Revisiting fits to $B^0\to D^{*-}{\ell }^{+}{\nu }_{\ell }$ to measure $|V_{cb}|$ with novel methods and preliminary LQCD data at nonzero recoil

We present a study of fits to exclusive $B^0\to D^{*-}{\ell }^{+}{\nu }_{\ell }$ measurements for the determination of the Cabibbo-Kobayashi-Maskawa matrix element magnitude $|V_{cb}|$, based on the most recent Belle untagged measurement. Results are obtained with the Caprini-Lellouch-Neubert (CLN) and Boyd-Grinstein-Lebed (BGL) form factor parametrizations, with and without the inclusion of preliminary lattice QCD (LQCD) measurements of form factors at nonzero hadronic recoil from the JLQCD Collaboration. The CLN and BGL fits are also studied in different scenarios with reduced theoretical assumptions and at higher-order expansions, respectively. To avoid bias from high systematic error correlations, we employ a novel technique in the field of $B$-physics phenomenology with a toy Monte Carlo method using a Cholesky decomposition of the covariance matrix. Using additional input from lattice QCD calculations of form factors at nonzero recoil, in collaboration with JLQCD, allows for well-defined fit results with reduced model dependence in CLN and BGL. The results obtained are consistent between different configurations, ultimately providing a method for a more model-independent exclusive measurement of $|V_{cb}|$. Using preliminary LQCD inputs and assuming a value of $\mathcal{F}(1)=0.904\pm 0.012$, we find $|V_{cb}|=(38.48\pm 0.32\pm 0.97\pm 0.46)\times {10}^{-3}$ in BGL(2,2,2) and $(38.42\pm 0.35\pm 1.05\pm 0.46)\times {10}^{-3}$ in CLNnoHQS, where the errors are statistical, systematic, and related to $\mathcal{F}(1)$, respectively.

Figure 1

The measured binned yields (data points) with statistical uncertainties for each observable of the $B^0\to D^{*-}{\ell }^{+}{\nu }_{\ell }$ decay overlaid with both the CLN (cyan) and BGL(1,0,2) (orange) parametrization fit results. The statistical and systematic uncertainties in the fits are determined via the toy MC method with the Cholesky decomposition and added in quadrature for these graphs. As the analysis uses a forward-folding approach, the yields predicted by the fit results are also binned. The results from the CLN and BGL parametrizations are in agreement with each other and with the data.

Figure 2

2D contours of pairs of fit parameters for CLN (left) and BGL(1,0,2) (right) at 68.3%, 95.4%, and 99.7% confidence levels. Note that, from Eq. (17), the correlations for ${\tilde{a}}_0^f$ and $\mathcal{F}(1){\eta }_{\mathrm{EW}}|V_{cb}|$ are equal in BGL, and, therefore, only the latter is shown.

Figure 3

${\eta }_{\mathrm{EW}}^2|V_{cb}{|}^2{\mathcal{F}}^2$, $h_{A_1}(w)/h_{A_1}(1)$, $R_1$, and $R_2$ as a function of the hadronic recoil $w$ for the CLN, CLNnoR, and CLNnoHQS scenarios. The solid lines correspond to the central values from the fit with no toy MC, and the uncertainty bands correspond to the combination of the statistical and systematic uncertainties obtained from the standard deviation of the distribution of the function values at each point in $w$.

Figure 4

The form factors and ratios ${\eta }_{\mathrm{EW}}^2|V_{cb}{|}^2{\mathcal{F}}^2$, $h_{A_1}(w)/h_{A_1}(1)$, $R_1$, and $R_2$ as a function of the hadronic recoil $w$ for BGL(1,0,2) and BGL(1,1,2). The central values and uncertainty bands are calculated with the same method as Fig. 3.

Figure 5

The normalized form factor $h_V(w)/h_V(1)$ as a function of the hadronic recoil $w$ for BGL(1,1,2) with (orange) and without (cyan) LQCD constraints and BGL(2,2,2) with LQCD constraints (pink). The enlarged window within the plot is used to emphasize the region around the LQCD inputs for $h_V$. The central values and uncertainty bands are calculated with the same method as Fig. 3.

Figure 6

The form factors and ratios ${\eta }_{\mathrm{EW}}^2|V_{cb}{|}^2{\mathcal{F}}^2$, $h_{A_1}(w)/h_{A_1}(1)$, $R_1$, and $R_2$ as a function of the hadronic recoil $w$ for BGL(1,0,2), BGL(1,1,2), and BGL(2,2,2) using additional LQCD constraints. The LQCD input has been overlaid in the plot for $h_{A_1}(w)/h_{A_1}(1)$. The central values and uncertainty bands are calculated with the same method as Fig. 3.

Figure 7

The form factors and ratios ${\eta }_{\mathrm{EW}}^2|V_{cb}{|}^2{\mathcal{F}}^2$, $h_{A_1}(w)/h_{A_1}(1)$, $R_1$, and $R_2$ as a function of the hadronic recoil $w$ for CLN, CLNnoR, and CLNnoHQS using additional LQCD constraints. The LQCD input has been overlaid in the plot for $h_{A_1}(w)/h_{A_1}(1)$. The central values and uncertainty bands are calculated with the same method as Fig. 3.

Figure 8

The measured binned yields (data points) with statistical uncertainties for each observable of the $B^0\to D^{*-}{\ell }^{+}{\nu }_{\ell }$ decay overlaid with the BGL(2,2,2) (cyan) parametrization fit results, using additional LQCD constraints. The statistical and systematic uncertainties in the fits are determined in the same way as Fig. 1. The results from the fit are in agreement with the data.

Figure 9

Summary of the $\mathcal{F}(1){\eta }_{\mathrm{EW}}|V_{cb}|$ values from the various fit scenarios covered in this paper, where the PDG values of $|V_{cb}|$ from Sec. 1 have been multiplied by the relevant values listed in Table 1.

Figure 10

The measured binned yields (data points) with statistical uncertainty for each observable of the $B^0\to D^{*-}{\ell }^{+}{\nu }_{\ell }$ decay overlaid with CLN fit results for different covariance matrices $\mathcal{C}$, used in Eq. (18). Results are shown for the total covariance matrix being equal to the statistical covariance matrix alone (cyan) and the sum of both the statistical and systematic covariance matrices in a naive way (orange).