New Partial Symmetries from Group Algebras for Lepton Mixing

Rong, Shu-Jun

doi:https://doi.org/10.1155/2020/3967605

Advances in High Energy Physics

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 3967605 | https://doi.org/10.1155/2020/3967605

New Partial Symmetries from Group Algebras for Lepton Mixing

Shu-Jun Rong¹

Academic Editor: Edward Sarkisyan-Grinbaum

Received13 Nov 2019

Revised06 Jan 2020

Accepted29 Jan 2020

Published08 Feb 2020

Abstract

Recent stringent experiment data of neutrino oscillations induces partial symmetries such as and 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 symmetry. Lepton mixing matrices from group algebras are of the trimaximal form with the reflection symmetry. Accordingly, elements of group algebras are equivalent to . Comments on group algebras are given. The predictions of broken from the group with the generalized symmetry are also obtained from elements of group algebras.

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 .

After symmetry breaking from vacuum expectation values of scalar multiplets, is reduced to in the charged lepton section and in the neutrino section. Accordingly, the mass matrix of charged leptons is invariant under some unitary transformation, i.e.,

So we have

The counterparts for Dirac neutrinos are written as

For Majorana neutrinos, they read

So residual symmetries and can determine the lepton mixing matrix 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 . Although some large groups could give a viable , the Dirac -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 [23–25] and [26–43]. Here denotes a generalized transformation (). For symmetries, an unfixed unitary rotation is contained in the mixing matrix. Even so, they may predict some mixing angle, Dirac phase, or correlation of them. If the residual symmetry is (, ) or (, ) with , the Dirac phase would be trivial or maximal in the case that the residual flavor group is from small groups and [30, 32, 39]. Here, the symmetries of the charged lepton sector and those of neutrinos are marked with the subscripts and , respectively. To obtain a more general phase, one can choose the residual symmetry (, ) [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 is the set of all linear combinations of elements of the group with coefficients in the field . A general element of is denoted as

is an algebra over 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 , 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 and 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, seems strange. It is not a group element in general. The choice of seems random. To realize the characteristic of the novel construct, we study a minimal case with the group algebra. We find that in the group algebra is equivalent to the symmetry in the case of Dirac neutrinos. Furthermore, the maximal or trivial Dirac phase could be obtained from in the group algebra. Although we cannot prove that the equivalence holds for 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 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 of is not unitary in general even if the representation of the group elements is unitary. In order to keep the representation of 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 is the phase of the term and is the zero matrix. Up to a global phase, by a redefinition of the matrix or , X can be parameterized as [46] where is the imaginary factor and and satisfy the constraints

So and are generators of groups. can be rewritten as with , .

Let us make some necessary comments here: (a)For Majorana neutrinos, the residual symmetry is . It can be broken to the partial symmetry . depends on a continuous parameter . It is not a symmetry in general. So is used for the description of residual symmetries of charged leptons and Dirac neutrinos(b)With a special choice of group elements and the parameter , could become a generator of a large cyclic group. An example is given in Ref. [46](c)The mixing matrix from is dependent on a parameter . Furthermore, is equivalent to in the case of group algebras. This interesting observation still holds for some elements of group algebras(d)Although is dependent on the parameter , some mixing angle or phase may be independent of . We may separate impacts of discrete group elements and in special cases

3. A Minimal Case for Group Algebra

For illustration, we consider a minimal case that the group algebra is constructed by elements of the group . Although the 3-dimensional representation of group algebras is reducible, it can be viewed as the special case of 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 . Then, we show equivalence of elements of group algebras and the residual symmetry . Comments on group algebras are also made. Finally, we discuss general residual symmetries of the charged lepton sector.

