A new class of invariants in the lepton sector
aa r X i v : . [ h e p - ph ] J a n IPM/P-2007/067November 4, 2018A new lass of invariants in the lepton se torArman Esmaili †§ § † Department of Physi s, Sharif University of Te hnologyP.O.Box 11365-8639, Tehran, IRAN § Institute for Studies in Theoreti al Physi s and Mathemati s (IPM)P.O.Box 19395-5531, Tehran, IRANAbstra tWe onstru t a new set of ombinations from the mass matri es of the harged lep-tons and neutrinos that are invariant under basis transformation, hereafter the invari-ants. We use these invariants to study various symmetries and neutrino mass texturesin a basis independent way. In parti ular, we show that by using these invariants theansatz su h as µ − τ ex hange and re(cid:29)e tion symmetries, various texture zeros and(cid:29)avor symmetries an be expressed in a general basis.PACS numbers: 14.60.Lm; 14.60.Pq; 11.30.Hv1armanmail.ipm.ir2yasamantheory.ipm.a .ir Introdu tionThere is a onsensus among neutrino physi ists that the re ent neutrino data from the solarand atmospheri neutrino observations [1℄, the KamLAND rea tor experiment [2℄ and thelong baseline experiments [3℄ an be explained only through neutrino os illation. As is well-known the os illation s enario is based on the fa t that the neutrinos are massive and domix. The three by three mass matrix of the a tive neutrinos, m ν , introdu es new parametersto the standard model. Among these parameters, six parameters in prin iple show up inthe neutrino os illation probabilities: two mass-square splittings, three mixing angles andone Dira CP-violating phase. To this list, one should add the mass s ale of neutrinos (i.e.,the mass of the lightest neutrino) whi h does not appear in the os illation probability andhas to be derived in other types of experiments su h as the beta de ay experiments or the osmologi al observations.So far the nature of neutrinos (Majorana vs. Dira ) is not known. However, the majorityof neutrino mass models predi t a Majorana type neutrino mass matrix at low energies.Throughout this paper, we shall assume that neutrinos are of Majorana type whi h impliesthat their mass matrix is symmetri . If neutrinos are of Majorana type, in addition tothe above parameters the mass matrix will ontain two more physi al degrees of freedom:two more CP-violating phases whi h are alled Majorana phases. These two phases donot appear in the os illation probabilities. Even if neutrinos are proved to be of Majoranatype, extra ting the Majorana phases is going to be quite hallenging if possible at all [4℄.Moreover, only a ombination of the phases an be measured. That is, separately extra tingea h of the Majorana phases will not be possible with the present methods.Sour es of CP-violation are asso iated with the phases of the neutrino mass matrix;however, one has to be aware that by rephasing the neutrino (cid:28)elds the phases of the elementsof the mass matrix also hange. Sin e the seminal work by Jarlskog [5℄, de(cid:28)ning invariantsunder (cid:28)eld rephasing and basis transformation has proved to be very useful for studyingthe CP-violation in both quark and lepton se tors. An in omplete list of the papers thathave attempted to study the CP-violation in the lepton se tor by de(cid:28)ning invariants is[6, 7, 8, 9, 10℄. As is well-known, be ause of the presen e of the extra CP-violating phases,2he number of the independent invariants is more than what we have in the quark se tor(or for the ase of Dira neutrinos). Re ently, the ne essary and su(cid:30) ient onditions forCP-violation has been systemati ally formulated in terms of the rephasing invariants in themass basis of the harged leptons [8, 9℄. It is also possible to study the CP-violation in termsof the ombinations of neutrino and harged lepton mass matri es that are invariant undergeneral basis transformation [7℄. In this paper, we introdu e a new lass of invariants undergeneral basis