Phases of Flavor Neutrino Masses and CP Violation Masaki Yasu`e∗
arXiv:1207.2318v3 [hep-ph] 20 Oct 2012
Department of Physics, Tokai University, 4-1-1 Kitakaname, Hiratsuka, Kanagawa 259-1292, Japan P DG For flavor neutrino masses Mij (i, j = e, µ, τ ) compatible with the phase convention defined by Particle Data Group (PDG), if neutrino mixings are controlled by small corrections to those with P DG sin θ13 =0 denoted θ13 δMeτ and sin θ13 δMτPτDG , CP-violating Dirac phase δCP is calculated byP sin DG∗ P DG∗ P DG P DG P DG∗ P DG to be δCP ≈arg Mµτ / tan θ23 + Mµµ δMeτ + Mee δMeτ − tan θ23 Meµ δMτPτDG∗ (mod π), where θij (i, j=1,2,3) denotes an i-j neutrino mixing angle. If possible neutrino mass P DG hierarchies are taken into account, the main source of δCP turns out to be δMeτ except for the inverted mass hierarchy with m ˜ 1 ≈ −m ˜ 2 , where m ˜ i = mi e−iϕi (i=1,2) stands for a neutrino mass mi accompanied by a Majorana phase ϕi for ϕ1,2,3 giving We two CP-violating MajoranaP phases. P DG P DG P DG can further derive that δCP ≈ arg Meµ − arg Mµµ with arg Meµ ≈ arg MeτDG for the P DG P DG normal mass hierarchy and δCP ≈ arg Mee − arg Meτ + π for the inverted mass hierarchy with m ˜1 ≈ m ˜ 2 . For specific flavor neutrino masses Mij whose phases arise from Meµ,eτ,τ τ , these P DG phases can be connected with arg(Mij ) (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 with m ˜1 ≈ m ˜ 2 and for the degenerate mass pattern satisfying the inverted mass ordering and that Majorana CP-violation becomes maximal as |arg (Mτ τ )| 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: 12.60.-i, 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 δCP and two CP-violating Majorana phases φ2,3 [7]. Neutrino mixings are parameterized by the Pontecorvo-MakiNakagawa-Sakata (PMNS) unitary matrix UP MN S [8], which converts the massive neutrinos νi (i = 1, 2, 3) into the flavor neutrinos νf (f = e, µ, τ ). The standard description of UP MN S adopted by the Particle Data Group (PDG) [9] DG 0 0 is given by UPPMN S = Uν K with c12 c13 s12 c13 s13 e−iδCP Uν0 = −c23 s12 − s23 c12 s13 eiδCP c23 c12 − s23 s12 s13 eiδCP s23 c13 , iδCP iδCP c23 c13 −s23 c12 − c23 s12 s13 e s23 s12 − c23 c12 s13 e K 0 = diag(1, eiφ2 /2 , eiφ3 /2 ),
(1)
where cij = cos θij and sij = sin θij with θij representing a νi -νj 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]: ∆m221 [10−5 eV2 ] = 7.62 ± 0.19, sin2 θ12
δP C +1.20 = 0.80 −0.80 , π
∗ Electronic
+0.06
∆m231 [10−3 eV2 ] = 2.55 −0.09 , +0.016 +0.034 +0.022 = 0.320 −0.017 , sin2 θ23 = 0.427 −0.027 0.613 −0.040 ,
address:
[email protected]
(2) +0.0029
sin2 θ13 = 0.0246 −0.0028 ,
(3) (4)
