Combined analysis of semileptonic $B$ decays to $D$ and $D^*$: $R(D^{(*)})$, $|V_{cb}|$, and new physics

TL;DR

This paper performs a comprehensive analysis of semileptonic B decays to D and D* mesons, incorporating uncertainties and higher-order corrections to improve predictions of decay rates, CKM matrix element |V_cb|, and potential new physics effects.

Contribution

It introduces a novel combined fit method including subleading Isgur-Wise functions and uncertainties, providing more precise predictions for decay rates and new physics constraints.

Findings

Predicted B to D(*) tau nu decay rates with improved accuracy.

Quantified uncertainties in form factor ratios at order Λ_QCD/m_c,b.

Assessed potential impacts of new physics on semileptonic decay rates.

Abstract

The measured $\bar{B}\to D^{(*)} l \bar\nu$ decay rates for light leptons ($l=e,\mu$) constrain all $\bar{B}\to D^{(*)}$ semileptonic form factors, by including both the leading and ${\cal O}(\Lambda_{\text{QCD}}/m_{c,b})$ subleading Isgur-Wise functions in the heavy quark effective theory. We perform a novel combined fit to the $\bar{B}\to D^{(*)} l \bar\nu$ decay distributions to predict the $\bar{B} \to D^{(*)} \tau\bar\nu$ rates and determine the CKM matrix element $|V_{cb}|$. Most theoretical and experimental papers have neglected uncertainties in the predictions for form factor ratios at order $\Lambda_{\text{QCD}}/m_{c,b}$, which we include. We also calculate ${\cal O}(\Lambda_{\text{QCD}}/m_{c,b})$ and ${\cal O}(\alpha_s)$ contributions to semileptonic $\bar{B}\to D^{(*)} \ell \bar\nu$ decays for all possible $b \to c$ currents. This result has not been available for the tensor current form factors, and for two of those, which are ${\cal O}(\Lambda_{\text{QCD}}/m_{c,b})$, the corrections are of the same order as approximations used in the literature. These results allow us to determine with improved precision how new physics may affect the $\bar{B}\to D^{(*)} \tau\bar\nu$ rates. Our predictions can be systematically improved with more data; they need not rely on lattice QCD results, although these can be incorporated.

Tables

Table 1. Table 1: Summary of theory and data inputs for each fit scenario. All use the HQET predictions to order 𝒪 ​ ( Λ QCD / m c , b ) 𝒪 subscript Λ QCD subscript 𝑚 𝑐 𝑏 \mathcal{O}(\Lambda_{\rm QCD}/m_{c,b}) and 𝒪 ​ ( α s ) 𝒪 subscript 𝛼 𝑠 \mathcal{O}(\alpha_{s}) , as well as the unitarity constraints.

Fit	QCDSR	Lattice QCD			Belle Data
		$ℱ\left(1\right)$	$f_{+,0}\left(1\right)$	$f_{+,0}\left(w>1\right)$
${\text{L}}_{w=1}$	—	$✓$	$✓$	—	$✓$
${\text{L}}_{w=1}+\text{SR}$	$✓$	$✓$	$✓$	—	$✓$
NoL	—	—	—	—	$✓$
$\text{NoL}+\text{SR}$	$✓$	—	—	—	$✓$
${\text{L}}_{w\ge 1}$	—	$✓$	$✓$	$✓$	$✓$
${\text{L}}_{w\ge 1}+\text{SR}$	$✓$	$✓$	$✓$	$✓$	$✓$
${\text{th:}\text{L}}_{w\ge 1}+\text{SR}$	$✓$	$✓$	$✓$	$✓$	—

Table 2. Table 2: Summary of the results for the fit scenarios considered. The correlations are shown in Appendix  B .

	${\text{L}}_{w=1}$	${\text{L}}_{w=1}+\text{SR}$	NoL	$\text{NoL}+\text{SR}$	${\text{L}}_{w\ge 1}$	${\text{L}}_{w\ge 1}+\text{SR}$	${\text{th:}\text{L}}_{w\ge 1}+\text{SR}$
${\chi }^2$	$40.2$	$44.0$	$38.7$	$43.1$	$49.0$	$53.8$	$7.4$
dof	$44$	$48$	$43$	$47$	$48$	$52$	$4$
$\left|V_{cb}\right|\times {10}^3$	$38.8\pm 1.2$	$38.5\pm 1.1$	—	—	$39.1\pm 1.1$	$39.3\pm 1.0$	—
$𝒢\left(1\right)$	$1.055\pm 0.008$	$1.056\pm 0.008$	—	—	$1.060\pm 0.008$	$1.061\pm 0.007$	$1.052\pm 0.008$
$ℱ\left(1\right)$	$0.904\pm 0.012$	$0.901\pm 0.011$	—	—	$0.898\pm 0.012$	$0.895\pm 0.011$	$0.906\pm 0.013$
${\overline{\rho }}_{\ast }^2$	$1.17\pm 0.12$	$1.19\pm 0.07$	$1.06\pm 0.15$	$1.19\pm 0.08$	$1.33\pm 0.11$	$1.24\pm 0.06$	$1.24\pm 0.08$
${\hat{\chi }}_2\left(1\right)$	$-0.26\pm 0.26$	$-0.07\pm 0.02$	$0.36\pm 0.62$	$-0.06\pm 0.02$	$0.13\pm 0.22$	$-0.06\pm 0.02$	$-0.06\pm 0.02$
${\hat{\chi }}_2^{\prime }\left(1\right)$	$0.21\pm 0.38$	$-0.00\pm 0.02$	$0.14\pm 0.39$	$-0.00\pm 0.02$	$-0.36\pm 0.28$	$-0.00\pm 0.02$	$-0.00\pm 0.02$
${\hat{\chi }}_3^{\prime }\left(1\right)$	$0.02\pm 0.07$	$0.05\pm 0.02$	$0.18\pm 0.19$	$0.04\pm 0.02$	$0.09\pm 0.07$	$0.05\pm 0.02$	$0.04\pm 0.02$
$\eta \left(1\right)$	$0.30\pm 0.04$	$0.30\pm 0.03$	$-0.56\pm 0.80$	$0.35\pm 0.14$	$0.30\pm 0.04$	$0.30\pm 0.03$	$0.31\pm 0.04$
${\eta }^{\prime }\left(1\right)$	$0\text{(fixed)}$	$-0.12\pm 0.16$	$0\text{(fixed)}$	$-0.11\pm 0.18$	$0\text{(fixed)}$	$-0.05\pm 0.09$	$0.05\pm 0.10$
$m_b^{1S}\text{[GeV]}$	$4.70\pm 0.05$	$4.70\pm 0.05$	$4.71\pm 0.05$	$4.70\pm 0.05$	$4.71\pm 0.05$	$4.71\pm 0.05$	$4.71\pm 0.05$
$\delta m_{bc}\text{[GeV]}$	$3.40\pm 0.02$	$3.40\pm 0.02$	$3.40\pm 0.02$	$3.40\pm 0.02$	$3.40\pm 0.02$	$3.40\pm 0.02$	$3.40\pm 0.02$

Table 3. Table 3: The predictions for the form factors f + , 0 subscript 𝑓 0 f_{+,0} at w = 1.0 ,  1.08 ,  1.16 𝑤 1.0 1.08 1.16 w=1.0,\,1.08,\,1.16 using the synthetic data results of Ref.  Bailey  et al. ( 2015 ) . The correlations can be found in Table VII in Ref.  Bailey  et al. ( 2015 ) .

Form factor	$w=1.0$	$w=1.08$	$w=1.16$
$f_{+}$	$1.1994\pm 0.0095$	$1.0941\pm 0.0104$	$1.0047\pm 0.0123$
$f_0$	$0.9026\pm 0.0072$	$0.8609\pm 0.0077$	$0.8254\pm 0.0094$

Table 4. Table 4: The R ​ ( D ) 𝑅 𝐷 R(D) and R ​ ( D ∗ ) 𝑅 superscript 𝐷 R(D^{*}) predictions for our fit scenarios, the world average of the data, and other theory predictions. The fit scenarios are described in the text and in Table  1 . The bold numbers are our most precise predictions.

