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Leptogenesis and neutrino parameters

We calculate the baryonic asymmetry of the universe in the baryogenesis-via-leptogenesis framework, assuming first a quark-lepton symmetry and then a charged-neutral lepton symmetry. We match the results with the experimentally favoured range. In the first case all the oscillation solutions to the solar neutrino problem, except the large mixing matter solution, can lead to the allowed range, but with fine tuning of the parameters. In the second case the general result is quite similar. Some related theoretical hints are discussed.


I. INTRODUCTION
Strong indications for nonzero neutrino mass and mixing come from solar and atmospheric neutrino experiments. In fact, if interpreted in terms of neutrino oscillations, such experiments, together with the tritium beta decay endpoint, imply small neutrino masses [1].
In the minimal standard model (MSM) the neutrino is massless because there are no right-handed neutrino singlets and there is no Higgs scalar triplet. The simplest way to get a mass for the neutrino field is by adding the right-handed state ν R , the analogue of the quark state u R in the leptonic sector, in which case it becomes possible to build both a Dirac mass term m ν ν L ν R and a Majorana mass term (1/2)m R ν c L ν R for the right-handed neutrino. The Dirac mass m ν is expected to be of the same order of magnitude of the quark or charged lepton masses, while the Majorana mass m R is not constrained. A popular mechanism to obtain a very small neutrino mass is the seesaw mechanism [2], where the right-handed neutrino mass is very large and as a consequence a very light left-handed Majorana neutrino appears, with a mass m L ≃ m 2 ν /m R . The MSM plus the right-handed neutrino (which we would like to call SM) is also a minimal scenario to produce a baryonic asymmetry in the universe, according to the Fukugita-Yanagida baryogenesis-via-leptogenesis mechanism [3,4]. In this framework the out-of-equilibrium decays of right-handed neutrinos generate a leptonic asymmetry which is partially transformed into a baryonic asymmetry by electroweak sphaleron processes [5].
The baryonic asymmetry depends on both the Dirac and the right-handed Majorana neutrino mass matrices. Therefore, assuming a quark-lepton symmetry or a charged-neutral lepton symmetry, we should be able to determine the value of the baryonic asymmetry, and to match it with the experimental bounds coming from nucleosynthesis in the standard big bang theory. This is the main subject of the present paper, already discussed by several authors [6][7][8][9]. However, our approach is quite different, more general and direct. We scan over the neutrino parameter space, using several forms for the Dirac mass matrices, and taking into account the vacuum and matter solutions to the solar neutrino problem. A graphical representation of the results is given, from which one can eventually infer approximate bounds on neutrino parameters. Both the nonsupersymmetric (SM) and the supersymmetric (SSM) cases are considered.
Section II is about neutrino oscillation data, from which one may obtain light neutrino masses and mixings. Section III deals with the quark-lepton symmetry, which allows to get the Dirac and heavy neutrino mass matrices. In section IV, after a short collection of the relevant formulas of the baryogenesis-via-leptogenesis mechanism, the calculation of the baryonic asymmetry is carried out, based on the content of sections II and III. In section V the same calculation is done assuming a charged-neutral lepton symmetry. Finally, in section VI, we give our conclusions and a brief discussion.

II. NEUTRINO PARAMETERS
From the phenomenological point of view the baryonic asymmetry also depends on which solution for solar neutrinos is taken into account. Therefore, in this section we summarize the neutrino oscillation data that we will use in our analysis. For atmospheric neutrinos the best fit is [10]  The latest day-night and spectral data favour the LMA and LOW solutions, but do not exclude the others [12]. A further information on neutrino oscillations comes from the CHOOZ experiment [13] which gives sin θ c 0.16 for ∆m 2 c > 1 × 10 −3 eV 2 . Therefore, neutrinos do have masses and mixings, and a unitary matrix U αi (α = e, µ, τ ; i = 1, 2, 3) relates the mass eigenstates ν i to the weak eigenstates ν α , It is clear that ∆m 2 s ≪ ∆m 2 a . According to ref. [14] we assume where the numbering corresponds to the family index. Moreover, we work with the hierarchical spectrum of light neutrinos, m 1 ≪ m 2 ≪ m 3 . Then, m 2 3 ≃ ∆m 2 a , m 2 2 ≃ ∆m 2 s , and for m 1 we take 10 −4 m 2 < m 1 < 10 −1 m 2 . The mixing matrix U (the MNS matrix [15]) can be written as the standard parametrization of the CKM matrix (including one phase δ ′ ) times a diagonal phase matrix D = diag(e iϕ 1 , e iϕ 2 , 1) [16,17]. Hence, it depends on three angles and three phases. From neutrino oscillation data we can determine the three angles [14,16].
For |U e3 |, related to the result of the CHOOZ experiment, we use the bound while U e2 and U µ3 are obtained from the best fits of atmospheric and solar neutrinos.
Then we are left with five free neutrino parameters: |U e3 |, δ = arg(U e3 ), m 1 , ϕ 1 , ϕ 2 . Choosing values for the free parameters leads to a complete determination of light masses and the mixing matrix U. These will be used in the following section, together with the quark-lepton symmetry, to obtain the heavy neutrino mass matrix.

III. SEESAW MECHANISM WITH QUARK-LEPTON SYMMETRY
The Lagrangian for the relevant lepton sector is (for simplicity we do not write the 1/2 factor in the Majorana terms) where M e is the mass matrix of charged leptons, M ν is the mass matrix of Dirac neutrinos, and M R the mass matrix of right-handed Majorana neutrinos. The effective Lagrangian of the seesaw mechanism is with the light neutrino mass matrix M L given by Setting where D L , D e are diagonal matrices, we obtain the MNS matrix as Inverting eqn. (3) we get the heavy neutrino mass matrix Now, assuming a quark-lepton symmetry, we take the pair of hermitian matrices with one phase α = π/2 in M e12 . These lepton mass matrices are obtained in the following way. We take the five texture zero model for the quark mass matrices M u and M d from ref. [18], which studies the phenomenologically viable textures. The quark mass matrices are related to the lepton mass matrices by an approximate running factor m b /m τ from the high scale where the quark-lepton symmetry should hold [19], and in addiction a factor −3 is included in M e22 in order to have a good relation between charged lepton and down quark masses [20]. For five texture zeros the matrices M u , M d lead to the simple meaningful relations In this way, by means of eqns.(4)- (8), we can calculate M R and then its eigenvalues . The quark-lepton symmetry is usually obtained within unified theories such as SU (5) and mostly SO (10), where quarks and leptons belong to the same multiplets. In particular, the factor −3 in M e22 is due to suitable Yukawa couplings of these multiplets with the 45 (in SU (5)) or 126 (in SO (10)) Higgs representations. However, here we can also regard the quark-lepton symmetry as a phenomenological feature.

IV. THE BARYONIC ASYMMETRY
A baryonic asymmetry can be generated from a leptonic asymmetry [3]. In order to study this baryogenesis-via-leptogenesis mechanism we diagonalize M e : and also M R by means of U R M R U T R = D R : Due to electroweak sphaleron effect, the baryonic asymmetry Y B is related to the leptonic asymmetry Y L by [22] with where N f is the number of families (three) and N H the number of Higgs doublets (one in the SM and two in the SSM; a ≃ 1/3 in both cases). Remember that where n B,B,γ are number densities. The leptonic asymmetry can be written as [4] Y L = d ǫ 1 g * where, in the SM, the CP-violating asymmetry ǫ 1 is given by [23,24] ǫ 1 = 1 with and a factor 4 is included in ǫ 1 [23], due to more decay channels. For a hierarchical spectrum of heavy neutrinos f ≃ −3M 1 /2M j , g ≃ −3M 1 /M j ≃ 2f , with a very good accuracy. Eqn. (14) arises from the interference between the tree level and one loop decay amplitudes of the lightest heavy neutrino, and includes vertex and self-energy corrections. The latter may be dominant if M 1 and M j are nearly equal, so that an enhancement of the asymmetry may occur.
A good approximation for d, the dilution factor, is inferred from refs. [25][26][27]: for k 10 6 , for 10 k 10 6 , and for 1 k 10, 0 k 1, respectively, where the parameter k is and M P is the Planck mass. In the SSM the critical value 10 6 for k is lowered, but in our calculation k remains always much smaller. The presence of the dilution factor in eqn. (13) takes into account the washout effect produced by inverse decay and lepton number violating scattering.
We make a random extraction of the free neutrino parameters for a total of tuning of the parameters. Notice that we plot Log 10 Y B , which is negative. However, the trend is clear. For example, we find the phase δ tuned around π − α, which corresponds to δ ′ = α, sin θ e3 < 0. Moreover, according to ref. [9], we find an enhancement of the asymmetry for |U e3 | ≃ V D12 U µ3 , where V D = U νL U † eL is the mixing matrix in the Dirac sector (the analogue of V CKM = V u V † d ). Since V D12 ≃ (1/3) m d /m s ≃ 0.07, we get the maximum of Y B around |U e3 | ≃ 0.07 · 0.7 ≃ 0.05.
In a similar way one can explain why the SMA solution shows an enhancement, contrary to the LMA solution. In fact, the further condition is U e2 ≃ V D12 U µ2 , which is compatible with the SMA but not with the LMA. The two conditions correspond to the decoupling of M 1 from m 3 and m 2 , respectively. Note that in ref.
[9] V D12 ≃ 0.21 because there the factor −3 in our M e22 is absent. In this case one has a different enhancement value for U e2 , |U e3 |, so that the SMA is also excluded.
The presence of the factor −3 allows the SMA to be reliable for leptogenesis, for the matrix texture (7), (8).
In the supersymmetric case the calculated baryonic asymmetry is increased by a factor nearly 6. In fact, going from the SM to the SSM, there is a factor 4 due to ǫ 1 , a factor 1/ √ 2 due to g * (for d ∼ 1/k) and a factor 2 due to g(x): Hence, in the present context, the SSM works better for leptogenesis, with respect to the SM.
To test the dependence of Y B on matrix texture, let us consider a second pair of hermitian matrices [18], with four texture zeros, which is formed by the same matrix M e as in eqn. (8), but with a further phase π/2 in M e23 , and Note that this form for M ν has entries 1-2 and 2-3 filled in, with respect to matrix (7), while M ν13 = 0. Results useful are in figs. 5, 6, for the SMA, LOW and VO solutions, while the LMA gives very small asymmetry as in the foregoing case. For SMA, LOW to work one has |U e3 | ≃ 0.01, and for VO also δ ≃ π/2.

V. THE CASE OF A CHARGED-NEUTRAL LEPTON SYMMETRY
If there is a charged-neutral lepton symmetry, the Dirac neutrino mass matrix is related to the charged lepton mass matrix rather than to the up quark mass matrix: for example (four texture zeros). This form for M e is obtained by analogy to M d in ref. [18]. There are some theoretical (left-right) models [29] with a charged-neutral lepton symmetry, along with an up-down symmetry. However, again, we can also assume it as a phenomenological hypothesis. The position of phases is somewhat arbitrary. We put phases π/2 in M e12 , M e23 in order to have a mixing in the Dirac sector similar to the previous case. The value of the baryonic asymmetry Y B is quite analogous to the case of a quark-lepton symmetry, see figs. 7, 8. However, the maximum level of asymmetry is now reached for |U e3 | ≃ 0.
Also for this case, we have checked the dependence on matrix texture by using the hermitian matrices formed by six texture zeros with a single phase π/2 in M e13 , but we have found no relevant difference with the asymmetry generated by matrix (20).

VI. CONCLUSION AND DISCUSSION
The baryonic asymmetry Y B has been calculated using a random extraction for five of the nine neutrino parameters (three light masses; three angles and three phases in the mixing matrix) and assuming quark-lepton symmetry or charged-neutral lepton symmetry for the Dirac mass matrices. Other parameters have been checked. As a result, for quark-lepton symmetry, we find that the SMA, VO, and LOW solutions for solar neutrinos are able to generate enough asymmetry, especially in the supersymmetric case, but with fine tuning and selected values of the parameters |U e3 | and δ. For charged-neutral lepton symmetry the general results are similar.
Let us discuss some related theoretical issues. Unified theories such as SO (10), or left-right models such as SU(3) c ×SU(2) L ×SU(2) R ×U(1) B−L , naturally contain heavy Majorana neutrinos, generated at the unification or left-right scale [30], but also contain other particles, for example additional gauge bosons. Usually these particles are much heavier than the lightest heavy Majorana neutrino, so that they are sufficiently decoupled from the leptogenesis process, as confirmed in ref. [31]. In Since the quark-lepton symmetry is natural in unified models, while the chargedneutral (up-down) symmetry is natural in left-right models, in the present context the preferred solution for solar neutrinos could be the SMA. However, neutrino data slightly favour the LMA solution [32]. A possible alternative for the baryogenesisvia-leptogenesis mechanism to work more extensively is by means of horizontal symmetries [33].