1 Introduction

hep-ph/***

## 1 Introduction

By now all three lepton mixing angles have been measured with a certain degree of precision [1] (see also [2, 3])

 sin2θ13=0.0218(9)+0.0010(1)−0.0010,sin2θ12=0.304+0.013−0.012,sin2θ23={0.452+0.052−0.028(0.579+0.025−0.037) (1)

for a normal ordering (NO) and in brackets for an inverted ordering (IO) of the neutrino masses, respectively. Assuming that neutrinos are Majorana particles and none of them to be massless, the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix that encodes lepton mixing does not only contain the three mixing angles , and , but also three phases: the Dirac phase and the two Majorana phases and . The former can be measured in neutrino oscillation experiments, while a linear combination of the latter can be accessible in neutrinoless double beta decay experiments (for a review on leptonic CP violation see [4]). No direct signals of CP violation have been observed in the lepton sector and recent global fits only show a weak indication (below the significance) of a non-trivial value of the Dirac phase [1].

Many approaches have been pursued in order to describe the data on lepton mixing. A particularly promising Ansatz assumes the existence of a flavor symmetry , usually finite, non-abelian and discrete, that is broken to different residual groups and in the charged lepton and neutrino sectors, respectively (for reviews see [5, 6]). In (the most predictive version of) such a framework all three mixing angles together with the Dirac phase can be fixed by the symmetries of the theory , and , if the three generations of left-handed (LH) leptons are assigned to an irreducible three-dimensional representation 3 of the flavor group , i.e. to a representation that cannot be decomposed further in .111Since lepton mixing like quark mixing only regards LH fields, we do not need to specify in this approach how right-handed (RH) charged leptons, and possibly RH neutrinos, transform under , unless we construct an explicit model in which this approach is realized. We note that this approach does not constrain lepton masses and thus all statements made on the predictive power regarding lepton mixing angles and the Dirac phase are valid up to possible permutations of rows and columns of the PMNS mixing matrix. For neutrinos being Majorana particles surveys of finite, non-abelian and discrete subgroups of and , see [7] and references therein, have shown that symmetries giving rise to mixing angles which are in good agreement with experimental data in general lead to trimaximal (TM) mixing [8] (and thus ) and a trivial Dirac phase .

An extension of this approach that involves also CP as symmetry has been proposed in [9] (see also [10, 11] as well as [12]). In this case, also the CP symmetry acts in general in a non-trivial way on the flavor space [13] and conditions have to be fulfilled in order for the theory to be consistent. The residual groups and are chosen as follows: is an abelian subgroup of with three or more elements and is the direct product of a group contained in and the CP symmetry. Thus, the form of these groups is similar to the one in the approach without CP. The advantages of the extension with CP are threefold: Majorana phases are also predicted, the Dirac phase does not need to be trivial, if the lepton mixing angles are accommodated well, and the mixing pattern contains one free parameter . The latter allows for a richer structure of patterns that are in good agreement with the experimental data. In particular, mixing does not need to be TM. At the same time, being governed by only one free parameter all mixing parameters are strongly correlated. Such correlations can be testable and/or distinguishable at future facilities [14]. The value(s) of this free parameter that admit(s) a reasonable agreement with the experimental data is (are) not fixed by the approach itself, but has (have) to be achieved in a concrete model (see [15] for several successful models with different symmetries and CP). This approach has already been studied for a variety of flavor symmetries: and [9, 15], [16], [17] as well as and with general [18].222Note that a variant of this approach has been considered for in which the residual symmetry in the neutrino sector is a Klein group contained in the flavor group and a CP symmetry [19].

Here we would like to consider as flavor group. This group has already been employed as flavor symmetry [20, 21, 22, 23]. In particular, it has been shown to give rise to the so-called “golden ratio” (GR) mixing pattern, , and with so that .333Different versions of the GR mixing pattern are known in the literature that lead to different predictions for the solar mixing angle in terms of the golden ratio [22, 24, 25]. These are based on different flavor symmetries. Very recently, predictions of CP phases have been discussed in a scenario with as flavor group and a CP symmetry [26]. Since the authors assume a Klein group and a CP symmetry to be preserved in the neutrino sector, the mixing angles are fixed to the values of the GR mixing pattern, while possible values of the two Majorana phases depend on the CP transformation that is preserved. The latter is not constrained to correspond to an automorphism of the flavor group and thus results obtained in [26] differ from ours. In [27] an Ansatz has been pursued in which two CP symmetries are present as residual symmetries in the neutrino sector. The combination of these two leads in general to a symmetry acting on the flavor space only. Under certain conditions this can be a transformation belonging to a finite, non-abelian and discrete group. Therefore, if the latter is the alternating group results of our study can also be achieved using the Ansatz with two CP transformations in the neutrino sector.

