# Lepton Mixing Patterns fromPSL2(7) with a Generalized CP Symmetry

Authors: Shu-Jun Rong
Journal: Advances in High Energy Physics
 (2020)

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## Abstract

Lepton mixing patterns from the modular groupPSL2(7) with generalized CP symmetries are studied. The residual symmetries in both charged lepton and neutrino sectors are Z2×CP. Seven types of mixing patterns at the 3σ level of the new global fit data are obtained. Among these patterns, three types of patterns can give the Dirac CP phase which is in the 1σ range of the global fit data. The effective mass of neutrinoless double-beta decay for these patterns is also examined.


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## Body

## 1. Introduction

CP violation in the hadron sector was observed in 1964 [1]. Whether there is a counterpart in the lepton sector is still a mystery. Recent neutrino oscillation experiments show that the 1-3 mixing angle of leptons is nonzero [2–5]. It intrigues experiments to detect the Dirac CP-violating phase. Particularly, some fit results [6, 7] hint that this phase is around −π/2. In the theoretical respect, how to predict nontrivial lepton CP phases is interesting. In order to obtain mixing parameters of leptons, discrete flavor symmetries are widely used [8–41]. However, if no perturbation is considered, only finite groups of large orders could accommodate the results of new experiments [35]. Furthermore, they give a trivial Dirac CP-violating phase [35]. In order to improve predictions of flavor groups, some efforts have been made in generalizations of symmetries [42, 43]. Particularly, an intriguing method called generalized CP (GCP) symmetry was introduced [44–66]. In this scenario, the leptonic lagrangian satisfies both flavor and GCP symmetries. After spontaneous symmetries breaking, the residual flavor and GCP symmetries constrain the structures of mass matrices of leptons. Then, information on leptonic mixing angles and CP phases is obtained. From groups S4 and A5 with GCP symmetries, a trivial or maximal Dirac CP phase is obtained [46, 56]. The maximal Dirac phase satisfies the 1σ constraint from the new recent global fit data in case of inverted mass ordering [67]. However, it is not in the 1σ range for the normal mass ordering. S4 and A5 are small modular groups. We want to know whether a large one could give a more fit CP phase.In this paper, we study the predictions of the modular groupPSL2(7) with the GCP symmetry in the case of Majorana neutrinos. We suppose that residual symmetries in neutrino and charged lepton sectors are both Z2×CP. Here, CP denotes a GCP symmetry. After examinations of combinations of residual symmetries, we find seven types of mixing patterns at the 3σ level of the fit data [67]. Among them, three types satisfy the 1σ constraint. So the group PSL2(7) with the GCP symmetry may serve as a candidate for explanations to experiment accommodable mixing patterns. We note that lepton mixing patterns from large finite modular group have been studied in the recent Refs. [32, 68, 69]. In Refs. [32, 68], no GCP symmetry is considered. In Ref. [32], residual GCP symmetries are considered either in the neutrino sector or the charged lepton sector. Namely, there is only one unfixed parameter in the lepton mixing matrix. Here, we consider the case that residual GCP symmetries constrain both charged leptons and neutrinos. So two parameters are contained in our mixing patterns.This paper is organised as follows. In Section2, the framework for the application of the group PSL2(7) with the GCP symmetry is introduced. In Section 3, the results from examination of the residual symmetries are presented. Finally, a summary is made.


## 2. Framework

In this section, we describe the basic facts of the groupPSL2(7) and introduce the method of deriving lepton mixing patterns from the residual flavor and GCP symmetries.


### 2.1. Group Theory ofPSL2(7)

#### 2.1.1. Generic Facts

The groupPSL2(7) is also named ∑168. It could be constructed with two generators which satisfy following relations [68]:
(1)S2=T7=E,ST3=ST−1ST4=E,where E is the identity element. This group has 6 conjugacy classes listed as follows [68]:
(2)1C1:E,21C2:S,56C3:ST,42C4:ST3,24C71:T,24C72:T3,where iCj denotes that the class contains i elements of order j. Accordingly, there are 6 irreducible representations, namely,
(3)1,3,3∗,6,7,8.Without loss of generality, we consider the 3-dimensional representation3 in the following sections. Accordingly, the generators could be expressed as [68]
(4)S=27s1s2s3s2−s3s1s3s1−s2,T=φ72000φ7000φ7∗3,where sk=sinkπ/7, φ7=ei2π/7.Resorting to the conjugacy classes, we can obtain abelian subgroups ofPSL2(7). These groups are candidates of the residual symmetries for leptons. In this paper, we consider the residual symmetry Z2×CP for leptons. So Z2 subgroups are relevant. There are 21 Z2 subgroups which are identified with the generators of them [68], i.e.,
(5)A1:S,A2:T2ST3ST,A3:TST3ST2,A4:T4ST3,A5:T3ST4,A6:T2ST4ST2,A7:ST2ST4ST2S,A8:ST4ST3S,A9:ST3ST4S,A10:T5ST2,A11:T2ST5,A12:T6ST,A13:TST6,A14:ST4ST4,A15:ST3ST3,A16:ST2ST,A17:ST5ST6,A18:T2ST3S2,A19:T5ST4S2,A20:ST3ST42,A21:ST4ST32.


#### 2.1.2. Automorphism ofPSL2(7)

An automorphism of a group is a transformation which permutate elements of the group. These transformations form a group, namely, the automorphism group. For the groupPSL2(7), the structure of the automorphism group is simple. It is listed as follows:
(6)ZPSL27=Z1,AutPSL27≅PSL2,Z7⋊Z2,InnPSL27≅PSL27,OutPSL27≅Z2=id,u,where Z, Aut, Inn, and Out denote the centre, the automorphism group, the inner automorphism, and the outer automorphism group, respectively. In detail, the inner automorphism group is composed of permutations of elements in the same conjugacy class. The outer automorphism group swaps conjugacy classes and representations. So it reflects the symmetries of the character table shown in Table 1. The unique nontrivial outer automorphism of the group PSL2(7) is
(7)u:24C71⟷24C72,3⟷3∗.Table 1
Character table of the groupPSL2(7) [68].
Rep.
1C1
21C2
56C3
42C4
24C71
24C72
1
1
1
1
1
1
1
3
3
-1
0
1
φ7∗+φ7∗2+φ7∗4
φ7+φ72+φ74
3∗
3
-1
0
1
φ7+φ72+φ74
φ7∗+φ7∗2+φ7∗4
6
6
2
0
0
-1
-1
7
7
-1
1
-1
0
0
8
8
0
-1
0
1
1The representation ofu could be obtained from its action on the generators S, T, i.e.,
(8)u:S⟷S,T⟷T∗=T6.In the 3-dimensional representation, the specific equations of the transformation read(9)XuS∗X−1u=S−1=S,XuT∗X−1u=T−1=T∗.The solution is(10)Xu=eiαdiag1,1,1.Since the global phase is trivial for the lepton mixing patterns, we chooseeiα=1 in the following sections. A general automorphism is the product of the inner and the outer one. It could be expressed as
(11)Xgi=ρ3giXu=ρ3gi,withgi∈PSL27,where ρ3gi is the 3-dimensional representation of the group element.


