Neutrino mixing and leptonic CP violation from S 4 flavour and generalised CP symmetries

Journal of High Energy Physics, Dec 2017

We consider a class of models of neutrino mixing with S 4 lepton flavour symmetry combined with a generalised CP symmetry, which are broken to residual Z 2 and Z 2 × H CP ν symmetries in the charged lepton and neutrino sectors, respectively, H CP ν being a remnant CP symmetry of the neutrino Majorana mass term. In this set-up the neutrino mixing angles and CP violation (CPV) phases of the neutrino mixing matrix depend on three real parameters — two angles and a phase. We classify all phenomenologically viable mixing patterns and derive predictions for the Dirac and Majorana CPV phases. Further, we use the results obtained on the neutrino mixing angles and leptonic CPV phases to derive predictions for the effective Majorana mass in neutrinoless double beta decay.

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Neutrino mixing and leptonic CP violation from S 4 flavour and generalised CP symmetries

Received: May S4 J.T. Penedo 0 2 S.T. Petcov 0 1 2 3 A.V. Titov 0 2 0 5-1-5 Kashiwanoha , 277-8583 Kashiwa , Japan 1 Also at: Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences , 1784 So a 2 Via Bonomea 265 , 34136 Trieste , Italy 3 Kavli IPMU (WPI), University of Tokyo We consider a class of models of neutrino mixing with S4 lepton avour symmetry combined with a generalised CP symmetry, which are broken to residual Z2 and HCP symmetries in the charged lepton and neutrino sectors, respectively, HCP being a remnant CP symmetry of the neutrino Majorana mass term. In this set-up the neutrino mixing angles and CP violation (CPV) phases of the neutrino mixing matrix depend on three real parameters | two angles and a phase. We classify all phenomenologically viable mixing patterns and derive predictions for the Dirac and Majorana CPV phases. Further, we use the results obtained on the neutrino mixing angles and leptonic CPV phases to derive predictions for the e ective Majorana mass in neutrinoless double beta decay. CP violation; Discrete Symmetries; Neutrino Physics - Neutrino HJEP12(07) Z2 Bulgaria. 1 Introduction 2 The framework 2.1 2.2 2.3 G 3.1 3.2 3.3 3.4 The PMNS matrix from Ge = Z2 and G Extracting mixing parameters and statistical analysis Results and discussion 3 Mixing patterns from Gf o HCP = S4 o HCP broken to Ge = Z2 and 4 5 Neutrinoless double beta decay Summary and conclusions A Symmetry of X B Conjugate pairs of S4 elements C Equivalent cases D Correspondence with earlier results 1 4 oscillation data in the recent years (see, e.g., [1]) is one of the most challenging problems in neutrino physics. It is part of the more general fundamental problem in particle physics of understanding the origins of avour in the quark and lepton sectors, i.e., of the patterns of quark masses and mixing, and of the charged lepton and neutrino masses and of neutrino mixing. The idea of extending the Standard Model (SM) with a non-Abelian discrete avour symmetry has been widely exploited in attempts to make progress towards the understanding the origin(s) of avour (for reviews on the discrete symmetry approach to the avour problem see, e.g., [2{4]). In this approach it is assumed that at a certain high-energy scale the theory possesses a avour symmetry, which is broken at lower energies to residual { 1 { symmetries of the charged lepton and neutrino sectors, yielding certain predictions for the values of, and/or correlations between, the low-energy neutrino mixing parameters. In the reference 3-neutrino mixing scheme we are going to consider in what follows (see, e.g., [1]), i) the values of certain pairs of, or of all three, neutrino mixing angles are predicted to be correlated, and/or ii) there is a correlation between the value of the Dirac CP violation (CPV) phase in the neutrino mixing matrix and the values of the three neutrino mixing angles,1 12, 13 and 23, which includes also symmetry dependent xed parameter(s) (see, e.g., [5{13] and references quoted therein). These correlations are usually referred to as \neutrino mixing sum rules". As we have already indicated, the sum rules for the Dirac phase , in particular, depend on the underlying symmetry form of the PMNS matrix [5{9] (see also, e.g., [10{13]), which in turn is determined by the assumed lepton favour symmetry that typically has to be broken, and by the residual unbroken symmetries in the charged lepton and neutrino sectors (see, e.g., [2{4, 7, 9]). They can be tested experimentally (see, e.g., [6, 10, 14{16]). These tests can provide unique information about the possible existence of a new fundamental symmetry in the lepton sector, which determines the pattern of neutrino mixing [5]. Su ciently precise experimental data on the neutrino mixing angles and on the Dirac CPV phase can also be used to distinguish between di erent possible underlying avour symmetries leading to viable patters of neutrino mixing. While in the discrete avour symmetry approach at least some of the neutrino mixing angles and/or the Dirac phase are determined (directly or indirectly via a sum rule) by the avour symmetry, the Majorana CPV phases 21 and 31 [17] remain unconstrained. The values of the Majorana CPV phases are instead constrained to lie in certain narrow intervals, or are predicted, in theories which in addition to a avour symmetry possess at a certain high-energy scale a generalised CP (GCP) symmetry [18]. The GCP symmetry should be implemented in a theory based on a discrete avour symmetry in a way that is consistent with the avour symmetry [19, 20]. At low energies the GCP symmetry is broken, in general, to residual CP symmetries of the charged lepton and neutrino sectors. In the scenarios involving a GCP symmetry, which were most widely explored so far (see, e.g., [19, 21{25]), a non-Abelian avour symmetry Gf consistently combined with a GCP symmetry HCP is broken to residual Abelian symmetries Ge = Zn, n > 2, or Zm Zk, m; k 2, and G HCP of the charged lepton and neutrino mass terms, respectively.2 The factor HCP in G stands for a remnant GCP symmetry of the neutrino mass term. In such a set-up, Ge xes completely the form of the unitary matrix Ue which diagonalises the product MeMey and enters into the expression of the PMNS matrix, Me being the charged lepton mass matrix (in the charged lepton mass term written in the left-right convention). At the same time, G xes the unitary matrix U , diagonalising the neutrino Majorana mass matrix M up to a single free real parameter | a rotation angle . Given the fact that the PMNS neutrino mixing matrix UPMNS is given by the product UPMNS = Uey U ; (1.1) 1Throughout the present study we use the standard parametrisation of the Pontecorvo, Maki, Nakagawa and Sakata (PMNS) neutrino mixing matrix (see, e.g., [1]). 2We note that in refs. [22, 23] the residual symmetry Ge of the charged lepton mass term is augmented with a remnant CP symmetry HCeP as well. { 2 { Parameter the global analysis of the neutrino oscillation data performed in [26]. all three neutrino mixing angles are expressed in terms of this rotation angle. In this class of models one obtains speci c correlations between the values of the three neutrino mixing angles, while the leptonic CPV phases are typically predicted to be exactly 0 or , or else =2 or 3 =2. For example, in the set-up considered in [19] (see also [21]), based on Gf o HCP = S4 o HCP broken to Ge = Z3T and G = ZS 2 HCP with HCP = fU; SU g,3 the authors nd: 2 3 sin2 13 = sin2 ; sin2 12 = 1 2 + cos 2 = data [26] and summarised in table 1, to be used in our further analysis,4 that the predictions quoted in eq. (1.2) for sin2 12 and sin2 23 lie outside of their respective currently allowed 2 ranges.5 Another example of one-parametric models is the extensive study performed in [28], in which the authors have considered two di erent residual symmetry patterns. The rst pattern is the one described above, and the second pattern has Ge = Z2 HCP as residual symmetries in the charged lepton and neutrino sectors, respectively. The authors have performed an exhaustive scan over discrete groups of order less than 2000, which admit faithful 3-dimensional irreducible representations, and classi ed HCeP and phenomenologically viable mixing patterns. 3S, T and U are the generators of S4 in the basis for its 3-dimensional representation we employ in this work (see subsection 3.2). 4The results on the neutrino oscillation parameters obtained in the global t performed in [27] di er somewhat from, but are compatible at 1 con dence level (C.L.) with, those found in [26] and given in table 1. 