Shedding New Light on ${\cal R}(D_{(s)}^{(\ast)})$ and $|V_{cb}|$ from Semileptonic $\bar{B}_{(s)}\to D_{(s)}^{(\ast)}\ell\bar{\nu}_{\ell}$ Decays

The flagship semileptonic $\bar{B}_{(s)}\to D_{(s)}^{(\ast)}\ell\bar{\nu}_{\ell}$ decay processes are of extraordinary phenomenological importance for the precise determination of the Cabibbo-Kobayashi-Maskawa (CKM) matrix element $|V_{cb}|$ and for probing the non-standard flavour-changing dynamics above the electroweak scale Glattauer et al. (2016); Abdesselam et al. (2017); Waheed et al. (2019); Aaij et al. (2020a, b); Prim et al. (2023). The persisting discrepancy between the exclusive and inclusive extractions of $|V_{cb}|$ at the level of $2\,\sigma\sim 3\,\sigma$ has triggered enormous efforts on uncovering the ultimate mystery of this intriguing puzzle in the Standard Model (SM) and beyond (see for instance Altmannshofer et al. (2019); Gambino et al. (2020); Boyle et al. (2022); Kronfeld et al. (2022); Di Canto and Meinel (2022)). Moreover, the state-of-the-art theory predictions for the two lepton-flavour-universality (LFU) ratios ${\cal R}({D^{(\ast)}})$ Bernlochner et al. (2022) appear to be in tension with the HFLAV-averaged experimental measurements Amhis et al. (2022) (at the $3\,\sigma$ level when combined together) as well, thus stimulating intensive investigations on a wide range of other interesting LFU observables Bernlochner and Ligeti (2017); Bernlochner et al. (2018); Cohen et al. (2018); Das and Dutta (2022); Gubernari et al. (2022); Bernlochner et al. (2023); Patnaik and Singh (2023); Patnaik et al. (2023); Penalva et al. (2023). Apparently, disentangling the potential new physics (NP) signals from the unaccounted theoretical uncertainties in the SM computations in a robust manner will be indispensable for the unambiguous interpretation of these emerged flavour anomalies. It remains important to remark that the updated LHCb measurements for the electron-muon universality rations ${\cal R}(K^{(\ast)})$ in the flavour-changing neutral current loop processes $B\to K^{(\ast)}e^{+}e^{-}$ and $B\to K^{(\ast)}\mu^{+}\mu^{-}$ Aaij et al. (2022a, b) do not necessarily lead to conclude the tauon-muon universality in the neutral current $b\to s\ell\bar{\ell}$ transitions Singh Chundawat (2023); Alok et al. (2023), let alone in the flavour-changing charged current tree decays $\bar{B}_{(s)}\to D_{(s)}^{(\ast)}\ell\bar{\nu}_{\ell}$ Algueró et al. (2023). Actually, introducing new CP-violating couplings in the weak effective Hamiltonian for $b\to s\ell\bar{\ell}$ can bring about the significant space to violate the electron-muon universality, while accommodating the new LHCb result for the ${\cal R}(K)$ ratio simultaneously Fleischer et al. (2023).

The model-independent descriptions of the heavy-to-heavy bottom-meson decay form factors in the low recoil regime can be naturally formulated by adopting the heavy quark effective theory (HQET) based upon an expansion in powers of $\Lambda_{\rm QCD}/m_{b,\,c}$, yielding a tower of the non-perturbative Isgur-Wise (IW) functions. In order to better constrain the desirable shape of these form factors, an attractive prescription to derive the unitarity constraints for the form-factor expansion coefficients has been developed from the fundamental field-theoretical principles Boyd et al. (1995, 1996, 1997); Caprini and Macesanu (1996); Caprini et al. (1998); Bourrely et al. (2009). Additionally, we have witnessed the substantial progress on the encouraging lattice QCD calculations of the $\bar{B}\to D^{(\ast)}\ell\bar{\nu}_{\bar{\ell}}$ form factors Bailey et al. (2015a); Na et al. (2015); Bazavov et al. (2022) and the $\bar{B}_{s}\to D_{s}^{(\ast)}\ell\bar{\nu}_{\bar{\ell}}$ form factors Bailey et al. (2012); McLean et al. (2020); Harrison and Davies (2022) at non-zero recoil. By contrast, the continuum QCD determinations of such fundamental heavy-to-heavy form factors at large recoil are mainly achieved by evaluating only the leading-order QCD contributions at the twist-four accuracy Faller et al. (2009); Gubernari et al. (2019), apart from the currently available higher-order QCD computations of the $\bar{B}\to D$ form factors Wang et al. (2017); Gao et al. (2022).