Scenario	$R\left(D\right)$	$R\left(D^{\ast }\right)$	Correlation
${\text{L}}_{w=1}$	$0.292\pm 0.005$	$0.255\pm 0.005$	41%
${\text{L}}_{w=1}+\text{SR}$	$0.291\pm 0.005$	$0.255\pm 0.003$	57%
NoL	$0.273\pm 0.016$	$0.250\pm 0.006$	49%
$\text{NoL}+\text{SR}$	$0.295\pm 0.007$	$0.255\pm 0.004$	43%
${\text{L}}_{w\ge 1}$	$0.298\pm 0.003$	$0.261\pm 0.004$	19%
${\text{L}}_{w\ge 1}+\text{SR}$	$0.299\pm 0.003$	$0.257\pm 0.003$	$\mathrm{𝟒𝟒}$%
${\text{th:}\text{L}}_{w\ge 1}+\text{SR}$	$0.306\pm 0.005$	$0.256\pm 0.004$	33%
Data Amhis et al. (2016)	$0.403\pm 0.047$	$0.310\pm 0.017$	$-23\%$
Refs. Aoki et al. (2016); Bailey et al. (2015); Na et al. (2015)	$0.300\pm 0.008$	—	—
Ref. Bigi and Gambino (2016)	$0.299\pm 0.003$	—	—
Ref. Fajfer et al. (2012)	—	$0.252\pm 0.003$	—

Table 5. Table 5: The correlations of the “ L w = 1 subscript L 𝑤 1 \text{L}_{w=1} ” fit scenario.

	$\left|V_{cb}\right|$	$𝒢\left(1\right)$	$ℱ\left(1\right)$	${\overline{\rho }}_{\ast }^2$	${\hat{\chi }}_2\left(1\right)$	${\hat{\chi }}_2^{\prime }\left(1\right)$	${\hat{\chi }}_3^{\prime }\left(1\right)$	$\eta \left(1\right)$	$m_b^{1S}$	$\delta m_{bc}$
$\left|V_{cb}\right|$	$1.00$	$-0.16$	$-0.18$	$0.30$	$-0.13$	$0.28$	$0.11$	$0.04$	$-0.01$	$0.00$
$𝒢\left(1\right)$	$-0.16$	$1.00$	$0.06$	$-0.11$	$0.03$	$-0.04$	$-0.09$	$-0.23$	$0.00$	$-0.00$
$ℱ\left(1\right)$	$-0.18$	$0.06$	$1.00$	$0.18$	$-0.00$	$0.08$	$0.21$	$-0.02$	$0.01$	$-0.00$
${\overline{\rho }}_{\ast }^2$	$0.30$	$-0.11$	$0.18$	$1.00$	$0.67$	$-0.47$	$0.82$	$0.13$	$-0.16$	$0.01$
${\hat{\chi }}_2\left(1\right)$	$-0.13$	$0.03$	$-0.00$	$0.67$	$1.00$	$-0.87$	$0.82$	$-0.11$	$0.07$	$-0.01$
${\hat{\chi }}_2^{\prime }\left(1\right)$	$0.28$	$-0.04$	$0.08$	$-0.47$	$-0.87$	$1.00$	$-0.47$	$0.01$	$0.01$	$-0.00$
${\hat{\chi }}_3^{\prime }\left(1\right)$	$0.11$	$-0.09$	$0.21$	$0.82$	$0.82$	$-0.47$	$1.00$	$-0.12$	$0.12$	$-0.02$
$\eta \left(1\right)$	$0.04$	$-0.23$	$-0.02$	$0.13$	$-0.11$	$0.01$	$-0.12$	$1.00$	$-0.52$	$0.05$
$m_b^{1S}$	$-0.01$	$0.00$	$0.01$	$-0.16$	$0.07$	$0.01$	$0.12$	$-0.52$	$1.00$	$0.00$
$\delta m_{bc}$	$0.00$	$-0.00$	$-0.00$	$0.01$	$-0.01$	$-0.00$	$-0.02$	$0.05$	$0.00$	$1.00$

Table 6. Table 6: The correlations of the “ L w = 1 + SR subscript L 𝑤 1 SR \text{L}_{w=1}{+\text{SR}} ” fit scenario.

	$\left|V_{cb}\right|$	$𝒢\left(1\right)$	$ℱ\left(1\right)$	${\overline{\rho }}_{\ast }^2$	${\hat{\chi }}_2\left(1\right)$	${\hat{\chi }}_2^{\prime }\left(1\right)$	${\hat{\chi }}_3^{\prime }\left(1\right)$	$\eta \left(1\right)$	${\eta }^{\prime }\left(1\right)$	$m_b^{1S}$	$\delta m_{bc}$
$\left|V_{cb}\right|$	$1.00$	$-0.12$	$-0.32$	$0.48$	$-0.02$	$0.02$	$0.14$	$0.05$	$0.02$	$-0.02$	$0.00$
$𝒢\left(1\right)$	$-0.12$	$1.00$	$0.14$	$-0.05$	$0.04$	$0.01$	$-0.14$	$-0.23$	$0.09$	$-0.00$	$-0.00$
$ℱ\left(1\right)$	$-0.32$	$0.14$	$1.00$	$0.04$	$-0.07$	$-0.01$	$0.24$	$-0.02$	$-0.11$	$-0.03$	$0.01$
${\overline{\rho }}_{\ast }^2$	$0.48$	$-0.05$	$0.04$	$1.00$	$-0.09$	$-0.04$	$0.57$	$0.32$	$0.08$	$-0.45$	$0.04$
${\hat{\chi }}_2\left(1\right)$	$-0.02$	$0.04$	$-0.07$	$-0.09$	$1.00$	$-0.03$	$0.17$	$-0.06$	$-0.20$	$0.04$	$-0.00$
${\hat{\chi }}_2^{\prime }\left(1\right)$	$0.02$	$0.01$	$-0.01$	$-0.04$	$-0.03$	$1.00$	$0.06$	$-0.02$	$-0.09$	$0.01$	$-0.00$
${\hat{\chi }}_3^{\prime }\left(1\right)$	$0.14$	$-0.14$	$0.24$	$0.57$	$0.17$	$0.06$	$1.00$	$0.08$	$0.38$	$-0.03$	$0.00$
$\eta \left(1\right)$	$0.05$	$-0.23$	$-0.02$	$0.32$	$-0.06$	$-0.02$	$0.08$	$1.00$	$-0.14$	$-0.48$	$0.05$
${\eta }^{\prime }\left(1\right)$	$0.02$	$0.09$	$-0.11$	$0.08$	$-0.20$	$-0.09$	$0.38$	$-0.14$	$1.00$	$0.08$	$-0.01$
$m_b^{1S}$	$-0.02$	$-0.00$	$-0.03$	$-0.45$	$0.04$	$0.01$	$-0.03$	$-0.48$	$0.08$	$1.00$	$0.01$
$\delta m_{bc}$	$0.00$	$-0.00$	$0.01$	$0.04$	$-0.00$	$-0.00$	$0.00$	$0.05$	$-0.01$	$0.01$	$1.00$

Table 7. Table 7: The correlations of the “ NoL ” fit scenario.

	${\overline{\rho }}_{\ast }^2$	${\hat{\chi }}_2\left(1\right)$	${\hat{\chi }}_2^{\prime }\left(1\right)$	${\hat{\chi }}_3^{\prime }\left(1\right)$	$\eta \left(1\right)$	$m_b^{1S}$	$\delta m_{bc}$
${\overline{\rho }}_{\ast }^2$	$1.00$	$-0.22$	$-0.18$	$-0.03$	$0.46$	$-0.22$	$0.01$
${\hat{\chi }}_2\left(1\right)$	$-0.22$	$1.00$	$-0.41$	$0.94$	$-0.92$	$0.33$	$-0.03$
${\hat{\chi }}_2^{\prime }\left(1\right)$	$-0.18$	$-0.41$	$1.00$	$-0.19$	$0.08$	$-0.02$	$-0.00$
${\hat{\chi }}_3^{\prime }\left(1\right)$	$-0.03$	$0.94$	$-0.19$	$1.00$	$-0.88$	$0.32$	$-0.03$
$\eta \left(1\right)$	$0.46$	$-0.92$	$0.08$	$-0.88$	$1.00$	$-0.35$	$0.02$
$m_b^{1S}$	$-0.22$	$0.33$	$-0.02$	$0.32$	$-0.35$	$1.00$	$0.00$
$\delta m_{bc}$	$0.01$	$-0.03$	$-0.00$	$-0.03$	$0.02$	$0.00$	$1.00$

Table 8. Table 8: The correlations of the “ NoL + SR NoL SR \text{NoL}{+\text{SR}} ” fit scenario.

