Revisiting Leptogenesis in a SUSY SU(5) x T' Model of Flavour

We investigate the generation of the baryon asymmetry of the Universe within a SUSY SU(5) x T' model of flavour, which gives rise to realistic masses and mixing patterns for quarks and leptons. The model employs the see-saw mechanism for generation of the light neutrino masses and the baryon asymmetry is produced via leptogenesis. We perform detailed calculations of both the CP violating lepton asymmetries, originating from the decays of the heavy Majorana neutrinos operative in the see-saw mechanism, and of the efficiency factors which account for the lepton asymmetry wash-out processes in the Early Universe. The latter are calculated by solving numerically the system of Boltzmann equations describing the generation and the evolution of the lepton asymmetries. The baryon asymmetry in the model considered is proportional to the J_{CP} factor, which determines the magnitude of CP violation effects in the oscillations of flavour neutrinos. The leptogenesis scale can be sufficiently low, allowing to avoid the potential gravitino problem.


I. INTRODUCTION
In the present article we consider the generation of the baryon asymmetry of the Universe in the SUSY model of flavour based on the SU (5) × T ′ symmetry, which was developed in [1,2]. The model possesses a number of appealing features which makes it worthwhile to investigate whether it provides also a viable scenario for the baryon asymmetry generation.
The group T ′ is the double covering of the symmetry group of the tetrahedron A 4 (see, e.g., [3]). It was realised by a number of authors (see, e.g., [4]) that the T ′ symmetry can be used for the description of masses and mixing of both leptons and quarks. The SU (5) × T ′ model of flavour of interest accounts successfully for the pattern of quark masses and mixing, including the CP violation in the quark sector [2]. It is free of discrete gauge anomalies [5] and gives rise to realistic masses and mixing of the leptons as well.
The SU (5) × T ′ model proposed in [1,2] we will discuss in the present article, includes three right-handed (RH) neutrino fields N lR , l = e, µ, τ , which possess a Majorana mass term. The light neutrino masses are generated by the type I see-saw mechanism and are naturally small. The light neutrino mass spectrum is predicted [6] to be with normal ordering and is hierarchical (throughout this article we use the definitions and the conventions given in [7]). The neutrino masses m j , j = 1, 2, 3, are functions of two real parameters of the model [2,6]. The latter can be determined by using the values of the two neutrino mass squared differences, ∆m 2 21 and ∆m 2 31 , or of ∆m 2 21 and the ratio r = ∆m 2 21 /∆m 2 31 , obtained in the global analyses of the neutrino oscillation data. Using the best fit values of ∆m 2 21 = 7.58 × 10 −5 eV 2 and ∆m 2 31 = 2.35 × 10 −3 eV 2 , found in the analysis performed in [8], we have [6]: The values of m 1 , m 2 and m 3 are essentially fixed: the uncertainties corresponding to the 3σ ranges of allowed values of ∆m 2 21 and r are remarkably small [6].
The part of the Pontecorvo, Maki, Nakgawa and Sakata (PMNS) neutrino mixing matrix (see [7]), resulting from the diagonalisation of the Majorana mass term of the left-handed flavour neutrino fields ν lL (x), l = e, µ, τ , which is generated by the see-saw mechanism, is of the tri-bimaximal form [9]. The latter is "corrected" by the unitary matrix originating from the diagonalisation of the charged lepton mass matrix M e (for a general discussion of such corrections see, e.g., [10][11][12]). Since the model is based on the SU (5) GUT symmetry, the charged lepton mass matrix is related to the down-quark mass matrix M d . The model exploits the Georgi-Jarlskog approach for obtaining viable relations between the masses of the muon and the s-quark [1,2]. The Cabibbo angle is given by the "standard" expression: θ c ∼ = m d /m s , m d and m s being the masses of the d-and s-quarks. As a consequence, in particular, of the connection between M e and M d , the smallest angle in the neutrino mixing matrix θ 13 , is related to the Cabibbo angle [2]: Here we implicitly assumed the "standard" parametrisation of the PMNS matrix [7]: where c ij = cos θ ij and s ij = sin θ ij , 0 ≤ θ ij ≤ π/2, δ and α j1 , j = 2, 3, are the Dirac and the two Majorana CP violation phases [13], 0 ≤ δ ≤ 2π and, in general [14], 0 ≤ α j1 /2 ≤ 2π. We will use this parametrisation in the discussion which follows. The CP violation, predicted by the model, can entirely be geometrical in origin [2]. This interesting aspect of the SU (5) × T ′ model we will consider is a consequence of one of the special properties of the group T ′ , namely, that its group theoretical Clebsch-Gordon (CG) coefficients are intrinsically complex [15]. The only dominant source of CP violation in the lepton sector of the model is the Dirac phase δ. The two Majorana phases present in the PMNS neutrino mixing matrix, α 21 and α 31 , are predicted to leading order to have CP conserving values. In the standard parametrisation of U PMNS we have: α 21 ∼ = 0 and α 31 ∼ = π. Higher order corrections induce small CP violating deviations of the order of few degrees from these CP conserving values of the two phases [6].