In view of the noticeable significance of implementing the large-recoil theory predictions in the numerical fit of the exclusive bottom-meson decay form factors Arnesen et al. (2005); Cui et al. (2023); Bigi et al. (2017a); Gambino et al. (2019); Bordone et al. (2020a, b); Jaiswal et al. (2017, 2020); Biswas et al. (2022); Cheung et al. (2021), accomplishing the next-to-leading-order (NLO) computations to the semileptonic $\bar{B}_{(s)}\to D_{(s)}^{\ast}\ell\bar{\nu}_{\bar{\ell}}$ form factors will therefore be in high demand for pinning down the obtained uncertainties of their shape parameters from the combined $z$-series fitting procedure. To achieve this goal, we will first establish the NLO factorization formulae for the appropriate bottom-meson-to-vacuum correlation functions at leading power (LP) in the soft-collinear effective theory (SCET) framework and then construct the large logarithmic resummation improved light-cone sum rules (LCSR) for the considered form factors. In particular, we will report on a novel observation of the LP ${\cal O}(\alpha_{s})$ contribution to the longitudinal form factors due to the unsuppressed charm-quark mass dependent pieces in the SCET Lagrangian, when applying the preferable power-counting scheme $m_{c}\sim{\cal O}\left(\sqrt{\Lambda_{\rm QCD}\,m_{b}}\right)$ Boos et al. (2006a, b). Phenomenological implications of the simultaneous Boyd-Grinstein-Lebed (BGL) expansion fitting Boyd et al. (1995, 1996, 1997) to both the SCET sum rules predictions and the lattice simulation data points will be further explored with the focus on the updated extractions of the LFU quantities ${\cal R}(D_{(s)}^{(\ast)})$ and the CKM matrix element $|V_{cb}|$.

We adopt the customary definitions of the bottom-meson decay form factors $\{V,\,A_{0},\,A_{1},\,A_{2}\}$ as displayed in Gao et al. (2020). Implementing the matching program ${\rm QCD}\to{\rm SCET_{I}}$ for these form factors enables us to derive the factorization formulae in terms of the SCET form factors

The newly introduced form factors ${\cal V}$, ${\cal A}_{0}$, ${\cal A}_{1}$ and ${\cal A}_{12}$ can be expressed in terms of the linear combinations of the conventional form factors as displayed in (1) of the Supplemental Material. The explicit definitions of $\xi_{\|,\,\perp}$ and $\Xi_{\|,\,\perp}$ take the same form as the ones for the exclusive heavy-to-light transitions Beneke and Yang (2006), while the remaining two effective form factors $\xi_{\|,\,m_{c}}$ and $\xi_{\perp,\,m_{c}}$ can be defined by

The analytic expressions for the short-distance coefficients $C_{i}^{(\rm A0)}$ and $C_{i}^{(\rm B1)}$ (determined from Beneke et al. (2004); Hill et al. (2004)) as well as $C_{i}^{({\rm A1},\,m_{c})}$ are collected in (2) of the Supplemental Material. We will therefore dedicate the next section to the transparent computations of the effective form factors at ${\cal O}(\alpha_{s})$, including further the tree-level determinations of the next-to-leading power (NLP) corrections.

In analogy to the strategy for computing the heavy-to-light bottom-meson decay matrix elements De Fazio et al. (2006, 2008) (see also Khodjamirian et al. (2005, 2007)), we can derive the LCSR for $\xi_{\|}(n\cdot p)$ by exploring the particular ${\rm SCET_{I}}$ correlation function

where the manifest representations of $j_{\xi\xi,\|\nu}^{(0)}$, $j_{\xi q,\|\nu}^{(2)}$, ${\cal L}_{\xi q}^{(1)}$ and ${\cal L}_{\xi q}^{(2)}$ have been presented in Beneke and Feldmann (2003); Gao et al. (2020). The remaining effective Lagrangian densities and the ${\rm SCET_{I}}$ weak current are presented in (3) of the Supplemental Material. Since the charm-quark mass dependent term ${\cal L}_{\xi m_{c}}^{(0)}$ describes the unsuppressed interaction between the collinear fields Leibovich et al. (2003), the third term in the correlation function (4) will result in the power-enhanced contribution to the correlation function (4). Moreover, the spectator-quark mass contributions from the second and the fourth terms on the right-hand side of (4) can further give rise to the LP effects (see also Leibovich et al. (2003); Böer (2018); Cui et al. (2023)).

Matching the determined spectral representation of $\Pi_{\nu,\,\|}^{\rm(A0)}$ with the corresponding hadronic dispersion relation leads to the desired NLO sum rules

where the two decay constants ${\cal F}_{B_{q}}$ and $f_{D_{q}^{\ast},\|}$ are defined with the conventions of Gao et al. (2020). The lengthy expressions for $\phi_{B,\,{\rm eff}}^{-}$, $\phi_{B,\,{\rm eff}}^{+,\,m_{c}}$ and $\phi_{B,\,{\rm eff}}^{+,\,m_{q}}$ are summarized in the Supplemental Material. Remarkably, the ${\rm SCET_{I}}$ diagram (b) in Figure 1 of the Supplemental Material does not lead to the power-enhanced contribution (but does generate the LP contribution) to the sum rules of $\xi_{\|}$ after implementing the continuum subtraction. Applying the established computational strategy further allows for the construction of the SCET sum rules for the effective form factors $\xi_{\|,\,m_{c}}$ and $\Xi_{\|}$ (shown in (8) and (12) of the Supplemental Material) by investigating the appropriate correlation functions. Along the same vein, we can readily derive the NLO sum rules for the transverse form factors $\xi_{\perp}$, $\xi_{\perp,\,m_{c}}$ and $\Xi_{\perp}$ as collected in the Supplemental Material.

We then proceed to construct the subleading-power sum rules for the $\bar{B}_{q}\to D_{q}^{\ast}$ form factors from four distinct sources: I) the off-light-cone corrections from the two-body nonlocal HQET matrix elements, II) the yet higher-twist corrections from the three-particle $B_{q}$-meson distribution amplitudes Geyer and Witzel (2005); Braun et al. (2017), III) the subleading-power corrections from the effective matrix element of $(\bar{\xi}W_{c})\,\Gamma\,[i\,\not{D}_{\rm\top}/(2m_{b})]\,h_{v}$ Beneke and Feldmann (2003), IV) the higher-order terms from expanding the hard-collinear charm-quark propagator. The tree-level LCSR for the power-suppressed terms ${\cal V}^{\rm NLP}$, ${\cal A}_{0}^{\rm NLP}$, ${\cal A}_{1}^{\rm NLP}$ and ${\cal A}_{12}^{\rm NLP}$ will be presented explicitly in a forthcoming longer write-up.

We are now in a position to explore phenomenological implications of the newly determined SCET sum rules with the three-parameter ansätz for the bottom-meson distribution amplitudes Beneke et al. (2018) (see also Bell et al. (2013); Wang and Shen (2015); Wang (2016); Wang and Shen (2018); Shen et al. (2020); Wang et al. (2022); Feldmann et al. (2022)), which fulfills simultaneously the non-trivial equations-of-motion constraints Beneke et al. (2018), the sum rule determinations of the two inverse moments $\lambda_{B_{d,s}}$ Braun et al. (2004); Khodjamirian et al. (2020) and the HQET parameters $\lambda_{E,\,H}^{2}$ Grozin and Neubert (1997); Nishikawa and Tanaka (2014); Rahimi and Wald (2023), and the asymptotic behaviours at small quark and gluon momenta Braun et al. (2017). It is perhaps worth mentioning an attractive method for the first-principles determination of the $B$-meson light-cone distribution amplitude based upon the large momentum effective theory and the lattice simulation technique Wang et al. (2020). The leptonic decay constants of the pseudoscalar bottom mesons have been extracted from the lattice calculations precisely Aoki et al. (2022). We further employ the QCD sum rule computations Pullin and Zwicky (2021) for both the longitudinal and transverse $D_{q}^{\ast}$ decay constants. The allowed intervals of the intrinsic LCSR parameters $M^{2}=n\cdot p\,\omega_{M}$ and $s_{0}=n\cdot p\,\omega_{s}$ are consistent with the previous determinations Khodjamirian et al. (2009); Duplancic and Melic (2015); Li et al. (2020); Khodjamirian et al. (2021). The choices for the additional parameters in our numerical studies are identical to the ones summarized in Cui et al. (2023).

Inspecting the numerical features for the distinct classes of higher-order contributions indicates that the newly determined NLL QCD corrections can reduce the corresponding tree-level LP predictions by an amount of ${\cal O}(20\,\%)$. As displayed in Figure 1, the particular charm-quark mass dependent contribution $\hat{\xi}_{\|}^{\,m_{c}}$ in (5) appears to bring about the ${\cal O}(5\,\%)$ enhancement of the LP prediction of the form factor ${\cal A}_{0}$ with $q^{2}\in[-3.0,\,2.0]\,{\rm GeV^{2}}$. We further note that the effective form factor $\xi_{\|,\,m_{c}}$ defined in (2) can only result in the negligible impact on the numerical prediction for ${\cal A}_{0}$ in the large recoil region due to the kinematical suppression of the multiplication hard coefficient. It remains important to remark that the yielding uncertainties of our ${\rm NLL\oplus NLP}$ LCSR predictions arise, on the one hand, from the SCET computations of the bottom-meson-to-vacuum correlation functions and, on the other hand, from the extraction of the ground-state charmed meson contribution with the parton-hadron duality ansätz and the Borel transformation. In order to determine the mean values and theoretical uncertainties of the $\bar{B}_{q}\to D_{q}^{\ast}$ form factors, we employ the statistical procedure discussed in Sentitemsu Imsong et al. (2015); Leljak et al. (2021) by simultaneously scanning the complete set of input parameters (displayed in Table I of the Supplemental Material) in the adopted intervals with the prior distribution. It is worthwhile mentioning further that the systematic uncertainty due to the parton-hadron duality ansätz has been addressed in a wide range of QCD computations (e.g., Chibisov et al. (1997); Shifman (2000); Bigi and Uraltsev (2001); Cata et al. (2005); Beylich et al. (2011); Jamin (2011); Dingfelder and Mannel (2016); Boito et al. (2018); Pich (2021); Pich and Rodríguez-Sánchez (2022)), leading to the encouraging observation on the smallness of duality violations in the inclusive hadron production in $e^{+}\,e^{-}$ annihilations Pich (2021), in the hadronic decays of the $\tau$-lepton Pich and Rodríguez-Sánchez (2022), and in the semileptonic bottom-meson decays Dingfelder and Mannel (2016). In an attempt to obtain more conservative predictions of our SCET sum rules, we nevertheless increase the default intervals of the sum-rule parameters $M^{2}$ and $s_{0}$ by a factor of two in the ultimate error estimates. As elaborated further in the Supplemental Material, one of the principal benefits from our LCSR analysis consists in the yielding strong correlations between the LCSR data points at distinct $q^{2}$-values, which are expected to be particularly insensitive to the duality approximation.

In order to extrapolate the LCSR predictions for the $\bar{B}_{q}\to D_{q}^{\ast}$ form factors towards the large momentum transfer, we will apply the BGL parametrization Boyd et al. (1995, 1996, 1997) as widely employed in the form-factor determinations (see, for instance Bigi et al. (2017b, a); Gambino et al. (2019)) and then perform the binned $\chi^{2}$ fit of our LCSR predictions at $q^{2}\in\left\{-3.0,\,-2.0,\,-1.0,\,0.0,\,1.0,\,2.0\right\}\,{\rm GeV}^{2}$, in combination with the available lattice QCD results. Moreover, we will implement the strong unitarity constraints on the BGL coefficients $b_{n}^{i}$ by including all the two-body $\bar{B}_{q}^{(\ast)}\to D_{q}^{(\ast)}$ channels. Consequently, we will adopt the HQET relations between these form factors at ${\cal O}(\alpha_{s},\,1/m_{b},\,1/m_{c}^{2})$, allowing us to express the $\bar{B}_{q}^{\ast}\to D_{q}^{(\ast)}$ form factors in terms of the corresponding $\bar{B}_{q}\to D_{q}^{(\ast)}$ form factors, the NLP IW functions $\hat{\chi}_{2,\,3}^{(q)}(\omega)$ and $\hat{\eta}^{(q)}(\omega)$ Neubert (1994), and the ${\cal O}(1/m_{c}^{2})$ IW functions $\hat{\ell}_{1,...,6}(\omega)$ Falk and Neubert (1993). The determined intervals of the normalization and the slop parameters of these IW functions from Bordone et al. (2020b) will therefore be employed.

Combining the obtained LCSR results for the $\bar{B}_{q}\to D_{q}^{\ast}$ form factors with I) the updated SCET sum rules for the $\bar{B}_{q}\to D_{q}$ form factors Gao et al. (2022), II) the lattice results for the $\bar{B}\to D$ form factors at $\omega=\left\{1.00,\,1.08,\,1.16\right\}$ from the FNAL/MILC Collaboration Bailey et al. (2015a) and the synthetic data points at $\omega=\left\{1.01,\,1.06\right\}$ from the HPQCD analysis Na et al. (2015), III) the unquenched lattice results for the $\bar{B}\to D^{\ast}$ form factors at $\omega=\left\{1.03,\,1.10,\,1.17\right\}$ from the FNAL/MILC Collaboration Bazavov et al. (2022), IV) the lattice determinations of the $\bar{B}_{s}\to D_{s}$ form factors at $\omega=\left\{1.0,\,1.06,\,1.12\right\}$ from the HPQCD analysis McLean et al. (2020), V) the lattice computations of the $\bar{B}_{s}\to D_{s}^{\ast}$ form factors at $\omega=\left\{1.0,\,1.04,\,1.08\right\}$ Harrison and Davies (2022), we display in Table 1 the resulting $z$-series coefficients from the combined BGL fitting with the truncation $n=3$. In order to better understand the phenomenological significance of including the LCSR data points in the constrained BGL fit, we further carry out an alternative fit to the “only lattice QCD” data points of the $\bar{B}\to D^{(\ast)}$ form factors Bailey et al. (2015a); Na et al. (2015); Bazavov et al. (2022) and the $\bar{B}_{s}\to D_{s}^{(\ast)}$ form factors McLean et al. (2020); Harrison and Davies (2022). It needs to be stressed that our major objective is to investigate whether the inclusion of the large-recoil LCSR data points in the BGL fit strategy allows us to increase effectively the precision of the yielding form factors. It is evident from Figure 2 that taking into account the ${\rm NLL\oplus NLP}$ LCSR results will indeed be highly beneficial for pining down the uncertainties of the $\bar{B}\to D^{(\ast)}$ form factors at small momentum transfer. As the third fit model, we also carry out the statistical analysis by taking into account the experimental data points for $\bar{B}_{(s)}\to D_{(s)}^{(\ast)}\ell\bar{\nu}_{\ell}$ together with the lattice QCD results and the LCSR predictions.

We continue to present the correlated numerical predictions for ${\cal R}({D})$ and ${\cal R}({D^{\ast}})$ in Figure 3, confronting with the available measurements from the BaBar Lees et al. (2012a, 2013), Belle Huschle et al. (2015); Caria et al. (2020); Hirose et al. (2017), LHCb Aaij et al. (2023a, b) and Belle II Collaborations Kojima et al. (2023). Importantly, our determinations of ${\cal R}({D})$ and ${\cal R}({D^{\ast}})$ by encompassing the LCSR results in the numerical fit deviate from the HFLAV-averaged measurements by approximately $2.6\,\sigma$, in contrast with the $3.3\,\sigma$ discrepancy between the arithmetic average of the previous SM predictions Bigi and Gambino (2016); Bordone et al. (2020a); Gambino et al. (2019) and the state-of-the-art experimental average Amhis et al. (2022). Subsequently, we carry out the simultaneous fit for $|V_{cb}|$ and $|V_{ub}|$ by taking the determined $\bar{B}_{(s)}\to D_{(s)}^{(\ast)}$ form factors and the updated predictions of the $\bar{B}\to\pi$ form factors in Cui et al. (2023) in combination with the lattice data points for the $\bar{B}_{(s)}\to D_{(s)}^{(\ast)}$ form factors Bailey et al. (2015a); Na et al. (2015); Bazavov et al. (2022); McLean et al. (2020); Harrison and Davies (2022) and for the $\bar{B}\to\pi$ form factors Flynn et al. (2015); Bailey et al. (2015b, c) and the experimental measurements for $\bar{B}_{(s)}\to D_{(s)}^{(\ast)}\ell\bar{\nu}_{\ell}$ Glattauer et al. (2016); Waheed et al. (2019); Aaij et al. (2020a, b) and for $\bar{B}\to\pi\ell\bar{\nu}_{\ell}$ del Amo Sanchez et al. (2011); Lees et al. (2012b); Ha et al. (2011); Sibidanov et al. (2013); Adamczyk et al. (2022). The yielding intervals of $|V_{cb}|$ and $|V_{ub}|$ are

which coincide with the previous exclusive extractions for $|V_{cb}|$ (for instance, Bigi et al. (2017a); Grinstein and Kobach (2017); Gambino et al. (2019); Bordone et al. (2020a, b); Jaiswal et al. (2017, 2020); Biswas et al. (2022); Martinelli et al. (2022a, b, c); Iguro and Watanabe (2020)) and for $|V_{ub}|$ (e.g., Leljak et al. (2021); Biswas et al. (2021); Biswas and Nandi (2021); Martinelli et al. (2022d)). Unsurprisingly, the obtained numerical predictions (6) remain to be in tension with the world averages of the inclusive determinations $|V_{cb}|=(42.2\pm 0.8)\times 10^{-3}$ and $|V_{ub}|=(4.13\pm 0.12^{+0.13}_{-0.14}\pm 0.18)\times 10^{-3}$ Workman et al. (2022) at the level of $2.5\,\sigma$ and $1.5\,\sigma$, respectively. In an attempt to understand qualitatively the systematic uncertainties due to the truncated BGL expansions, we increase the expansion order to $n=4$ and repeat the numerical fit procedure for the form factors, yielding the theory predictions for both the LFU ratios ${\cal R}(D^{(\ast)})$ and the CKM matrix elements $|V_{cb}|$ and $|V_{ub}|$ only marginally different from the achieved results with $n=3$.

In conclusion, we have endeavored to accomplish for the first time the complete next-to-leading-order QCD computations of the $\bar{B}_{q}\to D_{q}^{(\ast)}\ell\bar{\nu}_{\ell}$ form factors at large recoil and identified explicitly the unsuppressed charm-quark-mass dependent contributions in the heavy quark expansion. Taking into account the obtained LCSR data points in the BGL $z$-series fit of the $\bar{B}_{q}\to D_{q}^{(\ast)}\ell\bar{\nu}_{\ell}$ form factors enabled us to enhance significantly the achieved accuracy of the large-recoil theory predictions for the $\bar{B}\to D^{(\ast)}$ form factors. Implementing the determined LCSR results in the statistical analysis turned out to be advantageous to mitigate the combined ${\cal R}({D})$ and ${\cal R}({D^{\ast}})$ tension between the SM predictions and the HFLAV-averaged measurements. Our computations of the heavy-to-heavy form factors near the maximal recoil will be of notable importance for obtaining the yet higher precision predictions of the $\bar{B}_{q}\to D_{q}^{(\ast)}\ell\bar{\nu}_{\ell}$ decay observables, when combined with the upcoming lattice determinations of the bottom-meson distribution amplitudes.

We are grateful to Thomas Mannel, Zi-Hao Mi, Ru-Ying Tang, Alejandro Vaquero, Chao Wang and Yan-Bing Wei for illuminating discussions. Y.M.W. acknowledges support from the National Natural Science Foundation of China with Grant No. 11735010 and 12075125, and the Natural Science Foundation of Tianjin with Grant No. 19JCJQJC61100.