2 where ∆m2ij = m2i − m2j with mi representing a mass of νi (i = 1, 2, 3). The quoted values in the case of the inverted mass ordering (∆m231 < 0) are not so different from Eqs.(2)-(4). There is another similar analysis with ∆m223 defined as ∆m223 = m23 − (m21 + m22 )/2 that has reported the slightly smaller values of sin2 θ23 = 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 δCP and φ2,3 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 UP 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: P DG P DG P DG Mee Meµ Meτ P DG P DG P DG Mµµ Mµτ M P DG = Meµ . (5) P DG P DG P DG Meτ Mµτ Mτ τ Since δCP is associated with sin θ13 , it is useful to divide M P DG into two pieces consisting of MθP13DG =0 giving sin θ13 = 0 and ∆M P DG inducing sin θ13 6= 0 [14]: P DG M P DG = MθP13DG , =0 + ∆M
(6)
with
MθP13DG =0 = ∆M P DG
P DG Mee P DG Meµ
P DG Meµ P DG Mµµ
P DG −t23 Meµ P DG Mµτ 1−t223 t23
,
P DG P DG P DG P DG −t23 Meµ Mµτ Mµµ + Mµτ P DG P DG 0 0 Meτ + t23 Meµ 0 0 0 = . 1−t223 P DG P DG P DG P DG P DG Meτ + t23 Meµ 0 Mτ τ − Mµµ + t23 Mµτ
(7)
It should be noted that Eq.(6) is just an identity. There are specific models giving MθP13DG =0 [15–18], whose predictions on CP-violation can be covered by our discussions. Noticing that M P DG = UP∗ MN S Mmass UP† MN S , where Mmass = diag.(m1 , m2 , m3 ), we can express MijP DG in terms of masses, mixing angles and phases including three Majorana phases ϕ1,2,3 that gives φi = ϕi − ϕ1 . Since sin θ13 acts as a correction parameter, ∆M P DG is redefined to be sin θ13 δM P DG : P DG P DG P DG , sin θ13 δMeτ = Meτ + t23 Meµ 1 − t223 P DG P DG P DG P DG sin θ13 δMτ τ = Mτ τ − Mµµ + , Mµτ t23 P DG from which δMeτ and δMτPτDG are calculated to be: c13 iδCP P DG δMeτ = m ˜ 3 − e−iδCP c212 m e ˜ 1 + s212 m ˜2 , c23 c12 s12 −iδCP P DG δMτ τ = (m ˜2 −m ˜ 1) , e s23 c23
(8)
(9)
where m ˜ i = mi e−iϕi (i = 1, 2, 3). To estimate CP-violating Dirac phase, let us consider M = M P DG† M P DG . The P DG P DG P DG quantity of s23 Meµ + c23 Meτ corresponding to ∆Meτ (= Meτ + t23 Meµ ) is also known to vanish at θ13 = 0 [19]. In fact, it is expressed in terms of observed masses and mixing angles to be: (10) s23 Meµ + c23 Meτ = c13 s13 e−iδCP m23 − c212 m21 + s212 m22 . On the other hand, Eq.(6) yields
1 P DG P DG P DG∗ Mµτ + Mµµ + s13 δMτPτDG δMeτ t23 P DG∗ P DG P DG∗ + Mee δMeτ − t23 Meµ δMτPτDG .
s23 Meµ + c23 Meτ = s13 c223
(11)
3 P DG∗ Since s13 δMτPτDG δMeτ in Eq.(11) can be safely neglected, CP-violating Dirac phase δCP is approximated to be: 1 P DG P DG P DG∗ P DG P DG∗ P DG∗ P DG∗ , (12) δMeτ + Mee δMeτ − t23 Meµ δMτ τ M + Mµµ δCP ≈ arg t23 µτ where an extra π should be added to δCP if m23 − c212 m21 + s212 m22 < 0. To discus more about δCP , since contributions of flavor neutrino masses to δCP depend on their magnitudes, we may include various constraints on MijP DG supplied by mass hierarchies: m21,2,3 : m21 < m22 ≪ m23 as normal mass hierarchy, m23 ≪ m21 < m22 as inverted mass hierarchy and m21 < m22 ∼ m23 as degenerate mass pattern with m21 < m22 ≈ m23 (or m23 ≈ m21 < m22 ). The magnitudes of masses are controlled by the ideal case of θ13 = 0 since corrections to the ideal case are O(sin2 θ13 ) [20]. For θ13 = 0, we have the following estimates of three masses and two mixing angles: P DG P DG P DG P DG P DG P DG P DG − t23 Mµτ − s212 Mµµ c212 Mee Mee + Mµµ − t23 Mµτ Meµ m ˜1 = = , − c212 − s212 2 c23 sin 2θ12 P DG P DG P DG P DG P DG P DG P DG − s212 Mee − t23 Mµτ c212 Mµµ Mee + Mµµ − t23 Mµτ Meµ m ˜2 = = , (13) + c212 − s212 2 c23 sin 2θ12 1 P DG m ˜ 3 = Mµµ + M P DG , t23 µτ
and tan θ23 = −
P DG Meτ , P DG Meµ
tan 2θ12 =
P DG Meµ 2 . P DG − t M P DG − M P DG c23 Mµµ 23 µτ ee
We are, then, allowed to use the following gross structure of MθP13DG =0 [21]: 0 0 0 P DG 0 1 1/t23 Mµµ MθP13DG , =0 ≈ 2 0 1/t23 1/t23
(14)
(15)
for the normal mass hierarchy (NMH) [14], and MθP13DG =0
2 0 0 P DG ≈ 0 1 −t23 Mµµ , 2 0 −t23 t23
for the inverted mass hierarchy with m ˜1 ≈ m ˜ 2 (IMH-1) [16], and −2 −2c23 tan 2θ12 2s23 tan 2θ12 P DG −2c23 tan 2θ12 Mµµ 1 −t23 MθP13DG , =0 ≈ 2 2c23 tan 2θ12 −t23 t23 for the inverted mass hierarchy with m ˜ 1 ≈ −m ˜ 2 (IMH-2) [22], and 1 0 0 P DG 0 cos 2θ23 − sin 2θ23 Mee , MθP13DG =0 ≈ 0 − sin 2θ23 − cos 2θ23
(16)
(17)
(18)
for the degenerate mass pattern with m ˜1 ≈ m ˜ 2 ≈ −m ˜ 3 (DMP) [23].1 Applying these estimates to Eq.(12), we reach P DG 1. for NMH, ignoring Mee,eµ,eτ ,
P DG∗ P DG , δCP ≈ arg Mµµ δMeτ
1
(19)
P DG does not vanish in the limit of m ˜1 ≈ m ˜2 ≈ m ˜3 Since Mµτ ˜1 = m ˜2 = m ˜ 3 because of the presence of s13 eiδCP , the simplest case of m P DG requiring fairly suppressed magnitude of Mµτ is not relevant. In other cases with m ˜ 1 ≈ −m ˜ 2 , relations among masses are complicated and seem to give no positive feedback to our discussions.
4 P DG 2. for IMH-1, ignoring Meµ,eτ ,
P DG P DG∗ + π, δCP ≈ arg Mee δMeτ
(20)
P DG P DG∗ P DG δCP ≈ arg Mee δMeτ − t23 Meµ δMτPτDG∗ + π,
(21)
P DG 3. for IMH-2, ignoring Mµµ,µτ,τ τ,
P DG 4. for DMP, ignoring Meµ,eτ ,
δCP ≈ arg
1 P DG P DG P DG∗ P DG∗ P DG∗ (+π) , δMeτ + Mee δMeτ Mµτ + Mµµ t23
(22)
with an extra π for the inverted mass ordering. P DG It is thus concluded that the main source of δCP is δMeτ except for IMH-2. This conclusion is inaccord with the P DG P DG P DG expectation from Eq.(9) that δMτ τ is suppressed unless m ˜ 1 ≈ −m ˜ 2 as in IMH-2. Since arg Meµ = arg Meτ P DG P DG is preserved for sin θ13 6= 0 especially in NMH ≈ arg Meτ is valid for sin θ13 = 0, we expect that arg Meµ P DG P DG = because the single term proportional to m ˜ 3 will dominate in Meµ,eτ . Using the approximation of arg δMeτ P DG P DG P DG arg Meτ + t23 Meµ ≈ arg Meµ,eτ , we can find more simplified relation from Eq.(19) in NMH:
P DG
with arg Meµ
P DG P DG δCP ≈ arg Meµ − arg Mµµ ,
(23)
P DG P DG + π, − arg Meτ δCP ≈ arg Mee
(24)
P DG . The similar relation is also found for IMH-1 and dictates from Eq.(20) that ≈ arg Meτ
P DG P DG does not serve as a good approximation. In fact, Eq.(24) using another choice of ≈ arg Meτ where arg Meµ P DG P DG is not numerically supported (See FIG.2-(a)). instead of arg Meτ arg Meµ 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 Mν 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 Mν , DG 3. to construct M P DG converted from M , whose eigenvectors yield UPPMN S.
