The Belle Collaboration

#
Measurement of the branching fraction and $\displaystyle C\\!P$ asymmetry for
$\displaystyle B\to\bar{D}^{0}\pi$ decays

T. Bloomfield  School of Physics, University of Melbourne, Victoria 3010  M.
E. Sevior  School of Physics, University of Melbourne, Victoria 3010  I.
Adachi  High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801
SOKENDAI (The Graduate University for Advanced Studies), Hayama 240-0193  H.
Aihara  Department of Physics, University of Tokyo, Tokyo 113-0033  S. Al Said
Department of Physics, Faculty of Science, University of Tabuk, Tabuk 71451
Department of Physics, Faculty of Science, King Abdulaziz University, Jeddah
21589  D. M. Asner  Brookhaven National Laboratory, Upton, New York 11973  V.
Aulchenko  Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090
Novosibirsk State University, Novosibirsk 630090  T. Aushev  Higher School of
Economics (HSE), Moscow 101000  R. Ayad  Department of Physics, Faculty of
Science, University of Tabuk, Tabuk 71451  V. Babu  Deutsches
Elektronen–Synchrotron, 22607 Hamburg  S. Bahinipati  Indian Institute of
Technology Bhubaneswar, Satya Nagar 751007  P. Behera  Indian Institute of
Technology Madras, Chennai 600036  J. Bennett  University of Mississippi,
University, Mississippi 38677  M. Bessner  University of Hawaii, Honolulu,
Hawaii 96822  T. Bilka  Faculty of Mathematics and Physics, Charles
University, 121 16 Prague  J. Biswal  J. Stefan Institute, 1000 Ljubljana  A.
Bobrov  Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090
Novosibirsk State University, Novosibirsk 630090  G. Bonvicini  Wayne State
University, Detroit, Michigan 48202  A. Bozek  H. Niewodniczanski Institute of
Nuclear Physics, Krakow 31-342  M. Bračko  Faculty of Chemistry and Chemical
Engineering, University of Maribor, 2000 Maribor, Slovenia  J. Stefan
Institute, 1000 Ljubljana  T. E. Browder  University of Hawaii, Honolulu,
Hawaii 96822  M. Campajola  INFN - Sezione di Napoli, 80126 Napoli  Università
di Napoli Federico II, 80126 Napoli  L. Cao  University of Bonn, 53115 Bonn
D. Červenkov  Faculty of Mathematics and Physics, Charles University, 121 16
Prague  M.-C. Chang  Department of Physics, Fu Jen Catholic University, Taipei
24205  V. Chekelian  Max-Planck-Institut für Physik, 80805 München  A. Chen
National Central University, Chung-li 32054  B. G. Cheon  Department of
Physics and Institute of Natural Sciences, Hanyang University, Seoul 04763  K.
Chilikin  P.N. Lebedev Physical Institute of the Russian Academy of Sciences,
Moscow 119991  H. E. Cho  Department of Physics and Institute of Natural
Sciences, Hanyang University, Seoul 04763  K. Cho  Korea Institute of Science
and Technology Information, Daejeon 34141  S.-J. Cho  Yonsei University, Seoul
03722  S.-K. Choi  Gyeongsang National University, Jinju 52828  Y. Choi
Sungkyunkwan University, Suwon 16419  S. Choudhury  Indian Institute of
Technology Hyderabad, Telangana 502285  D. Cinabro  Wayne State University,
Detroit, Michigan 48202  S. Cunliffe  Deutsches Elektronen–Synchrotron, 22607
Hamburg  S. Das  Malaviya National Institute of Technology Jaipur, Jaipur
302017  N. Dash  Indian Institute of Technology Madras, Chennai 600036  G. De
Nardo  INFN - Sezione di Napoli, 80126 Napoli  Università di Napoli Federico
II, 80126 Napoli  R. Dhamija  Indian Institute of Technology Hyderabad,
Telangana 502285  F. Di Capua  INFN - Sezione di Napoli, 80126 Napoli
Università di Napoli Federico II, 80126 Napoli  Z. Doležal  Faculty of
Mathematics and Physics, Charles University, 121 16 Prague  T. V. Dong  Key
Laboratory of Nuclear Physics and Ion-beam Application (MOE) and Institute of
Modern Physics, Fudan University, Shanghai 200443  S. Dubey  University of
Hawaii, Honolulu, Hawaii 96822  S. Eidelman  Budker Institute of Nuclear
Physics SB RAS, Novosibirsk 630090  Novosibirsk State University, Novosibirsk
630090  P.N. Lebedev Physical Institute of the Russian Academy of Sciences,
Moscow 119991  D. Epifanov  Budker Institute of Nuclear Physics SB RAS,
Novosibirsk 630090  Novosibirsk State University, Novosibirsk 630090  T.
Ferber  Deutsches Elektronen–Synchrotron, 22607 Hamburg  D. Ferlewicz  School
of Physics, University of Melbourne, Victoria 3010  B. G. Fulsom  Pacific
Northwest National Laboratory, Richland, Washington 99352  R. Garg  Panjab
University, Chandigarh 160014  V. Gaur  Virginia Polytechnic Institute and
State University, Blacksburg, Virginia 24061  A. Garmash  Budker Institute of
Nuclear Physics SB RAS, Novosibirsk 630090  Novosibirsk State University,
Novosibirsk 630090  A. Giri  Indian Institute of Technology Hyderabad,
Telangana 502285  P. Goldenzweig  Institut für Experimentelle Teilchenphysik,
Karlsruher Institut für Technologie, 76131 Karlsruhe  C. Hadjivasiliou
Pacific Northwest National Laboratory, Richland, Washington 99352  S. Halder
Tata Institute of Fundamental Research, Mumbai 400005  O. Hartbrich
University of Hawaii, Honolulu, Hawaii 96822  K. Hayasaka  Niigata University,
Niigata 950-2181  H. Hayashii  Nara Women’s University, Nara 630-8506  M. T.
Hedges  University of Hawaii, Honolulu, Hawaii 96822  M. Hernandez Villanueva
University of Mississippi, University, Mississippi 38677  W.-S. Hou
Department of Physics, National Taiwan University, Taipei 10617  C.-L. Hsu
School of Physics, University of Sydney, New South Wales 2006  T. Iijima
Kobayashi-Maskawa Institute, Nagoya University, Nagoya 464-8602  Graduate
School of Science, Nagoya University, Nagoya 464-8602  G. Inguglia  Institute
of High Energy Physics, Vienna 1050  A. Ishikawa  High Energy Accelerator
Research Organization (KEK), Tsukuba 305-0801  SOKENDAI (The Graduate
University for Advanced Studies), Hayama 240-0193  R. Itoh  High Energy
Accelerator Research Organization (KEK), Tsukuba 305-0801  SOKENDAI (The
Graduate University for Advanced Studies), Hayama 240-0193  M. Iwasaki  Osaka
City University, Osaka 558-8585  Y. Iwasaki  High Energy Accelerator Research
Organization (KEK), Tsukuba 305-0801  W. W. Jacobs  Indiana University,
Bloomington, Indiana 47408  E.-J. Jang  Gyeongsang National University, Jinju
52828  S. Jia  Key Laboratory of Nuclear Physics and Ion-beam Application
(MOE) and Institute of Modern Physics, Fudan University, Shanghai 200443  Y.
Jin  Department of Physics, University of Tokyo, Tokyo 113-0033  C. W. Joo
Kavli Institute for the Physics and Mathematics of the Universe (WPI),
University of Tokyo, Kashiwa 277-8583  K. K. Joo  Chonnam National University,
Gwangju 61186  A. B. Kaliyar  Tata Institute of Fundamental Research, Mumbai
400005  K. H. Kang  Kyungpook National University, Daegu 41566  G. Karyan
Deutsches Elektronen–Synchrotron, 22607 Hamburg  T. Kawasaki  Kitasato
University, Sagamihara 252-0373  C. H. Kim  Department of Physics and
Institute of Natural Sciences, Hanyang University, Seoul 04763  D. Y. Kim
Soongsil University, Seoul 06978  H. J. Kim  Kyungpook National University,
Daegu 41566  S. H. Kim  Seoul National University, Seoul 08826  Y.-K. Kim
Yonsei University, Seoul 03722  T. D. Kimmel  Virginia Polytechnic Institute
and State University, Blacksburg, Virginia 24061  P. Kodyš  Faculty of
Mathematics and Physics, Charles University, 121 16 Prague  T. Konno  Kitasato
University, Sagamihara 252-0373  A. Korobov  Budker Institute of Nuclear
Physics SB RAS, Novosibirsk 630090  Novosibirsk State University, Novosibirsk
630090  S. Korpar  Faculty of Chemistry and Chemical Engineering, University
of Maribor, 2000 Maribor, Slovenia  J. Stefan Institute, 1000 Ljubljana  E.
Kovalenko  Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090
Novosibirsk State University, Novosibirsk 630090  P. Križan  Faculty of
Mathematics and Physics, University of Ljubljana, 1000 Ljubljana  J. Stefan
Institute, 1000 Ljubljana  R. Kroeger  University of Mississippi, University,
Mississippi 38677  P. Krokovny  Budker Institute of Nuclear Physics SB RAS,
Novosibirsk 630090  Novosibirsk State University, Novosibirsk 630090  M. Kumar
Malaviya National Institute of Technology Jaipur, Jaipur 302017  R. Kumar
Punjab Agricultural University, Ludhiana 141004  K. Kumara  Wayne State
University, Detroit, Michigan 48202  Y.-J. Kwon  Yonsei University, Seoul
03722  K. Lalwani  Malaviya National Institute of Technology Jaipur, Jaipur
302017  J. S. Lange  Justus-Liebig-Universität Gießen, 35392 Gießen  I. S. Lee
Department of Physics and Institute of Natural Sciences, Hanyang University,
Seoul 04763  S. C. Lee  Kyungpook National University, Daegu 41566  C. H. Li
Liaoning Normal University, Dalian 116029  J. Li  Kyungpook National
University, Daegu 41566  Y. B. Li  Peking University, Beijing 100871  L. Li
Gioi  Max-Planck-Institut für Physik, 80805 München  J. Libby  Indian
Institute of Technology Madras, Chennai 600036  K. Lieret  Ludwig Maximilians
University, 80539 Munich  D. Liventsev  Wayne State University, Detroit,
Michigan 48202  High Energy Accelerator Research Organization (KEK), Tsukuba
305-0801  T. Luo  Key Laboratory of Nuclear Physics and Ion-beam Application
(MOE) and Institute of Modern Physics, Fudan University, Shanghai 200443  C.
MacQueen  School of Physics, University of Melbourne, Victoria 3010  M. Masuda
Earthquake Research Institute, University of Tokyo, Tokyo 113-0032  Research
Center for Nuclear Physics, Osaka University, Osaka 567-0047  T. Matsuda
University of Miyazaki, Miyazaki 889-2192  D. Matvienko  Budker Institute of
Nuclear Physics SB RAS, Novosibirsk 630090  Novosibirsk State University,
Novosibirsk 630090  P.N. Lebedev Physical Institute of the Russian Academy of
Sciences, Moscow 119991  M. Merola  INFN - Sezione di Napoli, 80126 Napoli
Università di Napoli Federico II, 80126 Napoli  F. Metzner  Institut für
Experimentelle Teilchenphysik, Karlsruher Institut für Technologie, 76131
Karlsruhe  K. Miyabayashi  Nara Women’s University, Nara 630-8506  R. Mizuk
P.N. Lebedev Physical Institute of the Russian Academy of Sciences, Moscow
119991  Higher School of Economics (HSE), Moscow 101000  G. B. Mohanty  Tata
Institute of Fundamental Research, Mumbai 400005  S. Mohanty  Tata Institute
of Fundamental Research, Mumbai 400005  Utkal University, Bhubaneswar 751004
H. K. Moon  Korea University, Seoul 02841  R. Mussa  INFN - Sezione di Torino,
10125 Torino  M. Nakao  High Energy Accelerator Research Organization (KEK),
Tsukuba 305-0801  SOKENDAI (The Graduate University for Advanced Studies),
Hayama 240-0193  Z. Natkaniec  H. Niewodniczanski Institute of Nuclear
Physics, Krakow 31-342  A. Natochii  University of Hawaii, Honolulu, Hawaii
96822  L. Nayak  Indian Institute of Technology Hyderabad, Telangana 502285
M. Nayak  School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978
M. Niiyama  Kyoto Sangyo University, Kyoto 603-8555  N. K. Nisar  Brookhaven
National Laboratory, Upton, New York 11973  S. Nishida  High Energy
Accelerator Research Organization (KEK), Tsukuba 305-0801  SOKENDAI (The
Graduate University for Advanced Studies), Hayama 240-0193  H. Ono  Nippon
Dental University, Niigata 951-8580  Niigata University, Niigata 950-2181  Y.
Onuki  Department of Physics, University of Tokyo, Tokyo 113-0033  P. Oskin
P.N. Lebedev Physical Institute of the Russian Academy of Sciences, Moscow
119991  P. Pakhlov  P.N. Lebedev Physical Institute of the Russian Academy of
Sciences, Moscow 119991  Moscow Physical Engineering Institute, Moscow 115409
G. Pakhlova  Higher School of Economics (HSE), Moscow 101000  P.N. Lebedev
Physical Institute of the Russian Academy of Sciences, Moscow 119991  S. Pardi
INFN - Sezione di Napoli, 80126 Napoli  H. Park  Kyungpook National
University, Daegu 41566  S.-H. Park  High Energy Accelerator Research
Organization (KEK), Tsukuba 305-0801  S. Paul  Department of Physics,
Technische Universität München, 85748 Garching  Max-Planck-Institut für
Physik, 80805 München  T. K. Pedlar  Luther College, Decorah, Iowa 52101  R.
Pestotnik  J. Stefan Institute, 1000 Ljubljana  L. E. Piilonen  Virginia
Polytechnic Institute and State University, Blacksburg, Virginia 24061  T.
Podobnik  Faculty of Mathematics and Physics, University of Ljubljana, 1000
Ljubljana  J. Stefan Institute, 1000 Ljubljana  V. Popov  Higher School of
Economics (HSE), Moscow 101000  E. Prencipe  Forschungszentrum Jülich, 52425
Jülich  M. T. Prim  University of Bonn, 53115 Bonn  M. Ritter  Ludwig
Maximilians University, 80539 Munich  A. Rostomyan  Deutsches
Elektronen–Synchrotron, 22607 Hamburg  N. Rout  Indian Institute of Technology
Madras, Chennai 600036  M. Rozanska  H. Niewodniczanski Institute of Nuclear
Physics, Krakow 31-342  G. Russo  Università di Napoli Federico II, 80126
Napoli  D. Sahoo  Tata Institute of Fundamental Research, Mumbai 400005  Y.
Sakai  High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801
SOKENDAI (The Graduate University for Advanced Studies), Hayama 240-0193  S.
Sandilya  Indian Institute of Technology Hyderabad, Telangana 502285  A.
Sangal  University of Cincinnati, Cincinnati, Ohio 45221  L. Santelj  Faculty
of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana  J. Stefan
Institute, 1000 Ljubljana  T. Sanuki  Department of Physics, Tohoku
University, Sendai 980-8578  V. Savinov  University of Pittsburgh, Pittsburgh,
Pennsylvania 15260  G. Schnell  Department of Physics, University of the
Basque Country UPV/EHU, 48080 Bilbao  IKERBASQUE, Basque Foundation for
Science, 48013 Bilbao  J. Schueler  University of Hawaii, Honolulu, Hawaii
96822  C. Schwanda  Institute of High Energy Physics, Vienna 1050  A. J.
Schwartz  University of Cincinnati, Cincinnati, Ohio 45221  Y. Seino  Niigata
University, Niigata 950-2181  K. Senyo  Yamagata University, Yamagata 990-8560
M. Shapkin  Institute for High Energy Physics, Protvino 142281  C. Sharma
Malaviya National Institute of Technology Jaipur, Jaipur 302017  C. P. Shen
Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and Institute
of Modern Physics, Fudan University, Shanghai 200443  J.-G. Shiu  Department
of Physics, National Taiwan University, Taipei 10617  B. Shwartz  Budker
Institute of Nuclear Physics SB RAS, Novosibirsk 630090  Novosibirsk State
University, Novosibirsk 630090  F. Simon  Max-Planck-Institut für Physik,
80805 München  A. Sokolov  Institute for High Energy Physics, Protvino 142281
E. Solovieva  P.N. Lebedev Physical Institute of the Russian Academy of
Sciences, Moscow 119991  M. Starič  J. Stefan Institute, 1000 Ljubljana  Z. S.
Stottler  Virginia Polytechnic Institute and State University, Blacksburg,
Virginia 24061  J. F. Strube  Pacific Northwest National Laboratory, Richland,
Washington 99352  M. Sumihama  Gifu University, Gifu 501-1193  K. Sumisawa
High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801
SOKENDAI (The Graduate University for Advanced Studies), Hayama 240-0193  T.
Sumiyoshi  Tokyo Metropolitan University, Tokyo 192-0397  W. Sutcliffe
University of Bonn, 53115 Bonn  M. Takizawa  Showa Pharmaceutical University,
Tokyo 194-8543  J-PARC Branch, KEK Theory Center, High Energy Accelerator
Research Organization (KEK), Tsukuba 305-0801  Meson Science Laboratory,
Cluster for Pioneering Research, RIKEN, Saitama 351-0198  K. Tanida  Advanced
Science Research Center, Japan Atomic Energy Agency, Naka 319-1195  Y. Tao
University of Florida, Gainesville, Florida 32611  F. Tenchini  Deutsches
Elektronen–Synchrotron, 22607 Hamburg  K. Trabelsi  Université Paris-Saclay,
CNRS/IN2P3, IJCLab, 91405 Orsay  M. Uchida  Tokyo Institute of Technology,
Tokyo 152-8550  K. Uno  Niigata University, Niigata 950-2181  S. Uno  High
Energy Accelerator Research Organization (KEK), Tsukuba 305-0801  SOKENDAI
(The Graduate University for Advanced Studies), Hayama 240-0193  Y. Usov
Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090  Novosibirsk
State University, Novosibirsk 630090  S. E. Vahsen  University of Hawaii,
Honolulu, Hawaii 96822  R. Van Tonder  University of Bonn, 53115 Bonn  G.
Varner  University of Hawaii, Honolulu, Hawaii 96822  K. E. Varvell  School of
Physics, University of Sydney, New South Wales 2006  A. Vossen  Duke
University, Durham, North Carolina 27708  E. Waheed  High Energy Accelerator
Research Organization (KEK), Tsukuba 305-0801  C. H. Wang  National United
University, Miao Li 36003  E. Wang  University of Pittsburgh, Pittsburgh,
Pennsylvania 15260  M.-Z. Wang  Department of Physics, National Taiwan
University, Taipei 10617  P. Wang  Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing 100049  M. Watanabe  Niigata University, Niigata
950-2181  S. Watanuki  Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405
Orsay  O. Werbycka  H. Niewodniczanski Institute of Nuclear Physics, Krakow
31-342  J. Wiechczynski  H. Niewodniczanski Institute of Nuclear Physics,
Krakow 31-342  E. Won  Korea University, Seoul 02841  X. Xu  Soochow
University, Suzhou 215006  B. D. Yabsley  School of Physics, University of
Sydney, New South Wales 2006  W. Yan  Department of Modern Physics and State
Key Laboratory of Particle Detection and Electronics, University of Science
and Technology of China, Hefei 230026  S. B. Yang  Korea University, Seoul
02841  H. Ye  Deutsches Elektronen–Synchrotron, 22607 Hamburg  J. H. Yin
Korea University, Seoul 02841  C. Z. Yuan  Institute of High Energy Physics,
Chinese Academy of Sciences, Beijing 100049  Z. P. Zhang  Department of Modern
Physics and State Key Laboratory of Particle Detection and Electronics,
University of Science and Technology of China, Hefei 230026  V. Zhilich
Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090  Novosibirsk
State University, Novosibirsk 630090  V. Zhukova  P.N. Lebedev Physical
Institute of the Russian Academy of Sciences, Moscow 119991  V. Zhulanov
Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090  Novosibirsk
State University, Novosibirsk 630090