The Dirac phase δ in the PMNS matrix is induced effectively by the complex CG coefficients of the group T ′ . As we shall see in Section 2, δ can take two values in the model considered. One was identified in [2] and is equal approximately to δ ∼ = 5π/4 = 225 • , the precise value being The second possible value of δ is given to leading order, as will be discussed in Section 2, by The tri-bimaximal mixing value of the solar neutrino mixing angle θ 12 , which corresponds to sin 2 θ 12 = 1/3, is corrected by a quantity which, as it follows from the general form of such corrections [10][11][12]), is determined by the angle θ 13 and the Dirac phase δ: In the SU (5) × T ′ model considered, θ 13 is related to the Cabibbo angle, eq. (2). The rephasing invariant associated with the Dirac phase δ [16], J CP , which determines the magnitude of CP violation effects in neutrino oscillations [17], predicted by the model to leading order reads [6,10,11]: For δ ∼ = 5π/4, eq. (4), the correction to the TBM value of sin 2 θ 12 given in eq. (6), is negative and sin 2 θ 12 ∼ = 0.299, where we have used eq. (2) and sin θ c = 0.22. This value lies within the 1σ allowed range, found in the global data analysis [8]. We also have, including the higher order corrections [2]: J CP ∼ = −9.66 × 10 −3 . If δ ∼ = π 4 , eq. (5), the correction to the TBM value of sin 2 θ 12 is positive and sin 2 θ 12 ∼ = 0.37. According to the analyses performed in [8] and in [18], the current neutrino oscillation data imply respectively sin 2 θ 12 ∼ < 0.36 and sin 2 θ 12 ∼ < 0.374 at 3σ. Thus, the case of δ ∼ = π 4 is disfavored by the data. For the J CP factor in this case we get: J CP ∼ = +9.95 × 10 −3 . Since the neutrino masses, the neutrino mixing angle and the CP violating phases in the PMNS matrix have essentially fixed values, the model provides also specific predictions [6] for the sum of the three neutrino masses, as well as for the effective Majorana mass in neutrinoless double-beta decay (see, e.g., [19]): It should be clear from the preceding discussion that the SU (5) × T ′ model of flavour of interest is remarkably predictive: the values of the neutrino masses, the type of the neutrino mass spectrum, the values of the neutrino mixing angles and the CP violating phases in the neutrino mixing matrix, as well as the effective Majorana mass in neutrinoless double beta decay, obtained in the model are essentially free of ambiguities. The predictions for sin θ 13 , sin 2 θ 12 , δ and J CP can be tested directly in the upcoming neutrino oscillation experiments. The value of sin θ 13 one gets in the model, for instance, is relatively small, sin θ 13 ∼ = 0.058 1 . It lies outside the 2σ, but within the 3σ, ranges of allowed values of sin θ 13 , determined in the global analyses of the current neutrino oscillation data [8,20]. The results of the three reactorν e experiments on θ 13 , Double Chooz [21], RENO [22] and Daya Bay [23], which are currently taking data, can provide a critical test of the model.
In the present article we investigate the prediction of the SU (5) × T ′ model of flavour proposed in [1,2] for the baryon asymmetry of the Universe. The latter is generated in the model via the leptogenesis mechanism [24,25]. The dominant source of CP violation in the lepton sector and in leptogenesis is the Dirac phase δ 2 . Therefore there is a direct connection between the baryon asymmetry of the Universe and the CP violation in neutrino oscillations.