In the present paper we analyze the scenario with the flavor symmetry and a CP symmetry comprehensively, since we consider all CP symmetries that correspond to involutive ‘class-inverting’ automorphisms of , all possibilities for , i.e. , as well as , and all possible subgroups of as residual flavor symmetry in the neutrino sector. All these combinations of symmetries together with all possible permutations of the rows and columns of the PMNS mixing matrix are subject to an analytical and a numerical study. In particular, we perform a analysis using the results of the mixing angles from the global fit [1]. As outcome we only find four patterns that admit a reasonable agreement with the experimental data, i.e. at the level or better, for a particular choice of the parameter . Two of these four patterns predict a maximal Dirac phase together with maximal atmospheric mixing, while the other two ones lead to a trivial Dirac phase and in general non-maximal . Majorana phases are trivial for all four patterns. Thus, two out of these four lead to no CP violation. This fact can be traced back to the existence of an accidental CP symmetry, common to the charged lepton and neutrino sectors, as we discuss. As regards the reactor and the solar mixing angles, we note that is in general accommodated well, whereas is subject to non-trivial constraints in all four cases: two of the four patterns give rise to a lower bound , the value of the GR mixing pattern, one incorporates TM mixing and thus and the remaining one entails an upper limit , a value that is associated with a different version of the GR mixing pattern [24].

The paper is organized as follows: in section 2 we recapitulate the approach with a flavor and a CP symmetry as well as the main features of the group . We also discuss the admitted CP transformations and relegate further details regarding their relation to the automorphisms of and their nature to appendix A. Section 3 contains the analytical study of all patterns that can lead to a good agreement with the experimental data as well as the results of our analysis. We summarize our main results in section 4. Besides appendix A we include appendix B that contains our definitions of mixing angles, CP phases and corresponding CP invariants , and .

## 2 Approach

We briefly recapitulate the essential ingredients of the approach [9] and summarize the necessary information on the group . We list the candidates of generators of residual flavor symmetries in the charged lepton and neutrino sectors as well as the CP transformations. At the end of this section we also comment on the possible presence of an accidental CP symmetry common to charged leptons and neutrinos. Since we focus on the case in which the generations of LH leptons are assigned to an irreducible three-dimensional representation of the flavor group, the CP transformation as well as the elements of the flavor symmetry are represented by (unitary, complex) three-by-three matrices in the following.

Let us consider a theory with a flavor group combined with a CP symmetry that in general also acts non-trivially on the flavor space [13, 12]. The CP transformation is a unitary and symmetric matrix

 XX†=XX⋆=1. (2)

The requirement that , the CP transformation associated with the CP symmetry preserved in the neutrino sector and subject to the condition in (4), must be a symmetric matrix has been shown in [9] to be necessary, since otherwise the neutrino mass spectrum would be partially degenerate and, consequently, inconsistent with experimental observations [1]. In order to ensure a consistent combination of the flavor and CP symmetry we require that the subsequent action of the CP transformation, an element of the flavor group and the CP transformation is equivalent to the action of an (in general different) element of the flavor group

 (X−1AX)⋆=A′, (3)

with and representing (different) elements of .444We use throughout this paper lowercase letters for the abstract elements of the flavor group and capital letters for the matrix representatives (in the representation ). As shown in [12, 10, 11], a CP transformation corresponds to an automorphism of the flavor group. In particular, our request that fulfills (2) renders this automorphism involutive. Following the discussion in [11] this automorphism should be class-inverting, i.e. the image of the element under the automorphism has to lie in the same class as the inverse of . This is guaranteed for the three-dimensional representation by the fulfillment of the condition in (3). As we show in appendix A the automorphisms corresponding to the CP transformations we consider in our analysis are also class-inverting when acting on the other representations of the flavor group .

