Measurement of the form factors of the decay B^{0}\to D^{*-}\ell^{+}\nu_{\ell} and determination of the CKM matrix element |V_{cb}|

Measurement of the form factors of the decay and determination of the CKM matrix element

W. Dungel Institute of High Energy Physics, Vienna    C. Schwanda Institute of High Energy Physics, Vienna    I. Adachi High Energy Accelerator Research Organization (KEK), Tsukuba    H. Aihara Department of Physics, University of Tokyo, Tokyo    T. Aushev École Polytechnique Fédérale de Lausanne (EPFL), Lausanne Institute for Theoretical and Experimental Physics, Moscow    T. Aziz Tata Institute of Fundamental Research, Mumbai    A. M. Bakich School of Physics, University of Sydney, NSW 2006    V. Balagura Institute for Theoretical and Experimental Physics, Moscow    E. Barberio University of Melbourne, School of Physics, Victoria 3010    M. Bischofberger Nara Women’s University, Nara    A. Bozek H. Niewodniczanski Institute of Nuclear Physics, Krakow    M. Bračko University of Maribor, Maribor J. Stefan Institute, Ljubljana    T. E. Browder University of Hawaii, Honolulu, Hawaii 96822    P. Chang Department of Physics, National Taiwan University, Taipei    Y. Chao Department of Physics, National Taiwan University, Taipei    A. Chen National Central University, Chung-li    P. Chen Department of Physics, National Taiwan University, Taipei    B. G. Cheon Hanyang University, Seoul    R. Chistov Institute for Theoretical and Experimental Physics, Moscow    I.-S. Cho Yonsei University, Seoul    K. Cho Korea Institute of Science and Technology Information, Daejeon    K.-S. Choi Yonsei University, Seoul    Y. Choi Sungkyunkwan University, Suwon    J. Dalseno Max-Planck-Institut für Physik, München Excellence Cluster Universe, Technische Universität München, Garching    A. Drutskoy University of Cincinnati, Cincinnati, Ohio 45221    S. Eidelman Budker Institute of Nuclear Physics, Novosibirsk Novosibirsk State University, Novosibirsk    H. Ha Korea University, Seoul    J. Haba High Energy Accelerator Research Organization (KEK), Tsukuba    H. Hayashii Nara Women’s University, Nara    Y. Horii Tohoku University, Sendai    Y. Hoshi Tohoku Gakuin University, Tagajo    W.-S. Hou Department of Physics, National Taiwan University, Taipei    T. Iijima Nagoya University, Nagoya    K. Inami Nagoya University, Nagoya    M. Iwabuchi Yonsei University, Seoul    Y. Iwasaki High Energy Accelerator Research Organization (KEK), Tsukuba    N. J. Joshi Tata Institute of Fundamental Research, Mumbai    T. Julius University of Melbourne, School of Physics, Victoria 3010    J. H. Kang Yonsei University, Seoul    T. Kawasaki Niigata University, Niigata    H. J. Kim Kyungpook National University, Taegu    H. O. Kim Kyungpook National University, Taegu    J. H. Kim Korea Institute of Science and Technology Information, Daejeon    M. J. Kim Kyungpook National University, Taegu    Y. J. Kim The Graduate University for Advanced Studies, Hayama    K. Kinoshita University of Cincinnati, Cincinnati, Ohio 45221    B. R. Ko Korea University, Seoul    P. Krokovny High Energy Accelerator Research Organization (KEK), Tsukuba    T. Kuhr Institut für Experimentelle Kernphysik, Karlsruher Institut für Technologie, Karlsruhe    T. Kumita Tokyo Metropolitan University, Tokyo    A. Kuzmin Budker Institute of Nuclear Physics, Novosibirsk Novosibirsk State University, Novosibirsk    Y.-J. Kwon Yonsei University, Seoul    S.-H. Kyeong Yonsei University, Seoul    M. J. Lee Seoul National University, Seoul    S.