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On the “Low” and “High” Energy CP Violation in Leptogenesis
Nuclear Physics B (Proc. Suppl.) 188 (2009) 329–331 www.elsevierphysics.com
On the “Low” and “High” Energy CP Violation in Leptogenesis S. T. Petcov a
a ∗
Scuola Internazionale Superiore di Studi Avanzati, and INFN, Trieste, Italy
The CP violation necessary for the generation of the baryon asymmetry of the Universe Y B in the “flavoured” leptogenesis scenario, based on the simplest type I see-saw model of ν-mass generation, can arise from the “low energy” neutrino mixing matrix U and/or from the “high energy” part of neutrino Yukawa couplings. The latter can mediate CP-violating phenomena only at some high energy scale. We review briefly some of the results on the the interplay between the contributions in YB due to these two sources of CP violation in thermal leptogenesis with three heavy right-handed Majorana neutrinos having hierarchical mass spectrum.
1. Introduction It is well established at present that [1,2] lepton flavour effects can play a very important role in the leptogenesis mechanism [3,4] of generation of the baryon asymmetry of the Universe, YB . In the regime in which the lepton flavour effects in leptogenesis are significant (“flavoured” leptogenesis), the CP violation necessary for the generation of the observed matter-antimatter asymmetry can be provided exclusively [5] by the Dirac and/or Majorana [6] CP-violating phases in the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing matrix U . In the case of three hierarchical heavy right-handed (RH) Majorana neutrinos Nj , j = 1, 2, 3, and CP violation (CPV) due to the Majorana phases in U , this typically requires that the mass of the lightest RH Majo10 rana neutrino, N1 , satisfies M1 > ∼ 4 × 10 GeV [5]. One can have successful leptogenesis also if the only source of CP violation is the Dirac phase δ in U , provided [5] | sin θ13 sin δ| ≥ 0.09, θ13 being the CHOOZ angle. In thermal leptogenesis with “hierarchical” spectrum of the heavy Majorana neutrinos Nj , CPV lepton asymmetry is produced in out-of-equilibrium lepton number and CP-nonconserving decays of the lightest one, N1 . The lepton asymmetry is converted into a baryon asymmetry by (B − L)-conserving but (B + L)violating sphaleron interactions [4]. The CP violation necessary for the generation of the baryon asymmetry YB in “flavoured” lepto∗ Also at: Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria
0920-5632/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2009.02.076
genesis can arise both from the “low energy” neutrino mixing matrix U and/or from the “high energy” part of the matrix of neutrino Yukawa couplings, λ, which can mediate CP-violating phenomena only at some high energy scale. In the orthogonal parametrisation [7], the “high energy” part of λ is represented by a complex orthogonal matrix R, √ R RT =RT R=1, and the masses of Nj : √ −1 M R mU † , where M and m are λ = v diagonal matrices formed by the masses of Nj and of the light Majorana neutrinos νj , M ≡ Diag(M1 , M2 , M3 ), m ≡ Diag(m1 , m2 , m3 ), Mj > 0, mk ≥ 0, and v = 174 GeV is the vacuum expectation value of the Higgs doublet field. The matrix R, as is well-known, does not affect the “low” energy neutrino mixing phenomenology. We review briefly some of the results derived in [8] on the possible interplay in “flavoured” leptogenesis between contributions in YB due to the “low energy” and “high energy” CP violation, originating from the ν-mixing matrix U and the R-matrix, respectively. 2. Baryon Asymmetry from “Low” and “High” Energy CP Violation Consider the simplest leptogenesis scenario based on the type I see-saw model with three heavy right-handed Majorana neutrinos having hierarchical mass spectrum. Suppose that the baryon asymmetry YB is produced in the “twoflavour” regime in leptogenesis [1,2]. This regime < T ∼ is realised at temperatures 109 GeV ∼ < 1012 GeV. Under the assumptions made, M1 ∼
S.T. Petcov / Nuclear Physics B (Proc. Suppl.) 188 (2009) 329–331