The residual symmetry in the neutrino sector is assumed to be the direct product of a symmetry contained in the flavor group and the CP symmetry. Thus, the matrix representing the generator of the former symmetry and the CP transformation have to fulfill

 XZ⋆−ZX=0, (4)

which is a particular case of the condition in (3). We note that the presence of the residual symmetries given by and implies the existence of a second CP transformation in the neutrino sector that fulfills the same conditions in (2)-(4) as the CP transformation [9].

In the charged lepton sector, in contrast, we take as residual group an abelian subgroup of the flavor symmetry that offers the possibility to distinguish among the three generations of charged leptons, i.e. this group has to have at least three different elements. The residual group can be described with a set of generators , that commute.

The derivation of lepton mixing in this scenario has already been presented in detail in [9] and we only mention it briefly. In the charged lepton sector, the residual group generated by constrains the charged lepton mass matrix , here given in the right-left basis,

 Q†im†lmlQi=m†lml. (5)

The matrices are diagonalizable by the unitary matrix , i.e. is diagonal, which is determined up to the ordering of its columns and possible overall phases of the single columns. As a consequence, also the combination

 U†em†lmlUe is diagonal. (6)

Thus, diagonalizes the charged lepton mass matrix as regards LH charged leptons. In the neutrino sector, the light neutrino mass matrix is subject to the following conditions555We do not need to specify the generation mechanism of neutrino masses unless we construct an explicit model. So, this mass matrix can arise from integrating out heavy RH neutrinos, from Higgs triplets acquiring a vacuum expectation value, etc..

 ZTmνZ=mνandXmνX=m⋆ν, (7)

if the and CP symmetry are imposed. Without loss of generality we can choose a basis such that

 X=ΩΩTandΩ†ZΩ=% diag((−1)z1,(−1)z2,(−1)z3), (8)

with being unitary and . Since generates a symmetry, two of the three parameters have to coincide. The matrix combination is then real and block-diagonal and thus can be diagonalized by a rotation in the -plane through an angle that is determined by the matrix entries of . Their actual values are in general not predicted in this approach and thus is taken to be a free parameter in the interval between and in the following. The -plane is fixed by the degenerate sub-sector of , i.e. the two and that are equal. The positiveness of the light neutrino masses (a vanishing neutrino mass can also be included) is ensured by the diagonal matrix with entries and on its diagonal. So, the contribution to lepton mixing from the neutrino sector is given by

 Uν=ΩRij(θ)Kν, (9)

and the PMNS mixing matrix resulting from this approach reads

 UPMNS=U†eΩRij(θ)Kν. (10)

It is important to note that this mixing matrix is only determined up to permutations of its rows and columns (and unphysical phases), since this approach does not make any predictions concerning the mass spectrum of charged leptons and neutrinos. For example, if not embedded in a model context, see e.g. first reference in [15], one cannot predict whether neutrinos follow NO or IO. It is also worth to emphasize that a mixing matrix of the form in (10) has one column that is determined by group theory only and that does not depend on the free parameter .

We note also that two tuples and lead to the same physical results, if the generators of the symmetries are related by a similarity transformation

 ~Ω†Q′~Ω=Q,~Ω†Z′~Ω=Zand~Ω†X′~Ω⋆=X. (11)

The alternating group describes the even permutations of five distinct objects. It is isomorphic to the icosahedral rotation group . It has 60 different elements, organized in five conjugacy classes, and, thus, possesses five irreducible representations: , , , and . All representations apart from the singlet are faithful.666A representation is called faithful, if all elements of the group are represented by different matrices in this representation. So, in the case of the 60 group elements are represented by 60 different matrices in the representations , , and . The group can be generated with two generators and that fulfill the relations

 s2=e,t5=eand(st)3=e, (12)

with denoting the neutral element of . Since we would like to assign LH leptons to triplets, we are particularly interested in the representations and . The explicit form of the generators and in the representation can be chosen as [21]

 S=1√5⎛⎜ ⎜⎝1√2√2√2−ϕ1/ϕ√21/ϕ−ϕ⎞⎟ ⎟⎠andT=⎛⎜⎝1000eiΦ000e4iΦ⎞⎟⎠, (13)

with

 ϕ=12(1+√5)andΦ=2π5. (14)

The generators in are easily obtained from and in (13) by using the combination and as generators, see also [22]. This shows immediately that the set of all matrices describing the representations and is the same and thus all conclusions obtained in a comprehensive study of mixing using the representation also hold for . Consequently, it is irrelevant for our analysis whether LH leptons are in or of and, without loss of generality, we assume in the following that LH leptons transform as of .

