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UdeM-GPP-TH-11-203 → → ¯ Measuring γ with B Kππ and B KKK Decays 2 1 0 Nicolas Rey-Le Lorier a,1 and David London b,2 2 n a: Cornell University, Ithaca, NY 14853, USA a J b: Physique des Particules, Université de Montréal, 1 C.P. 6128, succ. centre-ville, Montréal, QC, Canada H3C 3J7 1 ] h (January 13, 2012) p - p Abstract e h [ We present a method for cleanly extracting the CP phase γ from the Dalitz plots 3 of B+ K+π+π−, B0 K+π0π−, B0 K0π+π−, B0 K+K0K−, and B0 1v K0K0K¯→0. The B dK→ππ and B dK→KK¯ decays adre→related by flavor SUd(→3) 8 → → symmetry, but SU(3) breaking is taken into account. Most of the experimental 8 0 measurements have already been made – what remains is a Dalitz-plot analysis of 9. Bd0 → K0K0K¯0 (or Bd0 → KSKSKS ). We (very) roughly estimate the error on 0 γ to be 25%. This is somewhat larger than the error in two-body decays, but 1 ∼ it would be the first clean measurement of γ in three-body decays. Furthermore, 1 : at the super-B factory, it is possible that γ could be measured more precisely in v i three-body decays than in two-body decays. X r a PACS numbers: 11.30.Er, 13.20.He [email protected] [email protected] In the past, most of the theoretical work looking at clean methods for extracting weak-phase information in the B system focused on two-body decays. This is essentially because (i) final states such as ψK , π+π−, etc. are CP eigenstates, and S (ii) if there is a second decay amplitude, with a different weak phase, it has been possible to find methods to remove this “pollution,” and cleanly get at the weak phases. On the other hand, in three-body B decays, final states such as K π+π− S are not CP eigenstates – the value of its CP depends on whether the relative π+π− angular momentum is even (CP +) or odd (CP ). Furthermore, even if the CP of − the final state were determined in some way, one still has the problem of removing the pollution due to additional decay amplitudes. For these reasons, it has generally been thought that it is not possible to obtain clean weak-phase information from three-body decays [1]. Recently, it was shown that this is not true. By doing a diagrammatic analysis of the three-body amplitudes, one can resolve these two problems [2]. First, a Dalitz- plot analysis can be used to experimentally separate the CP + and components − of the three-particle final state. Second, one can often remove the pollution of additional diagrams and cleanly measure the CP phases. In fact, in Ref. [3], it was shown how to extract the weak phase γ from B Kππ decays. We briefly describe → this method below. In B Kππ decays, the isospin state of the ππ pair must be symmetric (anti- → symmetric) if the relative angular momentum is even (odd). As we will see below, it is the symmetric case which is most interesting. Here there are six possible decays: B+ K+π+π−, B+ K+π0π0, B+ K0π+π0, B0 K+π−π0, B0 K0π+π−, → → → d → d → and B0 K0π0π0. The first step is to express the amplitudes for these processes in d → terms of diagrams. The diagrams are as in two-body B decays [4]: the color-favored and color-suppressed tree amplitudes T and C, the gluonic-penguin amplitudes P tc and P , and the color-favored and color-suppressed electroweak-penguin (EWP) uc amplitudes P and PC . (We neglect annihilation- and exchange-type diagrams.) EW EW Furthermore, for three-body decays, it is necessary to “pop” a quark