Decays of $B$, $B_s$ and $B_c$ to $D$-wave heavy-light mesons

We study the weak decays of $\bar{B}_{(s)}$ and $B_c$ into $D$-wave heavy-light mesons, including $J^P=2^-$ ($D_{(s)2},D'_{(s)2},B_{(s)2}, B'_{(s)2}$) and $3^-$~($D^*_{(s)3}, B^*_{(s)3}$) states. The weak decay hadronic matrix elements are achieved based on the instantaneous Bethe-Salpeter method. The branching ratios for $\bar{B}$ decays are $\mathcal{B}[\bar{B}\to D_{2}e\bar{\nu}_e] = 1.1^{-0.3}_{+0.3} \times 10^{-3}$, $\mathcal{B}[\bar{B}\to D'_2e\bar{\nu}_e]=4.1^{-0.8}_{+0.9} \times 10^{-4}$, and $\mathcal{B}[\bar{B}\to D^*_3e\bar{\nu}_e]=1.0^{-0.2}_{+0.2} \times 10^{-3}$, respectively. For semi-electronic decays of $\bar B_s$ to $D_{s2}$, $D'_{s2}$, and $D^*_{s3}$, the corresponding branching ratios are $1.7^{-0.5}_{+0.5}\times 10^{-3}$, $5.2^{-1.5}_{+1.6}\times 10^{-4}$, and $1.5^{-0.4}_{+0.4}\times 10^{-3}$, respectively. The branching ratios of semi-electronic decays of $B_c$ to $D$-wave $D$ mesons are in the order of $10^{-5}$. We also achieved the forward-backward asymmetry, angular spectra, and lepton momentum spectra. In particular the distribution of decay widths for $2^-$ states $D_2$ and $D'_2$ varying along with mixing angle are presented.

We use M µ to denote hadronic transition element D ( * ) J |J µ |B , which can be described with form factors. The general form of the hadronic matrix element depends on the total angular momentum J of the final meson. For D 2 (D 2 ) and D * 3 the form factors are defined as M µ = e αβ P α (s 1 P β P µ + s 2 P β P µ F + s 3 g βµ + is 4 µβP P F ) if J = 2, e αβγ P α P β (h 1 P γ P µ + h 2 P γ P µ F + h 3 g γµ + ih 4 µγP P F ) if J = 3.
In above equation, we used the definition µνP P F = µναβ P α P β F where µναβ is the totally antisymmetric Levi-Civita tensor; g µν is the Minkowski metric tensor; e αβ and e αβγ are the polarization tensor for J = 2 and 3 mesons, respectively, which are completely symmetric; s i and h i (i = 1, 2, 3, 4) are the form factors for J = 2 and 3, respectively. To state it more clearly, we will use s i , t i , and h i to denote the form factors for transitionsB to D 2 , D 2 , and D * P µ X P ν X M 2 X − g µν , where s denotes the polarization state; f P and f V are the corresponding decay constants.
Then the |A| 2 can be expressed by hadronic tensor H µν , which is just the same with that in the corresponding semi-leptonic decays, and light meson tensor X µν as where X µν has the following expression X µν = X|(qΓ µ u)|0 X|qΓ ν u|0 * = P µ X P ν X f 2 P X is a pseudoscalar meson,

Several observables
One of the interested quantity inB semi-leptonic decay is the angular distribution of the decay width Γ, which can be described as where M is the initialB mass; m 2 ν = (p + p ν ) 2 is the invariant mass square of andν; p * and p * F are the three momenta of and D ( * ) J in the ν rest frame, respectively; θ is the angle between p * and p * F ; |p * | = 1 2m ν λ 1 2 (m 2 ν , M 2 , M 2 ν ) and |p * F | = 1 2m ν λ 1 2 (m 2 ν , M 2 , M 2 F ), where we have used the Källén function λ(a, b, c) = (a 2 + b 2 + c 2 − 2ab − 2bc − 2ac), M and M ν are the lepton mass and anti-neutrino mass, respectively. Another quantity we are interested is the forward-backward asymmetry A F B , which is defined as A F B = Γ cos θ>0 − Γ cos θ<0 Γ cos θ>0 + Γ cos θ<0 .
The decay width varying along with charged lepton 3-momentum |p | is given by where E denotes the charged lepton energy in the rest frame of initial state meson. The non-leptonic decay width of theB meson is given by where p represents the 3-momentum of the final D ( * )

Hadronic transition matrix element
The hadronic transition matrix element D ( * ) J |J µ |B plays an key role in the calculations ofB semi-leptonic and non-leptonic decays. In this section we will give details to calculate the hadronic transition matrix element by Bethe-Salpeter method in the framework of constituent quark model.

Formalism of hadronic transition matrix element with Bethe-Salpeter method
According to the Mandelstam formalism [54], the hadronic transition amplitude M µ can be written by Beter-Salpeter (BS) wave function as where Ψ B (q, P ) and Ψ D (q , P F ) are the BS wave functions of theB meson and the final D ( * ) J , respectively;Ψ is defined as γ 0 Ψ † γ 0 ; q and q are respectively the inner relative momenta ofB and D ( * ) J system, which are related to the quark (anti-quark) momentum p ( ) 2). And here we defined the symbols α i = m i m 1 +m 2 and α i = m i m 1 +m 2 , where m i and m i are masses of the constituent quarks in the initial and final bound states, respectively (see Fig. 1). Here inB decays we have m 1 = m b , m 1 = m c , m 2 = m 2 = m d . As there is a delta function in above equation, the relative momenta q and q are related by q = q −(α 2 P −α 2 P F ).
In the instantaneous approximation [32], the inner interaction kernel between quark and antiquark in bound state is independent of the time component q P (= q · P ) of q. By performing the contour integral on q P and then we can express the hadronic transition amplitude as [37] where we have used the definitions q ⊥ ≡ q − P ·q M 2 P and q ⊥ ≡ q − P ·q M 2 P . Here ψ denotes the 3-dimensional positive Salpeter wave function (see appendix B). ψ B and ψ D denote the positive Salpeter wave functions forB and D The positive Salpeter wave function for 1 S 0 (0 − ) state can be written as [55] where we have the following constraint conditions, The definition ω i ≡ m 2 i − q 2 ⊥ (i = 1, 2) is used. The derivation of Eq. (17) and (18) can be found in appendix B. So there are only two undetermined wave function k 1 and k 2 here, which are the functions of q ⊥ . The positive Salpeter wave function for 3 − ( 3 D 3 ) state with unequal mass of quark and anti-quark has the following forms [56] In above equation n i (i = 1, 2, · · · , 8) can be expressed with 4 wave functions u i (i = 3, 4, 5, 6) as below, , In above Salpeter positive wave functions ψ B and ψ D , the undetermined wave functions k 1 , k 2 for 0 − and u i (i = 3, 4, 5, 6) for 3 − can be achieved by solving the full Salpeter equations numerically (see appendix B). The positive Salpeter wave functions for 1 D 2 [41], and 3 D 2 [56] states can be seen in appendix C. e µνα is the symmetric polarization tensor for spin-3 and satisfies the following relations [57] e µνα g µν = 0, e µνα P F µ = 0, where Ω abc;µνα and we have used the definition g µν . Inserting the initialB wave function ψ B ( 1 S 0 ) (Eq. 17) and final D * 3 wave function ψ D ( 3 D 3 ) (Eq. 19) into the hadronic transition amplitude Eq. (16), after calculating the trace and performing the integral in Eq. (16) we achieve the form factors h i forB →D * 3 transition defined in Eq. (3). When performing the integral over q in the rest frame of the initial meson, the following formulas are used.
where g µν T are defined as (g µν − P µ P ν P 2 ) and P µ F ⊥ = (P µ F − P F ·P M 2 P µ ). From above equations we can easily obtain the following expressions of C i , where η is the angle between q and P F . The physical 2 − D-wave states D 2 and D 2 are the mixing states of 3 D 2 and 1 D 2 states, whose wave functions are what we solve directly from the full Salpeter equations. Here we will follow Ref. [58] and Ref. [59], where the mixing form for D-wave states is defined with the mixing angle α as In the heavy quark limit (m Q → ∞), the D mesons are described in the |J, j basis, where m Q denotes the heavy quark mass and j denotes the total angular momentum of the light quark. The relations between |J, j and |J, S for L = 2 are showed by Then the mixing angle for L = 2 can be expressed as α = arctan 2/3 = 39.23 • . So in this definition D 2 corresponds to the |J P , j = |2 − , 5/2 state and D 2 corresponds to the |2 − , 3/2 state. In this work the same mixing angle will also be used for 2 − states D Here the mixing angle is the ideal case predicted by the HEQT in the limit of m Q → ∞. The dependence for decay widths varying over the mixing angle can be seen in equations (29) and (30).
The wave functions of 1 D 2 and 3 D 2 states can be achieved by solving the corresponding Salpeter equations directly. Then the amplitude for physical 2 − states can be considered as the mixing of the transition amplitudes for 1 D 2 and 3 D 2 states, namely By using Eq. (27), replacing the final state's wave function ψ D ( 3 D 3 ) by ψ D ( 1 D 2 ) and ψ D ( 3 D 2 ), and then repeating the above procedures for 3 D 3 state, we can get the form factors s i for D 2 and t i for D 2 defined in Eq. (3).

