# New Partial Symmetries from Group Algebras for Lepton Mixing

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

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## Abstract

Recent stringent experiment data of neutrino oscillations induces partial symmetries such asZ2 and Z2×CP to derive lepton mixing patterns. New partial symmetries expressed with elements of group algebras are studied. A specific lepton mixing pattern could correspond to a set of equivalent elements of a group algebra. The transformation which interchanges the elements could express a residual CP symmetry. Lepton mixing matrices from S3 group algebras are of the trimaximal form with the μ−τ reflection symmetry. Accordingly, elements of S3 group algebras are equivalent to Z2×CP. Comments on S4 group algebras are given. The predictions of Z2×CP broken from the group S4 with the generalized CP symmetry are also obtained from elements of S4 group algebras.


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## Body

## 1. Introduction

Discoveries of neutrino oscillation [1–3] opened a window to physics beyond the standard model. In order to explain possible patterns of lepton mixing parameters, discrete flavor symmetries were extensively investigated in recent decades [4–21]. The general route on this approach is as follows. First, suppose that the Lagrangian of leptons is invariant under actions of some finite group Gf.After symmetry breaking from vacuum expectation values of scalar multiplets,Gf is reduced to Ge in the charged lepton section and Gν in the neutrino section. Accordingly, the mass matrix of charged leptons is invariant under some unitary transformation, i.e.,
(1)Xe+Me+MeXe=Me+Me.So we have(2)Ue+Me+MeUe=diagme2,mμ2,mτ2,Ue+XeUe=diageiα1,eiα2,eiα3.The counterparts for Dirac neutrinos are written as(3)Xν+Mν+MνXν=Mν+Mν,Uν+Mν+MνUν=diagm12,m22,m32,Uν+XνUν=diageiβ1,eiβ2,eiβ3.For Majorana neutrinos, they read(4)XνTMνXν=Mν,UνTMνUν=diagm1,m2,m3,Uν+XνUν=diag±1,±1,±1.So residual symmetriesXe and Xν can determine the lepton mixing matrix UPMNS≡Ue+Uν up to permutations of rows or columns.However, mixing patterns based on small flavor groups cannot accommodate new stringent experiment data, especially the nonzero mixing angleθ13. Although some large groups could give a viable θ13, the Dirac CP-violating phase from them is trivial [22]. In order to alleviate the tension between predictions of flavor groups and experiment constraints, one can resort to partial symmetries. Namely, the lepton mixing matrix is partially determined by symmetries such as Z2 [23–25] and Z2×CP [26–43]. Here CP denotes a generalized CP transformation (GCP). For Z2 symmetries, an unfixed unitary rotation is contained in the mixing matrix. Even so, they may predict some mixing angle, Dirac CP phase, or correlation of them. If the residual symmetry is (Z2e×Z2e, Z2ν×CPν) or (Zne, Z2ν×CPν) with n≥3, the Dirac CP phase would be trivial or maximal in the case that the residual flavor group is from small groups S4 and A5 [30, 32, 39]. Here, the symmetries of the charged lepton sector and those of neutrinos are marked with the subscripts e and ν, respectively. To obtain a more general CP phase, one can choose the residual symmetry (Z2e×CPe, Z2ν×CPν) [44, 45]. Then, the lepton mixing matrix contains two angle parameters to constrain by experiment data.In this paper, we explore a new construct to describe partial symmetries which was proposed recently in Ref. [46]. The partial symmetry is expressed by an element of a group algebra. According to Ref. [47], a group algebra KG is the set of all linear combinations of elements of the group G with coefficients in the field K. A general element of KG is denoted as
(5)∑g∈Gagg.KG is an algebra over K with the addition and multiplication defined, respectively, as
(6)∑g∈Gagg+∑g∈Gbgg=∑g∈Gag+bgg,∑g∈Gagg∑h∈Gbhh=∑g∈G,h∈Gagbhg·h,where the operation “·” denotes the multiplication of group elements. The product by a scalar is defined as
(7)a∑g∈Gagg=∑g∈Gaagg.From the above definitions, we can see that a group algebra describes the superposition of symmetries expressed by group elements. Similar to the residual symmetryZ2×CP, the elements of a group algebra with continuous superposition coefficients may also describe partial symmetries of leptons. They may be used to predict the lepton mixing pattern. For simplicity, we consider the group algebra constructed by two group elements in this paper. Namely, the residual symmetry is expressed as
(8)Xe,ν=x1e,νA1e,ν+x2e,νA2e,ν,where A1e,ν and A2e,ν are elements of a small group. Through equivalent transformations, the superposition coefficients are dependent on a real parameter in a special parametrization. So we can obtain clear relations between mixing parameters and the adjustable coefficient. In spite of the economy of the structure, Xe,ν seems strange. It is not a group element in general. The choice of Ai seems random. To realize the characteristic of the novel construct, we study a minimal case with the S3 group algebra. We find that X in the S3 group algebra is equivalent to the symmetry Z2×CP in the case of Dirac neutrinos. Furthermore, the maximal or trivial Dirac CP phase could be obtained from X in the S4 group algebra. Although we cannot prove that the equivalence holds for X in a general algebra, we may have more choices in the realization of partial symmetries.This paper is organised as follows. In Section2, we show an economical realization of group algebras. In Section 3, we study a minimal case with an S3 group algebra. Finally, we give a conclusion.


