The rare semi-leptonic Bc decays involving orbitally excited final mesons

The rare processes Bc→Ds*Jμμ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {B}_c\to {D_{(s)}^{\left(*\right)}}_J\mu \overline{\mu} $$\end{document}, where D(s)(*)J stands for the final meson Ds0*(2317), Ds1(2460, 2536), Ds2*(2573), D0*(2400), D1(2420, 2430) or D2*(2460), are studied within the Standard Model. The hadronic matrix elements are evaluated in the Bethe-Salpeter approach and furthermore a discussion on the gauge-invariant condition of the annihilation hadronic currents is presented. Considering the penguin, box, annihilation, color-favored cascade and color-suppressed cascade contributions, the observables dBr/dQ2, ALPL, AFB and PL are calculated.


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The rare decays b → s(d)ll have particular features. These transitions are of the singlequark flavor-changing neutral current (FCNC) processes, which are forbidden at tree level in the Standard Model (SM) but mediated by loop processes. Hence, within the SM, the b → s(d)ll amplitudes are greatly suppressed. The situation is different for the standard model extensions, where many new particles beyond the SM are predicted. These new particles can virtually entry the loops relevant to FCNC processes or induce the transitions at tree level, which makes that the observables predicted in the standard model extensions may significantly deviate from the ones in the SM. This sensitive nature to the effects beyond the SM can be exploited as a tool for stringently testing the SM and indirectly hunting the New Physics (NP).
In the previous works [23,24], the process B c → D * s0 (2317)ll was calculated including only the b → sll effects, whose typical Feynman diagrams are Box and Penguin (BP) diagrams, as plotted in figures 1 (a, b). However, besides the BP effects, the Annihilation (Ann) diagrams, as shown in figure 1 (c), also make un-negligible contributions. On one hand, both BP and Ann diagrams are of order O(α em G f ) and the ratio of their CKM matrix elements is |V * cb V cs(d) |/|V * ts(d) V tb | ∼ 1. On the other hand, from figure 1 (c), we see that the color factors of Ann diagrams are 3 times larger than those of BP diagrams. Thus, when the decay B c → D * s0 (2317)ll is analyzed, it is necessary to include the Ann effects. In addition to the BP and Ann effects, the process B c → D * s0 (2317)ll is also influenced by resonance cascade processes, such as B c → D * s0 (2317)J/ψ (ψ(2S)) → D * s0 (2317)ll. Their typical Feynman diagrams are illustrated in figures 1 (d, e). Transition amplitudes of these diagrams in the area m 2 ll ∼ m 2 J/ψ (ψ(2S)) always become much larger than the BP and Ann ones. Hence, to avoid overwhelming the BP and Ann contributions, the regions around m 2 ll ∼ m 2 J/ψ (ψ(2S)) should be experimentally removed. In ref. [23], the regions [25], which are defined through comparing the BP and color-suppressed (CS) cascade contributions, are employed. However, in the B c → D * s0 (2317)ll process, both the color-favored (CF) and CS diagrams exist. Furthermore, the CF transition amplitudes are expected to be larger  s(d)J ll process. In annihilation diagrams (c) the photon can be emitted from each quark, denoted by , and decays to the lepton pair. than the CS ones by a 3 times larger color factor approximately. Thus, it is necessary to redefine these regions with both CF and CS cascade influences.
So in this paper, we investigate B c → D * s0 (2317)ll transition including BP, Ann, CS and CF contributions. In addition, in order to give a more comprehensive discussion on the semi-leptonic rare decays of B c , the processes B c →D s1 (2460, 2536)ll, B c → D * s2 (2573)ll and B c → D ( * ) J ll are also analyzed. In our calculations, the low-energy effective theory is employed [26]. Within this method, the short distance information of transition amplitude is factorized into the Wilson coefficients, while the long distance effects are described by the matrix element which is an operator sandwiched by the initial and the final states. The Wilson coefficients in the SM can be attained perturbatively. But the matrix elements are of non-perturbative nature and in this paper we calculate them with the Bethe-Salpeter (BS) method [27]. In this method, the BS equation [28,29] is employed to solve the wave functions for mesons, while the Mandelstam Formalism [30] is used to evaluate hadronic matrix elements. With such method, the hadronic matrix elements keep the relativistic effects from both the wave functions and the kinematics. In our previous paper [31], within the BS method, we calculated the B c → D ( * ) s,d ll rare transitions, whose final mesons are of S-wave states, and checked the gauge-invariance condition of the annihilation hadronic currents. In this paper, we investigate the processes B c → D

