Extraction of $|V_{cb}|$ from two-body hadronic B decays

We propose a method of extracting the Cabibbo-Kobayashi-Maskawa matrix element $|V_{cb}|$ from two-body hadronic decay processes of $B\to DK$ with precisely determined form factors of $B$ meson semi-leptonic decays. The amplitude $\mathcal{M}(\bar{B}^0 \to D^+ K^-)$ which does not include the effect of hadronic final state interactions can be theoretically evaluated by using factorization and form factors of semi-leptonic B decays. We can obtain all the amplitudes in an isospin relation $\mathcal{A}(B^-\to D^0K^-) =\mathcal{A}(\bar{B}^0\to D^+K^{-})+\mathcal{A}(\bar{B}^0\to D^{0}\bar{K}^0)$ including the effect of hadronic final state interactions as well as $|V_{cb}|$ using the experimental data of branching fractions of these three processes with a truncation of the states which contribute to the hadronic final state interactions. The extracted value of $|V_{cb}|$ is $(37\pm 6)\times 10^{-3}$. The decay processes of $B\to DK^{*}$ and $B\to D^{*}K$ can also be used in the same way and the extracted values of $|V_{cb}|$ are $(41\pm 7)\times 10^{-3}$ and $(42\pm 9)\times 10^{-3}$, respectively. This method becomes possible by virtue of recent precise determinations of the form factors of semi-leptonic B decays. The uncertainties of $|V_{cb}|$ by this method are expected to be reduced by the results of future B-factory experiments and lattice calculations.


Introduction
The precise determination of Cabibbo-Kobayashi-Maskawa (CKM) matrix elements [1,2] is one of the approaches to test the Standard Model and to search the physics beyond the Standard Model. The Standard Model predicts the unitarity relation of CKM matrix i=u,c,t which gives a triangle in a complex plane. The existence of the physics beyond the Standard Model may violate this relation. The sides and angles of this triangle will be precisely measured using various decay processes of B mesons in future B-factories [3,4]. In this work we focus on the determination of |V cb |.
There are mainly two methods to extract the value of |V cb | from semi-leptonic B meson. The method using inclusive decay data gives |V cb | = (42.00 ± 0.65) × 10 −3 [5], and the method using exclusive decay data gives |V cb | = (38.71 ± 0.75) × 10 −3 [6]. Though the difference of these values are within 3.3σ, it is a problem in understanding non-perturbative physics of QCD. 1 In fact, it has been pointed out that the proper parameterization of form factors is important [8,9]. In order to analyze exclusive decay processes B → D ( * ) lν with a small amount of data, the Caprini-Lellouch-Neubert (CLN) parameterization of form factors [10] is precise enough. However, with much more data recently provided by Belle collaboration, not only q 2 -distributions but also angular-distributions [11], the Boyd-Grinstein-Lebed (BGL) parameterization of form factor [12] is better than the CLN parametrization, because the CLN parametrization may include about 10% errors from the absence of O(1/m 2 c,b ) corrections [9]. Since the accuracy of recent lattice QCD results [13,14] is typically of the order of 1%, we need to use theoretical frameworks with correspondingly high precisions. 2 In this work we intend to provide another method to extract |V cb |, which may give new information to the above conflict in future. We propose that the hadronic decays of B mesons, especially two-body decays of B → DK, B → DK * and B → D * K, can be used to extract precise value of |V cb | in future. The amplitude M(B 0 → D + K − ) which does not include the effects of hadronic final state interactions can be theoretically evaluated by using the factorization, the form factors of semi-leptonic B decays and decay constant of K − meson. The form factors of semi-leptonic B decays are precisely determined by the latest Belle data [11,16] and the latest lattice QCD results [13,14] with the BGL parameterizations in [8]. The isospin symmetry provides a relation including the effects of hadronic final state interactions. We can extract these three amplitudes as well as the value of |V cb | by using the amplitude M(B 0 → D + K − ) and the experimental values of three branching fractions [17]. In this procedure we need to truncate the states which contribute final state interactions: not including all the possible states, but including only two-body DK states. The processes of B → DK * and B → D * K can also be used in the same way. For B → D * K, we use the form factor obtained by the CLN parameterization with latest data by Belle collaboration [11]. 1 It has been pointed out that this problem can not be solved by New Physics [7]. 2 The error with CNL parameterization comes from an excessive reduction of the number of parameters in form factors by using heavy quark symmetry. In fact improvements are possible by including higher order corrections (for example, see [15]). We emphasize that this method becomes possible only with recent precise determination of all the form factors of semi-leptonic B decay. More precise experimental data of the branching fractions of two-body hadronic B decays give more precise value of |V cb |. This method can be understood as an intermediate approach between inclusive and exclusive determination of |V cb |, since it requires to use several exclusive B-decay modes. It may be possible that this method will play an important role in the problem of |V cb | determinations with the results of future B-factory experiments and future precise lattice calculations, if the validity of the truncation of the states in final state interactions is established. In other words, once the value of |V cb | is precisely determined with semi-leptonic decays without any conflicts, this method will provide useful information to understand the final state interactions in two-body hadronic B decays.
In the next section we investigate the amplitudes of B → DK processes in detail, and propose a procedure to extract the value of |V cb |. We also show that the same procedure applies to the processes of B → DK * and B → D * K. In section 3 we provide the numerical analyses of extracting the value of |V cb | from two-body hadronic B decays by our procedure. In section 4 we provide a summary and discussion.

Two-body hadronic decays of B mesons
Consider the hadronic two-body decay processesB → M 1 M 2 , where M 1 and M 2 indicate D mesons and K or π mesons, respectively. The quark-level Feynman diagrams of these decays are classified into four topological types [18,19]. The amplitudes from the diagrams corresponding to each topological type are called as follows. (1) Tree amplitudes T : the diagrams have b → c weak current with the light degrees of freedom as spectator antiquarks ofB and M 1 mesons, and the W boson decays into the light quark-antiquark pairs which constitute M 2 meson (see Fig.1). (2) Color-suppressed amplitudes C: the W boson decays into the light quark-antiquark pairs, and the antiquark is included in M 1 meson as the spectator of c quark, and the quark constitutes M 2 meson with the light degrees of freedom inB meson (see Fig.2).
(3) Exchange amplitudes E: the exchange of the W boson changes the flavor of spectator of B, and light quark-antiquark pair creation from gluons completes two mesons (see Fig.3). (4) W-annihilation amplitudes A: theB meson decays to a W boson and the W boson decays into a charm antiquark and a light quark, and they become constituents of M 1 and M 2 with a light quark and and light antiquark from gluons, respectively (see Fig.4). In this paper we do not consider the process which contains the contribution of A, since it does not include |V cb | and it is rather relevant to |V ub |.
In table 1 Table 1: Two-body hadronic decays and their amplitudes. Note that the decay modē B 0 → D + K − is the only mode which is described by the diagram of Tree topology only. The contributions of penguin diagrams are also listed. diagrams with different topologies contribute to the amplitudes for each decay process. We see that the amplitude ofB 0 → D + K − consists of a single diagram of topology T . The penguin diagrams (see Fig.5) contribute only to the amplitude of B → DD. For example, the amplitude ofB 0 → D + D − s consists of T with a pollution by a penguin diagram. Until the size of the contribution of the penguin diagram is clarified, we can not useB 0 → D + D − s to extract |V cb | precisely.
We focus on the two-body decayB 0 → D + K − which is described only by the diagram of topology T . The effective weak Hamiltonian [20] for the decay is where η EW is the electroweak correction which represents the effects of short-distance QED correction. The factors C 1,2 are the Wilson coefficients and O 1,2 are the current-current operators: where α and β are color indices. The amplitude is given by the matrix element where p, p and p K are 4-momenta ofB 0 , D + and K − , respectively. The momentum p K satisfies The factor a 1 (µ) = C 2 (µ) + C 1 (µ)/3 represents the effects of shortdistance QCD correction including short-distance non-factorizable QCD effects. The amplitude A(B 0 → D + K − ) includes also the effects of non-factorizable hadronic final state interactions (or rescattering effects) which are non-perturbative QCD effects. Now, we introduce the amplitude M(B 0 → D + K − ) which does not include the effects of hadronic final state interactions. The amplitude is given by factorizing the matrix element in eq.(6), because only the diagram of topology T contributes. The final state is written by two independent asymptotic states of D + and K − mesons, because we have temporarily neglected the effects of final state interactions, or long-distance non-factorizable QCD effects. The amplitude is written as The B → D part of the matrix element is given by where f ± (q 2 ) are form factors of semi-leptonic decay ofB 0 . Another matrix element is described as where f K ± = 155.6 ± 0.4MeV [17] is the decay constant of K ± mesons. The absolute value of the amplitude is written as where and the function of f 0 (q 2 ) is precisely determined in [8] for all possible q 2 region. Notice that this amplitude depends only on one form factor f 0 (q 2 ). If we could neglect the effect of hadronic final state interactions, the value of |V cb | could be straightforwardly extracted from the data of the decay rate in [17], since the decay rate is simply described as where We obtain the value of |V cb | = (32.0±1.9)×10 −3 which is inconsistent with the values determined by the inclusive and exclusive methods with semi-leptonic decays. This result indicates the failure of "naive factorization" and shows that the effect of hadronic final state interactions can not be ignored and it is important to extract |V cb | from hadronic two-body B decays. 3 In order to consider the effect of hadronic final state interactions, we introduce a relation between decay amplitudes which follows from isospin symmetry. The amplitudes of B − → D 0 K − ,B 0 → D + K − andB 0 → D 0K 0 are related by isospin symmetry as We expect that this relation should be satisfied within 1% accuracy, because the isospin breaking effect should be proportional to (m d − m u )/Λ QCD ∼ 0.02 or α/π ∼ 0.002. We can represent this relation as a triangle on a complex plane (see Fig.6). The isospin decomposition of these amplitudes are given by is satisfied, because for each isospin channel there is only one final state. If we include further the states like DKππ, for example, the effect of final state interactions can not be represented only by simple phases and the magnitudes of |A 0 | and |A 1 | are also affected [23]. This truncation of the states, or neglecting inelastic final state interactions, is the main theoretical assumption in our method, except for isospin symmetry.
There is no justification of this assumption, since it has been known that the inelastic final state interactions is important in B decays in general [24,25]. To be precise we need to describe α 0 |A 0 | and α 1 |A 1 | instead of naive |A 0 | and |A 1 | in eq.(18), where α 0 and α 1 parametrize the changes of magnitudes of the amplitudes by neglecting the effects inelastic final state interactions. A rough estimate α 0 ∼ α 1 ∼ 0.8 can be obtained by using the results of a global fit of the amplitudes and strong phases in [18], which means about 20% errors in our final results. This is a large error which is comparable to the error from the present measurements of branching fractions. We certainly need to discover some methods to calculate α 0 and α 1 from the first principle, but we leave this task for a future work because of the large experimental errors in the measurements of branching fractions at this moment in time. Considering the other way around, if the value of |V cb | will be precisely extracted by other methods, our method will give a good place to investigate the final state interactions in two-body hadronic B decays.
Once the formula of eq.(18) has been accepted, we can extract |V cb V * us | from the values of |A 0 | and |A 1 | which, as well as cos δ s , can be extracted from the measurements of three decay rates. Now we are going to extract |V cb | from the experimental values of decay fractions of corresponding three decay modes. From eqs. (15), (16) and (17) the ratios of decay fractions can be described as where the coefficients K 1 and K 2 are kinematical factors of Eqs. (18), (19) and (20) are used to describe |A 0 | and |A 1 | in terms of |M(B 0 → D + K − )| as where and R i ≡ R i /K i with i = 1, 2. From eqs. (19) and (20) cos δ s is described only by directly observable quantities as From eq.(24) the absolute value of the amplitude |A(B − → D 0 K − )| = |A 1 | is given by where is a known quantity. Finally, we get |V cb V * us | 2 from the above equation and the value of decay rate Γ(B − → D 0 K − ) as where For B → DK * and B → D * K, we can extract |V cb V * us | 2 in the same way. The only major differences are the concrete forms of the amplitudes M(B 0 → D + K * − ) and M(B 0 → D * + K − ).
with the factorization procedure. The first matrix element in the amplitude is given in eq. (8).
The second matrix element in the amplitude is simply described as where f K * ± and µ (p K * ) are the decay constant and the polarization vector of K * ± mesons, respectively. The polarization vector µ (p K * ) satisfies (p K * ) · p K * = 0. Then, we have Notice that this amplitude depends only on the form factor f + (q 2 ) instead of f 0 (q 2 ) in case of B → DK. ForB 0 → D * + K − , the amplitude is given by with the factorization procedure. The first matrix element in this amplitude is described as [26] where µ (p ) is the polarization vector of D * meson satisfying (p ) · p = 0 and V (q 2 ), A 1 (q 2 ), A 2 (q 2 ) and A 0 (q 2 ) are form factors. Even though there are many form factors, we have a simple expression as Notice that this amplitude depends on only the form factor A 0 (q 2 ) .

Numerical analyses and results
In our analysis we use the experimental data, masses and branching fractions in [17] and the form factors f 0,+ (q 2 ) in [8]. We use the value of the electroweak correction η EW = 1.0066 in [27] and the short-distance QCD correction a 1 (µ) = 1.038 at leading order with Λ  [20]. The accuracy of a 1 (µ) is of the order of 1%. We do not consider the effect of isospin symmetry breaking expecting that the effect is very small within 1%.
From B → DK using eq.(29) and the experimental data in Table 2, we obtain |V cb | = (37 ± 6) × 10 −3 and cos δ s = 0.60 ± 0.14. Notice that the value of |V cb | is consistent with that determined by both the inclusive and exclusive methods with semi-leptonic decays. The uncertainty of |V cb | is about 30% which is dominated by the experimental errors of the ratios, Table 3 shows sources of uncertainty of |V cb |. We find that the precise measurements of branching fractions, B(B 0 → D + K − ), B(B 0 → D 0K 0 ) and B(B − → D 0K − ), play an important role in the precise determination of |V cb | in our method. We note that the value of |V cb | is determined by using the Input Value Reference  form factor which does not employ the CLN parameterization but the BGL parameterization.
To compare the strong phase shift cos δ s with the one in the previous work [18], we convert cos δ s to their cos δ c , where δ c is defined as the phase difference between A(B 0 → D + K − ) and A(B 0 → D 0K 0 ). Our result cos δ c = 0.43 ± 0.16 is consistent with that in [18] within errors. From B → DK * we can obtain the value of |V cb | and the strong phase cos δ s in the same way. Using eq.(32) and the experimental data in Table 4, we obtain |V cb | = (41 ± 7) × 10 −3 and cos δ s = 0.82 ± 0.20. This value of |V cb | is also consistent with both the inclusive and exclusive results. Notice that cos δ s is larger (δ s is smaller) than that in B → DK. This suggests that the effect of hadronic final state interactions between a pseudo-scalar meson and a vector mesons is less important than that in case of two pseudo-scalar mesons. The corresponding value of cos δ c = −0.07 ± 0.28 is also consistent with that in [18] within errors. We have used the form factor with the BGL parameterization in [8]. The decay constant of charged vector meson f K * ± is determined by the branching ratio of τ → K * − ν τ [28]. Since the branching fraction is described as by using the measured values of B(τ → K * − ν τ ) = (1.20±0.07)×10 −2 , m τ = 1776.86±0.12MeV [17] f K * ± 205.6 ± 6.0 MeV see text f + (m 2 K * ± ) 0.696 ± 0.012 [8] 0.622 ± 0.062 see text Table 5: Inputs for determination from B → D * K.