The group has several subgroups. In particular, the group contains 15 elements that generate a symmetry which give rise to five distinct Klein groups. These are, like the generating elements themselves, all conjugate to each other. Since we make explicit use of these elements we mention them here

 v1=s, v2=st2st3st2, v3=t2st3st2, v4=t4st, v5=st3st2s, v6=t2st3sts, v7=tst4, v8=st2st3s, v9=stst3st2, v10=st2st, v11=t2st3, v12=tst3st2s, v13=tst2s, v14=t3st2, v15=st2st3st.

They form the Klein groups

 K1={v1,v2,v3,e},K2={v4,v5,v6,e} K3={v7,v8,v9,e},K4={v10,v11,v12,e}andK5={v13,v14,v15,e}, (15)

see also [22]. Furthermore, there are ten and six subgroups. Also these are all conjugate to each other. For the complete list of generating elements of these subgroups see appendix B in [22]. So, three different types of groups, , and , can function as residual symmetry in the charged lepton sector.

The form of the CP transformations we consider is

 X=VX0 (16)

with

 X0=⎛⎜⎝100001010⎞⎟⎠ (17)

and is the matrix representative of a generating element, i.e. , or . So, we find in total 16 different possible CP transformations. All of them fulfill (2). Notice that the CP transformation gives rise to the so-called - reflection symmetry [28] whose phenomenological consequences have been studied in detail in the literature.

As one can check, it holds for

 (X−10SX0)⋆=Sand(X−10TX0)⋆=T. (18)

Thus, for the generators and in the representation the condition in (3) is valid for . As a consequence, the corresponding automorphism is the trivial one

 s→sandt→t. (19)

The CP transformations correspond to different inner automorphisms of the group (i.e. automorphisms whose action on the elements of the group can be represented by a similarity transformation with a(nother) group element) that map the generators and in the following way

 s→vsv−1andt→vtv−1. (20)

The automorphism group of is the symmetric group and the group of inner automorphisms is isomorphic to itself. As we show in appendix A the 16 different CP transformations we consider correspond to the 16 class-inverting involutive automorphisms of . We thus discuss all CP transformations that can be consistently imposed according to [11] and that fulfill the requirement in (2).

A last condition that needs to be examined is the constraint in (4), namely whether the generator commutes with the chosen CP transformation. We find that for each four different possible CP transformations are admitted: for being one of the non-trivial elements of the Klein group , , the CP transformation with belonging to the same Klein group (this time is included) is a viable choice. Taking into account that the generator and the CP transformation automatically imply the existence of a further CP transformation , , we can reduce the number of independent choices of CP transformations for each generator to two.

Eventually, we mention that it can happen that an accidental CP symmetry , common to the charged lepton and the neutrino sectors, exists that leads to trivial CP phases. This can be checked by searching for a transformation that fulfills the following constraints

 Y⋆m†lmlY=(m†lml)⋆% andYmνY=m⋆ν. (21)

These are equivalent to the conditions involving the generators of the different symmetries

 QiY−YQTi=0, (22) ZY−YZ⋆=0andXY⋆−YX⋆=0, (23)

with being a diagonal and real matrix in the neutrino mass basis, i.e. is diagonal and real. For details see [9].

## 3 Lepton mixing

In order to study mixing comprehensively, we analyze all possible combinations of residual symmetries in the charged lepton and neutrino sectors, i.e. all possible and ( generators and CP transformations ). These can be expressed as tuples of generators of and and for . As mentioned in section 2, can be either a , or a Klein group, while one of the 15 different symmetries that each can be consistently combined with four different CP transformations specify the residual group . Instead of computing the mixing pattern for all these we can greatly reduce the number of cases that we need to study by applying similarity transformations as well as the fact that for a pair and also the pair and leads to the same mixing pattern. We allow for all possible permutations of rows and columns of the mixing matrix. Furthermore, we consider both possible neutrino mass orderings NO or IO in our (numerical) analysis. We exclude all patterns that cannot accommodate the experimental data on lepton mixing angles at the level or better for (a) certain choice(s) of the free parameter . As a consequence, we end up with in total only four cases. These we call in the following Case I to III and Case IV-P1/Case IV-P2. We first study the mixing patterns analytically for each possible residual symmetry in the charged lepton sector and then show the results of a analysis of these patterns, since they can accommodate the experimental data best. We briefly comment on a fifth mixing pattern that can fit the data also well apart from the solar mixing angle whose value turns out to be slightly smaller than the lower bound on [1].

