CP Violation in Predictive Neutrino Mass Structures

We study the CP violation effects from two types of neutrino mass matrices with (i) $(M_\nu)_{ee}=0$, and (ii) $(M_\nu)_{ee}=(M_\nu)_{e\mu}=0$, which can be realized by the high dimensional lepton number violating operators $\bar \ell_R^c\gamma^\mu L_L (D_\mu \Phi)\Phi^2$ and $\bar \ell_R^c l_R (D_\mu{\Phi})^2\Phi^2$, respectively. In (i), the neutrino mass spectrum is in the normal ordering with the lightest neutrino mass within the range $0.002\,{\rm eV}\lesssim m_0\lesssim 0.007\,{\rm eV}$. Furthermore, for a given value of $m_0$, there are two solutions for the two Majorana phases $\alpha_{21}$ and $\alpha_{31}$, whereas the Dirac phase $\delta$ is arbitrary. For (ii), the parameters of $m_0$, $\delta$, $\alpha_{21}$, and $\alpha_{31}$ can be completely determined. We calculate the CP violating asymmetries in neutrino-antineutrino oscillations for both mass textures of (i) and (ii), which are closely related to the CP violating Majorana phases.

In the present paper, we investigate a relevant question: to what extent can the conditions (M ν ) ee = 0 and (M ν ) ee = (M ν ) eµ = 0 restrict the neutrino mass matrix structure, especially the leptonic CP violating phases, based on measured quantities from oscillation experiments?
With the predicted neutrino mass parameters, the next question is how to test the above two texture-zero structures by measuring all the relevant parameters in the neutrino mass matrices, especially the non-trivial Majorana phases. Previous studies showed that neutrinoantineutrino oscillations gave us prospective approaches to probe the Majorana phases [68][69][70][71][72][73][74][75][76][77], which is impossible for the conventional (anti)neutrino-(anti)neutrino oscillation experiments. We find that once the possible regions of these phases are depicted for the present two textures, the associated CP violating asymmetries of the neutrino-antineutrino oscillations can be predicted. As will be shown later, by appropriately choosing the (anti)neutrino beam energy and baseline length, some of the asymmetries can be of O(1).
This paper is organized as follows. In Sec. II we study the implications of the texturezero conditions (M ν ) ee = 0 and (M ν ) ee = (M ν ) eµ = 0 to the unknown neutrino mass parameters, including the lightest neutrino mass and the three CP violating phases, based on the existing data. With the preferred values of these parameters, we predict the CP violating asymmetries in the neutrino-antineutrino oscillations for both textures in Sec. III.
In Sec. IV, we give a short summary. where the charged lepton mass matrix is diagonal, the neutrino mass matrix defined in the can be decomposed as follows, where m 1,2,3 are three neutrino masses. V is the charged current leptonic mixing matrix [78,79], conventionally expressed in the standard parametrization as [80,81] where s ij ≡ sin θ ij , c ij ≡ cos θ ij , δ is the Dirac phase, and α 21,31 represent two Majorana phases within the range [0, 2π]. The values of θ 12 , θ 23 , θ 13 , ∆m 2 21 and |∆m 2 32 | have already been obtained from the neutrino oscillation experiments [81], so that the rest four unknown neutrino parameters are the three CP phases, δ, α 21 and α 32 , and the lightest neutrino mass, m 0 . Note that only the absolute value ∆m 2 32 has been acquired, which leaves us two possible orderings: the normal ordering for ∆m 2 32 > 0 with (m 1 , m 2 , m 3 ) = (m 0 , m 2 0 + ∆m 2 21 , m 2 0 + ∆m 2 31 ), and the inverted one for ∆m 2 32 < 0 with (m 1 , m 2 , m 3 ) = ( m 2 0 − ∆m 2 31 , m 2 0 − ∆m 2 31 + ∆m 2 21 , m 0 ). M ν can have some special approximate texture-zero forms when it is generated by some gives the leading contribution to neutrino masses, then (M ν ) ℓℓ ′ should be approximately proportional to the sum of charged lepton masses, m ℓ + m ℓ ′ , with ℓ and ℓ ′ = e, µ, τ . Consequently, (M ν ) ee should be much smaller than other elements. Similarly, if O 9 =l c R ℓ R (D µ Φ) 2 Φ 2 dominates over other LNV operators, (M ν ) ℓℓ ′ will be proportional to m ℓ m ℓ ′ . It turns out that not only (M ν ) ee but also (M ν ) eµ are expected be greatly suppressed due to the hierarchy in the charged lepton masses. In other words, the neutrino mass matrices obtained from these LNV effective operators are characterized by the special zero textures (M ν ) ee = 0 and (M ν ) ee = (M ν ) eµ = 0 1 , the implications of which will be discussed in detail in the following two subsections. 1 Besides the relative smallness of the element (M ν ) ee already argued in the main text for the two highdimensional effective operators of O 7 and O 9 , its absolute value is further constrained by the neutrinoless double beta (0νββ) decay processes [82][83][84][85][86][87][88]. Note that these two effective operators give the dominant contributions to the 0νββ decay at tree level, while the Majorana mass terms arising from O 7(9) begins at one-(two-)loop level. Due to the absence of the loop suppression, these two operators are more sensitive to the 0νββ decay processes, which constrain the cutoff scales and Wilson coefficients of the effective operators greatly and lead to the conclusion that (M ν ) ee < 10 −13 eV. For further details, please refer to Refs. [32] and [38].