pair from the vacuum. The diagrams are written with subscripts, indicating that the popped quark pair is between two (non-spectator) final-state quarks (subscript ‘1’), or between two final-state quarks including the spectator (subscript ‘2’). (For B Kππ ¯ → decays, the popped quark pair is uu¯ or dd. Under isospin, these amplitudes are equal.) In addition, some time ago it was shown that, under flavor SU(3) symmetry, there are relations between the EWP and tree diagrams in B Kπ decays [5, 6]. → In Ref. [3], it was shown that similar EWP-tree relations hold for B Kππ decays. → Taking c /c = c /c for the Wilson coefficients (which holds to about 5%), these 1 2 9 10 take the simple form ′ ′ ′ ′ P = κT , P = κT , EW1 1 EW2 2 P′C = κC′ , P′C = κC′ , (1) EW1 1 EW2 2 1 where (s) 3 λ c +c κ | t | 9 10 , (2) ≡ −2 λ(us) c1 +c2 | | with λ(s) = V∗V . p pb ps Now, the EWP-tree relations assume SU(3) symmetry (and the approximate ratio of Wilson coefficients). The expected error due to SU(3)-breaking effects is ¯ ′ O(30%). However, the dominant diagram in b s¯ decays is P , so that EWPs → tc and trees are subleading effects. Thus SU(3) breaking is subdominant – the net theoretical error due to the use of the EWP-tree relations is only O(5%). This is consistent with the error estimates given in Ref. [5] (for EWP-tree relations in B Kπ). → In addition, there is an important caveat. Under SU(3), the final state in B → Kππ involves threeidentical particles, so that thesix permutations ofthese particles (thegroupS )mustbetakenintoaccount. Thatis, thethreeparticlesareinatotally 3 symmetricstate, atotallyantisymmetric state, oroneoffourmixedstates. However, the EWP-tree relations hold only for the totally symmetric state. Thus, the analysis must be carried out for this state. Now, the expressions for the B K(ππ) sym → amplitudes in terms of diagrams hold even under full SU(3) symmetry [3]. It is therefore only necessary to produce observables for the totally symmetric states. This is doable, and below we present the details of how this is carried out. With the above EWP-tree relations, the six B K(ππ) amplitudes can be sym → written in terms of 5 effective diagrams (i.e. linear combinations of the diagrams) [3]. There are therefore 10 theoretical parameters in the amplitudes3: 5 magnitudes of effective diagrams, 4 relative (strong) phases, and γ. On the other hand, there are 11 experimental observables. Given that B+ K0π+π0 is not independent (its amplitude is proportional to that of B0 K+π0π→−), these are the branching ratios and direct CP asymmetries of B+ dK→+π+π−, B+ K+π0π0, B0 K+π0π−, B0 K0π+π−, and B0 K0π0π→0, and the indire→ct CP asymmdet→ry of B0 Kd0π→+π− (the indirect CPd a→symmetry of B0 K0π0π0 will essentially be imposdsib→le d → to measure). Since there are more observables than theoretical parameters, γ can be extracted by doing a fit4. The disadvantage of this method is that it involves the decays B+ K+π0π0 → and B0 K0π0π0. With two π0 mesons in the final state, both of these decays d → will be extremely difficult to measure. We are therefore motivated to see if the 3In fact, the expressionfor any indirect CP asymmetry contains another theoretical parameter – the phase ofB0-B¯0 mixing, β. However,its value canbe takenfromthe indirectCP asymmetry d d in B0 J/ψK [7]. 4Tdh→ereisacSomplicationinthatthediagramsaremomentumdependent,asaretheobservables. Inobtainingthebest-fit“values”ofthediagrams,onewilldeterminethemomentumdependenceof their magnitudes and relativestrongphases. Onthe other hand, γ is independent of the particles’ momenta. Later in the paper, we detail how such a fit is done. 2 B Kππ method can be modified, avoiding these two decays. As we show below, → ¯ this can indeed be done – things can be considerably improved by using B KKK → decays. The use of these decays is quite natural since they, like B Kππ, are also ¯ → b s¯ transitions. → First, consider B Kππ decays with the ππ pair in a symmetric isospin state. → We leave aside B+ K+π0π0, B0 K0π0π0 and B+ K0π+π0 (since, as men- → d → → tioned above, its amplitude is not independent). The amplitudes of the remaining three processes are 2A(B0 K+π0π−) = T′eiγ +C′eiγ P′ P′C , d → sym 1 2 − EW2 − EW1 √2A(B0 K0π+π−) = T′eiγ C′eiγ P˜′ eiγ +P˜′ d → sym − 1 − 1 − uc tc 1 2 1 + P′ + P′C P′C , 3 EW1 3 EW1− 3 EW2 √2A(B+ K+π+π−) = T′eiγ C′eiγ P˜′ eiγ +P˜′ → sym − 2 − 1 − uc tc 1 1 2 + P′ P′C + P′C . (3) 3 EW1− 3 EW1 3 EW2 In the above, P˜′ P′ +P′. (As B Kππ is a ¯b s¯ transition, the diagrams are ≡ 1 2 → → written with primes.) Here we have explicitly written the weak-phase dependence (this includes γ and the minus sign from V∗V [P˜′ and EWPs]), while the diagrams tb ts tc contain strong phases. Second, consider B KKK¯ decays. For the case in which the final KK pair is in a symmetric isosp→in state, there are four such processes: B+ K+K+K−, B+ K+K0K¯0, B0 K+K0K−, and B0 K0K0K¯0. Here, B+→ K+K+K− and→B+ K+K0K¯d0→are not independentd–→their amplitudes are pr→oportional to those of B→0 K+K0K− and B0 K0K0K¯0, respectively. These are d → d → √2A(B0 K+K0K−) = T′ eiγ C′ eiγ Pˆ′ eiγ +Pˆ′ d → sym − 2,s − 1,s − uc tc 2 1 2 1 + P′ P′ + P′C P′C , 3 EW1,s − 3 EW1 3 EW2,s − 3 EW1 A(B0 K0K0K¯0) = Pˆ′ eiγ Pˆ′ (4) d → sym uc − tc 1 1 1 1 + P′ + P′ + P′C + P′C , 3 EW1,s 3 EW1 3 EW2,s 3 EW1 where Pˆ′ P′ +P′. In the above, certain diagrams are written with the subscript ≡ 2,s 1 ‘s.’ This indicates that the popped quark pair is ss¯. When the diagram has no subscript s (the penguin or EWP diagrams), this means that the popped quark pair ¯ is uu¯ or dd, but the virtual particle decays to ss¯. We now assume flavor SU(3) symmetry. This has two consequences. First, the ¯ amplitudewithapoppedss¯quarkpairisequaltothatwithapoppeduu¯ordd. That is, we no longer need the subscript s on diagrams. This means that the diagrams in B KKK¯ decays are the same as those in B Kππ decays. Second, the → → EWP-tree relations of Eq. (1) hold. 3 Thus, under SU(3) the amplitudes of Eqs. (3) and (4) take the form 2A(B0 K+π0π−) = T′eiγ +C′eiγ κ(T′ +C′) , d → sym 1 2 − 2 1 √2A(B0 K0π+π−) = T′eiγ C′eiγ P˜′ eiγ +P˜′ d → sym − 1 − 1 − uc tc 1 2 1 ′ ′ ′ + κ T + C C , 3 1 3 1 − 3 2 (cid:18) (cid:19) √2A(B+ K+π+π−) = T′eiγ C′eiγ P˜′ eiγ +P˜′ → sym − 2 − 1 − uc tc 1 1 2 ′ ′ ′ + κ T C + C , 3 1 − 3 1 3 2 (cid:18) (cid:19) √2A(B0 K+K0K−) = T′eiγ C′eiγ P˜′ eiγ +P˜′ d → sym − 2 − 1 − uc tc 1 1 2 ′ ′ ′ + κ T C + C , 3 1 − 3 1 3 2 (cid:18) (cid:19) A(B0 K0K0K¯0) = P˜′ eiγ P˜′ d → sym uc − tc 2 1 1 ′ ′ ′ + κ T + C + C . (5) 3 1 3 1 3 2 (cid:18) (cid:19) Note that this implies that A(B+ K+π+π−) = A(B0 K+K0K−) . Fur- → sym d → sym ther, we reiterate that the above expressions for the amplitudes hold also for the totally symmetric final state, to which the EWP-tree relations apply. We now define the following five effective diagrams: ′ ′ ′ T T T , a ≡ 1 − 2 ′ ′ ′ T C +T , b ≡ 2 2 P′ P˜′ +T′ +C′ , a ≡ uc 2 1 1 2 1 ′ ˜′ ′ ′ ′ P P +κ T + C C , b ≡ tc 3 1 3 1 − 3 2 (cid:18) (cid:19) ′ ′ ′ C κ(C C ) . (6) a ≡ 1 − 2 The amplitudes can be written in terms of these five diagrams: 2A(B0 K+π0π−) = T′eiγ +T′eiγ C′ κT′ , d → sym a b − a − b √2A(B0 K0π+π−) = T′eiγ P′eiγ +P′ , d → sym − a − a b √2A(B+ K+π+π−) = P′eiγ +P′ C′ , → sym − a b − a √2A(B0 K+K0K−) = P′eiγ +P′ C′ , d → sym − a b − a 1 A(B0 K0K0K¯0) = P′eiγ T′eiγ C′eiγ d → sym a − b − κ a ′ ′ ′ ′ P +κT +κT +C . (7) − b a b a As with the B Kππ method, five effective diagrams corresponds to 10 theo- → retical parameters: 5 magnitudes of diagrams, 4 relative phases, and γ. But there 4 are 11 (momentum-dependent) experimental observables: the decay rates and direct asymmetries for the four decays B0 K+π0π−, B0 K0π+π−, B0 K+K0K− and B0 K0K0K¯0 (we ignore Bd+→ K+π+π− sdin→ce its amplitudde→is not inde- pendendt)→, and the indirect asymmetri→es of B0 K0π+π−, B0 K+K0K− and B0 K0K0K¯0. With more observables thdan→theoretical padra→meters, γ can be d → extracted from a fit. We now present the details of how the fit is carried out. Consider the decay B P P P , in which the three pseudoscalar mesons P (i = 1-3) have momenta 1 2 3 i → p . From these, we can construct the three Mandelstam variables: i s (p +p )2 , s (p +p )2 , s (p +p )2 . (8) 12 1 2 13 1 3 23 2 3 ≡ ≡ ≡ These are not independent, but obey s +s +s = m2 +m2 +m2 +m2 . (9) 12 13 23 B 1 2 3 Experimentally, the Dalitz plot of this decay is measured. Its events are given in terms of two Mandelstam variables, say s and s . Now, the great advantage of a 12 13 Dalitz-plot analysis is that it allows one to extract the full amplitude of the decay. We write (B P P P ) = c eiθjF (s ,s ) , (10) 1 2 3 j j 12 13 M → j X where the sum is over all decay modes (resonant and non-resonant). c and θ are j j the magnitude and phase of the j contribution, respectively, measured relative to one of the contributing channels. The distributions F , which depend on s and s , j 12 13 describethedynamicsoftheindividual decayamplitudes, andtakedifferent(known) forms for the various contributions. The key point is that a maximum likelihood fit over the entire Dalitz plot gives the best values of the c and θ . Thus, the decay j j amplitude can be obtained, up to an overall normalization. This normalization is fixed by the constraint of the measured partial rate [7]: 1 1 Γ = 2ds ds . (11) (2π)332m3 |M| 12 13 B Z With this, the decay amplitude (s ,s ) is known. 12 13 M As will be seen below, we rely heavily on (s ,s ). In particular, we use it 12 13 M to obtain the observables for the B P P P decay. As such, the errors on these 1 2 3 → observables come entirely from the uncertainty in (s ,s ). While, as noted 12 13 M above, it is possible to obtain the best-fit values of the Dalitz-plot variables c and j θ , there are errors associated with these values. This is due to two sources. First, j one has the statistical error in the experimental Dalitz plot. Second, there is a systematic uncertainty related to the choice of the F in Eq. (10). In addition, there j is a statistical error in the overall normalization [coming from Eq. (11)]. All of these 5 must be carefully taken into account in order to obtain conservative errors on the Dalitz-plot variables. As noted earlier, the EWP-tree relations hold only for the totally symmetric SU(3) decay amplitude. But this can be found from the above: 1 = [ (s ,s )+ (s ,s )+ (s ,s ) fully sym 12 13 13 12 12 23 M √6 M M M + (s ,s )+ (s ,s )+ (s ,s )] . (12) 23 12 23 13 13 23 M M M Using this, it is possible to compute the B P P P observables. However, recall 1 2 3 → that the method involves a fit using the observables from several different decays (B0 K+π0π−, B0 K0π+π−, B0 K+K0K− and B0 K0K0K¯0). All d → d → d → d → observables must involve the same Mandelstam variables. On the other hand, the numbering of final-state particles is arbitrary, so that s for one decay might equal 12 s for a different decay. All of this makes it somewhat confusing to ensure that 13 observables in different decays have thesame Mandelstam variables. For thisreason, it is useful at this stage to change notation (but the physics is unchanged). In any decay there are three Mandelstam variables. We define s++, s+ and s− to be the largest, second-largest, and smallest of these, respectively. The identities of the particles which are associated with s++, s+ and s− are irrelevant (e.g. s++ can correspond to s , s or s ). This is consistent with the assumption of SU(3) and 12 13 23 the fully symmetric decay amplitude. With these Mandelstam variables, we have 1 fully sym = [ (s++,s+)+ (s+,s++)+ (s++,s−) M √6 M M M + (s−,s++)+ (s−,s+)+ (s+,s−)] . (13) M M M Since s++, s+ and s− are not independent, this gives the fully symmetric amplitude as a function of two Mandelstam variables, say s and s . ++ + The observables are obtained as follows. First, one forms the totally symmetric SU(3) decay amplitudes as in Eq. (13) for each B P P P decay ( ) and 1 2 3 fully sym its CP conjugate ( ¯ ). Second, using the→se, for specific vaMlues of s and fully sym ++ M s , one computes the partial rates: + 1 1 Γ = (s ,s ) 2 , s++,s+ (2π)332m3 |Mfully sym ++ + | B 1 1 Γ¯ = ¯ (s ,s ) 2 . (14) s++,s+ (2π)332m3 |Mfully sym ++ + | B These allow the computation of the CP-averaged branching ratio and direct CP asymmetry: 1 ¯ BR = Γ +Γ , s++,s+ Γ s++,s+ s++,s+ B (cid:16) (cid:17) ¯ Γ Γ A = s++,s+ − s++,s+ . (15) s++,s+ Γ +Γ¯ s++,s+ s++,s+ 6 Third, for those decays in which the final state is accessible to both B0 and B¯0 d d mesons, one has an indirect (mixing-induced) CP asymmetry. It is given by ¯ (s ,s ) S = Im e−2iβ Mfully sym ++ + . (16) s++,s+ " fully sym(s++,s+)# M Asdiscussed earlier, inallcases, theerrorontheobservables isfoundbypropogating the errors onthe Dalitz-plotvariables. These include bothstatistical and systematic effects. Now, given that the method assumes flavor SU(3) symmetry, one would like to know how SU(3) breaking affects the analysis, and what is its size. Leaving aside the EWP-tree relations, in which SU(3)-breaking effects are subdominant, there are two areas where the breaking may be significant. First, under SU(3), the diagrams in B KKK¯ and B Kππ are the same. Since both decays are ¯b s¯ transitions, the→difference betwe→en them is that B KKK¯ decays have an ss¯q→uark pair in the ¯ → ¯ final state, hadronizing to KK, while B Kππ decays have uu¯ or dd, hadronizing → to ππ. This is essentially the same for each diagram. (The SU(3)-breaking effect associated with an ss¯pair being popped from the vacuum may not be exactly equal to that when ss¯ is produced in the decay of a virtual particle, but the difference is small.) Thus, including SU(3) breaking, the amplitudes of Eq. (7) can be written 2A(B0 K+π0π−) = T′eiγ +T′eiγ C′ κT′ , d → sym a b − a − b √2A(B0 K0π+π−) = T′eiγ P′eiγ +P′ , d → sym − a − a b √2A(B+ K+π+π−) = P′eiγ +P′ C′ , → sym − a b − a √2A(B0 K+K0K−) = (1+f ) P′eiγ +P′ C′ , d → sym SU(3) − a b − a h 1i A(B0 K0K0K¯0) = (1+f ) P′eiγ T′eiγ C′eiγ d → sym SU(3) a − b − κ a (cid:20) ′ ′ ′ ′ P +κT +κT +C ] , (17) − b a b a where f is the SU(3)-breaking factor. Second, under SU(3), π’s and K’s are SU(3) identical particles, so that there is no difference between the Mandelstam variables for the processes B KKK¯ and B Kππ. There is therefore an SU(3)-breaking → → effect between the fully symmetric decay amplitudes for the two types of decay. However, it can be included in f . SU(3) The addition of f brings the number of unknown theoretical parameters SU(3) to 11. In principle, these can all be determined from a fit to the 11 experimental observables, albeit with discrete ambiguities. However, we can do better. Above it was noted that, in the limit of perfect SU(3), A(B+ K+π+π−) = A(B0 K+K0K−) . This means that f can be determin→ed by a compsyamrison of tdhe→se sym SU(3) two decays. In particular, τ (B0 K+K0K−) +B d → sym = (1+f )2 . (18) τ (B+ K+π+π−) SU(3) 0 sym B → 7 In fact, this comparison can be performed now since the decays have been measured: B+ K+π+π− inRef.[8], B0 K+K0K− inRef.[9]. Now, sincetheEWP- → d → tree relations [Eq. (1)] have been used to derive the expressions for the amplitudes, Eq. (18) holds only for totally symmetric states. Using the technique described above, one can obtain A(B+ K+π+π−) and A(B0 K+K0K−) . → fully sym d → fully sym Inordertogetthebranchingratios, wecomputetheintegralofthesquareofthefully symmetric amplitudes over the Dalitz plot (taking care to avoid sextuple counting). Doing this gives (B+ K+π+π−) = 0.19 (B+ K+π+π−) , fully sym B → B → (B0 K+K0K−) = 0.50 (B0 K+K0K−) . (19) B d → fully sym B d → From Ref. [10], we have τ /τ = 0.93, (B+ K+π+π−) = (51.0 2.9) 10−6 and 0 + (B0 K+K0K−) = (24.7 2.3) 1B0−6. E→q. (18) then gives ± × B d → ± × f = 0.17 0.06 . (20) SU(3) ± (This error does not include the errors in the parameters obtained from the Dalitz- plot analyses of the two decays.) We can now put all the pieces together to describe how the fit is to be performed. The fully symmetric amplitudes for the decays B0 K+π0π−, B0 K0π+π−, B0 K+K0K− and B0 K0K0K¯0 are given ind E→q. (17). Theydar→e a function d → d → of 10 unknown parameters, including γ. The value of f is taken from Eq. (20). SU(3) The 11 observables and their errors are computed as described above – the (fully symmetric) branching ratios and direct CP asymmetries are given in Eq. (15), and the indirect CP asymmetries in Eq. (16). Note that these are for specific values of s and s . One has a different set of observables for each (s ,s ) pair. With 10 ++ + ++ + unknowns and 11 constraints, one can now perform the fit. This will determine the magnitudes and relative strong phases of the five effective diagrams, as well as γ, all for the chosen values of (s ,s ). This is to be repeated for each independent ++ + (s ,s ) pair5. This has two effects. First, one will be able to fix the momentum ++ + dependence of the diagrams. Second, and more importantly, since γ is momentum independent, one can average over all the (s ,s ) fits. This will reduce its error, ++ + perhaps considerably. Now, we already have experimental information about most of the required B Kππ and B KKK¯ decays. In particular, the measurements of the Dalitz plo→ts of B0 