-H. Lee Korea University, Seoul    J. Li University of Hawaii, Honolulu, Hawaii 96822    A. Limosani University of Melbourne, School of Physics, Victoria 3010    C. Liu University of Science and Technology of China, Hefei    Y. Liu Department of Physics, National Taiwan University, Taipei    D. Liventsev Institute for Theoretical and Experimental Physics, Moscow    R. Louvot École Polytechnique Fédérale de Lausanne (EPFL), Lausanne    A. Matyja H. Niewodniczanski Institute of Nuclear Physics, Krakow    S. McOnie School of Physics, University of Sydney, NSW 2006    H. Miyata Niigata University, Niigata    R. Mizuk Institute for Theoretical and Experimental Physics, Moscow    G. B. Mohanty Tata Institute of Fundamental Research, Mumbai    T. Mori Nagoya University, Nagoya    E. Nakano Osaka City University, Osaka    M. Nakao High Energy Accelerator Research Organization (KEK), Tsukuba    Z. Natkaniec H. Niewodniczanski Institute of Nuclear Physics, Krakow    S. Neubauer Institut für Experimentelle Kernphysik, Karlsruher Institut für Technologie, Karlsruhe    S. Nishida High Energy Accelerator Research Organization (KEK), Tsukuba    O. Nitoh Tokyo University of Agriculture and Technology, Tokyo    T. Nozaki High Energy Accelerator Research Organization (KEK), Tsukuba    T. Ohshima Nagoya University, Nagoya    S. Okuno Kanagawa University, Yokohama    S. L. Olsen Seoul National University, Seoul University of Hawaii, Honolulu, Hawaii 96822    G. Pakhlova Institute for Theoretical and Experimental Physics, Moscow    C. W. Park Sungkyunkwan University, Suwon    H. Park Kyungpook National University, Taegu    H. K. Park Kyungpook National University, Taegu    M. Petrič J. Stefan Institute, Ljubljana    L. E. Piilonen IPNAS, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061    M. Prim Institut für Experimentelle Kernphysik, Karlsruher Institut für Technologie, Karlsruhe    M. Röhrken Institut für Experimentelle Kernphysik, Karlsruher Institut für Technologie, Karlsruhe    M. Rozanska H. Niewodniczanski Institute of Nuclear Physics, Krakow    S. Ryu Seoul National University, Seoul    H. Sahoo University of Hawaii, Honolulu, Hawaii 96822    K. Sakai Niigata University, Niigata    Y. Sakai High Energy Accelerator Research Organization (KEK), Tsukuba    O. Schneider École Polytechnique Fédérale de Lausanne (EPFL), Lausanne    A. J. Schwartz University of Cincinnati, Cincinnati, Ohio 45221    K. Senyo Nagoya University, Nagoya    M. E. Sevior University of Melbourne, School of Physics, Victoria 3010    J.-G. Shiu Department of Physics, National Taiwan University, Taipei    J. B. Singh Panjab University, Chandigarh    P. Smerkol J. Stefan Institute, Ljubljana    A. Sokolov Institute of High Energy Physics, Protvino    S. Stanič University of Nova Gorica, Nova Gorica    M. Starič J. Stefan Institute, Ljubljana    T. Sumiyoshi Tokyo Metropolitan University, Tokyo    S. Suzuki Saga University, Saga    S. Tanaka High Energy Accelerator Research Organization (KEK), Tsukuba    G. N. Taylor University of Melbourne, School of Physics, Victoria 3010    Y. Teramoto Osaka City University, Osaka    K. Trabelsi High Energy Accelerator Research Organization (KEK), Tsukuba    S. Uehara High Energy Accelerator Research Organization (KEK), Tsukuba    T. Uglov Institute for Theoretical and Experimental Physics, Moscow    Y. Unno Hanyang University, Seoul    S. Uno High Energy Accelerator Research Organization (KEK), Tsukuba    P. Urquijo University of Melbourne, School of Physics, Victoria 3010    G. Varner University of Hawaii, Honolulu, Hawaii 96822    K. E. Varvell School of Physics, University of Sydney, NSW 2006    K. Vervink École Polytechnique Fédérale de Lausanne (EPFL), Lausanne    C. H. Wang National United University, Miao Li    P. Wang Institute of High Energy Physics, Chinese Academy of Sciences, Beijing    M. Watanabe Niigata University, Niigata    Y. Watanabe Kanagawa University, Yokohama    R. Wedd University of Melbourne, School of Physics, Victoria 3010    K. M. Williams IPNAS, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061    E. Won Korea University, Seoul    Y. Yamashita Nippon Dental University, Niigata    D. Zander Institut für Experimentelle Kernphysik, Karlsruher Institut für Technologie, Karlsruhe    Z. P. Zhang University of Science and Technology of China, Hefei    P. Zhou Wayne State University, Detroit, Michigan 48202    V. Zhulanov Budker Institute of Nuclear Physics, Novosibirsk Novosibirsk State University, Novosibirsk    T. Zivko J. Stefan Institute, Ljubljana
July 25, 2019
Abstract

