Measurements of branching fractions and $q^2$ distributions for $B \to \pi \ell \nu$ and $B \to \rho \ell \nu$ Decays with $B \to D^{(*)} \ell \nu$ Decay Tagging

We report measurements of the charmless semileptonic decays $B^0 \to \pi^- / \rho^- \ell^{+} \nu$ and $B^+ \to \pi^0 / \rho^0 \ell^{+} \nu$, based on a sample of $2.75 \times 10^8$ $B \bar{B}$ events collected at the $\Upsilon(4S)$ resonance with the Belle detector at the KEKB $e^+e^-$ asymmetric collider. In this analysis, the accompanying $B$ meson is reconstructed in the semileptonic mode $B \to D^{(*)} \ell \nu$, enabling detection of the signal modes with high purity. We measure the branching fractions ${\mathcal B}(B^0 \to \pi^- \ell^+ \nu) = (1.38\pm 0.19\pm 0.14\pm 0.03) \times 10^{-4}$, ${\mathcal B}(B^0 \to \rho^- \ell^+ \nu) = (2.17\pm 0.54\pm 0.31\pm 0.08) \times 10^{-4}$, ${\mathcal B}(B^+ \to \pi^0 \ell^+ \nu) = (0.77\pm 0.14\pm 0.08\pm 0.00) \times 10^{-4}$ and ${\mathcal B}(B^+ \to \rho^0 \ell^+ \nu) = (1.33\pm 0.23\pm 0.17\pm 0.05) \times 10^{-4}$, where the errors are statistical, experimental systematic, and systematic due to form-factor uncertainties, respectively. For each mode we also present the partial branching fractions in three $q^2$ intervals: $q^2<8$, $8 \leq q^2<16$, and $q^2 \geq 16$ GeV$^2/c^2$. From our partial branching fractions for $B \to \pi \ell \nu$ and recent results for the form factor from unquenched Lattice QCD calculations, we obtain values of the CKM matrix element $|V_{ub}|$.


Introduction
Exclusive B → X u ℓν decays proceed dominantly via a b → uW − tree process and can be used to determine |V ub |, one of the smallest and least known elements of the Cabibbo-Kobayashi-Maskawa matrix [1]. However, the need to translate the observed rate to a |V ub | value using model-dependent decay form-factors (FF) has resulted in large theoretical uncertainties. The recent release of FF results for B → πℓν calculated by unquenched Lattice QCD (LQCD) [2,3] makes possible the first model-independent determination of |V ub |. Since LQCD results are available only in the high q 2 region (≥ 16 GeV 2 /c 2 ), a clean measurement of the partial B → πℓν branching fraction in the same high q 2 region is needed.
There have been several measurements in the past by CLEO, BaBar and Belle for the B → πℓν, ρℓν, ηℓν and ωℓν modes [4,5,6,7,8,9]. The analyses in these measurements utilize the method, originally developed by CLEO, where the B decays are reconstructed by inferring the undetected neutrino mass from missing energy and momentum ("ν-reconstruction method") [4]. In the Bfactory era, we will improve the statistical precision by simply applying the ν-reconstruction method using a large amount of data. However, the poor signal-to-noise ratio will limit the systematic uncertainty of the measurement.
In this paper we present measurements of B 0 → π − /ρ − ℓ + ν and B + → π 0 /ρ 0 ℓ + ν decays using B → D ( * ) ℓν decay tagging. We reconstruct the entire decay chain from the Υ(4S), Υ(4S) → B sig B tag , B sig → π/ρℓν and B tag → D ( * ) ℓν with several D ( * ) sub-modes. The back-to-back correlation of the two B mesons in the Υ(4S) rest frame allows us to constrain the kinematics of the double semileptonic decay. The signal is reconstructed in four modes, B 0 → π − /ρ − ℓ + ν and B + → π 0 /ρ 0 ℓ + ν. Yields and branching fractions are extracted from a simultaneous fit of the B 0 and B + samples in three intervals of q 2 , accounting for cross-feed between modes as well as other backgrounds. We have applied this method to B → π/ρℓν decays for the first time, and have succeeded in reconstructing these decays with significantly improved signal-to-noise ratios compared to the ν-reconstruction method. Inclusion of charge conjugate decays is implied throughout this paper.

Data Set and Experiment
The analysis is based on data recorded with the Belle detector at the KEKB collider operating at the center-of-mass (c.m.) energy for the Υ(4S) resonance [10]. The Υ(4S) dataset that is used corresponds to an integrated luminosity of 253 fb −1 and contains 2.75 × 10 8 BB events.
The Belle detector is a large-solid-angle magnetic spectrometer that consists of a silicon vertex detector (SVD), a 50-layer central drift chamber (CDC), an array of aerogel thresholdČerenkov counters (ACC), a barrel-like arrangement of time-of-flight scintillation counters (TOF), and an electromagnetic calorimeter comprised of CsI(Tl) crystals (ECL) located inside a super-conducting solenoid coil that provides a 1.5 T magnetic field. An iron flux-return located outside of the coil is instrumented to detect K 0 L mesons and to identify muons (KLM). The detector is described in detail elsewhere [11]. Two inner detector configurations were used. A 2.0 cm beam pipe and a 3-layer silicon vertex detector was used for the first sample of 152 × 10 6 BB pairs, while a 1.5 cm beam pipe, a 4-layer silicon detector, and a small-cell inner drift chamber were used to record the remaining 123 million BB pairs [12].
A detailed Monte Carlo (MC) simulation, which fully describes the detector geometry and response and is based on GEANT [13], is applied to estimate the signal detection efficiency and to study the background. To examine the FF dependence, MC samples for the B → π/ρℓν signal decays are generated with different form-factor models: a quark model (ISGW II [14]), light cone sum rules (LCSR for B → πℓν [15] and B → ρℓν [16]) and quenched lattice QCD (UKQCD [17]). We also use unquenched lattice QCD (FNAL [2] and HPQCD [3]) for B → πℓν and a relativistic quark model (Melikhov [18]) for B → ρℓν. To model the cross-feed from other B → X u ℓν decays, MC samples are generated with the ISGW II model for the resonant components (B → πℓν and B → ρℓν components are excluded in this sample) and the DeFazio-Neubert model [19] for the non-resonant component. To model the BB and continuum backgrounds, large generic BB and qq Monte Carlo (based on Evtgen [20]) samples are used.

Event Reconstruction and Selection
Charged particle tracks are reconstructed from hits in the SVD and the CDC. They are required to satisfy track quality cuts based on their impact parameters relative to the measured interaction point (IP) of the two beams. Charged kaons are identified by combining information on ionization loss (dE/dx) in the CDC,Čherenkov light yields in the ACC and time-of-flight measured by the TOF system. For the nominal requirement, the kaon identification efficiency is approximately 88% and the rate for misidentification of pions as kaons is about 8%. Hadron tracks that are not identified as kaons are treated as pions. Tracks satisfying the lepton identification criteria, as described below, are removed from consideration.
Neutral pions are reconstructed using γ pairs with an invariant mass between 117 and 150 MeV/c 2 . Each γ is required to have a minimum energy of 50 MeV. K 0 S mesons are reconstructed using pairs of tracks that are consistent with having a common vertex and that have an invariant mass within ±12 MeV/c 2 of the known K 0 S mass.
Electron identification is based on a combination of dE/dx in the CDC, the response of the ACC, shower shape in the ECL and the ratio of energy deposit in the ECL to the momentum measured by the tracking system. Muons are identified by their signals in the KLM resistive plate counters, which are interleaved with the iron of the solenoid return yoke. The lepton identification efficiencies are estimated to be about 90% for both electrons and muons in the momentum region above 1.2 GeV/c, where leptons from prompt B decays dominate. The hadron misidentification rate is measured using reconstructed K 0 S → π + π − and found to be less than 0.2% for electrons and 1.5% for muons in the same momentum region.
For the reconstruction of B tag → D ( * ) ℓν, the lepton candidate is required to have the correct sign charge with respect to the D meson flavor and a laboratory momentum (p lab ℓ ) greater than 1.0 GeV/c. The D meson candidates are reconstructed by using seven decay modes of D + : D + → K − π + π + , K − π + π + π 0 , K 0 S π + , K 0 S π + π 0 , K 0 S π + π + π − , K + K 0 S , K + K − π + ; and ten decay modes of D 0 : The candidates are required to have an invariant mass m D within ±2.5σ (σ is a standard deviation) of the nominal D mass, where the mass resolution σ is dependent on the decay mode. D * mesons are reconstructed in the modes D * + → D 0 π + , D + π 0 and D * 0 → D 0 π 0 by combining a D meson candidate and a charged or neutral pion. Each D * candidate is required to have a mass difference ∆m = mD π − mD within ±2.5σ of the nominal values.
For the reconstruction of B sig → X u ℓν, the lepton candidate is required to have the right sign charge with respect to the X u system and p lab ℓ greater than 0.8 GeV/c. The X u system may consist of one pion or two pions ( N π + = 1 or N π + = N π 0 = 1 for aB 0 tag and N π 0 = 1 or N π + = N π − = 1 for a B − tag). The event is required to have no additional charged tracks or π 0 candidates. We also require that the residual energy from neutral clusters be less than 0.15 GeV (E neut < 0.15 GeV). The two leptons on the tag and the signal sides are required to have opposite charge. The loss of signal due to B 0 −B 0 mixing is estimated by MC simulation.
We then impose a constraint based on the kinematics of the double semileptonic decay in the Υ(4S) rest frame. In the semileptonic decay on each side, B 1(2) → Y 1(2) ν (Y 1 = D ( * ) ℓ and Y 2 = X u ℓ), the angle between the B 1(2) meson and the detected Y 1(2) system θ B 1(2) is calculated from the relation, P * ν 2 = (P * B − P * Y ) 2 = 0 (P * : 4-momentum vector) and the known p * B (the absolute momentum of the mother B meson). This means that the B 1(2) direction is constrained on the surface of a cone defined with the angle θ * B 1 (2) around the direction of the Y 1(2) system, as shown graphically in Fig. 1. The back-to-back relation of the two B meson directions then implies that the real B direction is on the intersection of the two cones when one of the B systems is spatially inverted. Denoting θ * 12 the angle between the p * , and with the coordinate definition in Fig. 1, where the p * Y 1 and p * Y 2 are aligned along the z-axis and in the y − z plane, respectively. If the hypothesis of the double semileptonic decay is correct and all the decay products are detected except for the two neutrinos, x 2 B must range from 0 to 1. Events passing a rather loose cut x 2 B > −2.0 are used for signal extraction at a later stage of the analysis. Since the direction of the B meson is not uniquely determined, we calculate, , using the beam energy (E * beam ), energy (E * Xu ) and momentum (p * Xu ) of the X u system and neglecting the momentum of the B meson in the c.m. system. The signal Monte Carlo simulation finds that the q 2 resolution depends on the reconstructed q 2 and is in the range 0.32-0.95 GeV 2 /c 2 .
According to Monte Carlo simulation, the largest backgrounds originate from B → X c ℓν and non-signal B → X u ℓν decays, where some particles escape detection. There are sizable contributions from cross talk between theB 0 and B + tags. The contribution from qq processes is found to be negligible. For events selected as described above, the signal MC simulation indicates that the total detection efficiency (ǫ total ), averaged over electron and muon channels, is 1.98 × 10 −3 for B 0 → π − ℓ + ν and 0.76 × 10 −3 for B 0 → ρ − ℓ + ν, 1.49 × 10 −3 for B + → π 0 ℓ + ν and 1.78 × 10 −3 for B + → ρ 0 ℓ + ν assuming the LCSR FF model. Here, ǫ total is defined with respect to the number of BB pairs, where one B decays into the signal mode, and includes the loss of signal due to B 0 −B 0 mixing. Because of the loose lepton momentum cut (p lab ℓ > 0.8 GeV/c), the variation of efficiency with different FF models is relatively small. Table 1 gives the matrix ǫ(q 2 rec. , q 2 true ), the efficiency for a signal event generated with true q 2 in the bin q 2 true to be reconstructed in the bin q 2 rec. . Table 1 Detection efficiency matrix ǫ(q 2 rec. , q 2 true ) based on the LCSR model in units of 10 −3 . To check the validity of the method, we apply the procedure described above to reconstruct B 0 sig → D * − ℓ + ν followed by D * − →D 0 π − ,D 0 → K + π − for aB 0 tag and B + sig →D * 0 ℓ + ν followed byD * 0 →D 0 π 0 ,D 0 → K + π − for a B − tag, with the same requirement on the tagging side. Figures 2-a) and c) show the M Kππ distributions that are obtained in data and expected from MC. As a result, we obtained 224.7 ± 15.4 (295.9 ± 17.6)B 0 (B − ) tagged events. These values are in good agreement with expected values 224.5 ± 9.5 (288.6 ± 11.7) calculated from the branching fractions [21] and efficiencies obtained from MC. Here, we use B(B + →D * 0 ℓ + ν) calculated from B(B 0 → D * − ℓ + ν) and the liftime ratio [21]; The ratio of the reconstructed to expected value, R = 1.00±0.08±0.05 (1.03± 0.07 ± 0.05) where the first error is statistical error and the second is due to the uncertainty of the branching fractions from [21], is consistent with unity.

Systematic Errors
Tables 3 and 4 summarize the experimental systematic errors on the branching fractions. The experimental systematic errors can be categorized as originating from uncertainties in the signal reconstruction efficiency, the background Table 2 Extracted branching fractions for each signal mode with different FF models in units of 10 −4 : the total branching fraction and the partial branching fractions in three q 2 intervals. χ 2 /n and the associated probability for this χ 2 indicate the quality of the fit for the FF shape to the observed q 2 distribution. Mode 0.77 ± 0.14 0.19 ± 0.06 0.39 ± 0.10 0.20 ± 0.08 2.9/2 0.24 ISGW II 0.77 ± 0.14 0.19 ± 0.06 0.39 ± 0.10 0.20 ± 0.08 7.8/2 0.02 UKQCD 0.77 ± 0.14 0.19 ± 0.06 0.38 ± 0.10 0.20 ± 0.08 2.5/2 0.28 FNAL 0.77 ± 0.14 0.19 ± 0.06 0.39 ± 0.10 0.20 ± 0.08 2.8/2 0.25 HPQCD 0.77 ± 0.14 0.19 ± 0.06 0.39 ± 0.10 0.20 ± 0. The effect from the uncertainty on the signal reconstruction efficiency is evaluated based on the efficiency calibration with the B sig → D * ℓ + ν sample, discussed above. The error is taken to be that on the ratio of observed to expected number of the calibration signals (9.3% for B 0 → π − /ρ − ℓ + ν, 9.2% for B → π 0 /ρ 0 ℓ + ν). This gives the largest contribution to the systematic error. Note that this error is dominated by the statistics of the calibration signals, as explained above. Therefore, accumulation of additional integrated luminosity in the future will help to reduce this uncertainty. We also include residual errors for the reconstruction of the signal side: 1% and 2% for the detection of each charged and neutral pion, respectively, and 2% for the charged pion selection and 2.1% for the lepton selection.
The systematic error due to the uncertainty on the inclusive branching fraction B(B → X u ℓν), which is used to constrain B → X u ℓ + ν background, is estimated by varying this parameter by its ±1σ error. The uncertainty in the BB background shape after our pion multiplicity selection requirements (N π + = 1 or N π + = N π 0 = 1 for aB 0 tag and N π 0 = 1 or N π + = N π − = 1 for a B − tag) is studied in the simulation by randomly removing charged tracks and π 0 according to the error in detection efficiency (1% for a charged track, 2% for π 0 ), and also by reassigning identified charged kaons as pions according to the uncertainty in the kaon identification efficiency (2%). The resultant changes in the extracted branching fractions are assigned as systematic errors. We find a significant uncertainty in the high q 2 region (q 2 > 16 GeV 2 /c 2 ) for B → ρℓ + ν due to the poor signal-to-noise ratio. We also vary the fraction of B → D * * ℓν decays in the BB background MC by the error quoted in [21] to test the B → X c ℓν model dependence in the BB background shape. To assess the uncertainty due to the production rate of K 0 L , we vary the production rate in the MC simulation by the uncertainty in the inclusive branching fraction for B → K 0 X quoted in [21].
For the normalization, we consider the uncertainty in the number of B 0B0 and B + B − pairs: the ratio of B + B − to B 0B0 pairs (f + /f 0 ), f + /f 0 = 1.029 ± 0.035 [23] , the mixing parameter (χ d ), χ d = 0.186 ± 0.004 [21], and the measured number of BB pairs (N BB , 1.1%). Table 3 Summary of systematic errors (%) for B(B 0 → π − /ρ − ℓ + ν).  Table 4 Summary of systematic errors (%) for B(B + → π 0 /ρ 0 ℓ + ν). The dependence of the extracted branching fractions on the FF model has been studied by repeating the above fitting procedure with various FF models for the signal mode and also for the cross-feed mode (B → πℓν ↔ B → ρℓν). We consider the models listed in Table 2. For the extracted B(B → π − ℓ + ν (π 0 ℓ + ν)), the standard deviation among the models is < 1.7 (0.9)% for B → πℓ + ν and < 1.9 (0.5)% for B → ρℓ + ν. For B(B → ρ − ℓ + ν (ρ 0 ℓ + ν)), the standard deviation is < 2.9 (3.6)% for B → ρℓ + ν and < 1.0 (1.3)% for B → πℓ + ν. The total error due to FF model dependence is the quadratic sum of the maximum variations with the signal and cross-feed FF models. Table 5 summarizes our measurements of the total and partial branching fractions for the four signal modes. Each branching fraction is obtained by taking the simple average of the values obtained from the FF models shown in Table 2. The errors shown in the table are statistical, experimental systematic, and model dependence due to form-factor uncertainties. The obtained branching fractions for B 0 → π − /ρ − ℓ + ν are consistent with the existing measurements by CLEO [6] and BaBar [9]. The overall uncertainty on our result for B 0 → π − ℓ + ν (17%) is comparable to those on CLEO and BaBar results based on ν-reconstruction. Our results for B 0 → ρ − ℓ + ν have the smallest uncertainty. Table 5 Summary of the obtained branching fractions. The errors are statistical, experimental systematic, and systematic due to form-factor uncertainties.
In this work, the B 0 → π − ℓ + ν/B + → π 0 ℓ + ν and B 0 → ρ − ℓ + ν/B + → ρ 0 ℓ + ν signals are extracted separately, which allows us to test the isospin relations. From the obtained branching fractions and the B meson lifetimes in [21], the ratios of decay rates are found to be, where the first and second errors are statistical and systematic errors, respectively. Both ratios are found to be consistent with the isospin relations; The obtained branching fractions in Table 5 can be used to extract |V ub | using the relation, whereΓ thy is the form-factor normalization, predicted from theories. In Ta region above 16 GeV 2 /c 2 , where the LQCD calculations are most reliable. The table provides also the results in the region below 16 GeV 2 /c 2 , so that one can deduce |V ub | based on other approaches such as LCSR calculations [15,16].
In this paper we calculate |V ub | based on the B → πℓ + ν data in the high q 2 region and the form factor predicted by recent unquenched LQCD calculations. Their predictions (Γ thy ) for the q 2 ≥ 16 GeV 2 /c 2 region areΓ thy (B 0 → π − ℓ + ν) = 1.83 ± 0.50 ps −1 (FNAL) [2] andΓ thy (B 0 → π − ℓ + ν) = 1.46 ± 0.35 ps −1 (HPQCD) [3]. We use τ B 0 = 1.532 ± 0.009 ps and τ B + = 1.638 ± 0.011 ps [21], and we use isospin symmetry to relateΓ thy for B 0 → π − and B + → π 0 transitions. The results for B 0 → π − ℓ + ν and B + → π 0 ℓ + ν are then averaged, weighted by their respective statistical errors.  Table 6 summarizes the results, where the first and second errors are the experimental statistical and systematic errors, respectively. The third error is based on the error onΓ thy quoted by the LQCD authors. These theoretical errors are asymmetric because we assign them by taking the variation in |V ub | whenΓ thy is varied by the quoted errors. The values are in agreement with those from inclusive B → X u ℓν decays [24].
To summarize, we have measured the branching fractions of the decays B → πℓν and B → ρℓν in 2.75×10 8 BB events using a method which tags one B in the mode B → D ( * ) ℓν. Our results are consistent with previous measurements, and their precision is comparable to that of results from other experiments. The ratios of results for neutral and charged B meson modes are found to be consistent with isospin. The partial rates are measured in three bins of q 2 and compared with distributions predicted by several theories. From the rate in the region q 2 ≥ 16 GeV 2 /c 2 and recent results from LQCD calculations, we extract |V ub |: |V ub | π − ℓ + ν+π 0 ℓ + ν (q 2 ≥16 GeV 2 /c 2 ) = (3.60 ± 0.41 ± 0.20 +0.62 −0.41 ) × 10 −3 (FNAL LQCD), (5) |V ub | π − ℓ + ν+π 0 ℓ + ν (q 2 ≥16 GeV 2 /c 2 ) = (4.03 ± 0.46 ± 0.22 +0.59 −0.41 ) × 10 −3 (HPQCD LQCD). (6) The experimental precision on these values is 13%, currently dominated by the statistical error of 11%. By accumulating more integrated luminosity, a measurement with errors below 10% is feasible. With improvements to unquenched LQCD calculations, the present method may provide a precise determination of |V ub |.
We thank the KEKB group for the excellent operation of the accelerator, the KEK cryogenics group for the efficient operation of the solenoid, and the