|YB | generated via thermal leptogenesis can be written as [1,2] |YB | ∼ τ ) + = 3 × 10−3 |τ η(0.66m 2 η(0.71m 2 )|, where 2 ≡ e + μ , l being the CPV asymmetry in the l flavour (lepton charge) produced in N1 -decays 2 , l = e, μ, τ , and η(0.66m τ ) and η(0.71m 2 ) are the corresponding efficiency factors [2], m 2,τ being the associated 3 wash-out mass parameters , m 2 = m e + m μ , m l = | j mj R1j Ulj∗ |2 . For complex R we have: 1/2 3/2 ∗ 3M1 Im( jk mj mk Ulj Ulk R1j R1k ) . l = − 2 16πv 2 i mi |R1i | In what follows we use the standard parametrisation of the 3-neutrino mixing matrix: U = V (θ12 , θ13 , θ23 , δ) diag(1, ei
α21 2
, ei
α31 2
),
where V (θ12 , θ13 , θ23 , δ) is a CKM-like matrix, θ12 and θ23 being the solar and atmospheric neutrino mixing angles, δ = [0, 2π] is the Dirac CPV phase and α21 , α31 are two Majorana CPV phases [6]. All numerical results discussed below are obtained for the best fit values of the solar and atmospheric neutrino oscillation parameters [9–11]: Δm2 =Δm221 = 7.65×10−5 eV2 , sin2 θ12 =0.30, |Δm2A |=|Δm231(32) |= 2.4×10−3 eV2 , sin2 θ23 =0.5. The 3σ limit on the CHOOZ angle θ13 , sin2 θ13 < 0.056, is taken into account as well. We shall summarise next some of the results in [8]. Consider the case of NH spectrum, m1 m2 m 3 ∼ = (Δm2A )1/2 . Suppose for simplicity that m1 =0, and R11 =0 (N3 decoupling). Suppose further that R12(13) =|R12(13) |eiϕ12(13) are complex and that δ=0,π. The R-phases ϕ12,13 will be a source of “high energy” CPV if ϕ12,13 = kπ/2, k=0,1,... Using the orthogonal2 2 ity condition R12 + R13 = 1, it is possible to express ϕ12,13 in terms of |R12,13 |2 : cos 2ϕ12 =(1 − |R13 |4 + |R12 |4 )/(2|R12 |2 ), cos 2ϕ13 =(1 + |R13|4 − |R12 |4 )/(2|R13 |2 ), sgn(sin 2ϕ12 )=−sgn(sin 2ϕ13 ). Under the conditions of m1 =0 and δ=0,π, the “low energy” CPV can only be due to the Majorana phase difference α32 =α31 − α21 = 0, π. The analysis of the case of NH spectrum and CPV due to the Majorana phase α32 and the 2 The
expression for YB we have given is normalised to the entropy density, see, e.g. [5]. 3 Approximate analytic expression for η(m ) is given in [2].
8 7 6
|YB| ⋅ 1010
330
5 4 3 2 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
|R13|
Figure 1. The “high” and “low” energy CPV terms in YB and the total |YB | as functions of |R13 | (blue, green and red lines) in the case of NH spectrum, for α32 =π/2, sin2 θ23 =0.5, θ13 =0, |R12 | ∼ = 1 and M1 =1011 GeV (from [8]).
R-phases ϕ12,13 showed [8] that there exists a relatively large region of the relevant parameter space in which the predicted value of the baryon asymmetry exhibits a strong dependence on the Majorana phase α32 provided the latter lies in the interval 0 < α32 < π (if sin 2ϕ12 < 0), or 3π < α32 < 4π (when sin 2ϕ12 > 0). The regions typically correspond to 0.05 < ∼ |R13 | < ∼ 0.5, |R13 | < |R12 | ≤ 1 (Fig. 1), and to |R12 | > 1, |R13 |2 ∼ |R12 |2 − 1. Depending on the value of α32 , we can have, e.g. either |YB | 8.6 × 10−11 or YB compatible with the observations in the indicated regions. The effects of the “low energy” CPV due to α32 can be non-negligible in leptogenesis also for 0.5 ≤ |R13 | ≤ |R12 | ≤ 1. In the regions of the parameter space where the Majorana phase effects are significant, the contributions to YB due to the “high energy” CPV and that involving the “low energy” CPV phase α32 typically have opposite signs and tend to compensate each other. This mutual compensation can be complete and we can have YB = 0 for certain values of the relevant parameters, in spite of the fact that each of the two contributions can be sufficiently large to account by itself for the observed YB . It was found also [8] that in the regions of the parameter space with significant interplay between the contributions in YB due to the “high”
S.T. Petcov / Nuclear Physics B (Proc. Suppl.) 188 (2009) 329–331
and “low” energy CPV, the predicted value of 1.6 1.4
|YB| ⋅ 1010
1.2 1 0.8 0.6 0.4 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
|R12|
Figure 2. The “high” and “low” energy CPV terms and the total |YB | versus |R12 | (blue, green and red lines) for IH spectrum, α21 =π/2, θ13 =0.2, δ=π, |R11 |=1, M1 =1011 GeV. The “high energy” term is strongly suppressed (from [8]).