This article describes a determination of the Cabibbo-Kobayashi-Maskawa matrix element  from the decay using 711 fb of Belle data collected near the  resonance. We simultaneously measure the product of the form factor normalization and the matrix element as well as the three parameters , and , which determine the form factors of this decay in the framework of the Heavy Quark Effective Theory. The results, based on about 120,000 reconstructed decays, are , , and . The branching fraction of is measured at the same time; we obtain a value of . The errors correspond to the statistical and systematic uncertainties. These results give the most precise determination of the form factor parameters and to date. In addition, a direct, model-independent determination of the form factor shapes has been carried out.

pacs:

The Belle Collaboration

I Introduction

The study of the decay  is important for several reasons. The total rate is proportional to the magnitude of the Cabibbo-Kobayashi-Maskawa(CKM) matrix element  Kobayashi:1973fv (); Cabibbo:1963yz () squared. Experimental investigation of the form factors of the decay can check theoretical models and possibly provide input to more detailed theoretical approaches. In addition, is a major background for charmless semileptonic  decays, such as , or semileptonic  decays with large missing energy, including . Precise knowledge of the form factors in the  decay will thus help to reduce systematic uncertainties in these analyses.

This article is organized as follows: After introducing the theoretical framework for the study of  decays in Section II, the experimental procedure is presented in detail in Section  III. This is followed by a discussion of our results and the systematic uncertainties assuming the form factor parameterization of Caprini et al. Caprini:1997mu () in Section  IV. Finally, a measurement of the form factor shapes is described in Section V.

This paper supersedes our previous result Abe:2001cs (), based on a subset of the data used in this analysis.

Ii Theoretical framework

ii.1 Kinematic variables

The decay  ref:0 () proceeds through the tree-level transition shown in Fig. 1. Below we will follow the formulation proposed in reviews  Neubert:1993mb (); Richman:1995wm (), where the kinematics of this process are fully characterized by four variables as discussed below.

Figure 1: Quark-level Feynman diagram for the decay .

The first is a function of the momenta of the and mesons, labeled and defined by

(1)

where and are the masses of the and the mesons (5.279 and 2.010 GeV/, respectively Amsler:2008zzb ()), and are their four-momenta, and . In the  rest frame the expression for reduces to the Lorentz boost . The ranges of and are restricted by the kinematics of the decay, with corresponding to

(2)

and to

(3)

The point  is also referred to as zero recoil.

The remaining three variables are the angles shown in Fig. 2:

  • , the angle between the direction of the lepton and the direction opposite the meson in the virtual  rest frame;

  • , the angle between the direction of the  meson and the direction opposite the meson in the rest frame;

  • , the angle between the plane formed by the decay and the plane formed by the  decay, defined in the meson rest frame.

Figure 2: Definition of the angles , and for the decay , .

ii.2 Four-dimensional decay distribution

Three helicity amplitudes, labeled , , and , can be used to describe the Lorentz structure of the  decay amplitude. These quantities correspond to the three polarization states of the , two transverse and one longitudinal. When neglecting the lepton mass, i.e., considering only electrons and muons, these amplitudes are expressed in terms of the three functions , , and as follows Neubert:1993mb ():

(4)

where

(5)
(6)

with and . The functions and are defined in terms of the axial and vector form factors as,

(7)
(8)

By convention, the function is defined as

(9)

The axial form factor dominates for . Furthermore, in the limit of infinite - and -quark masses, a single form factor describes the decay, the so-called Isgur-Wise function Isgur:1989vq (); Isgur:1989ed ().

In terms of the three helicity amplitudes, the fully differential decay rate is given by

(10)

with  . Four one-dimensional decay distributions can be obtained by integrating this decay rate over all but one of the four variables, , , , or . The differential decay rate as a function of is

(11)

where

and is a known phase space factor,

A value of the form factor normalization is predicted by Heavy Quark Symmetry (HQS) Neubert:1993mb () in the infinite quark-mass limit. Lattice QCD can be utilized to calculate corrections to this limit. The most recent result obtained in unquenched lattice QCD is  Bernard:2008dn ().

ii.3 Form factor parameterization

A parameterization of form factors , , and can be obtained using heavy quark effective theory (HQET). Perfect heavy quark symmetry implies that , i.e., the form factors and are identical for all values of and differ from only by a simple kinematic factor. Corrections to this approximation have been calculated in powers of and the strong coupling constant . Various parameterizations in powers of have been proposed. We adopt the following expressions derived by Caprini, Lellouch and Neubert Caprini:1997mu (),

(12)
(13)
(14)

where . In addition to the form factor normalization , these expressions contain three free parameters, , , and . The values of these parameters cannot be calculated in a model-independent manner. Instead, they have to be extracted by an analysis of experimental data.

