Nonleptonic $B_c$ decay rates from model independent relations

TL;DR

This paper analyzes nonleptonic B_c decays to P-wave charmonia and light mesons using a model-independent approach, providing predictions for branching ratios and exploring the production of exotic states like _{c1}(3872).

Contribution

It introduces a universal form factor framework for B_c to P-wave charmonia transitions at leading order, enabling model-independent predictions of decay rates.

Findings

Predicted ratios of branching fractions for various decay modes.

Estimated production rates for the exotic _{c1}(3872) state.

Identification of universal functions governing form factors.

Abstract

Nonleptonic $B_c$ decays to $P$-wave charmonia and light $\pi^+$, $K^+$, and $\rho^+$, $K^{*+}$ mesons are analysed using factorization. The hadronic form factors parametrizing the $B_c \to \chi_{cJ}(h_c)$ matrix elements are expressed in terms of universal functions at the leading order of an expansion in the relative velocity of the heavy quarks in the $B_c$ rest-frame and in $1/m_Q$. Several ratios of branching fractions are evaluated, and when experimental information can be used, single branching fractions are presented. Both the $1P$ and $2P$ charmonia are considered. If the exotic candidate state $\chi_{c1}(3872)$ corresponds to $\chi_{c1}(2P)$, it should be produced in nonleptonic $B_c$ decays with predicted abundances with respect to the other states in the charmonium $2P$ spin four-plet.

Tables

Table 1. Table 1: Input parameters [ 42 ] .

$G_F=1.16637\left(1\right)\times {10}^{-5}{\text{GeV}}^{-2}$	$\left|V_{cb}\right|={0.04182}_{-0.00074}^{+0.00085}$
$m_{B_c}=6.274\text{GeV}$	$\left|V_{ud}\right|=0.97435\pm 0.00016$
${\tau }_{B_c}=0.510\pm 0.009\text{ps}$	$\left|V_{us}\right|=0.22500\pm 0.00067$
$m_{B_s^0}=5.367\text{GeV}$	$\left|V_{cs}\right|=0.97349\pm 0.00016$
$m_{J/\psi }=3.097\text{GeV}$
$m_{h_c\left(1P\right)}=3.525\text{GeV}$	$m_{h_c\left(2P\right)}=\{\begin{array}{cc}3.897\text{GeV}\text{<cite>[<a>44</a>]</cite>}& \\ 3.902\text{GeV}\text{<cite>[<a>43</a>]</cite>}& \\ 3.934\text{GeV}\text{<cite>[<a>45</a>]</cite>}& \\ 3.956\text{GeV}\text{<cite>[<a>46</a>]</cite>}& \end{array}$
$m_{{\chi }_{c0}\left(1P\right)}=3.415\text{GeV}$	$m_{{\chi }_{c0}\left(2P\right)}=3.860\text{GeV}$
$m_{{\chi }_{c1}\left(1P\right)}=3.511\text{GeV}$	$m_{{\chi }_{c1}\left(2P\right)}=3.872\text{GeV}$
$m_{{\chi }_{c2}\left(1P\right)}=3.556\text{GeV}$	$m_{{\chi }_{c2}\left(2P\right)}=3.930\text{GeV}$
$m_{\pi }=139.57\text{MeV}$	$f_{\pi }=130.2\text{MeV}$
$m_K=493.677\text{MeV}$	$f_K=155.7\text{MeV}$
$m_{\rho }=770\text{MeV}$	$f_{\rho }=209\text{MeV}$
$m_{K^{\ast }}=892\text{MeV}$	$f_{K^{\ast }}=205\text{MeV}$

Table 2. Table 5: B c → J / ψ → subscript 𝐵 𝑐 𝐽 𝜓 B_{c}\to J/\psi form factor A 0 ​ ( m π 2 ) subscript 𝐴 0 superscript subscript 𝑚 𝜋 2 A_{0}(m_{\pi}^{2}) from different models, and corresponding values of ℛ ℛ \mathcal{R} and ℬ ​ ( B c + → J / ψ ​ π + ) ℬ → superscript subscript 𝐵 𝑐 𝐽 𝜓 superscript 𝜋 \mathcal{B}(B_{c}^{+}\to J/\psi\pi^{+}) .

