A 3 X 2 texture for neutrino oscillations and leptogenesis

In an economical system with only two heavy right handed neutrinos, we postulate a new texture for $3 \times 2$ Dirac mass matrix $m_D$. This model implies one massless light neutrino and thus displays only two patterns of mass spectrum for light neutrinos, namely hierarchical or inverse-hierarchical. Both the cases can correctly reproduce all the current neutrino oscillation data with a unique prediction $m_{\nu_e \nu_e} = \frac{\sqrt{\Delta m^2_{solar}}}{3}$ and $\sqrt{\Delta m^2_{atm}}$ for the hierarchical and the inverse-hierarchica cases, respectively, which can be tested in next generation neutrino-less double beta decay experiments. Introducing a single physical CP phase in $m_D$, we examine baryon asymmetry through leptogenesis. Interestingly, through the CP phase there are correlations between the amount of baryon asymmetry and neutrino oscillation parameters. We find that for a fixed CP phase, the hierarchical case also succeeds in generating the observed baryon asymmetry in our universe, plus a non-vanishing $U_{e3}$ which is accessible in future baseline neutrino oscillation experiments.


Introduction
The origin of the observed baryon asymmetry in our universe, ratio of number of baryons to photons [1] η B = n B − nB n γ = 6.1 ± 0.2 × 10 −10 , is one of the major problems in cosmology. This number has been deduced from two independent observations. (1) From the existing abundance of light elements formed after big bang [2]. (2) Precision measurements of cosmic microwave background [1].
Leptogenesis [3,4] may explain this observed asymmetry between matter and antimatter content of the universe. In explaining this asymmetry one first creates a tiny lepton asymmetry in the early universe. This lepton asymmetry is recycled into observed baryon asymmetry above the electroweak scale via sphaleron interactions [5]. This is possible since sphaleron interactions remain in thermal equilibrium above the electroweak scale, they violate B + L, and since they conserve B − L.
It is widely believed that the lepton asymmetry is formed by out-of-equilibrium, lepton number violating, CP violating decay of heavy right handed neutrinos. Existence of heavy right handed neutrinos also give a natural framework for explaining smallness of neutrino mass via see-saw [6] mechanism. If there are no symmetry structures in the theory to make the right handed neutrinos stable, they must decay. They have non-vanishing Yukawa couplings with Higgs scalars and left handed doublets, complex in general, for them to do so. Therefore we study lepton asymmetry generated by the CP violating decays of heavy right handed neutrinos (with Majorana mass) at the early stage of our universe. Since leptogenesis involves no new interactions apart from those required for see-saw mechanism to succeed, we may expect that the Physics of neutrino oscillations would clarify some deep mystery of cosmology such as the observed asymmetry between matter and antimatter with which it is linked.
We know from pioneering works of Sakharov[7] that CP violation is an essential in- Any model of leptogenesis is required to reproduce these masses and mixing angles.
It is indeed interesting to see that, via see-saw mechanism, existing neutrino data can give desired mass spectrum of heavy right handed neutrinos plus right magnitudes of primordial lepton asymmetry. There are many studies of this kind where there are three heavy right handed neutrinos and generated lepton asymmetry depends on the form of the Yukawa texture [16]. In this paper, we examine the system with only two right handed neutrinos. As discussed above, this system is the minimum one to bring physical CP phases in the lepton sector. Number of free parameters in the neutrino sector is much reduced compared to the usual three right handed neutrino case. However the system still contains an enough number of free parameters to reproduce the current neutrino oscillation data. We introduce a texture for 3 × 2 Dirac neutrino mass matrix by which the number of free parameters is further reduced. With a small number of free parameters, we investigate neutrino oscillation parameters and the amount of baryon asymmetry through leptogenesis. We will see correlations between them through a CP phase.
This paper is organized as follows. In the next section, we introduce 3 × 2 Dirac mass matrix and a texture for it. We begin with the CP invariant case and apply a simple ansatz to the light neutrino mass matrix so as to reproduce the current best fit values for the neutrino oscillation parameters. In Sec. 3, we introduce a single CP phase and examine baryon asymmetry generated through leptogenesis and neutrino oscillations.
With four input parameters (two light and two heavy neutrino mass eigenvalues), all neutrino oscillation parameters as well as the baryon asymmetry through leptogenesis are shown as a function of only the CP phase. Also, the averaged neutrino mass relevant to neutrino-less double beta decay experiments and the Jarlskog invariant characterizing CP violation in the lepton sector are presented as a function of the CP phase. We see correlations among these outputs, and presents some predictions for a fixed CP phase.
The last section is devoted to conclusions.

