Leptonic Dirac CP Violation Predictions from Residual Discrete Symmetries

Assuming that the observed pattern of 3-neutrino mixing is related to the existence of a (lepton) flavour symmetry, corresponding to a non-Abelian discrete symmetry group $G_f$, and that $G_f$ is broken to specific residual symmetries $G_e$ and $G_\nu$ of the charged lepton and neutrino mass terms, we derive sum rules for the cosine of the Dirac phase $\delta$ of the neutrino mixing matrix $U$. The residual symmetries considered are: i) $G_e = Z_2$ and $G_{\nu} = Z_n$, $n>2$ or $Z_n \times Z_m$, $n,m \geq 2$; ii) $G_e = Z_n$, $n>2$ or $Z_n \times Z_m$, $n,m \geq 2$ and $G_{\nu} = Z_2$; iii) $G_e = Z_2$ and $G_{\nu} = Z_2$; iv) $G_e$ is fully broken and $G_{\nu} = Z_n$, $n>2$ or $Z_n \times Z_m$, $n,m \geq 2$; and v) $G_e = Z_n$, $n>2$ or $Z_n \times Z_m$, $n,m \geq 2$ and $G_{\nu}$ is fully broken. For given $G_e$ and $G_\nu$, the sum rules for $\cos\delta$ thus derived are exact, within the approach employed, and are valid, in particular, for any $G_f$ containing $G_e$ and $G_\nu$ as subgroups. We identify the cases when the value of $\cos\delta$ cannot be determined, or cannot be uniquely determined, without making additional assumptions on unconstrained parameters. In a large class of cases considered the value of $\cos\delta$ can be unambiguously predicted once the flavour symmetry $G_f$ is fixed. We present predictions for $\cos\delta$ in these cases for the flavour symmetry groups $G_f = S_4$, $A_4$, $T^\prime$ and $A_5$, requiring that the measured values of the 3-neutrino mixing parameters $\sin^2\theta_{12}$, $\sin^2\theta_{13}$ and $\sin^2\theta_{23}$, taking into account their respective $3\sigma$ uncertainties, are successfully reproduced.


Introduction
The discrete symmetry approach to understanding the observed pattern of 3-neutrino mixing (see, e.g., [1]), which is widely explored at present (see, e.g., [2][3][4][5]), leads to specific correlations between the values of at least some of the mixing angles of the Pontecorvo, Maki, Nakagawa, Sakata (PMNS) neutrino mixing matrix U and, either to specific fixed trivial or maximal values of the CP violation (CPV) phases present in U (see, e.g., [6][7][8][9][10] and references quoted therein), or to a correlation between the values of the neutrino mixing angles and of the Dirac CPV phase of U [11][12][13][14][15]. 1 As a consequence of this correlation the cosine of the Dirac CPV phase δ of the PMNS matrix U can be expressed in terms of the three neutrino mixing angles of U [11][12][13][14], i.e., one obtains a sum rule for cos δ. This sum rule depends on the underlying discrete symmetry used to derive the observed pattern of neutrino mixing and on the type of breaking of the symmetry necessary to reproduce the measured values of the neutrino mixing angles. It depends also on the assumed status of the CP symmetry before the breaking of the underlying discrete symmetry.
The approach of interest is based on the assumption of the existence at some energy scale of a (lepton) flavour symmetry corresponding to a non-Abelian discrete group G f . Groups that have been considered in the literature include S 4 , A 4 , T , A 5 , D n (with n = 10, 12) and ∆(6n 2 ), to name several. The choice of these groups is related to the fact that they lead to values of the neutrino mixing angles, which can differ from the measured values at most by subleading perturbative corrections. For instance, the groups A 4 , S 4 and T are commonly utilised to generate tri-bimaximal (TBM) mixing [18]; the group S 4 can also be used to generate bimaximal (BM) mixing [19]; 2 A 5 can be utilised to generate golden ratio type A (GRA) [21][22][23] mixing; and the groups D 10 and D 12 can lead to golden ratio type B (GRB) [24] and hexagonal (HG) [25] mixing.
The flavour symmetry group G f can be broken, in general, to different symmetry subgroups G e and G ν of the charged lepton and neutrino mass terms, respectively. G e and G ν are usually called "residual symmetries" of the charged lepton and neutrino mass matrices. Given G f , which is usually assumed to be discrete, typically there are more than one (but still a finite number of) possible residual symmetries G e and G ν . The subgroup G e , in particular, can be trivial, i.e., G f can be completely broken in the process of generation of the charged lepton mass term.
The residual symmetries can constrain the forms of the 3 × 3 unitary matrices U e and U ν , which diagonalise the charged lepton and neutrino mass matrices, and the product of which represents the PMNS matrix: Thus, by constraining the form of the matrices U e and U ν , the residual symmetries constrain also the form of the PMNS matrix U . In general, there are two cases of residual symmetry G ν for the neutrino Majorana mass term when a portion of G f is left unbroken in the neutrino sector. They characterise two possible approaches -direct and semi-direct [4] -in making predictions for the neutrino mixing observables using discrete flavour symmetries: G ν can either be a Z 2 × Z 2 symmetry 1 In the case of massive neutrinos being Majorana particles one can obtain under specific conditions also correlations between the values of the two Majorana CPV phases present in the neutrino mixing matrix [16] and of the three neutrino mixing angles and of the Dirac CPV phase [11,17]. 2 Bimaximal mixing can also be a consequence of the conservation of the lepton charge L = Le − Lµ − Lτ (LC) [20], supplemented by a µ − τ symmetry.
(which sometimes is identified in the literature with the Klein four group), or a Z 2 symmetry. In models based on the semi-direct approach, where G ν = Z 2 , the matrix U ν contains two free parameters, i.e., one angle and one phase, as long as the neutrino Majorana mass term does not have additional "accidental" symmetries, e.g., the µ − τ symmetry. In such a case as well as in the case of G ν = Z 2 × Z 2 , the matrix U ν is completely determined by symmetries up to re-phasing on the right and permutations of columns. The latter can be fixed by considering a specific model. It is also important to note here that in this approach Majorana phases are undetermined.
In the general case of absence of constraints, the PMNS matrix can be parametrised in terms of the parameters of U e and U ν as follows [26]: HereŨ e andŨ ν are CKM-like 3 × 3 unitary matrices and Ψ and Q 0 are given by: where ψ, ω, ξ 21 and ξ 31 are phases which contribute to physical CPV phases. Thus, in general, each of the two phase matrices Ψ and Q 0 contain two physical CPV phases. The phases in Q 0 contribute to the Majorana phases [16] in the PMNS matrix (see further) and can appear in eq. (2) as a result of the diagonalisation of the neutrino Majorana mass term, while the phases in Ψ can result from the charged lepton sector (U † e = (Ũ e ) † Ψ), from the neutrino sector (U ν = ΨŨ ν Q 0 ), or can receive contributions from both sectors.
For all the forms ofŨ ν considered in [11] and listed above we have i) θ ν 13 = 0, which should be corrected to the measured value of θ 13 ∼ = 0.15, and ii) sin 2 θ ν 23 = 0.5, which might also need to be corrected if it is firmly established that sin 2 θ 23 deviates significantly from 0.5. In the case of the BM and HG forms, the values of sin 2 θ ν 12 lie outside the current 3σ allowed ranges of sin 2 θ 12 and have also to be corrected.
The requisite corrections are provided by the matrix U e , or equivalently, byŨ e . The approach followed in [11][12][13][14] corresponds to the case of a trivial subgroup G e , i.e., of G f completely broken by the charged lepton mass term. In this case the matrixŨ e is unconstrained and was chosen in [11] on phenomenological grounds to have the following two forms: These two forms appear in a large class of theoretical models of flavour and theoretical studies, in which the generation of charged lepton masses is an integral part (see, e.g., [17,[32][33][34][35][36][37]).
In the case of G e = Z 2 (G ν = Z 2 ) the matrix U e (U ν ) is determined up to a U (2) transformation in the degenerate subspace, since the representation matrix of the generator of the residual symmetry has degenerate eigenvalues. On the contrary, when the residual symmetry is large enough, namely, G e = Z n , n > 2 or Z n × Z m , n, m ≥ 2 and G ν = Z 2 × Z 2 (G ν = Z n , n > 2 or Z n ×Z m , n, m ≥ 2) for Majorana (Dirac) neutrinos, the matrices U e and U ν are fixed (up to diagonal phase matrices on the right, which are either unphysical for Dirac neutrinos, or contribute to the Majorana phases otherwise, and permutations of columns) by the residual symmetries of the charged lepton and neutrino mass matrices. In the case when the discrete symmetry G f is fully broken in one of the two sectors, the corresponding mixing matrix U e or U ν is unconstrained and contains in general three angles and six phases. Our article is organised as follows. In Section 2 we describe the parametrisations of the PMNS matrix depending on the residual symmetries G e and G ν considered above. In Sections 3, 4 and 5 we consider the breaking patterns 1), 2), 3) and derive sum rules for cos δ. At the end of each of these sections we present numerical predictions for cos δ in the cases of the flavour symmetry groups G f = A 4 , T , S 4 and A 5 . In Section 6 we provide a summary of the sum rules derived in Sections 3 -5. Further, in Sections 7 and 8 we derive the sum rules for the cases 4) and 5), respectively. In these cases the value of cos δ cannot be fixed without additional assumptions on the unconstrained matrix U e or U ν . The cases studied in [14] belong to the ones considered in Section 7, where the particular forms of the matrix U e , leading to sum rules of interest, have been considered. In Section 9 we present the summary of the numerical results. Section 10 contains the conclusions. Appendices A, B, C, D and E contain technical details related to the study.

