A measurement of γ from the decays Bd0(t)→D(*)+D(*)− and Bd0→Ds(*)+D(*)−

Recently, it was proposed to use measurements of B0 d (t) → D(∗)+D(∗)− andB0 d → D(∗)+ s D(∗)− decays to measure th CP phaseγ . In this Letter, we present the extraction of γ using this method. We find that γ is favored to lie in one of the ranges[19.4◦–80.6◦](+0◦ or 180◦), [120◦–147◦](+0◦ or 180◦), or [160◦–174◦](+0◦ or 180◦) at 68% confidence leve (the (+0◦ or 180◦) represents an additional ambiguity for each range). These constraints come principally from the vector final states; the vector–pseudoscalar decays improve the results only slightly. Although, with present data, the c disappear for larger confidence levels, this study does demons trate the feasibility of the met hod. Strong constraints on γ can be obtained with more data.  2004 Elsevier B.V. Recently, two of us(A.D.,D.L.) proposed a method for extracting the CP phase γ from measurements o B0 d (t) → D(∗)+D(∗)− andB0 d → D(∗)+ s D(∗)− decays[1]. The technique is quite straightforward. Consider pseudoscalar–pseudoscalar (PP) decay B0 d → D+D−. The amplitude for this decay recei v s several contributions described by tree, exchange, b̄ → d̄ penguin and color-suppressed electroweak penguin diagrams [2]: A = (T + E + Pc)V ∗ cbVcd + PuV ∗ ubVud + ( Pt + P EW ) V ∗ tbVtd = T + E + Pc − Pt − P EW ) V ∗ cbVcd + ( Pu − Pt − P EW ) V ∗ ubVud (1) ≡Act eiδct +Aute eiδut . E-mail addresses: justin@hep.caltech.edu (J. Albert),datta@physics.utoronto.ca (A. Datta),london@lps.umontreal.ca (D. London). Open access under CC BY license. 0370-2693 2004 Elsevier B.V. doi:10.1016/j.physletb.2004.11.046 Open access under CC BY license. 336 J. Albert et al. / Physics Letters B 605 (2005) 335–343

. The technique is quite straightforward. Consider the pseudoscalar-pseudoscalar (PP) decay B 0 d → D + D − . The amplitude for this decay receives several contributions, described by tree, exchange,b →d penguin and color-suppressed electroweak penguin diagrams [2]: Open access under CC BY license.
Open access under CC BY license.
Here, A ct ≡ |(T + E + P c − P t − P C EW )V * cb V cd |, A ut ≡ |(P u − P t − P C EW )V * ub V ud |, P i is theb →d penguin amplitude with an internal i-quark, and we have explicitly written out the strong phases δ ct and δ ut , as well as the weak phase γ . The second line is obtained by using the unitarity of the CKM matrix, V * ub V ud + V * cb V cd + V * tb V td = 0, to eliminate the V * tb V td term. The amplitudeĀ D for the decayB 0 d → D + D − can be obtained from the above by changing the signs of the weak phases.
There are three observables which can be obtained from a time-dependent measurement of this decay: B (the branching ratio), a dir (the direct CP asymmetry) and a indir (the indirect CP asymmetry). In terms of the above parameters, these can be written where δ ≡ δ ut − δ ct . Here, β is the phase of B 0 d -B 0 d mixing, which has been measured in the CP asymmetry in B 0 d (t) → J/ψK S [3]. However, these observables still depend on four unknown theoretical parameters: the two magnitudes A ct and A ut , one relative strong phase δ, and the weak phase γ . We therefore have three observables, but four theoretical unknowns. Thus, in order to obtain weak phase information, it is necessary to add some theoretical input. 1 This input comes from the decay B 0 d → D + s D − , which receives tree,b →s penguin and color-suppressed electroweak penguin contributions [2]: (The primes on the amplitudes indicate ab →s transition.) Here, the last line arises from the fact that V * ub V us /V * cb V cs 2%, so that the piece proportional to V * ub V us is negligible. The measurement of the total rate for B 0 d → D + s D − therefore yields A ct . We now make the SU(3) flavour assumption that where sin θ c is the Cabibbo angle. With this assumption, the knowledge of A ct can be used to give us A ct . This in turn means that the three observables in B 0 d → D + D − depend on only three theoretical unknowns: A ut , δ, and γ . We can therefore now solve for γ (up to discrete ambiguities).
