$B\to \pi l\nu$ semileptonic form factor from three-flavor lattice QCD: A model-independent determination of $|V_{\mathrm{ub}}|$

We calculate the form factor $f_{+}(q^2)$ for $B$-meson semileptonic decay in unquenched lattice QCD with $2+1$ flavors of light sea quarks. We use Asqtad-improved staggered light quarks and a Fermilab bottom quark on gauge configurations generated by the MILC Collaboration. We simulate with several light-quark masses and at two lattice spacings, and extrapolate to the physical quark mass and continuum limit using heavy-light meson staggered chiral perturbation theory. We then fit the lattice result for $f_{+}(q^2)$ simultaneously with that measured by the BABAR experiment using a parameterization of the form-factor shape in $q^2$, which relies only on analyticity and unitarity in order to determine the Cabibbo-Kobayashi-Maskawa matrix element $|V_{\mathrm{ub}}|$. This approach reduces the total uncertainty in $|V_{\mathrm{ub}}|$ by combining the lattice and experimental information in an optimal, model-independent manner. We find a value of $|V_{\mathrm{ub}}|\times {10}^3=3.38\pm 0.36$.

Figure 1

Pion mass (left plot) and $B$-meson mass (right plot) versus minimum timeslice in 2-point correlator fit. The red (filled) data points show the fit ranges selected for use in the $B\to \pi \ell \nu$ form-factor analysis.

Figure 2

Pion correlator fit corresponding to the red data point in the left-hand graph of Fig. 1. The left plot shows the fit (red line) to the zero-momentum pion propagator on a log scale, while the right plot shows the deviation of the fit from the data point for each timeslice. On both plots the dashed vertical line indicates $t_{\min }$. Single-elimination jackknife statistical errors are shown.

Figure 3

$B$-meson correlator fit corresponding to the red data point in the right-hand graph of Fig. 1. The left plot shows the fit (red line) to the $B$-meson propagator on a log scale, while the right plot shows the deviation of the fit from the data point for each timeslice. On both plots the dashed vertical line indicates $t_{\min }$. Single-elimination jackknife statistical errors are shown.

Figure 4

Single-elimination jackknife error versus block size in the zero-momentum pion propagator at $t=6$. The statistical errors in the errors are calculated with an additional single-elimination jackknife loop. The red line is an average of the errors for block sizes 5–8 and is only to make it easier to see that the statistical error plateaus after a block size of 5; it is not used in the form-factor analysis.

Figure 5

Comparison of pion energy $E_{\pi }$ (left plot) and amplitude $Z_{\pi }$ (right plot) with the prediction of the continuum dispersion relation. We also show a power-counting estimate for the size of momentum-dependent discretization errors, which are of $\mathcal{O}({\alpha }_s(a|{\overrightarrow{p}}_{\pi }|{)}^2)$, as dashed black lines.

Figure 6

Determination of the form factors $f_{\parallel }$ (left plot) and $f_{\perp }$ (right plot) from plateau fits to the ratios defined in Eqs. (40) and (41). The statistical errors on the data points are from a single-elimination jackknife. The statistical errors in the plateau determination are from separate fits of 500 bootstrap ensembles.

Figure 7

Chiral-continuum extrapolation of $f_{\parallel }$ (upper) and $f_{\perp }$ (lower) using constrained NLO $\mathrm{HMS}\chi \mathrm{PT}$ plus all NNLO analytic terms with $g_{B^{*}B\pi }=0.51$ and $r_1=0.311\mathrm{fm}$. The square symbols indicate $a\approx 0.09\mathrm{fm}$ lattice data points, while the circular symbols indicate $a\approx 0.12\mathrm{fm}$ coarse data points. The black curve is the chiral-continuum extrapolated form-factor symmetrized bootstrap statistical errors only.

Figure 8

Normalized form factor $f_{+}$ at momentum $\overrightarrow{p}=2\pi (1,1,0)/L$ as a function of $\kappa$ on the $a{m}_l/a{m}_s=0.02/0.05$ coarse ensemble. The central data point corresponds to the tuned ${\kappa }_b$, and the thick red line shows a linear fit to the three data points. The two dashed vertical lines indicate the upper and lower bounds on ${\kappa }_b$.

Figure 9

Experimental data for the $B\to \pi l\nu$ form factor times the CKM element $|V_{\mathrm{ub}}|$ from the BABAR Collaboration [42]. The left plot shows $|V_{\mathrm{ub}}|\times f_{+}$ versus $q^2$, while the right plot shows $|V_{\mathrm{ub}}|\times f_{+}$ multiplied by the functions $P_{+}{\varphi }_{+}$ and plotted against the new variable $z$. Both the 2-parameter fit (dashed blue line) and 3-parameter fit (solid red curve) have good ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$’s.

Figure 10

Lattice calculation of the $B\to \pi l\nu$ form factor. The left plot shows $f_{+}$ vs $q^2$, while the right plot shows $P_{+}{\varphi }_{+}{f}_{+}$ vs $z$. The inner error bars indicate the statistical error, while the outer error bars indicate the sum of the statistical and systematic added in quadrature. A 3-parameter $z$ fit is needed to describe the lattice data with a good ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$

Figure 11

Model-independent determination of $|V_{\mathrm{ub}}|$ from a simultaneous fit of lattice and experimental $B\to \pi \ell \nu$ semileptonic form-factor data to the $z$ parameterizaton. The left plot shows $P_{+}{\varphi }_{+}{f}_{+}$ vs $z$, while the right plot shows $f_{+}$ vs $q^2$. Inclusion of terms in the power-series through $z^3$ yields the maximum uncertainty in $|V_{\mathrm{ub}}|$; the corresponding 4-parameter $z$ fit is given by the red curve in both plots. The circles denote the Fermilab-MILC lattice data, while the stars indicate the 12-bin BABAR experimental data, rescaled by the value of $|V_{\mathrm{ub}}|$ determined in the simultaneous $z$-fit.