Preliminary Considerations
As was already mentioned in the Introduction, the residual symmetries of the charged lepton and neutrino mass matrices constrain the forms of the matrices U e and U ν and, thus, the form of the PMNS matrix U . To be more specific, if the charged lepton mass term is written in the left-right convention, the matrix U e diagonalises the hermitian matrix M e M † e , U † e M e M † e U e = diag(m 2 e , m 2 µ , m 2 τ ), M e being the charged lepton mass matrix. If G e is the residual symmetry group of M e M † e we have: where g e is an element of G e , ρ is a unitary representation of G f and ρ(g e ) gives the action of G e on the LH components of the charged lepton fields having as mass matrix M e . As can be seen from eq. (13), the matrices ρ(g e ) and M e M † e commute, implying that they are diagonalised by the same matrix U e .
Similarly, if G ν is the residual symmetry of the neutrino Majorana mass matrix M ν one has: where g ν is an element of G ν , ρ is a unitary representation of G f under which the LH flavour neutrino fields ν lL (x), l = e, µ, τ , transform, and ρ(g ν ) determines the action of G ν on ν lL (x).
It is not difficult to show that also in this case the matrices ρ(g ν ) and M † ν M ν 8 commute, and therefore they can be diagonalised simultaneously by the same matrix U ν . In the case of Dirac neutrinos eq. (14) is modified as follows: The types of residual symmetries allowed in this case and discussed below are the same as those of the charged lepton mass term. In many cases studied in the literature (e.g., in the cases of G f = S 4 , A 4 , T , A 5 ) ρ(g f ), g f being an element of G f , is assumed to be a 3-dimensional representation of G f because one aims at unification of the three flavours (e.g., three lepton families) at high energy scales, where the flavour symmetry group G f is unbroken.
At low energies the flavour symmetry group G f has necessarily to be broken to residual symmetries G e and G ν , which act on the LH charged leptons and LH neutrinos as follows: where g e and g ν are the elements of the residual symmetry groups G e and G ν , respectively, and l L = (e L , µ L , τ L ) T , ν lL = (ν eL , ν µL , ν τ L ) T .
The largest possible exact symmetry of the Majorana mass matrix M ν having three nonzero and non-degenerate eigenvalues, is a Z 2 × Z 2 × Z 2 symmetry. The largest possible exact symmetry of the Dirac mass matrix M e is U (1) × U (1) × U (1). Restricting ourselves to the case in which G f is a subgroup of SU (3) instead of U (3), the indicated largest possible exact symmetries reduce respectively to Z 2 × Z 2 and U (1) × U (1) because of the special determinant condition imposed from SU (3). The residual symmetries G e and G ν , being subgroups of G f (unless there are accidental symmetries), should also be contained in U (1) × U (1) and Z 2 × Z 2 (U (1) × U (1)) for massive Majorana (Dirac) neutrinos, respectively.
If G e = Z n , n > 2 or Z n × Z m , n, m ≥ 2, the matrix U e is fixed by the matrix ρ(g e ) (up to multiplication by diagonal phase matrices on the right and permutations of columns), U e = U • e . In the case of a smaller symmetry, i.e., G e = Z 2 , U e is defined up to a U (2) transformation in the degenerate subspace, because in this case ρ(g e ) has two degenerate eigenvalues. Therefore, U e = U • e U ij (θ e ij , δ e ij )Ψ k Ψ l , where U ij is a complex rotation in the i-j plane and Ψ k , Ψ l are diagonal phase matrices, The angle θ e ij and the phases δ e ij , ψ 1 , ψ 2 and ψ 3 are free parameters. As an example of the explicit form of U ij (θ a ij , δ a ij ), we give the expression of the matrix U 12 (θ a 12 , δ a 12 ): where a = e, ν, •. The indices e, ν indicate the free parameters, while "•" indicates the angles and phases which are fixed. The complex rotation matrices U 23 (θ a 23 , δ a 23 ) and U 13 (θ a 13 , δ a 13 ) are defined in an analogous way. The real rotation matrices R ij (θ a ij ) can be obtained from U ij (θ a ij , δ a ij ) setting δ a ij to zero, i.e., R ij (θ a ij ) = U ij (θ a ij , 0). In the absence of a residual symmetry no constraints are present for the mixing matrix U e , which can be in general expressed in terms of three rotation angles and six phases.
Similar considerations apply to the neutrino sector. If G ν = Z n , n > 2 or Z n × Z m , n, m ≥ 2 for Dirac neutrinos, or G ν = Z 2 × Z 2 for Majorana neutrinos, the matrix U ν is fixed up to permutations of columns and right multiplication by diagonal phase matrices by the residual symmetry, i.e., U ν = U • ν . If the symmetry is smaller, G ν = Z 2 , then Obviously, in the absence of a residual symmetry, U ν is unconstrained. In all the cases considered above where G e and G ν are non-trivial, the matrices ρ(g e ) and ρ(g ν ) are diagonalised by U • e and U • ν : In what follows we define U • as the matrix fixed by the residual symmetries, which, in general, gets contributions from both the charged lepton and neutrino sectors, Since U • is a unitary 3 × 3 matrix, we will parametrise it in terms of three angles and six phases. These, however, as we are going to explain, reduce effectively to three angles and one phase, since the other five phases contribute to the Majorana phases of the PMNS mixing matrix, unphysical charged lepton phases and/or to a redefinition of the free parameters contained in U e and U ν . Furthermore, we will use the notation θ e ij , θ ν ij , δ e ij , δ ν ij for the free angles and phases contained in U , while the parameters marked with a circle contained in U • , e.g., θ • ij , δ • ij , are fixed by the residual symmetries. In the case when G e = Z 2 and G ν = Z n , n > 2 or Z n × Z m , n, m ≥ 2 for massive Dirac neutrinos, or G ν = Z 2 × Z 2 for Majorana neutrinos, we have: where (ij) = (12), (13), (23) and The unitary matrix U • contains three angles and three phases, since the additional three phases can be absorbed by redefining the charged lepton fields and the free parameter δ e ij (see below). Here Ψ • j is a diagonal matrix containing a fixed phase in the j-th position. Namely, The matrix Q 0 , defined in eq. (3), is a diagonal matrix containing two free parameters contributing to the Majorana phases. Since the presence of the phase ψ • j amounts to a redefinition of the free parameter δ e ij , we denote (δ e ij − ψ • j ) as δ e ij . This allows us to employ the following parametrisation for U : where the unphysical phase matrix Ψ • j on the left has been removed by charged lepton rephasing and the set of three phases {δ • kl } reduces to only one phase, δ • kl , since the other two contribute to redefinitions of Q 0 , δ e ij and to unphysical phases. The possible forms of the matrix U • , which we are going to employ, are given in Appendix B.
For the breaking patterns G e = Z n , n > 2 or Z n × Z m , n, m ≥ 2 and G ν = Z 2 , valid for both Majorana and Dirac neutrinos, we have: where (ij) = (12), (13), (23), and the two free phases, which contribute to the Majorana phases of the PMNS matrix if the massive neutrinos are Majorana particles, have been included in the diagonal phase matrix Q 0 . Notice that if neutrinos are assumed to be Dirac instead of Majorana, then the matrix Q 0 can be removed through re-phasing of the Dirac neutrino fields. Without loss of generality we can redefine the combination δ ν ij − ψ • i + ψ • j as δ ν ij and the combination Ψ • i Ψ • j Q 0 as Q 0 , so that the following parametrisation of U is obtained: In the case of G e = Z 2 and G ν = Z 2 for both Dirac and Majorana neutrinos, we can write with (ij) = (12), (13), (23), (rs) = (12), (13), (23). The phase matrices Ψ • i are defined as in eq. (19). Similarly to the previous cases, we can redefine the parameters in such a way that U can be cast in the following form: where Q 0 can be phased away if neutrinos are assumed to be Dirac particles. 9 If G e is fully broken and G ν = Z n , n > 2 or Z n × Z m , n, m ≥ 2 for Dirac neutrinos or G ν = Z 2 × Z 2 for Majorana neutrinos, the form of U reads where the phase matrices Ψ 2 and Ψ 3 are defined as in eq. (16). Notice that in general we can effectively parametrise U • in terms of three angles and one phase since of the set of three phases {δ • kl }, two contribute to a redefinition of the matrices Q 0 , Ψ 2 and Ψ 3 . Furthermore, under the additional assumptions on the form of U (θ e 12 , θ e 13 , θ e 23 , δ e rs ) and also taking {δ • kl } = 0, the form of U given in eq. (25) leads to the sum rules derived in [11,14]. In the numerical analyses performed in [11,13,14], the angles θ • ij have been set, in particular, to the values corresponding to the TBM, BM (LC), GRA, GRB and HG symmetry forms.
Finally for the breaking patterns G e = Z n , n > 2 or Z n × Z m , n, m ≥ 2 and G ν fully broken when considering both Dirac and Majorana neutrino possibilities, the form of U can be derived from eq. (25) by interchanging the fixed and the free parameters. Namely, The cases found in eqs. (20), (22), (24), (25) and (26) are summarised in Table 1. The reduction of the number of free parameters indicated with arrows corresponds to a redefinition of the charged lepton fields. In the breaking patterns considered, it may be also possible to impose a generalised CP (GCP) symmetry. An example of how imposing a GCP affects the sum rules is shown in Appendix D. In the case in which a GCP symmetry is preserved in the neutrino sector we have for the neutrino Majorana mass matrix [39]: Since the matrix X i is symmetric there exists a unitary matrix Ω i such that X i = Ω i Ω T i and Ω T i M ν Ω i is real. Therefore when GCP is preserved in the neutrino sector, the phases in the matrix U ν can be fixed. An alternative possibility is that GCP is preserved in the charged lepton sector, which leads to the condition [39]: Since (X e i ) T = X e i , the phases in the matrix U e are fixed, because (Ω e i ) † M e M † e Ω e i is real. The fact that the matrices X i , if GCP is preserved in the neutrino sector, or X e i if it is preserved in the charged lepton sector, are symmetric matrices can be proved applying the GCP transformation twice. In the first case, eq. (27) allows one to derive the general form of X i [40][41][42]: while in the latter case Equations (29) and (30) imply that X i and X e i are symmetric matrices. 10 We note finally that the titles of the following sections refer to the residual symmetries of the charged lepton and neutrino mass matrices, while the titles of the subsections reflect the free complex rotations contained in the corresponding parametrisation of U , eqs. (20), (22), (24), (25) and (26). 3 The Pattern G e = Z 2 and G ν = Z n , n > 2 or Z n × Z m , n, m ≥ 2 In this section we derive sum rules for cos δ for the cases given in eq. (20). Recall that the matrix U e is fixed up to a complex rotation in one plane by the residual G e = Z 2 symmetry, while U ν is completely determined (up to multiplication by diagonal phase matrices on the right and permutations of columns) by the G ν = Z 2 × Z 2 residual symmetry in the case of neutrino Majorana mass term, or by G ν = Z n , n > 2 or Z n × Z m , n, m ≥ 2, residual symmetries if the massive neutrinos are Dirac particles. At the end of this section we will present results of a study of the possibility of reproducing the observed values of the lepton mixing parameters sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 and of obtaining physically viable predictions for cos δ in the cases when the residual symmetries G e = Z 2 and G ν = Z n , n > 2 or Z n × Z m , n, m ≥ 2, originate from the breaking of the lepton flavour symmetries A 4 (T ), S 4 and A 5 . Employing the parametrisation of the PMNS matrix U given in eq. (20) with (ij) = (12) and the parametrisation of U • given as we get for U (see Appendix B for details): The results derived in Appendix B and given in eq. (212) allow us to cast eq. (33) in the form: From eqs. (35) and (36) we get the following correlation between the values of sin 2 θ 13 and sin 2 θ 23 : sin 2 θ 13 + cos 2 θ 13 sin 2 θ 23 = sin 2 θ • 13 + cos 2 θ • 13 sin 2 θ • 23 .
In order to obtain a sum rule for cos δ, we compare the expressions for the absolute value of the element U τ 2 of the PMNS matrix in the standard parametrisation and in the parametrisation defined in eq. (34), From the above equation we get for cos δ: cos δ = cos 2 θ 13 (sin 2 θ • 23 − cos 2 θ 12 ) + cos 2 θ • 13 cos 2 θ • 23 (cos 2 θ 12 − sin 2 θ 12 sin 2 θ 13 ) sin 2θ 12 sin θ 13 | cos θ • 13 cos θ • 23 |(cos 2 θ 13 − cos 2 θ • 13 cos 2 θ • 23 ) For the considered specific residual symmetries G e and G ν , the predicted value of cos δ in the case A1 discussed in this subsection depends on the chosen discrete flavour symmetry G f via the values of the angles θ • 13 and θ • 23 . The method of derivation of the sum rule for cos δ of interest employed in the present subsection and consisting, in particular, of choosing adequate parametrisations of the PMNS matrix U (in terms of the complex rotations of U e and of U ν ) and of the matrix U • (determined by the symmetries G e , G ν and G f ), which allows to express the PMNS matrix U in terms of minimal numbers of angle and phase parameters, will be used also in all subsequent sections. The technical details related to the method are given in Appendices B and C.
We note finally that in the case of δ • 12 = 0, the symmetry forms TBM, BM, GRA, GRB and HG can be obtained from U for specific values of the angles given in Table 2. In this case, the angles θ • ij are related to the angles θ ν ij defined in Section 2.1 of ref. [14] as follows: Mixing