Iii Experimental procedure

iii.1 Data sample and event selection

The data used in this analysis were taken with the Belle detector unknown:2000cg () at the KEKB asymmetric-energy  collider Kurokawa:2001nw (). 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 threshold Cherenkov 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 superconducting solenoid coil that provides a 1.5 T magnetic field. An iron flux-return located outside of the coil is instrumented to detect mesons and to identify muons (KLM). The detector is described in detail in Ref. unknown:2000cg (). Two inner detector configurations were used. A 2.0 cm beampipe and a 3-layer silicon vertex detector were used for the first sample of million pairs, while a 1.5 cm beampipe, a 4-layer silicon detector and a small-cell inner drift chamber were used to record the remaining million pairs svd2 ().

The data sample consists of 711 fb taken at the  resonance, or about million  events. Another 88 fb taken at 60 MeV below the resonance are used to estimate the non- (continuum) background. The off-resonance data is scaled by the integrated on- to off-resonance luminosity ratio corrected for the  dependence of the  cross section.

This data sample contains events recorded with two different detector setups as well as two different tracking algorithms and large differences in the input files used for Monte Carlo generation. To ensure that no systematic uncertainty appears due to inadequate consideration of these differences, we separate the data sample into four distinct sets labeled A (141 fb), B (274 fb), C (189 fb) and D (107 fb), where the number in parentheses indicates the integrated luminosity corresponding to the individual samples.

Monte Carlo generated samples of decays equivalent to about three times the integrated luminosity are used in this analysis. Monte Carlo simulated events are generated with the Evtgen program Lange:2001uf (), and full detector simulation based on GEANT Brun:1987ma () is applied. QED final state radiation in  decays is added using the PHOTOS package Barberio:1993qi ().

Hadronic events are selected based on the charged track multiplicity and the visible energy in the calorimeter. The selection is described in detail elsewhere Abe:2001hj (). We also apply a requirement on the ratio of the second to the zeroth Fox-Wolfram moment Fox:1978vu (), , to reject continuum events.

iii.2 Event reconstruction

Charged tracks are required to originate from the interaction point by applying the following selections on the impact parameters in the and directions:  cm and  cm, respectively. In addition, we demand at least one associated hit in the SVD detector. For pion and kaon candidates, the Cherenkov light yield from the ACC, the time-of-flight information from TOF, and from the CDC are required to be consistent with the appropriate mass hypothesis.

Neutral  meson candidates are reconstructed in the decay channel. We fit the charged tracks to a common vertex and reject the  candidate if the -probability is below . The reconstructed  mass is required to lie within  MeV/ of the nominal mass of 1.865 GeV/  Amsler:2008zzb (), corresponding to about 2.5 times the experimental resolution measured from data.

The  candidate is combined with an additional charged pion (oppositely charged with respect to the kaon candidate) to form a  candidate. Due to the kinematics of the decay, the momentum of this pion does not exceed 350 MeV/. It is therefore referred to as the “slow” pion, . No impact parameter or SVD hit requirements are applied for . Again, a vertex fit is performed and the same vertex requirement is applied. The invariant mass difference between the and the  candidates, , is required to be less than 165 MeV/. This selection is tightened after the background estimation described below. Additional continuum suppression is achieved by requiring that the  momentum in the c.m. frame be below 2.45 GeV/.

Finally, the  candidate is combined with an oppositely charged lepton (electron or muon). Electron candidates are identified using the ratio of the energy detected in the ECL to the track momentum, the ECL shower shape, position matching between track and ECL cluster, the energy loss in the CDC, and the response of the ACC counters. Muons are identified based on their penetration range and transverse scattering in the KLM detector. In the momentum region relevant to this analysis, charged leptons are identified with an efficiency of about 90% while the probability to misidentify a pion as an electron (muon) is 0.25% (1.4%) Hanagaki:2001fz (); Abashian:2002bd (). Lepton tracks have to be associated with at least one SVD hit. In the laboratory frame, the momentum of the electron (muon) is required to be greater than 0.30 GeV/ (0.60 GeV/). We also require the lepton momentum in the c.m. frame to be less than 2.4 GeV/ to reject continuum. More stringent lepton requirements are imposed later in the analysis.

For electron candidates we attempt bremsstrahlung recovery by searching for photons within a cone of degrees around the electron track. If such a photon is found, it is merged with the electron and the sum of the momenta is taken to be the lepton momentum.

iii.3 Background estimation

Figure 3: Result of the fits to the (, , ) distributions in the mode (left) and mode (right) of the sub-sample B. The bin boundaries are discussed in the text. The points with error bars are continuum-subtracted on-resonance data. Where not shown, the uncertainties are smaller than the black markers. The histograms are, top to bottom, the signal component, background, signal correlated background, uncorrelated background, fake component and fake component.

Because we do not reconstruct the other  meson in the event, the momentum is a priori unknown. However, in the c.m. frame, one can show that the  direction lies on a cone around the -axis ref:2 () with an opening angle defined by:

(15)

In this expression, is half of the c.m. energy and is . The quantities , and are calculated from the reconstructed  system.