## 2. Realization of a Group Algebra

An element of a group algebra is constructed by the superposition of elements of a group. Here, we consider the elements of group algebras obtained from two group elements. We note that the representation matrix ofX is not unitary in general even if the representation of the group elements is unitary. In order to keep the representation of X unitary, we set extra constraints on coefficients and group elements, namely,
(9)x12+x22I+x1x2∗A1A2++x2x1∗A2A1+=I,x12+x22I+x2x1∗A1+A2+x1x2∗A2+A1=I,where the signal “∗” denotes the complex conjugation. An economical solution to the constraint equations is
(10)x12+x22=1,eiαA1A2++e−iαA2A1+=O,e−iαA1+A2+eiαA2+A1=O,where α is the phase of the term x1x2∗ and O is the zero matrix. Up to a global phase, by a redefinition of the matrix A1 or A2, X can be parameterized as [46]
(11)Xθ=cosθA1+isinθA2,where i is the imaginary factor and A1 and A2 satisfy the constraints
(12)A1A2+=A2A1+,A1+A2=A2+A1.SoA1A2+ and A1+A2 are generators of Z2 groups. X can be rewritten as X=A1eiθB with B=A1+A2, B2=I.Let us make some necessary comments here:(a)
For Majorana neutrinos, the residual symmetry isZ2×Z2. It can be broken to the partial symmetry Z2. X depends on a continuous parameter θ. It is not a Z2 symmetry in general. So X is used for the description of residual symmetries of charged leptons and Dirac neutrinos
(b)
With a special choice of group elementsAi and the parameter θ, X could become a generator of a large cyclic group. An example is given in Ref. [46]
(c)
The mixing matrix fromXθ is dependent on a parameter θ. Furthermore, Xθ is equivalent to Z2×CP in the case of S3 group algebras. This interesting observation still holds for some elements of S4 group algebras
(d)
AlthoughX is dependent on the parameter θ, some mixing angle or CP phase may be independent of θ. We may separate impacts of discrete group elements and θ in special cases


## 3. A Minimal Case forS3 Group Algebra

For illustration, we consider a minimal case that the group algebra is constructed by elements of the groupS3. Although the 3-dimensional representation of S3 group algebras is reducible, it can be viewed as the special case of S4 group algebras. In this section, we first consider the special case that the mass matrix of charged leptons is diagonal. So the lepton mixing matrix is just dependent on the residual symmetry Xν. Then, we show equivalence of elements of S3 group algebras and the residual symmetry Z2×CP. Comments on S4 group algebras are also made. Finally, we discuss general residual symmetries of the charged lepton sector.