transformation. As we shall see, this new set of invariants is very helpful forstudying the symmetries of the neutrino mass matrix.Among the nine parameters of the neutrino mass matrix, the two mass-square splittingsand two of mixing angles are so far measured. There are various running and plannedexperiments as well as proposals to measure the remaining parameters. However, as alludedto before, even in the most optimisti ase, with the present experiments and proposals, wewill not be able to extra t all the neutrino parameters [4℄. Motivated by this fa t varioustheoreti al onje tures have been made to re onstru t the neutrino mass matrix. Most ofthese onje tures are based on symmetries that are apparent only in a parti ular basis.Examples are texture zeros, µ − τ ex hange and re(cid:29)e tion symmetries [8, 11℄ and various(cid:29)avor symmetries [12℄. All these symmetries are de(cid:28)ned in the mass basis of the hargedleptons. Using the invariants de(cid:28)ned in the present paper, we an formulate these symmetriesin a basis-independent way.This paper is organized as follows. In se t. 2, we introdu e a new lass of ombinationsthat are invariant under general basis transformation. In se t. 3, we use the invariants forformulating the symmetries of the neutrino mass matrix in a basis-independent way. Wespe ially dis uss the µ − τ ex hange and re(cid:29)e tion symmetries, (cid:29)avor symmetry onserving L µ − L τ harge and texture zero ansatz. A summary of results is given in se t. 4.3 New lass of invariantsConsider the following transformation on the harged leptons and neutrinos: ℓ Lα → U αβ ℓ Lβ ,ν Lα → U αβ ν Lβ ,ℓ Rα → V αβ ℓ Rβ , (1)where U and V are arbitrary unitary matri es and α and β are the (cid:29)avor indi es. The massterm of harged leptons and the e(cid:27)e tive mass term for Majorana neutrinos at the low energyare of the form: − L = ( m ℓ ) αβ ℓ Rα ℓ Lβ + 12 ( m ν ) αβ ( ν Lα ) c ν Lβ + H . c . (2)In order for the Lagrangian to remain invariant under transformations shown in Eq. (1), themass matri es have to transform as follows m ℓ → V m ℓ U † ,m ν → U ∗ m ν U † . (3)(Noti e that we have here used the assumption that neutrinos are of Majorana nature. ForDira neutrinos with ν Rα → W αβ ν Rβ , m ν would transform as W m ν U † so all the following dis- ussion should have been re onsidered.) It an be readily shown that under transformationsin Eq. (1) m ν ( m † ℓ m ℓ ) n → U ∗ m ν ( m ℓ † m ℓ ) n U † , ( m Tℓ m ∗ ℓ ) m m ν → U ∗ ( m Tℓ m ∗ ℓ ) m m ν U † , (4)where m and n are arbitrary integer numbers. In general, a linear ombination of these ombinations also transforms in the same way: X i [ a i m ν ( m † ℓ m ℓ ) n i + b i ( m Tℓ m ∗ ℓ ) m i m ν ] → U ∗ X i [ a i m ν ( m † ℓ m ℓ ) n i + b i ( m Tℓ m ∗ ℓ ) m i m ν ] U † , (5)where a i and b i are arbitrary onstants. In the above relation m i and n i an take positive aswell as negative integer numbers. Thus, the determinant of this matrix will transform intoitself times Det[ U † ]Det[ U ∗ ] , whi h is a pure phase. As a result, the ratio of any pair of su h4eterminants is invariant under the transformations shown in Eq. (1): Det[ P i ( a i m ν ( m † ℓ m ℓ ) n i + b i ( m Tℓ m ∗ ℓ ) m i m ν )]Det[ P i ( a ′ i m ν ( m † ℓ m ℓ ) n ′ i + b ′ i ( m Tℓ m ∗ ℓ ) m ′ i m ν )] is invariant . (6)Moreover, Det h X i ( a i m ν ( m † ℓ m ℓ ) n i + b i ( m Tℓ m ∗ ℓ ) m i m ν ) i Det h X i ( a ′ i m ν ( m † ℓ m ℓ ) n ′ i + b ′ i ( m Tℓ m ∗ ℓ ) m ′ i m ν ) i! ∗ is also invariant. Noti e that m nν ≡ m ν ( m † ν m ν ) n − transforms exa tly in the same form as m ν under basis transformations (see, Eqs. (1,3)). As a result, if we repla e any of m ν appearing inEq. (6) with m nν , the ombination will maintain its invarian e under transformation (1). Theabove ombinations present an in(cid:28)nite number of invariants. However the × Majorananeutrino mass matrix ontains nine degrees of freedom; so all of these invariants annot beindependent. It