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in the recent report on the detailed analysis of solar neutrino oscillations @xcite done by the super - kamiokande collaboration @xcite , it has been suggested that solutions with large mixing angles are favored while solutions with small mixing angles are disfavored at the 95@xmath18 confidence level . neutrino oscillations controlled by such large mixing angle have also been observed and confirmed for atmospheric neutrinos @xcite . both observed oscillations are characterized by the same property of neutrino mixings , namely , large neutrino mixings , which are theoretically consistent with bimaximal mixing scheme @xcite . the difference arises in their oscillation scales denoted by @xmath19 for atmospheric neutrinos and @xmath20 for solar neutrinos , which are specified by @xmath21 ev@xmath22 and @xmath23 ev@xmath22 , thereby , indicating a hierarchy of @xmath24 . neutrino oscillations arise for massive neutrinos @xcite . since neutrino masses of @xmath25 ev are implied by @xmath19 , models for neutrino oscillations must be equipped with mechanisms generating such tiny masses of neutrinos @xcite , which include breaking of the lepton number conservation for majorana neutrino masses . it has been long known @xcite that such lepton number breaking interactions are inevitably contained in the minimal @xmath26 model for electroweak interactions ( called the 331 model ) @xcite because a charged lepton ( @xmath27 ) and its antiparticle ( @xmath28 ) together with a neutrino ( @xmath29 ) are placed in a triplet of @xmath30 as @xmath31=(@xmath32 , @xmath33 , @xmath34)@xmath35 . as a result , the model provides a charged - lepton mass term included in @xmath36 , which simultaneously contains @xmath37 as a neutrino mass term . namely , the need for majorana mass term is linked to the existence of the massive charged leptons . therefore , the 331 model is suitable for incorporating majorana neutrino masses @xcite .    with this nice feature of the 331 model in mind , we introduce the bimaximal mixing scheme into the model to accommodate observed neutrino oscillations . one of the possible scenario for bimaximal mixing scheme is to employ a new @xmath38 symmetry based on @xmath1 ( @xmath39 @xmath10 ) @xcite . neutrino mixings are predicted to be maximal for solar neutrinos with @xmath20=0 but arbitrary for atmospheric neutrinos with @xmath40 . the hierarchy of @xmath24 is trivially satisfied since @xmath20=0 . one has to next explain how a nonvanishing @xmath20 is induced while preserving @xmath24 . an elaborated mechanism is to generate @xmath20 as radiative effects @xcite , whose generic smallness explains the relative smallness of @xmath20 over @xmath19 @xcite .    in this article , we construct our interactions that induce phenomenologically consistent neutrino and charged - lepton mass matrix , @xmath41 and @xmath42 , with appropriate radiative effects for @xmath20 . since the model describes both matrices in a unified form , masses for charged leptons and neutrinos are correlated to each other . the mass term of @xmath36 transforms as @xmath43 and @xmath44 , which requires @xmath45 and/or @xmath46 as higgs scalars . the coupling of the triplet higgs scalar to @xmath36 is antisymmetrized with respect to the @xmath30-index . the spin and statistics , then , requires antisymmetrized flavor indices carried by @xmath31 , in turn , generating antisymmetrized @xmath42 with eigenvalues corresponding to @xmath47 , which contradicts with the observed hierarchy of @xmath48 . the triplet higgs scalar alone is not consistent with the observed lepton mass spectrum . therefore , the sextet scalar , which provides symmetrized mass terms , is the key ingredient of the 331 model @xcite .- singlet vector - like heavy @xmath49electron " without introducing a sextet scalar @xcite has been advocated to describe consistent mass spectra of neutrinos and charged leptons @xcite . ]    because the sextet higgs scalar contains both an @xmath50-doublet scalar for charged leptons and an @xmath50-triplet scalar for neutrinos , we naturally expect @xmath49degeneracy `` in the structure of their mass matrices . the charged leptons should contain a diagonal mass term such as the @xmath15-mass , which forces @xmath41 to have a diagonal term . however , @xmath41 should not contain diagonal terms to be consistent with bimaximal mixing scheme based on @xmath10 . to avoid this @xmath49degeneracy '' , we use the conservation of the @xmath10 quantum number supplemented by an appropriate @xmath3 parity as well as a specific breaking pattern of @xmath10 yielding a residual discrete symmetry . the collaboration of these symmetries allows neutrinos to develop no diagonal mass terms . as a result , atmospheric neutrino oscillations are controlled by those tree - level off - diagonal masses while solar neutrino oscillations are to be induced by one - loop effects . and a discrete symmetry have been done in ref.