Form factors
To solve the Salpeter equations, in this work we choose the Cornell potential as the inner interaction kernel as before [55], which is a linear scalar potential plus a vector interaction potential as below ) .
In above equations, the symbol ⊗ denotes that the Salpeter wave function is sandwiched between the two γ 0 matrices. The model parameters we used are the same with before [35], which read The free parameter V 0 is fixed by fitting the mass eigenvalue to experimental value. With the numerical Salpeter wave function we can obtain the form factors.

Numerical Results and Discussions
Firstly we specify the meson mass, lifetime, CKM matrix elements and decay constants used in this work. For the mass ofB,B s , and B c mesons we take the values from PDG [60]. We follow the mass predictions and J P assignments of Ref. [21] for D-wave charm and charm-strange mesons. For D-wave bottom mesons B 2 , B 2 , and B * 3 we use the average values of Ref. [61] and Ref. [59]. Predictions of Ref. [59] and Ref. [62] are averaged to achieve the mass of D-wave bottom-strange mesons B s2 , B s2 , and B * s3 . These mass values we used can been seen below In the calculation of non-leptonic decays, the decay constants we used are [48,60] For the theoretical uncertainties, here we will just discuss the dependence of the final results on our model parameters λ, Λ QCD in the Cornell potential, and the constituent quark mass m b , m c , m s , m d and m u . The theoretical errors, induced by these model parameters, are determined by varying every parameter by ±5%, and then scanning the parameters space to find the maximum deviation. Generally, this theoretical uncertainties can amount to 10% ∼ 30% for the semi-leptonic decays. The theoretical uncertainties show the robustness of the numerical algorithm.

Lepton spectra and A F B
The distribution ofB and B − c decay width Γ varying along with cos θ for e and τ modes can be seen in Fig. 4, from which we can see that, forB decays, the distribution of semi-electronic decay widths are much more symmetric than that for the semi-taunic mode. These asymmetries over cos θ can also be reflected by the forward-backward asymmetries A F B , which are showed in Tab. I. We can see that A F B is sensitive to lepton mass m and is the monotonic function of m . Considering the absolute values of A F B , we find that forB → D θ cos The spectra of decay widths forB and B − c varying along with |p |, the absolute value of the three-momentum for charged leptons, are showed in Fig. 5. This distribution is almost the same for J , the momentum spectrum ofD 2 is sharper than that of D 2 andD * 3 .
GeV | e p |

Branching ratios of semi-leptonic decays
The semi-electronic decay widths we got are Γ(B →D 2 eν e ) = 4.9 × 10 −16 GeV, Γ(B →D 2 eν e ) = 1.8 × 10 −16 GeV, and Γ(B → D * 3 eν e ) = 4.5 × 10 −16 GeV. The branching ratios ofB to D-wave charmed mesons are listed in Tab. II. We have listed others' results for comparison if available. Our results are about 5 times greater than that in Ref. [23]. It's noticeable that our results for decays into D 2 and D 2 are in the same order, while in the results of QCD sum rules [23] B(B→D 2 ) is about 25 times larger than B(B →D 2 ). The branching ratios for semi-leptonic decays ofB s into D s2 , D s2 and D * s3 are listed in Tab. III. Our results forB s to D-wave charm-strange mesons are also much larger than the results of QCD sum rules in Ref. [24].
The branching ratios for B c to D-waveD
J ] for B − c decays are in the order of 10 −1 . This big difference is mainly due to the phase space. By simple integral over the phase space, we can find that, the phase space ratio of semi-taunic decay over semi-electronic decay for B − c meson is about 30 times larger than that forB orB s meson.
The decay widths forB (s) or B c to 2 − states mesons are dependent on the mixing angle α, which can be showed by Fig. 6(a) and Fig. 6(b). This dependence forB decays can be described by the following equations Our fit results give that the parameters are as The tiny differences in parameters for D 2 and D 2 come from the small difference between m D 2 and m D 2 . In Fig. 6(c) and Fig. 6(d), we also show the ratios Γ(B→D 2 eν) Γ(B→D 2 eν) and Γ(B − c →D 2 eν) Γ(B − c →D 2 eν) , which are very sensitive to the mixing angle.

Non-leptonic decay widths and branching ratios
The non-leptonic decay widths are listed in Tab. VI, where we have kept the Wilson coefficient a 1 in order to facilitate comparison with other models. The corresponding branching ratios are listed in Tab. VII, where we have specified the values a b 1 = 1.14 for b → c(u) transition and a c 1 = 1.2 for c→d(s) transition [48]. From the non-leptonic decay results we can see that, with the same final D meson, the ρ mode has the largest branching ratio and can reach 10 −3 order inB (s) decays, and 10 −6 order in B c decay. When the light mesons have the same quark constituents, the width for decay into vector meson (ρ, K * ) mode is about 2 ∼ 3 times greater than its pseudoscalar meson (π, K) mode.

Summary
In this work we calculated semi-leptonic and non-leptonic decays ofB We also present the angular distribution and charged lepton spectra forB and B c decays. The 2 − states D 2 and D 2 are the mixing states of 1 D 2 − 3 D 2 , so we present the dependence of the decay width varying along with the mixing angle. Based on our results, the semi-leptontic and non-leptonic branching ratios forB (s) decays to the D-wave charm and charm-strange mesons have

A. Expressions for N i s in the Hadronic Tensor H µν
The hadronic tensor N i (i = 1, 2, 4, 5, 6) forB to D 2 meson are Here p F denotes the three-momentum of final D systems and E F = M 2 F + p 2 F . ForB to D 2 the relations between N i and form factors t k (k = 1, 2, 3, 4) have the same form with that for D 2 , just s k are replaced with t k . Both s k and t k are functions of q 2 ⊥ .

B.1. Salpeter equations
Salpeter wave function ϕ(q ⊥ ) is related to BS wave function Ψ(q) by the following definition where the 3-dimensional integration η(q ⊥ ) can be understood as the BS vertex for bound states, and V (|q ⊥ − k ⊥ |) denotes the instantaneous interaction kernel.
C. Positive Salpeter wave function for 1 S 0 , 1 D 2 and 3 D 2 The positive Salpeter wave function and its constraint conditions for 1 D 2 state [41] are displayed in C.1 and C.2. And the undetermined wave function are f 1 and f 2 .
The positive Salpeter wave function of 3 D 2 state [56] and constraint conditions can be written as

(C.4)
Here we also only have two undetermined wave function v 1 and v 2 .
In above equations C.1 ∼ C.4 the indeterminate wave functions, such as f 1 and f 2 in ψ D ( 1 D 2 ), v 1 and v 2 in ψ D ( 3 D 2 ), which are functions of q 2 ⊥ and can be determined numerically by solving the coupled Salpeter eigen equations B.5 and B.6.