Origin of Constrained Maximal CP Violation in Flavor Symmetry

Current data from neutrino oscillation experiments are in good agreement with $\delta =-\frac{\pi}{2}$ and $\theta_{23}^{} = \frac{\pi}{4}$. We define the notion of"constrained maximal CP violation"(CMCPV) for predicting these features and study their origin in flavor symmetry. We derive the parametrization-independent solution of CMCPV and give a set of equivalent definitions for it. We further present a theorem on how the CMCPV can be realized. This theorem takes advantage of residual symmetries in the neutrino and charged lepton mass matrices, and states that, up to a few minor exceptions, $(|\delta |, \theta_{23}) =(\frac{\pi}{2}, \frac{\pi}{4})$ is generated when those symmetries are real. The often considered $\mu$-$\tau$ reflection symmetry, as well as specific discrete subgroups of O(3), are special cases of our theorem.


Introduction
While a coherent picture in leptonic mixing has emerged, important measurements are still lacking. In particular, the Dirac CP angle δ and the exact value of the atmospheric neutrino mixing angle θ 23 are of great interest. Whether θ 23 is maximal or departs sizably from π 4 has important ramifications for flavor symmetry models [1]. The CP phase also has model building impact, and the question of whether the lepton sector violates CP has conceptual significance in connection to the matter-antimatter asymmetry via leptogenesis [2]. While maximal atmospheric mixing is compatible with data since the observation of atmospheric neutrino oscillations, recently first hints towards a Dirac CP angle δ = − π 2 have arisen from the appearance and disappearance measurements of T2K [3] when combined with reactor antineutrino data. Indeed, global fits [4,5,6] confirm a mild preference for this particular value of CP phase.
With these in mind, it is tempting to study the origin of such values of δ and θ 23 within theories of flavor symmetry. In particular, the so-called µ-τ reflection symmetry [7,8,9,10] was often considered in the literature in this respect. It transforms the neutrino fields as (ν e , ν µ , ν τ ) → (ν * e , ν * τ , ν * µ ) , leading to |δ| = π 2 and θ 23 = π 4 in the standard parametrization of the PMNS mixing matrix [11,12]. In our study, we demonstrate that these two features arise as the outcome of "Constrained Maximal CP Violation" (CMCPV), which we will establish in a parametrizationindependent way by maximizing the Jarlskog invariant under a minimal constraint.
The framework we will discuss is that a flavor symmetry group G is broken such that the neutrino and charged lepton mass matrices are invariant under certain subgroups of G . We will propose and prove a general theorem revealing that if the residual flavor symmetries are real 1 , then the CMCPV is generated. There are a few minor exceptions to this theorem which we will clarify in Sec. 3. The µ-τ reflection symmetry is actually a special case of this theorem, which can be shown explicitly after a simple basis transformation. We further deduce some corollaries from the theorem which are practically useful in understanding and building models for the CMCPV. For instance, specific subgroups of O(3) can generate CMCPV, so do the models with certain groups under which all neutrino fields transform as triplets. As an illustration, we will present a simple model to explicitly realize the CMCPV. This paper is organized as follows. In Sec. 2, we will establish our definition of CMCPV in a parametrizationindependent way, and give a set of equivalent descriptions. Various physical implications (such as leptonic unitarity triangles) from CMCPV are further discussed. In Sec. 3, we present our theorem for the origin of CMCPV and derive its corollaries which are important for practical model buildings. We will study applications in Sec. 4, and finally we conclude in Sec. 5. Some elaborated mathematical proofs are presented in Appendices A and B.