This cosine is also a powerful discriminator between signal and background: Signal events should lie in the interval , although – due to finite detector resolution – about 5% of the signal is reconstructed outside this interval. The background, on the other hand, does not have this restriction.

The signal lies predominantly in the region defined by 144 MeV/ 147 MeV/ and GeV/ ( GeV/) for electrons (muons). The region outside these thresholds can be used to estimate the background level.

We therefore perform a fit to the three-dimensional ) distributions. The range between and is divided into 30 bins. The () range is divided into five (two) bins, with bin boundaries at MeV ( GeV/ for electrons and GeV/ for muons).

The background contained in the final sample has the following six components:

  1. : background from  decays with or and from non-resonant events, where the lepton has been correctly identified;

  2. correlated background: background from processes other than decays in which the and the lepton originate from the same  meson, e.g, , ;

  3. uncorrelated background: the and the lepton come from different  mesons and the lepton is not from a  decay;

  4. fake lepton: the charged lepton candidate is a misidentified hadron while the  candidate may or may not be correctly reconstructed;

  5. fake : the  candidate is misreconstructed; the lepton candidate is identified correctly, but it is not from a  decay;

  6. continuum: background from processes.

To model the component, which consists of a total of four resonant (, , , ) and one non-resonant mode for both neutral and charged decays, we reweight the branching ratios of each subcomponent to match the values reported by the Particle Data Group Amsler:2008zzb (). For the resonant parts, only products of branching ratios are available and consequently we reweight these products. The shape of the momentum distributions is also reweighted in 22 bins of to match the predictions of the LLSW model Leibovich:1997em (); Urquijo:2006wd ().

All of the background components are modeled by MC simulation except for continuum events; these are modeled by off-resonance data. For muon events, the shape of the fake lepton background is corrected by the ratio of the pion fake rate in the experimental data over the same quantity in the Monte Carlo, as measured using  decays. The lepton identification efficiency is corrected by the ratio between experimental data and Monte Carlo in events Hanagaki:2001fz (); Abashian:2002bd (). The (, , ) distribution in the data is fitted using the TFractionFitter algorithm Barlow:1993dm () within ROOT Brun:1996ro (). The fit is done separately in each of eight subsamples defined by the experiment range and the lepton type. The results are given in Table 1. Figure 3 shows plots of the projections in for subsample B.

In all fits, the continuum normalization is fixed to the on- to off-resonance luminosity ratio, corrected for the  dependence of the  cross section. In general, the normalizations obtained by the fit agree well with the MC expectations except for the  component and the fake component, which are overestimated in the MC. After the background determination only candidates satisfying the requirements , and GeV/ ( GeV/) for electrons (muons) are considered for further analysis.

A, A, B, B,
Num. Candidates 14802 14203 29217 26894

Signal events
11609 181 11139 190 23029 280 21002 258
Signal () 78.43 1.22 78.43 1.34 78.82 0.96 78.09 0.96

()
5.63 0.78 4.02 0.86 4.32 0.66 3.90 0.60

Signal correlated ()
1.07 0.17 1.41 0.25 1.33 0.16 1.71 0.19

Uncorrelated ()
7.24 0.35 6.01 0.40 7.19 0.31 6.31 0.29

Fake ()
0.36 0.17 1.99 0.34 0.50 0.17 2.10 0.23

Fake ()
2.59 0.12 2.81 0.13 3.07 0.11 2.96 0.10

Continuum ()
4.68 0.54 5.32 0.59 4.77 0.38 4.93 0.40
C, C, D, D,
Num. Candidates 22056 20428 15871 14719

Signal events
17301 240 15513 235 12365 189 11469 205
Signal () 78.44 1.09 75.94 1.15 77.91 1.19 77.92 1.39

()
5.15 0.71 5.22 0.71 4.54 0.72 4.67 0.86

Signal correlated ()
1.56 0.27 2.07 0.37 2.01 0.26 2.73 0.43

Uncorrelated ()
6.35 0.35 6.01 0.33 7.33 0.38 6.30 0.40

Fake ()
0.75 0.18 2.26 0.28 0.30 0.19 1.68 0.38

Fake ()
2.86 0.12 2.69 0.11 2.89 0.13 2.80 0.14

Continuum ()
4.88 0.45 5.81 0.51 5.02 0.53 3.89 0.49

Table 1: The signal yield and the signal and background fractions (given in ) for selected events passing the requirements , and GeV/ ( GeV/) for electron (muon) channels.

iii.4 Kinematic variables

Figure 4: Reconstruction of the  direction. Refer to the text for details.

