Angular Analysis of the Decay $\Lambda_b \to \Lambda (\to N \pi) \ell^+\ell^-$

We study the differential decay rate for the rare $\Lambda_b \to \Lambda (\to N \pi)\ell^+\ell^-$ transition, including a determination of the complete angular distribution, assuming unpolarized $\Lambda_b$ baryons. On the basis of a properly chosen parametrization of the various helicity amplitudes, we provide expressions for the angular observables within the Standard Model and a subset of new physics models with chirality-flipped operators. Hadronic effects at low recoil are estimated by combining information from lattice QCD with (improved) form-factor relations in Heavy Quark Effective Theory. Our estimates for large hadronic recoil -- at this stage -- are still rather uncertain because the baryonic input functions are not so well known, and non-factorizable spectator effects have not been worked out systematically so far. Still, our phenomenological analysis of decay asymmetries and angular observables for $\Lambda_b \to \Lambda (\to N \pi)\ell^+\ell^-$ reveals that this decay mode can provide new and complementary constraints on the Wilson coefficients in radiative and semileptonic $b \to s$ transitions compared to the corresponding mesonic modes.


Introduction
Rare decays based on radiative or semi-leptonic b → s transitions offer various possibilities to test the predictions for flavour-changing neutral currents (FCNCs) in the Standard Model (SM) against new physics (NP) extensions (for comprehensive summaries of theoretical and experimental aspects, see e.g. [1][2][3]). In the past -notably during the "B-factory" era [4] -the main phenomenological focus was on inclusive distributions (B → X s γ and B → X s + − ) or exclusive decay observables (e.g. in B → K * γ, B → K ( * ) + − , . . . ) for mesonic decays. With the recent b-physics program at LHC (and here, in particular, the dedicated LHCb experiment) not only more precise measurements of radiative B-and B smeson decays, but also information on baryonic modes like Λ b → Λ + − with reasonable accuracy will become available [5]. (For an incomplete list of previous phenomenological studies, see e.g. [6][7][8][9] and references therein). Exclusive hadronic decays, by definition, are theoretically challenging because the calculation of decay amplitudes induces a number of hadronic uncertainties related to long-distance QCD dynamics. For b → s + − transitions this includes hadronic transition form factors, which parametrize the "naively" factorizing contributions from b → sγ and b → s + − operators. In addition, systematic uncertainties related to non-factorizable effects appear, where the short-distance dynamics is induced by hadronic b → s operators, while the radiation of the photon or charged lepton pair is linked to the long-distance hadronic transition. Baryonic transitions, at first glance, seem to suffer from even larger hadronic uncertainties than their mesonic counterparts since transition form factors and hadronic wave functions are only poorly known, and the analysis of the spectator dynamics is more complicated [10]. Recent progress with respect to Λ b → Λ form factors includes lattice-QCD results in the heavy-mass limit [11] (valid at low and intermediate recoil), and a sum-rule analysis of spectator-scattering corrections to form factor relations at large recoil [12]. Also, a better theoretical understanding of the Λ b wave function (in the form of light-cone distribution amplitudes) has been achieved recently [13][14][15]. However, as we will also argue in this paper, exclusive modes like Λ b → Λ + − do provide interesting phenomenological potential. Exploiting the full set of angular observables that can be derived from the analysis of the subsequent Λ → N π decay, one may obtain information on the underlying short-distance weak interactions that is complementary to the analogous mesonic decay observables (see e.g. [16][17][18][19][20][21][22]). This is mainly a consequence of the fact that the subsequent weak decay Λ → N π is parity violating, while the strong decay K * → Kπ in the mesonic counterpart case, B → K * + − , is not. 1 Furthermore, within such an analysis, independent information on the hadronic parameters themselves can be extracted from experimental data. In particular, form-factor relations that arise in the infinite-mass limit for the heavy b-quark, can be tested (notice that the number of independent form factors for Λ b → Λ transitions in that limit is smaller than for B → K * transitions: 2 at low recoil [23], and one at large recoil [12,24]). The outline of this paper is as follows. In the next section, we will briefly summarize our notation and conventions regarding the short-distance operator basis in the weak effective Hamiltonian. In section 3 we derive the necessary expressions to describe the Λ b → Λ(→ N π) + − differential decay rate. 2 To this end, we define the relevant kinematic 1 The background from direct Λ b → N π + − decays is expected to be low: First, the direct decay is relatively suppressed by |V td /Vts|. Second, the signal with an intermediate Λ baryon can be distinguished by requiring a sizable displacement between the Λ b + − and N π decay vertices. 2 We will restrict ourselves to the case of unpolarized Λ b baryons, as the Λ b polarization in the LHCb setup has been measured to be small [25], and polarization effects in the symmetric ATLAS and CMS variables, define the general set of Λ b → Λ form factors in the helicity basis, and discuss the hadronic couplings appearing in the Λ → N π decay. From this we derive expressions for the angular observables in terms of transversity amplitudes. Section 4 is dedicated to the phenomenological analysis of interesting observables. Specifically, we consider the fraction of transverse dilepton polarization, the leptonic, baryonic and mixed forward-backward asymmetries, and a number of certain ratios of angular observables where either shortdistance effects or form-factor uncertainties drop out to first approximation. We also work out the simplifications that arise from the heavy-quark expansion at low or large recoil. We conclude this section with numerical predictions for a selection of observables, making use of the available theoretical and phenomenological information on the relevant input parameters, and compare to presently available experimental data in the low-recoil region. We conclude with a summary and outlook. Some technical details are summarized in the appendices.