### 3.1. Mixing Patterns fromS3 Group Algebra in the Case of the Diagonal Mass Matrix Me+Me

The 3-dimensional reducible representation of the groupS3 is expressed as
(13)I=100010001,S12=010100001,S13=001010100,S23=100001010,S123=001100010,S132=010001100.According to the unitary conditions of Equation (12), viable nontrivial realizations of Xν are listed as
(14)X1ν≡S23eiθS12,X2ν≡S23eiθS13,X3ν≡S12eiθS13,X4ν≡S12eiθS23,X5ν≡S13eiθS12,X6ν≡S13eiθS23,X7ν≡S123eiθS12,X8ν≡S123eiθS23,X9ν≡S123eiθS13,X10ν≡S132eiθS12,X11ν≡S132eiθS23,X12ν≡S132eiθS13.All thesesXν correspond to the same lepton mixing matrix up to permutations of rows, columns, or trivial phases. We consider X1ν as a representative, whose expression is
(15)X1ν≡cosθisinθ000eiθisinθcosθ0.It is diagonalized as(16)Uν+X1νUν=diageiθ1,eiθ,eiθ2,where eiθ1≡1−s2/4−is/2, eiθ2≡−1−s2/4−is/2, s≡sinθ. The matrix Uν reads
(17)Uν=eiθ1−ceiθ−θ1isN113eiθ2−ceiθ−θ2isN21N1131N2eiθ−θ1N113eiθ−θ2N2,where c≡cosθ, Nj≡2+1+c2−2ccosθ−2θj/s2, j=1,2. It is of trimaximal form with the μ−τ reflection symmetry [27, 48–50], i.e., Uα2=1/3 with α=e,μ,τ and Uμj=Uτj with j=1,2,3. The lepton mixing matrix UPMNS is equal to Uν up to permutations of rows or columns. Given the recent global fit data of neutrino oscillations[51], viable mixing matrices are
(18)U1≡Uν,U2≡S23U1,U3≡U1S13,U4≡U2S13.Note thatU3θ=U1θ+π and U4θ=U2θ+π. Furthermore, according to the standard parametrization [52]
(19)UPMNS=c12c13s12c13s13e−iδCP−s12c23−c12s13s23eiδCPc12c23−s12s13s23eiδCPc13s23s12s23−c12s13c23eiδCP−c12s23−s12s13c23eiδCPc13c23eiα1000eiα20001,where sij≡sinθij and cij≡cosθij, δCP is the Dirac CP-violating phase, α1 and α2 are Majorana phases, and U1 and U2 are interchanged through the following transformation: θ23→π/2−θ23 and δCP→δCP+π. So without loss of generality, we can just consider U1. Lepton mixing angles and the Dirac CP phase are listed as
(20)sin2θ13=1+c2−2ccosθ−2θ22+s2−2ccosθ−2θ2,sin2θ12=13cos2θ13,sin2θ23=12,δCP=−signsπ2,where s≠0. Dependence of sin2θ13 and sin2θ12 on the variable θ is shown in Figure 1. From the figure, we can see that sin2θ12 is a slowly varying function of the parameter θ. So the parameter space of θ is mainly constrained by sin2θ13. According to the function χ2 defined as
(21)χ2=∑ij=13,23,12sin2θij−sin2θijexpσij2,where sin2θijexp are best global fit values from Ref. [51] and σij are 1σ uncertainties; best fit data of θ, sin2θij, and δCP are listed in Table 1. They are in the 3σ ranges of the global fit data.Figure 1
Dependence of functionssin2θ13 and sin2θ12 on the variable θ. Two dashed lines in (a) and (b) label the 3σ range of the mixing angle from the recent global fit [51]. For sin2θ13, we take the 3σ range in the normal mass ordering.
(a)
(b)Table 1
Best fit data of the parametersθ, sinθij, and δCP.
Order
χmin2
θbf
sin2θ13bf
sin2θ23bf
sin2θ12bf
δCPbf
Normal
4.856
±0.131π
0.0216
0.5
0.341
∓π/2
Inverted
5.855
±0.132π
0.0220
0.5
0.341
∓π/2


### 3.2. Equivalence of Elements ofS3 Group Algebras and Z2×CP

The neutrino mass matrixMν+Mν which is invariant under the action of X1ν is of the form
(22)Mν+Mν=meemeμmeμ∗meμ∗mττmeμ+mee−mττmeμmeμ∗+mee−mττmττ,where mee and mττ are real and Immeμ=1/2mee−mττtanθ. Obviously, Mν+Mν follows the residual symmetry Z2×CP, i.e.,
(23)CMagic+Mν+MνCMagic=Mν+Mν,S23+Mν+MνS23=Mν+Mν∗,where(24)CMagic≡131−2−2−21−2−2−21,CMagic2=I.Correspondingly, forX1ν we have
(25)CMagic+X1νCMagic=X1ν,S23+X1νS23=X2ν.S23 works as the GCP for the mass matrix Mν+Mν on the one hand. On the other hand, it acts as an equivalent transformation for symmetries X1ν and X2ν. So X1ν is equivalent to the residual symmetry Z2Magic×CP.


