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in the last few years there has been another wave of excitement regarding the question of neutrino masses . this is largely due to the many new experiments testing neutrino oscillations , most notably the positive indications obtained by super kamiokande on atmospheric neutrino oscillations @xcite . similar indications come from other experiments @xcite . the solar neutrino experiments have for many years provided independent evidence for neutrino oscillations @xcite . accelerator and reactor experiments have also played an important role . they have furnished strict bounds on neutrino oscillation parameters @xcite . in the case of the lsnd experiment @xcite at los alamos evidence for @xmath2 oscillation has been reported . see refs @xcite for recent reviews . it is hoped that new experimental results can be used to determine the neutrino squared mass differences and mixing angles . in turn , these may help to infer the neutrino mass matrix . this is presumably a possible gateway to a more fundamental theory beyond the standard model . of course this is a highly speculative area , and even though there are many imaginative proposals @xcite , it seems fair to say that the the true answer is essentially unknown . in order to make progress in this direction , it seems useful to investigate various plausible ansatze for the neutrino mass matrix . from this point of view we propose the ansatz for the 3 generation neutrino mass matrix , @xmath3 : @xmath4 and investigate its consequences . we are considering the neutrinos to be represented by 2-component spinors so that , in the most general situation , @xmath3 is an arbitrary symmetric complex matrix .    as we will see in section ii , eq . ( 1.1 ) can be motivated from an so(10 ) grand unified model @xcite , in which it may be derived with some assumptions . physically , eq . ( 1.1 ) corresponds to the well known approximate signature of grand unification that @xmath5 . furthermore we will see in sections iv and v that eq . ( 1.1 ) can be straightforwardly combined with experimental information to get an idea of the neutrino masses themselves as well as the `` texture '' of @xmath6 . relevant matters of notation are discussed in section iii while a summary is presented in section vi . in the so(10 ) grand unification model each generation contains one light massive two component neutrino and also a very heavy one which is `` integrated out '' according to the `` seesaw mechanism '' @xcite . the effective @xmath7 neutrino mass matrix takes the form : @xmath8 where @xmath9 , @xmath10 and @xmath11 are respectively the mass matrices of the light neutrinos , heavy neutrinos and heavy - light mixing ( or `` dirac matrix '' ) . generally the second , seesaw , term is considered to dominate . here however we shall assume the first term to be the dominant one . this is necessary for the present derivation of eq . ( 1.1 ) to hold . also , a rough order of magnitude estimate for the second term would be @xmath12 or about @xmath13 ev . thus , the seesaw term could be negligible if neutrino masses turn out to be appreciably larger than this value . now in so(10 ) , higgs mesons belonging to the 10 , 120 and 126 representations can contribute to the fermion masses at tree level . one has @xcite for the down quark , charged lepton and light neutrino mass matrices , @xmath14 where @xmath15 , @xmath16 , @xmath17 , @xmath18 , @xmath19 are numbers representing higgs meson vacuum values . s(10 ) , a(120 ) and s(126 ) are the matrices of the yukawa type constants which couple the fermions to the 10 , 120 and 126 higgs mesons respectively ; the matrices s(10 ) and s(126 ) must be symmetric while a(120 ) is antisymmetric . finally , @xmath20 is a renormalization factor for comparing the quark masses with the charged lepton masses at a low energy scale rather than at the grand unified scale ; @xmath21 is a similar factor for the neutrino masses . with the stated assumption that the @xmath22 term dominates in eq . ( 2.1 ) we get @xmath23 which clearly also holds when any number of 10 s or 120 s are present but only a single 126 . the matrices appearing in eq . ( 2.3 ) are so far essentially unrestricted complex ones . to proceed , we make the further assumption that the matrices are hermitian . then @xmath24 and @xmath25 