The Case with
Using the parametrisation of the PMNS matrix U given in eq. (20) with (ij) = (13) and the following parametrisation of U • , we get for U (for details see Appendix B): The results derived in Appendix B and presented in eq. (212) allow us to recast eq. (44) in the following form: Hereδ 13 = α − β, where sinθ 13 , α and β are defined as in eqs. (213) and (214) after setting i = 1, j = 3, θ a 13 = θ e 13 , δ a 13 = δ e 13 , θ b 13 = θ • 13 and δ b 13 = δ • 13 . Using eq. (45) and the standard parametrisation of the PMNS matrix U , we find: Thus, in this scheme, as it follows from eq. (47), the value of sin 2 θ 23 is predicted once the symmetry group G f is fixed. This prediction, when confronted with the measured value of sin 2 θ 23 , constitutes an important test of the scheme considered for any given discrete (lepton flavour) symmetry group G f , which contains the residual symmetry groups G e = Z 2 and G ν = Z n , n > 2 and/or Z n × Z m , n, m ≥ 2 as subgroups.
As can be easily demonstrated, the case under discussion coincides with the one analysed in Section 2.2 of ref. [14]. The parameters θ ν 23 and θ ν 12 in [14] can be identified with θ • 23 and θ • 12 , respectively. Therefore the sum rule we obtain coincides with that given in eq. (32) in [14]: The dependence of cos δ on G f in this case is via the values of the angles θ • 12 and θ • 23 .