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This paper is organized as follows. In section 2, we introduce the transition amplitudes corresponding to BP, Ann, CS and CF contributions and specify the involved hadronic matrix elements. Within section 3, we calculate these hadronic matrix elements through the Bethe-Salpeter method and express the results in terms of form factors. In section 4, using these form factors, we compute the observables, including dBr/dQ 2 , A LP L , A F B and P L . Section 5 is devoted to the discussions on the theoretical uncertainties. Finally, we summarize and conclude in section 6.

Transition amplitudes of BP, Ann, CS and CF contributions
In this section, we briefly review the transition amplitudes corresponding to BP, Ann, CS and CF effects. A more detailed introduction can be found in our previous paper [31].
According to low-energy effective theory [26], the transition amplitude describing the b → s(d)ll (or equivalently, BP) contribution is, where Q = P i − P f and P i(f ) stands for the momentum of the initial (finial) meson. V tb and V ts(d) denote the CKM matrix elements. C 10 is the Wilson coefficient. C eff 7,9 are the combinations of the Wilson coefficients which are multiplied by the same hadronic matrix elements. The numerical value of C 10 and the explicit expressions of C eff 7,9 can be found in ref. [32]. The hadronic matrix elements W µ and W T µ are defined as where the definition σ µν = (i/2)[γ µ , γ ν ] is used. Based on the effective theory [26] and the factorization hypothesis [33], the transition amplitude describing the Ann effects is [31] where C 1,2 are the Wilson coefficients, whose values can be found in ref. [32]. The annihilation hadronic current W µ ann is defined as W µ (2.4) p q 1−4 and m q 1−4 are momenta and masses of the propagated quarks, respectively.

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For the CS and CF cascade resonance effects, the transition amplitudes are [31] where M V and Γ V are the mass and full width of the resonance meson, respectively. Γ(V → ll) denotes the branching width of the transition V →ll. The resonance meson V stands for the particle J/ψ or ψ(2S). The CF hadronic current W µ CF is defined as Consequently, the total transition amplitude is

Hadronic transition matrix elements in the BS method
In section 2, the transition amplitudes of the B c → D ( * ) (s)J ll processes are introduced and the hadronic matrix elements W (T ) , W ann and W CF are defined. In this section, within the BS method, we show how to calculate these hadronic matrix elements. In section 3.1, we express the hadronic currents as the integrals of the wave functions. Section 3.2 is devoted to showing the wave functions of the mesons which are involved in this paper. Using these wave functions, we calculate the hadronic currents in section 3.3 and parameterize the results in terms of form factors in section 3.4. In section 3.5, we present the numerical results of the form factors.

General arguments on hadronic currents
In this part, we rewrite the hadronic currents as the integrals of the wave functions and present some general arguments.
According to the Mandelstam formalism [30], W (T ) can be expressed as the integrals of the 4-dimensional BS wave functions. In the spirit of the instantaneous approximation [34], the integrations with respect to q 0 i , where q i represents the relative momentum between the quark and anti-quark of the initial meson, can be performed first. And then we have [27,31]

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where the hadronic tensors Y µν V,A are defined as The term ϕ ++ i(f ) in eqs. (3.1)-(3.2) denotes the positive energy part of the initial (finial) wave function [34] and will be specified in the next subsection. In this paper we ignore the negative-energy parts since they give negligible contributions.