may each be brought to diagonal form by unitary transformations . thus the right hand side of eq . ( 2.3 ) may be evaluated to yield approximately , @xmath26 according to a well known numerical success , based on the observation that @xmath27 , of grand unification @xcite . note that we have not needed to assume that the mass matrix has any zero elements . where , in addition , a special combined fritzsch - stech ansatz was used . here we are not making any special ansatz of this type for the mass matrices . ] even if the cancellation on the right hand side of eq . ( 2.4 ) is not perfect , it should still be a good approximation . in an so(10 ) model where the mass matrices are hermitian , @xmath28 will be real symmetric . we will investigate this case and also the possibility that the more general case holds . our plan is to combine the ansatz eq . ( 1.1 ) with experimentally obtained results on neutrino oscillations in order to learn more about @xmath29 itself . for this purpose it may be helpful to set down our notation @xcite for the pieces of the effective @xmath30 theory involving neutrinos and to make some related remarks . the free lagrangian containing three two component massive fields is : @xmath31 where @xmath32 is the ( not yet diagonalized ) neutrino mass matrix of the underlying theory to be identified with the matrix in eq . note that we are free to multiply the first mass term in eq . ( [ lfree ] ) by an overall arbitrary phase which is a matter of convention . it is possible@xcite to find a unitary matrix @xmath33 which brings @xmath0 to real , positive , diagonal form in the following way : @xmath34 the mass diagonal fields @xmath35 are then @xmath36 similarly , the column vector of left handed negatively charged leptons in the underlying theory , @xmath37 is related to the mass diagonal fields @xmath38 by @xmath39 where @xmath40 .    combining factors from eq . ( 3.3 ) and eq . ( 3.4 ) we obtain the unitary mixing matrix , @xmath41 for the charged current weak interaction , @xmath42 this appears in the lagrangian term , @xmath43 where a conventional four component dirac notation with @xmath44 diagonal is being employed and @xmath35 has only the first two components non zero . next we parameterize @xmath41@xcite . it is possible to restrict @xmath45 by adjusting an overall phase which can be absorbed in @xmath46 . then we write @xmath47 where @xmath48 with @xmath49 and , for example @xmath50 . \label{3.9}\ ] ] eq . ( 3.7 ) contains the eight parameters needed to characterize an arbitrary unitary unimodular matrix . from the standpoint of eq . ( 3.6 ) it can be further simplified by using the freedom to rephase @xmath51 without changing the free part of the charged lepton lagrangian . on the other hand , the form of the mass terms in eq . ( [ lfree ] ) shows that the neutrino fields can not be rephased . thus a suitable minimal parameterization for @xmath41 in ( 3.6 ) is @xmath52 involving three  angles " , @xmath53 and three  phases " , @xmath54 . note the identity @xmath55 this identity may be used to transfer two of the phases @xmath54 in eq . ( 3.7 ) to a diagonal matrix on the right of @xmath41 as , for example , @xmath56 where @xmath57 , which may be used instead of eq . ( 3.10 ) . we also need the formula for the amplitude of neutrino oscillation . for the case when a neutrino , produced by a charged lepton of type @xmath15 ,  oscillates " to make at time @xmath58 , a charged lepton of type @xmath16 , we have @xmath59 where the sum goes over the neutrinos of definite mass , @xmath60 . inserting the parameterization eq . ( 3.12 ) into eq . ( 3.13 ) shows that the effect of the factor @xmath61 cancels out . thus for ordinary oscillations , @xmath41 is parameterized by three angles and one cp violating phase as for the ckm quark mixing matrix . on the other hand , the two additional cp violating phases @xmath62 and @xmath63 show up if one considers neutrino - antineutrino oscillations @xcite or neutrinoless double beta decay @xcite . the formula for the probability , @xmath64 is gotten by taking the squared magnitude of eq . ( 3.13 ) and replacing the exponential factor @xmath65 by @xmath66 , where @xmath67 is the neutrino energy and @xmath68 is the oscillation distance . for practical reasons it is very important to take account of the experimental uncertainties in @xmath67 and @xmath68 . the simplest approximation@xcite is to define @xmath69 and assume that one can smear @xmath64 with a gaussian distribution in @xmath16 . @xmath70 is defined as the mean value and @xmath71 as the standard deviation appropriate to the particular physical setup . then we find for the smeared probability @xmath72 , \label{3.14}\end{aligned}\ ] ] where @xmath73 and @xmath74 . notice that when @xmath75 , @xmath76 is independent of the sign of @xmath77 . since @xmath78 provides only two real equations for 12 real parameters , it is clear that it has a relatively small amount of predictivity . in particular it can not say much about the texture ( e.g. possible zeroes ) of @xmath0 which is suppposed to derive from a deeper theory than the standard model . on the other hand , we shall see that our ansatz is complementary to the results which should emerge from analysis of neutrino oscillation experiments . together , they should enable us to actually ( with some conditions ) reconstruct @xmath0 . first , in this section , we shall consider @xmath0 to be hermitian so that the argument in favor of @xmath78 presented in section ii holds without any further assumptions . since @xmath0 is symmetric it must be real . it can be brought to real diagonal form via a real rotation @xmath79 as @xmath80 . however there is no guarantee that all eigenvalues of @xmath0 will be positive . we can make them all positive by rephasing the diagonal fields with negative eigenvalues by a factor @xmath81 ( see eqs . ( 3.1 ) and ( 3.2 ) ) . this means that the general diagonalizing matrix @xmath33 in eq . ( 3.2 ) now takes the form @xmath82 where @xmath83 with @xmath84 for a positive eigenvalue and @xmath85 for a negative eigenvalue . we notice that eq . ( 4.1 ) is of the form eq . ( 3.12 ) for which we already noticed that the factor @xmath86 cancels out in the neutrino oscillation formula eq . ( 3.14 ) . furthermore only the square of the mass is relevant in eq . ( 3.14 ) . thus we choose to work in this section with some negative masses and no factor @xmath86 in eq . ( 4.1 ) .    to avoid confusion , we remark that the factor @xmath86 in eq . ( 4.1 ) does _ not _ introduce any cp violation in the theory @xcite since @xmath0 is real in any event . now let us suppose that an experimental analysis of all neutrino oscillation experiments is made based on a formula like eq . ( 3.14 ) ( or one which treats the experimental uncertainties in a more sophisticated way ) . furthermore assume that the cp violating phase @xmath87 in eq . ( 3.12 ) is negligible . then we should know the _ magnitudes _ of the squared neutrino mass differences @xmath88 where @xmath89 and @xmath90 can be either positive or negative . then , assuming the leptonic theory to be cp conserving , our ansatz would imply @xmath91 where eq . ( 3.2 ) was used . ( 4.2 ) and ( 4.3 ) comprise three equations for the three neutrino masses @xmath92 , @xmath93 and @xmath94 . we can solve to get : @xmath95 , \\ \nonumber m_2 & = & \frac { { b } - { \left ( m_1 \right)}^2}{2m_1 } , \\ \nonumber m_3 & = & - m_1 - m_2 . \label{4.4}\end{aligned}\ ] ] this leads to a limited number of solutions , depending on sign choices . if we make the further assumption that the charged lepton mixing matrix @xmath96 in eq . ( 3.4 ) is approximately the unit matrix ( this is expected to be a reasonable but not perfect approximation ) we can identify @xmath97 which would be obtained from experiment . then , using the masses found in eq . ( 4.4 ) , we could reconstruct @xmath0 as @xmath98    to proceed , we need only insert the experimental results for @xmath89 , @xmath90 and @xmath41 in ( 4.4 ) and ( 4.5 ) . of course , it is presumably the task of the next decade to solidify the experimental determination of these quantities . we can , at the moment , only give a preliminary discussion . for this purpose we will use the results of a recent preliminary analysis of all neutrino experiments by ohlsson and snellman @xcite . these authors found , by a least square analysis , a best fit for ( our notation ) @xmath99 , @xmath100 and the leptonic mixing matrix @xmath41 . they used the formula eq . ( 3.14 ) with a suitable choice of @xmath70 and @xmath101 for each experiment . furthermore , they made the simplifying assumption that @xmath41 is real . finally they only searched for a fit