Lepton Mixing Predictions including Majorana Phases from Flavour Symmetry and Generalised CP
Abstract
Generalised CP transformations are the only known framework which allows to predict Majorana phases in a flavour model purely from symmetry. For the first time generalised CP transformations are investigated for an infinite series of finite groups, . In direct models the mixing angles and Dirac CP phase are solely predicted from symmetry. flavour symmetry provides many examples of viable predictions for mixing angles. For all groups the mixing matrix has a trimaximal middle column and the Dirac CP phase is 0 or . The Majorana phases are predicted from residual flavour and CP symmetries where can take several discrete values for each and the Majorana phase is a multiple of . We discuss constraints on the groups and CP transformations from measurements of the neutrino mixing angles and from neutrinoless doublebeta decay and find that predictions for mixing angles and all phases are accessible to experiments in the near future.
1 Introduction
The question of the origin of neutrino masses and mixing parameters is of fundamental importance. One approach are socalled direct models of neutrino masses [1] where a discrete nonAbelian family symmetry group is broken to a group in the Neutrino sector, and a subgroup in the charged lepton sector. In such a model the lepton mixing angles and the lepton Dirac CP phase are completely fixed by symmetry.
Recently such direct models have been analysed with the help of the group database GAP [2, 3]. The only flavour groups that can produce viable mixing parameters in a direct model belong to the group series or are subgroups of such groups. The group theory of groups has been analysed in [4]. The consequences for neutrino mixing from a flavour symmetry in direct models have been studied in detail in [5] for arbitrary even . Some examples of groups or subgroups have previously been studied in [6, 7, 8, 9, 10, 11, 12, 13, 14].
In the Standard Model, violation of CP occurs in the flavour sector. Promoting CP to a symmetry at high energies which is then broken allows to impose further constraints on mass matrices of charged leptons and Majorana neutrinos. In this case the interplay between CP and flavour symmetries has to be carefully discussed[15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. For direct models, especially with a flavour group from , CP symmetries have not been studied in detail yet.
In this paper we examine a class of generalised CP (gCP) transformations consistent with groups for arbitrary . We will start by defining flavour and generalised CP transformations and stating their effect on mass matrices. In the following section we review and develop the general theory of gCP transformations in the presence of flavour symmetries in a general context. Afterwards we specialise on direct models with as a flavour group, where we compute the lepton mixing matrix including Majorana phases for arbitrary even for all possible breaking patterns of the flavour group and of gCP. Here we also analyse the constraints from measurements of the mixing angles and from neutrinoless doublebetadecay on these models. In the last section we conclude.
2 Generalised CP Transformations, Flavour Symmetries, Automorphisms and the Character Table
In this section we review the interplay between flavour symmetries and CP symmetries which has especially been discussed in [15, 18, 20, 22, 25] and use general arguments to show that for a class of groups , of which is an example, physical CP transformations correspond to with a real number.
2.1 Generalised CP transformations and flavour symmetries
Consider a theory where generations of fermions are assigned to multiplets of representations of a flavour group and that is invariant under transformations of the multiplets under the group
(2.1) 
where is the representation matrix for in the representation .
Further consider the group being broken to a Klein subgroup subgroup in the neutrino sector and an abelian subgroup with in the charged lepton sector. If these subgroups remain unbroken at all energies, in the lowenergylimit constraints on the mass matrices of charged leptons and neutrinos are imposed. Lefthanded doublets transform under the same representation . The charged lepton mass matrix has to fulfill
(2.2) 
with being the representation matrix of in the representation . The Majorana neutrino mass matrix is constrained by
(2.3) 
with .
Define generalised CP (gCP) by
(2.4) 
where is the representation of according to which transforms. ^{3}^{3}3Other Authors consider transformations of the type where can be different. In [15] has been shown that only gCP transformations where actually make observables (e.g. particle decays) conserve CP. is a unitary matrix. We need to find all matrices that are “allowed” in coexistence with a flavour group . The aforesaid will be made a more precise statement in the following section, where the conditions for the existence of gCP transformations as well as their properties will be discussed.