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Using eqs. (3.3)-(3.7), we now discuss the gauge invariant condition of the Ann hadronic currents calculated in BS method. One may note that examining whether W ann satisfies the gauge invariant condition is equivalent to checking whether W ann · Q is zero. If we multiply eqs. (3.3)-(3.6) by Q µ , it is obvious that (W 1ann · Q) + (W 2ann · Q) cancels (W 4ann · Q)+(W 3ann · Q). Hence, we have W ann ·Q = 0. This implies that the Ann hadronic currents in BS method indeed satisfy the gauge invariant condition. We stress that there is no need to specify the initial or final state in the process of obtaining W ann · Q = 0. Thus, our conclusion is quite general.
For W CF , in this paper, we do not go into any details of their calculations, because W CF s involved in the B c → D (s) µμ) calculation.) The decay constants of the scalar and axial-vector mesons can be found in ref. [35]. But due to the angular momentum conservation condition, the longitudinal decay constants of the tensor mesons are zero. Hence, we have W CF (B c → D * s2 (2573)(D * 2 (2460))µμ) = 0.

Wave functions in BS method
In BS method, the meson is considered to be a bound state of two constituent quarks and can be described by the BS wave functions [28]. In the framework of instantaneous approximation [34], the time component of the BS wave functions' arguments can be integrated out and the BS equations are reduced to the Salpeter equations. By means of solving the Salpeter equations, we obtain the wave function [35][36][37][38] for each meson.

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(3) Wave Functions of D s1 (2460, 2536) and D 1 (2420, 2430). Unlike the mesons introduced above, D s1 (2460, 2536) and D 1 (2420, 2430) can not be described by the pure (2S+1) L J states. Based on [40,41], we consider them as the mixtures of the 1 P 1 and 3 P 1 states, namely, where α = θ − arctan( 1/2) and β = θ s − arctan( 1/2). Based on the experimental observation [42] and the discussions in ref. [41], the mixing angle θ = 5.7 • is used in this paper. Besides, according to the analysis in the quark potential model [43], θ s = 7 • is employed. From eq. (3.10), the wave functions of D s1 (2460, 2536) and D 1 (2420, 2430) can be constructed from the ones of 1 P 1 and 3 P 1 states. In the BS method, the positive energy wave functions of 1 P 1 and 3 P 1 states [39] are where ǫ A µ is the polarization vector of the axial-vector meson. The explicit expressions of b 1−4 and c 1−4 can be found in ref. [39] and their numerical values can be obtained by solving the Salpeter equations [35]. In the processes of solving the Salpeter equations, the masses of 1 P 1 and 3 P 1 states, namely, M D (s) 1 P 1 and M D (s) 3 P 1 , are required. In analogy to the case of η 1 − η 8 mixing [44], we determine them from the following relationships [45,46], where M D 1 (2420,2430) and M D s1 (2460,2536) stand for the physical masses and we take them from ref. [22].
(4) Wave function of B c . The B c meson is considered as a 1 S 0 state, whose the positive energy wave function can be written as [36], where the parameters e 1−4 can be found in ref. [36].

Calculations of hadronic matrix elements
In this part, we calculate the hadronic currents through the formalism introduced above.
Since W µ s have been investigated extensively in our previous papers [39,[47][48][49][50][51], here we do not introduce the W µ calculations but pay more attentions to W µ T,ann s. Please recall that W µ T s have been expressed in combinations of Y µν V,A s within eq. (3.1), while in eqs. (3.3)-(3.6), W µ ann s are written in terms of F i,f 0(±) s. Hence, in order to obtain W µ T,ann , it is convenient to compute Y µν V,A s and F i,f 0(±) s first of all. From their definitions in eq. (3.2) and eq. (3.7), we see that the calculations of Y µν V,A s and F i,f 0(±) s are channel-dependent and the channels under our consideration include P → S, T, A transitions, where P, S, T, A are the abbreviations for pseudo-scalar, scalar, tensor, axial-vector mesons, respectively.