To calculate the four kinematic variables defined in Eq. 1 and Fig. 2, , and – which characterize the  decay, we need to determine the  rest frame. The  direction is already known to lie on a cone around the -axis with an opening angle in the c.m. frame, Equation (15). To initially determine the  direction, we estimate the c.m. frame momentum vector of the non-signal meson by summing the momenta of the remaining particles in the event ( ref:2 ()) and choose the direction on the cone that minimizes the difference to , Fig. 4.

To obtain , we exclude tracks passing far from the interaction point. The minimal requirements depend on the transverse momentum of the track, , and are set to  cm ( cm,  cm) or  cm (50 cm, 20 cm) for a track  MeV/ ( MeV/,  MeV/). Track candidates that are compatible with a multiply reconstructed track generated by a low-momentum particle spiraling in the central drift chamber are also checked for and only one of the multiple tracks is considered. Unmatched clusters in the barrel region must have an energy greater than 50 MeV. For clusters in the forward (backward) region, the threshold is at 100 MeV (150 MeV). We then compute (in the laboratory frame) by summing the 3-momenta of the selected particles,

(16)

where the index stands for all particles passing the conditions above, and transform this vector into the c.m. frame. Note that we do not introduce any mass assumption for the charged particles. The energy component of is determined by requiring to be .

With the  rest frame reconstructed in this way, the resolutions in the kinematic variables are found to be about 0.025, 0.049, 0.050 and 13.5 for , , and , respectively.

Iv Analysis based on the parameterization of Caprini et al.

iv.1 Fit procedure

Our main goal is to extract the following quantities: the product of the form factor normalization and , (Eq. 11), and the three parameters , and that parameterize the form factors in the HQET framework (Eqs. 1214). For this, we perform a binned  fit to the , , and  distributions over nearly the entire phase space. Instead of an unbinned fit, we fit the one-dimensional projections of , , and . This avoids the difficulty of parameterizing the six background components and their correlations in four dimensions. In addition, the one-dimensional projections have sufficient statistics in each bin. However, this approach introduces bin-to-bin correlations that must be accounted for.

The distributions in , , and are divided into ten bins of equal width. The kinematically allowed values of are between and , but we restrict the fit range to values between and . In each sub-sample, there are thus 40 bins to be used in the fit. In the following, we label these bins with a single index , . The bins correspond to the bins of the  distribution, to , to , and to the distribution.

The number of produced events in the bin , , is given by

(17)

where is the number of  mesons in the data sample, and are the and branching ratios into the final state under consideration Amsler:2008zzb (), is the  lifetime Amsler:2008zzb (), and is the width obtained by integrating Eq. 10 in the kinematic variable corresponding to from the lower to the upper bin boundary (the other kinematic variables are integrated over their full range). This integration is numerical in the case of and analytic for the other variables. The expected number of events is related to as follows

(18)

Here, is the probability that an event generated in the bin  is reconstructed and passes all analysis cuts, and is the detector response matrix, i.e., it gives the probability that an event generated in the bin  is observed in the bin . Both quantities are calculated using MC simulation. is the number of expected background events, estimated as described in Sect. III.3.

Next, we calculate the variance  of . We consider the following contributions: the Poissonian uncertainty in ; fluctuations related to the efficiency, estimated by a binomial distribution with repetitions and known success probability ; a similar contribution related to using a multinomial distribution; and the uncertainty in the background contribution . This yields the following expression for ,

(19)

The first term is the Poissonian uncertainty in . The second and third terms are the binomial and multinomial uncertainties, respectively, related to the finite real data size, where () is the total number of decays (the number of reconstructed decays) into the final state under consideration (, or ) in the real data. The quantities and are calculated from a finite signal MC sample ( and ); the corresponding uncertainties are estimated by the fourth and fifth terms. Finally, the last term is the background contribution , calculated as the sum of the different background component variances. For each background component defined in Section III.3 we estimate its contribution by linear error propagation of the results determined in Section III.3. For continuum, we estimate the error in the on-resonance to off-resonance luminosity ratio to be 1.0%. These variances give the diagonal elements of the covariance matrix .

In each sub-sample we calculate the off-diagonal elements of the covariance matrix  as , where is the relative abundance of bin  in the 2-dimensional histogram obtained by plotting the kinematic variables against each other, is the relative number of entries in the 1-dimensional distribution, and is the size of the sample. Covariances are calculated for the signal and the different background components in the MC samples, and added with appropriate normalizations.

The covariance matrix is inverted numerically within ROOT Brun:1996ro () and, labeling the electron and muon mode in each sub-sample with the index ,  functions are calculated,

(20)

where is the number of events observed in bin  in data sample . We sum these two functions in each sub-sample and minimize the global with MINUIT James:1975dr ().