If the theory at the lowenergy end is invariant under residual gCP transformations with matrices for charged leptons and for neutrinos then the mass matrices will be constrained by
(2.5) 
for charged leptons and by
(2.6) 
for Majorana neutrinos.
If (), no new constraints on the neutrino (charged lepton) mass matrix follow but it being real. With from only the same constraints as for follow for the mass matrix. This means only that are not in allow for a mass matrix that is not real and at the same time impose new constraints on it.
2.2 The consistency equation
We would like to know which transformations of the type
(2.7) 
can be applied to the theory without destroying the invariance under , i.e. which matrices can appear in Eq.(2.7) that preserve symmetry under ? Consider performing a gCP transformation followed by a flavour transformation followed by the inverse gCP transformation. From invariance of the theory under follows that the matrix is allowed in a gCP transformation if for every there is a such that
(2.8) 
Eq. (2.8) is called the consistency equation and an that fulfills it is called consistent with .
If is a faithful representation, which is equivalent to saying that is injective, one can define a bijective mapping between the elements of the group:
(2.9) 
(One can drop the index on because for all faithfull irreps the mapping generated by Eq.(2.9) will be the same). For faithful representations , is an automorphism of the group .
2.3 Inner and outer automorphisms
Group automorphisms come in two kinds: Inner and outer automorphisms. Inner automorphisms are such automorphisms where for all one single group element exists such that
(2.10) 
All inner automorphisms are given by , where is the center of G, i.e. all elements of that commute with every other group element. Outer automorphisms are all automorphisms that are not inner.
An inner automorphism will map each element into its original conjugacy class. An outer automorphism however is not inner which means that there is at least one for which with all (compare with the definition of inner automorphisms before Eq. (2.10)), i.e. there is at least one which is not mapped back into its original conjugacy class. Also if is in the class and it is mapped onto which is in the class , every element in is mapped on an element in by .
This proves also that an automorphism that maps each element back into its original conjugacy class is inner, as well that an automorphism that maps elements from at least two conjugacy classes on each other is outer. ^{4}^{4}4 An outer automorphism also generates mappings between different representations of . For two representations and define (2.11) with which follows (2.12) The outer automorphism acting inside the group thus interchanges columns of the character table while when acting between representations via interchanges rows of the character table. We call a symmetry of the character table (2.13) any transformation of the type (2.14) with permutation matrices and that leaves invariant, i.e. (2.15) and where only classes of the same size and elementorder are interchanged, i.e and for and . An outer automorphism will always generate a nontrivial symmetry of the character table, just as a symmetry of the character table always gives rise to an outer automorphism: Define the automorphism by the action on the conjugacy classes, a corresponding permutation of the representations is always given by any outer automorphism via .
We will now return to the automorphism (2.9) that is induced by the consistency equation (2.8). If is real and then will be an inner automorphism. This is also true if .
If on the other hand is an outer automorphism it follows that a matrix that could mediate á la Eq. (2.9) is not in (if it exists).
One could ask now if there can be a matrix that is not in for that only connects elements within the same conjugacy class, i.e. that generates an inner automorphism? As for an inner automorphism there always is a single such that the automorphism is given by it follows that
(2.16) 
For a real matrix multiplying by from the right and by from the left yields
(2.17) 
As can be every element of , commutes with every group element. One can now apply Schur’s Lemma ^{5}^{5}5To be precise one uses the second part of Schur’s Lemma which states that an operator that in some representation commutes with every group element is proportional to the identity. to find that
(2.18) 
where to keep unitary. As was supposed to not be in this is in contradiction to the assumptions. For real this proves that inner automorphisms correspond to . For real representations, there is always a basis where this is the case, i.e where is real for every .