###### Abstract

We measure the branching fractions and $\displaystyle C\\!P$ asymmetries for
the decays $\displaystyle B^{0}\to\bar{D}^{0}\pi^{0}$ and $\displaystyle
B^{+}\to\bar{D}^{0}\pi^{+}$, using a data sample of $\displaystyle 772\times
10^{6}$ $\displaystyle B\kern 1.79993pt\bar{\kern-1.79993ptB}{}$ pairs
collected at the $\displaystyle\Upsilon(4S)$ resonance with the Belle detector
at the KEKB $\displaystyle e^{+}e^{-}$ collider. The branching fractions
obtained and direct $\displaystyle C\\!P$ asymmetries are
$\displaystyle\mathcal{B}(B^{0}\to\bar{D}^{0}\pi^{0})=[2.70\pm
0.06~{}\text{(stat.)}\pm 0.10~{}\text{(syst.)}]\times 10^{-4}$,
$\displaystyle\mathcal{B}(B^{+}\to\bar{D}^{0}\pi^{+})=[4.53\pm
0.02~{}\text{(stat.)}\pm 0.15~{}\text{(syst.)}]\times 10^{-3}$,
$\displaystyle{\cal A}_{C\\!P}(B^{0}\to\bar{D}^{0}\pi^{0})=[+0.42\pm
2.05~{}\text{(stat.)}\pm 1.22~{}\text{(syst.)}]\%$, and $\displaystyle{\cal
A}_{C\\!P}(B^{+}\to\bar{D}^{0}\pi^{+})=[+0.19\pm 0.36~{}\text{(stat.)}\pm
0.57~{}\text{(syst.)}]\%$. The measurements of $\displaystyle\mathcal{B}$ are
the most precise to date and are in good agreement with previous results, as
is the measurement of $\displaystyle{\cal
A}_{C\\!P}(B^{+}\to\bar{D}^{0}\pi^{+})$. The measurement of
$\displaystyle{\cal A}_{C\\!P}$ for $\displaystyle B^{0}\to\bar{D}^{0}\pi^{0}$
is the first for this mode, and the value is consistent with Standard Model
expectations.

###### pacs:

13.25.Hw, 12.15.Hh

††preprint:  Belle Preprint 2021-17 KEK Preprint 2021-15

##  I Introduction

The branching fraction ($\displaystyle{\cal B}$) of the color-suppressed decay
$\displaystyle B^{0}\to\bar{D}^{0}\pi^{0}$ CC  is measured Abe  _et al._
(2002); Coan  _et al._ (2002); Blyth  _et al._ (2006) to be about a factor of
four higher than theory predictions made using the “naive” factorization
model, where final-state interactions (FSIs) are neglected Beneke  _et al._
(2000); Neubert and Stech (1998). This has led to a number of new theoretical
descriptions of the process Neubert and Petrov (2001); Chua  _et al._ (2002);
Chua and Hou (2005); Rosner (1999); Deandrea and Polosa (2002); Chiang and
Rosner (2003); Chua and Hou (2008); Leganger and Eeg (2010) that include FSI’s
and also treat isospin-related amplitudes of color-suppressed and color-
allowed decays. The $\displaystyle B^{0}\to\bar{D}^{0}\pi^{0}$ process has
been shown to have large non-factorizable components Leganger and Eeg (2010),
so precise measurements of its properties are valuable in comparing different
theoretical models used to describe it. Many of these models predict a
substantial strong phase in the final state. A non-vanishing strong phase
difference between two amplitudes is necessary to give rise to direct
$\displaystyle C\\!P$ violation Bigi and Sanda (2009). The direct
$\displaystyle C\\!P$-violation parameter, $\displaystyle{\cal A}_{C\\!P}$,
for the $\displaystyle B\to\bar{D}^{0}\pi$ decay is defined as:

${\cal A}_{C\\!P}=\frac{\Gamma(\kern 1.79993pt\bar{\kern-1.79993ptB}{}\to
D^{0}\pi)-\Gamma(B\to\bar{D}^{0}\pi)}{\Gamma(\kern
1.79993pt\bar{\kern-1.79993ptB}{}\to D^{0}\pi)+\Gamma(B\to\bar{D}^{0}\pi)},$
(1)

where $\displaystyle\Gamma$ is the partial decay width for the corresponding
decay.