The generation of the baryon asymmetry in the SU (5) × T ′ model of interest was studied in [27]. However, the authors of [27] limited the discussion of the baryon asymmetry generation to the calculation of the CP asymmetries in the additive lepton charges, ǫ ℓ i , generated in the heavy Majorana neutrino decays, ℓ = e, µ, τ , i = 1, 2, 3. They based their conclusions on the results obtained for these asymmetries. In the present article we perform a complete calculation of the baryon asymmetry, i.e., we calculate not only the asymmetries ǫ ℓ i , but also the corresponding efficiency factors which account for the effects of the CP asymmetry wash-out processes, taking place in the Early Universe. The efficiency factors are computed by solving numerically the Boltzmann equations, which describe the evolution of the CP violating asymmetries in the Early Universe. The results we obtain for the lepton asymmetries ǫ ℓ i do not agree with those found in [27] and our results for the baryon asymmetry contradict the claims made in [27].

II. INGREDIENTS
In the L − R convention in which the neutrino mass terms are written with the RH neutrino fields on the right, the superpotential of the model leads [1,2] to the following neutrino Dirac mass matrix, and to the following Majorana mass matrix of the RH neutrinos, In eqs. (10) and (11), ξ 0 , η 0 , ζ 0 , ζ ′ 0 and s 0 are dimensionless real parameters, Λ is the scale above which the T ′ symmetry is exact,Ỹ ν is the matrix of the neutrino Yukawa couplings in the basis in which the charged lepton and the RH neutrino mass matrices are not diagonal, and v u is the vacuum expectation value of the "up" Higgs doublet field of the SUSY extensions of the Standard Model. Thus, the neutrino Dirac mass matrix in the model, M D , is real and symmetric. As can be easily shown, it is diagonalised by the tri-bimaximal mixing (TBM) matrix: We have: where all elements in the diagonal matrix M diag D are real. The RH neutrino Majorana mass matrix M RR is diagonalised by the unitary matrix S: where and M j are the masses of the heavy Majorana neutrinos N j (possessing definite masses), C being the charge conjugation matrix. Thus, to leading order, the masses of the three heavy Majorana neutrinos N j coincide, The effective Majorana mass matrix of the left-handed (LH) flavour neutrinos, M ν , which is generated by the see-saw mechanism, is also diagonalized by the TBM matrix (12), Here Q = i diag(1, 1, ±i) is the matrix which determines, as we shall see, the leading order values of the two Majorana phases in the PMNS matrix, and m k > 0, k = 1, 2, 3, are the masses of the three light Majorana neutrinos, where In what follows we will ignore the overall unphysical factor i in Q. The values of m j given in eq. (1) correspond to [6] The charged lepton mass matrix M e is not diagonal; it is diagonalised by a bi-unitary transformation: where V eR and U e are unitary matrices and M d e = diag(m e , m µ , m τ ), m ℓ being the mass of the charged lepton ℓ, ℓ = e, µ, τ . The matrix U e , which enters into the expression for the PMNS matrix, The charged lepton mass matrix M e (with the corresponding mass term written in the R-L convention in terms of the chiral charged lepton fields l ′ aR and l ′ aL ) has the following form [2]: It is related to the down-type quark mass matrix M d via the well-known SU (5) relation: M e = M T d , with the factor (-3) in M e replaced by 1 in M d . The up-type quark mass matrix in the model has the form [2]: In eqs. (20) and (21), φ 0 , ψ ′ 0 , φ ′ 0 , ψ 0 , y d and y t are real dimensionless parameters, y d and y t are Yukawa couplings and v d is the vacuum expectation value of the "down" type Higgs doublet of the SUSY extension of the Standard Model.