### 3.3. Comments on Equivalence of Elements ofS4 Group Algebras and Z2×CP

For theS4 group with the GCP, the residual symmetries Z2×CP could bring maximal or trivial Dirac CP phase. We have seen that Xν≅Z2×CP in S3 group algebras gives a maximal CP phase.In fact, the equivalence can still hold for someX in S4 group algebras which are not elements of S3 group algebras. The trivial CP phase could be obtained from X. Here, we give an example of X from S4 group algebras with a different representation. Three generators of S4 which satisfy the relation [32]
(26)S2=V2=SV2=TV2=I,T3=ST3=I,STV4=I,are expressed as [32]
(27)S=13−1222−1222−1,T=1000ω2000ω,V=∓100001010,where ω=ei2π/3. A nontrivial example of the S4 group algebra element could be X=TVcosθ+isinθSTV. Its specific expression is of the form [46]
(28)Xθ=i3sinθ−cosθ23eiπ/6sinθ−23e−iπ/6sinθ−2i3sinθ23eiπ/6sinθ13e−iπ/6sinθ−e−i2π/3cosθ−2i3sinθ−13eiπ/6sinθ−ei2π/3cosθ−23e−iπ/6sinθ.If we takeXν=Xθ and suppose that the mass matrix of charged leptons is diagonal, we can obtain the lepton mixing matrix written as
(29)U=diag1,ω,ω2·−23c113−23s116c1+12s11316s1−12c116c1−12s11316s1+12c1,where c1≡cosθ1, s1≡sinθ1, and θ1 is a parameter constrained by the mixing angle θ13. So the mixing pattern is of trimaximal form with a trivial Dirac CP-violating phase. For Xθ, we can verify that the following relation holds, i.e.,
(30)C1+XθC1=Xθ,where C1=T+ST, C12=I, and T+C1T=C1∗. So C1 and T are a Z2 symmetry and the corresponding CP transformation, respectively. Following the methods used in GCP [30], the lepton mixing matrix from the residual symmetry Z2×CP can be expressed as Ua=ΩR13θ1P, where Ω and R13θ1 are expressed, respectively, as
(31)Ω=−23130ω6ω3−ω2ω26ω23ω22,R13θ1=cosθ10sinθ1010−sinθ10cosθ1,P is a phase matrix which can be neglected in our case of Dirac neutrinos. In particular, the matrix Ω satisfies the relations as follows
(32)Ω+C1Ω=diag−1,1,−1,T=ΩΩT.We can check that the matrixUa from the Z2×CP is just the U shown in Equation (29). So Xθ is equivalent to the symmetry Z2×CP generated by C1 and T. Furthermore, let us consider the element X′θ≡T+XθT. The lepton mixing matrix from X′θ is U′=T+U. Since T is a phase matrix, U′ is equivalent to U. So the CP transformation interchanges the equivalent elements Xθ and X′θ. Therefore, the observation from the case of the S3 algebra still holds in this example of the S4 group algebra.