Hadronic matrix elements of P → S processes
First, we introduce the details of the Y µν V,A (P → S) estimations. We have expressed Y µν V,A s as the overlapping integrals of ϕ ++ i,f s in eq. (3.2). In the P → S processes, the initial wave The expressions of ϕ ++ 1 S 0 and ϕ ++ 3 P 0 are given in eq. (3.13) and eq. (3.8), respectively. Substituting eqs. (3.8), (3.13) into eq. (3.2), the hadronic matrix elements Y µν V,A s can be obtained. In light of the forbidden where the definition of q a has been given in section 3.1, while q b is the relative momentum of the final meson. Due to the spectator approximation, the retarded relationship between q a and q b reads [27] q Now we turn to the discussions of F i,f 0(±) (P → S)s. In eq. (3.7), F i0(±) s are written in

Hadronic matrix elements of P → T processes
Here we deal with Y µν V,A in the P → T precesses. The calculations of Y µν V,A (P → T ) are similar to the ones of Y µν V,A (P → S), except replacing the final wave function ϕ ++ 3 P 0 by ϕ ++ 3 P 2 . The expression of ϕ ++ 3 P 2 can be found in eq. (3.9). Hence, we have The expressions of F αµν V l and F αµν Ak , where l = 1, . . . , 7 and k = 1, 2, 3, are presented in appendix A.
Next, we pay attentions to F i0(±) (P → T )s. From eq. (3.7), we see that F i0(±) (P → T )s are the same as F i0(±) (P → S)s, due to the identical initial meson B c in the decays P → S, T . The discussions of F i0(±) (P → S)s have been performed in section 3.3.1.
But for F f 0(±) (P → T )s, the situations are different. They should be calculated through eq. (3.7), with the final wave functions ϕ ++ f being ϕ ++ 3 P 2 . After factoring the polarization tensor out, we have

Hadronic matrix elements of P → A processes
Due to the mixing nature of the final mesons as formulated in eq. (3.10), the calculations of Y µν V,A (P → A)s and F i,f 0(±) (P → A)s are different from the cases of P → S and P → T . In order to obtain Y µν V,A (P → A)s and F i,f 0(±) (P → A)s, first of all, we compute Y µν V,A (P → A3 P 1 , 1 P 1 )s and F i,f 0(±) (P → A3 P 1 , 1 P 1 )s. And then, based on the mixing relationships in eq. (3.10), we combine the results of P → A3 P 1 and P → A1 P 1 .

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For Y µν V,A (P → A3 P 1 , 1 P 1 )s, we calculate them from eq. (3.2), with the initial wave function ϕ ++ i being ϕ ++ 1 S 0 and the final one ϕ ++ f being ϕ ++ 3 P 1 , 1 P 1 . The expressions of ϕ ++ 3 P 1 , 1 P 1 are given in eq. (3.11), while the initial ones ϕ ++ 1 S 0 is shown in eq. (3.13). The results of Finally, with the results above and the mixing relationship in eq. (3.10), we can calculate the hadronic matrix elements of the physical processes from ,

The definitions of form factors
In the previous parts, we show how to calculate the hadronic currents. In order to show their results conveniently, here we parameterize the hadronic matrix elements in terms of the form factors. In this paper, we do not define the form factors of W CF s, because as introduced in section 3.1, W µ CF (P → S, A) can be obtained from W µ CF (P → P, V ) by some trivial replacements, while W µ CF (P → T ) = 0. Hence, in the following paragraphs, we pay more attentions to the form factors of W (T ) and W ann s.
In the case of the P → Sll transitions, according to the Lorentz symmetry and the gauge invariant condition of the Ann currents discussed in section 3.1, we have where P + ≡ P i + P f and F S z , F S 0 , F S T , B S z are form factors.

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Similarly, for P → T ll transitions, the definitions are shown as T T zann and V T ann are the form factors. As to P → All decays, the definitions take the following forms, , T A zann and V A ann are the form factors.

Numerical results of form factors
In this part, we present the numerical results of form factors and the according discussions.

Parameters in the calculations
Here we specify the involved parameters. First, the masses and the lifetimes of B c and D ( * ) (s)J are required in our calculations and we take their values from ref. [22]. Second, the BS-inputs are also needed, which include the Cornell-Potential-Parameters (CPPs) and the masses of the constituent quarks. The CPPs can be found in ref. [57]. The masses of the constituent quarks are taken as m b = 4.96 GeV, m c = 1.62 GeV, m s = 0.5 GeV and m d = 0.311 GeV [47].

