B ! P and B ! V form factors from B -meson light-cone sum rules beyond leading twist

: We provide results for the full set of form factors describing semileptonic B meson transitions to pseudoscalar mesons (cid:25) , K , (cid:22) D and vector mesons (cid:26) , K (cid:3) , (cid:22) D (cid:3) . Our results are obtained within the framework of QCD Light-Cone Sum Rules with B -meson distribution amplitudes. We recalculate and con(cid:12)rm the results for the leading-twist two-particle contributions in the literature. Furthermore, we calculate and provide new expressions for the two-particle contributions up to twist four. Following new developments for the three-particle distribution amplitudes, we calculate and provide new results for the complete set of three-particle contributions up to twist four. The form factors are computed numerically at several phase space points using up-to-date input parameters, including correlations across phase space points and form factors. We use a model ansatz for all contributing B -meson distribution amplitudes that is self-consistent up to twist-four accuracy. We (cid:12)nd that the higher-twist two-particle contributions have a substantial impact on the results, and dominate over the three-particle contributions. Our numerical results, including correlations, are provided as machine-readable (cid:12)les in the supplementary material. We discuss the qualitative phenomenological impact of our results on the present b anomalies.


Details on the computation and analytical results
We construct the LCSRs for the full set of form factors in the hadronic matrix elements of local flavour-changing b → q 1 currents. For B → P transitions, with P = π, K,D, we discuss the three independent non-vanishing form factors f B→P , which are defined via: with η representing the polarization of the vector meson, and we use the Bjorken-Drell convention with 0123 = +1. If the final state is either a π 0 or a ρ 0 , the l.h.s. of eqs. (2.1)-(2.6) have to be multiplied with a factor of √ 2. Note that A B→V 3 is redundant, since it is a linear combination of the two other axial form factors (2.7) However, the decomposition of the matrix elements including the A B→V 3 form factor is convenient for the extraction of the form factors within a sum rule approach, as discussed below.
The matrix elements defined in eq. (2.1) and eq. (2.4) exhibit apparently unphysical singularities at q 2 = 0. These are removed by the identities In addition, the algebraic relations between σ µν and σ µν γ 5 give rise to the identity It is common to replace the form factors A B→V 2 and T B→V 3 with the linear combinations , (2.10) ; (2.11) where λ( is the Källén function. The linear combinations A B→V 12 and T B→V 23 correspond to form factors for the transition into a longitudinal vector state and therefore simplify the structure of angular coefficients in the differential decay rate of the semileptonic B-meson decays.
The starting point for the construction of the B-LCSRs is the correlation function of two quark currents J ν int ≡q 2 (x)Γ ν 2 q 1 (x) and J µ weak (0) ≡q 1 (0)Γ µ 1 h v (0). The various choices of spin structures Γ 1,2 and quark flavours q 1 and q 2 for the form factors extracted in this article are shown in table 1. The correlator (2.12) is calculated in the framework of heavy quark effective theory (HQET), i.e. the b-quark field is replaced by the HQET field h v . In the kinematic regime q 2 ≤ m 2 b + m b k 2 /Λ had. and k 2 −Λ 2 had. , the dominant contributions -4 - Table 1. Summary of the various combinations of weak and interpolating currents used to extract the form factors. We abbreviate σ µ{q} ≡ σ µν q ν .
to the correlator eq. (2.12) arise at light-like distances x 2 0 [7]. This motivates a systematic expansion of the time-ordered product in terms of bi-local operators with lightlike separationq 2 (x)Γ[x, 0]h v (0), where the [x, 0] denotes a gauge link that renders the bi-local operators gauge invariant. The expansion of the q 1 propagator up to next-toleading power in x 2 near the light-cone x 2 0 gives rise to two-particle and three-particle contributions to the correlator. Four-particle contributions are not taken in account in this work. The two-particle contributions can be summarized as where α, β are spinor indices. The three-particle contributions involve a further gluon field: (2.14) -5 -

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In the above G λρ ≡ g s (λ a /2)G a λρ (x) denotes the gluon field strength tensor, and is the momentum-space representation of the next-to-leading-power term in the light-cone expansion of the quark propagator [45], in which u is the position of the gluon field as a fraction of the light-like distance [0, x] . The B-meson to vacuum matrix elements of the non-local heavy-light currents, appearing in eq. (2.13) and eq. (2.14), are parametrized in terms of B-meson Light-Cone Distribution Amplitudes (LCDAs). The full expressions of these non-local matrix elements for the B-meson LCDAs used in this work are collected in appendix A. Previous works [6][7][8] calculate the correlation functions up to twist three for the two-particle LCDAs, and use an incomplete set of three-particle LCDAs. In our work we go beyond the accuracy of the previous calculations by including all contributions to the correlator from two-and three-particle Fock states up to twist four [23].
In order to construct the sum rule, one has to insert a complete set of hadronic states in eq. (2.12), thereby obtaining a hadronic dispersion relation for Π µν : with M = P, V . The last term involving the spectral density ρ(s) on the r.h.s. of eq. (2.16) captures the contributions arising from excited and continuum states, s h 0 is the corresponding threshold for the lowest-mass excited or continuum state. The local M to vacuum matrix elements are proportional to decay constants f P and f V : (2.17) The B → P and B → V matrix elements have been already introduced in eqs. (2.1)-(2.6). Again, if the initial state of the equation above is a π 0 or a ρ 0 , the l.h.s. receives an additional factor of √ 2. Using the formulas given in eq. (A.1) and eq. (A.2), we can cast the two-and threeparticle terms in eq. (2.13) and eq. (2.14) into an integral form similar to eq. (2.16). Using semi-local quark hadron duality to subtract the continuum contributions we obtain the sum rule. In the process, we apply a Borel transformation from k 2 to M 2 , which removes surfaces terms in the integrals and improves the numerical stability of the sum rule. The latter is achieved by accelerating the convergence of the twist expansion, and by reducing the sensitivity to the duality approximation. The sum rule can then be written in the following form for all the form factors F and final states M = P, V considered here: where we use the auxiliary variable s and its derivative In eq. (2.18), the expressions involving powers of differential operators should always be read as d dσ We further abbreviateσ ≡ 1 − σ and σ 0 ≡ σ(s 0 , q 2 ), where s 0 is an effective threshold parameter not to be confused with s h 0 , from which it differs in general. The functions I (F ) n can be represented as integrals involving the two-particle and three-particle LCDAs: (2.20) with σ = ω/m B in eq. (2.20) and σ = (ω 1 + uω 2 )/m B in eq. (2.21), respectively. The coefficients C (F,ψ) , as well as the normalization factors K (F ) of eq. (2.18) are listed in the appendix B. In the cases F = f B→P we can construct the sum rule for the form factor directly, whereas for the remainder of the cases F denotes one of the following linear combinations of form factors: Here f B→P − is given by Our results for the analytical expressions are always provided for a generic final state meson P, V with valence quark content (q 1q2 ). To the precision we work at, only the mass m 1 of the quark field q 1 enters the expressions. We fully reproduce the two-particle leading-twist contributions proportional to φ + and φ − given in ref. [7]. Furthermore, we extend the two-particle results adding the terms containing g + and g − , that take in account corrections up to twist five. The results for three-particle contributions in refs. [7,8] are obtained for only a subset of the three-particle -7 -

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LCDAs: ψ 3p = ψ V , ψ A , X A , and Y A . When artificially restricting the LCDAs to the same subset, we reproduce the results of refs. [7,8]. Our results for the coefficients C (F,ψ) provide for the first time the complete results for the two-and three-particle contributions up to and including twist four.

LCSR results
We implement the sum rules for the full set of B → P and B → V form factors as part of the EOS software [46], which is an open source project for the evaluation of flavour observables [47]. Our implementation is agnostic of the concrete parametrization of the various LCDAs entering the sum rules. This is achieved by computing all contributing integrals numerically. For this work, the LCDAs implemented in EOS conform to the exponential model put forward in ref. [23]. However, further LCDA models can readily be added to EOS, in order to challenge the (implicit) dependency of the sum rules on the LCDA model. Realistically, this can only be done after measurements of the photo-leptonic decay B − → γ −ν ; see [48] for a recent update of the theoretical framework for the extraction of the LCDA model parameters.
In order to obtain numerical predictions for the form factors and to estimate the theory uncertainties due to the input parameters, we follow the statistical procedure used in ref. [16]. Within a Bayesian framework we first define an a-priori Probability Density Function (PDF) for the input parameters. A summary of the process-specific elements of this PDF is given in table 2. The universal elements of this PDF can be summarized as the following independent Gaussian PDFs for the B-meson to vacuum matrix elements:  We use the f B value from the most precise LQCD analysis available [49], our own estimate of 1/λ B,+ and the λ 2 E,H from ref. [50]. Two classes of the process-specific parameters deserve a more detailed discussion: the Borel parameters M 2 and the duality thresholds s 0 .
Borel parameters. The Borel parameters M 2 P and M 2 V for the pseudoscalar and vector final states are taken from previous studies [6][7][8]. As usual in QCD sum rules, a window should be chosen for the Borel parameter M 2 such that: a) M 2 is not too large, to ensure that excited and continum state contributions to the correlation function are exponentially suppressed; and b) M 2 is not too small, to ensure that the impact of higher-twist contributions are suppressed by powers of 1/M 2 .
We explicitly confirm that the central values of the form factors for B → K * ,D ( * ) transitions exhibit a plateau in their M 2 dependence. For the remaining form factors we find no such plateau, which, however, does not preclude us from applying the sum rules with  [17,49,[54][55][56][57][58][59][60][61]. The Borel parameters are the same as the ones used in [7,8].
some increased systematic uncertainties. Based on the variations of the form factors under change of the Borel parameters, we assign a systematic uncertainty as a percentage of the central value as follows: For π, K and ρ final states the systematic uncertainties can be further reduced through a simultaneous analyses of the form factors and the light-meson decay constants within the framework of QCD sum rules, since both analyses have the Borel parameters and the thresholds in common. This effect has been previously shown in the case of LCSRs with π LCDAs [16]. For K * andD ( * ) final states the uncertainty arising from the variation of the Borel parameter can be included in the statistical procedure. Given the present knowledge of the B-meson LCDA parameter(s), these uncertainties are presently subleading to the parametric uncertainties due to thresholds and LCDA parameters. We leave both of these improvements to future work.
Power corrections. Using the full set of LCDAs up to twist-four accuracy, the authors of ref. [23] expect to account for the contributions of HQET operators up to and including 1/m b corrections. This expectation is based on the observation that an increase by two units of collinear twist corresponds to a suppression by a factor of 1/m b [62]. Moreover, fourparticle LCDAs also start to contribute at the twist-four level and are presently unknown. Given the small size of the three-particle contributions to the sum rules we do not expect sizeable contributions from the four-particle terms, which we ignore throughout. The corrections at order 1/m 2 b are presently unknown, and we estimate them based on naive dimensional arguments at ∼ 5%. We add this uncertainty in quadrature to the systematic uncertainty incurred by the Borel parameters. can in principle be determined by closely following the procedure carried out in ref. [16]. First, one defines a prior interval with uniform probability for the threshold parameters. In this step one also varies the LCDA parameters 1/λ B,+ , λ 2 E and λ 2 H to determine the correlations between thresholds and LCDA model parameters. In a next step, the a-priori PDF is challenged with a theoretical likelihood. The pseudo-observables that are constrained through the likelihood are the "first moments" of the form factors' correlation function. These moments warrant a more careful definition: for any form factor F we differentiate its scalar-valued correlator Π F (q 2 ; M 2 ) with respect to −1/M 2 and normalize it to Π F . The resulting ratio is a pseudo-observable that is expected to yield the final state's mass square m 2 P or m 2 V , respectively, within the accuracy of the light-cone OPE for the correlation function.
We carry out this procedure for the K * andD ( * ) final states. Within the likelihood, we impose that the theory prediction for the first moments match the square of the respective final state hadron mass. We impose relative uncertainties of 5% on these predictions, in order to account for the impact of 1/m 2 b corrections to the correlators. The added uncertainties are considerably larger than in the B → π analysis [16]. We think our more conservative treatment is warranted as we expect the "first moments" to exhibit a substantial but difficult-to-quantify dependence on the B-meson LCDA model. For some of the threshold parameters we find a marked non-gaussianity for the two-dimensional joint posterior PDF of a single threshold parameter and 1/λ B,+ .
For pseudo-Goldstone bosons such as the π and K, and for the ρ meson with its substantial decay width, the first moments are not expected to reproduce the meson mass squares. As an exercise, we attempt anyway to apply the procedure described above and find it to be too unstable to determine the duality threshold for any of these states. We therefore adopt the thresholds used in ref. [7], which are determined (λ B,+ independently) from two-point QCD sum rules of the π, K and ρ decay constants.
In our analysis of form factors to K * andD ( * ) final states a further complication arises from the fact that the first moments of the correlation functions exhibit a noticable but mild q 2 dependence. We choose to study this effect as follows: for each form factor, the theory likelihood includes the form factor's first moment for seven values of q 2 in the range −15 GeV 2 to 0 GeV 2 , with increments of 2.5 GeV 2 . We make a linear ansatz for the q 2 dependence of the threshold parameters s We then determine the two parameters s  Table 3. Detailed budget of the φ ± , g + , g WW − and three-particle contributions to our LCSR results for the form factors at q 2 = 0. state hadrons, we find only negligible impact due to our treatment of q 2 dependence of the threshold parameters when comparing to the dominant uncertainties incurred by the B-meson LCDA parameters. We therefore proceed with the assumption of q 2 -independent duality thresholds. However, we remark that this problem needs to be revisited once the parametric uncertainties due to the LCDA model-dependence are under better control.
For theD andD * final states, we find that increasing q 2 to positive values increases the uncertainty in the prediction of the first moments substantially. In fact, for q 2 5 GeV 2 we find very broad intervals that include s 0 = 0 at 68% probability. This increase in uncertainty is accompanied by a substantial growth of relative contributions (to ∼ 50% and beyond) due to the higher-twist two-particle terms. This clearly poses a problem for the calculation of the B →D ( * ) form factors at positive q 2 . It remains to be seen if this effect is due to the modelling of the LCDAs, or indicates an earlier-than-expected breakdown of the Light-Cone OPE at positive q 2 .

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Predictions. Based on the procedure discussed above, we obtain threshold parameters for the individual form factors. A summary of these parameters and their uncertainties are listed in table 2. We then proceed to produce posterior-predictive distributions for the form factors at five different q 2 points: q 2 = {−15, −10, −5, 0, +5} GeV 2 . Note that the form factors A B→V 0 and T B→V 2 are linearly dependent on the remaining form factors at q 2 = 0, and therefore this particular point is dropped from the predictions for these two quantities. For heavy final states M = D, D * we remarked previously that the threshold computation becomes unstable for q 2 > 0. We therefore drop the point q 2 = +5 GeV 2 for these two final states. The resulting Probability Density Functions (PDFs) of the form factors at the various q 2 points are most readily communicated in form of machine readable files, containing the mean values and covariance matrices of a multivariate Gaussian density. The results are included in the EOS software [46] as of version v0.2.3 as YAML files, defining the following named constraints: We provide a detailed budget of the individual contributions to the form factors at q 2 = 0 in table 3. We also compare our results and their uncertainties, including all sources of systematic uncertainties, with results in the literature in table 4.
Our numerical results can subsequently be used to fit concrete parametrizations of the respective form factors. We carry out such fits for the BSZ parametrization [17] in the next subsection.

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For each final state we perform two fits. The first fit includes only the information at small q 2 values, obtained from the LCSRs, within the likelihood. Within all plots in appendix C, the results of this fit are displayed as a dark gray band. For the second fit, we add further information from lattice QCD analyses of the form factors at large values of q 2 as available [64][65][66][67][68][69][70]. Due to the absence of lattice QCD analyses of the B → ρ transitions there is no combined fit for the respective form factors. Results arising from the second fit are displayed as blue bands, throughout this work.
For the LCSR-only fits we have four data points and three parameters per form factor, equivalent to one degree of freedom (three data points and two parameters in the case of ). Given the large uncertainties and small number of degrees of freedom, it is not surprising that we find a p value 3%, our a-priori threshold, in each of these fits. For the combined fits to LCSR and LQCD inputs, we find p values very close to one, indicating an excellent fit in each of these analyses.
As for the LCSRs, the posterior PDFs of our fits are most readily provided as machine readable files containing the mean values and covariance matrices of a multivariate Gaussian density. The results are included in the EOS software [46]  Moreover, we provide our results also through machine-readable JSON files in the same format as used in ref. [17]. These files are part of the supplementary material of this article. Moreover, our results will be available by default to users of the flavio software [71] from the next release on.

Selected phenomenological implications
We will briefly discuss the impact of our results for the form factors on the present b anomalies.
Overview of the lowest-lying resonances in the individual b → {u, d}, b → s and b → c transitions, and the association to the respective form factors. The masses above enter the parametrization of the form factors eq. (3.4) as the resonance mass parameter m R,F . The B u,d,s masses have been taken from ref. [17], to ensure interoperability of their and our results. The B c resonance masses have been taken from ref. [72].
Several studies [82][83][84][85][86][87][88][89][90][91][92] come to the conclusion that a negative shift to the short-distance coupling C µ 9 , and potentially to some couplings that vanish in the SM, can explain simultaneously the deviations in all anomalous b → s + − measurements; see [2] for a recent review and the definition of the low-energy Lagrangian. In the case of e vs µ universality with a lower dilepton mass cut q 2 ≤ 1 GeV, the SM predictions of the LFU ratios are insensitive to the hadronic form factors [93][94][95]. We will therefore not discuss them here any further. Instead, we will discuss the qualitative impact of our results on fits of the b → s + − short-distance couplings to the available data on exclusive B → K * µ + µ − decays, which have presently the biggest impact in global b → sµ + µ − fits.
Assuming the global fits to correctly account for non-local effects arising from fourquark operators in the B → K * µ + µ − amplitudes [98], the data leads to two possible conclusions [84,86]: 1. the ratio of form factors V B→K * /A B→K * 1 deviates from the ratio predicted by symmetry relations at large kaon energies [96,97] as well as a-priori predictions from extrapolations of lattice QCD result [99] and light-meson LCSRs [17], leading to global fits with border-line goodness of fit; or 2. there is a New Physics (NP) shift to the short-distance coefficient C 9 corresponding to ∼ 25% of its SM value.
This interpretation has been strengthened recently by a proof-of-concept analysis in which the non-local matrix elements are further constrained in shape due to their properties following from analyticity and unitarity [98]. The particular solution to obtaining a good fit in the absence of NP effects requiress the ratio V B→K * /A B→K * 1 to not only deviate in value from the large-energy limit prediction, but also in shape. We show explicitly in figure 1 that our predictions are compatible with the symmetry limit at large energies; with extrapolations of lattice QCD results within their large uncertainties; and with the rather precise results obtained from LCSRs with K * LCDAs.   Figure 1. The ratio V (q 2 )/A 1 (q 2 ) for B → K * transitions. The green, red and gray lines and shaded areas correspond to the central values and the 68% probability envelope of the form factors obtained from fits to only LQCD results, fits to LCSR results from ref. [17], and fits to our LCSR results, respectively. The dashed line correspond to the large-energy limit for this ratio [96,97]. The dotted line corresponds to the central value of the SM fit to B → K * µ + µ − data from ref. [98]. The exclusive semileptonic decays B →D ( * ) ν are of great phenomenological interest. One the one hand, they can be used to extract the magnitude of the Cabibbo-Kobayashi-Maskawa (CKM) matrix element V cb . Its determinations from exclusive and inclusive B decays has been famously in tension with each other for the last decade. On the other hand, the exclusive decays allow to test the SM through LFU ratios R(D) and R(D * )

Lattice
Both the extraction of |V cb | and testing the SM through LFU violation require accurate predictions of the relevant form factors. The Heavy-Quark-Expansion, in combination with data, can help in this particular case of heavy-to-heavy flavour-changing quark transitions; see refs. [100,101] and references therein for dispersive bounds, and ref. [102] for the SM prediction of R(D * ). It has been recently argued that strict adherence to the so-called CLN parametrization [101] is, at least partially, responsible for the exclusive-vs-inclusive tension [4,5,103] when determinig V cb from semileptonic B →D * transitions. In the case of B →D, recent lattice QCD analyses yield V cb values that are compatible with both the inclusive and the B →D * determinations.

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LCSR determinations of B →D * form factors [8] play an important role in some of the phenomenological analyses [4,5], e.g. through form factor ratios at q 2 = 0. With our updated results for the form factors, we are in the position to also update these ratios and also to provide parametric correlations between them. The ratios under discussion are labelled R 0 , R 1 and R 2 (see e.g. ref. [101] for their definitions), which are functions of the recoil parameter w, with m B m * D w = p · k. At maximal recoil w max , corresponding to q 2 = 0, one has: Moreover, at q 2 = 0 the equation of motion implies that only two of these ratios are linearly independent. Based on our correlated results for the form factors we obtain R 0 (q 2 = 0) = 1.117 ± 0.061 , R 1 (q 2 = 0) = 1.151 ± 0.114 , The correlation coefficients ρ between R 1 and R 2 reads Our correlated results are compatible with the previous LCSR determinations of R 1 (q 2 = 0) and R 2 (q 2 = 0) [8] at less than one standard deviation. We can also use our results to calculate the values of the LFU observables R(D) and R(D * ) in the SM and beyond. Using the correlated results for the form factor parameters obtained in section 3.2 from the fit to only our LCSR results, we obtain:  Our result for R(D) is dominated by the precise LQCD inputs [68] beyond zero recoil, and the agreement with the LQCD prediction R(D) = 0.300 ± 0.008 is therefore not surprising.

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Our result for R(D * ), on the other hand, is supported by two LQCD inputs for the A B→D * 1 form factor at only the zero recoil point. We find excellent agreement with the values obtained using heavy quark symmetry relations, i.e.:

R(D * )
SM, [4] = 0.260 ± 0.008 . (4.10) As a closing remark, we wish to emphasize that the predictions in the framework of heavy quark symmetry relations are complicated by the proliferation of matrix elements associated with 1/m c corrections that are not present in our LCSR-derived results.

Summary and outlook
We have presented a comprehensive update of Light-Cone Sum Rules (LCSRs) results for the full set of form factors relevant to semileptonic B decays. Our update includes, for the first time, a consistent treatment of all two-particle and three-particle Light-Cone Distribution Amplitudes (LCDAs) up to twist four. Moreover, our work also updates the numerical inputs across the board. We have implemented our analytical results agnostic of the concrete expressions for the distribution amplitudes, thereby ensuring that our analysis can be readily repeated once our knowledge of either the properties of the amplitudes or their parameters improves in the future. The relevant computer code is publicly available [46] under an open source license as part of the EOS software [47]. Moreover, all of our numerical results are available as machine-readable files. For form factors that are common to our and a previous LCSR analysis using light-meson LCDAs [17] we have ensured interoperability of the data files with the supplementary material of this article.
Within our analyses we find sizable contributions from two-particle states at the twistfour level, which exceed the twist-three and twist-four three-particle contributions by one order of magnitude. Our analysis has been carried out strictly in the framework of Heavy-Quark Effective Theory, which enables us to be agnostic of the final state quark flavour, thereby facilitating the analysis. However, it also precludes us from using the O (α s ) corrections to the leading-twist two-particle results obtained in the framework of SCET Sum Rules for massless [21] and massive [22] pseudoscalar final states. The logical next step is therefore to extend our present framework with these radiative corrections, and to check if the combined twist and α s expansion of the non-local operators in the light-cone OPE is well behaved. Particularly, we wonder if the instability of inferring the duality thresholds for π, K and ρ final states can be overcome by including the radiative corrections, or by including contributions at the twist-five and twist-six levels; see ref. [63] for recent efforts in the latter direction.
Finally, we have selected two phenomenological applications connected to the present B anomalies to highlight the usefulness of our results. Our finding weaken the interpretation of the B → K * µ + µ − angular anomalies as effects of our lack of knowledge of hadronic form -18 -

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factors. Furthermore, we have updated the form factor ratios R 1 and R 2 , relevant for V cb extractions and predictions of the LFU ratio R(D * ). Our results permit for the first time to account for correlations among the relevant B →D * form factors, and are in agreement with previous results at less than one standard deviation.

Acknowledgments
This work is supported by the DFG within the Emmy Noether Programme under grant DY-130/1-1 and the DFG Collaborative Research Center 110 "Symmetries and the Emergence of Structure in QCD". We are very grateful to Sébastien Descotes-Genon, Paolo Gambino and Javier Virto for helpful discussions, and to Martin Jung, David Straub and Javier Virto for comments on the manuscript. We are also grateful to the authors of [36,37] for discussing their results prior to publication.

A B-meson distribution amplitudes
In this appendix we collect formulas relevant to the parametrization in terms of momentumspace of B-LCDAs of the non-local matrix elements in eqs. (2.13) and (2.14). The twoparticle B-LCDAs are defined via while, for the three-particle B-LCDAs, we have where a gauge link is implied in the above, and the derivatives are abbreviated as ∂ µ ≡ ∂/∂l µ ,is the momentum-space representation where l µ = ωv µ in the two-particle case and l µ = (ω 1 + uω 2 )v µ in the three-particle case. Throughout, these derivatives are understood to act on the hard-scattering kernel. In addition, we define the following shorthand notation:φ where ψ 3p represents any of the three-particle LCDAs. We use 0123 = +1 in both the definition of the form factors and in eq. (A.2), which matches the conventions of refs. [12,17]. Accounting for the different convention, we reproduce the results of refs. [7,8]. The "traditional" basis of three-particle LCDAs can related to a basis of LCDAs with definite twist as follows [23]: Note that we adopt the same nomenclature for the LCDAs as in ref. [23], except for renamingψ 4,5 → χ 4,5 such that our notation involving barred LCDAs (see eq. (A.3)) becomes more legible. It is possible to invert these relation. We obtain: A parametrization of the set of three-particle LCDAs at the twist-five and twist-six level has been recently suggested [63]. This set includes three twist-five LCDAs and one twistsix LCDA. However, to obtain the full set of three-particle LCDAs one has to expand the position-space non-local matrix elements around the light-cone x 2 0 in a consistent manner. Including the terms ∝ x 2 for the structures multiplied by φ 3 , φ 4 , ψ 4 and χ 4 , -20 -

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the full set of momentum-space matrix elements at the twist-six level can be obtained from eq. (A.2) by using eq. (A.5) in combination with the replacements The twist of the new g ψ 3p functions corresponds to the twist of their partner ψ 3p plus two units of twist. Up to the twist-six level we therefore find twelve independent three-particle LCDAs: one at twist three, three at twist four, four at twist-five, and further four at twistsix; in variance with the ansatz of ref. [63]. Our argument here is in full analogy to the approach to the off-the-light-cone contributions for two-particle LCDAs in form of g + and g − introduced in ref. [23].
In order to evaluate numerically the form factors, we use the exponential models proposed in ref. [104] and adapted in ref. [23] to the LCDAs φ + , φ − , g + , φ 3 , φ 4 , ψ 4 and χ 4 . Since g − receives contributions from the three-particle DA ψ 5 , for which no model is given in ref. [23], we approximate g − in the Wandzura-Wilczek (WW) limit. We use where in the second line we use the Grozin-Neubert relation 2λ B,+ = 4Λ/3.

B Coefficients of the LCSR formula
Here we list all the coefficients of eq. (2.18). The normalization factors are: , In the next subsections we give the C (F,ψ) n coefficients of eqs. (2.20) and (2.21). For all the form factors, the following relations hold among the three-particle contributions: The coefficients of eq. (2.20), for the two-particle DAs, are listed in the following. For f B→P + we find the non-vanishing coefficients: 3) For f B→P +/− we find: ,

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For χ 4 : The coefficients of eq. (2.20), for the two-particle DAs, are listed in the following. For V B→V we find: we find: we find:

30
we find:

(B.21)
For T B→V 1 we find: For T B→V

C Plots of the form factors
This appendix is dedicated to illustrate our numerical results for the form factors in relation to previous results obtained from LCSRs with B-meson LCDAs [6][7][8] and to results obtained from LQCD.