A new bridge between leptonic CP violation and leptogenesis

Flavor effects due to lepton interactions in the early Universe may have played an important role in the generation of the cosmological baryon asymmetry through leptogenesis. If the only source of high-energy CP violation comes from the left-handed leptonic sector, then it is possible to establish a bridge between flavored leptogenesis and low-energy leptonic CP violation. We explore this connection taking into account our present knowledge about low-energy neutrino parameters and the matter-antimatter asymmetry observed in the Universe. In this framework, we find that leptogenesis favors a hierarchical light neutrino mass spectrum, while for quasi-degenerate and inverted hierarchical neutrino masses there is a very narrow allowed window. The absolute neutrino mass scale turns out to be m<0.1 eV.


Introduction
The possibility of relating low-energy neutrino physics with the baryon asymmetry of the Universe (BAU) produced via the mechanism of thermal leptogenesis [1] has received a great deal of attention in the last few years [2]. In the simplest extension of the standard model (SM) where heavy neutrino singlets are added to the particle content, light neutrino masses arise through the seesaw mechanism [3]. Besides providing a natural mechanism to suppress neutrino masses, the seesaw mechanism puts at our disposal the necessary ingredients to explain the matter-antimatter asymmetry observed in our Universe. Indeed, the out-of-equilibrium decays of heavy Majorana neutrinos, under the presence of CPviolating interactions, produce a lepton asymmetry which is partially converted into a baryon asymmetry by the (B + L)-violating electroweak sphaleron interactions [4].
Recently, it has been noted that charged-lepton flavor effects play a crucial role on the dynamics of the thermal leptogenesis mechanism [5,6,7,8]. In particular, for temperatures below ∼ 10 12 (10 9 ) GeV the interactions mediated by the τ (µ) are non-negligible and, therefore, their effects should be properly taken into account in the computation of the final value of the BAU. In the limit of hierarchical heavy Majorana neutrinos M 1 ≪ M 2 < M 3 , the leptogenesis temperature is typically around T ∼ M 1 . Consequently, depending on the actual value of M 1 and on which charged-lepton Yukawa interactions are in equilibrium, one has different possible scenarios.
In the one-flavor limit where all the charged leptons are equally treated, one can show that a necessary condition for the mechanism of leptogenesis to work is the presence of a nonvanishing high-energy CP violation in the right-handed neutrino sector. In the flavored leptogenesis perspective, this remains true if M 1 10 12 GeV, which corresponds to the temperature above which all the charged lepton Yukawa interactions are out of equilibrium. In this temperature regime, the CP asymmetry generated in the decays of the heavy Majorana neutrinos is summed up over all flavors and its CP -violating part does not depend in general on the low-energy CP -violating quantities which could be potentially measured in future neutrino experiments. Therefore, in the one-flavor approximation, the observation of low-energy leptonic CP violation does not necessarily imply the existence of a nonvanishing BAU.
One may ask whether the above conclusions remain valid when flavor effects are accounted for. In this letter we show that, in a general class of models where CP is an exact symmetry in the high-energy right-handed neutrino sector, it is indeed possible to establish a direct link between low-energy leptonic CP violation and the generation of the cosmological baryon asymmetry. In these models, the baryon asymmetry only depends on the left-handed leptonic CP phases, which in turn are determined by the low-energy Dirac and Majorana neutrino phases [7]. We shall also briefly address the question on the possibility of naturally preserving CP as a good symmetry of the right-handed neutrino sector.
We work in the simple framework of the SM extended with three right-handed neutrinos N i (i = 1, 2, 3) with hierarchical heavy Majorana masses M 1 ≪ M 2 < M 3 . Working in the basis where the charged-lepton Yukawa couplings and the heavy Majorana neutrino mass matrix are diagonal, the relevant Dirac neutrino Yukawa interaction is Y iα N i ℓ α H, where ℓ α (α = e, µ, τ ) are the SM lepton doublets and H is the Higgs doublet. We take advantage of the so-called Casas-Ibarra parametrization [9] where U is the low-energy leptonic mixing matrix, which diagonalizes the effective neutrino mass matrix m ν in such a way that Here m i are the effective light neutrino masses and v ≡ H 0 ≃ 174 GeV. The matrix R in Eq. (1) is an orthogonal matrix, in general complex. In what follows, we consider a class of seesaw models where R is real, corresponding to the cases where CP is conserved in the right-handed neutrino sector [7].
In this special case, the flavored CP asymmetries generated in the decays N 1 → ℓ α H are simply given by [7] where I αkj ≡ Im( U * αk U αj ). Summing up over all flavors, ε 1 = α ε α , one recovers the standard one-flavor result [10]. It is straightforward to show that if R is real, then ε 1 = 0 due to the unitarity of U. Thus, at temperatures where all lepton flavors are out of equilibrium and the one-flavor approximation is valid, no lepton asymmetry can be generated. This in turn implies an upper bound on the lightest heavy Majorana neutrino mass, M 1 10 12 GeV, for the present scenario to be viable.
Therefore, for QD neutrinos, the quantities ε α turn out to be suppressed by the absolute neutrino mass scale m, contrarily to what occurs in the case of a complex R, where the upper bound on the flavor asymmetries is proportional to m [6]. We also remark that for a heavy Majorana mass M 1 < 10 9 GeV, the above asymmetries will be typically too small to account for the BAU. Thus, in the present framework, flavored leptogenesis could be viable if 10 9 GeV M 1 10 12 GeV.
1 For the matrix U δ we adopt the parametrization used in Ref. [12]. Since in this mass window only the τ Yukawa coupling is in thermal equilibrium, the final value of the baryon asymmetry per entropy density can be written as [7] where Y 2 is a combined density coming from the indistinguishable e and µ asymmetries.
The individual flavor densities Y 2 and Y τ can be found by solving the corresponding system of Boltzmann equations. In the mass region (6), it suffices to consider the leptonic CP asymmetry ε τ , since ε 2 ≡ ε e + ε µ = −ε τ when R is real.
In Fig. 1 we present the correlation between the low-energy Dirac (δ) and Majorana (α) phases for different neutrino spectra: normal hierarchy (upper panels) and inverted  Fig. 2 for the case of quasi-degenerate neutrinos. In the latter case, the correlation between the low-energy Majorana phases, α and β, is shown for maximal low-energy Dirac CP violation, i.e. when δ = π/2. We also remark that all the results can be easily extrapolated to any value of M 1 in the range of Eq. (6), since as can be readily seen from the maximal CP asymmetries in Eqs. (5), there is essentially a linear dependence between this parameter and the final baryon asymmetry.
Comparing the results of Figs. 1 and 2 with the experimentally observed value Y B = (8.7 ± 0.3) × 10 −11 [13], it becomes clear that for IH and QD neutrinos there is a very small allowed window for leptogenesis to be viable in the present framework (i.e when no CP violation arises from the right-handed neutrino sector). Moreover, as expected from the expression of the CP asymmetries given in Eq. (5), we find an upper bound on the absolute neutrino mass scale. In Fig. 3 we present the regions of the (m 1 , M 1 )-plane where flavored leptogenesis can produce the observed cosmological baryon asymmetry.
The solid (dot-dashed) line corresponds to a vanishing (thermal) initial N 1 abundance.
From the figure we obtain the lower bound M 1 > 10 10 GeV (3 × 10 9 GeV), as well as the upper bound m 1 0.1 eV.