Fitting the quark sector observables and charged lepton masses one finds that [2] two of the three angles, present in the "standard-like" parametrisation of the matrix U e , are extremely small, sin θ e 13 ∼ = 1.3 × 10 −5 , sin θ e 23 ∼ = 1.5 × 10 −4 , while the third satisfies: Thus, to a very good approximation one can set θ e 13 = θ e 23 = 0, and in this approximation U e takes the form [6]: It follows from the above discussion that in the basis in which the charged lepton and the RH neutrino mass matrices are diagonal, the matrix of neutrino Yukawa couplings Y ν has the form: In the same basis, the Majorana mass term for the LH flavour neutrinos, generated by the see-saw mechanism, is given by: where Using the approximate expression for U e = Φ R 12 (θ e 12 ), with Φ = diag(1, e iϕ , 1) and R 12 (θ e 12 ) given by eq. (23), we get: where c e 12 = cos θ e 12 , s e 12 = sin θ e 12 . As was shown in [6,12], the phase ϕ in eq. (27) and the Dirac phase δ in eq. (3) are related as follows: Comparing the expressions in the left-hand and right-hand sides of the equation M † e M e = U e (M d e ) 2 U † e and assuming first following [2] that φ 0 ψ ′ 0 ψ 0 ζ ′ 0 < 0 and y ′ ψ 0 ζ 0 < 0 (with cos θ e 12 sin θ e 12 > 0), one finds [6]: The (21)) allows to get the best description of the quark masses and mixing, possible in the model considered. However, one gets similar description also in the case of φ 0 ψ ′ 0 ψ 0 ζ ′ 0 > 0 and y ′ ψ 0 ζ 0 > 0 3 . In this latter case we get for ϕ and δ: Numerically, for ϕ = π/4 and s e 12 = 0.22/3 (see eq. (22)), the PMNS matrix, eq. (27), reads: Taking into account the corrections due to the non-zero values of the angles θ e 13 and θ e 23 in U † e on finds [2]: Obviously, the differences between the approximate and the "exact" matrices (31) and (32) are negligibly small. The leading order predictions of the SU (5)×T ′ model for sin θ 13 , sin 2 θ 12 and sin 2 θ 23 were given in the Introduction (see eqs. (2), (4), (6) and the related discussions). They can be obtained by comparing eqs. (3) and (27) and using eq. (22).
Equations (31) and (32) allow to determine the values of the Majorana phases α 21 and α 31 . In the parametrisation in which the PMNS matrix is written in eqs. (27), (31) and (32) they are fixed by the matrix Q = diag(1, 1, ±i) and read α 21 /2 = 0 and α 31 /2 = π/2 or 3π/2. Thus, α 21 and α 31 are CP conserving. Note, however, that the parametrisation of the PMNS matrix in eqs. (27), (31) and (32) does not coincide with the standard one. Thus, in order to get the values of the Dirac and Majorana phases δ and α 21 /2 and α 31 /2 of the standard parametrisation of the PMNS matrix, one has to bring the expressions (31) or (32) in a form which corresponds to the "standard" one in eq. (3). This can be done by using the freedom of multiplying the rows of the PMNS matrix with arbitrary phases and by shifting some of the common phases of the columns to a diagonal phase matrixQ. The results for the "approximate" and "exact" numerical matrices, eqs. (31) and (32), is: and [6] U ≃ whereQ a = diag(e −i3.14 • , e i1.45 • , −1) andQ e = diag(e −i3.55 • , e i1.63 • , −1). Now comparing eq. (33) and eq. (34) with eq. (3) we can obtain the "approximate" and "exact" values of the Dirac and the two Majorana phases of the standard parametrisation of the PMNS matrix, predicted by the model. For the Dirac phase, for instance, we find, respectively, δ ∼ = 226.7 • and [2] δ ∼ = 226.9 • . Note that the Majorana phases α 21 /2 and α 31 /2 in the standard parametrisation are not CP conserving [6]: due to the matrixQ a (orQ e ) they get small CP violating corrections to the CP conserving values 0 and π/2 or 3π/2. The high precision provided by the expression (27) for the PMNS matrix is more than sufficient for the purposes of our investigation and we will use it in our further analysis. This allows to get simple analytic results for the CP violating asymmetries, relevant in leptogenesis, which in turn makes transparent and easy to interpret the results we are going to obtain.
Equation (25), as is well known, allows to express Y ν in terms of U , D ν , D N and an orthogonal (in general, complex) matrix [26] R, R T R = RR T = 1: From eqs. (24) - (35) and (13), we obtain the following exact expression for the matrix R: Using the explicit forms of Q = diag(1, 1, ±i), M diag D , U T BM , S and D N = M diag(1, 1, 1) we get: The same expression for the matrix R was obtained in [27]. Thus, in the SU (5) × T ′ model considered, the R matrix is real, i.e., CP conserving [28] (see also [29]), and symmetric, R * = R, R T = R, and the elements R k3 = R 3k = 0, k = 1, 2. We note that the signs of the entries in the 1-2 sector of R depend on the signs of X and Z: the signs in eq. (37) correspond to X > 0 and Z < 0 (see eq. (19) and the related comments).