@xcite . ]    in the following section , we first introduce particles and their interactions and discuss how the required vacuum alignment is realized . section iii deals with the diagonalization of @xmath41 , which reflects rotation effects due to the diagonalization of @xmath42 . the bimaximal condition on the neutrino masses determines a pattern of the matrix elements for the electron mass . one - loop radiative mass of @xmath51 is also calculated . the discussions in sec.iv include to examine @xmath20 by utilizing the correlation between @xmath41 and @xmath42 and to confirm that the partial @xmath49degeneracy " between @xmath41 and @xmath42 suppresses the rotation effects to yield the lma solution based on the radiative effects . the final section is devoted to summary and discussions . the model contains neutrinos and charged leptons in triplets of @xmath52 ( @xmath53=1,2,3 ) and the greek letters of @xmath54 , respectively , stand for the three families and three @xmath30-indices . ] with @xmath55 for @xmath56 and @xmath57 for @xmath58 , where the superscript @xmath59 of @xmath60 stands for the charge conjugation , and two ( anti-)sextet higgs scalars @xmath61 with @xmath10=(0 , 2 ) for ( @xmath9 , @xmath11 ) . since @xmath9 can couple to @xmath62 ( @xmath8=2,3 ) , @xmath63 mixing terms are generated by @xmath64 and are to be responsible for the bimaximal structure . on the other hand , @xmath65 ( and also @xmath66 ) are generated by @xmath67 via its coupling to @xmath68 ( @xmath69=2,3 ) . since @xmath68 also supply couplings to @xmath70 and @xmath71 , if @xmath72 is also induced , then it disturbs the nice feature of the bimaximal mixing scheme based on @xmath10 . therefore , in order for the present approach to be acceptable , the simultaneous alignment of @xmath73 and @xmath67 should be dynamically allowed . it is demonstrated that this vacuum alignment of @xmath74 and @xmath75 is indeed possible to occur as a result of the emergence of a residual discrete symmetry specific to @xmath38 . the 331 model is specified by the @xmath76-charge . let @xmath77 be the @xmath76 quantum number , then the hypercharge , @xmath78 , is given by @xmath79 and the electric charge @xmath80 is given by @xmath81 , where @xmath82 is the @xmath83 generator with tr@xmath84 @xmath85 . let us first summarize the whole particle content in our model , where the quantum numbers are specified by ( @xmath30 , @xmath76 ; @xmath38 ) : @xmath86 for leptons , @xmath87 for quarks @xmath88 for triplet higgs scalars , and @xmath89 for ( anti-)sextet higgs scalars . these higgs scalars have the following vacuum expectation values ( vev s ) : @xmath90 and quarks and leptons will acquire masses via these vev s , where the orthogonal choice of the vev s in eq.([eq : vev_triplet ] ) and the vanishing vev of @xmath91 in eq.([eq : vev_sextet ] ) are to be ensured by appropriate higgs interactions . the pure @xmath30-anomaly is cancelled in a vectorial manner . the anomalies from triplets of the three families of leptons and of the three colors of the third family of quarks are cancelled by those from antitriplets of three colors of the first and second families of quarks @xcite . other anomalies including @xmath76 are also cancelled . the scale of the 331 model is set by @xmath92 , yielding @xmath93 and we expect the magnitude of @xmath92 is of order tev .      to realize phenomenologically viable interactions , we further impose a @xmath3 parity on the model , where @xmath3=+ for @xmath56 , @xmath94 and @xmath11;=@xmath95 for @xmath58 and @xmath9;=@xmath8 for @xmath96 and @xmath97 and similarly for quarks . hereafter , we restrict ourselves to the lepton sector . the yukawa interactions for leptons are , then , controlled by the following lagrangian : @xmath98 where @xmath99 s denote the yukawa couplings with @xmath100 . the possible term of the @xmath101-type is forbidden by @xmath3 . the higgs interactions are described by self - hermitian terms composed of @xmath102 ( @xmath103=@xmath94 , @xmath96 , @xmath97 ) and @xmath104 , which include the potential term of @xmath105 : @xmath106 where @xmath107 s represent coupling constants and @xmath108 and by the non - self - hermitian terms in @xmath109 where @xmath110 denotes a mass scale and @xmath111 stand for coupling constants . there is a leptonic number associated with @xmath31 , which can be taken to be @xmath112 for @xmath113 and 0 for all others . since the @xmath114 conservation is broken both spontaneously by @xmath113 and explicitly by @xmath115 , there is no harmful nambu - goldstone boson . although the lepton number can not be globally assigned to @xmath30-multipulets , to argue majorana masses for neutrinos , it is useful to define the lepton number @xmath116 to classify interactions @xcite in the similar way used for the electric charge defined in the standard @xmath117 model . following the classification lately done in ref.