Conclusion
We have studied the correlation between low-energy leptonic CP violation and thermal leptogenesis in a class of seesaw models where CP is an exact symmetry of the highenergy right-handed neutrino sector. In this case, and taking into account the role played by flavor effects on the dynamics of the leptogenesis mechanism, one can establish a correlation between low-energy CP violation and the cosmological baryon asymmetry. We have shown that for hierarchical light neutrino masses a successful flavored leptogenesis requires M 1 > 10 10 GeV (3×10 9 GeV) for a zero (thermal) initial abundance of the lightest right-handed neutrino N 1 . On the other hand, values of M 1 close to the upper bound of 10 12 GeV (above which the present scenario is not viable) are needed for the inverted hierarchy and quasi-degenerate neutrino masses, thus leaving a very narrow window for these neutrino mass spectra. We find an upper bound on the absolute neutrino mass scale: m 0.1 eV.
Regarding the correlation between the low-energy CP -violating phases and the value of the BAU, our analysis has shown that there are certain combinations of phases which are excluded for all values of M 1 (see plots). Furthermore, in this class of models, the observation of low-energy leptonic CP violation would in general indicate that the observed baryon asymmetry could have indeed been created through the leptogenesis mechanism.
Future information about low-energy CP violation either from neutrino oscillations or neutrinoless double beta decay searches could further constrain the present scenario.
Finally, we briefly address the question of how to construct, in the seesaw framework, a model where CP violation only occurs in the left-handed neutrino sector. In general, once one allows for CP violation through the introduction of complex Yukawa couplings, CP arises both in the left-handed and right-handed sectors, so both matrices U δ and R are complex. The simplest way of restricting the number of CP -violating phases is through the assumption that CP is a good symmetry of the Lagrangian, only broken by the vacuum. A model can actually be constructed, where one has in a natural way U δ complex while R is real. Let us consider the seesaw framework and impose CP invariance at the Lagrangian level. Now we introduce three Higgs doublets, together with a Z 3 symmetry under which the left-handed fermion doublets ψ Lj transform as ψ Lj → e i2πj/3 ψ Lj and the Higgs doublets as φ j → e −i2πj/3 φ j , while all other fields transform trivially. It can be readily shown that there is a region of parameters where the vacuum violates CP through complex vacuum expectation values φ 0 i = v i e iθ i . Due to the Z 3 restrictions on Yukawa couplings, the combination YY † is real, thus implying a real R (cf. Eq. (1)), but a complex U δ is generated. The drawback of such a scheme is that in order to generate the required baryon asymmetry, leptogenesis would have to take place at the TeV scale.