Results and discussions on form factors
From the aforementioned parameters and the derivations in section 3.3, the form factors can be evaluated. In the following paragraphs, we will show and discuss them.
In figure 3 (a), the form factors of W µ (T ) (B c → D * s0 (2317)) are presented. These form factors are all positively related to Q 2 . This behavior can be understood from the facts that 1) as shown in eqs. (3.1)-(3.2), our hadronic currents W µ (T ) s are obtained from the integrals over the overlapping regions of the initial and final wave functions and 2) due to the retarded relationship in eq. (3.15), the overlapping regions grow with increase in the variable Q 2 .
In recent years, W µ (T ) (B c → D * s0 (2317)) have also been calculated in the three-point QCD sum rules [23] and light-cone quark model [24]. The definitions of the W µ (T ) form factors in refs. [23,24] are different from the ones in this paper. But if the same definitions are taken, the absolute values of our form factors are comparable with theirs. Figure 3 (b) shows the form factors of W µ ann (B c → D * s0 (2317)). We see that B S z are complex. The reason is that in the calculations of the W ann , the quark propagators are involved, as shown in eqs. (3.3)-(3.6). In order to deal with these propagators, we separate them into two parts: the principal value terms and δ function ones. The real part of B S z comes from the principal value terms, while its imaginary part is caused by δ function terms. 2 Figures 4 (a, b) (2573)) also increase monotonically as Q 2 grows. This similarity comes from the facts that both W µ (T ) (B c → D * s0 (2317)) and  (2317)). To see how this happens, one should recall that the Ann currents W ann are the sums of the terms W ann1,...,ann4 s. In the case of W µ ann (B c → D * s0 (2317)), the four terms all contribute. But as to W µ ann (B c → D * s2 (2573)), the vanishing decay constant of the final meson forbids the W ann1,ann2 contributions and leaves only W ann3,ann4 terms. Compared with the sums of W ann1 and W ann2 , the contributions of W ann3 and W ann4 are fairly suppressed. 3 Thus, we see the smaller W µ ann (B c → D * s2 (2573)) form factors in figures 4 (c, d).
In figures 5 (a, b) and figures 6 (a, b), we plot the BP form factors of B c → D s1 (2460, 2536)ll. First, we see that the form factors of W µ (T ) (B c → D s1 (2460, 2536)) are not of the same sign. To understand this feature, recall that in order to calculate The monotonicity of the BP form factors and complexity of the Ann form factors can also be found in the case of Bc → D ( * ) (s) µμ processes [31]. And in ref. [31], there is a more detailed discussion on them. 3 The reason of this suppression is that Wann3 and Wann4 correspond to the diagrams where the virtual photons are emitted from the final quarks. Under the non-relativistic limit, the propagated quarks of these diagrams are highly off-shell and therefore when calculating the amplitudes of these diagrams, the denominators are considerably large. Even though the relativistic effects are included, this kind of suppression is still not obviously ameliorated.

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ated and then we mix the results according to the mixing relationship in eq. (3.22). The form factors of W (T ) (B c → D s 1 P 1 , 3 P 1 ) are all of the same sign. But in the mixing step, we need to evaluate the sums and differences of the W (T ) (B c → D s 1 P 1 , 3 P 1 ) form factors. Hence, as illustrated in figures 5 (a, b) and figures 6 (a, b), the form factors with the different signs emerge.
Second, from figures 6 (a, b), one may note that the absolute values of V A , A A 1 , T A 1 and T A 2 are much smaller than those of A A 0,2 and T A 3 . This feature implies that the hadronic matrix element W (T ) (B c → D s1 (2536) ⊥ ) obtained in the BS method is suppressed significantly compared with W (T ) (B c → D s1 (2536) ). Here D s1 (2536) ⊥( ) stands for the final meson D s1 (2536) which is transversely (longitudinally) polarized.

The observables
In the previous section, we calculate the hadronic matrix elements within the BS method and express the results in terms of the form factors. Using these form factors, the total amplitude M Total in eq. (2.7) can be estimated. From the obtained total amplitude, in this section, we evaluate the physical observables.

