$B_{(s)} \rightarrow D^{**}_{(s)}$ form factors in HQEFT and model independent analysis of relevant semileptonic decays with NP effects

The form factors of $B_{(s)}$ decays into P-wave excited charmed mesons (including $D^*_0(2300)$, $D_1(2430)$, $D_1(2420)$, $D^*_2(2460)$ and their strange counterparts, denoted generically as $D^{**}_{(s)}$) are systematically calculated via the QCD sum rules in the framework of heavy quark effective field theory (HQEFT). We consider contributions up to the next leading order of heavy quark expansion and give all the relevant form factors, including the scalar and tensor ones only relevant for possible new physics effects. The expressions for the form factors in terms of several universal wave functions are derived via heavy quark expansion. These universal functions can be evaluated through QCD sum rules. Then, the numerical results of the form factors are presented. With the form factors given here, a model independent analysis of relevant semileptonic decays $B_{(s)} \rightarrow D^{**}_{(s)} l \bar{\nu}_l$ is performed, including the contributions from possible new physics effects. Our predictions for the differential decay widths, branching fractions and ratios of branching fractions $R(D^{**}_{(s)})$ may be tested in more precise experiments in the future.


I. INTRODUCTION
The B (s) to charmed meson semileptonic decays are important for measurements of the CKM matrix element |V cb | and are also probes for new physics (NP) beyond the standard model (SM). Despite being a charged current channel, some intriguing hints of discrepancies have been observed by several experimental collaborations. Measurements of the ratios of branching fractions, show a 3.3σ tension with their SM expectations when the D and D * results are combined [1], which may imply violation of lepton flavor universality. To further confirm or rule out these hints, it is necessary to investigate the additional decay modes mediated by the same parton level transition, not only because these decays can give complementary information, but also because they constitute important backgrounds to R(D ( * ) ) measurements. Moreover, better theoretical control of these modes will help improve the determinations of |V cb | and understand the composition of inclusive B (s) → X c lν l decays in terms of relevant exclusive channels.
In the quark model, these mesons can be viewed as constituent quark-antiquark pairs with a total orbital angular momentum L = 1. (Note that the structures of D * s0 (2317) and D s1 (2460) are not completely clear, and we simply interpret them as the lightest orbitally excited states of quark-antiquark pairs for consistency here.) For a hadron containing a single heavy quark, the heavy quark is approximately decoupled. Therefore, the above excited charmed mesons can be classified by the total momentum and parity of the light degrees of freedom j P l . The first and last two mesons for D * * (s) (c.f. Eqs. (2), (3)) have j P l = 1 2 + and j P l = 3 2 + respectively, which are denoted as D 1/2+ (s) and D
For exclusive semileptonic decays, the non-perturbative contributions can be parameterized in terms of form factors. The B (s) → D * * (s) form factors were initially estimated in the Isgur-Scora-Grinstein-Wise (ISGW) quark model and its improved version ISGW2 [6][7][8]. They have also been calculated via the covariant light-front quark model (LFQM) [9,10]. Some model independent predictions for these decays can be obtained based on heavy quark symmetry. In Ref. [5,[11][12][13][14], with the available experimental results as inputs, the semileptonic B (s) decays into excited charmed mesons and relevant form factors were investigated in the usual heavy quark effective theory (HQET), including the next leading order corrections of heavy quark expansion and NP effects. The form factors of B → D * * decays were also studied using QCD sum rules in HQET [15,16]. Additionally, the B → D 1 (2430), D 1 (2420), D * 2 (2460) form factors were evaluated via light cone sum rules (LCSR) and applied to the analysis of relevant semileptonic decays [17][18][19].
Because heavy quark-antiquark coupling effects in the finite mass corrections are not considered in HQET [20][21][22], the B → D * * form factors were calculated to the next leading order of heavy quark expansion in heavy quark effective field theory (HQEFT) with QCD sum rules [23,24]. In HQEFT, all the odd powers of the transverse momentum operator D / ⊥ in the effective current are absent, and thus the forms of the operators become similar to those in the effective Lagrangian. For this reason, fewer universal wave functions are involved. In this study, we intend to give a systematic calculation for the B (s) → D * * (s) form factors using QCD sum rules in HQEFT and perform a model independent analysis of relevant semileptonic decays, including the contributions from possible NP effects.
The remainder of this paper is organized as follows. In Section II, we give the definitions of form factors and derive their expressions in terms of several universal wave functions using heavy quark expansion to the next leading order in HQEFT. These universal functions can be evaluated via QCD sum rules. The numerical results and discussions of the form factors are presented in Section III. Based on these form factors, we predict the differential decay widths, branching fractions, and ratios of branching fractions R(D * * (s) ) for all the relevant semileptonic decays in Section IV. Section V presents our summary.

A. Definitions of form factors
As in Ref. [14], we consider the B (s) → D * * (s) matrix elements of operators with all possible Dirac structures, i.e.

B. Formulation via heavy quark expansion and QCD sum rules in HQEFT
Now, let us derive the expressions for the form factors using heavy quark expansion and QCD sum rules in HQEFT following similar procedures detailed in Ref. [23,24]. The hadronic matrix elements can be expanded over the inverse of heavy quark mass, i.e. 1/m Q .
To the next leading order, where Γ is an arbitrary combination of Dirac matrices, and P (′) where Similarly, for the B (s) → D 3/2+ (s) decays, The universal wave functions ζ, τ , χ b(c) i (i = 0, 1, 2), and η b(c) , and the spin wave functions for the initial and final state mesons M v , K v ′ and F µ v ′ have the following forms: For the B (s) → D
, the expressions agree with those in Ref. [23,24]. The values of (s) can be extracted by fitting the meson masses. As detailed in APPENDIX A, where the spin average masses of j P l = 1 Additionally, the binding energies As found from Eqs. Ref. [23,24]. It is found that For the decay constants f 1 From Eqs.(78)-(86), we easily obtain As mentioned in Ref. [23,24], the QCD higher order corrections are not included in the 1 (2) are generally expected to be very small and can be safely neglected, supported by the relativistic quark model and QCD sum rule study [24][25][26][27][28]. However, as pointed out in Ref. [23], under the condition η b(c) i = 0(i = 1, 2, 3), the resulting branching fraction for the B → D * 2 lν l decay seems to exceed the CLEO upper limit when including 1/m Q contributions. Considering this, the wave functions , and a 2 = 0.67.

III. NUMERICAL RESULTS AND DISCUSSIONS OF THE FORM FACTORS
With Eqs.(38)-(93), we are now in a position to calculate the B (s) → D * * (s) form factors. For the masses of heavy mesons, which have been well established in experiments, we use the latest values given by the particle data group (PDG) [2], Furthermore, we adopt the masses estimated in the context of effective theory for the heavy mesons predicted by the quark model but not observed in experiments [29], For the masses of heavy quarks, we take the values [24] The condensates in Eqs.(78)-(86) have the typical values [24,30] qq = For the values of η b , η c i (i = 1, 2, 3) at q 2 = q 2 max in Eqs.(93), based on the analysis in Ref. [23], we choose for which the branching fractions for the B (s) → D * (s)2 lν l decays can be significantly suppressed and the corresponding results for decays with D (s)1 in the final states are largely unaffected.
From Eqs.(78)-(92), it is easily observed that the sum rules for the universal wave func- for the B (s) → D 1/2 (s) decays and decays, additional uncertainties induced by the values of η b , η c i (i = 1, 2, 3) at q 2 = q 2 max are also included, and the maximum total uncertainties can reach 90%. Note that the uncertainties quoted here merely indicate variations in our results within the chosen ranges of the relevant parameters mentioned above. The form factors g P , g V 1 , and f V 1 approach zero at q 2 = q 2 max . The form factor g + is equal to zero in the entire q 2 region up to the next leading order of heavy quark expansion. In addition, the form factors g V 2 = −g T 3 = 0 and g S = −g V 3 = g A under the condition that the contributions from chromomagnetic operators are neglected. The uncertainties of k A 2 , k T 2 , and k converted to meet the current definitions of form factors using the formulae in APPENDIX B.
We can see that large differences exist among the form factors given by different groups.
Overall, our results are in better agreement with the values obtained in HQET with available experimental measurements as inputs, i.e. the HQET+EXP. method [13,14].    Decays  Decays   0 with respect to T for different 's 0 ' parameters at q 2 = 0 (left column) and q 2 max (right column).     Ref.    Ref. f with With this effective Hamiltonian, the differential decay widths w.r.t. q 2 for the B (s) → D * * (s) lν l decays can be obtained. Concretely [14], for the B (s) → D * (s)0 lν l decays, For the B (s) → D ′ (s)1 lν l decays, The corresponding formulae for the B (s) → D (s)1 lν l decays can be obtained from Eqs.(109), where and m l is the mass of charged lepton in the final state.
In this study, we adopt the single operator scenario, i.e. consider the contributions from O T one by one and assume the corresponding Wilson coefficients to be real. Additionally, we assume that only the third generation leptons are relevant to NP for simplicity. The fitted values for the Wilson coefficients obtained in Ref. [31] are as follows: C S L = 0.08 ± 0.02, C S R = −0.05 ± 0.03, First, to observe the effects of each NP operators intuitively, we calculate the differential decay widths normalized to the SM widths for the B → D * * τν τ decays, as shown in FIG.8.
It is easily found that the NP effects are most significant in the moderate q 2 region, around q 2 ∈ [4.5, 6.5]GeV 2 . For the B → D * 0 τν τ decay, all operators except O S R give positive contributions to the differential decay width, and the operator O V L has a maximal contribution.
The operator O V L also has a maximal contribution, and the contribution of O V R is almost zero for the B → D ′ 1 τν τ decay. In contrast, the operators O S R and O T have maximal contributions for the B → D 1 τν τ and B → D * 2 τν τ decays, respectively. For the former decay, only the operator O V R has a negative contribution to the differential decay width, and the contributions of O S L and O S R are nearly zero for the later decay. For the B s → D * * s τν τ decays, the corresponding behaviors of the differential decay widths normalized to the SM widths are similar.
FIG. 8: Differential decay widths normalized to the SM widths for the B → D * * τν τ decays.
Integrating the differential decay widths over q 2 in the entire physical region and using the lifetimes of B (s) as inputs, we can obtain the branching fractions Similar to Eq.(1), the corresponding ratios of branching fractions The numerical results of Br and R(D * * (s) ) for the B (s) → D * * (s) lν l decays are shown in TABLE IX-XII. For specificity, we give the SM results of these two observables for the B − → D * * 0 l −ν l andB 0 s → D * * + s l −ν l decays in TABLE IX and X, respectively. For comparison, the values given by current experiments [1,2] and several previous theoretical analyses [5,14,23,24] are also listed. (Note that the original values from Ref. [1,2] are modified using the expected absolute D * * decay branching fractions, as detailed in Ref. [5], and the theoretical results of Br forB 0 decays in Ref. [5,24]

V. SUMMARY
In this study, we calculate the B (s) → D * * (s) form factors systematically using QCD sum rules in the framework of HQEFT and perform a model independent analysis of the corresponding semileptonic decays, including the contributions from possible NP effects. We consider contributions up to the next leading order of heavy quark expansion and give all the relevant form factors, including the scalar and tensor ones only related to the NP effects.
Expressions for the form factors in terms of universal wave functions are derived via heavy quark expansion, and several relations among the form factors are obtained, i.e. g − = g T , and k A 2 = k T 3 . We find that the form factor g + is equal to zero in the entire physical region of q 2 , and g P , g V 1 , f V 1 approach zero at q 2 = q 2 max . Neglecting the contributions from chromomagnetic operators for the B (s) → D