III. RADIATIVELY INDUCED LEPTOGENESIS
As we have seen, the three heavy Majorana neutrinos N j are degenerate in mass at the scale M X at which the Majorana mass matrix of the RH neutrinos is generated. We will assume that this scale does not exceed the GUT scale, M GUT = 2 × 10 16 GeV: M X ≤ M GUT . Actually, in the SUSY SU (5) × T ′ model considered, we have M X = M GUT . Given the fact that the R matrix is real and CP conserving, the baryon asymmetry can only be generated in the regime of flavoured leptogenesis [30,31]. Since the mass splittings |∆M ij (M FLG )| thus generated are exceedingly small, we expect the baryon asymmetry to be generated in the regime of resonant flavoured leptogenesis [28,36].
In the case of resonant flavoured leptogenesis, the CP violating asymmetry in the lepton charge L l , l = e, µ, τ , generated in the out of equilibrium decays of the heavy Majorana neutrino N j taking place at the scale M F LG , is given by [28]: Here where Y ν is defined in eqs. (35), and The parameter δ N ji describes the deviation from complete degeneracy of the masses of the the heavy Majorana neutrinos N j and N i . All quantities which appear in eqs. (38) - (41) should be evaluated at the leptogenesis scale M F LG . The baryon asymmetry is generated in the regime of resonant leptogenesis if at M F LG the following condition is fulfilled: We have discussed above the asymmetry generated in the decays of the heavy Majorana neutrinos N i into the Higgs and lepton doublets. A lepton flavour asymmetry ǫl i is also generated from the out-of-equilibrium decays of N i in the Higgsino and slepton doubletsl. Similarly, the sneutrinosÑ i generate CP asymmetries ǫ l i and ǫl i with, respectively, ℓ andl in the final state. As can be shown, one has neglecting soft SUSY breaking terms: ǫ ℓ i = ǫl i = ǫ l i = ǫl i . It follows from eq. (39) that the necessary conditions for a successful resonant flavoured leptogenesis include: i) the presence of CP violating phases in the matrix of neutrino Yukawa couplings Y ν ; ii) non-vanishing off-diagonal elements of the matrix Y † ν Y ν : (Y † ν Y ν ) ij = 0 for i = j; iii) non-degeneracy of the heavy Majorana neutrino masses M i : δ N ji = 0, i = j. The first requirement is fulfilled by the presence of the CP violating phases in the neutrino mixing matrix U . The second and third general requirements are satisfied, as we are going to discuss next, owing to the RG corrections in the quantities M i and Y ν , which have to be included when the latter are evaluated at the leptogenesis scale M F LG . The RG running of the heavy Majorana neutrino masses M i depends on the quantity Y † ν Y ν [33]. It proves convenient to work at the scale M X in a basis of the heavy Majorana neutrino fields in which the matrix Y † ν Y ν is diagonal. This can be achieved by performing an orthogonal transformation of N j . The latter can be done without affecting the heavy Majorana neutrino mass term since at the scale of interest the heavy Majorana neutrinos N j are degenerate in mass. The change of basis, where O is an orthogonal matrix, implies the following change of the matrix of neutrino Yukawa couplings: Y ′ ν = Y ν O. Using eq. (35) and the facts that D N = M diag(1, 1, 1) and the matrix R is real and orthogonal, there always exists an orthogonal matrix O such that RO, and correspondingly Y ′ † ν Y ′ ν , are diagonal matrices. Taking into account the explicit form of the matrix R in the model considered, eq. (37), in what follows we will use It is easy to verify that RO = diag(−1, 1, −1) ≡ I, I jj = η j , with η 2 = − η 1,3 = 1. The matrix of neutrino Yukawa couplings Y ′ ν is given by: In the new basis we have where Γ ′ i is the N ′ i total decay width. The expression for the CP violating asymmetry ǫ ℓ i in the new basis in which Y ′ † ν Y ′ ν is diagonal at M X can be obtained from eqs. (38) -(40) by replacing Y ν and Γ i with Y ′ ν and Γ ′ i , respectively. Note, however, that in the new basis we have I ℓ ij = 0. Thus, the CP violating asymmetries ǫ ℓ i will be zero unless non-diagonal elements of Y ′ † ν Y ′ ν are radiatively generated at the leptogenesis scale M F LG < (≪)M X .