Effective Hamiltonian for b → s + − Transitions
The effective weak Hamiltonian for b → s + − transitions (|∆B| = |∆S| = 1) in the SM (see e.g. [26][27][28]) contains radiative (b → sγ) and semi-leptonic (b → s + − ) operators as well as hadronic operators (b → sqq, b → sg). For the radiative and semileptonic operators, we find it convenient to use the normalization convention with the SM operators For simplicity, we only take into account the factorizable quark-loop contributions of the hadronic operators O 1−6 and O g 8 , which can be lumped into effective Wilson coefficients C 7 eff and C 9 eff (q 2 ). We ignore non-factorizable effects, which are expected to play a nonnegligible role, particularly at large hadronic recoil [29,30]. Notice that the radiative operators O 7( ) contribute to the semi-leptonic decay through photon exchange and its electromagnetic coupling to a lepton pair such that with q µ = p µ − k µ being the momentum transfer to the lepton pair. In this notation the Wilson coefficient C 7 is suppressed by m s /m b in the SM. Potential NP contributions to b → sγ and b → s + − , on the one hand, modify the Wilson coefficients of the SM operators above and, on the other hand, feed into new effective operators, H eff NP,naive = N 1 i=S( ),P ( ),9 ,10 ,T,T 5 detectors will average out. Here the SM operators are completed to form a basis of dimension-six operators by: which contains the chirality-flipped counterparts of the SM operators O 9 and O 10 together with scalar, pseudoscalar, tensor and pseudotensor operators.

Kinematics
We assign particle momenta and spin variables for the baryonic states in the decay according to: Here, s i are the projections of the baryonic spins onto the z-axis in their rest frames. It is convenient to define sums and differences of the hadronic and leptonic momenta in the final state, The discussion of the kinematics is similar to what has been already worked out for semileptonic four-body B-meson decays, see e.g. [16,17,31,32]. We end up with four independent kinematic variables, which can be chosen as the invariant mass q 2 , the helicity angles θ Λ and θ , and the azimuthal angle φ, see Fig. 1, which are defined in the relevant Lorentz frames in appendix D. Furthermore, we introduce a set of virtual polarization vectors ε µ (λ = t, +, −, 0) with q · ε(±) = q · ε(0) = 0, which in the dilepton rest frame take the canonical form as shown in appendix D. In terms of the leptonic angle θ and the relativistic lepton velocity, the polarization vectors obeȳ Similarly, the corresponding hadronic kinemetic variables appear in the following Lorentzscalar products, We also define (3.6)