### 2.2. Approach

#### 2.2.1. GCP Compatible withPSL2(7)

The GCP transformation acts on the flavor space as(12)Φ⟶XΦC,where Φ is a multiplet of fields, X is a unitary matrix, and ΦC is the CP conjugation of Φ. In contrast, the flavor group acts on the fields as
(13)Φ⟶ρgiΦ,withgi∈PSL27.Accordingly, the consistence condition of GCP is [45]
(14)X−1ρgX∗=ρg′.Therefore,X is an automorphism of the flavor group PSL2(7). These GCP transformations form an automorphism group CP. The general theory satisfies the symmetry PSL27⋊CP. After fermions obtain masses from the vacuum expectation values of scalar fields, the original symmetry PSL27⋊CP is broken to Ge⋊CPe in the charged lepton sector and Gv⋊CPv in the neutrino sector. Thus, the mass matrices of charged leptons and Majorana neutrinos satisfy the relations
(15)ρ+gememe+ρge=meme+,withge∈Ge,ρTgνmνρgν=mν,withgν∈Gν.The CP transformationX follows the relations
(16)Xe+meme+Xe=meme+∗,Xe−1ρgeXe∗=ρge′,XνTmνXν=mν∗,Xν−1ρgνXν∗=ρgν′.Since masses of leptons are nondegenerate, theCP transformation X should be a symmetric unitary matrix [56], i.e.,
(17)Xα=XαT,XαXα∗=E,withα=e,νnosum.SoXα can be decomposed as Xα=ΩαΩαT. This kind of CP transformations is called Bickerstaff-Damhus automorphism (BDA) [70, 71]. For the group PSL2(7), all BDAs in the 3-dimensional representation are listed as follows:
(18)T3ST3,TST4ST,TST5ST,ST3ST3S,STST4STS,STST5STS,E,Ti,STiS,withi=1,2,3,4,5,6,T2ST2j,ST2ST2Sj,T2ST5ST2j,withj=1,2,3.


#### 2.2.2. Mixing Patterns from Residual SymmetriesZ2×CP

Once the residual symmetries are fixed, the lepton mixing pattern could be obtained up to permutations of rows and columns. In the direct method, the mixing matrix is completely determined by the symmetries. In the semidirect method, only several elements of the matrix are certain because of degeneracy of the eigenvalues of the residual symmetries. We concern on the semidirect method in this paper. The residual symmetry isZ2e×CPe,Z2ν×CPν in charged lepton and neutrino sectors, respectively. The consistence equation is written as
(19)Xρ∗ge,νX∗=ρge,ν,withge,ν∈Z2.Accordingly, the lepton mixing matrixUPMNS≡Ue+Uν is obtained from the matrix [46]
(20)Ue,ν=Ωe,νOθe,vPe,ν,where Re,ν is a rotation matrix with an angle parameter θe,ν, and Pe,ν is a phase matrix, i.e.,
(21)Pe,ν=diag1,ij,ik,withj,k=0,1,2,3.BecausePe gives nonphysical phases, it is omitted in the following sections.


#### 2.2.3. Similar Transformations

In order to obtain viable mixing patterns, all possible combinations of residual symmetries should be examined. However, if two combinations are connected by a similarity transformation, namely,(22)ρge,ν′=Vρge,νV+,Xe,ν′=VXe,νVT,they would correspond to the same mixing matrix. Therefore, we could just examine nonequivalent combinations. In the following sections, Z2ν is fixed on the subgroup Z2S which is generated by the group element S. The consistent GCP transformations for Z2S are listed as follows:
(23)X1=E,X2=S,X3=T2ST5ST2,X4=T5ST2ST5=T2ST5ST2∗,where X1 and X2 correspond to the equivalent mixing patterns, so do X3 and X4. Z2e, Xe can be obtained from the similar transformations. In detail, for generators of Z2 subgroups, we have ρAi=ViSVi+ with Vi listed as follows:
(24)V2=001φ73000−φ70,V3=001φ7∗3000−φ7∗0,V4=1000φ73000−φ7,V5=1000φ7∗3000−φ7∗,V6=00−1−100010,V7=0−10001−100,V8=001φ7∗000−φ720,V9=001φ7000−φ7∗20,V10=1000φ72000φ73,V11=1000φ7∗2000φ7∗3,V12=1000φ7000φ7∗2,v13=1000φ7∗000φ72,v14=01000−φ73φ700,v15=01000−φ7∗3φ7∗00,v16=001φ72000−φ730,v17=001φ7∗2000−φ7∗30,V18=01000−φ7∗φ7200,V19=01000−φ7φ7∗200,V20=01000−φ72φ7300,V21=01000−φ7∗2φ7∗300.Particularly,V1 is the identity matrix. So a general combination of the residual symmetries is of the form
(25)Z2eAi,XejAi,Z2νS,XνkS,with j, k=1,3. The corresponding lepton mixing matrix is written as
(26)UPMNS=OTθeΩj+Vi+ΩkOθνPν,where
(27)Oθe,ν=1000cosθe,νsinθe,ν0−sinθe,νcosθe,ν,Ω1=r1−sinθ1−cosθ1sinθ2r20cosθ2r3cosθ1−sinθ1sinθ2,Ω3=Ω1⋅1000eiπ/4cosϕei3π/4sinϕ0−eiπ/4sinϕei3π/4cosϕ,with
(28)r1=227sin2π7sin3π7,r2=227sinπ7sin2π7,r3=227sinπ7sin3π7,θ1=arcsinr3r12+r32,θ2=arcsinr2cosθ1r12+r22cos2θ1,ϕ=arcsin11+x12,where x1 is a real root of the equation
(29)x12−48x10+323x8−608x6+323x4−48x2+1=0,x1≃±0.449807.


## 2.1. Group Theory ofPSL2(7)

### 2.1.1. Generic Facts

The groupPSL2(7) is also named ∑168. It could be constructed with two generators which satisfy following relations [68]:
(1)S2=T7=E,ST3=ST−1ST4=E,where E is the identity element. This group has 6 conjugacy classes listed as follows [68]:
(2)1C1:E,21C2:S,56C3:ST,42C4:ST3,24C71:T,24C72:T3,where iCj denotes that the class contains i elements of order j. Accordingly, there are 6 irreducible representations, namely,
(3)1,3,3∗,6,7,8.Without loss of generality, we consider the 3-dimensional representation3 in the following sections. Accordingly, the generators could be expressed as [68]
(4)S=27s1s2s3s2−s3s1s3s1−s2,T=φ72000φ7000φ7∗3,where sk=sinkπ/7, φ7=ei2π/7.Resorting to the conjugacy classes, we can obtain abelian subgroups ofPSL2(7). These groups are candidates of the residual symmetries for leptons. In this paper, we consider the residual symmetry Z2×CP for leptons. So Z2 subgroups are relevant. There are 21 Z2 subgroups which are identified with the generators of them [68], i.e.,
(5)A1:S,A2:T2ST3ST,A3:TST3ST2,A4:T4ST3,A5:T3ST4,A6:T2ST4ST2,A7:ST2ST4ST2S,A8:ST4ST3S,A9:ST3ST4S,A10:T5ST2,A11:T2ST5,A12:T6ST,A13:TST6,A14:ST4ST4,A15:ST3ST3,A16:ST2ST,A17:ST5ST6,A18:T2ST3S2,A19:T5ST4S2,A20:ST3ST42,A21:ST4ST32.