As will be shown later, in the model considered a non-zero baryon asymmetry can be produced only in the regime of 3-flavoured leptogenesis, i.e. for M < (1 + tan 2 β) × 10 9 GeV ∼ < 4.9 × 10 12 GeV, where we have used the constraint tan β ∼ < 70 (see 4 , e.g., [37]). Taking into account that m i ∼ < 5 × 10 −2 eV and v = 174 GeV, we get |( In the new basis, the running of the heavy Majorana neutrino masses M i is governed by the following equation [33]:  [34,35] lead for |(Y ′ † ν Y ′ ν ) ii | ≪ 1 to the following expression for the mass splitting parameter δ N ji : For M X /M = 2 × 10 6 , 2 × 10 5 and 2 × 10 4 , we gett = 0.092, 0.077, 0.063. The corresponding values of δ N ji are given in Table I.  The elements of the matrix of neutrino Yukawa couplings Y ′ ν also evolve with the scale µ when the latter diminishes from M X to M F LG ∼ = M . This change is governed by the RG equations for (Y ′ ν ) ℓi , whose general form was given in [32][33][34][35]. In the case considered by us we have at M X : In this case the RG equations for (Y ′ ν ) ℓi [35] simplify considerably and read: where y q , q = u, c, t, and y ℓ , ℓ = e, µ, τ , are the charge 2/3 quark and charged lepton Yukawa couplings, g 1,2 are the U (1) Y and SU (2) gauge couplings of the Standard Model and we have neglected terms ∝ Y ′ † ν Y ′ ν . The quantities which appear in the square brackets in the r.h.s. of eq. (48) evolve with the scale µ as it decreases from M X , but the effects of their evolution are subdominant for the problem under study and we will neglect them. Thus, we will use their values at the scale M X , which will be assumed to be close, or equal, to M GUT .
We are interested in the quantities (Y ′ † ν Y ′ ν ) ij , i = j, at the scale M F LG ∼ = M , which enter into the expression for the CP violating asymmetry ǫ ℓ i . Since these quantities are zero at M X , they can get non-zero values at M F LG due only to the term involving the charged lepton Yukawa coupling y 2 ℓ in the RG equation (48) [34,35]. The solutions of the RG equations (48) in the leading logarithmic approximation lead to the following result: where y τ = (m τ /v d ) ∼ = (m τ /v) 1 + tan 2 β is the τ Yukawa coupling, m τ being the τ mass, and v = v 2 u + v 2 d = 174 GeV. Neglecting relatively small effects, the quantities in the r.h.s. of eq. (49) can be taken at the scale M X . Note that even though at M X the off-diagonal elements of Y ′ † ν Y ′ ν are zero, they have non-zero values at the leptogenesis scale M F LG due to the radiative corrections.
Using the result obtained for (Y ′ † ν Y ′ ν ) ij , eq. (49), and eqs. (38), (44) and (45), we get for the CP violating asymmetry: It follows from the expression (50) for ǫ ℓ i we have derived that ǫ e i +ǫ µ i +ǫ τ i = 0, i = 1, 2, 3. This result is a consequence of the fact that the R matrix in the model considered is CP conserving (see, e.g., [28]). One can easily convince oneself using the explicit expression for the PMNS matrix (27) that we also have: ǫ τ i = 0, i = 1, 2, 3. The same conclusion is reached also if one uses the PMNS matrix in which the higher order corrections have been included 6 , eq. (32) or (34). Thus, in the SUSY SU (5) × T ′ model of interest the baryon asymmetry can be generated only in the regime of 3-flavoured leptogenesis [31]. 5 For a matrix of neutrino Yukawa couplings Yν such that Re[(Y † ν Yν ) ij ] = 0, i = j, the RG equations for Yν have a singularity in the case of degenerate in mass heavy Majorana neutrinos [32,34,35]. As a consequence, the quantity (Y † ν Yν ) ij , i = j, that enters into the expression for the CP violating asymmetry ǫ ℓ i , does not vary continuously with the scale when the latter changes from M X to M F LG . This fact was not taken into account in the calculation of the asymmetries ǫ ℓ i performed in [27]. Since in the basis in which we work we have (Y ′ † ν Y ′ ν ) ij = 0, i = j, at M X at which M j = M , j = 1, 2, 3, the indicated problem does not appear when we consider the RG evolution of (Y ′ ν ) ℓi and of (Y ′ † ν Y ′ ν ) ij . 6 It is claimed in [27] that ǫ τ i = 0, which does not correspond to the result ǫ τ i = 0 we obtain. The latter is not difficult to verify.
The requirement that the baryon asymmetry is generated in the 3-flavoured thermal leptogenesis regime combined with the upper limit on tan β implies: M ∼ < 4.9 × 10 12 GeV. As is not difficult to show, we have for M ∼ < 10 13 GeV: For M = 10 13 GeV, (δ N ji ) 2 is bigger by a factor of 10 than the term in the right-hand side of the above inequality. Neglecting the correction due to the latter, we get a rather simple expression for the asymmetry ǫ ℓ i : where we have used eqs. (47) and (50). Expression (52) for ǫ ℓ i does not depend explicitly on the masses of the heavy Majorana neutrinos and on the RG factor t. Thus, the CP-asymmetries ǫ ℓ i are entirely determined by the τ Yukawa coupling and the low-energy neutrino mixing parameters, i.e., the neutrino masses, the neutrino mixing angles and CP violating phases in the neutrino mixing matrix. They depend weakly on scales M X and M , e.g., via the running of the τ Yukawa coupling. The asymmetries ǫ ℓ i depend quadratically on the τ Yukawa coupling and thus on tan 2 β. This dependence is crucial for having a viable thermal leptogenesis in the SU (5) × T ′ model of flavour under consideration.
The final lepton number asymmetry, which is partially converted into a non-zero baryon number asymmetry by the fast sphaleron interactions in the thermal bath in the Early Universe, receives a contribution from the outof-equilibrium decays of the three heavy Majorana neutrinos (sneutrinos) N ′ i (Ñ ′ i ), which are quasi-degenerate in mass. The amount of matter-antimatter asymmetry predicted by the model is computed numerically by solving the corresponding system of Boltzmann equations. We report below the relevant set of Boltzmann equations in supersymmetric leptogenesis [38,39] for the lepton flavour (lepton charge) asymmetriesŶ The entropy density, s, and the expansion rate of the Universe, H(T ), are given by: where g * = 228.75 [31] and m P l ≃ 1.22 × 10 19 GeV is the Planck mass. In the case in which the soft SUSY breaking terms are negligible, the thermal rates in (54)-(56) satisfy the conditions [31]: γ i X = γĩ X and γ i,ℓ X = γ i,l X = γĩ ,ℓ X = γĩ ,l X . As a good approximation, supersymmetric leptogenesis proceeds as a manifest generalization of the standard leptogenesis scenario of the type I see-saw extension of the SM. Indeed, new effects due to different supersymmetric equilibration mechanisms between particle and sparticle number densities provide typically only relatively small corrections [39], which can be safely neglected for the purposes of the present study.
The dominant contribution to the production and damping of the lepton asymmetries is generally provided by decays and inverse decays of N ′ i [40], whose thermal averaged rates are where K 1 (z) is a modified Bessel function of the second kind. We neglect in (54)-(56), for simplicity, thermal corrections to the CP asymmetries and the decay/scattering rates [40]. We do not include either the ∆L = 2 washout of the flavour lepton asymmetries in the Boltzmann equations listed above because they are subdominant at the temperatures at which the 3-flavoured leptogenesis takes place. 8 The final baryon number density (normalized to the entropy density of the Universe) is: In order to have successful leptogenesis, the CP asymmetries ǫ ℓ i (ℓ = e, µ) should be sufficiently large and should have the correct sign. According to eq. (53), the sign of ǫ e i = −ǫ µ i and, consequently, of Y B , is fixed by the value of the rephasing invariant associated to the Dirac phase δ, J CP . Numerically, from (53), we get for tan 2 β ≫ 1: where we have used eq. (1) and y 2 τ ≃ 10 −4 tan 2 β. Taking, more explicitly, tan β = 10, one easily obtains: ǫ e 1 ≃ −sgn(sin δ) 1.4 × 10 −6 , ǫ e 2 ≃ sgn(sin δ) 7.0 × 10 −7 ǫ e 3 ≃ sgn(sin δ) 1.3 × 10 −7 , which, in general, is the right order of magnitude of the CP asymmetry in order to have a successful leptogenesis. Notice that sgn(sin δ) is equal either to (−1) or to (+1), depending on whether the Dirac phase δ ∼ = 5π/4 or δ ∼ = π/4, which are the two approximate values δ can have in the model considered (see eqs. (29) and (30)). Taking into account eq. (60), expression (59) can be recast in the form: where Y eq N ′ = 45/(π 4 g * ) ≃ 2 × 10 −3 , ǫ ≡ 10 −6 and η B > 0 is, by definition, the efficiency factor of the asymmetry. It follows from eqs. (7) and (62) that for δ = 226.93 • ≃ 5π/4, the baryon asymmetry has the wrong sign. Thus, the observed value of the baryon asymmetry can be obtained in the model considered only for δ ≃ π/4.
