A massless neutrino and lepton mixing patterns from finite discrete subgroups of
Abstract
Finite discrete subgroups of as possible flavour symmetries for a massless neutrino with predictive mixing angles are studied. This is done by assuming that a residual symmetry appropriate for describing a massless neutrino is contained in . It is shown that all the groups admitting three dimensional faithful irreducible representation and generated from a specific set of matrices imply only one of the three flavour compositions for the massless state namely, unmixed, maximally mixed with equal probabilities and bimaximally mixed with probabilities and their permutations. This result holds irrespective of the order of and the choice of within it. All of these lead to unfavorable leading order prediction for the solar mixing angle. Neutrino mixing pattern is then numerically investigated in case of subgroups of with order less than 512 and it is found that only one of these can lead to a massless neutrino and leading order predictions for all the mixing angles close to their experimental values. Ways to correct for the solar angle prediction are proposed and two concrete examples giving the observed mixing pattern are discussed.
1 Introduction
The orderly pattern found in leptonic mixing may be signaling the existence of some underlying flavour symmetry, see reviews (); Ishimori:2010au () for some recent reviews. The mixing pattern is quite wellknown by now but it is still difficult to fix a discrete symmetry responsible for it. A systematic approach to look for such symmetries has been pursued vigorously symmetry (). Basic idea in this approach is to (a) first obtain residual symmetries and of the neutrino and the charged lepton mass matrices based on the observed mixing pattern and (b) look for the flavour symmetry group which contain , as subgroups. is usually a symmetry in case of three massive Majorana neutrinos with unconstrained masses and is some discrete subgroup of . The advantage of this approach is that mixing pattern is completely fixed by the choice of and without any detailed knowledge of the underlying theory. Moreover, a general prescription can be formulated Lam:2011ag () which leads to the desired mixing from some underlying Lagrangian invariant under spontaneously broken to and . This basic framework has been used reviews (); Ishimori:2010au (); symmetry () to obtain zeroth order mixing patterns such as tribimaximal (TBM) and bimaximal (BM) and also to obtain some reasonable predictions on the mixing angles but flavour groups which can predict all the mixing angles in leading order within their values are very few and large with order Holthausen:2012wt ().
The above framework does not put any restrictions on neutrino masses. The present neutrino data are still consistent with the quasi degenerate neutrinos or a spectrum with a massless neutrino both in case of the normal and inverted mass hierarchies. It would be appropriate to look for the modification of the above scenario which yields either of these spectrum. The case of the quasi degenerate neutrino was considered in Hernandez:2013vya () while the flavour symmetries appropriate for obtaining a massless state were discussed in Joshipura:2013pga (). Both, the choices for the residual symmetries and the flavour groups become different in theses cases. If neutrinos are Majorana particles and one of them is massless then one can redefine the phase of the massless field. Thus, one of the symmetries of the neutrino mass matrix used as a member of in the usual approach gets replaced by a symmetry with the corresponding matrix having . As a result, the group embedding this also needs to be a discrete subgroup (DSG) of rather than that of . This idea was elaborated in Joshipura:2013pga () and the group series Ishimori:2010au () and as well as the group Ludl:2010bj () were studied as possible flavour symmetries of the massless neutrino. The groups were shown to lead to a neutrino mass matrix with  symmetry for arbitrary . The groups were shown to lead to democratic mixing while provided an example of symmetry leading to the BM mixing pattern. None of these groups could predict all three mixing angles correctly but both and could yield a good zeroth order predictions and perturbations leading to the correct mixing patterns were also studied in case of .
Aim of the present paper is to make an exhaustive search for an appropriate flavour group and see if a massless neutrino and realistic mixing angles can be obtained with some flavour symmetry. This search leads to a “nogo result” which demonstrates that the solar angle cannot be predicted correctly at the leading order in a very large class of groups to be specified as we go along. In all these groups, the flavour composition of the massless state gets determined irrespective of the nature of the group and for all possible choices of and within it. The allowed compositions are only three types. The trimaximal with probability to be in different flavour states as or bimaximal with probabilities and its permutations and trivial one and its permutations. This immediately results in wrong prediction for the solar angle in case of the normal hierarchy. In the case of inverted hierarchy, bimaximally mixed massless state provides a good leading order approximations to the reactor and atmospheric mixing angles but the solar angle turns out to be maximal as we will show later. This feature follows in a large class of groups which are DSG of having a three dimensional faithful irreducible representation (IR). In particular, this result holds for all but one group having order .
This paper is organized as the follows. We briefly review the idea of predicting a massless neutrino within the realm of the discrete symmetries in the next section. In section 3, we provide a simple analytic study of the large class of DSG of and discuss the allowed mixing patterns with a massless neutrino. The predictions of lepton mixing angles obtained by an exhaustive numerical scan of small DSG of are presented in section 4. We discuss in section 5 some alternatives which can lead to correct solar mixing angles as well and discuss specific scenario capable of reproducing all three mixing angles in the experimentally allowed range. Finally, we summarize our study in section 6.
2 Massless neutrino and flavour symmetries
We first review a framework relating flavour symmetries to neutrino mixing and its generalization which can also yield a massless neutrino. Assume that the neutrino (charged lepton) mass matrix in arbitrary basis is invariant under a set of unitary generators () which form a discrete group (). All are assumed to commute and are diagonalized by a matrix . Similarly, the commuting set is diagonalized by . The invariance of the leptonic mass matrices under the respective groups can be used to show that the neutrino mixing matrix is determined in terms of and symmetry ():
(1) 
where and are arbitrary diagonal phase matrices.
The minimal group can be constructed from the observation that mass terms for the Majorana neutrinos remain unchanged under a change of sign of any of the neutrino mass eigenstates. Thus neutrino mass matrix in the mass basis is trivially invariant under
(2) 
where Det is chosen +1. Any two of these define a symmetry. This implies that
(3) 
correspond to symmetry in a basis labeled by . This along with the
(4) 
with being some discrete phases can be used as residual symmetries to be embedded in a bigger group .
When one of the neutrinos is massless, then phase of the corresponding field can be changed without affecting neutrino mass term. In this case, one of the say can be replaced by Joshipura:2013pga ()
(5) 
with and . The forms a () group for even (odd) . This along with provide possible residual symmetries if one neutrino is massless. We are assuming here that the two massive neutrinos are nondegenerate. has to be chosen differently if this not the case Hernandez:2013vya (). We shall not entertain this possibility here.
Note that, all the eigenvalues of as defined above are different. This uniquely fixes all the mixing angles in and it is sufficient to take this as a residual symmetry of the neutrino mass matrix and look for embedding of this and into a bigger group . Arbitrariness in one of the mixing angles would remain if above is replaced by
(6) 
We shall use either or as possible in the following. As for , we shall choose defined in Eq. (4) which uniquely fixes if all phases are different. We then look for the discrete subgroups of which contain and or as elements. A complete classification of all discrete subgroups of is not available but our analysis would encompass all the known DSG of with a three dimensional faithful irreducible representation and many having (2+1)dimensional reducible representation Parattu:2010cy (); Ludl:2010bj ().
3 Massless neutrino and lepton mixing from discrete subgroups of
Let us first review here some available information on the DSG of which will be used in our analysis. Details can be found in GrimusLudl:groups () and specially in Ludl:2010bj (). All the DSG of which are also the subgroups of have been classified. The complete list can be found for example in GrimusLudl:groups (); Grimus:SU3 (). They are characterized in terms of some small set of matrices which are used to generate elements of various DSG of . The present knowledge on DSG of which are not the subgroups of is partial. We will concentrate in this section on such subgroups of admitting a faithful three dimensional irreducible representation. It is convenient to divide these groups in two categories:

Those which are generated by a specific set of generators and as given in Ludl:2010bj () and reproduced here in Table 1 and

those which are generated by more special textures labeled as in Table 1.
All the groups which can be generated using set (X) are not known but the groups with order generated by the set (X) and (Y) and which cannot be written as direct product of are listed by Ludl Ludl:2010bj (). One can obtain groups from this set as discussed by Ludl. There are 75 groups with order of which only 5 fall in the category (Y). In addition to these 75 groups, Ludl has also identified Ludl:2010bj () infinite series of groups called , , , which are expressible as semidirect product of DSG of with some cyclic group . All these series fall in the category (X).
3.1 Massless state: General flavour structure and implications
We now state and derive a general theorem regarding the structure of the massless state that holds
for a large class of DSG of .
Theorem: If and are elements of any group in category (X)
generated by combination of matrices listed in Table 1 and if all
the eigenvalues of are distinct then the only possible flavour compositions of the massless
state are (1) trivial and its permutation or (2) trimaximal with probability
or (3) BM with probability or permutations
thereof.
Proof: The proof of this theorem follows from few simple observations. We list these and
their implications in the following.

Let us define generalized textures with elements in set replaced by obtained by replacing nonzero entries in elements of by some arbitrary roots of unity. Thus, for example
(7) with . is clearly contained in . It is easy to show that product of any two elements in is an element in , e.g.
Also product of two elements in belongs to . This implies that all the elements of any group generated using elements in possesses only one of the six textures given in . It is useful to define subsets , and . A product of two elements of belongs to only while a product of any two elements of generates an element which is either in or in . A product of any two elements in can belong to either of ^{1}^{1}1The set of above textures is isomorphic to the group with mapped respectively to elements of Ishimori:2010au ()..

Since is diagonal it can either be used as (if one diagonal element is complex and two are unequal and real) or as (if all diagonal elements are different). Thus can either be used as a symmetry of the neutrinos or that of the charged leptons.

Eigenvalues of are given by
with . Thus at least two eigenvalues are complex and any element of with these two structures cannot be used as neutrino symmetry . can only be used as symmetry of the charged leptons.

Each element in contains a diagonal and two offdiagonal nonzero entries. Their eigenvalues are of two types. If the offdiagonal entries are complex conjugate of each other then eigenvalues are given by . This can be used as neutrino symmetry if . Otherwise elements in also have at least two complex eigenvalues and they would only be suitable as representing . Thus can be used as a symmetry of the neutrinos and/or charged leptons depending on its structure.
It follows from (24) that if flavour group belongs to category (X) then there are two possible choices for the neutrino symmetry and three for the the charged lepton symmetry apart from their cyclic permutations. We summarize below these choices and matrices which diagonalize them:

: and

: For example,
Other choices of are obtained from the cyclic permutation of above.

: and

: For example, and

: For example,
where is defined in Eq. (7) and
(8) 
. Other choices of are obtained from the cyclic permutation of above.
The above alternatives for and determine complete mixing pattern (apart from cyclic permutations) which is possible for all the flavour groups belonging to category (X). The detailed values of the mixing angles depend upon the values of complex parameters appearing above but flavour content of the massless state in all the cases is not sensitive to them. To see this, note that eigenstate of corresponding to a massless neutrino for all the choices listed above is given uniquely by or its cyclic permutations and let us consider ⟩ for definiteness. The corresponding flavour state in the basis with diagonal (and hence diagonal charged lepton mass matrix ) is given by ⟩⟩. Then three possible choices for listed above give three flavour mixing mentioned in the theorem stated above. (a) implies an unmixed massless ⟩ with probabilities or (b) implies trimaximally mixed ⟩ with probabilities or (c) implies bimaximally mixed ⟩ with probabilities . Other choices allowed by permutations of the chosen and only permute the above mentioned flavour compositions.
The above result has strong phenomenological implications due to the fact that the structure of the massless state determines one column of the mixing matrix. If only one neutrino is massless then the massless state has to be identified with the first (third) column of the neutrino mixing matrix in the standard convention in case of the normal (inverted) hierarchy. Then, for the trimaximal structure, one predicts for the inverted and for the normal hierarchy. Both of these differ from the experimental values and would need large corrections. The BM composition is more suitable to describe inverted hierarchy and taken as the third column of would predict and . These predictions signaling  symmetry have often been considered as a good zeroth order ansatz which can follow in a large number of groups in category (X) as we will show. But with either choice, one does not get the correct leading order value for the solar mixing angle in the whole class of groups in the category (X).
3.2 Mixing patterns with a massless neutrino
The foregoing discussion also allows us to determine all possible mixing matrices for all the groups in category (X). We list below nontrivial choices.

If and then has democratic structure:
(9) The above structure which can be realized for both normal and inverted hierarchy implies . It fails to predict two of the mixing angles correctly at the leading order and would need very large corrections.

If is to be nondiagonal then it is given by the choice (S.b). With as in (T.b), one gets as
(10) where, . Note that the first column determining massless state is fixed for the normal hierarchy while one can permute the second and third columns by interchanging the order of the two real eigenvalues of . Similarly, changing the order of the three rows amount to reordering the eigenvalues of . Thus there are six possible choices for the reactor mixing angle and minimum of this should be identified with and the corresponding column as the third column of the PMNS matrix. Explicitly,
(11) where sign correspond to two columns and rows are labeled by
is defined in Eq. (7). The also gets determined by the entries in the same column as that of . Moreover, it is seen from Eq. (10) that
independent of the values of complex parameters showing that all the groups in category (X) lead to the absence of the Dirac CP violation even if they predict nonzero . Alternative choice also give prediction similar to above but other possible choices of , namely give unacceptable predictions for mixing angles in case of the normal hierarchy.

Unique choice for in case of the inverted hierarchy is given by
(12) The corresponding choice for giving nontrivial pattern is given by in (T.c). The resulting mixing matrix is given by:
(13) Note that the has the BM values and independent of the values of the group dependent complex parameters. In this sense, the BM mixing pattern is more universal and follows in many groups constructed from the category (X).
The case (2) discussed above can lead to reasonable values for the mixing angles , and we will explore various possibilities numerically in the next section. The is predicted either or close to . This needs to be changed and we shall discuss possible ways to modify this in section 5.
Before closing this section, we wish to emphasize generality of the result derived here. Firstly, this result is valid for any group in category (X) independent of its order. Secondly, large number of the finite DSG of fall in category (X) to which the theorem derived here is applicable. In addition to 70 of the 75 groups with order , six infinite series of groups also fall in this category. These are generated by the following specific matrices of Table 1:
defined above is a matrix with determinant 1 and can be written as independent of . It is clear from the structure of the generators that the group series either contain diagonal elements or elements with two complex eigenvalues. Thus these groups provide an example of the case (1) discussed before Eq. (9) and only mixing pattern possible within these groups is democratic mixing. Of these, were already discussed in Joshipura:2013pga ().
The above theorem does not however hold for the groups in category (Y). These are not easily amenable to analytic discussion but we study them numerically and show that one of these can give quite satisfactory leading order predictions for the mixing angles.
4 A numerical scan of discrete subgroups of
We now study numerically the predicted values of the reactor and atmospheric mixing angles for all the DSG of of order . These include all the 70 groups of category (X) and 5 groups of category (Y). It is clear form the discussions in the last section that if the groups of category (X) are used as flavour symmetry for a massless neutrino then sizable corrections at least in the solar angle would be needed which in general may change the other angles also. It is thus appropriate to work out leading order predictions for other angles. While the predicted solar angle is almost universal for all the groups under study, the other angles depend on the details of the group and we present this numerically. In our analysis, we choose a particular group and look at its suitability to be a flavour symmetry with a massless neutrino and work out all possible values of the predicted mixing angles.
4.1 Results of the groups in category (X)
First, we restrict ourselves to the groups of category (X) and of order tabulated by Ludl Ludl:2010bj (). We demand that (1) this groups should contain at least one element to be identified as with eigenvalue with and appropriate to describe a massless neutrino, (2) should have all three different eigenvalues and (3) does not commute with . These requirements are essential but quite restrictive and rule out large number of groups as possible . Consider all the groups generated only from the matrices , and alone for arbitrary values of the integers defining these matrices. As mentioned in the last section, the above structures close among themselves as result the groups in question contain either diagonal elements or elements with at least two complex eigenvalues. The neutrino symmetry in this case would come only from the diagonal elements and one would get only democratic mixing which is not of much interest. There are 48 such groups among the 70 listed groups and the same argument would apply to any higher order groups generated using any of these three matrices. Requirement (1) is also restrictive and we find numerically that of the remaining 22 groups, 16 get ruled out by this requirement. This leaves only six groups which can be used as suitable flavour symmetry. We numerically find all the elements which can be used as and within these allowed groups and determine the resulting mixing matrix. The allowed groups and predicted values of the mixing angles are listed in Table 2.
Group  Classification  Generators  

