Roy Aleksan Centre d Etudes Nucleaires, Saclay, DAPNIA/SPP, F Gif-sur-Yvette, CEDEX, France

Transcription

1 CSUHEP -1 DAPNIA--95 LAL--83 Measring the Weak Phase γ in Color Allowed B DKπ Decays Roy Aleksan Centre d Etdes Ncleaires, Saclay, DAPNIA/SPP, F Gif-sr-Yvette, CEDEX, France Troels C. Petersen LAL Bat 8, Orsay BP 34, France Abner Soffer Department of Physics, Colorado State University, Fort Collins, CO 853, U.S.A. (September 18, ) Abstract We present a method to measre the weak phase γ in the three-body decay of charged B ± mesons to the final states DK ± π. These decays are mediated by interfering amplitdes which are color-allowed and hence relatively large. As a reslt, large CP violation effects that cold be observed with high statistical significance are possible. In addition, the three-body decay helps resolve discrete ambigities that are sally present in measrements of the weak phase. The experimental implications of condcting these measrements with three-body decays are discssed, and the sensitivity of the method is evalated sing a simlation. 1 Typeset sing REVTEX

2 I. INTRODUCTION CP violation is crrently the focs of a great deal of attention. Since the start of operation of the B-factories, the standard model description of CP violation via the Cabibbo- Kobayashi-Maskawa (CKM) matrix [1] is being tested with increasing precision. BaBar [] and Belle [3] have recently pblished measrements of the CKM parameter sin(β), where β =arg( V cd Vcb /V tdvtb ), verifying the CKM mechanism to within the experimental sensitivity. Althogh improved measrements of sin(β) andb s B s mixing will probe the theory with greater scrtiny dring the next few years, the measrement of the other angles of the nitarity triangle are necessary for a comprehensive stdy of CP violation. Important constraints on the theory will be obtained from measrements of the CKM phase γ =arg( V d Vb /V cdvcb ). A promising method for measring this phase in the B system has been proposed [4]. Althogh this method involves color-allowed decays and hence offers favorable rates, it makes se of B s mesons, which are not prodced at B-Factories operating at the Υ(4S) resonance. By contrast, the extraction of γ sing the B and B d system generally involves decays which are highly sppressed or difficlt to reconstrct. In addition, these methods are generally sbject to an eight-fold ambigity de to a-priori nknown strong phases [4,5]. As a reslt, obtaining satisfactory sensitivity reqires very high statistics and necessitates the se of as many decay modes and measrement methods as possible. One important class of theoretically clean measrements will make se of decays of the type B DK. Grona and Wyler [6] have proposed to measre sin γ in the interference between the b cs decay B + D K + and the color-sppressed b cs decay B + D K +. Interference between these amplitdes takes place when the D meson is observed as one of the CP-eigenstates D 1, 1 ( D ± D ), (1) which are identified by their decay prodcts, sch as K + K or K s π. Several variations of this method have been developed [7 1], inclding addressing the effects of dobly Cabibbosppressed decays of the D meson [11] and mixing and CP-violation in the netral D meson system [1], as well as insights to be gained from charm factory measrements [13]. A serios difficlty with measring γ sing B DK is that the b cs amplitde is expected to be extremely small. To a large degree, this is de to the color-sppression associated with the internal spectator diagram throgh which this amplitde proceeds. In the factorization model, color-sppression of the amplitde is parameterized by the phenomenological ratio a /a 1. This ratio is measred to be abot.5 [14] by comparing decay modes which depend only on color-allowed amplitdes with those that depend on both color-allowed and color-sppressed amplitdes. With this vale of a /a 1, one expects the amplitde ratio A(B + D K + )/A(B + D K + ) to be only abot.1. The small branching fraction B(B + D K + ) is therefore very difficlt to measre with adeqate precision, reslting in a large statistical error in the measrement of sin γ. The recent observation of the color-sppressed decays B D ( ) π, D η,andd ω by Belle [15] and CLEO [16] has raised the possibility that a /a 1 may be effectively larger in some modes. However, significant sppression is still expected for internal spectator diagrams.

3 This difficlty has led to attempts to address the problems presented by color sppression. Dnietz [7] proposed to apply the method to the decays B DK, making se of the fact that the decay K K + π tags the flavor of the B. In this mode, both the b cs and the b cs amplitdes are color sppressed, and hence of similar magnitdes, albeit small. Jang and Ko [9] and Grona and Rosner [1] have devised a method in which the small branching fraction of the color-sppressed decay B + D K + does not have to be measred directly. Rather, it is essentially inferred by sing the larger branching fractions of the decays B D K +, B D K and B D 1, K. Qantitative stdy sggests that the varios alternative methods are roghly as sensitive as the method of Ref. [6], and are ths sefl for increasing statistics and providing consistency checks [5]. II. MEASURING γ WITH COLOR-ALLOWED B DKπ DECAYS In this paper we investigate a way to circmvent the color sppression penalty by sing B ± decay modes which cold potentially offer significantly large branching fractions, as well as large CP asymmetries. Similar modes involving netral B decays can also be sed. For example the final state D (D )K ± π can be analyzed with the same techniqe as described here. Some other decays (sch as B D K s π + ) need a different treatment and will be discssed elsewhere [17]. The particlar decays which are considered here are of the type B D ( ) K ( ) π(ρ). These three body final states may be obtained by popping a q q pair in color allowed decays. Althogh modes where one or more of the three final state particles is a vector can also be sed, for clarity and simplicity only the mode B ± D K ± π is discssed here. b + B a) s c π K s + D π D B + s K π π b c + B c) d) D K + + B b b b) c s c + K D + B + b B + b a) c) c s D K + π D c s + K π + B B + b b b) d) c s c π s π D K + K + D FIG. 1. Feynmann Diagrams for the decay B + D K + π involving the CKM matrix element prodct V cb V s FIG.. Feynmann Diagrams for the decay B + D K + π involving the CKM matrix element prodct V b V cs Figs. 1 and show the diagrams leading to the final states of interest. As can be seen, the leading diagrams (Fig. 1a and a) are both color-allowed and of order λ 3 =sin(θ c )inthe Wolfenstein parameterization [18], where θ c is the Cabibbo mixing angle. De to the absence of color sppression, both interfering amplitdes are large, avoiding the complications which 3

4 arise de to the small magnitde of the b cs amplitde in the two-body decays. As a reslt, observable CP-violating effects in the three-body decays are expected to be large, and the b cs amplitde is more easily measred from the relatively large branching fraction B(B + D K + π ), which is now sbject to significantly less contamination from dobly Cabibbo-sppressed D meson decays than the corresponding two-body modes. However, shold the b cs amplitde be nexpectedly small, one cold still carry ot the analysis described in this paper by taking dobly Cabibbo-sppressed decays into accont [5,11]. + A( D cp K + B π ) A( B + D K + π ) - γ A( D π ) A( π ) cp K B B D K δ A( B + D K + - π ) = A( B D K - π ) FIG. 4. Illstration of the triangle relations in the decays B ± D K ± π and B ± D K ± π. FIG. 3. Two points on the Dalitz plot of the decays B + D K + π and B + D K + π. Let s examine how one cold observe CP violation and measre the angle γ in these decays. We first consider the case of very large statistics, and then discss how one wold proceed when the data sample is limited. Since we are dealing with a three-body decay, we se the Dalitz plot of the system DK ± π (see Fig. 3). Selecting a particlar point i in this representation, Eq. (1) implies the relations A i (B + D1, K+ π )= 1 ( Ai (B + D K + π ) ±A i (B + D K + π ) ) A i (B D 1, K π )= 1 ( Ai (B D K π ) ±A i (B D K π ) ). () Let s write the amplitdes corresponding to the transitions in Fig. 1 and as A i (B + D K + π )=A Ci e iδ C i, A i (B + D K + π )=A U i e iδ U i e iγ, A i (B D K π )=A Ci e iδ C i, A i (B D K π )=A U i e iδ U i e iγ, (3) where γ is the relative phase of the CKM matrix elements involved in this decay, and A C (A U )andδ C (δ U ) are the real amplitde and CP-conserving strong interaction phase of 4