Texture and a simple ansatz in CP invariant case
Without loss generality, we begin with a reference basis in which the charged lepton mass matrix m l , Dirac mass matrix m D and the right handed Majorana mass matrix M R are written as where all parameters are real and 0 < M 1 < M 2 . We have three physical CP phases in m D by re-phasing. There is no triplet Higgs in the model, so left handed neutrinos do not have a Majorana mass at the beginning.
After see-saw mechanism, the light neutrino mass matrix becomes Note that the system with 3 × 2 Dirac mass matrix leads to Therefore, at least, one mass eigenvalue of light neutrinos is zero. Concerning the current best fit values of neutrino oscillation data, we can conclude that only two patterns of diagonalized mass matrix for light neutrinos are possible. One is the so-called hierarchical case, with m 2 = ∆m 2 12 and m 3 = ∆m 2 12 + ∆m 2 23 . The other is the so-called inversehierarchical case, with m 1 = −∆m 2 12 + ∆m 2 23 and m 2 = ∆m 2 23 . Now we introduce a texture for the Dirac mass matrix as c 1 = 0 and δ 2 = 0, and m D becomes a more simple form, with a single CP phase φ. The texture reduces the number of free parameters into six and allows us to analyze the correlations between the amount of baryon asymmetry and neutrino oscillation parameters with the single CP-phase. Similar textures have been discussed in Ref. [8]. The explicit form of the light neutrino mass matrix is given by Six parameters in the Dirac mass matrix and two heavy neutrino masses, eight parameters in total, correspond to physics of neutrinos.
We first tackle only neutrino oscillations in the CP invariant case, φ = 0. CP violation will be introduced in the next section. As our stating point, we impose a simple ansatz that m ν is diagonalized by the so-called tri-bimaximal mixing matrix [12], In fact, the tri-bimaximal mixing matrix is in excellent agreement with the current best fit values in Eq. (2). This ansatz strongly constrains the parameters in Eq. (8). For the hierarchical case, solving Eq. (10) with Eq. (9) For the inverse-hierarchical case, we find while for the inverse-hierarchical case, Here we have used an approximation m 1 ≃ m 2 ≃ ∆m 2 23 for ∆m 2 12 ≪ ∆m 2 23 , in the inverse-hierarchical case. Therefore we have a chance of testing this relations in future neutrino-less double beta decay experiments with the sensitivity m νeνe ≥ 10 −3 eV.

