$B\to \pi \ell \nu$ and $B_s\to K\ell \nu$ form factors and $|V_{ub}|$ from $2+1$-flavor lattice QCD with domain-wall light quarks and relativistic heavy quarks

We calculate the form factors for $B\to \pi \ell \nu$ and $B_s\to K\ell \nu$ decay in dynamical lattice quantum chromodynamics (QCD) using domain-wall light quarks and relativistic $b$-quarks. We use the ($2+1$)-flavor gauge-field ensembles generated by the RBC and UKQCD collaborations with the domain-wall fermion action and Iwasaki gauge action. For the $b$-quarks we use the anisotropic clover action with a relativistic heavy-quark interpretation. We analyze data at two lattice spacings of $a\approx 0.11$, 0.086 fm with unitary pion masses as light as $M_{\pi }\approx 290\mathrm{MeV}$. We simultaneously extrapolate our numerical results to the physical light-quark masses and to the continuum and interpolate in the pion/kaon energy using SU(2) “hard-pion” chiral perturbation theory for heavy-light meson form factors. We provide complete systematic error budgets for the vector and scalar form factors $f_{+}(q^2)$ and $f_0(q^2)$ for both $B\to \pi \ell \nu$ and $B_s\to K\ell \nu$ at three momenta that span the $q^2$ range accessible in our numerical simulations. Next we extrapolate these results to $q^2=0$ using a model-independent $z$-parametrization based on analyticity and unitarity. We present our final results for $f_{+}(q^2)$ and $f_0(q^2)$ as the coefficients of the series in $z$ and the matrix of correlations between them; this provides a parametrization of the form factors valid over the entire allowed kinematic range. Our results agree with other three-flavor lattice-QCD determinations using staggered light quarks, and have comparable precision, thereby providing important independent cross-checks. Both $B\to \pi \ell \nu$ and $B_s\to K\ell \nu$ decays enable determinations of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{ub}|$. To illustrate this, we perform a combined $z$-fit of our numerical $B\to \pi \ell \nu$ form-factor data with the experimental measurements of the branching fraction from BABAR and Belle leaving the relative normalization as a free parameter; we obtain $|V_{ub}|=3.61(32)\times 10^{-3}$, where the error includes statistical and all systematic uncertainties. The same approach can be applied to the decay $B_s\to K\ell \nu$ to provide an alternative determination of $|V_{ub}|$ once the process has been measured experimentally. Finally, in anticipation of future experimental measurements, we make predictions for $B\to \pi \ell \nu$ and $B_s\to K\ell \nu$ differential branching fractions and forward-backward asymmetries in the Standard Model.

Figure 1

Effective masses of the pion (upper left), kaon (bottom left), $B$ meson (upper right) and $B_s$ meson (bottom right) on the $a\approx 0.086\mathrm{fm}$ ensemble with $a{m}_l=0.004$. Shaded bands show the correlated fit results with jackknife statistical errors over the fit ranges used. All results are shown in lattice units.

Figure 2

Comparison of pion (top) and kaon (bottom) energies (left) and amplitudes (right) with continuum-limit expectations on the $a\approx 0.086\mathrm{fm}$ ensemble with $a{m}_l=0.004$. The dashed lines show a power-counting estimate of the leading $\mathcal{O}((a\overrightarrow{p}{)}^2)$ momentum-dependent discretization errors.

Figure 3

Three-point correlation function used to obtain the $B\to P$ form factors. The single and double lines correspond to light- and $b$-quark propagators, respectively. For $B\to \pi \ell \nu$, the spectator-quark mass ($m_{l^{\prime }}$) is the same as the light sea-quark mass, while for $B_s\to K\ell \nu$, the spectator-quark mass is close to the physical $m_s$. The light daughter-quark mass ($m_l$) is always equal to the light sea-quark mass. Black and grey circles denote local and smeared operators, respectively.

Figure 4

Unimproved three-point ratio $R_{3,0}^0$ for ${\overrightarrow{p}}_{\pi }=0$ with several source-sink separations $t_{\text{snk}}$ on the $a\approx 0.086\mathrm{fm}$ ensemble with $a{m}_l=0.004$.

Figure 5

$\mathcal{O}({\alpha }_{s}a)$-improved ratios $R_{3,i}/p_P^i$ (left) and $R_{3,0}$ (right) with $t_{\text{snk}}=26$ on the $a\approx 0.086\mathrm{fm}$ ensemble with $a{m}_l=0.004$. Plots for $B\to \pi l\nu$ are on the top and $B_s\to Kl\nu$ are on the bottom. Fit ranges and fit results with jackknife statistical errors are shown as horizontal bands.

Figure 6

Chiral-continuum extrapolation of the $B\to \pi \ell \nu$ (upper plots) and $B_s\to K\ell \nu$ (lower plots) form factors from correlated fits using NLO SU(2) hard-pion/kaon HM$\chi \mathrm{PT}$. Fits of $f_{\perp }$ are on the left and of $f_{\parallel }$ are on the right. In each plot, the colors distinguish between data points on the five different ensembles: circles and squares correspond to the $a\approx 0.11\mathrm{fm}$ data and triangles and diamonds to $a\approx 0.086\mathrm{fm}$ data. The colored fit curves show the interpolation/extrapolation in pion/kaon energy: the fit function is evaluated at the unphysical sea-quark masses and nonzero lattice spacings on the different ensembles, such that the curves should go through the data points of the same color. The continuum, physical-quark-mass form factors are shown as a function of pion/kaon energy by the black lines with gray error band. The vertical dashed line on the left-hand side of each plot shows the physical pion or kaon mass.

Figure 7

Visualization of the error budgets for the $B\to \pi \ell \nu$ (upper plots) and $B_s\to K\ell \nu$ (lower plots) form factors. Error budgets for $f_{\perp }$ are on the left and of $f_{\parallel }$ are on the right. The curves from bottom-to-top show the increase in the total percentage error as we add each individual source of error in quadrature. In each plot, the left y-axis label shows the squared error, while the right y-axis label shows the error in the form factor. For readability, we have combined all of the sources of uncertainty that we estimate to be below $\sim 1\%$ into a single entry labeled “other systematics.” The three vertical lines in each plot show the location of the synthetic data points used in the subsequent extrapolation to $q^2=0$. Detailed error budgets at these $q^2$ values are given in Table 6.

Figure 8

Relative change of the form-factor central value under the considered fit variations for $B\to \pi \ell \nu$ (upper) and $B_s\to K\ell \nu$ (lower). In each plot, the shaded band shows the statistical uncertainty of the preferred fit. The three vertical lines show the location of the synthetic data points used in the subsequent extrapolation to $q^2=0$.

Figure 9

RHQ parameter dependence of the $B_s\to K$ form factors $f_{\perp }$ (left) and $f_{\parallel }$ (right) on the $24^3$ ensembles with $a{m}_l=0.005$ using the unimproved heavy-light vector current in Eq. (10). The slopes are normalized using the form factors obtained at the central set of RHQ parameters. From left to right, the plots show the dependence on $m_{0}a$, $c_P$, and $\zeta$. The colored lines show the results of a linear fit to the three data points at each momentum. The black vertical lines indicate the tuned values of the RHQ parameters. The shaded vertical bands indicate the systematic errors in the RHQ parameters due to the lattice-scale uncertainty. For clarity, data points at equal RHQ parameter values are plotted with a slight horizontal offset.

Figure 10

Valence strange-quark mass dependence of the $B_s\to K$ form factors $f_{\perp }$ (top) and $f_{\parallel }$ (bottom) on the $a\approx 0.11\mathrm{fm}$ ensemble with $a{m}_l=0.005$. The slopes are normalized by the form factors obtained with the strange-quark mass used in our production simulations. The colored lines show the results of a linear fit to the three data points at each momentum. The black vertical line with error band shows the total (statistical plus systematic) uncertainty in the physical strange-quark mass [16]. For clarity, data points at equal strange-quark masses are plotted with a slight horizontal offset.

Figure 11

Fits of the $B\to \pi \ell \nu$ (upper plots) and $B_s\to K\ell \nu$ (lower plots) lattice form factors to the $z$-expansion versus truncation $K$ without constraints (left) and with the kinematic and heavy-quark constraints (right). The black open symbols show the synthetic data points with statistical (inner) and statistical $\oplus$ systematic (outer) error bars. The curves with colored bands show the fit results with errors for different truncations $K$. We do not show unconstrained fits with $K=4$ in the left-hand plot because we only have three synthetic data points; the inclusion of the kinematic and heavy-quark constraints allows us to perform the $K=4$ fit shown in the right-hand plot. In the right-hand plot, we do not show the $K=2$ combined fit because of the poor fit quality.

Figure 12

Preferred $K=3$ fit of the $B\to \pi \ell \nu$ (upper plots) and $B_s\to K\ell \nu$ (lower plots) lattice form factors to the $z$-expansion including the kinematic and heavy-quark constraints versus $z$ (left) and versus $q^2$ (right). The black open symbols show the synthetic data points with statistical (inner) and statistical $\oplus$ systematic (outer) error bars. The solid curves with error bands show the fit results for $f_{+}(q^2)$ and $f_0(q^2)$.

Figure 13

Comparison of fits to the $B\to \pi \ell \nu$ (left) and $B_s\to K\ell \nu$ (right) lattice form-factor data using the BCL and BGL parametrizations. The black open symbols show the synthetic data points with statistical (inner) and statistical $\oplus$ systematic (outer) error bars. The gray and red curves with error bands show the BCL and BGL fits, respectively.

Figure 14

Left: SU(3)-breaking ratios $R_{+}(q^2)$ (red) and $R_0(q^2)$ (blue). The dashed horizontal lines show the expected size of SU(3) breaking from simple power counting. Right: ratios of scalar to vector form factor $f_0(q^2)/f_{+}(q^2)$ for $B\to \pi \ell \nu$ (red) and $B_s\to K\ell \nu$ (blue). The black diagonal line shows the prediction in the heavy-quark symmetry limit. Errors are shown in shaded bands. In the left plot, errors shown are statistical only.

Figure 15

Shape parameters $b_{+}^{(1)}/b_{+}^{(0)}$ and $b_{+}^{(2)}/b_{+}^{(0)}$ from $K=3$ BCL fits to the $B\to \pi \ell \nu$ form factor $f_{+}^{B\pi }(q^2)$ (filled ellipse) and from experimental measurements of the $B\to \pi \ell \nu$ branching fraction [2, 3, 4, 5] (patterned and empty ellipses). The colored ellipses show the constraints from the individual experiments, while the black ellipse shows the constraint from all experiments. For each determination, the inner and outer contours show the 68% and 95% allowed confidence limits, respectively.

Figure 16

Model-independent determination of $|V_{ub}|$ from a combined fit of experimental measurements of the $B\to \pi \ell \nu$ branching fraction [2, 3, 4, 5] and our lattice result for the $B\to \pi \ell \nu$ form factor $f_{+}(q^2)$ to the BCL $z$ parametrization, Eqs. (44) and (45), with $K=3$. The left plot shows $(1-q^2/m_{B^{*}}^2)f_{+}(q^2)$ versus $z$ (where the experimental data have been rescaled by the value of $|V_{ub}|$ determined in the fit), while the right plot shows $\mathrm{\Delta }\mathcal{B}/\mathrm{\Delta }{q}^2$ versus $q^2$ (where the lattice points have been rescaled by $|V_{ub}|$). In both plots, the filled black circles show the lattice data, while the open colored symbols show the experimental data. The black curve with gray error band shows the fit result.

Figure 17

Standard-Model predictions for the differential decay rate divided by $|V_{ub}{|}^2$ for $B_{(s)}\to P\ell \nu$ decays to muon (left) and $\tau$-lepton (right) final states using our form-factor determinations in Tables 11 and 12.

Figure 18

Standard-Model predictions for the differential branching fraction for $B_s\to K\mu \nu$ (left) and $B_s\to K\tau \nu$ (right) using our determinations of the $B_s\to K\ell \nu$ form factors in Table 12. Each plot shows predictions for $d\mathcal{B}/d{q}^2$ using our determination of $|V_{ub}|$ in Eq. (55) from exclusive $B\to \pi \ell \nu$ decay as well as using the determination of $|V_{ub}|$ from inclusive $B\to X_u\ell \nu$ decay [66]. In these plots, the outer error band denotes the total uncertainty in $d\mathcal{B}/d{q}^2$, while the inner error band removes the error contribution from $|V_{ub}|$.

Figure 19

Standard-Model ratio of differential branching fractions ${\mathcal{R}}_P^{\tau /\mu }(q^2)$ using our determinations of the $B\to \pi \ell \nu$ and $B_s\to K\ell \nu$ form factors in Tables 11 and 12.

Figure 20

Standard-Model predictions for the forward-backward asymmetry for $B_s\to K\mu \nu$ (left) and $B_s\to K\tau \nu$ (right) using our determinations of the $B_s\to K\ell \nu$ form factors in Table 12. The plots show predictions for ${\mathcal{A}}_{FB}$ using our determination of $|V_{ub}|$ in Eq. (55) from exclusive $B\to \pi \ell \nu$ decay as well as using the determination of $|V_{ub}|$ from inclusive $B\to X_u\ell \nu$ decay [66]. In these plots, the outer error band denotes the total uncertainty in $d\mathcal{B}/d{q}^2$, while the inner error band removes the error contribution from $|V_{ub}|$.

Figure 21

Standard-Model predictions for the normalized forward-backward asymmetry ${\overline{\mathcal{A}}}_{FB}$ for decays to muon (left) and $\tau$-lepton (right) final states using our form-factor determinations in Tables 11 and 12.

Figure 22

Determinations of $|V_{ub}|$ from Table 14. For points with double error bars, the inner error bars are experimental while the outer error bars show the total experimental plus theoretical uncertainty added in quadrature.

Figure 23

Top: theoretical calculations of the $B\to \pi \ell \nu$ form factors from light-cone sum rules [70, 71], NLO perturbative QCD [72], and ($2+1$)-flavor lattice QCD [26, 27]. Bottom: theoretical calculations of the $B_s\to K\ell \nu$ form factors from QCD models [73, 74, 75] and ($2+1$)-flavor lattice QCD [30]. In all plots, the predictions for $f_{+}(q^2=0)$ are displayed with a slight horizontal offset for clarity.