5We have used the best t value of sin2 13 to obtain the prediction of sin2 12 leading to the quoted conclusion. Using the 2 allowed range for sin2 13 leads to a minimal value of sin2 12 = 0:340, which is above the maximal allowed value of sin2 12 at 2 C.L., but inside its 3 range. { 3 { (1.2) (1.3) HJEP12(07) Theoretical models based on the approach to neutrino mixing that combines discrete symmetries and GCP invariance, in which the neutrino mixing angles and the leptonic CPV phases are functions of two or three parameters have also been considered in the literature (see, e.g., [29{32]). In these models the residual symmetry Ge of the charged lepton mass term is typically assumed to be a Z2 symmetry or to be fully broken. In spite of the larger number of parameters in terms of which the neutrino mixing angles and the leptonic CPV phases are expressed, the values of the CPV phases are still predicted to be correlated with the values of the three neutrino mixing angles. A set-up with Ge = Z2 and G = Z2 HCP has been considered in [32]. The resulting PMNS matrix in such a scheme depends on two free real parameters | two angles and e . The authors have bined with HCP, broken to all possible residual symmetries of the type indicated above. Models allowing for three free parameters have been investigated in [29{31]. In, e.g., [30], the author has considered Gf = A5 combined with HCP, which are broken to Ge = Z2 and HCP. In this case, the matrix Ue depends on an angle e and a phase e, while the matrix U depends on an angle . In these two scenarios the leptonic CPV phases possess non-trivial values. The speci c correlations between the values of the three neutrino mixing angles, which characterise the one-parameter models based on Ge = Zn, n > 2, or Zm Zk, m; k 2, and G = Z2 HCP, do not hold in the two- and three-parameter models. In addition, the Dirac CPV phase in the two- and three-parameter models is predicted to have non-trivial values which are correlated with the values of the three neutrino mixing angles and di er from 0, , =2 and 3 =2, although the deviations from, e.g., 3 =2 can be relatively small. The indicated di erences between the predictions of the models based on Ge = Zn, n > 2, or Zm Zk, m; k 2, and on Ge = Z2 symmetries make it possible to distinguish between them experimentally by improving the precision on each of the three measured neutrino mixing angles 12, 23 and 13, and by performing a su ciently precise measurement of the Dirac phase . In the present article, we investigate the possible neutrino mixing patterns generated by a Gf = S4 symmetry combined with an HCP symmetry when these symmetries are broken down to Ge = Z2 and G HCP. In section 2, we describe a general framework for deriving the form of the PMNS matrix, dictated by the chosen residual symmetries. Then, in section 3, we apply this framework to Gf = S4 combined with HCP and obtain all phenomenologically viable mixing patterns. Next, in section 4, using the obtained predictions for the neutrino mixing angles and the Dirac and Majorana CPV phases, we derive predictions for the neutrinoless double beta decay e ective Majorana mass. Section 5 contains the conclusions of the present study. 2 The framework We start with a non-Abelian avour symmetry group Gf , which admits a faithful irreducible 3-dimensional representation . The three generations of left-handed (LH) leptons are assigned to this representation. Apart from that, the high-energy theory respects also the GCP symmetry HCP, which is implemented consistently along with the avour symmetry. { 4 { At some avour symmetry breaking scale Gf o HCP gets broken down to residual symmetries Ge and G of the charged lepton and neutrino mass terms, respectively. The residual avour symmetries are Abelian subgroups of Gf . The symmetries Ge and G signi cantly constrain the form of the neutrino mixing matrix UPMNS, as we demonstrate below. 2.1 element of Gf of order two, generating the Z2ge subgroup. The invariance of the charged f1; geg, ge2 = 1 being an lepton mass term under Ge implies (ge)yMeMey (ge) = MeMey : Below we show how this invariance constrains the form of the unitary matrix Ue, diagonalising MeMey: UeyMeMey Ue = diag(me2; m2 ; m2) : Lets e be a diagonalising unitary matrix of (ge), such that ye (ge) e = (ge)d 0 0 1 C ; A { 5 { This result is obtained as follows. The diagonal entries of (ge)d are constrained to be 1, since this matrix must still furnish a representation of Z2 and hence its square is the identity. We have assumed that the trace of (ge) is 1, for the relevant elements ge, as it is the case for the 3-dimensional representation of S4 we will consider later on.6 Note that we can take the order of the eigenvalues of (ge) as given in eq. (2.3) without loss of generality, as will become clear later. Expressing (ge) from eq. (2.3) and substituting it in eq. (2.1), we obtain (ge)d ye MeMey e (ge)d = ye MeMey e : This equation implies that ye MeMey e has the block-diagonal form 6For the other 3-dimensional irreducible representation of S4 the trace can be either 1 or +1, depending on ge. Choosing +1 would simply imply a change of sign of (ge)d, which however does not lead to new and, since this matrix is hermitian, it can be diagonalised by a unitary matrix with a U (2) transformation acting on the 2-3 block. In the general case, the U (2) transformation can be parametrised as follows: cos e of the charged lepton elds, and we will not keep it in the future. Thus, we arrive to the conclusion that the matrix Ue diagonalising MeMey reads with Ue = e U23( e; e)y PeT ; U23( e; e) = B0 sin e e i e C ; 0 cos e and Pe being one of six permutation matrices, which need to be taken into account, since in the approach under consideration the order of the charged lepton masses is unknown. The six permutation matrices read: (2.7) (2.8) (2.9) (2.10) P123 = B0 1 0C ; P132 = B0 0 1C ; P213 = B1 0 0C ; P231 = B0 0 1C ; P312 = B1 0 0C ; P321 = B0 1 0C : 0 1 0 0 0 0 0 1 0 1 0 Note that the order of indices in Pijk stands for the order of rows, i.e., when applied from the left to a matrix, it gives the desired order, i-j-k, of the matrix rows. The same is also true for columns, when Pijk is applied from the right, except for P231 which leads to the 3-1-2 order of columns and P312 yielding the 2-3-1 order. In the neutrino sector we have a G HCP residual symmetry. We will denote the Z2 symmetry of the neutrino mass matrix as Z element of Gf , generating the Z 2g subgroup. HCP = fX g is the set of remnant GCP g 2 f1; g g, with g 2 = 1 being an unitary transformations X forming a residual CP symmetry of the neutrino mass matrix. HCP is contained in HCP = fXg which is the GCP symmetry of the high-energy theory consistently de ned along with the avour symmetry Gf .7 The invariance under G of the 7It is worth to comment here on the notation HCP we use. When we write in what follows ual avour Z g 2 symmetry (see eq. (2.13)). However, when writing G = Z g 2 HCP = fX 1; X 2g, we mean a set of GCP transformations (X 1 and X 2) compatible with the residHCP, HCP is intended to be a group generated by X 1. Namely, following appendix B in [19], HCP is isomorphic to fI; X 1g, where I is the unit matrix and both of them acting on ('; ' )T . Then, Z2g is isomorphic to fI; G g, where g 2 contained in it. The same logic applies to the notation HCP, and, as has been shown in appendix B of [19], the full symmetry group is a semi-direct product Gf o HCP. Note that these notations are widely used in the literature. neutrino mass matrix implies that the following two equations hold: In addition, the consistency condition between Z 2g and HCP has to be respected: (g )T M XT M (g ) = M ; X = M : X (g ) X 1 = (g ) : To derive the form of the unitary matrix U diagonalising the neutrino Majorana mass HJEP12(07) matrix M as U T M U = diag(m1; m2; m3) ; mj > 0 being the neutrino masses, we will follow the method presented in [32]. Lets 1 be a diagonalising unitary matrix of (g ), such that y 1 (g ) 1 = (g )d Expressing (g ) from this equation and substituting it in the consistency condition, eq. (2.13), we nd (g )d y1 X 1 (g )d = y1 X 1 ; meaning that y1 X 1 is a block-diagonal matrix, having the form of eq. (2.5). Moreover, this matrix is symmetric, since the GCP transformations X have to be symmetric in order for all the three neutrino masses to be di erent [19, 21], as is required by the data. In appendix A we provide a proof of this. Being a complex (unitary) symmetric matrix, it is diagonalised by a unitary matrix 2 via the transformation: y2 ( y1 X 1 ) 2 = ( y1 X 1)d : )d is, in general, a diagonal phase matrix. However, we can choose 1 )d = diag(1; 1; 1) as the phases of ( y1 X 1 )d can be moved to the matrix . With this choice we obtain the Takagi factorisation of the X (valid for unitary Since, as we have noticed earlier, y1 X 1 has the form of eq. (2.5), the matrix can be chosen without loss of generality to have the form of eq. (2.5) with a unitary 2 matrix in the 2-3 block. This implies that the matrix = 1 2 also diagonalises (g ). Indeed, mass matrix, eq. (2.12), and nd that the matrix a symmetric matrix, since the neutrino Majorana mass matrix M is symmetric. A real X = T ; { 7 { sin 0 sin cos 1 C : A U = R23( ) P Q ; Finally, the matrix U diagonalising M reads where P is one of the six permutation matrices, which accounts for di erent order of renders them positive. Without loss of generality Q symmetric matrix can be diagonalised by a real orthogonal transformation. Employing eqs. (2.19) and (2.11), we have (g )d T M (g )d = T M ; implying that T M is a block-diagonal matrix as in eq. (2.5). Thus, the required orthogonal transformation is a rotation in the 2-3 plane on an angle : (2.20) (2.21) (2.22) (2.23) (2.24) Assembling together the results for Ue and U , eqs. (2.7) and (2.22), we obtain for the Q = diag(1; ik1 ; ik2 ) ; with Thus, in the approach we are following the PMNS matrix depends on three free real parameters8 | the two angles e and and the phase e . One of the elements of the PMNS matrix is xed to be a constant by the employed residual symmetries. We note nally that, since R23( + ) = R23( ) diag(1; 1; 1), where the diagonal matrix can be absorbed into Q , and U23( e + ; e) = diag(1; 1; 1) U23( e; e), where the diagonal matrix contributes to the unphysical charged lepton phases, it is su cient to consider e and 2.2 in the interval [0; ). Conjugate residual symmetries and fZ2ge0 ; Z2g0 g are conjugate to each other under h 2 Gf if In this subsection we brie y recall why the residual symmetries G0e and G0 conjugate to Ge and G , respectively, under the same element of the avour symmetry group Gf lead to the same PMNS matrix (see, e.g., [19, 22]). Two pairs of residual symmetries fZ2ge ; Z2 g g h ge h 1 = ge0 and h g h 1 = g0 : (2.25) 8It should be noted that the matrix 2 in eq. (2.17) with ( y1 X 1) d = diag(1; 1; 1), and thus the matrix = 1 2 in eq. (2.18), is determined up to a multiplication by an orthogonal matrix O