Study of {\boldmath $CP$}-Violating Asymmetries in $B^0 \to \pi^+\pi^-$ Decays
Aomori
% insert abstract here We present a measurement of $CP$-violating asymmetries in $B^0 \rightarrow \pi^+\pi^-$ decays based on a $41.8~{fb}^{-1}$ data sample collected at the $\Upsilon(4S)$ resonance with the Belle detector at the KEKB asymmetric-energy $e^+e^-$ collider. We fully reconstruct one neutral $B$ meson as a $B^0 \rightarrow \pi^+\pi^-$ $CP$ eigenstate and identify the flavor of the accompanying $B$ meson from its decay products. From the asymmetry in the distribution of the time intervals between the two $B$ meson decay points, we obtain the $CP$-violating asymmetry parameters $\spipi = \sresult$ and $\apipi = \cresult.$
apipi.tex
%% pi+pi- CPV paper: Version 2002062701
%%
%   This file is part of the APS files in the REVTeX 4 distribution.
%   Version 4.0 of REVTeX, August 2001
%
%   Copyright (c) 2001 The American Physical Society.
%
%   See the REVTeX 4 README file for restrictions and more information.
%
% TeX'ing this file requires that you have AMS-LaTeX 2.0 installed
% as well as the rest of the prerequisites for REVTeX 4.0
%
% See the REVTeX 4 README file
% It also requires running BibTeX. The commands are as follows:
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%  1)  latex apssamp.tex
%  2)  bibtex apssamp
%  3)  latex apssamp.tex
%  4)  latex apssamp.tex
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%%% You also need to do the following:
%%%   * Comment out widetext, use eqnarray and \nonumber 
%%%     (for the first line) for eq:likelihood
%%%   * Change the figure size to 0.6
%%%   * Put preprint numbers and the Belle logo
\documentclass[aps,prl,preprint,tightenlines,superscriptaddress,showpacs,byrevtex]{revtex4}
%
%%% Use this for PRL submission 
%%% You also need to do the following:
%%%   * Comment out widetext, use eqnarray and \nonumber 
%%%     (for the first line) for eq:likelihood
%%%   * Change the figure size to 0.6
%%%   * Comment out preprint numbers and the Belle logo
%\documentclass[aps,prl,preprint,superscriptaddress,showpacs,byrevtex]{revtex4}
%
%%% Double-column style
%%% You also need to do the following:
%%%   * Use widetext for eq:likelihood, comment out \nonumber
%%%   * Change the figure size appropriately (should be less than 0.5)
%%%   * Comment out preprint numbers and the Belle logo
%\documentclass[aps,prl,twocolumn,superscriptaddress,showpacs,preprintnumbers,amsmath,amssymb]{revtex4}
%
% Some other (several out of many) possibilities
%\documentclass[preprint,showpacs,preprintnumbers,amsmath,amssymb]{revtex4}
%\documentclass[preprint,aps]{revtex4}
%\documentclass[preprint,aps,draft]{revtex4}
%\documentclass[prb]{revtex4}% Physical Review B

\usepackage{graphicx}% Include figure files
\usepackage{dcolumn}% Align table columns on decimal point
\usepackage{bm}% bold math

%\nofiles
\newcommand{\ra}{\rightarrow}
\newcommand{\myindent}{\hspace*{2cm}}  % My indent for equation environments

%\bibliographystyle{apsrev}

\begin{document}

\vspace*{-3\baselineskip}
\resizebox{!}{3cm}{\includegraphics{belle.eps}}

\preprint{KEK Preprint 2002-6}
\preprint{Belle Preprint 2002-8}

%%%%%%%%%%%%%%%%%%%%%%
% New commands
%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\svalue}{-1.21}
\newcommand{\sstaterr}{^{+0.38}_{-0.27}}
\newcommand{\ssyserr}{^{+0.16}_{-0.13}}
\newcommand{\sresult}{\svalue~\sstaterr({\rm stat})~\ssyserr({\rm syst})}
%
\newcommand{\cvalue}{+0.94}
\newcommand{\cstaterr}{^{+0.25}_{-0.31}}
\newcommand{\csyserr}{\pm 0.09}
\newcommand{\cresult}{\cvalue~\cstaterr({\rm stat})~\csyserr({\rm syst})}
\newcommand{\avalue}{\cvalue}
\newcommand{\astaterr}{\cstaterr}
\newcommand{\asyserr}{\csyserr}
\newcommand{\aresult}{\cresult}
\newcommand{\rt}{\rightarrow}
\newcommand{\pipi}{\pi\pi}
\newcommand{\spipi}{{\cal S}_{\pipi}}
\newcommand{\apipi}{{\cal A}_{\pipi}}
\newcommand{\skpi}{{\cal S}_{K\pi}}
\newcommand{\akpi}{{\cal A}_{K\pi}}
\newcommand{\taub}{{\tau}_{B^0}}
\newcommand{\bz}{B^0}
\newcommand{\bzb}{\overline{B}{}^0}
\newcommand{\dmd}{\Delta m_d}
%
\title{Study of
     {\boldmath $CP$}-Violating Asymmetries in $B^0 \to \pi^+\pi^-$ Decays}
% repeat the \author\address pair as needed

\affiliation{Aomori University, Aomori}
\affiliation{Budker Institute of Nuclear Physics, Novosibirsk}
\affiliation{Chiba University, Chiba}
\affiliation{Chuo University, Tokyo}
\affiliation{University of Cincinnati, Cincinnati OH}
\affiliation{University of Frankfurt, Frankfurt}
\affiliation{Gyeongsang National University, Chinju}
\affiliation{University of Hawaii, Honolulu HI}
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\affiliation{Hiroshima Institute of Technology, Hiroshima}
\affiliation{Institute for Cosmic Ray Research, University of Tokyo, Tokyo}
\affiliation{Institute of High Energy Physics, Chinese Academy of Sciences, Beijing}
\affiliation{Institute of High Energy Physics, Vienna}
\affiliation{Institute for Theoretical and Experimental Physics, Moscow}
\affiliation{J. Stefan Institute, Ljubljana}
\affiliation{Kanagawa University, Yokohama}
\affiliation{Korea University, Seoul}
\affiliation{Kyoto University, Kyoto}
\affiliation{Kyungpook National University, Taegu}
\affiliation{IPHE, University of Lausanne, Lausanne}
\affiliation{University of Ljubljana, Ljubljana}
\affiliation{University of Maribor, Maribor}
\affiliation{University of Melbourne, Victoria}
\affiliation{Nagoya University, Nagoya}
\affiliation{Nara Women's University, Nara}
\affiliation{National Kaohsiung Normal University, Kaohsiung}
\affiliation{National Lien-Ho Institute of Technology, Miao Li}
\affiliation{National Taiwan University, Taipei}
\affiliation{H. Niewodniczanski Institute of Nuclear Physics, Krakow}
\affiliation{Nihon Dental College, Niigata}
\affiliation{Niigata University, Niigata}
\affiliation{Osaka City University, Osaka}
\affiliation{Osaka University, Osaka}
\affiliation{Panjab University, Chandigarh}
\affiliation{Peking University, Beijing}
\affiliation{Princeton University, Princeton NJ}
\affiliation{RIKEN BNL Research Center, Brookhaven NY}
\affiliation{Saga University, Saga}
\affiliation{Seoul National University, Seoul}
\affiliation{Sungkyunkwan University, Suwon}
\affiliation{University of Sydney, Sydney NSW}
\affiliation{Tata Institute of Fundamental Research, Bombay}
\affiliation{Toho University, Funabashi}
\affiliation{Tohoku Gakuin University, Tagajo}
\affiliation{Tohoku University, Sendai}
\affiliation{University of Tokyo, Tokyo}
\affiliation{Tokyo Institute of Technology, Tokyo}
\affiliation{Tokyo Metropolitan University, Tokyo}
\affiliation{Tokyo University of Agriculture and Technology, Tokyo}
\affiliation{Toyama National College of Maritime Technology, Toyama}
\affiliation{University of Tsukuba, Tsukuba}
\affiliation{Utkal University, Bhubaneswer}
\affiliation{Virginia Polytechnic Institute and State University, Blacksburg VA}
\affiliation{Yokkaichi University, Yokkaichi}
\affiliation{Yonsei University, Seoul}

