Minimal seesaw model with tri/bi-maximal mixing and leptogenesis

We examine minimal seesaw mechanism in which the masses of light neutrinos are described with tri/bi-maximal mixing in the basis where the charged-lepton Yukawa matrix and heavy Majorana neutrino mass matrix are diagonal. We search for all possible Dirac mass textures which contain at least one zero entry in $3 \times 2$ matrix and evaluate the corresponding lepton asymmetries. We present the baryon asymmetry in terms of a single low energy unknown, a Majorana CP phase to be clued from neutrinoless double beta decay.


I. INTRODUCTION
Thanks to the accumulating data from experiments on the atmospheric and solar neutrinos experiments [1,2,3], we are now convinced that neutrinos oscillate among three active neutrinos. Interpreting each experiment in terms of two-flavor mixing, the mixing angle for the oscillation of atmospheric neutrinos is understood to be maximal or nearly maximal: sin 2 2θ atm ≃ 1, , whereas the one for the oscillation of solar neutrinos is not maximal but large: sin 2 θ sol ≃ 0.3, [6]. The upper bound for θ reac , sin θ reac 0.2, was obtained from the non-observation of the disappearance of ν e in the Chooz experiment [4] with ∆m 2 ≤ 10 −3 eV 2 . The unitary mixing matrix is defined via ν a = 3 j=1 U aj ν j (a = e, µ, τ ), where ν a is a flavor eigenstate and ν j is a mass eigenstate. Including data from SNO [3] and KamLand [5], the range of the magnitude of the MNS mixing matrix is given by [7,8,9,10], at the 90% confidence level. The existing data also show that the neutrino mass squared differences which induce the solar and atmospheric neutrino oscillations are ∆m 2 sol ≃ 7 +10 −2 × 10 −5 eV 2 and ∆m 2 atm ≃ 2.5 +1.4 −0.9 × 10 −3 eV 2 , respectively. It can be readily recognized that the central values of elements in the mixing matrix in Eq.(1) are pointing an elegant form, which is called tri/bi-maximal mixing [11], which consists of sin θ sol = 1 √ 3 and sin θ atm = 1 √ 2 . There are some literatures [12] which proposed textures of the mass matrix based on the particular mixing type U T B .
On the other hand, the baryon density of our universe Ω B h 2 = 0.0224 ± 0.0009 implied by WMAP(Wilkinson Microwave Anisotropy Probe) data indicates the observed baryon asymmetry in the Universe [13,14], where n B , nB and n γ are number density of baryon, anti-baryon and photon, respectively. The leptogenesis [15] has become a compelling theory to explain the observed baryon asymmetry in the universe, due to increasing reliance on the seesaw mechanism from experiments. Theory for lepton asymmetry requires two heavy right-handed neutrinos or more. For that reason, a class of models with two heavy right-handed neutrinos and 3×2 neutrino Dirac mass matrix is called the minimal neutrino seesaw models(MNSMs) which were intensively studied by several authors recently [16,17], especially for simple Dirac mass textures that make prediction compatible with solar and atmospheric neutrino data. The main framework of our work is seesaw mechanism in bottom-up approach. We launch our analysis by taking U T B for mixing of light neutrinos and then investigate the structure of 3×2 Dirac matrix. That is, our concern remains on the combination of tri/bi-maximal mixing and MNSMs, in order to study the phenomenological implication of the high energy theory based on the low energy theory. One advantage of our framework is that physical observables can be explained in minimal terms of physical parameters. In Section II, we present the light neutrino mass matrix in terms of the mixing given in Eq.(2) and mass squared differences measured in experiment. The mass matrix reconstructed in that way will constrain Dirac mass matrix. In subsections, depending on the type of mass hierarchy, possible 3 × 2 Dirac matrices will be examined carefully. In Section III, leptogenesis will be discussed in details based on the Dirac matrices investigated before. In section IV, we will present numerical results on leptogenesis in our scheme and a relationship between leptogenesis and neutrinoless double beta decay as well as the lower bound of M 1 will be discussed focusing on the contribution from a single Majorana phase.