in the range @xmath102 , @xmath103 . this range corresponds to mass difference choices for which the msw effect @xcite for solar and atmospheric neutrinos is not expected to be important and so greatly simplifies the analysis . thus there is no guarantee that the solution of @xcite is unique . altogether they fit sixteen different solar neutrino , atmospheric neutrino , accelerator and reactor experiments , including lsnd . the best fit is : @xmath104 for the squared mass differences and @xmath105 \label{4.7}\ ] ] for the lepton mixing matrix @xmath41 . as discussed above we will identify @xmath106 here , keeping @xmath41 real but allowing for negative masses . the best fit matrix k was obtained to be similar but not identical to the `` bimaximal mixing '' matrix @xcite .    with the best fit squared mass differences in eq . ( 4.6 ) , our model predicts , from the first of eq . ( 4.4 ) , eight different possibilities . these correspond to four different sign configurations for @xmath89 and @xmath90 times the two possible signs for @xmath92 . however , only two of these eight are essentially different ; these are @xmath107 the other solutions correspond to interchanging whichever of @xmath108 and @xmath109 is greater ( which has only a negligible effect since they are almost degenerate ) or reversing the signs of all masses . physically it is clear what is happening : the smallness of @xmath110 compared to @xmath111 in eq . ( 4.2 ) forces @xmath112 . then we have either @xmath113 with , using the constraint eq . ( 4.3 ) , @xmath114 or @xmath115 with @xmath94 very small . since we have assumed the neutrinos to be of majorana type for our plausibility argument in section ii , their interactions will violate lepton number . then they should mediate neutrinoless double beta decay @xmath116 @xcite . such a process has not yet been observed and an upper bound has been set for the relevant quantity @xmath117 the best upper bound at present is @xcite @xmath118 ev , reflecting some uncertainty in the estimation of the needed nuclear matrix elements .    substituting the best fit for the matrix k from eq . ( 4.7 ) together with our results in eqs . ( 4.8 ) and ( 4.9 ) into eq . ( 4.10 ) yields predictions for the two cases : @xmath119 both solutions seem to be acceptable , the type i case marginally but the type ii case definitely . note that the small value for @xmath120 in the type ii case is due to the best fit prediction @xcite @xmath121 and also to the fact that @xmath93 is negative . the same value would clearly result if we made @xmath93 positive and set @xmath122 as discussed around eq . ( 4.1 ) above . finally , let us reconstruct the underlying neutrino mass matrices for each of the two cases . we use eq . ( 4.5 ) based on the assumption that @xmath0 is real and also our ansatz to find ( in units of ev ) : @xmath123 , \\ { \rm type } \hskip 0.2 cm { \rm ii } : \quad m_\nu & = & m_\nu^t \approx \left [ \begin{array } { c c c } 6.7 \times 10^{-5 } & -0.9199&0.5078\\ -0.9199&0.0654&0.0410\\ 0.5078&0.0410&-0.0654\\ \end{array } \right ] . \label{4.13}\end{aligned}\ ] ] the type i matrix does not have an excellent candidate for a `` texture '' zero . however the small value of @xmath124 in the type ii case is certainly suggestive . these matrices lead to neutrino masses and a mixing matrix which give a best fit to all present data . it will be interesting to see if either of them hold up in the future .    incidentally , on comparing eqs . ( 4.13 ) and ( 4.14 ) it is amusing to observe the large difference in two mass matrices `` generated '' in the same way except with respect to how @xmath125 is satisfied . it seems interesting to also investigate the ansatz @xmath126 when @xmath0 is no longer restricted to be real . this also raises the problem of constructing the unitary diagonalizing matrix @xmath33 in eq.(3.2 ) , in terms of the experimentally measured lepton mixing matrix @xmath127 . for simplicity , as before , we will make the approximation that the charged lepton diagonalizing matrix @xmath96 is the unit matrix .    apart from an overall ( conventional ) phase we may write @xmath128 where the 2-parameter quantity @xmath129 was defined in eq . ( 3.8 ) . since @xmath127 has four parameters @xmath33 in eq . ( 5.1 ) is described by eight parameters . as mentioned before , the two parameters in @xmath61 are not measurable in neutrino oscillation experiments but show up when one considers @xmath130 . the two parameters in @xmath131 may be eliminated , for experimental purposes , by rephasing the charged leptons . however , for the theoretical purpose of reconstructing the underlying neutrino mass matrix @xmath0 , their existence can not be ruled out . ( they also do not contribute to @xmath132 . )    for the purpose of relating the ansatz on @xmath0 to the physical neutrino masses in @xmath133 , we note    @xmath134    for further simplicity of the analysis we adopt the special case @xmath135 and also identify @xmath127 with the real best fit in eq . ( 4.7 ) ; our ansatz now reads    @xmath136    with the redefinitions @xmath137 and @xmath138 , eq . ( 5.3 ) becomes @xmath139 this may be conveniently visualized as the vector triangle shown in fig . [ trianglefig ] .    combining eqs . ( 5.4 ) and ( 4.2 ) gives four real equations for the five unknown quantities ( @xmath140 ) . thus we have ( for each set of ( @xmath141 ) sign choices ) a one parameter family of solutions . it is convenient to choose this parameter to be @xmath94 . then @xmath92 and @xmath93 may be found from the equations ( 4.2 ) , provided that solutions exist . in this way all three sides of the triangle in fig.[trianglefig ] are determined . the angles may finally be found as @xmath142    we also need to investigate the constraint arising from the non - observation of @xmath132 . ( 4.10 ) now becomes , with eq . ( 5.1 ) as the mixing matrix @xmath143    using the ansatz constraint eq . ( 5.4 ) , eq . ( 5.6 ) may be rewritten as @xmath144 m_2 e^{-i\beta_2 } + \left [ ( k_{exp\hskip .2 cm 13})^2 - ( k_{exp\hskip .2 cm 11})^2   \right ] m_3 \right|.\ ] ] this form is very convenient when identifying @xmath127 with the best fit solution in eq . ( 4.7 ) . in the present context such an identification corresponds to cp violation for the @xmath132 process but not for usual neutrino oscillations . since the ( 11 ) and ( 12 ) matrix elements are equal in eq . ( 4.7 ) we find the simple result @xmath145 thus if the upper bound on @xmath146 is conservatively identified as in the @xmath147 ev range , we should have in this case @xmath148 in the present complex case there is a continuum of possible solutions labelled by those values of @xmath94 satisfying eq . ( 5.9 ) , rather than just the two possibilities found in eq . ( 4.8 ) and eq . ( 4.9 ) . actually , the continuum separates roughly into two classes similar to either eq . ( 4.8 ) or eq . ( 4.9 ) . in the generalized type i class , @xmath94 is of the order @xmath149 ev while @xmath92 and @xmath93 are related to nearly oppositely directed vectors in fig . [ trianglefig ] and are also of the order @xmath90 . in the generalized type ii class , @xmath90 is negative ; @xmath92 and @xmath93 correspond to vectors of order @xmath150 which are oppositely directed to each other , while @xmath94 ranges from very small to order @xmath100 . given the bound eq . ( 5.9 ) from the non - observation of @xmath130 , there are important limitations on the allowed @xmath94 values for type i solutions . in this case @xmath90 is positive so the equation @xmath151 will only allow solutions for @xmath152 ev . this range is barely compatible with eq . thus the type ii case where @xmath153 and @xmath154 seems most probable .    as an example of a solution for complex @xmath0 , consider choosing @xmath155 ev . then a solution is obtained with ( compare with fig . [ trianglefig ] )    @xmath156    the matrix @xmath33 , which diagonalizes @xmath0 is obtained from eq . ( 5.1 ) , with the approximation @xmath157 , and with now : @xmath158 this factor introduces cp violation in the @xmath132 process but not in ordinary neutrino oscillations .    finally the underlying neutrino mass matrix , @xmath159 is `` reconstructed '' as ( in units of ev ) : @xmath160.\ ] ] this is structurally similar to the real type ii solution displayed in eq . ( 4.14 ) , although the suppression of the ( 11 ) element is not so pronounced . notice that @xmath94 is considerably smaller than the almost degenerate pair @xmath92 and @xmath93 . furthermore @xmath92 and @xmath93 are large enough to possibly play some role in astrophysics . we investigated the ansatz @xmath78 for the underlying ( pre - diagonal ) three generation neutrino mass matrix . it was motivated by noting that in an so(10 ) grand unified model where @xmath0 was taken to be real ( cp conserving ) , it corresponds to the well known unification of b quark and @xmath161 lepton masses . while not very predictive by itself it yields information complementary to what would be gotten from a complete three flavor analysis of all lepton number conserving neutrino oscillation experiments . specifically from the specification of the magnitudes of two neutrino squared mass differences and also of the leptonic mixing matrix we can , with some assumptions , find the neutrino masses themselves and `` reconstruct '' @xmath0 . this determination can be sharpened by consideration of the constraints imposed by non - observation of neutrinoless double beta decay . for the purpose of testing our ansatz we employed the results of a reasonable best fit to all present neutrino experiments ( including lsnd ) by ohlsson and snellmann @xcite . this fit will inevitably be improved in the next few years as new experiments are completed . they were able to fit the data without assuming any cp violation . this agrees with assuming @xmath0 to be real . we found two essentially different solutions in that case . the first features two neutrinos having approximately equal mass 0.608 ev and a third neutrino of mass 1.217 ev . this solution is on the borderline of being ruled out by non - observation of @xmath162 . the second solution has two neutrinos with approximately degenerate mass 1.054 ev and a third neutrino with a mass @xmath163 ev . this solution is very safe from being ruled out by @xmath164 experiments . it also features a reconstructed @xmath0 which has an extremely small ( 11 ) element . note that , for both solutions , even though @xmath0 is real there are some ( cp conserving ) factors of @xmath81 in the mixing matrix when all masses are taken to be positive . alternatively one may have no @xmath81 s in the mixing matrix while allowing some masses to be negative . the latter form is useful for seeing intuitively how @xmath165=0 is possible . the case of matching the above best fit data to a complex @xmath0 was also considered . in this situation there are cp violating phases in the lepton mixing matrix which affect the @xmath132 process but do not affect ordinary total lepton number conserving neutrino oscillations . such phases could also be measurable in principle with the observation of a decay like @xmath166 . the case of complex @xmath0 allows a larger number of solutions . with a simplifying assumption there is a one parameter family of allowed neutrino mass sets . roughly , these fall into one of the two types already encountered for real @xmath0 .    a question of some interest is whether the neutrinos are massive enough to play a role in cosmology . the relevant criterion @xcite for this to occur is usually stated as @xmath167 ev . for the type ii solutions with complex @xmath0 we have found the largest mass sum to be about 4.5 ev corresponding to @xmath168 ev and @xmath169 ev . however this is on the very border of acceptability for non observation of @xmath132 . super - kamiokande collaboration , y. fukuda _ et al . * 81 * , 1562 ( 1998 ) . super - kamiokande and kamiokande collaborations , t. kajita , in _ proceedings of the xviiith international conference on neutrino physics and astrophysics ( neutrino 98 ) _ , takayama , japan , 1998 [ nucl . phys . b ( proc . suppl . ) * 77 * , 123 ( 1999 ) ] . soudan collaboration , w.w.m . et al . _ , b * 391 * , 491 ( 1997 ) ; 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we propose a simple ansatz for the three generation neutrino mass matrix @xmath0 which is motivated from an so(10 ) grand unified theory . 
 the ansatz can be combined with information from neutrino oscillation experiments and bounds on neutrinoless double beta decay to determine the neutrino masses themselves and to reconstruct , with some assumptions , the matrix @xmath1 . 
 = 17pt    # 1=#1pt = by -by -makecaption#1#2 tempboxa tempboxa > = = # 1 # 2 to    = 13pt su-4240 - 719 + hep - ph/0004105 +    * complementary ansatz for the neutrino mass matrix *    = 12pt    deirdre black amir h. fariborz + 0.5 cm salah nasri joseph schechter + _ department of physics , syracuse university , syracuse , ny 13244 - 1130 , usa . 
 _ + 0.5 cm    * abstract *