {e, µ, τ } is the flavour of the given neutrino flavour field, ν αL , and ν iL is the left-handed component of the ith massive neutrino. Throughout this work we employ the conventional PDG parametrisation [14]: where c ij ≡ cos θ ij , s ij ≡ sin θ ij , δ is the Dirac phase and α 21 , α 31 are the Majorana phases [43] with best-fit values of θ 12 , θ 23 , θ 13 and δ given in Table 1. Neutrino oscillation experiments also allow to measure with precision the two independent neutrino mass squared differences ∆m 2 12 and ∆m 2 31 (∆m 2 32 ) which are given in Table 1. In order to accommodate them, the three neutrino masses m 1 , m 2 m 3 can be arranged into two possible orderings, normal ordering (NO) for m 1 < m 2 < m 3 and inverted ordering (IO) m 3 < m 1 < m 2 . The ordering is not yet known although data show some mild preference for normal ordering [44].
A simple means of explaining the smallness of neutrino masses is the type I seesaw framework [5][6][7][8] in which heavy Majorana neutrinos are added to the SM particle spectrum. We shall work within a realisation of this framework which incorporates three heavy Majorana neutrinos, N i (i ∈ {1, 2, 3}), such that after electroweak symmetry breaking, when the Higgs has developed a vacuum expectation value (vev) v ≈ 174 GeV, the neutrino mass terms of the Lagrangian are given by are the light flavour (heavy mass eigenstates) neutrino fields and Y is the neutrino Yukawa matrix which couples the heavy Majorana neutrinos N 1,2,3 to the leptonic and Higgs doublets. In Eq.
The mass terms in Eq. (2.3) are written in the basis where the charged lepton Yukawa couplings are flavour diagonal, which we use throughout the present study. We shall also work, without loss of generality, in a basis in which M is diagonal and positive.
To first order in the seesaw expansion, the tree-level light neutrino mass matrix is given by m tree generally suppressed by additional loop factors and couplings and generally do not experience fine-tuned cancellations, instead they contribute in the usual way to the perturbative expansion. We define a fine-tuning measure, F, to quantify the level of cancellation: (2.8) The one-loop correct light neutrino mass matrix of Eq. (2.6), m ν , may be transformed into a positive diagonal form (denoted by a caret) using the Takagi factorisation such thatm ν = diag(m 1 , m 2 , m 3 ), with m i the mass corresponding to ν i . By analogy with the method of Casas and Ibarra [49], we parametrise the Yukawa matrix to include the relevant radiative corrections as [50] where R is a 3 × 3 complex orthogonal matrix. Explicitly, we choose to work with the parametrisation: where c i = cos w i , s i = sin w i and the complex angles are given by w i = x i + iy i (i ∈ {1, 2, 3}).

C and CP Properties of Majorana Neutrinos
As we focus on the possibility that low-scale CP phases are responsible for the BAU, the C and CP properties of neutrinos will be crucial in understanding the structure of the Rmatrix which results in the CP conservation of the high-scale phases. In the type I seesaw, the light (ν i ) and the heavy (N i ) neutrino mass states are both Majorana in nature and thus satisfy the following conditions: where C denotes the charge conjugation matrix.
Following [26], we express the CP -conjugated neutrino fields in terms of the CP operator U CP as where x is the parity-transformed coordinate and iρ N i = ±i and iρ ν i = ±i are the CP parities of the respective Majorana fields. The conditions for CP invariance impose the following restrictions on the elements of the matrix of neutrino Yukawa couplings (setting the unphysical phases in the CP transformations of the lepton and Higgs doublets to unity) is given by, and on the elements of the PMNS matrix [51]: From the parametrisation of the Yukawa matrix of Eq. (2.10), this imposes the following conditions on the elements of the R-matrix [26]: The leptogenesis scenarios considered in this work have CP violation provided only by the phases of the PMNS matrix. This corresponds to imposing the condition of Eq. (2.16) onto the R-matrix but not the condition Eq. (2.15) on U . In these scenarios the values of the Dirac and Majorana phases of the PMNS matrix determine the success of leptogenesis. One should bear in mind, however, that there are certain intuitively unexpected possibilities for CP violation in (non-resonant) leptogenesis even when the PMNS-and R-matrices are CP -conserving, i.e., conditions (Eq. (2.15)) and (Eq. (2.16)) are individually fulfilled and the elements U lj and R jk are real or purely imaginary [26]. 3 CP violation due to the Dirac phase δ can only be practically investigated in neutrino oscillation experiments. There has been a slight statistical preference from the existing data for maximally CP -violating δ ∼ 270 • . This hint has been obtained from the combination of results from long-baseline experiments such as T2K [52] and NOνA [53] with reactor experiments like Daya-Bay [54], RENO [55] and Double-Chooz [56]. In principle, the difference in oscillation probabilities [57][58][59], J CP is the analogue of the Jarlskog invariant for the lepton sector, which gives a parametrisationindependent measure of CP violation in neutrino oscillations, L is the distance travelled by the neutrinos, E the neutrino energy and ∆m 2 ij ≡ m 2 i − m 2 j . In the case of CP -invariance 3 This unusual possibility is realised when ρ N i and ρ ν j are fixed by conditions (Eq. (2.14)) and (Eq. (2.15)), but the product of the so fixed values of ρ N i and ρ ν j differs from the value of ρ N i ρ ν j in (Eq. (2.16)) [26]. Under these conditions the low energy PMNS matrix U and the high-scale R-matrix are individually CPconserving, but the interplay between the two in leptogenesis is CP -violating.
we have δ = 0, π and therefore J CP = 0. By measuring, for example, A e,µ CP , one can determine J CP which has the following expression in the standard parametrisation of the PMNS matrix: J CP = 1 4 sin 2θ 12 sin 2θ 23 cos 2 θ 13 sin θ 13 sin δ . (2.21) The best-fit value and 1σ uncertainty of J CP reported in [44] are In the longer term, the next generation of neutrino oscillation experiments such as DUNE [61] and T2HK [62], will be able to measure the Dirac CP -violating phase δ with greater precision and determine whether CP -symmetry is indeed violated in the lepton sector. Information on CP -violating Majorana phases can, in principle, be obtained in neutrinoless double beta decay experiments [63][64][65] (see, however, also [66]). These experiments are the most sensitive probes of the possible Majorana nature of massive neutrinos. They can also provide information on the neutrino mass ordering [67] (see also [65]). The rate of neutrinoless double beta decay is given by (see, e.g., [68]) where G 01 is a kinematic factor and A denotes the amplitude which has the following form The amplitude is dependent on the nuclear matrix elements M 0νββ for which M 0νββ MeV (see, e.g., [68,69]), which shall always be the case in this work. The mixing elements V ei for the heavy states are O (m D /M ) and thus the second term of Eq.
, the second term is negligible in comparison with the first and we find [70] (see e.g., [51]): where m ν is the neutrinoless double beta decay effective Majorana mass in the case of 3neutrino mixing. In the case of CP -invariance we have α 21 = kπ, α 31 = qπ, k, q = 0, 1, 2, . . . [71][72][73]. 4 The most stringent upper bound on | m ν | was reported by the KamLAND-Zen collaboration [74] searching for neutrinoless double beta decay of 136 Xe: (reported by the GERDA-II collaboration), T 0ν 1/2 > 1.5 × 10 25 yr (from the combined results of the Cuoricino, CUORE-0, and CUORE experiments), and T 0ν 1/2 > 1.07 × 10 26 yr (from the KamLAND-Zen collaboration), with all limits given at the 90% CL. Most importantly, a large number of experiments of a new generation aim at sensitivities to | m ν | ∼ (0.01 ÷ 0.05) eV (see, e.g., [69,75] The GERDA-II and KamLAND-Zen experiments have already provided the best lower limits on the double beta decay half-lives of 76 Ge and 136 Xe. The experiments listed above aim to probe the ranges of predictions of | m ν | corresponding to neutrino mass spectra of quasi-degenerate type and with inverted ordering (see, e.g., [14]).
The primary focus of this work is to answer the question: at what scales can low-energy CP -violating phases produce the observed BAU? We shall show that the scale of successful leptogenesis in the case of interest may indeed vary across many orders of magnitude from 10 6 − 10 13 GeV. The observation of low-scale leptonic Dirac CP violation, in combination with the positive determination of the Majorana nature of the massive neutrinos, would make more plausible, but will not be a proof of, the existence of high-scale thermal leptogenesis. These remarkable discoveries would indicate, in particular, that thermal leptogenesis could produce the BAU with the requisite CP violation provided by the Dirac CP -violating phase in the neutrino mixing matrix.

CP -Conserving
The cancellation of large tree-level and large one-loop light neutrino mass matrices occurs as a result of relations between the magnitudes and phases of the R-matrix elements which lead to the following structure for the Dirac mass matrix: We may rewrite the tree and one-loop masses in terms of this relatively simple matrix m D √ f , such that where the commutativity of the diagonal matrices M and f has been exploited and This ensures that the sum of the tree-level and one-loop masses is (2.31) Due to the relative smallness of the elements of ∆, the matrix m ν may be considerably smaller than m tree . Immediately, we have which is an explicit expression of the fine-tuned cancellation.
As the R-matrix structure of Eq. (2.27) is required for successful leptogenesis at intermediate scales, we are tasked with the problem of finding the R-matrices which assume this form and obey the CP -invariance conditions of Eq. (2.16). We intend to translate the conditions in Eq. (2.27) and Eq. (2.16) into constraints on x i and y i . However, we know a priori from the work of [76] that one must have y 2 ∼ 0 • and y 1 180 • , y 3 180 • to produce the relative magnitudes of the elements of R in Eq. (2.27), crucial to the successful production of the observed baryon asymmetry.
We begin with the elements and which result from the expansion of the R-matrix parametrised as in Eq. (2.11). The condition of Eq. (2.27) that R 22 ≈ −R 31 implies that sin w 2 ≈ 1, which in turn imposes sin x 2 ≈ 1 and y 2 ≈ 0 • . In order to simplify future expressions, we promote the condition on y 2 to the exact equality y 2 = 0 • . With conditions on x 2 and y 2 determined, we now examine According to the condition Eq. (2.16), R 13 (like all the elements of R) must be purely real or imaginary and thus we should choose one of, cos x 1 = 0 or sin x 1 = 0. We exclude the possibility of y 1 = 0 for the reason given above. Likewise, consider and select cos x 3 = 0 or sin x 3 = 0 by the same reasoning.
In summary, we have the following set of constraints cos x 2 ≈ 0 and y 2 = 0, which lead to an R-matrix of purely real and imaginary components and are therefore good candidates for CP -invariant R-matrices. We shall make use of these conditions in considerations where enhancement of the R-matrix is necessary for successful leptogenesis.

The Effects of Flavour and Scale in Leptogenesis
In addition to explaining the smallness of neutrino masses, the type I seesaw provides a framework under which the matter-antimatter asymmetry of the Universe is explicable. The heavy Majorana neutrinos N i may undergo out-of-equilibrium, C-/CP -and leptonnumber-violating decays in the early Universe. The resulting leptonic matter-antimatter asymmetry is then partially converted into a baryonic asymmetry by SM sphaleron processes which violate B +L but conserve B −L. The baryon asymmetry, which quantifies the excess of matter over antimatter in the Universe, is defined by where n B , n B and n γ are the number densities of baryons, antibaryons and photons respectively. This quantity has been measured using two independent methods. There is the measurement of the baryon-to-photon ratio from Big-Bang nucleosynthesis (BBN), a process which occurs when the temperature of the Universe drops below T 1 MeV [77]: In complement, there is the determination of η B from Cosmic Microwave Background radiation (CMB) data [78] for which the relevant cosmological period is that of recombination, for which T 1 eV: η B CMB = (6.02 − 6.18) × 10 −10 .
Throughout our numerical study we apply the latter, more precisely measured value.
In the simplest scenario, thermal leptogenesis describes the time evolution of a lepton asymmetry as a result of the CP -violating decays of the heavy Majorana neutrinos. In these processes, the lepton and anti-lepton states are where, at tree-level, the projection coefficients are expressed as for i ∈ {1, 2, 3} and α ∈ {e, µ, τ }. The simplest scenario for leptogenesis is the single flavour regime, in which, the leptons resulting from the decay of each heavy Majorana neutrino N i are always found in the coherent superposition of flavours described by the corresponding |i . This condition is valid if flavour-dependent interactions mediated by the SM charged lepton Yukawa couplings are negligible. This is usually a sufficiently good approximation for temperatures T 10 12 GeV, when the charged lepton Yukawa interactions proceed at a slower rate than the expansion of the Universe. However, in this work, we shall demonstrate that such an approximation fails in certain regions of the model parameter space.
The single flavoured Boltzmann equations for thermal leptogenesis provide a semiclassical description of the time evolution of the heavy neutrino densities, n N i (i ∈ {1, 2, 3}), and the lepton asymmetry, n B−L . 5 Introducing the parameter z ≡ M 1 /T , which increases monotonically with time, these kinetic equations are written (2.41) In general, the decay parameter D i , describing the decay of N i is defined in terms of the heavy neutrino decay rate Γ i ≡ Γ i N i → φ † l i (with φ and l i the Higgs and lepton doublets), the CP -conjugate rate, Γ i , and Hubble rate, H [11]: Likewise, the washout factor is defined in terms of the heavy neutrino inverse decay rate Γ ID i and the CP -conjugate inverse decay rates Γ Finally, the CP -asymmetry parameter (i) is defined as 6 As discussed, the light neutrino masses may have an accidental cancellation in the treelevel mass that makes it comparable to the one-loop mass. However, there is no reason to expect an accidental cancellation like this to occur in the CP -asymmetry. This is because the cancellation in the light neutrino masses was due to the presence of terms of the form Y M −1 Y T while in the CP -asymmetries, the Yukawa matrix is multiplied by their conjugates in combinations of Y † Y and therefore a similar cancellation cannot occur. As such, the higher-order corrections to the CP -asymmetry should not be any more significant in our case than they usually are. Eq. (2.41) describes the time evolution of lepton number density asymmetry, n B−L , from an initial value (usually it is assumed that there is a vanishing initial abundance) to a final value, n B−L (z final ). The final leptonic asymmetry is then partially converted into a baryonic asymmetry through sphaleron processes. This is expressed quantitatively by the relation η B ≈ a/f × n B−L ≈ 10 −2 n B−L [11], where a = 28/79 describes the partial 5 All number densities are normalised to a volume containing a single heavy Majorana neutrino in ultra-relativistic thermal equilibrium. 6 Note that the Yukawas enter only in the combination Y † Y and hence, from the Eq.  conversion of the B − L asymmetry into a baryon asymmetry by sphaleron processes, and f ≡ n rec γ /n * γ = 2387/86 accounts for the dilution of the asymmetry due the change of photon densities (n γ ) between leptogenesis (n γ = n * γ ) and recombination (n γ = n rec γ ). If the era of leptogenesis is lowered below T ∼ 10 12 GeV, the interactions of the tau charged lepton, mediated by its SM Yukawa coupling, come into thermal equilibrium. The effect is that the τ -component of each |i experiences relatively rapid interactions with the early Universe plasma. The left-handed τ component is rapidly converted to a righthanded τ via scattering with the Higgs. Similarly, the reverse process repopulates the τ asymmetry density at the same rate. This rate is determined by the imaginary part of the thermal self-energy of τ , (Λ τ ), which by the optical theorem determines the mean free path of the τ state (see Fig. 1). When this process is sufficiently rapid, the coherent superposition of flavours is destroyed and the τ component can no longer contribute at the level of amplitudes to the decay and inverse decay processes with e and µ (which form a single coherent flavour state which we shall refer to as τ ⊥ such that τ |τ ⊥ = 0). Instead τ undergoes decay and inverse decay as a separate decoherent state. Correspondingly, the kinetic equations must separately describe the time evolution of n τ τ and n τ ⊥ τ ⊥ and the total baryon asymmetry is simply the sum: n B−L = n τ τ + n τ ⊥ τ ⊥ . This is the two-flavoured regime and the Boltzmann equations are given by where p iα ≡ |C iα | 2 , p iα ≡ |C iα | 2 are the projection probabilities expected from the decoherence -the classical measurement of τ by the early Universe plasma. Furthermore, the CP -asymmetries are for α = τ ⊥ , τ . Analogously, there exists the possibility that the out-of-equilibrium decays of the heavy Majorana neutrinos occur at temperatures where the SM muon Yukawa interactions has thermalised T ∼ 10 9 GeV and the three-flavoured Boltzmann equations are the relevant kinetic equations. This possibility was explored in [76] and it was shown that thermal leptogenesis can be lowered to T ∼ 10 6 GeV.
It has been shown [22,23,[79][80][81] that the density matrix equations produce a more physically accurate description of leptogenesis where the density matrix may be expressed as where |α are states of definite lepton flavour, defining the flavour basis. Using this description, the diagonal elements, n αα , are the differences of the normalised densities of α and α particles such that n B−L = Tr n. The off-diagonals describe the degree of coherence between the flavour states. The advantage of a density matrix description is that decoherence effects are easily incorporated and as such, the dynamical process by which different flavour states decohere is readily incorporated into the equations. This allows for a single set of equations with solutions which transition between flavour-regimes as appropriate. Furthermore, the equations should remain accurate even in the regions of transition where the interactions leading to decoherence are not infinitely fast.
Explicitly, the density matrix equations of leptogenesis are [22,23,[79][80][81] which generalise the notion of the projection probability and appear in the double commutator structure. The double-commutator structures in Eq. (2.49) give rise to an exponentially damping term proportional to (Λ α )/Hz for the equations describing the off-diagonal elements of n. In the flavour basis, if these terms are dominant, the density matrix is driven towards a diagonal form. The CP -asymmetry parameters are [21,22,79,81,82] (i) The diagonal components of the (i) matrix simplify to the following form 3 Leptogenesis in the regime 10 9 < M 1 (GeV) < 10 12 In this section, we explore the possibility that successful leptogenesis derives solely from the CP -violating PMNS phases and the mass scale is between 10 9 ≤ M 1 (GeV) ≤ 10 12 , which generally corresponds to the two-flavour regime. Historically, the link between lowenergy CP violation and the baryon asymmetry was first established in this regime and thus our main purpose in this section is to revisit the scenario with more robust numerical methods than have previously been applied. We shall perform a comprehensive exploration of the parameter space for a model with three heavy Majorana neutrinos in both the normal ordered and inverted ordered scenarios. We shall then investigate a subset of scenarios in which only the Dirac or only the Majorana phases are varied.

Results of Parameter Exploration
In this particular explorations of the parameter space, we fix M 1 and vary M 2 and M 3 such that M 3 > 3M 2 > 9M 1 , ensuring that resonant regimes are avoided [13,82,[84][85][86][87]. We choose to set M 1 = 10 10 GeV, this being typical of the mass window under consideration. We fix, x 1 = 90 • and x 3 = 180 • and y 2 = 0 • such that there is a complete leptonic CPsymmetry when δ = 0 • , α 21 = 180 • and α 31 = 0 • . 7 With the specified parameters fixed or constrained as stated, we explore the parameter space using a flat prior and log-likelihood function evaluated at a point p = (δ, α 21 , α 31 , m 1,3 , M 2 , M 3 ) (varying m 1 or m 3 for normal or inverted ordering respectively) by  Figure 2: The two-dimensional projections for leptogenesis with M 1 = 10 10 GeV and CP violation provided only by the phases of the PMNS matrix. The NO case is coloured blue/green and the IO one is orange/red. The contours correspond to 68% and 95% confidence levels. This plot was created using SuperPlot [83].
to define regions of 1σ and 2σ agreement with the observed value of the asymmetry. In addition we impose a bound on the sum of neutrino masses of 1 eV which is consistent with the tritium beta-decay experiments [88][89][90] but more conservative than recent constraints from Planck [78]. In the numerical work of this section we allow only for the two lightest heavy Majorana neutrinos to decay (an excellent approximation) and we neglect lepton numberchanging scattering processes, spectator effects [20,91], thermal corrections [92,93] and the inclusion of quantum statistical factors [94][95][96][97] which typically introduces an O (10%) error [27,[98][99][100]. To solve the density matrix equations we use the Python interface [101] to the LSODA algorithm [102] that is available in Scientific Python [103]. Due to the high-dimensionality of the parameter space we found the use of Multinest [104][105][106] (more explicitly, pyMultiNest [107], a wrapper around Multinest written in Python) particularly useful. In Fig. 8 we display the two-dimensional posterior probability plots for the CP -violating PMNS phases in the normal ordered and inverted ordered cases.
The results of this parameter search are shown in the form of two-dimensional projections in Fig. 19. For points in these regions of parameter space for which η B = η B CM B , the fine-tuning is F ≈ 0.23 which corresponds only to a very slight enhancement of the 228 447 570 2.82 × 10 10 1.00 × 10 13 3.16 × 10 13 90 18 180 0  For the case of an NO light neutrino mass spectrum, we find that the observed baryon asymmetry may be obtained to within 1σ (2σ) with δ between [95,265] . Both of these scenarios comfortably incorporate the measured bounds on δ (Table 1). In what follows, we provide some explanation of these results and plots by introducing an analytical approximation which we use to study the scenarios where only the Dirac or only the Majorana phases provide CP violation.