II. DIRAC MASS MATRICES IN MINIMAL SEESAW
In general, a unitary mixing matrix for 3 generations of neutrinos is given bỹ where R's are rotations with three angles and a Dirac phase δ and the P = Diag 1, e iϕ/2 , e iϕ ′ /2 with Majorana phases ϕ and ϕ ′ is a diagonal phase transformation. The mass matrix of light neutrinos is given by where U ≡Ũ P −1 andm 2 ≡ m 2 e iϕ andm 3 ≡ m 3 e iϕ ′ . If the U T B in Eq.(2) is adopted for the U in Eq.(5), the light neutrino mass is which orients toward a minimal model of neutrino sector by removing an angle and the Dirac phase. With SNO and KamLand, data have narrowed down the possible mass spectrum of neutrinos into two types, Normal Hierarchy (N H), m 1 m 2 < m 3 , and Inverse Hierarchy (IH), m 3 < m 1 m 2 for MSW LMA. Those two types include the possibility of zero mass for a neutrino, which is necessarily followed by relegating one of the Majorana phases to the unphysical. In other words, the minimal model with the physical observables which the present experimental data guarantee can be obtained by U T B and dictating zero mass to one generation of neutrinos, where the non-zero physical parameters in the model consist of 2 masses, 2 angles, one Majorana phase.
When only two of three neutrinos are massive, by accommodating the experimental results ∆m 2 sol = m 2 2 − m 2 1 and ∆m 2 atm = |m 2 3 − m 2 2 | to the two types of mass hierarchies, one can obtain the following expressions for mass eigenvalues, Phase transformation P = Diag 1, e iϕ/2 , 1 now can replace the phase transformation in Eq.(4) without loss of generality, whether N H or IH, so that one can singlem 2 out in order to investigate the CP violating contribution of the Majorana phase.
Effective neutrino mass models with one zero mass eigenvalue involved in three active neutrinos can be generated naturally from MNSMs. In the basis the mass matrix M R of right-handed neutrinos N R = (N 1 , N 2 ) is diagonal, the model is given from which the light masses are derived through the seesaw mechanism, On the other hand, the matrix m D is found as the solution to the seesaw mechanism in bottom-up approach once one launches the analysis with the light neutrino masses M ν . Let M 1 and M 2 be the masses of right-handed neutrinos and M ij the elements of the matrix M ν . The Dirac matrix, is resulted in with where the i is 2 or 3, the σ i is a sign ±, and the sign of a 1 is fixed as positive. The solution in eq. (12) was first derived and formulated in Ref. [17]. It is clear that only 5 parameters out of 6; a 1 , b 1 , a i , b i , can be specified in terms of the elements of M ν . There are various ways to decrease the number of parameters in Dirac matrix, posing one or more zeros or posing equalities between elements for the matrix texture. It is known that texture zeros or equalities among matrix entries can be generated by imposing additional symmetries to the theory. In this paper, we focus on posing one or more zeros in Dirac matrix, and show that only one-zero textures are allowed for N H and only one-zero and two-zero textures are allowed for IH, accompanied with the low energy mixing U T B . On the other hand, from Eq.(11), one can recognize that, if there exists a kind of symmetry between entries in M ν , the Dirac matrix also has a symmetry in certain entries inherited from the symmetry of the M ν . So, there are a number of patterns with equalities among the entries in Dirac matrices obtained with one or two zeros, as a consequence of maximal mixing.

A. Normal Hierarchy
With m 1 = 0, the neutrino mass M ν in Eq.(6) reduces to where d = 3m 3 /2m 2 , which, using Eq.(11), gives rise to Dirac matrix with the following entries: where σ 2 σ 3 = −1. Depending on the position of texture zero, the types of Dirac matrices can be classified as follows; The matrix in Eq. (12) features the equalities between M 22 and M 33 and between M 12 and M 13 as a consequence of the maximal mixing of atmospheric neutrinos. In the case with b 2 = 0, it can be recognized that the ratios of a 1 to a 2 and a 3 to a 2 inherit those of M 12 to M 22 and M 23 to M 22 , respectively. b 3 = 0 or a 3 = 0 cases will not be presented as an independent case, since it can be made by exchanging b 3 with b 2 and a 3 with a 2 from N H 2-a and 2-b.

B. Inverse Hierarchy
With m 3 = 0, where x ≡m 2 /m 1 , which, using Eq. (11), gives rise to Dirac matrix with the following entries: where σ ≡ σ i and σ 2 σ 3 = 1. The equality M 22 = M 23 = M 33 in Eq. (14), which is again a consequence of the maximal mixing of atmospheric neutrinos, constrains the elements of the Dirac matrix such that a 2 = a 3 , b 2 = b 3 . Hence, texture with a single zero included appears only if a 1 = 0 or b 1 = 0, while texture with two zeros appears if a 2 = a 3 = 0 or b 2 = b 3 = 0 .
Listed are all the cases with one or more texture zeros in Dirac matrix derivable from the light neutrino mass with U T B , whether N H or IH. In the following, the eligibility of each case to generate the CP asymmetry for leptogenesis will be examined.

III. LEPTOGENESIS
The baryon asymmetry Eq.(3) can be rephrased The n γ is the photon number density and the s is entropy density so that the number density with respect to a comoving volume element is taken into account. The baryon asymmetry produced through sphaleron process is related to the lepton asymmetry [18,19] by where a ≡ (8N F + 4N H )/(22N F + 13N H ), for example, a = 28/79 for the Standard Model(SM) with three generations of fermions and a single Higgs doublet, N F = 3, N H = 1. The purpose of this work is to estimate whether the Yukawa interaction which produces the light neutrinos with the mixing Eq.(2) through the seesaw mechanism can also generate a sufficient lepton asymmetry for the observed baryon asymmetry. The generation of a lepton asymmetry requires the CP-asymmetry and out-of-equilibrium condition. The Y L is explicitly parameterized by two factors, ε, the size of CP asymmetry, and κ, the dilution factor from washout process.
where g * ≃ 110 is the number of relativistic degree of freedom. The ε i is the magnitude of CP asymmetry in decays of heavy Majorana neutrinos [20,21], where i denotes a generation. When one of two generations of right neutrinos has a mass far below that for the other generation, i.e., M 1 < M 2 , the ε i in Eq. (19) is obtained from the decay of M 1 [22,23,24], where v = 174 GeV and f (M 2 /M 1 ) represents loop contribution to the decay width from vertex and self energy and is given by for the Standard Model. For large value of x, the leading order of f (x) is (−3/2)x −1 .
It is convenient to consider separately the factor that depends on Dirac matrix in ε 1 in Eq. (20) at this stage.
where a's and b's are defined in Eq. (11). From a number of types of matrices with a texture zero derived in Eq. (13) and Eq. (15), only 6 different non-zero values of ∆ 1 's can be evaluated. Those particular Dirac matrices to contribute the imaginary parts are the matrix with b 2 = 0 and that with a 2 = 0 for N H, and the matrix with b 1 = 0, that with a 1 = 0, that with b 2 = b 3 = 0, and that with a 2 = a 3 = 0 for IH. For N H, if a 1 = 0, or b 1 = 0, the (m † D m D ) 12 vanishes from the trivial relation between entries. Applying Eq.(13) and Eq.(15) for Eq. (22), one can find that each type of Dirac matrix gives rise to ∆ 1 as follows; where m 2 and m 3 are given in terms of ∆m 2 sol and ∆m 2 atm in Eq. (7), where m 1 and m 2 are given in terms of ∆m 2 sol and ∆m 2 atm in Eq. (8). Thus, for M 2 ≫ M 1 case, the CP asymmetry in Eq. (20) reduces to ε 1 ≈ 3/(16πv 2 )M 1 ∆ 1 , which is now parameterized by the lightest mass of heavy neutrino M 1 and Majorana phase ϕ. The sign of ε 1 depends on the position of a texture zero in a row of Dirac matrix.
The κ in Eq.(18) is determined by solving the full Boltzmann equations. The κ can be simply parameterized in terms of K defined as the ratio of Γ 1 the tree-level decay width of N 1 to H the Hubble parameter at temperature M 1 , where K ≡ Γ 1 /H < 1 describes processes out of thermal equilibrium and κ < 1 describes washout effect [19] [25], κ ∼ 1 2 √ K 2 + 9 for 0 K 10.
The decay width of N 1 by the Yukawa interaction at tree level and Hubble parameter in terms of temperature T and the Planck scale M pl are Γ 1 = m † D m D

11
M 1 /(8πv 2 ) and H = 1.66g 1/2 * T 2 /M pl , respectively. At temperature T = M 1 , the ratio K is which reduces to, using the Dirac matrices in Eq.(10), where all fixed numbers are included in a factor of order. As done for the ∆ 1 's, one can apply Eq.(13) and Eq. (15) for Eq.(32) to find dilution factor κ when the decay width is determined by Yukawa couplings in each type of Dirac matrix. For the six types of Dirac matrices that are eligible for the CP asymmetry as in Eqs.(23)-(28), the ratio K for each case is which shows that the dilution factor also depends on the phase ϕ, but it does not significantly affect the order of magnitude. Out of all the types of Dirac matrices examined, there is no such a case that Yukawa couplings originate decays of neutrinos N 1 which satisfy the out-of-equilibrium condition K < 1 at T = M 1 . The washout effect of asymmetry is most suppressed with the Dirac matrix of type N H 2 − b, where, depending on ϕ, the dilution factor ranges from 0.010 to 0.013, the amount of asymmetry survived from washout is at most about 1%. When T < M 1 , the Boltzmann equations still depict the finite value of κ as M 1 /T increases for the universe evolution [20][21] [26].