### 3.4. Discussion on General Residual Symmetries of the Charged Lepton Sector

We have studied the case that the mass matrixMe+Me is diagonal. The corresponding symmetry of the charged lepton sector is U1×U1×U1, namely, Xe=diageiα1,eiα2,eiα3. Now we discuss a more general case that Xe is expressed by an element of the S3 group algebra. Because all the elements listed in Equation (14) give the same mixing matrix up to permutations of rows or columns, we can take X1e=S23eiθeS12. Then, the matrix Ue is of the form
(33)Ue=eiθ1e−ceeiθe−θ1eiseN1e13eiθ2e−ceeiθe−θ2eiseN2e1N1e131N2eeiθe−θ1eN1e13eiθe−θ2eN2e,where eiθ1e≡1−se2/4−ise/2, eiθ2e≡−1−se2/4−ise/2, se≡sinθe, ce≡cosθe, and Nje≡2+1+ce2−2cecosθe−2θje/se2, j=1,2. With respect to the mixing matrix UPMNS≡Ue+Uν, we have an element UPMNSαi=1. Obviously, it does not satisfy the constraint of the global fit data of neutrino oscillations. So the combination of the residual symmetries (X1e,X1ν) does not give a realistic lepton mixing patten in the case of S3 group algebra. Furthermore, if θe is equal to 0, X1e is reduced to S23. The corresponding matrix Ue becomes
(34)Ue′=0−sinθ′cosθ′−12cosθ′2sinθ′212cosθ′2sinθ′2,where θ′ is an angle variable from the degeneracy of the eigenvalues of S23. Then, UPMNS contains a zero element. This observation still holds when S23 is replaced by S12 or S13. So the combination (Z2e,X1ν) is not a viable choice for the residual symmetries of leptons. We can also check that UPMNS from the combination (Z3e,X1ν), where Z3e is generated by S123 or S132, does not satisfy the constraint of the global fit data of neutrino oscillations either. It contains an element which is equal to 1. Therefore, when the residual symmetry of the neutrino sector is X1ν in the S3 group algebra, we can only take Xe=diageiα1,eiα2,eiα3.


## 3.1. Mixing Patterns fromS3 Group Algebra in the Case of the Diagonal Mass Matrix Me+Me

The 3-dimensional reducible representation of the groupS3 is expressed as
(13)I=100010001,S12=010100001,S13=001010100,S23=100001010,S123=001100010,S132=010001100.According to the unitary conditions of Equation (12), viable nontrivial realizations of Xν are listed as
(14)X1ν≡S23eiθS12,X2ν≡S23eiθS13,X3ν≡S12eiθS13,X4ν≡S12eiθS23,X5ν≡S13eiθS12,X6ν≡S13eiθS23,X7ν≡S123eiθS12,X8ν≡S123eiθS23,X9ν≡S123eiθS13,X10ν≡S132eiθS12,X11ν≡S132eiθS23,X12ν≡S132eiθS13.All thesesXν correspond to the same lepton mixing matrix up to permutations of rows, columns, or trivial phases. We consider X1ν as a representative, whose expression is
(15)X1ν≡cosθisinθ000eiθisinθcosθ0.It is diagonalized as(16)Uν+X1νUν=diageiθ1,eiθ,eiθ2,where eiθ1≡1−s2/4−is/2, eiθ2≡−1−s2/4−is/2, s≡sinθ. The matrix Uν reads
(17)Uν=eiθ1−ceiθ−θ1isN113eiθ2−ceiθ−θ2isN21N1131N2eiθ−θ1N113eiθ−θ2N2,where c≡cosθ, Nj≡2+1+c2−2ccosθ−2θj/s2, j=1,2. It is of trimaximal form with the μ−τ reflection symmetry [27, 48–50], i.e., Uα2=1/3 with α=e,μ,τ and Uμj=Uτj with j=1,2,3. The lepton mixing matrix UPMNS is equal to Uν up to permutations of rows or columns. Given the recent global fit data of neutrino oscillations[51], viable mixing matrices are
(18)U1≡Uν,U2≡S23U1,U3≡U1S13,U4≡U2S13.Note thatU3θ=U1θ+π and U4θ=U2θ+π. Furthermore, according to the standard parametrization [52]
(19)UPMNS=c12c13s12c13s13e−iδCP−s12c23−c12s13s23eiδCPc12c23−s12s13s23eiδCPc13s23s12s23−c12s13c23eiδCP−c12s23−s12s13c23eiδCPc13c23eiα1000eiα20001,where sij≡sinθij and cij≡cosθij, δCP is the Dirac CP-violating phase, α1 and α2 are Majorana phases, and U1 and U2 are interchanged through the following transformation: θ23→π/2−θ23 and δCP→δCP+π. So without loss of generality, we can just consider U1. Lepton mixing angles and the Dirac CP phase are listed as
(20)sin2θ13=1+c2−2ccosθ−2θ22+s2−2ccosθ−2θ2,sin2θ12=13cos2θ13,sin2θ23=12,δCP=−signsπ2,where s≠0. Dependence of sin2θ13 and sin2θ12 on the variable θ is shown in Figure 1. From the figure, we can see that sin2θ12 is a slowly varying function of the parameter θ. So the parameter space of θ is mainly constrained by sin2θ13. According to the function χ2 defined as
(21)χ2=∑ij=13,23,12sin2θij−sin2θijexpσij2,where sin2θijexp are best global fit values from Ref. [51] and σij are 1σ uncertainties; best fit data of θ, sin2θij, and δCP are listed in Table 1. They are in the 3σ ranges of the global fit data.Figure 1
Dependence of functionssin2θ13 and sin2θ12 on the variable θ. Two dashed lines in (a) and (b) label the 3σ range of the mixing angle from the recent global fit [51]. For sin2θ13, we take the 3σ range in the normal mass ordering.
(a)
(b)Table 1
Best fit data of the parametersθ, sinθij, and δCP.
Order
χmin2
θbf
sin2θ13bf
sin2θ23bf
sin2θ12bf
δCPbf
Normal
4.856
±0.131π
0.0216
0.5
0.341
∓π/2
Inverted
5.855
±0.132π
0.0220
0.5
0.341
∓π/2


## 3.2. Equivalence of Elements ofS3 Group Algebras and Z2×CP

The neutrino mass matrixMν+Mν which is invariant under the action of X1ν is of the form
(22)Mν+Mν=meemeμmeμ∗meμ∗mττmeμ+mee−mττmeμmeμ∗+mee−mττmττ,where mee and mττ are real and Immeμ=1/2mee−mττtanθ. Obviously, Mν+Mν follows the residual symmetry Z2×CP, i.e.,
(23)CMagic+Mν+MνCMagic=Mν+Mν,S23+Mν+MνS23=Mν+Mν∗,where(24)CMagic≡131−2−2−21−2−2−21,CMagic2=I.Correspondingly, forX1ν we have
(25)CMagic+X1νCMagic=X1ν,S23+X1νS23=X2ν.S23 works as the GCP for the mass matrix Mν+Mν on the one hand. On the other hand, it acts as an equivalent transformation for symmetries X1ν and X2ν. So X1ν is equivalent to the residual symmetry Z2Magic×CP.


## 3.3. Comments on Equivalence of Elements ofS4 Group Algebras and Z2×CP

For theS4 group with the GCP, the residual symmetries Z2×CP could bring maximal or trivial Dirac CP phase. We have seen that Xν≅Z2×CP in S3 group algebras gives a maximal CP phase.In fact, the equivalence can still hold for someX in S4 group algebras which are not elements of S3 group algebras. The trivial CP phase could be obtained from X. Here, we give an example of X from S4 group algebras with a different representation. Three generators of S4 which satisfy the relation [32]
(26)S2=V2=SV2=TV2=I,T3=ST3=I,STV4=I,are expressed as [32]
(27)S=13−1222−1222−1,T=1000ω2000ω,V=∓100001010,where ω=ei2π/3. A nontrivial example of the S4 group algebra element could be X=TVcosθ+isinθSTV. Its specific expression is of the form [46]
(28)Xθ=i3sinθ−cosθ23eiπ/6sinθ−23e−iπ/6sinθ−2i3sinθ23eiπ/6sinθ13e−iπ/6sinθ−e−i2π/3cosθ−2i3sinθ−13eiπ/6sinθ−ei2π/3cosθ−23e−iπ/6sinθ.If we takeXν=Xθ and suppose that the mass matrix of charged leptons is diagonal, we can obtain the lepton mixing matrix written as
(29)U=diag1,ω,ω2·−23c113−23s116c1+12s11316s1−12c116c1−12s11316s1+12c1,where c1≡cosθ1, s1≡sinθ1, and θ1 is a parameter constrained by the mixing angle θ13. So the mixing pattern is of trimaximal form with a trivial Dirac CP-violating phase. For Xθ, we can verify that the following relation holds, i.e.,
(30)C1+XθC1=Xθ,where C1=T+ST, C12=I, and T+C1T=C1∗. So C1 and T are a Z2 symmetry and the corresponding CP transformation, respectively. Following the methods used in GCP [30], the lepton mixing matrix from the residual symmetry Z2×CP can be expressed as Ua=ΩR13θ1P, where Ω and R13θ1 are expressed, respectively, as
(31)Ω=−23130ω6ω3−ω2ω26ω23ω22,R13θ1=cosθ10sinθ1010−sinθ10cosθ1,P is a phase matrix which can be neglected in our case of Dirac neutrinos. In particular, the matrix Ω satisfies the relations as follows
(32)Ω+C1Ω=diag−1,1,−1,T=ΩΩT.We can check that the matrixUa from the Z2×CP is just the U shown in Equation (29). So Xθ is equivalent to the symmetry Z2×CP generated by C1 and T. Furthermore, let us consider the element X′θ≡T+XθT. The lepton mixing matrix from X′θ is U′=T+U. Since T is a phase matrix, U′ is equivalent to U. So the CP transformation interchanges the equivalent elements Xθ and X′θ. Therefore, the observation from the case of the S3 algebra still holds in this example of the S4 group algebra.


## 3.4. Discussion on General Residual Symmetries of the Charged Lepton Sector

We have studied the case that the mass matrixMe+Me is diagonal. The corresponding symmetry of the charged lepton sector is U1×U1×U1, namely, Xe=diageiα1,eiα2,eiα3. Now we discuss a more general case that Xe is expressed by an element of the S3 group algebra. Because all the elements listed in Equation (14) give the same mixing matrix up to permutations of rows or columns, we can take X1e=S23eiθeS12. Then, the matrix Ue is of the form
(33)Ue=eiθ1e−ceeiθe−θ1eiseN1e13eiθ2e−ceeiθe−θ2eiseN2e1N1e131N2eeiθe−θ1eN1e13eiθe−θ2eN2e,where eiθ1e≡1−se2/4−ise/2, eiθ2e≡−1−se2/4−ise/2, se≡sinθe, ce≡cosθe, and Nje≡2+1+ce2−2cecosθe−2θje/se2, j=1,2. With respect to the mixing matrix UPMNS≡Ue+Uν, we have an element UPMNSαi=1. Obviously, it does not satisfy the constraint of the global fit data of neutrino oscillations. So the combination of the residual symmetries (X1e,X1ν) does not give a realistic lepton mixing patten in the case of S3 group algebra. Furthermore, if θe is equal to 0, X1e is reduced to S23. The corresponding matrix Ue becomes
(34)Ue′=0−sinθ′cosθ′−12cosθ′2sinθ′212cosθ′2sinθ′2,where θ′ is an angle variable from the degeneracy of the eigenvalues of S23. Then, UPMNS contains a zero element. This observation still holds when S23 is replaced by S12 or S13. So the combination (Z2e,X1ν) is not a viable choice for the residual symmetries of leptons. We can also check that UPMNS from the combination (Z3e,X1ν), where Z3e is generated by S123 or S132, does not satisfy the constraint of the global fit data of neutrino oscillations either. It contains an element which is equal to 1. Therefore, when the residual symmetry of the neutrino sector is X1ν in the S3 group algebra, we can only take Xe=diageiα1,eiα2,eiα3.


## 4. Conclusion

We have studied a new structure to describe partial symmetries of charged leptons and Dirac neutrinos. The residual symmetry is expressed by an element of group algebras. In our construction, a specific lepton mixing pattern corresponds to a set of equivalent residual symmetries which are expressed by elements of group algebrasXi. These equivalent symmetries Xi can be interchanged through a transformation which corresponds to a residual CP symmetry. For S3 group algebras and a special case of S4 group algebras, we found that Xi is equivalent to a residual symmetry Z2×CP. The corresponding lepton mixing matrix is trimaximal. It is a difficult mathematical problem for us to determine whether Xi is equivalent to Z2×CP in general cases. Even so, observations from simple examples could still give us some interesting clues: (a) The parameter in partial symmetries may be viewed as a quantity to measure how discrete symmetries are mixed in the residual symmetry. (b) A partial symmetry dependent on a continuous parameter may be equivalent to a discrete symmetry with GCP. (c) The elementary residual CP transformation could be a permutation matrix or a diagonal phase matrix. A general one may be a finite product of elementary ones. Therefore, despite stringent experiment data, we could still construct some novel partial symmetries to obtain viable lepton mixing patterns.



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*Source: 3967605-2020-02-08.xml*