New Partial Symmetries from Group Algebras for Lepton Mixing

Recent stringent experiment data of neutrino oscillations induces partial symmetries such as Z2 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.


Introduction
Discoveries of neutrino oscillation [1][2][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][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][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 G f . After symmetry breaking from vacuum expectation values of scalar multiplets, G f is reduced to G e in the charged lepton section and G ν in the neutrino section. Accordingly, the mass matrix of charged leptons is invariant under some unitary transformation, i.e., The counterparts for Dirac neutrinos are written as For Majorana neutrinos, they read So residual symmetries X e and X ν can determine the lepton mixing matrix U PMNS ≡ U + e U ν 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 angle θ 13 . Although some large groups could give a viable θ 13 , 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 Z 2 [23][24][25] and Z 2 × CP [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43]. Here CP denotes a generalized CP transformation (GCP). For Z 2 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 (Z 2e × Z 2e , Z 2ν × CP ν ) or (Z ne , Z 2ν × CP ν ) with n ≥ 3, the Dirac CP phase would be trivial or maximal in the case that the residual flavor group is from small groups S 4 and A 5 [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 (Z 2e × CP e , Z 2ν × CP ν ) [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 K½G is the set of all linear combinations of elements of the group G with coefficients in the field K. A general element of K½G is denoted as K½G is an algebra over K with the addition and multiplication defined, respectively, as where the operation "·" denotes the multiplication of group elements. The product by a scalar is defined as From the above definitions, we can see that a group algebra describes the superposition of symmetries expressed by group elements. Similar to the residual symmetry Z 2 × 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 where A 1e,ν and A 2e,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, X e,ν seems strange. It is not a group element in general. The choice of A i seems random. To realize the characteristic of the novel construct, we study a minimal case with the S 3 group algebra. We find that X in the S 3 group algebra is equivalent to the symmetry Z 2 × CP in the case of Dirac neutrinos. Furthermore, the maximal or trivial Dirac CP phase could be obtained from X in the S 4 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 Section 2, we show an economical realization of group algebras. In Section 3, we study a minimal case with an S 3 group algebra. Finally, we give a conclusion.

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 of X 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, where the signal " * " denotes the complex conjugation. An economical solution to the constraint equations is where α is the phase of the term x 1 x * 2 and O is the zero matrix. Up to a global phase, by a redefinition of the matrix A 1 or A 2 , X can be parameterized as [46] 2 Advances in High Energy Physics where i is the imaginary factor and A 1 and A 2 satisfy the constraints So A 1 A + 2 and A + 1 A 2 are generators of Z 2 groups. X can be rewritten as X = A 1 e iθB with B = A + 1 A 2 , B 2 = I. Let us make some necessary comments here: (a) For Majorana neutrinos, the residual symmetry is Z 2 × Z 2 . It can be broken to the partial symmetry Z 2 . X depends on a continuous parameter θ. It is not a Z 2 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 elements A i 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 from XðθÞ is dependent on a parameter θ. Furthermore, XðθÞ is equivalent to Z 2 × CP in the case of S 3 group algebras. This interesting observation still holds for some elements of S 4 group algebras (d) Although X 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

A Minimal Case for S 3 Group Algebra
For illustration, we consider a minimal case that the group algebra is constructed by elements of the group S 3 . Although the 3-dimensional representation of S 3 group algebras is reducible, it can be viewed as the special case of S 4 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 X ν . Then, we show equivalence of elements of S 3 group algebras and the residual symmetry Z 2 × CP. Comments on S 4 group algebras are also made. Finally, we discuss general residual symmetries of the charged lepton sector.
According to the unitary conditions of Equation (12), viable nontrivial realizations of X ν are listed as 13 , 23 , 13 , 12 , 3 Advances in High Energy Physics All theses X ν correspond to the same lepton mixing matrix up to permutations of rows, columns, or trivial phases. We consider X 1ν as a representative, whose expression is It is diagonalized as where e iθ 1 ≡ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi where c ≡ cos θ, N j ≡ 2 + ð1 + c 2 − 2c cos ðθ − 2θ j ÞÞ/s 2 , j = 1, 2.
It is of trimaximal form with the μ − τ reflection symmetry [27,[48][49][50], i.e., U α2 = 1/ ffiffi ffi 3 p with α = e, μ, τ and jU μj j = jU τj j with j = 1, 2, 3. The lepton mixing matrix U PMNS is equal to U ν up to permutations of rows or columns. Given the recent global fit data of neutrino oscillations [51], viable mixing matrices are Note that U 3 ðθÞ = U 1 ðθ + πÞ and U 4 ðθÞ = U 2 ðθ + πÞ. Furthermore, according to the standard parametrization [52] where s ij ≡ sin θ ij and c ij ≡ cos θ ij , δ CP is the Dirac CP-violating phase, α 1 and α 2 are Majorana phases, and U 1 and U 2 are interchanged through the following transforma-tion: θ 23 → π/2 − θ 23 and δ CP → δ CP + π. So without loss of generality, we can just consider U 1 . Lepton mixing angles and the Dirac CP phase are listed as sin 2 θ 12 = 1 3 cos 2 θ 13 , where s ≠ 0. Dependence of sin 2 θ 13 and sin 2 θ 12 on the variable θ is shown in Figure 1. From the figure, we can see that sin 2 θ 12 is a slowly varying function of the parameter θ. So the parameter space of θ is mainly constrained by sin 2 θ 13 . According to the function χ 2 defined as where ðsin 2 θ ij Þ exp are best global fit values from Ref. [51] and σ ij are 1σ uncertainties; best fit data of θ, sin 2 θ ij , and δ CP are listed in Table 1. They are in the 3σ ranges of the global fit data.