|YB | can exhibit strong dependence i) on sin2 θ23 when the latter is varied in the range (0.36 - 0.64) allowed by the data, and, ii) on whether the Dirac phase δ = 0 or π, if sin2 θ13 is sufficiently large. Very different results were obtained [8] for IH neutrino mass spectrum 4 m3 m1 < m2 , m3 =0 and R13 =0 (decoupling of N3 ). Now R11(12) =|R11(12) |eiϕ11(12) , ϕ11,12 playing the role of “high energy” CPV phases. The “low energy” CPV can be due to the Dirac phase δ and/or Majorana phases α21 . In this case there are large regions of values of the corresponding parameters, for which the contribution to YB due to the “low energy” CPV Majorana phase α21 , or Dirac phase δ (for α21 = (2k + 1)π), is comparable in magnitude, or exceeds, the purely “high energy” contribution in YB , originating from CPV generated by the R-matrix. Moreover, in certain significant subregions of the indicated regions, the contribution to YB due to the “high energy” CPV is subdominant. We have found 4 For
NH (IH) spectrum, the νe(μ) → νμ(e) (¯ νe(μ) → ν¯μ(e) ) transitions of the Earth core crossing atmospheric ν’s can be maximally amplified (P =1) by the neutrino oscillation length resonance [12] (NOLR), while the ν¯e(μ) → ν¯μ(e) (νe(μ) → νμ(e) ) transitions are suppressed. The NOLR effect can be used, e.g. to determine the type of ν-mass spectrum [12], see also T. Schwetz, these Proceedings.
331
also that for (− sin θ13 cos δ) > ∼ 0.1, the “high energy” term in YB is strongly suppressed by the factor (|Uτ 1 |2 − |Uτ 2 |2 ) (Fig. 2). The “high energy” phases ϕ11 and ϕ12 in this case can have large CP-violating values. Nevertheless, if the indicated inequality is fulfilled, the purely “high energy” contribution to YB due to the CPV Rphases would play practically no role in the generation of baryon asymmetry compatible with the observations. One would have successful leptogenesis in this case only if the requisite CP violation is provided by the Majorana and/or Dirac phases in the neutrino mixing matrix. To summarise, the results obtained in [8] show that the CP violation due to the “low energy” Majorana and Dirac phases in the neutrino mixing matrix can play a significant role in the production of baryon asymmetry compatible with the observation in “flavoured” leptogenesis even in the presence of “high energy” CP violation generated by additional “high energy” (R-) phases in the neutrino Yukawa couplings. Acknowledgements. It is a pleasure to thank G.L. Fogli, E. Lisi and their colleagues for organizing such a scientifically interesting Workshop. REFERENCES 1. A. Abada et al., JCAP 0604 (2006) 004; E. Nardi et al., JHEP 0601, 164 (2006). 2. A. Abada et al., JHEP 0609 (2006) 010. 3. M. Fukugita and T. Yanagida, Phys. Lett. B 174 (1986) 45. 4. V.A. Kuzmin, V.A. Rubakov, M.E. Shaposhnikov. Phys.Lett. B 155 (1985) 36. 5. S. Pascoli, S.T. Petcov, A. Riotto, Phys. Rev. D 75 (2007) 083511; Nucl. Phys. B 774 (2007) 1. 6. S.M. Bilenky, J. Hosek, S.T. Petcov, Phys. Lett. B 94 (1980) 495. 7. J.A. Casas, A. Ibarra, Nucl. Phys. B 618 (2001) 171. 8. E. Molinaro, S.T. Petcov, arXiv:0803.4120 and Phys. Lett. B 671 (2009) 60. 9. A. Bandyopadhyay et al., Phys. Lett. B 608 (2005) 115, and arXiv:0804.4857. 10. G.L. Fogli et al., arXiv:0806.2649. 11. T. Schwetz, M. Maltoni, arXiv:0812.3161. 12. S.T. Petcov, hep-ph/9805262; M. Chizhov, S.T. Petcov, Phys.Rev.Lett. 83 (1999) 1096 and 85 (2000) 3979; Phys.Rev.D 63 (2001) 073003 (hep-ph/9903424); M. Chizhov et al., hep-ph/9810501.