We have tested this fit procedure using generic MC data samples. All results are consistent with expectations and show no indication of bias.

iv.2 Investigation of the efficiency of low momentum tracks

The tracking efficiency of the Belle experiment is reproduced well by MC simulations for tracks with momenta above 200 MeV/, which we refer to as “high momentum tracks”. However, a significant portion of the momentum spectrum of the slow pions emitted in the decay lies below this boundary. For low momenta, the effects of interactions with the detector material such as multiple scattering and energy loss become important and might lead to a deviation between data and MC in the reconstruction efficiency.

We use one half of the reconstructed sample to obtain corrections to the MC reconstruction efficiency in the low momentum range, measured using real data. The second half is used to perform the analysis with a statistically independent sample. The results of the background estimation shown in Table 1 are those obtained in the samples used for the analysis. Both of the samples contain about 120,000 signal events.

The sample used to investigate the efficiency of low momentum tracks is divided into a total of six bins in . The bin borders of the first five are 50 MeV, 100 MeV, 125 MeV, 150 MeV, 175 MeV and 200 MeV. The region beyond 200 MeV/ defines the sixth bin. By subtracting the background, we obtain an estimate of the signal in data and form the ratio with the signal in MC in each bin, .

The high momentum range is used as normalization, no efficiency correction is applied there. In the lower momentum bins we obtain the ratios , which are identical to the ratio of reconstruction efficiencies in the bins and the high momentum region, . We calculate this set of ratios for the electron and muon modes and form the weighted average, separately for each of the four sub-samples. These values are applied as weights when filling the MC histograms to correct the reconstruction efficiency.

Most systematic uncertainties cancel out in the ratios . Only the uncertainties in the various background components give a small systematic contribution to the uncertainty.

This procedure assumes that the distribution of events in the spectrum is identical for data and MC. However, one of the aims of the analysis is to measure the form factor parameters that govern this distribution. Therefore, an iterative procedure is adopted: we calculate one set of corrections, apply them and perform the analysis to determine and the form factor parameters. We then calculate a new set of corrections using these results and repeat the analysis. The changes of the parameters during this iterative procedure are small and vanish after the third iteration. We assign an additional systematic uncertainty to our results based on the stability of the corrections against changes in the form factor parameters. As will be shown in Table 3, this is a negligibly small contribution.

Sample A total sample
1.248 0.102 0.022 1.285 0.114 0.028 1.259 0.076 0.019

1.317 0.099 0.041 1.577 0.131 0.036 1.436 0.078 0.030

0.804 0.076 0.017 0.768 0.093 0.020 0.795 0.058 0.015

34.8 0.5 1.2 34.6 0.6 1.2 34.7 0.4 1.2

/ n.d.f
32.2 / 36.0 31.6 / 36.0 70.9 / 76.0

0.651 0.676 0.643

Sample B
total sample
1.169 0.079 0.011 1.167 0.088 0.016 1.168 0.059 0.011

1.411 0.079 0.026 1.449 0.090 0.028 1.427 0.059 0.022

0.902 0.054 0.011 0.859 0.061 0.013 0.882 0.041 0.010

34.4 0.4 1.1 33.9 0.4 1.1 34.2 0.3 1.1
/ n.d.f 22.7 / 36.0 36.5 / 36.0 60.7 / 76.0

0.958 0.443 0.900

Sample C
total sample
1.226 0.088 0.011 1.262 0.101 0.016 1.239 0.066 0.011

1.363 0.086 0.026 1.480 0.107 0.033 1.411 0.066 0.023

0.891 0.062 0.012 0.851 0.076 0.015 0.876 0.048 0.012

34.4 0.5 1.1 33.9 0.5 1.1 34.2 0.3 1.1

/ n.d.f
38.6 / 36.0 38.2 / 36.0 81.4 / 76.0

0.352 0.370 0.314
Sample D total sample
1.321 0.102 0.019 1.174 0.106 0.020 1.247 0.073 0.014

1.448 0.109 0.041 1.230 0.089 0.031 1.330 0.069 0.027

0.791 0.081 0.019 0.931 0.071 0.015 0.864 0.053 0.014

35.4 0.6 1.2 35.7 0.6 1.2 35.6 0.4 1.2

/ n.d.f
25.1 / 36.0 42.0 / 36.0 70.1 / 76.0

0.913 0.226 0.669

Table 2: The fit results for the four sub-samples. The first two columns show results obtained by investigating only the or the channel, the third column is obtained by minimizing the sum of the values calculated for each channel. Where given, the first error is statistical, and the second is systematic.
Figure 5: Result of the fit of the four kinematic variables in the sub-sample B. The electron and muon modes are added in this plot. The points with error bars are continuum subtracted on-resonance data. Where not shown, the uncertainties are smaller than the black markers. The histograms are, top to bottom, the signal component, background, signal correlated background, uncorrelated background, fake component and fake component.

iv.3 Results of the fits and investigation of the systematic uncertainties in the subsamples

After applying all analysis cuts and subtracting backgrounds, a total of signal events are used for the analysis, divided into a total of four experimental sub-samples as mentioned above. The result of the fit to these data is shown in Fig. 5 and Table 2. The per degree of freedom, , of all fits is good. Table 2 also gives the probabilities or P-values, .