	lattice QCD [49]	relativistic quark model [17]	NRQCD (NLO) [50]
$A_0\left(m_{\pi }^2\right)$	$0.478\pm 0.031$	$0.449$	$0.947$
$ℛ$	${46.252}_{-6.464}^{+6.530}$	$48.315$	$10.871$
$ℬ\left(B_c^{+}\to J/\psi {\pi }^{+}\right)$	$\left(7.6\pm 1.0\right)\times {10}^{-4}$	$6.9\times {10}^{-4}$	$29.7\times {10}^{-4}$

Equations

$${\mathcal{H}}_{eff}=\frac{G_F}{\sqrt{2}}{V}_{cb}^{\ast }{V}_{uq}\left(C_1(\mu )Q_1(\mu )+C_2(\mu )Q_2(\mu )\right)+\text{h}.c.$$

$$Q_1$$

$$Q_2$$

$${\delta }_{\alpha \lambda }{\delta }_{\rho \beta }=\frac{1}{C}(T_a{)}_{\alpha \beta }(T_a{)}_{\rho \lambda }+\frac{1}{N_c}{\delta }_{\alpha \beta }{\delta }_{\rho \lambda },$$

$$Q_2=\frac{1}{N_c}{\bar{u}}_{\alpha }{\gamma }^{\mu }(1-{\gamma }_5)q_{\alpha }{\bar{b}}_{\beta }{\gamma }_{\mu }(1-{\gamma }_5)c_{\beta }+\frac{1}{C}{\bar{u}}_{\alpha }(T_a{)}_{\alpha \beta }{\gamma }^{\mu }(1-{\gamma }_5)q_{\beta }{\bar{b}}_{\rho }(T_a{)}_{\rho \lambda }{\gamma }_{\mu }(1-{\gamma }_5)c_{\lambda }.$$

$${\mathcal{H}}_{eff}=\frac{G_F}{\sqrt{2}}{V}_{cb}^{\ast }{V}_{uq}\left(C_0(\mu )Q_0(\mu )+C_8(\mu )Q_8(\mu )\right),$$

$$Q_0$$

$$Q_8$$

$$C_0=C_1+\frac{1}{N_c}{C}_2=a_1,C_8=\frac{1}{C}{C}_2.$$

$$\Gamma (B_c^{+}\to c\bar{c}(P)M_{P(V)})=\frac{|\vec{q}|}{8\pi m_{B_c}^2}|\mathcal{A}(B_c^{+}\to c\bar{c}(P)M_{P(V)}){|}^2,$$

$$\mathcal{A}(B_c^{+}\to c\bar{c}(P)M_{P(V)})=⟨c\bar{c}(P)M_{P(V)}|{\mathcal{H}}_{eff}|B_c^{+}⟩.$$

$$\mathcal{A}(B_c^{+}\to c\bar{c}(P)M_{P(V)})=\frac{G_F}{\sqrt{2}}{V}_{cb}^{\ast }{V}_{uq}{a}_1(\mu )⟨c\bar{c}(P)|\bar{b}{\gamma }_{\mu }(1-{\gamma }_5)c|B_c^{+}⟩⟨M_{P(V)}|\bar{u}{\gamma }^{\mu }(1-{\gamma }_5)q|0⟩.$$