	${\overline{\rho }}_{\ast }^2$	${\hat{\chi }}_2\left(1\right)$	${\hat{\chi }}_2^{\prime }\left(1\right)$	${\hat{\chi }}_3^{\prime }\left(1\right)$	$\eta \left(1\right)$	${\eta }^{\prime }\left(1\right)$	$m_b^{1S}$	$\delta m_{bc}$
${\overline{\rho }}_{\ast }^2$	$1.00$	$-0.15$	$-0.07$	$0.57$	$0.44$	$-0.11$	$-0.31$	$0.03$
${\hat{\chi }}_2\left(1\right)$	$-0.15$	$1.00$	$-0.02$	$0.07$	$-0.15$	$-0.09$	$0.02$	$-0.00$
${\hat{\chi }}_2^{\prime }\left(1\right)$	$-0.07$	$-0.02$	$1.00$	$0.03$	$-0.07$	$-0.05$	$0.00$	$-0.00$
${\hat{\chi }}_3^{\prime }\left(1\right)$	$0.57$	$0.07$	$0.03$	$1.00$	$0.17$	$0.16$	$0.00$	$-0.00$
$\eta \left(1\right)$	$0.44$	$-0.15$	$-0.07$	$0.17$	$1.00$	$-0.40$	$0.09$	$-0.01$
${\eta }^{\prime }\left(1\right)$	$-0.11$	$-0.09$	$-0.05$	$0.16$	$-0.40$	$1.00$	$0.02$	$-0.00$
$m_b^{1S}$	$-0.31$	$0.02$	$0.00$	$0.00$	$0.09$	$0.02$	$1.00$	$0.01$
$\delta m_{bc}$	$0.03$	$-0.00$	$-0.00$	$-0.00$	$-0.01$	$-0.00$	$0.01$	$1.00$

Table 9. Table 9: The correlations of the “ L w ≥ 1 subscript L 𝑤 1 \text{L}_{w\geq 1} ” fit scenario.

	$\left|V_{cb}\right|\times {10}^3$	$𝒢\left(1\right)$	$ℱ\left(1\right)$	${\overline{\rho }}_{\ast }^2$	${\chi }_2\left(1\right)$	${\chi }_2^{\prime }$	${\chi }_3^{\prime }$	$\eta \left(1\right)$	$m_b^{1S}$	$\delta m_{bc}$
$\left|V_{cb}\right|\times {10}^3$	$1.00$	$-0.30$	$-0.16$	$0.18$	$-0.13$	$0.28$	$0.07$	$0.01$	$0.00$	$0.00$
$𝒢\left(1\right)$	$-0.30$	$1.00$	$0.08$	$-0.28$	$-0.04$	$-0.04$	$-0.16$	$-0.23$	$0.01$	$-0.00$
$ℱ\left(1\right)$	$-0.16$	$0.08$	$1.00$	$0.38$	$0.18$	$-0.10$	$0.32$	$0.00$	$0.01$	$-0.00$
${\overline{\rho }}_{\ast }^2$	$0.18$	$-0.28$	$0.38$	$1.00$	$0.64$	$-0.44$	$0.80$	$0.18$	$-0.22$	$0.01$
${\hat{\chi }}_2\left(1\right)$	$-0.13$	$-0.04$	$0.18$	$0.64$	$1.00$	$-0.79$	$0.89$	$-0.17$	$0.21$	$-0.03$
${\hat{\chi }}_2^{\prime }\left(1\right)$	$0.28$	$-0.04$	$-0.10$	$-0.44$	$-0.79$	$1.00$	$-0.48$	$0.05$	$-0.13$	$0.02$
${\hat{\chi }}_3^{\prime }\left(1\right)$	$0.07$	$-0.16$	$0.32$	$0.80$	$0.89$	$-0.48$	$1.00$	$-0.12$	$0.18$	$-0.03$
$\eta \left(1\right)$	$0.01$	$-0.23$	$0.00$	$0.18$	$-0.17$	$0.05$	$-0.12$	$1.00$	$-0.54$	$0.05$
$m_b^{1S}$	$0.00$	$0.01$	$0.01$	$-0.22$	$0.21$	$-0.13$	$0.18$	$-0.54$	$1.00$	$0.01$
$\delta m_{bc}$	$0.00$	$-0.00$	$-0.00$	$0.01$	$-0.03$	$0.02$	$-0.03$	$0.05$	$0.01$	$1.00$

Table 10. Table 10: The correlations of the “ L w ≥ 1 + SR subscript L 𝑤 1 SR \text{L}_{w\geq 1}{+\text{SR}} ” fit scenario.

	$\left|V_{cb}\right|$	$𝒢\left(1\right)$	$ℱ\left(1\right)$	${\overline{\rho }}_{\ast }^2$	${\hat{\chi }}_2\left(1\right)$	${\hat{\chi }}_2^{\prime }\left(1\right)$	${\hat{\chi }}_3^{\prime }\left(1\right)$	$\eta \left(1\right)$	${\eta }^{\prime }\left(1\right)$	$m_b^{1S}$	$\delta m_{bc}$
$\left|V_{cb}\right|$	$1.00$	$-0.27$	$-0.28$	$0.33$	$-0.05$	$0.00$	$0.21$	$0.03$	$0.13$	$-0.01$	$0.00$
$𝒢\left(1\right)$	$-0.27$	$1.00$	$0.24$	$-0.26$	$0.07$	$0.02$	$-0.22$	$-0.22$	$-0.27$	$-0.02$	$0.00$
$ℱ\left(1\right)$	$-0.28$	$0.24$	$1.00$	$0.15$	$-0.06$	$-0.02$	$0.25$	$-0.03$	$-0.19$	$-0.01$	$0.00$
${\overline{\rho }}_{\ast }^2$	$0.33$	$-0.26$	$0.15$	$1.00$	$-0.13$	$-0.08$	$0.72$	$0.35$	$0.13$	$-0.49$	$0.05$
${\hat{\chi }}_2\left(1\right)$	$-0.05$	$0.07$	$-0.06$	$-0.13$	$1.00$	$-0.04$	$0.25$	$-0.10$	$-0.11$	$0.06$	$-0.01$
${\hat{\chi }}_2^{\prime }\left(1\right)$	$0.00$	$0.02$	$-0.02$	$-0.08$	$-0.04$	$1.00$	$0.07$	$-0.05$	$-0.06$	$0.04$	$-0.00$
${\hat{\chi }}_3^{\prime }\left(1\right)$	$0.21$	$-0.22$	$0.25$	$0.72$	$0.25$	$0.07$	$1.00$	$0.16$	$0.19$	$-0.06$	$0.01$
$\eta \left(1\right)$	$0.03$	$-0.22$	$-0.03$	$0.35$	$-0.10$	$-0.05$	$0.16$	$1.00$	$0.05$	$-0.48$	$0.05$
${\eta }^{\prime }\left(1\right)$	$0.13$	$-0.27$	$-0.19$	$0.13$	$-0.11$	$-0.06$	$0.19$	$0.05$	$1.00$	$0.04$	$0.00$
$m_b^{1S}$	$-0.01$	$-0.02$	$-0.01$	$-0.49$	$0.06$	$0.04$	$-0.06$	$-0.48$	$0.04$	$1.00$	$0.01$
$\delta m_{bc}$	$0.00$	$0.00$	$0.00$	$0.05$	$-0.01$	$-0.00$	$0.01$	$0.05$	$0.00$	$0.01$	$1.00$

Table 11. Table 11: The correlations of the “ th: L w ≥ 1 + SR subscript th: L 𝑤 1 SR \text{th:}\text{L}_{w\geq 1}{+\text{SR}} ” fit scenario.

	$𝒢\left(1\right)$	$ℱ\left(1\right)$	${\overline{\rho }}_{\ast }^2$	${\hat{\chi }}_2\left(1\right)$	${\hat{\chi }}_2^{\prime }\left(1\right)$	${\hat{\chi }}_3^{\prime }\left(1\right)$	$\eta \left(1\right)$	${\eta }^{\prime }\left(1\right)$	$m_b^{1S}$	$\delta m_{bc}$
$𝒢\left(1\right)$	$1.00$	$0.00$	$-0.15$	$0.01$	$-0.00$	$-0.02$	$-0.25$	$-0.40$	$0.01$	$-0.00$
$ℱ\left(1\right)$	$0.00$	$1.00$	$-0.00$	$-0.00$	$-0.00$	$-0.00$	$-0.00$	$-0.00$	$-0.00$	$-0.00$
${\overline{\rho }}_{\ast }^2$	$-0.15$	$-0.00$	$1.00$	$-0.27$	$-0.13$	$0.81$	$0.08$	$-0.07$	$-0.24$	$0.02$
${\hat{\chi }}_2\left(1\right)$	$0.01$	$-0.00$	$-0.27$	$1.00$	$0.00$	$0.01$	$0.01$	$0.03$	$-0.01$	$0.00$
${\hat{\chi }}_2^{\prime }\left(1\right)$	$-0.00$	$-0.00$	$-0.13$	$0.00$	$1.00$	$-0.01$	$-0.01$	$0.01$	$0.01$	$-0.00$
${\hat{\chi }}_3^{\prime }\left(1\right)$	$-0.02$	$-0.00$	$0.81$	$0.01$	$-0.01$	$1.00$	$-0.02$	$-0.09$	$0.04$	$-0.00$
$\eta \left(1\right)$	$-0.25$	$-0.00$	$0.08$	$0.01$	$-0.01$	$-0.02$	$1.00$	$0.11$	$-0.48$	$0.04$
${\eta }^{\prime }\left(1\right)$	$-0.40$	$-0.00$	$-0.07$	$0.03$	$0.01$	$-0.09$	$0.11$	$1.00$	$0.07$	$-0.01$
$m_b^{1S}$	$0.01$	$-0.00$	$-0.24$	$-0.01$	$0.01$	$0.04$	$-0.48$	$0.07$	$1.00$	$0.00$
$\delta m_{bc}$	$-0.00$	$-0.00$	$0.02$	$0.00$	$-0.00$	$-0.00$	$0.04$	$-0.01$	$0.00$	$1.00$

Equations

$$R(D^{(\ast )})=\frac{\Gamma (\overline{B}\to D^{(\ast )}\tau \bar{\nu })}{\Gamma (\overline{B}\to D^{(\ast )}l\bar{\nu })},l=\mu ,e,$$

$$O_S=\bar{c}b,O_P=\bar{c}{\gamma }^{5}b,O_V=\bar{c}{\gamma }^{\mu }b,O_A=\bar{c}{\gamma }^{\mu }{\gamma }^{5}b,O_T=\bar{c}{\sigma }^{\mu \nu }b,$$

$$m_{V,P}=m_Q+\bar{\Lambda }-\frac{{\lambda }_1}{2m_Q}\pm \frac{(2J_{P,V}+1){\lambda }_2}{2m_Q}+\dots ,$$

$$\frac{⟨D^{(\ast )}|\bar{c}\Gamma b|\overline{B}⟩}{\sqrt{m_{D^{(\ast )}}{m}_B}}=-\xi (w)$$

$$\displaystyle+\varepsilon_{c}\,\text{Tr}\big{[}\bar{H}^{(c,1)}_{v^{\prime},v}\,\Gamma\,H^{(b)}_{v}\big{]}+\varepsilon_{b}\,\text{Tr}\big{[}\bar{H}^{(c)}_{v^{\prime}}\,\Gamma\,H^{(b,1)}_{v,v^{\prime}}\big{]}\Big{\}}\,,$$

$$H_{v}^{(Q)}=\frac{1+\not{v}}{2}\,\big{(}V_{v}^{(Q)}\not{\epsilon}-P_{v}^{(Q)}\gamma_{5}\big{)}\,.$$

$$H_{v,v^{\prime }}^{(Q,1)}$$

$$\displaystyle+\frac{1-\not{v}}{2}\,\Big{\{}V_{v}^{(Q)}\big{[}\not{\epsilon}\hat{L}_{5}(w)+\epsilon\cdot v^{\prime}\hat{L}_{6}(w)\big{]}-P_{v}^{(Q)}\,\gamma_{5}\,\hat{L}_{4}(w)\Big{\}}\,.$$

$$w=v\cdot v^{\prime }=\frac{m_B^2+m_{D^{(\ast )}}^2-q^2}{2m_B{m}_{D^{(\ast )}}},v=\frac{p_B}{m_B},v^{\prime }=\frac{p_{D^{(\ast )}}}{m_{D^{(\ast )}}}.$$

$${\hat{L}}_1$$

$${\hat{L}}_4$$

$$\bar{c}b$$

$$\bar{c}{\gamma }^{5}b$$

$$\bar{c}{\gamma }^{\mu }b$$

$$\bar{c}{\gamma }^{\mu }{\gamma }^{5}b$$

$$\bar{c}{\sigma }^{\mu \nu }b$$

$$\displaystyle\qquad\quad+C_{T_{4}}(v^{\prime\mu}v^{\nu}-v^{\prime\nu}v^{\mu})\big{]}b_{v}\,,$$

$$⟨D|\bar{c}b|\overline{B}⟩$$

$$⟨D|\bar{c}{\gamma }^{5}b|\overline{B}⟩$$

$$⟨D|\bar{c}{\gamma }^{\mu }b|\overline{B}⟩$$

$$⟨D|\bar{c}{\sigma }^{\mu \nu }b|\overline{B}⟩$$

$$⟨D^{\ast }|\bar{c}b|\overline{B}⟩$$

$$⟨D^{\ast }|\bar{c}{\gamma }^{5}b|\overline{B}⟩$$

$$⟨D^{\ast }|\bar{c}{\gamma }^{\mu }b|\overline{B}⟩$$

$$⟨D^{\ast }|\bar{c}{\gamma }^{\mu }{\gamma }^{5}b|\overline{B}⟩$$

$$⟨D^{\ast }|\bar{c}{\sigma }^{\mu \nu }b|\overline{B}⟩$$

$$h_{-}=h_{A_2}=h_{T_2}=h_{T_3}=0,$$

$$h_{+}=h_V=h_{A_1}=h_{A_3}=h_S=h_P=h_T=h_{T_1}=\xi .$$

$$\hat{h}(w)=h(w)/\xi (w).$$

$${\hat{h}}_{+}$$

$${\hat{h}}_{-}$$

$${\hat{h}}_S$$

$${\hat{h}}_T$$

$${\hat{h}}_V$$

$${\hat{h}}_{A_1}$$

$${\hat{h}}_{A_2}$$

$${\hat{h}}_{A_3}$$

$${\hat{h}}_P$$

$${\hat{h}}_{T_1}$$

$${\hat{h}}_{T_2}$$

$${\hat{h}}_{T_3}$$

$$[{\overline{m}}_b(\mu )-{\overline{m}}_c(\mu )]⟨D|\bar{c}b|\overline{B}⟩=⟨D|\bar{c}/qb|\overline{B}⟩,$$

$$-[{\overline{m}}_b(\mu )+{\overline{m}}_c(\mu )]⟨D^{\ast }|\bar{c}{\gamma }^{5}b|\overline{B}⟩=⟨D^{\ast }|\bar{c}/q{\gamma }^{5}b|\overline{B}⟩,$$

$$m_{Q}=\overline{m}_{Q}(\mu)\bigg{[}1+{\hat{\alpha}_{s}}\bigg{(}\frac{4}{3}-\ln\frac{m_{Q}^{2}}{\mu^{2}}\bigg{)}+\ldots\bigg{]}\,.$$

Full text

I introduction

Heavy quark symmetry Isgur and Wise (1989, 1990) plays an essential role in understanding exclusive semileptonic $b\to c\ell\bar{\nu}$ mediated transitions, by providing relations between hadronic form factors. At leading order in $\Lambda_{\rm QCD}/m_{c,b}$, the symmetry also determines the absolute normalization of form factors at the “zero recoil” point, $v_{B}=v_{D^{(*)}}$, corresponding to maximal invariant mass, $q^{2}$, of the outgoing lepton pair. Incorporating small corrections to the symmetry limit permits a (hadronic) model-independent determination of $|V_{cb}|$ from exclusive decays. Recently, the Babar Lees et al. (2012, 2013), Belle Huschle et al. (2015); Abdesselam et al. (2016a, b), and LHCb Aaij et al. (2015) measurements of the $|V_{cb}|$-independent ratios

[TABLE]

renewed interest in these decays. The world average of $R(D)$ and $R(D^{*})$ is in tension with the SM expectation at the $4\sigma$ level Amhis et al. (2016). This is intriguing as it occurs in a tree-level SM process, while most new physics (NP) explanations require new states at or below one TeV Freytsis et al. (2015).

Besides the search for new physics, understanding $b\to c\ell\bar{\nu}$ mediated semileptonic decays as precisely as possible is also important for future improvements of the determinations of the CKM elements $|V_{cb}|$ and $|V_{ub}|$, both from exclusive and inclusive $B$ decays, which exhibit some tensions Amhis et al. (2016). Depending on the particular measurement, some decay modes contribute to the signals, some to the backgrounds. Future progress is essential for increasing the scale of new physics probed by the Belle II and LHCb experiments Charles et al. (2014).

The main uncertainty in predicting $R(D^{(*)})$ comes from the fact that the ${\,\overline{\!B}{}\to D^{(*)}\tau\bar{\nu}}$ decay rates depend on certain form factors, that only give $m_{l}^{2}/m_{B}^{2}$ suppressed contributions to the differential rates for the precisely measured light lepton channels. Using heavy quark effective theory (HQET), however, all $\,\overline{\!B}{}\to D^{(*)}$ form factors are described by a single Isgur-Wise function in the $m_{c,b}\gg\Lambda_{\rm QCD}$ limit. At order $\Lambda_{\rm QCD}/m_{c,b}$, only three additional functions of $q^{2}$ are needed to parametrize all form factors.

We perform the first combined fit to ${\,\overline{\!B}{}\to D^{(*)}l\bar{\nu}}$ differential rates and angular distributions, including $\mathcal{O}(\Lambda_{\rm QCD}/{m_{c,b}},\,\alpha_{s})$ terms in HQET, to constrain both the leading and three subleading Isgur-Wise functions. This fit constrains all form factors, up to higher order corrections, with uncertainties suppressed by $\mathcal{O}(\Lambda_{\rm QCD}^{2}/m^{2}_{c,b}\,,\,\alpha_{s}\Lambda_{\rm QCD}/m_{c,b}\,,\,\alpha_{s}^{2})$. We extract $|V_{cb}|$ and form factor ratios under various fit scenarios, that include or omit lattice QCD and/or QCD sum rule inputs, and which provide checks of previously untested theory assumptions or results. Most prior theoretical and experimental studies neglected HQET relations for the form factors at order $\Lambda_{\rm QCD}/m_{c,b}$ or the correlations of the uncertainties in the deviations from the heavy quark limit. Our fits fully incorporate these. These fits also allow precise predictions of the ${\,\overline{\!B}{}\to D^{(*)}\tau\bar{\nu}}$ rates and $R(D^{(*)})$. Our predictions can be systematically improved with more ${\,\overline{\!B}{}\to D^{(*)}l\bar{\nu}}$ data, and need not rely on lattice QCD results. A similar approach to analyze $\,\overline{\!B}{}\to D^{**}l\bar{\nu}$ decays was recently carried out in Ref. Bernlochner and Ligeti (2017).

We also compute, for all possible $b\to c$ currents, the $\mathcal{O}(\Lambda_{\rm QCD}/m_{c,b})$ and $\mathcal{O}(\alpha_{s})$ contributions to the form factors. While the $\mathcal{O}(\Lambda_{\rm QCD}/m_{c,b})$ corrections to the vector and axial-vector matrix elements have been known for over 25 years Luke (1990); Neubert and Rieckert (1992), the corrections for the tensor current form factors are not explicitly available in past literature. Two of these form factors vanish in the heavy quark limit, and receive unsuppressed corrections to the partial results, also of order $\Lambda_{\rm QCD}/m_{c,b}$, used previously in the literature.

Section II contains the HQET calculations of the form factors, including order $\Lambda_{\rm QCD}/m_{c,b}$ and $\alpha_{s}$ contributions, corresponding expressions for form factor ratios, and some details of our numerical evaluations in the $1S$ scheme to avoid known bad behaviors in the perturbation expansions. In Section III we review analyticity constraints on the form factors, parametrizations of the Isgur-Wise functions, and develop several fit scenarios consistent with HQET, which we apply to the data. The results for $|V_{cb}|$, form factor ratios, and $R(D^{(*)})$ are discussed. Section IV concludes.

II Elements of HQET

II.1 Matrix elements to order $\Lambda_{\rm QCD}/m_{c,b}$ and $\alpha_{s}$

We are concerned with matrix elements $\langle D^{(*)}|\,O_{\Gamma}\,|\,\overline{\!B}{}\rangle$, where a full operator basis is

[TABLE]

with $\sigma^{\mu\nu}\equiv(i/2)[\gamma^{\mu},\gamma^{\nu}]$. (The sign convention is fixed by $\sigma^{\mu\nu}\gamma^{5}\equiv-(i/2)\epsilon^{\mu\nu\rho\sigma}\sigma_{\rho\sigma}$, which implies $\text{Tr}[\gamma^{\mu}\gamma^{\nu}\gamma^{\sigma}\gamma^{\rho}\gamma^{5}]=+4i\epsilon^{\mu\nu\rho\sigma}$.) The construction of the HQET expansion to order $\mathcal{O}(\Lambda_{\rm QCD}/m_{c,b})$ and $\mathcal{O}(\alpha_{s})$ was developed in the early ’90s Manohar and Wise (2000); Neubert (1994); we summarize here the central elements to establish our conventions.

The HQET allows model independent parametrization of the spectroscopy of heavy mesons and some hadronic matrix elements between them. The ground state heavy quark spin symmetry doublet pseudoscalar ($P$) and vector ($V$) mesons correspond to the light degrees of freedom (the “brown muck”) in a spin-$\frac{1}{2}$ state combined with the heavy quark spin. They form two states with angular momentum $J_{V,P}=\frac{1}{2}\pm\frac{1}{2}$. Their masses can be expressed as

[TABLE]

where $m_{Q}$ is the heavy quark mass parameter of HQET, $\bar{\Lambda}={\cal O}(\Lambda_{\rm QCD})$, $\lambda_{1,2}={\cal O}(\Lambda_{\rm QCD}^{2})$, etc. To evaluate matrix elements relevant for semileptonic decays, it is simplest to use the trace formalism Falk et al. (1990); Bjorken (1990); Falk (1992). Including $\Lambda_{\rm QCD}/m_{c,b}$ corrections, the $B\to D^{(*)}$ matrix elements can be written as Falk and Neubert (1993)

[TABLE]

where $\varepsilon_{c,b}=\bar{\Lambda}/(2m_{c,b})$ and $\Gamma$ is an arbitrary Dirac matrix. The pseudoscalar and vector mesons can be represented by a “superfield”, which has the right transformation properties under heavy quark and Lorentz symmetry,

[TABLE]

The $\Lambda_{\rm QCD}/m_{c,b}$ corrections can be parametrized via Falk and Neubert (1993)

[TABLE]

It is convenient to use the dimensionless kinematic variable $w$ instead of $q^{2}=(p_{B}-p_{D^{(*)}})^{2}$,

[TABLE]

In Eq. (II.1) and hereafter, we absorb into the leading order Isgur-Wise function a heavy quark spin symmetry conserving ${\cal O}(\Lambda_{\rm QCD}/m_{c,b})$ subleading term, which does not affect any model independent predictions of HQET, via $\xi(w)\to\xi(w)+2(\varepsilon_{c}+\varepsilon_{b})\chi_{1}(w)$. The function $\chi_{1}$ parametrizes the matrix element of the time ordered product of the kinetic operator in the subleading HQET Lagrangian, $O_{\rm kin}=\bar{h}_{v}\,(iD)^{2}\,h_{v}/(2m_{Q})$, with the leading order current. It satisfies $\chi_{1}(1)=0$ Luke (1990), and hence $\xi(1)=1$ is maintained. Reparametrization invariance Luke and Manohar (1992) ensures that this redefinition of $\xi(w)$ is RGE invariant.

The $w$-dependent $L_{1\ldots 6}$ functions are Falk and Neubert (1993)

[TABLE]

Here the $\hat{\chi}_{2,3}$ terms in $\hat{L}_{1,2,3}$ originate from the matrix elements of the time ordered product of the leading order current with the chromomagnetic correction to the Lagrangian, $O_{\rm mag}=(g_{s}/2)\,\bar{h}_{v}\sigma_{\mu\nu}G^{\mu\nu}h_{v}/(2m_{Q})$. Luke’s theorem implies $\hat{\chi}_{3}(1)=0$ Luke (1990). The $\hat{L}_{4,5,6}$ terms arise from $\Lambda_{\rm QCD}/m_{c,b}$ corrections in the matching of the $\bar{c}\,\Gamma b$ heavy quark current onto HQET, \bar{c}\,\Gamma b\to\bar{c}_{v^{\prime}}\big{[}\Gamma-i\overleftarrow{\not{D}}\,\Gamma/(2m_{c})+\Gamma\,i\overrightarrow{\not{D}}/(2m_{b})+\ldots\big{]}b_{v}.111Our definitions of the subleading Isgur-Wise functions, $\chi_{1,2,3}$, $\eta$, and hence $\hat{L}_{1\ldots 6}$, are dimensionless due to factoring out $\bar{\Lambda}$, as done, e.g., in Refs. Neubert (1994); Grinstein and Ligeti (2002) but not in Refs. Luke (1990); Manohar and Wise (2000); the correspondence is obvious. The QCD sum rule calculations Neubert et al. (1993a, b); Ligeti et al. (1994) also compute these functions with the dimensionless definitions.

The perturbative corrections to the heavy quark currents may be computed by matching QCD onto HQET Falk et al. (1990); Falk and Grinstein (1990); Neubert (1992). At $\mathcal{O}(\alpha_{s})$, the following operators are generated

[TABLE]

where the $C_{\Gamma_{i}}$ are functions of $w$ and $z=m_{c}/m_{b}$, and ${\hat{\alpha}_{s}}=\alpha_{s}/\pi$. (We follow the notation of Ref. Manohar and Wise (2000), while Ref. Neubert (1994) uses $C_{i}={\hat{\alpha}_{s}}C_{V_{i}}+\delta_{i1}$ and $C_{i}^{5}={\hat{\alpha}_{s}}C_{A_{i}}+\delta_{i1}$.) Evaluating these contributions using the leading order trace in Eq. (II.1) leads to $\mathcal{O}(\alpha_{s})$ modifications of the coefficients of the Isgur-Wise function, $\xi(w)$. In this paper we neglect $\mathcal{O}(\alpha_{s}\,\varepsilon_{c,b})$ corrections, which can also be included straightforwardly (and should be, if NP is established).

The $\alpha_{s}$ corrections for all five currents were computed in Ref. Neubert (1992). Appendix A contains their explicit expressions, at arbitrary matching scale $\mu$. The vector and axial-vector currents are not renormalized in QCD, but the corresponding heavy quark currents have non-zero anomalous dimensions, leading to $\mu$-dependence for $C_{V_{1}}$ and $C_{A_{1}}$ for $w\neq 1$. The scalar, pseudoscalar, and tensor currents are renormalized in QCD, and thus $C_{S}$, $C_{P}$, and $C_{T_{1}}$ are also $\mu$-dependent. In the $\overline{\rm MS}$ scheme with dimensional regularization, the remaining $C_{\Gamma_{j}}$ ($j\geq 2$) are scale independent.

II.2 $\,\overline{\!B}{}\to D^{(*)}$ form factors

We use the standard definitions of the form factors. For $\,\overline{\!B}{}\to D$ decays,

[TABLE]

while for the $\,\overline{\!B}{}\to D^{*}$ transitions,

[TABLE]

The $i$, $-1$, and $w+1$ factors are chosen such that in the heavy quark limit each form factor either vanishes or equals the leading order Isgur-Wise function,

[TABLE]

Using Eqs. (II.1) and (II.1), one can compute all form factors to order $\mathcal{O}(\Lambda_{\rm QCD}/m_{c,b})$ and $\mathcal{O}(\alpha_{s})$. It is convenient to factor out $\xi(w)$, defining

[TABLE]

By virtue of Eq. (II.1), the ${\,\overline{\!B}{}\to Dl\bar{\nu}}$ form factors only depend on two linear combinations of subleading Isgur-Wise functions, $\hat{L}_{1}$ and $\hat{L}_{4}$,

[TABLE]

For the ${\,\overline{\!B}{}\to D^{*}l\bar{\nu}}$ form factors we obtain

[TABLE]

In Eqs. (II.2) and (II.2), the relations for the SM currents — that is, $h_{+}$, $h_{-}$, $h_{V}$, $h_{A_{1}}$, $h_{A_{2}}$, and $h_{A_{3}}$ — agree with the literature, e.g., Refs. Falk and Neubert (1993); Neubert (1994). Because of Luke’s theorem, the $\mathcal{O}(\Lambda_{\rm QCD}/m_{c,b})$ corrections to $h_{+}$, $h_{S}$, $h_{A_{1}}$, and $h_{T_{1}}$ vanish at zero recoil. To the best of our knowledge, the expressions for $h_{T}$ and $h_{T_{1,2,3}}$ cannot be found in the literature. For $h_{T_{2}}$ and $h_{T_{3}}$, which start at order $\Lambda_{\rm QCD}/m_{c,b}$, the partial results used in the literature (e.g., Ref. Tanaka and Watanabe (2013)) kept and left out terms, which are both order $\mathcal{O}(\Lambda_{\rm QCD}/m_{c,b})$.

The scalar and vector matrix elements in $\,\overline{\!B}{}\to D$ transitions, and the pseudoscalar and axial vector ones in $\,\overline{\!B}{}\to D^{*}$, are related by the equations of motion

[TABLE]

in which $\overline{m}_{Q}(\mu)$ are the $\overline{\rm MS}$ quark masses at a common scale $\mu$, obeying

[TABLE]

One can verify using $m_{b}=m_{B}-\bar{\Lambda}+\mathcal{O}(\Lambda_{\rm QCD}^{2}/m_{b})$ and $m_{c}=m_{D^{(*)}}-\bar{\Lambda}+\mathcal{O}(\Lambda_{\rm QCD}^{2}/m_{c})$ that the form factor expansions in Eqs. (II.2) and (II.2) satisfy these relations, including all $\mathcal{O}(\varepsilon_{c,b})$ and $\mathcal{O}(\alpha_{s})$ terms. We emphasize that this only holds using the $\overline{\rm MS}$ masses at the common scale $\mu$. Using $\overline{m}_{b}(\overline{m}_{b})$ and $\overline{m}_{c}(\overline{m}_{c})$ Patrignani et al. (2016) in Eqs. (16), as done in some papers, is inconsistent.

We prefer to evaluate the scalar and pseudoscalar matrix elements using Eqs. (II.2) and (II.2) instead of Eq. (16), because the natural choice for $\mu$ is below $m_{b}$ (or sometimes well below, as in the small-velocity limit Shifman and Voloshin (1988); Boyd et al. (1996a)). In the $\overline{\rm MS}$ scheme fermions do not decouple for $\mu<m$, introducing artificially large corrections in the running, compensated by corresponding spurious terms in the $\beta$-function computed without integrating out heavy quarks Manohar (1997).

II.3 Decay rates and form factor ratios

The ${\,\overline{\!B}{}\to D^{(*)}l\bar{\nu}}$ differential rates have the well-known expressions in the SM,

[TABLE]

where $r_{D^{(*)}}=m_{D^{(*)}}/m_{B}$ and $\eta_{\rm EW}\simeq 1.0066$ Sirlin (1982) is the electroweak correction. In addition,

[TABLE]

and the form-factor ratios are defined as

[TABLE]

In the heavy quark limit, $R_{1,2}(w)=1$ and $\mathcal{F}(w)=\mathcal{G}(w)=\xi(w)$, the leading Isgur-Wise function. It is common to fit the measured ${\,\overline{\!B}{}\to D^{*}l\bar{\nu}}$ angular distributions to $R_{1,2}(w)$. To $\mathcal{O}(\varepsilon_{c,b},\,\alpha_{s})$, the SM predictions are

[TABLE]

To include the lepton mass suppressed terms, one sometimes defines Fajfer et al. (2012); Tanaka and Watanabe (2013) additional form factor ratios

[TABLE]

All contributions of $R_{0,3}(w)$ are proportional to $m_{\ell}^{2}$. (Ref. Fajfer et al. (2012) defines $R_{3}=h_{A_{3}}/h_{A_{1}}$.) They are not linearly independent from $R_{1,2}(w)$, as there are only three form factor ratios in $B\to D^{*}\ell\bar{\nu}$ in the SM. In the heavy quark limit, $R_{3}(w)=R_{0}(w)=1$. At $\mathcal{O}(\varepsilon_{c,b},\,\alpha_{s})$, the SM predictions are

[TABLE]

II.4 The $1S$ scheme and numerical results

The $C_{\Gamma}$ coefficients defined in Eq. (II.1) are functions of $w$ and $z=m_{c}/m_{b}$, and thus depend on the quark masses. As is well known, the pole mass of a heavy quark contains a leading renormalon ambiguity of order $\Lambda_{\rm QCD}$, and so does the HQET parameter $\bar{\Lambda}$, as they are ill-defined beyond perturbation theory. The ambiguity is canceled by a corresponding ambiguity in the perturbation series, connected to factorial growth of the coefficients of ${\hat{\alpha}_{s}}^{n}$ Neubert and Sachrajda (1995); Luke et al. (1995); Beneke and Braun (1994); Bigi et al. (1994); Beneke et al. (1994). The cancellation comes about as a non-analytic term connected to the asymptotic nature of the perturbation series, $e^{-c/\alpha_{s}(M)}\sim(\Lambda_{\rm QCD}/M)^{c\beta_{0}/(4\pi)}$, where $\beta_{0}=(11-2n_{f}/3)$ is the first coefficient in the expansion of the $\beta$ function. For example, Eq. (21) implies at zero recoil, $R_{1}(1)\simeq 1+4{\hat{\alpha}_{s}}/3+\varepsilon_{c}+\varepsilon_{b}-2\varepsilon_{b}\eta(1)$, where the order ${\hat{\alpha}_{s}}^{2}\beta_{0}$ terms are also known Grinstein and Ligeti (2002). The leading renormalon corresponding to the worst behavior of the ${\hat{\alpha}_{s}}^{n}$ power series is canceled by the ambiguity in $\bar{\Lambda}$ within the $\varepsilon_{c}+\varepsilon_{b}$ term. The $-2\varepsilon_{b}\eta(1)$ term, however, does not contribute to this leading renormalon cancellation, as the only participating terms are those $\bar{\Lambda}/m_{c,b}$ terms not multiplied by any subleading Isgur-Wise functions.

The $\alpha_{s}$ perturbation series is known to be poorly convergent for many $B$ decay processes already at $\mathcal{O}(\alpha_{s}^{2})$, when expressed in terms of the pole mass. To ensure the order-by-order cancellation of the fastest factorially growing terms, it is convenient to reorganize the perturbation series in terms of a suitable short-distance mass scheme, instead of the pole mass. We use the $1S$ scheme Hoang et al. (1999a, b); Hoang (2000), which has been tested in the calculations of numerous observables. (Using the $\overline{\rm MS}$ mass yields a poorly behaved perturbation series, for the reasons mentioned at the end of Sec. II.2. Other possible short-distance mass schemes include the PS mass Beneke (1998) or the kinetic mass Czarnecki et al. (1998).)

The $1S$ scheme defines $m_{b}^{1S}$ as half of the perturbatively computed $\Upsilon(1S)$ mass. It is related to the pole mass as $m_{b}^{1S}=m_{b}\,(1-2\alpha_{s}^{2}/9+\ldots)$ Hoang et al. (1999a, b); Hoang (2000), so that we may treat the pole mass as the function $m_{b}(m_{b}^{1S})=m_{b}^{1S}(1+2\alpha_{s}^{2}/9+\ldots)$. Neglecting higher order terms, as done throughout this paper, is a good approximation in all cases where they are known, including the evaluation of $R_{1,2}$ Grinstein and Ligeti (2002). We adopt the inputs Ligeti and Tackmann (2014),

[TABLE]

from fits to inclusive $B\to X_{c}l\bar{\nu}$ spectra and other determinations of $m_{b}^{1S}$. We eliminate $m_{c}$ using $m_{c}=m_{b}(m_{b}^{1S})-\delta m_{bc}$, and extract $\bar{\Lambda}$ via

[TABLE]

Here $\overline{m}_{B}=(m_{B}+3m_{B^{*}})/4\simeq 5.313$ GeV is the spin-averaged meson mass, and we use $\lambda_{1}=-0.3$ GeV2 Ligeti and Tackmann (2014). Enforcing the cancellation of the leading renormalon is equivalent to using $m_{b}(m_{b}^{1S})\to m_{b}^{1S}$ everywhere in Eqs. (II.2) and (II.2), except in the $\bar{\Lambda}/m_{c,b}$ terms that are not multiplied by subleading Isgur-Wise functions.

We match the QCD and HQET theories at scale $\mu^{2}=m_{b}m_{c}$, corresponding to $\alpha_{s}\simeq 0.26$. The $1S$ scheme then yields, for example, the following SM predictions for $R_{1,2}(1)$

[TABLE]

For $R^{\prime}_{1,2}(1)$ we obtain

[TABLE]

For completeness, the similar relations for $R_{0,3}$ are

[TABLE]

III Combined fit to $B\to D^{*}l\bar{\nu}$ and $B\to Dl\bar{\nu}$

III.1 Parametrization of the $w$ dependence

Unitarity and analyticity provide strong constraints on the shapes of the ${\,\overline{\!B}{}\to D^{(*)}\ell\bar{\nu}}$ form factors Boyd et al. (1996b, 1997); Caprini et al. (1998); Boyd and Savage (1997); Hill (2006); Bourrely et al. (2009). It is common to employ a parametrization of the ${\,\overline{\!B}{}\to D\ell\bar{\nu}}$ form factor $\mathcal{G}(w)$, defined in Eq. (19), via the conformal mapping $\mathpzc{z}(w)=(\sqrt{w+1}-\sqrt{2})/(\sqrt{w+1}+\sqrt{2})$. Unitarity constraints yield, e.g., $\mathcal{G}(w)/\mathcal{G}(1)\simeq 1-8\rho^{2}\mathpzc{z}+(51.\rho^{2}-10.)\mathpzc{z}^{2}-(252.\rho^{2}-84.)\mathpzc{z}^{3}$, in which $\rho^{2}=-\mathcal{G}^{\prime}(1)/\mathcal{G}(1)$ is a slope parameter Caprini et al. (1998). The convergence of this expansion may be optimized by parametrizing it in a way that minimizes the range of the expansion parameter, via

[TABLE]

For ${\,\overline{\!B}{}\to Dl\bar{\nu}}$, $|z_{*}|\leq 0.032$. The unitarity constraints suggest a form factor parametrization of the form

[TABLE]

Here $w_{0}=2a^{2}-1\simeq 1.28$ is defined such that $\mathpzc{z}_{*}(w_{0})=0$, while $V_{21}\simeq 57.$ and $V_{20}\simeq 7.5$ are obtained numerically from Ref. Caprini et al. (1998). The uncertainty in the coefficient of the $\mathpzc{z}_{*}^{2}$ term in Eq. (30) may be sizable Caprini et al. (1998). However, the impact of this term on the physical fit results is expected to be small.

The leading order Isgur-Wise function, $\xi(w)$, may be extracted from the parametrization in Eq. (30) by using Eqs. (II.2) and (13). Keeping terms to $\mathcal{O}(\varepsilon_{c,b}(w-1))$, we can approximate the subleading Isgur-Wise functions as

[TABLE]

since $\hat{\chi}_{3}(1)=0$. One finds at $\mathcal{O}(\varepsilon_{c,b},\,\alpha_{s})$,

[TABLE]

where $\Xi=64a^{4}\bar{\rho}_{*}^{2}-16a^{2}-V_{21}$. The slope parameter $\bar{\rho}_{*}^{2}=-\xi^{\prime}(w_{0})/\xi(w_{0})$ is related to the slope $\rho_{*}^{2}=-\mathcal{G}^{\prime}(w_{0})/\mathcal{G}(w_{0})$ via

[TABLE]

Enforcing $\xi(1)=1$, one may directly extract $\xi(w_{0})$ via evaluation of Eq. (III.1) at the zero recoil point, $\mathpzc{z}_{*}(w=1)=(1-a)/(1+a)$, and thereby obtain a properly normalized parametrization for $\xi(w)$. Since $\eta(1)$ does not appear in Eq. (III.1), this implies that constraining $\xi(w)$ in itself does not constrain $\eta(1)$, which is the largest unknown contribution in $R_{1,2}(1)$.

This expression for $\xi(w)$, combined with the HQET expansions in Eqs. (II.2) and (II.2), allows one to parametrize all $\,\overline{\!B}{}\to D^{(*)}$ form factors in terms of six parameters: $\bar{\rho}_{*}^{2}$, $\hat{\chi}_{2}(1)$, $\hat{\chi}^{\prime}_{2}(1)$, $\hat{\chi}^{\prime}_{3}(1)$, $\eta(1)$ and $\eta^{\prime}(1)$. The normalizations of the form factors are also fixed by Eq. (III.1), thus $|V_{cb}|$ may be determined from a global fit to overall rates without using lattice results.

III.2 QCD sum rule inputs

The subleading Isgur-Wise functions have only been calculated using model dependent methods, and are not yet available from lattice QCD. The two-loop QCD sum rule (QCDSR) calculations Ligeti et al. (1994); Neubert et al. (1993a, b) imply that the subleading Isgur-Wise function $\eta(w)$ is approximately constant. The functions $\hat{\chi}_{2,3}$, which parametrize corrections from the chromomagnetic term in the subleading HQET Lagrangian, are small, in agreement with quark model intuition.

The QCD sum rule results are obtained at a fixed scale. The scale dependence can be removed from $\hat{\chi}_{2,3}$ by defining “renormalization improved” functions, $\hat{\chi}^{\text{ren}}_{2,3}$ Neubert (1994). These are obtained by multiplying the results of Refs. Neubert et al. (1993a, b) for $\hat{\chi}_{2,3}$ by $[\alpha_{s}(\Lambda)]^{3/\beta_{0}}\sim 1.4$, where $\Lambda\sim 1\,\text{GeV}$ and $\beta_{0}=9$ for three light flavors. For these renormalized subleading Isgur-Wise functions, we use

[TABLE]

These central values reproduce $\hat{L}_{1\ldots 6}$ in Ref. Caprini et al. (1998), often used to predict $R_{1,2}$ and $R(D^{(*)})$.

We assign relatively large uncertainties, to permit assessment of possible pulls of the experimental data from these QCDSR predictions. Replacing $\hat{\chi}_{2,3}$ with $\hat{\chi}_{2,3}^{\text{ren}}$, the Wilson coefficient of the chromomagnetic operator receives a corresponding $\alpha_{s}(\mu)^{3/\beta_{0}}$ factor at the matching scale $\mu=\sqrt{m_{b}\,m_{c}}$, partly canceling the above $[\alpha_{s}(\Lambda)]^{3/\beta_{0}}$ enhancement. For ease of comparison with the literature we ignore this, as it can be viewed as a higher order correction, and is in any case covered by the large assigned uncertainties. We ignore correlations in the QCDSR results (arising from the common calculational method), which is conservative.

Using Eqs. (34) in Eqs. (21) yield expressions for $R_{1,2}(w)$ as polynomials in $(w-1)$, with the coefficients and their uncertainties correlated by HQET. In Ref. Caprini et al. (1998), the central values in Eq. (34) were used to write $R_{1,2}(w)$ as quadratic polynomials, without quoting any theory uncertainties on their slopes and curvatures. It subsequently become standard practice in experimental $|V_{cb}|$ and $R_{1,2}$ measurements to fit for $R_{1,2}(1)$, while fixing $R^{\prime}_{1,2}(1)$ and $R^{\prime\prime}_{1,2}(1)$ to their quoted central values Caprini et al. (1998). Such an approach is inconsistent with the simultaneous use of the HQET constraints and the QCDSR results. For example, the present world average central value, $R_{1}(1)\simeq 1.4$, cannot simultaneously satisfy the HQET prediction for $R_{1}(1)$ in Eq. (II.4) and the QCDSR expectation $\eta(1)>0$, which holds at the $3\sigma$ level, and is used elsewhere in the same fit. A consistent treatment of these form factor ratios is absent from the derivations of the state-of-the-art predictions for $R(D^{(*)})$ in the SM (except for LQCD $R(D)$ predictions) and in the presence of new physics Fajfer et al. (2012); Tanaka and Watanabe (2013).

We now proceed to assess the importance of obeying the HQET relations between different form factors, and of including the uncertainties in the QCDSR predictions in Eq. (34). These effects will be important in the future, to systematically improve the SM predictions.

III.3 Fit scenarios

A simultaneous fit of the six parameters $\bar{\rho}_{*}^{2}$, $\hat{\chi}_{2}(1)$, $\hat{\chi}^{\prime}_{2}(1)$, $\hat{\chi}^{\prime}_{3}(1)$, $\eta(1)$, and $\eta^{\prime}(1)$ to the ${\,\overline{\!B}{}\to D^{(*)}l\bar{\nu}}$ rates can be carried out with the present data. Such a fit fixes both the shapes and normalizations of the ${\,\overline{\!B}{}\to D^{(*)}l\bar{\nu}}$ rates, without any theory input other than the HQET expansion. However, one expects large uncertainties at present, because of the limited experimental precision and the number of subleading HQET parameters. One may instead use QCD sum rule predictions and/or lattice QCD results to constrain the fit, increasing sensitivity to $\bar{\rho}_{*}^{2}$. The fit propagates the uncertainties on the subleading Isgur-Wise functions into the fit result, and allows the data to further constrain the subleading contributions.

Our fit relies on the HQET predictions and unitarity constraints to determine the ratios and shapes of the form factors. The form factors at zero recoil, $\mathcal{G}(1)$ and $\mathcal{F}(1)$, have been computed in lattice QCD (LQCD), providing state-of-the-art predictions for the normalizations of the ${\,\overline{\!B}{}\to D^{(*)}l\bar{\nu}}$ rates. The most precise lattice QCD predictions at zero recoil are Bailey et al. (2014, 2015)

[TABLE]

where we combined the quoted systematic and statistical uncertainties. Although these normalizations may be expected to drop out of the predictions for $R(D^{(*)})$, they do influence the fit to the differential decay distributions and hence the resulting form factor ratios. Making use of these lattice constraints leads to our first fitting scenario:

• $\blacksquare$

Rescale the $\,\overline{\!B}{}\to D$ and $\,\overline{\!B}{}\to D^{*}$ form factors in the fit by $\mathcal{G}(1)_{\text{LQCD}}/\mathcal{G}(1)$ and $\mathcal{F}(1)_{\text{LQCD}}/\mathcal{F}(1)$, respectively, such that the rates at $w=1$ agree with the lattice predictions. We refer to this fit as “$\text{L}_{w=1}$”.

Measurements of the rate normalizations are, however, subject to relatively large systematic uncertainties. For example, the calibration of the hadronic tagging efficiency produces systematic uncertainties of the order of a few percent Abdesselam et al. (2017). To compare the best-fit shapes without lattice constraints and such systematic effects, we consider a second scenario:

• $\blacksquare$

Allow the normalizations of the ${\,\overline{\!B}{}\to Dl\bar{\nu}}$ and ${\,\overline{\!B}{}\to D^{*}l\bar{\nu}}$ rates to float independently. This approach only uses ${\,\overline{\!B}{}\to D^{(*)}l\bar{\nu}}$ shape information to constrain the form factors, but no theory input for the normalizations at zero-recoil, and is independent of lattice information. We refer to this fit as “NoL”.

For each fit, we apply (relax) the QCDSR constraints, exploring a “constrained” (“unconstrained”) fit. The QCDSR constrained fits are denoted with a suffix “$+\text{SR}$”. Both $\text{L}_{w=1}$ and NoL fits alter the overall normalizations the ${\,\overline{\!B}{}\to Dl\bar{\nu}}$ and ${\,\overline{\!B}{}\to D^{*}l\bar{\nu}}$ rates, but leave the HQET expansions of the form factors unchanged. Thus, they can be considered as introducing an extra source of heavy quark symmetry breaking in the normalizations (to effectively account for higher order effects), while still preserving the form factor relations independently in Eqs. (II.2) and (II.2).

Since lattice QCD predictions are also available for $w\geq 1$ for the ${\,\overline{\!B}{}\to Dl\bar{\nu}}$ form factors $f_{+}(w)$ and $f_{0}(w)$, it is possible to obtain a prediction for the slope parameter, $\bar{\rho}_{*}^{2}$, from them. This leads to a third fit approach, namely:

• $\blacksquare$

Extract $\xi(w)$, including the slope parameter $\bar{\rho}_{*}^{2}$, by fitting to the $w\geq 1$ lattice QCD data for $B\to D$, and apply it simultaneously with the LQCD normalization of $B\to D^{*}$ at $w=1$. We refer to this fit as “$\text{L}_{w\geq 1}$”.

In a “theory only” version of this fit, denoted by “$\text{th:}\text{L}_{w\geq 1}{+\text{SR}}$”, one fully constrains the ${\,\overline{\!B}{}\to D^{(*)}l\bar{\nu}}$ differential rates without any experimental input; the only fit is to lattice data and QCDSR constraints. For the “$\text{L}_{w\geq 1}{+\text{SR}}$” fit, we combine the $w\geq 1$ $B\to D$ and $w=1$ $B\to D^{*}$ lattice data with QCDSR constraints and the experimental information, to include all available information and explore possible tensions. We summarize the inputs of the various fit scenarios pursued in this paper in Table 1.

All fits explored in this paper use the unitarity constraints. The consequences of relaxing the unitarity constraints between the slope and the curvature terms in Eq. (30) will be explored in detail elsewhere Bernlochner et al. (2017a).

III.4 Data and fit details

To determine the leading and subleading Isgur-Wise functions and $|V_{cb}|$, we carry out a simultaneous fit of the available ${\,\overline{\!B}{}\to D^{(*)}l\bar{\nu}}$ spectra. There are only two measurements Glattauer et al. (2016); Abdesselam et al. (2017) which provide kinematic distributions fully corrected for detector effects. The measured recoil and decay angle distributions are analyzed simultaneously by constructing a standard $\chi^{2}$ function. Common uncertainties (tagging efficiency, reconstruction efficiencies, number of $B$-meson pairs) should be treated as fully correlated between the two measurements and we construct a covariance using Table IV in Ref. Glattauer et al. (2016) and Table IV in Ref. Abdesselam et al. (2017). While Ref. Glattauer et al. (2016) provides a full breakdown of the total uncertainty for each measured $w$ bin, Ref. Abdesselam et al. (2017) only provides a breakdown for the total branching fraction. To construct the desired covariance between both measurements, we thus assume that there is no shape dependence on the tagging and reconstruction efficiency uncertainty of Ref. Abdesselam et al. (2017). Comparing this with the mild dependence on these error sources in Ref. Glattauer et al. (2016), this seems a fair approximation of the actual covariance. To take into account the uncertainties of $m^{1S}_{b}$ and $\delta m_{bc}$, we introduce both as nuisance parameters into the fit, assuming Gaussian constraints with uncertainties given in Eq. (24). The $\chi^{2}$ function is numerically minimized and uncertainties are evaluated using the usual asymptotic approximations by scanning the $\Delta\chi^{2}=\chi^{2}_{\rm scan}-\chi^{2}_{\rm min}$ contour to find the $+1$ crossing point, which provides the 68% confidence level. The constraints from lattice QCD predictions and/or QCD sum rules are incorporated into the fit assuming (multivariate) Gaussian errors and are added to the $\chi^{2}$ function.

The full fit results are shown in Table III.4. The “$\text{L}_{w=1}$” unconstrained fit, i.e., using only the lattice normalizations at $w=1$, yields

[TABLE]

to be compared with the current world average Patrignani et al. (2016) $|V_{cb}|=(42.2\pm 0.8)\times 10^{-3}$ and $|V_{cb}|=(39.2\pm 0.7)\times 10^{-3}$, from inclusive and exclusive $b\to c\,l\bar{\nu}_{l}$ decays, respectively. The uncertainties of the subleading Isgur-Wise parameters are sizable. There is no sensitivity to disentangle $\eta^{\prime}(1)$ from $\bar{\rho}_{*}^{2}$, so we fix $\eta^{\prime}(1)$ to be zero for all QCDSR unconstrained fits. Including the QCDSR constraints in the “$\text{L}_{w=1}{+\text{SR}}$” fit yields

[TABLE]

resulting in almost the same $|V_{cb}|$ value. The normalization of $\eta(1)$ is comparable between these two fits, at about half the value of the QCDSR expectation. Both fits have reasonable $\chi^{2}$ values, corresponding to fit probabilities of 64% each.