Improved measurements of D meson semileptonic decays to \pi and K mesons

Improved measurements of meson semileptonic decays to and mesons

D. Besson University of Kansas, Lawrence, Kansas 66045, USA    T. K. Pedlar    J. Xavier Luther College, Decorah, Iowa 52101, USA    D. Cronin-Hennessy    K. Y. Gao    J. Hietala    Y. Kubota    T. Klein    R. Poling    A. W. Scott    P. Zweber University of Minnesota, Minneapolis, Minnesota 55455, USA    S. Dobbs    Z. Metreveli    K. K. Seth    B. J. Y. Tan    A. Tomaradze Northwestern University, Evanston, Illinois 60208, USA    J. Libby    L. Martin    A. Powell    C. Thomas    G. Wilkinson University of Oxford, Oxford OX1 3RH, UK    H. Mendez University of Puerto Rico, Mayaguez, Puerto Rico 00681    J. Y. Ge    D. H. Miller    I. P. J. Shipsey    B. Xin Purdue University, West Lafayette, Indiana 47907, USA    G. S. Adams    D. Hu    B. Moziak    J. Napolitano Rensselaer Polytechnic Institute, Troy, New York 12180, USA    K. M. Ecklund Rice University, Houston, Texas 77005, USA    Q. He    J. Insler    H. Muramatsu    C. S. Park    E. H. Thorndike    F. Yang University of Rochester, Rochester, New York 14627, USA    M. Artuso    S. Blusk    S. Khalil    R. Mountain    K. Randrianarivony    N. Sultana    T. Skwarnicki    S. Stone    J. C. Wang    L. M. Zhang Syracuse University, Syracuse, New York 13244, USA    G. Bonvicini    D. Cinabro    M. Dubrovin    A. Lincoln    M. J. Smith    P. Zhou    J. Zhu Wayne State University, Detroit, Michigan 48202, USA    P. Naik    J. Rademacker University of Bristol, Bristol BS8 1TL, UK    D. M. Asner    K. W. Edwards    J. Reed    A. N. Robichaud    G. Tatishvili    E. J. White Carleton University, Ottawa, Ontario, Canada K1S 5B6    R. A. Briere    H. Vogel Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA    P. U. E. Onyisi    J. L. Rosner University of Chicago, Chicago, Illinois 60637, USA    J. P. Alexander    D. G. Cassel    J. E. Duboscq    R. Ehrlich    L. Fields    L. Gibbons    R. Gray    S. W. Gray    D. L. Hartill    B. K. Heltsley    D. Hertz    J. M. Hunt    J. Kandaswamy    D. L. Kreinick    V. E. Kuznetsov    J. Ledoux    H. Mahlke-Krüger    J. R. Patterson    D. Peterson    D. Riley    A. Ryd    A. J. Sadoff    X. Shi    S. Stroiney    W. M. Sun    T. Wilksen Cornell University, Ithaca, New York 14853, USA    J. Yelton University of Florida, Gainesville, Florida 32611, USA    P. Rubin George Mason University, Fairfax, Virginia 22030, USA    N. Lowrey    S. Mehrabyan    M. Selen    J. Wiss University of Illinois, Urbana-Champaign, Illinois 61801, USA    R. E. Mitchell    M. R. Shepherd Indiana University, Bloomington, Indiana 47405, USA
July 13, 2019

Using the entire CLEO-c event sample, corresponding to an integrated luminosity of and approximately 5.4 million events, we present a study of the decays , , , and . Via a tagged analysis technique, in which one is fully reconstructed in a hadronic mode, partial rates for semileptonic decays by the other are measured in several bins. We fit these rates using several form factor parameterizations and report the results, including form factor shape parameters and the branching fractions %, %, %, and %, where the first uncertainties are statistical and the second are systematic. Taking input from lattice quantum chromodynamics (LQCD), we also find and , where the third uncertainties are from LQCD.

preprint: CLNS 09/2049preprint: CLEO 09-02thanks: Deceased

CLEO Collaboration

I Introduction

Semileptonic decays are an excellent environment for precision measurements of the Cabibbo-Kobayashi-Maskawa (CKM) Kobayashi and Maskawa (1973); Cabibbo (1963) matrix elements. However, because these decays are governed by both the weak and strong forces, extraction of the weak CKM parameters requires knowledge of strong interaction effects. These can be parameterized by form factors. Techniques such as lattice quantum chromodynamics (LQCD) offer increasingly precise calculations of these form factors, but as the uncertainties in the predictions shrink, experimental validation of the results becomes increasingly important. Because the magnitudes of CKM matrix elements and are tightly constrained by CKM unitarity, semileptonic decays of mesons provide an excellent testing ground for the new theoretical predictions. The relevance of tests using charm decays is increased by the similarity of meson decays to those of mesons, where QCD calculations are critical to extractions of in exclusive semileptonic decays Artuso et al. (2009).

We present a study of the decays , , , and (with charged conjugate modes implied throughout this article) using 818 of data collected by the CLEO-c detector. Taking advantage of the fact that mesons produced near the resonance are produced solely as part of pairs, we follow a tagged technique pioneered by the Mark III Collaboration Adler et al. (1989) and used in semileptonic analyses of smaller portions of CLEO-c data Ge et al. (2008); Huang et al. (2005a); Coan et al. (2005). Hadronically decaying tags are first reconstructed; one then looks for the decays of interest in the remainder of each event. This strategy suppresses backgrounds and provides an absolute normalization for branching fraction measurements.

For semileptonic decays such as those of interest here, in which the initial and final state hadrons are pseudoscalars and the lepton mass is negligibly small, the strong interaction dynamics can be described by a single form factor , where is the invariant mass of the lepton-neutrino system. The rate for a semileptonic decay with final state meson is given by


where is the Fermi constant, is the relevant CKM matrix element, is the momentum of the daughter meson in the rest frame of the parent , and is a multiplicative factor due to isospin, equal to 1 for all modes except , where it is . The primary measurements described here are the partial decay rates in seven bins each for and and nine bins each in and . We fit the using several parameterizations of , extracting form factor shape parameters, measurements of , and branching fractions. Taking estimates of from theory, we also extract and .

The remainder of this article is organized as follows: the CLEO-c detector and event reconstruction are described in Sec. II. Measurements of partial rates and their systematic uncertainties are detailed in Secs. III and IV, respectively. Extractions of branching fractions, form factor shapes, and CKM parameters are reviewed in Sec. V, and Sec. VI summarizes our results.

Ii Detector and Event Reconstruction

The CLEO-c detector has been described in detail elsewhere Kubota et al. (1992); Peterson et al. (2002); Artuso et al. (2005). The 53-layer tracking system, composed of two drift chambers covering 93% of the solid angle and enclosed by a superconducting solenoid operating with a 1-Tesla magnetic field, provides measurements of charged particle momentum with a resolution of at 700 MeV. The tracking chambers also supply specific-ionization () information, which is combined with input from the Ring-Imaging Ĉerenkov (RICH) detector to provide excellent discrimination between charged pions and kaons. A 7784 crystal cesium-iodide calorimeter covering 95% of the solid angle provides photon energy resolution of 2.2% at GeV, with a mass resolution of about 6 MeV, and contributes to positron identification.

The entire CLEO-c data sample has an integrated luminosity of 818 , equivalent to approximately 5.4 million events. The data were collected at center-of-mass energies near 3.774 GeV with a RMS spread in beam energy of approximately 2.1 MeV. Events collected at this energy, approximately 40 MeV above the production threshold, are composed primarily of , , and non-charm continuum final states.

GEANT-based gea () Monte Carlo (MC) simulations are used to determine reconstruction efficiencies, develop line shapes for yield extraction fits, and conduct tests of the analysis procedure. Final state radiation (FSR) is simulated using PHOTOS Barberio and Was (1994) version 2.15 with the option to simulate FSR interference enabled. A sample of generic events, generated using EvtGen Lange (2001) and corresponding to approximately 20 times the data luminosity, was generated using input from Ref. Yao et al. (2006), combined with CLEO-c results using the initial 281 data sample where appropriate. This sample, along with samples of simulated (,, or ), , and events corresponding to five times the data luminosity, is referred to as “generic MC” for the remainder of the article. We also use a sample of events in which the meson decays to one of the four studied semileptonic modes and the decays to one of the hadronic final states used in tag reconstruction. This sample is referred to as “signal MC.” In both the generic and signal MC samples, the semileptonic decays are generated using the modified pole parameterization Becirevic and Kaidalov (2000) (see Sec. V.1) with parameters fixed to those measured in the initial 281 of CLEO-c data Ge et al. (2008); Cronin-Hennessey:2008 (). All simulations are corrected for small biases in the positron, charged hadron, and identification efficiencies.

Charged pions and kaons are identified from drift chamber tracks with momentum greater than 50 MeV/ and with , where is the angle between the track and the beam axis. Charged track reconstruction efficiencies are approximately 84% for kaons and 89% for pions; lost tracks within are almost exclusively due to particle decay in flight and material interaction in the drift chambers. Pions and kaons are distinguished using a combination of specific ionization measurements and, if the track momentum is greater than 700 MeV/, RICH detector information. For all other tracks, hadron identity is determined using specific ionization information only. Given a properly reconstructed track, hadron identification efficiencies are approximately 95%, with misidentification rates of a few percent. Identical hadron selection criteria are used in tag and semileptonic reconstructions.

Neutral pion candidates are reconstructed via . Photon candidates are identified from energy depositions in the calorimeter greater than 30 MeV using shower shape information. The invariant mass of each pair of photon candidates is calculated using a kinematic fit that assumes the photons originate at the center of the detector. This mass is required to be within three standard deviations () of the nominal mass, where is determined from the kinematic fit. The resulting energy and momentum from the fit are used in further event analysis. Efficiencies for reconstruction vary from 40% at a momentum of 100 MeV/ to 60% at 900 MeV/. If multiple neutral pion candidates are reconstructed opposite the tag, the candidate with the mass closest to the nominal mass is chosen.

Neutral kaon candidates are reconstructed via using vertex-constrained fits to pairs of oppositely charged intersecting tracks. The invariant mass of the candidate is required to be within 12 MeV/ of the nominal mass. This procedure results in a mass resolution of 2 – 2.5 MeV/ and a reconstruction efficiency of about 94%. If multiple candidates are reconstructed opposite a tag, the candidate with mass closest to the nominal mass is chosen.

Positron candidates are identified from tracks with momentum greater than 200 MeV/ and within the solid angle . Positrons are selected using a combination of specific ionization, calorimetry, and RICH detector information. The efficiency for positron identification is about 50% at the low momentum threshold of 200 MeV/, rises sharply to 92% at 300 MeV/, and varies by a few percent as a function of momentum beyond 300 MeV/. Roughly 0.1% of charged hadrons satisfy the positron identification criteria. Positron momentum resolution is degraded by FSR. We reduce this effect by identifying bremsstrahlung photon candidates in the calorimeter within of the positron candidate track and adding their 4-momenta to that of the positron candidate. Such photons must have energy greater than 30 MeV and no associated track reconstructed in the drift chamber.

Tag candidates are reconstructed in three decay modes (, , and ) and six decay modes (, , , , , and ). Backgrounds are suppressed by requiring that satisfy the requirements given in Table 1. These cuts correspond to approximately , with depending on the decay mode. Backgrounds are further reduced using the beam-constrained mass, , where is the beam energy and is the total measured momentum of the tag candidate. The tag candidates must satisfy GeV, while tag candidates are required to have GeV. In events with multiple tag candidates, the one candidate per mode and per flavor with reconstructed energy closest to the beam energy is chosen.

Mode Requirement (GeV)  
Table 1: requirements for tag reconstruction.

Semileptonic candidates are formed from positron and hadron candidate pairs. Although the semileptonic neutrino daughter is not detected, its energy and momentum can be inferred from the missing energy and momentum of the event:




where the energy and momentum of the hadron-positron system are constructed from the measured energy and momenta of the hadron, positron and any bremsstrahlung photon candidates. The tag momentum is formed from the measured tag momentum with the magnitude constrained using the beam energy and mass: . All momentum vectors are boosted to the center-of-mass frame by correcting for the small net momentum due to the beam crossing angle ( mrad).

Semileptonic backgrounds are reduced by requiring that the variable , defined as


satisfy GeV for each candidate. Additionally, the positron and hadron from the semileptonic decay are required to have opposite charge in and candidates, while the positron and tag are required to have opposite charge in and .

Semileptonic candidates are partitioned into several bins, where the reconstructed is determined based on measurements of the positron and neutrino:


The neutrino energy is taken to be the missing energy of the event, while the neutrino momentum is equated with the missing momentum scaled so that . Because does not require measurements of the tag decay, it is a better measured quantity than ; constraining the neutrino momentum in this manner thus improves the resolution in . and candidates are divided into seven bins, with boundaries defined by [0, 0.3), [0.3, 0.6), [0.6, 0.9), [0.9, 1.2), [1.2, 1.5), [1.5, 2.0), and [2.0, ) GeV. In the and modes, nine bins are used, defined by [0, 0.2), [0.2, 0.4), [0.4, 0.6), [0.6, 0.8), [0.8, 1.0), [1.0, 1.2), [1.2, 1.4), [1.4, 1.6), and [1.6, ) GeV.

Iii Extraction of Partial Rates

In order to measure the partial rates we first determine the number of observed tags (“tag yields”) in each tag mode . This is related to the number of tags produced in mode , , by


where is the reconstruction efficiency for tag mode . We then determine the number of events with both a tag and semileptonic candidate. These “signal yields” are determined separately for each tag mode and bin . The signal yields are related to the number of tag-semileptonic combinations produced in each bin, , by


where are the elements of a matrix that describes the efficiency and smearing across bins associated with tag and semileptonic reconstruction. As the number of tag-semileptonic combinations produced is a function of the number of tag decays and the differential semileptonic decay rate, , we can rewrite Eq. (7) as


where is the lifetime and the integration is over the width of bin . Combining the above equations and solving for the differential rate, we obtain a simple formula for extracting the partial rates:


The small correlation between signal and tag yields that arises from the signal yields being a subset of the tag yields is neglected.

The following sections describe the extraction of the tag yields , tagging efficiencies , signal yields , and signal smearing and efficiency matrices . With all of these numbers in hand, we then extract the partial rates.

iii.1 Tag Yields and Efficiencies

The tag yields are obtained separately for each mode by performing fits to beam-constrained mass distributions. The fitting procedure has been described in detail in Dobbs et al. (2007) and involves an unbinned likelihood maximization. True tag decays are modeled using a function specially designed to take into account the natural line shape, beam energy resolution, momentum resolution, and initial state radiation (ISR) effects. Tag backgrounds are modeled using an ARGUS function Albrecht et al. (1990), modified so that the power parameter is allowed to float Dobbs et al. (2007). The fits, shown in Fig. 1, are performed over a wide GeV window. Tag yields, given in Table 2, are obtained by subtracting the backgrounds estimated by the fits from event counts in data inside the narrower signal regions. Also shown in Table 2 are tagging efficiencies, which are obtained by fitting generic MC distributions with the same procedure used in data.

Figure 1: distributions in data (points), with fits (solid lines) and background contributions to fits (dotted lines). The vertical lines show the limits of the signal regions.
Mode Yield Efficiency(%)  
Table 2: Tag yields and statistical uncertainties in data and tag reconstruction efficiencies.

iii.2 Signal Yields and Efficiencies

Signal yields are extracted from distributions of , defined in Eq. (4). Events in which both the tag and semileptonic decay have been correctly reconstructed, leaving only an undetected neutrino, are expected to peak at , with the shape of the distribution being approximately Gaussian due to detector resolution. Misreconstructed events and background modes generally have non-zero values. Properly reconstructed decays are separated from backgrounds using an unbinned maximum-likelihood fit, executed independently for each semileptonic mode, each tag mode, and each bin. A sample of the distributions is shown in Figs. 25.

For the fit, the distribution of signal candidates is taken from signal MC samples. While the resolution in data is approximately 12 MeV for the modes with only charged tracks in the final state (, , and ) and 25 MeV for , the distributions are slightly narrower in MC simulation. To accommodate the poorer resolutions in data, the MC distributions are convolved with a double Gaussian with parameters fixed for each semileptonic mode to the values that maximize the fit likelihoods summed over all bins and tag modes. The smearing functions are dominated by central Gaussians with widths of approximately 6 MeV in , , and and 13 MeV in . The secondary Gaussians have normalizations of 3% – 7% of the central Gaussian and have widths of 30 – 35 MeV. The overall normalization of the corrected signal distribution is allowed to float in each fit.

The background distributions used in the fits are taken from generic MC samples. Backgrounds arise from both non- and events. The and modes are subject to large backgrounds from and , respectively. To allow for variations in and fake rates between the data and MC simulations, the normalizations of these components are fixed to the values that minimize the fit likelihood summed over all bins and tag modes. The mode is also subject to a large background from events; the normalization of this background is fixed using the known branching fraction and the tag yields in data and MC samples. The remaining backgrounds occur due to misreconstruction of either the semileptonic or tag decay, although the largest backgrounds are composed of events with a correctly reconstructed tag but misreconstructed semileptonic decay. In each semileptonic mode, all of the backgrounds not discussed above are combined into a single background distribution with the normalization allowed to float in each fit. The normalization of the non- distribution is fixed using the ratio of luminosities in data and MC samples.

Because each bin and tag mode is treated separately, the total numbers of fits for , , , and are 21, 27, 42, and 54, respectively. Figs. 25 show four individual fits, as well as the summed fit results for all bins and tag modes.

Figure 2: distributions in data (points) for , with fit results (histograms) showing signal (clear) and background components: (darkest gray), (lightest gray), other (medium gray), and non- (black).
Figure 3: distributions in data (points) for , with fit results (histograms) showing signal (clear) and background components: (gray) and non- (black).
Figure 4: distributions in data (points) for , with fit results (histograms) showing signal (clear) and background components: (light gray), other (dark gray), and non- (black).
Figure 5: distributions in data (points) for , with fit results (histograms) showing signal (clear) and background components: (gray) and non- (black).
Rec True (GeV)
(GeV) [0,0.3) [0.3,0.6) [0.6,0.9) [0.9,1.2) [1.2,1.5) [1.5,2.0) [2.0,)
41.25(34) 1.19(8) 0.02(1) 0.00(0) 0.00(0) 0.00(0) 0.00(0)
0.76(6) 42.57(36) 1.55(10) 0.01(1) 0.00(0) 0.00(0) 0.00(0)
0.04(1) 1.12(8) 44.65(38) 1.54(10) 0.02(1) 0.00(0) 0.00(0)
0.02(1) 0.08(2) 1.09(8) 44.77(41) 1.37(10) 0.03(1) 0.00(0)
0.01(1) 0.03(1) 0.09(2) 1.33(9) 46.11(44) 0.91(8) 0.00(0)
0.01(1) 0.02(1) 0.02(1) 0.11(3) 1.20(10) 47.01(40) 0.74(8)
0.00(0) 0.00(0) 0.01(1) 0.02(1) 0.04(2) 0.56(6) 47.28(48)
Rec True (GeV)
(GeV) [0, 0.2) [0.2, 0.4) [0.4, 0.6) [0.6, 0.8) [0.8, 1.0) [1.0, 1.2) [1.2, 1.4) [1.4, 1.6) [1.6, )
19.70(4) 0.80(1) 0.03(0) 0.00(0) 0.00(0) 0.00(0) 0.00(0) 0.00(0) 0.00(0)
0.45(1) 19.80(5) 1.03(1) 0.03(0) 0.00(0) 0.00(0) 0.00(0) 0.00(0) 0.00(0)
0.01(0) 0.54(1) 20.58(5) 1.12(1) 0.03(0) 0.01(0) 0.00(0) 0.00(0) 0.00(0)
0.01(0) 0.02(0) 0.61(1) 21.32(6) 1.12(2) 0.03(0) 0.01(0) 0.00(0) 0.01(0)
0.01(0) 0.01(0) 0.03(0) 0.63(1) 21.92(6) 1.03(2) 0.01(0) 0.00(0) 0.00(0)
0.00(0) 0.01(0) 0.02(0) 0.03(0) 0.59(1) 21.64(7) 0.95(2) 0.01(0) 0.00(0)
0.00(0) 0.00(0) 0.01(0) 0.01(0) 0.01(0) 0.51(1) 21.08(9) 0.88(3) 0.01(0)
0.00(0) 0.00(0) 0.00(0) 0.00(0) 0.00(0) 0.01(0) 0.39(1) 20.05(11) 0.79(4)
0.00(0) 0.00(0) 0.00(0) 0.00(0) 0.00(0) 0.00(0) 0.00(0) 0.25(1) 16.72(17)
Table 3: Selected efficiency matrices in percent for and . The column gives the true bin , while the row gives the reconstructed (“Rec”) bin . The elements account for the reconstruction efficiencies of both the tag and the semileptonic decay. The statistical uncertainties in the least significant digits are given in the parentheses.
Rec True (GeV)
(GeV) [0, 0.3) [0.3, 0.6) [0.6, 0.9) [0.9, 1.2) [1.2, 1.5) [1.5, 2.0) [2.0, )
22.44(20) 0.83(5) 0.02(1) 0.00(0) 0.00(0) 0.00(0) 0.00(0)
1.23(5) 21.69(21) 1.02(5) 0.01(1) 0.00(0) 0.00(0) 0.00(0)
0.03(1) 1.62(6) 21.23(22) 1.14(6) 0.01(1) 0.00(0) 0.00(0)
0.02(1) 0.03(1) 1.75(7) 21.12(23) 1.05(6) 0.00(0) 0.00(0)
0.02(1) 0.03(1) 0.06(1) 1.61(7) 19.72(25) 0.65(4) 0.00(0)
0.02(1) 0.03(1) 0.04(1) 0.13(2) 1.47(7) 20.50(22) 0.49(5)
0.17(2) 0.19(2) 0.31(3) 0.47(4) 0.70(5) 1.65(7) 22.81(27)
Rec True (GeV)
(GeV) [0, 0.2) [0.2, 0.4) [0.4, 0.6) [0.6, 0.8) [0.8, 1.0) [1.0, 1.2) [1.2, 1.4) [1.4, 1.6) [1.6, )
5.06(3) 0.21(1) 0.00(0) 0.00(0) 0.00(0) 0.00(0) 0.00(0) 0.00(0) 0.00(0)
0.11(0) 4.98(3) 0.24(1) 0.00(0) 0.00(0) 0.00(0) 0.00(0) 0.00(0) 0.00(0)
0.00(0) 0.15(1) 5.09(3) 0.25(1) 0.01(0) 0.00(0) 0.00(0) 0.00(0) 0.00(0)
0.00(0) 0.00(0) 0.16(1) 5.12(3) 0.28(1) 0.00(0) 0.00(0) 0.00(0) 0.00(0)
0.00(0) 0.00(0) 0.00(0) 0.15(1) 5.13(4) 0.26(1) 0.00(0) 0.00(0) 0.00(0)
0.00(0) 0.00(0) 0.00(0) 0.01(0) 0.15(1) 5.14(4) 0.24(1) 0.00(0) 0.00(0)
0.00(0) 0.00(0) 0.00(0) 0.00(0) 0.01(0) 0.12(1) 5.29(5) 0.22(1) 0.01(0)
0.00(0) 0.00(0) 0.00(0) 0.00(0) 0.00(0) 0.01(0) 0.12(1) 5.34(7) 0.26(3)
0.00(0) 0.00(0) 0.01(0) 0.01(0) 0.01(0) 0.01(0) 0.01(0) 0.08(1) 5.45(11)
Table 4: Selected efficiency matrices in percent for and . The column gives the true bin , while the row gives the reconstructed (“Rec”) bin . The elements account for the reconstruction efficiencies of both the tag and the semileptonic decay. The statistical uncertainties in the least significant digits are given in the parentheses.

The signal efficiency matrix elements , as defined in Eq. (7), are obtained from signal MC simulations and corrected for previously determined deviations from positron, charged hadron, and identification efficiencies in data. Each gives the fraction of events generated in bin with tag mode that are reconstructed in bin with the same tag. The efficiency matrix thus accounts for reconstruction of both the signal and tag decays. The efficiencies include the branching fraction Amsler et al. (2008) and efficiencies include the fraction of the and branching fraction Amsler et al. (2008). In total, there are eighteen efficiency matrices – one for each tag and semileptonic mode combination. Tables 3 and 4 provide four examples of these matrices. The diagonal elements, giving the efficiency for the tag and semileptonic decays to be reconstructed in the correct bin, vary from 5% – 50% depending on semileptonic mode, tag mode, and . The neighboring off-diagonal elements, giving the efficiencies for the tag and semileptonic decay to be reconstructed in the wrong bin, range between 1% and 10% of the diagonal elements. The signal yields summed over tag modes both before and after correction by these matrices are shown in Table 5.

(GeV) [0, 0.3) [0.3, 0.6) [0.6, 0.9) [0.9, 1.2) [1.2, 1.5) [1.5, 2.0) [2.0, )
Raw yield 251(17) 232(16) 204(15) 194(15) 161(13) 173(14) 159(13)
Corrected yield 858(60) 795(59) 636(51) 612(50) 495(45) 532(46) 505(45)
Partial rate (ns) 1.39(10) 1.22(9) 1.02(8) 0.98(8) 0.79(7) 0.84(7) 0.80(7)
(GeV) [0, 0.2) [0.2, 0.4) [0.4, 0.6) [0.6, 0.8) [0.8, 1.0) [1.0, 1.2) [1.2, 1.4) [1.4, 1.6) [1.6, )
Raw yield 2751(54) 2541(51) 2325(49) 2034(46) 1662(42) 1243(36) 904(31) 496(23) 167(13)
Corrected yield 11290(233) 10026(222) 8629(203) 7432(186) 5867(163) 4485(143) 3393(125) 1983(98) 781(68)
Partial rate (ns) 17.82(36) 15.83(35) 13.91(32) 11.69(29) 9.36(26) 7.08(22) 5.34(19) 3.09(15) 1.28(11)
(GeV) [0, 0.3) [0.3, 0.6) [0.6, 0.9) [0.9, 1.2) [1.2, 1.5) [1.5, 2.0) [2.0, )
Raw yield 148(13) 141(13) 124(12) 122(12) 100(11) 107(12) 96(13)
Corrected yield 799(80) 748(82) 665(80) 640(77) 570(80) 625(82) 460(86)
Partial rate (ns) 0.71(7) 0.66(7) 0.56(6) 0.57(6) 0.48(6) 0.54(7) 0.37(7)
(GeV) [0, 0.2) [0.2, 0.4) [0.4, 0.6) [0.6, 0.8) [0.8, 1.0) [1.0, 1.2) [1.2, 1.4) [1.4, 1.6) [1.6, )
Raw yield 1704(42) 1511(40) 1389(38) 1229(36) 912(31) 809(29) 514(24) 275(17) 124(12)
Corrected yield 10090(282) 8732(271) 7934(256) 6951(240) 5101(207) 4511(190) 2812(152) 1412(106) 625(73)
Partial rate (ns) 17.79(47) 15.62(45) 14.02(43) 12.28(40) 8.92(34) 8.17(32) 4.96(25) 2.67(18) 1.19(13)
Table 5: Signal yields, both raw () and corrected for smearing and reconstruction efficiency (), and partial rates (). Statistical uncertainties in the least significant digits are given in parentheses.

iii.3 Partial Rate Results

Using the tag yields, tag efficiencies, signal yields, and signal efficiency matrices, we solve Eq. (9) for the partial rates in each bin and tag mode, . The procedure detailed in Lefebvre et al. (2000) is used to calculate uncertainties and correlations in the inverted efficiency matrices. We then average the resulting over tag modes, obtaining . Statistical covariance matrices detailing the uncertainties on the are also calculated and are available in the Appendix. Within each semileptonic mode, there are small correlations across bins that arise from the smearing in . Both the individual and tag-averaged partial rates are shown in Fig. 6. The tag-averaged partial rates are also given in Table 5.

Figure 6: Partial rates for each semileptonic mode. The points show measurements in each tag mode; the histograms show the partial rates averaged over all tag modes.

iii.4 Tests of Partial Rate Results

Calculating the partial rates separately for each tag mode allows for a test of the consistency of results across different tag modes. To quantify the tag mode agreement, we calculate a for each semileptonic mode:


where is the statistical uncertainty on . This quantity is expected to have a distribution, with mean and variance , where the number of degrees of freedom is given by . Table 6 gives the measured , the number of degrees of freedom, and the probability for each mode. These values show that the semileptonic rates agree well across tag modes.

Semileptonic mode  
Table 6: of partial rates across tag modes, with number of degrees of freedom and probability.
Figure 7: Distributions of (upper), the cosine of the angle between the virtual and the positron, positron momentum (middle) and hadron momentum (lower) in events satisfying MeV MeV. Signal and background shapes are taken from MC simulations and scaled using the parameters of the signal yield fits.

As a test of the signal yield fits, we compare the observed and predicted distributions in three variables: the lepton momentum, the hadron momentum, and the angle of the virtual in the rest frame relative to the positron in the rest frame. Fig. 7 shows these distributions for data and MC candidates with MeV, with signal and background MC distributions normalized using the fits. All of the distributions show good agreement between data and MC simulations.

We also check consistency between isospin conjugate pairs. Isospin symmetry implies that total rates for and are approximately equal, while the total rate for is approximately twice that of . After correcting for phase space differences, our partial rates summed over all bins agree with these expectations within 1.4 standard deviations. Because there are small differences in phase space, it is convienient to compare not rates, but form factors, as shown in Fig. 8. We obtain the at the center of bin using


where is the size of bin , and are from Particle Data Group fits assuming CKM unitarity Amsler et al. (2008), and the effective in bin is given by


where and are calculated using the three parameter series parameterization with parameters measured in the data (see Sec. V.3).

Our measured form factors in each bin are seen to be in good agreement with the LQCD calculations Aubin:2004ej (), but with significantly smaller uncertainties, as shown in Fig. 8.

The procedure for measuring partial rates is tested using the generic MC sample, from which events are drawn randomly to form mock data samples, each equivalent in size to the data sample. In each case, the measured partial rates are consistent with the input rates and the distributions of the deviations are consistent with Gaussian statistics.

Figure 8: comparison between isospin conjugate modes and with LQCD calculations Aubin:2004ej (). The solid lines represent LQCD fits to the modified pole model Becirevic and Kaidalov (2000). The inner bands show LQCD statistical uncertainties, and the outer bands the sum in quadrature of LQCD statistical and systematic uncertainties.

Iv Systematic Uncertainties in Partial Rates

Our determinations of the are subject to a variety of systematic uncertainties. Tables 7 and 8 list each source of systematic uncertainty and its contribution to the total uncertainty in each of the partial rates. Because we are interested in measuring form factor shapes that vary with , it is important that we not only understand the uncertainties in the individual partial rates but also their correlations across . For each semileptonic mode and each significant source of systematic uncertainty, we construct an (where is the number of bins studied for the mode in question) covariance matrix that encapsulates both of these pieces of information. We now describe how each of the covariance matrices is estimated.

Tag line shape 0.40 0.40 0.40 0.40 0.40 0.40 0.40
Tag fakes 0.40 0.40 0.40 0.40 0.40 0.40 0.40
Tracking efficiency 0.48 0.48 0.48 0.48 0.48 0.49 0.51
ID 0.21 0.11 0.05 0.03 0.02 0.02 0.04
ID 0.37 0.38 0.38 0.39 0.33 0.18 -0.14
FSR 0.18 0.11 0.09 -0.02 -0.10 -0.20 -0.24
Signal shape 0.56 0.46 0.58 0.49 0.50 0.56 0.49
Backgrounds 0.39 0.43 0.60 0.61 0.58 0.52 0.76
MC form factor 0.06 -0.05 -0.05 -0.05 -0.07 -0.11 -0.04
smearing 0.84 -0.11 -0.26 -0.16 0.30 -0.60 -0.28
D Lifetime 0.37 0.37 0.37 0.37 0.37 0.37 0.37
All systematic 1.44 1.13 1.27 1.22 1.22 1.31 1.30
Statistical 6.84 7.29 7.90 8.06 8.87 8.42 8.63
Tag line shape 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40
Tag fakes 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40
Tracking efficiency 0.69 0.72 0.75 0.79 0.84 0.92 1.04 1.26 1.22
ID 0.19 0.13 0.11 0.10 0.09 0.10 0.11 0.11 0.21
ID 0.41 0.42 0.43 0.45 0.47 0.48 0.44 0.33 0.21
FSR 0.12 0.08 0.07 0.01 -0.10 -0.15 -0.23 -0.28 -0.32
Signal shape 0.16 0.12 0.12 0.14 0.12 0.11 0.09 0.14 0.21
Backgrounds 0.14 0.04 0.12 0.09 0.08 0.08 0.04 0.10 0.33
MC form factor 0.02 -0.02 -0.02 -0.01 -0.01 -0.01 0.00 0.02 -0.08
smearing 0.62 -0.11 0.07 -0.12 -0.06 -0.51 0.08 -0.62 -2.05
D Lifetime 0.37 0.37 0.37 0.37 0.37 0.37 0.37 0.37 0.37
All systematic 1.26 1.10 1.12 1.15 1.20 1.36 1.35 1.63 2.55
Statistical 2.03 2.19 2.31 2.47 2.73 3.14 3.63 4.90 8.43
Table 7: Summary of partial rate () uncertainties in percent for and . The sign gives the direction of change relative to the change in the first bin.
Tag line shape 0.40 0.40 0.40 0.40 0.40 0.40 0.40
Tag fakes 0.70 0.70 0.70 0.70 0.70 0.70 0.70
Tracking efficiency 0.25 0.25 0.25 0.24 0.24 0.24 0.23
ID 1.06 0.98 1.04 1.22 1.83 2.14 1.96
ID 0.32 0.32 0.34 0.32 0.27 0.13 -0.22
FSR 0.14 0.20 0.08 -0.05 -0.14 -0.22 -0.21
Signal shape 1.72 0.93 1.91 -1.24 3.51 2.43 3.26
Backgrounds 0.92 0.82 -1.01 0.72 0.74 1.38 -6.04
MC form factor 0.15 -0.03 -0.07 -0.06 -0.10 -0.15 0.57
smearing 1.69 0.28 -1.74 1.45 -0.17 -1.22 -1.41
D Lifetime 0.67 0.67 0.67 0.67 0.67 0.67 0.67
All systematic 3.01 1.97 3.17 2.63 4.18 3.89 -7.38
Statistical 9.25 10.23 11.24 11.28 13.44 12.38 17.98
Tag line shape 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.40
Tag fakes 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70
Tracking efficiency 0.76 0.77 0.78 0.79 0.81 0.83 0.87 0.91 0.96
ID 2.00 1.96 1.90 1.83 1.71 1.51 1.25 1.35 1.89
ID 0.42 0.43 0.43 0.45 0.48 0.48 0.44 0.33 0.20
FSR 0.17 0.13 0.08 0.01 -0.11 -0.16 -0.23 -0.24 -0.28
Signal shape 0.20 0.22 0.17 0.20 0.23 0.26 0.38 0.26 0.47
Backgrounds 0.13 0.13 0.11 0.11 0.14 0.15 0.27 0.23 1.46
MC form factor 0.03 -0.02 -0.02 -0.02 -0.02 -0.01 0.01 0.02 0.08
smearing 0.63 -0.24 -0.02 0.29 -1.06 0.75 -0.67 -0.78 -1.11
D Lifetime 0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.67 0.67
All systematic 2.52 2.42 2.36 2.33 2.47 2.23 2.08 2.16 3.03
Statistical 2.63 2.90 3.04 3.23 3.82 3.98 5.04 6.88 10.63
Table 8: Summary of partial rate () uncertainties in percent for and . The sign gives the direction of change relative to the change in the first bin.

Tag reconstruction biases enter both the numerator and denominator of our partial rate formulation in Eq. (9), and therefore largely cancel. However, there are two sources of systematic uncertainty related to tag yields. One source originates in the line shapes used to extract tag yields in data; we estimate this by using alternate line shapes and find an uncertainty of 0.4% for partial rates in both and modes. The selection of one tag per mode also introduces a systematic uncertainty, primarily due to possible mismodeling of MC fake rates. Based on estimates of tag-fake rates in data and MC samples, we assign a systematic uncertainty of 0.4% to the partial rates in modes and 0.7% to those in modes, where a greater fraction of tags contain ’s. As the uncertainties associated with tag yields are independent of the kinematics of the semileptonic decay, they are fully correlated across bins.

Systematic uncertainties associated with semileptonic track, , , , and reconstruction are all studied in a similar manner: we choose fully hadronic events containing a particle of type , where , , , or , and reconstruct all particles in the event except for . We then form missing mass squared distributions, which peak at for correctly reconstructed events. We then tally the fraction of events with the appropriate in which was successfully reconstructed, after correcting for backgrounds. By doing this in bins of missing momentum, we compare the data and MC efficiencies as a function of particle momentum.

In the case of and track reconstruction and finding, no evidence of bias in the efficiencies is found. Biases of less than 1% are observed in and identification efficiencies. We also find reconstruction efficiencies to be approximately 6% lower in data than in MC simulations, the bias being roughly constant across momentum. About half of this discrepancy has been traced to incorrect modeling of the lateral spread of photon showers in the calorimeter and the energy resolution; the other half is of unknown origin. We reweight the MC distributions to correct for all reconstruction biases.

Systematic uncertainties in the particle reconstruction efficiencies are often correlated across momentum bins. When this is the case, we construct a covariance matrix binned in particle momentum by noting that efficiencies binned in particle momentum () are related to efficiencies binned in semileptonic () via


where is a matrix giving the fraction of type particles that are part of a semileptonic decay in a given bin that are also in a given momentum bin; this matrix is estimated using signal MC simulations. The fractional covariance matrix binned in , , is then given by


where is the momentum-binned fractional covariance matrix. This equation is used to obtain the binned systematic covariance matrices associated with track, , , , and reconstruction.

The covariance matrices for all remaining systematic uncertainties – those associated with positron identification, FSR, background and signal shapes used to obtain signal yields, form factor parameterization in MC simulations, and smearing in – are estimated by the following procedure: for each source of systematic uncertainty, we vary the analysis in a manner that approximates the uncertainty on the effect in question and remeasure the partial rates . The covariance matrix elements for this source can then be estimated via


where denotes the difference between the partial rate in bin measured using the varied analysis and the rate using the standard analysis technique. In most cases, we make several variations to the analysis and sum the resulting covariance matrices. Where it is possible to vary some parameter by positive and negative values, we average the results of the positive and negative variations.

Positron identification efficiencies as a function of positron momentum are measured in MC simulations and in data using radiative Bhabha () and two-photon () events. Since the positrons in these events are relatively isolated, we embed these positrons into hadronic events, and determine the decrease in efficiency. Biases of around 1.5% are observed, and the MC signal efficiency matrices and distributions are corrected for these biases. To estimate the systematic uncertainty due to positron identification, we shift the corrections by the uncertainties on their measurement and remeasure the partial rates using efficiencies and distributions obtained with the shifted corrections.

FSR affects the partial rate measurements primarily by causing mismeasurements of positron momentum. FSR in the MC simulations is modeled using PHOTOS version 2.15, which models FSR significantly better than earlier versions. To estimate the systematic uncertainty due to FSR, we reweight the efficiency matrices and distributions so that the energy and angular distribution of photons reconstructed in the neighborhood of the positrons match those measured in data, and remeasure the partial rates. We have also studied systematic uncertainties associated with ISR, which are found to be negligible.

The signal shapes used to model semileptonic candidates in the signal yield fits are taken from signal MC distributions convolved with a double Gaussian. The systematic uncertainty associated with this procedure is estimated by varying the widths of the Gaussians and the normalization of the wider one within their uncertainties and remeasuring the signal yields. In the mode, there is also evidence of a possible shift between data and signal MC distributions. In this mode only, we apply a systematic uncertainty equal to the change in rates when a shift is applied.

The background lineshapes in the signal yield fits introduce systematic uncertainties in three ways. The first arises from our choice to fix the normalization of several background shapes, namely the small non- background in all modes, the and backgrounds to and the background to . The systematic uncertainties associated with these backgrounds are estimated by varying the normalizations within their uncertainties. In the case of the and backgrounds, where the normalizations are those that minimize the fit likelihoods summed over all and tag modes, we vary the normalization to values that increase the likelihood by unity. Secondly, the choice to combine many background modes into one shape using fixed relative normalizations may result in incorrect background shapes. We estimate this systematic uncertainty by varying the normalization of several of the largest components of the combined shapes based on branching fraction uncertainties. Finally, incorrect MC fake rates may also lead to inaccurate background shapes. Our technique is most sensitive to positron and fake rates. Using estimates of hadron-to-positron fake rates studied in and and fake rates studied in , we estimate this systematic uncertainty by increasing the fake rates in MC simulations to match those found in data.

The use of efficiency and smearing matrices binned in reduces the dependence of our results on the used to generate signal events in the MC simulations. However, we are still sensitive to non-linear effects within