Parametrization-Independent Formulation of Constrained Maximal CP Violation
What we mean by "constrained maximal CP violation" (CMCPV) is not merely |δ| = π 2 in the standard parametrization of the PMNS matrix [11,12], but both the |δ| = π 2 and θ 23 = π 4 . In general, a parametrization-independent definition of the maximal CP violation should be given in terms of Jarlskog invariant J [13], rather than the CP angle δ , because δ is not rephasing invariant. Furthermore, we will clarify shortly that naively maximizing J without constraint is already excluded by experimental data. Hence, introducing the new concept of CMCPV is essential for studying the viable maximal CP violation. For this purpose, we first formulate the CMCPV in a parametrizationindependent form. The Definition 1 is parametrization-independent because it does not invoke any explicit form of the PMNS matrix. Note that this is a constrained maximization problem. The Jarlskog invariant J [13] can be regarded as a function of the PMNS matrix U. We are looking for the maximal values of the function J(U) , where U is not an arbitrary unitary matrix but a constrained one. We impose this constraint on U by requiring the absolute values of its elements in the first row, (|U e1 |, |U e2 |, |U e3 |), be fixed to certain given values. (Actually, fixing any two of them in the first row is enough, due to the unitarity condition |U e1 | 2 + |U e2 | 2 + |U e3 | 2 = 1.) The reason that we fix absolute values of the elements of U in its first row, rather than any other rows or columns, will become clear shortly (cf. footnote-2).
This constraint is necessary because without it J would reach its maximal value as allowed by unitarity, |J| = 1 6 √ 3 , which is equivalent to all unitarity triangles being equilateral. The corresponding U in this case is just the Wolfenstein mixing matrix [14], with ω = e i2π/3 , which has been excluded by oscillation data. In the standard parametrization [12], the Jarlskog invariant is given by J = 1 8 sin δ cos θ 13 sin 2θ 13 sin 2θ 23 sin 2θ 12 .
Indeed, if we compute its maximum by ∂ θ J = 0 with θ = {θ 12 , θ 13 , θ 23 , δ}, we obtain the Wolfenstein mixing, (θ 13 , θ 12 , θ 23 , |δ|) = arctan 1 √ 2 , π 4 , π 4 , π 2 . This includes the desired values of (θ 23 , δ), but gives unrealistic (θ 12 , θ 13 ). Hence, the Wolfenstein mixing is already excluded by experimental data. To derive acceptable maximization of J , we observe that if we fix θ 12 and θ 13 (to their best-fit values for instance) and then maximize J , we still obtain (|δ|, θ 23 ) = π 2 , π 4 . This is in fact consistent with the above parametrization-independent Definition 1 of CMCPV , because fixing (θ 12 , θ 13 ) corresponds to fixing the absolute values of the elements in the first row of U under the standard parametrization. Note that this is the allowed minimal constraint we could impose on the Jarlskog invariant: fixing any other row or column of the PMNS matrix and then maximizing J will not lead to experimentally acceptable results. 2 Hence, the above Definition 1 gives a minimal parametrization-independent definition of viable maximal CP violation.
It is worth noting that the Jarlskog invariant can be further cast into a manifestly parametrization-independent form [15], The above Eq. (3a) is a general expression with (α, β) = e, µ, τ and (i, j) = 1, 2, 3, where α β and i j . The Eq. (3b) corresponds to the case of (α, β) = (e, µ) and (i, j) = (2,3). It is clear that the Jarlskog invariant can be fully determined by any 4 independent matrix elements {|U αi |, |U α j |, |U βi |, |U β j |} with α β and i j , as shown in Eq. (3a). According to our above Definition 1 of CMCPV and using Eq. (3b), we can maximize Jarlskog invariant by imposing the extremal conditions with the two elements (|U e1 |, |U e3 |) of first row fixed, ∂J 2 /∂|U µ1 | 2 = 0 and ∂J 2 /∂|U µ3 | 2 = 0 . From these two equations, we can derive the solution of CMCPV, which is the maximum of J under the minimal constraint of Definition 1, and is manifestly parametrization-independent. The detail of this derivation is presented in Appendix A.
When adopting the standard parametrization [12] of PMNS mixing matrix U, we can use Eq. (4) to immediately deduce the explicit realization of CMCPV, (|δ|, θ 23 ) = π 2 , π 4 , which is proven in Appendix A. Furthermore, we find that once we realize (|δ|, θ 23 ) = π 2 , π 4 under the standard parametrization, the PMNS matrix exhibits an interesting feature, which we explain as follows. The standard parametrization of the PMNS matrix is expressed as [12], where we have used the common notations (s i j , c i j ) = (sin θ i j , cos θ i j ). Under a rephasing we obtain For (δ, θ 23 ) = ± π 2 , π 4 , we find that U ′ becomes Eq. (8) explicitly reveals that the first row is real, while the second row and the third row are complex conjugates of each other. We call this feature the "row conjugation equality" (RCE). It is easy to see that the reverse is also true: holding RCE will result in (|δ|, θ 23 ) = π 2 , π 4 . Even though we have demonstrated RCE by using the standard parametrization (5) [12], we stress that the RCE form should be independent of parametrizations (up to trivial rephasing). We can use any other parametrizations [16] [17] and maximize Jarlskog invariant under the same constraint as in Definition 1. Then, we find that the mixing matrix always exhibits the RCE form, up to a trivial rephasing. In fact, we see that any specific RCE form does obey |U µ j | = |U τ j | as in our parametrization-independent general solution (4) of the CMCPV.
For later usage, let us introduce the following lemma on the RCE. (3) is an arbitrary orthogonal matrix.

Lemma [ O(3) invariance of RCE ]: If a unitary matrix V has the form of RCE, then after a right-handed real transformation V
The proof of this statement is delegated to Appendix B.1.
Another interesting feature of RCE concerns leptonic unitarity triangles (LUTs) of the PMNS matrix U, in connection to its column orthogonality, where the column indices i j . We call these unitarity triangles the column triangles. It is evident that if U has the form of RCE, then all column triangles should be isosceles triangles because under the RCE the two sides U * µi U µ j and U * τi U τ j have equal length, |U * µi U µ j | = |U * τi U τ j | . Unitarity triangles are intrinsically connected to CP violation because all these triangles have the same area, which equals half of the absolute value of Jarlskog invariant J. The LUTs are less studied than the unitarity triangles in quark sector since measuring the LUTs and thus the leptonic CP violation is much harder. Nevertheless, the LUTs can be directly measured in principle via oscillation experiments [18]. Furthermore, the LUTs can provide a geometrical formulation of the CMCPV. Since we define CMCPV as J reaching its maximum under certain constraints, it also means that the area of the LUT reaches its maximum under those constraints. How do these constraints appear in our current geometrical picture? The constraint in our Definition 1 is that the first row of U is fixed, namely, |U * e1 U e2 |, |U * e2 U e3 | and |U * e3 U e1 | are fixed, which means that the e-sides of the column triangles are fixed. Hence, the Definition 1 is equivalent to saying that each column triangle reaches its maximal area with its e-side fixed. This provides a geometrical formulation of the CMCPV.
Note that for a triangle with its e-side fixed and its perimeter (the sum of the lengths of its three sides) bounded from above, its area reaches the maximum if and only if it is an isosceles triangle. This is clear from geometrical intuition. In Ref. [19], we proved that a unitarity triangle must always have its perimeter equal or less than 1. This is a necessary and sufficient condition for a triangle to be unitarity triangle, and requires the perimeter of each unitarity triangle to be bounded from above, which ensures the area of each unitarity triangle to have a maximum. With these, we give a geometrical formulation of the CMCPV: it corresponds to the maximal area of the LUT by fixing its e-side, and such LUT is an isosceles triangle.
Finally, we summarize the analysis of this section into the following theorem.
After setting up the above preliminaries, we are ready to study the origin of CMCPV in flavor symmetry in the next section.