### 2.1.2. Automorphism ofPSL2(7)

An automorphism of a group is a transformation which permutate elements of the group. These transformations form a group, namely, the automorphism group. For the groupPSL2(7), the structure of the automorphism group is simple. It is listed as follows:
(6)ZPSL27=Z1,AutPSL27≅PSL2,Z7⋊Z2,InnPSL27≅PSL27,OutPSL27≅Z2=id,u,where Z, Aut, Inn, and Out denote the centre, the automorphism group, the inner automorphism, and the outer automorphism group, respectively. In detail, the inner automorphism group is composed of permutations of elements in the same conjugacy class. The outer automorphism group swaps conjugacy classes and representations. So it reflects the symmetries of the character table shown in Table 1. The unique nontrivial outer automorphism of the group PSL2(7) is
(7)u:24C71⟷24C72,3⟷3∗.Table 1
Character table of the groupPSL2(7) [68].
Rep.
1C1
21C2
56C3
42C4
24C71
24C72
1
1
1
1
1
1
1
3
3
-1
0
1
φ7∗+φ7∗2+φ7∗4
φ7+φ72+φ74
3∗
3
-1
0
1
φ7+φ72+φ74
φ7∗+φ7∗2+φ7∗4
6
6
2
0
0
-1
-1
7
7
-1
1
-1
0
0
8
8
0
-1
0
1
1The representation ofu could be obtained from its action on the generators S, T, i.e.,
(8)u:S⟷S,T⟷T∗=T6.In the 3-dimensional representation, the specific equations of the transformation read(9)XuS∗X−1u=S−1=S,XuT∗X−1u=T−1=T∗.The solution is(10)Xu=eiαdiag1,1,1.Since the global phase is trivial for the lepton mixing patterns, we chooseeiα=1 in the following sections. A general automorphism is the product of the inner and the outer one. It could be expressed as
(11)Xgi=ρ3giXu=ρ3gi,withgi∈PSL27,where ρ3gi is the 3-dimensional representation of the group element.


## 2.1.1. Generic Facts

The groupPSL2(7) is also named ∑168. It could be constructed with two generators which satisfy following relations [68]:
(1)S2=T7=E,ST3=ST−1ST4=E,where E is the identity element. This group has 6 conjugacy classes listed as follows [68]:
(2)1C1:E,21C2:S,56C3:ST,42C4:ST3,24C71:T,24C72:T3,where iCj denotes that the class contains i elements of order j. Accordingly, there are 6 irreducible representations, namely,
(3)1,3,3∗,6,7,8.Without loss of generality, we consider the 3-dimensional representation3 in the following sections. Accordingly, the generators could be expressed as [68]
(4)S=27s1s2s3s2−s3s1s3s1−s2,T=φ72000φ7000φ7∗3,where sk=sinkπ/7, φ7=ei2π/7.Resorting to the conjugacy classes, we can obtain abelian subgroups ofPSL2(7). These groups are candidates of the residual symmetries for leptons. In this paper, we consider the residual symmetry Z2×CP for leptons. So Z2 subgroups are relevant. There are 21 Z2 subgroups which are identified with the generators of them [68], i.e.,
(5)A1:S,A2:T2ST3ST,A3:TST3ST2,A4:T4ST3,A5:T3ST4,A6:T2ST4ST2,A7:ST2ST4ST2S,A8:ST4ST3S,A9:ST3ST4S,A10:T5ST2,A11:T2ST5,A12:T6ST,A13:TST6,A14:ST4ST4,A15:ST3ST3,A16:ST2ST,A17:ST5ST6,A18:T2ST3S2,A19:T5ST4S2,A20:ST3ST42,A21:ST4ST32.


## 2.1.2. Automorphism ofPSL2(7)

An automorphism of a group is a transformation which permutate elements of the group. These transformations form a group, namely, the automorphism group. For the groupPSL2(7), the structure of the automorphism group is simple. It is listed as follows:
(6)ZPSL27=Z1,AutPSL27≅PSL2,Z7⋊Z2,InnPSL27≅PSL27,OutPSL27≅Z2=id,u,where Z, Aut, Inn, and Out denote the centre, the automorphism group, the inner automorphism, and the outer automorphism group, respectively. In detail, the inner automorphism group is composed of permutations of elements in the same conjugacy class. The outer automorphism group swaps conjugacy classes and representations. So it reflects the symmetries of the character table shown in Table 1. The unique nontrivial outer automorphism of the group PSL2(7) is
(7)u:24C71⟷24C72,3⟷3∗.Table 1
Character table of the groupPSL2(7) [68].
Rep.
1C1
21C2
56C3
42C4
24C71
24C72
1
1
1
1
1
1
1
3
3
-1
0
1
φ7∗+φ7∗2+φ7∗4
φ7+φ72+φ74
3∗
3
-1
0
1
φ7+φ72+φ74
φ7∗+φ7∗2+φ7∗4
6
6
2
0
0
-1
-1
7
7
-1
1
-1
0
0
8
8
0
-1
0
1
1The representation ofu could be obtained from its action on the generators S, T, i.e.,
(8)u:S⟷S,T⟷T∗=T6.In the 3-dimensional representation, the specific equations of the transformation read(9)XuS∗X−1u=S−1=S,XuT∗X−1u=T−1=T∗.The solution is(10)Xu=eiαdiag1,1,1.Since the global phase is trivial for the lepton mixing patterns, we chooseeiα=1 in the following sections. A general automorphism is the product of the inner and the outer one. It could be expressed as
(11)Xgi=ρ3giXu=ρ3gi,withgi∈PSL27,where ρ3gi is the 3-dimensional representation of the group element.


## 2.2. Approach

### 2.2.1. GCP Compatible withPSL2(7)

The GCP transformation acts on the flavor space as(12)Φ⟶XΦC,where Φ is a multiplet of fields, X is a unitary matrix, and ΦC is the CP conjugation of Φ. In contrast, the flavor group acts on the fields as
(13)Φ⟶ρgiΦ,withgi∈PSL27.Accordingly, the consistence condition of GCP is [45]
(14)X−1ρgX∗=ρg′.Therefore,X is an automorphism of the flavor group PSL2(7). These GCP transformations form an automorphism group CP. The general theory satisfies the symmetry PSL27⋊CP. After fermions obtain masses from the vacuum expectation values of scalar fields, the original symmetry PSL27⋊CP is broken to Ge⋊CPe in the charged lepton sector and Gv⋊CPv in the neutrino sector. Thus, the mass matrices of charged leptons and Majorana neutrinos satisfy the relations
(15)ρ+gememe+ρge=meme+,withge∈Ge,ρTgνmνρgν=mν,withgν∈Gν.The CP transformationX follows the relations
(16)Xe+meme+Xe=meme+∗,Xe−1ρgeXe∗=ρge′,XνTmνXν=mν∗,Xν−1ρgνXν∗=ρgν′.Since masses of leptons are nondegenerate, theCP transformation X should be a symmetric unitary matrix [56], i.e.,
(17)Xα=XαT,XαXα∗=E,withα=e,νnosum.SoXα can be decomposed as Xα=ΩαΩαT. This kind of CP transformations is called Bickerstaff-Damhus automorphism (BDA) [70, 71]. For the group PSL2(7), all BDAs in the 3-dimensional representation are listed as follows:
(18)T3ST3,TST4ST,TST5ST,ST3ST3S,STST4STS,STST5STS,E,Ti,STiS,withi=1,2,3,4,5,6,T2ST2j,ST2ST2Sj,T2ST5ST2j,withj=1,2,3.


### 2.2.2. Mixing Patterns from Residual SymmetriesZ2×CP

Once the residual symmetries are fixed, the lepton mixing pattern could be obtained up to permutations of rows and columns. In the direct method, the mixing matrix is completely determined by the symmetries. In the semidirect method, only several elements of the matrix are certain because of degeneracy of the eigenvalues of the residual symmetries. We concern on the semidirect method in this paper. The residual symmetry isZ2e×CPe,Z2ν×CPν in charged lepton and neutrino sectors, respectively. The consistence equation is written as
(19)Xρ∗ge,νX∗=ρge,ν,withge,ν∈Z2.Accordingly, the lepton mixing matrixUPMNS≡Ue+Uν is obtained from the matrix [46]
(20)Ue,ν=Ωe,νOθe,vPe,ν,where Re,ν is a rotation matrix with an angle parameter θe,ν, and Pe,ν is a phase matrix, i.e.,
(21)Pe,ν=diag1,ij,ik,withj,k=0,1,2,3.BecausePe gives nonphysical phases, it is omitted in the following sections.


### 2.2.3. Similar Transformations

In order to obtain viable mixing patterns, all possible combinations of residual symmetries should be examined. However, if two combinations are connected by a similarity transformation, namely,(22)ρge,ν′=Vρge,νV+,Xe,ν′=VXe,νVT,they would correspond to the same mixing matrix. Therefore, we could just examine nonequivalent combinations. In the following sections, Z2ν is fixed on the subgroup Z2S which is generated by the group element S. The consistent GCP transformations for Z2S are listed as follows:
(23)X1=E,X2=S,X3=T2ST5ST2,X4=T5ST2ST5=T2ST5ST2∗,where X1 and X2 correspond to the equivalent mixing patterns, so do X3 and X4. Z2e, Xe can be obtained from the similar transformations. In detail, for generators of Z2 subgroups, we have ρAi=ViSVi+ with Vi listed as follows:
(24)V2=001φ73000−φ70,V3=001φ7∗3000−φ7∗0,V4=1000φ73000−φ7,V5=1000φ7∗3000−φ7∗,V6=00−1−100010,V7=0−10001−100,V8=001φ7∗000−φ720,V9=001φ7000−φ7∗20,V10=1000φ72000φ73,V11=1000φ7∗2000φ7∗3,V12=1000φ7000φ7∗2,v13=1000φ7∗000φ72,v14=01000−φ73φ700,v15=01000−φ7∗3φ7∗00,v16=001φ72000−φ730,v17=001φ7∗2000−φ7∗30,V18=01000−φ7∗φ7200,V19=01000−φ7φ7∗200,V20=01000−φ72φ7300,V21=01000−φ7∗2φ7∗300.Particularly,V1 is the identity matrix. So a general combination of the residual symmetries is of the form
(25)Z2eAi,XejAi,Z2νS,XνkS,with j, k=1,3. The corresponding lepton mixing matrix is written as
(26)UPMNS=OTθeΩj+Vi+ΩkOθνPν,where
(27)Oθe,ν=1000cosθe,νsinθe,ν0−sinθe,νcosθe,ν,Ω1=r1−sinθ1−cosθ1sinθ2r20cosθ2r3cosθ1−sinθ1sinθ2,Ω3=Ω1⋅1000eiπ/4cosϕei3π/4sinϕ0−eiπ/4sinϕei3π/4cosϕ,with
(28)r1=227sin2π7sin3π7,r2=227sinπ7sin2π7,r3=227sinπ7sin3π7,θ1=arcsinr3r12+r32,θ2=arcsinr2cosθ1r12+r22cos2θ1,ϕ=arcsin11+x12,where x1 is a real root of the equation
(29)x12−48x10+323x8−608x6+323x4−48x2+1=0,x1≃±0.449807.


## 2.2.1. GCP Compatible withPSL2(7)

The GCP transformation acts on the flavor space as(12)Φ⟶XΦC,where Φ is a multiplet of fields, X is a unitary matrix, and ΦC is the CP conjugation of Φ. In contrast, the flavor group acts on the fields as
(13)Φ⟶ρgiΦ,withgi∈PSL27.Accordingly, the consistence condition of GCP is [45]
(14)X−1ρgX∗=ρg′.Therefore,X is an automorphism of the flavor group PSL2(7). These GCP transformations form an automorphism group CP. The general theory satisfies the symmetry PSL27⋊CP. After fermions obtain masses from the vacuum expectation values of scalar fields, the original symmetry PSL27⋊CP is broken to Ge⋊CPe in the charged lepton sector and Gv⋊CPv in the neutrino sector. Thus, the mass matrices of charged leptons and Majorana neutrinos satisfy the relations
(15)ρ+gememe+ρge=meme+,withge∈Ge,ρTgνmνρgν=mν,withgν∈Gν.The CP transformationX follows the relations
(16)Xe+meme+Xe=meme+∗,Xe−1ρgeXe∗=ρge′,XνTmνXν=mν∗,Xν−1ρgνXν∗=ρgν′.Since masses of leptons are nondegenerate, theCP transformation X should be a symmetric unitary matrix [56], i.e.,
(17)Xα=XαT,XαXα∗=E,withα=e,νnosum.SoXα can be decomposed as Xα=ΩαΩαT. This kind of CP transformations is called Bickerstaff-Damhus automorphism (BDA) [70, 71]. For the group PSL2(7), all BDAs in the 3-dimensional representation are listed as follows:
(18)T3ST3,TST4ST,TST5ST,ST3ST3S,STST4STS,STST5STS,E,Ti,STiS,withi=1,2,3,4,5,6,T2ST2j,ST2ST2Sj,T2ST5ST2j,withj=1,2,3.


## 2.2.2. Mixing Patterns from Residual SymmetriesZ2×CP

Once the residual symmetries are fixed, the lepton mixing pattern could be obtained up to permutations of rows and columns. In the direct method, the mixing matrix is completely determined by the symmetries. In the semidirect method, only several elements of the matrix are certain because of degeneracy of the eigenvalues of the residual symmetries. We concern on the semidirect method in this paper. The residual symmetry isZ2e×CPe,Z2ν×CPν in charged lepton and neutrino sectors, respectively. The consistence equation is written as
(19)Xρ∗ge,νX∗=ρge,ν,withge,ν∈Z2.Accordingly, the lepton mixing matrixUPMNS≡Ue+Uν is obtained from the matrix [46]
(20)Ue,ν=Ωe,νOθe,vPe,ν,where Re,ν is a rotation matrix with an angle parameter θe,ν, and Pe,ν is a phase matrix, i.e.,
(21)Pe,ν=diag1,ij,ik,withj,k=0,1,2,3.BecausePe gives nonphysical phases, it is omitted in the following sections.


## 2.2.3. Similar Transformations

In order to obtain viable mixing patterns, all possible combinations of residual symmetries should be examined. However, if two combinations are connected by a similarity transformation, namely,(22)ρge,ν′=Vρge,νV+,Xe,ν′=VXe,νVT,they would correspond to the same mixing matrix. Therefore, we could just examine nonequivalent combinations. In the following sections, Z2ν is fixed on the subgroup Z2S which is generated by the group element S. The consistent GCP transformations for Z2S are listed as follows:
(23)X1=E,X2=S,X3=T2ST5ST2,X4=T5ST2ST5=T2ST5ST2∗,where X1 and X2 correspond to the equivalent mixing patterns, so do X3 and X4. Z2e, Xe can be obtained from the similar transformations. In detail, for generators of Z2 subgroups, we have ρAi=ViSVi+ with Vi listed as follows:
(24)V2=001φ73000−φ70,V3=001φ7∗3000−φ7∗0,V4=1000φ73000−φ7,V5=1000φ7∗3000−φ7∗,V6=00−1−100010,V7=0−10001−100,V8=001φ7∗000−φ720,V9=001φ7000−φ7∗20,V10=1000φ72000φ73,V11=1000φ7∗2000φ7∗3,V12=1000φ7000φ7∗2,v13=1000φ7∗000φ72,v14=01000−φ73φ700,v15=01000−φ7∗3φ7∗00,v16=001φ72000−φ730,v17=001φ7∗2000−φ7∗30,V18=01000−φ7∗φ7200,V19=01000−φ7φ7∗200,V20=01000−φ72φ7300,V21=01000−φ7∗2φ7∗300.Particularly,V1 is the identity matrix. So a general combination of the residual symmetries is of the form
(25)Z2eAi,XejAi,Z2νS,XνkS,with j, k=1,3. The corresponding lepton mixing matrix is written as
(26)UPMNS=OTθeΩj+Vi+ΩkOθνPν,where
(27)Oθe,ν=1000cosθe,νsinθe,ν0−sinθe,νcosθe,ν,Ω1=r1−sinθ1−cosθ1sinθ2r20cosθ2r3cosθ1−sinθ1sinθ2,Ω3=Ω1⋅1000eiπ/4cosϕei3π/4sinϕ0−eiπ/4sinϕei3π/4cosϕ,with
(28)r1=227sin2π7sin3π7,r2=227sinπ7sin2π7,r3=227sinπ7sin3π7,θ1=arcsinr3r12+r32,θ2=arcsinr2cosθ1r12+r22cos2θ1,ϕ=arcsin11+x12,where x1 is a real root of the equation
(29)x12−48x10+323x8−608x6+323x4−48x2+1=0,x1≃±0.449807.