The Case with
In the case with (ij) = (23), as can be shown, cos δ does not satisfy a sum rule, i.e., it cannot be expressed in terms of the three neutrino mixing angles θ 12 , θ 13 and θ 23 and the other fixed angle parameters of the scheme. Indeed, employing the parametrisation of U • as , we can write the PMNS matrix in the following form: Using the results derived in Appendix B and shown in eq. (212), we can recast eq. (50) as Comparing eq. (51) and the standard parametrisation of the PMNS matrix, we find that sin 2 θ 13 = sin 2 θ • 13 , sin 2 θ 23 = sin 2θ 23 , sin 2 θ 12 = sin 2 θ • 12 and cos δ = ± cosδ 23 . It follows from the preceding equations, in particular, that since, for any given G f compatible with the considered residual symmetries, θ • 13 and θ • 12 have fixed values, the values of both sin 2 θ 13 and sin 2 θ 12 are predicted. The predictions depend on the chosen symmetry G f . Due to these predictions the scheme under discussion can be tested for any given discrete symmetry candidate G f , compatible, in particular, with the considered residual symmetries.
We have also seen that δ is related only to an unconstrained phase parameter of the scheme. In the case of a flavour symmetry G f which, in particular, allows to reproduce correctly the observed values of sin 2 θ 12 and sin 2 θ 13 , it might be possible to obtain physically viable prediction for cos δ by employing a GCP invariance constraint. An example of the effect that GCP invariance has on restricting CPV phases is given in Appendix D. Investigating the implications of the GCP invariance constraint in the charged lepton or the neutrino sector in the cases considered by us is, however, beyond the scope of the present study.

Results in the Cases of
The cases detailed in Sections 3.1 -3.3 can all be obtained from the groups A 4 (T ), S 4 and A 5 , when breaking them to G e = Z 2 and G ν = Z n (n ≥ 3) in the case of Dirac neutrinos, or G ν = Z 2 × Z 2 in the case of both Dirac and Majorana neutrinos. 11 We now give an explicit example of how these cases can occur in A 4 .
In the case of the group A 4 (see, e.g., [45]), the structure of the breaking patterns discussed, e.g., in subsection 3.1 can be realised when i) the S generator of A 4 is preserved in the neutrino sector, and when, due to an accidental symmetry, the mixing matrix is fixed to be tri-bimaximal, U • ν = U TBM , up to permutations of the columns, and ii) a is preserved in the charged lepton sector. The group element generating the Z 2 symmetry is diagonalised by the matrix U • e . Therefore the angles θ • 12 , θ • 13 and θ • 23 are obtained from the product The same structure (the structure discussed in subsection 3.2) can be obtained in a similar manner from the flavour groups S 4 and A 5 (A 4 , S 4 and A 5 ).
We have investigated the possibility of reproducing the observed values of the lepton mixing parameters sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 as well as obtaining physically viable predictions for cos δ in the cases of residual symmetries G e = Z 2 and G ν = Z n , n > 2 or Z n × Z m , n, m ≥ 2 12 (Dirac neutrinos), or G ν = Z 2 × Z 2 (Majorana neutrinos), discussed in subsections 3.1, 3.2 and 3.3 denoted further as A1, A2 and A3, assuming that these residual symmetries originate from the breaking of the flavour symmetries A 4 (T ), S 4 and A 5 . The analysis was performed using the current best fit values of the three lepton mixing parameters sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 . The results we have obtained for the symmetries A 4 (T ), S 4 and A 5 are summarised below.
We have found that in the cases under discussion, i.e., in the cases A1, A2 and A3, and flavour symmetries G f = A 4 (T ), S 4 and A 5 , with the exceptions to be discussed below, it is impossible either to reproduce at least one of the measured values of sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 even taking into account its respective 3σ uncertainty, or to get physically viable values of cos δ satisfying | cos δ| ≤ 1. In the cases A1 and A2 and the flavour groups A 4 and S 4 , for instance, the values of cos δ are unphysical. Using the group G f = A 5 leads either to unphysical values of cos δ, or to values of sin 2 θ 23 which lie outside the corresponding current 3σ allowed interval. In the case A3 (discussed in subsection 3.3), the symmetry A 4 , for example, leads to (sin 2 θ 12 , sin 2 θ 13 ) = (0, 0) or (1,0).
(cos δ = 0.992). In the part of the 3σ allowed interval of sin 2 θ 12 , 0.321 ≤ sin 2 θ 12 ≤ 0.359, one has −0.992 ≤ cos δ ≤ −0.633 (0.992 ≥ cos δ ≥ 0.633). 4 The Pattern G e = Z n , n > 2 or Z n × Z m , n, m ≥ 2 and G ν = Z 2 In this section we derive sum rules for cos δ in the case given in eq. (22). We recall that for G e = Z n , n > 2 or Z n ×Z m , n, m ≥ 2 and G ν = Z 2 of interest, the matrix U e is unambiguously determined (up to multiplication by diagonal phase matrices on the right and permutations of columns), while the matrix U ν is determined up to a complex rotation in one plane.

The Case with
Combining the parametrisation of the PMNS matrix U given in eq. (22) with (ij) = (13) and the parametrisation of U • as we get for U (the details are given again in Appendix B): The results derived in Appendix B and reported in eq. (212) allow us to recast eq. (54) in the form: Hereδ 13 = −α−β and we have redefined P 13 (α, β)Q 0 as Q 0 , where P 13 (α, β) = diag(e iα , 1, e iβ ) and the expressions for sin 2θ 13 , α and β can be obtained from eqs. (213) and (214), by setting i = 1, j = 3, θ a 13 = θ • 13 , δ a 13 = δ • 13 , θ b 13 = θ ν 13 and δ b 13 = δ ν 13 . Using eq. (55) and the standard parametrisation of the PMNS matrix U , we find: It follows from eq. (58) that in the case under discussion the values of sin 2 θ 12 and sin 2 θ 13 are correlated. A sum rule for cos δ can be derived by comparing the expressions for the absolute value of the element U τ 2 of the PMNS matrix in the standard parametrisation and in the one obtained using eq. (55): From this equation we get The dependence of the predictions for cos δ on G f is in this case via the values of θ • 12 and θ • 23 .

The Case with
Utilising the parametrisation of the PMNS matrix U given in eq. (22) with (ij) = (23) and the following parametrisation of U • , we obtain for U (Appendix B contains the relevant details): The results given in eq. (212) in Appendix B make it possible to bring eq. (62) to the form: Using eq. (63) and the standard parametrisation of the PMNS matrix U , we find: Equation (66) implies that, as in the case investigated in the preceding subsection, the values of sin 2 θ 12 and sin 2 θ 13 are correlated. The sum rule for cos δ of interest can be obtained by comparing the expressions for the absolute value of the element U τ 1 of the PMNS matrix in the standard parametrisation and in the one obtained using eq. (63): From the above equation we get for cos δ: The dependence of cos δ on G f is realised in this case through the values of θ • 12 and θ • 13 .
It follows from the expressions for the neutrino mixing parameters thus derived that, given a discrete symmetry G f which can lead to the considered breaking patterns, the values of sin 2 θ 13 and sin 2 θ 23 are predicted. This, in turn, allows to test the phenomenological viability of the scheme under discussion for any appropriately chosen discrete lepton flavour symmetry In what concerns the phase δ, it is expressed in terms of an unconstrained phase parameter present in the scheme we are considering. The comment made at the end of subsection 3.3 is valid also in this case. Namely, given a non-Abelian discrete flavour symmetry G f which allows one to reproduce correctly the observed values of sin 2 θ 13 and sin 2 θ 23 , it might be possible to obtain physically viable prediction for cos δ by employing a GCP invariance constraint in the charged lepton or the neutrino sector.