To estimate the systematic uncertainties in these results, we consider contributions from the following sources: uncertainties in the background component normalizations, uncertainty in the MC tracking efficiency, errors in the world average of and as well as in the components Amsler:2008zzb (), uncertainties in the shape of the distribution of events based on the LLSW model Leibovich:1997em (), uncertainties in the  lifetime Amsler:2008zzb (), and the uncertainties in the total number of  mesons in the data sample.

To calculate these systematic uncertainties, we consider 300 pseudo-experiments in which one of 15 parameters is randomly varied, using a normal distribution. The entire analysis chain is repeated for every pseudo-experiment and new fit results are obtained, in total for 4500 variations. One standard deviation in the pseudo-experiment fit results for a given parameter is used as the systematic uncertainty in this parameter.

The parameters varied in the pseudo-experiments are as follows:

  1. The corrections on the tracking efficiencies for low momentum tracks are varied within their respective uncertainties. To obtain the most conservative estimate, the uncertainties in different momentum bins are assumed to be fully correlated. Therefore, this component corresponds to a single parameter in the toy MC.

  2. The lepton identification efficiencies are varied within their respective uncertainties Hanagaki:2001fz (); Abashian:2002bd ().

  3. The normalization of the continuum background is not correlated with any of the other backgrounds, it is therefore varied individually within the uncertainty on the on- to off-resonance luminosity ratio, which is 1.0%.

  4. The normalizations of the remaining five background components are varied within the uncertainties listed in Table 1, while taking into account the correlations found in the background estimation described in Section III.3.

  5. Uncertainties in the composition of the D** component are accounted for by varying each of the components contributing to the background within the uncertainty reported by the Particle Data Group Amsler:2008zzb (). For the resonant modes, this is the uncertainty in the branching fraction products ; for the non-resonant mode, this is the uncertainty in .

  6. In addition, the shape of the distributions of the components is varied according to the LLSW model Leibovich:1997em () and the uncertainties on the model parameters as determined in Ref. Urquijo:2006wd ().

  7. The number of events is obtained from the product of the number of events in the sample with the branching fraction of to a pair. We vary the fraction within its uncertainty Amsler:2008zzb (). This affects both the overall normalization and the background distributions.

The uncertainties in the reconstruction of the high momentum tracks, the branching ratios and , the number of events in the sample, and the  lifetime affect only , not the form factors. Therefore, their uncertainties are considered by analytical error propagation.

iv.4 Averaging the results of the subsamples

Figure 6: Plots of the result of the averaging procedure. Projections in  vs.  (top left),  vs.  (top middle),  vs.  (top right),  vs.  (bottom left),  vs.  (bottom middle) and  vs.  (bottom right) are shown. The red dot (solid line) shows the position (1 ellipse) of the average, the blue rectangle (dashed line) the position (1 ellipse) of the sub-sample A, the green triangle (dash-dotted line) the position (1 ellipse) of the sub-sample B, the magenta diamond (dash-double dotted line) the position (1 ellipse) of the sub-sample C and the cyan cross (dash-triple dotted line) the position (1 ellipse) of the sub-sample D.

To obtain the average of the four sub-samples, which have been measured independently, we use the algorithm applied by the Heavy Flavor Averaging Group TheHeavyFlavorAveragingGroup:2010qj () to obtain the world average for from semileptonic decays. This algorithm combines both the statistical and the systematic uncertainties. The correlations of some of these errors between different samples is considered. For example, the uncertainty on the branching fraction will lead to a fully correlated systematic uncertainty in each analysis.

The average is obtained with the MINUIT package James:1975dr () by using a minimization. Here, gives the total number of fit parameters, in our case . When calculating the average of measurements of the four fit parameters , a total of values are available as inputs, which we label as . In general this number can be labeled as . Each measurement corresponds to one of the parameters , which defines a primitive map . The statistical covariance matrix of each measurement is known, as well as the correlation between the samples. The latter are zero in our case. This information allows one to construct a -dimensional covariance matrix containing the statistical uncertainties and to obtain the statistical part of the to be minimized: