Branching fraction and form-factor shape measurements of exclusive charmless semileptonic $B$ decays, and determination of $|V_{ub}|$

A precise measurement of the CKM matrix CKM element $|V_{ub}|$ will improve our quantitative understanding of weak interactions and CP violation in the Standard Model. The value of $|V_{ub}|$ can be determined by the measurement of the partial branching fractions of exclusive charmless semileptonic $B$ decays since the rate for decays that involve a scalar meson is proportional to $|V_{ub}f_{+}(q^{2})|^{2}$. Here, the form factor $f_{+}(q^{2})$ depends on $q^{2}$, the square of the momentum transferred to the lepton-neutrino pair. Values of $f_{+}(q^{2})$ can be calculated at small $q^{2}$ ($\lesssim$ 16 $\mathrm{\,Ge\kern-1.00006ptV^{2}\!}$) using Light Cone Sum Rules (LCSR) LCSR ; LCSR2 ; singlet and at large $q^{2}$ ($\gtrsim$ 16 $\mathrm{\,Ge\kern-1.00006ptV^{2}\!}$) from unquenched Lattice QCD (LQCD) HPQCD06 ; FNAL . Extraction of the $f_{+}(q^{2})$ form-factor shapes from exclusive decays PlusCC such as $B^{0}\rightarrow\pi^{-}\ell^{+}\nu$ Jochen ; Simard ; Belle , $B^{+}\rightarrow\pi^{0}\ell^{+}\nu$ Jochen , $B^{+}\rightarrow\omega\ell^{+}\nu$ Wulsin and $B^{+}\rightarrow\eta^{(\prime)}\ell^{+}\nu$ Simard may be used to test these theoretical predictions PDG10 . Measurements of the branching fractions (BF) of all these decays will also improve our knowledge of the composition of charmless semileptonic decays. This input can be used to reduce the large systematic uncertainty in $|V_{ub}|$ due to the poorly known $b\rightarrow u\ell\nu$ signal composition in inclusive semileptonic $B$ decays. It will also help to constrain the size of the gluonic singlet contribution to form factors for the $B^{+}\rightarrow\eta^{(\prime)}\ell^{+}\nu$ decays singlet ; singlet2 .

In this paper, we present measurements of the partial BFs $\Delta{\cal B}(B^{0}\rightarrow\pi^{-}\ell^{+}\nu,q^{2})$ in 12 bins of $q^{2}$, $\Delta{\cal B}(B^{+}\rightarrow\pi^{0}\ell^{+}\nu,q^{2})$ in 11 bins of $q^{2}$, $\Delta{\cal B}(B^{+}\rightarrow\omega\ell^{+}\nu,q^{2})$ and $\Delta{\cal B}(B^{+}\rightarrow\eta\ell^{+}\nu,q^{2})$ in five bins of $q^{2}$, as well as the BF ${\cal B}(B^{+}\rightarrow\eta^{\prime}\ell^{+}\nu)$. From these distributions, we extract the total BFs for each of the five decay modes. Values of these BFs were previously reported in Refs. Jochen ; Simard ; Belle ; Wulsin , and references therein. In this work, we carry out an untagged analysis (i.e. the second $B$ meson is not explicitly reconstructed) with the loose neutrino reconstruction technique Cote whereby the selections on the variables required to reconstruct the neutrino are much looser than usual. This results in a large candidate sample. Concerning the $B^{+}\rightarrow\pi^{0}\ell^{+}\nu$ and $B^{+}\rightarrow\omega\ell^{+}\nu$ decay modes, this is the first analysis using this technique.

We assume isospin symmetry to hold, and combine the data of the $B^{+}\rightarrow\pi^{0}\ell^{+}\nu$ and $B^{0}\rightarrow\pi^{-}\ell^{+}\nu$ channels thereby leading to a large increase, of the order of 34%, in the effective number of $B^{0}\rightarrow\pi^{-}\ell^{+}\nu$ events available for study. We refer to such events as $B\rightarrow\pi\ell^{+}\nu$ decays. The values of the BFs obtained in the present work are based on the use of the most recent BFs and form-factor shapes for all decay channels in our study. In particular, the subsequent improved treatment of the distributions that describe the combination of resonant and nonresonant $b\rightarrow u\ell\nu$ decays results in an increase of 3.5% in the total BF value of the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu$ decays. This increase is significant in view of the total uncertainty of 5.1% obtained in the measurement of this BF.

We now optimize our selections over the entire fit region instead of the signal-enhanced region, as was done previously Simard . The ensuing tighter selections produce a data set with a better signal to background ratio and higher purity in the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu$ and $B^{+}\rightarrow\eta^{(\prime)}\ell^{+}\nu$ decays. As a result, we can now investigate the $B^{+}\rightarrow\eta^{(\prime)}\ell^{+}\nu$ decays over their full $q^{2}$ ranges. The present analysis of the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu$ decay channel makes use of the full BABAR data set compared to only a subset in Ref. Jochen . As for the $B^{+}\rightarrow\omega\ell^{+}\nu$ decay channel, it uses the unfolded values of the partial BFs and a selection procedure that is significantly different from the one in Ref. Wulsin . The unfolding process is used to obtain the distribution of the true values of $q^{2}$ by applying the inverse of the detector response matrix to the distribution of the measured values of $q^{2}$. Each element of this matrix is constructed in MC simulation for each bin of $q^{2}$ as the ratio of the number of true events to the total number of reconstructed events. The current work provides results for five decay channels using the same analysis method.

In this work, we compare the values of $\mbox{$\Delta\cal B$}(q^{2})$ for the $B\rightarrow\pi\ell^{+}\nu$ mode to form-factor calculations LCSR ; LCSR2 ; HPQCD06 ; FNAL in restricted $q^{2}$ ranges to obtain values of $|V_{ub}|$. Values of $|V_{ub}|$ with a smaller total uncertainty can also be obtained from a simultaneous fit to the $B\rightarrow\pi\ell^{+}\nu$ experimental data over the full $q^{2}$ range and the FNAL/MILC lattice QCD predictions FNAL . Such values were recently obtained by BABAR Jochen ($|V_{ub}|=(2.95\pm 0.31)\times 10^{-3}$) and Belle Belle ($|V_{ub}|=(3.43\pm 0.33)\times 10^{-3}$). These results are consistent at the 2$\sigma$ level, when taking into account the correlations, but display a tension with respect to the value of $|V_{ub}|$ measured PDG10 in inclusive semileptonic $B$ decays, $|V_{ub}|=(4.27\pm 0.38)\times 10^{-3}$. This study attempts to resolve the tension by analyzing the data using the most recent values of BFs and form factors.

We use a sample of $462\times 10^{6}$ $B\bar{B}$ pairs, corresponding to an integrated luminosity of 416.1 $\mbox{\,fb}^{-1}$, collected at the $\Upsilon{(4S)}$ resonance with the BABAR detector ref:babar at the PEP-II asymmetric-energy $e^{+}e^{-}$ storage rings. A sample of 43.9 $\mbox{\,fb}^{-1}$ collected approximately 40 $\mathrm{\,Me\kern-1.00006ptV}$  below the $\Upsilon{(4S)}$ resonance (denoted “off-resonance data”) is used to study contributions from $e^{+}e^{-}\rightarrow$ $u\bar{u}/d\bar{d}/s\bar{s}/c\bar{c}/\tau^{+}\tau^{-}$ (continuum) events. Detailed Monte Carlo (MC) simulations are used to optimize the signal selections, estimate the signal efficiencies, obtain the shapes of the signal and background distributions and determine the systematic uncertainties associated with the BF values.

MC samples are generated for $\Upsilon(4S)\rightarrow B\bar{B}$ events, continuum events, and dedicated signal samples containing $B^{0}\rightarrow\pi^{-}\ell^{+}\nu$, $B^{+}\rightarrow\pi^{0}\ell^{+}\nu$, $B^{+}\rightarrow\omega\ell^{+}\nu$ and $B^{+}\rightarrow\eta^{(\prime)}\ell^{+}\nu$ signal decays, separately. These signal MC events are produced with the FLATQ2 generator bad809 . The $f_{+}(q^{2})$ shape used in this generator is adjusted by reweighting the generated events. For the $B\rightarrow\pi\ell^{+}\nu$ decays, the signal MC events are reweighted to reproduce the Boyd-Grinstein-Lebed (BGL) parametrization BGL , where the parameters are taken from Ref. Simard . For the $B^{+}\rightarrow\omega\ell^{+}\nu$ decays, the events are reweighted to reproduce the Ball parametrization Ball05 . For the $B^{+}\rightarrow\eta^{(\prime)}\ell^{+}\nu$ decays, the signal MC events are reweighted to reproduce the Becirevic-Kaidalov (BK) parametrization BK , where the parameter $\alpha_{BK}=0.52\pm 0.04$ gave a reasonable fit to the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu$ and $B^{+}\rightarrow\eta\ell^{+}\nu$ data of Ref. Simard . The BABAR detector’s acceptance and response are simulated using the GEANT4 package Geant4 .

To reconstruct the decays $B^{0}\rightarrow\pi^{-}\ell^{+}\nu$, $B^{+}\rightarrow\pi^{0}\ell^{+}\nu$, $B^{+}\rightarrow\omega\ell^{+}\nu$ and $B^{+}\rightarrow\eta^{(\prime)}\ell^{+}\nu$, we first reconstruct the final state meson. The $\omega$ meson is reconstructed in the $\omega\rightarrow\pi^{+}\pi^{-}\pi^{0}$ decay channel. The $\eta$ meson is reconstructed in the $\eta\rightarrow\gamma\gamma$ ($\eta(\gamma\gamma)$) and $\eta\rightarrow\pi^{+}\pi^{-}\pi^{0}$ ($\eta(3\pi)$) decay channels while the $\eta^{\prime}$ is reconstructed in the $\eta^{\prime}\rightarrow\eta\pi^{+}\pi^{-}$ decay channel, followed by the $\eta\rightarrow\gamma\gamma$ decay ($\eta^{\prime}(\gamma\gamma)$). The $\eta^{\prime}\rightarrow\rho^{0}\gamma$ decay channel suffers from large backgrounds and we do not consider it in the present work.

Event reconstruction with the BABAR detector is described in detail elsewhere ref:babar . Electrons and muons are mainly identified by their characteristic signatures in the electromagnetic calorimeter and the muon detector, respectively, while charged hadrons are identified and reconstructed using the silicon vertex tracker, the drift chamber and the Cherenkov detector. The photon and charged particle tracking reconstruction efficiencies are corrected using various control samples. The average electron and muon reconstruction efficiencies are 93% and 70%, respectively, while the corresponding probabilities that a pion is identified as a lepton are less than $0.2$% and less than $1.5$%, respectively.

The neutrino four-momentum, $P_{\nu}=(|\vec{p}^{{}_{*}}_{miss}|,\vec{p}^{{}_{*}}_{miss})$, is inferred from the difference between the momentum of the colliding-beam particles $\vec{p}^{{}_{*}}_{beams}$ and the vector sum of the momenta of all the particles detected in the event $\vec{p}^{{}_{*}}_{tot}$, such that $\vec{p}^{{}_{*}}_{miss}=\vec{p}^{{}_{*}}_{beams}-\vec{p}^{{}_{*}}_{tot}$. All variables with an asterisk are given in the $\Upsilon{(4S)}$ frame. To evaluate $E_{tot}$, the total energy of all detected particles, we assume zero mass for all neutral candidates, and we use the known masses for the charged particles identified in the event. If the particle is not identified, its mass is assumed to be that of a pion.

In this analysis, we calculate the momentum transfer squared as $q^{2}=(P_{B}-P_{meson})^{2}$ instead of $q^{2}=(P_{\ell}+P_{\nu})^{2}$, where $P_{B}$, $P_{meson}$ and $P_{\ell}$ are the four-momenta of the $B$ meson, of the $\pi$, $\omega$, $\eta$ or $\eta^{\prime}$ meson, and of the lepton, respectively, evaluated in the $\Upsilon{(4S)}$ frame. With this choice, the value of $q^{2}$ is unaffected by any misreconstruction of the neutrino. To maintain this advantage, $P_{B}$ must be evaluated without any reference to the neutrino. It has an effective value since the magnitude of the 3-momentum $\vec{p}^{{}_{*}}_{B}$ is determined from the center-of-mass energy and the known $B$ meson mass but the direction of the $B$ meson cannot be measured. It can only be estimated.

To do this, we first combine the lepton with a $\pi$, $\omega$, $\eta$ or $\eta^{\prime}$ meson to form the so-called $Y$ pseudoparticle such that $P_{Y}=P_{\ell}+P_{meson}$. The angle $\theta_{BY}$, between the $Y$ and $B$ momenta in the $\Upsilon{(4S)}$ frame, can be determined under the assumption that the only unobserved decay product is a neutrino, i.e., $B\rightarrow Y\nu$. In this frame, the $Y$ momentum, the $B$ momentum and the angle $\theta_{BY}$ define a cone with the $Y$ momentum as its axis and with a true $B$ momentum lying somewhere on the surface of the cone. The $B$ rest frame is thus known up to an azimuthal angle $\psi$ about the $Y$ momentum. The value of $q^{2}$ is then computed, as explained in Ref. DstrFF , as the average of four $q^{2}$ values corresponding to four possible angles, $\psi$, $\psi+\pi/2$, $\psi+\pi$, $\psi+3\pi/2$ rad, where the angle $\psi$ is chosen randomly. The four values of $q^{2}$ are weighted by the factor $\sin^{2}\theta_{B}$, $\theta_{B}$ being the angle between the $B$ direction and the beam direction in the $\Upsilon{(4S)}$ frame. This weight is needed since $B\bar{B}$ production follows a $\sin^{2}\theta_{B}$ distribution in the $\Upsilon{(4S)}$ frame. We require that $|\cos\theta_{BY}|\leq 1$. We correct for the reconstruction effects on the measured values of $q^{2}$ (the $q^{2}$ resolution is approximately 0.6 $\mathrm{\,Ge\kern-1.00006ptV^{2}\!}$) by applying an unregularized unfolding algorithm to the measured $q^{2}$ spectra Cowan .

The selections of the candidate events are determined in MC simulation by maximizing the ratio $S/\sqrt{(S+B)}$ over the entire fit region, where $S$ is the number of correctly reconstructed signal events and $B$ is the total number of background events. The continuum background is suppressed by requiring the ratio of second to zeroth Fox-Wolfram moments FW to be smaller than 0.5. Radiative Bhabha and two-photon processes are rejected by requirements on the number of charged particle tracks and neutral calorimeter clusters BhabhaVeto . To ensure all track momenta are well measured, their polar angles are required to lie between 0.41 and 2.46 rad with respect to the electron beam direction (the acceptance of the detector). For all decays, we demand the momenta of the lepton and meson candidates to be topologically compatible with a real signal decay by requiring that a mass-constrained geometrical vertex fit mass of the tracks associated with the two particles gives a $\chi^{2}$ probability greater than 1%. In the fit, the external constraints such as reconstructed tracks are treated first, followed by all four-momenta conservation constraints. Finally, at each vertex, the geometric constraints and the mass are combined. These combined constraints are applied consecutively.

To reduce the number of unwanted leptons and secondary decays such as $D\rightarrow X\ell\nu$, ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$, $\tau$ and kaon decays, the minimum transverse momentum is 50 $\mathrm{\,Me\kern-1.00006ptV}$ for all leptons and 30 $\mathrm{\,Me\kern-1.00006ptV}$ for all photons, and all electron (muon) tracks are required to have momenta greater than 0.5 (1.0) $\mathrm{\,Ge\kern-1.00006ptV}$ in the laboratory frame. The momenta of the lepton and the meson are further restricted to enhance signal over background. We require the following:

• •

  for $B\rightarrow\pi\ell^{+}\nu$ decays:
  $|\vec{p}^{{}_{*}}_{\ell}|>2.2$ $\mathrm{\,Ge\kern-1.00006ptV}$ or $|\vec{p}^{{}_{*}}_{\pi}|>1.3$ $\mathrm{\,Ge\kern-1.00006ptV}$ 
  or $|\vec{p}^{{}_{*}}_{\ell}|+|\vec{p}^{{}_{*}}_{\pi}|>2.8$ $\mathrm{\,Ge\kern-1.00006ptV}$;

• •

  for $B^{+}\rightarrow\omega\ell^{+}\nu$ decays:
  $|\vec{p}^{{}_{*}}_{\ell}|>2.0$ $\mathrm{\,Ge\kern-1.00006ptV}$ or $|\vec{p}^{{}_{*}}_{\omega}|>1.3$ $\mathrm{\,Ge\kern-1.00006ptV}$ 
  or $|\vec{p}^{{}_{*}}_{\ell}|+|\vec{p}^{{}_{*}}_{\omega}|>2.65$ $\mathrm{\,Ge\kern-1.00006ptV}$;

• •

  for $B^{+}\rightarrow\eta\ell^{+}\nu$ decays:
  $|\vec{p}^{{}_{*}}_{\ell}|>2.1$ $\mathrm{\,Ge\kern-1.00006ptV}$ or $|\vec{p}^{{}_{*}}_{\eta}|>1.3$ $\mathrm{\,Ge\kern-1.00006ptV}$ 
  or $|\vec{p}^{{}_{*}}_{\ell}|+|\vec{p}^{{}_{*}}_{\eta}|>2.8$ $\mathrm{\,Ge\kern-1.00006ptV}$;

• •

  for $B^{+}\rightarrow\eta^{\prime}\ell^{+}\nu$ decays:
  $|\vec{p}^{{}_{*}}_{\ell}|>2.0$ $\mathrm{\,Ge\kern-1.00006ptV}$ or $|\vec{p}^{{}_{*}}_{\eta^{\prime}}|>1.65$ $\mathrm{\,Ge\kern-1.00006ptV}$ 
  or $0.69|\vec{p}^{{}_{*}}_{\ell}|+|\vec{p}^{{}_{*}}_{\eta^{\prime}}|>2.4$ $\mathrm{\,Ge\kern-1.00006ptV}$.

These cuts primarily reject background and reduce the signal efficiencies by less than 5%.

To remove ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$ decays, we reject any combination of two muons, including misidentified pions, if the two particles have an invariant mass consistent with the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass [3.07-3.13] $\mathrm{\,Ge\kern-1.00006ptV}$. We do not apply a specific ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ veto for ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow e^{+}e^{-}$ decays, since we find no evidence for any remaining such events in our data set. We restrict the reconstructed masses of the meson to lie in the interval:

• •

  for $B^{+}\rightarrow\pi^{0}\ell^{+}\nu$ decays: $0.115<m_{\pi^{0}}<0.150$ $\mathrm{\,Ge\kern-1.00006ptV}$,

• •

  for $B^{+}\rightarrow\omega\ell^{+}\nu$ decays: $0.760<m_{\omega}<0.805$ $\mathrm{\,Ge\kern-1.00006ptV}$,

• •

  for $B^{+}\rightarrow\eta\ell^{+}\nu$ decays: $0.51<m_{\eta}<0.57$ $\mathrm{\,Ge\kern-1.00006ptV}$,

• •

  for $B^{+}\rightarrow\eta^{\prime}\ell^{+}\nu$ decays: $0.92<m_{\eta^{\prime}}<0.98$ $\mathrm{\,Ge\kern-1.00006ptV}$.

Backgrounds are further reduced by $q^{2}$-dependent selections on the cosine of the angle, $\cos\theta_{thrust}$, between the thrust axes thrust of the $Y$ and of the rest of the event; on the polar angle, $\theta_{miss}$, associated with $\vec{p}_{miss}$; on the invariant missing mass squared, $m^{2}_{miss}=E^{2}_{miss}-|\vec{p}_{miss}|^{2}$, divided by twice the missing energy ($E_{miss}=E_{beams}-E_{tot}$); on the cosine of the angle, $\cos\theta_{\ell}$, between the direction of the virtual $W$ boson ($\ell$ and $\nu$ combined) boosted in the rest frame of the $B$ meson and the direction of the lepton boosted in the rest frame of the $W$ boson; and on L2, the momentum weighted Legendre monomial of order 2. The quantity $m^{2}_{miss}/2E_{miss}$ should be consistent with zero if a single neutrino is missing. The phrase “rest of the event” refers to all the particles left in the event after the lepton and the meson used to form the Y pseudoparticle are removed.

The $q^{2}$-dependent selections are shown in the panels on the left-hand side of Fig. 1, and their effects are illustrated in the panels on the right-hand side of the same figure, for $B^{0}\rightarrow\pi^{-}\ell^{+}\nu$ decays. A single vertical line indicates a fixed cut, a set of two vertical lines represents a $q^{2}$-dependent cut. The position of the two lines corresponds to the minimum and maximum values of the selection, shown in the left-hand side panels. The functions describing the $q^{2}$ dependence are given in Tables 8-13 of the Appendix for the five decays under study. For $B^{+}\rightarrow\eta\ell^{+}\nu$ decays, additional background is rejected by requiring that $|\cos\theta_{V}|<0.95$, where $\theta_{V}$ is the helicity angle of the $\eta$ meson bad809 .

The kinematic variables $\Delta E=(P_{B}\cdot P_{beams}-s/2)/\sqrt{s}$ and $m_{ES}=\sqrt{(s/2+\vec{p}_{B}\cdot\vec{p}_{beams})^{2}/E_{beams}^{2}-\vec{p}_{B}^{\,2}}$ are used in a fit to provide discrimination between signal and background decays. $\sqrt{s}$ is the center-of-mass energy of the colliding particles. Here, $P_{B}=P_{meson}+P_{\ell}+P_{\nu}$ must be evaluated in the laboratory frame. We only retain candidates with $|\Delta E|<1.0~\mathrm{\,Ge\kern-1.00006ptV}\mbox{ and }m_{ES}>5.19~\mathrm{\,Ge\kern-1.00006ptV}$, thereby removing from the fit a region with large backgrounds. Fewer than 6.6% (12.5%, 7.2%, 7.4%, 1.9%) of all $\pi^{-}\ell\nu$ ($\pi^{0}\ell\nu$, $\omega\ell\nu$, $\eta\ell\nu$, $\eta^{\prime}\ell\nu$) events have more than one candidate per event. For events with multiple candidates, only the candidate with the largest value of $\cos\theta_{\ell}$ is kept. The signal event reconstruction efficiency varies between 6.1% and 8.5% for $B^{0}\rightarrow\pi^{-}\ell^{+}\nu$ decays, between 2.8% and 6.0% for $B^{+}\rightarrow\pi^{0}\ell^{+}\nu$ decays, between 1.0% and 2.2% for $B^{+}\rightarrow\omega\ell^{+}\nu$ decays, and between 0.9% and 2.6% for $B^{+}\rightarrow\eta\ell^{+}\nu$ decays ($\gamma\gamma$ channel), depending on the value of $q^{2}$. The efficiency is 0.6% for both $B^{+}\rightarrow\eta\ell^{+}\nu$ ($\pi^{+}\pi^{-}\pi^{0}$ channel) and $B^{+}\rightarrow\eta^{\prime}\ell^{+}\nu$ decays. The efficiencies are given as a function of $q^{2}$ in Tables 23-27 of the Appendix.

Backgrounds can be broadly grouped into three main categories: decays arising from $b\rightarrow u\ell\nu$ transitions (other than the signal), decays in other $B\bar{B}$ events (excluding $b\rightarrow u\ell\nu$) and decays in continuum events. The “other $B\bar{B}$” background is the sum of different contributions, where more than $75$% are from $B\rightarrow D/D^{*}/D^{**}$ decays. For the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu$, $B^{+}\rightarrow\pi^{0}\ell^{+}\nu$ and combined $B\rightarrow\pi\ell^{+}\nu$ modes, for which there is a large number of candidate events, each of the first two categories of background is further split into a background category where the pion and the lepton come from the decay of the same $B$ meson (“same-$B$” category), and a background category where the pion and the lepton come from the decay of different $B$ mesons (“both-$B$” category).

We use the $\Delta E$-$m_{ES}$ histograms, obtained from the MC simulation, as two-dimensional probability density functions (PDFs) in an extended binned maximum-likelihood fit Barlow to the data to extract the yields of the signal and backgrounds as a function of $q^{2}$. This fit method incorporates the statistical uncertainty from the finite MC sample size into the fit uncertainty. The $\Delta E$-$m_{ES}$ plane is subdivided into 34 bins for each bin of $q^{2}$ in the fits to the $\pi^{-}\ell^{+}\nu$, $\pi^{0}\ell^{+}\nu$ and $\pi\ell^{+}\nu$ candidate data where we have a reasonably large number of events, and into 19 bins in the fits to the $\omega\ell^{+}\nu$, $\eta\ell^{+}\nu$ and $\eta^{\prime}\ell\nu$ decay data. The $\Delta E$-$m_{ES}$ distributions for the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu$ decay channel are shown in Fig. 2. The binning used in this case is also displayed in the figure. We use variable bin sizes because we want to have a large number of small bin sizes in the signal-enhanced region to better define this specific region. The signal-enhanced region is the region of the $\Delta E$-$m_{ES}$ plane with a large proportion of signal events. It is delimited in our work by the boundaries: $-0.16<\Delta E<0.20$ $\mathrm{\,Ge\kern-1.00006ptV}$ and $m_{ES}$ $>$ 5.268 $\mathrm{\,Ge\kern-1.00006ptV}$ (see Fig. 2). To allow the fit to converge quickly we cannot have too many bins in the overall $\Delta E$-$m_{ES}$ plane. Hence the bins outside the signal-enhanced region will have a larger size. The actual size is dictated by the need to have a good description of the smooth backgrounds. The parameters of the fit are the scaling factors of the MC PDFs, i.e., the factors used to adjust the number of events in a PDF to minimize the $\chi^{2}$ value of the fit.

Given the sufficient number of events for the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu$ and combined $B\rightarrow\pi\ell^{+}\nu$ decay modes, the data samples can be subdivided in $12$ bins of $q^{2}$ for the signal and two bins for each of the five background categories. The use of two bins for each background component allows the fit to adjust for inaccuracies in the modelling of the shape of the background $q^{2}$ spectra. The boundaries of the two background bins of $q^{2}$ for the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu$ and $B\rightarrow\pi\ell^{+}\nu$ decays are: [0-18-26.4] $\mathrm{\,Ge\kern-1.00006ptV^{2}\!}$ for the $b\rightarrow u\ell\nu$ same-$B$ category, [0-22-26.4] $\mathrm{\,Ge\kern-1.00006ptV^{2}\!}$ for the $b\rightarrow u\ell\nu$ both-$B$ category, [0-10-26.4] $\mathrm{\,Ge\kern-1.00006ptV^{2}\!}$ for the other $B\bar{B}$ same-$B$ category, [0-14-26.4] $\mathrm{\,Ge\kern-1.00006ptV^{2}\!}$ for the other $B\bar{B}$ both-$B$ category and [0-22-26.4] $\mathrm{\,Ge\kern-1.00006ptV^{2}\!}$ for the continuum category. In each case, the $q^{2}$ ranges of the two bins are chosen to contain a similar number of events. In the fit to the data, we determine for each bin of $q^{2}$, the signal yield, the $b\rightarrow u\ell\nu$, the other $B\bar{B}$ and the continuum background yields in each bin of $\Delta E$-$m_{ES}$.

Note, however, that the scaling factors obtained for each background are constrained to have the same value over their ranges of $q^{2}$ defined above. We thus have a total of 22 parameters and $(12\times 34-22)$ degrees of freedom in the fit to the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu$ data and to the combined $B\rightarrow\pi\ell^{+}\nu$ data. The limited number of events for the other signal modes reduces the number of parameters, and hence the number of $q^{2}$ bins, that can be used for the fits to converge. Table 1 shows the number of bins of $q^{2}$ used for each signal mode as a function of the fit category.

As an initial estimate in the fit, the MC continuum background yield and $q^{2}$-dependent shape are first normalized to match the yield and $q^{2}$-dependent shape of the off-resonance data control sample. This results in a large statistical uncertainty due to the small number of events in the off-resonance data. To improve the statistical precision, the continuum background is allowed to vary in the fit to the data for the $\pi\ell\nu$, $\omega\ell\nu$ and $\eta\ell\nu(\gamma\gamma)$ modes where we have a relatively large number of events. The fit result is compatible with the measured distribution of off-resonance data. Whenever a background is not varied in the fit, it is fixed to the MC prediction, except for the continuum background which is fixed to its normalized yield and $q^{2}$-dependent shape using the off-resonance data. The background parameters which are free in the fit, typically require an adjustment of less than 10% with respect to the MC predictions. The initial agreement between MC and data is already good before we do any fit. After the fit, the agreement becomes excellent, as can be seen in Fig. 3 for a number of variables of interest. The values of the scaling factors, obtained in this work, are presented in Table 14 of the Appendix for each decay channel. The full correlation matrices of the fitted scaling factors are given in Tables 15-22 of the Appendix.

We refit the data on several different subsets obtained by dividing the final data set based on time period, electron or muon candidates, by modifying the $q^{2}$, $\Delta E$ or $m_{ES}$ binnings, and by varying the event selections. We obtain consistent results for all subsets. We have also used MC simulation to verify that the nonresonant decay contributions to the resonance yields are negligible. For example, we find that there are 30 nonresonant $\pi^{+}\pi^{-}\pi^{0}\ell\nu$ events out of a total yield of $1861\pm 233$ events for the $B^{+}\rightarrow\omega\ell^{+}\nu$ decay channel.

For illustrative purposes only, we show in Figs. 4,  5, and 6, $\Delta E$ and $m_{ES}$ fit projections in the signal-enhanced region for the $B^{0}\rightarrow\pi^{-}\ell^{+}\nu$, $B^{+}\rightarrow\pi^{0}\ell^{+}\nu$ and combined $B^{0}\rightarrow\pi^{-}\ell^{+}\nu$ and $B^{+}\rightarrow\pi^{0}\ell^{+}\nu$ decays, respectively, in two ranges of $q^{2}$ corresponding to the sum of eight bins below and four bins above $q^{2}$ = 16 $\mathrm{\,Ge\kern-1.00006ptV^{2}\!}$, respectively. More detailed $\Delta E$ and $m_{ES}$ fit projections in each $q^{2}$ bin are shown in Figs. 13 and 14 of the Appendix for the combined $B\rightarrow\pi\ell^{+}\nu$ decays. The data and the fit results are in good agreement. Fit projections for $B^{+}\rightarrow\omega\ell^{+}\nu$ and $B^{+}\rightarrow\eta^{(\prime)}\ell^{+}\nu$ decays, over their $q^{2}$ ranges of investigation, are shown in Fig. 7. Table 2 gives the fitted yields in the full $q^{2}$ range studied for the signal and each background category as well as the $\chi^{2}$ values and degrees of freedom for the overall fit region. The yield values in the $B^{+}\rightarrow\eta\ell^{+}\nu$ column are the result of the fit to the combined $\gamma\gamma$ and $3\pi$ modes.

Systematic uncertainties on the values of the partial branching fractions, $\mbox{$\Delta\cal B$}(q^{2})$, and their correlations among the $q^{2}$ bins have been investigated. These uncertainties are estimated from the variations of the resulting partial BF values (or total BF values for $B^{+}\rightarrow\eta^{\prime}\ell^{+}\nu$ decays) when the data are reanalyzed by reweighting different simulation parameters such as BFs and form factors. For each parameter, we use the full MC dataset to produce new $\Delta E$-$m_{ES}$ distributions (“MC event samples”) by reweighting the parameter randomly over a complete Gaussian distribution whose standard deviation is given by the uncertainty on the parameter under study. One hundred such samples are produced for each parameter. Each MC event sample is analyzed the same way as real data to determine values of $\mbox{$\Delta\cal B$}(q^{2})$ (or total BF values for $B^{+}\rightarrow\eta^{\prime}\ell^{+}\nu$ decays). The contribution of the parameter to the systematic uncertainty is given by the RMS value of the distribution of these $\mbox{$\Delta\cal B$}(q^{2})$ values over the one hundred samples.