The calculations of observables
In this part, we employ the helicity amplitude method [32] to calculate observables.
First of all, we need to split the total transition amplitudes as where M µ 1(2) can be determined by matching eq. (2.7) to the equation above. And then by projecting M µ 1 (2) to the helicity components ǫ µ H (t, 0, ±1), the helicity amplitudes can be obtained, that is [32], The explicit expressions of ǫ µ H (t, 0, ±1) are specified in appendix B. Finally, according to the derivations in ref. [32], the differential branching fractions dBr/dQ 2 , the forward-backward asymmetries A F B , the longitudinal polarizations of the JHEP09(2015)171 final mesons P L and the leptonic longitudinal polarization asymmetries A LP L can be expressed in terms of helicity amplitudes, which are where h denotes the helicity of l − , while the denotation λ = (

Numerical results of the observables
Within figures 11-18, the numerical values of the observables are presented in the solid (or dash-dot) lines, while their theoretical uncertainties are illustrated in the pale green (or pink) areas. In this part, we lay stress on the introductions of numerical results of the observables. And in next section, the systematic discussions on the theoretical uncertainties will be shown. When the numerical values of observables are calculated in this paper, we have considered the BP, Ann, CS and CF diagrams. In order to show their influences clearly, for each channel, we plot 1) the observables where only BP contributions are considered, 2) the ones where BP and CS effects are contained, 3) the ones with BP and Ann influences and 4) the ones including the BP, Ann, CS and CF diagrams. In the following paragraphes, their comparisons and discussions will be presented.
For dBr/dQ 2 which includes only BP contributions, as shown in the dash-dot line of figure 11 (a), we see that dBr/dQ 2 is biggest around Q 2 ∼ 10.5 GeV 2 and suppressed

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considerably at the end points. This is similar to the result in ref. [24] but quite different from the one in ref. [23]. If the Ann effects are added, as plotted in the dash-dot line of figure 11 (b), dBr/dQ 2 is enhanced un-negligibly around Q ∼ 12.5 GeV 2 .
For dBr/dQ 2 which contains BP and CS effects, as plotted in the solid line of figure 11 (a), because of the Breit-Wigner propagators in C CS 9 , the significant enlargements emerge around the resonance regions. If the Ann and CF diagrams are included, as displayed in solid line of figure 11 (b), dBr/dQ 2 around Q 2 ∼ M 2 J/ψ continues enlarging. But in light of the node structure of the ψ(2S) wave function, which leads to the cancelations in the W CF (B c → D * s0 (2317)ψ(2S) → D * s0 (2317)µμ) calculation, dBr/dQ 2 around Q 2 ∼ M 2 ψ(2S) changes imperceptibly. This feature can also be found in the processes B c → D (s) µμ [31].
In Once the BP, Ann, CS and CF contributions are all considered, as seen in solid line of figure 11 (d), one may find that A LP L ∼ −1 in the low Q 2 region. This is due to the cancelations between Ann and CF transition amplitudes. Figures 12 (a-h) depict observables of the B c → D * s2 (2573)µμ transition. Considering W CF (B c → D * s2 (2573)) = 0 as discussed in section 3.1, the B c → D * s2 (2573)µμ process does not receive any contributions from the CF diagrams. Hence, in figures 12 (a-h), we do not illustrate the observables which include CF effects.