\author{K.~Abe}               % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{K.~Abe}               % TohokuGakuin
\affiliation{Tohoku Gakuin University, Tagajo}
% \author{N.~Abe}               % TIT
% \affiliation{Tokyo Institute of Technology, Tokyo}
% \author{R.~Abe}               % Niigata
% \affiliation{Niigata University, Niigata}
\author{T.~Abe}               % Tohoku
\affiliation{Tohoku University, Sendai}
\author{I.~Adachi}            % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{Byoung~Sup~Ahn}       % Korea
\affiliation{Korea University, Seoul}
\author{H.~Aihara}            % Tokyo
\affiliation{University of Tokyo, Tokyo}
\author{M.~Akatsu}            % Nagoya
\affiliation{Nagoya University, Nagoya}
% \author{K.~Asai}              % Nara
% \affiliation{Nara Women's University, Nara}
% \author{M.~Asai}              % Hiroshima
% \affiliation{Hiroshima Institute of Technology, Hiroshima}
\author{Y.~Asano}             % Tsukuba
\affiliation{University of Tsukuba, Tsukuba}
\author{T.~Aso}               % Toyama
\affiliation{Toyama National College of Maritime Technology, Toyama}
\author{V.~Aulchenko}         % BINP
\affiliation{Budker Institute of Nuclear Physics, Novosibirsk}
\author{T.~Aushev}            % ITEP
\affiliation{Institute for Theoretical and Experimental Physics, Moscow}
\author{A.~M.~Bakich}         % Sydney
\affiliation{University of Sydney, Sydney NSW}
\author{Y.~Ban}               % Peking
\affiliation{Peking University, Beijing}
\author{E.~Banas}             % Krakow
\affiliation{H. Niewodniczanski Institute of Nuclear Physics, Krakow}
% \author{S.~Banerjee}          % Tata
% \affiliation{Tata Institute of Fundamental Research, Bombay}
\author{I.~Bedny}             % BINP
\affiliation{Budker Institute of Nuclear Physics, Novosibirsk}
\author{S.~Behari}            % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{P.~K.~Behera}         % Utkal
\affiliation{Utkal University, Bhubaneswer}
% \author{D.~Beiline}           % BINP
% \affiliation{Budker Institute of Nuclear Physics, Novosibirsk}
\author{A.~Bondar}            % BINP
\affiliation{Budker Institute of Nuclear Physics, Novosibirsk}
\author{A.~Bozek}             % Krakow
\affiliation{H. Niewodniczanski Institute of Nuclear Physics, Krakow}
\author{M.~Bra\v cko}         % Ljubljana
\affiliation{University of Maribor, Maribor}
\affiliation{J. Stefan Institute, Ljubljana}
\author{J.~Brodzicka}         % Krakow
\affiliation{H. Niewodniczanski Institute of Nuclear Physics, Krakow}
\author{T.~E.~Browder}        % Hawaii
\affiliation{University of Hawaii, Honolulu HI}
\author{B.~C.~K.~Casey}       % Hawaii
\affiliation{University of Hawaii, Honolulu HI}
\author{P.~Chang}             % Taiwan
\affiliation{National Taiwan University, Taipei}
\author{Y.~Chao}              % Taiwan
\affiliation{National Taiwan University, Taipei}
\author{B.~G.~Cheon}          % Sungkyunkwan
\affiliation{Sungkyunkwan University, Suwon}
\author{R.~Chistov}           % ITEP
\affiliation{Institute for Theoretical and Experimental Physics, Moscow}
\author{S.-K.~Choi}           % Gyeongsang
\affiliation{Gyeongsang National University, Chinju}
\author{Y.~Choi}              % Sungkyunkwan
\affiliation{Sungkyunkwan University, Suwon}
\author{M.~Danilov}           % ITEP
\affiliation{Institute for Theoretical and Experimental Physics, Moscow}
\author{L.~Y.~Dong}           % IHEP
\affiliation{Institute of High Energy Physics, Chinese Academy of Sciences, Beijing}
% \author{R.~Dowd}              % Melbourne
% \affiliation{University of Melbourne, Victoria}
\author{J.~Dragic}            % Melbourne
\affiliation{University of Melbourne, Victoria}
\author{A.~Drutskoy}          % ITEP
\affiliation{Institute for Theoretical and Experimental Physics, Moscow}
\author{S.~Eidelman}          % BINP
\affiliation{Budker Institute of Nuclear Physics, Novosibirsk}
\author{V.~Eiges}             % ITEP
\affiliation{Institute for Theoretical and Experimental Physics, Moscow}
\author{Y.~Enari}             % Nagoya
\affiliation{Nagoya University, Nagoya}
\author{C.~W.~Everton}        % Melbourne
\affiliation{University of Melbourne, Victoria}
\author{F.~Fang}              % Hawaii
\affiliation{University of Hawaii, Honolulu HI}
\author{H.~Fujii}             % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{C.~Fukunaga}          % TMU
\affiliation{Tokyo Metropolitan University, Tokyo}
\author{M.~Fukushima}         % ICRR
\affiliation{Institute for Cosmic Ray Research, University of Tokyo, Tokyo}
\author{N.~Gabyshev}          % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{A.~Garmash}           % BINP+KEK
\affiliation{Budker Institute of Nuclear Physics, Novosibirsk}
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{T.~Gershon}           % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{B.~Golob}             % Ljubljana
\affiliation{University of Ljubljana, Ljubljana}
\affiliation{J. Stefan Institute, Ljubljana}
\author{A.~Gordon}            % Melbourne
\affiliation{University of Melbourne, Victoria}
% \author{K.~Gotow}             % VPI
% \affiliation{Virginia Polytechnic Institute and State University, Blacksburg VA}
% \author{H.~Guler}             % Hawaii
% \affiliation{University of Hawaii, Honolulu HI}
\author{R.~Guo}               % Kaohsiung
\affiliation{National Kaohsiung Normal University, Kaohsiung}
\author{J.~Haba}              % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{H.~Hamasaki}          % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{K.~Hanagaki}          % Princeton
\affiliation{Princeton University, Princeton NJ}
\author{F.~Handa}             % Tohoku
\affiliation{Tohoku University, Sendai}
\author{K.~Hara}              % Osaka
\affiliation{Osaka University, Osaka}
\author{T.~Hara}              % Osaka
\affiliation{Osaka University, Osaka}
\author{N.~C.~Hastings}       % Melbourne
\affiliation{University of Melbourne, Victoria}
\author{H.~Hayashii}          % Nara
\affiliation{Nara Women's University, Nara}
\author{M.~Hazumi}            % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{E.~M.~Heenan}         % Melbourne
\affiliation{University of Melbourne, Victoria}
% \author{Y.~Higasino}          % Nagoya
% \affiliation{Nagoya University, Nagoya}
\author{I.~Higuchi}           % Tohoku
\affiliation{Tohoku University, Sendai}
\author{T.~Higuchi}           % Tokyo
\affiliation{University of Tokyo, Tokyo}
% \author{T.~Hirai}             % TIT
% \affiliation{Tokyo Institute of Technology, Tokyo}
\author{T.~Hojo}              % Osaka
\affiliation{Osaka University, Osaka}
\author{T.~Hokuue}            % Nagoya
\affiliation{Nagoya University, Nagoya}
\author{Y.~Hoshi}             % TohokuGakuin
\affiliation{Tohoku Gakuin University, Tagajo}
\author{K.~Hoshina}           % TUAT
\affiliation{Tokyo University of Agriculture and Technology, Tokyo}
\author{S.~R.~Hou}            % Taiwan
\affiliation{National Taiwan University, Taipei}
\author{W.-S.~Hou}            % Taiwan
\affiliation{National Taiwan University, Taipei}
\author{S.-C.~Hsu}            % Taiwan
\affiliation{National Taiwan University, Taipei}
\author{H.-C.~Huang}          % Taiwan
\affiliation{National Taiwan University, Taipei}
\author{T.~Igaki}             % Nagoya
\affiliation{Nagoya University, Nagoya}
\author{Y.~Igarashi}          % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{T.~Iijima}            % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{H.~Ikeda}             % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
% \author{K.~Ikeda}             % Nara
% \affiliation{Nara Women's University, Nara}
\author{K.~Inami}             % Nagoya
\affiliation{Nagoya University, Nagoya}
\author{A.~Ishikawa}          % Nagoya
\affiliation{Nagoya University, Nagoya}
\author{H.~Ishino}            % TIT
\affiliation{Tokyo Institute of Technology, Tokyo}
\author{R.~Itoh}              % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{M.~Iwamoto}           % Chiba
\affiliation{Chiba University, Chiba}
\author{H.~Iwasaki}           % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{Y.~Iwasaki}           % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{D.~J.~Jackson}        % Osaka
\affiliation{Osaka University, Osaka}
\author{P.~Jalocha}           % Krakow
\affiliation{H. Niewodniczanski Institute of Nuclear Physics, Krakow}
\author{H.~K.~Jang}           % Seoul
\affiliation{Seoul National University, Seoul}
% \author{M.~Jones}             % Hawaii
% \affiliation{University of Hawaii, Honolulu HI}
% \author{R.~Kagan}             % ITEP
% \affiliation{Institute for Theoretical and Experimental Physics, Moscow}
% \author{H.~Kakuno}            % TIT
% \affiliation{Tokyo Institute of Technology, Tokyo}
% \author{J.~Kaneko}            % TIT
% \affiliation{Tokyo Institute of Technology, Tokyo}
\author{J.~H.~Kang}           % Yonsei
\affiliation{Yonsei University, Seoul}
\author{J.~S.~Kang}           % Korea
\affiliation{Korea University, Seoul}
\author{P.~Kapusta}           % Krakow
\affiliation{H. Niewodniczanski Institute of Nuclear Physics, Krakow}
\author{N.~Katayama}          % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{H.~Kawai}             % Chiba
\affiliation{Chiba University, Chiba}
\author{H.~Kawai}             % Tokyo
\affiliation{University of Tokyo, Tokyo}
% \author{Y.~Kawakami}          % Nagoya
% \affiliation{Nagoya University, Nagoya}
\author{N.~Kawamura}          % Aomori
\affiliation{Aomori University, Aomori}
\author{T.~Kawasaki}          % Niigata
\affiliation{Niigata University, Niigata}
\author{H.~Kichimi}           % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{D.~W.~Kim}            % Sungkyunkwan
\affiliation{Sungkyunkwan University, Suwon}
\author{Heejong~Kim}          % Yonsei
\affiliation{Yonsei University, Seoul}
\author{H.~J.~Kim}            % Yonsei
\affiliation{Yonsei University, Seoul}
\author{H.~O.~Kim}            % Sungkyunkwan
\affiliation{Sungkyunkwan University, Suwon}
\author{Hyunwoo~Kim}          % Korea
\affiliation{Korea University, Seoul}
\author{S.~K.~Kim}            % Seoul
\affiliation{Seoul National University, Seoul}
\author{T.~H.~Kim}            % Yonsei
\affiliation{Yonsei University, Seoul}
\author{K.~Kinoshita}         % Cincinnati
\affiliation{University of Cincinnati, Cincinnati OH}
% \author{S.~Kobayashi}         % Saga
% \affiliation{Saga University, Saga}
\author{S.~Koishi}            % TIT
\affiliation{Tokyo Institute of Technology, Tokyo}
\author{H.~Konishi}           % TUAT
\affiliation{Tokyo University of Agriculture and Technology, Tokyo}
% \author{K.~Korotushenko}      % Princeton
% \affiliation{Princeton University, Princeton NJ}
\author{S.~Korpar}            % Ljubljana
\affiliation{University of Maribor, Maribor}
\affiliation{J. Stefan Institute, Ljubljana}
\author{P.~Kri\v zan}         % Ljubljana
\affiliation{University of Ljubljana, Ljubljana}
\affiliation{J. Stefan Institute, Ljubljana}
\author{P.~Krokovny}          % BINP
\affiliation{Budker Institute of Nuclear Physics, Novosibirsk}
\author{R.~Kulasiri}          % Cincinnati
\affiliation{University of Cincinnati, Cincinnati OH}
\author{S.~Kumar}             % Panjab
\affiliation{Panjab University, Chandigarh}
% \author{T.~Kuniya}            % Saga
% \affiliation{Saga University, Saga}
% \author{E.~Kurihara}          % Chiba
% \affiliation{Chiba University, Chiba}
\author{A.~Kuzmin}            % BINP
\affiliation{Budker Institute of Nuclear Physics, Novosibirsk}
\author{Y.-J.~Kwon}           % Yonsei
\affiliation{Yonsei University, Seoul}
\author{J.~S.~Lange}          % Frankfurt
\affiliation{University of Frankfurt, Frankfurt}
\affiliation{RIKEN BNL Research Center, Brookhaven NY}
\author{G.~Leder}             % Vienna
\affiliation{Institute of High Energy Physics, Vienna}
\author{S.~H.~Lee}            % Seoul
\affiliation{Seoul National University, Seoul}
\author{A.~Limosani}          % Melbourne
\affiliation{University of Melbourne, Victoria}
\author{D.~Liventsev}         % ITEP
\affiliation{Institute for Theoretical and Experimental Physics, Moscow}
\author{R.-S.~Lu}             % Taiwan
\affiliation{National Taiwan University, Taipei}
\author{J.~MacNaughton}       % Vienna
\affiliation{Institute of High Energy Physics, Vienna}
\author{G.~Majumder}          % Tata
\affiliation{Tata Institute of Fundamental Research, Bombay}
\author{F.~Mandl}             % Vienna
\affiliation{Institute of High Energy Physics, Vienna}
\author{D.~Marlow}            % Princeton