K+π→0π−, B0 K0π+π− and B0 K+K0K− are described in Refs. [11], d → d → d → [12], and [9], respectively. On the other hand, we do not yet have the Dalitz plot of B0 K0K0K¯0. The branching ratio and CP asymmetries of B0 K K K d → d → S S S are given in Ref. [17]. While the use of the final state K K K is excellent – S S S 5The two pairs (s++,s+)1 and (s++,s+)2 are considered as independent if fully sym((s++,s+)1) and fully sym((s++,s+)2) do not overlap when one takes into |M | |M | account the errors on the Dalitz-plot parameters of Eq. (10). 8 it is proportional to the fully symmetric state of K0K0K¯0 – the observables are momentum independent. That is, an integration over the Dalitz plot has been performed. However, the method described in this paper requires the momentum- dependent observables. Once the Dalitz plot for B0 K K K is known, this d → S S S method for extracting γ can be carried out. Eventhoughalltheexperimental dataisnotyet available, wecanstillattemptto estimatetheprecisionwithwhichγ canbeobtained. ConsiderfirstB0 K+K0K−. d → According totheBaBarmeasurement inRef.[9], thelargest contributions tothisde- caycomefromtheφK0 andf K0 resonances, andthe(K+K−) K0,(K+K0) K− 0 NR NR and (K−K0) K+ non-resonant pieces. They find NR φK0 : c = 0.0085 0.0010 , j ± f K0 : c = 0.622 0.046 , 0 j ± (K+K−) K0 : c = 1 (fixed) , NR j (K+K0) K− : c = 0.33 0.07 , NR j ± (K−K0) K+ : c = 0.31 0.08 , (21) NR j ± where c is defined in Eq. (10). The errors, which are statistical only, range from 7% j to 25%. The above method describes how to obtain (B0 K+K0K−) from the amplitude given in Ref. [9], and from this the BM0fullyKsy+mK0Kd →− observables. d → Afull numerical analysis isneeded to do this, properly taking into account the errors on the c above, as well as the errors on the θ and F of Eq. (10), and the other j j j resonances. However, a rough guess is that the errors on the observables will be about 20%. Similarly, we (guess)timate that the errors on the observables of the otherdecays, includingthoseofB0 K0K0K¯0, willbe 20%. Inordertoobtainγ, d → ∼ a fit to the observables must be performed, taking into account the SU(3)-breaking factor of Eq. (20) (the error on f will increase once the errors in the Dalitz- SU(3) plot parameters are included), and one must average over the independent (s ,s ) ++ + pairs. It is impossible to predict with any accuracy what the error on γ will be, but an error of O(25%) does not seem unreasonable. How does this compare with the precision on γ measured in two-body decays? The answer is: not that badly. The standard way of directly measuring γ uses B D0/D¯0K decays within the GLW [14] or ADS [15] methods. The latest mea→surement yields γ = (68+10)◦ [16], i.e. the error is 15%. To be sure, our −11 ∼ estimated error of O(25%) on the value of γ as extracted from three-body decays is worse than 15%. However, it is still roughly the same size, and if a full analysis were done, the real error might turn out to be smaller than our estimate. More to the point, when the Dalitz-plot measurements are done at the super-B factory, the Dalitz-plot parameters will be obtained with a smaller statistical error. This will have two effects. First, the error on γ will be reduced for each (s ,s ) pair. ++ + Second, one will have more independent (s ,s ) pairs, so the error will be further ++ + reduced when one averages over all the (s ,s ) fits. (See the discussion following ++ + 9