## 3. Results

### 3.1. Viable Mixing Matrices from Combinations of Residual Symmetries

Given the recent global fit data of neutrino oscillations [67], we examine the predictions of combinations of residual symmetries of the form Z2eAi,XejAi,Z2νS,XνkS with the χ2 function defined as
(30)χ2=∑ij=13,23,12sin2θij−sin2θijexσij2,where sin2θijex is the best fit data from Ref. [67] and σij is the 1σ error. The viable combinations at the 3σ level (up to equivalent ones) are listed as follows.Type Ia:(31)Z2eST4ST4,X3e=V16T2ST5ST2V16T,Z2νS,X1ν=E,Type Ib:(32)Z2eST4ST4,X1e=V16V16T,Z2νS,X3ν=T2ST5ST2,Type Ib∗:
(33)Z2eST3ST3,X1e=V17V17T,Z2νS,X3ν=T2ST5ST2,Type IIa:(34)Z2eT4ST3,X1e=V4V4T,Z2νS,X1ν=E,Type IIa∗:
(35)Z2eT3ST4,X1e=V5V5T,Z2νS,X1ν=E,Type IIb:(36)Z2eT4ST3,X1e=V4V4T,Z2νS,X3ν=T2ST5ST2,Type IIb∗:
(37)Z2eT3ST4,X1e=V5V5T,Z2νS,X3ν=T2ST5ST2.The corresponding mixing matrices are dependent on permutations of rows and columns. For every combination, the matrix which fits the data best is listed as follows.(38)UIa=S13OTθeΩ3+V16+Ω1OθνPν,UIb=S12OTθeΩ1+V16+Ω3OθνPνS12,UIb∗=UIb∗=S12OTθeΩ1+V17+Ω3∗OθvPνS12,UIIa=S13OTθeΩ1+V4+Ω1OθνPν,UIIa∗=UIIa∗=S13OTθeΩ1+V5+Ω1OθνPν,UIIb=S12OTθeΩ1+V4+Ω3OθνPν,UIIb∗=UIIb∗=S12OTθeΩ1+V5+Ω3∗OθνPν,where
(39)S12=010100001,S13=001010100.We note that the mixing matrices exceptUIa are paired through the complex conjugation. The predictions of the matrices in a pair are identical except the signs of the CP phases. So we can consider UIa, UIb, UIIa, and UIIb as representatives.


### 3.2. Mixing Angles and CP Invariants

Employing the parametrization of the form(40)UPMNS=c12c13s12c13s13e−iδ−s12c23−c12s13s23eiδc12c23−s12s13s23eiδc13s23s12s23−c12s13c23eiδ−c12s23−s12s13c23eiδc13c231000eiα/2000eiβ/2+δ,where sij≡sinθij, cij≡cosθij, δ is the Dirac CP-violating phase, and α and β are Majorana phases; we could obtain lepton mixing angles and the CP invariants Jcp [72], J1, and J2 defined as
(41)Jcp≡ImU11U13∗U31∗U33=18sin2θ13sin2θ23sin2θ12cosθ13sinδ,J1≡ImU11∗2U122=sin2θ12cos2θ12cos4θ13sinα,J2≡ImU11∗2U132=sin2θ13cos2θ13cos2θ12sinβ.Their specific forms are listed as follows.UIa:
(42)sin2θ13θe,θν≃0.3125−0.1758cos2θν+0.02073cos2θν−2θe−0.1654cos2θe+0.2786cos2θν+2θe+0.065086sin2θν−0.02338sin2θν−2θe−0.08839sin2θe+0.03824sin2θν−2θe,sin2θ23θe,θν≃0.002647cos2θecos2θν+0.05023cos2θesin2θν+0.9531cos2θesin2θν+0.02677sin2θecos2θν+0.2579sin2θesin2θν+0.27067sin2θecos2θν+0.150sin2θesin2θν+0.07994sin2θesin2θν+0.02361sin2θesin2θν1−sin2θ13θe,θν,sin2θ12θe,θν=sin2θ23θe+π/2,θν+π/21−sin2θ13θe+π/2,θν+π/21−sin2θ13θe,θν,Jcpθe,θν=0,J1θe,θν=0,J2θe,θν=0.UIb:
(43)sin2θ13θe,θν≃0.2438cos2θecos2θν+0.1871cos2θesin2θν+0.01618cos2θesin2θν+0.264sin2θecos2θν+0.1659sin2θesin2θν−0.16566sin2θesin2θν+0.712sin2θecos2θν−0.187sin2θesin2θν+0.13245sin2θesin2θν,sin2θ23θe,θν≃0.375−0.3307cos2θν1−sin2θ13θe,θν,sin2θ12θe,θν≃0.375+0.2194cos2θe−0.2475sin2θe1−sin2θ13θe,θν,Jcpθe,θν≃0.001543+0.008446cos2θν+0.006903cos4θν+0.002733cos2θν−4θe−0.003911cos4θν−2θe+0.003405cos2θν−2θe+0.000384cos4θν−4θe−0.002896cos2θe+0.001353cos4θe−0.008932cos2θν+2θe−0.002234cos4θν+4θe−0.00114cos4θν+2θe−0.00323cos2θν+4θe−0.00951sin2θν+0.000585sin4θν+0.002935sin2θν−4θe−0.000292sin4θν−2θe−0.02566sin2θν−2θe+0.000304sin4θν−4θe+0.00328sin2θe−0.00591sin4θe−0.000519sin2θν+2θe+0.00715sin4θν+4θe−0.00775sin4θν+2θe+0.00388sin2θν+4θe,J1θe,θν≃±0.05167−0.00391cos2θν−0.00304cos2θν−4θe−0.01371cos2θν−2θe+0.01037cos2θe+0.006185cos4θe+0.04114cos2θν+2θe−0.00491cos2θν+4θe+0.02521sin2θν−4θe−0.01546sin2θν−2θe−0.1169sin2θe+0.0153sin4θe−0.0464sin2θν+2θe−0.0407sin2θν+4θe,J2θe,θν≃±0.00726cos2θν−4θe−0.0248cos2θν−2θe+0.05787cos2θν+2θe−0.05079cos2θν+2θe−0.07308sin2θν−0.000875sin2θν−4θe+0.02199sin2θν−2θe+0.0513sin2θν+2θe+0.00612sin2θν+4θe.UIIa:
(44)sin2θ13θe,θν≃0.616cos2θecos2θν−0.133cos2θesin2θν+0.0769cos2θesin2θν−0.1328sin2θecos2θν+0.01762sin2θesin2θν−0.1322sin2θesin2θν+0.0769sin2θecos2θν−0.1322sin2θesin2θν+0.4155sin2θesin2θν,sin2θ23θe,θν≃0.2963+0.0502cos2θν−0.1185cos2θν−2θe−0.0502cos2θe−0.101cos2θν+2θe−0.01325sin2θν+0.01325sin2θe+0.000294sin2θν+2θe1−sin2θ13,sin2θ12θe,θν=sin2θ23θe+π/2,θν+π/21−sin2θ13θe+π/2,θν+π/21−sin2θ13θe,θν,JCPθe,θν≃0.00919cos2θν−2θe−0.01601cos2θν+2θe+0.06163sin2θν+2θe,J1θe,θν≃±−0.00475−0.0474cos2θν+0.0144cos2θν−4θe+0.0203cos2θν−2θe+0.0541cos2θe−0.0461cos4θe+0.0188cos2θν+2θe−0.03285cos2θν+4θe+0.002649sin2θν−0.0101sin2θν−4θe+0.02699sin2θν−2θe+0.02349sin2θe+0.05424sin4θe−0.01937sin2θν+2θe−0.04581sin2θν+4θe,J2θe,θν=J1θe,θν+π2.UIIb:
(45)sin2θ13θe,θν≃0.2963−0.1324cos2θν+0.199cos2θν−2θe−0.0502cos2θe−0.05315cos2θν+2θe+0.1464sin2θν−0.00634sin2θν−2θe+0.1325sin2θe−0.01973sin2θν+2θe,sin2θ23θe,θν≃0.4074+0.2648cos2θν−0.2927sin2θ1−sin2θ13,sin2θ12θe,θν=sin2θ13θe,θν+π/21−sin2θ13θe,θν,JCPθe,θν≃0.0261cos2θν−2θe+0.02447cos2θν+2θe+0.04sin2θν−2θe+0.01804sin2θν+2θe,J1θe,θν≃±0.01355+0.01357cos2θν+0.02594cos2θν−4θe+0.00115cos2θν−2θe−0.02127cos2θe+0.0643cos4θe−0.03256cos2θν+2θe+0.0213cos2θν+4θe−0.03385sin2θν−0.04857sin2θν−4θe−0.02714sin2θν−2θe−0.001069sin2θe+0.03351sin4θe−0.007202sin2θν+2θe−0.002316sin2θν+4θe,J2θe,θν=J1θe,θν+π2.The sign ofJ1 and J2 is uncertain. It is dependent on the index j and k in the diagonal phase matrix Pν.


