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B_c Decays and Lifetime in QCD Sum Rules

In the framework of three-point QCD sum rules, the form factors for the semileptonic decays of B_c -->B_s(B_s^*) l \nu are calculated with account for the Coulomb-like alpha_s/v-corrections in the heavy quarkonium. The generalized relations due to the spin symmetry of HQET/NRQCD for the form factors are derived at the recoil momentum close to zero. The nonleptonic decays are studied using the assumption on the factorization. The B_c meson lifetime is estimated by summing up the dominating exclusive modes in the c -->s transition combining the current calculations with the previous analysis of b -->c decays in the sum rules of QCD and NRQCD.


Introduction
For better understanding and precise measuring the weak-action properties of heavy quarks, governed by the QCD forces, we need as wide as possible collection of snapshots with hadrons, containing the heavy quarks. Then we can provide the study of heavy quarks dynamics by testing the various conditions, determining the forming of bound states as well as the entering of strong interactions into the weak processes. So, a new lab for such investigations is a doubly heavy long-lived quarkonium B c recently observed by the CDF Collaboration [1] for the first time.
This meson is similar to the charmonium and bottomonium in the spectroscopy, since it is composed by two nonrelativistic heavy quarks, so that the NRQCD approach [2] is well justified to the system. The modern predictions for the mass spectra ofbc levels were obtained in refs. [3] in the framework of potential models and lattice simulations. The arrangement of excitations is close to what was observed in the charmonium and bottomonium. However, the feature of B c -mesons is an absence of annihilation into light quarks, gluons and leptons due to QCD and QED, that implies the higher excitations decay into the low lying levels and ground state due to the emission of photons and pion pairs. The measured value of B c mass yet has a large uncertainty in agreement with the theoretical expectations.
The production mechanism for the B c -meson was studied in refs. [4]. The most simple picture takes place for the production in the e + e − -annihilation, where the universal perturbative fragmentation functions can be analytically calculated for the S-, P-and D-wave levels in the framework of factorization for the hard production of quarks and their soft binding into the hadron, which can be reliably described in the potential models. In hadron collisions, the fragmentation regime takes place at the transverse momenta p T ≫ m Bc , and at p T ∼ m Bc the subleading terms in 1/p T or higher twists have to be taken into account. This can be calculated in the framework of factorization approach by a careful evaluation of complete set of diagrams in the given α s -order, O(α 4 s ). The non-fragmentational contributions dominate at p T ∼ m Bc [4].
The measured B c lifetime τ [B c ] = 0.46 +0.18 −0.16 ± 0.03 ps, agrees with the estimates obtained in the framework of both the OPE combined with the NRQCD evaluation of hadronic matrix elements [5,6,7] and potential quark models, where one has to sum up the dominating exclusive modes to calculate the total B c width [8,9] τ OPE,PM [B c ] = 0.55 ± 0.15 ps.
The accurate measurement of B c lifetime could allow one to distinguish various parameter dependencies such as the optimal heavy quark masses, which basically determine the theoretical uncertainties in OPE.
At present, the calculations of B c decays in the framework of QCD sum rules were performed in [10,11,12,13]. The authors of [10,11] got the results, where the form factors are about 3 times less than the values expected in the potential quark models, and the semileptonic and hadronic widths of B c are one order of magnitude less than those in OPE. The reason for such the disagreement was pointed out in [12] and studied in [13]: in the QCD sum rules for the heavy quarkonia the Coulomblike corrections are significant, since they correspond to summing up the ladder diagrams, where α s /v is not a small parameter, as the heavy quarks move nonrelativistically, v ≪ 1. The Coulomb rescaling of quark-quarkonium vertex enhances the estimates of form factors in the QCD sum rules for the B + c → ψ(η c )l + ν decays, where the initial and recoil mesons are both the heavy quarkonia. In the framework of NRQCD at the recoil momentum close to zero one derives the spin symmetry relations for the form factors of semileptonic B c decays [14,13]. In the strict limit of v 1 = v 2 , where v 1,2 denote the four-velocities of initial and recoil mesons, respectively, the authors of [14] found a single relation between the form factors 4 . In [13] the soft limit v 1 · v 2 → 1 at v 1 = v 2 was considered, and the generalized spin symmetry relations were obtained for the B c → ψ(η c ) transitions: four equations, including that of [14]. Moreover, the gluon condensate term was calculated in both QCD and NRQCD, so that it enforced a convergency of the method.
In the present paper we calculate the B c decays due to the c → s weak transition in the framework of QCD sum rules, taking into account the Coulomb-like α s /v-corrections for the heavy quarkonium in the initial state. In the semileptonic decays the hadronic final state is saturated by the pseudoscalar B s and vector B * s mesons, so that we need the values of their leptonic constants entering the sum rules and determining the normalization of form factors. For this purpose, we reanalyze the two-point sum rules for the B mesons to take into account the product of quark and gluon condensates in addition to the previous consideration of terms with the quark and mixed condensates. We demonstrate the significant role of the product term for the convergency of method and reevaluate the constants f B as well as f Bs . Taking into account the dependence on the threshold energy E c of hadronic continuum in thebs system in both the value of f Bs extracted from the two-point sum rules and the form factors in the three-point sum rules, we observe the stability of form factors versus E c , which indicates the convergency of sum rules.
