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A Predictive Ansatz for Neutrino Mixing and Leptogenesis

We propose a non-SO(10) modification of the Buchmueller-Wyler ansatz for neutrino mixing and leptogenesis, in which charged lepton, Dirac and Majorana neutrino mass matrices have fewer free parameters. Predictions of this new ansatz for three light neutrino masses, three lepton flavor mixing angles, the neutrinoless double-beta decay and the cosmological matter-antimatter asymmetry are all in very good agreement with current experimental and observational data.


I. INTRODUCTION
The atmospheric and solar neutrino oscillations observed in the Super-Kamiokande [1] and SNO [2] experiments have provided rather convincing evidence that neutrinos are massive and lepton flavors are mixed. To interpret the smallness of neutrino masses and the largeness of lepton flavor mixing angles, as indicated by current solar and atmospheric neutrino data, many theoretical models and phenomenological ansätze have been proposed [3]. Among them, the one proposed by Buchmüller and Wyler [4] is of particular interest, because (a) it is based on the simplest SO (10) lepton-quark mass relations [5] and the seesaw mechanism [6]; (b) it leads to quite specific predictions for the light neutrino masses, lepton flavor mixing angles, CP violation, and the neutrinoless double-β decay; and (c) it is able to predict the cosmological matter-antimatter asymmetry via a very attractive mechanism -leptogenesis [7].
The present paper aims to propose a non-SO(10) modification of the Buchmüller-Wyler ansatz, in order to account for current experimental data in a more accurate and more flexible way. To see why our effort makes sense, let us consider the following relation obtained by Buchmüller and Wyler [4]: where ǫ is a small expansion parameter of the Dirac neutrino mass matrix; (∆m 2 sun , ∆m 2 atm ) and (θ sun , θ atm ) are the mass-squared differences and mixing angles of solar and atmospheric neutrino oscillations, respectively. From current Super-Kamiokande [1] and SNO [2] data, one obtains ∆m 2 sun = (3.3 − 17) × 10 −5 eV 2 , tan 2 θ sun = 0.30 − 0.58 (2) at the 90% confidence level [8] 1 ; and ∆m 2 atm = (1.6 − 3.9) × 10 −3 eV 2 , sin 2 2θ atm > 0.92 (3) at the same confidence level [11]. Then it is straightforward to get ǫ > 0.1 from Eqs. (1), (2) and (3). This lower bound of ǫ is not consistent with ǫ ≈ m u /m c ≈ 0.04 − 0.08 obtained from the SO(10) relation between Dirac neutrino and up-type quark mass matrices (i.e., M D = M u [4]). Hence one may wonder whether a modification of the Buchmüller-Wyler ansatz is possible, so as to avoid any inconsistency with data or any fine-tuning of the parameter space. It is obvious that the Buchmüller-Wyler ansatz will get much flexibility to accommodate the present data on solar and atmospheric neutrino oscillations, if the relevant SO (10) lepton-quark mass relations are suspended. In this spirit, we propose a non-SO(10) modification of the Buchmüller-Wyler ansatz, in which charged lepton, Dirac and Majorana neutrino mass matrices have fewer free parameters. The predictability of such a new ansatz is expected to be more powerful. It is worthwhile to emphasize that we are following a purely phenomenological approach. We hope that the good agreement of our results with current experimental and observational data may shed light on some appropriate ways of model building, either within or beyond grand unified theories.
The remainder of this paper is organized as follows. In section 2, we specify the textures of charged lepton, Dirac and Majorana neutrino mass matrices, from which one may see both similarities and differences between the new ansatz and its original version. Section 3 is devoted to explicit predictions of this new ansatz for light neutrino masses, lepton flavor mixing angles, CP violation, neutrino oscillations, and the neutrinoless double-β decay. We calculate the lepton asymmetry in section 4, and translate it into the baryon asymmetry via leptogenesis. Finally, a brief summary is given in section 5.

II. LEPTON MASS MATRICES
A simple extension of the standard model is to include one right-handed neutrino in each of three lepton families, while the Lagrangian of electroweak interactions keeps invariant under the SU(2) L × U(1) Y gauge transformation [7]. After spontaneous symmetry breaking, the lepton mass term can be written as where ν c α ≡ Cν T α with C being the charge-conjugation operator (for α = e, µ, τ ); and M l , M D and M R stand respectively for the charged lepton, Dirac neutrino and Majorana neutrino mass matrices. We expect that the scale of M l and M D is characterized by the gauge symmetry breaking scale v ≈ 175 GeV. The scale of M R may be much higher than v, because right-handed neutrinos are SU(2) L singlets and their mass term is not subject to the electroweak symmetry breaking. As a consequence, the 3 × 3 light neutrino mass matrix M ν arises from diagonalizing the 6 × 6 neutrino mass matrix and takes the seesaw form [6] M Given specific textures of M D and M R , one can calculate the mass eigenvalues of M ν . The phenomenon of lepton flavor mixing at low energy scales stems from a non-trivial mismatch between diagonalizations of M l and M ν . In contrast, the lepton asymmetry at high energy scales depends on complex M D and M R [7]. Now let us propose a non-SO(10) modification of the Buchmüller-Wyler ansatz [4] for neutrino mixing and leptogenesis. First of all, we assume M l and M D to be symmetric matrices, just like M R . Second, we assume that the (1,1), (1,3) and (3,1) elements of M l , M D and M R are all vanishing in a specific flavor basis, in analogy to a phenomenologicallyfavored texture of quark mass matrices M u and M d [12]. Note that we do not invoke any direct relationship between (M D , M l ) and (M u , M d ) (such as M D = M u and M l = M d in the SO(10) grand unified theory [4]). Instead, we assume that the non-zero elements of M D and M l can be expanded in terms of the Wolfenstein parameter λ ≈ 0.22 [13]. It is well known that the mass spectra of charged leptons and quarks are hierarchical [14]: We conjecture that M D might have a similar hierarchy as M d , but its dominant mass eigenvalue should be close to the electroweak symmetry breaking scale v ≈ 175 GeV. To be more explicit, we take whereλ ≡ λe iω ; (x, y) are real and positive coefficients of O(1); and m 0 ≈ v holds. It is easy to check that three mass eigenvalues of M D have the hierarchy λ 4 : λ 2 : 1, and those of M l have the hierarchy shown in Eq. (7). The hierarchical structure of M l implies that its contribution to lepton flavor mixing is very small and even negligible. Thus we expect that large lepton mixing angles observed in solar and atmospheric neutrino oscillations are essentially attributed to the light neutrino mass matrix M ν in our ansatz. Because the (1,1), (1,3) and (3,1) elements of both M R and M D have been assumed to be vanishing, M ν must have the same texture zeros via the seesaw relation in Eq. (6) [15]. To generate a sufficiently large mixing angle in the ν µ -ν τ sector to fit current Super-Kamiokande data on atmospheric neutrino oscillations [1], the (2,2), (2,3), (3,2) and (3,3) elements of M ν should be comparable in magnitude. This requirement is actually strong enough to constrain the texture of M R in a quite unique way, as first observed by Buchmüller and Wyler [4]. For our purpose, we obtain 2 where z is a real and positive coefficient of O (1), and M 0 ≫ v holds. The texture of M ν turns out to be where (1), and 2ϕ ≡ 2ζ − 5ω. Note that an overall phase factor e i(5ω−π) has been omitted from the right-hand side of Eq. (10), since it has no contribution to lepton flavor mixing and CP violation at low energy scales. We remark that the (2,2), (2,3), (3,2) and (3,3) elements of M ν in Eq. (10) are all of O (1), from which a large mixing angle (around π/4) can be obtained for the ν µ -ν τ sector. It is due to such a prerequisite that the hierarchy of M R can almost uniquely be fixed through the seesaw relation between M R and M ν . In other words, we essentially require little information about the ν e -ν µ sector of M ν to arrive at Eq. (9) 3 . As the ν µ -ν τ sector of M ν is relatively insensitive to the renormalization effects from one scale to another [16], we expect that our phenomenological constraints on the texture of M R at high energy scales make sense. To generate a large mixing angle in the ν e -ν µ sector to fit current Super-Kamiokande [1] and SNO [2] data on solar neutrino oscillations, the condition must be satisfied [4,17]. Some instructive constraints on the parameter space of x, z, ω and ζ can be drawn from Eq. (11), as one will see below.