Origin of Constrained Maximal CP Violation
In this Section, we will trace CMCPV to the "residual symmetries", i.e., the subgroups of the original flavor symmetry group that remain intact after the full group is broken.
Consider that the flavor symmetry group G is broken down to two residual symmetries G ν and G ℓ for neutrinos and charged leptons, respectively. They are defined as follows, where M ν is the Majorana mass matrix of neutrinos, and M ℓ M † ℓ is the effective mass matrix of left-handed charged leptons. Thus, the mixing matrices U ν and U ℓ (which diagonalize M ν and M ℓ M † ℓ , respectively) are directly determined by S and T [20], Here the matrices D S and D T are diagonal matrices. Since Eq. (11) demonstrates a direct connection between flavor symmetry and lepton mixings, it also attracted extensive studies [21,22,23,24] via the approach of symmetry and group theory, without resorting to explicit mass matrices or a fundamental Lagrangian. With this general massindependent approach, we will analyze the origin of CMCPV. Roughly speaking, our theorem states that the CMCPV can be realized if the residual symmetries are real. In rigorous manner, we formulate this theorem in the following form.

Theorem 2 [Origin of CMCPV].
If the residual flavor symmetries in the lepton sector (including charged leptons and neutrinos) are real and fully determine the mixing pattern, then the CMCPV always holds, up to a few minor exceptions: (i). one of the three mixing angles in the PMNS matrix is zero; (ii). neutrinos are not Majorana fermions; (iii). the residual symmetry for charged leptons is a Klein group, i.e., It is clear that the exception (i) is already excluded by current oscillation data, and the exception (ii) is not a concern for most neutrino theories. The exception (iii) is less trivial, but can be easily evaded in model-buildings. Besides, in Theorem 2, for the residual symmetries being real, we mean that there always exists a basis under which these symmetries become real.
To illustrate this theorem explicitly, we first consider a simple (unrealistic) example. A rotation of 120 • around the axis (1, 1, 1) T can be represented by Suppose that this R corresponds to the residual symmetry group G ℓ . Thus, we have where U W is the Wolfenstein mixing matrix defined in Eq. (1). According to the relation (11), we have U ℓ = U W , which shows that U † ℓ exhibits RCE. If we further assume that the neutrino mass matrix is diagonal (and thus U ν = I ), then the PMNS matrix U = U † ℓ U ν gives, (|δ|, θ 23 ) = ( π 2 , π 4 ), because Theorem 1 states that the RCE always leads to CMCPV.
Two remarks are in order for this example. One is that both residual symmetries are real. This is explicit for G ℓ . The form of G ν is, G ν ⊃ {diag(1, 1, −1), diag(1, −1, 1)}, since neutrino mass matrix is taken to be diagonal. The other point is that U † ℓ exhibits RCE. In general, the validity of Theorem 2 (excluding its exceptions) implies the following important points: 1). real G ν leads to real U ν , which will be explicitly proven in Appendix B.2; 2). real G ℓ leads to RCE in U † ℓ , which will be explicitly proven in Appendix B.3; 3). if U † ℓ has RCE and U ν is real, then the PMNS matrix U = U † ℓ U ν has RCE. The three points above are combined to prove Theorem 2. The last point is just based on the Lemma given above Eq. (9), and the first two points can be understood by the following reasoning. (We delegate the mathematical proofs to Appendix B.) Since both G ℓ and G ν contain only real transformations, they can be geometrically regarded as rotations in 3-dimensional Euclidean space. (Here a trivial minus sign between the determinants of SO (3) and O (3) does not matter.) For such a rotation represented by a matrix R , the rotation axis is one of its eigenvectors with the corresponding eigenvalue equal to 1. The remaining two eigenvectors must be complex conjugate to each other, which is a general property of SO(3) matrices. (The explicit forms of the two eigenvectors are given in Appendix B.3.) Hence, if R ∈ G ℓ , then the eigenvectors of R are the columns of U ℓ , which implies two columns in U ℓ are conjugate to each other, and thus U † ℓ has RCE. There are differences in the neutrino sector, because we consider the neutrinos as Majorana particles here. Hence, the residual symmetry has to be constructed with Z 2 's, i.e., G ν = Z 2 ⊗ Z 2 , which geometrically correspond to two rotations of 180 • . These are special rotations in the sense that only such rotations may commute with rotations around different axes. For rotations of 180 • , the eigenvalues are (1, −1, −1), cf. Eq. (B.7). Due to a partial degeneracy of the eigenvalues, the neutrino mass matrix M ν should be determined by two Z 2 -rotations with orthogonal axes. Each axis determines one column of U ν , so U ν only contains real column vectors. This in turn implies that G ℓ cannot be Z 2 ⊗ Z 2 , which is the exceptional case (iii) pointed out in Theorem 2: if G ℓ = Z 2 ⊗ Z 2 , then U ℓ will be real and no CP violation exists. Now, it is also easy to understand why Theorem 2 requires neutrinos to be Majorana fermions, since the symmetries Z 2 ⊗ Z 2 are needed for G ν .
Theorem 2 further leads to a series of corollaries which we will discuss as follows.

Corollary A [ O(3) Subgroups ]: If an O(3) subgroup G contains sufficient residual symmetries that can fully determine a mixing matrix, then it leads to the CMCPV after avoiding the three exceptions listed in Theorem 2.
This is manifest because the constraint which requires the residual symmetries to be subgroups of O(3) makes G ℓ and G ν automatically real. According to Theorem 2, this leads to the CMCPV. Examples of such residual symmetries include popular groups like A 4 , S 4 , and A 5 , corresponding to tetrahedral, octahedral, and icosahedral symmetries, respectively.
We should comment on the phrase "sufficient residual symmetries" in Corollary A. As is well-known, the maximal residual symmetries in the charged lepton and neutrino sectors are U(1) ⊗ U(1) ⊗ U(1) and Z 2 ⊗ Z 2 ⊗ Z 2 , respectively [20]. But when seeking flavor groups to unify the residual symmetries, it is not necessary to cover those large groups. For charged leptons, the minimal choice is to take a Z 3 subgroup from those U(1)'s, which is in fact sufficient to determine the mixing U ℓ . For the neutrino sector, the minimal sufficient residual symmetry should be Z 2 ⊗ Z 2 . So, this smaller set of residual symmetries should be included in the O(3) subgroup for the Corollary A.
However, in some models, especially those based on A 4 , sometimes the flavor group does not contain sufficient residual symmetries, and the so-called accidental symmetries are present to fully determine the mixings. Those accidental symmetries depend on the detailed dynamics of the model (instead of the flavor group), so the Corollary A does not apply. But, if the accidental symmetry is a real symmetry, then Theorem 2 applies and there is still CMCPV.

Corollary B [ Real M ν ]:
If G ℓ is real and M ν is real or M ν can be written as M ν = z 1 I + z 2Mν , whereM ν and I are real and identity matrices, respectively, and (z 1 , z 2 ) are complex numbers, then there is CMCPV, after evading the three exceptions listed in Theorem 2.
We first consider the case that M ν is real. Then, as a real symmetric matrix, M ν can be diagonalized by a real orthogonal matrix, which implies U ν and G ν are real. Hence, according to Theorem 2, we have CMCPV. Multiplying M ν by an overall complex phase will not change U ν . Thus, if M ν = z 2Mν with realM ν , then this means that M ν is essentially real, up to an overall complex phase factor. Hence, this case also leads to CMCPV. Next, consider M ν = z 1 I +M ν with realM ν . This means that M ν is real up to subtracting a constant from all diagonal elements. In this case, for S (∈ G ν ) satisfying S TM ν S =M ν , we have S T M ν S = S T (z 1 I +M ν )S = z 1 I +M ν , which shows that M ν is also invariant under S . Hence, M ν has invariance under real G ν and there is CMCPV. Finally, combining the two cases above, we have thus proven the Corollary B for the general form M ν = z 1 I + z 2Mν .
The form M ν = z 1 I + z 2Mν has important applications in model buildings for CMCPV. Typically, for building flavor symmetry models, at least one flavon φ is introduced to couple with neutrinos ν and forms a Yukawa term ννφ , which contributes to neutrino masses if the vacuum expectation values (VEV) φ 0 , where φ is a scalar field acting as a multiplet under the flavor symmetry. Neutrinos are commonly considered as flavor triplets in many models, so a νν term or νν ξ term will usually show up, where ξ is a flavor singlet. These terms will contribute to M ν as a diagonal mass term z 1 I , where z 1 is complex because the coefficients (Yukawa couplings) of these terms are complex in general. The ννφ term will contribute in a form of z 2Mν if φ is real (up to an overall complex phase) and the Clebsch-Gordan (CG) coefficients for the term are real.
Note that here we only consider the case with all 3 generations of neutrinos unified into a triplet of the flavor group. Otherwise, the νν term would not be diagonal. 3 The flavon φ can be in any non-trivial representation. We should point out that both real φ and real CG coefficients are very common in many groups. For instance, in the 3-dimensional representation of A 4 , the CG coefficients for 3 ⊗ 3 ⊗ 3 → 1 are real in both the real basis (used for example in [25]) and the complex basis (used for example in [26]). This corollary does not apply to groups with inherent complex CG coefficients, like T ′ [27] or ∆(27) [28]. As for real φ , if φ is a real scalar field by definition, then φ is real. If φ has to be a complex field, then as known from minimization of scalar potentials in many models, it is still common to have VEV alignment according to φ ∝ (1, 1, 1), (1, 0, 0), etc., which is real. If φ is however complex, then in general CMCPV does not follow. For those "real ννφ models", where only the ννφ term makes a non-trivial contribution (not proportional to the unit matrix) to M ν , we have the following corollary.

Corollary C [ Real ννφ Models ]: Real ννφ models always lead to CMCPV , if the three minor exceptions listed in Theorem 2 do not happen.
For a demonstration of the above general discussion, we will build a simple "real ννφ model" in the following Section 4.3.

Applications
In this section, we apply our theorems and corollaries to various situations and understand why CMCPV is realized in certain cases. We will illustrate how to achieve the CMCPV in model buildings. There are extensive recent literature [29] studying specific models of (|δ|, θ 23 ) = ( π 2 , π 4 ) .

µ − τ Reflection Symmetry
The µ-τ reflection symmetry was studied before [7,8,9,10], which is sometimes also called the generalized µ-τ symmetry. This symmetry is defined in the flavor eigenbasis (with M ℓ diagonal), Imposing this symmetry leads to the following form of the neutrino and lepton mass matrices, whereM ν is symmetric andM 2 ℓ ≡M ℓM † ℓ is diagonal. Note that the elements r 1,2 are real, but z 1,2 are complex in general. The operation (14) will transform ν TM ν ν to its Hermitian conjugate. We can directly check that the Lagrangian term, L ⊃ ν TM ν ν + h.c., is invariant under the transformation (14) if and only ifM ν takes the form of Eq. (15).
Let us make a transformation, with Thus, we derive .
where we have defined notations, z j ≡ z j1 + iz j2 , ( j = 1, 2), and a ≡ m 2 e , b ± ≡ 1 2 (m 2 µ ± m 2 τ ) . The quantities (z j1 , z j2 ) and (a, b ± ) are all real. Note that M ν is a real matrix, and the charged lepton sector has an SO(2) residual symmetry, This is because RM 2 ℓ R † = M 2 ℓ holds for θ ∈ [0, 2π) . Since M ν and G ℓ are all real, this will lead to CMCPV according to our Corollary B. Note that real M ν implies that G ν is real.
The µ-τ reflection symmetry is certainly not the only possibility to generate CMCPV. From Eq. (18), we see that M ν is real, while Corollary B shows that M ν can have a more general form including complex numbers. Hence, the µ-τ reflection symmetry is just a special case of real residual symmetries, although this is not manifest before the transformation of basis in Eq. (16).