Λ b → Λ Helicity Form Factors
The hadronic form factors for Λ b → Λ transitions are most conveniently defined in the helicity basis [12]. For the vector current, this yields three independent helicity form factors f V i (q 2 ), entering the corresponding helicity amplitudes where we slightly changed notation compared to [12]. The kinematic functions s ± are defined as Analogous expressions are obtained for the axial-vector current, Restricting ourselves to the SM operator basis and its flipped counterpart, only the q ν projections of the tensor and pseudotensor currents appears, which lead to another four independent form factors, and The spinor matrix elements for given combinations of spin orientations are summarized in appendix E. For the nonzero helicity amplitudes, we then obtain in case of the vector current Similarly, in case of the axial-vector current we have For the tensor and pseudotensor currents, the non-vanishing helicity amplitudes read and From these relations the advantage of using form factors defined in the helicity basis becomes evident.

Hadronic Couplings in Λ → N π
In the SM the decay Λ → N π is described by the effective Hamiltonian The hadronic matrix element which determines the Λ → N π decay can be parametrized as [33] p As a consequence of the equations of motion, only two independent hadronic parameters appear which we have denoted as ξ and ω. 3 They can be extracted from the Λ → pπ − decay width and polarization measurements.
In terms of the kinematic variables introduced above (see also appendix D), the helicity amplitudes for the secondary decay can be written as where we abbreviate The corresponding helicity contributions to the decay width can be defined as which yield Here the Λ → N π decay width is given as [33] and the parity-violating decay parameter α reads

Angular Observables
The angular distribution for the 4-body decay can be written as a 4-fold differential decay width, which can be decomposed in terms of a set of trigonometric functions, (3.28) Here the first line corresponds to a relative angular momentum (L, M ) between the N π system and the dilepton system of (L, M ) = (0, 0). The lines two to four correspond to L = 1, with the third component M = 0 in the second line, and |M | = 1 in lines three and four. This implies that each line of eq. (3.28) can be decomposed in terms of associated Legendre polynomials P |M | l (cos θ ), where 0 ≤ l ≤ 2 holds for the dilepton angular momentum l on the basis of angular momentum conservation. This agrees exactly with our results eq. (3.28). In particular, (c) no further terms can arise from dimension-six operators which are absent in our calculation; i.e., scalar and tensor operators.
The coefficients in the decomposition eq. (3.28) are refered to as angular observables and depend on the dilepton invariant mass. In our notation, they are denoted as K nλ ≡ K nλ (q 2 ), with n = 1, . . . , 4, and λ = s, c, ss, cc, sc. In terms of the transversity amplitudes for Λ b → Λ transitions and the decay parameter α in Λ → N π defined above, we find and The angular observables contain all the relevant information about the short-and longdistance dynamics in the SM or its extension by chirality-flipped operators in the effective Hamiltonian.

Simple Observables
For the experimental analyses one can construct weighted angular integrals of the differential decay width, to obtain different decay distributions in the dilepton invariant mass q 2 as linear combinations of angular observables (or ratios thereof).
(a) The simplest of these distributions is just the differential decay width in q 2 , which corresponds to ω X ≡ 1.
(b) The fraction of transverse or longitudinal polarization of the dilepton system is obtained as This is achieved by the weight functions The well-known forward-backward asymmetry with respect to the leptonic scattering angle, normalized to the differential rate, is defined as The analogous asymmetry for the bayonic scattering angle reads (e) Finally, one can also study a combined forward-backward asymmetry via

Exploiting Form-Factor Symmetries at Low Recoil
In the limit of low hadronic recoil to the (N π) system (i.e. large invariant lepton mass , the number of independent form factors reduces as a consequence of the HQET spin symmetry [23]. In our notation the form-factor relation with the two leading Isgur-Wise functions ξ 1 and ξ 2 read (cf. [12]) In phenomenological analyses, one can make use of these relations in the following way: • At the moment, lattice-QCD estimates for the Λ b → Λ form factors exist in the HQET limit [11]. These results, together with their respective uncertainties, can be implemented as a theoretical prior distribution for a Bayesian analysis of the experimental data (within the OPE/factorization approximation).
• Measuring suitably defined combinations of angular observables (see below), one can obtain experimental information about the size of corrections to the form-factor relations (4.8), in a similar way as has been proposed for the well-studied B → K * (Kπ)µ + µ − decay [20,22]. This will result in independent posterior distributions for the individual form factors, as found in [34][35][36][37].
• For the time being, priors for the individual form factors can be generated by estimates of Gaussian distributions for the subleading form factor contributions, which are defined in appendix A. In the long run, we expect lattice data for all individual form factors and their ratios.
In the following discussion, we will pursue a simplified approach that keeps the four contributing vector and axialvector form factors as independent hadronic quantities, and relates the four contributing tensor form factors via (4.8), including perturbative corrections at 1-loop (so called "improved Isgur-Wise relations, see appendix A). In this approximation each transversity amplitude is proportional to a single form factor and can be written as and where we neglect 1/m b corrections. The combinations of Wilson coefficients that appear in this way are given by Here the parameter κ = κ(µ) contains the radiative QCD corrections to the form factor relations such that together with the product of Wilson coefficients and the b-quark mass the above expressions for the transversity amplitudes are renormalization-scale independent (in a given order of perturbation theory).
The simplification for the transversity amplitudes directly translates to the angular observables. It turns out that these are sensitive to the following combinations of short-distance parameters, that also appear in the angular observables for B → K * (→ Kπ) + − decays [20,22], and two new bilinears of Wilson coefficients Re {(C 79 ± C 7 9 )(C 10 ± C 10 ) * } (4.14) which contribute as a consequence of parity violation in the secondary weak decay. 4 Here we abbreviate We observe that in this approximation K 3sc = 0 even in the presence of chirally-flipped operators.
For the simple observables introduced in the previous subsection we then obtain and and We will present numerical estimates for these observables in the SM (integrated over q 2 in the low-recoil region) in section 4.4.
Future experimental data will also allow to simultaneously test the short-distance structure of the SM against NP, and to extract information on form-factor ratios. In the presence of both SM-like and chirality-flipped operators, we find one ratio of angular observables where the form factors cancel in the given approximation, 20) and two ratios of angular observables which only depend on form factors, We also find ratios that are only functions of the Wilson coefficients and a single ratio of form Re If, on the other hand, we assume the absence of chirality-flipped operators (including C 7 → 0 in the SM), we obtain ρ ± 3 → 2 ρ 2 = 2 Re {C 79 C * 10 } and ρ 4 → This further implies K 3s → 0, and one also finds two more ratios of observables which are free of form factors, The additional short-distance free observables in the SM operator basis are given by (4.25)

Simplifications at Large Recoil
The number of independent form factors reduces further in the limit of large recoil energy [39,40] where the leading contributions 5 can be identified using soft-collinear effective theory (SCET [41,42]). In our notation this simply implies the equality of all helicity form factors, Including α s corrections to the soft form factors, the modification of the form-factor relations (4.26) can be described by a vertex factor, and a single q 2 -dependent function ∆ξ Λ that emerges from spectator scattering, see appendix C of [12]. The values of ξ Λ and ∆ξ Λ have been estimated from sum rules with Λ b distribution amplitudes in [12]. Due to the complexity of the baryonic transition and -compared to the mesonic case -the poor theoretical knowledge on the baryonic wave functions, these estimates have a large uncertainty. With sufficient experimental information, however, one can again try to constrain the form-factor values from the data itself.
To this end, we first recall that the function ∆ξ Λ drops out in the following sums, More precisely, one has to distinguish soft overlap contributions [12,24] and hard spectator interactions [10]. In contrast to the analogous B-meson transitions, the former are suppressed by one power of the bquark mass compared to the latter. On the other hand, the hard spectator term now only starts at second order in the strong coupling αs and therefore numerically appears to be a sub-dominant effect. In any case, both contributions fulfill the form-factor symmetry relations to first approximation.
, and the µ-dependence of the tensor form factors related to the anomalous dimension of the tensor operators is explicit. The corresponding differences between form factors will be proportional to ∆ξ Λ . We remind the reader that in the largerecoil region we also expect sizeable corrections from non-factorizable (i.e. not form-factor like) contributions, which formally enter at the same order as ∆ξ Λ . These have not been calculated or estimated at present, but would be required for a consistent extraction of ∆ξ Λ . The phenomenological strategy would then be the following.
• Take (4.27) as a theoretical constraint on the form factors.
⊥ → 0 as the central value for a theory prior on the form-factor differences with a conservative estimate for the theory uncertainty (say, of the order 20-30%, independently for each individual form factor difference).
• The posterior distributions for the form-factor differences, resulting from the comparison with experimental data, would provide an estimate for the size of factorizable and non-factorizable spectator effects.
As a first step, we may ignore the spectator effects altogether. Including the known factorizing contributions from the relevant hadronic operators in b → s + − processes [43], the transversity amplitudes can then be written as where the quantities τ i,± can be expressed in terms of Wilson coefficients C i , form-factor ratios R i and perturbative functions F (7,9) 29) and the function Y 9 (q 2 ) captures the one-loop virtual quark-loop contribution which can be absorbed into C 9 → C eff 9 (q 2 ). The form-factor ratios occuring in this limit are simply given by The three observable forward-backward asymmetries, A ,Λ, Λ FB (q 2 ), develop a characteristic q 2 -behaviour (see numerical discussion below). In particular, we find that within the SM A FB (q 2 ) and A Λ FB (q 2 ) cross zero, where to first approximations the roots q 2 0, and q 2 0, Λ are the same, This expression is well known from other exclusive and the inclusive b → s + − decays.
On the other hand, A Λ FB (q 2 ) does not cross zero in the SM.

Numerical Analysis
In order to translate our theoretical results into numerical predictions, we are going to determine predictive probability distributions for the Λ b → Λ(→ N π) + − observables.
All central values and uncertainty ranges that we quote are based on these distributions. In the course of our work, we extend the EOS flavor program [44] through implementation of the relevant decay observables at both large and low hadronic recoil.

Transition Form Factors
Within our simplified factorization approach, the essential hadronic input functions are the Λ b → Λ transition form factors. For the numerical analysis we need probability distributions for the individual helicity form-factor values at different values of q 2 . To this end we take into account the available lattice data in the heavy-quark limit and an estimate of the soft form factor from a sum rule in the large-recoil limit, which are combined using the parametrization eq. (B.2). Corrections to the heavy-quark limit are allowed for as well.
The specific steps of our analysis aim toward a Bayesian analysis once the knowledge of the transition form factors improves. These steps are presented in detail in appendix C. Our setup includes eight form factor parameters x, see eq. (C.6). The main result of the fits is the posterior P ( x|Estimates), which is defined in eq. (C.13). The posterior is central to the computation of the numerical results that follow.

Results
Theory uncertainties in the computation of the observables arise, beyond the transition form factors, also from variations of CKM matrix elements, the Λ → N π coupling α, and the masses of the charm and bottom quark; see table 1 for a summary.
For the CKM matrix elements, we use uncorrelated Gaussian distributions for our analysis. Their parameters follow from the marginalised posterior distributions as obtained by the UTfit collaboration in their "Tree Level Fit" analysis [45]. For α and the quark masses, we use world averages as provided by the Particle Data Group [46]. Further input parameters, such as the hadron masses and lifetimes, are fixed to the central values of their respective world averages [46]. For the Λ b lifetime, we use the more recent world average by the Heavy Flavor Averaging Group [47], which includes the two recent measurements by the LHCb collaboration [48,49].
The observables of interest in our analysis are the branching ratio B, the three forwardbackward asymmetries A FB , A Λ FB and A Λ FB , and the fraction of longitudinal lepton pairs F 0 . For the numerical evaluation, we first calculate q 2 -integrated angular observables K nλ . All observables of interest are then computed from these pre-integrated angular observables, which we denote as O for any observable O(q 2 ). Our nominal choice of the integration region is 15 GeV 2 ≤ q 2 ≤ 20.28 GeV 2 , in order to minimize the uncertainties from quarkhadron duality violation (see the discussion for the mesonic counterpart in [50]). 6 We obtain numerical estimates for the observables O through uncertainty propagation. For this, we compute 8 · 10 5 variates of the predictive distribution P ( O),  In addition, we compare our result for the somewhat larger q 2 bin 14.18 GeV 2 ≤ q 2 ≤ 20.28 GeV 2 at 68% probability, parameter value and 68% interval unit source λ 0.2253 ± 0.0006 [45] A 0.806 ± 0.020 [45] ρ 0.132 ± 0.049 [45] η 0.369 ± 0.050 [45] α 0.642 ± 0.013 [46] m c (m c ) 1.275 ± 0.025 GeV [46] m b (m b ) 4.18 ± 0.03 GeV [46]  We find good agreement between our prediction and the measurement, which is not surprising given the substantial uncertainties that affect both the prediction of the branching ratio and the experimental measurement of the same.

Summary and Outlook
In the present article we have investigated the phenomenological potential of the rare decay Λ b → Λ + − with a subsequent, self-analyzing Λ → N π transition. From the kinematics of the primary and secondary decay we have worked out the fully differential decay width that follows from the Standard Model (SM) operator basis for radiative b → s transitions and its chirality-flipped counterpart which may be relevant for physics beyond the SM. Similar to the corresponding mesonic decay, B → (K * → Kπ) + − , the differential decay width can be expressed in terms of 10 angular observables. In the (naive) factorization approximation, these can be conveniently expressed in terms of short-distance Wilson coefficients and hadronic transition form factors in the helicity or transversity basis. Exploiting the simplifications that arise in the heavy b-quark mass limit -noteably the form-factor relations that arise in the framework of heavy-quark effective theory for low recoil, or soft-collinear effective theory for large recoil -we have discussed the phenomenological consequences for some interesting observables: In the SM the fraction of transverse dilepton polarization, and various forward-backward asymmetries in the leptonic or baryonic variables show a characteristic dependence on the leptonic invariant mass q 2 which can be confronted with experimental data.
Numerical predictions for these observables have been obtained on the basis of a careful statistical analysis of the presently available estimates of hadronic input parameters and their (correlated) uncertainties. We have also identified a number of ratios of angular observables where either the short-distance Wilson coefficients or the long-distance form factors drop out to first approximation. In particular, as a consequence of the parityviolating nature of the secondary decay, we have found that the angular analysis of the Λ b → Λ(→ N π) + − decay is sensitive to combinations of Wilson coefficients that cannot be directly tested in B → K * (→ Kπ) + − decays. Future experimental information on these ratios can thus be used to complement the on-going search for new physics from rare radiative b → s transitions. Compared to the mesonic counterpart decays, B → K ( * ) + − , both the theoretical and experimental situation is not yet competitive: Detailed experimental information, in particular for the large-recoil region, is still lacking at the moment; from the theoretical side, a systematic analysis of non-factorizable hadronic effects (i.e. not form-factor like) is still missing. Both issues are expected to be (at least partially) solved in the future, and the decay Λ b → Λ(→ N π) + − can thus play an important role in the flavour-physics program at the Large Hadron Collider.

A Corrections to HQET Form Factor Relations
In the low-recoil region, the Λ b → Λ form factors are related by HQET spin symmetries in the heavy-quark limit. We follow the analysis in [52] and take into account sub-leading terms in α s and 1/m b appearing in the matching of QCD currents to HQET. For the vector and axial-vector currents this amounts tō and for the tensor and pseudotensor currents one has Here the leading-power matching coefficients at NLO read The hadronic matrix elements of these currents can then parametrized in terms of leading and sub-leading Isgur-Wise functions, denoted as ξ n ≡ ξ n (v · k) and χ m ≡ χ m (v · k), respectively: Here, the independent Dirac structures are given by for the leading-power terms, and mass [GeV 2 ] 5.367 unknown 5.415 5.829 Table 2. List of low-lying B s resonances for the transition form factor Λ b → Λ as taken from [46] for the terms at subleading power. For the physical form factors this translates into

B Form Factor Parametrisation
The form factors for Λ b → Λ transitions can be treated in a similar way as those for e.g. B → K ( * ) transitions. To this end, one considers the analytic continuation into the complex t plane, where q 2 = Re {t}. For 0 ≤ q 2 ≤ t − (semileptonic domain) the form factors describe the semileptonic decay region, Λ b → Λ. For q 2 ≥ t + they describe the pair-production process |0 → Λ b Λ (pair production domain). Here t ± = (m Λ b ± m Λ ) 2 , and |0 denotes the hadronic vacuum. Inbetween the two domains (i.e., for t − ≤ q 2 ≤ t + ) exists a region where the unphysical process Λ b → ΛH contributes. Here the low-lying resonance H with mass m H denotes any hadron with one bottom quark and one strange anti-quark that fulfills A summary of these resonances and their spin-parity quantum numbers is given in table 2.
The contribution of the corresponding poles in the complex plane can be included in the parametrization of the form factors as follows (see e.g. [53,54]), for λ = 0, ⊥. Here the resonance factor is defined as and the remaining q 2 -dependence is obtained from a Taylor series in the variable which corresponds to a conformal mapping of the complex t plane onto the unit disc |z| ≤ 1.
Given the currently available numerical precision for the form factors, it is sufficient to truncate the expansion after the first order. For the auxiliary parameter t 0 we choose t 0 = 12 GeV 2 .

C Statistical Analysis of the Transition Form Factors
This appendix is dedicated to details of our fit of the hadronic transition form factors. Specifically, we undertake the following steps.
(a) The available lattice results [11] provide information on the HQET form factors ξ 1 and ξ 2 in the interval q 2 ∈ [13.5, 20.5] GeV 2 (low recoil energy). This includes correlations between form-factor values at different values of q 2 which -non-surprisingly -are sizeable (up to 100% for form factors related to the same current at adjacent points). For a stable numerical analysis we therefore only consider the two q 2 values at the very boundaries of the simulated range which are expected to have minimal correlation.
To translate this into estimates for the helicity form factors f V,A ⊥,0 (q 2 ), we allow for corrections to the HQET limit from subleading form factors χ n=1...6 , see appendix A. With no detailed information on the latter -following the principle of maximum entropy [55] -we use Gaussian priors, i.e. a normal distribution centered around zero with variance σ = 1 from naive power-counting. Notice that by construction this leads to prior distributions with flat q 2 -dependence.
The present lattice analysis do not provide correlation information between the two individual HQET form factors. Given the large correlations between individual q 2 points, we expect these correlations to be sizeable, too. As a consequence, we cannot completely determine the correlation between the data points for, say, the physical vector form factor. Notice that the correlation is further enhanced by the flat priors for the subleading contributions χ n . This issue can be solved as soon as improved lattice results for the physical form factors beyond the HQET limit will be available.
The result of this part of the analysis is summarized in table 3. However, we find that the estimates correlations in excess of 97% lead to a degeneracy of the parameters in our fits. Hereafter, our incomplete estimates of the correlation between q 2 points are therefore disregarded. , based on probability distributions obtained from lattice points [11] in the HQET limit and Gaussian priors for the subleading IW form factors. See text for details.  Table 4. Estimates of mean values and standard deviations for ratios of helicity form factors R Γ,Γ λ , based on probability distributions obtained from lattice points [11] in the HQET limit and Gaussian priors for the subleading IW form factors. See text for details.
(b) As explained in the main text, the form factors for tensor and pseudotensor currents enter the observables in the form of ratios with the corresponding vector or axialvector form factors. We therefore derive probability distributions for the ratios (λ = 0, ⊥, Γ = T, T 5,Γ = V, A) again based on the lattice results [11] at the boundaries q 2 = 13.5 GeV 2 and q 2 = 20.5 GeV 2 . In the heavy-quark limit these ratios are unity. Deviations are estimated by the same implementation of sub-leading corrections as described in (a).
The result of this part of the analysis is summarized in table 4.
Notice, that the ratios R Γ,Γ λ are only very poorly constrained towards smaller values of q 2 . More precise inputs on these quantities directly from the lattice are desirable.
The high degree of correlation between q 2 points leads us to parametrize these ratios through linear interpolation in q 2 , and one random number r for their uncertainty. We use with fixed parameters a Γ,Γ λ through d Γ,Γ λ as given in table 4. In the above, r Γ,Γ λ ∼ N (0, σ Γ,Γ λ ).
(c) The above results can be continued to the low-q 2 region by taking into account information from sum-rule analyses (see e.g. [12,56,57]) to determine probability distributions for the helicity form factors f V,A ⊥ (q 2 = 0) and f V,A 0 (q 2 = 0) at maximal recoil. We have f where we describe the soft form factor by a normal distribution ξ Λ (0) ∼ N (µ = 0.38, σ = 0.19) where, for concreteness, we have used the estimates from a SCET sum rule obtained in [12]. We further allow for two independent (ad-hoc) correction factors ζ λ for the two helicities λ = 0, ⊥ whose probability distributions are modelled as N (µ = 0, σ = 0.25).
The helicity form factors at q 2 = 0 are to good approximation multivariate-normally distributed. The mean values and standard deviations read The form factors are strongly correlated, and we find for the correlation matrix Based on the inputs discussed above in points (a) and (c), we can now fit the helicity form factors for the vector and axial vector currents to our theory estimates. We use the parametrization given in eq. (B.2), which features two parameters f Γ λ (0) and b Γ λ per form factor. The overall set of fit parameters therefore reads In our fit we use a uniform prior P 0 ( x) that is supported on for all λ =⊥, 0 and Γ = V, A. The likelihood for the form factor estimates P (Estimates| x) factorizes into P (Estimates| x) = P (HQET| x) × P (SSR| x) . (C.8) The HQET components are where µ HQET,Γ,λ and Σ HQET,Γ,λ can be read off table 3, and F Γ λ is an abbreviation for where µ SSR and Σ SSR can be read off eq. (C.4) and eq. (C.5). In the above, we denote the form factor parametrization as )) . (C.12) As usual the posterior P ( x|Estimates) follows from Bayes' theorem as P ( x|Estimates) = P (Estimates| x)P 0 ( x) d xP (Estimates| x)P 0 ( x) . (C.13) We find the best-fit point This is a good fit, with the largest pull value being 0.05σ. We find χ 2 = 5.4 · 10 −3 , and with N d.o.f. = 4 degrees of freedom (from 12 observations reduced by 8 fit parameters) this yields a p value of > 0.99.