In the Standard Model (SM), $\displaystyle B^{0}\to\bar{D}^{0}\pi^{0}$
transitions proceed mainly via the tree-level diagram of Fig.1a. An exchange
diagram (Fig.1b) with the same CKM factors is also present, but, due to OZI
suppression, is expected to have a much smaller amplitude. As such, direct
$\displaystyle C\\!P$ violation in this mode is expected to be small, even in
the presence of a strong phase difference from FSI. Measurements of notable
$\displaystyle C\\!P$ violation in this decay would be of significant interest
and could hint at contributions from Beyond-the-Standard-Model (BSM) physics
diagrams. Recently the BaBar and Belle collaborations performed time-dependent
$\displaystyle C\\!P$-violation analyses of the related modes
$\displaystyle\bar{B}^{0}\to D^{(*)}_{CP}h^{0}$, where $\displaystyle
h\in{\pi^{0},\eta,\omega}$ and $\displaystyle D^{(*)}_{CP}$ refers to
$\displaystyle D$ or $\displaystyle D^{*}$ in a $\displaystyle CP$ eigenstate
Abdesselam  _et al._ (2015). They measure the $\displaystyle C\\!P$-violation
parameters $\displaystyle C(=-{\cal A}_{CP})$, $\displaystyle{\cal S}_{CP}$
and $\displaystyle\phi_{1}$alp , and obtain $\displaystyle
C(B^{0}\to\bar{D}^{*0}h^{0})=(-2\pm 8)\times 10^{-2}$. This value is
consistent with the expectation of small $\displaystyle{\cal A}_{CP}$ for
$\displaystyle B^{0}\to\bar{D}^{0}\pi^{0}$. However, this result does not
exclude larger values up to 0.1, which would be much larger than SM
predictions.

$\displaystyle\bar{b}$$\displaystyle
V_{cb}$$\displaystyle\bar{c}$$\displaystyle d$$\displaystyle d$$\displaystyle
u$$\displaystyle\bar{d}$$\displaystyle(c1)!0.5!(c3)-(1cm,0)$$\displaystyle
V^{*}_{ud}$$\displaystyle W^{+}$$\displaystyle
B^{0}$$\displaystyle\bar{D}^{0}$$\displaystyle\pi^{0}$a)

$\displaystyle\overline{b}$$\displaystyle V_{cb}$$\displaystyle
V^{*}_{ud}$$\displaystyle\overline{c}$$\displaystyle\overline{d}/\overline{u}$$\displaystyle
d/u$$\displaystyle d$$\displaystyle
u$$\displaystyle(c1)!0.5!(c3)-(1.5cm,0)$$\displaystyle W$b)$\displaystyle
B^{0}$$\displaystyle\bar{D}^{0}$$\displaystyle\pi^{0}$

$\displaystyle b$$\displaystyle V^{*}_{ub}$$\displaystyle
u$$\displaystyle\bar{d}$$\displaystyle\bar{d}$$\displaystyle\bar{c}$$\displaystyle
d$$\displaystyle(c1)!0.5!(c3)-(1cm,0)$$\displaystyle V_{cd}$$\displaystyle
W^{-}$$\displaystyle\bar{B}^{0}$$\displaystyle\bar{D}^{0}$$\displaystyle\pi^{0}$c)

Figure 1: Tree-level Feynman diagrams for a) color-suppressed $\displaystyle
B^{0}\to\bar{D}^{0}\pi^{0}$ decay, b) W exchange diagram, and c) color-
suppressed and doubly Cabibbo-suppressed
$\displaystyle\bar{B}^{0}\to\bar{D}^{0}\pi^{0}$ decay.

A high precision measurement of $\displaystyle{\cal B}$ and
$\displaystyle{\cal A}_{CP}$ for $\displaystyle B^{0}\to\bar{D}^{0}\pi^{0}$
has further utility in addition to comparison to theoretical predictions, as
it is a common control mode for use in rare charmless $\displaystyle B$ decays
with a $\displaystyle\pi^{0}$. The precise measurement of properties of a
control mode is important to provide validation and refinement of analysis
techniques.

In this paper, we present new measurements of $\displaystyle
B^{0}\to\bar{D}^{0}\pi^{0}$ using the full data sample of
$\displaystyle(772\pm 10.6)\times 10^{6}\bar{B}B$ pairs ($\displaystyle
711~{}\text{fb}^{-1}$) collected with the Belle detector at the KEKB
asymmetric-energy $\displaystyle e^{+}e^{-}$ ($\displaystyle
3.5{\rm{\,Ge\kern-1.00006ptV}}$ on $\displaystyle
8.0{\rm{\,Ge\kern-1.00006ptV}}$) collider KEK  operating near the
$\displaystyle\Upsilon(4S)$ resonance. We also present corresponding
measurements of $\displaystyle B^{+}\to\bar{D}^{0}\pi^{+}$ decays, which
proceed via a simple spectator diagram with no color suppression.

##  II Belle Detector

The Belle detector Bel  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 (ECL) consisting of CsI(Tl) crystals. All these
detector components are 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 with resistive plate chambers to detect $\displaystyle
K^{0}_{\scriptscriptstyle L}$ mesons and to identify muons. Two inner detector
configurations were used: a 2.0 cm beam-pipe and a 3-layer SVD (SVD1) were
used for the first sample of $\displaystyle 152\times 10^{6}$ $\displaystyle
B\kern 1.79993pt\bar{\kern-1.79993ptB}{}$ pairs, while a 1.5 cm beam-pipe, a
4-layer SVD (SVD2), and small cells in the inner layers of the CDC were used
to record the remaining $\displaystyle 620\times 10^{6}$ $\displaystyle B\kern
1.79993pt\bar{\kern-1.79993ptB}{}$ pairs Natkaniec  _et al._ (2006).

We reconstruct $\displaystyle B^{0}\to\bar{D}^{0}\pi^{0}$ candidates from the
subsequent decays of the $\displaystyle\bar{D}^{0}$ and the
$\displaystyle\pi^{0}$ mesons. We employ two reconstruction modes:
$\displaystyle B^{0}\to\bar{D}^{0}(\to K^{+}\pi^{-})\pi^{0}$ ($\displaystyle
B_{2b}$) and $\displaystyle B^{0}\to\bar{D}^{0}(\to
K^{+}\pi^{-}\pi^{0})\pi^{0}$ ($\displaystyle B_{3b}$). The
$\displaystyle\pi^{0}$ mesons are reconstructed from their decay to two
photons.

The flavor of the neutral $\displaystyle B$-meson ($\displaystyle B^{0}$ or
$\displaystyle\bar{B}^{0}$) is determined by the charge of the reconstructed
kaon ($\displaystyle K^{+}$ or $\displaystyle K^{-}$). This method of flavor
tagging is not perfect, as the wrong-sign doubly Cabibbo-suppressed decays
(DCS) $\displaystyle\bar{B}^{0}\to\bar{D}^{0}\pi^{0}$ (Fig.1c) will result in
wrongly tagged flavor. The same effect occurs with charm DCS decays
$\displaystyle D^{0}\to K^{+}\pi^{-}$ and $\displaystyle D^{0}\to
K^{+}\pi^{-}\pi^{0}$, and charm mixing, although the charm mixing effect is
negligibly smaller. These effects can be calculated using the ratio of wrong-
sign (Cabibbo-suppressed) to right-sign (Cabibbo-favored) decay rates:

$R=\frac{\cal{B}_{WS}}{\cal{B}_{RS}},$ (2)

where $\displaystyle R(\bar{B}^{0}\to\bar{D}^{0}\pi^{0})$ has not been
measured and thus is approximated with $\displaystyle
R({B}^{0}\to{D}^{+}\pi^{-})=(2.92\pm 0.38\pm 0.31)\times 10^{-4}$Das  _et al._
(2010), while $\displaystyle R(D^{0}\to K^{+}\pi^{-})=(3.79\pm 0.18)\times
10^{-3}$ and $\displaystyle R(D^{0}\to K^{+}\pi^{-}\pi^{0})=(2.12\pm
0.07)\times 10^{-3}$ are taken from the PDG Zyla  _et al._ (2020). These
effects lead to the true value of $\displaystyle\cal{B}$ being
$\displaystyle(0.314\pm 0.008)\%$ lower than the measured value, and the true
$\displaystyle{\cal A}_{CP}$ being $\displaystyle(3.05\pm 0.08)\times 10^{-5}$
higher than the measured value. In the case of $\displaystyle\cal{B}$ this is
corrected for; however, for $\displaystyle{\cal A}_{CP}$ the correction is
significantly smaller than our uncertainty and it is neglected.

Photon candidates are mainly taken from clusters in the ECL but additionally
are reconstructed from $\displaystyle e^{+}e^{-}$ pairs resulting from photon
conversion in the inner detector. Photons from $\displaystyle\pi^{0}$ decay
must have an energy greater than $\displaystyle 50$
$\displaystyle(100){\rm{\,Me\kern-1.00006ptV}}$ in the barrel (endcap) region
of the ECL. The invariant mass of the two-photon combination must lie in the
range $\displaystyle
104{{{\rm\,Me\kern-1.00006ptV\\!/}}c^{2}}<M_{\gamma\gamma}<165{{{\rm\,Me\kern-1.00006ptV\\!/}}c^{2}}$,
corresponding to $\displaystyle\pm 3\sigma$ around the nominal
$\displaystyle\pi^{0}$ mass Zyla  _et al._ (2020). We subsequently perform a
mass-constrained fit with the requirement $\displaystyle\chi^{2}<50$.

Charged tracks originating from a $\displaystyle B$ decay are required to have
a distance-of-closest-approach with respect to the interaction point of less
than 4.0 cm along the $\displaystyle z$-axis (the direction opposite the
positron beam), and of less than 0.3 cm in the plane transverse to the
$\displaystyle z$-axis. Charged kaons and pions are identified using
information from the CDC, ACC, and TOF detectors. This information is combined
to form a $\displaystyle K-\pi$ likelihood ratio $\displaystyle
R_{K/\pi}=L_{K}/(L_{K}+L_{\pi})$, where $\displaystyle L_{K}(L_{\pi})$ is the
likelihood of the track being a kaon (pion). Track candidates with
$\displaystyle R_{K/\pi}>0.6$ $\displaystyle(<0.4)$ are classified as kaons
(pions). The typical kaon (pion) identification efficiency is 83% (88%), with
a pion (kaon) misidentification probability of 7% (11%).

Two kinematic variables are used to distinguish signal from background: the
beam-energy-constrained mass, $\displaystyle M_{\rm
bc}\equiv\sqrt{E_{\text{beam}}^{2}-|\vec{p}_{B}|^{2}c^{2}}/c^{2}$, and the
energy difference $\displaystyle\Delta E\equiv E_{B}-E_{\text{beam}}$. Here,
$\displaystyle\vec{p}_{B}$ and $\displaystyle E_{B}$ are the momentum and
energy, respectively, of the $\displaystyle B$-meson candidate evaluated in
the center-of-mass (CM) frame, and $\displaystyle E_{\text{beam}}$ is the beam
energy in the CM frame.

Due to energy leakage in the ECL, the reconstructed $\displaystyle\pi^{0}$
energy is typically lower than its true value. To compensate for this, we
rescale the reconstructed $\displaystyle\pi^{0}$ momentum to give
$\displaystyle E_{\pi^{0}}\equiv E_{\rm beam}-E_{D}$, specifically:

$\displaystyle\displaystyle\vec{p}_{\pi,\text{corr}}$
$\displaystyle\displaystyle=\vec{p}_{\pi}\times\frac{\sqrt{(E_{\text{beam}}-E_{D^{0}})^{2}-M_{\pi^{0}}^{2}c^{4}}}{|p_{\pi}|c}.$
(3)

Using this we calculate a new $\displaystyle B$-meson momentum,
$\displaystyle\vec{p}_{B,\text{corr}}$, then calculate a corrected
$\displaystyle M_{\rm bc}$ (from now on simply referred to as $\displaystyle
M_{\rm bc}$),

$M_{\rm
bc}\equiv\frac{\sqrt{E_{\text{beam}}^{2}-|\vec{p}_{B,\text{corr}}|^{2}c^{2}}}{c^{2}}.$
(4)

Rescaling $\displaystyle M_{\rm bc}$ in this way improves the mass resolution
and removes some correlations between $\displaystyle M_{\rm bc}$ and
$\displaystyle\Delta E$. This procedure is only applied to the
$\displaystyle\pi^{0}$ that is the direct daughter of the $\displaystyle
B^{0}$.