If is complex one has to deal with complex conjugation: Assume there is a matrix such that by applying complex conjugation and this matrix on an element of , the element is mapped into the class of its inverse, :
(2.19) 
This can be thought of as an automorphism mapping followed by an automorphism that maps onto another element in the same class. As in the second step every element is sent into the original class, this second mapping is an inner automorphism and therefore by definition a single group element exists which inverts this step such that
(2.20) 
For this reason we assume in the following that the matrix maps elements directly onto their inverses. Using this, the general mapping induced by the consistency equation is given by:
(2.21) 
This mapping can be seen as an automorphism mapping on followed by an automorphism given by :
(2.22) 
If both and are contained in , will map in the same conjugacy class as . For , maps elements into the class of the inverse and is contained in the group. We will not consider further.
Analogous to real irreps above one can now ask if there can be matrices that are not in but that with will map in the conjugacy class of ? This would be equivalent to being an inner automorphism which would mean that for each group element there is a single such that
(2.23) 
Again we can use Schur’s Lemma and find there is such that
(2.24) 
with to make unitary. This contradicts . We have proved now that if then if and only if will be in the conjugacy class of . In [15] the authors show that only gCP transformations that map elements into the class of its inverse element make observables conserve CP. We have proved here that such transformations are given by . ^{6}^{6}6 We would now be able to find all by reading off all automorphisms from the symmetries of the character table that do not map the class of on the class of . (This would often contain the identity transformation on the character table.) In the following we will specialise to be .
3 gCP Symmetries and groups
In this section we consider gCP transformations where for . First we derive the gCP transformations that are consistent with and . Afterwards we state the constrained mass matrices and the lepton mixing matrix. After this we discuss constraints from measurements of lepton mixing angles and from neutrinoless doublebeta decay for arbitrary .
If we want to break the flavour symmetry to and subgroups, the residual flavour and residual gCP transformations are not independent, as they still have to fulfill the consistency equation. If e.g. in one sector and are unbroken, then also must be unbroken. Thus the allowed residual gCP transformations have to map elements from the Klein group in consideration into said Klein group.
The Klein subgroups of are given by [5]
(3.25)  
(3.26)  
(3.27)  
(3.28) 
where . The group Eq. (3.25) will produce a mixing matrix with , we will not consider it further. The bottom three Klein subgroups will generate the same mixing matrix, thus it is sufficient to only consider the mixing matrices generated by group Eq.(3.26). The allowed matrices in the lowenergylimit have to be contained in . A matrix is allowed if for a Klein subgroup holds that for each also . For said Klein subgroup one finds that the allowed matrices are given by the representation matrices for
(3.29) 
with and .
Without loss of generality, lefthanded doublets are assigned to the representation (c.f.[5]). Invariance of the mass matrix under the Klein subgroup in consideration plus invariance under one of the transformations from Eq. (3.29) constrains the Majorana neutrino mass matrix to
(3.30) 
where the values of and can be found in table (1). In principle, several gCP transformations can remain unbroken. However, the phases are already fixed by one single unbroken transformation. Leaving a second gCP transformation unbroken with incompatible constraints on the phase will force the corresponding mass parameters to be zero. The masses of neutrinos are and . Thus or will result in a pair of degenerate neutrino states. It is not possible to have without or . Leaving a second gCP transformation unbroken is never physically viable.
The neutrino mass matrix Eq.(3.30) will be diagonalised by a unitary matrix via . A matrix such that the diagonalised mass matrix is real and positive is given by
(3.31) 
for and by
(3.32) 
for .
For charged leptons, the allowed gCP transformations with have to be consistent with and are given by
(3.33) 
where . Especially when 3 divides there is a huge number of allowed matrices. But, as the charged lepton mass matrix is already invariant under transformations with and transformations with force it to be zero (for ) or produce no new constraint (for ), the only transformations that produce physical constraints are given by
(3.34) 
For the mass matrix of charged leptons is restrained to
(3.35) 
with all parameters being real or for to
(3.36) 
with complex and real. Both charged lepton mass matrices can be diagonalised by
(3.37) 
Above charged lepton mass matrices only differ by unphysical phases which can be absorbed into the charged lepton fields.
