Neutrino masses and mixing from flavour antisymmetry
Abstract
We discuss consequences of assuming () that the (Majorana) neutrino mass matrix displays flavour antisymmetry, with respect to some discrete symmetry contained in and () together with a symmetry of the Hermitian combination of the charged lepton mass matrix forms a finite discrete subgroup of whose breaking generates these symmetries. Assumption () leads to at least one massless neutrino and allows only four textures for the neutrino mass matrix in a basis with a diagonal if it is assumed that the other two neutrinos are massive. Two of these textures contain a degenerate pair of neutrinos.Assumption () can be used to determine the neutrino mixing patterns. We work out these patterns for two major group series and as . It is found that all and groups with even contain some elements which can provide appropriate . Mixing patterns can be determined analytically for these groups and it is found that only one of the four allowed neutrino mass textures is consistent with the observed values of the mixing angles and . This texture corresponds to one massless and a degenerate pair of neutrinos which can provide the solar pair in the presence of some perturbations. The wellknown groups and provide examples of the groups in respective series allowing correct and . An explicit example based on and displaying a massless and two quasi degenerate neutrinos is discussed.
I Introduction
Orderly pattern of neutrino mixing appears to hide some symmetry, discrete or continuous. It is possible to connect a given mixing pattern with some discrete symmetries of the leptonic mass matrices. Such symmetries may however be residual symmetries arising from a bigger symmetry in the underlying theory. One can obtain a possible larger picture by assuming that these symmetries are a part of a bigger group operating at the fundamental level whose breaking leads to the symmetries of the mass matrices. There is an extensive literature on study of possible residual symmetries of the mass matrices and of the groups which harbor them Lam:2008rs (); Lam:2008sh (); Lam:2009hn (); Lam:2011ag (); Toorop:2011jn (); deAdelhartToorop:2011re (); Altarelli:2012ss (); Holthausen:2012wt (); Hu:2012ei (); Hernandez:2012ra (); Hernandez:2012sk (); Holthausen:2013vba (); Holthausen:2013vba (); Lavoura:2014kwa (); Fonseca:2014lfa (); Hu:2014kca (), see Altarelli:2010gt (); King:2013eh (); Smirnov:2011jv () for reviews and additional references.
Starting point in these approaches is to assume the existence of some symmetries (usually a ) and (usually ) of the (Majorana) neutrino and the charged lepton mass matrices
(1)  
(2) 
Matrices diagonalizing the symmetry matrices can be related to the mixing matrices in each sector. The structures of these matrices can also be independently fixed if one assume that and represent specific elements of some discrete group in a given three dimensional representation. In this way, the leptonic mixing can be directly related to group theoretical structures. This reasoning has been used for the determination of the neutrino mixing angles in case of the three nondegenerate neutrinos Lam:2008rs (); Lam:2008sh (); Lam:2009hn (); Lam:2011ag (); Toorop:2011jn (); deAdelhartToorop:2011re (); Altarelli:2012ss (); Holthausen:2012wt (); Hu:2012ei (); Hernandez:2012ra (); Hernandez:2012sk (); Holthausen:2013vba (); Holthausen:2013vba (); Lavoura:2014kwa (); Fonseca:2014lfa (); Hu:2014kca (), two or three degenerate neutrinos Hernandez:2013vya (); Joshipura:2014qaa () and one massless and two nondegenerate neutrinos Joshipura:2013pga (); Joshipura:2014pqa ().
The residual symmetries may arise from spontaneous breaking of if the vacuum expectation values of the Higgs fields responsible for generating leptonic masses break but respect . We wish to study in this paper consequences of an alternative assumption that the spontaneous breaking of leads to an which displays antisymmetry instead of symmetry, i.e. assume that eq.(2) gets replaced by
(3) 
but (1) remains as it is. These assumptions prove to be quite powerful and are able to simultaneously restrict both the mass patterns and mixing angles when embedding of into is considered. We shall further assume that belong to some finite discrete subgroup of with Det Then the first consequence of imposing eq.(3) is that Det, i.e. at least one of the neutrinos remains massless. Since cases with two (or three !) massless neutrinos are not phenomenologically interesting, we shall restrict ourselves to cases with only one massless neutrino. Then as a second consequence of eq.(3), one can determine all the allowed forms of in a given basis for all possible contained in . There exist only four possible (and their permutations) consistent with eq.(3) in a particular basis with a diagonal . Two of these give one massless and two nondegenerate neutrinos and the other two give a massless and a degenerate pair of neutrinos which may be identified with the solar pair.
We determine all the allowed textures of the neutrino mass matrix in the next section. Subsequently, we discuss groups and and identify those which can give correct description of mixing using flavour antisymmetry. In section IV, we introduce as neutrino residual symmetry and present an example in which neutrino mass matrix gets fully determined group theoretically except for an overall scale. We discuss a realization of the basic idea with a simple example based on the group in section V. Section VI contains summary and comparison with earlier relevant works.
Ii Allowed textures for neutrino mass matrix
We shall first consider the case of only one satisfying eq.(3) and subsequently generalize it to include two. The unitary matrix can be diagonalized by another unitary matrix :
where is a diagonal matrix having the form:
(4) 
Unitarity of implies that are some roots of unity. They are related by the condition which we assume without lose of generality. We now go to the basis with a diagonal . Defining , eq.(3) can be rewritten as:
(5) 
It follows that a given element is nonzero only if the factor in bracket multiplying it is zero. This cannot happen for an arbitrary set of and one needs to impose specific relation among them to obtain a nontrivial . We now argue that only two possible forms of and their permutations lead to neutrino mass matrices with two massive neutrinos. The third mass will always be zero as a consequence of eq.(3) and the assumption that belongs to . These forms of are given by:
(6) 
is an arbitrary root of unity. This can be argued as follows. Assume that at least one offdiagonal element of is nonzero which we take as the 12 element for definiteness. In this case, eq.(5) immediately implies the first of eq.(II) as a necessary condition. One can distinguish three separate cases of this condition^{1}^{1}1 case corresponds to permutation of the case with . (I) (II) and (III) . The entire structures of get determined in these cases from condition eq.(5) as follows:
(7) 
where . This structure implies one massless and two degenerate neutrinos with a mass . In case of (II),
(8) 
This case corresponds to one massless and two nondegenerate neutrinos. In the third case one gets
(9) 
which implies a massless and a pair of degenerate neutrinos.
The cases (I,III) lead to the same mass spectrum but different mixing patterns. in eq.(7) is diagonalized as with
(10) 
The arbitrary rotation by an angle originates due to degeneracy in masses. The texture II, eq.(8) is diagonalized by a unitary rotation in the 12 plane while the one in eq.(9) by a similar matrix with the angle .
The permutations of entries in give equivalent structures and are obtained by permuting entries in . The case which is not equivalent to above textures follows with a starting assumption that one of the diagonal elements of say, . In this case one requires with . The case with gives which is already covered. implies the condition in (5). This leads to a new texture
(11) 
For one gets permutation of or and for only 11 element of is non zero and two neutrinos remain massless. Thus conditions eq.(II) and their permutations exhaust all possible textures of consistent with the antisymmetry of , eq.(3) and two massive neutrinos. Any admitting an element with these sets of eigenvalues will give a viable choice for flavour antisymmetry group. Note that texture III (IV) can be obtained from I(II) by putting to zero. But the residual symmetries in all four cases are different. Because of this, the embedding groups can also be different. We therefore discuss all these cases separately.
The mixing matrix in texture I contains two unknowns and apart from an overall complex scale . This is a reflection of the fact that the corresponding is a symmetry and contains two degenerate eigenvalues . These unknown can be fixed by imposing another residual symmetry commuting with and satisfying eq. (2) or (3). We shall discuss such choices in section IV.
Iii Group theoretical determination of mixing
The physical neutrino mixing matrix depends on the structure of and . The latter can be determined if the symmetry as in eq.(1) is known. We now make an assumption that satisfying eq.(3) and as in eq.(1) are elements of some discrete subgroup (DSG) of denoted by . The DSG of have been classified in Miller (); Fairbairn:1964sga (); Bovier:1980gc (). They are further studied in Luhn:2007yr (); Luhn:2007uq (); Escobar:2008vc (); Ludl:2009ft (); Ludl:2010bj (); Zwicky:2009vt (); Parattu:2010cy (); Grimus:2010ak (); Grimus:2011fk (); Grimus:2013apa (); Merle:2011vy (). These can be written in terms of few presentation matrices whose multiple products generate various DSG. Two main groups series called and Grimus:2013apa () constitute bulk of the DSG of . Of these, we shall explicitly study two infinite groups series and which are examples of the type and respectively. See King:2013vna (); Ding:2014ora (); Hagedorn:2014wha () for earlier studies of neutrino mixing using the groups and and neutrino symmetry rather than antisymmetry.
Eq. (1) implies that commutes with . Thus, the matrix diagonalizing the former also diagonalizes and corresponds to the mixing matrix among the left handed charged leptons. Similarly, the matrix diagonalizing gets related to the structure of . In this way, the knowledge of and can be used to determine the mixing matrix
(12) 
This is the strategy followed in the general approach and we shall also use this to determine all possible mixing pattern for a given consistent with eqs.(1) and (3).
Not all the groups can admit an which will provide a legitimate antisymmetry operator , i.e. an element with eigenvalues specified by eq.(II). Our strategy would be to determine a class of groups which will have one or more allowed and then look for all viable within these groups. There would be different mixing patterns associated with each choice of and it is possible to determine all of them analytically for and groups.
iii.1
The groups are isomorphic to , where denotes the semidirect product. The group theoretical details for are discussed in Luhn:2007uq (); Ishimori:2010au (). For our purpose, it is sufficient to note that all the elements of the group are generated from the multiple product of two basic generators defined as:
(13) 
with . Here generates one of the groups and generates in the semidirect product . The other group is generated by . The above explicit matrices provide a faithful three dimensional irreducible representation of the group and multiple products of these matrices therefore generate the entire group whose elements can be labeled as:
(14) 
All elements of are obtained by varying over the allowed range in the above equation. Thus each matrices have elements giving in total elements corresponding to the order of . The eigenvalue equation for the nondiagonal elements and is simply given by . These elements therefore have eigenvalues with . These are not in the form of eq.(II) required to get the neutrino antisymmetry operator . Thus has to come from the diagonal elements. This requires that should be such that matches the required eigenvalues of given by eq.(II) or their permutations. This cannot happen for all the values of variables and one can easily identify the viable cases. It is found that

can match any of only for even . Thus only groups with contain neutrino antisymmetry operator .

The eigenvalue set is always contained as a diagonal generator for all groups and can be chosen as . Hence the texture I with two degenerate and one massless neutrino can follow in any . The smallest such group is which is one of the most studied flavour symmetry from other points of view Ma:2001dn (); Babu:2002dz (); Altarelli:2005yx (); Gupta:2011ct (); Ma:2015pma (); He:2006dk (); He:2015gba (); Hirsch:2007kh (); Dev:2015dha (); He:2015afa (); He:2015gba ().

The set arises only for N multiple of 4, i.e. in case of groups , . These groups also contain a satisfying the second of eq.(II). Thus textures are possible for all groups.

The set with and the associated texture III is viable in with
Let us now turn to the mixing pattern allowed within the groups. has to be a diagonal operator identified above. Then can be any other diagonal operator or any of or . In the former case, , where denotes a identity matrix. The neutrino mixing in this case coincides with diagonalizing any of the four textures of giving . None of the allowed are suitable to give the correct mixing pattern with a nonzero . Thus, needs to be any of the nondiagonal element . The matrices diagonalizing are given by
(15) 
where,
(16) 
The final mixing matrix depends upon the choice of specific texture for . Consider the texture I which arises within all the groups. in this case is given by eq.(10) and . Since a neutrino pair is degenerate, the solar mixing angle remains undetermined in the symmetry limit. This is reflected by the presence of an unknown angle in eq.(10). In this case, the neutrino mass hierarchy is inverted and the third column of needs to be identified with the massless state. It is independent of the angle . We get for ,
(17) 
with for the group . take discrete values in above equation while and are unknown quantities appearing in the neutrino mixing matrix eq.(10). The entries in can be permuted by reordering the eigenvalues of . We will identify the minimum of with . If the minimum of the remaining two is identified with then one will get a solution with the atmospheric mixing angle . In the converse case, one will get a solution . The experimental values of the leptonic angles are determined through fits to neutrino oscillation data Capozzi:2013csa (); Forero:2014bxa (); GonzalezGarcia:2014bfa (). Throughout, we shall specifically use the fits presented in Capozzi:2013csa () for definiteness. The texture I corresponds to the inverted hierarchy and the best fit values and 3 ranges appropriate for this case are given Capozzi:2013csa () by:
(18) 
Let us mention salient features of results following from eq.(17)

It is always possible to obtain correct by choosing unknown quantities and of . This should be contrasted with situation found in Joshipura:2014qaa () which used neutrino symmetry instead of antisymmetry to obtain a degenerate pair of neutrinos. As discussed there, none of the groups could simultaneously account for the values of within 3.

It is possible to obtain more definite predictions by choosing specific values of and or . In contrast to and which are unknown, the choice of is dictated by the choice of and it is possible to consider any specific choice of in the range . Consider a very specific choice of real ,i.e. and a residual symmetry corresponding to putting in eq.(17). This equation in this case gives a prediction which holds for all values of . This relation is equivalent to a maximal which lies within the 1 range of the global fits Capozzi:2013csa (). then can be chosen to get the correct . Since the specific choice is allowed within all the groups, all of them can predict the maximal and can accommodate correct .

The relation does not hold for a complex even if . Such choices of give departures from maximality in . It is then possible to reproduce both the angles correctly by choosing . This is nontrivial since a single unknown determines both and for a specific choice of group (i.e. ) and a residual symmetry (i.e. and ). The resulting prediction can be worked out numerically by varying over the allowed integer values and over continuous range from to . Values of obtained this way are depicted in Fig.(1). This is obtained by requiring that lies within the allowed 1 range. The phase is put to zero. It is seen form the Figure that all the groups always allow maximal as already discussed. But solutions away from maximal are also possible for . The minimal group capable of doing this is . The next group can lead to near to the best fit values of the parameters. Specifically the choice , within the group and gives and to be compared with the best fit values and in Capozzi:2013csa ().

can only be zero or 1 and is real for the smallest group . In this case, one immediately gets the prediction for .  symmetry is often used to predict the maximal . This is not even contained in which has only even permutations of four objects. Still the use of antisymmetry rather than symmetry allows one to get the maximal and it also accommodates a nonzero within . This should be contrasted with the situation obtained in case of the use of symmetry condition eq.(2) instead of (3). It is known that in this case group gives democratic value for , see for example deAdelhartToorop:2011re ().
We now argue that the other three textures though possible within groups do not give the the correct mixing pattern. Texture II has one massless and in general two nondegenerate neutrinos. This texture can give both the normal and the inverted hierarchy. The mixing matrix is blockdiagonal with a matrix giving mixing among two massive states. Given this form for and a general as given in eq.(III.1), one finds that the case with inverted hierarchy leads to the prediction while the normal hierarchy gives instead . Neither of them come close to their experimental values.
The texture (III) having degenerate pair corresponds to the inverted hierarchy. in this case is block diagonal with an unknown solar angle. Given the most general form, eq.(III.1) for one obtains once again the wrong prediction ruling out this texture as well. Likewise, texture IV also gets ruled out. This corresponds to a diagonal with and has the universal structure .
To sum up, all the groups contain a neutrino antisymmetry operator and allow a neutrino mass spectrum with two degenerate and one massless neutrino and can reproduce correctly two of the mixing angles . The values for the solar angle and the solar scale have to be generated by small perturbations within these group. We shall study an example based on the minimal group in this category in section V.
iii.2 groups
groups are isomorphic to with . The group in the semidirect product is generated by in eq.(13) and a matrix
(19) 
The matrices provide a faithful irreducible representation of Escobar:2008vc () and generate the entire group with elements. elements generated by give the subgroup. The additional elements are generated from the multiple products of with elements of . These new elements can be parameterized by:
(20) 
Here . Since is a subgroup of , the neutrino mass and mixing patterns derived in the earlier section can also be obtained here. But the new elements allow more possibilities now. In particular, they allow more elements which can be used as neutrino antisymmetry . To see this, note that the eigenvalues of are given by . This can have the required form, eq.(II) when or . The eigenvalues in respective cases are or and one gets the textures I or IV by using any of as neutrino antisymmetry with and respectively. Similarly, possible choices of the charged lepton symmetry also increases. It can be any of the six types of elements: , as before or . Important difference compared to is that the texture I can now be obtained for both odd and even values of by choosing any of the with as neutrino antisymmetry. Texture IV still requires and hence even for its realization. We determine mixing matrix for each of these textures and discuss them in turn.
iii.2.1 Texture I
The residual anti symmetries which lead to texture I can be either
(1) or (2) where
.
The residual symmetry
of can be any elements in the group which we divide in
three classes:
, and . Here and in the following, we use symbols
and to collectively denote and . We use the basis as specified in eqs.(III.2,III.1) for . Then the
neutrino mixing matrix is given by in case (1) while it
is given by in case (2). This
follows by noting that the texture given in eq.(7)
holds in a basis with diagonal but in the chosen group
basis of eq.(III.2) is nondiagonal in case (2).
The neutrino mass matrix in this basis is thus given by
where
diagonalizes . The matrix which
diagonalizes
is then given by where diagonalizes
. Explicitly,
with
(21) 
We have chosen the ordering of columns of in such a way that the first column always corresponds to the eigenvalue . With this ordering one gets the texture I given in eq.(7) when is used as neutrino antisymmetry.
The matrices diagonalizing in three cases above are given in the same basis by in cases (A),(B),(C) respectively where are given in eq.(III.1). Thus we have six (four) different choices for () giving in all 24 leptonic mixing matrices . We list these choices and the corresponding matrices in Table I.
Case  