### 3.1 Ge=Z5: Case I and Case II

If we consider as a symmetry, we find six different categories of tuples with being a generator of a group, taking into account the mentioned operations in order to relate different tuples . We can show that for each of these categories we can find a representative tuple with . Thus, the mixing matrix resulting from the charged lepton sector is the unit matrix, up to permutations of its columns and unphysical phases. Out of these six representative tuples four lead to a mixing pattern that we dismiss, because the column that is fixed by group theory is not compatible with the data at the level or better [1]. The two remaining representatives, which we can choose as

 (Q,Z,X)=(T,T2ST3ST2,SX0)(Case I) (24)

and

 (Q,Z,X)=(T,ST2ST,X0)(Case II), (25)

give rise to a mixing matrix with a column whose components have the absolute values

 ⎛⎜ ⎜⎝sinφcosφ/√2cosφ/√2⎞⎟ ⎟⎠≈⎛⎜⎝0.5260.6020.602⎞⎟⎠, (26)

where we defined for convenience

 tanφ=1/ϕ. (27)

This column can only be identified with the second one of the PMNS mixing matrix, if good agreement with the experimental data should be achieved. Notice that the ordering of the components in this column as well as its position within the PMNS mixing matrix are not fixed by the approach we are using, since no constraints on the lepton masses are imposed.

We first derive the mixing pattern for the tuple in (24). As explained and we can take to be

 Ω\footnotesizeI=1√2⎛⎜⎝√2cosφ−√2isinφ0sinφicosφ−1sinφicosφ1⎞⎟⎠, (28)

that fulfills (8) for and chosen as in (24). Since and of the (diagonal) combination are equal, see (8) for definition, the correct indices of the rotation matrix in (9) are . Thus, the PMNS mixing matrix reads

 UPMNS=Ω\footnotesizeIR13(θ)Kν. (29)

We can extract the mixing angles from (29) in the usual way and find

 sin2θ12=22+(3+√5)cos2θ,sin2θ13=110(5+√5)sin2θ, sin2θ23=12−√2(5+√5)sin2θ7+√5+(3+√5)cos2θ(Case I). (30)

Furthermore, we can derive the following exact sum rule among the solar and the reactor mixing angles

 sin2θ12=sin2φ1−sin2θ13≈0.2761−sin2θ13≳0.276. (31)

This sum rule can also be directly obtained from . Using for its best fit value we find for the solar mixing angle which is within the range, see (1). This value coincides with the one obtained from a analysis, see table 1. The non-trivial lower bound on the solar mixing angle, see (31), is also nicely seen in figure 1 (and implicitly also in the - plane in figure 2). For the atmospheric mixing angle a simple approximate relation to the reactor mixing angle is given by

 (32)

if we use again the best fit value . The first subleading term arises at and can thus be safely neglected in this estimate. We can clearly see having a look at the different symbols in figures 1 and 2 that values of lead to , while larger values, , entail . This is also confirmed by the two different ‘best fitting’ values and that are obtained for NO and IO, respectively, see table 1.

Note that the formulae for the mixing angles and remain invariant, if we replace by , while the relative sign in the expression for changes. The same effect can be achieved, if we consider the PMNS mixing matrix in (29) with second and third rows exchanged.

As one can check, all CP invariants , and vanish exactly and thus an accidental CP symmetry must be present. Indeed, there is one, namely

 Y=⎛⎜⎝100010001⎞⎟⎠, (33)

that fulfills the conditions in (21)-(23) for the tuple shown in (24). The vanishing of can also be confirmed by verifying that the condition presented in [27] is fulfilled.

