U_{PMNS} = U_ell^dagger U_nu

We consider corrections to vanishing U_{e3} and maximal atmospheric neutrino mixing originating from the relation U = U_ell^dagger U_nu, where U is the PMNS mixing matrix and U_ell (U_nu) is associated with the diagonalization of the charged lepton (neutrino) mass matrix. We assume that in the limit of U_ell or U_nu being the unit matrix, one has U_{e3} = 0 and theta_{23} = pi/4, while the solar neutrino mixing angle is a free parameter. Well-known special cases of the indicated scenario are the bimaximal and tri-bimaximal mixing schemes. If U_{e3} \neq 0 and theta_{23} \neq pi/4 due to corrections from the charged leptons, |U_{e3}| can be sizable (close to the existing upper limit) and we find that the value of the solar neutrino mixing angle is linked to the magnitude of CP violation in neutrino oscillations. In the alternative case of the neutrino sector correcting U_{e3} = 0 and theta_{23} = pi/4, we obtain a generically smaller |U_{e3}| than in the first case. Now the magnitude of CP violation in neutrino oscillations is connected to the value of the atmospheric neutrino mixing angle theta_{23}. We find that both cases are in agreement with present observations. We also introduce parametrization independent"sum-rules"for the oscillation parameters.


Introduction
The low energy neutrino mixing implied by the neutrino oscillation data can be described by the Lagrangian (see, e.g., [1]) which includes charged lepton and Majorana neutrino mass terms.When diagonalizing the neutrino and charged lepton mass matrices via m ν = U * ν m diag matrices in U = U † ℓ U ν corresponds to Eq. ( 3) or ( 4), and is "perturbed" by the second matrix leading to the required PMNS matrix.Following this assumption, corrections to bimaximal [9,10,11,12,13] and tri-bimaximal [14,15,13] mixing have previously been analyzed.For instance, scenarios in which the CKM quark mixing matrix corrects the bimaximal mixing pattern are important for models incorporating Quark-Lepton Complementarity (QLC) [16,17,18] (for earlier reference see [19]).Corrections to mixing scenarios with θ 12 = π/4 and θ 13 = 0 were considered in [20] (motivated by the L e − L µ − L τ flavor symmetry [21]) and in [12].The case with θ 23 = π/4 and θ 13 = 0 has been investigated in Refs.[22,23,13,24].Up to now in most analyzes it has been assumed that U ν possesses a form which leads to sin 2 θ 23 = 1/2 and θ 13 = 0.However, the alternative possibility of θ 23 = π/4 and θ 13 = 0 originating from U ℓ is phenomenologically equally viable.We are aware of only few papers in which that option is discussed [11,12,25,26].A detailed study is still lacking in the literature.In the present article we perform, in particular, a comprehensive analysis of this possibility.We also revisit the case of U e3 = 0 and θ 23 = π/4 due to corrections from U † ℓ and derive parametrization independent sum-rules for the relevant oscillation parameters.We point out certain "subtleties" in the identification of the relevant phases governing CP violation in neutrino oscillations with the Dirac phase of the standard parametrization of the PMNS matrix.
Our paper is organized as follows: Section 2 briefly summarizes the formalism and the relevant matrices from which the neutrino mixing observables can be reconstructed.We analyze the possibility of U ν leading to sin 2 θ 23 = 1/2 and θ 13 = 0 and being corrected by a non-trivial U ℓ in Sec. 3. In Sec. 4 the alternative case of U ℓ causing sin 2 θ 23 = 1/2 and θ 13 = 0 and being modified by a non-trivial U ν is discussed.Section 5 contains our conclusions.

Formalism and Definitions
We will use the following parametrization of the PMNS matrix: where ) is a 3 × 3 orthogonal matrix of rotations on angle θ ij in the ij-plane.We have also defined Hereby we have included the Dirac CP violating phase δ and the two Majorana CP violating phases α and β [3,4].In general, all phases and mixing angles of U are functions of the parameters characterizing U ν and U ℓ .It can be shown that [27,10] after eliminating the unphysical phases, U can be written as U = Ũ † ℓ U ν , where in the most general case U ν and Ũℓ are given by Here we have defined c ℓ,ν ij = cos θ ℓ,ν ij and s ℓ,ν ij = sin θ ℓ,ν ij .Thus, Ũν and Ũℓ contain one physical CP violating phase each1 .The remaining four phases are located in the diagonal matrices P and Q.Note that Q is "Majorana-like" [10], i.e., the phases σ and τ contribute only to the low energy observables related to the Majorana nature of the neutrinos with definite mass.Typically that are specific observables associated with |∆L| = 2 processes, like neutrinoless double beta decay (A, Z) → (A, Z + 2) + e − + e − (see, e.g., [28,29]).In the following we will be interested in models and the phenomenological consequences that result if Ũν corresponds to Eq. ( 3), while Ũℓ contains comparatively small angles, and vice versa.It proves convenient to introduce the abbreviations sin θ ℓ,ν ij = λ ij > 0 for the small quantities we will use as expansion parameters in our further analysis.
Turning to the observables, the sines of the three mixing angles of the PMNS matrix U are given by The expressions quoted above are in terms of the absolute values of the elements of U, which emphasizes the independence of parametrization.In the case of 3-ν mixing under discussion there are, in principle, three independent CP violation rephasing invariants, associated with the three CP violating phases of the PMNS matrix.The invariant related to the Dirac phase δ is given as which controls the magnitude of CP violation effects in neutrino oscillations and is a directly observable quantity [30].It is analogous to the rephasing invariant associated with the Dirac phase in the Cabibbo-Kobayashi-Maskawa quark mixing matrix, introduced in Ref. [31].In addition to J CP , there are two rephasing invariants associated with the two Majorana phases in the PMNS matrix, which can be chosen as2 [32,33] (see also [29]): The rephasing invariants associated with the Majorana phases are not uniquely determined.Instead of S 1 defined above we could also have chosen The Majorana phases α and β, or β and (β − α), can be expressed in terms of the rephasing invariants in this way introduced [29], for instance via cos The expression for, e.g., cos α in terms of S ′ 1 is somewhat more cumbersome (it involves also J CP ) and we will not give it here.Note that CP violation due to the Majorana phase β requires that both S 1 = Im{U e1 U * e3 } = 0 and Re{U e1 U * e3 } = 0. Similarly, S 2 = Im{U * e2 U e3 } = 0 would imply violation of the CP symmetry only if in addition Re{U * e2 U e3 } = 0. Finally, let us quote the current data on the neutrino mixing angles [2,8]: sin 2 θ 12 = 0.30 +0.02, 0.10 −0.03, 0.06 , sin 2 θ 23 = 0.50 +0.08, 0.18 −0.07, 0.16 , |U e3 | 2 = 0 +0.012, 0.041 −0.000 , where we have given the best-fit values as well as the 1 σ and 3 σ allowed ranges.
3 Maximal Atmospheric Neutrino Mixing and U e3 = 0 from the Neutrino Mass Matrix In this Section we assume that maximal atmospheric neutrino mixing and vanishing |U e3 | are realized in the limiting case, where U ℓ corresponds to the unit matrix.We can obtain θ ν 23 = −π/4 and θ ν 13 = 0 by requiring µ-τ exchange symmetry [5,23] of the neutrino mass matrix in the basis in that the charged lepton mass matrix is diagonal.Under this condition we have where m 1,2,3 are the neutrino masses.The indicated symmetry is assumed to hold in the charged lepton mass basis, although the charged lepton masses are obviously not µ-τ symmetric.However, such a scenario can, for example, be easily realized in models with different Higgs doublets generating the up-and down-like particle masses.For the sines of the "small" angles in the matrix U ℓ we introduce the convenient notation sin θ ℓ ij = λ ij > 0 with ij = 12, 13, 23.We obtain the following expressions for the observables relevant for neutrino oscillation in the case under consideration: Setting in these equations θ ν 12 to π/4 (to sin −1 1/3) reproduces the formulas from [10] (also [15]).