The observables of
Within figures 12 (a, b), we plot dBr/dQ 2 s as the functions of Q 2 . First, we see that dBr/dQ 2 (B c → D * s2 (2573)µμ)s are much bigger than dBr/dQ 2 (B c → D * s0 (2317)µμ)s around the Q 2 ∼ 0 GeV 2 point. To understand this behavior, note that 1) from eq.   12 (a, b), one may note that dBr/dQ 2 including the BP and Ann effects deviates imperceptibly from the one with only BP contribution. This is because that as plotted in figures 4 (c, d), the Ann form factors are quite small, which suppresses M ann considerably so that the Ann contributions are much less than the BP ones. Hence, as illustrated in figures 12 (a, b), dBr/dQ 2 s show the insensitivities to the Ann diagrams. Figures 12 (c, d)  In figures 12 (e, f), we display A F B s of the B c → D * s2 (2573)µμ process. In figure 12 (e), we see that A F B s are positive over the high Q 2 domain (except the resonance regions), while due to suppressions from the γ penguin diagrams, A F B s turn negative in the low Q 2 region. Once the Ann influences are take into account, likewise for dBr(B c → D * s2 (2573)µμ)/dQ 2 s and A LP L (B c → D * s2 (2573)µμ)s, A F B s behave insensitively to Ann effects. Figures 12 (g, h) show the results of P L (B c → D * s2 (2573)µμ)s. When only the BP diagrams are contained, P L is positively related to Q 2 in the low Q 2 region but inversely to Q 2 in the high Q 2 domain. If the Ann effects are added, P L s change negligibly. These sensitive behaviors imply that the CF and Ann contributions play important roles in the B c → D s1 (2460)µμ process. Therefore, when the observables of B c → D s1 (2460)µμ transition are calculated, besides the BP and CS Feynman diagrams, it is necessary to include the CF and Ann diagrams.
Second, we see that when only BP Feynman diagrams are included, A F B ∼ 0 and P L ∼ 1 within the area Q 2 ∈ [1, 6] GeV 2 . To see how this happens, we note that as concluded in section 3.5.2, the hadronic current (2536) ). This implies that, if only BP effects are considered, the transverse helicity amplitudes in the B c → D s1 (2536)µμ decay are considerably suppressed compared with the longitudinal ones, namely, H Hence, according to the expressions of A F B and P L in eq. (4.3), over the domain Q 2 ∈ [1, 6] GeV 2 , |A F B | has a quite small value, while P L almost equals one.
Third, if the Ann and CF influences are contained, the B c → D s1 (2536)µμ observables show the insensitivities. This is because the decay constant of D s1 (2536) is fairly small, which suppresses M ann and M CF strongly so that the BP contributions are quite bigger than the others. Hence, as illustrated in figures 14 (a-h), when the Ann and CF diagrams are added, there are no obvious deviations in the B c → D s1 (2536)µμ observables outside the resonance regions.

The experimentally excluded regions and integrated branching fractions
Using the results of dBr/dQ 2 s, as shown in figures 11-18 (a, b), now we define the experimentally excluded regions. According to the sensitivities to the CF effects, the decays B c → D ( * ) (s)J µμ fall into two categories. The first category includes B c → D * 0 (2400)(D * s0 (2317))µμ,

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Modes  The second category contains B c → D * 2 (2460)(D * s2 (2573))µμ, B c → D s1 (2536)µμ and B c → D 1 (2420)µμ transitions, which are not sensitive to the CF contributions. So their experimentally excluded area is defined as Region : Q 2 > 7 GeV 2 . (4.6) Based on the experimentally excluded regions introduced above, the integrated branching fractions are calculated and shown in table 1. As seen in table 1, the branching fractions including BP and Ann effects are comparable with the ones containing both BP, Ann, CF and CS contributions. This implies that our experimentally excluded regions defined in eqs. (4.5), (4.6) are workable.

