Phases of Flavor Neutrino Masses and CP Violation

For flavor neutrino masses M^{PDG}_{ij} (i,j=e,mu,tau) compatible with the phase convention defined by Particle Data Group (PDG), if neutrino mixings are controlled by small corrections to those with sin(theta_{13})=0 denoted by sin(theta_{13})deltaM^{PDG}_{e tau} and sin(theta_{13})deltaM^{PDG}_{tau tau}, CP-violating Dirac phase delta{CP} is calculated by using these corrections. If possible neutrino mass hierarchies are taken into account, the main source of delta{CP} turns out to be deltaM_{e tau}^{PDG} except for the inverted mass hierarchy with {m}_1 approx -{m}_2, where {m}_i=m_ie^{-i varphi_i} (i=1,2) stands for a neutrino mass m_i accompanied by a Majorana phase varphi_i for varphi_{1,2,3} giving two CP-violating Majorana phases. We can further derive that delta_{CP} approx arg(M_{e mu}^{PDG})-arg(M_{mu mu}^{PDG}) with arg (M_{e mu}^{PDG}) approx arg(M_{e tau}^{PDG}) for the normal mass hierarchy and delta_{CP} approx arg(M_{ee}^{PDG})-arg(M_{e tau}^{PDG})+pi for the inverted mass hierarchy with {m}_1 approx {m}_2. For specific flavor neutrino masses M_{ij} whose phases arise from M_{e mu,e tau,tau tau}, these phases can be connected with arg(M_{ij}^{PDG}) (i,j=e,mu,tau). As a result, numerical analysis suggests that Dirac CP-violation becomes maximal as |arg(M_{e mu})| approaches to pi/2 for the inverted mass hierarchy with {m}_1 approx {m}_2 and for the degenerate mass pattern satisfying the inverted mass ordering and that Majorana CP-violation becomes maximal as |arg(M_{tau tau})| approaches to its maximal value around 0.5 for the normal mass hierarchy. Alternative CP-violation induced by three CP-violating Dirac phases is compared with the conventional one induced by delta{CP} and two CP-violating Majorana phases.

, CP-violating Dirac phase δCP is calculated to be δCP ≈arg M P DG * µτ / tan θ23 + M P DG * µµ δM P DG eτ + M P DG ee δM P DG * eτ − tan θ23M P DG eµ δM P DG * τ τ (mod π), where θij (i, j=1,2,3) denotes an i-j neutrino mixing angle. If possible neutrino mass hierarchies are taken into account, the main source of δCP turns out to be δM P DG eτ except for the inverted mass hierarchy withm1 ≈ −m2, wheremi = mie −iϕ i (i=1,2) stands for a neutrino mass mi accompanied by a Majorana phase ϕi for ϕ1,2,3 giving two CP-violating Majorana phases. We can further derive that δCP ≈ arg M P DG eµ − arg M P DG µµ with arg M P DG eµ ≈ arg M P DG eτ for the normal mass hierarchy and δCP ≈ arg M P DG ee − arg M P DG eτ + π for the inverted mass hierarchy withm1 ≈m2. For specific flavor neutrino masses Mij whose phases arise from Meµ,eτ,ττ , these phases can be connected with arg(M P DG ij ) (i, j=e, µ, τ ). As a result, numerical analysis suggests that Dirac CP-violation becomes maximal as |arg(Meµ)| approaches to π/2 for the inverted mass hierarchy withm1 ≈m2 and for the degenerate mass pattern satisfying the inverted mass ordering and that Majorana CP-violation becomes maximal as |arg (Mττ )| approaches to its maximal value around 0.5 for the normal mass hierarchy. Alternative CP-violation induced by three CP-violating Dirac phases is compared with the conventional one induced by δCP and two CP-violating Majorana phases.