On the “Low” and “High” Energy CP Violation in Leptogenesis S. T. Petcov a
a ∗
Scuola Internazionale Superiore di Studi Avanzati, and INFN, Trieste, Italy
The CP violation necessary for the generation of the baryon asymmetry of the Universe Y B in the “flavoured” leptogenesis scenario, based on the simplest type I see-saw model of ν-mass generation, can arise from the “low energy” neutrino mixing matrix U and/or from the “high energy” part of neutrino Yukawa couplings. The latter can mediate CP-violating phenomena only at some high energy scale. We review briefly some of the results on the the interplay between the contributions in YB due to these two sources of CP violation in thermal leptogenesis with three heavy right-handed Majorana neutrinos having hierarchical mass spectrum.
1. Introduction It is well established at present that [1,2] lepton flavour effects can play a very important role in the leptogenesis mechanism [3,4] of generation of the baryon asymmetry of the Universe, YB . In the regime in which the lepton flavour effects in leptogenesis are significant (“flavoured” leptogenesis), the CP violation necessary for the generation of the observed matter-antimatter asymmetry can be provided exclusively [5] by the Dirac and/or Majorana [6] CP-violating phases in the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) neutrino mixing matrix U . In the case of three hierarchical heavy right-handed (RH) Majorana neutrinos Nj , j = 1, 2, 3, and CP violation (CPV) due to the Majorana phases in U , this typically requires that the mass of the lightest RH Majo10 rana neutrino, N1 , satisfies M1 > ∼ 4 × 10 GeV [5]. One can have successful leptogenesis also if the only source of CP violation is the Dirac phase δ in U , provided [5] | sin θ13 sin δ| ≥ 0.09, θ13 being the CHOOZ angle. In thermal leptogenesis with “hierarchical” spectrum of the heavy Majorana neutrinos Nj , CPV lepton asymmetry is produced in out-of-equilibrium lepton number and CP-nonconserving decays of the lightest one, N1 . The lepton asymmetry is converted into a baryon asymmetry by (B − L)-conserving but (B + L)violating sphaleron interactions [4]. The CP violation necessary for the generation of the baryon asymmetry YB in “flavoured” lepto∗ Also at: Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria
0920-5632/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2009.02.076
genesis can arise both from the “low energy” neutrino mixing matrix U and/or from the “high energy” part of the matrix of neutrino Yukawa couplings, λ, which can mediate CP-violating phenomena only at some high energy scale. In the orthogonal parametrisation [7], the “high energy” part of λ is represented by a complex orthogonal matrix R, √ R RT =RT R=1, and the masses of Nj : √ −1 M R mU † , where M and m are λ = v diagonal matrices formed by the masses of Nj and of the light Majorana neutrinos νj , M ≡ Diag(M1 , M2 , M3 ), m ≡ Diag(m1 , m2 , m3 ), Mj > 0, mk ≥ 0, and v = 174 GeV is the vacuum expectation value of the Higgs doublet field. The matrix R, as is well-known, does not affect the “low” energy neutrino mixing phenomenology. We review briefly some of the results derived in [8] on the possible interplay in “flavoured” leptogenesis between contributions in YB due to the “low energy” and “high energy” CP violation, originating from the ν-mixing matrix U and the R-matrix, respectively. 2. Baryon Asymmetry from “Low” and “High” Energy CP Violation Consider the simplest leptogenesis scenario based on the type I see-saw model with three heavy right-handed Majorana neutrinos having hierarchical mass spectrum. Suppose that the baryon asymmetry YB is produced in the “twoflavour” regime in leptogenesis [1,2]. This regime < T ∼ is realised at temperatures 109 GeV ∼ < 1012 GeV. Under the assumptions made, M1 ∼
S.T. Petcov / Nuclear Physics B (Proc. Suppl.) 188 (2009) 329–331