3.1. Mixing Patterns from Group Algebra in the Case of the Diagonal Mass Matrix

The 3-dimensional reducible representation of the group is expressed as

According to the unitary conditions of Equation (12), viable nontrivial realizations of are listed as

All theses correspond to the same lepton mixing matrix up to permutations of rows, columns, or trivial phases. We consider as a representative, whose expression is

It is diagonalized as where , , . The matrix reads where , , . It is of trimaximal form with the reflection symmetry [27, 48–50], i.e., with and with . The lepton mixing matrix is equal to up to permutations of rows or columns. Given the recent global fit data of neutrino oscillations[51], viable mixing matrices are

Note that and . Furthermore, according to the standard parametrization [52] where and , is the Dirac -violating phase, and are Majorana phases, and and are interchanged through the following transformation: and . So without loss of generality, we can just consider . Lepton mixing angles and the Dirac phase are listed as where . Dependence of and on the variable is shown in Figure 1. From the figure, we can see that is a slowly varying function of the parameter . So the parameter space of is mainly constrained by . According to the function defined as where are best global fit values from Ref. [51] and are uncertainties; best fit data of , , and are listed in Table 1. They are in the ranges of the global fit data.

(a)

(b)

3.2. Equivalence of Elements of Group Algebras and

The neutrino mass matrix which is invariant under the action of is of the form where and are real and . Obviously, follows the residual symmetry , i.e.,

where

Correspondingly, for we have

works as the GCP for the mass matrix on the one hand. On the other hand, it acts as an equivalent transformation for symmetries and . So is equivalent to the residual symmetry .

3.3. Comments on Equivalence of Elements of Group Algebras and

For the S₄ group with the GCP, the residual symmetries could bring maximal or trivial Dirac phase. We have seen that in S₃ group algebras gives a maximal phase.

In fact, the equivalence can still hold for some in group algebras which are not elements of group algebras. The trivial phase could be obtained from . Here, we give an example of from group algebras with a different representation. Three generators of S₄ which satisfy the relation [32] are expressed as [32] where . A nontrivial example of the group algebra element could be . Its specific expression is of the form [46]

If we take and suppose that the mass matrix of charged leptons is diagonal, we can obtain the lepton mixing matrix written as where , , and is a parameter constrained by the mixing angle . So the mixing pattern is of trimaximal form with a trivial Dirac -violating phase. For , we can verify that the following relation holds, i.e., where , , and . So and are a symmetry and the corresponding transformation, respectively. Following the methods used in GCP [30], the lepton mixing matrix from the residual symmetry can be expressed as , where and are expressed, respectively, as

is a phase matrix which can be neglected in our case of Dirac neutrinos. In particular, the matrix satisfies the relations as follows

We can check that the matrix from the is just the shown in Equation (29). So is equivalent to the symmetry generated by and . Furthermore, let us consider the element . The lepton mixing matrix from is . Since is a phase matrix, is equivalent to . So the transformation interchanges the equivalent elements and . Therefore, the observation from the case of the S₃ algebra still holds in this example of the group algebra.

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

We have studied the case that the mass matrix is diagonal. The corresponding symmetry of the charged lepton sector is , namely, . Now we discuss a more general case that is expressed by an element of the 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 . Then, the matrix is of the form where , , , , and , . With respect to the mixing matrix , we have an element . Obviously, it does not satisfy the constraint of the global fit data of neutrino oscillations. So the combination of the residual symmetries () does not give a realistic lepton mixing patten in the case of group algebra. Furthermore, if is equal to 0, is reduced to . The corresponding matrix becomes where is an angle variable from the degeneracy of the eigenvalues of S₂₃. Then, contains a zero element. This observation still holds when is replaced by or . So the combination () is not a viable choice for the residual symmetries of leptons. We can also check that from the combination (), where is generated by or , 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 in the group algebra, we can only take .

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

Acknowledgments

This work is supported by the National Natural Science Foundation of China under grant nos. 11405101 and 11705113, the Guangxi Scientific Program Foundation under grant no. Guike AD19110045, and the Research Foundation of Guilin University of Technology under grant no. GUTQDJJ2018103.

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Copyright

Copyright © 2020 Shu-Jun Rong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

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