@xcite , the @xmath116-number can be introduced in the form of @xmath118 where @xmath119 = 2 for @xmath120 ; = 4/3 for @xmath97 ; = 2/3 for @xmath121 ; = 1/3 for @xmath122 ; = 0 for @xmath123 and @xmath124 ; = @xmath952/3 for @xmath125 , @xmath94 , @xmath96 and @xmath113 ; = @xmath952 for @xmath126 , which yields @xmath116 = ( 1 , 1 , @xmath127 ) for @xmath122 as usual . the assignment for leptons and higgs scalars is shown in table [ tab : lnumber0 ] . among our interactions constrained by @xmath10 , only @xmath128 breaks the @xmath116 conservation ; therefore , it is this interaction that gives the nonvanishing @xmath129 . a few comments are in order to support our choice of vev s of @xmath130 and @xmath131 :    1 . the nonvanishing vev of @xmath132 is triggered by the @xmath116-violating @xmath128 for @xmath133 since @xmath134 , @xmath135 and @xmath136 ; 2 . the nonvanishing vev of @xmath137 is triggered by the @xmath116-conserving @xmath138 for @xmath139 since @xmath135 and @xmath136 and by the @xmath116-conserving @xmath140 for @xmath141 since @xmath134 ; 3 .   the orthogonal choice of vev s of @xmath94 , @xmath96 and @xmath97 as in eq.([eq : vev_triplet ] ) is supported by @xmath105 if all @xmath107 s are negative . it is because @xmath105 gets lowered if @xmath94 , @xmath96 and @xmath97 develop vev s . so , one can choose vev s such that @xmath142 @xmath143 0 , @xmath144 @xmath143 0 and @xmath145 @xmath143 0 . thus , these vev s are dynamically aligned by our higgs potential . on the other hand , we can not argue whether @xmath146 acquire vev s or not since there are no triggers such as an @xmath116-violating @xmath147 and an @xmath116-conserving @xmath148 , which can , respectively , align @xmath91 and @xmath149 to be nonvanishing . however , so far these interactions are forbidden by the @xmath10 conservation . to invoke such dynamical alignment , thus , requires @xmath10-breaking interactions .    to introduce @xmath10-breaking effects calls for careful consideration because of the specific alignment of vev s of @xmath11 , _ i.e. _ @xmath73 and @xmath67 . its breaking readily induces dangerous interactions that creates @xmath91 such as tadpole interactions for @xmath74 . owing to the vacuum alignment of @xmath73 , @xmath10-breaking interactions should conserve @xmath116 . there is one candidate that can keep this specific alignment , which is an @xmath116-conserving @xmath150 : @xmath151 where @xmath152 represents a breaking of the @xmath10 conservation . since @xmath38 is broken by the unit of @xmath153=4 , there is still a residual conservation due to a @xmath3 symmetry of @xmath154 to be denoted by @xmath155 , which is shown in table [ tab : lnumber0 ] together with @xmath10 and @xmath3 . this @xmath155 symmetry can be used to constrain possible forms of @xmath10-breaking interactions . the mass terms for @xmath14 and @xmath15 are created by @xmath67 , which is now supported by @xmath156 for @xmath157 . therefore , @xmath156 serves as a trigger for @xmath67 . the vev of @xmath91 would be created by the following @xmath116-violating interactions of    * tr(@xmath158 ) since @xmath64 , * @xmath147 since @xmath159 , * @xmath160 since @xmath134 , @xmath135 and @xmath136 , * @xmath161 and @xmath162 since @xmath135 , @xmath136 and @xmath163 and * tr(@xmath164 ) , tr(@xmath165 ) and tr(@xmath166 ) since @xmath163 . fortunately , all of these interactions are forbidden by the interplay between @xmath3 and @xmath155 . tabulated in table [ tab : lnumber ] are @xmath3 and @xmath155 for non - self - hermitian higgs interactions allowed by @xmath0 , where the reader can recognize how these dangerous interactions are excluded . at this moment , we reach the plausible situation , where the alignment of @xmath167 is dynamically supported . the resulting mass matrices for charged leptons and neutrinos take the form of @xmath168 where @xmath169 are expressed in terms of vev s : @xmath170 leading to the proportionality of @xmath171 and @xmath172 stand for one - loop radiative masses to be estimated in the next subsection . we simply assume @xmath173 ( @xmath69=2,3 ) so as to reproduce the hierarchy of @xmath174 by the mechanism discussed in ref.@xcite . since the magnitude of neutrino masses @xmath175 is controlled by @xmath129 , let us explain how naturally small @xmath129 is obtained . an estimate of @xmath176 can be found in the analyses of ref.@xcite applied to our higgs potential . it calls for @xmath133 as has been expected in the vacuum alignment . the smallness of @xmath129 is , then , ascribed to either @xmath177 @xcite or @xmath178 @xcite , the latter of which is known as the type ii seesaw mechanism @xcite , where we use the mechanism due to @xmath177 by simply expecting that @xmath92 is of order tev . one will find that @xmath179a few@xmath180 provides neutrino masses about @xmath181 ev . this is in fact a fine - tuning of @xmath182 to reproduce neutrino masses so as to meet the value of @xmath181 . however , we have already known this tiny magnitude , which is give by @xmath183 . the relative smallness of @xmath184 is linked to the underlying chiral symmetry recovered at the limit of @xmath184 @xmath185 0 . this feature dictates the so - called @xmath49naturalness " @xcite . similarly , since the @xmath116 conservation is respected in the limit of @xmath186 , the process of taking @xmath177 is also regarded to be natural although both cases require other new physics to explain why such tiny values are realized . our program to discuss patterns of neutrino oscillations is based on the use of the proportionality of @xmath175 to @xmath187 indicated by eq.([eq : nupropto ] ) . since to generate the electron mass and to reproduce @xmath19 will , respectively , determine @xmath187 and @xmath175 , which yield the estimation of @xmath188 . the magnitude of @xmath129 derived in eq.([eq : vev_nu ] ) now determines that of @xmath182 once appropriate values of vev s are given and , then , @xmath189 ( @xmath69=2,3 ) to be defined in eqs.([eq : neutrino_ee ] ) and ( [ eq : neutrino_mutau ] ) are calculable for this estimated @xmath182 . in the end , solar neutrino oscillations controlled by @xmath189 are found to be consistent with the lma solution . we first discuss the diagonalization of @xmath42 . the following unitary matrix @xmath190 is obtained under the approximation of @xmath173 ( @xmath69=2,3 ) and transforms @xmath42 into @xmath191=@xmath192=diag.(@xmath193 , @xmath66 , @xmath65 ) : @xmath194 up to the second order of @xmath195 with @xmath196 where @xmath69=@xmath197 for @xmath198 , @xmath199 and @xmath200 . the masses of @xmath201 , @xmath14 and @xmath15 are given by @xmath202 where @xmath203    to be consistent with the observed mass pattern of the charged lepton mass matrix , we should present a mechanism generating plausible mass textures such as the hierarchical pattern @xcite and the democratic pattern @xcite . however , in the present context of the model , we are not aimming at clarifying all the physical aspect of the lepton sector but , instead , at clarifying the correlation of neutrino mass matrix with charged lepton mass matrix . here , we simply parameterize @xmath204 by adopting two distinct ways to realize @xmath48 : hierarchical and democratic mass textures and discuss their influences in neutrino mixings . the resulting parameterization is given by    1 . the hierarchical pattern of @xmath205 and @xmath206 leading to @xmath207 with @xmath208 ( @xmath209 ) and 2 .   the democratic pattern of @xmath210 and @xmath211 where @xmath212 and @xmath213 specify @xmath214 and @xmath215 ( @xmath216 ) , leading to @xmath217 . therefore , we have found the estimate of @xmath218 that reproduces @xmath184 . neutrinos acquire radiative masses of @xmath189 ( @xmath69=2,3 ) generated by one - loop interactions corresponding to diagrams depicted in fig.[fig : loopdiagrams_ee ] and fig.[fig : loopdiagrams_mutau ] . it is convenient to shift the basis for charged - leptons from the original one for @xmath42 to the intermediate one , where @xmath14 and @xmath15 have diagonal masses of @xmath219 and @xmath220 , respectively . the corrections due to @xmath221 are neglected because of @xmath222 . in this basis , the couplings of @xmath9 to the lepton triplets are described by @xmath223 ( @xmath8=2,3 ) with ( @xmath224 , @xmath225)=(@xmath226 , @xmath227 ) and @xmath228 are the rotated @xmath229-couplings similarly defined as ( @xmath230 , @xmath231)=@xmath232 , @xmath233 ) , which appear in the figures .    the result of the calculation of @xmath189 ( @xmath69=2,3 ) is given by @xmath234v_\eta \nonumber \\ & & + 2\lambda_2\lambda_1 v^{(0)}_\ell   f^2_{1i } m_{\ell^i } \left [ i(m^2_{\ell^i } , m^2_{s^{(0)+ } } , m^2_{s^{(0)- } } , m^2_{\eta^+ } ) + i(m^2_{\ell^i } , m^2_{s^{(0)- } } , m^2_{s^{(0)+ } } , m^2_{\eta^- } ) \right ] v_\eta v_\rho v_\chi , \label{eq : neutrino_ee } \\ \delta m^{rad}_{ij}&= & 2\lambda_2\lambda_b   v^{(0)}_\ell   f_{1i}f_{1j } m_{\ell^i } \left [ i(m^2_{\ell^i } , m^2_{s^{(+)+ } } , m^2_{s^{(0)- } } , m^2_{\eta^+ } ) + i(m^2_{\ell^i } , m^2_{s^{(+)- } } , m^2_{s^{(0)+ } } , m^2_{\eta^- } ) \right ] v_\eta v_\rho v_\chi , \label{eq : neutrino_mutau}\end{aligned}\ ] ] for @xmath69=2,3 and @xmath184=0 in this basis , with @xmath235 and @xmath236 as in eq.([eq : unitaryentries ] ) , where @xmath237 the majorana mass @xmath238 for @xmath51 is induced by effective couplings corresponding to fig.[fig : loopdiagrams_ee ] : @xmath239 for @xmath103=@xmath96 or @xmath97 , and @xmath240 with similar terms scrambled by possible permutations of the fields . both couplings conserve @xmath10 . on the other hand , the same effective coupling as eq.