IV. DISCUSSION
Based on the formulation of the leptogenesis derived in the previous section, we numerically analyze baryon asymmetry for each case classified as N H or IH. For the numerical calculation, we take ∆m 2 sol = 7.0 × 10 −5 eV 2 and ∆m 2 atm = 2.5 × 10 −3 eV 2 as inputs. Consider a model with neutrino masses in normal hierarchy. In Fig.1, we plot the baryon asymmetry Y B as a function of the Majorana phase ϕ for N H2 − b. The different curves correspond to M 1 = 2.0 × 10 11 to 2.0 × 10 13 GeV for fixed M 2 /M 1 = 5. We note that we can choose any reasonable M 2 /M 1 value which can protect L-violating processes with N 1 from the wash-out when T < M 2 . As expected from Eq. (20), the value of Y B for a fixed ϕ increases with M 1 . The horizontal line in Fig.1 presents the current cosmological observation of Y B given in Eq. (16). From the analysis, we see that the current observation on Y B constrains the lower bound of M 1 , which turns out to be M 1 2.0 × 10 11 GeV. It is clear that the CP asymmetry in high energy is almost proportional to the imaginary part of Majorana CP contribution in low energy from Eqs.(22)-(28). Thus, the plots show that the lower bound of M 1 to generate the observed baryon asymmetry should be raised if the imaginary contribution of low energy phase is decreased as the ϕ approaches 0 or π. In all aspects of the prediction of Y B , N H2 − a and N H2 − b are quite similar to each other except an overall factor. The Y B for N H2 − b is enhanced from both the enhancement of CP asymmetry, ∆ 1 (b)/∆ 1 (a) ≃ 3.6, and the suppression of wash-out effect, κ(b)/κ(a) ≃ 4.5. The lower bound of M 1 with ϕ = π/2 is pulled down to 2.0 × 10 11 GeV for N H2 − b, whereas that for N H2 − a is 3.2 × 10 12 GeV. Suppressing a certain Yukawa coupling by putting a texture zero can vary the amount of the asymmetry by order of magnitude.
In Fig.2, we plot Y B as a function of the Majorana phase ϕ for IH1 − a. The different curves correspond to M 1 = 5.5 × 10 13 to 5.5 × 10 15 GeV for fixed M 2 /M 1 = 5. As in N H, we obtain a lower bound on M 1 5.5 × 10 12 GeV for IH. The prediction of Y B for IH with the same value of M 1 is smaller than that for N H because ∆ 1 for IH is proportional to m 2 2 − m 2 1 which corresponds to the solar mass squared difference, while ∆ 1 for N H is proportional to m 2 3 − m 2 2 which corresponds to the atmospheric mass squared difference. We expect from Eqs. (25)(26)(27)(28)) that the predictions of Y B 's for other cases of IH are almost the same as that for IH 1-a because m 1 m 2 /(2m 2 1 + m 2 2 ) ∼ 1/3.
Although the Majorana phase is not detectable through neutrino oscillations, it may affect the amplitude of neutrinoless double beta decay. Thus, one can anticipate that there may exist a correlation between leptogenesis and neutrinoless double beta decay in our scenario where the heavy Majorana neutrino mass matrix and the charged-lepton Yukawa matrix are both diagonal. The neutrinoless double beta decay amplitude is proportional to the effective Majorana mass | m ee | which can be written in the form: where ϕ i are Majorana CP-violating phases. The | m ee | depends on the CP phase ϕ only with inverted hierarchy, so that one can draw a simple correlation between leptogenesis and neutrinoless double beta decay only for the particular case. In Fig.3, we present a correlation between Y B and | m ee | for IH 1-a. The inputs are taken to be the same as in Fig.2. As the value of | m ee | approaches to that with ϕ = π/2, the asymmetry is enhanced and the bound of M 1 becomes lower. The lower bound of M 1 as a function of Majorana phase or that of effective Majorana mass is obtained from the current cosmological observation of Y B . In Fig.4, we present a correlation between the lower bound of M 1 and | m ee |. We examined the minimal seesaw mechanism of 3 × 2 Dirac matrix by starting our analysis with the masses of light neutrinos with tri/bi-maximal mixing in the basis where the charged-lepton Yukawa matrix and heavy Majorana neutrino mass matrix are diagonal. We found all possible Dirac mass textures which contain one zero entry or two in the matrix and evaluated the corresponding lepton asymmetries. The baryon asymmetry can be presented in terms of low energy observables, where only one Majorana CP phase among them remains yet unknown. The numerical work exhibits the dependence of both the size of baryon asymmetry and the lower bound of M 1 upon the low energy CP phase to be clued from neutrinoless double beta decay.
Note added : After completing this wrok, we have been noticed that similar analysis for the hierarchical case in supersymmetric seesaw model appeared in Ref. [27].