### 3.3. Constraints on (θe, θν) and Best Fit Data

The parameter space of (θe, θν) is shown in Figure 1. The best fit data of lepton mixing angle and CP phases are listed in Table 2. We make some comments on the numerical results:Figure 1
Parameter space for the mixing patterns constrained by the global fit data [67] at the 3σ level in the normal mass ordering (NO). The parameter spaces for θ12 are the strips with blue boundaries, and those for θ23 are with green boundaries. The strips for θ13 are tiny, i.e., almost reduced to black curves. Their intersection areas are signed by the red dots. The parameter spaces in the case of inverted mass ordering (IO) are similar. So they are not shown here.Table 2
Best fit data of lepton mixing angles and CP phases. “N” and “I” denote the normal ordering of neutrino masses and the inverted ordering, respectively.
Patterns
(θebf, θνbf)
χmin2
sin2θ13
sin2θ23
sin2θ12
sinδ
sinα
sinβ
Ia(N)
(0.18725π, 0.1644π), (-0.0297π, 0.72267π)
0.000016
0.0216
0.547
0.320
0
0
0
Ia(I)
(0.18596π, 0.16544π), (-0.0297π, 0.72172π)
0.021
0.0220
0.549
0.318
0
0
0
Ib(N)
(-0.4028π, 0.3393π), (-0.8663π, -0.3393π)
2.33
0.0217
0.563
0.345
0.813
0.952
0.43
Ib(I)
(-0.4037π, 0.340π), (-0.8654π, -0.340π)
1.97
0.0221
0.565
0.343
0.817
0.953
0.42
IIa(N)
(0.3694π, 0.0767π)
0.064
0.0216
0.549
0.316
0.997
0.769
0.588
IIa(I)
(0.3701π, 0.0745π)
0.057
0.022
0.553
0.316
0.998
0.774
0.547
IIb(N)
(-0.2345π, 0.06297π)
0.79
0.02156
0.5507
0.304
0.800
0.617
0.747
IIb(I)
(-0.2348π, 0.06145π)
0.81
0.0220
0.5546
0.304
0.742
0.616
0.730(i) In a period of the parameters (θe, θν), there are two best fit points for the mixing matrices UIa and UIb. For UIb, these two points give the same magnitude of sinδ with different signs. For the mixing matrices UIIa and UIIb, there is only one best fit point in a period. sinδ for them are both positive. Even so, UIIa∗ and UIIb∗ could give negative sinδ while the mixing angles are kept the same. (ii) The best fit value of θ23 from the global fit data [67] is in the second octant. Accordingly, our fit data is in the same octant. In the case of normal mass ordering (NO), the best fit value of δ in UIb and UIIb∗ could be around −0.3π. It is in the 1σ range of the global fit data. In the case of inverted mass ordering (IO), the best fit value of δ in UIIa∗ is around −0.5π. It is also in the 1σ range of the global fit data.


### 3.4. The Effective Mass of Neutrinoless Double-Beta Decay<mee>

Although the residual symmetries (Z2e×CPe, Z2ν×CPν) cannot restrain the masses of neutrinos mi with i=1,2,3, they may affect the effective mass of neutrinoless double-beta decay <mee> through the mixing angles and Majorana phases. Here, <mee> is expressed as
(46)mee≡m1U112+m2U122+m3U132.Employing the lepton mixing matrixUθebf,θνbf and the best fit data on Δm122,Δm132 [67], we plot <mee> against the mass of the lightest neutrino m0 in Figure 2. For (θe, θν) taken from the 3σ range around the best fit data, the curves of <mee> in every pattern are shown in Figure 3. We make some comments on the main results shown in these figures.
(i)
In the case of IO, these patterns give stringent constraints on the ranges of<mee>. Particularly, <mee> for patterns with the indexes (1, 0) and (1, 1) is independent of the parameters (θe, θν).
In the range ofm0 favored by cosmology, <mee> is around 0.045 eV for patterns Ia, IIa, and IIb. For pattern Ib, it is 0.04 eV. These values approximate the upper limit from the global fit data at the 3σ level. They are in the reach of future double-beta decay experiments [73]. Interestingly, similar observations still hold for the patterns from the group S4 with GCP [62]
(ii)
In the case of NO, the variance of<mee> is noticeable for every pattern. Particularly, for patterns Ia and Ib, <mee> with the indexes (1, 1) could reach the upper limit at the 3σ level. Even so, it is not accessible to near future experiments
(iii)
In both NO and IO case, without the precise constraint on the Dirac CP phase, these four mixing patterns cannot be discriminated by future double-beta decay experiments because of the overlaps of ranges of<mee>Figure 2
The effective mass of neutrinoless double-beta decaymee against the mass of the lightest neutrino m0 with the best fit data (θebf, θvbf). The bound on m0 from cosmology is taken from Ref. [74]. The constraint on mee is from Ref. [75]. The best fit data of Δm122 and Δm132 is from Ref. [67]. The legends for every pattern are shown in the top left panel. The indexes (j, k) are defined in Equation (21). The best fit data (θebf, θvbf) for pattern Ia and that for pattern Ib take the second one in Table 2, respectively.Figure 3
The effective mass of neutrinoless double-beta decaymee against the mass of the lightest neutrino m0 in the 3σ ranges of (θe, θν). The conventions follow those in Figure 2. The dashed boundary lines at the 3σ level are obtained from the global fit data [67].


