document: the lep experiments at the resonance of @xmath1-boson have tested the standard model ( sm ) at quantum level , measuring the @xmath1-decay into fermion pairs with an accuracy of one part in ten thousands . 
   the good agreement of the lep data with the sm predictions have severely constrained the behavior of new physics at the @xmath1-pole . 
   taking these achievements into account one can imagine that the physics of @xmath1-boson will again play the central role in the frontier of particle physics if the next generation @xmath1 factory comes true with the generated @xmath1 events several orders of magnitude higher than that of the lep . 
   this factory can be realized in the gigaz option of the international linear collider ( ilc)@xcite . 
   the ilc is a proposed electron - positron collider with tunable energy ranging from @xmath12 to @xmath13 and polarized beams in its first phase , and the gigaz option corresponds to its operation on top of the resonance of @xmath1 boson by adding a bypass to its main beam line . 
   given the high luminosity , @xmath14 , and the cross section at the resonance of @xmath1 boson , @xmath15 , about @xmath16 @xmath1 events can be generated in an operational year of @xmath17 of gigaz , which implies that the expected sensitivity to the branching ratio of @xmath1-decay can be improved from @xmath18 at the lep to @xmath19 at the gigaz@xcite . in light of this , the @xmath1-boson properties , especially its exotic or rare decays which are widely believed to be sensitive to new physics , should be investigated comprehensively to evaluate their potential in probing new physics .    among the rare @xmath1-decays , the flavor changing ( fc ) processes were most extensively studied to explore the flavor texture in new physics @xcite , and it was found that , although these processes are severely suppressed in the sm , their branching ratios in new physics models can be greatly enhanced to @xmath19 for lepton flavor violation decays @xcite and @xmath20 for quark flavor violation decays @xcite . besides the fc processes 
   , the @xmath1-decay into light higgs boson(s ) is another type of rare process that was widely studied , e.g. the decay @xmath21 ( @xmath22 ) with the particle @xmath0 denoting a light higgs boson was studied in @xcite , the decay @xmath23 was studied in the two higgs doublet model ( 2hdm)@xcite and the minimal supersymmetric standard model ( mssm)@xcite , and the decay @xmath4 was studied in a model independent way @xcite , in 2hdm@xcite and also in mssm@xcite . 
   these studies indicate that , in contrast with the kinematic forbidden of these decays in the sm , the rates of these decays can be as large as @xmath18 in new physics models , which lie within the expected sensitivity of the gigaz . in this work , 
   we extend the previous studies of these decays to some new models and investigate these decays altogether . 
   we are motivated by some recent studies on the singlet extension of the mssm , such as the next - to - minimal supersymmetric standard model ( nmssm ) @xcite and the nearly minimal supersymmetric standard model ( nmssm ) @xcite , where a light cp - odd higgs boson @xmath0 with singlet - dominant component may naturally arise from the spontaneous breaking of some approximate global symmetry like @xmath24 or peccei - quuin symmetry @xcite . 
   these non - minimal supersymmetric models can not only avoid the @xmath25-problem , but also alleviate the little hierarchy by having such a light higgs boson @xmath0 @xcite . 
   we are also motivated by that , with the latest experiments , the properties of the light higgs boson are more stringently constrained than before . 
   so it is worth updating the previous studies . 
   so far there is no model - independent lower bound on the lightest higgs boson mass . in the sm 
   , it must be heavier than @xmath26 gev , obtained from the null observation of the higgs boson at lep experiments . 
   however , due to the more complex structure of the higgs sector in the extensions of the sm , this lower bound can be significantly relaxed according to recent studies , e.g. , for the cp - odd higgs boson @xmath0 we have @xmath27 gev in the nmssm @xcite , @xmath28 gev in the nmssm @xcite , and @xmath29 gev in the lepton - specific 2hdm ( l2hdm ) @xcite . with such a light cp - odd higgs boson , the z - decay into one or more 
   @xmath0 is open up . noting that the decay @xmath30 is forbidden due to bose symmetry , we in this work study the rare @xmath1-decays @xmath6 ( @xmath22 ) , @xmath31 and @xmath4 in a comparative way for four models , namely the type - ii 2hdm@xcite , the l2hdm @xcite , the nmssm and the nmssm . in our study 
   , we examine carefully the constraints on the light @xmath0 from many latest experimental results . 
   this work is organized as follows . in sec . 
   ii we briefly describe the four new physics models . in sec . 
   iii we present the calculations of the rare @xmath1-decays . in sec . 
   iv we list the constraints on the four new physics models . in sec . 
   v we show the numerical results for the branching ratios of the rare @xmath1-decays in various models . finally , the conclusion is given in sec . 
   as the most economical way , the sm utilizes one higgs doublet to break the electroweak symmetry . as a result , 
   the sm predicts only one physical higgs boson with its properties totally determined by two free parameters . in new physics models , 
   the higgs sector is usually extended by adding higgs doublets and/or singlets , and consequently , more physical higgs bosons are predicted along with more free parameters involved in . 
   the general 2hdm contains two @xmath32 doublet higgs fields @xmath33 and @xmath34 , and with the assumption of cp - conserving , its scalar potential can be parameterized as@xcite : @xmath35,\end{aligned}\ ] ] where @xmath36 ( @xmath37 ) are free dimensionless parameters , and @xmath38 ( @xmath39 ) are the parameters with mass dimension . after the electroweak symmetry breaking , the spectrum of this higgs sector includes three massless goldstone modes , which become the longitudinal modes of @xmath40 and @xmath1 bosons , and five massive physical states : two cp - even higgs bosons @xmath41 and @xmath42 , one neutral cp - odd higgs particle @xmath0 and a pair of charged higgs bosons @xmath43 . noting the constraint @xmath44 with @xmath45 and @xmath46 denoting the vacuum expectation values ( vev ) of @xmath33 and @xmath34 respectively , we choose @xmath47 as the input parameters with @xmath48 , and @xmath49 being the mixing angle that diagonalizes the mass matrix of the cp - even higgs fields . 
   the difference between the type - ii 2hdm and the l2hdm comes from the yukawa coupling of the higgs bosons to quark / lepton . in the type - ii 2hdm 
   , one higgs doublet @xmath34 generates the masses of up - type quarks and the other doublet @xmath33 generates the masses of down - type quarks and charged leptons ; while in the l2hdm one higgs doublet @xmath33 couples only to leptons and the other doublet @xmath34 couples only to quarks . 
   so the yukawa interactions of @xmath0 to fermions in these two models are given by @xcite @xmath50 with @xmath51 denoting generation index . 
   obviously , in the type - ii 2hdm the @xmath52 coupling and the @xmath53 coupling can be simultaneously enhanced by @xmath54 , while in the l2hdm only the @xmath53 coupling is enhanced by @xmath55 . 
   the structures of the nmssm and the nmssm are described by their superpotentials and corresponding soft - breaking terms , which are given by @xcite @xmath56 where @xmath57 is the superpotential of the mssm without the @xmath25 term , @xmath58 and @xmath59 are higgs doublet and singlet superfields with @xmath60 and @xmath61 being their scalar component respectively , @xmath62 , @xmath63 , @xmath64 , @xmath65 , @xmath66 and @xmath67 are soft breaking parameters , and @xmath68 and @xmath69 are coefficients of the higgs self interactions .    with the superpotentials and the soft - breaking terms 
   , one can get the higgs potentials of the nmssm and the nmssm respectively . like the 2hdm , 
   the higgs bosons with same cp property will mix and the mass eigenstates are obtained by diagonalizing the corresponding mass matrices : @xmath70 where the fields on the right hands of the equations are component fields of @xmath71 , @xmath72 and @xmath61 defined by @xmath73 @xmath74 and @xmath75 are respectively the cp - even and cp - odd neutral higgs bosons , @xmath76 and @xmath77 are goldstone bosons eaten by @xmath1 and @xmath78 , and @xmath79 is the charged higgs boson . 
   so both the nmssm and nmssm predict three cp - even higgs bosons , two cp - odd higgs bosons and one pair of charged higgs bosons . in general , the lighter cp - odd higgs @xmath0 in these model is the mixture of the singlet field @xmath80 and the doublet field combination , @xmath81 , i.e. @xmath82 and its couplings to down - type quarks are then proportional to @xmath83 . 
   so for singlet dominated @xmath0 , @xmath84 is small and the couplings are suppressed . as a comparison 
   , the interactions of @xmath0 with the squarks are given by@xcite @xmath85 i.e. the interaction does not vanish when @xmath86 approaches zero . just like the 2hdm where we use the vevs of the higgs fields as fundamental parameters , we choose @xmath68 , @xmath69 , @xmath87 , @xmath88 , @xmath66 and @xmath89 as input parameters for the nmssm@xcite and @xmath68 , @xmath54 , @xmath88 , @xmath65 , @xmath90 and @xmath91 as input parameters for the nmssm@xcite . 
   about the nmssm and the nmssm , three points should be noted . 
   the first is for the two models , there is no explicit @xmath92term , and the effective @xmath25 parameter ( @xmath93 ) is generated when the scalar component of @xmath59 develops a vev . 
   the second is , the nmssm is actually same as the nmssm with @xmath94@xcite , because the tadpole terms @xmath95 and its soft breaking term @xmath96 in the nmssm do not induce any interactions , except for the tree - level higgs boson masses and the minimization conditions . and 
   the last is despite of the similarities , the nmssm has its own peculiarity , which comes from its neutralino sector . 
   in the basis @xmath97 , its neutralino mass matrix is given by @xcite @xmath98 where @xmath99 and @xmath100 are @xmath101 and @xmath102 gaugino masses respectively , @xmath103 , @xmath104 , @xmath105 and @xmath106 . 
   after diagonalizing this matrix one can get the mass eigenstate of the lightest neutralino @xmath107 with mass taking the following form @xcite @xmath108 this expression implies that @xmath107 must be lighter than about @xmath109 gev for @xmath110 ( from lower bound on chargnio mass ) and @xmath111 ( perturbativity bound ) . like the other supersymmetric models , @xmath107 as the lightest sparticle acts as the dark matter in the universe , but due to its singlino - dominated nature , it is difficult to annihilate sufficiently to get the correct density in the current universe . 
   so the relic density of @xmath107 plays a crucial way in selecting the model parameters . 
   for example , as shown in @xcite , for @xmath112 , there is no way to get the correct relic density , and for the other cases , @xmath107 mainly annihilates by exchanging @xmath1 boson for @xmath113 , or by exchanging a light cp - odd higgs boson @xmath0 with mass satisfying the relation @xmath114 for @xmath115 . for the annihilation , @xmath54 and @xmath25 
   are required to be less than 10 and @xmath116 respectively because through eq.([mass - exp ] ) a large @xmath87 or @xmath25 will suppress @xmath117 to make the annihilation more difficult . 
   the properties of the lightest cp - odd higgs boson @xmath0 , such as its mass and couplings , are also limited tightly since @xmath0 plays an important role in @xmath107 annihilation . 
   the phenomenology of the nmssm is also rather special , and this was discussed in detail in @xcite . 
   in the type - ii 2hdm , l2hdm , nmssm and nmssm , the rare @xmath1-decays @xmath118 ( @xmath22 ) , @xmath3 and @xmath4 may proceed by the feynman diagrams shown in fig.[fig1 ] , fig.[fig2 ] and fig.[fig3 ] respectively . for these diagrams , the intermediate state @xmath119 represents all possible cp - even higgs bosons in the corresponding model , i.e. @xmath41 and @xmath42 in type - ii 2hdm and l2hdm and @xmath41 , @xmath42 and @xmath120 in nmssm and nmssm . in order to take into account the possible resonance effects of @xmath119 in fig.[fig1](c ) for @xmath2 and fig.[fig3 ] ( a ) for @xmath11 , we have calculated all the decay modes of @xmath119 and properly included the width effect in its propagator . as to the decay @xmath121 , two points should be noted . 
   one is , unlike the decays @xmath6 and @xmath11 , this process proceeds only through loops mediated by quarks / leptons in the type - ii 2hdm and l2hdm , and additionally by sparticles in the nmssm and nmssm . 
   so in most cases its rate should be much smaller than the other two . 
   the other is due to cp - invariance , loops mediated by squarks / sleptons give no contribution to the decay@xcite . in actual calculation , this is reflected by the fact that the coupling coefficient of @xmath122 differs from that of @xmath123 by a minus sign ( see eq.([asqsq ] ) ) , and as a result , the squark - mediated contributions to @xmath121 are completely canceled out .    with regard to the rare decay @xmath11 , we have more explanations . in the lowest order , this decay proceeds by the diagram shown in fig.[fig3 ] ( a ) , and hence one may think that , as a rough estimate , it is enough to only consider the contributions from fig.[fig3](a ) . however , we note that in some cases of the type - ii 2hdm and l2hdm , due to the cancelation of the contributions from different @xmath119 in fig.[fig3 ] ( a ) and also due to the potentially largeness of @xmath124 couplings ( i.e. larger than the electroweak scale @xmath125 ) , the radiative correction from the higgs - mediated loops may dominate over the tree level contribution even when the tree level prediction of the rate , @xmath126 , exceeds @xmath20 . on the other hand 
   , we find the contribution from quark / lepton - mediated loops can be safely neglected if @xmath127 in the type - ii 2hdm and the l2hdm . in the nmssm and the nmssm , besides the corrections from the higgs- and quark / lepton - mediated loops , loops involving sparticles such as squarks , charginos and neutralinos can also contribute to the decay . 
   we numerically checked that the contributions from squarks and charginos can be safely neglected if @xmath127 . 
   we also calculated part of potentially large neutralino correction ( note that there are totally about @xmath128 diagrams for such correction ! ) and found they can be neglected too . since considering all the radiative corrections 
   will make our numerical calculation rather slow , we only include the most important correction , namely that from higgs - mediated loops , in presenting our results for the four models . 
   one can intuitively understand the relative smallness of the sparticle contribution to @xmath11 as follows . 
   first consider the squark contribution which is induced by the @xmath129 interaction ( @xmath130 denotes the squark in chirality state ) and the @xmath131 interaction through box diagrams . because the @xmath132 interaction conserves the chirality of the squarks while the @xmath133 interaction violates the chirality , to get non - zero contribution to @xmath11 from the squark loops , at least four chiral flippings are needed , with three of them provided by @xmath131 interaction and the rest provided by the left - right squark mixing . 
   this means that , if one calculates the amplitude in the chirality basis with the mass insertion method , the amplitude is suppressed by the mixing factor @xmath134 with @xmath135 being the off diagonal element in squark mass matrix . 
   next consider the chargino / neutralino contributions . since for a light @xmath0 , its doublet component , parameterized by @xmath84 in eq.([mixing ] ) , 
   is usually small , the couplings of @xmath0 with the sparticles will never be tremendously large@xcite . 
   so the chargino / neutralino contributions are not important too . in our calculation of the decays 
   , we work in the mass eigenstates of sparticles instead of in the chirality basis . 
   for the type - ii 2hdm and the l2hdm , we consider the following constraints @xcite :    * theoretical constraints on @xmath136 from perturbativity , unitarity and requirements that the scalar potential is finit at large field values and contains no flat directions @xcite , which imply that @xmath137 * the constraints from the lep search for neutral higgs bosons . we compute the signals from the higgs - strahlung production @xmath138 ( @xmath139 ) with @xmath140 @xcite and from the associated production @xmath141 with @xmath142 @xcite , and compare them with the corresponding lep data which have been inputted into our code . 
   we also consider the constraints from @xmath138 by looking for a peak of @xmath143 recoil mass distribution of @xmath1-boson @xcite and the constraint of @xmath144 mev when @xmath145 @xcite . 
   + these constraints limit the quantities such as @xmath146 \times br ( h_i \to \bar{b } b ) $ ] on the @xmath147 plane with the the subscript @xmath148 denoting the coupling coefficient of the @xmath149 interaction . 
   they also impose a model - dependent lower bound on @xmath150 , e.g. , @xmath151 for the type - ii 2hdm ( from our scan results ) , @xmath152 for the l2hdm@xcite , and @xmath153 for the nmssm @xcite . 
   these bounds are significantly lower than that of the sm , i.e. @xmath154 , partially because in new physics models , unconventional decay modes of @xmath155 such as @xmath156 are open up . as to the nmssm , another specific reason for allowing a significantly lighter cp - 
   even higgs boson is that the boson may be singlet - dominated in this model . 
   + with regard to the lightest cp - odd higgs boson @xmath0 , we checked that there is no lower bound on its mass so long as the @xmath157 interaction is weak or @xmath155 is sufficiently heavy . * the constraints from the lep search for a light higgs boson via the yukawa process @xmath158 with @xmath22 and @xmath61 denoting a scalar @xcite . 
   these constraints can limit the @xmath159 coupling versus @xmath160 in new physics models . * the constraints from the cleo - iii limit on @xmath161 and the latest babar limits on @xmath162 . 
   these constraints will put very tight constraints on the @xmath163 coupling for @xmath164 . in our analysis , we use the results of fig.8 in the second paper of @xcite to excluded the unfavored points . * the constraints from @xmath165 couplings . 
   since the higgs sector can give sizable higher order corrections to @xmath165 couplings , we calculate them to one loop level and require the corrected @xmath165 couplings to lie within the @xmath166 range of their fitted value . 
   the sm predictions for the couplings at @xmath1-pole are given by @xmath167 and @xmath168 @xcite , and the fitted values are given by @xmath169 and @xmath170 , respectively@xcite . 
   we adopt the formula in @xcite to the 2hdm in our calculation . * the constraints from @xmath171 leptonic decay . 
   we require the new physics correction to the branching ratio @xmath172 to be in the range of @xmath173 @xcite . 
   we use the formula in @xcite in our calculation . + about the constraints ( 5 ) and ( 6 ) , two points should be noted . 
   one is all higgs bosons are involved in the constraints by entering the self energy of @xmath171 lepton , the @xmath174 vertex correction or the @xmath175 vertex correction , and also the box diagrams for @xmath176@xcite . 
   since the yukawa couplings of the higgs bosons to @xmath171 lepton get enhanced by @xmath54 and so do the corrections , @xmath54 must be upper bounded for given spectrum of the higgs sector . generally speaking , the lighter @xmath0 is , the more tightly @xmath54 is limited@xcite . 
   the other point is in the type - ii 2hdm , @xmath177 , b - physics observables as well as @xmath178 decays discussed above can constraint the model in a tighter way than the constraints ( 5 ) and ( 6 ) since the yukawa couplings of @xmath171 lepton and @xmath179 quark are simultaneously enhanced by @xmath54 . 
   but for the l2hdm , because only the yukawa couplings of @xmath171 lepton get enhanced ( see eq.[yukawa ] ) , the constraints ( 5 ) and ( 6 ) are more important in limiting @xmath54 . 
   * indirect constraints from the precision electroweak observables such as @xmath180 , @xmath181 and @xmath182 , or their combinations @xmath183 @xcite . 
   we require @xmath184 to be compatible with the lep / sld data at @xmath185 confidence level@xcite . 
   we also require new physics prediction of @xmath186 is within the @xmath187 range of its experimental value . 
   the latest results for @xmath188 are @xmath189 ( measured value ) and @xmath190 ( sm prediction ) for @xmath191 gev @xcite . in our code , we adopt the formula for these observables presented in @xcite to the type - ii 2hdm and the l2hdm respectively . 
   + in calculating @xmath180 , @xmath181 and @xmath182 , we note that these observables get dominant contributions from the self energies of the gauge bosons @xmath1 , @xmath192 and @xmath193 . 
   since there is no @xmath194 coupling or @xmath195 coupling , @xmath0 must be associated with the other higgs bosons to contribute to the self energies . 
   so by the uv convergence of these quantities , one can infer that , for the case of a light @xmath0 and @xmath196 , these quantities depend on the spectrum of the higgs sector in a way like @xmath197 at leading order , which implies that a light @xmath0 can still survive the constraints from the precision electroweak observables given the splitting between @xmath150 and @xmath198 is moderate@xcite . * the constraints from b physics observables such as the branching ratios for @xmath199 , @xmath200 and @xmath201 , and the mass differences @xmath202 and @xmath203 . 
   we require their theoretical predications to agree with the corresponding experimental values at @xmath187 level . + in the type - ii 2hdm and the l2hdm , only the charged higgs boson contributes to these observables by loops , so one can expect that @xmath198 versus @xmath54 is to be limited . 
   combined analysis of the limits in the type - ii 2hdm has been done by the ckmfitter group , and the lower bound of @xmath204 as a function of @xmath87 was given in fig.11 of @xcite . 
   this analysis indicates that @xmath198 must be heavier than @xmath205 at @xmath185 c.l . 
   regardless the value of @xmath54 . in this work 
   , we use the results of fig.11 in @xcite to exclude the unfavored points . as for 
   the l2hdm , b physics actually can not put any constraints@xcite because in this model the couplings of the charged higgs boson to quarks are proportional to @xmath206 and in the case of large @xmath54 which we are interested in , they are suppressed . in our analysis of the l2hdm , we impose the lep bound on @xmath198 , i.e. @xmath207@xcite . * the constraints from the muon anomalous magnetic moment @xmath208 . 
   now both the theoretical prediction and the experimental measured value of @xmath208 have reached a remarkable precision , but a significant deviation still exists : @xmath209 @xcite . in the 2hdm , @xmath208 gets additional contributions from the one - loop diagrams induced by the higgs bosons and also from the two - loop barr - zee diagrams mediated by @xmath0 and @xmath155@xcite . 
   if the higgs bosons are much heavier than @xmath25 lepton mass , the contributions from the barr - zee diagrams are more important , and to efficiently alleviate the discrepancy of @xmath208 , one needs a light @xmath0 along with its enhanced couplings to @xmath25 lepton and also to heavy fermions such as bottom quark and @xmath171 lepton to push up the effects of the barr - zee diagram@xcite . 
   the cp - even higgs bosons are usually preferred to be heavy since their contributions to @xmath208 are negative . 
   + in the type - ii 2hdm , because @xmath54 is tightly constrained by the process @xmath210 at the lep@xcite and the @xmath178 decay@xcite , the barr - zee diagram contribution is insufficient to enhance @xmath208 to @xmath187 range around its measured value@xcite . 
   so in our analysis , we require the type - ii 2hdm to explain @xmath208 at @xmath211 level . 
   while for the l2hdm , @xmath54 is less constrained compared with the type - ii 2hdm , and the barr - zee diagram involving the @xmath171-loop is capable to push up greatly the theoretical prediction of @xmath208@xcite . 
   therefore , we require the l2hdm to explain the discrepancy at @xmath187 level . 
   + unlike the other constraints discussed above , the @xmath208 constraint will put a two - sided bound on @xmath54 since on the one hand , it needs a large @xmath54 to enhance the barr - zee contribution , but on the other hand , too large @xmath54 will result in an unacceptable large @xmath208 . * 
   since this paper concentrates on a light @xmath0 , the decay @xmath212 is open up with a possible large decay width . 
   we require the width of any higgs boson to be smaller than its mass to avoid a too fat higgs boson@xcite . 
   we checked that for the scenario characterized by @xmath213 , the coefficient of @xmath214 interaction is usually larger than the electroweak scale @xmath125 , and consequently a large decay width is resulted . 
   for the nmssm and nmssm , the above constraints become more complicated because in these models , not only more higgs bosons are involved in , but also sparticles enter the constraints . 
   so it is not easy to understand some of the constraints intuitively . 
   take the process @xmath199 as an example . in the supersymmetric models , besides the charged higgs contribution , chargino loops , gluino loops as well as neutralino loops also contribute to the process@xcite , and depending on the susy parameters , any of these contributions may become dominated over or be canceled by other contributions . as a result , although the charged higgs affects the process in the same way as that in the type - ii 2hdm , charged higgs as light as @xmath215 is still allowed even for @xmath216@xcite .    since among the constraints 
   , @xmath208 is rather peculiar in that it needs new physics to explain the discrepancy between @xmath217 and @xmath218 , we discuss more about its dependence on susy parameters . in the nmssm and the nmssm 
   , @xmath208 receives contributions from higgs loops and neutralino / chargino loops . for the higgs contribution , it is quite similar to that of the type - ii 2hdm except that more higgs bosons are involved in@xcite . for the neutralino / chargino contribution , in the light bino limit ( i.e. 
   @xmath219 ) , it can be approximated by@xcite @xmath220 for @xmath221 with @xmath222 being smuon mass . so combining the two contributions together 
   , one can learn that a light @xmath0 along with large @xmath54 and/or light smuon with moderate @xmath87 are favored to dilute the discrepancy .    because more parameters are involved in the constraints on the supersymmetric models , we consider following additional constraints to further limit their parameters :    * direct bounds on sparticle masses from the lep1 , the lep2 and the tevatron experiments @xcite . * the lep1 bound on invisible z decay @xmath223 ; the lep2 bound on neutralino production @xmath224 and @xmath225@xcite . 
   * dark matter constraints from the wmap relic density 0.0975 @xmath226 0.1213 @xcite . 
   note that among the above constraints , the constraint ( 2 ) on higgs sector and the constraint ( c ) on neutralino sector are very important . this is because in the supersymmetric models , the sm - like higgs is upper bounded by about @xmath227 at tree level and by about @xmath228 at loop level , and that the relic density restricts the lsp annihilation cross section in a certain narrow range .    in our analysis of the nmssm , we calculate the constraints ( 3 ) and ( 5 - 7 ) by ourselves and utilize the code nmssmtools @xcite to implement the rest constraints . 
   we also extend nmssmtools to the nmssm to implement the constraints . 
   for the extension , the most difficult thing we faced is how to adapt the code micromegas@xcite to the nmssm case . 
   we solve this problem by noting the following facts :    * as we mentioned before , the nmssm is actually same as the nmssm with the trilinear singlet term setting to zero . 
   so we can utilize the model file of the nmssm as the input of the micromegas and set @xmath229 . * since in the nmssm , the lsp is too light to annihilate into higgs pairs , there is no need to reconstruct the effective higgs potential to calculate precisely the annihilation channel @xmath230 with @xmath61 denoting any of higgs bosons@xcite . 
   we thank the authors of the nmssmtools for helpful discussion on this issue when we finish such extension@xcite . 
   with the above constraints , we perform four independent random scans over the parameter space of the type - ii 2hdm , the l2hdm , the nmssm and the nmssm respectively . 
   we vary the parameters in following ranges : @xmath231 for the type - ii 2hdm , @xmath232 for the l2hdm , @xmath233 for the nmssm , and @xmath234 for the nmssm .    in performing the scans , we note that for the nmssm and the nmssm , some constraints also rely on the gaugino masses and the soft breaking parameters in the squark sector and the slepton sector . 
   since these parameters affect little on the properties of @xmath0 , we fix them to reduce the number of free parameters in our scan . for the squark sector 
   , we adopt the @xmath235 scenario which assumes that the soft mass parameters for the third generation squarks are degenerate : @xmath236 800 gev , and that the trilinear couplings of the third generation squarks are also degenerate , @xmath237 with @xmath238 . for the slepton sector , we assume all the soft - breaking masses and trilinear parameters to be 100 gev . 
   this setting is necessary for the nmssm since this model is difficult to explain the muon anomalous moment at @xmath239 level for heavy sleptons@xcite . 
   finally , we assume the grand unification relation @xmath240 for the gaugino masses with @xmath241 being fine structure constants of the different gauge group .    with large number of random points in the scans , 
   we finally get about @xmath242 , @xmath243 , @xmath244 and @xmath242 samples for the type - ii 2hdm , the l2hdm , the nmssm and the nmssm respectively which survive the constraints and satisfy @xmath245 . analyzing the properties of the @xmath0 indicates that for most of the surviving points in the nmssm and the nmssm , its dominant component is the singlet field ( numerically speaking , @xmath246 ) so that its couplings to the sm fermions are suppressed@xcite . 
   our analysis also indicates that the main decay products of @xmath0 are @xmath247 for the l2hdm@xcite , @xmath248 ( dominant ) and @xmath247 ( subdominant ) for the type - ii 2hdm , the nmssm and the nmssm , and in some rare cases , neutralino pairs in the nmssm@xcite .    in fig.[fig4 ] 
   , we project the surviving samples on the @xmath249 plane . 
   this figure shows that the allowed range of @xmath54 is from @xmath250 to @xmath251 in the type - ii 2hdm , and from @xmath252 to @xmath253 in the l2hdm . 
   just as we introduced before , the lower bounds of @xmath254 come from the fact that we require the models to explain the muon anomalous moment , while the upper bound is due to we have imposed the constraint from the lep process @xmath255 , which have limited the upper reach of the @xmath256 coupling for light @xmath61 @xcite(for the dependence of @xmath256 coupling on @xmath54 , see sec . 
   this figure also indicates that for the nmssm and the nmssm , @xmath54 is upper bounded by @xmath257 . 
   for the nmssm , this is because large @xmath87 can suppress the dark matter mass to make its annihilation difficult ( see @xcite and also sec . 
   ii ) , but for the nmssm , this is because we choose a light slepton mass so that large @xmath54 can enhance @xmath208 too significantly to be experimentally unacceptable . 
   we checked that for the slepton mass as heavy as @xmath258 , @xmath259 is still allowed for the nmssm .    in fig.[fig5 ] and fig.[fig6 ] , we show the branching ratios of @xmath260 and @xmath261 respectively . 
   fig.[fig5 ] indicates , among the four models , the type - ii 2hdm predicts the largest ratio for @xmath260 with its value varying from @xmath262 to @xmath263 . 
   the underlying reason is in the type - ii 2hdm , the @xmath264 coupling is enhanced by @xmath54 ( see fig.[fig4 ] ) , while in the other three model , the coupling is suppressed either by @xmath265 or by the singlet component of the @xmath0 . 
   fig.[fig6 ] shows that the l2hdm predicts the largest rate for @xmath266 with its value reaching @xmath5 in optimum case , and for the other three models , the ratio of @xmath261 is at least about one order smaller than that of @xmath267 . 
   this feature can be easily understood from the @xmath268 coupling introduced in sect . 
   we emphasize that , if the nature prefers a light @xmath0 , @xmath260 and/or @xmath269 in the type - ii 2hdm and the l2hdm will be observable at the gigaz . then by the rates of the two decays 
   , one can determine whether the type - ii 2hdm or the l2hdm is the right theory . 
   on the other hand , if both decays are observed with small rates or fail to be observed , the singlet extensions of the mssm are favored .    in fig.[fig7 
   ] , we show the rate of @xmath3 as the function of @xmath270 . 
   this figure indicates that the branching ratio of @xmath121 can reach @xmath271 , @xmath272 , @xmath273 and @xmath274 for the optimal cases of the type - ii 2hdm , the l2hdm , the nmssm and the nmssm respectively , which implies that the decay @xmath121 will never be observable at the gigaz if the studied model is chosen by nature . 
   the reason for the smallness is , as we pointed out before , that the decay @xmath121 proceeds only at loop level .    comparing the optimum cases of the type - ii 2hdm , the nmssm and the nmssm shown in fig.5 - 7 
   , one may find that the relation @xmath275 holds for any of the decays . 
   this is because the decays are all induced by the yukawa couplings with similar structure for the models . in the supersymmetric models , 
   the large singlet component of the light @xmath0 is to suppress the yukawa couplings , and the @xmath0 in the nmssm has more singlet component than that in the nmssm . 
   next we consider the decay @xmath11 , which , unlike the above decays , depends on the higgs self interactions . in fig.[fig8 ] we plot its rate as a function of @xmath270 and this figure indicates that the @xmath276 may be the largest among the ratios of the exotic @xmath1 decays , reaching @xmath277 in the optimum cases of the type - ii 2hdm , the l2hdm and the nmssm . 
   the underlying reason is , in some cases , the intermediate state @xmath119 in fig.[fig3 ] ( a ) may be on - shell . 
   in fact , we find this is one of the main differences between the nmssm and the nmssm , that is , in the nmssm , @xmath119 in fig.[fig3 ] ( a ) may be on - shell ( corresponds to the points with large @xmath278 ) while in the nmssm , this seems impossible . 
   so we conclude that the decay @xmath11 may serve as an alternative channel to test new physics models , especially it may be used to distinguish the nmssm from the nmssm if the supersymmetry is found at the lhc and the @xmath11 is observed at the gigaz with large rate . 
   before we end our discussion , we note that in the nmssm , the higgs boson @xmath0 may be lighter than @xmath279 without conflicting with low energy data from @xmath178 decays and the other observables ( see fig.[fig4]-[fig8 ] ) . in this case , @xmath0 is axion - like as pointed out in @xcite . 
   we checked that , among the rare @xmath1 decays discussed in this paper , the largest branching ratio comes from @xmath280 which can reach @xmath281 . since in this case , the decay product of @xmath0 is highly collinear muon pair , detecting the decay @xmath280 may need some knowledge about detectors , which is beyond our discussion . 
   in this paper , we studied the rare @xmath1-decays @xmath2 ( @xmath7 ) , @xmath282 and @xmath4 in the type - ii 2hdm , lepton - specific 2hdm , nmssm and nmssm , which predict a light cp - odd higgs boson @xmath0 . in the parameter space allowed by current experiments , the branching ratio can be as large as @xmath5 for @xmath118 , @xmath8 for @xmath3 and @xmath9 for @xmath4 , which implies that the decays @xmath2 and @xmath283 may be accessible at the gigaz option . 
   since different models predict different size of branching ratios , these decays can be used to distinguish different model through the measurement of these rare decays . 
   this work was supported in part by hastit under grant no . 
   2009hastit004 , by the national natural science foundation of china ( nnsfc ) under grant nos . 10821504 , 10725526 , 10635030 , 10775039 , 11075045 and by the project of knowledge innovation program ( pkip ) of chinese academy of sciences under grant no . 
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summary: various new physics models predict a light cp - odd higgs boson ( labeled as @xmath0 ) and open up new decay modes for @xmath1-boson , such as @xmath2 , @xmath3 and @xmath4 , which could be explored at the gigaz option of the ilc . in this work 
   we investigate these rare decays in several new physics models , namely the type - ii two higgs doublet model ( type - ii 2hdm ) , the lepton - specific two higgs doublet model ( l2hdm ) , the nearly minimal supersymetric standard model ( nmssm ) and the next - to - minimal supersymmetric standard model ( nmssm ) . 
   we find that in the parameter space allowed by current experiments , the branching ratios can reach @xmath5 for @xmath6 ( @xmath7 ) , @xmath8 for @xmath3 and @xmath9 for @xmath4 , which implies that the decays @xmath10 and @xmath11 may be accessible at the gigaz option . 
   moreover , since different models predict different patterns of the branching ratios , the measurement of these rare decays at the gigaz may be utilized to distinguish the models .