All candidates satisfying $\displaystyle
M_{\text{bc}}>5.25{{{\rm\,Ge\kern-1.00006ptV\\!/}}c^{2}}$ and
$\displaystyle-0.2{\rm{\,Ge\kern-1.00006ptV}}<\Delta
E<0.2{\rm{\,Ge\kern-1.00006ptV}}$ are retained for further analysis. We find
that $\displaystyle 16\%$ $\displaystyle(47\%)$ of events have more than one
$\displaystyle B^{0}$ candidate in the $\displaystyle B_{2b}$ ($\displaystyle
B_{3b}$) reconstruction modes. In these cases, we select one of the
reconstructed $\displaystyle B^{0}$ mesons based on the mass difference
$\displaystyle\Delta m(X)=m_{\text{PDG}}(X)-M(X)$, where $\displaystyle
m_{\text{PDG}}(X)$ is the mass reported by the Particle Data Group (PDG) Zyla
_et al._ (2020) for particle $\displaystyle X$, and $\displaystyle M(X)$ is
the reconstructed mass. The best candidate is selected as the $\displaystyle
B^{0}$ or $\displaystyle B^{+}$ that minimizes $\displaystyle\Delta m(D^{0})$.
If there are multiple candidates with the same minimal $\displaystyle\Delta
m(D^{0})$, the one that minimizes $\displaystyle\Delta m(\pi^{0})$ is
selected. Monte Carlo simulation (MC) studies show that this procedure selects
the correct $\displaystyle B^{0}$ in 96% (86%) of cases for the $\displaystyle
B_{2b}$ ($\displaystyle B_{3b}$) samples.

##  III Belle detector and signal selection

Backgrounds to our signal are studied using MC simulation. These simulations
use EvtGen Lange  _et al._ (2001) and PYTHIA Sjöstrand  _et al._ (2006) to
generate the physics interactions at the quark level, and employ GEANT3 Brun
_et al._ (1987) to simulate the detector response.

The largest background arises from $\displaystyle e^{+}e^{-}\to
q\bar{q}\;(q\in\\{u,d,s,c\\})$ continuum events. A neural network Feindt and
Kerzel (2006) is used to distinguish the spherical $\displaystyle B\bar{B}$
signal from the jet-like continuum background. It combines the following five
observables based on the event topology: a Fisher discriminant formed from 17
modified Fox-Wolfram moments SFW ; the cosine of the angle between the
$\displaystyle B$-meson candidate direction and the beam axis; the cosine of
the angle between the thrust axis Brandt  _et al._ (1964) of the
$\displaystyle B$-meson candidate and that of the rest of the event (all of
these quantities being calculated in the CM frame); the separation along the
$\displaystyle z$-axis between the vertex of the $\displaystyle B$-meson
candidate and that of the remaining tracks in the event; and the tagging
quality variable from a $\displaystyle B$-meson flavor-tagging algorithm
Kakuno  _et al._ (2004). The training and optimization of the neural network
are performed with signal and continuum MC samples. These are divided into
five training samples and one verification sample. The output of the neural
net ($\displaystyle C_{\rm NN}$) has a range of $\displaystyle(-1,1)$, with
$\displaystyle 1$ being the most signal-like and $\displaystyle-1$ being the
most background-like.

In order to maximally use $\displaystyle C_{\rm NN}$ information, we impose
only a loose requirement on $\displaystyle C_{\rm NN}$ and use $\displaystyle
C_{\rm NN}$ as a variable in the fit. We require $\displaystyle C_{\rm
NN}>-0.05$ for both the $\displaystyle B_{2b}$ and $\displaystyle B_{3b}$
modes. This results in 86% background reduction and 87% signal efficiency. To
facilitate modelling $\displaystyle C_{\rm NN}$ analytically with Gaussian
functions, we transform it to an alternative variable $\displaystyle C_{\rm
NN}^{\prime}$ via the formula

$C_{\rm NN}^{\prime}=\text{log}\left(\frac{C_{\rm NN}-C_{\rm
NN}^{\text{min}}}{C_{\rm NN}^{\text{max}}-C_{\rm NN}}\right),$ (5)

where $\displaystyle C_{\rm NN}^{\text{min}}$ is the minimum value of
$\displaystyle-0.05$, and $\displaystyle C_{\rm NN}^{\text{max}}$ is the
maximum value of $\displaystyle C_{\rm NN}$ obtained from the signal MC sample
used to verify the training.

There is a significant background arising from $\displaystyle b\to c$
transitions, which we refer to as “generic $\displaystyle B$” decays. The main
components of the generic $\displaystyle B$ background are incorrectly
assigned tracks, combinatorial backgrounds, $\displaystyle
B^{0}\to\bar{D}^{0}\rho^{0}$, and $\displaystyle B^{0}\to\bar{D}^{0*}\pi^{0}$
with either $\displaystyle\bar{D}^{0*}\to\bar{D}^{0}\gamma$ or
$\displaystyle\bar{D}^{0*}\to\bar{D}^{0}\pi^{0}$. These are investigated with
MC simulations of $\displaystyle B\bar{B}$ decays. To reduce this background,
signal candidates are selected within $\displaystyle\pm 3$ standard deviations
of the mean values for $\displaystyle M_{\rm bc}$, $\displaystyle\Delta E$,
and the reconstructed $\displaystyle\pi^{0}$ and $\displaystyle D^{0}$ mass
distributions. Low final-state momentum events are excluded with selection
criteria on the lab-frame momentum of the final-state particles:
$\displaystyle P(K^{\pm})$, $\displaystyle P(\pi^{\pm})$, $\displaystyle
P(\pi^{0}_{B^{0}})$, and $\displaystyle P(\pi^{0}_{\bar{D}^{0}})$. These
requirements are listed in Table 1.

Table 1:  Requirements on kinematic variables employed in the reconstruction of $\displaystyle B^{0}\to\bar{D}^{0}\pi^{0}$ decays, to minimize generic $\displaystyle B$-decay background. The subscript identifies the origin of the particle, and the $\displaystyle\bar{D}^{0}$ decay mode is shown in square brackets. Variable | Selected Range
---|---
$\displaystyle M_{\rm bc}$ | $\displaystyle 5.253\text{ -- }5.288{{{\rm\,Ge\kern-1.00006ptV\\!/}}c^{2}}$
$\displaystyle\Delta E$ | $\displaystyle-0.2\text{ -- }0.2{\rm{\,Ge\kern-1.00006ptV}}$
$\displaystyle M(D^{0})$ | $\displaystyle 1.841\text{ -- }1.882{{{\rm\,Ge\kern-1.00006ptV\\!/}}c^{2}}$
$\displaystyle M(\pi^{0}_{B^{0}})[K^{+}\pi^{-}]$ | $\displaystyle 104.1\text{ -- }163.1{{{\rm\,Me\kern-1.00006ptV\\!/}}c^{2}}$
$\displaystyle M(\pi^{0}_{B^{0}})[K^{+}\pi^{-}\pi^{0}]$ | $\displaystyle 105.8\text{ -- }164.4{{{\rm\,Me\kern-1.00006ptV\\!/}}c^{2}}$
$\displaystyle M(\pi^{0}_{\bar{D}^{0}})[K^{+}\pi^{-}\pi^{0}]$ | $\displaystyle 107.9\text{ -- }162.3{{{\rm\,Me\kern-1.00006ptV\\!/}}c^{2}}$
$\displaystyle P(K^{\pm})$ | $\displaystyle 0.3\text{ -- }3.5{{{\rm\,Ge\kern-1.00006ptV\\!/}}c}$
$\displaystyle P(\pi^{\pm})$ | $\displaystyle 0.3\text{ -- }3.5{{{\rm\,Ge\kern-1.00006ptV\\!/}}c}$
$\displaystyle P(\pi^{0}_{B^{0}})$ | $\displaystyle 1.5\text{ -- }3.5{{{\rm\,Ge\kern-1.00006ptV\\!/}}c}$
$\displaystyle P(\pi^{0}_{\bar{D}^{0}})$ | $\displaystyle 0.2\text{ -- }3.5{{{\rm\,Ge\kern-1.00006ptV\\!/}}c}$

There is also a very small background component from $\displaystyle b\to u$
and $\displaystyle b\to s$ transitions that consists mainly of combinatorial
background, with some non-resonant $\displaystyle B$ decays to the same final
states ($\displaystyle K^{+}\pi^{-}\pi^{0}$, $\displaystyle
K^{+}\pi^{-}\pi^{0}\pi^{0}$, $\displaystyle K^{+}\pi^{-}\pi^{+}$ and
$\displaystyle K^{+}\pi^{-}\pi^{+}\pi^{0}$). These are studied using a large
MC sample corresponding to 50 times the number of $\displaystyle B\bar{B}$
events recorded by Belle. We refer to these background events as “rare.” The
yield of these rare events is fixed when fitting for the signal yield based on
the most recent branching fractions from the PDG Zyla  _et al._ (2020).

After all selections have been made for $\displaystyle
B^{0}\to\bar{D}^{0}\pi^{0}$, the reconstruction efficiencies are
($\displaystyle 27.53\pm 0.04$)% for the $\displaystyle B_{2b}$ mode and
($\displaystyle 9.43\pm 0.02$)% for the $\displaystyle B_{3b}$ mode. Including
intermediate branching fractions, the overall efficiencies are ($\displaystyle
1.09\pm 0.01$)% and ($\displaystyle 1.36\pm 0.01$)%, respectively.

For $\displaystyle B^{+}\to\bar{D}^{0}\pi^{+}$, the reconstruction efficiency
after all selections is calculated to be ($\displaystyle 33.08\pm 0.04$)% for
the $\displaystyle B_{2b}$ mode (($\displaystyle 1.31\pm 0.05)$% including
intermediate branching fractions) and ($\displaystyle 9.05\pm 0.02$)% for the
$\displaystyle B_{3b}$ mode (($\displaystyle 1.30\pm 0.05)$% including
intermediate branching fractions).

##  IV Fitting strategy

The signal yield and $\displaystyle{\cal A}_{C\\!P}$ are extracted via an
unbinned extended maximum-likelihood fit to the variables $\displaystyle
M_{\rm bc}$, $\displaystyle\Delta{}E$, and $\displaystyle C_{\rm
NN}^{\prime}$. There are four categories of events fitted: $\displaystyle
B^{0}\to\bar{D}^{0}\pi^{0}$ or $\displaystyle B^{+}\to\bar{D}^{0}\pi^{+}$
signal events ($\displaystyle s$), continuum events ($\displaystyle c$),
generic $\displaystyle B\bar{B}$ events ($\displaystyle b$), and rare
$\displaystyle B$-decay backgrounds ($\displaystyle r$). These events are
described by probability density functions (PDFs) denoted as $\displaystyle
P^{s}$, $\displaystyle P^{c}$, $\displaystyle P^{b}$, and $\displaystyle
P^{r}$, respectively. Separate PDFs are constructed for the $\displaystyle
B_{2b}$ and $\displaystyle B_{3b}$ reconstruction modes, which are fitted as
two separate data sets. The data are further divided into events tagged as
$\displaystyle B^{0}$ and $\displaystyle\bar{B}^{0}$, defined as having flavor
$\displaystyle q=+1$ and $\displaystyle q=-1$, respectively, based on the
charge of the kaon.