Estimations of the theoretical uncertainties
In the previous section, the numerical results of the B c → D ( * ) (s)J µμ observables are discussed. In this part, we discuss their theoretical uncertainties.
In this paper, we estimate the theoretical uncertainties of the observables including two aspects. First, the theoretical errors from hadronic matrix elements are considered. Recall that our hadronic currents are calculated in the BS method and the obtained form factors are dependent on the numerical values of the BS inputs. In order to estimate the according JHEP09(2015)171 systematic uncertainties, we calculate the observables with changing the BS inputs by ±5%. Second, the systematic errors aroused by the factorization hypothesis are included. In the derivations of M Ann,CS,CF , the factorization hypothesis [33] is employed. In this method, in order to include the non-factorizable contributions, the number of colors N c in the expression (C 1 /N c + C 2 ) or (C 1 + C 2 /N c ) is treated as an adjustable parameter which should be determined by fitting the experimental data [58][59][60][61]. But since that the present experimental data on B c meson is still rare so that this parameter can not be obtained at the moment, we calculate the observables with N c = 3 but change the numerical values of N c within the region [2, ∞] for estimating systematic uncertainties brought by factorization hypothesis.
Actually, in recent years, several methods, dealing with the non-factorizable contributions more systematically, have been devoted to investigating the B c decays, such as perturbative QCD approach(PQCD) [62,63] and QCD factorization (QCDF) [64]. However, the channels in which the PQCD and QCDF are workable must have energetic final particles. Moreover as to B c → D ( * ) (s)J ll, the finial mesons have small recoil momenta in the high Q 2 domain. Hence, in this paper, we choose to employ the factorization method [33]. Similar situations can also be found in the calculations of B c → D ( * ) (s) ll [32,[65][66][67][68][69][70][71][72][73] in which the factorization method has to be used extensively to account for the non-factorizable effects.
Here we stress that using the factorization assumption to deal with the non-factorizable effects is a temporary way in the early stage of investigating the rare B c decays. A more systematical method is important and necessary. Hence, more work in the future is required.

Testing the hadronic matrix elements
In the previous subsection, by changing the BS inputs within ±5%, we estimate the theoretical uncertainties from hadronic currents. Strictly speaking, this only measures parts of the uncertainties, because the systematic uncertainties from the approximations made within the BS method are not considered. Considering that this kind of uncertainties are rather difficult to be systematically estimated, in fact, we do not control the hadronic uncertainties confidently. 4 Hence, testing whether the hadronic currents are properly evaluated is important.
From eq. (2.1), we see that within the transition amplitude M BP , the hadronic currents are multiplied by the Wilson coefficients C eff 7,9 , C 10 which are sensitive to NP. This makes that from the observables of B c → D ( * ) (s)J ll, it is quite involved to tell whether each hadronic current is correctly estimated. Hence, in order to test them, it is beneficial to analyze the channels in which the short distance interactions are not sensitive to NP and the hadronic matrix elements are similar or identical to the ones participating in B c → D

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In order to examine the BS inputs and the approximations of the BS method, we should pay attentions to the B c,s,u,d → D ( * ) s,d,u (η c , J/ψ)µν decays whose finial mesons are of S-wave states. In our previous papers [48,75], the observables of the processes B (s) → D ( * ) (s) µν are estimated and the results are in good agreements with the experimental observations [22]. In ref. [76], the B c → J/ψ(η c )µν are analyzed and we expect that these channels can be tested by the future experimental data. If our results deviate from the future data, constraining our BS inputs or modifying BS method is required.
In this work, we take all the D ( * ) (s)J mesons as the conventional charmed(-strange) mesons. However, there are still controversies on the natures of D * s0 (2317) and D s1 (2460) mesons (A recent review on this problem can be found in ref. [77].) For examining whether D * s0 (2317) and D s1 (2460) mesons are pure cs states, we need to lay stress on their electromagnetic and strong decays. If the future data implies that this assumption is not suitable, we should modify our wave functions describing D ( * ) (s)J mesons.

Conclusion
In this paper, including the BP, Ann, CS and CF contributions, we re-analyze the process B c → D * s0 (2317)µμ and first calculate the decays B c → D s1 (2460, 2536)µμ, B c → D * s2 (2573)µμ and B c → D 1. If only BP effects are considered, our results on the B c → D * s0 (2317)µμ transition are agreeable with the ones in ref. [24] but quite different from the ones in ref. [23]. Once Ann, CS and CF Feynman diagrams are contained, the B c → D * s0 (2317)µμ observables change considerably, as shown in figures 11 (a-d).
2. As plotted in figures 14, 18 (a-h), the observables of the B c → D s1 (2536)(D 1 (2430))µμ processes behave quite sensitively to the Ann and CF influences. This makes that when these channels are analyzed, besides the BP and CS diagrams, it is necessary to include the Ann and CF ones. Here we present the explicit expressions of F α V 1−7 and F α A1−3 .

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