$B \rightarrow T$ transition form factors in light-cone sum rules

We present a new calculation of the semileptonic tree-level and flavor-changing neutral current form factors describing $B$-meson transitions to tensor mesons $T=D_2^*,K_2^*,a_2,f_2$ ($J^{P}=2^{+}$). We employ the QCD Light-Cone Sum Rules approach with $B$-meson distribution amplitudes. We go beyond the leading-twist accuracy and provide analytically, for the first time, higher-twist corrections for the two-particle contributions up to twist four terms. We observe that the impact of higher twist terms to the sum rules is noticeable. We study the phenomenological implications of our results on the radiative ${B} \to K_2^{*}\gamma$ and semileptonic ${B} \to D_2^* \ell {\bar \nu}_\ell$, ${B} \to K_2^{*}\ell^+\ell^-$ decays.


I. INTRODUCTION
B-meson decays represent a promising area for checking the gauge structure of the Standard Model (SM), looking for physics beyond it, as well as precise determination of the elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix.
If these results are confirmed by the forthcoming experiments, it will be an unambiguous discovery of existence of new physics (NP).
With respect to these experimental observations one expects that if NP exists at the quarklevel b → c transition, then such discrepancies should also be seen in B-meson transitions to tensor mesons in addition to B decays to pseudo-scalar or vector-mesons 1 .
In regard to seeking NP effects, the B-meson decays to tensor mesons have the following advantage: tensor mesons have additional polarizations compared to the vector mesons and therefore this could provide additional kinematical quantities that are sensitive to the existence of NP. As a result, B-meson decays to tensor mesons could provide a complementary platform to search for new helicity structures, that deviate from the SM ones.
The main ingredients of B → T transitions are the relevant form factors. In this work, the form factors of B → T transitions are calculated within the light-cone QCD sum rules (LCSRs) [23,24] (for reviews see e.g. [25]) using B-meson Light-Cone Distribution Amplitudes (LCDAs).
Note that the light-cone sum rules have successfully been applied to a wide range of problems of hadron physics. The recent applications of LCSRs with heavy meson and heavy baryon distribution amplitudes are discussed in detail in many works (see for example [26][27][28][29][30][31][32][33][34] and references therein).
It should be noted that the B → T (J P = 2 + ) form factors have previously been calculated by several groups using various methods [35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50]. For example, the B → f 2 (1270) form factors have recently been calculated in [38] within the LCSRs framework using the f 2 (1270) meson DAs. Also, for the light tensor meson final states B → f 2 , a 2 , K * 2 , f 2 the form factor calculation has been carried out previously by [39] within LCSRs employing tensor-meson DAs, and in [40] using perturbative QCD approach. Within three-point QCD sum-rule approach, a sub-set of the relevant form factors under consideration was estimated in Ref. [35] for B → f 2 , a 2 , K * 2 transitions, and in Ref. [37] for B → D * 2 . The LCSRs analysis carried out in Ref. [41] computes the relevant form factors for B → f 2 , a 2 , K * 2 transitions considering only the φ + ,φ B-LCDAs with vanishing virtual quark masses regarding the f 2 , a 2 final states. Our analysis here extends previous works by providing new results for the full set of B → T (D * 2 , K * 2 , a 2 , f 2 ) transition form factors up to and including twist-four accuracy of B-LCSRs as well as takes into account the finite virtual quark mass effects in the results of the form factors. Moreover, we provided results for the tensor form factors in B → D * 2 transitions for the first time. We should further mention that the tensor isosinglet final state f 2 (1270) considered in this study, in principle, possesses a mixing pattern with the other isosinglet tensor meson of the same quantum numbers f 2 (1525) in the form where the mixing angle δ has been found to be small indicating that f 2 could be considered nearly as a pure 1 √ 2 (uū + dd) state (∼ 98.2%) while f 2 is nearly a pure ss state (for details, see Refs. [51] and [52]). We will therefore assume no mixing between f 2 with f 2 when studying the B → f 2 form factors in this paper (see e.g. [35,38] for similar assumptions in regard to analyses of B → f 2 form factors).  of two quark currents j µν int =q 2 (x)Γ µν 2 q 1 (x) and j ρ weak (0) =q 1 (0)Γ ρ 1 h v (0), where h v denotes the Heavy Quark Effective Theory field instead of a b-quark. The spin structures of Γ 1,2 together with various choices of quark flavors q 1 and q 2 for the form factors extracted in this paper are given in Table I. The interpolating current for tensor mesons (with valence quark content q 1q2 ) is, in general, given by where When regard to two-particle contributions, which we are interested in this work, it suffices to take the first terms in the covariant derivatives, because the second terms involving the fields A ν (x) only contribute to three-particle effects 3 .