The physics parameters are determined via a simultaneous fit to the four data
sets. The total likelihood is given by

$\displaystyle\displaystyle{\cal L}$
$\displaystyle\displaystyle=\frac{e^{-\sum_{j}N^{j}}}{\prod_{q,d}N_{q,d}!}\times\prod_{q,d}$
$\displaystyle\displaystyle\left[\prod_{i=1}^{N_{q,d}}\left(\sum_{j}f^{j}_{d}N^{j}P^{j}_{q,d}\left(M_{\text{bc}}^{i},\Delta
E^{i},C_{\rm NN}^{\prime i},q\right)\right)\right].$ (6)

where $\displaystyle N_{q,d}$ is the number of events with flavor tag
$\displaystyle q$ for the data set $\displaystyle d$ ($\displaystyle
d\in{B_{2b},B_{3b}}$), and $\displaystyle N^{j}$ is the number of events in
the $\displaystyle j^{\rm th}$ category ($\displaystyle j\in{s,c,b,r}$)
contributing to the total yield. The fraction of events in the data set
$\displaystyle d$ for category $\displaystyle j$ is $\displaystyle f^{j}_{d}$,
with $\displaystyle f^{j}_{3b}=1-f^{j}_{2b}$. The PDF $\displaystyle
P^{j}_{q,d}$ corresponds to the $\displaystyle j^{\rm th}$ category in the
$\displaystyle d$ data set for flavor $\displaystyle q$, measured at
$\displaystyle M_{\rm bc}^{i}$, $\displaystyle\Delta E^{i}$, and
$\displaystyle C_{\rm NN}^{\prime i}$ for the $\displaystyle i^{\rm th}$
event.

The PDF for each component is given by:

$\displaystyle\displaystyle P^{j}_{q,d}(M_{\text{bc}},\Delta E,C_{\rm
NN}^{\prime},q)=$ $\displaystyle\displaystyle\left(\frac{1-q\times{\cal
A}^{j}_{C\\!P}}{2}\right)$ $\displaystyle\displaystyle\times
P^{j}_{d}(M_{\text{bc}},\Delta E,C_{\rm NN}^{\prime}),$ (7)

The model accounts for a possible direct $\displaystyle C\\!P$ asymmetry,
$\displaystyle{\cal A}^{j}_{C\\!P}$, and the fractions of signal and
backgrounds expected in each reconstruction mode $\displaystyle B_{2b}$
($\displaystyle B_{3b}$). In Eq. (IV), the fraction $\displaystyle f^{s}_{d}$
is determined via MC studies of the $\displaystyle B_{2b}$ and $\displaystyle
B_{3b}$ modes and is fixed in the fit to data, and $\displaystyle{\cal
A}^{j}_{C\\!P}(j\in{c,b,r})$ is fixed based on studies of detector bias using
sideband data (see Section V).

The 20 free parameters in the fit are: the number of signal events
$\displaystyle N^{s}$, signal asymmetry $\displaystyle A^{s}_{C\\!P}$, the
number of continuum events ($\displaystyle N^{c}$) and generic $\displaystyle
B$-decay events ($\displaystyle N^{b}$), fractions of backgrounds expected in
each reconstruction mode $\displaystyle f^{j}_{d}$ ($\displaystyle
j\in{c,b,r}$), shape parameters of the continuum $\displaystyle M_{\rm bc}$
and $\displaystyle\Delta E$ PDFs, and the mean and width of the $\displaystyle
B_{2b}$ signal $\displaystyle M_{\rm bc}$ and $\displaystyle\Delta E$ PDFs.
The number of rare background events ($\displaystyle N^{r}$) is fixed to that
expected from MC studies. The effects of these assumptions are included in the
systematic uncertainties.

The PDFs used for the $\displaystyle M_{\rm bc}$ and $\displaystyle\Delta E$
distributions for the various event types are as follows.

  * •

Signal: for the $\displaystyle B_{2b}$ mode, the $\displaystyle M_{\rm bc}$
PDF is a Crystal Ball function Cry , while the $\displaystyle\Delta E$ PDF is
the sum of a Crystal Ball function and a Gaussian with the same mean. The
Gaussian component is small and included to handle the tails of the
distribution. For the $\displaystyle B_{3b}$ mode, there is a strong
correlation between $\displaystyle M_{\rm bc}$ and $\displaystyle\Delta E$,
and no analytic 2D PDF could be found to fit the data satisfactorily. Instead
a 2D kernel density estimation (KEST) PDF Cranmer (2001) is used.

  * •

Generic $\displaystyle B$-decay background: similarly to $\displaystyle
B_{3b}$ signal, there exist complex correlations between $\displaystyle M_{\rm
bc}$ and $\displaystyle\Delta E$, so a 2D KEST PDF obtained from MC
simulations in both modes.

  * •

Continuum background: $\displaystyle M_{\rm bc}$ is fitted as an ARGUS
function Albrecht  _et al._ (1990), and $\displaystyle\Delta E$ as a 3rd-order
Chebyshev polynomial, in both modes.

  * •

Rare $\displaystyle B$-decay background: as with generic $\displaystyle B$ and
$\displaystyle B_{3b}$ signal, the $\displaystyle M_{\rm bc}$ and
$\displaystyle\Delta E$ distributions for rare $\displaystyle B$-decay
background are modelled with a 2D KEST PDF in both modes. This PDF is
determined using MC simulations corresponding to 50 times the luminosity of
the Belle data set.

To fit $\displaystyle C_{\rm NN}^{\prime}$, three summed Gaussians are used
for all components except continuum background, which employed two summed
Gaussians. The PDFs for all event types are summarized in Table 2.

Table 2:  Functional forms for PDFs employed by the different event categories for fits. Category | $\displaystyle M_{\rm bc}$ | $\displaystyle\Delta E$ | $\displaystyle C_{\rm NN}^{\prime}$
---|---|---|---
Signal $\displaystyle B_{2b}$ | Crystal Ball | Crystal Ball | 3 Gaussians
|  | \+ Gaussian |
Signal $\displaystyle B_{3b}$ | 2D KEST PDF | 3 Gaussians
Generic $\displaystyle B$ | 2D KEST PDF | 3 Gaussians
Continuum | ARGUS |  $\displaystyle 3^{\rm rd}$ Order | 2 Gaussians
|  | Cheby. Poly. |
Rare $\displaystyle B$ | 2D KEST PDF | 3 Gaussians

The fitting procedure and accuracy of the various PDF models are extensively
investigated using MC ‘pseudoexperiments’. In these studies, the signal and
rare $\displaystyle B$ background events are selected from large samples of
simulated events. Events for $\displaystyle e^{+}e^{-}\to q\bar{q}$ and
generic $\displaystyle B$-decay are generated from their respective PDF
shapes. For $\displaystyle B^{0}\to\bar{D}^{0}\pi^{0}$, we observe a small
bias of $\displaystyle(+0.6\pm 0.3)$% in signal yield, and
$\displaystyle(+0.04\pm 0.05)$% in $\displaystyle{\cal A}_{CP}$. We correct
for this bias in our final measurements and include a corresponding systematic
uncertainty for it. No significant bias is observed for $\displaystyle
B^{+}\to\bar{D}^{0}\pi^{+}$, i.e., only $\displaystyle(+0.06\pm 0.16)$% in
signal yield and $\displaystyle(-0.02\pm 0.02)$% in $\displaystyle{\cal
A}_{CP}$.

##  V $\displaystyle B^{+}\to\bar{D}^{0}\pi^{+}$

We first select and fit a sample of $\displaystyle B^{+}\to\bar{D}^{0}\pi^{+}$
decays. This sample is not color-suppressed and thus has much larger
statistics and lower background than the sample of $\displaystyle
B^{0}\to\bar{D}^{0}\pi^{0}$ decays. As well as ensuring the fitted
$\displaystyle{\cal B}$ and $\displaystyle{\cal A}_{CP}$ are consistent with
existing measurements, this mode is used to obtain calibration factors for the
fixed shape parameters of the PDFs used to fit $\displaystyle
B^{0}\to\bar{D}^{0}\pi^{0}$ decays, to account for any differences between MC
and data. In addition, this mode provides a data-driven estimation of the
systematic uncertainty associated with the $\displaystyle{\cal A}_{CP}$
correction for a detection asymmetry (discussed below).

To account for potential differences in the distribution of fitting variables
between MC and data, additional parameters (calibration factors) are included
in the fit to enable small adjustments in the fitted PDF shapes. These
calibration factors are applied as mean shifts and width factors to the
$\displaystyle C_{\text{NN}}^{\prime}$ Gaussians.

To account for small differences between MC and data in the width of the
$\displaystyle\Delta E$ distribution, the 2D KEST PDF for $\displaystyle
M_{\rm bc}$ and $\displaystyle\Delta E$ in $\displaystyle B_{3b}$ is modified
slightly. This is done by modifying each $\displaystyle\Delta E$ data point in
the MC data set with a random shift based on a Gaussian distribution with a
mean of 0 GeV and a width of 7 MeV (chosen after testing with a range of
widths) and generating a new KEST PDF from this modified data sample.

Fits to the $\displaystyle B^{+}\to\bar{D}^{0}\pi^{+}$ sample are performed to
determine the signal yield, $\displaystyle{\cal A}_{CP,\text{raw}}$, the
continuum background yield, the generic $\displaystyle B$-decay background
yield, and the calibration factors. The rare $\displaystyle B$-decay
background yield is fixed to the value expected from MC simulations. From the
fit we obtain $\displaystyle N_{\rm sig}=84537\pm 306$ and $\displaystyle{\cal
A}_{CP,\text{raw}}=(1.97\pm 0.36)\%$. The uncertainties listed are
statistical. $\displaystyle{\cal A}_{CP,\text{raw}}$ is the output of the fit
without a correction to account for sources of bias. Figure 2 shows the fits
to data in $\displaystyle M_{\text{bc}}$, $\displaystyle\Delta E$ and
$\displaystyle C_{\rm NN}^{\prime}$.

Figure 2: Projections of the $\displaystyle B^{+}\to\bar{D}^{0}\pi^{+}$ fit
results into the signal region ($\displaystyle 5.275<M_{\rm
bc}<5.285{\rm{\,Ge\kern-1.00006ptV}}$, $\displaystyle-0.05<\Delta
E<0.05{\rm{\,Ge\kern-1.00006ptV}}$, $\displaystyle-1<C_{\rm NN}^{\prime}<6$ )
for $\displaystyle M_{\rm bc}$ (left), $\displaystyle\Delta E$ (middle) and
$\displaystyle C_{\rm NN}^{\prime}$ (right) split into the $\displaystyle
B_{2b}$ mode (top) and $\displaystyle B_{3b}$ (bottom). The blue short-dashed
curve shows the signal PDF, red dotted curve shows the $\displaystyle
B\bar{B}$ background PDF, green dash-dotted curve shows the continuum
background PDF, pink long-dashed curve shows the (almost negligible) rare
background PDF, black line is the fit result, points are data. Also shown
underneath each graph is the residual pulls between the data points and fitted
PDF.

To account for possible bias in $\displaystyle{\cal A}_{CP}$, we perform an
analysis over a “sideband” region of data, defined as $\displaystyle
0.1{\rm{\,Ge\kern-1.00006ptV}}<\Delta E<0.4{\rm{\,Ge\kern-1.00006ptV}}$ and
$\displaystyle 5.255{{{\rm\,Ge\kern-1.00006ptV\\!/}}c^{2}}<M_{\rm
bc}<5.27{{{\rm\,Ge\kern-1.00006ptV\\!/}}c^{2}}$. This region consists almost
entirely of continuum events and has an expected $\displaystyle{\cal A}_{CP}$
of zero. Counting the number of events in this region we find
$\displaystyle{\cal A}_{CP,\text{sideband}}=(1.78\pm 0.38)\%$. We subtract
this value from $\displaystyle{\cal A}_{CP,\text{raw}}$ to correct for the
detection asymmetry bias.

The branching fraction is calculated as:

$\mathcal{B}=\frac{N_{\rm
sig}}{N_{B^{\pm{}}}}\times\text{mean}\left(\frac{f^{s}_{2b}}{\epsilon_{2b}},\frac{f^{s}_{3b}}{\epsilon_{3b}}\right),\\\
$ (8)

where $\displaystyle N_{B^{\pm{}}}$ is the number of charged $\displaystyle
B$-mesons in the data set, based on the PDG average value of
$\displaystyle{\cal B}(\Upsilon(4S)\to{B}^{+}B^{-})=(51.4\pm 0.6\%)$Zyla  _et
al._ (2020); $\displaystyle f^{s}_{d}$ is the fraction of signal events in
data set $\displaystyle d=2b\text{ or }3b$ ($\displaystyle f^{s}_{2b}=0.51$,
$\displaystyle f^{s}_{3b}=0.49$); and $\displaystyle\epsilon_{d}$ is the
product of the reconstruction efficiency, the $\displaystyle\bar{D}^{0}$
branching fraction $\displaystyle\mathcal{B}_{d}$, and small corrections for
particle identification (PID), and charged track and $\displaystyle\pi^{0}$
reconstruction efficiencies (see Section VII), for mode $\displaystyle d$. The
$\displaystyle\pi^{0}\rightarrow\gamma\gamma$ branching fraction is accounted
for in the MC simulation. The resulting values for
$\displaystyle\epsilon_{2b}$ and $\displaystyle\epsilon_{3b}$ are
$\displaystyle(1.19\pm 0.03)\times 10^{-2}$ and $\displaystyle(1.16\pm
0.05)\times 10^{-2}$, respectively. The mean is calculated as a generalized
weighted mean Rice (2007)Cox  _et al._ (2006), taking into account correlated
and uncorrelated uncertainties in a covariance matrix. This approach is used
because the difference in systematic uncertainties between the two
$\displaystyle\bar{D}^{0}$ decay modes leads to the need to weight them in
order to calculate the final branching fraction and uncertainty correctly.
Finally, the correction due to DCS decays discussed in Section II is made.

The results for $\displaystyle{\cal B}$ and $\displaystyle{\cal A}_{CP}$ for
$\displaystyle B^{+}\to\bar{D}^{0}\pi^{+}$ are:

$\displaystyle\displaystyle{\cal B}$ $\displaystyle\displaystyle=(4.53\pm
0.02\pm 0.15)\times 10^{-3},$ (9) $\displaystyle\displaystyle{\cal A}_{CP}$
$\displaystyle\displaystyle=(0.19\pm 0.36\pm 0.57)\%.$ (10)

The uncertainties quoted are statistical and systematic, respectively. The
systematic uncertainties associated with the measurement of
$\displaystyle{\cal B}$ and $\displaystyle{\cal A}_{CP}$ are explained in
detail in Section VII, and the contributions of each of these are listed in
Table 5 and 6, respectively. These results are in agreement with the PDG
values Zyla  _et al._ (2020) of $\displaystyle{\cal B}=(4.68\pm 0.13)\times
10^{-3}$ and $\displaystyle{\cal A}_{CP}=(-0.7\pm 0.7)\%$. As a cross-check,
we determined $\displaystyle{\cal B}$ and $\displaystyle{\cal A}_{CP}$ for
each of the $\displaystyle B_{2b}$ and $\displaystyle B_{3d}$ modes
separately, and for just the SVD1 data set. All are in agreement within
statistical uncertainties. The fitted yield for each respective category is
listed in Table 3.

Table 3:  Fitted number of signal and backgrounds events for the two reconstruction modes ($\displaystyle B_{2b}$ and $\displaystyle B_{3b}$) of $\displaystyle B^{+}\to\bar{D}^{0}\pi^{+}$. Uncertainties are statistical only. Category |  $\displaystyle B_{2b}$ mode ($\displaystyle\times 10^{4}$) |  $\displaystyle B_{3b}$ mode ($\displaystyle\times 10^{4}$)
---|---|---
Signal | $\displaystyle 4.27\pm 0.02$ | $\displaystyle 4.18\pm 0.02$
Continuum | $\displaystyle 0.70\pm 0.01$ | $\displaystyle 1.78\pm 0.3$
Generic B | $\displaystyle 3.58\pm 0.03$ | $\displaystyle 3.87\pm 0.03$
Rare |  $\displaystyle 0.03$ (fixed) |  $\displaystyle 0.05$ (fixed)

##  VI $\displaystyle B^{0}\to\bar{D}^{0}\pi^{0}$

After applying the calibration factors determined from studies of the
$\displaystyle B^{+}\to\bar{D}^{0}\pi^{+}$ mode to the PDFs, we fit the signal
$\displaystyle B^{0}\to\bar{D}^{0}\pi^{0}$ PDFs to data and find
$\displaystyle 4448\pm 97$ signal events and $\displaystyle{\cal
A}_{CP,\text{raw}}=(1.48\pm 2.05)\%$. The uncertainties quoted are
statistical. As was the case for $\displaystyle B^{+}\to\bar{D}^{0}\pi^{+}$,
$\displaystyle{\cal A}_{CP,\text{raw}}$ is the value returned from the fit
without a correction for sources of bias. Figure 3 shows the signal-enhanced
projections of the fits. Figure 4 shows signal-enhanced projections of
$\displaystyle M_{\rm bc}$ separated into $\displaystyle B^{0}$ and
$\displaystyle\bar{B}^{0}$ decays.

Using Eq. 8, the PDG value $\displaystyle{\cal
B}(\Upsilon(4S)\to\bar{B}^{0}B^{0})=(48.6\pm 0.6\%)$Zyla  _et al._ (2020), the
fraction of signal events in data set $\displaystyle d$, $\displaystyle
f^{s}_{2b}=0.45$, $\displaystyle f^{s}_{3b}=0.55$, and the efficiencies
$\displaystyle\epsilon_{2b}=(1.00\pm 0.03)\times 10^{-2}$ and
$\displaystyle\epsilon_{3b}=(1.21\pm 0.07)\times 10^{-2}$, we determine the
branching fraction to be:

${\cal B}(B^{0}\to\bar{D}^{0}\pi^{0})=(2.70\pm 0.06\pm 0.10)\times 10^{-4},$
(11)

where the quoted uncertainties are statistical and systematic, respectively.
The fitted yield for each respective category is listed in Table 4.

Table 4:  Fitted number of signal and backgrounds events for the two reconstruction modes ($\displaystyle B_{2b}$ and $\displaystyle B_{3b}$) of $\displaystyle B^{0}\to\bar{D}^{0}\pi^{0}$. Uncertainties are statistical only. Category |  $\displaystyle B_{2b}$ mode ($\displaystyle\times 10^{3}$) |  $\displaystyle B_{3b}$ mode ($\displaystyle\times 10^{3}$)
---|---|---
Signal | $\displaystyle 2.01\pm 0.04$ | $\displaystyle 2.44\pm 0.05$
Continuum | $\displaystyle 4.26\pm 0.06$ | $\displaystyle 16.47\pm 0.22$
Generic B | $\displaystyle 4.76\pm 0.10$ | $\displaystyle 8.39\pm 0.18$
Rare |  $\displaystyle 0.15$ (fixed) |  $\displaystyle 0.47$ (fixed)

The $\displaystyle{\cal A}_{CP}$ correction for the $\displaystyle
B^{0}\to\bar{D}^{0}\pi^{0}$ decay is measured in the same way as for the
$\displaystyle B^{+}\to\bar{D}^{0}\pi^{+}$ mode. A sideband region of data is
defined as $\displaystyle 0.1{\rm{\,Ge\kern-1.00006ptV}}<\Delta
E<0.4{\rm{\,Ge\kern-1.00006ptV}}$ and $\displaystyle
5.255{{{\rm\,Ge\kern-1.00006ptV\\!/}}c^{2}}<M_{\rm
bc}<5.27{{{\rm\,Ge\kern-1.00006ptV\\!/}}c^{2}}$. Events in this region consist
almost entirely of continuum with an expected $\displaystyle{\cal A}_{CP}$ of
zero. In this region we find $\displaystyle{\cal
A}_{CP,\text{sideband}}=(1.02\pm 0.64)\%$, and we subtract this value and the
fit bias ($\displaystyle 0.04\%$) from $\displaystyle{\cal A}_{CP,\text{raw}}$
to correct for detector bias.

Figure 3: Projections of the $\displaystyle B^{0}\to\bar{D}^{0}\pi^{0}$ fit
results into the signal region ($\displaystyle 5.275<M_{\rm
bc}<5.285{\rm{\,Ge\kern-1.00006ptV}}$,$\displaystyle-0.12<\Delta
E<0.07{\rm{\,Ge\kern-1.00006ptV}}$,$\displaystyle-1<C_{\rm NN}^{\prime}<6$ )
for $\displaystyle M_{\rm bc}$ (left), $\displaystyle\Delta E$ (middle) and
$\displaystyle C_{\rm NN}^{\prime}$ (right) split into the $\displaystyle
B_{2b}$ mode (top) and $\displaystyle B_{3b}$ (bottom). The blue short-dashed
curve shows the signal PDF, red dotted curve shows the $\displaystyle
B\bar{B}$ background PDF, green dash-dotted curve shows the continuum
background PDF, pink long-dashed curve shows the (almost negligible) rare
background PDF, black line is the fit result. Also shown underneath each graph
is the residual pulls between the data points and fitted PDF.

The direct $\displaystyle C\\!P$-violation parameter is thus measured to be:

${\cal A}_{C\\!P}(B^{0}\to\bar{D}^{0}\pi^{0})=(0.42\pm 2.05\pm 1.22)\%$ (12)

The uncertainties quoted are statistical and systematic, respectively.

Figure 4: Projections of the $\displaystyle B^{0}\to\bar{D}^{0}\pi^{0}$ fit
results for $\displaystyle M_{\rm bc}$ into the signal region
($\displaystyle-0.12<\Delta
E<0.07{\rm{\,Ge\kern-1.00006ptV}}$,$\displaystyle-1<C_{\rm NN}^{\prime}<6$ )
for $\displaystyle B^{0}\to\bar{D}^{0}(K^{+}\pi^{-})\pi^{0}$ (top left),
$\displaystyle\bar{B}^{0}\to D^{0}(K^{-}\pi^{+})\pi^{0}$ (top right),
$\displaystyle B^{0}\to\bar{D}^{0}(K^{+}\pi^{-}\pi^{0})\pi^{0}$ (bottom left),
$\displaystyle\bar{B}^{0}\to D^{0}(K^{-}\pi^{+}\pi^{0})\pi^{0}$ (bottom
right). The blue short-dashed curve shows the signal PDF, red dotted curve
shows the $\displaystyle B\bar{B}$ background PDF, green dash-dotted curve
shows the continuum background PDF, pink long-dashed curve shows the (almost
negligible) rare background PDF, black line is the fit result. Also shown
underneath each graph is the residual pulls between the data points and fitted
PDF.

##  VII  Systematic Uncertainties

The systematic uncertainties associated with the measurement of
$\displaystyle{\cal B}$ and $\displaystyle{\cal A}_{CP}$ are as follows.

  * •

Number of $\displaystyle B\bar{B}$ pairs: the uncertainty associated with the
measured number of $\displaystyle B\bar{B}$ pairs in the full data set
collected at Belle is 1.37% Bevan  _et al._ (2014).

  * •

$\displaystyle{\cal B}(\Upsilon(4S)\to B^{0}\bar{B}^{0})$: uncertainty from
the branching fraction $\displaystyle{\cal
B}(\Upsilon(4S)\to\bar{B}^{0}B^{0})=(48.6\pm 0.6\%)$ Zyla  _et al._ (2020).

  * •

DCS mode correction: the uncertainty due to the correction for doubly Cabibbo-
suppressed decays is 0.01% for both $\displaystyle B^{0}\to\bar{D}^{0}\pi^{0}$
and $\displaystyle B^{+}\to\bar{D}^{0}\pi^{+}$ (see Section II).

  * •

Charged track efficiency: the uncertainty associated with a possible
difference in efficiency between MC and data for charged-track reconstruction
is found to be 0.35% per track using partially reconstructed $\displaystyle
D^{*+}\to D^{0}(\to\pi^{+}\pi^{-}\pi^{0})\pi^{+}$ events Bevan  _et al._
(2014).