The spin symmetries of leading terms in the lagrangians of HQET [17] for the singly heavy hadrons (here B ( * ) s ) and NRQCD [2] for the doubly heavy mesons (here B c ) result in the relations between the form factors of semileptonic B c → B ( * ) s decays. We derive two generalized relations in the soft limit v 1 · v 2 → 1: one equation in addition to what was found previously in ref. [14]. The relations are in a good agreement with the sum rules calculations up to the accuracy better than 10%, that shows a low contribution of next-to-leading 1/m Q -terms.
We perform the numerical estimates of semileptonic B c widths and use the factorization approach [18] to evaluate the hadronic modes. Summing up the dominating exclusive modes, we calculate the lifetime of B c , which agree with the experimental data and the predictions of OPE and quark models. We discuss the preferable prescription for the normalization point of nonleptonic weak lagrangian for the charmed quark and present our optimal estimate of total B c width. We stress that in the QCD sum rules to the given order in α s , the uncertainty in the values of heavy quark masses is much less than in OPE. This fact leads to a more definite prediction on the B c lifetime.
The paper is organized as follows. Section 2 is devoted to the general formulation of three-point sum rules for the B c decays with account of Coulomb-like corrections. The analysis of two-point sum rules for the leptonic constant of singly heavy meson with the introduction of term allowing for the product of quark and gluon condensates is presented in Section 3, where we also show the convergency of three-point sum rules with respect to a dependence on the threshold energy of continuum in the heavy-light system. We estimate the form factors of semileptonic B c → B ( * ) s decays. The relations between the form factors of semileptonic decays as follows from the spin symmetry of HQET and NRQCD are derived in Section 4 in the soft limit of zero recoil. Section 5 contains the description how the nonleptonic decays modes are calculated and the B c lifetime is evaluated. We discuss the optimal estimation of lifetimes for the heavy hadrons in Section 6. In Conclusion we summarize the results.
2 Three-point sum rules for the B c meson.
In this paper the approach of three-point QCD sum rules [19] is used to study the form factors of semileptonic and nonleptonic decay rates for the c → s transition in decays of B c meson. From the two-point sum rules we extract the values for the leptonic constants of mesons in the initial and final states. In our consideration we use the following notations: and where P and V represent the scalar and vector mesons, and m 1 , m 2 are the quark masses. The hadronic matrix elements for the semileptonic c → s transition in the B c decays can be written down as follows: where q µ = (p 1 −p 2 ) µ and ǫ µ = ǫ µ (p 2 ) is the polarization vector of B * s meson. V µ and A µ are the flavour changing vector and axial electroweak currents. The form factors f ± , F V , F A 0 and F A ± are functions of q 2 only. It should be noted that since the leptonic current l µ =lγ µ (1 + γ 5 )ν l is transversal in the limit of massless leptons, the probabilities of semileptonic decays are independent of f − and F A − (the τ + ν τ mode is forbidden by the energy conservation). Following the standard procedure for the evaluation of form factors in the framework of QCD sum rules [20], we consider the three-point functions whereq 1 (x)γ 5 q 2 (x) andq 1 (x)γ ν q 2 (x) denote interpolating currents for B s and B * s , correspondingly. J V,A µ are the currents V µ and A µ of relevance to the various cases. The Lorentz structures in the correlators can be written down as The form factors f ± , F V , F A 0 and F A ± are determined from the amplitudes Π ± , Π V , Π A 0 and Π A ± = 1 2 (Π A 1 ± Π A 2 ), respectively. In (8)-(10) the scalar amplitudes Π i are the functions of kinematical invariants, i.e. Π i = Π i (p 2 1 , p 2 2 , q 2 ). The leading QCD term is a triangle quark-loop diagram in Fig. 1, for which we can write down the double dispersion representation at q 2 ≤ 0 where Q 2 = −q 2 ≥ 0. The integration region in (11) is determined by the condition where λ(x 1 , x 2 , The expressions for spectral densities ρ pert i (s 1 , s 2 , Q 2 ) are given in Appendix A. Now let us proceed with the physical part of three-point sum rules. The connection to hadrons in the framework of QCD sum rules is obtained by matching the resulting QCD expressions of current correlators with the spectral representation, derived from a double dispersion relation at q 2 ≤ 0.
Assuming that the dispersion relation (13) is well convergent, the physical spectral functions are generally saturated by the ground hadronic states and a continuum starting at some effective thresholds s th 1 and s th where the resonance term is expressed through the product of leptonic constant and form factor for the transition under consideration, so that where M 1,2 denote the masses of hadrons in the initial and final states. The continuum of higher states is modelled by the perturbative absorptive part of Π i , i.e. by ρ i . Then, the expressions for the form factors F i can be derived by equating the representations for the three-point functions Π i in (11) and (13), which means the formulation of sum rules. For the heavy quarkoniumbc, where the relative velocity of quark movement is small, an essential role is taken by the Coulomb-like α s /v-corrections. They are caused by the ladder diagram, shown in Fig. 2. It is well known that an account for this corrections in two-point sum rules numerically leads to a double-triple multiplication of Born value of spectral density [19,21]. In our case it leads to the finite renormalization for ρ i [13], so that E 1 E 2 p q p 1 · · · p n · · · k 1 k n Figure 2: The ladder diagram of the Coulomb-like interaction.
where v is the relative velocity of quarks in thebc-system, To the moment, the procedure of calculations is completely described.

Numerical results on the form factors and the semileptonic decay widths
We evaluate the form factors in the scheme of spectral density moments. This scheme is not strongly sensitive to the value of thebc-system threshold energy. In our calculations Eb c c = 1.2 GeV. The two-point sum rules for the B c meson with account for the Coulomb-like corrections give α c s (bc)=0.45, which corresponds to f Bc =400 MeV [21]. The quark masses are fixed by the calculations of leptonic constants f Ψ and f Υ in the same order over α s . The requirement of stability in the sum rules including the contributions of higher excitations, results in quite an accurate determination of masses m c = 1.40 ± 0.03 GeV and m b = 4.60 ± 0.02 GeV, which are in a good agreement with the recent estimates in [22], where the quark masses free off a renormalon ambiguity were introduced. The values of leptonic constants f Ψ , f Υ linearly depend on the Coulomb-exchange α s . We find α c s (cc) ≃ 0.60, α c s (bb) ≃ 0.37, which obey the remormalization group evolution with the appropriate scale prescription, depending on the quarkonium contents. In this way, we can extract the above values of α c s (bc) and f Bc . Note, that the heavy (Q 1Q2 )-quarkonia constants obey the scaling relation [21,9] where n denotes the radial excitation number of nS-level, and c is independent of heavy quark flavors. The leptonic constant for the B s meson is extracted from the two-point sum rules. The Borel improved sum rules for the B meson leptonic constant [23] have the following form: where we use qq = −(0.23 · GeV) 3 , m 2 0 = 0.8 GeV 2 , αs π G 2 = 1.77 · 10 −2 GeV 4 as the central values, and M B =5.28 GeV. The K-factor is due to α s -corrections. We expect it is large, but we suppose the appearance of the same factor in evaluating the α s -corrections to the heavy-light vertex in the triangle diagram. For this factor we have the following expression [23]: where we suppose that the scaleΛ is equal to 1.25 GeV. The dependence of K-factor on the Borel parameter τ and the threshold energy E c is shown in Fig. 3. The K-factor is not sensitive to E c changing in the range 1.0 ÷ 1.5 GeV. We see that NLO corrections to the leptonic constant are about 40%. Using the Padé approximation, we find that higher orders corrections can be about 30%. So, we hold the K factor in conservative limits 1.4 ÷ 1.7. It is quite reasonable to suppose its cancellation in evaluating the semileptonic form factors due to the renormalization of heavy-light vertex in the triangle diagram.
The contribution of quark condensate term is not sensitive to the variation of qq in the limits from -(0.23 · GeV) 3 to -(0.27 · GeV) 3 (this variation corresponds to the renormalization group evolution and the insertion of α s -corrections to this term, so that K can be putted as the overall factor).
In the limit of semi-local duality [24,25] τ → 0 we get the relation:Λ(µ) = 3 4 ω 0 (µ) (the contribution of the quark condensate term to this equation is about ∼ 15% ). We introduce the renormalization invariant quantities ForΛ ren dual we haveΛ ren dual = M B − m b = 0.63 GeV, and we obtain that in the semi-local duality the threshold energy ω ren 0,dual = 0.84 GeV. Neglecting the quark condensate term in the leptonic constant we have Since in the three-point sum rules we use the scheme of moments and search for a stable region, in the general Borel scheme for f B we have to consider the stability at τ = 0 with the extended region of resonance contribution. We expect, that the sum rules in (20) with the redefined ω ren andΛ ren , as mentioned, have a stability point at where E c is already not equal to ω ren 0,dual . Demanding a low deviation ofΛ fromΛ ren = 0.63 GeV, we find that sum rules in Eq. (23) can lead to the results, which are in a good agreement with the semi-local duality if E c = 1.1 ÷ 1.3 GeV (see Fig. 4). Then the optimal value of Borel parameter τ = τ m ≃ 6.5 GeV −1 . We write down where R denotes the average value of e −ωτm . So, we find the E 3/2 c -dependence of f B √ M B , whereas the contribution of condensate is numerically suppressed, as expected from the semi-local duality. The results of general Borel scheme calculations of f B √ M B ignoring the overall K-factor are presented in Fig. 4. We observe two stability regions. The stability region at τ = 2 ÷ 4 corresponds to that of considered in [23]. The results for the leptonic constant f B obtained from this region [23] is about 1.5 greater than the value obtained from the stability region at τ = 6 ÷ 7. The second region appears only when we introduce the term with the product of quark and gluon condensates. The similar situation has been observed in the NRQCD sum rules for doubly heavy baryons [26]. The product of quark and gluon condensates was not taken into account in [23], and therefore, the intermediate stability point was observed only. Fixing the optimal values of f 2 B M B in Eq.(24) from Fig. 4, we can invert the sum rules to study the dependence ofΛ on τ , as shown in Fig. 5, where the optimal values ofΛ agree with the semi-local duality and the estimateΛ ren = M B − m b . Note that the intermediate stability point τ ∼ 4 exhibit a low variation ofΛ close to 0.4 GeV, which was obtained in [26,23], and usually given by the potential models (see, for instance, [27]).
The calculated physical quantity should be independent of parameters in the sum rule scheme. However, we truncate both the operator product expansion and the perturbative series for the Wilson  coefficients by fixed orders in the opreator dimension and α s , respectively. This fact leads to that the results depend on the scheme parameters, say, the Borel variable. Moreover, the physical part of sum rules is modelled by the contributions of resonances and a continuum term starting at a threshold, that introduces the dependence on the threshold value and suggests that the stability of results can be improved by the terms of excited resonaces in addition to the ground state.
Let us, at first, consider the results on the leptonic constant in the limit of semi-local duality, which requires the stability at τ → 0 and corresponds to the duality region containing the ground state, only. The sum rules show that the condensate contributions are given by the polinomials over τ , and the leading correction at τ → 0 is the term with the quark condensate. Then, we expect that the region of stability will extend at τ > 0 if we will add the higher condensates with the appropriate ratios of their values. The central values of condensates as mentioned above correspond to the results shown in Fig. 4. The stability of semi-local sum rules can be improved by a variation ofΛ and ω 0 , which is not important for the current discussion. In order to clarify this statement we present the results of semi-local sum rules for the leptonic constant of B meson in Fig. 6, where we putΛ = 0.54 GeV. Note, that the value of leptonic constant is the same as we have found in the pertubative limit of semi-local duality. Then, we state that the value of leptonic constant obtained at τ → 0 agrees with the value corresponding to the stability at τ ≈ 7 GeV −1 , i.e. in the second point of local extremum in Fig. 4. In order to confirm, we present also the result for the other ratio of condensate values (the lower value of mixed condensate and the upper value of gluon condensate in the regions mentioned below) in Fig. 7. We see that the semi-local duality is completely broken at unappropriate choice of condensate values. Second, we study the general Borel sum rules in the scheme, where we do not require the stability at τ → 0, which means that we extend the duality region to incorporate possible excitation contributions.  Then, we expect that the estimate of leptonic constant should result in the same value obtained in the semi-local duality. We see that this can be reached at the same values of condensates involved as well as in the same region of τ , which is again given by the second extremum (see Figs. 4 and 6). Certainly, the extension of duality region leads to a failure of stability at low τ , which is caused by additional terms given by the excitations beyond the control of the method.
At higher values of τ the stability would be reached by introduction of higher condensates contributing as the polinomials of higher powers. Of course, we can achieve the better stability in the general Borel scheme by adjusting the condensate values, but this will destroy the semi-local duality, that indicates the divergency of the method, while the convergency demands an appropriate choice of condensate values as we put.
The criterion on the covergency of both the semi-local duality and general Borel sum rules was ignored in [23], where the greater value of leptonic constant was obtained (see Fig. 7).
The same notes can be done in the discussion onΛ. For the sake of comparison, we present the results in Figs. 5 and 8, corresponding to the inverted sum rules for the leptonic constant given in Figs. 4 and 6.
The physical meaning of gluon condensate as it stands in the operator product expansion taken between the observed states is independent of the scheme of calculations. In the sum rules the gluon condensate contributes in the region, where the excited states are suppressed, while the ground state corresponding to the current under consideration dominates. So, the condensate essentially determines the binding energy of quarks in the hadron containing the heavy quark. The definition of sum rules scheme includes the parameter determining the threshold energy of continuum contribution in addition to the resonance term. So, the difference between the schemes of semi-local duality and usual Borel representation is due to the variation of duality region. So, the semi-local duality means the duality for the region contaning the ground state only, while the usual Borel (or moment) scheme explore the extended region containing several hadronic states: the ground state and its excitations. However, the gluon condensate essentially contributes at the scheme parameters, when the excitations are suppressed. Thus, its value is the same for both schemes used, i.e. for the semi-local duality and in the Borel scheme. The value of gluon condensate has been varied in the range αs π G 2 = (1.5 ÷ 2) · 10 −2 GeV 4 . As for the mixed condesate we have used the range m 2 0 = (0.72 ÷ 0.88) GeV 2 . The variation of these parameters does not change the qualitative picture for the lepton constant as it has been discussed, while the numerical uncertainty of its value is less than 7%.
Numerically, multiplying the result taken from Fig. 4, by the K-factor we find the value f B = 140 ÷ 170 MeV, which is in a good agreement with the recent lattice results [28] and the estimates in the QCD SR by other authors [29]. So, we can conclude that the 1/m b -corrections are not valuable for f B . The uncertainty of estimates is basically connected with the higher orders in α s . For the vector [30,29]). For the leptonic constant of B s meson we explore the following relation f Bs f B = 1.16, which expresses the flavor SU(3)-symmetry violation for B mesons [25].
Remember, in sum rules the heavy quark masses are fixed by the two-point sum rules for bottomonia and charmonia with the precision of 20 MeV. In our consideration the quark masses are equal to m b =4.6 GeV, m c =1.4 GeV, and we use m s =0.15 GeV, which agrees with the various estimates [31]. The uncertainties in the values of form factors are basically determined by the variation of b-quark mass, while changing the other quark masses in the ranges m s = 0.14 ÷0.16 GeV and m c = 1.35 ÷1.45 GeV, results in the uncertainty less than 2%. In Figs. 9, 10 and 11 we present the results in the scheme of spectral density moments.   We have investigated the dependence of form factors on thebs threshold energy of continuum in the range E c = 1.1 ÷ 1.3 GeV. The characteristic forms of this dependence are shown in Figs. 12 and 13. We see that the optimal choice for thebs system threshold energy is 1.2 GeV. In Table 1 we present the results of sum rules for the form factors in comparison with estimates in the framework of potential models [9,32]. We see a good agreement of estimates in the QCD sum rules with the values in the quark model. For the sake of completeness the quark model expressions for the form factors are given in Appendix B.
Method In [10] the form factors were derived using the similar SR technique but without the Coulomb-like corrections in thebc system, which enhance the form factors about three times, as we have found.
Let us discuss the uncertainties in the sum rules and other approaches. So, the potential models [8,15] with similar choices of parameters result in the form factor values, which are slightly modeldependent. The corresponding accuracy is about 10%. Then, we expect that the potential models give good reference points for the appropriate numerical values.
The accuracy of sum rules under consideration is basically determined by the variation of heavy quark masses. Indeed, the significant α s correction to the leptonic constant of B s meson should cancel   E c ,GeV f + Figure 13: The dependence of f + on thebs threshold energy E c , determining the region of resonance contribution.
the same factor for the renormalization of quark-meson vertex in the triangle diagram. The dependence on the choice of threshold energy in thebs-channel can be optimized and, hence, minimized. The variation of threshold energy in thebc-channel give the error less than 1%. The effective coulomb constant is fixed from the two-point sum rules for the heavy quarkonium, and its variation is less than 2%, which gives the same uncertainty for the form factors. The heavy quark masses are determined by the two-point sum rules for the heavy quarkonia, too. However, their variations result in the most essential uncertainty. Summing up all of mentioned variations we estimate δf /f ≃ 5%. The enhancment of form factors in the decays of heavy quarkonium considered in the framework of sum rules was also found in the decays B c → ψlν [12,13]. It is important to stress that the current consideration removes the contradiction between the estimates in OPE, potential models and the values in the SR. We have calculated the form factors in the SR consistent with the values in the OPE and potential models.
We use the three-point sum rules to determine the dependence on q 2 of form factors in vicinity of q 2 = 0, where the method works even in the approximation by a bare quark loop [20]. Naively, we expect a simple pole form so that the first derivative f ′ can be evaluated in the framework of sum rules to estimate the "pole" mass M pole , which may deviate from the value of physical mass of corresponding bound state. Of course, the bare quark loop approximation cannot be justified at q 2 ∼ m 2 pole , while the three-point sum rules in the form of double dispersion relation cannot be explored in the region of resonances at q 2 > 0 because of so-called "non-Landau" singularities indicating the presence of strongly bound states in the q 2 -channel. That is why we calculate the form factors as well as their Borel transforms at q 2 < 0.
Numerically, we have found M pole =1.3÷1.4 GeV for the form factors with the B * s decay modes, and M pole =1.8÷1.9 GeV for the decay form factors with the B s modes.

The symmetry relations
At the recoil momentum close to zero, the heavy quarks in both the initial and final states have small relative velocities inside the hadrons, so that the dynamics of heavy quarks is essentially nonrelativistic. This allows us to use the combined NRQCD/HQET approximation in the study of mesonic form factors. The expansion in the small relative velocities to the leading order leads to various relations between the different form factors. Solving these relations results in the introduction of an universal form factor (an analogue of the Isgur-Wise function) at q 2 → q 2 max . We consider the soft limit where v µ 1,2 = p µ 1,2 / p 2 1,2 are the four-velocities of heavy mesons in the initial and final states. The study of region (26) is reasonable enough, because in the rest frame of B c meson (p µ 1 = ( p 2 1 , 0)), the four-velocities differ only by a small value | p 2 | (p µ 2 = (E 2 , p 2 ), whereas their scalar product w deviates from unity only due to a term, proportional to the square of | p 2 |: w = 1 + | p 2 | 2 . Thus, in the linear approximation at | p 2 | → 0, relations (26) are valid and take place. Here we would like to note, that (26) generalizes the investigation of [14], where the case of v 1 = v 2 was considered. This condition severely restricts the relations of spin symmetry for the form factors and, as a consequence, it provides a single connection between the form factors. In the soft limit of zero recoil we find for the four-velocity of spectator b-quark, and v µ for the decaying c-quark. The matrix element of J µ =Q 1 Γ µ q 2 with the spin structure Γ µ = {γ µ , γ 5 γ µ } has the form where Γ 1 determines the spin state in the heavy meson Q 1Q3 (in our case it is pseudoscalar, so that Γ 1 = γ 5 ), Γ 2 determines the spin wave function of quarkonium in the final state Γ 2 = {γ 5 , ǫ µ γ µ } for the pseudoscalar and vector states, respectively (H = P, V ). The 'propagator of the light quark' 5 is taken in a general form where B, C, D are the functions of w. The quantity h in (29) at w → 1 is an universal factor independent of the spin state of meson. So, for the form factors, discussed in our paper, we have where c ǫ = −2, 1+C . The rest coefficients c 1,2 depend on the C and D parameters. We have the symmetry relations for the following form factors 6 : Equating the second relation in (35), for example, we obtaiñ where all form factors are taken at q 2 max . SubstitutingB in first and third relations, we get f + ≃ 2.0 and f − ≃ −8.3. These values have to be compared with the corresponding form factors obtained in the QCD sum rules: f + (q 2 max ) = 1.8 and f − (q 2 max ) = −8.1, where we suppose the pole like behaviour of form factors (see Eq. (25)). Thus, we find that in the QCD sum rules, relations (35) are valid with the accuracy better than 10% at q 2 = q 2 max . The deviation could increase at q 2 < q 2 max because of variations in the pole masses governing the evolution of form factors. However, in B + c → B ( * ) s l + ν decays the phase space is restricted, so that the changes of form factors are about 50%, while their ratios develop more slowly.

Nonleptonic decays and the lifetime
The hadronic decay widths can be obtained on the basis of assumption on the factorization for the weak transition between the quarkonia and the final two-body hadronic states. For the dominant nonleptonic decay modes B + c → B ( * ) s π + (ρ + ) the effective Hamiltonian can be written down as where where i, j run over the colors. The factors C ± (µ) account for the strong corrections to the corresponding four-fermion operators caused by hard gluons. The review on the evaluation of C ± (µ) can be found in [34]. In the present paper, dealing with the QCD sum rules in the leading order over α s , we explore the C ± (µ)-evolution to the leading log accuracy. The B + c → B s π + amplitude, for example, takes the form where a 1 (µ) = 1 2Nc (C + (µ)(N c + 1) + C − (µ)(N c − 1)) at N c = 3 being the number of colors. In our calculations we put the following light meson parameters: m π + =0.14 GeV, m ρ + =0.77 GeV, f π + =0.132 GeV, f ρ + =0.208 GeV. The results are collected in Table 3.  It is worth noting that the sum of widths for transitions B + c → B s (B * s )π + (ρ + ) is 10% larger than the width for the transition B + c → B s (B * s ) + light hadrons, which is calculated using the simple formula , where we neglect the contributions given by the modes with the factor a 2 instead of a 1 . In addition, the deviation between these estimates can be caused by the corresponding 'bag' factor appearing in the formulation of factorization approach and vacuum saturation in the connection of leptonic form factors to the hadronic ones. The modern lattice estimates show that the 'bag' parameters are about 7% less than 1 [28].
In the parton approximation we could expect which results in the estimate very close to the value obtained as the sum of exclusive modes at µ > 0.9 GeV. The deviation between these two estimates slightly increase at mc 2 < µ < 0.9 GeV. Concerning the comparison of hadronic width summing up the exclusive decay modes with the estimate based on the quark-hadron duality, we insist that the deviation between these two estimates is less that 10% and, hence, it cannot be treated as an essential argument against the validity of our calculations.
We estimate the lifetime using the fact that the dominant modes of the B c meson decays are the c → s, b → c transitions with the B ( * ) s and J/ψ, η c final states respectively, and the electroweak annihilation 7 .
We stress that in the B c → B s decays caused by the weak decays of charmed quark the possible hadronic final states are the charged mesons π, ρ and K or the multi-particle states like ππ, πππ or Kπ. First, the states with the kaon are suppressed by the Cabibbo angle, and we neglect their contributions in the total nonleptonic width of B c (the corresponding error of estimates is about 4%). The method for the calculation of multi-particle branching fractions was offered by Bjorken in his pioneering paper devoted to the decays of hadrons containing heavy quarks [35]. He supposed a simple relation for the yields of pions, as given by the Poisson distribution with the average value of pions determined by the energy release. The Bjorken's model of multi-particle yields in the decays does not distinguish the resonant and continuum final states as well as B s and B * s . So, we check that the ratio of R 2π = Γ(B + c → B ( * ) s π + )/Γ(B + c → B ( * ) s ρ + ) ≈ 0.82 calculated in the framework of sum rules is close to the estimate R 2π = Γ(B + c → B ( * ) s π + )/Γ(B + c → B ( * ) s π + π 0 ) ≈ 0.85 given by Bjorken. Then, we see that the non-resonant multi-particle states are suppressed in comparison with the resonance yields. The same fact can be found, once we consider the Kπ, Kππ and K * π, Kρ branching fractions in the decays of D mesons as measured experimentally. This consideration confirms us that we take into account all of significant nonleptonic decay modes of B c . In order to estimate the contribution of non-resonant 3π modes of B c decays into B ( * ) s we use the Bjorken's technique, i.e. the Poisson distribution with the average value corrected to agree with the non-resonant 3π-modes in the decays of D mesons. We have chosen the following branching ratios: BR(D + → K − π + π + ) = 9.0 ± 0.6% and BR(D + → K − π + π + π 0 | non−resonant ) = 1.2 ± 0.6%. So, we have foundn ≈ 1 8 , which means that We see that the neglected modes contribute to the total width of B c as a small fraction in the limits of uncertainty envolved.  The width of beauty decay in the sum rules was derived using the similar methods in [13]: Γ(B + c → cc + X) = (28 ± 5) · 10 −14 GeV. The width of the electroweak annihilation is taken from [9] as 12 · 10 −14 GeV.
In Fig. 14 we present the B c meson lifetime calculated in the QCD SR under consideration. We also show the results of the lifetime evaluation in the framework of Operator Product Expansion in NRQCD [6,7].
In contrast to OPE, where the basic uncertainty is given by the variation of heavy quark masses, these parameters are fixed by the two-point sum rules for bottomonia and charmonia, so that the accuracy of SR calculations for the total width of B c is determined by the choice of scale µ for the hadronic weak lagrangian in decays of charmed quark. We show this dependence in Fig. 14, where mc 2 < µ < m c and the dark shaded region corresponds to the scales preferred by data on the charmed meson lifetimes. The discussion on the optimal choice of scale in hadronic decays is addressed in the next section.

Discussion on the lifetimes of heavy hadrons
At present the ordinary prescription for the normalization point of lagrangian generating the nonleptonic decays of heavy quark Q, is µ ≃ m Q . The motivation is the following: the characteristic scale is determined by the energy release given by the heavy quark mass. Therefore, we can argue the operator product expansion in the inverse powers of m Q , wherein we can factorize the Wilson coefficients taken in the perturbative QCD and the matrix elements of operators over the hadronic states with µ usually posed to m Q . This prescription is in a qualitative agreement with the current data on the measured lifetimes of charmed and beauty hadrons and their branching ratios for the semileptonic decay modes, say.
Let us consider this issue in more details. To the moment, the analysis of decays in the QCD sum rules is restricted by the leading order (LO) in α s (except the Coulomb-like corrections in the heavy quarkonia). The corresponding parameters, the heavy quark masses, are also fixed to the same order (they have to be reevaluated in next-to-leading order). Therefore, for the sake of consistency, we use the LO expressions, which are given by the partonic approximation improved by 1/m Q -corrections in HQET or NRQCD. In this way, we can write down the following formulae: 1. The semileptonic branching fraction of D 0 meson is given by 2. The difference between the total widths of charmed mesons is determined by the Pauli interference in decays of D + , so that [36] where θ c is the Cabibbo angle, f D is the leptonic constant of D meson, and the 'bag' constants are defined by The similar expression can be derived for the beauty mesons [36] Γ 3. The semileptonic branching fraction of B 0 meson is given by where the fraction of 0.22 is due to the τ ν τ -contribution, while the value of k denotes the fraction of b → ccs transition in the nonleptonic decays. In the same way, the average yield of charm in the decays of beauty mesons is equal to As for numerical applications, one usually puts MeV, and B =B = 1 naively motivated by nonrelativistic potential models. At k = 0.4 [37], this set (marked as SETMQ column) results in the estimates shown in Table 4 in comparison with the experimental data [33].  First, we note that the semileptonic width of D 0 is well described to the given order and at chosen value of m c , while its branching ratio is in a valuable contradiction with the data indicating a more higher enforcement of nonleptonic modes. Second, the qualitative agreement of predictions with the measured differences of Γ[D + ] − Γ[D 0 ] and Γ[B + ] − Γ[B 0 ] is mainly based on the assumption ofB ≈ 1. Recent consideration of charmed baryon lifetimes by M.Voloshin [38] clearly drawn a conclusion that the naive picture of color structure as given by the potential models (i.e. the purely antisymmetric color-composition of valence flavors) is significantly broken. A similar statement was obtained in the description of D * meson production at HERA, where the authors of [39] found that the O(α em α 3 s )calculations for the differential cross sections of cq-pair composing the meson, are able to reproduce the measured spectra, if we introduce the valuable contribution by the color-octet state in addition to the singlet one. So, the four-quark singlet-operator results in which is reduced to O (1) = |Ψ (1) (0)| 2 in the framework of nonrelativistic potential model, where with ǫ α denoting the polarization vector and Ψ (1) (q) being the wave function. The term of color-octet was parameterized by which is, in a similar way, can be represented as O (8) = |Ψ (8) where thec and q fields represent the rapid valence quarks, and n a is a random color-vector determined by soft degrees of freedom inside the meson (i.e. by the quark-gluon sea). We have n a n b = δ ab in the production, while n a n b = 1 8 δ ab in decays. The data on the D * production in DIS give O (8) /O (1) ≃ 1.3. We can analogously expect that in the D meson the ratio of four-quark matrix elements is close to that in D * . So, O (8) [D]/O (1) [D] ≃ 1, where O (1,8) [D] can be obtained from the above expressions for D * by the substitution of γ 5 γ µ for γ µ and removing the transverse projector. Then we straightforwardly find PuttingB = 2 into the SETMQ we get the values given in the SETMQ 2 column in Table 4. So, the choice of µ = m Q is in a deep contradiction with the observed differences of total widths for the heavy mesons at the most reasonable value ofB = 2.
In this position we argue the following: There are other physical scales in the problem, which are characteristic for two hadronic systems in the decay process. The first system is the decaying hadron. The second is the transition current c → s orb →c, where the form factor behaviour versus the transferred momentum is determined by the cs andbc states. Those hadronic systems have the following scales, being the average squares of heavy quark momentum p 2 , which are phenomenologically equal to where according to the potential modelsΛ ≃ 0.4 GeV is the binding energy of heavy quark (i.e. the constituent mass of light quark), T ≃ 0.45 GeV is the average kinetic energy in the system. T determines the 2S-1S splitting and it is approximately independent of quark flavors. Analogously, we put whereΛ s =Λ + (m Ds − m D ) =Λ + (m Bs − m B ) ≃ 0.5 GeV. For the b → c current we put Let us suppose that the decay scale is given by the following combinations: Bs = µ bs · µ cb , AtB = 2, this set of parameters with f B ∼ = f D ∼ = 175 MeV, and k = 0.18 is represented by the SETH column in Table 4. We see a good agreement with the data. The result of µ B and k variations is also shown in Fig. 15. We stress that the theoretical OPE prediction for the contribution of b → ccs mode, k = 0.4, is significantly overestimated, to our opinion. Indeed, looking at the semileptonic decay with τ + ν τ in the final state we see that at the same values of form factors as in modes with light leptons the heavy lepton mode is suppressed because of the restricted phase space, so that the reduction factor equals 0.22. The same effect has to take place in the b-quark decays with two charmed quarks in the final state. So, since the sum of D and K masses is greater than the τ mass we could expect even the greater suppression than for the heavy lepton mode. Nevertheless, OPE operates with the quark masses at so moderate release of energy, that certainly leads to the overestimation of phase space in the decay of b → ccs. We use more realistic values for k close to 0.2.
Next, the contribution of four-quark operators are suppressed in decays of Λ b baryon. So, one expects that the deviation of its total width from that of B mesons is about 2-3% [40]. To the moment, the experimental result is far away from this expectation (see Table 4). We can reproduce the data on the Λ b lifetime, if we introduce the light diquark scale where T /2 corresponds to the half tension of color string inside the diquark. Then we pose which result in a good agreement with the experimental data.
To finish this discussion we draw the conclusion: a probable way to reach the agreement between the theoretical predictions and available experimental data on the lifetimes and inclusive decay widths of heavy hadrons is to suggest the different normalization points in the effective nonleptonic lagrangian for the heavy quark weak decays as dependent on the hadron. This assumption provide us with quite acceptable results to the leading order in α s . The variation of normalization point shows the sensitivity of calculations to the higher orders in the QCD coupling constant, which indicates, first, the necessity to proceed with the higher orders, and, second, the appropriate choice of scale can allow us to decrease the scale-dependent higher orders terms.
Thus, we suppose that the preferable choice of scale in the c → s decays of B c is equal to µ 2 Bc = µ cb · µ cs ≈ (0.85 GeV) 2 , and at a 1 (µ Bc ) = 1.20 in the charmed quark decays we predict τ [B c ] = 0.48 ± 0.05 ps.

Conclusion
We have investigated the semileptonic decays of B c meson due to the weak decays of charmed quark in the framework of three-point sum rules in QCD. We have found the important role played by the Coulomb-like α s /v-corrections. As in the case of two-point sum rules, the form factors are about three times enhanced due to the Coulomb renormalization of quark-meson vertex for the heavy quarkonium B c . We have studied the dependence of form factors on the threshold energy, which determines the continuum region ofbs system. The obtained dependence has the stability region, serving as the test of convergency for the sum rule method. The HQET two-point sum rules for the leptonic constant f Bs and f B * s have been reanalyzed to introduce the term caused by the product of quark and gluon condensates. This contribution essentially improves the stability of SR results for the leptonic constants of B mesons, yielding: f B = 140 ÷ 170 MeV. We have studied the soft limit for the form factors in combined HQET/NRQCD technique at the recoil momentum close to zero, which allows us to derive the generalized relations due to the spin symmetry of effective lagrangian. The relations are in a good agreement with the full QCD results, which means that the corrections to the form factors in both relative velocity of heavy quarks inside thebc quarkonium and the inverse heavy quark masses are small within the accuracy of the method.
Next, we have studied the nonleptonic decays, using the assumption on the factorization of the weak transition. The results on the widths and branching fractions for various decay modes of B c are collected in Tables.
Finally, we have estimated the B c meson lifetime, and showed the dependence on the scale for the hadronic weak lagrangian in decays of charmed quark τ [B c ] = 0.48 ± 0.05 ps.
Our estimates are in a good agreement with the theoretical predictions for the lifetime in both the potential models and OPE as well as with the experimental data.