Neutrino Mixing and Leptonic CP Violation
[2mm] from Flavour and Generalised CP Symmetries
[8mm] J. T. Penedo111E-mail: email@example.com, S. T. Petcov222Also at: Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria. and A. V. Titov333E-mail: firstname.lastname@example.org
SISSA/INFN, Via Bonomea 265, 34136 Trieste, Italy
Kavli IPMU (WPI), University of Tokyo, 5-1-5 Kashiwanoha, 277-8583 Kashiwa, Japan
We consider a class of models of neutrino mixing with lepton flavour symmetry combined with a generalised CP symmetry, which are broken to residual and symmetries in the charged lepton and neutrino sectors, respectively, being a remnant CP symmetry of the neutrino Majorana mass term. In this set-up the neutrino mixing angles and CP violation (CPV) phases of the neutrino mixing matrix depend on three real parameters — two angles and a phase. We classify all phenomenologically viable mixing patterns and derive predictions for the Dirac and Majorana CPV phases. Further, we use the results obtained on the neutrino mixing angles and leptonic CPV phases to derive predictions for the effective Majorana mass in neutrinoless double beta decay.
Understanding the origin of the pattern of neutrino mixing that emerged from the neutrino oscillation data in the recent years (see, e.g., ) is one of the most challenging problems in neutrino physics. It is part of the more general fundamental problem in particle physics of understanding the origins of flavour in the quark and lepton sectors, i.e., of the patterns of quark masses and mixing, and of the charged lepton and neutrino masses and of neutrino mixing.
The idea of extending the Standard Model (SM) with a non-Abelian discrete flavour symmetry has been widely exploited in attempts to make progress towards the understanding the origin(s) of flavour (for reviews on the discrete symmetry approach to the flavour problem see, e.g., [2, 3, 4]). In this approach it is assumed that at a certain high-energy scale the theory possesses a flavour symmetry, which is broken at lower energies to residual symmetries of the charged lepton and neutrino sectors, yielding certain predictions for the values of, and/or correlations between, the low-energy neutrino mixing parameters. In the reference 3-neutrino mixing scheme we are going to consider in what follows (see, e.g., ), i) the values of certain pairs of, or of all three, neutrino mixing angles are predicted to be correlated, and/or ii) there is a correlation between the value of the Dirac CP violation (CPV) phase in the neutrino mixing matrix and the values of the three neutrino mixing angles 111Throughout the present study we use the standard parametrisation of the Pontecorvo, Maki, Nakagawa and Sakata (PMNS) neutrino mixing matrix (see, e.g., ). , , and , which includes also symmetry dependent fixed parameter(s) (see, e.g., [5, 6, 7, 8, 9, 10, 11, 12] and references quoted therein). These correlations are usually referred to as “neutrino mixing sum rules”. As we have already indicated, the sum rules for the Dirac phase , in particular, depend on the underlying symmetry form of the PMNS matrix [5, 6, 7, 8, 9] (see also, e.g., [10, 11, 12]), which in turn is determined by the assumed lepton favour symmetry that typically has to be broken, and by the residual unbroken symmetries in the charged lepton and neutrino sectors (see, e.g., [2, 3, 4, 7, 9]). They can be tested experimentally (see, e.g., [6, 10, 14, 13]). These tests can provide unique information about the possible existence of a new fundamental symmetry in the lepton sector, which determines the pattern of neutrino mixing . Sufficiently precise experimental data on the neutrino mixing angles and on the Dirac CPV phase can also be used to distinguish between different possible underlying flavour symmetries leading to viable patters of neutrino mixing.
While in the discrete flavour symmetry approach at least some of the neutrino mixing angles and/or the Dirac phase are determined (directly or indirectly via a sum rule) by the flavour symmetry, the Majorana CPV phases and  remain unconstrained. The values of the Majorana CPV phases are instead constrained to lie in certain narrow intervals, or are predicted, in theories which in addition to a flavour symmetry possess at a certain high-energy scale a generalised CP (GCP) symmetry . The GCP symmetry should be implemented in a theory based on a discrete flavour symmetry in a way that is consistent with the flavour symmetry [17, 18]. At low energies the GCP symmetry is broken, in general, to residual CP symmetries of the charged lepton and neutrino sectors.
