Measurement of the inclusive electron spectrum from B meson decays and determination of |V_{ub}|

Measurement of the inclusive electron spectrum from meson decays and determination of

J. P. Lees    V. Poireau    V. Tisserand Laboratoire d’Annecy-le-Vieux de Physique des Particules (LAPP), Université de Savoie, CNRS/IN2P3, F-74941 Annecy-Le-Vieux, France    E. Grauges Universitat de Barcelona, Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain    A. Palano INFN Sezione di Bari and Dipartimento di Fisica, Università di Bari, I-70126 Bari, Italy    G. Eigen University of Bergen, Institute of Physics, N-5007 Bergen, Norway    D. N. Brown    Yu. G. Kolomensky Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA    H. Koch    T. Schroeder Ruhr Universität Bochum, Institut für Experimentalphysik 1, D-44780 Bochum, Germany    C. Hearty    T. S. Mattison    J. A. McKenna    R. Y. So University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1    V. E. Blinov    A. R. Buzykaev    V. P. Druzhinin    V. B. Golubev    E. A. Kravchenko    A. P. Onuchin    S. I. Serednyakov    Yu. I. Skovpen    E. P. Solodov    K. Yu. Todyshev Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090, Novosibirsk State University, Novosibirsk 630090, Novosibirsk State Technical University, Novosibirsk 630092, Russia    A. J. Lankford University of California at Irvine, Irvine, California 92697, USA    J. W. Gary    O. Long University of California at Riverside, Riverside, California 92521, USA    A. M. Eisner    W. S. Lockman    W. Panduro Vazquez University of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA    D. S. Chao    C. H. Cheng    B. Echenard    K. T. Flood    D. G. Hitlin    J. Kim    T. S. Miyashita    P. Ongmongkolkul    F. C. Porter    M. Röhrken California Institute of Technology, Pasadena, California 91125, USA    Z. Huard    B. T. Meadows    B. G. Pushpawela    M. D. Sokoloff    L. Sun Now at: Wuhan University, Wuhan 43072, China University of Cincinnati, Cincinnati, Ohio 45221, USA    J. G. Smith    S. R. Wagner University of Colorado, Boulder, Colorado 80309, USA    D. Bernard    M. Verderi Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS/IN2P3, F-91128 Palaiseau, France    D. Bettoni    C. Bozzi    R. Calabrese    G. Cibinetto    E. Fioravanti    I. Garzia    E. Luppi    V. Santoro INFN Sezione di Ferrara; Dipartimento di Fisica e Scienze della Terra, Università di Ferrara, I-44122 Ferrara, Italy    A. Calcaterra    R. de Sangro    G. Finocchiaro    S. Martellotti    P. Patteri    I. M. Peruzzi    M. Piccolo    M. Rotondo    A. Zallo INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy    S. Passaggio    C. Patrignani Now at: Università di Bologna and INFN Sezione di Bologna, I-47921 Rimini, Italy INFN Sezione di Genova, I-16146 Genova, Italy    B. Bhuyan Indian Institute of Technology Guwahati, Guwahati, Assam, 781 039, India    U. Mallik University of Iowa, Iowa City, Iowa 52242, USA    C. Chen    J. Cochran    S. Prell Iowa State University, Ames, Iowa 50011, USA    H. Ahmed Physics Department, Jazan University, Jazan 22822, Kingdom of Saudi Arabia    A. V. Gritsan Johns Hopkins University, Baltimore, Maryland 21218, USA    N. Arnaud    M. Davier    F. Le Diberder    A. M. Lutz    G. Wormser Laboratoire de l’Accélérateur Linéaire, IN2P3/CNRS et Université Paris-Sud 11, Centre Scientifique d’Orsay, F-91898 Orsay Cedex, France    D. J. Lange    D. M. Wright Lawrence Livermore National Laboratory, Livermore, California 94550, USA    J. P. Coleman    E. Gabathuler    D. E. Hutchcroft    D. J. Payne    C. Touramanis University of Liverpool, Liverpool L69 7ZE, United Kingdom    A. J. Bevan    F. Di Lodovico    R. Sacco Queen Mary, University of London, London, E1 4NS, United Kingdom    G. Cowan University of London, Royal Holloway and Bedford New College, Egham, Surrey TW20 0EX, United Kingdom    Sw. Banerjee    D. N. Brown    C. L. Davis University of Louisville, Louisville, Kentucky 40292, USA    A. G. Denig    M. Fritsch    W. Gradl    K. Griessinger    A. Hafner    K. R. Schubert Johannes Gutenberg-Universität Mainz, Institut für Kernphysik, D-55099 Mainz, Germany    R. J. Barlow Now at: University of Huddersfield, Huddersfield HD1 3DH, UK    G. D. Lafferty University of Manchester, Manchester M13 9PL, United Kingdom    R. Cenci    A. Jawahery    D. A. Roberts University of Maryland, College Park, Maryland 20742, USA    R. Cowan Massachusetts Institute of Technology, Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA    R. Cheaib    S. H. Robertson McGill University, Montréal, Québec, Canada H3A 2T8    B. Dey    N. Neri    F. Palombo INFN Sezione di Milano; Dipartimento di Fisica, Università di Milano, I-20133 Milano, Italy    L. Cremaldi    R. Godang Now at: University of South Alabama, Mobile, Alabama 36688, USA    D. J. Summers University of Mississippi, University, Mississippi 38677, USA    P. Taras Université de Montréal, Physique des Particules, Montréal, Québec, Canada H3C 3J7    G. De Nardo    C. Sciacca INFN Sezione di Napoli and Dipartimento di Scienze Fisiche, Università di Napoli Federico II, I-80126 Napoli, Italy    G. Raven NIKHEF, National Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam, The Netherlands    C. P. Jessop    J. M. LoSecco University of Notre Dame, Notre Dame, Indiana 46556, USA    K. Honscheid    R. Kass Ohio State University, Columbus, Ohio 43210, USA    A. Gaz    M. Margoni    M. Posocco    G. Simi    F. Simonetto    R. Stroili INFN Sezione di Padova; Dipartimento di Fisica, Università di Padova, I-35131 Padova, Italy    S. Akar    E. Ben-Haim    M. Bomben    G. R. Bonneaud    G. Calderini    J. Chauveau    G. Marchiori    J. Ocariz Laboratoire de Physique Nucléaire et de Hautes Energies, IN2P3/CNRS, Université Pierre et Marie Curie-Paris6, Université Denis Diderot-Paris7, F-75252 Paris, France    M. Biasini    E. Manoni    A. Rossi INFN Sezione di Perugia; Dipartimento di Fisica, Università di Perugia, I-06123 Perugia, Italy    G. Batignani    S. Bettarini    M. Carpinelli Also at: Università di Sassari, I-07100 Sassari, Italy    G. Casarosa    M. Chrzaszcz    F. Forti    M. A. Giorgi    A. Lusiani    B. Oberhof    E. Paoloni    M. Rama    G. Rizzo    J. J. Walsh INFN Sezione di Pisa; Dipartimento di Fisica, Università di Pisa; Scuola Normale Superiore di Pisa, I-56127 Pisa, Italy    A. J. S. Smith Princeton University, Princeton, New Jersey 08544, USA    F. Anulli    R. Faccini    F. Ferrarotto    F. Ferroni    A. Pilloni    G. Piredda INFN Sezione di Roma; Dipartimento di Fisica, Università di Roma La Sapienza, I-00185 Roma, Italy    C. Bünger    S. Dittrich    O. Grünberg    M. Heß    T. Leddig    C. Voß    R. Waldi Universität Rostock, D-18051 Rostock, Germany    T. Adye    F. F. Wilson Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom    S. Emery    G. Vasseur CEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, France    D. Aston    C. Cartaro    M. R. Convery    J. Dorfan    W. Dunwoodie    M. Ebert    R. C. Field    B. G. Fulsom    M. T. Graham    C. Hast    W. R. Innes    P. Kim    D. W. G. S. Leith    S. Luitz    V. Luth    D. B. MacFarlane    D. R. Muller    H. Neal    B. N. Ratcliff    A. Roodman    M. K. Sullivan    J. Va’vra    W. J. Wisniewski SLAC National Accelerator Laboratory, Stanford, California 94309 USA    M. V. Purohit    J. R. Wilson University of South Carolina, Columbia, South Carolina 29208, USA    A. Randle-Conde    S. J. Sekula Southern Methodist University, Dallas, Texas 75275, USA    M. Bellis    P. R. Burchat    E. M. T. Puccio Stanford University, Stanford, California 94305, USA    M. S. Alam    J. A. Ernst State University of New York, Albany, New York 12222, USA    R. Gorodeisky    N. Guttman    D. R. Peimer    A. Soffer Tel Aviv University, School of Physics and Astronomy, Tel Aviv, 69978, Israel    S. M. Spanier University of Tennessee, Knoxville, Tennessee 37996, USA    J. L. Ritchie    R. F. Schwitters University of Texas at Austin, Austin, Texas 78712, USA    J. M. Izen    X. C. Lou University of Texas at Dallas, Richardson, Texas 75083, USA    F. Bianchi    F. De Mori    A. Filippi    D. Gamba INFN Sezione di Torino; Dipartimento di Fisica, Università di Torino, I-10125 Torino, Italy    L. Lanceri    L. Vitale INFN Sezione di Trieste and Dipartimento di Fisica, Università di Trieste, I-34127 Trieste, Italy    F. Martinez-Vidal    A. Oyanguren IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain    J. Albert    A. Beaulieu    F. U. Bernlochner    G. J. King    R. Kowalewski    T. Lueck    I. M. Nugent    J. M. Roney    N. Tasneem University of Victoria, Victoria, British Columbia, Canada V8W 3P6    T. J. Gershon    P. F. Harrison    T. E. Latham Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom    R. Prepost    S. L. Wu University of Wisconsin, Madison, Wisconsin 53706, USA
Abstract

Based on the full BABAR data sample of 466.5 million pairs, we present measurements of the electron spectrum from semileptonic meson decays. We fit the inclusive electron spectrum to distinguish Cabibbo-Kobayashi-Maskawa (CKM) suppressed decays from the CKM-favored decays, and from various other backgrounds, and determine the total semileptonic branching fraction = %, averaged over and mesons. We determine the spectrum and branching fraction for charmless decays and extract the CKM element , by relying on four different QCD calculations based on the heavy quark expansion. While experimentally, the electron momentum region above 2.1 is favored, because the background is relatively low, the uncertainties for the theoretical predictions are largest in the region near the kinematic endpoint. Detailed studies to assess the impact of these four predictions on the measurements of the electron spectrum, the branching fraction, and the extraction of the CKM matrix element are presented, with the lower limit on the electron momentum varied from 0.8 to the kinematic endpoint. We determine using each of these different calculations and find, = (De Fazio and Neubert), (Bosh, Lange, Neubert, and Paz), (Gambino, Giordano, Ossola, and Uraltsev), (dressed gluon exponentiation), where the stated uncertainties refer to the experimental uncertainties of the partial branching fraction measurement, the shape function parameters, and the theoretical calculations.

pacs:
13.20.He, 12.15.Hh, 12.38.Qk, 14.40.Nd

BABAR-PUB-16/006

SLAC-PUB-16855



thanks: Deceased

The BABAR Collaboration

I Introduction

Semileptonic decays of mesons proceed via leading order weak interactions. They are expected to be free of non-Standard-Model contributions and therefore play a critical role in the determination of the Cabibbo-Kobayashi-Maskawa (CKM) quark-mixing matrix sm () elements and . In the Standard Model (SM), the CKM elements satisfy unitarity relations that can be illustrated geometrically as triangles in the complex plane. For one of these triangles, asymmetries determine the angles, normalizes the length of the sides, and the ratio determines the side opposite the well-measured angle . Thus, precise measurements of and are crucial to studies of flavor physics and violation in the quark sector.

There are two methods to determine and , one based on exclusive semileptonic decays, where the hadron in the final state is a or meson, the other based on inclusive decays , where refers to either or , i.e., to any hadronic state with or without charm, respectively.

The extractions of and from measured inclusive or exclusive semileptonic meson decays rely on different experimental techniques to isolate the signal and on different theoretical descriptions of QCD contributions to the underlying weak decay processes. Thus, they have largely independent uncertainties, and provide important cross-checks of the methods and our understanding of these decays in general. At present, these two methods result in values for and that each differ by approximately 3 standard deviations pdg2014 ().

In this paper, we present a measurement of the inclusive electron momentum spectrum and branching fraction (BF) for the sum of all semileptonic decays, as well as measurements of the spectrum and partial BF for charmless semileptonic decays. The total rate for the decays is suppressed by about a factor 50 compared to the decays. This background dominates the signal spectrum except near the high-momentum endpoint. In the rest frame of the meson, the electron spectrum for signal extends to , while for the background decays the kinematic endpoint is at . In the rest frame, the two mesons are produced with momenta of which extends the electron endpoint by about . The endpoint region above , which covers only about of the total electron spectrum, is more suited for the experimental isolation of the charmless decays.

To distinguish contributions of the CKM suppressed decays from those of CKM-favored decays, and from various other backgrounds, we fit the inclusive electron momentum spectrum, averaged over and mesons produced in the decays pdg2014 (); hfag14 (). For this fit, we need predictions for the shape of the spectrum. We have employed and studied four different QCD calculations based on the heavy quark expansion (HQE) ope (). The upper limit of the fitted range of the momentum spectrum is fixed at 3.5 , while the lower limit extends down to 0.8 , covering up to of the total signal electron spectrum. From the fitted spectrum we derive the partial BF for charmless decays and extract the CKM element . While the experimental sensitivity to the spectrum and to is primarily determined from the spectrum above , due to very large backgrounds at lower momenta, the uncertainties for the theoretical predictions are largest in the region near the kinematic endpoint. Studies of the impact of various theoretical predictions on the measurements are presented.

Measurements of the total inclusive lepton spectrum in decays have been performed by several experiments operating at the resonance pdg2014 (). The best estimate of this BF has been derived by HFAG hfag14 (), based on a global fit to moments of the lepton momentum and hadron mass spectra in decays (corrected for decays) either with a constraint on the -quark mass or by including photon energy moments in decays in the fit. Inclusive measurements of have been performed at the resonance, by ARGUS argus (), CLEO cleo (); cleo2 (), BABAR babarVub () and Belle belle-vub (), and experiments at LEP operating at the resonance, L3 l3-vub (), ALEPH aleph-vub (), DELPHI delphi-vub (), and OPAL opal-vub (). Among the measurements based on exclusive semileptonic decays pdg2014 (), the most recent by the LHCb experiment at the LHC is based on the baryon decay  lhcb-vub ().

This analysis is based on methods similar to the one used in previous measurements of the lepton spectrum near the kinematic endpoint at the resonance argus (); cleo (). The results presented here supersede the earlier BABAR publication babarVub (), based on a partial data sample.

Ii Data Sample

The data used in this analysis were recorded with the BABAR detector detector () at the PEP-II energy-asymmetric collider. A sample of 466.5 million events, corresponding to an integrated luminosity of 424.9  lumi (), was collected at the  resonance. An additional sample of 44.4  was recorded at a center-of-mass (c.m.) energy 40 below the  resonance, i.e., just below the threshold for production. This off-resonance data sample is used to subtract the non- background at the  resonance. The relative normalization of the two data samples has been derived from luminosity measurements, which are based on the number of detected pairs and the QED cross section for production, adjusted for the small difference in center-of-mass energy.

Iii Detector

The BABAR detector has been described in detail elsewhere detector (). The most important components for this study are the charged-particle tracking system, consisting of a five-layer silicon vertex tracker and a 40-layer cylindrical drift chamber, and the electromagnetic calorimeter consisting of 6580 CsI(Tl) crystals. These detector components operated in a 1.5 T magnetic field parallel to the beam. Electron candidates are selected on the basis of the ratio of the energy deposited in the calorimeter to the track momentum, the shower shape, the energy loss in the drift chamber, and the angle of signals in a ring-imaging Cerenkov detector. Showers in the electromagnetic calorimeter with energies below 50  which are dominated by beam background are not used in this analysis.

Iv Simulation

We use Monte Carlo (MC) techniques to simulate the production and decay of mesons and the detector response geant4 (), to estimate signal and background efficiencies, and to extract the observed signal and background distributions. The sample of simulated generic events exceeds the data sample by about a factor of 3.

The MC simulations include radiative effects such as bremsstrahlung in the detector material and QED initial and final state radiation photos (). Information from studies of selected control data samples on efficiencies and resolutions is used to adjust and thereby improve the accuracy of the simulation. Adjustments for small variations of the beam energy over time have been included.

In the MC simulations, the BFs for hadronic and meson decays are based on values reported in the Review of Particle Physics pdg2014 (). The simulation of inclusive charmless semileptonic decays, , is based on calculations by De Fazio and Neubert (DN) dFN (). This simulation produces a continuous mass spectrum of hadronic states . To reproduce and test predictions by other authors this spectrum is reweighted in the course of the analysis. Three-body decays with low-mass hadrons, , make up about of the total charmless rate. They are simulated separately using the ISGW2 model isgw2 () and added to samples of decays to nonresonant and higher-mass resonant states , so that the cumulative distributions of the hadron mass, the momentum transfer squared, and the electron momentum reproduce the inclusive calculation as closely as possible. The hadronization of with masses above is performed according to JETSET jetset ().

The MC-generated electron momentum distributions for decays are shown in Fig. 1 for individual decay modes and for their sum. Here and throughout the paper, the electron momentum and all other kinematic variables are measured in the  rest frame, unless stated otherwise. Above 2 GeV/c, the significant signal contributions are from decays involving the light mesons , , , , and , in addition to some lower mass nonresonant states .

Figure 1: MC-generated electron momentum spectra in the  rest frame for charmless semileptonic decays. The full spectrum (solid line) is normalized to 1.0. The largest contribution is from decays involving higher-mass resonances and nonresonant states () (dash-three-dotted). The exclusive decays (scaled by a factor of five) are: (dash-dotted), (dashed), (dotted), (long-dashed), (long-dash-dotted).

The simulation of the dominant decays is based on a variety of theoretical prescriptions. For and decays we use form factor parametrizations hqet (); CLN (); GL (), based on heavy quark effective theory. Decays to pseudoscalar mesons are described in terms of one form factor, with a single parameter . The differential decay rate for is described by three amplitudes, with decay rates depending on three parameters: , , and . These parameters have been measured by many experiments; we use the average values presented in Table 1.

For the simulation of decays to higher-mass resonances, , i.e., two wide states , , and two narrow states , , we have adopted the parametrizations by Leibovich et al. llsw_pr () and the HFAG averages hfag14 () for the BFs. For decays to nonresonant charm states , we rely on the prescription by Goity and Roberts gr () and the BABAR and Belle measurements of the BFs hfag14 (). The simulations of these decays include the full angular dependence of the rate.

Table 1: Average measured values hfag14 () of the form factor parameters for and decays, as defined by Caprini, Lellouch, and Neubert CLN ().

The shapes of the MC-generated electron spectra for individual decays are shown in Fig. 2. Above 2 the dominant contributions are from semileptonic decays involving the lower-mass charm mesons, and . Higher-mass and nonresonant charm states are expected to contribute at lower electron momenta. The relative contributions of the individual decay modes have been adjusted to the results of the fit to the observed spectrum (see Sec. VI.2.2).

Figure 2: MC-generated electron momentum spectra for semileptonic decays to charm mesons, with the total rate (solid line) normalized to 1.0. The individual components are: (dash-dotted), (dashed), + (dotted). The highly suppressed signal spectrum (long dashed) is shown for comparison.

The difference between the measured exclusive decays and the inclusive rate for semileptonic decays to charm final states is BZT (). The decay rate for was measured by BABAR Dpipi (). Based on these results it was estimated that decays account for up to half the difference between measured inclusive and the sum of previously measured exclusive branching fractions. Beyond these observed decays, there are missing decay modes, such as and . Candidates for the 2S radial excitations were first observed by BABAR babarDprime () and recently confirmed by LHCb lhcbDprime (). We have adopted the masses and widths ( and ) measured by BABAR babarDprime (), and have simulated these decays using the form factor predictions BZT (). Both and may contribute by their decays to to decays. The decay rate for was measured by Belle Belle_Dpipi () and LHCb LHCbDpipi (), LHCb also measured the decay rate for . We account for contributions from , , and decays to final states.

The main sources of secondary electrons are semileptonic charm meson decays and decays. The momentum distribution was determined from this data set and the MC simulation was adjusted to reproduce these measured spectra. The momentum spectra of and mesons produced in decays were measured earlier by BABAR  Dspectra () and the MC simulated spectra were adjusted to reproduce these measurements.

V Calculations of Decay Rate

While at the parton level the rate for decays can be reliably calculated, the theoretical description of inclusive semileptonic decays is more challenging. Based on HQE the total inclusive rate can be predicted with an uncertainty of about , however, this rate is very difficult to measure due to very large background from the CKM-favored decays. On the other hand, in the endpoint region where the signal to background ratio is much more favorable, calculations of the differential decay rates are much more complicated. They require the inclusion of additional perturbative and nonperturbative effects. These calculations rely on HQE and QCD factorization neubert94 () and separate perturbative and nonperturbative effects by using an expansion in powers of and a nonperturbative shape function (SF) which is a priori unknown. This function accounts for the motion of the quark inside the meson, and to leading order, it should be universal for all transitions of a quark to a light quark kolya94 (); neubert94a (). It is modeled using arbitrary functions for which low-order moments are constrained by measurable parameters.

For the extraction of , we rely on , the partial BF for decays measured in the momentum interval , and , the theoretical predictions for partial decay rate normalized by , measured in units of ps:

(1)

Here ps is the average of the and lifetimes pdg2014 (). is the total predicted decay rate and refers to the fraction of the predicted decay rate for the momentum interval .

In the following, we briefly describe four different theoretical methods to derive predictions for the partial and total BFs. In the original work by De Fazio and Neubert dFN () and Kagan and Neubert kagan_neubert () the determination of relies on the measurement of the electron spectrum for and on the radiative decays to derive the parameters of the leading SF. More comprehensive calculations were performed by Bosch, Lange, Neubert, and Paz (BLNP) blnp1 (); blnp2 (); blnp3 (); blnp4 (); neubert_loops (); neubert_extract (); blnp_nnlo (). Calculations in the kinetic scheme were introduced by Gambino, Giordano, Ossola, Uraltsev (GGOU) ggou1 (); ggou2 (). BLNP and GGOU use and decays to derive the parameters of the leading SF. Inclusive spectra for decays based on a calculation of nonperturbative functions using Sudakov resummation are presented in the dressed gluon exponentiation (DGE) by Andersen and Gardi gardi (); dge1 (); dge2 (); dge3 ().

We assess individual contributions to the uncertainty of the predictions of the decay rates by the different theoretical approaches. For this purpose, the authors of these calculations have provided software to compute the differential rates and to provide guidance for the assessment of the uncertainties on the rate and thereby . We differentiate uncertainties originating from the SF parametrization, including the sensitivity to , the -quark mass, from the impact of the other purely theoretical uncertainties. The uncertainty on , the -quark mass, has a large impact. Weak annihilation could contribute significantly at high-momentum transfers (). The impact of weak annihilation is generally assumed to be asymmetric, specifically, it is estimated to decrease by  V3 ().

v.1 DN calculations

While the calculations by BLNP are to supersede the earlier work by DN, we use DN predictions for comparisons with previous measurements based on these predictions and also for comparisons with other calculations.

The early DN calculations dFN () predict the differential spectrum with corrections to leading order in HQE. This approach is based on a parametrization of the leading-power nonperturbative SF. The long-distance interaction is described by a single light-cone distribution. In the region close to phase-space boundaries these nonperturbative corrections to the spectrum are large. The prediction for the decay distribution is obtained by a convolution of the parton model spectrum with the SF. The SF is described by two parameters and which were determined from the measured photon energy moments in decays kagan_neubert (). We use BABAR measurements babar_photons () of the SF parameters, and with correlation.

DN predict the shape of the differential electron spectrum, but they do not provide a normalization. Thus to determine the partial rates , we rely on the DN predictions for , the fractions of decays in the interval , and an independent prediction for the normalized total decay rate ps dge1 () (the current value of  pdg2014 () is used to calculate ). Earlier determinations of can be found in V1 (); V2 (); V3 (); V4 (); V5 (); V6 (); V7 ().

The uncertainty on due to the application of the shape function is derived from variations of and , as prescribed by the authors. The estimated total theoretical uncertainty on is about (for ).

v.2 BLNP predictions

The BLNP calculations incorporate all known perturbative and power corrections and interpolation between the HQE and SF regions blnp1 (); blnp2 (); blnp3 (). The differential and partially integrated spectra for the inclusive decay are calculated in perturbative theory at next-to-leading order (NLO) in renormalization-group, and at the leading power in the heavy quark expansion. Formulas for the triple differential rate of and for the photon spectrum are convolution integrals of weight functions with the shape function renormalized at the intermediate scale . The ansatz for the leading SF depends on two parameters, and ; subleading SFs are treated separately.

The SF parameters in the kinetic scheme are determined by fits to moments of the hadron mass and lepton energy spectra from inclusive decays and either additional photon energy moments in decays or by applying a constraint on the -quark mass, . These parameters are translated from the kinetic to the SF mass scheme neubert_loops ().

The impact of the uncertainties in these SFs are estimated by varying the scale parameters and choices of different subleading SF. The next-to-next-to-leading order (NNLO) corrections were studied in detail blnp_nnlo (). In extractions of , the choice   introduces for the NNLO corrections significant shifts to lower values of the partial decay rates, by , while at the same time reducing the perturbative uncertainty on the scale . At NLO, small changes of the value of impact the agreement between the NLO and NNLO results. We adopt the authors’ recommendation and use values   and  , as the default. The results obtained in the SF mass scheme with the constraint and   are and  hfag12 (). The contours for different choices of these parameters are presented in Fig. 3.

Figure 3: The shape function parameters and in the kinetic scheme (HFAG 2014): fit to data with constraint on the -quark mass (solid line, solid triangle); fit to data ( GeV, ) (dotted line, solid square). Translation of fit to data with constraint on the -quark mass (short dashed line, open triangle); translation of fit to data with  GeV, (dash-dotted line, open square). The previous BABAR endpoint analysis babarVub () was based on a fit (long dashed line, open circle). The contours represent .

In the BLNP framework, the extraction of is based on the predicted partial rate  neubert_extract () for decays and the measurement of . The predictions for total decay rate are:

(2)
(3)
(4)
(5)

The estimated SF uncertainty and total theoretical uncertainty on are about and , respectively (for  ).

v.3 GGOU predictions

The GGOU calculations ggou1 (); ggou2 () of the triple differential decay rate include all perturbative and nonperturbative effects through and . The Fermi motion is parametrized in terms of a single light-cone function for each structure function and for any value of , accounting for all subleading effects. The calculations are based on the kinetic mass scheme, with a hard cutoff at .

The SF parameters are determined by fits to moments of the hadron mass and lepton energy spectra from inclusive decays, and either including photon energy moments in decays or by applying a constraint on the -quark mass. The results obtained in the kinetic scheme with the constraint are and  hfag12 (). The contours for the resulting SF parameters are presented in Fig. 3.

The uncertainties are estimated as prescribed in  ggou2 (). To estimate the uncertainties of the higher order perturbative corrections, the hard cutoff is varied in the range  . Combined with an estimate of of the uncertainty in corrections, this is taken as the overall uncertainty of these higher order perturbative and nonperturbative calculations. The uncertainty due to weak annihilation is assumed to be asymmetric, i.e., it tends to decrease . The uncertainty in the modeling of the tail of the distribution is estimated by comparing two different assumptions for the range .

The extraction of is based on the measured partial BF , and the GGOU prediction for the partial normalized rate . The predictions for the total decay rate are,

(6)
(7)

The estimated uncertainties on for the SF and the total theoretical uncertainty are about and , respectively (for  GeV/c).

v.4 DGE predictions

The DGE gardi () is a general formalism for inclusive distributions near the kinematic boundaries. In this approach, the on-shell calculation, converted to hadronic variables, is directly used as an approximation to the decay spectrum without the use of a leading-power nonperturbative function. The perturbative expansion includes NNLO resummation in momentum space as well as full and corrections. The triple differential rate of was calculated dge1 (); dge3 (). The DGE calculations rely on the renormalization scheme.

Based on the prescriptions by the authors dge3 (), we have estimated the uncertainties in these calculations and their impact on . The theoretical uncertainty is obtained by accounting for the uncertainty in and  pdg2014 (). The renormalization scale factor =1.0 is varied between 0.5 and 2.0, and the default values of are changed to to assess the uncertainties in the nonperturbative effects.

DGE predict the shape of differential electron spectrum, but do not provide a normalization. Thus we rely on the DGE predictions for , the fraction of decays in the interval , and an independent prediction for the normalized total decay rate, ps dge1 () to derive (the current value of  pdg2014 () is used to calculate ).

The estimated total theoretical uncertainty on for DGE calculations is about (for  ).

Vi Analysis

vi.1 Event Selection

To select events with a candidate electron from a semileptonic meson decay, we apply the following criteria:

  • Electron selection: We select events with at least one electron candidate in the c.m. momentum range and within the polar angle acceptance in the laboratory frame of . Within these constraints the identification efficiency for electrons exceeds . The average hadron misidentification rate is about .

  • Track multiplicity: To suppress background from non- events, primarily low-multiplicity QED processes, including pair production and annihilation ( represents a or quark), we reject events with fewer than four charged tracks.

  • suppression: To reject electrons from the decay , we combine the selected electron with other electron candidates of opposite charge and reject the event if the invariant mass of any pair is consistent with a decay, .

If an event in the remaining sample has more than one electron that passes this selection, the one with the highest momentum is chosen as the signal candidate.

To further suppress non- events we build a neural network (NN) with the following input variables which rely on the momenta of all charged particles and energies of photons above 50  detected in the event:

  • , the ratio of the second to the zeroth Fox-Wolfram moments foxw (), calculated from all detected particles in the event [Fig. 4(a)].

  • , where the sum includes all detected particles except the electron, and is the angle between the momentum of particle and the direction of the electron momentum [Fig. 4(b)].

  • , the cosine of the angle between the electron momentum and the axis of the thrust of the rest of the event [Fig. 4(c)].

Figure 4: The number of events before the NN selection, as a function of (a) , (b) , (c) : on-resonance data (triangles), the sum of simulated events and off-resonance data (solid histogram), MC simulated events (dashed histogram), and off-resonance data (dotted histogram). For comparison, the distributions for events with a signal decay (dash-dotted histogram) are shown (scaled by a factor of 50). Ratio = [ (MC) + off-resonance]/on-resonance.

The distribution of the NN output is shown in Fig. 5. Only events with positive output values are retained, this selects of and non- events. The positive output corresponds the selection with maximum significance level.

This selection results in an efficiency of % for decays; the dependence on the electron momentum is shown in Fig. 6.

Figure 5: The distribution of events as a function of the NN output: On-resonance data (triangles), the sum of MC simulated and off-resonance data (solid histogram), MC simulated events (dashed histogram), off-resonance data (dotted histogram). For comparison, the distributions for events with a signal decay (dash-dotted histogram), are shown (scaled by a factor of 50). Ratio = [ (MC) + off-resonance]/on-resonance.
Figure 6: Selection efficiency for events with a decay as a function of the MC-generated electron momentum. The error bars represent the statistical uncertainties only.

vi.2 Background subtraction

The selected sample of events from the on-resonance data contains considerable background from events and non- events. The background is dominated by primary electrons from semileptonic decays and secondary electrons from decays of charm mesons and mesons. Hadronic decays contribute mostly via the misidentification of charged particles. Non- events originate from annihilation and lepton pair production, especially .

vi.2.1 Non- background

To determine the momentum-dependent shape of the non- background, we perform a binned fit to the off-resonance data in the momentum interval 0.8 to 3.5 , combined with on-resonance data in the momentum interval 2.8 to 3.5 , i.e., above the endpoint for electrons from decays. Since the c.m. energy for the off-resonance data is lower than for the on-resonance data, we scale the electron momenta for the off-resonance data by the ratio of the c.m. energies.

The relative normalization for the two data samples lumi (); detector () is

where and refer to the c.m. energy squared and integrated luminosity of the on- and off-resonance data. The statistical uncertainty of of is based on the number of detected pairs used for the measurement of the integrated luminosity; the relative systematic uncertainty on the ratio is estimated to be .

The binned for the fit to the electron spectrum for selected non- events is defined as

(8)

Here and refer to the number of selected events in the off- and on-resonance samples for momentum bins and , respectively, represents the set of free parameters of the fit, and is the uncertainty on . To fit the momentum spectrum, we have chosen an exponential expression of the form,

(9)

We perform three different fits and they all describe the data well, see Table 2. The results of the fit to both on- and off-resonance data are shown in Fig. 7. In the fit to the full on-resonance data spectrum, the constraint on the ratio is applied.

Data
Off-resonance only
Off- and on-resonance
Off- and on-resonance
with constrained
Table 2: Results of the fit to the non- background
Figure 7: The combined fit to off-resonance data in the momentum interval of 0.8 to 3.5 and to on-resonance data in the momentum interval of 2.8 to 3.5 , with the constraint on ; (a) comparison of off-resonance data (solid squares), on-resonance data (open circles) and fitted function; (b) : off-resonance data (solid histogram), on-resonance data (dotted histogram)

In Fig. 8(a) the data and the result of this fit to the non- background are compared to the full spectrum of the highest momentum electron in selected events observed in the on-resonance data sample. By subtracting the fitted non- background we obtain the inclusive electron spectrum from decays, shown in Fig. 8(b). Above 2.3 , an excess of events corresponding to the expected signal decays is observed above the background.

Figure 8: Electron momentum spectra in the rest frame: (a) on-resonance data (solid squares), scaled off-resonance data (solid triangles), the solid line shows the results of the fit to the continuum component using both on-resonance and off-resonance data. (b) On-resonance data with non- background subtracted (open squares), MC without decays (open triangles).

vi.2.2 Background and fit to the electron spectrum

Figure 9: The simulated contributions to events as a function of the momentum for electron candidates (a) all events (solid histogram), primary electrons (dashed histogram), secondary electrons (dotted histogram), misidentified hadrons (dash-dotted histogram). (b) Primary electrons: (solid histogram), (dashed histogram), (dotted histogram), (long-dash histogram), (long-dash-dotted histogram), signal decays (dash-dotted histogram). (c) Secondary electrons from: (solid histogram), (dashed histogram), (dotted histogram), (dash-dotted histogram), (long-dash histogram), conversion (long-dash-dot histogram), other (dash-three-dot histogram).

The background spectrum is composed of several contributions, dominated by primary electrons from various semileptonic decays, and secondary electrons from decays of , and mesons or photon conversions. Hadronic decays contribute mostly via charged-particle misidentification, primarily at low momenta. The MC simulated contributions from different background sources are shown in Fig. 9.

We estimate the total background by a simultaneous fit to the observed inclusive electron spectra in off- and on-resonance data to the sum of the signal and individual background contributions. For the individual signal and background contributions, we rely on the MC simulated shapes of the spectra (including some corrections), and treat their relative normalization as free parameters in the fit. For this extended fit, we expand the definition as follows,

(10)

with

Here and refer to the number of selected events in the off- and on-resonance samples for momentum bins and , respectively, and is the set of free parameters of the fit.

The first sum refers to the on-resonance data. The electron spectrum is approximated as , where the free parameters are the BFs for the individual contributions representing the signal decays, the background of primary electrons from semileptonic decays (, , , and ), and secondary electrons from decays of and mesons ( fitted as a scale factor relative to the MC input). Smaller contributions to the background are fixed in the fit, for example, electrons from and decays, photon conversions, and hadrons misidentified as electrons. Their simulations and rates rely on independent control samples and are well understood.

The momentum spectra are histograms taken from MC simulations. The array refers to the form factor parameters , , , and and other fixed parameters such as form factor parameters for and decays. is the statistical uncertainty of the number of simulated events in the -th bin. is the covariance matrix for the detection of electrons among charged tracks. It includes electron identification and misidentification of pions, kaons, protons and antiprotons, studied with large data control samples. only includes the uncertainty for the shape of the momentum distribution due to particle identification (PID) uncertainties. The uncertainty of the relative normalization due to PID uncertainties is about and is taken as a systematic uncertainty. The last two terms of Eq. (10) refer to quantities that are well known.

In the fit, and contributions are highly correlated. The BF for is constrained to  pdg2014 (). The luminosity ratio is constrained to the value  lumi (); detector ().

The fit is performed in the momentum range from 0.8 to 3.5, in bins of 50. At lower momenta, the data determine the relative normalization of the various background contributions, allowing for an extrapolation of these backgrounds into the endpoint region. This fitting procedure was chosen in recognition of the fact that the current BFs for the individual decays are not sufficiently well measured to perform an adequate background subtraction. The shape of the signal spectrum is fixed in the fit to one of the theoretical predictions, its normalization is a free parameter. In a given momentum interval, the excess of events above the sum of the fitted backgrounds is taken as the number of signal events.

To reduce a potential systematic bias from the theoretically predicted shape of the signal spectrum in a region where these calculations are less reliable, events in the interval from 2.1 to 2.7 are combined into a single bin. The lower limit of this bin is chosen so as to retain sensitivity to the steeply falling background distributions, while containing a large fraction of the signal events in a region where the background is low. The upper limit at 2.7  is chosen to limit the non- background at higher momenta where the signal contributions become very small compared to the non- background.

The fits are performed separately for the different theoretical predictions of the signal spectrum, introduced in Sec. V. The results of these fits are shown in Table 3, and for the fit with the GGOU signal spectrum, the correlation matrix is presented in Table 4. The differences of the correlation matrices for the fit with DN, BLNP, GGOU and DGE signal spectra are small. The difference in the fit results for the BF is primarily due to the difference between the various predictions for the fraction of the signal spectrum in the high-momentum range. The fitted BFs for the dominant decays agree reasonably well with expectations pdg2014 ().

DN BLNP GGOU DGE
constraint constraint
Table 3: Results of the fit to the electron spectrum, with the non- background subtracted and with all entries in the interval from 2.1 to 2.7 in a single bin, for four different theoretical predictions of the spectrum. Fitted BFs(%), averaged over charged and neutral mesons, for the signal , the background (constrained), , , , +, and scale factors relative to reweighted MC inputs for secondary , and the luminosity ratio (constrained) are presented. The contributions to the