\affiliation{Princeton University, Princeton NJ}
% \author{T.~Matsubara}         % Tokyo
% \affiliation{University of Tokyo, Tokyo}
% \author{S.~Matsui}            % Nagoya
% \affiliation{Nagoya University, Nagoya}
\author{T.~Matsuishi}         % Nagoya
\affiliation{Nagoya University, Nagoya}
\author{S.~Matsumoto}         % Chuo
\affiliation{Chuo University, Tokyo}
\author{T.~Matsumoto}         % Nagoya
\affiliation{Nagoya University, Nagoya}
\author{Y.~Mikami}            % Tohoku
\affiliation{Tohoku University, Sendai}
\author{W.~Mitaroff}          % Vienna
\affiliation{Institute of High Energy Physics, Vienna}
\author{K.~Miyabayashi}       % Nara
\affiliation{Nara Women's University, Nara}
\author{Y.~Miyabayashi}       % Nagoya
\affiliation{Nagoya University, Nagoya}
\author{H.~Miyake}            % Osaka
\affiliation{Osaka University, Osaka}
\author{H.~Miyata}            % Niigata
\affiliation{Niigata University, Niigata}
% \author{L.~C.~Moffitt}        % Melbourne
% \affiliation{University of Melbourne, Victoria}
\author{G.~R.~Moloney}        % Melbourne
\affiliation{University of Melbourne, Victoria}
% \author{G.~F.~Moorhead}       % Melbourne
% \affiliation{University of Melbourne, Victoria}
\author{S.~Mori}              % Tsukuba
\affiliation{University of Tsukuba, Tsukuba}
\author{T.~Mori}              % Chuo
\affiliation{Chuo University, Tokyo}
\author{A.~Murakami}          % Saga
\affiliation{Saga University, Saga}
\author{T.~Nagamine}          % Tohoku
\affiliation{Tohoku University, Sendai}
\author{Y.~Nagasaka}          % Hiroshima
\affiliation{Hiroshima Institute of Technology, Hiroshima}
% \author{Y.~Nagashima}         % Osaka
% \affiliation{Osaka University, Osaka}
\author{T.~Nakadaira}         % Tokyo
\affiliation{University of Tokyo, Tokyo}
% \author{T.~Nakamura}          % TIT
% \affiliation{Tokyo Institute of Technology, Tokyo}
\author{E.~Nakano}            % OsakaCity
\affiliation{Osaka City University, Osaka}
\author{M.~Nakao}             % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
% \author{H.~Nakazawa}          % Chuo
% \affiliation{Chuo University, Tokyo}
\author{J.~W.~Nam}            % Sungkyunkwan
\affiliation{Sungkyunkwan University, Suwon}
% \author{S.~Narita}            % Tohoku
% \affiliation{Tohoku University, Sendai}
% \author{Z.~Natkaniec}         % Krakow
% \affiliation{H. Niewodniczanski Institute of Nuclear Physics, Krakow}
\author{K.~Neichi}            % TohokuGakuin
\affiliation{Tohoku Gakuin University, Tagajo}
\author{S.~Nishida}           % Kyoto
\affiliation{Kyoto University, Kyoto}
\author{O.~Nitoh}             % TUAT
\affiliation{Tokyo University of Agriculture and Technology, Tokyo}
\author{S.~Noguchi}           % Nara
\affiliation{Nara Women's University, Nara}
\author{T.~Nozaki}            % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{S.~Ogawa}             % Toho
\affiliation{Toho University, Funabashi}
\author{F.~Ohno}              % TIT
\affiliation{Tokyo Institute of Technology, Tokyo}
\author{T.~Ohshima}           % Nagoya
\affiliation{Nagoya University, Nagoya}
% \author{Y.~Ohshima}           % TIT
% \affiliation{Tokyo Institute of Technology, Tokyo}
\author{T.~Okabe}             % Nagoya
\affiliation{Nagoya University, Nagoya}
% \author{T.~Okazaki}           % Nara
% \affiliation{Nara Women's University, Nara}
\author{S.~Okuno}             % Kanagawa
\affiliation{Kanagawa University, Yokohama}
\author{S.~L.~Olsen}          % Hawaii
\affiliation{University of Hawaii, Honolulu HI}
\author{W.~Ostrowicz}         % Krakow
\affiliation{H. Niewodniczanski Institute of Nuclear Physics, Krakow}
\author{H.~Ozaki}             % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{P.~Pakhlov}           % ITEP
\affiliation{Institute for Theoretical and Experimental Physics, Moscow}
\author{H.~Palka}             % Krakow
\affiliation{H. Niewodniczanski Institute of Nuclear Physics, Krakow}
\author{C.~W.~Park}           % Korea
\affiliation{Korea University, Seoul}
\author{H.~Park}              % Kyungpook
\affiliation{Kyungpook National University, Taegu}
\author{K.~S.~Park}           % Sungkyunkwan
\affiliation{Sungkyunkwan University, Suwon}
\author{L.~S.~Peak}           % Sydney
\affiliation{University of Sydney, Sydney NSW}
\author{J.-P.~Perroud}        % Lausanne
\affiliation{IPHE, University of Lausanne, Lausanne}
\author{M.~Peters}            % Hawaii
\affiliation{University of Hawaii, Honolulu HI}
\author{L.~E.~Piilonen}       % VPI
\affiliation{Virginia Polytechnic Institute and State University, Blacksburg VA}
\author{E.~Prebys}            % Princeton
\affiliation{Princeton University, Princeton NJ}
\author{J.~L.~Rodriguez}      % Hawaii
\affiliation{University of Hawaii, Honolulu HI}
\author{F.~J.~Ronga}          % Lausanne
\affiliation{IPHE, University of Lausanne, Lausanne}
% \author{N.~Root}              % BINP
% \affiliation{Budker Institute of Nuclear Physics, Novosibirsk}
\author{M.~Rozanska}          % Krakow
\affiliation{H. Niewodniczanski Institute of Nuclear Physics, Krakow}
\author{K.~Rybicki}           % Krakow
\affiliation{H. Niewodniczanski Institute of Nuclear Physics, Krakow}
% \author{J.~Ryuko}             % Osaka
% \affiliation{Osaka University, Osaka}
\author{H.~Sagawa}            % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{S.~Saitoh}            % Chiba
\affiliation{Chiba University, Chiba}
\author{Y.~Sakai}             % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{H.~Sakamoto}          % Kyoto
\affiliation{Kyoto University, Kyoto}
% \author{H.~Sakaue}            % OsakaCity
% \affiliation{Osaka City University, Osaka}
\author{M.~Satapathy}         % Utkal
\affiliation{Utkal University, Bhubaneswer}
\author{A.~Satpathy}          % KEK+Cincinnati
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\affiliation{University of Cincinnati, Cincinnati OH}
\author{O.~Schneider}         % Lausanne
\affiliation{IPHE, University of Lausanne, Lausanne}
\author{S.~Schrenk}           % Cincinnati
\affiliation{University of Cincinnati, Cincinnati OH}
\author{C.~Schwanda}          % KEK+Vienna
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\affiliation{Institute of High Energy Physics, Vienna}
\author{S.~Semenov}           % ITEP
\affiliation{Institute for Theoretical and Experimental Physics, Moscow}
\author{K.~Senyo}             % Nagoya
\affiliation{Nagoya University, Nagoya}
% \author{Y.~Settai}            % Chuo
% \affiliation{Chuo University, Tokyo}
\author{M.~E.~Sevior}         % Melbourne
\affiliation{University of Melbourne, Victoria}
\author{H.~Shibuya}           % Toho
\affiliation{Toho University, Funabashi}
\author{B.~Shwartz}           % BINP
\affiliation{Budker Institute of Nuclear Physics, Novosibirsk}
% \author{A.~Sidorov}           % BINP
% \affiliation{Budker Institute of Nuclear Physics, Novosibirsk}
\author{V.~Sidorov}           % BINP
\affiliation{Budker Institute of Nuclear Physics, Novosibirsk}
\author{J.~B.~Singh}          % Panjab
\affiliation{Panjab University, Chandigarh}
\author{S.~Stani\v c}         % Tsukuba
\altaffiliation{on leave from Nova Gorica Polytechnic, Slovenia}
\affiliation{University of Tsukuba, Tsukuba}
\author{A.~Sugi}              % Nagoya
\affiliation{Nagoya University, Nagoya}
\author{A.~Sugiyama}          % Nagoya
\affiliation{Nagoya University, Nagoya}
\author{K.~Sumisawa}          % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{T.~Sumiyoshi}         % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{K.~Suzuki}            % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{S.~Suzuki}            % Yokkaichi
\affiliation{Yokkaichi University, Yokkaichi}
\author{S.~Y.~Suzuki}         % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{S.~K.~Swain}          % Hawaii
\affiliation{University of Hawaii, Honolulu HI}
\author{H.~Tajima}            % Tokyo
\affiliation{University of Tokyo, Tokyo}
\author{T.~Takahashi}         % OsakaCity
\affiliation{Osaka City University, Osaka}
\author{F.~Takasaki}          % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{M.~Takita}            % Osaka
\affiliation{Osaka University, Osaka}
\author{K.~Tamai}             % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{N.~Tamura}            % Niigata
\affiliation{Niigata University, Niigata}
\author{J.~Tanaka}            % Tokyo
\affiliation{University of Tokyo, Tokyo}
\author{M.~Tanaka}            % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
% \author{Y.~Tanaka}            % Nagasaki
% \affiliation{Nagoya University, Nagoya}
\author{G.~N.~Taylor}         % Melbourne
\affiliation{University of Melbourne, Victoria}
\author{Y.~Teramoto}          % OsakaCity
\affiliation{Osaka City University, Osaka}
\author{S.~Tokuda}            % Nagoya
\affiliation{Nagoya University, Nagoya}
\author{M.~Tomoto}            % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{T.~Tomura}            % Tokyo
\affiliation{University of Tokyo, Tokyo}
\author{S.~N.~Tovey}          % Melbourne
\affiliation{University of Melbourne, Victoria}
\author{K.~Trabelsi}          % Hawaii
\affiliation{University of Hawaii, Honolulu HI}
\author{W.~Trischuk}          % Princeton
\altaffiliation{on leave from University of Toronto, Toronto ON}
\affiliation{Princeton University, Princeton NJ}
\author{T.~Tsuboyama}         % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{T.~Tsukamoto}         % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{S.~Uehara}            % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{K.~Ueno}              % Taiwan
\affiliation{National Taiwan University, Taipei}
\author{Y.~Unno}              % Chiba
\affiliation{Chiba University, Chiba}
\author{S.~Uno}               % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{Y.~Ushiroda}          % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
% \author{S.~E.~Vahsen}         % Princeton
% \affiliation{Princeton University, Princeton NJ}
\author{K.~E.~Varvell}        % Sydney
\affiliation{University of Sydney, Sydney NSW}
\author{C.~C.~Wang}           % Taiwan
\affiliation{National Taiwan University, Taipei}
\author{C.~H.~Wang}           % Lien-Ho
\affiliation{National Lien-Ho Institute of Technology, Miao Li}
\author{J.~G.~Wang}           % VPI
\affiliation{Virginia Polytechnic Institute and State University, Blacksburg VA}
\author{M.-Z.~Wang}           % Taiwan
\affiliation{National Taiwan University, Taipei}
\author{Y.~Watanabe}          % TIT
\affiliation{Tokyo Institute of Technology, Tokyo}
\author{E.~Won}               % Seoul
\affiliation{Seoul National University, Seoul}
\author{B.~D.~Yabsley}        % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{Y.~Yamada}            % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{M.~Yamaga}            % Tohoku
\affiliation{Tohoku University, Sendai}
\author{A.~Yamaguchi}         % Tohoku
\affiliation{Tohoku University, Sendai}
\author{H.~Yamamoto}          % Tohoku
\affiliation{Tohoku University, Sendai}
% \author{T.~Yamanaka}          % Osaka
% \affiliation{Osaka University, Osaka}
\author{Y.~Yamashita}         % NihonDental
\affiliation{Nihon Dental College, Niigata}
\author{M.~Yamauchi}          % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
% \author{S.~Yanaka}            % TIT
% \affiliation{Tokyo Institute of Technology, Tokyo}
\author{J.~Yashima}           % KEK
\affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{P.~Yeh}               % Taiwan
\affiliation{National Taiwan University, Taipei}
\author{M.~Yokoyama}          % Tokyo
\affiliation{University of Tokyo, Tokyo}
\author{K.~Yoshida}           % Nagoya
\affiliation{Nagoya University, Nagoya}
\author{Y.~Yuan}              % IHEP
\affiliation{Institute of High Energy Physics, Chinese Academy of Sciences, Beijing}
\author{Y.~Yusa}              % Tohoku
\affiliation{Tohoku University, Sendai}
% \author{H.~Yuta}              % Aomori
% \affiliation{Aomori University, Aomori}
\author{C.~C.~Zhang}          % IHEP
\affiliation{Institute of High Energy Physics, Chinese Academy of Sciences, Beijing}
\author{J.~Zhang}             % Tsukuba
\affiliation{University of Tsukuba, Tsukuba}
% \author{H.~W.~Zhao}           % KEK
% \affiliation{High Energy Accelerator Research Organization (KEK), Tsukuba}
\author{Y.~Zheng}             % Hawaii
\affiliation{University of Hawaii, Honolulu HI}
\author{V.~Zhilich}           % BINP
\affiliation{Budker Institute of Nuclear Physics, Novosibirsk}
\author{D.~\v Zontar}         % Tsukuba
\affiliation{University of Tsukuba, Tsukuba}

\collaboration{The Belle Collaboration}
\noaffiliation



\date{\today}% It is always \today, today,
             %  but any date may be explicitly specified

\begin{abstract}
% insert abstract here
We present a measurement of
$CP$-violating asymmetries in $B^0 \rightarrow \pi^+\pi^-$ decays
based on
a $41.8~{\rm fb}^{-1}$ data sample collected at the $\Upsilon(4S)$ resonance
with the Belle detector at the KEKB asymmetric-energy $e^+e^-$ collider.
We fully reconstruct one neutral $B$ meson
as a  $B^0 \rightarrow \pi^+\pi^-$
$CP$ eigenstate and identify
the flavor of the accompanying $B$ meson
from its
decay products.
From the asymmetry in the
distribution of the time intervals between the two $B$ meson decay points,
we obtain the $CP$-violating asymmetry parameters
$\spipi = \sresult$
and
$\apipi = \cresult.$
\end{abstract}

% insert suggested PACS numbers in braces on next line
\pacs{PACS numbers: 11.30.Er, 12.15.Hh, 13.25.Hw}

\maketitle
% Introduction
Kobayashi and Maskawa (KM) proposed, in 1973, a model
where $CP$ violation is
incorporated as an irreducible complex phase in the
weak-interaction quark mixing matrix~\cite{KM}.
%%MH This model is now one of the key components of the Standard Model (SM)
%%MH and is supported by many experimental results.
Recent measurements of the $CP$-violating parameter $\sin 2\phi_1$ 
by the Belle~\cite{CP1_Belle} and BaBar~\cite{CP1_BaBar} collaborations 
established $CP$ violation in the neutral $B$ meson system that
is consistent with KM expectations.
Measurements of other $CP$-violating parameters provide important
tests of the KM model. 
In this Letter 
we describe a measurement of $CP$-violating
asymmetries
in the mode $\bz \to \pi^+\pi^-$~\cite{CC}; 
these are sensitive to the parameter 
$\sin 2\phi_2$~\cite{alpha}.

