Analysis of the semileptonic Bc→D10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_c\rightarrow D_1^0$$\end{document} transition in QCD sum rules and HQET

We investigate the structure of the D10(2420[2430])(JP=1+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{1}^0(2420[2430])(J^P=1^+)$$\end{document} mesons via analyzing the semileptonic Bc→D10lν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_{c}\rightarrow D_{1}^0 l\nu $$\end{document} transition in the frame work of the three-point QCD sum rules and the heavy-quark effective theory. We consider the D10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{1}^0$$\end{document} meson in three ways: as a pure |cu¯⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|c\bar{u}\rangle $$\end{document} state, as a mixture of the two |3P1⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|^3P_1\rangle $$\end{document} and |1P1⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|^1P_1\rangle $$\end{document} states with a mixing angle θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document}, and as a combination of the two mentioned states with mixing angle θ=35.3∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta =35.3^\circ $$\end{document} in the heavy-quark limit. Taking into account the gluon condensate contributions, the relevant form factors are obtained for the three above conditions. These form factors are numerically calculated for |cu¯⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|c\bar{u}\rangle $$\end{document} and the heavy-quark limit cases. The obtained results for the form factors are used to evaluate the decay rates and the branching ratios. Also for mixed states, all of the mentioned physical quantities are plotted with respect to the unknown mixing angle θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document}.

Heavy-light mesons are not charge conjugation eigenstates and so mixing can occur among states with the same J P and different mass that are forbidden for neutral states [22]. So the mixing of the physical D 1 and D 1 states can be parameterized in terms of a mixing angle θ , as follows: where we used the spectroscopic notation 2S+1 L J for introduction of the mixing states. Considering | 3 P 1 ≡ |D 1 1 and | 1 P 1 ≡ |D 1 2 with different masses and decay constants [22,29], we can apply these relations, beyond the heavyquark model, for axial vectors D 1 (2420) and D 1 (2430) mesons with two different masses, i.e., |D 1 (2420) = sin θ |D 1 1 + cos θ |D 1 2 , The masses and decay constants of the D 1 1 and D 1 2 states are presented in Tables 1 and 2. In the heavy-quark limit where the quark mass m c → ∞, both axial vector D 0 1 (2420) and D 0 1 (2430) mesons can be produced and identified with |P 3/2 1 and |P 1/2 1 , respectively. It is useful to change from the L-S basis 2S+1 L J to the j-j coupling basis L j J , where j is the total angular momentum of the light quark. The relationship between these states is given as [22,29,30] These relations occur for the mixing angle θ = 35.3 • in Eq.
(1). But note that the value of the mixing angle can be positive equal to θ = 35.3 • or negative corresponding θ = −54.7 • if the expectation of the heavy-quark spin-orbit interaction is positive or negative, respectively [22]. The B c → D * 0 lν [32] and B c → Dll/νν [33] have been studied via three-point QCD sum rules (3PSR). In this work, we analyze the semileptonic B c → D 0 1 (2420 [2430])lν decays in 3PSR and heavy-quark effective theory (HQET). To this aim, we consider the structure of the D 0 1 meson in three conditions: 1. The D 0 1 meson as a pure state (|cū ). 2. The D 0 1 meson as a mixture of two states of the | 1 P 1 and | 3 P 1 with a mixing angle θ [see Eq. (2)]. 3. The D 0 1 meson as a combination of two | 1 P 1 and | 3 P 1 states with the mixing angle θ = 35.3 • in the heavyquark limit [see Eq. (3)].
Taking into account the gluon condensate corrections, as the important term of the non-perturbative part of the correlation function, the form factors of the B c → D 0 1 transition are obtained within 3PSR for the conditions 1, 2 and within the HQET approach for the condition 3. For the conditions 1 and 3, the form factors of the B c → D 0 1 (2420[2430]) transitions are a function of the transferred momentum square q 2 . So, we plot these form factors and decay widths of these decays with respect to q 2 . Also the branching ratios for these cases are evaluated. But it should be remarked, when we consider the D 0 1 as a mixture of two states with mixing angle θ in the region −180 ≤ θ ≤ 180, that the transition form factors of the B c → D 0 1 (2420 [2430]) decays are functions of two variables, θ and q 2 . Since the decay width of the B c → D 0 1 transition is related to the form factors, it is a function of the mixing angle θ and q 2 , too. For a better analysis, we plot the form factors and the decay widths of the B c → D 0 1 (2420 [2430]) in three dimensions. In this case, the branching ratios are shown with respect to the mixing angle θ . Detection of these channels and their comparison with the phenomenological models like QCD sum rules could give useful information as regards the structure of the D 0 1 meson and the unknown mixing angle θ . This paper is organized as follows. In Sect. 2, we calculate the form factors for the B c → D 0 1 transition in 3PSR for above conditions 1 and 2. In Sect. 3, the transition form factors are evaluated via HQET approach for condition 3. Finally, Sect. 4 is devoted to the numeric results and discussions.

Sum rules method
In this section, we study the transition form factors of the semileptonic B c → D 0 1 lν decay by QCD sum rules mechanism. To this aim, first, we consider the D 0 1 meson as a pure state. The B c → D 0 1 lν process is governed by the tree level b → ulν transition and c quark is the spectator, at quark level (see Fig. 1).
The three-point correlation function is considered for the evaluation of the transition form factors in the framework of the 3PSR. The three-point correlation function is constructed from the vacuum expectation value of time ordered product of three currents as follows: where J We can obtain the correlation function of Eq. (4) in two respects. The phenomenological or physical part is calculated saturating the correlation by a tower of hadrons with the same quantum numbers as interpolating currents. The QCD or theoretical part, on the other side is obtained in terms of the quarks and gluons interacting in the QCD vacuum. To derive the phenomenological part of the correlation given in Eq.
c c c c c c γ ν γ 5 γ ν γ 5 γ ν γ 5 γ ν γ 5 γ ν γ 5 γ ν γ 5 g g g g g g g g g g g g  This procedure leads to the following representations of the above-mentioned correlation: + higher resonances and continuum states.
The general expression for the hadronic matrix element of the weak current with the definition of the transition form factors is given by the formula: where  Form factor Value and ε is the four-polarization vector of the D 0 1 meson. Also the following matrix elements are defined in the standard way in terms of the leptonic decay constants of the D 0 1 and B c mesons: where f D 0 1 and f B c are the leptonic decay constants of D 0 1 and B c mesons, respectively. Using Eqs. (6) and (8) in Eq. (5) and performing summation over the polarization of the D 0 1 meson, we get the following result for the physical part: The coefficients of the Lorentz structures i μναβ p α p β , g μν , P μ p ν , and q μ p ν in the correlation function μν will be chosen in determination of the form factors f V (q 2 ), f 0 (q 2 ), f 1 (q 2 ), and f 2 (q 2 ), respectively. So the Lorentz structures in the correlation function can be written down as where each i function is defined in terms of the perturbative and non-perturbative parts as With the help of the operator product expansion, in the deep Euclidean region where p 2 (m b +m c ) 2 and p 2 m 2 c , the vacuum expectation value of the expansion of the correlation function in terms of the local operators is written as follows [32,34]: where (C i ) μν are the Wilson coefficients, G αβ is the gluon field strength tensor, and are the matrices appearing in the calculations. The non-perturbative part contains the quark and gluon condensate diagrams. We consider the condensate terms of dimension 3, 4, and 5. It is found that the heavyquark condensate contributions are suppressed by inverse of the heavy-quark mass and can be safely omitted. The light u quark condensate contribution is zero after applying the double Borel transformation with respect to the both variables p 2 and p 2 , because only one variable appears in the denominator. Therefore in this case, we consider the two gluon condensate diagrams with mass dimension 4 as an important term of the non-perturbative corrections, only, i.e., The diagrams for contribution of the gluon condensates are depicted in Fig. 2. To obtain the contributions of these diagrams, the Fock-Schwinger fixed-point gauge, x μ A a μ = 0, are used; here A a μ is the gluon field. The procedure of the evaluation of such as diagrams in Fig. 2 has been discussed in Ref. [32] completely.
Using the double dispersion representation, the bare-loop contribution is determined: By replacing the propagators with the Dirac-delta functions (Cutkosky rules) we have and the spectral densities ρ per i (s, s , q 2 ) are found as where For the heavy quarkonium bc, where the relative velocity of quark movement is small, an essential role is taken by the Coulomb-like α s /v-corrections [35]. It leads to the finite renormalization for ρ per i , so that with where α C s is the coupling constant of effective coulomb interactions. Also v is the relative velocity of quarks in the bc- The value of α C s for B c meson is [35] α C s [bc] = 0.45.
By performing the double Borel transformations over the variables p 2 and p 2 on the physical parts of the correlation functions and bare-loop diagrams and also equating two representations of the correlation functions, the sum rules for the f i (q 2 ) are obtained: where i = V, 0, 1 and 2, s 0 , and s 0 are the continuum thresholds in the pseudoscalar B c and axial vector D 0 1 channels, respectively, and the lower bound integration limit of s L is as follows: The explicit expressions for C 4 i are presented in Appendix A.
Now, we would like to consider the form factors related to the B c → D 0 1 transition when the D 0 1 meson is a mixture of the two | 1 P 1 and | 3 P 1 states. To this aim, first the f B c →D 1 1(2) i (q 2 ) are obtained from the above equations, where f D 1 1 = (183 ± 25) MeV, and f D 1 2 = (89 ± 7) MeV [29]. Then by straightforward calculations, the f form factors of B c → D 0 1 (2420) transition are found as follows:

Numerical Analysis
Now, we present our numerical analysis of the form factors f i (q 2 ) (i = V, 0, 1, 2) via 3PSR and HQET. From the sum rules expressions of the form factors, it is clear that the main input parameters entering the expressions are gluon condensates, element of the CKM matrix V ub , leptonic decay constants f B c , f D 0 1 (see Table 3), f D 1 1 , and f D 1 2 , Borel parameters M 2 1 and M 2 2 as well as the continuum thresholds s 0 and s 0 .
The sum rules for the form factors contain also four auxiliary parameters: Borel mass squares M 2 1 and M 2 2 and continuum thresholds s 0 and s 0 . These are not physical quantities, so the form factors as physical quantities should be independent of them. The parameters s 0 and s 0 , which are the continuum thresholds of B c and D 0 1 mesons, respectively, are determined from the condition that guarantees the sum rules to practically be stable in the allowed regions for M 2 1 and M 2 2 . The values of the continuum thresholds calculated from the twopoint QCD sum rules are taken to be s 0 = (45-50) GeV 2 [41] and s 0 = (6-8) GeV 2 [42]. We search for the intervals of the Borel mass parameters so that our results are almost insensitive to their variations. One more condition for the intervals of these parameters is the fact that the aforementioned intervals • Pure state or |cū state If the D 0 1 meson is the pure |cū state, using Eqs. (7) and (21), the values of the form factors at q 2 = 0 are presented in Table 4. In this case, the values of the transition form factors at q 2 = 0 for B c → D 0 1 (2420)lν decay are the same as those for B c → D 0 1 (2430)lν. Our calculations show that The sum rules for the form factors are truncated at about 9 GeV 2 , so to extend our results to the full physical region, we look for a parametrization of the form factors in such a way that in the region 0 ≤ q 2 ≤ (m B c − m D 0 1 ) 2 GeV 2 , this parametrization coincides with the sum rules predictions. Our numerical calculations show that the sufficient parametrization of the form factors with respect to q 2 is as follows [44]: The values of the parameters a, b, and m fit are given in Table 5. Figure  By using the expressions for the form factors, the differential decay width d /dq 2 for the process B c → D 0 1 lν in terms of H i can be presented as follows: H ± and H 0 are defined as where ±, 0 refer to the D 0 1 helicities. Note that in the limit of vanishing lepton mass (in our case the electron and muon) the f 2 (q 2 ) form factor does not contribute to the decay width formula.
To calculate the branching ratios of the B c → D 1 (2420, 2430)lν decays, we integrate Eq. (36) over q 2 in the whole physical region and use the total mean life time τ B c = (0.46 ± 0.07) ps. Our numerical analysis shows that the contribution of the non-perturbative part (the gluon condensate diagrams) is about 13 % of the total and the main contribution comes from the perturbative part of the form factors. The value for the branching ratio of these decays is obtained as presented in Table 6. The function of decay width of B c → D 0 1 (2420, 2430)lν decays with respect to q 2 is shown in Fig. 5. ) 2 GeV 2 and θ = ±N π/6, N = 1, 2, 3. Using Eq. (36), we denote the variation of the decay widths with respect to q 2 and θ in the same regions for each decay in Fig. 10. Also the branching ratios only in terms of the mixing angle θ are shown in Fig. 11.
• Compound state in the heavy-quark limit Eventually, we study the structure of the D 0 1 meson as a mixture of two | 3 P 1 and | 1 P 1 states with the mixing angle The values of the HQET form factors at q 2 = 0 are presented in Table 7.
Also using Eq. (36) and the HEQT form factors, we evaluated the branching ratios of the B c → D 0 1 (2420 [2430])lν decays as given in Table 8. Figure 14 depicts the dependence of the decay widths of these decays on the q 2 in HQET approach.

Conclusion
In summary, we analyzed the semileptonic B c → D 0 1 (2420 [2430])lν decays in the framework of the 3PSR and HQET approach. First, we assumed the D 0 1 (2420) and D 0 1 (2430) axial vector mesons as the pure |cū state. In this case, the related form factors were computed. The branching ratios of these decays were also estimated. Second, the D 0 1 (2420[2430]) mesons were considered as a combination of two states | 3 P 1 ≡ |D 1 1 and | 1 P 1 ≡ |D 1 2 with different masses and decay constants. We evaluated the transition form factors and the decay widths of these decays with respect to the mixing angle θ and the transferred momentum square q 2 . The dependence of the branching ratios on θ was also presented. Finally, we obtained all of the mentioned physical quantities in the HQET approach. Any future experimental measurement on these form factors as well as decay rates and branching fractions and their comparison with the obtained results in the present work can give considerable information as regards the structure of these mesons and the unknown mixing angle θ . + 5Î [0,1]