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The use of early data on B ρπ decays → Helen R. Quinn and João P. Silva∗ Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309, USA (February 1, 2008) ThispaperreviewstheDalitzplotanalysisforthedecaysB0(B¯0) ρπ π+π−π0. Wediscuss → → whatcanbelearnedaboutthetenparametersinthisanalysisfromuntaggedandfromtaggedtime- integrateddata. Wefindthat,withtheimportantexceptionoftheinterestingCP violatingquantity α, the parameters can be determined from this data sample – and, hence, they can be measured at CLEO as well as at the asymmetric B factories. This suggests that the extraction of α from the time-dependent data sample can be accomplished with a smaller data sample (and, therefore, sooner) thanwouldberequiredifalltenparametersweretobeobtained fromthattime-dependent datasamplealone. Wealsoexploreboundsontheshiftofthetrueangleαfromtheanglemeasured 0 from charged ρfinalstates alone. Thesemay beobtained priortomeasurements of theparameters 0 describing theneutral ρ channel, which are expected tobe small. 0 2 11.30.Er, 13.25.Hw, 14.40.-n. n a J I. INTRODUCTION 8 2 The B factories at SLAC and KEK are now collecting data and expect to produce measurements of the CP 1 asymmetry in the mode B J/ψK within one year,measuring sin2β. Preliminaryresults onthis mode from CDF S v → [1] already indicate that it is unlikely that a discrepancy with the Standard Model will be found from this result 0 alone. ThismeansthattestsoftheStandardModelmechanismforCP violationwillrelyuponourabilitytomeasure 9 furtherCP-violatingparameters,suchasthe angleα=π β γ ofthe Unitaritytriangle,withsufficientaccuracyto 2 − − 1 besensitivetoStandardModelrelationships. Thiswillrequirethatwemastertheremovaloftheoreticaluncertainties 0 due to penguindiagramsin atleastoneofthe availablechannels. So farthere aretwosets ofcandidatedecaymodes, 0 ππ [2] and ρπ [3], for which analyses to extract α using isospin relationships have been suggested. The first suffers 0 fromrelativelysmall branchingratios[4]and fromthe experimentaldifficulty ofmeasuringthe π0π0 branchingratio. / h SomeofthemodesforthesecondcasehaverecentlybeenobservedatCLEO[5]. Theseresultsareencouraging. They p are close to model-dependent predictions and, hence, appear to reinforce estimates that the analysis suggested by - Snyder and Quinn will require of order 180 fb−1, or six years of running at BABAR design luminosity, to complete p [6]. e h This paper revisits that analysis and reviews what intermediate steps can be made with earlier data samples. In : particularwestressthatmanyoftheparametersrelevanttotheextractionofαcanbe predeterminedfromuntagged, v andfromtaggedbuttime-integrateddatasamples. Thebenefitofthisisthat,oncethisisdone,onlytheCP-violating i X parameter α remains to be fit to the full taggedand time-dependent Dalitz plot. Presumably it will then be possible r to perform the ρπ analysis with smaller data samples than those needed if the full set of ten parameters is fit to the a tagged,time-dependentsamplealone(ashasbeendonetodateinMonteCarlostudiesofthis mode). Wealsodiscuss bounds on the shift in α from penguin contributions. These can be obtained, by methods similar to those previously suggestedfortheππ modes[7–9],eveniftheρ0π0 amplitudeistoosmalltobemeasureddirectly. Wewillalsodiscuss a bound which applies only in the ρπ case, having no parallel in the ππ case. This paper presents a purely theoretical discussion, with no simulations and no attempt to address issues of backgrounds or of other modes that may contribute to the π+π−π0 Dalitz plot [10]. Certainly these issues will be important. The larger untagged data sample will also be the first place to explore the issues of further contributions to the Dalitz plot. The approach of fixing as many parameters as possible from the untagged and from the tagged but time-integrated data samples, combined with improvements in machine luminosity, such as those already under study for PEPII, may make the extraction of a reliable value for α a reality on a somewhat faster time scale than suggested by the estimates based on fits to tagged data only. ∗Permanent address: InstitutoSuperior deEngenharia deLisboa, Rua Conselheiro Em´ıdio Navarro, 1900 Lisboa, Portugal. 1 II. NOTATION A. Isospin decomposition We are interested in the decays from B+ and B0 into ρπ final states. The decay amplitudes can be classified accordingtoisospin: B+,B0 formanisospindoublet; the finalstateρπ canhaveisospinI =0,I =1,andI =2 f f f { } components. In general, the final state with I = 0 can only be reached with operators having ∆I = 1/2; the final f state with I = 1 can be reached with operators having ∆I = 1/2 or ∆I = 3/2; and the final state with I = 2 can f f be reached with operators having ∆I =3/2 or ∆I =5/2. We denote the isospin amplitudes by A . Thus, ∆I,If 1 3 1 1 1 1 a =a B+ ρ+π0 = A A + A A , +0 → 2 2 3/2,2− 2√2 3/2,1 √2 1/2,1 −√6 5/2,2 r (cid:20) (cid:21) (cid:0) (cid:1) 1 3 1 1 1 1 a =a B+ ρ0π+ = A + A A A , 0+ → 2 2 3/2,2 2√2 3/2,1− √2 1/2,1 −√6 5/2,2 r (cid:20) (cid:21) (cid:0) (cid:1) 1 1 1 1 1 a =a B0 ρ+π− = A + A + A + A + A , +− → 2√3 3/2,2 2 3/2,1 2 1/2,1 √6 1/2,0 2√3 5/2,2 (cid:20) (cid:21) (cid:0) (cid:1) 1 1 1 1 1 a =a B0 ρ−π+ = A A A + A + A , −+ → 2√3 3/2,2− 2 3/2,1− 2 1/2,1 √6 1/2,0 2√3 5/2,2 (cid:20) (cid:21) (cid:0) (cid:1) 1 1 1 a =a B0 ρ0π0 = A A + A . (1) 00 → √3 3/2,2− √6 1/2,0 √3 5/2,2 (cid:20) (cid:21) (cid:0) (cid:1) Similar relations hold for the B¯0 decay amplitudes, which we denote by a¯ , a¯ , a¯ , a¯ , and a¯ , respectively. −0 0− −+ +− 00 Note that our notation here is that a¯ is the amplitude for the B¯0 to decay to a ρ of charge i and a π of charge j. ij The CP relationships are thus CP(a )=a¯ . The B¯0 isospin components are A¯ . ij −i,−j ∆I,If In the Standard Model, there are tree-level amplitudes, gluonic penguin amplitudes, electroweak penguin amplitudes, and final state rescattering effects. The tree level b uu¯d decays have both ∆I = 1/2 and ∆I = 3/2 → components. In contrast, the gluonic b d penguins are pure ∆I =1/2,because the gluonis pure I =0. Therefore, → the isospin amplitudes A and A only receive contributions (and, thus, weak phases) from the tree-level dia- 3/2,1 3/2,2 grams. There are no diagrammatic∆I =5/2contributions atthis order;such effects ariseonly from electromagnetic corrections to the weak-decay diagrams. Apriori,thereisnohierarchyamongthe∆I =1/2and∆I =3/2isospinamplitudes. However,thecombinationof isospin amplitudes involved in the decay B0 ρ0π0 is generally argued to be suppressed because the tree-level and → gluonic penguin diagrams can only contribute to B0 ρ0π0 through a color-suppressedrecombination of the quarks in the final state. This argument is not theoretically→rigorous because B0 ρ0π0 may be fed from other topologies → through strong final state rescattering. Eventually experiment will tell us whether these effects are important. InwhatfollowswewillneglecttwocontributionswhicharesuppressedintheSM.Thefirstcontributionarisesfrom the electroweak penguin diagrams, which have the same weak phase structure as the QCD penguin diagrams, but contribute to both ∆I = 1/2 and ∆I = 3/2 amplitudes. As a result, the electroweak penguin contributions are not removed by the isospin-based analyses. However, they are expected to be very small in these channels [6,11]. The secondcontribution is due to a possible ∆I =5/2isospin component, included within square bracketsin Eqs. (1). In the SM, this component comes from electromagnetic rescattering effects and, thus, it is suppressed by α 1/127.1 Both effects could become relevant for B0 ρ0π0, should this amplitude turn out to be very small. ∼ → Henceforth, we will only consider the tree-leveland gluonic penguin diagrams. These give contributions with weak phases V∗V ub ud =eiγ = e−i(β+α), V∗V − | ub ud| V∗V tb td =e−iβ. (2) V∗V | tb td| 1Notice, however, that this effect is important in K ππ decays because there one has a strong hierarchy among the decay → amplitudes, ReA2/ReA0 1/22, encoded into the∆I =1/2 rule[13–15]. ∼ 2 Thefirstoftheseexpressionsistheweakphaseofthetree-levelcontributions. Herewefollow[12]andusetheunitarity relationship to rewrite the penguin contribution as a dominant term proportional to the second of the CKM factors in Eq. (2) plus a sub-dominant term proportional to the first