is straightforward to show that there is a set of invariants whi h all theother invariants an be written in terms of them. We will ome ba k to this point at theend of this se tion. In the following we give a on rete example for su h a (cid:16) omplete(cid:17) set ofinvariants. We will amply use these invariants in formulating the symmetries of the neutrinomass matrix in se t. 3.Let us de(cid:28)ne the following ombination of mass matri es: P ≡ m − ℓ tr [ m − ℓ ] , P ≡ m † ℓ m ℓ tr [ m † ℓ m ℓ ] , P ≡ I − P − P , (7)where I is the three by three identity matrix and m − ℓ ≡ m − ℓ ( m − ℓ ) † . It is straightforwardto show that when the eigenvalues of m † ℓ m ℓ are hierar hi al, P i a t as proje tion operators.In this se tion we will not use this property, but this feature will play an important roll inthe dis ussion of se t. 3.Now using these operators, let us de(cid:28)ne E ≡ Det[( P + P ) m ν − m ν P ]Det[ m ν ] , E ≡ Det[( P + P ) m ν − m ν P ]Det[ m ν ] , E ≡ Det[( P + P ) m ν − m ν P ]Det[ m ν ] , (8)5nd F ≡ Det[( P + P ) m ν − m ν P ]Det[ m ν ] , F ≡ Det[( P + P ) m ν − m ν P ]Det[ m ν ] , F ≡ Det[( P + P ) m ν − m ν P ]Det[ m ν ] . (9)It is worth mentioning that only three out of the six ombinations de(cid:28)ned in Eqs. (8,9) areindependent; that is three of them an be written in terms of the other three ombinations.It an be shown that the ombinations de(cid:28)ned in Eq. (6) an be in general written as a ombination of three E i and F i (e.g., E , E and F ).Let us de(cid:28)ne invariants G i and H i by repla ing Det[ m ν ] in the denominators of Eqs. (8,9)with Det[ m ν ] . That is G i ≡ Det[ m ν ]Det[ m ν ] E i and H i ≡ Det[ m ν ]Det[ m ν ] F i . (10)It an be shown that four out of the six invariants G i and H i are independent. Dependingon the problem in hand, one an hoose four independent invariants out of all these twelveinvariants to perform various analyses; for example, {E , E , F , G } . An example of theappli ation of su h (cid:16) omplete(cid:17) set of invariants is formulating CP symmetry in the leptonse tor. Both in the quark and lepton se tors, using the Jarlskog invariant ( J ) to test CP-violation is an established and widely used te hnique [5℄. As is well-known, the Majorananeutrino mass matrix ontains more than one sour e of CP-violation and therefore morethan one invariant will be ne essary to he k CP-invarian e [6, 7, 8, 9℄. Suppose we takea (cid:16) omplete(cid:17) set of invariants for testing the CP-violation. If any of these independentinvariants is omplex, the lepton se tor is not CP-invariant (i.e., at least one of the threepossible CP-violating phases is di(cid:27)erent from zero). One may ask whether the opposite isalso orre t. That is if all these invariants are real, an we on lude that CP is onserved?Sin e the equations are non-linear, if we take only three invariants, we may in general (cid:28)nd aspe i(cid:28) solution in addition to the trivial CP-invariant one. However, this spe i(cid:28) solutionmay not be ompatible with the neutrino data. On e we examine the realness of the fourthindependent invariant, this solution an be ex luded.6n the rest of this paper, we show how the ombinations E i and F i an fa ilitate the basisindependent analysis of the symmetries of neutrino se tor.3 Symmetries of neutrino mass matrix in terms of theinvariantsThere are onje tures (su h as µ − τ re(cid:29)e tion and ex hange symmetries [8, 11℄, various(cid:29)avor symmetries [12℄ and texture zero matri es for the neutrino mass matrix [13℄) that are ustomized to be used for the lepton se tor. Any model that a ommodates su h onje turesalso has to reprodu e the hierar hy of the harged lepton masses. In this se tion, we use thisproperty to formulate the onje tures in a model-independent fashion.In the harged lepton mass basis the operators P , P and P take the following forms P = 1 m − e + m − µ + m − τ m − e m − µ
00 0 m − τ , (11) P = 1 m e + m µ + m τ m e m µ
