Lattice QCD calculation of the $B_{(s)}\to D_{(s)}^{*}\ell \nu$ form factors at zero recoil and implications for $|V_{cb}|$

We present results of a lattice QCD calculation of $B\to D^{*}$ and $B_s\to D_s^{*}$ axial vector matrix elements with both states at rest. These zero recoil matrix elements provide the normalization necessary to infer a value for the CKM matrix element $|V_{cb}|$ from experimental measurements of ${\overline{B}}^0\to D^{*+}{\ell }^{-}\overline{\nu }$ and ${\overline{B}}_s^0\to D_s^{*+}{\ell }^{-}\overline{\nu }$ decay. Results are derived from correlation functions computed with highly improved staggered quarks (HISQ) for light, strange, and charm quark propagators, and nonrelativistic QCD for the bottom quark propagator. The calculation of correlation functions employs MILC Collaboration ensembles over a range of three lattice spacings. These gauge field configurations include sea quark effects of charm, strange, and equal-mass up and down quarks. We use ensembles with physically light up and down quarks, as well as heavier values. Our main results are ${\mathcal{F}}^{B\to D^{*}}(1)=0.895\pm 0.010_{\mathrm{stat}}\pm {0.024}_{\mathrm{sys}}$ and ${\mathcal{F}}^{B_s\to D_s^{*}}(1)=0.883\pm 0.012_{\mathrm{stat}}\pm 0.028_{\mathrm{sys}}$. We discuss the consequences for $|V_{cb}|$ in light of recent investigations into the extrapolation of experimental data to zero recoil.

Figure 1

Fractional $h_{A_1}(1)$ variation with the number of exponentials used in the fit function on Set 6, showing fits done using each individual source/sink smearing (local $l$, or Gaussian with 2 radii, $g2$ and $g4$) as well as the full $3\times 3$ matrix fit.

Figure 2

Fit to our data using staggered chiral perturbation theory. Finite volume corrections are included in the data points, visible only for the physical pion mass points. The blue line and grey band are the continuum chiral perturbation theory result and error extrapolated from our lattice data. The error band includes systematic errors coming from matching uncertainties and hence has a much larger error than any of the data points, which are only shown with their statistical error.

Figure 3

Plot showing the $a^2$ dependence of our $B\to D^{*}$ data. Finite volume corrections are included in the data points, visible only for the physical pion mass points. The blue line with grey error band shows the physical result for the form factor determined by the fit described in the text.

Figure 4

Lattice spacing dependence of our results for the $B_s\to D_s^{*}$ zero recoil form factor. The blue line with grey error band shows the physical result for the form factor determined by the fit described in the text.

Figure 5

Comparison of the prefactor $Q_F(q^2)$ for the BGL and BCL series expansions of form factor $F=f$, $F_1$, and $g$, from top to bottom. Curves are normalized by $Q_F(0)$, which is given in the legend.

Figure 6

Values of $I=|{\overline{\eta }}_{\mathrm{EW}}{V}_{cb}|h_{A_1}(1)$ obtained from different fit ansätze (see text).

Figure 7

Comparison of fit results to experimental data [16]. The binned fit results are slightly offset from the bin midpoints for clarity. See Appendix pp7 and Ref. [16] for definitions.

Figure 8

Comparison of the $|V_{cb}|$ from (41) with the latest determinations from $B\to X_c\ell \nu$ [19, 20] and $B\to D\ell \nu$ [33].

Figure 9

Plots of $N_{\exp }$ fit behavior on all 8 ensembles (see Table 1). In each plot 4 sets of data points are shown: the full fit including all $3\times 3$ source-sink combinations, and, for comparison, separate “diagonal” fits where only one type of source-sink smearing is used. (The notation is defined in Sec. 3.) A significant improvement is seen in the full fit. All diagonal fits show good agreement for $N_{\exp }\ge 4$, but with the increased precision, sometimes 5 or 6 exponentials are needed to get a good $3\times 3$ matrix fit.

Figure 10

Pion mass dependence of the finite volume corrections to $h_{A_1}(1)$, as determined from staggered chiral perturbation theory [60], with parameters corresponding to the physical-mass lattices used here. The curves for the heavier-mass lattices used here show much smaller finite volume corrections, of $O(0.1\%)$. The vertical blue line is the physical pion mass and the solid point at the end of each curve is at the measured value of the pion mass on each lattice.