On the Colour Suppressed Decay Modes B^0 -->D_s^+ D_s^- and B_s^0 -->D^+ D^-

We point out that the decay modes B^0 -->D_s^+ D_s^- and B_s^0 -->D^+ D^- have no factorized contribution. At quark level these dacays can only proceed through the annihilation mechanism, which in the factorized limit give zero amplitude due to current conservation. In this paper, we identify the dominating non-factorizable (colour suppressed) contributions in terms of two chiral loop contributions and one soft gluon emission contribution. The latter contribution can be calculated in terms of the (lowest dimension) gluon condensate within a recently developed heavy-light chiral quark model. We find braching ratios BR(B^0 -->D_s^+ D_s^-) = 7*10^-5 and BR(B^0_s -->D^+ D^-) = 1*10^-3.


I. INTRODUCTION
There is presently great interest in decays of B-mesons, due to numerous experimental results coming from Ba Bar and Belle, and later at LHC.
It has been shown [1] that some classes of B-meson decay amplitudes exhibit QCD factorization. This means that, up to α s /π (calculable) and Λ QCD /m b (not calculable), their amplitudes factorize into the product of two matrix elements of weak currents. In a previous paper [3], it was pointed out that the decay mode D 0 → K 0 K 0 was zero in the factorized limit due to current conservation. However, there are in that case nonfactorizable (colour suppressed) contributions in terms of chiral loops and soft gluon emission modelled by a gluon condensate.
In this paper we report on the following observation: The decay modes B 0 → D + s D − s and B 0 s → D + D − have no factorized (colour non-suppressed) contributions. At quark level, these decays a priori proceed through the annihilation mechanism bs → cc and bd → cc, respectively. However, within the factorized limit the annihilation mechanism will give a zero amplitude due to current conservation, as for D 0 → K 0 K 0 . But there are nonzero factorized contributions through the axial part of the weak current if at least one of D-mesons in the final state is a vector meson D * . Such contributions are, however, proportional to the numerically non-favourable Wilson coefficient C 1 , which we will neglect in this short paper. In contrast, the typical factorized decay modes which proceed through the spectator mechanism, say B 0 → D + D − s , are proportional to the numerically favourable Wilson coefficient C 2 . If the mesons in this amplitude are also allowed to be vector mesons, such amplitudes will generate non-factorizable (∼ 1/N c ) chiral loop contributions to the process B 0 d → D + s D − s due to K 0 -exchange. These will be considered in the present paper. There are also non-factorizable (∼ 1/N c ) contributions due to soft gluon emission. Such contributions can be calculated in terms of the (lowest dimension) gluon condensate within a recently developed Heavy Light Chiral Quark Model (HLχQM) [4], which is based on Heavy Quark Effective Theory (HQEFT) [5]. This model has been applied to processes with B-mesons in [6,7]. The gluon condensate contributions is also proportional to the favourable Wilson coefficient C 2 .
In the next section (II), we shortly present the four quark Lagrangian at quark level.
In section III we present our analysis of chiral loop contributions within the heavy light chiral perturbation theory. In section IV we give the calculation of non-factorizable matrix elements due to soft gluons expressed through the (model dependent) quark condensate. In section V we give the results and conclusion. Throughout the paper, we will give formulae and figures for the decay mode B 0 → D + s D − s . The treatment of B 0 s → D + D − will proceed analogously.