$$⟨M_P(q)|\bar{u}{\gamma }^{\mu }(1$$

$$⟨M_V(q,{ϵ}_V)|\bar{u}{\gamma }^{\mu }(1$$

$$⟨S(p^{\prime })|\bar{b}{\gamma }^{\mu }{\gamma }_{5}c|B_c(p)⟩$$

$$⟨A(p^{\prime },ϵ)|\bar{b}{\gamma }^{\mu }c|B_c(p)⟩$$

$$⟨A(p^{\prime },ϵ)|\bar{b}{\gamma }^{\mu }{\gamma }_{5}c|B_c(p)⟩$$

$$⟨T(p^{\prime },\eta )|\bar{b}{\gamma }^{\mu }c|B_c(p)⟩$$

$$⟨T(p^{\prime },\eta )|\bar{b}{\gamma }^{\mu }{\gamma }_{5}c|B_c(p)⟩$$

$$f_0^S(0)$$

$$A_0^{A(T)}(0)$$

$$\Gamma (B_c^{+}\to {\chi }_{c0}{M}_P)$$

$$\Gamma (B_c^{+}\to A{M}_P)$$

$$\Gamma (B_c^{+}\to {\chi }_{c2}{M}_P)$$

$$\Lambda =\frac{a_1(m_b{)}^2{G}_F^2}{32\pi m_{B_c}^3}.$$

$$\Gamma (B_c^{+}\to {\chi }_{c0}{M}_V)$$

$$\Gamma (B_c^{+}\to A{M}_V)$$

$$\Gamma (B_c^{+}\to {\chi }_{c2}{M}_V)$$

$$f_0^{{\chi }_{c0}}(q^2)$$

$$A_0^{{\chi }_{c1}}(q^2)$$

$$A_0^{h_c}(q^2)$$

$$A_0^{{\chi }_{c2}}(q^2)$$

$$f_{+}^{{\chi }_{c0}}(q^2)$$

$$V^{{\chi }_{c1}}(q^2)$$

$$A_1^{{\chi }_{c1}}(q^2)$$

$$A_2^{{\chi }_{c1}}(q^2)$$

$$V^{{\chi }_{c2}}(q^2)$$

$$A_1^{{\chi }_{c2}}(q^2)$$

$$A_2^{{\chi }_{c2}}(q^2)$$

$$V^{h_c}(q^2)$$

$$A_1^{h_c}(q^2)$$

$$A_2^{h_c}(q^2)$$

Full text

1 Introduction

The decays of the $B_{c}$ meson, a hadron with the quarkonium structure and only weak transitions, represent a unique opportunity to analyse the properties of strong and weak interactions. The semileptonic decays are induced by both beauty and charm quark transitions, and give access to the $V_{cb}$, $V_{ub}$, and $V_{cs}$, $V_{cd}$ elements of the Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix. Several studies have been devoted to them, namely to the processes with excited states [1, 2, 3, 4, 5]. The $b\to c$ transitions, considered in the present analysis, involve $\lvert V_{cb}\rvert$ for which a tension between determinations from inclusive and exclusive $B$ decays is under scrutiny [6, 7]. Properties of the Standard Model (SM) as lepton flavour universality can be tested in semileptonic $B_{c}$ decays [8, 9, 10, 11, 12]. Rare semileptonic $B_{c}$ modes induced by flavour changing neutral current transitions of both beauty and charm quark also probe the SM and its extensions [13]. In such processes the hadronic uncertainty is in the $B_{c}$ form factors parametrizing the matrix elements of current operators.

For nonleptonic $B_{c}$ decays the theoretical description is more difficult than for the semileptonic modes, since it deals with the hadronic matrix elements of four-quark operators. The factorization approach represents a simple way to treat such operators in the decay amplitudes. For $b\to c\bar{q}_{1}q_{2}$ and neglecting the annihilation topology, the matrix elements of the four-quark operators is factorized in the product of two terms, one describing the $W$ production of the light meson and parametrized by the decay constant, the other one the $B_{c}$ to charmonium current matrix element.111 Examples of such calculations are in [14, 15]. Invoking the heavy quark spin symmetry, it has been observed that relations can be established among the $B_{c}$ to charmonium form factors, namely the two $B_{c}\to S$-wave $(\eta_{c},J/\psi)$ modes, the four $B_{c}\to P$-wave charmonium $(h_{c},\chi_{cJ})$ modes, etc., in a region of the phase space close to zero recoil [16, 17]. Such relations have been worked out through an expansion in the relative three-velocity of the heavy quarks in $B_{c}$ and in $1/m_{Q}$, including NLO terms [18, 5]. They can constrain the hadronic uncertainties and establish model-independent connections among different processes.

The present study is focused on nonleptonic $B_{c}$ decays to $1P$ and $2P$ charmonium plus a light meson $M$: $B_{c}\to\chi_{cJ}(h_{c})M$. They are of interest for several reasons. For example, the analysis of new physics contributions in processes like $B_{c}^{+}\to K^{+}\,K^{-}\,\pi^{+}$ must consider $B_{c}^{+}\to\chi_{c0}\,\pi^{+}$ as an intermediate state [19]. Moreover, the $B_{c}$ nonleptonic transitions probe the structure of states as $\chi_{c1}(3872)$ (previously denoted as $X(3872)$), whose multiquark/conventional quarkonium nature is under investigation [20, 21, 22, 23, 24, 25, 26, 27]. The nonleptonic $B_{c}$ processes provide additional information with respect to the semileptonic ones [28, 29]. Of particular interest is to exploit the model independent relations among the $B_{c}\to P$-wave charmonium form factors [5].

The plan of the paper is as follows. In Sec. 2, using the effective Hamiltonian inducing the $b\to c\bar{q}_{1}q_{2}$ transition with $q_{1,2}$ light quarks, we evaluate the decay widths of $B_{c}$ to $P$-wave charmonium and a light pseudoscalar/vector meson. In Sec. 3 we use the LO relations for the form factors in [5] to predict several ratios of branching fractions. In Sec. 4 we comment on a recent measurement of the ratio $\displaystyle\frac{\mathcal{B}(B_{c}^{+}\to B_{s}^{0}\pi^{+})}{\mathcal{B}(B_{c}^{+}\to J/\psi\pi^{+})}$ [30], suggesting a violation of factorization in the $B_{c}^{+}\to B_{s}^{0}\pi^{+}$ mode. Then we conclude.

2 Nonleptonic $B_{c}$ decays in the factorization approach

The effective Hamiltonian governing the nonleptonic $B_{c}$ decays to charmonium and a light meson is given by [31, 32]

[TABLE]

where

[TABLE]

$\alpha,\beta$ are color indices, and $q$ identifies the down type quark of the final light meson. $\mu$ represents the scale dependence of the Wilson coefficients which encode the short-distance physics from energy greater than $\mu$. It is cancelled in the amplitude by the $\mu$ dependence of the operators matrix elements. In our computation of $B$ decays we are setting $\mu=m_{b}$. Strong interaction effects are taken into account considering the Wilson coefficient $C_{2}(\mu)\neq 0$. Using the Fierz color rearrangement relation

[TABLE]

with $C=\frac{1}{2}$ and $N_{c}=3$, the $Q_{2}$ operator can be expressed as

[TABLE]

The effective Hamiltonian is rewritten as

[TABLE]

in terms of the operators

[TABLE]

$Q_{0}$ and $Q_{8}$ are color-singlet and color-octet operators contributing to the decay. The Wilson coefficients in (5) are related to the ones in (1):

[TABLE]

We consider the processes $B_{c}^{+}\to c\bar{c}(P)\,M_{P(V)}$, where $c\bar{c}(P)$ is one of the P-wave charmonium, and $P$,$V$ a light pseudoscalar or a vector meson. The decay width

[TABLE]

with $\lvert\vec{q}\rvert=\frac{1}{2m_{B_{c}}}\sqrt{\lambda(m_{B_{c}}^{2},m_{c\bar{c}}^{2},m_{P(V)}^{2})}$ the three-momentum of one of the final mesons in the $B_{c}$ rest-frame and $\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2xz-2yz$ the Källén function, involves the matrix element

[TABLE]

In naive factorization the contribution of the color-octet operator is discarded and the matrix element is factorized as

[TABLE]

The parametrization is used for the current-particle matrix elements,

[TABLE]

and for the matrix elements in terms of form factors:

[TABLE]

with $q=p-p^{\prime}$, and $S=\chi_{c0}$, $A=\chi_{c1},h_{c}$ and $T=\chi_{c2}$. $\epsilon_{V}$, $\epsilon$ and $\eta$ are the polarization vector and tensor of the light vector meson, and of the charmonium axial-vector and tensor state, respectively. The conditions are imposed

[TABLE]

The light meson selects the form factors contributing to the decay width: $f_{0}$ and $A_{0}^{A(T)}$ at $q^{2}=m_{P}^{2}$ for a light pseudoscalar, all the others at $q^{2}=m_{V}^{2}$ for a light vector meson. For the transitions to a pseudoscalar light meson we have, using factorization:

[TABLE]

where

[TABLE]

In case of a light vector meson we have:

[TABLE]

