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the violation of the cp symmetry is now established in the b meson sector and the parameter @xmath3 is measured with precision by babar  @xcite and belle  @xcite . however , the measurement of the other angles of the unitarity triangle are necessary for a more comprehensive study of cp violation . these measurements suffer from several difficulties related to the uncertainties of the theory , the limited statistics and/or ambibuities in their extraction . we have recently proposed a new method  @xcite involving three - body b meson decays , which circumvent these problems and allows to determine the angle @xmath4 .    in this paper we extend this method to the weak phase @xmath5 , which can be identified to the combination @xmath6 of the unitarity triangle angles  @xcite . important constraints on the theory will be obtained from the measurement of this phase . in the following , we describe a new _ time dependent _ method involving the entire area of the dalitz plot of the 3-body decays @xmath7 . + the method involves no reconstruction into @xmath8 eigenstates and avoids problems with interfering dcsd . [ sec : method - infinite - stat ] we investigate a way to circumvent the color suppression penalty by using @xmath1 decay modes which potentially offers significantly large branching fractions and cp asymmetries . the particular decays considered here are of the type . these three body decays may be obtained by popping a @xmath10 pair in color allowed decays . although modes where one or more of the three final state particles is a vector can also be used , for clarity and simplicity only the mode @xmath7  is discussed here .    to    the diagrams leading to the final states of interest are shown in fig . [ fig : diagrams ] . as can be seen , the diagrams ( fig . [ fig : diagrams]a and  [ fig : diagrams]c ) are both color - allowed and of order @xmath11 in the wolfenstein parameterization  @xcite , where @xmath12 is the cabibbo mixing angle . the general formalism describing how the weak angle is extracted from observation of the time - dependent cp violation has been first presented in @xcite . however , in the present analysis additional information is used to resolve the inherent ambiguities related to the method .    selecting a particular point @xmath13 in the dalitz plot , let us write the amplitudes corresponding to the transitions in fig . [ fig : diagrams]a and b on the one hand and  [ fig : diagrams]c and d on the other hand as @xmath14 and their cp conjugates @xmath15 where @xmath16 is the relative phase of the ckm matrix elements involved in this decay , and @xmath17 ( @xmath18 ) and @xmath19 ( @xmath20 ) are the real amplitudes and cp - conserving strong interaction phases . + the time dependent probability density function ( pdf ) of the transition @xmath21 is @xmath22 where @xmath23 , @xmath24 with @xmath25 = @xmath26 and @xmath27 . we have used the approximation @xmath28 . + the pdf of the transition @xmath29 is @xmath30 from the total number of events in each of the four final states ( [ eq : btodm ] ) and ( [ eq : btodp ] ) and a global fit of their time dependence , it is possible to extract the quantities , @xmath31 where @xmath32 and @xmath33 . one obtains : @xmath34 hence , in the limit of very high statistics , one would extract @xmath35 for each point @xmath13 of the dalitz plot . however , for every point of the dalitz plot , @xmath36 is obtained with an eight - fold ambiguity in the range @xmath37 $ ] , because of the invariance of the @xmath38 terms in eq . ( [ eq : btodm ] ) and ( [ eq : btodp ] ) under the three symmetry operations @xmath39 when the multiple measurements made in different points of the dalitz plot are combined , some of the ambiguity will be resolved , in the likely case that the strong phase @xmath40 varies from one region of the dalitz plot to the other . this variation can either be due to the presence of resonances or because of a varying phase in the non - resonant contribution . in this case , the exchange symmetry @xmath41 is numerically different from one point to the other , which in effect breaks this symmetry and resolves the ambiguity . similarly , the @xmath42 symmetry is broken if there exists some _ a - priori _ knowledge of the dependence of @xmath40 on the dalitz plot parameters . this knowledge is provided by the existence of broad resonances , whose breit - wigner phase variation is known and may be assumed to dominate the phase variation over the width of the resonance . to illustrate this , let @xmath13 and @xmath43 be two points in the dalitz plot , corresponding to different values of invariant mass of a particular resonance . one then measures @xmath44 at point @xmath13 and @xmath45 at point @xmath43 , where @xmath46 is known from the parameters of the resonance . it is important to note that the sign of @xmath46 is also known , hence it does not change under @xmath42 . therefore , should one choose the @xmath42-related solution @xmath47 at point @xmath13 , one would get @xmath48 at point @xmath43 . since this is different from @xmath45 the @xmath42 ambiguity is resolved . this is illustrated graphically in eq . [ eq : ssign ] , where @xmath49 is the weak phase : @xmath50 bw \downarrow & & \downarrow bw \\[-0.5ex ] \sin(\phi \pm ( \delta\delta_i \!+\ ! \alpha_{ij } ) )   & \stackrel{\ssign}{\not\!\!\!\longleftrightarrow}\ ! &   \sin(\pi - \phi \mp ( \delta\delta_i \!-\ ! \alpha_{ij } ) ) \end{array } \hspace*{-1ex } \label{eq : ssign } \vspace*{-2ex}\ ] ] thus , broad resonances reduce the initial eight - fold ambiguity to the two - fold ambiguity of the @xmath51 symmetry , which is never broken . fortunately , @xmath51 leads to the well - separated solutions @xmath36 and @xmath52 , the correct one of which is easily identified when this measurement is combined with other measurements of the unitarity triangle . since experimental data sets will be finite , extracting @xmath16 will require making use of a limited set of parameters to describe the variation of amplitudes and strong phases over the dalitz plot . the consistency of this approach can be verified by comparing the results obtained from fits in a few different regions of the dalitz plot , and the systematic error due to the choice of the parameterization of the data may be obtained by using different parameterizations . a fairly general parameterization assumes the existence of @xmath53 breit - wigner resonances , as well as a non - resonant contribution : @xmath54 where @xmath55 represents the dalitz plot variables , @xmath56 is the breit - wigner amplitude for a particle of spin @xmath57 , normalized such that @xmath58 , @xmath59 and @xmath60 ( @xmath61 and @xmath62 ) are the magnitude and cp - conserving phase of the non - resonant @xmath63 ( @xmath64 ) amplitude , and @xmath65 and @xmath66 ( @xmath67 and @xmath68 ) are the magnitudes and cp - conserving phase of the @xmath63 ( @xmath64 ) amplitude associated with resonance @xmath43 . the functions @xmath69 and @xmath70 may be assumed to vary slowly over the dalitz plot , allowing their description in terms of a small number of parameters . the decay amplitudes of @xmath71 mesons are identical to those of eqs . ( [ eq : amp - param ] ) , with @xmath16 replaced by @xmath72 . the decay amplitudes of eqs . ( [ eq : amp - param ] ) can be used to conduct the full data analysis . given a sample of @xmath73 signal events , @xmath16 and the other unknown parameters of eq . ( [ eq : amp - param ] ) are determined by minimizing the negative log likelihood function @xmath74 where the amplitude @xmath75 is given by one of the expressions of eqs . ( [ eq : amp - param ] ) , or their cp - conjugates , depending on the final state @xmath76 , and @xmath77 are the dalitz plot variables of event @xmath13 .    in what follows , we discuss important properties of the method by considering the illustrative case , in which the @xmath63 decay proceeds only via a non - resonant amplitude , and the @xmath64 decay has a non - resonant contribution and a single resonant amplitude . for concreteness , the resonance is taken to be the @xmath78 . we take the a priori @xmath55-dependent non - resonant phases to be @xmath79 . [ sec : simulation ] to study the feasibility of the analysis using eq . ( [ eq : chi2 ] ) , we conducted a simulation of the decays @xmath80 and @xmath81 . events were generated according to the pdf in eqs . ( [ eq : btodm ] ) and  ( [ eq : btodp ] ) , with the parameter values given in table  [ tab : reference ] . in this table and throughout the rest of the paper , we use a tilde to denote the `` true '' parameter values used to generate events , while the corresponding plain symbols represent the `` trial '' parameters used to calculate the experimental @xmath82 . for simplicity , additional resonances were not included in this demonstration . however , broad resonances that are observed in the data should be included in the actual data analysis . .parameters used in the simulation . the value of @xmath83 is chosen so as to roughly agree with the measurement of the corresponding branching fraction  @xcite . [ cols="^,^,^,^",options="header " , ]     in fig . [ fig : sig - gamma - vs - gamma ] we present @xmath84 , the statistical error on @xmath36 as a function of @xmath85 and @xmath86 , obtained by fitting simulated event samples . each point is obtained by repeating the simulation 250 times , to minimize sample - to - sample statistical fluctuations . all the parameters of table  [ tab : reference ] were determined by the fit , and the arrow in these figures indicates the point corresponding to the parameters of table  [ tab : reference ] . one observes that @xmath84 does not depend strongly on the value @xmath36 . as expected , a strong dependence on @xmath87 is seen . however , it should be noted that @xmath88 changes very little for all values of @xmath89 above 0.4 . this suggests that the potential for a significantly sensitive measurement is high over a broad range of parameters . we have shown how @xmath36 may be measured in the color - allowed decays , focusing on the simplest mode @xmath7 . the absence of color suppression in the @xmath63 amplitudes is expected to result in relatively large rates and significant cp violation effects , and hence favorable experimental sensitivities . while the dalitz plot analysis implies some experimental complication , it should reduce the eight - fold ambiguities , which are a serious problem with other methods for measuring @xmath36 . as a result of these advantages , this method is likely to lead to a measurement of @xmath36 even with the current generation of @xmath90  factory experiments . the authors thank francois le diberder for his fruitful ideas and help with simulation . this work was supported by cea and the danish research academy . 9 babar collaboration ( b. aubert _ et al . _ ) , phys . rev . * 86 * ( 2001 ) , 2515 . belle collaboration ( b.  abe _ et al . _ ) , phys . rev . * 87 * ( 2001 ) , 091802 . r. aleksan , t. petersen and a. soffer , phys . d. * 67 * ( 2003 ) 096002 . r. aleksan , b. kayser and d. london , phys . lett . * 73 * , ( 1994 ) 18 . l. wolfenstein , phys . * 51 * ( 1983 ) 1945 . r. aleksan , i. dunietz , b. kayser , f. le diberder , nucl . b361 ( 1991 ) 141 . the cleo collaboration , phys . 88 , 101803 ( 2002 ) .
we present a method to measure the weak phase @xmath0 in the three - body decay of neutral @xmath1 mesons to the final states @xmath2 . 
 these decays are mediated by interfering amplitudes which are color - allowed and hence relatively large . as a result , large cp violation effects can be observed with high statistical significance . 
 in addition , the three - body decay helps to resolve discrete ambiguities present in measurements of the weak phase . 
 the experimental implications of conducting these measurements are discussed , and the sensitivity of the method is evaluated using a simulation .