Exclusive semileptonic $B \to \pi \ell \nu_\ell$ and $B_s \to K \ell \nu_\ell$ decays through unitarity and lattice QCD

The Cabibbo-Kobayashi-Maskawa (CKM) matrix element $\vert V_{ub}\vert$ is obtained from exclusive semileptonic $B \to \pi \ell \nu_\ell$ and $B_s \to K \ell \nu_\ell$ decays adopting the unitarity-based dispersion matrix approach for the determination of the hadronic form factors (FFs) in the whole kinematical range. We use lattice computations of the relevant susceptibilities and of the FFs in the large-$q^2$ regime in order to derive their behavior in the low-$q^2$ region without assuming any specific momentum dependence and without constraining their shape using experimental data. Then, we address the extraction of $\vert V_{ub}\vert$ from the experimental data, obtaining $\vert V_{ub}\vert = (3.62 \pm 0.47) \cdot 10^{-3}$ from $B \to \pi$ and $\vert V_{ub}\vert = (3.77 \pm 0.48) \cdot 10^{-3}$ from $B_s \to K$, which after averaging yield $\vert V_{ub}\vert = (3.69 \pm 0.34) \cdot 10^{-3}$. These results are compatible with the most recent inclusive value $\vert V_{ub} \vert_{incl} = 4.13\,(26) \cdot 10^{-3}$ at the 1$\sigma$ level. We also present purely theoretical estimates of the ratio of the $\tau/\mu$ decay rates $R^{\tau/\mu}_{\pi(K)}$, the normalized forward-backward asymmetry $\bar{\mathcal{A}}_{FB}^{\ell,\pi(K)}$ and the normalized lepton polarization asymmetry $\bar{\mathcal{A}}_{polar}^{\ell,\pi(K)}$.


I. INTRODUCTION
Since many years the heavy-to-light semileptonic transitions are very intriguing processes mainly because a long-standing tension affects the inclusive and the exclusive determinations of the CKM matrix element |V ub |.The most recent version of the FLAG report [1] quotes for the exclusive estimate of |V ub | the value |V ub | excl • 10 3 = 3.74 (17) from B → π ν decays, while the inclusive determination performed by HFLAV [2] reads ), implying a ∼ 3σ discrepancy between them.However, a recent measurement of the inclusive value of |V ub | made by Belle [3] has changed the picture.In fact, the collaboration has presented the result of an average over four theoretical calculations (BLNP [4], DGE [5,6], GGOU [7], ADFR [8,9]), which reads where the first two errors represent the statistical and systematic uncertainties respectively, the third one denotes the theoretical model uncertainty and the fourth one is their sum in quadrature.The FLAG review [1] quotes the inclusive value |V ub | incl • 10 3 = 4.32 (29), which does not include the Belle result (1), but takes into account in the error the spread among various theoretical calculations.The FLAG inclusive value differs from the exclusive one by 1.7 standard deviations.The last PDG review [10] includes both the recent Belle result and the spread among various theoretical calculations.For the exclusive and inclusive determinations of |V ub | the PDG [10] quotes the values |V ub | excl • 10 3 = 3.70 (10) exp (12) th = 3.70 (16) from B → π ν decays and |V ub | incl • 10 3 = 4.13 (12) exp ( +13 −14 ) th (18) model = 4.13 (26), which differ by 1.4 standard deviations.New analyses of the exclusive b → u transitions, claiming that their exclusive determinations of |V ub | are consistent with the estimate (1) at the 1 ÷ 1.5 σ level, also appeared [11][12][13].Note that the latter results were obtained by adopting for the hadronic Form Factors (FFs) the Bourrely-Caprini-Lellouch (BCL) [14] or the Bharucha-Straub-Zwicky (BSZ) [15] parameterizations or the Padé approximants [16].
In this work our aim is to re-examine the b → u transition through the Dispersive Matrix (DM) method, originally proposed in Ref. [17] and recently reapprised in Ref. [18].
The DM method can be applied to any semileptonic decays once lattice QCD (LQCD) computations of the relevant susceptibilities and of the FFs are available.As for the susceptibilities, we present here their computation for b → u transitions following the same strategy and the same gauge ensembles considered in the case of the b → c transition in Ref. [19].The FFs, instead, are taken from the results of the RBC/UKQCD [20] and FNAL/MILC [21] Collaborations for the B → π ν decays, and from RBC/UKQCD [20], HPQCD [22] and FNAL/MILC [23] Collaborations for the B s → K ν decays.As already done for the analysis of the exclusive B → D ( * ) decays [24,25], we stress that only LQCD computations of the FFs for small values of the recoil will be used to determine the shape of the FFs in the whole kinematical range without making any assumption on their momentum dependence.Moreover, the experimental data are not used to constrain the shape of the FFs, but only to obtain the final exclusive determination of In this way, our calculation of the FFs allows to obtain pure theoretical estimates of several quantities of phenomenological interest, namely the τ /µ ratio of the decay rates R τ /µ π(K) , which is important for testing Lepton Flavour Universality (LFU), the normalized forward-backward asymmetry Ā ,π(K) F B and the normalized lepton polarization asymmetry Ā ,π (K)  polar .
The paper is organized as follows.In Section II we review the main properties of the DM method [18].In Section III we apply our procedure to predict the FFs of interest in the whole kinematical range relevant for the semileptonic B → π and B s → K decays.
The non-perturbative computation of the unitarity bounds for the b → u (and as a by-product for the c → d) transition is based on suitable lattice two-point correlation functions, evaluated using the gauge configurations produced by the Extended Twisted Mass Collaboration (ETMC), and it is presented in the Appendix A. Then, the experimental data are used to determine |V ub | from a bin-per-bin analysis in the case of the B → π ν decays and from the total branching ratio for the B s → K ν decays.In Section IV we investigate the issue of LFU by evaluating the ratio of the τ /µ decay rates R τ /µ π(K) from theory.We determine also the forward-backward Ā ,π(K) F B and lepton polarization Ā ,π(K) polar asymmetries.Finally, in Section V we summarize the main results of this work and sketch possible future developments in the extraction of |V ub | from exclusive semileptonic decays.