A comment on the CP phases is in order.The relevant Dirac CP violating phase(s) can be identified from the expression for the rephasing invariant J CP : these are φ or (ω − ψ), depending on the relative magnitude of λ 12 and λ 13 .However, within the approach we are employing, a Dirac CP violating phase appearing in J CP does not necessarily coincide with the Dirac phase in the standard parametrization of the PMNS matrix.For illustration it is sufficient to consider the simple case of λ 12 = 0 and λ 13 = λ 23 = 0. Working to leading order in λ 12 , it is easy to find that in this case the PMNS matrix can be written as where P = diag(e iφ , e iφ , e iω ) and Q = diag(1, e iσ , e iτ ).The phase matrix P can be eliminated from U by a redefinition of the phases of the charged lepton fields.The Majorana phases α and β ′ ≡ (β + δ) can be directly identified (modulo 2π) with σ and τ .It is clear from the expressions ( 5) and ( 14) for U, however, that the phase φ does not coincide with the Dirac phase δ of the standard parametrization of U. Actually, the phase φ could be directly identified with the Dirac CP violating phase of a different parametrization of the PMNS matrix, namely, the parametrization in which Ũ in Eq. ( 14) is given by mixing can be maximal, or very close to maximal, for instance if ω − φ = π/2.Note that λ 12 and λ 13 in the expressions for sin 2 θ 12 , |U e3 | and J CP are multiplied by cosines and/or sines of the same phases φ and (ω − ψ), respectively.This means that if the terms proportional to λ 12 (to λ 13 ) dominate over the terms proportional to λ 13 (to λ 12 ) -we will refer to this possibility as λ 12 (λ 13 )-dominance5 -we have [10,13,23]: where γ = φ or (ψ−ω) is the CP violating phase (combination) appearing in the expression for J CP , J CP ∝ sin γ.The relation ( 16) implies a correlation of the initial 12-mixing in U ν with |U e3 | and the observable CP violation in neutrino oscillations.If Ũν is a bimaximal mixing matrix, we have sin 2 θ ν 12 = 1/2 and cos γ has to take a value close to one (while |U e3 | has to be relatively large) in order to obtain sufficiently non-maximal solar neutrino mixing.Consequently, in the case of λ 12 (λ 13 )-dominance, CP violation would be suppressed even though |U e3 | can be sizable.On the other hand, if Ũν is a tri-bimaximal mixing matrix, we have sin 2 θ ν 12 = 1/3 which already is in good agreement with the present data.Hence, |U e3 | cos γ has to be relatively small.Consequently, CP violation can be sizable if |U e3 | has a value close to the existing upper limit.This interesting feature has first been noticed in Ref. [15].Generally, in the case of λ 12 (λ 13 )-dominance we get from Eq. ( 13): where γ = φ (γ = ψ − ω) for λ 12 -dominance (λ 13 -dominance).The following "sum-rule" holds as well: where the minus (plus) sign represents a positive (negative) cosine of the relevant Dirac CP violating phase.The sign ambiguity is unavoidable because the CP conserving quantity sin 2 θ 12 can only depend on the cosine of a CP violating phase, whereas any CP violating quantity like J CP can only depend on the sine of this phase.Knowing the cosine of a phase will never tell us the sign of the sine6 .Note that since all parameters in Eq. ( 18) are rephasing invariant quantities, it can be applied to any parametrization of the PMNS matrix U and of the matrix Ũν .If Ũν is a bimaximal (tri-bimaximal) mixing matrix, we get respectively.The first relation has been obtained also in Ref. [18].Obviously, one has to choose here the negative sign.
In Fig. 1 we show the allowed parameter space for the exact equations in the cases of sin 2 θ ν 12 = 1/2 (bimaximal mixing), 1/3 (tri-bimaximal mixing) and 0.2.We have chosen the   12 we have plotted these observables only once.The chosen ranges of the λ ij lead from Eq. ( 13) to a lower limit of |U e3 | > ∼ 0.09/ √ 2 ≃ 0.06, as is seen in the figure.Improved future limits on the range of sin 2 θ 12 and, in particular, on the magnitude of |U e3 | can give us valuable information on the structure of U ℓ .The allowed parameter space of sin 2 θ 23 is roughly half of its allowed 3 σ range.The interplay of θ ν 12 and leptonic CP violation in neutrino oscillations mentioned above results in the "falling donut" structure when J CP is plotted against sin 2 θ 12 .We can also directly plot the sum-rule from Eq. ( 18), which is shown in Fig. 2. As a consequence of varying the observables in Eq. ( 18) we can extend the parameter space to smaller values of |U e3 |.In fact, if U ν corresponds to tri-bimaximal mixing, U e3 is allowed to vanish.Equation ( 13) can be used to understand the results in Fig. 2: if, for instance, we have sin 2 θ ν 12 = 1/2, the experimental upper limit of (sin 2 θ 12 ) max = 0.4 implies that |U e3 | ≥ 1/2 − (sin 2 θ 12 ) max ≃ 0.1.On the other hand, for sin 2 θ ν 12 = 0.2, and therefore sin 2θ ν 12 = 0.8, we have with (sin 2 θ 12 ) min = 0.24 that |U e3 | ≥ ((sin 2 θ 12 ) min − 0.20)/0.8≃ 0.05, which is in agreement with the figure.A more stringent limit on, or a value of, |U e3 | 2 < ∼ 0.01 would strongly disfavor (or rule out) the simple bimaximal mixing scenario.The equations given up to this point are also valid if neutrinos are Dirac particles.We will discuss now briefly the observables describing the CP violation associated with the angles take the quoted extreme values and some form of perturbation leads to non-zero θ 13 and non-maximal θ 23 .It is hoped that this perturbation is imprinted in correlations between various observables.Future precision experiments can tell us whether there are such correlations, which can then be used to identify the perturbation and to obtain thereby valuable hints on the flavor structure of the underlying theory.In this paper we have studied one interesting class of perturbations: because the observable lepton mixing matrix is a product of the diagonalization matrices of the charged lepton and neutrino mass matrices, U = U † ℓ U ν , we assumed that in the limit of one of these matrices being the unit matrix, maximal θ 23 and zero θ 13 would result.When the second matrix deviates from being the unit matrix, i.e., has a CKM-like form, we investigated the effects on the CP conserving and CP violating observables.Free parameters are the small angles of the "correction matrix", the 12-mixing angle of the leading matrix, and various phases.Scenarios like bimaximal mixing, tri-bimaximal mixing or Quark-Lepton Complementarity are special cases of our analysis.We consistently worked only with rephasing invariants in order to avoid the subtleties of identifying CP phases within different parameterizations.We should stress here also that our analysis is independent of the neutrino mass ordering and hierarchy.In the first scenario we have considered, the neutrino sector alone is responsible for zero θ 13 and maximal θ 23 .Requiring the neutrino mass matrix to obey a µ-τ symmetry can generate such a mixing pattern.Figures 1 and 2 illustrate the results.We find that |U e3 | will typically be non-zero, proportional to the sine of the largest angle in U ℓ , and in most of the cases will be well within reach of up-coming experiments.If U ν is bimaximal, |U e3 | should satisfy |U e3 | > ∼ 0.1 in order for sin 2 θ 12 to be within the 3 σ interval allowed by the current data.There is no similar constraint on |U e3 | in the case of tri-bimaximal U ν : even a vanishing value of |U e3 | is allowed.Atmospheric neutrino mixing can be maximal.There is a correlation between the solar neutrino mixing, the magnitude of |U e3 | and CP violation in neutrino oscillations, given by sin 2 θ 12 = sin 2 θ ν We find that both scenarios are in agreement with the existing neutrino oscillation data, have interesting phenomenology and testable differences.Future higher precision determinations of sin 2 θ 12 and sin 2 θ 23 , and more stringent constraints on, or a measurement of, |U e3 | can provide crucial tests of these simplest scenarios, shedding more light on whether any of the two scenarios is realized in Nature.

Figure 3 :
Figure 3: Correlations resulting from U = U † ℓ U ν if U ν is CKM-like and U † ℓ has maximal θ ℓ 23 and vanishing θ ℓ 13 , but free θ ℓ 12 .The results shown correspond to three representative values of θ ℓ 12 .The currently allowed 1 σ and 3 σ ranges of the observables are also indicated.