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in the framework of discrete flavor symmetries one of the most used ansatz for the lepton mixing matrix @xmath0 is the tribimaximal mixing ( tbm ) matrix  @xcite , which despite the non - zero value for @xmath2 recently confirmed by the t2k collaboration  @xcite , double chooz  @xcite , daya bay  @xcite and reno  @xcite experiments , can still be used as a good approximation . contributions from the charged lepton sector and/or from renormalization effects , can generate a non - zero value for @xmath2 in agreement with the experimental data . for a classification of models predicting tbm mixing see  @xcite and references therein , and for an overview of flavor symmetry models for neutrino mixing , based on non - abelian discrete symmetries that give the tbm pattern , see ref . @xcite . other ansatzs for the neutrino mass matrix have been explored . one such case consists on the so - called @xmath3-zero textures where , for example , taking the case of majorana neutrinos , it has been found that only two independent zero entries are allowed by current data  @xcite . for dirac mass matrices the situation is different ( since the mass matrices are not symmetric ) as shown in  @xcite , where a study of the minimal mass matrices with as many zeros as possible determined that up to five zero entries are allowed . in both of these cases the neutrino mass matrices were expressed in the charged lepton diagonal basis . fritzsch - like  @xcite textures have also been considered for both the charged lepton and neutrino mass matrices in  @xcite , where dirac neutrinos are considered , and in  @xcite , where all possible hermitian six - zero fritzsch - like as well as non fritzsch - like textures for lepton mass matrices have been investigated , for both majorana and dirac matrices . for a recent overview of all possible cases of fritzsch - like as well as non fritzsch - like matrices as possible textures of the fermion mass matrices see  @xcite .    in this paper we take the following approach : assuming that the neutrino mass matrix is diagonalized by the tbm matrix , we determine the textures for the charged lepton mass matrix that leads to a lepton mixing matrix in agreement with the experimental data . we find both the number and location of zeros in the charged lepton mass matrix as well as the relative sizes of the non - zero entries in the corresponding yukawa matrix . the results obtained can be of use to model builders interested in obtaining flavor models with tbm mixing in the neutrino sector . in section  [ sec : setup ] we present the general idea and setup of our analysis . the @xmath3-zero textures for @xmath4 are obtained in section  [ sec : zeros ] . section  [ sec : analysis ] contains a description of the numerical analysis performed , as well as our results and discussion . finally , section  [ sec : conclusions ] contains our conclusions . [ sec : setup ]    the lepton mixing matrix @xmath0 is given by @xmath5 where @xmath6 relates left - handed leptons in the weak basis @xmath7 to the mass basis leptons @xmath8 : @xmath9 ; and @xmath10 corresponds to the same basis transformation for left - handed neutrinos @xmath11    the @xmath0 matrix can be parametrized as follows  @xcite : @xmath12 where @xmath13 , @xmath14 and @xmath15 is the dirac cp violating phase . since the @xmath16 matrices in are unitary , we can express @xmath6 as    @xmath17    in our approach , we demand that @xmath10 be the tbm matrix , namely @xmath18 and so , if @xmath19 is the matrix that diagonalizes the neutrino mass matrix @xmath20 and @xmath6 the one that diagonalizes the _ squared _ charged lepton mass matrix @xmath21 , we want to determine the form of @xmath22 such that @xmath23 has values in the allowed experimental range . as described above the matrix @xmath6 diagonalizes @xmath24 and thus satisfies @xmath25 where @xmath26 and @xmath27 .    at this point @xmath28 is an arbitrary mass matrix with unknown values and we parametrize it as @xmath29 in general the mass matrix @xmath28 has complex entries , however , in this first approach we consider the particular case of a real mass matrix and thus assume all parameters in @xmath22 to be real . in this particular case , eq .   becomes a set of equations for the entries of @xmath30 : @xmath31    there are nine variables and only six equations encoded in eq .  . thus we have an incomplete set of equations . to proceed we can start by making some assumptions on the _ form _ of the lepton mass matrix . just as an example take @xmath28 to be a lower triangular matrix ( by making @xmath32 ) . this gives @xmath33 the number of variables has now been reduced to six variables , and with the six independent equations ( as we show later in the paper ) it is easy to find solutions for the parameters in eq . ( [ muem ] ) in agreement with the experimental data . we are interested in determining the textures with the maximum number of zeros . the main motivation for this is that having such textures can be useful to model builders interested in obtaining them using an underlying flavor symmetry . with this in mind , we explore the textures with the maximum number of zeros such that eq . has real solutions in the allowed experimental range for the mixing angles . note that our assumption of real mass matrices implies that in our case the cp violating phase satisfies @xmath34 or @xmath35 . we also assume that the majorana cp violating phases are zero but note however , that since they can be factorized from @xmath0 as a diagonal matrix , their inclusion would not affect the magnitudes of the @xmath0 entries . we are concerned with the possible textures for the non - diagonal mass matrix for charged leptons ( as we said before , assuming that the neutrino mass matrix is diagonalized by the tbm matrix ) . we assume that the entries are real and so @xmath36 . zeros in the entries of our matrices may be due to some underlying symmetry(ies ) and because of that we will look for the textures with the highest possible number of zeros .    as discussed above , we have a system of six coupled , quadratic equations and ( in general ) nine variables . thus there is a need for some extra constraints on the variables . for matrices with no zeros at all we need three of this extra constraints , for matrices with one zero we need only two . we begin our analysis with three - zero mass matrices requiring no extra constraints . a combinatorial analysis shows that , for the three - zero matrices , we have @xmath37 ways to place the three zeros and thus @xmath38 different matrix forms . nevertheless , it is well known that in forming the real product @xmath39 , any permutation of the columns of the matrix involved does not change the resulting product matrix . thus , this reduces the number of matrices to @xmath40 . from these textures we discard those with a row or column made of zeros because they are singular . there are @xmath41 cases of this and we are left with only @xmath42 different textures . another condition that has to be satisfied , in view of the non - zero values for the charged lepton masses , is @xmath43 .    denoting the possible mass matrices by @xmath44 , where indexes stand for the @xmath45 texture of the non - diagonalized charged lepton matrix with @xmath3 zeros , for @xmath4 we obtain : @xmath46    as none of the matrices shown above is obtained from another by means of permuting columns , these are all the possible three - zero textures to be analyzed . we are interested in identifying the textures which provide solutions to eq .  . in order to find which textures provide solutions we performed ( for each texture ) a scan over the whole allowed experimental range for the mixing angles taking the values @xmath47 and @xmath48 for the cp - violating phase .    to carry out the numerical analysis we used the experimental data at 3@xmath49 from the global neutrino data analysis  @xcite @xmath50 with normal ( inverted ) hierarchy , and the recently daya bay results @xcite ( confirmed at 5@xmath49 ) @xmath51 which can be rewritten as @xmath52 .    for the charged leptons masses we use the values given in  @xcite @xmath53    for instance by taking the central values of the mixing angles and those of the charged lepton masses ( setting @xmath47 ) we obtain @xmath54 and from eq . we get for @xmath6 @xmath55    if we use these values for the example of the lower triangular matrix presented in section  [ sec : setup ] , equating eq .   to the explicit value of the right hand side of eq .  , and using eq . and @xmath56 evaluated at the central value for the lepton masses , we find : @xmath57 + where @xmath58 is in mev and the yukawa matrix @xmath59 shown above is obtained from : @xmath60 being @xmath61 gev the vacuum expectation value ( vev ) of the higgs field  @xcite ( here we are assuming only one higgs doublet contributes to the mass matrix . in general in this paper we assume either that , or if there are more su(2 ) doublets involved , that they all acquire vev s of @xmath62  gev ) .    to perform the complete scan we used a grid of dimensions @xmath63 in the experimental range of the three mixing angles . for each combination of the mixing angles that provides solutions to eq . , we select only the real solutions . once those are determined , we then also determine the order of magnitude of the yukawa mass entries by finding their minimum and maximum values . from all the possible three - zero textures ( shown in eq .  ) only the following provide ( real ) solutions : @xmath64 , @xmath65 , @xmath66 , @xmath67 , @xmath68 , @xmath69 , @xmath70 , @xmath71 , @xmath72 , and @xmath73 .    for the textures @xmath74 , @xmath75 , @xmath76 , @xmath77 , @xmath78 , and @xmath79 we did not find any solution . the reason is that all these textures lead to an extra constriction because the matrix product @xmath80 has a zero entry . those entries are @xmath81 , @xmath82 , @xmath83 , @xmath84 , @xmath85 , and @xmath86 . as mentioned above , once we have performed all calculations ( including the normalization of the mass matrix by the vev of the higgs field ) we find the maximum and minimum orders of magnitude of the entries for each texture of the yukawa matrix . they are shown here ( up to @xmath87 coefficients ) :     @xmath88      as part of the analysis we performed , we searched for the solution volume of each three - zero texture , that is , the set of points given by the three mixing angles that make possible to find real solutions to the entries of the @xmath28 matrix . in the case of textures @xmath65  @xmath73 the solution volume fills the complete space given by the experimental intervals of the three mixing angles ( the experimentally allowed parameter space ) . this happens for both @xmath47 and @xmath48 . the most interesting case is the texture @xmath64 , which has the zeros on the major diagonal ( and its permutations ) . the relevance of this case is that the angle @xmath89 is now very restricted . thus , we took a closer look within a smaller interval for @xmath89 using a finer grid of @xmath90 , and analyzing a @xmath89 interval of @xmath91 $ ] , we find that the solution volume of @xmath64 , for both @xmath47 and @xmath48 , is a thin curved surface ( presented in figure  [ fig1 ] for @xmath48 ) ; the only difference between these two cases is a displacement of the interval allowed for @xmath89 , which is @xmath92 $ ] for @xmath47 , and @xmath93 $ ] for @xmath48 . these intervals are near but exclude the central value of @xmath89 .     for @xmath48 , found with a grid of @xmath90 . similar results are obtained for @xmath47.,width=340 ]      the yukawa matrices in   can explicitly be written as @xmath94 then , the diagonal entries @xmath95 are just the sum of the square of the elements of the i - th row of @xmath96 :    @xmath97    the textures obtained in subsection  [ subsec : results ] present some interesting features : first note that in all textures the largest upper bound ( @xmath98 ) always appears in the second row . thus , from eq .  , @xmath99 . this is a little higher than @xmath100 ) . note also that ( except for the @xmath101 ) the third row has always an upper bound of @xmath102 and then , from eq . , we can deduce that @xmath103 . this is quite close to the value of the ratio @xmath104 . more precisely and using the explicit values of the yukawa matrix of eq . we find    @xmath105    we thus find that there are extra conditions on the entries of @xmath96 in terms of the charged leptons masses . because of equation ( [ equ ] ) , we can expect that these conditions may be due to an internal structure of the mixing matrix @xmath6 for the charged leptons . in what follows we show that there is indeed a parametrization for this matrix that leads to specific relations between the entries in the yukawa matrices and the charged lepton masses that reproduce the above results , including the small deviations in the orders of magnitude .    motivated by the fact that the charged leptons are analogous to the down type quarks in terms of representations of the standard model gauge group , we consider a ckm - like parametrization for the @xmath6 mixing matrix in terms of the three angles @xmath106 , @xmath107 and @xmath108 , taking values between @xmath109 and @xmath110 , and all phases set to zero in order to use real values for the @xmath6 matrix , as we have done in the whole analysis . then , we can write : @xmath111    using @xmath112 instead of @xmath0 ( to avoid agglomeration of indexes ) one can easily find that :    @xmath113    @xmath114    @xmath115     + here the @xmath116 are , as before , the parameters of @xmath0 ; different from @xmath117 , which parametrize @xmath1 . we evaluated equations ( [ e1])-([e3 ] ) with @xmath118 $ ] , and took all the points that matched the experimental intervals given by ( [ experim1 ] ) and ( [ experim2 ] ) . the points ( @xmath119 ) are presented in figure  [ fig2 ] .        in figure  [ fig2 ] the blue region is made of those values of the angles that solve equation ( [ e1 ] ) within the experimental values , the purple points are those that solve equation ( [ e1 ] ) and ( [ e2 ] ) within the experimental ranges , and the dark yellow region is the set of points that satisfies the three equations ( [ e1 ] ) , ( [ e2 ] ) and ( [ e3 ] ) within all the experimental values . we then obtain a small region given by two small angles and a large one :    @xmath120\\ \theta^{l}_{13}&=&[0.07 - 0.16]\nonumber\\ \theta^{l}_{23}&=&[1.43 - 1.57\approx(\pi/2)]\nonumber\end{aligned}\ ] ]    now we use the parametrization of the @xmath1 matrix given by equation ( [ para ] ) and the values of its angles to make an analysis of the entries in a general charged lepton mass matrix , of the form given by ( [ gen ] ) . let us remember that @xmath1 appears in ( [ equ ] ) . using the values given in ( [ values ] ) , one can built a simpler form for @xmath1 . indeed , according to equation ( [ para ] ) and using the fact that @xmath121 and @xmath122 are small , we can use a first - order approximation for their respective rotation matrices . the size of the angle @xmath123 allows us to use a first order approximation too , but around @xmath124 . then , @xmath125 and @xmath126 . thus , we end up with :    @xmath127     + using ( [ apro0 ] ) in ( [ equ ] ) , we obtain the matrix @xmath128 in terms of @xmath106 , @xmath107 and @xmath129 . comparing the result with ( [ mue ] ) , we can see that the variables from the first row of the mass matrix must satisfy :    @xmath130    the variables of the second row of ( [ gen ] ) satisfy :    @xmath131    and the third row of ( [ gen ] ) must have a contribution of the form    @xmath132    these expressions explain the observations considered at the beginning of the present subsection , including the corrections to the order of magnitude estimates , and represent the main result of this analysis .    taking our specific example above , dividing equations ( [ row1])-([row3 ] ) by @xmath133 as in ( [ overhiggs ] ) , and using the values @xmath134 , @xmath135 , @xmath136 as well as the central values for the charged lepton masses we have : @xmath137 these are the values from ( [ yuk ] ) , which are relations that exhibit the dependence of the entries of @xmath96 on the masses and the mixing angles of @xmath1 . it is easily observed that for the first row , although the leading term ( zeroth order on the angles ) is @xmath138 , the other contributions play a more important role than this one , because the masses of the @xmath139 and @xmath140 leptons are much greater than that of the electron and even overcome the small size of their coefficients ( which are powers of the angles ) . finally , we note that these expressions are independent of the texture chosen for the yukawa matrix and also independent of the two values of the dirac phase used in the pmns matrix . all the yukawa textures found exhibit the relations given by equations ( [ row1])-([row3 ] ) , so an important feature that we want to point out is that the matrices we have found show a very large mixing ( nearly @xmath110 ) between the second and third generations of the charged leptons . we have already explored the textures with three zeros and found that it is possible to have solutions in the allowed ranges for many of them . now we are interested in checking if this is also possible for textures with four zeros . to start , we list the possible textures with four zeros and their equivalent ( column ) permutations , discarding those with a row or column completely filled with zeros ( thus avoiding singular matrices ) : @xmath141    we have also calculated the product @xmath80 for each texture and we can observe that most of them render one or two zero entries . as in the previous case , this increases the difficulty in finding solutions to the system of equations in eq .  . moreover , since we are assuming only real values in the mass matrix entries , for the four - zero textures we are left with @xmath142 equations and only @xmath143 variables to satisfy them and the system is over - constrained . the zero entries in the product @xmath80 for each texture are the following : @xmath144    the only textures which give non - zero entries in @xmath80 are @xmath145 , and @xmath146 . those would be the more interesting cases to study if we allow complex parameters ( this would imply also to allow the @xmath15 cp violating phase to take all its possible values ) . @xmath147 , @xmath148 and @xmath149 also render non - zero entries in @xmath150 but they are singular . we performed a scan over all the allowed experimental range for the mixing angles trying to solve the system of equations for each particular texture , in the case of real parameters , without success . the only possibility is to consider cases where one of the zero entries in the four - zero texture gets slightly turned on . this case could correspond then to one of the three - zero texture above where one of the entries is much smaller than all others . analyzing the three - zero textures and the relative size of their entries ( presented above ) , we observe that except for @xmath64 , all of them contain entries in the range @xmath151 @xmath152  mev , and that the smallest one is at least two orders of magnitude smaller than the next one . it is then possible , from a model building perspective , to consider a four - zero texture where one of the zeros is _ lifted _ ( perhaps due to radiative corrections , or higher order operator mixings ) such that at the end one ends up with one of the successful three - zero textures above . here we show which four - zero texture could be used in such a scenario @xmath153    as a final remark we comment that even though at this point we considered only real matrices for the charged lepton mass matrices , it is possible to extend some of our results to the case of complex matrices with factorizable phases , i.e. mass matrices that can be written as @xmath154 , where @xmath155 are matrices with real entries and @xmath156 a diagonal phase matrix , @xmath157 . then , if @xmath158 is the orthogonal matrix which diagonalizes @xmath159 we can write , under our assumption @xmath160 , @xmath161 . nevertheless , from the property of orthogonality and @xmath162 we can infer that the only possible values for the factorizable phases are integer multiples of @xmath35 . we checked exactly the same three - zero textures as before , but allowing complex entries and a cp - violating phase @xmath163 and found solutions for the same textures as in the real case . the difference with our previous results is that in these cases solutions are found in all the allowed volume of angles , even for @xmath101 , and the order of magnitude of the entries are generally larger . in the four - zero texture we again find no solutions and due to the fact that the entries in the three - zero textures are larger in this case , it is not possible to easily start from a four - zero texture and lift one zero ( here we are thinking of a naturally small correction to the zero entry in question ) . we analyze several textures for the charged lepton mass matrix under the assumption that the neutrino mass matrix is diagonalized by the tbm matrix , and with the intention of maximizing the number of zeros in them . requiring the @xmath0 mixing matrix to be consistent with the allowed experimental range for mixing angles ( fixing the cp violating phase ) , and with ( the central values of ) the charged lepton masses , we explore the maximum number of zeros possible in a texture under the assumption of real charged lepton mass matrix entries . we find that there are ten three - zero textures which provide @xmath0 values in agreement with and also determine the size range for their entries . among the successful textures , the one with zeros in the diagonal shows an interesting behaviour in the sense that in order to work , it requires the mixing angle @xmath89 to lie in a very restricted range which , albeit consistent with the experimental range , it excludes the central experimental values . a general analysis of the successful textures showed that there are relations between their entries and the charged lepton masses . through a ckm - like parametrization of the @xmath1 mixing matrix we are able to obtain the texture - independent specific relations in terms of the three rotation angles in @xmath1 . we find no solutions for four - zero textures but observe that it is possible to consider cases where one the zeros is lifted in such a way that the remaining mass matrix corresponds to some of the successful three - zero textures . finally , we explore a possible extension to consider the cases with complex factorizable textures . we find similar results , i.e. solutions available for the three - zero textures and no solutions for the four - zero - texture , with the difference that the typical entry size of the charged lepton mass matrix is generally larger in this case .    while preparing this paper , a new preprint appeared in the arxiv  @xcite , where an analysis of the charged lepton mixing matrix was performed using the tbm matrix form as well as the bimaximal mixing matrix ( bm ) for the neutrino mixing matrix . they consider the charged lepton mixing matrix to be nearly the identity but deviated from it by one and two parameters for the bm and the tbm case respectively . the parameters are the cabbibo angle @xmath164 and @xmath165 for the tbm case , assuming that the @xmath6 has its own ckm - like parametrization .    in our case the assumption of @xmath6 as a matrix near the identity is not used . instead , we determine this matrix by allowing the three rotation angles to vary over the whole experimental range and find that in fact , since @xmath166 , the matrix is not close to the identity    also , in our context , the goal included to find the textures that are diagonalized with eq .  . this is not explored in  @xcite and we feel it is a useful result that can be used by model builders interested in flavor symmetries . k.  abe _ et al . _ [ t2k collaboration ] , `` indication of electron neutrino appearance from an accelerator - 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[ arxiv:1108.1376 [ hep - ph ] ] .
assuming that the neutrino mass matrix is diagonalized by the tribimaximal mixing matrix , we explore the textures for the charged lepton mass matrix that render an @xmath0 lepton mixing matrix consistent with data . 
 in particular we are interested in finding the textures with the maximum number of zeros . 
 we explore the cases of real matrices with three and four zeros and find that only ten matrices with three zeros provide solutions in agreement with data . 
 we present the successful yukawa textures including the relative sizes of their non - zero entries as well as some new and interesting relations among the entries of these textures in terms of the charged lepton masses . 
 we also show that these relations can be obtained directly from a parametrization of the charged lepton mixing matrix @xmath1 .