$B_s\to D_s\ell \nu$ form factors for the full $q^2$ range from lattice QCD with nonperturbatively normalized currents

We present a lattice QCD determination of the $B_s\to D_s\ell \nu$ scalar and vector form factors over the full physical range of momentum transfer. The result is derived from correlation functions computed using the highly improved staggered quark (HISQ) formalism, on the second generation MILC gluon ensembles accounting for up, down, strange and charm contributions from the sea. We calculate correlation functions for three lattice spacing values and an array of unphysically light $b$-quark masses, and extrapolate to the physical value. Using the HISQ formalism for all quarks means that the lattice current coupling to the $W$ can be renormalized nonperturbatively, giving a result free from perturbative matching errors for the first time. Our results are in agreement with, and more accurate than, previous determinations of these form factors. From the form factors we also determine the ratio of branching fractions that is sensitive to violation of lepton universality: $R(D_s)=\mathcal{B}(B_s\to D_s\tau {\nu }_{\tau })/\mathcal{B}(B_s\to D_s\ell {\nu }_l)$, where $\ell$ is an electron or a muon. We find $R(D_s)=0.2993(46)$, which is also more accurate than previous lattice QCD results. Combined with a future measurement of $R(D_s)$, this could supply a new test of the Standard Model. We also compare the dependence on heavy quark mass of our form factors to expectations from heavy quark effective theory.

Figure 1

Effective energies and amplitudes, defined from Eqs. (17) and (18) on the fine ensemble. Grey bands show the fit result for the ground-state energies and amplitudes including their uncertainty. The fit used the correlator data on time slices between $t=2$ and $t=N_t-2$.

Figure 2

Tests on the correlator fits on the fine ensemble. The y-axis shows the best fit result for $J_{00}^{nn}$, with the appropriate current, heavy mass and $q^2$ specified. At $N_{\mathrm{test}}=1$ we give our final result, reproduced by the light grey band for ease of comparison. $N_{\mathrm{test}}=2$ and 3 give the results of setting $N_{\exp }=4$ and 6 respectively. $N_{\mathrm{test}}=4$ gives the result of setting $t_{\mathrm{cut}}=3$ for all 2-point correlators (in the final fit $t_{\mathrm{cut}}=2$ for all correlators). $N_{\mathrm{test}}=5$ gives the value when the prior width on the $J_{00}^{nn}$ parameters is doubled. $N_{\mathrm{test}}=6$ gives the results from the output from a fit to the appropriate correlators from that heavy mass and $q^2$ value only, and therefore not including correlation with results from other masses and momentum values.

Figure 3

A comparison of the relativistic dispersion relation for our $D_s$ mesons on gluon field ensemble sets 1 and 3. We plot the square of the “speed of light” against the square of the spatial momentum of the $D_s$ in lattice units. Values on the coarser lattices, set 1, show a small deviation from 1 that is reduced on the finer lattices.

Figure 4

$f_0^s(q_{\max }^2)$ against $M_{{\eta }_h}$ (a proxy for the heavy quark mass). The circle, square and triangular points show lattice results, i.e., outputs from the correlator fits. The grey band shows the result of our fit at $a=0$ and physical $l$, $s$ and $c$ masses. Notice that the $y$-axis scale does not begin at zero. We also include the result from a previous lattice calculation, which used the NRQCD discretization for the $b$ quark with a nonrelativistic expansion of the current through $\mathcal{O}(\mathrm{\Lambda }/m_b)$ and $\mathcal{O}({\alpha }_s)$ matching to continuum QCD [35]. Sets of gluon field configurations listed in the legend follow the order of sets in Table 1.

Figure 5

Results of tests of the $f_0^s(q_{\max }^2)$ fit. The top three blue points show $f_0^s(q_{\max }^2)$ at continuum and physical $b$ mass, if data from the fine, superfine or ultrafine ensembles are not used in the fit. The fourth and fifth blue points show the result if data at the highest/lowest $a{m}_{h0}^{\mathrm{val}}$ value on each ensemble are removed. The point labeled $N_{fit}=3$ is the result of extending the sum in Eq. (28) so that it truncates at 3 rather than 2 in each of the $i$, $j$, $k$ directions. The points labeled $+{\log }^2(M_{{\eta }_h}/M_{{\eta }_c})$ represents the result of adding a ${\rho }_2{\log }^2(M_{{\eta }_h}/M_{{\eta }_c})$ term in the first set of brackets in Eq. (28), where ${\rho }_2$ is a new fit parameter with the same prior distribution as $\rho$. Similarly for the $+\log (M_{{\eta }_h}/M_{{\eta }_c})/M_{{\eta }_h}$ point. The point labelled “no log” results from omitting the factor $(1+\rho \log (M_{{\eta }_c}/M_{{\eta }_h}))$. The lowest point shows the value from the fit result for $R_0^s(q_{\max }^2)$, multiplied by the experimental value for $\sqrt{M_{B_c}}$ [28] and the result of our determination of $f_{B_c}$ at the physical point detailed in Appendix A of [49].

Figure 6

$P{f}_{0,+}^s$ in $z$-space, where $P$ is the appropriate pole function for each form factor given in Eq. (26). The colored points show lattice results, i.e., outputs from the correlator fits. The colors correspond to the legend given in Fig. 7. Sets listed in this legend follow the order of sets in Table 1. The lowest grey band shows the result of our fit at $a=0$ and physical $l$,$s$, $c$ and $b$ masses. Each of the higher grey bands show the fit form evaluated at the heavy quark masses, lattice spacings and $l$,$s$ and $c$ masses of each of our sets of lattice data.

Figure 7

$f_{0,+}^s(q^2)$ plotted against $q^2$. The colored points show lattice results, i.e., outputs from the correlator fits. The grey band shows the result of our fit at $a=0$ and physical $l$,$s$, $c$ and $b$ masses. Sets listed in the legend follow the order of sets in Table 1.

Figure 8

Final result for $f_{0,+}^s(q^2)$ against $q^2$ at the physical point.

Figure 9

Error budget for $f_{0,+}^s(q^2)$ as a function of $q^2$.

Figure 10

Results for $f_{0,+}^s(q^2)$ against $q^2$ at the physical point, comparing the ratio method (from Appendix pp2) and the direct method (from Sec. 3b).

Figure 11

Our final result for $f_{0,+}^s(q^2)$ compared to form factors calculated using an NRQCD action for the $b$ quark [35]. Part of the NRQCD band is shaded darker than the rest ($q^2⪆9.5{\mathrm{GeV}}^2$) to signify the region where lattice results were directly calculated. The NRQCD form factors in the rest of the $q^2$ range are the result of an extrapolation using a BCL parametrization.

Figure 12

Differential decay rates for the $B_s\to D_s\mu {\nu }_{\mu }$ and $B_s\to D_s\tau {\nu }_{\tau }$ decays, calculated using the form factors determined in this work.

Figure 13

Form factor values at $q_{\max }^2$ and $q^2=0$ plotted against $M_{{\eta }_h}$, a proxy for the heavy quark mass.

Figure 14

Form factor ratios against $M_{{\eta }_h}$, a proxy for the heavy quark mass. $S_1^s$ and $V_1^s$ are defined in Eqs. (37) and (38). The colorful points are NLO HQET expectations from [25], derived with input from QCD sum rules.

Figure 15

We show two quantities derived from the form factor slopes as a function of $M_{{\eta }_h}$. $1/\beta (m_h)$ is defined in Eq. (42) and $\delta$ in Eq. (43). Our results are shown by the grey bands. The blue band shows the leading order HQET expectation for $\delta$ given in Eq. (45).

Figure 16

$R_0^s(q_{\max }^2)=f_0^s(q_{\max }^2)/(f_{H_c}\sqrt{M_{H_c}})$ against $M_{{\eta }_h}$ (a proxy for the heavy quark mass). The grey band shows the result of the extrapolation at $a=0$ and physical $l$,$s$ and $c$ masses. Sets listed in the legend follow the order of sets in Table 1.

Figure 17

$R_{0,+}^s(q^2)=f_{0,+}^s(q^2)/(f_{H_c}\sqrt{M_{H_c}})$ plotted against $q^2$. The grey band shows the result of our fit at $a=0$ and physical $l$,$s$, $c$ and $b$ masses. Sets listed in the legend follow the order of sets in Table 1.