Equivalence of Elements of S 3 Group Algebras and
where m ee and m ττ are real and Im ðm eμ Þ = ð1/2Þðm ee − m ττ Þ tan θ. Obviously, M + ν M ν follows the residual symmetry Z 2 × CP, i.e., where 4 Advances in High Energy Physics Correspondingly, for X 1ν we have S 23 works as the GCP for the mass matrix M + ν M ν on the one hand. On the other hand, it acts as an equivalent transformation for symmetries X 1ν and X 2ν . So X 1ν is equivalent to the residual symmetry Z Magic 2 × CP.

Comments on Equivalence of Elements of S 4 Group
Algebras and Z 2 × CP. For the S 4 group with the GCP, the residual symmetries Z 2 × CP could bring maximal or trivial Dirac CP phase. We have seen that X ν ≅ Z 2 × CP in S 3 group algebras gives a maximal CP phase.
In fact, the equivalence can still hold for some X in S 4 group algebras which are not elements of S 3 group algebras. The trivial CP phase could be obtained from X. Here, we give an example of X from S 4 group algebras with a different are expressed as [32] S = 1 3 where ω = e i2π/3 . A nontrivial example of the S 4 group algebra element could be X = ðTVÞ cos θ + i sin θðSTVÞ. Its specific expression is of the form [46] X θ ð Þ = If we take X ν = XðθÞ and suppose that the mass matrix of charged leptons is diagonal, we can obtain the lepton mixing matrix written as where c 1 ≡ cos θ 1 , s 1 ≡ sin θ 1 , and θ 1 is a parameter constrained by the mixing angle θ 13 . So the mixing pattern is of trimaximal form with a trivial Dirac CP-violating phase. For XðθÞ, we can verify that the following relation holds, i.e., where C 1 = T + ST, C 2 1 = I, and T + C 1 T = C * 1 . So C 1 and T are a Z 2 symmetry and the corresponding CP transformation, respectively. Following the methods used in GCP [30], the lepton mixing matrix from the residual symmetry Z 2 × CP can be expressed as U a = ΩR 13 ðθ 1 ÞP, where Ω and R 13 ðθ 1 Þ are expressed, respectively, as 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 We can check that the matrix U a from the Z 2 × CP is just the U shown in Equation (29). So XðθÞ is equivalent to the symmetry Z 2 × CP generated by C 1 and T. Furthermore, let us consider the element X ′ðθÞ ≡ T + XðθÞT. The lepton mixing matrix from X ′ðθÞ is U ′ = T + U. Since T is a phase matrix, U ′ is equivalent to U. So the CP transformation interchanges the equivalent elements XðθÞ and X ′ðθÞ. Therefore, the observation from the case of the S 3 algebra still holds in this example of the S 4 group algebra.
3.4. Discussion on General Residual Symmetries of the Charged Lepton Sector. We have studied the case that the mass matrix M + e M e is diagonal. The corresponding symmetry of the charged lepton sector is Uð1Þ × Uð1Þ × Uð1Þ, namely, X e = diag ðe iα 1 , e iα 2 , e iα 3 Þ. Now we discuss a more general case that X e is expressed by an element of the S 3 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 X 1e = S 23 e iθ e S 12 . Then, the matrix U e is of the form Obviously, it does not satisfy the constraint of the global fit data of neutrino oscillations. So the combination of the residual symmetries (X 1e , X 1ν ) does not give a realistic lepton mixing patten in the case of S 3 group algebra. Furthermore, if θ e is equal to 0, X 1e is reduced to S 23 . The corresponding matrix U e becomes where θ ′ is an angle variable from the degeneracy of the eigenvalues of S 23 . Then, U PMNS contains a zero element. This observation still holds when S 23 is replaced by S 12 or S 13 . So the combination (Z 2e , X 1ν ) is not a viable choice for the residual symmetries of leptons. We can also check that U PMNS from the combination (Z 3e , X 1ν ), where Z 3e is generated by S 123 or S 132 , 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 X 1ν in the S 3 group algebra, we can only take X e = diag ðe iα 1 , e iα 2 , e iα 3 Þ.

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 algebras X i . These equivalent symmetries X i can be interchanged through a transformation which corresponds to a residual CP symmetry. For S 3 group algebras and a special case of S 4 group algebras, we found that X i is equivalent to a residual symmetry Z 2 × CP. The corresponding lepton mixing matrix is trimaximal. It is a difficult mathematical problem for us to determine whether X i is equivalent to Z 2 × 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.

Data Availability
The global fit data supporting this research paper are from previously reported studies, which have been cited. The processed data are freely available.

Conflicts of Interest
The author declares that there is no conflict of interest regarding the publication of this paper.