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the factorization calculation@xcite has successfully reproduced the decay rate for many two - body channels of @xmath0 decay . it has been argued@xcite that the factorization should become exact in the large @xmath7-quark mass limit and that the only deviation from the factorization is short - distance corrections of @xmath8 . for some channels of @xmath0 decay , however , the factorization calculation appears to be in clear disagreement with current measurement@xcite . theoretically the factorization and the final - state interaction ( fsi ) are closely tied together@xcite . if the factorization is a good approximation , the fsi phase should be small in general . if the fsi phase turns out to be large in experiment , it is a warning sign against the factorization . some _ initial state _ interaction can also affect the factorization@xcite . the observed smallness of the upper bounds on the fsi phases of the two - body @xmath9 processes such as @xmath10 is consistent with absence of long - distance fsi@xcite . on the other hand , purely phenomenological suggestions were recently made about possibility of large fsi phases in @xmath11 , @xmath12@xcite . at present , little is known in experiment about fsi phases of decays other than the charm producing channels . the factorization clearly fails in the @xmath13 decay in spite of the relatively small value of @xmath14 . the observed relative fsi phase in @xmath15 is close to @xmath16@xcite though one would naively expect the factorization to work reasonably well if @xmath17 is really the controlling parameter of its corrections . in most of the two - body decay channels of @xmath18 so far analyzed@xcite . ] in two - body @xmath0 decays the _ c.m . _ energy between a fast quark and a spectator grows only like @xmath19 with @xmath20 , not linearly in @xmath20 . therefore , @xmath19 must be large in order for perturbative qcd to be applicable . even if one accepts the asymptotic validity of the factorization in theory@xcite , one should test in experiment whether or not the @xmath0 mass of 5.3 mev is high enough to be in the asymptopia of its final states .    in this paper , we extend the basic premise of the factorization and explore its consequences in three - meson decay channels of @xmath0 decay in which one of mesons is soft . we propose simple selection rules which will provide a further test of the fractorization at the @xmath0 mass scale . our approach is purely phenomenological and we do not attempt here a rigorous justification of the selection rules by perturbative qcd . when we compare the selection rules with the soft - pion theorem of chiral symmetry , it appears as if they were incompatible . the origin of this superficial incompatibility is traced to difference in kinematics : the selection rules hold when the _ c.m . _ energy @xmath19 between a fast quark and a soft quark is sufficiently large , while the soft - pion theorem holds when the same _ c.m . _ energy is equal to a light quark mass , _ i.e. , _ negligibly small . the purpose of our selection rules is to probe if @xmath19 in @xmath0 decay is in the qcd asymptopia or not . we state the factorization of two - body decays as follows : when a @xmath0 meson decays with a current - current interaction through the quark decay process , @xmath21 either @xmath22 or @xmath23 forms one meson and the remaining @xmath24 or @xmath25 forms the other meson with a soft quark @xmath26 from the light spectator cloud around @xmath27 . ( see fig.1a . ) the formation of a meson by two hard quarks through a color - singlet current is given by a decay constant while the formation of the other meson is described by a transition form factor of the other current . it is important that once combinations of @xmath28 have been chosen for meson formation , no flavor exchange occurs thereafter , not even a charge exchange . any flavor - changing process is supposedly a higher order process of perturbative qcd . therefore the final mesons retain their flavor contents , including charges , determined by the short - distance quark decay process . this leads us to various constraints of the factorization which relate different isospin amplitudes of two - body decays for a given decay operator .    what happens if three mesons are produced ? in the case that all final mesons are energetic , each meson should contain one of the three final quarks of eq.([qqq ] ) picking up one soft quark from the spectator cloud . ( see fig.1b . ) in the case that two of the mesons are energetic and one is very soft , the energetic mesons are formed just as in the two - body decay and the third one emerges from the spectator cloud with energy no larger than @xmath29 . in this case the energetic mesons retain the history of the weak interaction while the soft meson is made of whatever left in the spectator cloud . this is the natural extension of the factorization from two - body decays to three - body decays . we call this picture as the factorization for three - body decays and explore its consequences here . the _ c.m . _ energy of the fsi between an energetic meson and the soft cloud is @xmath30 , which is the same as the _ c.m . _ energy of @xmath25 and @xmath26 of the two - body decay in fig.1a . , but our selection rules do not depend on a soft interaction between them . ] if @xmath19 is large and short - distance qcd dominates in fsi , flavor exchange between any pair of final mesons would be a higher order correction . in this case , the three quarks produced by the @xmath27-quark can be tracked down into the flavor contents of the final mesons . we illustrate derivation of our selection rule in the decay , @xmath32 which proceeds through the tree - operator @xmath33 and its gluon corrections . the boldface letter @xmath34 in eq.([dpipi ] ) represents a cloud of light quark - antiquarks and gluons around @xmath27 which carries the net flavor of the @xmath35-quark ; @xmath36 we refer to @xmath34 as the spectator @xmath35-quark cloud . when the decay products @xmath37 of @xmath27 enter three mesons one each , all mesons are energetic . in this case we can not find a simple flavor selection rule of the factorization . the reason is as follows : even if no flavor exchange occurs between final mesons , the process @xmath38 occurs perpetually inside the cloud prior to meson formation . this annihilation process has the same effect as a flavor exchange between final mesons since each meson picks up one soft quark from the cloud . therefore , no simple testable selection rule results in this case .    on the other hand , if one of the pions is soft , it must be generated by the spectator cloud without involving a hard quark . the three quarks from @xmath39 must go into @xmath40 and the energetic @xmath41 . and @xmath42 are negligible . ] the hard @xmath43 can form either @xmath6 , @xmath44 , or @xmath40 while the hard @xmath42 can form either @xmath6 or @xmath44 . even though @xmath42 or @xmath45 picks one soft quark from the cloud , the resulting meson is energetic . after one quark ( @xmath43 or @xmath35 ) is removed from the spectator cloud * d * of total charge @xmath46 , the remainder of the cloud is either negatively charged or neutral . that is , the soft pion can be either @xmath5 or @xmath44 , but not @xmath6 . we thus conclude that @xmath6 can not be soft in @xmath47 . this selection rule is depicted in fig.2 . in fact , the selection rule holds irrespective of the decay interaction . it depends only on the flavor content of the spectator * d * : _ when one quark is taken away from a cloud of charge @xmath46 , the remainder can not be positively charged . _ the key is that mesons carry quantum numbers of quark - antiquark no matter what their fock - space compositions are . the only assumption we have made here is that the energetic mesons do not exchange a flavor with the soft meson by short - distance qcd dominance in fsi .    our prediction should be tested in the form , @xmath48 where @xmath49 for the soft pion . the bands of the @xmath12 and @xmath50 resonances should be excluded in testing eq.([sel ] ) . if , contrary to the factorization , elastic or inelastic rescattering is important at @xmath51 ( @xmath52 gev ) , @xmath53 can be fed through the unsuppressed decay modes , for instance , through @xmath54 followed by the forward charge exchange rescattering @xmath55 or through @xmath56 followed by the backward elastic rescattering @xmath57 , _ i.e. , _ the forward rescattering of two units of charge exchange . thus the selection rule is sensitive to fsi . the fsi of our concern is the soft - hard process at _ c.m . _ energy of @xmath30 that involves a hard quark , since the soft - soft fsi is no more than a different pick of soft quarks from the cloud . selection rules similar to eq.([sel ] ) can be derived for many other three - body decay modes . considering their relative simplicity , it is worthwhile exploring such selection rules for better understanding of strong interaction dynamics in @xmath0 decay . the @xmath40 meson can be replaced by any meson or meson resonance in eq.([dpipi ] ) ; @xmath58 , where @xmath59 denotes a neutral meson or meson resonance such as @xmath60 , @xmath44 , or @xmath61 . the selection rule reads : @xmath62 we can apply the same reasoning to @xmath63 by replacing the spectator @xmath34 by @xmath64 ; @xmath65 where @xmath66 is a positively charged meson or meson resonance such as @xmath67 , @xmath6 , and @xmath68 . for @xmath69 , we can write the selection rule @xmath70    if we include @xmath71 in the spectator cloud , we obtain corresponding selection rules , @xmath72 furthermore , @xmath73 since the rest energy of a kaon exceeds @xmath74 , formation of any kaon from the spectator cloud is kinematically suppressed . eqs.(8 ) to ( [ kaon4 ] ) mean that the factorization - forbidden modes are even further suppressed than the modes in the denominators . the lightest meson resonance that the spectator cloud can produce is the @xmath75-meson . since its rest mass far exceeds @xmath74 , @xmath76 is suppressed no matter what charge state the @xmath75-meson is in . the cloud simply does not have enough energy to produce @xmath75 even at rest .    to be concrete , consider @xmath77 . then the lack of energy in producing a resonance from the cloud leads us to @xmath78 where @xmath75 can be in any charge state . in eq.([selrho ] ) @xmath75 may be replaced by @xmath79 or some other resonance . similarly , @xmath80    though we have listed only the selection rules that are relatively simple and clean , one can write down many more selection rules by the same reasoning .    since meson or meson resonance production of energy higher than @xmath29 from the cloud is highly suppressed kinematically , we may raise the softness limit considerably from @xmath81 to , for instance , @xmath82 in actual testing of the selection rules , provided that one can avoid resonance bands of quasi - two - body decays . if we raise it too high , however , the tail of the energy spectrum of a decay product quark enters the `` soft meson '' . to find a compromise , we need some numerical study . our selection rules are based on the postulate that a soft meson must be produced entirely from the spectator cloud . therefore , they would fail if a soft meson contains a significant contribution from the low - energy tail of a decay product of @xmath83 . for this reason we would like to know how much the low - energy tail contributes when we define the soft meson by @xmath84 . if the tail is indeed negligible , then we should ask how much we can raise the upper limit of softness in actual test without violating the selection rules substantially .    to be concrete , consider @xmath85 again . suppose that we define the soft pion by its momentum @xmath86 . estimate of the matrix element involves uncertainties in soft meson formation . if we use the quark - hadron duality for the low - energy tail region and replace the meson wavefunctions by their average values , we can obtain a crude estimate of the decay rate ratio as @xmath87 where @xmath88 . the factor @xmath89 , which arises from difference in meson formation probability , is given by the ratio of the pion wavefunction @xmath90 on the light - cone as @xmath91 where @xmath92 . the value of @xmath89 is sensitive to @xmath93 near the end point . let us extrapolate @xmath94 smoothly to @xmath95 . then for @xmath96 @xmath97 the right - hand side is a fraction of 1% for @xmath98 mev . it is indeed small due to the phase space suppression of the soft @xmath43-quark and a smaller meson formation probability for a quark pair of highly uneven energies . from this estimate we expect that accuracy of our selection rules should be very high . when @xmath99 is raised to @xmath100 , the ratio in eq.([q1 ] ) goes up to 8% . in carrying out experimental test , therefore , we may be able to raise the softness upper limit from @xmath74 to @xmath100 . the main background in testing our this selection rule will be a spillover from the resonance bands of @xmath61 and @xmath101 due to errors in pion measurement . in the zero four - momentum limit of the soft pion , we can relate three - body decay matrix elements to two - body decay matrix elements by the soft - pion theorem of chiral symmetry . it can be expressed with @xmath102 mev as @xmath103|b\rangle + k_{\mu}m^{\mu},\ ] ] where the @xmath104 term gives the correction to the soft - pion limit@xcite . for soft @xmath6 emission , @xmath105 should be replaced by @xmath106 . upon extrapolation to the physical pion at rest @xmath107 , the correction term @xmath104 is of @xmath108 where @xmath109 is a typical excitation energy of the @xmath0 system . according to the traditional wisdom , the born diagram of @xmath110 is likely the most important correction ( see fig.3a ) since it is nearly degenerate with @xmath0 ( @xmath111 mev ) . this correction can be of @xmath112 or even dominate over the zero - four - momentum limit for a physical soft pion . it is easy to see that the @xmath113 contribution to @xmath104 satisfies the selection rule eq.([sel ] ) . in fact , this is the case for the contributions from all born diagrams . the reason is that the quark diagrams for the born diagrams of fig.3a are topologically equivalent to those of the three - body factorization process . the only difference between them is whether a soft pion is emitted from the cloud before or after weak interaction takes place . if we keep only the soft - pion limit , we obtain @xmath114 eq.([s2 ] ) results from the vanishing commutator , @xmath115=0 $ ] . in contrast , the zero four - momentum limit of the factorization - forbidden decay @xmath116 is related to @xmath117 . while @xmath118 is @xmath119 for @xmath120 , it is the leading term for @xmath121 . eqs.([s1 ] ) and ( [ s2 ] ) look puzzling at the first sight . if we work out the soft - pion limit for @xmath122 , our three - body factorization predicts @xmath123 while the soft - pion theorem gives for the tree - operator @xmath124 and @xmath125 @xmath126 the situation looks equally bizzare in @xmath127 through the penguin operators . since the penguin operators of @xmath128 are all singlets of chiral su(2)@xmath129su(2 ) , their commutators with the axial isospin charges are zero . that is , the soft - pion limit should disappear for all matrix elements of @xmath130 as @xmath131 for the penguin operators . it appears as if the selection rules contradicted the soft - pion theorem of chiral symmetry . we can trace the origin of this apparent discrepancy to the _ c.m . _ energy between a fast quark ( @xmath132 ) from @xmath27-decay and a soft quark ( @xmath133 ) in the cloud . in the _ physical _ soft pion limit where our selection rules should hold , @xmath134 while in the zero four - momentum limit @xmath135 in order for the factorization to work , the right - hand side of eq.([cm1 ] ) must be large enough to be in the asymptopia of qcd . the same variable is equal to the light quark mass in the soft - pion theorem . their predictions apply to different kinematical regions . the chiral lagrangian has been successful for which the soft - pion theorem gives the leading terms . however , extrapolation to the physical region of @xmath0-decays is a long way because of large @xmath0 mass ( cf eqs.([cm1 ] ) and ( [ cm2 ] ) ) and involves large incalculable corrections . we can understand what physical process the soft - pion theorem represents@xcite . the commutator of the axial charge @xmath136 with weak decay operator @xmath137 arises from its commutator with the currents in @xmath137 . in the case of @xmath138 at @xmath139 with @xmath140 , for instance , the commutator with @xmath141 survives to give @xmath142 =    ( \overline{b}_{l\alpha}\gamma_{\mu}c_{l\beta } ) (     \overline{u}_{l\beta}\gamma^{\mu}u_{l\alpha }    -\overline{d}_{l\beta}\gamma^{\mu}d_{l\alpha}),\ ] ] where the right - hand side is the @xmath143 partner of @xmath121 . this commutator describes production of states of charge @xmath144 through @xmath141 of @xmath121 and subsequent fragmentation of the soft @xmath6 from them . emission of a zero - four - momentum pion from the energetic quark pair @xmath141 is not suppressed by asymptotic freedom since no large momentum transfer occurs . after the @xmath6 emission , these states ( @xmath145 ) and the operator @xmath146 together produce the final state @xmath147 . ( see fig.3b . ) for soft @xmath5 emission , the commutator vanishes so that there is no diagram corresponding to fig.3b . thus physics in the zero - four - momentum limit is quite different from what we have introduced as an extension of the factorization in this paper . an important question is which picture gives a better description of the three - body @xmath0 decay in which one pion is soft . is the approximation of @xmath148 better than the zero four - momentum approximation ? the issue should be settled by experimental test of the selection rules proposed here . the dynamics underlying the factorization can be tested with the selection rules in three - body decays in which one of mesons is soft . the selection rules test whether the soft - hard fsi at _ c.m . _ energy of @xmath30 is in the qcd asymptopia or not . the rules are intuitive and do not depend on decay operators nor on form factors . if the selection rules are verified in experiment , we would obtain better understanding of physics underlying the factorization and strong interactions in @xmath0 decay in general . this work was supported in part by the director , office of science , division of high energy and nuclear physics , of the u.s . department of energy under contract de  ac0376sf00098 and in part by the national science foundation under grant phy9514797 . d. fakirov and b. stech , nucl . phys . * b 232 * , 315 ( 1978 ) ; n. cabibbo and l. maiani , phys . 73b * , 418 ( 1978 ) ; m. wirbel , b. stech , and m. bauer , z. phys . c * 29 * , 637 ( 1985 ) . bjorken , nucl . b(proc . suppl . ) * 11 * , 325 ( 1991 ) ; m.j . dugan and b. grinstein , phys . 255b * , 583 ( 1991 ) ; h.d . politzer and m.b . wise , phys . 257b * , 399 ( 1991 ) ; c.h . chang and h .- n . li , phys . d*55 * , 5577 ( 1997 ) ; t.w . yeh and h .- n . li , phys . rev . d*56 * , 1615 ( 1997 ) ; y .- y . keum , h .- li , and a.i . sanda , hep - ph/0004004 ; hep - ph/0004173 . m. beneke , g. buchalla , m. neubert , and c.t . sachrajda , phys . lett . * 83 * , 1914 ( 1999 ) and hep - ph/0006124 . m. beneke , g. buchalla , m. neubert , c.t . sachrajda , hep - ph/0007256 ; d. cronin - hennessy _ et al . _ ( cleo collaboration ) , hep - ex/0001010 . for a viewpoint different from perturbative qcd , m. suzuki , in _ proceedings of the third international conference on b physics and cp violation _ , taipei , december , 1999 ( world scientific , singapore , 2000 ) , to be published . a. deandrea , r. gatto , m. ladisa , g. nardulli , and p.santorelli , hep - ph/0002038 . m. neubert and b. stech , hep - ph/9705292 ; m. suzuki , phys . rev . d*58 * , 111504 ( 1998 ) . hou and k .- c . yang , phys . lett * 84 * , 4906 ( 2000 ) ; y.f . zhou , y.l . ng , and c.q . geng , hep - ph/0006225 . m. bishai _ et al._(cleo collaboration ) , phys . lett . * 78 * , 3261 ( 1997 ) . j. jousset _ et al . _ , d * 41 * , 1389 ( 1990 ) : see m. suzuki , hep - ph/0006296 for recent references . s.l . adler and r.f . dashen , _ current algebra and applications to particle physics _ , benjamin , new york , 1968 .
extending the dynamics underlying the factorization calculation of two - body decays , we propose simple selection rules for nonresonant three - body @xmath0 decays . 
 we predict , for instance , that in the dalitz plot of @xmath1 , practically no events should be found in the corner region of @xmath2 as compared with the corner of @xmath3 . 
 we also predict that there should be very few three - body decay events containing one soft meson resonance and two energetic mesons or meson resonances . 
 the selection rules are quite different from the soft - pion theorem , since they apply to different kinematical regions . for @xmath4 , 
 the latter predicts that the decay matrix element vanishes in the zero four - momentum limit of @xmath5 instead of @xmath6 . since this marked difference from the soft - pion theorem 
 is directly related to the issue of short - distance qcd dominance in two - body b decays , experimental test of the selection rules will shed light on strong interaction dynamics of @xmath0 decay .