  * •

$\displaystyle\pi^{0}$ detection efficiency: the ratio of data to MC
efficiency for $\displaystyle\pi^{0}$ reconstruction is based on a study of
$\displaystyle\tau^{-}\to\pi^{-}\pi^{0}\nu_{\tau}$ decays Ryu  _et al._
(2014). This ratio is $\displaystyle(96\pm 2)\%$ per $\displaystyle\pi^{0}$.

  * •

MC statistics in efficiency calculation: uncertainty associated with the
reconstruction efficiency is based on the binomial statistics of the MC data
set used. This is 0.094% for $\displaystyle
B^{0}\to\bar{D}^{0}(K^{+}\pi^{-})\pi^{0}$, 0.18% for $\displaystyle
B^{0}\to\bar{D}^{0}(K^{+}\pi^{-}\pi^{0})\pi^{0}$, 0.075% for $\displaystyle
B^{+}\to\bar{D}^{0}(K^{+}\pi^{-})\pi^{+}$, and 0.19% for $\displaystyle
B^{+}\to\bar{D}^{0}(K^{+}\pi^{-}\pi^{0})\pi^{+}$.

  * •

$\displaystyle\bar{D}^{0}$ subdecay branching fraction and $\displaystyle
A_{CP}$: from the PDG average Zyla  _et al._ (2020).

  * •

PID efficiency: systematic error associated with a small difference in PID
efficiency between MC and data. This is based on an inclusive $\displaystyle
D^{*+}\to D^{0}(K^{-}\pi^{+})\pi^{+}$ study Bevan  _et al._ (2014). The
uncertainty is calculated as 1.3% for $\displaystyle
B^{0}\to\bar{D}^{0}(K^{+}\pi^{-})\pi^{0}$, 1.3% for $\displaystyle
B^{0}\to\bar{D}^{0}(K^{+}\pi^{-}\pi^{0})\pi^{0}$, 2.2% for $\displaystyle
B^{+}\to\bar{D}^{0}(K^{+}\pi^{-})\pi^{+}$, and 2.2% for $\displaystyle
B^{+}\to\bar{D}^{0}(K^{+}\pi^{-}\pi^{0})\pi^{+}$.

  * •

Signal decay mode yield ratio $\displaystyle f^{s}_{d}$: the ratio between the
$\displaystyle D^{0}$ decay modes in signal, $\displaystyle f^{s}_{d}$, is
fixed based on the expected yields from MC. To account for the uncertainty, we
perform two fits, varying the fixed value by $\displaystyle\pm 1\sigma$ (based
on MC statistics of the simulation). This variation gives changes of
$\displaystyle[-0.38,+0.31]$% and $\displaystyle[-0.08,+0.19]$% in
$\displaystyle\mathcal{B}$ for $\displaystyle B^{0}\to\bar{D}^{0}\pi^{0}$ and
$\displaystyle B^{+}\to\bar{D}^{0}\pi^{+}$ respectively. The uncertainty in
$\displaystyle{\cal A}_{CP}$ is $\displaystyle[-0.02,+0.03]$ for
$\displaystyle B^{0}\to\bar{D}^{0}\pi^{0}$ and $\displaystyle<0.01$ for
$\displaystyle B^{+}\to\bar{D}^{0}\pi^{+}$.

  * •

$\displaystyle C_{\text{NN}}^{\prime}$ calibration factors: we fit with and
without the calibration factors applied to the PDFs. The difference between
the yields and $\displaystyle{\cal A}_{CP}$ of these fits is quoted as the
uncertainty. The uncertainty in $\displaystyle\mathcal{B}$ is 0.34% and 0.06%
for $\displaystyle B^{0}\to\bar{D}^{0}\pi^{0}$ and $\displaystyle
B^{+}\to\bar{D}^{0}\pi^{+}$, respectively. For $\displaystyle{\cal A}_{CP}$ it
is 0.06 and $\displaystyle<0.01$, respectively.

  * •

Modification of the $\displaystyle B_{3d}$ $\displaystyle M_{\rm
bc}\times\Delta E$ KEST PDF: the uncertainty from the $\displaystyle\Delta E$
modification to the $\displaystyle\bar{D}^{0}\to K^{+}\pi^{-}\pi^{0}$
$\displaystyle M_{\rm bc}\times\Delta E$ KEST PDF is evaluated by comparing
the fit results obtained using the corrected and uncorrected PDF. The
difference in the fitted yields of 0.63% for $\displaystyle
B^{0}\to\bar{D}^{0}\pi^{0}$ and 0.24% for $\displaystyle
B^{+}\to\bar{D}^{0}\pi^{+}$ is quoted as the uncertainty. For
$\displaystyle{\cal A}_{CP}$ this is 0.06 and $\displaystyle<0.01$,
respectively.

  * •

2D KEST PDFs: $\displaystyle B\bar{B}$, rare, and $\displaystyle B_{3b}$
signal $\displaystyle M_{\rm bc}\times\Delta E$ PDFs all use a fixed 2D KEST
PDF. To estimate the uncertainty from this, an ensemble test is performed
running 1000 fits over the data, with each fit using a different Gaussian-
fluctuated KEST PDF based on bin statistics. The uncertainty is quoted as the
RMS of the resulting yield and $\displaystyle{\cal A}_{CP}$ distributions.
This contributes 0.35% and 0.05% to the uncertainty in
$\displaystyle\mathcal{B}$ for $\displaystyle B^{0}\to\bar{D}^{0}\pi^{0}$ and
$\displaystyle B^{+}\to\bar{D}^{0}\pi^{+}$, respectively. For
$\displaystyle{\cal A}_{CP}$ the contribution is 0.15 and
$\displaystyle<0.01$, respectively.

  * •

Fixed rare $\displaystyle B$-decay background yield: the uncertainty due to
the rare B yield is the quadratic sum of the statistical uncertainty based on
the size of the MC dataset and the uncertainty in the branching fractions used
to generate the MC. For modes with three body final states ($\displaystyle
K^{+}\pi^{-}\pi^{0}$ and $\displaystyle K^{+}\pi^{-}\pi^{+}$), this latter
component is taken from the uncertainty in the PDG branching fractions
$\displaystyle\mathcal{B}({B}^{0}\to K^{+}\pi^{-}\pi^{0})=(37.8\pm 3.2)\times
10^{-6}$ and $\displaystyle\mathcal{B}({B}^{+}\to
K^{+}\pi^{-}\pi^{+})=(51.0\pm 2.9)\times 10^{-6}$ Zyla  _et al._ (2020). For
$\displaystyle\bar{D}^{0}\to K^{+}\pi^{-}\pi^{0}$ modes, this latter component
is taken as the difference between the PDG values for the decays with
experimentally measured branching fractions and the branching fractions used
in the MC generator (or the uncertainty on the PDG value if that is larger).
To estimate the effect on signal yield, the data is refitted, varying the rare
yield by $\displaystyle\pm 1\sigma$. The uncertainty in
$\displaystyle\mathcal{B}$ is 0.47% for $\displaystyle
B^{0}\to\bar{D}^{0}\pi^{0}$ and $\displaystyle 0.03$% for $\displaystyle
B^{+}\to\bar{D}^{0}\pi^{+}$.

  * •

Fit bias: the uncertainty in the fit bias obtained from the signal MC ensemble
tests is quoted as an uncertainty. For $\displaystyle{\cal B}$ this is 0.30%
and 0.16% for $\displaystyle B^{0}\to\bar{D}^{0}\pi^{0}$ and $\displaystyle
B^{+}\to\bar{D}^{0}\pi^{+}$, respectively, and for $\displaystyle{\cal
A}_{CP}$ it is 0.05 and 0.02, respectively.

  * •

$\displaystyle{\cal A}_{CP}$ detector bias correction: uncertainty on the
correction made to $\displaystyle{\cal A}_{CP}$ is the statistical uncertainty
on the $\displaystyle{\cal A}_{CP,\rm{sideband}}$ measurement, summed in
quadrature with the deviation of the $\displaystyle{\cal A}_{CP}$ of the
$\displaystyle B^{+}\to\bar{D}^{0}\pi^{+}$ mode from the expected value of
$\displaystyle{\cal A}_{CP}=0$. This is 0.66 for $\displaystyle
B^{0}\to\bar{D}^{0}\pi^{0}$ and 0.42 for $\displaystyle
B^{+}\to\bar{D}^{0}\pi^{+}$.

  * •

Fixed background $\displaystyle{\cal A}_{CP}$: uncertainties from background
$\displaystyle{\cal A}_{CP}$ being fixed in fits are estimated by varying them
by $\displaystyle\pm 1\sigma$ (based on sideband data) and comparing the
$\displaystyle{\cal A}_{CP}$ in the resultant fits. This is found to be 0.49
for $\displaystyle B^{0}\to\bar{D}^{0}\pi^{0}$ and 0.03 for $\displaystyle
B^{+}\to\bar{D}^{0}\pi^{+}$. This is correlated with the $\displaystyle{\cal
A}_{CP}$ detector bias correction uncertainty.

In order to accurately calculate the uncertainty in
$\displaystyle\mathcal{B}$, the $\displaystyle\bar{D}^{0}$ decay-mode-
dependent factors are combined in a generalized weighted mean as shown in Eq.
8. The absolute uncertainties for charged track efficiency,
$\displaystyle\pi^{0}$ detection efficiency, reconstruction efficiency, PID
efficiency, and $\displaystyle\bar{D}^{0}$ branching fraction are combined
into a covariance matrix, $\displaystyle\Sigma$, that accounts for their
correlations between the two-body and three-body modes. For $\displaystyle
B^{0}\to\bar{D}^{0}\pi^{0}$,
$\displaystyle\Sigma=\big{[}\begin{smallmatrix}1.47&2.40\\\
2.40&6.76\end{smallmatrix}\big{]}$, and for $\displaystyle
B^{+}\to\bar{D}^{0}\pi^{+}$
$\displaystyle\Sigma=\big{[}\begin{smallmatrix}1.17&1.05\\\
1.05&4.03\end{smallmatrix}\big{]}$. The combined value, which we call “mean
efficiency”, is calculated as
$\displaystyle\bar{\epsilon}=\sigma^{2}_{\bar{\epsilon}}(\mathcal{J}^{T}\Sigma^{-1}\left[\frac{f^{s}_{2b}}{\epsilon_{2b}},\frac{f^{s}_{3b}}{\epsilon_{3b}}\right]^{T})^{-1}$,
with variance
$\displaystyle{\sigma^{2}_{\bar{\epsilon}}=(\mathcal{J}^{T}\Sigma^{-1}\mathcal{J})^{-1}}$
(where $\displaystyle\mathcal{J}=[1,1]^{T}$) Rice (2007)Cox  _et al._ (2006).
The relative uncertainty on this is found to be 2.43% for $\displaystyle
B^{0}\to\bar{D}^{0}\pi^{0}$, and 2.54% for $\displaystyle
B^{+}\to\bar{D}^{0}\pi^{+}$.