CMCPV from Geometrical Symmetry Breaking
As another example illustrating our theorem, we revisit a model from Ref. [23], which predicted |δ| = π 2 and θ 23 = π 4 (as well as θ 13 ≃ π 4 − θ 12 ). We will show that this model fulfills the criteria for CMCPV. This model identifies a Z 4 rotation around the x-axis as G ℓ , and the product reflections Z 2 ⊗Z 2 as G ν , where one Z 2 reflects y → −y and the other Z 2 transforms (x, z) → −(z, x) . These rotations are subgroups of the octahedral symmetry O h , and can be shown by simple geometrical picture. This group setting generates the bimaximal mixing, θ 12 = θ 23 = π 4 and θ 13 = 0. The necessary deviation from this leading order scheme was generated by slightly tilting the axis of Z 4 rotation by a small angle (defined as √ 2ǫ ) that turns out to be related to nonzero θ 13 . We also verified that this geometrical symmetry breaking can arise from certain flavon models. For example, we may set up a concrete realization, where a flavor triplet φ is responsible for mass-generation of the charged leptons and the Yukawa terms involving φ are SO(3) symmetric in the 3-dimensional flavor space. With these, the geometrical breaking is connected to the VEV misalignment of flavons. After the axis tilt, the residual symmetry of charged lepton mass matrix is represented by [23], The neutrino mass matrix is still invariant under the original reflections Z 2 ⊗ Z 2 , as represented by [23], Since all the residual symmetries are real, Theorem 2 applies and the model should realize CMCPV. This is indeed the case, as found in Ref. [23].

A Real ννφ Model
As stated in Corollary C, the real ννφ models should always produce CMCPV. In the following, we will build such a model as an explicit illustration.
We use A 4 ⊗ Z 2 as flavor symmetry group and introduce two scalar fields φ ℓ and φ ν , in addition to the SM Higgs doublet H. The relevant particle content of this model is summarized in Table 1. The Lagrangian for the lepton-neutrino sector contains, L ⊃ y e (Lφ ℓ )He c + y µ (Lφ ℓ ) ′ Hµ c + y τ (Lφ ℓ ) ′′ Hτ c + y ν1 (LL)HH + y ν2 (φ ν LL)HH + h.c., (22) where L stands for the left-handed lepton doublet of SU(2) L and H is the Higgs doublet. Since φ ℓ and φ ν do not have any charge other than the Z 2 assignment, they can be real fields. Consider that they acquire the following alignment of VEVs, where similar to [30], we have introduced a small perturbation on the usual VEV alignment in φ ν , with ǫ 2 , ǫ 3 ≪ 1 . Similar VEV alignment was already considered in the literature [30], but its further elaboration is irrelevant to the current illustration purpose of realizing CMCPV on the ground of residual symmetries [20]; it is also fully beyond the main goal of this short Letter. For ǫ 2 = ǫ 3 = 0 , one would obtain the tri-bimaximal mixing; and the small (ǫ 2 , ǫ 3 ) should produce the necessary corrections [30]. Note that for ǫ 2 = ǫ 3 = 0 , although φ ν is real as required by our definition of real ννφ models, one of the mixing angles, θ 13 , is zero, which just matches the exception-(i) of our Theorem 2. Hence, CMCPV does not follow in this case. For ǫ 2,3 0 , the charged leptons and neutrinos acquire masses as follows, where the mass parameters (a, b, c, d) are complex in general. This type of lepton and neutrino mass matrices are often studied in the literature [31]. For instance, diagonalizing the lepton mass matrix M † ℓ M ℓ gives the masseigenvalues (m e , m µ , m τ ) ∝ (|a|, |b|, |c|) . [This also shows that using the observed mass values (m e , m µ , m τ ) does not fully fix the parameters (a, b, c) themselves; while inputting the model parameters (a, b, c) can fully accommodate the observed lepton masses.] The focus of our paper is on the origin of leptonic mixings from flavor symmetry. Thus, by diagonalizing M ℓ and M ν , we derive the following lepton and neutrino mixing matrices, and From U = U † ℓ U ν , it is straightforward to extract the PMNS parameters in the standard parametrization, These results show that, apart from model-specific deviations of (θ 13 , θ 12 ) from their tri-bimaximal values, we have realized the CMCPV, as expected from Corollary C. From Eq. (27b), we can determine the perturbative parameters (ǫ 2 , ǫ 3 ) in terms of (sin θ 13 , tan θ 12 ) via Taking θ 13 ≃ 9 • and θ 12 ≃ 34 • , we derive (ǫ 2 , ǫ 3 ) ≃ (−0. 18, 0.21) . In general, we can extend the real ννφ models to type-I neutrino seesaw. In this case, we may introduce three right-handed neutrinos ν R in the 3-dimensional representation of A 4 . Thus, the neutrino Dirac mass matrix will be proportional to unit matrix, m D ∝ I , while the heavy Majorana mass matrix M R shares similar structure with M ν in Eq. (24). Hence, we find that the seesaw mass matrix of light neutrinos M ν ∝ M −1 R .

