$|V_{ub}|$ from $B\to \pi \ell \nu$ decays and ($2+1$)-flavor lattice QCD

We present a lattice-QCD calculation of the $B\to \pi \ell \nu$ semileptonic form factors and a new determination of the CKM matrix element $|V_{ub}|$. We use the MILC asqtad ($2+1$)-flavor lattice configurations at four lattice spacings and light-quark masses down to $1/20$ of the physical strange-quark mass. We extrapolate the lattice form factors to the continuum using staggered chiral perturbation theory in the hard-pion and SU(2) limits. We employ a model-independent $z$ parametrization to extrapolate our lattice form factors from large-recoil momentum to the full kinematic range. We introduce a new functional method to propagate information from the chiral-continuum extrapolation to the $z$ expansion. We present our results together with a complete systematic error budget, including a covariance matrix to enable the combination of our form factors with other lattice-QCD and experimental results. To obtain $|V_{ub}|$, we simultaneously fit the experimental data for the $B\to \pi \ell \nu$ differential decay rate obtained by the BABAR and Belle collaborations together with our lattice form-factor results. We find $|V_{ub}|=(3.72\pm 0.16)\times {10}^{-3}$, where the error is from the combined fit to lattice plus experiments and includes all sources of uncertainty. Our form-factor results bring the QCD error on $|V_{ub}|$ to the same level as the experimental error. We also provide results for the $B\to \pi \ell \nu$ vector and scalar form factors obtained from the combined lattice and experiment fit, which are more precisely determined than from our lattice-QCD calculation alone. These results can be used in other phenomenological applications and to test other approaches to QCD.

Figure 1

Lattice spacings and light-quark masses used in this analysis. The area of each filled disk is proportional to the number of configurations in the ensemble. Open black circles indicate ensembles use in the analysis of Ref. [5]; those with $(a,m_l^{\prime }/m_h^{\prime })=(0.12\mathrm{fm},0.4)$ and (0.09 fm, 0.4) used in Ref. [5] (open circles without disks) are not used in this analysis.

Figure 2

Illustrative diagram for the three-point correlation functions.

Figure 3

Dispersion-relation tests on the ensemble $a\approx 0.12\mathrm{fm}$, $m_l^{\prime }/m_h^{\prime }=0.1$. Left: Fit results of $E_{\pi }^2(\mathit{p})$ from two-point functions are compared with the continuum dispersion relation $|\mathit{p}{|}^2+M_{\pi }^2$. Right: Wave-function overlaps $⟨0|O_{\pi }|\pi (\mathit{p})⟩$ at different momenta are compared. The dotted lines show the expected size of the leading momentum-dependent discretization errors based on power counting, which are of $\mathcal{O}({\alpha }_s^2|\mathit{p}{|}^2{a}^2)$.

Figure 4

Pion (left) and $B$-meson (right) mass in lattice units from the two-point correlator fits versus different choices of $t_{\min }$ for the $a\approx 0.12\mathrm{fm}$, $m_l^{\prime }/m_h^{\prime }=0.1$ ensemble. The $t_{\min }$ marked in red (filled) is that of our preferred fit. The pion and $B$-meson correlators are fit to $2+0$ and $1+1$ states with $t_{\max }=30,21$, respectively.

Figure 5

Fitted $M_{\pi }$ and $M_B$ in lattice units versus the block size for the $a\approx 0.12\mathrm{fm}$, $m_l^{\prime }/m_h^{\prime }=0.1$ ensemble (top) and for the $a\approx 0.045\mathrm{fm}$, $m_l^{\prime }/m_h^{\prime }=0.2$ ensemble (bottom). The dashed line shows the ratio of the fit errors from the blocking and nonblocking data, which can be read off from the right $y$ axis.

Figure 6

Ground-state energy (bottom) and the first excited-state splitting (top) of the $B$ meson on the $a\approx 0.12\mathrm{fm}$, $m_l^{\prime }/m_h^{\prime }=0.1$ ensemble, from different fits to local, 1S-smeared, and both two-point correlators.

Figure 7

Fits of ratios $R_{\perp }/h_{\perp }$ (top left), $R_{\parallel }/h_{\parallel }$ (bottom left) and $R_T/h_T$ (bottom right) for the $a\approx 0.12\mathrm{fm}$, $m_l^{\prime }/m_h^{\prime }=0.1$ ensemble. The open data points with error bars are the ratios constructed from two- and three-point correlators with various momenta. The fit results are shown as solid lines with dashed error widths. The shaded bands indicate the range of data employed in the fit. The resulting lattice form factors $f_J$ are shown by the colored bars on the left of the plots.

Figure 8

Left: Fitted ratio $R_{\perp }/h_{\perp }$ on the $a\approx 0.12\mathrm{fm}$, $m_l^{\prime }/m_h^{\prime }=0.1$ ensemble with momentum $\mathit{p}=(2\pi /L)(1,0,0)$ for different combinations of $t_{\min },t_{\max }$ of the three-point correlator in the combined fit, where circles are results of varying $t_{\min }$ with fixed $t_{\max }=12$ and squares are those of varying $t_{\max }$ with fixed $t_{\min }=6$. Right: The same $R_{\perp }/h_{\perp }$ upon variations of $t_{\min }$ of the two $B$-meson two-point correlators (local and $1S$-smeared) in the combined fit, where $t_{\max }$ is given in Table 5. The filled points in both plots show the preferred fit ranges, and the dashed lines indicate the $p$ values of the fits.

Figure 9

Fitted lattice form factors $f_J$ at different momenta: From top to bottom, $(2\pi /L)(0,0,0)$ ($f_{\parallel }$ only), $(2\pi /L)(1,0,0)$, $(2\pi /L)(1,1,0)$ and $(2\pi /L)(1,1,1)$. Each compares the following different fit schemes: Separate (squares) or combined (circles) fits of the three lattice form factors $f_J$ using Eq. (2.21), a fit including the full first excited states as defined in Eq. (2.22) (upward-pointing triangles) and the simple plateau fit (downward-pointing triangles).

Figure 10

The normalized slopes $-(\partial \ln f/\partial \ln {\overline{m}}_2)$ of the three different form factors $f_{\parallel ,\perp ,T}$ at several momenta for $B\to \pi$ (left) and $B\to K$ (right) semileptonic decays. The horizontal shaded error bands in each plot are the results of correlated fits to all momenta for each form factor.

Figure 11

Chiral-continuum extrapolation of lattice form factors $f_{\perp }$ (upper left), $f_{\parallel }$ (lower left), and $f_T$ (lower right) as functions of $E_{\pi }$, where all quantities are in $r_1$ units. The colors denote the lattice spacings: 0.12 fm (gold), 0.09 fm (green), 0.06 fm (blue), and 0.045 fm (violet). The symbols denote the light-quark masses $m_l^{\prime }/m_h^{\prime }$: 0.05 (diamond), 0.1 (circle), 0.15 (square), 0.2 (downward-pointing triangle), and 0.4 (upward-pointing triangle). The colored lines correspond to the fit results evaluated at the parameters of the ensembles. The physical-mass continuum-limit curve is shown as a black curve with cyan error band.

Figure 12

Discretization effects in the form factor $f_{\perp }$ (left) and $f_{\parallel }$ (right) at a few kinematic points. The plots show the form factors on the $m_l^{\prime }=0.2m_h^{\prime }$ ensembles at each lattice spacing versus $a^2$ for various pion momenta (a slight extrapolation/interpolation is applied to adjust the raw data to the same $E_{\pi }{r}_1$). The range $E_{\pi }{r}_1\in [0.28,1.2]$ is used in the $q^2$ extrapolation to the full kinematic range.

Figure 13

The function $(1-q^2/M_{B^{*}}^2)f_{+}$ obtained from the chiral-continuum fits with NLO logarithms and analytic terms through NLO (hatched black band), NNLO (solid cyan band) and NNNLO (dashed red lines). Errors shown are fit statistical errors only. The vertical line indicates the lowest $q^2$ of the lattice data.

Figure 14

Variations in the chiral-continuum extrapolation from different fit Ansätze. The shaded area shows the fit error from the preferred NNLO SU(2) fit. The other curves show the systematic deviations from the NNLO SU(2) fit under the variations discussed in the text.

Figure 15

Heavy-quark discretization effects in the chiral-continuum fit. The solid blue, solid black, and dashed red curves show the NNLO fit error, the preferred fit error, and the NNLO fit error added to a 1.5% power-counting estimate of heavy-quark discretization errors, respectively.

Figure 16

Subdominant systematic errors over the range of simulated lattice momenta. Error estimates are described in the text.

Figure 17

The distribution of the errors for $f_{+}$ (left) and $f_0$ (right) as a function of $q^2$. The different bands in the plot show the contribution of the error source to the sum of squared errors (left $y$ axis). The corresponding error can be read off from the right $y$ axis.

Figure 18

Comparison of $f_{+}$ (left) and $f_0$ (right) from this work with previous lattice-QCD calculations by HPQCD [4], Fermilab/MILC [5], and RBC/UKQCD [13].

Figure 19

Eigenspectrum of the kernel function $K(z,z^{\prime })$ for $f_0$ and $f_{+}$.

Figure 20

The $z$ expansion of the form factors $f_{+}$ and $f_0$ without any constraints (left) and with the kinematic constraint. The fits use $N_z=4$.

Figure 21

Bootstrap-sample distribution defined in Eq. (5.5) for the $z$-expansion coefficients of $f_{+}$ (left) and $f_0$ (right), compared with the heavy-quark scaling estimate 0.013 (red vertical line in the left plot) and the unitarity bound (red vertical line in the right plot). The 1000 samples of fits use only the kinematic constraint and $N_z=4$.

Figure 22

Effects of truncating the $z$ expansion for $f_{+}$. The areas indicated by the hatched band, colored shaded band, and the lines are results with $1\sigma$ errors for $N_z=3,4,5$ $z$ fits, respectively.

Figure 23

$z$-fit results for the form factors $f_0$ and $f_{+}$ as functions of $z$.

Figure 24

Comparison of $f_{+}$ (left) and $f_0$ (right) from the $z$-expansion fit results of this work with recent theoretical calculations using light-cone sum rules (LCSR) [62] and lattice QCD [13].

Figure 25

The lattice form-factor ratio $(f_0/f_{+})/(1-q^2/M_{B^{*}}^2)$ versus $q^2$ (cyan shaded band) compared with the prediction in the soft-pion limit from heavy-quark symmetry and $\chi \mathrm{PT}$ [63] (hatched band). The width of the hatched band reflects only the uncertainty from $g_{B^{*}B\pi }=0.45(8)$ and not other theoretical errors.

Figure 26

The contour plot of the slope $b_1/b_0$ and curvature $b_2/b_0$ of the form factor $f_{+}$. The open ellipses are the $1\sigma$ contour of the slope and curvature constructed from the three-parameter $z$ fit to individual experimental data. The gold-filled ellipse is from the combined fit of all experimental data. The cyan-filled ellipse is from the four-parameter $z$ fit to lattice form factors.

Figure 27

Left: Comparison of vector form factor $f_{+}(z)$ from $z$ expansion fits to only the lattice-QCD data (cyan band), and to only experimental data including all four measurements (gold band). Right: The similar plot for the partial branching fraction $dB/d{q}^2$. The fits including lattice results use $N_z=4$, while the experiment-only fit uses $N_z=3$. The experimental data points and the experiment-only $z$-fit result in the left plot have been converted from ${(\mathrm{\Delta }\mathcal{B}/\mathrm{\Delta }{q}^2)}^{1/2}$ to $f_{+}$ using $|V_{ub}|$ from the combined fit. The lattice-only fit result (cyan band) and the combined-fit result (red band) in the right plot are converted from the form factor with the same $|V_{ub}|$.

Figure 28

Determinations of $|V_{ub}|$. The squares are obtained from $B\to \pi \ell \nu$ decay using theoretical form factors from this analysis, our earlier work [5] (now superseded, but with updated experimental input from HFAG 2014 [6]), a three-flavor lattice calculation by RBC/UKQCD [13], light-cone sum rules (orange square) [62], and HPQCD [4] (using the $q^2>16{\mathrm{GeV}}^2$ experimental data only). The blue upward-pointing triangle is obtained from ${\mathrm{\Lambda }}_b\to p\ell \nu$ decay using lattice-QCD form factors from Ref. [68] and experimental data from LHCb [69]. The black diamond shows the inclusive determination using $B\to X_u\ell \nu$ decays [6] with the theoretical approach of Ref. [15]. Also shown is the expectation from CKM unitarity [70] (green-filled circle). For the exclusive determinations from $B\to \pi \ell \nu$ decay (squares), all four experimental results [7, 8, 9, 10] are used, except in the LCSR $z$ fit, where only the more recent BABAR [8] and Belle [10] data are used.

Figure 29

The form factors $f_{+}(z)$ (left) and $f_0(z)$ (right) from the $z$ fits to the lattice data (cyan band), to all experiments (gold band), and to the lattice data plus experiments (red band). The fits including lattice results use $N_z=4$ and the kinematic constraint, while the experiment-only $z$ fit uses $N_z=3$. The experiment-only $z$-fit result has been converted from ${(\mathrm{\Delta }\mathcal{B}/\mathrm{\Delta }{q}^2)}^{1/2}$ to $f_{+}$ using $|V_{ub}|$ from the combined fit.