## 3.1. Viable Mixing Matrices from Combinations of Residual Symmetries

Given the recent global fit data of neutrino oscillations [67], we examine the predictions of combinations of residual symmetries of the form Z2eAi,XejAi,Z2νS,XνkS with the χ2 function defined as
(30)χ2=∑ij=13,23,12sin2θij−sin2θijexσij2,where sin2θijex is the best fit data from Ref. [67] and σij is the 1σ error. The viable combinations at the 3σ level (up to equivalent ones) are listed as follows.Type Ia:(31)Z2eST4ST4,X3e=V16T2ST5ST2V16T,Z2νS,X1ν=E,Type Ib:(32)Z2eST4ST4,X1e=V16V16T,Z2νS,X3ν=T2ST5ST2,Type Ib∗:
(33)Z2eST3ST3,X1e=V17V17T,Z2νS,X3ν=T2ST5ST2,Type IIa:(34)Z2eT4ST3,X1e=V4V4T,Z2νS,X1ν=E,Type IIa∗:
(35)Z2eT3ST4,X1e=V5V5T,Z2νS,X1ν=E,Type IIb:(36)Z2eT4ST3,X1e=V4V4T,Z2νS,X3ν=T2ST5ST2,Type IIb∗:
(37)Z2eT3ST4,X1e=V5V5T,Z2νS,X3ν=T2ST5ST2.The corresponding mixing matrices are dependent on permutations of rows and columns. For every combination, the matrix which fits the data best is listed as follows.(38)UIa=S13OTθeΩ3+V16+Ω1OθνPν,UIb=S12OTθeΩ1+V16+Ω3OθνPνS12,UIb∗=UIb∗=S12OTθeΩ1+V17+Ω3∗OθvPνS12,UIIa=S13OTθeΩ1+V4+Ω1OθνPν,UIIa∗=UIIa∗=S13OTθeΩ1+V5+Ω1OθνPν,UIIb=S12OTθeΩ1+V4+Ω3OθνPν,UIIb∗=UIIb∗=S12OTθeΩ1+V5+Ω3∗OθνPν,where
(39)S12=010100001,S13=001010100.We note that the mixing matrices exceptUIa are paired through the complex conjugation. The predictions of the matrices in a pair are identical except the signs of the CP phases. So we can consider UIa, UIb, UIIa, and UIIb as representatives.


## 3.2. Mixing Angles and CP Invariants

Employing the parametrization of the form(40)UPMNS=c12c13s12c13s13e−iδ−s12c23−c12s13s23eiδc12c23−s12s13s23eiδc13s23s12s23−c12s13c23eiδ−c12s23−s12s13c23eiδc13c231000eiα/2000eiβ/2+δ,where sij≡sinθij, cij≡cosθij, δ is the Dirac CP-violating phase, and α and β are Majorana phases; we could obtain lepton mixing angles and the CP invariants Jcp [72], J1, and J2 defined as
(41)Jcp≡ImU11U13∗U31∗U33=18sin2θ13sin2θ23sin2θ12cosθ13sinδ,J1≡ImU11∗2U122=sin2θ12cos2θ12cos4θ13sinα,J2≡ImU11∗2U132=sin2θ13cos2θ13cos2θ12sinβ.Their specific forms are listed as follows.UIa:
(42)sin2θ13θe,θν≃0.3125−0.1758cos2θν+0.02073cos2θν−2θe−0.1654cos2θe+0.2786cos2θν+2θe+0.065086sin2θν−0.02338sin2θν−2θe−0.08839sin2θe+0.03824sin2θν−2θe,sin2θ23θe,θν≃0.002647cos2θecos2θν+0.05023cos2θesin2θν+0.9531cos2θesin2θν+0.02677sin2θecos2θν+0.2579sin2θesin2θν+0.27067sin2θecos2θν+0.150sin2θesin2θν+0.07994sin2θesin2θν+0.02361sin2θesin2θν1−sin2θ13θe,θν,sin2θ12θe,θν=sin2θ23θe+π/2,θν+π/21−sin2θ13θe+π/2,θν+π/21−sin2θ13θe,θν,Jcpθe,θν=0,J1θe,θν=0,J2θe,θν=0.UIb:
(43)sin2θ13θe,θν≃0.2438cos2θecos2θν+0.1871cos2θesin2θν+0.01618cos2θesin2θν+0.264sin2θecos2θν+0.1659sin2θesin2θν−0.16566sin2θesin2θν+0.712sin2θecos2θν−0.187sin2θesin2θν+0.13245sin2θesin2θν,sin2θ23θe,θν≃0.375−0.3307cos2θν1−sin2θ13θe,θν,sin2θ12θe,θν≃0.375+0.2194cos2θe−0.2475sin2θe1−sin2θ13θe,θν,Jcpθe,θν≃0.001543+0.008446cos2θν+0.006903cos4θν+0.002733cos2θν−4θe−0.003911cos4θν−2θe+0.003405cos2θν−2θe+0.000384cos4θν−4θe−0.002896cos2θe+0.001353cos4θe−0.008932cos2θν+2θe−0.002234cos4θν+4θe−0.00114cos4θν+2θe−0.00323cos2θν+4θe−0.00951sin2θν+0.000585sin4θν+0.002935sin2θν−4θe−0.000292sin4θν−2θe−0.02566sin2θν−2θe+0.000304sin4θν−4θe+0.00328sin2θe−0.00591sin4θe−0.000519sin2θν+2θe+0.00715sin4θν+4θe−0.00775sin4θν+2θe+0.00388sin2θν+4θe,J1θe,θν≃±0.05167−0.00391cos2θν−0.00304cos2θν−4θe−0.01371cos2θν−2θe+0.01037cos2θe+0.006185cos4θe+0.04114cos2θν+2θe−0.00491cos2θν+4θe+0.02521sin2θν−4θe−0.01546sin2θν−2θe−0.1169sin2θe+0.0153sin4θe−0.0464sin2θν+2θe−0.0407sin2θν+4θe,J2θe,θν≃±0.00726cos2θν−4θe−0.0248cos2θν−2θe+0.05787cos2θν+2θe−0.05079cos2θν+2θe−0.07308sin2θν−0.000875sin2θν−4θe+0.02199sin2θν−2θe+0.0513sin2θν+2θe+0.00612sin2θν+4θe.UIIa:
(44)sin2θ13θe,θν≃0.616cos2θecos2θν−0.133cos2θesin2θν+0.0769cos2θesin2θν−0.1328sin2θecos2θν+0.01762sin2θesin2θν−0.1322sin2θesin2θν+0.0769sin2θecos2θν−0.1322sin2θesin2θν+0.4155sin2θesin2θν,sin2θ23θe,θν≃0.2963+0.0502cos2θν−0.1185cos2θν−2θe−0.0502cos2θe−0.101cos2θν+2θe−0.01325sin2θν+0.01325sin2θe+0.000294sin2θν+2θe1−sin2θ13,sin2θ12θe,θν=sin2θ23θe+π/2,θν+π/21−sin2θ13θe+π/2,θν+π/21−sin2θ13θe,θν,JCPθe,θν≃0.00919cos2θν−2θe−0.01601cos2θν+2θe+0.06163sin2θν+2θe,J1θe,θν≃±−0.00475−0.0474cos2θν+0.0144cos2θν−4θe+0.0203cos2θν−2θe+0.0541cos2θe−0.0461cos4θe+0.0188cos2θν+2θe−0.03285cos2θν+4θe+0.002649sin2θν−0.0101sin2θν−4θe+0.02699sin2θν−2θe+0.02349sin2θe+0.05424sin4θe−0.01937sin2θν+2θe−0.04581sin2θν+4θe,J2θe,θν=J1θe,θν+π2.UIIb:
(45)sin2θ13θe,θν≃0.2963−0.1324cos2θν+0.199cos2θν−2θe−0.0502cos2θe−0.05315cos2θν+2θe+0.1464sin2θν−0.00634sin2θν−2θe+0.1325sin2θe−0.01973sin2θν+2θe,sin2θ23θe,θν≃0.4074+0.2648cos2θν−0.2927sin2θ1−sin2θ13,sin2θ12θe,θν=sin2θ13θe,θν+π/21−sin2θ13θe,θν,JCPθe,θν≃0.0261cos2θν−2θe+0.02447cos2θν+2θe+0.04sin2θν−2θe+0.01804sin2θν+2θe,J1θe,θν≃±0.01355+0.01357cos2θν+0.02594cos2θν−4θe+0.00115cos2θν−2θe−0.02127cos2θe+0.0643cos4θe−0.03256cos2θν+2θe+0.0213cos2θν+4θe−0.03385sin2θν−0.04857sin2θν−4θe−0.02714sin2θν−2θe−0.001069sin2θe+0.03351sin4θe−0.007202sin2θν+2θe−0.002316sin2θν+4θe,J2θe,θν=J1θe,θν+π2.The sign ofJ1 and J2 is uncertain. It is dependent on the index j and k in the diagonal phase matrix Pν.


