Heavytolight scalar form factors from Muskhelishvili–Omnès dispersion relations
Abstract
By solving the Muskhelishvili–Omnès integral equations, the scalar form factors of the semileptonic heavy meson decays \(D\rightarrow \pi \bar{\ell }\nu _\ell \), \(D\rightarrow {\bar{K}} \bar{\ell }\nu _\ell \), \({\bar{B}}\rightarrow \pi \ell \bar{\nu }_\ell \) and \({\bar{B}}_s\rightarrow K \ell \bar{\nu }_\ell \) are simultaneously studied. As input, we employ unitarized heavy meson–Goldstone boson chiral coupledchannel amplitudes for the energy regions not far from thresholds, while, at high energies, adequate asymptotic conditions are imposed. The scalar form factors are expressed in terms of Omnès matrices multiplied by vector polynomials, which contain some undetermined dispersive subtraction constants. We make use of heavy quark and chiral symmetries to constrain these constants, which are fitted to lattice QCD results both in the charm and the bottom sectors, and in this latter sector to the lightcone sum rule predictions close to \(q^2=0\) as well. We find a good simultaneous description of the scalar form factors for the four semileptonic decay reactions. From this combined fit, and taking advantage that scalar and vector form factors are equal at \(q^2=0\), we obtain \(V_{cd}=0.244\pm 0.022\), \(V_{cs}=0.945\pm 0.041\) and \(V_{ub}=(4.3\pm 0.7)\times 10^{3}\) for the involved Cabibbo–Kobayashi–Maskawa (CKM) matrix elements. In addition, we predict the following vector form factors at \(q^2=0\): \(f_+^{D\rightarrow \eta }(0)=0.01\pm 0.05\), \(f_+^{D_s\rightarrow K}(0)=0.50 \pm 0.08\), \(f_+^{D_s\rightarrow \eta }(0)=0.73\pm 0.03\) and \(f_+^{{\bar{B}}\rightarrow \eta }(0)=0.82 \pm 0.08\), which might serve as alternatives to determine the CKM elements when experimental measurements of the corresponding differential decay rates become available. Finally, we predict the different form factors above the \(q^2\)regions accessible in the semileptonic decays, up to moderate energies amenable to be described using the unitarized coupledchannel chiral approach.
1 Introduction
Exclusive semileptonic decays play a prominent role in the precise determination of the Cabibbo–Kobayashi–Maskawa (CKM) matrix elements, which are particularly important to test the standard model (SM) – any violation of the unitarity of the CKM matrix would reveal new physics beyond the SM (see for instance the review on the CKM mixing parameters by the Particle Data Group (PDG) [1]). Experimental and theoretical efforts have been devoted to multitude of inclusive and exclusive semileptonic decays driven by electroweak charge currents. For instance, the \(K_{\ell 3}\) decays and those of the type \(H\rightarrow {\phi }\,{\bar{\ell }}\,\nu _\ell \) and \(H\rightarrow {\phi }\,{\ell }\,\bar{\nu }_\ell \) (hereafter denoted by \(H_{\ell 3}\) or \(H\rightarrow \phi \)), where \(H\in \{D,\, {\bar{B}}\}\) is an open heavyflavor pseudoscalar meson and \(\phi \in \{\pi ,\, K,\, {\bar{K}},\, \eta \}\) denotes one of the Goldstone bosons due to the spontaneous breaking of the approximate chiral symmetry of Quantum Chromodynamics (QCD), are important in the extraction of some of the CKM matrix elements. Experimentally, significant progresses have been achieved and absolute decay branching fractions and differential decay rates have been accurately measured [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. On the theoretical side, determinations of the form factors in the vicinity of \(q^2=0\) (with \(q^2\) the invariant mass of the outgoing lepton pair) using lightcone sum rules (LCSR) have significantly improved their precision [12, 13], and have reached the level of twoloop accuracy [14]. Meanwhile, improvements have been made by using better actions in lattice QCD (LQCD), which have allowed to extract CKM matrix elements with significantly reduced statistical and systematical uncertainties [15, 16, 17, 18, 19, 20, 21]. As a result of this activity in the past decade, lattice calculations on the scalar form factors in heavytolight semileptonic transitions have been also reported by the different groups (see the informative review by the Flavour Lattice Averaging Group (FLAG) [22]).
The extraction of the CKM mixing parameters from \(K_{\ell 3}\) and/or \(H_{\ell 3}\) decays relies on the knowledge of the vector [\(f_+(q^2)\)] and scalar [\(f_0(q^2)\)] hadronic form factors that determine the matrix elements of the charged current between the initial and final hadron states.^{1} Various parameterizations, such as the Isgur–Scora–Grinstein–Wise updated model [23] or the series expansion proposed in Ref. [24], are extensively used in LQCD and experimental studies. In this work, we will study the scalar form factors in \(H_{\ell 3}\) decays by using the Muskhelishvili–Omnès (MO) formalism, which is a model independent approach to account for \(H \phi \) coupledchannel rescattering effects. The coupledchannel MO formalism has been extensively applied to the scalar \(\pi \pi \), \(\pi K\) and \(\pi \eta \) form factors, see, e.g., Refs [25, 26, 27, 28, 29]. It builds up an elegant bridge to connect the form factors with the corresponding Swave scattering amplitudes via dispersion relations. The construction of those equations is rigorous in the sense that the fundamental principles, such as unitarity and analyticity, and the proper QCD asymptotic behaviour are implemented. The first attempts to extend this method to the investigation of the scalar \(H\rightarrow \phi \) form factors were made in Refs. [30, 31, 32, 33, 34, 35, 36, 37, 38], but just for the singlechannel case. A similar dispersive MO approach has been also employed to study the semileptonic \({{\bar{B}}} \rightarrow \rho l \bar{\nu }_l\) [37, 39, 40, 41] and \({\bar{B}}_s \rightarrow {\bar{K}}^* l \bar{\nu }_l\) [37] decays and the possible extraction of the CKM element \(V_{ub}\) from data on the fourbody \({\bar{B}} \rightarrow \pi \pi l \bar{\nu }_l\) and \({\bar{B}}_s \rightarrow {\bar{K}} \pi l \bar{\nu }_l\) decaymodes.
The study of heavylight form factors using the MO representation incorporating coupledchannel effects has not been undertaken yet. This is mainly because of the poor knowledge on the \(H\phi \) interactions up to very recent years. However, a few intriguing positiveparity charmed mesons, like the \(D_{s0}^*(2317)\), have been recently discovered [1], giving support to a new paradigm for heavylight meson spectroscopy [42] that questions their traditional \(q{\bar{q}}\) constituent quark model interpretation. Hence, the study of the \(H\phi \) interactions aiming at understanding the dynamics of these newly observed states has become an interesting subject by itself, see, e.g., Refs. [43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54] for phenomenological studies and [55, 56, 57, 58, 59, 60, 61] for LQCD calculations. For the \(D_{\ell 3}\) decays, several LQCD results on the relevant form factors have been recently reported, see, e.g., Refs. [16, 17, 21]. This situation makes timely the study of the scalar \(D\rightarrow \phi \) form factors by means of the MO representation incorporating our current knowledge of \(D\phi \) interactions. The extension to the \(H=\bar{B}\) case is straightforward with the help of heavy quark flavour symmetry (HQFS). Based on HQFS, the low energy constants (LECs) involved in the \(D\phi \) interactions or \(D\rightarrow \phi \) semileptonic form factors are related to their analogues in the bottom sector by specific scaling rules. It is then feasible either to predict quantities in the bottom (charm) sector by making use of the known information in the charm (bottom) case or to check how well HQFS works by testing the scaling rules.
In the present study, we construct the MO representations of the scalar form factors, denoted by \(f_0(q^2)\), for the semileptonic \(D\rightarrow \pi \) and \(D\rightarrow \bar{K}\) transitions, which are related to the unitarized Swave scattering amplitudes in the \(D\phi \) channels with strangeness (S) and isospin (I) quantum numbers \((S,I)=(0,\frac{1}{2})\) and \((S,I)=(1,0)\), respectively. These amplitudes are obtained by unitarizing the \({\mathcal {O}}(p^2)\) heavymeson chiral perturbative ones [62], with LECs determined from the lattice calculation [56] of the Swave scattering lengths in several (S, I) sectors. The scheme provides an accurate description of the \(D\phi \) interactions in coupled channels. For instance, as it is shown in Ref. [52], the finite volume energy levels in the \((S,I)=(0,1/2)\) channel calculated with the unitarized amplitudes, without adjusting any parameter, are in an excellent agreement with those recently reported by the Hadron Spectrum Collaboration [60]. In addition, it is demonstrated in Ref. [42] that these well constrained amplitudes for Goldstone bosons scattering off charm mesons are fully consistent with recent high quality data on the \(B^ \rightarrow D^+\pi ^\pi ^\) final states provided by the LHCb experiment [63].
The unitarized chiral scattering amplitudes are used in this work as inputs to the dispersive integrals. However, these amplitudes are valid only in the low Goldstoneboson energy region. Hence, asymptotic behaviors at high energies for the phase shifts and inelasticities are imposed in the solution of the MO integral equations. The Omnès matrices obtained in this way incorporate the strong final state interactions, and the scalar form factors are calculated by multiplying the former by polynomials. The (a priori unknown) coefficients of the polynomials are expressed in terms of the LECs appearing at nexttoleading order (NLO) in the chiral expansion of the form factors [64, 65].
The scheme employed in the charm sector is readily extended to the bottom one. Afterwards, the LECs could be either determined by fitting to the results obtained in the LQCD analyses of the \(D\rightarrow \pi (\bar{K})\) decays carried out in Refs. [16, 17, 21] or to the LQCD and LCSR combined \(\bar{B}\rightarrow \pi \) and \(\bar{B}_s\rightarrow K\) scalar form factors reported in Refs. [15, 18, 19, 20] and [12, 13, 14], respectively. In both scenarios LQCD and LCSR results are well described using the MO dispersive representations of the scalar form factors constructed in this work.
However, our best results are obtained by a simultaneous fit to all available results, both in the charm and bottom sectors. As mentioned above, all of the LECs involved in the \(\bar{B}_{(s)}\phi \) interactions or \(\bar{B}_{(s)}\rightarrow {\bar{\phi }}\) semileptonic transitions are related to those in the charm sector by making use of the heavy quark scaling rules [65], which introduce some constraints between the polynomials that appear in the different channels. Thus, assuming a reasonable effect of the HQFS breaking terms, a combined fit is performed to the \(D\rightarrow \pi /\bar{K}\) and the \(\bar{B}\rightarrow \pi \) and \(\bar{B}_s\rightarrow K\) scalar form factors, finding a fair description of all the fitted data, and providing reliable predictions of the different scalar form factors in the whole semileptonic decay phase space, which turn out to be compatible with other theoretical determinations by, e.g., perturbative QCD [66, 67]. The results of the fit allow also to predict the scalar form factors for the \(D\rightarrow \eta \), \(D_s\rightarrow {K}\) and \(D_s\rightarrow \eta \) transitions in the charm sector, and for the very first time for the \(\bar{B}\rightarrow \eta \) decay. In some of these transitions, the form factors are difficult for LQCD due to the existence of disconnected diagrams of quark loops.
Based on the results of the combined fit, and taking advantage of the fact that scalar and vector form factors are equal at \(q^2=0\), we extract all the heavylight CKM elements and test the secondrow unitarity by using the \(V_{cb}\) value given in the PDG [1]. We also predict the different form factors above the \(q^2\)regions accessible in the semileptonic decays, up to energies in the vicinity of the involved thresholds, which should be correctly described within the employed unitarized chiral approach.
This work is organized as follows. In Sect. 2.1, we introduce the definitions of the form factors for \(H_{\ell 3}\) decays. A general overview of the MO representation of the scalar form factors is given in Sect. 2.2, while the inputs for the MO problem and the solutions are discussed in Sect. 2.3. In Sect. 2.4, we derive the scalar as well as vector form factors at NLO in heavymeson chiral perturbation theory and then perform the aforementioned matching between the MO and the chiral representations close to the corresponding thresholds. Section 3 comprises our numerical results and discussions, with details of the fits given in Sects. 3.1 and 3.2. With the results of the combined charmbottom fit, in Sect. 3.3, we extract the related CKM elements and make predictions for the values of \(f_+(0)\) in various transitions. Predictions for flavourchanging \(b\rightarrow u\) and \(c\rightarrow d, s\) scalar form factors above the \(q^2\)regions accessible in the semileptonic decays are given and discussed in Sect. 3.4. We summarize the results of this work in Sect. 4. Finally, the heavyquark scaling rules of the LECs involved in the \(H\phi \) interactions are discussed in Appendix A, while some further results for \(b\rightarrow u\) form factors, obtained with quadratic MO polynomials, are shown in Appendix B.
2 Theoretical framework
2.1 Form factors in \(H_{\ell 3}\) decays
In the next two subsections, Sects. 2.2–2.3, we give specific details on the MO representation of the form factors. For brevity, in some occasions we will focus on the formalism for the case of the charm sector (\(H=D\)). The extension to the bottom sector (\(H=\bar{B}\)) is straightforward using HQFS, though some aspects are explicitly discussed in Sect. 2.3.2b.
2.2 Muskhelishvili–Omnès representation
2.3 Inputs and MO solutions
2.3.1 The \((S,I)=(1,0)\) sector
In Fig. 2, we show the solution of the MO integral equation (20), with the input specified above, and the contour condition \(\Omega (q^2_{\mathrm{max}})={\mathbb {I}}\), with \(q^2_{\mathrm{max}}=(m_DM_K)^2\). We display results only up to \(s=s_m\) that would be later used to evaluate the scalar form factors entering in the \(D\rightarrow {\bar{K}}\) and \(D_s \rightarrow \eta \) semileptonic transitions. Note that the imaginary parts are zero below the lowest threshold \(s_{\mathrm{th}}=(m_D+M_K)^2\), and how the opening of the \(D_s\eta \) threshold produces clearly visible effects in the Omnès matrix. At very high energies, not shown in the figure, both real and imaginary parts of all matrix elements go to zero, as expected from Eq. (26).
2.3.2 The \((S,I)=(0,1/2)\) sector
2.3.2.1 a. Charm sector:
2.3.2.2 b. Bottom sector:
Note also that now \(\lim _{s\rightarrow \infty } \sum _{i=1}^3 \delta _i(s) = 4\pi > 3\pi \), which implies a slightly faster decreasing of the MO matrix elements at high energies.
2.4 Chiral expansion of the form factors and the MO polynomial
2.4.1 Form factors in heavy meson chiral perturbation theory
Strangenessisospin coefficients appearing in the chiral expansion of the form factors
(S, I)  (1, 0)  \((0,\frac{1}{2})\)  

Channel  DK  \(D_s\eta \)  \(D\pi \)  \(D\eta \)  \(D_s {\bar{K}}\) 
\({\bar{B}}\pi \)  \({\bar{B}} \eta \)  \({\bar{B}}_s\bar{K}\)  
\({\mathcal {C}}\)  \(\sqrt{2}\)  \( \sqrt{\frac{2}{3}}\)  \(\sqrt{\frac{3}{2}}\)  \(\frac{1}{\sqrt{6}}\)  1 
2.4.2 Matching
It is worth noting that the NLO LECs \(\beta _1^P\) and \(\beta _2^P\) determined from a fit to data using the MO scheme would have some residual dependence on the matching point. To minimize such dependence, we have chosen \(s_0=q_{\mathrm{max}}^2\), where the momentum of the Goldstone bosons is close to zero and higher order chiral corrections are expected to be small. Different choices of the matching point, within the chiral regime, will amount to changes in the fitted (effective) \(\beta _1^P\) and \(\beta _2^P\) LECs driven by higher order effects.
Properties of the \(D_{s0}^*(2317)\) pole from the unitarized chiral amplitudes derived in Ref. [56]
\(\sqrt{s_{p}}\) [MeV]  \(g_{DK}\) [GeV]  \(g_{D_s\eta }\) [GeV] 

\(2315^{+18}_{28}\)  \(9.5^{+1.2}_{1.1}\)  \(7.5^{+0.5}_{0.5}\) 
Goldstone  Charm sector  Bottom sector  

\(M_\pi \)  139  \(m_D\)  1869.6  \(m_B\)  5279.5 
\(M_K\)  496  \(m_{D_s}\)  1968.5  \(m_{B_s}\)  5366.8 
\(M_\eta \)  547  \(f_D\)  147.6  \(f_B\)  135.8 
\(F_0\)  92.4  \(f_{D_s}\)  174.2  \(f_{B_s}\)  161.5 
\(m_{D^*}\)  2008.6  \(m_{B^*}\)  5324.7  
\(m_{D_s^*}\)  2112.1 
3 Numerical results and discussion
So far, the theoretical MO representations of the scalar form factors have been constructed. In this section, we want to confront the soobtained form factors to the LQCD and LCSR results. In what follows, we first fit to the \({\bar{B}} \rightarrow \pi \) and \({\bar{B}}_s \rightarrow {K}\) scalar form factors, where we expect the \(1/m_P\) corrections to the chiral expansion in Eq. (41) to be substantially suppressed. Next, we will carry out a combined fit to all the data in both charm and bottom sectors, by adopting some (approximate) heavyquark flavor scaling rule [65] for the \(\beta _1^P \) and \(\beta _2^P\) LECs in Eq. (41). Using the results of our combined fit, we will: (i) determine the CKM elements, \(V_{cd}\), \(V_{cs}\) and \(V_{ub}\), (ii) predict form factors, not computed in LQCD yet, and that can be used to overconstrain the CKM matrix elements from analyses involving more semileptonic decays, and (iii) predict the different form factors above the \(q^2\)regions accessible in the semileptonic decays, up to moderate energies amenable to be described using the unitarized coupledchannel chiral approach.
Masses and decay constants used in this work are compiled in Table 3. In addition, the mass of the heavylight mesons in the chiral limit, see Eq. (35), is set to \(\mathring{m}=(m_P+m_{P_s})/2\), for simplicity the same average is used to define \({\bar{m}}_P\) in the Appendix A and in the relations given in Eq. (53). The \({\mathcal {P}}{\mathcal {P}}^*\phi \) axial coupling constant \({\tilde{g}}\) in Eq. (39) can be fixed by calculating the decay width of \(D^{*+}\rightarrow D^0\pi ^+\) [1], which leads^{13} to \(g\sim 0.58\) and hence \({\tilde{g}}_{D^*D\pi }\sim 1.113\) GeV. In the bottom sector we use a different value for g, around 15% smaller, consistent with the lattice calculation of Ref. [78], where \(g\sim 0.51\) (or \({\tilde{g}}_{{\bar{B}}^*{\bar{B}}\pi }\sim 2.720\) GeV) was found. Note that the difference is consistent with the expected size of heavyquarkflavor symmetry violations. In addition, there exist sizable SU(3) corrections to the overall size of the \(\mathcal{P}^*\) pole contribution to both \(f_+\) and \(f_0\) form factors. Thus, such contribution is around \(\sim 20\%\) smaller for \({\bar{B}}^* {\bar{B}}_s K \) than for \({\bar{B}}^*{\bar{B}}\pi \) [32, 38]. According to [79] this suppression is mainly due to a factor \(F_\pi /F_K\sim 0.83\) [22]. We will implement this correction in the pole contribution to \(f_0\) in Eq. (41) when the Goldstone boson is either a kaon or an eta meson (for simplicity, we also take \(F_\eta \approx F_K\)), and both in the bottom and charm sectors.
3.1 Fit to the LQCD+LCSR results in the bottom sector
The UKQCD Collaboration [20] provides data for both \({\bar{B}}\rightarrow \pi \) and \({\bar{B}}_s\rightarrow K\) form factors together with statistical and systematic correlation matrices for a set of three form factors computed at different \(q^2\) (Tables 8 and 9 of this reference). In the case of HPQCD [18] \({\bar{B}}_s\rightarrow K\) and FLMILC [19] \(B\rightarrow \pi \) form factors, we have read off four points from the final extrapolated results (bands) given in these references, since in both cases, originally only four momentum configurations (\(0\rightarrow 0\), \(0\rightarrow 1\), \(0\rightarrow \sqrt{2}\) and \(0 \rightarrow \sqrt{3}\)) were simulated. Finally, we also include in the fit the five \(B\rightarrow \pi \) points provided by the HPQCD Collaboration in the erratum of Ref. [15].
In addition, as we discussed before, the \({\bar{B}} \pi \)scalar form factor decreases by a factor of five in the \(q^2\)range accessible in the decay, and the LQCD results around \(q_{\mathrm{max}}^2\) and the LCSR predictions in the vicinity of \(q^2=0\) are not linearly connected at all. In the current scheme, where only rankone MO polynomials are being used, this extra needed curvature should be provided by the \(q^2\)dependence of the MO matrix, \(\Omega \), whose behaviour near \(q^2=0\), far from \(q^2_{\mathrm{max}}\), is not determined by the behaviour of the amplitudes in the chiral regime. Indeed, it significantly depends on the highenergy input.^{15} This is an unwanted feature, source of systematic uncertainties. To minimize this problem, in the next subsection we will perform a combined fit to transitions induced by the \(b\rightarrow u\) and \(c \rightarrow d,s\) flavourchanging currents. The latter ones describe \(D\rightarrow \pi \) and \(D\rightarrow {\bar{K}}\) semileptonic decays for which there exist recent and accurate LQCD determinations of the scalar form factors. Moreover, in these latter transitions the \(q^2\)ranges accessible in the decays and the form factor variations are much limited, becoming thus more relevant the input provided in the chiral regime.
To finish this subsection, we would like to stress that given the large value found for \(\chi ^2/dof\), statistical errors should be taken with some care. Indeed, one can rather assume some systematic uncertainties affecting our results, that could be estimated by considering in the best fit alternatively only the HPQCD or the UKQCD and the FLMILC sets of predictions. We will follow this strategy to obtain our final results for the CKM matrix elements and form factors at \(q^2=0\) from the combinedfit to charm and bottom decays detailed in the next subsection.
3.2 Extension to the charm sector and combined fit
First we need to incorporate the \(c \rightarrow d,s\) input into the merit function \(\chi ^2\), which was defined in Eq. (49) using only bottom decay results. In the last 10 years, LQCD computations of the relevant \(D\rightarrow \pi \) and \(D\rightarrow \bar{K}\) semileptonic decay matrix elements have been carried out by the HPQCD [16, 17] and very recently by the ETM [21] Collaborations. Compared with the former, the latter corrects for some hypercubic effects, coming from discretization of a quantum field theory on a lattice with hypercubic symmetry [80], and uses a large sample of kinematics, not restricted in particular to the parent D meson at rest, as in the case of the HPQCD simulation. Moreover, it is argued in Ref. [21] that the restricted kinematics employed in the simulations of Refs. [16, 17] may obscure the presence of hypercubic effects in the lattice data, and these corrections can affect the extrapolation to the continuum limit in a way that depends on the specific lattice formulation. This might be one of the sources of the important discrepancies found between the \(D\rightarrow \pi \) form factors reported by the HPQCD and ETM Collaborations in the region close to \(q_\mathrm{max}^2=(m_DM_\pi )^2\), as can be seen in the left top panel of Fig. 9.
Here, we prefer to fit to the most recent data together with the covariance matrices provided by the ETM Collaboration. This analysis is based on gauge configurations produced with \(N_f=2+1+1\) flavors of dynamical quarks at three different values of lattice spacing, and with pion masses as small as 210 MeV. Lorentz symmetry breaking due to hypercubic effects is clearly observed in the ETM data and included in the decomposition of the current matrix elements in terms of additional form factors. Those discretization errors have not been considered in the HPQCD analyses, and for this reason we have decided to exclude the results of these latter collaboration in our fits.
The results for the bottom scalar form factors are almost the same as the ones shown in Fig. 8, while the ETM \(c \rightarrow d,s\) transition form factors are remarkably well described within the present scheme. As in the former bestfit to only the \({\bar{B}}_{(s)}\) results, the large value obtained for \(\chi ^2/dof\) is mainly due to the existing tension between the LQCD results from different collaborations in the \({\bar{B}} \rightarrow \pi \) decay.
Results from the bottomcharm combined fit, with \(\chi ^2\) defined in Eq. (55) and a total of 38 degrees of freedom. The first set of errors in the bestfit parameters is obtained from the minimization procedure, assuming Gaussian statistics, while the second one accounts for the uncertainties of the LECs quoted in Ref. [56] that enter in the definition of the chiral amplitudes. The LECs \(\beta _0\) and \(\beta _1^B\) (\(\beta _2^B\)) are given in units of GeV (GeV\(^{1}\))
Correlation matrix  

\(\frac{\chi ^2}{dof} =2.77\)  \(\beta _0\)  \(\beta _1^B\)  \(\beta _2^B\)  \(\delta \)  \(\delta ^\prime \)  
\(\beta _0\)  0.152(14)(13)  1.000  0.502  0.499  \(\) 0.490  0.311 
\(\beta _1^B\)  0.22(4)(4)  0.502  1.000  0.995  \(\) 0.965  0.848 
\(\beta _2^B\)  0.0346(16)(15)  0.499  0.995  1.000  \(\) 0.958  0.845 
\(\delta \)  0.138(21)(18)  \(\) 0.490  \(\)0.965  \(\) 0.958  1.000  \(\)0.942 
\(\delta ^\prime \)  \(\) 0.18(4)(2)  0.311  0.848  0.845  \(\) 0.942  1.000 
The HQFS breaking parameters \(\delta \) and \(\delta ^\prime \) turn out to be quite correlated and their size is of the order \(\Lambda _{\mathrm{QCD}}/m_c\). As expected, \(\delta \) presents also a high degree of correlation with \(\beta _{1}^B\) and \(\beta _2^B\), and on the other hand, the combined fit does not reduce the large correlation between these two latter LECs, while the central values (errors) quoted for them in Table 4 are compatible within errors with (significantly smaller than) those given in Eq. (51), and obtained from the fit only to \(b \rightarrow u\) transitions. In addition, the values quoted for \((\beta _{1}^B,\beta _2^B)\) in Table 4 perfectly lie in the straight line of Eq. (52), deduced from the fit to only bottom form factors carried out in the previous Sect. 3.1. Indeed, the straight line that one can construct with the results of Table 4 in the \((\beta _1^B, \beta _2^B)\)plane is practically indistinguishable from that of Eq. (52). All this can be seen in the left plot of Fig. 10, where both straight lines are depicted, together with the statistical 68% CL ellipses and the onesigmarectangle bands obtained by minimizing the merit function given in Eq. (49) or alternatively in Eq. (55), and considering only bottom or bottom and charm scalar form factors, respectively.

The combined charmbottom analyses (solid lines) provide large curvatures of \(\chi ^2\) as a function of \(\beta _1^B\), hence leading to better determinations of this latter LEC, always in the 0.2 GeV region, as we also found in Eq. (49) from the best fit to only the bottom results. A value for \(\beta _1^B\) close to this region, taking into account errors, is also found from a fit where only the bottom form factors are considered, but without including the HPQCD \({\bar{B}} \rightarrow \pi \) results (dashedgreen curve). Only the dashedred line (fit only to the bottom results, but without including in this case the UKQCD and FLMILC \({\bar{B}} \rightarrow \pi \) scalar form factors) turns out to be incompatible with the combined fit presented in Table 4. Thus, we find some arguments to support the range of values quoted in Table 4 for the parameters \((\beta _1^B, \beta _2^B)\) that appear in the heavy meson chiral perturbation theory (HMChPT) expansion of the scalar form factors at the bottom scale.

The existing tension between HPQCD, and UKQCD and FLMILC sets of \({\bar{B}} \rightarrow \pi \) form factors leads to large values of \(\chi ^2\). Thus, as mentioned above, statistical errors should be taken with some care, and some systematic uncertainties would need to be considered in derived quantities, as for instance in the values of the form factors at \(q^2=0\) or in the CKM mixing parameters. We note that this source of systematics also induces variations on the fitted parameters in Table 4 which range between 50 (\(\beta _0\) and \(\beta _2^B\)) and 100% (\(\beta _1^B\), \(\delta \) and \(\delta ^\prime \)) of the statistical errors quoted in the table.
3.2.1 Further considerations
We have also obtained results using constant and quadratic MO polynomials. In the first case, the dispersive representations of the form factors should be matched to the LO chiral ones, where the terms driven by the \(\beta _1^P\) and \(\beta _2^P\) LECs are dropped out. The first consequence is that bottom and charm sectors are no longer connected since, in addition, we are not enforcing the heavyquark scaling law for the decay constants. To better describe the data, one might perform separate fits to bottom and charm form factors with free \({\vec {\alpha }}_0\) parameters in Eq. (34). Fits obviously are poorer, and moreover, they do not necessarily provide reliable estimates of the form factors at \(q^2_{\mathrm{max}}\), since the fitted parameters are obtained after minimizing a merit function constructed out of data in the whole \(q^2\)range accessible in the decays. For charm decays, the description of the \(D\rightarrow \pi \) form factor is acceptable, while that of the \(D\rightarrow {\bar{K}}\) is in comparison worse, mostly because the \(s\)dependence induced by the \(D_{s0}^*(2317)\) can not be now modulated by the MO polynomial. In the bottom sector, as one should expect, the simultaneous description of LQCD and LCSR form factors in the vicinity of \(q^2_{\mathrm{max}}\) and \(q^2=0\), respectively, becomes poorer. Indeed, since the LQCD input has a larger weight in the \(\chi ^2\) than the LCSR one, the latter form factors are totally missed by the new predictions, which now lie below the lower error bands of the LCSR results.
The consideration of quadratic MO polynomials solves this problem, as shown in Fig. 12 of Appendix B. Indeed, it is now possible to improve the description of the \({\bar{B}}\rightarrow \pi \) LCSR form factors, providing still similar results in the \(q^2_{\mathrm{max}}\)region, where the LQCD data are available. Thus, for instance, we get \(f_+^{\bar{B}\rightarrow \pi }(0)=0.248 (10)\) using the new fit to be compared with 0.211(10) obtained using the parameters of the fit of Eq. (51) (form factors displayed in Fig. 8). Nevertheless, as we will see in the next subsection, there exist some other systematic errors, which practically account for the latter difference, and thus this source of uncertainty will be considered in the determination of the CKM matrix element \(V_{ub}\). In addition, though the \(\chi ^2/dof\) obtained with quadratic MO polynomials is better, it is still large (around 3.7) due to the tension between the \({\bar{B}} \rightarrow \pi \) LQCD results from different collaborations. Moreover \(\beta _1^B\) and \(\beta _2^B\) are still fully correlated, and the quadratic terms of the MO polynomials that multiply the elements \(\Omega _{ij}\), \((i=1,2,3, ~j=2,3)\) of the matrix displayed in Fig. 6 are almost undetermined (see the large errors in the parameters \(\alpha _{2,3}\) and especially \(\alpha _{2,2}\) given in Table 6). Finally, the centralvalue predictions, that will be shown in Sect. 3.4, for the form factors above \(q^2_{\mathrm{max}}\) and to moderate energies amenable to be described using the unitarized coupledchannel chiral approach, are not affected by the inclusion of quadratic terms in the MO polynomials, though errors are enhanced. For all of this, we consider our best estimates for the form factors those obtained using rankone polynomials.
We do not discuss quadratic terms in the charmsector because rankone MO polynomials led already to excellent reproductions of the form factors (see Fig. 9), in part due to the smaller \(q^2\)range involved in these decays. Moreover a correct charmbottom combined treatment will require the matching at nexttonexttoleading order (NNLO) in the chiral expansion, which is beyond the scope of this work.
3.3 Extraction of CKM elements and predictions
Taking advantage that scalar and vector form factors are equal at \(q^2=0\), the results of the combined charmbottom fit presented in the previous subsection can be used to extract the vector form factor, \(f_+\), at \(q^2=0\) for various semileptonic decays studied in this work. Moreover, given some experimental input for the quantity \(V_{Qq} f_+(0)\), with \(Qq=bu,cd\) or cs, we can extract the corresponding CKM matrix element using the present MO scheme. Measurements of the differential distribution \(d\Gamma (H\rightarrow {\bar{\phi }}\ell {\bar{\nu }}_\ell )/dq^2\) at \(q^2=0\) will directly provide model independent determinations of \(V_{Qq} f_+(0)\),^{19} while measurements of the total decay width could be used to estimate this latter quantity only after relying on some model for the \(q^2\)dependence of \(f_+\).
3.4 Scalar form factors above the \(q^2_{\mathrm{max}}\)region
It is worth recalling here the relation between the results obtained for the form factors and the scattering amplitudes used as input of the MO representation. If we focus, for instance, on the charm form factors, the lightest opencharm scalar resonance, called \(D^*_0(2400)\) by the PDG [1], lies in the \((S,I)=(0,1/2)\) sector. In Refs. [43, 45, 47], two different states, instead of only one, were claimed to exist in the energy region around the nominal mass of the \(D^*_0(2400)\). These studies were based on chiral symmetry and unitarity. This complex structure should be reflected in the scattering regime of the form factors. Indeed, this can be seen in the first row of panels of Fig. 11, where form factors for different semileptonic transitions are shown above the \(q^2_{\mathrm{max}}\)region. As discussed in Sect. 2.3, here we use the \({\mathcal {O}}(p^2)\) HMChPT amplitudes obtained in Refs. [56, 62], which also successfully describe the (0, 1 / 2) finitevolume energy levels reported in the recent LQCD simulation of Ref. [60] (see Ref. [52] for details) and are consistent with the precise LHCb data [63] for the angular moments of the \(B^\rightarrow D^+\pi ^\pi ^\) [42]. These chiral amplitudes predict the existence of two scalar broad resonances, instead of only one, with masses around 2.1 and 2.45 GeV, respectively [42, 52], which produce some signatures in the \(D\rightarrow \pi \), \(D\rightarrow \eta \) and \(D_s\rightarrow K\) form factors at around \(q^2=4.4\) and 6 GeV\(^2\), as can be appreciated in Fig. 11. The effect of this twostate structure is particularly visible in the \(D_s\rightarrow K\) form factor. Note that this twostate structure should have also some influence in the region below \(q^2_{\mathrm{max}}\), where we have fitted the LQCD data. Below \(q^2_{\mathrm{max}}\), the sensitivity of the form factors to the details of the two resonances is however smaller than that of the energy levels calculated in the scattering region, since the former ones are given below the lowest threshold, while the latter ones are available at energies around and above it. Nonetheless, the success in describing the LQCD results for the \(D\rightarrow \pi \) scalar form factor clearly supports the chiral input, and the predictions deduced from it, used in the current scheme. If better determined form factors were available in all of the channels, perhaps the two state structure for the \(D^*_0(2400)\) could be further and more accurately tested.
A similar pattern is found in the bottom sector [42, 52], as expected from the approximate heavyflavor symmetry of QCD. The twostate structure is clearly visible, more than that in the charm sector, in the corresponding form factors (three bottom plots of Fig. 11), and it has a certain impact in the form factors close to \(q^2_{\mathrm{max}}\), where LQCD results are available.
4 Summary and outlook
We have studied the scalar form factors that appear in semileptonic heavy meson decays induced by the flavourchanging \(b\rightarrow u\) and \(c\rightarrow d, s\) transitions using the MO formalism. The coupledchannel effects, due to rescattering of the \(H\phi \) (\(H=D,\bar{B}\)) system, with definite strangeness and isospin, are taken into account by solving coupled integral MO equations. We constrain the subtraction constants in the MO polynomials, which encodes the zeros of the form factors, thanks to lightquark chiral SU(3) and heavyflavor symmetries.
The \(H\phi \) interactions used as input of the MO equations are well determined in the chiral regime and are taken from previous work. In addition, some reasonably behaviors of the amplitudes at high energies are imposed, while appropriate heavyflavor scaling rules are used to relate bottom and charm form factors. We fit our MO representation of the scalar form factors to the latest \(c\rightarrow d , s\) and \(b\rightarrow u\) LQCD and \(b\rightarrow u\) LCSR results and determine all the involved parameters, in particular the two LECs (\(\beta _1^P\) and \(\beta _2^P\)) that appear at NLO in the chiral expansion of the scalar and vector form factors near \(q^2_{\mathrm{max}}\), which are determined in this work for first time. We describe the LQCD and LCSR results rather well, and in combination with experimental results and using that \(f_0(0)=f_+(0)\), we have also extracted the \(V_{ub}\), \(V_{cd}\) and \(V_{cs}\) CKM elements, which turn out to be in good agreement with previous determinations from exclusive decays.
We would like to stress that we describe extremely well the recent ETM \(D\rightarrow \pi \) scalar form factor, which largely deviates from the previous determination by the HPQCD Collaboration, providing an indication that the Lorentz symmetry breaking effects in a finite volume, due to the hypercubic artifacts, could be important in the LQCD determination of the form factors in semileptonic heavytolight decays, as claimed in Ref. [21]. As it is also pointed out in the previous reference, this is a very important issue, which requires further investigations, since it might become particularly relevant in the case of the determination of the form factors governing semileptonic \({\bar{B}}\)meson decays into lighter mesons.
We have also predicted the scalar form factors, which are in the same strangenessisospin multiplets as the fitted \(D\rightarrow \pi \), \(D\rightarrow {\bar{K}}\), \({\bar{B}} \rightarrow \pi \) and \({\bar{B}}_s \rightarrow K\) ones. Our prediction of the form factors in such channels (\(D\rightarrow \eta \), \(D_s\rightarrow K\), \(D_s\rightarrow \eta \), and \(B\rightarrow \eta \)) are difficult for LQCD simulations due to the existence of disconnected diagrams. These form factors are related to the differential decay rates of different semileptonic heavy meson decays and hence provide alternatives to determine the CKM elements with the help of future experimental measurements.
Moreover, we also find that the \(D\rightarrow \eta \) scalar form factor is largely suppressed compared to the other two components (\(D\rightarrow \pi \), \(D\rightarrow {\bar{K}}\)) in the threechannel \((0,1/2)\)multiplet, which is similar to what occurs for the \(K\rightarrow \eta \) strangenesschanging scalar form factor in Ref. [26].
Our determination of the form factors has the advantage that the constraints from unitarity and analyticity of the Smatrix have been taken into account, as well as the stateoftheart \(H\phi \) chiral amplitudes. Thus, our predictions for the flavourchanging \(b\rightarrow u\) and \(c\rightarrow d, s\) scalar form factors above the \(q^2\)region accessible in the semileptonic decays, depicted in Fig. 11, should be quite accurate^{23} and constitute one of the most important findings of the current research. Indeed, we have shown how the form factors in this region reflect details of the chiral dynamics that govern the \(H\phi \) amplitudes, and that give rise to a new paradigm for heavylight meson spectroscopy [42] which questions the traditional \(q{\bar{q}}\) constituent quark model interpretation, at least in the scalar sector.
As an outlook, the scheme presented here will also be useful to explore the \(H\phi \) interactions by using the lattice data for the scalar form factors in semileptonic decays of \(\bar{B}\) or D mesons. As pointed out in Ref. [73], more data are needed to fix the LECs in the NNLO potentials. Since the dispersive calculation of \(D\phi \) and \(\bar{B}\phi \) scalar form factors depend on the scattering amplitudes of these systems, the LQCD results for the form factors can be used to mitigate the lack of data and help in the determination of the new unknown LECs.
One might also try to extend the MO representation to a formalism in a finite volume with unphysical quark masses, such that comparisons to the discretized lattice data could be directly undertaken. On the other hand, the chiral matching of the form factors can be carried out at higher order to take into account the expected sizable corrections in SU(3) HMChPT. Moreover, this improved matching will in practice suppose to perform additional subtractions in the dispersive representations of the form factors, and it should reduce the importance of the highenergy input used for the \(H\phi \) amplitudes. The high energy input turns out to be essential to describe the scalar \({\bar{B}} \rightarrow \pi \) form factor near \(q^2=0\), and it represents one of the major limitations of the current approach.
Both improvements would lead to a more precise and modelindependent determination of the CKM matrix elements related to the heavytolight transitions.
Footnotes
 1.
The contribution of the scalar form factor to the decay width is suppressed since it vanishes in the limit of massless leptons. However, both scalar and vector form factors take the same value at \(q^2=0\), and thus an accurate determination of the \(q^2\)dependence of the scalar form factor can be used to constrain the vector one in this region.
 2.
Here also \(D^0 \rightarrow K^\) and \(D^+ \rightarrow {\bar{K}}^0\) form factors are related by an isospin rotation.
 3.Taking into account that the \(\Omega (s)\) matrix should have only a righthand cut and it should be real below all thresholds, Eq. (18) is equivalent towith \(H(s)= \left( {\mathbb {I}}+ 2i T(s)\Sigma (s)\right) \). Furthermore since \(H(s)H^*(s)={\mathbb {I}}\), though H(s) is not the Smatrix in the coupledchannel case, it follows
with \(\exp {2i\phi (s)}=\mathrm{det}\left[ H(s)\right] \). This is to say that the determinant of the matrix \(\Omega (s)\) satisfies a singlechannel Omnèstype relation [71], which is extensively used in this work to check the accuracy of the numerical calculations. Note that above all thresholds, \(\mathrm{det}\left[ H(s)\right] = \mathrm{det}\left[ S(s)\right] \) and therefore in the elastic case (\(\eta _i \rightarrow 1\) \(\forall i\)), \(\phi (s)=\sum _{i=1}^n, \delta _i(s)\), with n the number of channels.
 4.
 5.
We will specifically discuss the situation for these transitions below.
 6.
For example, let us consider Omnès matrices \(\Omega \) and \({\bar{\Omega }}\) normalized to \(\Omega (0) = {\mathbb {I}}\) or \({\bar{\Omega }}(s_n) = A\) (\(s_n\) the normalization point, \(s_n \leqslant s_\text {th}\) and A a real matrix), respectively. The matrix \({\bar{\Omega }}(s)\) is readily obtained from \(\Omega (s)\) as \({\bar{\Omega }}(s) = \Omega (s) \Omega ^{1}(s_n) A\). The form factors can then also be written as \({\vec {\mathcal {F}}}(s) = \Omega (s) {\vec {\mathcal {P}}}(s) = {\bar{\Omega }}(s) A^{1} \Omega (s_n) {\vec {\mathcal {P}}}(s) \equiv {\bar{\Omega }}(s) {\vec {\mathcal {\bar{P}}}}(s)\), where the definition of \(\mathcal {\vec {\bar{P}}}(s)\) reabsorbs the constant matrix \(A^{1} \Omega (s_n)\).
 7.
The Tmatrix is obtained from Eq. (30).
 8.
The values of the involved LECs in the \(\bar{B}\phi \) interactions are determined from their analogues in the charm sector by imposing the heavyquark mass scaling rules [52] discussed in the Appendix A.
 9.
In general, the MO matrix in the chiral domain, between the \(q^2_{\mathrm{max}}\) and scattering (below \(s_m\)) regions is rather insensitive to the high energy behaviour of the amplitudes.
 10.
These are \(\beta _1^P\) and \(\beta _2^P\), to be introduced in Sect. 2.4, that appear in the chiral expansion of the form factors at NLO.
 11.
This not strictly true in the case of \({\bar{B}}_{(s)}\)decays since, as discussed above, different asymptotic conditions have been assumed in the bottom sector and the Omnès matrix elements are expected to decrease slightly faster than 1 / s.
 12.
Note that the first term in the bracket of Eq. (46) should have a more general form, \(\frac{\vec {\beta }_0}{ss_p}\), with \({\vec {\beta }}_0\) a vector with two independent components, \({\vec {\beta }}_0 = (\beta _0^a, \beta _0^b)^T\). The specific form in Eq. (46) reduces the number of free parameters, by forcing \(\beta _0^a/\beta _0^b = g_{DK}/g_{D_s \eta }\). On the other hand, this has the effect that the form factors \(f_0^{DK}\) and \(f_0^{D_s \eta }\) are not exactly independent of the choice of the point \(s_n\) where one normalizes the Omnès matrix, \(\Omega (s_n) = {\mathbb {I}}\). Nonetheless, we have checked that this choice, varying \(s_n\) from zero to \(q^2_\text {max}\), has no practical effect in the determination of \(\beta _0\), which indicates that our assumption is reasonable. We also remark that this discussion has no effect at all in the (0, 1 / 2) sector.
 13.
Errors on g determined from the decay \(D^{*+}\rightarrow D^0\pi ^+\) are very small of the order of 1%.
 14.
This can be easily understood since these LECs enter in the definition of \({\vec {\alpha }}_0\) and \({\vec {\alpha }}_1\) in the combinations \(\beta _1m_P^2\beta _2\) and \(\beta _1m_P(m_P2M_\phi )\beta _2\), which are identical up to some small SU(3) corrections.
 15.
The results displayed in Fig. 8 might suggest that the present approach hardly provides enough freedom to simultaneously accommodate the near \(q^2=0\) (LCSR) and \(q^2_{\mathrm{max }}\) (LQCD) determinations of the \({\bar{B}} \pi \) scalar form factor. The situation greatly improves when only the HPQCD, among all LQCD calculations, \({\bar{B}}\pi \) results are considered in the \(q^2_{\mathrm{max }}\) region, being then possible to find an excellent combined description of the LCSR and HPQCD results with \(\chi ^2=9.65\) for a total of 18 degrees of freedom (see dashedred curve in the right plot of Fig. 10), which leads to \(\chi ^2/dof= 0.5\). The parameters \(\beta _{1,2}^B\) come out still to be almost totally correlated as in Eq. (51), and moreover they lie, within great precision, in the straight line of Eq. (52), but in the \(\beta _1^B\sim 0.7\) GeV region.
 16.
Notice that the particle charges are not specified in the notation used in Lattice QCD, for instance, \(D^0\rightarrow \pi ^\) in Eq. (6) is simplified to \(D\rightarrow \pi \), to be used below and denoted by \(D\pi \) in the lattice paper.
 17.
The \(D\rightarrow \pi \) and \(D \rightarrow {\bar{K}} \) scalar form factor covariancematrices have troublesome small eigenvalues, as small as \(10^{6}\) or even \(10^{9}\). Due to this, the fitting procedure could be easily spoiled since a tiny error in the fitting function yields a huge \(\chi ^2\) value (specific examples can be found in Ref. [81]). We have used the singular value decomposition (SVD) method to tackle this issue, which is widely used by a number of lattice groups [82, 83, 84].
 18.
Thus, the ranges marked by the circles show the statistical errors of \(\beta _1^B\) in each fit.
 19.
Neglecting the lepton masses.
 20.
Determinations from leptonic and semileptonic decays, as well as from neutrino scattering data in the case of \(V_{cd}\), are used to obtain the PDG averages.
 21.
Fitting only to the \(b\rightarrow u\) data and not considering UKQCD and FLMILC sets of \(\bar{B}\rightarrow \pi \) results, we find \(f_0^{\bar{B}\rightarrow \pi }(0)\sim \) 0.27, even in better agreement with the LCSR determination.
 22.
Indeed, the existence of the \(D_{s0}^*(2317)\) was suggested in [34] by fitting the single channel MO representation of the \(D\rightarrow {\bar{K}}\) scalar form factor, constructed out the unitarized LO chiral elastic DK amplitude, to LQCD results of the scalar form factor below \(q^2_{\mathrm{max}}\).
 23.
Note that for the moderate \(q^2\)values shown in Fig. 11, the form factors are largely insensitive to the highenergy input in the MO dispersion relation, and they are almost entirely dominated by the lowenergy (chiral) amplitudes.
Notes
Acknowledgements
DLY would like to thank YunHua Chen and Johanna Daub for helpful discussions on solving the MO problem. We would like to thank the authors of Ref. [21] for providing us the covariance matrices and J. Gegelia for comments on the manuscript. P. F.S. acknowledges financial support from the Ayudas para contratos predoctorales para la formació de doctores program (BES2015072049) from the Spanish MINECO and ESF. This research is supported by the Spanish Ministerio de Economía y Competitividad and the European Regional Development Fund, under contracts FIS201451948C21P, FIS201784038C21P and SEV20140398, by Generalitat Valenciana under contract PROMETEOII/2014/0068, by the National Natural Science Foundation of China (NSFC) under Grant No. 11747601, by NSFC and DFG though funds provided to the SinoGerman CRC 110 “Symmetries and the Emergence of Structure in QCD” (NSFC Grant No. 11621131001), by the Thousand Talents Plan for Young Professionals, by the CAS Key Research Program of Frontier Sciences under Grant No. QYZDBSSWSYS013, and by the CAS Center for Excellence in Particle Physics (CCEPP).
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