00 0 m τ , (12)and P = − P − P . (13)Using the hierar hy of the harged lepton masses, m µ /m τ = 3 . × − ≪ m e /m µ = 2 . × − ≪ m e /m τ = 7 . × − , we an write P = + O (cid:16) m e m µ , α (cid:17) , (14)7 = − α
00 0 α + O (cid:16) m e m µ , α (cid:17) , (15) P = α
00 0 1 − α + O (cid:16) m e m µ , α (cid:17) , (16)where α ≡ m µ /m τ .In the harged lepton mass basis, to the (cid:28)rst order of the parameter α , E i and F i de(cid:28)nedin Eqs. (8,9) an be written as E = m ee ( m µτ − m µµ m ττ )Det[ m ν ] , E = m µµ ( m eτ − m ee m ττ )Det[ m ν ] (1 − α ) , E = m ττ ( m eµ − m ee m µµ )Det[ m ν ] (1 − α ) , (17)and F = m µτ ( m ee m µτ − m eµ m eτ )Det[ m ν ] (1 − α ) , F = m eτ ( m µµ m eτ − m eµ m µτ )Det[ m ν ] (1 − α ) , F = m eµ ( m eµ m ττ − m eτ m µτ )Det[ m ν ] (1 − α ) , (18)where m αβ are the elements of m ν .Noti e that orre tions to these formulae omes from the next to leading order terms whi hare of the order ∼ − .In the subse t. 3.1 we dis uss the µ − τ symmetries of the neutrino mass matrix and onserved L µ − L τ (cid:29)avor symmetry in terms of the invariants. Then, in subse t. 3.2, wedis uss the so- alled texture zero matri es in an arbitrary basis for the lepton (cid:28)elds.3.1 µ − τ symmetries of m ν In this subse tion, we study the µ − τ re(cid:29)e tion and ex hange symmetries by using E i and F i . The aim is to formulate these symmetries in a basis independent way. We (cid:28)rst study the8 − τ ex hange symmetry, and we then turn our attention to the µ − τ re(cid:29)e tion symmetry[8, 11℄.The µ − τ ex hange symmetry (the symmetry under ν µ ↔ ν τ ) implies m eµ = m eτ , m µµ = m ττ , (19)where m αβ are the entries of m ν in the harged lepton mass basis. From Eqs. (17,18), these onditions an be expressed in terms of E i and F i in the following form (cid:12)(cid:12)(cid:12)(cid:12) F − F F + F (cid:12)(cid:12)(cid:12)(cid:12) ≪ (cid:12)(cid:12)(cid:12)(cid:12) E − E E + E (cid:12)(cid:12)(cid:12)(cid:12) ≪ . (20)The above inequalities are basis independent riteria for the µ − τ ex hange symmetry. Inany basis to he k for the µ − τ ex hange symmetry, we an immediately ompute the ombination in the Eq. (20); if the ondition in this equation is satis(cid:28)ed, the neutrino massmatrix is symmetri under the µ − τ ex hange.Now let us onsider the µ − τ re(cid:29)e tion symmetry. In the mass basis of harged leptons,the symmetry under ν µ ↔ ν ∗ τ , (with proper rephasing) implies m eµ = m ∗ eτ , m µµ = m ∗ ττ , m ee = m ∗ ee , m µτ = m ∗ µτ . (21)It is straightforward to show that these equalities, expressed in terms of E i and F i , impliesthe following inequalities (cid:12)(cid:12)(cid:12)(cid:12) F − F ∗ F + F ∗ (cid:12)(cid:12)(cid:12)(cid:12) ≪ , and (cid:12)(cid:12)(cid:12)(cid:12) E − E ∗ E + E ∗ (cid:12)(cid:12)(cid:12)(cid:12) ≪ . (22)As dis ussed in se t. 2, the above ratios are invariant under general basis transformationEq. (1). Thus, we have found some basis independent riteria for testing the µ − τ re(cid:29)e tionsymmetry; if either of the inequalities in Eq. (22) does not hold in a given basis, the neutrinomass matrix is not symmetri under the µ − τ re(cid:29)e tion.The relations in Eqs. (20) and (22) are the ne essary but not su(cid:30) ient onditions for µ − τ ex hange symmetry and µ − τ re(cid:29)e tion symmetry, respe tively. If the inequalitiesin Eqs. (20) or (22) do not hold in a given basis, we an on lude that lepton se tor is notsymmetri under the orresponding transformations. But the reverse is not orre t; that9s in ertain very spe ial ases it is possible to satisfy the onditions in Eqs. (20) or (22)without having the orresponding symmetry.Another symmetry proposed in the literature, whi h has some ommon features with the µ − τ ex hange symmetry, is the (cid:29)avor symmetry with onserved L µ − L τ [12℄. In the hargedlepton mass basis, the onservation of L µ − L τ implies the following form for the neutrinomass matrix × × × , (23)where (cid:16) × (cid:17) means that the orresponding entry is nonzero. In the limit of approximationmade in Eqs. (17) and (18), for a mass matrix of the form Eq. (23), the invariants E i and F i take the following values E = E = F = F = 0 , E = − and F = − α. (24)[Corre tions to these values are of the order ∼ O ( α ∼ − ) .