Results in the Cases of
The schemes discussed in Sections 4.1 -4.3 are realised when breaking G f = A 4 (T ), S 4 and A 5 , to G e = Z n (n ≥ 3) or Z 2 × Z 2 and G ν = Z 2 , for both Dirac and Majorana neutrinos. As a reminder to the reader, we investigate the case of Z 2 × Z 2 when it is an actual subgroup of G f . As an explicit example of how this breaking can occur, we will consider the case of G f = A 4 (T ). The other cases when G f = S 4 or A 5 can be obtained from the breaking of S 4 and A 5 to the relevant subgroups as given in [46] and [47], respectively.
In the case of the group A 4 (see, e.g., [45]), the structure of the breaking patterns discussed, e.g., in subsection 4.1 can be obtained by breaking A 4 i) in the charged lepton sector to any of the four Z 3 subgroups, namely, , and ii) to any of the three Z 2 subgroups, namely, , in the neutrino sector. In this case the matrix U • = U TBM gets corrected by a complex rotation matrix in the 1-3 plane coming from the neutrino sector.
The results of the study performed by us of the phenomenological viability of the schemes with residual symmetries G e = Z n , n > 2 or Z n × Z m , n, m ≥ 2 and G ν = Z 2 , discussed in subsections 4.1, 4.2 and 4.3, and denoted further as B1, B2 and B3, when the residual symmetries result from the breaking of the flavour symmetries A 4 (T ), S 4 and A 5 , are described below. We present results only in the cases in which we obtain values of sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 compatible with their respective measured values (including the corresponding 3σ uncertainties) and physically acceptable values of cos δ.

5.1
The Case with U 12 (θ e 12 , δ e 12 ) and U 13 (θ ν 13 , δ ν 13 ) Complex Rotations (Case C1) Similar to the already considered cases we combine the parametrisation of the PMNS matrix U given in eq. (24) with (ij) = (12) and (rs) = (13), with the parametrisation of U • given as and get the following expression for U (as usual, we refer to Appendix B for details): Utilising the results derived in Appendix B and reported in eq. (212), we can recast eq. (72) in the form Hereδ = α e − β e + α ν + β ν and we have redefined the matrix Q 0 by absorbing the diagonal phase matrix P 13 (−β ν , −α ν ) = diag(e −iβ ν , 1, e −iα ν ) in it. Using eq. (73) and the standard parametrisation of the PMNS matrix U , we find: The sum rule for cos δ of interest can be derived by comparing the expressions for the absolute value of the element U τ 2 of the PMNS matrix in the standard parametrisation and in the one obtained using eq. (73): |U τ 2 | = | cos θ 12 sin θ 23 + sin θ 13 cos θ 23 sin θ 12 e iδ | = | sin θ • 23 | .
Given the assumed breaking pattern, cos δ depends on the flavour symmetry G f via the value of θ • 23 . Using the best fit values of the standard mixing angles for the NO neutrino mass spectrum and the requirement | cos δ| ≤ 1, we find that sin 2 θ • 23 should lie in the following interval: 0.236 ≤ sin 2 θ • 23 ≤ 0.377. Fixing two of the three angles to their best fit values and varying the third one in its 3σ experimentally allowed range and considering all the three possible combinations, we get that | cos δ| ≤ 1 if 0.195 ≤ sin 2 θ • 23 ≤ 0.504.

5.2
The Case with U 13 (θ e 13 , δ e 13 ) and U 12 (θ ν 12 , δ ν 12 ) Complex Rotations (Case C2) As in the preceding case, we use the parametrisation of the PMNS matrix U given in eq. (24) but this time with (ij) = (13) and (rs) = (12), and the parametrisation of U • as to get for U (again the details can be found in Appendix B): The results derived in Appendix B and reported in eq. (212) allow us to rewrite the expression for U in eq. (80) as follows: whereδ = α e − β e + α ν + β ν , and also in this case we have redefined the matrix Q 0 by absorbing the phase matrix P 12 (−β ν , −α ν ) = diag(e −iβ ν , e −iα ν , 1) in it. From eq. (81) and the standard parametrisation of the PMNS matrix U we get: cos 2θe 13 sin 2θν 12 + cos 2θν 12 sin 2θe Given the value of sin 2 θ • 23 , eq. (83) implies the existence of a correlation between the values of sin 2 θ 23 and sin 2 θ 13 .

23
fixed by G f and the assumed symmetry breaking pattern, as well as of the phase parameter δ of the scheme. Predictions for cos δ can only be obtained whenδ is fixed by additional considerations of, e.g., GCP invariance, symmetries, etc. In view of this we show in Fig. 1 cos δ as a function of cosδ for the current best fit values of sin 2 θ 12 and sin 2 θ 13 , and for the value sin 2 θ • 23 = 1/2 corresponding to G f = S 4 . We do not find phenomenologically viable cases for A 4 (T ) and A 5 . Therefore we do not present such a plot for these groups.

5.3
The Case with U 12 (θ e 12 , δ e 12 ) and U 23 (θ ν 23 , δ ν 23 ) Complex Rotations (Case C3) We get for the PMNS matrix U , utilising the parametrisations of U shown in eq. (24) with (ij) = (12) and (rs) = (23) and that of U • given below (further details can be found in Appendix B), With the help of the results derived in Appendix B and especially of eq. (212), the expression in eq. (87) for the PMNS matrix U can be brought to the form whereδ = β e − α e + α ν + β ν and, as in the preceding cases, we have redefined the phase matrix Q 0 by absorbing the phase matrix P 23 (−β ν , −α ν ) = diag(1, e −iβ ν , e −iα ν ) in it. Using eq. (89) and the standard parametrisation of the PMNS matrix U , we find: The sum rule for cos δ of interest can be derived, e.g., by comparing the expressions for the absolute value of the element U τ 1 of the PMNS matrix in the standard parametrisation and in the one obtained using eq. (89): For cos δ we get: cos δ = sin 2 θ 12 sin 2 θ 23 − sin 2 θ • 13 + cos 2 θ 12 cos 2 θ 23 sin 2 θ 13 sin θ 13 sin 2θ 23 sin θ 12 cos θ 12 .
In this case, in contrast to that considered in the preceding subsection, cos δ is predicted once the angle θ • 13 , i.e., the flavour symmetry G f , is fixed. Using the best fit values of sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 for the NO neutrino mass spectrum, we find that physical values of cos δ satisfying | cos δ| ≤ 1 can be obtained only if sin 2 θ • 13 lies in the following interval: 0.074 ≤ sin 2 θ • 13 ≤ 0.214. Fixing two of the three neutrino mixing parameters sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 to their best fit values and varying the third one in its 3σ experimentally allowed range and taking into account all the three possible combinations, we get that | cos δ| ≤ 1 provided 0.056 ≤ sin 2 θ • 13 ≤ 0.267.

5.4
The Case with U 13 (θ e 13 , δ e 13 ) and U 23 (θ ν 23 , δ ν 23 ) Complex Rotations (Case C4) The parametrisation of the PMNS matrix U , to be used further, is found in this case from the parametrisations of the matrix U given in eq. (24) with (ij) = (13) and (rs) = (23) and that of U • shown below (see Appendix B for details), The results presented in eq. (212) of Appendix B allow us to recast eq. (95) in the form: Hereδ = β e −α e −α ν −β ν and we have absorbed the phase matrix P 23 (α ν , β ν ) = diag(1, e iα ν , e iβ ν ) in the matrix Q 0 . Using eq. (97) and the standard parametrisation of the PMNS matrix U , we find: Comparing the expressions for the absolute value of the element U µ1 of the PMNS matrix in the standard parametrisation and in the one obtained using eq. (97), we find |U µ1 | = | sin θ 12 cos θ 23 + sin θ 13 sin θ 23 cos θ 12 e iδ | = | sin θ • 12 | .
The predicted value of cos δ depends on the discrete symmetry G f through the value of the angle θ • 12 . Using the best fit values of the standard mixing angles for the NO neutrino mass spectrum and the requirement | cos δ| ≤ 1, we find that sin 2 θ • 12 should lie in the following interval: 0.110 ≤ sin 2 θ • 12 ≤ 0.251. Fixing two of the three neutrino mixing angles to their best fit values and varying the third one in its 3σ experimentally allowed range and accounting for all the three possible combinations, we get that | cos δ| ≤ 1 if 0.057 ≤ sin 2 θ • 12 ≤ 0.281.