on the right. The matrix 2 O must be unitary since it diagonalises a complex symmetric matrix, which implies that O must be unitary in addition of being orthogonal, and therefore must be a real matrix. Equation (2.19) restricts further this real orthogonal matrix O to have the form of a real rotation in the 2-3 plane, which can be \absorbed" in the R23( ) matrix in eq. (2.24). { 8 { At the representation level this means (h) (ge) (h)y = (ge0) and (h) (g ) (h)y = (g0 ) : Substituting (ge) and (g ) from these equalities to eqs. (2.1) and (2.11), respectively, we obtain (ge0)yMe0Me0y (ge0) = Me0Me0y and (g0 )T M 0 (g0 ) = M 0 ; where the primed mass matrices are related to the original ones as Me0Me0y = (h) MeMey (h)y and M 0 = (h) M (h)y : HJEP12(07) As can be understood from eq. (2.12) (or eq. (2.13)), the matrix M 0 will respect a remnant CP symmetry HC0P = fX0 g, which is related to HCP = fX g as follows: X0 = (h) X (h)T : Obviously, the unitary transformations Ue0 and U 0 diagonalising the primed mass matrices are given by thus yielding Ue0 = (h) Ue and U 0 = (h) U ; U P0MNS = Ue0y U 0 = Uey U = UPMNS : 2.3 Phenomenologically non-viable cases Here we demonstrate that at least two types of residual symmetries fGe; G g = fZ2ge ; Z2g HCPg, characterised by certain ge and g , cannot lead to phenomenologically viable form of the PMNS matrix. Type I: ge = g . In this case, we can choose e = P , with P123 or P132. Then, eq. (2.24) yields data [26, 27]. UPMNS = Pe U23( e; e) P R23( ) P Q : This means that up to permutations of the rows and columns UPMNS has the form of eq. (2.5), i.e., contains four zero entries, which are ruled out by neutrino oscillation Type II: ge; g Gf . Now we consider two di erent order two elements ge 6= g , which belong to the same Z2 Z2 = f1; ge; g ; ge g g subgroup of Gf . In this case, since ge and g commute, there exists a unitary matrix simultaneously diagonalising both (ge) and (g ). Note, however, that the order of eigenvalues in the resulting diagonal matrices will be di erent. Namely, lets 1 be a diagonalising matrix of (g ) and (ge), and lets 1 diagonalise (g ) as in eq. (2.15). Then, y 1 (ge) 1 can yield either diag( 1; 1; 1) or diag( 1; 1; 1), but not diag(1; 1; 1). Hence, e diagonalising (ge) as in eq. (2.3), must read e = 1 P ; with and P = P213 or P312 if P = P231 or P321 if y y 1 (ge) 1 (ge) 1 = diag( 1; 1; 1) ; 1 = diag( 1; 1; 1) : (2.33) (2.34) { 9 { (2.26) (2.27) (2.28) (2.29) (2.30) (2.31) (2.32) Taking into account that eq. (2.5), we obtain = 2, with 2 of the block-diagonal form given in UPMNS = Pe U23( e; e) P T 2 R23( ) P Q ; where P T 2, depending on P , can take one of the following forms: 0 0 B or 1 A 0 0 B 0 1 C ; A As a consequence, UPMNS up to permutations of the rows and columns has the form HJEP12(07) (2.35) (2.36) (2.37) (3.1) (3.2) (3.3) (3.4) (3.5) containing one zero element, which is ruled out by the data. 3 Mixing patterns from Gf o HCP = S4 o HCP broken to Ge = Z2 and G = Z2 HCP 3.1 Group S4 and residual symmetries S4 is the symmetric group of permutations of four objects. This group is isomorphic to the group of rotational symmetries of the cube. S4 can be de ned in terms of three generators S, T and U , satisfying [33] S2 = T 3 = U 2 = (ST )3 = (SU )2 = (T U )2 = (ST U )4 = 1 : From 24 elements of the group there are nine elements of order two, which belong to two of ve conjugacy classes of S4 (see, e.g., [21]): 3 C2 : fS ; T ST 2 ; T 2ST g ; 6 C20 : fU ; T U ; SU ; T 2U ; ST SU ; ST 2SU g : Each of these nine elements generates a corresponding Z2 subgroup of S4. Each subgroup can be the residual symmetry of MeMey, and, combined with compatible CP transformations, yield the residual symmetry of M . Hence, we have 81 possible pairs of only residual avour symmetries (taking into account remnant CP symmetries increases the number of possibilities). Many of them, however, being conjugate to each other, will lead to the same form of the PMNS matrix, as explained in subsection 2.2. Thus, we rst identify the pairs of elements fge; g g, which are not related by the similarity transformation given in eq. (2.25). We nd nine distinct cases for which fge; g g can be chosen as fS; Sg ; fU; U g ; fS; T U g ; fT U; Sg ; fT U; U g : fT 2ST; Sg ; fS; U g ; fU; Sg ; fSU; U g ; is conjugate to fge; g g with ge = g being one of the remaining ve elements from conjugacy class 6 C20 given in eq. (3.3). The pairs fT 2ST; Sg, fS; U g, fU; Sg and fSU; U g are conjugate to ve pairs each, and fS; T U g and fT U; Sg to eleven pairs each. Finally, fT U; U g is conjugate to 23 pairs. As it should be, the total number of pairs yields 81. The complete lists of pairs of elements which are conjugate to each of these nine pairs are given in appendix B. The cases in eq. (3.4) do not lead to phenomenologically viable results. The rst two of them belong to the cases of Type I (see subsection 2.3). The remaining four belong to Type II, since S4 contains ZS 2 ZT ST 2 = f1; S; T ST 2 ; T 2ST g and ZS 2 2 Z2U = f1; S; U ; SU g subgroups (see, e.g., [34]). Thus, we are left with three cases in eq. (3.5). We have chosen g in such a way that it is S, U or T U for all the cases in eq. (3.5). Now we need to identify the remnant CP transformations X compatible with each of these three elements. It is known that the GCP symmetry HCP = fXg compatible with Gf = S4 is of the same form of Gf itself [20], i.e., X = (g); g 2 S4 : Thus, to nd X compatible with g of interest, we need to select those X = (g), which i) satisfy the consistency condition in eq. (2.13) and ii) are symmetric in order to avoid partially degenerate neutrino mass spectrum, as was noted earlier. The result reads:9 X X X = 1 ; (S) ; U ; (SU ) ; T ST 2U ; (T 2ST U ) for g = S ; = 1 ; (U ) ; S ; (SU ) = U ; (T ) ; ST S ; (T 2ST U ) for g = U ; for g = T U : (3.6) (3.7) (3.8) (3.9) A GCP transformation in parentheses appears automatically to be a remnant CP symmetry of M , if X which precedes this in the list is a remnant CP symmetry. This is a consequence of eqs. (2.11) and (2.12), which imply that if X is a residual CP symmetry of M , then (g )X is a residual CP symmetry as well. Therefore, we have three sets of remnant CP transformations compatible with ZS, namely, HCP = f1; Sg, 2 fU; SU g and fT ST 2U; T 2ST U g, two sets compatible with Z2U , which are HCP = f1; U g and fS; SU g, and two sets consistent with Z2T U , which read HCP = fU; T g and fST S; T 2ST U g. Taking them into account, we end up with seven possible pairs of residual symmetries fGe; G g = fZ2ge ; Z2 g HCPg, with fge; g g as in eq. (3.5). In what follows, we will consider them case by case and classify all phenomenologically viable mixing patterns they lead to. Before starting, however, let us recall the current knowledge on the absolute values of the PMNS matrix elements, which we will use in what follows. The 3 ranges of the absolute values of the PMNS matrix elements read [35] 00:796 ! 0:855 0:497 ! 0:587 0:140 ! 0:1531 jUPMNSj3 = B@0:245 ! 0:513 0:543 ! 0:709 0:614 ! 0:768C A 0:244 ! 0:510 0:456 ! 0:642 0:624 ! 0:776 (3.10) 9For notation simplicity we will not write the representation symbol , keeping in mind that X = g meas X = (g) with g 2 Gf . for the neutrino mass spectrum with normal ordering (NO), and for the neutrino mass spectrum with inverted ordering (IO). The ranges in eqs. (3.10) and (3.11) di er a little from the results obtained in [27]. First, we present an explicit example of constructing the PMNS matrix in the case of ge = S, g = T U and HCP = fU; T g, which is the rst case out of the seven potentially viable cases indicated above. We will work in the basis for S4 from [36], in which the matrices for the generators S, T and U in the 3-dimensional representation read 2 1 2 1 2 CA ; 1 0 Be 23i C ; (3.11) (3.13) (3.14) (3.15) where ! = e2 i=3. For simplicity we use the same notation (S, T and U ) for the generators and their 3-dimensional representation matrices. We will follow the procedure described in subsection 2.1. The matrix e which diagonalises (ge) = S (see eq. (2.3)) is given by The matrix , such that T = U (see eq. (2.18)), reads Using the master formula in eq. (2.24), we obtain that up to permutations of the rows and columns UPMNS has the form of 1=p2 IO spectra. where \ " entries are functions of the free parameters count the current data, eqs. (3.10) and (3.11), the xed element with the absolute value 0:707 can be (UPMNS) 2, (UPMNS) 3, (UPMNS) 2 or (UPMNS) 3. Note that j(UPMNS) 2j = 0:707 is outside the 3 range in the case of the NO neutrino mass spectrum, while j(UPMNS) 2j = 0:707 is at the border of the 3 allowed ranges for both the NO and Moreover, from j(UPMNS) 2j = 1=p2 we obtain a sum rule for cos : cos = Let us comment now on the following issue. Once one of the elements of the PMNS matrix is xed to be a constant, we still have four possible con gurations, namely, a permutation of two remaining columns, a permutation of two remaining rows and both of them. For instance, in the case considered above, except for Pe = P = P213, we can have a xed (UPMNS) 2 with (Pe; P ) = (P213; P231), (P312; P213) and (P312; P231). These combinations of the permutation matrices will not lead, however, to di erent mixing patterns by virtue of the following relations: PMNS matrix are related to the free parameters , e and e as follows: Let us consider as an example the rst possibility, i.e., Pe = P j(UPMNS) 2j = 1=p2. In this case the mixing angles of the standard parametrisation of the sin2 13 = j(UPMNS)e3j2 = 1 24 Indeed, e.g., in the case of (Pe; P ) = (P312; P231), de ning ^ = + =2, ^e = e + =2 and absorbing the matrix diag ( 1; 1; 1) in the matrix Q , we obtain the same PMNS matrix as in the case of (Pe; P ) = (P213; P213): UPMNS = P213 U23(^e; e) ye R23(^ ) P213 Q : The phases in the matrix diag ei e; 1; e i e