The Wilson coefficient $a_{1}(\mu)$ is evaluated at the scale $m_{b}$. $a_{1}(\mu)$ is a combination of the coefficients $C_{1}(\mu)$ and $C_{2}(\mu)$, whose two-loop computation is described in [33]. The naive factorization approach does not take into account nonfactorizable effects, e.g. the final-state interaction, the gluon exchanges between quarks of final mesons, which are expected to be sizable if both the final hadrons are heavy and the phase-space is small. On the basis of the Bjorken’s colour transparency argument [34], if the mesons do not have enough energy to escape from the colour field of each other, nonfactorizable interactions emerge. It is possible to partially encode such effects replacing the coefficient $a_{1}(\mu)$ ($a_{2}(\mu)$ for colour suppressed processes), by the effective coefficient $a_{1}^{eff}(\mu)$ ($a_{2}^{eff}(\mu)$), treated as a phenomenological parameter [31, 35]. Since we are considering processes involving one light final meson, we neglect the nonfactorizable terms and use $a_{1}(m_{b})=1.025$. A recent use of factorization for nonleptonic decays of $B$ and $D$ mesons can be found in [36]. Factorization and nonfactorizable contributions in nonleptonic $B$ meson decays, and their effects in the heavy-quark limit, are reviewed in [37]. For $B_{c}$ decays such terms have been considered in the PQCD approach (in [38, 2, 39] and in the references quoted therein) and in the QCD factorization [1, 40, 41].

3 Observables

The hadronic form factors of $B_{c}$ to the $P$–wave charmonium 4-plet $\chi_{cJ},h_{c}$ can be expressed near the zero-recoil point in terms of universal functions, performing an expansion in QCD in the relative velocity of the heavy quarks and in $1/m_{Q}$ [5]. At the leading order in the expansion a single universal function $\Xi(q^{2})$ parametrizes all such form factors. The following results are based on the extrapolation of the universal form factor from the zero-recoil point to the $q^{2}$ value corresponding to mass of the final light meson. Even though the next to leading order (NLO) in the expansion is presented in the analysis [5], the number of universal functions parametrizing these contributions does not allow us to construct useful observables. Therefore, the NLO is ignored and the analysis is performed considering a single universal function.

The relations for the form factors relevant for the transition to a light pseudoscalar meson are:

[TABLE]

The relations for the form factors relevant for the decays to a light vector meson are:

[TABLE]

Remarkably, at LO some form factors vanish: a consequence of Eq. (28) is that the $B_{c}$ transition to $\chi_{c1}$ and a light pseudoscalar is suppressed.

We use the relations (27)-(40) and the parameters in Table 1. The decay constants $f_{\rho}$ and $f_{K^{*}}$ are obtained from $\mathcal{B}(\tau^{-}\to\rho^{-}\nu_{\tau})=(25.19\pm 0.3)\times 10^{-2}$ and $\mathcal{B}(\tau^{-}\to K^{*-}\nu_{\tau})=(1.20\pm 0.07)\times 10^{-2}$ [42]. For the mass of the not yet discovered $h_{c}(2P)$ there are predictions from different models, a few results are presented in Table 1. We use $m_{h_{c}(2P)}=3.902\,\textrm{GeV}$ obtained in [43], since in such a model the masses of the $\chi_{cJ}(2P)$ triplet agree with measurements.

$\pi^{+}$ and $K^{+}$ modes

Using Eqs. (27)-(30) in Eqs. (LABEL:eq:GammaChic0P)-(22) we have for the $B_{c}$ transition to $\pi^{+}$:

[TABLE]

with $\Lambda_{\pi}=\Lambda\frac{f_{\pi}^{2}\lvert V_{cb}\rvert^{2}\lvert V_{ud}\rvert^{2}}{16m_{B_{c}}^{3}}$, where $\Lambda$ is given in Eq. (23). From the measurement $\mathcal{B}(B_{c}^{+}\to\chi_{c0}\pi^{+})=(2.4^{+0.9}_{-0.8})\times 10^{-5}$ and from Eq. (41) the value $\Xi(m_{\pi}^{2})=0.243^{+0.045}_{-0.040}$ is obtained, allowing us to predict the branching fractions to $\chi_{c2}(1P)$ and $h_{c}(1P)$:

[TABLE]

For the $2P$ excitations there are no measurements that can be exploited to determine the universal form factor. However, since this quantity cancels in the ratios of different channels involving charmonia belonging to the same 4-plet, the predictions in Table 2 can be derived.

The modes with $\chi_{c1}(1P)$ and $\chi_{c1}(2P)$ are suppressed: the observation of such a suppression for $\chi_{c1}(3872)$ would be in favour of the identification of $X(3872)$ as an ordinary charmonium state.

For the branching ratios with $K^{+}$ in the finale state the predictions are also in Table 2.

$\rho^{+}$ and $K^{*+}$ modes

Exploiting Eqs. (24)-(26) and Eqs. (31)-(40) we obtain the results in Table 3 for the $\rho^{+}$ modes, and in Table 4 for the $K^{*+}$ modes.

