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the hadronic b decays are important for testing the standard model ( sm ) , and also for uncovering the signal of new physics . understanding non - leptonic @xmath1 meson decays is crucial for the ckm matrix elements measurements and cp violation detection . especially the direct cp violation is expected to be measured in hadronic b decays . recently , both belle and babar claimed to find direct cp asymmetry in @xmath2 and @xmath3 decays @xcite . it is the start of direct cp measurement in b physics .    in the theoretical side , there are much more complication in direct cp asymmetry predictions . the direct cp violation in hadronic decays requires at least two decay amplitudes with different weak phase and strong phase . in the standard model , the weak phase comes from the so called ckm matrix , which is well defined . although the weak phase from sm is clean , the strong phase requires hadronic matrix element calculation , which is usually model dependent . the simplest case is two - body hadronic @xmath1 meson decays , for which bauer , stech and wirbel ( bsw ) proposed the naive factorization assumption ( fa ) in their pioneering work @xcite . considerable progress , including generalized fa @xcite and qcd - improved fa ( qcdf ) @xcite , has been made . on the other hand , technique to analyze hard exclusive hadronic scattering was developed by brodsky and lepage @xcite based on collinear factorization theorem in perturbative qcd ( pqcd ) . a modified framework based on @xmath0 factorization theorem was then given in @xcite , and extended to exclusive @xmath1 meson decays in @xcite . the predictions of branching ratios agree well with experiments in most cases , thus it is difficult to tell from experiments that which method is better than others . however , the strong phase , which is important for the cp violation prediction , is quite sensitive to various approaches . the mechanism of this strong phase is quite different for various method , and give quite different results . the recent experimental results @xcite can make a test for the validity of these approaches .    in this paper , we will first introduce the factorization approach in next section . the qcd factorization and improved pqcd approach based on @xmath0 factorization are then introduced in section [ sec3 ] and section [ sec4 ] , respectively . in section [ sec5 ] , we upgrade the numerical results of these approaches using newest parameters . we compare the three major approaches to show the difference of direct cp asymmetry predicted by them . finally the summary is presented . the calculation of non - leptonic decays involves the short - distance lagrangian and the calculation of hadronic matrix elements which are model dependent . the short - distance qcd corrected lagrangian is calculated to next - to - leading order @xcite . in 1987 , bauer , stech and wirbel first calculate the hadronic @xmath1 and @xmath4 decays using the naive factorization approach @xcite . in their factorization method , the hadronic matrix element is expressed as a product of two factors @xmath5 . the first factor is proportional to the @xmath6 form factor , while the second one is proportional to the decay constant of @xmath7 meson . all the perturbative channel dependent part is described by the wilson coefficients of four quark operators . these effective four - quark operators are induced by the weak interaction in the quark level , which gives the short distance contribution to the non - leptonic decays of b mesons . the effective hamiltonian for the charmless non - leptonic b decays is @xcite @xmath8   , \end{aligned}\ ] ] where @xmath9 and @xmath10 denotes the ckm factors . the operators @xmath11 are tree level current operators . the operators @xmath12 are qcd penguin operators . @xmath13 arise from electroweak penguin diagrams , which are suppressed by @xmath14 . only @xmath15 has a sizable value whose major contribution arises from the @xmath16 penguin . the @xmath17 s are the wilson coefficients of four quark operators with qcd corrections . although this is a very simple method , later experiments show that many of the decay branching ratios explained well by the fa @xcite , especially for the color enhanced decays class i , iii and class iv @xcite . when charmless @xmath1 meson decays are considered , efforts have been made to generalize the fa approach @xcite . to explain the non - factorizable dominated class ii and class v decays , phenomenological parameter @xmath18 is introduced . most of the branching ratios of hadronic @xmath1 decays agree well with experiments by @xmath19 @xcite . an effort is also made to predict the cp violation parameters in different hadronic @xmath1 decays @xcite . since direct cp violation requires a strong phase difference between decay amplitudes in addition to a weak phase difference , the precision of strong phase calculation is essential . in naive fa , the hadronization is described by the form factor only , therefore , no strong phase is given . ( 130,160)(0,20 ) ( 60,72)(0,72 ) ( 60,72)(60,145 ) ( 90,72)(30,72,71 )    ( 120,72)(170,72)54 ( 60,72)5.5 ( 170,72)(230,72 ) ( 170,72)(170,145 )    ( 65,135)@xmath20 ( 96,105)@xmath21 ( 5,78 ) @xmath22    in generalized fa @xcite , the next - to - leading order wilson coefficients contain strong phases generated by the charm quark loop , where charm quark can be on shell . the diagram shown in fig.[fig1 ] is also called bss mechanism @xcite . the size of this kind of strong phase is sensitive to the momentum of the gluon connecting to the charm quark loop . however , in fa , this momentum is not well defined , since all hadronic dynamics are defined only by form factors . in ref . @xcite , the authors use @xmath23gev@xmath24 to give the cp asymmetry parameters of many channels . in 1999 , beneke , buchalla , neubert , and sachrajda ( bbns ) proposed a formalism for two - body charmless @xmath1 meson decays @xcite . in this approach , they expand the hadronic matrix element by the heavy b quark mass and @xmath25 @xmath26.\label{bbnsf}\ ] ] here @xmath27 is a local operator in the effective hamiltonian and @xmath28 are bilinear quark currents . by neglecting the power corrections of @xmath29 , one need only calculate the order @xmath25 corrections including the vertex corrections for the four quark operators and the non - factorizable diagrams . these diagrams are shown in fig.[figbbns ] . the first 6 diagrams have already been included in the generalized fa approach as next - to - leading order qcd corrections to local four quark operators . what new are the last two non - factorizable diagrams , which has a hard gluon line connecting the four quark operator and the spectator quark . they claimed that factorizable contributions , for example , the form factor @xmath30 in the @xmath31 decays , are dominated by long distance contributions . hence , it is treated in the same way as fa , and expressed as products of wilson coefficients and form factor @xmath30 . the non - factorizable contributions calculated perturbatively , are written as the convolutions of hard amplitudes with three @xmath32 meson wave functions . annihilation diagrams are neglected as in fa . values of form factors at maximal recoil @xmath33 and non - perturbative meson wave functions are all treated as input parameters . it is easy to see from eq.([bbnsf ] ) , that this equation is only applicable for those color enhanced decay modes , where the factorizable contribution dominates the final results . while for the color suppressed modes , where the non - factorizable contributions are not small , the expansion of eq.([bbnsf ] ) is not right , since the large non - factorizable contribution is grouped into the next - to leading order term of eq.([bbnsf ] ) . the numerical results show that the theory and experiments agree well for those class i and iv decays , which are color enhanced and dominated by the factorizable contribution . this also agrees with the fa result , since the dominant part in eq.([bbnsf ] ) is the same as the fa . the success of qcd factorization is that one can calculate the sub - leading @xmath34 non - factorizable contribution ( second term in eq.([bbnsf ] ) ) using perturbative qcd . while in the fa , one has to input a free parameter @xmath18 to accommodate the non - factorizable contribution .    in the qcd factorization calculations , people found that there exist endpoint divergence in the annihilation diagram calculations @xcite . logarithm divergence occurred at twist 2 contribution , and linear divergence exists in twist 3 contribution . if not symmetric wave function , like @xmath35 meson , there is also soft divergence in the non - factorizable diagrams . it is very difficult to treat these singularity in the bbns approach . a cut - off is introduced to regulate the divergence , thus makes the qcd factorization approach prediction parameter dependent especially in the annihilation diagrams .    as for the strong phase in bbns , like in fa , it mainly comes from the bss mechanism . here the momentum of the inner gluon is well defined , since wave functions are introduced . however , it predicts too small strong phase , because of the small gluon momentum . hence , small direct cp asymmetry is predicted . there is also another source of strong phase from the annihilation diagrams , but strongly depends on the cut - off parameter . the strong phase in qcd factorization can be large due to this cut - off .    in a word , the qcd factorization approach is at least one step forward from naive factorization approach . it gives systematic prediction of sub - leading non - factorizable contribution for the class i and class iv decays , which are dominated by the factorizable contribution . problem remained is the endpoint singularity in higher order calculations , but may be solved with sudakov form factors like pqcd approach . in this section , we will introduce the idea of pqcd approach . the three scale pqcd factorization theorem has been developed for non - leptonic heavy meson decays @xcite , based on the formalism by brodsky and lepage @xcite , and botts and sterman @xcite . in the non - leptonic two body b decays , the @xmath1 meson is heavy , sitting at rest . it decays into two light mesons with large momenta . there must be a hard gluon to kick the light spectator quark ( with small momentum ) in the b meson to form a fast moving light meson . so the dominant diagram in this theoretical picture is that one hard gluon from the spectator quark connecting with the other quarks in the four quark operator of the weak interaction . unlike the usual factorization approach , the hard part of the pqcd approach consists of six quarks rather than four . we thus call it six - quark operators or six - quark effective theory . there are also infrared ( soft and collinear ) gluon exchanges between quarks . summing over those leading soft contributions gives a sudakov form factor , which suppresses the soft contribution . therefore , it makes the pqcd reliable in calculating the non - leptonic decays . with the sudakov resummation , we can include the leading double logarithms for all loop diagrams , in association with the infrared contribution . there are three different scales in the b meson non - leptonic decays @xmath36 , @xmath37 and @xmath38 . the first scale describe the intrinsic electroweak decay of @xmath1 meson , through charged current . the second scale @xmath37 denote the scale of energy release in the decay . since the @xmath39 quark decay scale @xmath37 is much smaller than the electroweak scale @xmath40 , the qcd corrections to the four quark operators are non - negligible , which are usually summed by the renormalization group equation . this has already been done to the leading logarithm and next - to - leading order for years @xcite . the third scale @xmath38 involved in the @xmath1 meson exclusive decays is usually called the factorization scale , with @xmath39 the conjugate variable of parton transverse momenta . the dynamics below @xmath38 scale is regarded as being completely non - perturbative , and can be parameterized into meson wave functions . the meson wave functions are not calculable in pqcd . but they are universal , channel independent . we can determine them from experiments , and they are constrained by qcd sum rules and lattice qcd calculations . above the scale @xmath38 , the physics is channel dependent . we can use perturbation theory to calculate channel by channel . with all the large logarithms resummed , the remaining finite contributions are absorbed into a perturbative b quark decay sub - amplitude @xmath41 . therefore the three scale factorization formula is given by the typical expression , @xmath42 , \label{eq : factorization_formula}\ ] ] where @xmath43 are the corresponding wilson coefficients , @xmath44 are the meson wave functions and the variable @xmath45 denotes the largest mass scale of hard process @xmath46 , that is , six - quark effective theory . the quark anomalous dimension @xmath47 describes the evolution from scale @xmath45 to @xmath38 . since logarithm corrections have been summed by renormalization group equations , the above factorization formula does not depend on the renormalization scale @xmath48 explicitly . as shown above , in the pqcd approach , we keep the @xmath0 dependence of the wave function . in fact , the approximation of neglecting the transverse momentum can only be done at the non - endpoint region , since @xmath49 is qualified at that region . at the endpoint , @xmath50 , @xmath0 is not small any longer , neglecting @xmath0 is a very bad approximation . by , keeping the @xmath0 dependence , there is no endpoint divergence as occurred in the qcd factorization approach , while the numerical result does not change at other region . furthermore , the sudakov form factors suppress the endpoint region of the wave functions . the main input parameters in pqcd are the meson wave functions . it is not a surprise that the final results are sensitive to the meson wave functions . fortunately , there are many channels involve the same meson , and the meson wave functions should be process independent . in all the calculations of pqcd approach , we follow the rule , and we find that they can explain most of the measured branching ratios of b decays . for example : @xmath51 decays , @xmath52 , @xmath53 decays @xcite , @xmath54 decays @xcite , @xmath55 decays @xcite , the form factor calculations of @xmath56 , @xmath57 @xcite , @xmath58 decays @xcite , @xmath59 decays @xcite etc . we emphasize that non - factorizable ( fig.[fig2](c)(d ) ) and annihilation diagrams ( fig.[fig2](e - h ) ) are indeed sub - leading in the pqcd formalism as @xmath60 . this can be easily observed from the hard functions in appendices of ref.@xcite . when @xmath61 increases , the @xmath1 meson wave function enhances contributions to factorizable diagrams . however , annihilation amplitudes , being independent of b meson wave function , are relatively insensitive to the variation of @xmath61 . hence , factorizable contributions become dominant and annihilation contributions are sub - leading in the @xmath62 limit @xcite . although the non - factorizable and annihilation diagrams are sub - leading for the branching ratio in color enhanced decays , they provide the main source of strong phase , by inner quark or gluon on mass shell . the bss mechanism should also be present in the pqcd approach . however this mechanism of strong phase is small , since it is at next - to - leading order @xmath63 corrections . in numerical analysis , we use recently updated decay constants and form factors for the fa and qcdf approaches . the branching ratios changes a little bit , since they are sensitive to these parameters . but the direct cp asymmetry changes very little . in the pqcd approach , we use the full set of light meson wave functions , including two twist 3 distribution amplitudes , where only one used in the previous papers @xcite . the threshold resummation for the endpoint of the hard part calculation is also newly included @xcite . the numerical results for the branching ratios and cp asymmetries change only a little bit . .direct cp asymmetries calculated in fa @xcite , qcdf @xcite and pqcd @xcite for @xmath64 and @xmath54 decays together with the averaged experimental results at percentage . [ cols="<,^,^,^,^",options="header " , ]      as discussed in the previous section , the strong phase generated from pqcd approach is quite different from the fa and qcdf approaches . the direct cp asymmetry in sm is proportional to the sine of the strong phase difference of two amplitudes . therefore the direct cp asymmetry will be different if strong phase is different . the predicted cp asymmetry by the three methods are shown in table [ cp ] . it is easy to see that the fa @xcite and qcdf @xcite results are quite close to each other , since the mechanism of strong phase is the same for them . recently the two b factories measure some channels with non - zero direct cp asymmetry @xcite , which are shown in table [ cp ] @xcite . it is claimed that direct cp has been found in @xmath2 and @xmath65 decays with more than @xmath66 signal . it is easy to see that our pqcd results of direct cp asymmetry @xcite agree with the experiments , especially for the experimentally well measured channels @xmath67 and @xmath68 decays . although fa and qcdf are not yet ruled out by experiments , but the experiments at least tell us that the dominant strong phase should come from the mechanism of pqcd not qcdf . charm quark loop mechanism , which gives the central value of strong phase in qcdf , is argued next - to leading order in pqcd . this argument is now proved by b factories experiments .    in summary , branching ratios of both fa and qcdf depend on the values of form factors . the strong phase generated from these approaches is dominantly from perturbative bss mechanism . in the pqcd approach , dominant by short distance contribution , the form factors are sensitive to meson wave functions . by including the @xmath0 dependence and sudakov suppression , there is no endpoint divergence . in the pqcd formalism , annihilation amplitudes are of the same order as non - factorizable ones in powers of @xmath69 , which are both @xmath70 . the strong phase comes mainly from the annihilation and non - factorizable diagrams in pqcd approach , which is quite different from the fa and qcdf approaches . the experimentally measured direct cp asymmetry implies that pqcd gives at least the dominant strong phase than other approaches . this will be further tested by experiments . m. bauer , b. stech , m. wirbel , z. phys . c * 29 * , 637 ( 1985 ) ; _ ibid . _ * 34 * , 103 ( 1987 ) . a. ali , g. kramer and c.d . l , phys . d58 , 094009 ( 1998 ) ; c.d . l , nucl . 74 , 227 - 230 ( 1999 ) cheng , phys . b * 335 * , 428 ( 1994 ) ; h.y . cheng , z. phys . c * 69 * , 647 ( 1996 ) . m. beneke , g. buchalla , m. neubert , and c.t . sachrajda , phys . * 83 * , 1914 ( 1999 ) ; nucl . phys . * b591 * , 313 ( 2000 ) . lapage and s.j . brodsky , phys . b * 87 * , 359 ( 1979 ) ; phys . rev . d * 22 * , 2157 ( 1980 ) ; c .- s . huang , phys . fortis et phys . nucl . 4 ( 1980 ) 761 . 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there are different approaches for the hadronic b decay calculations , recently . in this paper 
 , we upgrade three of them , namely factorization , qcd factorization and the perturbative qcd approach based on @xmath0 factorization , by using new parameters and full wave functions . although they get similar results for many of the branching ratios , the direct cp asymmetries predicted by them are different , which can be tested by recent experimental measurements of b factories .