By expanding the right-hand side of Eq. (1) with the standard parametrization of V in Eq. (2), the condition (M ν ) ee = 0 can be transformed into the following relation, where the phase ∆ ≡ α 31 −2δ is defined, which will be used to replace α 31 as an independent Majorana phase hereafter. Note that this equation excludes the inverted ordering at more than 2σ significance by current oscillation experiment results [81], so that we only need to consider the normal-ordering neutrino mass matrix from now on. and ∆, differentiated by the positive or negative sin α 21 , which are shown in Fig. 1 as red or blue curves/shadows. Another interesting observation is that the obtained α 21 is limited around π, which can be understood directly from Eq. (3). Since s 2 13 is very small, the third term in Eq. (3) can be neglected, and the first two terms must balance each other to achieve the constraint of the vanishing (M ν ) ee , which only requires α 21 ∼ π in order to reverse the sign of the second term. Moreover, α 21 is precisely predicted to be 1.1π or 0.9π when m 0 is located within 0.004 eV m 0 0.005 eV 2 , no matter how experimental errors vary. Finally, we remark that (M ν ) ee = 0 does not provide any constraint on δ, which is only contained in ∆. If one focus on the real M ν , then δ can be taken as 0 or π, for the cases m 0 = (m 0 ) min and (m 0 ) max . Therefore, there are 4 independent real neutrino mass matrices for (M ν ) ee = 0.
2 Similar results are also given in Ref. [76]. In this subsection, we will concentrate on the case with (M ν ) ee = (M ν ) eµ = 0. Note that such constraints correspond to two complex equations, which enable us to uniquely solve for the remaining four parameters (m 0 , δ, α 21 , ∆) in the neutrino mass matrix undetermined from the current oscillation experiments. Now we sketch the procedure of deriving these quantities in terms of the measured observables [28]. The first step is to write down the two conditions in the parametrization independent form: with which we can obtain the following useful formulas with Since neither |X| nor |Y | depends on the two Majorana phases and m 0 , we can determine the Dirac phase δ from Eq. (6). By substituting the obtained Dirac phase into Eq. (7), we can solve for m 0 . Finally, two Majorana phases can be fixed with Eq. (8). In the standard parametrization, the solution is expressed by [28] cos δ = s −1 with r ≡ ∆m 2 21 /(∆m 2 32 + ∆m 2 21 /2). Note that for each value of m 0 , we can obtain two solutions of the CP violating phases (δ, α 21 , and ∆), which can be connected with each other by the replacements of δ → 2π − δ, α 21 → 2π − α 21 , and ∆ → 2π − ∆.  [81]. Moreover, in the present texture-zero case, when Dirac phase δ is taken to be the CP conserving values, such as δ = 0 or π, the two Majorana ones can only be CP conserving values too, i.e., α 21 and ∆ should be 0 or π. However, with the results shown in Fig. 2, it is interesting that the CP conserving cases are excluded at the 1σ level. Finally, if the Dirac phase is taken to be of the maximal CP violating value with δ = π/2 (−π/2), the Majorana phases are predicted to be α 21 = 0.88π (1.12π) and ∆ = 1.42π (0.58π) with the experimental central values for the mixing angles. nos [71]. However, even in this case, it is still practically impossible to observe these neutrino-antineutrino oscillations, as will be shown below.
The general formulas for the neutrino-antineutrino oscillation probabilities P (ν α →ν β ) and P (ν β →ν α ) in the three-flavor framework are [71] where K andK are the kinetic factors with |K| = |K| and L is the neutrino traveling length. Now it is interesting to estimate the neutrino-antineutrino oscillation probabilities for different channels to see if they have the potential to be observed under the present experimental status, especially the Mössbauer neutrinos advertised in Ref. [71]. By assuming the kinematic factor K ∼ O(1), electron antineutrino energy E ∼ 18.6 keV, and oscillation baseline length L ∼ 300 m, we can obtain the largest ν e −ν e oscillation probability to be P (ν e →ν e ) ∼ O(10 −13 ) for m 0 = 0.0065 eV. The largest probabilities for other oscillation channels, such as P (ν e → ν µ ), would be of the similar order. In the view of these simple exercises, it seems impossible to observe these oscillations practically in the foreseeable experiments.
It is obvious that P (ν α →ν β ) and its CP conjugate process P (ν α →ν β ) can have different values when V is complex, which is the origin of CP violation in the lepton sector. Therefore, we can define the CP asymmetry parameter A αβ by where f = (L/E)(∆m 2 21 /π). In the following, we shall use the obtained CP violating phases from the previous two texture-zero neutrino mass matrices to predict the oscillation probabilities and the associated CP violating asymmetries in some neutrino-antineutrino oscillation channels of great phenomenological interest, and then see how these measurements can help us to probe or constrain the whole picture of neutrino masses.  Table I.  Therefore, the ν e -ν e oscillation is the most prospective channel to probe this neutrino mass texture.

IV. SUMMARY
We have studied the CP violating asymmetries and related LNV processes such as the  It is interesting to consider other probes to the Majorana character of the neutrino masses, such as rare LNV meson decays. It is well-known that ordinary channels with Majorana neutrino mass insertions are too small to be observed in the near future. However, it is remarkable that the effective operators, such as O 7 and O 9 , would give new leading-order contributions. For concreteness, let us consider the process B + → π − µ + µ + . If Majorana neutrino masses are induced by O 9 , the dominant channel to this process is given by the Feynman diagram in Fig. 4a, as this tree-level diagram does not involve the tiny Majorana neutrino masses which arise at two-loop level via O 9 . However, with the model parameters fixed by the observed neutrino masses as in Ref. [38], a simple estimation shows that the typical branching ratio for this process is to be of O(10 −25 ). Other LNV rare meson decays, like K + → π − µ + µ + , would have even smaller branching ratios. Similar results can also be obtained for O 7 from Fig. 4b. As a result, it seems also impossible to measure such LNV meson decays practically.

ACKNOWLEDGMENTS
The work was supported in part by National Center for Theoretical Science, National and National Tsing Hua University (104N2724E1).