|YB | generated via thermal leptogenesis can be written as [1,2] |YB | ∼ τ ) + = 3 × 10−3 |τ η(0.66m 2 η(0.71m 2 )|, where 2 ≡ e + μ , l being the CPV asymmetry in the l flavour (lepton charge) produced in N1 -decays 2 , l = e, μ, τ , and η(0.66m τ ) and η(0.71m 2 ) are the corresponding efficiency factors [2], m 2,τ being the associated 3 wash-out mass parameters , m 2 = m e + m μ , m l = | j mj R1j Ulj∗ |2 . For complex R we have: 1/2 3/2 ∗ 3M1 Im( jk mj mk Ulj Ulk R1j R1k ) . l = − 2 16πv 2 i mi |R1i | In what follows we use the standard parametrisation of the 3-neutrino mixing matrix: U = V (θ12 , θ13 , θ23 , δ) diag(1, ei
α21 2
, ei
α31 2
),
where V (θ12 , θ13 , θ23 , δ) is a CKM-like matrix, θ12 and θ23 being the solar and atmospheric neutrino mixing angles, δ = [0, 2π] is the Dirac CPV phase and α21 , α31 are two Majorana CPV phases [6]. All numerical results discussed below are obtained for the best fit values of the solar and atmospheric neutrino oscillation parameters [9–11]: Δm2 =Δm221 = 7.65×10−5 eV2 , sin2 θ12 =0.30, |Δm2A |=|Δm231(32) |= 2.4×10−3 eV2 , sin2 θ23 =0.5. The 3σ limit on the CHOOZ angle θ13 , sin2 θ13 < 0.056, is taken into account as well. We shall summarise next some of the results in [8]. Consider the case of NH spectrum, m1 m2 m 3 ∼ = (Δm2A )1/2 . Suppose for simplicity that m1 =0, and R11 =0 (N3 decoupling). Suppose further that R12(13) =|R12(13) |eiϕ12(13) are complex and that δ=0,π. The R-phases ϕ12,13 will be a source of “high energy” CPV if ϕ12,13 = kπ/2, k=0,1,... Using the orthogonal2 2 ity condition R12 + R13 = 1, it is possible to express ϕ12,13 in terms of |R12,13 |2 : cos 2ϕ12 =(1 − |R13 |4 + |R12 |4 )/(2|R12 |2 ), cos 2ϕ13 =(1 + |R13|4 − |R12 |4 )/(2|R13 |2 ), sgn(sin 2ϕ12 )=−sgn(sin 2ϕ13 ). Under the conditions of m1 =0 and δ=0,π, the “low energy” CPV can only be due to the Majorana phase difference α32 =α31 − α21 = 0, π. The analysis of the case of NH spectrum and CPV due to the Majorana phase α32 and the 2 The
expression for YB we have given is normalised to the entropy density, see, e.g. [5]. 3 Approximate analytic expression for η(m ) is given in [2].
8 7 6
|YB| ⋅ 1010
330
5 4 3 2 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
|R13|
Figure 1. The “high” and “low” energy CPV terms in YB and the total |YB | as functions of |R13 | (blue, green and red lines) in the case of NH spectrum, for α32 =π/2, sin2 θ23 =0.5, θ13 =0, |R12 | ∼ = 1 and M1 =1011 GeV (from [8]).
R-phases ϕ12,13 showed [8] that there exists a relatively large region of the relevant parameter space in which the predicted value of the baryon asymmetry exhibits a strong dependence on the Majorana phase α32 provided the latter lies in the interval 0 < α32 < π (if sin 2ϕ12 < 0), or 3π < α32 < 4π (when sin 2ϕ12 > 0). The regions typically correspond to 0.05 < ∼ |R13 | < ∼ 0.5, |R13 | < |R12 | ≤ 1 (Fig. 1), and to |R12 | > 1, |R13 |2 ∼ |R12 |2 − 1. Depending on the value of α32 , we can have, e.g. either |YB | 8.6 × 10−11 or YB compatible with the observations in the indicated regions. The effects of the “low energy” CPV due to α32 can be non-negligible in leptogenesis also for 0.5 ≤ |R13 | ≤ |R12 | ≤ 1. In the regions of the parameter space where the Majorana phase effects are significant, the contributions to YB due to the “high energy” CPV and that involving the “low energy” CPV phase α32 typically have opposite signs and tend to compensate each other. This mutual compensation can be complete and we can have YB = 0 for certain values of the relevant parameters, in spite of the fact that each of the two contributions can be sufficiently large to account by itself for the observed YB . It was found also [8] that in the regions of the parameter space with significant interplay between the contributions in YB due to the “high”
S.T. Petcov / Nuclear Physics B (Proc. Suppl.) 188 (2009) 329–331
and “low” energy CPV, the predicted value of 1.6 1.4
|YB| ⋅ 1010
1.2 1 0.8 0.6 0.4 0.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
|R12|
Figure 2. The “high” and “low” energy CPV terms and the total |YB | versus |R12 | (blue, green and red lines) for IH spectrum, α21 =π/2, θ13 =0.2, δ=π, |R11 |=1, M1 =1011 GeV. The “high energy” term is strongly suppressed (from [8]).