In a similar manner we can study the lepton mixing that can be obtained for the second tuple , the one in (25). The form of the matrix that fulfills (8) for and can be chosen as

 Ω\footnotesizeII=1√2⎛⎜ ⎜⎝−√2cosφ−√2sinφ0−e−3iΦsinφe−3iΦcosφ−e−7iΦ/4−e−2iΦsinφe−2iΦcosφe−3iΦ/4⎞⎟ ⎟⎠. (34)

Like in the preceding case, also here the necessary rotation is in the (13)-plane. Thus, taking into account that is trivial, the PMNS mixing matrix is of the form (up to permutations of rows and columns and unphysical phases)

 UPMNS=Ω\footnotesizeIIR13(θ)Kν. (35)

The predictions for the solar and the reactor mixing angles are the same as for the first tuple, namely

 sin2θ12=22+(3+√5)cos2θ,sin2θ13=110(5+√5)sin2θ(Case II). (36)

Thus, also the sum rule and the estimate given in and below (31) hold. In figure 1 the non-trivial lower bound on is visible also for Case II. The results for the atmospheric mixing angle and the Dirac phase instead are different, since both of them are predicted to be maximal

 sin2θ23=12and|sinδ|=1%(CaseII). (37)

The sign of depends on whether . The corresponding Jarlskog invariant reads777For the Jarlskog invariant vanishes. If this happens, one of the mixing angles becomes either or and thus the Dirac phase becomes unphysical. Clearly, these values of are highly disfavored by experimental data.

 JCP=−120√2√5+√5sin2θ. (38)

Like in the first case also in this case the Majorana phases are trivial. We can check that the conditions found in [27] for and are fulfilled in the case at hand. Furthermore, we can verify with the help of the formulae given in [27] that both Majorana phases must be trivial.

We notice that the replacement of the parameter by does not change the form of the mixing angles, while the sign of and, consequently, of changes. The very same result is achieved, if we exchange the second and third rows of the mixing matrix in (35). Consequently, we expect to find two best fitting values of the parameter . This is confirmed by the results of the analysis in table 1. As expected, the sum of these two best fitting values equals .

One of the categories initially dismissed by the criterion that the absolute values of the group theoretically fixed column of the PMNS mixing matrix should agree at the level or better with the values given in [1] might still be interesting in a concrete model in which (small) corrections can lead to the agreement with experimental data. A representative of this case is the tuple . The absolute values of the elements of the fixed column are . Thus, this column can be identified with the first one of the PMNS mixing matrix. This pattern fails to describe the data well without corrections mainly because of the tight relation between the solar and the reactor mixing angle that can be derived. We find that leads for to a too small solar mixing angle . At the same time, the atmospheric mixing angle is maximal. The Dirac phase is also maximal, whereas both Majorana phases are trivial. So, this case shows strong similarities to Case II with the representative tuple shown in (25).

### 3.2 Ge=Z3: Case III

If we do the same analysis for the case , we find eight categories of tuples . We can always fix . In this case is not trivial anymore and is of the form

 Ue=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝−√7+3√53(5+√5)−√(5−√5)152√3(5+√5)2√3(5+√5)−12−130√75+30√5−12+130√75+30√52√3(5+√5)12−130√75+30√512+130√75+30√5⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (39)

Using the representatives of the eight different categories we see that indeed only one of these can lead to a mixing that is compatible with experimental data. The column that is fixed by group theory is in this case TM [8]

 1√3⎛⎜⎝111⎞⎟⎠≈⎛⎜⎝0.5770.5770.577⎞⎟⎠ (40)

and has therefore to be identified with the second column of the PMNS mixing matrix. Immediately, we know that the solar mixing angle has a lower bound, , see the sum rule in (46), the result of the analysis in table 1 and the lower bound in figure 1. We use as representative the tuple

 (Q,Z,X)=(T2ST2,ST2ST3S,X0)(Case III). (41)

A possible admitted form of the matrix is

 (42)

As can be seen, the form of the matrix is quite similar to the ones used in the other cases, see (28) and (34). The matrix is composed as follows

 Uν=Ω\footnotesizeIIIR13(θ)Kν, (43)

since, as in the cases above, and of the matrix combination are equal. The PMNS mixing matrix is then given by

 UPMNS=U†eΩ\footnotesizeIIIR13(θ)Kν. (44)

We can extract the following results for the solar and the reactor mixing angles

 sin2θ12=12+sin2θandsin2θ13=13(1−sin2θ)(Case III) (45)

that fulfill the exact – and well-known – sum rule [8]

 sin2θ12=13(1−sin2θ13)≳13. (46)

If we use as best fit value for , we arrive at which is the value that is also obtained in our analysis, see table 1. The atmospheric mixing angle as well as the Dirac phase are, as in Case II, maximal

 sin2θ23=12and|sinδ|=1%(CaseIII). (47)

Again, the actual sign of depends on the choice of . The form of the Jarlskog invariant is

 JCP=cos2θ6√3. (48)

If vanishes, equals . Since at the same time one mixing angle becomes or , the Dirac phase turns out to be unphysical for . As happened for Case I and II, also here the mixing pattern that is well compatible with experimental data gives rise to trivial Majorana phases.