It is worthwhile at this point to comment on two major differences of the present ansatz from its original version [4]: (a) We do not assume any SO(10) lepton-quark mass relations. Therefore the pattern of M D can be taken as Eq. (8) with a structural hierarchy weaker than before. This modification will lead to a ratio of ∆m 2 sun to ∆m 2 atm at the percent level (i.e., ǫ 2 in Eq. (1) is replaced by λ 2 ), consistent very well with current experimental data. In this sense, we would say that the phenomenological success of this new ansatz may compensate for the theoretical cost for having discarded the simplest SO(10) mass relations.
(b) The number of free parameters in M D and M R is reduced from ten [4] to five (x, z, M 0 , ω, and ζ). To do so, we have expanded M D in terms of the complex parameterλ and M R in terms of the real parameter λ. Such a treatment is plausible, since the Majorana neutrino mass matrix M R is a priori independent of the Dirac neutrino mass matrix M D . While |λ| = λ holds by definition, it actually reflects the requirement of a large mixing angle in the ν µ -ν τ sector of M ν , which imposes strict constraints on M D and M R via the seesaw mechanism. Because of the reduction of free parameters, the new ansatz is expected to have more powerful predictability.

III. NEUTRINO MIXING
The symmetric neutrino mass matrix M ν in Eq. (10) can be diagonalized by a 3 × 3 unitary matrix V , where m 1 , m 2 and m 3 are physical (real and positive) masses of three light neutrinos. As pointed out above, the contribution of M l to lepton flavor mixing is expected to be very small and even negligible. Therefore the matrix V , which links the neutrino mass eigenstates (ν 1 , ν 2 , ν 3 ) to the neutrino flavor eigenstates (ν e , ν µ , ν τ ), can well describe the dominant effects of lepton flavor mixing at low energy scales. Current experimental data on solar, atmospheric and reactor neutrino oscillations [1,2,18] strongly suggest that |V e3 | ≪ 1, |V e1 | ∼ |V e2 | and |V µ3 | ∼ |V τ 3 | hold. Then a parametrization of V needs two big mixing angles (θ x and θ y ) and one small mixing angle (θ z ) [19], in addition to a few complex phases. After some lengthy but straightfoward calculations, we obtain 4 in which s a ≡ sin θ a and c a ≡ cos θ a (for a = x, y, z), α ≡ ω + ζ, β ≡ 5ω − ζ = ζ − 2ϕ, and 2γ ≡ arg(z ′ e i2ϕ − 1). The explicit expressions of three mixing angles (θ x , θ y , θ z ) are In addition, three neutrino massses are given by We can see that θ z is as small as we have expected, and a normal neutrino mass hierarchy m 1 : m 2 : m 3 ∼ λ : λ : 1 shows up. It is worth mentioning that our instructive results for (m 1 , m 2 , m 3 ) and (θ x , θ y , θ z ) will not get dramatic variations, if arbitrary coefficients of O(1) are taken for those non-zero elements in M D and M R . The reason is simply that the hierarchical structures of M D and M R guarantee a stable texture of M ν , from which the light neutrino masses and flavor mixing angles can straightforwardly be derived. For instance, a replacement λ 4 =⇒ Aλ 4 with |A| ∼ O(1) for the (2,3) and (3,2) elements of M R does not affect the pattern of M ν in the leading-order approximation [4]. Given a replacementλ 2 =⇒ Bλ 2 with |B| ∼ O(1) for the (2,3) and (3,2) elements of M D , the only variation of M ν is that its corresponding (2,3) and (3,2) elements change from 1 to B. In this case, we find and It is obvious that Eqs. (14) and (15) can be reproduced, respectively, from Eqs. (16) and (17) with the choice |B| = 1. Therefore, small deviations of |B| from unity do not give rise to significant changes of the results obtained in Eqs. (14) and (15). Note that an arbitrary coefficient of O(1) for the (3,3) (14) and (15) are discussed in order.