II. EFFECTIVE NON-LEPTONIC LAGRANGIAN AT QUARK LEVEL
Based on the electroweak and quantum chromodynamical interactions, one constructs an effective Lagrangian at quark level in the standard way: where all information of the short distance (SD) loop effects above a renormalization scale µ is contained in the (Wilson) coefficients C i . In our case there are two relevant operators for q = d, s. Penguin operators may also contribute, but have small Wilson coefficients. We Performing perturbative QCD corrections within Heavy Quark Effective Theory (HQEFT) [5], the effective Lagrangian (1) can be evolved down to the scale µ ∼ Λ χ ∼1 GeV [8,9], where one finds |a 1 | ≃ 0.4 and |a 2 | ≃ 1.4. The b, c, and c quarks are then treated within HQEFT.
In order to study non-factorizable contributions at quark level, we may use the following relation between the generators of SU(3) c (i, j, l, n are colour indices running from 1 to 3): where a is the color octet index. Then the operators Q 1,2 may, by means of a Fierz transformation, be written in the following way : where the operators with the "tilde" contain colour matrices: To obtain a physical amplitude, one has to calculate the hadronic matrix elements of the quark operators Q i within some framework describing long distance (LD) effects.
As an example of a typical factorized case we choose the amplitude for B 0 → D + D − s obtained from (1) and (2): which will in section III be compared with our chiral loop contributions. This term is proportional to the D-meson decay constant times the Isgur-Wise function (forB → D transition) and is vizualized in figure 1. (1) and (2) is vizualized in figure 2, and is given by Unless one or both of the D-mesons in the final state are vector mesons, this matrix element is zero due to current conservation: The genuine non-factorizable part for B 0 → D + s D − s can be written in terms of coulored 5 currents (see eqs. (4) and (5)): We observe that the annihilation mechanism amplitude in the non-factorizable case has the numerically favourable Wilson coefficient C 2 . This amplitude is, within the HLχQM vizualized later in figure 4 .

III. HEAVY LIGHT CHIRAL PERTURBATION THEORY
Our calculations will be based on HQEFT [5], which is a systematic 1/m Q expansion in the heavy quark mass m Q . The heavy quark field Q(x) = b(x) (eventually c(x) or c) is replaced with a "reduced" field Q v (x) for a heavy antiquark. These are related to the full field Q(x) in the following way: where P ± are projecting operators P ± = (1 ± γ · v)/2. The Lagrangian for heavy quarks is: where D µ is the covariant derivative containing the gluon field. In [7] the 1/m Q corrections were calculated for B − B -mixing. In this paper these will not be considered.
Integrating out the heavy and light quarks, the effective Lagrangian up to O(m −1 Q ) can be written as [4,10] where H (±) a is the heavy meson field containing a spin zero and spin one boson: The fields P 6 where f is the bare pion coupling, and Π is a 3 by 3 matrix which contains the Goldstone bosons π, K, η in the standard way. The axial chiral coupling is g A ≃ 0.6. Eqs. (12), (13), and (14) will be used for the chiral loop contributions.
The simplest way to calculate the matrix element of four quark operators like Q 1,2 in eq. (1) is by inserting vacuum states between the two currents, as indicated in section II.
This vacuum insertion approach (VSA) corresponds to bosonizing the two currents in Q 1,2 separately and multiply them, i.e. the factorized case. Based on the symmetry of HQEFT, the bosonized current for decay of the bq system is [4,10]: where Q (+) v is a heavy b-quark field, v is its velocity, and H neglected (see [4,5]): where C v,γ are Wilson coefficients due to perturbative QCD for scales µ < m Q (Q = b, c for H = B, D). We take µ = Λ χ , which is the scale where perturbative QCD are matched to our hadronic matrix elements.
For the W -boson materializing to aD we obtain the bosonized current wherev is the velocity of the heavyc quark and H (−) c is the corresponding field for theD meson.
For the b → c transition, we obtain the bosonized current where ζ(ω) is the Isgur-Wise function for theB → D -transition, and v ′ is the velocity . Note also that from conservation of momentum we find the relation between the heavy quark velocities: For the weak current for DD production (corresponding to the factorizable annihilation mechanism) we obtain where (20) is a complex function, and not so well-known as for the b → c transition. In the factorized limit, the matrix elements of the four quark operators are obtained by multiplying the bosonized currents above.
In the following we will consider explicitely the decay mode B 0 → D + s D − s . The analysis of B 0 s → D + D − proceed the same way. To calculate the chiral loop amplitudes we need the (factorized) amplitudes for B * 0 s → D + s D * − and B 0 → D * + D * − , which proceed through the spectator mechanism as in figure 1. The point is that the leading chiral coupling obtained from (12) is a coupling between a pseudoscalar meson H, vector meson H * a light pseudoscalar M (= π, K, η). Using the bosonized currents in eqs. (17) and (18), we obtain the following chiral loop amplitude for the process B 0 → D + s D − s from the figure 3: where the factorized amplitude for the process The quantity R χ is a sum of contributions R χ 1,2 from the left and right part of figure 3 respectively. In the MS scheme the results for R χ 1,2 are Adding these two contributions we find : As usual, the 1/N c suppression is due to f 2 ∼ N c . The function r(x) is also appearing in loop calculations [8,9] of the anomalous dimension in HQEFT (for x > 1 and x < −1 respectively): which means that the amplitude gets an imaginary part. Numerically, we find

IV. NON-FACTORIZABLE SOFT GLUON EMISSION
The genuine non-factorizable part (see eqs. (4), (5) and (9) ) can, within the framework presented in this section, be written in a quasi-factorized way in terms of matrix elements of coulored currents: where a G in the bra-kets symbolizes emision of one gluon (from each current) as vizualized in figure 4. We observe that the annihilation mechanism amplitude in the non-factorizable case has the numerically favourable Wilson coefficient C 2 .
In order to calculate the matrix elements (28), we will use a model which incorporates emision of soft gluons modelled by a gluon condensate. This will be the Heavy Light Chiral Quark Model (HLχQM) recently developed in [4]. This model belongs to a class of models extensively studied in the literature [11,12,13,14,15,16]. For details we refer to ref. [4].
The Lagrangian for the HLχQM is The first term is given in equation (11). The light quark sector is described by the Chiral Quark Model (χQM), having a standard QCD term and a term describing interactions between quarks and (Goldstone) mesons: Here m is the SU(3) invariant constituent light quark mass, and χ is the flavour rotated quark fields given by Feynman diagram techniques as in [4,6,7,17,18]. They may also be calculated by means of heat kernel techniques as in [15,16,19].
The interaction between heavy meson fields and heavy quarks are described by the following Lagrangian [4]: where G H ∼ √ 2m/f is a coupling constant. In [4] it was shown how (12) could be obtained from the HLχQM. Performing this bosonization of the HLχQM, one encounters divergent loop integrals which will in general be quadratic-, linear-and logarithmic divergent [4]. Also, as in the light sector [18] the quadratic and logarithmic integrals are related to the quark condensate and the gluon condensate respectively.
Within the model, one finds the following expression for the Isgur-Wise function [4] ζ(ω) amplitude analogously to R χ in (21) and (25) for chiral loops. Numerically, we find that the ratio between the two amplitudes in (37) and (22) is which is of order one third of the chiral loop contribution in eq. (25).

V. DISCUSSION AND RESULTS
Our amplitude is complex as expected. In the chiral loop amplitude these are due to physical cuts (exchanges of physical particles) to the one-loop order we consider in this paper.
The Wilson coefficients turn complex when the c-quark is treated [8,9] within HQEFT. This is also the case for the matrix elements that these Wilson coefficients should be matched to.
There is a potential problem with a quark model without confinement that the amplitude may get an imaginary part due to production of free quarks. Still, within HQEFT one can hardly distinguish m c from M D because of the reparametrization invariance. Thus, at the present stage, it is not clear how well our model describes imaginary matrix elements, and we will not go into such details here, as the numerical consequences turn out to be minor.
Adding the amplitudes R χ and R G and multiplying with the Wilson coefficient [8,9] a 2 ≃ 1.33 + 0.2i, we obtain the quantity: Dropping the imaginary parts of the three quantities would give instead the value ≃ 0.25.
Anyway, we have found that the amplitude for B 0 → D + s D − s is of order 15 − 20% of the factorizable amplitude for B 0 → D + D − s , before the different KM-factors are taken into account. We obtain the branching ratios The difference between the branching ratios is mainly due to the difference in KM factor.
Taking into account the comments above, we end up with the conclusion that The ongoing searches at Belle might soon give the limit on the rate B 0 → D + s D − s , while the detection of the B 0 s mode might be presently more difficult due to troubles with B 0 s identification.
S.F. thanks P. Križan and B. Golob for fruitful discussion