℄ The above riterion for the(cid:29)avor symmetry onserving L µ − L τ , is a ne essary but not a su(cid:30) ient riterion for thissymmetry. That is, if a model does not satisfy this riterion in one basis, the model doesnot respe t the L µ − L τ onserving (cid:29)avor symmetry; but the reverse is not orre t.Noti e that a mass matrix of form Eq. (23) annot a ommodate the present data ofneutrino experiments be ause it predi ts two degenerate mass eigenvalues. In order to a - ommodate the present data, the L µ − L τ onserving symmetry has to be broken; i.e., L µ − L τ an be only an approximate symmetry. Thus, in pra ti e the exa t equalities, (cid:16) = (cid:17),in Eq. (24) has to be repla ed by (cid:16) ≃ (cid:17).3.2 Texture zero matri es in terms of invariantsAmong the various onje tures that an be imposed on the neutrino mass matrix, the texturezero ansatz have re eived more attention in the literature [13, 14℄. In these s enarios, ertainentries of the neutrino mass matrix in the harged lepton mass basis are onje tured to beequal to zero. Su h an assumption an originate from a more fundamental theory or anunderlying symmetry [15℄. On the other hand, most of the models of the neutrino mass10atrix are built based on some symmetry that is apparent only in a ertain basis whi h maynot orrespond to the mass basis of the harged leptons. In this se tion, we show that byusing the ombinations E i and F i de(cid:28)ned in Eqs. (8,9), we an express the onditions fortexture zero in a basis-independent way.As shown in [13℄, a neutrino mass matrix m ν with three or more zero entries is not ompatible with the data. Moreover, among the (cid:28)fteen possible two zero texture massmatri es, only seven of them an be made ompatible with the present neutrino data [13℄.The two zero textures are labeled A , A , B , B , B , B and C . We have listed them inTable 1. The non-vanishing entries in this Table are denoted by × . These textures have ertain predi tions for the values of the neutrino parameters. For example, the A and A textures are ompatible with data only for normal hierar hi al s heme ( m ≪ p ∆ m atm ).From the Eqs. (17,18), we readily observe that if one of the diagonal or o(cid:27)-diagonalelements of the mass matrix vanishes, some of E i or F i will also go to zero (e.g., m ee = 0 = ⇒E → m µτ = 0 = ⇒ F → and so on). Remember that under basis transformation, F i and E i are invariant. Thus, if an E i or an F i vanishes in a parti ular basis, it will vanish inall bases. To be pre ise, there is a orre tion of order of α ∼ m e /m µ ∼ − to E i and F i shown in Eqs. (17,18). As a result, when a mass matrix element vanishes, ertain F i and E i be ome mu h smaller than the rest but not exa tly zero.11able 1: Two zero texture mass matri esLabel Neutrino mass matrix m ν A × × ×× × × A × × × × × × B × × × × × × B × × × ×× × B × × ×× × × B × × × × × × C × × ×× ×× × E i and F i (leading order) for textures in Table 1Label Values of E i and F i A E ≈ E ≈ F ≈ F ≪ E ≈ F A E ≈ E ≈ F ≈ F ≪ E ≈ F B E ≈ F ≪ E ≈ F , E ≈ F B E ≈ F ≪ E ≈ F , E ≈ F B E ≈ E ≈ F ≈ F ≪ E ≈ F B E ≈ E ≈ F ≈ F ≪ E ≈ F C E ≈ E ≪ F ≈ F ≈ F − E = 0 Ea h of the texture zeros implies a ertain pattern for E i and F i . For example, for the A texture we (cid:28)nd E ≈ E ≈ F ≈ F ≪ E ≈ F . The patterns of E i and F i for the rest of thetextures are summarized in Table 2. Here, A ≈ B means | A − B | / | A + B | . O ( α ∼ − ) .Going to higher orders of α makes the analysis uselessly umbersome, espe ially that in mostmodels that predi t texture zeros the vanishing elements of m ν re eive a small orre tion(due to