(1) The hierarchy of three light neutrino masses allows us to determine the absolute value of m 3 from the observed mass-squared difference of atmospheric neutrino oscillations ∆m 2 atm ≡ |m 2 3 − m 2 2 | ≈ m 2 3 . Using the recent Super-Kamiokande data listed in Eq. (2), we obtain Given m 0 ≈ v for the Dirac neutrino mass matrix M D , the mass scale of three heavy Majorana neutrinos turns out to be We observe that this mass scale is not far away from the scale of grand unified theories Λ GUT ∼ 10 16 GeV.
(2) The small parameter δ defined in Eq. (11) can well be constrained, if we take account of current experimental data on the mass-squared difference of solar neutrino oscillations ∆m 2 sun ≡ |m 2 2 − m 2 1 | shown in Eq. (2). As the ratio R ≡ ∆m 2 sun /∆m 2 atm is given by we obtain δ ≈ 0.21 − 1.2 for λ ≈ 0.22. Note that δ > 0.5 is apparently in conflict with our original assumption δ ∼ O(λ) in Eq. (11). Therefore the reasonable range of δ should be δ ≈ 0.21 − 0.50, which leads in turn to R ≈ (0.85 − 2.5) × 10 −2 . Subsequently we fix δ = √ 2λ ≈ 0.31 as a typical input. (3) Using δ = √ 2λ, we explicitly obtain To a good degree of accuracy, the mixing factors of solar, atmospheric and reactor neutrino oscillations are associated respectively with θ x , θ y and θ z [20]. From Eq. (14) or Eq. (21), we get Note that we have kept the δ-induced correction to sin 2 2θ atm , in order to illustrate its small departure from unity (maximal mixing). The typical results in Eq. (22) are in good agreement with current Super-Kamiokande [1], SNO [2] and CHOOZ [18] data. (4) Due to the mass hierarchy of three light neutrinos, our ansatz predicts a relatively small value for the effective mass term of the neutrinoless double-β decay: which seems hopeless to be detected in practice. Indeed the present experimental upper bound is m ee < 0.35 eV at the 90% confidence level [21]. (5) CP or T violation in normal neutrino oscillations is measured by a universal and rephasing-invariant parameter J [22], which can be calculated as follows: where sin 2γ ≈ 2x δ sin 2ϕ − z δ sin(2ϕ + ω) .
Because of x ∼ z ≫ δ ∼ λ, a significant cancellation on the right-hand side of Eq. (25) is naturally expected. There exists an interesting parameter space, in which Considering Eq. (11), one may easily check that δ = √ 2λ does hold for the chosen values of x, z, ω and ζ. It is particularly amazing that sin 2γ = 1 holds in this case. Therefore we obtain J ≈ λ/(4 √ 10) ≈ 2%. CP violation at the percent level could be measured in the future at neutrino factories [23].