([eq : inducedmajorana_ee2 ] ) but with the replacement of @xmath241 , thus of the @xmath10-breaking type , yields @xmath172 ( @xmath69=2,3 ) for @xmath242 , which correspond to fig.[fig : loopdiagrams_mutau ] . now , let us turn to examining the neutrino sector . the neutrino mass matrix is also transformed into @xmath243 by @xmath190 of eq.([eq : unitarymatrix ] ) to maintain diagonal weak currents and is calculated , up to the second order of @xmath244 , to be : @xmath245 where @xmath246 with @xmath247 it is obvious to see that @xmath243 satisfies tr(@xmath243)=tr(@xmath41)=@xmath248 and tr(@xmath249)= tr(@xmath250)=2(@xmath251 . since @xmath252 , the diagonal neutrino masses given by ( @xmath253 , @xmath254 , @xmath255 ) are calculated to be : @xmath256 up to the second order of @xmath257 equivalent to the second order of @xmath244 , where @xmath258 with @xmath259 and @xmath260 as the mixing angle for atmospheric neutrinos and @xmath261 the masses of @xmath262 and @xmath263 can be further converted into : @xmath264 , \label{eq : eachmassterms3}\end{aligned}\ ] ] where other irrelevant terms involving @xmath172 are omitted . the mass of @xmath265 is obtained by noticing the relation of tr(@xmath243)=@xmath248 that yields @xmath266 from which we find that @xmath267 to enhance the bimaximal structure in @xmath243 is achieved by setting @xmath268 , where @xmath269 can be approximated to be : @xmath270 because @xmath271 . the condition of @xmath268 thus , requires , for the hierarchical case with @xmath272 and @xmath273 , @xmath274 and , for the democratic case with @xmath275 , @xmath276 the essence of the present research lies in the fact that the bimaximal mixing scheme for neutrinos , in turn , determines how the electron mass is derived . the bimaximal condition of eq.([eq : bimaximalcond1 ] ) combined with eq.([eq : masses23_1 ] ) yields @xmath277 while eq.([eq : bimaximalcond2 ] ) combined with eq.([eq : masses23_2 ] ) yields @xmath278 and @xmath279 where @xmath280 . the neutrino oscillation parameters of @xmath19 and @xmath281 are expressed in terms of our neutrino masses as @xmath282 leading to @xmath283 with @xmath284 the ratio @xmath285 arises from the radiative effects and @xmath286 receives the first order contributions from the rotation due to the diagonalization of @xmath287 . we are now in a position to estimating @xmath288 to meet the lma solution . before going to its explicit evaluation , we list our assumptions on numerical values of the relevant mass parameters :    * @xmath289 (= @xmath290=174 gev ) for @xmath291 , * @xmath292 as the simplest case to keep @xmath293 for weak boson masses , * @xmath294 and @xmath295 ,    where @xmath92 , presumably of order tev , is left unspecified and is to be measured in the unit of 10@xmath296 . let us first reproduce @xmath21 ev@xmath22 , which requires that @xmath297 ev@xmath22 from eq.([eq : atm_m2 ] ) . the relation of @xmath298 ( @xmath8=2,3 ) together with eqs.([eq : bimaximalchargedcond1]-[eq : bimaximalchargedcond3 ] ) for @xmath187 , then , leads to @xmath299 where three terms to be placed in the parenthesis , respectively , denote the cases with ( hierarchical masses with @xmath300 , democratic masses with @xmath301 , democratic masses with @xmath302 ) . in the following , the same notation is used . since @xmath129 is expressed in terms of @xmath182 and vev s as estimated in eq.([eq : vev_nu ] ) , we find the constraint on @xmath182 given by @xmath303 therefore , the magnitude of @xmath182 is not arbitrary but fixed to be @xmath304 by vev s in order to obtain @xmath21 ev@xmath22 . next , we calculate @xmath20 , which depends on the sum of @xmath238 and @xmath305 given by eqs.([eq : neutrino_ee ] ) and ( [ eq : neutrino_mutau ] ) with @xmath306 . to find numerical values of @xmath189 , we use the following property of the integral @xmath307 of eq.([eq : integral ] ) : @xmath308 by using the assumptions on mass parameters stated above , we know that the second term in eq.([eq : neutrino_ee ] ) is proportional to @xmath309 and all other terms including @xmath172 of eq.([eq : neutrino_mutau ] ) are proportional to @xmath310 . in eq.([eq : neutrino_ee ] ) , the first term thus dominates the second term because of @xmath311 for @xmath312 and @xmath313 . the other terms proportional to @xmath310 , the third and fourth terms together with @xmath305 , are more suppressed by the factor of @xmath314 compared with the first term since @xmath315 are at most of order unity . therefore , the first term in @xmath238 gives the most dominant contributions to @xmath20 and those from all other terms can be safely neglected . the radiative mass of @xmath238 saturated by the mediating @xmath15 lepton contribution turns out to be : @xmath316 where , in the democratic cases , the factor of @xmath317 is due to @xmath318 in eq.