5.5
The Case with U 23 (θ e 23 , δ e 23 ) and U 13 (θ ν 13 , δ ν 13 ) Complex Rotations (Case C5) The parametrisation of the PMNS matrix U , which is convenient for our further analysis, can be obtained in this case utilising the parametrisations of the matrix U given in eq. (24) with (ij) = (23) and (rs) = (13) and that of the matrix U • given below (for details see Appendix B), The expression in eq. (103) for U can further be cast in a "minimal" form with the help of eq. (212) in Appendix B: whereδ = β e − α e − α ν − β ν and we have absorbed the matrix P 13 (α ν , β ν ) = diag(e iα ν , 1, e iβ ν ) in the phase matrix Q 0 . Using eq. (105) and the standard parametrisation of the PMNS matrix U , we find: We note that, given G f , the values of sin 2 θ 12 and sin 2 θ 13 are correlated. This allows one to perform a critical test of the scheme under study once the discrete symmetry group G f has been specified. The sum rule for cos δ of interest can be derived, e.g., by comparing the expressions for the absolute value of the element U τ 2 of the PMNS matrix in the standard parametrisation and in the one obtained using eq. (105): This leads to cos δ = cos 2 θ 13 (cos 2 θ • 12 sin 2θe 23 − sin 2 θ 23 ) + sin 2 θ • 12 (sin 2 θ 23 − cos 2 θ 23 sin 2 θ 13 ) sin 2θ 23 sin θ 13 | sin θ • 12 |(cos 2 θ 13 − sin 2 θ • 12 ) Similar to the case C2 analysed in subsection 5.2, cos δ is a function of the known neutrino mixing angles θ 13 and θ 23 , of the angle θ • 12 fixed by G f and the assumed symmetry breaking pattern, as well as of the phase parameterδ of the scheme. Predictions for cos δ can be obtained ifδ is fixed by additional considerations of, e.g., GCP invariance, symmetries, etc. In view of this we show in Fig. 2 cos δ as a function of cosδ for the current best fit values of sin 2 θ 13 and sin 2 θ 23 , and for the value sin 2 θ • 12 = 1/4 corresponding to G f = S 4 and A 5 . We do not find phenomenologically viable cases for A 4 (T ). Therefore we do not present such a plot for these groups.

5.6
The Case with U 23 (θ e 23 , δ e 23 ) and U 12 (θ ν 12 , δ ν 12 ) Complex Rotations (Case C6) We show below that in this case cos δ coincides (up to a sign) with the cosine of an unconstrained CPV phase parameter of the scheme and therefore cannot be determined from the values of the neutrino mixing angles and of the angles determined by the residual symmetries. Indeed, using the parametrisation of the matrix U given in eq. (24) with (ij) = (23) and (rs) = (12) and the parametrisation of U • as follows (see Appendix B for details), we get for U : The results derived in Appendix B in eq. (212) make it possible to recast eq. (112) in the form: U = R 23 (θ e 23 )P 2 (δ)R 13 (θ • 13 )R 12 (θ ν 12 )Q 0 , P 2 (δ) = diag(1, e iδ , 1) .
Hereδ = α e − β e − α ν − β ν and, as in the preceding cases, we have redefined the phase matrix Q 0 by absorbing the phase matrix P 12 (α ν , β ν ) = diag(e iα ν , e iβ ν , 1) in it. Using eq. (113) and the standard parametrisation of the PMNS matrix U , we find: Comparing the absolute value of the element U τ 1 allows us to find that cos δ = ± cosδ. It follows from eq. (114) that for a given flavour symmetry G f , the value of sin 2 θ 13 is predicted. This allows to test the phenomenological viability of the case under discussion, since the value of sin 2 θ 13 is known experimentally with a relatively high precision.
A comment, analogous to those made in similar cases considered in subsections 3.3 and 4.3, is in order. Namely, for a non-Abelian flavour symmetry G f which allows to reproduce correctly the observed values of sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 , it might be possible to obtain physically viable prediction for cos δ by employing GCP invariance in the charged lepton or the neutrino sector.

23
fixed by G f and the assumed symmetry breaking pattern, as well as of the phase parameter δ of the scheme. Predictions for cos δ can only be obtained whenδ is fixed by additional considerations of, e.g., GCP invariance, symmetries, etc. In view of this we show in Fig. 3 cos δ as a function of cosδ for the current best fit values of sin 2 θ 12 and sin 2 θ 13 , and for the value sin 2 θ • 23 = 1/2 corresponding to G f = S 4 . We do not find phenomenologically viable cases for G f = A 4 (T ) and A 5 .  6) and (8). The solid (dashed) line is for the case when sin 2θ e 12 sin 2θ ν 12 is positive (negative).
Given the assumed breaking pattern, cos δ depends on the flavour symmetry G f via the value of θ • 23 . Using the best fit values of the standard mixing angles for the NO neutrino mass spectrum and the requirement | cos δ| ≤ 1, we find that sin 2 θ • 23 should lie in the following interval: 0.537 ≤ sin 2 θ • 23 ≤ 0.677. Fixing two of the three angles to their best fit values and varying the third one in its 3σ experimentally allowed range and considering all the three possible combinations, we get that | cos δ| ≤ 1 if 0.496 ≤ sin 2 θ • 23 ≤ 0.805.

12
fixed by G f and the assumed symmetry breaking pattern, as well as of the phase parameter δ of the scheme. Predictions for cos δ can only be obtained whenδ is fixed by additional considerations of, e.g., GCP invariance, symmetries, etc. In view of this we show in Fig. 4 cos δ as a function of cosδ for the current best fit values of sin 2 θ 23 and sin 2 θ 13 , and for the value sin 2 θ • 12 = (r + 2)/(4r + 4) ∼ = 0.345 corresponding to G f = A 5 . We do not find phenomenologically viable cases for G f = A 4 (T ) and S 4 .  For the symmetry group A 4 we find that none of the combinations of the residual symmetries G e = Z 2 and G ν = Z 2 provide physical values of cos δ and phenomenologically viable results for the neutrino mixing angles simultaneously.

Summary of the Results of Sections 3, 4 and 5
The sum rules derived in Sections 3, 4 and 5 are summarised in Tables 3 and 4. The formulae for sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 , which lead to predictions for the indicated neutrino mixing parameters once the discrete flavour symmetry G f is fixed, are given in Tables 5 and 6. In the cases in Tables 5 and 6 in which cos δ is unconstrained, a relatively precise measurement of sin 2 θ 12 , sin 2 θ 13 or sin 2 θ 23 can provide a critical test of the corresponding schemes due to constraints satisfied by the indicated neutrino mixing parameters.
A general comment on the results derived in Sections 3, 4 and 5 is in order. Since we do not have any information on the mass matrices, we have the freedom to permute the columns of the matrices U e and U ν , or equivalently, the columns and the rows of the PMNS matrix U . The results in Tables 3 and 4 cover all the possibilities because, as we demonstrate below, the permutations bring one of the considered cases into another considered case. For example, consider the case of U = U 13 (θ e 13 , δ e 13 )U • U 23 (θ ν 23 , δ ν 23 )Q 0 . The permutation of the second and the third rows of U is given by π 23 U = π 23 U 13 (θ e 13 , δ e 13 )π 23 π 23 U • U 23 (θ ν 23 , δ ν 23 )Q 0 , where we have defined Since the combination π 23 U 13 (θ e 13 , δ e 13 )π 23 gives a unitary matrix U 12 (θ e 13 , δ e 13 ), the result after the redefinition, θ e 13 → θ e 12 , δ e 13 → δ e 12 and π 23 U • → U • , yields which represents another case present in Table 4. It is worth noting that the freedom in redefining the matrix U • follows from the fact that U • is a general 3 × 3 unitary matrix and hence can be parametrised as described in Section 2 and in Appendix B. All the other permutations should be treated in the same way and lead to similar results.