Since flavor neutrino masses in M P DG are expressed by measured quantities, useful information on phases of Mν can be extracted from M P DG . We start with the following neutrino mass matrix Mν , which has three complex flavor neutrino masses Meµ , Meτ and Mτ τ . This choice of phases is suggested by Eq.(7) and yields |Mee | Meµ Meτ Mν = Meµ |Mµµ | |Mµτ | . (25) Meτ |Mµτ | Mτ τ The mass matrix M physically equivalent to Mν can be obtained by including the freedom of three charged-lepton phases denoted by θe,µ,τ and is expressed to be: e−i(θe +θµ ) Meµ −e−i(θe +θτ ) Meτ e−2iθe |Mee | (26) M = e−i(θe +θµ ) Meµ e−2iθµ |Mµµ | e−i(θµ +θτ ) |Mµτ | . −i(θe +θτ ) −i(θµ +θτ ) −e Meτ e |Mµτ | e−2iθτ Mτ τ One has to diagonalize Eq.(26) to give m1,2,3 . Since Eq.(26) contains six phases associated with six complex masses, the relevant UP MN S , UP′ MN S , should contain six phases, among which three phases are redundant [24–26]. We use three phases denoted by δ associated with the 1-3 mixing, γ associated with the 2-3 mixing and ρ associated with
5
δ CP
(a)
PDG arg ( M ePDG µ ) − arg ( M µµ )
δ CP
(b)
arg ( M ePDG ) − arg ( M µµPDG ) τ
P DG P DG FIG. 1: The predictions of δCP for the normal mass hierarchy (NMH): (a) δCP ≈ arg Meµ − arg Mµµ or (b) δCP ≈ P DG P DG arg Meτ − arg Mµµ .
δ CP
(a)
arg ( M eePDG ) − arg ( M ePDG µ )+π
δ CP
(b)
arg ( M eePDG ) − arg ( M ePDG ) +π τ
P DG P DG FIG. 2: The predictions of δCP for the inverted mass hierarchy with m ˜1 ≈ m ˜ 2 (IMH-1): (a) δCP ≈ arg Mee −arg Meµ + P DG P DG π or (b) δCP ≈ arg Mee − arg Meτ + π.
the 1-2 mixing and another three phases denoted by α1,2,3 as Majorana phases to define UP′ MN S [24]. After three redundant phases ρ, γ and ϕ1 are removed from UP′ MN S , Eq.(26) is modified into: 2iρ ei(ρe +γµ ) Meµ ei(ρe −γτ ) Meτ e e |Mee | (27) M P DG = ei(ρe +γµ ) Meµ e2iγµ |Mµµ | ei(γµ −γτ ) |Mµτ | , i(ρe −γτ ) i(γµ −γτ ) e Meτ e |Mµτ | e−2iγτ Mτ τ where ρe = ρ − θe , γµ = γ − θµ and γτ = γ + θτ , which can be diagonalized by UP MN S of Eq.(1) with δCP = δ + ρ, φ2 = ϕ2 − ϕ1 and φ3 = ϕ3 − ϕ1 for ϕ1 = α1 − ρ and ϕ2,3 = α2,3 . As a result, phases of Meµ , Meτ and Mτ τ are expressed in terms of arg MijP DG (i, j =e, µ, τ ) as follows: P DG P DG arg Mee + arg Mµµ P DG , arg (Meµ ) = arg Meµ − 2 P DG P DG arg Mee − arg Mµµ P DG arg (Meτ ) = arg Meτ , (28) − 2 P DG P DG arg (Mτ τ ) = arg MτPτDG + arg Mµµ − 2 arg Mµτ .