running or et .). Thus, in the following we onsider the leading order patterns for E i and F i . That is to perform the analysis, we will repla e “ ≈ ” with “ = ” and set E i and13 i for ea h pattern that a ording to Table 2 are mu h smaller than the rest equal to zero.These patterns an be onsidered as a test for the two zero textures. That is, by om-puting E i and F i in any given basis, one an he k if a ertain pattern an be the ase. Itis obvious that if the pattern asso iated with a ertain texture does not hold, m ν in the harged lepton mass basis will not have the format of that parti ular texture. In the follow-ing, we explore whether the opposite is also true. The question is as follows. Suppose thata ertain pattern of E i and F i listed in Table 2 is realized. Can we then on lude that m ν in the harged lepton mass basis has the format of the texture orresponding to that parti -ular pattern? To answer this question we he k if the equations listed in Table 2 have anysolution ompatible with the neutrino data other than the parti ular texture zero solution orresponding to them. To perform the analysis we use the standard parametrization of theneutrino mass matrix presented by the parti le data group [16℄.Let us (cid:28)rst dis uss the A and A textures. Noti e that among the textures listed inTable 2, only for the A and A textures we have F = E = 0 . In the following, we (cid:28)rst he k if, despite m ee , m µτ = 0 , we an have F = E = 0 and then he k for solutions with m ee = 0 , m µτ = 0 and m ee = 0 , m µτ = 0 . It is straightforward but rather umbersome toshow that, assuming m ee , m µτ = 0 , the only solution of F = E = 0 is m = 0 , s = 0 .(In fa t, there is another solution whi h requires m = − m ( c − c s s tan θ e iδ ) / ( s + s c s tan θ e iδ ) but this is not ompatible with neutrino data.) It is straightforward toshow that m = s = 0 implies E , E = 0 so not all of the onditions for the A and A textures an be ful(cid:28)lled. Thus, so far we have on luded that if the pattern asso iated tothe A or A textures holds (if E = E = F = F = 0 or E = E = F = F = 0 ) at leastone of the ee or µτ entries must be nonzero. On the other hand, m ee = 0 and E = F = 0 with m µτ = 0 implies m eµ m eτ = 0 whi h is the ondition for the A or A textures. Thesetwo textures an be distinguished by omputing E and E and he king whi h one vanishes.Finally, m ee = 0 , m µτ = 0 and E = F = 0 implies m µµ m ττ = 0 whi h is not ompatiblewith the data [13℄. In sum, we have proved that the equations E = E = F = F = 0 ( E = E = F = F = 0 ) are both ne essary and su(cid:30) ient onditions for the A ( A )texture provided that the mass matrix a ommodates the present neutrino data.Now let us dis uss the B i textures. As shown in Table 2, the onditions for B are14 = F = 0 , E = F and E = F . Noti e that E = F = 0 automati ally implies E = F and E = F . Thus to he k if the onditions shown in the third row of Table 2 guarantee theformat of texture B , it is su(cid:30) ient to solve E = F = 0 . It an be shown that E = F = 0 (the onditions for B ), in addition to m µµ = m eτ = 0 , have another solution whi h yieldsthe following relations ( | m | − | m | = | m | cos δ (4 s ) / ( s c ) | m | − | m | = | m | ( s − c ) /c . (25)As a result, s cos δ/ ( s − c ) ∼ (∆ m sol ) / (∆ m atm ) ≪ . Moreover, the se ond equationof Eq. (25) an be onsidered as a lower bound on | m | ; with the present data [17℄, thisequation gives the bound | m | > . eV at 3 σ . Noti e that with the present un ertaintieson the neutrino data this solution is still a eptable. Thus, the onditions listed in the thirdrow of Table 2, in addition to texture B have another solution whi h is ompatible withthe present neutrino data. Future measurements of the neutrino mass s ale [4℄, θ and sgn( | m | − | m | ) may enable us to test the se ond equation in Eq. (25). In parti ular,the NO ν A [18℄ and T2K [19℄ experiments an measure θ and the absolute