Numerical analysis in CP violating case
In this section we will introduce non-zero CP phase φ in the texture of Eq. (8). Except the CP phase, parameters in the Dirac mass matrix are described in terms of four free parameters as in Eq. (11) or in Eq. (12). With non-zero CP phase, we can obtain baryon asymmetry through leptogenesis. In addition, the Dirac mass matrix becomes complex and resultant neutrino oscillation parameters are deviating from those in the CP invariant case. We will see correlations among resultant neutrino oscillation parameters and the amount of baryon asymmetry created via leptogenesis.
Let us first consider leptogenesis. Primordial lepton asymmetry in the universe is generated through CP violating out-of-equilibrium decay of the lightest heavy neutrinos, which is characterized by the CP violating parameter ǫ [13], This formula for asymmetry is valid in the basis where the right handed neutrino is diagonal. Here, v = 246 GeV is the vacuum expectation value of Higgs field, and Sphaleron processes will convert this lepton asymmetry into baryon asymmetry and, as a result, the baryon asymmetry is approximately described as Here κ < 1 is the efficiency factor, that parameterizes dilution effects for generated lepton asymmetry through washing-out processes. To evaluate the baryon asymmetry precisely, numerical calculations [14] are necessary. We use a fitting formula of the efficiency factor given in terms of effective light neutrino massm such that [15] κ = 2 × 10 −2 0.01eṼ m To understand these results, it is useful to give explicit formulas for leptogenesis in terms of parameters in the texture of Eq. (8). The CP violating parameter and the effective massm are, respectively, written as In the hierarchical case, the parameters fixed in Eq. (11) gives Here we have used an approximation formula F (M 2 2 /M 2 1 ) ≃ 3M 1 /M 2 , assuming M 1 ≪ M 2 . Our result is independent of M 2 as long as M 1 ≪ M 2 . To obtain a formula for the inverse-hierarchical case, we use an approximation, m 1 = −∆m 2 12 + ∆m 2 23 ≃ ∆m 2 23 (1 + 0.5(∆m 2 12 /∆m 2 23 )). Thus the parameters in Eq. (12)  If M 1 is very large, for example M 1 ≥ 10 15 GeV, we can give sufficient baryon asymmetry even with the suppression. However, in thermal leptogenesis re-heating temperature after inflation would be larger than the lightest heavy neutrino mass. It would be difficult to achieve such a quite high reheating temperature in usual reheating scenarios. We need some other mechanism such as a resonant leptogenesis [17] to enhance the primordial lepton asymmetry.
Now we analyze the neutrino oscillation parameters in the case of non-zero CP phase.
Parameters c i in the texture are fixed as discussed in the previous section, and lead to the tri-bimaximal mixing matrix in the CP invariant case. When we switch CP phase on, the Dirac mass matrix becomes complex and, as a result, output oscillation parameters are deviating from the CP invariant case. In particular, we will find non-vanishing U e3 .
Substituting parameters given in Eq. (11) or Eq. (12) into the light neutrino mass matrix of Eq. (9), we find that m ν is independent of M 1 and M 2 even for non-zero parameters are functions of only the CP phase φ.
In the hierarchical case with inputs m 2 = 9.59 × 10 −3 eV and m 3 = 4.56 × 10 −2 eV, resultant oscillation parameters are depicted in Fig. 2 They are all consistent with observations. Non-vanishing U e3 is our prediction, whose value would be covered in future baseline neutrino oscillation experiments. As can be seem from Eq. (9), the ν e ν e element of m ν is independent of the CP phase and we obtain the same result as Eq. (13), numerically, In the inverse-hierarchical case with input parameters m 1 = 4.46 × 10 −3 eV and m 2 = 4.56 × 10 −2 eV, resultant oscillation parameters are depicted in Fig. 5 and 6. In this case, we find ∆m 2 23 ≃ 2.05 × 10 −3 eV 2 and sin 2 θ 23 = 1, (almost) independent of the CP-phase. Although the inverse-hierarchical case cannot provide the observed baryon asymmetry, output oscillation parameters are consistent with the current data for a small CP phase φ ≤ 1.04(rad). Again, the ν e ν e element of m ν is independent of the CP phase, and we obtain the same result as Eq. (14), numerically, This is an order of magnitude larger than the value in the hierarchical case.
It is also interesting to see a correlation between the baryon asymmetry through leptogenesis and the leptonic CP violating phase (Dirac phase) [18]. CP violation in the lepton sector is characterized by the Jarlskog invariant [19], where δ is the Dirac phase. The Jarlskog invariant as a function of the CP phase φ is depicted in Fig. 7 for (a) the hierarchical and (b) the inverse-hierarchical cases, respectively. We obtain a small but non-vanishing J CP correlating with other outputs. In the hierarchical case, we find for φ = 0.668(rad).

Conclusions
Neutrino oscillation experiments have explored neutrino masses and mixing patterns.
Tiny neutrino masses compared to the ordinary quark masses are naturally explained by the see-saw mechanism with heavy right handed neutrinos. Right handed neutrinos play the important role to generate the baryon asymmetry in our universe through leptogenesis.
Leptogenesis requires CP violation in the lepton sector. For CP to violate we must have at least two right handed neutrinos. Keeping this minimal possibility in mind we have introduced only two heavy right handed neutrinos and studied a 3 × 2 Dirac type mass matrix m D . Without loss of generality one can choose a reference basis where both charged lepton mass matrix as well as the heavy right handed Majorana mass matrices are real and diagonal. In this basis, three physical CP phases appear in m D .
After the see-saw mechanism we obtain an effective 3 × 3 Majorana mass matrix for light neutrinos. As a result from the 3 × 2 Dirac mass matrix m D , light neutrino mass spectrum should contain (at least) one zero mass eigenvalue. This fact allows only two patterns for neutrino mass spectrum, normal hierarchical or inverse-hierarchical.
We have chosen a simple texture for m D in our reference basis. and ∆m 2 23 in the hierarchical and the inverse-hierarchical cases, respectively. These results can be tested in next generation experiments of neutrino-less double beta decay.
We have worked in the context of a specific texture. However, as an extension to our approach one can introduce small c 1 and check whether the solutions reported in this article get drastically modified. This is so because, often in real world models, one may be able to restrict c 1 such that it is very small yet not exactly zero. If a real c 1 is introduced in the complex case (φ = 0), and its magnitude is of order 10% of the rest of the c i s, we see a 50% variation in ǫ and U e3 . However, ∆m 2 solar , ∆m 2 atm , θ 12 , θ 23 remain almost the same.     Figure 3: Neutrino mixing angles as a function of the CP phase in unites of π. The region between two horizontal lines in Fig. 3(a) is consistent with the current best fit values in Eq. (2). The entire region shown in Fig. 3(b) is allowed.