5 the transitions of Fig. 1 (Fig. ). Let s note here that we have sed the Wolfenstein parameterization at order O(λ 3 ). Shold one se the fll expansion, a small weak phase of order λ 4 wold be present. Indeed the angle which is measred is γ =arg(v d Vb/V cs Vcb) = γ + ξ, whereξ =arg( V cd Vcs /V dvs ). The angle ξ is one of the angles [19] arising from the nitarity relation V d Vs + V cd Vcs + V td Vts =. The amplitdes in Eqs. 3 can be obtained from the measrements of the B decay widths Eq. () implies Γ i (B + D K + π )=Γ i (B D K π )=A C i Γ i (B + D K + π )=Γ i (B D K π )=A U i. (4) Γ i (B + D1,K + π )=A Ci + A U i ± A Ci A U i cos( δ i + γ) Γ i (B D1, K π )=A Ci + A U i ± A Ci A U i cos( δ i γ), (5) where δ i δ U i δ Ci. Ths, by measring the widths in Eq. 4 and 5, one extracts sin γ from sin γ = 1 ( ) 1 CC ± (1 C )(1 C ), (6) where C cos( δ i + γ) andc =cos( δ i γ). Hence in the limit of very high statistics, one wold extract sin γ for each point i of the Dalitz plot and therefore obtain many measrements of the same qantity. This wold allow one to obtain a large set of redndant measrements from which a precise and consistent vale of sin γ cold be extracted. We note that in or treatment we disregard dobly Cabibbo-sppressed D decays [11] since, de to the lack of color-sppression, their effect is small, and can be dealt with [5] in any case. In every point of the Dalitz plot, γ is obtained with an eight-fold ambigity, which is a conseqence of the invariance of the cos( δ i ±γ) terms in Eq. (5) nder the three symmetry operations [5] : γ δ, δ γ S sign : γ γ, δ δ S π : γ γ + π, δ δ + π. (7) However, an important benefit is gained from the mltiple measrements made in different points of the Dalitz plot. When reslts from the different points are combined, some of the ambigity will be resolved, in the likely case that the strong phase δ i varies from one region of the Dalitz to the other. This variation can either be de to the presence of resonances or becase of a varying phase in the non-resonant contribtion. In this case, the exchange symmetry is nmerically different from one point to the other, which in effect breaks this symmetry and resolves the ambigity. Similarly, the S sign symmetry is broken if there exists some a-priori knowledge of the dependence of δ i 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 assmed to dominate the phase variation over the width of the resonance. To illstrate this, let i and j 5

6 be two points in the Dalitz plot, corresponding to different vales of the invariant mass of the decay prodcts of a particlar resonance. For simplicity we consider only one resonance. One then measres cos( δ i ± γ) atpointi and cos( δ i + α ij ± γ) atpointj, whereα ij is known from the parameters of the resonance. It is important to note that the sign of α ij is also known, hence it does not change nder S sign. Therefore, shold one choose the S sign -related soltion cos( δ i γ) atpointi, one wold get cos( δ i + α ij γ) atpoint j. Since this is different from cos( δ i + α ij ± γ), the S sign ambigity is resolved. This is illstrated graphically in Eq. 8: cos( δ i ± γ) BW cos( δ i + α ij ± γ) S sign cos( δi γ) BW S sign cos( δ i + α ij γ) (8) Ths, broad resonances redce the initial eight-fold ambigity to the two-fold ambigity of the S π symmetry, which is not broken. Fortnately, S π leads to the well-separated soltions γ and γ + π, the correct one of which is easily identified when this measrement is combined with other measrements of the nitarity triangle. III. THE FINITE STATISTICS CASE Since experimental data sets will be finite, extracting γ will reqire making se of a limited set of parameters to describe the variation of amplitdes and strong phases over the Dalitz plot. The consistency of this approach can be verified by comparing the reslts obtained from fits of the data in a few different regions of the Dalitz plot, and the systematic error de to the choice of the parameterization of the data may be obtained by sing different parameterizations. A fairly general parameterization assmes the existence of N R Breit-Wigner resonances, as well as a non-resonant contribtion: A ξ (B + D K + π )= A C e iδ N R C + A Cj B sj (ξ) e iδ C j e iδ C(ξ) j=1 A ξ (B + D K + π )= A U e iδ N R U + A U j B sj (ξ) e iδ U j e iδ U (ξ) e iγ, (9) where ξ represents the Dalitz plot variables, j=1 B sj (ξ) b sj (ξ) e iδ j(ξ) (1) is the Breit-Wigner amplitde for a particle of spin s j, normalized sch that (b sj (ξ)) dξ =1, A U and δ U (A C and δ C ) are the magnitde and CP-conserving phase of the non-resonant b cs (b cs) amplitde, and A U j and δ U j (A Cj and δ Cj ) are the magnitdes and CP-conserving phase of the b cs (b cs) amplitde associated with resonance j []. The fnctions δ C (ξ) andδ U (ξ) may be assmed to vary slowly over the Dalitz plot, allowing their description in terms of a small nmber of parameters. Eq. (1) again implies 6