After removing an overall phase to render the top left entry real, the physical mixing matrix is given by (For and c.f. Eq.(3.31) and Eq.(3.32)):
(3.38) 
where the factor on the last column only appears in . As the ordering of the mixing matrix is arbitrary at this point, we would like to fix it by requiring that the smallest entry of the matrix has to be the topright entry, i.e. . For small the first row and third column are in the right place in the above matrix.
As this matrix is now in the PDG convention, the values of Majorana phases and as well as the Dirac CP phase for this ordering of the mixing matrix can be read off the matrix. Recall that the PDG convention is in terms of , , the Dirac CP violating phase and further Majorana phases contained in .
The Majorana phase is then given by
(3.39) 
With table [1] follows that
(3.40) 
or
(3.41) 
The values of all CP phases depend on the ordering of Eq.(3.38) which needs to be changed for higher values of . The possible values of the CP phases can be found in table (2). There, denotes the mixing matrix after reordering such that the entry with the smallest absolute value is in the top right corner. As for every the second and third row can be interchanged, which results in changing by while changing the prediction for and and thus the prediction for . The Dirac CP phase is hence predicted to be or , and since the lepton mixing matrix has the trimaximal form for the second column, referred to as TM2, this leads to the mixing sum rules for , respectively, as previously noted in [5] (for a review of sum rules see [1]).
The prediction of also depends on the order of these rows. In the table (2) the second row of the mixing matrix after reordering it is indicated in the column . Improved measurements of will constrain this freedom of interchanging the second and third row.

0  




0  

















The key observable for Majorana phases is neutrinoless double beta decay (). The effective mass of neutrinoless doublebeta decay is given by
(3.42) 
with
(3.43) 
for normal ordering and
(3.44) 
for inverted ordering, where is the mass of the lightest neutrino and
(3.45) 
The absolute values of the entries of the mixing matrix after reordering are periodic in which is why one can simplify the analysis by defining in this way.
There are 8 cases to distinguish for combinations of phases. Adding a multiple of will not change the effect of or . For this reason, for both Eq.(3.40) and Eq.(3.41) the 12 cases in table (2) reduce to 8 cases of values for
(3.46) 
that are given by
(3.47) 
The by far most stringent constraint on comes from the measurement of . The current 3 sigma range for from [30] yields values of in the range .
It is generally fine to not only consider in this range but even because changing by only changes by , which is included in the four cases discussed above.
In order to understand predictions of groups for decay on a general level, in figure (1) the effective mass of is plotted against the mass of the lightest neutrino for all combinations of and . In these plots, models defined by some values of and correspond to single fine lines. takes 11 values, starting with the 3 sigma lower bound and increases in 10 equal steps until it reaches the 3 sigma upper bound. takes values .
and are not varied, as doing so only would almost unnoticeably broaden each single line. Instead we used the best fit value from [30]:
(3.48) 
(3.49) 
In figure (1), Magenta lines correspond to predictions assuming inverted hierarchy, red lines to normal hierarchy. Dashed blue and yellow lines indicate the currently allowed three sigma region for normal and inverted hierarchy, respectively. The three sigma ranges for mixing angles are taken from [30]. The upper bound eV is given from measurements by the EXO200 experiment [31]. Planck data in combination with other CMB and BAO measurements [32] provides a limit on the sum of neutrino masses of eV from which the upper limit on the mass of the lightes neutrino can be derived.
The main features of the results from figure (1) are as follows:

For inverted hierarchy there is no particular structure visible. Additionally, the predicted values for are well within the reach of e.g. phase III of the GERDA experiment of eV [33].

For normal ordering, it follows from figure (1) that for the values of and considered is always a lower limit on which means that these parameters are accessible to future experiments.

Further for normal ordering, in the very low region, predicted values of are closer to the upper end of the blue three sigma range.

With the current data, no combination of and is favoured. Only for values of eV and eV it would be possible to distinguish different values of and .
The necessary precisions on and are unfortunately outside of the range of any projected experiments known to the authors. Nevertheless, the red curves corresponding to fixed values of and are often close to the blue dashed three sigma range. With increasingly precise knowledge of the values of the mixing angles, especially , the three sigma ranges will shrink, perhaps making it possible to draw conclusions about and without an overly precise measurement of or of the mass of the lightes neutrino.
To recapitulate, the following assumptions went into producing these results: There are 3 lefthanded doublets of leptons, which in turn transform as a triplet under a group. The neutrinos are Majorana fermions and is broken to a subgroup in the neutrino sector and to in the charged lepton sector. The mixing angles are solely predicted from the aforementioned assumptions. There is a generalised CP symmetry which is consistent with which is broken to one element in each sector. From this gCP symmetry the Majorana phases are predicted.
If one of the mixing angles would be found to be incompatible with any of the predictions this would mean that either is not broken to (or to , as the predictions for the mixing angles would be the same) or that the flavour group is not or that one of the more fundamental assumptions is wrong. The neutrinos could still be Majorana fermions as could still be broken completely.
4 Conclusions
In this paper we have examined the interplay of groups and generalised CP transformations (gCP) in a direct model for three generations of Dirac charged leptons and Majorana neutrinos. We find that gCP transformations that actually are physical CP transformations have . Leaving a single gCP transformation unbroken will constrain the mixing matrix such that all phases, Dirac and Majorana are predicted and depend only on the group, the residual group (parametrised by ) and the residual gCP transformation (parametrised by ) in the neutrino sector. Leaving two or more gCP transformations unbroken is not physically viable.
Comparing the predictions for the mixing angles with experimental data we find that the strongest constraint on is imposed by the relatively precise measurement of . The smallest group where lies within three sigma of the central value has . Furthermore, since the Majorana CP violating phases are predicted, we have studied predictions for neutrinoless doublebeta decay. We find that for inverted ordering, the predicted is within the reach of upcoming experiments like GERDA III. For normal ordering, measuring down to eV could exclude large regions of and , depending on the value of .
In conclusion, this paper represents the first time that an infinite series of finite groups has been examined for generalised CP transformations that are consistent with it. We emphasise the important role of among the subgroups of with triplet irreducible representations and hope that this study will help to shed some light on the mystery of neutrino mixing. If the Dirac CP phase is measured to differ from 0 or , or the mixing angles deviate from the sum rules , respectively, then this would mean that in general a potential flavour group cannot be broken to , as in the case of the direct approach assumed here. However the semidirect approach, in which a subgroup is preserved, would remain a possibility for theories based on .
Acknowledgments
The authors would like to thank Alex Stuart for many very useful discussions. The authors acknowledge partial support from the European Union FP7 ITNINVISIBLES (Marie Curie Actions, PITN GA2011 289442). SFK acknowledges support from the STFC Consolidated ST/J000396/1 grant.
References
 [1] S. F. King and C. Luhn, Rept. Prog. Phys. 76 (2013) 056201 [arXiv:1301.1340 [hepph]]; S. F. King, A. Merle, S. Morisi, Y. Shimizu and M. Tanimoto, arXiv:1402.4271 [hepph].
 [2] C. S. Lam, Phys. Rev. D 87 (2013) 013001 [arXiv:1208.5527 [hepph]].
 [3] M. Holthausen, K. S. Lim and M. Lindner, Phys. Lett. B 721 (2013) 61 [arXiv:1212.2411 [hepph]].
 [4] J. A. Escobar and C. Luhn, J. Math. Phys. 50 (2009) 013524 [arXiv:0809.0639 [hepth]].
 [5] S. F. King, T. Neder and A. J. Stuart, Phys. Lett. B 726, 312 (2013). [arXiv:1305.3200 [hepph]].
 [6] R. d. A. Toorop, F. Feruglio and C. Hagedorn, Phys. Lett. B 703, 447 (2011). [arXiv:1107.3486 [hepph]].