The Case of Fully Broken G e
If the discrete flavour symmetry G f is fully broken in the charged lepton sector the matrix U e is unconstrained and includes, in general, three rotation angle and three CPV phase parameters. It is impossible to derive predictions for the mixing angles and CPV phases in the PMNS matrix in this case. Therefore, we will consider in this section forms of U e corresponding to one of the rotation angle parameters being equal to zero. Some of these forms of U e correspond to a class of models of neutrino mass generation (see, e.g., [17,[32][33][34][35][36]) and lead, in particular, to sum rules for cos δ.
We give in Appendix C the most general parametrisations of U under the assumption that in the case of fully broken G e one rotation angle in the matrix U e vanishes. The second case in Table 14 with θ • 13 = 0 have been analysed in [11,13,14], while the third case with U 12 (θ e 12 , δ e 12 )U 13 (θ e 13 , δ e 13 ) has been investigated in [14].

13
− κ cosδ cos θ 13 sin θ 23 sin 2θ • 12 sin θ • 13 cos 2 θ • 13 − cos 2 θ 13 sin 2 θ 23 where κ = 1 ifθ 23 belongs to the first or third quadrant, and κ = −1 otherwise. As in the previous case, cos δ is a function ofδ. For θ • 13 = 0 the sum rule in eq. (166) reduces to the one derived in [14]. In this case we consider the following parametrisation of the PMNS matrix (see Appendix C, third case in Table 14): We find that: Since, as can be shown, |U µ2 | is a function of the parameters θ e where κ = 1 ifθ 13 belongs to the first or third quadrant, and κ = −1 otherwise. In this case the sum rule for cos δ has been derived first in [14] assuming θ • 13 = 0, but as we can see this result holds also for any fixed value of θ • 13 , since the parametrisation given in eq. (175) and the corresponding one in [14] are the same after a redefinition of the parameters.
The sum rules derived in Section 7 are summarised in Table 7.

E1
In this section we summarise the numerical results obtained in the cases of the discrete flavour symmetry groups A 4 (T ), S 4 and A 5 , which have been already discussed in subsections 3.4, 4.4 and 5.10. In Tables 9 -11 we give the values of the fixed angles, obtained from the diagonalisation of the corresponding group elements which lead to physical values of cos δ and phenomenologically viable results for the "standard" mixing angles θ 12 , θ 13 and θ 23 . In the cases when the standard mixing angles are not fixed by the schemes in Tables 9 -11, we use their best fit values for the NO spectrum quoted in eqs. (6) - (8). For the cases in the tables marked with an asterisk, physical values of cos δ, i.e., | cos δ| ≤ 1, cannot be obtained employing the best fit values of the neutrino mixing angles θ 12 , θ 13 and θ 23 , but they can be achieved for values of the relevant mixing parameters allowed at 3σ. Note that unphysical values of cos δ, | cos δ| > 1, occur when the relations between the parameters of the scheme and the standard parametrisation mixing angles cannot be fulfilled for given values of sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 . Indeed the parameter space of sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 is reduced by these constraints coming from the schemes.
The cases in Table 10 marked with an asterisk are discussed below. Firstly, using the best fit values of sin 2 θ 12 and sin 2 θ 13 we get a physical value of cos δ in the case C3 for the minimal value of sin 2 θ 23 = 0.562, for which cos δ = −0.996. For C8 with sin 2 θ • 23 = 1/2 and 3/4, using the best fit values of the neutrino mixing angles for the NO spectrum, we have cos δ = −1.53 and 2.04, respectively. The physical values of cos δ can be obtained, using, e.g., the values of sin 2 θ 23 = 0.380 and 0.543, for which cos δ = −0.995 and 0.997, respectively. In the parts of the 3σ allowed range of sin 2 θ 23 , 0.374 ≤ sin 2 θ 23 ≤ 0.380 and 0.543 ≤ sin 2 θ 23 ≤ 0.641, we have −0.938 ≥ cos δ ≥ −0.995 and 0.997 ≥ cos δ ≥ 0.045, respectively. Secondly, in the case B1 we obtain cos δ = −0.990 employing the best fit value of sin 2 θ 13 and the maximal value of sin 2 θ 23 = 0.419. Finally, utilising the best fit value of sin 2 θ 13 , we get physical values of cos δ in the cases A1 and A2 for the minimal value of sin 2 θ 12 = 0.348, for which cos δ = −0.993 and 0.993, respectively. Note that for the cases in which sin 2 θ 23 is fixed, the predicted values are within the corresponding 2σ range, while in the cases in which sin 2 θ 12 is fixed, the values of sin 2 θ 12 = 0.341 and 0.317 are within 2σ and 1σ, respectively. The value of sin 2 θ 12 = 0.256 lies slightly outside the current 3σ allowed range.
For the symmetry group A 5 we find that the residual symmetries • (G e , G ν ) = (Z 2 , Z 2 ) in the cases C2, C6 and C7; • (G e , G ν ) = (Z 3 , Z 2 ) in the cases B2 and B3; • (G e , G ν ) = (Z 5 , Z 2 ) in the case B3; • (G e , G ν ) = (Z 2 × Z 2 , Z 2 ) in the cases B1 and B3; • (G e , G ν ) = (Z 2 , Z 3 ) or (Z 2 , Z 5 ) in the case A3; • (G e , G ν ) = (Z 2 , Z 2 × Z 2 ) in the cases A1, A2 and A3 do not provide phenomenologically viable results for cos δ and/or for the standard mixing angles θ 12 , θ 13 and θ 23 . We will describe next the cases in Table 11 marked with an asterisk, apart from those which have also been found for G f = S 4 and discussed earlier. Using the best fit values of sin 2 θ 12 and sin 2 θ 13 we get a physical value of cos δ in the case C4 for the minimal value of sin 2 θ 23 = 0.487, for which cos δ = −0.997. Instead using the best fit values of sin 2 θ 13 and sin 2 θ 23 one gets the physical values of cos δ = −1 for the maximal value of sin 2 θ 12 = 0.277. Employing the best fit value of sin 2 θ 13 we find a physical value of cos δ in the case B2 with residual symmetries (G e , G ν ) = (Z 2 × Z 2 , Z 2 ) for the minimal value of sin 2 θ 23 = 0.518, for which cos δ = −0.996. Similarly for the cases A1 and A2 with residual symmetries (G e , G ν ) = (Z 2 , Z 5 ), the values of cos δ = −0.992 and 0.992 are obtained using the minimal value of sin 2 θ 12 = 0.321.
The values of sin 2 θ • ij in Table 11 used to compute cos δ and sin 2 θ ij are the following ones: 1/(4r 2 ) ∼ = 0.      Tables 3 -6, have been obtained using the best fit values of the other standard mixing angles for the NO spectrum quoted in eqs. (6) - (8). In the cases marked with an asterisk, the predicted values of cos δ, obtained for the best fit values of the neutrino mixing angles θ 12 , θ 13 and θ 23 , are unphysical; physical values of cos δ can be obtained for values of the neutrino mixing parameters sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 lying in their respective 3σ allowed intervals. See text for further details.
In the present article we have employed the discrete symmetry approach to understanding the observed pattern of 3-neutrino mixing and, within this approach, have derived sum rules and predictions for the Dirac phase δ present in the PMNS neutrino mixing matrix U . The approach is based on the assumption of the existence at some energy scale of a (lepton) flavour symmetry corresponding to a non-Abelian discrete group G f . The flavour symmetry group G f can be broken, in general, to different "residual symmetry" subgroups G e and G ν of the charged lepton and neutrino mass terms, respectively. Given G f , typically there are more than one (but still a finite number of) possible residual symmetries G e and G ν . The residual symmetries can constrain the forms of the 3×3 unitary matrices U e and U ν , which diagonalise the charged lepton and neutrino mass matrices, and the product of which represents the PMNS neutrino mixing matrix U , U = U † e U ν . Thus, by constraining the form of the matrices U e and U ν , the residual symmetries constrain also the form of the PMNS matrix U . This can lead, in particular, to a correlation between the values of the PMNS neutrino mixing angles θ 12 , θ 13 and θ 23 , which have been determined experimentally with a rather good precision, and the value of the cosine of the Dirac CP violation phase δ present in U , i.e., to a "sum rule" for cos δ. The sum rule for cos δ thus obtained depends on residual symmetries G e and G ν and in some cases can involve, in addition to θ 12 , θ 13 and θ 23 , parameters which cannot be constrained even when G f is fixed. For a given fixed G f , unambiguous predictions for the value of cos δ can be derived in the cases when, apart from the parameters determined by G f (and G e and G ν ), only θ 12 , θ 13 and θ 23 enter into the expression for the respective sum rule.
In the present article we have derived sum rules for cos δ considering the following discrete residual symmetries: i) G e = Z 2 and G ν = Z n , n > 2 or Z n × Z m , n, m ≥ 2 (Section 3); ii) G e = Z n , n > 2 or Z n × Z m , n, m ≥ 2 and G ν = Z 2 (Section 4); iii) G e = Z 2 and G ν = Z 2 (Section 5); iv) G e is fully broken and G ν = Z n , n > 2 or Z n × Z m , n, m ≥ 2 (Section 7); and v) G e = Z n , n > 2 or Z n × Z m , n, m ≥ 2 and G ν is fully broken (Section 8). The sum rules are summarised in Tables 3, 4, 7 and 8. For given G e and G ν , the sum rules for cos δ we have derived are exact, within the approach employed, and are valid, in particular, for any G f containing G e and G ν as subgroups. We have identified the cases when the value of cos δ cannot be determined, or cannot be uniquely determined, from the sum rule without making additional assumptions on unconstrained parameters (cases A3 in Section 3 and B3 in Section 4 (see also Table 3); cases C2, C5, C6, C7 and C9 in Section 5 (see also Table 4); the cases discussed in Sections 7 and 8). In the majority of the phenomenologically viable cases we have considered the value of cos δ can be unambiguously predicted once the flavour symmetry G f is fixed. In certain cases of fixed G f , G e and G ν , correlations between the values of some of the measured neutrino mixing parameters sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 , are predicted, and/or the values of some of these parameters, typically of sin 2 θ 12 or sin 2 θ 23 , are fixed. These correlations and "predictions" are summarised in Tables 5 and 6. We have found that a relatively large number of these cases are not phenomenologically viable, i.e., they lead to results which are not compatible with the existing data on neutrino mixing. We have derived predictions for cos δ for the flavour symmetry groups G f = S 4 , A 4 , T and A 5 using the best fit values of sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 , when cos δ is unambiguously determined by the corresponding sum rule. We have presented the predictions for cos δ only in the phenomenologically viable cases, i.e., when the measured values of the 3-neutrino mixing parameters sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 , taking into account their respective 3σ uncertainties, are successfully reproduced. These predictions, together with the predictions for the value of one of the mixing parameters sin 2 θ 12 and sin 2 θ 23 , in the cases when it is fixed by the symmetries, are summarised in Tables 9 -11. The results derived in the present study show, in particular, that with the accumulation of more precise data on the PMNS neutrino mixing parameters sin 2 θ 12 , sin 2 θ 13 and sin 2 θ 23 , and with the measurement of the Dirac phase δ present in the neutrino mixing matrix U , it will be possible to critically test the predictions of the current phenomenologically viable theories, models and schemes of neutrino mixing based on different non-Abelian discrete (lepton) flavour symmetries G f and sets of their non-trivial subgroups of residual symmetries G e and G ν , operative respectively in the charged lepton and neutrino sectors, and thus critically test the discrete symmetry approach to understanding the observed pattern of neutrino mixing.  T , satisfying S 2 = T 5 = (ST ) 3 = 1. We employ the basis defined in [47], which for the 3-dimensional representation of the generators S and T is summarised in Table 12. We conclude this appendix by noting that a list of the Abelian subgroups of A 4 , T , S 4 and A 5 can be found in [49], [17], [46] and [47], respectively.