7 A ξ (B + D1,K + π )= 1 ) (A ξ (B + D K + π ) ±A ξ (B + D K + π ). (11) ThedecayamplitdesofB mesons are identical to those of Eqs. (9) and (11), with γ replaced by γ. The decay amplitdes of Eqs. (9) and (11) can be sed to condct the fll data analysis. This is done by constrcting the probability density fnction (PDF) P (ξ) = A ξ (f), (1) where the amplitde A ξ (f) is given by one of the expressions of Eq. (9), Eq. (11), or their CP-conjgates, depending on the final state f. Given a sample of N e signal events, γ and the other nknown parameters of Eq. (9) are determined by minimizing the negative log likelihood fnction N e χ log P (ξ i ), (13) i=1 where ξ i are the Dalitz plot variables of event i. IV. RESONANCES AND AMBIGUITIES It is worthwhile to consider the resonances which may contribte to the D K ± π final state. Obvios candidates are broad D and Ds states. However, only the ones which can decay as D D π or Ds + D K + are relevant for the final state of interest. This exclde the 1 + states, which wold decay to D π or D K. Frthermore, since the Ds + is essentially prodced throgh a W +,the + state is forbidden as well. Ths, one does not expect a large contribtion from these states. A promising candidate might be the broad D recently observed by the Belle collaboration [1]. We note that inclding sch resonances in the analysis does not raise particlar difficlties and wold frther enhance the sensitivity of the γ measrement. Similar argments can be made for higher excited K states. One also expects narrow resonances, sch as the D (7) and a narrow Ds + state, to be prodced. However, as seen in the Dalitz plot of Fig. 5, these resonances do not overlap, and hence do not interfere. In addition, interference between a very narrow resonance and either a broad resonance or a non-resonant term is sppressed in proportion to the sqare root of the narrow resonance width. Therefore, narrow resonances contribte significantly to the CP violation measrement only if both the b cs and b cs amplitdes proceed throgh the same resonance. This scenario is favorable, bt is not necessary for the sccess of or method, and will therefore not be focsed on in the rest of this stdy. In what follows, we discss important properties of the method by considering the illstrative case, in which the b cs decay proceeds only via a non-resonant amplitde, and the b cs decay has a non-resonant contribtion and a single resonant amplitde. For concreteness, the resonance is taken to be the K ± (89). We take the ξ-dependent nonresonant phases to be δ C (ξ) =δ U (ξ) =. Under these circmstances, the PDF of Eq. (1) depends on for cosine terms that are measred in the experiment: 7

8 c ± cos(δ U ± γ) c ± K cos(δ U δ K δ K (ξ) ± γ), (14) where δ K (ξ) istheξ-dependent K Breit-Wigner phase of Eq. (1). The cosines c ± (c ± K ) arise from interference between the non-resonant (resonant) b cs amplitde and the non-resonant b cs amplitde. The phases δ U, δ K,andγ are all a-priori nknown. However, it is important to note that δ K is flly determined from the interference between the resonant and non-resonant contribtions to the relatively high statistics decay mode B + D K + π as a fnction of the Dalitz plot variables. Therefore, δ K is obtained with no ambigities, and with an error mch smaller than those of δ U or γ. Conseqently, the only relevant symmetry operations are : γ δ U, δ U γ S sign : γ γ, δ U δ U S π : γ γ + π, δ U δ U + π S K + ex : γ δ U δ K, δ U γ + δ K S K ex : γ δ U + δ K, δ U γ + δ K. (15) As discssed above, only S π is a symmetry of all for cosines of Eq. (14), and is therefore flly nresolved. The transformation properties of the cosines nder any combination of the remaining for operations that can lead to an ambigity are shown in Table I. TABLE I. Invariance of each of the cosines of Eq. (14) nder combinations of the symmetry operations of Eq. (15), exclding S π. Fll invariance (approximate invariance) is indicated by a ( ). Operation c + K c K c + c Non-resonant regime S sign S sign Resonant regime S K + ex S K ex S K + ex S K ex Even when one of the b cs amplitdes is small enogh that the determination of δ K becomes difficlt, one effectively has δ U δ U δ K, δ K, and the determination of δ K is again not aproblem. 8