The conclusion is that the production of the state $\chi_{c1}$ is suppressed, and of the state $h_{c}$ is enhanced compared to the other charmonia. The effect of changing the $h_{c}(2P)$ mass is shown in Figs. 1 and 2.

A comment on the mass of $\chi_{c0}(2P)$ is in order. We use the mass of $\chi_{c0}(3860)$ even though another state, $X(3915)$ with $J^{PC}=0^{++}$ has been observed and there are hints on its charmonium nature [47]. Our choice respects the hyperfine splitting hierarchy, which would be violated if $X(3915)$ is identified with $\chi_{c0}(2P)$.

4 Comment on a recent LHCb measurement

The LHCb collaboration has measured the ratio of the branching fractions of nonleptonic $B_{c}^{+}\to B_{s}^{0}\pi^{+}$ and $B_{c}^{+}\to J/\psi\pi^{+}$ decays [30]:

[TABLE]

where the last uncertainty is due to the knowledge of the branching fractions of the intermediate state decays. Using naive factorization for the two amplitudes we get

[TABLE]

where $f_{0}$ and $A_{0}$ enter in the parametrization of $B_{c}\to B_{s}$ and $B_{c}\to J/\psi$ matrix elements, respectively. The ratio is:

[TABLE]

Using the parameters in Table 1 and setting $a_{1}(m_{c})=1.089$, the ratio (47) depends only on the form factors, which are determined, e.g., by lattice QCD computations, QCD sum rules, quark models, approaches based on nonrelativistic QCD (NRQCD).

The result $f_{0}(m_{\pi}^{2})=0.625\pm 0.010$ from lattice QCD [48], allow us to obtain $\mathcal{B}(B_{c}^{+}\to B_{s}^{0}\pi^{+})=0.0348\pm 0.0013$. For the $J/\psi$ transition, different determinations of $A_{0}(m_{\pi}^{2})$ produce the results in Table 5.

All methods fail to reproduce Eq. (45), the smallest result corresponding to NRQCD. In the improved relativistic quark model [51] the obtained $\mathcal{R}$ is also similar to the one in Table 5. On the other hand, for the $J/\psi$ modes the ratio

[TABLE]

is measured [52]. Using the form factors in [49] we obtain $\mathcal{R}_{J/\psi}=0.0493$. Therefore, the discrepancy between the values in Table 5 and the measurement in (45) should be attributed to the $B_{s}^{0}$ channel.

The ratio analogous to Eq. (45) with a kaon in the final state, using lattice QCD form factors, gives

[TABLE]

and

[TABLE]

Combining $\mathcal{B}(B_{c}^{+}\to J/\psi K^{+})$ and $\mathcal{B}(B_{c}^{+}\to J/\psi\pi^{+})$ we have

[TABLE]

in agreement with the measurement $\mathcal{R}_{J/\psi}^{NL}=0.079\pm 0.007\pm 0.003$ [53].

The conclusion is that factorization works quite well for the $J/\psi$ channel. Deviations emerge in the $B_{s}^{0}$ channel, for which the argument of colour transparency fails due to the small momentum of pion and kaon in the final state.

5 Conclusions

Applying naive factorization we have analyzed the nonleptonic $B_{c}$ decays to the $P$-wave lowest-lying and first radial excitations of the charmonium, and $\pi^{+}$, $K^{+}$, $\rho^{+}$ and $K^{*+}$. Using the LO relations among the $B_{c}\to\chi_{cJ}(h_{c})$ form factors [5], several ratios of branching fraction are predicted. The $\chi_{c1}$ channel, both for the $1P$ and $2P$ state, is always suppressed: this should be observed if $\chi_{c1}(3872)$ is an ordinary charmonium. On the other hand, $h_{c}$ production is enhanced compared to the other states in the multiplet. The results are model-independent, as they are obtained by an expansion of the form factor expressions.

An analysis of the recent LHCb measurement in Eq. (45) shows that the $B_{c}$ branching fractions involving $J/\psi$ are reproduced, while for $B_{c}\to B_{s}\pi^{+}$ the nonfactorizable contributions are sizeable, as it could be expected on the basis of the color transparency argument.

Acknowledgements. I thank Pietro Colangelo, Fulvia De Fazio, Francesco Loparco and Martin Novoa-Brunet for discussions. This study has been carried out within the INFN project (Iniziativa Specifica) QFT-HEP.