are unphysical, and we have disregarded them. We list in table 2 the matrices e and residual symmetries fGe; G g = fZ2ge ; Z2g seven pairs, namely, fGe; G g = fZ2S; ZT U 2 for all seven phenomenologically viable pairs of HCPg. It turns out, however, that four of these HCPg with HCP = fU; T g and fST S; T 2ST U g, and fGe; G g = fZ2T U ; Z2S HCPg with HCP = fU; SU g and fT ST 2U; T 2ST U g, lead to the same predictions for the mixing parameters. We demonstrate this in appendix C. (3.16) (3.17) (3.18) (3.19) (3.20) (3.21) (3.22) S 1 p 0p p p fU; T g 1 p 0 0 p 2 p2e i3 C p2e i3 1 A 2 2 2 p 3 0 p 3 11 2 2 1 1 i i e 6 1 i e 3 1 A B1 e i6 e i3 C 1 p 0 0 0 p21 1 1 0 A 3 p p2i 21 p2i 1 A 3 p2i 1 dictated by the residual symmetries Ge = Z2ge and G = Z g 2 HCP for all seven phenomenologically viable pairs of Ge and G . For each pair HCP = fX 1; X 2g of remnant GCP transformations, the given matrix provides the Takagi factorisation of the rst element, i.e., X 1 = Extracting mixing parameters and statistical analysis In this subsection we perform a statistical analysis of the predictions for the neutrino mixing angles and CPV phases for each of the four distinctive sets of the residual avour and CP 10X 2 is instead factorised as X 2 = ~ ~ T , with ~ = diag(1; i; i), as follows from X 2 = (g ) X 1 = derive predictions for the three neutrino mixing angles and the three leptonic CPV phases, which, in many of the cases analysed in the present study is impossible to obtain purely analytically. Once a pair of residual symmetries and the permutation matrices Pe and P are specied, we have the expressions for sin2 ij in terms of , e and e of the type of eqs. (3.16){ (3.18). Moreover, employing a sum rule for cos analogous to that in eq. (3.19) and computing the rephasing invariant JCP = Im (UPMNS)e1 (UPMNS) 3 (UPMNS)e3 (UPMNS) 1 ; which determines the magnitude of CPV e ects in neutrino oscillations [37] and which in the standard parametrisation of the PMNS matrix is proportional to sin , 1 8 JCP = sin 2 12 sin 2 23 sin 2 13 cos 13 sin ; we know the value of for any , e and e. Similarly, making use of the two charged lepton rephasing invariants,11 associated with the Majorana phases [38{41], I1 = Im f(UPMNS)e1 (UPMNS)e2g and I2 = Im f(UPMNS)e1 (UPMNS)e3g ; and the corresponding real parts R1 = Re f(UPMNS)e1 (UPMNS)e2g and R2 = Re f(UPMNS)e1 (UPMNS)e3g ; which in the standard parametrisation of the PMNS matrix read: I1 = sin 12 cos 12 cos2 13 sin ( 21=2) ; I2 = cos 12 sin 13 cos 13 sin ( 31=2 R1 = sin 12 cos 12 cos2 13 cos ( 21=2) ; R2 = cos 12 sin 13 cos 13 cos ( 31=2 ) ; (3.27) ) ; (3.28) we also obtain the values of 21 and 31 for any the values of sin2 ij and the CPV phases. We require sin2 ij to lie in the corresponding 3 ranges given in table 1. The obtained values of sin2 ij and certain value of the 2 function constructed as follows: can be characterised by a 4 X i=1 2 (~x) = i2 (xi) ; where ~x = fxig = (sin2 12; sin2 13; sin2 23; ) and i2 are one-dimensional projections for NO and IO taken from [26].12 Thus, we have a list of points (sin2 12, sin2 13, sin2 23, , 11In their general form, when one keeps explicit the unphysical phases j in the Majorana condition C j T = j j, j = 1; 2; 3, the rephasing invariants related to the Majorana phases involve j and are invariant under phase transformations of both the charged lepton and neutrino elds (see, for example, eqs. (22){(28) in [38]). We have set j = 1. IO of 2 IO NO footing. A discussion on this issue can be found in [26]. 12We note that according to the latest global oscillation data, there is an overall preference for NO over 3:6. Nevertheless, we take a conservative approach and treat both orderings on an equal (3.23) (3.24) (3.25) (3.26) (3.29) HJEP12(07) 21, 31, 2). To see the restrictions on the mixing parameters imposed by avour and CP symmetries we consider all 15 di erent pairs (a; b) of the mixing parameters. For each pair we divide the plane (a; b) into bins and nd a minimum of the 2 function in each bin. We present results in terms of heat maps with colour representing a minimal value of 2 in each bin. The results obtained in each case are discussed in the following subsection. In this subsection we systematically go through all di erent potentially viable cases and summarise their particular features. All these cases can be divided in four groups corresponding to a particular pair of residual symmetries fGe; G g In each case we concentrate on results for the ordering for which a better compatibility with the global data is attained. Note that results for NO and IO di er only i) due to the fact that the 3 ranges of sin2 13 and sin2 23 depend slightly on the ordering and ii) in the respective 2 landscapes. Moreover, we present numerical results for the Majorana phases obtained for k1 = k2 = 0, where k1 and k2 are de ned in eq. (2.23). However, one should keep in mind that all four (k1; k2) pairs, where ki = 0; 1, are allowed. Whenever k1(2) = 1, the predicted range for 21(31) shifts by . The values of the ki are important for the predictions of the neutrinoless double beta decay e ective Majorana mass (see, e.g., [38, 42{44]), which we obtain in section 4. Group A: fGe; G g = fZ2T U ; Z2S sponding matrices e and HCPg with HCP = f1; Sg. Using the correfrom table 2 and the master formula for the PMNS matrix in eq. (2.24), we nd the following form of the PMNS matrix (up to permutations of rows and columns and the phases in the matrix Q ): UPAMNS = 1 0 Bp p 6 e i6 p 3 ei p 3 e i 1 2 ce p 2 cee i3 + 2 see i e a1 ( ; e; e) a2 ( ; e; e)CC ; 2 see i3 ei e a3 ( ; e; e) a4 ( ; e; e)A with ce cos e, se sin e, c cos , s sin and a1 ( ; e; e) = a2 ( ; e; e) = hp3c + 2 hp3s a3 ( ; e; e) = p p p From eq. (3.30), we see that the absolute values of the elements of the rst row are xed. Namely, the modulus of the rst element is equal to 1=p2, while the moduli of the second and third elements equal 1=2. Taking into account the current knowledge of the mixing j(UPMNS) 1j = j(UPMNS) 2j = 1=2 and j(UPMNS) 3j = 1=p2. parameters, eqs. (3.10) and (3.11), this implies that there are only two potentially viable cases: i) with j(UPMNS) 1j = j(UPMNS) 2j = 1=2 and j(UPMNS) 3j = 1=p2, and ii) with Case A1: j(UPMNS) 1j = j(UPMNS) 2j = 1=2, j(UPMNS) 3j = 1=p2 (Pe = P213, P = P321). In this case we obtain This means that only a narrow interval sin2 23 2 [0:510; 0:512] is allowed using the 3 region for sin2 13. From the equality j(UPMNS) 1j = 1=2, which we nd to hold in this case, it follows that cos satis es the following sum rule: cos = 1 where the mixing angles in addition are correlated among themselves. We nd that sin2 13 is constrained to lie in the interval (0:0213; 0:0240(2)] for NO (IO) and, hence, sin2 23 in [0:5109; 0:5123(4)]. This range of values of sin2 23 is not compatible with its current 2 range. Moreover, sin2 12 is found to be between approximately 0:345 and 0:354, which is outside its current 2 range as well. What concerns the CPV phases, the predicted values of are distributed around 0, namely, 21 around , 21 2 (0:93 ; 1:07 ), while the values of 31 ll the whole range, i.e., 31 2 [0; 2 ). These numbers, presented for the NO spectrum, remain practically unchanged for the IO spectrum. However, the global minimum 2min of the 2 function, de ned in eq. (3.29), yields approximately 22 (19) for NO (IO), which implies that this case is disfavoured by the global data at more than 4 . Case A2: j(UPMNS) 1j = j(UPMNS) 2j = 1=2, j(UPMNS) 3j = 1= = P321). This case shares the predicted ranges for sin2 12, sin2 13, 21 and case A1, but di ers in the predictions for sin2 23 and . Again, there is a correlation p between sin2 13 and sin2 23: which, in particular, implies that sin2 23 2 [0:4877(6); 0:4891], which is not compatible with its present 2 leads to the following sum rule: range. We also nd that j(UPMNS) 1j = 1=2. This equality cos = 4 sin2 12 sin2 23 + 4 cos2 12 cos2 23 sin2 13 It is worth noting that we should always keep in mind the correlations between the mixing angles in expressions of this type. The values of in this case lie around , in the interval [0:89 ; 1:11 ]. As in the previous case, the global minimum of 2 is somewhat large, 2 min disfavoured. 18:5 (15) for NO (IO), meaning that this case is also with Group B: fGe; G g = fZ2T U ; Z2S HCPg with HCP = fU; SU g. For this choice of the residual symmetries, the PMNS matrix reads (up to permutations of rows and columns and the phases in the matrix Q ): 0 B 1 p 6 e i3 p 2 cee i6 + 2 i see i e 2 i ce + p 2 see i6 ei e p 3 (c + s ) e i3 p 3 (s c ) e i3 1 b1 ( ; e; e) b3 ( ; e; e) b2 ( ; e; e) CC ; b4 ( ; e; e) A b1 ( ; e; e) = (3s c ) cee i6 b2 ( ; e; e) = b3 ( ; e; e) = b4 ( ; e; e) = (3c + s ) cee i6 p 2 2 i c ce (3s 2p2 i s ce + (3c + s ) see i6 ei e : p 2 2 i c see i e ; p 2 2 i s see i e ; c ) see i6 ei e ; (3.41) (3.42) (3.43) (3.44) (3.45) HJEP12(07) predicted to be 1=p2. Thus, we have four potentially viable cases. Equation (3.41) implies that the absolute value of one element of the PMNS matrix is p Case B1: j(UPMNS) 2j = 1= = P213). Note that from eqs. (3.10) and (3.11) it follows that this magnitude of the xed element is inside its 3 range for NO, but slightly outside the corresponding range for IO. Hence, we will focus on the results for NO. The characteristic feature of this case is the following sum rule for cos : cos = which arises from the equality of j(UPMNS) 2j to 1=p2. The pair correlations between the mixing parameters in this case are summarised in gure 1. The colour palette corresponds to values of 2 for NO. As can be seen, while all values of sin2 13 in its 3 range are allowed, the parameters sin2 12 and sin2 23 are found to lie in [0:250; 0:308] and [0:381; 0:425) intervals, respectively. The predicted values of span the range [0:68 ; 1:32 ]. Thus, CPV e ects in neutrino