II. THE DM METHOD
In this Section we review the main properties of the non-perturbative DM approach to the description of the semileptonic FFs, proposed in Ref. [18] and already applied to the study of B → D ( * ) ν decays in Refs.[24,25].

A. The unitarity bounds on the FFs
The dispersion relation for a given spin-parity channel can be written in a compact form as [26-28] where f (z) is the FF of interest, φ(z, q 2 0 ) is a kinematical function (whose definition depends on the spin-parity channel), χ(q 2 0 ) is related to the derivative of the Fourier transform of suitable Green functions of bilinear quark operators [27] and q 2 0 is an auxiliary value of the squared 4-momentum transfer.Hereafter, we will refer to the functions χ(q 2 0 ) as the susceptibilities.
By introducing the inner product defined as [17,29] where ḡ(z) is the complex conjugate of the function g(z), Eq. ( 2) can be also written as As for the B → D ( * ) case [24,25], in this work we limit ourselves to the case q 2 0 = 0 and we postpone the discussion of the phenomenological implications of the choice q 2 0 = 0 to a forthcoming work.Following Refs.[17,29] we introduce the set of functions where z is the integration variable of Eqs. ( 2)-(3) and z(t) is the complex conjugate of the conformal variable z(t), defined as with t ≡ q 2 being the squared 4-momentum transfer and Using the Cauchy's theorem one has .
The central ingredient of the DM method is the matrix [17,29] where t 1 , . . ., t N are the values of the squared 4-momentum transfer at which the FF f (z) is known.In the DM method we consider only values f (z(t i )) (with i = 1, 2, ...N ) computed nonperturbatively on the lattice.
The important feature of the matrix M is that, thanks to the positivity of the inner products, its determinant is positive semidefinite, i.e. det M ≥ 0. This property is not modified when the first matrix element in Eq. ( 7) is replaced by the susceptibility χ(q 2 0 ) through the dispersion relation (2).Thus, using also the fact both z and f (z) can assume only real values in the allowed kinematical region for semileptonic decays, the original matrix (7) can be replaced explicitly by where corresponding to the given set of values z i .Furthermore, in order to simplify the notation we indicate z and the corresponding unknown value φf as z 0 and φ 0 f 0 ≡ φ(z 0 )f (z 0 ), respectively, so that the index i now runs from 0 to N .
By imposing the positivity of the determinant of the matrix (8) it is possible to compute explicitly the lower and the upper unitarity bounds for the FFs of interest, namely [18] where Unitarity is satisfied only when γ ≥ 0, which implies χ ≥ χ DM .Since χ DM does not depend on z 0 , the above condition is either never verified or always verified for any value of z 0 .This means that the unitarity filter χ ≥ χ DM represents a parameterizationindependent test of unitarity for a given set of input values f j of the FF f .
We remind also an important feature of the DM approach.When z 0 coincides with one of the data points, i.e. z 0 → z j , one has β → f j and γ → 0. In other words the DM method reproduces exactly the given set of data points.This is at variance with what may happen using truncated BCL parametrisations, since there is no guarantee that such parametrizations can reproduce exactly the set of input data.Thus, it is worthwhile to highlight the following important feature: the DM band given by Eqs. ( 9)-( 14) is equivalent to the results of all possible fits which satisfy unitarity and at the same time reproduce exactly the input data.
B. The kinematical constraint at q 2 = 0 In the semileptonic B → π and B s → K decays, there are two FFs the vector f + (q 2 ) and the scalar f 0 (q 2 ) one, which are related at zero 4-momentum transfer by the following kinematical constraint (KC) As in Ref. [17], we consider where in terms of Eq. ( 9) one has f +(0),lo(up We now consider the FF at zero 4-momentum transfer to be uniformly distributed in the range given by Eq. (15).The resulting value is considered as a new input at t N +1 = 0.
Thus, for each of the two FFs we consider a new matrix, M KC , that has one more row and one more column with respect to M in order to contain the common value In order to predict the DM bands for f +,0 (q 2 ) in the whole kinematical range, we consider the matrix M KC at any value of the momentum transfer and, by using the explicit forms ( 9)-( 14), we get the corresponding unitarity bounds.
Finally, as discussed in Refs.[18,24,25], we use the mean values, the uncertainties and (when available) the correlations of the LQCD computations of the FFs and the susceptibilities to construct a multivariate Gaussian distribution for generating a sample of bootstrap events to each of which the DM method is applied.
In this Section we apply the DM method to the study of the semileptonic B → π and B s → K decays.First we describe the state of the art of the LQCD computations of the relevant FFs, which are limited to large values of the 4-momentum transfer q 2 , and then we apply the DM method to get the FFs in the whole kinematical range accessible to experiments.To this end another nonperturbative input is used, namely the values of the longitudinal and transverse vector susceptibilities, χ 0 + (0) and χ 1 − (0), whose determination based on suitable lattice two-point correlation functions for the b → u transition is illustrated in the Appendix A. Finally, we compare our theoretical results with the experimental data in order to extract |V ub | from the semileptonic B → π and B s → K channels.

A. State of the art of the LQCD computations of the FFs
The FFs entering semileptonic B → π decays have been studied by the RBC/ UKQCD [20] and the FNAL/MILC [21] Collaborations.In the case of the B s → K transition several LQCD computations of the FFs are available, namely from the RBC/UKQCD [20], HPQCD [22] and FNAL/MILC [23] Collaborations.For both channels the lattice computations of the FFs are available in the large-q 2 region, 17 GeV 2 The authors of Ref. [20] provide synthetic LQCD values of the FFs (together with their statistical and systematic correlations) at three values of q 2 in the large-q 2 regime, namely q 2 = {19.0,22.6, 25.1} GeV 2 for the B → π transition and q 2 = {17.6,20.8, 23.4} GeV 2 in the case of the B s → K transition.These data can be directly used as inputs for our DM method.In the other works [21][22][23] the results of BCL fits of the FFs extrapolated to the continuum limit and to the physical pion point are available.Thus, from the marginalized BCL coefficients we evaluate the mean values, uncertainties and correlations of the FFs at the three values of q 2 given in Ref. [20].The LQCD results used as inputs for our DM method are collected in Tables I and II for the B → π and B s → K decays, respectively.In the next future new LQCD computations of the FFs are expected to become available [30,31].
For both B → π and B s → K decays we have also combined all the LQCD determinations of the FFs corresponding to the same values of the momentum transfer.We have followed the procedure already applied in Ref. [24] to the B → D * case: starting from N computations of the FFs with mean values x corresponding to a given value q 2 i of the squared 4-momentum transfer, the combined LQCD average x i and uncertainty σ i are given by (see Ref. [32]) where ω (k) represents the weight associated to the k-th calculation ( N k=1 ω (k) = 1).Since the uncertainties of the various lattice computations are comparable, in what follows we assume the same weight for all the computations, i.e. we consider the simple choice ω (k) = 1/N .The results of Eqs. ( 17)-( 18) are shown in the last columns of the Tables I and II for both the B → π and the B s → K cases, respectively.Moreover, the covariance matrix C of the combined data can be easily evaluated in terms of the covariance matrices C (k) of each single LQCD computation as where the indices i and j run over the number of values of the 4-momentum transfer at which the LQCD computations of the FFs have been performed, namely in this work i, j = 1, 2, 3.

B. Theoretical expression of the differential decay width
For the semileptonic B → π ν and the B s → K ν decays the vector f and scalar f π(K) 0 (q 2 ) FFs are related to the matrix elements of the weak vector current where . We remind that the two FFs in Eq. ( 20) are constrained at zero momentum transfer by the kinematical relation f Mean values and uncertainties of the LQCD computations of the FFs f π +,0 (q 2 ) obtained at three selected values of q 2 from the results of the RBC/UKQCD [20] and FNAL/MILC [21] Collaborations.For the RBC/UKQCD computations the first error is statistical while the second one is systematic.The last column contains the results of the combination procedure given in Eqs. ( 17)- (18) with ω (k) = 1/N .II.Mean values and uncertainties of the LQCD computations of the FFs f K +,0 (q 2 ) obtained at three selected values of q 2 from the results of the RBC/UKQCD [20], HPQCD [22] and FNAL/MILC [23] Collaborations.For the RBC/UKQCD computations the first error is statistical while the second one is systematic.The last column contains the results of the combination procedure given in Eqs. ( 17)- (18) with ω (k) = 1/N .

A direct computation of the two-fold differential decay width within the Standard
Model gives the final expression where G F is the Fermi constant, p π(K) the 3-momentum of the π(K) meson in the B (s) -meson rest frame, m the mass of the produced lepton and θ represents the angle between the final charged lepton and the B (s) -meson momenta in the rest frame of the final state leptons.By integrating out the dependence on the angle θ one gets where explicitly C. Application of the DM method to the description of the FFs The kinematical functions φ 0 and φ + corresponding to the scalar and vector FFs of the B (s) → π(K) decays are given by [27] where z ≡ z(t = q 2 ) is defined in Eq. ( 5) and n I is an isospin Clebsh-Gordan factor equal to n I = 3/2 for the B → π decays and to n I = 1 for the B s → K case.In order to take into account the B * pole in the transverse channel, the transverse kinematical function φ + is modified as with m B * = 5.325 GeV from the PDG [10].
The evaluation of the unitarity bound χ(0) ≥ χ DM (see Eq. ( 12)) requires the knowledge of the susceptibilities χ(0) appearing in the DM matrix (8).For the b → u transition we have been computed them nonperturbatively using suitable two-point lattice correlators, as described in the Appendix A. The nonperturbative values for the susceptibilities relevant for the scalar f 0 (q 2 ) and vector f + (q 2 ) FFs are respectively after subtraction of the contribution of the B * -meson bound state (see Appendix A).
We now apply the DM method to the B → π decay using as inputs the lattice data of Table I corresponding to the three sets labelled RBC/UKQCD, FNAL/MILC and combined.A total of 5 • 10 4 events are generated using the multivariate Gaussian distribution including the correlations among the LQCD data.It turns out that the unitarity bounds for both f π 0 and f π + as well as the KC f π 0 (0) = f π + (0) ≡ f π (0) are satisfied by 98 ÷ 100% of the events and, therefore, neither the skeptical nor the iterative procedures described in Refs.[18,24,25] need to be applied.In Figs. 1 and 2 we show the resulting bands of the two FFs.The extrapolation to q 2 = 0, which is crucial in order to analyze the experimental data, reads The above results exhibit large uncertainties due to the long extrapolation from the high-q 2 region of the input data down to q 2 = 0. We stress again that our results do not depend on any parameterization of the shape of the FFs.This is at variance with what happens with the BCL parameterizations of Refs.[20,21], where the extrapolated mean values and uncertainties of the FFs at q 2 = 0 are plagued by instabilities with respect to the order of the truncation of the expansion.
The scalar f π 0 (q 2 ) (left panel) and vector f π + (q 2 ) (right panel) FFs entering the semileptonic B → π ν decays computed by the DM method as a function of the 4-momentum transfer q 2 using the LQCD inputs from RBC/UKQCD [20] and FNAL/MILC [21] Collaborations (see Table I).For both FFs the red and blue bands correspond to the DM results obtained at 1σ level using the RBC/UKQCD data (red circles) and FNAL/MILC (blue squares) data, respectively.
In the right panel the vector FF is multiplied by the factor As for the semileptonic B s → K decays few differences have to be considered with The bands of the scalar f π 0 (q 2 ) (left panel) and vector f π + (q 2 ) (right panel) FFs entering the semileptonic B → π ν decays computed by the DM method at 1σ level using as lattice inputs the combined LQCD data of Table I, shown as green diamonds.In the right panel the vector FF is multiplied by the factor respect to the B → π case besides the obvious changes in the masses of the mesons involved.First, in the kinematical functions (24) the isospin factor n I is now equal to unity instead of 3/2 as in the B → π case.This is due to the fact that in the B s → K decays only the strange quark can be the spectator quark of the transition.Second, following Refs.[20,22,23] a modification like the one in the Eq. ( 25) has to be applied also to φ 0 (z, 0) due to the presence of a scalar resonance B * (0 + ) with a mass close to 5.68 GeV, expected from the lattice results of Ref. [33], lying below the pair production threshold located at M Bs + M K 5.86 GeV.For the susceptibilities χ 0 + (0) and χ 1 − (0) we adopt conservatively the same values of the B → π case.
We apply the DM method using as inputs the various sets of LQCD data of Table II.
A total of 5 • 10 4 events are generated using the multivariate Gaussian distributions including the correlations among the LQCD computations.As in the B → π case, the unitarity bounds for both f K 0 and f K + as well as the KC f K 0 (0) = f K + (0) ≡ f K (0) are satisfied by 98 ÷ 100% of the events.The DM bands for the FFs corresponding to the use of the combined LQCD data of Table II are shown in Fig. 3.Note the impact of the KC at q 2 = 0 on the extrapolation of the FFs in the low-q 2 region.
3. The bands of the scalar f K 0 (q 2 ) (left panel) and vector f K + (q 2 ) (right panel) FFs entering the semileptonic B s → K ν decays computed by the DM method at 1σ level using as inputs the combined LQCD data of Table II, shown as green diamonds.In the left panel the scalar FF is multiplied by the factor (1 − q 2 /m 2 B * 0 ) with m B * 0 = 5.68 GeV, while in the right panel the vector FF is multiplied by the factor (1 − q 2 /m 2 B * ) with m B * = 5.325 GeV.
The extrapolation of the FFs to q 2 = 0 reads The above results can be compared with the recent LCSR estimate of Ref. [34], which is It can be seen that the results based on the RBC/UKQCD and FNAL/MILC data differ respectively by 1.7 and 2.4 standard deviations from the LCSR estimate, while the results based on the HPQCD data and the combined LQCD ones are in agreement thanks to larger mean values and uncertainties.
The DM results presented so far indicate clearly that for both the B → π and B s → K channels the extension of direct LQCD computations of the FFs toward values of q 2 lower than ∼ 17 GeV 2 is crucial for improving the precision of their extrapolation to q 2 = 0 without resorting to the use of the experimental data.Collaborations [35][36][37][38] have measured the differential branching ratios (BRs) in different bins of the 4-momentum transfer q 2 .Instead, for the B s → K decays only the ratio of the total branching fractions of the semileptonic B s → K and B s → D s decays is available at present [39].
For the extraction of the CKM matrix element we follow the procedure used in Refs. [24,40] in the case of several semileptonic heavy-meson decays characterized by the production of a final pseudoscalar meson.In what follows, we will distinguish the two different channels that have been measured by the experiments, i.e.B 0 → π − + ν and B + → π 0 + ν with = e, µ.Starting from the Eq. ( 22), for the generic i-th bin in q 2 we have [38] |V where ∆B| exp i is the experimental branching fraction and ∆ζ i the corresponding theoretical decay width (without |V ub | therein) in the given bin.Since the isospin coefficient C v is equal to 2 for the B + → π 0 + ν decays and to 1 for the B 0 → π − + ν transitions.Finally, τ B v is the lifetime of the decaying B-meson.
Our procedure can be summarised as follows: • using the mean values and the covariance matrices available for the of the FFs and of the susceptibilities computed in LQCD we generate a multivariate Gaussian distribution of events of input data to each of which the DM method is applied for obtaining the subset of events passing the unitarity filter and satisfying the KC at q 2 = 0 (see Section III C); • for each of the surviving events we evaluate the vector FF f + (q 2 ) at several values of q 2 , which allow to perform the partial integration needed to calculate the theoretical differential decay width ∆ζ i in each of the experimental q 2 -bins (in the massless lepton limit); • from the resulting distribution of values of ∆ζ i we evaluate the mean values ∆ζ i for each bin and the corresponding covariance matrix; • through multivariate Gaussian distributions we generate N boot events of the measured differential branching fraction ∆B| exp i for each bin in q 2 and experiment [35][36][37][38] and, independently, N boot events of the theoretical decay widths ∆ζ i ; • we compute N boot values |V ub | i for each q 2 -bin and each experiment through Eq. ( 28); • using the distributions of values of |V ub | i we calculate the corresponding mean values |V ub | i and covariance matrix C ij among the bins for each experiment; • we evaluate the CKM matrix element |V ub | for the n-th experiment (n = 1, . . ., 6 for the semileptonic B → π decays) as the best constant fit over all the bins of the given experiment, i.e. through the following formulae where the indices i, j run over all the q 2 -bins of the n-th experiment.
In Fig. 4 BaBar 2011 average Belle 2011 average Refs.[35][36][37][38] specified in the insets of the panels as a function of q 2 .The theoretical DM bands of the FFs correspond to the use of the combined LQCD data of Table I  As shown in Figs. 1 and 2, the form factor f + (q 2 ), which is the only one contributing to the decay rate in the limit of massless leptons, may become numerically very small (in absolute value) below q 2 ≈ 10 GeV 2 .Since the theoretical decay rate appears in the denominator of Eq. ( 28), the resulting values of |V ub | for the bins corresponding to q 2 10 GeV 2 exhibit a tendency to larger values.However, the uncertainties are quite large for those bins (due to the present uncertainties of the input lattice data for f + (q 2 ) and to the long extrapolation to low values of q 2 ) and, therefore, for each experiment the average ( 29) is dominated by the contributions of the large-q 2 bins.Direct lattice calculations at smaller values of q 2 will allow in the future to clarify this point.
Our final results for |V ub |, evaluated making use of the averaging procedure given by Eqs. ( 17)-( 18), read which are consistent with the latest exclusive determination |V ub | excl • 10 3 = 3.70 (16) from PDG [10].Our uncertainties are larger than the PDG one, because we do not mix the theoretical calculations of the FFs with the experimental data to constrain the shape of the FFs in order to avoid possible biases.We are currently investigating strategies to improve the precision of the determination of |V ub | within our DM approach.
The LHCb Collaboration has observed for the first time the semileptonic B s → K ν decays [39] and measured the ratio of the branching fractions of the B 0 s → K − µ + ν µ and the B 0 s → D − s µ + ν µ processes, R BF (high) = (3.25 ± 0.21 +0.16 −0.17 ± 0.09) where the first error is statistical, the second one is systematic and the third one is due to the uncertainty on the In order to obtain an exclusive estimate of |V ub | we make use of the life time of the where the first error is statistical, the second one is systematic and the third one is due to limited knowledge of the normalization branching fractions.
Then, we use the FFs obtained with our DM method to compute the differential decay width dΓ/dq ) and high (q 2 ≥ 7 GeV 2 ) q 2 -bins using the DM bands for the theoretical FFs.
averages of the two bins, carried out following Eqs.( 17)-( 18) for each set of FFs, read which are consistent with our results (30), obtained from the analysis of the B → π ν decays, and with the latest exclusive determination |V ub | excl •10 3 = 3.70 ( 16) from PDG [10].
We remind that the PDG uncertainty results from analyses in which theoretical calculations of the FFs and experimental data are mixed in order to constrain the shape of the FFs.
In this Section we give pure theoretical estimates of various quantities of phenomenological interest, which are independent of |V ub |, namely the ratio of the τ /µ decay rates and the normalized lepton where with m being the lepton mass ( = τ, µ) and The forward-backward asymmetry is defined as where from Eq. ( 21) one has Then, the normalized forward-backward asymmetry Ā ,π(K) F B is given by We compute also the lepton polarization asymmetry A ,π(K) polar defined as where [42] dΓ The normalized lepton polarization asymmetry Ā ,π(K) polar is given by In Tables V and VI we collect our theoretical estimates of the quantities (34)(35)(36) for each set of LQCD computations of the FFs in the case of the B → π and B s → K decays, respectively.Within the our results are consistent with recent estimates [11,43,44] which still has a large uncertainty compared to our theoretical ones.Note that the uncertainty on the above ratio expected by Belle II at 50 ab of luminosity [46] is δR τ /µ π 0.09, which will be comparable to our present theoretical uncertainties.
Since it is likely that experimental measurements of R and Ā ,π(K) polar will be carried out in limited regions of the phase space, we provide in Appendix B our theoretical estimated of these quantities in three selected q 2 regions.

V. CONCLUSIONS
In this work we have analysed the available lattice and experimental data concerning the semileptonic B → π and B s → K decays.We have obtained new exclusive estimates of the CKM matrix element |V ub | in a rigorous model-independent way in order to shed a new light onto the tension between its inclusive and exclusive determinations.This has been achieved by evaluating the semileptonic FFs according to the non-perturbative and model-independent DM method proposed in Ref. [18] and by computing for the first time non-perturbatively the susceptibilities relevant for the unitarity bounds in the b → u transition.
Our results for |V ub | can be summarized as where the quantities Π j (Q 2 ) with j = {0 + , 1 − , 0 − , 1 + } are the vacuum polarization functions corresponding to definite spin-parity channels (see for more details Refs.[18,19]), Q is an Euclidean 4-momentum, j 0 (x) = sin(x)/x and j 1 (x) = [sin(x)/x−cos(x)]/x are spherical Bessel functions and the Euclidean correlators C j (t) are given by with u(x) representing the light-quark u field.Note that the longitudinal (first) derivatives (A1) and (A3) are dimensionless, while the transverse (second) ones (A2) and (A4) , where E is an energy.
As shown in Ref. [18], Eqs.(A1)-(A4) are obtained in the Euclidean region Q 2 ≥ 0, but they can be easily generalized also to the case Q 2 < 0. In the Euclidean region Q 2 ≥ 0 a good convergence of the perturbative calculation of the above derivatives is expected to occur far from the kinematical regions where resonances can contribute.In the case of the b → u weak transition this means down to Q 2 = 0 [17] and, indeed, this is the value of Q 2 that has been generally employed in the evaluation of the dispersive bounds for heavy-to-light [14,17] and also for heavy-to-heavy [27,28,[49][50][51]] semileptonic form factors.By contrast, with a non-perturbative determination of the two-point correlation functions we can use the most convenient value of Q 2 at disposal, namely the value which will allow the most stringent bounds on the semileptonic form factors.In this work we will limit ourselves to the usual choice Q 2 = 0, which will allow the comparison with perturbative results, and we will leave the investigation of the choice Q 2 = 0 to a future, separate work.
At Q 2 = 0 the derivatives of the longitudinal and transverse polarization functions correspond to the second and fourth moments of the longitudinal and transverse Eu-     [56,57]), but the ratios of the transverse factors (A23) appearing in Eq. (A21) turn out to be almost insensitive to such high-order corrections.
Thanks to the large correlation between the numerator and the denominator in Eq. (A21) the statistical uncertainty of the ETMC ratios R j (n; a 2 , m ud ) is much smaller than those of the separate susceptibilities and it may reach the permille level, as it is shown in Fig. 6 in the case j = 1 − and n = 5 as an illustrative example.r-parameters should differ only by discretization effects (at least of order O(a 2 ) in our maximally twisted setup).This means that the value of R j (n) should be independent of the choice of the Wilson r-parameters.The conclusion is that the Anstaz (A25) is not sufficient for describing the lattice data, since discretization effects beyond the order O(a 2 ) should be taken into account.Following Ref. [19] a possible option is to add a term proportional to a 4 , namely Since our lattice setup includes only three values of the lattice spacing, it would be reasonable to expect that Eq. (A26) would require the use of a (gaussian) prior on the two parameters D 1 and D 2 .However, at variance with the case of the b → c transition analyzed in Ref. [19] there is no need to introduce a prior for describing the discretization effects for the ETMC ratios of the b → u transition.This is clearly illustrated in Fig. 7, where the introduction of a discretization term proportional to a 4 is greatly beneficial for obtaining fits with good quality for both r-combinations.
It turns out that for m h (n) 2.5 GeV the ratios R j (n) corresponding to the two combinations (r, −r) and (r, r) of the Wilson r-parameters and extrapolated to the physical pion point and to the continuum and infinite volume limits agree within the errors, while for m h (n) 2.5 GeV the agreement deteriorates and holds only within ∼ 2.5 standard deviations.Consequently, we enforce that the extrapolated values R j (n) must be independent of the specific r-combination by performing the following combined extrapolation where now only the coefficients D The quality of the combined fitting procedure (A27) is always good for all channels and heavy-quark masses (χ2 /(d.o.f.) 0.7 with 30 data points and 7 free parameters for each of the 32 fits).The results obtained for R j (n) are shown in Fig. 8.

Extrapolation to the b-quark point
The important feature of the ETMC ratio method is that the extrapolation to the physical b-quark point of the ratios R j can be carried out taking advantage of the fact that by construction Thus, we fit the lattice data for the ratios R j (n) adopting the following Ansatz which contains 2M parameters to be determined by a χ 2 -minimization procedure 2 .We respectively.Such limits can be improved by removing the contributions of the bound states lying below the pair production threshold.
Note that in the low-q 2 region, when the τ -lepton is involved, the minimum value of q 2 is equal to m 2 τ .
The results, based on the combined LQCD data of Tables I and II used as inputs for our DM method, are collected in Tables VIII and IX for the B → π and B s → K decays, respectively.It can be seen that large (and even quite large) uncertainties affect the low-q 2 intermediate-q and the normalized lepton polarization asymmetry Ā ,π(K) polar evaluated in the three selected q 2 -bins in the case of the semileptonic B → π ν decays with = µ, τ adopting the combined LQCD data of Table I as inputs for our DM method.
theoretical predictions of some of the quantities in the low-q 2 bin.This is related to the large uncertainties of the hadronic form factors at low values of q 2 (see Figs. 2-3), which are a consequence of the present uncertainties of the input lattice data and of the long extrapolation to low values of q 2 .Direct lattice calculations at smaller values of q 2 will low-q 2 intermediate-q allow in the future to reach a more significative precision also in the low-q 2 bin.

D
. New estimate of |V ub | In order to obtain |V ub | we use our results for the FFs in the whole kinematical range and the experimental data.For the semileptonic B → π decays the BaBar and the Belle we show our results for |V ub | for each of the semileptonic B → π experiments, together with the mean values(29), adopting the DM results for the FFs obtained using as inputs the combined LQCD data of TableI.For each experiment the corresponding

2 )FIG. 4 .
FIG.4.Bin-per-bin estimates of |V ub | obtained using Eq.(28) for each of the six experiments of

1 +FIG. 5 .
FIG.5.The susceptibilities χ 0 + (left top panel), χ 0 − (right top panel), χ 1 − (left bottom panel) and χ 1 + (right bottom panel) corresponding to the gauge ensemble B25.32 for the h → u transitions as a function of the heavy-quark mass m h given by Eq. (A19) up to n = 9, i.e. up to m h (9) 3.9GeV.The red circles correspond to the choice of opposite values (r, −r) of the two valence-quark Wilson parameters, while the blue squares refer to the case of equal values (r, r).

FIG. 6 .
FIG. 6. Light-quark mass dependence of the ratio of the susceptibilities corresponding to Eq. (A21) for j = 1 − and n = 5 for the two combinations (r, −r) (left panel) and (r, r) (right panel) of the Wilson r-parameters.The solid lines represent the results of the fitting function (A25) evaluated in the continuum and infinite volume limits, while the dashed ones correspond to the fitting function evaluated at each value of β and for the largest value of L/a.The crosses represent the value of the ratio extrapolated at the physical pion point (m ud = m phys ud ) and in the continuum and infinite volume limits.

FIG. 7 .
FIG.7.The same as in Fig.6, but here the solid and dashed lines represent the results of the fitting function (A26) evaluated in the continuum and infinite volume limits and at each value of β (for the largest value of L/a).

1 +FIG. 8 .FIG. 9 .
FIG. 8.The susceptibility ratios R 0 + (left top panel), R 0 − (right top panel), R 1 − (left bottom panel) and R 1 + (right bottom panel) after extrapolation to the physical pion point and to the continuum and infinite volume limits based on the combined fit (A27) of the data corresponding to the two combinations (r, −r) and (r, r) of the Wilson r-parameters, versus the inverse heavyquark mass 1/m h .The dashed lines are the results of the fitting procedure (A29), described in the next subsection, and the crosses represent the values of the ratios R j at the physical b-quark point, shown as a vertical dotted line.

TABLE III .
(29)nputs.The black dashed bands represent the correlated weighted averages(29)for each experiment, shown in TableIII.The correlated weighted averages(29)for each of the six experiments of Refs.[35-  38].The theoretical DM bands of the FFs correspond to the use of the combined LQCD data of Table I as inputs.

TABLE IV .
(22)ording to the formula (except |V ub | 2 ) given in Eq.(22).Our results for |V ub | are collected in Table IV. Vlues of |V ub | • 10 3 extracted from the B s → K ν decays measured at LHCb in the low (q 2 ≤ 7 GeV 2 based on the BCL or BSZ parameterizations of the FFs.

TABLE V .
(36)theoretical values of the quantities (34)-(36)in the case of the semileptonic B → π ν decays with = µ, τ adopting the RBC/UKQCD, the FNAL/MILC and the combined LQCD data of TableIas inputs for our DM method.

TABLE VI .
The same as in TableV, but in the case of the semileptonic B s → K decays adopting the RBC/UKQCD, the FNAL/MILC, the HPQCD and the combined LQCD data of TableIIas inputs for our DM method.As for the experimental side, only one measurement of R the pole quark mass m pole h is given in terms of the M S mass m h (m h ) by

TABLE VIII .
The theoretical values of the ratio of the τ /µ decay rates R

TABLE IX .
The same as in TableVIII, but in the case of the semileptonic B s → K ν decays with = µ, τ adopting the combined LQCD data of TableIIas inputs for our DM method.