The the efficiency factor η B in eq. (62) can be computed by solving the full system of Boltzmann equations (54)-(56). We note that in the model considered the parameter η B does not depend on the leptogenesis scale M F LG ∼ M . This can be easily understood if one considers, for simplicity, the solution of the Boltzmann equations where only decay and inverse decay processes are included: as we have already mentioned, this is a good approximation in thermal flavoured leptogenesis. In this case, from eqs (45), (57) and (58) one has: Therefore, the Boltzmann equations do not explicitly depend on the heavy Majorana neutrino mass scale M within the indicated approximation. We verified numerically that the dependance of η B and Y B on M is relatively weak also if we take into account the scattering processes. This implies that, in the class of SUSY see-saw models of the type considered in this paper, the leptogenesis scale M F LG can be lowered sufficiently in order to avoid the potential gravitino problem 9 .
From eq. (62) and the numerical value of Y B thus computed, we get an efficiency factor η B ≃ 0.07. Obviously, one can get a value of Y B closer to the mean best fit valueȲ obs B = 8.77 × 10 −11 for a somewhat smaller value of tan β. We would like to conclude with the following remarks. As we have shown, the correct sign of the baryon asymmetry in the SU (5) × T ′ × Z 12 × Z ′ 12 model considered [1,2] can be obtained only in the case of δ ∼ = π/4. As has already been discussed in the Introduction, for this value of the Dirac phase δ we have sin 2 θ 12 ∼ = 0.37, while the current neutrino oscillation data imply at 3σ sin 2 θ 12 ∼ < 0.36 [8], or sin 2 θ 12 ∼ < 0.374 [18], depending on the details of the analysis. For δ ∼ = 5π/4, the value of sin 2 θ 12 ∼ = 0.299 predicted by the model lies within the 1σ interval of values suggested by the data, but the predicted baryon asymmetry of the Universe has the wrong sign 10 (see eq. (62)). If sin 2 θ 12 ∼ = 0.37 would be definitely excluded by future data, one would have to modify the SU (5) × T ′ model of flavour we have considered in the present article. One possible "minimal" modification could be to lift the degeneracy in mass of the the three heavy Majorana neutrinos (sneutrinos) at the scale M X , at which the flavour symmetry is spontaneously broken. This could be achieved, e.g., by replacing the chiral superfield S in the SU (5) × T ′ × Z 12 × Z ′ 12 invariant superpotential of [2] with a new chiral supermultiplet χ, which is a Standard Model singlet and is charged only under the discrete group Z ′ 12 , with charge ω 2 . The model, therefore, has the same gauge and flavour symmetry groups and the same number of fields as the one discussed in [2]. In this new scenario, the flavour structure of the 9 The Davidson-Ibarra bound [41] does not apply in the radiative leptogenesis scenario discussed by us. 10 Our result for the sign of the baryon asymmetry in the case of δ = 226.93 • ≃ 5π/4 contradicts the claim made in [27]. superpotential naturally generates a Majorana mass matrix (term) for the heavy RH neutrinos at the scale M X . The latter is still diagonalised by the tri-bimaximal mixing matrix U T BM , but has non-degenerate eigenvalues. The low energy phenomenology, as well as the generation of the baryon asymmetry of this class of models is therefore worthwhile investigating, but such an investigation lies outside the scope of the present work.