We note that the formulae of the mixing angles remain invariant, if we replace with . Thus, we expect (at least) two best fit solutions. This expectation is confirmed by our analysis, see table 1. Replacing with leads, at the same time, to an additional sign for and thus . Similarly, the exchange of the second and third rows of the PMNS mixing matrix in (44) does not alter the results for the mixing angles, but changes the sign of the Jarlskog invariant and thus of .

### 3.3 Ge=Z2×Z2: Case IV-P1 and Case IV-P2

For the remaining possibility we find that all admitted combinations of , with and describing the residual symmetry can be classified in four different categories of tuples . Thus, it is sufficient to calculate the mixing pattern for one representative of each category. Note that we can always choose a representative for which , see (15), i.e. and can be chosen as and . So, the form of the matrix is

 Ue=1√2⎛⎜⎝√2cosφ0−√2sinφsinφ−1cosφsinφ1cosφ⎞⎟⎠. (49)

It turns out that only one category of tuples is capable of accommodating the experimental values of the mixing angles well for a particular choice of the parameter , while the other three ones fail to do so. In particular, two out of these three lead to patterns with only one non-vanishing mixing angle, since the generators are diagonalized by the same matrix as the generator . A representative of the category that allows for good agreement with the data is

 ({Q1,Q2},Z,X)=({S,T2ST3ST2},ST2ST,X0)(Case IV). (50)

We can check that the column that does not depend on the free parameter has components with absolute values of the form

 12⎛⎜⎝ϕ11/ϕ⎞⎟⎠≈⎛⎜⎝0.8090.50.309⎞⎟⎠. (51)

Thus, this column can be identified with the first one of the PMNS mixing matrix. We call this situation Case IV-P1. We note that we can exchange the second and third components of the vector in (51), i.e. we can exchange the second and third rows of the resulting PMNS mixing matrix, and also obtain good agreement with the experimental data. This situation is denoted by Case IV-P2 in the following. The crucial change occurs in the predicted value of the atmospheric mixing angle, see (56) and (60).

A unitary matrix that fulfills the conditions in (8) for and is

 Ω\footnotesizeIV=1√2⎛⎜ ⎜⎝√2sinφ−√2cos2φ√2cosφsinφe−iΦ/2cosφe−iΦ/2(e15iΦ/4+cosφ)sinφe−iΦ/2(e15iΦ/4cosφ−sin2φ)−e−2iΦcosφe−2iΦ(e15iΦ/4−cosφ)sinφe−2iΦ(e15iΦ/4cosφ+sin2φ)⎞⎟ ⎟⎠. (52)

Since the diagonal matrix reveals equal and in the convention of (8), the form of the neutrino mixing matrix is given by, up to permutations of columns and unphysical phases,

 Uν=Ω\footnotesizeIVR23(θ)Kν. (53)

Taking into account the contribution to leptonic mixing coming from the charged lepton sector that is encoded in the matrix in (49), the PMNS mixing matrix is of the form, up to permutations of rows and columns and unphysical phases,

 UPMNS=U†eΩ\footnotesizeIVR23(θ)Kν. (54)

We find for the mixing angles the following expressions

 sin2θ12=(5−2√5)cos2θ1+(5−2√5)cos2θ,sin2θ13=18(5−√5)sin2θ(Case IV) (55) sin2θ23=3(5−√5)+(9−5√5)cos2θ+8sin2θ25−3√5+5(3−√5)cos2θ(Case IV-P1). (56)

For this case, as mentioned above, the absolute values of the elements of the first column of the PMNS mixing matrix are ordered in the same way as in (51). From (55) we can derive an exact sum rule relating the solar mixing angle to the reactor one

 sin2θ12=1−3+√58(1−sin2θ13)≈0.331, (57)

if we insert the experimental best fit value . We note that in this case a non-trivial upper bound on exists, namely . This can also be directly derived from the constraint that . This bound is marked with a (dashed red) vertical line in figure 1. Furthermore we can obtain an approximate relation of the atmospheric mixing angle and

 sin2θ23≈110(5−√5)+25√5+2√5sinθ13≈0.276+1.23sinθ13, (58)

using that lies in the interval (and thus ).888 For we can derive a similar relation that shows, however, that the measured values of the atmospheric and the reactor mixing angle cannot be accommodated well at the same time. Subleading corrections are of order at maximum. For , we find

 sin2θ23≈0.459. (59)

This estimate is consistent with the result of our analysis, see table 1.