PACS numbers: 13.15.+g,14.60.Pq,14.60.St Various experimental evidences of neutrino oscillations provided by the atmospheric [1], solar [2,3], reactor [4,5] and accelerator [6] neutrino oscillation experiments have indicated that neutrinos have tiny masses and their flavor states are mixed with each other. Nowadays, to study CP violation in neutrinos is one of the important issues to be addressed in order to further understand neutrino physics. The recent observation on the nonvanishing reactor neutrino mixing [5] has opened the possibility that details of Dirac CP-violation can be experimentally clarified in near future. Theoretically, effects of CP-violation are described in terms of three phases, one CP-violating Dirac phase δ CP and two CP-violating Majorana phases φ 2,3 [7]. Neutrino mixings are parameterized by the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) unitary matrix U P MN S [8], which converts the massive neutrinos ν i (i = 1, 2, 3) into the flavor neutrinos ν f (f = e, µ, τ ). The standard description of U P MN S adopted by the Particle Data Group (PDG) [9] is given by U P DG P MN S = U 0 ν K 0 with where c ij = cos θ ij and s ij = sin θ ij with θ ij representing a ν i -ν j mixing angle (i, j=1,2,3). The best fit values of the observed results in the case of the normal mass ordering are summarized as [10]: where ∆m 2 ij = m 2 i − m 2 j with m i representing a mass of ν i (i = 1, 2, 3). The quoted values in the case of the inverted mass ordering (∆m 2 31 < 0) are not so different from Eqs.(2)- (4). There is another similar analysis with ∆m 2 23 defined as ∆m 2 23 = m 2 3 − (m 2 1 + m 2 2 )/2 that has reported the slightly smaller values of sin 2 θ 23 = 0.365 − 0.410 [11]. In this note, we would like to address the issue of leptonic CP-violation with the emphasis laid on the role of phases of flavor neutrino masses and to find possible correlations between phases of flavor neutrino masses and δ CP and φ 2,3 of CP-violation. CP-violating phases arise from complex flavor neutrino masses. However, because of the freedom of choosing charged-lepton phases, phases of neutrino masses are not uniquely defined. Namely, different phase structure gives the same effects of CP-violation. We first discuss how to relate phases of flavor neutrino masses to observed quantities. To do so, we use a neutrino mass matrix M P DG , whose phases are so chosen that the corresponding eigenvectors giving U P MN S show the phase convention defined by PDG, which is nothing but Eq.(1). Next, we give theoretical and numerical estimation of phases of flavor neutrino masses and present possible correlations with CPviolating phases. Also discussed is alternative CP-violation characterized by three CP-violating Dirac phases [12], which has an advantage to discuss property of neutrinoless double beta decay [13].
We start with discussions based on M P DG defined to be: Since δ CP is associated with sin θ 13 , it is useful to divide M P DG into two pieces consisting of M P DG θ13=0 giving sin θ 13 = 0 and ∆M P DG inducing sin θ 13 = 0 [14]: with It should be noted that Eq. (6) is just an identity. There are specific models giving M P DG θ13=0 [15][16][17][18], whose predictions on CP-violation can be covered by our discussions.
Noticing that M P DG = U * P MN S M mass U † P MN S , where M mass = diag.(m 1 , m 2 , m 3 ), we can express M P DG ij in terms of masses, mixing angles and phases including three Majorana phases ϕ 1,2,3 that gives φ i = ϕ i − ϕ 1 . Since sin θ 13 acts as a correction parameter, ∆M P DG is redefined to be sin θ 13 δM P DG : from which δM P DG eτ and δM P DG τ τ are calculated to be: ) is also known to vanish at θ 13 = 0 [19]. In fact, it is expressed in terms of observed masses and mixing angles to be: On the other hand, Eq. (6) Since s 13 δM P DG τ τ δM P DG * eτ in Eq. (11) can be safely neglected, CP-violating Dirac phase δ CP is approximated to be: where an extra π should be added to δ CP if m 2 3 − c 2 12 m 2 1 + s 2 12 m 2 2 < 0. To discus more about δ CP , since contributions of flavor neutrino masses to δ CP depend on their magnitudes, we may include various constraints on M P DG ij supplied by mass hierarchies: m 2 1,2,3 : m 2 1 < m 2 2 ≪ m 2 3 as normal mass hierarchy, m 2 3 ≪ m 2 1 < m 2 2 as inverted mass hierarchy and m 2 1 < m 2 2 ∼ m 2 3 as degenerate mass pattern with m 2 . The magnitudes of masses are controlled by the ideal case of θ 13 = 0 since corrections to the ideal case are O(sin 2 θ 13 ) [20]. For θ 13 = 0, we have the following estimates of three masses and two mixing angles: We are, then, allowed to use the following gross structure of M P DG θ13=0 [21]: for the normal mass hierarchy (NMH) [14], and for the inverted mass hierarchy withm 1 ≈m 2 (IMH-1) [16], and for the inverted mass hierarchy withm 1 ≈ −m 2 (IMH-2) [22], and for the degenerate mass pattern withm 1 ≈m 2 ≈ −m 3 (DMP) [23]. 1 Applying these estimates to Eq.(12), we reach 1. for NMH, ignoring M P DG ee,eµ,eτ , 1 Since M P DG µτ does not vanish in the limit ofm 1 =m 2 =m 3 because of the presence of s 13 e iδ CP , the simplest case ofm 1 ≈m 2 ≈m 3 requiring fairly suppressed magnitude of M P DG µτ is not relevant. In other cases withm 1 ≈ −m 2 , relations among masses are complicated and seem to give no positive feedback to our discussions.