and τ τ = (290.3 ± 0.5) × 10 −15 s [17] we obtain f K * ± = 205.6 ± 6.0MeV. From B → D * K, in the same way, we obtain |V cb | = (42 ± 9) × 10 −3 and cos δ s = 0.80 ± 0.19 using eq.(35) and the experimental data in Table 5. This value of |V cb | is again consistent with those obtained by inclusive and exclusive determinations within errors. The value of strong phase supports the previous suggestion that the effect of hadronic final state interactions is less important in case with a vector meson in final state. The corresponding value cos δ c = 0.63±0.24 is also consistent with that in [18] within errors. The form factor A 0 (q 2 ) is not given by the BGL parameterization, because there are no experimental data of the differential decay rate of B → D * τ ν τ and also no lattice QCD calculations for the form factor. We have to use the form factor A 0 which is given by the CLN parameterization instead of the BGL parameterization by fully utilizing heavy quark symmetry. The CLN parameterization based on the heavy quark effective theory gives where and w and z are kinetic variables defined as The value of h A 1 (1) has been obtained by the unquenched lattice QCD calculation [29]. The value of R 0 (1) can be obtained by using the relation based on heavy quark symmetry [30,31] if we know the value of R 2 (1), where r = m D * /m B . The values of R 2 (1) and ρ 2 D * are determined by Belle collaboration [11] from semi-leptonicB 0 → D * + l −ν l decay as R 2 (1) = 0.91 ± 0.08 and ρ 2 D * = 1.17 ± 0.15. In this way we obtain the value R 0 (1) = 1.08 with the uncertainty of 10% considering unknown O(1/m 2 c ) corrections. Our results are summarized in Table 6.

Conclusions
We have proposed a method of extracting the value of |V cb | from hadronic two-body B meson decays. The recent precise determination of the form factor f 0 (q 2 ) of semi-leptonic B Mode cos δ s |V cb | × 10 3 B → DK 0.60 ± 0.14 37 ± 6 B → DK * 0.82 ± 0.20 41 ± 7 B → D * K 0.80 ± 0.19 42 ± 9 Table 6: Summary of our results. meson decays in [8] allows us to perform this method with B → DK decay processes. The main theoretical assumption in our method, except for isospin symmetry, is that the effect of inelastic final state interactions is small. The small effect of non-factorizable spectator quark scattering has also been neglected, which should be included in case with more precise experimental data. Specifically, we have neglected the possible states except for DK two-body states in final state interactions. The quantitative investigation of this truncation is a future work which belongs to the efforts to understand non-perturbative QCD physics in hadronic decays. The effect of isospin symmetry breaking is not included, since it is negligibly small in the present precision of experimental data. In future when the errors of branching fractions will be smaller and close to 1% accuracy as well as relevant form factors, we need to include the effect of isospin symmetry breaking. We have used form factors of semi-leptonic B meson decays which are determined by using the BGL parameterization in [8,9] for the extraction of |V cb | from B → DK and B → DK * . In the extraction of |V cb | from B → D * K we had to use the CLN parameterization and heavy quark symmetry to obtain the form factor A 0 (q 2 ), which may contain possibly large uncertainties from higher order corrections in heavy quark expansions.
Our final results are summarized in Table 6. The extracted values of |V cb | have about 30% uncertainties and they are consistent with the values from both inclusive and exclusive semileptonic decays within errors. These consistent results show that our method is reasonable at least in the present precision. The experimental errors of the hadronic branching fractions, in particular B(B 0 → D 0K 0 ), B(B 0 → D 0K * 0 ) and B(B 0 → D * 0K 0 ), dominate the uncertainty of |V cb |. We can expect that the uncertainty becomes smaller by the results of future experiments and lattice calculations. It may be possible that this method will be the third one competing conventional and established methods from inclusive or exclusive semi-leptonic B decays, if the problem of inelastic final state interaction is appropriately treated.
We have also examined the effects of hadronic final state interactions in two-body hadronic decays. The extracted strong phase shifts are consistent with the previous works of [18,32,33]. The strong phase in B → DK is larger than that in B → DK * and B → D * K which involve the vector meson in final states (see Table 6). It is known in general that the final state interaction is more important for B → P P decays than B → P V decays, where P and V indicate pseudoscalar and vector mesons. Here, we must note that the definition of our phases are not exactly the same in [18,32,33], and they coincide in the limit of negligible contribution of inelastic final state interactions. This fact will give a way to investigate the magnitude of the effect of inelastic final state interactions in future. If the magnitude of |V cb | will be precisely extracted by other methods in future, our method will give a good place to investigate the final state interactions in two-body hadronic B-decays.