([eq : bimaximalchargedcond2 ] ) and the factor 1/2 arises from @xmath319 . the rotated couplings of @xmath228 used in these calculations are approximated to be ( @xmath229 , @xmath320 , @xmath321 ) ( @xmath322 ) for @xmath323=@xmath127 for @xmath8=2 and @xmath323=1 for @xmath8=3 .    by inserting the above result of eq.([eq : numerics_rad1 ] ) into @xmath285 shown in eq.([eq : ratiosolarrad ] ) with @xmath324 ev , we reach the following estimate : @xmath325 where @xmath326 is used in eq.([eq : numerics_rad1 ] ) . this estimate together with eq.([eq : lambda2_estimate ] ) for @xmath182 yields @xmath327 where @xmath328 is bounded from below by @xmath280 . by taking @xmath329{\cos\phi } , 2.0 - 3.6 \right),\end{aligned}\ ] ] we obtain @xmath281=(@xmath330 ) ev@xmath22 relevant to the lma solution to the solar neutrino problem and @xmath331 since @xmath296=174 gev , @xmath92 roughly lies in the range of @xmath332 tev , which is the right order of @xmath92 (= @xmath333(1 tev ) ) as anticipated for the scale of @xmath0 @xmath185 @xmath117 . for @xmath334 tev as a typical value , namely , @xmath335=2.3 , we find that @xmath336 leading to @xmath337 ev@xmath22 for @xmath338 ev@xmath22 . finally , we consider effects from @xmath286 , which may disturb the behavior specific to the lma solution indicated by @xmath285 . since @xmath286 is controlled by @xmath265 as in eq.([eq : ratiosolarrad ] ) , it can be found that the leading terms of @xmath339 contained in @xmath265 turns out to vanish . the cancellation can be seen from the insertion of the leading terms of @xmath340 into the equation of @xmath341 appearing in the numerator of @xmath265 in eq.([eq : thirdmass ] ) that turns out to vanish . the next nonvanishing contributions to @xmath265 are given by the second order terms in @xmath341 that result in the third order contributions to @xmath265 , which is beyond our approximation because possible third order terms for @xmath257 ( @xmath69=2,3 ) , which have not been included , directly give the third order contributions as can be seen from the definition of @xmath265 in eq.([eq : neutrinomassparam ] ) . therefore , we expect that @xmath286 becomes at most @xmath342 , which is about @xmath343 ) as shown in eqs.([eq : bimaximalchargedcond1])-([eq : bimaximalchargedcond3 ] ) , leading to @xmath344 . this estimate of @xmath286 should be compared with eq.([delta_sol_lma2 ] ) for the radiative case and in fact gives more suppressed contributions . it is , then , concluded that we can safely neglect @xmath286 and that the model can provide the lma solution to the solar neutrino problem as indicated by @xmath285 . similarly , the third neutrino @xmath345 will also acquire masses at most of order @xmath346 , which is about @xmath347 ev . the radiative correction to @xmath255 arising from @xmath348 is estimated to be about @xmath349 since @xmath350 and @xmath351 , we find that @xmath352 ev for @xmath353 tev . thus , altogether we observe that the bound on @xmath255 is given by @xmath354 ev . the cancellation of the diagonalization effects on the neutrino masses of @xmath355 is a reasonable consequence because the ( 1,2 ) and ( 1,3 ) entries of @xmath41 are exactly proportional to those of @xmath42 ( see eqs.([eq : leptontextureh])-([eq : leptontextureb2 ] ) ) so that the rotation effects due to @xmath244 , which induce @xmath356 , @xmath357 , @xmath358 , are partially cancelled . however , the diagonalization effects manifest themselves in the matrix element @xmath359 that connects @xmath51 to @xmath345 . the chooz and paloverde experiments imply that @xmath360 @xcite . for the mass matrix @xmath243 of eq.([eq : numassesweak ] ) , this element is calculated to be @xmath361 \nonumber \\ & = & -\frac{1}{m_\nu}\left [ \left ( c_\alpha \delta m^\nu_{13 } + s_\alpha \delta m^\nu_{12 }    \right)\delta u_\mu - \left ( c_\alpha \delta m^\nu_{12 } - s_\alpha \delta m^\nu_{13 } \right)\delta u_\tau \right ] , \label{eq : ue3}\end{aligned}\ ] ] leading to @xmath362 with @xmath363 for eq.([eq : bimaximalchargedcond2 ] ) allowing @xmath364 , which includes the typical value of @xmath365 . our model predicts that @xmath366 , which are consistent with the current experimental bound . we have successfully demonstrated that the minimal 331 model with the global @xmath1 and @xmath3 symmetry , which involves two ( anti-)sextet higgs scalars , @xmath113 , indeed explains solar neutrino oscillations based on the lma solution . the lma solution is explicitly obtained by the use of the plausible parameter values for masses and couplings . the model has the following properties , which do not depend on the parameterization , that    1 . the global @xmath1 symmetry is explicitly broken by @xmath150 down to a discrete @xmath155 symmetry defined by @xmath154 that forbids the @xmath116-breaking interactions for @xmath11 , the absence of which is the essence of our discussions to be consistent with the bimaximal neutrino mixing scheme based on @xmath1 ; 2 .   