B Appendix: Parametrisations of a 3 × 3 Unitary Matrix
Parametrisations of a 3×3 unitary matrix W (see, e.g., [50][51][52]) can be obtained, e.g., from one of the six permutations of a product of three complex rotations and diagonal phase matrices, e.g., as follows: where we have assumed ij = kl = rs. It is worth noticing that sometimes it is convenient to use the parametrisations of W of the following form: As shown in [50], the number of distinctive parametrisations of a CKM-like matrix is nine. We have defined the phase matrices Ψ i in eq. (16) and the complex rotation matrix in the i-j plane U ij ≡ U ij (θ ij , δ ij ) in eq. (17). The latter can be always parametrised as a product of diagonal phase matrices and the rotation matrix R ij ≡ R ij (θ ij ) = U ij (θ ij , 0), i.e., where P i (δ) are diagonal matrices defined as follows: P 1 (δ) = diag(e iδ , 1, 1) , P 2 (δ) = diag(1, e iδ , 1) , P 3 (δ) = diag(1, 1, e iδ ) .
Finally, in the case iv) from eqs. where f (θ) = (sin 2 θ − sin 2θ). Therefore the general results derived in Sections 4.1 and 4.2 with the choices as in i), ii), iii) and iv) and the additional restriction of the parameters due to the presence of GCP allow one to find the formulae derived in [10].

E Appendix: General Statement
In this appendix we prove the general statement that Z 2 symmetries preserved in the neutrino and charged lepton sectors can lead to phenomenologically viable predictions, only if their generators do not belong to the same Z 2 ×Z 2 subgroup of the original flavour symmetry group.
We compute the form of U • in a model independent way. Given a Z 2 × Z 2 symmetry with elements Z 2 × Z 2 = {1, g 1 , g 2 , g 3 } and a unitary matrix V such that V † g 1 V = diag(1, −1, −1), V † g 2 V = diag(−1, 1, −1), V † g 3 V = diag(−1, −1, 1), we consider first the case of G e = Z 2 = {1, g i } and G ν = Z 2 = {1, g j } with i, j = 1, 2, 3 for all the cases C1 -C9 in Table 4. In the case C1 (C2) we find that the matrix U • reads defined up to permutations of the 1st and 3rd (1st and 2nd) columns and the 1st and 2nd (1st and 3rd) rows. These permutations are not relevant because they correspond to a redefinition of the free parameters in the transformations U 12 (θ e 12 , δ e 12 ), U 13 (θ ν 13 , δ ν 13 ) and phase matrices contributing to the Majorana phases or removed with a redefinition of the charged lepton fields. In the case C3 (C6) we find that the matrix U • reads defined up to permutations of the 2nd and 3rd (1st and 2nd) columns and the 1st and 2nd (2nd and 3rd) rows. For the case C4 (C5) we find that the matrix U • reads U • = π 12 ≡    defined up to permutations of the 2nd and 3rd (1st and 3rd) columns and the 1st and 3rd (2nd and 3rd) rows. The freedom in permuting the columns and rows as we described above does not have physical implications because it represents the freedom to perform a fixed U (2) transformation in the degenerate subspace of the generator of the corresponding Z 2 symmetry. For the other cases we find similar results. Namely, U • = diag(1, 1, 1) for i = j and U • = π 23 (13) for i = j for case C7, U • = diag(1, 1, 1) for i = j and U • = π 23 (12) for i = j for case C8, U • = diag(1, 1, 1) for i = j and U • = π 13 (12) for i = j for case C9.