CKM factor in Eq. (2).2 In what follows we always subsume this second term within the amplitudes we refer to as ∆I =1/2 tree amplitudes. Since we do not calculate these quantities, but rather discuss extracting them from fits to experiment, this makes no difference to our analysis. Note,however,thatiftheamplitudesextractedinthiswayaretobecomparedtothosecalculatedinanygivenmodel, then the penguin contributions to our so-called “tree” amplitudes must be taken into account. The interference between the two amplitude contributions in Eq. (2) depends on the weak phase α. However, we will show that, as is usual with direct CP-violation effects, any sensitivity to α is masked by an unknown coefficient with large theoretical uncertainty. Another weak phase arises from the interference between the B0 B0 mixing, − q/p=exp[ 2i(β θ )], and the tree-level diagrams. For example, d − − qA¯ 3/2,2 =e−2i(β−θd)e−2iγ =e2i(α+θd), (3) pA 3/2,2 where θ parametrizes a possible new physics contribution to the phase in B0 B0 mixing. In the SM, θ = 0 and d d − the two phases coincide; other models have θ = 0 and a difference arises. The phase probed by the Snyder–Quinn d 6 method is α+θ [14]. d B. B0 π+π−π0 decay amplitudes → The ρ-mediated B0 π+π−π0 decay amplitudes may be written as → A=a B0 π+π−π0 =f a +f a +f a , + +− − −+ 0 00 → A¯=a (cid:0)B0 π+π−π0(cid:1) =f a¯ +f a¯ +f a¯ , (4) + +− − −+ 0 00 → (cid:16) (cid:17) where f and f are Breit-Wigner functions representing ρ± and ρ0, respectively, and also include the cosθ angular ± 0 dependencesofthehelicityzeroρdecays. ThecrucialobservationmadebySnyderandQuinnisthattheBreit-Wigner functions contain CP-even phases, and that the interference between the different Breit-Wigner shapes across the Dalitz plot provides experimental sensitivity to the weak and strong phases contained in Eqs. (4). There is some systematic uncertainty associated with the specific choice made for the shape of the functions f and f . This ± 0 uncertainty affects primarily the corners of the Dalitz plot where the tails of two such functions interfere; these are the regions of the Dalitz plot from which one extracts the interference between two distinct decay amplitudes. It will prove convenient to rewrite Eqs. (4) as A=f a +f a +f a , c c d d n n A¯=f a¯ +f a¯ +f a¯ , (5) c c d d n n where f +f f f f + − + − 0 f = , f = − , f = , (6) c d n 2 2 2 and a =a +a , a =a a , a =2a , c +− −+ d +− −+ n 00 − a¯ =a¯ +a¯ , a¯ =a¯ a¯ , a¯ =2a¯ . (7) c +− −+ d +− −+ n 00 − Notice that the CP-conjugate of a is not a¯ , but rather a¯ . d d d − As discussed in the appendix, we parametrize the amplitudes a , a , and a as c d n 2The term is sub-dominant in that it is a difference of up and charm quark contributions and, hence, it vanishes in thelimit that these two quark masses are taken to be equal. 3 a =Te−iα(1 z r ), c 0 − − a =Te−iα(z +r ), d 1 1 a =Te−iα(z+r ), (8) n 0 where z and z are CP-even, while r and r are CP-odd. The quantity Tz contains the CP-even contributions to 1 0 1 1 the final state with I = 1, summing the tree amplitude and the penguin amplitude multiplied by cosα. Tr (Tr ) f 1 0 contains the CP-odd part, given by the penguin contributions to the final state with I = 1 (I = 0), multiplied by f f isinα. Similarly, the amplitudes contained in qA¯ /p can be written as f q a¯ =e2iθdTeiα(1 z+r ), c 0 p − q a¯ =e2iθdTeiα( z +r ), d 1 1 p − q a¯ =e2iθdTeiα(z r ). (9) n 0 p − We have chosen to define strong phases so that T is a real positive quantity. Notice that a +a = Te−iα, and c n q/p(a¯c +a¯n) = e2iθdTeiα. Therefore, the imaginary (real) part of the ratio of these quantities, measures the sine (cosine) of the phase in Eq. (3). If a and a¯ are indeed color suppressed, then z and r will be smaller than 1, z and r . In that case, the 00 00 0 1 1 CP-violating difference a 2 a¯ 2 | c| −| c| = 2Rer +2Re(zr∗) (10) 2T2 − 0 0 will also be small. In connection with Eqs. (4) and (5), there is no advantage of one parametrization with respect to another. We could equally well have chosen to write the amplitudes in a new set of basis functions f ,f ,f , given by 1 2 3 { } f +f +2f f f f +f f + − 0 + − + − 0 f = , f = − , f = − . (11) 1 2 3 6 2 3 In this basis, the amplitudes would become A=f (a +a )+f a +f (a a /2) 1 c n 2 d 3 c n − A¯=f (a¯ +a¯ )+f a¯ +f (a¯ a¯ /2). (12) 1 c n 2 d 3 c n − Eqs.(4)highlighttheintermediateρπstates;Eqs.(5)highlighttheCP structureoftheintermediatechargedρπstates, but still treat the intermediate ρ0π0 separately;and Eqs. (12) highlight the extraction of a +a =Te−iα. Although c n each basis has its pedagogical advantages, they have no experimental significance. What one does experimentally is a maximum likelihood fit of the variables T, z , r , z, and r to all the data in the Dalitz plots. 1 1 0 C. Observables in the decay rates In the B system we have q/p 1 and ∆Γ Γ. Therefore, d | |∼ ≪ Γ[B0 f]+Γ[B0 f] A 2+ A¯ 2, (13) f f → → ∝| | | | q Γ[B0 f] Γ[B0 f] A 2 A¯ 2 cos∆mt 2Im A¯ A∗ sin∆mt. (14) → − → ∝ | f| −| f| − p f f (cid:18) (cid:19) (cid:0) (cid:1) One may rewrite these expressions using λ =qA¯ /(pA ).3 The three terms in Eqs. (13) and (14) become 1+ λ 2, f f f f 1 λ 2, and Imλ , respectively. The untagged decay rate in Eq. (13) probes only the combination 1 + |λ |2, f f f − | | | | regardless of whether these measurements are time-dependent or time-integrated. Due to the anti-symmetric nature 3Notice, however,that some authors use theopposite sign convention for q/p and, thus,for λ . f 4 ofthe B0 B0 pairsproducedatthe Υ(4s),tagged,time-integratedmeasurementsperformedatfacilities workingon this reson−ance can probe 1+ λ 2 and 1 λ 2, but not Imλ . f f f If f is a CP eigenstate and| th|e decay−am|pli|tudes A and A¯ are defined in such a way that the rates A 2 and f f f A¯ 2 includeallthephasespaceintegrations,then1+ λ 2 isCP-even,while1 λ 2 andImλ areCP-od|d. |Under f f f f | | | | −| | these conditions, the ratioof Eq.(14) to Eq. (13)is the famous CP-violating asymmetry. It is sometimes stated that one canonly probe Imλ /(1+ λ 2). While this is true if one looksonly at theCP-violating asymmetry,one cansee f f | | fromEqs.(13)and(14)that,inprinciple,thereisenoughinformationinthetwodecayratesforacleandetermination of Imλ . The caveat is that disentangling Imλ from 1 λ 2 requires that all these quantities be affected by the f f f ±| | same normalization. This may not be true once the experimental cuts, in particular possible cuts on the time t, are folded into the analysis. Using Eqs. (5) we find A2 A¯2 = a 2 a¯ 2 f 2+2 Re f f∗ a a∗ a¯ a¯∗ , (15) | | ±| | | i| ±| i| | i| i j i j ± i j i i<j X(cid:0) (cid:1) X (cid:2) (cid:0) (cid:1)(cid:3) whereiandjcantakethevaluesc,d,andn. Thenotationi<jmeansthatthe(i,j)pairsarenotrepeated. Untagged decaysprobetheobservablescorrespondingtothe+sign. Tagged,time-integrateddecaysprobetheobservableswith both signs (or, what is the same, they measure A2 and A¯2 separately). Since the functions f are quite distinct in i their Dalitz plot structure one can treat the coe|ffic|ient of|ea|ch different pair f f∗ as a separate observable.4 We may i j define the nine untagged observables by A2+ A¯2 | | | | =T2 U f 2+2 URRe f f∗ 2 UI Im f f∗ , (16) 2  ii| i| ij i j − ij i j  i i<j i<j X X (cid:0) (cid:1) X (cid:0) (cid:1)   where a 2+ a¯ 2 i i U = | | | | , ii 2T2 Re a a∗+a¯ a¯∗ UR = i j i j (i=j), ij 2T2 6 (cid:0) (cid:1) Im a a∗+a¯ a¯∗ UI = i j i j (i=j). (17) ij 2T2 6 (cid:0) (cid:1) We will refer to these observables genericallyas the U-observables. We may define similar quantities with U replaced by D to describe the corresponding observables obtained for the difference A2 A¯2. These D-observables are | | −| | obtained from Eqs. (17) by replacing the ‘+’ signs with ‘-’ signs. Using Eqs. (8) and (9), the untagged observables are a 2+ a¯ 2 U = | c| | c| = 1 z 2+ r 2, (18) cc 2T2 | − | | 0| a 2+ a¯ 2 U = | d| | d| = z 2+ r 2, (19) dd 2T2 | 1| | 1| a a∗+a¯ a¯∗ UR +iUI = c d c d =r∗ zr∗ r z∗, (20) cd cd 2T2 1 − 1 − 0 1 a a∗ +a¯ a¯∗ UR +iUI = c n c n =z∗ z 2 r 2, (21) cn cn 2T2 −| | −| 0| a a∗ +a¯ a¯∗ UR +iUI = d n d n =z r∗+r z∗, (22) dn dn 2T2 1 0 1 a 2+ a¯ 2 U = | n| | n| = z 2+ r 2. (23) nn 2T2 | | | 0| 4Note, however, that the errors in the extraction of these various observables are correlated. Thus, our observables, while distinctarenottechnicallyindependent. Adiscussion oferrorcorrelations isbeyondthescopeofthispaper,butwill ofcourse be important in theapplication of this approach to data. 5 Thus the U contain nine different functions of the four complex parameters z ,r . The requirement that the combi- i i i nation U +2UR +U equals 1 yields the overall normalization T. cc cn nn Similarly, the tagged, time-integrated measurements will provide the additional observables a 2 a¯ 2 D = | c| −| c| = 2Rer +2Re(zr∗), (24) cc 2T2 − 0 0 a 2 a¯ 2 D = | d| −| d| =2Re(z r∗), (25) dd 2T2 1 1 a a∗ a¯ a¯∗ DR +iDI = c d− c d =z∗ zz∗ r r∗ (26) cd cd 2T2 1 − 1 − 0 1 a a∗ a¯ a¯∗ DR +iDI = c n− c n =r∗ 2Re(zr∗), (27) cn cn 2T2 0 − 0 a a∗ a¯ a¯∗ DR +iDI = d n− d n =z z∗+r r∗, (28) dn dn 2T2 1 1 0 a 2 a¯ 2 D = | n| −| n| =2Re(zr∗). (29) nn 2T2 0 NotethatD +2DR+D =0;thusonlyeightdifferentcombinationsofT,z ,r ,z,andr getfixedbymeasurements cc cn nn 1 1 0 of the D-observables. Muchcanbe learnedaboutthe interplaybetweenthe DalitzplotanalysisandCP-violationbylookingatEqs.(18) through (29). From the definitions of r and r in Eqs. (A3), we know that these quantities involve the product of a 0 1 penguin contribution with sinα. Any nonzero value for r and/or r signals direct CP-violation. It is true that such 0 1 quantities do not by themselves allow us to determine the size of CP-violation, because sinα appears multiplied by anunknownparameter;thetheoreticalcalculationofthisparameterisplaguedbylargehadronicuncertainties. (This is as expected for any observable probing direct CP-violation.) As expected, the quantities D , D , DR , DI , and D are CP-odd. Surprisingly, the quantities UR, UI , UR, cc dd cn cn nn cd cd dn and UI are also CP-odd, despite the fact that they are obtained by looking for untagged decays. How does this dn come about? The reasonis that there is a source of sensitivity to CP-violationinduced in the Dalitz plot analysisby the fact that ρ and ρ are CP-conjugate of each other. Said otherwise, when one performs a CP-transformation + − on f a , one obtains f a¯ and not a quantity proportional to f . As pointed out before, this means that a CP + +− − −+ + transformationona yields a¯ . Therefore,the quantities linear ina anda¯ havepeculiar CP-properties;UR, UI , UR, and UI are CPd -odd, w−hidle DR, DI , DR , and DI are CP-evedn. d cd cd dn dn cd cd dn dn Additional observablesare obtained in the tagged, time-dependent decays, which containa sin∆mt term givenby q Im A¯A∗ =T2 I f 2+ II Re f f∗ + IRIm f f∗ . (30) p  ii| i| ij i j ij i j  (cid:18) (cid:19) i i<j i<j X X (cid:0) (cid:1) X (cid:0) (cid:1)   Here q I =Im a¯ a∗ /T2, ii p i i (cid:18) (cid:19) q II =Im a¯ a∗+a¯ a∗ /T2 (i=j), ij p i j j i 6 (cid:20) (cid:21) q (cid:0) (cid:1) IR =Re a¯ a∗ a¯ a∗ /T2 (i=j). (31) ij p i j − j i 6 (cid:20) (cid:21) (cid:0) (cid:1) As before, i and j take the values c, d, and n, and the notation i < j means that no (i,j) pair gets repeated in the sum. As a result, we have nine new observables. These observables depend on sin2(α+θ ) and cos2(α+θ ), with d d coefficients given in table I. We stress that the uncertainties associatedwith the exact shape chosenfor the ρ resonancesare likely to affect the U , D , and I coefficients less than they affect the coefficients U , D , and I with i=j ii ii ii ij ij ij 6 III. ANALYSIS This sectionreviews whatcan be learnedin the various experimentalsearches. Eightof the ten parametersneeded to extract α can be fit with untagged decays alone. This greatly increases the data sample that will be available for 6 determining these parameters, since at the B-factories tagging efficiencies are estimated to be of order 0.3 (or less). Further,manyquestionsaboutbackgroundsandthecontributionsoftheotherresonancestothethreepionDalitzplot canalsobegintobeansweredusingthislargerdatasampleofuntaggedevents;thoughtheymustalsobere-examined in the tagged data sample, where non-B background will presumably be reduced. Fitting to tagged, time-integrated events, fixes one further parameter and gives eight additional measurements that depend on combinations of the eightparametersalreadyfixed,thusimprovingtheprecisionoftheirdetermination. OnlytheimportantCP-violating CKM-related parameter α+θ remains to be fit to the tagged time-dependent Dalitz plot data. Our conclusion is d that these preliminary steps can and should be performed at both symmetric and asymmetric B-factories. A. Observables from untagged decays The observables U through U can be combined to yield T, z, r , z , r and arg(z r∗). Therefore, the cc nn 1 | 1| | 0| 1 0 nine observables present in untagged decays, U through U , allow us to measure eight quantities and give one cc nn mathematically redundant piece of information, which of course serves to further constrain that combination of observables. The result of this analysis is that the large data sample of untagged decays is extremely important for the final determination of α in the B ρπ channels. One may use untagged decays to measure all relevant quantities except → one angle—the angle between z (or r ) and z (or r )—and the CP-violating phase α+θ . Moreover, one will be 1 0 1 d sensitive to direct CP-violation through r and r . 0 1 | | | | B. Observables from tagged time-integrated decays The subset of events corresponding to tagged, time-integrated decays will provide measurements of the additional observablesD throughD . SinceU andD areexpectedtobesmall,itisinterestingtonotethattheremaining cc nn nn nn 16 U- and D-observables are sufficient to fix the nine parameters T, z , r , z, and r , and give seven mathematically 1 1 0 redundant pieces of information. That means that one does not need to probe quantities quadratic in a and a¯ n n (that is, one does not need to have an observable ρ0π0 branching fraction) in order to determine all the| ob|servab|les| attainablewiththesemeasurements. ShouldU andD bemeasured,theywillprovidetwofurthermathematically nn nn redundantpiecesofinformation. Allthesenineparametershavemodel-independentinformationthat,likeabranching fraction,canbetakenfromoneexperimentandusedasinputinanother. Therefore,theseexperimentscanandshould beperformedbothatCLEOandattheasymmetricB-factories. Theadvantageofthisisthatfewerparametersremain to be determined from the relatively small time-dependent data sample. C. Observables from tagged time-dependent decays Onceadatasampleoftagged,time-dependenteventsisavailable,theirratescontainasin∆mtterm,allowingfora measurementof I through I . (One expects that, for some time to come, this data sample will be small compared cc nn to the data samples discussed above.) The I-observables depend on sin2(α+θ ) and cos2(α+θ ), with coefficients d d givenintableI.We havealreadyseenthatthe combinationofuntaggedandtaggedtime-independent decaysyieldT, z , r , z, and r . These must now be combined with the observables in I through I . A combination of any pair 1 1 0 cc nn of these I-observables yields sin2(α+θ ) and cos2(α+θ ) independently. A fit to all the observables determines d d 2(α+θ ) (up to discrete ambiguities) and provides additional mathematically redundant pieces of information. d Weshouldpointoutthat,asexpected,onecannotextractα+θ unlesssomeinformationisknownaboutquantities d linear in a and a¯ . However, since these amplitudes into neutral ρ0π0 might be color suppressed, it is interesting n n to ask what one may learn while only bounds, rather than measurements, on a and a¯ are known. In section IV n n | | | | we will show that bounds on these quantities can be combined with measurementsof quantities involving the charged ρ±π∓ channels in order to determine α+θ , up to an error that decreases with a and a¯ . d n n | | | | IV. TWO USEFUL BOUNDS Inthissectionwesupposethatthe bandsinthe Dalitz plotcorrespondingto B ρ±π∓ havebeenmeasuredwhile → only bounds on the pieces linear and quadratic in a and a¯ are known. We will show that one may still find bounds n n on α+θ , with an error that decreases as a and a¯ decrease. d n n | | | | 7 A. One useful bound As we have seen before, a measurement of quantities which are independent of a and a¯ is not enough for a n n determination of α+θ . In particular, we can see from Eqs. (18), (24) and the first entry in Table 1 that measuring d the parameters U , D , and I , which refer only to the decays into ρ±π∓, is enough to determine5 cc cc cc Im qa¯c Icc = pac =sin[2(α+θ +δ )], (32) Uc2c−Dc2c (cid:16)pqaa¯cc (cid:17) d α p (cid:12) (cid:12) where we have defined (cid:12) (cid:12) (cid:12) (cid:12) 1 z+r 0 2δ =arg − . (33) α 1 z r (cid:18) − − 0(cid:19) Unfortunately, δ is neither zero nor calculable. α However, as we will now show, bounds on a and a¯ are enough to constrain δ . As these bounds decrease, the n n α | | | | deviation of Eq. (32) from sin2(α+θ ) also decreases. d z+r 0 1-z+r r 0 0 z 1-z 2 δ α -r0 z-r0 1-z-r 0 FIG.1. Triangleconstructionfor1 z r0andz r0. Thesecomplexnumbersarerelatedtoa¯c,ac,an,anda¯n,respectively. − ± ± To prove this, we use Fig. 1 to derive 2r 2 = 1 z+r 2+ 1 z r 2 21 z+r 1 z r cos(2δ ), (34) 0 0 0 0 0 α | | | − | | − − | − | − || − − | and 2r z+r + z r . (35) 0 0 0 | |≤| | | − | Using these equations and the expressions for a , a¯ , a , and a¯ in Eqs. (8) and (9), we find n n c c | | | | | | | | (a + a¯ )2 (a a¯ )2 sin2δ | n| | n| − | c|−| c| . (36) α ≤ 4a a¯ c c | | Notice that, since we assume a and a¯ to be known, we only need to know bounds on quantities linear in a and c c n | | | | a¯ in order to get bounds on a and a¯ . The observables quadratic in a and a¯ are not needed. We have thus n n n n n proved that one can determine|(α|+θ )|by|measuring only quantities from charged ρ±π∓ final states, up to an error d that decreases as the bounds on the color suppressed amplitudes B ρ0π0 decrease. → It is interesting to compare the bound in Eq.(36), with similar bounds obtainedpreviously for the B ππ decays → [7–9]. WenoteasimilaritybetweentheparametrizationoftheB ππ decaysandtheparametrizationoftheB ρπ → → decays, related through a 2A B0 π0π0 , n ↔ → ac √2A(cid:0) B0 π+(cid:1)π− , (37) ↔ → (cid:0) (cid:1) 5Notice that √Uc2Icc−cDc2c ∼ UIcccc 1+ 2DUc2c2cc . Therefore, measurements of Icc and Ucc determine sin[2(α+θd+δα)], up to an error that is of second order in D(cid:16)cc (secon(cid:17)d order in r0). 8 and similarly for the complex conjugated channels. After these substitutions and some straightforward algebra we can write a bound similar to that in Eq. (36) as 2 (B0 π0π0)+ (B0 π0π0) 1 B → B → cos(2δππ)≥ 1 a2 1−2(cid:18)p (B0 π+π−)+q(B0 π+π−) (cid:19) , (38) − dir  B → B →    p     where (B0 π+π−) (B0 π+π−) a = B → −B → . (39) dir (B0 π+π−)+ (B0 π+π−) B → B → Ifthere isonly dataonuntaggedB π0π0 decays,wemaystill obtainaboundby substituting the squaredquantity ontherighthandside(RHS)ofEq.→(38)by2 (B0 π0π0)+2 (B0 π0π0),aboundpreviouslyderivedbyCharles B → B → [8]. If, in addition, there is no data on a , then we may obtain a weaker bound by setting a to one on the RHS of dir dir Eq. (38) [7–9]. In this form, the bound depends only on untagged data and it is related to a bound obtained earlier by Grossman and Quinn [7] by using B± π+π0 decays instead of B π+π− decays on the RHS of Eq. (38). → → B. A bound from interference effects One may find a much cleaner bound by rewriting Eq. (33) in the form 1+x 2δ =arg , (40) α 1 x − where r D +DR +iDI x= 0 = cc cn cn. (41) 1 z − U +UR +iUI − cc cn cn We find 2Imx tan(2δ )= . (42) α 1 x2 −| | If the quantities z and r are of order one, then we expect to be able to measure them. The case of interest here is 0 when z 1, r 1, and the best we can do is place bounds on quantities linear in these variables, while we expect 0 ≪ ≪ to be able to measure U and I . In this case, we can constrain the allowed values for tan(2δ ) by combining the cc cc α measurement of U with the bounds on UR, UI , DR and DI . (In effect, to gain any information, we need the cc cn cn cn cn bounds on the magnitudes of the latter quantities to be smaller than the value for U .) cc Notice that Eq. (41) involves UR, UI , DR and DI which are linear in a and a¯ . That means that here one is cn cn cn cn n n using the interference between the tails of two different ρπ channels. This has two consequences. Firstly, this bound, which is extremely powerful, has no analogue for the B ππ decays. Secondly, since the bound depends on the → interferenceeffects,itmaybesensitivetotheassumptionsmentionedaboveabouttheexactshapeoftheρ-resonances. V. SOME EXPERIMENTAL ISSUES Thisanalysishastakenapurelyformalapproachandhasnotevaluatedalltherelevantexperimentalquestions. The quantitieswhichwecall“mathematicallyredundant”areactuallyadditionaldatasamplesthatcontributetofixingthe parameters. The parameterswe define areallintrinsic tothe physicalprocessandhence are,likebranchingfractions, expected to be the same in any experiment. One issue that will be important experimentally is that the errors on the variousparameters are highly correlatedand must be treated correctlyin establishing bounds suchas those from Eq. (42) and the allowed range for α. Moreover, backgrounds, efficiencies, and cuts will differ in the different data samplesandmustbeinvestigatedseparatelyineachcase. Agoodknowledgeofthevariationsinefficienciesacrossthe Dalitzplotisalsoessentialforthisanalysis. TheextractionoftheparameterT issensitivetotheknowledgeofoverall efficiencies; it maybe better to simply define a T measuredseparatelyfor eachdata sample thanto depend onthe eff 9 accuracy with which the overall efficiency and luminosity for each data sample is known. Other caveats have been mentioned, such as the need to explore the sensitivity of the results to reasonable changes in the assumed ρ-shape parameterization, and the recognition of the possible large backgrounds in the untagged sample. All of these issues andmany more willonly be settled by examiningthe data.6 None of themappear to us to invalidate the expectation that it will be very valuable, at least in the early years of study of these modes, to use the parameters determined from the time-integrated experiments when studying the CP violating effects in the time-dependent data. VI. CONCLUSIONS The B ρπ decays are described by ten parameters, c.f. Eqs. (8) and (9). We have shown that untagged data → canbe usedto extracteightofthese ten parametersand taggedtime-integrateddata allowsevaluationofone further parameter. ThisleavesonlytheoneCP-violatingparameterα(α+θ ifnewphysicscontributesanextraphaseθ to d d the mixing) to be determined from time-dependent data. The parameters in question are defined in an experiment- independent fashion and hence the values measured in a time-integrating experiment, such as can be pursued at CLEO, can be used as input to fits of the time-dependent data sample – thereby possibly allowing a measurement of the parameter α+θ earlier than could be achieved without this input. d We have also shown that, if the neutral ρ contributions are small, then, prior to the time when the statistics are sufficienttoprovideameasurementofthese effects,boundsontheircontributionwillallowboundsontheshiftofthe angle measured from the rates and interference of the two charged ρ channels from the true value α. ACKNOWLEDGMENTS WeareindebtedtoY.GaofordiscussionsconcerningtheB ρπ measurementsatCLEO,andtoY.Grossmanfor → discussionsregardingtheboundsonδ andforreadingthismanuscript. ThisworkissupportedbytheDepartmentof α EnergyundercontractDE-AC03-76SF00515. TheworkofJ.P.S.issupportedinpartbyFulbright,InstitutoCamo˜es, and by the Portuguese FCT, under grant PRAXIS XXI/BPD/20129/99and contract CERN/S/FIS/1214/98. APPENDIX: PARAMETRIZING THE DECAY AMPLITUDES In this appendix we parametrize the B0 π+π−π0 decay amplitudes by breaking them into CP-even and CP- → odd components. This will allow us to see more clearly what quantities may be measured with the various types of experiments. We may decompose the isospin amplitudes into tree-level and penguin contributions as A = T eiγ, 3/2,2 3/2,2 − A = T eiγ, 3/2,1 3/2,1 − A = T eiγ +P e−iβ, 1/2,1 1/2,1 1/2,1 − A = T eiγ +P e−iβ, (A1) 1/2,0 1/2,0 1/2,0 − where we have used the fact that A and A only receive contributions from the tree-level diagrams. For 3/2,1 3/2,2 convenience,we have included an explicit minus signin the definitions of the tree-levelamplitudes. Substituting into Eqs. (1) and (7), and dropping the ∆I =5/2 amplitude, we find 1 2T 2P 2P eiβa =Te−iα + 1/2,0 + 1/2,0 cosα+i 1/2,0 sinα , c 3 3 T 3 T 3 T " r r r # T T P P eiβa =Te−iα 3/2,1 + 1/2,1 + 1/2,1 cosα+i 1/2,1 sinα , d T T T T (cid:20) (cid:21) 2 2T 2P 2P eiβa =Te−iα 1/2,0 1/2,0 cosα i 1/2,0 sinα, (A2) n "3 −r3 T −r3 T − r3 T # 6Onecangetsomeideaoftheimpactofsomeofthembysimulationsbasedonmodelvaluesfortheparameters. Suchstudies are in progress [16]. 10