There is an alternative CP violation [12] induced by three CP-violating Dirac phases but without explicitly referring
6
M ee
10
−2
eV
(a)
(b)
δ CP
arg ( Mττ )
δ ′(= δ + ρ )
(c)
φ
arg ( Mττ )
FIG. 3: The predictions of (a) δCP as a function of arg (Meµ ), (b) δCP as a function of arg (Mτ τ ) and (c) φ (=ϕ3 − ϕ2 ) as a function of arg (Mτ τ ) for the normal mass hierarchy (NMH).
δ CP
(a)
(b)
δ CP
arg ( M eµ )
(c)
φ
arg ( Mττ )
arg ( Mττ )
FIG. 4: The same as in FIG.3 but for the inverted mass hierarchy with m ˜1 ≈ m ˜ 2 (IMH-1) and φ = ϕ2 − ϕ1 .
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 UPRV MN S is known to have an P DG advantage to discuss property of Mee to be measured in (ββ)0ν -decay [13], which is given by P DG (29) Mee = e−iϕ1 c213 c212 m1 + s212 m2 e−2iρ + s213 m3 e2iδ . All three CP-violating Dirac phases are physical and observable and are related to δCP and φ2,3 as δCP = δ + ρ + τ , φ2 = 2ρ and φ3 = 2 (ρ + τ ) leading to δ = δCP −
φ3 , 2
ρ=
φ2 , 2
τ=
φ3 − φ2 . 2
(30)
If m1 = 0, CP-violating Majorana phase is ϕ3 − ϕ2 (= φ) and τ and δ + ρ are determined to be δ + ρ = δCP −
φ , 2
τ=
φ , 2
(31)
P DG where Mee only depends on δ + ρ, while if m3 = 0, CP-violating Majorana phase is φ2 and ρ and δ + τ are determined to be
δ + τ = δCP −
φ2 φ2 . ρ= , 2 2
(32)
7
δ CP
(a)
δ CP
arg ( M eµ )
(b)
φ
arg ( Mττ )
(c)
arg ( Mττ )
FIG. 5: The same as in FIG.3 but for the inverted mass hierarchy with m ˜ 1 ≈ −m ˜ 2 (IMH-2).
P DG where Mee only depends on ρ. Our results of numerical calculations are listed in FIG.1-FIG.6. Shown in FIG.1 and FIG.2 are predictions on δCP from the simplified relations Eqs.(23) and (24). In the remaining figures, FIG.3-FIG.6, each of which corresponds to each mass pattern, predictions on δCP are depicted as functions of either arg (Meµ ) or arg (Mτ τ ), which exhibit a certain correlation with δCP . On the other hand, any predominant correlation between δCP and arg (Meτ ) in each case cannot be found. Other correlations with CP-violating Majorana phases are also shown in the figures. For the sake of simplicity, calculations have been done for m1 = 0 eV for NMH and m3 = 0 eV for IMH-1 and IMH-2. For DMP, m1 =0.1 eV (m3 =0.1 eV) for the normal (inverted) mass ordering is adopted. The parameters used are
∆m221 [10−5 eV2 ] = 7.62, sin2 θ12 = 0.32,
∆m231 [10−3 eV2 ] = 2.53, sin2 θ23 = 0.45, sin2 θ13 = 0.025.
(33) (34)
In Ref.[10], the suggested best fit value of δCP /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 δCP P DG P DG P DG , and of Eq.(24) using arg Meτ and arg Meτ to predicted values of Eq.(23) for δCP using both arg Meµ P DG is which supports the validity of our predictions. However, FIG.2-(a) for IMH shows that δCP using arg Meµ P DG P DG not a suitable approximation and implies that the assumption of arg Meµ ≈ arg Meτ is not numerically supported. From Eqs.(23) and (24), we obtain arg (Meµ,eτ ) related to δCP as P DG P DG arg (Meµ ) ≈ δCP − arg Mee − arg Mµµ /2, (35) for NMH, and
P DG P DG /2 + π, + arg Mµµ arg (Meτ ) ≈ −δCP + arg Mee
(36)
for IMH-1. We have also checked that the approximated expressions of δCP , Eqs.(19)-(22), numerically well reproduce actual values of δCP . For the results of arg (Meµ,τ τ ) with respect to δCP and CP-violating Majorana phases, their suggested features are summarized as follows: • For NMH with m1 = 0, where φ = ϕ3 − ϕ2 , FIG.3 indicates that P DG − – δCP following the thick line approximated to be δ ≈ 2 arg (M ) is realized by requiring arg M CP eµ ee P DG arg Mµµ ≈ δCP because of Eq.(35), – |arg (Mτ τ )| . 0.5, – φ has a simple dependence on arg (Mτ τ ): φ = 0, π if arg (Mτ τ ) = 0 and φ = ±π/2 if | arg (Mτ τ ) | reaches its maximal value of around 0.5.
8
δ CP
(a)
arg ( M eµ )
δ CP
(b)
δ CP
(d )
φ3
arg ( Mττ )
(e)
δ CP
arg ( M eµ )
arg ( Mττ )
(c)
φ2 φ3
(f)
φ2
FIG. 6: The same as in FIG.3 for (a) and (b) but (c) φ3 as a function of φ2 for the degenerate mass pattern with m ˜1 ≈ m ˜ 2 ≈ −m ˜3 (DMP) satisfying the normal mass ordering. Lower figures (d)-(f) show results for the inverted mass ordering.
• For IMH-1, where φ = ϕ2 − ϕ1 , FIG.4 indicates that – δCP → ±π/2 as arg (Meµ ) → ±π/2, – |arg (Mτ τ )| . 0.1, – φ ≈ 0 is set by the condition of m ˜1 ≈ m ˜ 2. • For IMH-2, where φ = ϕ2 − ϕ1 , FIG.5 indicates that – π/3 . |arg (Meµ )| . π/2, – |arg (Mτ τ )| . 0.6 for |δCP | . π/2, – |arg (Mτ τ )| . 0.2 if φ approaches toward ±π/2. – φ ≈ ±π is set by the condition of m ˜ 1 ≈ −m ˜ 2. • For DMP, FIG.6 indicates that – π/4 . |δCP | . 3π/4 as arg (Meµ ) → ±π/2 for the normal mass ordering, – δCP → ±π/2 as arg (Meµ ) → ±π/2 for the inverted mass ordering, – φ2 ≈ 0 and φ3 ≈ ±π for both mass orderings, which are linked to the fact that the sign of m ˜ 3 is different from that of m ˜ 1,2 .
9
M ee
(a)
10 −2 eV
δ ′(= δ + ρ )
(b)
δ′
arg ( M eµ )
FIG. 7: The predictions of (a) |Mee | as a function of δ ′ (= δ + ρ) and (b) δ ′ as a function of arg (Meµ ) for NMH.
The phases of Meµ,τ τ are taken to run from −π/2 to π/2. It should be noted that arg (Meµ )-δCP for IMH-1 (FIG.4(a)) and for DMP with the inverted mass ordering (FIG.6-(a)) have the quite similar shape to each other, showing that the maximal Dirac CP-violation signalled by δCP = ±π/2 is realized by arg (Meµ ) ≈ ±π/2 and, at the same time, arg (Mτ τ ) ≈ 0 is necessary IMH-1. Pfor DG Also estimated is |Mee | (= Mee ) as the effective neutrino mass mββ in (ββ)0ν -decay: • 0.002 . |Mee | [eV] . 0.004 for NMH, • |Mee | [eV] ≈ 0.05 for IMH-1, • 0.02 . |Mee | [eV] . 0.04 for IMH-2, • 0.095 . |Mee | [eV] . 0.1 for DMP. The results are consistent with naive estimation from Eqs.(15)-(18). Namely, the magnitude of mββ is suppressed P DG for NMH. To analyze Mee itself, it is useful to adopt UPRV 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 P DG by UPRV MN S and those by UP 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 UPRV MN S that we can observe are expected to arise for NMH. Obvious one as shown in FIG.7 (a) is that |Mee | exhibits a clear correlation with δ ′ (= δ + ρ) for NMH as in Eq.(31). Another one is shown in FIG.7 (b), where Meµ and δ ′ show a clear correlation that δ ′ is scattered around the line δ ′ = arg (Meµ ) (mod π). DG The corresponding prediction by UPPMN S includes δCP as in FIG.3 (a) and shows that δCP is scattered in the entire region, which indicates no correlation with arg (Meµ ) although the scattered points tend to form a straight line. P DG To summarize, we have derived a general formula to calculate δCP expressed in terms of the corrections δMeτ and δMτPτDG to neutrino mixings with θ13 = 0. The formula is given by Eq.(12): 1 P DG∗ P DG∗ P DG P DG P DG∗ P DG δCP ≈ arg Mµτ + Mµµ δMeτ + Mee δMeτ − t23 Meµ δMτPτDG∗ , (37) t23 P DG and δMτPτDG are described in terms where an extra π should be added if m23 − c212 m21 + s212 m22 < 0. These δMeτ P DG P DG P DG P DG P DG P DG P DG P DG /t23 . of M as sin θ13 δMeτ = Meτ + t23 Meµ and sin θ13 δMτ τ = Mτ τ − Mµµ + 1 − t223 Mµτ Their mass dependence is then determined to be: P DG δMeτ =
c13 iδCP ˜ 1 + s212 m ˜2 , m ˜ 3 − e−iδCP c212 m e c23
δMτPτDG =
c12 s12 −iδCP (m ˜2 −m ˜ 1) . e s23 c23
(38)
Other useful findings are P DG 1. The main source of δCP is δMeτ except for IMH-2 because δMτPτDG is not suppressed if m ˜ 1 ≈ −m ˜ 2 , and
10 P DG P DG P DG P DG for NMH and ≈ arg Meµ with arg Meτ − arg Mµµ 2. δCP is well predicted to be arg Meµ P DG P DG + π for IMH-1. − arg Meτ arg Mee
For the specific neutrino masses, whose phases are adjusted to arise from Meµ,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 m1 = 0 eV (m3 = 0 eV) for NMH (IMH) and m1 = 0.1 eV (m3 = 0.1 eV) for DMP with the normal (inverted) mass ordering. It is, then, numerically indicated that δCP tends to satisfy δCP ≈ 2 arg (Meµ ) requiring P DG P DG ≈ δCP in NMH. In the inverted mass hierarchies, we have observed that − arg Mµµ the relation of arg Mee |arg (Mτ τ )| . 0.1 for IMH-1 and π/3 . | arg (Meµ ) | . π/2 for IMH-2. CP-violating Majorana phase φ2 (φ3 ) for DMP is limited to locate around 0 (±π) owing the mass relation of m ˜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 (Mτ τ )| reaches its maximal value of ≈ 0.5. If Majorana CP-violation tends to be maximal, we have also found that |arg (Mτ τ )| . 0.2 for IMH-2. Dirac CP-violation gets maximal as arg (Meµ ) → ±π/2 for IMH-1 and DMP with the inverted mass ordering and arg (Mτ τ ) ≈ 0 is also satisfied for IMH-1. Another parameterization of UP 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 δCP = δ + ρ + τ , φ2 = 2ρ and φ3 = 2 (ρ + τ ). There are some P DG advantages of choosing UPRV MN S over UP MN S found in the present analysis: 1. The oscillation behavior of |Mee | is well traced for NMH as already pointed out [12] and is useful to determine δ ′ (= δ + ρ) from |Mee |; 2. In NMH, δ ′ is scattered around the line of δ ′ = arg(Meµ ) (mod π/2) while δCP is scattered in the entire region. It is in principle possible to know an allowed range of arg(Meµ ) from δ ′ to be extracted from |Mee | 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.
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