If we permute the second and the third rows, the form of the reactor as well as of the solar mixing angle is the same, while the atmospheric one turns out to be , i.e. here the atmospheric mixing angle reads

 sin2θ23=1−3(5−√5)+(9−5√5)cos2θ+8sin2θ25−3√5+5(3−√5)cos2θ(Case % IV-P2). (60)

So, in this case the approximate sum rule relating and is given by

 sin2θ23≈110(5+√5)−25√5+2√5sinθ13≈0.724−1.23sinθ13. (61)

Again, we assume (and for lying in the interval see footnote 8). For we get

 sin2θ23≈0.541 (62)

which is consistent with the value obtained from the fit, see table 1.

All CP phases are trivial. This points towards an accidental CP symmetry in the theory. This is clear, since the CP transformation is not only present in the neutrino sector, but also – as one can check explicitly – in the charged lepton one.

Note that there is no evident symmetry as regards the parameter in the formulae of the mixing angles: while and remain invariant, if we replace the parameter with , this is not the case for the atmospheric mixing angle and thus we expect in general only one value of for which the mixing angles can be accommodated best. This is confirmed in our numerical analysis, see table 1.

### 3.4 Numerical discussion

In the following we present our results of a analysis for the different cases, Case I through IV-P2. The function is defined in the usual way

 χ2=χ212+χ213+χ223 (63) with χ2ij=(sin2θij−(sin2θij)bfσij)2forij=12,13,23. (64)

are the mixing angles derived in the different cases, e.g. see (30) for Case I, that depend on the continuous parameter , ranging from to , are the best fit values and the errors reported in (1). Note that these errors also depend on whether is larger or smaller than the best fit value. Since the global fit results for the mixing angles (slightly) differ for the case of NO or IO, we consider these two separately and calculate for all patterns the function under the assumption of NO being realized in nature and for IO. In particular, in doing so we do not take into account the fact that NO is slightly disfavored by compared to IO [1]. A mixing pattern is considered to agree reasonably well with the experimental data, if and all mixing angles are within the intervals given in (1).999 The upper limit on , , is chosen, since it results from summing three gaussian errors for one degree of freedom. The functions and are minimized at the best fitting point(s) and we only report the global minimum/a in table 1 for each case.101010 We have tested the validity of our analysis by constructing a likelihood function to fit the various cases that uses the one-dimensional projections provided in [1]. These results are consistent with those in table 1, up to the fact that the roles of the local and the global minimum in Case I become exchanged for NO. However, the difference between these two minima turns out to be statistically insignificant. Since the indication of a preferred value of the Dirac phase coming from global fit analyses is rather weak [1], i.e. below the significance, we do not include any information on in the function in (63).

Our findings for the different cases are summarized in table 1. As one can see, these results agree well with our analytical estimates and observations made in subsections 3.1-3.3. In particular, the sum of the two best fitting values of for NO and IO in Case I, and , approximately equals , since the formulae for the mixing angles and are invariant under the transformation , while turns into . Similarly, the sum of the two best fitting points and ((almost) the same for NO and IO) equals in Case II. Also related to the symmetry properties of the formulae for the solar and the reactor mixing angles is the observation in Case III that the two best fitting points (for NO and IO), and , sum up to . Case IV does not reveal such a symmetry in the parameter and thus we discuss in this case the results corresponding to two different permutations, Case IV-P1 and Case IV-P2, that are related by the exchange of the second and third rows of the PMNS mixing matrix. This allows us to accommodate as well as . In other cases the discussion of this permutation is already implicitly included in our analysis. In Case I and Case II we also confirm the estimate made for the solar mixing angle that is bounded from below , while the lower bound in Case III is . The upper bound found in Case IV-P1 and IV-P2 is obeyed as well, see table 1 and figure 1.