Lattice QCD form factor for $B_s\to D_s^{*}l\nu$ at zero recoil with nonperturbative current renormalization

We present details of a lattice QCD calculation of the $B_s\to D_s^{*}$ axial form factor at zero recoil using the highly improved staggered quark (HISQ) formalism on the second-generation MILC gluon ensembles that include up, down, strange and charm quarks in the sea. Using the HISQ action for all valence quarks means that the lattice axial vector current that couples to the $W$ can be renormalized fully nonperturbatively, giving a result free of the perturbative matching errors that previous lattice QCD calculations have had. We calculate correlation functions at three values of the lattice spacing, and multiple $b$-quark masses, for physical $c$ and $s$. The functional dependence on the $b$-quark mass can be determined and compared to heavy quark effective theory expectations, and a result for the form factor obtained at the physical value of the $b$-quark mass. We find ${\mathcal{F}}^{B_s\to D_s^{*}}(1)=h_{A_1}^s(1)=0.9020(96{)}_{\mathrm{stat}}(90{)}_{\mathrm{sys}}$. This is in agreement with earlier lattice QCD results, which use NRQCD $b$ quarks, with a total uncertainty reduced by more than a factor of 2. We discuss implications of this result for the $B\to D^{*}$ axial form factor at zero recoil and for determinations of $V_{cb}$.

Figure 1

Tests of the stability of correlator fits for $J_{00}^{nn}$ from fitting the two- and three-point correlators at heavy mass $a{m}_{h0}^{\mathrm{val}}=0.5$ on the fine ensemble. $N_{\text{test}}=1$ gives our final result. $N_{\text{test}}=2$ gives the results when all priors are broadened by 50%. $N_{\text{test}}=3$ and 4 give the results of setting $N_{\exp }=4$ and 6, respectively. $N_{\text{test}}=5$, 6 give the result of setting $t_{\text{cut}}=2,4$ respectively for all correlators. $N_{\text{test}}=7$ gives the result without marginalizing out the $n=5$ excited state. $N_{\text{test}}=8$ gives the result of changing the SVD cut from ${10}^{-3}$ to ${10}^{-2}$. $N_{\text{test}}=9$ gives the result from a fit containing only $a{m}_{h0}^{\mathrm{val}}=0.5$ correlators and hence with a smaller covariance matrix. This allows us, as a test, to use a reduced SVD cut of ${10}^{-5}$.

Figure 2

$h_{A_1}^s(1)$ against $M_{{\eta }_h}$ (a proxy for the heavy quark mass). The grey band shows the result of the extrapolation to $a=0$ at physical $l,s$ and $c$ masses; the black star shows our result at the physical $b$-quark mass. Gluon field ensembles listed in the legend follow the order of sets in Table 1. Solid lines simply join the points on a given ensemble for added clarity. The red inverted triangle gives the determination of the same quantity from a previous study using the NRQCD action for the $b$ quark [41].

Figure 3

The $D_s^{*}$ meson mass obtained on each of our gluon field ensembles, given as a difference from one half the ${\eta }_c$ meson mass. Errors include statistical and lattice spacing uncertainties. The grey band gives the experimental result [46].

Figure 4

The $H_s$ meson mass obtained on each of our gluon field ensembles, given as a difference to one half of the ${\eta }_h$ meson mass. Errors include statistical and lattice spacing uncertainties. The grey band gives a fit to the heavy-quark mass dependence as discussed in the text, with black stars giving our results at $m_h=m_c$ and $m_h=m_b$. The inverted red triangles give the corresponding experimental values [46].

Figure 5

Results of testing the fit to $h_{A_1}^s(1)$ results. The top black point gives our baseline fit result in the continuum and at physical $b$-quark mass. The top three blue points show the corresponding value if results from the fine, superfine or ultrafine ensembles are dropped from the fit. The fourth and fifth blue points show the result if instead results at the highest/lowest $a{m}_{h0}^{\mathrm{val}}$ value on each ensemble are removed. $N_{\text{nuisance}}=3$ shows the result of truncating each sum in ${\mathcal{N}}_{\mathrm{disc}}$ (28) at 3 rather than $2.+1/m_b^3$ results from adding an extra term to (20) of the form $p/M_{{\eta }_h}^3$ where $p$ is a fit parameter with the same prior as $l_{V,A,P}^s$. In this case the Bayes factor falls by a factor of 7, suggesting that the results do not contain a cubic dependence on the heavy mass. The next two points show the results of including specific implementations of ${\eta }_A$ described in Sec. 2e (rather than the value 1). In the upper variant parameter $\rho$ is given prior $0\pm 1$. The lower variant shows the result of using the one-loop expression for ${\eta }_A$ [Eq. (22)], with $m_c/m_h$ replaced with $M_{{\eta }_c}/M_{{\eta }_h}$. $A+1/m_b{m}_c+1/m_b^2$ is the result of replacing $1+l_V/m_c^2$ in the fit with a fit parameter $A$ with prior distribution $1\pm 1$. The fact that this does not affect the fit shows that mistuning of the charm quark mass is a negligible effect here. The points with labels beginning ${ϵ}_h=\text{show}$ the result of replacing the heavy mass proxy $M_{{\eta }_h}/2$ with $M_{H_s}$ and the MRS quark mass [Eq. (23)], respectively. The bottom point labeled ratio with $f_{H_c}$ is the result of an alternative extrapolation described in Appendix A.

Figure 6

Comparison of lattice QCD results for $h_{A_1}^s(1)$ and $h_{A_1}(1)$. Our results for $h_{A_1}^{(s)}(1)$ are marked (HISQ, HPQCD) and for $h_{A_1}(1)$ are marked (HPQCD). Those marked (NRQCD,HPQCD) are from [41] and the value marked (Fermilab, Fermilab/MILC) is from [39].

Figure 7

More detailed comparison of lattice QCD results for $h_{A_1}(1)$ (left side) and $h_{A_1}^s(1)$ (right side). Raw results for $h_{A_1}(1)$ are from [41] and [39] and are plotted as a function of valence ($=\text{sea}$) light quark mass, given by the square of $M_{\pi }$. On the right are points for $h_{A_1}^s(1)$ from [41] plotted at the appropriate valence mass for the $s$ quark, but obtained at physical sea light quark masses. The final result for $h_{A_1}(1)$ from [39], with its full error bar, is given by the inverted blue triangle. The inverted red triangles give the final results for $h_{A_1}(1)$ and $h_{A_1}^s(1)$ from [41]. Our results here are given by the black stars.

Figure 8

The ratio $h_{A_1}^s(1)/(f_{H_c}\sqrt{M_{H_c}})$ plotted against $M_{{\eta }_h}$ (a proxy for the heavy quark mass). Gluon field ensembles listed in the legend follow the order of sets in Table 1. The grey band shows the result of the fit described in the text, evaluated at $a=0$ and physical $l$, $s$ and $c$ quark masses to give the physical heavy quark mass dependence of the ratio. At $m_h=m_b$ we obtain the result given by the black star. For comparison with our previous fit for $h_{A_1}^s(1)$ the inverted red triangle shows our result from Eq. (33) converted to a ratio using the value for $f_{B_c}$ from Fig. 9 and $M_{B_c}$ from experiment [46].

Figure 9

The heavy-charm pseudoscalar meson decay constant, $f_{H_c}$, plotted against $M_{{\eta }_h}$ (a proxy for the heavy quark mass). Gluon field ensembles listed in the legend follow the order of sets in Table 1. The grey band shows the result of the fit described in the text, evaluated at $a=0$ and physical $l,s$ and $c$ quark masses to give the physical heavy quark mass dependence of the decay constant. At $m_h=m_b$ we obtain the result given by the black star. The red triangle shows the result from a previous heavy-HISQ determination of $f_{B_c}$ on $n_f=2+1$ gluon field ensembles [68].