oscillations due to the phase suppressed. The Majorana phases instead are distributed in relatively narrow regions around 0, so the magnitude of the neutrinoless double beta decay e ective Majorana mass (see section 4 and, e.g., [38, 42{44]) is predicted (for k1 = k2 = 0) to have a value close to the maximal possible for the NO spectrum. Namely, 21 2 [ 0:16 ; 0:16 ] and 31 2 ( 0:13 ; 0:13 ). In addition, is strongly correlated with which in turn exhibit a strong correlation between themselves. Finally, 2 both NO and IO, i.e., this case is compatible with the global data at less than 3 .13 min 13The apparent contradiction between the obtained value of 2 min 7, which suggests compatibility also for IO, and the expectation of min > 9, according to eq. (3.11), arises from the way we construct the 2 2 function (see eq. (3.29)), which does not explicitly include covariances between the oscillation parameters. Case B2: j(UPMNS) 2j = 1= of j(UPMNS) 2j is compatible at 3 = P213). Note that this value with the global data in the case of IO spectrum, but not in the case of NO spectrum, as can be seen from eqs. (3.10) and (3.11). Thus, below we present results for the IO spectrum only. As in case B1, the whole 3 range for sin2 13 is allowed. The obtained ranges of values of 21 and 31 are the same of the preceding case. The range for sin2 12 di ers somewhat from that obtained in case B1, and it reads sin2 12 2 [0:250; 0:328].14 The predictions for sin2 23 and di erent. Now the following sum rule, derived from j(UPMNS) 2j = 1=p2, holds: are cos = 1 min The values of are concentrated in [ 0:38 ; 0:38 ]. For sin2 23 we nd the range (0:575; 0:636]. The correlations between the phases are of the same type as in case B1. We summarise the results in gure 2. Finally, 6 in the case of IO and 2 min 12:5 for NO, which re ects incompatibility of this case at more than 3 for the NO spectrum. This occurs mainly due to the predicted values of sin2 23, which are outside its current 2 range for NO. p Case B3: j(UPMNS) 3j = 1= 2 (Pe = P213; P 1=p2, the angles 13 and 23 are correlated as in case A1, i.e., according to eq. (3.35). For IO this leads to sin2 23 2 [0:5097; 0:5124] due to the fact that the whole 3 range = P321). Since j(UPMNS) 3j = of sin2 13 is found to be allowed, as can be seen from gure 3. Note that this range is outside the current 2 range of sin2 23. In addition, we nd that the whole 3 range of the values of sin2 12 can be reproduced. In contrast to case A1, j(UPMNS) 1j does not equal 1=2, but depends on in the following way: j(UPMNS) 1j2 = 1 sin 2 4 : (3.48) From this equation we nd cos = 1 explicitly (not only via 12, 23 and 13). With this relation, any value of between 0 and 2 is allowed (see gure 3). The Majorana phases, however, are constrained to lie around 0 in the following intervals: 21 2 [ 0:23 ; 0:23 ] and 31 2 ( 0:18 ; 0:18 ). Moreover, both phases and the same peculiar way with the phase . The correlation between is similar to those in cases B1 and B2 (cf. gures 1 and 2). Due to the predicted values of sin2 23, which belong to the upper octant, IO is preferred over NO, the corresponding 2 min being approximately 5 and 8:5. 14This di erence is related to the fact that the current 3 range of sin2 23 for IO, which reads [0:384; 0:636], is not symmetric with respect to 0:5. The asymmetry of 0:02 translates to increase of the allowed range of sin2 12 by approximately 0:02. This can be better understood from the top right plots in gures 1 and 2. 20 15 10 5 30 25 20 15 10 5 30 25 20 15 10 5 25 20 15 10 5 30 25 20 15 10 5 20 15 10 5 30 25 20 15 10 5 25 20 15 10 5 25 20 15 10 5 30 25 20 15 10 5 2α0.8 3α0.8 2α0.8 0.4 0.0 χ χ 2 χ 2 χ 2 χ 2 25 20 15 10 5 30 25 20 15 10 5 25 20 15 10 5 25 20 15 10 5 25 20 15 10 5 2 HJEP12(07) mixing parameters in case D3. The values of all the three mixing angles are required to lie in their respective 3 ranges. Colour represents values of for the IO neutrino mass spectrum. We turn now to the possibilities to discriminate experimentally between the di erent cases listed in tables 3{6 using the prospective data on sin2 12, sin2 13, sin2 23 and . The rst thing to notice is that the predicted ranges for sin2 12, sin2 13, sin2 23 and in cases A1 and A2 practically coincide with the predictions respectively in cases D4 and D5. However, cases A1, D4 and cases A2, D5 are strongly disfavoured by the current data: for the NO (IO) neutrino mass spectrum A1 and D4 are disfavoured at 4:7 (4:4 ), while A2 and D5 are disfavoured at 4:3 (3:9 ). In all these cases sin2 12, in particular, is predicted to lie in the interval (0.345,0.354) compatible with the current 3 range and, given the current best t value of sin2 12 and prospective JUNO precision on sin2 12, it is very probable that future more precise data on sin2 12 will rule out completely these scenarios. We will not discuss them further in this subsection. It follows also from tables 5 and 6 that the combined results on the best t values of sin2 12, sin2 23 and we have obtained in the di erent viable cases (excluding A1, A2, D4 and D5) di er signi cantly. Assuming, for example, that the experimentally determined best t values of sin2 12 and sin2 23 will coincide with those found by us for a given viable case, it is not di cult to convince oneself inspecting tables 5 and 6 that the cited prospective 1 errors on sin2 12 and sin2 23 will allow to discriminate between the di erent viable cases identi ed in our study. More speci cally, considering as an example only the case of NO neutrino mass spectrum, the prospective high precision measurement of sin2 12 will allow to discriminate between case C1 and all other cases B1{B4, C2{C5, D2 and D3. The same measurement will make it possible to distinguish i) between case B1 and all the other cases except B2, ii) between case B2 and all the other cases except B1, B3 and B4, and similarly iii) between case B3 and all the other cases except B2, B4, C4 and C5. However, the di erences between the best t values of sin2 23 in cases B1, B2 and B3 (or B4) are su ciently large, which would permit to distinguish between these three cases if sin2 23 were measured with the prospective precision. It follows from table 5, however, that it would be very challenging to discriminate between cases B3 and B4: it will require extremely high precision measurement of sin2 23. These two cases would be ruled out, however, if the experimentally determined best t value of sin2 23 di ers signi cantly from the results for sin2 23, namely, 0.511 and 0.489, we have obtained for sin2 23 in the B3 and B4 cases. In the remaining cases C2{C5 and D2{D3, the results we have obtained for sin2 12, as table 6 shows, are very similar. However, the predictions for the pair sin2 23 and di er signi cantly in cases C2 or D2, and C3 or D3. The cases within each pair would be ruled out if the experimentally determined values of sin2 23 and di er signi cantly from the predicted best t values. Thus, the planned future high precision measurements of sin2 12 and sin2 23, together with more precise data on the Dirac phase , will make it possible to critically test the predictions of the cases listed in tables 3{6. A comprehensive analysis of the possibilities to distinguish between the di erent viable cases found in our work in the considered S4 model can only be done when more precise data rst of all on sin2 12 and sin2 23, and then on , will be available. HJEP12(07) 0.34 1 i s 2θn0.30 corresponding best t points in cases B1{B4, C1{C5, D2 and D3 for the NO neutrino mass spectrum. The values of all the three mixing angles are required to lie in their respective current 3 ranges. We schematically summarise in gure 12 the predicted 3 allowed regions in the plane (sin2 23; sin2 12) for all currently viable cases from gures 1{11. In this gure we also present the best t point in each case used in the preceding discussion. When future more precise data on sin2 23 and sin2 12 become available, the experimentally allowed region in the (sin2 23; sin2 12) plane will shrink, and only a limited number of cases, if any, will remain viable. It will be possible to distinguish further between some or all of the remaining viable cases with a high precision measurement of . Finally, we note that the sum rules for sin2 23 (sin2 12 in case C1) and/or cos obtained in the present study follow from those derived in [7] for certain values of the parameters sin2 ij , xed by Gf = S4 and the residual Z2ge and Z g 2 avour symmetries, and the additional constraints provided by the GCP symmetry HCP. Note that in [7] only avour symmetry, without imposing a GCP symmetry, has been considered. As we have seen in subsection 2.1, a GCP symmetry does not allow for a free phase coming from the neutrino sector, which is present otherwise. This, in turn, leads to the fact that in certain cases the parameter sin ^ij (see eq. (213) in [7]), which is free in [7], gets xed by the GCP symmetry. Thus, we nd additional correlations between ij and between ij and cos in these cases. We provide the correspondence between the phenomenologically viable cases of the present study and the cases considered in [7] in appendix D. (p.f.e.) 10 1 10 1 10 1 ( 3) A2 ( 3) B1 ( 2) B2 ( 2) B3 ( 3) B4 ( 3) C1 (e2) C2 ( 1) C3 ( 1) C4 ( 2) C5 ( 2) D2 ( 1) D3 ( 1) D4 ( 2) D5 ( 2) 0:89{1:11 0:88{1:12 0:68{1:32 0:69{1:31 0{2 0{2 0{2 0{2 = 0{2 0{2 3:81{4:25 3:84{4:25 5:10{5:12 5:10{5:12 4:88{4:90 4:88{4:90 g In this case the magnitude of the xed element is 1=2. 31= 0{1 0{1 0{1 0{1 predicted values of all the three mixing angles lie inside their respective 3 allowed ranges. The cases presented here correspond to Ge = Zge and G 2 = Z g 2 which the magnitude of the xed element is 1=p2 (p.f.e. denotes its position in UPMNS). For each HCP with fge; g g = fT U; Sg, for case, the upper and lower rows refer to NO and IO, respectively. 