The solution for γ is as follows. We introduce a fourth (nonindependent) observable, a R : One can obtain a R from measurements of B, a dir and a indir , up to a sign ambiguity. One can then easily obtain Given the knowledge of 2β, the assumption in Eq. (4) therefore allows us to obtain γ . The leading-order theoretical error in this technique is given simply by the SU(3)-breaking ratio of decay constants f D s /f D [1]. (There are other errors, such as the neglect of the E amplitude in A ct , but these are expected to be smaller.) This ratio has been computed quite precisely on the lattice: f D s /f D = 1.20 ± 0.06 ± 0.06 [5]. With this value, the theoretical error in this method is rather small, so that γ can be extracted from measurements of is unavailable, so we cannot apply the above method to this decay. However, the vector-vector (VV) decays B 0 d (t) → D * + D * − and B 0 d → D * + s D * − have been measured. The method can be applied in a similar way to these decays, but with some additional complexity as described below. The main theoretical error is given by f D * s /f D * . A modification of this method can be used when the final state is not self-conjugate, as is the case for vectorpseudoscalar (VP) final states. 2 Consider the decay B 0 d → D * + D − . Following Eq. (1), its amplitude can be written (Although we use the same symbols, the amplitudes and strong phases are not the same as those for B 0 d → D + D − .) Now consider the decay of aB 0 d meson to the same final state,B 0 d → D * + D − . The amplitude for this decay is not simply related to that for B 0 The measurement of B 0 d (t) → D * + D − still yields three observables, B, a dir and a indir , but now they take more complicated forms. The three observables for the D * + D − final state are: where δ ≡ δ ut − δ ct ,δ ≡δ ut −δ ct and ≡ δ ct −δ ct . For the D + D * − final state, the observables arẽ 2 A similar method is described in [6]. However, there is a mistake in that method, due to the fact that the hadronization is different for the Considering that β has been experimentally determined, the 6 observables are written in terms of 8 theoretical unknowns: A ut , A ct ,Ã ut ,Ã ct , γ , δ,δ and . We therefore need 2 assumptions to extract information. These come from using the decays B 0 The measurements of the branching ratios for these decays allow us to extract A ct andÃ ct , respectively. With motivation similar to that for the PP mode above, we assume that With these assumptions, the measurements of the branching ratios for B 0 give us A ct andÃ ct . We now have 6 observables and 6 theoretical unknowns, thus we can solve for γ (up to discrete ambiguities). The main theoretical error is the deviation from unity of f D * s /f D * and f D s /f D in the first and second assumption above, respectively.
In this Letter we extract γ via this method, using BaBar and Belle data on the VV and VP modes [7][8][9][10][11][12][13]. We determine the value of γ to be in one of the ranges , where the (+0 • or 180 • ) represents an additional ambiguity for each range. The first of these ranges, [19.4 • -80.6 • ], is the range that is favored by other, external information on the unitarity triangle (assuming the Standard Model) [14][15][16]. As we will see, the constraints on γ come principally from the data on VV modes; at present, the VP decays add only a small amount. Note also that if we consider a larger C.L. (e.g., 90%), the constraints on γ disappear. Thus, even though we obtain limits on γ at 68% C.L., our main purpose here is to demonstrate the feasibility of the method. With more data, the constraints on γ will be correspondingly stronger. Eventually this method can be used to obtain a precision determination of γ .
We begin with the analysis of vector-vector decays. In order to extract γ , however, we must make additional assumptions. VV final states come in three transversity states, 0, , and ⊥. The amplitudes 0 and are CP-even, while ⊥ is CP-odd. The data shows that the D * D * final state is almost entirely CP-even, i.e., the ⊥ amplitude is negligible [11]. Unfortunately, at present experiments cannot distinguish between the 0 and amplitudes. This has the following effect. For a single transversity σ , the three observables in B 0 d (t) → D * + D * − can be written similarly to Eq. (2): where δ σ ≡ δ ut σ − δ ct σ . However, for all three observables, what is measured is the sum of the helicities 0 and : Because the parameters A ct , A ut and δ are different for the two helicities 0 and , there are too many theoretical unknowns to apply the method. We would like to cast the expressions for the observables in Eq. (13) in the same form as those in Eq. (2). In order to do this, we must relate the parameters for the 0 and helicities. Specifically, we assume that the strong phases are equal: δ 0 = δ ≡ δ. We also assume that the amplitudes are proportional to one another (with the same proportionality constant, c): A ct,0 = cA ct, , A ut,0 = cA ut, . That is, our assumptions are: With these definitions, the observables in Eq. (13) take exactly the same form as Eq. (2), and the method for extracting γ can be applied. In fact, these assumptions are theoretically reasonable. The amplititudes for B 0 d → D * D * andB 0 d →D * D * are given by (15) A VV,σ = A ct,σ e iδ ct σ + A ut,σ e iδ ut σ e iγ ,Ā VV,σ = A ct,σ e iδ ct σ + A ut,σ e iδ ut σ e −iγ , with σ = 0, , ⊥ being the three transversity states. We now define the ratios of amplitudes: Our assumptions are equivalent to k ct = k ut , i.e., both amplitudes and phases are equal. Below, we investigate the extent to which these relations hold true. We note that, in general, we can write any amplitude in terms of factorizable and nonfactorizable pieces: where we denote the factorizable contributions by an index 'F '. The quantities r σ , ρ σ , s σ and λ σ parametrize the ratios of nonfactorizable and factorizable amplitudes. Consider first only the factorizable contributions. In this case, the ratios of factorizable amplitudes are The factorizable amplitude for the decayB 0 d →D * D * is given by [17] (19) where X = X 1 + X 2 , and In the above, a j = c j + c j −1 /N c , where the c j are Wilson coefficients. From Eqs. (19) and (20), we can read off the individual factorizable amplitudes: However, note that the strong phases come from a u,c 4,10 , and appear only in the factors X 1,2 in Eq. (20). These factors are independent of transversity. That is, the relative strong phases between the factorizable ct and ut amplitudes are independent of the polarization state, leading to Arg(k F ct ) = Arg(k F ut ). Furthermore, the expressions in Eq. (21) above lead to Thus, we have k F ct = k F ut , i.e., the factorizable contributions satisfy our assumptions.
We now consider the nonfactorizable contributions. If these pieces are independent of polarization (at least for the σ = 0, states), then in Eq. (17) we will have This leads to k ct = k ut , so that our assumptions will be satisfied. Our assumptions are therefore invalid only to the extent that the nonfactorizable pieces are transversity-dependent. In the heavy-quark limit with m b,c → ∞, there is only one universal form factor resulting from the spin symmetry of the theory. This implies that the (factorizable) P σ D * 's in Eq. (21) are proportional for different polarization states. In other words, the various transversity amplitudes are related to one another. It is likely that these relations remain true in the presence of nonfactorizable corrections. This then implies Eq. (23). We therefore expect deviations from Eq. (23) to be suppressed by O(1/m c,b ). In all, the net correction to our assumptions is O (1/m c,b ) times the ratio of nonfactorizable and factorizable effects. We expect this to be small, so that the assumptions in Eq. (14) are justified.
Finally, we note that our assumptions can be tested. In the presence of nonfactorizable effects of the form in Eq. (23), Eq. (19) can be rewritten as where X T = X 1T + X 2T , and As X T is common to both σ = 0, states we find the relative phase between these transversity amplitudes is 0 or π . This prediction can be checked through an angular analysis of B → D * D * decays. Note also that the assumptions in Eq. (14) are not in fact required in the original method of Ref. [1]. Eventually it will be possible to experimentally separate out the 0 and the components, making such assumptions unnecessary.
With the assumptions of Eq. (14), we can determine the value of γ from the VV decays, using the method above. We use the measurements of the B → D ( * )D( * ) and B → D ( * ) SD ( * ) branching fractions, measurements of the B → D ( * )D( * ) CP asymmetries, and the world-average values of sin 2β (0.736 ± 0.049) [3] and sin 2 θ c (0.0482 ± 0.0010) [18]. We take 2β to lie in the first quadrant.
We use a toy Monte Carlo (MC) method to determine the confidence intervals for γ . We consider 500 values for γ , evenly spaced between 0 and 2π . For each value of γ considered, we generate 25000 toy MC experiments, with inputs that span the range of the experimental errors of each quantity. For each experiment, we generate random values of each of the experimental inputs according to Gaussian distributions, with means and sigmas according to the measured central value and total errors on each experimental quantity. We make the assumption that the ratio f D * s /f D * is equal to f D s /f D = 1.20 ± 0.06 ± 0.06 [5]. An additional theory error of 10% is included to take into account the assumptions of Eq. (14), as well as smaller errors such as the neglect of the exchange diagram, subdominant SU(3)-breaking terms, etc. We then calculate the resulting values of A ct , a dir , a indir , and B, given the generated random values (based on the experimental values). Inputting the quantities a dir , a indir , and B, along with β and the value of γ that is being considered, into Eq. (6), we obtain a residual value for each experiment, equal to the difference of the left-and right-hand sides of the equation. One thus obtains an ensemble of residual values from the 25000 experiments. A likelihood, as a function of γ , can be obtained from χ 2 (γ ), where χ 2 ≡ (µ/σ ) 2 , in which µ is the mean of the above ensemble of residual values and σ is the usual square root of the variance. The value of χ(γ ) is then considered to represent a likelihood which is equal to that of a value χ standard devations of a Gaussian distribution from the most likely value(s) of γ . We define the "exclusion level", as a function of the value of γ , as follows: the value of γ is excluded from a range at a given C.L. if the exclusion level in that range of γ values is greater than the given C.L.      Fig. 2 shows a check on the confidence distribution to ensure that it accurately describes the level of uncertainty on the measured value of γ .
We now turn to the VP decays B 0 The advantage of the VP method is that no additional assumptions of the type described in Eq. (14) are needed. The disadvantage is that, as we will see, the data are such that no information on the most likely regions of γ can be obtained from the VP modes.
In order to implement the VP method, we proceed as follows. We first use the expressions for B, a dir ,B and a dir in Eqs. (9) and (10) to analytically solve for the theoretical unknowns A ut ,Ã ut , δ and δ , up to a four-fold ambiguity. We then have the two remaining equations for a indir andã indir and two theoretical unknowns, γ and . We now consider 200 values for each of γ and , each evenly spaced between 0 and 2π . For each of the 200 × 200 possible combinations of values of γ and , we generate 5000 toy MC experiments, with inputs that span the range of the experimental errors of each quantity.
Similar to the toy Monte Carlo γ determination for the VV mode, we generate random values of each of the experimental inputs according to Gaussian distributions, with means and sigmas according to the measured central value and total errors on each experimental quantity. Making the assumption that f D * s /f D * is equal to f D s /f D = 1.20 ± 0.06 ± 0.06 [5], we again obtain a confidence distribution as a function of γ . The result is shown in Fig. 3. As can been seen in this figure, present data on VP decays alone do not lead to useful constraints on γ .
Finally, we can combine information from the VV and VP modes. The result is shown in Fig. 4. From the combined fit, we see that γ is favored to lie in one of the ranges represents an additional ambiguity for each range). The first of these ranges is that favored by fits to the unitarity triangle, assuming the standard model. The ranges come principally from data on vector-vector decays, although the vectorpseudoscalar decays do improve the constraints slightly. Note that, if we consider a larger confidence level, there are no constraints on γ at present. However, this study demonstrates the feasibility of the method-with more data, we will be able to obtain strong constraints on γ .