Dependence of η B on the Dirac and Majorana Phases
In the scenario 10 9 < M 1 (GeV) < 10 12 , it is appropriate to apply the two-flavour Boltzmann equations (Eq. (2.46)). These equations have the following analytical solution [108] where it is assumed that the dominant contribution to the final asymmetry is from the lightest of the heavy Majorana neutrinos and that leptogenesis occurs in the strong washout regime. We denote the z for which the washout becomes less than one as z d , W (z d ) < 1, is the decay parameter for N 1 , and the z for which leptogenesis is initiated as z 0 . As we are interested in those scenarios in which CP violation derives only from the phases of the PMNS matrix, we have the supplementary condition Tr (1) = 0 (or τ ⊥ τ ⊥ ) which we may use to simplify the solution to   At the benchmark point for normal ordering defined in Table 2, which we will use in the further analyses in the present section, we have:  for CP violation from δ, α 21 and α 31 respectively. For the case in which δ provides the CP violation in Eq. (3.5), this phase gives a subdominant contribution to |Y τ 1 |. As can be shown, P 0(1) τ τ is similarly weakly dependent on the phases. Thus, the phase dependence of the solutions of Eq. (3.3) does not come predominantly from the flavour factor ∆F but from the CP -asymmetry (1) τ τ . However, in the case of α 21 providing the CP violation, the two terms of Eq. (3.5) are similar in magnitude and we may get a strong enhancement in ∆F . The final case where α 31 provides the CP violation is intermediate and should experience a slight phase-dependent enhancement from ∆F .
(3.6) Thus, given the approximate phase-independence of ∆F , we obtain a sinusoidal dependence of η B on δ, with η B = 0 when δ = 0 • or 180 • . Keeping all other parameters fixed, we find that for M 1 = 2.82 × 10 10 GeV no value of δ can produce the observed baryon asymmetry of the Universe, the maximum value of η B as a function of δ is 4.07 × 10 −11 . We might scale the heavy Majorana neutrino masses by a constant value, as when the two-flavour approximation of Eq. (3.3) is valid, the factor (1) τ τ scales in proportion with this constant and thus so does η B . In doing so, we find that the final asymmetry rises until M 1 = 7.08 × 10 11 GeV, where η B takes maximum value 4.01 × 10 −10 . After this, the simple scaling fails as one begins to enter the transition to what is usually the single-flavour regime.
Performing a detailed numerical parameter exploration we find that purely Dirac phase CP violation leads to successful leptogenesis for M 1 = 5.13 × 10 10 GeV, M 2 = 2.19 × 10 12 GeV and M 3 = 1.01 × 10 13 GeV. This is illustrated in Fig. 3 in which the plotted η B comes from solving the full density matrix equations. In this case, we have: (3.7) Given the different order of magnitude of the two terms in the expression for Y τ 1 , the baryon asymmetry should exhibit dependence on δ only from (1) τ τ and not from ∆F . Our theoretical expectations are borne out by the approximate sinusoidal dependence of η B on δ seen in Fig. 3.

CP Violation from the Majorana Phase α 21
Here, we set δ = α 31 = 0 • but allow CP violation from α 21 . Setting all other parameters to their benchmark values we find (1) τ τ = 3.14 × 10 −7 cos α 21 2 . (3.8) It follows from this expression for (1) τ τ that at the CP -conserving values for α 21 = 0 • , 360 • we have (1) τ τ = 0 (see also Fig. 4). This corresponds to the case of CP -conserving R-matrix, CP -conserving PMNS matrix, but CP -violating interplay between the R and PMNS matrix elements in leptogenesis [25]. In a similar way to the previous subsection, we find that no value of α 21 can achieve successful leptogenesis using this combination of phases and the benchmark values from Table 2. Thus, we find it necessary to scale all of the heavy Majorana neutrino masses by a common factor such that M 1 = 3.05 × 10 10 GeV, may allow for successful leptogenesis. With this scaling we obtain the results plotted in Fig. 4. The deviation from pure (co)sinusoidal behaviour is explained by the α 21 -dependence of ∆F . For α 21 < 360 • , ∆F varies relatively slowly exhibiting a global minimum at α 21 = 180 • , resulting in a slightly modified sinusoidal dependence through this point in η B . A strong peak exists for ∆F around α 21 = 540 • , which results in the peak of η B occurring before 720 • , as would be expected from the dependence of (1) τ τ . The small sign-changing fluctuation around the zero at α 21 = 540 • is a feature that does not appear in the solution of twoflavour Boltzmann equations and thus cannot be explained in terms of the analytic solution Eq. (3.3). However, the extra zeros of η B that are seen in Fig. 4 are due only to accidental cancellations and do not correspond to cases of CP -symmetry (unlike those at α 21 = 180 • and α 21 = 540 • ). Here it is assumed that the light neutrino mass spectrum has normal ordering. Contours correspond to 68% and 95% confidence levels. This plot was created using SuperPlot [83].

CP Violation from the Majorana Phase α 31
We set δ = 0 • and α 21 = 180 • such that CP violation is provided by α 31 . Using the benchmark values for the other parameters from Table 2 we find: Again we find that without scaling the heavy Majorana neutrino masses, no value of α 31 corresponds to successful leptogenesis. At M 1 = 5.13 × 10 10 GeV we obtain the first point for which the observed baryon asymmetry is created and this is plotted in Fig. 5. We see that analytical expectation of a sinusoidal dependence of the baryon asymmetry (η B ∝ τ τ ∝ sin(α 31 /2)) from Eq. (3.9) is present. ∆F exhibits a broad peak around α 31 = 360 • which results in the slight shift to the centre of the otherwise sinusoidal peaks.

The Case of N 3 Decoupled
In this section, we review the case that the heaviest Majorana neutrino, N 3 , physically decouples. We restrict ourselves to normal ordered light neutrino masses. The resultant scenario with two relevant heavy Majorana neutrinos is the simplest (minimal) type I framework compatible with all neutrino data. In this scenario only two of the light neutrinos have non-zero masses since m 1 = 0. For normal ordering, the R-matrix may be parametrised as [109][110][111] The resulting neutrino Yukawa matrix thus has Y α3 = 0, consistent with the premise that N 3 has decoupled. We choose to take θ in Eq. (3.10) to be real in order to have the condition 228 516 100 10 11 10 12 25.05 Table 3: A benchmark point for leptogenesis with M 1 = 10 11 GeV and N 3 decoupled with a normal ordered light mass spectrum.
of Eq. (2.16) satisfied. We assume further that at least one of the three phases in the PMNS matrix has a CP -violating value.
As with the previous sections, we have performed an exhaustive exploration of the parameter space where again we are primarily concerned with the situation in which CP violation is provided only by the PMNS phases. We choose to fix M 1 = 10 11 GeV and M 2 = 10 12 GeV such that the parameter space to explore is described by p = (δ, α 21 , α 31 , θ). In Fig. 6, we present the two-dimensional posterior projection for the case of normal ordering. Here, it is seen that with M 1 = 10 11 GeV, for normal ordering, successful leptogenesis may produce a baryon asymmetry with 1σ (2σ) agreement with the observed value for δ ∈ [95, 315] • , (δ ∈ [25, 360] • ).
In Table 3, we provide a benchmark point for normal ordered leptogenesis, with purely low-energy CP violation and N 3 decoupled. At this point, the observed BAU is produced with a corresponding fine-tuning of F = 0.23. In Fig. 7, we illustrate a similar scenario, in which the CP violation is provided only by δ (α 21 = 180 • , α 31 = 0 • ), and where the observed baryon asymmetry is produced near δ = 270 • . We conclude that, even for the minimal type I seesaw scenario with two heavy Majorana neutrinos exhibiting hierarchical mass spectrum, it is possible to generate the observed value of the baryon asymmetry with the requisite CP violation provided exclusively by the Dirac phase δ, and/or by the Majorana phase α 21 or α 31 .
Furthermore, we note that, in performing a similar investigation for the inverted ordering scenario, we find no point in the parameter space which corresponds to successful leptogenesis with N 3 decoupled in this mass window 8 with real R-matrix. If, however, e.g., R 11 R 12 = ±i|R 11 R 12 | (R 13 = 0 in the case of interest), we can have successful leptogenesis with the CP violation provided by the Dirac and/or Majorana phases in PMNS matrix also for the IO spectrum. These conclusions are in agreement with the results of [25] wherein one may find a detailed discussion of the cases considered in the present subsection.
Finally, in [25] the following necessary condition for successful leptogenesis in the case of NO spectrum with the requisite CP violation provided exclusively by the Dirac phase δ was obtained: | sin θ 13 sin δ| 0.09 . We recall that this condition was derived by using values of the the CP -conserving Rmatrix elements maximising the lepton asymmetry and assuming that the transition from two-flavour to one-flavour regime starts at T ∼ = 5 × 10 11 GeV, i.e., that at M 1 5 × 10 11 GeV the two-flavour regime is fully effective.
4 Leptogenesis in the regime M 1 < 10 9 GeV Successful thermal leptogenesis at intermediate scales may be accomplished through the combination of flavour effects and fine-tuned Yukawa matrices with F O(10) [76]. In Section 2.3, we first review these fine-tuned scenarios and then proceed to determine the subset among them in which the R-matrix is CP -conserving while the PMNS matrix contains CP -violating phases. In Section 4.1 we present and analyse the results of a comprehensive search of the model parameter space for regions with successful leptogenesis compatible with these subsets where we have numerically solved the density matrix equations, for two-decaying heavy Majorana neutrinos, exactly. Following this, we consider in detail the scenarios in which CP violation is due solely to the Dirac phase in Section 4.2.1, or due only to the Majorana phases in Section 4.2.2 and Section 4.2.3. In Appendix B, we display results for M 1 = 10 9 GeV, where O(10) fine-tuning is also required.
We present an analytic approximation of the baryon asymmetry to find that the detailed dependence of the baryon asymmetry on the low energy phases may be roughly explained by the features of Y τ 1 and Y µ1 . We reiterate that we apply these approximation simply to illustrate the qualitative behaviour of the solutions but we numerically solve the density matrix to produce all plots in this paper.

Results of Parameter Exploration
The options of Eq. (2.37) are satisfied by sixteen distinct R-matrices which may be divided into four classes according to the corresponding parity vectors ρ ν , ρ N (see Appendix A for definitions and further details). All such matrices are identical except for the placement of factors ±1 or ±i. The phenomenological implications of each will be qualitatively similar except for the precise positions in parameter space that certain features occur. As we are primarily concerned with demonstrating the viability of leptogenesis with the O(100) finetuned Yukawa matrices (of the type in Eq. (2.28)) then we shall focus our numerical efforts on just one possible R-matrix of the set of sixteen. Namely, we choose a scenario corresponding to cos x 1 = 0, cos x 3 = −1 such that ρ ν = ±(+1, −1, +1) T , ρ N = ±(+1, +1, −1) T . For the numerical analysis, we follow the same procedure outlined in Section 3.1 with one additional constraint. At values of F 1000, higher-order corrections to the light neutrino mass become important. For this reason we fix y 1 = y 3 = 180 • and thereby avoid these problematic regions of the model parameter space.
In the parameter searches of this section, we consider two cases, in one we fix M 1 ∼ 10 6 The normal ordered case is coloured blue/green and inverted ordering orange/red and contours correspond to 68% and 95% confidence levels. This plot was created using SuperPlot [83].
GeV (Fig. 8) and in the other we fix M 1 ∼ 10 8 GeV (Fig. 9). We note that M 2 = 3.5M 1 and M 3 = 3.5M 2 and m 1 = 0.21 eV. 9 This allows for a comparison of the effects of two different degrees of fine-tuning, with the former corresponding usually to F ∼ 500. This is close to the maximum fine-tuning (and correspondingly, the smallest non-resonant leptogenesis scale) for which second-order radiative corrections to the mass can be ignored [76].

Dependence of η B on Dirac and Majorana Phases
In this section, we use the benchmark point given in Table 4, in order to analytically study leptogenesis from low-energy CP violation in the case that the lightest heavy Majorana neutrino has mass M 1 < 10 9 GeV, such that a relatively high degree of fine-tuning in the light neutrino masses is required. We choose this benchmark point as it allows us to accurately neglect the contributions from decays of the other heavy Majorana neutrinos and thus simplify the analysis. With M 1 < 10 9 GeV, leptogenesis occurs in the three-flavour regime for which the threeflavoured Boltzmann equations are a good approximation to the density matrix equations and have approximate analytical solution [108]: where we take into account the decays of only the lightest heavy Majorana neutrino. As we are most interested in the scenarios in which CP violation is due to PMNS phases only, i.e. Tr (1) = 0, then we can re-express Eq. (4.1) as where the asymmetry depends on the low-energy phases via (1) τ τ and (1) µµ and from the difference of the inverse flavour projections: However, for the case of Table 4, the two and three-flavour regime Boltzmann equations, to a high degree of accuracy give the same value of η B . Given the comparative simplicity of the two-flavour solution Eq. (3.3), we choose to use this for the practical purpose of simplifying the analysis. For the benchmark parameter values listed in Table 4, we may find analytical approximations for the CP -asymmetries (1) αα . Under the relatively good approximation, that m 1 = m 2 10 , the asymmetry is given by (4.4) Selecting x 1 = (2k 1 + 1)π/2 and x 3 = k 3 π for k 1 , k 3 ∈ Z, such that cos x 1 = 0 and | cos x 3 | = 1 and cos x 3 = (−1) k 3 is satisfied, we find the CP -asymmetry (1) αα to be 5) where, at our benchmark point, the coefficient of [U * α3 (U α1 + iU α2 )] has magnitude approximately equal to 3.7 × 10 −6 . This form is particularly useful in order to isolate the dependence of the CP -asymmetry on the PMNS phases in the factor [U * α3 (U α1 + iU α2 )].

CP Violation from the Majorana Phase α 21
Alternatively, consider the case of CP violation from α 21 , where δ = 0 • , α 31 = 0 • and all other parameters are set to their benchmark values of Table 4. The variation of η B with α 21 in this scenario is plotted on the left of Fig. 11. On the right of the same figure, we parametrically plot η B against | m ν | with parameter α 21 . The baryon asymmetry η B vanishes at the CP -conserving values of α 21 = 180 • and 540 • . However, as is seen in Fig. 11, η B = 0 at the CP -conserving values of α 21 = 0 • , 360 • and 720 • since at these values the interplay between the CP -conserving R and PMNS matrices leads to CP violation in leptogenesis [25]. The efficiency function ∆F , when plotted as a function of α 21 , exhibits a very strong narrow peak at α 21 = 180 • and a much less pronounced peak at α 21 = 540 • . As a consequence, the corresponding η B is modified from the simple cosine curve expected from the dependence of Thus, there is a sharp transition around α 21 = 180 • . We can conclude then that the strong peak in ∆F is what has allowed the observed baryon asymmetry of the Universe to be reproduced. This peak originates in an accidental cancellation of terms in the function P 0(1) τ τ . There is no a priori reason to expect such a cancellation and it should be understood as a feature of the fine-tuned solutions that are being here studied. Thus, we see that the flavour-effects introduce a pair of accidental zeros of η B , one in the range [180, 540] • and the other in [0, 180] • .

CP Violation from the Majorana Phase α 31
Finally, consider the case of CP violation from α 31 where δ = 0 • , α 21 = 180 • and for which the baryon asymmetry is plotted in the left plot of Fig. 12 and on the right we show the parametric dependence of the effective neutrino mass | m ν | with α 31 against that of η B . The baryon asymmetry η B vanishes at the CP -conserving values of α 31 = 0 • , 360 • and 720 • .
The efficiency function ∆F in this case is qualitatively similar to that for the case of δ: CP violation only strongly peaks at α 31 close to 0 • and to 720 • . Thus, we do not observe, in the left plot of Fig. 12, the simple dependence, η B ∝ sin (α 31 /2) as may be expected from the expression for τ τ , (4.8) Rather, we find an enhanced positive peak near α 31 = 0 • and an enhanced negative peak near α 31 = 720 • .

Summary of Fine-Tuned Solutions with High Energy CP -Symmetry
The fine-tuned solutions we have discussed in this section share the property that they enhance η B through the production of a peak in the efficiency factor ∆F . In each projection the PMNS matrix cancels from the denominator such that all phase-dependence comes from that of |Y α1 | 2 in the numerator. In this subsection, we may safely use the usual Casas-Ibarra parametrisation (obtained by the replacement of f with √ M −1 in Eq. (2.10)), to obtain The absolute value |Y α1 | is extremised when each term in the parentheses in Eq. (4.10) has a common complex phase or when terms may differ in complex phase by π. This occurs at CP -conserving values of the PMNS phases and so the enhancement expected in the functions ∆F is likely to occur at CP -conserving phases also. As an example, around the benchmark point of Table 4, we find that with only α 21 contributing to CP violation The absolute value of this function has extrema when α 21 = 180 • or α 21 = 540 • -the CPconserving values. Moreover, the cancellation that occurs at α 21 = 180 • is strong because of the similarity in magnitude of the two terms in Eq. (4.11). As a result of this there are strong peaks in ∆F which enhance η B . This is a way in which the solutions are found to be fine-tuned as there is no reason to expect these two terms to be so similar in size. At these same points, the asymmetries (1) αα are vanishing as CP is a symmetry in the leptonic sector. Thus η B , being proportional to the product of (1) τ τ and ∆F , is strongly enhanced on either side of the CP -invariant points (for instance, around α 21 = 180 • in the left plot of Fig. 11). Thus the fine-tuned solutions tend to achieve large η B of one-sign on one side of a CP -invariant point and large η B of the opposite sign on the other side. Similarly, this effect persists when all phases may contribute together to CP violation ( Fig. 8 and Fig. 9). Thus, successful leptogenesis tends to occur near α 21 ∼ 180 • , α 31 ∼ 0 • , 720 • when leptogenesis is achieved with fine-tuned light neutrino masses, as it is at intermediate scales (M 1 10 9 GeV). Note that although we made these arguments based on the two-flavoured Boltzmann equations, very similar conclusions are reached based on considerations of ∆F τ e and ∆F µe for the solutions of the three-flavoured Boltzmann equations. For this reason, one expects similar behaviour to hold even for lower values M 1 such as in Fig. 8.
Furthermore, one may usually argue that Dirac-phase leptogenesis suffers a suppression not present in Majorana phase leptogenesis due to the factors of s 13 that appear in the CPasymmetries as shown in Eq. (4.6). However, for Dirac-phase leptogenesis η B ∝ sin δ∆F (δ), where the maximum absolute value of ∆F (δ) is ∼ 408, whereas for α 21 leptogenesis, η B ∝ cos α 21 2 ∆F (α 21 ) with maximum absolute value ∼ 77. Thus, what is more relevant when leptogenesis occurs intermediate scales, is the degree of enhancement from ∆F that occurs due to fine-tuning.
Finally, as we observe in Fig. 9, the contours for α 21 , α 31 show a strong dependence on α 31 − α 21 . A rough explanation of this is given by the dependence of (1) τ τ on the Majorana phases. With δ fixed at its benchmark value, but α 21 and α 31 free to vary, this CP-asymmetry is given by In previous studies in which a connection between low-energy CP violation (CP -conserving R) and leptogenesis was established [26], the scale of leptogenesis was limited to M 1 ≤ 5 × 10 11 GeV. This allowed for the use of the two-flavour Boltzmann equations as shown in Eq. (2.46) where the CP -asymmetries τ ⊥ τ ⊥ appear separately. The expectation had been that for M 1 10 12 GeV, the single-flavour Boltzmann equation Eq. (2.41) would be appropriate. In this equation, the CP -asymmetries appear only in the factor Tr (1) = 0 and hence no baryon asymmetry may be produced. In Section 5.1 we argue that even at high scales M 1 10 12 GeV, if R is CP -conserving, then flavour effects are significant and that the density matrix equations do not reduce to the single flavour Boltzmann equations. Hence we conclude that viable leptogenesis may result from low energy CP violation. Finally, in Section 5.2 we proceed to numerically analyse this possibility in detail. 2σ region 1σ re gi on Figure 13: The two-dimensional projections for high-scale leptogenesis with M 1 = 10 13 GeV with CP violation provided only by the phases of the PMNS matrix. The NO case is coloured blue/green and the IO one is orange/red. The contours correspond to 68% and 95% confidence levels. This plot was created using SuperPlot [83].

Flavour Effects with M 1 10 12 GeV and High Energy CP -Symmetry
In Appendix C, we demonstrate that the complete formal solution of the density matrix equations (Eq. (2.49)), with one decaying heavy Majorana neutrino is In a typical leptogenesis scenario, if M 1 10 12 GeV, flavour effects are negligible and one obtains the well-known result: which may be found by solving the single flavour Boltzmann equation. However, with a CP -conserving R-matrix, such that CP violation is provided solely by the PMNS phases,  Table 5: The benchmark values for high-scale leptogenesis with normal ordering. Here we have m 1 = 0.0159 eV and y 1 = y 2 = y 3 = 0 • . one has Tr (1) = 0 and so the λ term in Eq. (5.1) becomes the dominant one: If this is the case, then the baryon asymmetry is produced purely through flavour-effects from (Λ τ )/Hz. The physical effect of Tr (1) = 0 is that opposite asymmetries are produced in the τ and τ ⊥ flavours due to the decay of N 1 : τ ⊥ τ ⊥ . However, with flavour effects, the lepton asymmetries (1) τ τ and (1) τ ⊥ τ ⊥ produced in the decay experience differing washouts such that n τ τ = −n τ ⊥ τ ⊥ and η B = n τ τ + n τ ⊥ τ ⊥ = 0. It is an asymmetry produced by this method that is described in Eq. (5.4). The obvious question at this point is whether this can ever be large enough to produce the observed baryon asymmetry when M 1 10 12 GeV. The density matrix equations may be conveniently expressed in terms of the vectors as where In terms of these quantities, the formal solution, with flavour effects neglected is: Although flavour effects in high-scale leptogenesis may be negligible, this solution may not be accurately applied for finding η B in the case that Tr (1) = 0. This is because there is a strong cancellation of components of the density matrix when computing η B = n τ ⊥ τ ⊥ +n τ τ , such that the errors made in neglecting flavour effects are dominant. For this reason, we make use of it only for finding the approximate behaviour of individual components of the density matrix and avoid applying it to situations where this cancellation occurs.
If the heavy Majorana neutrino masses M i are scaled by a common factor x, such that M i → xM i , then: (1) scales in proportion with x, D 1 and W 1 do not scale with x and (Λ τ ) /Hz varies inversely with x. Consequently, according to Eq. (5.9), n αβ (z) scales in proportion to x, (with increasing precision for larger x since we can better neglect the thermal widths). In λ the scaling of (Λ τ )/Hz cancels that of n τ τ ⊥ and so λ does not scale with x if M 1 10 12 GeV. Thus, at sufficiently large values of M 1 , η B , given by Eq. (5.4), asymptotically approaches a non-zero constant. This is shown in Fig. 14 (d) over a range of M 1 values in which the ratios M 1 /M 2 and M 2 /M 3 are fixed. The curve increases ever more slowly for larger M 1 as the approximation leading to Eq. (5.9) becomes ever more precise. This may be interpreted as the transition region between the two flavour regime and the single flavour having grown infinitely large. 12 In each of the plots of Fig. 14, we see a dip in the density matrix solution curve near 10 12 GeV. This feature is due to the difference in sign of the two-flavour solutions, for which η B ∝ In Appendix D, we discuss the robustness of the plateau that forms for large heavy Majorana neutrino masses when the effects of scattering and when a more realistic treatment of the right-handed taus are incorporated. In the next section we explore the parameter space of the three heavy Majorana neutrino type I seesaw with regard to the solutions of Eq. (D.1) with CP -conserving R-matrix and M 1 10 12 GeV.

Results of Parameter Exploration
At values of M 1 10 12 GeV, fine-tuning through large elements of the R-matrix is not required for successful leptogenesis. Thus, in this section we analyse the parameter space corresponding to real, and therefore CP -conserving, R-matrices (y i = 0 • ), using the same numerical technique as described in Section 4.1. In order to perform this analysis we fix M 1 = 10 13 GeV and again require M 3 > 3M 2 > 9M 1 in order to avoid the resonant regime. With a much higher value of M 1 , one would need a correspondingly a higher temperature of inflation. For this reason, we choose to illustrate the possibility of successful thermal leptogenesis at just one order of magnitude beyond the two-flavour to single-flavour transition temperature of 10 12 GeV. In Fig. 13, we display the two-dimensional projection plots for both normal ordering and inverted ordering.
In the NO case, we find that the observed baryon asymmetry may be obtained to within 1σ (2σ) with δ between [240, 331] • ([0, 360] • ). In the IO one, the 1σ (2σ) range is [50,304] • ( [20,352] • ). In what follows, we analyse these results by considering separately the cases of purely Dirac or purely Majorana CP violation. For brevity we consider only the case of NO spectrum.

Dependence of η B on the Dirac and Majorana Phases
As the final value of the baryon asymmetry becomes approximately constant for M 1 10 12 GeV (see Fig. 14) with a CP -conserving R-matrix, then one can can use the value of η B that is predicted by the two-flavour Boltzmann equations (2FBE) at the start of the transition M 1 ∼ 10 12 GeV as a proxy for the full solution of the density matrix equations (DME). That is,  11) in order to gain an analytical understanding of the numerical solutions.
As in the analysis of Section 4.1, we investigate the cases where CP violation comes from precisely one of δ, α 21 or α 31 . Unlike the fine-tuned scenario previously considered, P 0(1) τ τ and consequently ∆F are approximately constant with the PMNS phases as would be expected from our discussion of the fine-tuned solutions in Section 4.2.4. Hence the phase-dependence of the η B can be understood by reference to (1) τ τ alone. This may also be understood by reference to the Yukawa couplings when CP violation comes only from δ, α 21 or α 31 respectively: (5.12) The difference in scale of the two terms means that the cancellation is never strong for any value of the phase and so the peaks in ∆F are not large. In this high scale case, ∆F is approximately constant and thus the plots of η B exhibit a nearly pure sinusoidal variation given by the CP -asymmetries below.

Dirac Phase CP violation
In this case, with a real R, we have α 21 = α 31 = 0 • such that δ is the sole provider of all CP violation. The asymmetry is given by  Table 5, with α 21 = α 31 = 0 • , allowing CP violation only from δ, the largest η B achieved is a factor ∼ 9 smaller than the observed value. This is large enough that even enormously larger scales of the heavy masses cannot make δ-only leptogenesis a viable option. An alternative for producing the observed baryon asymmetry of the Universe with CP violation only from δ is to work with an R-matrix containing both zero and purely imaginary components which are CP -conserving and may potentially be large in magnitude. If for example, we choose x i = 0 • such that all w i are either purely imaginary or zero, and take y 2 = 0 • also, then by setting α 21 = 180 • and α 31 = 0 • , all CP violation will be due to δ. Varying y 1 and y 2 together in this setup, we find that y 1 = y 2 = 169 • is the smallest value for which the observed baryon asymmetry of the Universe is produced. With all other parameters equal to the values in Table 5, this corresponds to F = 105. Hence a noticeable degree of fine-tuning is required even at high scales to make δ the sole contributor to CP violation with viable leptogenesis. In Fig. 15, we plot the variation of η B with pure δ CP violation for this fine-tuned scenario in the left plot, and on the right we parametrically plot η B against J CP as a function of δ.
The corresponding η B , plotted in the left plot of Fig. 16 is thus a factor of O(10) higher and of opposite sign than in the previous case without fine-tuning. Thus, we obtain the observed baryon asymmetry of the Universe (or higher) for values of α 21 between about 450 • and 650 • . In the right plot of Fig. 16 is η B for the same scenario parametrically plotted against the effective neutrino mass with parameter α 21 .

CP Violation from the Majorana Phase α 31
Finally, we turn to the scenario in which CP violation is provided entirely by α 31 , plotted on the left in of Fig. 17 for which (1) τ τ = −3.22 × 10 −5 sin Similarly to the case discussed in the preceding subsection, we see that (1) τ τ = 0 at the CP -conserving values of α 31 = 180 • , 540 • . This again corresponds to the case of CP violation in leptogenesis due to the interplay of the CP -conserving R-matrix (y i = 0) and CP -conserving PMNS matrix [25].
Compared with the previous scenario, there is a sign flip and an enhancement by a factor ∼ 1.6 of the resulting baryon asymmetry of the Universe. Thus, the observed BAU is achieved and exceeded for smaller values of α 31 , between about 50 • and 300 • . On the right of Fig. 17, we display a parametric plot for the same scenario with η B against the effective neutrino Majorana mass with the parameter α 31 .

Conclusions
In this paper we have investigated the connection between leptogenesis and low energy leptonic CP violation over a large range of scales 10 6 < M 1 (GeV) < 10 13 . We summarise our main findings below: • Firstly, we revisited the question of the possibility of successful thermal leptogenesis at scales 10 9 < M 1 (GeV) < 10 12 . At such scales, tau-Yukawa interactions are in equilibrium, such that there are sufficiently frequent interactions between the leptons and the early Universe plasma causing decoherence between the tau flavour from the other flavour components. We show that successful leptogenesis is indeed possible in this range of scales in the case that the PMNS phases provide all of the CP violation in the model. By performing parameter explorations at M 1 = 10 9 GeV and M 1 = 10 10 GeV, we found that some degree of fine-tuning, F ∼ 10, is required for these particular mass scales (with the degree of fine-tuning diminishing as one goes to higher values of M 1 ).
• By demanding pure Dirac phase or pure Majorana phase CP violation, we found that each phase alone can produce the correct CP -asymmetry, with the cases of Majorana phases requiring, in general, a somewhat lower value of M 1 than those required for the Dirac phase.
• If leptogenesis takes place at scales M 1 10 9 GeV, then all three of the leptonic flavour components involved in leptogenesis will decohere. For masses in this range (M 1 ∼ 10 6 GeV and M 1 ∼ 10 8 GeV), we determined the regions of parameter space in which low energy leptonic CP violation, provided by either the Dirac or the Majorana phases individually, leads to successful intermediate scale leptogenesis. At these scales a large amount of fine-tuning (F ∼ O (100)) is required between the tree-level and oneloop neutrino masses. We restricted ourselves to fine-tuning such that F < 1000 and in doing so have found an approximate lower bound of M 1 ≈ 3 × 10 6 GeV (consistent with the conclusions of [76]).
• We studied the possibility of pure Dirac phase CP violation and showed that M 1 8 × 10 6 GeV in order to produce the observed baryon asymmetry. Similarly for the purely Majorana phase CP violation, for α 21 , we obtain a bound M 1 ≈ 4.5 × 10 6 GeV whereas for α 31 we obtained M 1 ≈ 3 × 10 6 GeV. Observables depending on the Dirac and Majorana phases, for example J CP or m ν , may be well within experimental bounds in the same parts of parameter space in which leptogenesis is successful. The Dirac phase δ is only very weakly constrained, with the tightest constraint being δ ∈ [16,263] • , for 1σ agreement with the observed BAU, which comes from assuming normal ordering and M 1 = 1.29 × 10 8 GeV.
• If leptogenesis takes place at high scales, with M 1 10 12 GeV, interactions between the leptons and the early Universe plasma only very weakly decohere the tau flavour from the other flavour components. Normally, this leads to the conclusion that the single flavour Boltzmann equations are an appropriate description of the process. However, we have demonstrated that, if CP violation arises only in the low energy leptonic sector, the effects of decoherence cannot be neglected. Therefore, one should not ignore the phenomenology of high-scale leptogenesis with purely low-energy CP violation.
• We explored the parameter space at M 1 ∼ 10 13 GeV, finding regions in which thermal leptogenesis is a viable explanation of the BAU. We found that the strongest constraint on δ is for normal ordering, for which we require δ ∈ [240, 331] • to produce a baryon asymmetry within 1σ of the observed value. With only Dirac phase CP violation, we have concluded that it is not possible produce the observed baryon asymmetry of the Universe unless one introduces significant fine-tuning in the light neutrino masses. We argued that there is no scale of the heavy Majorana neutrino masses beyond M 1 10 12 GeV for which Dirac phase leptogenesis may be made to work without this fine-tuning. However, with pure Majorana phase violation, we found that successful leptogenesis is possible with essentially no fine-tuning.
The results of this article underscore the significance of understanding leptonic CP violation through experimental searches for Dirac and/or Majorana leptonic CP violation. We have departed from previous literature by concluding that low energy leptonic CPviolating phases may always be relevant to the production of the baryon asymmetry in the leptogenesis scenario. It has commonly been thought that their relevance was limited to the window of masses 10 9 M 1 (GeV) 10 12 . However, we have shown this window to be significantly wider: Dirac and Majorana phases may be crucial to leptogenesis even at scales as low as M 1 ∼ 10 6 GeV, or as high as M 1 10 12 GeV.  Figure 19: The two-dimensional projections for leptogenesis with M 1 = 1.00 × 10 9 GeV with CP violation provided only by the phases of the PMNS matrix. The normal ordered case is coloured blue/green and inverted ordering orange/red and contours correspond to 68% and 95% confidence levels. This plot was created using SuperPlot [83].
In Fig. 19 we present results for M 1 = 10 9 GeV. We find that a fine-tuning of the light neutrino masses F ≈ 14 at the best-fit points. In the normal ordered case, we find that the observed baryon asymmetry may be obtained to within 1σ (2σ) with δ between [0, 360] • ([0, 360] • ). While for inverted ordering, the 1σ (2σ) range is [25,360] • ([0, 360] • ). This is significantly higher than the case for which M 1 = 10 10 GeV where the fine-tuning is considerably less at F ≈ 0.23. In the normal ordered case, we find that the observed baryon asymmetry may be obtained to within 1σ (2σ) with δ between [95,265]

C Single Flavour Boltzmann Equations from Density Matrix Equations
In this appendix we find the conditions under which the density matrix equations (Eq. (2.49)) approximate to the single flavour Boltzmann equations. We begin by analysing the criteria under which the single flavour Boltzmann equation emerges as an approximation from the density matrix equations, which, written in the (τ ⊥ , τ )-basis are τ ⊥ τ D 1 n N 1 − n eq N 1 − 1 2 W 1 (n τ ⊥ τ + C * 1τ C 1τ ⊥ (n τ ⊥ τ ⊥ + n τ τ )) − Im (Λ τ ) Hz n τ ⊥ τ .
(C.2) As n B−L = n τ τ + n τ ⊥ τ ⊥ , we find an equation for the evolution of n B−L by adding the second and third equations together, obtaining dn B−L dz = D 1 n N 1 − n eq N 1 Tr (1) − W 1 |C 1τ | 2 n τ τ + |C 1τ ⊥ | 2 n τ ⊥ τ ⊥ + 2 [C 1τ ⊥ C * 1τ n τ τ ⊥ ] . (C.3) If this were to reproduce the single flavour limit, then we should find that the coefficient of is equal to n B−L in the limit that Im(Λ τ )/Hz is small. Recalling that |C 1τ ⊥ | 2 + |C 1τ | 2 = 1, then one should expect, that in the limit of small thermal widths, 2 [C 1τ ⊥ C * 1τ n τ τ ⊥ ] = |C 1τ ⊥ | 2 n τ τ + |C 1τ | 2 n τ ⊥ τ ⊥ . (C.5) In order to demonstrate this equality, first we show that the z-derivative of 2 [C 1τ ⊥ C * 1τ n τ τ ⊥ ] equals the z-derivative of |C 1τ ⊥ | 2 n τ τ + |C 1τ | 2 n τ ⊥ τ ⊥ meaning that the quantities themselves 13 Here we neglect terms involving factors e −y 1 or e −y 3 such that, as given, these matrices are not strictly orthogonal. may differ only by a constant. Then we note that, since at z = z 0 the quantities are equal, then they must be equal for all z.
Neglecting (Λ τ )/Hz, as we expect this to be small in the single-flavour regime, then we need only show that and then it is demonstrated that the coefficient of W 1 in Eq. (C.3) is approximately equal to n B−L and thus the single flavour equations are recovered. The relation of Eq. (C.7) can be put into a more suggestive form if we use |C 1τ ⊥ | 2 = 1 − |C 1τ | 2 to re-express it thus The right-hand side of this equation is merely the trace of the CP -asymmetry tensor Tr (1) in the (τ ⊥ , τ )-basis. Thus, we suspect that the left-hand side is merely the trace expressed in an unfamiliar basis. This can be confirmed to be the case by construction of the unitary matrix then, by explicit calculation it can be seen that the left-hand side is the result of summing the diagonals (evaluating the trace in a particular basis) of Thus, we may conclude that, if we set (Λ τ ) = 0, we are left with and so d dz (|C 1τ | 2 n τ τ + |C 1τ ⊥ | 2 n τ ⊥ τ ⊥ + 2 [C 1τ ⊥ C * 1τ n τ τ ⊥ ]) = dn B−L dz . (C.12)

D Robustness of the High-Scale Plateau
In the transition region, the approximation that left-handed τ leptons are produced and destroyed at the same rate by flavour effects is somewhat inaccurate. In fact we should consider a slightly more accurate version of the density matrix equations in which the asymmetry density of right-handed τ leptons, n τ R is computed. Then, the density matrix equations are Hz (n τ τ − 2n τ R ). (D.1) The simpler set we previously considered result from the assumption that (Λ τ )/Hz is large enough to enforce n τ τ = 2n τ R . Clearly this is inaccurate for the situation under consideration where M 1 10 12 GeV. We should now append to λ(z) an extra term such that Hz n τ τ ⊥ (z ) − 2 (Λ τ ) Hz (n τ τ (z ) − 2n τ R (z )) .
(D.2) Now in this solution, there is a term in n τ τ (Λ τ )/Hz which scales approximately as xx −1 = x 0 and a term n τ R (Λ τ )/Hz in which, it may be shown n τ R ∝ x and thus λ (z) exhibits a approximate invariance under a scaling x as does λ(z).
It may be added that scattering effects can be incorporated by modifying the decay function D 1 (z) → D 1 (z) = D 1 (z) + S 1 (z) and the washout W 1 (z) → W 1 (z) = j(z)W 1 (z) [11]. The new decay function D 1 (z) which depends on a scattering part S 1 (z) is still multiplied by zero in the Tr = 0 case and is thus unimportant. The new washout function is multiplied by j(z) which depends on M 1 through log(M 1 /m H ). Thus, the plateau demonstrated in Fig. 14 picks up some unimportant logarithmic dependence on M 1 in addition to the small variation when scattering is neglected. In the numerical calculations of Section 5, the effects of scattering are included.