3:45{3:54 2:14{2:40 5:05{5:12 0{0:11 1:89{2 0{0:16 0:83{1 0{0:08 0:92{1 3:45{3:54 2:14{2:42 5:05{5:12 0{0:11 1:89{2 0{0:17 0:83{1 0{0:08 0:91{1 3:45{3:54 2:13{2:40 4:88{4:95 3:45{3:54 2:13{2:42 4:88{4:95 0{0:15 0:86{1 0{0:09 0:91{1 0{0:15 0:85{1 0{0:09 0:91{1 0{0:26 0:74{1 0{0:17 0:84{1 0{0:25 0:74{1 0{0:16 0:84{1 0{1 0{1 0{1 0{1 0{0:48 0:52{1 0{0:48 0:52{1 0{0:47 0:52{1 0{1 21= ( 3) A2 ( 3) B1 ( 2) B2 ( 2) B3 ( 3) B4 ( 3) 3:54 3:53 3:54 3:53 2:74 2:75 2:83 2:83 2:95 2:95 2:93 2:97 2:18 2:19 2:18 2:20 2:17 2:18 2:17 2:17 2:15 2:15 2:16 2:16 5:11 5:11 4:89 4:89 3:99 4:01 6:09 6:09 5:11 5:11 4:89 4:89 1:96 1:95 1:05 1:04 1:09 1:07 1:89 1:89 1:36 1:36 1:38 1:31 0:97 0:97 0:03 0:02 0:94 0:96 0:07 0:07 0:80 0:80 0:19 0:16 0:43 0:89 0:01 0:67 0:96 0:97 0:05 0:05 0:85 0:85 0:13 0:11 min 22:0 19:0 18:5 15:0 7:0 7:0 12:5 6:0 8:5 5:0 6:5 4:5 (p.f.e.) 10 1 10 2 2 2 min, for the viable cases, i.e., those cases for which the predicted values of all the three mixing angles lie inside their respective 3 allowed ranges. The cases presented here correspond to Ge = Zge 2 and G = Z HCP with fge; g g = fT U; Sg, for which the magnitude of the 1=p2 (p.f.e. denotes its position in UPMNS). For each case, the upper and lower rows refer to NO xed element is and IO, respectively. 4 Neutrinoless double beta decay As we have seen, in the class of models investigated in the present article the Dirac and Majorana CPV phases, and 21, 31, are (statistically) predicted to lie in speci c, in most cases relatively narrow, intervals and their values are strongly correlated. The only exception is case C1, in which the exact predictions 21 = 0 or These results make it possible to derive predictions for the absolute value of the neutrinoless double beta (( )0 -) decay e ective Majorana mass, hmi (see, e.g., refs. [1, 42{44]), as a function of the lightest neutrino mass. As is well known, information about jhmij is provided by the experiments on ( )0 -decay of even-even nuclei 48Ca, 76Ge, 82Se, 100M o, 116Cd, 130T e, 136Xe, 150N d, etc., (A; Z) ! (A; Z + 2) + e + e , in which the total lepton charge changes by two units, and through the observation of which the possible Majorana nature of massive neutrinos can be revealed. If the light neutrinos with de nite mass j are Majorana fermions, their exchange between two neutrons of the initial nucleus (A; Z) can trigger the process of ( )0 -decay. In this case the ( )0 -decay amplitude has the following general form (see, e.g., refs. [42{44]): A(( )0 ) = G2F hmi M (A; Z), with GF, hmi and M (A; Z) being respectively the Fermi constant, the ( )0 -decay e ective Majorana mass and the nuclear matrix element (NME) of the process. All the dependence of A(( )0 ) on the neutrino mixing parameters is contained in hmi. The current best (p.f.e.) 10 1 10 2 ( 2) C5 ( 2) D2 ( 1) D3 ( 1) D4 ( 2) D5 ( 2) 2:56 2:56 3:15 3:14 3:11 3:08 3:00 3:00 3:01 2:99 3:13 3:15 3:11 3:06 3:54 3:53 3:54 3:53 2:16 2:16 2:16 2:16 2:16 2:17 2:14 2:14 2:15 2:17 2:15 2:17 2:17 2:16 2:18 2:20 2:19 2:19 4:25 5:85 4:19 4:24 5:92 5:93 5:95 5:95 4:21 4:26 4:20 4:23 5:91 5:96 5:11 5:11 4:89 4:89 1:32 1:36 1:86 1:88 1:15 1:13 1:69 1:69 1:25 1:22 1:88 1:87 1:14 1:12 1:96 1:95 1:05 1:04 3 3 7 3 3 3 7 3 0:93 0:94 0:07 0:06 0:81 0:81 0:15 0:13 0:43 0:43 0:61 0:50 0:97 0:97 0:03 0:03 3 3 3 3 0:64 0:73 0:96 0:96 0:05 0:04 0:88 0:88 0:10 0:09 0:65 0:66 0:38 0:69 0:98 0:98 0:02 0:01 3 3 7 3 3 3 3 3 3 3 7 7 min 7:0 7:0 4:5 5:5 8:5 1:5 8:5 2:0 0:5 0:5 4:5 5:5 8:5 1:5 22:0 19:0 18:5 15:0 3 3 7 7 g 2 In this case the magnitude of the xed element is 1=2. HCP with fge; g g = fT U; U g. A1 A2 B1 B2 B3 B4 C1 C2 C3 C4 C5 D2 D3 D4 D5 3 2 NO IO NO IO 3 3 7 7 3 3 7 7 3 3 3 3 3 3 7 3 3 3 7 7 3 3 7 7 3 3 7 7 3 3 3 3 ranges of the three neutrino mixing angles for both types of the neutrino mass spectrum. experiments searching for ( )0 -decay of 136X e and 76Ge, respectively: limits on jhmij have been obtained by the KamLAND-Zen [54] and GERDA Phase II [55] jhmij < (0:061 0:165) eV [54] and jhmij < (0:15 0:33) eV [55] ; (4.1) both at 90% C.L., where the intervals re ect the estimated uncertainties in the relevant NMEs used to extract the limits on jhmij from the experimentally obtained lower bounds on the 136Xe and 76Ge ( )0 -decay half-lives (for a review of the limits on jhmij obtained )0 -decay experiments and a detailed discussion of the NME calculations for )0 -decay and their uncertainties see, e.g., [56]). It is important to note that a large number of experiments of a new generation aims at a sensitivity to jhmij (0:01 0:05) eV, which will allow to probe the whole range of the predictions for jhmij in the case of IO neutrino mass spectrum [57, 58] (see, e.g., [56, 59] for reviews of the currently running and future planned ( )0 -decay experiments and their prospective sensitivities). The predictions for jhmij (see, e.g., [38, 42{44]), jhmij = 3 i=1 X miUe2i = m1 cos2 12 cos2 13 + m2 sin2 12 cos2 13ei 21 + m3 sin2 13ei( 31 2 ) ; (4.2) m1;2;3 being the light Majorana neutrino masses, depend on the values of the Majorana phase 21 and on the Majorana-Dirac phase di erence ( 31 2 ). For the normal hierarchical (NH), inverted hierarchical (IH) and quasi-degenerate (QD), neutrino mass spectra jhmij is given by (see, e.g., [1, 60]): jhmij = jhmij = q q q m221 sin2 12 cos2 13 ei 21 + m231 sin2 13 ei( 31 2 ) (NH) ; m223 cos2 13 cos2 12 + sin2 12 ei 21 (IH); jhmij = m0 cos2 12 + sin2 12 ei 21 (QD) ; (4.3) (4.4) (4.5) and thus, m2 = (m21 + ( m231) 21 = 0:0506 eV. The IH spectrum corresponds to m3 where m0 = m1;2;3. We recall that the NH spectrum corresponds to m1 10 3 eV, m3 = (m23 + m1 = (m23 + m223 m221) 21 = ( m223 m221) 21 = 0:0497 eV, m2 = (m23 + m223) 21 = / sin2 13,15 and / m231(23)=m20. Clearly, the values of the phases ( 31 1 ( m223) 2 = 0:0504 eV. In the case of QD spectrum we have: m1 = m2 = m3 = m0, m2 j m231(23), m0 > 0:10 eV. In eqs. (4.3) and (4.4) we have assumed that the contributions respectively / m1 and / m3 are negligible, while in eq. (4.5) we have neglected corrections 21 2 ) and 21 determine the ranges of possible values of jhmij in the cases of NH and IH (QD) spectra, respectively. Using the 3 ranges of the allowed values of the neutrino oscillation m1 < m2, and therefore, m2 < 1m3, m231) 2 = parameters from table 1, we nd that: 10 3 eV < jhmij < 4:33 10 3 eV in the case of NH spectrum; i) 0:79 ii) p m223 cos2 13 cos 2 12 < jhmij < p 10 2 eV in the case of IH spectrum; m223 cos2 13, or 1:4 10 2 eV < jhmij < 5:1 leading term / (cos2 12 cos 2 12 0:29 at 3 . iii) m0 cos 2 12 < jhmij < m0, or 2:9 10 2 eV < jhmij < m0 eV, m0 > 0:10 eV, in the case of QD spectrum, where we have used the fact that at 3 C.L., cos 2 12 0:29. 15The term / sin2 13 gives a subleading contribution because even in the case of 21 = , when the sin2 12) has a minimal value, sin2 13 cos 2 12 since sin2 13 0:0242 while In what follows, we obtain predictions for jhmij using the phenomenologically viable neutrino mixing patterns found in subsection 3.4. In gures 13{16 we present jhmij as a function of the lightest neutrino mass mmin (mmin = m1 for the NO spectrum and mmin = m3 for the IO spectrum) in cases B1{B4, C1{C3, C4 and C5, and D2 and D3. The solid and dashed lines limit the found allowed regions of jhmij calculated using the predicted ranges for 12, 13, 21, ( 31 2 ). In the left panels we require the predicted values of sin2 12, sin2 13 and sin2 23 to lie in their corresponding experimentally allowed 3 intervals, while in the right panels we require them to be inside the corresponding 2 ranges. The mass squared di erences m231(23) in the case of NO (IO) spectrum are varied in their appropriate ranges given in table 1. The light-blue (light-red) areas in the left and right panels are obtained varying the neutrino oscillation parameters 12, 13, and 2 NO (IO) ranges, respectively, and varying the phases 21 and ( 31 2 ) in the interval [0; 2 ). The horizontal brown and grey bands indicate the current most stringent upper limits on jhmij, given in eq. (4.1), set by KamLAND-Zen and GERDA Phase II, respectively. The vertical grey line represents the prospective upper limit on mmin < 0:2 eV from the KATRIN experiment [61]. Several comments are in order. Firstly, for given values of (k1; k2) and a given ordering we nd jhmij to be inside of a band, which occupies a certain part of the allowed parameter space. Secondly, we note that most cases are compatible with both 3 and 2 ranges of all the mixing angles for both neutrino mass orderings (see table 7). There are several exceptions. Namely, cases B2, C3, C4 and D3, in which, due to the correlations imposed by the employed symmetry, the predictions for sin2 23 for the NO spectrum are not compatible with its 2 allowed range (see tables 3 and 4). Moreover, there is incompatibility for both orderings of cases B3 and B4 with the allowed 2 ranges of sin2 23 (see table 3), and of case C1 with the 2 range of sin2 12 (see table 4). Thirdly, the predictions for jhmij compatible with the 3 ranges of all the mixing angles are almost the same for the following pairs of cases: (B1, B2), (B3, B4), (C2, C3), (C4, C5) and (D2, D3). As discussed at the end of subsection 3.4, the cases in each pair share some qualitative features, in particular, the allowed ranges of 12, 13, 21 and ( 31 2 ) are approximately equal. We note also that case C1 stands out by having relatively narrow bands for jhmij due to the predicted values of 21 = k1 and ( 31 2 ) = k2 . Finally, the results shown in gures 13{16 and derived using the predictions for the CPV phases and the mixing angles 12 and 13 in the case when the predicted values of all the three mixing angles 12, 13 and 23 are compatible with their respective 3 experimentally allowed ranges, can be obtained analytically in the limiting cases of NH, IH and QD spectra using eqs. (4.3){(4.5), the values of m231(23) quoted in table 1 and the results on sin2 12, sin2 13, , 21 and tables 3 and 4. 5 Summary and conclusions In the present article we have derived predictions for the 3-neutrino (lepton) mixing and leptonic Dirac and Majorana CP violation in a class of models based on S4 lepton avour symmetry combined with a generalised CP (GCP) symmetry HCP, which are broken to 10-1 〈 | 10-3 100 10-1 〈 | 10-3 100 10-1 〈 | 10-3 10-140-4 GERDA-II 10-2 m min [eV] Cases B3, B4; 3σ GERDA-II KamLAND-Zen I I 10-1 100 10-140-4 10-3 10-2 10-1 100 10-1 〈 | 10-3 100 10-1 〈 | 10-3 GERDA-II KamLAND-Zen GERDA-II NO IO I I K K 10-140-4 10-3 10-1 100 10-140-4 10-3 10-2 10-140-4 10-3 10-2 10-1 100 mmin. The lines limit the allowed regions of jhmij calculated using the predictions for the relevant mixing angles and the CPV phases obtained in cases B1{B4 and compatible with the 3 (left panels) and 2 (right panels ) ranges of all the three mixing angles. The light-blue (light-red ) areas are obtained varying the neutrino oscillation parameters 12, 13, m231(23) for NO (IO) in their allowed 3 and 2 ranges in the left and right panels, respectively, and the phases 21 and ( 31 2 ) in the interval [0; 2 ). The horizontal brown and grey bands indicate the current upper bounds on jhmij quoted in eq. (4.1) set by KamLAND-Zen [54] and GERDA Phase II [55], respectively. The vertical grey line represents the prospective upper limit on mmin < 0:2 eV from KATRIN [61]. Cases B3 and B4 are compatible with the 3 ranges of the mixing angles, but not with their 2 ranges. 10-1 V 〈 | 10-3 100 10-1 〈 | 10-3 100 10-1 〈 | 10-3 GERDA-II KamLAND-Zen 10-2 m min [eV] Case C2; 3σ GERDA-II KamLAND-Zen 10-2 m min [eV] Case C3; 3σ GERDA-II KamLAND-Zen 10-140-4 10-3 100 10-1 〈 | 10-3 100 10-1 〈 | 10-3 10-1 100 NO IO 10-2 gure 13, but for cases C1{C3. Case C1 is compatible with the 3 ranges of the mixing angles, but not with their 2 ranges. GERDA-II 10-2 m min [eV] Case C5; 3σ GERDA-II 10-140-4 10-3 10-1 100 10-140-4 10-3 10-1 〈 | 10-3 100 10-1 〈 | 10-3 I GERDA-II KamLAND-Zen GERDA-II 10-1 〈 | 10-3 100 10-1 〈 | 10-3 10-2 HCP symmetries in the charged lepton and neutrino sectors, respecg = f1; g g and HCP = fX g, 1 being the unit element of S4. 2 The massive neutrinos are assumed to be Majorana particles with their masses generated by the neutrino Majorana mass term of the left-handed (LH) avour neutrino elds lL(x), l = e; ; . We show that in this class of models the three neutrino mixing angles, 12, 23 and 13, the Dirac and the two Majorana CP violation (CPV) phases, and 21, 31, are functions of altogether three parameters | two mixing angles and a phase, e , and e The S4 group has 9 di erent Z2 subgroups. Assuming that the LH avour neutrino Z g 2 and charged lepton elds, lL(x) and lL(x), l = e; ; , transform under a triplet irreducible unitary representation of S4, we prove that there are only 3 pairs of subgroups Z2ge and which can lead to di erent viable (i.e., compatible with the current data) predictions for the lepton mixing. For these three pairs, fge; g g = fS; T U g, fT U; Sg and fT U; U g, where S, T and U are the generators of S4 (see eq. (3.1)) taken here in the triplet repre GERDA-II KamLAND-Zen 10-2 m min [eV] Case D3; 3σ GERDA-II 10-140-4 10-3 10-1 100 10-140-4 10-3 10-1 100 Case D2; 2σ 10-1 〈 | 10-3 100 10-1 〈 | 10-3 I K HJEP12(07) GERDA-II 10-2 Case D3; 2σ GERDA-II 10-1 〈 | 10-3 100 10-1 〈 | 10-3 10-2 = XT ) are satis ed in the following cases: sentation of S4 (eq. (3.12)). In what concerns the residual GCP symmetry in the neutrino sector, HCP = fX g, we show that the constraints on X (following from the conditions of 2g and HCP and of having non-degenerate neutrino mass spectrum, i) for g = S, if HCP = f1; Sg, fU; SU g or fT ST 2U; T 2ST U g; ii) for g = U , if HCP = f1; U g or fS; SU g; iii) for g = T U , if HCP = fU; T g or fST S; T 2ST U g. HCP = fU; T g and HCP = fST S; T 2ST U g in the case of g However, HCP = fU; SU g and HCP = fT ST 2U; T 2ST U g in the case of g = T U , are shown to lead = S, and to the same predictions for the PMNS neutrino mixing matrix. Thus, we have found that e ectively there are 4 distinct groups of cases to be considered. We have analysed absolute value equal to 1=p2 them case by case and have classi ed all phenomenologically viable mixing patterns they lead to. In all four groups of cases the PMNS neutrino mixing matrix is predicted to and e contain one constant element which does not depend on the three basic parameters, e , . The magnitude of this element is equal to 1=p2 in the \Group A" cases of fGe; G g = fZ2T U ; Z2S HCPg with HCP = f1; Sg, and in the \Group B" cases of HCPg with HCP = fU; SU g; and it is equal to 1=2 in the \Group HCPg with HCP = f1; U g, and in the \Group D" cases HCPg with HCP = fS; SU g. In the approach to the neutrino avour and GCP symmetries employed by us, the PMNS matrix is determined up to permutations of columns and rows. This implies that theoretically any of the elements of the PMNS matrix can be equal by absolute value to 1=p2 in the Group A and Group B cases, and to 1=2 in the Group C and Group D cases. However, the data on the neutrino mixing angles and the Dirac phase imply that, taking into account the currently allowed 3 ranges of the PMNS matrix elements (see eqs. (3.10) and (3.11)), only 4 elements, namely, (UPMNS) 2, (UPMNS) 3, (UPMNS) 2 or (UPMNS) 3, can have an 0:707, and only 5 elements, namely, (UPMNS)e2, (UPMNS) 1, (UPMNS) 1, (UPMNS) 2 or (UPMNS) 2, can have an absolute value equal to 1=2. It should be added that i) j(UPMNS) 2j = 0:707 lies outside the respective currently allowed 3 range in the case of NO neutrino mass spectrum, ii) j(UPMNS) 2j = 0:707 is slightly outside the 3 allowed range for the IO spectrum, and that iii) the value of j(UPMNS) 2j = 1=2 is allowed at 3 only for the IO spectrum. We have derived predictions for the six parameters of the PMNS matrix, 12, 23 and 13, , 21 and 31, in the potentially viable cases of Groups A{D. This was done for both NO and IO neutrino mass spectra in the cases compatible at 3 with the existing data. We have performed also a statistical analysis of the predictions for the neutrino mixing angles and CPV phases for each of these cases. We have found that in certain cases the predicted values of the neutrino mixing angles are ruled out, or are strongly disfavoured, by the existing data (see subsection 3.4 for details). These are: i) in Group A, the cases of j(UPMNS) 3j = j(UPMNS) 3j = 1=p2 (strongly disfavoured); 1=p2 (strongly disfavoured), and ii) in Group D, the cases of j(UPMNS)e2j = 1=2 (ruled out), j(UPMNS) 2j = 1=2 (strongly disfavoured), and j(UPMNS) 2j = 1=2 (strongly disfavoured). The results of the statistical analysis in the viable cases are presented graphically in gures 1{11. The predicted ranges of the neutrino mixing parameters and the their corresponding best t values are summarised in tables 3{6. Given the di erence in the currently allowed 2 ranges of sin2 23 (see table 1), the prediction for the allowed values of sin2 23 in certain phenomenologically viable cases makes the IO (NO) spectrum statistically somewhat more favourable than the NO (IO) spectrum. At the same time, we have found that in a large number of viable cases the results we have obtained for the NO and IO spectra are very similar. As a consequence of the fact that, in the class of models we consider, the six PMNS matrix parameters, 12, 23, 13, , 21 and 31, are tted with the three basic parameters, HJEP12(07) and e, it is not surprising that we have found that there are strong correlations i) between the values of the Dirac phase and the values of the two Majorana phases 21 and 31, which in turn are correlated between themselves ( gures 1, 2, 6{9), and depending on the case ii) either between the values of 12 and 13 ( gure 5), or between the values of 23 and 13 ( gures 3 and 4) or else between the values of 12 and 23 ( gures 1, 2, 6{11). In certain cases our results showed strong correlations between the predicted values of 23 and the Dirac phase and/or the Majorana phases 21;31 ( gures 8{11). 1=2, and iii) Group D with j(UPMNS) 1j = 1=2, or j(UPMNS) 1j = 1=2, the cosine of the Dirac phase satis es a sum rule by which it is expressed in terms of the three neutrino mixing angles 12, 23 and 13. Taking into account the ranges and correlations of the predicted values of the three neutrino mixing angles, is predicted to lie in certain, in most of the discussed cases rather narrow, intervals (subsection 3.4). In the remaining viable cases of Groups B and C, cos was shown to satisfy sum rules which depend explicitly, in addition to 12, 23 and 13, on one of the three basic parameters of the class of models considered, e or . In these cases, as we have shown, cos can take any value. We have derived also predictions for the Majorana CPV phases viable cases of Groups B, C and D (subsection 3.4). With one exception | the case of j(UPMNS)e2j = 1=2 of Group C | the values of 21 and 31, as we have indicated earlier, are strongly correlated between themselves. In case C1 there is a strong linear correlation between 31 and . Using the predictions for the Dirac and Majorana CPV phases allowed us to derive predictions for the magnitude of the neutrinoless double beta decay e ective Majorana mass, jhmij, as a function of the lightest neutrino mass for all the viable cases belonging to Groups B, C and D. They are presented graphically in gures 13{16. All viable cases in the class of S4 models investigated in the present article have distinct predictions for the set of observables sin2 12, sin2 23, sin2 13, the Dirac phase and the absolute value of one element of the PMNS neutrino mixing matrix. Using future more precise data on sin2 12, sin2 23, sin2 13 and the Dirac phase , which will allow also to determine the absolute values of the elements of the PMNS matrix with a better precision, will make it possible to test and discriminate between the predictions of all the cases found by us to be compatible with the current data on the neutrino mixing parameters. Future data will show whether Nature followed the S4 oHCP avour + GCP symmetry \three-parameter path" for xing the values of the three neutrino mixing angles and of the Dirac (and Majorana) CP violation phases of the PMNS neutrino mixing matrix. We are looking forward to these data. Acknowledgments We would like to thank F. Capozzi, E. Lisi, A. Marrone, D. Montanino and A. Palazzo for kindly sharing with us the data les for one-dimensional 2 projections. This work was supported in part by the INFN program on Theoretical Astroparticle Physics (TASP), HJEP12(07) by the research grant 2012CPPYP7 under the program PRIN 2012 funded by the Italian Ministry of Education, University and Research (MIUR), by the European Union Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grants 674896 and 690575, and by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan (S.T.P.). A Symmetry of X If the neutrino sector respects a residual GCP symmetry HCP = fX g, the neutrino mass matrix satis es eq. (2.12), namely, (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) where Explicitly, = diag(ei 1 ; ei 2 ; ei 3 ) and the Uij (#ij ; ij ) are complex rotations in the i-j plane. X~ = U23(#23; 23) U13(#13; 13) U12(#12; 12) ; 0 sin #23 ei 23 sin #23 e i 23 C ; ei( 2 23) m2 cos #13 sin #23 = e i( 2 23) m3 cos #13 sin #23 ; ei( 1 12) m1 cos #13 sin #12 = e i( 1 12) m2 cos #13 sin #12 : XT M X = M : d X~ = X~ d ; The GCP transformation matrices X must be unitary due to the GCP invariance of the neutrino kinetic term. In what follows we show that these matrices are additionally constrained to be symmetric if the neutrino mass spectrum is non-degenerate, as is known to be the case. Expressing M from eq. (2.14) and substituting it in eq. (A.1) yields where d diag(m1; m2; m3) and X~ U y X U is unitary. as follows: Being 3 3 unitary, X~ can be parametrised as the product of three complex rotations with a straightforward generalisation to (ij) = (12); (13). Imposing eq. (A.2) produces the following relations: From the non-degeneracy of the neutrino mass spectrum it follows that sin #13 = sin #23 = sin #12 = 0. Thus, X~ is constrained to be diagonal and hence symmetric, X~ T = X~ . This nally implies that also XT = X , i.e., a phenomenologically relevant X must be symmetric. As detailed in subsection 2.2, residual avour symmetries Z2ge and Z2g which are conjugate toeachotherleadtothesameformofthePMNSmatrix. For Gf = S4, thereareninegroup elements of order two, given in eqs. (3.2) and (3.3), which generate Z2 subgroups. The resulting 81 pairs of elements fge;g g can themselves be partitioned, under the conjugacy relation of eq. (2.25), into the following nine equivalence classes: fS;Sg, fTST2;TST2g, fT2ST;T2STg; fT2ST;Sg, fTST2;Sg, fT2ST;TST2g, fS;T2STg, fS;TST2g, fTST2;T2STg; fS;Ug, fS;SUg, fTST2;T2Ug, fT2ST;TUg, fTST2;ST2SUg, fT2ST;STSUg; fU;Sg, fSU;Sg, fT2U;TST2g, fTU;T2STg, fST2SU;TST2g, fSTSU;T2STg; fSU;Ug, fU;SUg, fST2SU;T2Ug, fSTSU;TUg, fT2U;ST2SUg, fTU;STSUg; fTST2;Ug, fT2ST;T2Ug, fTST2;SUg, fT2ST;ST2SUg, fTST2;STSUg; fS;TUg, fS;STSUg, fS;T2Ug, fTST2;TUg, fS;ST2SUg, fT2ST;Ug, fT2ST;SUg, fU;TST2g, fT2U;T2STg, fSU;TST2g, fST2SU;T2STg, fSTSU;TST2g; fTU;Sg, fSTSU;Sg, fT2U;Sg, fTU;TST2g, fST2SU;Sg, fU;T2STg, fSU;T2STg, fSTSU;T2Ug, fSU;TUg, fST2SU;SUg, fT2U;SUg, fU;STSUg; fTU;Ug, fSTSU;Ug, fSTSU;SUg, fTU;SUg, fT2U;Ug, fTU;T2Ug, fST2SU;Ug, fU;TUg, fTU;ST2SUg, fSU;STSUg, fU;T2Ug, fT2U;TUg, fU;ST2SUg, fSU;T2Ug, fST2SU;STSUg, fST2SU;TUg, fSTSU;ST2SUg, where in boldface we have identi ed a representative pair of elements for each class, matching the choice made in eqs. (3.4) and (3.5). C Equivalent cases value of one element is 1=p2. For Pe = Pe0 and P UP0MNS would be equivalent, if the products ye following way: A necessary condition for two matrices UPMNS and UP0MNS to be equivalent is the same magnitude of the xed element. Indeed, in the four cases under consideration the absolute = P0, the two matrices UPMNS and and 0ey 0 could be related in the y = diag(ei 1;ei 2;ei 3)U23( e; e) 0ey 0 R23( ) diag(1;ik;ik); e (C.1) with i, e and e, being xed phases and angles, respectively, and k is allowed to be 0;1;2 or 3. Indeed, if this relation holds, from eq. (2.24) we have UPMNS = PeU23( e; e)diag(ei 1;ei 2;ei 3)U23( e; e) 0ey 0R23( )diag(1;ik;ik)R23( )P Q = Pediag(ei 1;ei 2;ei 3)U23( e;~e)U23( e; e) 0ey 0R23(^ )P Q^ ; (C.2) HJEP12(07) ge g HCP S T U fU; T g S T U fST S; T 2ST U g T U S fT ST 2U; T 2ST U g arctan arctan 3 3 + 2 6 arccot 5=p3 1 + 2p2=3 p arctan q arctan 3 p 6 2 =7 p 2 2 =6 q p arctan p arccot 2p2 + p arctan p =4 1 arctan 6 =13 =2 arccot (2) arctan (2) p 5 3 arctan 2=p5 =4 3 e e k with Now, using we obtain with the same. , , e , e , and k for which eq. (C.1), proving the equivalence of the PMNS matrix in a given case to the PMNS matrix in the reference case of ~e = e + 2 3 ; ^ = and Q ^ = P T diag(1; ik; ik) P Q : U23( e; ~e) U23( e; e) = diag(1; ei ; e i ) U23(^e; ^e) ; where (see appendix B in [7]) = arg ncos e cos e sin e sin e ei( e ~e)o ; = arg nsin e cos e e i~e +cos e sin e e i e o ; cos ^e = cos e cos e sin e sin e ei( e ~e) ; sin ^e = sin e cos e e i~e +cos e sin e e i e and ^e = UPMNS = Qe Pe U23(^e; ^e) 0ey 0 R23(^ ) P Q^ ; Qe = Pe diag ei 1 ; ei( 2+ ); ei( 3 ) PeT being the matrix of unphysical phases. Thus, up to this matrix, UPMNS and U P0MNS are Taking fGe; G g = fZ2T U ; Z2S HCPg with HCP = fU; SU g as a reference case and denoting the corresponding diagonalising matrices as 0e and 0 , we nd the values of i , and k for which eq. (C.1) holds, if e and are the diagonalising matrices in one of the three remaining cases under consideration. We summarise these values in table 8. D Correspondence with earlier results The sum rules for cos or sin2 23 (sin2 12 in case C1) can formally be obtained from the corresponding sum rules derived in [7]. In certain cases, this requires an additional input which is provided by the residual GCP symmetry HCP considered in the present article. (C.3) (C.5) (C.6) Below we provide the correspondence between the phenomenologically viable cases of the present study and the cases considered in [7]. i) Cases B1, C4 and D4 of the present study correspond to case C8 in [7], since for all these cases (UPMNS) 2 is xed. The sum rule for cos in case B1, eq. (3.46), follows from that of case C8 in [7] (see table 4 therein) for sin2 23 = 1=2, while the sum rule fZ2ge ; Z2 g avour symmetries (see table 10 in [7]). g in eq. (3.57), valid in cases C4 and D4, can be obtained from the same sum rule found in [7], but for sin2 23 = 3=4. As should be, these two values of sin2 23 follow from Gf = S4, when it is broken to two di erent non-equivalent speci c pairs of residual ii) Cases B2, C5 and D5 correspond to case C1 in [7], since for all of them (UPMNS) 2 is xed. The sum rule for cos in case B2, eq. (3.47), follows from that of case C1 in [7] (see table 4 therein) for sin2 23 = 1=2, while the sum rule in eq. (3.58), valid in cases C5 and D5, can be obtained from the same sum rule found in [7], but for 23 = 1=4. Again, these values of sin2 23 are xed uniquely by Gf = S4 and the speci c choice of the residual symmetries considered in the present article.16 iii) Cases A1 and B3 of the present study correspond to case C2 in [7], since for these cases (UPMNS) 3 is xed. The expression for sin2 23 in eq. (3.35) follows from the corresponding expression for case C2 in table 6 of [7] with sin2 23 = 1=2. This value is in agreement with table 10 of [7]. Moreover, the sum rule for cos in eq. (3.37) in case A1 can be obtained from the sum rule for case C217 in table 4 of [7] with 23 = 1=2 and sin2 ^ 12 = 1=2. The value of sin2 ^12, which was an arbitrary free parameter in [7], is xed by the GCP symmetry employed in the present study. Finally, we note that the expression for cos in eq. (3.49) valid in case B3 can formally be obtained from the corresponding expression in case C2 of table 4 in [7] setting 12 = =4. iv) Analogously, cases A2 and B4 correspond to case C7 in [7]. Equation (3.38) can be obtained from the corresponding formula in table 6 of [7] for sin2 agrees with the result in table 10 therein. The sum rule in eq. (3.40) follows from that in case C7 in table 4 of [7] with sin2 the value of sin2 ^12, which in [7] is a free parameter, here is 23 = 1=2 and sin2 ^ 12 = 1=2, where again xed by the GCP symmetry. Similarly to the previous clause, eq. (3.51) can formally be derived from the corresponding expression in case C7 of table 4 in [7] setting ^12 = =4. 23 = 1=2, which v) Case C1 corresponds to case C5 in [7], in which all possible residual avour symmetries Ge = Z2 and G = Z2 have been considered. The expression for sin2 12 in eq. (3.53) follows from that of case C5 in table 6 in [7] with sin2 12 = 1=4. This value of sin2 16Note that the value of sin2 23 = 1=2 is not present in table 10 of [7], since in this reference the best t values of the mixing angles for the NO spectrum quoted in eqs. (6){(8) therein have been used, and employing them, one obtains cos 2:76. 17We would like to point out a typo in eq. (85) in [7]: cos2 23 should read cos 23. This typo, however, does not a ect the corresponding sum rule for cos in eq. (86) and in table 4 of [7]. is found for Gf = S4 and the speci c choice of the residual symmetries (see table 10 in [7]). Moreover, eq. 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J.T. Penedo, S.T. Petcov, A.V. Titov. Neutrino mixing and leptonic CP violation from S 4 flavour and generalised CP symmetries, Journal of High Energy Physics, 2017, 22, DOI: 10.1007/JHEP12(2017)022