The KM model predicts
$CP$-violating asymmetries in the time-dependent
rates for  $B^0$ and $\bzb$
decays to a common $CP$ eigenstate, $f_{CP}$~\cite{Sanda}.
In the decay chain $\Upsilon(4S)\to \bz\bzb \to f_{CP}f_{\rm tag}$,
where one of $B$ mesons decays at time $t_{CP}$ to $f_{CP}$ 
and the other decays at time $t_{\rm tag}$ to a final state
$f_{\rm tag}$ that distinguishes between $B^0$ and $\bzb$, 
the decay rate has a time dependence
given by~\cite{CPVrev}
\begin{eqnarray}
\label{eq:R_q}
{\cal P}_{\pi\pi}^q(\Delta{t}) = 
\frac{e^{-|\Delta{t}|/{\taub}}}{4{\taub}}
\left[1 + q\cdot 
\left\{ \spipi\sin(\dmd\Delta{t})   \right. \right. \nonumber \\
\left. \left.
   + \apipi\cos(\dmd\Delta{t})
\right\}
\right],
\end{eqnarray}
where $\taub$ is the $B^0$ lifetime, $\dmd$ is the mass difference 
between the two $B^0$ mass
eigenstates, $\Delta{t}$ = $t_{CP}$ $-$ $t_{\rm tag}$, and
the $b$-flavor charge $q$ = +1 ($-1$) when the tagging $B$ meson
is a $B^0$ 
($\bzb$).
The $CP$-violating parameters $\spipi$ and $\apipi$ 
defined in Eq.~(\ref{eq:R_q}) are expressed by
%%MH 
$\spipi = 2Im \lambda/(|\lambda|^2 + 1)$ and
$\apipi = (|\lambda|^2 - 1)/(|\lambda|^2 + 1)$,
%\begin{eqnarray}
%\spipi = \frac{ 2Im \lambda } {|\lambda|^2 + 1}~{\rm and }~
%\apipi = \frac{ |\lambda|^2 - 1}{|\lambda|^2 + 1},
%\end{eqnarray}
%%MH
where $\lambda$ is a complex 
parameter that depends on both $\bz\bzb$
mixing and on the amplitudes for $\bz$ and $\bzb$ decay to 
$\pi^+\pi^-$. In the SM, to a good approximation,
$|\lambda|$ is equal to the absolute value
of the ratio of the $\bzb$ to $\bz$ decay amplitudes.
%%MH Therefore $|\lambda|\neq 1$, or equivalently $\apipi \neq 0$,
%%MH indicates direct $CP$ violation.
%
%%MH
%In the case of $B^0 \rightarrow (c\overline{c}) K_S^0~CP$ 
%eigenstate decays,
%the $CP$-violating parameters are rather precisely expressed as
%${\cal S}_{(c\overline{c}) K_S^0} = \sin 2\phi_1$ and
%${\cal A}_{(c\overline{c}) K_S^0} = 0$.
%This is due to the fact that
%the tree amplitude with $W$ emission dominates 
%the $b \rightarrow s$ penguin amplitude with
%associated $c\overline{c}$ production, which is small and has
%the same weak phase. 
%For the $B^0 \rightarrow \pi^+\pi^-$ decay, we would have
%%MH
We would have
$\spipi = \sin 2\phi_2$ and
$\apipi =0$, or equivalently $|\lambda| = 1$,
%%MH $\apipi =0$
if the $b \to u$ tree amplitude were dominant.
The situation is complicated
by the possibility 
of significant contributions from gluonic $b\to d$
penguin amplitudes that
have a different weak phase and additional strong 
phases~\cite{pipipenguin}.
As a result, 
$\spipi$ may not be equal to $\sin2\phi_2$ and
direct $CP$ violation, $\apipi \neq 0$,  may occur.

This measurement 
%%MH of $CP$-violating parameters in $B^0 \to \pi^+\pi^-$ 
is based on a $41.8~{\rm fb}^{-1}$ data sample,
which contains 44.8 million $B\overline{B}$ pairs, 
collected  with
the Belle detector at the KEKB asymmetric-energy
$e^+e^-$ (3.5 on 8~GeV) collider~\cite{KEKB}
operating at the $\Upsilon(4S)$ resonance.
At KEKB, the $\Upsilon(4S)$ is produced
with a Lorentz boost of $\beta\gamma=0.425$ nearly along
the electron beamline ($z$).
Since the $B^0$ and $\bzb$ mesons are approximately at 
rest in the $\Upsilon(4S)$ center-of-mass system (cms),
$\Delta t$ can be determined from the displacement in $z$ 
between the $f_{CP}$ and $f_{\rm tag}$ decay vertices:
$\Delta t \simeq (z_{CP} - z_{\rm tag})/\beta\gamma c
 \equiv \Delta z/\beta\gamma c$.

%%MH
The Belle detector~\cite{Belle} is a large-solid-angle
%The Belle detector is a large-solid-angle magnetic
%%MH
spectrometer that
consists of a silicon vertex detector (SVD),
a central drift chamber (CDC), an array of
aerogel threshold \v{C}erenkov counters (ACC), 
%%MH
time-of-flight
%a barrel-like arrangement of time-of-flight
%%MH
scintillation counters (TOF), and an electromagnetic calorimeter
comprised of CsI(Tl) crystals (ECL)  located inside 
a super-conducting solenoid coil that provides a 1.5~T
magnetic field.  An iron flux-return located outside of
the coil is instrumented to detect $K_L^0$ mesons and to identify
muons (KLM).  
%%MH The detector is described in detail elsewhere~\cite{Belle}.

The $B^0 \to \pi^+\pi^-$ event selection is described
in detail elsewhere~\cite{pipi}.  
We use oppositely charged track pairs
%%MH We use oppositely charged pairs of well measured tracks 
that are positively identified as pions according to the
combined information from the ACC and the CDC $dE/dx$ measurement.
Candidate $B$ mesons are reconstructed using
the energy difference 
$\Delta E\equiv E_B^{\rm cms} - E_{\rm beam}^{\rm cms}$
and the beam-energy constrained
mass $M_{\rm bc}\equiv\sqrt{(E_{\rm beam}^{\rm cms})^2-(p_B^{\rm cms})^2}$,
where $E_{\rm beam}^{\rm cms}$ is the cms beam energy,
and $E_B^{\rm cms}$ and $p_B^{\rm cms}$ are the cms energy and momentum
of the $B$ candidate.
The signal region is defined as 
%%MH 5.271 GeV/$c^2$ $< M_{\rm bc} <$ 5.287 GeV/$c^2$
5.271 $< M_{\rm bc} <$ 5.287 GeV/$c^2$
and $|\Delta E |< $ 0.067 GeV, corresponding to $\pm 3\sigma$ from
the central values.
In order to suppress background from the $e^+e^- \rightarrow q\overline{q}$
continuum ($q = u,~d,~s,~c$),  we form signal and background
likelihood functions, ${\cal L}_S$ and ${\cal L}_{BG}$, 
from two variables. One is a Fisher
discriminant determined from six modified Fox-Wolfram
moments~\cite{SFW};
the other is 
the $B$ flight direction in the cms, 
with respect to the $z$ axis ($\cos\theta_B$).
We determine ${\cal L}_S$ from Monte Carlo (MC)
and ${\cal L}_{BG}$ from data, and
require ${\cal L}_S/({\cal L}_S+{\cal L}_{BG}) > 0.825$
for candidate events.
%%MH The likelihood ratio ${\cal L}_S/({\cal L}_S+{\cal L}_{BG})$
%%MH for candidate events is required to be greater than 0.825.
Figure~\ref{fig:DeltaE} shows the $\Delta E$ distribution for
$\pi^+\pi^-$ candidates.  
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[h]
\begin{center}
\resizebox{0.6\textwidth}{!}{\includegraphics{pipi_exp7-15_defit1.eps}}
%\resizebox{0.35\textwidth}{!}{\includegraphics{pipi_exp7-15_defit1.eps}}
\end{center}
\caption{ $\Delta E$ distribution for $\pi^+\pi^-$ 
event candidates that are in the
 $M_{\rm bc}$ signal region.}
\label{fig:DeltaE}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%
The signal yield is extracted by fitting
the $\Delta E$ distribution with a Gaussian $\pi^+\pi^-$ signal function, 
plus contributions from misidentified $B^0\rt K^+\pi^-$ events, 
three-body $B$-decays, and continuum background.
From the fit, we obtain
73.5 $\pm$ 13.8(stat) $\pi^+\pi^-$ events, 
28.4 $\pm$ 12.5(stat) $K^+\pi^-$ events, and
98.7 $\pm$ 7.0(stat) continuum events in the 
%%MH 98.7 $\pm$ 7.0(stat) continuum background events in the 
signal region. The $K^+\pi^-$ contamination is consistent
with the $K\rt\pi$ misidentification probability measured
independently. The contribution from
three-body $B$-decays is negligibly small in the signal region.

Leptons, charged pions, and kaons
that are not associated with the reconstructed
$B^0 \rightarrow \pi^+\pi^-$ decay are used to identify
the flavor of the accompanying $B$ meson.
%%MH We apply the same method used for the $\sin 2\phi_1$ 
%%MH measurement~\cite{CP1_Belle}.
We use two parameters, $q$ and $r$, to represent the tagging information.
%%MH
The first, $q$, is already defined in Eq.~(\ref{eq:R_q}).
%%MH
%The first, $q$, corresponds to the sign of the $b$ quark charge 
%where $q = +1$ for $\overline b$ and hence
%$B^0$, and $q = -1$ for $b$ and $\bzb$.
The parameter $r$ is an event-by-event,
MC-determined flavor-tagging dilution factor 
that ranges from $r=0$ for no flavor
discrimination to $r=1$ for {unambiguous flavor assignment.
It is used only to sort data into six $r$ intervals
%%MH 
(boundaries at 0.25, 0.5, 0.625, 0.75 and 0.875).
%%MH defined as $0 < r \le 0.25$, $0.25 < r \le 0.5$, $0.5 < r \le 0.625$, 
%%MH $0.625 < r \le 0.75$, $0.75 < r \le 0.875$ and $0.875 < r \le 1$.
The wrong tag fractions for the six $r$ intervals, $w_l\ (l=1,6)$,
%%MH
%are determined directly 
are determined 
%%MH
%%MH
%from the data by fitting the time evolution of
%exclusively reconstructed, self-tagged
%$B^0\to D^{*-}\ell^+\nu$, $D^{(*)-}\pi^+$, and
%$D^{*-}\rho^+$ decays.
from the data;
%%MH
we use the same $w_l$ values
that were used for the $\sin 2\phi_1$ measurement~\cite{CP1_Belle}.

The vertex positions for the $\pi^+\pi^-$ and $f_{\rm tag}$ decays are
reconstructed using tracks 
with associated hits in the SVD.
Each vertex position is also constrained by
the interaction point profile smeared in the
$r$-$\phi$ plane by the average transverse $B$ meson decay length.
The $f_{\rm tag}$ vertex
is determined from all well reconstructed tracks, 
excluding the $B^0 \to \pi^+\pi^-$ candidate and
tracks that form a $K_S^0$ candidate.
%%MH excluding the $B^0 \to \pi^+\pi^-$ candidate.
%%MH Tracks that form a $K_S^0$ candidate are not used.
%%MH The MC simulation
%%MH indicates that the typical vertex-finding efficiency 
%%MH is 92\% and 91\% for $\pi^+\pi^-$ and $f_{\rm tag}$, respectively, while
%%MH the typical vertex rms resolution in the $z$ coordinate is
%%MH $75~\mu{\rm m}$ for $B^0 \to \pi^+\pi^-$ decays and
%%MH $140~\mu{\rm m}$ for $f_{\rm tag}$ decays.

%MH
The proper-time interval resolution for the signal, $R_{\rm sig}(\Delta t)$, 
is obtained by convolving a sum of
two Gaussians (a {\it main} component,
%%MH due to  the SVD vertex resolution and charmed meson lifetimes,
plus a {\it tail} component caused by poorly reconstructed tracks)
with a function that takes into account  
the cms motion of the $B$ mesons.
We use the same parameters
as those used for the $\sin 2\phi_1$ measurement;
the fraction and the typical width of the main Gaussian
are $0.97\pm 0.02$ and 1.49 ps, respectively~\cite{CP1_Belle}.
%%MH 
% The fraction in the main Gaussian
%The main Gaussian fraction
%is determined to be $0.97\pm 0.02$ from a
%study of $B^0\to D^{*-}\pi^+$, $D^{*-}\rho^+$,  $D^-\pi^+$,
%$J/\psi K^{*0}$, $J/\psi K_S^0$ and 
%$B^+\rightarrow \overline{D}{}^0\pi^+$,  $J/\psi K^+$ events.
%The means
%($\mu_{\rm main}$, $\mu_{\rm tail}$)
%and widths
%($\sigma_{\rm main}$, $\sigma_{\rm tail}$)
%of the Gaussians are
%calculated event-by-event from the
%$f_{CP}$ and $f_{\rm tag}$ vertex fit error matrices
%and the $\chi^2$ values of the fit;
%typical values are $\mu_{\rm main}=-0.24~{\rm ps}$,
%$\mu_{\rm tail}=0.18~{\rm ps}$
%and
%$\sigma_{\rm main}=1.49~{\rm ps}$, $\sigma_{\rm tail}=3.85~{\rm ps}$.
%%MH
The background resolution function $R_{q\overline{q}}(\Delta t)$,
which is dominated by continuum background,
has the same functional
form but the parameters are obtained from a sideband region
in $M_{bc}$ and $\Delta E$.
% 
Using these resolution functions,
%%MH Using the same resolution parameterization for the above samples,
we perform a $\bz$ lifetime measurement that yields
$\taub = 1.49\pm 0.21$(stat) ps for $B^0 \to \pi^+\pi^-$ candidates,
which is consistent with the world average value~\cite{PDG}.

%CP fitting
We determine $CP$ violation parameters
by performing an
unbinned maximum-likelihood fit of a $CP$-violating
probability density function (pdf) to the 
$\Delta t$ distributions.
%%MH observed $\Delta t$ distributions.
We define the likelihood value for each event as a
function of $\spipi$ and $\apipi$:
%\begin{widetext}
%\begin{equation}
\begin{eqnarray}
P_i =
\int 
[\{f_{\pi\pi}^l{\cal P}_{\pi\pi}^q(\Delta t^\prime, w_l;\spipi, \apipi) +
f_{K\pi}^l{\cal P}_{K\pi}^q(\Delta t^\prime, w_l)\}
\cdot R_{\rm sig}(\Delta t_i-\Delta t^\prime) 
\nonumber \\
+ f_{q\overline{q}}^l{\cal P}_{q\overline{q}}(\Delta t^\prime)
\cdot R_{q\overline{q}}(\Delta t_i-\Delta t^\prime)]d\Delta t^\prime.
\label{eq:likelihood}
\end{eqnarray}
%\end{equation}
%\end{widetext}
%
Here $f_{\pi\pi}^l$, $f_{K\pi}^l$, and 
$f_{q\overline{q}}^l$ ($= 1 - f_{\pi\pi}^l - f_{K\pi}^l$) are the fractions 
of $\pi^+\pi^-$ signal,  
$K^+\pi^-$ background, and continuum background in flavor-tagging interval
$l$, respectively.  
These fractions are determined on an event-by-event basis
as a function of $\Delta E$ and $M_{\rm bc}$, properly normalized
by the average signal and background fractions in the
signal region.
The average fractions of $q\overline{q}$ background
for six $r$ bins $(l=1,6)$ are 0.632, 0.505, 0.462, 0.440, 0.322 and 0.117.
For higher $r$ values where we are more sensitive to the
asymmetry, the fraction of continuum background decreases;
the ratio of $\pi^+\pi^-$ signal events to background
$K^+\pi^-$ events is the same for all $r$ bins.
The pdfs for $\pi^+\pi^-$ (${\cal P}_{\pi\pi}^q$), 
$K^+\pi^-$ (${\cal P}_{K\pi}^q$), and continuum background 
(${\cal P}_{q\overline{q}}$),
are convolved with their respective resolution functions. 
We use the same vertex resolution 
function for $\pi^+\pi^-$ and $K^+\pi^-$ candidates.
For the $\pi^+\pi^-$ signal,
the pdf is given by Eq.~(\ref{eq:R_q}) with $q$ replaced by
$q(1-2w_l)$, to account for the dilution due to
wrong flavor tagging.
The pdf for the $K^+\pi^-$ background is
${\cal P}_{K\pi}^q(\Delta t,w_l)
={e^{-|\Delta t|/\taub}}/{4\taub}
\{ 1 + q\cdot(1-2w_l)
  \akpi\cos(\dmd\Delta{t}) \}$,
where $\akpi$ is the $\bzb \to K^-\pi^+$ and
$B^0 \to K^+\pi^-$ decay rate asymmetry.
We fix $\akpi = 0$~\cite{pipi}, 
and $\taub$ and $\dmd$ to their world average
values~\cite{PDG}.
%%MH Although the $K^+\pi^-$ background contamination is low, a possible
%%MH deviation of $\akpi$ from zero is included in the systematic error.
The pdf used for the $q\overline{q}$ background is
%%MH The pdf used for the $q\overline{q}$ background distribution is
${\cal P}_{q\overline{q}}(\Delta t)
=\{f_\tau e^{-|\Delta t|/\tau_{\rm bkg}}/2
\tau_{\rm bkg}+(1-f_\tau)\delta(\Delta t)\}/2,$
where $f_\tau$ is the background fraction
with an effective lifetime $\tau_{\rm bkg}$ and $\delta$ is the Dirac delta
function.  We determine
$f_{\tau} = 0.011\pm 0.004$ and $\tau_{\rm bkg}= 2.7^{+1.0}_{-0.7}$ ps
from the sideband data.
%
In the fit,
$\spipi$ and $\apipi$ are free parameters 
%%MH that are
determined by maximizing
the likelihood function
${\cal L}=\prod_i P_i$, where the product is over all
$B^0 \rightarrow \pi^+\pi^-$ candidates.

The result of the fit to the
162 candidates (92 $B^0$- and 70 $\bzb$-tags) that 
remain after flavor tagging and vertex reconstruction is:
\begin{eqnarray}
\spipi &=& \sresult;
 \nonumber \\
\apipi &=& \aresult.
 \nonumber
\end{eqnarray}
The result is 1.3$\sigma$ away from the physical boundary
$\spipi^2+\apipi^2 = 1$, which is consistent with
a statistical fluctuation.
The correlation between $\spipi$ and $\apipi$ is found to be 0.28.
%%MH Our result has better precision than the previous 
%%MH measurement~\cite{apipi_BaBar}.
In Figs.~\ref{fig:asym}(a) and (b), we show the
$\Delta t$ distributions for
$B^0$- and $\bzb$-tagged events together
with the fit curves;
the background-subtracted $\Delta t$ distributions
are shown in Fig.~\ref{fig:asym}(c).
Figure~\ref{fig:asym}(d) shows the background-subtracted
$CP$ asymmetry between
the $B^0$- and
$\bzb$-tagged events 
as a function of $\Delta t$. The result
of the fit is superimposed and shown by the solid curve.
%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[!htbp]
\begin{center}
\resizebox{0.6\textwidth}{!}{\includegraphics{pipi_cpfit3.eps}}
%\resizebox{0.45\textwidth}{!}{\includegraphics{pipi_cpfit3.eps}}
\end{center}
\caption{
The $\Delta t$ distributions 
 for the $B^0 \rightarrow \pi^+\pi^-$ candidates in the
signal region: 
(a) candidates with $q = +1$, i.e. the tag side is identified as $B^0$;
(b) candidates with $q = -1$; 
(c) $\pi^+ \pi^-$ yields after background subtraction. The errors
are statistical only and do not include the error of the
subtracted background obtained by a fit. The rightmost
(leftmost) bin ranges from 5 to 10 ps ($-5$ to $-10$ ps);
(d) the $CP$ asymmetry for $B^0 \rightarrow \pi^+\pi^-$
after background subtraction. The
point in the rightmost bin has a large negative value
that is outside of the range of the histogram;
(e) the raw asymmetry for $B^0 \to \pi^+\pi^-$ sideband events.
In Figs. (a) through (c), the curves show the
results of the unbinned maximum likelihood fit.
In Fig. (d), the solid curve shows the resultant $CP$ asymmetry,
while the dashed (dotted) curve is the contribution from
the cosine (sine) term.
}
\label{fig:asym}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%

The systematic error on $\spipi$ is primarily due to
uncertainties in the background fractions ($\pm 0.09$) and
a possible fit bias near the physical boundary ($^{+0.11}_{-0.02}$). 
For $\apipi$, the background fractions ($\pm 0.06$) and the wrong-tag
fractions ($\pm 0.06$) are the two leading components.
We find that uncertainties in 
$f_{\pi\pi}^l$, $f_{K\pi}^l$ and $f_{q\overline{q}}^l$
in Eq.~(\ref{eq:likelihood})
%%MH the parameters used in the aforementioned signal and background functions,
%%MH $f_{\pi\pi}^l$, $f_{K\pi}^l$ and $f_{q\overline{q}}^l$,
account for the largest systematic errors.
We add their contributions in quadrature to obtain the above values,
where each contribution is obtained by
varying a parameter by its error and repeating the fit.
Other sources of systematic error are
uncertainties in the resolution function, physics parameters
($\dmd$, $\taub$, and $\akpi$) and the background modeling.
A value of $\akpi = -0.06\pm 0.08$ is obtained from the self-tagged 
$B^0\to K^+ \pi^-$   
sample~\cite{pipi}; this introduces a systematic error of
$<0.01$ for $\spipi$ and
$^{+0.02}_{-0.01}$ for $\apipi$.

We validate our fitting procedure using a large ensemble of MC
pseudo-experiments wherein events are generated with nominal pdfs and the
observed number of events. For various input values of
$\spipi$ and $\apipi$, we confirm that there is no bias in
the fit results.
%%MH and the errors are consistent with 
%%MH the differences between the inputs and fit values.
We find that the average expected errors,
0.52 for $\spipi$ and 0.35 for $\apipi$,
are larger than our measurements.
However, the probability of obtaining errors that are smaller
than our measurement is 5.4\% for $\spipi$ and
23.6\% for $\apipi$; the results are within 
the expected range of statistical
fluctuations. 
The measured likelihood is
in good agreement with the average likelihood value
obtained in the pseudo-experiments.

We perform a number of cross checks. 
We examine the event yields and $\Delta t$ distributions for
$B^0$- and $\bzb$-tagged events in the sideband region and find
no significant asymmetry as shown in Fig.~\ref{fig:asym}(e).
%%MH no significant difference between the two samples. 
%%MH Figure~\ref{fig:asym}(e) shows the observed raw asymmetry
%%MH in the sideband region.
We select $B^0 \to K^+\pi^-$ candidates,
which have the same track topology as $B^0 \to \pi^+\pi^-$,
by positively identifying charged kaons.
A fit to 309 candidates ($209 \pm 16$ signal events)
yields 
$\taub = 1.73\pm 0.15$(stat) ps and $\dmd = 0.57 \pm 0.08$(stat) ps$^{-1}$;
these are consistent with the world average values~\cite{PDG}.
%%MH $CP$ asymmetries are obtained to be
$\akpi$ is $0.07 \pm 0.17$, in agreement with the
counting analysis mentioned above
and $\skpi = 0.15 \pm 0.24$, which is consistent with zero.
We also select $B^0 \to D^-\pi^+$, $D^{*-}\pi^+$ and $D^{\*-}\rho^+$
candidates using the same event shape criteria.
%%MH These are non-$CP$ eigenstate self-tagged modes,
%%MH where neither 
Neither mixing-induced nor direct $CP$-violating asymmetry is observed
as expected.
As an additional test of the consistency of the background treatment,
we add events from the $B^0 \to \pi^+\pi^-$ sideband and
adjust their $\Delta E$ and $M_{\rm bc}$ values.
A fit to this background-enriched control sample,
which has a background fraction comparable to the $B^0 \to \pi^+\pi^-$ sample,
yields
${\cal S} = 0.08 \pm 0.06$ and
${\cal A} = 0.03 \pm 0.04$,
both consistent with a null asymmetry.

We determine the statistical significance from the likelihood
function, taking into account the boundary of the physical region
as well as the effect of the systematic error.
The Feldman-Cousins frequentist approach~\cite{FeldmanCousins}
gives a 99.6\% confidence level (C.L.) for 
$-1 \leq \spipi < 0$, equivalent to 
a $2.9\sigma$ significance for a Gaussian error.
A similar analysis yields a significance of $2.9\sigma$ for 
$0 < \apipi \leq 1$.
The 95\% C.L. intervals are found to be 
$-1.00 \leq \spipi < -0.39$
and
$+0.30 < \apipi \leq +1.00$,
respectively~\cite{ToyMC}. 

In summary, we have measured the
$CP$ violation parameters in $B^0 \rightarrow \pi^+\pi^-$ decay.
Our result for $\spipi$ indicates that mixing-induced $CP$
violation
is large. The large $\apipi$ term is an indication of direct
$CP$ violation in $B$ meson decay, and suggests that there is
a large hadronic phase and interference between the tree and
penguin amplitudes~\cite{apipi_BaBar}. 
In this case the precise determination of $\sin 2\phi_2$ from $\spipi$
requires additional measurements
including the branching fractions for the decays
$B^0 \rightarrow \pi^0 \pi^0$~\cite{GL}. 

%\section*{Acknowledgments}
We wish to thank the KEKB accelerator group for the excellent
operation of the KEKB accelerator.
We acknowledge support from the Ministry of Education,
Culture, Sports, Science, and Technology of Japan
and the Japan Society for the Promotion of Science;
the Australian Research Council
and the Australian Department of Industry, Science and Resources;
the National Science Foundation of China under contract No.~10175071;
the Department of Science and Technology of India;
the BK21 program of the Ministry of Education of Korea
and the CHEP SRC program of the Korea Science and Engineering Foundation;
the Polish State Committee for Scientific Research
under contract No.~2P03B 17017;
the Ministry of Science and Technology of the Russian Federation;
the Ministry of Education, Science and Sport of the Republic of Slovenia;
the National Science Council and the Ministry of Education of Taiwan;
and the U.S.\ Department of Energy.

\begin{thebibliography}{99}

\bibitem{KM}
M.~Kobayashi and T.~Maskawa, Prog. Theor. Phys. {\bf 49}, 652 (1973).

\bibitem{CP1_Belle}
 K.~Abe {\it et al.} (Belle Collab.),  Phys. Rev. Lett. {\bf
 87}, 091802 (2001); hep-ex/0202027, submitted to Phys. Rev. D.

\bibitem{CP1_BaBar}
 B.~Aubert {\it et al.} (BaBar Collab.),  Phys. Rev. Lett. {\bf 87}, 
 091801 (2001); hep-ex/0201020, submitted to Phys. Rev. D.

\bibitem{CC}
Throughout this Letter, %% whenever a mode is quoted 
the inclusion of the charge conjugate mode decay is implied
unless otherwise stated.

\bibitem{alpha} Another naming convention, 
$\beta$ ($= \phi_1$) and $\alpha$ ($= \phi_2$), is
also used in the literature.

\bibitem{Sanda}
A.~B.~Carter and A.~I.~Sanda, Phys. Rev. Lett. {\bf 45}, 952 (1980); 
A.~B.~Carter and A.~I.~Sanda, Phys. Rev. D {\bf 23}, 1567 (1981); 
I.~I.~Bigi and A.~I.~Sanda, Nucl. Phys. {\bf 193}, 85 (1981).

\bibitem{CPVrev}
A general review of the formalism is given in
I.I.~Bigi, V.A.~Khoze, N.G.~Uraltsev, and A.I.~Sanda, ``$CP$ Violation''
page 175, ed. C.~Jarlskog, World Scientific, Singapore (1989). 

\bibitem{pipipenguin}
M.~Gronau, Phys. Rev. Lett. {\bf 63}, 1451 (1989); 
D.~London and R.~Peccei, Phys. Lett. B {\bf 223}, 257 (1989);
M.~Beneke, G.~Buchalla, M.~Neubert and C.~T.~Sachrajda,
Nucl. Phys. B {\bf 606}, 245 (2001);
Y.~Y.~Keum, H-N.~Li and A.~I.~Sanda, Phys. Rev. D {\bf 63}, 054008 (2001);
M.~Ciuchini {\it et al.}, Phys. Lett. B {\bf 515}, 33 (2001);
M.~Gronau and J.~L.~Rosner, Phys. Rev. D {\bf 65}, 013004 (2002).

\bibitem{KEKB}
 E.~Kikutani ed., KEK Preprint 2001-157 (2001), 
 to appear in Nucl. Instr. and Meth. A.

\bibitem{Belle}
A.~Abashian {\it et al.} (Belle Collab.),
Nucl. Instr. and Meth. A {\bf 479}, 117 (2002).

\bibitem{pipi}
 K.~Abe {\it et al.} (Belle Collab.), Phys. Rev. Lett. {\bf 87}, 101801 (2001); a Phys. Rev. D paper is in preparation.

\bibitem{SFW}
 The Fox-Wolfram moments were introduced in
 G.~C.~Fox and S.~Wolfram, Phys. Rev. Lett. {\bf 41}, 1581 (1978).
 The Fisher discriminant used by Belle is described in \cite{pipi} and
 K.~Abe {\it et al.} (Belle Collab.), Phys. Lett. {\bf B 511}, 151 (2001).

\bibitem{PDG}
D.~E.~Groom {\it et al.} (Particle Data Group), 
Eur. Phys. J. {\bf C15}, 1 (2000).

\bibitem{FeldmanCousins}
G.~J.~Feldman and R.~D.~Cousins, Phys. Rev. D {\bf 57}, 3873 (1998).

\bibitem{ToyMC}
An ensemble of MC pseudo-experiments indicates
a 1.6\% probability to measure
$CP$ violation at or above the one we observe
when the input values are $\spipi = \apipi =0$.

\bibitem{apipi_BaBar}
$\spipi = 0.03^{+0.53}_{-0.56}({\rm stat})\pm 0.11({\rm syst})$ and
$C_{\pi\pi} = -0.25^{+0.45}_{-0.47}({\rm stat})\pm 0.14({\rm syst})$
in B.~Aubert {\it et al.} (BaBar Collab.),  Phys. Rev. D {\bf 65}, 051502R (2002).
Note that the sign convention for the coefficient of the cosine term 
$C_{\pi\pi}$ defined in the above reference is opposite to our convention,
i.e. $C_{\pi\pi} = -\apipi$.

\bibitem{GL}
M.~Gronau and D.~London, Phys. Rev. Lett. {\bf 65}, 3381 (1990);
M.~Gronau, D.~London, N.~Sinha and R.~Sinha, Phys. Lett. B {\bf 514}, 315 (2001).

\end{thebibliography}
%
\end{document}
%
% ****** End of file template.aps ******






hep-ex0204002.extracted_bib
\begin{thebibliography}{99}

\bibitem{KM}
M.~Kobayashi and T.~Maskawa, Prog. Theor. Phys. {\bf 49}, 652 (1973).

\bibitem{CP1_Belle}
 K.~Abe {\it et al.} (Belle Collab.),  Phys. Rev. Lett. {\bf
 87}, 091802 (2001); hep-ex/0202027, submitted to Phys. Rev. D.

\bibitem{CP1_BaBar}
 B.~Aubert {\it et al.} (BaBar Collab.),  Phys. Rev. Lett. {\bf 87}, 
 091801 (2001); hep-ex/0201020, submitted to Phys. Rev. D.

\bibitem{CC}
Throughout this Letter, %% whenever a mode is quoted 
the inclusion of the charge conjugate mode decay is implied
unless otherwise stated.

\bibitem{alpha} Another naming convention, 
$\beta$ ($= \phi_1$) and $\alpha$ ($= \phi_2$), is
also used in the literature.

\bibitem{Sanda}
A.~B.~Carter and A.~I.~Sanda, Phys. Rev. Lett. {\bf 45}, 952 (1980); 
A.~B.~Carter and A.~I.~Sanda, Phys. Rev. D {\bf 23}, 1567 (1981); 
I.~I.~Bigi and A.~I.~Sanda, Nucl. Phys. {\bf 193}, 85 (1981).

\bibitem{CPVrev}
A general review of the formalism is given in
I.I.~Bigi, V.A.~Khoze, N.G.~Uraltsev, and A.I.~Sanda, ``$CP$ Violation''
page 175, ed. C.~Jarlskog, World Scientific, Singapore (1989). 

\bibitem{pipipenguin}
M.~Gronau, Phys. Rev. Lett. {\bf 63}, 1451 (1989); 
D.~London and R.~Peccei, Phys. Lett. B {\bf 223}, 257 (1989);
M.~Beneke, G.~Buchalla, M.~Neubert and C.~T.~Sachrajda,
Nucl. Phys. B {\bf 606}, 245 (2001);
Y.~Y.~Keum, H-N.~Li and A.~I.~Sanda, Phys. Rev. D {\bf 63}, 054008 (2001);
M.~Ciuchini {\it et al.}, Phys. Lett. B {\bf 515}, 33 (2001);
M.~Gronau and J.~L.~Rosner, Phys. Rev. D {\bf 65}, 013004 (2002).

\bibitem{KEKB}
 E.~Kikutani ed., KEK Preprint 2001-157 (2001), 
 to appear in Nucl. Instr. and Meth. A.

\bibitem{Belle}
A.~Abashian {\it et al.} (Belle Collab.),
Nucl. Instr. and Meth. A {\bf 479}, 117 (2002).

\bibitem{pipi}
 K.~Abe {\it et al.} (Belle Collab.), Phys. Rev. Lett. {\bf 87}, 101801 (2001); a Phys. Rev. D paper is in preparation.

\bibitem{SFW}
 The Fox-Wolfram moments were introduced in
 G.~C.~Fox and S.~Wolfram, Phys. Rev. Lett. {\bf 41}, 1581 (1978).
 The Fisher discriminant used by Belle is described in \cite{pipi} and
 K.~Abe {\it et al.} (Belle Collab.), Phys. Lett. {\bf B 511}, 151 (2001).

\bibitem{PDG}
D.~E.~Groom {\it et al.} (Particle Data Group), 
Eur. Phys. J. {\bf C15}, 1 (2000).

\bibitem{FeldmanCousins}
G.~J.~Feldman and R.~D.~Cousins, Phys. Rev. D {\bf 57}, 3873 (1998).

\bibitem{ToyMC}
An ensemble of MC pseudo-experiments indicates
a 1.6\% probability to measure
$CP$ violation at or above the one we observe
when the input values are $\spipi = \apipi =0$.

\bibitem{apipi_BaBar}
$\spipi = 0.03^{+0.53}_{-0.56}({\rm stat})\pm 0.11({\rm syst})$ and
$C_{\pi\pi} = -0.25^{+0.45}_{-0.47}({\rm stat})\pm 0.14({\rm syst})$
in B.~Aubert {\it et al.} (BaBar Collab.),  Phys. Rev. D {\bf 65}, 051502R (2002).
Note that the sign convention for the coefficient of the cosine term 
$C_{\pi\pi}$ defined in the above reference is opposite to our convention,
i.e. $C_{\pi\pi} = -\apipi$.

\bibitem{GL}
M.~Gronau and D.~London, Phys. Rev. Lett. {\bf 65}, 3381 (1990);
M.~Gronau, D.~London, N.~Sinha and R.~Sinha, Phys. Lett. B {\bf 514}, 315 (2001).

\end{thebibliography}