 [7] R. de Adelhart Toorop, F. Feruglio and C. Hagedorn, Nucl. Phys. B 858, 437 (2012). [arXiv:1112.1340 [hepph]].
 [8] G. J. Ding, Nucl. Phys. B 862, 1 (2012). [arXiv:1201.3279 [hepph]].
 [9] S. F. King, C. Luhn and A. J. Stuart, Nucl. Phys. B 867, 203 (2013). [arXiv:1207.5741 [hepph]].
 [10] C. S. Lam, Phys. Rev. D 87, no. 5, 053012 (2013). [arXiv:1301.1736 [hepph]].
 [11] S. F. King and C. Luhn, JHEP 0910, 093 (2009). [arXiv:0908.1897 [hepph]].
 [12] I. de Medeiros Varzielas and G. G. Ross, JHEP 1212, 041 (2012). [arXiv:1203.6636 [hepph]].
 [13] R. Krishnan, J. Phys. Conf. Ser. 447, 012043 (2013). [arXiv:1211.3364 [hepph]].
 [14] M. Holthausen and K. S. Lim, Phys. Rev. D 88, 033018 (2013). [arXiv:1306.4356 [hepph]].
 [15] M. C. Chen, M. Fallbacher, K. T. Mahanthappa, M. Ratz and A. Trautner, [arXiv:1402.0507 [hepph]].
 [16] I. Girardi, A. Meroni, S. T. Petcov and M. Spinrath, JHEP 1402, 050 (2014). [arXiv:1312.1966 [hepph], arXiv:1312.1966].
 [17] G. J. Ding and Y. L. Zhou, [arXiv:1312.5222 [hepph]].
 [18] F. Feruglio, C. Hagedorn and R. Ziegler, JHEP 1307, 027 (2013). [arXiv:1211.5560 [hepph]].
 [19] F. Feruglio, C. Hagedorn and R. Ziegler, Eur. Phys. J. C 74, 2753 (2014). [arXiv:1303.7178 [hepph]].
 [20] C. C. Nishi, Phys. Rev. D 88, 033010 (2013). [arXiv:1306.0877 [hepph]].
 [21] C. Luhn, Nucl. Phys. B 875, 80 (2013). [arXiv:1306.2358 [hepph]].
 [22] W. Grimus and M. N. Rebelo, Phys. Rept. 281, 239 (1997). [hepph/9506272].
 [23] G. J. Ding, S. F. King and A. J. Stuart, JHEP 1312, 006 (2013). [arXiv:1307.4212].
 [24] G. J. Ding, S. F. King, C. Luhn and A. J. Stuart, JHEP 1305, 084 (2013). [arXiv:1303.6180 [hepph]].
 [25] M. Holthausen, M. Lindner and M. A. Schmidt, JHEP 1304, 122 (2013). [arXiv:1211.6953 [hepph]].
 [26] R. Kuchimanchi, Phys. Rev. D 86, 036002 (2012). [arXiv:1203.2772 [hepph]].
 [27] R. Kuchimanchi, Phys. Rev. D 82, 116008 (2010). [arXiv:1009.5961 [hepph]].
 [28] R. Kuchimanchi, Eur. Phys. J. C 74, 2726 (2014). [arXiv:1209.3031 [hepph]].
 [29] M. S. Boucenna, S. Morisi, E. Peinado, Y. Shimizu and J. W. F. Valle, Phys. Rev. D 86, 073008 (2012). [arXiv:1204.4733 [hepph]].
 [30] F. Capozzi, G. L. Fogli, E. Lisi, A. Marrone, D. Montanino and A. Palazzo, [arXiv:1312.2878 [hepph]].
 [31] M. Auger et al. [EXO Collaboration], Phys. Rev. Lett. 109, 032505 (2012). [arXiv:1205.5608 [hepex]].
 [32] P. A. R. Ade et al. [Planck Collaboration], [arXiv:1303.5076 [astroph.CO]].
 [33] S. F. King, A. Merle and A. J. Stuart, JHEP 1312, 005 (2013). [arXiv:1307.2901 [hepph]].