value of ∆ m with very high a ura y. The a ura y in the measurement of sin θ an rea h 1%. Ifthese experiments establish that θ is lose to maximal, the lower bound on | m | will bewithin the rea h of the KATRIN experiment [20℄. For relatively large values of θ (i.e., s > . ), more futuristi experiments su h as the T2KK setup [21℄ an help us to solvethe o tant-degenera y and derive information on sgn( | m | − | m | ) and δ . Su h informationmakes the solution ompletely testable. In summary, the onditions listed in the third row ofTable 2, in addition to texture B have another solution ompatible with the present data.Forth oming data may ex lude this solution.Now let us study the onditions for texture B whi h are listed in the fourth row of Table2. The onditions E = F = 0 automati ally yield E = F and E = F so it will besu(cid:30) ient to study the onsequen es of E = F = 0 . Similarly to the ase of texture B , E = F = 0 for m eµ , m ττ = 0 implies ( | m | − | m | = −| m | cos δ (4 s ) / ( s c ) | m | − | m | = −| m | ( s − c ) /c . (26)A dis ussion similar to the one after Eqs. (25) holds here, too. That is, the onditions15 = F = 0 other than texture B has another solution whi h is ompatible with thepresent data but an be ex luded by the forth oming NO ν A [18℄ and T2K [19℄ experiments.Now let us dis uss the B and B textures whose ne essary onditions are E = E = F = F = 0 . From Eqs. (25,26), we readily see that if E = E = F = F = 0 , there is no solutionwith m eµ , m eτ , m µµ , m ττ = 0 . That is some of these entries should vanish. Considering thedi(cid:27)erent on(cid:28)gurations of vanishing entries, we (cid:28)nd that E = F = E = F = 0 implieseither B or B .Finally, let us dis uss the ondition for texture C whose onditions are listed in thelast row of Table 2. Noti e that for m µµ , m ττ = 0 , E = E = 0 automati ally yields F = F = F − E . In addition to m µµ = m ττ = 0 , the equations E = E = 0 have anothersolution whi h implies | m | −| m | | m | = cos 2 θ s , | m | −| m | | m | ≃ cos (2 θ ) c s c s s . (27)The above relations in turn implies ∆ m atm / ∆ m sol = ( s c / cos 2 θ )(1 / s c )[cos (2 θ ) /s ] ≫ . The above relation might be tested by forth oming measurements. If these experiments donot on(cid:28)rm Eq. (27), the aforementioned solution will be ruled out and the texture C willbe the only solution of E = E = 0 .In sum, we have listed the ne essary onditions for di(cid:27)erent texture zero s enarios in Table2. We have shown that in the ase of texture A and A the onditions listed respe tively inthe (cid:28)rst and se ond rows of this Table are su(cid:30) ient to establish these textures. Moreover, inthe ase of textures B and B , the onditions listed in the Table have no solution ompatiblewith the neutrino data other than these textures. However, the onditions for B , B and C an have another solution whi h might be ruled out by improving the neutrino data.4 SummaryIn this paper, we have studied the symmetries of the lepton se tor in a basis independentway by de(cid:28)ning a new lass of basis invariants onstru ted out of the lepton mass matri es.16e have fo used on the symmetries of the e(cid:27)e tive mass matrix of neutrinos at low energies(below the ele troweak s ale) under the assumption that neutrinos are Majorana parti les.As is well-known, even in the most optimisti ase through the present methods andproposals, the neutrino mass matrix annot be fully re onstru ted. Motivated by this ob-servation, various neutrino mass matrix ansatz have been developed in the literature. Mostof these onje tures are based on symmetries and onditions that are apparent only in aparti ular basis. We have shown that by using the invariants de(cid:28)ned in this paper su h sym-metries and onditions an be formulated in a basis independent way. We have in parti ularfo used on the µ − τ ex hange and re(cid:29)e tion symmetries, (cid:29)avor symmetry with onserved L µ − L ττ