⟦48,30⟧  0.0447  0.349  
⟦162,10⟧  0.0201  0.399  
0  0.5  
⟦192,182⟧  0.0447  0.349  
0  0.5  
⟦384,571⟧  0.0114  0.424  
⟦432,260⟧  0.0447  0.349  
⟦486,125⟧  0.0201  0.399  
0  0.5 
We have restricted ourselves only to choices which give . In this case one gets only four possible forms of in case of the normal neutrino mass hierarchy. They are listed as below:
(14) 
Note that there exist two predictions for in each of the above pattern as the second and third row can be interchanged though in Table 2 we select its first octant () values as they are favored at in case of normal hierarchy by a recent global fit Capozzi:2013csa (). It is seen from the above equations that

Structure displays  symmetry.

predicts close to the best fit value as per the global fits in Capozzi:2013csa (). The predicted requires significant corrections.

fares better and predicts within and within of their best fit values.
All the structures have trimaximally mixed massless state and predict . Situation changes if one were to impose lepton number as an additional symmetry and assume that neutrinos are Dirac particles. All the residual symmetries considered here and many more will be allowed choices in this case but now none of the neutrinos would be forced to be massless. In this case, one has the freedom to interchange the first and the second columns of all the structures obtained here. The group ⟦162,10⟧ in this case will predict all the mixing angles correctly within 3. This group has already been identified as a possible group for the Dirac neutrinos Holthausen:2013vba ().
The above results are obtained for the normal hierarchy. The inverted hierarchy allows only the BM pattern as argued analytically in the previous section. All the groups listed in Table 2 predicts BM mixing for the inverted hierarchy. The group is the smallest among them and has been studied in detail in Joshipura:2013pga () where an explicit model giving satisfactory mixing pattern is discussed.
4.2 Results of the groups in category (Y)
We now discuss the groups in category (Y) with order listed by Ludl Ludl:2010bj (). There are only five such groups having 3dimensional faithful IR. The corresponding generators are given by matrices  and reproduced here in Table 1. Consecutive pairs of matrices with generate groups of order 216, 324, 432, 432, 432 respectively. We have numerically generated all the elements of these five groups and find that four of the five groups do not contain any element with one complex and two real eigenvalues. They therefore do not qualify as possible flavour groups for a massless neutrino in this approach. The group generated by (classified in GAP GAP4 () as ⟦432,239⟧) is the only one which contain such elements and we analyze it further.
The group ⟦432,239⟧ generated numerically from the products of contain 372 elements with all eigenvalues unequal. Of these, 72 have two real eigenvalues and would be appropriate as a member of . The corresponding charged lepton symmetry group may be generated from any of the 372 elements. Taking only the noncommutative set of and , all the mixing angles are fully predicted with any of these choices due to their completely different eigenvalues. We have numerically constructed all possible mixing matrices using elements in these sets. We find that this group allows only two possible structures for the massless state. These have flavour content or and permutations of these. Since the former now differs from the trimaximal structure predicted in class (X) groups, there is a possibility of better agreement with the data. We explore it further by demanding that the reactor mixing angle should satisfy . It turns out that only two possible mixing structures satisfy this restriction, one for the normal and one for the inverted hierarchy. These are respectively given by,
(15) 
and
(16) 
The predicted mixing angles are:
(17) 
The prediction in case of the normal hierarchy does not reproduce all the mixing angles within 3. But it is fairly close to the observed ones with . It is thus conceivable that small nonleading order effect can correct the predictions. The same predictions can be obtained for all massive neutrinos if they are of Dirac type^{2}^{2}2The same pattern is obtained for Dirac neutrinos in Hagedorn:2013nra () using the group which is an subgroup of ⟦432,239⟧ discussed here.. The predictions in case of the inverted hierarchy are similar in some sense to the BM mixing. The predicted and are the same as in the BM mixing. The solar angle predicted here requires large corrections which are similar in magnitude to the case of the BM mixing but now in opposite direction.
The above mixing matrices can arise with several different choices of and from elements of the group. We give here an example for the normal hierarchy which happens to be the simplest choice: