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the rare @xmath2 decays are being measured at @xmath2 factories in kek and slac . the generalized factorization approach has been applied to the theoretical treatment of non - leptonic @xmath2 decays @xcite . it is a great success in explaining many decay branching ratios @xcite . the factorization approach ( f.a . ) is a rather simple method . some efforts have been made to improve their theoretical application @xcite and to understand the reason why the f.a . has gone well @xcite . one of these method is the perturbative qcd approach ( pqcd ) , where we can calculate the annihilation diagrams as well as the factorizable and nonfactorizable diagrams . the rare decays @xmath5 are pure annihilation type decays . in the usual f.a . , this decay picture is described as @xmath2 meson annihilating into vacuum and the @xmath6 and @xmath7 mesons produced from vacuum then afterwards . to calculate this decay in the f.a . , one needs the @xmath8 form factor at very large time like momentum transfer @xmath9 . however the form factor at such a large momentum transfer is not known in f.a . this makes the f.a . calculation of these decays unreliable . the annihilation amplitude is a phenomenological parameter in qcd factorization approach ( qcdf)@xcite , and the qcdf calculation of these decays is also unreliable . in this paper , we will try to use the pqcd approach , where the annihilation amplitude is calculable , to evaluate the @xmath5 decays . by comparing the predictions with the experimental data , we can test the pqcd evaluation of the annihilation amplitude . a @xmath10 boson exchange causes @xmath11 or @xmath12 , and the @xmath13 quarks included in @xmath14 are produced from a gluon . this gluon attaches to any one of the quarks participating in the @xmath10 boson exchange . this is shown in figure [ fig : need1gluon ] . in the rest frame of @xmath2 meson , both @xmath15 and @xmath16 quarks included in @xmath14 have @xmath17 momenta , and the gluon producing them also has @xmath18 . this is a hard gluon . one can perturbatively treat the process where the four quark operator exchanges a hard gluon with @xmath19 quark pair . it is just the picture of pqcd approach .    in the next section , we explain the framework of pqcd briefly . in section [ sc : formula ] , we give the analytic formulas for the decay amplitude of @xmath20 decays . in section [ sc : neval ] , we show the predicted branching ratio from the analytic formulas and discuss the theoretical errors . it is found that the prediction are is in good agreement with the data , and pqcd approach correctly gives the annihilation amplitude in consequence . finally , we conclude this study in section [ sc : concl ] . pqcd approach has been developed and applied in the non - leptonic @xmath2 meson decays @xcite for some time . in this approach , the decay amplitude is separated into soft(@xmath21 ) , hard(@xmath22 ) , and harder(@xmath23 ) dynamics characterized by different scales . it is conceptually written as the convolution , @xmath24 , \label{eq : convolution1}\ ] ] where @xmath25 s are momenta of light quarks included in each mesons , and @xmath26 denotes the trace over dirac and color indices . @xmath27 is wilson coefficient which results from the radiative corrections at short distance . in the above convolution , @xmath27 includes the harder dynamics at larger scale than @xmath28 scale and describes the evolution of local @xmath29-fermi operators from @xmath30 , @xmath10 boson mass , down to @xmath31 scale , where @xmath32 . @xmath22 describes the four quark operator and the spectator quark connected by a hard gluon whose @xmath33 is on the order of @xmath34 , and includes the @xmath35 hard dynamics . therefore , this hard part @xmath22 can be perturbatively calculated . @xmath36 is the wave function which describes hadronization of the quark and anti - quark to the meson @xmath37 . while the @xmath22 depends on the processes considered , @xmath36 is independent of the specific processes . determining @xmath36 in some other decays , we can make quantitative predictions here . we consider the @xmath2 meson at rest for simplicity . it is convenient to use light - cone coordinate @xmath38 to describe the meson s momenta , where @xmath39 and @xmath40 . using these coordinates we can take the @xmath2 , @xmath6 , and @xmath7 mesons momenta as @xmath41 , @xmath42 , and @xmath43 , respectively , where @xmath44 and we neglect the @xmath7 meson s mass @xmath45 . putting the light ( anti-)quark momenta in @xmath2 , @xmath6 and @xmath7 mesons as @xmath46 , @xmath47 , and @xmath48 , respectively , we can choose @xmath49 , @xmath50 , and @xmath51 . then , integration over @xmath52 , @xmath53 , and @xmath54 in eq.([eq : convolution1 ] ) leads to @xmath55 , \label{eq : convolution2}\end{gathered}\ ] ] where @xmath56 is the conjugate space coordinate of @xmath57 , and @xmath58 is the largest energy scale in @xmath22 , as the function in terms of @xmath59 and @xmath56 . the large logarithms ( @xmath60 ) coming from qcd radiative corrections to four quark operators are included in the wilson coefficients @xmath27 . the large double logarithms ( @xmath61 ) on the longitudinal direction are summed by the threshold resummation@xcite , and they lead to @xmath62 which smears the end - point singularities on @xmath59 . the last term , @xmath63 , contains two kinds of logarithms . one of the large logarithms is due to the renormalization of ultra - violet divergence @xmath64 , the other is double logarithm @xmath65 from the overlap of collinear and soft gluon corrections . this sudakov form factor suppresses the soft dynamics effectively @xcite . thus it makes perturbative calculation of the hard part @xmath22 applicable at intermediate scale , i.e. , @xmath28 scale . we calculate the @xmath22 for @xmath20 decays in the first order in @xmath66 expansion and give the convoluted amplitudes in next section . in order to calculate analytic formulas of the decay amplitude , we use the wave functions @xmath67 decomposed in terms of spin structure . in general , @xmath67 having dirac indices @xmath68 are decomposed into 16 independent components , @xmath69 , @xmath70 , @xmath71 , @xmath72 , @xmath73 . if the considered meson @xmath37 is @xmath2 or @xmath6 meson , to be pseudo - scalar and heavy meson , the structure @xmath72 and @xmath73 components remain as leading contributions . then , @xmath67 is written by @xmath74 where @xmath75 is color s degree of freedom , @xmath76 is the corresponding meson s momentum , and @xmath77 are lorentz scalar wave functions . as heavy quark effective theory leads to @xmath78 , then @xmath2 meson s wave function can be expressed by @xmath79   \phi_b(x , b).\ ] ] according to ref . @xcite , a pseudo - scalar meson moving fast is parameterized by lorentz scalar wave functions , @xmath80 , @xmath81 , and @xmath82 as @xmath83 where @xmath84 . we ignore the difference between @xmath85 quark s mass and @xmath6 meson s mass in the perturbative calculation . this means , putting @xmath86 , the terms proportional to @xmath87 are neglected . in this approximation , the contributions of eq.([pt ] ) are of higher power than those of eqs.([pv ] , [ ps ] ) by @xmath88 because of the factor @xmath89 in eq.([pt ] ) , and we neglect the @xmath90 component in the @xmath6 meson s wave function . in addition , the eq.([pv ] ) , eq.([ps ] ) , and the relations @xmath91 with equations of motion lead to @xmath92 therefore the @xmath6 meson s wave function can be expressed by one lorentz scalar wave function , @xmath93   \phi_{d_s}(x , b),\ ] ] where @xmath94 is defined by @xmath95 the wave function @xmath96 for @xmath97 meson is normalized by its decay constant @xmath98 @xmath99    in contrast to the @xmath2 and @xmath100 mesons , for the @xmath7 meson , being light meson , the @xmath101 component remains because the factor corresponding to @xmath89 in eq.([pt ] ) is @xmath102 . then , @xmath7 meson s wave function is parameterized by lorentz scalar wave functions @xmath103 as @xmath104_{\alpha\beta}\end{aligned}\ ] ] where @xmath105 , @xmath106 , and @xmath107 . in the numerical analysis we will use @xmath103 which were calculated from qcd sum rule @xcite . they will be shown in section [ sc : neval ] . we first consider the neutral @xmath109 decay @xmath108 . the effective hamiltonian at the scale lower than @xmath110 related to this decay is given as @xcite @xmath111 , \\ o_1 = ( \bar{b}d)_{v - a } ( \bar{u}c)_{v - a } , \quad   o_2 = ( \bar{b}c)_{v - a } ( \bar{u}d)_{v - a},\end{gathered}\ ] ] where @xmath112 are wilson coefficients at renormalization scale @xmath113 , and summation in @xmath114 color s index @xmath115 and chiral projection , @xmath116 , are abbreviated to @xmath117 . the lowest order diagrams contributing to @xmath108 are drawn in fig.[fig : diagrams1 ] according to this effective hamiltonian . as stated above , @xmath5 decays only have annihilation diagrams . we get the following analytic formulas by calculating the hard part @xmath22 at first order in @xmath66 . together with the meson wave functions , the amplitude for the factorizable annihilation diagram in fig.[fig : diagrams1](a ) and ( b ) results in @xmath118 , @xmath119 , \label{eq : fa}\end{gathered}\ ] ] where @xmath120 is the group factor of @xmath114 gauge group , and @xmath121 , and the functions @xmath122 , @xmath123 , @xmath124 are given in the appendix . the explicit form for the wave functions , @xmath96 , is given in the next section . the amplitude for the nonfactorizable annihilation diagram in fig.[fig : diagrams1](c ) and ( d ) results in @xmath125 , \label{eq : ma1}\end{gathered}\ ] ] where @xmath126 dependence in the numerators of the hard part are neglected by the assumption @xmath127 . because such a @xmath126 behaves as the cut off , the resultant branching ratio is smaller than it given in ref . @xcite where the whole @xmath126 in the hard part are neglected . ] . the total decay amplitude for @xmath108 decay is given as @xmath128 where the overall factor is included in the decay width with the kinematics factor . the decay width is expressed as @xmath129 the decay width for cp conjugated mode , @xmath130 , is the same value as @xmath108 , just replacing @xmath131 with @xmath132 . since there is only one kind of ckm phase involved in the decay , there is no cp violation in the standard model . the effective hamiltonian related to @xmath1 decay is given as @xmath133 , \\ o_1 = ( \bar{b}d)_{v - a } ( \bar{c}u)_{v - a } , \quad   o_2 = ( \bar{b}u)_{v - a } ( \bar{c}d)_{v - a}.\end{gathered}\ ] ] the amplitude for the factorizable annihilation diagram results in @xmath134 . the amplitude for the nonfactorizable annihilation diagram results in @xmath135 . \label{eq : ma2}\end{gathered}\ ] ] thus , the total decay amplitude @xmath136 and decay width @xmath137 for @xmath1 decay is given as @xmath138 the decay width for cp conjugated mode , @xmath139 , is the same value as @xmath1 . similar to the @xmath109 decay , there is also no cp violation in this decay within standard model . in this section we show numerical results obtained from the previous formulas . at the beginning , we give the branching ratios predicted from the same parameters and wave functions that are adopted in the other works . secondly , we show the theoretical errors due to uncertainty of some parameters .    for the @xmath2 meson s wave function , there is a sharp peak at the small @xmath140 region , we use @xmath141,\ ] ] which is adopted in ref . this choice of @xmath2 meson s wave function is almost a best fit from the @xmath142 , @xmath143 decays . for the @xmath6 meson s wave function , we assume the same form as @xmath100 meson s one except for the normalization @xcite @xmath144 since @xmath85 quark is much heavier than @xmath15 quark , this function is peaked at @xmath85 quark side , i.e. small @xmath140 region , too . the wave functions @xmath103 of the @xmath7 meson are expanded by gegenbauer polynomials , @xmath145 where @xmath146 . the coefficients @xmath147 calculated from qcd sum rule are given in ref.@xcite , and their values are @xmath148 for @xmath149 . in addition , we use the following input parameters : @xmath150 with these values and eq.([eq : normalization ] ) we get the normalization factor @xmath151 gev . we show the decay amplitudes calculated with the above parameters at table [ tb : amplitudes ] . for the neutral decay @xmath108 , the dominant contribution is the nonfactorizable annihilation diagrams , where the contribution @xmath152 is proportional to the wilson coefficient @xmath153 , which is of order one . the factorizable annihilation diagram contribution is proportional to @xmath154 , which is one order magnitude smaller . for the charged decay @xmath155 , it is the inverse situation . the wilson coefficient in @xmath156 is @xmath157 , which is smaller than the one in @xmath158 , @xmath159 . both amplitudes @xmath160 and @xmath136 are at the same order magnitude here . the propagators of inner quark and gluon in figure 2 and 3 are usually proportional to @xmath161 . one may suspect that these amplitudes are enhanced by the endpoint singularity around @xmath162 . this can be explicitly found in eq.([eq : propagator1 ] , [ eq : propagator2 ] ) , where the bessel function @xmath163 diverges at @xmath162 or @xmath164 . however this is not the case in our calculation . first we introduce the transverse momentum of quark , such that the propagators become @xmath165 . secondly , the sudakov form factor @xmath166 $ ] suppresses the region of small @xmath167 . therefore there is no singularity in our calculation . the dominant contribution is not from the endpoint of the wave function . as a proof , in our numerical calculations , for example , an expectation value of @xmath66 in the integration for @xmath152 results in @xmath168 , which gives the dominant contribution to @xmath108 decay . therefore , the perturbative calculations are self - consistent . .amplitudes(@xmath169 gev ) with parameters eqs . ( [ eq : parm1]-[eq : shapewv ] ) . [ cols="^,>,^ , > " , ]     it is found that uncertainty of the predictions on pqcd is mainly due to @xmath170 and @xmath171 . below we show the limits of the branching ratio within the suitable ranges on @xmath170 and @xmath171 . the appropriate extent of @xmath170 can be found in ref.@xcite , @xmath172 @xmath173 in the wave functions @xmath174 are given as functions with respect to @xmath170 , @xmath175 , and some input parameters , @xmath176 , @xmath177 in ref.@xcite . within eq . ( [ eq : m0k ] ) , the branching ratios normalized by the decay constants and the ckm matrix elements result in @xmath178 from the @xmath179 transition form factor @xmath180 , the appropriate extent of @xmath171 can be obtained . @xmath180 calculated from pqcd at @xmath181 gev in the region @xmath182 is consistent with @xmath180 by qcd sum rules given in ref . the branching ratios calculated at the region of eqs . ( [ eq : wb ] ) are found within @xmath183 in the factorizable contribution , the @xmath2 meson s wave function is integrated and normalized by the decay constant @xmath184 . thus , in the factorizable dominant decay , @xmath1 , its branching ratio is insensitive to the change of @xmath171 . in two - body hadronic @xmath2 meson decays , the final state mesons are moving very fast , since each of them carry more than 2 gev energy . there is not enough time for them to exchange soft gluons . the soft final state interaction is not important in the two - body @xmath2 decays . this is consistent with the argument based on color - transparency @xcite . we thus neglect the soft final state interaction in the pqcd approach . the pqcd with sudakov form factor is a self - consistent approach to describe the two - body @xmath2 meson decays . although the annihilation diagrams are suppressed comparing to other spectator diagrams , but their contributions are not negligible in pqcd approach @xcite .    in this paper , we calculate the @xmath108 and @xmath1 decays , which occur purely via annihilation type diagrams . the branching ratios are still sizable . the @xmath108 decay has been observed in the @xmath2 factories  @xcite . this is the first channel measured in @xmath2 decays via annihilation type diagram . the fact that the predicted branching ratio is in good agreement with the data means that pqcd approach gives the annihilation amplitude correctly , the soft final state interaction is probably small in @xmath2 decays , and it is one of the evidences to justify pqcd approach . we thank our pqcd group members : c .- h . chen , y .- y . keum , e. kou , t. kurimoto , h .- li , t. morozumi , s. mishima , m. nagashima , a.i . sanda , n. sinha , r. sinha , m.z . yang and t. yoshikawa for fruitful discussions . this work is partly supported by national science foundation of china under grant ( no . 90103013 and 10135060 ) . the research of k.u . is supported by the japan society for promotion of science under the predoctoral research program . this work was supported by a grand - in aid for special project research ( physics of cp violation ) and by a grand - in - aid for science research from the ministry of education , culture , sports , science and technology , japan ( no . 12004276 ) . the definitions of some functions used in the text are presented in this appendix . in the numerical analysis we use one loop expression for strong coupling constant , @xmath185 where @xmath186 and @xmath187 is number of active flavor at appropriate scale . @xmath188 is qcd scale , which we use as @xmath189 mev at @xmath190 . we also use leading logarithms expressions for wilson coefficients @xmath191 presented in ref.@xcite . then , we put @xmath192 gev , @xmath193 gev , and @xmath194 gev . the function @xmath195 , @xmath196 , and @xmath197 including wilson coefficients are defined as @xmath198 where @xmath199 and @xmath200 , @xmath201 , and @xmath202 result from summing both double logarithms caused by soft gluon corrections and single ones due to the renormalization of ultra - violet divergence . the above @xmath203 are defined as @xmath204 where @xmath205 , so - called sudakov factor , is given as @xcite @xmath206 ,   \label{eq : sudakovexpress}\end{aligned}\ ] ] @xmath207 is euler constant , and @xmath208 is the quark anomalous dimension . the functions @xmath124 , @xmath209 , and @xmath210 in the decay amplitudes consist of two parts : one is the jet function @xmath62 derived by the threshold resummation@xcite , the other is the propagator of virtual quark and gluon . they are defined by @xmath211 where @xmath212 , and @xmath213s are defined by @xmath214 we adopt the parametrization for @xmath215 of the factorizable contributions , @xmath216^c,\quad c = 0.3,\ ] ] which is proposed in ref .  @xcite . in the nonfactorizable annihilation contributions , @xmath215 gives a very small numerical effect to the amplitude@xcite . therefore , we drop @xmath215 in @xmath209 and @xmath210 . the hard scale @xmath58 s in the amplitudes are taken as the largest energy scale in the @xmath22 to kill the large logarithmic radiative corrections : @xmath217      m. wirbel , b. stech , m. bauer , z. phys . c29 , 637 ( 1985 ) ; 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the rare decays @xmath0 and @xmath1 can occur only via annihilation type diagrams in the standard model . 
 we calculate these decays in perturbative qcd approach . 
 we found that the calculated branching ratio of @xmath0 agreed with the data which had been observed in the kek and slac @xmath2 factories . while the decay @xmath1 has a very small branching ratio at @xmath3 , due to the suppression from ckm matrix elements @xmath4 . 
 ( 0,0)(-320,-320 ) ( 0,32)*hep - ph/0210206 * ( 0,15)bihep - th-2002 - 47 ( 0,0)dpnu-02 - 31