The higher Fock state contributions to the correlation function arise when expanding the position-space virtual-quark q 1 propagator in x 2 near the light-cone x 2 0. In the present work, we focus on the two-particle contributions, while higher Fock state contributions are beyond our current scope. We summarize the two-particle Operator-Product-Expansion (OPE) contributions where p = k −l, and l describes the momentum of the spectator quark inside the B-meson with α,β being spinor indices. In Eq. (11), the B-meson to vacuum matrix elements are non-perturbative objects that are expressed in terms of B-meson LCDAs, whose explicit definitions are relegated to Appendix A.
The hadronic correlator Π µνρ reads: The decay constant f T is defined via 4 : The spin sum for the tensor mesons is given by: The form-factors are extracted by matching independent Lorentz structures appearing in both correlators Π µνρ OPE (q, k) and Π µνρ had (q, k). For the particular choice of the weak currents as in Table I, the correlator Π µνρ OPE (q, k) can be split as: 3 In a recent comprehensive work with B-LCDAs [31] for B → P, V transitions it has been shown that compared to two-particle contributions, the relative impact of the three-particle contributions to the form-factors is only at percent level or less (for details see [31]). The same conclusion was also drawn e.g. in Ref. [41] for B → T transitions. We therefore feel safe to neglect such effects in the present analysis. 4 Note that this definition implies fT to be dimensionless.
where the ellipsis stand for terms involving other Lorentz structures. The extraction of the B → T form factors is then achieved as follows: •Ṽ : we considered terms with Lorentz-structure µραβ q ν q β k α in Eq. (15).
The choice of these structures is dictated by the fact that they contain contributions coming purely from tensor mesons.
Following the formulation introduced in Ref. [31], we write down the sum rule for all the B → T form factors in a form of a master-formula 5 as: where In Eq. (17), χ = √ 2 (χ = 1) for the light unflavored states f 0 2 , a 0 2 (for other states), and the differential operator is understood to act as Using the first relation in Eq. (18) one obtains where s 0 is an effective threshold parameter to be determined and supplied as an input.
The two-particle LCDAs appear in the definitions of the functions I (F ) n [31]: Further, we give our results for generic final state tensor meson T (q 1q2 ), where q 1 = c, s, u At this stage, a remark on our form factor results is in order. We compared our analytical results related to the two-particle contributions at the leading-twist limit to those of Ref. [41].
We observe the followings: first, we see that the surface-term contributions 7 given in Eq. (17) 6 The theoretical approach presented in this work, together with our form factor results, is generic and can also be readily applied to other tensor mesons with J P = 2 + by making obvious replacements. 7 Surface-terms arise after performing continuum subtraction. We observed and stress that the numerical impact of the surface terms on the form factor results could be sizable. of our paper have not been taken into account in the work of [41]. Nonetheless, when we still compare our analytical results to [41], after also dropping the mentioned surface-term effects in our results, we then reproduce the analytical results for their form factors called V, A 1 , T 1 ,Ã 3 and T 3 . Next, forÃ 2 of Ref. [41] we reproduce their results for φ + terms, while for theφ terms we have a disagreement. Last, for the T 2 form factor of Ref. [41] we have a complete disagreement.
The disagreement in the T 2 form factor of Ref. [41] is particularly interesting because while in our case the conditionT B→T (0) is exactly fulfilled (as required by equation-of-motion conditions), the analytical results given in Eqs. 20-21 of Ref. [41] (arXiv v6) for these two form factors seem not to satisfy this condition.

A. Input
In this section we collect the input parameters used in our numerical estimates. We use up-todate input parameters.
The meson masses entering our numerics are quoted from the latest PDG averages [53]: Moreover, the quark masses m q 1 (q 1 = c, s, u(d)) appearing in the C (F,ψ 2p ) n coefficients of Appendix B together with b-quark mass are defined in MS scheme, for which we use [53] Table II. For the non-perturbative parameters entering the explicit expressions of B-LCDAs we use: [55] 0.0406 ± 0.0023 [56] 0.050 ± 0.002 [57] 0.0185 ± 0.0020 [58] TABLE II. Tensor mesons' decay constants used in our numerical results.

B. LCSRs Results
We obtained results for the full set of B → T form factors within LCSRs up to q 2 = 0 GeV 2 .
Our LCSRs results involve, besides other input, free parameters introduced by the method; the continuum threshold s 0 and Borel mass parameter M 2 , which we determine by fulfilling some physical criteria. First, the working interval of the Borel parameter M 2 is determined following a standard criteria, i.e demanding that both the power corrections and the continuum contributions in the sum rules should be suppressed. Next, the working region of the continuum threshold is determined by defining so-called first-moments for each form factor and respective final state by differentiating the OPE correlator with respect to −1/M 2 and normalizing it to itself. These first-moments are then expected to give the mass squares m 2 T of the respective final state mesons. Imposing ±5% uncertainty on the mass of each final state tensor meson, we were then able to find validity window for s 0 too.
Based on these discussions, we determined the following working regions for s 0 and M 2 for the considered transitions: With these working regions for M 2 and s 0 's, the smallness of the sub-leading twist-4 contributions compared to the leading twist-2 ones as well as the suppression of higher state contributions are satisfied simultaneously. In where are the fit parameters that are constrained and presented in Table III for each form factor and final state transition separately. Beside, m R,F quantities describe the mass of the resonances associated with the quantum numbers of the respective form factor F , whose values can be found in Ref. [31] (for details see Table 5 of [31] and references therein). Note that the kinematical conditions given in Eqs. (7)-(8) impose the following relations among the fit parameters which are respected in our numerical results presented in Table III.
The uncertainties in the values of the form factors of Table III are due to the variation of various input parameters involved in the LCSR calculation. In particular, the non-perturbative parameters λ B , λ 2 H , λ 2 E of B-LCDAs together with the continuum threshold s 0 are mostly responsible for these errors.
In order to estimate the uncertainties of the results presented in this work, such as the form factors, decay rates etc., we followed a Monte Carlo based analysis as performed e.g. in Refs. [62,63]. For this analysis, randomly selected data sets of thousands of data points are generated for any input parameter and its given uncertainty. This led us to determine the mean and corresponding standard deviations of our results.

D. Illustrations
The q 2 dependence of the complete set of B → T form factors is depicted in Figs. 2,3,4 and 5. In these plots, comparing the leading-twist central results (empty red-circles) with the corresponding new results including twist-four terms (dotted-blue curves) we see that the relative impact of the calculated higher twist terms for the two-particle contributions could be sizable 10 (in particular for light tensor meson transitions) and therefore should be included in the estimations of the form factors. The magnitude of the central values of the form factors based on the leading-twist terms, is observed to decrease due to the calculated higher twist terms.
In Table IV,  (0) using Eq. (6) and Eq. (7). We observe that our numerical results for at q 2 = 0, given in the top-left pane of Table IV, severely differ 11 from the corresponding values quoted in Ref. [37], which use three-point sum rules. On the other side, concerning the light tensor transition form factors, our numerical results are in agreement with some of the existing results in the literature, which use various calculation methods. 10 For the B → T form factors under consideration, in the charmed case the relative impact of the calculated highertwist terms is observed to be relatively less significant when compared to light final state transitions. In our opinion, this could mainly be related to the presence of the heavy mass scale mc in the problem. 11 A remark on this point is in order. Our definition for theÃ3 −Ã0 form factor is related to the form factor b− of Ref. [37] (arXiv v3) in the following way: (2mT /q 2 )(Ã3−Ã0)/mB = −b−. At q 2 = 0,Ã3−Ã0 should exactly be zero according to the equation-of-motion condition given in Eq. (7) of our paper. However, the b−(q 2 = 0) form factor of Ref. [37] is seen to differ from zero (see Table 2 of the mentioned reference), in explicit violation of this condition.

IV. PHENOMENOLOGICAL ANALYSES
In this section, using our new results for the relevant form factors we give SM predictions for some selected observables. We considered the decay channels B → D * 2 ν, B → K * 2 γ and For the D * 2 (2460) mode, currently experimental data only on the decay chain B(B → D * 2 ν)B(D * 2 → Dπ) is available, form factors as [37,68] dΓ where λ(a, b, c) = a 2 + b 2 + c 2 − 2ab − 2ac − 2bc is the Källén function. We presented the q 2 dependence of B → D * 2 ν form factors up to and including twist-four accuracy in Table III. Using these results together with the input parameters G F = 1.167 × 10 −5 GeV −2 and V cb = 0.0405 [52], we obtain the following predictions and 0.16 ± 0.04 [37] .
The variance of our predictions from those of Ref. [37] is due to aforementioned discrepancy in the estimation of the form factors (see Table IV).
B. SM predicition for B → K * 2 γ We continue with a phenomenological analysis on exclusive rare radiative decay of B meson to radially excited tensor meson K * 2 (1430). The branching ratio of this radiative mode has been measured by several experiments: which gives the PDG average of (1.24 ± 0.24) × 10 −5 [53]. In the SM, B → K * 2 γ decay is governed by the electromagnetic dipole operator O 7 , and its matrix elements between initial B and final K * and is given by [72] where V ij are the CKM matrix elements, α is the fine-structure constant and C 7 (m b ) is the Wilson coefficient associated with O 7 . Since the inclusive radiative decay B → X s γ is accurately measured by several experiments [73,74], it is more convenient 12 to consider the ratio of exclusive to inclusive branching ratios [72] where the world average of the inclusive decay is given by the Heavy Flavour Averaging Group [19] as B(B → X s γ) = (3.32 ± 0.16) × 10 −4 , which is compatible with the theoretical estimate [75].
We determine the experimental ratio R exp  . They read which are in agreement within the quoted error budget.
are the only ones contributing to B → K * 2 + − . The related Wilson coefficients are discussed thoroughly in the literature (for details, see e.g. [76][77][78] and references therein). In terms of 12 Considering this ratio, one avoids most of the parametric uncertainties.
the Wilson coefficients and the form factors defined in Eqs.
(2)-(5), the general expression of the differential decay width for B → K * 2 + − can be written as [79]: where λ ≡ λ(m 2 B , m 2 K * 2 , q 2 ), and the individual quantities F i read The new input parameters entering the decay rate prediction here are taken as V tb = 0.77 +0.18 −0.24 [52], V ts = 0.0406 ± 0.0027 [52], C ef f 7 (m b ) = −0.306 [80], C ef f 9 (m b ) = 4.344 [76,81] and C 10 = −4.669 [76,81]. Using the calculated LCSR results for the form factors we obtain B(B → K * 2 e + e − ) = (7.72 ± 4.28) × 10 −7 , Our predictions are compatible with the references given within the error budget. Furthermore, in analogy to Eq. (33) we also give our prediction for the LFU ratio: As a final remark before summary, we would like to stress that the results presented in this work include only factorizable contributions and non-factorizable (non-local cc-loop) effects are not taken into account in this work. Analysis of such non-factorizable contributions lies beyond the scope of this paper and we plan to come back to discuss this point separately in the future.

V. CONCLUSION
The study of semileptonic B-meson decays involving tensor mesons can provide additional information on physics beyond the Standard Model due to the rich polarization structure of the tensor mesons. In connection to that we calculated the B → D * 2 , K * 2 , a 2 , f 2 (J P = 2 + ) transition form factors within light-cone sum rules using B-meson distribution amplitudes, including the twist-four terms. We find that the calculated higher-twist terms have a noticeable impact on the sum rules. Using the obtained results for the form factors we estimate the decay rates of B → D * 2 ν, B → K * 2 γ and B → K * The two-particle momentum-space projector can be expressed in terms of B-LCDAs (up to twist-four) as where v µ is the four-velocity of the B-meson, and ∂ µ ≡ ∂/∂l µ with l µ = ωv µ in the two-particle case. The above momentum-space derivatives are understood to act on the hard-scattering kernel of Eq. (11). Moreover, we abbreviatē In our numerical estimates for the form factors we follow the local duality model 13 proposed in Ref. [83] for the two-particle B-LCDAs φ + , φ − , and g + . The explicit expressions for φ + , φ − , and g + in this model are given in Eqs. 5.22-5.23 of Ref. [83].
For g − no model expression is available yet; we therefore use the Wandzura-Wilczek (WW) In the local duality model considered in this work, Eq. (A3) explicitly yields: where θ(x) is the heavy-side step function.
The parameters λ 2 E , λ 2 H and λ B appearing in the explicit expressions of B-LCDAs are provided as input in Sec. III A.
ForṼ B→T we obtain: we obtain: ForT B→T

23A
we obtain: ForT B→T