In the scenarios involving a GCP symmetry, which were most widely explored so far (see, e.g., [17, 19, 20, 21, 22, 23]), a non-Abelian flavour symmetry consistently combined with a GCP symmetry is broken to residual Abelian symmetries , , or , , and of the charged lepton and neutrino mass terms, respectively 222We note that in refs. [20, 21] the residual symmetry of the charged lepton mass term is augmented with a remnant CP symmetry as well. . The factor in stands for a remnant GCP symmetry of the neutrino mass term. In such a set-up, fixes completely the form of the unitary matrix which diagonalises the product and enters into the expression of the PMNS matrix, being the charged lepton mass matrix (in the charged lepton mass term written in the left-right convention). At the same time, fixes the unitary matrix , diagonalising the neutrino Majorana mass matrix up to a single free real parameter — a rotation angle . Given the fact that the PMNS neutrino mixing matrix is given by the product
all three neutrino mixing angles are expressed in terms of this rotation angle. In this class of models one obtains specific correlations between the values of the three neutrino mixing angles, while the leptonic CPV phases are typically predicted to be exactly or , or else or . For example, in the set-up considered in  (see also ), based on broken to and with 333, and are the generators of in the basis for its 3-dimensional representation we employ in this work (see subsection 3.2)., the authors find:
It follows, in particular, from the results on the neutrino oscillation parameters — best fit values, and allowed ranges — obtained in the latest global fit of neutrino oscillation data  and summarised in Table 1, to be used in our further analysis 444The results on the neutrino oscillation parameters obtained in the global fit performed in  differ somewhat from, but are compatible at confidence level (C.L.) with, those found in  and given in Table 1., that the predictions quoted in eq. (1.2) for and lie outside of their respective currently allowed ranges 555We have used the best fit value of to obtain the prediction of leading to the quoted conclusion. Using the allowed range for leads to a minimal value of , which is above the maximal allowed value of at C.L., but inside its range..
|Parameter||Best fit value||range||range|
Another example of one-parametric models is the extensive study performed in , in which the authors have considered two different residual symmetry patterns. The first pattern is the one described above, and the second pattern has and as residual symmetries in the charged lepton and neutrino sectors, respectively. The authors have performed an exhaustive scan over discrete groups of order less than 2000, which admit faithful 3-dimensional irreducible representations, and classified phenomenologically viable mixing patterns.
Theoretical models based on the approach to neutrino mixing that combines discrete symmetries and GCP invariance, in which the neutrino mixing angles and the leptonic CPV phases are functions of two or three parameters have also been considered in the literature (see, e.g., [27, 28, 29, 30]). In these models the residual symmetry of the charged lepton mass term is typically assumed to be a symmetry or to be fully broken. In spite of the larger number of parameters in terms of which the neutrino mixing angles and the leptonic CPV phases are expressed, the values of the CPV phases are still predicted to be correlated with the values of the three neutrino mixing angles. A set-up with and has been considered in . The resulting PMNS matrix in such a scheme depends on two free real parameters — two angles and . The authors have obtained several phenomenologically viable neutrino mixing patterns from combined with , broken to all possible residual symmetries of the type indicated above. Models allowing for three free parameters have been investigated in [27, 28, 29]. In, e.g., , the author has considered combined with , which are broken to and . In this case, the matrix depends on an angle and a phase , while the matrix depends on an angle . In these two scenarios the leptonic CPV phases possess non-trivial values.
The specific correlations between the values of the three neutrino mixing angles, which characterise the one-parameter models based on , , or , , and , do not hold in the two- and three-parameter models. In addition, the Dirac CPV phase in the two- and three-parameter models is predicted to have non-trivial values which are correlated with the values of the three neutrino mixing angles and differ from , , and , although the deviations from, e.g., can be relatively small. The indicated differences between the predictions of the models based on , , or , , and on symmetries make it possible to distinguish between them experimentally by improving the precision on each of the three measured neutrino mixing angles , and , and by performing a sufficiently precise measurement of the Dirac phase .
In the present article, we investigate the possible neutrino mixing patterns generated by a symmetry combined with an symmetry when these symmetries are broken down to and . In Section 2, we describe a general framework for deriving the form of the PMNS matrix, dictated by the chosen residual symmetries. Then, in Section 3, we apply this framework to combined with and obtain all phenomenologically viable mixing patterns. Next, in Section 4, using the obtained predictions for the neutrino mixing angles and the Dirac and Majorana CPV phases, we derive predictions for the neutrinoless double beta decay effective Majorana mass. Section 5 contains the conclusions of the present study.
2 The Framework
We start with a non-Abelian flavour symmetry group , which admits a faithful irreducible 3-dimensional representation . The three generations of left-handed (LH) leptons are assigned to this representation. Apart from that, the high-energy theory respects also the GCP symmetry , which is implemented consistently along with the flavour symmetry. At some flavour symmetry breaking scale gets broken down to residual symmetries and of the charged lepton and neutrino mass terms, respectively. The residual flavour symmetries are Abelian subgroups of . The symmetries and significantly constrain the form of the neutrino mixing matrix , as we demonstrate below.
2.1 The PMNS Matrix from and
We choose to be a symmetry. We will denote it as , being an element of of order two, generating the subgroup. The invariance of the charged lepton mass term under implies
Below we show how this invariance constrains the form of the unitary matrix , diagonalising :
Lets be a diagonalising unitary matrix of , such that
This result is obtained as follows. The diagonal entries of are constrained to be , since this matrix must still furnish a representation of and hence its square is the identity. We have assumed that the trace of is , for the relevant elements , as it is the case for the 3-dimensional representation of we will consider later on 666 For the other 3-dimensional irreducible representation of the trace can be either or , depending on . Choosing would simply imply a change of sign of , which however does not lead to new constraints. The conclusions we reach in what follows are then independent of the choice of 3-dimensional representation. . Note that we can take the order of the eigenvalues of as given in eq. (2.3) without loss of generality, as will become clear later.
This equation implies that has the block-diagonal form
and, since this matrix is hermitian, it can be diagonalised by a unitary matrix with a transformation acting on the 2-3 block. In the general case, the transformation can be parametrised as follows:
The diagonal phase matrix is, however, unphysical, since it can be eliminated by rephasing of the charged lepton fields, and we will not keep it in the future. Thus, we arrive to the conclusion that the matrix diagonalising reads
and being one of six permutation matrices, which need to be taken into account, since in the approach under consideration the order of the charged lepton masses is unknown. The six permutation matrices read:
Note that the order of indices in stands for the order of rows, i.e., when applied from the left to a matrix, it gives the desired order, --, of the matrix rows. The same is also true for columns, when is applied from the right, except for which leads to the -- order of columns and yielding the -- order.
In the neutrino sector we have a
residual symmetry. We will denote the symmetry of
the neutrino mass matrix as
, with being an element of ,
generating the subgroup.
is the set of remnant GCP unitary
forming a residual CP symmetry of the neutrino mass matrix.
in which is the GCP symmetry
of the high-energy theory consistently
defined along with the flavour symmetry 777It is worth to comment here on the notation we use.
When we write in what follows , we mean a set of GCP transformations ( and ) compatible with the residual
flavour symmetry (see eq. (2.13)).
However, when writing ,
is intended to be a group generated by .
Namely, following Appendix B in ,
is isomorphic to ,
where is the unit matrix and
In addition, the consistency condition between and has to be respected:
To derive the form of the unitary matrix diagonalising the neutrino Majorana mass matrix as
being the neutrino masses, we will follow the method presented in .
Lets be a diagonalising unitary matrix of , such that
Expressing from this equation and substituting it in the consistency condition, eq. (2.13), we find
meaning that is a block-diagonal matrix, having the form of eq. (2.5). Moreover, this matrix is symmetric, since the GCP transformations have to be symmetric in order for all the three neutrino masses to be different [17, 19], as is required by the data. In Appendix A we provide a proof of this. Being a complex (unitary) symmetric matrix, it is diagonalised by a unitary matrix via the transformation:
The matrix is, in general, a diagonal phase matrix. However, we can choose as the phases of can be moved to the matrix . With this choice we obtain the Takagi factorisation of the (valid for unitary symmetric matrices):
Since, as we have noticed earlier, has the form of eq. (2.5), the matrix can be chosen without loss of generality to have the form of eq. (2.5) with a unitary matrix in the 2-3 block. This implies that the matrix also diagonalises . Indeed,
where we have used eq. (2.15).
We substitute next from eq. (2.18) in the GCP invariance condition of the neutrino mass matrix, eq. (2.12), and find that the matrix is real. Furthermore, this is a symmetric matrix, since the neutrino Majorana mass matrix is symmetric. A real symmetric matrix can be diagonalised by a real orthogonal transformation. Employing eqs. (2.19) and (2.11), we have
implying that is a block-diagonal matrix as in eq. (2.5). Thus, the required orthogonal transformation is a rotation in the 2-3 plane on an angle :
Finally, the matrix diagonalising reads
where is one of the six permutation matrices, which accounts for different order of , and the matrix renders them positive. Without loss of generality can be parametrised as follows:
Thus, in the approach we are following the PMNS matrix depends on three free real parameters 888It should be noted that the matrix in eq. (2.17) with , and thus the matrix in eq. (2.18), is determined up to a multiplication by an orthogonal matrix on the right. The matrix must be unitary since it diagonalises a complex symmetric matrix, which implies that must be unitary in addition of being orthogonal, and therefore must be a real matrix. Equation (2.19) restricts further this real orthogonal matrix to have the form of a real rotation in the 2-3 plane, which can be “absorbed” in the matrix in eq. (2.24). — the two angles and and the phase . One of the elements of the PMNS matrix is fixed to be a constant by the employed residual symmetries. We note finally that, since , where the diagonal matrix can be absorbed into , and , where the diagonal matrix contributes to the unphysical charged lepton phases, it is sufficient to consider and in the interval .
2.2 Conjugate Residual Symmetries
In this subsection we briefly recall why the residual symmetries and conjugate to and , respectively, under the same element of the flavour symmetry group lead to the same PMNS matrix (see, e.g., [17, 20]). Two pairs of residual symmetries and are conjugate to each other under if
At the representation level this means
where the primed mass matrices are related to the original ones as
Obviously, the unitary transformations and diagonalising the primed mass matrices are given by
2.3 Phenomenologically Non-Viable Cases
Here we demonstrate that at least two types of residual
characterised by certain and ,
cannot lead to phenomenologically viable form of the PMNS matrix.
Type I: . In this case, we can choose , with or . Then, eq. (2.24) yields
Type II: . Now we consider two different order two elements , which belong to the same subgroup of . In this case, since and commute, there exists a unitary matrix simultaneously diagonalising both and . Note, however, that the order of eigenvalues in the resulting diagonal matrices will be different. Namely, lets be a diagonalising matrix of and , and lets diagonalise as in eq. (2.15). Then, can yield either or , but not . Hence, diagonalising as in eq. (2.3), must read
Taking into account that , with of the block-diagonal form given in eq. (2.5), we obtain
where , depending on , can take one of the following forms:
As a consequence, up to permutations of the rows and columns has the form
containing one zero element, which is ruled out by the data.
3 Mixing Patterns from
3.1 Group and Residual Symmetries
is the symmetric group of permutations of four objects. This group is isomorphic to the group of rotational symmetries of the cube. can be defined in terms of three generators , and , satisfying 
From 24 elements of the group there are nine elements of order two, which belong to two of five conjugacy classes of (see, e.g., ):
Each of these nine elements generates a corresponding subgroup of . Each subgroup can be the residual symmetry of , and, combined with compatible CP transformations, yield the residual symmetry of . Hence, we have 81 possible pairs of only residual flavour symmetries (taking into account remnant CP symmetries increases the number of possibilities). Many of them, however, being conjugate to each other, will lead to the same form of the PMNS matrix, as explained in subsection 2.2. Thus, we first identify the pairs of elements , which are not related by the similarity transformation given in eq. (2.25). We find nine distinct cases for which can be chosen as
The pair is obviously conjugate to and , while is conjugate to with being one of the remaining five elements from conjugacy class given in eq. (3.3). The pairs , , and are conjugate to five pairs each, and and to eleven pairs each. Finally, is conjugate to 23 pairs. As it should be, the total number of pairs yields 81. The complete lists of pairs of elements which are conjugate to each of these nine pairs are given in Appendix B.
The cases in eq. (3.4) do not lead to phenomenologically viable results. The first two of them belong to the cases of Type I (see subsection 2.3). The remaining four belong to Type II, since contains and subgroups (see, e.g., ). Thus, we are left with three cases in eq. (3.5).
We have chosen in such a way that it is , or for all the cases in eq. (3.5). Now we need to identify the remnant CP transformations compatible with each of these three elements. It is known that the GCP symmetry compatible with is of the same form of itself , i.e.,
Thus, to find compatible with of interest, we need to select those , which i) satisfy the consistency condition in eq. (2.13) and ii) are symmetric in order to avoid partially degenerate neutrino mass spectrum, as was noted earlier. The result reads 999For notation simplicity we will not write the representation symbol , keeping in mind that meas with .:
A GCP transformation in parentheses appears automatically to be a remnant CP symmetry of , if which precedes this in the list is a remnant CP symmetry. This is a consequence of eqs. (2.11) and (2.12), which imply that if is a residual CP symmetry of , then is a residual CP symmetry as well. Therefore, we have three sets of remnant CP transformations compatible with , namely, , and , two sets compatible with , which are and , and two sets consistent with , which read and . Taking them into account, we end up with seven possible pairs of residual symmetries , with as in eq. (3.5). In what follows, we will consider them case by case and classify all phenomenologically viable mixing patterns they lead to.
Before starting, however, let us recall the current knowledge on the absolute values of the PMNS matrix elements, which we will use in what follows. The ranges of the absolute values of the PMNS matrix elements read 
for the neutrino mass spectrum with normal ordering (NO), and
3.2 Explicit Forms of the PMNS Matrix
First, we present an explicit example of constructing the PMNS matrix in the case of , and , which is the first case out of the seven potentially viable cases indicated above. We will work in the basis for from , in which the matrices for the generators , and in the 3-dimensional representation read
where . For simplicity we use the same notation (, and ) for the generators and their 3-dimensional representation matrices. We will follow the procedure described in subsection 2.1. The matrix which diagonalises (see eq. (2.3)) is given by
The matrix , such that (see eq. (2.18)), reads