1A  
1B  
1C  
2A  
2B  
2C 
Not all of 24 mixing matrices listed in Table I give independent predictions for the third column of which determines and . We discuss the independent ones below.
The choice (1A) giving has one of the entries zero and thus
cannot lead to correct or .
The choice (1C) involves only elements belonging to the
subgroup and its predictions are already discussed in the previous
section. The remaining choices give new predictions.
The case (1B) leads to three different . One obtained
with contain a zero entry in the third column and can be
used only as a zeroeth order choice. One gets the following result in (1B)
if
(22) 
The ordering of the entries can be changed by rearranging the eigenvectors of of appearing in . We have chosen here and below an ordering which is consistent with the values of the parameters when is equated with the standard form of the mixing matrix.The result in the third case with can be obtained from above by the replacement . All the three entries above follow for all the choices of and the phase . The case (1B) in this way gives a universal prediction. Two of the are equal within this choice and they correspond to and . Equality of the two then implies a independent prediction . in the above case is then given by and can match the experimental value with appropriate choice of the unknown . Since the choice of within (1B) is possible only for even it follows that all the groups lead to a prediction of the maximal atmospheric mixing angle and can accommodate the correct .
The choice (2A) also gives the same result for as (1B) with an important difference. The neutrino residual symmetry used in this choice is allowed for all and not necessarily . Thus one gets a universal prediction of the maximal for all within all groups.The smallest group in this category is the permutation group which contain symmetries appropriate for both the cases and .
There are two independent structures within nine possible choices contained in case (2B). The example of the first one is provided by the choice and . The elements in the third column of mixing matrix are given in this case by
(23) 
While this choice does not give universal prediction as in the case (1B) discussed above it still leads to a prediction for which is independent of the unknown angle and phase :
This follows from eq.(III.2.1) when is identified with . The predicted now depends only on the group theoretical factors .
Unlike (1B), both the maximal and nonmaximal values are allowed for in this case. The former occurs whenever . The latter occurs for other choices. It is possible to find values of parameters which lead to a nonmaximal within the experimental limits. The minimal such choice occurs for , i.e. the group which leads as shown in Table II to a within the 2 range as given in Capozzi:2013csa (). The next example of the group fairs slightly better.
The other prediction of the case (2B) is obtained with and . One obtains in this case
(24) 
In this case, is necessary nonmaximal if is to be small but nonzero. We may identify, with and fix . This determines the other two entries of for a given . For one obtains either or . Thus all the groups with this specific choice give results close to the 3 range in the global fits. This prediction can be improved by turning on or choosing different . An example based on the group giving close to the best fit value Capozzi:2013csa () is shown in the table.
Group  Predictions  

Maximal for all  