Present address: ]Forschungszentrum Jülich, Institute for Advanced Simulation,
Jülich Supercomputing Centre, 52425 Jülich, Germany Present address: ]Higgs
Centre for Theoretical Physics, School of Physics & Astronomy, The University
of Edinburgh, EH9 3FD, UK RBC and UKQCD Collaborations

#  $B\to\pi\ell\nu$ 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

J. M. Flynn  School of Physics and Astronomy, University of Southampton,
Southampton SO17 1BJ, UK  T. Izubuchi  RIKEN-BNL Research Center, Brookhaven
National Laboratory, Upton, NY 11973, USA  Physics Department, Brookhaven
National Laboratory, Upton, NY 11973, USA  T. Kawanai  [  RIKEN-BNL Research
Center, Brookhaven National Laboratory, Upton, NY 11973, USA  Physics
Department, Brookhaven National Laboratory, Upton, NY 11973, USA  C. Lehner
Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA  A.
Soni  Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA
R. S. Van de Water  Theoretical Physics Department, Fermi National Accelerator
Laboratory, Batavia, IL 60510, USA  O. Witzel  [  Center for Computational
Science, Boston University, Boston, MA 02215, USA

###### Abstract

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 $M_{\pi}\approx 290$ 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.

###### pacs:

11.15.Ha 12.38.Gc 13.20.He 14.40.Nd

††preprint:  FERMILAB-PUB-15-019-T

##  I Introduction

Semileptonic $B$-meson decays play an important role in the search for new
physics in the quark-flavor sector. Tree-level decays that occur via charged
$W$-boson exchange are used to obtain the Cabibbo-Kobayashi-Maskawa (CKM)
matrix elements $|V_{ub}|$ and $|V_{cb}|$, while flavor-changing neutral-
current decays provide sensitive probes for heavy new particles that may enter
virtual loops. Decays involving $\tau$ leptons are especially sensitive to
charged Higgs bosons that arise in many new-physics models (see e.g. Ref.
Celis:2014pza  and references therein).

The decays $B\to\pi\ell\nu$ and $B_{s}\to K\ell\nu$ probe the quark-flavor-
changing transition $b\to u$. In the Standard Model, the differential decay
rate for these processes in the $B_{(s)}$-meson rest frame is given by

$\displaystyle\frac{d\Gamma(B_{(s)}\to P\ell\nu)}{dq^{2}}$
$\displaystyle=\frac{G_{F}^{2}|V_{ub}|^{2}}{24\pi^{3}}\,\frac{(q^{2}-m_{\ell}^{2})^{2}\sqrt{E_{P}^{2}-M_{P}^{2}}}{q^{4}M_{B_{(s)}}^{2}}\bigg{[}\left(1+\frac{m_{\ell}^{2}}{2q^{2}}\right)M_{B_{(s)}}^{2}(E_{P}^{2}-M_{P}^{2})|f_{+}(q^{2})|^{2}$
$\displaystyle+\,\frac{3m_{\ell}^{2}}{8q^{2}}(M_{B_{(s)}}^{2}-M_{P}^{2})^{2}|f_{0}(q^{2})|^{2}\bigg{]}\,,$
(1)

where $P$ denotes the light pseudoscalar pion or kaon and
$q\equiv(p_{B}-p_{P})$ is the momentum transferred to the outgoing charged-
lepton-neutrino pair. The vector and scalar form factors $f_{+}(q^{2})$ and
$f_{0}(q^{2})$ parametrize the hadronic contributions to the electroweak decay
and must be calculated nonperturbatively, such as with lattice QCD. Given an
experimental measurement of the branching fraction and a theoretical
calculation of the form factor(s), these decays enable a determination of the
CKM matrix element $|V_{ub}|$. (The contribution from $f_{0}^{B\pi}(q^{2})$ in
Eq. (1) can be neglected for light leptons $\ell=e,\mu$ given the current
experimental and theoretical precision.) To date, both the BaBar and Belle
experiments have measured ${\mathcal{B}}(B\to\pi\ell\nu)$ delAmoSanchez:2010af
; Ha:2010rf ; Lees:2012vv ; Sibidanov:2013rkk , and the experimental
uncertainty will continue to improve with the collection of data at Belle II.
The decay $B_{s}\to K\ell\nu$ has not yet been measured, but we anticipate a
result from LHCb in the next few years.

The CKM matrix element $|V_{ub}|$ places a constraint on the apex of the CKM
unitarity triangle CKMfitter ; UTfit ; Laiho:2009eu . Its value, however, is
under scrutiny because of the long-standing $\sim 3\sigma$ disagreement
between $|V_{ub}|$ obtained from exclusive $B\to\pi\ell\nu$ decay and
$|V_{ub}|$ obtained from inclusive $B\to X_{u}\ell\nu$ decays, where $X_{u}$
denotes all charmless final states with up quarks CKMfitter ; UTfit ;
Antonelli:2009ws ; Laiho:2009eu ; Beringer:2012zz ; Aoki:2013ldr . The value
of $|V_{ub}|$ can also in principle be obtained from leptonic $B\to\tau\nu$
decay, but the current determination from this process lies in between those
from exclusive and inclusive semileptonic decays, and is not as precise
Aoki:2013ldr . Further, $B\to\tau\nu$ is sensitive to charged-Higgs boson
exchange, and therefore does not provide a clean Standard-Model determination
of $|V_{ub}|$. Thus the decay $B_{s}\to K\ell\nu$, once measured
experimentally, will provide an important new determination of $|V_{ub}|$.

In this paper we present a new calculation of the semileptonic form factors
for $B\to\pi\ell\nu$ and $B_{s}\to K\ell\nu$ in (2+1)-flavor lattice QCD.
Preliminary results were presented in Refs. Kawanai:2012id ; Kawanai:2013qxa .
This is the second in a series of $B$-meson matrix-element calculations that
uses the same lattice actions and ensembles, and our analysis follows a
similar approach to our earlier work on $B$-meson decay constants
Christ:2014uea . We use the gauge-field ensembles generated by the RBC and
UKQCD collaborations with the domain-wall fermion action and Iwasaki gluon
action which include the effects of dynamical $u,d$, and $s$ quarks
Allton:2008pn ; Aoki:2010dy . For the bottom quarks, we use the Columbia
version of the relativistic heavy-quark (RHQ) action introduced by Christ, Li,
and Lin in Ref. Christ:2006us , with the parameters of the action that were
obtained nonperturbatively in Ref. Aoki:2012xaa . We renormalize the lattice
heavy-light vector current using the mostly nonperturbative method introduced
in Ref. ElKhadra:2001rv , in which we compute the bulk of the matching factor
nonperturbatively Aoki:2010dy ; Christ:2014uea , with a small correction, that
is close to unity, evaluated in lattice perturbation theory Lehner:2012bt ;
CLehnerPT . We also improve the lattice heavy-light current through
${\mathcal{O}}(\alpha_{s}a)$.

We analyze data on five sea-quark ensembles with unitary pions as light as
$\approx$ 290 MeV and two lattice spacings of $a\approx$ 0.11 and 0.086 fm. 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 ($\chi$PT) for heavy-light meson
form factors Becirevic:2002sc ; Bijnens:2010ws , which applies when the
pion/kaon energy is large compared to its rest mass. For $B\to\pi\ell\nu$
($B_{s}\to K\ell\nu$), we directly simulate in the momentum region $q^{2}_{\rm
max}>q^{2}\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{$\mathchar
536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar 318\relax$}}19.0$ GeV2
($q^{2}_{\rm max}>q^{2}\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{$\mathchar
536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar 318\relax$}}17.6$ GeV2). Both
statistical errors and discretization errors increase at lower $q^{2}$, which
corresponds to larger pion/kaon energies. To extend our results beyond the
momenta accessible in our simulations, we extrapolate our results to $q^{2}=0$
using a model-independent $z$-parametrization based on analyticity and
unitarity Boyd:1994tt ; Bourrely:2008za . Our results can be combined with
current and future experimental measurements of the experimentally measured
$B\to\pi\ell\nu$ and $B_{s}\to K\ell\nu$ branching fractions to obtain the CKM
matrix element $|V_{ub}|$.

There are two earlier published (2+1)-flavor calculations of the
$B\to\pi\ell\nu$ semileptonic form factor in the literature by the HPQCD and
Fermilab/MILC collaborations Dalgic:2006dt ; Bailey:2008wp ; updates of these
works are in progress Bouchard:2013zda ; Du:2013kea . In addition, HPQCD
recently obtained the first results for the $B_{s}\to K\ell\nu$ form factor in
Ref. Bouchard:2014ypa . Both groups use the MILC collaboration’s asqtad-
improved staggered gauge-field ensembles Bernard:2001av ; Bazavov:2009bb , so
their results are somewhat correlated. The differences between the two sets of
calculations lie in the choices of light valence- and $b$-quark actions. For
the $b$ quarks, HPQCD uses the NRQCD action Lepage:1992tx  while Fermilab/MILC
uses a relativistic formulation similar to ours. Specifically, they use the
Fermilab interpretation of the isotropic clover action ElKhadra:1996mp  with
the tadpole-improved tree-level value of the clover coefficient $c_{SW}$. The
more recent HPQCD calculation uses the HISQ action for the light valence
quarks to reduce taste-breaking discretization effects, while in the other
work asqtad valence quarks are used.

Our form-factor calculation with domain-wall light quarks and RHQ $b$ quarks
has the advantage that discretization errors from the light quarks and gluons
are simpler, such that the SU(2) heavy-light meson $\chi$PT expressions are
continuum-like. Further, as compared to the Fermilab/MILC calculation, we tune
the coefficient of the clover term in the $b$-quark action nonperturbatively
and improve the heavy-light vector current through
${\mathcal{O}}(\alpha_{s}a)$, whereas Fermilab/MILC only improve it through
${\mathcal{O}}(a)$. Thus, for similar values of the lattice spacing,
discretization errors from the heavy-quark action and current are smaller in
our calculation. Our new results for the $B\to\pi\ell\nu$ and $B_{s}\to
K\ell\nu$ form factors therefore enable important independent determinations
of the CKM matrix element $|V_{ub}|$.

This paper is organized as follows. Section II provides an overview of the
lattice calculation. First we define the needed matrix elements and form
factors in Sec. II.1. Next we present the lattice actions and parameters in
Sec. II.2. Then, in Sec. II.3 we describe the renormalization and improvement
of the heavy-light vector current operator. Section III presents the numerical
analysis. First, in Secs. III.1 and III.2 we fit lattice two-point and three-
point correlators to extract the needed meson masses and matrix elements,
respectively. Then, in Sec. III.3 we extrapolate our numerical data to the
physical light-quark masses and continuum, and interpolate in the pion/kaon
energy, using SU(2) hard-pion $\chi$PT. Section IV provides complete error
budgets for $f_{+}(q^{2})$ and $f_{0}(q^{2})$ at three momentum values that
span the range accessible in our numerical simulations; for clarity, we
discuss each source of systematic uncertainty in a separate subsection. In
Section V we extrapolate our form-factor data to $q^{2}=0$ using a model-
independent $z$ parametrization. We present our 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 model-independent parametrization
of the form factors valid over the entire allowed kinematic range. We
illustrate the phenomenological utility of our form-factor results in Sec. VI.
First, in Sec. VI.1, 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 to determine $|V_{ub}|$. Next, in Sec.
VI.2, we make predictions for Standard-Model observables for the decay
processes $B\to\pi\ell\nu$ and $B_{s}\to K\ell\nu$ with $\ell=\mu,\tau$ in
anticipation of future experimental measurements. Section VII concludes with a
comparison of our results with other lattice determinations, and with an
outlook for the future.

##  II Lattice calculation

Here we present the setup of our numerical lattice calculation.

###  II.1 Form factors

The $B\to\pi\ell\nu$ and $B_{s}\to K\ell\nu$ semileptonic form factors
parametrize the hadronic matrix element of the $b\to u$ vector current
$\mathcal{V}^{\mu}\equiv\overline{u}\gamma^{\mu}b$:

$\displaystyle\langle P|\mathcal{V}^{\mu}|B_{(s)}\rangle$
$\displaystyle=f_{+}(q^{2})\left(p_{B_{(s)}}^{\mu}+p_{P}^{\mu}-\frac{M_{B_{(s)}}^{2}-M_{P}^{2}}{q^{2}}q^{\mu}\right)$
$\displaystyle+f_{0}(q^{2})\frac{M_{B_{(s)}}^{2}-M_{P}^{2}}{q^{2}}q^{\mu}\,,$
(2)

where $f_{+}(q^{2})$ and $f_{0}(q^{2})$ are the vector and scalar form
factors, respectively. It is convenient in lattice simulations to instead
calculate the form factors $f_{\parallel}(E_{P})$ and $f_{\perp}(E_{P})$,
which are defined by

$\langle
P|\mathcal{V}^{\mu}|B_{(s)}\rangle=\sqrt{2M_{B_{(s)}}}\left[v^{\mu}f_{\parallel}(E_{P})+p^{\mu}_{\perp}f_{\perp}(E_{P})\right]\,,$
(3)

where $E_{P}$ is the outgoing light pseudoscalar meson energy, $v^{\mu}\equiv
p_{B_{(s)}}^{\mu}/M_{B_{(s)}}$ is the $B_{(s)}$-meson velocity, and
$p^{\mu}_{\perp}\equiv p_{P}^{\mu}-(p_{P}\cdot v)v^{\mu}$. In the
$B_{(s)}$-meson rest frame, which we will use for our simulations,
$f_{\parallel}$ and $f_{\perp}$ are proportional to the hadronic matrix
elements of the temporal and spatial vector currents:

$\displaystyle f_{\parallel}(E_{P})$ $\displaystyle=\frac{\langle
P|\mathcal{V}^{0}|B_{(s)}\rangle}{\sqrt{2M_{B_{(s)}}}}\,,$ (4) $\displaystyle
f_{\perp}(E_{P})$ $\displaystyle=\frac{\langle
P|\mathcal{V}^{i}|B_{(s)}\rangle}{\sqrt{2M_{B_{(s)}}}}\frac{1}{p^{i}_{P}}\,.$
(5)

The vector and scalar form factors can be easily obtained from $f_{\parallel}$
and $f_{\perp}$ via

$\displaystyle f_{+}(q^{2})$
$\displaystyle=\frac{1}{\sqrt{2M_{B_{(s)}}}}\Big{[}f_{\parallel}(E_{P})+\big{(}M_{B_{(s)}}-E_{P}\big{)}f_{\perp}(E_{P})\Big{]}\,,$
(6) $\displaystyle f_{0}(q^{2})$
$\displaystyle=\frac{\sqrt{2M_{B_{(s)}}}}{M^{2}_{B_{(s)}}-M^{2}_{P}}\Big{[}\big{(}M_{B_{(s)}}-E_{P}\big{)}f_{\parallel}(E_{P})$
$\displaystyle+\left(E^{2}_{P}-M^{2}_{P}\right)f_{\perp}(E_{P})\Big{]}\,.$ (7)

###  II.2 Actions and parameters

Table 1: Lattice simulation parameters Allton:2008pn ; Aoki:2010dy . The columns list the lattice volume, approximate lattice spacing, light ($m_{l}$) and strange ($m_{h}$) sea-quark masses, residual chiral symmetry breaking parameter $m_{\rm res}$, physical $u/d$\- and $s$-quark mass, unitary pion mass, number of configurations analyzed and number of sources. The tildes over $a\widetilde{m}_{u/d}$ and $a\widetilde{m}_{s}$ denote that these values include the residual quark mass.  $\left(\frac{L}{a}\right)^{3}\times\left(\frac{T}{a}\right)$ |  $\approx a$(fm) |  $a^{-1}$ [GeV] |  $am_{l}$ |  $am_{h}$ | $am_{\rm res}$ | $a\widetilde{m}_{u/d}$ | $a\widetilde{m}_{s}$ |  $M_{\pi}$[MeV] | # configs. | # time sources
---|---|---|---|---|---|---|---|---|---|---
$24^{3}\times 64$ | 0.11 | 1.729(25) | 0.005 | 0.040 | 0.003152 | 0.00136(4) | 0.0379(11) | 329 | 1636 | 1
$24^{3}\times 64$ | 0.11 | 1.729(25) | 0.010 | 0.040 | 0.003152 | 0.00136(4) | 0.0379(11) | 422 | 1419 | 1
$32^{3}\times 64$ | 0.086 | 2.281(28) | 0.004 | 0.030 | 0.0006664 | 0.00102(5) | 0.0280(7) | 289 | 628 | 2
$32^{3}\times 64$ | 0.086 | 2.281(28) | 0.006 | 0.030 | 0.0006664 | 0.00102(5) | 0.0280(7) | 345 | 889 | 2
$32^{3}\times 64$ | 0.086 | 2.281(28) | 0.008 | 0.030 | 0.0006664 | 0.00102(5) | 0.0280(7) | 394 | 544 | 2

We use the $(2+1)$-flavor domain-wall fermion and Iwasaki gauge-field
ensembles generated by the RBC and UKQCD collaborations Allton:2008pn ;
Aoki:2010dy . We perform measurements at five different light sea-quark masses
$m_{l}$ and at two lattice spacings of $a\approx 0.11$ fm ($a^{-1}\approx
1.729$ GeV) and $a\approx 0.086$ fm ($a^{-1}\approx 2.281$ GeV). The light
sea-quark masses $m_{l}$ correspond to pion masses of $289\ {\rm MeV}\lesssim
M_{\pi}\lesssim 422\ {\rm MeV}$. The up and down sea-quark masses are
degenerate and the strange sea-quark mass $m_{h}$ is tuned within 10% of its
physical value. The spatial volumes are approximately $(2.6\ {\rm fm})^{3}$,
such that $M_{\pi}L\geq 4$. We summarize the simulation parameters in Table 1.

In the valence sector we use for the light quarks the domain-wall action
Shamir:1993zy ; Furman:1994ky  and generate propagators with periodic boundary
conditions in space and time and with the same domain-wall height
($M_{5}=1.8$) and extent of the fifth dimension ($L_{s}=16$) as in the sea
sector. We generate both unitary light valence-quark propagators with the same
mass as the light sea quarks and propagators with a mass close to the physical
strange quark. On the coarser ensembles we choose $am_{s}=0.0343$ and on the
finer ensembles $am_{s}=0.0272$.

For the bottom quarks, we use the Columbia version of the relativistic heavy
quark (RHQ) action Christ:2006us  to control heavy-quark discretization errors
introduced by the large lattice $b$-quark mass. We use the anisotropic
$\mathcal{O}(a)$ improved Wilson-clover action with the following three
parameters: the bare-quark mass $m_{0}a$, clover coefficient $c_{P}$, and
anisotropy parameter $\zeta$. In this work we use the RHQ parameters tuned
nonperturbatively in Ref. Aoki:2012xaa  to reproduce the experimentally
measured $B_{s}$-meson mass and hyperfine splitting; we list their values in
Table 2.

Table 2: Tuned RHQ parameters on the $24^{3}$ and $32^{3}$ ensembles Aoki:2012xaa . The errors listed for $m_{0}a$, $c_{P}$, and $\zeta$ are from left to right: statistics, heavy-quark discretization errors, the lattice scale uncertainty, and the uncertainty in the experimental measurement of the $B_{s}$-meson hyperfine splitting, respectively. | $m_{0}a$ | $c_{P}$ | $\zeta$
---|---|---|---
$a\approx 0.11$ fm | 8. | 45(6)(13)(50)(7) | 5. | 8(1)(4)(4)(2) | 3. | 10(7)(11)(9)(0)
$a\approx 0.086$ fm | 3. | 99(3)(6)(18)(3) | 3. | 57(7)(22)(19)(14) | 1. | 93(4)(7)(3)(0)

We reduce autocorrelations between our lattices by shifting the gauge fields
by a random 4-vector before creating the sources for the valence-quark
propagators used in the 2-point and 3-point correlation functions. This random
4-vector shift is equivalent to placing the sources at random positions in
spacetime but simplifies the subsequent analysis. On the finer ensembles, we
double the statistics by using two sources per configuration separated by half
the lattice temporal extent.

###  II.3 Operator renormalization and improvement

Table 3: Operator renormalization factors and improvement coefficients. The flavor-conserving $Z$ factors were obtained nonperturbatively Aoki:2010dy ; Christ:2014uea . The $\rho$ factors and improvement coefficients $c_{i}^{n}$ were computed at one loop in mean-field improved lattice perturbation theory and are evaluated at $\alpha_{s}^{\overline{\rm MS}}(a^{-1})$ CLehnerPT . | $Z_{V}^{ll}$ | $Z_{V}^{bb}$ | $\alpha_{s}^{\overline{\rm MS}}(a^{-1})$ | $\rho_{V_{0}}$ | $\rho_{V_{i}}$ | $c_{t}^{3}$ | $c_{t}^{4}$ | $c_{s}^{1}$ | $c_{s}^{2}$ | $c_{s}^{3}$ | $c_{s}^{4}$
---|---|---|---|---|---|---|---|---|---|---|---
$a\approx 0.11$ fm | 0.71689(51) | 10.039(25) | 0.23 | 1.02658 | 0.99723 | 0.0558 | -0.0099 | -0.00079 | 0.0018 | 0.0485 | -0.0033
$a\approx 0.086$ fm | 0.74469(13) | 5.256(8) | 0.22 | 1.01661 | 0.99398 | 0.0547 | -0.0095 | -0.0012 | 0.00047 | 0.0480 | -0.0020

To match the lattice amplitudes to the continuum matrix elements, we multiply
by the heavy-light renormalization factor $Z_{V_{\mu}}^{bl}$:

$\langle P|\mathcal{V}_{\mu}|B_{(s)}\rangle=Z_{V_{\mu}}^{bl}\langle
P|V_{\mu}|B_{(s)}\rangle\,,$ (8)

where $\mathcal{V}_{\mu}$ and $V_{\mu}$ are the continuum and lattice current
operators, respectively. Following Ref. ElKhadra:2001rv  we calculate the
renormalization factor $Z_{V_{\mu}}^{bl}$ using a mostly nonperturbative
method in which we express $Z_{V_{\mu}}^{bl}$ as the following product:

$Z_{V_{\mu}}^{bl}=\rho_{V_{\mu}}^{bl}\sqrt{Z_{V}^{bb}Z_{V}^{ll}}.$ (9)

Most of the heavy-light current renormalization comes from the flavor-
conserving factors $Z_{V}^{bb}$ and $Z_{V}^{ll}$. The remaining factor
$\rho^{bl}_{V}$ is expected to be close to unity because most of the radiative
corrections, including contributions from tadpole graphs, cancel Harada:2001fi
.

Both flavor-conserving renormalization factors $Z_{V}^{bb}$ and $Z_{V}^{ll}$
were computed nonperturbatively in previous works. We computed $Z_{V}^{bb}$
for our earlier calculation of $B$-meson decay constants from the matrix
element of the $b\to b$ vector current between two $B_{s}$ mesons
Christ:2014uea . We can also take advantage of the fact that for domain-wall
fermions $Z_{V}^{ll}=Z_{A}^{ll}$ up to corrections of $\mathcal{O}(am_{\rm
res})$ and use the determination of $Z_{A}^{ll}$ from Ref. Aoki:2010dy . The
flavor off-diagonal renormalization factor $\rho^{bl}_{V}$ is calculated at
${\mathcal{O}}(\alpha_{s})$ in mean-field improved lattice perturbation theory
Lepage:1992xa  and evaluated at the $\overline{\rm MS}$ coupling
$\alpha_{s}^{\overline{\rm MS}}(\mu=a^{-1})$. Our perturbative computation
extends the work of Ref. Aoki:2002iq  to bilinears with one relativistic heavy
quark in the Columbia formulation and one domain-wall light quark. For
$\alpha_{s}^{\overline{\rm MS}}$, we use Eq. (167) of Ref. Aoki:2002iq , which
does not take into account sea-quark effects. Because sea-quark effects enter
at two loops, however, and the rest of the computation is performed at one
loop, the error introduced by setting $N_{f}=0$ is of the same size as other
remaining truncation errors. Further details of the perturbative calculation
will be provided in a forthcoming publication CLehnerPT . Table 3 shows the
renormalization factors used in this work.

We reduce discretization errors in the heavy-light vector current by improving
it through $\mathcal{O}(\alpha_{s}a)$. We compute the matrix element of the
tree-level heavy-light vector current

$V_{\mu}^{0}(x)=\overline{q}(x)\gamma^{\mu}Q(x),$ (10)

plus matrix elements of these additional single-derivative operators

$\displaystyle V_{\mu}^{1}(x)$ $\displaystyle=$
$\displaystyle\overline{q}(x)2\overrightarrow{D}_{\mu}Q(x),$ (11)
$\displaystyle V_{\mu}^{2}(x)$ $\displaystyle=$
$\displaystyle\overline{q}(x)2\overleftarrow{D}_{\mu}Q(x),$ (12)
$\displaystyle V_{\mu}^{3}(x)$ $\displaystyle=$
$\displaystyle\overline{q}(x)2\gamma_{\mu}\gamma_{i}\overrightarrow{D}_{i}Q(x),$
(13) $\displaystyle V_{\mu}^{4}(x)$ $\displaystyle=$
$\displaystyle\overline{q}(x)2\gamma_{\mu}\gamma_{i}\overleftarrow{D}_{i}Q(x),$
(14)

where the covariant derivatives are defined by

$\displaystyle\overrightarrow{D}_{\mu}Q(x)=\frac{1}{2}($ $\displaystyle
U_{\mu}(x)Q(x+\hat{\mu})$ $\displaystyle-
U_{\mu}^{\dagger}(x-\hat{\mu})Q(x-\hat{\mu})),$ (15)
$\displaystyle\overline{q}(x)\overleftarrow{D}_{\mu}=\frac{1}{2}($
$\displaystyle\overline{q}(x+\hat{\mu})U_{\mu}^{\dagger}(x)$
$\displaystyle-\overline{q}(x-\hat{\mu})U_{\mu}(x-\hat{\mu})).$ (16)

The temporal and spatial $\mathcal{O}(a)$-improved vector-current operators
are given by the following sums:

$\displaystyle V_{0}^{\rm imp}(x)$ $\displaystyle=$ $\displaystyle
V_{0}^{0}(x)+c_{t}^{3}V_{0}^{3}(x)+c_{t}^{4}V_{0}^{4}(x),$ (17) $\displaystyle
V_{i}^{\rm imp}(x)$ $\displaystyle=$ $\displaystyle
V_{i}^{0}(x)+c_{s}^{1}V_{i}^{1}(x)+c_{s}^{2}V_{i}^{2}(x)$ (18) $\displaystyle\
\ \ \ \ \ \ \ \ \ \ \ +c_{s}^{3}V_{i}^{3}(x)+c_{s}^{4}V_{i}^{4}(x)\,.$

We calculate the coefficients $c_{t}^{n}$ and $c_{s}^{n}$ at one loop in mean-
field improved lattice perturbation theory CLehnerPT ; the values of the
coefficients evaluated at $\alpha_{s}^{\overline{\rm MS}}(a^{-1})$ are shown
in Table 3.

##  III Analysis

Here we present our determinations of the form factors $f_{+}(q^{2})$ and
$f_{0}(q^{2})$ for $B\to\pi\ell\nu$ ($B_{s}\to K\ell\nu$) at large values of
$q^{2}\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{$\mathchar
536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar 318\relax$}}19.0~{}{\rm GeV}^{2}$
($q^{2}\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{$\mathchar
536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar 318\relax$}}17.6~{}{\rm
GeV}^{2}$) accessible in our numerical simulations.

Our analysis proceeds in three steps: First, in Sec. III.1, we fit the pion,
kaon, and $B_{(s)}$-meson 2-point correlation functions to obtain the ground-
state meson masses. The results for these meson masses then enter our 3-point
correlator fits in Sec. III.2 to obtain the lattice form factors
$f_{\parallel}(E_{P})$ and $f_{\perp}(E_{P})$ at fixed values of the pion/kaon
energy $E_{P}$. In Sec. III.3, we interpolate the renormalized values for
$f_{\parallel}(E_{P})$ and $f_{\perp}(E_{P})$ in energy, and extrapolate to
the physical light-quark masses and the continuum limit, using SU(2) hard-pion
$\chi$PT formulated for heavy-light mesons. To avoid possible biases due to
analysis choices, we use the same fit functions in the correlator and chiral
fits for both processes $B\to\pi\ell\nu$ and $B_{s}\to K\ell\nu$, and fitting
ranges that are as close as possible.

We propagate statistical errors throughout the analysis via a single-
elimination jackknife procedure. We avoid a direct dependence on the lattice
scale by carrying out our analysis in units of the $B_{s}$-meson mass. The
$B_{s}$-meson mass plays a special role because we tuned parameters of the
$b$-quark action to match the experimental value. Thus we can obtain the form
factors in physical units after the chiral-continuum extrapolation by
multiplying by $M_{B_{s}}^{\rm exp.}$ to the appropriate power. With this
approach, the uncertainty on the lattice scale enters only indirectly via the
values of the RHQ parameters.

We use the Chroma software library for lattice QCD to compute our numerical
data for the lattice 2-point and 3-point correlation functions Edwards:2004sx
.

Table 4: Time ranges used in two-point and three-point fits to determine the lattice meson masses and form factors. For the three-point fits, we use the same range for all operators and momenta. |  [$t_{\rm min}$, $t_{\rm max}$]
---|---
| 2-point fits | 3-point fits
| $M_{\pi}$ | $M_{K}$ | $M_{B}$ | $M_{B_{s}}$ | $f_{\parallel}^{B\pi}$ | $f_{\perp}^{B\pi}$ | $f_{\parallel}^{B_{s}K}$ | $f_{\perp}^{B_{s}K}$
$a\approx 0.11$ fm | $[12,23]$ | $[12,23]$ | $[7,30]$ | $[10,30]$ | [6,10] | [6,10] | [6,10] | [6,10]
$a\approx 0.086$ fm | $[16,30]$ | $[16,30]$ | $[9,30]$ | $[13,30]$ | [8,13] | [8,13] | [8,13] | [8,13]

###  III.1 Two-point correlator fits

To obtain the lattice $B_{(s)}\to P$ amplitude, we first calculate the
following two-point correlation functions:

$\displaystyle C_{P}(t,\vec{p}_{P})$ $\displaystyle=$
$\displaystyle\sum_{\vec{x}}e^{i\vec{p}_{P}\cdot\vec{x}}\langle\mathcal{O}_{P}^{\dagger}(\vec{x},t)\mathcal{O}_{P}(\vec{0},0)\rangle,$
(19) $\displaystyle C_{{B_{(s)}}}(t)$ $\displaystyle=$
$\displaystyle\sum_{\vec{x}}\langle\mathcal{O}_{B_{(s)}}^{\dagger}(\vec{x},t)\tilde{\mathcal{O}}_{B_{(s)}}(\vec{0},0)\rangle,$
(20) $\displaystyle\tilde{C}_{{B_{(s)}}}(t)$ $\displaystyle=$
$\displaystyle\sum_{\vec{x}}\langle\tilde{\mathcal{O}}_{B_{(s)}}^{\dagger}(\vec{x},t)\tilde{\mathcal{O}}_{B_{(s)}}(\vec{0},0)\rangle,$
(21)

where $\mathcal{O}_{P}=\overline{q}\gamma_{5}q$ and
$\mathcal{O}_{B_{(s)}}=\overline{Q}\gamma_{5}q$ are interpolating operators
for the light pseudoscalar and ${B_{(s)}}$-meson, respectively. Both pions and
kaons are simulated with a point source and point sink, whereas $b$-quark
propagators are generated with a gauge-invariant Gaussian smeared source
Alford:1995dm ; Lichtl:2006dt  to reduce excited state contamination. We
employ the same smearing parameters optimized in Ref. Aoki:2012xaa  and denote
a smeared source in Eqs. (19)-(21) with a tilde above the operator.

We obtain the pion or kaon energy and ${B_{(s)}}$-meson mass from the
exponential decay of the correlators in Eqs. (19) and (20). The correlator in
Eq. (21) is used to normalize the $B_{(s)}\to P$ three-point function. We work
in the ${B_{(s)}}$-meson rest frame such that only pions or kaons carry
nonzero momentum. In our analysis we use data with discrete lattice pion
momenta through $2\pi(1,1,1)/L$ and kaon momenta through $2\pi(2,0,0)/L$. We
average the results for all equivalent momenta, i.e. with different spatial
directions but the same total $|\vec{p}_{P}|$. We effectively double our
statistics by folding the two-point correlators at the temporal midpoint of
the lattice, thereby averaging forward- and backward-propagating states.

At sufficiently large lattice times, the ground-state masses and energies can
be determined from simple two-point correlator ratios. We define the light
pseudoscalar-meson effective energy and $B_{(s)}$-meson effective mass as

$\displaystyle E_{P}(t,\vec{p}_{P})$ $\displaystyle=$
$\displaystyle\cosh^{-1}\left[\frac{C_{P}(t,\vec{p}_{P})+C_{P}(t+2,\vec{p}_{P})}{C_{P}(t+1,\vec{p}_{P})}\right],\quad$
(22) $\displaystyle M_{B_{(s)}}(t)$ $\displaystyle=$
$\displaystyle\cosh^{-1}\left[\frac{C_{B_{(s)}}(t)+C_{B_{(s)}}(t+2)}{C_{B_{(s)}}(t+1)}\right].$
(23)

We perform correlated, constant-in-time, fits to these expressions, choosing
fit ranges without visible excited-state contamination that lead to acceptable
$p$ values. Figure 1 shows example meson-mass determinations on our fine
ensemble with $am_{l}=0.004$. To minimize bias, we use the same fit range for
all ensembles with the same lattice spacing (although different for light and
heavy-light mesons); these fit ranges are given in Table 4. The resulting
pion/kaon and $B_{(s)}$-meson masses on all ensembles are given in Tables 15
and 16, respectively.

In the continuum limit, the pion and kaon energies should satisfy the
dispersion relation $E_{P}^{2}=M_{P}^{2}+\vec{p}_{P}^{2}$ and the amplitudes
of the two-point functions $Z_{P}=|\langle 0|\mathcal{O}_{P}|\pi\rangle|$
should be independent of the momentum $\vec{p}_{P}$. We obtain the amplitudes
from correlated plateau fits to

$Z_{P}(t)=\sqrt{\frac{2E_{P}C_{P}(t,\vec{p}_{P})}{e^{-E_{P}t}}}$ (24)

using the same fit ranges as for the masses. Figure 2 compares the measured
pion and kaon energies and amplitudes with continuum expectations on the
$a\approx 0.086$ fm, $am_{l}=0.004$ ensemble. The measured kaon energies and
amplitudes agree remarkably well with the predictions from the continuum
dispersion relation, to within 5% even at the largest momentum
$a\vec{p}_{K}=2\pi(2,0,0)/L$. Although the pion data is not precise enough to
draw strong quantitative conclusions, the measured energies and amplitudes
still agree with continuum expectations within the large statistical
uncertainties for all momenta. Dispersion-relation plots for the other
ensembles show similar behavior.

The kaon data, for which both the energies and amplitudes are statistically
well resolved, provides an accurate measure of momentum-dependent
discretization effects, while the pion data provides only a rough cross-check.
On all ensembles, the measured pion and kaon energies both agree within
statistical errors with the predictions from the continuum dispersion
relation, and the measured pion and kaon amplitudes agree with the zero-
momentum result. Thus, in our determinations of the lattice form factors
$f_{\parallel}$ and $f_{\perp}$ in the next section, we use pion and kaon
energies calculated from the continuum dispersion relation (rather than the
measured values) to reduce the statistical uncertainties. Although we do not
use the amplitudes obtained from Eq. (24) in our subsequent form-factor
determinations, the observed momentum independence of $Z_{P}$ provides further
support for this strategy.

Figure 1: Effective masses of the pion (upper left), kaon (bottom left), $B$
meson (upper right) and $B_{s}$ meson (bottom right) on the $a\approx 0.086$
fm ensemble with $am_{l}=0.004$. 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 $a\approx 0.086$
fm ensemble with $am_{l}=0.004$. The dashed lines show a power-counting
estimate of the leading $\mathcal{O}((a\vec{p})^{2})$ momentum-dependent
discretization errors.

###  III.2 Three-point correlator fits

To extract the desired $B_{(s)}\to P$ hadronic amplitudes, we calculate the
following three point correlation functions:

$C_{3,\mu}^{\rm imp}(t,t_{\rm snk},\vec{p}_{P})\\\
=\sum_{\vec{x},\vec{y}}e^{i\vec{p}_{P}\cdot\vec{y}}\langle\tilde{\mathcal{O}}_{B_{(s)}}^{\dagger}(\vec{x},t_{\rm
snk})V^{\rm imp}_{\mu}(\vec{y},t)\mathcal{O}_{P}(\vec{0},0)\rangle,$ (25)

where the improved lattice temporal and spatial lattice vector currents
$V_{\mu}^{\rm imp}$ are defined in Eqs. (17) and (18). As shown in Fig. 3, we
fix the location of the pion or kaon at the temporal origin and the location
of the $B_{(s)}$ meson at time $t_{\rm snk}$, and vary the location of the
current operator over all time slices in between. In our calculations, the
mass of the light daughter quark ($l$) is always equal to the light sea-quark
mass. For $B\to\pi$ decay, the spectator-quark mass ($l^{\prime}$) also equals
the light sea-quark mass. For $B_{s}\to K$ decay, the spectator-quark mass is
close to that of the physical strange quark. We use a Gaussian-smeared
sequential source for the $b$ quark in the $B_{(s)}$ meson to reduce excited-
state contamination. We insert discrete nonzero momentum at the local current
operator through $\vec{p}_{\pi}=2\pi(1,1,1)/L$ for $B\to\pi$ and
$\vec{p}_{K}=2\pi(2,0,0)/L$ for $B_{s}\to K$ (recall that the $B_{(s)}$ meson
is at rest). To improve statistics, we compute the three-point correlators
with both positive and negative source-sink separations ($\pm t_{\rm snk}$);
we also average over equivalent spatial momenta.

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 to the
light sea-quark mass. Black and grey circles denote local and smeared
operators, respectively.

The lattice form factors $f_{\parallel}^{\rm lat}$ and $f_{\perp}^{\rm lat}$
are obtained from the following ratios of correlation functions far away from
both the pion/kaon source and the $B_{(s)}$-meson sink:

$\displaystyle f^{\rm lat}_{\parallel}(\vec{p}_{P})$ $\displaystyle=$
$\displaystyle\lim_{0\ll t\ll t_{\rm snk}}R_{3,0}(t,t_{\rm snk},\vec{p}_{P}),$
(26) $\displaystyle f^{\rm lat}_{\perp}(\vec{p}_{P})$ $\displaystyle=$
$\displaystyle\lim_{0\ll t\ll t_{\rm
snk}}\frac{1}{p^{i}_{\pi}}R_{3,i}(t,t_{\rm snk},\vec{p}_{P}),$ (27)

with

$R_{3,\mu}(t,t_{\rm snk},\vec{p}_{P})\\\ =\frac{C_{3,\mu}^{\rm imp}(t,t_{\rm
snk},\vec{p}_{P})}{\sqrt{C_{2}^{P}(t,\vec{p}_{P})\tilde{C}_{2}^{B_{(s)}}(t_{\rm
snk}-t)}}\sqrt{\frac{2E_{P}}{e^{-E_{P}t}e^{-M_{B}(t_{\rm snk}-t)}}},$ (28)

where we use the continuum dispersion relation and measured light
pseudoscalar-meson mass $M_{P}$ to construct the energy $E_{P}$. To determine
the optimal source-sink separation for $C_{3,\mu}(t,t_{\rm snk},\vec{p}_{P})$,
we carried out a dedicated study. We computed the unimproved ratio
$R_{3,\mu}^{0}$ for several values of the source-sink separation on one
$a\approx 0.086$ fm and one $a\approx 0.11$ fm ensemble, choosing those with
the lightest sea-quark mass because they are most sensitive to excited-state
contamination. Figure 4 shows the ratio $R_{3,0}^{0}$ for $B\to\pi$ with
$\vec{p}_{\pi}=0$ for several source-sink separations on the $a\approx 0.086$
fm ensemble. All plateaus overlap within statistical uncertainties in the
region far from both the source and the sink. The results for the ratios
$R_{3,0}^{0}$ and $R_{3,i}^{0}$ at nonzero momenta and on the $a\approx 0.11$
fm ensemble look similar. Because the statistical errors increase with larger
source-sink separation, we chose $t_{\rm snk}=26$ (20) on the $a\approx 0.086$
fm ($a\approx 0.11$ fm) ensembles. This corresponds to approximately the same
physical distance for the two lattice spacings.

Figure 4: Unimproved three-point ratio $R_{3,0}^{0}$ for $\vec{p}_{\pi}=0$
with several source-sink separations $t_{\rm snk}$ on the $a\approx 0.086$ fm
ensemble with $am_{l}=0.004$.

Figure 5 shows the ${\mathcal{O}}(\alpha_{s}a)$-improved ratios $R_{3,0}$ and
$R_{3,i}/p_{P}^{i}$ for different momenta on the $a\approx 0.086$ fm ensemble
with $am_{l}=0.004$. Results for other ensembles look similar. We perform
correlated, constant-in-time, fits to these ratios using fit ranges without
visible excited-state contamination that lead to acceptable $p$ values. To
minimize bias, we use the same fit range for all momenta and ensembles with
the same lattice spacing; these fit ranges are given in Table 4. The complete
fit results for the three-point ratios are given in Tables 17 and 18.

Figure 5: ${\mathcal{O}}(\alpha_{s}a)$-improved ratios $R_{3,i}/p_{P}^{i}$
(left) and $R_{3,0}$ (right) with $t_{\rm snk}=26$ on the $a\approx 0.086$ fm
ensemble with $am_{l}=0.004$. 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.

Finally, we obtain the renormalized $B_{(s)}\to P\ell\nu$ form factors
$f_{\parallel}$ and $f_{\perp}$ in the continuum after multiplying by the
heavy-light renormalization factors $Z_{V_{\mu}}^{bl}$ given in Table 3:

$\displaystyle f_{\parallel}(\vec{p}_{P})$ $\displaystyle=$ $\displaystyle
Z_{V_{0}}^{bl}f^{\rm lat}_{\parallel}(\vec{p}_{P}),$ (29) $\displaystyle
f_{\perp}(\vec{p}_{P})$ $\displaystyle=$ $\displaystyle Z_{V_{i}}^{bl}f^{\rm
lat}_{\perp}(\vec{p}_{P}).$ (30)

###  III.3 Chiral-continuum extrapolation

We extrapolate the renormalized lattice form factors to the physical light-
quark mass, and interpolate in the pion or kaon energy using next-to-leading
order (NLO) SU(2) chiral perturbation theory for heavy-light mesons
(HM$\chi$PT) in the “hard-pion” limit. In the SU(2) theory, the strange-quark
mass is integrated out, and only the light-quarks’ degrees-of-freedom are
included. Therefore the chiral logarithms for $B\to\pi\ell\nu$ ($B_{s}\to
K\ell\nu$) depend on the pion mass and the pion (kaon) energy. The SU(2) low-
energy constants depend upon the value of the strange-quark mass, as well as
on the value of the $b$-quark mass for $B$-meson form factors. “Hard-pion”
$\chi$PT, which was introduced by Flynn and Sachrajda for the light-
pseudoscalar-meson decay $K\to\pi\ell\nu$ in Ref. Flynn:2008tg  and later
extended to heavy-light-meson decays by Bijnens and Jemos in Ref.
Bijnens:2010ws , applies in the kinematic regime where the pion or kaon energy
is large compared to its rest mass. Almost all of our lattice simulation data
is in this hard-pion (or kaon) regime. We can obtain the expressions for the
$B\to\pi\ell\nu$ and $B_{s}\to K\ell\nu$ form factors in hard-pion/kaon
$\chi$PT by taking the limit of the continuum expressions from Ref.
Becirevic:2002sc  as $M_{\pi}/E_{P}\to 0$, where $P=\pi,K$ denotes the final-
state pseudoscalar meson.

The NLO SU(2) $\chi$PT full-QCD expressions for the $B\to\pi\ell\nu$ and
$B_{s}\to K\ell\nu$ form factors in the hard-pion/kaon limit are functions of
the pion mass $M_{\pi}$, pion or kaon energy $E_{P}$, and lattice spacing $a$.
They have two general forms:

$\displaystyle f_{\rm no\ pole}^{B_{(s)}P}(M_{\pi},E_{P},a^{2})$
$\displaystyle=$ $\displaystyle c_{\rm np}^{(1)}\left[1+\left(\frac{\delta
f_{\parallel}}{(4\pi f)^{2}}+c_{\rm
np}^{(2)}\frac{M_{\pi}^{2}}{\Lambda^{2}}+c_{\rm
np}^{(3)}\frac{E_{P}}{\Lambda}+c_{\rm
np}^{(4)}\frac{E_{P}^{2}}{\Lambda^{2}}+c_{\rm
np}^{(5)}\frac{a^{2}}{\Lambda^{2}a_{32}^{4}}\right)\right]$ (31)
$\displaystyle f_{\rm pole}^{B_{(s)}P}(M_{\pi},E_{P},a^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{E_{P}+\Delta}c_{\rm p}^{(1)}\left[1+\left(\frac{\delta
f_{\perp}}{(4\pi f)^{2}}+c_{\rm p}^{(2)}\frac{M_{\pi}^{2}}{\Lambda^{2}}+c_{\rm
p}^{(3)}\frac{E_{P}}{\Lambda}+c_{\rm
p}^{(4)}\frac{E_{P}^{2}}{\Lambda^{2}}+c_{\rm
p}^{(5)}\frac{a^{2}}{\Lambda^{2}a_{32}^{4}}\right)\right]\,,$ (32)

one with a pole at $E_{P}=-\Delta=M_{B^{*}}-M_{B_{(s)}}$ and one without. Here
the $B^{*}$ resonance corresponds to a state with flavor $bu$ and quantum
numbers $J^{P}=0^{+}$ for $f_{\parallel}$ and $1^{-}$ for $f_{\perp}$. The
experimentally measured vector-meson mass is $M_{B^{*}}=5.3252(4)$ GeV
Beringer:2012zz . The scalar $B^{*}$ meson has not been observed
experimentally, but its value has been estimated theoretically using heavy-
quark and chiral-symmetry arguments to be $M_{B^{*}}(0^{+})=5.63(4)$ GeV
Bardeen:2003kt , while the $0^{+}$-$0^{-}$ splitting has been estimated in
(2+1)-flavor lattice QCD to be $M_{B^{*}}(0^{+})-M_{B}\sim 400$ MeV
Gregory:2010gm . In our chiral-continuum extrapolations we include the effects
of resonances below the $B\pi$ and $B_{s}K$ production thresholds, i.e.
$q^{2}<(M_{B_{(s)}}+M_{P})^{2}$. For $B\to\pi\ell\nu$, the $B^{*}$ meson lies
below the $B\pi$ production threshold, so we include a pole in the fit for
$f_{\perp}^{B\pi}$ taking $\Delta_{\perp}^{B\pi}=45.78$ MeV from experiment
Beringer:2012zz . The predicted value of $M_{B^{*}}(0^{+})$ is well above
$M_{B}+M_{\pi}$, however, so we do not include a pole in the fit of
$f_{\parallel}^{B\pi}$. For $B_{s}\to K$, both $M_{B^{*}}$ and
$M_{B^{*}}(0^{+})$ are below $M_{B_{s}}+M_{K}$, so we include a pole in the
fits for both $f_{\perp}^{B_{s}K}$ and $f_{\parallel}^{B_{s}K}$, taking
$\Delta_{\perp}^{B_{s}K}=-41.6$ MeV from experiment Beringer:2012zz  and
taking $\Delta_{\parallel}^{B_{s}K}=263$ MeV from the model estimate in Ref.
Bardeen:2003kt . The precise value of $M_{B^{*}}(0^{+})$ has little impact on
the fit because the pole location is so far outside the semileptonic region,
but we vary its value by a generous amount when estimating the chiral-
continuum extrapolation error in Sec. IV.1.

The one-loop chiral logarithms are the same for $f_{\parallel}$ and
$f_{\perp}$, but differ for $B\to\pi\ell\nu$ and $B_{s}\to K\ell\nu$:

$\displaystyle\delta f^{B\pi}$ $\displaystyle=$
$\displaystyle-\frac{3}{4}\left(3g_{b}^{2}+1\right)M_{\pi}^{2}{\rm
log}\left(\frac{M_{\pi}^{2}}{\Lambda^{2}}\right)$ (33) $\displaystyle\delta
f^{B_{s}K}$ $\displaystyle=$ $\displaystyle-\frac{3}{4}M_{\pi}^{2}{\rm
log}\left(\frac{M_{\pi}^{2}}{\Lambda^{2}}\right)\,,$ (34)

where $g_{b}$ is the $B^{\ast}B\pi$ coupling constant. At tree level, the mass
of a pion composed of two domain-wall quarks is given in terms of the light-
quark mass by

$\displaystyle M_{\pi}^{2}=2\mu(m_{l}+m_{\rm res})\,,$ (35)

where $\mu$ is a leading-order low-energy constant.

We include a term proportional to $a^{2}$ in the chiral fit functions Eqs.
(31) and (32) to account for the dominant lattice-spacing dependence. To make
the $a^{2}$ analytic term dimensionless with an expected coefficient of
${\mathcal{O}}(1)$ in $\chi$PT, we normalize it using the lattice spacing on
the finer $32^{3}$ ensembles $a_{32}$. Discretization errors from the domain-
wall and Iwasaki actions are of ${\mathcal{O}}\left(a\Lambda_{\rm
QCD}\right)^{2}$; using $\Lambda_{\rm QCD}=500$ MeV,111Recent three- and four-
flavor lattice-QCD calculations typically give values for
$\Lambda_{\overline{{\rm MS}}}$ in the range of about 300–400 MeV
Maltman:2008bx ; Aoki:2009tf ; McNeile:2010ji ; Bazavov:2012ka ;
Blossier:2013ioa . The 2013 Flavor Lattice Averaging Group (FLAG) review
quotes the range $\Lambda^{(3)}_{\rm{\overline{MS}}}=339(17)$ MeV for three
active flavors Aoki:2013ldr . To be conservative, we take a slightly larger
value $\Lambda_{\rm QCD}=500$ MeV for the power-counting estimates throughout
this work. we estimate these to be about 5% on the $32^{3}$ ensembles. The
remaining discretization errors – light-quark and gluon discretization errors
in the heavy-light current, and heavy-quark discretization errors from both
the action and current – are expected from power counting to be much smaller.
In Secs. IV.5 and IV.6, we estimate their sizes to be below 2%. We therefore
expect light-quark and gluon discretization errors from the action to dominate
the scaling behavior of the form factors, such that including an $a^{2}$ term
in the fit will largely remove these contributions. We will add the remaining
subdominant discretization errors a posteriori to the systematic error budget
after the chiral fit.

In addition to the pion masses and pion/kaon energies, several parameters
enter the expressions in Eqs. (31) and (32). For completeness, we compile the
values of the fixed parameters in our chiral fits in Table 5. We use the
lattice spacings and low-energy constant $\mu$ obtained in Ref. Aoki:2010dy
from the RBC/UKQCD analysis of light pseudoscalar meson masses and decay
constants. We use the PDG value of $f_{\pi}=130.4(2)$ MeV Beringer:2012zz ,
and take $\Lambda_{\chi}=1$ GeV for the scale in the chiral logarithms. We use
the $B^{\ast}B\pi$ coupling constant $g_{b}=0.57(8)$ obtained in our companion
analysis also using the RBC/UKQCD domain-wall+Iwasaki ensembles and the RHQ
action for the $b$-quarks Flynn:2013kwa .

Table 5: Constants used in the chiral and continuum extrapolations of the $B\to\pi\ell\nu$ and $B_{s}\to K\ell\nu$ form factors Aoki:2010dy ; Beringer:2012zz ; Flynn:2013kwa .  $a$ |  $\approx 0.11$ fm |  $\approx 0.086$ fm
---|---|---
$a^{-1}$ | 1.729 GeV | 2.281 GeV
$a\mu$ | 2.348 | 1.826
$f_{\pi}$ | 130.4 MeV
$g_{b}$ | 0.57
$\Lambda_{\chi}$ | 1 GeV

We perform correlated chiral-continuum fits to the data calculated on all five
sea-quark ensembles listed in Table 1 using the full-QCD NLO SU(2) hard-
pion/kaon HM$\chi$PT expressions. For $B\to\pi\ell\nu$, we include discrete
lattice momenta up to $\vec{p_{\pi}}=2\pi(1,1,1)/L$, which corresponds to
$\approx 0.78$ GeV on the coarser ensembles. For $B_{s}\to K\ell\nu$, where
the statistical errors are smaller, we include momenta up to
$\vec{p}_{K}=2\pi(2,0,0)/L$, or $\approx 0.91$ GeV on the coarser ensembles.
For the pion masses in Eqs. (31) and (32), we use the tree-level expression in
Eq. (35). We obtain the physical form factors after the chiral-continuum fit
by setting the light quark mass to the physical average $u/d$-quark mass
$a_{32}m_{ud}=0.00102(5)$ Aoki:2010dy  and the lattice spacing to zero.

Figure 6 shows the resulting fits, which all have good $\chi^{2}/{\rm dof}$
and $p$-values. We do not observe any statistically-significant lattice-
spacing dependence for any of the form factors, and cannot resolve the
coefficients of the $a^{2}$ terms in the four fits. Dropping the $a^{2}$ term
altogether does not reduce the fit quality, and we consider this alternate fit
as one of many possibilities when estimating the systematic uncertainty due to
the chiral-continuum extrapolation in Sec. IV.1. We observe a mild sea-quark
mass dependence for $f_{\parallel}^{B_{s}K}$, and cannot resolve any sea-quark
mass dependence in the other form factors. Dropping the term proportional to
$M_{\pi}^{2}$ reduces the $p$-value of the $f_{\parallel}^{B_{s}K}$ to $\sim
5\%$, which is still acceptable, and does not impact the quality of the other
fits. Again, we consider this alternative when estimating the chiral-continuum
extrapolation error. Finally, we do not see any evidence for the onset of
chiral logarithms given that our lightest pion $M_{\pi}\approx 290$ MeV is
still quite heavy, and consider fits without the logarithms in Eqs. (31) and
(32) among the alternate fits for assessing the systematic uncertainty.

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$\chi$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 $a\approx 0.11$ fm data and triangles and diamonds to
$a\approx 0.086$ 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.

As a consistency check of our chiral-continuum extrapolation, we can use our
$B\to\pi\ell\nu$ form-factor fit results to obtain a rough estimate for the
$B^{\ast}B\pi$ coupling at lowest order in the $1/m_{b}$ expansion of
HM$\chi$PT. From our preferred fits of $f_{\parallel}$ and $f_{\perp}$, we
find that the ratio of leading-order coefficients gives

$\displaystyle g_{b}\approx c^{(1)}_{\rm p}/c^{(1)}_{\rm np}=0.35(15)\,$ (36)

where the error is statistical only (and does not include omitted higher-order
corrections in the chiral and $1/m_{b}$ expansions). The value for $g_{b}$ in
Eq. (36) is consistent with our independent determination of the
$B^{\ast}B\pi$ coupling in Ref. Flynn:2013kwa , and mostly independent of the
input value of $g_{b}$ in the chiral logarithms.

We also considered chiral-continuum extrapolations of the $B\to\pi\ell\nu$
form factors using NLO SU(3) HM$\chi$PT, in which the logarithms have explicit
strange-quark mass dependence, but were unable to obtain good fits for
$f_{\parallel}^{B\pi}$. Fits of $f_{\parallel}^{B\pi}$ to NLO SU(2) “soft-
pion” $\chi$PT, in which the logarithms have explicit dependence on the pion
energy, also failed to describe the data. All fits tried led to acceptable
$p$-values for the case of $f_{\perp}^{B\pi}$ due to the fact that the shape
is largely dictated by the $B^{*}$ pole term in the denominator. Finally, we
tried supplementing the NLO expressions for the $B\to\pi\ell\nu$ form factors
with NNLO analytic terms. The resulting partly NNLO fits yielded form-factor
results consistent with those from our preferred fits, but with significantly
larger uncertainties due to the fact our data could not resolve any of the
higher-order terms.

##  IV Estimation of systematic errors

We now discuss the sources of systematic uncertainty in our determinations of
the $B\to\pi\ell\nu$ and $B_{s}\to K\ell\nu$ form factors. Each uncertainty is
discussed in a separate subsection. We visually summarize the error budgets
for the form factors versus $q^{2}$ in Fig. 7, and provide a detailed
numerical error budget for the form factors at three representative $q^{2}$
values within the range of simulated lattice momenta in Table 6. The form
factors at these three points will be used later in Sec. V for the
extrapolation to $q^{2}=0$ via the $z$ expansion.

Table 6:  Error budgets for the $B\to\pi\ell\nu$ and $B_{s}\to K\ell\nu$ form factors at three representative $q^{2}$ values in the range of simulated lattice momenta. For convenience, we also show the corresponding pion or kaon energy, $E_{P}$. Errors are given in %. The total error is obtained by adding the individual errors in quadrature. | $f_{+}^{B\pi}$ | $f_{0}^{B\pi}$ | $f_{+}^{B_{s}K}$ | $f_{0}^{B_{s}K}$
---|---|---|---|---
$E_{P}$ [GeV] | 0.85 | 0.50 | 0.27 | 0.85 | 0.50 | 0.27 | 1.07 | 0.77 | 0.53 | 1.07 | 0.77 | 0.53
$q^{2}$ [GeV2] | 19.0 | 22.6 | 25.1 | 19.0 | 22.6 | 25.1 | 17.6 | 20.8 | 23.4 | 17.6 | 20.8 | 23.4
$f(q^{2})$ | 1.21 | 2.27 | 4.11 | 0.46 | 0.68 | 0.92 | 0.99 | 1.64 | 2.77 | 0.48 | 0.63 | 0.81
Statistics | 7.9 | 5.9 | 12.4 | 7.3 | 4.6 | 3.3 | 4.1 | 3.4 | 3.2 | 3.4 | 2.7 | 2.6
Chiral-continuum extrapolation | 6.3 | 5.0 | 6.2 | 10.9 | 7.6 | 5.8 | 3.2 | 2.8 | 2.5 | 5.0 | 4.9 | 5.1
Light-quark mass $m_{ud}$ | 0.3 | 0.2 | 0.2 | 0.4 | 0.3 | 0.2 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1 | 0.1
Strange-quark mass $m_{s}$ | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.1 | 0.1 | 0.1 | 0.0 | 0.0 | 0.0
Lattice-scale uncertainty | 2.0 | 2.0 | 2.0 | 2.2 | 2.2 | 2.2 | 2.0 | 2.0 | 2.0 | 2.2 | 2.2 | 2.2
RHQ parameter tuning | 0.9 | 0.9 | 0.8 | 1.0 | 1.0 | 1.0 | 0.9 | 0.9 | 0.9 | 1.0 | 1.0 | 1.0
Renormalization factor | 0.8 | 0.8 | 0.7 | 1.6 | 1.6 | 1.7 | 0.9 | 0.8 | 0.8 | 1.6 | 1.6 | 1.7
Finite volume | 0.5 | 0.4 | 0.3 | 0.7 | 0.5 | 0.4 | 0.2 | 0.2 | 0.2 | 0.2 | 0.1 | 0.1
Heavy-quark discretization errors | 1.8 | 1.8 | 1.8 | 1.7 | 1.7 | 1.7 | 1.8 | 1.8 | 1.8 | 1.7 | 1.7 | 1.7
Light-quark & gluon discretization errors | 1.1 | 1.1 | 1.1 | 1.1 | 1.1 | 1.1 | 1.3 | 1.3 | 1.3 | 1.3 | 1.3 | 1.3
Isospin breaking | 0.7 | 0.7 | 0.7 | 0.7 | 0.7 | 0.7 | 0.7 | 0.7 | 0.7 | 0.7 | 0.7 | 0.7
Total (%) | 10.6 | 8.4 | 14.3 | 13.6 | 9.6 | 7.6 | 6.2 | 5.5 | 5.3 | 7.1 | 6.7 | 6.8

In cases where the estimation of a systematic uncertainty requires the
explicit variation of simulation parameters, we use the $a\approx 0.11$ fm
ensemble with $am_{l}=0.005$, and take the dependence of that ensemble to be
representative of all ensembles. We choose this ensemble because it has very
high statistics, and therefore allows us to most reliably measure the
dependence of the form factors on on the input parameters. We expect the
behavior of the form factors on this ensemble to provide conservative bounds
on the errors since it has the largest lattice spacing and heaviest kaons.

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.

###  IV.1 Chiral-continuum extrapolation

We estimate the systematic uncertainty due to the chiral-continuum
extrapolation of the $B\to\pi$ and $B_{s}\to K$ form factors by varying the
chiral-continuum fit Ansätze. We consider the following fit alternatives:

  * •

standard HM$\chi$PT including explicit $E_{P}$ dependence in the chiral
logarithms

  * •

omitting the term proportional to $a^{2}$ in Eqs. (31) and (32)

  * •

omitting the term proportional to $M_{\pi}^{2}$ in Eqs. (31) and (32)

  * •

omitting terms proportional to $a^{2}$ and $M_{\pi}^{2}$ in Eqs. (31) and (32)

  * •

analytic fits omitting the chiral logarithms in Eqs. (31) and (32)

  * •

analytic fits omitting the chiral logarithms and the term proportional to
$a^{2}$ in Eqs. (31) and (32)

  * •

varying the value of $f_{\pi}$ in the coefficients of the chiral logarithms
from $f_{0}$= 112(2) MeV Aoki:2010dy  in the chiral limit to $f_{K}=156.1(8)$
MeV Beringer:2012zz

  * •

varying the $B^{\ast}B\pi$ coupling in the coefficients of the chiral
logarithms $g_{b}=0.57(8)$ by plus/minus one standard deviation Flynn:2013kwa

  * •

varying the scalar pole mass $M_{B^{*}}(0^{+})=5.63$ GeV in $f_{0}^{B_{s}K}$
by plus/minus 100 MeV

  * •

omitting the data point at zero momentum

  * •

omitting the data point at the highest momentum $\vec{p}=2\pi/L(2,0,0)$ for
$f_{+/0}^{B_{s}K}$

  * •

excluding ensembles with pion masses $M_{\pi}\gtrsim 400$ MeV

Figure 8 shows the relative changes of the form-factor central values under
each fit variation

$\displaystyle\Delta f_{i}=|f_{i}^{\rm pref.}-f_{i}^{\rm alt.}|/f_{i}^{\rm
pref.}\,,$ (37)

where $i=\\{0,+\\}$. We take the largest difference between our preferred fit
and any of the alternate fits as systematic uncertainty due to the chiral-
continuum extrapolation. We do not use fits with $p$-values below 1% or those
that cannot resolve the coefficients within statistical uncertainties for our
error estimate. Thus we exclude the fit omitting ensembles with pion masses
$M_{\pi}\gtrsim 400$ MeV and the fit using standard “soft-pion” HM$\chi$PT.

For each form factor, we obtain the largest difference from our preferred fit
using the following variation:

$\displaystyle f_{+}^{B\pi}:\ $ analytic for $18.7\ {\rm GeV}^{2}\leq
q^{2}\leq 22.7\ {\rm GeV}^{2}$ and omitting the $M_{\pi}$ term elsewhere,
$\displaystyle f_{0}^{B\pi}:\ $ analytic, $\displaystyle f_{+}^{B_{s}K}:\ $
analytic, $\displaystyle f_{0}^{B_{s}K}:\ $ omitting the $a^{2}$ and $M_{\pi}$
terms,

We therefore use these fits to obtain the $q^{2}$-dependent chiral-continuum
extrapolation errors quoted in Table 6 and shown in Fig. 7.

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 $q^{2}=0$.

###  IV.2 Lattice-scale uncertainty

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
$am_{l}=0.005$ 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.

We tuned the parameters of the $b$-quark action to reproduce the experimental
value of the ${B_{s}}$ meson mass, and carry out our analysis in terms of
dimensionless ratios over $M_{B_{s}}$ to remove all explicit dependence on the
lattice scale. We then obtain the form factors and momentum transfers in
physical units by multiplying by the appropriate power of $M_{B_{s}}=5.367(4)$
GeV. The uncertainties in the form factors due to to the experimental error on
$M_{B_{s}}$ are negligible.

We do, however, still need to consider the implicit dependence on the lattice
spacing through the parameters of the $b$-quark action. We estimate the size
of this dependence, by computing the form factors $f_{\perp}$ and
$f_{\parallel}$ for seven sets of RHQ parameters. We then calculate the slopes
with respect to the parameters – $\Delta f/\Delta m_{0}a$, $\Delta f/\Delta
c_{P}$ and $\Delta f/\Delta\xi$ – for all momenta used in the analysis. Next
we multiply each slope by the uncertainty in the corresponding RHQ parameter
due to the lattice spacing from Table 2, e.g. $\Delta f/\Delta
m_{0}\times\sigma_{m_{0}a}^{a}$. Finally, we add the individual contributions
from the three RHQ parameters in quadrature to obtain the total systematic
error due the lattice spacing.

We examined the slopes with respect to the RHQ parameters for both $B\to\pi$
and $B_{s}\to K$, and found them to be consistent. We therefore base our
estimates for the systematic uncertainty due to the lattice spacing on the
slopes obtained for the $B_{s}\to K$ form factors because the smaller
statistical errors in $B_{s}\to K$ enable the slopes to be resolved more
precisely. Figure 9 shows the slopes of the $B_{s}\to K\ell\nu$ form factors
with respect to the $\\{m_{0}a,c_{P},\zeta\\}$ on the $a\approx 0.11$ fm
ensemble with $am_{l}=0.005$. For this slope estimate, we use the unimproved
heavy-light vector current from Eq. (10). We find the largest slopes at
$\vec{p}=2\pi(2,0,0)$ for $f_{\perp}$ and $\vec{p}=2\pi(1,1,0)$ for
$f_{\parallel}$. Following the procedure outlined above, we estimate lattice-
spacing errors in $f_{\perp}$ and $f_{\parallel}$ of $1.9$ % and $2.2$ %,
respectively. In the continuum this corresponds to errors on $f_{+}$ ($f_{0}$)
of $2.0$% ($2.2$%) which we take for both $B_{s}\to K$ and $B\to\pi$.

###  IV.3 Light- and strange-quark mass uncertainties

Here we estimate the error in the form factors due to the uncertainty in the
light-quark mass and the mistuning of the strange sea quark. For clarity we
discuss separately each place where the light- or strange-quark mass enters
the analysis.

####  IV.3.1 $u/d$-quark mass uncertainty

We obtain the physical form factors $f_{\perp}$ and $f_{\parallel}$ after the
chiral-continuum fit by evaluating Eqs. (31) and (32) at the physical average
$u/d$-quark mass $a_{32}\tilde{m}_{ud}^{\rm phys}=0.00102(5)$. We estimate the
error in the form factors due to the light-quark mass uncertainty by varying
$\tilde{m}_{ud}^{\rm phys}$ by plus/minus one sigma. For $B\to\pi$ the central
value shifts by $0.2-0.3$% for $f_{+}^{B\pi}$ and $0.2-0.4$% for
$f_{0}^{B\pi}$, while for $B_{s}\to K$ both $f_{+}^{B_{s}K}$ and
$f_{0}^{B_{s}K}$ change by $0.1$%.

####  IV.3.2 Strange sea-quark mistuning

Our preferred chiral-continuum fit employs $SU(2)$ chiral perturbation theory,
in which the strange quark mass is integrated out, so our fit function has no
explicit dependence on $m_{s}$. Further, at each lattice spacing, results for
the form factors are only available at a single value of the strange sea-quark
mass, so we cannot directly compute the strange sea-quark mass dependence of
$f_{\parallel}$ and $f_{\perp}$. We therefore study the light sea-quark mass
dependence and use it to bound the strange sea-quark mass dependence. We
cannot resolve any light sea-quark mass dependence within statistical
uncertainties, and expect the strange sea-quark mass dependence to be even
smaller. Thus we take the error due to mistuning the strange sea-quark mass to
be negligible.

####  IV.3.3 Valence strange-quark mass uncertainty

Figure 10: Valence strange-quark mass dependence of the $B_{s}\to K$ form
factors $f_{\perp}$ (top) and $f_{\parallel}$ (bottom) on the $a\approx 0.11$
fm ensemble with $am_{l}=0.005$. 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
Aoki:2010dy . For clarity, data points at equal strange-quark masses are
plotted with a slight horizontal offset.

The $B_{s}\to K$ form factors have explicit strange valence-quark mass
dependence. The strange-quark masses employed in our simulations differ
slightly from the physical, tuned values
$a_{24}\tilde{m}_{s}^{phys}=0.0379(11)$ and
$a_{32}\tilde{m}_{s}^{phys}=0.0280(7)$ Aoki:2010dy . To study the valence
strange-quark mass dependence, we calculated the $B_{s}\to K$ form factors on
the $a\approx 0.11$ fm, $am_{l}=0.005$, ensemble with two additional
spectator-quark masses of $a_{24}\tilde{m}_{s}=0.033$ and $0.043$. Figure 10
shows the valence-quark mass dependence of the $B_{s}\to K$ form factors; we
observe the largest slopes for $f_{\parallel}$ at $p=(0,0,0)$ and for
$f_{\perp}$ at $p=(1,0,0)$. Multiplication of these measured slopes by the
discrepancy between the simulated and tuned strange-quark masses,
$\Delta({m_{s}})\equiv(\tilde{m}_{s}-\tilde{m}_{s}^{\rm phys})=0.004$, leads
to estimates for the error due to mistuning the valence strange-quark mass of
about $0.1\%$ for $f_{+}$ and below this for $f_{0}$ (which we consider as
negligible).

###  IV.4 RHQ parameter uncertainty

We compute the semileptonic form factors using the nonperturbatively tuned RHQ
parameters obtained in Ref. Aoki:2012xaa  and given in Table 2. The RHQ
parameters have four significant sources of uncertainty: statistics, heavy-
quark discretization errors, lattice scale, and the experimental inputs. We
already discussed the uncertainty due to the lattice scale in Sec. IV.2. We
follow the same approach for propagating the uncertainty in the RHQ parameters
due to heavy-quark discretization errors and experimental inputs. We multiply
the estimated slopes of the form factors with respect to changes in $m_{0}$,
$c_{P}$, and $\zeta$ (shown in Fig. 9) by the uncertainties in the
corresponding parameters due to heavy-quark discretization errors and
experimental inputs. Adding the contributions from the three RHQ parameters
and the two uncertainty sources in quadrature, we obtain error estimates for
$f_{+}^{B\pi}$ of 0.8–0.9%, $f_{0}^{B\pi}$ of 1.0%, $f_{+}^{B_{s}K}$ of 0.9%,
and $f_{0}^{B_{s}K}$ of 1.0%.

We neglect the statistical uncertainties in the RHQ parameters in our final
analysis, after checking that they have a negligible impact on the form
factors. On the $a\approx 0.11$ fm, $am_{l}=0.005$ ensemble, we computed the
form factors with seven sets of RHQ parameter values. We then used the
approach detailed in Refs. Aoki:2012xaa ; Christ:2014uea  to interpolate to
the tuned RHQ parameters. This procedure automatically propagates the
statistical errors in the RHQ parameters via the jackknife. Indeed we find
that the statistical errors obtained from the two procedures are identical.
Thus we do not need to perform the more complicated and computationally
expensive procedure of interpolating to the tuned RHQ parameters in our
analysis.

###  IV.5 Heavy-quark discretization errors

Table 7: Percentage errors from mismatches in the action and current for the bottom quark on the $24^{3}$ and $32^{3}$ ensembles. For this estimate, we calculate the mismatch functions for the nonperturbatively tuned parameters of the RHQ action from Table 2. We estimate the size of operators using HQET power counting with $\Lambda_{\rm QCD}=500$ MeV and the coupling constant $\alpha_{s}^{\overline{\rm MS}}(1/a)=1/3$ on the $24^{3}$ ensemble and $0.22$ on the $32^{3}$ ensembles. To obtain the total, we add the individual errors in quadrature, including each contribution the number of times that operator occurs. Contribution $E$ is counted twice, and 3 is counted twice for $f_{\parallel}$ and four times for $f_{\perp}$. The definitions of operators “E,” “$X_{1},$” “$X_{2}$,” “$Y,$” and “3” and expressions for the mismatch functions are given in Appendix B of Ref. Christ:2014uea . |  ${\mathcal{O}}(a^{2})$ error |  ${\mathcal{O}}(a^{2})$ errors |  ${\mathcal{O}}(\alpha_{s}^{2}a)$ error |  |
---|---|---|---|---|---
| from action | from current | from current | Total (%)
| $E$ | $X_{1}$ | $X_{2}$ | $Y$ | 3 | $f_{\parallel}$ | $f_{\perp}$
$a\approx$ 0.11 fm | 0.55 | 0.67 | 1.27 | 1.34 | 1.48 | 2.97 | 3.64
$a\approx$ 0.086 fm | 0.42 | 0.46 | 0.85 | 0.91 | 0.55 | 1.65 | 1.82

The RHQ action gives rise to nontrivial lattice-spacing dependence in the form
factors in the region $m_{0}a\sim 1$. To estimate the size of the resulting
discretization errors, we use the same power-counting approach as in our
companion papers on bottomonium masses and splittings Aoki:2012xaa  and
$B_{(s)}$-meson decay constants Christ:2014uea .

We tune the parameters of the operators in the dimension-5 RHQ action
nonperturbatively, such that the leading heavy-quark discretization errors
from the action are of ${\mathcal{O}}(a^{2})$. We use an
${\mathcal{O}}(\alpha_{s}a)$-improved vector current and calculate the
improvement coefficient to 1-loop; therefore the leading heavy-quark
discretization errors from the current are of
${\mathcal{O}}(\alpha_{s}^{2}a,a^{2})$. Because we use the same actions and
simulation parameters as in our earlier calculation of the $B_{(s)}$-meson
leptonic decay constants Christ:2014uea , the numerical error estimates are
almost identical in the two works. The same operators contribute in both
cases, but enter a different number of times for the spatial and temporal
vector currents. Table 7 quotes the estimate of heavy-quark discretization
errors from the five different operators in the action and current on the
$24^{3}$ and $32^{3}$ ensembles, and we refer the reader to Sec. V. E. and
Appendix B of Ref. Christ:2014uea  for details. We take the size of heavy-
quark discretization errors in our calculation of the $B\to\pi\ell\nu$ and
$B_{s}\to K\ell\nu$ semileptonic form factors to be the estimate on our finer
$a^{-1}=2.281$ GeV lattices, which is $\sim 1.7$% for the lattice form factors
$f_{\parallel}$ and $\sim 1.8$% for $f_{\perp}$. These lead to errors in the
continuum form factors $f_{+}$ and $f_{0}$ of $\sim 1.8$% and $\sim 1.7$%,
respectively.

###  IV.6 Light-quark discretization errors

The dominant discretization errors from the light-quark and gluon sectors are
of $\mathcal{O}((a\Lambda_{\rm QCD})^{2})$ from the action, and is about 5%
using $\Lambda_{\rm QCD}=500$ MeV. We remove this $\mathcal{O}((a\Lambda_{\rm
QCD})^{2})$ error by including a term proportional to $a^{2}$ in the chiral-
continuum extrapolation (see Eqs. (31) and (32)). Then the leading light-quark
and gluon discretization errors in the heavy-light vector current are of
$\mathcal{O}(\alpha_{s}a\tilde{m}_{q},(a\tilde{m}_{q})^{2},\alpha_{s}^{2}a\Lambda_{\rm
QCD},(ap)^{2})$ where $\tilde{m}_{q}$ denotes the bare lattice mass. The first
entry of $\mathcal{O}(\alpha_{s}a\tilde{m}_{q})$ leads to estimated errors of
$\sim 0.1\%$ in $B\to\pi$ and $\sim 0.6\%$ in $B_{s}\to K$ form factors on the
$32^{3}$ ensembles. The second is negligible ($<0.1\%$) and the third is
estimated to be $\sim 1.1\%$ in both $B\to\pi$ and $B_{s}\to K$ form factors.
Adding these contributions in quadrature, we estimate the total uncertainty
from light-quark and gluon discretization errors in the heavy-light current to
be $1.1$% in $B\to\pi$ and $1.3$% in $B_{s}\to K$.

We do not observe any evidence of sizable momentum-dependent discretization
errors in our data. Figure 2 shows that the pion and kaon energies and
amplitudes are consistent with continuum expectations, and smaller than power-
counting estimates of $\mathcal{O}((ap)^{2})$. Thus we do not include a
systematic error due to momentum-dependent discretization errors.

###  IV.7 Renormalization factor

We renormalize the lattice form factors using a mostly nonperturbative
approach in which we separate $Z_{V_{\mu}}^{bl}$ into three components. We
consider the uncertainties from these three multiplicative factors separately,
and then add them in quadrature to obtain the total error on the form factors.

For $Z_{V}^{ll}$, we use the nonperturbatively determined value of the axial-
current renormalization factor $Z_{A}$ in the chiral limit from Ref.
Aoki:2010dy . We can neglect the statistical uncertainty in $Z_{A}$ (which is
only $0.02$% on the finer ensembles) and the difference between $Z_{V}^{ll}$
and $Z_{A}$ (which is about $\mathcal{O}(am_{\rm res})\sim 7\times 10^{-4}$ at
$a\approx 0.086$ fm). For $Z_{V}^{bb}$, we use the nonperturbative
determination from Christ:2014uea . The statistical uncertainty in
$Z_{V}^{bb}$ on the finer ensemble is $0.15$%. We conservatively estimate the
perturbative truncation error in $\rho_{V}^{bl}$ to be the full size of the
1-loop correction at the finer $a\approx 0.086$ fm lattice spacing, which
leads to $1.7$% for $\rho_{V_{0}}$ and $0.6$% for $\rho_{V_{i}}$. These are
significantly larger than what we would estimate for two-loop contributions
from naive power counting. Taking $\alpha_{s}\sim 0.23$ on the coarser lattice
spacing and a coefficient of $1/(4\pi)^{2}$ from two loop-suppression factors,
we would obtain an estimate of 0.03%. Even with a coefficient of $1/\pi^{2}$,
we would obtain an estimate of 0.5%, which is slightly smaller than the
perturbative uncertainty that we assign to $\rho_{V_{i}}$ and 3 times smaller
than the error we assign to $\rho_{V_{0}}$. Because we use the values of
$\rho_{V_{\mu}}$ and $Z_{V}^{ll}$ in the chiral limit, we must consider the
errors due to the nonzero physical up, down, and strange-quark masses. The
leading quark-mass dependent errors in $\rho_{V_{\mu}}$ and $Z_{V}^{ll}$ are
$\mathcal{O}(\alpha_{s}a\tilde{m}_{q})$ and
$\mathcal{O}((a\tilde{m}_{q})^{2})$, respectively, but these are already
accounted for in our estimate of light-quark and gluon discretization errors
(see Sec. IV.6). Thus we do not count them again here.

Perturbative truncation errors are by far the dominant source of uncertainty
in the renormalization factor, and the quadrature sum of the three error
contributions is 1.7% for $f_{\parallel}$ and 0.6% for $f_{\perp}$.

###  IV.8 Finite-volume errors

We compute the form factors on a finite-sized lattice. We estimate the effect
of the finite spatial volume using one-loop finite-volume SU(2) hard-pion
$\chi$PT, in which loop integrals are replaced by a sum over lattice sites.
Only a single integral enters the NLO SU(2) hard-pion ChPT expression. The
correction to Eqs. (33) and (34) to account for the finite spatial volume is
given by a sum over modified Bessel functions Arndt:2004bg ; Aubin:2007mc :

$M_{\pi}^{2}{\rm log}\left(\frac{M_{\pi}^{2}}{\Lambda^{2}}\right)\rightarrow
M_{\pi}^{2}{\rm
log}\left(\frac{M_{\pi}^{2}}{\Lambda^{2}}\right)+\frac{4M_{\pi}}{L}\sum_{\vec{r}\neq
0}^{L^{3}}\frac{K_{1}(|\vec{r}|M_{\pi}L)}{|\vec{r}|}.$ (38)

From Eq. (38), the ensemble with the lightest quark mass receives the largest
correction. For $B\to\pi$ we find corrections to $f_{\perp}$ ($f_{\parallel}$)
of 0.3-0.4% (0.6–0.8%), while for $B_{s}\to K$ corrections to
$f_{\perp}$($f_{\parallel}$) of 0.2–0.3% (0.4–0.5%). These result in the
following errors on the continuum form factors: 0.3–0.5% for $f_{+}^{B\pi}$,
0.4–0.7% for $f_{0}^{B\pi}$, 0.2% for $f_{+}^{B_{s}K}$, and 0.1–0.2% for
$f_{0}^{B_{s}K}$.

###  IV.9 Isospin breaking

Our $B\to\pi\ell\nu$ and $B_{s}\to K\ell\nu$ form factors are calculated in
the isospin limit. The form factors of the charged and neutral $B$
($B_{s}$)-mesons, however, differ due to both the masses and the charges of
the constituent light $u$ and $d$ quarks. The leading quark-mass contribution
to the isospin breaking from the valence-quark masses is of
$\mathcal{O}((m_{d}-m_{u})/\Lambda_{\rm QCD})\sim 0.5$%, which is obtained
using the determination of the quark masses $(m_{d}-m_{u})=2.35(8)(24)$ MeV
from FLAG Aoki:2013ldr  and $\Lambda_{\rm QCD}=500$ MeV. The difference
between the $u$\- and $d$-quark masses in the sea sector should have a
negligible effect on the $B\to\pi\ell\nu$ ($B_{s}\to K\ell\nu$) form factor
because the sea quarks couple to the valence quarks through $I=0$ gluon
exchange, and they give only the uncertainty of
$\mathcal{O}(((m_{d}-m_{u})/\Lambda_{\rm QCD})^{2})\sim 0.003$%. The
electromagnetic contribution to the isospin breaking is expected to be
$\mathcal{O}(\alpha_{s})\sim 1/137\sim 0.7\%$ which is the typical size of
1-loop QED corrections. We therefore take 0.7% as the uncertainties due to the
isospin breaking and electromagnetism effects.

###  IV.10 Correlation matrices

In the next section we fit synthetic lattice data generated at three values of
$q^{2}$ to the $z$-expansion to extend it to the full kinematic range. Thus,
in addition to the systematic uncertainties on the individual $q^{2}$-bins, we
also need the correlations between $q^{2}$ values. Although it is
straightforward to obtain the statistical correlations further explanation is
needed for the systematic error correlations.

The chiral-continuum extrapolation error is estimated by varying the fit
function and parametric inputs. This procedure does not provide any
information on correlations of the resulting systematic error between
different $q^{2}$-bins. Alternate chiral-continuum fits to our data with
different fit functions do, however, exhibit highly similar statistical
correlations between $q^{2}$-bins. Hence we take the (normalized) statistical
correlation matrix from our preferred fit and multiply it by the estimated
chiral-continuum extrapolation error at each $q^{2}$ value (For off-diagonal
elements of the correlation matrix we use the product
$\sigma_{q^{2}_{i}}\sigma_{q^{2}_{j}}$.) We follow the same procedure to
estimate the correlations between the $q^{2}$-dependent systematic error due
to the light-quark mass uncertainty. We take the remaining systematic errors
for which we do not assume any $q^{2}$ dependence to be 100% correlated.

Tables 8 and 9 present the normalized statistical and systematic correlation
matrices, which enable the full reconstruction of the total covariance
matrices using the values for $B\to\pi\ell\nu$ and $B_{s}\to K\ell\nu$ form
factors and their errors from Table 6.

Table 8:  Normalized statistical (upper) and systematic (lower) correlation
matrices for the $B\to\pi\ell\nu$ form factors at three representative $q^{2}$
values.

|  | $f_{+}^{B\pi}$ | $f_{0}^{B\pi}$
---|---|---|---
|  $q^{2}$ [GeV2] | 19.0 | 22.6 | 25.1 | 19.0 | 22.6 | 25.1
| 19.0 | 1.000 | 0.868 | 0.045 | 0.663 | 0.586 | 0.541
$f_{+}^{B\pi}$ | 22.6 | 0.868 | 1.000 | 0.239 | 0.591 | 0.654 | 0.616
| 25.1 | 0.045 | 0.239 | 1.000 | 0.176 | 0.188 | 0.283
| 19.0 | 0.663 | 0.591 | 0.176 | 1.000 | 0.822 | 0.836
$f_{0}^{B\pi}$ | 22.6 | 0.586 | 0.654 | 0.188 | 0.822 | 1.000 | 0.941
| 25.1 | 0.541 | 0.616 | 0.283 | 0.836 | 0.941 | 1.000

|  | $f_{+}^{B\pi}$ | $f_{0}^{B\pi}$
---|---|---|---
|  $q^{2}$ [GeV2] | 19.0 | 22.6 | 25.1 | 19.0 | 22.6 | 25.1
| 19.0 | 1.000 | 0.897 | 0.245 | 0.702 | 0.663 | 0.645
$f_{+}^{B\pi}$ | 22.6 | 0.897 | 1.000 | 0.427 | 0.639 | 0.725 | 0.719
| 25.1 | 0.245 | 0.427 | 1.000 | 0.289 | 0.342 | 0.448
| 19.0 | 0.702 | 0.639 | 0.289 | 1.000 | 0.840 | 0.840
$f_{0}^{B\pi}$ | 22.6 | 0.663 | 0.725 | 0.342 | 0.840 | 1.000 | 0.948
| 25.1 | 0.645 | 0.719 | 0.448 | 0.840 | 0.948 | 1.000

Table 9:  Normalized statistical (upper) and systematic (lower) correlation
matrices for the $B_{s}\to K\ell\nu$ form factors at three representative
$q^{2}$ values.

|  | $f_{+}^{B_{s}K}$ | $f_{0}^{B_{s}K}$
---|---|---|---
|  $q^{2}$ [GeV2] | 17.6 | 20.8 | 23.4 | 17.6 | 20.8 | 23.4
| 17.6 | 1.000 | 0.868 | 0.828 | 0.799 | 0.754 | 0.702
$f_{+}^{B_{s}K}$ | 20.8 | 0.868 | 1.000 | 0.783 | 0.677 | 0.799 | 0.764
| 23.4 | 0.828 | 0.783 | 1.000 | 0.615 | 0.703 | 0.708
| 17.6 | 0.799 | 0.677 | 0.615 | 1.000 | 0.828 | 0.755
$f_{0}^{B_{s}K}$ | 20.8 | 0.754 | 0.799 | 0.703 | 0.828 | 1.000 | 0.974
| 23.4 | 0.702 | 0.764 | 0.708 | 0.755 | 0.974 | 1.000

|  | $f_{+}^{B_{s}K}$ | $f_{0}^{B_{s}K}$
---|---|---|---
|  $q^{2}$ [GeV2] | 17.6 | 20.8 | 23.4 | 17.6 | 20.8 | 23.4
| 17.6 | 1.000 | 0.939 | 0.921 | 0.865 | 0.843 | 0.808
$f_{+}^{B_{s}K}$ | 20.8 | 0.939 | 1.000 | 0.913 | 0.794 | 0.860 | 0.835
| 23.4 | 0.921 | 0.914 | 1.000 | 0.760 | 0.806 | 0.801
| 17.6 | 0.865 | 0.794 | 0.760 | 1.000 | 0.889 | 0.840
$f_{0}^{B_{s}K}$ | 20.8 | 0.843 | 0.860 | 0.806 | 0.889 | 1.000 | 0.983
| 23.4 | 0.808 | 0.835 | 0.801 | 0.840 | 0.983 | 1.000

##  V Form-factor results

In this section we extrapolate our $B\to\pi$ ($B_{s}\to K$) form-factor
results from large $q^{2}$, where we have our (synthetic) data, to $q^{2}=0$
using a model-independent parametrization based on the general properties of
analyticity, unitarity, and crossing symmetry. We first give the expressions
for the $z$-parametrizations used in our analysis in Sec. V.1; we use the
parametrization of Bourrely, Caprini, and Lellouch (BCL) for our preferred
results, but also consider the parametrization of Boyd, Grinstein, and Lebed
(BGL) as a cross-check. Next, in Sec. V.2 we extrapolate our synthetic lattice
data to $q^{2}=0$; we present our preferred results for $f_{+}$ and $f_{0}$ in
Tables 11 and 12 as coefficients of the $z$-expansion and the matrix of
correlations between them.

Use of the $z$-parametrization to describe semileptonic form factors has
several advantages over other functional forms used in the literature
Becirevic:1999kt ; Ball:2004ye . Because the absolute value of $|z|$ is small
in the semileptonic region, and the $z$-coefficients are constrained to be
small by unitarity and heavy-quark symmetry, one needs only the first few
terms in the expansion to accurately describe the form factor shape with a
negligible truncation error. Moreover, as the precisions of both the lattice
calculations and experimental measurements improve, one may easily include
higher-order terms in $z$ as needed. Finally, comparisons of the $z$-expansion
parameters resulting from fits to different theoretical or experimental data
sets enable a meaningful quantitative comparison of the shapes, while a
combined fit to lattice and experimental data enables a clean determination of
$|V_{ub}|$. The $z$-expansion has therefore been adopted as the preferred
method for obtaining exclusive $|V_{ub}|$ by experimentalists on Babar and
Belle, the Heavy Flavor Averaging Group, and the Particle Data Group
delAmoSanchez:2010af ; Ha:2010rf ; Lees:2012vv ; Sibidanov:2013rkk ;
Beringer:2012zz ; Amhis:2012bh .

###  V.1 $z$-expansions of semileptonic form factors

The $B\to\pi\ell\nu$ and $B_{s}\to K\ell\nu$ form factors are analytic
functions of $q^{2}$ except at physical poles and branch cuts above the
production threshold. Therefore, given a suitable change of variables, they
can be expressed as a convergent power series (see, e.g., Boyd:1994tt ;
Lellouch:1995yv ; Boyd:1997qw ; Bourrely:1980gp ; Arnesen:2005ez ;
Bourrely:2008za ). Unitarity and heavy-quark power counting bound the size of
the series coefficients. In the literature, the new variable is called $z$,
and the class of functions are called $z$-parametrizations. Two such
parametrizations commonly used to extrapolate the $B\to\pi\ell\nu$ form factor
are by Boyd, Grinstein, and Lebed (BGL) Boyd:1994tt  and Bourrely, Caprini,
and Lellouch (BCL) Bourrely:2008za .

The change of variables from $q^{2}$ to $z$ is given by

$z(q^{2},t_{0})=\frac{\sqrt{1-q^{2}/t_{+}}-\sqrt{1-t_{0}/t_{+}}}{\sqrt{1-q^{2}/t_{+}}+\sqrt{1-t_{0}/t_{+}}},$
(39)

where $t_{+}\\!\\!\equiv\\!\\!(M_{B_{(s)}}+M_{\pi})^{2}$ and
$t_{-}\\!\\!\equiv\\!\\!(M_{B_{(s)}}-M_{\pi})^{2}$. This transformation maps
the semileptonic region $0<q^{2}<t_{-}$ onto a unit circle in the complex $z$
plane. The $B\to\pi\ell\nu$ and $B_{s}\to K\ell\nu$ form factors can then be
expanded as a simple power series in $z$:

$P_{i}(q^{2})\phi_{i}(q^{2},t_{0})f_{i}(q^{2})=\sum_{k=0}^{\infty}a_{i}^{(k)}(t_{0})z(q^{2},t_{0})^{k},$
(40)

where $i=\\{0,+\\}$ for the scalar and vector form factors, respectively. The
free parameter $t_{0}$ in Eq. (39) determines the range of $|z|$ in the
semileptonic region, and hence can be chosen to accelerate the series
convergence. The “Blaschke factor” $P_{i}(q^{2})$ must be chosen to vanish at
any subthreshold poles to preserve the correct analytic structure of
$f_{i}(q^{2})$. For $f_{+}$, the relevant state is the $J^{P}=1^{-}$ meson,
while for $f_{0}$, the relevant state is the $J^{P}=0^{+}$ meson. As discussed
earlier in Sec. III.3, the scalar $B^{*}$ meson has not been observed
experimentally, but is predicted to have a mass well above the $B\pi$
production threshold Bardeen:2003kt ; Gregory:2010gm . Thus the functions
$P_{i}$ for $B\to\pi\ell\nu$ are typically taken in the literature to be

$\displaystyle P^{B\pi}_{+}(q^{2})=z(q^{2},M_{B^{*}})\,,\quad
P^{B\pi}_{0}(q^{2})=1\,.$ (41)

Finally, the outer function $\phi_{i}(q^{2},t_{0})$ can be any analytic
function of $q^{2}$; different choices for $\phi_{i}$ correspond to different
$z$-parametrizations.

The form factors that describe $B\to\pi\ell\nu$ ($B_{s}\to K\ell\nu$) in the
range $0<q^{2}<t_{-}$, when analytically continued, also describe $B\pi$
($BK$) production for $q^{2}>t_{+}$. The coefficients of the $z$-expansion are
therefore bounded by the fact that the rate of production of $B\pi$ ($BK$)
states is less than the production rate of all states coupling to the $b\to u$
current. In Ref. Boyd:1994tt , Boyd, Grinstein, and Lebed choose the outer
function $\phi_{i}$ so that the unitarity constraint on the series
coefficients takes a particularly simple form:

$\sum_{k=0}^{N}\left(a_{i}^{(k)}\right)^{2}\mathrel{\hbox to0.0pt{\lower
3.0pt\hbox{$\mathchar 536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar
316\relax$}}1,$ (42)

where this holds for any value of $N$. The explicit functions for
$\phi_{+}^{\rm BGL}$ and $\phi_{0}^{\rm BGL}$ and their numerical values can
be found in Ref. Arnesen:2005ez . When using the BGL parametrization for
subsequent $z$-fits, we use $t_{0}=0.65t_{-}$ as in Ref. Arnesen:2005ez , such
that $-0.341<z<0.216$ ($-0.144<z<0.148$) for $B\to\pi l\nu$ ($B_{s}\to
K\ell\nu$) decay.

In Ref. Bourrely:2008za , Bourrely, Caprini, and Lellouch (who only discuss
$f_{+}$) choose a simpler outer function $\phi_{0}^{\rm BCL}=1$. They also
point out that the BGL form-factor parametrization does not obey the known
asymptotic behavior near the $B\pi$ production threshold ${\rm
Im}f_{+}(q^{2})\sim\left(q^{2}-t_{+}\right)^{3/2}$ (which is due to angular
momentum conservation). Therefore, at $q^{2}=t_{+}$ ($z=-1$), the derivative
of the form factor must satisfy

$\left.\left[\frac{df_{+}}{dz}\right]\right|_{z=-1}=0\,.$ (43)

BCL use this constraint on the derivative of the form factor to remove an
independent degree of freedom from the series expansion in $z$. Thus they
arrive at the following parametrization for the vector form factor:

$f_{+}(q^{2})=\frac{1}{1-q^{2}/m_{B^{*}}^{2}}\sum_{k=0}^{K-1}b_{+}^{(k)}\left[z^{k}-(-1)^{k-K}\frac{k}{K}z^{K}\right]\,,$
(44)

where we label the BCL series coefficients $b_{k}$ to distinguish them from
the BGL coefficients $a_{k}$. There is no analogous constraint to Eq. (44) on
the value or derivative of $f_{0}$ at any $z$, so one cannot remove a further
degree of freedom in the series expansion for the scalar form factor. We
therefore use the following functional forms for the scalar form factors:

$\displaystyle f_{0}^{B\pi}(q^{2})$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{K-1}b_{0}^{(k)}z^{k}\,,$ (45) $\displaystyle
f_{0}^{B_{s}K}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{1-q^{2}/m_{B^{*}}^{2}}\sum_{k=0}^{K-1}b_{0}^{(k)}z^{k}\,,$
(46)

where we include a pole at the theoretically predicted value
$M_{B^{*}}(0^{+})=5.63$ GeV for $B_{s}\to K\ell\nu$ Bardeen:2003kt . Equation
(45) has been called the “simplified series expansion” in the literature
Bharucha:2010im . To minimize the error from truncating the $z$-expansion for
the $B\to\pi\ell\nu$ form factor, BCL choose $t_{0}=t_{\rm
opt}\equiv\left(M_{B}+M_{\pi}\right)\left(\sqrt{M_{B}}-\sqrt{M_{\pi}}\right)^{2}$,
such that the magnitude of $|z|\leq 0.280$ is minimized in the semileptonic
range. With the analogous choice for $B_{s}\to K\ell\nu$, $|z|\leq 0.146$ for
the semileptonic range.

Although the functional form of the BCL parametrization is simpler than that
of BGL, the unitarity constraint on the coefficients is more complicated
Bourrely:2008za :

$\displaystyle\sum_{j,k=0}^{K}B_{jk}(t_{0})b_{i}^{(j)}(t_{0})b_{i}^{(k)}(t_{0})\mathrel{\hbox
to0.0pt{\lower 3.0pt\hbox{$\mathchar 536\relax$}\hss}\raise
2.0pt\hbox{$\mathchar 316\relax$}}1\,,$ (47) $\displaystyle
B_{jk}(t_{0})=\sum_{n=0}^{\infty}\eta_{n}(t_{0})\eta_{n+|j-k|}(t_{0}),$ (48)

where $\eta_{i}$ is the Taylor coefficients in the expansion of the outer
function

$\displaystyle\Psi(z)$ $\displaystyle=$
$\displaystyle\frac{M_{B^{*}}^{2}}{4(t_{+}-t_{0})}\phi_{i}(q^{2}(z),t_{0})\frac{(1-z)^{2}(1-z_{*})^{2}}{(1-zz_{*})^{2}},$
(49) $\displaystyle z_{*}$ $\displaystyle=$ $\displaystyle
z(M_{B}^{2},t_{0}),$ (50)

around $z=0$. The values of $B_{jk}$ for the $B\to\pi\ell\nu$ and $B_{s}\to
K\ell\nu$ form factors with the choice $t_{0}=t_{\rm opt}$ are given in Table
10.

Table 10:  Matrix elements $B_{jk}(t_{0})$ that enter the unitarity bound on the BCL series coefficients for the choice $t_{0}=t_{\rm opt}$. The remaining coefficients can be obtained from the relations $B_{j(j+k)}=B_{0k}$ and the symmetry property $B_{jk}=B_{kj}$. To derive these results we use the outer functions $\phi_{+}$ and $\phi_{0}$ in Eq. (7) of Ref. Arnesen:2005ez  with inputs from Ref. Bourrely:2008za , giving $\chi^{(0)}_{+}=5.03\times 10^{-4}$ and $\chi^{(0)}_{0}=1.46\times 10^{-2}$. | $B_{00}$ | $B_{01}$ | $B_{02}$ | $B_{03}$ | $B_{04}$ | $B_{05}$
---|---|---|---|---|---|---
$f_{+}^{B\pi}$ | 0.0197 | 0.0042 | -0.0109 | -0.0059 | -0.0002 |  0.0012
$f_{0}^{B\pi}$ | 0.1062 | 0.0420 | -0.0368 | -0.0406 | -0.0201 | -0.0057
$f_{+}^{B_{s}K}$ | 0.0115 | 0.0004 | -0.0076 | -0.0007 |  0.0018 |  0.0004
$f_{0}^{B_{s}K}$ | 0.0926 | 0.0137 | -0.0484 | -0.0174 | -0.0003 |  0.0024

For the $B\to\pi\ell\nu$ vector form factor, Becher and Hill Becher:2005bg
use heavy-quark power counting to provide an estimate for the sum of the
coefficients:

$\sum_{k=0}^{N}\left(a_{+}^{(k)}\right)^{2}\sim\left(\frac{\Lambda}{m_{b}}\right)^{3}\,,$
(51)

where $\Lambda$ is a typical hadronic scale. Taking $\Lambda\sim 1000$ MeV,
this would imply $\sum a_{k}^{2}\sim 0.01$, which is well below the bound from
unitarity. Experimental measurements delAmoSanchez:2010af ; Ha:2010rf ;
Lees:2012vv ; Sibidanov:2013rkk  and previous lattice calculations
Bailey:2008wp  confirm this expectation. This argument also applies to the
$B_{s}\to K\ell\nu$ vector form factor, where we emphasize that Eq. (51) is
only a rough constraint due to the imprecise scale $\Lambda$ and omitted
higher-order corrections in the OPE and $1/m_{b}$.

###  V.2 Extrapolation of lattice form factors to $q^{2}=0$

We now extrapolate our results for the $B\to\pi\ell\nu$ and $B_{s}\to
K\ell\nu$ form factors to $q^{2}=0$ using the $z$-expansion. We first generate
synthetic data points in the range of simulated data from the output of the
chiral-continuum extrapolation. Recall that the continuum, physical quark-mass
form factors are obtained from fits to Eqs. (31) and (32) by fixing
$M_{\pi}^{2}$ to the physical value and $a^{2}\to 0$. After these
replacements, the physical form factors depend upon three independent
functions of the pion or kaon energy $E_{P}$. We therefore generate three
synthetic data points each for $f_{0}$ and $f_{+}$ in order to ensure that the
covariance matrix is not singular. In anticipation of the $z$-fit, we choose
the points to be evenly spaced in $z$ (rather than $q^{2}$). The $q^{2}$
values and error budgets for the synthetic lattice data are given in Table 6.

We fit our synthetic lattice data for the $B\to\pi\ell\nu$ and $B_{s}\to
K\ell\nu$ form factors including statistical and systematic correlations
between $q^{2}$ values. For our preferred fit we use the BCL parametrization
with the kinematic constraint $f_{+}(0)=f_{0}(0)$ and use the theoretical
estimate from heavy-quark power counting to constrain the sum of the
coefficients of the vector form factor via Bayesian priors. We study the
central values and errors of the series coefficients as a function of the
truncation $K$ such that our final form-factor results include the truncation
error. The complete $z$-fit results are given in Appendix B. We also compare
to results using the BGL parametrization as a check.

We first perform separate fits of $f_{+}$ and $f_{0}$ without imposing any
constraints on the sum of coefficients. The results for $B\to\pi\ell\nu$ are
given in the top two panels of Table 19, and for $B_{s}\to K\ell\nu$ in the
upper two panels of Table 20. The separate fits of $f_{+}$ and $f_{0}$ for
$K=2,3$ are shown in the left-hand plots of Fig. 11 for $B\to\pi\ell\nu$
(upper) and $B_{s}\to K\ell\nu$ (lower). The synthetic lattice data points are
correlated, and one must include a term quadratic in $z$ to obtain a good fit
(recall that for $f_{+}$ the expression with $K=2$ includes a term
proportional to $z^{2}$ that is related to the $z^{0}$ and $z^{1}$ terms). The
normalizations $b_{i}^{(0)}$ are well determined by the lattice data, with
central values that are stable within errors when going from $K=2$ to $K=3$.
This is important because the normalization of the vector form factor plays a
key role in the determination of $|V_{ub}|$ (see Sec. VI.1). We cannot go
beyond $K=3$ because we have only three synthetic data points.

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.

In the separate fits to $f_{+}$ and $f_{0}$ with $K=3$, the kinematic
constraint $f_{+}(0)=f_{0}(0)$ is automatically satisfied within
uncertainties, but with large errors. We can therefore impose the kinematic
constraint $f_{+}(0)=f_{0}(0)$. The results of the combined fits are given in
the third panels of Tables 19 and 20. As expected, the constrained fits with
$K=2$ for both $f_{+}$ and $f_{0}$ have poor $p$-values, but the remaining
fits tried are all of good quality. Adding the kinematic constraint (and only
considering the good fits) has little impact on the results for the
normalizations and even on the slopes ($b^{(1)}_{i}/b^{(0)}_{i}$). It reduces
the error on the curvatures ($b^{(2)}_{i}/b^{(0)}_{i}$) as compared to the
separate fits with $K=3$, however, and consequently improves the determination
of $f_{+}(q^{2}=0)$.

Even with the kinematic constraint, however, the slopes and curvatures of the
form factors are still not well determined by the lattice data, with errors
ranging from 25% to as much as 300%. For all fits considered, the sum of the
coefficients $\sum B_{jk}b_{j}b_{k}$ satisfy the unitarity constraint.
Further, for $f_{+}$, the sum $\sum B_{jk}b_{j}b_{k}$ is also consistent with
expectations from heavy-quark power counting, Eq. (51), but with large
uncertainties. We can therefore use theoretical guidance from heavy-quark
power counting to further improve our lattice form-factor determination.
Keeping the kinematic constraint, we also constrain the sum of the
coefficients of the $B\to\pi\ell\nu$ and $B_{s}\to K\ell\nu$ vector form
factors with Bayesian priors based on their estimated size from heavy-quark
power counting. For the hadronic scale in the heavy-quark estimate we take
1000 MeV, with a generous uncertainty of $\pm 500$ MeV. Thus for the prior
central value we use $\overline{B}=0.01$, and for the Gaussian prior width we
use $\sigma_{B}=0.03$. We implement the Bayesian fit by minimizing the
augmented $\chi^{2}$ Lepage:2001ym ,

$\chi^{2}_{\rm aug}=\chi^{2}+\chi^{2}_{\rm prior}\,,$ (52)

where

$\chi^{2}_{\rm prior}=(\overline{B}-\sum
B_{jk}b_{j}b_{k})^{2}/\sigma_{B}^{2}\,.$ (53)

The results for different truncations $K$ are given in the bottom panels of
Tables 19 and 20. The inclusion of the heavy-quark constraint improves the
determinations of the slopes and curvatures, and leads to a reduction in the
absolute error on $f_{+}^{B\pi}(0)$ by about a factor of 2 for
$B\to\pi\ell\nu$ for $K=3$. The improvement in the error on
$f_{+}^{B_{s}K}(0)$ is smaller but non-negligible, about 25%.

After implementing the heavy-quark constraint, we are able to include an
additional parameter in our fits and can consider expansions with $K=4$. This
enables us to study the stability of the central values and errors of the
parameters with truncation $K$, and thus assess the systematic uncertainty
associated with truncating the $z$-expansion. The central values and errors
for the normalizations and slopes are stable when increasing the truncation
from $K=3$ to $K=4$, in most cases changing only in the last decimal place
(except for the slope of $f_{0}^{B_{s}K}(q^{2})$, for which the results are
still consistent within uncertainties). The combined fits of $f_{+}$ and
$f_{0}$ imposing the kinematic and heavy-quark constraints are shown versus
the truncation $K$ in the right-hand plots of Fig. 11 for $B\to\pi\ell\nu$
(upper) and $B_{s}\to K\ell\nu$ (lower). The central fit curves for $K=3$ and
$K=4$ lie almost on top of each other, while the widths of the error bands and
the uncertainties in $f_{+}(0)$ increase only slightly in going to $K=4$. Thus
we conclude that the $K=3$ constrained fit includes the systematic uncertainty
due to truncating the series in $z$.

We therefore take as our preferred fits for $B\to\pi\ell\nu$ and $B_{s}\to
K\ell\nu$ the results from the fit with $K=3$ for both $f_{+}$ and $f_{0}$
including the kinematic and heavy-quark constraints. This is the highest
truncation $K$ for which we still have more data points than fit parameters,
and the uncertainties are comparable to the $K=4$ fits. Figure 12 shows our
preferred fits for $B\to\pi\ell\nu$ (upper plots) and $B_{s}\to K\ell\nu$
(lower plots) plotted versus $z$ (left) and versus $q^{2}$ right.

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 $f_{0}(q^{2})$.

As a cross-check, we compare our preferred fit using the BCL parametrization
to the analogous fit (also imposing the kinematic and heavy-quark constraints,
and to the same order in $z$) using the BGL parametrization. Figure 13
overlays the results of the BCL and BGL fits for $B\to\pi\ell\nu$ (left) and
$B_{s}\to K\ell\nu$ (right). The fits to the different series expansions are
consistent, indicating that our quoted form-factor uncertainties encompass the
error due to truncating the $z$-expansion. The error bands from the BCL fits
are narrower because the BCL form for $f_{+}$ relates the coefficient of
highest-order term in $z$ to the coefficients of the lower-order terms.

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.

Tables 11 and 12 present our final results for the $B\to\pi\ell\nu$ and
$B_{s}\to K\ell\nu$ form factors as coefficients of the $z$-expansion and the
matrix of correlations between them. These results are model independent and
valid over the entire semileptonic region of $q^{2}$. As we illustrate in the
next section, they can be used in combined fits with experimental data to
obtain the CKM matrix element $|V_{ub}|$, or to make predictions for Standard-
Model observables for these decay processes.

It is interesting to compare ratios of these form factors to predictions from
approximate symmetries of QCD. In the SU(3) limit ($m_{d}=m_{s}$), the form
factors for $B\to\pi\ell\nu$ and $B_{s}\to K\ell\nu$ should be identical. Thus
the ratios $R_{i}(q^{2})=f_{i}^{BK}(q^{2})/f_{i}^{B\pi}(q^{2})-1$, for
$i=\\{+,0\\}$, provide a measure of SU(3)-breaking in $B\to{\rm light}$
semileptonic form factors. Figure 14, left, plots these ratios for the full
kinematic range. The results for $f_{+}$ and $f_{0}$ are similar. The
deviations from unity are consistent with expectations from simple power
counting of $(m_{s}-m_{d})/\Lambda_{\rm QCD}\sim 20\%$, but with large
uncertainties.

At large recoil (low $q^{2}$) and in the heavy-quark symmetry limit
($m_{b}/\Lambda_{QCD}\to\infty$), the $B\to\pi\ell\nu$ and $B_{s}\to K\ell\nu$
processes are each described by a single independent form factor as follows
Beneke:2000wa :

$\displaystyle f_{0}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{m_{B_{(s)}}^{2}-q^{2}}{m_{B_{(s)}}^{2}}f_{+}(q^{2})\,.$
(54)

This expression reduces to the kinematic constraint
$f_{+}(q^{2}=0)=f_{0}(q^{2}=0)$ at $q^{2}=0$. Figure 14, right, plots the
ratio $f_{0}(q^{2})/f_{+}(q^{2})$ for the full kinematic range. The results
are similar for $B\to\pi\ell\nu$ and $B_{s}\to K\ell\nu$. They agree exactly
with the prediction from Eq. (54) at $q^{2}=0$ by construction because we
imposed the kinematic constraint in our preferred $z$-fit, but are consistent
with heavy-quark-symmetry expectations throughout the low $q^{2}$ region.

Table 11:  Central values, errors, and correlation matrix for the parameters of our preferred fit to the $B\to\pi\ell\nu$ form factors $f_{+}$ and $f_{0}$ to the BCL $z$-parametrization in Eqs. (44) and (45) using $t_{0}=t_{\rm opt}\equiv\left(M_{B}+M_{\pi}\right)\left(\sqrt{M_{B}}-\sqrt{M_{\pi}}\right)^{2}$ in Eq. (39) and pole mass $M_{B^{*}}=5.3252(4)$ GeV in $f_{+}(q^{2})$. |  | Correlation matrix
---|---|---
| value | $b^{(0)}_{+}$ | $b^{(1)}_{+}$ | $b^{(2)}_{+}$ | $b^{(0)}_{0}$ | $b^{(1)}_{0}$ | $b^{(2)}_{0}$
$b^{(0)}_{+}$ | 0.412(39) | 1.000 | 0.337 | -0.076 | 0.679 | 0.045 | 0.100
$b^{(1)}_{+}$ | -0.511(184) | 0.337 | 1.000 | 0.150 | 0.222 | 0.698 | 0.581
$b^{(2)}_{+}$ | -0.524(612) | -0.076 | 0.150 | 1.000 | 0.029 | 0.436 | 0.659
$b^{(0)}_{0}$ | 0.520(60) | 0.679 | 0.222 | 0.029 | 1.000 | -0.258 | -0.224
$b^{(1)}_{0}$ | -1.657(182) | 0.045 | 0.698 | 0.436 | -0.258 | 1.000 | 0.564
$b^{(2)}_{0}$ | 2.146(682) | 0.100 | 0.581 | 0.659 | -0.224 | 0.564 | 1.000
Table 12:  Central values, errors, and correlation matrix for the parameters of our preferred fit to the $B_{s}\to K\ell\nu$ form factors $f_{+}$ and $f_{0}$ to the BCL $z$-parametrization in Eqs. (44) and (46) using $t_{0}=t_{\rm opt}\equiv\left(M_{B_{s}}+M_{K}\right)\left(\sqrt{M_{B_{s}}}-\sqrt{M_{K}}\right)^{2}$ in Eq. (39) and pole masses $M_{B^{*}}=5.3252(4)$ GeV and $M_{B^{*}}(0^{+})=5.63$ GeV in $f_{+}(q^{2})$ and $f_{0}(q^{2})$, respectively. |  | Correlation matrix
---|---|---
| value | $b^{(0)}_{+}$ | $b^{(1)}_{+}$ | $b^{(2)}_{+}$ | $b^{(0)}_{0}$ | $b^{(1)}_{0}$ | $b^{(2)}_{0}$
$b^{(0)}_{+}$ | 0.338(24) | 1.000 | 0.255 | 0.146 | 0.873 | 0.603 | 0.423
$b^{(1)}_{+}$ | -1.161(192) | 0.255 | 1.000 | 0.823 | 0.311 | 0.954 | 0.770
$b^{(2)}_{+}$ | -0.458(1.009) | 0.146 | 0.823 | 1.000 | 0.346 | 1.060 | 0.901
$b^{(0)}_{0}$ | 0.210(17) | 0.873 | 0.311 | 0.346 | 1.000 | 0.556 | 0.479
$b^{(1)}_{0}$ | -0.169(202) | 0.603 | 0.954 | 1.060 | 0.556 | 1.000 | 0.965
$b^{(2)}_{0}$ | -1.235(880) | 0.423 | 0.770 | 0.901 | 0.479 | 0.965 | 1.000

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.

##  VI Phenomenological applications

In this section we present two phenomenological applications of our form-
factor results.

First, in Sec. VI.1, we use our results for the $B\to\pi\ell\nu$ form factors
to determine the CKM matrix element $|V_{ub}|$. We fit recent experimental
measurements of the $B\to\pi\ell\nu$ differential branching fraction to the
$z$-parametrization to obtain the slope $b^{(1)}_{+}/b^{(0)}_{+}$ and
curvature $b^{(2)}_{+}/b^{(0)}_{+}$. Confirming that the lattice and
experimental shapes are indeed consistent, we then perform a combined $z$-fit
of our numerical $B\to\pi\ell\nu$ form-factor data with the experimental
measurements to obtain a model-independent determination of $|V_{ub}|$. This
method can also be applied to the decay $B_{s}\to K\ell\nu$, once it has been
observed experimentally, to provide an alternate determination of $|V_{ub}|$.

Next, in Sec. VI.2, we make predictions for Standard-Model observables for the
decay processes $B\to\pi\ell\nu$ and $B_{s}\to K\ell\nu$ for both
$\ell=\mu,\tau$ final-state charged leptons. (Here we use $\mu$ to indicate
both muon and electron final states, for which the Standard-Model predictions
are indistinguishable at the current level of precision.) We show results for
the differential branching fractions, forward-backward asymmetries, and
$\mu/\tau$ ratios (which are independent of $|V_{ub}|$). We only calculate
observables that depend upon $|V_{ub}|$ for $B_{s}\to K\ell\nu$ decays, using
the value determined previously in Sec. VI.1. Once the experimental error on
the branching fraction is commensurate with the theoretical form-factor
uncertainties, our $B_{s}\to K\ell\nu$ form-factor results will enable a
sufficiently precise determination of $|V_{ub}|$ to illuminate the discrepancy
between $|V_{ub}|$ from inclusive $B\to X_{u}\ell\nu$ and exclusive
$B\to\pi\ell\nu$ semileptonic decays.

###  VI.1 Determination of $|V_{ub}|$ from $B\to\pi\ell\nu$

For the determination of $|V_{ub}|$, we include the two most recent
experimental measurements from BaBar, which are the untagged 6-bin (“BaBar
2010”) and 12-bin (“BaBar 2012”) analyses in Refs. delAmoSanchez:2010af ;
Lees:2012vv . Because the 12-bin analysis uses more data and different
candidate selections and cuts than the 6-bin analysis, the statistical
correlations between the two data sets are quite small, and we treat the two
data sets as statistically uncorrelated. There are some correlations between
the systematic uncertainties in the two analyses, but these are estimated to
be sufficiently small that they have a tiny impact on $|V_{ub}|$
DingfelderPrivateComm . We therefore treat the two BaBar analyses as fully
independent. We also include the two most recent experimental measurements
from Belle, which are the untagged analysis in Ref. Ha:2010rf  (“Belle 2010”)
and the full-reconstruction tagged analysis in Ref. Sibidanov:2013rkk  (“Belle
2013”). The tagged and untagged data sets have little overlap. Further, the
dominant systematic error in the tagged analysis is from the uncertainty in
the tagging calibration, which is not present for the untagged analysis. Thus
we treat the Belle tagged and untagged analyses as independent. The BaBar and
Belle data sets are statistically independent. The only commonality to the
BaBar and Belle analyses is the use of the same event generation
DingfelderPrivateComm . Because the event generation is not a significant
source of uncertainty in the analyses, we treat the systematic uncertainties
as uncorrelated between the BaBar and Belle data sets.

We first fit the experimental measurements to the BCL $z$-parametrization to
obtain the shape parameters ($b_{+}^{(i)}/b_{+}^{(0)}$). For these fits, we do
not impose any constraint on the sum of the coefficients $\sum
B_{mn}b_{m}b_{n}$. Fits with truncation $K=2$ are sufficient to obtain good
$\chi^{2}/{\rm dof}$ for three of the four data sets, but we perform fits with
$K=3$ in order to enable comparison of both the slopes and curvatures with
those of the form factor $f_{+}^{B\pi}(q^{2})$ obtained in the previous
section. The numerical results for the $K=3$ fits to the individual
experimental data sets, as well from a combined fit to all experiments, are
given in Table 21. The fit to the BaBar 2010 data set has a somewhat large
$\chi^{2}/{\rm dof}$ that stems from the highest $q^{2}$ bin, for which the
error on the measured differential branching fraction is small but the central
value is low with respect to the other points. The inconsistency of the BaBar
2010 data leads the fit to all four experimental measurements to have a
somewhat low, but still reasonable, $p$-value of 5%. Figure 15 shows the
constraints on the slope ($b_{+}^{(1)}/b_{+}^{(0)}$) versus curvature
($b_{+}^{(2)}/b_{+}^{(0)}$) from the different experimental measurements, as
well as from the combined fit to all four measurements. The three most recent
measurements agree at the 2$\sigma$ level, but display some tension with the
BaBar 2010 result. Combining the information from all four experimental
analyses improves the determination of the shape parameters significantly.

Because we do not impose any constraint on the sum of the coefficients $\sum
B_{mn}b_{m}b_{n}$, we can check to see whether the experimental data is
compatible with expectations from heavy-quark power counting for the size of
the series coefficients. Taking the determination of $|V_{ub}|=3.63(12)\times
10^{-3}$ from CKM unitarity UTfit , we find a value for $\sum
B_{mn}b_{m}b_{n}\sim 0.02$ from the fit to all experimental data. This is
consistent with the prediction from Eq. (51) taking a reasonable value for the
heavy-quark scale $\Lambda\sim 1.1$ GeV, and validates the prior central value
and width that we used to constrain $\sum B_{mn}b_{m}b_{n}$ in our preferred
$z$-fit of the lattice form factors in the previous section.

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 delAmoSanchez:2010af ;
Ha:2010rf ; Lees:2012vv ; Sibidanov:2013rkk  (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.

Finally, before we fit the experimental and lattice data together to obtain
$|V_{ub}|$, it is important to check that their shapes are consistent. Figure
15 also shows the determination of the slope and curvature from our
calculation of $f_{+}^{B\pi}(q^{2})$ (see Table 11). The shapes of the lattice
form factors and the experimental data are in good agreement, but the shape
(as well as the overall normalization) is determined more precisely by
experiment. This suggests that the error on $|V_{ub}|$ can be minimized by
performing a combined fit to the lattice and experimental data, as we now
show.

Table 22 shows the results for the BCL coefficients and $|V_{ub}|$ obtained
from a combined fit of the experimental measurements for the $B\to\pi\ell\nu$
differential branching fraction and the lattice determination of the form
factor $f_{+}^{B\pi}(q^{2})$, leaving the relative normalization $|V_{ub}|$ as
a free parameter to be determined in the fit. As in the experiment-only
$z$-fits above, we do not constrain the sum of the coefficients $\sum
B_{mn}b_{m}b_{n}$. We present results from separate fits to each experimental
data set, as well as from a fit including all experimental data. The results
for $|V_{ub}|$ from fits to the different experimental data sets agree within
about $1\sigma$, and the $p$-value of the $K=3$ fit to all data is 6%. We also
show results for truncations $K=3,4,5$ to study the uncertainty due to
truncating the expansion in $z$. The errors on $|V_{ub}|$ remain the same size
as the number of fit parameters increase, and the central value for the fit
including all experimental data is unchanged. We take our final result

$|V_{ub}|=3.61(32)\times 10^{-3}$ (55)

from the fit to all experimental data with $K=3$. The quoted error on
$|V_{ub}|$ is the total uncertainty, and includes both the theoretical error
from the form factor and the experimental error (as well as the uncertainty
from truncating the $z$-expansion). Figure 16 shows the preferred $K=3$ BCL
$z$-fit used to obtain $|V_{ub}|$ plotted as
$(1-q^{2}/m_{B^{*}}^{2})f_{+}(q^{2})$ vs. $z$ (left) and as
$\Delta{\mathcal{B}}/\Delta q^{2}$ vs. $q^{2}$ (right).

Figure 16: Model-independent determination of $|V_{ub}|$ from a combined fit
of experimental measurements of the $B\to\pi\ell\nu$ branching fraction
delAmoSanchez:2010af ; Ha:2010rf ; Lees:2012vv ; Sibidanov:2013rkk  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
$(1-q^{2}/m_{B^{*}}^{2})f_{+}(q^{2})$ vs. $z$ (where the experimental data
have been rescaled by the value of $|V_{ub}|$ determined in the fit), while
the right plot shows $\Delta{\mathcal{B}}/\Delta q^{2}$ vs. $q^{2}$ (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.

Although we cannot precisely disentangle the error contributions, we can
estimate the contribution to the error on $|V_{ub}|$ from the lattice form-
factor determination. Our most precise synthetic data point has a total
statistical plus systematic uncertainty of 8.4%. If we assume that this is the
lattice contribution to the 8.9% error $|V_{ub}|$ in Eq. (55), this suggests
that the experimental error contribution is approximately 2.8%.

The combined $z$-fit optimally combines the available information from lattice
and experiment in a model-independent manner, thereby providing a
determination of $|V_{ub}|$ that is both reliable and precise. We can quantify
this statement by comparing the error on $|V_{ub}|$ from the simultaneous fit
to the error obtained from the previously standard approach. One can determine
$|V_{ub}|$ by relating the measured partial branching fraction in an interval
$[q^{2}_{\text{min}},q^{2}_{\text{max}}]$ to the normalized partial decay rate
calculated from the form factor as follows:

$\Delta{\mathcal{B}}(q^{2}_{\rm min},q^{2}_{\rm
max})/|V_{ub}|^{2}=\tau_{0}\Delta\zeta(q^{2}_{\rm min},q^{2}_{\rm max}),$ (56)

where

$\Delta\zeta(q^{2}_{\rm min},q^{2}_{\rm
max})\equiv\frac{G_{F}^{2}}{24\pi^{3}}\int\limits_{q^{2}_{\textrm{min}}}^{q^{2}_{\textrm{max}}}dq^{2}{\vec{p}_{\pi}}^{\,3}|f_{+}(q^{2})|^{2}\,.$
(57)

The momentum range $q^{2}>16$ GeV2 is typically used for comparison with
lattice-QCD calculations because it is directly accessible in simulations and
avoids (or at least minimizes) the need for an extrapolation. From our
preferred BCL parametrization (cf. Table 11) we obtain

$\Delta\zeta^{B\pi}(16~{}{\rm GeV}^{2},q^{2}_{\rm max})=1.77(34)\ {\rm
ps}^{-1}\,.$ (58)

Table 13 shows the determinations of $|V_{ub}|$ obtained from combining
$\Delta\zeta(16~{}{\rm GeV}^{2},q^{2}_{\rm max})$ above with the different
experimental measurements of $\Delta{\mathcal{B}}(16~{}{\rm
GeV}^{2},q^{2}_{\rm max})$, and with their average, via Eq. (56). The results
agree with those from the simultaneous $z$-fits, but with larger errors. In
particular, the error on $|V_{ub}|$ obtained using the average
$\Delta{\mathcal{B}}(16~{}{\rm GeV}^{2},q^{2}_{\rm max})$ from all experiments
is 10.0%, to be compared with the 8.5% error from our combined $z$-fit.
Separate $z$-fits to the lattice and experimental data lead to a similar error
on $|V_{ub}|$. The error on the normalization of the form factor $b_{0}$ in
Table 11 is 9.4%, while the error on the normalization of the experimental
branching fraction from the $K=3$ fit to all experimental data $b_{0}|V_{ub}|$
is 2.2%. Adding these in quadrature leads to a total error of 9.7%. Thus we
conclude that the combined $z$-fit of all lattice and experimental data is
indeed the best approach for minimizing the uncertainty on $|V_{ub}|$.

Table 13:  Determinations of $|V_{ub}|$ from a comparison of the measured $B\to\pi\ell\nu$ partial branching fractions with the normalized partial decay rate $\Delta\zeta^{B\pi}(16~{}{\rm GeV}^{2},q^{2}_{\rm max})=1.77(34)$ calculated from our preferred BCL paramterization of the vector form factor $f_{+}^{B\pi}(q^{2})$. | $\Delta{\mathcal{B}}(16~{}{\rm GeV}^{2},q^{2}_{\rm max})\times 10^{7}$ | $|V_{ub}|\times 10^{3}$
---|---|---
All | 368(19) | 3.69(37)
BaBar 2010 delAmoSanchez:2010af  | 319(34) | 3.44(38)
BaBar 2012 Lees:2012vv  | 369(32) | 3.70(39)
Belle 2010 Ha:2010rf  | 398(30) | 3.84(40)
Belle 2013 Sibidanov:2013rkk  | 386(51) | 3.78(44)

###  VI.2 Standard-Model predictions for $B\to\pi\ell\nu$ and $B_{s}\to
K\ell\nu$ observables

The Standard-Model differential decay rate for $B_{(s)}\to P\ell\nu$ is given
in Eq. (1). Using the experimentally measured lepton and meson masses
Beringer:2012zz , we obtain predictions for the differential decay rate
divided by $|V_{ub}|^{2}$. These are plotted for the muon and $\tau$-lepton
final states in Fig. 17, where we use “muon” to denote decays to either of the
light charged leptons ($\ell=\mu,e$) throughout this section. Integrating the
differential decay rates over the kinematically-allowed $q^{2}$ range
gives222In practice, the full kinematic range may not be accessible
experimentally, in which case the limits of integration here and throughout
this section will need to be changed accordingly.

$\displaystyle{\Gamma(B\to\pi\mu\nu)/|V_{ub}|^{2}}$ $\displaystyle=$
$\displaystyle{6.2(2.5)}\ {\rm ps}^{-1}\,,$ (59)
$\displaystyle{\Gamma(B\to\pi\tau\nu)/|V_{ub}|^{2}}$ $\displaystyle=$
$\displaystyle{4.3(1.2)}\ {\rm ps}^{-1}\,,$ (60) $\displaystyle\Gamma(B_{s}\to
K\mu\nu)/|V_{ub}|^{2}$ $\displaystyle=$ $\displaystyle 4.55(1.08)\ {\rm
ps}^{-1}\,,$ (61) $\displaystyle\Gamma(B_{s}\to K\tau\nu)/|V_{ub}|^{2}$
$\displaystyle=$ $\displaystyle 3.52(0.60)\ {\rm ps}^{-1}\,,$ (62)

with errors of about 25–40% and 15–30% for the $\mu$ and $\tau$ final states,
respectively. We also use the determination of $|V_{ub}|$ from our calculation
of the $B\to\pi\ell\nu$ form factors (Eq. (55)) to make predictions for the
$B_{s}\to K\ell\nu$ differential branching fractions for $\ell=\mu,\tau$.
These are plotted in Fig. 18. For comparison, we also show the prediction for
$d{\mathcal{B}}/dq^{2}$ using the determination of $|V_{ub}|$ from inclusive
$B\to X_{u}\ell\nu$ decay HFAGSemiPDG13 . The form-factor uncertainties are
sufficiently small for $q^{2}\mathrel{\hbox to0.0pt{\lower
3.0pt\hbox{$\mathchar 536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar
318\relax$}}13{\rm\ GeV}^{2}$ that, given an experimental measurement of the
branching fraction in this region with commensurate precision, one can
distinguish between the curves corresponding to $|V_{ub}|_{\rm excl.}$ and
$|V_{ub}|_{\rm incl.}$. Thus we anticipate that $B_{s}\to K\ell\nu$
semileptonic decay will eventually play an important role in addressing the
current “$|V_{ub}|$ puzzle.”

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 $d{\mathcal{B}}/dq^{2}$ 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
HFAGSemiPDG13 . In these plots, the outer error band denotes the total
uncertainty in $d{\mathcal{B}}/dq^{2}$, while the inner error band removes the
error contribution from $|V_{ub}|$.

Semileptonic decays to $\tau$ leptons may be particularly sensitive to new
physics associated with electroweak symmetry breaking due to the large $\tau$
mass, or more generally sensitive to any Standard-Model extensions with new
scalar currents. Moreover, the ratio of $\mu/\tau$ differential decay rates
Meissner:2013pba

${\mathcal{R}}^{\tau/\mu}_{{P}}(q^{2}){\equiv}\frac{d\Gamma({B_{(s)}\to
P}\tau\nu)/dq^{2}}{d\Gamma({B_{(s)}\to P}\mu\nu)/dq^{2}}$ (63)

provides a precise test of the Standard Model that is independent of the CKM
matrix element $|V_{ub}|$. Figure 19 shows the predictions for the ratios of
differential branching fractions using our determinations of the
$B\to\pi\ell\nu$ and $B_{s}\to K\ell\nu$ form factors in Tables 11 and 12.
Integrating over the kinematically allowed ranges, we obtain the following
Standard-Model predictions for $R{{}^{\tau/\mu}_{P}}\equiv\Gamma({B_{(s)}\to
P}\tau\nu)/\Gamma({B_{(s)}\to P}\mu\nu)$:

$\displaystyle R{{}^{\tau/\mu}_{\pi}}$ $\displaystyle=$
$\displaystyle{0.69(19)}\,,$ (64) $\displaystyle R{{}^{\tau/\mu}_{K}}$
$\displaystyle=$ $\displaystyle{0.77(12)}\,.$ (65)

Figure 19: Standard-Model ratio of differential branching fractions
${\mathcal{R}}^{\tau/\mu}_{P}(q^{2})$ using our determinations of the
$B\to\pi\ell\nu$ and $B_{s}\to K\ell\nu$ form factors in Tables 11 and 12.

The three-body final state in ${B_{(s)}\to P}\ell\nu$ decay also enables one
to construct and study observables that depend on the kinematics of the decay
products. Such angular observables are particularly sensitive to possible
right-handed currents. An example is the forward-backward difference333This
quantity is sometimes referred to as an “asymmetry” in the literature, but
does not satisfy the convention that its magnitude is bounded by unity.

${\cal A}_{FB}^{{B_{(s)}\to
P\ell\nu}}(q^{2})\equiv\left[\int^{1}_{0}-\int^{0}_{-1}\right]d\cos\theta_{\ell}\frac{d^{2}\Gamma({\overline{B}_{(s)}\to
P}\ell\nu)}{dq^{2}d\cos\theta_{\ell}}\,$ (66)

where $\theta_{\ell}$ is the angle between the charged-lepton and
$B_{(s)}$-meson momenta in the $q^{2}$ rest frame. In the Standard Model, the
forward-backward difference is given by Meissner:2013pba

$\displaystyle{\cal A}_{FB}^{{B_{(s)}\to P\ell\nu}}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{G_{F}^{2}|V_{ub}|^{2}}{32\pi^{3}M_{B_{s}}}\left(1-\frac{m_{\ell}^{2}}{q^{2}}\right)^{2}|\vec{p}_{K}|^{2}$
(67) $\displaystyle{\times}$
$\displaystyle\frac{m_{\ell}^{2}}{q^{2}}(M_{B_{s}}^{2}-M_{K}^{2})f_{+}(q^{2})f_{0}(q^{2})\,.$

Figure 20 shows the Standard-Model predictions for ${\cal A}_{FB}^{{B_{s}\to
K\ell\nu}}(q^{2})$ with $\ell=\mu,\tau$ using our determination of $|V_{ub}|$
from $B\to\pi\ell\nu$ decay (Eq. (55)) and the inclusive determination of
$|V_{ub}|$ from $B\to X_{u}\ell\nu$ decay HFAGSemiPDG13 . Again, the
theoretical errors are sufficiently small at large $q^{2}$ that an
experimental measurement with commensurate precision could distinguish between
the two predictions. Integrating over the full kinematic ranges we obtain

$\displaystyle{\int_{m_{\mu}^{2}}^{q_{\rm
max}^{2}}dq^{2}{\mathcal{A}}^{B\to\pi\mu\nu}_{\rm FB}(q^{2})/|V_{ub}|^{2}}$
$\displaystyle=$ $\displaystyle{0.028(19)\ {\rm ps}^{-1}}\,,$ (68)
$\displaystyle{\int_{m_{\tau}^{2}}^{q_{\rm
max}^{2}}dq^{2}{\mathcal{A}}^{B\to\pi\tau\nu}_{\rm FB}(q^{2})/|V_{ub}|^{2}}$
$\displaystyle=$ $\displaystyle{1.08(35)\ {\rm ps}^{-1}}\,,$ (69)
$\displaystyle\int_{m_{\mu}^{2}}^{q_{\rm
max}^{2}}dq^{2}{\mathcal{A}}^{B_{s}\to K\mu\nu}_{\rm FB}(q^{2})/|V_{ub}|^{2}$
$\displaystyle=$ $\displaystyle 0.0175(87)\ {\rm ps}^{-1}\,,$ (70)
$\displaystyle\int_{m_{\tau}^{2}}^{q_{\rm
max}^{2}}dq^{2}{\mathcal{A}}^{B_{s}\to K\tau\nu}_{\rm FB}(q^{2})/|V_{ub}|^{2}$
$\displaystyle=$ $\displaystyle 0.93(18)\ {\rm ps}^{-1}\,,$ (71)

where the $\mu$ results are much smaller than the $\tau$ results due to
helicity suppression from the small muon mass.

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 ${\cal A}_{FB}$ 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
HFAGSemiPDG13 . In these plots, the outer error band denotes the total
uncertainty in $d{\mathcal{B}}/dq^{2}$, while the inner error band removes the
error contribution from $|V_{ub}|$.

Figure 21: Standard-Model predictions for the normalized forward-backward
asymmetry $\overline{\cal A}_{FB}$ for decays to muon (left) and $\tau$-lepton
(right) final states using our form-factor determinations in Tables 11 and 12.

The normalized forward-backward asymmetry is particularly interesting because
it removes the ambiguity from $|V_{ub}|$, and the theoretical form-factor
uncertainties cancel to some degree:

$\overline{\mathcal{A}}^{{B_{(s)}\to P\ell\nu}}_{\rm
FB}{\equiv}\frac{\int_{m^{2}_{\ell}}^{q_{\rm
max}^{2}}dq^{2}{\mathcal{A}}^{{B_{(s)}\to P\ell\nu}}_{\rm
FB}(q^{2})}{\int_{m^{2}_{\ell}}^{q_{\rm max}^{2}}dq^{2}\,d\Gamma({B_{(s)}\to
P}\ell\nu)/dq^{2}}\,.$ (72)

The Standard-Model predictions for $\overline{\cal A}_{FB}^{{B_{(s)}\to
P\ell\nu}}(q^{2})$ with $\ell=\mu,\tau$ are given in Fig. 21. The percentage
uncertainty in $\overline{\cal A}_{FB}^{{B_{(s)}\to P\ell\nu}}(q^{2})$ is
indeed smaller than in ${\cal A}_{FB}^{{B_{(s)}\to P\ell\nu}}(q^{2})$.
Integrating $\overline{\cal A}_{FB}^{{B_{(s)}\to P\ell\nu}}(q^{2})$ over the
full kinematic range we obtain

$\displaystyle\overline{\mathcal{A}}^{{B\to\pi\mu\nu}}_{\rm FB}$
$\displaystyle=$ $\displaystyle 0.0044(13)\,,$ (73)
$\displaystyle\overline{\mathcal{A}}^{{B\to\pi\tau\nu}}_{\rm FB}$