## 3.3. Constraints on (θe, θν) and Best Fit Data

The parameter space of (θe, θν) is shown in Figure 1. The best fit data of lepton mixing angle and CP phases are listed in Table 2. We make some comments on the numerical results:Figure 1
Parameter space for the mixing patterns constrained by the global fit data [67] at the 3σ level in the normal mass ordering (NO). The parameter spaces for θ12 are the strips with blue boundaries, and those for θ23 are with green boundaries. The strips for θ13 are tiny, i.e., almost reduced to black curves. Their intersection areas are signed by the red dots. The parameter spaces in the case of inverted mass ordering (IO) are similar. So they are not shown here.Table 2
Best fit data of lepton mixing angles and CP phases. “N” and “I” denote the normal ordering of neutrino masses and the inverted ordering, respectively.
Patterns
(θebf, θνbf)
χmin2
sin2θ13
sin2θ23
sin2θ12
sinδ
sinα
sinβ
Ia(N)
(0.18725π, 0.1644π), (-0.0297π, 0.72267π)
0.000016
0.0216
0.547
0.320
0
0
0
Ia(I)
(0.18596π, 0.16544π), (-0.0297π, 0.72172π)
0.021
0.0220
0.549
0.318
0
0
0
Ib(N)
(-0.4028π, 0.3393π), (-0.8663π, -0.3393π)
2.33
0.0217
0.563
0.345
0.813
0.952
0.43
Ib(I)
(-0.4037π, 0.340π), (-0.8654π, -0.340π)
1.97
0.0221
0.565
0.343
0.817
0.953
0.42
IIa(N)
(0.3694π, 0.0767π)
0.064
0.0216
0.549
0.316
0.997
0.769
0.588
IIa(I)
(0.3701π, 0.0745π)
0.057
0.022
0.553
0.316
0.998
0.774
0.547
IIb(N)
(-0.2345π, 0.06297π)
0.79
0.02156
0.5507
0.304
0.800
0.617
0.747
IIb(I)
(-0.2348π, 0.06145π)
0.81
0.0220
0.5546
0.304
0.742
0.616
0.730(i) In a period of the parameters (θe, θν), there are two best fit points for the mixing matrices UIa and UIb. For UIb, these two points give the same magnitude of sinδ with different signs. For the mixing matrices UIIa and UIIb, there is only one best fit point in a period. sinδ for them are both positive. Even so, UIIa∗ and UIIb∗ could give negative sinδ while the mixing angles are kept the same. (ii) The best fit value of θ23 from the global fit data [67] is in the second octant. Accordingly, our fit data is in the same octant. In the case of normal mass ordering (NO), the best fit value of δ in UIb and UIIb∗ could be around −0.3π. It is in the 1σ range of the global fit data. In the case of inverted mass ordering (IO), the best fit value of δ in UIIa∗ is around −0.5π. It is also in the 1σ range of the global fit data.


## 3.4. The Effective Mass of Neutrinoless Double-Beta Decay<mee>

Although the residual symmetries (Z2e×CPe, Z2ν×CPν) cannot restrain the masses of neutrinos mi with i=1,2,3, they may affect the effective mass of neutrinoless double-beta decay <mee> through the mixing angles and Majorana phases. Here, <mee> is expressed as
(46)mee≡m1U112+m2U122+m3U132.Employing the lepton mixing matrixUθebf,θνbf and the best fit data on Δm122,Δm132 [67], we plot <mee> against the mass of the lightest neutrino m0 in Figure 2. For (θe, θν) taken from the 3σ range around the best fit data, the curves of <mee> in every pattern are shown in Figure 3. We make some comments on the main results shown in these figures.
(i)
In the case of IO, these patterns give stringent constraints on the ranges of<mee>. Particularly, <mee> for patterns with the indexes (1, 0) and (1, 1) is independent of the parameters (θe, θν).
In the range ofm0 favored by cosmology, <mee> is around 0.045 eV for patterns Ia, IIa, and IIb. For pattern Ib, it is 0.04 eV. These values approximate the upper limit from the global fit data at the 3σ level. They are in the reach of future double-beta decay experiments [73]. Interestingly, similar observations still hold for the patterns from the group S4 with GCP [62]
(ii)
In the case of NO, the variance of<mee> is noticeable for every pattern. Particularly, for patterns Ia and Ib, <mee> with the indexes (1, 1) could reach the upper limit at the 3σ level. Even so, it is not accessible to near future experiments
(iii)
In both NO and IO case, without the precise constraint on the Dirac CP phase, these four mixing patterns cannot be discriminated by future double-beta decay experiments because of the overlaps of ranges of<mee>Figure 2
The effective mass of neutrinoless double-beta decaymee against the mass of the lightest neutrino m0 with the best fit data (θebf, θvbf). The bound on m0 from cosmology is taken from Ref. [74]. The constraint on mee is from Ref. [75]. The best fit data of Δm122 and Δm132 is from Ref. [67]. The legends for every pattern are shown in the top left panel. The indexes (j, k) are defined in Equation (21). The best fit data (θebf, θvbf) for pattern Ia and that for pattern Ib take the second one in Table 2, respectively.Figure 3
The effective mass of neutrinoless double-beta decaymee against the mass of the lightest neutrino m0 in the 3σ ranges of (θe, θν). The conventions follow those in Figure 2. The dashed boundary lines at the 3σ level are obtained from the global fit data [67].


## 4. Summary

For the groupPSL2(7) with GCP symmetries, the predictions of the residual symmetries Z2×CP in both neutrino and charged lepton sectors are examined. Seven types of viable mixing patterns at the 3σ level of the global fit data are obtained. Among them, six types are paired through the complex conjugation. Three types of patterns can give the Dirac CP phase which is in the 1σ range of the global fit data. With the parameters (θe, θν), the constraints of residual symmetries on the effective mass of neutrinoless double-beta decay are also examined. In the case of IO, every pattern can give the effective mass accessible to the future experiments.



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