if M 0 = 6.0 × 10 14 GeV and z = 1 + √ 2λ are typically taken. A lepton asymmetry may result from the interference between tree-level and one-loop amplitudes of the decay of the lightest heavy Majorana neutrino with mass M 1 [7]. This asymmetry can be expressed, in the physical basis where M R is diagonal and the Dirac neutrino mass matrix takes the form M D U * instead of M D , as [24] where v ≈ 175 GeV denotes the electroweak scale. In writing out Eq. (31), we have taken account of the strong mass hierarchy M 1 ≪ M 2 ≪ M 3 . With the help of Eqs. (8), (28) and (29), we get Once the parameters x, z, ω and ζ are specified, one will be able to predict the magnitude of ε 1 from Eq. (32). For the purpose of illustration, we adopt the specific parameter space given in Eq. (26) to evaluate the size of ε 1 . The result is or numerically ε 1 ≈ −5.2 × 10 −6 . To translate this lepton asymmetry into the baryon asymmetry of the universe [7], one needs to calculate a suppression factor κ induced by the lepton-number-violating wash-out processes [25]. Note that κ depends closely on the following quantity: which characterizes the out-of-equilibrium decay rate of the lightest heavy Majorana neutrino with mass M 1 . In Eq. (34), g * ≈ 100 represents the number of massless degree of freedom at the time of the decay, and M Pl ≈ 1.22 × 10 19 GeV is the Planck mass scale. Making use of the typical inputs taken above for Eq. (30), we arrive at K R ≈ 73. The suppression factor κ can then be calculated with the help of an approximate parametrization [25] obtained from integrating the Boltzmann equations (for 10 ≤ K R ≤ 10 6 ): Finally we get an instructive prediction for the asymmetry between baryon (n B ) and antibaryon (nB) numbers of the universe: where s denotes the entropy density, and c = −8/15 describes the fraction of ε 1 converted into Y B via sphaleron processes in the framework of three lepton-quark families and two Higgs doublets [26]. One can see that our result is consistent quite well with the observed baryon asymmetry, Y B ≈ (1 − 10) × 10 −11 [27]. Of course, one may go beyond the typical parameter space taken in Eq. (26) to make a delicate analysis of all measurables or observables, only if the condition in Eq. (11) is satisfied. It is remarkable that we can quantitatively interpret both the baryon asymmetry of the universe and the small mass-squared differences and large mixing factors of solar and atmospheric neutrino oscillations. In this sense, our ansatz is a complete phenomenological ansatz favored by current experimental and observational data, although it has not been incorporated into a convincing theoretical model. A number of different ansätze on leptogenesis and neutrino oscillations have recently been proposed [28,29], but some of them turn to be ruled out by the present Super-Kamiokande and SNO data.
Future neutrino experiments will test the present ansatz and help to distinguish it from other viable models in the following four aspects: (a) Our ansatz predicts the ratio of ∆m 2 sun to ∆m 2 atm (i.e., R) to be around 1%. If more accurate solar and atmospheric neutrino data yield R < 0.5% or R > 5%, our ansatz will somehow become disfavored.
(b) The prediction of our ansatz for the smallest lepton mixing angle is quite certain: . This result may easily be examined in a variety of long-baseline neutrino oscillation experiments [30]. Some other ansätze [28] have given quite different predictions for θ z , either much larger or much smaller than ours.
(c) The magnitude of m ee predicted by our ansatz (of order 10 −4 eV) is too small to be measured in any proposed experiments for the neutrinoless double-β decay [31]. If an unambiguous signal of the neutrinoless double-β decay is observed in the near future, the present ansatz will definitely be ruled out.
(d) In our ansatz, the rephasing-invariant parameter of CP violation (i.e., J ) is predicted to be at the percent level. This strength of leptonic CP or T violation could be detected in the far future at neutrino factories [23]. Therefore, another criterion to discriminate between our ansatz and other viable models is to see how large the CP-violating effects can be in neutrino oscillations.

V. SUMMARY
In summary, we have proposed a non-SO(10) modification of the Buchmüller-Wyler ansatz for neutrino mixing and leptogenesis. Its consequences on the light neutrino masses, lepton flavor mixing angles, the neutrinoless double-β decay, and the baryon asymmetry are all in good agreement with current experimental and observational data. In particular, an indirect connection between the lepton asymmetry at high energy scales and CP violation in neutrino oscillations has shown up in such a specific ansatz. We expect that phenomenological ansätze of this nature will get more stringent tests in the era of long-baseline neutrino oscillation experiments. On the theoretical side, a deeper understanding of the origin of fermion masses, flavor mixing and CP violation becomes more desirable than before.

ACKNOWLEDGMENTS
The author would like to thank H. Fritzsch for warm hospitality in Universität München, where part of this work was done. He is also grateful to W. Buchmüller for enlightening discussions in Bamberg, and to H. Fritzsch for helpful comments in Amsterdam.