It is thus concluded that the main source of δ CP is δM P DG eτ except for IMH-2. This conclusion is in accord with the expectation from Eq. (9) with arg M P DG eµ ≈ arg M P DG eτ . The similar relation is also found for IMH-1 and dictates from Eq. (20) that  2-(a)). To further enhance predictability based on our approach to CP-violations, we have to minimize the number of phases present in flavor neutrino masses, which can be as small as three. Therefore, a plausible program to discuss linkage between CP-violating phases and flavor neutrino masses is 1. to construct a reference mass matrix to be denoted by M ν with unique choice of phases of neutrino masses, 2. to construct a general mass matrix to be denoted by M that includes the ambiguity of charged-lepton phases to cover all phase structure, which is linked to M ν , 3. to construct M P DG converted from M , whose eigenvectors yield U P DG P MN S . Since flavor neutrino masses in M P DG are expressed by measured quantities, useful information on phases of M ν can be extracted from M P DG .
We start with the following neutrino mass matrix M ν , which has three complex flavor neutrino masses M eµ , M eτ and M τ τ . This choice of phases is suggested by Eq. (7) and yields The mass matrix M physically equivalent to M ν can be obtained by including the freedom of three charged-lepton phases denoted by θ e,µ,τ and is expressed to be: One has to diagonalize Eq. (26) to give m 1,2,3 . Since Eq.(26) contains six phases associated with six complex masses, the relevant U P MN S , U ′ P MN S , should contain six phases, among which three phases are redundant [24][25][26]. We use three phases denoted by δ associated with the 1-3 mixing, γ associated with the 2-3 mixing and ρ associated with the 1-2 mixing and another three phases denoted by α 1,2,3 as Majorana phases to define U ′ P MN S [24]. After three redundant phases ρ, γ and ϕ 1 are removed from U ′ P MN S , Eq.(26) is modified into: where ρ e = ρ − θ e , γ µ = γ − θ µ and γ τ = γ + θ τ , which can be diagonalized by U P MN S of Eq.(1) with δ CP = δ + ρ, φ 2 = ϕ 2 − ϕ 1 and φ 3 = ϕ 3 − ϕ 1 for ϕ 1 = α 1 − ρ and ϕ 2,3 = α  There is an alternative CP violation [12] induced by three CP-violating Dirac phases but without explicitly referring to Majorana phases. For the 2-3 mixing, it uses an analogous Dirac phase to δ instead of γ, which is denoted by τ [25], and τ is introduced as the same way as ρ is. This parameterization denoted by U RV P MN S is known to have an advantage to discuss property of M P DG ee to be measured in (ββ) 0ν -decay [13], which is given by All three CP-violating Dirac phases are physical and observable and are related to δ CP and φ 2,3 as δ CP = δ + ρ + τ , φ 2 = 2ρ and φ 3 = 2 (ρ + τ ) leading to If m 1 = 0, CP-violating Majorana phase is ϕ 3 − ϕ 2 (= φ) and τ and δ + ρ are determined to be where M P DG ee only depends on δ + ρ, while if m 3 = 0, CP-violating Majorana phase is φ 2 and ρ and δ + τ are determined to be In Ref. [10], the suggested best fit value of δ CP /π is 0.80 (-0.03) for normal (inverted) mass ordering although all values are allowed while, in Ref. [11], the allowed region at the 1σ range is 0.77-1.36 (0.83-1.47) for normal (inverted) mass ordering. As can be seen from FIG.1 and FIG.2, it is observed that the figures indicate the approximate proportionality of δ CP to predicted values of Eq.(23) for δ CP using both arg M P DG eµ and arg M P DG eτ and of Eq.(24) using arg M P DG eτ , which supports the validity of our predictions. However, FIG.2-(a) for IMH shows that δ CP using arg M P DG eµ is not a suitable approximation and implies that the assumption of arg M P DG eµ ≈ arg M P DG eτ is not numerically supported. From Eqs. (23) and (24), we obtain arg (M eµ,eτ ) related to δ CP as for NMH, and for IMH-1. We have also checked that the approximated expressions of δ CP , Eqs.  The results are consistent with naive estimation from Eqs.(15)- (18). Namely, the magnitude of m ββ is suppressed for NMH. To analyze M P DG ee itself, it is useful to adopt U RV P MN S parameterized by three Dirac phases, δ for the 1-3 mixing, ρ for the 1-2 mixing and τ for the 2-3 mixing as have been already noted. Differences between predictions by U RV P MN S and those by U P DG P MN S lie in the behavior of the CP-violating Majorana phases. Since these Majorana phases are constrained to be around 0 or ±π for IMH-1, IMH-2 and DMP, distinct differences cannot be expected. Notable features in predictions by U RV P MN S that we can observe are expected to arise for NMH. Obvious one as shown in FIG.7 (a) is that |M ee | exhibits a clear correlation with δ ′ (= δ + ρ) for NMH as in Eq.(31). Another one is shown in FIG.7 (b), where M eµ and δ ′ show a clear correlation that δ ′ is scattered around the line δ ′ = arg (M eµ ) (mod π). The corresponding prediction by U P DG P MN S includes δ CP as in FIG.3 (a) and shows that δ CP is scattered in the entire region, which indicates no correlation with arg (M eµ ) although the scattered points tend to form a straight line.

( )
To summarize, we have derived a general formula to calculate δ CP expressed in terms of the corrections δM P DG eτ and δM P DG τ τ to neutrino mixings with θ 13 = 0. The formula is given by Eq. (12): where an extra π should be added if m 2 3 − c 2 12 m 2 1 + s 2 12 m 2 2 < 0. These δM P DG eτ and δM P DG Other useful findings are For the specific neutrino masses, whose phases are adjusted to arise from M eµ,eτ,τ τ , the effects of CP-violation caused by each flavor neutrino mass are expressed in terms of M P DG according to Eq.(28). For the numerical calculations, we adopted m 1 = 0 eV (m 3 = 0 eV) for NMH (IMH) and m 1 = 0.1 eV (m 3 = 0.1 eV) for DMP with the normal (inverted) mass ordering. It is, then, numerically indicated that δ CP tends to satisfy δ CP ≈ 2 arg (M eµ ) requiring the relation of arg M P DG ee − arg M P DG µµ ≈ δ CP in NMH. In the inverted mass hierarchies, we have observed that |arg (M τ τ )| 0.1 for IMH-1 and π/3 | arg (M eµ ) | π/2 for IMH-2. CP-violating Majorana phase φ 2 (φ 3 ) for DMP is limited to locate around 0 (±π) owing the mass relation ofm 1 ≈m 2 ≈ −m 3 . Effects of Majorana CP-violation are expected to be suppressed for DMP. On the other hand, for NMH, Majorana CP-violation tends to be maximal as |arg (M τ τ )| reaches its maximal value of ≈ 0.5. If Majorana CP-violation tends to be maximal, we have also found that |arg (M τ τ )| 0.2 for IMH-2. Dirac CP-violation gets maximal as arg (M eµ ) → ±π/2 for IMH-1 and DMP with the inverted mass ordering and arg (M τ τ ) ≈ 0 is also satisfied for IMH-1.
Another parameterization of U P MN S utilizes three CP-violating Dirac phases δ, ρ, and τ , where the CP-violating phases in the PDG version are determined to be δ CP = δ + ρ + τ , φ 2 = 2ρ and φ 3 = 2 (ρ + τ ). There are some advantages of choosing U RV P MN S over U P DG P MN S found in the present analysis: 1. The oscillation behavior of |M ee | is well traced for NMH as already pointed out [12] and is useful to determine δ ′ (= δ + ρ) from |M ee |; 2. In NMH, δ ′ is scattered around the line of δ ′ = arg(M eµ ) (mod π/2) while δ CP is scattered in the entire region.
It is in principle possible to know an allowed range of arg(M eµ ) from δ ′ to be extracted from |M ee | if it is measured.
To say something more about the alternative CP-violation for NMH as well as IMH-1 and IMH-2, we have to include effects of two active CP-violating Majorana phases associated with three nonvanishing neutrino masses and results of CP-violation will be discussed elsewhere.

ACKNOWLEGMENTS
The author is grateful to T. Kitabayashi for reading manuscript and useful comments.