Table 5: Systematic uncertainties for $\displaystyle{\cal B}$ measurements. The mean efficiency results from combining charged track efficiency, $\displaystyle\pi^{0}$ detection efficiency, MC statistics in efficiency calculation, $\displaystyle\bar{D}^{0}$ subdecay properties, and PID efficiency in a general weighted mean calculation. Systematic | $\displaystyle B^{0}\to\bar{D}^{0}\pi^{0}$ | $\displaystyle B^{+}\to\bar{D}^{0}\pi^{+}$
---|---|---
No. $\displaystyle B\bar{B}$ | 1.37% | 1.37%
$\displaystyle{\cal B}(\Upsilon(4S)\to B^{0}\bar{B}^{0})$ | 1.23% | 1.17%
DCS mode correction | 0.01% | 0.01%
Mean efficiency | 2.43% | 2.54%
Fixed $\displaystyle f^{s}_{d}$ |  $\displaystyle{+0.31\atop-0.38}$% |  $\displaystyle{+0.19\atop-0.08}$%
Cal. Factors ($\displaystyle C_{\text{NN}}^{\prime}$) | 0.34% | 0.06%
$\displaystyle\Delta E$ KEST modification | 0.63% | 0.24%
KEST PDFs | 0.35% | 0.05%
Fixed Rare Yields | 0.47% | 0.03%
Fit bias | 0.30% | 0.16%
Bkg. $\displaystyle{\cal A}_{CP}$ | 0.01% | 0.05%
Total | 3.65% | 3.32%
Table 6: Systematic uncertainties for $\displaystyle{\cal A}_{CP}$ measurements. All numbers listed $\displaystyle\times 10^{-2}$. * denotes correlated uncertainties. Systematic for $\displaystyle{\cal A}_{CP}$ | $\displaystyle B^{0}\to\bar{D}^{0}\pi^{0}$ | $\displaystyle B^{+}\to\bar{D}^{0}\pi^{+}$
---|---|---
$\displaystyle\bar{D}^{0}$ Decay $\displaystyle{\cal A}_{CP}$ | $\displaystyle 0.35$ | $\displaystyle 0.35$
Fixed $\displaystyle f^{s}_{d}$ | $\displaystyle{+0.03\atop-0.02}$ | $\displaystyle<0.01$
Cal. Factors ($\displaystyle C_{\text{NN}}^{\prime}$) | $\displaystyle 0.06$ | $\displaystyle<0.01$
$\displaystyle\Delta E$ KEST modification | $\displaystyle 0.06$ | $\displaystyle<0.01$
KEST PDFs | $\displaystyle 0.15$ | $\displaystyle<0.01$
Fixed Rare Yields | $\displaystyle<0.01$ | $\displaystyle<0.01$
Fit bias | $\displaystyle 0.05$ | $\displaystyle 0.02$
Detector bias (signal)* | $\displaystyle 0.66$ | $\displaystyle 0.42$
Detector bias (background)* | $\displaystyle 0.49$ | $\displaystyle 0.03$
Total | 1.22 | 0.57

The values of all contributions to the branching fractions are listed in Table
5. The quadratic sum of these terms is quoted as the total systematic
uncertainty for $\displaystyle{\cal B}$. The values of all contributions to
the $\displaystyle{\cal A}_{CP}$ measurements are listed in Table 6. The
quadratic sum of these terms is quoted as the total systematic uncertainty for
$\displaystyle{\cal A}_{CP}$.

##  VIII Conclusions

Our measurements of

$\displaystyle\displaystyle\mathcal{B}(B^{0}\to\bar{D}^{0}\pi^{0})$
$\displaystyle\displaystyle=[2.70\pm 0.06~{}\pm 0.10~{}]\times 10^{-4},$ (13)
$\displaystyle\displaystyle{\cal B}(B^{+}\to\bar{D}^{0}\pi^{+})$
$\displaystyle\displaystyle=(4.53\pm 0.02\pm 0.15)\times 10^{-3}$ (14)

are the most precise to date. They agree with our previous measurements Blyth
_et al._ (2006)Kato  _et al._ (2018) within uncertainties, and supersede those
results. They are also in agreement with PDG values Zyla  _et al._ (2020).

Our result

${\cal A}_{C\\!P}(B^{0}\to\bar{D}^{0}\pi^{0})=(0.42\pm 2.05\pm 1.22)\%\\\ $
(15)

is the first reported for this mode. Our result

${\cal A}_{CP}(B^{+}\to\bar{D}^{0}\pi^{+})=(0.19\pm 0.36\pm 0.57)\%\\\ $ (16)

is the most precisely measured and agrees with our previous result Abe  _et
al._ (2006), which it supersedes.

##  IX Acknowledgments

We thank the KEKB group for the excellent operation of the accelerator; the
KEK cryogenics group for the efficient operation of the solenoid; and the KEK
computer group, and the Pacific Northwest National Laboratory (PNNL)
Environmental Molecular Sciences Laboratory (EMSL) computing group for strong
computing support; and the National Institute of Informatics, and Science
Information NETwork 5 (SINET5) for valuable network support. We acknowledge
support from the Ministry of Education, Culture, Sports, Science, and
Technology (MEXT) of Japan, the Japan Society for the Promotion of Science
(JSPS), and the Tau-Lepton Physics Research Center of Nagoya University; the
Australian Research Council including grants DP180102629, DP170102389,
DP170102204, DP150103061, FT130100303; Austrian Federal Ministry of Education,
Science and Research (FWF) and FWF Austrian Science Fund No. P 31361-N36; the
National Natural Science Foundation of China under Contracts No. 11435013, No.
11475187, No. 11521505, No. 11575017, No. 11675166, No. 11705209; Key Research
Program of Frontier Sciences, Chinese Academy of Sciences (CAS), Grant No.
QYZDJ-SSW-SLH011; the CAS Center for Excellence in Particle Physics (CCEPP);
the Shanghai Science and Technology Committee (STCSM) under Grant No.
19ZR1403000; the Ministry of Education, Youth and Sports of the Czech Republic
under Contract No. LTT17020; Horizon 2020 ERC Advanced Grant No. 884719 and
ERC Starting Grant No. 947006 “InterLeptons” (European Union); the Carl Zeiss
Foundation, the Deutsche Forschungsgemeinschaft, the Excellence Cluster
Universe, and the VolkswagenStiftung; the Department of Atomic Energy (Project
Identification No. RTI 4002) and the Department of Science and Technology of
India; the Istituto Nazionale di Fisica Nucleare of Italy; National Research
Foundation (NRF) of Korea Grant Nos. 2016R1D1A1B01010135, 2016R1D1A1B02012900,
2018R1A2B3003643, 2018R1A6A1A06024970, 2019K1A3A7A09033840,
2019R1I1A3A01058933, 2021R1A6A1A03043957, 2021R1F1A1060423, 2021R1F1A1064008;
Radiation Science Research Institute, Foreign Large-size Research Facility
Application Supporting project, the Global Science Experimental Data Hub
Center of the Korea Institute of Science and Technology Information and
KREONET/GLORIAD; the Polish Ministry of Science and Higher Education and the
National Science Center; the Ministry of Science and Higher Education of the
Russian Federation, Agreement 14.W03.31.0026, and the HSE University Basic
Research Program, Moscow; University of Tabuk research grants S-1440-0321,
S-0256-1438, and S-0280-1439 (Saudi Arabia); the Slovenian Research Agency
Grant Nos. J1-9124 and P1-0135; Ikerbasque, Basque Foundation for Science,
Spain; the Swiss National Science Foundation; the Ministry of Education and
the Ministry of Science and Technology of Taiwan; and the United States
Department of Energy and the National Science Foundation.

## References

  * (1) Throughout this Letter, the inclusion of the charge-conjugate decay modes is implied unless stated otherwise.
  * Abe  _et al._ (2002) K. Abe _et al._ (Belle Collaboration), Phys. Rev. Lett. 88, 052002 (2002).
  * Coan  _et al._ (2002) T. Coan _et al._ (CLEO Collaboration), Phys. Rev. Lett. 88, 062001 (2002).
  * Blyth  _et al._ (2006) S. Blyth _et al._ (Belle Collaboration), Phys. Rev. D 74, 092002 (2006).
  * Beneke  _et al._ (2000) M. Beneke, G. Buchalla, M. Neubert, and C. Sachrajda, Nucl. Phys. B 591, 313 (2000).
  * Neubert and Stech (1998) M. Neubert and B. Stech, Advanced Series on Directions in High Energy Physics 15, 294–344 (1998).
  * Neubert and Petrov (2001) M. Neubert and A. Petrov, Phys. Lett. B 519, 50 (2001).
  * Chua  _et al._ (2002) C.-K. Chua, W.-S. Hou, and K.-C. Yang, Phys. Rev. D 65, 096007 (2002).
  * Chua and Hou (2005) C.-K. Chua and W.-S. Hou, Phys. Rev. D 72, 036002 (2005).
  * Rosner (1999) J. Rosner, Phys. Rev. D 60, 074029 (1999).
  * Deandrea and Polosa (2002) A. Deandrea and A. Polosa, Eur. Phys. Jour. C 22, 677 (2002).
  * Chiang and Rosner (2003) C.-W. Chiang and J. Rosner, Phys. Rev. D 67, 074013 (2003).
  * Chua and Hou (2008) C.-K. Chua and W.-S. Hou, Phys. Rev. D 77, 116001 (2008).
  * Leganger and Eeg (2010) L. E. Leganger and J. O. Eeg, Phys. Rev. D 82, 074007 (2010).
  * Bigi and Sanda (2009) I. Bigi and A. Sanda,  _CP Violation_ , 2nd ed. (Cambridge University Press, 2009).
  * Abdesselam  _et al._ (2015) A. Abdesselam _et al._ (BaBar Collaboration, Belle Collaboration), Phys. Rev. Lett. 115, 121604 (2015).
  * (17) Another naming convention, $\displaystyle\beta$ ($\displaystyle=\phi_{1}$), $\displaystyle\alpha$ ($\displaystyle=\phi_{2}$) and $\displaystyle\gamma$ ($\displaystyle=\phi_{3}$) is also used in the literature.
  * (18) S. Kurokawa and E. Kikutani, Nucl. Instrum. Methods Phys. Res., Sect. A 499, 1 (2003), and other papers included in this Volume; T. Abe et al., Prog. Theor. Exp. Phys. (2013) 03A001 and following articles up to 03A011.
  * (19) A. Abashian et al. (Belle Collaboration), Nucl. Instrum. Methods Phys. Res., Sect. A 479, 117 (2002); also see the detector section in J. Brodzicka et al., Prog. Theor. Exp. Phys. (2012) 04D001.
  * Natkaniec  _et al._ (2006) Z. Natkaniec _et al._ (Belle SVD2 Group), Nucl. Instrum. Methods Phys. Res., Sect. A 560, 1 (2006).
  * Das  _et al._ (2010) A. Das _et al._ (Belle Collaboration), Phys. Rev. D 82, 051103 (2010).
  * Zyla  _et al._ (2020) P. Zyla _et al._ (Particle Data Group), PTEP 2020, 083C01 (2020).
  * Lange  _et al._ (2001) D. Lange _et al._ , Nucl. Instrum. Methods Phys. Res., Sect. A 462, 152 (2001).
  * Sjöstrand  _et al._ (2006) T. Sjöstrand, S. Mrenna, and P. Z. Skands, JHEP 05, 026 (2006).
  * Brun  _et al._ (1987) R. Brun _et al._ , CERN Report No. DD/EE/84-1 (1987).
  * Feindt and Kerzel (2006) M. Feindt and U. Kerzel, Nucl. Instrum. Methods Phys. Res., Sect. A 559, 190 (2006).
  * (27) The Fox-Wolfram moments were introduced by G. C. Fox and S. Wolfram in Phys. Rev. Lett. 41, 1581 (1978). The modified Fox-Wolfram moments used in this paper are described in S. H. Lee et al. (Belle Collaboration), Phys. Rev. Lett. 91, 261801 (2003).
  * Brandt  _et al._ (1964) S. Brandt, C. Peyrou, R. Sosnowski, and A. Wroblewski, Phys. Lett. B 12, 57 (1964).
  * Kakuno  _et al._ (2004) H. Kakuno _et al._ , Nucl. Instrum. Methods Phys. Res., Sect. A 533, 516 (2004).
  * (30) T. Skwarnicki, DESY F31-86-02, (1986) (PhD dissertation).
  * Cranmer (2001) K. Cranmer, Computer Physics Communications 136, 198 (2001).
  * Albrecht  _et al._ (1990) H. Albrecht _et al._ (ARGUS Collaboration), Phys. Lett. B 241, 278 (1990).
  * Rice (2007) J. Rice,  _Mathematical Statistics and Data Analysis_, Duxbury advanced series (Thompson/Brooks/Cole, 2007).
  * Cox  _et al._ (2006) M. G. Cox, C. Eiø, G. Mana, and F. Pennecchi, Metrologia 43, S268 (2006).
  * Bevan  _et al._ (2014) A. J. Bevan _et al._ , The European Physical Journal C 74, 3026 (2014).
  * Ryu  _et al._ (2014) S. Ryu _et al._ (Belle Collaboration), Phys. Rev. D 89, 072009 (2014).
  * Kato  _et al._ (2018) Y. Kato _et al._ (Belle Collaboration), Phys. Rev. D 97, 012005 (2018).
  * Abe  _et al._ (2006) K. Abe _et al._ (Belle Collaboration), Phys. Rev. D 73, 051106 (2006).