9 While none of the operations leaves all for cosines invariant, it is important to note cases where c ± K are approximately invariant nder + SK ex, S K ex, or their prodct. We define approximate invariance nder the operation S to be S app c ± K (δ K (ξ)) = c ± K ( δ K (ξ)). (16) Approximate invariance arises de to the fact that far from the peak of the K resonance, δ K (ξ) changes slowly as a fnction of the Kπ invariant mass, and takes vales arond and π. Therefore, for events in the tails of the Breit-Wigner, δ K (ξ) is almost invariant nder any S app satisfying Eq. (16). One can see that approximate invariance of one of the cosines c ± K implies minimal change in the χ of Eq. (13), which may reslt in a resolved yet clearly observable ambigity. Since both c ± K terms are only approximately invariant nder the prodct S K + ex S K ex, this ambigity is more strongly resolved than either S K + ex or S K ex. Observing that no single operation in the Table I is a good symmetry of all cosines, one identifies two different regimes: In the non-resonant regime, interference with the nonresonant b cs is dominant, and only and S sign may lead to ambigities. In the resonant regime, the K amplitde strongly dominates the b cs decay, and S K + ex and S K ex become the important ambigities. In the transition between these regimes, the operations of Table I do not lead to clear ambigities, as we have verified by simlation (See sec. V). Ths, while naively one may expect a 5 -fold ambigity, in practice the observable ambigity is no larger than eight-fold, with only the two-fold S π being flly nresolved, in the likely case of non-negligible resonant contribtion. This is demonstrated in Fig. 8. Frthermore, althogh one may write down more prodcts of the operations, S sign, S K + ex, and S K ex, only the prodcts listed in Table I reslt in fll or partial invariance of both cosines which dominate the same regime. The additional prodcts do not reslt in any noticeable ambigities. V. MEASUREMENT SENSITIVITY AND SIMULATION STUDIES To stdy the feasibility of the analysis sing Eq. (13) and verify the predictions of Sec. IV, we condcted a simlation of the decays B ± D K ± π, B ± D K ± π,andb ± D 1, K ± π. Events were generated according to the PDF of Eq. (1), with the base parameter vales given in Table II. In this table and throghot the rest of the paper, we se a tilde to denote the tre parameter vales sed to generate events, while the corresponding plain symbols represent the trial parameters sed to calclate the experimental χ. The only non-vanishing amplitdes in the simlation were the non-resonant amplitdes in the b cs and b cs decays, and the K resonant b cs amplitde. For simplicity, additional resonances were not inclded in this demonstration. However, broad resonances that are observed in the data shold be inclded in the actal data analysis. The simlations were condcted with a benchmark integrated lminosity of 4 fb 1, which each of the asymmetric B-factories plan to collect by abot 5. The final state reconstrction efficiencies were calclated based on the capabilities of crrent Υ(4S) detectors. We assmed an efficiency of 7% for reconstrcting the K ±, inclding track qality and particle identification reqirements, and 6% for reconstrcting the π. The prodct of reconstrction efficiencies and branching fractions of the D, smmed over the final states 9

10 TABLE II. Parameters sed to generate events in the simlation. The vale of ÃCK is chosen so as to roghly agree with the measrement of the corresponding branching fraction [], taking into accont the K + K + π branching fraction. Parameter Vale Parameter Vale γ 1. Ã U /ÃC.4 δ C (ξ) = δ U (ξ) Ã CK /ÃC 1. δ K 1.8 Ã CK 1 4 Γ B δ U.4 TABLE III. The nmbers of events obtained by averaging 1 simlations sing the parameters of Table II and the reconstrction efficiencies listed in the text. Mode Signal events per 4fb 1 B + D K + π = B D K π 61 B + D K + π = B D K π 5 B + D 1, K + π 186 B D 1, K π 34 K π +, K π + π,andk π + π π +, is taken to yield an effective efficiency of 6%. Using the CP-eigenstate final states K + K, π + π, K S π,andk S ρ, the effective efficiency for the sm of the D 1 and D final states is.8%. All efficiencies are frther redced by a factor of 1.7, in order to approximate the effect of backgrond. The nmbers of signal events obtained in each of the final states with the above efficiencies and the parameters of Table II are listed in Table III. In Figs. 6 throgh 8, we show the dependence of χ on the vales of γ and δ U. The smallest vale of χ is shown as zero. At each point in these figres, χ is calclated with the generated vales of the amplitde ratios A U /A C = ÃU /ÃC and A CK /A C = ÃCK /ÃC. We note that when these amplitde ratios are determined by a fit simltaneosly with the phases, the correlations between the amplitdes and the phases are generally fond to be less than %. Therefore, the reslts obtained with the amplitdes fixed to their tre vales are sfficiently realistic for the prpose of this demonstration. For each of these figres, we also show the one-dimensional minimm projection χ (γ) = min{χ (γ,δ U ), showing the smallest vale of χ for each vale of γ. Fig. 6 is a simlation obtained with the parameters of Table II, bt with A CK =. With no resonant contribtion, the eight-fold ambigity of the perfect non-resonant regime is clearly visible. This wold be the typical case for two-body final states. Fig. 7 is obtained with the parameters of Table II, bt with A C =. Withnononresonant b cs contribtion, the eight-fold ambigity of the perfect resonant regime is seen. The ambigities corresponding to approximate invariance are clearly resolved, with 1

11 Dalitz Plot of B ± D K ± π D s ** K * D * FIG. 5. Dalitz plots obtained from a simlation of B + and B decays into all final state, D K ± π, D K ± π,andd 1, K ± π. with the parameters of Table II. Along with non-resonant contribtions, the resonances K, D,andDs are shown. the dobly-approximate S K + ex S K ex ambigity resolved more strongly. Fig. 8 is obtained with the parameters of Table II and shows how efficient the method described in this paper cold be for extracting the angle γ. With eqal resonant and nonresonant b cs amplitdes, only the non-resonant regime ambigities are observed, de to the relative sppression of the resonant interference terms discssed in Sec. IV. Nonetheless, the c ± K terms are significant enogh to resolve all bt the S π ambigity. S sign is more strongly resolved, since it leaves neither of the c ± K terms invariant. In Figs. 9 throgh 1 we present σ γ, the statistical error in the measrement of γ, obtained by fitting simlated event samples sing the MINUIT package [3], as a fnction of one of the parameters of Table II. All the other parameters were kept at the vales listed in Table II. Each point in these plots is obtained by repeating the simlation 5 times, to minimize sample-to-sample statistical flctations. In all cases, all the parameters of Table II were determined by the fit. The arrows in these figres indicate the point corresponding to the parameters of Table II. The total nmber of signal events in all final states combined is the same for each of the data points. The error bars describe the statistical error at each point, which is determined by the nmber of experiments simlated. 11

12 π D Likelihood scan 5 45 Strong phase δ U 3π/ π π/ π/ π 3π/ π Weak phase γ 1 1D Likelihood scan S sign S π Tre S sign S π S π S π S sign S sign - Likelihood π/ π 3π/ π Weak phase γ FIG. 6. Top: χ as a fnction of γ and δ U, with the parameters of Table II and no resonant contribtion (ÃC K =). Bottom: Minimm projection of χ onto γ. One observes that σ γ does not depend strongly on δ K, and has a mild dependence on δ U. As expected, strong dependence on ÃU /ÃC is seen in Fig. 1. However, it shold be noted that σ γ changes very little for all vales of à U /ÃC above abot.4. This sggests that the likelihood for a significantly sensitive measrement is high over a broad range of 1

13 π D Likelihood scan 5 45 Strong phase δ U 3π/ π π/ π/ π 3π/ π Weak phase γ 1 1D Likelihood scan K*- Tre S K*+ K*- ex S π K*- Sex S π K*- S π Sπ S K*+ K*- ex K*+ - Likelihood π/ π 3π/ π Weak phase γ FIG. 7. Top: χ as a fnction of γ and δ U, with no non-resonant b cs contribtion (ÃC =). Thevaleδ K =1. is sed to ensre that ambigities do not overlap. All other parameters are those of Table II. Bottom: Minimm projection of χ onto γ. parameters. With the parameters of Table II, we find σ γ.3=13 with an integrated lminosity of 4 fb 1. 13

14 π D Likelihood scan 5 45 Strong phase δ U 3π/ π π/ π/ π 3π/ π Weak phase γ 5.5 1D Likelihood scan S sign S π S sign S π S sign S sign Likelihood Tre S π S π π/ π 3π/ π Weak phase γ FIG. 8. Top: χ as a fnction of γ and δ U, with the parameters of Table II. Minimm projection of χ onto γ. Bottom: VI. CONCLUSIONS We have shown how γ may be measred in the color-allowed decays B D ( ) K ( ) π(ρ), focsing on the simplest mode B ± D K ± π. The absence of color sppression in the 14

15 .35.3 Average σ γ (rad) Weak phase γ (rad) FIG. 9. The error in γ, σ γ, as a fnction of γ. b cs amplitdes is expected to reslt in relatively large rates and significant CP violation effects, and hence favorable experimental sensitivities. Althogh the Dalitz plot analysis reqired for this prpose constittes some experimental complication, it shold not pose a major difficlty, while being very effective at redcing the eight-fold ambigities that constitte a serios limitation with other methods for measring γ. Only the two-fold S π ambigity cannot be resolved solely by or method, reqiring additional constraints from other measrements of the nitarity triangle. As a reslt of these advantages, this method is likely to lead to relatively favorable errors and provide a significant measrement of γ, even with the crrent generation of B factory experiments. VII. ACKNOWLEDGMENTS The athors thank Francois Le Diberder for his fritfl ideas and help with simlation. This work was spported by the CEA and CNRS-INP3 (France), and by the U.S. Department of Energy nder contracts DE-AC3-76SF515 and DE-FG3-93ER

16 .35.3 Average σ γ (rad) Strong phase δ U (rad) FIG. 1. The error in γ, σ γ, as a fnction of δ U. REFERENCES [1] N. Cabibbo, Phys. Rev. Lett. 1, 531 (1963); M. Kobayashi and K. Maskawa, Prog. Theor. Phys. 49 (1973), 65. [] BaBar Collaboration (B. Abert et al.), Phys. Rev. Lett. 86 (1), 515. [3] Belle Collaboration (B. Abe et al.), Phys. Rev. Lett. 87 (1), 918. [4] R. Aleksan, I. Dnietz and B. Kayser, Z. Phys. C 54 (199), 653. [5] A. Soffer, Phys. Rev. D6 (1999), 543; A. Soffer, Proceedings, Snowmass 1 Workshop on the Ftre of High Energy Physics, 1. [6] M. Grona and D. Wyler, Phys. Lett. B65 (1991), 17. [7] I. Dnietz, Phys. Lett. B7 (1991), 75. [8] D. Atwood, G. Eilam, M. Grona, and A. Soni, Phys. Lett. B341 (1995), 37. [9] J.H. Jang and P. Ko, Phys. Rev. D58 (1998), [1] M. Grona and J.L. Rosner, Phys. Lett. B439 (1998), 171. [11] D. Atwood, I. Dnietz, and A. Soni, Phys. Rev. Lett. 78 (1997), 357. [1] J.P. Silva and A. Soffer, Phys. Rev. D61 (), 111. [13] A. Soffer, hep-ex/98118 (1998). [14] CLEO Collaboration (B. Barish et al.), CLEO CONF 97-1, EPS

17 .35.3 Average σ γ (rad) Strong phase δ K* (rad) FIG. 11. The error in γ, σ γ, as a fnction of δ K. [15] Belle Collaboration (B. Abe et al.), hep-ex/191 (1). [16] CLEO Collaboration (T.E. Coan et al., hep-ex/1155 (1). [17] R.Aleksan et al., in preparation. [18] L. Wolfenstein, Phys. Rev. Lett. 51 (1983) [19] R. Aleksan, B. Kayser and D. London, Phys. Rev. Lett. 73, (1994) 18. [] See, for example, J.C. Anjos et. al., Phys. Rev. D48, 56 (199). [1] Belle Collaboration (B. Abe et al.), Preprint BELLE-CONF-35 (). [] The CLEO Collaboration, Phys. Rev. Lett. 88, 1183 (). [3] F. James, M. Roos, Compt. Phys. Commn (1975). 17

18 .7.6 Average σ γ (rad) A U / A C FIG. 1. The error in γ, σ γ, as a fnction of ÃU /ÃC. 18