|YB | can exhibit strong dependence i) on sin2 θ23 when the latter is varied in the range (0.36 - 0.64) allowed by the data, and, ii) on whether the Dirac phase δ = 0 or π, if sin2 θ13 is sufficiently large. Very different results were obtained [8] for IH neutrino mass spectrum 4 m3 m1 < m2 , m3 =0 and R13 =0 (decoupling of N3 ). Now R11(12) =|R11(12) |eiϕ11(12) , ϕ11,12 playing the role of “high energy” CPV phases. The “low energy” CPV can be due to the Dirac phase δ and/or Majorana phases α21 . In this case there are large regions of values of the corresponding parameters, for which the contribution to YB due to the “low energy” CPV Majorana phase α21 , or Dirac phase δ (for α21 = (2k + 1)π), is comparable in magnitude, or exceeds, the purely “high energy” contribution in YB , originating from CPV generated by the R-matrix. Moreover, in certain significant subregions of the indicated regions, the contribution to YB due to the “high energy” CPV is subdominant. We have found 4 For
NH (IH) spectrum, the νe(μ) → νμ(e) (¯ νe(μ) → ν¯μ(e) ) transitions of the Earth core crossing atmospheric ν’s can be maximally amplified (P =1) by the neutrino oscillation length resonance [12] (NOLR), while the ν¯e(μ) → ν¯μ(e) (νe(μ) → νμ(e) ) transitions are suppressed. The NOLR effect can be used, e.g. to determine the type of ν-mass spectrum [12], see also T. Schwetz, these Proceedings.
331
also that for (− sin θ13 cos δ) > ∼ 0.1, the “high energy” term in YB is strongly suppressed by the factor (|Uτ 1 |2 − |Uτ 2 |2 ) (Fig. 2). The “high energy” phases ϕ11 and ϕ12 in this case can have large CP-violating values. Nevertheless, if the indicated inequality is fulfilled, the purely “high energy” contribution to YB due to the CPV Rphases would play practically no role in the generation of baryon asymmetry compatible with the observations. One would have successful leptogenesis in this case only if the requisite CP violation is provided by the Majorana and/or Dirac phases in the neutrino mixing matrix. To summarise, the results obtained in [8] show that the CP violation due to the “low energy” Majorana and Dirac phases in the neutrino mixing matrix can play a significant role in the production of baryon asymmetry compatible with the observation in “flavoured” leptogenesis even in the presence of “high energy” CP violation generated by additional “high energy” (R-) phases in the neutrino Yukawa couplings. Acknowledgements. It is a pleasure to thank G.L. Fogli, E. Lisi and their colleagues for organizing such a scientifically interesting Workshop. REFERENCES 1. A. Abada et al., JCAP 0604 (2006) 004; E. Nardi et al., JHEP 0601, 164 (2006). 2. A. Abada et al., JHEP 0609 (2006) 010. 3. M. Fukugita and T. Yanagida, Phys. Lett. B 174 (1986) 45. 4. V.A. Kuzmin, V.A. Rubakov, M.E. Shaposhnikov. Phys.Lett. B 155 (1985) 36. 5. S. Pascoli, S.T. Petcov, A. Riotto, Phys. Rev. D 75 (2007) 083511; Nucl. Phys. B 774 (2007) 1. 6. S.M. Bilenky, J. Hosek, S.T. Petcov, Phys. Lett. B 94 (1980) 495. 7. J.A. Casas, A. Ibarra, Nucl. Phys. B 618 (2001) 171. 8. E. Molinaro, S.T. Petcov, arXiv:0803.4120 and Phys. Lett. B 671 (2009) 60. 9. A. Bandyopadhyay et al., Phys. Lett. B 608 (2005) 115, and arXiv:0804.4857. 10. G.L. Fogli et al., arXiv:0806.2649. 11. T. Schwetz, M. Maltoni, arXiv:0812.3161. 12. S.T. Petcov, hep-ph/9805262; M. Chizhov, S.T. Petcov, Phys.Rev.Lett. 83 (1999) 1096 and 85 (2000) 3979; Phys.Rev.D 63 (2001) 073003 (hep-ph/9903424); M. Chizhov et al., hep-ph/9810501.