Conclusions
In this work, we stressed that a general parametrization-independent definition of the maximal CP violation should be constructed in terms of Jarlskog invariant J , rather than the CP angle δ (which is rephasing non-invariant). We pointed out that naively maximizing J without constraint is already excluded by oscillation data. We further demonstrated the crucial importance of introducing the new concept of constrained maximal CP violation (CMCPV) for studying the viable maximal CP violation. For this purpose, we constructed CMCPV in the Definition 1, and formulated it by a set of equivalent ways, as summarized in Theorem 1 (Sec. 2). We derived the parametrization-independent realization of the CMCPV via solution (4), which was proven to be the maximum of Jarlskog invariant under a minimal constraint on the PMNS matrix U (Sec. 2 and Appendix A). We found that the CMCPV just corresponds to (|δ|, θ 23 ) = ( π 2 , π 4 ) in the standard parametrization of the PMNS matrix (5). In Sec. 3 and Appendix B, we proved Theorem 2, stating that if the residual symmetries of neutrinos and charged leptons are real, then the CMCPV should be realized, up to a few minor exceptions. It was shown that the conditions for CMCPV are actually quite common, and we presented several sample models in Sec. 4, demonstrating that in particular the often considered µ-τ reflection symmetry is a special case of our theorem. We also note that the current formulation cannot be naively applied to the quark sector. The reason is that our Theorem 1 proves RCE to be essential for the CMCPV, but RCE cannot hold for the CKM mixing matrix due to experimental data. Namely, any two rows (or columns) in the CKM matrix cannot be conjugate to each other (up to rephasing).
If indeed the values of δ ≃ − π 2 and θ 23 ≃ π 4 continue to be favored by neutrino data, our general theorems and corollaries of CMCPV should be important, and provide strong guidelines for the model buildings with flavor symmetry.

Appendix A. Parametrization-Independent Solution of CMCPV
In this Appendix, we derive the general solution of CMCPV by using the manifestly parametrization-independent formula of Jarlskog invariant (3).
For G ν = Z 2 ⊗ Z 2 , we may use S 1 and S 2 to represent the transformations of the two Z 2 's, respectively. Thus, [S 1 , S 2 ] = 0 should hold, which implies that their rotation axes must be orthogonal. Hence, geometrically G ν contains two 180 • -rotations with orthogonal axes. These two axes can be represented by two normalized real vectors v 1 and v 2 with where v 1 and v 2 are column vectors, of the 3×1 matrix form. Taking v 3 = v 1 ×v 2 and U ν = (v 1 , v 2 , v 3 ), we see that U ν is a real matrix and can diagonalize S 1 and S 2 simultaneously in the way given by Eq. (11). Therefore, if G ν ⊃ {S 1 , S 2 } contains only real matrices, then U ν is real. The converse proposition that a real U ν leads to real G ν is also true, and can be readily proven.
Appendix B.3. Real G ℓ Leads to Complex U † ℓ with RCE We need to prove that any SO(3) matrix R can be diagonalized by U † R R U R , where the unitary matrix U R contains one real column and two other columns which are complex conjugate to each other. This can be explicitly proven as below.
The most general rotation in 3d Euclidean space which rotates the space around an axis n = (n 1 , n 2 , n 3 ) T by an angle φ is [24], where n · n = 1 and (s, c) = (sin φ, cos φ). We can directly verify that this matrix is diagonalized as We see explicitly that the first column of U R is real, and the second and third columns are conjugate to each other, i.e., U † R has RCE. Hence, if R ∈ G ℓ , then U † ℓ = U † R has RCE.