the @xmath116 conservation is explicitly broken by @xmath367 , which generates the vev of @xmath129 for neutrinos calculated to be @xmath368 , where @xmath133 is the trigger for @xmath369 ; 3 . the mechanism generating the diagonal electron mass via @xmath187 ( @xmath8=2,3 ) is linked to the requirement realizing the bimaximal neutrino mixing via @xmath175 owing to the proportionality of @xmath370 ;      1 . the smallness of @xmath371 ev for @xmath19 is realized by @xmath372 estimated to be @xmath373 ; 2 . solar neutrino oscillations are induced by * the one - loop radiative effects , which yield @xmath374 corresponding to the lma solution provided @xmath0 @xmath185 @xmath117 occurs at @xmath332 tev , * the effects of the simultaneous diagonalization of the neutrino and charged - lepton mass matrices , which are expected to yield @xmath375 and which certainly yield the @xmath359 matrix element estimated to be @xmath376 , + where , in the present model , the dominance of one - loop radiative effects yields the lma solution . it should be stressed that the choice of @xmath92=@xmath333(1 tev ) automatically leads to the lma solution . the present radiative mechanism generates the majorana mass of @xmath51 , @xmath238 , given by the effective coupling of @xmath377 . this coupling is induced by @xmath128 and @xmath138 as in fig.[fig : loopdiagrams_ee ] and is of the @xmath10-conserving type . the less - dominant radiative masses are induced by @xmath378 for @xmath379 with @xmath69=1 contributing to @xmath238 and for @xmath380 with @xmath69=2,3 contributing to @xmath172 . the @xmath380-terms involve the @xmath10-breaking coupling of @xmath150 as in fig.[fig : loopdiagrams_mutau ] . the splitting of @xmath381 and @xmath382 is provided by @xmath238 , which is about @xmath383 ev , leading to the lma solution . the mass of @xmath345 is provided by @xmath348 at most of order @xmath384 ev but the diagonalization due to @xmath287 may give @xmath255 @xmath385 @xmath347 ev .      1 .   for the hierarchical texture with @xmath386 , @xmath387 2 . for the democratic texture with @xmath388 , @xmath389 with @xmath390 and @xmath391 constrained by @xmath280 , 3 .   for the democratic texture with @xmath392 , @xmath393 with @xmath394 ,    where @xmath395=@xmath188 , @xmath396 and the asterisks stand for the suppressed entries to generate @xmath20 . the partial @xmath49degeneracy " between @xmath41 to @xmath42 suppresses the rotation effects on @xmath281 due to the diagonalization of @xmath41 and @xmath42 . as a result , the solar neutrino oscillations are , respectively , characterized by @xmath337 ev@xmath22 for @xmath353 tev . of course , if our mechanism explaining neutrino oscillations works further , one should next consider some reasons for the emergence of the approximate equality of @xmath397 in the hierarchical case and of the hierarchy of either @xmath301 or @xmath302 in the democratic case . the masses of higgs scalars are simply assumed to be about @xmath296 for those related to the weak boson masses for the order - of - magnitude estimate for radiative neutrino masses , namely , @xmath398 @xmath399 @xmath400 @xmath399 @xmath401 @xmath399 @xmath402 @xmath399 @xmath296 . on the other hand , the @xmath97 scalar that generates masses for the exotic quarks and gauge bosons is assumed to have @xmath403 @xmath399 @xmath92(@xmath404 @xmath296 ) because @xmath405 is related to the spontaneous breakdown of @xmath0 to @xmath117 . we have estimated @xmath92 to lie in @xmath332 tev in order to reproduce the observed neutrino mass pattern . it is further known that the 331 model contains peculiar particles , _ i.e. _ doubly - charged bilepton gauge bosons @xcite with masses of the order @xmath406 , that couple to @xmath407 . since @xmath92 can be @xmath332 tev , the bilepton gauge bosons can be as light as and even lighter than 1 tev . it is , therefore , quite conceivable that we will observe the bilepton gauge bosons with masses around 1 tev , which provide striking phenomena with clean @xmath408-@xmath408 jets @xcite . now equipped with the plausible mechanism that generates phenomenologically consistent masses for neutrinos and charged leptons , the 331 model may be a physically interested model of leptons . it not only explains three families of quarks and leptons because of the specific anomaly - cancellation mechanism used together with the asymptotic free condition on @xmath409 but also involves majorana mass terms demanded by charged - lepton mass terms . there remain dynamical questions about the smallness of the @xmath116-violating coupling of @xmath182 and the fine tuning of the lepton mass parameters to reproduce @xmath410 , which require some new physics beyond @xmath411 . it is also required to have extensive analyses on physics of the quark and gauge boson sectors as well to fully clarify potential power of the 331 model . d. v. ahluwalia , ; v. barger , p. pakvasa , t.j . weiler and k. whisnant , ; a. baltz , a.s . goldhaber and m. goldhaber , ; m. jezabek and y. sumino , ; r.n . mohapatra and s. nussinov , ; y. nomura and t. yanagida , ; i. starcu and d.v.ahluwalia , ; c. jarlskog , m. matsuda , s. skadhauge and m. tanimoto , ; q. shafi and z. tavartkiladze , ; ; p.h . frampton and s.l . glashow , ; c.h . albright and s.m . barr , ; h. georgi and s.l . glashow , ; r.n . mohapatra , a. prez - lorenzana and c. a. de s. pires , . t. yanagida , in _ proceedings of the workshop on unified theories and baryon number in the universe _ edited by a. sawada and a. sugamoto ( kek report no.79 - 18 , tsukuba , 1979 ) , p.95 ; ; m. gell - mann , p. ramond and r. slansky , in _ supergravity _ edited by p. van nieuwenhuizen and d.z . freedmann ( north - holland , amsterdam 1979 ) , p.315 ; r.n . mohapatra and g. senjanovi , . valle and m. singer , ; r. barbieri and r.n . mohapatra , ; j. liu , ; r. foot , o. f. hernndez , f. pisano and v. pleitez , ; v. pleitez and m.d . tonasse , ; ; ; j. liu and d. ng , ; p.h . frampton , p.i . krastev and j.t . liu , ; f. pisano , v. pleitez and m.d . tonasse , hep - ph/9310230 v2 ( feb , 1994 ) ; m. @xmath412zer , ; f. pisano , j.a . silva - sobrinho and m.d . tonasse , f. -z . chen , ; m.b . tully and g.c . joshi , hep - ph/9810282.v2 ( jan,1999 ) ; y. okamoto and m. yasu@xmath413 , ; n.a . long and d.v . soa , . t. kitabayashi and m. yasu , ; ; hep - ph/0102228 v3 ( feb . , 2001 ) to be published in phys . lett . b ( 2001 ) ; t. kitabayashi , hep - ph/0010341 ( oct . , 2000 ) ; hep - ph/0103195 ( mar . , 2001 ) to appear in phys . d ( 2001 ) . tully and g.c . joshi , hep - ph/0011172 to appear in phys . rev d ( 2001 ) ; j.c . montero , c.a . de s. pires and v. pleitez , . see for example , g.c . w. grimus and l. lavoura , ; w. grimus and h. neufeld , ; l. bento and j.w.f . valle , ; s.y . tsai , ; k.s . babu , r.n . mohapatra and l. rothstein , ; j.t . peltoniemi , a.yu . smirnov and j.w.f . valle , in ref . @xcite ; j.t . peltoniemi , d. tommasini and j.w.f . valle , ; j.t . peltoniemi and j.w.f . e. ma , ; ; ; ; ;    for recent discussions , see for example , a.s . joshipura and s.d . rindani , ; r.n . mohapatra , a. prez - lorenzana and c.a . de s. pires , in ref.@xcite . d. chang and a. zee , ; r. adhikari , e. ma and g. rajasekaran , . w. grimus and h. neufeld , ; w. grimus and l. lavoura , ; t. kitabayashi and m. yasu , in ref.@xcite ; t. kitabayashi , in ref.@xcite .    for supersymmetric models , see for example , e. ma , m. raidal and u. sarkar , ; j.c . romo , m.a . daz , m. hirsch , w. porod and j.w.f . valle , ; f. takayama and m. yamaguchi , ; o.c.w . kong , ; ; e.j . chun , ; s. davidson and m. losada , ; m. hirsch , m.a . daz , w. porod , j.c . romo and j.w.f . valle , . g. t hooft , in _ recent development in gauge theories _ , proceedings of the cargese summer institute , cargese , france , 1979 , edited by g. t hooft _ et al . _ , nato advanced study institute series b : physics vol . 59 ( plenum press , new york , 1980 ) .          for recent studies , see for example , p.h . frampton and x .- hu guan , ; f. pisano , j.a . silva - sobrinho and m.d . tonasse , ; b. dion , t. gregoire , d. london , l. marleau and h. nadeau , ; m.b . tully and g.c . joshi , ; p.h . frampton , ; p.h . frampton and a. ra , . .[tab : lnumber0 ] @xmath10 , @xmath119 , @xmath3 and @xmath155 for leptons and higgs scalars , where the lepton number @xmath116 is related to @xmath119 as @xmath116 = @xmath415 + @xmath119 . [ cols="<,^,^,^,^,^,^,^",options="header " , ]
in the minimal @xmath0 gauge model with a global @xmath1 ( @xmath2 ) symmetry and a discrete @xmath3 symmetry , it is found that the interplay between neutrinos and charged leptons contained in triplets of @xmath4=(@xmath5 , @xmath6 , @xmath7 ) ( @xmath8=1,2,3 ) naturally leads to the large mixing angle ( lma ) msw solution . 
 the model includes two ( anti)sextet higgs scalars , @xmath9 with @xmath10=0 and @xmath11 with @xmath10=2 , which , respectively , couple to @xmath12 for the electron mass and masses of atmospheric neutrinos and to @xmath13 for the @xmath14- and @xmath15-masses and one - loop radiative neutrino masses relevant to solar neutrinos . 
 this mechanism is realized by utilizing an additional residual discrete symmetry supplied by explicitly broken @xmath10 , which guarantees the absence of tree - level neutrino mass terms of the @xmath13-type . 
 pure rotation effects due to the diagonalization of neutrino and charged - lepton mass matrices are estimated to yield @xmath16 but the radiative effects supersede the rotation effects to yield @xmath17 as the lma solution . 
 epsf # 1#2#3#4#1 * # 2 * ( # 4 ) # 3    # = `` 0023 $ = ' ' 0024 % = `` 0025 = ' ' 705c    = `` 336e = ' ' 336f = `` 331a = ' ' 331b = `` 3328 = ' ' 3329 = " 0350